From b243e9830fd6028fcf31876927e3a102af501633 Mon Sep 17 00:00:00 2001 From: Pomax Date: Tue, 4 Jan 2022 11:50:23 -0800 Subject: [PATCH] deploy regeneration --- README.md | 9 + docs/chapters/matrixsplit/content.ja-JP.md | 2 +- .../abc/7a69dd4350ddda5701712e1d3b46b863.png | Bin 15097 -> 15019 bytes .../abc/eeec7cf16fb22c666e0143a3a030731f.png | Bin 17543 -> 17449 bytes ...g => 016f37843b2af2ae34dc8b2975505f3d.svg} | 0 .../1564fdc8d17a064fea88b9d508878e62.svg | 1 + .../28cc0f129fa0c028a1addd702e99f162.png | Bin 13331 -> 13465 bytes .../64fa084c1df7300f06af13f53681463c.svg | 1 + .../9a6755a1e31a990e8f072a6da98f811a.png | Bin 15632 -> 15849 bytes ...g => d412c72ed342c2036965ff251c8fb443.svg} | 0 ...g => e49d7bb45a18d14fbb12df4d91e2c67b.svg} | 0 ...g => e5db5e945606d3429e4475ff92283a9c.svg} | 0 .../7c9cce8142fa3e85bb124520f40645ff.png | Bin 12040 -> 11958 bytes .../da76341b841df1af8a39f797e85dfe3c.png | Bin 16446 -> 16311 bytes .../56533f47e73ad9fea08fa9bb3f597d49.png | Bin 11018 -> 10903 bytes .../5ce02cbdbc47585c588f2656d5161a32.png | Bin 11538 -> 11471 bytes .../fa4c587126e8097206b88d9ea51974ca.png | Bin 11265 -> 11187 bytes .../fe2663b205d14c157a5a02bfbbd55987.png | Bin 12372 -> 12253 bytes .../3fc083ea7bdcc6b021560f2f2491f8aa.png | Bin 9846 -> 9888 bytes .../537260c4aa9e98ffdea7c8120afbd427.png | Bin 11421 -> 11325 bytes .../12ec4a5039de2e2cc06611db5e826282.png | Bin 9054 -> 9112 bytes .../daad01218ba430e2355d151811aa971b.png | Bin 10603 -> 10650 bytes .../0215dc106e4ad51afe043c0176a595f6.png | Bin 13752 -> 13657 bytes .../0d9c2186423466a32bb8fbd187409f82.png | Bin 17029 -> 17001 bytes .../41167c64c51386414c6e62f0b45e6295.png | Bin 23234 -> 23184 bytes .../48a30189e74658737b3a8b28bb816f8a.png | Bin 12623 -> 12584 bytes .../ceaef2fbee05a58aa11835925403b4cd.png | Bin 12656 -> 12697 bytes .../fe3a8ca5706f286d916e36699e237e51.png | Bin 20317 -> 20257 bytes .../83fe2473e20ea68b768765129ee44ae4.png | Bin 30633 -> 30681 bytes .../c086e72bd8aaeab37436515ab251b2df.png | Bin 26779 -> 26682 bytes .../aa46749b9469341d9249ca452390d875.png | Bin 6980 -> 6956 bytes .../3a3ea30971de247da93034de614a63a4.png | Bin 14803 -> 14910 bytes .../08ca09aacb271735e063e7e8d941a195.png | Bin 12174 -> 11969 bytes .../12f78070db3292742f66cf3cd9c7ed92.png | Bin 16785 -> 0 bytes .../222d374252584ac37d967e3ea0f8f28b.svg | 1 - .../22bda43c696b48e49a2aa83f0c4a4a0e.png | Bin 12798 -> 0 bytes .../23d4cd81c759c5374773fae149207c5b.svg | 1 - .../474c72b5d1ad5c154d79312c15aee47a.svg | 1 - .../485d678b1d63ed7a4c7c8bb282467f02.svg | 1 - .../571246ffef745298873a78473257d871.png | Bin 16785 -> 0 bytes .../5ef0824098ef73efbd89accb8db408e8.png | Bin 17047 -> 0 bytes .../6e455a6733dbca47acc0a36e7e490ba6.svg | 1 - .../9e19c65e23c32c802d2d77edb9812122.svg | 1 - .../b5799015ebae3ab9b6712106b3b20ee6.svg | 1 - .../cf8fc61992d2a75416894dee46ee5ffe.svg | 1 - .../e1fbb58c3730b2f4a378e39c8dc5e61f.svg | 1 - .../610a69d8be6f86744ffb88d12eda701b.svg | 2 +- .../85c3dc0187f786b37f5fa5a7cc74d642.svg | 2 +- .../891d50a946936c9701adc855de12623d.png | Bin 16228 -> 15592 bytes .../9054528132317434ae2c0be27572d86b.svg | 2 +- .../a7dc2e51b90e89ec62e4a328d2b24635.svg | 2 +- .../acbde4be3cde3838b99b0ffc933f1f89.svg | 2 +- .../ad4d512cb92280ac88af531309cdbb8c.svg | 2 +- .../b985384d01cb32d422f5d1123707ebc8.svg | 1 - .../b996b7c1af4c9187004af7d04b3740a5.svg | 2 +- .../cb6686f1aff26d9f47ed4c695109fd5f.svg | 2 +- .../d880c4a1b3d7b651b054b008e952b493.svg | 2 +- .../e0d46f5fd4bf01e72f23495757f64448.svg | 2 +- .../f47561c3870425499e31c7527c38dc94.svg | 2 +- .../f65f4e30a9f7a08c5c0092a1a3853922.svg | 2 +- .../fe6cc524978eaa4f35d8de32c3b9ad94.svg | 1 - .../1e6e38f6403dbe4c8b80295a94fc6748.png | Bin 19022 -> 18898 bytes .../348694339257428a260144da4bbf80fc.png | Bin 20780 -> 20658 bytes .../1b8c5e574dc67bfb0afc3fb0a8727378.png | Bin 16616 -> 16602 bytes .../8332e5d34b7344bbee2a2e1f4521ce46.png | Bin 13998 -> 13880 bytes .../8d158a13e9a86969b99c64057644cbc6.png | Bin 9857 -> 9924 bytes ...g => be73034ac382b54863c7a18c2932cbbc.svg} | 0 .../c26d2655e8741ef7e2eeb4f6554fc7a5.png | Bin 30204 -> 29969 bytes .../392624cedf7c78aed6d4c6065a014b42.png | Bin 105629 -> 105439 bytes .../4f2647446363ca5d93b11e414fd976df.png | Bin 51683 -> 51523 bytes .../134baa1043d0849f31a1943d6d5bc607.svg | 1 - .../b155682162a5b6da6d40c7b531164a7e.png | Bin 32754 -> 32404 bytes .../df92f529841f39decf9ad62b0967855a.png | Bin 10260 -> 10266 bytes ...g => 87403e5b0da3dcc4ceca74fae058fe69.svg} | 0 ...g => de486728b41299269cc990ed09d5d286.svg} | 0 ...g => f00423aaceac6ade478aba2a761664d8.svg} | 0 ...g => 263d97ef5db7e823d8d374a2c91b3bf0.svg} | 0 ...g => 4319b2e361960c842a4308a610a35048.svg} | 0 ...g => 516214a32b170186c3f3f4d34c9fe8fe.svg} | 0 .../959762e39ae32407e914a687d804ff3a.png | Bin 7085 -> 6913 bytes .../2d17acb381ebdd28f0ff43be00d723c4.png | Bin 10944 -> 10991 bytes .../37948bde4bf0d25bde85f172bf55b9fb.png | Bin 9201 -> 9203 bytes .../fbfe9464c9653f5efcd04411e683faf9.png | Bin 23872 -> 23484 bytes .../fd68347a917c9b703ff8005287ac6ca4.png | Bin 20784 -> 20577 bytes .../0e0e4a2ee46bd89bcfde9f75cfe43292.png | Bin 10637 -> 10622 bytes .../6813bfc608aea11df1dda444b9f18123.png | Bin 9314 -> 9180 bytes .../17bf62e05a1fc3387b0c210f2decff45.png | Bin 14364 -> 14415 bytes .../8cc724d5343c65685d88c92b2d069a2a.png | Bin 13395 -> 13404 bytes .../726ece45630c43be14589c51f1606bd7.png | Bin 11254 -> 11076 bytes .../7196d3dec75d53f5df9d9c832ac3c493.png | Bin 16795 -> 16525 bytes .../9b70fb7b03f082882515e55c0a1eacff.png | Bin 15234 -> 15088 bytes .../b3f61036d8dc9888a6a64a1171583dd1.png | Bin 3194 -> 3173 bytes .../54e9ec0600ac436b0e6f0c6b5005cf03.png | Bin 8600 -> 8592 bytes .../8d158a13e9a86969b99c64057644cbc6.png | Bin 9857 -> 9924 bytes .../1039d0bb0e49cfb472c2fa37f9010190.png | Bin 21273 -> 21220 bytes .../2bd215d1db191b52a89a94727b6aa5ce.png | Bin 18910 -> 18875 bytes .../a54f990aa49fb9ea8200b59259f955f3.png | Bin 21273 -> 21220 bytes .../03b8e0849e7c8ba64d8c076f47fe2ec7.png | Bin 13267 -> 13212 bytes .../4c4738b6bf9f83eded12d680a29e337b.png | Bin 14829 -> 14776 bytes .../067a3df30e32708fc0d13f8eb78c0b05.png | Bin 7488 -> 7135 bytes .../43875f6ad588bfd04cdb65b591a62052.png | Bin 11356 -> 10685 bytes .../75f7b5b31e98444e13f17e5c3e5b7322.png | Bin 13234 -> 12654 bytes .../eab6ea46fa93030e03ec0ef7deb571dc.png | Bin 7742 -> 7453 bytes .../b8285282b8a0ce28fc2cc0392e9b607a.png | Bin 12354 -> 12267 bytes .../d0b09bb338bd42d4164ced871d1f77ba.png | Bin 9800 -> 9740 bytes .../11c1da2357004bb51cf0c591fc492115.png | Bin 13157 -> 13048 bytes .../f4a2fa1e0204c890b2bff07228ba678d.png | Bin 12730 -> 12694 bytes .../1b94c6ada011bd8e50330e31a851a62e.png | Bin 12902 -> 12584 bytes .../41522a397171423e8a465dc8c74f6e87.png | Bin 9904 -> 9971 bytes .../6fb33629373d7a731b6ac3f5365cb9f0.png | Bin 12902 -> 12584 bytes .../777f3814965c39ec3cdbb13eab0c4eeb.png | Bin 9904 -> 9971 bytes .../b3aebf7803f4430187c249a891095062.png | Bin 9904 -> 9971 bytes .../db04f805f42bdc9a1b7ec4d6b401d853.png | Bin 9904 -> 9971 bytes .../f6c55cbc66333b6630939f67fc20e086.png | Bin 12902 -> 12584 bytes .../fe41b628f46f7035d151a8210d30111f.png | Bin 12902 -> 12584 bytes .../536a94885dc1637c066d0ef4f6e4e650.png | Bin 25618 -> 25905 bytes .../c4874e1205aabe624e5504abe154eae9.png | Bin 11321 -> 11905 bytes .../fce5eb16dfcd103797c5e17bd77f1437.png | Bin 24540 -> 24296 bytes .../9ee5abc64b3fba71e284c70539279d74.png | Bin 13520 -> 12958 bytes .../ed91133976018ec032d9115344debb36.png | Bin 13010 -> 12416 bytes .../1cd7304fb8d044835bfbc305ca5e5d10.png | Bin 23917 -> 23810 bytes .../d6239520389637a3c42e76ee44d86c41.png | Bin 18210 -> 18180 bytes .../3d71e2b9373684eebcb0dc8563f70b18.png | Bin 9857 -> 9924 bytes ...g => 942e3b3cacc7f403ad95fcd4acce7d19.svg} | 0 .../524dd296e96c0fe2281fb95146f8ea65.png | Bin 30335 -> 30002 bytes .../2fc5c57e5d1ed0eaa1655edc31026252.png | Bin 12830 -> 12597 bytes .../e469af5bf27a2c27d1dd6fc62a78ac27.png | Bin 19663 -> 19403 bytes .../15225da473048d8c7b5b473b89de0b66.ascii | 19 +- .../131454dcbac04e567f322979f4af80c6.ascii | 7 +- .../3166afa345aec1abda432c39b68d39a0.ascii | 17 +- .../51a9d0588be822a5c80ea38f7d348641.ascii | 7 +- .../5924e162b50272c40c842fad14b8fa48.ascii | 11 +- .../8bd3e6fed5bf8d871d30221ae400fd93.ascii | 17 +- .../8c6662f605722fb2ff6cd7f65243a126.ascii | 5 +- .../8cd992c1ceaae2e67695285beef23a24.ascii | 11 +- .../8e7cfee39c98f2ddf9b635a914066cf6.ascii | 11 +- .../a0b99054cc82ca1fb147f077e175ef10.ascii | 11 +- .../00480d8ea1d0b86eb66939bced85e14b.ascii | 6 - .../016f37843b2af2ae34dc8b2975505f3d.ascii | 5 + .../1564fdc8d17a064fea88b9d508878e62.ascii | 5 + .../64fa084c1df7300f06af13f53681463c.ascii | 5 + .../6acf1a1e496f47c11e079a1d13f0a368.ascii | 6 - .../a75137c250be63877a30f4bda8d801f8.ascii | 6 - .../d412c72ed342c2036965ff251c8fb443.ascii | 5 + .../e49d7bb45a18d14fbb12df4d91e2c67b.ascii | 5 + .../e5db5e945606d3429e4475ff92283a9c.ascii | 5 + .../f6767b16ff8e04646f45fb9a1f3e4024.ascii | 6 - .../046bbb52e8c8ed617fdf3a4fd18d62e1.ascii | 11 +- .../0748ad25185548150b6c1c4c7039207e.ascii | 21 +- .../2f80643c66d8f1448b13537a7b24eb45.ascii | 6 - .../59ebc3a7c3547a50998d1ea3664fb688.ascii | 9 +- .../85620f0332fcf16f56c580794fd094c5.ascii | 9 +- .../a91fbfb7abc38ff712ef660d85679f2e.ascii | 25 +- .../b76753476ad6ecfe4b8f39bcf9432980.ascii | 3 +- .../f251e86158649c0e57f7a772ebff83b4.ascii | 9 - .../20e910bbea2e6eff511cb13cef18ef3b.ascii | 57 +- .../2421f47aa4fe1c0d830d53b2e6563c04.ascii | 11 +- .../2514e1aa0565840e33fde0b146e3efe2.ascii | 7 +- .../49af474c33ce0ee0733626ea3d988570.ascii | 7 +- .../89f8e37237d066fa70ccf6d37b3a4922.ascii | 7 +- .../a88566be442b67fb71f727de6bdb66df.ascii | 5 - .../c7af721e5e201fc3742bce67ff6cd560.ascii | 5 - .../cbdf5a61de10eeb6f23be077cf047ab5.ascii | 7 - .../e62558cdfd8abaf22511e8e68c7afb4a.ascii | 7 +- .../ee203de6e554936588eb93adead0a3e5.ascii | 5 - .../f0bf7d0f1931060cd801ff707f482c16.ascii | 7 +- .../2b6478075f2f9f5e5973e01b3b3a0c8b.ascii | 7 +- .../464b4ec0b67f248459792752be86d46d.ascii | 9 +- .../5f174bc5019245f467ca63ae84b90a4b.ascii | 3 +- .../674d251590411398d06fb99cba7920f7.ascii | 9 +- .../6959a552f2c90a2bcaa787c23e19f488.ascii | 11 +- .../7ed8b53100737cbf7d87aa6267395d2b.ascii | 7 +- .../88e3fae7aeef6d7614290587422542c9.ascii | 11 +- .../8e39a9e0c7469b4b45a260dd23bd4c6a.ascii | 19 +- .../9420fd9d7a8de30714e23b8f31b3aa6d.ascii | 15 +- .../ccbfd22cbccf633d182f7f451dee5164.ascii | 15 +- .../e61fd49e554a0ffc7d64893c75cd376d.ascii | 24 - .../f855cbf1d73e4bb7bccbbd4721d95f41.ascii | 11 +- .../fff37fa4275e43302f71cf052417a19f.ascii | 45 +- .../00357d2a2168fe313cd0b38d95a1a681.ascii | 14 - .../012a8ab7a4de935c1c8d61dcd14fc62c.ascii | 17 +- .../032409c03915a6ba75864e1dceae416d.ascii | 17 +- .../157b287d6b74109d8c8b634990ea6549.ascii | 25 +- .../1cef6bbf7b3d10e8c0aaecfac816cc86.ascii | 17 +- .../1f9fc156aeed9eb092573cd7446593d9.ascii | 10 - .../211dadbb9d0f6b2e381f18ea3c4d12fb.ascii | 25 +- .../389a1ea8c9e92df9a2b38718e34bae7b.ascii | 17 +- .../4c8684109149b0dc79f5583a5912fcd9.ascii | 17 +- .../4e0da16710a7339f04dd844c7705423e.ascii | 21 +- .../53f216327c0bbcf02b2a331fbf44d389.ascii | 21 +- .../574bed6665be06b309b8da722c616a41.ascii | 21 +- .../639ca0b74a805c3aebac79b181eac908.ascii | 25 +- .../65e589eafae8ff2f39392d8143d2845c.ascii | 21 +- .../7bab9dd3da654b05fa065076894e2d82.ascii | 17 +- .../8a2a00812363fe1a6cfa7f81b48d31d1.ascii | 12 - .../8d3a5ca7188f53b914229133b3dbe5fe.ascii | 10 - .../8de53f207d68b25854a5f0b924ac6010.ascii | 25 +- .../902c290a790b4d44d10236f4a1456cdc.ascii | 17 +- .../917b176a45959b026c56f81999505dc7.ascii | 21 +- .../9593c057c84ebf9beb70fd57a11c7e12.ascii | 10 - .../98ddf6415bd9827a6d899b21d0a5f736.ascii | 17 +- .../9ae99b090883023a485be7be098858e9.ascii | 17 +- .../a323848e706c473833cda0b02bc220ef.ascii | 25 +- .../a55773fdcdfd99947acc4f86ad2d4a3d.ascii | 25 +- .../a8158b35ec221cccff51a53cdc7f440b.ascii | 17 +- .../b59ff8d654e65df4c874901983208893.ascii | 21 +- .../b94a4dafc12ba7e4fbf3aff924f55464.ascii | 25 +- .../c0e30b49fbfce6f6b3c81eaa6ca5154f.ascii | 14 - .../c1f8861583b4176a9b607aa6a05f9356.ascii | 10 - .../d09e7466c267614c89ead28d6a900ba1.ascii | 10 - .../defc6fa4b51fa3c1945d15449f0f392d.ascii | 10 - .../e653724c11600cbf682f1c809c8c6508.ascii | 21 +- .../fe2e6fd487df224b2f55a601898ce333.ascii | 17 +- .../2f42c862a0a9d0764727d42b16cf68a0.ascii | 19 +- .../373248ec6a579bacf6c6a317e6db597a.ascii | 3 +- .../3e0594855ca99fb87dcc65a693e1ad22.ascii | 11 - .../674c42035da16a426ef7fe23277eea11.ascii | 3 - .../699e6ea5bffbe8fe377af21a2bd28532.ascii | 11 - .../9271faa4684904073127482aa91e3bce.ascii | 6 - .../b480dae093b572dc707b76eef7fbca04.ascii | 11 - .../06369b00338310df0a810c592485aa0a.ascii | 7 +- .../1829e42ea956ee4df0e45d9ac5334ef7.ascii | 3 +- .../222d374252584ac37d967e3ea0f8f28b.ascii | 34 - .../23d4cd81c759c5374773fae149207c5b.ascii | 8 - .../474c72b5d1ad5c154d79312c15aee47a.ascii | 17 - .../485d678b1d63ed7a4c7c8bb282467f02.ascii | 6 - .../5a23f3bc298c85540c6dd18e304d9224.ascii | 31 +- .../6e455a6733dbca47acc0a36e7e490ba6.ascii | 9 - .../7ab3da0922477af4cc09f5852100976b.ascii | 7 +- .../8237af1396bb567d70c8b5e4dd7f8115.ascii | 9 +- .../8b4e1d0a62380ed011f27c645ed13b28.ascii | 7 +- .../942c90bc8311e49d94059b3127fc78d5.ascii | 27 +- .../986ae9104e0bc52e95689ae7ae4504db.ascii | 13 +- .../9e19c65e23c32c802d2d77edb9812122.ascii | 6 - .../a04bd1558a76e60b8ca6e1fe4fa38c00.ascii | 7 +- .../b5799015ebae3ab9b6712106b3b20ee6.ascii | 8 - .../cf8fc61992d2a75416894dee46ee5ffe.ascii | 10 - .../d4bfb47b623c968e3231566c9705c6c4.ascii | 7 +- .../e1fbb58c3730b2f4a378e39c8dc5e61f.ascii | 5 - .../f9d15462df31186feef8c3d53c0f6163.ascii | 13 +- .../610a69d8be6f86744ffb88d12eda701b.ascii | 7 +- .../85c3dc0187f786b37f5fa5a7cc74d642.ascii | 9 +- .../9054528132317434ae2c0be27572d86b.ascii | 17 +- .../a7dc2e51b90e89ec62e4a328d2b24635.ascii | 9 +- .../acbde4be3cde3838b99b0ffc933f1f89.ascii | 7 +- .../ad4d512cb92280ac88af531309cdbb8c.ascii | 13 +- .../b985384d01cb32d422f5d1123707ebc8.ascii | 24 - .../b996b7c1af4c9187004af7d04b3740a5.ascii | 9 +- .../cb6686f1aff26d9f47ed4c695109fd5f.ascii | 45 +- .../d880c4a1b3d7b651b054b008e952b493.ascii | 13 +- .../e0d46f5fd4bf01e72f23495757f64448.ascii | 47 +- .../f47561c3870425499e31c7527c38dc94.ascii | 7 +- .../f65f4e30a9f7a08c5c0092a1a3853922.ascii | 33 +- .../fe6cc524978eaa4f35d8de32c3b9ad94.ascii | 10 - .../20b0be6397fbd726298de6ec70a8544b.ascii | 6 - .../3ec466bf6e1aff44b35b8e37cc86cc3e.ascii | 5 - .../501494295f07ba5049286489206d98f0.ascii | 7 - .../6890c4028109e7d30a0b4b89f6fbe292.ascii | 7 - .../80cdfeab6ed6038f0e550ef5c1dcb7dd.ascii | 7 +- .../87d587388add62445ec0f3e7d7295094.ascii | 7 - .../9df7dc66b51db8e3046e1f359874c38e.ascii | 5 - .../a337e3f97387b52d387fc01605314497.ascii | 3 +- .../b58fb122c5c8159938182c185f287142.ascii | 7 - .../be73034ac382b54863c7a18c2932cbbc.ascii | 5 + .../c2f2fe0ef5d0089d9dd8e5e3999405cb.ascii | 3 +- .../060acd6ff0a050fe4d98a7802a2b3a3f.ascii | 13 +- .../561ab3a938d655550de0abf458ac2494.ascii | 7 +- .../afd8cb8b0fe291ff703752c1c9cc33d4.ascii | 17 +- .../06605e008956609e8844ef95697c9096.ascii | 5 - .../097aa1948b6cdbf9dc7579643a7af246.ascii | 7 +- .../134baa1043d0849f31a1943d6d5bc607.ascii | 5 - .../17d5fbeffcdcceca98cdba537295d258.ascii | 7 +- .../2f82371abb7835f9b9d440dc5dd151a8.ascii | 9 +- .../31d659cbc72bf304abf4c9a75b6b81de.ascii | 7 - .../38bb81bdd3eaa72c2336514187aa374b.ascii | 19 +- .../409d10c3005b0c93489d72a5dba692d7.ascii | 7 - .../464dbfb5adb6233108053dfac6fa4fe5.ascii | 5 - .../4dd55c228a26bb50da912a45e8721024.ascii | 19 +- .../4e6e20c823c8cc72e0cc00e4ab5b7556.ascii | 9 +- .../505ab1ada6a187e9ba392d19739ac2c5.ascii | 5 - .../6f734d319a1cfe0de76574a65abb07e1.ascii | 13 +- .../7b0199bb515d2754c03d8f796b29febf.ascii | 5 +- .../7c6b50cee5dc685515943a199d7a65fc.ascii | 9 - .../8068231b915832938136d5833f74751d.ascii | 5 +- .../8928f757abd1376abdc4069e1aa774f2.ascii | 15 +- .../8a66af7570bac674966f6316820ea31b.ascii | 11 +- .../940455f4016ab1be6d46c6f176fd2f76.ascii | 5 - .../9651a687e1522b00bcba063881230902.ascii | 6 - .../989f2ad06ae308f71cef527a5594129a.ascii | 5 +- .../9dec10b81a61b456ca1550cd9b7ba513.ascii | 5 +- .../a6faaf6083c818431988fef49421cc47.ascii | 7 - .../b017da988c9a778a4ce6a6f4ea4790d4.ascii | 15 +- .../d39ca235454ced9681b523be056864d2.ascii | 7 +- .../db85a7a46b869c892662c26b6aea15a1.ascii | 7 +- .../dd303afb51d580fb2bf1b914c010f83d.ascii | 11 +- .../ea24b0e42f0a89464bda275ac8f9bacf.ascii | 7 +- .../ff701138fd7a6e35700a2e1ee3e9c020.ascii | 11 +- .../02cecadc92b8ff681edc8edb0ace53ce.ascii | 25 +- .../12fa7f83f055ef2078cc9f04e1468663.ascii | 7 +- .../153d99ce571bd664945394a1203a9eba.ascii | 3 +- .../171357d936dee742b43b9ffb7600c742.ascii | 7 +- .../18c6e782012234a2c7425204505c8888.ascii | 13 - .../2368534c6e964e6d4a54904cc99b8986.ascii | 3 +- .../2d733684f81b65a42c4cdb3f1e589c8b.ascii | 9 +- .../2e90e21710bf1bcbbfecbb464d27244a.ascii | 7 - .../2fc50617b6886534d1ab4638ed8a24ac.ascii | 7 - .../3cd7b36839a248eb35f0b678d7bf5508.ascii | 3 +- .../4eeb75f5de2d13a39f894625d3222443.ascii | 7 +- .../501494295f07ba5049286489206d98f0.ascii | 7 - .../50616f9c922967c0c9c179af9b091947.ascii | 5 - .../5d7af72e00fb0390af5281d918d77055.ascii | 6 - .../64c06c61727d0912a67c0f287a395e47.ascii | 19 +- .../67ca2710769505572e097ffb40de099f.ascii | 5 - .../6a3672344bb571eadb72669f60a93ff4.ascii | 11 +- .../8324bf1885267fe157bf316e261d1b30.ascii | 7 - .../869b60a8e6b992e6f62bc6a50b36deeb.ascii | 3 +- .../87403e5b0da3dcc4ceca74fae058fe69.ascii | 12 + .../897cfd8648720dc21463a9358cc65ab4.ascii | 14 - .../8f78fdb9ef54b1bc4dbc00f07263cc97.ascii | 7 +- .../a7b79877822a8f60e45552dcafc0815d.ascii | 7 +- .../a7c61e0e8b42010df6dab641c92ef13d.ascii | 6 - .../a992185a346518b5ca159484019b6917.ascii | 7 +- .../b58fb122c5c8159938182c185f287142.ascii | 7 - .../bbe9d45ab271549dea5ef54982fcaaa5.ascii | 7 - .../bec017d11e19b45df562c5a223761a69.ascii | 11 +- .../c3ac18fe4ba0606a15bc111e52b17a9a.ascii | 7 +- .../d575699ab7d13c62f47d3071c0b00da3.ascii | 5 - .../de486728b41299269cc990ed09d5d286.ascii | 5 + .../f00423aaceac6ade478aba2a761664d8.ascii | 8 + .../f29a9d52897d2060a0c8a37073ed04fc.ascii | 9 - .../f67d2d379ba6dfaa7f7686a7d1eae367.ascii | 5 - .../05c2d5954eb9dec5ce9f6eb7e89f1e0c.ascii | 8 - .../0cc876c56200446c60114c1b0eeeb2cc.ascii | 3 +- .../2493468e73b73f43eba8f66f0c189d1a.ascii | 7 - .../263d97ef5db7e823d8d374a2c91b3bf0.ascii | 7 + .../29695045f04fd06c75bfda7845121213.ascii | 7 +- .../2c47081c2a9c20d2110f13daa482a3ab.ascii | 8 - .../39330ef5591cf0f3205564ad47255d4f.ascii | 11 +- .../39d33ea94e7527ed221a809ca6054174.ascii | 7 +- .../4319b2e361960c842a4308a610a35048.ascii | 7 + .../4bf2d790d2f50bf7767c948e0b9f9822.ascii | 11 +- .../4def87a6683264d420f84562776f4b6c.ascii | 8 - .../516214a32b170186c3f3f4d34c9fe8fe.ascii | 7 + .../668d140df9db486e5ff2d7c127eaa9d4.ascii | 7 +- .../6914ba615733c387251682db7a3db045.ascii | 5 +- .../6aa5d4e20e83be1c95eaad792517dde9.ascii | 6 - .../6d58ec36bfb3fcff24248dc46889428a.ascii | 6 - .../79832780f9209be5569447c4d988e54b.ascii | 5 - .../7a44f3eaa167a5022e2281c62e90fff8.ascii | 8 - .../7acc94ec70f053fd10dab69d424b02a6.ascii | 9 +- .../7f74178029422a35267fd033b392fe4c.ascii | 11 +- .../855a34c7f72733be6529c3fb33fa1a23.ascii | 5 +- .../8986c536df8153b30197c3a5407d233a.ascii | 6 - .../9229934d71b0b02921bc92594ef11a98.ascii | 8 - .../9734aff037ac23a73504ff7cc846eab7.ascii | 9 +- .../9c921b7b8a8db831f787c1329e29f7cb.ascii | 9 +- .../a2891980850ddbb27d308ac112d69f74.ascii | 5 +- .../a5bb1312adc5e9e23bee6b47555a6e8f.ascii | 8 - .../adc7729f7872d71f3fbb1a79741ce10f.ascii | 7 - .../af40980136c291814e8970dc2a3d8e63.ascii | 9 +- .../b5977078d36d847fb299cbe3e7e2c3ba.ascii | 5 - .../dc48cdf8f492b44c7602eb64ce2b9986.ascii | 7 - .../f79dd2f2d992e22b8d057fdc641290b0.ascii | 5 - .../fa3ed9a4ab61d80ec175d29533b5728e.ascii | 5 - .../08cd4a8bf4557862c095066728e6ed5e.ascii | 5 +- .../4e0fa763b173e3a683587acf83733353.ascii | 5 +- .../5a7a12213ca36f2f833e638ea0174d4a.ascii | 5 +- .../b0eb0b24e7fa29c545ab1479d2df0554.ascii | 4 - .../dfd6ded3f0addcf43e0a1581627a2220.ascii | 5 +- .../e2e71b397009b51af8a3ee848bc727b4.ascii | 4 - .../f56f59f3c0b057c8ec79a477e4e38bec.ascii | 4 - .../fd520a6e2c7f39e90496e5cf494cce2e.ascii | 4 - .../1fab66c84e7df38a2edda147f939bd80.ascii | 11 +- .../2c398b492aadc90eb4e4853fc20b23e9.ascii | 7 - .../3125ab785fb039994582552790a2674b.ascii | 11 +- .../53e67a29f134bd561aca550a2091a196.ascii | 11 +- .../55e16ef652d30face0f6586b675a6c7b.ascii | 7 +- .../a6acf08f43aa1f48c08a40e76bdd2a31.ascii | 13 +- .../bf0ad4611c47f8548396e40595c02b55.ascii | 9 +- .../c4858be225d004441b2aefedacda89a3.ascii | 6 - .../d1c65d927825f20c3c358d1ff96ce881.ascii | 31 +- .../d31432533bd7940545d4a269eefbabf2.ascii | 9 +- .../2dbf3071d74e2ba37ab888aaa3c1a17c.ascii | 17 +- .../35299f4eb8e0bed76b68c7beb2038031.ascii | 17 +- .../75fae2d0a94eae4addf074c294855fc7.ascii | 27 +- .../8278b9bec92ae49927283396692b51d5.ascii | 15 +- .../852f0346f025c671b8a1ce6b628028aa.ascii | 5 +- .../a283e01df17f3d763ec89621f2af6c5c.ascii | 4 - .../be9e409d619ecd735b0fbc219bec6d07.ascii | 9 - .../d7d564126099bc0740058a7cdd744772.ascii | 11 +- .../e50243eaa99b5acc08533dd2e9b71a74.ascii | 7 +- .../ed68dcfb203517ca080fe48914769fb0.ascii | 3 +- .../f9f2258e59b038659087a5e87ba2e0af.ascii | 10 - .../1a64ed455c6dd2f8cacca5e5e12bdcc1.ascii | 9 +- .../1bae50fefa43210b3a6259d1984f6cbc.ascii | 13 +- .../67a5ea33d6c6558f7d954b18226f4956.ascii | 17 +- .../87cfac83cb8a4b0bee68ef006effc611.ascii | 5 +- .../8ecff6b8a37d60385d287ea2b26876db.ascii | 9 +- .../9a9a55f5b0323d9ea88f82fc6be58ad3.ascii | 7 +- .../b32cae2dfc47d5f36df0bc3defb7dfa8.ascii | 17 +- .../b9527f7d5a0f5d2d737eac118d69243e.ascii | 9 +- .../cdd88611833f3b178df91278359a4193.ascii | 17 +- .../ec118f296511c6e9ac8727be3703a7ce.ascii | 17 +- .../11505e0215ef026f2e49383ebb4a1abb.ascii | 1 - .../17e308aa6d459b1d06d3160cc8e2e786.ascii | 15 +- .../1bae50fefa43210b3a6259d1984f6cbc.ascii | 13 +- .../206f539367fa1aaefc230709e4f2068e.ascii | 13 +- .../266b71339b55ad3a312a9f41e6bcf988.ascii | 13 +- .../3b5e41808b6c3bc66f3da2f40651410e.ascii | 13 +- .../4549b95450db3c73479e8902e4939427.ascii | 13 +- .../4764868f43815e471bb1ea95a81e1633.ascii | 13 +- .../480ebd0234e2fe1adc94926e8ed4339c.ascii | 15 +- .../4ce218bc968cbd98da0ca6ab66d415ed.ascii | 13 +- .../55b45214ba90c96978cdc9cdfee24fef.ascii | 8 - .../598739d23cf8dbcdebe033e4c7f1d28a.ascii | 8 - .../5e008143622c66bb5e9cc4d5d6a8ea62.ascii | 11 +- .../63d1337d275abf7b296d500b9b5821fd.ascii | 8 - .../6a22184e6ca869d28f4a252b64f23eff.ascii | 13 +- .../7a1e792cd3fa6f3482459e154abf2e7d.ascii | 8 - .../8349aa18563bb43427ae2383f1e212ae.ascii | 13 +- .../9a4899b69e03cd4ad02c5eedffaa6a2f.ascii | 15 +- .../a899891096d82b7fdb23a90e6106b6df.ascii | 1 - .../aa17f7e82cf50498f90deb6a21a2489a.ascii | 13 +- .../baedd4067516584d425b93331b7ce04f.ascii | 8 - .../c32007be095224e0d157a8f71c62c90e.ascii | 13 +- .../c341532f693c2c1adfd298597bbfb5b5.ascii | 11 +- .../d9d04b9b6c66788d18832a383d6f7ea0.ascii | 8 - .../da4ebf090f84d9b5d48b0f1e79bb3e7b.ascii | 13 +- .../dceed84990aaf6878bcc67ddbaa8d8d9.ascii | 13 +- .../e079f44b56e07c8d7f83c17c8ebf1ecf.ascii | 13 +- .../e2622175dadafecc015f15c79ddf3002.ascii | 13 +- .../e58196b82b78f584779208cce88137f5.ascii | 15 +- .../ebf8d72c6056476172deeb89726b75c8.ascii | 17 +- .../f565e66677138927335535d009409c3d.ascii | 13 +- .../f63067c2c3042c374a58dfa7f692309e.ascii | 13 +- .../f690ff0502d9fd7d4697cc43d98afd5d.ascii | 17 +- .../2e65bc9c934380c2de6a24bcd5c1c7b7.ascii | 9 +- .../48887d68a861a0acdf8313e23fb19880.ascii | 5 +- .../6f12fcc00f4106bbc920d7451398d3b2.ascii | 6 - .../70262c533569a7da06cc1b950e932d6f.ascii | 4 - .../3c80407cfd0bd8c8ebea239272aeabe5.ascii | 3 +- .../57e62f3f2f7526b2cf7c1b276c17e472.ascii | 7 +- .../af4b584bb280cc941603255f62c9cc1a.ascii | 9 +- .../cf8e602eb0595cf4d9b851c6bda741af.ascii | 9 +- .../de8cdb128273beff2d98534b9f090b85.ascii | 3 +- .../4fe687c8a65265a2a755ba5841d0e31d.ascii | 5 +- .../8ffdd4a58cbd0fc24caef781f23a7950.ascii | 9 +- .../a5cd63b54be6b554290c38787cfbbabd.ascii | 5 +- .../f8182445c1cd7ae9f368b88fa7090e53.ascii | 19 +- .../1df6c055ae8e41a46bfdebc55a4f17c0.ascii | 5 +- .../33afd1a141ec444989c393b3e51ec9ca.ascii | 29 +- .../4d89f0042fc367f1614537c7f05389fb.ascii | 16 - .../58b19accb8a68c665ff5cbed610eea4e.ascii | 9 - .../8b15a314beca97071b0ccb22c969355d.ascii | 7 - .../afdd2dbe7690ed09ea91df63471b480b.ascii | 9 - .../b60cdba673c2c9fc84c800f07fd18145.ascii | 16 - .../c3d5f3506b763b718e567f90dbb78324.ascii | 15 +- .../da069c7d6cbdb516c5454371dae84e7f.ascii | 11 +- .../f02e359a5e47667919738fff69d2625b.ascii | 5 +- .../2249056953a47ab1944bb5a41dcbed8c.ascii | 9 +- .../a37252ff55837b918d9d64078ae92ae7.ascii | 5 +- .../ec93d3c42f0ae52a05d0aff9739675e5.ascii | 6 - .../056e25c397c524d80f378ce3823c7e78.ascii | 39 +- .../0cf0d5f856ec204dc32e0e42691cc70a.ascii | 9 +- .../0f5698b31598b2390e966fc5e43ab53e.ascii | 25 +- .../1f5b60d190a1c7099b3411e4cc477291.ascii | 5 +- .../46e64dc07502e14217ec83d755f736ee.ascii | 29 +- .../4ff41e183d60d5fd10a5d3d30dd63358.ascii | 31 +- .../56130afc4cb313e0e74cf670d34590f6.ascii | 6 - .../74e038deabd9e240606fa3f07ba98269.ascii | 3 +- .../8090b63b005bf3edb916b97bda317a0e.ascii | 5 - .../9fc4ecc087d389dd0111bcba165cd5d0.ascii | 7 +- .../ab7c087f7c070d43a42f3f03010a7427.ascii | 25 +- .../e9fc9c715bb55a702db68b2bb6da0a68.ascii | 9 - .../ff224ded6bbbc94b43130f5f8eeb5d29.ascii | 3 +- .../2ec04091c55fe31bf85ac28c5b6d95cb.ascii | 9 - .../55f079880a77c126c70106e62ff941d9.ascii | 9 - .../85d526fb17f9e859dcd7d40d22192e37.ascii | 5 - .../942e3b3cacc7f403ad95fcd4acce7d19.ascii | 8 + .../ceac4259d2aed0767c7765d2237ca1a3.ascii | 7 +- .../06bbc5c11ad3fd88ff93eb2c06177b66.ascii | 11 - .../3437a38af1218ca206e921e48678c07e.ascii | 19 +- .../35964d0485747082c0a8bedc0a16822b.ascii | 19 +- .../c3f06301f5ce610df1217bc633257297.ascii | 11 - .../c7f8cdd755d744412476b87230d0400d.ascii | 19 +- .../e0600b3be5b95f105a1cf2a2c0378b98.ascii | 11 - .../021718d3b46893b271f90083ccdceaf8.ascii | 5 +- .../058a76e3e7d67c03f733e075829a6252.ascii | 5 +- .../316e7fae61e10014000d770209779ab6.ascii | 3 - .../378d0fd8cefa688d530ac38930d66844.ascii | 5 +- .../4d23ee228c5b1cbc40e380496c2184d1.ascii | 3 - .../699459d89ca6622c90c1e42e4aa03f32.ascii | 3 - docs/index.html | 14009 +++++++++------ docs/ja-JP/index.html | 14293 +++++++++------ docs/ko-KR/index.html | 14467 +++++++++------ docs/news/2020-09-18.html | 425 +- docs/news/2020-11-22.html | 235 +- docs/news/index.html | 199 +- docs/news/rss.xml | 33 +- docs/ru-RU/index.html | 14527 ++++++++++------ docs/uk-UA/index.html | 14499 +++++++++------ docs/zh-CN/index.html | 14260 +++++++++------ package-lock.json | 1006 +- package.json | 2 +- .../markdown/processors/latex/latex-to-svg.js | 17 +- src/tex2utf/tex2utf.pl | 127 +- tmp-pdfcrop-1411-img.pdf | Bin 0 -> 3826 bytes tmp-pdfcrop-1411.tex | 0 508 files changed, 59429 insertions(+), 32670 deletions(-) rename docs/images/chapters/aligning/{a75137c250be63877a30f4bda8d801f8.svg => 016f37843b2af2ae34dc8b2975505f3d.svg} (100%) create mode 100644 docs/images/chapters/aligning/1564fdc8d17a064fea88b9d508878e62.svg create mode 100644 docs/images/chapters/aligning/64fa084c1df7300f06af13f53681463c.svg rename docs/images/chapters/aligning/{6acf1a1e496f47c11e079a1d13f0a368.svg => d412c72ed342c2036965ff251c8fb443.svg} (100%) rename docs/images/chapters/aligning/{f6767b16ff8e04646f45fb9a1f3e4024.svg => e49d7bb45a18d14fbb12df4d91e2c67b.svg} (100%) rename docs/images/chapters/aligning/{00480d8ea1d0b86eb66939bced85e14b.svg => e5db5e945606d3429e4475ff92283a9c.svg} (100%) delete mode 100644 docs/images/chapters/circles/12f78070db3292742f66cf3cd9c7ed92.png delete mode 100644 docs/images/chapters/circles/222d374252584ac37d967e3ea0f8f28b.svg delete mode 100644 docs/images/chapters/circles/22bda43c696b48e49a2aa83f0c4a4a0e.png delete mode 100644 docs/images/chapters/circles/23d4cd81c759c5374773fae149207c5b.svg delete mode 100644 docs/images/chapters/circles/474c72b5d1ad5c154d79312c15aee47a.svg delete mode 100644 docs/images/chapters/circles/485d678b1d63ed7a4c7c8bb282467f02.svg delete mode 100644 docs/images/chapters/circles/571246ffef745298873a78473257d871.png delete mode 100644 docs/images/chapters/circles/5ef0824098ef73efbd89accb8db408e8.png delete mode 100644 docs/images/chapters/circles/6e455a6733dbca47acc0a36e7e490ba6.svg delete mode 100644 docs/images/chapters/circles/9e19c65e23c32c802d2d77edb9812122.svg delete mode 100644 docs/images/chapters/circles/b5799015ebae3ab9b6712106b3b20ee6.svg delete mode 100644 docs/images/chapters/circles/cf8fc61992d2a75416894dee46ee5ffe.svg delete mode 100644 docs/images/chapters/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.svg delete mode 100644 docs/images/chapters/circles_cubic/b985384d01cb32d422f5d1123707ebc8.svg delete mode 100644 docs/images/chapters/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.svg rename docs/images/chapters/control/{20b0be6397fbd726298de6ec70a8544b.svg => be73034ac382b54863c7a18c2932cbbc.svg} (100%) delete mode 100644 docs/images/chapters/curvefitting/134baa1043d0849f31a1943d6d5bc607.svg rename docs/images/chapters/derivatives/{18c6e782012234a2c7425204505c8888.svg => 87403e5b0da3dcc4ceca74fae058fe69.svg} (100%) rename docs/images/chapters/derivatives/{5d7af72e00fb0390af5281d918d77055.svg => de486728b41299269cc990ed09d5d286.svg} (100%) rename docs/images/chapters/derivatives/{f29a9d52897d2060a0c8a37073ed04fc.svg => f00423aaceac6ade478aba2a761664d8.svg} (100%) rename docs/images/chapters/explanation/{4def87a6683264d420f84562776f4b6c.svg => 263d97ef5db7e823d8d374a2c91b3bf0.svg} (100%) rename docs/images/chapters/explanation/{a5bb1312adc5e9e23bee6b47555a6e8f.svg => 4319b2e361960c842a4308a610a35048.svg} (100%) rename docs/images/chapters/explanation/{7a44f3eaa167a5022e2281c62e90fff8.svg => 516214a32b170186c3f3f4d34c9fe8fe.svg} (100%) rename docs/images/chapters/weightcontrol/{55f079880a77c126c70106e62ff941d9.svg => 942e3b3cacc7f403ad95fcd4acce7d19.svg} (100%) delete mode 100644 docs/images/snippets/aligning/00480d8ea1d0b86eb66939bced85e14b.ascii create mode 100644 docs/images/snippets/aligning/016f37843b2af2ae34dc8b2975505f3d.ascii create mode 100644 docs/images/snippets/aligning/1564fdc8d17a064fea88b9d508878e62.ascii create mode 100644 docs/images/snippets/aligning/64fa084c1df7300f06af13f53681463c.ascii delete mode 100644 docs/images/snippets/aligning/6acf1a1e496f47c11e079a1d13f0a368.ascii delete mode 100644 docs/images/snippets/aligning/a75137c250be63877a30f4bda8d801f8.ascii create mode 100644 docs/images/snippets/aligning/d412c72ed342c2036965ff251c8fb443.ascii create mode 100644 docs/images/snippets/aligning/e49d7bb45a18d14fbb12df4d91e2c67b.ascii create mode 100644 docs/images/snippets/aligning/e5db5e945606d3429e4475ff92283a9c.ascii delete mode 100644 docs/images/snippets/aligning/f6767b16ff8e04646f45fb9a1f3e4024.ascii delete mode 100644 docs/images/snippets/arclength/2f80643c66d8f1448b13537a7b24eb45.ascii delete mode 100644 docs/images/snippets/arclength/f251e86158649c0e57f7a772ebff83b4.ascii delete mode 100644 docs/images/snippets/bsplines/a88566be442b67fb71f727de6bdb66df.ascii delete mode 100644 docs/images/snippets/bsplines/c7af721e5e201fc3742bce67ff6cd560.ascii delete mode 100644 docs/images/snippets/bsplines/cbdf5a61de10eeb6f23be077cf047ab5.ascii delete mode 100644 docs/images/snippets/bsplines/ee203de6e554936588eb93adead0a3e5.ascii delete mode 100644 docs/images/snippets/canonical/e61fd49e554a0ffc7d64893c75cd376d.ascii delete mode 100644 docs/images/snippets/catmullconv/00357d2a2168fe313cd0b38d95a1a681.ascii delete mode 100644 docs/images/snippets/catmullconv/1f9fc156aeed9eb092573cd7446593d9.ascii delete mode 100644 docs/images/snippets/catmullconv/8a2a00812363fe1a6cfa7f81b48d31d1.ascii delete mode 100644 docs/images/snippets/catmullconv/8d3a5ca7188f53b914229133b3dbe5fe.ascii delete mode 100644 docs/images/snippets/catmullconv/9593c057c84ebf9beb70fd57a11c7e12.ascii delete mode 100644 docs/images/snippets/catmullconv/c0e30b49fbfce6f6b3c81eaa6ca5154f.ascii delete mode 100644 docs/images/snippets/catmullconv/c1f8861583b4176a9b607aa6a05f9356.ascii delete mode 100644 docs/images/snippets/catmullconv/d09e7466c267614c89ead28d6a900ba1.ascii delete mode 100644 docs/images/snippets/catmullconv/defc6fa4b51fa3c1945d15449f0f392d.ascii delete mode 100644 docs/images/snippets/circleintersection/3e0594855ca99fb87dcc65a693e1ad22.ascii delete mode 100644 docs/images/snippets/circleintersection/674c42035da16a426ef7fe23277eea11.ascii delete mode 100644 docs/images/snippets/circleintersection/699e6ea5bffbe8fe377af21a2bd28532.ascii delete mode 100644 docs/images/snippets/circleintersection/9271faa4684904073127482aa91e3bce.ascii delete mode 100644 docs/images/snippets/circleintersection/b480dae093b572dc707b76eef7fbca04.ascii delete mode 100644 docs/images/snippets/circles/222d374252584ac37d967e3ea0f8f28b.ascii delete mode 100644 docs/images/snippets/circles/23d4cd81c759c5374773fae149207c5b.ascii delete mode 100644 docs/images/snippets/circles/474c72b5d1ad5c154d79312c15aee47a.ascii delete mode 100644 docs/images/snippets/circles/485d678b1d63ed7a4c7c8bb282467f02.ascii delete mode 100644 docs/images/snippets/circles/6e455a6733dbca47acc0a36e7e490ba6.ascii delete mode 100644 docs/images/snippets/circles/9e19c65e23c32c802d2d77edb9812122.ascii delete mode 100644 docs/images/snippets/circles/b5799015ebae3ab9b6712106b3b20ee6.ascii delete mode 100644 docs/images/snippets/circles/cf8fc61992d2a75416894dee46ee5ffe.ascii delete mode 100644 docs/images/snippets/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.ascii delete mode 100644 docs/images/snippets/circles_cubic/b985384d01cb32d422f5d1123707ebc8.ascii delete mode 100644 docs/images/snippets/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.ascii delete mode 100644 docs/images/snippets/control/20b0be6397fbd726298de6ec70a8544b.ascii delete mode 100644 docs/images/snippets/control/3ec466bf6e1aff44b35b8e37cc86cc3e.ascii delete mode 100644 docs/images/snippets/control/501494295f07ba5049286489206d98f0.ascii delete mode 100644 docs/images/snippets/control/6890c4028109e7d30a0b4b89f6fbe292.ascii delete mode 100644 docs/images/snippets/control/87d587388add62445ec0f3e7d7295094.ascii delete mode 100644 docs/images/snippets/control/9df7dc66b51db8e3046e1f359874c38e.ascii delete mode 100644 docs/images/snippets/control/b58fb122c5c8159938182c185f287142.ascii create mode 100644 docs/images/snippets/control/be73034ac382b54863c7a18c2932cbbc.ascii delete mode 100644 docs/images/snippets/curvefitting/06605e008956609e8844ef95697c9096.ascii delete mode 100644 docs/images/snippets/curvefitting/134baa1043d0849f31a1943d6d5bc607.ascii delete mode 100644 docs/images/snippets/curvefitting/31d659cbc72bf304abf4c9a75b6b81de.ascii delete mode 100644 docs/images/snippets/curvefitting/409d10c3005b0c93489d72a5dba692d7.ascii delete mode 100644 docs/images/snippets/curvefitting/464dbfb5adb6233108053dfac6fa4fe5.ascii delete mode 100644 docs/images/snippets/curvefitting/505ab1ada6a187e9ba392d19739ac2c5.ascii delete mode 100644 docs/images/snippets/curvefitting/7c6b50cee5dc685515943a199d7a65fc.ascii delete mode 100644 docs/images/snippets/curvefitting/940455f4016ab1be6d46c6f176fd2f76.ascii delete mode 100644 docs/images/snippets/curvefitting/9651a687e1522b00bcba063881230902.ascii delete mode 100644 docs/images/snippets/curvefitting/a6faaf6083c818431988fef49421cc47.ascii delete mode 100644 docs/images/snippets/derivatives/18c6e782012234a2c7425204505c8888.ascii delete mode 100644 docs/images/snippets/derivatives/2e90e21710bf1bcbbfecbb464d27244a.ascii delete mode 100644 docs/images/snippets/derivatives/2fc50617b6886534d1ab4638ed8a24ac.ascii delete mode 100644 docs/images/snippets/derivatives/501494295f07ba5049286489206d98f0.ascii delete mode 100644 docs/images/snippets/derivatives/50616f9c922967c0c9c179af9b091947.ascii delete mode 100644 docs/images/snippets/derivatives/5d7af72e00fb0390af5281d918d77055.ascii delete mode 100644 docs/images/snippets/derivatives/67ca2710769505572e097ffb40de099f.ascii delete mode 100644 docs/images/snippets/derivatives/8324bf1885267fe157bf316e261d1b30.ascii create mode 100644 docs/images/snippets/derivatives/87403e5b0da3dcc4ceca74fae058fe69.ascii delete mode 100644 docs/images/snippets/derivatives/897cfd8648720dc21463a9358cc65ab4.ascii delete mode 100644 docs/images/snippets/derivatives/a7c61e0e8b42010df6dab641c92ef13d.ascii delete mode 100644 docs/images/snippets/derivatives/b58fb122c5c8159938182c185f287142.ascii delete mode 100644 docs/images/snippets/derivatives/bbe9d45ab271549dea5ef54982fcaaa5.ascii delete mode 100644 docs/images/snippets/derivatives/d575699ab7d13c62f47d3071c0b00da3.ascii create mode 100644 docs/images/snippets/derivatives/de486728b41299269cc990ed09d5d286.ascii create mode 100644 docs/images/snippets/derivatives/f00423aaceac6ade478aba2a761664d8.ascii delete mode 100644 docs/images/snippets/derivatives/f29a9d52897d2060a0c8a37073ed04fc.ascii delete mode 100644 docs/images/snippets/derivatives/f67d2d379ba6dfaa7f7686a7d1eae367.ascii delete mode 100644 docs/images/snippets/explanation/05c2d5954eb9dec5ce9f6eb7e89f1e0c.ascii delete mode 100644 docs/images/snippets/explanation/2493468e73b73f43eba8f66f0c189d1a.ascii create mode 100644 docs/images/snippets/explanation/263d97ef5db7e823d8d374a2c91b3bf0.ascii delete mode 100644 docs/images/snippets/explanation/2c47081c2a9c20d2110f13daa482a3ab.ascii create mode 100644 docs/images/snippets/explanation/4319b2e361960c842a4308a610a35048.ascii delete mode 100644 docs/images/snippets/explanation/4def87a6683264d420f84562776f4b6c.ascii create mode 100644 docs/images/snippets/explanation/516214a32b170186c3f3f4d34c9fe8fe.ascii delete mode 100644 docs/images/snippets/explanation/6aa5d4e20e83be1c95eaad792517dde9.ascii delete mode 100644 docs/images/snippets/explanation/6d58ec36bfb3fcff24248dc46889428a.ascii delete mode 100644 docs/images/snippets/explanation/79832780f9209be5569447c4d988e54b.ascii delete mode 100644 docs/images/snippets/explanation/7a44f3eaa167a5022e2281c62e90fff8.ascii delete mode 100644 docs/images/snippets/explanation/8986c536df8153b30197c3a5407d233a.ascii delete mode 100644 docs/images/snippets/explanation/9229934d71b0b02921bc92594ef11a98.ascii delete mode 100644 docs/images/snippets/explanation/a5bb1312adc5e9e23bee6b47555a6e8f.ascii delete mode 100644 docs/images/snippets/explanation/adc7729f7872d71f3fbb1a79741ce10f.ascii delete mode 100644 docs/images/snippets/explanation/b5977078d36d847fb299cbe3e7e2c3ba.ascii delete mode 100644 docs/images/snippets/explanation/dc48cdf8f492b44c7602eb64ce2b9986.ascii delete mode 100644 docs/images/snippets/explanation/f79dd2f2d992e22b8d057fdc641290b0.ascii delete mode 100644 docs/images/snippets/explanation/fa3ed9a4ab61d80ec175d29533b5728e.ascii delete mode 100644 docs/images/snippets/extended/b0eb0b24e7fa29c545ab1479d2df0554.ascii delete mode 100644 docs/images/snippets/extended/e2e71b397009b51af8a3ee848bc727b4.ascii delete mode 100644 docs/images/snippets/extended/f56f59f3c0b057c8ec79a477e4e38bec.ascii delete mode 100644 docs/images/snippets/extended/fd520a6e2c7f39e90496e5cf494cce2e.ascii delete mode 100644 docs/images/snippets/extremities/2c398b492aadc90eb4e4853fc20b23e9.ascii delete mode 100644 docs/images/snippets/extremities/c4858be225d004441b2aefedacda89a3.ascii delete mode 100644 docs/images/snippets/inflections/a283e01df17f3d763ec89621f2af6c5c.ascii delete mode 100644 docs/images/snippets/inflections/be9e409d619ecd735b0fbc219bec6d07.ascii delete mode 100644 docs/images/snippets/inflections/f9f2258e59b038659087a5e87ba2e0af.ascii delete mode 100644 docs/images/snippets/matrixsplit/55b45214ba90c96978cdc9cdfee24fef.ascii delete mode 100644 docs/images/snippets/matrixsplit/598739d23cf8dbcdebe033e4c7f1d28a.ascii delete mode 100644 docs/images/snippets/matrixsplit/63d1337d275abf7b296d500b9b5821fd.ascii delete mode 100644 docs/images/snippets/matrixsplit/7a1e792cd3fa6f3482459e154abf2e7d.ascii delete mode 100644 docs/images/snippets/matrixsplit/baedd4067516584d425b93331b7ce04f.ascii delete mode 100644 docs/images/snippets/matrixsplit/d9d04b9b6c66788d18832a383d6f7ea0.ascii delete mode 100644 docs/images/snippets/molding/6f12fcc00f4106bbc920d7451398d3b2.ascii delete mode 100644 docs/images/snippets/molding/70262c533569a7da06cc1b950e932d6f.ascii delete mode 100644 docs/images/snippets/pointvectors/4d89f0042fc367f1614537c7f05389fb.ascii delete mode 100644 docs/images/snippets/pointvectors/58b19accb8a68c665ff5cbed610eea4e.ascii delete mode 100644 docs/images/snippets/pointvectors/8b15a314beca97071b0ccb22c969355d.ascii delete mode 100644 docs/images/snippets/pointvectors/afdd2dbe7690ed09ea91df63471b480b.ascii delete mode 100644 docs/images/snippets/pointvectors/b60cdba673c2c9fc84c800f07fd18145.ascii delete mode 100644 docs/images/snippets/polybezier/ec93d3c42f0ae52a05d0aff9739675e5.ascii delete mode 100644 docs/images/snippets/reordering/56130afc4cb313e0e74cf670d34590f6.ascii delete mode 100644 docs/images/snippets/reordering/8090b63b005bf3edb916b97bda317a0e.ascii delete mode 100644 docs/images/snippets/reordering/e9fc9c715bb55a702db68b2bb6da0a68.ascii delete mode 100644 docs/images/snippets/weightcontrol/2ec04091c55fe31bf85ac28c5b6d95cb.ascii delete mode 100644 docs/images/snippets/weightcontrol/55f079880a77c126c70106e62ff941d9.ascii delete mode 100644 docs/images/snippets/weightcontrol/85d526fb17f9e859dcd7d40d22192e37.ascii create mode 100644 docs/images/snippets/weightcontrol/942e3b3cacc7f403ad95fcd4acce7d19.ascii delete mode 100644 docs/images/snippets/whatis/06bbc5c11ad3fd88ff93eb2c06177b66.ascii delete mode 100644 docs/images/snippets/whatis/c3f06301f5ce610df1217bc633257297.ascii delete mode 100644 docs/images/snippets/whatis/e0600b3be5b95f105a1cf2a2c0378b98.ascii delete mode 100644 docs/images/snippets/yforx/316e7fae61e10014000d770209779ab6.ascii delete mode 100644 docs/images/snippets/yforx/4d23ee228c5b1cbc40e380496c2184d1.ascii delete mode 100644 docs/images/snippets/yforx/699459d89ca6622c90c1e42e4aa03f32.ascii create mode 100644 tmp-pdfcrop-1411-img.pdf create mode 100644 tmp-pdfcrop-1411.tex diff --git a/README.md b/README.md index d52ed1f9..0756e30d 100644 --- a/README.md +++ b/README.md @@ -18,6 +18,15 @@ Also note that you will need a TeX installation with several dependencies: on Wi - texlive-xetex - texlive-extra-utils +Specifically, you'll need `xetex`, `pdfcrop`, and the following fonts: + +- Linux Libertine +- XITS Math +- TeX Gyre +- TeX Gyre Math +- Arphic-ttf +- IPAex (not type1) + You'll also need [pdf2svg](https://github.com/dawbarton/pdf2svg/), which on linux can be installed just like everything else, but on Windows means that you'll need to run the build yourself, after which you'll need to put the .exe file somewhere sensible (like `C:\Program Files (x86)\pdf2svg`) add then add that dir to your PATH, so that `pdf2svg` can be invoked like any other CLI command. To make life easier, if your distro uses apt-get, just run this: diff --git a/docs/chapters/matrixsplit/content.ja-JP.md b/docs/chapters/matrixsplit/content.ja-JP.md index 4e90938d..630fb4cc 100644 --- a/docs/chapters/matrixsplit/content.ja-JP.md +++ b/docs/chapters/matrixsplit/content.ja-JP.md @@ -562,6 +562,6 @@ \end{bmatrix} \] -さて、これらの行列を見るに、後半部分の曲線の行列は本当に計算する必要があったのでしょうか？いえ、ありませんでした。片方の行列が得られれば、実はもう一方の行列も暗に得られたことになります。まず、行列***Q***の各行の値を右側に寄せ、右側にあった0を左側に押しのけます。次に行列を上下に反転させます。これでたちまち***Q'***が「計算」できるのです。 +さて、これらの行列を見るに、後半部分の曲線の行列は本当に計算する必要があったのでしょうか？いえ、ありませんでした。片方の行列が得られれば、実はもう一方の行列も暗に得られたことになります。まず、行列 ***Q*** の各行の値を右側に寄せ、右側にあった0を左側に押しのけます。次に行列を上下に反転させます。これでたちまち ***Q'*** が「計算」できるのです。 この方法で曲線の分割を実装すれば、再帰が少なくて済みます。また、数値のキャッシュを利用した単純な演算になるので、再帰の計算コストが大きいシステムにおいては、コストが抑えられるかもしれません。行列の乗算に適したデバイスで計算を行えば、ド・カステリョのアルゴリズムに比べてかなり速くなるでしょう。 diff --git a/docs/images/chapters/abc/7a69dd4350ddda5701712e1d3b46b863.png b/docs/images/chapters/abc/7a69dd4350ddda5701712e1d3b46b863.png index 2c8b018c6089d2a9c972099ee16dbe77f0cd03f6..6f8df6e9ce5a3b6c8104be497cf70beebaaba1e1 100644 GIT binary patch literal 15019 zcmb`uWk6JW+c!F-h;%EBgrdYqH_`&q(jW|iq=2L#-7O*rQle7Q(%qoa-Hjj(A|(pv zn&*D*_dOrahvS~Tw+w5oS@VzU`qhfiP*Wttzk!cHAP65Q$!Q@F806@G@o?cC%KYUT z_#ZAxMNtlMiT>w(V{QTh!Gw4qcVEXdZ71DZ|IOI^Ni!A}0dJb7&gxwjmW%;KYj1=K z*TWHp@+w1ztw$$zwR1Dtr^bJhlPU}T>FQQJ^y&TLfXnuVVhw|NSVufln&|l}bDB_d zv*eVci~D>5E@o!C=4#Dx)NiRzX1A(+PbAts#tlyl7dkDXk;&59+T_@n0k?DCVRwG? zq!=h$u1z+qsjW>Zm%GIP@5E9MJV7$!1@huXM5Mn;N(!y0s33~FF81u1PkI0{)Uqe3 zbzy;#r(qwHVfSHfc6LDbijCj!@UZAz*-1fGb|O4#6pH)o?5xJ~faTdjQ_N$(f0Xc} z=z7EPw4GTrT%w|ae|2rmT|czcL`jv-f2P{;CO0?H{QUgs&yR5xsMqkB8zZ?Et6$^y zkB__3L|g_ZC-D%TTSe7_^&XDc*w{>xk_;uy7jEiDhQOQd6g722=^)`1Vq%-2)y?eY~zpa(BwHo%@uI-q*I6JpqUhG|d zsH?ld#l>~HT|VHnIePo$--B*pw@nILTiboOqs^G}X}gjQO7e$xA9e=X*T*I%+Gl2H z5lj^LntFP<HO&cl4(4!OvGB z>%IOdCr5^rtWxah_GH<#Zcm=(M~z31LjL}J*_$Cj`|r|vN=}t3E-^!NrthDt#0=ss-x4^G_LIC0XI1NUH8t<^^RLQek+g>rju?fO5~Dx) zI-ixQl9GRY zeSIR|DiIlZU@1vuW#!lJ-(_TFW$Ra1;_=`$QEVb>Ys-m=iCOrDxf*Ui7)Rq4#n;~6 zU{&_^xq4Hjp-hl-p-P0r0t0Kvrh`Nr-pfn(#N zz((&NsG(V0Tw9I(3e*QdL>gQKlC7xN*wmpS|@XyP;)#yei>WVitHFdh0pkDXm z-Sih!CqC@=E4OdoW`B#mb8?cX&aZua^6u_}#m2RPfBtx#{`rIY`il81w;I>m(MJE zgX^%1*3qmZob+pdzk-u z#Ko%qo7JD?M|yfI7yr(0-Mt%Z%sr9c-rapAKRgQlqdZebQX)F}9 z^cq@0`xaP91PlpEtkAP?EmZHAI=uL^k$ccDwr4Hhub=YutJ0gq#FR8X_)rRKLk9T3 z8mA@9&CSi0^+U}^k8r4|saZKVEWW?FO(XfTrgc0pUq@FruzRJFR(9hqahxP?SE*Is zjk|^wc;!}o_dI(iCa9m+(s3p#JD%L;;J5<2Vf3PZ<$#2O;z>nxK5TWO2Sx|z9!!Cc z1N$HLsMF!vLu?!z=G(Wg!kqtHXqI|XX99%?vVGuSPIT5Z1P~ZU4?sNGU0C^z`O# z|4wHX5xEY343Sgy9y|!MxjHw5p%L4kSckc~N9Rx0pv55&t3Q4$edEaF*MHk|53#>f zy?~mouwh?UWhf0cVTo6Ag57Og?@p*!Y#gAKF9(fed3l-XZI7nDet7xQ0cd@g+BvGN zpB^NT(9z90FLHm-EkKVx^jFUpFH-1BK7GQ5e$@*2V`*o{LAP*crrP3Qxij6?5{d@u z*XlPVb#=u|Ngq{C*|^Z#873WfZr{HD;>C;lB*m96#mUIYNoi@Lt`IXUhB5lKs7w6| zhDE2EDM=qE{wuuI`C9Q|y~{dLLu2DmfgX?A@&i_~CstO=up=kSpAvH%h`Kc1(JgFQ ziIaj>Xq8MCS0@tu@PlKo#bQfq z4L}CAAu}^`T9-+UQ&!da(A>-Y)+_1UP<$t`|8$A-bH$C0>OgI3gC6qeQBkN&cPzcE zipn)DEiE{Zk?G#OVa^vS*;;@He*YY5FT~JO^zR5R#ix%@oVjQ;=I-`b@Cn=Awx4bC zq1Db+>&$tW`YJiudQ8eJL?+ICm-n@M)%S0j^eVv~gmU^1mcmI6mQCk6ge0-0;;Qp*80<<`cj2b zB}&>}e|>Tf&=HfU==7?*Uq5C<1l9*Dg4b4Tp5iT0d-y_n)X8G1Q&UJ{3$BQ^HW@m} zmo&NRnF?B3WK|9`p~1n}e0&w+fG2NpamfP$b3n~@jRXAT{hs}>MxU#-m*#Ly+|{yW zlmUwVU&RSYqvl_a6c1#3c zU}-0(zDCvJJlA<;I6OiGDA?fJ$TTk@XX?5Zj%z^dU95oPY2f`o8Z zd}d}Gn@SQICw`2RYy%8eSyJh?@>)O;1-mnyfma~pvF{nq`N>{4jNQSgw)ztEHTRLk zp)g}^#Yc}Q32WXV;5uc{F43qO4-c=U;l*JE>Q^bPUc$SuILSYGCI&7Hji1MCS65dT z0mR7yJ%KJ4jh+ijq9@mGB`Uvp^M;9^p8{$LJ+;$wpbX*AI?$C8fskxIDjHkwqGbKO z`bjcdE5f2X)_iZi5js{klqaOkFe?LUPZK3eUYxG{oGRx) zSy+_0tgES}3O<}+omlU(|pW$hf&r9(C-hwY2Uo+$yHFo|RHKS8Kjy|H#tPB#BJ z9M8Cy*Fh^kON-C*m&syVtI%iVY#A>v(c|r@An4X&72$}Y2_slA^NrrHEA++a>x0D2 zih`MmM@~;Ye{W3;JXv^1h%eC2Bk(Z%&_(f8{k0B!fhXU`bt8l_!w?=LIHAa#9hRf* zDRQ7P?fw{8qCPRl3ILK%Up5)cNC@+-V$3`|h<=gW)!FIh^;&=CFAbL3+X#~Fjo4GO zOd{nQae5jW{!kdGi;IgrH?rTNSyuV7b4d0IAc~J!Ss0p{nkF|m%Br_2<-|AF^zl>% zcpX%hg1l<>J!uBznKAsAmr>hOfKAujegKH+osQ=%fWyjYsvn&L#Qpno=)4f8eW^ z@2I;h^GJd{6sK*j(nY(`$k1>fV4wwTOwaAI?)^Q#OIn}v;~(ORMr-C<@@?W!XM?xc z&elHY@TJxBbS-8y?VtRdov(o{-eu(27y_OAC3JRxB-Q8F3rqqEscSQuLp1l zS|}qh8$&RI&)-Q3T)^CT9`*@o8XHGf+D)KlYn+o*hP&dGv!676MY6ItkECPetUYg3 zz-PlE3w&tA2P8TWwk?|8={I;H-M1$(gsS^1t>~_%sOF?EDc=dGZMW&{Fod(TrUs)? z5p*{(XO~cI;ag{i>t#OY?!h4;2A0)gMzR)-G&D3y<56@g{3z5lIDYuO#eSMFPyeAR{~eRswxlbnwTmyau`d{& zh5zRJi&tJ;OeE&9+W`;&eTxM2q7}4!6%Jp$hJL{*z3EI()X}3*pc{%CLZ;0UNRZ(I zgUUtm$FGi1C+1bjw59Wk6vf(DshK@f!~(ejIc)o zP0jSX5Lq`-^bRYeuBR^V_ynaB2!=KM@i1m3yAK6(y3$VR;X?u8c3wU{a-PQDWPorF zrk-Xlp@B4%#*6dj%bBGvanY!E$koIsQrcT`>U+lvdH;dY?Vwvk5z-2p0p!sAK3v5N zBaq#L`k-;^RZ0r1htPU1k9gC}STQaS1OoXCnY<%LJ484n?7mI2Hjr5|B4hLHSyu|* zQ>XpK`*~xwca5qD9zJ}Cu6W}mkK+ahH4ck=`)f5G5vvR7b>LR`k!hQl#6YpbL=Z8E zhrefPU%Ay!V8G)pWx^1{33LGXB7G z|85tw>G*^MtoKI;*?^@rZUvW=@BqxRPO^t;ZrX9`-)>?OC(`GSwM}!b|St9_!BCMf7 zyra7t%X(+I;b`3Ge?%n!YB%}#$_qAx2`V3Q)o9&xNKZaKr_if3j5%5GW1OjSXosb{ zKkG6uq|bnhvP={0n9aUVZ^{Kd2a z)8yFy-+Fl97AQus)VFDu}*sYGOdo=+MV2 zF$ww@P0~!f`68#K>mOlZ1D*OX`@zY?x$(&Cntu>N9D@McIKlWJuEo#n@4thedlQW0 zNKKC7ETMnqLmT$KYr}H~+>a*O=lTd`Ww2VY_WzwMtn>*jbSzw6c>LW^_xs}ITg9Nx znP~g8=G{H!;rWsYsuj1h0&Y%Dd<1XzQ=(JilcZI|B^_fyY;oH{>;)k*1F0XGO%ooV zBzt3bnVXgDDiOVCFbEhP0yqfd9U$R;7wdj#kuJy*4Jp5WS8h0& z_hz3<-nFKx${YQ9o77zT6^TqXo!xlheNRteG%3%T2$c{F;vy}nsGtyMC>gn^Joaj= zq8lhAZS?m?^ri|Ce^iW90B!X2r6jyO#Tfu}?}MJPN&md0|YiVl-t*x!8IqECuQpr3If0RP%N20qswxS$KI!NRA(3 zKluKB@5o33vMV_?l~y}nN7NEl+TroB;MonJQYH}|NB1SfoJ&F}O`F=RdXrn5&emwl zms6q|JMFBk&452y09Z@w;G>T0PL|IQINq8d=H}+E@jfX=l8&7DOuREux~3%Stbx22 zf0cMGK*y95_W-xmCqax;bE>!(01I=Xvh(ksnmg4AQ0PE>eR40QuU}wLpQlcb(xUJ3 z-yK2wNq;DcMq*s`_bnPHs~sQPc21E{rPlD8w_&G=x_w{V%F4=`HmE?9x^8Iw-kqZb zdXI@3IXI%{a}+FJE4B zJ#I?7ov7ScB6a!q(0*&YXnJ8tH2VEN^XM|DvQ*cve}iRd0E$xrpU8}HX$qUE%9Ezv zw|!3!uU}r3OnoCB0WetI@j4+?C`*u+_(l9w_^%o1*t$35y|YZ~e+m-WHo#-6&J{B6 z2#{lPd`fD=yD2G3+)Sc7UmG<0N}L0Y|J~%iG0a9qPTu}W%O)4~Y`!6dlA4;2l)P-t zUMRJNy;QQ0_*~`i@#JlHWyh4zP~7omKR;bVL)+&PNi}CslHFM^$y6RbwDj1U7Z783 zJ1Hw3uhapT`KH53$#GBZA-V%w6RVbS#3XsR)(-yNO%@; z(z-=44P+k$LDd!DjOmdq~ zb&`R~F$M0_3>cQM_i~gWraXh*2^csN-&4{}VFb z%j)Z?@lnRx-M$?+8EzuU)a#@oF5CR>hxe;T*BPB5sy9=I3giRJ5)#}@GezAv0gEHv zx2LrA;~DTwbIkpU=0YM#!4r`N^yr~8VPm`)7?B)4HyMMkV@9dV9F-XYH;zHYgmvA^ z=h+c+yZ}q}C)~|1M#*O!Cd?Ry63*37#}n`9E$`0NpB`pj_Wt-WDFMAB(={r&rmhZ6 z4yMa6jT8_VR_0X4g1hphk-U`JWmsY)0eFR7wy-XPEFWfS6%J-cl6J^8m76vtth(IV3+ljm4a_T(EG+yxd-Qx*MzY?wvsWUG_+*8>$Ml9u*% z*_p?v!FiC_lc^MKyT)@@_V7=U()FPTXkuN0h| zOjFP2cM!ygYffU?ZYLb(GPzzx%qMR3K48MXD?f%NJvlJAjJ(X_@Q&w3^s}w(A|-J9 za8m^xWaQ;>z^!vy?zoC53bfdy322;ajs@=`FE3A8PA=HAo8N9-@jx_oqUi9onBfC^ z9&7fl#P`^d4)~rqr)H1zCH_bu36VI>2>f>P6?V0>aMGI!EuEb>Hx*+$LatG#T@3LW z@QC8hZ8nL$`1QnowuW>-{7}_H*A6+m6dO-zQ95bKL1a1=u^5W`w{wCTj}S*oJAh5a zU+Ujg>ye^VSRpxT>9(DP4soD);bCLG8-2R&YS+Wc%8D)Fee9Zyp@_R!8#Ci1aLFIb zC>7df-afcKE3(*zW7dIZHYdu&f_ZwrkX7d~FugI&`lRuvu# zpuwHJ#o(*tV12%*_p^0gI+pl48*-|B$?J(PPLP@W3h^oXzSk0Fske~R2{#ENFBny{4(%|Iue^lV_ z*zDlYc~>9t9f}fpUBH%NceWNQ zLJk$P*|*%$r#0KC6>jPB)R`Qd5YMF$U5syiIDdX)Byr>55@8Py8`jGc$vHT10m&-b z^itHvEeMYY3%detraj-Ch=)J^1bNLC^Q$+L7^Y<5^mP1z^VF2T&5XPeVayRqS{m^` z_e4(aZSbJbir~c$l(9%k%ysha4^Ia&##OooN*+nyyEo|cr=q_@c(%s8aW?vWnd$b> z$FNFm=DXpK#EKKxAGU#R?^cCh$;PcS_OYo7_pSP0f6xw$#OS#;=NsMFyO`NJ*P!zO z4{K}NeqJeTE~-mOu6@i}#V+p_lV@Y!^>jwiyziYrm6-MEIebp7g*N#LaQWKOKaazT zidmB?@|?33B7Zex^1k=G8%oUhZ0C(=S14Rk7c=4L!=afQGDZBwJ#;@vRJPpFso*ad zy+tAPv3Zd4@_d5>w4-MXu0JsEs7DtR6|rr}bF^=Q7I7!yI+kxOk3t?b<72<7D`?g5 zpL*%$T7>i1*qBDanCN&^G!iWzMMOl9;9-YHM|TvIG<~<33qC5XOGZ>xzOLE7FJbH{ z?)lyJpq*H15oma&yYei`43);q7LpL*td&oJDxz98~2rs z+wq8nmm{J*f;)oUj zap#_bLwy*9Gh;TI`SBi)qgy3>miCrH{VvZJk|n&WTy`C2tWar^-NJnxHDJ<{w}%mT z-zjMb)H8j7b5kL@1)Nx`QLi%fOiA#XP#z{%hx#U?lcO0VZU{NfZy)DyV`*q;hRw?m z2cLUE?15QKj81^;8Q>w9M@_Um;iasCH`&Lh6HdKiD7b7xgBQmIZiW%j%P1>HjtLr68}O5_ zJX`1*h?Bt0(cMY|hsPPsq?o~=3q>acK;sN*Zk7T%gH@R!9PESQLt$2^mF)Ewt69qO zMY27Q-XKR%n{Gw-0byF+$j#iXqT43Y)7!@-8WrR1SviPSOIbhg)US8vDmHD51d*MB zl5zzINqIG?sT@}`Iu`M9UqTN4FFE3j^ph)Oc z4oJ)TViIl2u3PNJIlLfrH=OU*gPf4jshg5f@}^~>oiN_lZ=1mb#tm&%c2OxuAyK*_>5X3DyrB$G#7~*FY)DY1FTv#c&Y* z9Xu>cW1yl$gV7om7B=GOp!_%B;l&SqTegEB1)OJ*w7i1+h++pr39u#6LLr#2A|)g7 zOrWXK2;OE-(tmJQ@3V#mb61|}#r|`a>*sI#52FRi9 z2eYH2Pju5h)G36o;zw_1FMQ$vjd-OiCMY*I*WgH)^GTypa%5j*gq8X-gp(?JkhZCY zPEfVVvM?6we_kr|efxI*`Sa&*x162;c`)E1W49u%vBFG~HH_b2spmiXa0Ti;kb1%z_}~!~rmHWTbBa!EWGH*|=1yGf z1;MYwL*%nr;X^W4Uf>|#dwNKzsBk40*4UNE1qqrVrIX{;^4!7riE_jKz7twaSnErr z7p&cj+T1sCAP>dylAdP@nf+P|V!H-sB6uZ2$zcM;6YmbdjfN}|4hUGFd?eE^jF-G! zcBYNvA;D9C?YPK3ewKh8H6nT)1Fi3{6fdtxaJ+ooq;6U0+A%;96T-!9EHCegDfI;kW=>HfZSGmj1FeWMQ=q`d$tN_RW<~OX?WqnhkMALFO>6^8^ zeXtyh^R(i7Sn2xssniMo^~=m$y*yBvOhF};*)=sZ0Oi%rVpPYYg4xQ>n)4pl z2Dar~h)IPiq9vqTHoW=@3agQn_Ge645bomYfRsmFy|%XY3P08>xOZ0$_nYj*BSXq4 zAAWv*U;&-C-<_l17jnPDF1>!jT3l2lZ)Wx$hwlsFSFiX|Tz4+^^tptZrlt(->f1(g zMYgsPt)6+-q$kbWMT7hbS zl9XONzyNz_N=>|e-(R5X&24QoG$**hpV#ggjf7ZUicU$9X-SzU_|t%6!ee81e?Q0Y zZZM5}YILw&mx}ebjHB#!(Vbf3-O`7Cp!&aicf;_isy<=;<*3auYm?KGOxU`hfPlzq z0v!|$@U*_c{oiosVVB;@=z!EEeMYXvExO&fqM{?P4l3M5gUMG&Q1`%|gOZ-xj**j; zL-_6ETxd$%lgU|Xk~mXs9Sq`)5<95f4wt|Nj$I1ZGB-oSRy zzH+mvsVT3dq*_Zi9ekdv-fhgCa^Nsj?KWW;1-Z1J9tRT> zI?o1ksr&sqKl&yV>2y^U;uvBej8J9tV0&{DAL3@Idz=4BFfcJOwE*P_{$PH`S9VxJ~*J(aUjmo!QNGLb*-S}&`4afz4&68Dff@0J@;Cve>2{!+W= zas)?3x7W!oVXg7J`P_JoDXbQd$>bCjJ*=0X#O+<_blLVKn(4#v@92Bt>*m+OcklLEb;xnQO*mp%;uY@XsM%3Q_07iA)rl6YebV97)Y6iH zxolsgK>gZRUIr@_TplAX#{Uu=GNK^u@T-HqUOqZHZ|)0K?Scd~`B9O$;Dg%b|+@Qdn1mLvGMWOaK@l;V)Fmhq3+@m(D=i!d#E)2hhIRg5aw57gH84O z4`g1T5}?a{rKiu0txhNdH=Nk$Zsd}a_bD$+`utgh-~m-pLD3NlH1OEikqB5zWlF4V zV1RO2ErBT;T)yNAh|}H8_v{CCaB#5xHYpOOpg&X_z3iNIbSSt@8@|5y3ws?J;zxki zY2EC2Mx9v_W-;*@8L^NHgj|?80GhN;hiA`1l~`p!ma~b%Z_9VGCf2t=$BiJWm{(W` z?psNJLQ?VS1USo)k@(;bZwTiYoO|77=)6g+e+?bDgPa3gZ3&_SR^4d$GUaTD#4yaA zxPp!S{{8!wZGH5Y7gOV6Cudjnhfs&9X@-F_3_VktvFb}7A0${S5VvuC(R3BQgWMgp z%MG-AHaj<0&rh2KL%vL$W{Tbbgdp6f2|It#{((MudgV82d1)yCRGGzYM!#(x5!giG z$HI+JWysaV1Arp|a7lHQgo_4qv$7-5qEePdcyDhn(E#wmz(L}OtA=`HadaYIWRaTr zI&haHxJ<~ibO5$ld3kvEQhFCf0O0(e_gjOmpyQ0-9K_5|6ciL>Y2~ADgh0B-&`&W- z!whf=j95o*rZ0XM4djGL>FHx&eL@Th(s6_Scka{pcVHKdER}!xQZYxJt#uUv-~!!g zqoS@sFj+nc6Wbg+_j2q1BBVN-cO~arH_<$XIm}m-Lv*8?H2D)z^ldYAGKGS_|?@l8TJGlAxREA8R}f> z1V~CCTJnCo)?CatV)(yX&yaPI!PL_wi%LM7pbO42wbn0Odyj*Nf|CSpuebK0f;U3rl)`mm(P z_1_lkn$6ebThl}|HeUQLSU%L`^>(j!VO$%`l0|!6KVI{bWjMjLpZkOpBU!7FuKkPNzp@AvmW@veR;0>X&i+n`B+6DV(DLJu*yP?&ky zBHa~8;&&dbbYmdEla{r#WP?y_Be#i^nGw|J@&Qk^Oa3beNNn;t&Wod?sbFZJFF<1f z;VEgbC_(#PTwBYj-iB*9{~TmaU$L>VnXIrOM=Z95T$_aGYOTCV4gI+-Khc&Nw zUJv1MsO-FU<98j>AQTXY41H9fYcr54U`IuDSAQ_ujsve4HV?grhX({SC?JE6V^;V2 zZDJy`sA%0>8EDIBK|x*e?2};9+qcw^UqnCoAVg#V-qZgiUn{!B#l;ub*Tdi#Ioj@- z8Kd=lttU@dzkmNO>d}hbid2)bLUI4`iU4i54TyM!1v~UsNPq^avX9I>LPeuRCH)F( z(hQ$tgB@ou?a|yDg1b?IvBunCs_ZKj-;_y=rMbxYHfE}M{u@#)3i<>ocfP0nqZxeg zpaO&<+u+FzctQbdu&#y|B7kUm_P;^ao&JjK+FFA<((r3p1qEDB@r)`-#4S!v83;+U z@gy*b26uF*qUQ;st~-n;5PB_jn5lAKx1Xz{gb08?Z1kqniE<NR57EXXgn*uAXJ?zlX7fn7WO?>e$91XpvfjH#1|FNhpEnAO zMpgFrK?R>WYW}lX;CV9VVPmJ&wy7Z$h6TCsyd7UxA;y>c0gpD>!o#wX{y#pyAYN{p61i1X= z#Xo1rwIKX8Zt?5u`S?gMcyA*?g#Xw*y$l!+0VcoyaeKbwWy6WIuS;Y#OGc@Sm_$=lQgq?R5-;q zon}F#s5h5CLxt1Qi z5GSDuPvp=_`8M=d{#aYPCneQPv?cIH$RgVE>C^I-mKL>6UW$Q9AM{gC`Z6+@J$-$a z5Wn1p^=f~#IVQSZ=|)>!d2adhF!9Zs4iF@wlPn8ePNpZHzStbi--dF!4N0-PY;2v| z7G0%^9$pEQi^n@NcfkpJ;pO!Oq?MnuPO^^vYke&H ztiAayz8}8V>xms6C@GnN@HuvFI8);wMOWq7%y7|IRh4w~b10~?qQdo$smJfmr$3J% z%?hn&06Nnah|*Kt(MyA$m};>36+svkIiU9O$;ng2`>{ss3x1c4{UYf9&U;2z`YQRJ zE?uD_BO5l~4G0J*{PZc2DWJrUZa3;4+g85&cU0o%z!6A4 zW>;uAJmdqM&+2J^8uHy8iyas}*Jgr$P1Zv!?y$Z;T8h<8g!2A2Ai~E-LEvJ?ka~@d zFH|Bj;u}^eJSo;5e^U{3({M(=zeA%av!cN4!$V$latUIJ***|z+<`b)aw~ccLC4;~ zH6@H$CWi73&mI6unLa$Cx1(KIGM-tRb-!Ggo;D=7eorn}P4>iYV>tWgTz&PtPY7f2 z#N@ww?u^sF;{5(%4M<&z_h(34t2K^A_dd9Vo#G>N`wia>*?b)qkaofASLnG>H<9i$ z{ZZs``GgBR4B|du?Gh_Om%}-fubr!JY)p6f>lX?%*by=5@dzi;j;^jRkl(cPyMz>g zzP){6|BH8Cpp;F*b4W8WS7<&rzkly9uc%lK2({d?>8xuJDU0OqhAx;4{tlpY#-9Fu zt7zmgBs>?}^^CeJ{vyL<26Yk38QFMut|qW3NpPrz!Kn~ z>#T}xFTY|XElJ(lLmx`haWRFJdfsptKF7+Te@O6*~tlZdPVycxG4RQK~*!J#4|H9^uqTMkG`hl z4G!J7aib-SMxX@M9!4vBVPvDvi*nMFn*ZhBBWtY$VXVQy!PSv`x{hep=yadIHSI1N zqlNzA;l!T5SN()cD$8@HK3nUix)b*Jdh!_dlF6SU@bU2V~ZEqL< zm1Sc?Jbd`jc(MC|h-tkC#np!2l$n{CB%f{?kq(QNI@j0N>zSDq-ZC^ZyLk2L)!qao zj;z-KTC`urzwyP%!q)dXp>^F9Ns5@5n4O)S=GW}XJ3I5r)Ai4(y1TpAT_RwFC4LOY zr-c?57Y8BUzJ2@NZRM-+;;$ob5*8VH{TwYtH8r*5b;{*nG`f{R`U&5`=2UQRuh#I` z*y2Ka_;{5QD1wvN{EjpAp-k5t=Q>$^&_lZNh5M#V zOiV6o-?&eA+Q>Y9b@Pg^&(yl#bTVr_eVt$=xx2f2*eU{j^r^@mB>o;#+q0uT_+}xJ!VKgBjA&-%jZHv2e>+4~8`X5pAt%3Bip5hi~l6!DgmkG7qi{Mm(P-S0~hr$?d!YKGVg7#^{WjEn@aXzmuU&HVb&!+_wS zCYrduUElWU(fDO-GB7km!K=^hdMxXNmbhwY5xknirKEbr zCh3O`U?b3BL$VXRqWV90_?Is?+z^8MQJ0^%xcJVmUke46Vd}elX%6jKQ_TPC!S5S_ z2s3eFx*t?O-DLdr>zAIsKF)iap0Tbj4Ky0<+rN@5rMv}i7ebm@Rz|5F#q6iS^^LDM zH&;Pb^#W#2ORYMU_^6XED?74d&Qjoso!^;nBM)!y>bt05YwNSLgo}-+br)$BU0Ay;~{1vN3%x!$r&7 z9D#UWSvmS7)SZos2X_$bRF1BEZMIoK>fmQhcu2^sb@HAEULRcrBWC1rSmeAz`sL4= zMXkT02Qrjz85ks$5PPs%KL|BahSz-9fM`?HNNj4iU#2o>oS?%bgvy(ArQftZ%< z@@2xs&gjmlOV{T%HzS))e_WM(vPyniU%wR9PDH{g$6Q)kN=|^Qp)ohV_VMQKW|ePv zICD{vD;h1+oT-XW!ElerPr-M5+^En2QV1O;n9kFZlESL1C9CV|tmc~iB%lBKF!O6+ z0ma3`)7sXC6U(Em?fB2Kd+eJx!|qmARx$DMIpTQxS_DeOckbM& zet9Mx#UdMa^WBw04rmI~TWBUwniuez`e(Fz zgMx$cJEATTTU%QfO8uN^VDdScnjLO(nkc)Fe9cL%*u0^{>yM*Wy4GK>K(yR3L0iy8JBnb4 zu>md?$N{Z^#B|OFKkpeB7$|VWut65)REub?c3B*NL=Tl~H1RGUb?&E6N{=7&A=sem zK#(I4PEJnyPdv?(un{Z$>37HQlRVk@gUDD({S@9;ovsdMRXfjPL&n}&PIltBd^t~* z>2@CVXY2lss@a#}gIVhNJ;`HcCHJu)qaqL)85wZ;t5NS8S_E|;a&qvlIZi7W8q$6n z8oG7s7FMDbUW`V-%d_M8xVz>pkk{uH7l)R|Bakv4I|!u=aydDd`E+a|;WvPEWlbJbJ_-C`bZ>)(Z0r7(=qZzrFp!^XJc@ZukOJ z8r*$(x~o3j;6>N-?kYZ_aKYg2U3@Y!vcex6F$AKIdYg~-Rw!6mqn(=0XaN#1W)7w- z|7dS-U#}%3Bs@IcD5iiMK_g(u#?Mc**b!;xa@K_tU!rMiYwNu7^%AC#!+QoRWhngF z`VLEoOe6dDgNG02r)%5@h>6=mC^&`f$Au4%jw)EHQ~BaFp~5%!^bn^?yDxC%m|p3O zU=YOwj?O~f&cT5p8D3Q!%+lgUanq=$fQBqpm^IDruf4rnMn<`Vdm&L#YWDVmwa>Pr zU|LUq59uEMcxORQOB?jz16Kr#tno_qg|VX_OH0k)zh8oHNkM=9{YTkE2J*KcrGphq zgXg{yTn}L!{`xifbgervM0m$!h28wdMmXHUd#~?W%9$dQri~3|Ens6~YfTnqE25N) za3?!t3}mCBgtQI~QJI^Y^If(Pq1L>4(;p_o)m239{M0QVI5?SmyS7$(pVQO|-?Z=& zB~`0nIS&gvJNvp-_U+4<)M(WiH4piWUi1;kjL-QCbVoJ*GD~V zx1ow@DVd;^+kf=Di-Dv^#jC@{!-F@JqumDCUZ{0JFVBEOUY->T3u|>aPd|L^_W?mD z;UK$Mx}|Nd2SS^tSN`uICnVeXxjElZD&B>@YqJ5du~Z*%LOO0684=vQdp8;Bzl;rK zor;7(cqH(4o{E|p-of^4bVLM61fw_+tM?Ap@$a=PxsErDM?F0~g98J;P-xvgqHeuT zAm3~qz|wyK>xffaoCZ*3Fq94+-LwcYxkJR())tR+cqJ!o_$_nuD7eO-?regXi$@~l z4f1W+!`!&;`L4x`@MapvuTjE}a?1*>SdKj@A*?Y-PA6S1GKN!qp^Z#lMN2D!Vj>Z* z#xx3_jOD%ESP?DK(aPp^!Z>}P`+oIpV;r(Zs@8gGc8xhA-r&j}4KF;cefHRH=ijTh zeolPGuc)ZNRH_-TZQ1-%!3PoSFkUZTzFg=28^8l^qvGI?&2mMDeo9Coi|!4KF*FR* zjW3vxNaW$+VzfAI0M;g}utVgRFPh%xXMYa+eg68h$a>y~G)6DuQ8MzKL&p1%{Oi}R zm!!F8p`w>k&?9u|Hl03R;k}Zm6$Dtc>Ur}pIpCDxXiOO@u^Hm|tSu-S4>#1{qI|M3 z+BcnNBnEGSP1X#xSoqnNscE${7Qp23vWLWY7a2bch2}RlvcLg6vIIsk(RpBP zy$HPv+7Ac6VA)U&lfSvSxy9ye5pXLseE4n$lhYg|Bur%dyJ?N9>e=z4jwt{E%(6o) zvNZ6=X)?>dWVY}*6JOuh*!?0<3>ipHc#^qmnva?Y(iJBZm@8MV6y7ay)6lXHnSFV3 z5d$@P6GgEQQBh4!H}iTjdg!T%E`0m;t>neYQx7k%#r-jh&JxSE=EYq>TXPMs$Bqq; z`(y?5l~AZ`F;?L_kuK==cFWO%dkGQ_Ngbva&2&sRRRMxE`{9h2TH!-eRboFO0_QMX z8hbPM#d9!?4OvS0yHEJe3%9NDQXHp-eOg*UqxV}3!X23JLNvoij_g9JojeWsoRNVA z>#*}{>h(b=j|Jupj1cP!%eX=WR|@Gi2`QoK+S=M&cIH}WOG$n=eGC1>TL!QG zdwqm>W2_jHQ6s`^@-S`1d|NQ&&Vg!eE@Nmql;%)Dc}6n?&$B4Eym=+7tTHDT#Uvy+ zVCkDwIjR~MywwgT78A}ed9v%y!yyVo>!#N%z+--rs zJb&*jw7uK*ud1r5YV=DryKqKTc7H!lIDll|7mH3>>7;kAeUJUp(Oc!D;t9Xg&3cGn@8h+P z&QG48)6Y+>Nth)Eay?832X`S`u~UYUQ&0>Rxlr0&pN$GpuadeHfV}x@O4IP35VMEO zZ~Kq05zP%=e@fgp45_K9=OAyO;X#{^c5Xdv(7-9`P(-y-mBta6<=f2{4gf{-sRg@wz~y1K*}!XCWQoA{!D(Ui5B>DO{=h^-&4Yi&NS8 z&2N zx`t}@#h**&&wtPq7*}X+97G+wG7>cGcg~J$p0=z)p^C8dtuE7w&zVP3p9Ol;i(K6V zm}gYu%B`fN)LS-{nw?Q*{(E>lnGfH0H;%QQ+Ej#b?CP!Np^qmj4sU2Z~K`3 zG7Rt+3f_xx!}Vv}@@V+MwCDz(y`h$%1H2WrX3+28J0WX7u`I61+5QgchHPG9JHj8q zEJX?5Q}MX!nf|wF&q2y2!GcT46jV1^gnzuc*)G2G)J3iO-IjU+phBYtPjScZwSjR0 zcQY-TUh?z~l(BELM|IkxP(QVl?HGl{FfKkm$U6N6ZC+kpg0SVQuZxB5@RDD+L>}A?73Lo%i!#PF5~Q`- zoUB;)GhB~+tTY;))AktoKTX^4pA0CTArTRn^!}B??amPp4;Vtie!BXWvX;Fi-0nk| z+5_r4>~aH4sVOF&cBT!UG}}$*a*z@OV6n9H_3cRU9w6>rO`^?0fB64x^=wluSE2er zR^Wup|L%%et1lML>(tc9++05Izeg*Fn^m*+w!K+}d<@E-mNc~|1asR(N0FZl_@T&? zoUs(o@9u(M;1pb%4;xtP!bybmUzGkidgw9qj z^s7Da#la7k8o5D1!MO>1cd$yY_^3R|1I$PaFKD zpwK8B4x>@Xva&M0J9h#BEL)Bhnca_2Yv}5B!gVc+yCv5BEHeAU`u8L9e4ueCdA2=skn@1bV6?}jp}xol zHJNp}NHd!fC>3-t>&uWk`G)fqW9C^ODW%yqRQCqArkF>3tBCwRgFs81L8C)MFpx&Bs(>RKJ1CdTx#cJY^bfAMY%$pPfoXbf$ZgwiN2XxB#iSb zWRO$iQIKJErIMl|mW-?{d~w^v@6)kkWF?upw*&_qI{`D!5b2R6Ez+`K$Lz~6iQ^0PD45vVV3 zFYEEgJ+e7*F&GWsHLJ(!*g0BBw}SB>uXf2YWM>zso50dyw7dqEFZtttmR1>^}585xe%wh)0ELH_7Fb-p-)ri{s?gSbHy)Ij5zVSy));uQ{f8);#n1-HB(fz)nj`jrSj z8Wx*JhJ_KGuIc);LN(YQ)z;LsMe^C(E=6j7kS{Yg2-^^LnvsETyAR_kHekyJF)l?v z!b?pQm>}|y$Z@*bZ+hAkfHlrzPnnhF*^6!4=c$`@L&S z7jRzRGKjo_P4~W*UzeqIO^67spbQi2fDPM6C{bs{$b6YpScuL#m!_ zo72+Jw0nDdk9%!5bWTrC(_3j-e}^Dh{`s@F`-ri1ao6$fNEy|io_nJe8$T1R(=5-= zwtXVQtOZADb-5@@A^yvPg79<}7>L`g3Y62t<#y3j&d$yXj*dk$ivda*p7uD1R}>M7 zYyNEO?5$sV1O^4l=-$MTJdCHj{EeDAi#|I$+jz_vR?|Fi#<@TQDk)(DQ-pF1daFpw#CaMZXWHVfa1nuqHUh0a0n}WDUDM(3i+8s3fX(QBrzn4=wAdOY_=4| zrKJ=Y<_8#SB&-000t&@3T4)lSl*CwI(ZmuO8j47&+=z&dUfGGb2n)OGYD4hbhc|b2 zJ6Wy9ODuW%+l(#y3h3@swFEpIutbS?A3CQ)GFi{IBZbf>euQ zCS-}mhkrvgUO9ZOYB_)5!-w;B=WLX2rr3j%zGZH6ah{D-3F|_ALqp5T-tMoStBXKc zbN<~v|4WA08B}MTUmwXQAS7(5up8Tf)`v=A$mk2yu<2VLi)rHaIRt8=#eFQIe9ks3 zR;Q|pZ@85_O)|MkNgIUxqfvdi-^?ju?EInH(E`thO262$J=n^W9&d<%g4?IgjX=zNX zhpE!8^y6NYc4&k5XVxXJUtfe8v>CRE!&;(E^3jFS+J)BdOEWFi;%}F^W~^B0w=3Z^ ze1iGS2XU-(;a3~=bdNbnl%d`J@BtUv?ecCLF&u(^0QIBdsnya?L9lSYr_i&3Q7`ZwNuW}dtYClQa~q^LP|rWtE&t2 z&pgcUrHKf;; zGta(Pz4*zUx>@aFSt#1>LU`PQ!K4;6G{EFQt#@s-Efw-;Aug6XE+!8=WSLzoJ{0(( z7qpfTbuS^1WwqsbUDI@Z@YGoGe8ujK)yazd+SA`_Ly~UGB)&Nh2?6A&pZ`6Ra+;mA zacnrO?iaRv{=}mny0Z%|E-sSLeyLx3NZvxwG3CLO*oC%a4iYepiFkl!!^0cx=;P(bDE$ad=4HSwZ#ey^8s@OjAbi`6fr=0>VMR6`Bqe@Fd9A@d?Tp6&pm&L z5gBqzH)1M4yG!*-CPkzQ7wVc%)o7?o&}$RZpz9SE3Eqy)<@sq-EwWfR0dS%B{ynIE zxD=}^?6$0xBJVS^x>rgA11gOSO@PD<1%pIYRW;{vQJ-WpmNJq$xr9)auhqM^lj0*M zftqlRWs)Z#^kK`T2rtjKwWqr+ZP4uAANh30Vs}f1j|IU&=<7w!}4)jbV%v2Np)vpiaTUqWKE@m1p17$ub7x`Xu0=Z#4ZEE`1Sq~=B^^pz z$FUZQvM4Gla&~c11icfHd;a3j_h%RaPc@u21K5Nd4|zd~xMn`M-%lh)PAlpwYZFV1sFocDL0Xvs7g6x|Y~Am^dt~ zU-70I`sN(_c(-Hu<83Q%l9piX>V_@9qo!dF=i3~$BV#xs4DYRNPBFrUg+=jSTAiCW zMSF-I-SjJju@P6u1JA8r$CXJ*D^FLB4-XZth|fxm{Eol&%r*JS|PZSy=d~ z;Ldt}^N=o%aoQ5Fj2&9{Ojw}aG9^4|H&9SOoSvP<#Kk@Q^5{AV-dx2))`gCvAxTfM zi}!TpLf`jl&HZ}#-p6OI*yloPt8aVwyz@#_%}|8~6a_^^#bup9VanL6P4*7lrQ*A7 z#dHXDUhmTxZ!s}3Hc`>fYqVD+={-&y*hHx$z3H9Z15e&wR?s3Tv88$6ZZ^w85#G*s z_3FpCW{Fqbvx*~{C_5jN%hA@0kRxoFOElQm?#c?>vqJg4r5-kGB(a~wU0TK-98^GY zlvv^4|2TvOOnV?)laOmjY0>CCF|6&T7H1TeTp{8yOr2q36k-|2%ds7Ibg(w@cug`PEHtYE$X(mDtIeskD3n zVN2|iXIEdp!;fGXdU8TVJNm@yiSZw_q=_#~0M+M^8{XbBuCA_nGhPhA2n1s$#tWdY zZxyv1xu9}@vt-`in$k^GO}M&<)0;ZQ#b(`)t*F@ib3&omTtQQF$RXz9t$X(fLsc?M z!;(ra_Hyt5)3iGDF}l5gOX97`!-{GPgI9kQl@1RNoo5w{jE1P*|FplqqVxU}5xq?H z+QH*Cy!X@fQ7#=Eahfm`LCC4=(m=B3dw$aAiJ3RP!M(*IAE2Ud&ok5H6YEP&E3vO9 zBI?H?o8%aMuxXb>Z>{93CE~Yx`QG7i{~vq>No%0qclDat=aL~+$eD4W7K?N zhHlI3?5uaU$8&MQt`>jtRJrdc6tm;3goUFg)DhdM(3p}-y`GEGK#ZqIyMJoWS&T>j zcw;1G1rgs?f;& zc!TGCHh*GN?VRz0lu<5;J~)LrJc(W%ScJfdb7_)^*m36JuDEJ*^WfIs76YQ6u7gI> zn|gc0XaFraqIn}?qtsj9#p>Z zx@T8EZc;6`{6t|JjcP6wlo?CzcsNttT>$c<6_~ERRLZK=1bSW#o#h88Q!}$mpi9qf zY~;H)yYDUc9ga2q4b)DS$be*C_jI%M@7YPDZEOjN)-qOpG;lbACY2=lcZ(%90&?DX zUT|Gs3i;#S&TBW;3{!A`RslK1`FQ{C&a>^Hq4U4N9g6;JUTfT4osmNu^nWCY00Upvz3vxv-LgyTV)gP;8|b$qCt)D_@i4s)P z!n9UDf7-`Bb7Sz0$h3VzQO!0ybdiRJM)K)TdWWzJkKH4bT}2#acX4mWUkOy<@H61Q zbB@8Ipg(?N`UMP02cS&nt!XKstWE2lUIg>Un^2vQRSmd#Zw2pe;V@^pM}2#;Ot^tYK1)5wqd0PV9x zs+lCrP0PvSyY`3c%71K{|EmD!Iq|-68Rt0!bX$&P%mt+C>N}qM-=u|m7I%$PMfw+a zg{{9bW0W|MuwcU4%Wa0F`Au2_jNT?Bxay)}wETJ(Y)MFd1?B4xJ=C~)Q`z3WV835r z?LP21MR&x26A8M=^e8BT;AV3s8f_pDO7lNFTh(lnWz+dzuUemLzG<>vC~#YVGeiIU z$@(zi<;#~>Cq93y8RF~>xN&V>Bmh>#Ja|Offj*G7k6X)LVDzYyid`2Z2zFW{HHBE#VrnwZJGPLCoUDG{scNOdho|m;QZ~gkrTL zSWv>ijP;l?*s}w~c#mIZ&Sl)j(H^i7 zhlVV(C5L@jS9Pi;3ziws$$D;>R1)30N?B+bgTi*c-tgD;hR}4%r9rdRnT3k=bz4yS z20685U+;5R63`0pe3A{_*w}~+i@htP7-$fL*I{U_gdZ zK|<~USEI)+s?O>^nut{Q0`dIP`!l9}cr$u&TZ6i(l7|3{x;BDKLnyXDxgVea8 zL14c!K+NL3soTs=NmyiB^NL+Lh(0!_@3x*E0Lg$Bqw<9EXWXSPkX=?8)@Q1`YK=3E30|Le?B$V2jPpm(}etU?S!FfgFU=*&&hk=c&>_(XHz zmGtE>mHm+Ig~-vrBwy3zqv2x~Da(-nZ@^PTK}m_t%Nw4|rfnpwv%kMDJd6`>x5%Nq z5EvAcSzb;JolD`<8dgJt#i}o!GHZ+2dAX#wWjkr9sTDHxny02rtXY;NE!;Zti;C24 z+#m$Xiv3+rWR%m|%FW&C$dx4+;{lBZDcQ>)f<-`{TL`2xA9${%dPlryOx9{E!#`M_%0VTQxN?OKKv4@7wFHw`&Zm23(yZ#_L-rwlF6J{7`B&Y_t zSxH&hW<+ezv0yn?^uem$2aCGG{ffu(WD!w2!BNj`grLp#g)=oY2!a`md0lIm$yj83 zqoc_V9mY9>*;qRH(H}~RSR7vjD|`!0_G$#>g_DP;V~f>Gq~mpZIvIEYlIAt(BirRq z@*VV=+u8~HpKia>OrLF%N(RH)Nl0Fe%OYOlF~S2#{gl+y`IQy*>uVHhR)exsc+ErY zglXQqJBY*Mzgtlq^md;^a=X5Mt@H|S`A>Bd>F1Gb%WMz8p97%?RvGp5_wN);OyPSg z{Y;nsz_F09FcCRL0o8wE2ktwgQ(l{cpA!k$8tR=Lvg{W^z?+_ zZ2?_qRO>>g*2|O4YrQE_RM0r!BC=$~fjSL_f?ap2#)-As{8)>Ut|g3Zbm!*B(xY=!(w=)8F&qS~ z51)k$YwODD)?(C_+?L0uXSylU&&&F##=i9QY(7G~DWD_NM~S0;Nfs7zjRt9Oz4G_> zH+%926SU%n0z3*+f4GVY3Id{|b@3lPV8A2^(gI_x$y#0)2M3{#j{5N-AUwZ(DF;G( zmwUSKW!i15d@>NH508&IWMwab5alRt4xSO1ri+ea-@cB$_JHCA9FVWSzjE@_=Z?K> zk9ewnewq9GNEDjDwDt+%ziU{mNI+%4tFX&k$-ip;{J9i}`5c^_Q}w3JjQ#Hx?itGD zH}+w@oQ)lSJrifZUolbC1Zx(~E5zmL4I1%TH}Jw>>zWZO$Yu_|ZbW_YECm7jL4!uZ z#e6!d_cppHWpAAaB8h!xrQ%iQ)C9Q8I>9AY_DcUMZc0juA%4P=s}!tH?YD6@poD6W zFQa+TnD+Svw3F_-3mqM5$;PW{FfH7cR48nOKi)-KcWfrB-bau8zdR6ui4-JCKB87r zSI_#diCxyoZUevxeFN`?YHMZ*1JdQm6BQt|fRE4Gqvz%2%}Sl@jRaK6MvfWiTjLQ4 zh{Ej~_Qa@65TbT=grD)_X837`-mnu0O-)UOETf%5^9*xp_xRfc#PMCqy4u>z#>N>m zg9{EUxA>^O(9p&>w4uAAjVeFO#1qHOl?obp_rQ;;`81Uhm zQQy8$!U3@c^Qpop1vXy|E)iw5&0tt1b}c;ZDiv=IcXug(C>b6dO}d<42^?*JxMeIV z)A)w>?t*HhE0~uI?%YX)pA_Mz3it{8anlF|QDAOiAy^c|^bLbBKZv6~qeQ^r2~xOS z0oT(fPXg-d>e^@5scC6jM@DFrG6tse$OBq-KHR=6I*c8KRk zxLq_fLfS(@4fuhEY<*M`uIs!PhyGN<1hU?vz8e{OZ1VEyFuutxSI|qlacJcO!2?bF4AW z|CE!b?=Gc0aIMvDD^cBU03urB;Y_TTO(SbF4cE738uEwQ8#(nLR^TVYQUTeZqJToHXP6aZ3aSdc zxO+k%CoH^!pzpBop5(jDw_d=-#eFi)iGpwcd9jWfbBmA=0>Fxadxs#UKYsjJK~Il- zrIfJscBu`S(F-uPEqsX+0LzzAr9FeS3lk$FC^lFi!cS(tDiN@s#4%(AZ#D)ohlPjF zudPii+_>rWE4JqcA3whmh{?pAVs7bI!Fs-gHKhm-$5>V1Zq+{Ljf1%nf5QlvVgMMLzyBYy`tPFn3OOa7Saz_0 zf{kpv(t(NI3cOuv4<2v<`+f^_N>HN*eFZSL$JHL1j|PpR@dYF|gfECE&}o3Bk4;$k zwcrH;C9sQ#Ak5*u1EG0zIq(`J-i!g z1_H60m*?#L#0v-Uqqrd`s6-gI*vJnnFORX45_gcYuBi_MD=fs}%8$GrMh*_Wzw7y{ z5-+5g85zlURx2hajXy7GkdON6zjz@_vhRE6Y~Cf1yWOsFqvWPJOPa^7s$^W8c*DUe zA*oewu7u#4_hpv$Md`Hfp3R@Y@2ms{Wlc^F@@e}zy|~v11sj!sSr$=buZ_dtx9B(O+`j zgL@L>2$0P5bYFX=3=bPX*=m7S9(FaLyJL9)6p%72G`&0<024jBvvYCG!e@j@R6>H* z#>OVaT;SC}@b_@S^!`q8^*}oE`?llEoF>m|;V|x7EMea~f3$atk%bggz6hpkq|go< zTVc$!5Mk@H3|8yuV1`)0ND+wpy9@1r+sOa`kpo17oLwGL@vw*X|E#_OMpejE#9`)&D0*A1b-M~zW9p)ta9b#uD~dlfq9!I}?&2B{}Y zUjs3BdbCGy@#4twBuK@&_LJp=2#ltJv19?1F#69EYs{Q}yowYHH#^1%XI7-eVuHN{VlPX!VX*eCVO z%`sjdgl|)m{FkJQzvOiO#~z#8W3cfGUhrV+J267q>gm=r1JE#dKXl1W@BgWiGG(JC zsvv3uH~;R}>)W|LfA;qD>uSKAv0m=21;RYIaZ8}P;^O7a?_scdt-*B&`sr|$PAw`v zF0N&4>|HVy7Xc0%a^6?b9}k>y7FjW0Q80Q!T1|t-< z8_eeyH@1S1l6xin`m>^17)3xqGYQ~*3IXlv)Ra((w^sqoxff-Jw-#Q2ol=mSTQ$pm zsVn9uX#WEtcMS~C;H+)~N!JLr(g^VLqau)_rw$53r&5vW(6$_a<){Q)gUef6UxP>( zTL1oOI!AUSd`*DX`)A#DieAK!oGL%%L3$P_c;^pb#(@s18946{$WJj9UmK0BH zw`BO3Az#{Adlr@vsq@pP9WDO&Q)cPN(PKr2CGeZ2baD~A7)T&gBu`O5DUsvyf~66+ zg%EuAiq6=f|KEo(TP&1i?-Raad(vWsr_)rfrmOjHlfv4Q)kA0YWjV`+nb9hLg%y2`bj3AO>p>L8+YNtg$hr(h{j3pOml9STBn|FUCvvMNr~N5X zu|lTRukYNw`xy+CWnjl~F`i+htr)pr74h)Vqq3O>FHu%j)&p2|4caJdp_7(XX}N3H z-nDmhh=9V4va*^0|F(O_gP(uo<9X9OGDl2d2UbRDy0+bqz8DQIK?#ZYGmuk_7jI_Y z1~Z@BXqD3}ZLxUMVWF9N2R%kp4-X5oy)_fRk-L2?qx0m67z{?{jAXRc7rVU8o`}*) zAD)l8>0c}DCs2B39b&V=qm6?54J_2uXvl>0RuQruJHGCllW{}8MoT1-Yry@TKx|_M z;2dJ*dV%$?cTJ~(o8Rj|a~+4znE?6y<9_wBkCwMwdUj4uCF9h{NL<688L!^X&R}Q{ zOUui}8h-RW2dxW`^2V@(T@NtiWx$72EQsX~f$qQ?ao9L{zrEMLvNv)q$~f=4QKltp1Lym(tW?fSDQu)LJOtl0H6$;wi77@MbeZme%4uC|(v zl}R}qX~7`eU8EgO_@l^lFe7R+#QEC1!E+uOfgqsgpma>#rYX2lZG0;86%sneuOInO z;s^g_dB~o}PjHwT?qRkJ$gDoj+Am+o&&xU@-AktlDX#^cY}-2k+JO}?DB#QU>^DI7oW3Eh84+X)bLjkOlGRp zZXqq_u4S`BJ^id4OltEM^Y!hWomG;CBec<-m_lj~D2K7wCab*sZ6wpRC_-Fp+Dn)E zU>nP)D$~2**pJ@E@m}str0t{^d6c;+JvuV7)@W4t9u`^D?YsvjmG+jrJM4o$N2S-jX>;Ju+(?$K?S|k$cxB25zzW$h- zENG{k-}P8`GQD}N&NAZOuuOQT%}b%Fn;2RG-0cYo^x^UrMWuXhTrnExdmNYiplf?Z zBqV&$=A>Eu(g*g@Nkw+HSJcDCDA)vtK*00VIyW>xc>kBqCUM}kNBDK7A$@_l2xu19 zn)(q}tAT@yu-hxEDyqU%y}7oCJH0E_z(3S zyn#w8&Aqgdrsh(|YQFWDgluyzaZ~2{J-IFigzU z1C#1~eV+~^vA(!-=hycupC&K7frFWGNB)5aWINm6m;$hc^>00U_N=abAJFWXZ2OynjABK;(}tlfr@F z>+75Dwun|3nYXmG^oI6Ppzr?vesxXs1ZPj4%PT|A4F;;3r*w27pFTbExw&%Psb6~% z6B9$gWv8EOsq1|-D%E^_c`#62Q}LLM2!lpGor;}3uCbBNAN%y;B1f^{?`#uqX=$nK zwqXvXCw1SuIp3Q!CY^8dD=QseO?n?Rg~ClCu6cgE9hQ*rc&RfY=gSw=cPDec8NYuk z+`fG~I5Kjzq;|H@)b)qoU20E}PX7B#=>aT1qsYuo_ZE=v-evJRwTmR-R@`(X;rAHX z;E)w^apBk;FI!8{^C4$oKrc}Zlgp=MUC!6 zcOlAE{GeDs{(Otg=;w9V$BzeJMlIx0xdumeIOq|FN4Lpi|7ctHEAl*M8zVC{Nd9MFysLg8HuZC^;^y$<5_O|TqY?EwF1jU=~>PCjYqz`d9K$?0!KRX z34c$W{gOrX(Qgr~!CdLb%Jg`5P%xK=a;0hG;?@>#4)X;oJWp(9J$IXU&Up2|BN-YQ zt&bGoR#a3xAY%J6bhgr)fJ;Ci(crTEb~q0smPX!mZ~g~+O$ExofB%qBG3D4vhAloY z=-}cg`Y^#F?~_#CdDPL-(K<$~NfjxgtNXRS6Y#adT4UG``B zknZr*#pzO160a*0{KL2R{mzW<&6UeaM$^>6!2#lQ|N2EMEHYg0w0_6$!((E9lnMmXr{jsf7Dn{6Sdbo!VPJ^?{WC|eQhnLuC8lL z(qzcjucAFYGjt4&H2h7xXlRaIzmykXBV}Y{s*lah&3~nEMqvq@BGG23pBx@iJ%1iP zTBJr?+TgZNAtWRem=PNj6C4^!K~EnFAsXA9o0pettyE#Q_Q!SZ852|F;h|GXBbL#l zcrgrK!&>R7*khtrT`{qH<7Ikl6BR~(oxWMDd3$*gKzv+ZUyt_prE7lY!4|J2ol-o~7#`{Gk%wxjcK29`R?%c>L0>2y16&COsjTIiXS`Hr zDJM#xaQNWE`U%ZN4NeFigaRzw`cySTe}BJro!$Fizhsd7K77EKUf0pqmZ_P8&%#=j zxM=4pE-Wn>opM7ih#`QO3}-RweCq#{iK+jyHMzf#LSc>j@uV5X&!3T`y_8XWhJQY2 zy+bUhFeK0Q^>zQCprw_S*H%`~;T-ZEjVQUfyYG)E$&Xr4Qc~WDlTukv#P)-{mIP@g z(ORv7R~^-0zh8fe15bJiDLOjZ*w|RzxQ*y@q68Iw=J+O@2Am&1egrb!bgreQK2Px} z<|%sWuRm6-A(PIRRAKZp*?MhrQySu#C#`>Zub_q_q9R2-6zT=KFzV3o@czGl*(yb< zqy3vsJgI4Ej9>Ba@x?SXiTFK_k@Yu53Pi=l$7!jMnmkXa$;od82pdcg!rS_SA5uz4 zN)}Yhc3fq8WpnWH%~1H32>sV(VR1!Jg;F@IWzxqjs_~giqDZ)7{QZ&Jm`~O+d@;X6 zs(3Hx!Gza@6(IZ`+b;n9@k{d|Dx~J?b4w)5CwyJY-LV)o46D*I)f&xS^$Xv2n=c}x zDaAFH%h)AYq9rVz($EC8whAlLtK@de8XbBxO**P1@q2Q(&Utg%+uOT&dZx%C#igWZ znZJDrL9eK!#Gn)V=~Meiff9zGTD8eco?*XeiE5?8ij>Rtlu2~U*wd3F2`W>CqEYP@ zpS0lMV4@(Fu|pXON=l77yR1n(LQC~>{aX(oK2)tnd=Ht=DY=1x!BMlXit&Ib1CLTZwGuF$SlD6VG)U>wZ}Ero$=+B&*uUWQ|&s;KZ(mB91}u6yk-!x4&!@ zEhP~T6#m6}rSp+yiE1dCq6`(z9Xtlkg@pzDvi^5lNy<+5u%ef`~7VAK`@eJ2*O;z!$?J_C*G2!lLMw(ZKiE+V0+K2%;qK5FOfj`z0==aCvog zaeJGQGSC+BTJhU1cx|T6{zGPF=Dnu>ywU3I-~WDbet!O5OH>Z*!_f1$;qaodsp-%+ z=S{Kt1WI^uagS}RM^*)J16@ZuDjhXWP+kS>ul>=+=)H_(XDWC)lH^X!dX|nj1~pU+ z4C}*-DTbFQ#Z_zN$55N^-o1;Af)bmNp=awk=5Y4Uq_L?9qeU?C_}a=^9=70^WGEe$ zTTqY;5<@=P`1yJM_-1#1f1XGXW-6b1%+QcxkcTgZfjp)o6{fgS@r?2c zwkA(uGzf==w3C;sq!DXHr>8S{Qmep1S9n@!taYk(8OS-qb+(AK#_KEYdNif?CNTFJ z$l)3PAqn|2gzao&^#ij1{>e$gI#JP6PimZo_HMHB#hD7eIV>$3 z3+j!Ij3`4iD>WFnP7KgcMbES<*RXl^RC>Vs>8Pm@-^*ap6JO(X=3r=OXy@um^YUfP zWThErNR&gO5fLnULsnj5%7DN{xbMxy+;gp(@YGZi92}hA7S$0uIdBR9uVRw&u$?z; zlh!9wR-jvi)t}9Oz&hQk8~~L26p4j}<+*0nmx@1BjEoUbFnZ%iaA8f@&;a|az%|K` zD8<6Rx3%ST#-$vr4dCM94kt1tFKy>jzycTx88B>FJvNE^|CH*T*(}V)n9~w*y-r$v+L?6_I5vD z3CzQFO{bdBLKZNsB@STre!iUTxmP4^BHkyag zFDdDCcIAqC-@>Byxp2;)9Y9y^GtD~=iInD#=sRYt<}mNv;|l-y>I@rBlH!h62_moE z%h&wX7l`q~Kv=241P_WgB?m_hY^^0?Yd=}Aa6Wrh?#&xcHaAkL@mfkM^iixPBK5qG zF_oTfCN(r7=c+SLtGwN$JFvg>a~;mVQiI_*;t&x~Z_aSdW3NHoLqEH6TJ&{5-47&%fMbz(c_C|*@K7=_EBq;R$k2y!+%9z6wA%Ai&{@p?x?{ME4x)lqgH z59ht(odB2Z?>~A(5NG%L>S9_Y&B#*>vUdN7@>(7aCQtihgDbP^jJ>#0!@>(#qmhuQ zjTr(R)D#Zx8Tv@E9CB`MjnELhE|T)0^wDE~Y-1CXEM@wl6_FAPdH21iJJ;8b0t1El zudIKq@!QQKk@7gb%2F8d&`Y{gLy~kyb*ge|7B;}%#f6%JqV>?fsG`YwdOliwW82b5 z&=ywN^^bGH>yuC1`n}O@Bq1tLx|BofS%R=vL{fvuY)ICu{B_IL@-uu4C&~?ARYjmq z7_#7(%n&&K#ja}*IoZM3O~p7{Nzj9go?G&v1DhSOpZEs*JrOL#9N*hRFgDfi?6qMD z+70lZuvHUd20^%%s5e4_jUXnqkyI}l{kIe)=&+XN%2CbmUq6i)A#yajxAy7NKW4XopxMi4qo0Q@hEp9pph=&|>rFh)3C2aFXD_QK?jiEPpq_ zA32D45rFLR?%Y6EG-b*ZUJ%sEb}H2}xbsEiFQfDNlie;mGmsSC?%~YM%LY0#A`pN%Z_q9?mgEmUjD0wji_q)P>Fn;+Mh`_n`6rF`~Yf z#{_X_)|bpKlgjmw1|@l^LT#DYDr>?Rp6KxKpF!lEtAK%?9z&>t;bel%1v=*V@X7!D z+<8;~(#H$ZB;_%y?=?&EGBe-J&+Df+?>6l=CB$$$ z{+{)&mEB@zgD&y%Y`}LFHo4Rl=QE&Rq|QiEOZ0IPd^n>4lS)OS%WG@EozTnHjrF!z zj?d3xc61q{NcHW^+d+W$*IY@o1}aN3NCu~EDPtVdM#$SfBWpA#@?#f_Q;itgPx5vfFmWT;|ghp45y;DR5>g&2hsCmC^%d zZ=D`b+pa?bYw~=SfG#!RLLQ6c`dAr)YH58iI{^7}BAe$%EMp%9$=hHhMa5PqnR@9S z8wDQI7HdoWv?{?1Yrri4(IKg>PEe#)R$A)~C{B_!Pj_t zy@$la(vb754F|dSIz2I_kbp_q6I+Ob=Ea$SaQgZSE*2D0~Oj z1HtcXnB*51haYcGJ6@jJBgv)nNs5XhYnI?6#YqK3MPVrjOJqSzm|@N=jz8lcEQt&QA7HcBF|l}-W_!bK7ur(pOpR_LQrCX{BAoxOo?<8s+bH28W1 zK*Vfkrk?guXrS+ji&XDM5W5k*sto|Hw&EEAC3RH6Yt#(i+47SD8WR(f#)E#|#=TZ_ zjuii~^}D&k^7(RUv1uO3go%#34v@L_qbmjzr7T_*B!XC5wUSE2cMB~1aG8Hg) zfBnFcj5eVC_UFV2Zg+QgB;@=&qR2QgQ9FL7$Y8VdAN85{wVQP3z_q`t| zjNfVtETx?NKFuh4eUfuJanAjc>GHnx^r|9%S6XQT+R*Kp^s+MM$;nA$vDbq;zX;p* z?~~u^3J<;gePE!hUqEYlZmeW4Ur@s-s?*2!F_ZQ@Fu2AYp?JvZ<;K=Mx;_Q*lnIp~ zdq-3}UH{UW5GWNIgZxL2A6FW7+&|u$yz}Z`o&8^#Ptm{W;;826>qwoYl?xG-Bu_p= zS1_ojr-w$5%&XkToVFa+C~hZzTtmXbDEaySQ15O5QKb9 zfVKMOx@%LHMUQ(gj->(;~%XnPq*afOx9#d`btN~pvD zSs49FWI-i&BWJ?IBnwp2D;WjAxvbdl|p(QEiCWGOU5=zKRd_xw8L|Xj?W=nj;jymd@dN*1~T#;*ZL7p4S->9IaWS- zrekAcL+U?V-@6rG^OrAQDikQIUL+(W1c4MHjB4oUkSR|FefXFQ&@#}t?=J+AsaD3O zq*&H5L7NMmd(qX)LBR~zh`~Y?1`eB+TNda{u^vbv-|$rclSlo)D$X zc$ocn(5u%;k8p@7J3il{V`MY|kl+P@03y^uLd#`M49-qis2?UNS7M?%-E3}Q+7o_{ zJwU*?PoDH=zYars;POt&9Jl3ShgWPY%Wk|-rRWp{k~829il>-->ofh0#t;7DZpafe zbEeqN#{hc74h;*NZ1Utp90>ZjP+&aM0U?5NLQ6|KpJbMvB_8zzQDBck$twJT#BhWs zzZyJpr3CT_ z8ZTp516BTBD59sUDYojsm#W~F2I@EMsh&1;6Ql$JiG$JI+q)D*=A(acyxrc}`LBne z?02aSZ(BWZN9n%T?vRq)76KjeaS@PIX)DBJ|Z=XTJ?>Au@^@?5um zaN}-J>GIQeS=)E)?PFM}Fd`nsdMH$gQ$2eY;JH&D^g0Zm@cs)GldJP1?Rtj*dBLj@ zdmPMZp>OYp^W{M=BGL6&f97$tVe%#grzG-ynF={8V$t_< zV_VD9zUA?gbb5-4MUr&Gas>wkomo#nNSkd>RYRIDw0EeQx=r4L=)LBQnNkJ3lh@Kc zBi#>HHJ($s=dP^dQF`$khs#n9XLpF^3_BiroF#s94JON)wJ|L#`6h0B_;>JEGJEe< zztWP6ZPQVFpCT#;Q4r?)23~^gc54}c>?YVTSt3E2Rp$;mIz7y4j_Y`f(uq;w73Nl& zRf%;xyo1amEsD42fv>rS-eM6FitYKZ00r+wXYnz7nTaw-m|y+#)t!t-T=S?gOjGuQG!^99}0AGHS}a zvq);6^VnpZ^*WH%ZNdxsdNY9rJ)`*qr6D7vZbRUhFq)CA9648?@a+dg60xfRKZM^DQ+0)m22l=9_F zfZs#(1?{-!+X{b{qq(@0eh!od+dgBVw=2iVADfhjY zo+Ds;iw%PM<>9crX_=#=V_RR}rBoF6OIB9s!73LXe=f)vr|_hadu7iu+ZLphfk701 z>blZ{r|Y(G7pUn@M6fM=bq{DNHpKX8vpPgM5p{gj6>8dhLgV;7et2jv}C_5l0t+cBA?gD~EwmrGB(2y}G`dX<}bUB%cOG#&k z=op#R`Pvl~oSOcLiBptAJ+)pU!yBVT3_8SQWO~fESURED%C+TCn_PLQ*A)NwWm&B$ zaJG!=0u{OC`iME!-BLDMQ@!(ONHNeKWsJKi`?G=hSO%yVM$n5p&r&MBFF5p%|6OCP z1rinGb%42p&jp5qqrv?5dn~?y2UZ6%aESOD zFyB?p=OFF;z;=#^GdJ~ZwOVp+BEi~2v@xHN76wA{C=$1VO?IQm0E@9PetKrFI;D4Dp=?*24h zgdQd&fplN0+R19AA(Vb=_C+R{PVWp`+WdZV>RpAtI8BdcyyJ-FS;IKkV_shDy2Jg@^T9Hllb&t%!tDqgMfC6EvLRuyu?7;ufODC&>ky6xTxuD82z{iNP1zsje;YbnojYG& z%-y`S9D5Gs2~rZr0N+74IeB!+m#9oYT_2#V#Ky)NDP>qi^ybN~h>#Fll}|&lgpP0N z^6bD1C^cwC3a0qtZlA*8HwPM6r@>{VcFu=2Rf#^X%64w{BVqs4ldR9?cEFx1xr>7Q zCp$h^q)1K+Wx{UHA+p1fHjT&G5cC{GD|@*ncvCS=K$^g)HJ|3XVX_2p(d%$fe0h0! z4=gPR5K=il3)&CotlJ0Z&j0ms&h3F=pkix`UJ*Pd((%t(i~|`4H#S!iKHJ;TvJ}uo zNf=R~mKJ65oh`>A*ykm1t}Nyz=zeWkmRMh3X-hR|`=$F8=z})iswy@}M-!E1SUTUFgo;K1g|!aL3xbsY0PbQjE9gn# z$OZl2MxCa5Yd7)Tt@=+LkyXU6+-;`7@&TtX2-`1>*EJ?Ct^@RRcgW=QePvwJNykz$ zMz#F7TH6oJQ8$J}Kuwqj8f2C1<`hMq|L74yb|)t%k65;6w3>J|S*$TqlkWt)gs7O< zZEO+}88}YBBmj9u$jQmsbLjChOsboH$WBB@>PN&T?W6hK9pQH==xJK+51kmsC7=^R zkSjpXF0ZVxIIhYdCFS@xTXGLKM71hSqilD zebQ%VXTW=FK=F$byylXkx%)%c#>r+SqHLN>FCQFOmBKL^$~^_dcOSfc{4OjOxBq^C z%=yfN^7=$T*p<-JS#-Yx@wW>MI*04S(BMq%jo0OAvPQnrp)uig`uiRHSrb6T1A?Ht z0Zid`x)4OjZtD4KkN zDFQ4GP|Z(X@J5$bTTO zu(hJ61_tL^0%trU&deR7Yb%=cZ|Kds&0@=79$ z(phSjTcYc)mu!iQKHS0%2@3f#gE$_bW0_-j9)w?353(h>;<#iJ&d*1_P}^`du5iT{s3r zBJ$H;Ft>E}=>#!mW!Co%`R2|7fmTSkHAvcb&JIi;ulvxg*l=s2Uu`IaHA(jQ#HD4T-JK-I8s1Ux~YH ze%OAm;i0LpL~$QW5#H@_)e^d~zjT8ebZEbG6y8OJ;WkUrcxk8Zx!MbxlUNkd<-Jm} zI(-w48?F?&bl{jl969;#YQ_986(U4}`ElzE@)<|DwR}by3xbZ~=)1MKwkEEsivNb` zP3mAOi@WS6%SQ^$Tc<$Jvf9nR(yX>H$+|c?D681UIL2L$d%N$fl#T5n5`SJ8&N)Gp>x z2fCKTPrtm5wR;vX4#v4X7^7}S#OpJ2p4VdNTjg^cny-kJ>FZ6Msb{d!CTRPtnsw@5 ze3orl`@)@`ByYvQnSvJ5&Qe~G2frtS1$tA!5><>&J3NY~Q-1-{I4ps1V z>+U$0)3YEMXF6S`{=>K1Z4;5?Go+)*v*at26A_$&qj8K@3vxT{DAOg`Xt#cruGzDc zB`!RCf7(_2mJu^W?)$fI-(GqgzC}>p#zIsBH6Arn>;eJ;|CN`ex0jbip_@lI(8ME2 z763Qof|LQhQAESex3HF*t6i`FJ$(Gwj5bDPI^Ggt6k7}31c#ujNM0GVq5ET6wprtw zi0*FUn=`}g>@0%SL4csZcD^Ghz3+ibD%GxU1qd`*XHV&SeUMS4lRu394do1~tD<{) z&ije)_oxjU;@C=b?EnnlCYTbyeoS*0Ge%3!P{gYF4#(4{??8vJ+ZdrkAV9DZAwhBr z+&%=qA~nqi1iA;lEhHksq~{YI5EzJDjgT8PGzj7Lecw;v@}=;Sk&z7m)_ec&wuVt> zxDcw4wsz`5GBi;Cx+tS2L}kE&V1!Ng-O6yi^!@Bnb3*PT#jdE!t~D3fmr&BZF9B_;0y8kTgLU zl}Wd;3kaj&s3-)B2-M3EAZL*EtZGVkVF3WBy%%!x^7f`SWV@c6Wz>&QR&LzbN(7TH|joQ7Ek;4X;LPjK?G}&5I*_x9`=U zKm9GDIu`rMNX_u2!?FYrwGRycftC4=uhj~^FjPHR3k!Pi&LLbmK!kaDc_9K7`ho{R zn55ZokFU-Y)_;=u8iNpl;EdyTs)3wqD;)s9IAwR)u-k03+!f8?Nlo5kq;!aiJ#R?O zlv^6mTudG#OBBQwo2q@SEnj zWigj0&-zQ_?t8n}bLBi9Df*qjK4KP}^M7#?GzEUb>WNc@6)wvqIt}q)6D4>VJ3jsd z(KaEB)$pDFZFCvTmnSP??fLnN9H+6jiEFEM)PGZ7Mi{)fVL%c}nW_Kt;A#VMsVITU3qgt8O}Za<7z z3czR91^zq-a2{espf#Ux$_Oq&ejOH%jN=xkX4%+3IvRJC;Ap2xOF;A_*pU*ML@N2Q zqD*lng|0-dB8&Z^C=eRlZhKDwaOGB2ee$_FpHx=2BByPbjJS`tZEkyzQCt?4tjv?57X*f#2Yl3hahXQSZ#7fJ}$(@ zSK1d=DZu`e!E8Fl6_hPLK+RDRCg_J>?IHAURNKkpNBOZ(-Hc{M5{ESr_|5ZWQ#ic9 zz+K~OCmHrCR1(*~yzyf}yR6*(zYDva8r97)nU5^0W@v{p2iLiGTs|U|sXD2A+6@4x zIO)qx?92JsaQggB#XkmreGr?hrfVXAcjb_xN0vMQGY+XfT+QB_93BDWaP3q)V?0aq zrIo;xp^(1Ni1yWR>H?~u$XmKpdVTCky8F;vAUHbMrNM8`mH>@a=`=ox8qz7sY=tK; z2Hd`L2SNf1oKz_3>FHc_netx*rw7ckomB1Qm9eF0XpsFfB&mKJ@Q$b%?B+f{uiDRP z(mTqXYi}17dHDq)*Ci&>f(eKQt|}ON2>>gFFU+~E*diGGVmlhDk(KGNr$MVOj6By z=e97-@`Ll4*57DuzlH7I9G|NhZQ9>&izX)l5rKvC#nx|UdwT=OidJ*YO%-I4Ll3}0 zZ2AgZDbd?BFCjXaa`TtG|KeiWN-m#eGahb^N8S2x?}OtJhP3Zluij~|-u3Ov=-(bd zSVe3%e@=SZ(f(uKMI}Z})0CEbP=GJB#XsmEATz0_8-hX zU@kiXDd$7ZzY{757!aL;(T+PN-?k;mp~@MrDiw~hWO~1-STG`BH4P`%rDb7>mKQkl z2Qdo@aNpiSTS?^O7zujB_j1uant3=DYLs+l)$&m?IDyxuU+&X}leSvI(A01QI1yB) z9P}PCvzh)PfIINMINk?Kjs__+RWx|fKNHR`Zi%)i_>P$H*gT5M9$epdan7~O#L{JA zUsEiA*z9UhOmHI)K{$dK8nsjRUsm*vDsJGy$HBsu_?c;6I3_;gzszt%%p>YY0aV8! z;2={OdBkk2t=Ba@-B6$t$MzCDrtQ?AX!>6BwOsx~hl1HTJRA!gCjvr2DxVnHVW&m( zRZ_05u3^XcT!wN|=4JNVDu?Pwav1`B;71t|FvM5&!$n5oP?5PyxkZ$QXtLz0w~bO~n}VWxtW{H)pdvAyDB zM&jpgR7^|`NGy2F%H!w4F?lfZ(9ZM%kJ-4fq(!Yv7mBS+C;B7{Y?w<>!MSkf`{0+l zT>=QZr6v7h0Ho?S^l|V?B)nozVBGl88w{I^AG_ShFA$dKg$g#^FVKw1Zd+6${E`}P zs+)g%@MP1V!@Y*9a zWQs9%T?w$h{#%xW0f_FtzPtKuff!)a0pUEd$nwbIRv#(C%r80mO4M>O;A>^&WQ{d( zKtMoTq?w6HG?06cSPj>CQXgiI{GV3O--Y3)64L% z63?No!Vn(DyO9z7%JICsK%|cu8FRGFljuDVZO{8|r6eSn4W~%M=;YEAuIT{nqm?y( z?4xf%-u$-P$HtiI=JHdOw*>_S0SO0(_ALJ*9Okvi$$p73J~nFR>8@+J)olR?|#0lYw1w}5p1ta_4?{vqNmh?cklO*8n(Z+55jMem){&bD?Q4MS;MTH-P9?ZE# zNX2!ha-%uzrbjk-67KAnz~7+1LHnYd1Y;^-BUdW%2HxgiwQmW0-$;Ny2f zdB+&n0k0(jsSfc{P&R5V5I%JqLK?@5(!+bWpiz4 z=n#%~+THWgQrpA$^#33Z?hG7Lu?dXb6Fm{Ih@ZlR<-b%B9 zRnTm~HD0XivYZNw)rD}1NybuxH~$_WVJlqTuVxg{;1`D!E^3I2SQ?xZ7>c2Yot&vXNl*%ap~!D1L^#5K-jcSl(Q5v z6=>~TT=HPCfL*$W1guo$$;;CEZve)m)>O-Mt&hVLj_RS|eLG%CN*~u_+@vP`LQQK- z^H)<)Au2hTF}ML_gX(|_s|(+5S*>?8DLeY5WLtyYtN0Y99IP@Bz;8k10{-1x=gFVX zzIX^j3u5fkbXw!$+OlZha-%^d!w55?WBp6fyO65OBa`l!XNEA{w+QORBaoB|j+#O1 z#swY*e;Rx~EkIteUZ%n%AcAQaJ1HE%d-#wXlT6@wjm;l=u(}%Tdl#KE`eXCD?mQ~d ztp3^)$nDAA6lBsHAF}hOAq?gZ!m_g|?7=<@T$9@Nx5CNJB7qAS(_|bg)nR}yrWA`H zG8wE)^eBVUKmkU5aDKf4=hevNNA?NUlQd!hueymQPuG%W_48{>4;!OGpV|%2Uk@wy zwPskkp`yd|1`VLqz?fh$KjRvm(H6N*-`v6&P)h+QBGdR)zWy21Nu_UVeZmBR}sXcYL!0)WZ%i6zS$fB)>R$QrOH4 z=;w{Py832i((^)Nr`e8<*MFUOVAwDqEG+ExfZ~EU@HpyC9@TJpkY2;Um4*tnt^yif zqKv_*-5GFAx5Gs(tu1@Z9OrI14(Ncr?6U(i z7ekvzi}h>iMO1z&yJjJ_@y#f4QqO!q7ANw?%XVP_UH%bdCJJU|`6`<~Q`u$@g;CSw zXux80LpTL{7X|~Lh%-3|UJxShugAQc?*EJ{O#70FrZpEUS@O~w=v7*@j`*qljY2~@ z+EO7mezji`3k?mOsJ2YV?ZaO2Q`y#2LChQg2ZpgW=XKCW6vPAm4N!uWlif$(DJHFnd U^B>`Ne;`S|mJ=-%*7y5=0Apu?MF0Q* literal 17543 zcmb8X1yq&o_bs}mq`Q<3DFNw{ZjllUKpLe}BqXG}MQKSv2?+^lL{d6Ly1PMIK+f9V zFV6qmaqbxRj&C@=!DjDwzwh&`XRSHsnu{<^btOFPo7e~h0`H!(ycPm+jRg4*69Zl$ z%U`I5|6!QkSCU6uA^*#4_!y5s&?D~2%RF>X+kXE_C;rKwlO_|a`JU92(w8LaOFDIS zc7CgM*;G~U|LI)URb@9xNKPKy=zVT_e!T6G(PmmE;4{(xXf-N2`d4b=f=KoF8{6OobS#@qL%2-39Pzw2OT#$?`~{#m$0BKF6#YL`uN zmHYPx2L=$bm=~9q&8r7QBqXRpsdf$&cKCM5BD}Y5-BMOj@ec_JS)nkfQ;3U;^UJ-T zXu8lEwzt3ET;4Akv9e-LD7$w)dqw;A?_WwmYf{5k%Vh5p)D;ymeia);RN9QXTC#uI zwfV|?SjqM?P1G5K-=a_c{_%`Yzvcz{!D@d;rc8ipmyOLcz8mW4 z;#g%mk%kQ_v9YmXGYt{f5F;a(4|GIuV(1Ry-oKBksHmVFkFx!%#=Prw+byqcoS z?4OZAN6w`e$RHOyH*S&9wm0A6boKWV|IDH{1zj?OfIBh5X|a8NFh_}vgX8BT|+1AYC-&Q4|OyRJoYuPG>|za|m-udP|8iMe3Ed-u-$bdN#( zz1Rpt#@DaH1qB84iQRXnx;r}&3?EuDBzcp?U-9wr^FMg_FmUp#!{VevQ`^#y_@jhJ z`41O1MhZnd4%rkVDYN2F`rhAFJeUZ-PC$!55R>rFZxpOK&bBG|J;>p689FBYlq4b& z78d4yb$L=|+>C;ThGsnfJqYlq|a;PUwlw`F(BP1~~QK4My^87>z;xbP)+5hLyheZZ;u3_gRKy%o?_TAu_GuXG||U<-S<|U zGUhOj9$TNBuI>#(_jz=$i~Wuy9uxnDX(xAlI!W9hS-)HNAAg%Rz)t2wxpm8kp2M0~ zT)Yc`cu~W8yfuz1EG!IL*xyGxi7%|7p%I@jE+ioEt-n8}?7Ft+aplMKviW^?_vLp_ zV&A-3(EKY}AR#3sg**lHp|X~i=w02yOw3E#QLzYLX~NullaDfJa_d*da;DR0!#$>Q z8XIY;gzeEH+P-|fukKfy`bd-K`tmZJUrSA(uP{$TwbtePnjGBV!IEKRZ~xCT)?Yn!2%Jt(we-uFp-8pP&D!xOis| z*ZH8mt&QRC-O23QEK}L6q-&|YiYhADC@3h72j-&c?>$gGk8Ru2B~qP!FJWV%#m&8_ zNs{yyo0^_Jniy;Hk%GJ?SmL%b6-IN{H8U>{qdW0Vhd(+FLYn{nuk?|^CuU=%Cde>d zSXkKGsoQl`HnX3BE#Dd|%en889n7tlquC?PNuEvN@=!;h~_S{`;m> zSP5k{A3V5*9}?i}>x+N#PUV)qfq`0;u*ae0+r&g0L!aYui=?}5+z^=oiH08e!x!3l zs?EW8R09J8zEM=drd@B@S)?X4XMX>F!OX&vVyi?4QSw$D_7j5v4HD}wPopf&Z@s;T zL;ao5J%O^k{+ev^&d&U&UX9I_A`qKJE@F2YPZ^O3z1POO zoO^?k{mVVCL8HPRq3`s z*}<&xWGZji*PMu-9@xm^!6lYjGZ-S9l?l58dya9`qvq~jty|7t@q7q_j079Ksb}hN zbF_bJpP)v6CI>}{%0C<{x5KQjCo1cr;<;>F;7|*%+e)5K8d6q;UN z)>2n}$;y&BxO6)FJw#bb8cwEpU6c=b{odYQmP^VLG*}J~YN!Kvcz6aWHC7th+9dCP zyh0%(QHhb2a3-Oo>Io%U@pCgeNF z@1IO1q1Yu~Aiw)->yU(S9wTv>W%s{6k8jgERiTutCBsJQ>XBxl;RmZWY4SNa^*B_X z@$~d`fTl$w(^Pi-v5qr598_scZ0s)^{ApMmEN|@tDlP0Mw6(Q!MA6caE38)9)}!Xx zBOy>1pz2 ziI5$`uJpHr5D<}*n}1!s%ZR+6014y+85%}IBC7i{XNi0x-fm4!=JnT@puAHF>grON=(V~oAsn($ zs$lheISe@G)Vq#(hqO7b8sz4LhbSvN%LTYMNix-D*Oid|v0FLc8UC|=; z4u>^by1NU9lT*KZ;geTVn(R_&m}>B(fDcp^5=3K@k%af2oi4_dnfKl#loc2q8Xn$n zzD`HkJ6d8SMM_3yrN?qSH`CxrDdLzpoxa;Sz)KRYR^z;igz5uBLnvYL*=LjWulV5` zZ3K2x9U8*`7&G3vL%_wwg~Zh>{TU8S{gatoBrO98SZ#S}Qa*+s6|xm`wE)E6Q1U$s zo$(4J(MvQuA_5|3Ke6eeZGwCA)N!?srp~RBn(QWO!bEv~!>?bD8?1CW z$@QK-#rre2vx5&OiqoK$^nN1uNX$ptjuAf=$7D9m7W~pI-dw zBJ-Lpq=-$6ycf?R+LpPZA$1uzd%5Fw?Al@i1;6aZhJ^6NOvAj#86U=M69F3f_#{_6 zl99OW($aVdqHM2d+WPmUrw)Jg8_apHsO($Ct^Vv8v%#4)8bo{Hut|UKe9O1`SI4g1 zoUbqTl>mMqLo*oStKj9B^Y8?Anj~INvYEqZ&%&Rh&4;RDkVo3A*P;?!{|FG$5d;Q~ zjf(PX=0B`!3HRPIWFz zy9)8R6??zukN5XSVZL<>c}5_pgPd^Fo#W$C<&{vq|BUt5aBFI6`aZ}}{Lil+M5a40 z1{zT3YK3i%mY4xXI(BWFhlMb~-A<;0q&}UBjUM|BUHq!vJ8hHktNpa>Y@_%5?ydve z7cJbE;@V+aYPrQVNjk?bOoGnN&PW0P>rM){#v}W2bxOp9;yQxcuwJ>XReo)TI=s#D zcxvwqYLI2lLzIr_Yc(}hZha^l8ykC&+yJ$WPm+D~g`7@qvX(|g!>y0_{vE-3mlheo z_wL=pcj+r*f{Z!zJo3GBPC&e?6Ssb~@5gkHt{Hm~JV;Ci-`_ghr!%8P)4J5^QYWhh z!*X!2m`j^|)ABT*WD3muSKXpn$g29f>bg!Wet9`aeG^+}t%R#PH8C;d@8x+wo2knm z{r?0g|D@rgM`Qp=2I2T!q83jatG`Dv^cplZvNM>0i>s!KkAJX$%4P?MsEzX{zp$96 zJ+6%MQ;O9f3fo0$xla!{E_QvcS^}|1wzk8yAqSfBvGh(^#%BaC5|^}G6e(gY`?@++14b*N1rlf)qzGAX6mtLS8D z&P~-g1Eqe`VBjPMIW=-fP~s$CTxg3(;(HzrS5|>5ADSNbLZEPtPo*nf>{A<8)VxdVq8C+Wg|I zt{EhhGWhRXAn)0QltvtQRV=*_{Zd}e+xIS1JkALO!oJW=}Gz$l%WnP)K;Z|GYf?^JfD2hWm_hgxV04wbudp zsHv#|gf=O9wR^BtZfQ`79d40s(wbfwr_8KVYJ9EAcHGp7@Iz4%C*oUAGJmpTuT>VW zv1byp__wz!?wxtPzxE{JqOcfaXByRFMRhJrO^LmP0iRYpw9Kr7QA}*Y^r--bEmcM8 za6qYhe-!)%ts<~>Fz&}Qq*ejs|Cz&+TC#_#9+;7UiUQ+%7l2^nqF%>u1unk5t@71B z{2qbtelnFEtrSMF(e*TDj8IjUmemZOKIKAr1ZTtMDEAO#Bg@A9jtDH`WA?2VHO8QP;xlaGh zfrNX)9d9hHSmxyo(x9+xjh9~s2K9C9G36);36npMq&VS^!OQs{u1evyza7%{pTlBF7HwM~TO4oLODB z#{5!JZgLtmQsas?|FJn!M}5IxZ`URID2dTN9Ny zP-6rbnV7h~@VoCBN772%taIHS8l#c$sOY0qh2%0cG*tTPz&s*-`6VFjfc!jziOB)ymA5=iQnH22qe@{M$_Z+Dx-*YLerS z;OYH6VHpYMmeqsL{LVRQ&isx$)O%;Ezr3S^I+tH~9S$nFZ+`wtNfr2*?UAF4i3yND zdU>}nz4K8CJPsChe~3PyXqtL@;laTegtBwHy9AOMF){%dx;`t4sYRT|7w*0t<$udL zjdy1nLc_zqL+srsV@^}xDv*n>f!e?rhY7p~ zSYy==9B=i3w)~=^R^-`-m-tLI#yFG=a|r`bMZ{zo3GX{GG!bG4p=iX=L>W5>>8EY& z@#bZkF3bTPOA~)Z3@5um1+=*2u$i_tgl~Z#>i*hd3IhHSKeis#E$I$7_RU-1GR#igxpXT_=vM#P)$ zm#5tTJ+==>`efjTYRZ1QF~pM~S?7Mc71h9p4>N14DNQBmKw6WRmGyU_>FUE!jKT2t zFFL2j5pmt3Oi4}M|J{A(sqJQ?W~AqlY2+pq!bigmqkgk-V4V8;DxfQbZ)=Y#4FgeM za==QN8PWc)r<5)i)#`eM3v&k4e$u`W)ug)(C+P|gmN0!c{T0P6Et>;Fr8$^cq}y7X ztt>4Oh`_fA*2VYEk6IRmXXTrOXMAn6NjxNKM#si14J`DY`VD`aG;I*04Xza}QIAzKzmjj76PCr2ND$WhY0fl+@*JcZ_8 z<*9*jfyqEFkyBp3Gd^zI(Js3+e=TW``swH_6p+EGsYpP_DTlV+Rv%RSHyPQh?YTqu zpXcP}JNrHM@u6NnFeLzom z!{r&90kaA6@?x9!2W$275~~OclZuNv-rWfMRH4M3`1G|+!H?Cjb=lPoiB3+K+D;Ixce@F8;v#ac+Y#r)O?V$4`TKmZxrNM*!zcP47SC8VXJ z<2s5Yoo?adEJhZL7aAI!5)u!HNCJu;71GZg%r7nmR9A~M2oX?=_@17+_u3q3?H=FX z!80fUaGVkUCYF_$!DZj{;gctB?!j60z=Gtwyu|Ap8sd_YLdz}st!>r+tCihMb#ij* z?CTRM?34U^YKD7*??bne8bpAM%sb4S7I(~(qm4Is*za$;QqoGjj@;U^^*Z1F`YAuZ znZI8=;PCLJV~sv&O-R|+zf(C)zbsqM5Oij%;k@Q)gR96EKwr z9D#JU=!T6R?7eqw!P*)rotd8OOtUOce|4Zk7CI=e3u}WpI@OL0v=SbH|5nMqE}V(h0;WvP)3UlHh(WvWKXcX6;_EQrp4q=eNDsZ_bdH9`|aCu zj~|m`#zN>f zdffo9WpKQ2tB&t_Ki5&W6tT|Dt*ovtB)Tg+>xG*6`Sy47aw1}46h!BbA2P3BOI}=D z#HObgm%-=!K);K(orOY^DrEQlczbewd)w}D^25N)Ock^;eZR0v|G9Y;s`UifwpMij zeb+l;=#-qCcroy91|p@*3`v@j#ye*6pfBD`g?D}`DcSvLE+>aN^ZMMb{-E#f+`*ao z8IyK_<7Y$cXjZDl7S#C(xx^-ZJTek0S-ImkFw=SCnSBSVOZhKYkS zT&An7J!~~a1WNVS+1FJ4;s?0tyays$N9GbfRksz7=KZ=bod>Gk<7sld>+J4sdFs#* z8Hwqia^T2R6@3E#^v?()S&Sz+7%l$(L!B!zdRc4FN2o8L((&Wo2n7#*d41{4z7?ZD$*&H&PFPcgKK; zs3y;|Eu(dG1>Kijm*1BsdpCIy7Tw)t+~U+A+MX^%+!3%k#H;^OK|ZLK#B&Xa9gs%| zPAsxdgebM27UR^bz?|Q6>+CA6o8%K zfu0c|olE3f{Q7;2snvd^zPjC;-iqh-8+SHACMa{9|Bk=zd9o9gmX?M;3d&}%8=m+k zbcaM7=Mb*j6VHG77MhnEx+5+m)@Z40fc410n(xhC{iXP*k)a5gJ?JiHKGNuePSfH-jaWIoR z?K2WmV)DGSeDtK|^`+QfU{{NaF;YUWRur=sJAIhZuEi1)VY*SUesW+92@ga6G#aYk z^qK~uYplYGq#7f;l1}U;W(=*wchD|YsAW+^HT@zcpkmH+i?Am#AMpu6*>g#duv_PA%9a3Kz zZ0TgH<#QC-iCPg>^|9U{xc0?7fzLOizrQKsgFeVikQ92h5jNQPb0Xf2rs{eQ|zl7ZnR#sU(5( z-#VOa?>LBpPkt*Hal-{|;DV(F281-7=%X!a`54b*?rqm7Z%O{1%DBuD=s zPH%UM|5^W7$IMJXCn^8xl_-mP+SnuE?Us>3J2MtN6CB%SfC&4b^qUUnsj*;% zy&+tZB9<9oY-z>hU_Q?4`1KdMgi-n09~UY|4N)W7zC+)kwu-z3Q|e+5wLON;;gaMy zslDBRS{Ad)emWdF$i`4OS+*S$I38B(=n`;G;z7Rj*wPh@mXC?Nwz;_(VbT)Jz|DV`<^HV>!Owa=Da6KA z;Slaa9EJvA9iqwBnx~dJx4LS0kf2^*aWxVbcNv!f5zY`;;Zl+1N4>G=y zi51%1rk8(?6hHQk9B(l5UFh7FwGE|Xrbt_ZCRcX4d?8R54?5I=k&b!k0 zKH_A?v;V`9FhMGF^@Vb1tENvxV4x|DM{818qadTb2BQrbIr+v1JE?dxELvJxbC(Ee zf4L9@iJp~#LYAiS=C99CRhy-*&NKp?Y$mlQI6KVpNf)i$6+&{KHr)nbkKDFsI;ojl zLB@okj~aB)4NJNRD!VIng&QjWEs6IruMvn6a2Lp8bP5Pru=iQAR#1m2yWWzDdre(7 zMTZ<~r0J^wv?D$;7Dcq?md_o6dGCYm*;yu>LBg`KR0UR8+W-yo4` zX@xO5N^lkvs7d6{Ya(85feYO~$L$HhRozn8ImF(@0iwmUP%B>`3)RqbTTo2|4P{Q< zjwD=x3v=I`X=s+J*^rjTLxE?6o&|2(8?XKl}Q$HVnlY$9u+=LeoA`oc*;|1MUunP-QB(a_k1_tA2EDg_59cU zXQ79q5_>A#iDs~cV_%-*1j;f%aL3q9R=J2SF}JzC#L0KCw3G%=^!n<2Hs`|!G@r{& zqy4_SyGhK<@3Y8BDadgWIVSbk^N4TW3?CcQ3E&l@#!CFc2WktC)0u|_Fd}GiqJk!j zY`NbpqGLeAJT!F+<$aTp5d$!0@qD*Q3C`!A!}S2SnO7l18En7W^4pg77dw!=^Kr30 z(%EpdIl2h-Y5|JWVMihY?-pNV?mK8KbJ3)RA*;-{`~pr zQoP0l$vl}s7I zWEz2JHlUGvbu{v1zi##t9|F21{!{IM4!^8# zz2|tEfW?3o6cluNkZ~2s^|Xo}GSky4TNypQltEwlU)i_0NS{1;f~03{ZEdHMvwtyy zWKB>tdgtb3IxTt-ht!e9y&z-n&o!eW@w}w>O|$k$Kb2g6Xk0_849-!CSvXtG7+oKN zYWT_8zU^}gOWK{1mVDxCzE$K;K{*F$7&BQFLlzS$CqkGYB^jjL>Gb#FRcm{@Uga7o z1_KN9;Z)VSLM4J^Wx$-F1Q6y&!K1=XAaROEg)&6N8S_+cXxLuL_7nV(Go-%nq7H&N)v1< z?~`AwJi0#D@u}&dcs=i-WvIFm=Qrt}uN7S>CMu&aSpV(r;jxf$b-~(rx`2ZOaDdb~ zAmiE0>adQCe0zUa6RPoBugNMqXiZfEC;>B6)ZCmg+e5U~E^CGdZ1bD*_y8pWJ!(x} zCN(uHmJ~K?d^RjP8hXfY?DnsKFB^?By zpJAQKqj%rynF<*imsUk5nd9>^!nrn#er@3@?%_H^_MVncN&j1+x%Hd)nUHz5jv+2$ zZFh#Q+{?mmwnxd;RS5cdj=*bjEP6$75{llc-Mc3PMi3x{$fdKLt|enqif#o_p9dNq zT7#|G+k7;l&!w%#%yo+H%|4n7;92fDYh*E-~xEXmOy4EH;uuz z*IlF#SA==8v19W5o64b*50V)`=TWgg#fgQ}z#}6z);BN^Ews&4zuNH{a;=;G9^CFFUGm}cDxTr5#1pD<-iTu@tIpab|G8gErvm*nAKA)pmUAf5?X zY+#1nM1MeE&Q7xKw~oxaLtCVmQ+hfaY>033j>x8iYN`-kAR)B*7C&aZmVNMms4q=4 z09+hECBM(k(g7jeJ3hup>>jz{zz~>Oql@eA-rOREnDJx4jJoL?`}#qwXZ`?dffV=F zU)(DyPuw9YVt1dHh%2CgJYpWc4`yq>6dB|_oY%@Fq_yX!q@%@rA|^CsXYx?>XMl6= z7p(v*m1VmV=#9t6xKiR`sYJ-44{?l&t`+iw?+OE#64WRRV<@00n|#T*xw*M1Ja3|+ zqUek5HNP}u8hgsE&*PCzlGqZSelZ|gS_~pA7IshHx#<)i`ue+@)A^iBGfqkUZ)~U> z{q6>#%PRU_Ix&RN*Moyj{kxP;+E;jub@k}jEh3z(@B>7-t8cwo_#SYxx)y=QN4796 zjvF2(y8oOk(3ge~H%v9?aS--raWWlF-_0EqBVh{1X@j}l9Teg11DEH-0za5j5}ye* z+kPECy_i`|u~GeH6rdWEDkvDbRGm|DU4{v@ie~+CB)HwzE&u@A>({S86&A|9^(5Wt z+*wm2X^TcdfHja_SeOE`w~Vy3Z`jV$m5k5Zx6X-w-g~xp;208e`~EQ5pxZ!V*69~) zGMRvf_$Gza*{nXqLB!O;6lgER2WJzb=@m%}1eW*O;NJz*G z^|0B>%@(3rBSB;*A~HHqL7@_>&34DOdVO7{rWvo?%lsO&`o8b&&C4N02b~5FL8Pt& z9taSGq3aQH?<=P{sAC2bA67TM%if8;EcKIN+n$8}};0)(fJHp>+QuKL6vle54ZYWAx7`8NDIkK-1FF zDuw0%5PX^liaUxx`F|DZ28@?m1oWEceEC9*h)qro12x3IWu6`o2KfE14+y1aUVbEy zJDFU4)E^CEn;F=bkR~$F@d^qHuT?0hc7v`~C#hpO*-uPPj^X3uQ~&qRCWbR2 zm#zC^dv~`)SABP9r`U}LlHfdKm+x?Ou&)FU4QVm6vM%~uoznr12!S3)>9}o_G(f=S z-d;Qs7S(T8m$O$c`5GX^(;*$r2w%`_pvBPwhS`#)zhdC;yM_RXud}~jc=9+#>Tf9Y zr#5Y~=%ilPvtSem&Foa2p$O&NSarnDSaeq0s{EfKfWjB&ryNWMf@PP;MAN9wmQGXbn?;U7d zpxzy>4_`-Uf=Lf+g@lwe5PDh&Y48aYXDj$fkv0I`g^Oy^4cfoL%aAcUzu2(Pn7v_6oTB?y0DR z_ffb&s;`c zQKzPqpH<2?j`}I1ZDZQ`ty<=TLH7Dbvg_Rhi7>0S_9k?+fA%yA2n*Z%)lgLp)ZtVF zM)a%1C=&NI}$~!K$;{}F<{pA5O|$&pzwf6f?eWz$mU8t zD1&0CZ(BQ6ql6|REN47-C6MYJvUkUAP>b|4o2?CGQSzJPG81Ff*Vj8j-NF|FNb;G8 z&(0IH?%|%TqzIe{bb=@31)B<%F3&fq58Ir;Qur-9ps}a<4R`b#Lej^A(%G7(rtupE(BjeRu<*Wn>VWAG~Zim zgbj;1505w9Lq$naQPr}@K8r*Rk45&HufAh{)LBUCzqk{=KYvXimPp|ae8aDA9UV$c zoJb?M{gkwc2}9$>@B6S^OyK&4c4SafQGH^!T#K$wl|8P85QCnJGMmotsc)kNK^1?AFHxEksHsL??j$8e` zd~kP+d??-uq)`_ZBc< zu<7_%8JxL9e|<1ik?xR}y+N#g;na1WjGfeMKqH)1Utgcc_#1+mnb~-Mp$*wt{{8#+ z9>pQqXPO4oe!XfK;^aFo6j7Y|OS26F<_EXYJ6cDB$SMxmUBt$}d zfGS|1Z-umn4Za>gf{^me_U4)+xf3l^G)Xn-addS81lx!6F<;RAYME|BXP@_Tbjjiu zB0b17HO1II!FdEH9u&B-M(?{HwAhJBNN$jkv!E?=_W2V?G7_5bMClZ_zVl%BaBi@4 zS5QgJGBv$6KR*w~*-^}hM~^NlMOSCo|kf&n@SUE;*5@`pggbw_@GPlY>Ll3&`MHEBUh^@W_&cC5j*@QtfJJS^5 zP9=tsPTr`LVKcvImAs{jOm-onVpVcaM@n9|g-U!Y>$d5n-Cog229+P#eF5b$AR+>% z*s!6XyvU~~i5EGdlK5mb;(JA%xUd4B__9)DmFA&~6Q(bfzv9mdWnKox0dtvTwrM&J z7NNkfOId4~A6z7qG;$AQf^tN{`9*SS@-&G^ABJTJRr2jRG>ej|VBV(1ot~tyYpql?3FXsN76ADk@NELbBqpa_3T^^^N z)l7{sbQKn_OZ029MJ@k=`y06oY8$>o4bWCZB-I)zpL;QEbN9D}967o@&r;8pb^xq3 zrSBN&g0}zNs(Ol}6$&%@BgTMvGO4pA5~MC1NC<@J9hiXbYw|krL~c`H5+}nnZhE~(5m~iry@X0lbuFcqiNdh7dS$Yr=rW(De zpyFX5Y{yEkq4-IY;3UfOl0mOq_54y+>gs|YPDUwsBjLJcQ`JtPD%2s;4tGisAbXjp zQWBsSdAWZlnNpPOx}}DBXf9VUBxUv0c~za1lyt2q=nQ5^_KznWCO^G>yMr?AHP2ij zWn?*+tzckv zUurig1famCkB9>f8VE004>`6qRl@n@+FG$p%5*4sj1dW(y8ga1-lgt;BaG|n=39@W zs$U(j;8&P4KHd{xQLNbhb9CUS6t6ni+y^=-5jAyWYpZ;3CBVh(Cnbg=E*o#TBIMUG zz$q)7=`hQ*5D=d! z0mFBU8f~?&*r!p!EfGyHiHE0E!=mLS?=C_f9AakD#oYTkt0;{h^F~6ThB_T`s)yJ3 zn=iiWGrnX+FwYl53;##h-Vw48F^9NB2ItL$QfjdWCaMH0nO;Lm7~&p>m`sY1(vWk( zrHBtBBLUY7wAu2YHTh%y+%qTpfj=FcFg0s<}O7Fj9#Rqq&OaJJVLf0#=bgajGqUwk%Hf=_S#M{w=(Wgh)%J* zS_COyF1DntP%zqy=JjVx86HisVmsltR&gG92*F#4sNJ489u~b2BvXta>v%4Erl zL5Z{ECipFtbG1T_`X3)#3Wu1IA=OY=nbfp9@=my5z$=R8g4-B_WI>vmO`ZTmLyn$s z4l*aw(gFR436mzmwwlac$~JrCxLCr`=H#89BL_ij$4HYY!@yS~Wl!BJ81UjX`Hog= z)aeJlJSUOE_jf)t5XJ3KyikSjpuTzZPXe#5sTsVK#m5CZ^04#?jwq}Xn>yC?oh5fz z9TADV9ST4ILDJv3>3WXV6VWI!)?7@ciIzq(jl;>aV^>5~G;#`?XxBL*3RTI%*+NCUb*x{Iv zf|tdmcywe0Ig+5DppYbFchg-w{d4#tA@b7hJ>z$n0TTRPx(8g~?b}0Hb90pN2v1bR zpOcdIBHf_513IR=cd4wbtRxD8z?&~Sm|cD;3lJZvxNz?T;Zdar$nM|o&i(Z1&d&5^ zz`NV!15Hxcz7|3s0| zBg-4`=@UpTTjS`C&f2ckyDNFEMY`x>R+CJL-Mkik)SIF|=)ueoppwhS&INNJZMddp zX5bP*l{E5wRQQFEMEjbChN!s2a%foow}MBya|h@!S}CKipJqq}LZ`HsSN)31|I>v> zD?N#Ac$6VDGBN_jQS54cZXiq#3=F2{Z*1GGtgN6QqG|3113o}H=6HFDEc(+0t@%Ng z8Xq5L839KD+{V~{Iclf!wZ}ywVZEs*#s@1BI%NyYj7;eNUnj0AqG||-oNUdZ0f{`0 zw31$-;D4z<->IV+s`y&-a$O7Y4O}CRdvh`{yM`Q^!lB?6u;v#OB)jXn#glKpCI)w- z3jPjI-$DRjBikNe4+|s&+SU8QLrxY2k$A^QQC>a}xej?NS8w0GRo9T~`JcXy=%uA4 zd@J1Xid9-(&&Y^!c6Rpu5f%bYvq~bo4#QZZFQ4?>m?Omw!o3>UlV3o#y`{eQF#O;8151 z3hr2RA@oJ^I4F@wEd>--YXKja!!qAblcpcIt-5U|dCQ0uc}2T~#6;NR!3uw(|F{}INm%z|8ofTd z^`oT8|#(mYMaK!!XbpwBfKa`dO** zLtvyVdTC|lW@U0*oJ`~2KQ{4Q_uB$tCdt-TgC#x=rgo!wJ*^`v@oM#EAUl#ItD3)& z5JTM08l(L8e3gCTTC!r*TObz6X$jHEFh`{b12j;Qz~~{&P7~gtofq2TisnD6fbO)B<+WaI2S8 zyz?(=j$NDcot#XLa|Xu8@2o5=FSmV?U>VidLCx#7!QuRaRw-3Akj``Yy!2MIW{j%O znSp`9jmi%5a;`3ido+`amt__8$KRG5GRv?9U$PYNik&|fdG(uFhcjVNCwar7wzd{V zWEQ{qp}vUECx>HzlRBkiM zjY}tq92Ep_#``?s6Ha~Yu8(%wllE}zvvYFhmzRT}JiRB`>}iheciWv&c=qhWV14_= z=*jD(7c8Y&{Y=`UcH#LC3f`$a53$)UPcdnJK#Q1e*O2%0(ib`^ z7tL?V2>jwIVdoW9EqEvBx#N{K6fi@r@o|~AO+$Ob0=g>CVCaYW&Yc*j0WW|Z{vNu2 zC;5n45Gl~=*1Wg{uG?qJUGJX#cylX|aJb9j(@ucDzcO?Iz})lC_kl5r<;=X({(KZN z0c!*~7hlTg)Jcb?gr(WqLFZ2w!K%=9hhJZtN=svbUA*!2Mg0ESblJzD2AzifrPS1tir?sc#dZyz9wS? z9fv&j{av?|w_h8sf;T$edT0*+J?K1i2_cCGHVK|yaz^rqAFLT(rj-dlU^mq8sYRqQ z8I~ZxmSTJ^a%Nsm7_kVYs)0LN?Mq2t%i_MWp&>0pV5}6lSf0T|eaiX!<L6c&D(`y4D;%Erc->M1Z?0|A&l8l3`56bFu+ zRQ-g61oGcUqeeEbKO5-ltNy9nElLX^pksk{3Qe)ilg2WSp6+fYnA&~@v-21kcJ|$Y zvC&7vk|kvc_bifz`PPfPCP0o_SX`9DWOz3B&2I^wJ|*D?ELqfw;A*)Cgv5J$zgxR8 zLMs)j6F#)bjrKUayf)a+K-NLJ!eF$^AD&?4o{hhT{c3p${)W|$&EJfZoDA}v+V#7ktHDEj*-2<=PVC_4RX$U&x+Ee zd}v|t`29*{d-Atn$rx3Guk0(I;<6f+t1_QaP&D*R9a6`yKyG0slCG9n?bT!m&S3Mkeu(syY*L`mcA)TX@ z-wAT<5BU2Dm?165gR}&5ZR?u&-p7-pA!X%`NAEO@XwJ{i$M8s%#J_#`3M>&mim3UVnvE3@l*o=#{Rud#%L#Nfn4SLXe`m_{StWADGm z10Dhd_cxPz&E0gq6l=24lSka`y+25RobP-9f0~2oIL%ECCrH+b?LSonS*_-;P#N9% W$MsIfH{mafAnqxs%NNTU`~NSidY6s> diff --git a/docs/images/chapters/aligning/a75137c250be63877a30f4bda8d801f8.svg b/docs/images/chapters/aligning/016f37843b2af2ae34dc8b2975505f3d.svg similarity index 100% rename from docs/images/chapters/aligning/a75137c250be63877a30f4bda8d801f8.svg rename to docs/images/chapters/aligning/016f37843b2af2ae34dc8b2975505f3d.svg diff --git a/docs/images/chapters/aligning/1564fdc8d17a064fea88b9d508878e62.svg b/docs/images/chapters/aligning/1564fdc8d17a064fea88b9d508878e62.svg new file mode 100644 index 00000000..2fc121ea --- /dev/null +++ b/docs/images/chapters/aligning/1564fdc8d17a064fea88b9d508878e62.svg @@ -0,0 +1 @@ + \ No newline at end of file diff --git a/docs/images/chapters/aligning/28cc0f129fa0c028a1addd702e99f162.png b/docs/images/chapters/aligning/28cc0f129fa0c028a1addd702e99f162.png index 09d2497fc9d279dd3dc0525cad99d20ac4c5cb5d..667be9d598755bfd877e259e325ecfb7fe7dcd46 100644 GIT binary patch literal 13465 zcmc(GcRZH=`}RdfvMGDyCL~!6D^lBwLYPR#vt|RvD2^8CfBl z=e&GA-|_o>e}DX*f1c;{>GQf@Zufm%@9X_O&*MCf<2;3`D$A3eU^syw2&tk1Rvkfb zIS~Yh`Zxi+gB$a?8D0oXZpdSiL-fDYn%A)iavD*@Ue$0-S{!lH)0jCRT7b_D<_mbt))pInm{nOJiAI;uY zCh2nJwYulsyRz}@{2i^il*IloPYV{ZGBaVJqzsvH?ujN6{TE)jn>$GqS5)X&`@dyb zDE`Mk&DWjv0U;qF3IlXX=*z|b{R_v5K$)z?X0?!yB~2|YxCn-w9e)WcrGD!c4uXS& zgXXJ z4MIUrf7Xs`CnhoR1akcNaf%9R%oKZEn^|<5S!pu&^Y)4H@zl~sx1%<6x?JaP-@c6_ z?cX}SVHE^F$Iq{-kx^1oh7$M_p)WxivjidG3Fyf4I{rM0aOjuo0 zV{B=8)-D)}ooq0Y$zM*Nk)_uDF)$=B*ntG((MI5^g`9r-9J1>-5^u+7+kB&MGXU2ScaWQgn5N}7Q?1|x(= z!ig12RnOr$$ZvVb@gy;dk}`pZfEshY!fAo#vdvJy*;}tt4L^8Scx>_^btC%^B9?PR z%x!H8pRJbALRzd1JbDefFMFE2KX=KT2ba@5?H*NrVL1ot|V zu3F4m>eURTf4V}Qvo0tXAx4VWxwsTMhck;ikAI9|-&+5o;^Zx%@(py!rd zP1H^7s|XjRkooa?e?k%0mD4{u68w!DgV;nw=!VN}gEKPNY54R){0Yf1Y-~7!f`W7c zMu?}Ur;)WaF?m##PLYvRbxolx=eb8pTDZ2$3+NlL*A4ysG?J2%nk`k%iwx=M=|vAl z#1QF$0)1l}8&Xq>5h9}&dBz#v}e?g_bYIyFn;LpNRiX$&%-{g z&#EcK3*bylO!U>#oH`XaIH>b9E{;u1j3G2Mu2sL%@vNB3Qn=UdpB%&7C5f`KGS}^y z=f&)r1@fjpKDJlG=xQmwCNMqwMXJ_Y2 zwa!c5MZ@U?ae3lSK>BEo`mBs!-COoK^w-$gG~7=drKY0tBckT~y6qksQu3%iK&)KX-uTa-+J&1V z)o#=*EG%&bw{kRe-kK4an3!03-sF3Z#urcAb>@I%$A$j$dsD23(hnZ+O-@ZwGBD^) zc?^}=gfR*~_?l~r!Q2##k50CarF&HELpZRwwI0hFpZ*Y&Y8#*A?W^ zqemax+jBQ0u-Kg@BN@55w$o3ZJtN3cCxX3%tRMe9({9n5Nd%=T;Hp1CnrUmKO!=K_ zaIcsbFNh5d4UxJM%P-?IGg)M(FV*~j%-P@Fl*PoJUQ|Ni~xgajgCVPQN1 z0+ZFr=AzQlK!1P#Zkqva0Rf8L&1G%E*oq1fq%Qqh81i)Ne9ltosdHLJy_t%LZ(;4u z2_!l(@v;3>3l1{swW+U^Btm&25ch?!ErGAq*fE5VjHwP_%gHqki`7&-c_Q!)6y90M zkojwT&ruxs`0-<1a~Q3fuI^(pX0f_2U#Q{Rb8~Yd_LF$uzke5i-CkcFW4-y}@~O&2 zF*F$=@mHs!q~1s+d{NES%9n9rM2x@cB(`JRFZv6a0_}mEP>hIrvQBmY#jkp+5SGuJj8srnrTDeD2zAR>+~ue~ zh&awtT)%!@0HE>yPzjOEaM`IHDqu6jbV>>orT8&iKV`X3o?yD-N+BP~RNt4YEok*!tB!FGC0tc=EKBeWC4EZR#-9I+pE1ifpmqr>{efAzf*+OI3;XM-o!q!@sl$wrC zNNW^pMo|$(R#sM=m5z>1s#=Cz^z|%@ui2FMZEOH2^ABcTL1dsoop{j__ujpGB^Et0 zCvV_&kHK>I=c%zgN#ZVKL_|aiii(fQ%Y`|W;(fjMcj|k4)qT&M#@M*6jN|Yb)`sV5 z7h=T48AU}!E4=r`9*2eXyuEv)#H#N^XJ;pR^(zt*6VuapgoK1nb=+sqUY%-*=*d>g z;7OCo(MZqEJ`O8Xd-`-|P8=m>nY$PMLI{oR56DdY|JCPAWT3SX1x1*y&!0Ib^^})^pJaqbj zD?g=XVP+;CEuWsWmKM~U3SVMcKIt@7G^m-y9DU*Wy}sjA#|Yd>!4RXLNsfBzm$fw%A8si~^^{#hK(062h@#s2Jk*;Ak|9{=LS zfI&P%Kz%?-I$x}&wYrv8(3jULrpu!>B3?Te0jMaMnez=Ak7ZXXvQUQJx_z5oNJz-; z_q1Y?n9~VBZNJ>ybCoX3k!-R-r!g4Z{5$0|m7lL?s-M-<(;J_h^c$`5&=&gx`=$5Z z0sq6n&N+zJVE)=I+;ZD7xnMHp@r8x30{yCV>w$u}%Nrb@1>#lxzdF60e>eT}Cp%PH zNFcO2wS?3A&(2yz3@o0;?DO#OG^j}J2P}U$WZ_ay@`c)ri;FAZ_S>~IWg@{`GeV>3_>yIUlv6J=LkqR zHMlljvj5H#X9D0DEVZWf-dmL?VYqnwoOVG#d3kvO|D%wS?1~Dy)=1{Y@-gqQST2?H z8_)TUlaMs!7S`(CH3%vmSuH6mlZ#@J=+2NQM`m0?3-sRmUAWhYaxR@(Pf^&F?$;`; z>vLZS^{broOEQEd7(b|MrIYYZlzf@?LvtDyIuYe$OIhnAmBSPtb6d%uav zm_&T1r_CIfzSE=R@hGwC?|&^kRBAoj zmrJ{|xf}%vVgIf}S0J9kH%LV*|1+R|vB&0vU*FylCMPGyi$3H`_E-{)Pe`!bJ`tjz z%_8X;0w0-LDEy$May>=DqwK29TB^#AA3s($HUbL^`C$tK0oJFsR;NUqf6ANos#w{yO6-QZ{N0lxAWTr z9$yuQQ9w$N+`V)J7wLUH{ll~#u0a&l7hn5 z*B9Z{EjgZ(lcS)d)D%R@*!}LF3Ls2|?O3f$d5#MSr<}}itmZ;`Ny$k_2N^v*CMITP zLDLozv|dq#HqDw(P}9GAcdyIj%d49oySiG#=`VEWYV#Mgr`>z@>={Z!2}tOC<)c_8 zAesluY#6l)^g`i*Sg5&_175s%(QPx2?;rZ8x6iDvQk5reu-ull%ytYVo7ijD&?FFJ zj0$mNKX;Cp@$!9XAYlex8=54nl1%32=6lUDM!@lolagMuu;BLIT@?N`R7wohOXkI8 z8v*xqiz|nFYjR#(Vf9VYv+i%L^iR^#qS&#ux7O8@bBim_poRemT>o|jHdCLRI;iLN`=Zu;8@gvCRU@gUUA-{a3kz8sV0UHSaj?pT|CRE~$9(#gMw3mUvm;gIJ-^C2jWAR@aK_qEtWt7-$YuAx`9(5| z3q@1V7eP=O&wPq2diSnTp4AJFoSZ!EPC9{uy?y-)+uG3E@66fGo+VlsD3n)JBx-1A z5OA1Q(7An^Xe3<@74w>Udc2OTobvx8M`>>-UK}o$2bzJma5GDt?aUc`dwY93XXh3` z2Xyo5ci2|LA|mh+teo75@1fqAuU_FI0Caduna{ZU#KKFDzzJd9SW4Cf$7KbiJSMsId<*{-RCeShv2X?_|i#BbfI0TeUZ{yhWP-wc2~ zUwt_!ZU3;|UDYT3C!h?P$$~aR7fI-a@IX9Re0dcyzTZd4Ix`zW$C=wN4?G#&SD?q^ z8v}-|pt2%hd}82f#B0);K#*n9_YdCK(ti0qDAhyxZ@J9{yBTN|9ui za9^+cnsbZvhuRhY&}Si*Z>_w(fABGs5ch%NKqv0Z>r&4lWCU_sr`lEU%9SfcrmbYL z23Nm66%`XR0X27=oV>NxYggc2CzIoBHx5)e+I@HKJ)*DDCUE$yKxNTqDE8WQhV(QA zx@>1}Pg!{$(_5D;;ZfwaW)4ouX%P{fS!V^jy!?C=ZB6#JRtG=$NP)2HPQOOnku1S@ zaBv_~u6>>$R*cHzuBQI>YAfsN!R~SfFevoEx?kO7?7`>4yb%wSc?7g6m$gk&; zUI7n6LoPmj@gnT~!=LE@0fUuJxoU;SuP4S{&!Us`6oHrzMkOoAw}OI#c+k7xDg-_t zNBlz*({5_zM?HB$<$19Ea|<4Zf`ug#XzKIaD^&E4o<2Rfw>x$SJ9|OExB)M1nC+O! zg9nko%#52tD1jc=H3Sk*%*}-k7eP^_^N>UA5>* z59~fq855+za|#WB!Vd=wu(#BXBkV%=K@q`IUxj%6^Xr@cfT0g_jmKsS)IXj(B%UhpVF@3udi=+D1{qF&-<8>^MELIqa< z#pA@pvVd10%mjfHgfod!!D$cd^3TaJDuttg176-Zl$7SwtNv(s4Gs<-AEBbOyz1>O z2?!Mo|4)H7_~A*_D`stN-H{|p4N9!_b&5BtP$sgT8_4MB#E(9!6izfU2?z8(2l5Wc zr5w<2t$ba4DJiM6I~dIQ_uy-cx=xXzDCxRbI#7XN=AOpJQq%D2_$tP7PDmYYlcUTA zeR4>EV5n2dt(OOlOgdsU87G5Tz3bj+=B_jl^N$-I?mVxzoy1~&syw%aM%@?AqbUJg z?o=hS5F4CrNJK>Q_S#Q0*5F;{_zU+Ul?S8+ z=Y~qBg_Td|3XOoMyF$R1s(+}j$L45Y1O+Kk=ZmjRuAvdnEY1uE8M$?mOH9q3uH&rEbx|OK7 zxGB_(ElB-h`zg6!qcx1c-~+0vs^pv)nUs3Jd_lQ5?_HYFE)Ui$ZcWy9RrB%dqo+D- zuFx)alL7f$Sy>r3($fzbai!c{QOEmUj=H75{Pfp&c*`$^^d8*3yq;<(up~8L&61v$6_>E z?M9kdG%{|azUnV1pKxnRRP@YDD@e9gcyjGsZ}+mjJx|`-Z}Gsa0VJrtA!C-QoFWQ z$Sqa%Wp{RplUD~Gy0~a}i6>lf6XH}(3W3-E(X&-F>r!5~yC z?o_bfO1Qn~@%{Wr-%rW?!5?$(i9%)WhX9O6I4mUeb||6q=(d|?HIMQ48N`Cl8ZMVw z4^V)_@Y!E5M9Z1h&XI9KvTf^Zvu)!K}!qtRt!zmIfbiN zagHDiOdfW*a0*D(mmU*=v_%laD1rQ5VGlw+=T9|(igzgxb#N!Z^6mc9Rp$C`^@V0< z@w<1j@E$KO?;ECF;EaM!^BNnS6(t=6rV(a{uO z?^bjUEzGip%jF)(^cWxTB%3ZoV*qL=|yg z&)$h|@tD!k)rF)k?EPV{U1-n@*}U3&gTHU97Ls1;{kt^voFcO}Do8(sJ4Myh)btpH z82F$Bn>x2%me~wX0)rqe$R(xaH)yJMTU!~jE;Rm{t>*Vf23uJURLRWTeDYnFsFkNve^WU0 zU=oZ26o}93mPkV}{(;`3+}4mu+?fOdKKG9+#O{AHJOg1zH$;J+&Iy18htz2s zdf_9`3cCjDu8hR0*M_{v$jI2?Q);%R3e9X0L+r-ua7dWNG^_{ov_nHf(PZ9iI4wj8 zo|#UG1tm%;wzjtHW~|d0;B}TS{-W(D6L?q(Dk_B+4>KDM_s0&6*Jpd6Z{VAsZ&Mnp z16D{xQqrdzqbXP1r~ol<9loFRQaQNhQ>o?TbmGK`B5?GtQ$JV-e8NG?0UZ0K(MK^M zO)8*F00=-*O|#!g7wewu$OR)|0#2fj#rn%DQaFxBa2xl&@X*jlpf+1-KBq{rLjDJb zg@v`0&hx+pp_B+-wjJ$Il`+Gh97F1GkNV+{4$8_?82{9F;+)dkS7JdWg2R?Qba$ES zW@4syQ@SlE%-#jpG4I{;larH0mR~p_HRTFH?g^lgF;eAX3M?S{y3@|$vyT?D>ye=~ zH4@k`+*MvKN`)Q zi3i&+LnTF0#CxwoGKg9UeONyI6DleyU|?v@I)^AkMKX!{h0*f&d)RPd-YEUu0u^KB z=Henu@!mNL$DU>&G&*?y+gp4T8|M3R-<#tkJcV`ia;ixH9kF-Mjb*DNAE_p-c1QQ*^=)nA|t!&pbZRiP|7} zpKS0tT>F>E{rhj&Ci!L}@b`m6v!mnBghl0C7`poAJIK=7wlr&@QI zv|gVeCZ?pP4-E-9zI0N_%mf@&v~7o#mv00k5q2ls&%nPeR-BA649q6fWuy~z$n4Ii zh)He+CJ1>q`!y#Pex@g_DLh!qV-6<$3^r&?Y;2>xY^ahd*u@SHAHJ_V7xQdRB7S{w zm=-#Cfs&F$nK!@^8!WM;)+#i}Pd+Y>eU%1oc{7 z(pY5NxGqtx#T|DYtelAm;nDF2^;n-61rA`1l$E44XxA3}g_Y03W8>ggpqw}Z^U;J6 zYMs1;!`pg-SQW`_ynNkK84HVSSa4|2^{j@sciS1(4T^m{SLfW%o;{2BLKDd{EBO`d zX?RZZX>p~EU*D)fW)r-hvID*IXL&3NIxoZ{(2$Giq2!1M|D%3jU|`&tNsA{gBr>vP zZ+oo=C>|^xdcb0Si*1m*w*I@0%HVZ@Q62%-@$sW?t5NaBySlm2e~oMe-x_sbN4Xq0?^tNh#p9-x||AycN73niNd;tCEF z%|xpAz(N6sFKz7u2?g`6*FY9aAlGg5uZnc%Y&IOnRCuQ4-3H-05t}A z`Ur>)k_uD++!+v>cS=11G+n+oH{+u>*yq2Gt;B4p=oT-F?14~4TQML7sE=-Zc(D1w z&e5?6${`K{JQURVImz8$@2mz~bC$M1dm0%VKY|)26ZMihG_!S$H-X?H7(-oytk^> z=lQ#h3*wRk=yL$<6iFTQmU_T@+(@w^CNe)AW971rT{HJ(Y)Iiz^Y%U zxP!5O0uAFnx7UU?{A`A5U9b~|UK^*DMr(3=lxv25*}_7>NDPFd?Vq2YUjVYsS5kl_(A;Q(wCn+AH)6qZesj#HHI$mG$aYL}@Y^@btXG4= zH(`3`OQu0H|4CBjMxBBI=zF4#36LXrNc=@hDzv;Jb%5GvKMY!+Xr}|vzVMF-0v2%N z*w}~80V4}TseaR%dz7A$ih++?MM+6Hhf16lw89xKUAImP2vI_t%fO#rjk;ZdDsi~K z?9#r374X>JT)sDd^~QrdGkL0S_C)%c~S^eVQuK!Y>s1v7mQ3w;~(S zP=W?ysr#MBAxa>V=>IH@gq4((Fmw@-kp%??UW;hGAS47HNT(H7J5>U14<2=@XQ(@y z)XQ5M8Uau>CYostPv#L~gzf`*g+gz!vI}UX|Juz5+j3n4EkE+z%)#W$VtA~qtOa*q z+=zoh4WByAg$W=}nDb^L$Hzf1&N$v0;T|uaS^=;b? ze5g?TmG@Icl3*RMadRv1o5FB_+Gi}i60*N2l9ZAX1;+?}y5Uds7d`JnFaHB}uc@iU zD7PvqE6ahK)}5`!{xT9BoGPWZ5hyM%mj@?h6`BeQj!7W%H)>CzPr}!wFuhmH)VK7iPsu{$qA5-RJ4CS@YHjL+@>Tfa(EV33$5Rq^o$;`oBbyVAKHY zt^7XNcg$yctN>0))s!8$G(aN9=jY$Vxx*f1G80fMl>ml7VVGE4Bn`!;2X;d6DBL08 zjJJ5$IsvwNeS7;2Vtsh4MFZ8lA6CW>cfF2*}otF&K!!<1Zui{fnueFK-xIm96Fd?B{Ukt5B7b;RZ6PHBx zpa)<7`SZ>8zivpUC(z^AIsrP*Zl4OCBSxsg2-l))Y6bfDJ(W(@=Uw0-utP@C|4y$A zmNyT;T}#@!4nd$!K(EyC`Ew8S@|dK&t4Eh>9{wbQo>zf?0lN5G@N3^7C<(2sKB)kj zVn?gLXk1jE$tF%F<-UF#+zA*L6Z;hoA7}VvGAc1fG4dJ&?)VA zPgc(ffDSr~7(7RMkicW3UT9;P-M!ln&07V8 zg$HhF!;xqh%26sIC?_uED|@A$b2>(0{VUv`vhO({7e9X?!Lo=Fd~VX{Vtpo`IN6OX zb%4hCVsz`?kY^BaN-v6&U5H1Q_CgImcF38jtf)v@Q86^R3SCsC&>JJ%D;}-Ju&FQ$ zDsdDBPIn#ar*Dgs30r+V4m}-T^nkPZ!vBlta_ATuVgMfz*YO(M-O1Ro9*^5Fy|%Jd z6rGS-pN&D7(OuG~yL+@tbK4w${`^7b;-rsu&n8?bUgMVtG|i;6-A8(QD4|tGk$tr2 z2sHxg)SXsQWo2dfJlm&FpYV=0DY@CgU$M}j#tvFC*Tu;|dBJc}P}D+ zv6p+()8MubGCr10CJ(XoCp_Bf}?vdDK} zX)vF7V)!I%x~4HzJzjP-wg&uH_s*lC%iuP=SS}Z)2LlDnAPvwSN=QhuiBSD?8*?xW zaYO;+pgt8Pr9`utAJNL@R1!E>;~;V7OK-wQ6_}KG(BryqZVr}VOJF~`=PjB4D7!(> zcSYyhK6Z4RggylZ|L^Bpsa85 zR~)PLN>ky70$ghVlIgUI6ps@ty*6 z;BtZQ2ebeK>==vX?3(sJqB}5x0+sE=k&bA)ux*5kz>h6ZzU;vs{Yyevl5iqmZNbE5 z7|h51^F8h{`wD2C!E7op4^x5d8#+34VD_l0s-FJq7Gd?%{I#{UXb%_+ANIfAyussm z%p^^NM-ZAB==P*|{h@~z7xeC-`-x*~YYV>Hi4hE1gG>c*?EY$N)I(|`Xaz@Lg706g z4Ll$;Y{ML&a@;vtm7ZQd10!5$i{Qt%Z?nnKzBIRQBfi6nLlYg@Y86ux|L96oKZHcc zl;xCa<&co5x$@NPE}QG3ZiShd^cxMCHyRCR*pIPU52U+k(Lia2`9~PXfG3puJngRt z%~&M$9?Tvcx(BRryz5n8S5P<_-ld!iY9pkE8Dzw+vn^*e>4y)q$KTVt7$MDGz(nw8 z2j-?9vrZ0SUb(-y#~iEh=`cs#@?ubY!Sf9rl@}A#vHz8iG^=;j z*W*BJS1c?=_9rCtB|jV((f4{F>p|0w@|n5xWFW6de8<18aA8|sbjc?$@=IEpD{uY| zTUM+3S>-4O&AIQd7YB=vT`=$H){pHH@c45d*dLJ)aH*KG1c;R0EMn+I*bg)w+ok2L zbF#I4`q7L#rp<0-m=S$LpPYPfXsDsJ_4TJ+;M0^a>*}=7kArYfDP|qw)S)A}3k$^m z7|D%*mUjbSJRculNK}-MIWzpCqN-{N+#AIAl65~i`!rB!IM6l#qv{GNO5V)Wpx0r} z7M%}p87{jv`NCF9w6(L73@OWn!fpH;y4p67DCdIhCYgOfgIM6f~SsM+1gZ&@QcqV5IVw#*g<;`6mvbS z7Le!CA2g!Dw`GV*fhjvM?0vx{c%g!q4R$Kb3a1r3zX6D>OTy_>Tv8H{lEMl}hJ5uB z>H@kt4vza=DX{N77`y_1CaLmSY3U_&STrTY(1sbz<}){*o#y0}+fT$|9>FXbv`+({ z8@LmJfWLhCvZf^SR{(Qp2xN>7Kw;;&Ov1_FzGF{}pbG^jeEvNf028QsKVW|DQpvBa zJ6?*@8XFtK?2y_A_^XH@bVv*kx#Tm^mkcBZ)8JM|+@zoV*sO2swC0-;#ite`4IPC*zGn(yAtE^IhQ{aFDZoQ0;r-M00%PVlDwPNpm_lk-N3%xLia*Jxybo^M( zo9a>~J>?))h=gGLKZb85@~FuE{Ra;Ji8Y%L-<=GMDdz3wm`Z3Bo9&|5{iJNw16Nj4Wb|&(qYgc zDLMP+bKg7fntNxhnSW-USZ8_i8+-3h>}{Bu%1u%t1|k#+MS4pSqk%% z1XbakYvh5nIP{5M#Dz9}WLOS#e|h~p@=pCmYy9=k>+W_;t|oU{KwcjACksI>6)nQ0 z5|$-;l{zdIt8h~2Ank5lGufMQ69?5CLfWq=?T);vQAd`SqHAbaSR!{XdRzBj|Gmm^ z;b%jm(S4!>)bKz>`hWfq`}yed7-emXNw!!WR#SYVFZDM|34l=ZdqDpl~z~l&C6gFqi-lE z;G&wEn($XKSNr<;^OMpw8%R4EG6DgS-w407=Q^lWfo;QZyw#NOWC7kPM2p>*%vYaAS;b*W;a z3d2m3g<)HQ8-o;;9Hu@-W@d7{h>H`EFw)cvN>4`{R660Xv9T4ue@~rr9gW^F) zc=d{bgOf8oJsnqEoJ>)1KJ>j+4KAv-?xZ16;H8r%A4Wriiu2B^7Dp8<_P9x$6QlPm zE-v`#QzC<^S!}s&>UEC~mu+nCsO#H4UAEIHFg#!?GH||j?;d^W`}e^P3I8mC*%mCkb}wzkBcyZU0DU#q_D zR#i>l_8bph55tmB1`?_>UW&VIQ0qk*78a(cs(RDcS7vpxj@Zh|O6q8<{|p%!0m|b3 zeY3+oW2c3_5zbBJ&dya+_!9~>QRC_8@p^PruXb-Xe5ArL)q^jGG+X&)fMflU_Pi+e zlRt!;`sEk5FBBP61u#o`m<$!%qZ4=GF!KIE%W>gCV^Gy7_uec3Gb+K}_|qG{^uqmq%4>58g(xa+XD zzb{&=qNsSS-uKAF%Ib8LK@|JVD5>9Ga&kBUfq^sg^VnOloM~^~yy>8GcX!vV@!*k? zmS&4lWN^4Fy*Js@(-X4m^Y9`2rAxOmQu!!*1NR~cd#}TyU0ht&m)_OS&i7_iI?dp6 zU%s5C$-OafSJiZNyoyfdk(iymJ>z7C+C_%JXNnwWNJ(!%FpN(S3wS#8*LO;~F9uCd zPoEv;4+{^!?&amxymTi?)bvX{&-(g$!|s${^g{pJx&7@W9u{(P@|&qLkJkrvjG@%v zuGQ_be6ai0_aLHM=77Zb(Z(r{mC<0C$NR!tzo#GSyfd0upM3M?{G-E-b~@n)YBjRP z==7Tqtr!D<9($dUP zQBjGKp1ktcuO~%NQd0+t*bcpMglvkE-J1+1K8HSk9)+5ooekL9a*LD2L#Ka$IL*k= z?~wgLQL#UWo2d8uskM{n}Ax-i`+~IliXOPLkVMRDOpWrDk4^DrZLTC@a6wd2mIIPlo!%I_>21(tKu`%Hh#QKixT!(d&YLX`0;@U>e}twMA>(evVUCWLYv>$xOXqNps46< zH90yC>dyBcKW3J{y(gtsT3KH1d2glaim@}$}7iPauCS= z2U9kbG523~SO|WyDAiWgyhy`vbdGbWBqeWAN4qe!K<^|Hn~-o$US7T@UxyEt!zM0H zZ#z^VhvbdTz{T2~QClY$mp&iA(L5-Lp?Igs9uyco_S>F{FT43zmfYC!sP^@aFTwtL zeW-vjAAB+I5>&k~6(v`U&+ckuL&LSsWXW?MK780a+MRNq?_n!2sG{=l@Hj(6ByVST z<#lqBF>VSL+Y;Rvgg@NJM9IhZF64Y)+V#_bd`Nj1d2dBdMNMsb|9%K0eM?JA!{DIS z{Fn1+G$~M$el&))YjJG*Ak)o$`CxN@gRkW zxX8^df9n?E)9~<(U#WiSnVI-Me*7R^B7K>d_y&$$L_{RgMlq`|MBhGH!aX23_>_dl z@);l$BO@aXLqlg_8^s?!guH(*^g22D#(NBEqTNJZZ+Pgu#z{&nm+EZ86TH2Jx2fw$ zk=)-}fbvAq*}3PZkAAGJr}q@XZ>BFxsnX>Kk&CNqOhQ6vX{jJ&XG3f2DKauLC^Jt& zLrX%+&zOrW392+1+p%LA1bbzfkDLVtl4<2H}$XX)b{PA z;f;1yzc;WQC{Iov9wGVpm#aP3$UHqgx0b(AN6Q{pQn1FTL#Rfg(lYhAz&T)uRv zu`%#;iQ|-vA_tXnB0moCIo=?^zgw@gWdi=b^{~H@=R+;D2}V zZWbqZe0~!%3#h(vuvR~}w=wtW)2D_%o4OIm^xkwd;zq7OB zV|Ns@WIBQdqSXaCISns`tep0@tbswzW<<*b6c+M8K!@Vs5j6Do z-*tSaLqeyVqqXvq{dbO|nD5cH&hlg(*s9)Z(|{?z8LZgwFdYN!v`hzJal z$|emc5;}g=27yfTq+n@2f1I(j{PIQJvLSi#v@z+t@d$y|H% zTD;9R-6;?ey5Zzgw2#2*aj*T%o)@`2! zsK8WNvrvk>D<~*1)!46lxXbS_F7{}%>#CER8%?!{`eIKS9#T)B2AY3;ExrfGfgH)) zrxX3FBp!8)9|yysn=cOJ>ehOde2!kGj1hY+;ZF2$XSwmmmjnU# zMa|`jnscI}qVg4%6dY89$PKbx519rcfUr&%7eOQ6KbDXjctFA-0)!Vk-ji?=DE7x{ z>EG4S34QT`=F#?`P7Z|C)}&#vD>1iu3IsACzRi|LO6UYFus!yROo3rIJ#7|RS_4BvXo#?q zl9J-m(x%RrLdc1_&G$4;-=CqTFYWA9hU$ydnD_U)Y;>*T+-{;{roUOGw!ZI{35OEP zB<({p<$J&bYJq}@DWW5mE5mlUNWLPMnuOCwF)3C{%XxRzkb{HcJRrJZjmHfyFEOOp z6-M5~j5`1MZD}`Bj05zpmC>>G3E;+YvphbOUo7CNIh@O->@9i(Wn^{;0g9?=4xi+cCnt&BnN>e+}$w?4O z&e~RR?|reyiXLwXuZDwzgBL_u$%9@rlei21#>R&3`}?E{(JYaK)SPtRzb+p7rJq@F zDb3@yv9T%kTr;v8`#=f|1$h@9VGIjYS(csUZ-IsS6=*RrI=Jqc?`{aYrl6tG1rFSP_*>*!;cPj+2K%9Qc`p)9GKyI zQ}zKZOI&{39{Ki-4OR^Vm7@BB&0Ai>>gmFY$tK_oF^P#-_oX4ZTun2GIyz||w`R_e z{m+@Dr5g7P4SC#H6>j8eqWSp9p?J4KC3+Sciw7hP!2?+T{I~aS-$efel036M(>(I= zu_SO_e%qld#}pYipGHLBqk@8i3CF`6GPAO9P=`lHr&crF;oaEC$W|mJhK>E0q4J)q zzR<^?r4a_ffrK~UQ!`K$e$ys=x7G1`^FyPp%Z)tgtT)os9&8Og5g5G79rx+$R|k($ z8%4!4fX|~+H!=FdNhv8&s}^^av~@kt)5el-~QH^-`}Y}y@bQA5Yd38WOo9Jjg*W;aaT@-D5fwF7|jzChU>?p z0TnJ~*RI5Bjm<6&=8sgnU;e$hxzWbxXflwaPT*Sq{L{$5z`!)r6C@AgdGxc-O)WpW z@3uC{JTo)nb+kQP0z#|vl?X9hRC@%y(AH0FslYG-tbM7C{)-ncggd{cN6Dkn?Oo+e{zD7F031NEsEb_e)ITU~cwATj4!PMlWx1rZTi;LX4e+J*}C zd+Q#0_MYNGYZzKuUKlDe3)o2KDZd5sMU!oOntkqN3m>v)yx#`joNq6roh0X1r09F73=clgR!1v@V< zS^rzro4{$W{zMC+WDnQzpt$f`ePvQkk-9hUp%|U`>XjL!7>LhSSmydy$+hVYuGE$! z$B&oxt~w~tD{pzj6#0rttL6${3)E024DaD9&d6(oE;G zXU}@Aln<}Nl_9s*K+IR`PcL+T8c<;1TT~kXr7{VyD*d<@FHBZO%f(!OlF01Ovw!;G zlBN&DxypHt;AdA-D4;{_(ax9wDy9sHP*7=!67Jlhb|e09F*8tS`Z{KleCU-)9+%oY zM9L!wgT@^7jAGYcDs@vWVc6qehkrJaU`DorxFGTs*A-g8)<}Kyn@~_lXanHE6v#=U ztlvj%R)L{#6EkJV!aQJ7h-xY=Ev2BPJ;ln(`gd9TYknu;>+4Qe46A8CZPQjW3u=Jo zZT?_C_U!fR2sl_!M>DJ^d!gEAe`-o5LeBoTR#&az?zKo~STX{@;PcH-)ysbT^OMnS zc|;w#_dsyl){hU?*aZbO+`>&_HUtC&c{>2)3;;;e9- zQUGh?^XJbD%zPjq)YU0@4J!SixZ;ZBWMKh1Kb6}zCFgQ3_$Y?YC9n6_wg>d zreyrrg)+pb^>`n1<+lP(e6BbD@!SUHlc-g+?dut@I+!fjG4t%ETBB_q6Wh=H`OY*Z}p9-)HLqNTM?nRFu}bM{~!i ze#hcK?AhETF@?{+{6Lg3#kq5#z=~-3YS(c=NPk-mYWn>e%~%*zD%8v_VPFRi=18<0w$7 zwfl?t;2mAhV2}qOE6{m&Y=xCDr`z04k@aIioW9>bz990xsCs|vPf5zm@aEPQMp>EU ztao7vk(j#W4|8_q_#4*q)eI6A3{ z5!KMvmfc~W6uOlArDyR12ZzZ-wL4%?JHR4ln02XSGH^Rn;}WXuA&L3N$;&!L)4(s< zPqRMqmkT7Y&Xh&k5Y~BlwmgIC@u_+@IgL9dWuN~GUU_UC1f9e$B1X;{bB#Sgd zjmL^Sh+sq_fut3AXavi}pyeJsxCn)M#$h$8%EPr}(pq{*vantMgJa{wI5Pl`yrQCF zcz%6bYpcjUUX^t}r`@;GQwe;=ZJ=5CAMd-Nt^o}OuOP6X;0g%q*}C0HTaA2J+&4#w z7q>0n!)obiXo6t*9*;Q}FY@xXfhSAm>gsAYS{4?qN=;vk7|;i+HHc-nCw|j{1hTaR z@cN4W2ND2hgk^)>!I+#e^!Xk5E{Vd{Q~=|FuV23o;c`cu?p)2MU=T~YBz-R{69#=G zjJU}U_O>B-8HfIghQ9uO#76$d-n^Flf%RTLSQeltY=#}f^L5^tP1btTNqSs4KHjZ& z9acl*WovSSQVnZoc}N1@8f7&zS`N|5zNgM{<2r)jrwiZo$S$ldQ z_b!kY^#jKrQcuBofw~O*dh63Bn2}CPLxl?M8h7u`zRJj)M(2u@d7U>=>N*#F~|DWJgxl71V5-#uP->Rl6KzHJmv=wY0pw*>$DSn8&q^u0%;_{ol9jqv8k0%N7HWK3EBKn&k zZ*{BPE&<%Khy(MQs%K5Q>evP-N(SCjVT2rJ7Wxo~07gOzgdwt7AH8I6S_+EwPh$wn z9D&A6&62G-TN#kJrhx&?Id`1%GBPYkFG19CvQtgog2T+*TwOyW0E{H!38EOI9vgmo zDKCDNWQkyC-P)67oO>P{TVna;ENI>`Yaw=YpRu*IHDX4h+1X9+XP5KkgpoYe4h+l6 zx&S%`4573xQ_LEor9fWDoE*A_P@&n?p|u9I1raTX2iLPgu|QfVieW+d z9-Qaq&g(UE!B|v+1%|Maxj(;B9i5|?DMeJ58e6)%$)Q*%+`1L^A(Sc^Y6J#@IS>AJ zYa~h7r**rjkIX1UwLOt!>yyen%3&S2 ziRoaTmch5U9A>xQZ+`+s6UiGB2md?qsw2x;UV}5Tva)GoY-r|O@ESl*6a!uY(Lw5< zQQs|okjQdi_kfpH6h4CwXc!FXVFD5qT4iS;)BDoOZHLX)ryIfjeAZJ)vA~mzPYwT& zFo{useKy?c6i@}CdD8nAvxB9IMUd~|#!U#hIfU0CZx=y&*Oueur2u*w?!5Vzd-sDoZ?b-~;IOLKY4Vg=E>MzvV*W6if0 z2fH8y1Pu^X2C9w!;f`IJKAYef2~tkvjgAoC#TFe@AhR34d{LVBz$%6wA8x85zzDWg zC}M1^P|)Ki6U3hOWD@<$=fEbTprPS#yNt;+NBA=^)H7hhAzw=;VSLp7EIub1d^~8U zAg2pYqnW>a{i>j%LZ;&hIz9Px>^^g-`v%7|QNwCCLSx@Q7ofAhhffw0lbe^<1e-2# zThIXI2e&VHQR0NL+_!xOxKU70(7caVWDdON2$5tfdFaNhfqa0`530L?ajC7*jKOd8hpx zYmw*HygI6(*tA(#L)GwaYg_dhb2jYuSyEDdf+Sh0lsDDMpKibP$%ku%6F})>8Ywvv z6O)pz^Cm1!L^XrOa|3!aNbCaABe9DX6bo4CSXo6Y8jPI}FeBC2l!Mh0RUsjeiV8W5)o zSR&!=ER9d1dhUDcrl3Jgnt}-BE95agCZP8ia^Hii7&*Y!#}z<84FwC1)1cbz)+bSx z+|p7iKo#hPib=RpY|Ww3(oU=6;!ILr#l0oavW2u}kUicr4_~ugeGRthY+EFw-cp=m z1pY~q_Oc||+ISW9$B!S!-$Y|IAuZz-ri!4A&kX){8)V%2Zv6?fvEPw<>~5LcS8%&a zOq&Ukw^cX@0!~v4tce-u>S7Ub15o{Uv9kz<#Hz}CTi-`S2uMsyX68uyu@9yox$aE% z_lc{@{NbRYqH-IOjG;TjqTEtDw_bm|%?Gx46tz*9q9xeM0pa0F%}Zb%AWck^KSa=6 zQ!w%0Ps44ee;mA%NeVxI?X@JCF*IN?3Kyeh-*3nM|Dx1=?z33N`BNUQm@D3~%AT#~^ zd|l1Wc(O-Z)ZkRbKw)K(_7V1Wbp3UahUm4VC)w)iDz{;{%wt~&u4psP*pBnD9hwZ9 z`bo#oWoiZ!8ZoW_A_t1z7S+~DSy)&ga1kxNlc-?x^l?82%wM=K4xR+2*O}AxZHI(} zprN4wJ{}5n#h{YH@#7y(#4!MK=<0Gqt4$@ae@)Kz{q8gyLGrlMe ziIpg173IL^uV`{#{;FCxU<2}=+qZAyPn=(a8$>#{k3 zC`{R0ty#0lB#TL-yu*)kT!E=m%gf7K2l-$QmPWw#H4Mh06GN-~=+Ptm31=}jHXIOb zG}Rs}+@JO!#oJUg8!_|Dq4l6Um0t?pcvcbyNodiCPvOu$1K~;w9t`xH`xh$V&0DhR z#GLXgVvp9R<)n7UT_u=fAuSoG4f%20>gSxGKM}7<;fp=h?@ZjGu9$tnOq+JiSaVLQlXhc2;Bz$u;CQeIB zD+U7!V1Jc~62QlDRR;bYyR{cn#hh_rjiAyqv=^xfY5{VBUw$FgWW&+J*z1%yn z8hfrq6Wj-$?K8kx058=pm&0l0taQSOtHF3!U&r4p$=B$HksiO}KU~n?BIn}bDqkA> z(%pR;1;JdgC&N~h0C2^+G`n_QJwB9)NF%3sRXk^I% zk|G-9C}2B4F8$9=v47t#Wm#I#m(JBBLand_4(GE#^k2{8+SeEN_wZnnabKj_-&x5o z@qzVb%G90V9364q1I~uD({!bgLrFElr@r=HlI$EUZ5q@(US3{?+?(+3S7}!pF|m?l zCp%+~ND~XVzi!37@u}~uiaq+-Nx!u^p^s+>PsVUAK)?xCV2)bD&-FqtjMNRj z1M@0Vrx9}(>PfX*SR;@t#h@+j#0O^WGvJp%wj8r1vNbs^&o(=9siN=)Au6Eib2WN=^>dX)CQYh6iKK{VwA z7eV+b8iI@FzXMem2q9#%fzsa{N&qsJS+&oAZO||>A8291DuYQI=`0OGeSQ77Hrgd5Cc;Wm zG`Uatdqg)bX^}%2W*h8Z(id+y0l@sBp9;sX@(B0@l|UUi=e8u7)&2V)d>pj2;%-kN zBX=?UTQZ5LT~8(zRaM9BT9_*XU}XM#qbsHqe{%&GPvNSVfxbR5s9R?D8*th7mhjT} zIFn0N?I3gL+C)E6v*P$QRfku<^eJK z+Dabwtnr^Fl?9USjpc^j?d_*;Mlm&lDajb+zB*2?Q)D0)%_5Dzh75hG0k-|GwF1T| z?^>(0a%6($Da^$m@xn@Xyb18xYh`2L)w2C84aw9!y7+o`#!@QnhZi+ zP3^qN->pyul5uveh>b&o0!;hs4Syr}0s*tKf3_r05{RBiPo~wOLPp@3oKDD%3@~M!;Ur%bvoklxLq!fbh0pH9f}baib8Q}5I&95|NczHsCBM&{5|uUSGE|3jQ%EOwqvfm)em z+_@6eFRAJREGNc`4#7Q7$+tJ7St=j>DFs$eN!fcnf;({lVhB6jD&Dzs-$*$xN@KsI%3+@8ZKTir$SL2i*TY>I~8q|2;(a@M}8XaW- z9t?FDM*8F{6cO~HQ$~${e9Q==8KgDrLjwaS)YGR=k&l2odLGCZSak9qZXxf49U>hL z;K{%)oIZ8P)d%6b0e3;IhgMt_Y>$vjtC4*E{5ijMABV3ioJr;l1=)5Em@h=$Uh8tI zyU>t$c-by-XsD^|!`Y&5-@F`F^u%Bi6{e0ra1&4P!gl0@_D!BAB@v^%IAvI}%ggWh zPQjbITiWqZz+K(GF(6wO*f-J8(<4F^`(fsbPcGLEq$4_;ifiBr*ru`*@@ zlI<6_vxeGxS{}kEd5Zz!c$Xah;s3B?DLvG0DJ}F;nc$^zcHf(A)xrk>x{LQys7| zIu!JkLcM|hOV?0NVPR`a7$W53$#wN9_a?6P-y{$)Q8)_`#tr-iF}RDHs({U{8fT#}v^W;o72L z9yk=d-5KykKXr7-L6$fFx5KWlj_SZ;p@8hhM=?pbodIhX=5AU*%TZEL^n5=A2Oz|l z9NLmFze4)1u%N)i)bt5VkIglc%Lc)CTDXXzfx)wsl*yuQ=zl6+B;oXn`Dci=@M~=u$ z7?`|RWMzoRh1RWJ!2I zChz8?hKhCRKM5+{;^Y{8E82J!_1dfd2G5%QFTX6ve&ClTFWjk<5 z=U|`%Mz!$$6;!`QgdsOZW+fFB2bYxC*t_rBSqJ(?*~wYo8C$-~dIc728?-rC#m_=;jPNHu z@XyV?1XEaJtGnQn3cK$krIyI?2ZDGH_z@ibzj;X1)Yj(C&(9BD4a9Eg9%-NXH&|I+ zi@kw)W%&=y)RR_EQ)_^Zg*o1P7T}OG1F^TWP(7;xUUiG6@OK \ No newline at end of file diff --git a/docs/images/chapters/aligning/9a6755a1e31a990e8f072a6da98f811a.png b/docs/images/chapters/aligning/9a6755a1e31a990e8f072a6da98f811a.png index 51501137f5bad4ff1d8d2b43dacdffd230f9ba83..4c0ef1f979d32adf444afca9b94a539e5ef49a5b 100644 GIT binary patch literal 15849 zcmb8W2{_m9+AaPqA!DYDMM)|lLrOAaD3Ve#C9{f1WGIm#QxruK8qCTVg{a7^Qke=x zrlib6h>T}F_I~%??>X?mEBB9(QBug_JE3x>|c*`0IRseaBr?X1{clQ^hK-n}>Ku zuJhe1^XZ}y4DpdTH!)Wcy=kSK1l6eM^skpMug6R;yzaDSE4m!@i*;M1=H|~~adB&& z8fMl$vb9iXwo<4_o9>M=yZg7`(bP`8Pg6Gql$a@f4N2GZ2hLufGOP<5h)I2wppsDE zs(j+sl;MBb`pCPvztLh2i~)z`r%7+_qi&&qvd+V@!=a|NqDI z-@m98aNb8lU@Zqn*X-0_*t=6h)Op z%f5YmHa~v+xN})y(fl$6J6birMxvMw0=6V`7ZDZz_589|2IQD^#_1M~Re)dF>!i&SLW(h(o7G}i) zB96;z9cD1MHKJ^J?6!(Ve1Fmby0;%cR^{8Z_@9po(ip|s>nZE#uu#^?Z0fh#$C4E~LMibuA!GXv{#%{L^18qJW1ICotG(*uqLs9*~_~*}` z8|{@Cd>X6RoSUD&|G)v7#zy0Yy#6fuf`f;g_3Tf!$GV;_8febV$)Qkg7TlZKS@mC= zmr8GJ5_j$9#4pLu(vfi&TV6*3*a~DJSc0c)f?@B;0`xZetIioNucGff; zrc`=)dHD_D8ew;Z_n-RVlAN4(z#@S=Ai$FRjl7U@aAUYeSAYMB^GnIeLLctQFl;w? zvWk*t-^Qu0ukYyORNkC*Qm<|EmMuxjel+RXb+HNz1_lO&FJFG2|2xq;@cis2n!9)J z+I&uB#Rlk|k2@44zJkKJ!!)F;%RD(b*=EXlqhKRvt(A%{_0OL_$EK&%U0ilz(kyFb zUVPhyKX~a{Ow5TF(|5&9u1i`~m$tU@QdqdSXemL#!OBOEvK$EEUeR^g{BlF)iCh+b z{+qXNvwet>ragA-*y)ZE_bCy|hF!Zj7Z(?~rA}$Yc)nhowVNFJ%FY(ND(OIIdE)*X zV;^^S+ALmWVLi*^M|W)P*S0pRv97Yh%M-$E?Cjxp@6u2l9UYGzK4^Aq>k6Mu>Nm~j z{tWk;Mro$xJbil1^I>Ep_1?XEn_3K>7%T7FN4cZ9+1Jl+d#1_Jqdq@=oLWUgbGOa< z_m3s&)Y8&@tEj24fa<#s`UUNF)eQ(b5=YcwDYFPGwHbX!=S z?d>M6e^aD7Iy!JwB~Q1)INqEH^}ai*yVWTG0IKbxSmG|ygJTWeTwem&*RS8KeS&dx0HuECr+F=EnYG+JF6nReJ|7E*IaAK zi$jK#(y76g_f2jW;u<})oSdBjuUDg^x$|c?3#Qu_y}=Xp_E>%Y{@phxCr2|bGwphP z{n4Ec9YvaR{{H^vP3d|Fg~pi~{QOmGt)QK{ksewU^S=6o^h2Wg_QS3DzUzQ1Ch}xw zfLtQfK-ctjestl1^z`&1508ch1*Hd)aqr*1pFplrxw%U6##l+j20s9mX)Pyb`L}Pz zeY*fKE6LZ~s{MmBJdi4O9yK;*OwoDw*eNI|uX3l7k^p^xN>-?uae^)Zm&syNeSJ<~ zeV+d4{#o7ZN$Vit|HAyOQ%DVxmBVkQV}vYKuo@y!Li^TGELBT1qmmBu(y|4WPal?uc(LBq|y-WsmgA6L{;lr$#ojyT9 zJKta;S3i8vGCXmDuAPQ8ybKAgaUaHn6c)}2G(YQu1q;f%f!R~V#Kh#}<#D`Z5)|}9 z#!FtYyaM861x}qlJs$GUujm~=c#uv?S{l&ge?B1StvSK#<9MbJ8=V2CuwbJg4SS-k zrna`a9}ScLdB32y@ynlZcb8kYZry*pU}VJ0yX)Lm7d|yLngE34J$@yx56HC1r?Cl5 zlU-$0JI{YJTwZm0V1rD1iMPPbn>SSol+{ih)tZ=?NI1InbE~0vl-*PpR%zCF6sIsu zLnwVzz+^Cc*WlS6>3`_j?^Y0XB#bdu)vbaixmJ*G#-b(sJ8ULx%p{?gi7 zW+0AAfr{EGN9=6+R!K?LCr_R%FF_c2K|?q~StOF4?5(mTo|sr!sfG30ByO+1koTUF1fG5@_QT&JuXxO)xt_&k1Bg-Sq`5*Q~m+T44jVh$^BQc(b zO1%9?wBNe7&M?ZFe$3$?QFwL_Luqf_AIPCJINq9za2*JT;Lh`s86&1Do9eMJd%Bm`SlC^0iPIAlEZQIJ4G7K7?+n-E0 zxlvHc4J(y>;aBbYFKul}UJJ9!b05QEnf4&*_I-JpwEQ4y@?BZ)MfuhOhaKdMAGliP z-?#SqjT@|7T-T6}>DuX8!;j$@xCbDURJ?mfck$xI<%e;QhjmZ(f9m`8B6s;=On8{V z%))dh52Do;5r1s6l$4a?(xRuEyE{F3Py;znYUj?0%3T}^l5-tc{jV1*-$iU`NIAr{ zd`v|!__bTNZY}*g@oV{6G5;b=?|YrH_#aFeY8wnzRaYZ5a{|M&70&lh-%`9~@ag|! z(@dURTAaUJG*P?PcKxUF@|D2B>^B)_6&M&8D5-xW|GC{d3OMoZICpejbHlq`MmtRU z^n-#z5GoCa#ElCn>gwt{&o-+4`$c8}#gI##o(3dEsm>+iVW+z8E z2}gO-*ZEg%!in57jxH`7|9;WTdtoi8$MR{;!MEq&d|h=#V;o_i16T(=E}=F}?1xgqJD_N}`4Dl(Lzwt_XUU%$5b zRw%j4p(Etl8iDiA3y@gZM&?kJ8x}c2A zTFTgPL8qXiB0rMD+YcYqjvZrVV`Hm-b;;oy(3qNV`1tq?Po9kWdcHCF zpiSrN;(@`5^u;5b`PRNN}DgN|wh*9+r15EnXs%aI1szt$#t}|WDR&$baZrSLj&vdXeTX2cJJP#BM;@$ybN)N}KGd*i?@ekMCRJL5vf71)`#@87E)Il?S%S`-BKISG=rXV0F~ ze~yVnMmF04X`elNme=TQFZr>q&R}lD{Dtw8Ccj+%K{`FK&^50vjaCmdWym}^IlGYb zz9*~yzalPnq$NT_sL*3t9OXhyk*Tq9$b$##-uC{M+WmcQW~K|X!MU#tV%;=WyLWVh zurST|__*7|$K6lu+jfzSVB*_V`Ec*0uAe^xNm6NTHFBNL%iH-GtkhWP<^126>s6LP zRhIe8wsv+I`BY>?CmdRym;ipHzqPl^JUMM)u_8bv;S(8DiZ-|IGYqpQ!*e zj=vG`?gGlV{Tbo`dpkDkT-(vXAIdA^o0}`0tg{WzXp*yfbff3A!a~Nco*uvZyB!D} zPH!Q{cyiKZ{`dPGj?T^%398pX{;7;#U1YL+%57?D>bA5nEqQ6gc-rOM~_Z3IRN3k&1z+n$g8`4gxs5|E!SlbMwj9u-9!ph9SqyxSP% z(r8E3c+Z;)gUw=}v&{s+yFo0e=I7@pM;bI61(xQ2DM{MYZ$Qzb?n-s;;>BHoYXq1n zTSY`va?gAWi;cZ`zWt@e$9pmoGBW!+UwaxzuAE{Y|_ib%$_3z&I<8z}WRv65e zd^@gzmh3hA7CO670G*QJ0hIax)dc0`*CFGV?j0@iyVP-+7CVyOZxO~iJ~^qjw7Brp zwn;?m{w}7{iTh?|sQUe56+G^?6`+>x`}9~lrlmTBCuw#xifcphrBQytO`9$NJj7hh zz+{^x?}fDBg&fN&dU=mOnXTUESHzpVx|n@s2^nw(n$G0! zIx#?KB;dFk)#8t@YzlYp-0}Bb7&KeW%p6!!Qlf8nKPDytAk_Ccjs47r=&ShaMky(_ z_jkoq;}ku|7UxD)MIvpdqE8tQd`?S}JE;@l>+5Sf&n*Mq8FB!Uf!9not?0?;T7CMd z_kREW&4R*(Jl|-|<{i?~(yAz|8*;6+g@lB^$3P z*rj-t7ADz?ii!|ukEYan`~IE|zZbhG2lPZ?QF!IAl(^Yz1$Bo$KH617H>hGk{S8&E&>Jg%IaGJw=kW-c6p(D-&58}xxJ6t z@h0_HlCHCH*y1+uj^Va~?%`oEm+fjY2hw!EUlZ^uML34^zK=?B9c)%95L%6ewE0#0 zP*p``B@)pFm+i?{@7&=4Em#GK0{lpK6EKE=!l^T7ppuoR?l{5Q@^_<@NI*b;BOX??$-Q8#H3{G!uGyv~KFa-MfQ7ulyEpfYUUEx-(=0HHG+w$7zM#gO%G4G`tV`Gl1XlbMF z>vtGRM)zIaq8+Jv#v0s2h{k>b9+AgmLO(Hje1}C z;KAT}`yzc1Bmjc{3+G=e-Cl2PgUYj!Em*>`ay7y14}aDck0M+lv6&-`!O~P1{cM#l zT#(eyHVws8ElEY}?9BdhGnC839u&DOpctrNcWP?+*DWsCJP4yu+`c4Ii~kwDrLOZI zdR=VizCTkfW<7pu6n&DClIn-UI7x11`1!IA81Mh^;rf@CMwuzm($>sJ;uM*>{))Vv z_*J(f*K_~qbYBA%sCYTfs%}Q@ZHt52+HXph790$Xj4Tm{?k{2wAO5Z&EH~1cZ&#;! zP-W$;mjrwD3Td;l-wx74U z41A`8aCT~I8l^6}uMkoiW#6@hxnqx0iL|(XzfFm^lGj5iQV)%dkEiuAyB_A-|9i19`v#)bF-(06^ zJk$3;13`+Tf-K{Q$V*x}u>kHZ_r&Ee7i~;tqdo7!R0mMjE@g{JimnJNFYk3EzeF%Z zl>$|xQ?|C(QKj^tZjnKF z=d;@v78|DPv?B=NFfGV_^=hxBl~wnTAIXvKdMUMTD9koSywjcDC3F58H>gGW(%tvH z>75S{|7P8D6jFtJ`TAAwEaU9tud3()Fb(EIcSV&;ycWob1*y^hkWBt3g*QuTs$x8U-EEqUO#v$iB*F&Il(aC58X|?wk59;{C9ll@*n@w|8_*Or(cH ze`5-_^chCbIKAwHDUD8T`B;+=Etso{o13hrrly2-?K-vn`y<;PC(3E|PQ0?v%-*f9 zt1cK$YA2FEUjO}hzvYTl_MgG*lJB5d4>Ue%KuES+n4RL5b&Rji(tTT5xr(rLgeWF_ zi@u=0CA)Kb_8a`MLgsp4e7FwvL3c*>>vB|@Ip@Ag<(>U>pzrP^r(baC?T83g9-iPE zTsu?}j+Wv{z&f#sA6JWu6=>G*hXBq5A@V>cC7qgr0)zi9eV;bX=%c%unJLtK;>(6kT;ZsX$hI7s^aYZtc$xf-Qlsu%yVijsF45pgV~bNtAF4b zUkV%qQ2mYlyzlV6yW-mF)vHU@1w3`T<(WwH0Ngw(iibndYLOJUAEG6bjCwXLoU*j^wWtV0LvAGlmpI4E z^-@lVy&3$Tt-> z(K1(pMF1##uvDwtnR(JcW5{DslFal!jx)Ei+T?PG#;+`ylY@(E6FMBI#RJpQw$1(- zW-v1|BO_hIFTW2>!9{ zR7a!`0;CFKSl7yuX|{3DS5{Vb0pTHQQ~=bvQFftf)QXkoVdC9cijUg5TptDl*(4?P zOiN;0@eVa1wzS&RJFy?PdWwmPGC~odrVM>~N=0$}`5{J1o4&~SO?&%#kg1RNcN-{B z>mFNAk#+2%xI^>o9Y^A!-hKXCDHzqB-Xc~$c{hoRLtn0<%4vEVb0G0L6g7*EqRXVn zySaYv(Sq|P2`8mA7=>jvpz;MBu6XS^w;@tf@^lS{tgI~Qu6+Ia6>!0Zl2-!a1@F>( z86lxn*dY~1$D9Gt1N);cp_;Dwmiv8hkm2j0PhD{e9-M$tJaP2Uy1hhPL>ea+_ZN5?U zm!M2cxIFRmOWxUt`mC@Odp&1&E{@;whWu%c^xi5KNo9;~oe0hk4=PJpZ2zNs zG_e|dME?f&?!kva>W;+dC8eE_jJ6@aQ&=clG&k}J__06b(wQaHDuv!lim#UzJ&ruk zr3Md(R`irp#OqEGrK~dNzOZamU!WvXDC=(%FTlo)(Fz{Bz|YOkp55khDESwlH|Nr* zDaz#~LJe}z7R_&t*feY*krv4HYeRqgtKIrpMpHEcL3DI#+z3+#;nV z5ZrQWf@U#>BvAm)Jq)EhOZ}OYPnVvQN*RH7wv5`94=IgVRdNH2OH0(0P^0dL| zUOzhW(5OEoW5);=`jcxEXZGcD$Q5_s6ze4!bk+|b{epV7MeNvxbJ{fieb0EwyH zN0{D#AWAs@)FPcwl*Ftr*EdojdclgykixqM2Cnb!ymGwnuEv_Jk(%pd9Vwk&b7C(q z4qpd%vb7h~NLQqZ&_scf%Gs1*R`S}G`F46Mvb^H_m}>2dgh${Mf{0K^l=-&q)`G6k z2RRrxMN9E>PfxvL6Hh#l?b1`tL{7G_urQi=Js-2={L6l9PH_%Hey0^X`MmPZw&S|D z=JfSY(?P(y!LQ)HVapcv_YZj?0{k%tY?8cTlyAEkD|`*7;~HsU0tgICycF*3JS&JT zwQ6i^B=GY<$(vumdtMT$Xl$|{?T|z+iRk(L;!%W5$-4Bhpd>~N?+g`Cl6M{8AoGMk zZIgDWB3W1D>6wqb1gTz)zE;H+`=1Ym?;6Wj+3gUh(vlv}+cHNN<-?0`_0}=8d|Z zI#+du-4wD~j$^k9>PO3}>uYoJ^Rrx(Ox8O<=zj9YfNv^qoW~}&1hva3_$JbX{9Un= z`sYm(2I_mqpPBaJAhWDrABu)-0&0%*93!-Y9|b0aw6F{iAuT_)0rdFvBhi*S^;4#? zey5;9p8mO^;ehwqvuBAar|12ZxAE8GPNnSwHKrWu z?(av4pXwqSmWPK25s`^7)bPUDI^kp@uhZY(r!!9$Fk}AFvKIv1rv|G2YD|VH)3P7_ zwV^>5atBnq@{%h{x4~@zz{Jb&<)v$ezV-rZu_ofMAzqmZ6oCpUsR~*G&PZM#&~s+w z)#%t*x9XLd?rZQV{8FWL|FFR?SS4YfmR3l$bC^xdr%&urJIr>?zH1`r0oesuYpW|2e)GsqN$AcJ=nia2(|;5 zW}0l6+3Z1OX^CEh1YwfH?|%^q@I#!E02U2JSyxi2&{kSn+Dx+&eiHAfQ=gibt*m9s zu*Mk1+8jzO@B>7B#$y^DYYWh@QCYH-sfkA+5woGBx&{)3I7@0+KhyfXuW$95HEV|2 zUvZ5VPsi2|e@^%HU3q1?{nBzK0vy28aN*C%Mmi9%ndy?HP%M!Jbad`LPD0QjYn9?q zaqb$)I$2QO*vKZ}HM0@OU%li&qmk|R4%F2VeToNwF3_b)u#J2vv#YxfKagd`o~R(} zg#L`Zj+|=o_jj-3rAu#sgR3#Mgp-0Izlx_jRPXPymmuQo-iwS#^!q1%ehfH&!}NA| z_*Ddk1!`k-;$+U$J_z33;Z|Ye70bJ+m{E8~o$@nCeRuBM>H8t@8ShY0(J+=m03@9T zUYD08uzrAo^O5KT-|zdH`?y|=wfJ^Y(k4*Q$8xy_?p8w?KR+k5UI4}oK>>iuQVICD zf!ZoVoZ316Ld7G4vLAZRVST>^K0~Zpy)@fWLcA#un!a`Mn;p{9s`Or*5HK$BT91IJ zAPs|4*47N|oZ&`lf~-&m>x(@u5%fgWT>-^q9E^aW{a(YdtsDf-ycYjj%X`jp_|dMZ z#=FOs$xM<_@K=`}G$e=BOiWe8J%WVZePhRKpR}}E@5yXev{%Uqyl~+HH>y7n z?d*H35gMI4Z9lIjX9YC&>6O1u=Q>}nkGt}l&f#VjHEP*vP$SZ}A+8^GVJ)AyxH#oF zi$^v6L!CEO>XZ}hQF|E{yUQ#tjCxyIIu zrrKGjnSv(Ea1@!qRfo{NK|(oz11@>)i^Rhs-OM_4_b<$hSqycS2td0+jYs72$=|;O z&qmc!vzx^DB@3fpAZqfGgQzb!;^PhK0v2GK?yS!nKYbcU!(fM> z?1bv16xw7sunA@*UL?UQE=(SG8T!HsQo`y&ztR&e*lP$*qL>jo51I?(;M$4Evox22 z@26)USkxy9BDQ&1j_`t^PDLa#5whXh4#P)w&S9T9U`c>UDIO&i=+Q>6zr9?<+=J8D z^upnTbzN+CWiWft_3Pz$#yEN<<_(GamxVH!L?dX{0r1xdZrDHpnXiVPxiL}`$oB4# zyloA_6 zXQ!F#-F^ScgkeWMD(ieQ%CosT`g)F;oZIP6?^^_MAtNfzPWEr=3pItvC5tpk8l=WA zUFcw6QH8S!rmYWi>ThjMKu$xU%Dw9xH}iTqe|(Q0a%2_i%=!Wcv(Ip~^>MmtqWJb& zTDY`M$!i^E9RNkA!(&u|Fxo$V{-_kF%N@xmOGX0+HZ9nY_FGyA{!wbbd+a8B2gO$3k99)%3E)h9%~^b;%Vp;Em)fp5Q~A}!VG=Vg zBj0$?Y|3otQSzGSHpn)uf)^5Qv1{n@%>4b^-F~UFs_60rEN`}xGitMyA3iXms`gV8 zRMym_N8bXOoe`a!RjXFTgW(>yT%rddbmZF$KdkyZf>1+4V;^9tA>Yn`nBg>2Zcq1x zp*YZbV)7~wR9}cg$oxdL6i0%_l<5^^`H@rK%09avZ&)cd_DNf*tfGQyyMCrmXlUrJ zw)#Vu8N@ziI3SP2$_FC?h-yl#k;n;XSYh))+b3)TBLCe7+HYwoxHvzdO-v7{)limP zy?>wYXYE6V-Me=e7yb&+uz@#?ot z=)+AHv}=t63uF$c$!nl~fJl<=1Q;tko9bFxjD*!a){Z01U;N2=_RKdl^l;4FaTE+N z3x!8S_(4TDb?VeUQO+c&n`lf2Li9oXcJ1t^M`_~s-}O2z9#GX~g8!titzZYKY+o4P zTw0tiK>(>38ggTix(5eu;1tFq1l{>&(Y_+K2m-5!Uoyp`kB_X+vuk0785W(IpPzOn z%V1h5X@>e6GFJ8@@ai;&%njSH8V4V zj8CIdU|gGPUFV3rA%A6t#e0677IHjkouB!5j~oxuzDhnAQJ;K}J>leLs?+yiF-G6B zxw=(${g(12uiTbD{R&p;Erl1X@(c8~a@Z~?HTRAWI2AY6=^M;tgc1Rj@j?+lKmXX+ zSSflNK|@35piNStty784%2TjT$W&!bO>0qps3T!E;HOZ#B|`PkGZiZk>G_Cog!s&_ zcqIrhL|X9Mwr$gZDG?$tK96HbP9_4N2r^Y~?4!(qUm4m<-Y2xbxIeOdYM9mesA*4= zgPq+CAW-4G7vJ^E;9k8TbBTIPJ`%i|7h7TUBFUsJN)Z0E&;M-f<}NfHC~n;H9F8+9 z?(D2AF*MA_fEZ~53h02MciV!C;7|2SE22%p{-3jHXn7ral_}t{S3s+LD)Z8 zcx2+irlmE_(wXi?#dms8D`6&#IXpf}(u<;(;)1tnHJCLDESn6HA3d@}L7Qq;qIl)X zm3nwwh_<7Ci@yrTIREQCBjFztsJBJ5*mm8*e)yK8_=B~1O^}{>pq#P+<91A?PV5F9Oe)oKR-XCwuzsvQbGtu zqTKz%_G|piMB+uq52rS=lSo`^CA74((8{C%xRFEz!>)>j#YQ3zl4f$8=N}$8U7$RM zux>g)NtF#L(h!Nf`*)T|xfmUaLkX*D#(A*Ps{g|q3>lhR#+HK!k!X&Hd5S=!Nfytl zo40jvxK-vy+XHD;LrV*GTTaZU_K3{pVz`h4cGnQ8tF6F+kYwyURg6|3gw*e#(xkzR zHV~!jvc%=4$>AL(aS;(z;2UmkZX3kKnJF-XtVLegm!>NyD{GismDpjkk6! z(O=OQ=?yq;J7OB8$qqZY~oTAUwq=*dLABc&8HXeu|aJ$*=UXc6dT_2f8oDpK|t z8b%MKJwV}+>i%Cc;z?Cu{1g{{3V5TOZCcE-ZXIK=nojqp-H@AlP4}X#M`y$rb!O^$ zn~!iokTm-90}n8k72xOxYHVvM1XBC<)al2yb%0VewN;jumbPbnumzKva%;7WmzB9d z_}{_Am4_D~&%IaZwX(AE13o8#N|DA`)eSDajVZ{pgdu=%y#HVT-P;#0l)ZZwoB^LV zE8V`wzH}7E^>Vtd!TV1le~Bdj`1$iY#Mwf(alz;3+v%;Xtx4k%l1?@BlYDn$a?OPI zA%n2YN0{gz%DJk*t#lMdb+FZqkdVQle9=n;S6HQ_R3fq?9)nlAC;ig=x1h7LGg257 z3|jQK)1|jbpZVZ}UM#WtftA6tbgWN57zR1tW3&-mfU7i>n!*b`j;Onhi`~}}tBIQ6 zF_V$At$AtnSr|NzC`|-C8gudJH)4uq5HgANv$DF{a&i7IxPlHfH=-UnP*6~6E2D5W zy3e{d7+)89)nO{f;CU7h5FnFtD4uGzdt41qa-xBiM4E5oln4AiWRK9YC%$p$H-0F{ z0bATU2(_K1=7}q?FmP98sMKKh#(HikeMi8fdq9?>tGBmm%N2a^vrx_Z_s2yz=o987 zw%!s@IshG$xLS}C%Td*%4WN!@v$<`JsErNI;ce}bbZM}8vgB28z zPP%V&;oXvGJ;H^W-F{R&s-n4B-;Q?I4|fnRn}$Sc(r8EB-Sh2*Gq7?i^ng{Uxe0B7 zqs!KFM5aMt@N7YBKmaxFmDCb}NHfYkb>!cc#>H9F3eA?$%Wwu-SX%0wS%`&@SyorK z28Y|9xCY3Q)7?a~K+k%CAE&^o++2{wwPo}vaUAMl?TjkS6RAF%lsrb0OQk?lO-(n6 zTg&A`M)PRezjNENhg$@?>J<5n(Eu-k2_?4R{ZUJ`J zXoY(BH*Ureh#Z_|fkD_pqCr(KTqFN&7p=gR_ zCElpyS67)Ik))vBca4nP1oySc^IC(kKjPL3Iody~uwumu;$kB4JMeki{NyKX;}YC} zT2*Ck`2%aYt$5Ubd2lc}YXmI+t)5#eRb}}@{-x$B*(oEl3d?42td{%DGXL}+dV|yO zw>0(g2dfFXWB_rmsiAqK8KAP!MIiYXU>70m*j-L-T;#?IQfu?+5fedLhQB8tmXcQ-+>O%A9ZCzDb<Hn_b!*F|tLL6S`{4~6DwL=bWnRLWJ zVD#lliQ?kPteI8#zgE@)AYl12>;7?B;Xt;?;4~ET4I&~$hu6?L=f5!L85UO0O{q_; zQk@+2X4UE6L@HXAhlmi}}g9cgDmZzX0mjYEg1C_|l{30hmX>FJ*G zU6^a5qTIfH+cwt(UIQD@0V1 z?>{2f-qQGY^s?w(Maf4oFZG#_kP#D~ew}t!a&}tD>`WW+GZ1YoJTqI^i<3O=GmPVE2M(nB zzC(YVLIIDVjFm`>q*zPiYZyj(dAY=L4TB~zg@P1b2K`VZt7l|{8B`xdK}PJ#^i*aF z>6tEH6SoB5i%}s3pi<6x?V;X zd{Gy~1O*S{Lew!aJ~@}v0jKNYhOaJW)r+yPrjg4%{sk^Dd2;8nd774=wmD~HOo zvYNZ-{{w|-Ki%Ghi>J63cKm-xEg(<-`}O7VB|ewWGf8ndj<~x`Ik^9b+7lIX-~RkYh?$8o7!27Z6|6P}gUgM< z;Ls4_!zZ}0oh|Tx_~vTLSj++X-;3JpI1J_l<`VXTu3OTtQFp_~GyCV3B^O9>$zKSh zzj(p*;)LG`KIN|pS~{XSq8+DCf9mUxWBDK!AIDE;eWOLa^YzIg3qIbfcZe5>xmAgY zU%r0-?qhN15`~2~M>wA7^2b&0>+wTQK|i*?rdnN|NVIz6fcM5&Fqk@1o447#N_xiOp^$OZNs&%`sd?vef=J^-=TXKUI z|M)LIvacZ^BqSs!SL7Kb6;*7jsW3YS2S;RssR9;@W%fV(QVidph>D8p4k-g}59|x0`mWJ!vCp$GY1?z=_gX^l0`5>ajyhK~4{rk+! z1&-JpJ5%)2pOQT|VhJ(Ka`@2(;)l-`9e)=dLd8P(|NRh*>i_qJO@_1hA!S;3?(j}c zPsb)CgzfHnzMr|Pa*|r=?AfM{4w9EIUn*3pU^AbC#K8wMetMeB=!&kK8!kKY*Yy0>SOlD@Ln3W9{ zEAy}+AO~ml_ir3bMTJW+`L&U=0~K5x95YO6S=#S!p2?Ev>gdo}n1JQ|qOU0!FS*S) zKV11{L2r+Aaz)}*icB%V{hIyN4xy4;9rRz{c^$j={bNIOC=HJ6yVR_TQbsc~v)R5Z5(u^4H}|Xr zjLLFaCC=Xed0tnS_Kc`#iQ9_#$7doLV_yFL{v~%laX&f7#=)WIsj9OgJ%q zc>g}fc01Dl907dH($B zuUb*7t`jFtV3z6*3(2 zV`F2J75xwj64Zj+oV13MaQ&S-cfy87D(sqav@(_l3~ul)3gvdcxZqF8${N+!s1Ohw zY%x*)P`tu@sdke#{@J&?Z_LLct2Z>%)D*i@dXyc7pz)ysQ5i>ILrLMF^9t$q9XyY>( z+W4knVq%i~I}EmUVryl3Jxau7UROO?rrfA@XQS3*olkyuSutyehr?PGN-LlBY!(YA zC+E-6w+6qwa?V^Z(pgi)_A}NzR?JKfLM1rwm_Jc$*oTe|CMbUW0Sy5vN?BQ1>z8vb zVzD1AgxR&V+})+2WXC3;H4L#re-(`W>ej7WE`r0#P?nm|YP^1ZCPjpuLz@O}O7}QE zKI8T4N18K~RJoxgx_o*C87l&MrOJ6E3neKPLmdzvDkrtC)$-iYsm)7e&x>f7)+2+j zUiFL8hqq9Mx2Tfw#j5m62*YQB{T)zzJP&T0`*i9R@g<4hCuJ?N{%5(!eevSO`CL;? zcv>@f+6xyhJe;bu;^5Fe4(~)4Df-l*tbgPgVoDCd|I5hEG-)c2lWqsl^9)!nq)2; z`WO1DtLyT=zJd?ZT^0rvDlcF&btb8faTKV})n97uiMz_hsvN;Yvii5>Gee`jp4X-(jg5p}6PU z>wvn0{bNy4QHPIv4?XVW;^_pAfPRb$Sh7D3nxay%S*NwKy|hzD&RV z_Wr-F%z6EqdUI*q3Xp>}E?hz8e6a6jcp-1(Fj5^7~lK}%DHGQGBY{E_2jzASTjZqL4cI z#yIMjq`9O7U9qZ|u8rgF4E{}q2jdl1H0#Q*>MkXe`k;?f$%Cq|c z1=&&EzJIT8u9X0NI6Hm#6K36ifE&Af*Y6*h+&nx6hc!=9U*L`-8cU5?$8u0%#612- z67bILtS{6=KxQ|?YCLWRz}3Ho{&mI3D6DROhh%Ya@$h{=p%0gTr1~_rv>ZKcR7P1% z2a(wTdTcm#d;plbZ~ek59GCRk{>rpRNG zJ=HEYW!3|#hwly7^@gfWMd%7^5!N3iiX;HQw7~F-G9(3>C zy*qp=^jGsvzrKhgC1Wea9d` zT;6~Dh=nd7D<>Cd9~h$I01q}W`Ss1c+1?BSQnGD3SPJ7Un)n>@?90iGb^Gf=hnMoh zrR6d0H>F$`S^WI`0??&JjQ<9A5{!f-B|uE$ZVE#-Yh~j%YKI?fD1n1LEv$7 z+y~_fH6g4#O;_qU3I+y-GW$`Z!zH9fACj}lN0>DQ9d6?ogzb*@N8n zG=)%cn%=y5^6)wcjH#vpiVU01(da&vR59~(1)XHyA2F6TiD z)wWZ95%MC@OI?qVWE2zIbN|&nW0)bX=d zbajsafPmuDG`zL4?GJ%@UaTeR}47Uw}QqFIpnIJ}lDYoiPVIDCuGEz86 zou(Yk>gecr-o%7;u<#mznwna3b929Y-2O|AL`r)4pf~q^pf&?lZ)@$7W^ajwa&t>d zT2&PzWby3;JYCLxG49&+*dV{$w^e=B>6i^=hFV! zTndhkrqI^bM)1mJ=v701f7zJGx4Y7q+XGk4EOK>;q1*Jgy1TnuA5U60)XedKwk72_ zfvK?loPKvSZ;4Gqe(dF|f7HoOKRvp-x*!a-lX`Oy+-q!XG-``i%@P(9!``~Z1!Z6O z?l+aSg&_(I!s^X!ZPRalr0Cd=zw>&KB=5xx;OTv5XNx}p#fexITA#{_3jK(T_SDpW zOX7fn&ArX>49!&e+2N9m59LsGZ=QL6kZ8$R})CkCBp*F<<=p=9$pw{-*K5#vPkI%QkuAQxwanWpVXFY64PjlvD@>zqGl#-H?DKAAu#R9_;T+WLT z=e@nf?1px`l9vEl(hk0 z9K3#$(sw6xZZ(HcS+iC0iad)kk#<`?@pW``;I0D^(KJ*=H((&bHiH7BZ1Tp1<&lw* zlS@k!K0ZDIhQ%1v%%G57I!wWK09(Wdq(iEy*=?<&ZLiL7AJ3Nn)`7YLR5RAKpPy-XG*fIh zSXJtD_4TI!mfRkxC>;CxtP`G!T}X)H-2N74h203ZGrx?QV7x{9Bd*9V2G0}qN=r*c zEI%?rET#eSya@_k>@#B`BR zP_)5bu6^Gr5afgO>`atIX^?h}-TKVkH!RmGY#Scp5iD>2NX>vsr%-v45Ps(l2$gtRIy$r7bfx#*-EETNo*6^&0_zLAlygt|`m_s-%88*F z?-FQq+`4(Q{x5cr&^-FqRpd_J$+A12@DnBM`WCL>E=Pn8Lak8p^pu506q}k#0p&Op zi)&?N1tnsIZOjNoEeh_sy1Jgim@p999vj(e>iqmtPJ-y>n=S}75&N%F;?CHF`ETuU zh9%}lzkhu04{`_bXw*eh;a=R&pFcy3y$BCfz6mTo=E;-5%*>OdOlOWrNJ!wiT`!R|}hou^~D+dvyFrduPoepQYad+&M39_;>BsLT{`<5g~Y^0!bXAP{g}l{mjnBveY+kq{G6((8Hh| z0@hDk zY107nzj^Z};b@RH^nd+dcfy3;ZQ&w0$IVTu!eKlLR7d7N{pA`%udX|7El*l5BqxkP zzh7=W=J3K{{2kP}WRp(KbOC%CGS?qn$1%R+4gT=kpyXw&-rYJbONddqbm>mlpjaRD zwPN?*mH|OQ^P{B))gOC%sX)8x4=eS9aQBOgi-R$EydkMnF@byh1h`i-a$?zlll7fD+UAjNXliN78e)S5vY?h^q1Cm<#uuL z@xh3nwX|eiJ$PU#w?6pRZG}U>$RrXJ*@;NWG2hr&t&G;br_*`_|5Py(YG$Kf4VIRb zVeRb&&s;a2aGOR7=KWa=rug-1Yv=BSyET!8@kvP$z_wYT+8S5~8MtilP=}s~YcJdR zty7+&auU~{a)Ekb7+WB*GxMZOr}0@dy@aY65bW zD+TqtbJ?j#Cvvgpj8%zw^vEwJCdP_2ykc)Nonjr}-ot}Gbq7t*!Xpm$H|xUGToU?H zxre5e+{m89#rccd3{HR&h4KB(shr+e_5DdDv4uqvn=~~By3ijZnH**435+ zx4%|7k;$u_e4lz^8w58I25b0zS@{}lq60?a5MO0GOI_??;zm+3ncAKkyzrbtVR#6BDMkU&?a#DHe#%?XA(gy8bq(r{_x6 zCi^dO&n-um-5H@-z{M8o>R5Fkr!JJlViQH`Uc$GN$=U81NM7z}=` zv;^%yQ*+w*c@;pV1%MtxQc~NV{nOVf0-(6=tj+m{gb+ek8@&3dy`2b1jb5I;wl+1) zcPN>da`iqv(V0Gqk8eI&?V_Tt9`N?<+n!&PIBv3%04O~D8vF??07$?Hgua@qu+@|U z17eC=yx?U0Lp)Sc;{~rT|5>WDfWnB%a)iyB_~!n8S`-m#XUN`sJ|!t>kk$4HCM>j~ zIiPln-A?C+JAhUJHP^HyjP99~69-89T2>l=iOjO4_JBwC_m+G(futPYf3qF$oSK%# z-U=e?uaQcFx61x1guq>HcyDug@2wjMT(71*R4OuQ+<3E@K$xBY*4;+H=IINx-R#@@ zSm>_xkXi^JO(eUH<%&Pf`S9TbG3ZH=V|Bm*0F@IgZ;izS1+f4q25N;!q?FITXRWyC znKLQdx*AGKNB4iH9+*N@GYXrYSEL9o`w9Yv)654tU>1N+=1M@HnMO)8uy-KIEmzVn zl5%&xC@3IB)E#h2I8Zv2cH#7=6<|zoIaNOL_}TmQ-Q7j)trUcW=YeKsY_f^nrfh6# z(t>WeyzEHIB30ZVPE1T(Y~DiL!jxKV{W2?ym$kH@quOP`)ZRV{Fwxsd0!1TD0eXwR z%uBkqAcc(^! z6w>(lGt*MdZ*t6uGiP46Tj6-F4;WC<&}6vneWN+^Vb^PYen3S<#ka8VG&KzkohFU! z_T$b^e~hkP)q5}w9SlYjJnw+0c-TOH>3?gy3;ltLjV)SJemg*?YoWxz#Qi7`-BxIs zpyRB)ZwY4*flU7P{rglfoA-I3eu4Ivcve{BxPkclBYp)5JUjYU@~Pl;VwhRMj4C~y zqvp@zTeOii?!Oae*oI(gVFukJY^YpL4%&yo4;^uGa&jrSZ>@!Ft;2wTFIl1c`n;a! z)?{$`B0@4h$LrXjxrzh5nEw^*oEREPL@!|I3wnb&lv0hmM2~Y+ot#7zZGHJG5S1VZnM-0V(18w_%sIS{ei19?xrv~m6VUF5SzVmDIJ0vQbp<7{Ya zE6^Iv77d!y&#~IXv`nVuAyV-w*Ckdc@4j1GZluQ~NMSZB?wt|eF@q2Q$O6Kyi^dOS z1}+;EnX~|Wqhe(x2WXi#!cImFEhChMCmp^qDyd?-6aDd{I#OvsenHnd*xNyb^5rD> zpA;9TYYHN3TATYsXLDQj7a%jF*hx5yeSE;lJOO#P1qCcj^CJ7l2_|uCT3G5U3p2rd zfOAmzzuW$v=)t3qF^|gTVUW_FHb0dn6BrOM@$+-3VXcR_{Qi~&yvOPLzrSlBq6!1t zkcfzY{QUf$({muOaiL6)KDS@5d2a6raMcJ_IcAVY#0P`i6O0Ah^7QhBZ+)PJp8oQA z^?Bf24z z9GIZg{hy5H=H@U84c-3zL(+QwH{f>j5080kim;b_c2@`>FT#M|HI0won7ywOE+s8( z255wkgro_wKCe(xX{~Hws0is?F%TyewoE=_w<+=CvUNKz+}&Bb1N%$>1zb;dapVf< z{-_WD4-AGJ6L-yf0OC*p1nN|2x4a_M_5l0fvKWA(V?GC7p34mc!xeVNgi^H;3Lh@J zLjzsd|0*DTsQ%F17^+Wu;QF_pgJNF{4LIqX4{Jk114LViD*q~OJwSRYTH5CJ)RYT= zVt;?{L~n|Oq}3mnckG}CyTQ-|X)MsWmiuLK+&t+$VdgRX)>Q~a6;yL4y}M+$1fHm} z$F)EHVmK;!!djsgG%SD=%tJ*t+F(qGl!F$|;Jgi2sLfkjTc_r4H?CM4$>r*X0q{Bj zbN?HjD@TngY_I&6hvdfq!%=Cc=d)+D`l zoB@PcfVebV<&>dnHB~n0cC|^x$8u{EG zO9yczEGg;m=*yQy#2MdtN{<22v|iYv%>5KVM040Z#ym(SCnw1~(xb`@jEz|&?7tF% z^5D7FtGv9jLRHOWjj1K%j!gE)>cLD(JGbmoTaN^&x$rEU5T8gfgzP0ArJk#brmb0NJ(hG{Eja0{D>+ca3-|9!ywpsM0{nHk z_eG2InPL8&&!5ZTZT#@nAEV6EtKqBd@lP&dNb_COOXcfmqEkD)|msq&ZCyDvx=JF~iDmtyrYC#tz zD5~9H9&?)}+kaU&iTK?o&2y6@4s}W$PfxE5UUNtT`nr3dL2$=K_|1RoGb~cz528Ap zv=&;AdH=CHY`&YDLm|ceESFPcLjk@{f^ORc0Houy5U}A=d-?(z zn?j|MX8VT^sNMr4AOHi1=k7`y$cu4fyRrkyMoj7{DgHMBpMqe|%%F;y02JE>8(Cu6 z$qZb?B>R)sUobT$HumA>*aHf{2DH^&=^mc}=0f=ZYeM5Ohnj~#8c=x@hCWs;UdsT` zL+-%_;#g??AX9rm-NVeMvP6i54<3oL>b#KN9zFAJZ({_RBINhhbOHhc=e}}9zI9zP zhuc=&ogoHQKL~WwwBq6`vlSta&2QfhS($FLsQ1M|egM>^-G6^)2QSL(;M!$C4X8ta z#+DAco>1)?HLw-}7{jJr%dpH`X~5=4tMs>n;Gp(YpaCFH$gPg2>|$cNJl>pY>gvs| z<6glCP?ryxG)=XH=Xa*4WokfcFSZ|*Ns_*Y4>>wq`#^HcYjae_iiB(Z5?lk?A|N!p z8#iv$z{0@vZfH*QqpoBm8MgvtfjtZz>f>*Chepfq5Pr(mY=c-j>Sdl&0_i{n+#G=C z1l$23Z~D_l&F2+lGCky-0sx#9~#iN(SG-usRY+KeYqtUE#NXMh&zx!_E ziw&c!o3O!U?&O=cwh;&yBHRH%6b^ay0<`>RvhHV$uU`+e*R(;35_BplkF`@UH;eZ5 z#e7T<54TdzN#Nc#$SNzNU0GRC(a=x=u7G&~3kn@J2ykFwyoi_``JF(HJ-bm;@yy^u>!42#pt)mD%>KgqDStl!ye89}j~FJaPIoS$lgs zqo^fmO>M0R&}Z1F`CGn4pc8_Mi$ZqX=ioT}31|e0SFTXf3z?7tRjb_!jO$k{Y&iLs z`llkO)>>WQa!UEllnv4ABPTyli_{HxR?%0R1IIw~W@sB-$9QnIwznI=+ygrq5*gW& ztCPJ9aJ`q7m<7RBPzSGbqQ-YfvMR?( zTp02?E`F6Rv;9mh=do4~IYZ>*b6|aLNP3A-Jiy4m9bg{@LqI~VPr{}5K`YAtD8Y(% z{w8vq)!z8}N&*}#Zj*rw%Z(&L7|f%%I5e?3$;CAZ`J`xV&9}O`YHDLcg4`g9iA}@9 zXcBp)AGGK|T|T9&SD9GlJOY3cfZqW%-^-UT-yVM_2Lmh-AihfP+dDd@9mKGaJ&+7& zf&qp*WzgAxs6SK;CZCMcl)nHo4oXT&U&u`7xjrSEp?kSGIp<-#LrqQX2&rLQ<3K)bOk*~aO@i0lgYL`G56i>m0rMn335J1Rb z;o(~P`avKwwhU>gtD9O{;sZ7r9v%jH@~Px=Dsen`F_acCS_y)IHXvmrfS`Hw+}zyq z>Z$^uEVPlJo;Q8@Vl?~I29+;>%_|cPc;G)EfM^0yl|Csc2{qQ;g&PNr50+{PKnza1 z#&!IkD0Ay-_B5Ex`tXi1GUn{9rN|G!@F2NwSHsA%qzgrH~~Dv?ZFdxd$d{{sBiwB zh`4y+eG27EmoAkD6TnCTX7T}kez+~EK$ve@S#g3_3%pBVm~RskzCbKq_#Eud_N3tf zkMak9ZsE>>|GIO2ijmL^DzNa}jCZ7|D{od>f3-3vwTLKVA;O@bd|g^va`UX5GG5_433|xj?2mT4l$wugP_@nXcefGu z2dTUmw11eYO+gz)LuN4kq`M`3ZfL*(+Yp-jsHlXY)ZdzzY?uU6qGDnd6i10aQn z9SCF-zkWp`$pRWX;+D`wV&SczQ=3Bf30r}rn9{VH-}wR$8-=(PCnw(0yFY?Z&yQf> z0>mCE0gFMAI=ijND7CGtiyXLeSly3r-^8{Ik^SCG%sxyLs+ZE0oVzdtK}8JUUrUPI zgJl>jGy*a>0P(5#b}t8-BQgjY`+`H$Is%mP6Cxs1Z{EC_{al)DzCHy{z%D0u90TY$ zjZg)CgjsqV1AfL=y;Y{vP?%wq*|Gr1VAh|bwKbn#M7tS}^kGOqqzNE-1(j%U={7>( zmRJw)?e70s8sKKh?*P*Tn&j534_?FgUhv0fmz8CKZVCD#9zHn>jI;hS+a^xj4h;#J zy~NC*ZsN7B2ljwdCYs4IbRevv#$;-0iowIfqot?k+h3fS34~gX-~x(xU??D62Um`H zsFjfU8Xd@6_n)c(K<>6V>WYH_?tM`~0oT^f4h0+-hj1~#H-*g_11$GX!V0cw#(lL& z5BsMCxGv}PJClbk{#unIJE*0?c3;H6d;@|E4vd~=2l5$UGQChR2Q>(Q>`;#D-?AVd z+c5~n0ZqWLe~ES@vwQgp!C6^c;5ehCr40Z@q=oMp*f+ET9z&Od<|AABl5Y>4omdqU zd?t1joFqyrvF$gj?*C#2K!}(o+A^q6pul5+vIW>bfujyZb4tSQkHAr@?sEf3TR^}O zgpAG|Y*X0U+Vl9P{X%NFb4qO8jzjM!>y5kT!NechElyYm>`cP78b%yYCY0mc47Q68bS~J zm;H@EM8GlU2cizQW=aHd835}*q#g?R#E&1r01WDzn(!`OyogRH99m{m8Xz5ECDr- zmmu>92u1(y68zL`RBjyzS%NTSP*6~n^BgrmyJD|D_gnh6fD{Hn3+ySRhTV;7$YjdB zx5o;bcfHw^gYwn5A{MJl;~$ED<^Em@paEj^)>?7q!o9-lh2s`??&NMUcf-2I3c^RG znnmMr|LQBkXU-r{IkW`s{p3c|bS_ddGCG}U%@kQCsJx9(xvRV>s1EJOr*E2?zA(HQ zv#^P7WhUHrNS$!gh|14wj4<#?HtZeH^QxtKYy>ipia%y+`H~91#3~qP@ zf~vv+-u(Uj{X2muCX(c!-e{Ly{c6w;#OHJ-vlZRD6erpB+Bi3r2O7hll^_VN}o2KYX+Ru%vcp z{@js+_*1A<#bS-38UmOz->pCRSA>JgcwtsQG(?4j>G9*o@8RfEk5|moxQ-b>FcPm! zYQrGRkk*30C}=;+V4GQ3>)!%P4lV``<%Hw9awHwr87NFg7cwuig@LR6#pM~G4kk`Qy?2Hs znako~V6vUvk^4WrE&;y_D5=E3mONJ*R4}dsg_ct}>IJ}OUf%msp2Gy8q0vhQPOb;X znu4zTg{$Gi;UMH_N|6RD`w8UNm9^Pp1$!IRo>?`1%v<)i z0h2&q!5%f$94>g$xpWK1IjKW27+&5A7yAc4x~7X}e*BPBbHHM&n+^G%C~(K|mOGi4 zq!f+?>es+&07&hS30ftKZJ-fm}d%anKDY1&9VEZx4En&1fguyz`;PI6_sCv^Cc!8b6MOATFEd98k#7D1A%$}?L=48fcS9|dKSNID?V)c znGo!)onr&> z7zon8aP0iobI^c*T89mRCqjQ$4%8@pef{;Q1DFL(!<-8&5^XThq-10?v`Eahk$2y# z1St!Idu7m9poZcoE4Rtm$Vf|1!XJ0W4ayy0|T^&hhbr10f4CWqe4&IpwQ^(#LDMK|M%?ul#4N(!l{RoB%L4snvf5S zodwAoO5rkqLyMjl7uFZY^Yr*Lv$OHsSY)k9q&0_r4i%*y+DZTS2Xekfp% zEW^fH0_>6kQ@H+0bp?+D$SCtk=bC|SwMDZPxkhAfqDBxHsN`=0(29=Dd&{iW0=gaC zL`xLx0*zo=8!)NllMfA9)Qql+#(p?gQgi1+wBOnrrNC=8_r*s=2NQ$&br!)w6 z?)Uw^aqk^>-0=@_=-KSE_gZt!c;@rWNDVbbA_6)B6beP8tR#02g~Fgf{^H}p6{-(& zweSzFrHY~)>Kyr#+4w#YgJLMehWdu&D9^w7MO+d?hH*}Ajs;aChQ>9fEE>F3hDLHjQJG7I`1q#g z9QapAS^k``b1o>cum*DcdfIII(J=9)?HkmQ!1b4#9{T8Z0S#FzEB3;oqG1DVUhTLQ z&BimGe$~mOv{$cONAQQ=dTE!O4{P00#b?dorsNDEP0tiy4Ozkty}&AWnG3Fp-L{bN z`ud~Fq0D{V@a;6NW?XNGvWyI7c6N4re0=7nuF$)$^zYW|8;`mYM@L-gCwZOoELsK7 zrhZ4(qGDom!+K`&-Jh#QicJWqqjL|hb&O^@P#-&G@(ZAY6ECZ?adQ)1x9!7j7cei= zKXy`atUs1t`)amS9cWG9cf5R=l9F;J#muku%ggJJw&&!(qzb;Byyo1vm)z?kg=)zz zE7M#jG_-ecaI4>L%GcA-K#7QmxQH^K>4fm5PPeei%gZ(N^zgjBz2S99+f)4~lZY-TggPcQTE@@}`{F`M^3ztq#y)AE_W8I4-6 zPDx1#pP4aFx-qf;KEI^8pWEA)(SIM^KC?G!);-%EF6i`AltI)5WmMzjc+iOVZN0y3 zaIpOQwZ~F_cPZN1+62s7P$8kAS(%xr`L0-pgZld8bnWo6#%UhJaq7QRS65RL6BCO$ z=9H|YOL$ZFi~m70Ffa%>jPSgMkGi_LTs%A-?=$=e!2F0MCY+n)Z{NlcWq3B|^<=TH z^Jm?YGV@kU1_>`pxY@4&%)X|6xmn9W|P*jr-FjQdq*1^o4MWP@9;Eil$?Ts zWmn8a+Uqvxx%qiAa`K?%@9*cq5u%vZ*4MZH^h;UIwuOp%Zn5y0)Q5CN)68$pwkxTr z>Hc_o%Aa#u@+(g2Z0te9@87>&+}#xn4e3M~h%GHGU4HeD@fcPrm95j5dSVzD7(~Ry zb=OS$5tix4I8Ib&m6wxaU|^J%mt(S=M1Q>BOa5@Y949$B*<|G5L`L~$cy4ZP_M70~ z;K5P6%*pF>GFP>hMSu^Cnv{zb)+a&(pQX_K@_E^*uCKwOBJ*I*M@vi6KF`Ck zQe>T|fHJG@tA5G^goN8)uJ;GD&vF$F3=Rg3ntBtphY__eX81qrj$_6`VPawmJRB1c zy>-iKW#|JJKfm6wr@XxUdq>j-ccR(ZSr-qF@P>w4ZSC!v+S(zxDk*bAAGDP2+*uh? zG&g5?{rYvw{5IwJ=K6Z<@v&EugnQ1zhR)8;Q~?W2g;+WqBO{|2dJ$q0CULtOQ&(3Z zwG7GFKa2fVTQjY+l0Krak$}O$dxu9yS0;5mZZxc}LhhJ6LyOfvn#YAbwb)c_gCpaC z8Pl5%R!(_}O&UfXFy6$&!#h1*EkOmqJI5v_YP|i{e^i;%;vQ^F28V{?z+*Fa+zx2| zN~%I!I!)GxCMQ!1ps^z5f*=}YxhSiR`ntRErhWbpZA>*5RH|uek~#gX4TRh>Q@8j= zOZ;qWR#u6F{KbnGED{ooSy@?km`N^NSl0gD->+WwwZUU^?pJT()}KG^5N+EH(|?JP zfLAEdF^Hq4rq28DVcESwQc`j~S<_vucVvVbQdif`!smn+FG7C(y1(|bj)Iq$msZFw z{?odfPR07pr(*&%T3fti#r9i5O^wLeR=Cv6z*|MwP=Q{FZrww@=|uQgU@R#rN;~a) zz?ff9@MwR{02>zD#Vo4kIDxEA|iMwG_3&Y`@6gC zBGW$Ko$B)PxKuMF8PCqnz{|TJ)+Z#S9{*Aj_u9oz-w&It(#5iv1x8XBZQK|!QsWPz-*VVRHx5IYu|Hb$ME`c#<}lrx>5 zE=tWqV6F!#kGtFUz5XyF3^A#B?_N;dlf^lH|06=!fS~;lSLdIdb>G9;aQk5gh?XYF z^V28t5;{`B&))d%8@SXN!%uBp-E8yuU*{TMt_k=1_j5iiTvmIHFPRaSz^)k7zsP+S z(X&3)NFR(x(p75FKJ)u`*x{jPRTFbpOzs9Hc9$o5bYf52Hz;du`)teW1Jm$ozG?IH zb=HtwqQu(1oSMA)u$C9ayex9h_Wx79@CPmsP{>56X{3_MxCXU_kY-c-Tj}D$E-Yh! z2W4lo;bUVCjf@0Y&_c}$>Yi`y?x&m>W>8rg5&2j7!iXBOTwms2Ew&tgv;UB*L@&|H zHB~kOx7OG|~&DBqL)D;uTXG@YFI;7cszkw1zmpzgjFb0a`)?N2z*!^-UY zzZb-mZ!()yDnx|AVQ>9>(m#8=x5~xKE3czNg=*>Slz(*dZvY0?i(>DUp({DLxo%#s z!GZev)V7@yl!o|XijC`ndU})qfOK@o1%HMP}``#&6_7Hs<8#Ey;W ze>L+I0mQY80cc?B$OnFSjB81u^y=5TSk$cQ=<0%FR(Z!*(0<(RqE}K@)~qvibmV>L z=m>C2jxAx(y0EZtW@aWpUFvV;cQHq)r`x-`a@N+lV*Fa{u6-e-yvn)vQeS>J&|UG= z4YZ);xO9nCQj!TUO!lWesHg+qzM;zW3wQbg?tP#Hht~-)oN#k}8j(k^4(^&eGNRk* z`?9`$)G4hRZ_lBIlaUdY`E zw_g)js6c5H2?02?_<-k1^6+#}KA#d)`@Njks36t&L$~Vja50&pel`Qp+-b80kR@xVw##C zEg%5!YWVz4?j2{}eAUN~ui1P_4FuDdj<2bOOFU*v$eYaSg|f{`TG<>T#m`SdNX>&{ zX>hYwbR12{xK0+`IacG${P^+Xp%p!ff9&?4+;(nxIdW)7J1L{bj_uMV9Cn46Z+7XB z`5#U5axaG02R`sLORwg3Z(9lqbAn_~;qL~+v zKoZS|m*kL>Du)IKWh^j}|0#HLUPHq9LtWi?4(Bu7P~_HuH<4&$Gj@)SIPC(v4~XrE z_45_?D%r>*&Ea4xyc{nYNxZ^JUeT1`UG15c3L7&s8+qH#AXBMfW>zx7Y^5b_ArnEp zva+(2_!@!p12QHijO#DO0o4C$P~ifh;xtVN;3KQJ%|Wi2`tMzK{l7$z zO9(o}!XqB?pvMhF*7og9uQT$*ASI4$iA3b#%d8MouU^qXy!+NzlKyigK?z$p?Jj#* z2DBA9;ZP)I*VYO*h0(b1QVDGn6CxTK8s8cV0p#Ya{QRrvi`V3)9cj^E60aqwW|W+c zkXr|35g3q`mImvig~jZCn{4oy-PkBbp7a$%WiQW^WU}rF-a}!d?K8p3v*UWE|6End zy@!-SQ&Ur}t5;>Tv=VRcy^lk3oDzw+&mk$mF(GR!dgPJP*4CH8i0I$JBM}FZa5>p` zL^Deo!)hpz)nsO8W5P-QA0LDERmG5P;VlWgzYezuaU|V&wAjZe_GLK$N!zeSl4sAJ z37J^oAqLE=tfVyN(}M>v{BtuYKH#j~$*|Te0aiI#z(bUrcaS>5D)vS58S;HtJCr5Q zr%#s~sDfqT#Be`-`Xm5w@|Z&rsc1qje}3PMp}TPh9?9{~=hYG37>1ko{`*CN zUz}kUF7sVPhsVb&@Fck>E%zky-@gO*Z29qn4)F2%NbXH@>wkS~=tWvubU;9W+A2WK z0fy2{@TMX6zsGJdOcI9hU>Ura*sPgY zU0uC?r-~3Zo;Uj`iF@YH(Xmr|CbF8apdd_wL?`4}iNIl?BCv6BDNPsA3c;i6AHg0( zZ{D;3IC<>UiF{!&7Enk?Sl9}@P0>O|No42GA3^sueQ=vC#BF389JmPx2n1}t(1P3j zdkN4VpnV2L#=Xg?*0*p*gi>dR?3`+^^EQQJFyVMj4wv6&aH$T-!FmZ~vPeBPr{txj z1DcEk36LengAu-c(@0HATWz3vi)@O(w*O6Mj&dR#*Z>^5+y$uiur2}$qHq6H+baoa zX@vaz{H6MZPBkh|T0Z^vaz;kn^!eF=y<_6NO} z(E!W4zh%k7iZ#I_a**!W^7ogQV74X1Rtk_Xo6gU?Pd7Z8zk6^igtvvn3-cM*20)aw zPfnVE0U5!7KSUqRVM95Cb!uh{u>QILZB2&X@dd#BU{5Zv=S~N7JJFBky9Rzt9_yEr>;n=$Oc{TB{O(7?=hkBW_*-NM+J?ZX~-@O5fRbhXS1^)XwQ{amE#A$f5$*U0UwcfP>B8=kn6>1unG1|6@A3!~1P&-u2jj4Z=<{~|S?aUQJe5&?zfwWi}0ZYbM_hlfiINq&)> z&Mcf+#l^&U#0)K$Gkmh)Hr^lCSvf5rnIS!&xP-*swGr~0{}T$iX5hwue}od!XaMT2trdl(Z)&E-`xVsHU8EOr>KM#bv3T&{ z!>0SOgqW=*2ZUr)R1^y-zJ`$zVVK3t(h@#X9G;Yvlvf4Ub|;Nlz~=?aonO6hnj9Lb zw1NKvz?}t{PmHv7fEHw?i&7y!V_i;Aicdxn!de()- zMDp_G%ZSLxED1^!5E1DfQ>`lejo2O@9=l8V|{e^Cd}zX-6zQ*gwjJ3}2gX{`glNG{^_a>S0k9 zw9tIB@bM8(O-&s;b^p>A5*C)3ylKABJXSBWH`MLA#}Iw#gumUcib&S0QR0IKum(e; zqw{h8r*91>Ms)ml4&uuE&nt@*pR~|DZv1UCW@Ip{sX~}vYm9DqB)e>>W6XzYX>HBw zmZ0xg9nLm=@Zd_*>5j_Uc!gq-dX?)q^H&UR+3=fTdAT^Tp`ks>ciPrpGoYl;Uf;B< zS~{IIDhS2d$@uA#cEX`8b_b$JMn-0HuORDR=xt=XM@LUz_ITkAf*@U+%_Xu01O?)7 zFkhRNVHz!dmF-&o?)j+`=OH1X_O_>zSv8u<&(E&9TA`xq#?{l)_@;*!^F9BI%k%Yg zb;E`W^sFYT9izIjF)?1Jr%&+enVL>*lmQz5w`u#exX8U;WyOeac#mCOwGC{c1yxd2 zl>=g-zgg4qn2Te}%58Zd?>q)#R=t8*8$1FL|93b0p)TkRwbM}4mA9g!4= zM~fNDb3$0s7TLB73kyBR#|{w{6@t<{UGsrhWm}v6&LW#(iMf~F&ueZ|QAn^0^yAzC!QcD*Lt*oqcj}t^=jU*;k1A^H^z;)8^^9#DJnVk*1;V?3jq-UyW(Ea^8MN(1{-NA5m^TOv8E5OncUcPb%T&l;U z?ES|LGsXT3V!I?*Qim(4Ucqny|IC z6?Gqr_`Q0@Ge8GPt7ZB;LN{&%ezhqm#OhI-+TTq~A)*%!_-ZrHe151HH*&c#xGjg~ z#*G`SmoMW1ee2@t+5sFRLJ}V5UtN16LgG$LtpCxcZm^xMDpJlQOxe>@6bi&dXymCl zUUcWZwYr{m)iZs2`+Q3o)+5*O#r_Ouk*QoOMl&-r09`S(f->973FTs__=JSOtu2>W zvke+EpFoGv;@R)&Qn{ZOh|r5}S z2DC7j=G=sl(ccaKL|U|mU3mlDSKEGysZ3_Wt!!5^4(Xc3yHIlwjvLycwW&suHwl<$EF zk|lO~V=>+e49Eb;fCJIg&W-PRIl(9((ut@^duN}(~+hA`U{9t@Sy?qfF16bs8Q zQPI(5567<+8dl|h4|VG()kxh9IHgN?{+v7{JlqOQLo4c%`DsDr&K;N8ivF#QAe>SS zGGW6HA28l9sO?<__R{n3-e{TEo;?9MJ0yd5&qgEUDe{SHzel)m`J-a_EbSibOp&7@jfSdRMD(9}1|w zKhM*RpEt)OJW+lRA8s~|FD%}D<6~&G#%IdMlr@T(_* zeQSTe>-i-Wfycjkn1=HKHGDrm!)v~H{hPT{@x0otybl;chk3)Kj5i-+H@L62jT9LP z)~5+7c5G>+iVtklB>>%PFRA-37+@Cb>+az3%-aFado$fWMf9fuL&1204rmi^VIH(oe@BV(TZeFF~V;StdFn9Sj8+CM5{-|cR zSCPkN(7>FABC}LnSQswCTtkqK)w^+nT%cWSi{H$T0OM<%x9}czJIDbOG_pZQX0kh~ zuZeN+Xx<#4eA+eX@X;WyqRUB`3PN~m&_tEFF5iRhgN|PK;nEJSFJ+-pkQA}m^%@R-t2Aq z`0>8;*|8Tq@L;32iGzH6DzvJK$44lAHRoN|Oq<-*q^s|U^(*bj&UW8REjL^mBs|th zl#pD215#&&v>lywK%{rmd=V|Fv+h3T)h+!lY66)C(MxH;V^Tn*q{^9BRR$fteob~M6MS~tdqLOEa zRt#dU`25YM*MN(AU{mSWm-mku5F&n&Y@h~k%|H5G9ETS<)^7--fWo z{bEL*lN-2b>-=>e93Ii7=5uqCIE&DyL6iqxqw1E?Q5qnGF5R`c&lYn{^~G0qwxl~m zJ0VjLErPav3=e7o695c>ec;>Is8y*>)eMV?<%#a#eLU+ z)15f--<4 z+0*3XYn?${RlQQ&LZy-Rh!uwi99Z)x$-{`|bsT-&w*raB*?9n4KT%At$3dna;xbH@P(|vI(5H zDi8~A?rof8+xg$=a|_VGJWtkZfG5r0tQO)-y0bkWCp8DmX|3;x@bA9#Xt&jo4>vf1 zcb#!*IKv*tDmj-{S9c*Q2Y@51sgF+&C#KKBq|UbE6v*Ns(2@QTfCBpR<;xx2-BzH$ zHP(d7$EcN@a)N$BNkw&=g_Nt$T2xdt0BBB7PnHAtWE&K6YCx&_k|D*6C?mk5Y=QKM z=yR#!Ps^S9bC+65n$My@jbQ=77c6rK4Om<(y@IlCJ7mLcj1-Q)d>kc zN&;3&0!?G%n1~4cPP6&qjT_%Hpfb+F&br1cY`1$aXUu?YXx{jrJr)1+<$CGpJXImG zc1X>BhqHLbO+FW4O2-MQ7dQ>l=uR$2zRvCLZp}esOw<`qBZB$!dT+u?L661Q;3ugX*Z@90G1qxbqnS-Q;w3K1!i-QFd z{)bzvh)x33{^{GqN!A(9;tW|DhEHG!QuDW8`gt3+uSj|{*Hr#1DlnV6}~F!z0cG4Mk0EBqNaUd zfDH7>t-U>*xbF4{GIlRY0Z6g69-B-~(@n7;y`AoS=C1{@?bWMSq_nj7%*@Om_eP8m za31wUv}0>YNEK)#6smhQu7;4jj`|%=_v$mN-kD3Z5?(^kjl0$z`L%+=1&}-XPUtmKg_lmr%z zj`J$^tsD*zAz*!&< zoSmIP;#J(Pm17H(W|7v?qJU0k8|YfK)gKcR&=*jvsi}d`M(XRT;$@Q2kMX(9+5dK| z<1-c6{KenD-|d&*y?fXGZw%4V9^GT$Ms*e5KWPyY6_w_q47wE%P5+$!PRgv|V~ie2 z_tWieW|FR-UzaCc}2xuxuZ->x>|e3^fdEBVR~^DrsSLa z!5K-+g$5&~HU{nepGZExP6@Q_#Y+3y1x~iLQcz+!T5OUk56tIxiQi#0TmS$bJLKZX@wusdWtgvCx(H75Se*g2R~8W;PxQ^ z^0c+Jmv$y(B9zSq(d~H^ne`3I;4D3TeY!X8hyLxmfVEFw;Whn83)(xdbUmLx2gk+| zLA&`42p&KSGGMq)%i9K@9ZwR&V|;D`vBx0kL#v~sGgkYEz3F7*7GN-TR#ps94G@s3 zt4ocVS+AMyg7pH_w0CWIsjV)xN6aV|^6nky>0XKdu(Cu;saZ+Bmsm6ANN?|jeu)DH zpbSA>x)<^03;OrcQeN$Azm@=uJL8|8urS0t(3~P#zjO0n>aSnF9l+$^ z+!)-y=ul}2+%FgrH9iGFg_ZKT6Qs!a=g+fi>Ar&!8;ncT1_z$-`t{HC)MN?o{Tbj6 z!`{EY44#S6^L!Y565Y;)NfN)_sjaJPKP1B1(V{@)A?{;Ai>s`xjLa86TSyfL8rOU@ ze^Uzx;*Ws21*Qla2ghQrBT_)`aY4k=lHJhoa8^zZHbfrk{-}kJkWgk-6%{bZjpuXt z%Zuo%s=#SN>>^Q(f-V6AXf3MSY$85~E`SrMot&IJ7t=jlj&~o3pP&ATh>DVNaS;HO zQEb@h!Gl&{pg?QnLP^6#0N$;ZWu!lCqGSpd~g^$D!P{fVd_2=nrsF0>Hzf%4KswEN4PmnBZV6PSw-^tL|qQ zsC!ykWHdB*D45X5%npx19bTKNe_iFOKz7pUfp^s5bbkUS6d{K{}at?M%R{^m3=oFp(WgPh%1-IMof zH7XHT1ZxGB9YL+bCnS{B*QW*41A`hs3P*!_8Q4C{!o!26;8fK({y@qqcO#K7u^MJW zs1nI+2TkV<;*r0?TbVn4=6#@Tf;Y2(5S`abh(tnl=+ZIV1!dL5j6rdsnBt zR(U~U>CB41@ePS33Pc+yw7>G8r` zs`YT@GxPIU%*+|?^P%n=KO=!EpMhfwxFD zM{GFL5_s&CL|O!WV%D1&8nzC7jnt*c`uR}1IZjI+E>&~@{NMGFKdX!vVQ6IFvi-+z z(2XJNfpQ`z!q!X|kq;7wMq;drnlAaDpD=?tw?~Ye;a9kzt_sD{u;B)`btt92d|A~{ zcn-=maxnk7iVx&Pr*ub(sh9F211Y$8EN3xD;`*^Lql=KtH&bBQ*6t(e01ZK+pDGj2 zliy!%gO1E+`Y{4nPRs6Rj8P|qI5;>1o~+pvyGHlfpvaP5ych`rGMpvw?O-`981{hr z-w9y9l8#?al|D4`!t*$qsF)b-pJq$YpdUg-fp;tm5>QZFX6EORij$cSRZ<7m9tZBP z1o7`h_O^_i7A0qn(aJNL>=VM5!UPoA3B_H6RgKN0Xk>8IDrZ&cA-4UE;jjd!UcXtgWR5kOw+yhFTZl z#ktj_it=(qeQVtx)!SPV2;YrDz`^KfN*CcvQUZ#>RlASz4}ezpjp`!YGux$L_xe8R z$!hY`EMgZ#4NKA&u{8GH2L3Ro@=h)N&U`4E78rWbPaFJ0?ksGvG}lx?UC8Y1Cfq*q z(bw1i=E;f$fGInU}B(na&wDC{lu{^ zU2;O9_)AVAo;6`JZDL%zew~bh!s?zk7YI|81CyRv5Kgi{j^}hxG5e6h9Mtr_aa8op zS~mwS$ab>}dnmx0QAZ}KfiIirSvoYpEJX$txG?+InJQ?dp`(L?67};jZeU4pcp%um zJ#U49f?+k^@!~0PkZjx|PgWRBLj_3BWIr|tY@tGJLFh>?f7SReMecx1TRs9AI;Mu{ z?aoDNm3@YG0Yk_3Qd8%Hy?;eSAs9n|2@`z28Uhg~uIB9I&qeomCceKRTG+F{7f1XA ze*l&883*;{Wg-b-^oUx&k$xqcm{{{xh|B)?Il9DX;R))u_9qCalTwM6V z!oo8yqGQee&6agLJCC6i=RQkMNx|bwA>>PmK0Q5Eb4wbwDASMPKcXc1wDRyr^xNP;lpjDOIcZYA?a*u8xH+R?vp|uKu^WR#rln2 zZVlZ}9zQ0frzeEnqoN927M%NmJp?2#eKpAvDOR68-K$@(b^|xF^gUYD(OLWtRN4kb zFn>=X)5wH08w((==SJP43&^f7l9OdP$YERq+~x}3du9RifI&C)e<^9TxIdoaW8>pj z&Uj&XC3FDjCdd^R!YrVy=9Skxa}&F#W4pgL9uXV6zT@VBD27bnsn{9)c<_^X_D1a= z++*6%O&iuq@BA_F#S6-$RlqZC|3aEBOt+ju;^@wx@Nm45LW5vE)2H&*i~s;3FS}3p z>mkaeEMxywAdYf!IG|hfct7oFwV#BwooJM4T8AU!Oile^^Opd;fyihlV2 zi)9)b)gS$DaKgZVmWGpqE*xryhCz6zW!8re%8%HeTDzg7Sx8&j+hrSQy7OR~;zI)$ zGRLeQ>VG=VtO}Iw+;qIj-*<=i_C?zTzJQu=5tiKb{d;2X=g(HOVHRqyMO#0fY&2Y5 z1*J{^=25?BsY`SME3azJ7-o?QV;(jzJ!IHuN+?53mq`Ps6F4g2|ODR{%7HjJQ2dOf2&~ zafi9&qLptq;Vyj`Y5~9lf@@n}`YpG6;s4{P+CM|t1zA~vnjXaZGWeRIoQm0A$dFeK h6XWmyd#vryUpvq$6>ka?4ZS8xSzb-9SjIf)e*tEI*xvvE literal 12040 zcmZ{~1yogG^Dcf6q)S>rx4kZN)T3SK6yCo#< zY`<^)@A}{0y|`Q)_Bm(odFP#XW}fGn@VgpH1h~|=C=`l7MOpD43WY|B{Kv+GE99SM zE8rKVxtfw9>H_)qzBV@=g=k>!`kM3$#p4$;UzZ5$NREOfm{&< z_hW9Yj=bE}-xwd!t@aqVseO=Fc2e~&ufW}Qcx@!}S;25#0w+a+twG-OC5|rgNOH2* zJ~+fzcN(PH~hg< zr^tSIFwt$Xt?hP#u>8SCy5TjigLy(ydF&PZU>rt8Rt|M+Mou|1d2B5lTvGWk)w@Yg zk`C5JEzV93^tB(PS?;%e5jIP%D%1?>$WnBU5XP(O|D=KxRmbH)gb==T9t33ADIB(qO|NT3% zv{dNm=;&iLt9(p)dbI1>$V^i(o@pc_3kwD#BO{f_(}yW5BLxPz2aE&qTOIN2>ci}% z`#Fb`UPSTinwTgSR#vmoLL-zvx4}0C85#PkSFa{Xxbd7H_xYKF?PAZ@sZ`a}N^FKX z6_u3AKb2W_Q9S+i7DlyjRC)!%!1 zT90<-kf-Gv*HU|Ucx)dW+*R22D#QNqWuE?9(We&m#Nyw7?j)H($c!J zIa!OxlZaSd=@&%PGcy65ot=aR zEZkk2`OnoZ3YKC=aw{u6LRT_Be6T$KJys-v^4@8s{8?^KA0*Fc)$=y^+4wiJ@o&$b z`C3KtSNev)L!Svwc$qCFeEnT& zYKix;6W7CHl>W<(@b{iQcCJq zrSsBFpSlC5x!KvI++6mq&Q9~g%?S&Own($beQB1F{LlZa>wR;Wk^eKG##>TRA*M3L zZ8FyuWnsvZnDF{_G3q59_=)F~p*sMYS=vFwxn%atPOA%MKr&egE2 zKl@{lGWp@dhjFgz&tn^ZpG_?Z_DMGPrb&&P-Siku=Win+CH>VBPO5kRKKTt!PV>gb z#`Si3zx8dw_yfVG7inc_g=brPi#;j7-U{2YM9Z8kGyhr1O;_DW>0A0LbAI%K;M&dC zXMe`+Bl&wyz1H&5bhj31Z@OSxTU+x9o!SbUTUeBsx8kB?m6fsAH#X2v;^N}icz7>9 zfBrnA-`d)GG<9)il9ipE@$DN0Q!I8sKmf|WxR_5&LINEnub>c{nmVzvSCE*JlJatU z+d)iR+-$lbz7P=7yS`z=VK{q~gES9!%kmGNy(bQ zC4&@kXSTq=!0D|mB27(A2qx3E$g4;y3@2sZ{`EM0<|S`k!>EZL0Z-!R?Y}?jw%*9B zQX0FX)nA-ToIs7SGBP-EbaS&GjIMznKKn_@{jDkbv%iP!H6DAA@cOOZA|HQHXt+|m z-dnm=?voCcA&KAOcIe8Zf_kCBTur4%k673=)9@ieDrMwWdc8z!hB%)7EJlOFi#Vp( zdXNn^Dzhh7i#vUhii*ng{PY;fr==&_DpDioCPDbZ!opNy4(~fD^0^Q6FQyIR54`OU zl@30pR&}Ku9X{kqhum3>Q8(>1Gxxwl2~3RuqlW*lZ1>dh&bUrA7ju* zjDIC{`Gc8!l^g>xgPa-4Ep+2P>hx^GBR>#!o27MAETn&{rmTo zm6cKHefE3aw{BhL-R8-hA>+gb- z1!Y1Dp3xn7OMm(DWoW{#C=*OTg9?R(Iyz}@pqX=;C=VT_;L0v3b?hoEmR2RKvi)mvJfE&J)gi(a)W+wHhFU zprJ(VxYTHE3wMRF1b*b%&NlVjBPEHno z`En_=xiMQU5td0%rGMl4b-5=`IM&zKiRtKG#l^)Dc8NJJ;`O{Di-7n4#Sr13dB2^~zAJ%#bOG-+{BqxXWe^64xY_;0}ABg#$igk8% zmDo+(A{6`89zGcmR_W$UenEg$XJPEWXrDc4#hyo z;^X7f@vGv?qN|frQ?IrXVYk&?VOJnqlm(Y_R3>^hJk9urvZC|dg+Ekt-1|!=X zr_2)PaY1X%#m%i{Y&^MG2`L)`1H<%qcfn|KvuSLMIy*ZX)q=*torK!a4OeDqf`YQs zE3xHr(_T;SDkcFn%i&FB1vnRRWCZd4{wOjE3RSxHwl?$Lcj97D_EWsBbdk`~MkXau zZ6ABLw6rjAaC~eQ+NwQFdvy4cpG`bbeWV65+xFpMqSv)7WIt8pGm3k-ZK}VRHeO<3 zV^gK$HLjsrSXh8Iv1)!W1yX$ZF-z{jA#_p-y1EokpFWL=k7uq-_FB{QfwVQfwuYyz zt^IVOiu~{4)|Sg;ON(O7{(zclEs;xi5HV}GVkTj^%!ay-4%uQ~IpRf1zcHz0@Khq*Mk(@O>-x1HbrH~xotJ6=k zu}GZBa;rJ(-{A$@W127^|8U%4Aa?PLds_%;2wBm8uSR`SZ-EA13kpHJ1_3QM9^@xg z8ylNUZc@l!nPeGv!xk5eY^v7SKhZsX-QubJaih6)LHp z%vY)w58}uuEYv9UYx|cQScis(Rh~P$Ttn{qo;$uCxhKY$!12GwOz*fO9c|V44{XH1 zfTnp~5pbkgf4u21L9D%WP4M3>CWZhe!258JKUb4S zK{`|x-Eo(MKX16CysV3yyewb!siwwou|VAQ3fM5AXf{BB{MIoxCNwZGa4pZ^hx@JM z?Cv_@(@8#@c=iMzFizE{j~}rWLh!ZRXsBN_1;yBL^6=#FSqHcIX9HYHp_B5&aCUa? zn2OFh7#%-&g8wj2>7Xj3vhpe)m(!Wb`a%i;It_@Fw z+(x!tuBWGmgOf9p&pH&zbmZjZWU*h24C^V!o;Iuw zDoKonkgzZ#Gc!5>n^n78b70g!`1<-&=%AZ_|271pB*Cbgj>o%}>hHv~Acig@f5*nf z1vWIuLWq$gHfGe+&_e2h73;X&&<_BoT`e@Je}zNJCTo&@u0AifMZzd>3K-J;&#%W| z$Oz*02ms}V^}g4%bJVrqTQC0Oxro=V$s%dRNgxP}coN(E$v^O$exe(WXA%}Bmy?ro z;*LkIP0-4ZT?toIR#k0s?vu*h)GfsY-)tCGKPPzg>J^p1qcGSbPKcLM=bhPG>gm!( zVEHw$T>lC-#?jf~)OGbLFf&Hi?so+mB_-v*^HK1hPlQKDD=8`lLBOiQ4J7YBco6#e zK@k%Riz0Y~74DIC(&u-ME+ZpDU7eKz%lBCQUFi&{1YD~B64s7__#vW~&cgfq8V)VC zf4blK)vs1Pfq+iHDt$6G#U)`Oq3J{u-$wZ2@I-_z?8{B4C^(y&o5f$hYJ)eTnu3;V zw(2kFVq;?=*4JQPQs1PeMnOKfxt(_j?vZ;<%9CtwsV~>j_`d(7T;^9G7LaX87JQsn zogfXXTh3=zdS$xUoeVv%f0e-s>3*7|N7MW@^fxhefqhGJ3yCIeKvcZ0lRxTC7DT<@ zGY~_fda%P&DVpw zgawy0Q?L%O=F#!-O!DJ22nV0Ij4B$2EYYE%p=kh=icg>B!vnrP=?Yp7LO6oKoI5W7 zm;U$KzK7rWF*|@9$JrJf*Ybhf{x>))d*G^RMhcLIy5k*?IPUK%9JAZumChKq}P*T4X;jS7>H78`gBXuD1< zVrNtA>jJ8)B~DLIhbAI4-t+Ze`hRt@m^W_R5O)3Ofs2RdczWdM{dc1rj@e_NnZ)4e zJ1_NS)YZ{_J3iii72W>h9WAN>a7>nx;`{E_%Q!eVVUdwp++-*?3rF+un%k?Niw45> zYQtV7KdI>MRv#K2ZRX)65|p{1?bXxcBgQpjAZrr8frni9=HW5Dcl^Dx^PXEZ^-ONM zZ(v=W^!bwCxza?zG8rx@f!oEo56a)(kQP6viLi(UJ9sR6uy$=Bf19WG)x4w!q^qenU2!$jBpcs}JJ0KX%R8|>{@efOQ1h3k5N1BxsI5(4 za0$Qr!2ySxV}OEP;ZR3sC=nLwp}zj}>c>$JO;}f<#LPf~7IRsSF??K=_eJJEF)0*# zH|*`&=+543tGmkz32M4+{TK1o=cn6u9z6-8`T7_g6+%G6_Tk-KTf5F{2dHPF0OiXP5?&GW*WNg$qab2y+mR;Lh*=n(`W6=&X6wIuigx!1a;GJG2$>p@P zCjuh3<4gE)#b}WWmW73UqN1U>xw%7cp*;#sOiY|FNlQzc|17N6m-zZMgOCu}^XJc@ zNmf!(Q3<nuNzO?i^2mEADcP(e#lC5H_`X0?LJ2=pC{g<} zI-+B5X`7U{ExCh4RQ^{qmGrhoi_*I-4QoBgH8eCPHg`^s_bPx>0zgp=)Qx(Yq+Z4` z7Cse9P*4y96VrUDH`T$xAx3YoqXP$e^7zj@iCwLL@k;zId;tKP%p5S~-7?F>jO5SC z&7Ip{86>{1YyLKk?p|kSgU25|E}xmp&=k5PwJV#@8|&(BdwO=B0>m*}oV`?0?N%vC z7mYD1RnrQqx=2pxK{$DS3Jekpb$k5Qc7;K&HY@bIwF zNI^kCtIYWh5in#HM}z1;6l*!uJ!7E*U+7A71O|@k>Q%GJy}l2-EjWf6Y6?R%_NLuQ z{1N2yJM(Y52L?zWLMP}fyZ_U|BgBD}OW0e(qLZwpgo z^4{-P!X_XQN>WraNFkq8f-kEC`bK9f^7LVVmo67rWCX`CrTWqzf z)4`$fQj)&jOl|$ewUB$#y9qSWST2q=rj3D@mDSZ-fbv6o-;>M9J+zl_Ctxh2x-~6;BXr!EZPiHZ_nmz6m6^OCIW{&x*mguYbqPw`zw&i3 zYSIk!+MuuU)(1YVjED%_caWV#?S3O1qx8k8bBA6akRrrT?VNVZBd-Zy@YbB<*-&t{ zUmpNEP0V9A%jV5!L?(M;hk|bUXpu=fFyKh=J~S}+>|LqoLB__p4V2+KX&-9f?83sr z8X&qQPM01=}KM~gI1eK&-u{C#GBOr@|0YF^jlL@$z^Bh_Y)mYm>9kRumsyiiXe9Z|Nx4QF&=Hy9 z-~D4+h-rO|rBCYcT_YM>HBydi7Gq~PNt#ba3E4;a$j(^Iwgiq;TONs0S- zJC>)^8IGc>YdUs(S0b-Pa|prDQftzQO!=<6Nu`5Muf8YTctA`=6-GcK;^Gyw2O%%y zROvv3Bd4I5ZhBuEox_nooeg0%o^MS?U6Xj;(A=yru|a%UD@k(roHiC{`RlIP+$h6J zr^^}4w;LKdabpCiN>#0Q97)@#{%nkg;!*NI7^17zZz)Q-_~QS}%!GcBWg6ZBtkafr zCxxZ?A?LVB@4*^&`In}-T)Ew&9gFPMveCLUKwIkJM%P?VC8?>Y5neDOFAvwmd*?DH zJ{5zr^qnK0*br!G2Y>OKj;Ay40@^PXS5hQD6Q-ar4@;--?(S|;!2GyvVfN`jkxBMt zo4*f04&6dS93UONPgZhMUv*?|P9^sKpei=MS(Y-G>a6(M?B4y$7Ct`jGv!aOI9;4e zRnuhNqEBZ7%?SD?_qiz1F+)+s(nO6Q6}P@CS|bWrSlg$kruMdYZ3~wY5@so>NU>h# zHQ&N}m4Y2C4@w3Na9Rkl4YJJe1c_Fx^fk85H0De$19tZL;bF7xuYK}Q?0L_Z-}F&l z^ixuz-?}B+wZo~cy|dPduf#xt=IYwukCvy0M&8xe*FIKeJ2O4)zg72FBRge6@$BSL zQYoF1Vlvn*{Cxf4?WXJean zh;=N8Rv#XwreuTkg6Lg{RseOK`c_@^s(|k!UZlw~q}81{CXVUvyoq|N#7K&b@w47f=G8Tc5a8j=fs~c)^;I+xktZSoBv`jCKDc*f*BE+tXXoQrxDjpt(e1(Ic)uVm0%g=4n4~J< zZJ~Z+eNr}~)PHweB((HeD=6Ov5oc~tQ>GPqkNee?_D}JDjz!cv(Q;lN5#a1er>d<9i*aS$GO{3io0qOatBTh zNc|21-@T)y6?eh_(d1!;10yJ-?9N%@1Bcd6lRo~Xy>fqTs$K?BHKS$Dsnp(ZHGX?G zK6Ij@?ItWL%FN1oIh2V0HYko1-1h zo2rXiO827!r8Nk8$|?h9YUKqu!1sfpr6a_e22%X4mKGXM;!6;=IsyiJH-!fiE32x^ zLD8#lTEP9YH5JWUe@X&a;+DI+7;uZjBb3qY3QkS}K;}A}ow%w)tn=0#F+qFaxhY1p zLaIZ}b}0Y_jTn+}cT;w+m{3N)YokT^r_veTp7lcrUwGr;DtgPw!i!)x3o@chUqE(rmq@hrR zCCDrQaIvN!EQ8{*+`kZCr;u$H2gM(F{dwU9Gq2X?*gK=k=6X?j->thFLr3VF61F1X z3goG?ltu8KrZwUwppq-Pqv}6T38^|FP78VwwKK?8=!^XjBgcQPo!*-u2O~EBv zwK;3S)V7YnWKrlaWfL8hI>f^M;K756s;X}ug#ZpgSvHMCP~x3Cciea9!{-lt&rTYL zhREP?z~u)lFV1K{VE>i7I-qP|i$6={_#@#|o+uaizCn*3; z>e&_zgtTJB09PIOo#gB+3Wuyi4rvHwYb>2+nl7uq6~a-t?_jI;j8-sy^&kqtXGyBj zrDg97MaIoV@Uz$Dt~L>W<+l2VtT3h1qmfyO)(CPjr+I8AXXpQj6^8e}WtV>Z$PxUl z3cLRnCjlv8C^|I%6@Q#tI~#AzSg+$)NLf&=U43EE)QDWLna2e$LnO_XjT$%C`TQ+* zS-FReaS65-9V!9J0>84d5=JBbhkg;}7O(H8!n}5XoFzosC@X6P{`fBNpD2$1Eo)r8 z8Xgx%0@0m9LqbBLbF6!1U}>3O{9>WAI+Lyv8jssZj!g z1s9O$Fl+I!%!YiV$b<-pW@vL60NEj<4~B;CB<&M)#wvcP9?b7SEN7OKTzN^seNRP7 z_Uq^aU8#h0G+Rr{V9>RG*ZI&u06f#kr3dQu_wV07icDlaVZGSj-;aTo&A$GO0{AWt zF0K}^(Gi4S!wLs9xWWfI)ba}JA9PH7Xfm(D@ht(^Q#r*2kmi?u|KMOVq>M`_F~4)k zLZfO-$X#3p-|zu5qEPJY>_DU4Svd@esmoU??BN#>2m}olxPG#<#P+$ywl+-vqv)T2 z-ayv}749YzGc$CFQLJijrn`9SgX-&LLh!E!!cc^HM;t3KbaA#<0eUpU^vFw?kf=Fc zOo^L#2W@x9f$X<$-#BjEi1#)zGAh`vBF}eu||MfQVFF^ zpox~i{0>Ui+?>hSb0yo@mmRyGhg$T@9cV{Vy!tv%ia(w06-4tC>6hWUy1KfzlI z>itKSmW{2gmr2<*FrwSz^#%b7Lx`^5XKdG6bT4Q z6iEV>xH;;{fxvAr1L+EqTI=DIA1}@u5S$+B>AC7@C;MXelSAVr)d&m&2^VCb@4daZ zfm%Zv8t6G1+S{>Ui8+6(i8C=$vExS~qoQW!=Fnkm0FE`7#Rw&t-e>GSM3sr3o@O}2Hpbc$@1{++qW?=w=p!NgN#{OSy{nQQE+s0v~F~Srm}ti3ZHd> zNxd&{&QLbljXf5~f7f`B0^&kuhd^J)f~NsxeFf%jMk)f*ar!|hy0^TxW&tH&duONd zb5UJ{gxe;v*}%Yuz#syYG&c1l6wEkuH2e%&zKI*p1LIp}fb$W$hJweCqtbX8z9w&E zbgeC#J{olXrmn7+fx*i{xTKfM>Wn|KK72(h9yn3$);vB=1AcD-G`75`qqy|LzJ^2} zOARJ;VD8A_WZ!yytPGEq5DStPqQ%*=NXVC_PJ+*pYAhN^j&*c(XQ9PWzH{dmXd`g! zr|0Hg09My7-j{pa`wk7&{Z3rFC-q}uCzeuNV&Tx|{QMSxQjQyA!oVnF(b3Vt3Io#9 z=>cQB|8xE^ZTE=-3;$TpiW$T;AeInV>i)q&Y+BkYptfg!WGKW9p0M9}n>At(NRo+5 z##XDGFp762)0#tlD5#K<$%h~~95d$7lkxs!8(15C|t8=L(?x!MS)0c-CFDLyA#O3?@E--vDa3HT}0VF7tQ>l*N zFywP&WNbbtBVI)zw6o_2{BT8!5;tBtf8ksnj;5@Khd5C60e}o?CEbaUc??-u6!5R6 zHa2;ix+?UtW|_>OO|mo&IMv?ma-<+4BC4paw&;9)T@~)>LK7TLXj5s-Atpuz2Siy@ zvmN1e)_?L>8HXWnzOt<}{@&Gv2Q?L*ydr{2$|MI4umr8<)3PUHCI>F*0INq4W#(Tz z#V2CF3gz871^_nn#Xi`iQ1Q zLqtDSu}+cF>cady0}yX8*`d{OApf?e4$3Um#$pT|;35=g*;U5Mp+7X^_Ql|Z=-s3k z<6EP62Sz}gKdXW(!LT01E32#j0i$>P&&Gt{trs8@L@{ri%O~=hw0+Xf z1=9&Yx)RdwI*>OuHioK@u@)>u6^zqQ?NS?K1&p|Ucp>G0B;0zf!>g8f!sO#)gcHv` zI}6_YUHPK4w3P6WGSi507$MK!0LjVnaBRxtgJCol#uF(5AI(r3$g}GEzyaqxQ z4RXcJvS37dykTU-J3l)ch%}W8zs&yrp=2OCcg<(L(REc^Tw!M>^zrnnen@^k7czM1 z_{=l04kQLe?J`5&Qs5vEn2Uyn_Ax&n1J%-c{0%3LLOug+cGepW1r@k_XqH z1w%GNu89cb*7}*8>OXt-tY|;yIzA0GHNzHFF>ptmMq(f*kLhamYH1|shj1?5SCUcy z)~oV%#XuYXb&#{PU28#k$pj{VhiBn?k88}$p6!_h6iQ94%kD7$`r2n@DMT_vgU$jA z)t%;p9ukrXEIoAupVuD(R0B|BStG=_!7%b^7k%(i6=*Og`zWw46ZPL-tLlyULZ{&N zK9_%qBWYw*O2Z79>4NGGyEx=oNSUwo=OAjfcX#6?UqwfUmY0h{DsGr@a99~&Ma>Gj z8`S;Z=0cvB2wuPbNug)N)pFRy*%^j20u8qaKzPID6A+LG@X_7VGc$WH#IlQ|*!eNn z^UcWC>Mgd0)@*#F!@|Rh_juI-(;inyNSHiopns{UhnQQ3gvKQz z@&;sw2+K#0_yS~qtkk@~BC>+fhkGW~UWYu;Z`ux;{FTv$zP#fUBx+OOu2Xro04zM+ zU+$072ZOsOC)3V0?ENgF>7-UmtsoJDjJNAErTgc&s3m zufrTeYDhRzy^5)6Xp&1xO1dq}h9rGXO6SF6m0MuQ?H7!j{sa>x3V!|g(Ex(e&7p*E zb)_i^%Di7oOBG>sHf1R;KK=?!q(X*b789!&JO*xBOJ5&X);qts?e>W{Z-Fh)lzVYY zFz94vZyy~+BRUItCUu1C#*OCpaxdofc|oM~)Pn>GsuOE8@P-Hn2zm2vp+O`j78Wxb z8zulL;2UB7Z2w{?8ZA4(eYb4&f@FstKi-hn_7?(i-BHK zrbA!eK7e67Abr^;U2zBrt=QPuxL8<%$SEn+R$X+z!i2tSnSV`^tt=+N_@ zX^@;j%QBFQvl6fVcNC=P#tn|w8OrI%e98ZuJ_9}W{}||oLAC!(P*3uk!pJv;6x))u s^OzVL>%aN%!_|H}!vEhOc=!d@S;KzC3$9*47;Q(X+|p1ils66hU#%JnumAu6 diff --git a/docs/images/chapters/arcapproximation/da76341b841df1af8a39f797e85dfe3c.png b/docs/images/chapters/arcapproximation/da76341b841df1af8a39f797e85dfe3c.png index dd54a184dfcdd6de691bb9f40f8924746ca8c764..be4fbbc89d3276d606cea611df69a539de4b9050 100644 GIT binary patch literal 16311 zcma*O1yCMO@HPkp2<{TxC1`MW5AG6N0>RxSA-GF$m*B1m8XSTLcMa}tft&Yt|GT>H z>b|P0)K;i{cV}n1d%AnNpC?>JNd^^(5D5wj3RO;4QVj|U8XIyUz=J!urJFy%fAFUA zGLld)kbk-D#feZ*wVj7;A$5~#!8Z!$6ows8h=4%LuaY0$+urLT=NO&3;6V`bu zb~d;lyvp;#8au3$A}VKR%Z@)7Em<=+MqD#9sEEQj$wVL9gbH%a^AZ;BDS$H_9D1*z9|JhY1l)HHX-=;`O= zgM)+5TC3VOXrzLJxnf^s6%}c?xszi>o>T3Xnp8i0kT*A{j>tc|%z4Q)dcH_$d%74d znX%6H+(fMEe15=lZ?awJmS=63R8zw`oUPnq4Y)@p3wUDt`}Z#n4o;2ZhWPVtz;o!@ z^G&mGtjO7~S9S-!N3{c<_kUBM?>5i_2D^ii?-$1X`)`hyem)$O-;X-@_q3jN9A2B8 zuC{0LxrA(Qo1b(%+LL;&Ltl9deQ2_!U}33>j&*O%qRaNo@3TBAH>}he&KB}1si@fA z9!ea^7RpF5^hW&JwJzkaP-o%$^0>B}Zd0`pBXA;GK67z3rR*{;bX#LFjA!3|t@Ro2 z`gox}R4$RW(Wd@q!14Xelb_Mt+(W%KPB559Ml=x*yW4?=q2IM;w%1M?-Yx0gRsDo@ zqup=L^99QsHnV;dPC`_Uk0a~rU0)Tl_#6A&+^({&LZ4?>r@h-(tKTgh99ZY)=ffPx z8ZA8T6~z7ks|=5b*n~`}AXZqsc*^PS%;foDu9Mwrg1pILZ7haFfY#qG`du;WL$kfbT$KT_u35DDVwjG5NF5tnQ>bcC&r%{+$NIYLy>Pd~|v7T|uwZI&8$p z#r=Ev@EI>>rbsb^{_WchU<8l_9xeZ@A8ij<|MGfpc=+%PB4%o8YR_TLfel9tA|iPW z4IC#YC%X<-{;Zfro4Jb-j`4SvW5f{Mj-=TxHAN4Q`k=@qcwP+A zEi~B?L8%uhocqh6kqM8?Xd5Y7TbC&lv7|({p0q@M`-b!UeA}t6uKwzm6uyDv`}aua z=jZzV_kXu{cOL}}Ra7R5#tLL&*&S8|6A}~Ul$1WMws{frIS2Wiwx`^#`ozwZYd4>p zbq68#_V&ib$8Y@e=?%lwVM=!QvyHCyjwKTryV{>35c1}vO$;wpFILmnU*2r%i^Q+D znPU-td2-p@+J8pEtOq3TFS|L3ffezykf931y6r~U)D+WK7% z0gp>82IZ_|GuB*5Nl96=0v|_G@hp5t-=-uev-<)V^~S}!ZFg6tkCRnNV=&_(8@Ng)Crd;QkGbc*S{ zkEfl=pFX`M^Si>trd5DiRC+U97d85^$ClgUKq7$dsL%ZI<{0<`qNMk5pQQ^-ZjKh( zF2@9ze>3aV8>SWZd`uKwd_enrKIkMD3YbML?ahs`XL z{m;L6EUicEIRlnsnG5~RU~APFcfrsqWex(xq=^IwBO)R~d@V$X5}|0d9;b$#)iWuv zYyVC>78-4|kxec%Pm{C__q9oeQ&{PMD2W`};!pifeZD0pvwL4zojxjF{{!CeY}gQN zlKbo1&X*ONYKREt3_M8o_xB5ni#dRIcs_l3u2vWEooP4j3UnVH&*V$2s9=%?GuP5e zzLhBq$2ang|M7!YO-=0$rxgbH*zKYJl_hn6I>yxio4&)6%_(9?_11=CJBVO&@$EII zX$Y`^9VVi^X;*m1eHbW(^Dtj>5hpqRzd{)X0!$zVh@N5rs(Q8#EvhO=A5nbn$CK;p(pIaW z>4pmx9{%%qQR%(4XLb9w;>PdKgV2me*U>_;f6nA9>5*iSa5cX(6PAeAcV=)+6urFq zA;gmIOz$PgQq9~vASPw-)eg?10F~#YpI>$PM@Ed#a z8*(VM9OvJyI2y#*>LX&qm!(eh_h!Kn`C-I0$651h7mPuX?Rcol=efBm>gp1bK|>kD zx~j|c^uoj30iF9lmX2+GE7#9?OB?mxWkmf*>W7ai&STr^nxaQQHC>lhC5+AcyxFyK z$5^!v9UDkJG(GcyZ4Xs8I$AB+!eJ$$5XPZCiSStqN7UNa2%cInqWoMi?csqx3Z5#V z;$jtFE+HPO^t_Xyu-`=!m6Bp8noBm?;%u==uQFAkh$R3|+J&TXhWq{dKXP(>8}$lu za{Lo;PfR*_3k zeYDDDKB<+Up(3UK5H@oKW6A5_dYzoX|aE`eGUZFH111U`{wj^AN*Hw+1V#k z{UNAC^R;HL1O6Qv`K+wXR_Oms06Q(Yyt;#bz3El@r^bv*y@*2q!Ncs92 z-IP5omDh=~SUHClLHrC{ic=r1w&#@k3CL6|hLfSssssL&_cakt#92Lo z&>|@N98V*k+`zm0J;l%z#Cjbjyt?}O1+Z3$5S0CkYn}~x^uo;pvj`a{9&S;% z*iCEVIc`o#K7*W z8wR7Jhqjz}vWwSMmr+p_?X5HGcnF*>;c30dq~)gPmfjIWR}vOKDNbFTA1+|#UHrwp z5g*U~-rxT`eo@lQ@7FhKU{MdAL4o7hr1mczdz}@fBrdWq7-7SAXQ&K-7g2e7@Vl!NtALDxSj^4&_U7 zyFcchC^Fg#VMens*Ik(yOmcYdado&r!D1uY+NGo`R(ariB169Sc1Ko-qmwHkupSo& zS4-XClb9+s7B+VK##N;cwIXC_+xeQj(etP%6iQ0Uw(9b-GQF_LFRxzf)EGlYiDRSaC~fUe1?_)M;`3)Lhk0HSHZ8kjP%QEl$I=n)_T|UnDxg zSi;5dM}{s=nE#7N=M@U6JWTW2^MKMgBNG!f6;(kW8whY`U9WXp-R)X;HMOCWK>jPM+a&7%;L$|XVTcv=2h>=C5O)TQNAy}2~(u6Za&3`V$chcq7#br zQFC!6wY3SJ78x`zJ0qJ7#9X5(E}H!*l(SuJBall-<@|&-QM2v{j0!}#i4SA;%|3la z_k4=i(AzsZH&z#oLJ%J-)O+k?rt&!VZD%xN09@_St)B|6K#V54E86`-jgUZ{M~?)~ z2$#ZXkhBx}?&)&8J;NuOOe7l|Qr?*lViyye3P#4d)~Lb(}c3IR&R+8S;j<-o3%}P z&J?GmX``NQr@xSZ$H~?3S{3<@0ubBgIlcdQTH*Kawe~A5?m2_5!62bz2a0FO0vP~d zw$BNln!37=et?bSIQTdPDx2Cf&v&r*yh6$RMnTrHX453~-!JyfuQltrBwpb{0v)>3 z&lRibS9Aq>K9KXEci7Tj>B<|DbN)DR7R0#1XkH4|yHfa1T76rXJ;eXreUma3Elt%c z3672vuYg8ma}T-;HXNmn9eEx3{dO&^qGzm8t^Ptn_S&&-r54EQ$gf*7E59@q&ssP7 z+u`86ANmv|WWuU(G&3`k_s1$O?$KauR&*&5n^M~BPal;$<(@S89s_p2!a7uCa@T>MnipnCiv~gD5#hLu}(%1QQpE3q1d-E)^o;S8n*P+#xy?OfX-J zK+uS%!K{HlX1cnhVWJuHH6Nk~k|MVX8T}|lvmbaYEjzJ)SwU+O&OHeHk+F+bBx$7R zL|C6*q?QFr6A^`23Tcdd!jodG)H#9)CW_~QoffBcEhEeoc}_S6-WRBY1E#$Owk~%n zY6RJNL!WL_w{Jc?Wq1hP@%%OUk}!I~h)8c48Z93JZtdtdNQe)pc#&}JhN16gH~VpIJpz>`<`jn$2oqVL6Ir?# zncQNW53**&bzmtJ6&1N?6A2J;5!etrQaNQ1B6CGlbv0w;PJPzazR1dk%ee&AAc$uj zEWW3VasrbzN6=;aq~z1Bwp#+j=EaDB)!3*hDMJmr6djjm6ej1gh>cg?$TDAr2>*tw zaRx=1G`3ZEK065hpOQlQl)ldN32Xy=>|QOxLsVHA1H2;f$lN_{mpcu$%91Vjk$bCA zumm67ccxOO1yOl+YfTt;ZLaGvBrsW5Z5?s8-I5x7{u>5H#;|vN#?3}h;;nvH!9tCO z)iTwmzB6Vo!*T?#&DU>uYx&Asmdy}nG7Gv5t>)En;L01bLQ(`m}wG?5skCBV-XJFo#{voDd3KUh<(^Dw)WLge{rlB4EM8Oo1 z$nV1kM2yx9CxzKfxtCgZVQOlY@D*4Nn{i(uAA_7kkUs|IgP5nF(m|G{C-FUJkSwwZ zX2bWkx~Sf}JRrS74AVbIKWhiDY^F_P-g(v!EEqKA4^W<|>FRb(*QUS&DT;^5y)H3l z#(XfRyc+r;gjQWMpUKtbu83jIz`*b(Jss1QM;wWQ*8fj-&9oLyJ~v^!#-}YXOcZJpBSk1iq9~P|Bnp{a<-pg1vbrqFwJV*rZZ-QZX_k{W zsAWM`)0mjDb;|=|$oyNrP852YIMM|=>vMz@wC!K0O-T8WhWFKvk$&??WgtfewvI(y zeuE?W8)y7khswGhBUxqR5cwWO3FDlm11MNtHX+KZm)=^LWI%VzsMnUrq_lREzOs)+ zlbnW7HUC;%ClKVAY!Yg{zTvDvsaKQ!>M8ABR5W4+%I{job=9cLLAPms&v$M|kR z?;*u|l;8)L<^eeqG0ZfPbPQW<@r0ToXrhCgs2&+rj|b|i2PpH&6bskM@OAese#^dN zmY)3QMMfru0>l;yro3b`uIZzsZ}{s?hbpBS%pF9E$8_I11N=-_qg;N4WVeeY7OVI@ zA&`pnwNioOPSK!jSQM&9MGjDkX@AP-i%_~_p+D@PUhQc{c%4VRoJKN)4L(Wp7 zbN)pc!si!JF@wd&ow&F);J9Q5-N;<3VMSRmff3ww+^=cUrEA-l$8uU%gW&7(e0YhX zOhyK*1@CGRHKi0w+u{1;H)`Xetj69YPZ{n2J#YeA%61+&!d4@8%V~%;h z2raG<81~zW+Id9BT8UV~sr9kp1h0G3i7M$^ZdHS+b_!5qoY_bT6=ghisq#N0O(di- zHg?646xSe(MDhaewD31vA&ZFvU%6u`h8U(X<5LxAqq37FC_LIo`4&0XgTLqI#>vyI z;s6;CPUfIObAm@?VnQ&iDiW4$FSt$L#Q%*>Y&QC9Dl*6GCGe~6s&(1@!nuRiM%=x| z0P&OLIJM-2v=la*DNduMl2{n$EUn2TM)2x!w}JijSgq&gwd||H0X0Wq`~?y*{ml8U zYO0{rAdJP-`H(9(zG4Y!mGn4GYxA6LwQxGswRZ!ioWYW(3=`Ws*%n8GLy%n$1ZHYw zw)M_OP{+kZ+?{t)TIYQw@A%xhN{Ux7jjsx*hizR{kijp-$H|1rA>>CYgOjR`%Q*Z} zlF}z;rtCWYOd74DF$Ut&$-zll4QcgG5~PJJw@g0OUqT}3SipQqDQGDogeedONh|KX zy4?F5%##TDD6zQWf7qbz$#^FuA&cS#9iLG`^bfH#3bc~cdQ~zvtg1I=Z1=7&a2PxG-zQmsB$PRp4ZOAJR(EZbS z^rYxb`N&=F-v=5nlaaZ2k;@&e2gf-ZNg%Ae(4B1R9qpFa+C)-x1u-!E=ytxiVhcgf`uyHjWD_5J1{3(qGtZio z6gcn-RXNw7YONJi!EpGYcxCgzrX1(`?4U8{O=(RgzgxtoM5zw#08GkA$n zFjPR_u+H;oElCyhU}QyYI&TWw^zFaRH{AV|Wh@$F6=sIm5Fn#`Lbv!1RxW}6-?Ajy zJUqDktz)k+WuMdg;LP|nB(fj`P>S{*TSD=M99SS9nJq-_! zTRePYe0msxyxv>3;U7Ugy!Sdb_aX|p*=+s0@waKh2G!csO|@|DLM+unDVGMM-7dkS z-AG4W<`u)W?HgV`FTv1-lriL*xsB*lj{SkXeAhkr*h2nOl*v!zHm`RaBob>lM{9W_ z^^{iK-D3W3@AvE#dcvbC?OTlZAXCu(BUVqqJldb$&Y@vf6rY6^5vEGz1U`A{OZi}6 zeGO7;)7uMfn54X|=}4Ja5|jRK4%hJm8$G|C-Xc?Jc;hbWygDdBv3Aa)ubR;RM2un{ zeUp(F*WpC(T8K5b;j91M;>W?^V#SDiYi1^|e*z;*1{rnrBWu3ppK(<#G&@%(c)=5t zvWqc{tRm#WLR;QRY9>V?yq`5VZl7q}{SdRMu@4Zit-r=^6{DHVqG3s7+LQMd<`n$# zoBGK~O<2k*S4=CCGsuaW@E#$8)jS%{%^VbM5y9-Y^ik2236u&#)s6hRqJ}7&3I)PU zltg4^8=nwR!!cz;@}@KE3p{({iop)R^HUIvcaoUo%fEjRfg8!7$~SENijn9aqc`Wi z5=QYBYg(cv)8+F7b|>G(4D_aSGzGUX+kDlPJP2nl8vJWyd2D>}&EMYJ|J2NrB%dXkou&N3dq@ic7(9?2dtW4SCE2oO!LW zQq{~HTj0%v9FP}aXYzhMD+^P&+z@l7uP&(wF9W9oU*bEG!AUWMRYc*$OFT4?fYI^du;WgQf-_2VNJR>6D5rn`vP)Rw+Pv)9mbLY!UYqfcO z7RsN+#QcSm8nswH(3{G8M_?qr%dIz2a-mKN^0H!2{v&&olplI=UzKHU;DyT&O$&tN z1AHrVWr4APgqnP6=y%h%MDu~*tk#ppVVU~Uo)LKbUWXhqn@{G{KWz>j&7*_(xNzme z31qH+;d~o+ZyN|JoeGbLMQ>s>`7#JiM0;af<*k2x!m<=+y7hLrdVq5awO z2&jqtR%c*9Wl`9@lgMWSN+m$5pgKxen z2p32us7i41+^fT|KaLYc{tICHQuRB`DE?3PtIau;Y7c@F9m(Iv3e?#2Sc1iojKfbuzR zp(iOz<+Mho7BopXf1b(`2qO~CbW$$@(WeHaQa{Mynmk}~VDEZh>qR6(ZZus%dgl&=nI;fssHXi3 zDfiGGV5#<|YH(T}E&3n-v-FG6?@u# zjhBeeN=xD82;^h~JY?)1|n4slwd7N=#8)!X^t409|g$*CuT>@}D)h@o#LGlUXp zujD@Jzh`hW3VqHNWGPPc;OQLQd}(==MRqt2Dv40_Vg=o z_L%SR6Tpl7-gka+;rx4en9u)-7c^YxX!(mAh!cLrVFh2e1vcIZ7K!Dhe6Oe|2^$xA z$(#xBe>-1=@&>Gk7cKNF=*XHh>dEo@Bp)eey!n>+N_6sUSh#5ll{u?1RD7` zE_PDOQ<=*5bwO`w2yMQ+F1>^4FPvC#=-)c8xqewSK0uY1laoho5_kpAW{I^4dWAp4 zWSyOL?poL4y`OlkLti!ah#j17F+HrD%Kjb#MRe!8vkl=io=^EE$k94Sv(yI>5|+r( z?PN#!^M60?+nN4pWCfmNbNlaK=yz^Yv>Ghh8MhWGk#94&y@qeTr;19T_oWiXuB1!) z>7GrAXeA|GJijJ$JnV@jAKpA%NeOEu+2s7*FF!JB*Q}tn<)b+>LvnI5_hFC$(aajA z=>XPNY+2mQ>-ZNG6qzvu3m|ds>G{|=m+ClbDjh?D&nW`W05+OP?cAig&UZG}(9JLW zp}6IZ9Ut7Tcn7Sb!hc_CMcp7x$Cn9@L$`X&J>vWndfF>GeRQjSb$uT}T!RX#>=gE_ zl$3$Le}f|SlpLQ^w`E$wl`9mdH}>SYD!qllsUtkYK-r`f&~vOjzw&hMAnB4#KF7=0 z?%#{^Y@y}H&Hx(PqRV2vFHgQwkjVRvPO`MRdduN3=kH!=EY!TuapR8}JFZrzZI^4l zIIKLFg>WL3=bl1RMGbB3%h-U9K@yuPi}5c39JZcJ@Lxj`qENWkJjmeE(s-}EGrfht zAGRH*!tdGHC-$OJ&8D0h7e$KKyZN$xH=IaYAGKG_cE@%(?EQxoTaY7>^oW-NT*;>F zQ%4$zXkFItw1xfb?C88c3J!5>$)#ltN;NADs9_Cdv!b)zw-BZJuAut59q~fhoVI*@ z@qZxbU|~5q|0k|Gf79PN5&dQ~|NGkUPa&~pOeb0jaVICX56#b^vF;AymR06YK?-;V zmn10guNt|yJDxaW6B;<1R@}5RTpIbQK5cz8ak47$_!+N}z+>?1rQ?RK!)Mev3cjHE zsc&t_-puUg*0Ry_-*!$w-&aPm{sQ`^9aiPDgQ@CfzJZi1~ee*fk;@1)hmXiC?ToHoQljC(X`M_-Ym3ZibYYfn*_M0jCI&avvua;e9VEr_`qj2>y@0j}GQV}V zi#!mleek7foYkXLmoaAiT~KtB@fS)WF_F(mZPB8$?O9XnqTi;v?ykz`jj*!@WMU)~ zw0>dv^hax*IRG^Q3CWOF`!YFmsIvXm!(nGnsw)1y#jbc-;0mHQF0cMfsw@{o%cy6_9vM2Z(%Z4LYB5@{~WEUrvYtloO{mut#kN4tylUwoJXfBV*b!%S;j<^ zd_(29BqW4Z;lfAUR{LxbqR*f4b54JX>Oe6~p^|Ao3)iBH!&FdM4Ny!v9g5iA3d$uo@*br|6*aj_D-|}WNcCdy z-0!l+C48Hw$;lNXZf<)fOrD25F>o%C4BJ@mrl zE=D|0Uov)tq(_KmH|PI}JUiIG^w~kRoy{9T&wIF%&5gEp7zm`W9*` zUj4MHtm{2@U2%7!{EmRo8~IrA7dJXmMrPV5{>$oG{>XE|qfW<}&+1&|vG3EO{pp(G zWLd|N&M~6x`~{lB^Zo)_bu*U{=F>V|LuPs zX%v@`u<18$-+Xzt_+on03yU{qG)qHKS}&%n>2u03Ponw86;%lCb;K>(O5Aqd@=M}t zue;lq(W#e}#{Q-BN3G?xg3Am~(|=onEf;f={Mf4cNC~l`t+U(2!h=Owr}Yo$FA}C| zYNAgVupqgH&ZUqxXUy@vKm^T#%^|uRYL=?M#K88S9(d5-xdtT%U#7Ur$u1JDat`cN zNLNQoB-1dV@u=2qk5UJ$EIYV>%j0m-`zzh)BQsRfA8%`+&N6rUj4$LmcCR5|WZZ<8?3om$;c_z8!umB!kX# z_K(5N!4FEarrtPoEFL8M(b;8sbZmMA&7&*dy&nku0--+)y<^(FWGfz&b^aj6=WzjB zZSR?7bJ~#-ZW^#2CmkloV?{~@fx19iAP%3?Se3Gb`|(40>MrO+q{{d8Ot-uRzG>Xu zdd{7c_2;sw+PUSk6NnosKwX7P+tPfR_RET8Yuxyxg1xJhW&Tg(WvF>>dAzsOAfp2J z3Y-9*3f-axK?hK`-J6S0ao}DVS8Ah_$A5f!0UsMbWulS)%adu}DfJneB{OXna%)dC z&Qw6Q>mJtCG(KvV_cL)(7XvloSe9@faL`C=hAtv0qgnW8e=h@nQS}yLFWd7#lmZO4 ziAQ?-k38$O&zRW~odEs0Y++Q!FrN*w_;gI?<-8Z>V|E!SQPt2*2EL%?PVd%Ns{PStu3yK(ii{@bi( zoS!P*_4Rd$rng2vW@fWNZc|m9xdf%P?%a+b9ItlaM3$Z33aNm7?5m7{sm@;5yQdzZ zqvgiLoTmp>mh{cFKQLl@;HVov%SH)0oUCU3QTy?uSw_AigP2qwP{OI@`fX*qb~qEw zn8`KqzsN1In`t`zES@UnA=$_yjHJSPfmiR6QN>yHx->w`yo=x(ucow7PrBHNi&4ZPU3qRUMzKIvT)Bad$?tFowK&#%$r!ZPX&vZj+8$%gF@cP{C zCz`=hyfTq3_lMS;3>e-_CCdDQbKa+=I7Ssd<5tt^iiXTNqbK}HDk`1Zy8<7!o; zo7?YHkO~U?^1I@4HcFwc6KB(DH`@;h)FTh(N7mQ^$j;%VkvET@lv}x>TkG*sGBN5F zs?knGx&v01e~+9hm<=gcojS7XO^Cb!6{?jt@!9!F)6)XB{z;V$jHyCtdt8n4S(?%1 zpIcesSUiNT?FEW3GD6&1S8UhrSY%>IHf3U0dV2}51CJBPMS~hsuQ;U(i>ajt)b&*R zQ?Fuj;AegwX(W}rF_9Igp)spu%`bW;woZahT5cEe#m$}74G*u{McHSLhR>#n+!C9T3KdC7UdRTUGsp~KhFC;$L8@m6@FKGwr!{~}F4V7L)>fkuoK zL(6oNF;>>YOcJImslICc+H$RH-z;{WHlm_2wW3zCA-#FBj7m0)P&tV!)?0>;-_1NF zL`bhvvT`L_-F~nekLf+jx3im&(RmHGWTR!%vX3sl`@@k+bm7uh{k*4QlNU=JY~AN!>O#Yl%%nAx#N?s`g$kRMhJwW=bG)e z$1)slg<+VRW{t~Y2MRA)X})OL(F(ICGT5+i%fQiRMSi9JGWECXq@q7J`gGFmPU1Pu z<>+qwQDgD9?wQf$57YBZ!)kR5{wqedeR^r?Zz&}&M~@ZS->r7LfLcuw!-bDQE z0WSQ3XH@2W)bnhVUugeW?RqfA{5vDyJ}3i^)>t;`xJjtdmb_aqCcVXFdO?1& zY*bcl7*Q!2{=gQiq-&=FDJ+$qG`PTsbSrYjhtKGqtG+`h>*Zmq@{K|}wiMHzNKj^K z(EBJ^2V>4RUK@FI_ftkj8_F-2GAvnsG%hSzA$lu9U%t_BO%|bCrCTvD^9|O)Hy&h| z#A>196~pi&77zb16^N=*)79(KZbf%I`vt8ZkGA7?i*)kARlDYG+-Ihi+bnMO%VFFdF<()C8P`lfd0N!b(-VH?XZoff5FA`*IhDgXBV9y$)ii8 zRkk?Tjtq<3_tp>7vmmE%CgyU3(>bEC(xf%ljV?@bY!NMnYppLcV^ht|Z%9ZCRf>1v z4+RCsI9~2#0v`R5vU|pVdIl_Se!yxdfXW$w2KRvC!K_!ZB3H(s(+<=-eU8k&_{gCe za|n@YuJhbH_x zmU}Wk89laNp}=UM%{2v;3=`08&Q1y|w$LqLp?v04Q zFJw;CMT(;z7pHB(P{VD`{$nU0Ye{# zffhi&pls`GttqG#f8gL}So1h3FAGHx@x%Zm7i`@_%`bCyrG;Sk~IOi}YOm zoZ%!%iu#A~QjVYVM@kiL+-2o9s)WNlwh_e`B)DR(h2eKUK*v#VdOqe0b%CY~h8by$ zDn5THBJ9SO#gFwH4rueDl4;yVXk>dOGhfUdG^FQfOQ{;M-_{d%oAQPBz}N0Hby?*3 zw&&Sam;MVX3-GgG&E+#L$j_%|Wu-Gn977yH%(3{>xDN$T9q0SSIqZH9uBK+KWNkCJ zFYrz6^rog*r4>SQKmXzo_RvQyI?%R#JvDl11np67jES_St1I}tAKaUI)56j!*3(Sf ztnlPCG{N27;wPOi{{GLGIhXxJ4iJzux9$9|2wX;({p2+W8t{0j`Stgc4$)+dUa1%= z8hiNbrcL8kDg+LoudRXD0bB0GywUZJyT`$4i~N+UJsP337KM|@&R7&X_>c8z!z`^) zWG{CEZD-xc%f6SR0EVYkZ8-Wni7`#chubLN9~U;AQm++fLVEgd#_Bd0&JkqJm)%HE zU)Q)c*FW&*}HJHe$Jiv%k(RFX@s+rTTh$gn$Q7SXdae_O!zTAs38f@WkRWYOSAp z>HrWG*wwGr&j-kgOG?7r4ri}h?Tc#giG3W}&fx(g1HcvElHZUwR5uIxK*xTL4iDc5 zA+lcwfXhrFpJaeofdcSg0JI~F?%y?^0Py(%0LL8s{d;pK!-1BEC&lmarq*l#EmMgO zV93`#!^iJg*#ym=eqh1_m?t3oUBIz`BIb33up4sek;=6yFOm%!tT5$MSS(gr+@ODw z!E}k{Y0vIhHR{#X)q#-@{&)ok2b-IlL!q_ULOt~p8Jpkz(AXphkQUc z!jUq$KA2%L>R4{Q26hY1XrL_j^i_t;?^`0Bazav49|R!I=R(V%xg=N+iO*Wp)Rdf= zNeuLYE((Y>Aff-UVxM2|xR(44Myv!=%@ml$dqYD~Y+6p+5!Yen=6CK#^V~i^&`9{d zy;U!I2Qc9N*X2gFfH=@`nRN=`*U$nWd=!CQbF$-4{inaI0VaS=0i}@684i*qDmkn~ zTE&ax;n7ix!8nR`&m=3A=Kj-Di z5?~_3!@`XJ{!UtU8Xy7iI(&sRb~vjy41NHp2!Y#y*v1C-F%6+0O0X^K+PHuOm3CR zW@l&X0$%(fd&_Acy{dkdEITS!Qe*mu1PjJ=hFdZ#51E}Rze>9bVK*p!3B1{5;eIZN0 zb`b=}EWm6DaSgQoLXLy@mJM2`$u`a74wHW5W4*QxaQ~rbB&KT}er*@O8Bqw>g9tc2 zzH`}8$PReE|2)fxfkG{=q~J}II0SUSk;-NU56bjCq3HEKxAv=^Td1H2mIiK46)WG~ zYsdnC-S}th#Q&_bzP??xNl_>n$UX=36aW#Xze2ZOR!Ir1sT@XZkRul$qQEv`WMb5+ zSO@6j&51k-!HYqv)X|TpDpus%#kdG9Uoo)B2J1$`FsTN|#&*CS+5!A0NYx8)Uq)ID z>A_k7KM8h8^9#8-IEEl5P_um;dD9<7FlE|40byP&9n*qmAWQPT+EeOzfD3d46##rI zJ=SmW@m)YQ^z`)W;KAKse8XS%%^ZLT1(03Pkr;`j5WHpx3c)6B^1sL1%khBcL}YxH zIAHNW4q)G5DTlzmf+_9jFg)jZUGYog_wU~jPLq-UJ&lxr>*`xE#!Y_ zJir)HCO)Drh*GT&#Xni(&NDJHZhO_1ih=AL(T$FfECcv z8l;Lf<|OO5NORLor&EWO8UY@3#84oQGJ)aQ6uX^4?g9LMH`tFu9~|2Ia-Of~PM^g=tiU-8 z2DC!oNO*+s@Ng!b8i?=noI|!-X-N|Kx33%p;IqJIvIECXT8WQ^VjGl-BAPyn%&b%M ztF?22F2}bU;M-}z_gg^o3|eW*JFmzK2Yh+L0Kib&!mh;wv&nlU*RdhDG{G(YzO`#{ z(}27$(HPibDSKA!DwqF)M?SjL7~o#9QNFGqC#R1P2WARUC7^WlSaGtObi+c(Fu)E2 zj}5CSK>*I3J-qhs!@C*)rDVgkMVo_KC}I)+Oe&d}8vlO}pDXZ@`VQ>?_06PtNn*CM zqmC|wFVN8BY`qKkAgu~~LQ#O8z2XN}`b#t{JXaa>Ft{84JRC7l+VU_MrnWOK_Y@#! z5fFVV@vTW`MVvMzSvsGbYYpjxFu`p;-(%KkUy6JkR-AM&P7tT!VK9gWD&;e#?J5WW zXglG6aTqL4L_r(q9}XV!hJ4HiViz~~2N0=rN6bDVX9;>G0DU(v7|CAgz{CP!%u~uG zXzS>mvTnQbOX&oB2H$$~juZXRK`(56pbi7-n`DcH{7 zD=4yhtdd8+z4Ia>kO0#!rqA?~Q(IeGCTI!90G1f2ivb<)mk%Q%f?2=|8tAfaJMzv0 z`kGdGQoUv3e8by}%uF#T6f(o!&A!LMLHdut^)Ue^f#|ggcmsBH&$fL6IEq@m%I$6H z^Q$Rjr=K1BKZ!-MfA3Fy1vNL0o=|)jkOe#Vr+oQLZ2KBEQ04ol_K6;J5x}Jr>;dlM z?c2A|##2S5%t3{SDe@nuirzwj&ffL+RxufXPhNiwWa-LJ0`>D}_;K>+s3J^mQBjey zFETEmlcEN={L%tMv<7Fum_6*g`R2YBUl?+WBekzelIPZ*wS>)Q94ur~jSL~I85$~i zdn3{*-NlqBDzly^A6Jd%Iky|s0!sjVAz5YR*XxTmxA|JFOKsI!t-5WV4v}@03pFOM zfIr+5qH7%l8P!6EpHJ*DkSjvVFF+?^%NR8`|K_;nmoYTLBwFo}KPLeGSnqh`PK;L# zT(I5Ak7TiS@dx1*UFZq})Q*^-{+C>AlQ|_PN7>i6?e-ebNx>B8goLu%eQuj-4~Ju@ zoi*9!ok4_weSF{@G<&x`XpJa0R*2-1VPt=X$dqoHVUuBg7EqL zbT8~uu?!M0Y8smJ2?_K@-VT>trDPU^_g-FnpLvj)OTEVa-znCmu7`^NVEm3dw5;A7 z{SVRMJXGYVtJ;UWsRb~xL=_ZJc$OVs4aAUafg}-N-~qUvjg4*9(K}jt>v9>?_+n9s zxEuJg!RZqeif`RCH8qjh_#M}wknxyARdGNjIaB3OdE8JWBoK+mY`XvFJBgu(<$uEb zqE$}A$QTJq>%U&Il>nm?RR5V{-bQqPMHe1iY_wSts%~i^OdJZzr(bOptvt7oNuUO? zpc`UFcFilZRayThOF`he69C~5VEtqdb>XB`bwJG*P|{8veI^G7WkGYnsdYs-RJr2_ z@RtX+Ho)8FU>3{nOLkd{`-p%-9Vq-i%miA6RH$Ff&CA^&lN4Af_Wa*a6KV{A(11&6 zCmMRJdj)pla2$n%x4TnB4v^3N2Ms{u4r+J5+gA=KtN747_Lv^B$8#!TG=M9m3Q0R? zn3)yj*=wg~bp%@B05s_EI;oB35%|8ry-R@Ul)XI&;di|G>8_hMpSjG7kf={Emc z8#8k>NTJ>CB>RnKs|{QuS4s5FX<<*2=e3-qkI45A@L&-@gNmhv) G2mT)jQOLsp literal 16446 zcmaL9by$@__wS8_l1hVwba%HjNH@~mDJcyiAdPf)gLHT2CPliWrMv44&+~h)_nh;` zxpZG_+4sI@?wK`fty%N=8losKiGqlS2n7X&BK1j384BtZ#>)%-HTVry@p>cp@!Cj6 zQVi<(h$vS>enj@&%u85PZ~Ywq-gWMRD*x5hn}h4 zmzK@sPhUBBhE6NH*Q!?9)mhao7-h~ht0+!THzJA52V@}_4oM8QkIR~MbO?&{A0(5< zj>Me%w7IcdZ!aC+uJB!ZZU0&&8>*(?fKxBUN|)b}Q?5Y03J@@4sy(1-ZJRTzxg9|> zRG7oK5*g2!H6lfYhZP{{6(dHyfhD1Y{?UI`%ySJM2PqVK|~fh{BnP`#^>=a zt7>2FgJ|?u;!U0<1O%33LEn#>nwn3pJybU(sLr))aYofXj~<+MOLG<0-tmm6Zfk)! zpXB5yI5^^Q^u4O4R%a{pqsXN)e7hX=m$T0rU|_aR)_VT@`SaJcg~#oLI5|1_$yJIt zD=W+I4UyN+cY+V7nhBcKCI~VK^o9p>RfoR?ZvK83W+E;7G&(w3T|rfgOwsYM#ng1t zjr8i~;e6bEJJ}p+uF-)E4i1ih$1#AeqBTbUaqsgl{f>-Tec$xZP{f&XU5NMf^6c15 ze+2$KxKv3=Nyl-Bg8zESiT~+l0}}`5{IFqFSxakp`m_F_)a>$YTEsT6?@+ z9p9ZQOcwCuOy#ogH(+)?G<6A5Y>CGX1H&^Pm`w_8X8)1nbm#x z`$mx1BDyWFy?tddPt>fUzFvK7WF#(KzEDR+g={>9L#Di}EZ$TzfynDb_|_kZ@h4<> z*hDj7>+6cUq`ZPcqCz?^MT<`PheK=}90(+lDN*11Qth_u4N=_G)YMi^4rx^PTqlqgXap)=dm5`9bh~M|E{|s{8w^!&n>!Epj66FG5c%*X>mL zz`D=A!MxcFV`|rU0)w`A>$B>8T?c95S|oft{I`=Q6}x%3*zi?)=kO6yO@F*lCzHWt zpP{3zO_wR)sXjJ59JA_mJ}SSe`E}KBzNci`H+CkW*X<{kfEo~O$SUk1UmK{DMt zTh`=FruG|=4;)r;54(k#!_@>yTV-6H!pkG>w_AxXD~ERaYeSX$x!FMcAW(`|K8m# zlHswZZ*FxwxITQT-p~oG>#l;(rGbSxIyOEeSQ>@F^I={it z>oislm}3;d+P?Wk4|%v9_ca^O5E!hM_4XEM`Fgy4yOSyTvB6cTSk>cqVPRp7@xa^A zu&|ce>jmqI>}jhZ2qd?tD7Y^SXX6Vu)Zo}y!LMK8)zz#YZ~np*C}gf|+}7X9C$k0~ z9M}v)AUsan$cqj3=6ipNg=A_-*w|uEHwMo8@$H?@q_Pz++~^^WR2_+ox}iO5LKU-4 zR|lGD^5!kpT8}hzbo-kMpRX3d0EYwz8|;jw|I%v%kI)nZ3=}F6B5q@2)8Y*L%i-Bo zzpJ({-`&fKVRqU3Bfs0+@?1VRIA{PSH7+4RSzkYj9`Z}O;jP7Ux(ol&Zn8^jdAxWS z_9^g5KNAxr?d%vqA9vQnn2uP>=ph8$U;G+YeNw^fxh7NM!fUbI0w?#%hsiWDuwd4J5dcLCMI7-Ic5!*ahY+Eb(}^yfEz5+ zO_5=0i(78_`gOR4iuZ1Da-2dYq0*=?^uHt1xEUj((dNN{qvu8f+`V2@*2PY;TpD-u zj}IQfVEHD!3{A`GV;$MCeP}oSmKhZE)~YrYQma z)q>Bd?e5Wo;yu*?g6B6QCJSp2IEgFa_ukHBXSJkA7LA~w!g~c4CBwr)XA=KEVk<24 zHw9GD9%x~~%Edzh0gt%#bt61{{Akl;&{RxptdxR+WbWk6Vjqbl2*(qWR@T--V`Kd$ ztaW_)y1H^ME*!wCfo1$_IVOC~4;ABE*1NewSKu`T=&_MbPvJ&V>b@)GI=-T2klUEZ zj#pJL_?ezQx;I_=Lk&gz0(8&&@4v}h4SH;Jbom;qX=8G#oh(&eCE9fGrV{+ScOVlq z79AVAVgL09CP)v2iKyx6VGXVhY^meNSbh z;fqe`i-)kgI^VuFV7f{{7cNSa!I``_84H18RKS}* z>}6;Y;M6*IUK8UrU)-wCgecezHNXBT#ETOks!Yu(+6%C7lEgVD*FkEGZUrc^dke_T8DpE6R)Bn5l zz>EpFPg_m@*rI(P_34B4C)Q8Do1Wt`=u^tnkzd0w@JMCstCl1rG?UW^DW~&Q2qeB0 z`bJ*o^Ypf6e#x=JyBF6iySPhQO%0Jua@!@jpu~y*3YhPs&kz4o?+=`#1ons)i`}ya zMUu8(-xm2bFENhaN;r{S${Sa}srIATy_ZsUrOmQM%9%G3wm;6P^w1XK`3yClU&ouG zGGP+McdhTJgsRmx(vgXzUys^BvEQ zjBOVtg+3UGGJQ{vi7?I!GdWm=Q$rlHIz1iFkH^B-v)VJiG<&Qo+Gh5z7cgMZW81MO z>N~QqCCQ{yZH3@!51A}ti_vnICv0EEZ_fBndA~Z;(AX!qdxiCKnS3Hjp$tzJ^a8d% zj`|N}5e%q6j*0hNERnyvd1+~t-LIEDlR2#4NJ%lQaYBYEo&HTA($cBGBOw|0hricd zaNCIB0uJzm;CQa;yUYHJl7_}0Fl_g|;=;j^5u59ewJ5)M2c~o-}RlIqQ0eF-Fp8Pt*t5@Efh%9 zWoEyX_Vn~a5*2qa~SBN7G%Ab%5i;=d>~C4nVM3ZjR?WYW!YZU2RvfTx0$&mDAP; zWLA0t-x*VRB4%oA-%==OtHfX)3d$h<#I%UJJ667~@~iJVJUL`4m0pv`KbW|ldd5o? znqXLq=EXLV%h_~B!OBPZGtK#pKKkd+!JE?!DH$2jGWD;Xk1K7S@7c`4cicd}nlX-q z!$?j`TQcw<2*T#o;i9C72=v!g?*?;>=K9vl8Eqk%2#IV(w}7%1>w)n1J?-B2t-kQO zS-(rV+oX$r5>V$K5&aZmPE#z#%9D*GGj$~Joqv$jiKtp~T^w(1u5WM8Xl!hh-+@3N znl)w-<~W^4Mdjs`BqXo=dV9ra>F8QyXSL{}A|tPC@_#Saot~clnVxpv^8FetWewd0 z6}{}(aKGa$uBwWurlxi{b~s=2qseK<5u}ut&C4M2(qmy^357wqP{>Xw(>XNhfL(|0 z+K%9}d(|JVd03U!K?}xL*j=a`Z0t>YGnI@BEU2SFM?lc~!*7hIes9=MJ zjGDMOZ5Ot%a?|zBrxWZL_y@RF+{i#zU$!ovlp$H)=54OLUN3n2Tue%WG@XQzF>+~1 zXWZ<_R??T1l{E=$K0Mko9imJ<+V7!zq>#t83{H;@7%x;zHAp?f7%ZY zX`qgmUh%m<6h6)ExalUE909)%UOW`ZM(u!z$in)+T6bSJkX@aj>F@Kp@-8QQP*Eo=?zrq&I-813YW+ixxn=s&I;+v3K5s+ zx2hO0s_3$*5Kqu;(`P_Mc1lTQpvGbtOFEl_0u2QBPL zKWc^#!ngsI@UR9|NfSkHu!DJ2Po%_fNa+37 zLi=n;MA(K`?OPbQJr7 zmxPiz;8i{!qD+1ZZ2Sy~6}^9!c*2&Gr_PU!+WSTP=Z!?_#+W>z!bnn9`WTD(kUqVz zdG+tplYZuJoWBuwvLwq|xZq*N;C;DUB<$F@{<(lCC7#5Y!EDR#m|F02Uhg}7%j01N za1|&KU`)w=Dw6h+L-pn{9HxpCi)g^Uk`onWknoP{Scb=imX=*04G>5D$Id>3RvvZP zfd5LgGgL}@0Z9fC+miv?6J_+u!(?)R^aKN?z=O1BFkU9T0yA{qff4K%p(>SMdrT;ce(pY-@s-k)ryl8^LOVs_GM4DXj{=ozMjNHN9K*V zyd=-yN?{9}esZNCR@5w0x8Y3v90J2bnJ=QeR_h-`t{Td1Gxj=#uudaA;t@MRD4d7( z04}oaF6ec4Pfpdm^tVn%B2hzy{HfiEU1vRATpbn~<8Oal<+9uVXh3sp7ggfmUYTh2 z+igZYq78s7mqakESxw=AYcpyRMgbQg8z~GwxB5@YqMB} zDQA=DB#JdM4yhPBEQVPmlQIN|Pb4qg;h{^0zl>@iYV6zE84V+3yl%2^U|s)W78dX* z3!s&(=D5oaeJ7FfzM@>*HC}}3BBSro$ zET&-{h8p{dm$OXb+<_I?_(qZ$hlVv__=3>K9ao83NyJh_tnMYXSr zVu_T2Yrxl*+cDjxHO2*hc{TejSu-&bj7aE(1EwqsOIkxiFn&KAN_Ud1=n#=fp&J%s zKs95=Ifd{dgtavzl;)s*t$i(SlZ2W8m7i0BoX(5bre~wRf_7T}_H|4EiwnP9B8H^5 zcZH>PU>`Ard(`ZJgVH9aAN^+Lh^Iks(PQ$6y}C{CqITA8tS+u{uDXqQ zspw-+6NIvJa?%`(DR035C!56Ur|G*0OQWr8EttZ^X%I`-yyswZD=EAY=(x0sMxP-d&yli zN4TYB=klyJ(RhPdvJjCZ&qOq!8Tze=7fN~bgwKj-(pQkm)ai^mNmF=4Lq*CcjfPe6 z9HZ8*syY8LDF0Y9`<(k;G9DpSbt<5D>+@%b2n3scO&Vx$4s)lRd_e7};>NKf8>URC9wFIul0L^HfDvp;hTBYE3 z(mjLS&R;}nI@>^TI!NQ*D0gW}l@tYc1S5e|RSwM`6mfHTbgYDcE`yT>x>sSS7=k%} z@U^U{-B*eE(nV7mBsGA27dIoSsj#Vw%j}eq(#pci;Db7+xUUowJj*GY^!IPUQYsew z8DR&Gw|#GL z!M%n-^_~275lbPW4>euPG2N>>#$!(^{@Ts)`GIi!3aYO zD-(h(1`h05cwI$A8HXX7ZjKO!{VZ4k5YYT4Dac77cJzJXPpPUBVUjAFz&muVFamO?ea z6?m=bYRtg5Z~ysqc-}vGjlG{37YrDi$o+mvPAZ@Eh25oKcG-GA@T?H`l7C{3e2-+u zH52OS+p^Baiy&i)QUBZHSBYaDEX;5beBA5YRV#Wb`K@SPGu}{Cf};<}0ZFiE{Z%xB zcGq81y#i4gy3qqusF4||%B9&Ne@h&cSS_9M>S)1LcQNO6JDzPyZJX%vn%u(iiOBkg zS30G7-SP$)5@#!xD5jNG{nL@xQ8TkB>BpK3=PS>TGOe_xA63T~a6-D-ieh|U%P)e0 z2E&Pa*w>$JXVJMd0mUs04=`@>Oxn; z6d|BC5b?HUE|%+j_z*gZT-r*(qq|PKU?7odN>-8%%uL5Ubf`2yFbzD zLt>C4aSqk{8mW-VZ-uU#Y{LDI{9|ri zZABHzR?CU_ogFHlpFI(HfP4`6O;oD164zJ~BcDiAqv`b}*Clr(<~nu7K{d_hmpSeX zPk$Ng5R$caQY%VfdFv?&9OF*P!rzKsDT5~Q3cRMGA(0%|ee;N_9Kevjwtg@*k%`#4 z88QyA#s~=uyKaC8|3Hl;mMnfOcPx&W-f+#IP;^cibg%;9JFfhvDy?d?5^jXVg)84! zHD!uSN!2bLkBD!l9?cgYYJ*AaPfHe`_$g#m3KL_)%QJMI3boNI`Dr>}G5w33wdK_3Z zjtiVl3r3%iCDp^bR85fn{EK=eITlSeqS@XJ`~d^F3xO0(_P>G}wM@)F6}*VuaIAo= zi%*tHXjm?nUiL%E;6AnBKChb7wMm8Nig44$CX*mGpl2{YI_^T>uj&UVQ}mPHfs6!I zXgV9Co)8^ZQqs7A!tkRtA#b5SeClXF*Fa>|#R3O#4n@E@Ks(Rc{|?Rj$s8HdXDHLH z%m|xLw(8@Z^l#Gm6CT&tYB?;2h^z>deZhoGOn`$d0jCQ84DCIt8x}vo&6&)$E@RO5 zH=!@B6(43NPra5RG5jQ-)kK1A9f6AJmorspVN-wBhGlg3AoSAM#hm%?SAojiKhr)r z*&@$qkx+XIKE{PpKom<}GS+`D*HC-xvhBv{(HP3gcPnFkYg=LYqzCs=4h1Fcqjw6$ z^QAtxkR0{pLRIvb9Q*w*qf}O6X&t%<2x7={s8!}J zoEtW-m-9jFq(OTkbp3f|Bd`|8bHbIgOG~n2vp;TjjTKOCJkeMTv{6|MM0+#%B)VtD zDZ}KszBOaG_sYd7;(N1M(oGl-A3gmelaP}#R=yi~wTS+5{b(AhAu-yFDlf%C^bmWz zm4jB=#|v6!2i|l7p2o~y`+fD9?P9@F!4rvS>E()hn79&=xx3-mg>}cH`36Sg14L*n zR>)J^BbNUDb&p4SF8T>&AAOF#z)k=cq{)VMU(WSt1?A@r!n#VN~X4m|lv#Y5( zhA=hW?R0?1OA-Pa6E0S|GXF7RHyKc>BoUQgHAu+nf2Qfmc^X^D%Co@_tStsu8y5NT z>_4|KPUSFOM*tK|Xoz>47YuTpEp`MUj-GewM}G3K$n{2lA_k}9G;Y~Ar4gD^r80s3TA z*$XR#Unu!IDz~Z^%~{*xV5vC*sa!`+wslU!n)yFXU487igC(=*H{{(wPJx+du&3+IaVunx?^w9>(EBmjz#v$0 zN!WkcZUDbpCnu0x>H(OS`5lGf`XZ*kOVYB0WrQwDT7a; zc}>p9+ttd*QnchWm6-X}2lV&X$2&(I52LH)$oRmWa=W&qU$1z+3+of{5`I8XO@AU(9k4}n%udcP&>I%}>U|%> z&MTUO@-|I;Ti9$+y(wRfP3B5gY0#6WeH<+r_&~-(^(KERpkKu>JESX7q5U&y!TH@SP}-thz8{~GBqAcMeLQM9bl`KSy|8!$ndl^!~{z4F&`a+D9hp=`B;QhufWp zo7s0+iovs<)MyHPot8E>u*BtAS&QrrJFtb>*$#?kT~|SQKjHCtptXUU!F%qDb!k~X za63maD{uCfQYpL}jamy0eue+{>aJR_5?SNGY-D@;;|>f=R2A(2ZZTb5R>O2wr0QFD z7yN3`I-l!#3>)*L?+NB%T>nYh+1a_`%Io$q6t|Kg*i7C} z`SWK8hC;H`zhobT;cqiXt_RUKNa`*B;_;6qkV(np8p9d64f z`0<@;f$6r4ieJT>iSJg&xTqYHSllTU1Wu^djTbANcHPPtNj|8U3dAy667l7EtY$cI zS~ouW5>Zk;!?G31ZY%76ApXdiAhpqdI{DdiC`E9*bcX`M%D3VHbJ)wSUe?}UY7P7S zJ8CS{+DI~+&2oKw4-Zi;N-3G`9j1DjPDGbMR&m$xlRzNDxztf(?o=IK*|#doa~8fo z?0mYwfgf?MLDwI7rRuMIN!)2vqh3H1QrIDKfBGsbpWdD6b5G!V*HP$p(I@G4&`#yq zs-CTsh0jy6Wl!WC$@}ZE^r~ei!(7^mcD(GFL80xzm;bCgjNkp)+sk(hj{nj3`9e$f zt@jm;Bwfke#gD>PX(n9$fTWw`hX=2x$75d=4QF&?8<^8v-CsJc7&J5+0+!umE;?}> zAvseVaq*l6EFkJc~+fNnT%xhZ?o!82HNqx5)nI^|7t zz9K_}j)O5!CV2Bcak>6?>Au!ZtE@&7a<1?|TJkQx?yiLS!AFIhy+?}vCQli#wm#CA@X&;hL zQV*FrOHX3wQ9M4wQN=nO3&w4HO=4nb3s`!Kpp;L1|e0PN<&Q_u; z;+lVtKl${Tmy|yyrP(TGUGfTYoD^(PVqN+vlt{|d)-*Fv%J>hnnImO`b)C)yL=8$z zQbJKp8!~v~Mr&>L9&MTxT1D-%jES3D)RoTLH`-6Ph#QP!>+7bFI~N7$RWb>uAgCWb zU6MrJz3E?9hCmU&X}2KtX4KxN=jtGlhW{R}Vd-L34=H4bRAt%BZIz*=l23bE-(ngu zB}JE-SGIUOEkVlC)UxC8dF3FG|9K&kg)7y4ZCdcW?VM?^1|`*GEAZV4C4)OUV@qma zjrsYGZ(C4PQ?lUmz8YhS_eZFFtauao{U^;qJFer7+t+d|RZGr0y2}L^$O1WLtieg6 zW-q6kuAqJ$N2AiXli`WWWk-M=mV7_^9G7Q;GOOoJ1&Ap+Pev}x&(0ZxW;L_fN`WL& z=yr+Wr9BG0Scy8LIQN1Z54$zrt_XjXWhxxe@11#pDV|F}Kc|d>rz3{>|*GlliAdowkdZ zaw0$a^ZBcC9@HAm>o0 zRi~{_0}Y-RWgB&6HEJ!@me!Sq7C+t3hYA&o7NSgjoz}^}K5q`_Zae&)QnU$rcc*%RIISh7ic*$uUh zCO;1BT|!n{9t%pS;@O+D*0Or#YO>(EMvEo=ba$V4@1h_unSOH-1X3P3`GTq)j{AmVB^IZ`MHJ~|ib z%WEdwKIU4#=aC;8-YUH(C%5CmjE4w}A=fS7PgqN2seVc#v?Ik*7JB0|BViexZfwb~ z@~IaWE~fhauXJI@W037}+c*P07YiqaDSR8|1sn+_OqNR|ICo|NmqyZ*-kpM6(eS#$|YtDwZT` zxmce<$NZHYNi%(0?`RX3otW6z#;?C8LffMm{g&h%Lu*HrUoAsDJAhb0z&ov<##+wK=*s1Kq7U zPp88j8jq#-$?ol_F+$+fKhZgj%yC-nGmG`iFFj*G|BMi=c*G!KkC9Lo*xPz*2E|Cv z?^B<+N@5r;z}mRt6v5C@S}Ry=AhRb@8S|=$-B@6KQgq`_eD>VJDu0@O)Two=8c!TY z@`^%@)%(ItaIn-Ad%fzL31-4+N{)*q|DCq%(anP>?J~O56k|cJvRn)!L0QR;lxA7n z^q|I}_o;gSwgpu9985z?o{TdBgTE0YT|w&+cSZXX-sk7>iYALwBi|Ov6Hr84HmK6L z0W8UUoiz@d*~rNV+AiJm-2)*uLnAyWbT-(*iqD|0ADWR~A_&Y_(W@8zI&2};AiNa9 zb(QLGnWy{qja=#Tr&Bz=djHRr2D4ktH0 zGwolfhKfl$m6%C6xVRuI{eEC`kRc%B>dLw5bI$-kO0Vbp)oa@De6EMP@eV4>J??GO zk)yL?yLsW7vH%U6-re3GS(*#dM8dvMSKkFkKPlXJGnQzw<5 zWJkB+c!%Fxf68js`oXohu2nW;*TMC^l7}EbQhcr6vw!0>Qt{lk$5V0lk&R27Da-2Nuw(V

h9P@9Mn!RV?{>Udp+@ne+t$&}_W z66<#^j|Pf91!=w=)hKc?{+G{tDxH$PT4s__{w8)W>2iSv5%O1ViN)5+G67&X7JR6Os(97G;nM-?-3WOGQOtr*5v zNnniy2N#1bThz~HDP5M$=PJptYw6Bm%2`Z5;gb~8mKyC_FIuQ;Hf}^Uu==C~&*GH? z1SF|`{(KHFHiyl@Xn6;*hoMHLp*%72)7z)zk<@v2x4F$qqjDuSd7QM>bM2^_I6v421uo{xSN^ z89BJ#7uFI^wr+3Nu}AeVzDi`+e^cu!ns29B(AqDZN|Kbm(+>L8Mm zeQ%{OmxlZcwaKryY!VHG`9}YEjhx;RZsZaY855N4Jaw#$7CAhwnu((F!{#SBV@dW} z)xA#p31uRiJk}jj<1XMy*Ksq+kkKB6s6rOBjTUgp>%8QzWoja}$-XxlsBy?K=y

PyXWzaTg$9vS|fBo@7S4LqdkrPJddT*tk+ zmby?n?(LtNE?s=w`tW4s=@PV1XcpQd%&5t+SPVBT>m`*mJqZf6=Z$M56S^3`P3!#- z#()@HdL(4h>`Wbu^Y~Ck{o3ysuPE=ph#g|5c1M6t^uxjM9Ys5(pv@0mUG*ZmesqU< zp1Y$x-)1zHn(g#M<`dMrW1HMmpXGg0K#qE+}$l||pX zK5?p#D*0<4+|t!S?my>8{9$ZZ!B=#YPZ?zE);Zy5tTsj}3SlXYh z{ZtVJ&abkD_ z*=mxC=DZOi%2~?Z4XK$DaAt|fpXSmFI2R2s(t{RWzdaTh>4JGv^{{JW}zytIO7C;Png@w73&L1@4YeD;gBH4c0?mFq^pJe&;(r;qKQs}f4Nt+4(t`Nm#Fge=__6`4)TX1Ugxc5EK(w~``nDq4JVwm#g=9IlXtF!ZSKr$v5ZqydG znTaGLVDz6~<2-emX98e!)c$mciYS(aEf&eF5CebtI?#tUKLZU-WsJj z4iX$G@k-s(d)$ar)L0kecHnGie0;}E_ybuXh2xsbc@y9Ml^xkJ66{}Nma4_WR**UX z4iAw4?!aO}xSRpUkKGoJuIF=%ku(V(U+Tq$^AljXFB)e@z8I{DBWbuzLSH?a3)2F|Q#n4rVXNQj=7+XY6F)`$FR~cZiPXBJ+UxW6Hm| zj)q)R*9AtL3X#XV#IMF&rr3hs{2-iVL@T*Vp~go{Ozfl&2kqufX6Vp=OCo8mY1^lJ z0}ksK4-5kdqU8*Uo<_$c({aNI4BFh!59gjs)@Ix+JO1+{O$MBlu^|c{S0Pj_@6HRO zRwQ$y%>EQblT%PIv$90y^>P~-dwfuyu{6TXW>s8uMl7`H-bNOkH_0tt>E2zC-YcvK9$H3Dk;6RdF{i#3&15+P~1wjy5Oc<%xA(cR9iK zyaUZ!n_GWt1C^M-ND(NO;vt#elnTVQXRzZ{^rE_=fEveI`_gR5s#&=}P>a%5tX>Mw zwKr+0+B-0Sia}=RNHDN-mTB%GvAS-78!^S9&m=Kmg)&z5Mljn63iw!#}2WH^IA$ zd~F!B(G(8v5*IjHL@6m7F}#q;8q506_ltzW!D}ik0A3xiV2>%0-&#StW)%$HwmWheNsT{C2iSSp4a zh(exk1_W^hJlH8HC=Qle5&;5zzV?HN)#JhfY%duTA;wb{?9~Qc(401Ne=mC+n{+Z3 zLdm6C!Sg_Qopil9SZs(22!K6m+;|J1|F@lh`k!mCU-iq35JDYB9$p}bkPwxSc>VtU zd++xDGIj~Udw{q!1RVDZd`-aX91JuuY}PaMcI~%jPZx!PEaqbbF)=Z7wU!uMt6prG zzE4e`-CKb|X9y&c|4-GxT zJW{_pnD0^htlKC0PM}xYzO9ceieJPERPsFx`8heze#67kQAKJmbRI*!DJr}N>EB+Hqot;}xHz{+F8h@ctj{`KfE(Ym zjHM(e2edfi2cH{5pZmpC$CJOp;9R=tb~|+*JVvJX)%(%WHhUkJ%|WymMNe773Zjs@ zI<^A;DJ)nV<{9omVDe!U{|oKCUbBPqpCI|$yZJ`&=2eU9;Tgdyz*B)!a~PsXbdxZX zrHtKi0yT4>VX|4Qk0{LW2nB+js5Z}QM=&<^_(QvU=P!M;ZCLJS@;OC<`rv(`1gg%# zVPRyTQVuj2(3RG+jO8sy%_KhP(86EPzC_!;u(x%ydiy*_c9q3IlF$YLs-FDf;*eCk z<`8gez>NUrlV=|A2hRH!HCIij;qQ#fnwWt`_aD@9)&J z;orS8U+WG80t(cc`4J#?nQQsV^n5ql!Qaga)Iml@MjmxMsJ>v{L^?x;3=G)u1iFoC z=3fjGD0;9eI-Yo?q@+yD%p8FdEG8~))BZD@f0n66u#|SK76Xubfsc$D7D23EE3If> zpY3?!07gHO$`y5czJr0iz4++}TDXYq08nFi5&dNJZ-0CT@|3p(+?{uq`z&}7FFL7_ zk&&#(P$pDltR5}dg&g(LWESJsV6XW%2;c56ke9tc^;DT&=yh6hW@eP&`HexK5Uw+i z5@J?`XW4}g6zK{;zyOf&=78W63s_=1f4jc(-yhTfwh-t~Xj&4(ns^FntK9?Q16ahj7ugmGr{I{F2fD;8n%K*L& z{TiGI`HN(q$nHOb^kyVm^v56sg0UY6`wG;_iN#{b$3?`d^gf|dgZsYf^&k-@CSEB= z75#;SeuG3pNg})dBtobDE32d=^zr_>1}qCbk6j8N%NYasZPQ25@Cs#F8Xn28Uy-y-N;2ZuQ zC#O8J>+g{$C@Jq2)w)p0j5-m${OKWSye^?&*?j?SMu!n+C;jUOHuEu)-+59b0%iIy zv_h{R*467iJs{Ho?6mRq(eg)4Cd|QWz%|tqr||d-8-ndzUVeVarAQC_sT|^K!YVxr zH_#8}JkXPdn6P_@LLk`fycl|BEhk)?{D}HnaEQ_+CkNd*d&^Kh>V`_8pq1U=0bKK=N{=6 z%(cIv=%VFg84}uki{7AJgVV)7C1A?M6%~>Rru8=SB94x%Ajn&=r;s$}b^`9uK;yS7 ziIM&+()54Q^N1l6tFP_GU1vRe_OhUX545tff|{$hb>$w@ zy4NuN4G2Z5s8V=9|mDseb+fxW!zG`=FHm!+|Zj*N&RZG zQM?LkV=!_4AAe?KL{3_ebJY2J9(yOgXA7OQBDAP``+F*^>3St0I<2yj*=!_fGCu;) z=HJ7^k;hLX3Hg#Ztg(TY2D`KK(*YtF1~xbC=eqpX*0g+?#F(EyJI}TtG)zp#agvg$ zAo#merP?04nzQ{!P+?P%5%Dte?=0ByK)xZbp@B<76Tjpj$*CaNmRy**GIH2p9vh=9 z<>tl>Omg2fatOqQ;TQ@T{(*6UYXYExBn9fv^^WKJ4gqbm!RjT`vcFIF$L0&QmM?yO zD3+?mZn;_Bl6RqXD?yvz*Scf0q=D+hxqPYA_fM=7U;AVBLl^*I`nJWs%5lvdV8i!TVO|rF;PcdopRM-fbgsj zM}H|-u5_B#nmt9{wWV4Ag|81v0GVJAGoUTI=4A=~-_rSKBA4NJ?k>+^ Su;4!gfRYlI7b_Pr@c&=fFE=6p diff --git a/docs/images/chapters/arclength/56533f47e73ad9fea08fa9bb3f597d49.png b/docs/images/chapters/arclength/56533f47e73ad9fea08fa9bb3f597d49.png index 97953e459b537254a6cd5ed94d5f3f2ad3c5d2ec..c8540051585a665ed64029e84a18e12c0ae75423 100644 GIT binary patch literal 10903 zcmb_?bx_q|wCy3JJETEEKuWqp8YGlPL{cTBkw)n*0qG8DB}7W;Qa~Ch=}rL!0g1cM zy?@+!Gw;oPZ|04IGM@AMeYN-6Yp?Z1Jk(Ui$DzhSAQ1SfDhk>N1PUqg2MYuKhrD#T z0sg_TP*YYwTqFPFHWw!%5VsJj3bMMMnR|0S#_DI+mE4F~+K^pKA+m^UyIn=zPJR~?HnI@dxD%N(7YR1S-|B$Y^x zGVe`FO0U&<@J%_Dm4aK_TYsjfr-#SKpW2y4Dj%|&bVe$hy8dq-lkE;B>cINH8sUHZ z>aqWAn4))^wj>Nm8S(#VK%QA_q7GL2%0F5e(a}~7&MO%7GOvS09OoBTdI=W~Js;}m zgnjtHgNKJ_X7`i))xVRV^mL>5)eZQkq?7&zF+@wEw2S+5ePf=I^bS8o!&Z7N;}cx) zB1&-^6ys)LG#f<)rz&zd_~-@-rmlR4%Ewke|wvK{u=g&sJ=ub3wnlxgjChn_wGzq zI-eaqvwrb{SwbRXA`=c&zrqm3($W$+X=i8Ws;Vk8q2lv=Dbms0wY4?#<*o>JcJ_M~ z7EHCDK22+uCZq{iS^S!BP8E0KAR!@{(kwP9NqvLCDI!9#(i?}2vE}9E-2D6w_xq^$ z4AQj9v9tjN1u_C)-7P7rGqQ+ah4a)cL1a$# zfcfFW2TEQu#Ou=~%!6PzrvwyT_3(i^cTjL}IAq?w6)e->^pW&Aq1GwZl-C_6proXn z6wS%?94#!p<2h&B@Ay3(|Mbt6Kr;95n>LM+gf<10VhZbDyl*^xs++BEXo&vo*_!R| zo8;tXOC7(=?JqgUCzMx~6-Rp5quBPMJWQKWvt=lx7g~49(G4a=2eM`iq&L{w$PD!6S~5MW};oStdOo106|Zeb~O2Y-Ecef1Z+qth>>-h+00JiR(A zR(EP@K=I#6&s41q@oedTekp5tnF$?Ty|?1g{N~R_irFodiG0P!&U-W1#tjD)v!%N7 z@TRO(N=NG#^*vn?zo#@6x8Y^xXGhJ3_4`&|{H{ayUUDq0Z*0hq(^skg_0RPrtK`tr z(|h~YNz2esd#|yhL+)p@4?Ww1{k9^F$%neSx|J6=xVR|nlM85U>q0MIYQWOCJABiYWJ-B&@>NPozg^vM92K+~ z^@Y<}tZeOom%?{m#mDb&j#;q`3@FuPMGh=Ei5k{CrE%ZGX_L8@v=X7vS7uWPudO93 zETrwp6qAL4KEgnO@AoW%LkFJ;5pu99JSV51U=sYpHjr~IAt+c2*-@mCwH+GAu)xIU zEb>AO5{jQwSQy=Y=4D_=H(8xdyjNtTF_#fp0QckM#r_n+iq(5?oP{Ee*~@JKycb9P z81NmMo}O3cHe%$&#QPP|50IZ9?@+W2I@~d;vm0{faOwVb2a$4@ZSm@|Uz;HVRwBlx zN!#(+<1XK`J~?<^!q)aXuZIIxN5^_zVd430>;z==z5Vh<5*@u^DaEv-eXY;R!>bOT z!2#3FO~Y=kNv~{08>V_&RFv}7Io`cWlVn(JJsBCFR zVF=sLpcBRs!AL%I66t0=(NqUVM+q~j$m})z;>lKz6K23l!f%6f;n zgg#IBTwT<49Zy?FXK8)C$DIQ|G&%X^#l=OHQ{GF0r#gI$6)T6HPOh%Ekmw)fjUJXid3`i_7BP zUJU$iosLaaC?Nv@;J4k>t?7>uC??0zA)-Zr9mOqvqCP?(R$)OL{S7 zk$zB$)9+dE6Bt9&Penm%_&SI&I`&RrzC`Q##9!mVgWjoXOXuTlJ*W_BPbIfe)D8`! z{;q$gv74>K2Rcz9=%^uOrnr^G!Qu4_g|nV0XKR-X!>?4vb<-;0(3Jc)eUucaxJ@st({LwY!h2?n7Gc_*b2BQ9C z?nD(|*Z}hV{e347534`rdoGcrr2OOw z9Sm@Aa9}1WOis$=1_k@z@87VXEJa}+$=G(Ij@uA(ha=5s*7Lvqi5ZIYlPoL2@C@k%?h_K|_BEPLo zN<)K-kB@Ko`0+=0x;Qu2zpd>aoQ=Sb_!x-M$ozb!)6>&iyu5GrbiZC3K6;d=SFS(x z=>-MzArR2gsg_p~c=-5yFE`=n7|GGm(c6ZG;^LzarKv;_{X;{&k>s3J7Tq}Tp&lN! zAl8%5?ur0i&6WVbBoxHN0rSm11*Wa9?TY2l4XZ6~prWD0r=%EEibB*WtgWwGOqA)t z;;kgewb29w1pYv|zv$a*%D0eXvHiXNol?~4mb!)pAj5da&=4UbBjf4$@rx`=$Y7G+ z$2$fUk3M3ww6wt6gEvPCFago3pvZ~$0)=8>m6W7SPEO|K<1?2aXF_#!bgX#t`KFkd zn41w2i^2l~Q7azT6TnfxnQOAwyM;^;;98gjc5(|Xog;Zc ze6yu*ysPT}cC@@Q|JqM_nL-Dj^LcrhR$Dz_r=@G2g0o5tkQMo}Wf>8KDH8PskDRn4 zH3+jl*TWH)l6utZHhw^JGlhlaB%_RsLE#w$k`fu&s`;<^hoWL5CaUu0)>Cv+-gsS0 zoy1Jv0cUxaD_fb&EsnUwJ@&u3a|Fv05SUDT_Q1!%4Y}@u&&MB|o1;K|$g{kS|0Z7R zE+tI{qx9>HVXxBEH{KH7FD0U*)y?gGhVAKwqfgEHnE&0MVCN{iWD)0JEi{RYimGzm z;1Jgz^>^Ewp#U1zU2Q4xA}cNuyEJ8`_iyf#&zhU_%`&s4MA$X3=nqfY+3%E|y@jkV z9Q6D`Vz)q#j*0p0CsxK+0-dpT>xuge#hTqeLJx(Fa)UT*0J&1rm`@8Yu1AWr~;GN*_*Zvn6tQ;pESMk2NDcC%1(C1 z=sbMbwmY2-K&JO;19G{#yQ?C&xfxTG*888JQfKI^xZT~qDjHG-HsZ}R;fDJ^8^h}z zP22nWlJ|76)PQF{A21B0li*kg&PGO9-l3q7P<)Q5civ>3JxZ$d9KNDrH*92cvGp|n z^5S6$rX3Zlcboo&UDUZ4MfcZucfgJt4hxzA)zxB=QKgt%-K2ckWn~b$S~g;<$kEG4e%3;i;dl0 zs5)ppQ!56$N%ho5?8e><3J#(y=jj71MvG-(#}6^g%Jg|&u31J_1O5H-iL4siLa7nh zZ_GjbG)yczjE?pnP+y$>*wjtBR@@S)6ph%QD?jsMtuv)ULt7kUqn~fK+<#-B zQQ3Qby#LVPIuf5oIGp^!ydO@n=W_3g;bXtJW}j}Kf;T;kY-}`P1ooZF-BEysUr`1F zZ?rDO$p|yCg-w6F6LIlxCGMC~e4Lt2{G}Nz22+$yoYLZ8kuN-Iw$GhMA#`=^m7<-U z= z$Jvo82mtw}GMp&RdOPEe`X)Z}+FzQwS2#Fa%AWWOyH{{CF`+u`=;L7Ev2(MlKPl07 z=SwTR`-gMqw$bEO_Oi()X6}!@&ef3Dk&%ct%a{M`7g}&!*FArv7Bo;qd|L^zC98aW zh;HKEx8$_AyBiQ5o)nF*`E>Gl$LbJNjEagPB$ikTnRH6KSI}LZZIC<6f4TMD7YjE( zU*F!`JXwk$-D*J7^QrYHKW!S_s%pP&9d2LqG&JD=Mh z9_oxi=8U$5>#2{A3-Z25=G@ErTDmdBu-uoVlPbI?19?Gx>q8N1ZV@XmFN@&a^ggL*F;k5gz5bxYTIkTw*XLhT zb3ogq~;L!M<3$e5_l3D^2i4m2!MdCG&GExy6hGv(^FH~nFVxEwzk}W zUV|R=*6Fh8<^IIaq8JjRT6W@6OgkeZ>Xz#(pTeRd{vW~#M6wOgr7|6pkdOa%2Ch8S z;LMyY>4ggHFCL?izUtT@T|YuaO3I+p1fP(QaC2*GLOe(Eg!Y|h4SD6DsjL4eVO()B zTk>5~lgh5}r^e0RSEJYgIKRIi$6{Y1do3BsyVQi$>>ed_P9vjgKMGBGh>=Sd!Hq$_Na@;S+U^zk|9 zInQAkUwO}q`}_KE&d<+5aC*+qMXV(&i}IzpnV5{M3uZ}d;VUD`&W=1+;KQ6_c@abz zB92ON^isS}zd{Gd7(=wu-SzV)t$Zk+tf3+0Se`0(&Bu?#w6xKSW=*%T&BGL8hu<4F zl9-#Dd;Zd zLa(flATM+ulYq8uV<=~7aj^|Zph_K_v&X_KOm1#&==3y3<%KxT&dxw8z?PGj->UUu zA}u`P!NbK3g2qnz)j3Pd5_QOe-!&abj6GH6od5(xh~r%pP67i118ZAbiy6VK%9#TYpCgGcQ-!w2GR`LuL(gTs1RgG0F6Co6N_iM{d=)#})v%d>lTi=xH%EI2pUbZO-y`ThHScE$24F@{ovKGQYaQ;n|p zCN1h#0{Cfl&%cx4F-Q|P`w${WvhB9zgT9P;ec&)KCx;X?`;5}Bg=#69OQY`;8k)2@ zm6-k+w#p6ssQbfG2Tjbhrecxr<@t@4mcfJL82=`3ow_kb1@<3N*kPRkm5ze>`KX&a znl~63qj=5y$uS6MyCEdPgO4px5qq;V?cLop^Yb6uSF{!qS65q|w?^-9J*q*nTH-Wy zGExK*%K_}~y?nXS5-{Eymv(UM*4~#iyS0NK&&jW&4g$;KMm^oX6MF@t8NIqO^BXxs zy+>=BpuOJ!;6_e`3%-W9XHbtXDJ9Z5=BiZi#hp#qos?hTn(+6=7`I&bMSLktrZZHd zqb;C-nCO6Xwp3-JGpxSK>!XaE+!Z3_aXmd(aFyzMBsQX<@qPF24Z_2XoW~#=a=5ri zgFa?+aBxbuH|qfcu-vu+D-yLh^xsEzCUZ+S_+5*G0I9p>OrnfOX% zh|mxL=1|u=wKJxnoJ`uds;chrSj~XsjUO?bKWF{ryG?(Ar+lxzax9Aq*($n`+kIH*iTmm+R0<4C% z(~CW&%#@=~J?7AtetSK9Sn{QX?LoHx$B)cBCSL+U_KuKD&SwG{6G2Ws5*f$7TD`&E zo%r5U`#Qa#Q1_gj>%!>j*9<)h%GL0=9fnB#jZcFHLfL_VwUZSm$L#DSnG0jZgiEV} zG++zlS@ya!*+EM{>W!G&woBIf(ka!os1gGFhog)Hl;E zY#LaLii!XRgR-*dKA0p`>g=oCzt8J)V&^1EtDvML7Z4L99<1`ln4Aga&ii+9^L4Oo*0_2@NleX1{r#T560#aSw=~3 z-?o7xP-)uQv(yQ(CRGB+YHja{z*yXAlg{R1q;E-S*!!Xgl?03?Ti zee}G(l6uj&yU@xYA4U)Yby~mDgnRThTCgBG{(9Dj^75#$;s*tf>m4@XcmZPz#b0Y` z5*QmBFK%r`fFIL2KW|dDLfTP>0SN{xV`5=J*V58*a&l^`c_y>epAtvH@OOK{a&!74 z8`Aol{}KV{uHvgM_f#W@gY{WOMTOJGAe~d=eW#bZk)2^8`3()pYTi4WYyFspEk7}6 zM6M#=Kdxuu;K0$y&?dDW6&3nwf%^7D^YAxrZ(9xKBBt13KD-;TCX*f@AsLr+}`V+tej32J(O{Bi}6pV-ishY6+E!W|3FY{_vrjr zGWWms4km&}kFFP?7ZUaLeh&{NLvF|MOQti3Rr?=}m%n&PNj0vJMvb|LHA-1Cx^!|0({B+*(^*HCz1} zucD>~txaP>a&l;?PBD{=Osk3Wwx0U>KwIXQ0|Wx+9`Y_gS`aoVtJz?t*ztWi0qmY{$xsfSe$$ zxc#}${ca<%5fSYo3%+Q8E9%*jRB35x2!HTE3!e2SV_{*j=NJsvM)qYBVq>G+@|*>j zqXRNODTR*ei{U-~#UTlRp%ZMAVvX=!O+Q2Be7 zBgBh7fBtN^^6&KY0OjqF&r-*Ymb5e~83BRBh0h+2uPrR{hu0LTit!S6GBPsy8Nkl; z129c0k`NU=0F}P~T851+Oy>Cj<<E`w}laLTOII->1AFbk(l0qo%nwWuM93BxNd+*-8 zPg}rS4~~z!Q|_APmX=~m=MchLxgOWyfe(#9SXo&icNF&~S6c`!1+V9UncY+sir@96 zhZj9O(E~6uQl`fVXWTI{L9UT5*bdryZb3m1K$baZ`qpzzLO&`Vp&`IxDX)`;vAe{5i77vViJ--h!yBi+JF408#R@rp`j5K7q@_kI=Q$6LYpD%I3L~U zy0HvQ5=^li6a=Uz5IA=j`hWiX;#(neb&6(dYnyabdTt;D3kJ1%cw{8t-MiZk3oS9= z7bnYHuYMy%mH{O2H@WSUy|?P#8>p-@&^c+6^t+->WP6~bu8x;2<7aXwIVviu?WgT| z|MlUDGTF_W9iyW)zB8R+gz@R=ktHQ1M>gBr+t$-Hw*>_S(R)f824F92qtOtms;bD^ zlrCtaE&aak_%4{|4<9~!cYJVo*pB3>;CkozU0;d%p7O%Q7S0%*<4ZDn`q;asE%y`^ zZ*XvM9Gsloq%7EW5f>L{vhRiBV`yZwy!tJXQp|+~dWilU3c>gJkwb7SlMVJLxvGh* zRkgKUu;*r0R&2c4ylz+NiErK@I5{~HZJ=wvf!`6e-(2M zJfFY&-DkfyKxm49rkk0Q6BnYVvW~9kK~8KCCSl;v1~VfgiqFZeER-?ui$Wo}Q-(J7 z_f4dqPgJ|_PBFgpfrZP1=Yz=2%hTHXPRzg%_xkc2O4;2thO5Y(N6r!kx+hJrg5rx~ z5;8Siz#;G&xDmSEi9S9)od9yQgRPE@VDmFcNzv8S)uCEq8z~J7B0$V>`$U+;zGJEd zJ*Y|R>lncJdunR9pd3^P$L0PVgXRqST1bu@YD+Am55f?BNp&lbcyAIC0*f>L6&u4aeTF={L*j(_jfX9kaHyQc(h#edpCXSHB zJ6*()KwiG{$j@W{Cl#17>^wX@cUv!z%+Uf)^7P`&HD+h5)z9zc*60lg+|AkgyZizI zP@l=CrlxY?1_^LWvQvt2pgvZEneA)+DI;*1q}FkP9(iWSjvl)AQA23s$(x!XyW%r4 zaC86Mv;6q{JBO4ETzUa{z-)J_8W_u~PiveG8cpu{k3k$FPmhwD6$jZu!Lf`kPxh6fCSnxd)THJQ zd&b7J5D@xXXX*0~0Pn29>5AdAHTqDuw+Z24BLOOi1<*?*V}jfZ1X_igi3CWv6aXnx zN4Ha^*}V!CLFK^%WoKu;Q(tMg4&V&4h06r=j~+c@=i=%DGSz?Gfl?woH!x6PWDMAu zDe8>sIRB*`4jGhF)zs8fGq@*036$Y%l1%H}yLU^#A7T2mMe1p&h((vi2@X%aXO4XY zbmF&w!ohpjc8rsnn#4o#r~==qrF1}J4mx?y`u7YiV`Dsf`@JK+a{8wg;>o*bZoQtaR;O(M65rUo7UFWSeg2K4jMy~ zwR8gf{JE8trk2tj3Fg2YJJ1Z??YQqr2b4Q;U*{3ig(eQqq&Rrltmg zfK&xvJz6DX>+`P``4pZfX4cl>K)zQXR+Hv0o*^*=7=p$6KpMav2J9nTAxk;hu(!8| z4dv43`*VC8l9R&#R{Y}5PE-IIW42)Ws6x+5baFg@-Cv`^650EEToU8$jjV|xJwdxE z-b^WmE~tjCJ*fv>_sd7O+C1G6b=?NM&sy-R2{ zl-&&%#X{B%nBxiNemMQ9D@%{f#ZBiSR376zKRsmj12T63cb$}$Cn!h{Ia+vXQga#( z*9Wuu3Gwltab=6HN|{(&vtLh|`n@-h6t~d`kD(EPl8gpZL=MXOAbY_5ej&z-9~JGe z;;ljFd1CmQQY{|JTsFPA8n buTg|%xc&7S%~9Z{DMD3IQ{jW0S>S&F20ysY literal 11018 zcmb_?bx_q|xaJ|0Lw88G(j}pkASEdwAbk$fAR!{sNH<6-(%m854bt6>bT>%d@7%rn z$Ik5T?48-6W;oyP_thKE`#f*ZTV+{n3~~$z1cLoWPD%{|K_W!_Kp%tu5a+GdfIpAn z3bIm=2gFxuV@@;#LJN5#C86Q`Yk%R39^vH!+L4yzhdw;jIr?#LZH@d@$#IO_T;d|+ z&t7XG+F9bHI_x-d^uew~1+f5!oQaROeUyu%ik52 z@7zRA-g16991@OS8({i!DHbk8AN2}{J_=e_S9j9SN*XMC{LJ7kxI6aC%C~RGT?A>) zL=meK&CU*_gp$xa9AfaBAvXa}l1!r07q1P<%Iu3L^OlFsm-|~VI-M`Rh9ZPnjhZik=B(bMY(Ym_2La`N(pHX)f7#?pKJz>$AGx6tG)u6;3F zk%wXUD_E6>4%@)#BEVBR{ZdXYu%pR+*9Z>BA&no=uH0c1yUtV!GAK9?gor zT8j{}o*6}2sKfTk*|eUXy+DPKkv%(oR~iwFZgCVGjDb&DjnBi=H~f3kh+X%BZ?>XL zT_`*}oSKpMrz-G!#-f?CO?v~^`pYa4d ze0b}u%%j+7XRYktaSo8_ZBpS`&}L;0*oxR%PjbeF2^+xG24rO4a z)l}QS0Coh8xT!Ez?MNC-%~ncQmVuf1F)62>S6Uj~-1hjNKzxRylXV3w3IVJ)Z{C>C z*YM1-Oaxvi2Fm1ayGDrLU!k0C475iv$QqfNYHm9BruIbYagyHO-)~J6kamSrg?#@` zu+G$l$I2|VAK3ZI-u^luJX|8nA}c~RA{o}7z!Mlei)6Jsl>bXbQ{=;=k~<+DmwVPy3!F6M;T99`e6>@QF$B|7;;k7+qXOk^qS zpB7?+dnUJ;DgRMgY996`dZRokcobL>7Ms?Atohb+5qFn}eaC#MPG9rF3|UV}hm$8z zKiwH+>Ft+Jd)^jfzEJ4VD_C0KzT<~~n2CgrcdFBhV zBvzCCFk#{!AV`O)YioNQHc|LI;Uj79+V}*nDDM$wTU3lAWsb^2zY}XLuqX1T?AG{V z_tO}Lo}yv<8XAMc!(?=H^`(y68XrtdfYH4eM~TZN*4Ni>aLe-x#hU3BP*)#O~0 z+G-E@1f#3!c6LhRVdSjb98R?j4}-fM|5<%W`N7O=HQo@e-RF%MBJ5Ns9YR84_xbMU zzQ#Cp#1z!cgR!j*`S@UgLel{x#qm)PPD_oub&jqRfxQ?{&pNe}8e(<(W1qf%k^)|# zke0^BZ856%+}5BvdW`(%_SkFtjedN+dMAT*V|5;2=vT)nQ9rJU2$`>bCk1y|%ul2K zHY<@d&rRvo6;%TRn8d|tLP)NSN7BQVkIBoD!ez)soY_M|74fO5i%0GA67VX^-L4%R z_h(TqF5D4|uot3qd$@~$qNBI=#dEzA;3OtY7oZmR2t7KoJGyr4bPCx+#*>zb>CyfDSr(gm zXjm8>5yrbJWia_@K%o%)i2C*uu6DMCC9d|@n0=(g(D_42_%IF08meu3u1-LUYv7n3EE zkPz9D+bOnARp#r~d|EsxxK$hxx2v}-$6%M5^L3vwyThs4Z?E_CR%Y?|ttPR4n+CJoOy1Fa_WS8g6bP4h{~>`}+e@R`IgmyYwL{xXW#nm5fmHyQM^JTb)tx<=tJYuAW|26R9IfabSPOA z;G)h_So^ecR_mIb5Mp4;kXa!E1uzK$X=!O-BNFE3&*t(XJ+Uc;@CJVJzgEQJtL_G- ztwGPrOJFlwA(0~HKCSE*gHJ$!oc1=bt*tI)Ri)GSnZAdISRDKNZ=E5ezmqWXQURba zpDLzGPfy>W3<3MD1J>qvxJb27_t{vOnEth#95jwo9|OdX93ilFwe|IN*!j)`9)_QF zj%5CDqwAOHxw)y@L{YFb4SaljJ3BkCq$Db^G)s1a2C@RJ5*Y&ngCb)PH|3)f6GLLz zv?OWqO2O3LVA#L&-GIQrKpn)&W0=|5u?7bRx5o1b=koAg99_E=8Z?sE*49?)B!iWK zHTa(IPWcRNf`zvRsBc+~Ho3bBy?F6wLQ-8*BsugNn)i1>xO3y-xSPYyf!sGjyh_#l zy}Ns77u|&G`A&N?6H{H}rqMwE4+%1XJxu4XN&#uGHP+1nG8GkIfCMkdC@9w29$n*7 zGz9G*DQP;NRwb%P({IOZrs7w0m>a1f87| zDG=FC-|;v0_V`t{h7z)NJ7q&+yN?w5h9VPc3TCenuYMypjjo_zH9hkJ@hrXVp;1Q{ zXMnce=clf$%$buNs{~Tg9662OAGb_SB5iDNfN0&sx|s%-wA}9qYUJj#3O`s#0{)Qe z6+b0;d!}w4`voZzd@9XO!obTLb0JS2Ihd3LGcxkc%xvdP$pIg&UoPRk*3~5q4}Xn# zaCo{5v()uc?Z*&^e3f8_*y<6~uT|C5d==uuwOqo1g=(vfqJs-=`LoRi&X2Wg#cp}M z5?WT)$0iCk_>7&^@(Htw0R$(Y-HFxrk^4>E+WwriUhRo%?KW^N$zlT~NlneikFo!O zv;|cVqU012X=+avUAL7SA`Fs>tVXj|R>sMAqbf2GXl8BA-uU@%90wgDJGQmaLncxT z^b_Zlf;(sNLcDKpd*?oQcr>*g=bzFgQ!@zeKYnXW8I+0N0Y{gjzwRp7-G7OsPegoo zNgaFVssBRQ6?S=mH(wKlN|AVg7m|lCWxiwp%|JMh2*9;n3!n-2>Mm$XF}HC|P{WDFIa(%B zQ@i)AztI3kV<`#F_2A$>FiiikS1@|iSe1xe0iGc-F;XJVk;=d@ z_Ey>hguWb*iHP1IF5(Flm3x0e8UD+cb&jrY)qb1w#kG!&;dXT?AwErOa$h+9^Av*S zwWk0MR(i!F%GhN-MMvW+=Eclbp#S0snd(9FUch z^8bBh#vPcML#eGL`0#=I%VSFk2_%poS(R`xF}=#{sRcgH=e`wD5+_nVCW`2$;(YDp3By_-NXA%W6=Co!Gk{?qxZi1i^cg zln;07A#>FwUfS%YY&31<(N8ba-}7NQ?nntv&g0# zVdUlNFDOb9j&JWagT)>OrIQ9(tom6Jm3%z@ARxQyP2Iw$k@g5nC z1Z_fo2mB%*=dpf?dw#!Aey;uD15_p=8C_Hq1+lwO$eRDP)5d-js#)-Hi*CI6R?HP`oTfRNwGcubZVPe`Yt*TP>@R!QYR`Tu&Bi1!N z-(gA+a~G&{;?FD;qBrp&6p_X+A%WkdZx`0RJaSu%5S*qB(< z8q?{Y&7bK$BZ6P-&6w+T9$=}5CCasD4S!+I!otP|WCJW6$vY z`*)C}KGtps3JA!keYZR?n=i`{DF6MU##M{I{;;@sR>(t--?-AIz%Py?slAK zv$HcTKR>aHiwkQ{_;U?VZR*#$l=bdhZyg>+C?yK494yptO%^5Up@7g|$0oW%uc9*a zu0Rt8I6U)_bm_`d3;2*~Feo7`?|9H4XJ==nb}N#oj~-31@T61z`SYj7?V9)Y_7+6G z=aA2=%n$_c0uE+vZ|1k+-`8YBl$4>MVB@JnlaqV5qo-I=S&2_csp$IDas!NTK3Ya% zR!IgGW2SfRCmtlP(fm+Ma2*Fl&}^6;lXGu>WFT{2&2pda|;XD@oJ}bsS(mb zhBkIbcQ*r|Q;oaBL3KOo#jU0YX=!gqAqYZ6!=W@{Wz6#fJP5mXDI_f|ZQ0?8t1EgQ zzOgPvKzr-SdS99O2t5%gDIam%vuA*P0n`+kfHfH~-pBqU;`*GNoTcUEk{>=WM#F-B zVPIgiY@U{Zst#BnHs0DzYhN4(0zjXfoFw2VS>jPshYuzRXM8za1UE}g+Z7Vj*eLq% zVn2k&z4 zp=#ZL;%BoJ>7ua{X2J@6;Z`xoxzwY(nS+XiTL%4@*UO`t*@V7 znr`*St*H_k08x=s!$pmVdA=OzB_t#mjX#^1$1WAS^y264Y3Yf&5c;%%1@fr|FEbr~ zo;t3!bfI<`a#dB;*37A=`B*l}xmS@7G22G#&=4-wOGhI{>eL^ivMk9io$50IHhg@1 z>^c=tHm%V{-JR9hN^!e=9o~(>q&@)6VlqZ;4}iTMUF@?_iZ{u|wog(3+5w7rBaZs0 zC}&O+uQFp4-nMp(M^=NrJLMTLulJ8vQxy%TQ2KO3lZ@L)2&`Q#s5gSe+8i46ERIHCs95($LjCbF}ghX zY@L;g@9=4fmLAS|dnf4AZM-qeOV`_?!?R*1q+MZ-2>{FTpXhq)th9?N8%(LgMTVmp zvqmskq-B|lYPT%SdsuIbk+1L7sGk>aW_B{*)$Bn@0b;tOWye2aI5oA$!=mleUw~SX zH(1my)LBRId>9y>5`+s=fe|h|D-N~Grx_qH!$n-3QUF8KwK0ISHDU~{`}_%5t>oV} z7vRN!L0V0;y@KK-dwnOU^VR~hfXUj%^9p9bzH9TOd62~1=10c(I;w2)YxA}B*Z+Qi z4Uil5Cz4`|itZ%KJP(*9BE$(m@ir0$3{%TQ!5izG_MiM#ZYz5tI8?$HfYOsBN*Cxv zh4c@IceVK?;uAfmK3r-x5vGa_iNNg+Ayrvf8!NpW3^Y|MfFnchZX&#V+G&0L|CJ>L z%P%y05|50yvFi|FQkOW1YC~bsvxrE{XU0J-QWNw#psEy%`n%rvTCQj6s`=ptiG*Z- zS~(z6N3{=;R|AOIE*L~a$OLT-J_u7~>DD`8y!_(y?f0oR@T<&8XP_UQTzC~(y*4*C zHV%}c-ySk~FjW0*%5l8#1%@ClH8cnm6@N~WHCv06X;Z*?(ER)=nlP+qM^vi3@ z!Py@i99ZXQSb;n4K3Zw#@wjzX{P{d+-w_H6l_4U8AVM`DXC7mdqcSisU{MN1?#B>m zYHLHkeEH(IHH_5S+6tIVXvX--u-&^l$L()`jxyStE)(*+7n$3BmYJD}3ZxoY1fydy z##}W<=W9>?P7z|%;)N6#8VZ)kV0WBu`pbu&B!uVFhGN6l)5k!gK#R!X5%%@%QDUgx z!DoSSZDy_7t)ble-OYs1g;(T+#0f)nLBO_gu*2Feq);Bi%!I8q|YwV|+J;mp?V^05_ zTh-Lmc(t$4iiuJG`}fZ}M=DUw!V!2KSJM@8^TWNcPFGz57#!}an2p$UMg zh+js>;VESu1FxB_wvFiMkoqO&PR<}3-T{~#?LL#sgN3cJ9Ncf;zLi=|u-0-uq0JP_ z71XW!@#9H2m9W2n&1}o!BK1?1WR2~suV23&ou7A(W+_A^Bm@Q#KFxP@lbqJH4Mv9` z(gt9y0QKV)5rK=0;(mND`ddrB*I}hib$Vt-f}LayB(5tUslKY7M2`t4>#eCPMcQ}5(Y`ez2iTPl$F`Pn>GfkDj=4@XWb;+}tV4?0DCBC3@m5o8_ZhRL$ zn)xR{_Upj%@)4ESAuU0W-T4kNkdQ!Ff7;V?Rl<=dMc*Y$zwc9hNY@e;A^Wg5eA^P=BFi=!m=csb+FVvF; z;86WL^2x+PLqmJhF6J-}fnfYb^a219m8l%Kpv=bEteTUESPIw5>2+wQx>RS`wObP- zH+N~52}BN}(JM?(FFBh3j_pl>W)Y)qm30)GxU`|T`e8RPct$ZXvF{`tI&%H}{fvNS z4$DkUO`YFma|bZxjX|Re(+^&g;J+CWVyuFmom2EyGx^!^Oe7&p`gM*`e>XPdjgPP0 z!se1qSoEjLC-q%i>N+MS-i=mRsG~!+FSP9I3T>n&Mcx(d&d=)wMMp=2fGwBl$_y}= z^YsanL9<5~;NuHYv!lwvaR4FZ%FUy88<5SYDMQ_W6Tv4X4FIrt1>j+JllQ+963Bql z)4q7|q}p~##>M3&$U<2qC1)$!+vr?eTzm7igclbVphl8$aNt!_SI4KK(&x;6Mca}n zXv@l>`>wy7)36Ea)2B~lK)$i#$d%cyQ&v!5t1ez@a>pbkB^7%4a`j?=E>eyWiiH*P zqH%T6;@zo+-jt#HIdMZngOQODggPQ%d|cy5-Umnk=pdYqKFG?*AYUEHhI%+|DPow11moUmPxRxx=13c_OEx(mjwUI8vl52Y!O8W*wcRnzqFE)h#!1raIj(X znQ|s;YilDL8~u+a0RaJPI}-&UTD<}i2NEPlC8cXTPaOtY777d5!35j4Cf$QQJ+GQQ z?s~&mh8`y;BtU_@0U{X^#2=f&Up+^;bzwpO#fCohiya+KZis`Tx} zoGkzx36B5LUI(x$HGkT~QBHd4x4dkGD9m_x@HdB2x@UfyA#|tPlRhrfdqQ9%TaAV5KXJ0xw7^Ff6QdBtzEzVEzT;)2C~I(x`FT*U5@Cx3!I^|8lqr ziaqe^XDlqOh>nKK1AWK6X?#fLnkJ<%KY!(D!bwkWuOtxEv&02$7b&?68lDNN0I`jn z-_n=QY*6mOp{F-(??x@I@Uj}-AG}*ycc=%vR#8zAafl#>82#;uT2CegrWXu7T8YpG zt^jnO0zH~{9J&){;2hgQ6!@vN?E>l}rLf(sLSQs20LFk-k^^2RLio$U?WMki-xMbZ z5_7Ad&|(Wy)h`@fA4(E_s^66Td0@q^w`don-j>l(0{4Gs5+NiUN6im+xQ`w^dZ)~# zko?tGk}g{*k@;?Bex9<#pz+^ZMGzwFuTM;loJGJl060cB7yF|>&~T}-FflQ;-@k92 zm>?GXI3KB8T~Sf^+S!jB;{v(tNSu1<(q^fCaS8&XObaDiFet_h$G$n3}?uTf7mjIRb(( zqSNB?aH+&H5dx{HtxW@Wj(C=miV86pq2{5W;lwmio0)CO_kG?OM+Y#f4XAW0yStdc zU(zyBf+QlDk8SbuAJn!?);yqS7g3K4{`1SPlVOCZlfW7%q z*4xBo^2fz8Es~G($Q&ZDt&z`jahy>`qM$4ST*~$XFhMyZCZ-CPqwRr#j zz4>S+dJW(#R#sNzQ{ppgYH&DsW{ z?Wi|Ea$HG=z{lj|RXA1XoTLK?fZ-&Mq$CbP0hA zdH?wFzV@0moWn}-tB|np8fd;59qL3!N4IluU=@t0t$hK4!_nVAgxb2g40Lo7rZ^@` z@#P;Ho6F(73_x`TFCVpMA^=iF)Y(^%|H0A~JUxzAAL;4oMJ6SM0ayxRy5^Jm)pC9% zG_Pb)*Hr+0MpLYGxMJWUs`>Ur;cRyb7x+dem|-MaiQ@Uk`I=F;B;haqpwsH?{2U}QTtLyu z(+m+&QC!)6dH0M+@)^;$_LWz(crCyI0YT&sI-q_fC6xhf8mM$K8|5-GGQfU(L9Ua0 z(c|K2(pe~!U04G^_tkMX%`y0C2kSvhQC%$vS)Q-`gcuzX65@D|U(zFZ&~jBFiHOhtk~P%j{=tldQ~6~je=lw?2qR7cv4_i!wp|QIlH=MfK&wW@CvJ4LcFtTf($hgV;5X0)$3vhJ=TCkJj@cQsIshfLa) kSsm%7{9k%P10T@*s-iIZ2Ub3T=2plXX=SNmNuzK70d;#7n*aa+ diff --git a/docs/images/chapters/arclength/5ce02cbdbc47585c588f2656d5161a32.png b/docs/images/chapters/arclength/5ce02cbdbc47585c588f2656d5161a32.png index 98e4894a701adb1265f44e11b0ea54f2d0029f39..140898e0bb3c3267afb6d1f6eb028eef468a1bfb 100644 GIT binary patch literal 11471 zcmd6Nbx@Vz+vWkJ>(C9-DWQ~fNeF`Uq3cHo(%mRsB1%ekNh#fR00{x%-H~rq6>zaAu^^SJ`U-(Og(%5MYk8#~EczO%MBbtwkEmRx{fJt?-1{=mVDeqL*r>k$ zB6{MG;g#PaiBc4|9A=s?L7k!gv?almd-so;eyrt%A0GCN9%=6iU$#rm6&4m+{Ym{j zbk<^0={@$+I`cIBMNyIK@AOk@+HPK)gr`spQyZJ&q5Ri5Uc_;0?{vl5emIh1g??w2 zL9o%n$J@MKlM%^alwgs7pPa=hP#KssJOtW>#laE+)p^TDA5!%(3X?vBL`wyO{`(3E zF){CZlTk3h*K5F7_2J*WZ%SI)PeF?)1q55B2RyHv`u|~?Y;ze5Gt~d5H~t@wiko86 zf6u0^0-I46(^(rc!~36$`#&E2zwhIJe>7*%9XX`9xi5}xeRDX?>v#ppxXmw8$Z<)E zA-<1L|u9=yXJ-3&#V15W8=G#9pvJCMne+r^!+5_oSGej^W z<&Z_2m$pZ9P((axLK732ulyujTwUW66GJB^maBa)FBO%Wqcg>QJ7;Gp)_Y^aYik7{ zL4BM7??l0=XE&te7jBwf9>Sn?6By4*H$?S){V63c5G-%3`LfuEFyNEs1M3 z?oFD5**?q!re7B<1xZbY8Mcry5+=4`VN-1HQsxCdsAFJ8_NBfU8O`gH8cKbEm!}Zd zks<2YV62juXD+i_^TPUfw(HjK1}_Bi+8X!Mr=~m-_cgk)>)%pRIv4#}RMWje2pHwp zqIudnz;Fbx5YityY7;0)6H1MZjj$FkK5nC?km6#t7|OpfHT+LGm7U4Pb;_@7CSRjp z9+I#L@E91Nv;^D}DkpQpXMb7~5D^7;2BS~A5lnol$jPC{!Nny}EshjtXQzdK-I^*l zaB_EtkLJk6Cnce2YHCKngDSYQi;Ax9MMU5aloS+7;lOD7?+gQOtcJhQOGwae{m$s# z`1O^~VL|x2VIyHv(}|iv>FG@w7cI>uw46C!%Madn)Kg{F4W*>~^vs~Hb7n?sGmTn% z<@7^t?%H&=w5-#5@3V#GOY|@tim3K~Hwq#Z8wCS~T9IwT^!m|eL6Q$#|Jvmkq~$^+ zRkI{M`rmTrzKRy~Lgc#51-8SrQ``R`zaC-X@3=Yds|xVv^|?F@o|&<*nf^@*F}1~l z>6Z$UY3G>B)FjCvZ)`vq)ajGBQbTkrji1u5QVtDEb~byWviWNLDUP>w{WXkwax#Fo zYH_mh7G~fX6yEn%PL3%@A@0Kpg1lm;YM3&(ONQRuVx(i?j;KQO3!h&D zdRO;l$MP}>J3DKGYLPr}{qn@Dthk+Vh1>vtPB{fuX_Muky@ZV)KdC-$Z_`!#HM0KB zypP{=lc0P&CnhMy%Gz_ZSgLv{n{B+~>)m_LpRQXEruR2lypBtX{O@w3kx(SIL<(#* zj|5cu6`;{@av=w5u4m;?$|j=&Y^BIgG`{BZb##1|1OLg-&v(jDt$o+4&myOU)7>#Y zPemB{MATy+IjgeU{q^LZas$Wx*`N4dS>ALV%-1)n7U@5(mynn^36-KXsW8J{X;_=A~XeKFNtymJPzhU{ckT;cPGnu1qH(td)(#e9M87L`hWlK zEmqGv9^`577#fP%OCZh_C(0$!(9o!|8DpRp^}tE`=944;@^o`pcDcIJ&DYlaru3MYLOpLX< zb`qE(V24Ld-6p55j&OQ&@ZQP>){>5f#sZB-qE*B9yx9Gc)iM+Cg!smW2LHIbMe#JZ2ew~vZ<_01;ED&8 zXexXhseg64S&*^Z)$8Wn-|6%EUg#{eq{fu##yMrYMtwZD z*_M|WNlCG*yCT7RTDBS0&f;A9HRybNOs%er1!8qH793`kYm7j z^6GWZ)Ym0trqYd4QL3~NUh@$A++3C}I5P#Zbf(61PVvoY0mvh27HgPKkh%0XLJO2F z%?w93Fl+ zKBoyPHxM%Wot}!du7MAe88|*S3i@|XGur;3SfRN}omCl(itRx!CiWzbR(wr_1_DkY zOxW{~`gf*ya7Tv#uob~?g7)F?Xq45Rof6XIc}+fgTOL+cbl1(H&N};QUOx2)swLa; z7>XC2>j)5-&?+=Re3-~;ZFr;W*xr2!XkaD46eez}dt%*@Q|=8V0*(MiNphFDQa zsdFq(fq;^7z@5Xk3>fS2`FUtu9Db2ENba^}AU}*3zUjU>-_yHik)aySlBBCNZk3Ud zLAHMR_yFhg=Y;liH6Nm*v8PLl#f^-N-p|+J1_T6noo?c?{?{YyjEpFIQ{}QMDmc?6 zHaMc9q76+=UCYZ^U#}lG@^0$89<0HMUcbhs)18BonTrdDRW-e%)^6I?e+i(9A|^sD zFQ?1S;BdtLtn1J>0bP-y{(hBmhk9KyG1{l3$!#&Qd-MzpgyTgcL2qG9G!8a4Xw|UV zXV}GAn5%~R`a2JyUC2Y3;&k-%2(RPpGZ&Y5#Tag*vzPs<_lvCt7dhKfX@U+br}gx_ zPMn5@QI286(abh;32{l!w5Rt%qcB&AyZa?ZMY*|=>+1Z+s`FZhoN_*wjQ)rx9vnvf z&k^57XP-3+DL3`UrPjXAg#|(ef?zaUtu%Dt;I)Wp6E&ypQI@;Q_IPlS2>rSm_)6#D zOf6^C)TR_HD;_U;p7Oa`<=3b`H|wi1)BK!vi>F&r+&0We zLe$kr#JXl?u7Q7X;a0#of8|G`l13VP)SIWVG+uS0U*Vls|Tu}Mp zVtDqeXjMhUw+E9wATO^ulg9O}({pfOW%*mh>6D9bF5ppwNvp>{LH@AD`o+!_AExI6 z!Z$Rsw2jjKWCMfCSd83ttEs!E2Nu~7rH&7aDiTCI0;b_KA$q|JQXCwd(sUg*;*Wjn z(h3Ubz+-M}Y<=-(ez>{2=h*zQ8kT?`udY}5oZEgksMjfJ(9WT)k}H3Lj@~^uNEJ&t zb}v+G)*JJbiG^havEIiKsRj>mT8oqX*HLZJSLH-6LwD$DN=6u1Zc3$wBoVHUW< z5a0}V_xB|`>d(?%SaWlmw8bn3+{T{IXsIZcH~U>bKU`{!-We})&3K(*VQ&60JNqe@ zaZAd7e{TR#39xzMOgH9I;DeDvLn`SnUBf0D9tVqf?2}9QT|w~i@hdb-eu7O9kdlT4 z1wnYZ1ciYg+Qx!hZ)*CqtGl~kS_A5ka5gFKb^O$hZ-g`dxH8U-+5W*Ih zRwU2E1xzX`DjZ7wh`gAH7LQoq=CC%umsr?#0Yp!M8Hj`-si+J+jVBh0`8`7Rl%Jn? zFgg5ncw&p*_BxUCNJ%qbySwYnR!;uK%gffxJ$tZaDQ1REY$zD78nz5H~={b+@eP=zMF_VI=Br=A`~ zV&V<<%mzXb3kSfj^n83+#>RtS0=Ch@H)L){5>Rw}hnsEk$M30ytB}NfX1aUBRAX#D zvTJ^-8a;|Q?iz_e-5}uY?G0-dr2J`x>XLy~k~38<+PSnunpeiSEbq*wQ(wF4V^OF#rsFR1~w0;rk z?d(Ldva%BPyQ*410E5vWm%lh)kLv?fL^w}7wT>C}80}xl`JfmQo**}l(EK}57V3eA z(hCTn(uiJ!z`iSso3dxRAkNOul#`RQU=9!_5$g>-y&3M*_m@wMj6{a$CqI9N;^|IbkB(~wSTN1m?a#0`xNcEC ze}0GrO7Mh))N4rlLjZxQs%E2VXfY-V3Zs|T5d0GjClavu*KtK#JHW@pB(^3(l53@+vI`PC_SxAPfV*%8Mn*o{1fij! zU2yU0va;b#9dnc`NZEFznDhdLc4i37Pt@KJZ0tb}R^n*X?XhlXNyz}};V9D8>Bw5L zSZHYI=)+~Rw5w~!6sd5aapMdMI`T*s%Is`GcZf{RLS+l3;FE8Hs1?TA+!sMe9Y3w5 zJnE!YbvLJ`f~xtPP7Bu>z!XkSqY6GEZ{Y>_8s+*(O-&u+MJIjOPkvmx@1fnDU}tCl zq=jVrUakc`+?2V&EKTey47Tts*2Q{ydf4y`!wLgQR1u|gaYq|xfiunZK2%sGR4%T| z>!Udb0L1+Tn?||B3OZ z>s&Ih)frJyy=R&|QN+Z_oHMY-pni~)=azhggj3RNT5G166PW0D?`12EgGWoXeSx1+ ze#D2^&&}j2CqJz8$I}>Y$xZ~pxMXCyL{w7OM%+2%f)y)+&QGv9erLo(qZ1o7<86NC zDR|}@HFIyz#M73(N??7)Qdj3;Gg&IGny$FZ!3Z|=Qn#vH!g-sbjFD;Z?~5|@4^r7$reSl}-Wcx3k_5}r zVsMXHpV~leXNFl^0Ex|TDwn~CHwF(2JMjTcg^3WHxS#54hB1`xI-33A>i9!OS&l&M)-(8RgO`hMhmV)nQw9cPkgS7( zgHa$W-QoCMqM?JJWiz&y%z@gwM~;vQ@BECEA5OEMF$Hk5w|&Js&AwJw-x6p&_SCVj(0U3GL{R zg27;Hc-c?hw5Z3fO>$)}Fq(tl1CJ(S6cbCYO(i;aU+qHukF|C8s+UrX}Cl^mJcWDE7M}IpxJB zckFt{<=%XyuK-2ZAcZC~y-9tJ3OKx;#b!@CMmmk?uCA_?tu5Vh))oQ{icomZ=qR+U ztqs=d<8HqpYx@h3T*rHr?OpTpX<+q9Z+FdiN;HesK#o;ZR@QWv$WICh4>$XFcbzCf zm;8BGqcnw_iprulh5`e>$dz?&Wh~FP!EJ}o&(H610R#n(>nA5CV0C22R|vY&q40^1 zZ_>H|H{zF)GMXoW>6^gK0{)}`53X}8Y9)Gead++iNa($=8d?FqZCUCbd<_^h=lh!x zuN!_N?Q1BBvI*}_kU0bs`k|VOTDqNo`|9N$@_W&Tk zn9QVu{|ko&mAjigy*be_DNM|qtKG6jUOv8cFn?&NkxESLo7Gi5vg+!b(*E$`{?o0? z_-{8a(le|dqk{(BWP;;D!#0U_JS|&DBpZK@e*3hF~?^5hZi!H||lQdERRWd(?I%0z8gAiO%ZvIg;*o!3Q- zTNL4`FK*TA4#66EWM=ZuUJ`9COM3sMTi7@_2t7NyZSWBYrCsQu4`W}_v^loU-OgPOiJqP9G?Z`}-(6c{)-3C2azwkGE03;=8D z9_E~e6w{UkHc`{ijJ0^X?$1;Q4_C=40sIDdD_Elo^W*#x=l3g#oVpB5Oiuu;d>_t& z^?s>^!l=ouGYAOPAce1a>tv+TQK}fLn0Gjz6C6cA~z_^GA3^_SDCxEr`v|Pft z#l)!hrYpPmrXPwK#NsU`OEH9BzRb}qQUxZm%a%w?1WFn&FTYq^s^VAD_E)z@TM^OG z@RPrT01ix;aoip{{rV!6E|wZHG=yII4fT4uH}*q{gg@Fh{&!sf1LJAx@x2?(Mk-V_ z3VCKgT53!Bq~rcp=+R@2PxH=I8VdfDeQUe6B7HQ@i3!)z^fwuGR>OWOS+p{76;IPO z-|>x&kB7&N*uGy7rXy5}=djjhy}vt^-uUa^u|4(`tO7boN(3S2UExgmroYk?c~F|E zouL6^>u7QhNRgEM#+}A$o1;?)u5cPq%jAuXN46Fo?p;f*%Q^nHBc|3uDSKKNNN;07 z4RP+HNJ&mE%{3)$W=5c2vmMgrpHXF@0QK|B%^!b@KQo{`i6Y3k$9!@0famPouITz7 zaumO-&zgD7bpqvEY;es=uh*ABi|(*g zUIELum9%Oizh_{HdN$lN8EfvE7l}z8NS>guiyM(iT@7(6*T*d=3+_Yt#?Ky*xV_*_ z`Q_{S=gc;IAYdf#yy8%A*)l0oWD1JJ9Idn8;f+@e0c(gn-^&|oZDv1rVYYo`nZ_8W z>ldMZVQHEj45)(68U^>uV6|=o8oW6@Bln5H=IV z1i5*6jjyE{TYN9^9{D#QFeKh!eS=mw`OG;CCOrd1PZy9x8dPM_O{ug&qO_uDO1vzYK2nJOWXf9?SXPyr}D@3NKFZ zn>0bEj41tIgM*zl@0Moj?5T!^h8k4Ep`pS&EPfJnfB^3pNHn`TRTEiYC!lSE;Ng9` z0?G<-Kxl!Ju?i#`@OiKwOc?6wL?uUdh8j#via588!Ep zP3q~Mo?qCL>U>X$>hI5R^OEI{njLC!_1(hf~bu0e$h*< z=q?#}^pm-Q@Fx&Tew&YaKLr^Xx@fMi>40P~MI&5=IXfF@HplNHhcPB*s|^-pB58fL zf6Y8qnbqsgQMtVpm6LV8Dw~Yf*lR5x2pCTjkVo8r9s{MkCjIc^vr)b zfCQ1h?&jw9l+?}`9~Op0Da4KZcCN^M_#0b`_t(h$d?Q}o&qkSDGJIIroDc8$%N5w* z|Apy-9y{83B%*^~0NyN&)&W<_!O<-MwM#fg< zAR04kL;(=66~?VmfF%Vt!1?3n&()2M5bZLZvBeHKyW(NYnbMTh)YU7*`qka($XMS;9N*;h( zcW?mZge-bU$Y|(0x#r?~GgUo+q+~TT2p1L?owhz%8M*Mp6L${UF#r)0Z~&d-8%On+ z@v?c~76BQBF<;+jadS2XBvY2@9kx(2CRnov<&PgfJ|`rc9o4@Iu_24)WP$MdU15ZT zIRs}dE8ZdcDh-P?nIrQt%5>hy=uEj%3p;6*RIXx<$x-U5Kur5Tj{%Voo%9)HY?{!B zbH*n_hIlAF{rd5t5vh#Kr>>ODb)}k=UJ#bI=RKdu$Od+t{5`z7-fA^0Vq|39!o9nr z&MC4Es;bHxoi5s2kMtOy3HH<+UmF|`$D@fzdKMTKH^SE9pY|d3MRKsore11cVd=1} zE?L!&FEkfd2!FsNqsfK|TO2N60JbIqZtfGpI%W%&cIDSG*K~ynIhdPWo2w;mXjTiV ztR(U{T!=-(rG&7xEf{q^6QnCI#^B|Jhi+ao?Dd{QAUIx+y#VcrVww9Wt?q7+7cYp?MK52xw-@dFU4hPRELL=a?t>j#fNw*F z;lgLptHotR83dAN=eX;aq2SrjF0r{6^!jhg1RVj0mXnuf;O55r%Bt2|^KSh3;=mwr zEC(lH&8cr{)=#Fa%b!@w={sV@i4AS zq-Q87(=Ahwktw{%R>~+y1l~V1JX{Tghja;l$|BYKkVvBW)&6+g{*jRq=Y7e4SGdt+ zYC+A-;%K-(6!i5e!Q^`V-=M<<@vNq0d3Mtk!Y&(62nj1Jj+WT==SV`3PSC;TY#RwZc7 z(NIj>w|o2rX5hoU(3dY4>)T@j7?`O+zcZQvJIP&D{SJkblk@Jnh#hPjSmN=?U*-Iw zB9Nx`R!*Xw`zJ~<^BJ7vny$VJNBRqfq$bY8XU9PExMfV~yG#AMW6%{q+$75GQ{&TS~GjjOXcUEez zAKM{}58!zyf&U-1L*yp`H5&)yuyqmK@`SWBY;<(={DK1Kj3{{}Wo3ZPIiDa)$F5Bj zr+9gLn@tp}AD^6*suNCXe>Lg$$gHI_>5cXv4ILf6ot+&yIeABSH;SH~9wdk^;0g(sN>C04OL6-rMyqzi zkrmy7h>VO(toTX@L?v!52v;yX+ zsVgg{Q)}#^YJq_dw!>LwVk33}zJI)2W`h_xQcd~6g!AP|rEhu06pXIR~P3Ktg_QwIk; zkgchyW0;kbOy}$DArMDL#~2z>(w|ntT|Dh~A3^QR$ijjqd4KkmKMw3-b+tSE5#%5v zBZEp5RC5%--A=1&YI?vI)s};#{0gwu{$4W#OmdUAGu>#JtA@D(hnwq-GeNP9p z`!I@#Py((Oit(hXuC5p4Aj^Miv9#ipnVFeR9v;03?3&q?m2qHLcV8cRI6htP+0Hl; z1kkp;AhG@~1DyF2(W)qr6U?4(i5|ArXw#IOAP#4;uPIf(3bCrX~5ax`1BjsVhG z9R|bj_4R!WR(N=LRn^rcyhc7gA{v^S0F1B-sm>x^zI^F2trQ7ZZnM#BY2N40zu=YG zUWdxi>r@!B|D^8h>`W1~r@S~?jswthWqT|S*m**#+V#OZu-5)xzd%KY1WXf3Ow1%C ze|@Q^ug_6S8!BT~uvcB`ZV+&HdwEo5cquP0UsYegtk)TgN*8g*0y0o1 z;3=F=Hk1I-l4nrwK+b3Wq{;IL56G^mVllC?fGB=bV>5;f%muI}Xie^WVPF6FdDoiC9!R zK!-yIFOF)u@Q0orxsA=uH6YAwFEsvi6A26q+z}fb0k&o2GL1Ojn?`JpF~-nI(*FAO z>k7~}9E%y^0C<2Vm)P`l>K&fI*}q)PnS&hHhKCD{jQ1@YaE+-SmV=`;?=&Gnv9Y+1 zI$}ZsF+ew+zkXo?7fV#70&N~3?tBps&wIxx_6vKpU|0%!*x;&P$vhpL+8Oap?D%!%d1L8pE+#Dsu z>1atDTx2c?f;I~cd?4k44x}(tY|<4#+5nO(AXKc$;~)xzsqWrh`I`qIr7!@6$Gn;x zuzK>q+pF!Sxk0-~&Y!;kN#f7dJK_@&5zWlb(sOcRgRUY@`8e8GFi1v0p{UD5Hrs1t z1Gs8~#{uP|!2lM_6s*YW?uzfdy?x@hY2a2IDy|Rz{2zOofX9SXSMxn0NKmVQ8PErl zc@*U2LR6%IeEI72YpgJQx=+)UCb|*29^%Q`Cyo3zqf$5C>`?{VenY@`bMo?nL2c0q zC^8d21gdd+04^gVBNG!71oDU^fIb>SE3WTA?Y%x&`Srw0EOP;Nl9UziN~Q$-?D6zK zCl+YSSp{bDe4TSX)8~9Q6o{JIlzBfcwQFhb@9#+}`Ua-#BY&*p;^HP6U()U6Mg0O? z!K3v6%ms?0t}YpX73sY0Zv+7R1up$qukDx!LrB6dK`)Sup@D$`6D#XFuvM??6K2q9 zg$q7;Z2yrB!-=OYWaE7H40mT|2Q;cN7#bQvAWcn8BphDe%v8XZo;mOQVFn(goZKh| zP92Aw7lKX3-2)DIb#E^kgf3Yw()F)u67jz^moU43oO0~dF9!tI*9)1On*&)0J=?3B zbM_TJ9$wPyI8qR3EL#Ol7Q8}2z#2sp^i{LZT)j^=BF|1a=*q-iu?i{IVPa!rgWO1S zd9)neb!{S2A#{lWyWVbq>mFI@XIQ9RLQ7ws@bu zK2_fU=L%>{Q%6U9(98k}%FLwQKRAd_PmcjQO81{Xg!i|X;YCHPXn52Zk4FpcKxP3n z@IBVs+f(IKj{yX{zk6T+3(%`chf&FX#{)Lg6*NmN-ocOQMO2gu!W(esl`a`*T+($m zscQnLj|&iVrL!dbkMFKs8>vo0Fo{0L$0Gp+y`j085!g3q;!6HRCmA3HdI9s@_y8*= z4BD>1l3g2@J^Fw8-+BONekve90{B&RpO?xK$!6VQN|Kc))d{s0eRwcEJ*7*jMNs58 ztw_~4F4MmG#{U6OVBkGY*C&=h)DjjJ?vG`3^aEKkaqmCJE1eBatH@YbSP-rIm%d(J zD`3|)Ggb6r;^H}F27bGvAi$Xl1sL542R?`Me)5#C?nf zoSbDKm$#T%YFjUo5fiTfdYAn9eI0`08=3oC_O46OBaP#gKSbguK2ZEO+$eJnn z`Aj-%lf+brfdKB^6chLL>#Hl?v9Yo5@G`tNabz?!!ysZ5$s>TOg(~Q-uyMf-c@I n70}kkfA7hSdO(*vai%n+_%;YSlp#v5ROKpUOh5h)x|w$i5T%06CM19w0NZf z{D*G(T231BfcTTsT9^cZJb@@kOKN&#?9IFD6Nlem9IABDlR6OFReIlBC!%tjQc z^ZI}P>~X})&KN?VPRauQ)l%hxg}{TNuwWsinu?gPvK&(>OcOLxVku14qErGYolaI3 z7SpM=oKOPwn?v5#jQieb@*=%@b{n2=9whNA6k4L!X)piV6Oz3l1YOwwqs9M+w_>n? zw;Mx|N5AOR~PUIGszd;)uQW)|fUB}BkJ8K2dTlTpTE-6=!#O3c($IM@9^ z^6)4v+G*9RbFj=p0-xMkvh`_!wP_V}XQFp?HC~TS;KIF3=Ln_kiV?@q5N0)A#3xP1 z)bi@ptLS(g{d=NF=%ELhqtj8(M~20f-?=_6+iNkNCw<+&YdN(oj9bga zJ(es~&OB^?$n2@JdwA(_iC6P}>|f^Umvp_Z*&CeQMSm1p|J^XjV04q=3?U{qws$a_ zd1*0Yg>p?DorvGRRo_`y(6O`QWo2bA&G4bCWWV~Wyt8B+@=Yl=F^PWM%5c6pJuhbo zM#iomk@QT(v^(sPqqDR3WPFZu%+Sz~iM@S=o3!t)g2E9~7B@MEVNVDa@$X6_iM93h zu0MZ>h_H-Jkz?cRQd1$KG$i--AyTD-6^BATspWYwd;6IDzniwk?CXVMiC zU-iM9G9%-9=j@EhD?=EzO-kC@SN4{^gA=#TekcGuKgrB|c$gY9H8qckMG>x1=(i@E z=`Ph90%glpjo@gzM2)2qF+SVYOig8ynEFv0FHRW|OXVN(VkvyJAOAz!lBUg+|FA4BFG4>obfh$6N_K>OUhvh*dK_ekAhn^vt!N_lk(Q zrSuUKA{cFS=CGZ=M!CK9Ir_!X?wF}uyVbAGSU`!4f}*Cb9%MJ`wtY*ZAT!@AVs^1F zc6f5q4H8YGa-lBBLcv576pEE@w2{BAjtr<^IQUsyo4mqri_0+^Xi z2M)o&r;$eSZQU*nT3Rwb-Jho-ex@8EO}8veT?_VZbw5}T^YHdIyE)(6oT{Rq+=5$= zeN*zdI%YscMw%=xl96zZSTIigDh1X_6mfvr(BBP*~ zgp@c>+pwLTgYlX2uB_HXrML2d`+dfvIV0qSv^8W^c?#p!*Ml`1Mkbg~}%(&ttv z^~6iMV$M7glV*1J=WV81JlVUt$3W^Eauwz?y*x~_T9_j=^1ox4EM}7F4cW|G zF(u5)Z{_m6=B}~7gH3LY7fpXBPRR5{O-qw-%zVY5sTi@KplpcjeM1Hn;ttr+w9ZI* zV=$QAPE5r=*fUmm(WopJSOu!pH!)+Q7LU&DL1voX(zuls(|6{Xoj;Pi;qNubdQ7W+ zI=y~%35ikD)GYY$`04`M_n0UWl`ipmM<^;fwu7^e0GrgiL+iHQ$F@obH&@lyfBGX4 zF_hxyAwB-MbbqF956k^x1a2|p5<~5cDz$7Q|8?8U#`{hkUniwl%|+TWl@p*a_b+BvOZ284S(glCNX1A(T?$P63W}OADNvFd9OhpSbNk~l{aAg6e-N$+Ks5b%jlG8DV6|C`CJUhNzubz&Zo5+2CZkeB4c|$pZ zGA%7l(Btq49UWbRvW#V|%_N?Y-xY(!U~1=k8nDJN3p=~)ix-fI3IiM}DyoXH(f5e! zG@>2>$;l+c9{?C|nV6}ySzcNSI9ly@&Wsx7;o_21S0|Es`BF_)wL<|9tfG9qQ`ru8 z+tan3t`6PNJe7^75nI~Yij;{abK>>~AM*?i4Gs46DJtIelw80|E-o&_J8|>y*!U~} z%u$#urhj{L;NOuc;(pXi-dSz(>65F=wwt#>=LRc#v#{$njNP6T85tQy_L}5Vb{%b$ z@|_~)hW!HDgMTg@)5wPqB(}YQ5svl@{iiIFH|N*Wc80UZC%L~xo8{soIRzfjM@O+z zMLqc;FWA@vTU(ha`TiJ!&&S+YqutC*h2qHdXF0m;7INw{B}-wjACD_Y!E!uC=<#vL zYPUN9x+McvT7l>`Al3_gH%(f4z}l5x|!dxKj0c`OrxHhvB?Y5mH=CF}Vdz9?kbXTItou5O$T{Imt7X5_DgB z!Og~2w!AGIwxfxQm+cVZd9qUFWCjvX%}t9Y0ALy%Mr}Qxo)hXcIHG_}%4o#I`)&^0 zJI?pgYlUgIkzOAOM)$8C!L9$YobC!?KG$_ooqkyhFl53jiilnY>kYdBF=Ns+D-!RC zGA2~CN4@ zuYACJ6MK4kUS4fv6H@*hqOPb_>ke~E<*{V%>guW?%8tTX?#YvhhC41xS$k=AgSsB9 zQK-D!6^c{V|IJj&dM=H}(sAn#11frn{{oK5=adxEuI}#IKPMqkSjA4cxw#QBF(GLb z2049b5)u+~ZN5~`lrxO^*hz`i^YN&KLjb(!=fP$~Ykp`oEM6n$%JtE{(x@0ImnmYBD(jZHKmy(~z8<-oQC#Kb6p zFCdZ9ccETiKsZ@TJ2`RX<>i6jce}EzNd#lyhh=6`+r;;Si#0+0tIW3|%*5q-ABL2BoM#ZU}M3MbUZtH&9b|hV%MthYq|VJ)LQqtMUd6VSMtgM-X>s8?K8Pn|b9|gH{rK@?K%JdLP?ZzDev6KH zMWfK~Tok#;33Ys0WbFzjJSs6^Y~)U8jWr+jw{QOh2p&By@^O$Qo#BD$GLW=&4ri7U z5)k}wWswd&-;)|`yFtciogI0F2JB6$xG%-|zQLo!&DiV2 zH;8XsA4=DrYV&2V7>-#ICe4#A)iU_~^|k+F;pEE+E>Hu@6_gC(v&0NWzbRRQ6)01M z-vqRJYv&hj{?$`b=?$^!-I)xXYW1eJ7$k7a#2!^WS*QM8yYs^5;cm9}cLCUePD~7` z)f-nCQl;dO_Hum1Zbl_G%hpX%|~0~cj? zs)|y~yD8WA2EX)ieHlt1<1Fo2z#FBce=rhklXve>#f|PRH452f91kHaK0+9-J1h={ zGGfM&6Rm!Cls|s#aQO<7aE$h1}hup7<-gCY6e;N_Vyc-j9 z(^ML@Pk2xQ7vOVPXp!`})%7{*H3ddGDDpgsb98l;G0j__%ml|!rl6=85*>{*Q|r7Y zOxgu}wpZu7#iNCm(cKaq@I0whE`iNNS?5QFW&J`aoj=7I&jdw8Y+4YY5Kadiu+{6V zE1Xu#7d?FQb8qzP3uRs*Uts(eo<#YOh6cfq@G-;4+^vOHktVmjsI|3swX5n}umd*m zq(PI5WWD`-gL0^qq=$!~R)v1?Ih`;I<60&OIXOLO0;0*sC?3E3+wAV(vO7g6;=W(@ zK@@Qr7!t*9wl4Jhcb>^&g6FzTF3hH8W~ICD5c~3hz?+)VnPDwP%gD?eEqlweJKwDR zPf$~FK=1YI*TJo=V(55}L1ipJXESVpf*Ql(a0^iDLX;+pjG`j$UdlLKMMVX?s;V;( z1tYl)tXb5{nPRl)?+;mDUxy02q;@%)>abK?$V7L+eXc%CO^uhI(ui#b74@znJyT}F z!@-H}EKrDcIxbi_j2ACyY>7BGw`3JxEK zo#lgyi6BTv&pkaYk50FIaiN@9JDQksmp|x-cc>1V`t{s0>?Nbg&AF9+iw7mO zh#zWXPpVgky6DhKT59SO4h~}X{WTc(F$R>1^+4t4Dw2=i)(EZZEGcm4O!qCHhApI0 z=&a(uD)`nIIXwdd9{A{>M8dm4*x!V#Hp+Ziv|_fW1#T4LK9m&(%>nKAO}GJ|*s1#Y z(cIsjJS;79CGBWN#BuWRh1a`ty6x&9Auj*U68m-ZZo#bgm^$U zM?jcqe?gspjcd!fT`TrJ32X#Pu%jOlr=VgKkA%|K)!7*c%t5mx@cs`j$Um!x%TFtuVZsGPo1}i1Fv5ie7>Mjs zR9Xn55_YfqV(&|~iCr2j(@Q*>xu6OIj2xNhxc|CAx9%2_DJ-=5JGZiNID8Y5ScF+z zY_qgcAA{H+(V%91PmI?4qC!R?@|!ye(aV&?DyqF1*Z!M>F_eMG^>@7mX8l$B}2G2zSIXZyg(iF>@k7y_y*ciCGkcxgWqoXT(?bk=O`Z0#H&Gv3-yp6 zz3M$1-axS&ejJdNh6eZ&shv#Cbf@nYUq6R_-Ws!ns%vzs>0AB3X${gwAm_b*ls$X{BHuOy85?;j|D6iSp5=Si( zmX}A@(1a�uyO9Kdaq%-s|1$@yfeH`3o;rRaI4k<8sihP<3c%Xv(#r*4HxfqLTD~ zYYwNZHm=1bMupW=eSLkL0vrc9Ioq=B)l*;2$JnAD$&W8=TD~jr++A5&dGHGS^8R#9 z#BB8s)t%N$cEFT3l+qwSH(|ae*oQN?Cse}*=U zn>_)*1Wibb7of|?$iR4cHTT}KO7Z5_rKYB*(}Tvt$;pW?5x*D)u${D)7TI13!Gx7d zl?&*60IQQwSklZjDl$5%hKYqWdf5b%8Xql6)zQ^mIT?|VQ^d`j=0XM7LDs^85i}v7 zu0aW69afJoF63lo(eLl|x;NAaGE)-ZkB$5&kU=d1pN1~IYHIraePm7!M?$i+rQ7K= zC{~X9a|$!8u(HyLN9&T3k^?`5mA#*N*^Xo}DP~@ym6eqtmQJ-MB?g_Chxdr&hYvM% z$1q)&i+xg1V2zL0ra`>os@Cc)#?aD+Ec%Ir-QL>F41`;aW{HhL#k|)E#JFKdh^ zD==fI1Zc^~CJ-B>R;lVVVw7=<(%|FLFIuL=V-c^@!A!V5aV+Y+_d%2+x1o!GR^J<< z3VosQM(6gF=O~dO_=)mW7;HKIVsgw`_PrW2C|VTeV#ckl!EpBjuer9&=s0R`rWh5_ zjX3>9;pgMS2IVfi*449J8QgYOwZYUoR_@Wq()T8+A0;DWEnMdbohg^w0baj~#ND7& z)SBIqM@K&`E8flB#YuxZ2NeZHaYE&HQNyih0{-_tgCSQ3J2fl8>9==xj(6958DidF zXy$rgVa)TZ3H~zDY`q^DUK^X6S8!%b;E#s6FO(@NDOI^_sfUDze^10W`LQ)tcrFwjd( zkHESzUxwo?1HLO&(1~ucI6*h0)@Wq4FBYYnQ(i^IlJ%)44q!Y?OiUsoBayQ4;u0u( zW+p(MBA}v*W_bN+pknL^9|r>iT3uZol*<$qu|nA2N_GJO0R)GwprBwI#z09FDNTTh z8W9`&Pnc94r>c#%#k3b&T1IAhC|$rdoYDH})2BIrRC9E4TK>-shKd;(83>ZK%;MyyANQ zw}kYYVp>#XTOJu2-|}dFv;3=>DDz*RoHDVCdu!bGIoWKv{O&zzk57uPYU&uCamW}i zv9(+*cmr3m<|-#8*3;7){8EdV&L3)RLx$QBOC@x9cVU()-~gz%1YPA)ZN95Z)Rpr+ z(Y3YPlA-BwYa?IM5EY5okXk-Iak{xb)&L1|^=@SM!^Fj*f1+af2Sh|%t!bt61fcve zG~&t)g=ahE68D$HQ&Uqb{j^E7Vx_t=Y-Ds?PDmCbhH{^Wh%;5`^_J%vk1nrmkPKT* z>+SUI>ii$PY8)0(X#J+L0Wd>SjirIVIP;SKlJ+*I0O89)7DF@nD%pgW>*<(-U*7l? z3jhFBUP1$pr(NuB^EEOV`{rCN4#zbM>gxJ@;4)d)cJ}zIM-!>aDh>Sa9*jBV(>WI( z59?kbmyH%~ODi>(b+UwR?KWW4{}z>C>`s(NFKtmk5bA*E>{QEL()HFZ^}^0Rap;_3W5hS^>3uD(6|~wPjk|{OD!nq(+mK*W2m`0#oT-xNYjWT#6&FxludW zLHiXNlr-_1qfOovn&92lur@#mC}eNdkzPIe|d95PrvHF1h3mF0D{F5V4@pb zwunIAl1H}T-x?Sg2xx)GNG$!rJPpG(wvg;>jyl`vV`c3+B6STxf48UEBXS=;a9^Kp z1@qg5<-PqrvHh$BEiZoq=tCxs?(Q4Hq*Ap5tzI<{h=X1-yoh5r3HB3Sb#HgYo zhbngmI(9?RU#`L#gqXL>%UoDu-sd10NR`M~Mqxpwrujl&z|o?~lsZebL-h+aR)Mkt zQkVfM+TZ+R>`ZbH5^>Yy<{IV7M7!x1z8uh7z8#0RRIkQPOpwVd-r^Y= z-u*7u)AqSEQ3*XdZ8Qi>dEH1KU~e8(q8lBdfvYncTWBp9 z?iXhJbpdpkkdlH%Ma9aXLDAhMb=5T5Pm@bB0$<%)<0AI__33#q;F`oAZk;PMOx_q8 z(tK0Nj;pER6%-V7%*2BoC~ia}(*F7GbTCjNOy`?i^$WYB3RSXAfkvUz_<;dz(C;c7 z0+%Yp>nH)OZ*X@wKFA{CRug3ufK&bD`pRMNY-jTD^t1<57f#=6mx?i12%ndlV!$6a zv~WD?E>7+mjqaSOUK`nK>fsqH1C%?2%OdW5BmBT4X9&7;xaA0qVwNZ&Is3a0 zK#Gz7N_2Ux90(ki-(g`8@eazBz$I`gc`+A#l2H90)CPuzR)nczIb%q6CNXkyav+Po z7xVs>(U@FpUfPO~jxlGFm6bM0)!*`&98BuPs5TRq%UfeNzyl=n#|}d_>09rj9*cUA zL`7XDAJ^(btM=X$HME-5GZf!91k;(aff`FZY_CkZj$4v z73Jg>lgHV%5E+}}O=WEtEH$0AnJJc_JGb7O@zbZTBhTDisRPExA37%|QP9xbIla$` z4%h4#s^GuvX}*06$jn@Uw?DLn2|0@{ZC_)p4@RPZ_)~}sp;=bI{bEQ)EKXZw)P9Nv z8T#V-!eX7`@=~#Hu5lW{y`MdJC-t-X+E?-?Cnql?OcrZg!9j+CJ$=W%vZ7_SJ^uc~ zDmOO1^|PMeoB!OuR)^>5y?qPPK{lYTY)*H#LXw`vcMr&S9JcU6Ah^wlS%7BzQ+8JM zzP(J&DFr^^zC?UYD4d7--}t6u31uyWA1@h#+#I#d#NK z{4f@47V*1rQWT7)mE(WEPJO7JK;1>*x`hE!1S4)**x4KwpEkQ9(n&|@Uy zX@Wt-T)D(YVj0^H2N&zJPIzC~FMl?|TbioUBq0&)T>D{U`tK*Ips;QryA?wL83}RD zX-ssbgUaRgerLX1JluH$@3Gi5>B`EtT;~n;&ACRNFZ{mv0NXV9q+x_Hx1!$P-`!jS zY@V2TCjx*Nm&+#Lm*GTr*Jt6Q{)HB&jiFCrwk4#!()Vnyk{I1|;Knp$Y^fMP>Wn7m zSOG+F1K8sD#Nk&TadfZMLW>ZiA~gQ;ves(4ngJt?I8VX@bP+K?l){2ZlhV*H&NpCR zPbGasOdn`FAK{+c;=u|m)czlH6aT}}2E>TP|8Xl|Yr${DjDZ}M?@`v1pDpx%8`KCL zCQXWnz?7Dj-c-_8S|=nV3;l)+g)ulW?STwrEmyH7i z+gPbKhiqC<5c1dV#ZPssSg6RdPYB>-mBK8;dQS*A0ZaZy`DAyRXZ`z#xY-xp_g%bJ zzXQ^^%+TqHBHtZIcpitm|5FgjR?&1@a&tAUOFNP!W(K@8tuK~J`}*-@g@GAx#u~g%aYclWS#HCMpl?G-ZxGD`>KlTz~qI%Vfv1-(nrTE@Q{c5tk z(Wt{?DH-1Yaa?*4W&)M1v;3Hp6by3Z3Sy#Sc(|gtOZNV|SwEhjkkIn&)rp}0 zy(s7*;dZn9(>I{{5=8gTF%;zj5v!rG5phGqhkw|Bk_rO^V=f#{U~g}~IaY{Yq2Guc zX7bKk4!3&%Q2B-+{Z~s30)v7%5KT(Cd2STY>DcmGj{^*)#n(V$EjwM^=5#Hf_YsPu&s27|?DvY3OZl729C zkjxAQaiXH4pziMOi~<7MT4-pl*<5!w7nYQ~Rvit^%{*Y}#2gr|7ja%khli?{ zbx0jkGc(*^O;ezskgyc_CEp*P+{c0mZ>!%${t-(`7jWPf^Ex#KqCD%6kvPWH8C|<*a`|Eu7b`oCtG75% z2M1UCc)N(o&`yoak2%4mf_j9s8HJ z>)hAd%Y}!B7X!#&9-9d^S-M2cpOq_hOk~llL-{2-=3|q9$uRM9^-kOV}+jg%Tkk8iJ>*VfiLW@o7u78V)+lba>s&t9rU&aVIB)hlek4>%sL zDcs(kj8re*TpU#L-hzZF=(K|RiQOOyv~4hYdK84p>U+(LM=Kta?RSkuE$q^iy7JKdt`~ z7)3%PJuqE>iX8Z7`{w4LonAw%^=B4eVniT- z=mW74rpXd2O(*H*#&18@5Qr~+34y#jDHl8d?cA#qO>=Od4gL?}EiEmMV6vf1zme$u z`}dsO-0H&j7mNP*K=Zlz;$XS#9b4g#Kk;O5vXq;>;<&R?)J$7)| z)}NzWz>LLGM<5CUG=oeKx-(h%nU=NaaGD}4(JmT%RVi{~ zZ*8jacnwsLkdUhV1;9&yq{Au(X$CR>&eF1bY6=QSFhGH}F9(7#6;Z!i zDCkO3$dU^SpUu?S;dpy{H#9UDUmh-7&DK2!16)fhD?#2D`^%ld7+~ImbHV??x5jpw z2+WfpST0RFVn{$Dy9u@1M!xQfKwZ>h`wo`z8?Fs|W25eF%{q z_=~^4e@}n^D$vNPP4j-jO}bfzzSC9d!~jjx6yy%T24G~5^w%%?MZf;c4ro+7P`ZJ& z%W6!(E|1D}YlE7bML;@nI@^Bp^}o)q@*7tQGwVJWTH@GWY>x+s6eI@-T_UDjJWVfX z{+%A~eSs%x@GAj^3jphxdWTBxifX6i=RBUE>U!K>I;C9w!jU zQET_})zQ-fUJOPrEva+((d=uaK{=D9M$v1PHEcfXq(1LH@ zx&{YvH%7AM{QSg0Ia1Tt$Hl?HnFBL>3=9m`tK7IKa&mH^BT0#gYVU=?6+i(jfj$5# zVE-5q9jBXW@=u9?tzMY4M4f2yt9XLBhM@Fxsw`2@Ab{9FT1AEcX_mXc2j z8lWoD=}xR)UGrH03~jBar@J2khUGKPv5tA93lJPoss)pE@MDm07n=tRG|ea zNuv6UkcOs$v$ERmX!~y|_N-~IXHgV`gfAGF>pK0Z>HfA<4+FAbEjHiii~+hm)Q71m zGoXP5e*gY`SBadQ9HefH{&-pph|ggs22{k2ZKmcuing{kE)~BbzYmyIObPo%CG_Fx z+}zyK=4J%&*6!|Z6wsISgGMAVD=QB4q06O}ZBn3_1yW1JhtA>^5I)WB2VJA1gn+_I zx>JjXav`_hU$ZJID)I`WZG+1IzvTme1)>|1ZoPdhnDEe~tOVTC;qh_M*RKqqeP90f z@6Fc6Ti1<8$@||x;4jSB}=0H#hO~^LrdFAt4|~`QzkrBXcg+VU4%? zAlN}=A_VJ)0{;N=BsMKAZH@I`Y%m2V12PL@mIcI@8Te920FUMH5^ydsY^Xu;FxTju z(l`1pO$emYA4$)cSXhu%+w4J!EaH6tKX-R^A)lY0A08bc^i2o^_+(#$)9UYf2cts| z(L;}xS7V>;qCq_Z#~_0U0SWM3h~!n9l9DoGF6Oi%owD;n+|PA#3{=EJP|uXfX^XHrs<8c4!m z&MYj&j;J7qZQE*i!QclA8zs~~S)4J4Cf00wH S$_7Ku5Cs`!=?W?1p#KF|nIHE6 diff --git a/docs/images/chapters/arclength/fa4c587126e8097206b88d9ea51974ca.png b/docs/images/chapters/arclength/fa4c587126e8097206b88d9ea51974ca.png index e4ed18bb5c2940253b8cb093cd47d173ae52ab0b..4f894fb8076fe259a6ad4e20e0e63953a31279a7 100644 GIT binary patch literal 11187 zcmZWv1z1&Ew>^Y(3eq8-jZ%`5f|RtBij*KJsDyy@0YuUO=@JC#1_3F_ zH}{SA{`cN@&;8`=efC;wuQg+gF=w>)O=V&NIsz05MXai#po>CbP$2)~;lPy(Z)a-Z z9~^6SWd+n3@?Uo2>qHdlB1%<3R?jzMeZt?A!VZnQX+eg;g2(ZQ!~Xo><6!La$-*j^ ze7hX;F)u53UMZIt6RF0sNk^Z?O2e@dv`b^<{o;1CqlvKzZ+45LxcU}CtUdaSEERt%dT?`4swtaT}XU``sb%Eb=Y4(9#8Vdo+A8Bm(WaZ_te*E~MVKjJ2 zFJVPT>Zow>&dS%gxH$cLVeqxkE@)SSghCE)nJ65erR^Gr1|EyPtd$&IIROEMT-5DN z2@Pz_5Y3@^N#zNbeo6Rg2Tgx}KN%AfiKwV(lmaW4#fOF~{uS5MrB%qu$=!T>lr1eU z?yZl%F8aQgP(D94R$)had~(8X-+OWGN8Nz^%gIr8x#GuZ_MOSxTtY(61CG{f))upZ zY!ww1xvpOQGB_lu%|U6#W25A6Hhj;d^EtX>CE(9|iqpqn48H`kdhfXK$VeY_Ms;RF z0%>PYk6m3|UDu~iuRA3c_D0gWtLr6qCFFM*N;5s=tZOOGU^=8?IUG*i)YL*Cw=LrqPMjG8)< z^u_^-_rn3}#zcK2w@J0--r9FY^P1{vs>SY?t+|RZJITNI_BuRzq$1@bNpf6&ei*HE zpvZiBMBVZ0OYVbk6>d z!rMhon3%-$_G&Xo`B4tM*7`K~_Evpy-ukjn(-GtQiVA+qk0__{YT{e3 zZw@(Y$;n|-aT%geqmBNe{R0EUAMD1Mbaiz%H|l?}Uc88+<}vME?RoJgGjvE$I2tS= zRrl@Mrti`Du`G{UTbM*bk8BoI35Vo5vNNB-5785uh}Ti&1Dq{JoYWAIN`mlrBmMol$lon3Erv!$Mz4RtY9Bm!P|h59Ap1+$X-kx4 zXx2a~u~p&H`Eu6}e+q0cWPV{;F12>j*z9ji*qol878e#iCZ-h#yRD~J7I<>Vk0L^O zczcT&rt{mn)m85b)5UGrk&=?`{qB+W8A~Yl8h1Ribr7Q48)-UCx_jr&lGF{K4O8!( zr9PgNfnh=d0&BJWXBRG9C~pqMr7y8r3b2gjhc#cnetoAUjK3rB?C593pWlnCfg5X5 z{wKR5X-!SiN1LsraW19#Z!%|x1pi>K_2;XXyG_f@EiB}I_`n2~e{l5kjD(q)`5`=@ zeJx4MbLmOnZotEbe6Ew%ukQ}tN)>fwYG`QSyFaIL$IdP;F) zx-f%uAOpBVZr=!7n^OV1j;1E@c$pOzAwIqp4-bz9oI?_)!FxtY@3IgqT!EmVpz8bc zYI*+bvHaerbZCPOp5iMN{Bfl7V?& zV+8jp`zycW!Yh_tl~?f`@7fS3wSGM(a`6wUmvOEKWYl1 z&CM0y^L1OD`Z`_{s-w+^D}8dQx!mRxgt$d*e-6*?Ffu_?wCA$yljqjM#hQ3TTp<+| zblu&K*f_}$^E5a2dBfH6Tk_RYYF%m!bW+KzBeu8iqdvBh+GT5HEs*p`iVNEu9}qR( zWju$>-Fx%)(&80CbZk@5s>lAXpTCka4_fBLEivdb5}&Q;G%^*Ad>>Wsz$QKId49w17U$>B)tSbxIPyo3G+Mi*tvHsueBapXN?8|v z&BVAsW{p!`E6lO|doedZe>NqDo4fmSPgY)ylmGJavI>ORI(Er&-z{tKntc^#NNHVU zR8%-Q+EFWx_yU3k+3y7?H8oxA9q-HEy~}}IKpSJi-ObHSlGT)y6r0_Zq0#qu2~o46 z&P8wD1ecbUZY+W$MYUVQXKr!P=Z&QIn&Hms2q{FG(WaoFNU!cJX;RcEM8s#$o@pBy zISlQeL#=s_DHSU3`W2dkw?GKjF)*;(e4wSJ)z;Rw#7^$InxA6+2m*zEsd>r1@6DSe zGqbb18(Y}N0{r@(oi5 zc6LY-)GwX)_48W^9G?7qb9JUYCMquOoDGqxsw&RCq9I2}VoMvbz1}ZVECP*ceyn}3 zAZKQFJ&o`QX$gDc0tOkPcP(7+T00~{$YUXjF;rf4=C^NOxO6^x(k7O_*tm+|%a<=o zBSj@8khyVbZ2WwBojQkxE|{B}A54XkN)!+5rBgpu2`enTc1vV{3$a5&h`(auX?r@3 zLHZ&55nyuGr^kN|ex04V=B@7j`pLYzyQ}?H4~I0XGJrwCQ-#{md8B-SC6~X_PA)3k z*x0!DwH9+?GBFX6!h;9m9r28W*}DbKEG#TIzbYp5@7%d0ASlQ}hDSzAi=XZFaXOY( zP}|V(jmf+6avf?B=aIe-OsLOy1hCJYORI`2$mU~dll%7V8@gW5`?7=tU0PZiniO(N zJqC)?;2rj6XJ(8DS>?WlnyRWSBxBUnfR^+*RECs4)$hd~+}=a(0Z5Ti+1VG@M$55K zJHx8#>I1uObDzl-l$Cw_{o`_w@#*3^mYna7n|ydEZrhy^{p!^vyl}a^8h#xPO20^U ziYqt6Gzx<7g8-&v2J9K3_6O7Z+#Wy+{@TD7 z>FM7HDEt*6%7hO(R`taUGT?(pQ#rc06!2jnf8|Lf>Ql0Z8NPY@_Dc?XyhsEDFT%b| zRZj3QuHOEB%nq7>bgUOx$q*l70BY5u(Z(g_Ce`{03&Ht*dOa-?V?CtIN2}lbJX7VP z&OKMlS8a-A{A=hpMMZh8_nDAg$Mo{@soTy~bM`%R)c^kCF+xKrjj+vb1b%4)_TM{E1f0>qEwo%XzY@@B5c6 z>vB+%5rhj_AN+NHx%;K$=I-u1ns^e1c$L2KqYg`2TKe7i6fyGnTQr(36o+``zi&@A zs$2CVL3hG)b(ouo>3Zs_h;Jz3Bj^Xe*)cIOYh6M`?MVzF44{`g;^<*2bPwsz!_{>^ zAzUtqbracr_HMr>rLb{vk>C#yy*?$FXR1mfPhv4Hcq$;-Lx_&p8W+ei=>_+kBH<_nlai8Txu`rQaDE|o+0!&|q7_uJzY-{~?uQB;8vB36ShVAQe zTOSjSWe|S`dlZdK*qa&c4(m&e;*S8Qx_IeQ7@SJ<-^QXK>hdEvJ)Iu%cwP<9H;nPe z|6B*?k4H-St1Ppqi1Ql;p+px;+np3%i;h%&8`$g`NVEKX`#*yc07}EW+%T_3wtkU2 zOfB(e3?KRG_GkMpkQnrK6$LW;6hyx#YWZMT_mHpIWo1uVULiK-DGS!}<_#WD9MhT> z@sR-!fnoiTVQ11^KQuqTdaKr@l^yoJKiakFB)O$c0q)VT zq;$IfUeFASjP#gztce^z6l4O8R6dM=!<}jLg>=LR$X{z^bpT_ib7{3j4lmfaLkV@ezIg_D#{vjSm}BHlA;F8%(##O3YE?^Bk&CSia$|k=LI;Pd zrThVhp6^V`ufh^4JvM%bV`F1metyDYT4Q|+-r8aDU~ATQCi)7jcmY|ll81-jhr<=l z;}wIZ8CVq9j0*kFfPF@Ud;t`Np~R7)+;^6AAd9(8L}=#nvCLs4sfEM4QuX!q!N`d) z@`I^P-ln!zQfUp36i*7UB2t79Y2Em9X~cPy3w9I*xhXq07c=4dHr`0d-z z?h_H;^7vRt*1slk)_N?G#F15F5>U+a-Z(w5f!obrqHho+=lVPm;{s43XxBaU_H`z_ zWIgOW!|;`?hmS@_4R?P1%v#l7dzAg*LuQ%;E*Hxlg7TZrPJV4o`u{Szj<0KL%Z@^5 zfE&k4pDb&^rWkN&M74^0 zE|Dq4Gek!+`9;Aj1n@c>j#YH^KEDbYpD`ip_&VJh$?rJGF~+%!LCM}`B_t#?1-sY= z8M@Y`od=LxuF^52DfjAvnj_TNp2qATgjijZKLsbx1{evnjN3plnk6MxJTQ&fc*&{4sLEm z$cS(bVE@)Na1LcwtvFmJ)kFaKO>1oSmolQ`0HSj6@I+{^%R4)B)7StRk5FKh1@;|a zNHkjS!`o)fU!a-Z;l)W3($a?Wquz(~`}gm_k^B-m=kEiddH($QMRshRf-1U^$iOao4~ zkBI=)K{M&M%60*7hvRBv@ zqRx+k9IonJyTD_HZu6>da9iqS0XW{3B@O4#04LLvl#+slV(R`ZleK^VWMNU8he{EX z@8gRhd%S)7*092sSo8cu=hDPN-Po8hDmY8S)|2)Ow`Za$sL8-@hO>hP^7qpxPlAC~ zxJ|^jy1Urh2fy|B=y+yjLO3MIxVk1DlD*p$YqUf6aBOm8P||nv;&ZjjnSqB6iRR08;(m83}TmH_*Td=@*Uk_68eR1YF>4Iu@AVO_ir)zkUmJ4LF&Z<=0sq@677S z$jLDPY&%9{r*Z0M3kXqay13Zi_mID9&mmw?EK{ZMFBISX+Sys|x9jNi{k`ITss?)~ z;PA=&K*IeJAkRRilINmgla~HzS-QRo+Rg;CtE+3Cmx#-cC;(EB{k+fO9A~r6j;|;s zFs*OayqKahvumM{KG~s&XOxUU%o&n>s<;PxGM5o1pz^`N!HbfT^h2!V@67}2TG5%A znNiWvvOhlrNM5gT7+~izEJHCdF`++QIylE(b%Y{sRgLkIQ5Il5@jUVgeDNO^L04~rD!iekUHGOJoYL$Fl_@D~1 zw2KN01!{5olE7+xeSL%Zal+-Sw*4?eP_nYJh*eYZnA%syt6Y-1eLLxUK-Ky$pxDh2 zQ|qe&+73B1{K##H7JITjfkNO0a?O0~CG^91n20Z3#6kZ8U5@SdW5|5~e+) zo0v@p5hWEBotf4}GUP<$a(e9geR80K6@p zK^zl|?|7OtNJdYO5`57-@Xy_Y!$a#&@r+UuV!VG72tTV_sUBb;p8h;zjX3KzSFc`0 zG-qbMQMOehA;duQ}WOg)Ul+ml79NM@oP3J>a(?Ds4KI08H)2tx|dZ{CQttTt`R8LZePT{=@l>-+RAU9{X|vD=*ZyorIX?xPEw z`ctk!M<$Vkt9VV?sh;xV_Y`vT^2)td^?}~^ENz1X29EX#Ho$>hND$J_RB6x&CopzR zS!*l~4UM)jGd75Up}PX2dX2d&+NtUd%0fgO1!|eWISaM{1wJjKV4xbYiy4m(RI2?oxFmF!y*#wh>XXuY>o@&V5M~ zgS4O&OA~TgJ=t>Z^V5mK+mOX!FK5d?F|E3#6ZH3in2E#!r^yCdn~vCNG@~^;K_c#$ z1>1X&q+^++#&TcSb!RkJIt(Dxn$yc6i^gUY!r}e<_cu?Ee%XwcS((--pqr-`gBt-h z!R03@DUs0>SFAxnK|7prt^Y!lg~yNkho)QiwoA+*r3{x^B)oV*hs+7wMrmYrb+4bH ztc1avWOQ#N3W@c<=*aKw7?_#GK{Q&vadB_w)F^KfIX7fcw1PN@Iqcl?xZ%nRtl;Z3 ziAPtXQrmeD2ZS6F$j2y3qNAg!w4Ny(G_$n4svn6m{0g!IF)^{!AzgJrT^;Rsjq4{^ zVi`pA(BH%7$M1|-SuYE-CesPr!2>T?#`-OzsCW)!Lo5pmi?#8Zp2GTxFn@1acIQR% z@;N$kbr3XWTfzxRf|_3_<<)L05#=H@H)P0 z<)1dEBblw+A79AW{aA&ZLY(L%%7)0n!C|MWl(8a)?`=tmB0xj{(M2S`quj27jBjOS z1)?z7>1xwy%&NCv5&*-aUrh-oNBgsNi<$BdA6_>*=d|b~p)uCYCScWq9SYAmZ_Tv( zZnabM!}myWWjE058vv={;o~>C}ddWcwxB2B=UBV ze)IQ_nxCD&4HX$YO-_ygmUO(85Yz^$FNT4E!Mkx*hUxVt`*)?}<<%hJPlLeP`t$76 zhs(GU4@7@Pazuc9Ny_r_y<_T| zAxD^7HeWZV_$?9bzM%G)9svq(gFaULWKkyf`P68+4LD{Cf5yAdc31)ZP8GPTmM{@&xC)ErxOT1Odxf2jpR%xi_IpAq z6o2{g?bbOcbAU?6aXV$kVml~@_8WRpZSZm}ds9R{Rx&G1^TAxA7O1hq%Z;`SP%ER% z&CSi$ed~6(Q6K?}y?dJMwA?WjGCUSa;5n_IZ0=Cki>Ep%A(ydIEP>TT#l%!B zbdH~c+7HlBghRygKTl4c0l!jKRmF!8=5})E^`j|>8R+@u@86;I^^#DK*zK<2>(DkZ zz|Ggi50}HJGeruaM6?3ffPBbe+Eaz^Q9;5h@Hx`GVZNj66nVQP`+t^Pz36>w95oK9EkEIc6Q?m?!iJb*I@xui0y55lU0h@O7A17h6JNk&>)G?aMcSf3(#1qzZ3 zZn+^*xnZ%t(`-i^U|qSpck_jRQbF-0n7>2AsCRJU733I@|LGZpSC*HfQ&Va1upg1^ zn)@wZ784hjy_Uqv%S#GsBDjFuty>gub~tdRAu=d=?wWNidV#Td1y*th3BfBXjtF`` z+YdU6Tw1b+J&?=mv+T)~f(i>Js^QNEQ7-bp zH%mgH2u+b*fI+N-#>S2nqZ0zd{|D_+AIHZ{e8dBQY7l1lVARg z`ky{~W}UPjN~YY$X5(RMYI-qA4aWUuTv}L20L3Ml)yZ%L)(}Ik4lt{y85za(bOT?$ zAja|$fgz9ye8daxj=VGToqMVbNY1k+^Uc4CKKdh;@xC>fO4mv{SxbdZ>pH0gjg2AN z^XfuRRssZxRVFB&d<5jQ1wm?dWvB>{VpLSrN6@*5CP#d_GsHpOm3MHsYHe+esx*(( zv9L%;PNwqp_uoJ6Ah%xhy}%w462igD`w3zd2CC=A5tp2toTqHAu+u1CEVDFIC@yJ- ziECqc;uWo#nx+$O0IuPmKdWeDUaym1hk_Ec9#B-+e#1loxMvjFgv-WY=tgP=$d4Os}G%fEy(4had3yWW1rU)!GcqzVzstz2(Ui786`Tc7Ak=PD)YSzaG|tVK>6JI^VF|F|s3%XJWJ?~GQRQgA zA*2Fv4yyNHewJXFcmxD3fGYX#ex-!=3gcZbeD_NYgo&98Q2+Uoc*SOW?hB|owVuo8 z?|XQZL(2d*4vsZkOcio?*6bRN$I>Bgnp!zOGn1SC94ZK^HJ{!?jDgceh+nFN*R{XY z@+7iJk0m~bB50^UTdD%S4kx76*3%0cG505y%hC4Xn8W)$aK4|Bk{t&e*$P4_lEogK zBXol@GnCe609N(JQA7ViuF)92LntV@aO`=?2^~v)dAl8i`2__~Gd!oGqmz35fdC>- zPfip*dNK(x8)w~@1u{-;tr&or87Q!T+2PXB({BNvx>ZINsc2;#0tFo?5G_BNKmPOk z0MaFrm4Fq|@#D4}mz-693|i1Uip)gTs)hX4C&Ioq z%{G1Q)Xm_xv*E?=K`e59h)(4xVs1b6F$CE_kdCJ}HoCov!j^LQrpxzc!vhbdagqJ3 z8Zk$5Ak%|6ayniGsBN?`2puBLQ&zGeI$`c|g7s-}zSP=>~Y&NVl8;+O4 zp|K`&)OqO8KsJpT9{nF)lHFT_o(l+=qL-I2ATOkO#_DKq9eqTosG!j6%VK?D zr}VQdR4joKQ30YsZ^;3NZUs?-oQf*q$rCapVNk_Uv&P4@v|%`WB7|m=l?`4`po$NH z@(Yce9drrk znnDFb6A0#iwdk=mnYY$;+3-`MPiSYfX6~O?ePZ1>qg=%z#OH zD->1Qrqcw?RZ8^PI5`QgB@x`cd;0lvl$^t|B}&F^)440{7Akt~3WKCKk-NJ)NmV3L z=%P1^qo#vK3y@sZ)K9ibgTBdm{O0xNxqLaK*4r5`ndh5=+ol!-o@XwB)XGzBURYt9 zN!|^;FNE(Q+#m6xp`jsEaBOVk?jwbwy6^b3w6ho}&=4EH^DM$f^M`mhJb`KwN{0Qn zAi7v%ffQ8>@vC&FgSs1tBcB(^C@5Ytjb1XJ|8r|EgR$C|B{H+8(Cnw3UoR#9dsxaIIq%2u% zVQwA|B+3T5>rla9E+Fn91t##T0d4~w9Sk6{6FtyB1xOV>pL=?KX^&_#hR9?e<-N>p z)_xgBFUorLDn8Wzs2W8vJ5skon?#)4Yhs{ZBlC^Hty`hcaWXhGMEh`ZV1R<3pC4z@ z@WFC^%G9r42{7%dRAypg=)r+6z)w`KS$NTjuadGd8$UleP&5C%HCcWA}H z!y_)G&8G$w}dmF`vepa%9m0k2$qsO%b#?h*^NSKhg}uopbC#_9Ym?e6djl4m1V|87p5pp94>Io=PG$E#Cjt zobYneyfWzQ^i650vbuUqbL$m(FB%&ti~fgT$+Kz*O+HV|DQRe;pmQdcQIZO^37Q(~ zrAx0mk0_P@6PlomaT&Uy@&d}05+Z3s>{xNRpT-2EYWIJ1Kd9a1`6)GQwh(e% zzO1+}Xj~dNaKgVMW)lm*FF9I)mHm9QA@mK{#Fj(udmnT>%dE5t1nET`8vP!0m2zku zqc3TtK{UuUXhV5#?|)U}4b#wSyPrT>>^WTA3N%_GX0^nuJ{hVRCO@)t-`>enin9h2 zVILCEPk1ND@`@U!dG+d*Co~+q{iop|X7ztSMYf{XQU*RtBNqz`2B0a^nw-QJFC3F3 zpxNNxK2skB|9TjvS;=tvWl{^OJrutR*#UqWoE@g_RRR7 z_xt_+_UGUG>C@|eZr5{N=Qz&eJkBdjLrsyCh>-||LXj#fq3@wkIMm2r0(^KvoBOK@ z{=t8wqKHPFBLAh=<;0>;Y$#>)EiJc{)d_dq`_F66tqX?pQ=J`s9~z*KK1+$#Ncp8@ z-m708|JuMRO`F-q=AEnILhrJWOPSil%Sx@fGits1Sgp(Lyf5Y68PMFi{luq9{)nVh{R=7fi^G7AU)- zN*Ord)UWpmW#C%4%e6?9zBn zKn^u!-*eSlq@<+$GBT+Lhlii@^ZB(63_4!Ee5tFY^(OP8#~_wiPvFxhqeZjZ*v=&J z@cD&>&owoYuG5V;SFJ~ijrtA0xJDKj)#=YHt*)$$d<#F%U>5N{Cr9+$xfk$Zt=ntw z%{CsQosR^19iP7XQU)*Y&<^$=d0+$t1c=glHou)WW@Bf^F)}hjq0XN_&veZm6&V?c zLcu1*#Ke?7ta(}6*hCI~(wS}a$E~uTIkV867Hv>zn^96i2}>kgbKp=*l_I33ruN$I zRjl&D2#6E9DXr%r=2GT1AuM+2&z7CB^P zS=Sm)WtOqn66d9R))O@k?iD#wtB-nhwj)69(X>=~Su)m6{CrB=gNlsgOa3kph1TkyS(cZQ2gN^W_1 zN!8ZXT|D+Z+Gr&pCT>|7Eq$Go^m1#loAb&QwS~hK!>Jq3|DYc|;&^6d#ZDLgzL41R zRBai(BUYO#qAcdNZo*ALV6N6*YPe{ITKWE(Grzq2tuSW9s;Cb0$jl6EzWTE=x-(Jq z1w1UX>>ugPQiO-^WPMX3S!InDJhxMZ`up#19;Qlr>OOqPtaA6R#g!{pB7`wV$H#i` zwVytHGTYeLnDD^r`F5vDuQUaon^^FDoui&6A}cE!mzp}cda$x$U12}dY@~d}$^%3H z#d(S9f`~Z*ZexCa{>XesyyxK2#F`IJIXTMu^yE0durLbQ6COUkt3lV$kTxA9WfK)O z^(cg-9wdlaL`ix&TQ0WVNrol5b$4a7B`GP1MxS+TKmP4=kn$7NP6J_W*M)lVXK*$nU%3} zs|oG;+9d`v>mLciE3K@)<`FI{3lR*WA9OmqCel+;#ZS6ZmE-!Gn=^+UKR}qH_%B~p z*&Ls43bZIx$I$=X{b`q&mc}}};bwh&xU;E4(BDOlJ9xjqC@m_{{K*s142|!9ChIyT zeg4|CiFI{#t$;ZYUw-@ct?j5$1T&JY9IJ~)pCu+Dijk6VS=QQJt(+2ht;AIt-K8&` zvub!v&p`3y3!5?^*2tVcy$cUSZ<^k#f~eAeGy z(n2I`Zf?#Mc6TiTCXP&<>`LUWO}xG8L;o@L*$xT){goRnkl4CC-hp`m_& z|BB3xfw1;PSSdx?lkSW^ij3_2`j*N!6~PG{l$6g(SZ(d-$2>=QeSOBv%*^d)#KBRC z6q!K*0cQx#p1CGD*_v|w%k;870VgMALZgZ3lK;Qi;)moHx)7#z%~86)lL z`gSiTw~&P}-OPTiem9@5!)<+cH%@gR!v~b zO-xMWdv2tzuAcq=eQud8Vw(VP76&J1)9i-N^)&N~F)_bhy`ce@k!wDf$5&BNL8H;% zrCR5H_hvyIh?SbGv||NSbXc2|_JZ1=TlSRFFKSk_uYwNBD@ssM5JIA{r3F`5Sh()P zb?4s{QOq|&mY1J-9WLeQ)_YZtySFqq7yF+2yd5HaSEdje17S@;OB?*N(k`p4Olxy^ zqOFZEOCd5KCx-{0l#$EnBqfE3gjL3H^P0H$gH2B*B?2f_^Yy0(TZ;}m%MY$eN=F1s4?wYUk_X zI5|1x>z2r6xL;soG_kXr!XCVQNrYBZL@L~$BXMzYduQjfJ(MO79z3|0gB_P0RaKU&jv3Ag@m-f zn;IITd;9h+O7e`7z7P(x!No@<^n`?j`_qBsb6o}x9t3B|zqnLO(ueq+!=i;-YFA(cR$02VmnC5fR~vj_2140AKt_xqg3hwX^g|ca3*!|IQxs z4cGMIVhX_EDyK!V3Y(wYi63IYNN3KRF|oBpsv;*l`>j8<3=9mqRrVQou>MFEa+K(j z$jg5_y`re7Xx0>f7t`8TUt(!({l&g8VT=*SBa zm%wWO>~3^|394VpH3zv>?~jgsiIRsI6HIbi+13faFF{IbaZ!cws@^n}3vCaQ%NUOx z*hbjqC8{tq4L6Wav_+XkpFe;HZFzb5(qLonXT*4!6%|*woNk&m z-UALB6B2@-kd%~HV5Bpq9gXpO51IW<@?9G1-4#1FY_BaD`=zJ0w%Wg=OJtDc@j5#@ zVhY8Z!xhMXHxuE6G(0$Zb5TcBoQuZj^NXINjKk*}($ahwQI$Pi-BCU5WP z;8XPg^JSCss(NJo^QgUj`q!^!mkY2sAbojdvk`YFzkmPU-pL6$*FT@|s%FFTXhKLN;LEKaHM6R0 zZ_grY2E8C>ZEkAvx2?U11k9Z(&%G;BQc_?rPQZZ7pqLmcphqEa1S;g?bfMRgurWMg zjE13Mgu^VxVdAAirm#saAbU@iVl*TG87F6!?N%RsG4oRr-8{t|8t3_TS~pM6y`=7e zfj1aGywwGY%whnr{wq;T$T32j(J~OhG;B0P8$cvl`uk(C+G2F@bwvAn4X30M5)zCp zXfos#`>ln$`-g{!1(EaMRN7872N6=kZnI#wn`c&6S9vd8$`o?Pfwzs)tE%W-U0sXI z%jauoBNa^Lz7x+aEO4-~`N45mV1246Y0y9|S$zaKTeN*oJwP z>nbDn-K4Yf@`nk$0Yg;mB8D~2gbfFscCc?Na016|E33WsJ*xM0n`KwW-?$B(&<+~;ZG_b(xVZx3K! zRaIp&Qfz8infU+-M|NpxX3-~IbFfng3Z?UM85FX|+tk6ztVn8n+rbXcEG*#PxN!s4 zkc2gykrrmO^a)r;J5z{#?mXo9_qMiMy1Mjmw2N>wpN-jQ*+-jmZDOv^k1xf-WmP zXJjOFwl$iQk1q_X?M!ErNKv;Asli-Se|X8z&=8LFq&JSR*ZuUM+xHcC4al@VyvPVI zzB>=q5Bhd901jBNA8fAH(ba{Ch7xZoVvR1Fc*+tF<8=QgNu~86AyPe7` ztc>9eH_Oq7p@pczruB70uYYlEnmO@aKk%ZVqjQ`nRJi88$wfj+x-kVR0a!wZS5<8M zUrd*e4}Qm^&X$Xco$VqXN<&{?@>PLlHgv}vGBPaK;i8;dndj&WY)FBU1MiKerrB*c ztF~Bnm{(WFoGf97Nw%vG6E?%xPee@$+ge)sXPPx$y?Qlor!#jlTo`kBEIlAx#66&* zriKAYK-g|lDtOZ1;jxR4Pnq{|u0E94!F7+gxVWcJpWZA_zK{|g{$B0Av?CX)acr!n zTiU99bGLl*`0t*>_HRulDL0J_(T9VhtdTfB{au~Jw4iLd{uvKbiNAt_D(rqoTmqDN z>MI6QCG-3W%H(S(^A? zXcU;Ckn4(W?a3kbPN(8Pcr#jF%43_rb9bfbCGF)`DJc;kAAq2RA5JGAXKv2rCS}xm zz;#)NgYOENvV()t-*tJd>guMxJ}OE|N|A2i>mJ*@&CShR(F|tyP7gage4wnaZ*I!n zx`hMfjDngvYrXcNG4`+~Kgrb8bfm$z;ppw)w?xq=8d_RG0DYIG$d_21`xsjRvg|kK zl*hk3CkIJS=;~EWys-}(fnl1kY;<5>b;0VP3_@zqp^H2I_ z_eAlMfhdTZps~D>>|W~6ae$CT#7H3tiTI!73~n^s&$RB}51gAbH*D}>Mk2o4Dt?a5 z5|m89bnOED;EmZp108#S5nWtKey%N&VV>Ocv6pc zf#r}oo+9Z&golThAnkee*RNk5$Mfb9Ga%Om2L|Gv?6=A~4&=JFcXzi!OGI(u0x?ty zkV`}~5#8m^o5Q%fk1$o0n2Ds*{JsRJ6~H{`|IneUWSKnL2zK5Ecl*nlk66lGj z1a6vUrl&53)^1=ffMkw!5xAosA&iu@@biM0V$f?p8mtbBDP=G)(1aTSCttmO zjROeScAcYL5-J*|9*vD^k{hPjy{Po<$tic zHdJ5$=7=5W4VoV;^8olCsdnO7US39|uSefsv5PGB56r(8Ho5V8mA+WIoALk}?V5L- znNcTsd3n<_GdQ5wbj9mXW2FJ1t%eJOf!xZR9ywJTCtGc9BFuFA+21dMi=iLo*eC$y zGL4^EefjZ271_c5@tW_{Y7Kj4-8P5np+{6`9*j5=L1(gx^WHKomYVwrtsdP+COtItq9md;6ic4AvN*5Hfi(rQ>cD{cNYPY&>=$1Yv zeIjjN03rZXaC|co001$gPh|aJLP#l;J=q|KR$z}&F-QeUNed9X*D)~x>+AM~KN&QD zIwOJv08EM3fg`_ear#uWCZ(;VOM)}AqB$ml;a)4{%-G+xhG^NN8S(@v_sI&ssw7i$DW8pUK(t0WcC?|=Z6SoU*Uk5^F|R9H6w@c;EjANt!bb}=&nM|O92 zcf%G`u0zbVGg(IcmWd>#F6^e%tOP;WY^vUSA2Kk)q9e`}k}$=me?SL1uJ6Dm0S(vZ z+Nk;U%Lu?zG#u!yX7v3o$|Z8ZHzeA~SfoP1cNZE?kJl(^Y2VYfYQiK0X)pkyNLZxk z?do>unp;|yUOwgM6+ET>AH);IEK%g@;u0Jk9X)W@>+TK3>;`&zdXtS`-v|f^n*eR> zK`{nx>I{_H{;G!({x|)6kFmzrK=Vd&0pQULUL6&0)ts_?&) z)S$V>+*;4h-y+YSKZjb(cKI?f;^gH|MMw=qD=0LNjU|=VgP36t9j*D}$G>2nkRWDB zHR(1>3IV+Q)q$nYcYu)_q65YV!7wQuqP z;^G-xTuLgc0FZ~8`}#f`Po@1PD-TfyssmH1Ka~@$=f#b0o_#%6-aI5XZdn&7UST!e z^<2@>Q3Tphu012sLoPNp(06g5k|NG~uM%HuHzfnwN(8Gc3;Y4;pe2-&b9!zr+xZyu zaw(8WEG#T|1q8m;dG2WdU5SgM1C%m__7oaSTukJJ9NJd<&;?Da1Sz+56zWx9N}qs| zWEh*=9Va5~(ZYZiFJ4RV2Iq=Z9>bSp+Dwzehx>VV3zp?cK|GBM;@0KMz)5P+WKNQ4v~5 z4eaZRuGo$R{jT7WH`2Uj^k#+ghNwn3N#o)clA8!=RE%P}<=`OvFR8nzt3r#8nYmy< z`#k%hQh_OCI4&_UaTU2^tNxer^{YD}x&C~rq^dJ!3xVaXRx`H0fSbRVo;Kk(s-=gi z9|+0UA)#E?CL{GqANvc3{QC6>*kw`@5zPKZMn)Q2ThE;rG&e>fge+(*w#(Cz6B!Uq{*r)fksPeIC%k zgI>I7`TqT*jx>fI7~g}SprG`M3OdlA^z`(Qka_$g!5>CD2-^bzMo38LcSSi{QA28f zO;y!&PlkMguo*sRq7J{l60S@&u&TuI`@@VED%p#>$yaYdqb4j&slz`ppo~Hh=OK3i z3JsffSAPOVXMWF%xpP6nAUE|<6mSd}O#&(@3GARcjg0hm27K`IVUz;n5&OTpHgDqM zTyMPd8ThEVzP29<+DY{`k;<6yO0nC zQikA&B{1~@@VT_j0amd^TNaY$2A4Wn4cc(saX6qnk%G=9p z{pi3cJ3D*Y_hiwx*yq?|ZfxfuMUcV07TAfba(UBJvR?T?}PINpN1#oKSYZ$e6YJ(^*zkmP&*`V-AdG3f!2?E?ceFf3l*+j{{RX==(2s@FyvPg)}egAWpA6O1)sP*r97mZBi@ zxoO|qu&^*LT?9u#yB!}lv>q!Xq@$xFARzE-XlT%meyRTbW-nG~=~++uEvS|uK=UTu z7nzan4_N<5ou>qhn9ln8`XVQXGQlXZF3O(d*Ni5TmzR%Vk!FOwxb00mci8*$5cE4- zXd;3^nS^QvbteeKVr6^`H+o7lq1Mme9|se~91ip0Uj>zuF@}QMh@1r`Ms9q3P)$4o zf&vj;<}k17n}HpVj4qSwj&lUS6^Q|{kZ-5vR!vX7Pgy5_I{TnUT=&xD%dk8t(w~5W z;2#}Lespw1K~Mj&N69tl>5ag}?ljTct$r6@<`xuugMJaweSurLd3ZEWO|eq*sAZe< z-U0@WkAmv-Vpgr{{ch81n)iY`KvEn)b2hB=pnbkJ84M$F#0Q|ykuZn^ZJm5fU4Ckx zO;(oN-wlCH$;b%pF^Zm%kpmJK^nhz_>*;f4C%N}?96eMtwJ?0Fd!2@|G9kDY=?x&J*g?Q9 zJkP-x*o;%s<0MwcR0AAAeMOLGbKKgFxMt+BWxmywYH;~)hiBbF=1^$~FACKbNfTfsDQQL^a# zCJYfvN`xN@q7NgQgOrfH(1PZ83HHhx-W*i>3JlPD9K3xYm{9X&EjGAj(ipM>l+HF( zmas{fz7WP9=BL2Mp*y}n5&jK{5oi+qK$y|XQOD*$=h8!8tW{*ZJOO|OBU}&ZA zL0{qdWe?wsFlwG(0KV_B!D96A_*&l|L8QCm;^FDvkz<6jJS_vO0FawLeylJ|Y$*hy zt~e~+37T~o6)f!zYU|MbZ4dUE28~nD&?Vn59tXn(k#K0-w`&?GR*7<1VPQR}95E(K z?!Fly$h27=JNvTlz!=9P3u}4&?mWltjt6^&MktK&-@JVbmd1$?hAp(fSVBfdMo3tg z!S7-p3<=SKE_D=|qu5|*?Ow&j;mf2=tsX!{7=U{O0Cb>}c|2z!fv@vyHx%UhPu=xR zoFtq0Ao=q@&P=`=+g=f)rCZYr+QI+QU6%&wkkQq{S~wLz9>}!g4Qru-^=_mby+y71 zM8ppr7Q0k*OBck(Cq5y#2|jSsul9@nX8c zD`%>(h*vRGC={?U<{qDFuvh3Ah!hcVaXi$UxUr6u-KM~(VYo7cpDl6MbiA+c45Y7( zwY3&D`94f1ZFXpw=C$7MX|&cF4v1bA6}|s7=&}t{Yz*ALa~UrEHzga-(==Q`C>iG` z92Csuy#MIkkJCx?K&H@CaJPsMm2hKEG9vO@osq;MJ ze3zM(MLWs(&+rV4U0hu3_r>$#P@y!b-I_>3Qbp518L5nnh3=P%*<_~w5JW-S{6`R- z9T?EsKR9RtG=|>=>5aoLO4RL!Z%y9+6-awap;LNXl=KkW{vRr7@DQK``yFubt$zei zmhthtVn++cU%z4~&c_Fl%c=$@8_*(!g_4qeizybyZyfyms+(@mr-3Z+6{!09Mk}j| zZDJFBY|R5i|A6exNKZ!rc87+AEs5XlPU?skjISNitF*Gw69>|bTs8QnO?F=cRIkif zWDvZy19fN=^bbAgA8xpjg5|$Y;HQ{79B$$`lG+U)WBC- z_5<8~iaB7F@%kP@&U$6<*yrTn$->MGd#;70WgF;&BfiJOz9ldp?I_UG*2Vx+@*I3q zal)$#S41vMdjJ>DhY{xRmU^0u1>DW!mzI8L3xa2Y(wR?iD=IN5sokrnD+#o;7%3jQ z@X`Jp_0jgXmq(H2zFOQsUq2x%Ou4A0FD~BIiGaZbeE+eq@XzizupbHR%&T74>xQ7OT+~E?4fOEUfv}VW=XA#dB|aNKPZO( zo9xikd;{_R|J*mYEuNg5T;c*q=5L}W-qF!P2{maXplw{{o=K?!EnFX6vZ{l-F2!J9 z6&BpzJ6ro3X1U#>LnT-$M#grKO5jclz31U}Zgx~{5;9|-a+hKO!NeSH1vy%s>=gN0 zoPzR3qi-66uz-8F=C!#?)= z_S>_6>^ZyV>>iItWZt}Y-+lM~?(asRoQ(KWEFvs4G_!D5Z`CNei%^r%K71g1s}5FACW6%NXy8m zj5MZ;KH5JUGCp#+K#dpSy?T{9)%-tpZ(IY*$U8(HWnldu?f8Gabn_k^5b4M(1_qAI zdm;P(=0LFmyl1W2lRFD`pR{Hq8!vZp!$|piv!&N-VEb4YK6tvip7+t8DAYKp_eQ&T zyB?<^%;W_NkYgc0H^ar#LIgq+4U?aBSwv?M}9F-EObr`jfa~ z6FJ&8hK*nCOpUg8iKD3cJrp4AhD_X(z3{yOqsdZ9zH29Re;nQx*P3YQ2o~!%d4%Qgg3PvgT(xCzum}7G z>t&^36{;FK@M-8L^u>##i5Sbp2^$EiFFt&j^km z>+2ss#(|nO5GffS?eDkHpzCh%mC$6jN5Pyg*U8S#*ZHwYro#UI#mHCNfBEg(y=0z; zjg5_J1$xY0RaMV3hmOm7j6M}%-u)nhNgeh0?Z6<;kt@264hA;HxSNj7OyZz~oE-Ta z%#bvW>GqS65f^mNrt9h)kA(<%>#$IW+WWk}!h^T3rd|(w%n#@MY1p-2e{;d-=62t~ z%1ZRxOC7Uj?}zY@ZfDObEV)*iz|+yGI?(%BQFVx{xN_OHo*&9w9K!>+9R=R$P+v_#)g`pVw$OIIwEhH3rSv=uedwJiu5R z=gH>fJJq7OIJpi1cKm{*A<%NX{LlNttoZL*Jy!lCyfz958AWV=Z*1afqNpBCRQWIg z(UjNzG8ePEt8?SIiO<-kQKH+x^P1s-BohoAw-_NmsiC@4ihPyRn>zK9CC3ora}IP& zPEq*~a1j5&+Se3@1N_|vf4}Yrb+dm^MOBo{#cwmk7~1vzh0NBQ4Vk_Ya6`YobT%CQ zrE3S9Fn*s}66@EU?t@97QMKE1jFP!H(GJqtQx|8WQ^}V!T9L|uY&PFqUzW?PM*uXd z0XFvzAw#0y>O1NsiE?6H-9+eSiH9eYn++v5-SG5J8M6qYz`DpcBN_w*-} zoL#s=bt8IXrvKKT>uFS4b*Az^)<8s&15E7Vb;c36cA^muXp|Naq5t-+MFWu)G_Feq ziTQ9VZe(PdEQ1)^vE_fB47STg5L*o@Gb2oMnpt}2I4>t zY$#r)MvqN)Lu5ULonbT_zo+HqDq zQi+E|DPPU!Pf8c7kpyqQN!b`}w)EYu_V%}i7f0XP+w(i;1&hWuSH5?KUdsJa61_Mj z>#eZV8$HqKfCE*McA1n7A!IRYzXrd1w_;v^>|x=2O`})m8TvaYZ+jV}qPW54n&E(H^pWl=93Ym2rv9d{s7`$1h$wPaz zg(|fjj(>ZT)SSrCbg+q#K0jNYmy{}9MxmM)nGMQl)#zo2aqxP22(38zaRQzPcZAK_ zG%x1}i>_k+evGXwn179>? zV6A!6Y1O2H2aS?2cy$c^RILEI4Fyt!9jyKyDmN#)IQ2}Km|z+*{)evuwbcayYM+lf zRR%MP`q*fu`W&BD0ULqm<_4*{DEF?a`d#v)sL0lWOvTn#*n(-i{93&wNT;Z4dipZ# zy!1C_{?Nw7WLHj*ia8aUnE}I*PR62AaCiK0W=W;pPM$DmD&pXn5)yApY$2 z`F&s~z|C(~dfL!=yCiYd$cJHj4H+Z(uf=9-zf{PGI&vm`^SF*ZXu(`QTT5EDv1e8Z z3beBfy02RJ$!Z(gcxglDe(90*Rm$eGe7e&n*3m9W3nrP9b8z2IcKoBNHM?f#c`9b3 z?oYMOX|f6mqy>r&bUV|h;s_)Q`~*^<7-2v$GapGsnXA)4KFEjX|K{o*l+Lvw^*JTs z3ljd+pP5k(uJ&~oj_OHk;#2-zImAO!Wy<{O zHH!hM<2ufIkXZPm=(|OMcrD$`K23Nt-&&1>Sbwgl=*PhJ*UkB zoQ|NFVIG3;MJFND*s^+)2VP(7&e}>(GKnq_nHM!yu-xj$2FzRnrNkG8Oo%JgreT9_ zYid>hC>!j(6Q}bS_3UL_fAcqC*yT>b#&PPTwvKer{i6EeR4=bjV+Bb`nTB@ z5CR16jG%$!L2i$t20R^2Bhz?r|y)<0%5q3<;7?^Bemho1PNn}l=#e~7vW9&30Wq5V7z5{EY5U2Am%>#GT<6^XlO{NLu=?a| zC5j&ghZ{0y9SL(}o$w#;OKw_kHJh>je7}IQY&UW^mUh>IH`nHT3m@F62eI)S=HjAX zRzoCxesQt1`miHdP_oT`N3Fs2MNvI#Oumf;aPJxKQ7lZ{)@j@D441>y2M@Zhxwtbl zt0X+m^tll-#)DsYK|Bd7(^#Cg#ofO?Eg8vUyU*o5Wn37=q)V<*LDYV>pIJ)P2s9Ts zC`bmN{E>2#Mqq&DsKv9}>pdKg+iU01JCotI880MB7Px--CFx|8C#EZGU`_V&B;si2 zlCT_+2~Gnb8~N`U|I{D0q02xn`;o%>yMWp#`(%q3Wy!>FH4qqU;bc`rY21PhByEX4;%)UARV zAIvf6m;zMl3lP`+W6Iw=q#E^?#Qs1$Nux?Uknu!VM@B}TT^-T_lr>KYf}sVV*Vq4& z%MjA;47>JKM6kZR+lZ6R^v{amZJunNLWTx{f>a3M!>COW7<&CZE2|yS0pLN&`I?8A z*F?WJI(_@=uFV|9lM&HpN5m%)g{%{AC3a#E-Y0KE`sD`m+H00o@TR zGw5J{fk$?Y47|&Aes$HsSQ3gF6aOT0YufXW)5rv28A^$*hG$qDo^i;Cr)ozS6; zL_y;QS2#TLJ?*rxtOF~%4WEB*uIorHONeRO({#JVkR7H24)NR%EG!W&hdIMFhj5@C zR$9-SAf?}HJkW#pxZ!a?IhpU7Eyrm1Zr#wOVt(a*<)Mgj;@KFX^$vZ5;2%yP0U0F0 zRkL zn$!FI5aDpRHh|p#a9mClFx@myQsKS1=Rgi6g`XnEH-~9c{mI78W8QX^oN9)kpJpo-p-NH=mrj_}S>M z(QzJvh=D?FpWE= z$)ChVAbZ9NH1}M-1O>v3nywB=OdKhgny!A0<$Wj4L2kl!Dk9q3+h-0F?=~Q0O#W15 zvzIa6w|iJEkDse`&T(^diN$7m(;Y!sr5n}onJ-ei4r;&Hs;=X~4FJej43IzujruD#Hh5L z@0x4C+?;ge$MA8s2v>}f3?Epl6`dU)e;M_d80U?EfIy!67ZB!5kGwcu+1%zM?NQfa z*ci-2$FD*QS+Fti7&LJVkGE)#P*r0UUzDfch!)01KX^h2S*RHI*QBFcPFBfuD$WRVE6T{wo$#gw0(r*Pp8^EPbL-ZRc5Na zf4Jr+96+p;tJiGf{Dnywi%}OEs$QOpGGW!#eXsT`jQAU$ixwh}QQNyWMtjh_#^Iq< z9JGOpdr~HXLX1(zX@Kn2Bk^#d1XexHwme0-{CGO9fHsb~1hztCg3Dp8K^DEZ7I8Ga zd5gh5HF*oc@?O*8+Lv)w-NtPqN?n07RM9kQtMh5D+R(A90R=woQsST-wPZWNG1P!a(6yxBZ`VRo)M^QN#1|$4rcWI8sNqcz!EnVgBcmaaRHlz}#!E{TqFIuy{+824_JsA<&@(5<=7tLla} z4>AS5W;5X%4g>jTE7e@)omCxyfLydcs>-&alvdp}kTecJjJ(bNDUyd^@!rwm;@*ol zyia}~^YFMlN|K=Dgz_AtxDZa8!vMi%l3UO&xaR4_g9$??nmieGbO6rG_)R9acZRZU z&RuJR2uqp{+t(aVs6U;9=c}`1V!0oOeoT>3$&Z(aV>~#++TYl% zv4nvg^VJVO*A{Gx?w+*aqV`X4(#1lV@2dhsrgy8+Zmx zG%W_@gF7Opn%KA}N8cc=DiJ5d6&fUY4J-pDZ*T(kRhpz}f()uh;mFY{JdlJg+HJ5T zSyx?r+p;-EA2lqS-OV(ws=N4>v$Mb1+crARzN#Z7YHv`*JG2S-FHF4XPxPuCPm6(l zfok3CsJN0mWWowc95(H)(&E)-zjsnR9aU)^1?$ zu+pv44CxB{I<#4sDF&aM`Fp6cWXz{i%ddw5wK~8>`%?L*A3uKF1`^HU%yHD-rgFk? z8f$iVPI&mevU4bzq$)0iUqGNxAf6Veb2`LM)Sz%_c5!x=bwcxH(=u_`Q3-)GWZz-|A8wUvwAeH?5aR^U-A1Qo(z^41 zp~eexs8gj?V|leZOIm1rB7>wH^%{sRWwa?8kG3{tef{smKZY*zH8-nwdivW@D7$Q} z{FfzTex2Hsv0o~qY_NZDx67Re2P^W{Zi_i5+mnAPEN>5{EUiKA3@dgDmP}v;>E=#g zxS+Jw%piAiWVd+5s`vx?% zMN%DXzT@I?cf}PC-_6u*Dtz#eJ1Z@;J2K#g!D3|N3l7OFM- zc^*v~kukpF__GV423G-WGJy~V&CaNAi0@dq_6I8-VWii7jM}b%*>qEpszIE1RE3vs zuHhxN^EQ7{cz-Cm1X@+Pox?OLiM_+ai$I{(q7V+4xw^XAn5q!`t=d--HM7v4-~+r8 z?@Qv*5MUF*MrUX4U19v(c6z)BS+d?gZX%q0QJ5YV=6sZ5*cr-no(DaS6~0oeS`e^} z`JCIczPRwXE(2IG&_kKyI_?qbl+cym0Yf~Ox~-$#Oz9Es-e8)i$6ySdxDm~foOE0m zsB5%5c<{gs2yG0u!NRxuTW*d9Qu=ssw?0D_zQ1{j;%Cd@dmCc~X?HSjL=?Gf&o>Fy zIeT{1IEt52l0|$;mWk)Tbl0dN75U^f(?C0N+THQ)?OU%uDVQY=tCy1z;oTUn{O{fo zn+&Gzz2<|VH9Byr8TmkFLx(biJx_Knb^dI_9%VjPDJ1SH*7te*^y#OLNSP*vak^LZ^#jGrJ|z}fWvrh+=awc)(u&n?gr@YWs~$f2 zxiR(i)y>TfFbszhCwTLm>{M#_`C(i0H*~d1dhZfL)n&W)!{}pA;mxEU|6yL9?ip=Q z)Zp&i+cM`Dr%Hu)NLOKQUz^m|tuF)nisz}gZ*Py8tF-xg^B&5CU$3EFZ>|3^s8@I| zEL`F8sVoKl$G7?vc!_m&b!(v{Jy++MWzq3+X*#$d!|P38gP$+>==l1Yr9^b^l+I|P zMpfa@UOLAM-c00)2?&WEu7#Q$Z45nla3=ARBVwntKA3=ZNp!x^P_6!>P@%3JWz+2q zrRS}CWTAE)FfSOys3ty|2{%)$K}KO%R3>@imza#uLpGC;EY+ftcnCGX>F~$O0#`)X z7@c7xT{AV5tFJq3^iX4JFYxeO&R0)>W=JW2xkuT|);>j>cR$V6AQz>E!FkV*kn$lU zys1B)am!ISgjv9oWDuFod${o&boT?=3RTqhzeBgxKFi7J@IbA?q+~DfDTI}-k;-)^ z(ho4>-f==Bj(CT>22Aujf+RXalXAa)SYpZ2U>QPXn0$%nhg1Ak2^B7qRiG>HPJjBw zM6uS1z+&9vGsq7OC61X!L3WX@Qt$DL&9 zHypGHP%ra;Mk;{g>FMI+2bSqjY_6_~inIsxCvrSMAR|(Q`JoVMzv^lx`;RuvdY`yf zSUw6_PI@$ajiN4i{OFOvY^~oglZt3iB*>oH&R3#drMgPV&edy}z9V&>)B(r>%3$XG zsr>#cE0&Y|GlqcnZuZZQ-#TyUMcAm6b(tf7qX)(}{I$76`%>y!!Fuhg>4(PMe~v9|gZ2+c>&P zQGhv84Ea-(Oi1WM-U&9b?Dl_rBWyO4l$f|*(>8Q5GyFn&Fbcx@3Lw}HG5>`w1?7B> z-sb|g7fY4I*07;MZQBF_I>hr^O z0DWfEkPRvZQnrPSAtp%(UrVbWV87xhD_@B;6Ow>899y5{E51n+>e~Q-x_LuVo6M7p?`kBov+R7*WSLYk-d1b*8lpC<6wi-&eR}C37+9*sk5*o z;D1nh-}e3a^Y-@oK;-7g2ZE|BWXoQYXP0f0_38juHbJJHXQF4z1`;8d{%9OoKS6Fl^N7U%nDh{EU;it30n~@Nu<06Ug`9>8LEN zs^LzEhrWFEC?dkTMB|jjR0a+jJA|1-(Uz{wTgU&VZKpvpKrJ16C6poLr=;x}E(Hbz z@;e=W8&Yq+=CUF7@Nh|BGxZwF%iTIyrGzA%U?}F~Q?r-*-YMPKMi+AA%1$)WFSGthww9_NKi<#5D_gfY5#t*J-??OH^ZBPL&)gnBjk5vHOH5R@pOe# zFQN{nbcgba2-p79hIspB4ZFowG7UsDIkYaSCkpPkGerejdK_Xk@ggfiq{;xo#z{IC z7r(@_7?Ng+ajZ6f(=C@leRJZ$bX;?y61c%uL=<)n?y=O0f8tZ&`xT!xnkD%}EJUq6 z{WHVa;UHPL42?Etl3#p0^kYgK=(#A56z!d=&Dy>rWg^vt9ISl%?2}k*GC-6gv(sIu zoo1dgOj4-Z)T(LQ;5)Q2P-=`~bJ9chbgpNN%OO>#Ny&2g+_ylj1X_nlTVcF@~N2p7<3zT9&*Tx^|t_fBiz#NPA9h0Xm^ zrNhkWU|bh&HdoK{5X{_I6+T zuCUUMiw59Xx{)WoURpX3 z1sGg7j*(+?*qEa$Oj~d0r*gi|^>a*IXu46)hiF0dvP0_Q>pb0dUhlgZg1)jv@4yB?*hm)#LgN@+>H zU-?%;IVoIqo+XM);-aF@Nk~!_&-+c-7&I&2$)@rhZBHIHB3W5aG8|xhq`O-PRSUpj8JA1xJH)Bv-YF6m;D3)k?g7iUuz&FAHai zg;s+)1pnbvGJ#~1fkgQl2jg(YDX{BW4^0|qy(Xy?Xc)Wl=eWBctn_$UPNSS^M?B@_ z<&_Gx5~eCF4|nVKOc?B`d;krGcHO4PAQVcCB=4>#RbtSAk3c32$!&J_{R-)P1Gq4& zN&ln32yTej@;K(>KDR!*f92pJzeILdPwH3jwb0A5a3>;1E?IPLo znPR<1o5;QR^;W+pQgU)XKvn#Us61)omBx-|_=~-@fh4KESO&HRQel~6S4EXV zt-SX^mPxC9ak4dbi$eZ~bxAB`qGo$@4fsqt$$$UKVG+`I&o#g&i}lfgs5AlssYrw@ zvea0bkV&WiY=3z#LBM*79lW`Ogv48B=4Ybvuh>kmfr<_0>Rr&lvqb@Af*_3MadU3E zHImmB@|x}J0#&4VIMD#->Ix&t1P~oU%pKX{g9&B;`oPs5;oz76;rbKU0>F#08u#L( zf%E}Bc(6TH5lp9AXm|EczqPeBp4XZ2+cz-H&<;xXM!-}++$87Yy9#T9F=&*h!%uh1 zttKP6=H2j^n3zDy_a#lhU9HN7#QpN*;aeUaB3s+NkrIoM+z6hr^)%10v@~6dexOEd z40P(Kz#sy;Qrmd}C!Sdn@)J~fjZUj1vs9r>+_tE)RdVveif5!}%V zs=|9}VJwE7f9srWfo??sEd^z~JI3K0;&^()?zjd4#`W*yq!UTk zSZ0So&yg$S@&bsm2ipNf_wCubxQ9;&(H2lB>FbN*$x7=tJJVINTyt$k7nXl=z9hjm zv*XarMslAgaa!L`|Mcy?)nqBi7(RxP@OIXIMGdDk+AoWtfeAXPBAtf2mj~;Xl589G zZ*Q(adY(3r#DzyjCR<@KYBrqpSXS_weNff(<}lexl{-{BW|w5ED2zyYha$UVa%~Y?cks9t)|KY3FuUQ#K(7o>UW-2jl;)ds{^rUBx;rD8%I5;2A2)dJFU}eo9F}8Qu>B>cov9dL0Se$QBjX;|$Esmuf-*4J z8OdQXuk&uI?ynGfPYrMw_+QxJ`XC-Yemek`Jr`NmOXN_d_~ujvondE4jcv%$8vMt` z#s+;+S9wvD?fgyHI3*=zU{H`yQIq!l-dKjd$ud*2#OZ6J+IUdcO*ozRBukO$%94zl zsCQvQQ~srt0i=0=iK$qm!!MI2kg|Ieon<+a`#FL_6zwDI-&>}JqZa^@vdxBm)*lT^ z6MwYx$`B4H{fbiY$A0Y~OA1VyO1IGsbetHE*J)B=-oQh)p}gR|I6H!0nr;qf&s5tl zH})z1R$&kl68d*~x;|-|wmO_GeRa^s1V*pC!I$p)wAc;m3fsA}-Wgb(^Ii@xr05UI zLBJM5qG)7a5D_J2lrZYl!@5wPKYzBGCASFcUcNqe171p8$$VE12miz+(IG!IEyScloH@dmhB8N3< z9H@Xw^Z|6P3?#*Yd*8dm5^9;`?r^~ySQkwVNG=#>lZQ1us8Z@A!ic$B0cL_sSnqgi zG?~xkAZ&cEN^mq+-Us;XJXh!S8vL&-F^hpn?6~37xVxL1g6@2oDX!^YDwu^LJk{1s zh{!W=03v|b2UASq^7sG-0&qx{fc))UU4|7J-1bW%Te=??Qa1;Ne|6d0(b>{*Ylt)t z3V;J&6ZY2j_6DCY)q{+xzU>FO@GDNMS7oMyoiVC5%077baQ_G? z3o4xY^n_11lR~|`yqpl(p_WzC=gZ-OAQChil<2W0Ow>6u@xxDyIzwK^LvrAq^zuI+ z*xTFZ-Fp5#SO+CS!P~2-tgLr;SON?RqZv<@uZ6Ct9wcb0a`=AVzklalleYD&)pTXq z<0KCcYxW0IWje!QC#slDH3h={fraEF_ zK@Wr+;nUsOJk2V`W_~2n16O7!OA-&jxe&mcT$KVMU{5)EE#9Ed!E~Gn>-8V}4$#*Q z%$(eP)6Ipr9QfV;Y`~;C0$)1_MYNBYAivywDY3 z_+Uv^BPl8AAAoj85*>D}zK_A%vO(gRzbz~*z;g)R%PoYRGJ#nLP^3QDo$dP#n*EEb zAclkzaY)VTICu5t0`rkC)@vD&O$Ts{0Ohf}b3tz>S%!v$$4LSJO$x7bCm0?x=m^qh zesy;`nB&o3|C16;ug-c*FQUQJct&v-u(`Qk!~Q{c!GiH3T?CkF0PLP5bLi_vv2kB) zG=pX=;F$_p63-jYmV*CF-K?zhvnVj-ad1ZhydOR>0c7a(0=yG*(txj@-|YU}%8C(a zmCAsc6KUeSyYa@~9rv)gxg8JjzsRKB$Kpt2t?Tix7led4`8o|6F);OPvHJ4q4w6Io z2+aNBNZ!D}Kq>h2Js-ag1QNn(x6n-A5fwZ}YA}|s4ywR%l~(E?0E_)rDJYkboaq9z z8)P+^TIDC-0ha^~^}2X=bA{L?2|+wC*#0i{!~ROt{~Tl|Ee7N5-7Nq0k!9HbGUMKH ai{2E}T!Vb%*aGJD(IiD>L<)uUy#E_#W`9cn literal 12372 zcmb_@bx@UW6efy-2#C^12?zqxAaNTU zc6N3g#_{rg-+S)K=bYylA}1q?{_^ch1OxEjW8fUp%xNO>54QzoAgm!;{bZqmWlglWjIllC3sz z8~ZjSu`)Ikx0FK|o^>KwIZ?n%Nb%>*dzO7jGryJH^Dk}ULybpXd?X|hoLpOXu4yZK zi>!;HFzQeN$a^ZP)s2nVRMm0eRdJq>xW-lZc)g8+LX=sE7*EKDqcZIV2Rtk?e;0fj zDuEYY#hBj+{1FHZc`bmWAm#r$@24KQzgz$|I6#{f|AI^@T~bU;ZlWn&_{GubX!eEE zbrOel6sxh!B+@SzOxab#VwdAjqA${4{$IU}tW)U!Aqcq3`9ns***#A!#GMT0L zye{f9d!B@Te|LelO@&;KhPl-$-hFZGP>}yYSxB5=S zCyGv?U?n=-8rSiz^S$O$k1L4R-5Iw9J8A&xw+yk!vi1INkvy(XQC^POu5~#UX@;!~ z^GB4Jj7nr=yizDLc%G+Z`uSjynC2^O@l<|vc2;~aEH#6ZFJQjOGbuRuY9O9DJelj{ z@VLK_fW-hcJ-zetfWIdJ>-KJCo{L&*UfzX4J5aCl9prD@WWKUY%HOtw=3@8dNf_#X`oL$lBUk z%d-+CTau&ID#ExM>WcPuB6Oo)+u-t$**}m^wn!tPnnW!ct7J9T@t%iH<&Za9GER58 zC@q0`4G#%jO{(?#Mk;4jaWO`r`h{U%3>G%MS`wA?zmf8-p!qsS$$$aP=u;mY*=+S{ ztI@_An^#0LA{YDPG|C0c&tIwpM@(Z8=6~?^`uME~zeMHTyOyC!b0oX@450JlN z=>oU~%EZ$}1TRoZZSLxP)vM9g2TUobL?Yti1U~zuNXGqmhOAzyV#VZoEK0;v6Yz$D z{Es3G_v%EwauF?$#m&`qWn<&4hhU-fLpE@oi#iX2D(j{07GFzh`3%U(=A9pqTwZ>8 zWyN)lzHE&*J7F>$K~V^c=NoZDl1gcPAkYPUUR$+0)Ow2vhZgAXN zR_X|GIh%CzxWBR*O6C$DOhlipKA@f5NQ%J5irk~5Y@N(^x;`b466(Ouk-8#9e_J6^ zJUV)?79G~)$st#^zQbmw6nl3QVHzFWw{8aQIk%c;8n3c+FVb%A@JHe9O<;XJa?X>k zRlIh0A^MZwTRxG)J7uH)n|z}i`qnoJ=k**@)JVQ+>!lp&q%I!;gY~I(fmKMJvZZJm zU%%;S#`9<@OQ)S;9tA0(s+gY$&9_lKJH;IBHb0rrDW6QwId?j zeL80cWSBDI{7H$RSM*0GsLK!c2a;$=7n}pA*e(eX{EK1-i<-b)37zVv2n>I?TNsj! zncg^=jAh|{D^h_kdZM^BntPMiew!`(O2BILJk9H`@7>LXX^KoBNM*AULLJmHsq_6K z>E@$wjT55YOr;`?IITvbqs=7iGHY#Y4r{X%`Rm7{U2OrVR~vEJQpuNyO-){EuoI!a z7*p}JE<}^50_J+BPO}u5I&jeHU|l?(`@$Is?UEn-Boc9b2j4D)=B2~-NE(Y_9+RDc zx&?dE6b)|_HrC4dD-xbzQPRqE>m}j!zOR4xioWwmHG8FDk&;TPsKig@@;siJii(QD zXD{FaK9HqW8K?%!gfuSt@p(=$SubKzNzHs#ikTwdjRz-xS+IAra*%qA#NA$njTOv( zE>!GefTkV&Q)s zvBC4-J3534$Hk)`-zk&I({&2Ji-$Pkq^9nBZcsJ8{q#?Kr7gYT;U1w#>-o2Q8c+=8 z#3dv$jEC7zw~tVO0e?F}RH>NXBz_88p0jQ?9?J^XsLgN`agGL82hQ$p$FS?P*c@+s z6v`AiFilbTb#l_*yWA>ydEB3>(L_|}2ytenqN0jgBq`ze+Jqy!D&Fv;#)TOw9OhJR zHYw5dV|H*b3D?7;V8#NLqGtGZr-c1SNBdt8Yux_TmSfEZ8YRLZBoi>`A^ILZ~_|M)n=$bKulV zqjCz;^4XWvGka76M&`%(b?f+D>(RaJ7VKEKVc}Wtxpl~CXi&n0<9JdD@+W;3tbPlH z^iAZB`+40#%3M#>93UxFH*1S7V~EPDwbDE(ugJ;kSae!=o+6&;v3;+uM)7;{qidp3 zO}ZuJKL0VLSX)zw7}JXj6g!_o*n5lY*OT3L`0ykJ?ZjbCq(N zrgBY&tT@IpyY;%lQH@VF6>Hj1sX*n9%b|Xag-FP8KQL9eG@jo6zEIH)CG_bqpL#Nb zrU;Xx*|%MWk>byxAvUJ$gtw>T^hoIcy6X=_GN`#KWO}pN%+L8kGk`Y|6)HyRk!xrh ztiPuHX4MkLI5h^riOZbcflphLtaXL2PUMyoxTWO&t2N=McpY)RmBu|&dtzSaM4ahJ z%g{f3z?H~O{CTd5ccxLjp65|8!%`KOhr5gIDNEX>rhl3tJyzJy|5g%ymKqRlx^v1k zas@HR03RXSp0=j-@_O#F5}>RGTRZ+Cf<+-RuocD_XtUOZE^P2SR;QZ zuTl2};b6yZHQ%$*zhyF4)jXA3gCqPTiQeZGF%`3s6eEL1&)*0K5DfidVm|$GSeJ+G z_ooLBTu6T6J2CT68Dz(Q6_ldT&+Hb#J!7ShH`<=jg6n-TDQQmzSd05=qq?)k;oGD7 zxM8^An|~|KRoRQcoh%F*!Fuh1F9Ojb)i=|W#KZ{xO_~q)L{gMnNYPNW8U7$(jdxxo z_-teI1<37=i`sC1Ev8(cGF)N$@ERxfN}J&EuEU#yX8Xfsl0{chnYz5rIqna{kRN)j z-wS4QfXF#ki~j9FmJJt+L8rT1Z|WBmcVfz7-OlPT3Ejifycc8O`r;s->pTgK21A8l zCz&Rr)uk#GL)7l0JLnGkJmIF}f%?7c;U7DNOlTPxetpevyE<%7z}Um4BGES4-od6( zh)Q5J)@K(qs^p6#DZ!K-cVRX90rR-3^Y{0+TMI-lpA`}Ko=KWFYte3`STaYgJ}y!ljoWMK!P)<0`#<2CAf~mgkuzyG6WMrbx8Q zm?k~1PJkp+60z*_3pi_3SGfC6iXe@bLobOT{wF8HVZou0Xd2*I!F0y#?GgMWo739O znIhroiB8DwiO-&`ZwzwWFdGeYCtzxahKzKC;s}~9TvOcM-BrqL%oqa%Wc`Ky~d8C*tyFw-nK3 z=|wkFm6l%)=cqec@;1KQDNa)gr|%RpNQlQ*Db>SbX6DFME^Guvxe`RnK_f@t)k#ps zpG!^A(2yRM4>B|Oe7E`#$G1VVbU~@V%qGo3GN=pGtCcI`%S45LUESZ{JbU&`q1;Hn zP|@`t+yMfDxciU1?u-(|mK@xI64u#vdzA^Fb55Sh4L0Zuo9^uXz_`}UxG|CYA>so# z5ps31rTF_ZvzA9`-Qy7>RtGps;%vF7?bUJg(k_6`2DC}&diq%StVU=DO*bJ^@L+eP zyi-~-vf)xw#dWN1P{&v%Q+W?pg5F@$U1E}u#Io##d_B5!J13GhSXaTs!b+ot0=?y~ zuCC73*s^HWL!$&%GbG}?LdLQ`V!k;tSl9*Dk*f5%Ba(>ClNim|sE`rn@yYnSHI&9= zyK2Co_2UEC%F!xFG6_@+3{b;@$!yDTG0`x;wjZ3ZWr{pe#R&j2sUM4<9@%d#oq9BnQn8 zm{oFVA zQ3`hDI;EUbTx6{RO&-F36gC{Gq!T+@d=Xh48pJ@fz}y->nHxQp3~J>9y3x`17wn&O zn0%_M@!WWw&+MLOAzM&g_G?{GMHC+`tNTDL8>~E9NF{M1E zu;-|oS?&Ztx1V;V2Ga!szEP_%DeWlvzl1oGbQP#721i85PTztgM+;m+waMcpuRC-E zf|DNg^CyeaAGufp*3H+1H`~*p5n<9gmUGZOjNR|ZNKP4 zB0P+MnO*e6#l@wP&S^C3`(RPKLh_*Szl@=lPc2j@T{sW|2P{ILa#*(&sGiV#%_9;H zZTRCTL5@{n&HvqThf1$hg#?m@jde7$>%{5M)3)B52A~=f$=WN66(j+{e7O=GQ;@um zN3KA-_u+VuBaB9)<{1vH(j){YGURUsL2nEV7RldUt;c)Vlfpa0K8$1r&F>{H;4o;! z&ex){m`|AvC5yY0fS#_jP^&RiqxM+7)=s)m@lC?+x#gd4*7%Ar=vs?!#nMyiP9YxL z`h}s(qqXgMRlZmrdM&qHTraQt+q3CFHg84{gfj`OO!k{dEn!iVgk~ww9)Uu2?}2}l z`s24LAhvWL?onBdp_!)Rx*Z)RDd9a5jE$#A>`4Li>Q(a>+#n;hJMCSAA{mnSX$hc# z$W8|caiJQzabFD0#T_thPsv7q_juJ__qM~a!Qr@w{BMt|O^pVZl&cf(mHArkW&jyk z)+6}um;Y9pXt@{0PUXG`j~{dc4kaek!3p{qR!z4Si3YSz!cS({^y;rVjLP+^=St?n z-@kt!8I#qmHyZs@r&EvlsoQ`Nt5dzaTT7ih#!S|GMBFe2SN=p0$QD(Z)H&;fQ`p#@ zpoScjxE$f1@6K*|#D<*u#Ja#kFAkay?=HAGZIxmB(VTi^ro4gq(NT#( zuR$XFp5LI6fcQ-l(&LJFfkZo6MnzCwP$d3bsA6yo^I*qSlWY=y=ryUs$I|s9GMsoC9ilt+(1Rb-Aq$L zZOB5OBKcwXA$vn*wgQeu!-8Q8z~ZLkIe4Hu09IM~{%^um#)57efh(5E-$k!0najT1 z?Ti^<9pRy5P}SC0bV~K02SRa-++IyCqd#VMzt}J(^B>gkV86E6x}d@$;(0O5gNTCi zGv?Va@785w#^)vvo8$K`?taCL$-L$;3sqiR)MUtp%6=DmiCVhOmtZK8tn7=I-`Bdy zjAgpB3;N^I4a934`;Sx{(gkebnC~f7T{j6RjE8(8BG!hlPUNzJDK;nbp&p>+D4%84 zQi@p{OKWb{Z1BYvk8`upW42U7=QoAO{ya2!DA~54d3G0{(c3FH z1R(sLB)lRG+k)aWZA?5yx2TVC`ai<($k--INv4YheV;;^cb$5ok%v+n%SrGUKEubL z%tNV;KWxwPaPlX`QhA^oe&645;=dwoDlg2)B;Fyy+nV{%8Orsm*>#+GxY8VYhxr;o zG~zFfhDyZjZcnC(d^b+AVL#H^2F~zSns;hCV*dhTfMmmu&U%|Cr=QF!j5v_JPlIF3M@%rb_*WEeN_S<8IzkyS50Jw7ONhKJFrh9d48b27Kw)I}aC!?j# zQCqHP-W>a*vf7S=f6@pd&%^CW483~nU{do0gfyKw<<;r+Gv&fWfnu$+<+V&lkv2_E zYqgyjTPE`;$`f&XP`@-B-6gOTMB9-YrnizyO{7?u8IJ62#M!fm=PAy7 zKsPD!xfVuhB=I1wZf7l}Y1b@E5&ASFgoc-axZgL@ z-~SmMuBRQN#N#i;q#SshGp0Yjr}|qm#|x@2KHAGrz4tV|KPK~JhxTN&hqJP>)=7Rl zU;8Ew;#yS2Xr&m@;WpZ*6ev?T)emtMcj^hk0M126P@yxjU>CHr3S4(3zT+rV7JZNw z)s|&7Dkn)A+}=IPrFJvsm&>0#q8SFAY~|l4-7Y|)euAj#{MK2F2`diWN{AbbHk8>F zP4jpddY4XpzV30VLg!@6uM{Ty1vHa;4dsOnLnMD_PGI}UZtFc5)Dlnk=B9;U^&xNf z49f+F)kEU$T};n?JnznM=`=1;+|MmwCZmZit^`k5k^G*>%8D`$$1MJ&xnnvwO`FiK zgpm9`IX~}eZ@0yYiHoDD?atQcDp}jx+uJpVujtyGZ=D{qP+gol(8bjqz%sRSK8TA$ zo6=}J*(}*VrlzL$HbM?g_f8KFTR?v+_0x_5auC=rw2P+gS{szSpnr?@G*fZ@;?ih@SS{@uq1Ve%E5vzwgsDz9q; zTM|VqU9R!ra>Q(jE*+QMAND32dGCiiRt{^ebVEs3&j1$>Q{3ofo$cB3PjL*ww*Mxe z1w-Nnr9C~R6=t9C{aSxJ(zb>3mRM%C>gbI8CZp3zK9>v?KDSwJc{P;6Et~?`1IV(b zHt5{Pa=Hj4HKO(H(MHwSh#mngtrFK0d?fViU=~B{ty*$2l5Rl^AidCBm0tq4vxD*m zMVXw^1c^-YWe}fd9p-DDXhNJ$EZ6l1Vd+E%P{o@$9siMX&6Fj4G8Y-IK6uSTT>KMj zHH4=k-o@qfMxPcM5$8|oB$58L?b#mq_zkD6-iN)vj*nnM8l2fag9dha-ZXVLd3xLK zpNg9;ld=wea>ix{aOL~`TN;XEJ37hxu@YtRz`>7dO0LOCM?YGlMm9{n7t4N9zkb-CtSVYVSC_j8IMWS4xQv!lB^sCw~V ze}ijg=^QI7Wn=<)pu}=+`1Idd>d7y6vWcJjdU{NsyP=*lmFB3^#iX(BnDa^o>n4ky z)r?I|O(6G`B{;jlt8HtJqH536-QNIKOReZIoT~R2}y&8<7EFBFG%iG%<;;`M_7i;&epnha$Wpgm| z_U=OY`r^PZP6Hh`VYzh3qr!YT?(W8Wb)M_WVf7Unl1h^YHX`zQrURU4Oj*Dnw40WS zkI#$E5)$9jBX71v!O{vJN4~r{(1wr0B0#@jv0EA{F80Ddps~ADzge&J)~}+~#u61T za6ETh=l1p%c!z=hV+8)3;3Y*w1OvfEA(2=5x0jRDq2#=y+YrKhP{CvS&Z_ zl5fgruFw1E9C}Si)Jl8M+SV53R;5sm`>I~lhX`a+52h&B6A>Mj+{h@o z_Ex{S8V5MrLY*VxUo3IFA#{&U{4@uup-bqW^IhwcO?o-(fF*D3d}XmC{HKVP z1)4X^X&x-lmpJRAR*{ZM+&92*;?#5_lcN{aRttLfx3chA`^UBy zrj+ABfPzZc-xoVx<8{|O@Pct)m#JlQaP5!!1-h0yy;9vWinM_QGeG#%=!8!1$pzG~ z4FZxlhD&wZd8lOX{f*Ov!Dv7zCxvQn%n)C_s$gwb)%BBzk$RKI`xHpyaQQG_K4`0G ze$jMxOyikNz65kkdVv=2GwXu^d|`xSp4vFi8?UTy#Sj0PqCZA2W1Oq0+xzefqf9dA z!2MmbVHcAuZxXuE(5_&BuTElCL;!)^bx_OMQG)L!2CnL~AICgyPEDldTWKQ2$lWohKeV<3)oC-e4(KS@d= zrV(+M`x%22n!Ol9aXn=!i#hgZ{%4@jxf3x`QwLiu1hSg{YN+3^B?z~l;pDiqhA1O@0i~4 z0INt_n-Qz`{X4Y4ooxqIuDTm_FkNhI4FL2*2moNVX>{~dWMm_`;~&|q7yT>EgCv=I zeI(Xo_gor99}$gCO^PXj_d+i(&d|>=!8{NOib~@VL83i zDM7(u^{s6Ih#|q3Poo>+#cG%aLwoIqRaNRvC8r9L1T1z@3dd|*Pqr}9?KW}%CHUyq ztLbbXsYEgbj#dNkAr9}yVa{>LPY%+@TT^-YOU)2i<#`Z>$(U>#lXZ z08>XAkv00w!JWR{S#7!Irsjr^Dm6TVVIf+@FzseBT)IpDlP%fak1SF2DopX$XM|%7 zuAWa3BTppSHcS`kC&sZ7+35kJr#p}|9a*aHZ68p7AU}^ zfG|P9z)&g6;1oQ&*Ox}f@K{xsU7S3c^Gj=^=dV{R@ffpvN$ zZ!+IM@K^VEdD#?=;oEN4C}+?PQt9eiZdgVP<7puvVB9_K0(i7p8oFW3bce_BKd_$e zaYvI0s=X#@LM2W6Z83}J1RsM7iKZF#Mh$|N;eAT|YUJF3Enx94-XE0}AX|v6Z496o z_EmJ~&2}eq6%ueep(6Rw%$@p7!j~iXy#Od5R;e`m;Ciz8)$ztK25_LVZ-=vo60$a; zW%*j?=HiC=A3pCWDP@iW>M$osDPZU!e@#G<)}lWDIOmUN#uu0D=1SXPJ5>o zw(3`f>eKNo*))ndQdyey-+I%ucYgT0*fD5JA>=6?8Q)!7U!KqP5^+}ng`|pk+*$@= zSPK-Y5pU1emQTT7q`$4Mae9sY?8}#cnF>=Z$A1$P+I-$%PM7bfMvy~o)D>YZlP+Mm zJ!_x!c$&8Q(jY3s;|h-e;_#e8v}&WpmskxJP3BY|UTe?&lcPAKxH(n&>>ub<{OjiP zfVd;slEfgahy78cL!gVmPH?@-zX@z`&(&3{5(oI(A9k!EJ%RK8{!GE&S!-?2l+sXP zc^tMTNu@P~{C+pxm@5q{KkCF=2nnfe`9Y9$c;tF&=t4FhsqV=lz8B7s$l`LybA9F> z4RFIKB!x}2(8y@WLQp}0jgW&O$IX#(%vLhuwNr>~MQvHA_ zngvRB_P{5IW{416hI#*phzOwO$lUp=c+P$Q0EnZO{@Fiml{z-ir;F%Un?GQ#^~Xgp zyPZlloU7p|7pjRbHXLJ@^xT67vd&k-vQ-+d@t~v+vBUh$!j>TS0QvToMw`bjnk zpEQ5==1nxU+~)`n>Nm1}7V?kIZnt4~yi>d+|3{Ab`l7f8N~|tdb|@G+OEersxp4I_ zzjPp}I+j5bVz{s@P~!uhQE+i_Ve-1gZfo0Qa(DU$qD01{p-xsJXJad^Ib9KE=i>Hv zw*Ep^t;)mKIJNgwH^7EwHZL9ln*r^hSvzg58)N0*K(OA4QnlQOiakm7tQ7Fx9j9>X zYe|aX92XA;dyht-VYO46{w-9t}~_2wA%`N`zaMHL!evCltc%h!b){oKPcr* z#Jim>=6v>r#7iy0LmjVFIBEnJ8b_8L4P^cz%D) z@2fDY{H_SQ*B?smEKuk06AXg9!Bl{f3-_(=-O+vd;)@>+v04gjbf?#-wM)SIaKAE; z(9Han*X8ak5DbDN^S+w@{22G8t{XM7PZ6l{qcR<1G!^<&B-<-H@_MJewKVT@oAn-5 z25mk+Eeu@L`6f$B@B61p#fzbTD`5&XHgC|7)>E1VczhVh$i9eMKkmS|yiUo;I z-W8`gdY{_SU*|G=ZU%i#ZVq^y|Kk=9+5g+(&3|uoYk5hgM8>N_nF*Z+2H zs2?Z#@X!WKwic+Y+`3+1U>LXgqa;9_L<~F6ZEz%i(dka+eN}@+(if^#h88zn_Z{I; z#^!y^r)6xuEsOd^$YGVmb!Thn)KF5wL`g>{Rp~-tdoZJ`tK0XJOpv5b3T!crWr|`@ za&oGGC6f3ulhKSI8E+*o0!{DA^X{{q>DVI8dQh)hCFJCO{&$jwMqD0Sn9FH59^eO0 zO6UFgm&*eOU&a-|x{bQ)&0ckEOUq|UCMG%4g}O>iQqp3LW-m?Giv{P{+pCi$4&|5i z3q=i=t6>rv8p#>d&Q)Rc#KWfIvs0#p4~=td8<*}ARbC@JK*E%Qu&~tt)# zet(|JYQCoR`*Tz<=Vr$FsBAD@q`CXIGOW@4qReKct^U_SqdSM|@p|j@G{+9R04@+; zG=ju)y^4gzK&-Cn%x@FaQc-#At+eVzw=k!@zbiXCVWTDn2aApPir)uPd6Ob3L?Pa( z{N8-amz+d6IGsT9U??=la;{1$jcJ8WVwT!0MGaSmd&u?t%sf>pa}w4J5DzO>27-`)GUL#isJQ zD->yPfz7Wr@W$j57zsBJ3BcL;Pe%U`=dG$U$KlD|)oO*emunFi$%_rH_FEN-FqzRD zX+kIoFYu0e&7DL@(8NurRWkf>E=(0gN*b-h#E`~n5Vb~CZe zo1{{6U$YrR7?n(N50Jl9g$}^bWWd2@OfPxnEHCQY0+;zg%za+OECMCEYEhx-M}GF~ zu#3=|GtPUa3aQ5JERS{$!g~S$SoH7T@9Z}Eo_`h-d*OX|7PpoxE+vKLezBh|oy1-N zs9(>UbKtjY8ZN`BymXp%F)7^6$0x)5CkxI?Trlsm$!{0Z{9d=5{CB5V6rxar@f>N3 zg}PX<#{hbN_NJ>%!lt`(b%|J7VcJA%ceB^a>!1tqWbBF`(psV-Q71r|O! z)iS_TW9YLdr97wQ;YkLGpv3#Z3vhhY>Q!_gy8HtIvQ<4F?r*HgDJZyH4&Q}@gcNAh zMp-vq1p+pz+j#SuIHm*M6h^ z!9x8-6SLyZOew|v{rxj!WW{BJjaVJFt8tswbm z(UV2X%MyErm6a91lm7f5O9G5#I#obmJe*4T{=NUd$$Ts#BB?kAO)&`xx8>7`L=J0c zbu|YVaa>+qt!!Z#8|r`&fuS8eUg@bJg7a2&2QHw8~Hp7NhS|JdMK%Kws z-NeLtJjZt?^F?iJn82``BpZxmv1yfpGlavyKwAG5DIXdjQo@5nLsvm;v6+soH=$2v ziDQA7`qJca<+2g`&1Ef|18mkk19%hsOKSyWP(b=b);aDxhHEG`O&f@}L9ntx;vj`^ z2==iQ8eD9TH~K{@%_cd$?|FhSNOQnAVdyOPYyOmHe=NQFFEFr+0}qTfx;xi79}EGv z4)6`i?8^OypvLRGaN}ckLe- zXm`3j%4Yw+?3lx0q^1yh10pU23N-=+>Dlw=kufnH82q;_z0n^YKkSX7#7Iv43uFt4 zi1+|l;+h&xJ$-$^Gjbsql_vm2mRZcQOciOmFMSATX=%9$J!r3z{3q+{a}wPLehcs( za;@D)+wgEip?bC5dQW8irZg9>{YIbTQfa)~59CgXMh|dc%#jX;ohE9z6SU_yLGo!B zO3Xb1cL20w8`mDt@_Jt^YGVeqkdV+}H>o%0!?(F6Pj36ofiL1-7F%HTV{0s1@=?R2 zUYD&`82^2;+8K5s&4G_Lz+$l>VPr%OVRJZ5wlfq56i$2qOPj0(7uGGK(5y`Kh4+Os$H2Y@D zjcI^`fE8jHR*)+xD=W^e)tJXASXiQfo*>;{u>-)C8mNNdquaO2;W98m`50sw*}cud zBrYefHfJZV`oC$p1`u)Cl^FKpBBXlXyMiE>Ia=)mK8p)@ z3I`eMk>9PJ41s(B@+Jj2&7?~+e=&EHvM2o5YMcS6d4&w1ahb^ z;rl%fhzZs~}Toqt#2CW9M zFkE_;7(ulgj>7m0XoB+m_1O-q`4n9$Am;=G1!=UKctB1CY(d6jb%~;sQ~=3BKtLcl z&1)<%72*YxNjW?^A_G`zeJF*HpPwJ%wA-^gTRvN5NvQ34h69Azc7PyZka&FKd(6ck zF9P0nzS7!mYbeF&e(!H(JMb=GlF=qlcP)=2VT6BEg_1z1KU$3tNXn}e8Rw;Anr=Pc zr_c8xxCdvyRdVAFW=b>duS6})wY8T-t#*KYbIjHGO-Ouzr1nRv!oX+t2FUpMQsd+C zX_fvQZo>iE?s51JpUNgSRsG-_Ps_B`4Xcn&%F#Gk>%xl!mODSi;{~JI+5EqsaDiYB7 G{yzX5dB`yU diff --git a/docs/images/chapters/arclengthapprox/3fc083ea7bdcc6b021560f2f2491f8aa.png b/docs/images/chapters/arclengthapprox/3fc083ea7bdcc6b021560f2f2491f8aa.png index 6ec39c42151764bbdc3913a800c2dd1d60a9a0b9..938fe45b4241f68bc9e746ff9660ddab7e72c482 100644 GIT binary patch literal 9888 zcmb7q1yogCyY2#{rBS*O5ozgEKtf7FKzfs+(%lG1hk(MSMWj174YEN%q@-KAOS;d* z??30B``^+%l##{4wpLd2nSCPZRp~Qh82v0#?Mje7sNN#?yFhI+ly!jgN z2g5{BP6oQZ`T5+K69+-`kb=w;O}CV-X-^%^<5bDr;})z(+*305WC_Xhizt@Lc@&bX zbp;jm^am=na1?C!zSRhnnMDem7A?18yg45;%iPvd&gY5$N?GB~c8h_htx8$5jajBp z=3znzOD+ZMmKnKmQ?q1HBIF2z3nQmyN>=97dQP1TRjRpJl<*dt#LCjIt0R`WMYlsgfYzIEHcd4LD)nJ z{xDvDC6;~p>9H}L=pj6o^P??k7%T&kJjYX@RV4M|#V7Ewoaep5n>P=C4GzY{#`=$p zXsK3pvAE7Grs0I_xy}6grJl1M(IH`BW=7A*h#DFi%F4k}CAQPg1&!6aa#{R@zw3&- zztB!*RGA-_mWH3D9vT>Ui(1I8rM#iGmLiTtx%_mBjLWbs2#>1Ny!&qVzDH8g>a%L%ax?BL zG(7h6ZLsdEy=dPMVaH`GhQN^Ua1yyFYNC#4Ix>5EdzTGrQ5U7QwlRK6DrbR5=#wnhd9kraG%b?hqLts#ktbN50$1pc!u;+HR9#!oDZ95u11rbTpi zcEY+xN{piA=)6fqQt>GxG326Y-y4>6Aes(2lKG1b>VtxUFubnMXA$fhtE;88C7>5j z#5dQb{ZyD_PDV4yebqD>b@kYuqe%~gAKl#s6GoA;0byY{1>TNBN4>o&tA`%U_wQ#G z7u$^OjCYox)RieG%)f}MiNR1=^MU$`2eg4xY&G@=r&=!%NDVq?F=bb)5({b{a(?hrly(wulP)k zHpjo@<#m;(p3XL39hQ8_$avCe#@l_(#K;&_g~-Z!D)aOy-lP^5LwR|5+d!sV``Tbm zdj!R!!lS_k8xN1VVGpS79U5j^B$Z%W43k{j#|IjLgi=$Eec~oS>{2D8;YUYDFl!p! z(UIyScAe744aYN{TFyVeWM;OX?ynS0SliNVU5c5Qn6!bf!IxR~2imi*_`>}CQOhj* zXuHb?b5z4?9anHN2Ur6MO-73i=IduKi$g*~iH3)V1vVf;9v+_gmUn1c-PWNYA%r5S zy~`fUvkwBk8CL|Io@bOO>gpybp4Yz_7^S!W-`S5|>&|~dh&e6b3-@;l>`dw^n?0Ev; zlK|uqeDboBDLMqG0oI+s{*sZ6jg5Ww{=IvleG=zNz{+)}9J)DN-p~j;$h38I=xrf5 zo@M=l!6OP`_|+NF(MJ`C;qNi$$sa#5*VWY(9`&ff;RZE!ura#QckfX4KvW3>{;C&$ z>IfH_Mmay4v|XrONekdcCls)r2$T-SM}Z{HHy<4w9hrbQTeDGI&ptV(Co!}+UZJL| z8#N`HL6t!Sn}*hvU<;m6d$K8+hKmi;HGA@kLx*T*!I-W6w$zV(V~Y6leuj6?N)- zH(-Ig=Q>IJ>SV#~d<(I+izEfRTV!Zo$)!hE;Ohgu{aOXZSoS%R!I$;n$qts zPfDVERi(?WU(;V+SW-g3&(FWQzV5HjiO@qJ5Tk3Vwb{zaLGkf-yN{0y^&k*Gk+WAv zf=K72@GZfHD!ttGMaW>5*zFEFRPEw=*=^+Dm*d-~3~N1xA0|#eY>FDRLUu6mn8z;D;mA)TXMR7#=5Ec(P ze&QP1*x4Zq#-}MikQ36_>AIgz%E7@wYn$GDE%x)0IyE&lc4)u-a%+8^_>pBVX;D#; z#CX%d`ra4*P^EMU8W8EDCC*r@ttXccT-w{)gxVqzA)MfwwsO= z8-y3>AsBgh1}2s5Ll@#C?qi8MJ{;Gh&E+&~Slwa~BE^!7zPRvOTv_p~r=YFyK6kMd z{Kg^1%fP6>trQ|pe4B(Z_7*ZQg!vnQH2-rG!^YYgk^1ZqJj6d*Nk}+FGiQ*2g+;df zzJ>URX=6)%#~}6p*LeSLvwjmc4wWZPL5n}si9Te``(KkV9J`=}peBAZZb^J(**`UP zJ5>b}0)hCQZc&hI<`W$qc!C})O%QfWuzs@-&5IW=fR9mD>lX}56%2!O1w~j`7VQWb;onyhA@#H zB@((n_VyMBN8G>|+|M%IATTIMhWxezc=-63=f^D_Z`!xFZH7iiJ?m?4fTH_jMMYfb zbKxfb!Evyz_{=Pwe|dQ=a^%nzXn}y+wOT+>wX9yn(GeB;{XLRu;TMalDR|3YN#KO; zd^GTfnTi3|y5Ja!A~Q;W+HZ#(T5a2-`PY^NEC*Br1oZreY!attNj=u;8YQR@ILtcA zveJ-DB#dOM=a_c28qv|ktIvm>?%c2P@bam}8#MAj8J}&WO47KG0x8B1NpaGZP8f31 zwam9yL4HF+n#p}-wzCodzIfQ&r`D-@eJ4LZ|9d4KGO!by3Ar$>+T4=WOWPKt+JY#l zr1xC7Mo!gVWAD+8FfP6)hF$%mL^W)H~h9F*EUP2&e@U=lpU=autvS1Udu3rTh6q`vqwVSNA3uJ;&h<1k19x}rK;&qC zr*E^qREp>U>Khmt39_E77Ct;H>kGt+LNaj(5D&}+*nU!#Ksnk(aoIv`x^gz&xA>jr z+3&f4LCQ3v50J4=ozsKfH1Y84Y!)XcCkQezHRVVcRE2YYv+#_f7WN-C^z3lA441uMAqtDa5k0`bqj}0p;D?e9Ok}Aftm(?pjdxpblP`BumOu<4*Dog8r zn>-Ru%J4Qgczmh5C7cUY)CF=}h6p6EE`DSBP3_wz$21a1VS#f81K^a-o}Q|Db`cRu zaK4h0lb5=mebr>MU+rh8L?BGR`J?y0uFAAoU=PAZ|2m+e7uYoEk{&vM)o@vZmU1qSES>mDLju4-v@t^v@EU|4sZozD!?y8Up4}=bLJaEa>N3xsh2`eu5lLYIdt*rRSbqQ%2=XYYBCYT|jhrx$ zy-ZBYVlg~>_N;AvxFB`Ls24@ir^}dhm7Cmtnj5m-fo}Q1u^Hgxcf-a?pUeC(&m4jL z`s+jjcg9fMpQ4{#9~upu_e+zP?x}6L-8fi0`%8a-aR~rmu^9$JBJueXmmaakx&V?T zxS%a9uwuK7B7{g4o+PD(cyq36Q6m~gZR?d>RrQ(JX01J%I9KDXrDB8Z-eF8yNUIQH zQE{sEPIJ3td-%})WEZwMQKcB`VQMv-|DwQdr}22+bM`!K@NmphN}?+Ze{jI-grU^& zzTIzVaeoD^PaG3q@elTVP4_@{c6xeh=)S!*iHLTm3c^0Ul_07lH#mv=@l&FRCKaV$ ztJLo}Rl7%G`oEa4P2e9AguG+M=u>?nRPE+acRM6e!MeWY==m0k3O$7sf?-k%CWUiz z#QPm}Ae`i=ph#W|+s)rHwxp*~2!~10@?<|$nm#;TX_;wkp}T^V(%y>TxNq@4&ilA6 z{7UmSNo=5;h!PDEYHqQ>;d@POG|4lf=Y}>SSqvq{@DGY&BQO!lRQ$1Ky0N-pC*2B% zd!?R&+;05$-3gF^Df4^ygKB{z!zd+9uPN=0xvo*gt{c{j{!rW|DYw!WQIfL`oEbXa zyVwuuUsLjCSm{HHV{yN#wfafUa?3ac?AmcVWsPatpx*DN(&rlT)Y&TmlPU6nXK5YW z*y;;2VCn!AA-DdajI-lrG|BVCwT0^*Zi&?I54j@i$E6 z{XAJ2w|aAC6uJ{UChZf=;I&s0CoN8-CRaL$O46Eh8(Tye^Rjfa{bER;GdxOaWZ1l} zX4Chot;Hmop+k)gzQlrKoEl! zmt2eKsGfOTkjyh&0R`q)3;bS6P(B_>?JJUt(bi^yM&<0&UD`qAGGt` zPhr3$K&7CVvAJ_f9~2sXakJcIVd-jXw6pVfMi;G4887uy5PGr_5wLmY=9<3%Vgc~f!cNeLDDi|XvBd_H*3r3mMHt`jUlanT&srPxs&a= zzCDRKWgm>td-|l!cE{rzpvF4%oSDYn4c2==j zUk~*6`=z8%w+Zhh@UDs~84QeLU+f0AWGmk${v(nwn}HF4^6As3)$MIr4I`2tHXk#} zN{FBj0t|TJnPk$_S>H3HgG=9z^A<3Xy1?rcuztV8RH}fB@A@J1q!&Hl0EK1N*Hc9X z$%^H8<6o{K#72x?wMbT8wvuID13R(0h1NnE~~;EF2tk z5cH$&M-Y1G%jQ+NmzES);#ae3zE`NlZ7k(TO}3qefYRyucN4lT?Gg_d^J=MCmU~Am zq${SKlg^c*DGmk8xH(`aN8cu3>&2>)=srh}4mM{ZJFNyArYRY5YXRF>;n=eN-K|-h z0%L{ePU~{DX^~)(UrElWwDV3Qcj3!`D_bw*@kEEm6NA1bVPjJh697eGL>TU*lS`D< zD!Sf#Y#uD z=eHwAXsl+Nyjj`VovN^HUj75`8vaaij@R8eS!Ej3p<<+4bb$X%6Q^SPgtN}gKccLQ zsa%7BlZXnQBs;vD@6K>a@K49(pJfwQ^X=hdr7*PWx8tY~prsxIVvU%D1SH}(5IF3f zynfB5uA!l(r}w6Url+R|&Sw>iG_PL&Zy@gQ9=T*kbaL@M_XWI}gaog6B`k)Nmjfd( zWlv90fIa)i$E_Tbb8}f27Z>RT1V+}&K0VZCeDDC9O5iPdbwgbprMtU3fC`=e2GE{u z(2zH1=!5e-%X`^;*`Pw(a^ZXAt2b|uAh}AJFmkQYFDb&S*;g4P>N;-T@!T>MAE8LuW)>EVj~)${7aGh4PxOgyVqio1{|5?# zvJb)|A_56pi8YH$wE@k7p1^6~P+Qz_=T4%{!R46gj1Bfcu9(BPRf-9!8Tb2VDXlJ_ zBgE>7rHp%$_*q$5AKTb)LiGO)X5}xk=dDK+{8ArWl;!7dSQ9)&k2XN2kr&BqMAFv7 zc&#V+fbG+Zi_<`*UZ)N}llZ%dnV}D#KD`I@F`(qk%*|1_@i_HMjZq}7PE2zqO$zXh z*K+|FbTB^_D=5ZyJ5c-GN#v)E9hbhV16PXHn!lE~M0I45o4i2Q8Gl3ln2?~?YSxz~ z4*7Lp5Ujl?1aRqzkBT*|sTk zbMtMU%S+195;Vnxq~$k_TLQL-piV1V+_+BcjF1$guvVSZ_^Kjnz3n1a?1Nd@u3( zgkbQ#(DiBhjbh;LPE)=nmBg0Kt5VHsk7-T3i{sMMZG|t%Z6^`7$3ONboOeI3T{Trz zHTdG4-ixM93=9sUHxx!mceM{Gm0syZd8mzA(u^Nbla$bDxWLQfZV&K`y`? z3Z9`@tI~O#N2d9(T{GP(^%Bl8{Ob36j~g2=2A0o8ogm`wWU?gKjKXPz;dAY2iwNwr zrX?UT-qj3-@lJ{EK$8eGOnkfbr#M&(#4H+&t@ahRkO9U>sx9~+?DpkN zs(7&^h97XC=NBkza1o=|*h}*AL^C&K6YS!S{nX%U+QR_C{LH-#Lxrn27q4PA>I)=P*+%g3$?<}JhBJ!k)D`^FZ~%w$wWOr z+$CEpVv&!&d2|0q3^t^68yBxIrVXf;K%x_Le&x=gg!-Df%zeFm_9`B9>;Iz*hl~sj zcSeN{8HpgSqeQ58I&N$wF~`tr>CTN33mPa{{L-VD1ymb#G>~A7Rrf2D#cWuGb22gN7fk!dMKQ$ddrc3@M#wi-yNe{+*`zNrxrZE~ z-p|IOXs{k~JUyO65Fog(t%4XhAD!Q z32)W;Z+T>UCkwzoV*36ha-(=1q_r*Qks0}n^L-8pdHW}id#~1$*!O}b$B@|}R`q=- zS0r5-+~2~*qRds#ePr81rchz9EHD@A7NDx^r?iDz1^v8gk+Uck^Plm&f|7jv{E|mMx)1 zxVOfQ@*T!(uTR}Svt4^8^Q86uvY)dI*6&L$k6l&!xsVTv>E8LL`vdzI`{bQJVjBe1 zxB(W%VI>Vy(>_>E`51~pAwuY%$HS(uj`zrlSkv2EOXK#*0(AeYpUoa|(iNw0i%K&QZ7Wm@FB}c3nCC4HYyPM3tHzsca+Ly?>}G6| z7HOtXikl7v+zJs|^ZUhy2UPdOMqzM5h1QZ*vsHPME+b^oz^HMEF?%N z10WYGtdOa@+_3p0d6Tqt!rFVrt+(tP|L%zco~UInNusgX?^lO$u z!5X}X7smsH4L;s&PDdMwsSrW?vdGK)RE6Am`a}F(8P1pfz0s1i!SO`sK+?SiF>}$5 z>V|9EPXnq+W)mf>4p6tY{B*5_MfN)W%FLqCPrL`^JD6WZ`Y{+DKaS@=SH21i`Xmou z6c(y~cNfN+{Z}9f=w5Kh6l{o_!Qtiez|cY9c{$iI-TsV9DYSpL`P`2kaDf;KH+KON z(4WE~l_+KlAZzwlt!;V#8bcW_#2*VMVJ5n1y=6C_L~>&m?76LGL;vdW0v;ID$^GJv z&Glb%Nx1jM>seKQ24gFahk+v8TkcqvOvg#mrSDeqOW6%6ex>1>>aj9}d4#tl;;VR2-zu^J*F zFLEAxdr&V2g;*?HT&hMaH9}K3RO53gas$DKhs|~A6VPOtDs?%)LrF>bAucWelzevt zK6VD9d&9|ZV`yq=eF2#kwz6_fW8rNCp8zT}@$1*_xB2#(FIo%z8PcHKLlONvWE1O5 zBJS+WL*oOVfXP5bA7fbtibKH76N-n|>3#H)nNQ$mpu@w%85_=+KAbadD=RA$Xi7~h zsNZLUVnh1uoCi_(cC1dujAIoklQW0{W%CJ68r+BNmQkg=O6uhNS8A0s>!!Y!<d!!YH6Z!}>9=;FK$^eyUA3s9{EX+up?QkYe`mx*I)L0(?Gvw?L*$tKrESz69I z`lXf1y7BWN;29{jhrxPi4#Ka=xM-`l<^OfPV})|91u$Qra=7$qg@uxB6=?=)*Z^wQ ziC+ES;qe164S!NknC(F)|2mtlS(Iddw1scaG`@ zDuZ8ha+vt}?*LLzi~oNk#YGC^z&Zgb?gg$uKz=>Rf|+T(ra*sqxXq~fk zw*$#(S%pAk=I7(2x^QMhhlNS0sNe%CxrkP;(Ssjc1Sy3fgJ9iU`Za!ZSG&PN-e-K} z6&3cg-p5Zeo*VDXG=lpUm8W6=0)wx zSjMJ~cIS{|2;21GKRM|>l&WHTxBezio=2NcG&M=X!ooJIrHizf*w}t-FZ>tFeKK6) zWBxMjFSRXZ>6*~V1Pr{jwe>?{V(8CrqNl>iiy$ZR*Da#0E^KO|UFl6lM*AEvDHer< zV1c_9peBN}p!qLayPG#rDG1a>-%eCwAJ6(q{zGyXNxXQbpztkAG2wedpNzT*yryg9 z->|M4ST( zYG7bsb#3iiON-yv;6fR7V|dL29v&i41e8}*#WY@L9)4W~$omZQ)WSk`;549odULr0 zQVPH&W7qMEmb@bGPT=YQ969?F6PCEH{>jNV!iQb|$z>-lokE|%g&%NDAZ_LaV=y_3 z%@2_Iy1Umd^)&iidc2*gu^$rwDGJb_w7fjl(atOZBxFB-3kL^W$^vpmfo_VPkdoZJ z3X2}88|FJMZm$)WOJ^Mv3(+8#tqJqI>*ZuyMp4m8M^S~v+jC!m4lT9u@;yKT&&a`{ z?BwDSl{XWoD)B&9S9i+BNLxDrlti;{Dl)b4MMt&ahyEpo{YIAa#UDSCe*E}R=;Gt| zcpyb^ouvNq;>~Dr`$&;q3>?1d!olde%O{5l$l&|B&k?x* literal 9846 zcmbt)byU<}xA)N9(p>`r5(*Mh5>g{b2uPRG-9sxew1Xl;2ns6Q(p}OZ(v5(03DWf* zem>8-_qpq?_x~T{_Gw3L|usppAH`afe@)ED`-I=XcRa9xY%HX zGIy~Wd|_LtDJej%Z~i_sWWRwxSRg724|ROfwrBl3bbd8R?^(Mgb6CmkQ1gdiY9fSh zV_t^v*E-naMv5+U2`R;o^orXR>hTon>5lRz!fi%wkzs~%;!|NgW+mm!tO$-udaDrj zQcn46s0uE{-DQdHi3Y!_8lqQCTH4xZkm)Mt&h|(OWJ1Er zND8jZ+S;kSl}g9yz2S_j5%b4>bAjiMu=f(97Y)Vf1b&0B8ZJGbn@XQ7VBC;nXU7;f zawjq{Fes>etg3pqI3p>^`rz~rs#-aFI1kD^Vd;2#c_G4W+npp*vlTATQ zk{5ezY-|bd-a+fUcd5vb#&upQaCq(L*|>>cA%ld6G9mV>`ol3pT@gkpYEDkhGUs{e zq%qI1{K1Td)t1RC3A=fekmaW@Umhta%=zs~)hxw5D5#{JX>9O2?g0y3((>Cbt!N^f zot?FDcJ@BQf|v#zQW7&u$LN{(V$BDh-`UvQ?3dBY)$#c=q$FfF!VOEli{*E;c2M*CWQC( z!4wn~C;g2e#K)au_F`XG7RI03Jh<;Pg8)Xt#>R#o78XV$?)K>^J#)~7E1yv{DIlb* zrY3J~%{4VWJv2EP852X8e>h%b99>Yr=i%Y;p{|a;ySv*fBkQASM!*qmLPA2HBhwjO zjt)=saklO0!f`^@aqsuT_fjMSN8>|{G=$jEj`?|cebXEq94x%N1Wjk7#e2m;m%SHt z$K6~R)qnhUxk(D%5phK_1)i8sR=Z{)5J0|~Oz<_eo67ZBTK&Tm{R}ehuV*^G3c*fzubV5A`!|O8u4%6zU|r=&3}HhHTh$9)^Tp` zkF%m;n6R*LR~+L54habZ9v+@gfF|&>+m@Yi6%+AAMf|?Uv)<45m%FJvVK4eqI8*OC zlD}8y+HE{9x>)9tDTVXHrcq;*WYL~zUHiX!Z#7`IUL^e z%goH|+vPOeBn}9gdI0zMb!pN2{U0BhO6uuRO_ua6126XM23`9_Mn~VfE@)~pyNZmwAhTPe4(##*_uNY=BEH3UfciUf9QPI#4ZSsL|NJ-5uCY^6hmb0;Cj&E{G zOV6$EH#SPUUshL(rEW&jkeoxh<{JEh98Nx%wqIQZ4y}9K6%;hPGEuNHGn3WfNkky{ zOdE~d+}t>%rA;8pIy!F^87YPbADbeP$T5wn8po|d`+Opa_v(v_i=WENhwJ0xTHD8@ z3GbhD#2^qQHW7uQbN0mzP#~GiA)CG+%<%BYtjp5Fsfj$>`~axm@lN9JrOw`PBKm^B z`_=;tH>b<#`N5!!>0Hot0GUwZ)#`ii^wIm!%gf9Cl~m`RPbzQt-aRNX+d@M#6>9T3V!(K&xGXN(hC0)+u<+C600UER9Gq6O|&e|>fCwlQ*hvN52}*Vo7>jX@@GrvB=4k_k_7!%<3_glEr= zyZO}E*P<4)lGB5h`P4WVwhZ;rl$kR#cKLDA{A8-rV7kAJufz9ILSBi-JyN)v8!0ya)j~~bOi(X3|&#-f;r@aRHnymLNUtV6;ub58% z7yunGc{P-!)O)l&{W&{Z3HV3B97f!nz^h3=rnK#1Uh{ph4{fM=8uy^gb$!SFJ?nuk zK+7^Pu3^-A(G6wf|GQ(qs_C-d;@R`(pN-tQIlsvTOCC*F0?jSA5(vDVnli}l#W}w? z82nsP@+31W>p5_l{SInV1ft8aY8LvyYfDgO?!EV%ZD~Wkd6`u=`H+op0usp@FLPDx zdzz^j&B4z)5S{eHDj!p zHh*+1sUne&=Wr-J{&KHEfRj@(J@Cvn|6c-|yEzTNk6W%Skz8%8tpQ)i%w+T3EUe-i z3j%fnyi5tA?BrB*`p16ns?fMj_gfqzuU&<~7mW;la9fm~>&q9KG#I4&@2eaf1L=v* z`)fanXS$P{n_K3wu9dw@ZC%_3ym9DiYAUJa@NuuZo14{S+4GewgJHMWN9^U+gYViR zNTLy46dIDO&h@{;`;TgRvIJen47b<1-LvOhu(TV1zdfsTNFFt{v^K({@IKui=Jn<_ zF8=&kaiu>sTfaeee!k@pJX*Wrui8qz;eM@=EG^X4+uL?_xEWDV=<`!9%Y~AWF&23I z^nCvwf%A5ig}KFU?0_jmt>wfY(=xUD_~nrsoDHAve+gLrpF=sdwY3WCsdN;YCz_Gb z@^aJD(?}F5X9A;~X|vSCW^~DxnC$jTb2)`5GZZP3jRval2{#-x~uV_srQ@U}IzB`@{qSSSbYZD2^c#+`=C}e1OwCS9we$q2E;31V`Bq>=;)ddzGD5cv5GCv!NV(n4PkiT z)tB`6fG!*lNsW>)jAvAPWAJ?!2|zD#u5c4atLDphS=nn zh$9C9^^9#fn5m4!Q&}e`0%ArwNB6q4G%CoGKRr4@i%7g7K|F}DH#2e7dTDkp(R7^f zQtFY(pER3q8m2s*Ps^<>_K`tokd=W;LQTy}U6IVpGa>ICYe+CSX{Y@eUbO_D9<2H~ z%FJl$>SFt`y8xA{PvFN7YOJ|b7BkmGx^jyf#GCS6iR?!U$hQWg<&j9swhU{qOG`6R zK-1b6JEmV}%p*`J>Gt+NKqF+))Xr_8cn08@U0z+C_Mhob6}8+L$z9n65yKOGeZDS&uiILe= z(0%^=`MZs9SO4hZPF=pHW+^cx<)+KH@hBK!V`W7Hv8Y$E(BxNJim2%5{`q44QuKVi z;&0yb0TJgrbtIy5K3Q$Q;wATSzrd1j3;~uVMyB?TuM{3bZQ_F_&~$Nk&S!~e?GS#T z*}34z^*F_9uCltChF7oX>p^=B2()Wzd4ShE8Koyh4jVV@7i%-_$A8n-@;wRNlE_OG zNcG&mOy*zT*t~Xan=(L7J5zV}+`T&QpeCLkA0B3}sHiCO+|+&66xdK;+GJAcG|PiS z%pG5BWT5#|edg%hLr}ljO<5tGggT?j9Vhs&NE? zn(y@A00gkf7QeN><%u##t6ctKY4C#W8%I0a{f(L84$*mt_YB(dGQ{r)lCO*A>Djzp zg~6@_Z9R>dZ7&@XT_D+VyLFP7W@JLcebw z>Nom}^EX|%<#(d)S;y53KmDwgd=A5QHZ(DDnybvs#RG9{$?U>U2Cv(Zhx!`1@V1SE z620P5&E*yx#jX=; z{!>nlvW2DPucW4n&uL<=twx;+`J+9QdPb~5LPA$i6PXv02Q7jv~K}vl+ ziHL$i#oaqqACAAI3JTi29Y1cN1Tj_h=}Ky;c^kIkCX|!&A+X>*k*qU}kOub#&i3OXEGN;ya&HtbwiyIyTdYxVbX$NRq!v(;oY zq5~5+cfI)+T7)4X7Z;~_Oaj^2!R(N!y7Ld)TS5^^4x#w#DY(nf*k(mnl{NyBHw}Iv z)FzOlt&qF|!khw3sD^I+<7R3^W98N%xYuxSpg}RecDh_G!F|=k!d`f=&tcs+!QojVv$pz4x+(wzt!Fu!%q# zU+m5$IrC#2$6*RnM+cIvfpI)XaIvl-tCg@?(S17{@3zz0YyaY~i0z|GfD|UkFBX6s1gHL7h3q z9}2GUKBL@^QRd{~7eCB5V0zRg=C4ZSE3$E_SD6+jtS; zaqgtvN0Yf0-@7R>T$0f}!?|E7-mb`q7vNHckd@Y(b z4opl~MDD%$AQC=PVuu;rzBD|^6BYq}0#?E|EVo=3zr-ORyG_oL_>*LH`=F#{)Hdfi#RUoWolgZ8|@Xi1JNLnfC@Be6yFo zx^!OZtgV_sx3a=~bgn<>eCU(1gw--(zM*58^_$##I1PlY1M!~S=xqvU4`U91wLK$FA8b;EJWD&hL7KkM?x65KbhcY$&z zgq9xXnD`M@|Iu11WZ`dOc!G5m=6Xv~?k=qLOWO#trRwF$k>SF46Snj@A&p4S{GBXP zN^)~@U0`%J72l$J$BMANsQ&^>x%ZG3YCZ{UP3GD*&Bx~4%k0V@ki^HonB~D|e@F&3 zpElpftD3(*huVnZ4PH&hASb_=Wy6r9N)b2uBx&!6U1{7B+QRgRPeO4v>3+BBXU#K-=h~bjQI>ZVdI@9 zPn-Xko{naG;Ps)roTBqA?qgmSK8_zTI%MBs-{pC0V%O*@z|ZIisQ606I~ZtqGYPBXt8PfWb$`&J6qCU`LG&0o_NQ~0jX28o8@2IH9j`2HON$=S;Jic3y`K?ok&Ge;D~T-I>avtukHaRCKrk#~m zSNgw;tw*!%f6f6nYPAH8!}*8b^!x)yxR57AlViG4oS|^3r`BAHs{@LI%vZrz7Z<_y z%$s4}4LK7N`t7xXGyZF7@sW{{Hum=NMH18zaN@}sD3=5dF7uG_IYDA(PT$1e_h)2{wWIiP*X$ezS8&b#S5san%WSE3ZL=<9FHdvKaZ8& z{BdtMf0f=n0RfWH(a|00J9EG1{J!xTiaKLwW&!KDSq4RgLF|toS~sj7tuF4ot1pWg<6h7yp$<{L=7A6Ov^LrfSs6~AzG71Av-S|4*?q$DD0ek_U>bjho9 zp(T5;hCcz~6F&VC$h`jn5kRP3T!nJM6(oOj#&r$~q(s@^1gXfJE>nO=Ew!mQFYoU4 zby+XJyp|R@WMS5Oz5~F7Q2?H}II3#uD0o&bXM-9JfD(6yQr$5K1=%%3*3OPcJzWy7nMSx0-KKhLbi}SgSzZ08y0KHY z7`ySLRct4m8`o!>?x+;c)n;SOt$N^pjkW~Pf&sa=NlEg><5~6f3`wSeHJ#h#dgXs( ztG((*rK4TZIU(m>Di;2MCP!8`V(I4%S7)1=+S-;^S7V0GifU>E zF9y>cuP`4Kk1tJHXD9;i1+oCi-a|g)+FKAl>5oDIDV7F1Th~W+VVb|Pu` z3#qtyF~*8<1_CbBZxPnt7yp4+Q??n={O)zc!Ylo4MdJDM=hZv4n|<@sb&c{_LuLR* zC1%#h9-ge{?lkEKfNH7;34>TLz;#FL46rf;X@alM1eRi(tp|@TmneLj^W{CumA-vk zkGa%--_@n;Fj>~IeKuEJ?a=$~4&u-FLQMn{dcqqtdgcHMEQw!nL-JWHublSw?5?=ScJ|xq2ojl7|xfu-EX$UcL1P^K}=P zY;|I_IJ$+IzM((aIP0@JA4@NO$QFi~5vl_BC#)sHSX-+|=e^JOJ{3;p+<@h>p@>f= zjYmNlEQD8jlCH$nJX!t?*&KBC=C?_+0;?QwRL-ISw9Ux4MzoPlxQi{^Ctig1x6CyL ztvI>ypw#N_7t-53+u~*sykyO+0R(5G?61kG8Q;1a!Ht}^4HlMF%!Z*kQfWeWxQ~(! zTl3O2oQ<65RMN<)1GPyWZO=KT!CkKd)f<#qyR)l?+@Eu}D%;sKBOdH7gB||xr-Qyy zB9Dl5*A_UM36jY+`Sg0_fc`>)UWLJJ;Bmu*`|A5ECp@IysU++hv_13L{HrS$vC$jh zJo2AHBe)Pt(4TgcRW>bpo&tB>HCY;oZfLl2A_Fg6bYTj5wWn|G9Cyl}V*|p1%SnQ? zpCdbx8$-S`sPq9r!cIA1Y)^SUbtr_cC|iIfn)j>*K%vh{HEqO(!y>zXU)jCf7Xz{t zYGO!+$hyI-Y+^Z90PmFxDk!T+0!mwe{8Ru2S<04(ZDE3@!I8MZ8Ex&Oe&jg9fEs3V zJYxy;p;W?z@ZsXsCHT2D-83E+%))fTJ(BN43ii^CLnzyXqipe55bdQ4$M5T_VpoZf zFoMfj^333$<>F?H>-3~s_UqpYV#E}%Pby4=1R4=Jj?*q))At8ZkAx9}E5V++0)^#7 zXSgEYVm+K&Z8Q3kPr;#UuX41jWMN>iP!WHqKNx?Az9@`?`_hkGR2U*U+YZ=o;i3fb z*8IpbfSHg;1H1Z)ez_g%|Ik*J{lIo zBhiMHG++~Kb?7`WbX$(nyw71kFhugQVR)o0?2$y+G1Z5x;Y8~Vwfyx9tDcZ3U{9gPuc!mPqThGruy>sRN!K@o-wCZ=c> zOPI*dmgw%5HB3DQW0Aoqj4?8_620j#+uK5jcrEF35cq``t!5})f3=MohvIXg-NPwh zpOoVH8jYVtu{}>ZvwieATYh*Phj8*f&FX@c)YyS_L$FLgTD@d~m^e7$ryf*9wkkgf zjnq>jzXwCyBD-adeq-9M0JBoWe2nsO%fSzoqb! z9|j3iB6p`l2h5(MxC1pT(ld=JO_u%K+glnmzQv?-MSv!Zyc3x<*b|Dmy0zkh2gN4s8Q8wd@`kc_SmuLUmR zVHrNUW%AO3SYOHQ6Ge6{9En|R>O8`|5W!4(&*ofq)^yCWLAH>3BwM^Wo;~thN__OE zEiqs#4TG^qR?IRPHGcz~^}k2pnQbAYCTDj8(ExDQDaZED3Hrsk`SR}~!W2*nc{<9` zkuE_uTko=DC4)jceSsTFh2>>pa}tkh5^AdfUCm|Y|2dY;}cd6#uxxho8BVMXdj=1h3q zg07qCHwuamwL(>UYFL0(DuM7GBD!;=VN8jL>Jc^C7q+#LU>;k}M0EKt=;Y) zGq_{U4AuR|#hmcY1<}I){vKxjlJ)PPIm)c-A8{DcKvz+J4XsKLHXszwv9pdNOqVl| z_S^dn;-%`_CVEnCxSyZ7jl|3M=Rvab6dE9SdyPtq`7bDa{xJu>4Iu@@(a|xds3_c| zSd7?6d<9#sXp0_%c!VHy|L=bVrQ0OB_~&}lva zG7)#*(N6Wid58?;cNy5DU}CGw1i__2>H>(iHa z>Pm|;(KmT`o&<~=e_vVoUvTbd#Hs!D)nzqkvw%SUJG`ADjCTWX=T%sA;1`%Q(9H)N zGw#&wO3(N##opgcnaF=;WE7LgtH-m?Kvtt*^l5nA$XJAX6I4r;L9d9zZ0*yrVpD<5 zvBEcnhE+Ww7&xD*s>Xj-+S=Ow0PyO8wfpM8O_lbhtb7Aj!vQRvw6*0B;Tg{$&gSdu z3z`>X$;rt%czM-|8jcLaJ=R_$5VI+R!2P~%2Pc;=mY zJQrism3U9QzP_&8dy`Y$wXaJ)= z8@bso(~ea)y&z?V_dwrFws<1?>63|!6>e6Lb>6viN4NK;lVT?5`f_DgTU*-|RMa8g zN=hhYAtnC_Q~UZVC}4urYOhn~ip(_d^mAK^K_L}rt?1ceK@!kF@ozGxl#~>{ciYod zB-VrJ9iUq3V4|y|6S}|e2I>eP|5@7a6{yOqaGI6q1H}ycv4YYy0ipG^wPtM%spmU{ zmH*C&p16rdGFYIv7tS_|l|Uu9qfF-FCkd4jCuDD{d@!@IkqJmd%x8}Uv?}03s4@m- z;G2a2TW^anlu@K;ubfTZ)K?NcGQf`udt_1*!?_%;2)JyGGu# zo>aYF+f&agOmubM_5XSm`QBbWoL|&9z@~TwD0puzD_+8%kBw6Zc8ipjHu8>PWi=?V zL4JpFZL_-JXt;wWDR4dg&%UYu9J;AFwvURNMIyF9%`wRodnxTLaKVDrI=E#2zgw+t dUTMF^`JAl6Qg`eE@CgV+MNwU$NY3o#{{dM7|Kk7v diff --git a/docs/images/chapters/arclengthapprox/537260c4aa9e98ffdea7c8120afbd427.png b/docs/images/chapters/arclengthapprox/537260c4aa9e98ffdea7c8120afbd427.png index be0034c7146319fc5cc0088ec2f7ad78749111b0..3e923d93621cd7dff5eb4a218c1e11e794aa804d 100644 GIT binary patch literal 11325 zcmai41z1&0w?0UxgrqbGNQcr5A|Mh-m~|_S~F|SJMUVd>Z(uiaj0-05D31KqWlX81ce0oi-iG7NQ+j! zgFhI?&z{OdZjt}8nhN3|5PFD`yo{!6+MjtJ-Iwrrsr|Dy2(}UyTi9oMW=$PTt)!M8 zwN*9BuhW*CML190Cw8~(QWpolb^qQu8$s-|m%V@ErR^p9zVtPGUGUA-cV*4O38&Lw zgs$90chy| zc^fAw9CbhT`@@G1nTUlFcIF$%Nl8hwRZTid^y`UhYHGl3&mDFaE-tPlL2IfHA3h9? zjPSn@xpBY66-lX-{8U|Awjd+9o;>cF81`eCLl%9OZ3yHPg$DmmF6&7h_sFl zIT@GU+BjS~cX%mEY}Z$>RKd%uY0=f=bjKe4ha;+hL#NEJJqX`*JzfVAVB;eab>#ki zB30NvXVN{x(<`BQ#cH9EGCn>&7cq}R#S+tIXjtBKO7*(Q{qfb7n$Ir-k3WyIQ!gXQ zRZu%|4Ta{nF_@%allaV-T@=rrKYx32abCI;uKuUIFA?h9yW4oATJG9BVPdre`(~T=^(*^nIP}_BvxTlLjNH6N&*%{qrWd5u z4;9nyY~ICWcdlO8ZV3{hz$6?c=jJBz;>C+B?PB}Qp=_B%pTl0(9vCcm*rKC1j+M`0 zLxuUl1EY=pWMTUiw6O5-O4~)ro!Pp(ot>S#fAnt=`@vH3%E~ykHuGWKbUxI*y}eVF zCc$EpBO@>Fk&-S)3N6^dcV`fwFiG5vzbo>7u!P6y`J7traEO}JP4%KXb$517&YgGf z-u3kKv_cVw~y!74+%GLSxd*kXS(H~Ob><@ZdX zWEH|t`JTZ+1vxpnmOnfOjUl8Q+Dj*sP?wg)<`q}T1)p25IiK4bgUmSiUI%NY=f{!~ z&diG|cjzOkftb_D6WtbXi44y}rE*WtQlTmHX`w+dOZ&SGdp{Nv4J8zqGL{a#Pkw6& z+RO=?Rq7@UTbLFV74=)z@I!Wh9{PZ6EPVw0K zTy5?hsk7bNS?LTn@Og)ES-j^cI{oc!-&Cq?b1#L73S(>q7%59!@08eHV>^Y(4p?+5 z?D5~Oudlr`aY2u%LeXBw!%q#$*MZIDEJiB8VWr>|k1i{@J#Q9z$BiXJh(X89Q2i@3R>gp_wT>@{zn$mnvC zvBWE1wEA<`(Q#!BDT(4_)v~#?gvs4R4;kRmO!fj>)DV@MUczPE- zAPyDtiB%lC#jmd~)8*W)%P^;gO@3C+-}Ty$-^Bn-^El-M{=m2ep6UMk zVM0d6AmY4R3TNi2PiD6ROrm>abaZ;p=k^Na$bS8%4~qDtrK8i37)c|R@TUGlV7iDy zS8=EnE{fNUu6$ySn=zB_z$5L{ zWrVi?lcw%sY)HbAMDbYbA+nnOh91Eq?YbODZg;+K1ZmZQdIe@=&{{1v(>QGO+uN?% z))#pP!$qGQZn=F<_fhUr&jTk8fdNA^q}Y9PE#{TB{r4wv!y%Q3aFS9>j;p zGZKl4iJ?H^Sk)A)t+}WPFn63A7Hd4HZ(q*V=H{Y~ z7O3sjMa|oEhmx=w3Z^1$RpNH+rP^{&EYo=>k2ixWotGCaO!A5|idy93zW#zCD9csvsWlhhoUx&$RQzUw!9~>N}FOI;T>4Ug{TOF?T;z3eHo!CjjJ9nC| zc1;qw-xx>Save0xReejB@JJX)76^Xu;6WuYs;ZzMj9QCfM(sCm2EG;-PlItb*7|3* zD3;yc9z^?qh4~rm5%q}@x7SB1+3LJq6k=|N%zb@*CfL~6$i}zpGRv%^XeDNBo0@GC zx8Y#Vs(SeH3va#t;ICi5=6p`Xp_7ZX!2~qMJdOLZE#eE_SJnx!v2FYU0?7)wTW8n6 zs*~jzjan|p%hCSiw!l{JFP>w>DKeLQ`!)iorteqUUGVM8bwH{D_?hS zq6JrEWF#9mw*uIsL-_IV*;xY6TFt7mzkIHr6OTH^w*f3`CxzP_^`}C zYjuCQosEMdyod}ZU4VfZ0z7=fE5(!wi{UE0^Z# z9kNwjG&T6Z*JbYr%?7@9CGj=gWJHH=clIZb8*F#s{MGT}x61+8dM(>|1xGMo0+Gt9 z%d)05dgzN|Qzr@-xB!&JQrTwN$3@Jipjir+6RBQtFlQ)ol3g`5NiX7);yh6&$#%WR zGqUpV9?s^8dv#rLO!7q_%xPv;Yu70_#o_yoE&ahJZ*UBxP}4KJupEjPoO|1;9xNAd zdHJH;=?Z6kphMwM0jjBjg)QyvEosV6IWs#xWHFJG|KP530|mIaxZIG>O>gj%Rf53Y zP_=j}nm)K~X54IZj1EcUfhy?frCh;({17eFWCvHKkYFl6dth|{zj`q7+V*6212`&F z-fu#eD@m=d92hI7pB%6)D0Xb`?8pOurNR;igRg%O0T{(=f!KNctCpggT6Yls{l)}3 zF=xDIiQIt(<(Nwn7lIG~OpusEwSc6-Wl!Px^GMjFB(PFY(}095;#K-X0;j^@`X#E+8+eUFD=9E8R*1q1CIA68qVRO_$qV|1~ zQta-N*+(i@9~W)k$|cm6h$Gc!0fdVgvn_=14D zGgMp53A|(5c%aZ?EneQZwzig4Q=`>mgAp*8Ovq2I z$F+l^g0*l~a>oR(Sf{)TY~1_jtI=2U+#({BV6Ujq<&?ae32EJOTuFCXXoGp5R(a2& z9T5)G0XzDudi^3{xapKJq5`X2j2Jvqm>(IYNACdtRgdCPF>$$zm$7S`CIDf`6%!Vf?2>e>8HIt zAJ|e>M$nhYg9%BK@@a@Dq|am$!owE*h|UwN8d`40qZp_ zEKG7m&aAoy-+^prKuq??mTy{`x;c>`@JjreWBh-Y<`}O80$(LJJd0ahgTgcfWqM%` zrbT6wx*~wkAWaqX-Mfb><}lg!wWttH{>4#(4xAL!gcXg(?C;(*Jp+NV+0r@l0T~$? z`@eY5{xaC?EDn(yaO5&&+gy7ug;Wjj7=ITOCLPNmhcaZjm8L|8G~LhRI23r&3JQe+ z*c^X7t7598pmhvSc9S&S4?qhWOgxtdHKc(MpiianPn`in6=5< z&5CqMVPTMn6VA%R30vC|@H#rQam%=0Se?af9z59NAn}LMZeR zCMN@mB0Ep;O=Y(s^!rd##D?$sFm=Vf=oR#w8qxzdH{b4B@KH2@0~U72dY`njb72+M zGte~A@0rK+k_*LJvNk}|A?lA<%%zdVVP$K0A)t9#4L(Xju=E0QjQQz7zgSP2pqM}$ z!tBDge;2e-42}GRDXV$rfck_`FZ8@Th>}wFAhQ(^QXWgRgYq7-ewHym8?wGkT^+^W z1!c$ne^qfz68QT~r>Q#Vno`AI%N>BCpaV%wpbA8%3?5FE}=3v z9}{z8-o5n9_?MvgUz%u=#HaTz|B;gtFWr+g0Epn9ZOn)~L}SPuC1D|#QJZ?7JRukh zUWX;RsF+_wgt|HLDRM;-rblHGK8$K+_s@0-iSZj^h%!VD2vlzS>-CRMOccn5WV!w+ zU}yP=93e0sv|J#)6bs4FC~97QplDo(G|-VxN@|f7F$8pVE*GbWi5v{#dK>urJIJ?C z3RJ|YP*(M$v1EBXqTUSv)ny!7iATsyVkGt&eMk z3o3PR-cwo>VVfacAhFuL>dgRo)Gpu1`+m<;(MX9u;DX($J>R=&`9{Q*2$AWeDCK$;sfEKD|CNj^6LJJgy0(+3^iaw`U;61Pl_ znL3?pxwG1Zas&?sVm;*|A#R{1K;R6X{sHsLuHF~WQ6_bpocYOqp%F|2Y2jO0&wa3_vJv3g)*Q<9P#NNRDJ=!tG&BU>>B=hEX=3E)QWT8(5GrRKsThwZjK71+KM#q zIKYEIB``t6uXmgcM_G;=FWGMnF)3?Q6s;~sGks@mpYGMQ(EL`(As9e;W*j_EaM7Q< zxloDPLvo@YOv>t9&#CBjb(3}t0p6?!+$`6T< z5Z_;LMNJYVO%h-QgWv*#*fg&?5GedJOAeZ+0zDqj)|2$WFwI~DPAnRcVAwxSj`!ip zr$7uAjcDX^6GC*q)E1@63@Iv7IZY+n2z~4tWd(F({h3OC+gSBtQ%#^JdDddm^FOwJ z{+M#eM;O4ExBIdSbzR$8Dx5C4rL*HB>PrrDAhEG26;2pvTM~ zT;D2*9`Lk89DiW}2UKv@;U3!xny9q~67Dt%fIgW)pWs}I0x{FlF;TI#ZGo@4d+=5L zn0Y^+WGaYS-grm+#t|&iSDGq6(}2Vfl_ATUvN-Rf`m(Yzjoycx*7Ls>d0c;wfnik1 zYZsHs!hM}?j4^RLNuuzNsk;wu`2|ZRu?lYRnQ6@{rJJ2N`l;~KP#VoD4!5@|F&}3< z%j?V)F>Dt%_$KQ|&_w%(os_@j{3l!#+(4wW)$v5&cGHUtowK$0uKO}Qs1*Kk4y_nc zv@{z54;$i^UlgDdbfQ6|Z&o_(LN1}64k7Y|JY49ILL0L9=F{Ez@UXD3*k^*$;5Z9# znzh|WC#Qp8;q6IGR$5AaAo`br$>F zaAh|)FPN8kt=dt;z4P+&&LgYbj?BP$A}2rp?WXPLzCCYD&$EVBp=Q2oT~l4*t|{r@ zRh!tUov=JtvIf*dQ7((E-wN8_&|Y>aGl;mOoexJ3U95x`K356oH9f2^)#rX{Z(n(Z zyYuefSaYsVINE^|k$@yH{}W+8Z<+F{TNWO?@tH(ihF85CWvQ;IF`BKl_Fg1WlU$U_ zB}x*fJl|w8{B=*p9wS0XK63J+J?&$Eb(;f(>iIV$1mXhZN#bgq2J3U3U`mW)8ull|w3^SQJ*m54uP#?J@0brl=zQ=1PJk#XDoV@9py1%(K(KC~ zjs68zY3#fea*19P-HQOo3!`n;o;nziY2r7>qq>#Gfs4W-A|e*n*7(9ab$y8!z0r~g z*G)b}fRZSm;7O-Whxs@q4UcZi+kCy}VDqe#ViI~*-h$D5>qRe?di#m_0sigdK$_0e zxU{s0-CY}DYJ$vQHH1MQfUki0)BWN2P`gBzFHhKC)Qwp8$LCBocdWoq5nzYPZSD0H zEJi({nQXPEW{Hkxo&BRi?g2YX*%5UH ztIC^FjLC1{0Q@;I5fySe-{>}8^s=DBYMQ^r{)=ilPN{bI&W_dDI1|(!vb5CZ&&JM9 z+|1O})cR5lWv>zS`Y3a3^f5J`A0CGPkd}#L#=x>E)H!K7`LRb;J7SXpwP;0ez0!pBjSc>|-bXvGF!3Y2(JjaJL70^wlAowZNFbA`)b$XnTJ+#C z2`A$t%Y%s|weXOaIaR`5m!oshnk=7OaXz9w5x~xA_L6UodranH9@VK^8O_0ECQxCD zoiz+OIk|3nqadxSO7QjT*K31V><&P5#-^uB>eM&^x(@l4z!*B5V z_O)URW>M@?t}#EF_f-g<8C=!7ITAF4j;kO!tmycMYQKl+`t5<5WOjQBKwk1$GT^I@ zW#999xNm##MN75I9N*7rg+iF*DvU|u5Nm%Mit4CKqEVQH_p{em`=t=~?*Rm;l9!9> zx<(X%;`>9CUYDjWCvy9`2TPxpHP=5Zl#T#H|9ABgaHqt>6l;-*SG!F7Ns~}aetyG? z`MsJU0RQd z*UYzAtn&Az6Cy@|a|0HwUKwfOsP6u9QIc_yDs{6ZT9KZN6SCmJBdnwl2b01y8(5*E z!YBR2*6QTqB9QISs*r1gDc2WiQ0;VPoFgA#RfH8PTh@=YZbyG{I%_2Agu^Kz;cCyM zd%cLj=)FAIT=<55`stH;fq<2sfD~qlvS7cII1#rQ-1pDy)3xk~N&_$Xk12~@XEMjj z!HSm$(f%ZNvwd!Z?v=wXV!<5l|FR7MxY(dSU4ZffUF>p_6kX10w;w%OT!_PssJg-6 z*=ha}lg2}q!DL0Bl{2rBs`N0aq37X3DxM)uX6fkG>?zcCwKIwMjLy-Oq z{Qfj-YH`ItX5}YNeWp`j2OG@PRT3 zMRr>6mi0#|I z$ILr0+0P1VX%qwSdf6D^iRdGHe(b)q9DH$g?RkiuZV_o@bjyY3Mi*?0;VE~~o^KfgpP@; zgQc;e*U>!9kmeg}!&yN@*D~`);2S>dZb?R&lMa$rH6I+$a|S8B)7iz1+nmJsgR_Bj zKBh6sLjkzZBS?YTk;4S7)Sy~6uL*|TUXy3tIi&z3lzA-q6E8nwXNOtU^7X|L;Bt8v z#EA;j+2(|%RrmSsdkCKqVgKlm_qP0qj(eVJn|%ipYLf@f9oy%2qtQL){vY_hfy~xrS{tX-FkvZ+{KF^4MG7OTTKl?ITZs zXP|N$tf|=~#G~AmxR>_YI7y|>CJZtB^qS$Br?|Ys#(rmUNrcqhPf5bplZ<>13K#WK z2ZB;kr~p2EG@Gh1uNBGSK?0!{k|fXX%~wwy7xp~o-Qad5{E@-Ee3lYQ>gINBGA}A! zuxcw_wj@L%*7RHt0aC(VaVHh zdOAzkbj7{DQw7eBfF3+e>^zKL~U>xUDxBc{HUj*&-%JXuyoO>+8H78EYpFV z8S?O9MPfV(DXH)HuS|0#<&^?a!Vh$GvI@REmNl|~I_aW!jNK4-4?|7J@HhN!c1phHaA<9Vmu52E`fk-x*P^JrD2 zu;Z_ zu2E+#1TWRk2gKBq%36QUIjUsF;HAbbm&W0{ewe>)zx6UBQK>ahS=xx5gFtXMkhuVv zk~B(6O6u(K$e0hrcU8DNm`AUa48&Gf^i@iqydgialLCtZuR<6D=Wr!)RR05U$u(q=~9E9 zdLuH=f0;hlaXCPXJ z@XN_z)YQ}j45RVc?tEcU5emd=rWzZZ3ktt}4f}4!(uw=Lv1|L=~HW0cKCVD4QJ5D2~LEP&qMubH0VakjPPao(K^eG*QF0s%o}5HMGN`O+VN z-7|Nvguro^smebF=>3WRwlW3Af8z z2;N;&=-;exdY>y&()Osf>zHxa#2p=30R-LGb^rTwmLN^`&NnXefvA`S5D377>IMb` zkf}P`OtD~p%)2>-g%;J$$uWgQ+MT8YDUWq43{k);1t#HXQ1i~pYNxIQaJO&YzEupP z^PhcraYQc|Ok$ltTb=S_*ls-JEVU)*1-IYvhWxXCxZXdD#+8O($qAELA5|0cndEiJ9Wk`g?NhRFLur(%zkC`f{pdWq-KMab?Jvr0cq%o-nyO!-nQe6UAgfiE<) zy5AfOM)SVHXyzFJj81`%Y9H zMJXjMhMAVl-YX>|mP ziMmZJQ>foaG@aKK-OsF{TsFj`6vFM(RgyJbbc=~$01PQ!Zzv-(>#zC1`~l<6i}%;e zYG4HeH7F*Ac}0;kGSArk!)wbY*Q=m z@UTDiR|C7cPh=V6JivgTDf_>4cus7mlmI}VGe?}z z@T~owxRnW~WrVys6{g~m|A}!!rV!FRfB6QL=v^ZrHo2w>V+>rnV~ga|{QM=d{Evy#MQ9C2k}1g!On!fIfT9S96a4ug7twKi3k?ZzFx2{gX%b;(SR8cBZG z*g_a80cb>Q9E_~K-43l{)Ial0A5z$rmwnb4-#`2+-_U7DSIE4||}g(i4(0`pzp+8;@DjRWaMb*>qm@@jcLrJ?gLu9qfxpGciKu2l(Yh@ zRWCv_J}xjc_J{})lR6tTGBT=OT)y(}kT9U*#qk6UdUCGFK?o`(Wu)?LALH@q!tc?M zsltv#=t9tV@#CI>rZ9lrjJhHy8Rk-lpUh)mAGAL<*jy;7bX9*0_%uGfa(xttrVi|o zote9-$-b~G0~-Q3jND|K9%K{)qEm-hxPR?NT$lur~Nl07WHp%F)hjOD=KW_U+7^725ycT&j}MArL&VDLaL#m!S_K4!{TpL zMB@yo*8ntnDJeSsWIRhJv5B&AdY*E0<4PKExw&!aHfW>P{Gb6ZT^8KTTmCqS(g<2j z`RW+Bp+k7RE*(LdnoN^Me)-Y`fA7%+xW1;Q_dYkB)79@tW@ct8tY-yf_E6@3e~h6A#9Zg62hZ%5 zf1*DU5^6W(_2mm18`E@gxjc7W6j~E?KV}28>8x$NUTKg%x67^7|Dx+|HYS8rp@@~w zpFbOJjpU>5o*5#;I%xj_D=3gljwiBiXfnraP5z_WQ$u^6$ zWf0u!Vf%+U0}*_P;l<&4lBm;f-$1l~kOd&!0K`DqM5D4MFXei(caq&t$_=94u ztRM%0!GCg^3zHxaI>>W5X-)5p{rNycO^+6-qbVL9B)&TmlSm4R&$R>UUQF#5me%&{f0$C)f2F;npYlB~t>E?R^kUNKFYd(1FUSg9 zP{Pf~tgqg}o(#P<4S0isGf;kcdYl$9^k&F!VRPVU!)@sLt)xN+nFU^(tpJ07fM#Fb zJ9Y7Bh2uMRhG-PWRS!2c~G~)G`Mj5DDuMx0<=B+6_UF5zZ7Wwk)!0^-QQlm&x#T2 znk8R2dM`Fjb#XBNeqcbQtEY#7pP$h0@_;(!(nG<@>ha0xDLy6TdnnY9O7blkWMq6i z#AA1oo}C?&FFj^>yDyQkZ+KWiMh0d$r-@bk8T71Hspj~1W!z<)wME?%E+s-yyYhPwE#t}RX>T4-5=H-GK;TYzdAtcUG@gj`M;Nxyf={)5lN>Ob&urj znFbo^qVe6Lk2rL@*wX{g7a7m3a1@m0fvgYgOBgS>l*!wUo6=bcXw||sHq2;)v^ZS zXe56B(C56JC=cjf@V_)Fn6R5^JZ$L?xb8PO*)@RqA9s`L+1arvD=C>9)H;~;r*hjU zef{vYprER5!M8*wuD!jTgP%WXf1&v^!q)kST@73V4mW4p96~}VFEXAaQj2?!I7yw= zaX@N)&+Sq^eCQY(8@r-{3wCvN)ipO~Flcc7I)6Lol$3v-DN$8jJ+sj42bO0=Oib+1 z5sL2F|J|_R!(%2Uo?^|NGk+t;`267tXO%hs)g70O0rHgOy-N>-E5?$B@ce?a!Cr)_hVk)Ia&nVpHFen(@P&6HjhLYa3&RI`9?d=)y4v=);i8j z)55NLX}=M)2{}Rg1SqiQD$I~Ri+b3#mbcy>G~SuSeRgh1Z@!qdSIHl(7#n-#+0R+W z(?hnqvt$13*)xaJjlsPku|rGw{6v$*8)rQ|a(i1_)7y)E3wnC`@87@2z&$Jw765cM zW-kV8J`;?Ds_W_LS^h+Oyim<%uFly;zchvV$(tac=+PF$Mcv+?92VcFsvy%h=ez0h z`DV$<$#>ixYrm3aHU>Y#FO;`F_?A0vQMMl5#7W|os7c*_d?qi?2jO62i}t^mvGMF@ zC&H)=xc8l$oV2c^p7buO?H#q*VlBwOb1-W3j9uT{tPVlN()}`SplNPyu4iQmx5?1tcRh=(|R%QRGI?1~G08x7h zjO!M(8C>n`zy}e(qjuE&i|p!k@?F>|?Cx}U#{Y7@S4x;7R)skU{r5PzLU#F*Iq;m0 zmDN>0t)r`}E8|uuH8ph%Z_k;p1ST%-;pO?kfsyWiK>_i8hScXE8ZlZgNwY^w9!sp_*ccM=+!p*nSkRb9P6Cx6ASuWmCvEvRk! zMSn{J7pkb>vF0JWa&*^Hp>ho2oWpg4@AB1nsg28tdY9F`l)%kQFWrCuNexz_JKEru z?yiZ8CpQjM-R>W=C%^<|wnp=3=A2sxTT?!M{ImO}=jAfA#-6nQ_wOn{30Bs=GA()m zYePdSdpo<8?0S$6DKj}u)Zrbq)$I##}D=#hYyG*DZ$svh0{FHQ;qw=?XBx z1xA2nXJ!s^*S#KRURzs(Ktx4Fk2a-X*d+T`@6MggSi*1}{V!UGhA; zqlvv&V}(birz4yFE;}5AkRTl$9pGzU@6YXm&qu(Gmu?SRQ7vLjhVXZ*tgC07^b|y5 zXr*Wj>YUau!?YHuGD;6)R0~(@j2IGgnw#gH(z9WAn@{|21VCu601lTrcVO#Bd zwHeOZzw|IL@joZ-A0D;=yVc=j?bqJf_BilHnt&^_TIWT{hx>rj&x>dJi#O#puP$Qk z{afrN*b3BHbwMig+eo*$nxggF<0T;>>5*!Evy8~j#bwPXAG38nHS|(tx2iTUGh;mm z1mo@TLOoAUA>dtmQ7l=e7-1b|I77AV7anu1Sis`e+uPe%kp8|tQxGk2+)&!gEi7h0 zg0#JC(J!4`zj$(sb=$Ld>vQHGIH%y}C#F%XxmVVBVD8$GvA(f!5@=T1|D~|78f0AF zSozgAi)Zr+F}#Jmj*bgW=@OSvx4Y8^$q=|BJ%0SS5+u`F>%mU}t=wfN?muYBEWYRD z1U17RZkypd*KxJ+=B||2Q3|9T|CIgd%1*NY8SEXN-WgsVTE(`AJymh9>sL)+lh!)T ziwgPQxYwCKVP`gfLp&VYD)MdxJ9PS3-@stz?)voE`_271drIeryX$70WfaQcod{1V za+1m22N4kw@6$oydHqyuX@&f!!~_KIaHZ}>#$S7PsXNx$c6ZH?W(FYmo_60azZXeb z-mISAMyFr+e}kE#G&~tO>3bD4atLsgPoF3RgoN@+OR-~k<(@sG=j6n0XlO|IWtpYw zGX0X5zUu`=EQ*$v*2wO8v$0OegnG%ulfVb@=;&xXO3K|_EQt_+n4EahRi^8lX|Yg{ z2nco(vrVGb)ab1oJQd!Ov(+p8TTnLZvnQa7b(=m#vIW+A(Tp@jf zgU`IZpXlrNm=0&H+d5=~9yp2oYy|%ebZ1HQ=e~*!mdS1V@;r&EXS%`7rkpYey@5Bi zbYsJMAWMQ$=_^5ZcekyRQ>|aTgx|$d!Dpux3mP%cf%TvNB>-56m5VD}Cbxbx+w!m9 z{r&33#^K+-JytK~!^6i91|X}@Uj=N)xQ-5)hwE@`$_k%TYr`gQd=On$S>mjPlC0+R z0c2uUNeco#U} zJy>!I3XN}JNT@hVAT_JAC41i;_ws@b(P*_02=j*+N@HMPtoW^p8VCyt60@jf5w#Ff z0TkW`>9gcpTVJO?oO2$%IA*ColPYKA_W6?IQKQcZ^5VX+vy+qj^XKTlfB!ZtE7vZ` z)nr3}L`6lZ7i;3AR@2aQcXgR=4rOvpy--wa1KID<`XLte1|A<7HujdMYu-fq)|Ty$ z+S&`>eGnb_TCA%zk(nz?Cq3l^D9w66Lx~?lpUyI3h zhb9E?Y$JU7G?Mx0_k6tuU9n~~07Fq(L#CpcQHi*46{;`#qjq%%|60o$>S9;trMJwX zpa_b=QnOhW1aXE>{rp^0#o_PkJG;S@NH8>TowhC`1ey{Wth7elwD6h{rW5VD7E_1~+?>7){a%SB^j z_5{;h;R->pwMQZ#l*o+I$p!JIk;?>SY%EApxEI469$j=~kuu+U1c8lxkazkR2ObuU zG*zhJp{cPb?TK7>D4y5L2*HTI$J$Xk!~?5@=8hv+^W%|jZmTrmz`skUhGoLv5ZhWS z(SVPNN=#S?e*@>|hgBl5m`F$Oj(B2NF zSuzStdd7IQ^|EiORJJ!u0w+-QKP$b-e}%BxBVv@b5wt!6Y8K;ih6oLCP6-Nez#JuhwP##EOm+J3&!AGioXye%Jl60uPvTz09UE z^4}N0Yya!7zM;D`5c-zyu1+C@#sL z3;r1lo{i%{*@L1Fq_1wPyhQ=GsC*73o^qicG>wm^UtxVhN=az(^~H0UbKH8n{3s0* z$QOfcs@xJ!} zBUHmLKpf^!m{^dcuA>V1kM;iW(RVqZWclGtS9OCq3fQINoeYURwPWLPD**RjgZsgy zjg;QYkCsa7fD(uEQ_#^J0E~~tYPt6WW1fOB5mSG4WwOuI4uTOL*`+76)p)S;I8nWs za|Dy(R__l*24@%V2cNQh3j9YkDvD1(IL@J1+91^}>}+e^p@1t(+dJ>x&Wk!BGBHrV z2hcE%%i41=ofQBQbl?({bx4m=TifbA*om@+x`*LorU)-Um{mg`a}NWt)Iej%SW%;B z&x~oX#5^PQRlrWcfd>RcmW4TIq&g^KcMN&nh63;`phtA9%*-~E2gJP@V5&QkOn&-c zvs3G(6oa6r!K`^8+|9^u z&k}*A{D7t&_G_lw%?{!Q&kkxDm@Zdr; z{KP7psB4tM@oQ8f17|COKRagWKkf|IXYPXis6+btwJnh`YltZaEchz>e78aDG?w;o zTLlrLM&)hjFH0GKn(xY187WM3Gkd85eXo%@Vv~ZU4!y&V`d8(&YA5P3J9~P;&)}1&ZSYN!jo=qNbiEfnT7+TWfy3Vrlj%#|V^>E< zM@W!2X&A$Ue0wDqWy0^@L4F4aeEbq%1sDMF;fbPtS1lwwANrVoBBssa#+I5yW%dHLgMClOgyFbP5J0^D;67BM z@*hNL!npWODg-aw+M;RWaon0djzUB1v?vD>k0C)CryK1A4uQvm1XMBLe*}npJk35& z%8WzLyGNMeTzdG)A3xSlQHXb}pq*@?r0oiY=_gIWK~?%?2HnuMRw zN2SNn@86F?p$?ZcGUl@n19}4)8C1Yv=!<5sbCb3vpP(cerEpwgLhey(zSRs{1Yi3|fjd{usFtQ~$*#iz-sho~iw!oa|wuBo~9BmNg~ zkC(c-eNr|tB6quod*{UL+NOZ3Dk?6P*48G`F4ceRZ8VoX^8nQw0Jsw{gQMp|Rd$Z+ z`j2r==sV*@OMTJ`#zUP2e1CYPCz2D!iCYo9YY!q;LlE;q)Y6K~6hOy(!JHxHg>!y> z-u!bso?46q9DUHxT>_AbxLd9ei;IhqhK}2Uk)HURFo8AG1m3x*WJ}JMo-;Bs;?vN? z0m^~e%F0SnNy(Lh|6deIa+RzT}Ez0Go4S~~;rujgiHAA&wt zpDjMb-j?|75aE%cLIAez>+fI1)q#rS0%j1OmR4%XaKcUy2!zn_^5U~4D;@oe;XOM$ zd;0a2+4;@{{J4%oE4e3N4NkdbCanvBOYAy9C$1DBtIxK0Y{+;~1Zq82R#607Cbo|A zr{54+b}@3E$kR8xC9mHXqGLb=ToyM|2r*!80Re?jPD@G}k&;3Jfi~cHVQQ$W;}H-N z9s_o06&%;OEWT;7b#-;s<9P279dW}DzZ+r3Q15O`CgjR=TaxLL$tJbO#P4X;k1a%r zTQOM|#CEYhW4+S&pb+m$NDI5DgAUd_qq5+4oD~od6vXzvfAAcl$Aj1`Lvt zG#(xRlUZZ8##E<`j*d1Pt0^gUOqm46!tNJg9^|JZxw77HimX=JT)?$^H3T<^y9ELj zca{yO$>b+Tly}1g8g=REsoIb@R+l1<?gMCu5IX{8MRP@wIlt)!{P6YHdsa*`6w*U8V~0x%QoWQ2IJu zW1qU^GQYpW9cq-N5C?fr(tZ{pMdvWXH@X-f84(?YHmhQt0W-!;&c#gUU1p_93QC}h zIwjuOsgKi^y8o-fSCB3l$E2#oa$ny`KbTs~^!MPiw8B$*fZE8BBmCtIiSZ%Rjj4@i zjn>#lv@l){pPP&!)dT2g-J)x0?MI@gYtnXdZ#7j=+FsKpK72RoA8dH}5&;@;o3Oc^ zL_rz+2r~$3cyJdPBwf@crp}8W$(vK$Oh_s~oq&R%SBb$iYdq`GZQB*8K!fmbt+urG z+!U|*<^62zDWmtzHhsglXW=BoNddsmnkR$aPl@5tOG7x%Ge1yB>vEPI4@*#&B^7e>)VZzOA}`fpn0n;Suc z>R3xW)A7iAR2BZWkt$IXqb2c?Mc4B#s9$>H=w-S9N?8a*infuEIvEshOEqDD#+K+W zG&KvLa%*zK2h=$tSr47L3t6IFTk&omY`04ZQimKeGBBJDq$f~|67h7)nF}B) zry@>FK+UiRBR4MZb~2*QjrnPf?81s8%;fc{oXCmyk7W=c8|3_mn*_(}{b<<|lv_K9 zlCTVn=RrsB&CR!Jmp%&-E-HMXlQ7PZWMJSM&Om>Af^J&rEt;I`v4+Ou!S<7s0|LM+ zoj7!yJke}rJVK=QH1bR6^V@8Nt`J=r#Dom23`y*}tKKYC8(R>Jk_2FiZO6r(RPkr{_(@Wp zrF9pyyc7+mjIsjpoW}viCMq}0Q!(7>B1x3wbJJVfDk#IIkc++L0~5C?Bj`0g1u^wV z(*;kj{Y_-#Ukr#>yCw~KlWwl{!>tU0tW~D?B|u8{DxIa9uLbq$RWKxqG{sX2hen&No3_Ww zir&4eHyQYq)WEf25;*um;J#N1k(3KkZQ~a9wJ5TYxJR8YJ&3zC^zO>#Kxy12+vQ%? zsc}Ezu0RDrj*3bCE@`VVBG#KQh-VBqfjS`sp09BGY$H>LT4_CdTBC@@K;B>=tu8#dk?_-fYbe8Wp-e>>3@j; z%6wu(J3A{*C;m5QxDgyAm^>CJUQaw0{hu1skb^E*U< zU1zIXTDf>3HyujS{S{^KaAH+9;c_EzSyZyBdZwB+zR#OMPEc4 zhjsnN;%A#>KhlL-B-nmRB$qWReT0d-iQ}~~fG|5d0*V{*npz0Ml9+dEENu;LhRPeg zQYZy4R);)*lgs+fy_9i?(pUM;VzB$-Ts8IemJEm1#*Bn zzf(pbKO+Liw5@;)U_X8a{8VgoW3-Qw8*>)-dgLH(FO@9abD6$>Ugd3MG@sW~eNy9{ zH-tb*O#CQTJ_Jh-rDgl3XPIxCej5ULorbvRflB|F4-3DlT{z*b%aBU+ChajeE5#K} znU0T-BS20WHt46@@ilqg@p7BfKGJY286h7^zXDe;(=}7yV^{E53*Rum?pr5gq(lmn32r)`0r8o@ z>y1Rd3=@u5NBFS4UTy;7|CM0Ex)I5$#EUPFhU4m+4a(eQ}$jGPQFeKx2w)YVdiJ7!q z5UNNNueGwL;Ky?xr+2mUi9{<^Q}S|Mp+kB?V>hH=-Kn(B0a%6S({R(Mo zn+fbjr%03S?!c?H^P5%)1j4TUy)6`-5WcrT>Hp#UA0+(ppIJ4dI{Xjzw56p*$Lr?5 zSyLP}l;@BPk@Lq3X&=KrenbW^x$^bxZT*;27kVsu<*IWy10RNj@Q$pdB~wRdr_1er zodv+3@8Z!pNiY}n6BU`JTLYyaOaG0KQ@D(xS60j+PeC;egIgPLo%*FD8ilIsYE-@5-ML@CK7!=DD!$4qcvXx?NXWP#Pp}fvRFF<3M!st>a5ZYmT;6lL<4k@U( zG;6DRYFpYmIu`4{lS3ASWbV?Zhzb88ak0Jq?Q;{Jha`}YhyUo~aX0V<+di( z6mu}@7inH6AmG{_kg8rm&f&uY(=n0k>9jsx{OPg*MOEP44-b!MoL^`ujDs;xRz88^lNjIWpmv2_X zz7U`j5a%jJp?&R;i6jIRYyu<_16@X6lgX8kA-7@F{2J_9toqSUiHwpe2gi-C@K!jB z(#|?X@p!pZ!lRDyZFl)r?fh828O)AYZA+WiQ~kovRS%E2S)fKi$Hylo6rPDAts+S& zH<)wzsb{01yM~s>jET+z&~$rkg}b}Kn-|VqJ}*#Peiq2e)MJ6wD!vLG+q-3=Q~VcZ zM)AmF%G#$sV#|NzI8V~``>T35ip3YhXwz@(0folHg9x8P zduLi>emXa@9ox!*ZR8Sbz!T3b+D80~PU-CX@)>sr zqD(RL#mX|PVUbhd@XYAYcy55oP|J%BMl%K{BitJ^I&XrvUkuKIzc{JuxvML4pG+ z!%s0|#s42}1oUBqy;~C^c=QNL4j-@r6#mJuQ2tD@^h z93&fo|8g05z!8oB$%&58D4O}^*dzk2edoKzKB0|{M3eMP=w5_X&h>0~I5I`7pekw17 z7b+mT-t=xZsF0i%$%XAIY;5T8?Cfm#V_|1}2LTFtcO2bJRhFI48ldaX{up`4RkY|5 z&%#Kp3lj$$`?)AJ-|kr>dar3&hv$HVOG?1|ag@zc|~Ya1IY zN6VcMNR(WMl!KtKu*JpR49K^j7A)`PRz2@`7VEP&&D)q(>yp_$@avbfmR53(KPY

&+i3a zC@HN>wZ6UPQ-1Nn<2rzp%g|I$Pw(yb$Vjx5w6wm1=1XJG^Bt{JUP?;J^jF0{-f!E& zUh;yPzH{2@BUDuW{88J>8&DgCw=03J#+-h0hNNFt??BG)UUO|2zpUzkHv0*~K6RU$ zO_dp0D5r8+c|lzw`qMy%+AmAK#`?J0R2(-na%*6DTSv$J^Vuju79PN#iC;`xu(Gng zX$SzFOrX~e@@7+XWtIBUr!rQ=Z37^`oE$nPX6BXa&FtP{jY7j7&0@{Ss3`m&pm+#| z1EpG74GltAQ&SVw>E>{63a7!<>Ekj`c+LS#oNqG?Xs%2Xv`?AmyYF^zcJAGoD7hwV zA2Jsb6g1oUQ`{jm@cLHqX|**ED@`}%AMeAKTQ1Pqxp}p@ba1<{>jG2~QO@5({SypW zF*2gA4TxI+(|USch^w2MGhwZutp{`&?19D;E(^)qy=sTvIO=z`sVF!2d;>~KN^fBI zm#4$&A?!k;nM1E(aa~<PEx?Q|Lk&rN}m+$`Ro^MtamLl}C*?o$k<19?H(!_K3 zFI>nCDZDj}QB%Bbt~k9#(eNga&ZTF}?V&>DURuJ3i7j6{KaYu&4+<$pmLhC4Y^l}N zoO*hCmZsQ~S%Mqs$&LeVBAzoLA=EAGkv>`)^IuKn(wdYhd6v~0l_^sxshD|iy!>b! zL&!@JMx}E~|IfkCucD(*zp71Dp~M7;w|-MMGd4!=ZZ5IN6!sH*w0g?sJFH(YC^0f2 zl_*ti5-E93@?dyznm8Q(f9PO4dm&sIe!`QZ`FJE*Nr~??1n`wsuZf znEbW>?xtO-;gcs%Ts%E>Kiv2QQ-I;U>NIv%Kp-e8Dyph%!Cl&2`pnm_UzJrw#6c64v56nH5Wc=5`N#+KT*MJ6oVxPSjVjKP-YkQNkuRzZO~ zu(_E`Tj!eOwzp37Q9NE_Y{SsrRxkG2T-}~!81)dfvCG}XVXdCp`TckMk=R3Q_os7# zG8+deNnEDu6z!esWZ$PhC4_9GBCDju?yD0iQ!9S|iC3umY!_B(){h+?h@(=J`XOXJ zchpSJ*c9!1>i7!VD<4Ud%ezXFb8~Z6R#q<)5;}=jEN4Hb9-TRyU~_e??60<)`%<+@ zMOi@}n3#Q50A{S9?jYW6ru)I}rX!d46k}s!y}e=V5+7&% z7w1LQp2u7e60+(^mg`EAiH+n_6ZKrwXJTUdy|y;l6L28rySd17%xOyV4>3!n63-uWw6T)A>22eMj@N0ISeVrJR~)RaG(#4Gpv}yjCYwEG#Vz%AL+2 zpa|ZYRC#Xy{rlc=vGD>n9vk8#1lVm^On ztDGe5b4}sc`Sa(;Dm_jFrXQdXVKUNm2zq+@H*em+J7Hb+6!w15Z}03da&Y(#Z62kd z6Y-eWc9%9u8|CHWi<9$}92gkj=jVT!nCPb$aOKLCii(QgLv;r%f|>Cr2pO9OwSL$1 z@{KV~&uCTm52i%bD6t2VGgDJ8Q|*kkhX=k;&E?J$t!-_=nJJIrd(+hVs=S;zIl~m< zwl+6yzNDiPuem)taZ0#1wL6!Y@)=v| z8r*~ zLw? zL2LiF)Pu=mJ~dp<&d#@P-@fL%DRf5K{j98P;_u%sKAVeKo>jE&b6jFNUfoN&i@0{x&6#@b>nG zia0%9jKddFZ9P-7wic-I*=THOIezu(RaaNn*}*&xMn=Z!)mDy*s;Z09uX9SCKc|FC z&o}bR%X0+|nFXAAckSYRhtc=eokU4Tc7%-QBH}C`-_s6);L$?YU$>Hdm#=;N_;I+} zTTC%+7qR%QCZWCkW}0tVj)eJytN)3>pkF6+mX%9%A3V4M^T8)0lhBu`aZg7_C4x(C zb|B}JwY4>AmS;cnRC+A5!o+_1oO)w!xF{`K?&{1sX}i5iOWRqlT5)l6vmbb%m#g2< z6(^S95ko=?($=-G;7yY8lEiZFU@KTQ;&05CcDIC{+JOpKmR_f%q2ZU3B67+3xa|M+ z*jyTI(=+o|DR&xYyXMN@6ih$!BbZzAc0k7Vo`9erAwK?N%q63)FJ}I3(_KHl+NS)9 zvrWEInQS~w8QMQgZ2K9ZpbkiJ_wHSo)kp&2hO4Wn{ZRhf8^Ls8oY$fN1XkPmBHoSH z@@Cvg(DmBEI3-9rVmIZXns~(}r|W@z`gnBCoouthl-!%7^}QikXNoS zQczH|MsWAQz6&D|2r2>57eNUqg`F00yyf8`5f>M?IQD_i)rIEe<<)<0g&LP*RUEhE z#r*;byJWE#IJbY2g(Z?mZ0}BzdEn+|IoXUCGI*apu%$!k0Zf=C?$!14ku4Jtli%i7 zTfePwF95P)za3W~O!Pi04v!)}S1q-w|MLjwFe@u-ak%Jd!c}Kc>yIa=ds3XmZnIKT zF|UTk&=KX^!#S>Q@2&UjOom=2l}p{>{)+cnhhRdf@bf}CCZuP!#Lk>ZW&k0xNw3<5%Vn?BKyzwNIr-0%IIR-p0zQzo+A4p;SaBhn3<50w>rrBD6;bN zk(VxoN67zuzP9FS)1BB6C2%`F(Mr&PfWwDt`qcNc>FDT?Sb2N4}mL$apt1H9zk&rPex(}z31Vo%HaRpeSA<5e1!d1A!PM>s(#nFC^`zkPef zchdoRC*FO?%`QkJ#t&=Q&P0JuNPGW2N<)4Jhr=CfeQ~BUnETQ5f`Wn(N2-fRj0L~M zvWP_VO6~58@`*k^L8Y`rNfHtol{@`{Rj`@vipw?r5GCn2@@_n)@QNn~a!}fQ#JKIn8OxRs z7TD=u`}>x0zlst>0K?N?mjveM6|AjMTs9hq7agzk2;Ep1THoKAC^m06(%95A;Dj2M zN}wgQcFR)RuzV)Xx3IC@Kzw=b{I`1sM#ZiELu#hP!{f6&%EF z0?+7|Vp*a|{LAXfnaM76T6?yWS!sTs;G>bT@Z zjBf%6H>iGim`ffLJXX%+}WgDFhod9M8l(;x^VgBL$S!QkH(0~^11GZ zJ*#%NOa3iy#iec(waB19{TL~=Z*FZp@_V(7yD#+yMYIgl8`aPze&Db#U$k5|K2+!V zCS1s#e-x9~Go851xy}`@=2c0EjtN zT}*f1jKofS^~d;>4^>s`z&nzabrp4Q-te+sGQ>5Qn~#<`x=Qj2#h``)2Y%N@S9X1P zcK=?kBW*Vb*dq5iEr3iCQ`6DW(GEuH6f0y6Rr?HX?$|rkuojSw*7}o#6q)0Z_TCxs zN{v7_#c}(Un)FBssg)H#W2;Z`5<Q%F8N>ag_i(fA}rs_Z)-?()v{2zou45&twfj5 zaQ(gabm-y3hYTz%1G7r90f)O2-m5LA7HmqAr5+p!{I{S5h(j^t87C*KE+8XOZjuBzgFR^2M@Zf>s zr?U}YgUFvg2AR*grEk<`@DmNHZrt-WhsG5ysU1UOzMJMEY@`_g{WTMG$`n!G*GG8&UKFHff^5?jRmU?mrq2CW z^`>1x@lSUrb-qyvg{K`5LlF@)r$uqjb^B!SRlJ0vA74bD|r`4|d?0{BNTaC6hwFBH_#^(STt8I^_I@5>-7 zv}~h8h>D65<^6q3l19|@eZ0K~l9u|N`s~EL2G0AGJ_l$UTB`EhasulU?Ki55QdhUx z|GVKZT=+Oy&NqI4TZ#CjqY>yZ7_@R^bo9XN6S;!wHEL!g8e9pvm)Djmg%$-ZyRzxG z@$v<3@OALM1N**9k2Y3XtIBm9lSZsLQf=<00zuJnN_~E_HKc{d;~lrAbBi^}Tg-bF+Akjj=H&u!g;d^ZHDWEj*QtKfeu#SRgcEK&6B-ZUBt0p*unIa>&CgF3KD+_Mw~;Gjm^S|+#;l>C z0k)xoUSZGtW0QV0SjlezhyIdIW9P2(I>exwj+uie-`dx08$2WX@TG+PAW6Wh2>0L$ zwUP7$vZLfKn~Mfj#&uIr_RTFVtDB?Foox6*ZzMhQsZ*zb_S<05(EjH_U4JwlXBE<~ zhvh_RYQ7v->+?AUqF&3!n-$iNG8O>~xw5S$s0evNhPSq8;{;DYK#N0B}!C zO#D3O?!NMEe0uQv9z5Jl4UL(!2zjjgac_I$zI*fMjWh-Wbv{3}VKT}Y8#Dgqvv6rJ z*MKkL4TEQAE=JU0ScHyCwjE4Hm$yEl(tlSHY%V_%scM5bQDrkIlBGZx)~()|PWZDj zk#fbd_1$;~L$;_rv3}o{$tVF%i;9hH8~flk;G{x15GRLzMsvCmEOOWZd54xsh0EhR z36e$FfeXgQ#{Eue|BS%eGI1+B8XF7N4Kpnegq@*quo@wHLjSoLQ4aljdh;(hmz)Q7 zD%D%xeBGDF;$~;(-}`kr;BFNQk5CjAW8o-<*8`MY{8~KMpG6qSqr^mNpJ0BsqdNoDasN-; zwHT|3TK|)$PKklK|K};Q*paQW1bs))#H0;5Wa>*xL7@|1qNFcAaOkQ37d;6wnTeAV z4Gy~><}5VBhp4}Xo&+i@^xyR~^_deOU1u_j3mGFz*sj_7X4yr&V&ji{$1r$nLxij| z5#_fruR}6>CoWgNCcXUe;yg0nIv+@INLB;(OevN;Kg!+($WDOLo~3{6MW51*SDp;LAzdIA=kkWmrNg;9$p=?u>MEzDd>Z-@ondDqNa_ zY}HfF;OHP)XijUUiC)19P&ni`o?)}zqR|Q@GV3bZB1ygNV1GAG1oI@@pAzFJghUJd zifbmZcmKP)A<0bC7w9}b0V}AP1$>d zl?93I;XeRd6esBzf1!=FaRf#M`iS8*l4U!rkbBq-uDZN9CHPohB8I2umhS?1)Yo zU;j;GDBt9R=MuYZPjX9JTc*y&d_jYOrlz8f4vVR&>78X;M|+oBa-I2t041Y$^CZrv z05Pk>v;pWD=@cBWCsNRHn%4)C-C4E_|IbiE#Jw*51CXh66LobK8@NI*T}EUE5#=}8 zOa(m7k|T(#DGm`Kv$kb|wFSyh%VEm3tGb#1I&lb6hMXr8uo0ltHayf`X-6FA3)X`n8Y@sPcdX6I{SN>MSY(9FM z>@oDO{fD>LlHdQ+$0_3(_Rp~7A9du$CMMz1`EP5@vTSPr;q_1G$qb_6k(XU`|#9l}}eY(&E z^GT8iXW_t$Ca2@P>Ez_}^C;SDxqi6FWr_WeidLY&*{s4d&nDr>LI*Vuc37DRSt0`y z)0b`ZR`l4U2Wldq4~@~`7YW&IALbrXA^s4;19H3ZOK|2C~drbg5EhS$A&D}%n_ z88)MHebbIwlpHo<7$!Eh;cc`g2uNEdHPyky$9q4yrLf{` z%VUoN_R?&OtkR|7(oY?b=+{sLTwH~=^n2?mxW)0?b^g2GQx0>#PX9CXUlDaE0`er;u9- zsKu!%WozrV)FxDzT?b+G@)mj$Z1F$M4H)qG&(q4pq@>=3jSKGlxTWF_k^A#;&0s@S z3&weIc3cTGgp6d=;{5zA1B19*aE(z1QS|H8+ewhGYTdniz0@y_4>#!`iDvgj7OAE+ zfc{T-zZb|5QamqV&aHhJL_$IB-;f_o!AB+RUErjc>T1~%!@T)@l-zFiKO44pWH6mm z-cKsUfAchmH3W9>yStZHW^V4{Nat&1B%Rmr(z_AR74fNq9B`)<)i9tXmmV3eg8 zUQSjPaxM|F1UhCrDZ=phcoKxbIb+cbWo^?E^o+ zDI}>?@s8sL^w*l48m-!2^6UX-YwPIr*7$n4x`84JdG(SJzbv>Se`hAWF zDVFI~?SKT;(h9zn1JHCQvs;#XB19q^Ua`NuS5C zW4VdM#?Y2{2E<2_5ZPb-N!$5zT|fY+_t=ZSxE83t%$V9U?pR0twuV5+X)xNt%fWvvqD!yyPTgK1z} zhh+3EzAfnBBwKs3o*5~5N5_kIYu0XAN}_tc>gSa5^!9%Z zN;2%%YyD3Nr`|q_;}d5s0JLN#6wsE-1JRfl-xn z(!ctGQl5)!2#61nNaVBIvBGgvTYR{p#RE}_*M)^3)=xT(R}Ref08Bj(3CXU?%B$5g z^*+wc%?)>58F>Ur0a87&9n390ENp6LCZ%m)AR0_EQt%{~luIBqfZjl**olwrrXWQ{ zqrMi~-ZL@z1a+o}#a=)F)XhNp2$DNhz?r)+_mE4gM|$Dwc=@&P)&JJP)6~|6pbKJ= zs%6i=0RQ5Z&Tu}wopYf-xPKwjYBo~-!OYDa0rd@?#5`^-#R2+Kw2Et9vjv{Zcf06EDrJ8Eld zEA#d1*I!o*CF0=6nN8X>lT z0LgEEV@P&u%nc00QLxX@$*ORhdGRJQSLxjK#`I`Ny*he&1o`=oy*tf6Fy|h|C0DsD zL5e#B1)-Oy`cK^PL4=q03A3QOi2b0@k)uZ!o}|a9rlmq&v;0L+NGL2lonM^&&iE<< zoIFT0u?-`Z$y$E2;dxjhNLefE#z+b3&e|`p8XL$6pn1ar?HSL-VKPN%dqP-4arVEo z;=qaVL*@pBvI^Pc-~bo^c7#!xLsm}Rzv7NNTgV`hXAL(&pSRZb=g&j@cDmAQ#i_z$ zLAQARlEM4tmnkU|^9Q$X-J+56-4r*jbkAxjWnU5u2~(Inc0Ny5Yrn8=n#`Z%POa8Pox|iY?iAr&$58+|IgP!GIDdf zUjH#9DpBt8h4R7172JStti)dXZ{H+oKBVyRIz*+ZS3H2P+8|IW>dN^_=1=|)4wl17 literal 9054 zcmch7cTkgE_igA!dKc+nKxsjm^cI?iBJk2XC`u02o5dw zmzWTY&=)N>fIoz`57pt2YwS-hx-bm_;ecqul??o{wq~9iyz;&x+C6c<2}{ZJNWsU^ zVv8Xt(K8(=bByjk#(fRM_jTlv*4=Yv9H!J`4(5%??`iE#v{KH_Bggj zxy|pxa;1_4lB7A(U#y-Gefz8U#68quCBuK^a@{@Y0QP0**1EMbK>)5Bqb!`#{cuaJm;&8GJKRZiiA2g47~bajiO57V}inY zNdM;{I^XSI)SWH4kuh1x9mC(&+-93e*xA{KDEb3Rx&!trz4724j?T5Gog5u?_4UI&R|h$zq;6haUD0^+2}O~l z(2I$Sa|#Pn9vvNh`a}$O-Db*?4^n_A<~N3|jl@$^zgGNG{SYO`*h@$qO zF%wytMT;A(fs5Bc0FwLRL&uv^lSKXO6_&p{A#a`@I!6%6!et~jq>RwC80}o~5A?r< z%?VAV|7HfcJ5Do(_Vqy^Fp7R@>B0bq;WQZM`i-=SFm1WXV{gVtWt*d5=k}u63`m;+ zLX~-V`>Z*WTykNFbCEv0gm*e1jAK3|`BIpIRXSp+J6Yx9kabtW&E10o%$ZwGTZSm7 ziw@%9Mvax!u{q)2Y2bqZFwe{}sWb|59i)X(3a6AJBGM%KH{B~4Bm zFD?E0Wu9WC_j$$oX>n%q)YMceI=YCdY6q|nUQ%jKPW~dM=;1mav2&KRHz_?my}PH=#RdZv zRj3UnN-1BjeP(86^GZF5Md|6&PkTsL*Rs9u`UQv=<>hYm(pgK7Tp`oGjtPhI}xFjSPDl04blt>`+MMF9! zHRalcqW*!2YT=`!2KM&$UQOtN0zxGvB}iyno8r^jpF*{tKYw;V-L(-gu7H5k*U-^X z4-5<}3;GbCfm7_ zWy2a8?)&-0_+DySot~dBo*!*6-LsEDn%UTJuJmVwQ!4zW+S}Wkab>^n&QWI36ve3$ zrC2=nmJAva62im7V?FlyA3i?5wui##2ylPe7mi}gOD(3Q48g&{|GE4xR<_me@1ozm zV{b;Ikr&4y8PxBnRzV$!Iun z58>eCY(LGKu6N6vX$d%TY(8C#CnhFt>x`w{8<3oX{9$2!7r0NdGLRYF%^t!6qVjbA zn=nr9NK3N(IbWUg6fOk&ePG}0s;ml4@qAvd-i5TWGhcBac}E$;IC4LKCN0!?CqdzQ zesN)Uy1NjQl%%Gjf|taukn}G2oP?8;vr8WZ4p{SIdu_)Ggbeml*5oKXXu=(tFw!XB>+y0 zXoVZ;K@vr1cK!17LMeOS{zU%-7AYDK3&suZoPYoR-TfxqYP~ayKAjJzoZs0=?(Bry zKYq-~!vkMWpMS*!<&pBVV;dS8>e_WjB3~pYQwCoi83mrq`1XAJ7M_`zxySXnq5@WA z6*XdqdO2W1!^Wx*TvA@%og??WBIG*Qz|fHE&K(kdqcXUcS4~8D*mf6;vv}i~U(Jr^ z-&17Z_2tRwUN^f)z~R&WY#BycUTrZ;%?HPk*G3v%zE(DlntnASYhIJp4wODyKMC^n zVoj=_*z~5}*5#w~xqehuZHLl*XCiIws12|&iA5@Gr}dgmRaF%w?R|H6csSRvIPv=z z|APWSOVT48OhiBlFJ8e+QBhI*_DoZy(}WNi%Y7=*C;e?rJ1sW=3f@A~(9mKcwu>a~ z*k;ga_mPYEbpWCPc64@k+s!n3gT0TawIVdRiHT_ZqviK2zTH|;NeKtQ<0B)ZXoq(} z4Byg)SDUDA+JJQtB6jlY>KJ5vHrqhT2 zY!)HTBG!(vQBj1Ov*=q^K}Yl-K76R~`27Hc)9`xt?d~>MrAi$OtQM}XPwTTe`2qyD zO1_>Az*sWs^)MS-B4;^k$2*|df}7o!XAy=j3nKMm_WL@=ETOEjc>i zzk9a>Y$2(Ukx_|pB`y+)4EoIL6Ov@Ev+YE6Al6N~`M9ly7mMavnG)|C8=1?|$ILV| zG=9A@N5mL7{;`DhPg?10idz!SZC95Udwu-n^J|6gRvtVV<^Nl{M(~^!wnYL~zcpQt zUHzRqcY6Bz*1c{$_>g0ZGFJS|k0{NrsGwR~TO%eRX$Ld`=JiLn>4uDlEEz3rUQ5TQ zsgPAm!r`Hpfq}t1K%vg%-n4wzSo;VY+-?@wRxDsF&QnzcQ&UqY8DAm8Qd1D@cf8Gb zM73{1lVq)b>htR4bON|ac=?hl@bYK^RLF`^!4!m)Dyk;BdupY+!Dowk<Bb$xEcBJD-|@Qr|yvT_)h+uwT+(T_g((VDAnZ_hIx*x2=6$87uekM}CZ=m2p5 z;$d#Bmv1&oQnh-x5CW_)Efr=+8|6M3O}2J)t<%KfXp#Qu=~A-HjBixWPnLU#WqIqL zw}>cM+TMygy?psH6kxrnm}-~?f`S@4C#~um_-9MvmXUWgZJ zL?Z=SX^po(%tmX1;N0Ke*9BP;w?eu+JNR|2YXky$9Zfnygiw!=#|69&CHLH?(3j5x z-7(5gW{Hmb{Vjc|n3@LGD- zEVZizU*~>gXh^7_pg@5-egrBoKq8pitgP(S=~AjtF5Ii$GFtQT@(HK5 z^V8E@fM(Qv#97Z|?L6MOpSVLp0liAsXD(}>LIVa{VpdP_?TsMDE?Mr0?`5Us_YJAx z9V@r>FNbx;et!2M)+?bI?xM)LWkSh*78orAn;~i9WQ2G*0qaGxb=LF%nB@W{Zc8Xp zRREr8|Ky1+CW;&y9?`>gB#jt#6bA)AS>r@QfVGBBgiT~u*RM@w0jIW*`5}>@YTi8> zOH0cq<$b?ef@Ey%>=KhbJv~L9^fR4aT&R>9F15YDgFpZg%gf^dJdpGDZ35=Rf`@`u zR#tp;aYFGN^=?hio4&lg`5F093O24>hi%m>SG-Y}o<%iin-Y7u6^YZfA zzJ0qvy-jWWrRtk~IB9-Z|>=%yx_@p7y0$$h;Ghijz9fV=hQ$W8lk7ai;Pz0J&ASXg)g!~*93%!NfP z)hm5>Ea~X!Ir;dout5Fzi;}0fii*l31B21kb2;SN@NEIp8e$#UJ^bL4rnx-^+U1w$ zvXbim65G0wwZ58Tmsz#WKEv|Xu4MLPa6xW)c{T_PNFwtlA96*rA1Kn?oO?$4#bZA=>6Soo^jk`l z+}&>4xu?;yU~(G)v0m(q#Zrol9L%_<>Vp(i&|J8EACqLt#?b`dgfvk z#(^q5TUEF=0eZ+R*=h&vRg#{W$q&K@R(~h9E>1c+I%}|BAUyFpIy&-P zqcyw?$!W1FQe7Q;B;&FPGt;dh3KA|qBfiUDB!Dk@JO;zr4*JaP8yOji1vJBQ%b7J? zIj%320VqI;fGoT5W6f(^{?*JKK|xhhQ|953kpXQVp41z~(Xk-x*#iD>(lIcYUg-M# z26O#3O6zxly0=~KNztoUvG+Elq=on`r>JMG8>knL*$#bBHLkKJSzKHMtLuR}(?CT; zL>iDwgjn`!#-go9PEKyz>?e}*d|Qg=_HE4>ij;v#9t0gJ0Z*e8;Ht8RR&w73 zE(S;?!!q+{_cUU%UT9&Kht%1>0$K!6-S!*C^fjpLW^U+6g})1wKz6)oWaa9e6hlZ4 z^1O*pR=8pUyeE6S>b8fP1UapuW%`7J1J*m3->RQ`(@4+NJ0*+w=|E$4PLhh24!M%O z+Di`M5ELYLbu~0&i0ZMK{P*906;7Z3`0(uM(>C_&Q*{B$W>j@(?A5mw8{C@g^~rM| z8r#19ez>tQ6XOf$x?(CUHa=eE@#B1zK_2e>Y|qSC&XW{JEb=2IdM__mpqjv@K6shVfB9T?1+2rPvpH8S==zQ?DG&eyMBPk!;_ zoeezDDl+4&dI> z{+=D=yGHhKPCAoMZm)ms?BtMB{9>% zERKUSXZ5KWXHJ!um)9*axbL|_#ujuqBtIeAJuH5|7}x@be;zF_FXvjd2BX&tB`+&; znm3T-#C30^r2euF z_K*kHc-PC}u zOu3+H=F(09;26pcwucjmJno|_W()7RIq8Cej4O;4m6xjnFa!pTbFCef)Qu(zTBq+k z@`0x?zz>jfK_@2p@+qQ~8?ee_#3HEgcA#2jq90VRFxu6dVPH zkl=G*k*dY&%bP4*2nZ7FCm_0m*)qiBr7^$>VU`cg-dh{_@mk<6YAqGN3 zN-N?(^A0#<-0Dem@AI>hRIQ_Z{$(j85R@IH0Iv`@y?} zjLw09fd>BrC?O@AfUN_3S=R>l@X|oNaD-DpfJ{h;;$%2>^DX8waeV`^cns`$q?9@8 z&>Ol3;t=az&%1;9F8{vuyU^j{ckh9%eUpnl3XmhRiY;m5#N-K|@RSu7tLW;!q;KT% zaHY5jRonshG%;cKBUj0AJ+8-J${BU|1d0T#3!pIYuN=M(#$=^oqDX!mWOM?b(dH$U zNvuk=rj7#*Xg};fF@qc0o&aKtWhWdQ5Kvcub$U`#II*#?z3sY(+}b*-%CW;jD@e0$ z@jHb;-vBNLM8@-RT1R|fx0rgFKMz8etTq6^F3EO!r|I0=R19=jLsQdqE-%ZUaYQus zWShX;JXE+vF*})Z`}7RQ#3WQ1=kLW_o!c9Y0()vG1Qz_)Z-lWTTbdw&osy64I%Ey4 zJXE(edS_ZflgF-#gccZZ0noi4?-l+1Cs_NwO}D7SKfKgJp|UlzUg5ubjT3UoZ;q3j zPtel?3BLNW^&B*4rXonJpFk_ig!RkjQ5Ml$}SOV{Bu9; z^aT;y>9tz|Gr{Iuvi@xyrSb_pe$qUrRIsw6LHs|#(8}UyB4*VFG#|D1?IzfoE3`5* z^A;IlIKBmS@4L;8yr$b3QR_YZ2 z=O7IZK9MPa>HH-ZdvUZ8E#tebF-o7(43sNmq)>XRP^z@Fl*U_?$Eu2)R{w;@2Y6C| zJc2Of*-X&LVF3mD`B;2xZ2Jp5BH%HKL0p{Nk2fYL8HFu7D{L@4JIO~`#cUIyTNNKa zmTF3Y8TJkiOF-un{-`4?OiyQIW@g^@cUBKXmpIH}!vas#TgMhK8{l@5G;yW!nncGI zWpo0i(GYO-IRY#;tptPrM>-8CvQl@ z#+LLjT}bm6bxQMnEfj!766Qog!f+f)*nCCi%dI zy%AO}n&^=T5(sEDy$6ORP*ND6Kd@xYFzYU2yY)I4C~4*Qy4d>C;PE?dRFJwiNkI3v&nJgzV-$ zd>~{169Y3DL9p6%;G4GwvpRgw`Aw4q=R)?u`~f+0-<%Y|Cn8z`dL29U2Cz1^L}M`+ zjMi*Pd`y7$xKrC~DF8~lt^_3{QUvR#E-%|R{YgGrwnzr4J~=crp4V*FEq)&~5gq~m zv)HJ-V|w})X{7T0m9fq&_P`JX`1l-RVst`6LRp9EJh+AAw7D+W6Yx!!xu$;fUCR64 zE!5J|x*vF2Fx3^D!r(5NL^8Q{b2uoDk zF&I2h^wR(}h+a9gS=`=Eie?KSCZc3}_1}kZA>U;WK>w_X5Xc2@emF(E4X*X!pP}93 zV_>e8VXr6vNCP#5DgeoOJc$HNH-@d zRj@;;nWN;B@dX7?;MnJaMna~fTNDTJ?%tjf;9@`@zj}~ADo8UW9z_!R^QS3j_1tn$ zk`NcifjIuJ9Ywu5P5X%~d8`rhGCe)|{5$~P(7+nuDzvQ-#0+s)%~#$#T>GI0=KxMi zwZr!n89FY}tl<8JT^jQR{$4R7KUj24*sMaV9E5!zkW~GS&I7@jo_>`1twL z8ZEe*ejnQCnV673m@?DwjKbf4JFy@qOc?0Jfq=W5tP0b#!w z>%Bqcv69fcY95=Uw!Qe3N$e?APVo7&%;WBWl2yJVaIZYFvMdmH=M_XvgSXs<6Vek? znIVevZ_m&KV-1@xD$J=eO-84@eM@@dI-7$gawIy|v{dc->Jlk-WMXoX zFY7*184&lv3jc#Az-@kreXXmf#|3)V-c38XO5vctDdkxJ8nuFdyre;|q4TsE zKpe^Y_wQ%M1%-tf4eo-DZWu%nEIE{lg+)O2pkK5jL(~CZiK>6R{h-ZsBb}0{h(2MPU$*JOgu%ZRLLJnYkK)_f2 z+OkCe7zL8L2Nc8~$7Ol7we*0rvj-hgTl)W|2Tokif7ut9eRjwrcB#u}Nlc)9gS?zKe)GFcgSU zS0~3>)6=uF*aO^Hfi@U*`+ppYnScEb(BOgrNLZ=^P0zf-LY49ftW5ztiw!+~;5b)5 z8R!7EvUFm4O;gZ?g#7s*?j`GrFK>Zgh-H;1;j~X?WVF=SZkUB(eF!k~q|y91p}=?r z9ey&f9ruII#jw6IBwIc^12h1zvNR$B)FH_L^z7*>@RfnLA$)od7K;{7gG+53Ea8UzW@7MlN>_|iV4c) z?(MxbLh7=DnJXgwzny z4R?)yod4YC-gAsc=VQLT*Is+=wcht#JNS{3%w;@EJOl!9S@xmiV+7(1DfTbUMR;;8 zd%6n#xo9jeBZ)Z0{{C2>6^%gLLdZ(qS9_kgGVZ0T<}z`*j=n`Vee2EV_>Y{LA)jA9 zPqEZvv3)b7y_gwUG1%$4Z57^8YlHsqQ%ZnOpzmi&%wP^h{_h*iHgl%&Y$TRx7ZV!q z8BzujoTWS0xqtjh!I6=tvw@&ID`osryct+Sn|FkYRP*J&00P&AuydqiI}x>*v3c2%-T636Ho2~q+l_YLP22~B^$ zVB#j`+X{Zf>a<35enoh=q_lodKFi6=U$Z8%xFR8VGbX!v5g#9a{m-AG5mNcTgQ<%O zb^U+eeHe#Bnn>Y6y92~;iHa9ohdwLkz*usN?aO&#nZ2D7eVM3#e_4>-5o+1nk49;uG z_ce#SSHIl7@sdU3tf0H&S=3S5)Q{8$3JJXJsS%H66M~7&1Yhhd=iibwKiVtb+sl|* z&TAXiii}jIR8g^BoDfZV`nprhu&fZbf}1nm9PSFrkOvu#G{p zEtj}Bd~x6W)Gols&dOT06kWOPK{Y1u=U&UX=PVQVFm)8={nle-l_!V4@=uvVmIi&U zg?a}ruk2PAdvq5R6jasJB$Sz0>qP$?+)jLE9MM62&^O6`80|5KKIENI-Pqi8-&&aI?=d6g&xq?RVq&tgqwZ;&wgOfnDSwAm>DsYb)9n1XNXBcxryhh*QvnQ!qN|hNSM; zSw|8xQ_0-vOeOmtF`?ZEDZS#`25 z?LzV6eLv+{-mLHe7AB@!w6w3%(&!M=P1A@VVTubcYisXbxpL*XhsWTs+s^W^^V&pG zODGxo`O&v1dWB?>_@g6Fgj$Y4VJ*ThFp$Tjm4M*lxzP$+dNHvt6B8t)q`^T!_pb(KgCw!!m-{Y-3|%I+!u}79U?sSQx?MQ|DuEC{WmQ;3d^iIGxM zQ%myF_gt5naaB=K0RxAR$R=>$+$A)$wD`YyLo6X7q3LeO0B1ntWN#*XH8wihS2{g= zJuZ!jRl6+6wBtQrm4T5FZIZBygvwQCVZHiM=Y=0^2nh*& zs#7Ta#z#uan!yyZUiS5!t+kK1+p6v-rKImPCw~Gj?qbHOc``FI-@SX+|Ki}hAt(7z zts4iAd3RoEB%ksXDmodYgyy$I&h_?aC#j$pFMXjJPUal_wr>hk}yV2 z%EE%_5#Q9G)WDl`3=Az*dRkiXTsC%-SqJ-wGHtdmU%r^tORHkfDXnkg%jjs!OZ!t( zQ(s$JC{C-y_kNv2TxZwCIr5l&d;^C$22V;QC^*>Y`@1^`&!hD=M53U>>ulAmvrkgE zkN0LG_YMw9+&9f$QXbk2m$VEuzdCmbhlt3217jNmNA)`Q{I{nkNBdz^0uc)f7X4N( zNqxyy!bt_K+3m1Jy;kLX6|!=2#s^!Ah{U^Y1xn$gyV+mN9zTBERAD-dz;&!6C;Z@UZ< zq54UIE7`lOOz{IhpN_2%_mw7zIqLW_(+xr#qtx%;<#0(M&lbfs8h+HPb*->#p2L28 zLmjUc<#rDTzdcI!8Uz8WnwHibqu1EAWas7aTK~}2L0UKPn~3_>zDj#bhj-auJzL{L zf2o&fl2Tx4Y3abyctQCNTa&<*SyZ{p)+(FJSpAq!UTPP~9A^xnc01ddOCa>9EG8BBQdJt>TwoIhYB25=h%rk%3L zHg_UZKK7Z5OD2RT-%kGLH++Wo2VdF_dX!wW8O@1}i~H8_>KtZyewo#};WfC;sHDGN z?O=D!9*lkumLd;M4%cw;@UU)@htfg1uG^p^`hR{*6!S`e zYzNk&vDV&~lMC^arN?fqU!&o#&Ul{lmoAOgT&qPI(Mi(7I6<+oWO_Bu#V#zaim(aN zw;36EY=$`D{ax%!l8%nt4+02^VRB{o@!ijPdAAuD{2P7$4#H?e_w@#xctRQ)8WB-Z z+wqWbI0(~(Urr9JrNb#rG5p#Wz%{GC#83Al{epwBBi{4&785@`;yHJTu*`)eAA2S* zKU7e7I#gs-W{vJ1%rC>k#_+G9Vw13&q9cR0Wt8x_{{f`=#6(sjXH04(zu|hk1%uYcr9!4H%D&2Rn%AMlu7>MxSG}&u~qQx(xVX z)N`{-U~6e0^zTvm=QC=N=WhT&=~VVILu5I~%gW-O?)UgWBCUjlzJ-NWdYxDQge#C? zXMg{0Cop0Gh)QTX-<<@|4VRH|U$y96oo|!T@A-MBqdf=j-P%c5CI9^Ah^Z)Yvj!t1&*}L&tx6PeVdmY{Xbp4<=W@V&2R3%;d3OTuj0~@>&0zn;^ zj>hcIMk_x*Sj@5=m-_&xm%1A=KlUIkj#grgRt2LU^kTL>|1@55RQpMQ{Wx4{PqRDj zmO>rFdk&X$iqmeQuKjQg)dOLS9pyHR(kesaCHe3n&hf!^sSAtkTh@tN)DkKxSMIv4 z&W^iHmcz66)bTihMW!9sZ+a}yLiMqMR~clGXw_+A&d(^lQxT_{(h&}^d4y{f8es3wVDxgfrb z#G^jhX}&vk!0HYwtE9?RI1{}Vyyq`mXs3}Et+1brK%vBmME<*H(vK6}8t34oyKW_3hpsx6)10>zmJ{_08O6ks z0ZP}HKP9dJ;E|Pe?is_am(iriKcUjisSg65 zAV1^MXGJyk=`cS<59M7ieR(9|3aAs{y+0oIlzJ9PE{c(1fL2`2P6x2b*2j-8{rt{59v=nythWf8j*X84Hvb3^+r_)WbwwMtP8ujeSfnLyg7D~t zOPA1l-s{sXbvq;0>B++6rmN?=&&8W-)_d?#@!MF; zD%OEIjGQFB-&I^wc6BCEZEJ^R*zu($b zUsv}Wfd4~9MFx8MbAYt>X{QQVXNY2K2-)CcF%$lGNYQ1{>`p;}t z#;a2Iv-;xeQsgq5(CAnO`S*X0HhTm#+j@E;6BDUR4{PdCvHBjt{`h44i`%%EXlo?f zLzSya$s)476}EXhe?Iy=&D#De=!6>W6R}gW)_zH zdGXVBgVO^v5^bp$j$=k1ACB+Vxc}6O&+W`E& zuR9U$Fz`w0h~+4?9p$fiAL~mYgZQ-ZZ)Oo+ZU8g_SpH`@rqdASwe-N-Q5T*YUd}X4WZXcv$t=L&1=@ zH#hgfNI6T7=q@#sFb;&NsVNktOFdjB3vurQzIR!9s&b_|<-xDB}KBVB)#2z@%ri-FS`Gm~W97WUSW)-uwO# zKn_rsr6X^4X$FP9V7&Ygm7@8!2ng zTTy-dc<*rXv>Q?ed&<}l7cMFOQ&ZC`WMndOa(Li8P9WyZUTJn6|827FEy{4~{A!*N+`tP`|O>}`&1Bu6Qb*%bwgh;&? z3a$S_@jWrGs+Cm^*TdE@3X|={&8CCf3Y02ZTJ7sIZP@A$ut(%zAzkE8!#Si6t2>MY zB`PQ^%w|q$G(t5?X&tt#)OqnSxW^Xb>IVpu@*^4(PP$g(yl6~KZ(MMExXZIy9nXzy zl5C5(L61ZR#IR^^Km7&>nZEOu(PMc_%R4fWH%)tzMYk6F0)?FB$i+waG@!i4av01t zUZTl!`hcf`1{5|_=gtEzOZPcFsj!oYlWm&C>TB4;cyhevLrOx@1Us>SE;Q&1VPI@# zDg0HZ)DaSjwyZS8S4em|9xzHQr7~3IKra(PJ(c7(kqYdn!cLa=RZdx%8kfuEp$aAr zj!5|Fn|JTN0;}F%vG=j=?aCX;{Pc6t4VCV^?|5|c5I=YKBspvHIAP;6)AzZ~V&i$+ z2e5)%_53tQ^S?V{bzCeP-*4CqmLlpV551_|jCGi$xNNm8t#zX}l&m7JxX(t(ScP4& zboA-_)c7OyPZR!gy&YrBcD{l$O~g->c#bu?K@P2Cb9|zeKG7q7+KFA%BuX4uP~A-1 z+|H2)W0umVw6xdbzOFi%H-~mHwx5%mljh>?N@w6IBThHP5#$5`IW0A7)t995rCkqSz<(_QsB~o%NqwIh&V+KGtz`K zVx1%9))>7?^)e+}VFlJ3^gnNxnmu7_AF35t@lEW_{3Nkuw&k;T?#7aq-b56Hb@9`7 zlDhi(;bIe++&VEj%0Z-0ZO8ZTSnZ+3L^9#&mc>vRuD)kfL2UsSC-UM=y@oFF(|AI* zzE|VG3yv@br^X%c**X*WLfdG3#s=Af&3=4p%J2yL_QmV)hXwET@GHgN_SQMQMXvqc z-7vNgh+x+~pP9)vDI;IwaZNq%Wg#k{JIR4U{3gti;^??HI(ZV1#sp%Dl-Q1K^N3^| zloS#&vNj+R_W`ZS$FgVsrNa30J1dKQSww)gepjN7^Yce7#iXP-t*$D>A?#6?FI*5r zZQ#xCteGp-x{BxQkV{yFZ-z`Ox$G>DcZU;kn4`g zuN;{}b(?oL3>5>dEHP@nh=+$4S8aI4%JRe(iMiky z{?%ycI7&7&;)*2_7=y`hi78fwg9t9Q=%q^>ZD{*S#;$7!n=RflMAJ?7|Ec`fwWuGZi>BE%mL-Un!{y6p&bS zk>sZ3drn|8SaO`NZmnzA)7;$rJbL8-={(#8b9{Q79mj7=31k2PXLlN=F`es-$EG>g z${$Q#z-$1bM_6 z&#hh=klKGm>?Sze2D|+H{Z&Afg!u(sm&u`swE?IBv6F@Tv zKtrvsp`ZC37cWgeOz}Rr1GKKrVdZjnY*L}y2XFS{xIR-tAS*|^KBtxYH*|g&$fGgJ zJ_kQ85s5(lxBuBT3@YM+vv7d{N(pGJfZZ79;L^g<_o7aP?wyrTM;%1f!uD2s=WIqH z2e8eO5^e}p8Q9{*{%lgKa!P0_0JV2;ayn<>=yTa2!^7a3cgwH5#uCA|138l(XA$}O zqrdn^HgZy8-o4Y@%wGA)te*PdQbYI#RaLng4=U&|zi<@t@SE6J87X(6T^pTXpUJ>% zX9vEMu_v*iYwdIwBfW152nc`*obDs2xT?G~<=zo+<0uqnc zF^ty-zZAc%AE0WE)4ssQtkIW`@{RDHe>A}|GxuF1r`363yNd0EKHx>l+$Rt90cin+ zJj)d#L+Xs=rNEp?%NK{o){B2TG*9I56N*es0%gH;SsCsH;K4wRja^p0NTMBYGQ z^(Kf8J-2aFY= zE7a6sA3jilG`>7keC{*lmdl}r7c>ASCMH_fW~QeXhD#YK!vkh#pCaDAefwv3w{3EA za&HJ2Y*0i5^6uTy1bg`oauIT*lHbQ$I&O5-?*{B$R31ILCn14VthY~q5GH|crN;au z$pg+)YkPYzD2OoDGbbl{R@ShM4ad2z4>F+JF~{Ug#8+1CYTV!+7=_3#hD7M-6w``n#IN!GzxTSk;qR2H_YTjNBc+1t;ql^AD!%?D&<6r z*3hGmL~|owZvVG7kaJCrG7pF+;msfp*&D32d~6ORBBi2|htk>iN%2AMfaAujtk~(1 z)Ax=J_tWE?57c);9NSXo%YF29b;C+adCpxVfUI%h<$v~ys4BxlLX0f~%&Z)f?;7F*@qR^t~6r}6}pVY3_gvJ_p$3yiK z5Y93MC^+VGdc3#4?*K9Xky-1m%aUIjQ)`27PI?@c9v)nm7#hx-8p+}aN#p!Se@ISE z?cI8DULGq}KFBTn@*Sun%uFjN7zc`3FX-sPE{qCs_u$|!E-p$+N+Nz#JDEe<7@8Je z`}#`v4HAiM6ZQO|Fc8JZ#}m|)cTNIo#U&u%y(n9*U+=*Mz!od#s~70D_omCZZ?rQ4 zuMtU2O%0pA zf>pPQ__HuGOd^>(@UzyRTcrb(#7l#I^x?{_P-4QV7^lgUIoD zV97(gTvlE_cT@B6W0Gj3V$-kifm%u~#3f>Ez(R#V)6&uWu(}tX1`l7E~aBbz!iZ zeV|n9R5JUGJ1uZl>p_P!ypbFFuE1mOb|(q}MT z;*m{%@4cx@u#xV+V>BR9&VXnUyD!lM9kwzOB_$=kfbA4vmk{Xse)FWo_SUp%W8_Q| zgZ7q7dng09JK&)_3DWl7iqSE*!NZ}~ezH{bU`{@~ZZy!iF=P@M1+ppt+76J(hpQaY z*H0usX@u`;`X5~v3Idtn3I&Di)2A5@UqJN>#4<~4(&OZO19js^8kkZn8z=a2XI#Q8HdHEwzuK;T z>03U-T)|qnSHWBHk*`4Z&bD@i{`M98djL`ao|wP^RgdR*8d+GW9Y}ZsRH)L?6YU)6 z8U~p;fTluBPR_LHPYu0i3s-J~Z(Ly5b%ss<5%zs?q|K+|eTl&;)Ef&Su$#SKz+hp= z=onlBjUDwl{9({JjNF!LQ9o8lub6&$;{_cP*L;sJkDzpbm;j?i$i@Y!KY(uvz%rR( zy@$|{)fWB7dJ|${PF1T(g9Q*1ZzGX|)u=br#hNMbv{a2bhD#HE^trB1H(*;Q-Y@tW zc_k61Fga)l;^*JefA69*!V~%&7QB?NU%!6l;9$8LYvP7&XXJ=p=}p|1z>3#!+hTfp znlf_!Ke0OqpymA?#rUwUsdyfPhlOZT8X9G2BT-O{52M7xwzg`|;NVm>`QbA5T6q7* z?Hd@pO3`G~-(!183ckL$IKJN|-m0pyj){`jP9P%Vm}nlWb~u0jsA;&=JRDOic$N`v z3i;LNBy*GjaURUqY5Uda+uqf6!SM7XmG5xKH5K!=|4D}r>wNK>e<=w=*niy5|W5(gu5&++iuFY4p*0S z_ar5;#B~CsY6W^;v*} zOw7zVyYs87t5CfAYg@lB50@@Z)Zc|^3)c6d=w;$W+_?cbAzD?c27Ho%5_iUNo%3yM zfO`@_k&%VF)%Jy0?f*~%H4(6sr&;`(MWX=H*2U#Ol9(|85~w7P)U_#Y{m# z0XzZjcXf1h2quC~v9rDUV(*WUv55&rq@dRVq<;bL!oEc-zg|!ng2Td~A(fV%PPLrx znlwV0LwfC6%WQk}+FlJ6#J?mdG@(ERmw;npcn}2%jF2+?6IOsP0y++=<=`)Ge}8}H zrG8u~l`yAQR=I@{|4&&wQTh-?3gM#tXeAf#-@wPCZ8Q{Ec>K?wk3s6jhG{!yJW@HibF)W$(&7eWAe`RUuqkji?CFn>_Ysgzk{TKsq}NobDJkW2b*Y(SLVo;s z^wFwm>H(3cBxr%YE^5rz08qA4wWBpI-OC-A#$c?6LskEw8>=nJie{rOa z<*huhATXoh70D8k1~*_N);Bj*p&|G#BjdJ|%GKX+2EI?8p6p)4KqXXO{E_PGIO(-} z2P&;~5tu8dpb*9l+Ha_Sp=lj#~!d^)*N~?bjOY0x(oJ} z4qTG30Hmv)O%Ioa{>NK|;4-)w_(WSfC$8pV9gUOsW(cz1igZXShKH?b#NKe<++oY!wrrlh9k|Mu-$UV{F2 zeNpbpUA0SEmNj-xPURD1k?zWhOYsK6=tc=#8WgS3UBB)tc*Hb*`qFW@G}~eg61b@( z*XhM@F%j%UVvGKM!GvOL`2_~`1p|A&w`lz4ihnsj^wQG0tyk^XGfFHcoXfD&o|jB9 z|Gz#=C+|n-PV;}g2c1ez*l^>1wCu0m-qwzexuyQRoPLXH$G_f$hZDSh{d#D4_;z)Z zGXt~}_U`?A83l!`w6vL+qxbLLF)}gT6%4b8>KS+$s;eyCCn^{n*`IAUZnQ)jipG?2F6# zM(fw+tXt#>+Mb@AqSUJu)%~T$k{uuXwT$k##m#TG44_r@N+v>lk+jwGs zsX06MwLecoXD}nU82$P4ufw$F7cXA8d)Ye_R*tyw)aRaD8}A#(>a8U$@2jafo7i8= zk@OxwP0|=2dDZC%>s-2YiGqTnY^1wyAcNac*fKx<1>Y5)f5hhA5lEy3KmRzXB4<=@b~?Y80BUxYgpp;tX?Qpuf`FU9*<`fBTH?u*mo|lZH8G*&e4ATc zqFXcX**Kk+RI&OjoYFn6Y%O$|O?`cxk)7RJ@8fgpfv#(dvB>edc!lj`m{{u6&Y;1< zU)d=(&Z9Xwj00$$!Owe=A{|6xQUwJC-@kv4h>wQ|44WsI7#~ln^E#SYSda?Dr$mjN z4iy`3uMDdo1D4r3`4LHLXJ_|v_n`i8i8+i~G%P!tfvx_SlU6e1%Wn=rmp>Pl&bN+? zXmgQO=30B~uC#9aZYlS?L1xkK`biVE$NOZQn>o5sga!j9Ms z>g27gSn|}-7S2a$X?KN$g|XK+H`9uWih5ecg@l$y%9&yEZJl)p2nb|lWy_Z8@7u@G zN-*cDGUcPo&s6N~>>j5ojoDB3q)Xuu5SR_*Ye`5-^44Gy63B6Ia2|PK@!XPEs@G7~ z&}a#`M(%ogu#hlbQB)LBpjTsYLPiFe0~+gJ5>)uW%OE6_SX(R9vD%Tq$7MGo!D~H| z!Ef*@@y8G8nVFfKoSZuk9)tx21qo7ZEr_0JIoPYA)h(O*xXZ2e^{I@EjI`?mnj!u4 z$X5S}myhT4KRMd7v{6hEq4XE~)<__lf3Q72Qf^u4`Js3HXs#_bDl+nQO^u*gSMrO$ zer2Ls#%{ngnVU0G3Obt%m%OkVDoP!;-dmgOG7c9W_`LVHGai~|ik8h5jv5-$q*qMN zuNog>;Dw#`H0%f_?fGP#SBFT;_Fuj6B3_5AJE&1!R@P9QE4R%1eN-Uo%rg^G;>_Ec zn*;P}oXcgXSu?}EXW6BK)PFg3eb+>PrC_mW70Ky5xMtq!z_~b%HAaY(+F#ZaxUZj`!6?94Kb1p5<$0^pqH(!GNxB2 z?;yc@|HeCdwvLy<$ct(4qI8g161(hw4x$leg!}Ky{$h_SZN{&T2lOCs^WWUs`fCDH zlRC;yhP-X>9U(K|Agmm+Z>&D%QiHQK4#vEjk&MkDM zw7{QkYtoy@Fz>LjDK^aVBJcP0YfVDB`FdqtaKbVQdueIW@o3Cmo1^GFr=vu z>@O3M&z!ux-(yt{(EY(~G$|!?#PQYXv73;P(Be>Wda4eG0CZ?^Dv;Xylh4^H^qukV zEWEN}^#`-35A*pEI@I1$wTIVaCF1L~u&8w{~_M~69aAB-C360Ljz5=PjZ-A4FYmPAcHoV(h zrXj-(`FxzKy7@a?6!M)J3^xWkuf2bIw6?Lc6U`%!LaO{Ng%pveN=gy1j26ff(=}^# z#@(MDBQp-Ywj*JRe#d3jDIGV4U<;SX=;$X91g{(%xEz@!pTZK=JhYp+keZqb4=n!u z0E0rJ;bQl{e(niDp~!yy`h|;!#{o|$!V{*6{0xqJ3x*bSBZ8}tB z1dY%k>m2g&DJYm9wXJ_K`y(>Irpe?7S&_A-;(#yDxcmM4_qUe^38H14 z1{b1v43R1Q*RP=WckbR5%Uk$}T!-yhnG35u_3f?D{?uD&CIh->m+-qKyfHNj3U%R@xU<|PBjh=n}{CR&dJ=}G(l>uHj9LY{}>mCgawy3BmCXtz_@&o+*`*3I3 zYPiIIA;o6`9=ivRafqDm8^d(|^x7MT?a=-K0kLVqAETp-ja%?}Z6~bYV$4cB?)!Dv zBU|0#1W;=s{XMTR>5_=Uq<9|&XJpVh{~3D;_o3jvQW6r9ma><LU2 zW@JR9q;$=;Mj<_e*fy2RRV}d^=A7?J2~SE&N*(3!uN`br#qHc#IDo(DU=hzkByi=MNhbQ(8_=5ag5QklSzb^738_aErLPxmiutC54Ng z4t1k7ui_EYW~(v{mzohEIC*xKF!AGeVY9AtuX?(|Rg7Dd7nlDNQkMjt%TZ<0_gLjg zc3--$WmiWDD3dleIavX)c=AFg$DbsZ_0}I>vnp;SjA(mua+2p3nm6Fxdu?p>ScIqu-;U6m7_yRF^gV7GY+u;%yH*{Hz(s<0 zmIt|PCxo=D{o{R=FMAUn*T%z|BLA)) zA_Kl^dHn*ybv(l4qet16nEicMU^|`PgNe)CZM=3Ma}NAdXW&jJXXmW4vK#(|Y!g~q zp(2t!HGEhP9z0?G{?HdZ|Y4CG&D54a&*l2q)2^zp`C{BjxU*qhsDZj>$R(NcAXTL zkG8+Hwg&t9`bt{bz#@M7bf?3bnVGpmL;UHu&%x#k$#AMrz@h-9b?ZD2!WO!-OdC^p z)Ox$R!e?hqk|s(4W^oyQ#cB?xF6?y{tEZE{dY|Fa?&{ky-3Pi~zIv=-S{%BBr93@_ zrl+Sp59VU!lLVpx!NYEv+1VL#n=7l8gB~_NEfbTh zgAQ7q@37EEWv?qmH0AaAOCKOU&Q4FXzPw5u809_IYE0o%<6>fZ=d?V)b5_T*J{N0X zKi__Pz9S)fhd0go$0r_fMz+9Q6&lu1SzFsBPOerLmjy>nQh>?Fzp2meu(K;|z0%Oo zc>etPkM?#t28M;0>20s0-6j|zKBWNnfn!S;Wja7eR-3)N!n&FLjo+^B?wgx!Y+TR3 z<0e(Cja6}lg@s+E7D-ggQ=f))FzHAZ_OL(Rw?i+Pnwgba^fU2V4L!P?yJ#AEO#?>M7(z$HdJ1`7<%& z4E&XwRt8gB%gDmg=dM$(9V(j%AU>jU!u@@&O13?46naZ{S63xvW&6QujY9o7HlKql zTk{?1d!fYL4Mnr;*_8eL{Tm&8lS;)-%xinAZD;Z*KS;=e*NN@)t3FKC)=Iw3C-qN5}9&6{(dPIUtdzL{8ChsMT|$i=YG@bHumqx}8-T~4^O3JStB47{QtSs^*= zuS0tY@<{fVmw0%1j$ud6XU6K!?DwQU{H%I>P(C7 zDg}Lg{r8pOAt7`C{olR23_Jh`KzEg-n;X9^W+%q9>vJv&QjDL!KabAxxXWZyPfw&- zikJEx3GGOTpA0$dSkIk)jrK(TDBx;b9=lfhZaZpy;sNW0Qq(wxjRDto&Y0Fla2f*u zk*j3M8$tA-Mz4u#_00cI9D!S{x2b6^<_Lqq96&NYn0iZqLRSHrGG!aBf(;9~)9rUC zpUIcn$Vdlz>rWXOe&aPRD(qRlKYz+X<_LNF_B^DSY2YC&YXhMg_!OU1G)dJ*E5$7= zyy7T?hUibeH#bj|{L(Z|sfG%`bMP_*ruJxe)pD2LW^#9aQCiP3D>Jj$@Y@^7N*Q_i zS>U7#ZwW*`LCNmer5#C;sG+)f^JcM8<3)Y9xtp+&*0;A!er3p@h#@3CXqB4w%<0_qCow#h14ZsT z7Gq;$+o!c@J5YBBkp25-yhcV*(QIX?IA6Q`T^Oa{ra+-3Rl0hf+sV%0#`^jctnoQZ zOV=ao^Ul@+fVMQN9q$8^-QV9Qt)}d`fET6yqqB1!hQPqb7Yo&n)42KS%IYe)*9tVo z$jE4Ta=2rkU-&jZpXGEjwtjsnuMpw%;+3=j64PkQwL2iVkxC9jmQQ>b$?q`FU2W;z zt0;Q%eRp;A0E#h+QkUMw3HH2c2db&T|H?gSd0)Tan5Ctqho1ZF2HpqvoSmJ84j0oe zT)H&LK{DrTQI2}i7QYwYtE%`qpfQ4G2H`E&vT)z@mc*|HT=O#h5kq3#f6J#NUrgL3Byv$8T-pW^F_ zW!jZIo@O)}92_jfwR)RhF)gz(kf79c%aU=swo$=EtWX^5q-UCl}VAqCnoLum~t+r^s6#EX*s{ zJ3l}HoFpnSQQp*)J~A>ggp5Zv?*&W3^QidX-+p+itlAQVQfEImyQmj}oi63Za|bVJ zm2MAJx#a+~YkGI{1_{YlGr4Vdm4lghWk%`Cg>3AiszKB9F3&a9Em#9NP&25H9aUZ- z(h?HiS&b#M$0pBCs4rY)^_O1T)lXzCY4sT?$TbHovTzY6d*J7!EDi!; zS>JfPx)`LUPxWJj9E(E8HG*%_GfMPyBfPSm{CZZ~-G+=z=tzdG!TzZSDeqNg=DfqU zpc^-h)YLETjisM0aB-BEmm{+cx&kpEtdqS-+r=2ad10(8Fi3-R1^uu_=kIUw6+>eH~71Hcx)B=oX8c{j5n;ebIQXOYldQ=Ei9SwL^8~8 zXW3|gWvo(v`n3M!8dbb_|5jJ;em><^wsP+NN@>?nts56Go{ws?9Ury)Tan-hroFwkHW5M1@|1&Bh(YplFlepxR&_=NQMWD@6xXr8F#vBRDj4 zj=JmkEkG8sCtu5hpC&9dnjF}FI*mg4R99`B`kb{@GzM3xfyU{sj^?XCXHuL;a47BGZLx1VEWIbnO| zp6*s!jaG!xhzEQ<*t#SBn>C?(QF^H{M<_wbL^cQc_{t5s}{~faZ!Lq^G zPXI5Pb)Hp5A}=lOMaw>smAy>uy-#y+aDc!bnJXmEeHIDZq=`sEeJsB+(7Qu5ib0M>i%{CV>6 zCO#C$UF-IrKi|FV&El}1m5zC!_Y<gt;*XUBmUkG&7^@vV@H zG)qkIu(7e9+B!P?^kpeEx7((o2NeatlFGAZc5NqsIB(p%8408?j>DKcYT3p>>bquh zP+2*sWJOS?!YKF~KpND>wL$K$HmiAc5CgEa#!t7Wn~{h zqA=!_^ zca&{i* zWxhGDP-ui8QD7n1PUcc-OAo zC z2cCzy^G2XSf|wMQkRYd^fG>J>%r!hbtg5EQ0W|^mX3O!x*2dl*U$Dc&hW$k+y9{6$ z%DTGkKWCy%)XaDeAP#xh@;vUVBC`@{=4P3Ku$@x4pF{Z1qxAerxi3boR=<_|0 zdqL*6b9DR;?hA0!jtN7#Bg=R9`UO>;gQm8=l|1|~@%in%iDejPc5BPA^468Flnbiu zJ)AMmWbb1|^F%u?b<6DCRp6^5Q(}r{ym7gY>E;CWgXxK*mS-=Hmd|8cIB_O&4XzXJ z`hcQ8y|Usth^fD|$FTG2;2VyuKi`F=%*56bFh;~ic?{Q_Y3TGMU9`2d{yJ%pZB}1< zY*RD;USza}h&}0@!Y6H)(9iWJXm9yI$SycsN;`+bYI*yI zUvzR}-IoU&zXuNm`oB)0P87g-d#UkP=Fsvhu~8HKLZX4v%F68O>JjrJ%R+Z|;a+El zSHl+;@P>HpFT8lqN}Tw&p8_nKNwdtH2r&Rawz4np{*8w0rj{|powbODqKRn9e;`xz~?`GXd`w33n%rM!N+d-sSaC_?h`n88$| zc=!q{M^xTozdtU^`xHqI;L(8UNGj?r1X`_cTPz!w`?fiF3jkN9H|N^0_74ss6BF~h zDrM{Q^9_GQ9EJzJc&cV}$GyJqiOSmn8+&JG)9uBcp`oF%XNqWbgg+qJs6e+$Oizyz z0Y#_B2NDtz5fKqz0n@d1b+!EbxfFx(pFgZQDHe9$o&p{LMd%GswPMS`2VlJ*x;#Kn zBYFKnPU>4Ao6Tc2SKIi52uApI$6uYeHW7Ykay90&3PuHpzh$V%n zcWwV#DB*nm)A{*-dpQxEqcMMTKm5KkqP>;jsI!yFvpMK7xJrZ+6zxc*fqV+4h~wHs z5`t#|S%buu)9-*FJ`#P`*K)BBHrlzm%<&zd{&R1=h6O{&qo;wga@v+9QNst)aV05a zw^HvtMUY016!nXZnWmMos;J6I?*vHR>rEsEMNn}8CwB%DGvxfT@0kbZ$zgXzOIA(} z@5+_l#D~rf4(U_PKMjEsqLi3fSTZ2B9ePBH#nGf+iCSNOwYj;;UCC!V(GE(>18-87^$7q5b{GVb}#q&BWDq4SSS zOz2>3RaI5jCp^}Oh={Ns{o@E(uUq)DQ%jQ2|C`zMW-CgiDy^?K5?G4F=jP;eds0J9 z0M$oET6${D=S&bdF|u?*TkTwkQ#7-QP-DrxKjRSZM&E;oAU4Lk(U*-%SnkS`l~h(T(9!utL=b+Iy$8L_$jCtW*38n< zPpHzaQ?D<99b4_eLE{(DgwuON!fjYfvxZV4!4}Yf;l@q z0=tUKb=?q9g(P?g7v2|{U#AzacxbR_T4Xg|9Sw9A*(5M5Qf~9^LXzz34PrlRJ(;<$ z#(=1Sybr-VYnVOUt5@kBK8%9`k5(@sZs^m}tM}GJuaXss32GGRBA&!=h}+rudBnMe z(SX$)W5^*6c>*hhoUik15^2l?iPW!{%)S~xXut^PK_EJY;JZrk5ONnG2&uA&Iz5rn zXDO^jkqa_k83!L9-(6Z-n8~J*5tK@{g5AmxM}^gJ+UH9q2v)o)>iyh)+h}fY`_RXSvzQGmu40Uc6|9Ho;$$j{xFodN-||FziX1F8v&=djdwAD>#J9m17<%U|NX zOq}MIgf0I!Z{8G@mq#Cnr8R*99{_zqgI7d$Z_D4WfKL9q-b+CP=>j$rXv<=PLI|=9fATuK1Hd6G^*J3amTU${FVz(qt72okPZ27ai^$RxF;57K*39iN4 zs!>lGFI@xaGHQ_aZ;<`d3y6#P)Ih2a2V#=)4HBeD%>`_Z`|CzCzkkF06M-5pxeBua z@+us|F!|>Sz4-i<+kDW*2e_~SRsy}^=z*BMS84@VEbTv62T2a61F0XnZO&2-l6(-S zIYBi2Hc**}St<<-7UAFv%IL)2i{=DO(Oc=$MW+eGL^=Wn(z&V#GPSXJkeQhY^4uDOyySKusNW6FtohQ$S~|oMvI=!EVx7#8!JvP>q!6C0U7Xf&$}eB%>tT5fN`hW z=AW7~aqHE>1#M1F&xannRHj`D8Hy=loSdA9EjN|-(qJIK<`J&3lS5rM7@HtoT?3t1 zx5^&J*Uzt8=gn-^h&K84>tfhHT3cTZdS3(I50`Z;k>m`7Cj$z1GViXO(j zT+f~BRs|^H<~0|sw6x;&Q*@VPgjMpg=a{n5?)Y9-mO<_n`1ttR=vOb^St`p4=Vq_D zgHV_HhzlWg%5VXyT_&6{xeSQi$|)zM1g8#gkll0JTvc=d`MPK>~#Amn0y^1iUvQ5XQHT-w2H z-Pqm+#R&z4X>x<2a!|;1eTtHYe8Bp1);}j+v@v{O`z?O$>9{R-hl9z4T+mY_se=%%b5Na9CH~sUNIdHC{Or&8)T7@)cyRh@NkXn%*@Ao_QYKkX z#!t)UKlr&E1MDBq>&c6sc{2%9DPO9)-_L^I2x%2rg z`WQP-pscyNxfPqVUI)iD0j;H}DLq)ASMII{i&IwmKL8>1Skln{NA`xZJ}!A$`ud$Z zwwO$QqA}GDI4uJXPML?%(?8TNi*qybWwa zp344E&*v)Y>I1X;_%P)luMVy_jMZ7gAwC5K1%W|%IXUBRZ*V$92A?8h;`+Co{1oSj z<>IgGM6Q1y1cdzj|2@lBRPB9Itqo+&pD^t{A{TbVgIow>lfmrffgk(dj}@L$^Pe>* VcX*90!s$sYnI{So1!9J8{sT$574-lB diff --git a/docs/images/chapters/bsplines/0215dc106e4ad51afe043c0176a595f6.png b/docs/images/chapters/bsplines/0215dc106e4ad51afe043c0176a595f6.png index d3d4d56638ccb05e0a545de894842eaeb3890a90..036cdabf87ed6f51349217431e08324e1b70ec7b 100644 GIT binary patch literal 13657 zcmeHug;&(yxAxE-BMs6iBHi7Xl#W2+|-8@1F1d zz3bk)e(!qM{R8fDTrOumb3UJQ&OUoT`+1&;(bm);A*3fnAP^)suB+-I5E#kmzxcTD zPDqRRApC=CbxT7Pae@Am+fgB?Rn4U9CvG3v+uSommDjHB7R=< zB|pqtcuJJ|*bmgw)gllLgjOnV!;oHG&H4E#d`#Yza?~wuRI(xuk|c`%6}c{VcIw-M z0}ayIOcMLLbQuv}`Q<-GFLoB`x?~4syr&Gv)pS%D@9FC+myCX2%=2pZecpq9P04BL zjD?CnTaUtYXIe(TUb=oN@dzbHBF}>P$l2IZBXE>eRZ}Y!QK;*esPU1=8||2^Y-~3S zSyWXy2#6w3C=O()eu+jX+Sx z;Z)lVsXI7aS^D&h+e4ady2G1gL4^96@jX+f%an#>S4^b*@MYivCQYyeprV5`b@K@S>?py!_$(@uWS2RLe{wSw;v)9x z5dGniF~YU0qWC_qFZt*)xv7<2w&;Gi`FrzG^PWScEs6y{BUPp(9A&dx@8bkrgz~Kn za4>PPEfxydw?_{v1g*P?pG$e*=H=yOi8)cf^j*qcUB)-Yc>0FkKDmUq%y%ly->cB&mhz_4dU;h96?{YZN1||)r!*+l49V{6G~`jj6i`noaP$mE zT^B7oqAY(@T8_!xr78{MHN<&%e?5a?GNihbK;tFWhv|oc%)%Wh%OTah1bTIZaLb~C z62dOVRa}=T+agu7P0&`v-AX5O8s^8`Y`IvB7kJFWq4F zX2iNTLupT$;0y7#SHkXe@vBtS_`W#a^$cCpR2_bUu`{!6f_vh280$hZUI0 zgvb?h%%j;i!7cEM#8GCt;~86Yy-abt+bZHOiCO^-gVM0ylGYT1DenBJ#lk~FmW8TP zh5I`1nz4gMg75=*Df8U(Bt$DpONp01zff^;;s2bYKvpn<3Jb$bPfu@k7VoeZ&2MbH z%%53uEX9>Z5XPcdF~%+^NN!w4jT??JJ3EVtk5|2Sk1{_$pPZH!rTId7RK9Cn`bpa# zsmHB2u&{3Gl6lT~jaLRN#GjGKet7DF^D;)0smEw0;x_}u#A_GY4YAgW1oIa;PPOCi zbDadwCEYQ9{`}ed`E&R0uZ8aCr@nQQE)d$e?(^OFL_{hN9&pf$+K0@{%&dk9(h_UG zEkCWlSuxhJ{lBO^ob*(V-;esxPr)}7s59c}H9)6;3KGuy^HKQFdrxDp-JXZoVA_WJ>Q$oqwJvIoj=X<2CgYNm$@f-Ai6$GoIyz%0 z(0S(Y=Ap?HlZ(Z*_ZT($GY2sw1t#yDEk^?_y%(gZMo>Y3R=?^}aqL7l3qLuR# z?M`HjlMg&r%qtqF_96_%5&a}KELu@7yTt32=XQgYmocN}~y--L3aK6_0|{OQ!ziQ?Fzisz6-x; zmip!>cD%n99dvfg%E^giH}tX#{=2=9rj?(Yi?HsFZ-ZnwT&Tr-V>g5KrYHS3f^+8M z4xNo6_Z*qDT4)N!~ypOTe@qPnWzYE@-& zFxVE3SKzz5H0)SmqFbuxFv9na4wsB6KxF=|TsLOkxC-{5gYCknf$);v$d$+ud?qAu z+GvE$z>_>U16iUDm#nO;+!lI~&eP3|eSLlQ!?~doeXb=>uHCmfKRxOl9PB(j+TH#n zG0noog{vjIjjj1o)@WJk=Icuzy*IldT82xFL_{C{qU=jctJ)~{s${O6ZZ}Ze2SDT^ zbs{Ch-~1T4_^}@;bPdyzESj}Y6c4r(eO4ZCOvKbq2Qe5K8Tp(aEswmqK~c{>pqZyV zv+;4W+j}}N8Ya!o!-F4uG+^#_adt4jGE(666bZRXVmg4r?(=gcTU)Lvv%lg`<)(svJFj5s8_ON@zywV1eC12alZOG|n2B4K;JXMW0m zv#ZX1-fd|xTY3n;UILr236s<5ri-QSju#Ak<4qnXbv&I22pw)MyCkUuw@4$KnnuvVyR6XEADCGIdTRz5Jhc-^kbN1m$c8N)NJ zlQ@A>j|fqZW5{dtlGKhVjg_0GWM{`rHTe%PR!=#8eM5o{)gI>H%Mg$~@pK~EI_~Q>ym#)BDECC>Z=EUE z_50dTk1`TPtPUE@biVc$$n$w9q}(0B^2nCsCM74We1POs;DPC%A5~bpOP|7X%qMHj z3-7CtW8cma#ArD`zRE{UbcuqZ3hJbjyxEuVm{VS#7x<%seL#EC5u!_mlBzM-qyTbnq|J@+fz!Ksmki ziqf#D5-LFWU4v}U(IU;>$_N&t%;V1BixcxF(V6tpo;<=LA}MKU;VzR6vp=e=#WoCZ zd67bLn1JLg6?(2#ul#}z1*7Pf?PS9M;Q$knXQ06ZhlI9R~NE zg}(ez$FifwizhL@al5JpuWx+_c)nb}TNj~6_Ta$-3y2+bUV5YXB9@4TAAxXQcINRZ zLr{I>B|mgyX=fr07p`2&5jFQ;7kl;U6^ocy#`OA@&ib#Kj!-N-Xd#B7oIaNiY&dSt z%~Klgbksg|5Nai=UJ0&L4462W4o-slHoI2QqOPuvt6ypu`gi@;{FYO+PF<k`6S6D_Lk2#M>Ww4KSuW=*HZ3XAaoH z*tqP&eVQJGPfQ#EU!u^G4P97Rpies484%cBdgl7*XRISRHU<$jUu!DoEkU0x3+Jgu zIt(nVT$eocksUUai`i4XoNX0P=EAYHZHJvgp85*xrE=TOuC7sODnT(ZDvzZ>B>;<0 zMIG4g>h3&%AW*Qfwifc+v*$CZBe(8}S6j5p;F;8p9X_f1W@KKa81S4YR*+e+V0*87 z^nU7TFuK5Dk;#TE4rCTUE)wS8a7{^1k7)8guZz7Up1grJ7dV#%QW-5(fwc{4qK`9 zzH3wDE3+H4wGNkqE|$CvgFc)dQS_C-zJ9ylH(y*@n)_*&)5V<=fT_^(D|7GCWYGYp z@LqXdR;35#=KxRKMA_(xsugOh&_kAGPVNaC7yEIOpIA=6-aGi5Zug8pL7+?13EYDC zESP3I6LcR_eQxYWxIt)^y;$PI)j%ffDvqV8crjM&%`zQya?LDoUk|oA+Q4+=;qF|V zfYrFf_|{R~5c1?6A9lvpvKHIsqw1mFMOokF`big|<;cq;kK6Rm>Hww0ocvwiMFl)J zXjzgwT1d01t#g?>BKtg@?(5a%TWiOZSs4(2mfqC;ofC3p=hke;2sAR;@Nn-vf9rDW z-D(Z5P(ZbB5-ldGNYD{=_GEpzJ$z7V+#nrvD*3(CsH3F(x_B7NOvfTK+dCBwuED1v zu^o+#W0{aEQ7hz@QdXhO{(7NELGudA_u#POY39f$XfOxf+b~q!+THD_eb&-t)MYeyI)1yZ--n}I;GZuv z-~b*})#-#XoSd54UQ*s`4IGj4&uW~L*aORrANAzx6CwlSuyGM3?UDU=GlspsSVC9l zCjEj|#5R7S3gw7TNZ3y1AJp0~=w*1krHf*_)%gs4J!c1|sRrW=9-r}h{5a}sVJuq> z{yVZGBO^b~=O6P;zNd6ruE>xfokTD>tg%DifSp9#-Ewi}%9C-Em_hvZ#&y-PZ;g5$ zr*>b11^;}uL^)DN@)vyUWsKpNks)WE;c;u#TMBzqgL~!56~jskjOoDSp0%k$8F{n6 zLfxNeqeV_W_cBKRbFIacYgBnO>Bqa?1F@*AYwRSm zpM3i2C1z#F1-N6`2Jt=(bFXf9Gq}$DF3gEiZVN73kYl=kNQ=AA6oJIjuvp~2D)jh| zVLXEbDb(3y8TvDVW~}W;AJW8m-V;8re%=9{()MDec|T*!N<-7T9jeZrF6;hpSKJz& zBSP*x6SssScDVOd%SKaUtX-klPs-yd9qfes;S3f!*PNf9Khj`paDm_p|7lasPlgc8 zzI}MSR^bla6J6Q}vt_7{&jMcsTkMW(imBzJaB=o*Exd40QANhROBE^D}MX z!W{Vx7I3Z88>***cLCJfQ^iEc(6urK;yYJU|U@J1a;VE zi3go1Wbda>T~*dSx%YTSngjjqYmDoyN0?`_SImkZI1mtbkkHE72tMI63TruC6o944 z_fvm8>bUl8=f9BNzb_w=QfAT=u$5xs;NZ{#FiXGrx@Q4Rg*P4xCFO+ruzEDm{c)T37uDX5~CR-rz0CpTm$rEz#que$B6 z=$Or2SA7TTPIvbyK@lrv81XxVpaGi^J0ax#dr<(PDzI|*hR`2yPGX`dzT_uQq5y*SJx8Hd zWnd%S7G!2brSlwKhXSV|&~h3U5rJKAjnn}!zKyvgDzX>jhfjzB%n>RkX7)e&)$}y+R3`3Kk;hY~SF$>$KU@JymF!lDv;TGH2RA z)23IVFU-ZpR@{!cZIQ1M!TA+4FzG7JBVfYbJO9+}hO6}qS(pa>z7gILAxRH4_@HlAN67pSNWxCRJj%kZM=d@bA;&M!qRt4h9T< z`-L=>!r5Nh;BOQP(tXe8lHMZ4>sI*6^by~(kkDS|=Cb_U{g<*rFrbnRNtb)oX+j@J zzTV#J*_V*WQK;~Wi2-;fi~RP+4vrIMCMNNfm4ey2|N00Pi1-FBe(clnnrcNi6@YPiBJ-fMqxW@^6n~e8H#C$%Msk7;@WP^KI4-ixDhTX=*-0gbpREv>H!3VdIY#HQBShuVPGgHO59PstZDO5uDXft4N@XM08n& zjykHnF1YmMCJ{Bi2@2jX>$ZL<&LB$vNsJ4LoHXMj&wl;pjWT7IaV@Z6P}?-3R8>{I zIxthRS@6di&x{X{NMB59)Zp>ob^pxVUE6 z!OO7DjTP$a-T5l|1l)<*c$6$oMf(myOW{mZJ)YKmQFv;Hf91?`(>Q7O;2^roDutC5 zsq+xFZKT1=>AlM&J$dN!WsjL&4@JhZ@@?7|nn3BZCAD$#X|W@sRIvtsFByBAPDN&- zdMRKnljU!a!8ywRxU|@0X=w>O?2{{0NS!+{-m64i6}qB`b=6N~gXon3&5Nl&`JO*r ziBdRjo#*e=+c7AVBOFhjKr%TvUT)(PqqSl#6SP3q3BO@uYY4N;_pf|MHZU_i$^4XG z@5DSKJw4ZHtX$InZ|Pv)KM|ubbeBxKBi&YcIHY(Pl0}-yt?mr1*lNxQ2D zviz35f)|?Sl`m@A^fro{ukrJf7HYmo6@sUvqDJ9Se8co4kj^4AITCCFf=kkO=T-b$ z6sj2J_8jKs@hbz$g9w2``oPw<|9LD5g)>LBcf&91>nHL_Hscz{NQjJ9$C`e|IMk{XoNsN7w|RuR zZI9b=Ym5^}8Kq}{vU=15nK2P(j?*)yVU_G_T54)?N(!q79Lr5Z7PI!@4g^4i{5z_^ zlaMmW@_2Kw6)(dF%E;UhE=?@>3;=ca`Pm6;>|BmQ3TlNK@?FZnG=amHR|LekSnuAw zTS!z@-KT|ow<7)9LdGt9U`2VlIS@^+*0V!yfmmk?GY%|AXyOc+OUO}c zK5_x;>%y(+w(TJpSjkVH_HTxhr#QpSyDMHZPf$x>mSU`YdrP5v_wFM47L-Jt1NdPG z*kWE0ULmVa{D717+JCZSEY4iyUW^`&u-`X(=ZQL`X5n3QDkOtcm@IvhgjHMdy&V0W zr=o2@s{At)`H<>??sqwTNq)(#zSmlAv+X#IP}*Uz221@CDl{nqj45 z?5Pkr>8ERM7OoJjrF=7Xhx{e^aJ_~d^hAolZrqlKdb0MS)^3OiIF&rVXq=3jk&s_6 zVd0VPp^+j(W@Te~dv3^^dR*2_W$4C2Hk(g0;TcmG7Zzqh@t72ijOgOk+2Iu%7boQS zjfaw&x}Pzr95QVeBpulCEJ=4ZXpLFDFSGRlAw*C+Zo2kZhtl_L10IwPp$qKbAD@2; zKH9nt{rU`?ewa%t^y1j0(FNbvH47;?IXVAT`Kt!*KdVcs-v$A~>S$;2p9^7649;3j z^0ae-bNttJ^#d|{njl$R)i0g-&b5%?x4(+hq)V2es~7mJ&7u`eJlA{Jp`nM)*^x?b znuD?9Ot~k!K202xoS()1>S&g*O)S8{@_|1w)kucRahw$bZA;gl<=x{IvK^$2XOc7C zu;0Uj{njMMUir9YsVQL%_%}=ZAw8P*(Q@ zr$xt{yZJVHF$CxA_^*nhB0}IdXT|nxZsA%PTl<-AG}~$RNq%_>-BLb3!@Yw-Q&9Z7^7VfFendJ`tXNgfN5bm1?oy=`tv~2rpi$3kN|A){##Em zFP+eFH%A^vi5$+AL;*4IJE*8)v@mzF-i@E>(OxYXC?EN*Q-op*Ta(7e*; z%nCnv z%(h4Vc<(^*cXL|6L%2BUrQhn62XAbx_P>@O-Mu3w+*i1)Ll0?{$a6RIq5;Do(2c}1 z*XEdog$F^onzf_5LkdYRE~a9JxC0WqlSH3DuYd`70I#0Vl*hkc=Xc#XjJ#?puWD| z3Z$=2z|Bd$J-*FSL+g*@L~IAiWop;>nEd}_*$>rB56ixl-26K@f9Xk*FbAE5rij5s zpr@zk`O#4DT#5!$=%h=g5Zw{44kcQvu^K9uGZb^9noFI}P1GM?jB{ayp!sqz4B9N= z$YUFsGy)2)T%&5+EE(_e_`Y76n0!>Wh#lKQbNcV3=5cQh#3RcKG&c-A<6u40l`BHt zMB*;7FVWWqL+fi;>%s_>5!+qi9kG0#xft%3ihh35-%lTswBL_rw?GJtm65@={Rz&j zLa^}h5$6P-OL5(K8UfN~c4|t6o8)CQ`w4_)RnV~?t%c$WR^GpufDTZYJA$T2?cP>* zuLIEUC`giz1x_+4^Ka?{@B|!8`3peC5^t=-DE`W;dOIm3caF+nA9{gk0!NHvem@X0 z9VZXki3C`H9jUhcl*&ryD+3}i&)3~U!jg2+S#-BpV7@xSg3s*JZ! z?$o$IZ;=zQLjVG2RFcBEVbp_#GSFn*$4tHRK*coN?k1#F5Mw({aVw%PT1Zr{MR(E; z#`ccR4|gtbaB;1G>PGi~k2j|RPdZY@r2PN#0q}{F*{H*+C~Qn@KOjQN%F_N4H?A*8 zA%^wY*!j*SHSXc|=Z8?%CozS4NpW!4xJt_4oO@!FbxaS=Ib6W!G4NVf*Mpb~V5 zDbJsqoYg^F)NWN}u`-JwP|Yb-1Xa0Jx27~m6f zsB*WO{%&5~H48ejLBj=5Zq(e}x62Z5FmcJrF`^}9=who}CZ$1A?s)ik{!0r#>x0|) z@5_9;sPrsD59^ZTkMRM=hH2O6m+8K>5TvcN-(y?2)*CMMiCBlyNp3SiV(Opr1-dCz z@QIBd9BJt9p!F3Xp%agGmq8st@gpOdsghJ~ogNJx}xqw<%IbqR;E(U>E&dG$wkP?_u~Ol|>J@V&v5qC{z-I;HZ} zeeEp=j7}e9-a(`?-6Na86No}Jh%mG|j206i;F@2I{1!jlA2Uan*S(EN1~gZo?{5>+ zU)O~{F2%r>6NBy3Ms-HUZ#n7#hK!t78oflU02)(jCCNaLQ}Q%1`VNHbC1BShM+!9N z;{=+Mg|1CHqf-VhF>MIw($)~LV*MS*aa36xjY{YUeo&_}0-?y_60SZE0fnil-msRnaCC11YiqRFeCB<7x2s;l5)FrN>yE2pp- z=qG=EnOKjQhIeHZd)qN5*DP|xUR7Ir_|8eB)KK-Zo~=Rt-heL89$`d0NvOHC0z&b| z4FZrn@oL8t1ey;qLD+c*z5VZ9G=tuk#%&F<#q}FE-fov;A)u)<30Pzc+a!r<{&@aj z8LV*GJqNEn^?rRfR{fL^mY0cwucH5qK{?4*`Zl12)HFh znU9I4q^hiNm;$zx2A@eJ;obGEgQnecV}cI4edw!c1uYcIdQkbOxuLi`LOoVD!(8Gk zedB2L?TRN%m{6&skRg9Zm2-!Bu?<$=d%_)$lvzF=NDCGrA!JV?W10-_SyF21rQ6xU zIB8sW+JTqff*3LNaL7MBc_hg%O)f4}dhD;F8Vp84uQ##x^4jJ}$i?}IUz=y<&y-U-xq>#~ z3e*P+T9|bH^`5}B63ir7a{(PWx*CVx;#e)%#H!aon1Lny}lKXj5No*1lG~gp@b*gC6U%PIa`g!ruEa-AdYv0S=$AU;Z3GvSo;X!!M7gZ%LdJGu)8g}Mza3*j|RV{H8V<)1}vwp@t>r_svc@cpdx01%E{ zq}#MYT#`vgJPKyT8&Yan$b`CS(YhMi(E~0AFdJ<_He(eK(Auao>v$lHj(s$?JU`nE zM)QI{>OHujqCGbC0M^!!)QDVO>~dJFfk_J5OJI}Ub~8fmaumt|2^O=~E(8AjjAkk{ zLkX$9&9)lQ0BIfcLzS@RB`hiN4fU7c_}knPJ-%8h$3Z>HVRJgh~z7wDZjOA&t<$xp#$p#UJ>n;;=J`_ zrahAAk;ao3gPKfG;fMVYQQ_9T|_I>EecS?qyGGH&i*=0kG2q9t!NB`f}IP7AD~&fK@voV{f#s( zQ*U2H>)-Wp_v5|at2>C0n|7R^sh@YjJFd0?#XTT`Kge^*Rabq4%^O^z+Ly^$^y6;m za8(}AfGPze7T=0=XvG{Ud9PlrdN?kQ?!?iRUpem|QxgUjZmw32JSsLe_2jjQM07_e zINB^B^nbjwAp(IG1WfzJqorpnUkk}7331}p3l@&bb!ieeBneBpjMtj``){5C-3=t} zNiDsrpnA;Ap4T4)sMjg@oz~1^e4e-urDe!yQ{i?_7 z#s>4VT@QKpXj9NPC5~w6qLX6m$Kj8x0-LfB*ap7k~K60;pS`wX#~Y z8|_B4M}3)#FUdKY&=a@qFw(BE{dCuR%s-U1%mhtZg}#AygtCE;p5V?=(hw_swk##EF@kUFb=Wf<&`D<+=BYoCCgKG-ro)H$Y=?P9ag$SQ^H`iK-(xYq4 z!s23HT^%j(gFri8x_RqX2@%>l`j5-iwwew=RpD1wK(pX-^=o1R8WW@r#f~VFk_#Fd zn&{Tn2F6Yf{a80@>Dh*ru=y~oa!i`b^`zi;1;?mkei)EdHm@!fck|*0eQDY9K5E%) zctWdVx_#U8^X`go{w>w64MLM&J_LDzD#%A#_}J?S!T!e|nx>@r72Xzn8y(Q96=Pm@ zpCW<2l-l?GzzGnVgX3=Y+f_p-(0^OTYPjXPQbVL!2v2WhurVOdJ`~8T`r&8OB+gIsb zAunooX>(m{FgyG9ojJ(*l>Ce2!jc`!js+k2jx;Rbm>%8*={Y#H*%UZi9Pg*ggl|o+ zUvxnd2TvJYL{yX&kVHO9*>+HWsU;dtA4u$6_9+eqhjIQ-PbM z;~mvN@xZ)r_0M>T%;p$6A(0udfE%qwn7?9v-llmdaZq6#bR2A;0Ni;M_}!oP%Xxh| zDJ?6t7i~bMTiyhg61v>h+=f8Aa8!BX$L-W1`|%I!dag9-m{!LSavm<=z${ z-?@UW6M)kf6P4}>Y%%^ne-iRV`%VC@{ ztc6t0q`sN*jNSDRJCa%@TAk7$J~F>7pR>YGr6CgkmG(;p977yJ=SP>d!hef3XYHe# zItQpOG52L;Oy;cra4pZ){V-XjvDxo4D34~3&qwi9e7`rm>T43{lIqF2>;<0}9z#aP z_j9M*Tiz;X^#o!*%QkyOmh4C*@;nJ&6^VRYp^QRZqoO55Ahgf5j9)DsDP^jK>B9p1UmxYCdDk>^<8x%|OgY%fr^ot&D0Zz`xQUA?5 z3Q+#fzPzpLi@DG#-{;@cdl_ii&iP-B`G5L^|IKO)A+Nww0xrw`v#PAiTABZU!v9sw zob>(uH8E0T!cR|1Fjjhtj)YDqs%Jh$jyA={@FPq3is|>}U}@H%6^`_QJ6NrUd&@iR zB*N@hu3V^c8hR$fl5##PN4d*X-n2_BK&unoqphQ})6L|O8=sTLozK*Y!_ne3*BPgs z&_4fn=3d{;_qFFAK@AO>0xX5!?_Lpd#LUf4LktFo4p0SMOE3z?aW?^P`gBn*^U z3SwSABOu0okjB|?Lx6HOaPZ!hU9)9}kolc!I{FX4+Y6m%jyQMv3@OB}Eazimu|B^r zsG~=Hhq^s=EAp2a(}vS8N-NH>5tl-TF$2xwywc+6hG&e#ED6yo7Ut{=Rr013#4UnR zfjJB+p^?KLAxb4o@5tx7lR8#M-yo={B6*Eo+kSqjwK!cc%$o}r3z?hQRb54Q;W|CL z_}9npYE^axb4BaU8(iY%eATyw`6k0-W6B0Z-c7eT%i&z+a0MZszI>YN?0`sgODdFU zyiEQ2oj);!{5F=Dn3#LO>-OAmDLoRmtiRu=TZ6UJ#={B&8(`CeG{oWDC24 zB1f!UleBCTB22<_wwEyYYN*DL`}Cxt)Rh08)P$fyWisJR#>zr;mQyy*O*0%74t85* zEaUq{XTixGv&tG81bW%B(TRzqS@MBYJ;^K`5cqk9Zw;#MTpug5Vw#laf0=}>7# zNtY4Nov)?8BU2~Q+9^s>=p~WEH%oSY=;wxf)#O50cb;ju5d-D8Xf znnnB}Q|@&R$A5pBw}dGe8XC^{HI5XU317N$Wxjd;r<$*CweH8_L1&44qc?ghV{b2B z)_+0zd>}?4_^<7!XMzELzdNPm#G0eq+p;GQh@StwD@S;iR7VY+Iz&`A(0iv^ZQAytG;uMTOT_5 zQu!q|fS0`OzyODR*s{q-0KaxN||T-UEN^i~1HJiRt|F!)iU?$>% zyJJXEw>i77Q$Y9}TSAJ098;u)a(_R4x@@L_v0>Ey&tQt{&!xIh2rkU2-I#7+lk?kX zO&4`AD9|9Qe=>c;%gYP?M?lV|m}q7lh|eVLj?9+5FB0_IIZH9Lq0eW`Mbd9`GA@Nn zlT%8H_Ws&f+mnu*&L6Q~+wVV~_%T}AhLblj7}OtM^8IGTsf*fJOFm^LbNuM{H==Nh z&hBpe{mm)!YG*_;n^H2JbpSRbsZ_al8`Q-3L4Vew|EzzV|5oXMN>0Ya@%BmqdagHx z%On$1dEf}rISppD!J{j-l}Hf#_n3E|EwyDVHl1PRuiHipyUq7`aV(!QFO>Ib_O?1W}+YlcY*9HmJYi*3T+GU(L^ys(B^oP(& z*kz^{uih4@tZ?PMKRI)JuQU1eoFKmirgclNIu5d86?-q zPFmzhI`k3uD(u%fN_n-+@b*Ao>}Gs zwcKw9AysOiS(B#2j>*vSyMzR!^x-dGUOL(7VKaGw_;!V|NQ7p$rMD$X>*I8#^2)P{ zoOknc#dJ3^E#kLDXAP8Ic2D{*<*O&qV%8Zl?~4a#f$6%J>kE#^YAli}9eOw+-XToX zVckT$=8y}^HZ9b6D$3s+>^!Z#^8<8tSgz@uq3~6uyD+Ejf=ut}CAtU&T!f?y@h!AF z<}y!+XXdM(SU$^xj$V@*m!A+--5F@#neVZyn;3bib+PIE?3hL>sw6R&mQwUG4!2Tj zYo>d%+TD*i70TDBMtJkL8&z))4aCS=B&FXJ<%#E2_4XFSj8nj(nvmbLL>Yo|A=$zW zsjfbIxx?ofHs#>pz-~F&{WKkh7Bh(J-AU=am2#m6zlTLz`z#X1i=(y+jr{+R)>_{~Z1N z{CrnM8V80(74QkS7*X%bH; ziosX?x+vtwwKXbl8MZH`%c6^V!AfSO#!~Ax5o20vkp~&(Tcf`)B#_Jg#z$;C%O2{RiQ7s8$;@V(eo9%m+rD%0{!wOa1YOvP7!gdzDKCVkhVFX6=DD+s1%GmRnjydMg{QU&@J0UUDn1`yY(M=erpf5qiDGs}{k@-Jd3OC<1oH^Au;y`USaJ6t=eYH(H~Jr22ZXUs_7 zZga9eU?D~Gwyt*F=Xe_a?aw;Gn4iH+%C;qYJjkDx2&nW{aTwt3`Cl)JGFOj|jy4J9 zTRt3?M^A9ky75zqzZj(4GhK|^G~72CI)n_}xxMY_1uN(;&$c{u7Pz{)Y7Z5*3rcK) z)brBzukl9lT_tH%$hy+Gp=jY2>%qqRdk3!Sk`)im!=NMWSVbm(N}M%}(N1^QqzdVPKU%5Wa; zV3v&0SxxvcZf?Kme^dS1HI#07_-0gVt8$4|Bdt!Fkh-cW&QHKQ zyWFLZ7AO}4`c!76qY1;VnlHUvxNgoK=H8z!T464*cs{Y{-L_U~KlnxFAZ&S`(l|Az zP4w>POD89LL*JSL#Xl7=214pW(9y*%EZjbUi3RL*(Q)wLA>|n&HV(fVL>JpOppP?e zFbLTlR>B$7_P=vKc=D?E*|k@TFoDau=@=+;Z~6m^ot>ROCtqMn@K!TjECpLPCMy7D zXtS?W9wtID)5W!K5!;<-OHUav(4yGHGjt7j_|7$%85nM^&I>pHrZO}#+JS7c*PrYn z03ZyCUzBQID=vk6cvcp3VY>;6o*`lq?K9gqTff$cv*W7$ib_jM3$OwWt8|7gy|nwJ z$C83y}wG=tdkBRtm z%g+REMwad0kJfmrVqicC>AgFV;UR)c!#Ln4obhc~6NkN}0n6Ha?@0EU#Wb7lEE%8S z8dtM>6SZWogRc)ujdSjIScbD0mhNyiAgADZWON09P6kT)nGd;Zi zyLUSEP|>nSYh)B?R)u(|H5lM{JyRlugZ(f*Dhdy<$OXV^oe<$WUtB$A=ho?Nyeq)N z&c-cM3O^xFt2oIL>>!A~apP_vpk2F})`)<`RI^9v={#z(yONcamDjYi+93Ui1RmXv zLKA83blZQ^`>~L$A!ZwKAhTyJu1jP#UAMIpAmY1X@OyRig1*9^r=%Zy&AK`}ZP&k) zUJd%q^H`Ic%QgVZTWKJA@3GOQvu|vr@=5v%rJc;Zh`g!sTiz|ugY)q4w62%82&swf z-q$EVtbZ+!?U-#G6Q$faCYWleX%RTOvduQX+;c4Cp}Ok1d(6j>%p5y)J=?DTwI(T(PZoD zVSA7W``Gt1@g!RNRl8G8XbiL_w%d_yJR?upA9Rg3vORiWA929zU&-`WX+xyC@O_A6 z$bly*t2{j*lN4TtLnalWS{k|f&sBNqb8`WIfd*W7VBc;6eoO!^-hj=q zRp{==D_4D%NS6k)`5nY|H6dUy7;~;Uj{fME!R@ddqIE>^WaWyKL&U+`bE^-1y}hk_ zLrQ3rx-PgBohdjb|y1doD>m)#_w9xO}ZnY6pHewTO3wuv= z@v^Qm)HO$4jqx`A@}|h+#%Z)2Z-?hvsnPb+?LmPDUE(9Bj}C*yk!ZqEr9Q>;onLj+ z_CK2fzt#I(*=op_Q>9kd&6cGJuKU%s{7^}A>cM#P(s%KN4);F{^;uW;O*Z*I!b(rxDs&LW)V4X1=)m!(%zB5whp?&X+UC zJR)y6XtXT~9VhluD4m2u&*5;myD0Xiz!g|KHvh#cHlX`DpXe(RW$XE$dT{QFL5~zZ z?j|TqfhF>M;9%*w2c}RnJQFwd^zcby0i`#yrS!$a+lt0IX{zQLls5g&lSjXvwj=;<|Vqqnarxt z=cd#9rVCoE)z7a8NlB4VNpNs+5sZxSHLhRtHIf*;dkS1Me z#tkqQJZy@zTcJCd|jDYG5!dc>GXCU)JgM#lUCs)C4q~!om~^LBGRQy?lyOex3x$p!cp0 z$t~|NqA2*T^GBD}klQ>sl|1_UD|mEK1$hbod}&7%7&8bAH&;Ob#7v}&64zj1jIUQJ zha{f*%NeWiQM<s+qZ9_d@r=!KwjdfMWHJ;s#=HJu~YCk^YVBLR~HnJFH7b$PU$b`L$EOfj>K2oN0+!USkuY2HaCP*f7m!^+*pNTj1V zyLG?!Hyo=h@;)qXEM;nSXz&@~Fr>*aAP^*-#a7<~_v4$IDkU?krJcsFA2kSS z>Px|j|I%25>;m7OHy;yY3E6x|EG*>7ph6%_VBo6BB;*12Se@tiUPz^(-gqS*<73ung~ReR%b^m~=&g;R}1 zJ&7CzsoC8qoWv?L|u{3U+;O9RhiA<|3&%BI|7O^Z``c@a0%1t>Fq z0i??3F+HlWYV@0$l|Uk)Dg2iNCAL2HcGRS9-FgJtYPqZ!RNj=zVIOb-7anWy;qX zs%L2kJZv}=5Nu|lf;>Eg(8hr3z}dhE3z9)~9b#Jfw{cUmGyq@dqpMP5JSOWQK zzCD^m>@_Fn5edYNeKZ@b>Q}=HR?pvY#qRCxVbql@g<#X4wn!w9&ejV0Q|d{dr(O}R znf?uph=}l%%{Jhn-VTs|w~W!|n${H1I_B28)0Py+GIEyqLRgM<2i;v=99&##OSh3g z^p#&O#HYW-VE6Ofb^8#o(2Mx@(fXNMdcCZPBo2+nEF2u511zI_qNUI~C{_d+lT>Fk3Elg*oQRKb zIB_v+qxQ5SyIQ5xluJ>O6*#NMP~lB{AbHs7iahghd;I88`04Tfrz{kzNDeXwob&de zG2#)SdN4ER+mWHzeX42I2N$f={Xogas`wQz)pit^r<0k7G1e9#ukKY{}6U$BSaLuNZ-)hO{$`Gl$3 z(Tktb3e3u_tpO#Lb#n`_ans{DGdUx?dGR0%b<|CV@x#zR4!%%G07%Z<)zt->gsPX9 z=Z9AnDiZOkvxF$G2|@qAi(so#9n&`B!k(}duRytl}>$~It)@_ zJ<~tbvksEz{oeY?24$clL);1W;F6Y#3N|x&j|sZbQiz(Gy7v*P2tuv{FBMBo2UlKG zAz&`J`#8}* zB`ADawA+)Y!3BL?EX*Ge`&shZkv@mV>$h(rR;awvW`!Z;d$tFX3Jwe=5g#N(Nzh5fA(IUq`ypE7volFe?M?74=i?SCb>C zqJ{Ce@;QSej+d-j>W&Wy^0BaJ>@Z#cTVwVY^nar!QwrnXM=f3;{Y3^DBv;M;0-zaN zU^}4d7CjbLI-%MM{mp7xVja<1s0;v;459-YqNB)@rsPUVx{S}|JA+vt=iNW9|C+46 z=#aydK=w(Bz1Fd0((4)yEh^`|_oN#TsMuJ~r9Ga>AVz9VcP7|?(`|2mOEOYq!(TuD z2+qEe-93k>Kj}32R-K6q;v%kJ1U`^(ls`>PjoxHwba2@<0kFY^CdITrgSQmw9T07^ zp1BXU&I)6L%(usJcoI*Fd)T&f^4So^IKKt|h-``DVqwIpz`Z37etw;$+s4`TmO`X! z(u8gJZ3h91JqBM5KC#5PBVhcZ-023H7Q|SngoF;I^F#f9;CYsejkB-QOJhefM4SKK z?h9`Kdw(}{k(E^!fnErqCd0Smh6k#z9M(e71 z^IJt{D5w16)ux1U8>B%F`^xjwQW+;l$2{Pr51`W>{tz1ceb4K~gTcK3d3qV|OJd1) zW(tlKRhw7;=&wK#v=pecPVxxB)3jH%HN``}P95%yRGA)RTDO zps zoK&MXoU-0wgayEj!_l3ft5toqQimQaa(@3v`E3s`?0vC*+;3kwGaJjM5f-NSA#;bu zH$xVgh{Af`Wm3}ymqlua=cQu5y8Fs3^cOI}nn3gIEIiTgdz2Vr1&XVz|7Ic-Wry+N z;+e@=Waj}Pp+&9ceR!Zxs-S~arRS{b##AHS`QgxmO;5gE$D22A2CU@84nx*y5}#+G z4zUj3e$$>uyu@NQP6m;6cRkh3&0fj>PiZ5#XuiF3B)K#6A_?$w-=pM^0Z?g%%dF%u zGGzHs5Q|KL52frS>K_eF^F^}u5yrTHM6xEl$uE#ID+7fVU5@EnpbMi{3X|8%y2@ds zx{HsY9di^zMVtni9QB)D3xQoCs&E>z! zVFJFjcIyXu>;$&vjqfaG33S35pp1&xccONexw*N4L(GLEjiD~u+u!@}rG|FTze=-i z$4H2w$9@ubKjU>x#LqS?sD7&!A;b~RG;<2MFxNBbE)8h|_xXV7?9S)Uh60xFw~LmQ z_WH#!;sA!9h>VPsqj}Qw6jX6TLn@%?9%W|6U%sB%8q^v8_!l=JBLYulV7Cx@h_I6C zYHC*Wo6qHo3k$|a2cLvn5*0&_X3mbcT9$lz&$AVRX^Du4P>G41P@ur}L z(58qw53|FVDfv=~H|_mlS=7|>-j)Ju8At?Y;#O0DC@F#3p+$}MtH__ZJz1K|?0a3h zKunR2`20yegDymhfPys^K1zP=EM@SUKcqjTs)Auv3c->)Ch&}fq=e^>wAq`#LNH^N@{wN%a{@&uRU zB_l?xH!Zgg9iPwq!8mqcMrMvSPidB5vM)jbsa@E9epDF%9spahjN!LNS4wVH5m+_& z);}3pCBN#uK!}+faD#Jjy2DM6zsBMuNYiwfY^aWH^Dta|TsKEWt1@kg9)6oQ+`t`sb-=O|+Sc)yMX3VB89m6y7O&D&&*+?orXvc2I&a<_?P* zm*}TYsp|_2X)*@xK-DGyeypd+q*%n`XH#)shr>R*(+zT*U_9$8c~lw5yOKnaSPqeh z5ug0LWK*v7MFHwCGs0Jeoglf!084`Ry-vEPjH`dc0`-MxF)`TIL=IDi-}8$!_X z^b(@?Sb_ow7{ZNig|d|+&|R&FOL5xH2{rl?L_L=qQvMjy_6H9DH6TaMDvtubi5&z* zI0sMU_)gI$Cn1>PqA$OL_cLIgEM&Xwf=C9RQh5_-l>sHJQRMYU9dD4Sb`b`tylPJg znP_7w&PmjEzSvs^ty4v^v>mJ!mt*8q4ETojf4&hR%pT9Fc>3E&!tDpOMJX)|dF5;Q zv}BHlo_NPe0s}%{ecJ=SNbIa|m~2$?Ge?kygzkQ(=VA6I%skNYzX>s;gc|ani2Y8@ z{f%#;zH4Jd3ABRk&;(U@&bGHSv%4s{Y0e(0BCn$&+gn?x3fB%sRX-Ct_AGYyCr5Bz ziztz~DZ(mh|0jMt;PlT|QNImyOhADTEluujjo0dD?nEYO@~MXV9gjHFD*U&stM863 zd9?JTuB&c7@J1+dHnm^476%wtRa)K7fc>^Iq031z!M&D=N0L`artBN-0Kqv=z8AHq zb58;vSwVHR@n&IIG2}B)S@~?rWWDC_z^8NJmEmFC^m1j8XW*UuGxJoSRsX%q;08ZbaCG3=Mh{wVP zk0*S%t!X~~r9;H9;1wQxdRZm|AHC~X%1a3usE>(lnd7b%cSkQ{L~)>a(q-@Sy{oEn z)LWMWHx5Q6#ypHk>Ru~k-el+vIYdQwVwRq62t+KJk+f?@;ZDxQyW@~VxidxIh!|cl zz-;B8M>KRc&8W+-FPm50A;VaZ+S^qzvGnod$Ai^3IR;%;_lvSiN(A~}C@_Js1Eb1f zP|Lk<71)3*nr3g-K9QmhwTbHWN-zUg4$AENv0aZ!3ied zyrV%Xug-dVKwyGaomw55DEry==-&SRPVpi6htT~MfG=%jvR)jc)`3gPYV)8@9qbbr zRtVa(e9-uM2L`%6ztY)(+H5;kcthHf6$``uXUX}OHOXVOZoZoz%2KLy{PgS=Lr#(C z2V9PuJP$wBs(WP#%9++(eMO)5F>m1Gx> z;9fU>urnV33Cv@On-beL;!(duB_mRS4PPe{Uq;6}*CStZyVaHl_md*tn{k(sMt_e< z&{PSDiQ6%TGsjP~CDzT1z=WxLdWy*Uu3%c%T$q*fNIrg|MnQjh5ogu+MCdQsK|P+R z^XP#hIb30}ZV$)Cr;UxW!c~ntkq3ml|;3 zZI%YzH6Ex`zUF6EoAkAAtRO%~^xYjtj$9qA$e;@<1|}y7l0rh{5Z&~0egg8W6c|1d zq{taC<{SX}^_=S@!Z>*MCw(a3xW<6(iz$gwQWg9o?yC`UU%i1)z}OkD=)dR$CIOQ% zZr{E=0}Y4R_myP{`nwU>Xsu`>4z68Bn2dC-bUwI0Zx0P> zYXyFI_vz{{;zT$icJZO(zyhsMVQstFl9>AGGqc`M{41~wl1cCoJ0c=A=OGB<#3UqkmFBjU=3pCr2*Nd{u}u@QYKIdU2lhHp`SP`% zT*l<%x%qiahk&sHfr$Mz^fX^53iiRQmW2C?wk`iVFjxJ~#!BoWKKimccJcwZ>tiF|&>50`K zh|H9Xj5tV07~3cG7jRtE_4M=tMNg(o`QfH82;jZ}Z)lpE)5Af=n9`zRV+%mTH<{2t zQqO+eZw(zGTh7sYFVryeVI@ZHE1Bbasy;^TLYKnae>p+>C>O6!ZF(=N2AOClu4sVO zesFIykTus=pI0a}kf_b=A!1UPzAyYa|KnSkTe5rduUN01`L?Dk9=IOzej-v4RAwrU z=Ci=Ro=A`>%UH;1vZ&){eLVDV@9I{ICs9Jva~zTT5^dwH_(S9r@0^^tD*V}T8u5b0 zudn1AMV)4`l-)CncBL_VmV*kx$C!$9=`aSk@Ih=I^IPGcvtJQ1Fyn-ZpRx?VPE7{7 zU#;S^9;>}=VOw}Nyta&Xjb8Hi_1^|Pg?XeVss^4Nyu11kF1%}uF34nb)E^YhVexKW z+!qW`D?-I9^|{v_H!Ry(>PIiW*>m()lo2+w3=@-(RvtCOt@@HUQ!(@WXb%;FaW_8( zrb^VmCwJiuWY=e84su+oN71(HgQZ|wVAjh~OnUT)Qa3{kA0Tw3s@E@Wt`Xk$l*RWh zhg#KfU=&UhR2sk0F}CyLlfEiq+R@-o2biz9dGPW4NTR^l7(Gohw~nxxul$2MMXe+?0JP&q-7@l}f$Htj{zxI$mQq7cq^gOw<|EZ!s=)^$NLVV8?q z7(acAmtw)^9(7bj*M!0G&X2{3bUGRu4tAayy>6~uxs#fN6HITwH)W)qXUtx!BQVs; z-p~bpcj-HT7$4@%)m{OM8u)piRo~=5oGQ=~9wr@cHd27YPSB$E9uc_t!1+4^T5A`W z2BBwMNIQ7(;zS$c+}xf~sdu0>A;B=q$p6M`DDoGx;gytGK;Hj)k>$!=*0hTM)4MDG zFWv(YABlrEWM0G#{`3A!PvGMvFQSD1d^_g<#S{MP_iq0Gg#Ryh_(bW+me1*fZD9nw PiiM@6u7@mBwT=8=yEv!B diff --git a/docs/images/chapters/bsplines/0d9c2186423466a32bb8fbd187409f82.png b/docs/images/chapters/bsplines/0d9c2186423466a32bb8fbd187409f82.png index a67c5b4df34cd5d2c1d15ad7af0dd9242fae3bbb..e12342f9938babbf98758348c8d41965a76c1157 100644 GIT binary patch literal 17001 zcmeIabyQaC+b&8-H+dk#L41_$xymKxZk9|u@ZUX zCHEKq#Z7e=5t)&I0RPcv-zT1N4b=Wg-AP>}t2Gv&=E*88C3UH75Q#9rUoB%qzOX!G zoUI9{C@J*|ScWZSkS~HF??hr!R@Q7e%Ro}HcxFQQGkGYwqOx+ZkUR>dPKu37Muyjh z#=^>~K*Ox8%#P}QR9-86Mv6~H7AC7>ohu@I<+*mj5TlRff33LT-4Lqs-(HW48Hz$( zGXKwR_CJ2OI8qbt;&jLL|GDvHz1XDE|J@`1kE{Iu-|s*8u-HH(Y)oam{P1hK3>^mt$GJl^M(E%)k+xZ~h&BG7 zZz&PM!5C;QtsyuAIcg-Z5YEA*M@B|HTY!Z78jbhv5_ z?|qXx%)e=ITbEsD%ll;OX&D2!xw%^+nEaP=i@T>~Pye=yT%;unTVkhb=4RAo%eRS= zDjFD2v6J8YxGs=HpjOTB`Mq#bzVT0o>UM5X?`)ajox6=W<1excYYjNa@myS7GKcJB zh8VG~V&i7Hua7xN&|7cV50}xd_!*Ua5mU0R8hC45a^FDIs<@A@a&pFrSy*`BZjlZt z82&rmtLc9G_xbKuTpCtf{kD!DhA}tINTQRR_eD&_oK>O3va#i(`1$&l`t;ikcUoRu z#>n^_RYjf!HJmL*+_UBb&V~pbepQ8nVMW&X%$o_EhLv`dV^vRAH;22m;Cr-wfk77h zbDTPRe|{y+UJLi_6SKF?4|332OUU3jrc7cy(gScrGu|$58k~N}%tX7MLd2Qy@#~q$ z8$zB$e4fNFU1>*Jv5YcK`)j{aF61ucj!NH;Gi&Tk{362=#KvuP{)Rq|PDd_>U$i;~ zyLJEmeF={Z`cLKM@hK_M6BEY9wJuoHf+nGwxgi6#p+k0I1GdUmR_tIsUmEXe>*-x8 z|3#S4bLqJ%=6hW{_$o4puBnk2EHPzrTOC#@GOE-rF_9T8xJRFUfU07GyNv zkP(9NVMFVi%+6x|EH;4Gd8DdNr;(+ue^^hckgCZgA}Z>_~(ix3-=kBR`k;14d(p2a66 zMXaqoLEholuU|F3M%%*r)8)XP+KyFReS7_Bl+Q*j&YhI&L}O-U4wY@^^$c&x zt^`yo`ge*A*n2j%#VF1PB#?Zz$G5eRJ_!2#E#u(F`&*-pKH~mI3)J+ozBFnt=oN0? zCN!$B4FU_#dHnN(?T>eOZ?8SZc(1#>bAJBSu5xz7G~jbd^Ldo0&39%tHcYixMimni zMn=!y=oNND_(cz@RSOJ?w83#Q4!A;lXWGI%1emuzDl1Q1o{4`Obh`7&q#Kb5+e*9=Vl%J?3M=E`josuQr zlwy~6H;${9<>g<0^*f)@KDK_9`WDB{foc#WSIt>uG%66t{TYdgLu@u#=VsXKFH>yN z7`4AXp5t?{;kh?-&ly}>i5-7~g#Wo}h{I%E)`MyX@oH-;D`puP23~{07I0a_He6SS zi7>GVTS7==X|$6XtiD+}F}SyW>cy~5`qa10y(=tl$DKQbH}{B*+~m$*4_-8fiA+r` zt*$z^)l&0z?2fLkoPdk-!{gPG_?Is+%74+9PIt?2+_*v8)6?U*{5~o3=d#oJsh2N2 zj^{$+y*MyybWCis{!E3XAMXj;+S*Ph1uB#(yHnzERCuvEbb>BB%uAgXjA)0uVJlam zlnog!{7KiRo0$X~{$e23w)``9uDE`qZK~17d2LkK_irnX?LfAQyx-0~Wu$WOaM1(H zAMb8TxXk0iZbX2w%{3lODOp&ss>aX+HToPJS9X;enQK?*wUiG09+B-1u$u~a6FNI# zK|wDN_@N{~b-=jEmo|Y*OV!0iB;D%|E#f_3WUVh4r2dZCm#|#D+9T+P_37O^_J*xa zy3uM!rWZ_dFFxE!vFJdNscUL>Ry_V0myj@*)O79aS}9{ywuF?^W>)-xkh~r9nTyP| zSX*X;!~MSIXN#JIjM6k$uU_r`@}+I|b98)sd=TDc&PPtr() z%^Hs@ploOG}G5{G;!wUeBx*9KmJC47dpY6g?_w#KB_3RwvT2;0d}h zNw{vd$%mn*tsNLEcTV%1PP{YnGFR#2pPars-^Ah6_P?CyeBNk0FyXuZ=~BJNW+&L$ zK#jBI*VG%Aucvqjh^geNv>%3$ai|u*F}Vx-f!lrsGn+9c_}Wx??0Jo`fbZGf2y*&@ z&8NZ0BQO7W4>%f+{jX0P?weE3lT5tvLa>S2P-JXQdn@{}gH*kr z`*(lJ9~2dT-5b1GAtbMux@W_is-me}8J+LWpVS+s5=}L8ezK{SCg~OjMx2}O9VOls z0$H;yK}$9gqON?wQ1<=uM~Ab~g3{YrOhdBx0=WdkA;VE5A2K`9@XU)Xs0mjI`#(Gm zvh(jWZSW*(^xkV*%m{eF6mUw0_@bWPo>JuLt&e&Z$Ggj5tTG5v9-}fC`~B~)P(mgv?ZDPM;a6vWj!6QLR|K(xZ_5v zz*W)-zk^BS@33mfYyJnby@Jh!!@tx$e>@M^{3_>U=*Vpu7@9}!hxb+LYd;~W?DWu3 z!M%ZU8!bMQ2AX>J^){c)MrzXXmX{u=&lWKZQdCmzYc1fZ0|oa=?7o$~XDAGN9D(=3 zHv*^roqQV_p3ZTiH+MY`%X6yhPdvt}o_hWc!s)MbvtJvljP83j$Q+RRT>9{L2&4ka zA3uL$4aM8CYrE+vG#hQs5MR}Hy0c5Fcj_Y?b1Me*h7>ct&`w-VLqkKz@7NW2o@a~S zRqX8Ylj`1?l{!w#N*_!-qq+WsX3Fp2p*s6Zp2jKtwP1%R;xApV;>2EieY~|zT^z}1 zw`VNB@ya-_t{~I){?U!x7*)0>*Xe%z_(4iW7KFqDb#>CFQN8~2bHkEXy!VQKZ*?)@ zpz%Gdqrk(%)6ch&u?pLgVk75bS7aABA-KbQ)=+rw2wDkB3qBt{(wi!%CuH~Ek1$BN z6FwcQXl*Bvi{XDzrN=AHUBS-I{`+LJSuKK?q5qThEy#PHa zIL?gKXwv9C)5R3f!4@kWTv*6=I>&U8H<(0uUxt?Tty)XPuS?&|E($|$rgxRRWywCY z6XrYN*FGqiaOK`tfruVUgbFbqN78Z$yGgzU09e* z%40(r{`Gi&Jvulz7~ho5AS!FEIMkox-$ih_rSIZ)QR2LM$Yb(h*i0xB3LI2qWNQ>9 zFLM$>fJT~KRd=^KpJ@|41Z?>vfroJ!8KyJ#dJpwWP0QNdU-nLiV_>V7{#)MstRTJV z7{D(nC`ghf<)I2e_%+}Cw%jit$Joj7nhs|PTvrD1z|r&n_PE9CU0NsQheE|E(Wz?6 ztP2g#<dyYf$;`m>a92$^K*pZ)ndqt3fc6R#c$ft8J)pXgzo>qjS% z&h9_)ah>YsQ>|W~f*Isgf|JWz@Y<;`vwvng`WQPn2 z*c%{>ER9w2LY3_3Q|O*Y^?{#Lf}ehv?yE$MIBI!hCCj!*-f(?ePz9nV5t8_jh&N*5 z^DP2axaPiQh$cT${ICq%#RUh7YW$GwN{sM5Y-Kt36G3^38c%HUcN*IoW-=CGAGD4v zSN6PRy!Y24D=MztP8MoK*0>I9NbiqYLQXn)>R4F7q|@%*f$u|=)J$YR7{_dFVi{hQ zL1m%xD?l-!CvFW!D+>!Ki{FN78GK2h+5hZ|->6W&B}FQh zCI!P{%!XopwZ@+3M~f^;y}~x%>8Hijce&ss6d+w}?f%qC zs%^_s=L^1mhoD!I!1y!l;1e2Vh0fg&Jae~p=KAurqhiiGZ$lv$&jehYm?n-mFmGRj z6T~8*LgKFHX5+yZX;uGR`X7?I$v=9V233Nh(aDuHZgU7d6;lu`U87svS?ue&I6shk zGFE|yud*H77Hz@DAKA9^OvaaGa91c#_0V>%i;_=kUQ*uJGx40kn*a010uuMo zF^NE&zsWGcx;M0Zj~_oq#m06`)IOVB0sP1KjXRc3S`O(@)tdRr)JE)H4X1av)31j@ zg`)B)9~zVQ$+~VG&-a>0vwXeDoW;#vfz<+=5XjF+0D@j6?IaehpeU*OZPP_T( z_odv8yiYXD=wF>D(RTzy2hS$8I2ed=RG(i7JD3h&@|gCk&ZuL{Pk;37En=5#bFp&7 z^^<n;_R~pssjY*HmE>C%GgnNP|rAJaQQMEbMNQacNuti?eA&PAcUORuM zFV3kUD!PsruY3rGqAAP<|7?&}AFL%iam&RZ$7DyNT)P^mE}*C!bs|#f*nD=K z(f`mK9QdbJ8kv_G)NccgJ{}E6e^MOA*_cqF5*ma%i~<34D0tV-$8K$~C>t37(~g zNW)GThn8)c6;YRTg?ypNP5OJfOf-~`^rpUBVN6nN+i3x3tER~^KG(lP8hDms$P!+c zd)!InhED?*noF67%++D)q)jefc+EW*|NAG!F&m0?|CN~TYN$&is;B+wN*cDK#@+Sr zQEJ3+l%`c2eH1d^jB_FrX|C;=BS{Q!%=!+8H%#<%`gP;8E8v`~)y`T8cr zW6G;5HjNDH1>$|MLWoI8bL|~=YK0*+x4Cfd1uA_GK?q#_x}NVyhh2Fq0!9?so5Z&5 zpkqe5OPdV8KV2IrjNV&N(VOYUSLTn{#66;3% zl^g4t_lC#s3yhasueRLvZsZjZ_@4OY&GN`O0GJp>CXXI{;N|4(eE)NKqKZ0Q(AD~R zpUaKtRR8so55sdDd~!l(W{RHePTeVACae0Ag)sqPkk{3HnIXxBfBWXmn@b}lOz>9V z1`RqARz%i9?^C{bad=(E?C8=UgybQrAlmKMQ~+q@>B;nkyPOJFb_(b143x{Dh=TKTC#{e%-`-90NW}KrNsQ6&lvztp$n<0;l0LdgrX`Eo5Ss*?xyiYyQqZ5DA2iQF z-Jo(a@o)fU?}^TYthNw1_&NZz*tQUz-nZhR5jt@V@}&p6=$7bA*M!F|xw z2>a}>>5j8KP*Y^T+`4^vv1gkS8%0tRWz0Ql9Jpk@(L0FIvNhM`y!?ZMkY1u+Ymy8r z3Hlg;$Z#EHcvj#WF&C`#{q6^+w) zZ{6?`bG|VP?$vU_`uNm_7 z^xvFN^BD#Le(=5PAz4^>^VzziVS}ghn-kW0<+k*v5$o5In)f)dQAsta0!#On9Qp3j zaq?7NKuIFuy~~YFNb?)Ygy88+g%Bse=B#XN;O$;UWiq@A1IKbY&Z@l{u{1(Bd{!7lPr6bjYyyEUZDb@_+Lf!YUU#r(v!hF;@8{`s=lC~Ao! zXh&$X81nfKk`sCFSuFKu$*x6l?-J5oCxrHh&!ZXYDTwB&SkgpW*l+=adD0_!b}P4A zB;#0t&{m|%_)tMi-G6GS%<=(>&>E*141tGrdVkLU0j9KDO5=B|yf864+M}LD>p111 zy={8y5_`Yag#%slYAWQa_cy;!rpSdK%#Sb5B2t%+iB)PAXdUvh;yB*v!V)@=u z)^bjeME|N3MtyY_;+LS}F9?$TtvcsL1@$%gQLxqEU_3Ym=T71qbg zS$leJy0yl{wZ`qc9mn6j`VhWG1GKp`f~WrPiGw-I704+*VL-?**oa#^)94V z-vj(P47w!C!91OX5qqo}e|H66Lqh$7PfQzo1_4<{Kqg7IJa%t8&7pnUx-czqq2%o? z2_f%KNHNOp1=Qkz!rb#Tb~U;NH`1}-`}o%|p9Zr7bJu=XyW%c0AIn-*SDLKf6^k+& zC^GkSY?Q^6pmb+W)Yw`rX`b4bc$`Y{WnY74_?ck|mympZkN+>?@tbsmjl8mxC(s*1 z*qP^m#-nKPJ}N&JKa7h&$2I7{zaSc2_fDOoTmSBeJiX3Vi3{>(Ki)@^)UoD!L6cJk zzyyBm<=)TQ2l`6Q#w1D61X}tynD4F>;uJYi){zc1otoDN>_bG#g2yu1h{}7kGFryA zzd3FC(K3fI215gFl|?{WpMj7x$oR9s(#R5Y%vbU{P%RA&(iBuw**ZC@Q;sk4uyHo0Yz=C@gjf17EK7y#fy&|O6t0D(f?K=n#pZYka@)Aj|& zFtb2C9i2yH_mIKVhAUo!8cinLIu*}CTtSxYc&W{X%aC;ne%LFsl2x=t5+09pJ-&K333`@+}Q>;_sS2fz}yVZ_g8%s!nWd`qEYA zI-ko-P?9C8bklY-RbW<@u9asV;wLp15n`)z00x}7krl4YPqupu0R|A)uJ(syI`@GE z35INl>s@4cHT+WwdANRRQ8g6hl-ysrm7N`gUp-6=wzsmjMtC>IwJ6jVX!RK87Z&jR z8+r7@W*aO2KF>b8f6|7aMSFuj&oYPEhG@;9i@;K{Kv#0_>l#1)_<}C@tCKcR=Q& z!I~#i&2(_vn~89nx#-Cl*rZ4A+s?dQS`eDQ@_Teae*K<3+iky-L~O?{E0HY7A!wxi za)PU6Cfq}}Zr=u6M4{47RhBZcfz}JPNl)@D@@GWd@1|@K1G&FwzXrTA@m3| z03>cI%86u5Or}LA%xI(ryo6D&@oBFA4fpn36bcE z^OgYVml_o`k7Y{e61q8xeYLN7q<)`M*%n`1Lzjj;{)$+QKw2cV}yl; z;U2s`ABgnjV`atQOj!I=G3N4!;C#QN8C{?Si~9PL(ZE>Q zW851zZipQdSUi8x2Ocs5Rf|x{H*Nue@XwEe1|D_9u_#IJGk*U3Ij+Zwu|Q3ka%WH` zPA-N<7z0|};tNxu4(P0(Tt_$~MxR(Sy&Oy{0BC?wb{hW!ub7yKm8Q8Ee?S`j*-7$- z%(ZyIo9yJTUU@#6*8>JJD!Mn_c&jA{lhG^IOg{(nU4xs9vB15=kyGwit_u~;>@QaZ z1XK+T_tIyMmF_h6IZKHhi~lS!mE)$suYL-=!QRTy&gg``G$zZoOho%k8QSB~QuEr% zv5LoY@dvmDVv_Px-mCl*(A{eDB;pbfDCsFv43gOa==j5OvsDx}dS@Qo^=pP4Q})FL zlha)E2i=bykX+!MTk}+7+g|oqjRH8lY~Juv$}GYrd`mqet^ashsr znb9}(ujA86w zK=Yu-lc;9*@MJ_xU60&0+#tZALAxF8Y6zfauQ&OfjPe6^QovAsyc-#zEgmYKtJw&2 zctb2aXCd9MzhoZxilxwiX8xepeAMvlaVjmda#K=9Mr?BOO2E=$Lk6bG11zh99rNFx zgyg@F4RPTQw;NE{B%Pt@SPR383(CngkJ&H}=NK6Jf6CmKs+1Z0XH6vvmf*VTD|HJ) zKCEf~-n|JvSIhK=^-md%`enOQ!yFgqJGtbDd#SRV3SQgS5^;m0lQ`$y0;R~ zyG{t+d&dmpcPA{w4aK)s=pPRD1C`$O{#KIoD1>7!QPFZE)8XRp4N;|%=KZ^irIQdY zc@(7d2!4`qzM=_}(V5Crz{JLlk3HQVceL%#49XlCD_`#~8ORy#FiL;Qqbv*`zrq!^ zt-wLJ8cMYKv_BJFz_e*?rWlpS_vX!;U`+OdMq_2SNJT5rzOTw;E07-EtrP{XzarWv z(9d8i0A39$$l%2k)@)t>i!%=(|Mw_W2>f{h_i89BJ9pM{0LV#Xe;u8*oEx<9fC2Fs z6Q4ergWr%ROjcr0=W9S%sB38C|F8${-$m|=%366@oa(bt2ROxi=6pYj-OEpQ}+az(3IX_!+B%RccZl{vE=&d{Xo=bE74L<==eyr_%R@ zg@sWl)NemhR>6ADEmkWL+S$1|-m@#YLy@bd57pWp;GHTLvYyxFb0hH6^LF#&EW38^ z6fyhoxvp5BkG!lHfxs-z0>M=B!@l@-Er!?K8-tV52lOg;TG7bl0le8E7NtLWq0#(j zqki*IAgR}VeasF}5*+h;vo|FyeJ)m{##Bc8Xn?OzO}SgZE7zqD9rPAZJ)UK}tdbyo zfyGMwVSR8e?p3v)E!Bl3}C!t(x9@lX#cG*EyJ{_lm{%p(Tp0e0DBBA+Y z-K)W1Kjjda=SD+fl@K0cEx zY%lnl(6v(Z}%Qv2k zT!xF7ft6S(4)$MIkJyX)Y_YpEfSDrhxQ|rO?a&PO7d;S1DOa2B@*gnO_7e`v8>o%q z=ghxApJn+{tRYb>VJ78v;`Pb85I#&YB)n_8KdDQ)fAe7Ssid zR&v%6)*)D9&PFJ9>wwE5Z*JE>uLug8`^w77hc!ys{g&mw`hkTXFA3T}`-E+!AfroU z9q`_fj?mor@?^r^7-q4OfcguK%(azs!DXgBP_}?L!)d;oY8hEwxK$lcrkqW@_xMk)3|tGsnM z7n@WscM|fXLz))zv2@bMJ-8%ES%w13Cg}1o+px$12@p4Q*PjG_GpJl}j$Fbn7>a_} z2jorB*Rwin$yYQQaepm>W2Fjiz(cZ4 zF3f>`Lj5+MpVycBOC3XfC9GS|l>@n$Oxn@5SdY?Npa?e85cEkea zHRGOCviocow}hJ5jY;6{>K#H-!Jp7oymC;CEPFTDl%*6g3q0|k2}5Ol*Q0IgHBgEG zKgd_vm;O|(^$*72+wQRJ1^|1ke;8hU#gJI)z~d(f@+k~0>8(qD`Z6~EK{KdLeS?s@ z;;Ssvx;^7ukRMP%KbkcDFg#AfEStIUUUi^ErPP#LrqBbn*|5|MJu2E!E)yJ}0!JSJH>ie0dW0j-xMlj~ij4O)*^AR>a;_>A zzKDzu1RYs}`=?%$}uhi=2|$Rd@+UpGIr3@qqAni z4*iz%*6e59yGauKn)8C)YO@a_PxW?K&h!2LiM#*8{#6fo5@$C9E~F58^dBJX!|gZ0 zpr2yU{dm&OP@C|K%-Q48tBhzeK`%X&pwGi7nPYO=_VUEA666xsDj_9q6N-jpyKge zId?S*8Ud3A6d*Xip)^HBN7vU*?!L!+?Mgs?k%U{8j6O@kI%1mfgjX%EaE4BOZ|%Oj zahe}x<#Y*=^L)_-8nVHV=%(Ep z`h3T^pt>0wY{Ko}ZJ^DW}X4HSPE9#yZG+w^sQeIp9iuYx@6gCs#y z-d`3AyyZUu=BY-7;f7}u@pQv&6UyJGX zF!9^dM`Rul>>FyorQ1h1gb}-)>i{-CU@9z+p>F6)lRc)|TGQ%RH6=dndl?aqfno~) z|9HnIQw_9STIn5hjtw(aG6x~X+>$)Zj-5xv7OAH1D_v=e`jN5Erqax z5ZzF#R(px*{1u@gq=7Jic>mwOe^bT@h$&jUN2A6r+ntn2@esfQsCm}B<2;86?Ter6 z=p-_`qzG?}u->siAocN@fZesquc6#dT`*dT|`V#b{Thc+hdoq%59| z%UXY-SR1d7UCq7QeFTM3smG?t?R4p1PwO}67=UQ*h>6;|YvU?K7$?t;QSsOa0N9yN z(~<;{;Kg8sVIZ`j#SWRdj)e+V^Y@oUHVh+ycaap=>dIeSSsKmAlS$IT#R;liVAuK?v>l?!$*!z!5#h zH(Vv-rEXA{v01VhY|UNm-x8#_MDdhDp2qB`N7rr|{Ky%Um7oUrMv_LRKM8acC@qgf zxbVJR@G!AP;tm8v?r?BPGKq#zQxOC!Pz0tPF=g++uN&TXUFsCYjRM!jL-2$ zrR=DulXaAWjl1Ojr(0@+MGr=vX8c@J-Vc1z;YWev^w>ct}(X* zcYM~dR-S(UvZ z1yFr;ZfteZ_&j!PnZjWI%BHVpS~7bSfQko#X}3HoPxK;;FQKHQG>7DeD4)_jXCp;o z2ks0h{4~Y%@-wN!=LB^?4p7E2AV81#wD|zgCnAptXB{aL z1o`F5TR3JLK`4@ZZi5)8)PCe20p!9|2mAIbL}3XuGM4+=XwXnywSq9F{Oe62&AD*cDxNc%#KA2&NEr$yHb zhC?`PgkYEDQXDeS)eS~evq)Y;v{;9ShvSTZ49tE>1 zB$8aE-yon4Kr_(U;cyJu9-C7_DZPyNxAo$I#RRsF6k7Sws;6`o78X3d--hnrV=3YY z`_SBp!|<|NSY9y{NTZ(z02tijkv7QEHUqK+I&$D(cmTLu2_e4v3Q^HROQ;H>`|D4~ zE+uN{&)Cz~5-;6b)meQ=JWfPO7)Q$TJVEi730}Lm(q|k@kY6T?J2E1C2`HiGAmf>A z#3_CT@9YNw10jtN8bXvB8XCSgNjfvR{wggI^@M79fw<=$4Ks`KkXl$1_z9#3Qh}S2t81pHF97w0CcKwv^ji4!&aQck~K`2z#@7U=M zS>5tjrTx-C4h|yz2H8&zIJmy|*&fHp%%@t?9pBI5zpmDRg7rBrijj(FQiCB?#kwy+ zKOJTm5Q)0&RKpEu1xqcvh&OlD75-7{f)o?-O9+q@%-641X_R;vrPZ8&b{q6Apz6@| zp9sAISaUWgGUGVavzD$tNk=56=3kSAX`qM$(pr0&?mfkoHFqhbtOY4lGoq=0SsRu- z&2zOQC#xHALb1F=;p|!^@x9(bIwgkp*6Mc0IPPujS+wp!NB?JKy1BUkqGU19AhG-R z^a~g3y`jc-ue2R#8v?NLo~GuD-`-WzPm1sIYt}xodCIt6B4XGlzUQe)oBht{s@_9A zp~9kx(y342=N4dCG*G+&>VZO3OdK4PayE*sJ`ku@; zBw@MD(b}HHrABUu7ziBH9HjfeaEi^`P1AcsARS zVg`c}ljjIg2dem?e4DAqssq1R*<^XR z68+9Rog4`w4P0c<=x}Qe8CF8ncB4&xQZRv{*O;xz?C8QsuF55!KU+G9Eax0w#LDtI z+WeQzmU<)76M_tcn}AGWpQnkaR)Asz$*d`n=O;uHF(BA$XwTwj+qn!xQ1QN>Frmn=Ed(6#R z#MU_P)mC-QECL5f-v(WwSmTx7O56r|O_8149H?jsw-t&ykIffgIJ3WOGCo5UBy%jw zZVc!-v_&Bx(a8dC5}7ed4>Y<71bG>c(jz6+0o?CN$`p;==y@z|x7 z9!{qHSGi3*zzzVnnjik1fmyNSdsDGI?tqsd(oV1_B|xv80e@-OAG60LAvySRve1-7 z0I~q9Ju8!sT0s%$x*^Z1-Lnh|8u!yXzo}AKQ?hKieXn|ufXpQzB2=?~MW~)0^Lj8H zW#Y3g28lEqmh{~EOg&I|UldXLa^C&S%cVP%e4F!YiZ}^GG&!WgeE0r63jlM71}-~0 zyWw~WAa<{ec^3aC(j6}T?*`dV?MH;5&62;uA7-6o6TWA_E-g)u=q8ZigxY6|79ipQ zIGF`;9Kb1$-$iJ<|NX6ia308f1Oy^PeDxa=vnLdbED1+{bbnL^JPc9hP}uFa&1hi4 zB9>jai}^UhE?%6CB+39EmE(KBfIxqUJ8o}p3jqy@WDuBcLqHs)RAJ}fI4Vps+Xwly z<{#im!AK0JW-4mh|AffY|A+%_)L;Bis={%a@#1)-8K{_rq$Z5}Fvj z2Q|~BBVp=rBH-c-iFC;LlP?UDso!qcBMci@`t{gGy665<=nOCBN|Qdd1#L<2h!tVn z70q-$HN!NaQTMS-S_=B|+~ItE0;O=mR$v3sCJb?xK@HfBNJvf(oe|w_^Z79^!c^o9 z*?}Qv0fB)$6V%&ebaVtTIRp(e+x6>Iz#HiMOV-0wrDWR1A32zf6auIhrU(IU@!ZKc zJ@vizWRw&DASLLx{>~?8%})7js^q9XmzGULmCXCSj3^xgN`~Y`fN8f64B=+6H(#oI zBiYeFiCM-|(WLWdVF5<{2>y}Z0D!YPQZjwYB`3!unMQncbOf9xOr21`i0Qn`jd_;? z{au)Aq$kEfgFZunk4mXvsP61CD3O8v%LY<9?_LQbqPv7jG45^RaQv$LhGngKY7(k+ zUMQei$)D3j-$fWFXJBby^=l!u;hsxVk3x+oy|9+oBco0E3$NaM8<98KV6;H5k3QM> z*~p#vB}KdpkN^#=+d}6JipKO_WEMp<`5t=cznE5!s!W!A#j@?3-JE9PBV*;^f!N6v zI1|jb=X=f_dUN%7C_!jzHAtl1W`yzt@>$t=>$y!BI1TX|lB~Pa zn*JLx7+8+km+=V*;BdEe$6>I7vr!7KPj>%FqP=91rbN4RIR@+MQSQ_`53a;o<1H`) z9YaGJSEbyZ9U5lW@tZ^?rnN6(uG8ul-=F1ED3+*}d|H;CvHGytrE$U!=Ewey?_Q;l zpkWS$Zsv8pV3WJ_hODzkqJ8+6J8f@`1)ChCfe-ARNP78lHnNJl8|HcycoLCV^}Z@v znf2>iv0f4e9p+>)dxJN3J6)IhS$==eXSpy7Bt3k&2^|e6EvR2NUF}JS*wz(f-x}r1AeenE&6*f5Ut{a-n^8`9Hq`pe#M}|BmOg|2KQ}zyFN~ i|LKSSPsj7O188fpd#LQ4J>YjopxuPAxMD@K;Qs|MOfH-N literal 17029 zcmb`v1z43`*EPD4?(PmjX#@n6W)l_y0wN$Of~b_z-5VrT1O+KYKtQ^?QKUq=OOcdr z_~-q+?|0(A{_~yp#Pw_*iG9~vbIm#C7-Q`)ElqV&A|@gf3PpNN1FeHXVZ|U{1bFZp zzxvDH;U7GU8|rA(8S-ChZRRT!iUV~Gt)%Ogv^wGHb!X>P;qP6wv5$}RAEjB*5E)!P z7iYy-6jX#u5G zeD|x(R#W9ZzRmwV*Vr8TJE>;tQ(uio_d!yR`n*%6@dfO}F1)ul_1~y;2JIL`Bd_D? ztRJk!NrwcZ{bHl8OuQe-%|x)DWlN=qKX)d1_d4gB_&}K zij#{=iGc%+=D~E>mR60QP!Lg4;>v5^%e*AU6seUnVCQN1?=x<#d(;G}{$CzXfE|Rv z;F$ks7yEDD{3T2iuHta(`M)nrts5O*@V~m{|Mo2Z^Yi@&ZFq6Ms>t_)grMf$%^5nVyB>o1fs{Dbnb(uU%cO-!_*i?5eG=-(yo<0;9>b&1?SKiEE`~2>U+~2_-^(V*7 z;fas+_~T}ObtPu$X1}Z+r$=csnBuFk+p%K?}5ECU6gf-CmkGVw;Xv)zwS?gGCRcY7b_aku+e40v|9Yk zGolx``vo0J##%GMWi#TpOuzjT_~kKYu#9yX%g65~b{CcUQ+cy1Ih< zY=Q=CgIg{94f8^bh1K2M++scGUf{~e$aHje2ExVkbAsXj*E5uF-XuZdVrd|Uo0k`F zAm^_7&eDK(@xxmf^jbB|!qd@H!;keTWbSulrt6F(<})E=8!K8cF+mhTu7Zadf5?+G z`GigDBB=Q^r;iV|DxUwL624y*)EO_LlC7I=H&KOo@j?YIVqsySRcfoJo2`QzCvZ1J zgC{^ITTo0Qe9l4I?9y8^38vUrudqM#Vc;RS1O$i~85uh|Iu!Nw>Au{r4oOO4;o;#) zHL3N`(9lp+RzAlhX&W?J`uOhO+ZnS4dj_$toaFDcne5AzP1+>;0&9|YB4`gKJuaBs zQOw(^Jq^!sJ=nBbnW|@pS!yn{Yz#zs$>GjMuL?O#c3KOY?2dcF zO4l9Of{E!JPLJJ_q#e(VjErd9yot`y&+9F+x`8Y)F)^{dyx(i5IF9G;^4?ck z=`W0tdRd+E@%EYby?k^{5N(SwuJGdBWy*sGJew_bo^+?fOrNVS z@bcwLC$X~5C?De# zG55o{m^l)b#Quyzd{W9T$tX{MWxSG_`SO#7wn%0qL2a*3u_H?dQpmVZ_}UAIy-<&x zCG+nIVin#;mwnDoTvsNl>FsNG=pa!<`J5b@PdE6xtrWI<&B~Chnh6!jcL$`$otq_y z`bMO7{4?;ic_0?$Q!b9k-L-WJd~&vy8zQwqaMro49tHKSUtNy#9hAuU!3CA!t7bH% zJSZqA)%$4I?e}|Ehs9n_mMhN*NSP!W)Y-j*KI`So9{f&euk${t8M9oVxL_z?Hf;0P zmVI81B!=U`?c!#SA4Ga0ad#&b8baZ+v94(D2op4RPcW99zx|tyA|4blG<;fF47}rp^kLg7F zx<+>2WB*UL64R}Fl84Kl8WmD==+hM5ByqNv+|Joy^Sr&u3J+l^W9Q%h3yszZQlIRx zE4LgevO;QvS|n3&JDVqo@9_pcorp<9+0!wPkD-c#RnCtE4NC|fJ$l3-?Z7eVwRKD4 z=_t+F$-3{0?_Z3yU$4G)FnnLYQa*lWMsxvAKl{y9sa`F*gELkUm)znTr2hy;aji^E z)?AZXrv6X2nxP79d=to5%%0C;b$!*jz#!wqqxt?y@V(-P=5Q}rdO39P40wTQ0qfcr zKR8cseqVFkdSx;DraYbo#n+LyfxU)l1Uc0D=IiE{6U8jsNF?khS>Tyldlb$hc9w@e z7?s=kC~Ob1W6)eYe0-L_yWbDjxQinfsd8SPor_io`TF2+>|xAwBsz!<)@=T_H_q%&dY;&F?%159#7WD z)E%u|{rKtA?B0~G`b3rU##g=+ns><)?@IzpL{~30norvXHwM62BbFnyT(p!wJFgeq z&+HG&uRA0{LLin~{crzmg_*O{^|P3mnCZbi&Kai5PcX>et6f%`W|~7|V`BVTU(h%I zd0w^S&0|Q&HLBiFG3hOE@uj_`AC7-bj(frV0;PfG+GZDW7HL1YNwYscGKMQ0xd(Ia zQJ3^Q-4BF`HDCA^hm61*5z{FBJj0pZG!;8bXA?T4>$FR&#o|%phupLIP*3+`c-a z>;f9+mICQ>0cSQ=&6|^rC_;zIP$$lp7>dcw#!Oq+N@Ihf2ZZk~KMhBgCx8OCrR+?% zXXE5yXfD~U@8Z&)yMlm!db6~-1@9HL2$>$vR1=EOtc^V~MknApYE9-v^t)+_%it zJf^*=O4u;X77cz_kaJtUzZGX+MzA#TZ-!al=Uh5x$MU~1@&MLYtuCf*MS8wh| ze}5lTsNDP2bk(lwZNwrqsKSJ^uQ>d%8bS0XZdjhXD~;p|$9|CB08iaLETBJTe;1&9w`AFFQ?C4^Bng7|eULGobs5+ZF zPPFVMfzRoIwURRrKjG{9uY2gt!}wT4qHcT7s->0 zxOFdNFi4X4%u;sSBa&@jw*b_I->-H_d({64nFau>ojZqrAI`*N@pMe27%8#7EEiQg zhv}aUa(qQO`g*g=we-ozB@zZv)Y<7?y_n-HT7F1LRU<{cwyCiliI(&XM4U2 z9V*j z?+{EdQmVdw{fb2HukUZ~e2R7FEaZ%ldevMo;@9z7gZ%mRv_D2= z=GA~+8RWgBch@Ff$a(IjLRO+3=fn*UW%D2a{L^-Naf+2CJ+qqT7!|?j*i{9~s4ZH67y6pxlmiesgo;3Ev@m zu(CgZL2gb?EY_s#HjCh=|Vz>0X zbxq`qj8oP()n9T1Gz-fP-^Ay@3MyQq%7k`-+u6yUn8UQ<^W{NODf>ym9fNz`jLM&b z8n&Oc3KiZ+ZbWTZ;>zHPp4y=2r+=uIKCrg_@ObEp+s?pU1lf=~t2h2?CRb?rJJY-| zY#BWvg)X^hIDecF7?5-h{z&VOHQP?YOrjsu}r$nk;?84TV|Mazf`+HxnYA7!CZNqrvL+yeaw7NkX)oZQ9S<#uc)Y~xAK{F z&GwHQZmXq3vcs&n#So}-ad%CWGBw{{f8TX?;12HcMS+z^#;1d(Vv5HxoeKd6p5hYT zk`$fcX9Tx+sSmgR1Y%+1O-D*i{&fhYSwYq-tjl!Q^+{0%l2$hEH(_aOA41ly3i~eS zGjRBHczh?i>rYG>O{efgNC-|`T#T{H_qCZ^04b&@L-;iC1kITo*t3i1dKX2 z3g1lOYcYjy3u;4LxSAM&%lH@kE!1YNyw^lhF$RNa2*4$lo#N;_FbP zlbIsy6Rbj+%9~S5;!=OOB#>)ZN(3_) zd{h7M87DfKii#>=z3zxC=Wc!*RHFMUVwB$8W}{XCD(&t?$As7|(F}u(sa`h;!#*uu zDg~hBwm0F5tW)SVT8Sqp;{>lbA4ib8U`XcZ0P1`Z=@!2bv4an?-GZUJ#Mg&~+iE6Kn{l6(|7Q z&h^TDG-|}`WQ1r{p|q8Z3QCb=-$`onfgfXK{e1ej#696#C;5|buaN}>qJsCUem*`a zWLu6KdUabqEGCA+>u9xHb;-)-C85>T^E6LsOqQc}AZIB$j2Mf45m&xiu^73PUw_I3 z@zzx1zE#u980%o_wSh~ZFy#Hd`JqdWv@PG7*!!=p*92a1UNURJP?{`7bjXE}vqu5IrGM+Bo=c;$Q$cjN zz5{>w7JxPTA=f&uAag8j$=Gdp4LE|rhh0oincHu?r>q=;=BdDL;>z*yR!?x5z*PZA zz~p1=WmG2DCfiUOs+x-R*U48ogwQ`cEg8EWikrAE&`p$ZzMM!3NLQ9l*q9*h z^P(;<+FvnS*J5qF((Pa&eq(4vOxZ$(;#5OAB2gxHXh>AiOT))QUQk_7YfsqA*AmLR z&1lI*Xu1|Mx-qhIVSLGo%^{>>j|X#=$0WTaI^>dG9Bd2rwMIjo%=++Nf+(?;czsx3U+$-QT6HjLy#D3QP?9Kb>UMPNe6%P|#kX+LA$b~E2 zd&bJP9U7GKNe|+I&w5)r*63ut9DO0k?pT0wZPQjMS-GYquA0j{+R(Ayc7-9DcB)7< z$_Rvl%(YQ#UTswNq!F&U(Z+1=R(O@G$8#=H3L!t9Ox7H%U5+)tc^Xa!{Za)>5??1h zJgVQLc*3*+70w)2d)mh1W3QTnUx^8Fo|l!C!%genT+q74L-qZgB=KBZq>9gJ?Gani z;iVvba@B5ONT$RL947i^f{NX{EH?PGY-QCpeO%DjeSp>iiN`KX<;e__Lo_kCxgiZ$O0CjBrjlrd`8*6ljKer+sy)E^#!<|Mi-{_uMPEC};U z6P__zOy^b6Iyg{qOjaULc$m8j_d5eub4i!xo0?sqPk8~PNt)wa8*)7;lf+wmSzx4kRwV#yr+U`>~S&8a!_4?b0 zpP=9?4<~eXO}udP-_ZQRT>58SEJ#`L!;pg}EQ25B;&JuWqTXJzA}T5_uKdQ)XIis6 z`WO=)-sHJn>*|^a^~ooSEnJf*NRoWZ!d2#(%+x#iv*Tw7Kj7%-_|NkXKnvr2xZMbqi0CPK4r@q3Xj4paV6#UpRBiL9 zt7Hi-iK(nwZo_tVcEYF7ky>ufe?^WgzXgjDGOk~rft!x%DoW^kb(h2V%w~BqtbWlG za*qpF1vvXOh@PHpc8VZC9cI}xTbyfZ7nq>TN?m8_irCAtoaFJ7S$O(v4N-k+zA{{BP+*3|64hkQ{JCAPqxvQOm?{L- zH4=*3jwdo>UtCh@(kZml?k^AJtC!ji{O#b5cFUIGI_s@*e+I?X9N=7V*$ZXpQhJr^ zb;uob^V>xgWm`Nth+>U{{?g5;?smNu52fa%itAg>uaByH_M2{mEt!F=(3oTk>zP=7 zdN>3HDsv!gT&NPmLUr%K)Yn&h5K5%m%R@cH?0L@w-4(iDMbjKg9;(YP^eYa%?W&Va^Fb}L=|{$m7c0bhlte%X z0zK@JRy^H|$~HT=cOZIV5s;pl-XZ^@HNdL#%?wZ|d&)2>k0Fs8)OpF|>OYS4e&@qv zKz=VarFFqxfC>9!0OKH2>RMWeboyen2L!3jz-QXK%sl*=wgaEj;3UHZ<|?t?R2evw z_qkI|bKrc9!B{%}xz@QI^>O5S)_R3UXRZg~m~Zy`e`of4d{6JJ$lvp1+!Wt3m%Fn~ z58r zsg(~b29brPq>AA+B6myo!Xs!jBLR;o^1h<2PQhH~i|<4Lsi09=?*7TG`566|O-bL( zYiZ%;u#z6~MHof&lJ&iD`wmo+sM}vNau-z^w`8x)Yq`diMJ*?zG(seaqz>=iv#s#E zi7N8@nSa27HkVRI=Hu|m-7m(Cz+d!b-{B*7RiiPu@_f+`+Mj9Yms{7!!w8ko%LELU z9g}lmsvV1u6AucK3KYjO=ZmA)>(AKW*@Dqr7dIHVsuaS|0rK95W-zSh**>ggge+v? z9!Nnom|)xy4TUJm)#w zrL?Hxc!yKdx%dHGD^IJi_6Js0Gtj)(c&X0w-miMz2vJ6y_x98^+=WQGOVhx2uE4}~ z_T8c#LIg?O!5_H8AJjtcyNp}Y>9$X(QFhLgQ_?;-5WVsF}cwpYIA*<}Qr zC=OhB&_7=0Q)rD`v$}NeZOBWeNVTj1+wxk+)B*8Ro1jh3^XczY;|%TBCu zmR9RUmJ5?b29_tPrFK_F0^?0QvgbXzu!i!5G0~W`C6nXkTG|2A)WNy)f>IYm={T&d z$++%>cb6UA6^ax-RgwR@AHzq&@=+MGvzcnk2GZZAUShG~SqgZGdh-H7?Gqd!o z@Hc;t^=SONn93HWcro8#C`@YVN~AS@%r3a*uROhIwUZg>8|)KunR%4|1+u z1{o|EjPkngowayVA0^NlWP2I4{{p>343w1f)EqjEE9>ZUno)N3`gsw4R_u8!1D{fq z1xJlupOx*4XMd-apagvisOBQlymsxH^z%inw^4c}`Y4MAr(vDO`?#0!FpAx>eA^0( zlrlV2Tzw}!@~|At%g3~Vts{S~PUB@em`fPn8C+6Qa>OP;6_(1H8+__1m*k$rcjE#a z{Arc9W4V)(>F)Elqq7LdnrBr0W%>&yz{tXa3(%(VC(CBr5GDQFAxWz;wBvfy7JdZN z6?1+U2XMMjx89mQ18NqJKpXcl(2r3pGCIHJO0i2tGl2$~%WC*?jvgD0YTm;`aaZh>ievrD7C(tTWEIHV#8X!<{Ctf@jn##ew)3h4iX zj^o$kFSyF}Sto<2#X=04x^Kr!vAQ@=Nw#yg9b{y3%Vilj7R)B?L(mtM+_s8H$4*kF ze9u@32?=R`(o&x2LF?cEGv!pvTjMU>LTNF7`qBul$BU=IC?ljf|Kyv$i^Er9)wp?t zI)pd#rwVW#Rs%8ZginA%lSKtNexTG82;F%@i~TVIQVMVi+&pFT*`qf=yxn9;!=loW z#Mum8=oo1-ueEALUBm{3sIvZ-a)0?sv&-31Q!sIoVr+*Bo5v!jah=z_-p7(B5~A!9 z%?O!b)d=sOux0%z_me^(=%nJuj~`ljhLTdQs_52h8X66t(cskT=TfK)sNkM3ku3`m z<8^o&+%|%F@!KqX8J}I7Qc@zJn`bx>d7o1I+UL9ZQQyDcY|OvwtiyTcBA;tralCpW zs(5{d-$AMVu;<`Q(CJ5zJAisB8Qq-n;!EG`w~>HepkM??eu(pQE2Qe5)C(6P9YTu) zJ4e&H({H9So?n^7M;X_8(B78az|l;So@wjGWJ!Q}(F*)91wB1qMTlb{)?4{?P+u8_>wSQ{RP~6Funu z7F1MB_37Lz5N#@XtD?p1#kkXN;e70CYKC-maT~gz>TCfn{uhgji}+RK=TJXvF4E-D ziawd%EI)7V_t-Zo_rJUVG|kBm)lWyfvOMDK>vCp6MY<5zWZyuoeW3SQm$2M^%4hn@ z(xk0{PjN$Rw|Q-=xgU)RG$E?wxGhEaXB(i-%ZkX#vQ$=*%L|1W@P<^CGyP)!Znwz) zS5)z|&FTBOyUD~wgM0NNx^xeV87V0#?Xp$TR^0KO=A~8E3}1@gQ)wnrznL>W-Cbu} zIp>RVs5Blfw8V#D^H2CDX5(B%J8K#hdTTy8`#m%~G(^Ehwv^f<9S<&*;}W3#ocyS) z>#w{xe`XR;lK2QRGWMj$vJj@V@#LzT7Cxryp$ep*zenr!%`*;)=LmVY2wg}gf~&Nh zCpOt07QgMUpmsNKbCZrx`!KeYCs#rz(&NkMfF)c};JmDlAX1GRHv-kTy%Or{iI?`1 z0BUw8%MllkkR*167#j)SyiAQp$~0fLS1`6pKq@p=^#?uA=2Xc-_u@!cruJ|t2Oh>@ zrU#$+zA(MhvuAzf_NL{vp$bGxD9{UT#46~WJhtEBC@{W0lbE=KyQtqlSITmCAKO{|=3dV`nO-`Hg(fGzgRZI^)10DEM8dG|7dJgqYOL?ML%s zZ%&|*EnfI^RF6dmQwK-n%cim^IdUpBH8thT{>ltG%D7vwWOP5{r*x8`ihf(EFbM@$ z#UDMaCZ!>Xv9Q5^99zvzuelXy>hZ(2w0F5umN%B~&+{wgBtec3WKvBtLS4A%g9OO1 zKQeH}sZhLOz-w;9PicFLr9AKMx;CzqaXpT$Li14l56B^068_}a^Z?QVFl>|nWEzrp z&7}Gnk%5a6j~Q>S?2vo;BH^o;-4e0}u5-K$OZ}7Q@3Rx=eIO32<#uMj++RkGw$~(R zd1}IJ-o6bg1OO1$bK&)0(%5d>ti)4~?`1PH!QR>I7 z$G1A}y$Q5^oU<{sqT_ID`^VCwpE0DbW_ItEQ%>+QrBn2snC?s2mNJ=}n^!#hO@Q0z8gCno9{zh*LZH$dSwP5uWYMY^)6ZKf9;rve}AvvOk_8qx$3j0I z2H>xv5)x2%AP#1goH>FyGV4%DZ|^c~Dg%cVN8fb{02*yOO>|i^j0Ca2Wi|XBc)!NI zq2l=~TMtwr64SK)s$J2>t@b&&GVdgl{^3LCF_U!^b@`id z<&}@asVT{|HT(VinY|TS9n2lE#d%)M9xzZ5ulR23Yu!cTp!hVC6hZmz+<6zl$Vgz% zLZ-^x3fihN1IIa=yAIpRy0wn!en7#h2LatIbrpyIET5 z*p2)Ve1Jhw#o>GtzP>p0#yctpw5}Pz9S8(@FY8VPwMb}d;}82Ml`+#7y|yf60yx-c z4$y5@q*e;%xLf`Tf%It?(3-(HP;|5XU0Gs`uN-^a*}ckuY9eD23V*Z? zWeIjkZ~l+wg_^U)&lMHYLBYW@P{r96qb}?lS2$pSLBZFF+52CV!j^kSgXNQ19EIpJr9yigDc?CIA0zno*nZXrc9i(^}V?woD&TCi96 z_u0(X7@+N3pbs?=qyr}e#j91=a7@dLMb@i<-n_v$pCp3_HaQP7 zkh<8GA}&Zj87)EV2muYb zLq$%!y|dXw%q!nsR(+B5ROW&x?Q6H2PEMCZEGa9$g5To$iPd+Xelh@PpS=yb0`KGE z<6#o7eaO%%(QYV6n>Viog5RLpMTmvW{BbUIUkVWsQGxr`Lum8!TP!bSK2#pKhI_B+ zB)>%K0BJCx)hoMmdAL8^^z^d4JR3mNeJC1oc9)cmc7_wia=rK+v>~<+j|8Zs8I5f8c^#OO+Qi zBw5P-If`etkDy`Wci@dyXgV5!n2k*|DswNvWQi7!L;$;P*%8Cpx+aw?+OXijd5B9k zmIwWIIM=xn6$t-<<~Y-BK^|c2^9)OyU@U5SSz^z|N*+;8d7+5gool#uXI6p{lsK^)Wh&!xZ%0!n!9L zx~eaMhr!PRv7r8m`ea7>owl0e&CdOG-!tOs+I`M?v~EJz!)BmWHh$a?p&hY4{E7p8 zpoFEQ$)eyskUxqvTmk9j=YZdww%_ZAMe!npl-UA^qRB^59X*J;-1SaU;BY0Bx+5Xh z|C`9&5a}e-r^yW|MWG#r9e!FIds=+e|Rmn7vRfZtTsV z4(p1Wd6{sn3;!hx(8`lCf~^0=xH14xpF7ZEXAZ~vj>gq4q_F0NN+upTD3|MoetzXX zH~xx&gN??Jo2E$Z5`(V<-6$J~HH6f>W=lV_5gty#3%mjeqUPs5{%E@W#`^nrzudPZ z;e$z^Cy)a_c`Fld$&0g$1oP>zh*{u7vB`z^XWssYh;upIesp{Y@|KL#gS6GM;>!%K zOxmlaNl%CR;YeQn@H7u&+>^aS`W#e7F>4ZZTVRPn>ECNLV2=jnUef57fIm0;F0 z$vIaTVQMwmAlX;jQPAO6z-u1EH(|sq$H#{s78W)P!3=!P@76VpSM$NvZ^ZM*BIES2 z_fk%XFSFT6YxuCeW$o~N2kOe}h6S_s=d4IgO+Uw{#75_i+{-%j(ZkU0`2@ZLrveDK zA<(LMlq!H_)-+2B$R-egEkNjUEte@`tr$Myvh~rXp6LqLw4UwpHTC`6z9laL30=v} z0<-~Q@?~$3C>^F7uvo6TAQHqu2x1EWf12CL&Jgj_+bM6zT0ewaGzZFtV$E8;t7DpH z>sIygIGuP3;2aqAW*~@vw{~M{)=&1P+FQb?i6d)@*vh%~;*I4c7O`Q-dCBNiCT^Vk zCf%t=M%Hm-R`tnvMIT(teF$#fVc{FC?Ah^FaM{%HG%wc~CE+8Q&{YqpBlzg&7{C7c zpoRI`i&M=_O`@)A#x)=UK-ZX2^X|gm;Z+x~72&1$99~pbR$jA{$`xyPw?3F7sVd=tRer ze^wM+xEn7rojJ|@V8@CekB+Fn=acdnWc5gxsJYqJ7f4VSBP#=}fr#J|TxQ+<6>VV5 z@KbJFY81mpg545obEl+Zx>6fR@t-nWkHP)K_A8iS3&Si7>DAg#t*_4ma1~5Sg8LQ#Sm0q+zBE)H2yQT!n z#Jb(HEAKm!lEbcZ^DE`f4#nlq@XZF&Jf(G@tuT|Zy;$N6!V-W!33%pJ5SIAcfNFMx zGI}%da$c`XvlF&489jew?tDgfmIUZLJXB$)#-}cMSi(oO#W9G`X;6Ry+49P^hxbKV z%(1$Q%Vl28B+U66Z#ZO+Cx7xpvEUccKSUBgruF#qrs}b-moAd}^x&9>IqOOZLtL-(THq1w{3=>=vXDaTv z9gKlM$pxY{Xa>k|uN8vw1I$BdM~!{ii^~k<4B9)UkVWFf6ulDA-2O$cs6w1Wl*d$5 zR1iAz=G#lS_+%`9bzITpzmVkvo`pitUAQ`be!6`v6k7Q-NtGLJk_YEN8oML+MR{^T zc37A$nLG!~Mg|oQoXI|ij~F7`s9>@ZBhzjeRKP?hTignpP`YZ}3aB zhk-&s(u)X~$E$DJ{jjimfi14t{3ZZ3K_eBs1|PuM0@SGM467!` zb&oW0evvO>YkVwHtfGRKQ-55IgI658CR;bA$Iz6%j5$B#^&C<2SX{A{lS=MfXVMtIR{ZFR<0X8}+h?vtL3&Za2)f)}kf^ENvPlLSGH;I%p~Fj9cEvk6h*(^Nv~JR{bqS@|<`m2Ct= zR}ZQ2Fe%zD`c28^p(jbT!zYLl^uYt3WcOc;((g6NUDb6H`v0^P_s9ctWOKZ1dd9!Y z03J4$w`Az(g-fNUpy*nosVKSbaG9G}hvcL#t_lJMg@A$xm4{s&VkGZ*&_9}ZD(+bg=OdNGo} z#Mw{7!Ga@ED_Ou&%5nD4C#+o;k|y{Ko53z23TY6;g2qL}lkoEVjRNq$wY0sd!O7dn zq6NA$eaRCbK?>Mt68=lFxNgc<>$LD~AG|6E`ve(x67dne1r$>DWF!D8Ms~-YU%swe z9}+QM{8SdC^#R9>3UUgWHkp3?D?vL_kF(qLz8Eu(l|6;gXTHv&d5!1i$7`<&9`*Ag z@d$L@#T^5rJ*K1oT%&X7I1i2hA5EUqX9q@r4~(OJgs!x;wIL&>aJtV8U<+58MK#5S zk3Tvf;9L!`S;JM%0xs*5-c#b_nO$KkJ;jvbVj)xVSbps0D1sXVDqt)G&Y*I=5Q@8+ ztt~%r7*($8-KA&^(ZEK(wTWu(t5=!pk2i_aKdhf|nC-4S2^u5{E~1XlCdC1N7l>WK zk3tD?+X9Z*nCa+3Xa)2+#Kb5;dQdSme20JKqoNZuVy-E12da&;!}h~z zrkUzH(Vsn2RW@<(H2grw{+Etnm8hZ3LuC$zQU+N!QRo$dU}5QjGwA~2sD))p!Y0d* z;(H1r!nL<1@f+rNy7%=6B7b|`MOmSSBYjuYn0&(JPFAWewjD2ZX zyFdYPhKC4AFv!RTleh!t$|EOpIZM@*tLMpA&!2JcxM0uAr)yFE^oAz!XWF-Y{?=|e z-6TqGwN#Ka-F8Q;5!J@)^x!uE>4JUO^XLh?_5?A@0@rmDux-9MjA);dd9V5L_6_x) zTiO)re_aO^n{JCYfVUe(ZziX6STI@t7`Om3Hdzd8UEr3HVFbSRODfsdDa1{t9?b?W z2Y#J9h^6Jksn%q#l#Bv9w#V`9s3N$vMZw@Snr66IKh*s8Zrz{p59#RzPsc8o*iA5j zX|n+c7yhgpoAgI}>+N8+3_u*ZfHOf}=O0>Tl_48HoxN-^8Rl>BJwH*gS+WmXw|kRC zga2CUBIlDg{k5KZv;VGQhy(se#81;2PMdJmRTv!2yjf8`pV^`W@zZ}h z=0*F>^K%C_Y7MH~>%m`=i69$_bd)kaDBr>z5@(A1Vx8>XHUOOEP#87e_^7FSVe;=K zF!_D|=XYRvnZCpn8L)g$#NfWQQL^(w#zJLjeUX`5&(Ap-E5DanAc*TLpLs*4^#=Bk zG=V875VlmzLPOaODt*N_bJVj+V}$dBZ9^M_M)u|#v#na$x&g~pKCcO#*(H^892M14 zE&Omq^`vN5hE_YO_s!xjr~^6y=xG59&StDk2;urak$t_JB=60{dgXZosD_S!Qf2qX zpONeKR7?MD1=*vOVqov0 z60e-M2d8G^3z0euqKqEzZy@|QuvzR`pZaIegSmpbZ1)*080#(0F)B=;=qI^YW(mZc ze7`#1sD%Tjf%ZZ9(FqxvL4GrKogl9?7S1&)Cxh}7isouW_#>zu3(z6Gibzyj1E*MQRgLHB(p3I z<}pXHE36IfaqP*stS~@qgn-F>7EB5)$Q^>0zr0TUSH#T@_E_$kdlg2@FYRG22%q{2 zeaG+W=;3!iFiNX(Sq*4%-eUcU0 z%GT^Vnf&u7i>KX5GJ(I7J;@NR1a=A`79;R1C<({k?KraXdW{7;K0e0Dw3-xvL!Hu! z?>xAy3z9ER$=+L_q@@i8{}ct^?PN5#_4PSQTGXn;SAf%jFm^3+v^KRX`Z-(RH%F24l{C(zGlp*KL9Ch*K9c7MtR!oBF% zgOh{r__yT)h_HD&{>`=5w6DbO5tEX(B7?dzEcQJ`fgSPdf_#i@8Uc@C7q=Wv5sZ&H zcp9?qS6>D`BS`fH-3<7LkUS@M+4tUV&6A+EAmxcWtFe3BQ{b{!9&I#aC}*0H@N_YJ z37d0r4GPaeW1|Mc2`eWQ6nzH3CoQl!WTwyLcm$li$esz=- zF^|6B?!EUF-7{!qm}L#*buA|c;pI}p&^)@%11c9!%6f%TBG8&`yU z)>r54iTwEX?OT&So$-hTXn+3I?cUFN0zBu>bIQrFA_hq4nnZn$-O%cu)@5Vc(Ld6b zvI25XPlnK$4UkE75PeM3!0fib(PsG?FI8B~>(@a@(*nR8pkWg<2jQUnaX~RdegsrM z4Z8{lKa@+ekgbfBlfy5SL3lPSw~ItavM}5?}h?u;9f=o?KMIpP1;DVyiN`1`8NP{Rwwm^-R+eZVS zQsIhqtF|hKt+^ErGv|OsjfMfH1gkU}sH@&QBPqmY1{7G#SFh#r za|1CAh-z`P+Tl>m4cO8r?X@pT20K6g-MM3;qSSPBs1P?kX#Z)@giP8Mqd%4MO6O^! z>sx};0~oP+62g#eH5DfVPcKYn4W}iOr-MhB;qns(6VFvil%j{n736V%>@f#Fl{9dJ zC5A!(uV<=#nIvrd_V@RR*qb~Qz~uV7C#4-qk;Br@i%6~pd#KXGhrGyYgNKP=k|a7i z=**^pGFPhj@;j(I&rr=SF>|+BZLJU0J5wN}H1DKVw?qo(ynS^v6OHluC zyZHM3>q)!dnuy2g)Jb@J&r_CCSu=ytt0|_o=f8gc-U9BN_0vk{Wjg4oq5ltrwq*`B zVIcYtDBMj=O&sx^>!(}!@WbieSr}}>M05GWeoM6v{_xVzQ3VA|3%0=__X~rbb+4Eg zHb;)j|9&ksVm}-!DM+k$dIvX6vhs&n0d?iKiiAMe&1tw=haYUvQ@&)C`&}_hk|r)W z2d8=>s<#*a$m#D+y}jek=H@1B&owI^F)O7@UQ-p0pQ;UG1t(dSFCJfp9)tv?5^B{x0ah+;yi0Xae~pLqQ$$)=t|OK?=Rz$#7Vdv z6xOmDM!CNiJgUa3sP^8&e$B$dq%;0uF~zK(R<}M_e(3wF+w3$%@;Wi7(O9I)R^rm7 zaPs$1%quQA@@_D_c5&ZsVyJlU9<^Q#HK9?^GiK_qPg9rX!JmvrKQwCsp!d2k% z|NblloB`ki{%=0(;Q!+#|M?UJ_5 diff --git a/docs/images/chapters/bsplines/41167c64c51386414c6e62f0b45e6295.png b/docs/images/chapters/bsplines/41167c64c51386414c6e62f0b45e6295.png index f6c17060beb44236666a8ab56b4da079310003c5..b14eaa7ac2b0e9ac0ac08fbbdb1c7b67ef372711 100644 GIT binary patch literal 23184 zcma&ObyQVf^gW6JC`gyIbccX+NOw2V-QB5zbVxVI1?leYZbZ7fTe{x9pYJdJdgHwz z4#v3q+;jF>d+il-&P|AdoH!B!9s&#u43eaTh!PCUOAQ#97xZu7z$e6fBMIOaoT0S1 z2+T9|->;UuSQwZOFp?sID(-2AOKwR^F7Y>K*L91_Cx8Fxz6tpGk{+Ejtf{`Ja=$cc zaQ{goR0tBl_L4mAtNCXk4p@%^n+lQ*#8-V4Cl0?gMk2Q z*8lfQ;2LG%5^X=d2U78O8ZCsZvKO4SNu3t>?~LqaK6&Zs0LR z3lwvO-Z)tP&-G4Ye0>$i$#wK$JDGk@yP0}Em#WjHI$uWq{@;6^ndV@Ma^1CFUAdMv zY~y8kY$sdhQU;)!kiaIOLiV?Zlbwc1r4=z5>sMdGFt+?tBJw^J6bpZ^c=Qk4WbA() zv^fx~2_{Aqg1ZG)<;7q_(^1*?G1_L`usub z=)b5><(DJNvd65cPXSKPj@uL0ZEjHs5bD0gDvAKqODV9*BC_4AJjo8*1pH?e4&>-> z|K|8-9MJ@_HCm>ppbGsu zCykt^qz*v!$(g0%JjswCfAd4g1YZN{WgPAsE|e=`g1TOH^s@QRKl$;Iy^<4b$&9!K^$D|6yPO@Bb{I%3Fol=0%<28Q!9X z5J2@h*iK$u;3M!yGN?bQ8fj9XC%1^a+=#sY_=H9jS`agA_?G1E(|>~)W5*pkmK|9h zQTHE13sH$q^31*zh9D>^25+;V61ltpBX=y=2eeXhWN z&y^uYcfvJC4Iu30!| zd9nSucQgs32|$e}h5GZM=O3Q4#uT}HtNVfA3?lMYh95SA^kASBYHDh_A$DNTtG&(D zGfsbGZAGeqrR2&!Y@uMs7!YU_)hhnFK4hp7G?tN`W#_?$oX8kqmxOlXfN=%KhBO^& z2|rvCkrhqXXn#H?X7Y!RY*VTg4(O?@zvcrdZ|T?rfye$ZT)28`V=6(89w4;Eg+vaG zfcQyg^BV^fpQw<&O%9@;xQ90nbnI~BdJ1ARsvkHtJMR21Z>$ud4xkQ$L{rAvk*(wF zomDv3iRo8IWiUAYh(v|(|Mv=ErJJUtPY(Ho*|Zx!dhZANLN~oaRdMCtkjVYFs{BJZ zUes~x0X{~B?74+BB9ySZ9$%pTbSVzCZ>Gy{_DR&FaEQ0PfC|NJkXHqLs&%S%7Fj?( z6Lfv<$kTZOEU@3Y2detn|E{wUwvPS3gL26>U!3#UCo#DRMGHnIL-GcG98XFdM{j2f z1>ZsA$$A{{8v9+}=F;4h?FoC&;2T#9e%gS1V(6{qLM-Bgf4h#OCTSn2*x$H3bn$lA z8S{e^wfIMqzuZ_PAX`blaX6PU4OFoeK>8L^ zBez98m4%-CKIgtfq(m7_=nZtIQ86Vf3$Zb0L9UT95S}Ga)Fwz*NJhkepw>`8=ayr9 zew3sZ`P0q0MEMp*RMk8UkF=F5!bWNVjzr{00{92#TlVyUx{w>nnPP@6aRT1+LN?#YDq`bWRrmA~Pd!4~4Y>ekHHUIF`yaoi~d3o+TJy{ZRswsWRpU zf^!l3bERwKc7zKU<5xL}m+&#FpCpA#CehY?YHgPW%lsa#U>F;BNugO{BV2^yzvq*r zht$2j?_C{l{`{Ev7cmO4+)od_;p$~Z!rgp06I_{Uk($GTO`X9)i`Q!`(Y31uJF|fp zich05ajEoZ2qgSUwd0ble}%gRN~a6bIAQTd-%LEZ3b8kbHI|zUVXGBu2(qVABo0Ln zZS|X06>C&{XZTVJ^HYZMcG!&Hf8nUqp%zjG-NW9mQ%AeMum$BosI^LBxo9#daZ1bm zZ?1sl(7x|vcpdK!PkCu3^jwSyptLj8Eq2DyC`;sRS=F{F|F=&tEz^TgA#xOsJT!PT zDKE{YxZ{-#v25n`f+yTv-v_n%VFerNdz=qYY1Uff&RJji_?=vOYBo4hByQ_aXp_3(tQu@qSV)?%$mXbGFwg4vm_7SKP~TEob4Bt zv3Q&tH|sKO^hOwsW$-<82|T>Yd^|L}KiVKXJTnd{{|>TrPdF~CMzj0Jp+qJ#kiv_# zY6_<<-a2%0uOEcKQ8t&=??=7NZl-3>xiS&B^Z$|mb0C(g3S`JdA|}Q<{HQ+vnFEua zP%N2@MK6oSzKu0d9?&o~pT1Prb`XtV#r7e|dqcO`l8U=x1+MbmX3aD>-yD!hWsf*o zsKba5Yd9!vy4hE6)5^B~ar4XAl{mo$^$O|hZvvH7I^w5C~w!-iN&YUbC*ZesQzHmMbzz6 z(@AS8lh2;?rX$yM$rtslLlG{?Z>l7rj)%ljp%Cv?AEUlV3e-0$WDF#75@zcRFEiHa z3APsM6i_-bUJIC3)a)r1`Q6X!@nbld zbtg#_R1Mnp9kAlh)(h-l6?TB@mzjDB5vb>m%e7+WkuXc|Dr;Ie&n~+W3a}28=sEEY(qZ`07 zXjTSDGd66Rrv|bzrimImA52{9IrLzqxz1Z?I}T8M8deH+bD>=B-LdUCC_EWTyQlT^ z41U$N*Df&ZPq$wT42wED6HRvLg!+#Qgu{GvnsQp}KRJ>zDQqDkq=L+tn{$y7)i~?b z^_{%8o3Sv9_FYIYeT$e(A~nW{I-8+O;nq*?CCUYys_N#PtRj8*eDChvSqu4@+0Vo7 zj9>Af$3QaI{`z@vL5nfHuTJ@EK6<(8}e#;K7vn%JVB7p(kdzQ$YM(RQ{&aA?`!jre}3AM z)KSqRt%C2OM@5p!ccJ~1w~rt@N#<-5Y=oWjqJTQe=KhZi9!ZtDKiE^BHm9^@PEj^4 zGYDrZHK+^o&2x6;Y)mAK;X+B}U%?#bV?5iVEa9)$$ces#fp2m&FC+zDOyaGYo_+6! zSO}@4n$!i8`Czbs9b{NG0Wxe5qt5RYIusK?V8wen2pmf9xxG zT^rd*=x=Y`D|%Pk+$+7aI9Fthxt7eR>(1wPQVLLE5dKmSbx_J(D; zRNF=*_uI%zv@e-Lwc?>Du?$Iv2mLGt0d2s+xJiI%xmnp#gd5%;dF=I(-BIdEjmIy_ zD4!M!dSS`w{ph__Oh>2~835a3e|CH6Z7=fWhU?&A{4NFfrei)gD;b6c@rvju{WLe> zgnY7EbnR>r+mF%qtbwBa*5I@?Do_s9GU$P@Ll{aC_Af^|oRAitw7Utox-wc`=Eo#dK;m9YUz0SowLOq%Y<_m&D-;mx_3-`3GgAYqMsq8lEe$Ur_w_8kplAHwFOi$}f zH=9%f14$o^_{U*yHPWNKJU!hXFK0ZCYG!Vto^MbZ8yj!_=JEF=w*I4lHt!=r=Py#E ze_y6i+c3FI-cccO`m^|dOqyl4p2gi6b}%5&BE)=zFiL)#ijqN$T60UctKwJ?eD5(u z?>SdHkd>kS_D&4_-*oLMvuy11ZRYdZ09EGe)!57ja;xcLgMYgdRD^WsA*Uho-rwcD z*D+w6ot;map3j-IJ+|W09C~rRZ-?}@)0{?WjIOBqa2?0(#e%QY4d10N1#HIGWrcdWLL3a)aN&*l%yd5@~^&h2mxMek{LhNb?skIq14tp;kPj5dBbym_pojgB9 zi4%DQY;T*|Hf&+%%4h0Me*|?+GKU?Jx~>O>!2R4<{YmRRO#Q?8P{Yl7m}1<%-R?Eg zrGiYtu6s!k$TRYBF|vI z#~DA9y+6Ohi3M6AM#R#_NMkwOPbLaMh|moc0;|hQN+?iWHBSyy70$@3pzT20Bm#Id?DA^)O6d+(b@rvqpfgB zFv_(>;M1Qa5r4XND?Xlb>%B|<+3r6J16Vw`)Ao!~@rc$7wP3+-iJy*7 zo+5EsO;5XspUNqFrqBut&pG@qI38~L1acLqPk%6Zf75$d$59;-RKx;qAH(&Ntbs~- zszTHy#|e(-DldthA2fO*0w6F7xQdyxAt2ymctMR05phKUVKq=}ti-dd2)yC;Smqei zd=<7!+>C7xj%+3TrV3qa%tHwb8?E=p8(=z_&`u#)<}#FUq~H(m=kB_^f!d+;AFo18 z7MdGcfd0I9dtQYc%O2eT{$G&LxfaC%&L{ zAWE;%YVrI8O4O458!F4eIGSRerUV7*ZuQo4EOwiDm6{5dGO?jKw8rVafxb=1)i=ZY zR?Nm+7SmwuZF;#h@RYPt^Rxx#5WZyBE0ZP=`|^J+uUpb};g5D@sh@-_Grv1-*dZeF zI{Y|Ytf{PCrPjCM%-ygSh@T|j$B&3jzbYDom0$1K3wC9Cx|h&xQIHzbaIVti^~Z&M0?pZhoBaPWGWtFcx$>9%MJOEoDC8)X%3z%S<|t0ZkXG{9xw`oxLKZo5Fh-z zmZf-&H|?%q$;XZMw&n3^fnKLEZp`_CARavD-oJMNsBVDq_N9vA{@5snfdGf ztz+lOAr!wToFZKNH*Wqk-Xa=W?AwzI3gPc)X6_q(h0~!~8uXRcb1YCqVk1gG&dJG1 zwYMa`!l>_jGgkh*6Ork(#LfqCsL*=?BN?-uV|WA@usrk+K|PUX?TFPEulLitSv7*C zBp5zqu+uQ?e+jq3PtA2h2OJIj^ylE2;jx#W2`&Z-U=E1d5c0bE3b(~cvE-S!(199Q?T-}46doX?j| zd~xa)Mmm2W4(!nRjYRlW2aR(I1nKTVl43}UrW<1^YFlI1(NX9jk*$?w3@)wJ?%4b-yCg%jMTJ*9d|q#&$$Ph(z38!%zUX;Svgmt%B$7Q58cwkv z2I^Mh`D%-Y4S^>z#>6mbrj}~b5!A=V)VYW}Cu(9NBSRFm_+vAS6P6NEcq67*23O`v z3uIff%Tji3CGC+6zEnH!4SZGnx;6#OAHSs&SzPv2pw_A|xxYSVYcl66i$1PHC%^7D zh_j@@!}fD;O05v~sTQME)}}yzTiSHYt?T#b;(OCY?07Y2rfy822q61HR-&jNlc9tP z;{kNr`ZYKxrgBBL2})#88}}!R>PFi1l>hE>uECUgd*$+o9JFns@lU?cRS9)j2Rn_` z4EbVvq-ehWF#N!tq)rjAdyJFITUi-i#jdNiS( zps<#3a$=QEpf4EQ6Iv?z=Sw<1$D2`OfvcO}7s+K@A+qv@_r>bpIVLHdGa#VXcObQ}^C{xfLE$n@2pW z?8Rx3DQfSSmCiYb$DLrqZ)>V4+`vJAwQby^%#}$d^(vM7URVglKRDmFX~}qiahm%|}2|77pmWeYVscR6>#llE`#u$8>y05aK0rA<-&23$ zI6hy6YnG-x=k*>ozuje-Mr}p7hu&;QA!+kUWLNB^89 zqOY3iaWg`lTO-`KGewmN!!27;qpI#wzt}98ZVkqFZ#bxxYKP>i=CM9}-U8JVz@BGo ztZ+aRQm|Z40^h}G4nbvd+(mIz{8!MABda?*D>{3!IqVS6RncTRf~V`g1`92~w6wHn zSps+cfQ`)X@bKKBN$2?3bOIuZ#U;kyJ2P0k`5?OU{mk9{v;*D)?BClwzq`D`h$3!Ptud?inf>kd zi}cNTI=@1PUlKd-bZ5tVPEDE2!jV*s!^Mex;#Qun>WVe10`(P+nK4hmJ|jN&Z1LXV zS;?xYiKi=Tj){zntc*f)fxB^^RFTT?{n%|)U;#u3-N~#G%-Q9?!&d;v(NW2>Hn<-a8vC=jy#*-Wfhv~ zt+)~lXdN=0{- zcazWDXi@ijcrLeDsHI<6E+?~nzN>NNNoTm^T0VQYZ+mWfRQJ17cNG^G*If6(44Dzw z7(icxIh_Hm4k(1af@+}BVlp>0d`-n=G&Fpp_>_~K>iW#l8Xwo_l#rMSj|RC%_xFE5 zjzPF}ivP@%%0V^ISm1*|VFWgZOcHa}^xiuJU=zx^Wly>=946D1Rv%U36^`z%(m5)ZZq*bsk#b=;9sKorGC5M^ z@4Lr$H}k~_R~i{eJQ(4)>n0DUgU)8~?iV8*4Hu(49;@Hr-*Y({_w=H@>iR`rZoQDB zAf?2NjwwYWC6)q-?qI&yO+nPT_&FA@@b67%VO9ZmA!Ksl{^3!kD8Oc6*0`b=eS3{# zh}KX878BEXM3&zyrVii1T-SNTztqEim!BZzKFLA8=XNJ^Y{_k91c_tuKL!linJ3iy z0XVGL?+saVgtg@H#>~+HF3$T-b2~(>crQLxtsl`vK)O?^aOiITNR2>0#NLn9 zIDpMPkmyMrr&LeyBFz&uk*NE`wkPX;T=jaw^e0p=lYi3Z$*|DQeDSZF=Zw?k9jkch ztWW)~Aww-uf)tdXJ zmy58DOuMFZz7VqU_pzRkQt$*Gk$r)k1`ihigE7%0>W8EmbFvc>*#&aikub;%Ux#qp zf5SWMe~TLS0f2$`iPy>o$o>?WV;2STK7~psC@^iurfSYFLzDtb3SL>u6X(obIf*fk zmzh?BI|AYkz#faEs1xlX-le94pSm>UYHcn+hZWY_D^}WaHLsP&x(iCBlDg$Vml>Tt zsDH_FEU8p!6Y!@eRAp}(i5jM&le^}I$RMoRILy%JIkL#4j(APxAb8|PB2COI%=N{O z?PyjSzl9<*P!NocJK?m7b7Lq|OCi5}fCz$ilLl5~QTqrSiprONf*38_J>shr_xEk;f)l>sf@IehOux zwxeVZv2_31M%u23l2`)AeC@^1703HME7R&a+$gn2lUNJ`O$w!kSL9CjABPEEj4_L` zWkd;IPjT0vT(-YL8DCx=y>7kcYj~L_x7#=hU;>@itD3Si0BJ^fZ_()$Snwa`$7WTf z_7eo-qMSYq3XiA`@ci;h#(p2s zbz&-?n-20mWCF`Dr!GWH9rbrFt5B{gdAh*o-bJC(S^u#qk-q{jGMo6#ik@bnMPC?G zYd?zb>8rgzx@I8uqVS%=0Q`}wq@)C(u$lQTQa6F_r!}6yTxbh%Uq=wK&V0Rv7avWd z)OdA_K{p*}!fNICdTLO$x&CGS^V9vn&c&k7^^!@587?~zJ<-@G48~EL$b=Vvi~dw0 zi$9W`<>eVhGV*)sZ$kdl=EPBT! z*l@Q>a_-n1ZI3iT_)&$ERG#(jHDXi>!KK}q%n`7s5|#zI{oYI=vo@c3r@8Q7jH`!2 z)*WA;HCPLExFr%W6gfLj`T`9Duq7l~?7IGD+O4gl@<2U$z~vl3j<`YFC4=Y%>P{ai zIAcP-861weR&X^$#h5U6&U7=oz2a}tlE!BLp%epH1W;8%qT#xvrOkN(Ah7T1dZFQa zk-BpFZ&LnZ)+lPAhQOt;p&|MHq%9Mu*izXzI9Ofw|7zvEMz{&|JVmnK3W z&_zj1rzC}o8A}nmhb}hX&0<5~OBhskJ+iRhzBV<{(gHhm_CAEPbo?z_oT{f;>EEau z1-nkccKlyIsTE}s`rWXD?W7G-*^l9E#GyyyL# z5KhW|_kksZg!ye$)i-07A1gY^O#`$hKsN!vm6#x6-=r+zpeq`+0hW(87+vfiZNg%v zyd2yoSYXO4n2S*u$vD{vttD`j;}8%+A;0Ko8zG?XaaKKnclsI+jVrb-mujVLPo6}6 z4`=+(e(|w&JrQ+1vL6*m8+f{&^xRghTsDZ;dX^N3Ss{CA9g&X~^F|)M$QMr20nA!h zW(&U0_bdOk8veu&Fm2DmrE?Woai6FuJrN6pDkW$ve#)7?`IKl6M*x&x05T(Z(Na2Z zmXC}jpN%NL(Inj~HcCFD`e8=%mzgxeQudmS%Le{-$J#;SmXPda(O8nr7m&AhB~QqA z2|IST!xW+t_T&9TC!eXsQ(lIQQ*w}cBAtuX zzdpMuIC;l?Szai038x7r1D`fe(Hw6$`8Y#Mg}%RPu&w=#ygH#>?Mcek)EPjqjG-%b z$VdM3*rrKQiojPIVnxJTN(~}+M91Vr%~MOqb^Eth+J29hi{2L_Kqs*0b5dm;_Fa|K zE}lB&_wq_AMQDQH1g&DF?0UiL$nRkg6NSom1ciaQ=DVSRx%s=Hp%7|LRe1gEUl3v= zJM{wnuP^t60*{@s7ivFRBX+UVXu$=VtSfkhG1P5$mvB2+$?-NKW^Q-pH8~W?{Wba& ze@~Wc5=)n~8Af>v5zR7JVxyM|lpQE%mcgH_{P+$LpLxZJLytYR=itbuxxk`DPqwh| zmjwc{tf;c!j4DCOcVTr0Zk04-q`cBqSIc?=T1#TTUx02)1uRC&?ZwW;%|_JBQg}A& z=C8A2=7|_lPrbZ8{l~(%jTo@K?=BrChPMmpVOwXGTf7>M zY8UE(U}jnU{g+%>nmL{gWf?q742sYK^c3kXT<6s{yRXIX>|Z-q1~$>>jTgyYo4*!w ze;a=K9qBfMG})gE5Q#sIC%^~m_{V}%TPA-)b~#|xYR5&*3}4CdwFmdkxwO4)AgZaY zg<`SLHUGRer1x|_aqThnxan#DZ}dL&d&IC0k`wycud3!wTPJ($qHiB#zk8)LEIVo{&o zV#k;lDy)81j|tZE(Y_;xnDH0e(NbTW%)U4x-eh$Ywtj+%GKb-<(3G%=b{r`=aD5s!jO@+-zVLg^!NOgKTet2ix{4MGCeF2ZaRAh4&1>ae zSLZlcnUJ=lg8#db59Yck3;elrOUwsi_A?}h4X|V1sU6=BFxw!zZp}JT`&-O~<7z#E z5);(eP?8{}Y-Fn%EiZO9`IsRmh9bN;k+6>%218Z8cI@u1ch~yT1DLkAHuD7h&#zw$NDNh z`}$q3tJqwV_%{X^3hx{-_vDU@lxd{@&>KM3D%i9TBSdsaI!Boe^oG_t2fY@PE{7`_ z&Vvwhnl;Jo;`@_f&rMHOT=F~>SdlTi3@;)-2de4;)*n%j4)eOc$iXl;v(jQc)#I{%9VxHGm~k>471u9Etr!w=Qz#_qx-8 zJ0CYuV&F^hro%Fj8WwDuO60jzQ|J!6v0NcYRSsW>{scKXH_!Xq%(z%bb>fp3hmNz-g(|7atfG|law{Ouz6L}Kk7 z_bFS$n~3X;(}Txth0k*|1UhnTVc2_amDMenU28MJGH<^B)$CK5fmLgr74~{S;vIz( zWB2iHAC~$%|C2-*3Io;O?H5(hLlNMA0Z-|r4f5s%Bek-g>r1a+^3yN0S4+WYxV}9^ zA)*U(3uBUX||u4<^~hM)U|L~+>uLt<8(|^QIeoOJ6jV8e=dKPyGjXY%-o&iC$CWl^m%cM|Lml+gk!o zLfR0Kmu~IS%}JDMG8I&QKamOu1Jd~M(q5$K7<9_{sXptu$das1y>)jX2u)Sk^;a*) zl)|2CcBozr&66fL9E=^n2Oog-8Y^+wQP6x8Mr2$f#C9Zix*hRVctOz2x)QRu| zjjGQk#@Gzu;esD+^mAmT_RxPAzb(?wx_1t>49R=@nr45>Z*Xwv&>o9h3-ll`i0JF1{wNyL%hv9{lAQp<#{LZ-f|+9r;0Hhi$4M2Sg8 zpZFEq`!j4p*!wkaq|x%;+m`1shT}Kj2PW4BjV^bWo_zM_tn4!VpF;%VrzCqFwh9+6 z6>VgyXd`VuGfnon?~;^uF$3`$&XaWw7H8U_mH`!cBcyn=>n_*W`nq+?JQrKjryl=O z=%akz2kB95USr@P)|y<~?jF-bAMEn^m|`)ceOvaCKyLH>m!pjxrcv0P@k$cGH1ex@ z9>w~)kuv@rZ`X~k+Cb;xkVW_N}qgU5)Q7uvCHb7P-TmOgQ;CmV{7M+~axmFkBV*ILB@NI0GCb`WP z^`?y{4qc?IGMkF1psEU+sHpG}!>kIS&V+qFUSwB~N^f*7%^!u@zUyJuu~UHHor0=< z&HnFmtSY;mV>w;l(qq8(1SzcbLM2IvcI+hq1QT;-<#nPu`+4uI!2V*FSZz>!I7 zoNqdGcfYN0TTwqLBTAg!LAvqD${(Eg-MyN$jH9SFKsu-TVL6$ql|!_$NrFZ9Gm{|m zX8Q?SGUssE&|d{~?BCXr0j0kz5ekJj+*g8X-`1KK#CxV%YW-ebmJ7)s#}!9oU0K+X zRNOrx**dn#ne)0EnOM-6%=6og*#>3O;H1332>2f{bHI1p7qX$yKDkfMNh0B?mr$E% z~WN*lU+f2@|PzHvjAa z)g1lxqP|MSXKg}Te8DG<-oMrx1?a(mOOGhX$Dw898euP98iTGJ+9%-K)_?0G_DhQ< z6Mspptu}z;l}Hd}eG^52NItl7%MH5Z zE|tl48Jb^ZdxJ(09QCPso?C8WenkdvH=zYe!K}DV{X>e*)(Vk_4a@Y=Uw;-ChhL%a zw*UDpJ;7G8t}Yb3;RS>T8a0+!*NaXlpfUh5wj>rVe_5|daz`%63wWcT8}rCUx%^9^ z22Pc9v}H;$_tgGW(JQkipjbqNeZOTzqCa&~yaKe(4M5GL2{3YbK&^aS;2_}Q!bKvw z7wwFWjcn&Zi?g0bqirT*3D5^6mO_8I22+PvHA5VjtxH;>bu#y>rVlk^Vy?+=G z_FC1!bfxv*fet8+hBb>@1)>Z^;dbYLerOSpp43EhwtM^sQC^ z&GW@ON1_BAmpACi%m6nA=yoWsGsXQ|&}?TkonEgs4azZ3WQ)&mr=#qKe|Tq*Ora5t<<)+1g}9kXWMzN^p(?+L@H z^1R%SwOz5#fkC|WL>vgtb6&)qyS{qITCz-JINK8DNU`xaV(cO-s`??mHbhKlSIf>} zdpy1ndpzEBf!%G<0bV&*W-Uzbfi$H1^KOxX9sn+c$Lz=(sJgzbxT2>MkTYA_kC@Ul zXpb^tB7BnmpneFUtBKBCWm}9iA2~;D0s6!+Y=(6pme>RuwG%7g5fcE{s+rLpT5NE- zofLSS)XH;;vk^9H?k^SAq(>C1@qCZ&GVP2q9zv}FcYmG^dT&7Yyx+1VJs zxIe1faBQgG%T2wT5)6DguIM8&>Iy_!1dPk0VU&@dnX!x~doIwEg7a@KUZ0d3Xli{7 zcYa`moF8jSlB4gIzC1ePz{TUkuVekQq135yj=S&1|>a+&E7Ux&-fJ?QG(avDqCe|I$cK z`>FDLo@&7Z!jG3wk{(cdreKYIBGr!{;0&9gB|DUT%FFN!l^|Dgy9Gx-YfUBrdbh0~ zxt)xr9gQ+a#sfpd^w&OwS#O^*c0n;=T`EqU4bKEjTvc|Q!p#N8J|9>o+0%lu5IgdOX>Q+o4q)vLCyuFZCPAs6QC zEnq)DiPf)%(!J+)riPyBuuA{)wbo*i5_+No(A<&O>JV;BRT6HB;HD1Az9PPsmg?TW zPhRCQmMrp#Oqb)yiywMa2@!+3ioeuFs_Xg0pZH!krLBtOalY*_jHZxIV7B;^qntrQ zpC>cAx;64~`efJOP=VMf_tWLU`hJUws6kF$vVcv7SP@<4P=yVcpMSAt(-E3Dkgh^0 zS||r|;`e+HU|)R~SJz5F{3kH!jRw5MfQFSV8wXZnSF!@PO6L~bzFN5Bv4lb#4N8B{G9tG?)g^F;NjkRRNWe)Ndt_W3H_iwt)hIG+N>`{lptywrEMR8IU<0u_CP|83ngPZUT5bSjU`GsbU<&-4x7*nFq#CrV5NA@F079Yca zs@@7uU9IQwWz2q0-bw{3Bj@iWT7g2w zX?L8&w&{qSt3We|H1;K^bGiN*a{Rby_k5vOVfGgs?D_}}1Yo$tM@@3$>Bf0W7sIYQUdPpdfEJ?K{F0+|V%dWmjM+RJ`^TzTGt zs)rh`m%Xg648^>w+a;J}%>pg_8dANl&v@>yIk0HT!8zp4P%Juda%|JFVY$va!|TL1 z4jiZ=c&uM)a+{o`mj3bhBi90|9C(XCd3ssqmtI<0s+IS*jQ8@d^25W9fZ5S}jR_x% zHw5U{)$6)v9X^aSDB~q@k}6z378lyYl_hh@Doe&!^3I%L4=B}AdeCdv|4c~02Ad+% z1Znt@Ccf673+}k-g#Wnh*%zX?SD|sj%fOgoDzfX&AlB-nkEsdf367^V7-eu*{qIxJ zeUXIHZf-W$e8I8Vg|stzetdwpcngkR!FpXEm$*nhc7g~ka7O%mMhs~ATuS+?3YCzZ z<-_is>4w!|F(F@zQlR`D8Q_|b3`cplhO+c!Hkx1KvVmSy5qer1M1puabv2KFPbaG# z{!ljai*3UXhmFnsny1dmUC7oUHJ{z@{+jFOxrIGMbv zztVqLPp(#U_!BNincZ8PJ>sl{AC7^anv_4qo|OX;%Vs!R%@e#I8rz+%wlFMVpLCTe zq5V024+De80R6iFgrCGWzJ*eosf@bYA2OEX9P+mbC|T6t2`28$nXHuIJ}b-BB&_l^Pc#V=`s4t+(McC_?ke@mDJ74n)FN(9wxV=Wz>d zbY-quIC8(;N<0JUfs~0!exH-=vt<>6B8IFvLtUrC2!%@;JPS3_G?1?x&l(RHvpfY6 z3uSnpOF|u`50p||Z99xtCo54a58E8n3=E>y*2Vc&UYM?^BCAIYYRv0Was zJ{Ku$MCa4LQ%*fQi$J{jGsU=n$DNK01pGY!kuJAe;a8kN79)y+2jo!S-vZ^cmJ=T^ z9p_AyHK}fX%#dj(HBP(NC-KFC7Ue9gtcofsuhZq=OVkj-Pr6(X1VUN~z)@VZ5_!#vPB9Tf; zI23U$1q-`wj{}pN*JoFv$s?vfV~y10dejNzAmsCn9lbjjS>@%Z4h{~yK89};!~;l< zv#7P14!#!gab6oE%`0-zsTH>$&Q&>V_M^e*@%dU;Xuw%<#VFX?GP)kkLltO{+*~+; z$36Mqk>m}0URV0hPmkc-0=!r_svEuJh6@#K*(@_VpJBg@Ay_w$L4l(D@X$l1YM|E>_H@^-mgnDE&Iyt z#00U3w7taPT}~r<3_C!`s&GGB=V4dpSb6Qu`nv7r#{F=<22s6Cx95CoXkuxJ!G&D% z&G)Y$TC9#|3A^v4yNTG8zJ5{f-V-^WDFI&i@`HneB40Ru0weo%0?sHZCIW}H>vy~)5b)LBmw=qQ^HS5(i%3f& z0wq!8>8btM^zr3d?1ha&|2Gg`6liX1v4yL-36!&)#&_$#IZaEShWzt41gkt%>cx;*;b#XU6kmb^)eQ4yO2-yzJ-LU@v1&~-74zp(u7B)H|a@m zJwPa`psgJqfzM&MP-`RXsbT2l#RLA-1RTt4r3qrS))}&tYkl)gPBYd7nq@e?337sKLaS)Fma*6!XdH^n9BM46mB1+pn@Z$(8#HCnxI8M;!12S6JazP@f47Z>+|hDLIq zll*)C#MxS1?|Xy@ZBOkH^-~&@c;@uduMN9QX@ zEmRF$4q~0+wVyM$=b42lUCg?3zZ83DE#s0Djk3OyF&t^ssn#Se-$!c03IEsY4d?Gh;NKRoRLeFy@n%qpAe24=l0W~e{{=MMF zcJs)>j;+);By1$ws=08mfZ*5;8mKV)3Y7D5A~!kuqKNyV{2q*&0dw9zHimOKvN*B# z|CDm=;ZU#NK9_Ns+{QhO7;oas&id|ip7T4;@2~TmGtYP)&&+(^@B4k%TA%e<>s{;Z@DZ!*lf6Lr zi)Ym!<2GNruOr@kWqvJ)g&IzKnX3~Zm7JVx@8Tkfzw+q$JtG=26QJsS&!7JWGbuZA zziWDW;`FIgG;r`#?7BGq{;2Aj30b%k?pXt^xeJSno=}K3#^rhrLk|l|Btq@ks2?gnU-O90a0*ZXd4C1hzoCh9|TVFt@*biua$W zGNQCJH=b&7syM3q`CB$~Lm>Oi%8bVN_o%~eI&(wW_v0&`hqsM2OH{m2%QJ|ug$||! z%I76=+t?@N8)F97w;vw_+Jg+wmYcr8)#mQ)UBZ}Pp{|CZU0!LlxN97Da{AJ;3O8eJ zuFs>YC-(cbPOXc}Ck^;&c>xZ)KvxJFjjjPw&p9YIW_M&q6#leEMn-D0DmoT|C}>HPL;L|dob^^YPcvPLIYe6JxNgSr)$l+1xh52To{ z=a1J#e4Rb#E!|UUA@^6;e&>!Z#@K8|na$O4C*FKHInv(4Oa7&$rCwiGXTa+9q@<&P~6%;0qU21+e;YBO?yQ#l@D`B=)#&uSVAJmFm_lHAI7h3k~KE5HhDB zpsDfsQ}*iD=2zVVzwTd$nwf%<64HnbKR-WG84d=tHXZV(=g!tfrnYnbnTu~5V@z$T z$2@{Vbyk{A4#|zYXZH13Lb0!idNzb0YwYjB*V8_)bHiygF){3U#@D2f4#5!8PWa}U zCsOq7(7UUhi^d)YNAUosTZ4K=wxpyar18f6@}k(S>}=ZY&PxqvM@x@ILIGz+R#xn_ z?_aFKYpO=ycP9;mpSvGd*N6_*kC$>Rv?%(cZqG&tK+a?=gSxs~`WG}Dogfa00VX0pD8@ef;=u`*8V|M~K=9gG z*e9{ZEEbFG;DB@mFz452V+D=dp~ORBvZMB%?A{-j3;4^|xl*W@K)fNpY$YK+(A>MQ zv~JHc^`L~+fyH2mRX71Xk@CqOz?f){R(I4iG%T)PH-?h07*OA}+4E?>=~xmQ1Pn;> zjtM0wly@&R-~HVpJ=GA1Frkd>>_lj(Tez8C9Mn9cdv)`mfQ)W5)FM62(-Q%HLLd-g z@>`R?&u{SRI{H|=rW|sZ-^e^K|8wN0zuovNX*-1x@1BFHy3RfbaXaVZgTM?F7wSQo zOTKBfVBXJD#`%JQZ96&$kpoG=kp{4}4jOz1gTgCYtH z6nFPL4w{F^T2WD1Sz+Hk)cX3mz1oK#XPPQ2`6V>miQeA(5S)GRKyBg8o7cCJ9A9x-GDzjXt-J7U7|l)XNz2%2dTJjt|=BEKpS6Hl|`5t~gX&pYmd$YhmjXYbQS8W0Q{;|FDrs3f_KqfUq-q^slpw4V- zuJicR@aTRygWTR%eBNB~;)0~}v&1XfvdK?PimmgwRmS|Z0m&%9@F0c|g~Up?;Ehc) zE87!m->&LLuE|*+J(`i9pN`}UOpiFkEFG!sC}b2*ur#l2eLkhbxMC=)rL7Gm#A4PZ zAAdVVRm$Rpg}c_)FnRG{ZdYUQ&>8*-naAU+%&}=--6n&QG_%;J5AhyuL@C>du{kh$ z$y=}A*I$9Ck`C4Y%PPaB`j(+y(3b2uZNkkSKs9kVq(KEQkg=#I?BQSa@?bA_T{~5rxWo@Xo|M6O&=cu6)j0(!70=W$J2y zRmwdiCvjP0*O|Ef_mvL?0Paxl$^>df&&kncM8(K4VETKnea0Xb1XzWH zyf14!V->%*P>TVIED3M@3_S30^4>19hIqRX$+OlS{b^EeH4|r3R7sh9QmI@pAi>>T zR$pJAvNmf9p|21Ch&Tcq451))BAVt;^cU0PFCvP_+hA~re;qjvlR_f zRUJzahoh;f$=2{T!?&ik1D9!e+EF4;cCXS~ii0*=a4w(7vt@)uRq0kKkm5QE+H9?@ z(eBs3c_T=YnwnZxUTzu?a6mxTkh-FMd^7(}Q8N(=4TGwD2d zzRqgHTgWPbUp#~JWWXi2ACIIjpDmKl{xlrAmF!U3ZkDlUbF&8}De6b0X(_N+eT%_} zDMRM^(D|oyeF5^&?7TG*7xHzJQreRoRD@%YlbQ>1+4F4+dWa$V1NyBd5!FXmbcvz) zFFi0xmhH}2e`O^u3yBl}qe7Lbgz4ftMJRN3bsyOA{N*x@x%<}OKe^FsLiT%FMHt+>;R^4Vcv@& z@O_KcYc|L7ZsD$eE5ZpbFv+E!wBU_oRUAMPdieQkMGGiFPS!qtrr@qK8++y`VNLT8W_>F zIw@s~o@Kuuw#LR&w}8S9%V9WYFWwdVO;LgG^TaU+8) z(B7!`d7BJS?j5rHfl+eyiqS*xa5|kNIty#dA7g8wpXJH)@y!+VOiN8q_-ws{ zMS;9E*Pfm`qxP?^VX^nr;faT_H66H*Me05pRaYldr+?=uFB%fM|68Kx#V*Ud)iP3g zwKF1VX_)PG98Y_r{(gLB$`ftNLh1<%Ia}<_2bca{zR&|MFl)6 z4@^LoI-$&XoylL6#ZMo+97r!#h~*P`dI4n8{+^Wqdfj(6mh^d-pOk^8&I6%6vPvKZ z$zS!0m*y$YqpVg`i1O(mwreZFUkcZ6b4)69mZTBJlxKtlvl|la#DuME{-wr@kfxrr z93HVuFKq`4`KI`miJSQE`)qW42>a$^C5R68ep*t#T3>%e*C6n2bNy7#hKoiH>vTYf zY1Q$wp;3Gbc-QvDvn@kaJ~6y^J&kIJ_IwSeh_~4S*6Ng@T zv>vPpSNr1n>yOzpeb#S!!qHdBY6^wN=gMk7JtW|25mk?Vuw|>*yUZ9}JLrQJC_2{f z8(!ZrK8g%TXL8ZX^54xEJ@yP!2#7O<1==ai-0!KGX41HU=Pd6OiWdU0?u5=IV?oK8 zx2{gvQ>;e13G&X2zk^OA>lvFbO5CAT&Y-fVXmKu;Ag9$qNk_EbyJKscQLwtA!l0lt zv35fCu7MV|YQ1>-HN0BAWI-ySq=k$OfY_0dmRpEPkptxy4wTW+NcO7c%B7{{Onvk) zy_bu&2<&Dz6DHg%(w$%>fP%pkHeWr983^P1qNud(;sduUA<4nd0 zu7soTuC@QzwpM<1Iyl zqz;WGA5067`*3|d`Fy9 zJV!t<_b+q2Y}x6c+>$H^`-a=bbizE(J-OcgOl;Vme!-?)tXCWGy}U?>pLbhPp6P5S z;Lf7RYu~dIaPS>bIc_=9TiBG5Q6j4LaM2|Qnb1X23dr} z`$s|Qrm6WVq9fBr##%F!;16QMZGCcP4|D`1F2@NBbAXiP-#Z*VEFAepZur1!ZSvCk z#_NPB0Ea?Gx1AW;)u_c-%Z+6&_0=6SL*Qx`z87tM1yPn5aR?$@H=LU>(gio2p7p;V zHb_?`#knb&(dv#NcePRt+lA9S;p{vQ+<>B+cTo8TM@W983ovZ9K`$GV`1j}p|Dtv_ zB_j0T&fb8=h-SxUy^XMRYr0hCr+St^*wtQ?^Ho@gU*D_S&k$T5mW--JkRk6 zoepIlKrFP{#Tpew8KWpsCe~OED%GKLn`@VfhDPG(a@S1`ZY11G!e*$*kt&!4C(4GN zwzv>E01pT;HsNoviul}jzqNgeG%TyG?uE2xce1(Y+AlqvW8vDL)WnB(Q3S=~;~@N= zsqN3~Ki4-n$ok%LvNECn9CIqL?ZA1%!f5ewJfXtAp#L(5Rg(*NAr#8JqmWL|%}s)I zF62Ob>reV>Nx?s&Vq)@#4%K}KgFuB_K&Bl|wA1iBS=gR+!G;*%vzv2GO`LpV4J{s@ z0((pz(4?XpyubsiasGK8FRlCInd*?{87h@Zmy;*Y;w$)u_G|lw{&P(Y+|R4l#RJ~| zDEy^w&3Sn;W8j=2DQOU~dP@bacqA!8>DfXF|E@`Y<$Xp0bB*-hE&zu^vnvMxR;3Z1 zC^>+**^VhqFHde5JRY^%7L3_&0i#IYh^cn3m`Z5NGfu7wIYu{4=!I^_w1&S*2703& zN~ml^(&$v`R>0Sv+ZyiQyZxirK4>Z_INVD1lpS%%ul{EkE=G=LEvn^ZosS^o9F>B+aLa3|1o%b6PeTk_ zJGNJ=?;IYEW0OoZSH%gVIFVtsLMu3ttw>oBbz6{Oh7lj*U{=fu~WB4g0V6;o}Gf{on8O|C;8%U;HoZg@;Js z%a!9H+;ION*eZ*VR^GC(&85cM@;XsBH0qF2p%#2 E0c9)5>i_@% literal 23234 zcmb4rWl$Vl*d-AlxVr^+cTWhe!QI{6Ed+OWch|w4;K3odySw}Ly!&m{{@EW}nJOmC zbocG&K6=h+LVwDNA;99oLO?(uNQet7KtOy{gMj!z_XP_04H5rX67U1cKuSy);vM`i zr=vI?0^%!#gz$GIw~UijS1+yG)Pwg4Zc~rhh%)2F=3mA&%#Fl1s9&hlSF~L~i#3aM zivgd1!!+v~dWd%;)_k+eO|45NAEsSNUrk?M-C1i3kn#IgQZ(GmSwlY(S2U49UHtNf zz1;KbdtqIZBCr4d{t9(#)pDZEpZ~tcoJReFpN2A38Wk~^mZJu|g8y^THStJFseu3g z)BnF$uvg;xQ7B}}^7ZMK?|PK=pmkjLbyjz3byfR3Apkj##0X02i#(Q8GK2Bm$s(P4 zUBJ2T`}1fTdw8K#@*FQcaX`rz>Y4y#_W7Ba-a6xvN#{&0hTy7W;5@+p4h|0=Cr^{< zHG}e~oQIUY5JyD-#tBLDyjwKo{>Obi!GFK<-PdXMejnPSIY1kR1iW_zAYu1r}z8zjyZyH7L4a)NekiL|P9cAI}b^{1Tlq6)!d zO&5I!?r>hq6QkI>HU75Rba?+dW62&f`|0Jx-^MU)&u;=lgnOqGxL33PHu;DocJU82 z>%{IK5JkK+#S{wzYC4OX=IX7i3~+~OdB?5VROm;U!N>;a%$0l{p@LI5+|bI4_W~0^ zo%`EtnHz%y&a<4~i37MK!}$BS?~Pa_xbUkMy1~2LK805;xagl}Rv!>0NTsvafxo!M zcUmlZoRNy}GlBcA zi*2|XfCWt`)HEMmv@5eR%lPGLJG(yY*6$se5O)2m$$fz}bCuxf`+h~?+x3*iSQ~7RE8-U!K z@Za8wo49T`b8b652tE!xo78Txgp&tg7Zr4)3Mpk8r<2ub2D5oK=;~D@E6n8}C#%yD z_g#xTdLvE;7HhO?MG_wV5MXp9gi=+Nr;4)BWq^PAa|?2a&~B7A#z-(sxg zy5ZYh+W*d+{NJGxyKBAM7Jp9R36wL6&h%Q^o?~QXF@!>%;=z`}XtYL^vd!amd+@9l z7+T-_09og0gemofneRVChY7pj3Am2vYjZ!?=M4sjRLkjWvP6;xn0bK>Ni$wA*~h9G zqM~fG-?1tjGGQ# zTNq2F!Q*JpMyxKk8&y>dnj9#6XHL4`5RO=w930doCBQy^kfTH#(@cDbEhSc1u1kR| zr?~{qkBY86e=797l?;wJD$5*Wf#97qGZr+ZK;LfkOx!@Oj#;_Pl?F)9 z;G-`RN9hh%7}ETK#IOs|pNo8m$nwxwZ;dSUFB`nIy&Ng-!Df1r$`g2es6F-qbiJbM zv}0m{$hYL+!)uIgbm+>(74?KV6d#!&(@X<^UDLyvZUnNGAlS8nMjPp&rjJ9Qj^>j# zove(~WxDkn?=N>w{P4YlCbejTpo!b)_Mq=?jOI#=}^IBB5N#;TuQCZ2nJ*_W^g!N zcq=QVT&ry}GG(U<;V8n(=8u2OTRR(m} z+wmV?rgBhQ#u7&9DtqFOGXJ75hhx4nl9>r`2e48t%Zj}6Jssq)cwLVjwE9fWcjz!S zt^4q-<82>%-Gc#fY}xJ+fnCQ#Ea}FRC)2b6U*0uz)5Vuu$8g~I z@Nlz!MwT>rgy6#NQgnk}txgLJk)CZ0|6ho-^s znZv$ahs|4$mdzkMM08b%DYJwLpF7aWN=rPr@X}Vd*F~v^lagQWDqp>gMVvPQTf@+L zkoO6xNca6|9&8$`_0G(yDu%PRa{-3tjo%&xbba6B773`n#rp!D%%SLwpOhznspUvuJ(wxhd5)l@y!i)?@n!Pwz`o$SJ9;7CY#G&G;(P1R0SZW!M6YR+ zfmg`6Pe^>-q|tnT3MAqm?tR}~-!6&XV^`Z;=UJ4A1^Sq-;fpvYXB{Xamug?_TGF*X zw|T*LW)__84A>;lh*r_Jv1Z(sqH*hQEe5xbWxB2nb9Oz-bUySmPOq+x^&$ys^}>r3 ziu);>cD>zJBYF9(x$h6}Hs=w;7pxJV5CxB*l(^Dh%>E{pT{+iS6s@)?-T}M5q{X@oAv6U0_~ebHh1R6 zLz2G@$m`MJta%&p?REV9 zo)IqEYW4LPh1LWqi5QE{2+tL{=q+JFG+O%}M5C+b&C!($`&!Uj%TOY=ZQDg9G|fg< zkI7h7(qV)hEwQEyZqJ2$*1nY$`8RpsMq&N6@{CDdHO9I0{wOmH8TzJpZtK2&)0usq zH!;lus11T@WKEp9=SqEFjF)RJJJ<$?*UII^U@Lf1FmdH41xX#u~*Oj%5k(FKF4>p884V_QaJSasm$#0-PdQ z-@4$aA*%!wkJ$~%%f{AAVeP*8LSQx6`bKvaPVpbhWqCTrnw2RvL7w2>_TN#gQ=AAw zHGv$Ua}mXWBs%pBXXI&I`cp~vJZDrV(nLXTb}Doe#%lb8!g46SwWI^+Rf1BBxffoC zISDt>WpbTW&}y^DOnS;(c?<#;>REeh4UO+C>f=6^N4+2-t@hNmW3JOl1?Cl9^>fwU zhzwZKX9BTtNj2%>5U{{YEs2xAwiu#sasTZ@wPrp!op-e;D|HR- zFF8j?tD5*G)a95!A$b{dZ%z+p4jN4s_(ax?uMR>X9n}Lh((b}xy@nbb>8`t6Z`i<+ zPyg$aa(|xleZJ7JfYT3-HuN?G+t*LhB#1v$)eI^7_XtXbD-QzWn+k4!>7-M;l>+9E zrhLW;y}F?jY6ZPWmoRH2YgcD*vaebxTtPkWy)ub4efE4%a)L#f9~3N&4gk`<1r}p$ zNGx5a2pLmqjO~j2pykc8W~iIVl)*jdm2KjSq5~E#gJ9^q+a)Y{_uR^UcfltcBnD>9?|C5nugN8{D*HFsQ3U;l`8&kUchJ%vZk# znaqsC8*1`B@aKcftGR%Y6v3pTDKKVcFz99+n5RTT65a?!NM$VDJUSF{T=>Qq{=~oZ zt%;N3k@XJ_A3?P}IXaE3(bh%-2bzhxk#u-Obhh7Yqli)JMlh#n$Z5K9lx>x`^7JaW zIY5k6GBtCkpDPJ}(tNi`lgRv_h3Kx1VSFW+`=Lb%x>&gQ!o+}9xkz zLiiNRjSFbHKZZCFv)dYggFVX>3$0xO7M3L(M`SjNkqAh%WAQ+aMw zF??y*3aFwSCR&zMSI5_r#8)+g5Y*JlgZ}RW!gAloaVjdh)fm@QIA5zNjb_?%e%6CS zCp10{xQN-ey5okP7G=s};i&Dz1<`Sv%pf_(d0z6V?z%^Tt;Na+TmIr)&W8K@dyw@q zN2B#}P*rktNSJ8;gO`9%F0$XZUuS1FY?YS`uC`g2CimB$$eK*P$R$w}BGV}%wqwNBU=M3QQs;KsFz>dP zYPwKwj<(!r{Xf2{7wR zr*l*?!!kfF$%WQDM}KPmE!%D#c@4sBV2z6_IK$RfIHSqQLTy7kEm593+mc*l-G2Xf z9$DJ~42L;(B6-aubLc|#v}B3&~I&R zZN8n;uMQP%|J zme39f>+)6)aN4Vkz;L0v=R}JF*EdXYAK=Iq=}=rxyXMPv#bp)_4gxenD1 z-N}PQf$cr%yZEext|Mm!3|DzUsEgJG*KIgx{E&qMpk%;xjn7tEUIu+%ppgV$n7|VP zUcU23WZ1ywx2I$EPA_io0MfFXq&c+fz>cTuTvqQ(BWd6mq?Mha&J}keh*<2IL5C2Q z8~@P~dwKnk-|zsa;lMza2NlC3Pd{B0a5?k`YKxLQOOwBw39VuJM2We*F*cOzx;G~h{|$1vO6AtH6r**D zdif9gw$&B8UIZ=YS!M7pX1|`8F0gAzx&~&{XW|$7%47)DDaA93+`^N=p5P8SZjrTE zXyOx!%saRJjB(ZI)gGt2aBx6Vd^J3@8%dNVm7(Yg7q&q=-Bh%<=uC^miymwN!s616 zZ^daw=CglKTBJ6$=iSL>2NF$ z)C?wKaZ9zd;jrHPg&$s;_wt!Z(LsIXUhPWMgW>+isTir!<~NFJNqzHwfxH7TEwws1qG=%ziHPRKr(hdkle4j5*ZFe zluGyo$bwc*Sr^9!ulN-x$U;>2xY<&c0ZjwTZN?9~vvf zCLA(b{XMsR;i(?Qq)|e29jOVO@l{Ez_GW8okAqbh zgI!at*^apD_0-mq#l$Qxd&5o8n-R$Xvmz~mI!cyK+35I;4pc1s^Vm-47dB2oH?E(? zs=kQ`#kHhWKDv6Z5?`C2v=5f{SOaYpeD6k*rp@N#e)fJSWtQG_!4pA;q6~OgiRQn< zM&i4GpjEA^7~Z=tqW<7TpmXnI@_%ctE zCCh62=hG%b_1SFoFlhwTgR80M7l9`VoFiOXktKHk5Vy%X6)hi z8@LWylCpv8iLrFKxTYCLnL>JeMcpFZ_1;)-qqRz<20L9fIv*-9C_@u@#jLIA*tB1Y z!uOUurED60`aJlE?ZzHT&8m*DUr8^~&X7pGZPBzYAE5>A%!_2wQ+Qn&mYZy=m#{O} zs6eEHSCUqwuYpFh11MsHOwMa~ZiLCC-3xm&7IV4$vEmo2=RV0WXcTsX1g;wnEfy>` z`4=7?D?nK8T)Sg3>PXkBRXnB0DE@wE%?@a0J;_pkh3~t9gyQ9+LI-}t@ec!^3Scev^mutayR>J6Dz<=|PJ!S{t zrEgFV5vKt=_b0HCRyOr0g(936aos*p$yRzqf6|@@Mqo%)9e48mEYGu&RC>Zn(S246 z{|h%huYKP2&C=p5l0?bNCg0~KFgc-&Q{GRo_s@*emZDK|@E$5|&b8L_LPSGo5Df-j z6J&2#i|x2Kx;e@^VFaXKcDyEeZhXWf25m`CPhM4RXXV+ollE-iisXZOCQqF(d2uD* zJ!!9cx7Hy|&6x}jA@r^$kC7|JjvG|pdskmEI?QbVh2ZvmO!f^WQKgCknhW+{jg1^O z9k(iH9rrqfUKiiN@9P$=`UgXsKW5CbL?Rq!T1lUmxp^@=0nW+!hzTB=8Zu_^XG0YN+;( zWVf7C@`BIa+l$yMJ(+`cr)@zf7u^Si3bIy`v>8_aq$@nM;6^LXcYz)nFklACE9^M$ zIbY{}$N8+)%BGM!dR51rhRJA3&#bDhb|NPa(T=t2jJ*fzCtGrH>*_bRyb3e5mr~Q1 z(9915Y^wWK6G}Awzaink?dHS0@2je;z~fF?1SxJju(pBYTL*yRsnhHESk-pM1K6!6 zOjqb^b|qtmC-b-ZR$z}Sm0BHifr>+*&)VtxCmmH*ZFx zOdlO=wP2fJ!kDAFP*V;Np?pOo3HOp`u-wdOzDmh%}jm@Lda$R+f zAmB~1tZFHuP{rbTMdX>V?w^zaQWOL>c)A;zLl~HA2XLSqv@-hx^PEx93s|4!rEwVq5 zqaI=-k))%aX1v_L+lEEX z;XX|dI17Y6l4{VpLJo?*V_FL$2Ra+D! z9Oea3kLtDzr7xNs9o$$f<9QDx&SJdPF%pLFuWa5yZd&rKX%pS*y{wSAE4i;}mR0)G z1riEc1Rq4?!Oc#nXg=6F?B=3nC9qTGxmUzf6G$<~5C6J{= zJu*^@up_gW?93)zK9u!f312YBOyR*Y)?{6AMlLeDMPVHdh>FSXdmcQO#oCGWBui9N-Ja0;p1r*#oOe2(#;5u(5`y9U{0yc zKvP!u{H)eBzSZ&`5M@Sob}_~yNx9|@W~WnjJee!@g9KpE6sTA^CflulWO*hma#L7< zJU}scjPIwlwi9iOFp1*KYUt3s#i|Mc?!c(dwYElv=q5BSs%jr`8 zR)2W8L*R_CFLW`7D~PxwzA1j0*~(pk2)6g%B7tXW&p(f)bdxFEEFIAGXTgIWXdR;tdeMb$bTX@W1CNEI3lVRp?t&#iqIWRP=AJJ*lR+flq`wzm7}>n{{QUtIZs zjbi%CQKLw{9zN^zT)#$bu^d|rGu_F@)4e(o6na^?KiJ%r@PD)bn~qKY=+m>`Zi-(n zqkWBL%aq+Z8dk5_rvrQS3Eeh60N8Z)CAbN$I$X#N^#P_s)b$vXjnmJK)KWo2reD=J z>9H-{Rnv)r~|02mSn;1|xyWMs~% z$<&hFqY>o-w!IAP{vCYX+Wf|Nhc;3*So#?HRJ=9maXrZHu=HzPHa; z>w=j)S`e-2xtm8c|3ZBIw*O^*)V)vd<-I{4$}#<{{7N%>1RNQ;7RCxBd=1W zC84SLO7xs(KfH4F-Ab}3${cet+@GoiJxTdb9fH+gybc-AXIQF|5{t(V6&u1c_*7m2@-i~KM4di2{~mC`U# z1eYU0feLYI9;6=3q5zJ^92;+W*Y-;0?d=Ut>LTHom!)#vWow5KH2ayZgojRPK%NJ8 z*eI~d#gN^(W(J(BH8|W(Cv<3g}Wek|t0yG+RGc&B}t?b%g9ie_}wF z9SP^QN1eNPe1OD*8==9{<{*a^A;A08OJ|?~DP107j`0C6c&{*qb9kAai;YNjEOAcn{gGMz;Okr8OZw0cT1KnKT&Nq$~Zc>zVU$)y_mnp`=Xx4synx_T;vzbnD32ej?8!7v4ioHS>8-q+_kpw3 zwwAp#%jWmD=VOQ2BHp8d+Z5zb861~aeMXrYs`+!E5Z`&8q_iq)h%#;pB>voB+t zN13NBzDW;v%eKXBZEY6YA+h=PI@D!DjWRa|5mvcK9Zcd=Lt8EZ7tLvQ<~|>@5VfMZ zzxT_sq*e?m6E|8busCmp`m{4ZlamJP;D1dVk!Pz@bAc(|8{CE8TDq@&x$j2lh?&UX zj>n)?VFLIxx{)o`?8j00_23`&<+3ILlq`#aK?sN1FD~cl_;?39*t(Xp0~L@X9K$=V z@uOC~^#~+y5(w;u4Z5{zMpJnr86HQaUI5ro?aGtCWSp%1-)(~jszxhy?eQ!{r9g4l zm<=u>e}0L$WftM0pf64yw6}y+xqUMtYYPcqQJw79DpWEwZhiLt2T6b?v=xHy523Gt zXO-jV3YP9n_G0!M!~lS#LVCVWO{`j=s->{JO^44HTrxtPNLW@$)<0S5&?DhRNPkdt z1shPy43FlOGH6A8wJ|-&iOx=V{#U99=0Vq=*X{ub;$>I&&A8!=5W*)ocrkXkAQIm) z{=AMG6wdKm8V)jOc3OXGC}#BZYvyw3@}e-?Lgf04C}}5;dQ!jhytbH_7&H*M06?Cf zl}7~zlIS$*LwL`c{Q*)j*QSNb3l|WuOhLl}tbXiDLXo+}7vJDw8*b$|^Ti5ZPN~+I z{XPl(rTp0rT_9Y_t6IaVeP54Rr05VwLTGBNmPz8z$pNK`7qvw9D%i_9%`C^Fsc~qO z?KfP4?0E&%yb|yIF{E~lN3DDqFGn#YO)!i1@|7l;8kl$gjG*)#ay3tkFc0GQ9?h1g zvYH|W3i(xg?b0fDF@6t@o=Go`qcV0koDh90KSvmN)JQ!z?=ELI0ZEO}#17xbjdq0V z9|d@AW6ygaj`K8WK9IStxl;nCXgh0_it08;2D%AgsB~O3PQI*uf2#J{##=A36j;OL zG@LLeUoJ<}O(v;jmMP#3srG)r)adYcpR4YL&H!Y&7Y>HaFo<DTSzv%f*2+WRXjL|blx6!T-6d@A<(`4a;s{` z>C#E?T5L-wR|I-i0g*wUCW+gW!oP|&=```HwJ<@-bjH8up<}K zC_Gi09F@Lt>#t|7d;l&C)*i5T8OXrP00DQu;)vw9KVDc_83~Bp0D9r)SEe~BKvkus zo3*nCe|bwSg!Td`R&ejidbQS!R-@2sm14ICAi0WGqw~J8sK0Li&hj{-*6}zjB=sOQ zOr*QIBoPG+%0WdTg|0wuH8Ox1;a$XAC&?!CgddvWaH6OiAd!>z+wM{)uLn}6Y&;;$s^M6VjG;?n( znYU2<85e=ndV#I*^631s@vBR^bKbl{zU4$>TxrJ%?{(9j_Vm!=rt4J341jlAxu^U! zvRQ1Vw&0ROgV&N-HUP{+Xs@SaQtE3X2p2Ei8trN6wQD9kd{bcz#7m7Lxd~*t5&6jm z2Z7(&W&JKkJ!4%Z;ImTgx)sw|%~7^!yTaziI8l*#|GmV_M^Gn&xK|8*2D6%2x$b z83eKhGxp$d=KYHCU5Lr;fEC_)gkeIw5J8gK-=IvGSe-0=7{!LwK6s=*p=m5~=Dd5; zI^(y9a~U9Q0bT)Aw#>{<@h^HGymQ38ykW=kH_tCN@V5Ig5(e)!aa9DFmm9^6nouXK zh5C0Wj#QG{R^g<;yST9OEK$|WxC+HgKXNH zEZi2T7#YuDRSvwNui4MEbvw8&PcXr+(r`q*dc?yF-Z7Df+HP!E1X*9KvnbLs{D=hy zmm4LfhJI2N{uNQC{to1n(#MNd5+L8y2NZ4o3%XDB0jsF-==6{ zSz*yDP(?<*+G0{`K#7zK_t3=-71`~rCfs4DAS(Y*RALk9&owIv`(0GRr99}D-O+e{ zOk}i(S!$5xrZPBCWQK#CBr(OpWR2Fe_otA@u2i-?4X){^@7Qj8e&vCHuszLEe59|) zI3G*;3KFfuOo2)rJ4j}@hALI%al5mNhzgdz?d9Bwpi6GB_WdslB!1cNABsPfA&YhAR~gH7E<{&EJIOwud749DRv`5Hy6dsPE2^+gpxua0R^l5CVl` zM`bhleQ~=t9g*?Rdj$usLs^aTtQOGKTK+%)RLiEzXMiZU3+8YRmz$fL8vxWTqpn@g z+OUBn$vPg?&ze_{|HwA&LgGplS?U1MQ@;XK2X?-Z5ml;3%fqx67aL>R(b~Lp!^>1| z%ajsl+K~~Jtsh2CKS5DHfe93ons_ZG(f-(j3apCx`n?bvH{Syr^rHg4@`PRrj|i#M z@Q^$UC`IK%GU=}wlw7Eu@`FsoECG$xCN1CwJUkmVt&j%ZE~UwHVqq&ZvMV&CdTX;ev)Ifr)FD>v=S z+*T1+WFdWDVw4G3RUC)f(ULGrS zI&h4wASx>x=%GB5c;!Zrh6fUt{3BaQT4|1 zLOrJ3X0-K_JMbvCq9JaB8`CdbZ09xYDLwV{PN16VFVu0I;m~2Rzm*NXVhpA_=B9Dn z>$+lr7`glx_cMb&C1R?6sh$rMbKIWF_+b$kJE|B@%dITZlh1v*v6HWY4s1-1QY`;4 zT}LRqt?BemUXLdzH0B)TVP1U6;j zEQTc0*)R%^p1#JObLS0b%ZC-*583{|2JJU2ApkU4|M~8ePO~urjJYCs&+flrDeZGX zHYu-FH{8#1L9B6bW|?5EVcyG6=q+^X)pb|-WUlq_Rm#{DMIlX1E46v2*4TzJTE-pI z|MXz&+0w*v{QEGInkP-!!tGCc(g#$G!{+XTHHfNT#fyQ!OT9+idSd!s0p7Ix`$IE? zc;;;^Zfo~ntZhI<({pUImog`pUv{Ng5imY33nq@i&@H;iCcu)v_4(#g(WGsOI(lEB zYJ7g|W}Ck7EFj2rmkr0LJoQr!cHyi40miR8@15KDL%97egaDhr&6q?LpMSrptDIG;ooUI~ z5|imT2HT6s6|7qx1D`zB)b@{rI;f|$kEyQCPr7m&H@tt>ZCy5P95ye!9hfcEU0Rw6 zM`w6Twy(LcXHGU>7Ok7h+jopG=^`@ZJ1rQEe~M#J8^e$01=1cA6eOsmd@K>ixNARlwP9Uq?YDA(+t?ju$`Li@L9A> zI+*X0vm!tS?|u6rqO5e~5mgTgc!B}xa&QD08!S*@a*)sD7^NrB<-ZIv_%g;yY>vCz zl_IV@52_GA!*=G(1WVa(I5yjr@A69$QI@8!bXfrWXV9`uo3$E-D$Z&SqmG^}og9f` z0ryD}_sL#D6cusTbqsl&nB&x;b4AwKhbtO@buRJAAWZem zm6me7(h$D~>&D282-K{k(x%Y0+fdIl0(L>s|64y38UXjexBGE)U**!1;B&j|M;++M zyJMFp0xn9PFETfVnjHNF--+eP=2gFT3UpNewBLZ|`THDuXMHeE%i({B%Ii`%l<4r; zTXremW}Lh9wi{lTTrDUvH$CO~NeqWOUZGK#+EC#yZV6(c*KXFjAL|AgTaWHr+r$T0 zzQ>(-tA&;*F(PlJuySGrB}m_5Xy4*se!M%D2NhHx&WtR6CL!LNH-8+dLwMS<)l+kC z)fgwdOOMA}Okp;3d)N#H&yB5B*JHLmpfzfQR7GcGXUNc20k6PlW6gu}^&M*<7Vu_R zm}{xh$fuKJOVv702jzd-l`M5eZqP7{NqoEA_{pYKhPK~%emIU#AN$wZ;0<84ICXp; zjRD{`R)6YPk}>7vXrdORTWqvYFR~&{h?hft!m~BDg(#A5t;_7}WryFYZX_o#;MwwF zP0}WR5hHJAS=!SdU6ZJw6!v?ze@^wwb2`_+gs<<7&wqqY8h6wpWyW^~+qf~j1IYLX zolGrae;p=!o`g>zU7aSw^)<%Z?yMRO9r^q3VPzXpT=rsbFC-R;bkm<8w_e>*m#4{| zo$H2CC5I`{9X9XPqzSkCtp!hv1WfjB{W6T9y3K8>0s{#k3L}1c>QtQwZ=N#*85oV zR`#%@G)zN>uvshYnU?I>VVQ~rtXm7pL;c<7>(uR z;?`z?VS32ta?t!pc}vQ{xgH>4iZh-}j|P!04=xW|J9(tzoLlD)&n{qffr%qv+7Aze z!@3$Gjb?JcGTP`Jq3Mrv&bj!zD`Qalv8CNlr+x@a8a>dvU-j{?7^nbkI5Ams^*M=S z^_c;px(r?Tr{@U=$atqJVNu5YQW{vG%3OSATxCrQ3DR!Hrx9HvV@~NizX4{k z(LCihwK}R@tCM?e$R~NF%9VKuXBsYl+^{0OCj=6DVkOrW=3<8cysdTbdu^NMO(>rW ze zIRM#6vf=ImtenAc9QlJ+C-Lbp>TIpa7%o6;_XFZ+2}MQ4j+sYs^;NyqD0mocN?Zqf z>ELYUv9&Y?OKxT8ZSTs6`-V7Hxc21Z{e}mMj52>f{Q1$Pm#@KlXf&0z<>hKb9RPcQ zIp;x)BZB_?-2sJL7_P`*mYeOh?3M6`L}gs+Bi< zTJ=8!09~hFAlO?n*p%K6iUHD*Y!+*>>X-NJZ80W#z6s1)^}TgqP&p! ztX2m*hqE)N5$K27uNd+OH8eHdZ>Gc^*X}#TSWSV#MVMUmu4_3@@+F27Tu8r3T;{Y(#QwTEG=AD-92Qyp!2%>Iar;)&P7!=U@; zYrpObB#?B$wAzM0(v|~(PHm0vC!Zmt4T>#xL~c?|7CJgS_?bU!BardtJ}(^4(gPHz zFx!*pY|1^S)1v<%osx{fcP7hw%GUX^%NQ>m70?MA*^l8g>FxqoYOU^(P4tz8UD(cR zdwfXM|38L9$;N%#Zdc0xrg$xn-3f>$L-sAa$gNARc`Zi-0OvG)c$)9hId>9`e0%e$ z>U^>Q%y#vZt$Q!fi10d0vh3k3J!A3*8;JnrUY>KaqA}0-LdWNq>#a{-)mJn=>wwJ2 z+Z~%_4)CK849Vf-Qc3>+ymz;DkkIgSso_6CK$^-sCk*iL5o>5y=66S6Z;>L>L<<4U z?5WzRQC*#5#->qvofha&9n4gRXizmrYgUBZfQD$o-Zw`8{yfE}w=52plJO&4ogw3l9T0fw8fF z8_hur$mb3%&Ch3TEjQCrIIJe{x=+)|N5J{+0M^hBkQ!YBK7%-tZxzHUq8yTiJV4+) zb1xZDL`~}Z$x9oy!i#x)Z6_Un**Obc6@dmF*8ctVp4D_5|LTzscq#x}r6VQC-=KEr zPaxdk*_jETkhql%j8^neKLxoQplb^|~m%qB|N)0G^yYojups{#7hx3cDaRG8slP9LB$ZCXn^ zwU?xdL?-6QCc40j~Y#qI#>t$3YUN4&<+qtfiSt;envr5We0=m z6K4w}^#QWo+g4S(lUrZhJ9GeLBq0zYoU2Y~g$kuW&G&4yzRo9l{W8M_7Np6~K3PAU z5d3?5P4kf+*#mk-=_B}@sNboQGV-aY_#?&3CO>N=UmDM4)@Gn``qFqxfpv7)mb)iay(yc3b3AF zUiJ2PJ(u&2Mdl6Wb4R2rOe+^#_56tU{c5m2wwB`vm?yq;uGxcia=+ z?LHB@BUdNi_PrtXpw{$4No9Js(=o@n%rE;`ujN?V|z^3 zb6c0;TD!XuXt8ydIgSq~dBD>@piQJD!b*}=;y>}9ZUh9s3FrWDKZbIH=|A8xU>46i zt0X%0KY&&)h24__slqKxd}F}^0ibAh9SF>P+yPZD@N@?d88v{^7U}ZkdIDt75Bu`= zWOcDJTYY_!{u}2S&*KRed^vJ;-!1&QBXJWV|j6LFqm#X=4sK2 z(qa(PVXWC&dzY1yoA6tv4s+OuHuFj=y7VV{`GqFSw*({KItY!|nXO#u?2(@_!;&d2u^;{EJch%;dDsx(e}wbp zZ8{%K=q!!Sbu~jYr@fokDIuV#u5*=%5PBb42vbP>cnmPfyFX7G z0ARf3Q}%MB_t0|aKmipj_*ov?w>{g3V&5l94x1IV^Ms&4u#_zu5UGOU;QIRde_k(Y zt$jo~=ad`Hc6Sk&LO$H5Z$E*j1mr}DnTFSU;65EMB&-E0Z|_qBT36r+HF1IL{W(qB z#Qovk8{i?WI=4%Mv-WFQ3Yj#ebEiIJ_cU#=7PW#>rporgRs_d8ok3W{3uvZx0Xo0{ zNE*;%l3_^CixWNr)jqehDivz-*RIn^)-xb>iHSM3V(Cbx+my+Yg+3h^?M}>bYyMb2 zgD_Qv^r6%S{>wj`b{9Ypg;*~)UIMMuKc|yTx=dLRumpy-le+h=4o{@8BfegUw+!P& zJi3uTzrBzatRWz9b-{lZ;1}%g6(zQu&{7Um&#Ae38PfuD4Ln0DL(BYp7?ZpnGd2=? zT(L_`#`rxIXViFU-lE7#Dvt5sp(@ltQ137Mf|qNa=fkZoUyn{s`T*~XoXQs? zWn`2)bUEM%#GMW4poASW=5S*Be3nQDtf-=b4jmo+TlM^p{Hbrit9m{^CkH}ZUAW(nmKOvZmS|2PU}f0K`wpu68cH= z@iW=+v%ZrO)L-jaDv2D9H^OaW1)Sflf&$ob=bAkg(V(l7rh%2BT( zt`cXe`}^eQ+wC@Ej2CsUJe+|f&Hz48SVxEO?rarqV`MOzm4(Ht0iW7zp+?^gknA!8 zB&(8=(gxu6%B|%6m6}b6&`5-R08PIH*d)L_>q|gzTOrEhR4A4WxKx$31Dc4jt(XF8L(qPs0GPKe@CNwzXv-VmcD* zRBwV{P)9mtSC4aPNcV-E#-(-L`tmUQ&*-Q$@MjF*fM-)%&6}T|&;q>Dm}=DNFu^9G zr>7s9m=F^SW|7mSmO&Rm$HLm)=wsRw@NKsEOV^u---sTQ9-u72=5u2*8A}U%dh#gLj7yw97Lk`nk(89I zalbZE=d5qz*TJtc`(4=q*RsgWg!@3eE%oUJGlHdf%kkJfqjuzepO2rP0ATX^0H)%n zf&$5M>)Q#C(7H8kI|3Ul=SfINgbrPXp#PxWgh;b)h0CRFI75#wpz4T`OOz_njArs; zxx2fk2zYx;TKTxPt%8Mj3_9&0$+|x3b3```O<36p9K=eMe>#5TBbV(hYamW7eTfJk z7Rm&{kk2{RMLG=q{re}Ju`>iL2hGb9N@9PV?eu=G%ETnO|3OB^Xc+f_CmaevOw+Dz zYp3D;{LRaS(ib&e?&YRfupijQpK59$Kr!#$hN+Nh>(Fn|xV)K>>pi|T0H#PLZ1}p$ z)xCp1Vw6>SdV-%sjZ?U}luUs58c+}Fq4lt@sr|@M&G!jaEDApqXiC2d3KA(RE7w&V z+9DG2g@83?YHcp`sAy=3Ho-=E!(n}CoGSV|CGzirdP z6EcSjtV5{LT^$gK_W+DRFrb&osjE`}EYTl|MhSjS8Yu0srKNm6^lAx-$@-`1{dPR> zzLF9DtCTB`hq8V5cB7E7PPQ1zNZGP4Wf@C~Mr7Z&WTz2XvSb;?HkKom>@NyYDZI$e zTgkqp#-6N|N=f2(J)Pef5?M3KMKj|i=wE}U9vRzClEuG8q5cpg17JK`82aFvja+#y;Y8TPCTnIFkA?lY2e z)=G)6e6_s9fWmN6NTZ(_B8X*zC-Z$DRfR+D=nP%iI2^96&b8Og+U!bMSsB920Zfl6 z(oH;0W&fuF1RxcY#0=7$cD}6y)rShT!+C*pZZlW1q>7Mc7&pw5fuEoB-si+KfIm{y zVc+nz>Sv!JV08la%A@@Zf4|^`t~VC%a`HXcS z8bCC$s&%48Shd#U<~nE3vLGGca4rrrY`b#J&;#MS@t(9{NL<)q`4rY745+pZWauBM z`%6fJ4D>f7-nxZpoZ3B8msn-teZlcXSk;7Y3&fI}05UvBfU3K>JX8{5_vsqd4|yC2 zAij6XWF3M(dNYHl9ac=dxzdn{w;i&c#JzYEN~9`IC@f`U($&x?drgBSoob&QM- z!2V+@8N!7hN>$=Zvv}(Cjgas_%M~>=uRfHgGHFFitM!OgR7z5kJ&o;Yv)bm_n-yV`F1DX=(l7U`5zzH#Zl0rG^delpHjWvdzfn{U?B@ zkX6(sB}q8;!fg{BLZxgU$J$rLevea1zEg2v#?Iqa)>go5J%21~U?L~^<@=7|vU5YD zqj3|CQTbd`l@rhow~wRJ^(tgUaWK%H+1pItgxCxV@C{~zLx_ojXKvVypY~A_0J>We zB!{yQEVctzt=g&~s6j);d^*lq61~jVeU*{k`j$=NC!2)1sutnPQ>_{9#?=<#zS-?? zVSuTbnb^ywko%2|haoYHGb!)f*;+eXgL9H|qBstD)o*Y3N0FcJ|Nhhmmd*a0u)pQ?>#D-x>vGU{Crsv(u~`3YAkH|A z6@Ts~i}}3fxX+KYO|c8AL@UokaM?cfX$N!4xO0aLKsCKdQD9G`2WC|F>wNXVxei7s z(6InOB+rLyHfdmuUQbdmN0Lfa5+1aQpRex3yyRG~R!ug%EY3;%B8WN-E^!Q1y1VA0)VKHUdnE^Fh}q|~k7!irc`+D%OI??}xp-yaopcX~^9{BmK>}GLHI|mV99Qa&CPb5G|pB0If?TKB1dA z`{VQ(S4IY><-&k6ThSx(_qQ-Yl$cKA@?d2%@A38T^82sH>)WT6dO{Ly2dBsH6V(@8mj(me4E~x8eQBLi#~NdL za%RZ1v9Yo5)vGQbaZz$}69KIj49@IOd`>-P7gCD@ti$W;yNvv0YN-n`*jd)wGe)lls-226oDCR}jF^yT*x~qDBi`N9GY_Q# zJop}{ea1u9CIGQL=5$2Ez_>=pN5Nz=+0@c95ilF8tUPhq8!eeiOEz@gNRt+P=+#cm zGewPTo;>Wcv+^n+R4uUdr$UeZA-#&@`0=gBj~{RL2>3fq)Yv0VPF`MK*13ZLnSW8y zkZE%7LVNF#s1yL(8Jv_sRMJ zA>9n*OHxu&JJMds&BLSpF!z%Mj*^L~C2?cHG@BUfJcO8ff&_JIUnZaLDK{=BI6vr- zHTy)DYW^)u^9CT5>+bfZ=ann%u>CfF&Fl*^-K1SvSur#=*0ZtU&&!SWG{^F%v4Q_p`mT7 zs8EO)CUWvk$+!}%&glI*tl-IkpsvKdP&z(oJFQEXc<$f7{~2*z=T+`njiP`Dwl~z9e?L* z(rb|FKA7k^bb+suc#iO33{k?%Yc=@72Io+9;4+HxO6{(&pypUC<*>(c{TDyaWBcU? zS-Npo`Aw__iaHlAHVI*yj9IPl@|@i=KkUoC9ZKvL-w!Z2!0KVFl7JVV4*&;=Paodk z81m&1WKUcU+I%LCRW%=b+BrTQZ@O}*$fOX=*j{|A(O5eu$(tmaQco`PrFd&tzGD_rxpxX_@ylo#<)eAw7 zCZK3$%*&1L%@K@w6B{ z)5UAHMe8l<3Yi=_{=imFp32G-Qa>VO$dZ0jyIS|Ucj|C8j3W!>I1$%hT2Th&^x1{pDn;Q7UE0JlkrZV`JmblGD=jQt)`1f`WRNzCzTpLWEaFgp7?V z{_g1mIgUodfU=h8HqVoC9=JI=*?8*%pa#7b%ITo?w@#`j7*< zw01=65ffJhgaR5{3|Ef)MU>+18ctye2zytt;z@v)(9!b_2?d5XCmo<5_Hv6Rs5+B?$tTHvmZ0uT>{;HEw!g1;WG**YrP`+zEZtE^AAEPGIiQ>1T zQy=xIJj@BK$uMlO_r=cV&6lit%MT=RA5{@#A9_EZC4p}VE*(y2z_D?fBe&E0O#z@sWY)~!%-Mn$fM1Dc1=hLXcc`_Wq>D#v5}koRSEY4vJY zL+)(Audo##d-Y>L2=-h@~Wwb2k<7@To%nxZRz`g_ zmvMEzp%6z6Q3|igd;g-?DHjod16}i`!^I-7CZoDMA8aWUPdq7E{jFpz>!>XUwt_&t zVZCBOie;kbH9chK6;%9E%0(+h8y@++B=x!7eKgUrGAE~o4+87k;dL;@>jg_Cu+LjD z9kA1{7fwpRm_HrIQ4g3B{(R0S6QcEyEvCEl60C=MrpJ1^RDQMR$0t(oje=eKt2G_piE5Ov9D5?;#9g;UJb^35*8YYVX-2U zX9HC5xi8CzNANf1zFxhC+42(aU4y|5mA*?epMIR*9=TM6x)eAgO5q+=%V$^1|7h#z z`S^{K{8{Dx+gnxxGpqHgO|t>&pY!MHP?vsuoc9yC@f>j1GS5lJ)li*ZS1sd;;B3h6 zI^i`&7X*rTUc6~i5|@bkp=O8PV$AD_sqlNwnwS`(Kq_s!z=@@S_`%Na8uqB@d9VcT zqJu*sLmXtY?`p_oalvHK+^zIR%u&ZY7A#cqn4*R-n~Buabsq9^8iY85YXM;amH#AV z?=^?%wVyOE>zVx=de`s6r4z;K-O+m5pOA%GV7(iz#~^Z4gWXqH;hgyX3^hN{gI5K5z*30KFyv}&4*ZeA&gB>x12~|APe6nwDOQ@j0W}EH#T(W(+5SH ze0R-@3ddHy7w-fy4~H&^3o6dn3b9&oKasZcGKhF0ti~o`$wkGPTu!hl z>)=cC0JRPndX))?WSY6VAB9^rZlLq0>YJIVR843~;d z;5fM|n2O9Q0dQ*Bdm0YqX`*q-gi7;Tt5DlG+=vQCLH*qxnwyMtA+G2x6g@9v{JVE{ zsE+sBFZF(7JXK&~GfO9!`q zc`ES7B_!zlcfog}#e!QeR72S3%_ByjN8FNUcNRI7+IYT1n zmhc}?hMz@R&ma`73GatKZUTW2JvnJbDn`NPAUwnGbqn&ReI{tG-ZhRW&ZuC<)Z!}K zJUp8(lhns+%F7>>_uh7gA;C*;80^`~k+dIY zs0V-5ks%Z2I(B}$r~2B_U#^ZEjICdTVmx0-h%N2o%StX;*6BZXi~iaOjBi zWitLo3@w>N3VXkrgrYWv3CL+^2tpTTCvXfFy;A%9!O@^8rG61XgC371M_DLA4b1R$ zlA#NPn()Wz3d(@&7Z^t;4aQ3wcb`+BZJ3dFL}vzWVn9{D~ng*EV@U{{QX3{|}k|$Dbbi6I%pHQkg*( z(*jXaG4A_f{2v26`zRj52dn@3-|<4Q_WwL1|9<0_jGA#^6$FnzMQ)_!=Fj G*na{44Z`FA diff --git a/docs/images/chapters/bsplines/48a30189e74658737b3a8b28bb816f8a.png b/docs/images/chapters/bsplines/48a30189e74658737b3a8b28bb816f8a.png index 2dfeffd243da77b82b89cfebd5b43a8c3d2c52f8..60cc5dac86665834e823a1d6ba8d61c33a82701f 100644 GIT binary patch literal 12584 zcmeHuby(CJB zJ$~Q0&vWj%_uTK?zt7{SkIer4_TFpnwbr}d^%|?Eqe_HNhYx{3h}6|!1`r5ZGU_)j zHuyhe*bbRh0Y(@4%qSTgmAxAloPUJ-X(Be81(y=30DzfPM##ehHRLp^&$ zF=2sR+Pf@@HcdK0eeQf%xAYAqg+i~a2}#-5QX|n6VX#y|c?3d@3>y~;RquwdvazYt zu)ts(co>lg1PAniK&D?Yvmz^7(I>74I|iJ_<&>kNqvD&?_A);nf=}CSS-K4Q(@8`#C8h~HmxmcEM5!31g2TKJ0)uTxvlisn6C<%Hrt@ONlXIp%9`Hc;F`_E#M}>;CHr?>7F<8g=}=B6HNwZ(MMAF z9OC_f?qvoYcX1lgG{64o!%)4+1CuiwONYKgs_!)i7kO@t4Aj;lbkDPd9%7S~a-u_E z4Y*HZsmHm>7piu9yvMNS?lJd8J*w$b=Euov}t~GvgTL zQcm=H>MKo~$6eW=dBdSBA&gSpd=+Qs+xEk`u@0}E{ay^f%!9ZP@Ox2N=JhRJjt8H{ zU+z-*FWg%R3uK`pOG-iaPvGy9Kj4 zK9P**GBqYP!*523TLy=11f7g%SSYB%MHP%5-S6~9Vtr8~(7Mu2jY)}%k9Xf+R-beh zOG{5L{P>YVEQbhubV|v);H1XoM~i2YdBpy0OuXs%_O-NB@mSzbb#-+$EghIa1BdBt zs@}&-5`-$x@i8J5U#?Nxe5on4b-}X5=`Z>d{)W_R;(B<)qseW)6OBRA3wI!qu_uLH zH90>&>GCoVrmW1$!-Mzs?OQc9HF-rv^dWnZm*P)}i$`&z6!llH5mQ^yUsl3LRwHIy zRlU7MhhNFWf6WmuuBafqb?cTy!13+dR0QT99m9OKezEfN6Kzd3sIJ!JjSAP_LhlQE zLLA{PMiWJkjT?1)0*#v2#6U2bfbb zGNN_zBnux7BxXr^^ZxqyIKue7c^6!8I*oBUSBES3{0|p{TrdM_8w1QC39YTtX_=YD znwbJ?GcDrhM;kFyDznjJlVkop@L5kQ7D(BP@m;RdLg(^U)*d)h63%q}OIPgeSRK{@ z>&2KWPpRM#DNVyyg1%MI@L99)lVTyb(w+RU@%JgM4G?P zD7-kYFy

kO*xHII#uZIN9QRM~jmjn~2V9ucNV10wUi@Ctn}|38`Vewhh@zBDm7a zqRAZazlSQyCO%v_@|g9DVcI~u)ybRRK1Nw4=oSXvh&fS z-iqA=rxc|tdxpszbO;TUW$sU)jhL_sAi6p~iZGyUKOvCxUXKK0Y`_=0e$tcD>Ig+)Nq0 zjoFB?BitnOPJib-74+FOy(cTCOoNU4c`#_``QUiEo(fOOU3R?nH%=fNR-6sc-L`IL zDAmd94mz2M@tu8!qZC7&-ZGP;SUzrhy1PIl8z?dBw?wo#-9*kDe9DSVNE0gN@(aG$ z8{2p?8vv!GyuTWR--iB6%-nHmEN_sZ48LHrM7S|%+rJ}YYO6n6#KHCVK%q)xJWuNF z%aY)HWNy<&+| zEA?j(GQhE)Zq|;t9j@u$ee#2lT_vFtOj8qlbzUx@A2Pt`TcDjIhWsNSV^(GO2g&=E za=URUG1@Vo4h>952z6VQ)H_vVMPEVV7}4w(bEiK)-ifCbv4>xt?n!w5CIra_0j}}j zjLRo=4UO*Hd!BB;hiJ9(Wf>2Ce^v7Lmjv#M)iGzi6IfypySeGCl_N&<-n>pRo{C>^ zC57j7!ty5kq|78*S3U)SrYN3wl0e-T!lUx_2OWo&X6Nu}-kN9`_4Q$?+(OgptDZ0n z96|R*m}8x|yICr3p~2hT4eJ}axsV$ftB@=2z}+q!HBHSP9Ol4Kc^19Z>9R8J+h!jy zK`QCLzPi{u@x4wo*8Qm1C&%k_qQZI}MZ@Ct9rGPVnW{1Db$ ztk&D{*-Oc9#9x131Mgk!JE*u2af^X(;Z&#Vr;(8XVB+1&gOQ1BC`>%Y;B9Xt9(mvLkvm=*o(4@b-Lh+JH(`~$8krDFk-aC;@Q4%%#PPiJ&FPzSdatb?l;A4% zB{E9UFxkl4>9Yo#OY_RTtFR=>T637E#JzIfF21?hXQ#ULa>xli>1Tv0Qvjxv&t~VZ zx+f_qDf$jdocJ!ksLXq>#i;C_W)^-v&~D_yvlO=qLwX-%p6Z{g@bF%ng=OstsrmEb zJzL#Jwgvc>8S!53_KHR0=dY2kP zURz#j**WMz%}h_*j;K@;8Djh5sh8SLW+O%&(^~GIV-;;DhBwF3QQnj_t_$u7^H>{16=Jb=6n?CEkOb4+n>vH}P!#xkw;>#>Oa=Tx#;36)97&NT-6~0E(!REY| zjx!5B^+F22crbewuYc#|*23o)R*QUNwpAI5g@r{U<9`=izZ%a~owcJQlfaVQTV+@r zTDtOkU;6ZJmBBZ^mTNQ%6Wi!Bwi_@H&hja(45TC4`RLSjxhx5jEL(h7+1W8p=0fCv zbN1~b^aiXb&wE=uS^IiA`ks#_rub{(#_SgTM!B9(Zj$zDj{l^sIpEf z%@3w~+HoZoO+?T6uyjnXRkf}9eSOz{8kPougS~#xGQo;39Wu=CtozkXzFOQe`M`zS9Z0o2d7m^=vh4P@#vm z@U@5X+S=NAC$U8chx{NadNJotkYQ7Db8mm=VrlBT6vHkFxubkUJzkyg;nNIp6sc7h ztdYm?P3IpktK5m5lM9=MDWQ|vCN;BU^oP;VaF(%vIYV8UC;*61Sb{J~MMn=`E$by@ zzzS!x$){F2W*K3UZ1T_HoCLlrb_fS z3_4Hrhb1ARy_+Z%O!T&W)a(h9aT~n;7);+=)^SXu19(xP^!vhRuKeCK9>ewA2LB@` zRD`7$cZEH8kgGLgAR$O1q%>pveBW+`G$Qxqc$PbraU`e?Oz+v<;@x>4SzL1KQI-UE}t$tufVT4=^vaS{v2t-0@a~VmT31Z_{sPc z+2pzEzu?mtqN0r(l`YlqE_6aB>Bfbe;%d${Ob*!$#QM>=8}rgoMLzZ7#;wlH3+sMF7T+#wwd*9INaTY4A|Lt%oWT-|#S7}2xQ{#93J zBOxjPJzgI-bbLnADwECdOPcnZ>h0U(h3gCtIcWj&4|#WDX?qeG33;z-EH5w5Z#B&I z15d)QGv_z-rBK&u$cC>u%m0+JBfZMa9yb+~XUmeAAbzMalk@xj{ri3(we4-U?0CeI zH5c0U56Bmq6zspyXtF582KPu$L6!8CArno@1W~g33(kf1i|w9IDeQLZ32YgVUhTMu z6RtbX7A_vXN<^U}VKD9iU*a{X4OYiH{ArC&L-A^7cl_*Nbq_#^#jBIK7!|`1jXR^yi{cDfZJf|~<;o(fkX*h} zUgSdZY{?yCNpNq{X5!2roiD{)=wA6P;vH`OH0&QPt3lR}vi%kp;tndOP@s3PSLWT0 zWeQ`kz6X(mx0W8`KbmRwHf{A2s&|>{H!~S4V6c{)zZxa@QcQ*W?)+j{&u{6~8q`>q zLYZQi2iARv>G<+7yv}^|tsz(7`Jd9YsfMb}s~W4NcU*C7yt7rI1!SZr$aLyI327r! zlk4udeBFM!&o*vb3Wr<=CNSLFf489ERf`)Q+K96>&%fW@Jjx5U5P`a~HLJgDHZ7#Q zya8?syxbd*@K{C+S?WLJ1c6reC1No%zVLBTSbMqEcE%d^0;L)zG>9$tB3JOZk|-H~ zMBC%dpNOQS%q9`1&9qSs4(XSn`y_)4SD*P%6M(}-cy^G3TvcXkG}-!0B=~YqRr-9b z*rP!i6M}s0M0(O9k=AAyEic;8`y3Pa;NtXFq)-p8K7~-gvGZ;7kBHhz?$Vx~nlNi|j>I-?bYlfz2UlUfH2_~GoP}lk#2wcN2Y!_*Qrk_7!J#PO!6;}?vzLE! ziK9_M1`uR&%dH|lvaz}rT1gK^x4_lp7RFTILui4wWC5H))=$`u7Aa?mKV_I@_}=3hx!W5z zE<{AypuGQBB3V!dtx6|^vey1AOhriSN6FC4X!>scJDct9uLqzQv;Fp39l6n@g&CC(vK`fwu5$+s69nEcwYL*;Ll1)b ziyDz;I&3u>a&Yp`bhst?b{ArpZTFWydo;i}KF_uVEcmo+Fp^+LOx8Li5i*V*fmI1j z1XF+HuQP3U%;1{+} zqKcQ--NnVlYM(804i1h_Gfb9f4mMFdBP2{~Eo(J_M;-~xa!ls$jkhxp!6`#STDfu| zOgJP=kp^#J0KR)k=1sed-gnbI6uf`dppm8W(rsW_YWJJwokHu5V6aoz^8}63@0KQm zs)J1JDS9I&%Sy`HSkCTZX=!P2P;u$;rn7!$M)7{4iTQgM7R*K1g`0zo{|Io3?f~`m;4voydm8(sUhNAk9%cn12mMB!&Qi(7) zh9*6?1nsh_y*G@3ArP9Gq$$ocEH=yC-)=?EVn1?lSTgB2PY36%&)yHgV;`300yfuE z;j3|(5+kG)hJme?K_7-jrV#M$P*G5Hx;7L8!r|{`XfQamFkKY_C^8r0X!uE zY%#eM`ZfsDpC;en!?X?Gc|~35AZ^hDHjSydwBzY|dRQuyuk; zs2^RXjQhc>oP^{RC!^AR?{MF_6Qffa@zXh@h7Y`Lnp_4J>jPbhAB2j_5#zhK3^Rkh z7?$dw(E6k8LEKwq*;N}3;E|ARM|h@>n6z=!67W`lu` zheyG{AXSSIET#{Fr_j#Ej4q1iIQ4jgp{lG*9t9*&>qLE(_y9+XmJ>Iw`AeOMJc2t- zP+svl?xgPob31zI%aThTeN(|o##Rafh?1C}^8SO;#DK^T@cH5|B$_poQv1Hh`|(=v3Y{se@SAV?5^ zP-uK0&=}+@F{3k-FUJfxr*hBAfu%6x@;1d=Y(>$1kQ6h3Vqp3ezS1;=wt&qL_JYAs zMT0Oo=0e;+)gc{JpBoL4qL#?V} zblyiOICNf~9gG$##YnC?)tS4^2(#f+=0;JX+kbr-H8eD&`Q7h{f5A<~_0jP6jU!M; z2z#l)jF>SvU>~2(F#)KF1Dl^_9_|8VnxwMjDnmq9Cxo>qxr?F^g^x`Y4x z4k2Psv+${nf(2M*AeNl#(I!|h6$VG|k(xsDE|gL%sWR) zX)bBK^6|ClCJ|6~FT2Jg=wWeFY%J!s8-m68$$xWd6{C{)ykFj>&1c^q>!mRCDc(O3P1G~Ats(CUYfGG0Y5+{SsaDc}=7P6Da#*DJGI6>w3 zzakh^NgPAkK^oT-gGR!QrQZ1`o3AEp3;|Y+D4>F`*@t>2kw@T=^iNoa@c;Wa2|PWR z0hNp(wn=$-c$`Y(LO(}?Kzp~rk<<&@#wUC9y@fbHsGnxYprQs~W_Dckga*2W*$owY zdrnXlaiLzM2Bt`^(RB$;pqC;B_KKR@$!xjB<`fCynp$2(H0dptzMOEd;J}NmhQB)` zJ%N!JR=NnB;s^S590pNPP|S~;Sf>LhVLi0su>c55kSYLu(^9!ruPz(zmzN~+H>DU< ztp`$WR!y=>^AudXpK%9Q%(3oQGN=ii{xA6(9(#Q365Z`WB9(8&4^Esu6{_uz+& z#WfBiOaMiHLFICA(|6ATj}kAgf{l!g2a!-q5&#lfsz0jl!bv-Bu6Amie+r}4@!F)y zDznX|QjZ%|;?hbMZFE8KJgJS>@&INwt}yK?(dNE=v@^$y@rx)_|4FOg&?8|RK)bEY zwzVYWjgFTsyM_kSZt^UxQgA4%q^81>`W9^DFY&OU(_VVxC zU;qLRD( z8amBC>b>ZjjSJs=5_urSg-yhOS^IN{r;S1^P+8Wf;+_0quOs)vG=Nj&$N|JFdoiYG zp0>+y0tZkSRiihwlq6FF!sy(X$9DU9SaXRvs4vm6)gl|+M1sc3^l>rKC#!8DKt{Dp zStCacOQ2Puc`9mkK}3;?f~&|ckX1CUd92^VdGVkW+~@k%H3QjG){G^ws>?XN-_^WlLIVN~Dqv4UEQhLW|zU2G|t>(agBrRiqr>_>$+d0Gy~` zb4$2a7ZLL&N@d$;4cjIZMq?O_muOjs_N`C}M|!st!*3yq`q6eAoyaIx(czXs7Isd- z&b!PO#^x|!1(ol{7=Ty|7$$&Z$%V-N0DKI96rZPQMOBKpi)`;>-Th1lBUw!-SL=bq z3W5XnD@PGA5P5(CGj5xO1I}VMT2fSEBcx*21-_+%9E{Ae?Q%?!y11N9J!j#Q$K0B5Y-56WZwV)S42<`T#E))}m8 zY_UMXj{vy;%NVWlZD^UPKAJWG(NIIq%62pz1RUMfjvy7Jx3?EQ{_Z}i2?fH#BTz}} zk9U-ldSO8WYa~0&ycZ^ zfAC0+a^3$?&VzuwX%p$0R6F}55hO)(PLv@XbWJz1CPc{XawO9GBPqjr!nsA=GSf<%};zOKPG&XpHKj`ItK=d#3Inl z$(`~dMj`$l{gXHT8$aIf0e%M#>KIhbt7U00iHBaIsad#D7V)@1ecM%y6YPD}O+s+5 zEmkdngXrqwL^Uu_8gSQ^tGNmh*b9IZ??ph__4M>u3`;3=oS(RmQlQ)R_kEhd|BO&F zg}Q>S-sw)pO*j2+w}s&a<~gcALd9C(lFVb$yJZO@}9_%5Vwl|%c0bK8^vlzRqZ2PQxRDt|NS|x6CL}t*O=AE`kGz@Gw z*fAj#9fHZIkk)=b4sguKx}^lZcYTmpx1F{VCPHis%tD_E^AdwD5X2W6qquWTa8)ZY zGA%}L>L#puLeVgg9U&nUR8(SKQxuB5p@JEG>(JY+Uumh;U)v8Ti8`5`& zx<|Ml{o&JxpN@hs3R1O zh^(@f1B{IEw$2}Hz(7+n8_QFcD~=(!0TV0;v>8tfXebc0^bwr3FFS*}pLa(^xiNYy z)C@570ITm!dyd!u=QuJxFWEIXupewLocT_{#-_i5N6Rxf}uQm}TnJ z!kr4$J-)u(;6Hu)4H-)#A=e@xdB&U7+K*dwrYL5va*^=fn8ik_k{2W_(wTT&(XepD z7Eq9i-o@9Z7Q=$%hAc=&n?>7%%m$Sq(vVH4eu zLTeZ}q2q}fHGg5e${>N$%(EJg1r5Mzp`c4YU>B2$irAW_UVkACVjG;}Q5{2a|Lq;c zbdzF4y5a&BHP!xBSL$PhzCrv${>hI>7E;{#2ol-G9haCU5{W&9vF+>eS;zS|EDw`b_{L$2>tsd4-Q z%waS!W1K3}GfY(SBxaP14A^d3eYdlBxuyWpRr0ARTbliFW07?-sTw z8b5!J%SwaI(_yRD-EIOD+|I!Ic=>3pgRb=1@(YyrB{Ipx;gNI7 zzpbFh1%$dEAX`V9y`Ok({h}0kGa`RYe})L)CSIqg)5;HJ!^`~n8r$9Gr(ck-j8l81 z!e}@fVQUbJaL3o)j9h*{keBubfOH|C1BY+|0-Zx0VohIgFB}Azx3~%u=DW|EcHKEGB>9i&XbBE zq8DS4lA>oABFUg|g;L(I?TUK5xLuc6!SSHftQ4&*o$ps^1S38mIx=%^Ly$ zvtR?|iZaj$RGG2sQOeK$mTt_g^sj1~ab=JVJhlI*^PDuMTFtnbSn~CddRR~lOv#;G1tZ(bEYvcG+eo+koy9Xdd=hSoX)eYw4iW9$peH%L~1~OdR|RFr94OB zY;chOQH50$-Ts2F%4DJe+F}nrsUY2M@;rZZGSrJt3nc1&Q=71XO5Yr>B09x+jKBNzDt= zeOlA`NzgZvDOYF(1F6o$46a=TnTq{@Mtd(6X@&fhaDbzfvn9)OG~@< zy_6*Nw)$@78I+s2(_~($fCDo3jF}W-q-~^E@~%L~SSkB;VCe9$mG!zJW)q;IqH3m%DEx}Rc04*g}$ z5C22}&Sfm%J(vcVC-SSkl9?GT;InLMf1++LVEiy66ina$h+p0Cl3Y9T2kHQD;y_}D zlFlea6Dk1uD>XG0USmpH`aA+?{GJw4x7W&^CSJ^c-76Y;fUO;ez5d&YCV;Mx>23S^ zF1daqJXFtA55_}!&_V;mg{U6;=F}FV=??aEmox_S0o2+NWGWejH=5 ze?VyDF2107=I4Y) zP#{Dw1hbuI1OGfW65<8T$n|QGGz)`Z2C(T_0Hyi@Eo3gbs?9D{(eK#THI2v$GGGzp~ zvFuNB{}raNC--UAl==KBB_)T(w)+Mp81{k zz3bey&bs&hasNDv&2l+2d-i_z^Nmk@3D;0pCcve^g+L$#k5ypLArO=}}6 zXb}8^X{o9VgWMs1vRc0+Kp>2e$1pi<&$QjScb?j-H`2FPgKexr2`@-Uus?ItbkVpI zC926r=)(e@=W$9PWXOrh4P|@tcx6t|bUAe*^K>-b$&5+rdJG-C?`t3?^ zn0iiWJ*>&!lJSW1mcz!Ef|4dv49g1zh}WGfC@VTmr#HRY9>87)!-wSzA&Yy?|EB<$< zQZd3{EI9v<*ZIGHcrRQXT*c{`>!g(w@<`B*RnY(cdWHY8?Ei8*iy0fptDM1X|GBJ{ z)Z_nI@V|oNzdR)ixCZRo*RQhm4s+)GRHH>YT--)=_?_k&dvyqC{F6OQL|O7&vxn93 z#+Z|MIt2dn%d_RpFPTb@ALD%BHh{kMfV~g1Wh4{G@V|Z6U|c5NUsPOtIKWk&6>#>Q zWQ?x^l8tZ%=c-~V@)PwG4r(1FUs!KMaJV>QM?#fm*6f`xk2k&gF|bFiw>6Fxi38^u z9}Wp07~N>pXvwC34$Tk2#6@dYCBAu%PlW5;CeJV+!`q)WE;gOqN~oTE)McfVOB^e8 zN=3q;iVi`49e9v_)S2-*tbI226T|BTvDJVL%4~IQ1@KMcngsS-JmWm$8XHvHu;gw? zb_KaBmc^f!VMj*ngUmw>uZKkH`DwpYiqYlpKyavf7cnloU!Ah$49+&xeGCOzVFgmKY`^R z&cK;kq+Wsk6a_H`MGxqdSHnUJ?s}r;K)@+o2$wIG?PA6mtr?$fka>zyu%xZgTbQ+9 z((a^x9C<)_wJ5b4Qr{7|WisJ(y#m)-EFnm`Lyu&8jHXt#Cf`7;NP~$T5+eH!!s^c# za8^PoKx(;lEq|A{<1XQMxyi1rtD93%L9{-cu3&A=3SI<2(J~VW zSE@0T-K7~hZi+>&Kl$#j@n?1kA;IxbXEU6^RW^6?xIPP=bA+ve}h4T<** zM=H90-}o*qBO*fi@#DwoYFk3T^F^#QQD@Zq;;yn^)YIcWdWm+4^_%- z+>>ZXN>1+HbX{8cmhg3GNYTN8ce2DFDC8HOriKRm-hFZNgN4qGxn>bB^}X9rY`Hm|QBhN~Pdrn5AM2e{ zZTfW_PZYJn-zMy;fOfLEl>hV*YFDopyLwv3+E8kx^Rm*)>Z--pI3_A-AJNcz_ai8H z3}+8+%k-o)$=)X<1Vu&Rzp$~f1i_}Fs@h$knal5W{PJ^t{$%lb{netxZczEC$D(*- zdkm@O{$>Vx z42#4VU(VmiC)(D!Om-S%@foUt@c-BE=vcxdRbW#T^uuSKD-F$rqY` zeQ+e(E){;#$6~#E_fF*aM&oBE&K3*uPKR&q*3JJ|D(~!DTBO=^hewcluDn-{OG}H2 zmAR#QyFT3etJ=2G=fdT9^CvsBd)Xf+5bk$#xwx8a*7s4!mfvH~_|5Mc93G>(9`Fh0 z^Ml3WNsZp3QEL^-jpHV-*jKfNpY!tO5R-*CmT-++Tx|@E(=)*|?<2Y>sVEiWI7AdF zH8M6Z@u?uWV~G9sJ9?wKH=T}tN3;dnYG*74rMk$@qu=b#1VY9??jJD3!~r@+0n^$97dyssNymeV6bpM?f4ML zqq3h7^X_QPqAM6j)c5kCsn4NejorkPJqHDKOG_vS&)}98$)_I$F(N7Xl&(&94?D3K zj0IK3?S$R;X2T1$z94H2a=D3N`Q}58lO6t1-+%u+>4ap{4*T{*2eM)mTvTG>JY!+x z;>7dCJn7^rZwXY4q*y#y?Djn9#M<^T%2(AYo@m%_yK*{MkmELMqbt@el?N5(;2V49 zI37vyL`40(7asewjGkasO+3Z zO_jjXk2Yv&wHI3(gQid`Li_q#-nM!Pv1V-@ZU`TRd}@*6$$pW*A|WZi{qKj8x97{E zP74sFSo%2@-UyM<05mLQ;(Bi8B_ACBe70OS;)jTdi6JAGicaL~*Z0ha-GnmceNogE zzuxHiI2?a{SzxKdap(Px*-iO90(?=(`X73-R{h|u)(J!^br|<+3h}euD9{Gz#9gtU zNgnpFQa<>+t(vW|P~U#TQ)9SGp2?1lYPla9vY$p6!*@S={J!F+d^^y7 z%EyCsoT|B#@$v+VYes+WeLU^%Mf>CD&-d~1fdDNDNp`^n4a3jH4mqxrsaq~g8)vfl zyM>4@6^Hq%O+mt6rkk^(7P{kX)1E@{C7Y@*b>o|Id|LDro`tjUlhLlT)f(ESB|BZ zT5(8V8TA3#x#pdu)>zgyd?fhFd7M{m}U0Lw&t6k%TTFBZh<$ao^ zmiox0{>fChTOfI_BSo%@Dl2y`w@bYOQ;dSX>*KP06Au07kUy^&NHU)UpEz)lVPb%S z8U$LC0-0i{GCD~j>YJbTPQKfLuiI7k+&Ov#WDD+@1rV;!i(*~bNLaG>bv>HF)bZI3 z-^!5k9&Pq;ob}#s)o=9@KK4?-ih|+hqShHaN-Td6w$4)Dod10}V*?#RlnyO=q}S=) zgZEQ~w5GL&D3P+iL*&ggoIS%66|8#sc$Tf}DuGhxt|ah8lZ%YPZ|d`91G^V#IXk8z zXcgI>z5s3Dbme34UT{`bpUPiY=G&PN>|A1m#~o~)GW)wbeJG#9_;=UeQ0 z)SW%fu>H*_){N^D{O0P+>iYZ;xs^79DFFa2qd{h6nl#VfFxmN)1e{oKS@!jhh3xFS+4)tCMS*9#@=#~;P&cu~1oaA2g1dcj z9nQN%trDCn_Lj|m78p!w?Q5#3>g%tAt> zy2Z5uhIvHn8|XbFF=4(5_cJoZHbd*xDDn??pLT$_02IrGg8j+7rQ>47zTi4qU`)Kvs{iNE? zm+O_jwzhWqIufLG24VoXiWQ%bV7-;t6@IABEO~d3b^X(&Co1xN^Zj_75m{(Dw&ad3exOk1;@R2 zlJrdF6csUUkH5%#8>LnxQR5=8?CZ^1*(E_YX(^gdu>!z?h30Tr#a4+O)zx?_bwxn|_Huu0zs%jm zvxkoJ#`)luJ}ReX*Klaw#$FaPXuPn5!wTTckP&A>e3R{aW@?$z_W|AN=?FkueH8`$ zz=(~w@}={^kQQxo{yKM07-Grc}PN0AT=UB>#S8ZZU|X31Kq`N(mR>@N1^(*(7o)v%lntebMXv@8 zdSpll-EwF^25n4NGD}}>FweSt=lOfq@1<5|`s>P?hzYva*|7b*oJqgjdCM7_tIN|x z%IyR1#WKSk0)S3q0ZD|z=aefeA`ZeVm)ENknNKZ3mZ^!?+W9hUZEcrxW2L*sa+R(3 zB0dSqv35M_4$Vo9$coVFoDQN*QpC*&fW;lDCZ~8!nO>!cIMT0;WX7gY@z$&TeNIJQ zF;tbkw2bSie)#G2XRwn*dUbfe&0b|Ao|^-(`gLTaKTYiIibEteBuibJ`7}U3u|*5j zAEUk}8XW|pi{C1c4G;iPAt7O?p@~1W2effRVuM)fAq}fzvh1EANB%BC52&ehxk39Vr zPXY9RqrC?Yd!HiYM#*8THp7Fsw&2=}zt;mR>xnb{E+0H)BUOr`V)9UhC9`s1Oy~bv z_`M(aHd70v3lviY{eb*@!Q(u;*&r<0%#ZqLZ789cHK!OX9+7!JHe!TxL84clG#Wweh;#yR>k1!<-mbyh(}b$K2pn1P-vVy$rYlTBuh#Ke z%d~qrR%ZA$1DS5kZVf-aNY+Lv^hMo3&+q8h*}BC74zr>lsXt_{6ZJuG-ewpRCXPKx zb4mjdk97bZ#Xs>)I@AW&3-vo{21xpxM<9&W)`)VRH?px1U-4 zsBz=Hsg3a8(z6N2` zQS<8OibJ?cJYa%dyY5pGyb;j0t33&7@tja|c(8y+9WB`PN{1QoxkbuP#&`X}jd*orEqTthsRvgmi zU|PU~3(a+iz;2U$jC49bu;4Y4x8MP=8HsaUT^|nG2<7DF5|fe!3BR7q0`$_QUOBnb zK;L5EZ3mhHC7N!zNp$mm8!f0ddmj*G-$2!M?!WKB2uSFYJd$(?kG0QnMEz1@-P#w$ ztNe4;U2$Qd3&^ULDeViEi1w!-(y9l%?7 zbvZ)i5jX+r1#9qg>YVvy#a9xzOe3XLVIp~_^mZcLs!uLsa=e%B2R5OXF=J_hUmuT{noD zTsNU_^h*e`_mbAP}1d#>t5_P4MH0R*=spxpLlR7wGV~tl$+8E)8vNo#Rv5 z`vPIz&L;h0;fS&jprIB_R$BERoJ^T@>#Zm}^MCd**3oodj5iq%GiZ^doDFK{1J15+ z-F(h%(nJ=>D0>_Uhj1ed;23xm!i7-HHPOK?A7RKZ5@Y_wCa9 z$<#9Mc{mChO(QKdC)5i5wZy7F#&hkXZPwIw_SnlT-NNjtt;0c~ku31KTSoDOou-x+ zv(^_F%*~lX7hAu=9HlSjexzgD1?DY@Op~6g@*T#_w$z9@f5)!-_faNdO<%C8_n$sR zfapc!AOb@$4%WQ0Uuk+gAZ;}rfiUI7b%MvvjEx%A8s)V2cG7$B&7*FI`tPIQ0V7X+ zvGqC-gmjPj+-7TKWhDm}S3)7zKTBg$AukW?%7L$cX`=NNkVS9+)FPi)7>hGp($ua0 zD3zeMxA)&$2&49*S`c3e+-MYyN&>?=L{}FyLLLLL86~MH_B{*HF`xqL{=SzV8zu1dFAW%sD4)fW&+;V?bP3A?$8V*Ql zbMSD0aNU!INOfq!xwyD6)_#w8(GS5vEAk{y)Md4cnUDn%Avi2zd)_*(r6bbvbr`MhZjVE3Aj|ywqi+3D!&-T5ZL)g|(%hi^jt7G}glJ2kpZ{rk&UM|e`oXBes{h`< zYnKvwEkr>C&G^cM>fuYGw#zN9f4^h%)c0w@yixRwWBYvx_Z`49n=T^}2nPtJO3GNx znCx-HsNUunkRW${*Q$cHP!JAk0ks^F`Hnqo=$2;o-Fi9@iAw)ngcbUvl|1ix^83wt zmY6hC__doM&jA`#FcT18af?yw>}U<^9fmBpq45}JFhZ@pIfrxgi-W27b|@boA>aw2 z;Hyl8Kr2fZ?&Ug)3J@_fYfrSe2(5u+qD>+RVvfJ$MV=GY%^ zfnVqMsK<=R;L(aW zB!Wl&O+8&a<@ayh{p{=g2Hk{3SYiet=czC-~gN?V@vXTeLgH1gRp;N5; zZSLOt7u3|$$&zIn9H8f%_SuKhfZoxE49e9je)nm8AOUB3=V?tB6%@dywo5%`_s?>3 za^(Nj&!1}G@PDRV7&*22@%gbQ;G3&RlyH3Px^Z#|d>J`epeMZ8K+puDkgP@df;@Gz z{9W_v{Ggi4=H_$zN5$UvJrnWlY=o8v0h5uxIR1)qbv!!F?{IXqXjM)4q zWvwaF0GU7i^biN|>9FU8iOs?}n9er~ffC=N6J_!j{C?gv+?}0)b|*{}Rsn@)*Q1m< znl$9?^Hdm?PPe?P274)tSWm=SE>i@g}sZ$4q%K)ZdC&_I`ONr#i-&rU+8`1Japu zvk~q)6mJL+uNVQ=E%@Zj&q4mMguc#EXiCNK+(S2!>v@WJ-%Q@5^bf$MQYAtM-$exkNlWAu=4jRuSv_UADn5Oh(=QSce0>39rlI__@%+X?;N!cI*HA(?bFSQ9nxL<`0|gDQ}uRAwDyw)5b}2 zCVv3Fy>Gl9DY-%5b@nq$9_?56!SiByt2dq)o^dzW(DW1E#M^<0{4w~X(u@)wI?DdX z-$`njzNZFn=9_vTo3UK7C@Uf28O?0_NKHhWR5J&Pag=c7K2DRP#Hr3Qz zx+NV!gwkI_C<@1ZP0cWIpzHS=tU81^K=#7v15oo&v|9`U8d#kv+n`TV_)#`DH&bN% z+ggS+6Bg&ULtlsCO)8gaY@C}2lQ=&+#xB)6{sK)={*=A1@15)@Cx_DHz8e7)t89<` zd1T*zZ&JSYg(k60FT&>8qae=HEy|(p$}!JK2!!YyISX)e^Jv2B+K?!F*wMq=^!alV zQ?E_dr|IJ4lFOfIzv=BL{PMQ0P^%T@kmT?IA(EIKh9WF8$+&UPRP=OIfe`5wfgTwE zcqa>R2Wo}N{H#LjN_D}yF~PbyLgY~9hH6z)uj>h=&mr5BrQJ<#+b_1Fpt?jHCI+~^ z>{3sC#g>a6Alda1_?Sp!Zy@rM)iy(2K$N!zpVYkgYC+!r!JCBO~sE=ZP0cnx932%c%)1UVN?BUc1j8!E$- zKU|8l$D#$?Y0eYl2ab|muZ*U6?U%kC$<7q*l~7gxisEa(Cb_vh3HtS>FBpgP#a4Wm zW`#InsXmfeh$(;GHu4i@M!-XqK2xO?-;ZHP3qVIl$Bka~@#D^|iDCiUVY-!-m9OJW zA+*5GSxF~J@k%w{aEL=nE@&)WjmErBeO^PwhjxOQ6hb>}suzKbUUHNMygCa&AF0G# zSb*jTU<;Xg(Ui8rD;=npcFk;QTomZsi%5ws%7pp=v*kc{a=790DDy4~aC&N>FSH%B z94_~Fueww5hf;mbx5*b%v=Mh#>ieMX`Vpfo&pB%j~f zvI8=E$@~#e9W7gq0}KnczB_QaiKvE8Zc!2LL8v?S3~6E00M%EczxOr1+a@&We*;se@o)g$o^Us0n;rN--TVG`-cM5>5g>;R_E8Y#w?F-+ z^?qA?v<@hu#^T)g(ZDXgZBms1nhZIE9z?D~d>Ge0^>=LyQt)oxYr-omEIi*fyTffY zzDr%;NM>2@s6%5NA*Yk%l8K_P;)R9vyb`mI4)FiKt*R^ zXSX`rnI7nLKzD+z{K-5W*20!kz6M6;BovTr_`H zjqbyr@;|U^XI9yerbLNsXEg?uB~@V1pZh%ky^jJT_svY5+8*yF=XIW6Vi||x>`H^U zNF0eUI$Cg9IB+q57x77I3}0Ags-t=~DCt>w&yPG+0joazr)i?Zii(OZ^^cQ@T(LA; zN~qSxY+lYds~TReQZc%k6p&~mGD%~n|McZ7{lq;MpU0E&IYKk6c^Lqh^YVnAS;*9O zQu7dk5+Cz$#Q*lFNC5$aqRoNl8$Vq6=Pd0$?{WO#m)&UkQ9H$SNEkl)`PG8{UWe9P zX4i1Jhvih63DO7ulIa(LIp6%GeH5^Tm@<>*rF4(^#vMPtb)B+ebTZk&R*;Q9z>YVO zCZgikQ;%N%T7y8AIhmz>RuqtKF~FYUpqYe%?G{uiO90aX%acZqy;{$tLMY9AhylKq zjilh=y}B++*KxYpke^)jw)V!BXQjt#{re!ueEiiP&#uYto`n0 zU**}ee&A^cy!ydHC+$PE+!q}ruX)zrn3IDVEBP1u`Sa(ajc;uwB~L8q{I5jc{CuO) z3zFDG0ux)9#@ALV{sW8LjBezT47pW`7uCHN_dAeQY^?MpHqyTR`AjAnIK7QV1Y#i} zA^KG>aDa^*lEkHFdAQUo>UGRP4=jGKPLy%Xq0;*K2&78}ymO>wtm{`nvRoGxl*8)3 zvw69|9!`^#ML{GbN%I&lfv5yO$H|$=A+nLcsNSI$Y~6xO<7lh5$2EXJeINrtIb3xx zcqE-k2oQhIoX4MNFeZ|fl?6c#2AuBBI8T{)mE`yt%buQ z_i{KaM>3=UR+0wp#o_PQ<%`1-`!G{edfgKJ06;lRYD3!xV|N^@G?&X{ST+tl;}R0y zG8=wjL!f1|aH&NAKbx(ib|fU;<4Kldd6;zxMgp5!*XPFxPS@woC5E-Q0G2BQDraU4bCK&dViC1A8A0?@KPVycY%O3f0QaoYJzzgtRC{pY>t>oeVK-K!Q%rjuQve)>xSe4>+VW>p-2jweMB)(%bKtH?3?dN= zm_iHeXYi-{*U*W8=i*lVDNR zvE5yK6kV&%CMm1=8XQ?aH~B%+7W*TuNCZZ!qK>}z(R$Nbm^msh#>?ybSp{S_qd?*^ zGGYL!x?Uvomj&ZXNb2izv6`&k;KT%G3og(0W5HMj5}9s{7m|+K)ei0#eCS@31+HIJ zRaJ%enZwRpGo^U?JunE;4M0?d%jy%*3xQAs22Hv?uEHk9ILEaoJMnZQow)n@y!}7rX+tDOpvEjJIZO8SVfF zYfc|b?xW`gQ!(&ytUyVcu6JYr^gc@Ki+VO7n4m%_*x7MD*V00o-C)c|A$QF9%oHsE z5)ly*mz*31PBS}S2VOo0H#ZElPG_Js)G%*-A^0D7UxOpfC1ddX#7noWqFZHno88JahK zN3|ZGDEIYjH3i@hDNrJ3o`A^awA6zG3=2%mAe8X%a1|95Sz}|mrKKhFN_}vYc}|kx zV_MYDfJ0MPH#9z;B=N%sl3wn~1@7>)E)^_E5bU{B8H?BQy$HNYEe+ijDi9inH z=ZFo79AYVs%ZNV%!ZZX(0s?Bnq<0A*_x&;VkrQ^fczDixvtr1xyR58*h@YVta&R#8 zHesg-B9oYbA-17G6wKOnzY*+Ss_rn`X!k5#f)jT7V`!!u-$aIHBDyfZXkcI#C9-O3 z)+=UPs$mp)=yrRRlV8SMni+q8^N3kkSm%exBTX`54!jP%gUra^V$$brr`cO?$We#T zM(cjagYn3=R{AX#;Y5Ka*too+itocwl8Dw9Ruq34zvOYEu*pB z@Z5F}v{Z$kqVdXHvuOY24*En7vW!RW0i&17Loh`IP>$8j%1ML&+>KKc7*GW&Ew@1x zI?zs#pctkZ+&)TZmQ6V9FR@s9+@-e=1g6Q7)O0+~3-f3=@%$wbWx+B`WT%NXtd{$c zy?@ES^9A?617=G3?WWFP#K#a>If$Gi(Wf;U^isJEhaXIK=420f9<~&yv7n#@LzVV6 zXI1GbnR4<~3F6sV&Gh?Pz}{-E74*G{BQ~K$z5y&8E70vgzJ!gvgn^NT!=*?*%ds2)OO(jo zd#XPpwg!WRU>FBJYfmJMBZ-Eb_x|5TdjH$u@c)^^{-43~|MTAr+%jlzuq2V|s#Jo% Pkbpc^REL$xn+N^}Ie?7j9{YtA+2H^2EAt)-z%h)075fj|f!svvYA5VS3eft-n!4Hu@A4+2m15-^IyXhaN#Pr z6G7-Y1~3?Bi5fZV?MDv$dq405vg%4>pH-FL;(z9}ujfaPiPXdE!y{py7-Kk(eeJ(wv zp6KCIkdx18E0p&5qCOjDXT@|(2n6Cb7EK<3$mN%fic-0Q4J9FoZig_#;o4N-%e#0O zVNp>Ciy#B+Z(@nnK9#0rk7?E=}PRn+FW918qg#3w3%iE;CK$x@ED*Vi^2E#&3u%O!%VLb+t z31=OXR@>u6b?#fF@(Kz94%7GhQw8!?h2(=E9#X>Sr)UKcUE(L{&L71sG{5F;%9Hf_ z4U~VHVJ3b}iKCicdDjrkIH+E}6T-Z(a^!6;VHbMD&7ymz7nMm)sAzb^wj33kkkBNqaxsRYL5#V$WiZ$t>l(5cXRPtFajotN4jtTopI4ZbZ^01T?W4FhxT{7;B$e;I zg??&1fznqH6Z}IY7UI(f#LF|P8KJ}Xatg1Xnn2MeuGyVlykHX)6iiG`j_B`KUtM2sx4GZ`qN>g2 zK02iTg%BMP4ut9MT};6=0-AX2r=KZ&n;F1Q7Z;bPxH!b_tBR<78p3+x0HZ$hSwf|6 z5oq$JeZOM~>$5iG$@Anl7ZgaQ`E}iS^G2c`f6`bw-5L|~+NTTnduH(GsE4u6|5DKX z0ex;>9tk~td{h)PEFz-aoWIqaKU*#o8-|JYAcKhO9iHrwt%GANt$qnP`270yOGQ(& z=l9-BQd(NH%*_RcxVU(sQSH6`=5y2gj&$KffpVImF z?vr%sS2Wgx>2k)#bkkp7Mkdfph?gu#Dii8D5LZZFo$X`1^ z2u|AX9Z#_@66R92;VTUoAv&$+zZN?c((yiT~ac-w88bn|8v9akL&6{ef+ zbL72?N+WyBCK;#1GXLt_8mrn@e%{g z_Vog@Ma=^EB)=Y4CgyYtPoVD+-0*o0bTrc#nZ z`_!d=?)1CXyQKaY$I1k-hMnD%J(WNq`pd>+5&>p(3#)cjDgIiw4UIo*e1UQY^UWlT zjL4IXA*+G+!ke@8RbeK6?C(^=mpUR>dXreZ&*!`uUhc$gs$G@3@BIbMRr))pG;R1Mk$~*x7nimA#40$>6SXSXsWWW5zZAox z^4B!``Z>XKKYm#KZ$w$E7j(MP1Vef zsg^*Q2KTKHYM~d4mm66UTRP$NRL&=GbpVPJZ=I2$*4lVIQwb(ULn z9G;G8u5?C|c^`jgXMS^(V&Snl*C?96AQf}8(z9^PHOM~U(Cqr$b z{U(y0`a~$PR-71^0wyS(zZypr+Z6)5G)EzlXmdCRBj9>ZRPt>6aaRX+(bMm1u5onY zmE-#*uDG%nbTXp_pCjuC0*UE9Wxfh!H zLTa|3Y-Gc&0j1!;0RD7(I-PpD7)c}%eY-N_E7a*znc~$$fOPLKFIGwMT#n>p5Lfbf zPT_W4m}2ecLv`lS8$QCCHzMO{rZ_p85y^PS&qlS&Igt$V_Vzp)nUdcR4_&u??ZSu% z6@Okoab9z?rqH7=lP&y_POY&Tkc6+CjFF=mCO_76YR~pzAcE4&p9HvwlzypU@O-nM z+BoOjJqKD|x+0=y_kJKUs6CiCe2n39G-a}z$+P;Rab~nh(lYdD@fa{6iI3{u=~8|a z(b3W6Pd|&fEMq?K+~v@&y{qrMV9v3_k#zDC>;<2}AFhY@s1RkpD{9lnk4B@!=^QoJiU zv}caniBH7(0TUOloc&O8F9n~k<&U6<{ZtUWglG2izGPB&)rp5%cu}cT!KXf3S_qEp zmt}7nSdT3UU|d~ETe9=rl44&TWCt3XEv23{HA$l48c`$$4lx75rtsqA$M?qG&^6T4 z{pW%?aU9ZXiG`9NjYIArpVaPEwl(zNIZ*OeOI4H4t2BG=(L@kXhkP+K|AH!WS%_9(d3G?&Ye8=WkBNm4Ap;D+9#_b4?PV%D_~o@`Bn7z({Qbm znR=!qHDB}TsL28HEdR6h)gb1M1$XWXgxtq(K7HdxS5{KCJ4bB3yoVrqtwghD%7S!r zv|BZ_ygg9@3CitL?DZDHz-q*UNwU@Tr=H2EJ$xvKWRT(yy!oTr5&B2DmxWnt(tyQU zk*GEP$tkj0c>PX=t>XLYHr$N>TJj)sy{l+A{%WQ!Fzndz2I>I zGO5LQp*AX6GZ$x_A4Hv$K8QFJH%c7~{CJ=mh9p$}U1?ueDJ~RC#T4*} z?Ty9ZV%zh<2q*zTXVs9j(U!FrNX-C}p)#J$j8xFS7EKENw|(5jJ7>h zfD6ggx^L;6?$6V5)#J6bwLziKR$vvRb*9h9_8~ec-#Z|-bOAsW-PzemvzzQMr*+Q6 zLCcxc%tOl{TKi(+#ccT620*0xl2z#Sfdk;n_9I!BpHBr>@6#Y^UXzdE6k8q1)I`doZPm~K0-=6h}~uJ#qAOacgl`VCEr z?x8RyCRf+H$2F@thdf>{CX~P`t)*-f&3jsnzByf;Zbc;|DCy{o&N;>WiGUjiA093B znr0GOe|)^{Sr{Q8`zN($DR`|AYx?(^=6Lgx+3gi&z~vex>L4c*lau~^(%OtROgzqF zrHwtSzu#Z{2w0#WAoyZ2@;ebR=c@iE1EQ!kOW*!+v-$cChfrG)osl4{CY<7lTIHgP ztF^vE8|27p9plC316}{IKR@1adtYsRI{Iaum7|gIKzNG!k_DtJ0Ek>=`rb8=7c+we ztku+MyXQa~ey-uDSXeM)6EnmanEGI4Jn$%9+VADVm7SX6pDLTuE-#VYb}Yo^m;Lj0 zEfMQI1r2oj_`p{3_RmS(`MlrJtlQ9pv-RrGGc-YjdQ8xk5m{-AVS!`hz94o~R+FF@6TLWXIzZF_~(0;|J0h7ijvcFen z3M~PD`Jex|6Wx$E-B~$-eyu2R`sNgNXBNKi1Rltm0A8W;*P08?ru5e5N)n(M6ymO3 z2rydrAvXK_t#o7VOp<}vHF-l;3ks#BU-$T1F`oelmE^1nv+@*)hE^|=MEOuyou~35#U7s zRZ?luVwuuFU(5!KW+`P6-jLZazWExIGD~?#>2?5(nJlakLE8-E?Mh+HFCS7E`$&>b zG6H)=on?O=uVMtK{|(!t_9mYbvEs0%keaU)!6xWQX8{zFo`e0z9$?fZF6W#>dFJZR zjzkD!u)uLUVWoH)RxoS<);VrOZUXu*?n>>hN7<_dV9M=RRpisj)2~NZZEbB+CA{t> z^E_!cF<8~Zl#`w6<*z$7=66^d8KR)^p0jx-C_x0)a&nT{fIDPnT5r3+F>Y|{lb#htJMd5>ET##Lo zJ17w9aS#ZPaUINKd%VpGDaGbujnK=p{SsVLLO~rXSkCQ8DdI4PS#O*GFuyqnfI79x z@63ZP>T*vM6R~=;WAvDXgvgwlj<7xVrdd6V5VeXkk76y#h!uVm-7$=bIf%CZRf%O( zW8W4-&V6@3fvcUBhw;2cbjZwaPZ)qFQcxJD)N$v|oPh9Rc~IZXx`YM+)Q?%fB`c^; z*(|JesZE0~_>mnLJ1*Ro46$wXZX4wRe|^)%-ITU#q8w+2WNzcy+uH@p8s{A8r2VM? zrb*7`ZXm~wBCg6PmoqUVtT#+%;$vdtWE{K@yW7B8I~SF4&RCqkqzbd87-@ zTUFMpvmJKS_-aRN4akbDT7_Y#_fB5zVK=SQVJR{EH9?uIf81pdwaoWGK;bDZe0v3= zT$G&vuva2&p4$B77QOlhk$3|}#9dBs4F(LD(2xL^;-<&ICP-|chZnC=F-dp2wuW>? zQd+I|r)hC`AE_WlLFR#a1=F$4WWS+{57tLX(QxGn(jE?${z z(=2gGK#&avLZU|rtXYuG#{N{~WwghwNo2CaQ-DlJw5cV@=G1mx zeZAoBM091l)ecVR9O(ZGkL_5?kW$_>-jfv&SCQAFI)_c79qvswhi@X|Lw|j5uVZ@S zr|KLY>LS*1c=C{0LeB%Kh{Z&6)KZIKVEXYmhRq(r(xyih;~6j#y%@A*R#Y*wJ6lh- z5hzXksV92yZ(oa#24a<$ipkXc=FcSx9Dwkj?#VvuY=!z=`c=dDXE6P!bJ@F|fKR&1 z%Gzf46~)UfxwfvkGIMg`S`THuYmkEH;t9y!O@93v9oh33K(v{7P=~7w18!u3U-qv4 zNFqWGCMKH{)kK4Vurz6K=M?1tp>kueDM;9n z&}KA0dVCKxfyVNtR&eM>S^{rhJ2cUXI4%j?iV0TM_ReYG-CoX# zAYK{^&-#}Q1-oIldsuOCK}Mu|7k|yif|n`g{8UX}WPVyn{VbU~s|#D} z+TZddX85*YQEFVH^Uss!LEZv#oaTQPj|{8Lwd=|JqNDOVAxExKN@qdcpW6mABntl3 z0aG4}>RUZ4N%B4)@cQ>_WSD4qO2~GA)x71INqkU|5V}Jbk#kPml^Y)WkMCk)bm9P# zg=*4+PWt*(Xmxc}1^6Ec1UqUdYKI|Gn)hn4gIMNm134THA%JC(T*xvhO)|!J#ZrX6 z!HDXm<(Ksj<57tS!y$DJY`(7WJ+lLt9u5vJsR<4}TWwaH&&hh*mOIL+CtL|&jJXa4 zinKno>rz`!@0dGBB$Sf}60MDNoOia5@4prRc9k-9tM^BJqq zm0}DW7GlfayT7Jtofw}9-nZebs6MEK?cPqFf;O60;~5wvR69@Uojrjz$DNDiSia$AKd-V)2 zM;5{K=oaEOa+a2GQZlj%8=7nm&OS+1NdP_3o=jVgtriX>m8EH#;pQFJT(;ySy z>DwjSyzdTgCZ)VIr>H0;QDk2K4nT_a_4TCg2t5FpEku<*7!nx`ms=&-dU;WqJbavRI754(B%*Pc5D?&SD;9y5KY5xC+wrF=tRg{d(M$74 zA@9TH>G-nYX)*TmQDWa3>UjXgc|2a~g31jEoV(%o14E*x0D4&bc*}Wlv4X6y9b+S) z;+Nh29p(7l^9S~hBlq@D5ppSFiRxzDxyUG&Yne2eaoA(L&9@g1IfD@{A6z@{iywG< zz2I}o>m#N79z&el4JOEj=^ARh?YSue0C_iEw_K2q9s_0 zQ)hOS)|b_(ZtQ|8zv+M)4-)|Ui-iwckaC+5IFD%)6(HG@%DHooedrjT-5U)15=zv7 z<6e6M@nHpb4%dveU&WGV<}M4sMn*_2wtUNqZv7HGeAg!5 zF|wEhqjRW`rf3mRhAl7n&V1$N(LbuEg}AJAyBJVRPyxCj{hhd$VhQXR>ScUM3q~ga zZh+%n&cNeWH_lzwgkJS44pp~=3>JzAItaGey2ITOh2x)xOztR+DR>OEIx>=E+$bAR1q+}WWM_1^LD86U zBngvY9s8c`m>;p0iUq<$-yh@8`YR=6-{e7%n69=BrQm(4GR%5OPMmn+LZo}93mD0} z9-_J!05SA%8Fixv)zyjdn0=!Ir1wf8S%XzKU%d0oER?>R-ry^UBuLw*cILMorgqie zv!t#z^HRlJ;F+|1 zTl=Dl0Uc?32tdVz$r|w%UN?xIe_VKNdF}<3#FfR%mz}cu`F<_l9F64D|L|y%t_Yf_%y%DM<%t z`r;+W@78s}{RL^zIvGf?$&dZxyy;*Arhq}q3iJL*t6_KJ$L`zXxJye*;MnSiA29nL5B*WMYLf9?1c-=I2(pbR_?GNGG*eqy0MX0UJk>j z;3;%ZD+a01u+ftjMF)bG#!vIwReU^8idL646kqcE#b@okc0gi)YSrx3897jHs5Lu^ zd^l>uVbs0)wLjO0>c~*>n?nKsH_-Tt?GiRqCX)J1nE9D8^cl^~k@n1yK@ek=A&+yt z@!AcKkkCnT)@`5JO$cZWa#3X0 zIVh3jOD5!DF_XohZ-$P6kt8XG;%_a)&Fj}XLIE$)0q7V3=SA7w>8~Yh(`P#9j(Y-ctFyL=`Q{7i7wa|+PAX~a*dh`wTm8^8<$!m_UN>4LOGkRB!FH; zn{`$UZ0AmVssY0U{S0&OF39t0(xTgxn5F@rL$M!bKkeq5#DDM4#ZHtMpnCky54ah| zRlW^?taNfu2+pP~`*EF-T$f@9@kz&6iL!LT(4xgWa$lPseL5wuWBi$*#cZ96vlyLW zg(VgNwzSr|X8L4CuhDCqze2to%XM0EYh;pobS>L_))$qg6WioB`p^|hCBSh~W%A3C zmvEg>r}TxR&Kj+VeXD`lt8lXxN+J3yE9Jh|$LP1B2`a+Q$rTUI zcifBLAc$KbnqXpVIEtk$?F}x;J{4jGs2hbDUcY%WQe~|PLW6OxV^M=Qh~B*4<;_2} z;&p^>0^Nn4I>epcJ<5j{yAnn{`g$z!)(3CxpC~pk$#P6_fi6`g?4qs|g#M zr#6Y8Q31pP?GBCN6a`KT-%vmhg~>n!T?FAfvD~QPT`sp`ugQnlsD~tpIBwnKdh{4! z2$EIw5I@X3M^_U_>zP3r%Wb5;_iWd>GTPUxb%8rlgavFVf6+21q zHGLmHOp>#zDvdi{3XJ=OzGcz;V z?ag^h+eSiDXwQ<|@7MO2=g#= zjYzD2iVCdRGXQH&7P(b27B#x!_wWIX7M7Ff;t| z5iO5V4FQlxy!$p;*9NWaiXUFe5gw<%WC4C7A;&5^qY#@K*xl;jnnF!`3pjhrp-icZ zp}?C^EPM)7dl=PF-Suqj%((stYRGKR^+6$t*RNlfTMsdGc6aMv1OOCTs$2>U6rl?V zk@%t_i4C0-c5S9ZweSRTyzTw3n%wW0H2B=+0Z9zXk`#QVal$WuqSPGTEAgU^m$lYh zHC8=D?mIsT0g2ePIg;mcy!w!UPON-%=#B|b23ktj1{~=l5Xbc|I_vt@kuD?MhJ8 zd(BlGWpGp)K3eV)b6tgsIL*faxV`)zL?ri28nS`GwK)7YQhuX|OR#eP>%H0wkV}tR zF^EwjEyE^n0XQ7a^(TF1Jpzcbu015^CY^4|vhTo>=z}58A#{VI)^j*OECG_6*CnmC z!;(ZxUvJ>|pw_|Q;hGURklKuzswp^UnPYn4-}6e+V|(+?IPESwI}51sKs#Lt^cpf> z{eFV#jRSI22{ep{znDl=+m2y_18gc_@pH?4dA#T`sx$xlU;zn4ItpxWazN4Jy&cJ<5lr|x2d0!-C;y`LHo)Yz2wj8X3Ga(VpRxZV{D2*Iz&id-`&Df-f)4UyraW=_6OFo26K7+O)^|%GKyu`a1{7Qz z``WV9pTi_oi4;rGLsb7vjKwc3+_+{%HGu_KOO<`uaPiS9Z-k_oS;MQ=zQX0> zH-b-}RQ@^LxGBcishFQD1ahonJJIg$Mw-S&4VRUx^qt}JF_!Yj=HY#!wcLV&q)ACh zHY2&=n>2v>cz`4tywASd@++xUia6mu&hD0IO4eH!)y?Vc?A=eL?w8TM2K=(U?7bn+ z`Q!MrH&sdEvS=;lUq&q9VvNVz1w83~ycsOjL1g{DYPGoCe z2h~g8PDlp3K@FsfF%6kC+CsTNQK`YWW#J7& zR1+1rJ%I^2Tr(j*CXX@Z#g;$th?^mHR04Cx)4l^7Czq{JR-hcT1eF(XV5Ky1?ZEYZ zHwr$Pg6XA;4wsoeO;0j*+{b$zKyArdh`D)9xj-!7aNlAjHJ{(3G0Xvn^fHTK*YAOp?%#Loc}VIxqJHq{|z5H;kX zXgc{RkxC;<4z}*!rLH#1!d-ouw^i5CQN;;p@&PODXZYkd{XeH78bWvE8Q3D3p!`6$ zO@#ZUZ9nP{@u#EL3pQZ(^=0_3Xf7rz*-!)TE-^BAU>k(FMZoX3Jtj7GvhSl$V-_HH z5D?`y27xhI1O!M>jdYO1z_H)2-(cwfAW96Vt=A30@Rm+Lrd0!!x(?L8YIE4ypIf>A zYjW_Vv`v)a9ObYm;?zn?8!)Ge|0QYqe^23^8La`{a;Wb9FR%ScrU&Zz|3m!#-+RM< lF022a1^+)OfV(m){)2>t_A#3Kj@{_t1q^wS!Q zY`GfiD3QTOkt-k<5ur+kiA_u#*#TjO!E~s=f3iZ+LLwp% zrU696iEyFD1wN5e4LAZpcKs56Vc=6!6W#Lia_lMAH-?=`@MCN;wMs_uzh3~42nHhw z{eQgM|LI~+c1h%2gh!wK>+!T6%V?SsgDw7_zVu(7=l|p4SGlC%F4CkK|D0<431Z^E zANYR`k$)elyv4>)#^z)x?XkCvL9KnP*U|6xjiUz(6gt|E)D|E*<}-~R{Co8q3{Ok- zA#E2&8;AaPXZ~;=o~{^bVb;Ak__-sxg8Zt3{}rvRm|=A<7Z=yTkko#5;MsSQQ45m* z?GXvTYtK^o<)ceSu~i=FZU&qN7}WS{%4l{oW$@%0^r| zlmV_WVBNkV>d72}lh9AaGwuV{%a+i*r=4aDtyhkl?yuD5C@TNO$c~jfss_({H^R16 z#6Tg9DorWlnBmd>ZbzXz6PluxX8O*MTXDU$JvZ)&Vn%iuR6N(D6Ot3K#Cw?QsKjh^ zj}IYB_iPufxv)R^Yf(0~JT^q`d6{wnSJm>E$)R=MW4cmp6Z3$C7o;Ci2gE+)u;!r) z2AdPfh|vhSt@<5MP*A|c#VINIjO66yQFtgGPL%)s>zw@Vot_~B&xh~DWmzF%VYy1t z6arLus_NhXZD+FXbk?TWZ^5w>UzIvB2(IzCbj)PPMH8D3rHG1LnwX)l?5#X~d05dqXq0y7 z=9zu%+_Iwt%}JLeEZ|k{TSn(Bx6ZHasHsL_BUlPK>P=1i_sGe~C0sX&2pFUz{`|3h z`RY|3sKUE*wdJRQziR9ZtE?xW2BkHli@}}CZ8P6q9fD|M0%ysg`BBc@T`Z1XDonrB zV7Z6Z0by;;<$baZLOrBt3dbEKQ<`v|a{gfL&C=jmNBRziNC1dDr4rO4baFCYGVL1e zNhy5yl`b6h6af^KI64%F>();UTnZk!Z~}Uoo0~^RU7#EfJ8XLm8Yh?X2jA4u_h7|f zZxN&R@jV~56*C{sl$ox!p%fAla=bi#jr=kvC#ULz2kjc-3n5~5vqnz-C$Ft!YUg50 zN_bvcTEc{dNgF-(F(JoWV^|&wZj_g2`-FL;PxCVKf1sUR47{#bEdS;SmC;7Pxl}kK z-%;-BIdQ>Pli~dQz1363(JLz}MT4yC&i8V2a~-$G3%ULFZx4<-yniBl)C0`}Z200M zf`?%kIlcKyqb;wEVeSHi#zlTWk@-Jr~UNhORmFW%hM_=;s*C!1G|C# ztb)f}uKT~}7aBc!LvTnT5K!ql-}4$)yzD25p%o`Ue%=;{CTRJC3ZbHc21zDXgR>Ns zWB!xib;|=z?HQ|K7j@KSbd+n>_gWgx$^Vd_T1ydM`N)E{?Q=9JP9J~ntD*FYR)t=bSI;YhoSIJ7>EBBJ8m8jn=2Nl3;5OiKso%dMu9oadVFy@&0p7Xjq z@Qd%s*&gR<_JKpFV}p&RhE#^;M=6iJq83JLSt=%09FI6!wk@^;gQf@?Y7D#Z*>XiX zSs$wJ?|D0m^okQwQX+B`!_D@8HP`Jm?4;GLe#sfN60jO!0>`INs&78_MdN9`GaI`` zT1X;`irLRk4?g(p55Usv4o$7J!15_Sa6E z=x8|73@R(z*3=o(MD3MO_%gzug(1waIJ{Tu{!NGqqc7Q%kqbUD@#li;Jy&DJ zM;hF0w!F{Kcnxd1N}hc?*eP#?@$%x2nQryJ|I7c*m&c$An_kMj>;Bz^mZ$VYaJ!X= z+1`Bp`N`Ca#CPvPy%>iT8##Biub&Cd*Y8Q*qUVO0gB!1M9%noHF{|l+Xpj?c;JU#0I^qm6%ZJoYBznqpU9tfAcZP?NQd9Nly$l zm8f;{;;(Yz2uwnTn8y7UTAQgdC3SW6=3)FsDZD*YaW(y9rtQiC_U(iq{JWD!d5*U$ zh+D&pg6S85n}?&=v<&*>450X?h}qG2A8!UGCzE}Ya6&cji81vw!f>vq>kKDsUGo(b zO2yO3%Uj}e>};BtM=a=6EMte1MdVKh|2+BZnBmqgF^>IQoc^T{HQoBVhbpzUGeJ(P zJ7`G!6>4AAJhU^hVqg?W295>UcyXN{+bjyUX}pm0ZT+ zmO<|a4{+GEGGlzNPEGrh*fyuivqBagJts4Tc1hC*2N6PvNXgOMii~a;`oG2(N6QhG*MLzjc$-km6n4v znLdE4TGHx?sjEKwETg{?7iJ@t)Fn1l&q$vSqMg#y;((ydogHwT+C zmq<<9U79Tkh>&!eJ-#oeviTKGSLU zv~@4u)x&>2^DDK&B%qfJS?&y91x>ySGyuojYcEK^O#!`^@X?aQc7Rh-Wslg7e~rh! zF|rGsUoGAlJ9ugzVDHjHH4O2iK2V^|dnDe5xDd+QgB}7BhwP8L6uNvHDBTs#-D23YiXbU%sl<1;ZuIT`HvLA=_W7X&OGX*ehUE^)1ci5 zeDXKkpX|f0vC&u~-hKFx_vzE8o>k*4%4Zwjm~@RVl`A#U#Y{mvK{moV^;+t;uPq`{ zse-@q4=i1$M%x#VfbUV`g-O<`w~l)eLk_wrUzl8n3&{5_#rLj>M zO@etsy~-p7<$}@_a@s0PJF_SBTR(J~2$-yNMV_>{pY1JBNxAcg*-i(5T@=r-V-RUb z;dkughkHd*%Qk9lyQ`YvB{0=GE-TC@dSi`4MLRk>k!=t>@u$z9Coa-Jg;>oAXXr>1 zwQ&S(Ufi^n3>y@)?0Vzi+-@a^@TX_1TqR8TQ#+N;D{5qM@DS0ND3Dw*5s|8)jVw@X zg0TpD2HtUdwSU|u6JkXl^1r9^wG2J66ZY9YoV9~kwSp94^M%}Q&pNp5dE+E40|o{L zmC#G|C(&_~icZa+}QAd@!;< z|Cz4T7^lY&N>#=jLnr-V_`~s|qv`^|g8&qgwGylnfn$;JE4r-VaE?q*)C?()9#F{B zzX(F1prfm=YQPN)&^-bTx7`aO!C|H>k(gYKV%}(+IP}Z7{8lF;BQyEclK+uu5&k6>GfF@yWBB6wXt)xfDjvg{k{;(5xnVy&rW6-7 z;-hi-6b>B6N`42SkaTeoJbs+X^fcVOH_jC78yP^Lil$cxLzS1$LKKN}+HW1E#VFM2dntstPlFGL!*6=a!nqvwHVDq6 znti(wk5R=hWxlP&8<}qiWZs;1fl`oOCk(X|dWZMrko~FqQ72(XmAu50-J9sEeT~8ajsRYv-(C3k^^OdWUyshMy~X7%Q2<+;cSR7BxG4{2NFxY9jHyi}cX+*@G782R zM#F*Cu!vdR<_1b1$M0hRs?q*;{kzjzA2E&;3o0Po2pNjdj?hr6LuXug?DaX0D_tOZ z;{E-m9rwn=&Aj-P=Cfa`DC%B5sK?>M>d#U7`Ai6qiPCG7&NgK)YkCfw58U>B%74`A0#O=_FTq@lwv7Lmqx^;L<+*6zkr@S13QiY zJ8nyRU24?8jrLKKqbHU&91t$KyiwH!8lC3kO0|@pG!tXXu>39-H!0WDx;VZz99a}{ zEG{A`HVVkWayVnIVmJX37j$J1i^hCu#=Mj|>B~xwkC~*NKDfXHLW(-S^p)-B0mLft3`*YKlA0eSnB5E!A9evcsF<_KG=XEIw%pym0Towu zg#0-aeCLN>mvOaQf7Rt)GwTx^b$e#eL`<%|a!p6l_^z@YI=;$D#rjD$(nL(}w5mweu~x0b>s z`rJqoQfLTkfPq8p1}qR6ws0~iU@7v&6KV?-6Tz=?u?mtr9%eJm z$v!e7jry~Gv7Pn+)_3ZbcJ1MeRprW3sz$$s(1QaJ4~ssWXj29;yqNUf`BKyP{9Ko7I-kwC>^3epWYQQ`hI=t-%YnE^wt z{KE;IM9x;lR$@sO@?kFY&0f9AwtGZDs(Ao9(GChX1%p8Zr|uUyzKs|3#1Epg6i`5_ z!FIxVJ~la$0F0anmWE-fW>?TxEx#ShT1w>{J}5AN{l#sHU|`a;lL;-oZ`a+yPpP&5 zB_@W%$)N^G^~Qc)L;{^1pMtQ)zP2rLn8`kz-RN5{`R4>**zeS`P{e>hzS9? z>y#z33aeFEOuC5Xwthr}^%B{8Mycu(veS(}y=fd%s$e@h_;i8imEE`$tL=@9A?+p! z4dk#~8c+6yEawjmqxCMYN^YEaB|Ua|z%e0257b@nm)2f(PN>E_L~QRC#y;AA^e+vb zua+Ud;(Gk2-owPxDUrfCCRh_(z&3Zh6z+sPR%_;)%#9Io9_X8L>bf>a?%zwEWh-?$o|>ghZEp zLqqQyWMH`f2C>$>IkU$O(%#wMo%xm@*~kpDmiP~MAaUBRb^$Y0U4LhP!F#7vV;-Ma zrxSc*vQU>pJzWf6GJ^m_6f_hST8|2-zaoJ9pX}o!3N(#E*RhPf*2(ecCCNbXYujSf zG9T;FFbY&c_{F2Fa7emt1_1ObM}+Qkb1~I7of0T}c9z({3pV5g{XPsHCx%8EnsvQO zT4=}&E230k@`LYw7BQMzH%NY;+(Gp^8N^wxffM>lxLJ90x81Ho=U>7P50P&%gNn&- z`%ou-b!=GnT?76(xvBEcisZwHGxKkZsj-U(v=k)C6Uu`#*5%{Sv51(MWjB=9UThD8=(RSLlx;uD)?!K$me)e1sOQ#0K*9=JW>#lbP|y@J{PQWfe>GyVX6is zg!hepenn|CwJnLPc>u2?K;LF%mQg%VkAHPAY3R_tdWg&~Q5U=fNbXLFqPU)TS8tE` zLW3Lf;z2qz)_}gpLYs|Sf)*oW*EjqA8>j3DR!*?C;IYV(i9!L9WD@_2Z0k$(rnzZodIIHsni3Z9;IKon*NstDAu#`ZA%_GvjgfKc#M zV@!=V_;f0CNL+LP2sTJEV16XRUkzt=gAQXNmab!LOz>?c28deH$0Le;!^0mMWQx$h z8B%-GXgg)2iZ8g5B4rQ4`)gjzWGU`@^Uroxfy~h8zH2>eS4^V zr~lr|)9XIwZhPrNN(RLU!>0`c3ZmKq%|tLu0)*pjzsaVKmmoc)C!R5WWyQ>Px-1j*_FweEY9ehnfJd-KhvsgxC(PXOR>peu$K>gNC5 zH2nK!=$mG836H#|fy5f79~}nGZ7c=~h?@Z`wm-eQkN_Y=YdY&zKu4K237;0<-d$fD33}`q(aCr}?kJ0}=BPs_pQcMbFn zsDL;y6DBIYDW$0C^tXNWcU)IX{_`W#(Oe}GcjdKUyB~C0YJMN$^Q__3UbS^C^wm!-tighh~wI zdB2BrSL8=?VSp>p3_LBXgHnLy%{O1mb>Bu=$8)|@^SRv0Pmy%v(#(?OMUnuVp}Vp= z8WI9K6$T=5bpT|Q$Mn(8Ayc8-Qt9k$Y^HzL`l;^wNk6gBc?<9gl#*OwOcPC!KX`ms zn+0GBX5Xlm+3w@TW{~Fx7e1a>yR~eaQE7Q!H7pW7d&NL_8e^^U>2xn=9hyg9pkLt-m{tuv_W14&Ly!?DtloPX8P5HI^04|KQn*bY`>&nm(04~8h?y^1H@Glw4+wPlOt5d;DY(S!KE_1da z?c^1G(rP6_Q({<)q@oCrv`I-xteZ(eQH1czPuLAZl2rD;ql$kA0Hx)0qg3OTM^KQM z(f?XdIfg25etw?)PfUJONOeS<3gO7X$+N>Fr!j|55Dq>$7CmVv!yTc=C5+#L=}-1;92$4UN?x=woc(&kuL67A z4U`aze2nGS%L#>441@IG?DZZ$9dz^>tSFA6pbH*wP>Lk!7wmn5V_N|+a z*x?cfvI&+hx`rUZ5a&U-0^Dpr@+4fSC>GPLj z1QtVfqx^=)O=afYIPfzk)W(#9r$3kx&vK9{MMr7Z6;$-p!vI`x`Te8)<2@J0splTc zdGWGI1%}oKqWKA|(#|ma zP5_#aO#m=Zy7l#S5;yfH2nf)K6^|(KJ1(8$xf?_-CJ+6nz}Zw#T=ReUB~TH)sqy2VL)pS+;`JWt_+ z#+wWZr}Cb5$7mGF`%MumpSus9-99~)@;opBwp0z=WBc=-uiL99@@uR` zL%|vJ_ZD)jjWJpoIe*KzC z##_YM#f8m2?e>T6n*3<5QiNu7*%9WW+y();&8WT;XDUX4+-_XnrSq9Uuu8?NJ3%{9 z$0a#{6v%LJaFBZd@1pH0?mBpQud~~WlGsm6LS)yAmG`+UCblTlqJ%_HtCwjz5uRMe ztu&)z;kco$_bS;ondXDul0n9Msgv-2S9dh!j<5YikFy*BLbi+i2`d9y2u+4MbMURZjKPpC4n(vj zb{`aeP;L*}u^5qx?&f^Guzr~lPHp2YePE$G76F~yVvFx#2QL2-P~E$L(-00u4J5N1 zX6yoY3-Cu@k3wqea=F_eImOBhFqO3f)h^Ar(3=YS3y1Yir|tMvx2x0L=;w{@rpq0n zPphrTXTi?zx%|%l^a;^XhNhydj0vRHaHKDrrxLFWbTj}W_4n_s$B_~wJMgLe=%VVY zzTVJ!5_VtSb)t}T0_m#~E5eXPCG-e)ne*11a?#MCnf!&{1)G?|uNZ)jR?}?C+JV4} ztXU$C#Wj|lJ11tKG}T@FPCuM?8D8$^$m%*=?LC|}kC!hjX^AQVKpXT5<#aK-gyiIq z2j(2h$3bHPGOvS#qPWX82M+Ti(qi}?grCP9R{W_#I-pecf=p)G=r7X-96(rk^Z3w66S za|d9+qiwl>?!@=+uvS81N+9FnRVAlkK2r}dwkT(+T3CftC()g?w9mTiT8mCS@4#K0 z2#kdVsZlaE8H-fc^-I*($K4dkODf2Y@>;i;9vf=MgK0;k5(VJQh@ME0K05#$eD^!5xuM z&UT@J{6CcU#LP^B<`&b9#5Ykg5?@!b>D*Of7hNPXkPr%Z0Lp{et~V3kzm2CXI9tbVyMKA4o*m16|6YiADj62teFhg>={cEZ4HS?VK9y@&q{uFYB#CDEYj zl|aE9TIWF6@&s>r$L}h=g;y*ezUNF>J8fimsGZ(&)(Ob0w{L3K@a9oz3Y=7NoAz)rD}~ zoh1i#aAb?-ZiyAd7mzNOz$BcqrX~Rr7O<$qF-b|$Mn*;=G36!5{#4;t`mxx6M?Q(e zoqVhRG|q4d?$HLGV|XuEQXXH=$KolKH%)fblZnPdQv?DXS&pV!Z^^?68N-;ck@JV| z`JT@fa7^hO^$3slJJkERw|^Z)RVqNuISW@H&aXR;-UhO?)A6PbvPOf@?;5^;7X$Xi z#LrI%WGq14R8>_2`d=zoSy@e#S5#aQ>IHBlP)`YSN>uyTsB8*|gE1A2p4K%KnV!8GPRKzC))5?D=Q=7!AP8uk;N6cEEpzS8gRQ2yh+{(Q)deX2HSuH+xVe% z0qKNx54`xxpX<~8;FG8O{*KV->vTGSdFz1uW~>$m-@f$3@QOfF>>bictLPC8c2a

NETw}sNx!IVXEQ&jQyGZQDwZ(b;?UmWd7Kd*V!*;v%Q&D{c zLdj?}ONGpeiis@2^?_v9A`v0*$^ZUdV_y40M71w!Hn(S&BH;uZ`gjj5Z!`ilJ!0KX3c zNT!0*j#Sx!4!?K-3W4+S0Ujg_w4KA>=?)4+=$?P29XIMtZ@QsR>T!{#&p-2*+hbah zTyU22;ot(Xn7NGd^~NW`Ot-RjfR%bZb<|T+gecgOq#Y z=xr8q^ve?RG_u5`(Si#S1?AvYNS8y@y+o7>&l8wcI}4W|Bf3n%L@jcv0%T(xDAAx% zBZJ7H`Gn$^=PG`oAq!NEZcJCZel}T(_*P3h2xr|g-l|x4%L4ksTLRXNN)`WtB(Sw45W<^2(e?uucngYHc zm?_CgLp?(N%WW@7f`X!gQjnI=^343T@P-% zk|W$*fmA3kUyj`@oCk~0LYlgehM@B8Ud3QmCAlaATd{9h-)R!yB4Tebh=&UpLby`svkd7@Xwd% zsSAq-_8nKDeLSoxx%~<6dWwI(7}Zy9dj!iMQHNql-b*19K)%WcYW(vzI3~#^*plFT za76Ta;2TKP(6ayUKhk^FhPCs=@Xyap#GH6&#r?2D82Je8WBR{!faSEyzl?Xl#`*vK zvH#ysPLuovg+x8Ef4)1($jAsacD6HHL-{T#sUIT}OA;p7y%vc&KaIxFxQh^}=VipI<-Fx`Wrk!GgS{m}ZR>aNouPX7w$=o}w``ug4G}2(JgCH?WX91T3rPKlc_B_L*hI!X#ubXmvjp;0u zlN-D%t0LGO>R!r?ZfhdOBaP+Lr`!!_E#HKkgl!0;v1nfY+2*$k!$E$9x9jRsj<9x! z_Q&b%)Dcl2c;azbCFSJ>$DXcjwW&91>fRZCFha1GV`B|V!TR%fImqMbz8a^Sdd@+~ z{#+R%{fvpKR0l8C$}#lk1t3=mS?G#hAX8i@&rf~->S`#~#aiFi@~A_zC#McIqY;1`54ESK>M_ z0~;R(S)aOd7#Z&YUjpf7!>zxRF>DjZQzVD^ewt8uL~wZ$a+{a*$O8KLqh#!o^yV<; zV2FspjVAp2t;}>29=$Y=G-x4p@Ia(L4_q4qzrmEjOdX%v>z?LbheZAT8RSJ7?}?+I z<2n;Nb22PH(p4Jw@{d zvKD@?39bijj|gUC%mHxz~clg<*Gjeg8<(_Jb^8;U;S z=BD>q?U;7;FA?T3YcG8aH2U)B|0nOs40yXnCt%@lx)DcjaL5jHB{)DehM>;B2D$I$ zp94#4iR(W}J)|bjB+PVr=>&h_hD2TXnX#KD6i%9IVEsB;gVSX4McUbpG&MDy$e&+{ zcoqwOg>6_HnpQy+;Q4-P zqSqTDBCoO&I@qa0wZb7sng#F88rNxIOYEBEw(9FdxC7F?T!Ke4M@>3R`}g82GS1hfvkb(Y$s)fqW36ud!irS6ho` zu=?pvIyMALk{Pn}GCl_eL_ZHdbxVwy&_%}Z;v$5UrTSLZ~~Ax?JL3( zhLVgp7sPlfDFjwo7i)&SKu7S6R&^D-u7JAGwnQyQRzd^NfnqCt zjJz7{$?IF5HjfsY=3h~n$-?>j# zm{6+?vqrssS4)T$(JpwCB|y}8KcmC&VLGEVMF#EgTOD{UlE<+Vyqh# zco}4<3+|-VzgmOm?#y97S_(e4)o1h0yb&bR7E3iH5+uV&8@04sYQZNJa0nfB|BX?J z7qDz?9FJYDSmIbmG=kue_PV}O_r=$tGm+bieZpF^0jJ+yHtj=3N2mEC>W%8Xv-5;0 zC%>KB(;f!`7ry0nkD$YRG-?8mF}kkt*qdUicU*ZW-c+cVULQ)Nl~+&*k}c>Ky;=-8 z+a5C?P3I%xGzql$^GFA5go(_jZ^l2{8z^V+qfeHo$t`@g=d$`nR@X~YsMGTbMC(-5 z)N!+~Q)Z?2Y+!=Al7fN+C{$B*wxwp;>J`(Vf}KbXpn``$LzPb3+AVxW_qjUkpQ|&M ziA$S;r`MBkG3P4iU27^!tJAK~8Mo%Oo+`zjseD1}(iggD)D+WyLfFe`LTOqvV}_0M zq*#Oxet}}?ZS&FB{>A1u&gVNCF^-}XDfboVE(VPbvA7JXU0W^^C6kbYM75p7qMOF& zDK0ErU2yD#huYlUHlM3CCE~NyD($7~<9KiGO!5tBYzD0{e>Yb`#sMBA>)y*L+DP^k z8q|lyyRjXyk9Q3JrckGwbE@cO+1K`*Dgl2YqsC%i-#06ARd12O7v_I$s%)V2V|KYRakd_+wfHllNhh=imsO5AKNA7Z zY`Ju9bQM088d0lXb9M46Dl(Rq#Sw`(F0X^t0_j(RLm6=TFp5;`ZS>R5zB(DZ&=z;_ z;>yt2f6K|l`UNv}RpEHob}hG|aq57~dPA_|qp=&2tPzY4m(7j&MJJVdcb9I_d5m1% zJ_!X>fspyY=pTe@BKx5qn{illtKp#9uGgcX)L&_cP@ApCpP2B*Y@+tPQY1cKx>!9G zq>D*$|0rKLtZ5R|5l&IsB63CYK*n));uc2K@+&Y0vQJJ)}+0 zo?68kQj+X%Toi><3B_V;j3M)Geeb;@em!};$rPHMj|~@3OEe%mVc$8zuZ&#wE*MqhYwtx*tm+LWa?1WF`q=DWt__Frt zMiP&hn$POtJ6dlq0_R8DLmT;s|0)G!Rg%X@wf?WdIt)hl%b25=+4k8aB4Xf3!RM`a z?7X$fLRP?i*H!B97=$;;i7@Z3uVq(-q~RJm6*C1&o>FnD#DWH@61AU6;&?52y@XdP zPSkA!i&tC#FGco9_Uo+WNLTe>e5l~(UGIx@O~`De?Kbo}I~W|Ek^M4WSi24LK1(>J zZKxCE>%KHKW%}G@RYGoo9Q%pqbOz9kbJGovauR=aur)h zc{56j>cd#8@%37K;+qWNW>8u(py|E2gekTX1M1vO3*NhjGOz1-;f)dDlU1!G^T2CM5t= zZ#xh`^kXg7lRkqPYgs_?+wPj>8W1PJ1h5gj0~Ebp12t69h+>|7Nmy}g@wXA;a3WK- z&pm#_byDUKoAofNejGzG8u6DhVERp(CaBXUSM z+!?7i#EuXe)1`XO;aV2ypnH$D)`T`-gr9f@j4g>bn;MzEb^eHb8ah;!n#>)hSQM;S zRDKz0`H7g7w8@yDta@AUAaE-JKGxU<%`5Y!8#09yu<1Z&o;sf0{GJiaayC>~iad*IrmP4Al*b zTJ8nF+#i7GTJkz-XJs0!O;lf*cQCiSqF*z1;zf~Ty@E(S!c z(&ThChZ#BJI{wtID4NWXv6i^k7ZH$~-}X62ChSU{jrCjBI%T?bd8Md&yjzVrha`0( zgkQA3rU4wa?cP4_b{|4`slRH#a~&eo9!{xP>mcFrL;G?rA^p=qgT#^~dRX0MY{`BUco$T9y+)0Wj{7~fR*JmN zmRa}lU=5zygQ6!9bb|R1%s@m8E7cNI%$z`i3kMBq?YU_G3UYD&)iJeEIr&dAW8u}O z2NJ!z=rYmNrqVEq585y)2Iv<0U&0hjBA*g$pKgs@uZ2@O-JaWMX=yzx)1slKh9-$c zU>%`hVnV)J2@u~r9Y3txrTE@`A4Xh_%HGExiN5<9ch@D;DNut;eVM#Pn7mQ*#kmo8 zD_j%;N~wXg@&0=!31l7oMZ%?8UrULk)bv1&NDn*FVpFvGgZk1r{h;f$iOH84aK<13 zfdUr%pf8q#>2vw38?^XQjvY4$07wQKvJV0@7@nMrM-(e-*x^e|5s%29flfBK*0|!| zThsXvRbS8RdbCWw)aoHup;Hx{?ZkzGU?+t7CYxAcBPsY+pU%bUhHUQq_(PAzi*x_2 z5EkoWvk;-t5&8zX$2|e8Oqgocc9^AS(R2YOSO-L^Z!Ji{3F(c9LxhP`Z zBHP-pt@VZ{CMAVD-fxS`D=PLxVADdFhQi`v8tS#%zUuOCL%P+55*ixV?nle9uSQbx z3Jc*1WMg}H#zaCaN7Lci9PHCU<>vt{O+*>xjZ}@oQWHnZ8q>+Mr6+CfQ4f0OMw6;X zbeuaKljvV&QY5Ag;V)pW>1D7z%_J<;asyg#1Dqxv*=JgeM$_T5^Pj6 zSpBl~cvN6zHH^^cSt|QW2gPa!dT@ABZ_&DY!cb3sU#)Ba%oe6vBDMW3EAjpNu>1S_ zz(GtX?jkhMGCO%`Maz^A;yaDIO$KR%8pBA*9IfZW?PIdS6Ep?5^$LDvbZ4BMDA|R^ z>)D+w0k8_K5WM{+9|@mr=!;L5Yxg&2 zLtno1Br`o9|M*I2b8{2!eqCziIp%{PrW)}D9!=xVPj@Ebw&B&$f>p%#65z>saQgxO zD;d;|IJ#^IoESVk_1$nelML>>((5o!3FNTg?3UYzs|*?kg5gnXttV*k@bCZ$#0Hh9 zqf(_~zS!g}=zlNxa8VOr`YrS6&T_l3YNjA|?JQ`o$K7Uv78F|bHfJjPb*%NnJTh?i z&ehRzV%Mc! zkE?PVH*Z2fE5N9?RN?`MU_Ec$7x(jXj8sf5fx%kGCV?;>jYberG%XU0*u%9sXfP<2 zTHJa8J2u`K@CYUQg2$zqOilbZZ^z?<&#_4S)Rb1GL1Xo|{V(U-*21=?Ooaw4o0cGY zPd%R|dql0IcT<9lcAhWR&F#q3YjqVwjwL2%=P2WlAeB=s*KdUn$$*Sl=FZF>RXWv% zk-@<*ZTF`m>Wn?+vjSXP*p$A<(4Xyo1cO{aNh=!zH8^cQn0nCLqiGk^HHnDROp!X) zJ+7p$;>`lVL%aFZc=Ug-JzBNKB41 z#~u3`?*Xn|YyOp&e}xq>4>vO6rP0!c7io?|9gn&K(@zSL0WvL)$&z*0YF>rydQMAB zzjN!-_1(&2+y@>jv)SsGrVX#%SeW0VJU(1?o`7zoz_`=j^QisG*6(}5sEO%{dH8DK z6>0OAcaF_Mk}APDxyop;?yO2ON7P-%A{>HQ8vKh>h+6lpFkHi2<6?$aX2x7vzIs7( zK4y8lOzkX;INtZVB8FCmCWAA-bgs8XQcwDD)w)bLO?JN*aS|1}DZxRp8#VG_MNW?+F*>bTR(oGADQscS>yQW{-qDw`y7F( zeFdkGPY!^Sv6HitCrd(-zgmYqm413|;dC!gRBA>Dz_nwVu1X zR7hqmhbYYl@Vy>-{!`+|9yeW_Qm8V+Z_2DulQn=O&+L+y*6IHtAr2m^jsqUfa?XBiMh@QBpRkkVe;i>xzAjb!4BB*_>QOf?AF|HP ztqbEjJY6+^X)Jq0HX=UQ1*6*nHHV*oDg<3QC`%g|jvd|0?LN+fU@7=))^zs<0mt3tzM8S;!)bhX^ zA36vcNqR;cS-MMT1M%VydL?dVBm>VDRHLJ+rq;6Vzva{kK1o6r^*--vzgmuV+npGCaigGqJEp9WA#{fRxS(@{@~~>t&Ay^LxN~!19uybMx?+Hm-EMZf$u_ zB9@97mlTI3Bj0Dq_EGXG7lYevNsQZ`pVY{64gMTdhZiqiH0%u%*a2WGh@EBf{%=FJ z_VmE8Ol!R;)u`uWWANl`Oq`aF51*9Zu5?vKjIUh0Rh2R@GN7(E3kxOk|2-5H_LW4OS!HW5fv09eoy ziOaz0fA7hu?75^3UAiNhK*Qht5l?URVH$m-_PABDNu&2IdY$)$Ehr}#fS(%Nh!;1i zpl$%jc@_G)W>*D!w45~hxxRiBsC;>jD;;T~KJF0(sK%fkjiho60-O-v6?7@^*YTSCF4v-#*dm?dvkrWQ&I6zb7fe(X@qYf1!3E0|)}RNUO$4qgk{iCmn8q+JK4H(^+b z1WG~hx7iH~?abhyWn~c{>3~))4nm(o$FX~s?zA7$uy>#S@M}65Z;4y55~L;S8I z1DL)LOtG#uuQN{99rd<{+kJmZYQ$>yeVwI>eDQk?#be~!O#>78POq!-%1X3$?+ZyC z9fF6Ov4AM7EM88u5U;bJYY%rva^P=0JNC#;m!yOQIRw{Qp1H**egVQwH_NjRnrFR~ za&cr5;Lc4HtC0Qzd+^JnsvHEg8}19_6sTl&maxxR1nK-_Ew|)_j}p)Aq`Wfk#E^~>na-v^F! zP0q9k7-U3+JCfm$Wticx&b}aIl_8)LeGezgp4C#ZH?-vfxg%*$-#qcY*xR~dl9J+p zHKb=hUYQ`J(btOW*J!+LpR8Xv7^pSv9j`T&0x_cAw^OLVp1J`7{w);}k(jTbNcKLe z|6itSC&jn0?(nnZ%FU(UrbE3+Hw=Uqn``sp5%M@B>)Vlx25VuU!C~dthHrc;wVozD zv{BfK?<3qBj`=4SgUofOE#AwRRzbczn<{mz9%RPbjk{fR_K&gDv;~9MhJqw~ z&?yo2?eqcs%*XJ%v*}OYJ(V@BUR2%}_oNh}+Jbvwe)e+|f`9xcr_W=x%9sHOYr>T4 zvz>T2P$w8nWwPEE+2pc?0&W>%n4~wNSWl#8S>He`DWrig2a4~k=#%Fox%-u{jY;lx zG+}O?_+Nf1c48{3fLgZD=qTsx%ta#LK(@Kl2RaBo+8ekEv*++c^sx5cCbC4KtqABO zCRc|`j7&`3fzYt!2M-kDtQ4Rv62W(MBOi7ki{|mWbtMz`C(96Ue5;86gODOGf7&+h zYgWf!FThDZp|J^gi8S$m2e)|Qzxetw(ikTjP9m`Acm^C#a#mLC;-VqAGPoxvClYb0 zzFOyp;tC9U_s$fbY5D2m6k_I}GqopnU5AE;C6b<~j5oir0(h6B(YKmL0opH};-`x1 zcP7mxK?19j^?pdc7kj*av*WtY^NJ)M*f0sO!BfffB`8m z$+a``d;Rw5o1Z2CwhxVrNST?LA?4#z;#vY`q30yVhp%$)elrOKmYZCuGj{SHY%*&|Fv2N1fV*AMG2vU!MSeJa9BaVB@{6 zk%WL`W!WLlbYkFjMvniHQ=>)8)|WH-4B z`y7FQ2j9w7wVxA9&3OwkyP%l@#DukPP0Lbb1rGMm#_ysvOdV0v?V zTG80yJ!OAwS(wIY3IiMoA>ct9$7{V{3l}6%Wkk+5L5LV@jPY4GKjhN1^Ytet0=Q79| z6l&p00$G%5x&)^p<_7A_!f`);|Nb33J=%F=z;t6ERzQ>ja4nZL2Ts2wVQq^Jzg)5q zGWQ`9?7+>4z9)}7p_qSCA7|b5JNeg7r8>WrsI(7 zgDe7nQW(>q>vK6<8uGYL7|JgM;5~R-VI4bvf1X>D7JX}4Yxjd&uSy5VxovkpXVBbR$wLs(dkPhjI6*1H&QC(vJNM^Npwk0P{9L9YapYawhV&yrwjJjc+$OEOXY(geCk$e-QI>aFARe+1kC~PHP;y+{?LB2 zk}x=8Dy-b9r)k3RF+|16@wR1byBBaQQFlk3lpJlx!AOY;sBYe!Msts^-2VeiOP3ai z%`bpD)p38S)OI=lS`W1CN0^*}&S}`i=ufUQT_f4}jz+yPTRVP)u?c@xPrc+c=Olt0 z4ZtLM>KaIA?mW*3&rC=CCeV-QIi~C#4t@CC_O#{>+*x0Ilnxy#j1+2rWUs_!%~=X( zp)``{8G`Gyl1$t$DlE*ct;OQ8`quZoNU6r}rB^&oxsUd$`}jRC(~o)^+~ur0b}^rA z9y;FKr24f|jRIK+n&-M19lrkgMPN04Qq@^-14Xb@sG^Ipx--zY$%z^QJOImq$Hg_3 z`@?u@_qWGdh6(bs-H)2q3pRN+1=zwQ3a>$sprD`}|IBbS@}RdQGYf5)LjlH@hzaZf zazQ63HSr7Hl&hC<1X2bON=GZsBLJgt2 z>`Y#iUdPPyhqmVU-y%brn>OULaq0rtO#~7Z>_Z?ehZBFYG}(+wWPGYe<1Sc;GGoTV zTg98O8}P^|;&;m{8&46dCU&!#vN0%s0idulF6*ET;$L8+>ZTj?y-)!ckX$AY_cwaA zCeTn_<0DYIk*L*4Q=h2~-@kuP4X*m6ae5qk96RSZZv;8c$z&)p?y>wWEz$W7mAv!M z*L1kPx&}NuEsy@(OBUSJJUqC7f{|==Sdxxm=a%-F@R_66Gx`?&eX9u4vU`!eEJD-= z^)VqQC;sMadjs;W1ttkVDw~PbLCW4~`+{U8GDZtNMk*YmeVmkSnwzm{kOLG%lWIlYJVjt54)r-~Mg#yGU%0UJ=U1i8|3FUIVH?}+K zd=vvgu5M@;GdHJ~=APaqCP_%4+0RJCOFD;Efvami^hwHumt+EThjqPC`YFCZ12_*v zV`OFR19dtYEJ~uL##;-ctW}(#T>zr;%nd!}Vn%B%{a<_uApx~aEC722;2wkWi%sCy zhy-$@;Fu;?bsD%g;q*!=hQc(5BHN*7Ng#cJzv~MMSom$zCt0ofk$!ZKz zK~DACYAG9~W?zIuOc0wxiPy)ym=mC!*GWZtPUxJod-K!&B~XeMC~0hfOTp#Mo4V7D z_j)!jk^iV8_-)YuK|m!js11O_d;3Hs9SZ8NMU3D^*EdnQI9X!lWjuchzvs{Df&s3Qcey&7JMvUJe~fWl%x~~O z7ZMQ!i>P_`77M2h1~ADxt-DZO?sHh;Y!oq3 z)!%x%if8rCXo^UX!qx|XPu;5#g1&(P2(oc`+rf_#E(pOoVXT@K(=eP+ff|+Lpn3Lhndc7?jC*`Kb2cCFyE( z_4S#7#DN0%&}FxMuctLmrl;`T=9dv9lvRXt7j2okg2E@2t7XSX?yWB&ASzVX*snqL z_1m*CQq}#JzGwNhrw8wQz;?13cK9}b+h0Pd+NN|MWH7uRlN;ed^z z3jOoF%4Wnk=CFWA|@s#5_F<^qL%HuWNt`}sN#bj z+aJjZyjta#%)qM!>@jm6U!g*P9Z+B*$@Qh*JZ$y}Y zdRa!H7RW5r#WrZJS+3LKA0qONHoWIP+mW8WIXek>yb1t%i#H}Opx7ir5b^*ACBT=9 zX9&aU!^%s*AZg=c^GA8Vq6XYJaPaiLu4#V@DhCK|Ju89i{?IDiTGw68db=N*znq31 zrCla`hHzN{*>GsQG>)FX$-Kz7I}4W+dxy3{ev1(=dPyXby^|G1VGWp(0=f9NswKUh zkH3#H_8iH=$-o1Hhu1B}>!-Rt-tC1ef`nZQPFUpOdPC6tR~ks46t5%k3%271`V?@v zoK~OJ^->354Fif=rS^X!iiufiF3&HnX9XIu$ENe)O)fVmvN1%^snKsGXWvsHr3pII zC_lmd)Az04`gaDaK^gZ=>SZDg{^O(kR?*r^F67#kFB1)EGjakRUTp5rgi|4T`p#`w zd#gQfJH2Hw3zsTa|FPR{_lj7gDDe$79;hx(luqUccdxN2DdT~_(__aUO`bwGn)wtL zn4tPU+b`r!YWrnmE(}X7WXy9*!~W|IJzNo22hkV<2rr9QuO|AH$^A};8T2kDm@6Od zYaS9=F;f_NmCM-xqi6|duDgu=0pz~ zv~W1j{$Wfi4Jtu*iLd|JF4AGC1tNIj3iB#>-5oSSgh-IlsHyYzyyAi9%Iv6QvSEVi zvj*!Ql)QR7&;h7(8340Sb_(JF0UU*wRSyB_&x@72qdF^}O+xb`$T6=|#)fZyWT1&! z4tZySkB_hIwfFU}58DH5m4*(%he00N*t}HeY8=8$altx3dc+X(AY=)941hE0$56rf zIhF&tYoc!&+?qw!sM78EWA{f0f6@%R>?iSH{4X=pEf2U%|c2fbl`U`|DLGrx( zVRf=sQIjthPa)`gT|2iH))0nDxDF7B9I!l+@Y{KgY)|NA?iAXUhl&_sEUfK&)Pb)1 zc(){H9YT?lI<`>*#><@hLqcL=5Lkaf^f(~@yvy6)5jAU2V9OjyDAcyNALws!@cuE1 zp$;g$!vgF{u61r;(dypJ0f@;uaI4RcR&s8y7JC57T3K9LU`nube&=#FW%O-bg?HBlg?r8$JgmpLuYK(ALSOm0Zo&LX%r;bbhI9+I8 z*2-(p;k6@fad-jr{_)eNyb0@EU@ZgxPgo~pW%=1huq75>o1SoPe^@JP^46XJoh3y7 z1TMBb%s^6By;>-UC<%eg!bfdqpT{m<_|^v4hj);oi^j`Bz>_WU^@hL-4)Cr?8W~Xl zRXPmhwMalnUjGB8fM)$vKJE(=FWj?6IP~)Gh)aV!( z5cjv_BY-Y3Ub1VtY$4l|RuMLzCFIrvst+i`k$_V-UF!*>kz?uraZ9R{$}R)WD4oBm zqbp@#9ZQN5_Upn|U<6+|p994q_Ws@%0=fW&3{WIKbH(lHNY-V1uc&urkY`t2m>rG%>`T_N%#tuZ+}eDAv{0HJf^*2a ziN0g->fP^0;gna2ps4~?X9HL(20#}>XO^6|ec{bz+5`6sTt+Z<&?w4wEeE)3l4z+e zPP93>xrzq8Mu<7oZoVEF)B#xy4Gko~t+VJXGZ4 z^r@)NAp|k$LH0)%3^|)FPYGD{0)djSkrz%O>EKW;kNq}fa<>l{Q=P%xgD_&U1rs$# zxV|vJ3kV(z1(l9L;`v3?>0zgB>y41xUML6(Q^0r$dLEmAhl@V~C~`diy2cbRBeNKZ zg8snr!Kuz)GJ}@58835G(jn|-uc$28h>{N%J*D3h0hjE^6h8K|b+-;8YG&};Lt5lx zfNSJjKlK5LfR=>?6$9@%Rh!!&-+zt&5b%6#(67QuaBRl&5$O|Vo+c_I! zKvit0@nQE_ZCMTe4BKECAq_HqkFdcC^wD~aN@U%iQJ4;X`r?l(KV$s_`oS@@MH+%uGD92{XSLtQ;{E3f+rQ#L;5(iQ;-J&_TKYfMO*rEuTQrk&co; zogV1P3E1>Qf!(73RG+VJyH%kVUaO%CJ`saRU&+zU_^OO=Y-xtR$=*- z2oMKw0MVgHgsw9(Gxsz(ZyZm^QRac#IxG6~&9(YvJa=Cw7N!?jgK(rY@SOGmS{tTy z$$x2(eSGzl?Sa!h zFp6hniFj#_+cv;de-mlC%mGMMHCx2%z>7gO6KlTShSsxvIbpzqwnavMFAh4UH-^)m zvwOR(dG&WZ__SKvY3BQV!Y9pZ{E`J6FvTG6m4nbFZwV79@QtIYNC1%a^1k76}5zb@%&L@gO`P#3P~> zXtO$>n;H3CXn#%TL*@R^6kxntBaDo+C8}6@sIBdDTc!!lIF3?04qVTTj=RHNh&*0< z5dIS=t-n5vt%DLe036gOfMTI}QhX^|0rKTZxH4Nv^(Tx))yIs>&#z}OZf!&u|2ZHeg{4xy($5C;5g<>2o1}8JZm)2cgheFUMZ#x02`PVFALWsyeQ6P7+ z_-5>-&pp@#Mj*w3Mp-82lv%M>zKiIuNAqCzY-3}i&T^EbQojKOR5*+l*IgX zsBz#L0UZYMU>tdC*{*?uUQig9CS^` zEi=%Q87<1ZAF=mPfhzUyE?t+^|B;v`L?YWK&5eFe6gQ1~e)@)Io zVBr%BZ{m9LhFUy*+jGBvb##CTDg3WZ)0B&;WQfQ_#>}se*XG)N1Te@1ot6%ANP&3% z$13&~BO1#DJQsMGTIWoma^a%k(*A*Gh77OWr&*JRkFPi4H^2xUggTvXbc{FceQOF- zL5Zk(TcD{!0H746XlNX{@cQO(5;S?GlDV_IN}ZpcUKHE$4@ct_Pud3kSg@*%ZbV8* z_HqJbI#`n)Jv}{8sSp96f>1%g-?bo?0}PY<#jpjrGiZONYG9%?IOT~Fq4s52;M7L< z#q0IiK;D2%>SVALD=(4ghC!s8e{qT4_ngh%u z!T;g|$5)2_*O2`Bb2PkRyOY;`RU_ml@&TB*1}_tt3@zCcp2OBey_N7zBm9u( z^$b?V@E~B+f}0q^C{T!c2W|*L23EkULOOkj0S0Uz?C|g~FpvxY#N@Xy=?6{(Qp-9X zZeN2afdcL`NpN1AAcdUo&(^3i!2Xs1XKVj=u;_Ewc&oc`F63rAUmu$)f8d4()`4D7 zqg_Uxd*!);V}cNB`c(v=49Wmz`~Fs}+oOR`YW;bnL|j6`JQ%ft3lq`6_#=*4(#0qs z4W&a6;~al>ri# zAYrZ54N~+*;j`R7m>cQx5`<5^2U0^Sq2b$(jEBQW_khH{}#|`)(xGQzAEzaT`xVB+ztXVMlEaYcHEQx zQcbd9O@;1er2RCX6E=`xth0 z5XwmTLu3Hiy(6^0XDG1%L^nti0nlt3_bU{La0Iat1R+u&Dc5QX0s=WZ;K3_X#0v+K z9D&lRRpBPm>AlM5m1l#0nSTtfy}fAQuCeHpc;<4e=QYf<7vC z7I>Hc?T`;!_SqrwDiOB@(!zxjH#0YXO#YvWIAihE9U;zZr{X z_s5=fy~s=e)WWE|FAr8`gAK^Uyv{?8l??#^f(#ZBqYrZ85O*%PS)gD;#2H9`ty85> z&Bu2!U$y{LPo&_$@82~cR|<%feSqErv}z*g7y+)#Q%HIOUEx&^^e4NjP-Ue%Sh_Ya z`6o#OW;573oUvQjh5Go*e@|B=1B-zTCmSyVzQQ>t#G^DyHub!WiHXS*v?C`o299fB z*y$a{bSC^KFRTvjm*&++UN=1QVEWSi2ua?q+}HC(gG||NS3>VABD=h@WZ5kyXTLz~tV|FmxrAJ$D*u{MI2QHvR0 zt93>3%Cu1V33B7i8=FKd!`Y@8W+~f^xz@~(hVBl3d z%mMpdLK5lbc*(xT$%`=HWq}81E3fAdy0*JXTpF6yw+K6wAh)o+9isUz(OD&p_m7u# zwKKXh_zI}G^N6%pmkh`p^So?}+lycfC7N7O9f<;ZrlxYLGF2XfUQZe?Cb)Ti&Cl;C zoEnsYW^~NJdV+cq(<^lkjgXNIi0=+eOsg*!h31R7;nRGV%z=^S{W~n72fj@O3J-R4 zaRvhLFl2x1`ihCh@Mq>{S5&4D47<{l|K$b}9uQ%{^36=O0JjOoUS((4Q~n+32~*v` zfs9H=D1vDlFv4^8#7025k}TSMM?nQEQV^{j9ZV#3zrKtjSL||vCj+h%4X&FYY2b^F z*a6A(j~1ZVNSU#b5CYnGjp2WN4>2(YhhUhD%!+*6y%3gGIvwP>eDnV~BGtU%kNzbT z$!LWBPIjL&(Cc{^5A?K{tX3WQef{@??shP`FjvHjUVK9 zaQ>>*>o~%Ao#T3BIB{rx=&K!@R82*V(OcKu(6Zgneq=f{o)NN1-|i#@g;M_KfaehrCkI=D>;aj_-Po!nPx;19@W-YAi!0T*7d=0e$AVVvh)QO(r&n@grBM0AM$6FcIC8;v9huatjzwFbj7f2MaEg z{5OS)v^h4y2L6@)r_zI@5x~uzd-wOccsAJA9)1F=ry+hThLF}6c2%hcp(yg#-Em(0 znUi|n!m}!Gfhz)|e3X_;Stw^$e%)RSmlJsUV{2TzT<+@<*ok8XBp7AK$0N~X)(4+b z=IW!DwTHDd{`x>W1Ow*4_y_-9QTqUQ)RIRp1$_>6(z`ypQ%kM%cp+-VX*RqS%M<2=xTqjvgWNv6OM6-n7Z}%rzY7egA>O-YCvOYA{cw?I$_M?TrWpL= z9vnpb6Ks+=m%t1)v^Q+Rgw@Hz*O&x*C%?K}8X6j09D(hr2LirBKG;9uANvx;gjp$Bbb*5mC7iVWpK%;&R<$h`&~z`kLN8=Jp)&-#Oec}e z82;zlokY3Wuwo=~|MmT0iJ6xIBi%2#gK@V!a3-gapAPx!UEXng1453*EhZQ`<6BQA zp%?>76=ZBdL;NYu2XNQkb^nJc;Eb(>86csA{EvyQH@`7+wRJvSVm8aYM&2S{4V8xJ z?={OKeS|>7!i0ZE>Tx>>qQwWKK%zcXErRSj2vO`|1){{jQ?{++ACf#J2_R?5=O2|J z24sepD7V4VLUnDu;XKc`L85a3K}W~%cf5z>9^(iEoixb@eT0`*KOY-CJnA%x+{gw4tSsk0@JxP~w2_%1 ze74k;O!#_n^st4xzae;ElEmDM-@BQR&Tt1_p+NT#gKnUYw7>_x8;d=Hx>DcV%+M{o zF#$iUFSpf|&=7{J35W5klFv_N;WN*mE=g3h<9eQ~v)^>>RAdjoCwZTCdVPD7{Wz&0 zqq*om{^J^baD7YuN9w0^PjkuGt_PLiI>8-_2|KOf?!qtOgbw&4Q?bm0%wO8N$~lE7 zp;4pt?}wXkDsX1x0yzl~O;_zNwV0DFg2SHbI_zyGw}?#?RjbvNU}wb#hRJZ;RaJuD z)xlJ=V8!U6`b_skV7uJc&QzyP5kAiHnI~3l?T!qOMv*{~+izX>IC~TDQ*>vVvPI%4 z7RB2h_$=2{2_Lv!X6u+^oTyClXe~O@H4b{y?*qgh-g*qIQz^HIFI63GEv4U&+^fnl z588ab;pE$C)GnVYhSU>(jwC$7E1OJgZ(n`S@D418u8u-`bRIy{7b73QJ2d7iY==;znnz1qPu-0g zsgR{5*zh*4Wz`7=Zu)m?7tE+1hzd(hgImlFVJX{ z3i(mNxV9Rf^xL!K_ufU43cG9dWZ|EPvBcn5T`M1s&iCgXYs$F19zDj0qhS|Qe4muv zUbF-Uh%S$)@#tDX7Pt2lga89?Av6)T2Q2Cj4Rz2a%zwvP^m zZnDu>P&*Qb|JlTZcjrV!+4JY+zkLox>0|n#EtNzv)SbB057r9~`nmZEzAWHh*zMFU z64!#z8lRvUj%dSH!|lvW;QA!c#zTOZHX8Z#l;4ZFN3qq#CMKDZSS+SOEc}2sxOgGqiQyXa6?J=8w=w0^+OI*J{ zW$IktFXK_O`5iyg*us}O}Vm@xR za3?ltf&A%wvt@~_oSeZD$#@ElgS6!C_dgZ;T5DSBb54bg?A(eQfg?zk&eI-SNJ&ZU z`ue=WM(_OREx`5htvUP|*|4JN^e40P^KNDWBgP~0@A@y>7g#T_-PqI`AR!?ETuz zOE%1oIOvqo)7O`klH$_p20jCn$y#?VJNdT}Kz$ygw&>Xjkd$l6QA3b)&cW>@Ito71-|Z z`_!H(7oX0L5BqicWpqB5U;M0VXWEU6_P^M+!+OD9i+%5BD{w^JWoJkb+_CTXE}{AU z8Xs$%CWTl91er*OzRKqDVYfSFSiE<)UHeYEkM?}8fF;9~#rGbq7OxR_am3&6#oUAC zDa(4V{i*G3t1~OPKQXxb=YMb8SGHNUa)L|@v%dXWAAfyyb^O%Ao|KYgr9)9_N(|+X zb)6U-f`Msc>iOqSuh)b&znUW~m^V+i{JUha!wrEA3=F=RX-Z8ck7L%%j(3O<1PhcY zF)=WSF)Z*r>*rD+;|?6jy$(ErMSPR(g0)~BJFAtNUhmqhrX-OZvi|zk{QGvf_xJ5R zD$Bg~aC8ep!?M~Er6v_#e*W{bIv!Z`H7NvG&CPS|c8FF0E;Exmb+(hEIC5{*)^qdi z*8}&RW&qc4bKjA?V8;PG!Yv!Px$^QWv4(92R0|=pP18S3j3^P68!$d+&GEv!0l9K091ZMIH;C932i04ogu%MgtBGUJnlL9y=;B_zjKdU^4gv z*-Tkp1`Y=Om)(?~2nTl`PEkfu%RPN(&cj{vrmZ@l zn%6FI)@$s@gE}kJ+FnJ4bTs{H{`Eo!cQeigiVJaFFRrEvbsow`~fp{SFZa3 zxhkH$MA_u?;-I^K|9x>wsj(b`T?&;p2p6B4wH*3WiY^NM?l*C$x$u#=;5CN!f>t{4 z0%|zd|My!2BysgV@h{!Iudu()QGRV|*Hw=_Ijo5bjQ@8$W~0XLT;(9>VYGO1NW{>1 z3PI}rvUf8u)iS|@<|0E%r1$^(EqVCD_oNi&Tz00oSAYMO!bOat6vY3cUxP2q2xg3` z-$oY{X4V#fv@u>tU@?>)zMkoYO(pEw8GwY*sRGtV7XX&4FhE2^WV+NIjEi{B{wqbm zeFsuss3DMv~ssk|Uu4IXHUW~JHLa7lyj7iwNjGQzG#a-{WYZJ)7^ z(gnqZ<$yh-dlqniv9#e&`{l`Ymr5|MR8jEV7$<+`8b+0U3kwU^{Cu|x^Bwn^&X$v( z+lOo0fc?k&=h1dEwW)lsh^uWLjVQ#`5F zfLiPOL1D`W4KzIudmJRyL+RWbsb{bI2kSg`22&^rFp>LV15O5?eVzy5((yshmuXu+ z*x)Kc7?u0fX8%>%N$;L~EyDD6jaj}NDs4?ST0m=~QQ4FSuGCgs+{2(F&P@5H{B23o z#*xC-08B;Y z=3xw5sZkO)4sVA`YjE~My|oIkU9xw_(S8+3!rq#_mtA(K7c6;Cnjc|jrho(-vFblB zS4`Z~6eqy_T}^RxpU1$<0QImsLKZci2D&Kf>%yCXdu}vQqS2<}@-MX|7u0clQBKUj z!G%KhFn-`666F)MI6gt~)0WKaUe+)@L(s=}(9%$#r4^tjBQFVU1h5Qh?XDMTXpPkT zXnVF8C1nAOMHCwTV_!wby^cX-=BIt+q6^-|WHAOPsadiDJizepZnOI9&lLhoqr4Lm zV+IJ8v|pg11Mh}5ux3|r(F@TxSujJv7=iYS8rlvSSpjpW_M{QIdt1mhb*BwBxY1}C^zR;K;L^RU75Z`? zrc*%*jr}Pa8oN{5ja{X#n%}yOG%&xH;8!bHj%oUR)Pe1jhzt~31{FjVBonW(5YOR> z;?tqsrG3hMi7Cd`%L2aT3h-zRE&VUV`d@1M5Fo2dPTPAa4f(iYii06PSLzew328HD zGnwU^B*DQbO<@(Q>D>!-_~3;u7Qxi34@r>{ZX<)S+|GcsZUU_uJBh8Gz zD6U)#Jf1b=JT%yTkVX=4{Aa62lMjO8*#3Qm9K3QQ!DE18fI1!Uu~CdRNGk~1(=Rfw zz@8E!BVo>AmOJDcftkHg12Ze+UQnd*Zn-kqu_olh`9CI#3{8Tsd;Ngxlw0oQZxfe- z64njjt+u^v=+nG+pN`jy>MF$U$Hh4JrlzH3aAgHv8941v7G`|JOa?`SL~rcgI1&ro z`8eprwP=pYQ=`F0j=oq3I8&MUJ^&wy6XJNx^AMHRNp^~@^9`6L?^C`J-pSUgNd!z1 zVG1c$*MD3$JRCAMWth$v{qS_WaAe4DYhhqKLcIuJricvbcr*^Ln~A|LQLHE9NzAVI zRwYBo^!LE~RKeE|gj{#RkT8hDGKj7VJE;_G4M6ppTjz2-=l?VoL%Z`v)Y zx>#lLC0S#85XvSgezv)Ky>L^?gIwQtw#+?w8W9ehglR^_M=g)iw)q?j4F#S%5>LlW zhjqR?1RnA*j0sgLxaU##eN?H7*Yn;-PX{uDNgh70W%rArTyQXq?WySN+e8m|#vb2b zUQpV}_R4|~@gBW=Oq3o^uGRu)SLB|%rLi!Ep%;}Dl~lsO-m-g|F<-flsdPs^MZk*9 zSG*!E7%bcUFhW;LE69HCTidRy@!{(xqgbD-(;N_j;NUV;LaoR0mO606Wi&MKe-C~L zJw84zSeHMbqlo3ell##%7&R<HiuaL! zOx)5)8cD+0-R6K}roY82P%GFMOD5v(cu~aG+7rY5T z^rhCXwGA&RVW0I_&kz99VAo{)r1a9p{!73Tf#pmQ@EMxT(iRCbvbgwH)DI;<{YMg@Bhy;*Z zn05D(jTF_4Sn;YAPmB~*yW4bYuBD*#@BFGlnQL(CIo=%OHfe>6fArv!+0dMdVoVnr zN&c$IABPb79`%#9AQ`-@o{sTht<9D0c{zq^h1LtA^};Mvwd^uXx_&WQE=~P@`!Y^MH~Kr0mWixCteS># zTX|&h2eQ>@HQMwWD};tQ?5g=F&DSc1qmEY_xv}tYo~Jv{ZoM4~pnJVn%~t z-mqka4`(QxdT%8cB&T1abaYt@rh0y8byxw;cbt0&o(XdfYms+pk5>Na-l05_4x%7N z#(vf*y|vjWQEXLAO?&N+khnxKUYT!{ero!2VLtn>yD1|6v!X{|6KGSCa@Q?uHh zZ3GoMOr)T|ib^&!j5{>P8Zg!UiD;V_e3)-a2w4&}i)J_V=hU!39%M^lEhQBHpdN~R z?b&a0KHNLCt1iP0$C~sd!7qLKmbK90;qXs19WIiF7x`{ zT>Bj{g7y&4Rn9-AQ_)T^eXS6fFr|JG@+uZXGE=8Q;SG29Vy{cP8 zU6CL<+JIm2R=M!S$)x7YMn;%+&mUqMVw#Gp=4iJj*h_w&#@D}OOwf2t%#<#ixvKF0 zCSCCxhqtRp`M={F4R`;vYo`*9A9F}+nM~kk_0zASw+VIY@2-=7qkuz3T^M_{2<2~ClCkQM0&Y_+fyd-z(}yh&m@zD*V1BOo zZF!kyL2fl;WjdUf_nc0wKK6g&=;cSc`1RS%paWuh0qqx8q+bY1Pz9yI5)i-=;4rue zEr~naBk`tx>QZ~*#&kx>><5cnTUsH=&}CLu2pm*aSZV(ob)P8|>oHWOruf5Hol{Ua zb!x|i`I?kzm{8%5Jgxh@UFu%X?!&f1Uq&#Rc_E@9!F!9Jn|N?Fa}MHe9)d)Snb0Id z=f+p6{t)&%)F+tgB(s#{APHH+4Xpv`-0!qQgvFDI?bi6Fyu~5ZC5@Rn{*4^e&ZA~-Lm-JiG4uziUuJtVo&iwbn+AiTeBp%buWJORwKh4h$40FDLh0RLR`G+KzZu*bigPvZT@p4 zdI=ZspNnEUa!raStxi8gyL6vvbA5iVmZnWUF-iOBiAXHCfu|21lA7-mdUUD;_okYM zBIM2Ik?%(8Q7Ba#sj6O>KEx zNU;Sa4o>LUU@iich0jv82oTP_Z`-5k=yLPiHp5^Zq-pki4*CVYspt;aUh z=!|@+;>jGPg^}2c|C)WMP&40gwHJ?4)RT}S1w-jG{`c?SxyYl-^}m$sv2RRO&}$V< z{k8zfJpIhQy)q*P-q>5aQwA~A1q?Mk!Vv?a1~#fB7T|=N_IY`u`9^BKcpA&%Q)!uK zx1KfNT&IYniH{kKcN@D%^Fg__T{W$>RDzX9+_@(3F&AyC1uZ(vgi^k~Vn54_WA~b` zu;pW^gEa~}qxt8_ipM;0!P!2v}q zu!UGxE8-(KJnVQVAiB=$-Q=AhAWV1A=F^JxG65fLQpO?a_hMV1UX=xw*zLJFuk|S6 zN>6Oh?$?DF3Wr9#lkLft)u{+qIkDHjF%OT9DqgRv%=*D@&R1f^9bq?TEAjHwlKsgK zbCbV<;>Ih6>Q>XYfTB3}RERc7Sw;-cQ?;UW0){A+g*8iC2<4;AJ; z*gG>{amMmhc9S7PQSw*YYqzzA;`gsKv2UA zDzean%UyT%tF6b7pFMk~SMSW|d%lDs0&=*zKi{_L4#^+tSGS78E>y2iXB5;hx5M>TzFP~_xmiVQ_BfgN10`kQ}1V8O}j z^iHfR`rjb&47D&rA*RbTYAKSlSt1mhY99QzEcwbE_bMzrKIi&uP9sl(8{U ztczHQ87r038b>Q#?yw*>>oRFbD&o#X?XimYE?Le3J-{QItQ3KR{JAVEG5mRk$iU5~ zu~{QB5YH%Gnp@3iLqIqqb>Nno(I@xOO;Q|CKs4^{@C)I19+wtAu9uD@b`EFw?Tw_q zK6&%)L5G=J^uZ4OxFcS zEJr>olN@hNV9eCoNrV6C(@kHz`7=u93vPBJZV zom)K~KEQ4aTK5)BRd@ndUN~F$;;DRZ^NxL=HgEox0MXy0$*AHejD^^Q+2;!(8Wq0D zh9$0W26;tUwX{zX`Q>eV!$f8$r&}|{ZPrcstZozhHmDKm0IHfjxgG!heR!wLrm+&N zn1D%T(I18SZI_v(g~fw`-&_2Q!eWjR6Ed<{`~kQ$?H4ISx$|Qu*&}`HtJh`MEtjT# z3)tJdZ1290Hr6Yj>ZRH~eqI-}^}$@U zY;mBgXDo*Y>}SaXn6TTr@dR*N&M(-#Zp};K$=LGh_U2wv8t|OJ5csoj!Ek@+s-~uPpSi9mK_+ z-W2s*HGUZsenC3at#{ZgvKYm@1-3GL-)RSsPc^l}V=pYUy=Nqq3M z9l!z=z)t`d`T&`%dp=t-P)$!S3|uvGRk4c~QenjM6-uU^YnG#ORp+Ict^?1eQ~@VCm2?6B*;xZ# ztC9PU(IQsUUX@CMSPNYL!D1i|9Vw>6S62W}QU1af$|($N6!*eI56W(moTkv2MBF{5 zrgHh%>P#%O1X7>TPKFB2aCDGC*izV4lhM`7)R{X`Vxxd+9le{>+1dGpN508ZFbL3+*!wjN?PO-pNhz;pCGLsnv;l@PqU>@T9x;j#TugGry{Bm&6LnB=r-BNYL z#!ZSjRp&%MYtL`ru>06uMEi_K|1KR&3)qg5u!hT|5#e%;ee(LM1$N^w>pEix@`-ST zMunWbv)+hpcs7P0wQD2pgl;BDZDUUgdLyshnvEa1D4xd~{$xBBt@dRX13Z6gtr6@! zhQLjp{QMdZx+&CHw|^T609X}-!IA9~QZa7Vf zl>(U0bi*Dlzi>|RTmM@7zYMdG8WeC`gySX0n*Qn#8xRm+Bbv1$!0=Q-%36mdHASali*?}_V_v&mhWrkC4^hA+mqS83Z1EygiAND<9i^ zeQ_k)t3oXh(=qy4`A3d43MP#hrM+918z%W*MCuz~e+|dM-%9*k6St32z!Ay`+Mt9P z0GtZG6Ge36$40_V+9nma(oPoKQC~@O>>lnl4pw?}^vss4?PB2m<$LQ227 zz^$4zm?hL}&K28l@Bl!8!CL8rbGs8I#AHGkCV+fpEG#VahI#VH@R-pg&NIbuU1TA= zI#hfe+q22&#ful&Pd_|%-~5~+GoX^$r&GB#Sr+TG(rs8v#*)8evh^xJG?e3e55>9) z6WHmt!uBgEkd}UD%IX9bP~mZ80ifK+ePrFxRGt_R>iHkt6A*L0|I!(qFt5&CIfd-8 zE#yASP1*v=o0$c>WM89O3)p4sPgV%zsG*)M_=!i!(cfMS`zakJI|{0?CIPOLETHW^ zNT!{iaYoc{m&JMU~O(TA@c zo+LzI`*JI8r1CO4%y|^p7=%#L_+4{T`&}QfuIF1nf0fg$ge4Xms~#f5?nf2bLD~T@ z>?j_~A(OdpE^39JzWyjpx8f!DKK#xk;JQO z+0{Xl!ca`5j}hR z*F18u)P|c>DlY*T=(s9`TaDFHVewZB5X?W2+&gC6vn+CRJSOmF0~cB)14HLoxO~aR ziKp#s9bT5tDH-I#m8p5|_(<8TDuGr~gsdK8JuK{wqhaybVR0Dc^IBU;$h^`s&I>k0 zy(>A&4Xz9vX(Av}p)K z@~sfD<-u~-RJAoht?e)PK*=;s%+(^JCePl%!LH)6<_ID-op&$;Q?=HSLAa~OTU`ml z!|FJMo=EumT0Ulnph8>$rAzBjhVZtG6B4*7E{*~Bxyp2Xuj&wQDI_*4HT3fJb8HOw zIsLf0_eYgEO>ZtwrpisFKP!(7467ZQp*k+f7k8;PEB zGJ9Jnjc*t*_Z-uX5biz|q5sV*59dPYivGJ7De=N}@dUfTmXbAW z6&35LtE+$RX=rw-&^%J9P~##zx@I;yUPhXeJiW?o;enBfvoSN9fXdktlx+!5d=Dtg zcJ{77k-a9)W~4o!B+i*6<4ag}d2kIosUqM}EmzB4UPyd7@3Bthx?3|@naHHt%`0Bm zwc>WTGd`*|ZW$g@m~3}YW^v_S$Br7v^)gYF0W9XKmpB*vX@?;(JK ztQkG_hues>wR*euMud|qLt0<=GNrH=rg=|H*L2ORAI&~q6=OvNpiDgY)38S;DoWLI zeTV_q3NS6PEO~|OhLY7@KpAT~j1kF~_?-T-Oci$H0DC?1NimVsf=Tq{)4gl*%(Qv7 z`wj5@01F0Cx*|bhr;4n`1C{hJaF=Nye`9ZNAKatYqi;H6CEcsoa=ju6{Ehr`6v*M~ z8v77|gC-FEBE$%WKl(k`QEu9yrlA3^9fAJsv5){pI*=L^u`mQJ-gaY|7gda!yY5a0 zvuNg-fTOSbo%YHsHRH`QCcQR;C}2juLP84U@K{M-N*oDkW|=}UF>p${te(GyMHM{{6=%nlc#$`aZmk;@THi2 z%SaVh60?hp`9Dl|{3zDtw4HAx)67@t0{O1zua)E_V$=n+{fzzaII3{MZqf*Tq7{Em z8VQhSahrEzR%jNDWqSP)thW7y?>6T`q7|(Tv7ithTiFID)^hlT?Dl*G2a`&O5KxJ- z;8W)5>BlTj(5E+>;47KS{|E2hs68rrvo($e!uffR_^p2uo37Des#Nh?#<_rm8pbz+ z2Hmsrv+j;_fmT2AkdoP-(YXq_Q1>`NrNGjO=fT&O8?59U;;FAw))4_u&STEA%bBz5 zw+>g7fW041Kj^&%r);ZC=V8W>$yIOmpZgNmB0uYW8^AfXsItrk!X$Kdd+hJa$ z@#N?86cLMm^QZoL=ac<+p?F*;hE_dkjT}|Jb zzA6nug35iaTtRZVo{*8xH^@6u`=dScsjzDQcVO@3R^REaU;SuNE0dg?64bX(<9ihX z`DeeZGdD_BQN*)0w%97V<>e!f4f`Mdng*WmFi64!uz~aAG1X0(Jo3A7YyS^)!saSe z`SYVJ1_*zq^Nk+vfT!&9%S??&ZH^0tMHzSQDAb+)nTrg~6A2nRO#`o!@FYD^N

gaQ1BZ--rnOmV<2B8-o?%e}M{LlyOr!00e_g#8os5mfejCpeD}P2pv_-cgg=S8&?Aqcm>*;D12&LhlcSpb=;Xs5V9g9MJhqx##HS<_u z;ysTggkv-wF0)w~k5VW56IW=`nR)_%fy;GO@6*D2?5`gHjZ8K0F4%s$m%#RaK@|hf%E@YLP20I|L;$+w0CAVk5gq$yg^mEfz-F>D z+M&A$DUSu_gQsahrqQ92GfK)=t=|VC-*bM<$)TlA-IvI?PKakcqvh8+(N^RCHeyVK zis;xw<+ccXO;0Mfxy%rU@Az7to1!)sd#mH0xz}7+ytFxMNlfn((_a1fc!x_dDkm?V zOd5yQ))4nSP}=usoGOfu43eUSHm&aVa{FLsY&c+e&6L{x0_^tmJ61-Wre^dWY^siQ z!PlKCq1QMqH>1(A`Knpifo*Ed-&E;xIry8Dx3s2`Uk0)&${N#gk3hz>zMe8%+Caa8 zEs!i|0^H@J_)D)(CO|?_KC^oc!VE1XVfGUZx+QX7US8alhT9Wzy}@j?afuk%7KaV^ow^ENk$t!^dCL1cE84_q=aS&yGec4$Z=QZ zX74CGQz`}c-YYqyz=`l;?%x&MfDCT~fwoJfcxc&epDX`;n-it$pV{yfV_!03pA({} z?miUJ#OeCF&89X2tZaoan3N9w-iY6HY{Chnf(J7adJ}_25f5}p+uNl|^eW&Tyswb| zVkuvQ>?Vq|dj|$Oe|~vx`sn@QO&bQW$%xxXWCnR5l6eu{;M`!vpIw(!UTb`y80l0g zQ~}z(#{y3&G9BoyRg(X+~lw5i-Vs z$XIcJ?Zy$ihT9IOCzX8Y#aOB^3OD3A922EuD-uNUVTkScDCfh;6O@2;3mM1X+jXmd2Pcpb-hS5&rzz z_V-Gm_$mKoar})Z2a@G*rfl~R=x7-u8pgd{{&3$O#>qpSmY2BS|$#r+4sA@of1IrLb>WRk_Z$KwtI zx48hT1e%Qs;Qb8oIo0q@T5c8`*cVa5(rCIOZ5y+BdVSL_2{7Ivr}nIv0{x~fC@O0r zSk}(iKD+0Wq+8HjA0fC7@=ca6FE8u&6k~Q_;It%K=Ksz#lT_uRJ7A^P5mD#B3kkxd zET&W-&yxA@KuOKn`BClOub7>=U+#x{su39~230e=5CQ@@Q~+LIC-hJXQ*zDPW>*L} z$(gE2%*bL~Jr>5t#|K#ta>=tQWE$^%|Ea<*K2ShGor9^B*n1*RKPDWacP>(w7U_A4j*z;D|PsHc@fnwBXMYMjxrF{oZ=*RS44 z=t&K2tM6AdcUpeeoJCzK%7J)se`}&R6r|Nlh54CMMn;)n&@Wk)zU`}6i3dZ%d)WY1 zHOrU2Wy<_8Z*Bx~h3nwkz&AG2N#D8=$XBXZvX&Y|7nWj#|GZt;BN6^w2XYW@vrbfA z+bP`l@875F-aObLvSL@E)O^62vsZx9veJGW^s2NT+1UF4=}+U4iZ}2Mbe1H3JAPB# zw`&LPt3Tx9;^S#ug;wbfW{;`I{tvLCI5?)!<3yP@KS#LtsX6eq<dj;OS3N?c>qogF8lTRkUL1!xopOn(}i4=R(Fu4s8UO&uqKSZk-qa|XV*}MTHGHN-<$KY6LwpZ>oj zqA*{EgLjZu!6qvRhvpI&;HI0Kn~n{0X8DY-V#RL-uxZ4ipo~HP%V}lL9FU>t5%Cf; z4Hf>Cd$PlRO;wn418ou}#bZot`D?>Rc%Tdvja`8W#!O_D=CRP?yB?_Q(K;yXyrHAB zLHY@V%_D`HOatkH1fb}ZS*Ol87%Ftjw)43lL65Hm+JN)Zi~S#!v2h6r0nZ(ph#r4< zx>NmP^Ot2-7l3o)GcvHME)I!U?})mQv(JVz+wlL&`EJF%D6`y`(ID`ejEt;WUgsg; z@37f;bj=2j`MCJkAA_EUk10d3dnq=;kWUomUW zQ9>3c0N26GJwRy{ZoG_jb#gmz6jDnFN=!J2NuKVhvy`Kw zqqBYybK9NPzP0SuGteq@@#h{4lTd`qq{A`tSR;CpDMAKd6&`FVAqH2Q%Khu3Ve{ST znyH#sB;^k6?yJqWO%l4`G+CHX$ zpks&ys{8Il@WlF!wVFNQ16c#!Ci%~;TR^|{Ipqcvg}Da_49I^q3vNq44;2$1EZv+f zKxi5`_lM4|qo6u@)Ofj7Jk{(Y4C16~m-&SVb+?!tIfXXLvk*x8Yw-BR+>1x1pt6Hl z<$2#LyZFjPoU+MC^<23efZ@8E3?S^^?gTGcOgpH@Eaz;BvLfk#gqQo(Pxg&BLI8OI zKj1F|X?&57-4g`?B2QV5jc7-5DHVcA77$IrZ+5u){TOMV1Y$g)Ljid(l$Pp>Jk}!* z+Rly~s@-O`d-ag;0n#iAXN&Y#-3{*{VAY}sXiWXGbBEylXIqsz9 z?mHDdaTyuW;9`bo8w3a8Mt9MU{EOqO2FDr$Xz6@8k`t=6K)Xb@fUx!R$`@#^d-Ksi zxGMQYr=xxnZ4OI_`Qe*(IydL}O~dgAXO#`8oVT$GH2KX1OE(`g@p6BJVNpC@?51$& z1dPxAc3DeA$|qE*99)PX2bmSfHFG5ckxfO1>t{e5K^2GzT+_kW`LLTrJqDtT?Ns?a zKpsh%nwnx(C6)p zSb%xy0PgevB-}k*!kgdw`r1JtEe9_heozpR7ggIJQm~SI&eNF~`C&emQcOdP@i(}k zxWY9`KnZL!AoLS`Mvg*isDE$2?&?p&0f0$OfVDym8}Roo(p|K4ot(HIJ$i(TiCh6V zG1BNIhpI)OMt-CS?ZaD)nVdV(XY=8MNT18&w!S1bh??sN#pz)dwg$qLHu_kRfWZ&^ zE8n(D*%JiA`G#LhUY3=(l6_g`sBiMy?e zjSVX{wMZBO3U)TQi%#`NUVwV}=CU0!KCT6M4WSH3TE^za)IGnV=b+MpaQ!($)1bGA8L;4#RII zkV6>a@@&37REBN|ph$Z(RYkRTKkx!bYeRTsR`QSu(9*LDnDQzUxl$0cq+P6Iy4)EK zs?#t@*X5Z8H_kszXGRc=yD?p}%@`}_DgbFB@;_1JEvviacXD-?`@wu?!3rvr4)8&P zK@@UC!ZhwF#=iHOY;xLhf%qIcu3a?NOn86FfB1f2rOQQ{FE<#klhqGd@_F7<->rph~ z)MYKVLK+$xW=X%Z{>3Ua8ij~C3DGOYI5RCuZ%SELK#>#^O(ULJWBV&%ZGF9mp+%uv z*zFHxe{Zh|kP_Vy+|er@s_wtS{nAZnjhR1QXooC+=ghQAjLz_a8B84j_@JMh*Q&L$ zveM>5CIjpiI)|SCoUH|^I{J#BGgXQEOQ&cvWt$HEPKU>6_@S*7wwfZfqajXgoIqjoABa87fHTz#bZvi(P5w`}U%JE@;1u_pBdH%e6M_6LmKwPX_)}6qp zJd!Jq127DxzGphB_Vyg0w2J`3s1=~vkp=RGS5%4J%W;nHQPP%#7Fn}y$YMb8Vmh2D z3ISw>ZgWWWr&Eu*_dzYN48UU4TCxz!`MGZWdtBeX9dP5~u6)1Ll`xnKZuV_vq!KTE`1Dx!wL4 zK=TFqwwf4ab0-BCW=wR5?Oq_B?{j<)R%5%+Ocik71b7nxzgzEjkD{<8ak)&}5qG9* zFhI3F!FXkGleG5<$N%-cjp?yLx>Z%6laY{<3n>tU4zVxQLMEF5ZXrJ6t6c$Rj}xQ@ zfam~F_T@qfFj3EA#;^0;?H5PukOT$@<0$~}EDg|m8SziN#GzU#9&u!M)yuwa#aj$F zog}-L(5_ws>O#Gbt8yq^Fc4%8l4Xg$*-95r2LLa6m&Ki2OW0o0Qqm3m6}C@THt>I@Xa)DU!axYl2M zeI0B-iyDeZ>J=<5>K~^E%_;paP&5IPVm5#Do(n7-kBsS(#Z!M%k^I=6 zY8qtz9wEdYnVI0#{JQ7^OBxA!1`>qn6`8%R43 zBP<)4R5RIW=Bj|(hLnK7d$xmufE(ls`g3G@M5iqk0?A9`Ub021f3?+21A(t<;ipRt zg}uRopxg&4eW2PHaD9=Ec@u8e=T1-3c+rG1uu-rnB_Pd;ml*^T7QoIfe>{zgBmW5Y zH29G^nr0f>=VzY*9w|u2O~Gdq1W^(|K2!e77UIYtuwi%R>k5GS+rK9={it(VLHByB zk|~1LvJeaURC0g|BIUQI)G5*P2f<2?;cL+daB9zgKOYomIA(ux^Mr3+{AvE!5}>(T zNsPVyqoOcHjz^^3m&2Kg_@sF>03$l@wI>QZ1%;}}0U}^CotO!i%4F(s>T`<90=}sq zt3;s|VzHJhjDr-y$!3Vf-S=3+tLgE_3IYNGCkS)|1vWG6{lZ=k$dELa);d;KSCtzRE`TbNG$B7!e=@VRAZWjd; zHU@q-C+a@YY5N(+Z2(SU@gp!LCJtuv~otoV^+ajZuYJ(Hg%C zhaqc^i49y6xFnEk0#h{mp1^4JvxE$)5kUHmGZE86+5ri7K)HhfbfZwOP6!l{0A|^? z0J}Y#j!tt%ToPRN&&ki}=0*)i`X;0BQRQ!og`Gm0EjdXTBzHHfk2) ziL0??SINyM2ZG0Np5!*-spfLYg=aK1{VyO9F9`Ke5Tz7$#SF!v>0asxJpdcNa(=J^ zT;ZZXsdtc;nHe1v$G};bt@R~Y0fNbcy`jl7*O#^94pN*81b6#WEHv(an&IrW6|60p zJw&knA0L|(F-`@cp3xIIhZ_B8Kpf_Pb*9YLupyt_?lfB5m~UEP2>97_#U zV`uC!0?{L;83a=7sp|3wZp+-Uqast5K_u?VvOgZW?iFiRimASC*!&P3hJFE zVcsTVoaWP}HLv94Rrm&Dgdz@D1;631Ue z26d7X>B$hLFAoa_v*U_m?SVMgNuIxytvu0|Ns#46`0cbI&e0BDOA)%TB&_9OR)6sa zK+G!103S(5?BgJMm-rpjM0EWlS`wfB6V!_Xo~#9NG0s$=*%fGfUx9~*1&=v%ojmqm zAf{nrd)X-N(N-wPU?A_df`0*mW0=xQ?Y4IoQr0?!Q-$N`M_dAKf~I7ou&@0D){pMw z6htK8W1x}f3o5}ca7bZ8BE6Vg#H57LLP+$|-MX^jCgEAs|4FVvmZ;jXT=M6CTq8s@m9R>RTR>Ci5*J((XW z)u9`L`P=|+f%($t-plEL79%7VZ zpA)luDb)X1{J7!ujk0vFjFGwQz)>mwDBKIM5QUI$IpiSfPCf3k40YSXF)D|XmJ-%) zq8rUtQkR^RL$UPN+|jZL;zkyWZQ;W5$0Y+_?w}CyvJR=;4c?o{+bkcS7)Q+SMpPs8#)*?nJUiAZKi(B|#D7n_DI2WP*efy>Z zHC`N?^D5~-%~T3rDzxD8YFLoj`1?c}>f{!{Tf*<(ECrS(s(?<>LO$rd|09i8I{N3q z%74B1+H1?wyM~qrj!Ht-w4k?*IG*9JoL%|#m<*WqDx`3buazdg>IaC&UEj)Q+rp1AgS!0!Bq0xeOzmS~K#|T!ucP9CDs!;2DO%IZxFjYo%T%9C%H2Ys6 zIqMfYiu8sIU=zI|cP;FnyvI;Md&+d0eSK&#s}sICLOV+Z0zerL+&qiw1Zj z^fd`b;A;|QFkLVJ35~142;2g3QY&veL-??otT*hmt!55sd*KT>Q-#jEZ&1*;OG253 zzAPiYfkEnk*{H5Mv09+%nm5hOj=G}$1>fLs%>t4Sy=a0k1E4Tof@b%~7a}epQ||XBaGSc_P~VO?aF_=HFy-c}SOyB2rp9BYbwFVFT|C z^3uf*dRNwzgN$QZM-Dhcue)V|vtTFXiMrD*A<6dV*s{koBMee89JaIk+pQXw(f1NG zgXf7oa76#_Bg_Oz|1>1A>SIIu^}Zq>@0Q2c-7iUH*shl-z)|6C1&Q6D#4>cHAA^2- zAb*Q+ejf-7sD1??qc?+&s|uDxkm95?7aLvFhbaDB8del;>u3X!vG05}$d954tbyM` z0X_%V<5zp=8=`9D>TtbQ=f%vMc4!!Xsw%Q3>>`dzl>1EYdfE~2#!=L0Nl zvj$1258Wg7aoNw&xU$Rb33>4fZP>mkniC+R*TyEU5X=ZR^ujaHE(UQ{99+OYDaZ(V z+yuNhIQL$^n%{A%Ql^UPk|Y%<6LeN>NZU`gkpD9kYHs8CO);K`a`5e?d}}*F*{+wJ z^Kd=uJlEIV^a=sg!0xr^h_VPazXvT0W$V~Zya?@qZ-bePq+bP28#!qH4tsA-K^hsz zrD(6j>PDBJT~{}TcRlR=^xXqOFWTCIGK6t-c;YRoetvzeZ-xnOYX>y$mh9N=43}L6 zABSa<#J|?LRm>)GjN#(4BC?Mo^*1}Hm|8Dob&`Q+IvcsYyU{$Ym>0&;9D1jHUYDKh zGD{@qj$5wAZ=8=JFxr0PTOT-JAmm0%1HXWlQYhi2n${2aDtxslVy>POQvEBYO|Mq?Q+b##T^lZvTa8leG;Drd^qM^gpzO|pgS3Rn~rS|aLN zTk?e-6a6=$O!+hFm;IYv+p(VG#S@;Q8W&p6;*0T;GC9e%Ibps=@tm^nd@XMGitEQ$ zTWC%S_z{$Y3WUGwCLWeuvU+P2oV$@mA}9>W$=9z4ny=&s+*kV}qY^GQlqh`;7VcKk z>x0v%s&z_(WbOpV@{>vv7m?NX#qTGAFim5fPK{MTY+_mHjVbuA2y7h5c|HI9z}bC7 zjFS4|S#tW*7&(~jZ&SAz3yD;`a<#7-3#u%yd2_?BJ9^%75u?;G9n=IpkYoB?(|LRU z^K-_FtP308ieGA$t%<>JgTE0yBU3+89_*v^IY--z>bm!*!d5x_1Z^#5ow``%meZrg zyGd4u-hn-c;2ws{;6*L{9!+L9O!T%jiwBJV@YL){7lrHvOn(Yb%%7&WsrB%Fdo*~g zMUSoEg3FsLx34sHJP{QFt+qOWpI6xw*X9kU3`?$!k0M%dbpt!O4y8GFCRC5rXj#R?VDZBqmI-i>>| zEo2|o_*QBEa;rMJY-*QB`i1uV>9F%ETKkuZX)HKp+RHau724c}*MqV?xZd}NPsWd& zVvTOQ`X}tM3Z8vGQ8e6c$y-)Xk0(mXc)5mMuhiTg&jfc5Bj;c!f6r>Kxv7%AHXvN0 zO(xrLcyF||RifrOf=vQZ)%f%z>zSzTCXJZq_|r`}#T6Kt?sf42;|6wt9XlQ!VMu6b z&EBpL>pOJvqwdvbzR|CbvPXK_|c%9h;R@tsl zAJHaZ&bIjf+0UhEsrRdYY>p7V*K$m+Z{^bI%Ki+U4Xi%Jz`dWPB_$`e<=))37C8AD zx?6R@t4ZtCOsncbQp?}%j#&KMUG#m|=1i6!_VW%pX?*$m_2eeGJvkR|-?o1Hd*$n( zl~aHNa@$_)W!$>X@al>q;X+0JJF|ZK-G0N(bkg>1*3JH(4op4|`NW(+1H!M~zYn*O z3pbGpRS^mVGV1I1hpmsZeS1rRBSdd{>4O7|x%c<&wc=$qJyv;kV)d=7E#}qRmX{x$ z{Psqdo#E`t_cco*FTKMG?@k8=H21TX){zy86G?MRH$P zxZA$6t^E3@{?{&p3ditW>pWCfR_}=3yZ_bHxifxGyr=&2y|?Wv-7Rw7Ees7^d%xb^ z{^i+0^VFM@y;~dwGOic9cC@-dNz+@m1gq-|D-38=i1P0IdkI5>(`1 zN;Q&<-B%O&_U+qSz)hqCS#(ks`l|K(iSPy~jgjW<18@ zf8Kq6zxRFrM~9>1;Cb%*y3XsIpYwBG;SW?5FP^77kAs7A@vf4B1`ZBBJq`};tA7aK zH(JbR1mQmd6Jfc`x^fMy)SfK zYhxbv8LOK<$8UbS`qdW~+0OeIPWPa9zI=W%@MEa@iLHWy0(JRHJBv+Q$d*lWjCWL5 zdwb*X@UZo;hgp3D1rOe3GO~p@4E1g^TDCJ?C#94FzbeQ5Nzwf4*jQXP^>+&CuDN-6 z^Jp|W4i(j%hYC({Jlgw>GvSR+Q%&NLk(|Y7);D#be_?QrpJ>~5wG5G_c^H$BY@tX+ z?k{X^ZLsNAyQQ1PzJC4B#f$D^xomKsgUPk>$kWf6Y6%Y_`tH)3l{ofa%Qax^ykdW}8(Zs}(ZqJ%^ySI19 z`>mNo3ba1XMay8>w%fwm2lH{@GtTeg(+USTJzv}|9#K(@(c{#wj;r(CUs~n+{{4G) z&SHI`sseQ$9=y0E{22eO>4u>HDMOUh`2DiVf=4sysHPxW*JF=XmhxkO2SWKhkKpl4uUP$j=c6hoY| z-7)61cVZ?U>8Vrg#vVv;@txn%-r9IYG(1wX!a?63maRvJtZTSP|2m(^nNVUnmYX+U zl$y5n!4Jn$9cpx_FGu;hs1tp2C^4(Jcrr-XrAKPtX>;Z)$JC$G$)B0>*QHUS>gwv9 zNdlomA9dkr{)4s)9+O`K3CrA8G!T)gs1TJ|bp62k-@1i^lcgA)tCGY&Y8ddle7JI> z{%E8jK!!@hAzL<(U~Q^SqBu`8TWR?coPZLE)jn^bfye9py!fTT=ojc@;gm?s4tnRO zJhA$Rhxu#80ay$g*$+tNXQ5760e1stvzb`T=MS5a%=JR~&`WUodRrsH_G1t7HJC$p zj`zkLQM-fMact_(;p7aqLPRV0bQHZn$LY+F$xx*86Fl;oXwdy)cm3SCbH@ks(W9P{a9mZGr4-EWup#n) z@u6r+*SGF8i86r-+u|x_txpap2c9fZYztZSGrwfXfB<_Cm>({Sm&YUT6U)8$#=z{`gdlC1@RnmS;(Eo5HZ*7%R z=Ti_YCyqwKLlmV?V%bY`nS>;U;qsO9G(HAqZLDa~xQ3JNv%;3$)c!}mD>z<6JJw@D z>iv#fmVe%m#w?INDYv<8VX<^Ix@^?$C9PEQ$jIZl`T4s_N{u$M-`=hS%VG6Dm?caWvc03Bab@?{py|MS z_0eDr2U_oin6)`ZA)z&Ckz&fIZsS+TTYPH(-D4 zii48XzeCF!bkq(N8}>ZK#V@)hV$qq1x=M%Bl_JuE-2k2JuXeYUK5REwJUlqy^<3Bc z`xZXip+e*KS2wiv@*Jn@TfTjB0josm>$Y5XBLr2*4wKv@e!Me#m%QQ0ivTXJG27Ob z7G6X&C%eNO_4Us4cUV*s8^cH#ao+3K2on<1)|zB|ovQQAe^P!Ej7>nR;EnHkRp=Gw zd%ULKUUJatRya&jM|a&fF=5E?KV_fiB?j{%#5kr zW{^QxI3DCpLP{zhNh#zt8%&WXXnh6IilSi$kk(#6US1yQwQEXg;_jkOGdP1E9(C~5 zD@N1WjeQOWeSIAl7k7eUVr7j@PEKBRzo}K2h~9Ck0*mT{zjD-Ty^eltc=Vtu{$yQtC}k`t=uGI6 ztJy?T@P(>V?9uo3_DT~JJkou2s0#}_9UK8CySGhU!qY46cPJ>~u}06w7XyxNyH}BJ z(aR3B-bGTl0;SXO>dKWXqMqy2&d$yQMfyiCO0}u_v+t5hpZwAMVn6;(-*Yl}(8e5$ zQ*v*_6n60Qi@h$b?;4+Ks;I1ykretxiWdS+&3MS+eELqa+_x~8V4&E4Hr zlTtIwj_rMm#Dt!tn3$OPYAhCOW@h#(E>6M7h#pSZeEDbQ+GI_Fn#@U@L7mU+#)g`U z3qSL{IC=qr*o6f%Fs0(eJW!L-j=Am(ndmN$-K;E{6ESxX?bX%|MSnKKSyBFs^I){> zmm^x|9uPW{ru{5C6snY~FAALP80t64W##FF^Zwiv-B>h^%sBSWnsz3)HXaNr*G@jpG>U3CRZ6tM2g=2&_R zOY*?g_+<9Iz@SYbZ|&W|iBsJ6)hO}Ih_JBArzILc>&MT2;k-XnEEBOVE-ft$QouJ0 z7)y(fzms$Cji~pw*L|WYV%|Xx z&g|RE>jG9|pI08yxt8-maAXBt@O{~3(IIU>zUC>)eY!IZyQ4pf^o3);K*8PVPwFqk ztQ=oE7j>qwjV_=Qaj127WIsDhCYyJ}>z5YVZq2nSC@NZ>Vzy5=@Ch%KxS~}F#JsiIA{`%xG=*eiEQ0ij+(Rvn`%7pu{{%8B~ zcvt}`9i3W+v>%RD@4K_$R%amy%t1C$s9zHgx+gB^34Rdqz_9hMtvXPaorPGnKG+i| z`q@U-3{;cEMr~k2L&FZ1@HTb!-wfz>p_#SSv{2H0x>Q#24^~CJHJ*-I#Rm6XT(++o zHtcjuB=MON+_-TA5h*CU`9_42swxW?S0spZsV772!Gp^cQ$8GU)-5Kb3tnsABB{#g zYTQ=(^wi$q9i=}fVZ8GS6b*$szUq1BQ;}~uAH)V^X%PK2zu4*g8v57)Q6R~064c4H zgR4TXEPQHhv@A))krp&0M>9`#^#185y?O9-Jd}cTLiMi&VYZ4xur`Y`l2-a`!y}4;ORhxuP_vP+c zwMgPM#zTVWul;ntJ)NNh5=85Cq%AQtB9u{B2fdq3ql}E<*v!VCm zRq$#V^PjVhyD3OxJ?lT}R=7Z#;X2zCY@#clo|vcvTHp=8$yd+%_Grg_V=7JB@9-z+ zPGfy4r){yKvseID68s}GBmq68`T6@z~6p7c|? z`F4IbwfOXO2l3}h=?|YG0X4Rd-+v&Y;EVHed?^%pL=~UZJV*-sJ9aysi`>Fsr)dJeC|c% zmH&Lu&D&~RI?A4-FJeuBQ|)2?6+IiLfU^;29jfyWcEXgt{;ieREq-HT> z67}Xbh?_%X+ov6=S@!p`fit0WM$^*$cNLgq{{29oGrhurLYdRObos@b z&!4Xy$t`vAr70+E%Ohz?*o_=U`cJl3rt1UJ=R;NUoYffbGzMVt^Hh^m85p88kmE9W zfR=smRU~t?$_&ZCB zYp=N(ithe>iE$c1(?!`{&Vz&%ruBR+E~{R=cur&y)U%f2aF~^>DP543Ul^{r7B-&QQdh*_YXa2<&R0XPC3BxE*XiRkx#{y(3GU+J(#8BFM zEoeu&Vsg*=moHzU^oOseH#Bu782-=}T~8-QB%y?>!$~eDFTX>b5Er-bQ&drm-|)#3 z%BQ-EtA}i;Dwbqo%fb1jFn73NxEu>}b5yLA=O8t%&Zw!U;!5arp{C8l9?fTie!g>9`x^zMutBXQHuqaM;4_ zfAheN9JQ@hZ`x)2N6qh&WWHL6ztH_PMH59AdV131c7>#h98o|G`W#|fL7K1gK|yi} z^%NL$cWU2V7V&A6-)}!zi!OeUUD3bKQ(FEWyLq&cYwDGH~C5OKPcxHCy?AJdlM*URgmgk8=itzhJPFr>S$DocqI(H!ubSF|og_R^` z0#Vr*yzqQOy1lh$JPm?0t@EFMe=N|adItA1q1TqxNMjpmWrSqz2r zhIIiS3H~v1CBlxXHf>R7{0?qNTXrYHT2Uu&O&b*E_E<{%!IU++%H{FDG&DL}tV$B2HK)|2Wq>WKa<8LoDh-iM*R!BrG?CkG*vSnn!B1P)z zqq-C5KxKagaf7eQ@^SaQ@arPgOvSWH-WHp*^iOlA^WjITrR(_p))Q*OMNZq<#1bBf zV5kY2h@l4WH(hv;`8;Zo3olYPxBqzQ8o8eIS4(*G7)nPU` zq$!SlbG5j{Icr6b7ClRnluakm+P0&@>MV0|l(&UJI~`ZT5x@ScXrhXz*q?<-Cy%Et zP$0{3cYH=(k!JpV;oK+s;^ARt>d0!kXXn8Ud>t*@obt`>_eu>- zFCn>1(X~CA8(W#tn`Mke4Yo2A$stA|s@9*&Yjv5^@RDLwqUyZ5%|PIn{g$naVeZQb z`&(F-q+f%xQ4&U@{NUHl{B>((_tVZB8*h%;=Ei!Z(niZaz9+b$CY084@{-bAhVO|b zQ&{3(h$^)}<$9th33guwahukXKXwOapS7PE>bbu#V3^xqnILSxmS^I3QV#okcP9uj zJitPNf`SSH%GBuOro9O}Y>SOCGx)DlQNOdZZF{WBpU_YO%!YXshZ)=7pB?9k+WEF4 zvb{DDhqu_V>t4eoBs7@m3Bi9sP4Is#l>a+e|6&W zJE~%%7N1LZuEFFiF5?~1-_=oWlxoV#_)hPetHM>6gTGTyo=@}DpP9^|5IwPDw-Nba zD6J@d)w1rHb9Po~xayCyT_Rq*D#>T^)Iv2OkzL%`-Q{{9ZMnHLJbXQ_*lFOLzJfxh zO)nmHix^L<@M*(mgZ3nG;^!ZXS-mW4g#{NBop>b=**eo~;rKlND${MtLpM8F+4on= zU4w&UE3g!!b;9v!V-~SIYLd9h@r~ZQnk0O8{<^fZ^H-Jp4y{LOuh$=SyHQ5F1USCe zy^$=E(J}B{730)=@#hz=+S7Uv@zlSyaQ?t-bEYhQA|F2YZ@*VfJH$`Z)#fNtB|Bq6y4Vz7M*`B_W$dndfzcapUiUFVJ!6%6W;} zO;{m|{~NMy7u)MD_u`Xt2gS0&7p_T5Z*In3Pn`~UF}RVWI&YBf)#Oj1pzz|~0+Ck0 za!lfD9kzOfiFPr*Ceys^+7wmBk6;BbGsTUm&7GdjHaajIFS@_3S@NcK2s%BGry*#g z0m5CDq+nCX^r%&QH*Z>t-x-3Yx3ZgiT%>`zmp^EJaQpLW2ECfZ=xx{gVzH|C`Nn&b5>Mft2dV2)0+&%u|>W2YVz&z-Y~ z?tYNxNVvO7IPHI-EjmS&VUFhCkNf-Y`O)X8H9tQKo`L8T$Lk>6&164yNx;6yL&@rW z9Z!4BM)LoBT&ydldFE@0U6Oc{STD_fvwfuMPWO&=_Hvj`C=s0l&#gi&a14@v50h_o zP&?DjI#C!rw^(>9p5GnwrtenqJ^YZDOt_H_e7sPg1*y#(=I=subXwbr^h+!vVbNJy z@|FAPkF``ExymU@Z3PKuMUWB7^y31~Ej4N<3^)okWeU2ITc54eilp9k$tlxX(=O+p zzN?ppMxRiFW8LQ{mg2vLU-Y_o-iYpty;L_U&2iD=9fyQE${=xJad8 zGArk`lA{+2g)|8zrB7FWUtb^2HZ|grX76F-R&Sxx!)q%9gX>M3^9O&H4l}M^e5q+@ zNSM?uVt0*GpE7wh?Xuz$;vPPsx)Xa+3=J(ip+k|Ord$FW6&fhIXxi2^R$k(ohlphL z@_Y5|YXXY40g;R$M9+w2f?5<1Q70g2sx!@en|!&=-W2`H3LIn5D_$=vrJJGkk;V~U zMi%l-ZdamdAD1>kPScaa{FuRj6RJQco~~TI`lil8KvFW5Qpn~(gTEgh9v%@ZSurU! zktAOJ2-IXQ%7-GFkp#eFi*jpg>(*8q zT^WD?CZ?u$)zny}q|yLs^;jGE3h;pn3RSz3I2TFlg;VWWv2iXOF%!mvgIlMi*RyG( zEPnew%bl;R{bGNE`KgVYo10%d zc$$YkZZ0N(d|ygu*#|La7-8WA^rji38>`(fQz8Bu4dHe!IEq>f?^cc zYnra3zH95_7!wf}rU@U8&O}^vPm1b1Nm22o-*hj{=OxZjkJ*Yx+HyQP&%ii)vJd(E zmgHCv_gwHN;Z#Khg+=jPl=sno_iALu!aKlKT3T8b#JGiogpTwMkGFk)3*D=*Rs8kY znkky*!br-&V!7bvTx-NrzhX@TYRGf+qzqF+`$n<5>ev%6np~92Ziq4Tk8o~d56Ru? z=Y9;A<;j?Si_2Eac+5%OhVHNtT>1Tf8dkSFZz=O!viP7VF}XxbQqg2uc`0`LO-JfnV7eZFz>Bw3_l4sAJqwBgS1HcVgz+8q=( zbv`kbmX@}(JVb|3?HRhOsHp6&l`?3@LRSL7>-72alh&SNf+LkNCBufH#4=3m^z=R5 zGdiB+_Y%0%(0^FOGgG>UOAY^-c9i^182hV#cGd~+c@cH&?sx0b+WNDJ#iBbG=`L|- z$G=miXhTCn$MXesHc746adrv%Z#{VcoEW@^WX>6SKY?au3&+$9jLp$B65@t@S}lo< zx4h6aM=`Y*$RYJ1L80DGj}R5A_j0CL!|AL}(+NAAA=D}h3kw8~<&{Zrk@7Mnd(6UM~UB90q+*@KSFDEBw)t7w^ z(9TIAIth3f`Eq2wrS0H(0^*PyPZu8?X*!XL!lmS5Gy&A2E=Q5!FH zCe=cphv*vb)4Q=O&g&Cm20kmy1bV;pqyS}RU}O{n$QwEwNJ9cgB}F*$dn_v{6;%hc z%#x&hJ)$tsLhJy9RoTRZ!4=(|0~pvaYAwaF{yVg?M1B5PsY{v9ozGPZ-CjJGEPS!g zHGPT&GwdrE*CG0Cg5o}_vwd^O*JI4^ln5(B6y!p4IYl+=;s_qNv(&||`-Kxw6f+A8 z4S++EMI2S5X{Fd6eHeH|C*l6<;hiYzyC7%6OVn=xi8@0-;IcpINlHyk)Cg^kWs9eXr-z?IyXXuc8&ZD&sC@P6GQf@<&+rLdIx~a_M}{>B2?z{!N})p~iCKuP1yn@U zFRG2C2V(e-A0sw7~^ikpE zo{-95TB#C`HLFw6!x272JJ5=6{afc%0{8x3jCJ~K&B8u&UNs+VjAZtpIWEx zHiYK)k4sIWFo2GBLsz4{z1;}lG0CkKQg7)mut{jue84$H&pBx1s#Q3uWJv1RnxxXx zUp|cbe5GcUtl$rUg!akdSIZ2jR>a@wWtV`d{eyG;`t^guLvg_`a>3_QrTr+M0tg4~ zTSWt}*|X5UjFZZRc1+>p(&x~fhEAwLEX#eIBZ33KC1c~`zX8Hpx%G|8wbQZmF`LdO zGmqTDT>D|4HBiIMe<|T)V5oZO~K^5laR5 z6*?C=5A!vScQQ^FCY5a|XakO&5fcKa`g_nhQm z_;b^$CHL*5dfhIO@D&<1_z!$|6p>)y!`@R0=aCD|Vray3_^O2nxUEkl3E2)U967VI zvkO}H1w~ScXq4ML%3F+?xIZBn1IfX*LBsw_^Gz|YS0y#;3@6bDw$gBnzLIC~QG z!#2`vbA2o-PYEur)yZ<(H^x0Lc)*oLDBf1$JSFCT)ayn$?0S0Qw9T|OY6?k-FH(m+ zJ-HNi{<5 z3R>1iAHRMLIHs?sB#x($RI9}Nz>SlQ584&Qha592vkV6JVMDKCVxDzb^s_&n!*X-A zZr@dU+aj?6I=;tAeoeId&i9rEuM-^*L(f~;604XMGKO_U_?8p)*c%TY|9;zL=O~u@ zO0?3cPa~<^Z^n~OJixEszi+lPC+gvkq*N=t`2GjDQFJ0{3N3XAn0AGf^xXOL{50|{ z*4!ZU6_Lqo&C=BDj;x(J z^B2=zEHdFf<2}&fxZ99nG^zsY>wxt+-59ZhVzZJMoT;3N$vfe<5p8!HFcv~=Ztp@o z*(EwjPL+h{zW7W3FmJAj~%5V1XZ&YZ4%@qF5cjee)PMYev$U-Q_Gi`R@C z=*)LU67n;CC%)u1Lc6LX(6ao6l-I2sP+;;{*Fhe%(Q$r|54zcvbh0PW=Q_0K;eml1`+6-Wjk3RN5|!n3jzyIXQ{NX3bN- z`HMVj(?Ko8JoK>{vPO+x12@W0-i=ci+LE@Ddw|=qT3wV}+@oZO6aNkN=PWurHdp_b zkai2Iz*+-{$k5+Z^$TR2>jTT7`+n5H^Mq&q3~}VWoEOt1SgFpX-N{BS^5uTQddw0* zzq*26GLmn2LaGJ``_}H@9gQu9Xh}RNpT>!*18QkI@kreZJf^0N?<1(OOJug>$*QIX zr`Qtf{)=X2k5v>&oOR%=YA2EeEZKl9wsMW3)p1I)a$Wp6%l%KqDKU7qyY!!{e>1tZ z845aoRb!-FIiUE(0s=xMNVqN$0mPbX8b;Re$=rC$1QkUajM)MAVoQEO z^6*`x%TQM5wZ+BNlBd4wV%;Ls4-s=TNAgn_U^M{ctlN!1XW8Y#{LrIs^Ic zVo~Un{AJ*h+J^_aO z(T~o`LsL*7Ly2)KbwzL~C|D<7nN3MlE+uzzDmleYXLn3TMaspGX?OPLNC#Xx-5jPu+3ROWR!)P zCVd{k#Cvi)0XmHfQKrRQ{O@rfAZ; znNc7U@I$*2qbEg&P~fyBeD^pZ?*Rf>Vidz1<*&iA4gAqBT?Q|eo(;Yyt~*Sf@2HqR z*y8B7?yTH55Oq0oiB=E}t{lAimPuPQ9aNGIyr4kkl2}p6KbLDUENjz_pyjK=M3j%j zz6}^_Y1D2$BA4%4s{G-aW~BahOg0F?9YQLV8})!gqFCi zZ0n;a(#g5MRkNe2boW`EFVyqSG>BEf7ux_VVG`leZHnM&N*3D0puzu@8>q>}+O#D~ z{`?DczDQ`rtMjo|j``ai@Zy8``@K2n+|Ozul{A|MDu%OjbU4l5^5$KS*&HP;;XS?w z#*eG|+|H8>XTrvm%`Gfk=336d*Q&_r=pKJMb9}sZB-_`2pVdWI$T1|{@5+IR0rj+_ zU|O!1kTo7IE`l&IF%-!n#w+f<%>>jE1}3H!P>Nx#{41P~aT-ntcF@QUt7Ql(9v($l z%l9jRVD4}ASwChtRg3)Rb}hijq4@@N)hpNLzS#*=L38j@q`O;c)^;)v02PMAU9NiL zU(OI;yVW??nk{H`>l@Go{tMUr7sof}H$1gOC5q1RV5Mr$_td^|8RTol9xiDb?3hEX zer-52kVncVXtU`;xY(g?^~r{!G`rz!rLQ!Eyt(ps@7}#v&k8{nz)2n?KxwO+v(0d}+l*}O23~rV zI6MtR!YL~X4h7>6DzmVBxiazhCan9dU#JxujJ@Q>L)0KX-YA^xRv^J1Y3UdH{^wY^ z&8^@zR*Ng?Zb9tLwRx|6mv0;#m<|>(ARm4JApjU{6JGO?xmjVA8LSyW-D$JC!?V#c z5XN|G8aAs;inLHS@>u1$D96hj~UJymp1xt z$c}#6oN%34xHvbp@J!lipNrMYPvJa*7UX*US+j1qz%bYK7vBI%&x1Twntnco32pe! zcUGrgadkSGLjz=CkQk&`Wyx8PkseyUPO=tP?_C`w#qQ9ZwEt~c%mpJHDx1L%=Hek? zzBRvv6Zy{yJBAR;2&R3re^WzCAoHfFvQQ^Yj!u3_9n4>|>mK?G3eMzG{zhp5`lCT8Ml*fzJinn(4-U ze?WmLN| z(V$suAO*se?3s_2Z}30EsrR1dTkubFCCV<_lU5FV$r|PA^Ov_QkVU>+e;P)X>Alll z_~^p7=+H7t2#Cs9+|8B+m3z#0_%a*1Ors0xPCUXQgPTKlwNz@O{eRuARf48E(1-IN z4<&nZg}JiCi%+lCYC5jyg#N6H{KadMEJ=Zesb;Nqpq?So#BS?EtHVbrQ4AVuh)iRq z15VT64D<1;l}LWP*QJ>9qwqgGx6~s-(`H}3aw%p%&1`ljocf>@2HhAc+%z?p1(vC&xn3GKYrqqVPI>*(UneM9b(J8hi0``$dQS z_H)Y2FZ@lqm9~1jSxI=nyI%qr%0WED>oTw?5*} zCtb|8i9;l)a#Tlh|!Cqv7RCqR?X9PfUW= zue+0ltUo(Ca>b*#caNN}{wIzip^y@or!!v(=>%!yCBs@xZb(W>_Bj7_!>)0RJLV_& zYy21%wGbIBz0tZJ9;G1tM;~__goKgGgRHYyZt(|I+Sn!(hI4W42(9W>K+1( zQX}VH63D*}5VXOY?VU_+eN3DQ>-UH9A?D!I%(1g&C zM2_$o&{YxIXmZpIhH_ho*$1@JD&eEFV{rHzj;gi4MsYmij+O0ciTSgtZXp9TvnvSO5lJALTfcp~ zLPOI9BMNrnFT(U*AOGpkzH_JP<#p&%LN6o7r|YBS1W$VLQu4pB$-pyMr67Pw_!U&a>m4T{CQwxCVW} zZbcxErtus4lL;F3w9O90SV^O*$SbC3WMh@pGQ)?5r>6~Wku9K$O%)HOPm8iN^WJr! zJsRkOffo308^5IkGObTAv;c>(PG6wu)1SN96VUWEFqgE7CgLou-#N-hFHM@{^IOt{ z)7~3>CRV6@2EL!T99m?+Yj21A(-9O$lmQ&o2^j>QQq!-`PAFV*UOl)bC6rcjz?QyY z&Nyi!u8y7bm!Os|4D4+wL&}v0{~1*fppjSD&N8|ExQyV+m6=gXcYZD0hxuo-Sead2 zKwG1P(pXN9kKaDxI5@OLykr#e_PF%)ZlJST_2)){mwatlF@d+yKwRV59LEl+XpV*b zThHmo^{5@946bv$G_2EO<+fD9O5QTmnX1wfh;= zK_+)ocDUMy+T{(@;N0r2%ms;?FTX$sj^e;=?qT$~5&Zcth!AeutNN z1Ei6DcjWli*0Z)#^Q?9EP_Im1N@s`9d2`>WZtz~-tMT^q{L6*f;)TYQ-E@80(Rxh} zQ=cQSCc94ug3wYgGKF&qYE33qIhJUzBqO`<`NoOXcSC_8zh&h{GJmV z{44>b;h}0F zJE*0C!*&qGzl`AP09$86cMQ=!^<=UBDOR%~&VwZ?$P!hnR%b_txtnI?C8s;?+^Jqs zsDk#`7yCeEU4wri0^%)5jr+pQrOX6M#L}BSV~!JQHq;7dX43lQeKnqJ3n~I`y!_?H$J(4nh(VAg4j# z5sd~HBH}UIbQUDg+Dt5iM83p1erCrUd<*4x|T_`N#-Mwfm|DQq`y{v^;vu3GJ`VM*Phm95xRh<8Vndz0E8| zHLxW^XC<$+K%eUM>zReS;5t(ghA!1Th?_9;AbGf)1>z4Y*xC4|v=32rsiAiareZ^N|8K4BtQG@DF{Di5=e3-a`sWAbX+X{N>PP~QsKqF}? z>uKVqkTe`>-Swkn7^g;0@}UowNc*kV&bFUoEs?Vazv(PM!DC?7`|dU}8-%=^gG1+v z^bygAM`io}h&eSvdOJ6dAAQ$nlwT));WZyy{Of&*Hwlb5}^OPbE zDliBEj2see7y(3Ka_Vdk!$_R|yoZ#7nf`KmdeYtb^Pyh#n&$)7`4Vz;1P&G{%DqW? z-h$_e1PuBU3gZ%TiN*%by)xq78}O`azVhi_co~V;K>hDoiPmmvi31gS93JKh1m*`Y~tc{My$rp&&$NiC0Kj5K7&x zTT#Z=)_cM;nN?(L2@oS(tWvY6lX@Q3A67KI|GuRXu>KF}(ZE7vr1Y7DcCl_G{l>12 zzdKv_TlREbxqpW+Urls?rf zN)9>eT-POq#9~}!sOMI=h5NY4r^~5CLJR|fl|`N^OvHndP*|2fNfNfFRD+|y)y8CN zC@@Xa`d?e>4(8IG$v7S8y@weny!bN7DSg@T-P&y>n=L&NL9(0LL*YN>u%FK;zjy>S zr|-~HYa)A6f{hn}AR>af7`}n}rBzs}(aL%;zn%CBCuf?lZZ(y7} zEtOPXE`HRJR-x4h-r5+Yx<;N8a?v>Yt3MM%H>moH+pvuczrE{_Zp6u;5_4^b9?br{ zfulO<#VF&x-nl|41fym6;K~QmF0Zi?-B02VhS^^AQj^>L&I&D>k+k+aa^Iq0MuA05 z82s|*4g7p|ORs(V3!29_cjk`Y)fqI0?3LjwOeCGZ%ue2bJuIj@5OoWS60n@}9nEdK z2jg!i83CU>)*h=)dojY_Ie-NslsF`Wf{|DzW@cp82XSHj=3LD_zu0|A_&w};1L8Jh zt0m80U*W5*XlydmWlnZ0m5ID9>v^k%blH)Dog63~xNk?3g|x}&bWnZ$RqkEM9N&D( zfV92TmC|;wHNVxHf6$6Rtog4 zYiJB8YfMm8V9)%Q3y=cclG64uCt8xLZ&nkx6Y4#&#~73E3m*UvM0gg&GRM3Syaj0# znY3Y&2-@3dF>Xdi#u}dINX?;@B@UzV2z%Tit=K`|2F5!@xb&2spd`nEDr@JBipxk! zS{Wd|2b`vpK0+OTfA5A!_HNWwD!CNlL5$D*$K&D*oY zLDS#=2Tdm{g$XutZoPImPuvjzWz6&z zd_e^m%jXlaDcMW16xC{w9(qe(C@V`{V3<+fBPp*E`M>0+?tjToGix#%W96$a)EZ!7 zbY-$e1VDe7gO#XR9tBf^30)RZ(L~@>m3aIfqjo)oA$Q;48wU7wwJq)KBGdLp zU!UViS@!3tjFz!pAm=pN zTN@RdS|A4Q6ui41b_&CJT~BfGMg5PxilV!Iufmu*3o?NRyd*jp&~rcV)@|^YYB=7G zpuF@VY&A*pdQ;O(6`KAAu7Tgxfr6FfWBI@deF z3)6V6Ql-Wq%a_z*uV-eAaFD^i4xR=TWau~%nWcwxTOD-e=EL`mFr8tn99ox1wOuwJ>72sCmnja_;V|+! z<+rb&?tf?vtk*REPS`sEooaMg7(HA8(b#y)AG?#WGpKEd`{fb+j~_okSZ_anj!R5T zbX^ZP{9_F=ZEKHVye{drd3L|@-+jCt-DjjfS0)q7i@^R z!Q~mXO~w9a15OHlPj287&$zB7=1FXe%f>8HdphG(t?`d*)G~0z6g^c6yvf0VIFNv& z@o6i#vILS?o;z72Wuu+hcVoj|ykHO!*-x#|6GOOMXSiwN||mDI?Lca_`8t zxb7RE7OgPaB(x!~@FYD`-gr{H5@Ugvr}FG}#LR1D$;mgvR#g7UrUY~cz3y$xV^_b} zyXyOqX!CbYi1sH-v5oipg;9KX|K12lh=m4q$q=y|pf6KYyL}HwHBGz~$c8XtI}@Ww z>x@ZR2KLu$sIy>J+c;pa>3RrEIvW_I_y2nI@FC>QEx>l`sX0FcEsP$x6+!2VU6Q=~ z^9gfE#c^Py)whz=`rIcGbA$;VKUvY)|1nilC&~Uq(~-t*?8%|6=0k<{-xa32`{>SE z?*WgMxu`uh9v(va7Uc|C%tAlU^}}U_9}WeXCZpB~)bssn&( z-aZk^D0 zKRvWHkqSLJYUSaD+3IEc3KB{8n^3I!ie_G?P7-t|eT+k$?6e+!cNy1TOU}=RjcY4Q z9v@wjKRtcm#l|gtaxC_^m~X&vNA!sruEN?c3opbALBmP_F6Mw(W9Ud#FnG_y zLm21}4j1U$;Xa2+kHXyQpO>4Et4Lt{`!$$_O)Ce9wUA9w6@fJNnnNWnQ0dbHXnpqg z+;nLv<^w`Dh%rBLhb6P8OBs(37zYNdXmG5($!@hsH_wANH-qbJkc(YPN=jTEj^J{Z z9Vp*R;2xHWkmz^^U>d7rNT(syjBZ2X?dHf1a(#srr+4=g}R}mwp8MDswks{lD$~>cln`zJ`^YvPMqwNKk z(dZvGdfS%S4s+A`9~h^5&7CW%+k2u`!3h0+{hB73({qe>MhGZWlg)IC_8A7nIy>v} zf4`+ns2i_dxFYJJfUEF?7T(aQX6Z$Aoj(XUHcvdv=`_sm{lE6UJeunNUw0!^2qi;< zsX3V;7@RYgudOTYG=@=ktEQhUfJ>&pUCOgv3;ln`ECosQS6`zRa7XKi7{!$9hTy zkfcCcfMU_y4XOC$Q7WTKxBkg%-C z8-Dhb^Z=Cb5WOc@pjucQu~D3ImlYHoYj_snp!!lMBgcdWRltv0`W@+N-$zA8-EW)Z zxD1L_%#3!=ximF!xvZzZacU@8LJlk5GO_!M0cLjcU_6^6PS!eP-I*UYTR!UOC^FSw zmkf`U1joZFRc^opi#2#97~V$aw%@&QlX*}0kozrN=J;ktc98>Wg%_b^yngL^{ta4n zq599vQO}wSE(p;vFh7iF{9VY8b$aaB8Zgw&frJ=)4?azBJfc!Hr51VR*n_S{BY`cI zoCm?Ha@QBp=W)4edi5yTiCOT%OS?#R&SxY+;b^>R_dC2PB6~)Tw;tWGLXpV}DH1ofZ?LuV&cO9UsbxOL6$ZW=37(0HlG z5nJ|{uPjvIHf$%N*4CDrTmdc?vD8ry$qM_DHXU#eg%BmV!Onr;-@^#wkbwa=c+0wT zZ0s^IRVwf5@=RE~1_TUpKh$r2-Bwo+Gd1Y3Pv?8=yt1;+@$T4G!So~{e&G{VHlC7) zRaNOh1{ZnJ{ve4qtAG-O~R#&(yNF0a}Mg@jzJ5I=vQ#Js4nd_zM+Utiw@FiN6kLaY3J z@UDu=>8qJybHircd8NTJZiJ|O#(pL1th>!IkM9?3q^XQ^{+r!hlu46QLA$w3OxRtv z(cj>F_!{qyzm0@5jgYJne$N3A?@?Y0|Pbw)=~5vG;V(RGs1uVd0w5 z>Y44ixw%Eu84BUa0>cy{wP@#U75tjj?iMBN-sZ{_&g-zz;NI=N!Swi{g7l~M&Ae}m zMLb6u%}J>ZgVw6X$5BS2dQvH0@ zPRmf6z1j6YO*liE!Hs&(0Omcavk!CMSa#P1EJ;8HaOTrJal#rdmO)go$KIp>z9*H` z+1^r5@!Pk{!g6adQ>u2{)y+vk~|KqYZLMgHfi@!UKLc2M+Dc5`j~1-7Taju5s!bPe?s9WAWYHn!VLgHgN8c zTNue+{IS2Z+}ye5jJ|eDp@(R|e0^f`YquRJc)s5?ae_Y*PwG*VZ3%bFEp=AuM+&@BJMcht6+P9&^t8%F3sXY7961njg=Yx4V0{gKuYG zenj$FPieh4DTjNq9wLT@hKpk!rM-EjRN<1euE9Lmv1DW2lq~+#fJTH+qO4rGa>Ct6 z(6r86Oge}8g01Zfy_nRCNv7?h=F?3%-lkv7T=)9U?+V|1eYxk&T)-5I998m4a0{2} zDi#)p%!C&Ei|1qy%jjl|Md{Cs&R58nR~!CTWAtA>VP5PI=eG3e>_E%9Rx;J8hI5Kq zOqy;g^6u#18{lD!`5)B3D7tRu;Vh zSzNp}bw`hg@N!*~=?9~;p7{si_=D}$dq>~28`d>ei*u80-ot6aJr|$y1uWh~6$+Sq zd@VbBIXa3Omi=v?cc=G-hUT2^ecSRQC@Hye;LD0gp33`1j)>9`uG4-t12cI?eZ zk+;>pcpacv_4YvW-B8&^6jw@w5K?k&5cNKqqU9SxxgM?c}`j?EX`Q~ z=&86&&y?RADVq=W5lo|I4a|rkDGXmA;}8a5`6XGK57JPq#dOD4YuRmIJbrB6e&Ca~ zyz0Ps0ISbTS-cDzSN~w`Cf5a%YOj&s8Z9$dS>Mk}W;AwJc8#-+KFCkG8(O$CFh1yw zr3JwI+c7cOdFMsFti48Fw9+Zn8+2%rBuG_Hf4rHXDpK0TtU z&Y1-P-`eZkrL>xf3E|D--TU{CFv^4s_G^ZjA_QMuUwip5RUD<{P5tgq%yyti`7cf$ z*45SJmh%)VU1%SVpC>&I4fVWHUN-o0Ps+Nr=Vm`2B7gVh^W*sXRj%DI$k&OycQi`z z=O-r-fsFe!eIG?1>5KON+ScQ^n!i62z4nKXA6u+iDyOOwl4H*=G+i{jIGJI!YFfNlVEMyF^9%iNc8Zc^YH-7`8~ih8nJ?W|R<}i%8+fUR?C& z0ZK06jppJaPLw=gk1Q^9F8@I1_b4mtmRdl(^-mZZDWVNnNlZ1G#2fSTTO^7J>{ocqZ9bRD^0^)&imJC}Uy z!N9i{(aA=T=5wZ_f|PmXFaHQ=<7Y5^{Fh|DN6D+*;Y!yt@~X3SRg|d^&6c%1D*xtb zjFjW@Y$q%Kwl@dWLvy<$REuul9$g6ual`%=nJtWW+S3{D7Wa|rFx+tmo9p7}X)Jx* z19zJ^K|$V=rl!Z--Q9yIUg!wh&6d8I)eq$kEOClnl^uJirN+<=&7<^rjwP$=%h3r; z?q}9|x7)v%DiNKN*&2K8{+u=AGp(?*mD|l)b6ko-@;a}Et*fKX8&PNTR!wg>MBNv) z{n7r)@5_cKd{mlO`qth2;yY>WIp9nxxH$S@wlkm(FeTA)7`%4f2934js6JGIONr${ z4j)z8-PbK$=Q?zXQ`%Yfno&gF&$+>I+o>Q{iL=rrM*bxZBnzgv%ALAV__p9H5MtH% zfKByJp4Ce{+;Qou5-rQ6U(5sVG5f3fT@4BQ2ho7v8bEpuqho;L?q1DVLV4I#9`t&w zU>J?`w8G-zB8d_-rbunoUeCcncm%>k49GAP*xfH32y|(o1d=z5WExATfLIyJ?ulC& z6u7@&s=EhBDQ569&8f7<;DMX>A0s5`*>CGjsQR=yJnr#%8oLeS7~T4%iZm|llUvMm zSbCf>W^t2KQ?aDKYka&8Ms1$(avk(e5sDVv@gP?VHT!Uc8=9-b)b8Oe4a_QGbk>a($z3)6nz+5FC5#9hNo`o?b;cq*W z0qnGXTEtC$w#`j$>XQWE48q_QZfSLy7L^8QFBBJA>IgC6>yfOF@fTw8EQ%7Wf1SJh z^{e@a!9AIEjcGyEe2h+>OA{7CqvLY#Li`HdsC$9dpRM~*lxHPB!<}QtINLeiA*C9x zrLR=GMNsfEuJ8eLbs#VkhKx;3G0&gx1WNZBIpthadb`Z5-`MN^dzi`*r59$vn(@*% zQN&Ck)MW)M`rDm4x?QbB|J&kX@V`8c;TlZlD#brg!*Yl3m=3sefGC`C66vQ zHF}dXM9g_y!fTx#oEa9R7(~<1bwz6rIh~a{bQU}@nxaVtN!KfPLz}*=zri#yJG%o` zm{6;$jpm)L*#H5F$v$yzX8dBJR(71o^ylsDcJ1704oV0iF-B1j!Q*=@nK>Ztb%q%! zDc1no5H(O&TQ{9O>)Qrly^XxQwP@@{%{EG&V?%zk{Wc5(1lxVg43wuq8=Z3G%c<3h z6(M{T>$H`8i>g~wK0L1os8@c>%VOhK!uoNE*`u9aK)|_XA9-RjDxf*xcXJkf{9I9H zX!XGAP$uYin)%K!dP21=eqWPNNuqN^px-4dY>hilP$w!X4wlUI0N^)Jvak>UxqZeD zxwK~B%M%_M*R4T}tPK3JvT?igo_+JVSYR)7IXA+{%w+FzZM4OzP$~K@0}l`Ru)$qn zy}28WhiIzk4_q7pAbNOYx#)buZpYv6UDsnl^ds94@==S*tHewiQ&O}z>w)-Lm0zYO zyrY48|AdjRgnhHvLdmJr+>XQT{X`>F?7a_z>!Kn#kR#4^Cg50KF!Q6g zdRqpS6HwAbzaRHa(-|4=mN|XwHlkcBSKAW4X;ohP*^ionXpuoAfHgTCd)g&>5GfMbG&%{MH6+h1V) z`t_K#zJfeE?eez%T-WbMNmvW=ZDBbpTH>-=(CT~~Gx{-js`ka$#cvZ6&a57;jzbw< z#$OVPMG)Rt~-phrDO^BFucwbA~D1f8Ex&%sz>PXUixnP2Z{M}zqPw$A9)*E{%Jp%)EWo4R= zF`@*p!yTc9YZje5bQcgOVD9w_DpfC-8G;V2bCMUg&CAMKt3=)RszbHKj6y~b(~P}h z@qvTZ)H^J&NbcT&_4Cu&PCwplbx&N@G;umtm_5Wof6aeu*KZwK$IvTgQf)KZg9{Wn ze6vky$)p}Zi0+SN@NC#{o^{Lj@81bJ9`?kkmj=rA_ItKRhhf_PAXhH)D%&+w-_LzK zC}kD19obscILY^9sj(c}&#$wGqn3A0Ow;7=U$L4_DM0r2DRr8p;MJ>Em7m#bvG;za z!5+eCd~wki&Z$RYL`=l)YoY{UM z4aGJ@gm2s*2^1gZj$&~&pWeht3x1?L^6X{FF~a4>+hGaYup3Xjcf9&Y79 z5r6_5AG$jnLC!cdG10-Z``yjVGpk?=i9(FJu+DPt@s1bKZjn}`gcY@$0&N*G)ygIN zZ*xv^(6hld{w|`f4>vKx0!bF}=pHJ0&F$-e^XJ%Bh;2vxcS=~gdFJ8#J#naRKP71O zG^TADZr=z5&eAF_fSIq0r!152nbD)iLBh8~b97eo>uQ^D6ZY>g>+!`L@8#*-Yw_`$ zFhIiE`ZdQ)z*n$%7Pqm`?qN!kr% zQJXCkP9IZ0y|wuI?Wt>#k%X%%0$L=m{0EpG7+?r1BSVy2*y(%()D5k&e0vJs-y`&z zWm85#O^K0X_q?`Z@b`}nhgcS7Ll(w@6#QaMDU6h$0rrrtsFTi_OfSXG9ShBoUd`Wg z0|kK(x6Fkh4pmiE4i1h?kg&Xe4JaQyx*7zP74-C81t-y*M#-+@StVM$0(gDmGPB!{ z<^-OwsF1>OGK-A;hqM|7^R=H}t@v^B;POX{lN+@(V>ng0#*)6k2U=ZA%NnpKtdddE zao};@1Kxn#+6{KZzfu;;Tn4L_yr(*pj zz+}s4n_Zu}=d8&+S*{;7TReZe&NXBJO@8FJ@U~%uvB;Z&m&)K+z|0#J7-I6ZI8<*} zU}!y@;mr5m$KxX)g^^}ue}4r)cuC+!-qYOvw|@Wi_XFU5=h0@d&|GvswsV=cv&|U% zScWw@^f&D?avl0Z4Z6N>@N^WFtH1PlT6}`RIatW-iu%;fz)+33g$2rvH2sq&TQb4{ z4F+75mPzBMqxYDZZB%8u6Vb{)!*MA5$QkF+X&x=sP&;L{ppVOLomSwv)z$w z=L!D4@5<+Qzciq!Bs^Uz4)ZB%X;ofu47?Vj9q6)OZc+Z;`EIwSJkhTP1+;>W-+Oxx z`TEL1Jq$PROh{^ePEEn`ffG&WOe!HN{(Jq!i&TqdF-mnuF*#KbPKs7h%9`P$Q$dV} zn$r7)k5fuGO*1Rzw@g&58raChniFKGbem!t_KqG;pc->BA8sCbZ%MewCp~-^sCu3A zxSn3nmvdET_PR8j)VC1hmTOPPs$am+ccf+FS3Oxq@H!WT_I=MA=kIr%dm~^-=REkd zzhgN@Ns(z!3!O$~ca6wI96!auP>>h%0@Ejst9`t^3*^K5e*DliG(6H-;g#^Cya7ecfPqU zup^{3-Q)$YWa!XvMzWL?CkZD4Q0hqGbcs+mCi>8JRzH5Fz!$U}v~+J=TyI0ygZ?^7 zY3WeL?`h1@gs76?NZ*r~o#`1HYr@h7$qs?3Rnj^X)Y#y1^M|OvSse9GpC0;rG1B$l z@Ixfd=^_99GR3822a82dO7){d0D8M1I-M*1xq1oMv$5Ykc&kj74sM^*=B9bbI3YOk zU7R`Kz_G%SofTZ}Wj5#8G)aQ--xBFgSI+8F_mw8lhI$qki0!qkT6y&7(YcZAI-=Ag zg3h*W$e;tD*OZjZyKu7Kyspw;OQs2l-a+9tUH)j+HJUeRR>bY<_DF+^;##tO-LFw5 z0fEmO4bD6r>6KqmY#3hW^rSmL2L6! zwFS{~TIhrSBopazUHm05i~Z~=45_Ee2lv}k)Kru+9vRr_iuakZ1PrgdQn|j{`GF`i zY5s|lpw?rOVA^T;YHyET75)ASKte!DC>^p>FwO-KnIj6Ii!DL)k%?jZuNFTPMKa?#k=3q(^LnjB1JlTq~PM#epBc$ZEdk zr+U6qt?caVRIn{hN%`P!eFixLZc@BYzNljyf#f{-C*F?r4>N^@k)tMiX?6;UY%j{^ zzBo?y^PLhHslC)*qVt6*a`rN-PgzL&C8aVh-&UnU*HEcLG3WD{Lzx2SoLk*957v<% zmM)ll<522*jPGH?x`mi1piADSsF(<2g0(0;OSMu#VnPh7vZJOBnQ59y6qk>lKM#MQ z?CP7|Cr_RX9PPYd^UaQVx=AkWQ7PIo+DzM;?aBB}%!5+rhoYbNb7t0XPeSNlIT_I9 zwkXeB{c%F3J4D-%|HKMq>y=kqou4~0`p{aiPFR?nOk7@c+Qz-RTqmMIhlPP|?!+aU zqt%SP{o~`en<(6A-G~zlbo6ZS4D$_*irN8BFQ_@tR%;GW>G3j6%c#m%2j2Je1IX8A^Hlrqn~+;FXB zHmAuypqToPYrE-q~tf->I+F7XwGRid6#HVA81kcWprQ!L4xBEad~QX(g8$>D=JphA+f)H zSg>nkaBy%&>{eWy1_%`3=APt6b;54D6*q|Se-cEKJ2Z9?Rd_4%uB7sRTB`{G7!~}v zJlK9yT|)KnVYoc0fw7BD$eKcSAYAoP+@Z4_s842XGW8L@bm`Kcbf;3)%b}zazjOEQ z54>vsZ2k6_mBi!Zj2pLZ9fW#JSy}lXe|Zc3vhh|BJimQAfl5X^u7-n?vyo|w+LN2W zYv2mgX0Q7W9%rQNA(z@83kff--Hv;WSh|GuXbQf0AGRG5sPQEtWNDe6)^IN4&GV(>N<` zOwp~VC|1-FAVfyryjc#w?LEE>I(2|bl%S*VdrUT3Pds*%^z5RT;EY(DL4S>b^o`4x zX>oFgik21&7*x^kt4`{hAbU=@7viA`a)H72wc?wWg|@bKUdT%YQ2N0T26T{-Y<(DCEi0KYwy98VhFi}8Iu);(Xj*BWHw7CCiD*xF!G=ncF7^P2& zisFlXL*_*m=9N^B;620iWSQkA+!dABqckdgf*icOYs$XI#Ke2V z6=XCx2oc|n8Q+cT5qL(M0@=4HylKk&@iJp!zER9~E_s?vECUw?dbXR_b>QfZTv+wy zj+o!P&#w`$8)ap?Y3Wi2TBi8xlS>a98?VCu#;j*Jejg5J_y8<$=gytbNQXUq$d3(8 zuQhn|1-|c7 zjhF+}#V~l#*Q0_pz5-x?2a`SA3ir{|6-h^iJtyD9c;V;1r0r&e5jIXvBCw!Tw;5f= z4V(m17$js`wTKOG#{Fky;kfc%Ph0*CxO*gAD+4Unz5|&1gfmDO!OsdLnASE4UT}wu z6o+FXo=(6iD#VLXF1>GE<|Zt%va>g~&H5&q0hJLgnxI{n9rt+e#esGFwtf@t*9jGs z6?vWhvLH9@1A_G(^kNc7GLe6$re3uZG{o6X+qGf|AimIJoE+efnG-rfnu*!co~XzI&Hox&%&*`87YI)tOqh8F9k8e^Ay>GEwXRiGduq zWLY!8&8DQQge!$Oi);;_!pNr>QM`T4Ly+$y8et(wn6;K6o8^O6>a{po%_C)h&o2&Q zj!D4lx+W$yV0Eh>JxXiOS_VqthU;$*nws)r3M-rHkM@c<)QwE$1S<*#C4ZQ)Jw29i zB<#y_Bes0R6QfL(q%ROmeSis}C#=_aFnIv<^aKYM&14v*og#!6V7m~S8z8tgXpu=U zw7dlv5B7keKo<7zZz@SFH^)(BStuL{9xibt9)_L2JC~`QOvH^NKvXc+#-xRb8%L`A z6d{q;5|kz19uk_v|L<|nkzk2`;GW41sn4Gi+X^ZSw-)OdnY47JnP>LoB_E*{4{M)- zJ8pa9YDnP8dgn*G>0ulqzU}>MuZ7=#e|IT8FjReA)tfisIEpAMD+>o2 zh1Cxoknkm)v9=Zmb2^xn3yA+3>Sc+znqySe0d@K7Z)l}wm<;}w5eZRIqiZ~z{=3YQ z6b1sMeEY#5$tyZ&u@?pe+z~(!LN+3k{O`XIu0&+gTFb#vh1z9fTdn1*<|e|DiJ-BP zo|kTK#CJ&5H!)%HbPDM3$5?M_$jc|-p%6drBJQIax@i1+-Q&lV(Lw(=)4vy`KN_Z{ zbBl|AHZB!W7ytZmW=Tn3iHMOI^orB^rl|U>xh&$$599=jhi^1Q<_r{<85QRqcA$j`Q;NIYSD&4nRz8(7 zb2l0-t1D=p-Rnh_dKvZZXDyFZrVlRfdN=s489%Eilbfdu4-X$&K6)|vXja+j)F8kj z-JFk)Z*zOwWu$ENQ$lhw!QE%iiXJTu+s%$JP9&ELJehSvo ze7tSrf70ek)6qTC8ZlhuVx1XD_15kY7W`yJR%KlY)%G#A>nwV}mAs?%6$QsoolnL1 ztc;AzMM6T(7Fp&Axy!Ia;bb`2_fYked+n)K%h9?03Y#;nmkhI%61e_+RL5%k)TuR= znXd5o*+c;u2?+`2do7)Vr_+j*$9epbe-9I*X0<4}jg^~2FDpKHz|6)L`s0U!_uh)4 zr{{qH!~S-!RH@stc1&!n%}^1O{m8fPgM<4e$wK3<{gx&fVjfZJ^+!>#LEgJda&U`6 zugJoN(`2dBgBK6p-RjFxN=Vgv7ch9Xzh828R&3rrRO7*o>dnYAsBq9MG(LHCsIIOq zB2sQY+BRqspz`F&4GHg^_pXb-qa>1;LuC0FVINc}uM(ymWpAR+TDJ!sbNf9LC^)q5 zcy5@k)$JSatqeCW4dj0^su2u0*&Wns&&$i}&rzbH4>)$(TOE~u`jk^m>V$uFq^#!4 zLXJ|o_7~mnjnA>D`JY^&;xe3C87}?iFrF;ovp1K#T-;7?zVok-R?0thxYWA;vla(0 zA77#V+2mwr{lwYnzLa*Mab&C6Bj>1rKe7|pOT}P)8o->i=has`U8?_8IYTn-^yJ87&t|{cZF#8P zPYkwdp?uV#W{Ee=YkQ&CaZ(aPLqp@=M4cFHQG!IdN{V3IZG*pS;-lsEvo-7WEH`e1 z8`pUo?G{*^(Z~8J&6QgBmiVHbk&F7W@GLvy=SLkIaNhf$91hm5F$5fccWgMhpC;<& zd^(CtMEj#x>dd!ws5OErHZk$XLU)Rbl{hT-lTIQ1XoX|%uD*8uyRQZn3uBv0uwSs) z*;jfO15NzVA;^K-kCxAKUQ7}5;L1}?D>*`Q>J-pkqU6M2WMXb7*p@+bMdd-pI7 zHrp9KX?`Mw6~G-{z!9e*SMsyMZ%wBfpC6wdtz+ZixlA|V)D&=NT>bBU1-+6!!s{P8 zoY!-A?@w?W`rKF^U%g0b#n{u`-l0{rR zHRie!DG7=6-Mftx(e^_{(-hI&-8v?Ia_v!cBHlYU!291nZVnZxkgxxn7#xhPnjxuz z>PbU`nGb)nA*-157whQg;J5i5494pG?^mXsu}@U`CgJ3Bq$9*yRG@{2Jn zxeCtrHtH;aQ!fUtFL^p#lDb~E-6JM`_%93Ni(ysFb1b|Ceg=$_qkVQs$@DDwXybT; z7VPdo@Nx2(a*4;SGi02}`Z0ap``+5PQjPoSN8>tC>z>q1r`fia_V!1a_i)mE)$T-6e`{7ZVT?W?OY9uT0dX{Cvk@B3N!zH*dj97-MHQ9n zw{PDbv=ja$B_dAQmp+R8E5Nhk&!wRB^AEjDXW5xd=+k^5MkU z_Gmtk=d7_Dc8fw^_~-(+b~(5aCjFGV-deU$jipt8_TKU?9^x}y_L%KmhLC&^lGY`SDPx~r=zwT=#nsh*oN2Wa&4@^VV-qHR(sLy8f1&HeJgKn(C}B(%c~w7=@} zfF;6Ve1}6?gfQ0=kUwC1os;u5IF7NlIoLKtnRvM9cYlA{;WjF~c8Q{GJ&Yx;&~)Wf z9=xv}_5!jNmDFH|V%qbwyuKdL zYR=v5vFKL1{*Ro2q1QZGqRnT$P98S5F!|&+4`eb63kxIpmS?@HQf@{~oHc`7A-aT>2c~zc_oezupT8ShxJCUXL?p{=kFrYBxC2xhhn8&B`Xk zUN%Hb)oxe+I5A!w(W|xLgB_w_O}H+N}kZEG_wPGuJnL6;PM)#i|vmevv98!9%3e`+KcQ!zq{ zhB(3NcevBJcgJwa*8H>f*W`ZLAp5@VT#!ytDI@FMB#115x9kif^tD5#;xWuxs>YH-*jVFkGWWOm3xT?)1 z)^l`r^LvY2h)ZPX;AC-Tc345 zvhwWoNW^Q46`+VEIKK{^*wdjqi1`rFiOPP`$QdfNCWcMOwH^43_Pt*4Cs*a#XPrV- z*rZDE!{Is~AqezVKi)85jJXXOy`~X(l_us96CdA}EMT`XV*RMtOb!m_!C--rOMOCo z{CzN7YDr&VF!ww_W)QN#8C#Fh+nf9Q9)~g4NaR0Zf|cI1cuexE-hf~`p4HjY5c~V} z)PP7x^G>KVkFBjvZZd$h+b0lbdF)5{qv*u$hY(O)y<>W}$N2fLUk@)}VJ(hT@j;xO z4P)?UymjkUUzR+COqW)3`6$|??IA!;S12gl^su%Z$=FqESC&^`@0N}2m0`mvZfPQ> z#ocgB8mJh{Ha}Kn12-CT?z|m$OY7F)a!;yA?8e43h}idG4-TeY5aSaO^`N&AkFIbq z!mu4I=m!7QD>lQSj0kCqyjJSDY4*3kD1~p~_omK~XvMmbEqIpi-mvw`WJ3n5q3_e9 zywtj@YT}#NfDmXUeG|=EUlluyiwZhUq%QAr3)uY5?o56M03IXVYk^w-+Y@PkSdSk+ z7C$@Q7U`aC*iQ&27s%LLZZ}~6mS`#(79L*t_Tr?P>6w~T@yGEv@%H2)teCpv1zVKr zTwXehbG`=`eW*K8j@7z8A=;6)ks_t4u{ z-p0geSlIm_BO|k!s1?~>=*|LEtJ6LB)j;TQXEBzK6HBJwpoJM_#%v2y!@a7uL>}`>z%(KrYlfMB8vdktOz6#_ z6jz2esk)Q9v+V}iGz$zv4ID!<{HyoOx$2hV_@1C+6u$q*KA2Yz?@?#Wat4THKU_LG zJmiH))zsX4DW~2h!|!h#vD8!Pr(d|KxDJZE_{KaK0AignxGd6D%C-pn_Xcm(e-#^$ zN^vW4f7837_AZ^su?{C;qNvk0B#u1Mx#K}2mr?aYaF7qgH<;beOA(D=(cQ2cljfX$ zrO&LS#RoEFo;gh(T7J@oZrh ziz>SD3W-qWd3XpfTYUE*Zk=aAO)#y+4;#1XSLU-0x=_r~LG$+H4pwG#)_;CR!JX79 zu{w+P?-S_`5;h&-_uSLFq2BbWkQ*E4`&`Gb%Hl-}<{5keB(dOrgL~>V=)w;hCmD>r zFP@@Nj|akjQ^Te2*vF z$mGVXPb%@2hJlXXPaj^-!A`-6s>-iC*Mivl+T`u=7aRsmYa; zq&R`*3f$v`s06`$Q1tdGYro>yBMVm7bt0FAL{(+4qv?ZOZftnC#IXt>EBrDs36HFeaTF14~o^WyHc)*or zpWmBbIm1CTBz*3?uAPeru>HJE2^e_=ysz4{4OBLn$int%Mt2QzOBt}g;jqC+4<(PX} zr$SX-!T*f$i_sRI&VrjMGn4f-gIb4bTII%Cjp4t^v(K}D&Rlxxp5gi|==l29K7uXQLoW%1 ziBkA_p3Rf;+fsY9PJ5%@#mC3HsNQve3LY~&BEkh|4`B|Iv(r<;t5>8GxKOXeP4=W? z*N%@xGfA|xLwsI_;k=p1zG#ysMpo-9;?J&@95t)%>zirc%Nld`(zZn#&eQ+abrF%Ij~{Q0N~qZlFq`!;d+rW!v?U4Ki2GPE@$&w@ zxl23U<-5NZ=qM%fy~JkM>wN%ry+2N`b+2OUa7OBVIXRJ>be7bl^(vI;lpc@C@6*k2YXw2%$1}88 zxQ*RH%jLBp@2yxWkD~g0`9_H1N8T?s%Hy?wMPYu z{PcOKS?!?Swi4aWm$Z8avv&d{S+6I-kKZHVF!hs@d3aQ4j?&{^dRXOy)}>$7ZWl6o z6qt-X#N8Eg9{l=%!>Ml}`Z7Ea2}4fhIwm2c*G-o+i}Y3UHg6c=N^f3D#_5P=PY)ED z_$5IqYxw;9cLhHb5`{_19m>X&by9N;EuT!@eEZHZjeYGSgtz$~F(!UZsTk&hyLcq8 z{!?d>l93gjhSm9y?VTPS)Sik;M$$~L3i~XJd@kC$Onddm&LZZRzmId~LMIe4I+E6#e+LKk{=bC= zYJ`2FewUKp?tbG;D|Qsk{I}&1m*15TjEa_nUC7uosaQ>PuZsT?3-O?2b(t9^{quBA zSJFrIR*yfHdGE_MHdT!t$NF~q7mBfrf`UOKWtX@!GBQl+w?tkDHPrXK_r=MI{Us;c z@+DVg`mH>^VL2fY9dBA&mg@?hmJ+S`-u8(3Yd1#Jm>QRPY4jT=NaLq_@I&eNFq;>6U*_&3pnST z-4_q{Lv<(UV{=j>MWzmyGyQyYD}O<#TF*g_nm}u$@%h6-`mRJws(nd??`XWc`|ZC) zibbLv> zeIz5k{lH@$x+*BkXPvA%qfW#BVxgk7cw?SFCe=qQAbRhY%tku{cm{HB`SW{6mRL@Q zP)&_GK5oWE%rt}j3pP8g`CA7;tBS_E;ZfyZiOAo6IA7+g9x}DYNFiIiqFAjjnD!%V z8^f!y7@fOp-$jBTp~+wgpKna#9rU!gIeg|xMo@tl7Te`_AIC(^@$7tdr(vk3w{FrP3AS z_t2yBee(7~)?(g&kYhjh@UD&Bkc!FBU~IqR71>kFUv|2<7sXVIoHM}@nK--LxqggQ zL7?%(LjsMN>TKH`3w+dZr6rbjGO)CY7YJGv;Mr*41xjb9KKnp zq!X&Em!Eb21k*O(@#2jLeAIsMEE)DM)7}1~)+NcKE_jdL^YXrnfH=&if2y!yXRY_; z=W%TN(U$FwZx*Vwf@y6MgbW3-eR8jY8Wu4i+NFQd<^YW09@*_;7)BIa{^a@O7rktL zo1ikyPj*CV4<1x6WB?yKD$q0G9p(JaclGFKtNAdiR-%AFt+)Ek_DQXAG+>jOwqb)_ z_1vwpCzveql-A9mecY(OLbUmYP=aBL>zy^uk^E0bII5>b1_ZC&z(jr^a z#^b^Dr%>(R(Ry>s%PJxz;5j%sSCVr$si?M4=MR3CNH)kwqouAu@r@LB<@}v z{jj5H>fzx?hbs~sqNRw;|9&|h1hB&K&9~kV4U_WnAs(zKD;^7Jg4?24SGUyavZA7*tQ_4`{m}%osJC!m zUvt%ZeMR^y(YqQNq)?J~4|8&8Wap}21q67Wxb-WF-%m4mKW~(Wwy7*)$t?fXv+8^N&B!dcb^y zEmqD%-qH@jb$^L zLM1r0W*ng{e`R-2VbBUcdw(PA<#9lkd^4TLkg9^3!kx4>fD#0&q8Za?6Dk&GCJ+M4 z!}0%(>813^5xfx0jPT4{xs0;>IX7V#chhB~q?q}Pl7{Aip`l^=?C;;do!6AsZ=C6v zw1ZM0@Hfyv5fW)g-Bj&@e) zP>~-lr%Zawy(`oy#0Vw4nCG$1C(;`$v0Fl%yt9YmZrFuvxN9iEfFP@_DVH)2c4j%& zEbZGG^Z}`jg$-;|`;#i|oA}a9+VuYter%eM9eBMH%r*gfV=*|)0KLdQPV%rN3Gm!Z z2-Ju#qoYxOKHZArs964NuA26laL6O(vu7SKBp*I}*rL^$igjmK3oWewaL|=hir&%i-t!EK%{qpNWXaxm=tz? z{^%RI!?wMpM z88Zt@)oS_Z0UDK2h=#7sQOUFErhA^bEtX^cHLF?wqNUpQGXfwx;14Xz)ExB`xjx`6 z##702GHQ*G87$cNj)#PBA_x=4H-Pa13Jy5X`-X-Te+%fo2jTGP6`8Wx4HB_)6%YXB zhN=gmY02gv9uci6&<(ZlZ?f|8+Y2d0!!cCOe{s%2y_C{@c~YRxIb3~D!gfo z+Px7w~_A!Ii-1}7WD@Tp@T6|0Csx`z%o&Ma1 zUlOCTNt}>E(3lgp9r#G#BB_wT3pTO%JLeLZQtz;=mjH6~&^b%+>oqIBW7dK{;eTY0 zfzWI~aSj)o<3SUGQB;&Ba2wT&;=ejQJq?ld@YNxobx$^QDxd{~w4{t1JZbjI=toN| zmHvLwdlYCbGj{K2FWY1B!g?K^Xja=zQR2YrL5BGO=_CtVvzt0Hq5mvgNvx6L69aZ} z28FU790GdoVa|hhsfDpIG5xjU^vztV$=Wh(zc18mYrED z;~ji@f8I2g06Ns8y6-(&8Il{_LeFM`9aOk99>O_eKl_4t)1drmkNv6t!8G1Tg=5ur zsmcGe0KLH%$^HS8-jfJy=4Qj$32~AAEyq6}Nid*gVLtF#3*(D%T{3u+k^1D|U;@;I z58`e>ckXUa=7T}+mCf1qs?)<|x|3c?A&c8u1jas>%bz}{;82F3rlK$1DK?|KF@r;@ z3b;2JDj_8*-D*wipm6dy$V!-uEG#V6g>uI5N*SeNi+%D8^X&`)+2GB95awtNh254k zq0M&x;lnH3wHp^)7P>G^hl(D*z9vK@B_#y}W&20-_2I>_O;Q9=$^ObrIB=Xj2^ZK>UDW8KokD-M$@E((jO|5>Vx5T>po{Wr&Y-eo# z!qLk3T}fT-2qzQW%HroCfE)VX7A(O*VBFst_xHJ4SQr0bd!foP5%3x26ts=FKI;@0E1lDd8fzm z(rWg&_pWIZ_HnvbXpu_Fd#(Ib9?A%u#{YDI);^=FiJxaQi2THhwcVKPUG*7PPR&k@ zkH2V#hzyeosC%hm3R%`^X! z-E`9|QIgJ~jw1Eo{`j?!;P1^Ww~wulsq0TnoDL;zMBgrZ{bRZ$G)b#`??N1>jc8=J z&u@A#2ij3cbG#QWOwC3!@sPL@y?!({jF~Jb-V{c5T*h>Ai<#x?H~ULU@j%AS)B}>l zH#H!2{#q&}YpA_Cve9&@$SP9xl}k)IOYBPpk*3k!6xEsJ4_Qbmx+yz`lyaC)`50~Q zw%A8fW2qz$SxIv_G{zlsxX7^4Ri~Ss9Ampblp#Vz6msvZ6R)S#Tuco&Kma>Nc3vLY)sI-^ zEs+BSmwTm%p?^K4CIwEgcOmGD_*QlZpFz}iQv5X|s9C3C)fgqlk*6WNDTV;!3{zVa z(C-(MM93yvWPt)Pj@ABkTrRxO5Z6}a3C~j+_zc}%-_t6nZPt*w?akY_(%RbbG8NMl zPKu|ez&uEHbOis~>r(CW(4>59P0X_`0x+vLE4nENC-+f!B?Cm#CTLiud4H%1SKnQZ z<+UY$KIxZNVIVs6$*~C7#A)bwKn_TWVw&Oa zP!pHFe{#67lz*kLOa{j-fPc)jSv{AiqcO%xG;$fAC^TK-F$DGhUJ3y+y?rvn#j*Zk zM@RXi-#K?LyfAGnrOsXxX$!0F7_y}I`{OPIy*Xqv5Uv>zwzvjJ9?De~&-MfKU(I9h zrZm;rt8Q=3u|}=&rM0<@IbRA6ZVbH)S-FxJ7WIMp%BJCL+xKoEyPNT`;ybKy+ag`B z*>|1Ja&QR)(c3C{SDY9S5`d3!riyEVB4W-WQLx*Urh>|gZin9Yd0M_|u|w4&e?gX` z$>^%*BMl&rorvLrVCWE%l9HNsU9o}V zkf{F0;sd-<-L=2DQP)Yk4k}S}1%Y;4o%zPz z-qkXLZUes2ulLNG{>|2kWpbXhi56b!*k4ZR^^W@a)p)SrRD4_cPAt2%sH;%|`s9bT zgggX}N}uz%@q{>hE;>4z7zlT-C?#m=1Mjr*DdB{8L#zF^-i6&I5?;4kT`BS!4+x-L zq}rJf7Z+zO5z%_VbkJ=T0J-%-O_j(uTc;rI3C>omJ`&9@#v8sk{(jr)y(EL52Mg%L z(X6!MR~5@__lyST{iU|xzyF6DMXI)COH0eQZ{AG(5fWzR`s(YIC~&FV{splVxc};| zr+DN~X*@5thw-Xdc=X=*^+yG@i&vSsRKU>^#C-NR;e`$kmkO~qnf(s$78(Pkd$87D zJb^<(Z)4f`$cFatR5+s>Xpma}6GpaNT54)ZAXh6n>t?TPIs=~i09Y&6Je<7FW8!&< z6M1m=Bdo2CA2{nOo`xTxnH#^Q3c+K(* z-xs<8hL`;wl*#zsaa?ssKIsQus^>9D*Xh_Gx7elh))0Z^=kEv#E_lbQF4OY~_SV6# z8Hj<*iJJYpwu%#J*S6KBa0xY|lE=f&$(iQ8+p~d2A^yjPz%wpE!T9CEh6z_f5|^ML zKy(TUfrBPOX>DK8hu_~jC1Ixt|JYH_?N1!|(8?ou_`{WeWH8$W(CFBIp@hh@slde> zGggo(Uq(c9x)|!~>-YFArhQNM24N4z9*V;x{a0Q(^Ui>dqU|q?P9u+bF%xfZ?;a7F zDn*qtx)CadjD7y+61%kOLulrL7hJ?frl%hi>Yq^D?RSr=+Y8w`q}=RyXQ|5vp=j4+6?x6Pkn`?&>fB&aM|1K=$!DueD&sXVvE`P2q#HX zpRau5r&T!t9@!wzA=VkXMVU1R# zsLuYS_avmz6mI%9y_juuCoT6M8vXqRantV{9{d1EHx zO`L^j9E}`rfuemAsqYKz`cT6xpXhqANg1JSxbp&X+wVY*H}Mdj`5yzhQxY z-R|&LZQgl=r>7v|D5}!8wfTH)d^%PQiQIG$Qfr5RoT-`)y#~(UWHj3*ov(ez?G~-* z`AgylXOi)5O1LKqWeC6mcb9Vf7?~5G(>Id?T9fd|NG*8a?fQDiRuL4X0xXM*xnxye z9|&aXG|2w#7HJMG%k3xsn&KGRco(6TNJ$2P3v5vW1&Bt8g(67kl;ugB?B_d2liwKC znC?A=MUI^GJ#!ggQNCu&)Y`eXCj~5yffcmO?0!iHGaK82U_Z&hC%nK?NnYt^lq z?G?s5)_<}kCA0?&UAniwzklOA+Mx>PK|*N5S=7|;(*bjTsg-PBoY{-$yGlj>dFV%t zoL3NN*I0yy+~3RC-cnA)(9U0!b;viDs+`TuQf7$xnM4NLowhdq80`$8t_!ST25AY+ zL2kWcj(fZnVNVJ2ra|J++ElrYIV@%R6^Tg0-t=-mMm;?++C(6(TP|5zS(OBw`GaWA zO7vCu5j8))s9TA>!eq@LSd*LIbx%)S0eUC`VlyPDGcYjd7B!Z?4_MKKeqh=U`?6J>ZkiTepFcP=QOT;0_+4*gf}N8T?tmWb zAyf(dJhEdtpEQbWDN=h1NcuuiiOP_GInkh-cgVhneC3*;03A9G2#p z2_^cYI&U}RN`n`;8?gV)#3RHS80tce?|DAa>Ib|g&P`N(bYS~8xabbugBue_qM|}f zn6)`f2CBh@6kz!w@_)5j&GZ`;rfi7&Q80O;+&r$`mebLLZiH#U&qKCk^StX6>$y*?FvvU zD)K-rJbqm6lbWw5#M)@3`ajpfV{ghj zHmF!d(ro-zx`O(?o_hhPD$^UA&kKyEJGrqS z0^D>)Z085)Ar;@<(!#%Zk%vyM#oTh-lcg)|(r88V$}r%$uRshF*{f}R-+qZS)03N= z&Q75316g14{A!TqfPy;{JS4qPhx;l&zU1+hzlD%l?RR~MY~P9h2%uNl|6#B_;U}~* zJlnZB3kFYq8_CIYAP;*F;w*^po?(L~VIyUaw=9Ickl;y9iP_9%WjgE^PMrmKu|5!^ z5|vK)_X|&&Z1xj`<0aM8(*vd5qs!q6>K88-k6R0HTuS%C<)rL?Ii>;hFgU!~LaJ$Z zSZ=?t$~dpqq;0GD_CSd5{#vJR<>w`h4efM!?j30zAlo?ntI{)Xd9V zffUaJ>LNq?QL8It=3?y{Ed~cEM2yUG<{;JqNnwjC;U*&l!t++ah**!Y*MO3elFPs9 zkmN!|vLIvMw^J9%iZt^>bL5Gkr;e!rx{Cbtb6tgmPS^gAs|-X$W4(z50_a{U$_p=y z_EFq?eEn<7p`rI?+o*lfIU!MW{SmAgN+%bH4O~XYdKEnxl76q)GCEKEk0bv33b((LZW1gm zEJ!F4$w*c2bVf8?y&Y z{Whst`U2svgH=)>b>5^wjII&1_p-oAs=bK;Wc z(}geST_hF4h>!*+IkOy+=HA+Z^YIDo5TK71`!}-5OIY~Kbu3!LhkXJ_8PWz?n1*Z=JfzWg_t@l7#Sf!=REG<0WYC)GtH z6|n9ns+7!w1J67V+%%()O*cFPPQ|<#dff|KA|iU67K0044hkJH%F73wCi8Cs>B9Un zQX(=yax&BF&wAL_d_H2w2x0#XC|mV>?c1-YxE&zd%`vamA1U}RPzU`jxbE3X03r~L zd41{~dh+&qN74fetB;GqXzkCMl<#g?ZXT^;!U2%qLhe(Pn0OVD==Srp$iQe083n~V zlrS0ls#U-iN)Vl^3?1`(3tddv`T|e-n61COjQv$=EeRt%kEEz8%gnA$OJZEp`2vpgNsjZsY_;LaF5@h30(KjM%C&=t&q5OS)ZW{sCl zPIiUIN08sqVKnY^5b5*e$ft*dQV{W0ncT`VQ_1VdD7P;oF%myFf?X&=oSQETJ%gbV z@E+zpeEdK)sSxM+sZV*0QM;qgRc4uYY;D=D_b`|~jpqEFRw0n{W6TL}WjsH@L>zG* z;@xhc$zLEE1w&{d$$2mw)vT)YqA$zM`ipO6Ce7>QxO>Okc%Jh-{Wi3ybw`8e6%J{D zQ$r9!2Cqbyv!$g4GL4hG+fw82oFZ%D+8t8euYexYQr3g&qth&B!`RU?b$9!nX#eJ3 z_B}!D|3B9ru2xI}79s#XT@V(%HCul)HJ~$HvyN|gj_ChbGP*u>iz9B!EuF`MhxbS|1p7GT>?ohs8d$|HwAF}c6a7-wQKzkg&uCbd%DYi zb>&}AHH|u&U%owLGYP%S!Q<_n9oPSKpL=?A%8W_LM%a@qN(o-E`Dj^c?4(5^0}E0|fi@n8Jgv6-{^yQ~h;fnS545f*+3-97$eCutyqjmX|%)>&i|Rz9=DuuD%=; z8x~zUmvhBd;Pq6GuS0ApYInD0h=+U*#m#)-mR8Wo5liJfC2C>&+a&RPO)9u466-A30!yFKHD4hq&Oz4&c=T*DRMrFJW4wLfrGGKY>xV&*V7*$p;AFb zWj-JJeV|7MPgQvAwRCgT_)WBAkX!`7HZkc|)14ho2h;Ni-BHD%mRcZp>EP4{uyd*! za{a+{7z02)qCq2%G=O~?iQvd#4550Z&?<*gMRjp`8l7ZwKs(g_DD5kkpY@lTwICD! z=Me_Rvf`o0xffe>kXC!FJBWsclM6C1c& z$2ECOPWEn@ZIN3(xJeHJ9kBpe!~)}GFG;mY`TY6DV&212Ht{3Qm0@$yx1%<{3Fzru zLo-krrX9<<{d20FMJ;3qCc%J-3?7&>YZS`xXeA*LA}4t)#w5XfA>H$WYIUFbB-hO8 zlYh^8c__uNo!607)eGIz?F1!mOMZ+Pb>l`~EdfUyADh()T4-9ux#NP!LoWLr#tp zTdRBly%=B|TE1ao!O#uxedkTrF;z{kt4H3qPVx4)cCfTLKjo601av14$3K4QNq{`} zr&)24>!QZF&%xDNP-WHmG}n3+nT0I!JLK6KT|MK)Tc(nr^in)}DA0w8$?w@LBo+0j-n+ zboG&juanHDPoFsTzl8%71xUzTz;*+_{-DYID4WV8*ligAjaVAq-Hb30{^hfypM);* zm)CDaiJ6}37*Y{vaP4p=g8L(k3JTc%xdWlmP`%i!ypA!F=6j`6ky94bil|KWUu7ri zl;VD&0u0HjSQX>A4X3tHU7x=IAB+r|La&JMnqU{~xszpU#ApxxCu6#FAEML4PFLoN znOQ6>E9{01AQp=_yyhREcimxh&ivmYX%w8NuH1%j)(8|7L zd3@#%fcSyw6>)*)2PeyWe7XHNd#n5jIf`UMC31MAudJb4GIJlo8+tBQ?JDRPz};rJ ze!hF**bp*hdX?vb$*P6Bf-ZLLhANxDJ$Jqm%Ukk$iC)WyZ66rJ4cO0bFcEO(r(b5< zdRb@;z(sgvtnI*4J|=2vY?zTu6#=g?xT%M0>?!twVCAu41%~RirDjKY|ieK0a#rL!)qUCPPH5pgvt{GpS1!9TJQ?f2lNUb*Ps{i)5;f&;P>P4{RnRr*aeU16w2B^j`KpU{dnFKEbtjF-9Ud?)%-N!=*rOMZQ z-XwVIpuQIRU95UQj6$LCgwJ;ECuk*4$eEchT+&o)V8F?TvEjnJ>$=PiQxFK4fV8BS z@Jjj*md#WsX5SC$|m-c<1|z=o%j^=ZhEB8=Q&faK$oZZR`7w+FASD3AU$kC`Wgo4ltl_u8$ud@++I^|rw)f>{+`Qxg9{H); zHU2>R(W%@Nf-`;Q5IH_Z4GWME>DBv+Ku>rYq7Q5(C*>t%6b9yAIZf(Q0e1HD82x<9 zA`in-)G*u;jM(bS!~HO??Wv>nX*8;gwo|%WtSNBXdU$tU{gm(H6SkO=Ygb4-a-&7F z72t-@HkWzufIuCO=-KwtK-}4pBSzFfL3cb6U3k7m2#jz2uJiHe4#t4uvUcaU($4y1 z#&SVTk1H+7rY06{-~ho3+~MRm9kD9x&Kz_{(jy_ll=TU%0WWUx?4>o;8AHoyAtl7; z^4q3d>MJ)%|0hG*ThHtNW}`V)jMV@mw8i zdiKl7kM0Hg?$gz71XBK2~sf(;|<_o&((40qPf(+ z(rD59ycXSFcFaM_nP^ZwZ zDCiP05qfN{vaxwHbut#Imt!eU?qSNOt2^-! zczbvDQ58q)c<+JCkW22<%FfN@g(JTI`l@u5^Sp992?=)suj@xptqz(r$K50!$;DM8 zPsOwe5lS)1k0)Xw-_OOBC1{0l<0Jt)N@9tfD;zonb1*E82|lZ0>k~$-K$41fo%!9iuUe7 z1^~b6mmwmMdU|f=8P{00??c6a#WsF&i1Li9o_?p!w!DD4l;^)X@@!LekJ&*{si@^s z6urbxR7QX*40J*4$OZKbGC>DA1|+)FxPM)(jV56!@O%oNGN{_0@G11h{Q<;sqNJZ_ z!`Z1A1`LT>g1Q8mAGGfJuK)^W2kBva84*COjFd$T8v8`PBxC;#Y75E3e=HN=M=;Fk zs(145-xI5zR7GHD&@fTw1t1nC!=Dr&%L0wabi9EBq5&f6p8KL-hRpOW)Of*M9)B#m z40>zhVD`-$)0E47dMbl+3}#Fe9I4m@kMr;6Jb1Rnnw=y4@Xhve1UU=?%#ee6$qItS z3Wz_Tq=!#Xcx`tJeFCjwPa=F_0Z?y`Nx(@fEXsM!or0cTwaj(^u!(zt$J0bnw_6ay z*@cC#1||#G@%kToLQ^-M57Dc^S0*<9JW_)XQ*^*FVvbU?!Zp`n_7u`GpXJXuj-@ld z4td4y6rtWKRR?5EnVFfdukr-Jtlie$sN?9#ST_vvRk@&$cJ!dRtn8~l8`D>=V910u z{dfy3_`mghL!6MDx4{%Mf!ESZCVP9`#U@UU>`R%L*X~~qpGz6Ah26RB3(1Q0`gL_c zbRaeQH&o2>0{3$D(kclS78dfQ2$+L*{ihFls>d)CFJfX+P0le2BlR~7$~i&v1-ium zb{MCJ*96n_3gYsYC8wwFS+pAyA3s-Kg{LksRkR!DBS|Gm&_3Dd%m~KxXOpbcAH#8L z)>mFMIW_U|pr_aJ+Y30B7-qV}#PsGW7OQ6Dn<5K*R?f>q_@tbbfnP0^WUma9$6e62 zll6;hHGB6&`XOhK)~%oib2^_G30_|Ke&Od9zQRKie3Bpc**cF;6M3V5#3=F_#Plv# zJMRXGxMcYBn6hVxpnM%koRpvoZ_4t^mwIn{_7(h;O!#v9hs^_+=n5qOY${XPiw~ zI1vUl$;ioNf!YD95Ws*YF*yYV2`z0GoD@DWal^`XL_`D(8~a@4vm(U8BdT3;Db(hJ zEd){5;6d`|&z}<|d<5Zt%;lyylq?E^YWdH&hs<3VPNh;#eR7dWu#4u2?QxKjNW}L{ zOjb5r{t6wMTjRl8g83gmGKz_%z?;4q8#8Q+qU%IPt>N1d$TX)H@E>%fk_eRe+WFI{ zLq%x3Me1~u7()_0M+RT6z`g71E6(7%fr+s^SePdhMwB4#0Y{ddlk+wy$*WIgO2>O` zz?L5J8jK#$n837~L|8+E6d9ZHM<)Xkl8VjI$)=k1roJq|IrMp|Rn;sN{o>id_wexE z?5*B*e6Apq4Ui)%CWf@6v~;-AiDh$hQ+(%_%s0o$^xwZV*i=&$^0mL}mRNLHI?`7p zfTO|h3J*I$|G#}z!khn5u5u>CiZ8mw7Xl#|@PZ_)?KL%`{{()qWLE_D1!n26^8Nd` zVZ;o%P?7y@@3Pva$nYsk9v&WsRT%B=BA5n>`|Y0`g-<5J!uiE3#x9(#pMC#YF(YJ2 zPH6=AOei4}H+Qf1dWZz_tpEUTqhBXLYon;3 z0LOInYTvaM2A4UFs^j3Z7Vm%^tXZw8hh%=a5rmdTV=wu!lT!GGVSu~y{%sdGj)dK9 zIj%8v^Tl45L9XNVS~5c1p9{TtqHJx{tLhpW7j!94pUG!9o)Nz|6Haoe&{5yB{Uoyo zl56%-D*^M_wi$h?isIuV7m8h-08uRFgc~<*n1U4}FcT9Cx5)H+P(-eNvN9WJJ3+@v z3zBn>2Wr3{!R*#cn5P|I7|410xOW5d4$Lb!8*?wB3KE0Y zmhIanvtGHd*V$+_^L$x8@`3j8Y;;1syvovKM&z!o%*jCX*In|vElW*=(Qtx%@T~x2 z&*_ke_$r)EOGV^U8ORq-zT+{t+6Kh3sW>m4F3q44%uap>oNhCm9;2kT;ZqoW<@N^b z?$?=TgtabSgqd2xs~FI)*ZWYRM5R+P?>u;riW(xO|JKycwAWG4Sej-TUtLt}yi5dovI@voAc>%oSPo#aU>axi|&o;ZiL+gFUY zfe83DN~``pC^Z=LWY$QKm?6( zKI^7Rr}$we6-`Z3p0KSKnWSMS+Q%y%g(oR5B{w3<&TTG?+#*WxKKJKx#_ivQUi$u}l zsAAdEmPQxd>13j`c|v|S84w2nl8_~jH)IOx_Vgj*C_R1prMveG9OUNh+m`Uv6{3|A zr{OX?DiGFc9$a~w+uDvd;fYmHLj)X+|KY`Sf12ZC!||j?W%fo2Yax1!|4(mU9*yPO zuKl2tDJe3PAxeYkF@$6+q)ZKDo=S#DNrub~h73tEg;eyb%%w6pj^wbt{;<96TobzSFmp2v9{$A#CEX{@4h`P7+E*;~{a0`3nv zd0xCY`I=rrLgHd)P&nat0EKtuaa{$q2p$E~_8XAjaWdYFr`x)fe3lC!1_QgXld&ED z1Ad0R{yP#$J>0nwF(M+rbW-mhCE2I??yWop{d*Ix>N@g89LdY!GT+6KR%?N2xP?T) zzT;H@gBX|NPX15dw36u0w!Y$BvMdlD4qzAdbD}m@OYCN(t=ke{xn@*AmZL7w`G_;i znorFw;aVyxcw@rd5q=8-P&f)3=J(rSBLB$ zh(bHPgM&$JZkxRq#BWC%Icas~)i-kQRl7u)s@nYSSG{2MsKCy(At74~EUJ$F+=zYo*ypEsaeQnjYfIYK zqC3|A>Mo+1qvgQs>t&^H`6y706G$mjp}=&W0GFLN5wn z_6N(`{OVLz4`EOwAaE5vB3SCI=zobma^ghQt+5wz0!+7#}P<=vG?CB_&t@B8JRl(yOT zSLvf2dis0+dKoOR%Rm1qHb2Do+HY^8_Il5eiS!@KBBPO{;srf#FGZ~E;*etdQCHVJ z=SqKpOXSY$-tNkd?6zT%ho67idA2S zVblP|;HKm;`l8tb3WZ&wqDYVTYTcDD(4kp1dLM2)jvdAE2+px1o!l^D3WE+I!f$!L zQ`qR)wXFvN^9#4LM84U&i_UGtfAhI(oRSkpFWxnBw=~`>wQV^<-7hecZKi*^fKihe z3}=BGaott~aj)w2Yr^XxcBY$NFzPx8?Kffk0nto$@ni_)%DrHX6gEz}WM6vs&hYwM z+b0H7EH8ZcbX2FwH~2a!ZB934(ZN7Z>g)XM!^q=KZZtO%yRF2KR#?8fy$ki#hEHqH zu@)y1q9-)=;SEv_4TOrS>a}!}4J`J>3kzM-&AZMu&~}zBtfpt^e)5QGsrGyKc?XJQ z@vM$ep=i#yPr}28ABqdiG(%(qsDl}}b_#qeTh}RPyR! zIl|2eH@k-FhWhFh1GE@aitJ@YUDTBONg{bv&+Ml9#KmJ)M@GF7`ERIP+oR zrDcO}B67MxYs5;DpJi2#SJ5dgEWA%t@N(A9^Yd%8D3SWPq)b(1Z@QUzq6$bcGk?>NYVk6m;sSpCXDz<9D*SpCofQ8q%TL3f$zqwEoxSyWQl{e*_wL

Q?=x!0q3&NmzniHghT`LW$ixpET55%o1NYL*EYYHeY>@-ZLV{# zI*Qp#gZG2DYKIPi#6*aY`Y5rIXFl&Fyz9WbZ{izc*z5CK3dyrAvMMR~p5P^s&vz*o zKfiH5IC!2*;p?RSa*h?*X^Zdo?}^LpGrb?06CGW^;i?o&pNODgd`crbEY zvnu|lNwa>OmY~~shc+?~R7^B{J9iQ$Tc1!yP|ST1pp>4vjT#dsF4^WK*a>b>Vn4dB zx>})(9C+8RUGnoIxkQpWKVF8M^ml%Hd=k=Zq92VuY7ODhhkegu^6d_2$=qEXphQi- zwhDNNgoItGBd2)~bLB=|sl?OwPFi009I-H+>BeYglU)DyKoFnc8_|cViHE7a(bF?9 zwB%S-ZkM^k#>#r+$dU4jt7{7SVoWkLKW$=V%|S7-Y|d<}*ieQO z&S@x}sx`$*R8OAdN70_*KHhOH@T+rs1!bqO@Hb-7Vr&SI0x}-kAC`em2Uh=qpVqMOP}=&+)oIMtJv5qOtaR@e+_hj!m2EK3Xcwr{e}D}qgycv{_9*vYVJ~Kh+uYSEwx`2iVD=X`H!10GQHJ?0wd@m#8{qOv|JQ*zaO57y; z_|S-$a347LQuPi8G9GLhM!jx*eSJ-?-$yHAJP}MtL)kZ`xlH}?+5-)$Z>$2UXwIyt z@#~(-?G8UB8w(3OU53kn5qvp1Iz~*4fq_BH=L7T7O~Z;?#V3jvnm71*4HOvjmObDH zF*gs5M{1qBQWifAQ@s6#jk2W?BEJ~xQl;K)JNx=kU@F`vpPy}QYCH>PJ%J+t09c*A z_%Bb@_m-XCY7Qj|HZ;BG4g*6?qE=k`ZzvWEHTSCd2V3$UH)(!f8n!6iwp}0lW_g008{(xY1;;y<_`*v zJJwb<80R`ivERM>a&}^-m3(Qk&%!Uo&u{q1$|Krs@KiA6;EYVtWVGY zGmrda@Te~3?d6n%R)a^XUkLWt(euTAgo7)PzUx13ba$U%-Z;KFBD;d!;;LBEqTJN^ z^q}iMu5Sef{rR&V>JmfvssX&Iww8E`6vcNcq0=yEa&ckcRUSW#0b}K7G~x7_0`LN& z4ofL6*i$xaZ+)M%ulaY|$cQ86*IFi~hcGWA-~dwCmc6&IwSdfY9mr3@JwC(YxMvx{ zvfdmL5?z(dX6=p3*Jb|;nwb(lN9@b~mia!}&Hu%uNFqNu7sgpACbdfA5&|s$6iRZI#83J$TEEk z|IPYye)A@T6D^|H_s%k<(Y6U=L*$7E#ltyNa;a#9DbtEG9FAWsjb6&|ziP~^sI)O= zuD3l^zgfeF2{~N6ZGKOmp``785VW7&l~h$V)wWg6;sZ1a!%x6#%)~mIo_-A(VonYj z#T5tQH(T~7!k8jpiyaCIN!UIjkv&BvC)b1CJ4e*hXqt@|Pp!Lj=@Q;QZOX@*EVClf zC%Ld^bt?U3gPMX6XJT(}susN#5ieF607PPp)x|dE??g$ z((=7lMrF8D?HK7;PwUdV~fz9WQX-i=L4s)a{hrr$aShEtGTdN2H_)-y1ON=TsW zKo1sC9$@kSipSG{Ax7rIIV}%q6UD%u5&v<{K697d;Q>JD0e1Mv%x~=k-E+{m&$;o~FJrFHqa4RuU3rHNi z-xzrn6S0j?pgvlU?F7iy>#_DCBNz=Pdr#8A+%iuw56cuiMWS(n3AwgB!zdSs=ZxCy z4GuaZI+Oa^O*Vl3LoAto# zg^eGwNjwYzCX3>xYt&Qv4yKA9BRTv30XyT#R@5X+PoOlW&ucMlIp7%ztHXyFzJP;BTnQoz(-SXSe%}p9R zuAA>exM49FhHjc@E;D{r<1<_F`N{eDQL_E5DP0-)Gl8AFmy4XP)hkp~7ZkBO3s&|f z@W|NCP%gUihQ9;PokPNUE$*wyOX3V+D~i?;PflBNWNhp~Cnr&_nGwQH7wQ(mI2s<_ z+S7H(O1kKt0e5ZU+`(T)nCg)NnR&;0bW*q0v?abJcn8vF;?qqEGLXDe#mslNMgC6b zJ^0ghr%>f=i|r4eU#H*u`o?`~c)}s~^PT1~Gc|K_p7QM$icM7xx$M62&Bqjl5~p(O znisCgoe3GwR(MkOyMc548sJSd$iu85v2WU3?a;e_jzqaohGZCl?#QuYuP}-=wY5aM zsfm2+_U(&(_UYva2I$hHy$TBc5?x`LW8af0Bsdxq8%y)JOs>h{)*`%2pxXp62ZiQ@ zPxor?PM_ur(%bel7RHN2eNb51=UVdDsOS@Sp^4K}rBl=sCA(OivI<5mKFymb4Q2Z} zWZZTQEO=2L=SygRF;=|YsJ zdSKrztPMVK79P)f~QLr>2ajz&kso-jQ@g)zqv#4e3N>ci(BFHSS-? zsFiEpJ^1w|oa~f%+VCLv81vIZ`?|hQkw;B9K2kN2)_si*7P4EXA<#QIs*MfiFqENocDOORJ7iC_3E2~uPn)i zl{LXE0V> zVbc#beN{;l$P7Cgs9Pwi(?5FeA7ozeTXZ;#d7(2;wZwSacMY@GW%r$0Pj&5C3_hNV zy>OIcFC31sA6rA0Kv1jjD&P=J7_<@|`>yGD17ooJ-5{?X<_{1)-gdCiXWyfui5T)pWHw_u9>N2Tja(R-W^vfmw@db7OWX z`-D9C^96MbaiGsvRit9W4FyNc}=ei2d zoobtVt6fkv7TC|bQTmKFODkV^wmtabxcThYyUKzehW2@Mtm;&)NU+lhmXf9CI~j$( z3dH$vtA6mg7bo^sHq~TqN3u@mHXbkdc_ApgHMyh`|}mlapVdKZJ*CO z;T9%goBEQ`aEyvhfzEG_N~fJ+FV)8x-_`s#QqUaS#K8fb0X-rNWT`o)s@D-$e{m*y zPZ&C?(A4$}oj?X^Dnwq-<3SSCxd!K}UsEEQbER|qpcn_ta9nGZdP*e9dDtd(|G9Q`YwHENrkcnc zr=-5Hnp@kYmIstXWhEf6XarA7WTK;O0D=U64;K#)_K29|Op_86#4ln)u;?JUznp{J^asjOeq&o!6v?WTw6Oy|-j;}>ZKH^cySdL>o$#=ETN zWtzHExo8(H?5xR?<)qA|bne_az%g%YYAlhShVd%#M7^y7Y9B9uNrGT(;BQttN*5LC zV|@;UhF%>=h7k>%o%nXqL1Oq^ic%u5S!Vy~)3vU&%HJm&KDa1p{o?V|uXL);6FX*t z)|gJ;KQrV>cL}?APaUFbe z*5%r$c;xqG)RbTTkk!tn8F!#Ir<`$liKU-?f8Nfx^^CeC=|n9SXWG^4ojI>oSTYF! zoa%x#Qqx`mH8nLzO~Q$2;5M2@#+ucpOgDp*Iq~`UM2A-l`I#f9u+d4*Gag$Xo9+r* zpw*D7=sA$HZGA`Zm4kOGXmoto_+w9dcsKO+zxR#L8awcfpZygYM(e96oZwL7vBnc9zP(XwhJN{2_|x zVa(lD2lR50G@)he3^6SMU_p0dRCsY0`U_aqMEwCO2VN6aW1@h>CyB0!%WJv@k9y3X z#XoOzpGJvGi+3D^Wpa1yI1s_pHO@?@d?ntcbzSdlzq2=wJ(KRs%O97>RD&Z!?Ax`0 zNK5SmN1%y(;rCB_*a)32IhTs|HFk8*pM%{^k3jHK9Z56W7bi|*^ZwqT((=}c+ayIual<|La~zFyJi(VoP8tr=eHXDhxifWjIhp)(kN;xU2No(suE@2FjCfH2 zw{No^K71H6dky{)7&z~^)KRNJ^ya*>9o2RBt@inC=-dDPavv@ zzC1IV>R-Q^!DpX}vDjSJM(?H#OD(sQ%PjNhG-zn4#6`UoKH2T`*GTCdA3w(=0EA_v zy(kt@tiy9y1dE2u6%INhJfzqDiDdF3Ca;p*kA2P)BW2d#4G=5!A8i#`E`rgRg;k66 zbEb1IS@ks zrKEL(dg&Z>>D-ox#q0yFd(LxvEO3>M)A^0lXTPMPz84^Kaq3EGEguVo^VTqlH@0NA zJqds6Eh_h?fstse&Wv}~6Zz`<_uI(i55Eo3SWP#<2staBJ4facQJU>!E-yd{HaO=j zd!95i9<;a|=p$A^rMojCC^yN{<=~es3{q>n%e$CFLbfh!F*TtZy}Ws$%Rt5BnpXvr zvSxFMmb1d@_4c=lOG*F%6Q@*whkMH_d0<>N8vW6O0p+voej%Q+qbJ1=Ww!1uT z3;wzH%^%usCzv^8qIa_|DZ*{cI4+9Y1!D)czydk&Fb}$ zp(te?I5@>muEOq@TgSO;_wENkc;FP*@WR0$fW?0DtMoa_bCrfOZCNOX{YmpoY&j3j zt*o~CYYTP$eOL1U`KboyI}=KVO|qrADt{dJn@%4XTf9siT%(V@0#;|QK^4igvWDRq z5CAw?C=Zff*Y43s4V|#_)~2Qp$^ti_@6U3)v7*LWz~_LU^xOZLq3N0!H+%?YVH^@3!RG> zpSw<2p>|xE;VjN{tJkxvfp6mfJ&HIQKirQ}A@wQp8E5MMwy?t2@>g;JmY`$!vV?_& zMVPvPdI}4&hPw)0HE-WqLC*_l&QPo&BrNRWN^s(Ldqm>z*$*mu6ZzQDqZJ@{5Uycx z*~U2x#=N27;jZ{zEIFZ;m1tvSX12DZ{pBeb_2k67z8^a=P6lRXX6cJVb7oqLjoEjL zX4lX|b5YsangCWcvKLN}ztDWgga`^?VDA|n4gYAj4Svg;o${%!R&))ysr{2p!@yfU zlIhpjj8 zmxiaO4$3Pin6+nNP6XFtEZP!L>;^`nM|B=Fh~dP@|FgDX$-_JNRE~}Mznf_3{maGL zf*Zk+B%pghKO~~>LmbG0#t1lG&Oa&0!y*OMsZ>*`<}dxXbxP5<0?Z|#mfg1!UenXl zgQC=h_5^Wq!JXp>?EG%n$UbSHWDf)o0^Svj!C29jw8fD@1dl2|UluzIVWCH0^41cO zNlDuF_PY+=cn-qls+rMT;d!s#@J>mO^E@9v?m+8`1_xa=#3PwzdAb>sUmbwTdoFdH z0#gfcFk?3S;?U>o*dGD-@Ab}Ss#kCqijH1(AFsKqaJD5+>{bwuXL4Lz6msY_DIW)4 z83^DO!>(g#X+~?;xy4qZItwo`v>x9KH!im z+n0jGy(SXZ`2082eSL35C3Bgsx3^+xDVXsmJ`}hp;t&EODj^`<$ml6eFaS3=q^oNS zIQ!zr7lfPjSV;-eXQ_o<)W?dabd2+yv{4!Unk`#qQU>-UE-rp8Fz^%3Sb$t*QS%IN zj|9S^5w@EMJ`ivH&`&b6S z0H7U+|5|sRUmhHc!ZO6H$TM}L1u5g8$5_F@Tx;jj8)O^5hAo!gF)J;Uqg~Ir=L{iU(*EI8vr<<@PK9 z?;|hm!@OFz7s?R+V_w8y>1k-J#q`0^PCPK!$6-Ch=`_gX5VXWGg@trflF>#7^7E>` zJ_0OjXEeKd2L=w}1fk61%`#}jBKi={exPU+LrLV{H}Zzty-fwA%L4E+_IgZoZJ(Z+ zoYc_Ot+pheM=lDh#q8g0LW?EfT>>_-=_tPBrsPZ9kPMgV&QI5V&CPwl?t zt*si$oieGA01)MqK>qr70T+Qo*Nx@?Tr0`a*#& zDPW#!WCjEUxlDdLL_#}wKVS#MUPexds>0g;nYsEIyt(1GwY9GurH`ALoh?~4e7L^E zxE2i!OfakFX28k(u8&|hlG=nI0_u;=t7~*PgNSHFf?00u)8;5GiNkjA6t$4^5S(bC z6B(oxCS;OaH1Z@EslG+?*U{(h3LKn!rg8UJ8L%?6>oK#zlK8{wMb8<5%67{pBEnqm z{-GU_AUS`!u#lO>UK#p@s+(I6)KDn79bQ(<1i26mX18>AbUDH3#$COvQ)HPZ08`se zZ}t9hNwfBMr`g!qY1Xe_-_@fds=HF1vf`e5bthmu z%=0Ab!Qx`!GV=llzcJqh;yMIjWaaEZ%vx2fTH>b%l*oVQ1J)R(%}MD5@Ty+goBSc! zXbB$AoJn62{cCB+AiJ=z@Wg!MkD|EW=@ts^i0gI3#re#0LiLR-1L|JVEc_nz)w`7Lw6FKBrQf@tmR{8YR3YhhC6nKWC8_%kmo zrrpJf=W1+heDO8=M(68)ynPa$$<><7v3Bj+$IeAPJtq@h=0=N5i+YpN(HLIIXuGWn z%B#%gU1ob9;VlvwwAq#IEAK)F4E#SIfVTZFJ`l%k9NU2;ASEX!=gp{=V3ezwrxy}z zBO^}#-|44Th9EjR+JAm#V5}MoX(dA=q8jm|1ExCh` zOpKCkF!uF7WnurEoz#C%)9`=q11)hnLlS7BA)%rFpnfStz12%!Oe+f|DwXuCtgJ>~ z*M*e{_ovukVR<3|AgQ6s#6at7$plr#($I;OM+yq+qu%H~THhX#{P%>sNSL6ntfGJK z_WnQoRxFV{MFze_=tgX6c=^)FHvJC5(_hA;28%1nw7ILRE7n}^p9%LT2M*TY=47OW zQ}s$c_2myv^k4F3Ls7DG^HSU59Pt)iW|x#>Er|jNN3H_&S@~(euJf;d@Ho5VCJBPR tSv!CRDZYfV_Ar@_EGKkWx$Z49uPF{PPc&~zz}dT`LuxvzS^F)o{0Dh%qDlY& diff --git a/docs/images/chapters/canonical/c086e72bd8aaeab37436515ab251b2df.png b/docs/images/chapters/canonical/c086e72bd8aaeab37436515ab251b2df.png index f7677982217cd932c3ebad461aac77b8cbe93c34..875519925f8c101ff0ddb270930de9c6807414d3 100644 GIT binary patch literal 26682 zcmYg%1yoc~yEYwygh)3MN=t(@2uOE>bW3-4h?I1Lba!_s-6b`YbV@h>#_zlL{^J^4 zYv#;3d+)d3_j$tPWyN24)$~$xHhA8k?X4@@yoH z1&L}EMK2VE1?)*`P0{z4`UpKqUTqx${8zd)OPfo^TT36#y!lD4?2h@a93b0Hqit7b z{F#Rj9BIqVGkC9F`M%y4D1en;y1n!w>7Jx=!h7;&jyN8w)nKHe zg7;-H9^|S`PEVH+6!aB8H?O0)U8-qtIo5?r;j}rz^cadZA{h_QLiR{+ujpU` z!#<{$=3yBB)d4&L0t}>NY_Z~hZhCU|8&e7c+Hd^mud+dVNB`sTh3Xq5|JC$R*eanR z@tVe_NjFG_>9p2YG9-UB3bbAXKcN$e3TZU2m`+lU|6J<@t`%84vNgPMDD_a#B=;vn zg2zNO2w9ciIdMhkuxS|<=-4e2!+`5*!9ZO1Mf4CJ33Vvd%4&EegaLOfOcV;+-MzMU zw0nO#fgSmf*Ff<6u@F%x!oo$c1^Kr6-4BQ)oDfDUqR72zCccP>$i=KawyCNC4*Qc9 ze(%D_g+}{W6hWN>bffzPEmTZQjJsq_*p5S#PKz_N?Qz@HUTH56#QQ-s)5Eoe-xtcq z$47>)@j^W;Be|$M)HR@eG+=Frp~~}GlFy_B$!ZGyRZFk%~rUB zj%PnpT{c22!1~WW5{1U{@<7qv^$mWb7s8O_TNg~rwSqUBdmmH*k)=4Q-9F7h@+Z3{ zSQpe@je*Aa61o|Ht?w>{2`4oE(vFM~1N-G6sg%i8%lzaEOt|c6AdQbC#8B~ELtC5w zK5=~hLv1l83;e_5gfhETxdK3&aV}O-WtqU$uvd|`8635=`eK8M0@|SN0qJt*Ehnn!NDow6< z7G2=F2vI0v=+|Jh^w=BcWtCB)DXj=HB;mhgXuTqiL`W+4hHjSS|3vo{(Th@?-Gl z--|Z;W!b%$IB4k^65Ce{$oChq|3vr|p(Bf_BTyvdd{$xIwGeknn>-<&H4mEtNqC@_ ziIet40zGt&vSS7%*K3}uu|FrcxHpF%+Gc=IT}e9P%%@g6p|4M++UnxC790b#iqkeO z@SFb0_k_#ayvaF}3cLY|NF>|4$cms@asw?OEg$nNZbOWWl~RC zY`Y98;7-te#c!!&PShwoTwIS(aDY-0ifC)6z31T}h$IcUy~vg!bUkVkhQCnXP1n=R zd4-L=2^`q!C$FX^=Asb4sqB|-$8tXR_seqy#(8F*p1e>K6B8s=+o)7RE0>3h&^zPV z0pbVTBA*()e~j^YK+4fi%SAiZ5L|D~e^7da9TcIL64#RizHj=H#gn70Wz6|ynmVx2%f0x|-aUok{zo9u;DvHhhE;aVP z$taV5ETrO}(sg;gKvX)ILqBR&lG&G$nGsSP$g?&yvzzJYBZ26wHc8RMvY# zUgy>CDOfrsZ6~%nzu&}ja^}C{EFy9u;3=q}{G)v7>;=idTK3#{g|W&dZ>p(Dke!`P zGJy~s8*A9aUw?9WtajdHGiE1yj>C}UV&f0-K$rdG6Hc?sg<&Cx;#&Y;B)yKKe7A zb|I3!UcezBkZiKwY&@)Ae5+QTHgme%;^MN)HF`u7r(Me(O{>zM#1#{r%&LE7-MF1W zk)YxZjMVc{h`vHPZ2i#7n*`;GF^G1P8aA*tM_P%W^0g zB-_)IWu;USC3>VI8j&nwk(5SfCNnVRv#9?5UlrAg$eEFv(FY73&;A}PC20t(nWtQ#n~sb{?K<3!LqS0yAt^bjtZJ#2rdsP*<#wBthk2;R zOa_&}7*zj7e?tt9tq{pLo>7zk#mg1$F+wG}$y#!g_HC||VdEU2iuY^Mgy|!`+oqks=iVr*+z~Ykl+Z3v14fOb=_t|GaI%QAGY8a6v5Mi{<>8WDAF(syai-6sTLySnS64$7Dixp+ za#&?))R?G7nf^+M3_`(WF`uNgm@e$v97_Cqb+x_JvktiC5`@K130bv9drpKD=x zf#NT6dAx!yIFiEtB_Wx`_{DMS8Jh86T((MyYVDWBgkqHv`@QLR)N&aICwQ!;qq(x_ zJWDPsUV(vu7=(mF-|$(XPR?p{e|p~T77V9x$1W{tYc^QRZv${^$=3;%9b&(YpA3TE{xBL)A4Xg2#j$1a?xhav3xd(*NuhL=l*@0$Cb(T z$r@DY=S+cs#a|JG(0F|AY*692Ow=?qAwXi)Ajfq+hKy7j5B2x=>&;giCp;J}R6A+! zf5~J+o~;N;{H&V2V$3}FW=Wlt#FNYq9ak-XsFImh5~d@6r_$!2pf zzPGpcWLC@O?|wx`wa=3`qi!pbc6hyB6tF-&W*Jhe$=tB~2k*xlQ$eRH=;=a*o;|!s z)>`Z3h0|t*95FV_*>6l8_un11M-_nfGio<>0$B(xH#?ax)-ynPTpmDY2?xDMsCs>d zT5fX|X!{D$fOyX0F^g=Im&8r<Hq8$=Oh=m7Kn_dMITK8XhgL6eCU<(CmmI(HD@_2U~8z29(P$7r@5-c66 zbG0h~W&VTLZSAI$t!<@9dLHInDjG45V1aCQ&Rw*`;_IpO2uKR-bz0AV8c6V~M9BwIyn<#w8tLc6Ei`ue)Wr%zQ)9bI~z zW^BN^EbOG+=q~(U6IBAQ)Q2zN&AUHaX4oBwY_-%F)88*~dv%;C=b@^Kqtop8ZhDaU zk49X4YO19JZvoo|1Y4tB+vzid#yUASOs8SM+HGY`E02tD8d63V@6-T+&(i~E)-PDi zS~DRDiBMzuoI+;wz0?-q2CC^dI8BCSdzLhzf46xwY#9ay1x-|Z7I?frlY4gSasm&c z9o`S2p`n#=MV~wPp*DU+7@lp7SV3Bo*)3#E#N_ht~cC zVf_l%JtW)!ZBDf09`>2)d;2Q%L0KQOi_lAamcW;U9L#Q~MhIv`>bdSbjyq6=v(MJB zv)I7I=YF2$atujesk59znC2R2cG^#3x5TM28JRR%@oeu2F5>=M(SA#d5a0(?Zj)}V z!RE7IohlB4X0K&M+bSmCRoIia z^@%?)$egD#!K?up8QHf!8rEHOgF(_uBbhnd5(+uq+VJ7{@wFA7Wmp0}X@w$5^Tk86 z#HaBFuRG^eP2=ryK2zIY8BHZWb*U0#V)S?WnA9IfI9H57>UA@yR;oV15^c?>W-Fl8 zH}ivrn>!h-?xc^@=g%UfK#p`sx(MuloN4U37$`NV_jR_hS1=#(?(S}2(xlvy$8nqI z)58_-Nhhr5-=DZQk%{T)BZg6Yzaoi0oPMbQUfbj9C=b|?ucq0~z|SD6wI}GKZ+`0Z zaD{f!Eyuw#CHe>I4Kgoidq0gS%<~Goi5%ll%?TJ*!mwJk3XSpK3_-;gjv!zMM?}}u z)HGeDmB{aP`?^_+sWG!`!(sx9gk&^>&%+hOD=ya)GHhDqRfxy&2liNk-=TWwE}>?aZY8@z(O2*c-X8`ISoOy=gl$x#JI>a=+;;yu{w{UgeVBF)a<=Jt1JG z&_J|5H3s~3_G+JmnGlBHNSF%_U9{M5zAzq27~xbuisUnWi~NLc>w-$j#MF+YpipU> zU3>%A@E$K~tWlc_cc_JFAJoi}ILXP$38{iz?7Z88!=oFW(EY5}l#Swt2;-NJQmq|i z@!{JbCv?RE>X1Sw=x`}57n7EDQ2CAZc6e7SoKSpU1@7+mSlAK0*c)RK57Z}ELxj#0($4nIDK+-nHrqZLXoL+bE4q&*%TZN`cY4sR)Gnmt~Jv`0r{RM*t#q?pjn zjd>Ami*jZ+MIDZCW(&a%g22yexcJ|4FrR@P;>zQl}(GRxCd_uP7 z8=|3%QEFC^=jt-f`70xjdW-XTT?qkJSOi0LPaO6Aeb&k~5sQJr{H5w{@)Q$zm}2c3 zRU%_hr{GLHm203@YF8TrMgXa|JU%LX)cE$hH<<5C zHQ2RmsK8T9zVdfI{H~(?y}tf4Ysw(|yI37z)`MZuGKTJ+dLDLIO9Ay8@V0CS2?;V;SCUz$dUbNAF z6c1A!;;I>+n5z0-`}@q<3s@0+mfP9ZzshOt_PML$*24cb5(g<-r-Q%fjTlC&1C3wD zlP*yR{olYKLJ;#Q3Yx>a6H{i9Au^> z{IHS=UqE2z1Aj>AMwL}uX}3SwYu2BsRF4o3{S&?4jhG!iPYkmniOPGNL5P~4|17(` zOB49lBL` zOXU8GY?dfj%fx-DQ1S| z9pe=hcg|{KLUX5DA4}VLTG?M#;oDSH){1hDW6ABKynK04$d~PUVkPT{d-B0OF4W_Q zyv5C75+%1YK5eJ`Zx@}A)%|Ba6RUyMt>EO3DN0S!$t-AEgdiN_l{qS8jJl<>Uz5f?@){Z{L>r8m*zq%t2t*emoLtSdDB|gGAFe#~IO#nQ z(2hf@v9o^mFyF$z{XAeCYh7dbrZnZHUBTK>h~6hm6MHs-2vs^2HFFD=i<0!UF3{No zO;BPUDVN_JD3B!hl6Hq9R{0Kv&ZFZ;3mE!L)Rk2CDlNVjhPZ;c8y#^vlHHOvHgkxp z#KbePG(B4*mA2$;)~J}6?sTW-p#K>5K6yCPE35*7UNENKch{(d{NMp4#t$WP95OcA zok7f8BG5|`GS}feU%EJq4>zwNsbqBXgD-(5Jw%mPL;ttIj#s%C!XHj|In|zA zYzxK_DMRufMBidx3-kN68WT;2`V zVkKpn(m!#0tA@#QuH-W6mM*T;Q`fTRe>_r=xeHT9S2$<5xGnJO_oc-hWGr4u{SuW| z9iF5djV{0Ni<`McV}03!69!Q+geko+a^jK4)sQ;j-IyCY?>x&>z1*D8O8$5`ft5Du zF{nG^S7TBtwStD$GgtH^=*;lZt`dUYAFf(SuUR^6;eGR#h&LyB(MHfMho-psd#Y?R zt-dUX16$untI^81a3Ue!e?_fp-*TVT^!Hh^d;2zK<5kuDBHVG`2O=cF9I=yO)=DS3 zgP0lHBLil^D1~>T)S1Sc^bDIS43(a*dNb{g)SpP?$2cK-x&1IpEBKHnrTH5>nl#=Y z^_jdsAS?26P9MLh=aTef2CR+sG{c{4rQO$`IzJ?RuJBUe_SLV+elk*j;B9RvtqjKH z++J~Z6_9p^XUtHFzevuhyvE}4k!(Xuf)IK?K|>Z`Q3RCTP0!zXMzY=t-tiZ!od23T zSs`V134Pc%kZVYAxZS70`^$TO&)&)}(n4)HH@@Iuxol;!t#YCeEww%4Aj*xzllT@aFp1N54MmD{sm{b3^EbEHkB{@-^J!KyxWpK66Eo=_WYY%L0-Lyd z9p+#yRJ<|^R9t@D(M?}o$t)A{_0je<9)787li@U5lxcgs73sTdNV)W=q~X18bbshj z31U|+P4Urs8n7KAo`aw#!8j3?%vJTxI8?u*8j3sGi+l_*a=q*8CYU`6NNMOK9P2)S zw$Mt04f*oC8-h!s+QV{Pt(F#-kFjsO*)@G$Q~o^XL`Ml3XSr!WhD5n=M`w|#caW58 zE8I-OH@G$W5IE7MnXDCR+dF(oygPAH@4%QHuJO7u^KNBfe`?1rF5Hfz>~ptwAZP8l zHO;0*oNh-K{a8|-(>S%q%_8X(0iDJP&t#^AshLb&*qiC(d4Q7760=$Ha`G28QXq!K zPh{0X>2~=W?2=-}_l?esWAtjVh_VleOy7Lb=CoWajw^iqTzV!tR+}y?QgiyDFC8tp z3#t|N&`3*N^qYPRU}|cSndq3mvp7Z6whTjh-T_AcTs-!5O~9f_7kO8<*lpwp<>TC$ zm5yPHcGN_1s$u8rMVF|pz8&1tW37f5cYg2wdM^j~w2ZGK?~|h|Esa|u?zZA3EvO{m7 zGie{q>=v`vorpNiRls{R*QHJ6aY;1)&`qsKSC!dvsKcS}hM0m?Z>2{4>9r)tDK&fD zigoNjrnfstTjTf+VMj{XnTUX18ETX~Q;Hi{!?yzLU6H6C00xW8h z6(!5=>meEqrdZ3p6#{nBV(sI!hg+Go8K&uU>lTZXJvLV>=3$Rx&18kM>E$gc!|AmoS|nn@y8RXsU!&r(e8N5M71y)O_IKTc z9ZH~#2Gt5}WTEr^>P2E|BiN7W9nO)3{YRkf#R^&@nwEbn3DE7trda8euafGJp)dJi z>oAzv*=LTL_-;yXGgeX%OSXl#$(e17mv!Knx3~~pJ)7Cn81B#1e)BbWbITK^I zP+7U|FLOoF$MAD9Ng|F&?IH#SKde`@6oTpobME`8&$L@yVUkg!!bTxHnD<$RDt?U6BC@KaYz3?Y$mUYE+ka&-ZV zYcItfhhZgT)yuUG7P`Ov!C6CZM~Hwe1X1g=UQgQVA%fr^WLhH4>tHs>!ph1%FC{1E zqNd1SOLTRz)%Z*rYOF(AIK*X;BwR^R{S1$VggeQ&XJVxxzg<3FTo~Lw{MR+&R0=NSL))NqdLS z>GsXQ+*yCAnLD43XN^-U-}YI#P772cr(BA$P4~Z9C_2atFu*Ize4v>`YI8&hy)0p| z5}JU6^57?~{-)%2C8cma~ebw!2J$-l{UWM6L*W<)XkKQ_! z+o+lc$y4|4CHyuW|1-w$tU?(Ye#u$*uC3z;qf*?V?+VGUw!q=1#^E6vprR--M4xTz|21gayk_%$WImWD5@t$U-9&z# zvUO=8KR1hVWoyMj&L}P-Y*}Thi&UVoBEzX*GK@D!uOn&S6Kqs(wOic|^2ZSF7sHsc zBSnVGb=#;P%a~dzco*J8)LGJ2Zb18_)H}<1#CUf6yuV|mX*pH(`p-!BvQ>vPE(<3# zIW7!qz!7OT;2u`;#TTZ9;=vjo{3Z#7^SD zR#duB-KYUI%ZX{t0H0loTK?YiK;0(W=tH#qrZPaRa$vf;8tNteI#{H(Tj#Av-^a#UF_kmNVW=jTD*tYO_w+gAb|wc`#di;7f7Vzda$bc3~VX~pt?Yi>Ik0TW&f9%~{YLXPNqGfeIX z!WQF_Tripn+6cK%;>)MKaMvAUt;laO>REK)my&h7!tgp;1FeHZhaW9^=<-S)K6er0 zZfDaJB7`NiQcALpWtXS+9;`d;WHFqO=0Tfuy2)nC$1y44QPY92qYS=@?F>F(;CV$T zL(&qV_C!TRwC639P%g=gB5Oxlay4eI2*`HN`kLMN&I_D+{o?euQa%s8Gn2NNYkHv6 z2F{Gev2x5lNn8d(Z@?T0>eLQM%fw*(P`0}qJCahd~w$*vUwT=dq@oV12s|} zj2#)&MxRI`W2F^75%06(&6~*?U+dzU9hF$`_x5Sq1Z8k3@62UKf19VgPr3C^eqKesVVEUh-Fi=jA}I z#-Y;iK2ymwe0DBCDiBn3$XXvRoE+5H{`~>5@ciCMhR4pCX#c4%d1mY_Yc6~?-1Ak& zgVD{-47mj*LQeSTc47P0fn9P1w&&pvtqscmjETUGjHACu3m4*v<$Wh}8qz4VUp~I( z=TMrQ(i^k#Z|F(`;3!eLZF(nq>NN8Olu;{&5Qas{f^)QZ2`0*T-c!Rhw#7ot) ze_V~Gz;BQ4mi?kZW z$Gp6WGCE=SddW!&)#e+@>hSw#WmtUyz$)3Y#Gudflmr}lJk@MD1_n|dHv9F9O{+l(q9&+*Yp-#)#`xk)h-(7i_(mNv__rXs>qn?C_9*W=s7qcx&>JGy|jr0M1am5Q`HU(hp{NaRnI^ zu6=02{5PUD-24M0FKb*|pSaP!Gz}+k_M9gG@D(h~VB9+qN5|S8fA>tgnrmF{E-A7KVXRP*8a81eWjZ^TUj0h8j;f)R`~PrikTB74}_O zS_|C_ACJj+jFR0h-w)=^AJug77m&mLqGNUsa^gh^Y`?;XQxz3j+XnjP!qe}I;dAP zUPRQygnQ(oZmo#>UE19mrPQx~aHDkh= z{BUKO@cD2b{hjac^?J|5_7Vhs)s`ny_8a&Joe=c;y=F&%E!CBgjinw;zQ3!0eD#bx zsb;>tq3mY*#hjfmM}`@*h8j>RYG zSt*}Q2WNEe?CDHyu+}o9O6=ui0z=PE*EsC#$*;5>H@lB3v+P2pIzDf_aKt9f%_cZL zKDW+8`L9l#y~u#oR&STKpLiVkMG(GsJd{lS8LkrPBrr$d*PepK$5g(v&yK+OGsF0y z7GsZ{QL2WceD}Wyr)O#HcGG(EIQhe)tcEk`{d@tuKCoQWWb<6}vW?-R80uYHcp>p# zzn!k$FbjI49t788zUOPm^_lHHq0M}IFYQU0noH+S_bPBQj0>Ypy;`&k%k9SadW-P+ zG7vy=6DP^H@RU681_oxeV#wn7!Oem)GcEjW- zLAow@PRaJV-pVh9UB4TS%S{pJk@%2!mfVxn{fR13YQn$EaKUBy!tmGk4xn3HWwAKKT6j(T_@bOF6kS* zSFW^AB;(JkL+kZ(MUhDXn23`VA7&@oKYw=0`U{ZIg2I@#?WKAZel?8!xqsJ&jVwGM zP87;H(azWU7vW(34f(20miEV_YWnb<=_l7>uSn@*>_7@JqR@u-u9YeAb;c;Ax@90< zZZs3>PyTS^a{p$^Lr8Vtq{*_MziSFLqf18wGy%{R7xm#9?BP}eUkX7ISxfYXsbjm8OW zc!h^V>IKK3*?GQO>GDrMM@6#Kq8vO~m~>}Dk%Tb;gf12NWzd(!EFoABXwdral}+6Y zC>+haf;ltWyE#dpvo=W_u)OqsiR%N_LKUepOm2HoAc!DtD(j)eDAo=e7DVU$=LMpz z3wruV#;*=B%@kj}H>CU%77H@TUaF=gg-tvqQkjho0f5v%qJp&}YZt!QJDE|luriu{ zdr!$wt`wLUWL{z7;IQNK%`$Z$Ip=f=7tSTdCx*%HY=jn5#hr&7i&{|FALz-~LbIet z3!E8;Cu6ZWGRBN4AhK(Y?^w$(&nLH^&c@<)r;!coEOHrizD|fq{{EhXwTzXF-_fLTplMV08Z<^v^Tl;O;o2&T{Q)MdjqD)u(}NR42l1W_{~d zYL~=@1~hMycY%SxtpF?|(^L&j>m`Bb6W?$WIi_d&pV8Tq^U{)DZwGDe7*wEF{x;-j zy>s{+(wizzx0Xp9GHIc)wXS#>8FtB7pVN%QY$zL=$+&=D`QF)RS7{VxPcc;xj%D{< zl~})5DUs{ELur^8rm5-Q^923hCA`jXmkdfL7SM$^Orx8CP0#is?U!f)dW4rmDj#5t z6%(G$U067`aVQT(6^aC(_3l;mJX#xl^-L!&c`JKf^myvIYu z`*Vvd%nt)_(X#&+8sSl2dzH_Jcbv&8{>Uk6HN%XjQ&YA4Xtnfnzv>lV*XtoVFw1y` zLngHC&6L!9S>zbH6C{J3ktE=@r*zt%t+SBx_6frDI+_QQH8Mjos*E_(3_sIMm8Kb7 z`snwVh||R?;n^Y~uK<=R`ZES*;42)QUkCFwUwF`k-jAW`EeZnGQHjUI= zVzwbMGNBEjsHw4<1OtA6+xfXY<|W%n>eRG-hqg4?`^W}aU0pR$q(OoNm{eq`iO!MH z3BkbQCVjxApwJt9Eg>C<`?;3DS3!QdR3rAzrX!QbhZoA2j)$J~)hk8RVt}C@KjfI6 zalego56&eo zPB=u7I*XG(v9@deeQ2edk!9m-ECE7%r5Ko)YseZnwL9UVq2s)j`|K`-Z^XNsold6Y zX!*+D(S*n+4WCGp4Hd)Ax4N-%E<0jnQNyAFI0UUqv3#B+b}%ZwK{8o1uE(Zz^DZ_2 z^(w5CMqBeY_%8g?CrNZ7UqljLV1hNK`h93SE2NR6YJU^HhJosONZU-j|MgHe90G!7 zhqoMWw&KR+A<7%ELQe{kG!b3J${Cnvm0W7Hb3FKs>5HrD>NiuOR%7t<1sKa4_C@At zH`%wIPfFcf)~J<(p*?_G1p}Z91`f^^F@vg*t?va+|NTvLMmfxxN-5IAZPEM(Fj486 z6@*-E&FM4?2EEvGy1TrInrLrZc)Zz;j?ZgWE>>1(>L`%*E2T$82)KA3&7hU_39C>? zYZDof%Ndo#9ceHjXk6axH%9xtq89)?jTd7Fh5$f)T|iP&xAWnAzm!S-r0c`Q;Fbt_ zR~S}gMK~Yvmc8X%u_dIwY#Apr;YGgHe0BY?5vdOm|D*cyYgByyl{_i5!+4JB?wU^d z-+s>!t+Va1sroNkQ2ciXtWYFwe=tcr4yrsCPYo)x7wf--g@+pgaFTDWAlyo4Gc32Q zAm566iO1{*mtcV>WoPH3z&GO4m*9QpexbbCVix1hdqu?i)S6<=V;cFT<%3K2`cK>Q z!o6VVohfm?(F~QyqbLErevAu^W5`JldZj74=XLf%hg{~r4Gnb%Np4IR4JLd24QeiN z3B(~5c*rg9n)_VhM95J&PvyLYDxCu7o0*yUz>Y)A(`lM1lXQ-u{V|aR|AB~z__-vm zyPJ$brViC=89Dj5cx3;@X**{Jzl&@gEyP_aBKsQwJZPNch?jIN0YL6r52!${j|CcC z6-H3MehX}NTFdk)B|TgJnD?F@RaiJqRX+YWU#8NlrF#3gg5^S`=m%}zLY+!!VY^q? zY$=Nt$Em?=YBbd5kYxwsrDM|2&!0u_>f>pJA_#2v;+)billXPaGdRai9{`m@OVJ3Y z7l6zL9S^_0e;a!KGkl;-clpM!*|%ZaY6alvWIng1b|%R5L!xB>{1OU41vl?+&U%0T z#3Uw`k(b8+F7!rN?pf&-FH@@27y-_5FrE$ze3>>7Q;?U3T^)mKGnUNo59JR14+a#Z z$^&uD*;}Yo71F$9Q*q_y9-K%@m(xZ39+%r}IXRVWA|oLaBodP4?X~$<7DNX!Hs$yM zDN(p2M!jsqt*xzN0u}-lfyYL>@%)J9fR4K(%gis3QiQ0HWj263v8Qs}PgrT^BXq0U z);rX}_23!4OYQ@x^0N=(tV#vN?;17L+wV0|MxX}-?gd>F83*0kPQTv1Z1k%r7abcq zT5d6~{6QYWejLEt*3i|}l^_@nM#G}w;tp%w@a5V~{u5@oo;Rn;rRp&#@Xch0@k67d z)XdD$V2qv`*-i#&SJe0qw!p89u~aN#`Yc8ej^dfpH%TF?b5F@Mk5RGKI z)#ej59WU1CwvAr`=dyCLmb^-_GW2xj(bwIU606OY8a2jcM_84r9~m&~3ek#Z}ll;2$zwd^b8lzW+wAiCatVSj7QEVw^A znjUt;c$xTlTE|OQvsQkT_N@dra^n_U?Lxm)hxeY_RKBHqDyQ1wzr}A2D+DY;uk5gUj75z;vR++iF7~vcx?Ffv)k3s%>Lo5NXXHfJEs1AV;2Ce#& z-GWS*ame}S0m8ru-6nt^hp`2aSe*GLB>#e{Bf%6771$m-yJG)Tj+Y4N!B>zLRTBvB zOo4!NX>*OUsN*@i_(WYv$`@mkEol{f(0xef>B@=f{&39YLOqo5j?e3^xaGdp3MkdN z$E7m1S1QNN*vIgQR_SCIESln3L7+5M?RVw3!w{#5A34rnX8c{R_1TL$5ja}7EoPN# zcI3&2%F4tmrOGrbU6|M`^KnS5f3Ge*Qhh5up_l0Q=jG3pN@Ror5JOTZ-n)aE^WCWs zfP#7Na-o&n|_GSkt@cb{{V=z&C_|W1gZBEa1eFVke6$fBa z0R7N18C@cm%DttVM$^7dNfm z64v8vRi~-|8&F3kCnaQLE|T~vjF-5u1@>2k>M{ymGjI(@_KQggenifD6DQW%j3RYH zk>T21ZIKQQySn0Yp{h)unoG2Ue<84Ntb4wTlTX$d%7xLz4Q4yu_X;w{B2nob;O%?Km00u=+m*KQWG zftfz46M@^KD*Lj$ykc7_Y*3SmM9zlD!sZ}ZUnRiAwD@`dXW*pcsY^+29$jE#hz(A{ zkMshE0_3~k4o>$EDtdh`5QTS8!=L&j3F3We2i67QvZ27AtKDU-pCg0)zJ9POqp{zA z9VSbP24+JQg4Z0tYd|a z9XVnErtV&1pIIz;DEfPgZ_NY{1fIhogvAPQW{d~ccOn8qLOj?63kv?8ot;^Dk)esc zeSwNFSsDK8bE zDy^-pALY*o{pLH(SINJYrZRo_K=}M~o(Qowz?c$jSwLKXLqyc#BRylUS6gi%?FRN1;&uxdXSLc9Rk1;69p6O0yYSe>76nuxbhldOwyrwyBGLM1yi> z`pp!W_;QR(x0iU|J4$KFGk<>iUF`zp3e`x;g>nFO3)?r4ShI{>3#q!hyTf(%4v@#5 z(#UG{;5pDQVK-#P1Jug_mj|o0gQS>_j*g)+sDjIuv8J&!^JR_S@Y1lCTPs%1)^6`G zZ-8o@q?mY$!nHoNOJcR!WNI?4hj-yU_#(2bO&K++*M?Lt{CrO;ftU}A?dM(C(%(AT z)A?VBPfM0269SoBf(3IMCsbxmS~sg8b`}qFQ1_ZPs$0DdVqgkHj*WPOM~^x>wxCr* zWC&()HapFcUGIJ6>zV3eWGl}2hnMWxC5FRf-d{_~0A5O^t{Ws)a`?qo-XhF*@D!8)%?s{!NRMzJ^99}sf z^C8xkwO*j!oztuH+_NOL?+EqYyt)R9%V5E~_q?5?_=lE}J1-(HIz80Z>q_I=)7y`6 zut_haISyNfzXp%M*B>J4LXT$C?&oyUbiSMN@P%}ucap4%#6tL0N2{WC?Qk~rp0eai zsxbZ`5;CP1I}l8`vIiE}YcC<1oT**EdI)rU$E-ZJC{l<+3lN)HiRB6=W;SjCnbkcL ztLaP@z(~R1s$171;v@*AE0JOz0yo`;YU!Y}C_tzJ4WE6qm)>rRx_Ht#(5)k*b0dGQq|~fXQ*Y5|+MSS9$@f?O+!hpKSpN1= z;fGtJQgNi;h)XcMD5%U|YBnl~_h#*7$C4fWF1{{c%S19SxT9o5vD;PfztUYH+ zpyWLiJa6--|rOGBdI?o z8Hp-_zbrFwmz}gt)9zIL`sz&X2RcT?9uLnK_ef!<2>k_wVy$1i=o_WMhN(kw+nPa6 z&4T!r0z-p>kd99(%4+D^9}+hUt#hAzF*`5RT`cH)Tl_kFFY>=ce&=0yMsodtIxDrX z)KM=tvkzoth^7FXR!O`>#+!a3r2xO__lAw9qaSFIKzR?^50K*8zswl+LMkSsECe$n z@8SQ!YEa-WzZkjm8}+p}!N~r+blj-qH5Si0TK7oG2-&)6t={E+A4SdZm!*Q0?=!+H z=?csF@#sW+?gFCA7nWTa)(qRP5w+j9l}qa%n6qS- z)XfI80rT>E4K2%!5zr;SqX4K&d$VOjzR<8-j{Oj(mP!V3b**<6{N-;v>Ia+r8MGo* ztB>YuLI8J0F<%;=;>{a#K>O%()JTNs+Y0v9DH$yz_r@A>r{SsxtOhkAH0FfrZtjZwu67)@X8xn)uQ$v zGAjkv)5?8$;*-o44g~v!;W2Cadr%hO=j7znVLQtY0QXp9JcL8yzC#{TUpxAtA+Kw} z(@kw~9_*!u#d@u{475!g&UX}s?m*>j`*`ox;5grtB3rhg^T2PFZA_r&*(%9KLQMQp z><#^Av{-5(qg;FHkyNe0z4s9=faL?V1bRY20=)2!bm%qKfyo(}2iBUR*(ATiq zT8P!5mO3r<74ZrNJK3jPy6V5z^9qFgy%JsG}iJhx=o_1;ZUGX)%-_VFhmp8!BX@;p9U zpi?W?uKL-=^5lK+;KKLdVVaj&E|HI8((u1V<@f zn*80n0sGe0+XEc|({^-^ZS~i%=kphPc5d6UlNp}hSLOK0?f*!?v^x6omATM*<8C19 zl<+~60GTf=EDVtGwmd-My;xR>g@yG=N~-2oLsk|Ie3KBAqd72A#~(`%FvQ^WqGOd@ z135W46LWJZQ#G@$t@Tbxm;e<8=s~$|+up?eIjEm0si`eAFLdN%Z$8+4e0~HtE-C=` z+B3Z@_pw`7#7!t8?KjQz`qhIjFs$RU{IlKb4vNcR3pIj}bE`Y(e(DviTKVwf!(EYT zS&Gggd_(HW3gA zogk5Q(bw1j)0|Ey2???qA0Ph=&sS-vH2NqFEtSLH-gkCfG`Ta!_V4rJvWN{`NJ&ak z(9no03H}5??~*koZBPLG)JGyXJEo?f=+5RVXx3xP+Pq^|8`aEj#?`_#CLwwhIL#8z zML9X4f3Bq2VhsDrsbthE^`LPXwO2IoEImdX^a?JnS znUtCft6_+?X}<{u>^o9lP{CyY(o!PAO9I=IRiTo*nTpS#P`U}hqLG!aF_95;+SCSS z@Q=DD@CfR*?+G~lfMH~2O|aAEdBXxGuW<128qKe2Q7tTIjN2oDZZvT1`I~WP)U1T$ zIMfD-Ym&K1Xf?-;&Sfb(p@O6# zHtG(v`Jq6kdT6o(EO-vq_WDFWk!{I&@k_$oR(RP(!yh)s-fQi>_OqTjpZQE5r{ciiI7h}QtcmGyFVA{*n|dl<0yRr-|DCVI<&v*@ zliU<5AD(JE#{1xgN3oLW zzym9AT0aH|Wf;rJ+&w&Awm}6TBr;buufFc|hA2LM8wO5|+=e*i45pEhn*Do*UQ?x77=tn$>mvvB0ZgW`T@cho z*HTbV4@pl96{0=1@|niF$IJwU>OEt|XFWQ&p^3|B7y@_Wvt@#EK@;mlt(8*8@XHHm2ZE-M{^`>kD!JylIjwBUE~(qOsj6C@!pb|7zIP?G$ig-+(QA1 z!1(q%*_x!!MHx+^89_fY@J~1*67TQtcOPKzp+qFaBqaO>_wYsK=d?!kjE5oYDrxK7 zj{>it!(T!%W^yEaey=6Ud`Yb|ZV|*GVf+ZT#`B@fv)K6fPi|}dy?q;rQe^b>-$a)D zt!VEH7ri1i*$IZVKXIyT1w1*^2Vp0}*QU^RzY`u7YmisoGR2;!sd|7HR`DOR2g$Jr zT~LU3{<+a6uU0*+R zzt^^pzP?S&D{Mo}FD^<~-o4{CL`ZI3|NjFq-Y$m`Apbj!!1)GBidtiA;Nn5ee_ zMOI#!=H^d`tr;Q`#FAEBNuKPB-D4}zJj5dmcUzlW?FqrKII@ug;#A($!Z`XjQu}h^ zR{`|74~H&D3wl`q86pMNwT;8>Vmy#zP?5h{mK{sN?|k(ao#S3Nr>(j1ID;bTu`GR* z=0)?Rb29Au?qZ-3uN=J?c_a~r4mwxj1jZIMNp-Hiod#Ax_$|4TgFG)YW>ZT1tf%bA zLwFYxP&6_(&}*nwu+nIE1wqZs(Qm9LB)uWHpRzh-X~bWMC4W*`LAp^H?QEBDt-)Wz zuwOJR;;47^#y&`FVMdAq5($l&N0q37NupStkk?w`*--~@jIUL8l;!odL6`+3>zkDR%~hiByTJ)t#q(^ zd@}h(`()}%VdZJ?vw}MciNOX5q!Hl`8SwW>-|l6q+Kezs)aStG&j@WO0`p$JOoHdy z242FgZ%zYqQIZ<`^h`u_R8g?rX3|JDeP+4daT8<0QOTkN^Pmg6#_Dz0HGwnCgh45} zcXm#Tt5ly!#wi@e80m0f-Pay8exO3T3+6=bTJI%QxQt0 za9=0Hf?-(Gvtm-^6in7|#+NSwu`bo+x8-C590lX%2cz**Nd3NVE=uiv$PbG|_&;ceu5SILaa z>pAfn|BWKX3WvKi><%eP^dvE*^iiQ(dml?(nm7yvX$ijvJ z9T&{S{q8g3o>;RkBp*`DH7)%SxYc;m@=U6VW)udiDv9H3bG24D5KtZ%dEN5dfg7Od z8pqji-yOaE^L>5Ym8?fNOY<#A?gxb(d^-krep1&7?0J&9#(hKud5`-VabhKe#`ULj zDc+!-`){lGsd>Jn5KZn)H}zFa+{^J^%W>z8O(s@Z_%N; zPuzu3UWMZ_c#{eVV@IkeCY`|CE!ZFa;pt6R^!@jY>6~f?a)iZNMU!YT5O>*HRPy zFeV1S`OFl2iJ$J3EQgVp58pZ284bct!Q)U1bqOG->bPs>#l^4_K-rYJ@?H;VA}uL{ zZbKM$^A2kXOVCIAfjlO-+cZhALiE3_DyYNr$={XVv!<*My9xGp%&ho%dPRH%jlHPm z-MOx|D76nAkSQ~d0lA{MjZ_kkPi?H6N%q&s*vwC`+7x!Y*Vs~qR z#Iet5N)8dV38uaozDOWJf<`Dq-+qY&<@(HuxH%aj$3!=*@sZ{IcVKx@Ha0f40YebT z?Isw@*ZMk%4PU@t)PL_y_IctBq;q_sQoZc(U|=84Sz|{5HPvcj&_hqTL%x+2e>g6W zG@{kST_aeTv&q0Re%ep*+{ua8rTyd%L?w+!Ko1oi2QJYI0d_0PD{qcA4Ety-bgz#-Qz5V{y z9-)ugOfqVm9qt&(U8M>8+eQ)ms6<^0ilChQ`PmF-DQtE#|0khIO1>9)n?BK=Nt8HO zI&Gb*xldRTh2o_-;X`3MKE7TdR!BPK6E~!hQn}3X{AZbWLL0dvdw2#sY2kFICnpwy znS!VN@YLzKIfCHcl4@Y1baDuissWkn1Cm_?+n}TH(XLBS)-c~!N6j^Y_hlEuh%H$= zC-(A2s!u0`FP-kxjjYS+^Bu1jAqmhV>@D%$yt;u&Np$qEF;ojB^OR9+X~^UEoClvwO-m|MWbGi>BfhUA6RwEw z*57?}H&0n$oH{Do{D7_!#+rYhnvVu6E(Tes;a8l^ziWT1$0E|=g&h1{&#pNwd6fgX~{=7FJkri z3nDF15rTgH%xvQkxH`JJeV~{Phr_!pYt3s72H|iE;LjpJ66kmT{tK-L-zcxAkw5dM z?ECB+i(dr6e-3L_!aU+9J4i8Zt$vK(>^8PtAj%=25+?a)IbDShD_q0;b<=FE<)=jH z*6seVxA%#}3~C<6S**WPl0HWstS#@fB#3Y2$5us6=ct69+&W!AfQe9wCyJ zw0-1S;?Zg>{~-u=b>6#%rv3*Cj6qj!5XYH84kK=W(HXvFK6(pOG!gFuz#q7Q^w|Mp zU(qOX@ug?QMcB-V;Mlowkm5Z&`KWu=?>;p)-mH)Wb#8mw<8;DqPB z>`UCnKz`1O%@yhg@oX`^x;}&9m}mIXa0vRC-I&Mf2a$d?#TGZ)87;lt>M-eqDJMBn z0M?)gR^e;Ggp^F+1ur;H>$wp~62n3K<1>dy$k$(;#aUcsUnTQ_V z=w5L(Z#3NA-6hKM>$a?2KJl|kV{n9#U``qZ(9zLx@$&YqxGuZ5Kt$&~GIZKYEYmEG zF9_oK8?FH@=-Of|^!Txswl?|chmaHZ);%GJcZ$?sZ#6YFQThZHqaejs!kMbdSfVs+ z3}HdRntN~iO*Zj&WEspHo-dC~#m z`NCO{Z&@NF!A*;Pu`txG3*=j#43lP-A@*Y0-R}U@p=Ag=eK<81S)h%Yr&*mclhh9C z=eS(}&2DQ-Cd}Kvvs|#3k49y1mJ8l8u3j$PNCT!|b6kn(RbLF%)gN1#2~&qg4n|R* z{d{r13t-TM)tSgdk_+V*E9otV$$z|#Qvpep^8VT;p)JJAxdIsD)B?T+f{%|s)9l8k zUP4ha0Tf{IKT8gP)Jn-@bCU~#tor&2+2|!jN$E7gT_$4%GE8}Vb$4>QFHv}CyfIiJ&bttORRQ01wB;V9 zN8g2hls+!EPGRt}RNyzgOVesQT8rS1FX#+!U#7y4Sb8*1KZ)9xFlRx# zf(V-rMSKsimbZhl6F*)@65nQsSnhYS8NnsO<|Ej{N#&@)&v02h!9M^0bntk5g8AUn zz}7)TRkX9bVzo>(u)Rtou~s{77)n9@;2!6Z@;%aM#@&nf|GAq zk9`0BskvEnB{EFlhd_oeyL7d+3;1SfK^W!HRk@ORLMiger7-%oVEzXCx{A^yT^(rfA3?y z`G7he`}(`Lq2#skI3&{kl-i}z0?P)7-ptdG%3$=N`>_}GFNT}4{-o+umGfXlxDRm^ ztnai&ePd`{jX5>#p^3!65nB&ORm^Cs*0Q$+NK%~e$k*KqiE5^#?0<+YaoRkGH5xOD zL-j&abv5C;s_U}(J+{U@BSS+PX@8{1;o?67>D5(lNt0C17VQEg@3)a0iF2?u023z* zVAoVgrO)2v6qefl4v~duX64a~b`On=!Uq;M6`!!k%D!0)jqpG0`4Ij` z?~TFHROjInC)0Whwk$_T8+1D#1L({yX}EuK6-`Baq!-0KmQl@USg-ZiRR0UR2?PS; zFu1iPxEPSpD)|_5s8VX?|Ai?m1ksr-;dv|toN4LkJXKOs>h>FCPp5YuWF_D?m4vPe zqoHHsj;V>{ct)b6!=n_)qoUxU3Ok&)#7>Ts``ry0q zn8N%Uvf`VtY=d{9K}vBPg=2?G578-#Yc?;p*6wlIu^*UMgw&>) zk+w&3Z{f%w(cTg>{OE9b{d$ZlJ(Rp<>DRlxvq>Ed_PB+}c=G2DNstp&gVmj}B4ylt z%X>v(3E^`wM_Ka`k1;oro-+DUW)R$eNS!o&^XxrQ*3bT%l)Nj2fwhRKwlP`fq z+~lDgNv=`?y#?^A1Y%b#*wQEvK>7j?MB2ypl#o21fRW6ON01kW++_`Vh=j+@YNfKw z`P@}wDARfyB49&k*&9X1WAr5sSgC0P!Q?^U0b&KDdKISaA9m-O)()Vz?t?I7A3u#u zVA~fp!ON}FufKMF(beWf8)ta15ZAMRA z5P$>{&!?&bu}V=q=_cN^PEJl*!|`6rj?T_DhwJccQL?~b{8N<~M`7{pbyE8jo!TZ* z3-oAehdR7_navO&OzvHWr!4`4pwu-55>dOui2aCD%w6r?FR1OUf zrz|Q(p<1HPd`CF4Vp~KJ3w}gGW6!LfZ*05qU89JtLD2h96NyQ1=RxUMvXY0#BZr?A ztaaMGb##2K%2>_FOAkq8joJF7&us;0 zSs@V-41^jj)r*hEZ%F&u^_lHbMP?3&nuto`1pmx5)=QO4z)z;lMsFA27st%1Iw>eJ znZMSeDBFFP`N!B;CjhI+VYBN(22#l9@^as;g16m^{6Al!K1JC^fqt~h~!65Sk#=)UDhh}ipf?Uk38i@7dCZkP7A41wU;R2^_jr}WLf zD}invS*eqTiQGx5XN3kukD0dP9YBb?ts_dkIR&19>!q-C>qUM8_1CD~nc6ThkKYfS z=S3?gWPSxm_Hh|EH%G;GWAhZRHcp#bp)O9^&KBRKf`krwg|sVSpyoRnsXp4@J2Ors zS;t?`{`r)oW!)Zb1tf#FuA&Hp{Q{irh4i@vL-oeS2GF(=fUpn){O&wJ7GLjw?#^?E z00<5@MPTP@-Gcn>$J)C$U2}m?OX8(LwNiFQcOaEZhdk}g`XCwP^XJc+lvoN{r4=tI zdb`4K8h2y3)LZMtwJPAxgsW0GHP8`MPsBs@!@*N+Ih_YJYP`~DhkW@%xm3#`}S&fXmZb*A+RC((t8om({0*-1dVw zw;d?%zG3Ppoy+`g!GPc^A4g2w{Bf4HDj65D3oj?f(ck)~0n3#2lm=By~IANUsMzrN!zw zLjWAvIXf3OHa`ATJ+_Xvg4es^ntvZ3hnPMJ>hkt!`&x0WK%Iw>ZkJ8&sPv`k8@}V4 z+Bin3yPDPJp#kJw)yv(Q5L?j{@3=B>7h~|fEPIzLyS6*bUgzvClrJh%{gKre0OeNg zYvacvB9E;435zxIMnEN^r^>IlMU8w=tm?g9Td5lH)YR0i7!{!Mj~`OV@zj_JsQEC> zf#I;@Y(w;IU3Y5gw%CmX;0(7D40i3DByL-eYQ z;lLW#&feY85%mb`@3Z`_twrhcvhc$_a|IaN=j-LZRl;H_pFv?#-MzKftz^Fw-1lG z9oN0?e_a>>-nIETCjE#E#+hEheP4hIyu7?7=b%&y;~%WBJE{r_Lb$8j$kT_ZPCxwl z7=U+7z7(NSL)aU2t zn7GTm$`ooAYSEiVOD;Sh6JE!+ezw0h5D~jpn)4BS!tfB%xUdwTl_pAJTdi{$B(WPj z3vw0#2a z)zbB8lk63mE_loT%(tW%Ik#W|1ZiQx0{B`R;BGqmq%|U~*~(%h(s~vBvDU5GDUdCv z03%Ld4}0sH-7p7z!jPwBU_jGUUD(!^eG86wOxrR-L(yYmV_Qys8vSZ?(Enyj6P)IlzCt<9ECy3Tu98Abl@EM1>PiqLe zp~YESBQID{ZBV<9@5rAwoPS(pPc+!uB!V7rb+}m1qI6J+W20ulog#}j_eP|ZnV)Lz zlQa%5VdcMp7?i4upM%}Se4OKtjwV=OS_2FW#Z3P*>B(B%*u$O&OvL|RRq_*Q4UgyZZnqhCWr6}I?<4~GS8>;QmP^S-TK zm4W`*CF?bDxUzBc{g0%NZ04n$Qyl?>M8|wxoQaO#z%{Fu(Qq;R9Uc#bfkT^*jDD`I zIjOVHb#j@tMps{SWr+)(pHuV$uIA&%bko@=X$@Va8;tykx#;g7mdO{2ObbayY1PRT z3~Kg)Y)Al*3u#aHrDHa?6vvvQ5n+g+9!$Re3SKeyJJFNpanC);D!ZK zuQ>YLz_bvh#=*xVT^U_c-$Y^L4fQQJAuyXZb5pmbP^w|#9!!*3J(2qe@I8$u8an+N zjTNm55d4rx?gx7}3TR5~3XM~q*vdt@qyZR8SP(|sv09#CJJ=p#s_qu!hj@iqPm0Z= z&)oe`5-R%5!u^6tuL61C;DkK?axA_TL7K^qCr_ge=)F{9FITMsxXK_nR;B)~7Ign|oqiiwo7sZ){{Mj4_Ct^8Br?u4h}_I?gH}p9iJO!qoKzgJ zHsB9p0Vfbn+6t;;|M!F2EdL)=uJKb|Be6G~5g_n74nMM1bhTAFfb&^ucAs|VBm4@KX5R>6aOyu5#TSFZ_*N?U~lig^Eyfr z!N5qtBt?Z(+_R3e-Mp1|?*6V>OzGxfQhQO0h?;~`=k3d<47t`AlM`UZM~CMjf@2AZ z3dTo6=!>C2HIXtCBElj=EsR6_2sy1EJc})XCzAA>D(aB*TCG;sm$c2vdTQY~aLVzB z*;t;fOj%zyxco>?4(aFj#ye~w7S)lIgb9YAz4n?d)PC_u&)Y^8-lp|~i!L=S4Ng!) zLnDWMYHErJKbm-=WF}|d3ADP};RSBK+TJ2@+#;;l>{kHd^IEM!>Bc1_lrc6o&Vj~2 zM~6c}xeI0=z5Qln1dgDqzf@J@^8(H9SjpDy=d^Y-DO{l2N8~@?qr~irv@{ua zrkd9&AK?Pmlfxk$1O-W*53~CtbW^#mJ60{Bgiza297yHNA8?m3@kXT%Lf`^Pw}lDA zF(88Sm8(@=?|D-S1r933QK5R;P+=2bZCZO|e=rQ_gp6B^Z3=z>>uj9pr5B{*k?t*5Nw->ihz|MP@ZoqM0US8Nv)@O~V zJhd_q$TyK5x$nY=NOH`sGMkv4I=E9ZG7$1C{uUGSh>7KT-B#o~w5#SetqDx*N`NUup4|7C^##NO|4XxA|V>~%ku6BWmN4weR5*PxHCBl&-i;25A!h~>Y5Lfz%w z3Y+kSP@#O#U^hRWJPG#sf2i;jv!evI`Upr_OiEn1?JxR5!zn(FyvG-TNI?44WXKdG zypse&cwf%DnuKrOrJqNzDR?PljOpyo8$iOIR)q5rPWYD?;aC2Hd6I(Xv0o1P*TPUj z)H&ghgH%6;1j}829j_U0F2ny@B_{u-;Bua9EG97WV%7TJGXL{T)PkjQvF~yv!)o#a z8dUx~8S)?5#p|#6fo~q~x>Ptuh28u_DV1q)r%~|l1Fqqa7htj1p^m;d8hq;iwh ziHd@X2c4bsb-f(x0=-NfHTWB#MU)8#FcpN(?#OgcS~Os7gnt6u!_m>kFCs*5t;6@$ z)~?maflr(RGm@S>2`7(VwLjJVQ#bQH%G^mIG92!n*Vj!UOB z;gOJs>RzvpPj#nb`iA3K4s8FN*B4&7;UJ=9kVwAgNhDKB37x@NFu@OmZHVa zuMJxE69zbVPB~P=ytJewI4LLRJo&9okMbj(X&P5EqM7i~ulqKudk^ysXG=>OAt52T z;pSR2VFe^`x^XG<{k2Vi-wJEYBn6=`dosD-{(N z_xzJX>#mf$fVkC$&xxayi_}GMapk6;o046?#KZ&!yy*dqz=ZZ$YO;gP^Le@@{jH@j`F_D`R9CiMQ+g^1{3qYiMhYH z%vfXdiU&PSwzTz7u6i*=V``QIDnlb}gM>#3!(&;m?lRbu%cj86^^9RQYO#-h>!@r( z2Q2e$u;Y{Yx?uB_oaU@H*zzPCvAJ*hO`Y=B_Tqn*dpB?*`21q{;MnX4o$_TyNNJ+@ zX$m`VYpm{+?Vl+%hrQ^+M-%ia%lGt0+aGI?Nz*K+kf=C59(sO$CzPGsZME zjHW}Pp)uO!{29Dtj71JWAAC2uo+gK5!H?t{KT_An$r|d_&vBD21>0gSyj$j#nsf??^d>&gR%1HReR_xE+rLNoaU&-lsS68gEl|9)TKYM*V)l7^2BBZ zpdLX{Q7D+8fKVg?rD7Sp zww*TDlZ@)C=0~pl{esbeBnJ7zB1tIu9K<%a-j_R^nu<-4qpBLCDUa0*Ovyx&7e?G` zHHv%ul~xz0`4 z!}T60yZO9LV>TOvfB@*S@_7S9e=3;tG%mhS1`gFc8@CQD_)e+W93gLS-iwP1LQZ>s zj#c*{sbm@^-)HyOg*oBD1WFr>9*=w%YB^f1(0N{;vCVK(*uV)m>6FsdzZqi7Cka^e)A ziC-KOI^sy&)akn!L3RK7bjE~>sMGEqkAOupULY2wv$xz}g#(!5vd3QbbcH5{e^C(y zG9jmb-?x0p#7`NXjDfVsQ0tvfc@iiZ4G4=3z(x_oRhfE0D4qhDW7`ur_iIb4`>*w@ zk--JRXR94}?AEw%FL$}U%<%B=h$Q@3NCfQ0<>$1jKmW{DYFhy7jPrsUe)QZONpHWN zlx|dwz@k+%22S^MwSF!tmGoPQeD2usLLF1O4A`>w)u{J*4=k(2H)t3r$CmiMVO@oa*IoKVzD8)iV!jQogDz(_cOp0aF zf490E!F~9!mE(1qFBXNZQmPPewFhD`8A{>>F}WP7*mgdcX0Y2>me*J2vT&wNFSWYR zvY3u^TeD9UhJ|0zjTYdi0d@4wz;=h+0 zR^1smk~I-%w$HXGd_Ln;kzU<|-h;Ey?ooGg7KFlY0LZt|FlQ{AhvzwK(6|7YhLGDi z{N?Gc&g;&W|LK@zU0l!I>^m+~V4*~8t=$^Gt7&n-e4SbB6o^j0lhB{bX}1O#>HGR% z*6Mm96$FFCs$0(YNjBq8qs@~0m6V`*eT6NvBbe7`W(G!qdt2t@D}Ge5CDfBx*R z^|(}HD1uDFUtuw5n@po>4XRAG6BiGi0_k*kVvVPq@z9PPuR2f3F6f~UaVL?9gx2`J zc=pE-4Fl^1tN>8a@7?n+x~)*%%JO{4!>b2VB{m?w4z#+`xVIbrlf^Rl8$g^X^!c7H zC_^-zN##)4t$JpCKRI)}*#k40jw-*NV_hxqQ{^h~>N0sL~UkeITe?pzA>8X~u~)=z)NPS0r15j-ALCP=V(HP2y1e ziCD+0pv^vZxZYO;!~UW(@N%~%Z*ESV3#d5Y@R!`7c37>T{%4KLcbyJac0 z-@UyDD{UNIZ?8Nc-YXytBg~V}yPk>iwXub^6%s zOioQ*IV6jb+!1~aK{&r0wqN%T_P!&flYYHKIm4yk*=RW1KH*a%1j zaJVp^?OeYkaZU*M2p|!+Sk0;(Rn4ZL`$4Q%hz1 z7nO*}W`Y^{fRVd!0dX|5_hPHYgT;K=zmRg#2JmZnt!+6qdkCIK(i}baw&LGD+M1n^ z`8_*c=QM~8e~*X&ua9OqLK~f0y0Ka7B=WyKS@Hq`spfAW?&(eI&W~(t$$5FCZ9A!7 zbcfD2dh3#7!^3M%qH0oxAGjp;W(ug7nBq-GGcK0wx=eUh>&)cYt>?kI-@W8qb?ynD zpPwJQ-Rg)@EMH1VOvOalS=$-E@BId>R-^UcoW5W7i8me|p7Z{APow9ERHaq}Cm`Ju zzo*3pD`KQ_J9 z!K%V=2Q_^GM-AM~L9qg_Y1h}zwp}l-t!`&y`xAxPzGD*}4g2sPJ{W;=_~^77zXR@B z@u$jXt$K5zSYkeFlgHZ=>Apy;2f*wG1_lh)y8{51sgf8|U(cEe>GAror`h88qmtuW z-?v}uXJ??2T;CzUs2)z+&b~@Y<{S2gA`)@4Y6n%5Zu((CFP{Zy;ISd3d=jN{_Pq=(rVVgfo0WKaV+8#A$9DDsZ1Z!#dG;jGe;tu z-`J89jaM5#kXaFjRFY^VsaZpT|S${rcrN-w!GZq^m+qe7h(trNnPnskU4BD7Mw2Yb5p1l{SI zUbQ1u>HMp5^rZ*7kk0(thd%qwKk%{y@GB8NjGm7eq?&A_4|-rF2M`<0 znbWgOB_XZxChLtMr+_4@T~}Y&o0|cp@_X4W{m2UivDb#T&cgowfrg~kdzNa&f4)1K z`EJdaA@602y1kRfJVsAaLC=*|3ceTC0DW(vZiX(*{?AdoLiyOO*HL@F1InLkWIs&< zNu)O~PBaIDl5)Nxd3aP%G=U^zX2-|HN}k*6SsZ-)u_(yrd$w0s_E#c7d-Gf7RpNIK z;^D=r_kZOegfdu&(hjn-4%mqan&pUfc1z7_yy2FA;RR5{h$INv69mg(4>9arf4RGO zdO8#4b5Z=^>f^q>L2I3rIq;D9p6ljwDmEiS8H^E@YQ(R5TnVDjcm>;%=LhZ}+R4xS zlWW9pH^3YJt~I$Q5kMl$g92Ho}e0&C_Uy#Oj)hpL8C?*55mmq3(+^V`ZcMOowe@ZUF>8uXe13g@wj3 zqOz8fdX{oH?2%^Qj~Q5e5>-vi$wQmKwI}4e1jTzkf$mQc`l< zp=?T>;$dWr1Ilh~+?5nmR7;a$I25ZnPRmO-LWZZJZl-55>ZnorHyKZTumj@%mO5wvpbV@KdH!b~p3HYCSu3 zU5k#jl|=8U`J8P#wT<~`FPD#(-u#!wvD$0Qy^TOS3ncc5{Ygg!1tldoDn5S3QO7;) z`7Xu$9TzU*_%WILBNlb~U01@V$lr2yK$UKt2r{-%qPNxMe9_AOZe_hEvXK)VAK52I z(r7kH{r)fr{o0rP45Xnw`>=5)!q|?*+Kj`0kA-|1TGm0jOV{4QvSxZUIvhm00<|PB zBD(Nfpo?QO@(K)hoX+peq%UWyruGo){$AJJ7F=Vb#(*w3HE;r5v8QC*4f&6hAXxhw z|DV6*dg1HrcMV~tNT3?mR>i5{R-RsaT2mn@bi^DOMb-5UD%B5*hrFlE%jTMU-Znn` zjUVbO^KxPj(pFyUh^8-ebGr-R$2;+zE5Pl(Ln_C_$Fw01dHB=dU>4{x;C`qaE(3M2 zK`$xoRqT;0us{7eu`*ciIJc*Cw_~@~07Fp?9%D1KGd+6BC1<=AGS?>75~q>y++AC~ z+dp==(Hs7HGq?BU{&OrQ*O;?aOEJ+=a#g$y+3Xl*A-OD9S9#Xm+=N%vxybIzQ`eXA zL--r-YUF679=`V80I+3GTHZHa@7r{W+)LDzT*)GpNxapJ1UF5q4=#f$&F+g0SA;wT z#8#`P1>JPK<9MqAh-`SJA58g&*a;R<}Q6@Q`Ddb|kXx{D;oiy{s~jO9q-A4+tR5f=qqBkb1N zhU}iQQ`yX@_u_2>5oUCFD4*e)4K?VcUpvR?NaXaN+Jd1TX6=}{E|aZ{UqzZbgBjd5v#|}O|#-E zdwu>}%YmnMI$UWu<6EkPp$1YFv9{mp+>vNXvBJMa=bcvOTeC9j{3wudxoH;l(v?MF z`sx)HD)QBIkArNr!|6jvtQ!|5AHXjY>!dmY;;?ekSS z4Hl3C?c`&VyW=e=r818gqnc;#8@q_s&zHME7TWk*pP4^TjjBJXt`BHtq)x`q#6hL1 zn#}@EhC@HD#z63vUtp4mj*5J*@t*cTQpO%|Beg5})?(IVSCw;2w(+$dk7+-Hm(+t8 zICdIuUWIX|g^@gD2bcHL)&GE*wKnJ|mwmnApC|b}6kFUrz6km2P)e@-q7D9TnXpFT zt9vRWI9n2%3BvKb(0UG^XZeK(L?GR@4|_B%EW>$xBK6}+ZfJa+VYl($%CARjrCbXF z+9mm(ybmQlOG4@8?8*t#hXh_4w7csH?%ruylI$&QKMh(DmFQ;%M*67Rs#nHqrA5jc zti^8J`A+*EXw{nhUMseBu#U{kvW1pTYqsdxJ5{51^gE!OBzC`;t)#NJ)!`RU1+uJY zRUJrON2~dy6LyrhvvO(g93S&l^2ITyN5=%ae+?h@l{qJB2L*u1W}`Cd#` zyQF@i&+p@7YUiMzVc$-`2omCo{r;VxQo9l{3j#dAlKI6mE5o~U7>s475OX1i|8!1Ireax=gxJ?Y2p`V8xotxBQQ}p-l=x{^NIB{(}BL~-wjUL3X+N{Btx`}zh zmaC^$ZXOAA1KxzH&H~(?MdWYsceH195ps~i;o5xiXigo?%ZTWE2VBO%Gv>114&7Dg zkG+hMu>pJL-cxAc4(*`4XT233gbQqQZ-DGE-(LJ35|>u$TK2&f4HpWMswmO%n9%Bj zdweFFTpG*dtQ*nyd+7syc9^`X^7{02bh1{+K6+uy%~kLH z5*-uc_36;^WeB|6SV4yRTCFZG`_u2W#hURN>?X5E#OBM@>oL@lUT7exnE{_y(T^V% zRr9*9N<^~4NblJ@ zHw3}(S^a+*ZA3AwKb_sEZC&wBV`R(;{4MWQifF5Nub_LQtvQf_F?7P+9QIUb(l}Pf z**!n6mwh(B_R+{0UvVX5k>qF~p%E_op2EypkjLLwsCv7m?)ty^OF zAAH8Jz68`~PnzBd zE{q;}7`a~xU$$7CLI*hk#B1!r2Nq(ymCZsV?Dyl%`4tH8xfXy4>xp1I7(G#TK74%# z%iLMSfCyla#k7>~BcRLl;kDrB3k4K&?_rNs_fMmS2s%Qj!5_B-;Sd7Ztkw@zHEIBi zjXye905Ls7XuL16on8FjG_?JH{)lQHikcM_7kje8`eS>|M%!oHs8-3$&;JS@O&1Xk zf+j;Xp2$cpMam3B5Js1iNwWGQm%5%U(vt9t?96;lJ#VcVcKn+iQ1b+Mhil%7oL3?OIPS7?y-Szx}uQ3sAUt1gUC&`&dWm!gPi<9}8{ zkEsg_VQFNTOmqJGLtI>56MilwBV4GgG}X&JUW&}jJ4m@IJmE@;hE`)sBr4H6lK!sC zL)KtopQ>6LA-boo@qc3kfiYlrDmW0c$N&bXmaHg;1}H-8oSO1RA9IC`7puN07b+9;tLe zdjg5+k%{_zoASHL_a!gvN%PPZDm5YfUZAB@_AC2K-hHXZ6`meZj(9?aEC3is+3kwD z1;sgR!Rh=8++L^~G@l)z*A}Cs^7Q4rX(E{tSJ{r?pTWHnf1SPc^cpjJuVVnrh<06) zdq=rz?|DcS{N-HVkH)Xw?r%QS%Kf&#Hj^-P%R4HTXK^2qqW4BAFY3+pW7Oo0{ko{P zy;HLGPZ$0Rcq(Lt)>J^j_J$(GB|uD-ki&Oh$%pxlKi}51v(tsp;`~eV4V9mPYC=0% zbrbLE#j> zYxtwH$^UXu#?PI|e+>MSY4fc$(|GX`60X&@>*`Oy@7Os(2~2~3vi|8XMC`(aH1E@R zlTA<*&de)nI?5o9uP>u@Xvk$ThRm5&zxAwkJSkj=(1j~$o@3}pE=rr*uy;6tsA>RN zq7vuL36~n^KNAH2m6CY!e02tmW>jR7S?y?$qaBa=9DAtC)aJtxj^8`9tk4|}`K#9X z^K=ywNB~70C{p#C;W6SJCi4xJ_d2j+uKuRL~KeaTUuBVfg)z7ZMP2l~eB@lP2FE-$B>!^bvCm1Hf2u!mn~LNP)gD za5@cAJ1RJreT5>*E#1Jr{jla+4j96ZHPKS-M~XU_ zG-HcuQcN;{rqUM(ZGTkB=*2E7aa7q)1LH<0C6(qqX9b zaDQyr2!(8*)pVOQW!_@L@yFcCf4ayKBn-zsoDO(8FdYFb5j2@E!0Qh3SjA20 z9_;$((AzNEfTM|e-A@1|pDVW*j#lJyq*mcnu$OA3j?~V|8QARD#CVh_4F#j!zWX#9 zu`TL`GO?3;Xee)3qTjfpX0GG8aJXt&F%9eoR)b7Hi(_h)6X4`Rmht zF0bT#AT;2JB%h-hjr(1GK9HDs|8-VMyiuco3oHjHD-1S^af}Wh1J=ADkLLbKIF^yc z{8BX`HYe$YEO0#tpdJMkndOi*IGJhG!`+@tX~)Pbl?O!V$@g8IOqwr{e1zi<pj`61#mSTQQnHm7`fkl5hC^cT6;1SK6}qYXgMYEb5*6tk1$ z56cz!o@J7+)$J#1{3sxHn8Ch9M>H#O|FAeifCgfSGkAbeIMJjKv zMf}&O5uhF7agKBupkU)JlZ);3Iem=oKLiXifk(MllkMa^8pYF|!}gk8d|%BB4o~kc zue(#|Jddm0e_V%?w_c1}9ii)%Fg349$P3gm6aksd`wOU(Wy$jtde7ktNy};3lD#2v*hdcUrpSpVOzx% zFt2dP&(w!AK&5hHo(ZdO?%RZRJjh}1W!y`F@M%u`YOpQ9a~KVZ40clxAt8(;l*?&7 z)9&4I02k35Lrl!(Jvzx~T2Y?+m4?!>gw!yWiSP{Yu4%gLWN1HqEg} zSJN=~Xdfo-9zq3IHT8n1OjC-FN&{tgDY+(AL;zI-mIl0=!l!rAy!f0C0Kf0E$xZMuRPYHTwIrCGO_Pa1+smX<#{NqXmZvx~o zyF4kZ9Z@BvBn-)~lTfvn?>C}r`1zu5AFKl zEinaI{qm$m4=w;M3p=BA!DIpBBQ=dqK!*GhDkf>|q|jS~et2_z!Wr9_=)n*oz+B8e z+}FqOXYG*MjX*53*Zjba%SHJv3KpmE_d$Vxjc|Np>>f%UZ*9OIdK6B8RQ6Z_ymx+?QEJmZ7K0y+?s)i9bbs@SRW_az#wdoF9L#jq%~>fkNj05(L5Nn{K+ zy!2U7_yny?m*A6Ja3KIyY%Aj;GM;;wE52dcA$Ih~ll+?%)`7HBaG~~mCsm<}5I|V7 zIP;r6=6+;^>+iJ9_kF;SIZPY5SbKNWVvxX4;U7}xGwP^>h=t-CZX1EXts8l+bpE`A zHx2(1W=1TtF~an4@FB2mz%QObv99jau<2eO6q2 z6-WEO$nUG1bJOubakYswy1>1EO_eTuCNp#jog!_cqXMQ=fma~+jp3gthj#7_mp#>CwoEDQPfGkestVJA z;2XFR&H!#?>cK1$01)c7t@q*dyvJbj6GfCVSx(t$d{lT-LYsiK0N2@t3lQSmttxNG zeUc5lcJ9D>c!wL(aHC_CExg0zL-%@lHJer|Sv1SSEgh4##2w946-b zR&}LWI-4lBV_+T@yor(uE~i3BHW9NZSNuZVyBeMMVao=YDm>>#XoY4xoUiq*ml~1$ zwHhoPhUuI3D--}q?m+-1WsaxRXoU9_67O64wW!Yag(2f{`mnVodjO2v@*n_f1K22K zG(Ow6*$Pe1VGO8h5yG#EM&}0ZTS)hhR=@Q<#InMpraSI9`~5jS=f=wb>_Gsw1YiIb zuUl*1H{EY}aKdWef$NNtl#~<{2tW*tE*Ck1QGCj#diE201sxe|Ef(XXhaFeQ2M!L; zh!J5}cEDNe`LEo?#bNBaUf2Pa zL7kPnuU`w=e`txMd~qL#{jzzkmi)m*iMWKb8?2ZHMrwvtL|=uVdhq%I;0)vc-->_! zfQ5H@FGc~hkKn_Z&m2D7J44Aq=`1}$w0g1G`@equ0{bsRk+W9)*Eg}HZ7xBuxX8t1 ze(!txcMQ7icKa@s;J?)0$!?lU3M7)MPt#rZ5W8c^P#Yc!b*8c~$b?@N6i~$l$n&Al z(b0b=eO4yqMiHW}GS}-Tz{FIE<0b^~meOI5C`=Z!pQA`lyZ(bkwFI8_{Q0F!_UnT0 zhCa&EX6lvvIghfS_7da@#?1=Kb2+s~K&%iA5dhe)N_8LB7cNPb`2p z`us?Hyo%J}*|Shr<75^P6$S7WDnrIT$lnP@LIBLg#<`CKd|txt&5^r2-=NwbJ`E@s zej&UOOh`z;1z0WFJZ>=S$?AF=D{gBhJ%P|^qox3TMjBwtr3v`?HiEcl=UQE?#J}ST z0Te-ovfwr$MXn34?(@Wh#RYCFI+nn-okNj2uIF0DnIZvPy?To&7=I!jSFrIP z(l|NpoA6++ixvjJHj8Bm0HV+zfQkWr5CGR1-~{Wq9q#FD_V8_Q%(A)2R`>pxh0|4r z=xvW={ud;WiU_;~KU}HGMFH&ChPmk>G70MiAOY@O^DCD$RNyP@^hsXH=GjQqSFxt{ zC+1TrTC`bO??p*g-k+js(lukRWp?{Al39{9@O0MOw6@rkib2oa`KCwT_Hrq`(@eBp z6XJQ8q-6qq|L~yL>TL1Xlh_xR|BbLphwT2KWCLITR_EgyUNprKXJicrSTEE<`!CtH zA^At)FhC3$YR`>n$C~tMa+O!c@;?srs*}M%OH^g;0~}@!ttK+ExP*cf-$g4lP3_s) z4(NBa>9?0^O-rZ#WV%xKbvc?p!sy$@INPkU|Nc)}tx z`u2I<+<9Z9^4)N;5S2&V6RnbYdqxo(=R-Ru(-hI zP~5Bagr2){_mWd139oxx+b%o#`f7hswKNq4i;X?NUTJZ^_y{O9iB>(PyIWA^lXBxYz~}~V zOzHs4CMqH#-}~V*i9vsLS0y1a(P?2IvT1Jw`#%Lr#EFRs>@Xf8yH-i0(fM-f4e=kT z@%c2GRwijP0Xkh({Y!}nDj{hZ2!7AtH-}{ocPwUkI$S%YwMw-VfBr5BGzFV3sdSpz5&dbryZcZmU8YSCRO(GxYiIOr^H#6L5O_4mGBxY_RH!y*6dSZk#?P z(h%ULt#E0j^Wy1wK+6I_sG7C~>Bf>Byx%5#gMjhHw^CKn#d=mxa z+>iYYsSLSGuUT9ir}2edKp4w)be;pleA=#zRE-zS=)51jG5k$V%biBlOh4Sf#T`H& zzyM|_xWb#WLwXnd(XR?l+SG&ve;W0wtP%zxR0PC5WaA;s8T1@BY1I^~+0r2%r_1qK zsg)-AQTTa2krd_cgaB82t(xX1aA+Nlc)K5OexkGBPD9nvYH0t0Mfn|xwXkM2E2Dah zhQ^TWtT<`}VBqa7+`Gp)<`Ya-*xt{-JXiCBjF4hDeYCjZB@??1 zx;Lfv6M!JuOLQnzy}a(2o|M`&O)7)*(-Z10;4hJu&B?T=rUV|gF&gdGkh^6syAQREhdY)tHgj(iWBT)QH0hGCPBF;e4#}7h+8H{`JEXkG!kp324obw z(C)=^t_nC1KYnR*s79FX^rEO9bvc=-PBhsf zk$A6XL{|WSnNfeKVU_DiQk~bE&rMF9SAm$s%uhzwidI+_Q_iHg1g8OzRD&h^-{kms zLpOmyW!p|XWI}~8+f7l8&gBLW4wH=Gr*PRYy6m{WBz}bc0Gxv5kE8D$Pt51ZXtT&# z46usVdxFw=J!&i4U3!A*9Q{mhFM<0&3pbS#j=1N6FdS$7fb&jIdTj8Hcbr)qkAOhh z)6?^{Utwy}pc`zyMu_WaFwqHg$C+O`5}@*&|B;QODIn?xnUg3JK)3_gbCY#uARRA0 z$IAt?Tqci8jg4_q@)ix0n7^qQW5MYBlg!ciP&H&(+d!#%899PZv zhii$m5za#67^BF4rm$M-b|Ta+wffVr>z!16$7&b0T#OiEK1%|8$eAl_C_9Nd$P4Ev zYOcLY6C8D(tPRb=;(KSMWs?dh4Q4}nu$=;~|=K-U_&frP3 zNOC>CU`ls1-BtiO9vozJui5&sBKAFL@-u22V%Lsh6)y4_d>|*P5RYso4HY_OQOd*X zbKPnr0-=*Xo~>*!Nfrvjqce)~zj{*-+u@x0lY7Nr8S|Dfmc?mJx4dMY4YwR>B+{lJ z)LfYs&_94i>>+C7oyEoC)?QN}>O;zfX6)2SY7C@naRO3F{;~W@nMeY!8qh<|E>i{7 z3M8c9w+Hm1sO>;nCH#5;gyBRS)(ZJIu*ks}RM%=d9z)>C=zRhIy42uKr|L>FbmZmL zhrQj-fT(}3g%8O$P7+mUAd(ZchlSPVr<%!%C+&9<`SnmFpy))0QuK2T=H4D3fQ4eP$f3*7E#;SO@&T7TT~Hku$B|n zfq%*XG(b+jc!YzU1U?#@OY#}8LgCA|*S4Wt$Md|^pUZo%i1+;9gxhjgrh!C1?8m6Y zy*^hty@!emgC1JBY{8vPi?oUbqB$n*D<1yYS^KNOO?yz-J|0_lXI;U!CsA*?ehIU6 zR2D%n3klyJgyo$5dCuA~z`FdN;fR#7i%d+ITtIJ~IE54DO%QUjvJC`DOq`te;?RvG zxG2xy!4N2rP65mhDM5YT2J3183KG!IXpF+t1>+7M4t)yCKh_0f#wm4R@Jr(>@Uj>< z90I~eYU+X01^{LqF};d@GN<1p?e<@={zOR`0`yrvKi4r1%gCoHiFS`86p`y4P>%Y()iSQ{ z1GISbNBcemh*%=!OcZi*n>S-EtuoJT$X{xNjgHa&ju*kcdPt$h3`Y@9sSU|ibE{-| zSoRF(v^50#&{@!~yKn7UTCKjmzAdzqLZy)htP1RJd!P>z5B>nbs;8$X!;`^dM{O`7 zH&?$iv>A9tMn)E_p{%_p#xl2hb$92+CooC8&b3~099B6tN-Xw~Bz&D%ZweY2In-?p zV5H8(spay=rH_J27{OwTdS;L8e;LS|G%004Q-DwT({v}6&KAvh;5JXxjy)_swtukX z{|n|kV%R0Jdm--+FUXb$!x*z{sDntoCuG_ZH!^3W_+)AcrraDq$aM6uw9DBDvx7+s zUKao@D3;02?`S5nSOOMZWIiZ3v$iDJv!n$|v_N@eZsOIatycg!Z%I?6T67 zHpt^f#L9dXTeq$D?0{ZD-F23Ajok}Bg=80^b$J8+^N*HrJ8TA~$UN9QAboH>3Eic- zZbF0n?5pZghhx+s2OY2+*jJI2H4N%K=yAw<p33-Henye1_& zYX1m8Yi5V9=sU{|mbGpuCmoF(Bc}G}X0c(8Rm>ly7nQ8yIh$KW@~qanxmxpRtO#zM04Ahne5Nm>Zb zQbT=pUobKU0;6S#bpa-xPN9 zKoxE^ri3UQSSu-M_md|X7*IzGBS~O%p$TYnE6*9c*z)ij(>6B?71*5S4K8mF!H>)c z+fpmKv2%<8sSS9y=ltaw@h{xJoV`KtQN%baSK!N66@(g(CEzQp^>=ycd*B^|=b-6> zvz0l_b+CV}NP?|xtfnczhU<<#(nzwn!u^R}1!G-nWJP8~3HZXG6 z^NOnG%eoUn%NGYB8d?*H3v@_%N{@&32u@DPV>EfP{rTIQCj?ZGJj*BhMFh_&HL_ws z{L%PT>EHjYQ7#6=KH}}1G(600ml?eEqd#AhI1C(K(2W#Ll_?BTG2NufWTIflOqJ=1 z>C5m#IS;3wn&YhS?QVEmAYrZbzAD&#}lEjz{jbzV4{kjY?;u$-xljWXxU)x$WouvrAKaU^_zMn(hc zwR@>o{R4CSGukXgGk?16iKg-TBV#`%De6p6gorFZfj@it+1!x#a@gUNTM^zfpS8UAr) zDoCD70)XHbX%3zjf9kfezrA`Hl>z+{Sk3Mi1=_7yhVmt9Cl4?Dg!v|T1~T6d&m_de z0{S3;s;5!E;8!@AOqjb>E86nE%&UA}w_x^bK6g95SG?!uP`vsWrtZFY^)4vFwu+@b zq+NNTLBt}mjuLsFq{8z0+t$QG^ucGCUFy&BNruq)dZH1Su^KMTF@9cpqO$!-uKPV^ z!2<0L&L)AHR%XII#I?ZVH)b@Zmn%Up_VQ6#-)S9by^%13U+ku6q?9YFPt%%n@>B+( z$7MukSlfjuLj< z|8W5f4I88NZfVHt7Dh{W)jG&-kr!!u(xU3WPDYvY8L~N{E+7Megq-$_Tba21lYyS56 z2JYK^(6j_GnZ><5osbZ4sin?c@`yxr=(D8a&sFQMojz?aBz3zNkg5B2%0-fiO}QDf z;(yj{b*_4gnx3Av?v3RlU_?VhBkp*8-ZWRv%>h`k0B87vz?ZZ{1TKF59DuD0F@Z@y zkZHw|U-;HkapNZEIP0Nxyz}RT3xMebPMf!o8?6_Fhg0ZVfS!5j6gqHQ6>}=NtUT$| zFDg|!=|I21#4jU>S72Axu31R33p~CTn_@u61V8X!8nnEJrT;Fsa__K74geuOZs+Gg$vw0*5 zc~bH~0wvp}inwnLP3!u8N{J!${AsieW zbbaTo$qAh8v)lqP1tmH89Rf>K-|w5Fxs}zHW6A%tIspVSR1y-qrL;qMAorZB(yjAi zxSVtbI`g!Hx>aRoiOz5N&3r(a19PY;h1jPtL;!?dMFjW(l&8Bh*!Am!DOStr@Oi?8 z!=Y3L(0dmjP;LZB-5$%2t*Yf}-Y++EjgFL3N$Wsci%OYN-{~Gurx8v3B+Ci(5wbru zSk3JM*+~nHh?$7S?A*r22CyLGr)6NY2hc)y%-vDJ$E@xGlzhI6N8|5UGC3bihR|y_ zZgcC)RnE>;`I3xPP2hb&FK5#)N&Ad07DFL)9MH6uB0w(hObk_CWK`JUC!D1#iX7XvV#rXi; z@9l--Z!Ag4dsAzdZ#MJ?Y)(1>w^JRtY;LioMjI~IV_cxyvsk%Aju6;cU({7y7L0An z4mQv%Hv)9IrE@xb1j}SEYY5RK%J2K;uZ6{JLOjsv^JKnQ|2eOpayaR;h=D=g$=`evCGGO6_lo}W zy#pW(ng7?>S4Tzlc5TlfB_K$Lgwi5PD&0tkk}4en0)phwAR%1>A|ah3g0wUaAYDUu zxAXu*4)ERcd!Fb0<6ZCjeQSLzS<7{1m~)?V_TKm2*WTAPfPT3>=lgIShPvq7qGMQ$ zWmKa2oXCC8d`TW3-)}uVK#{TQP@sQbhXmUs65116NAoGflbiiZ{$2*-!9BOfbJ+V9l5d_D|}^PU0pT_(t5>ygr^J^csUsms@T z1==O);ANWTQ9iYwb^RIIVzt(zKjNoD32V+jxG&L3$mo?AS>s(itZpLYY1$ZcSd`7$ zD(w*_4E`cu{q^Tv3KsEAX`gfDAg^A%zd9US<(~%iqbE~z`t?LWZ0lOA-QJ;a6Xe4e zP9v%amT-$NDy%fPd6r7~v#w}7W&Tjd5p}WO`q(p4@7G8}=jQ1Cdd7Y$d8xNZjj;c= zBbS_BhmHQSY|*4LB$fqqkY%8p~I+ zQNWp81)SJa39kgl<(Hv-fI&Oe9sjTpH#(8WSk=jiAMjaIY=_1ea}=VMFmDw1_*`bz zfh9cKWc9(mHIx*%6*)C^2Zk0+LsV%uE1IFbC#mP68}b#^hirdH5hsWG!bU81oT?CdFQb45VQE|oM2codRHEh15JQ4 z&ZNA>Lk1C21A=*$t!(GBvx8L;?^9j?gYWzV%vOWqS1--lBg+_iMEqyL{vdgIq@_O@ zPWixTrIQpilM;WXqF~fKM!A4gSIadHfJ~5VyiMV(%6>~`xD^PmP`3S92Rz{!h6Ti;Gs+d9cY(J^Rzj|)U1t=G?&+SJRk3BrbYb-yu^ zsAXIJ66%LY&0>YLWV)s^!IU(T%qjoX8(Q#ZR07aQ?eZrv8cAa@QsawEGrSMNe)fivvfvXDIoC}pZJjJ?ec{3(q;Ba?yG?_S5YO>s@x)F@wf2v>&eT`f zWqtlpy|ohz>r{F7(gEf#*xTPtb!Vif?;;2Op52p>0Gb`-_V{n?)9{eNUgjij2VB(h z{jPFO{4ymCvIEohW;co1fBO19PZN`qY_VVI4J)Qxc2oG6uYJ!&6GxMsa;E={eW%AB z(Z9xp4Cq@9Skphnt-kZpGi#=xbSC+>0ri>`_M-as90b-Z>sK@)WaYj0DJO0x#D8rX zqbsfGjk2nJlzDM^=qK^-nDU6|jfW_1#N*&Nfg;C`;%^a@l|{WyPcuN^Nl0t+oRgX@ z(0ux$-1jhwZYf#tW-v)eOL*a9#+oytqi20-eqaOvKbavkzxu!?jArZEcc_|T<|vp+ zAPUNSE0B?Xg0KO7lwVNCNq%8REKk%&-~g>{l-nKwezmkQ2RzQ5>R~C^S)uo6Zw6&Y z58}%OH%VQ8&ni-ryjN*&4CVkG_@;iSyu`D}YhvPfWy%c3gedr0{qfsEI+cFu*__z* z3e6LhG~!-f$XTpxU}cH$vJ+RO`}5(r(a8*ecDF1t#{)+8`waK>Pt7|dicuf4x18?X#wVkr zd@u7%Wl=pI&nl0;nJa-2#jRgRIrT`pVCsBe%Cj!eT7q!wP^VzRsd8+T(PgwltDwrS zg-nGIrx~l84T|MIv-dTyi}es)CXn*gfWr%GG7w>t&|_Y{E;fJ3GC4m9zXVzy^!qdp z!?f<-aj*6Uf~aU8aouG$nQ8X9zEn3LlDeR!7ip=yMynP3EaGrJWJo@7z_Ri8;8<+7 zUN$0)?MWgCRjSElJ-s?v+|W?H+Jb-f*tpGYY*!@Xw5_v^G;E?X8N7e05#;#BFsEXL z|1IK-Pd-@R^-0JcJcGP?uDe0PE0S>69&Y=5=ydOzIOD1kHKDq|(X=5@I!DrRxaii@ zq+%Q~5#BE?e?FKTj{`pqa zOjalY+iygdJH9SA#X&DKl{&F`pY`k;U*n994z?jpQ$G|HuuB7Py+(+ z?5T%2B%x{&Z;N71Hw*U*)S63tZ&lCY8ejfEy7;!O_(Drg-{|ta7bl3*TU1_H`uXJn z*)Qn0Jd>)cF}coJTl?P?H)0})^67z z6-0~ci^s>?;@cNzZ%4zlpERc$&@o!&oy{jJcPY~jp@YL`uRp{aZ)Y2k8`@x<gDUf6hHIluePV_-H<#@KG+zBw>Q!`7E-&n%LJ%hqj

WrxdBs5y z(Uh7HvV|1s^QWCW!S1ftC$D_UI<%-0Ivwc5#%g4cCBBb~~B{91}AyVm1mLw-flV;i4l&kb@26ipclW%DyPmD zqUr#%;v2!mA#&L2P|`A~M8d08iElmFIzlCBd~? zCWG&V1#ny~nrvuuTl4>jn~*vPIE?b{Mn}?sg*2XmM92du7(h9>xpTjNw@MNqh>L|* z&ngVh)I!XK05vUtd(sSjJ*u6x!)7+EA+q1~6#E9u5$!^NzJEt4K>5By0HGcB70$-0 z-Un{Ut5Y*0-%A(BZ}ck9bm66W=XbwQ=fhTVHITTe4NOknY>MFd4N|#6-9}FoujN=s zS6tNfZo}~D??whm^Z+M2@St5d`3W*&VDjek;2MP7q~)T}u!$CN4ADEgq}`9TH+jjI ziQe)r(0Yqz9TT4YRr9spJ0|Y|hTlRD`mB#_#s;RQ=#Xg8nb9dX2hv-r4BvWjn|Fd~ z1tPCs7eO-18d67K>!4L{c6kzA-K>acA>~Mg6VrWbI<|RpF)Gcx01de_lzgw6v>hiC z!WkaY@^Z<^P#xP`{x>SrN-8XQxp|z4sBU%_Dzv?^(e&4uc=C=Sc}Zg<17y|mVfw)x z?*@$A%7+VAtl709r&@y<_vNVPAXC#hBot!Zp(Y|5p9syVN~oFATiQ$xEp%D7de%Ca zjDLl_@w}y`i|DZM@O&cB^qUymO6AE94_3868!PKoWef0%Gh^@%#O9nSh#*2ua4Z1RrbxBdna>V zJVqtd`NLtue_OXuPJ4|f#~%9?E&c&m1P=Vp8FV{$8!inVQ9nwwQQx7qt&bczuAkXp zu2T}4KW!HRLm!f$0_TUg0UQ?x2WJgIuFTpv-Kymi%|!hO3)?Cuib@ZEsNz9KR0?;?YRisCvFaC%7rXV48Gu!2{K~*ni*|jt=^1EnbH`0Y14{b$ zsf-LX;~Kr*S0ZOnlhi3c<6K`lXw(~_pK_b>BH`hyT05mtxig+Ng-8eXo&80g?42R`vqm6i^?W(i4W?*>z{gUKv<~r5Fv%Y448y+q+^ry z(ule3ngez^fEU@AkI`p`n5iPhET7TN~R@58N_ssn9BRtWF zf339J;ceu1QiaQ3Oh>biW(^B=EVV)O(okaVLbK8*vpRr$9`YqI@kUH~x?S-Sl_zl| zVaGjya`ADzBccvRg-D^Hypk><#`M2uP~$!WZuQP?c~u#PB3GJci6*kLElv8w0c&d( ze)k$>EI9AOCnhA|jb0~Myg=rqy{3b=OYEi}kbF@$GE#l}mQ7bz*Ik@(YI>UGTe?I2 z?CIr|q*=)MUVx0*cYE-%h=|C_%1TmO{;48xNL(w#vzyf<_UqV{V9mWxrsCP0r7fJ|3~lU&g*8KF8lc zJ`}WHpo~FEAd={PIfxHPi7LF$oEW{f`FsxIGBC(85-7YRm!LcwZ?s>7|3Y-C-qS+T zoIicPFLG`J_y2ymtpaS9AkBG5`UV8ZqFHWFl%X&8eDUsr-u(RKuG`4ysD`0oA5eek zvrFzv5jv?$wc(`^kWv^R{#;YDV!swmH~ogqJ1Tg~oH_Hl&}7Ykq(XRDH4JviR{T6vmY-=wjuPp*IR)xp zHMn=NlqjNIr$l-tBWm#e6qX#rL{kL^Osb5z3asHhewa_d##fhBj#jJofoBM-t<+H% zH3$w!KOU8Mv?1zn22O!Ug=pUoSbxn~t?J0)Xw`rqsF2NlR8{YN1-HCizbQo#*<%bw;O&dDq#>06@} z`yf@ar8865@$&=`8rRb!22Sra% zg!$(%j&whc+hy?Ahns3#1t4CNDMQTb1j1mgu}wjfl_3Y0^0?rBYIL#24MwuZwi@~F z{!#RU6y(yWIg2H@4$&U^KMc#2BsmX=%GiAVi~hkU;db8O!N_+1Kknt0JEdQ=-H7SU zV`QF8wvsa;D+IOa-W6Bg`ydtDtPYL!X(LsH;!qRFF`aX9vkYRh$0$rS-68g;cj{yt zNB_>s@~^B^Ih}{;b~(Xp;yT zuRG9W^aeEFHF1_|3o>*fBx7Zh4Epwoon9E9Owf0&{RP2&cVu$L{w1TY{K>16Qxl4M z3e8@%P20*rB{TW$xuf{4?XU_c%^w=kdPYWZ3#2>P{7)B2d^?*1@Eq%BpC|G9&(^!? z_xFUwnSWNk3(c4g!BcCMyk5YQG64iWK|#TzX<}k})>wIQq_E?V!W~KPOIIlF%^_tD zNpHbnag&rV;0;HA0zxz*v4Dupyyap`(5d(DP}|MT&C_+M>#lDvQZQIgZEbC15Hr}p zA93*E`2pAJdzWm*9No}aoJH7m^dSCad1p(Qz|t86+VA*@&Kh@Wma*0%daE(w!OE8L zwJ5;7h2SI%y?7~!H$o=aEU0wt5hq?iHxI8@B8WmkfKIzwY6qR`J@qcS#%L&U0ItXq zb;Is~_LHaRwu>r11CUC?#<|rS0G8hT{7OY)E1!&_mNOmrY7bEz^Mgz#rtSb zab*Y&xJMuosDwirG$-I>3_PTWVQbecITa^3n74Mst(+FbGxi-@d_`HKF zf~2YJr-lANv76}BrRsAaw)WzQ#|oD zaj01URXZ?wKP*ewA#AaoA_Z&08Zs^(q>8aRz!C;sl;M1}{JcCi)A~G>!~>udsc0zu zOhx6Z7J`r{LEuq98ta69a&odr zr|ctO%Q;~{(WM9;r}Lw~GOKzoU$$OdUfh59@G;DLHD0?nqr@1Vetzy*48QUWkBAU) zTEWjiuW;Z+UaZ8aIRW$?BMJy`Z_bAgR}=K@7J;xMM&QqHuyJ_0k?ytAt?Sl#V`?2i z4=5T~0l45Gef!3E*m^&5NQDOqGLAK)`4S#A2^7VzU%z^j_yT~1V?+>89^_JCkCM#1(!SqtNUt@T1pw#I&P z>y&1WyJzjmm52d{CTFLawad%P!$dt`q-8eak07cJA93ljMQ_k@jvFaT>-H9wmd1b! zJ>Hu~a0<1Ow$480qZ4B{TzR0-dX#29Lue!FQiM2BoGxhI*o%rpu`gYa!6sC^6bQ`b{$XjMs3lrK?UY83*K|I}KjTTrirezMs|IH`M-_{sBO5ic>2epEfNiV22#?YdI z+d|$`8^D;c9)q9faxEpckF7ENiZ$ENP~_kLd>2T#kzz zZ8Jf|sE(``gRK!kfSC(rr0|(3v?5w=RYt$Y6?Ar6PjhoZjcQ{^Dc}MKXaexblrp2L zl9qe+2YKgu46GWifD!68xbp!=9~*lO6ifv-bP!V4;z0L$i?m)Wf4Hl`VXtjOM8th=?v;Z4^d-;%*gBEW)Z&?mp`ji>lx9qZR##M1%|g#n`9Wyn!wk4}B;xx`nTH2w-k?CK za7UURZ%_Vl88^)LdZ7TJj3G6hs(iOST^;SUTN?s|5a)a@Y*YIu&cBM)IPCXycW2kt zCI1N_f%QEemw%s_W;?^9VPwQOd;yJQl1c{+_Jd|TrX)jN5btRAU6$LUa&mJ60q6ll ziBBb~w{|}6w^7&{ot>T0OL`}L{!9jJ&-CmPBQpvdccR)pW73Dodmn0&D${NaI23F9 zt5!Hz(BJz(JiI{G*vNe`h!HO3bf!r?_O>{js~@&_4pcB2qCPo#ZZ6MY|Pv=g4E9&afTK1?z==# zJ;X8r?Q-Ba>c++hfY$(H=*^ItpDBjNT2aeYdIZ&d9V<3efx!ey8V(4EL;i>hK=(bP zc3bzi)MjUaNU*f|g#?h;gm8nb+RE+GGfY@wU|;~es6K8++e(cWsL~Y{IYsJ0vrx-K zG`%qM+H-3=I|~q##nJnJ{2VPEMJ_WKQl)(7#HE1)NZ%l)H|Y6Q<_2nG{M&AZ5e#LT{DqxW$|#;Tg{d7MW~aTfbh(WYi7I}h(83wuBj z`YK?#S&Ul4D~@wR)d%As1Pq8zl2>PnPr3 zj+55KySuuw%F92`UmaJ^_+0FOah;@b=@zLnxAi|E)b`PLMm}N}WCg=gBRr2r)x+6K z?lz}yML6HQotbPsq`X(gsARAAt}P&ceI=?Hvh!(!T`O|*Z?WRZJbYh-siiNpWL6#rsS{^WVl~$s4?l;s8V=q=F`yb%MQL_d!r~wl z&$&0;#5TNqbNUJ+`MC;M<6oK)93R>)Jf3QdQ|0>3o9{_-^Ov!#sVw-41U5<%#*f${ z2oElA%1Bi@i4ClI*`B65E-*fy>sA0+nXC7^Nq7zqp&(YrLUG%)3lxog!cUpvg~J7h z9G2G-;b^v?pdgHrFDNni`T1)ASxg5cZ(zv`CaI*TCmCck!zJ-=gm;EeRy%5_t zK-Q@#;_a9069wkpKyMga2kG=>c!lArP z>zZ|iEUX~_dFQ{NXP1)cCj$eP$lI|hDmkUCRLNFL!W|drKze*!&iqvJ)B$`((v$(( z&rI#ghB2w~!6i1D1Js_8*|#co~asx*%L`}|*9q;|pz>S0RPBSCEcRr_MHSCW;bQUAeoDkXyr4=et8&(frXlsa_+ zD5rD(Z{_sAs_bI0t-=1bUY%b0zzh!5L)ndN3i+5m|L0?G^HRx|+xWQJORZAti-Y`5 z6nqH*X@mh_u2dOT&hAEZKY&kQPMFw}R5Vq5HraUj#rQpPJDq8R2sF*P0mMT}ssdm; z`BpN@+oYsr@;I2o03`Tthd~i9{ZIer-HW4%f2w?`f#eN707sX#Re1S;8&c)o%~RzL zlIMSJ6jj0h3=;e=*unqL;s1IFYG82+|18dbOg{R4`oExs;pC07|12-_5jVItrT^vH e{`=w2*CYnE(E%z6{rM2^qx@7=zF5}ek}O%aMyGYo zjHEd=GrfNpI~dEhj&G*BN6)Vc{*dMOdnL<{`|IOvNtR_9Qz#ILL}0O4rt+y%!g{R7 zdYM&EH`U)9jYfqv!g{R7dYM;m%3(r8bRm;L0U#Za!r`zi`i2%FntPN*+1-ex}lpr%YObbqoJso$KO0XrJ$#%o9K zMLZcf>*G2}j|~eSgws-{P-oh;WqUuKeB~r?`iM`G@cHYDu<@EgWgDZkF<2}XSS*%{ zexepeYT+b4eLI#eX(M`(wTI)SIK*96+*Mk7d=%kfm=0aO%>qL1)0Ld%uC)tXddRs&I$e z(bVdn^c++)mc;k}>mW*tt;lnmQC(ukiUrxIDz+=oC8gS&43p7V9KB-^e5|S#CjFwb ze(jP4|Aw8-KgHVG`xWX=TfCtGY#%s)Si*#lp3gTJB!677EDtN{vN7O`ptWlp&-|?$ zW|JPbtSiIymlnaG=SiweF(<=B2VhmP8z%j^7qiOq)}o{TZ#dTfEY`TclK$p2qAxOn z-yVMj2mB{+Rn=uUY%n02NS^gMdYyz%Tv2@1Hb<6$1MPnN`PEMR-`~4&-|e-it8}Kj zeM*5jcYkR(DFY6Z0Y!H6c|T{>{I8*7@I{1U!)feH8)V3Mwr3~4w&!8Yw-@8dOYg%y ztFOhK8!KTk>0vhM&iuD|W5qe=OC$-XpYO)czp)H=-dKr$|J@;MeY;nouZBdLQ%5FA zS;iY3Lue??hQoCJFWqJ?Kv~W@9O--ttLERG#(xggEs}`gKU5N*4#6{ zwrPV04fT0vT^g>dx$u>otCr`Zrp$r+9yx$eB!Szm<>L%mHm8nEj`@Ocm<$&TlGElD z+>GP>+YpP7rJ*zR8;ghW?WW(usyE?R%fEV7kepUtWW&aFCD^#G1f>NRzd{HA#d%gd zaDUHoYukG#^-8QAsoC%X?B8j$b#DbjZ z-QSS!S`0wO@MueFn$mzrkG~8ZBz&vprj)OB!)3*AIgEI2M^DPvBJt)#nC$NJV>}k0 z(eJG&SP$>u-voE04LgUM@z&sBe828<&`Ig}QMvoGHF#p17vtfX{dtjcGCoXt#(yFh zjmBZ0#`fmJJo#6nbMQ^UU8!GSB8dB2pTqYS-G(fKHEruw6xqAUS!kR%EI;0d9dMt5OqJ$g!G`XEV*JPD#$C=zkk|Pf%CtcQDX_{ek1Sv+`q#tY4aM#gb|lUVDE) zk@ab2GCEB1Z6>TKcF(#_QT8hI`Su9tNd08Uc=*^0_-gehVA0PmglFTXD@*VK?~K)y zj17}xz97PJF^9>8?s^Q39)>Ju{$clyfrF3$TvM=A(LPo!%!Y3)h7R6$qJJ3~87BV7 z1X@SNVbF>B@=S&-I4s2&8EX?#Ppy)&jK{pMsJ-xjXm7zXhlYb zNz=e67Uwx()QS5Gk>^~3!O_FQ>Z#FNLx*A4Td+E(M(I`n0P9xfWBa~ArQ4t>%D6BY z2u6{VWq4d-FK6aC>o7dpEPu47+Wb$~4*ZMfYNgvaqoyni6Y(UxCqqiNLQ{~DGhpMg z41?gw&|~2)yno`SSXcLm(7I~TGTx0}b-sv)7TpD%il%Bmy0sgH*(QAY$^xZZp^M0< z8L&~K1=$vpknPP!`*S+8@OGOIHoFl&L`LrhSPbNjkXgRqzMfQ&L|^fr%jY1lqBoXrm)b~ssB=3wh9D7nH46l5A4VK!g|GTQoE44N`EKXdqapPc-}SyI!x)XGXZD5FoP$InidZ}|{*{y=W5>WjtS_inx|IvESqv!3G2>WwW_;f>m8Qew z=6xCgme3)8x8D*R*cAg;<+_JO{s zT2O9Bi+2>ru-CSBZhacwryI|_UR9DLRBYM|*Hs%7?ri!rdt@R3SW)PL#gMv7!IL>1 zCYj)^p(9wEw^->mrer~x4K3cxdhJN4rxV_18j*-bkbjItk&H!=h(_RjrV)`o?H&DU zGE9;(U^D7bnl-g|vxwxdlwv&Gm-cl<>k17bl8B)ut3>Hmrld5_jNw2O!3>%W>p#*2 zd9q3TNg3m9N7CQWENL!GWB@xlhY*NPD6+o8T8#0C_J{VoGu(oP+=Z&Xit}MQorL*i z4zzY<(0|d%`JRE2&i_klrnxZb@Q0w2B)H5*Mb>v%iV%#PRAgBp;-DGn;}hwNjxc|d3YT41y03!u$l7_j`@(3 zwR^Z7Nm7p_)Ig|c>S|7KB91DT6>f_%M490V-1l)NYCR4-{z`|^ZOmHR(_QG;+J~!uy%N@f29)1$4g81p0s!1gR>D?N zj(*xYpW}{-;n-8;FiV%$SYc@=F zkF?^F>;+1U1qHuy$k2!`XZTQ&8gdq&qA{W$(y53c&v3Rv^a003J_Icz28f1Dxz zLvL3I#}hczHHIvc9@TC;O0q1fIqRjFZGX<*lYz9|>3T5^OCf@hex=)(vE9Bltj^J1 zn4CEYtI<1Q1MTBD_)t3)ZJmL`eQ{Iz=45L zEGuxTW}KkiT!7)Rqe{0iBYS=ASdl$nGna%vs@9W*cMlIM-Nv-FJl+9kl^x~R77DMO zV>ROX>H>6*hS4+-fX8V)Gmc|V=YP|9$J==7eirG}9kvY>o3YBXNq8TINF;)(&VUUB z0zwy`O1;2D5++H8EfGZ^aN$1F)t{WOArS1D_IgTu|2}HhKr@y*t0w)uYUuA`^-8jV z=8k}n+UfdwL%k7nywHm^4=kM4n$ztsR|0agO-RUb1OfrPKXd|zd;S}V=yfM*Ji-=wuaLp6@}jZAAjHO%ysA>&Z66W^%)6kL|G zf3F()n@YVbrxT;0Ht38tWZ7nZGCrMOZ_6*bu;6nQD451`WFq==kDX^+h9qM&aRL>t znLkRQi5ak=?k@DcxCOm0Zht}Rkv0S;ls~cbQ8r5ff>G_g9LgIUfK_isp(R`CR%T2m zNmx+k(9F$i{qTDSFt2~=fo!5Y)eQ(2RwZh|GiELYH z=KZQBOdv^U#}?!%~P~R69=( z@AtQ(-d&}18^SD|=fc5uv6ps5>fbdCN4X7-GWETSC{Iq7-89MU^ zPf*^zS#?_+;D7Wv6YbmunvH{;>79D$kq5LkSGL@5733 z*DBpgTB#~b0KisOuG#?kFf9#9>I@FTrZ=O&oZ3BeI)4@CT3|Bi;XN5dWwE->o1A=S z7`e+_FlMRluUAT`W}BlmR&x$w@!&siRaUWm{`QR8;+(d+HXHA^_>^wr9KCN2qGWBp z(ru(2!(_%J*ewMZ53B#^q{)9gBg5p9x*Y5}qOOPj60sx(_V`eIMXu7Vq#eU#797?R z)eo1JA%A0khK9+i+H4%@7(*zcjyX2~Fu2bL*E|PIPUVMqr5nR!7P2hre}~SAU_TrN zEAlhwe9386vmVRlx$y4c5v5x>-_WRngJUjD_N>00K zc|Nx9Q_0PP@~M-rMGXuSnkJ{M45MLnzZ-UszkgkGQ|PB-ZG9g0wfZowf|qmokBvcZ z(Zf+@Q@RyJFid8_X)VWS3di8hRzgnVfd52BJSluW4yysHYO}F@uksIF_Psrf;w$Io zQ3=2>nH9UG2+;{Y;t6$L9zHtOjY5kXZj(dlR#NJQhGM+1Q~6i6p4>T%;)baYK*}@> zlYd!|BptG>Wf%>qZ$ttByzgtnn(P{-+eo=Ji?T5qilfD=+*pFhPz>S02y&Ka|3S)` zlEUGzJk=Snbgq3J_aooA0%bYtly2o5-);R9ZY#bL%d;})0l)Kke&--syMnm)wn|0T z@7g+u@s2PSeN%nUk)@03XTYkxUVioh#D6CGFZ#LZ?r%IEhBrEhhLR;P=`Ma-x7PnW zZdvcZ=YO~hqfsNu3od>FRCIrPcebJKj%r~wwfN`3ZO$yY?epLZ9#guFbG+|6hGnj* zDFsO;u$uL_<(e`)_EJaM*AAVGAk-T{{z~o6))j$aGD|s*8jOsoFHGJYZo#EFbARip z)z4m2ii7Qew2yS`+}4kRE3%<8sO<6OG-H^|lGU6GoupUQJlLe1#IBK}XvkftbSr6P zG3jyF#%et9zpY3lCm8{n0&-G@_vJnmUpIFz37^I=5g^xDi{Y_mrQ0~;@MtIUOioR| z(qg8rTV053rwLDNOV1UylkX10l7C}DR`uMOsCGUK69Mv^br>2wtaKY^yg76jSLW3z z-G*ZB`+O~)f3pWo$NVW@tL>>S%)fPZZ^p|M3=;wJol7wgIH+_R08Tw6{ExvySew5@ z=~fhz>oViJUtEMAH6B6lP&j4l^uITP*w_S0u9|zZOD4lG5g^yO2&2JHOn)RoO1FaM z(JokYMpWC2lx{^)t7@}x$Bk9E_u&H=@I{2zPsWot_{cFVxoaLI-P}I$dK$w-0KLwL zT<1az_?wh&1#b*AVSPcp(ru{1CoeC?t?SD0&0p?Ed(Zfpj|aO#=zn<#U0Zq)?3&p> z0L_nez)@yH(d9hxdfME$34b9CinG_CZ)7)0a~l-vAucEI_kkunwBl=u^`JI4UsZ~1 zhY|Na)Pyg8sv1kWCeZ$v7qXl@^`0h4!o1BDsN6L5J6C((975M?{aE+tsuXsvJv=fI zq$KxJ^bEhNXb-!FkD}a~j}j~YwfwMaRupHN zB0Ijdssx5f&Tq&v;5%^PD?jmY0*8Lnj;3F?;qsp?g}X*OFRV`&ZgT|k35kibKc1XOVqmuq=29yhWj2_cMo4-f7MMWb zcn|}-N6_`!017Y7!TR5>hRHSe=DSV>!$hE}!t1d8;P+5pbtfb#y$5~Ib??OW#j9b~ z@dy2UvX{9h{kWu;0E-?ifAnI!JB*333CKwqCZ`b&j}7@N+<&zZrID`=FC>ncmh_is>16qG}??n z$eWh+0uwC6V?Slt!N9A=XU598YI0H6Cx<|I7 zWAFuBvEXNrB)#zZ@niz`wf+&GELel}IZK7tCz>wKfPYO|A@$Vu{bN1t)#ElNWh6Azeg|!RThTYNTWICi`}g6Uz)^gs_7I40DzV5N|f6se^77R$-Vf^@t5$EC0|0ZJa=FAMC!ya zNrU2?)mT&Y1MF`5Ir@BiP+$3F7!9Y6DxU6q2aVnxkmXYYkR?gN=8CJa$@9-q24nFs z9&LXKP5u-3W&K^KuooZ@2q^Nk5StACF~B8unVyB!BQ9z%<-8I9f@ zh$JQuO~w&T#t}(OpwYVnouMHNMMv?t_YHjUyStoc&mQEE&hWO^t{o!?-4YDIQ$@6?kl%GfL{lFj0Wh z=6}KJns4ExeGhgXdlg6~&iZwoq{GLGR^ZOc_0XsJ>0_dq;!moAmboh?9aUhEbl6b1 zoIygG!k<_{%MrI8YobQzCpti*Zg)jKhS8=yDh)lYjvp9w9}eQCaj2Ekv~NNSWSF9}EWJ za5#i5%zCWHdihnee;H!|e_>F%0VM?@J`DQz0syLAt#Ddv74C>gyB2Ji?Dy`#e&2RL zK67v;J%lx$ZCF#cRoTXfG-^R^b0)&QIN*C2$y6N4bnNthsW=Y!9>zqtSLybMG-*MG z$*I6W%rp~9mSvm@9#o(sBJEqS+Z+WG_;`e+4-MHW>5ax#M3# zGJXE%)=7HYy7YT+m^})0rd^LddkVjKei+i3|Ca&WeN#E^x}`$NwkU@TX0sV)v-yf& zsKuF-m*Q}=pe3>e2l}@|I2$E@tcWFh;BmC%)#uH2z23nv9(`d1@p$^YAAj_P z5wu-fgNACGqW!U6e{Sk6*lmu^@*b>eTnC%f2>_^etwVv%h^O~`8YM-|FrZT54z=T8 zXJE#2Q1L_>Uwz;ZYO5@$EVZD%!iu%cE;LlmKU_$C7G#+CV+jn0r*Uh;G8lBr&-=M+ zmV5!PAN@Dn(0rdl-Dyi-co;8@9zr~A#D(Yc^*RYRx4Ln4e@ih2Mx!|Ho5XKk=!ey8 zz@3|G@UCmhVbCwgYXEs^L57J@ufy63Ck#62qMxfNy&iod&!KzxF|;~Au0VI%FdXya zf4cvO_CODAsNaBNIs>M0{x(OimvHknagWD&qe1;Bnep49jc5T8Df8Q_By7=B4cBpQ#R2<*x_zgy5 z0X(?oGpMnap8L72@2&me2%%$NawFWG2D!**KO{>e$RAa|i9z2LhEQ!0@_&9@> zEvO@t&WSJ_Cc|ZeL2{O-(uzCU zs&PkKe>JMzSH40B09BEtoKgBvR<|hq0opD5I}86}V7c zyaC-qPiMI^^^2sY@rCvuqs38=FE-za0-e|c+L9t8zV*o#Jh-zP-cw%lb@lZ#Wx>CP~dY*z_(&2MWcR<3$SHDF1n+=t) zSqymW)xoT;L)oz)!sM-!eoV!abNH1K>dV{U9eqJ~Pg<~h{1{#tJ&dm|yBm^}-5@#Z ze{**?Vf&wa2uJ7g=S51&`!MMbMG;A)V9Vg7>r6bQo8TLLMQ~T@7fM8M-|^q$>r3xO zp}~@~b!**LY*=25-@P=Dvo%SKybhC8TE^kgNvv`g&G`V`X>WuC9Rdq%&dCGaPoS-Q zHC8$sa=!M~ch+I&D}$Jhr*ghF&0SuHf5~_(j#8@;#TMhd>Qt4i$I!$cL0zd|d!Ppg z{N4Cy{Vj^DUtMOwibe;X`^$(T>(kuib(pxVCR|i$NO%m#;(^-Dz-k&UfzaDw?B+%Qp{m;xpNi9LPsIX`G45`^5`MxeX(GecR{oY-8aM>qwc7s|b zez+gouN&NQ30o<_>s{Z|By*%3H#8NW$4Ed4D(nDgBq=VDefWTCD zR_ZG$mDjhVx7{hTx3;~6CK_V;z>UIr*?T0CS8*WEOy#pGGy-~Iqi+`=V6#U zGrS+yIGSKHWHy5&m#ii|e->5Sam+iVbSrcPc@ZWP(FB5t6c)R(JN4XYZ9p*6m(?|t z@bsy@xT!?h`${fnb(0fE`xqu_mlt7DWHMlt%LKjPldczX*y<4s`?9)5uJwdZA(D(? znXN|YRxWAPVkeF;OjJ+vVG>QIV9-ghXK{OTmcv?yP{f<{b#m$Xf8hgYbFWhTCbi31 zUSEjr!3dHmo;Oi5O@~P`E#oEcI6~98nSEg~6(f;~B9Y9G`yOP-cyZ()ZZ2P=bSsx- zH5=e|8PPSsdAw?-=`iUSpF+9Kgd$U}hU`lywA3IJ?a$GA+0_vmfI+86gYEKDIeMh8$**qqaK zv*DG|L)cW-qI4S>X{xcI!y5unt#m6wd4InHzTa<$ES>(oF70>-o)2t;^QQMG+}Z4D z_Q*t%B&;cSz?9?HOBPzHF%`{^N8?``JBp2^%dh-}DzK!+e~M$?{5moT_xHlL^I@do zaiphXNKeO*ipSyG`7mN9wRfUllVKtQuodW0RhawxBJAcWOy)tD_#&g2PA9OWP#fRr zt4mFo2u@=vk7mOLjvj=Ji%%-ckTKPHM1hXxM{{8!19;gxhTwEUk@fACN=(JH|2oSz z#*g8;(&ehYe~R;VI-P{YHFlin&#$AC8L~vPG#4g)fd~rpIyg)PimY!lS0J1RVe;1a zacn5o)_I?%+Cp^p=g<8n@mMX=@$^MME`wMgk!W~JFve~1oWejJ%}{dNGycTA(A?H=fj zMo0z&B!dBZqY<9>Z$WWwJ)EWj>^&94^L=9&ji^6VG<7v6l1QP!X+z;0Ut1x9B z79$cHh22uASReDT&)4CMdxUYrh zJ-5Jre|R4Nz}2!2)~Z?rgF!SoY*<`u!$^4g+%sq1U=&WXL9rwA%h6PrI7|hq_)WB# zD-f9;(QKIP^>^WFmpZ1Y%|@-;0{>(j;b;;zi>ChSp40#AFnS*!!VUMYhNa8|0I*hj zV6FDd`1KNia+~Sg|4*k<*gY@_Yk?k(PAh7Pe=Mpw>!q1(&Vk`z&hB)*5}T*Q%binCqvN*#8U~>+q5~ktf{6@lUH<7 zf75B~`Q}lq{On>B)z9zgc1Z%N3e9M(bV33Uh$ZlB-zfH+3}9@Ab1M@OAC8Vbi35W> z;g4kR4qJ8n>%5G#3>+K@VO4n%)G|%bW-7<{)N!TTn3KJMF064a*32d03pINRvFGTx z(rwJz@$FtX8f~b#NqmSGTa0*DqZ__pe+2u7gQ#;@u%y@y0NCT*fwz3y0a*ed;Z^A& zT;th>^>tf?_hE>|V#ss`Y%mxUx_Bn_f{8Rtk_=NKhF~x@?|QRwsd{qCiqO=_S+A$W z@85;q965${MT=(qwQA_^O7&_=fS>jTh1AZ~=Zy@;(6eh0>+fGWt2JlaU#Si#e=Rm5 zjTC~xAoh-S;qc%Nq~!CDR^IaNK)HDpoaUL&)6BI$)z5%+0PvKGtCy+E)-xFewPZ=_fg!^%C^LybipZ{ys&|fC?ii(O5ikyJ1z=lHW+z+gC`FW51XFr-hSO<4~ z27i$=sFx{6sQ;4p#0644sB_Hyf4MA8%z%vy_%OI@CkA)z#ED}k5KgG?EqYc{86wl# zd!LIpJOYc}h;p+_=~m`Uk|Z?O+Hu?$Qo5Cl1WrsM&@qX|J97WnfJ}9rBrsOUo(f@7adNzul&AXXh(0SRqL|#FLXS8VZ$eCu8k_e;%xHYU@U| znKaiG;b`9^T9+0p-O3dm-{C{k-_=5I%Ki848t2LxUt1zc5^5ac$8@UZ$RzCTh3}Dv zk(&PN6O@Q2F!-BaBJAx|y1k3oOyvm2wDs!B{ed2=&4VymUSD`-9D%w;hGQ6O51{_` zDn-^Wu{B`5XA2B^BXp7he>%wkgWibso-HV`H3(l%HAg0a!XVX&IC zb>rH>KrdE18VIw+o?F#FTUR7iZFB1G z*^093Z-oEAnI|Y$uYLw?CysJl*=S`B4 z6h{8y$J#GxYhKKJsVYnWz*^&hwPt>I=H-ITT!E>WHpVxef4;xOgfdfh_sqFeSz?CK zpo4EXjJk^KzR7X6PQ5vf(iR5_?5g|cl~td^97&RN`sgI8{*$o#0^L~URNr~0`MP>pG2S|&u800o z@ia#F`cb~2e?;k4a*km#4|a2<>bE*FR?j0Bqx=1EG}&Qt zDnG<4*BB=AP-s#AYcG1k!>|`v^5XuI3t3HitX$;4n}^kNbaHCz5Xt=y7G66bJqH0#RyI%(4pXq z&10i+M22H1UXgz%x-%t3qft528L)J*eSP1ByJ#(Hir=AhD;N1%=XTsvxe==i^Y0#$ z3+>uHisSuZeDR)oMb__sd<0Y87?$0u{=2uce?|2(VAWo)%-M{1a`=j0o9q52lTmnM zqu5Y&RYq^_)B2ysoo#k}@L%3UXu1Hk?kkTa5Z&M2-6wF>M;nFJ)Z(89w>k6VvNysX z?pC^ui|m~^ftAh%&AkG2t}G@!?%Z61pFicz`Pz}87@|Wll&#h7Y+Vr;Ci7HmUjqNs ze+i}AxX2q59k{k+nbK|K)Sa7a(B2iu`AA3Kv%_#-?}E;tvd5QmjA1fQ7E=i%=n$D! z{?g=GX*rEOD2ZJcu?H(codF+W>I-DdAVfj^Ku}6-u|Fm{O++-}uOK{Oiw;VQBpFuaupS;XM90Gvlvn?Uz?|53=;u#x&o9GHDe^uu5>GSVWb^5xmPLOhAO=0`YL>2Q#J1W z-U0Lth0pzX*cZX6Cr9DiIf!uof84$S9Q{cz3O!boU&9lxXDzHDm048jT92WLKcl+j zTE%)urc-!v^dKHw^BKi@P@4~IszIs4gwOw=9Up&h1Dd*4W}Z{qgKs});c&FU*{`0TnW=w|vGFU#;By!;EjlF<|n|F9eF_n*LZ z|GpB=ru=>qdoH-m5vZ>0e;x3Sz5u5^_wz%BViV|`9K<(QY|HuDTA`C9=yVd&7rfIt zm5?#KX9C75Gm7eLFj({eP6tTH8^-Wk6X<_t7!}u*pzSBEFc$KNWF`y~f$H6Dc_Ik6ZsSXKQMay}^dY zEgp1l_d!ml0RWOtH|rCyG&6>YU`qzEGS;*f~}Uff5MBv6R?8SyKli*=m>(5+&mi+OoZ`=!8ft3zAY!~4&>zDjTR<7(Mf8`Xu()kPAUe=1X;uhic ziDrv4V6#-u;dn2GCtpW@;BmCne@J+L-#hX#{)UZ0>Z$GbkM*=ykK3HAk#OzO zucB*c7fw#>5nB2Ak#_6}b>YiP?-E*O#q+R}@+CBDcQ_WFAP0IjY@)LLhL z#pE-?`|!i=r|_*6A4a9Ta9{RB>clX~feP1Jv^IPVd%Nz#NT3}p^&f`OaQZ>99p2Zl z-S--FXC4=MNO~UIJR7mK?ysRUnTX)OyPrTus2|^7fBgy6TFVd&UjD%}qFm!PCr4bi zrMSKI|By&U@c3I>aLD@$9GN(RhkdUho=PB=KK)-jmB7Qk*U%dt!N_zFKkIuAAO7=w zu<6bC(Yk*?t+h<)k3uvf9+~7&fx(6i&0j*O_8xQ${2ovF?ng3JgVR%b{wk49;mZeq zfkZlrf6eZdxPSF086u>53=;(uSv^?4vu^#IQsmE;& zlTiU4ldc&hldc&K4mbmrh=kw_Sk_}b))P{be;Lyj{vZ24e;H|w3=#kU002ovPDHLk FV1l(nZk_-D diff --git a/docs/images/chapters/circleintersection/3a3ea30971de247da93034de614a63a4.png b/docs/images/chapters/circleintersection/3a3ea30971de247da93034de614a63a4.png index 7392de5b8f3ae4865116698d8500886125e0dc03..b68fd59a6c6421c041dcc0307031425545e3aef9 100644 GIT binary patch literal 14910 zcmbVz1yt2v*XBheq`SMM;Yx>qG>DXxbSOw8-QA5ycY_kr-CPhU0qJg(?uI%1-}ifG zzFBK#)*1!9@w?}q-#L5lv-k5nJ4{(o1|5Y21p;1&Fm)duh% zqVa1PNyx+F-|W_+cnE|FA}9Gu%{6^*!AI6!FLSK(yCDZK?rj&YC~dbGXq^J zQ+~$oy7Sm|NgvP1sDh9hdrC<{omS!I=S_d6bybT(qQ+Ph<}TlC8ct4lt}ZmLliPc$ z=ZSELrXe>Ddb%5fCyNJn7eVgy( zZ-v8Ujv&oK6vf6EBSu&*_LF<)gwwp^lU7APNWxLGlj> zz^0(UKte)tDEerxrhsgoySi#jM^FEXIbqUOj}GS*9S-%&m*grcDu<`1+;4+7U%>)n z8I&3EWEB(w6&M1t3<%uMgxL$f3}km93;_DGI?mQx$1Kdhxl&;vH@v*^r~4VodDfQye0QuY9Ry-vVAN`Td;4{fg^i6zPBF2oJU3T-ODZbMGd5WK zh%vLa*Yac4+g^U3yQDw0n6Cx(^;Ty-HR{?m>SplP)Cn*#QNXwygWjkfnP9vE%nya)I-W zqm?o4jU-n7HZfgkf`o&MTf2X=3X2F05BJw%Ew3D!GVqCtf3EZdvgmQ}a(!#dvFpb&Go}O zot>R_PEHfII;;d!jM7u(dJ>L~+(aheMlvU-QG%Wb2nd9Rg_Tqf%{Mym>=_#yV-XTg zrAydi0XIkmABMxm#tsb)?Kr;B=wA4SuJ3aMldc~a7?7~Dvx}M{GSRBDBGxF=RzGmY z346-U&Tbg_ij~09dTo0f8SJ{}*@UK3UCI4=(8v}o2ZvHpw-rv4nRsaC7kPRsYilP9 zotyF}CTOSxtR3t9@kb**_tlGg^Yin{`uZ3t3WaCK&mg}ocrYkMBHvY;PR#ViCWW5m zaN&xd=9iWRwnm%SJ$dpZHX|cyf1w#3Qurn4QGOObaa%=ZH@ymTx#=z_^z+NSI8757Mplm2&@Lg zHLNV3djZJrY16qn>r^uD+asAsGJ&d@J{?MChQMvP0=3Rwje3IusR#n2o>1&ZgK`zt z<#j-edeJ>UBJD;8%>LGeHky1+SBNtw|uuy-}o2#%fEO{V#EW`;rSeF9T~5+?c}y z9*^~sXJarBtgVPh+v4*VZz?S&n4cmb@R+APjz3=b1{oRom&ZfxXv#Bc_pfSxl6_7{ctEzD7oKIa+KnQhb=I}u)kB@I0=M?&9#?(pOqm@T- z7i{FfBu>>-?unuW71l<_RG`$7O2^?SO@nZcZ!mCF_|hhYKYM(fty+?;`pvM&L^M7z z@-gZ87cn^7V-EA(tkZM=dIO_`urKk)zvLU z2tPs$gkn>i2-xzGE4T3_$7A#^?CKJC_3%g&LXUV1mZr?Ctbr936^-NHiT-)p-~Y+w z6_d2JV-l92Lg1A^~CC(=?tOtd~Z_Aas2a5q;9s^NKK2pEN|(dns+q>C~Yia3XrhFo4n-lJ??i?dg(BVx6dgJYj~u!j^`U0ycU&s2VXtBv@1Le_P(EBD zO_u2(HR?I*>P_0xn%FGdW#grh#kyL{*d6}ROpL(Ec#w8kZ-*cJr_Dj)_lZBL zlLoLy7wuxg*!EnV!hn%$Y>(Z&o<5vIyrsK+Hsaz*L7F+|`|4Fds;<}|EZ8If&JUiG z$_~OSTn}gFFueMW#lQ&0!Gbl;g!nLZ@$jqO%fO*%2INOH4b9EnAWB0rX5iHnN`q5N z;_QUrb~wuW?#6DCXG|DGBCgQ%6(<)G=D?^qw>d;VDZ^KfjXl$@+4KbuaReJ&I=ajV zj1`KY1n=lhj*ow}d3iV#8L6-yG)r0-My?oMzdD_sh5t5E(L3+$5C`Rhx0P%&It$9@ z<#^69=;JQ3KinF@cheGf>2MS}P2$&Gk0H_b?NfZC%K?R;WAOunqot%71Htb5NjcV+ zwRpT_q^0DI1S&Zyn~ME(RswaQs4)^?Wurfkb%z^hEbE;$YhBC>q7kC;xwu1o=jU~w zKGk}_Ws=A>Hul>O-8Wnb3u@4FB(;<&hcyhZY3ov=>%Ck$a+3^^UTbDqGMbF3#@fe? z#+302RjyF3-b_eKi%`jvPT@g%Es-QZkT3qu?;-P^1JU~>mc*rm-C7i>EA?)WhK_9J zvK@cuv#8D}J{z^Oq?vOxjoYdx_6b)uo?{;r_0$wtdm7UnwkvCq_6e3?XME)*D_p26 zhC7D)HLT%zWweQCTK=MD4`FFReAtaVI9mr9d~OsLwImk@OGN|e+S@Ps2G^Md?j8P$ zPG{9oW=Z>mS7!933eiVwjhysheiB{&RVI<_Xo$n$k`{SsbYxfK#NB^SDdLHYiHXTC zB$W2vXVQOIVUV$m6J5#YjN6`0pn-RX1phMw2_-KD-(rJb%b4)#eJ@Q7V(zAVM{>9sf`yXP9xj|6 zM7AmAD2FaDPeNZbF@Y)fe6gTUd}tv<8pmAS^0|*JZ_^vfP>tZM-ck^rTY1;ga3GaF zAP_R7k{0DW0s=GYOOJDIc`Hk4GX@4VUH!-vq2M)*QMUYBmLQZczaY1em9A4%SLJg^ zo=cRpoK%>~IoOj~6(}?=eT)q4(j7uGE4{q@8i@xN9lQH@*`3Nr^$8*Zx_CRs4Rs!W zclObQ1rIp0#l=e<@~ffXbAI2hG>_)nwm9pTbgjr*M``*}Yv{-@kXS-Wdw{h<-``;6 zi^%h6?e6YcfpGjQo{VifGc1mNi+(MrmBaAv``M+;h)=RmhQ`Lj^*!b6@$712#>mQ# zb=vKzwyZ^{?NVFRaE1^mRdZrUco=pDxY_Bu%FFX0+oQ-4u1&=v_0-GXlKgW*CcBxH z)0EAW0se@8@HuE{;lVM@t?bN(nqf;SXRrUl_@dAFpr@6HTyS>&PmRSWd0wSQmrc)p z46NWL=`>u*jNvTL*!ZcB$Tcp~ zv^daI8D^5%UD47_cdJ|!Fsp5JJ)&uH+R=E9+4u9sj}7|>Vg0wgc>`qdD#dt2?Y6?k zf|^4{@~l%kt@?e3*(=GHcamyfS)3Nx+1SvPUY>oy=Bl<)Q$TxvuXM7~f7X+VZ8?~M zGWzN^+x2l7`}^qyZ5>WF-vow)kV08@q+vafw&r_s#Sa->^=s_~80H#XAdW5II@A8K z!Ew&t|5(#1o+BnLjkw@x4Y^z2`6JzlKB$zcq_^UK@!ZO6U!h!<7 zk*&8QQ;5pY(EwfgJ{BZ{&eS+dtbK{1_Q~RA1D9Ld?T5o-Ob&@0)!%dbv$d8W^W`)$ zodlatk(j9+j#YuB5S47PuNF#iKe1zge1-x$&Ptk^W8mRQ2$!QRPUap5QG!3#8Dh}u zX$~%fvm&lWXs$TuwQX#la8oMNSQ`<2kc^>tIT)LAmTf$SW}iS-?VTP3Tu5hLT&ql- zw1{__s1|!XMnNX1D!12zn3x#UgOX{y+@$)T%;oNGHd~V7YqtSW3&EjC9POic(lg`> z^gg65(c&X_d3luU^V;3o7&bWkc!nn0rT`8OpRhM7X<&_^{@_b0=|7-W$yae#6y)b; zyBsW<4*nDbIYe&Ia{}W(9R-IDM_ew2FmK z@Dnv=Emz2_$t&G^ELxn7+0sRQDDG}9_*~>Q{KCVcF%|`pG5Cq5+|nE0QZsPo3_N9= zkSA;K_66@1CvStyNXyNX{+@5wr_3AsXcDCLhAIbAm)&QWj(hU2KX3bohhgB@Jnw)l zBK&i*1SQALN$?$}yUG5Pf8oOX_#GG`Er-+uX9n;0xkJM;gj_J*iWeH*!Ok`nVa#9! zB!33r2{U^59_jU_3E>-ygUX1BiOH2^lWjhJe%yoR@1*ahuXWFkXlpd`67pblx;5C{ zXL>EJ*hm;;9oJ_&NBs#(c6o=0bt3iZmO82Mbs)K&6BFM2VfW&O?(q=bSGpWcPUfJJ|ey21dzFnwXOg^E#6uOlC)EdoiXv9ZtX5z_75-DtU<(2(W0hhpEk z<=jxx7yN8v)fkW{34azD^AM0T_EJpe_X7vwgsXLfA^(2TLf^t_v+2*D$12d0zRo$M zBh{&-kN(>l(j{uEgDG4@yjEIKQ`Y>vMBtb~8pB9t;u?@?XF8K@Nb{X8G9Jo6gjO#3 zJdl$MI{LP^`(#|KcCCFEwQeV_s=`oHsQJM%)J;1>D@=Y^^SH6jyVFofDf5Fr!r9%B z^-zby>|MXG>VK}}&-_HzebxO3rzjz=|9ykq$`k2mvXV0|ooF`cr!gH$DWWO-Tsvaz zY}#}SZ@NHW*;EWjo-TP?^s7iI3vr7*>@QJJPoyNS>4%ra$-ldZx`7`Ok?~K@mtf%v zIqwo>`rKb3&P5_q_<;7v=n7zQ4_ z!PAbWbKg470YK0EDutunNP zp_(7JjlIpz!ST^$C2ba(fqPisw2bMI;9eHZ!BXZnIB;feQh??OqWWK7U?Y$W}mG1TK_$VGYP;PP}_DGh(BuM?>R zQxyPaR7sj7!lIUuG7cRYwB>lTKZKC?_xA}>vIN$6oD34ZTW&56WB@EPTJSo_)Orb2 zHgv@50;6Q#)wf?1rGg6P(D@wPYR?Zm6a)-1eqW}yh1^2H_0@dg6okZ_EITr=d=gM~ zy`fk@ArtW=0w=ZP*6@&z{ZwxWI)1!1STEXdU;MT@K>gVh3dX3jT`He?_Y87K<{GR1 z6YY!ofEXAdNFEMJ2ZX>^k=sAwXt#ft#Os^ULzWG9AdR01D!xU7ZB$dxt_u%G@jMyg zOXjgmT&yJ9Bg;`{LW@-&cm>N(l~OCPx6i`Kse ziHmQYM>1c6BS=h3D|xYU*Lqi_q?}t!gAEDn)ULJ86y}eg2`rsN@Fr-+m0Bkcwe67K zH~MEBuh|(seE2Zk>cMw^R^dZTr(e+Ibhk2!06s-#MZW#JHfXlanlu=L97QC{HrCa? zAe}wT(%e(KL?bX|&ZORDA1i&nxZ4c&?5PA^G;;Vw2MziV; zMt23^&S>Eq(g4^O=6t(K7pPWCqA$4b-`^@XuI?_D;!;w=O3~5l6xIUut7hECDE{Rw-PpLoBjgdW7bH`yLk_OBg=GNj32Fqo{oyuX_E ziFdRtoxd!dxVyVxZf@@KpFie>Xlx_kEUNUlu-Tn&9A`Xopc4RPX(wZ489rDt|M>vB z){ZJua?Sr>u?89h?(}ro)}{;)Dp9(pNqy0x|Fsp0D6KV)23e~~{`lr0{e{KeX6W@(LpfG}P40d>*)W{*J$X915m z7cI~|h=i?;Cl@SMs2MVFkQ_CAklt`-DnxCuls-iEA5>h(zh--G>tGmZH1Spa5%?09 zh`NYiFUOLgN%B1y=9?8M#U6swd2LoPEITYO-R^wH%G@5D=wa-G-fI&14=B92*(2n* zjg_y;)Eh%tG2WJzW^f$v-qCzq>TQcT#xyfGk`-Ir9!(t6fTZYYb?EK<{DKNAPL3Fr z*o{toO&t@KnqD*b!zX0+`p$csgacF>OI{b`ZTHuf8JQM;;te)46eKTYKGqfe>jY)H zdvd*Eyf5WYzi%yvil0b^kVD$Wm7lp$SIxfh2RdfoC@ew+iwun-CIrOkiLfL9Sps0# zWUV(UbqZ88Ev=U4i-Z0BB}cR1d#SLH!B6bx&$&EPv!A|>Sg)?EsMw}E?Mxaiwz&Nm z;!()T#1y^7GBPp(&@EK9rS~@Rh7`<-v%jtM*%?z`5r>6&AAr6-n)6?j%W3}`5Z+LMr@kMsc^769ap8M&x%G%OE5=YCaEH0GKmL5Yo4{ zGNEYz1F3VR+sUT1NXjE%iKbbh&!fFnGYh;?C8ty@)GSo)-0aQcLA_$fnJh|Vc90>p z<=kuFqD507DP4*7?h4`Z!;MBL#bRF}IwiHRuugMyg0Hc3@g6*n%io4dP3 z)3_xI9mMY6V0W)&ByU1LwuxoLTOF@Y{|301>!KRH zZg@rRFHJn}ueJauFv*fwu4tCgi*+9>9n%avA{Vy(Ym`6X6Uu}8EN8X7IW4p1vU4Tg zYPkFtRNIxbm`N%@W|WEwow8cOZvg@DG&D3=*w}Dl7o-+$Zf@EmR3x;3URqtvO$mCV z2})9M&b!|1)k7UN2i?~WuRPZ`Hh2Uo{$8=Yt9okNdZtkwUD1ehz(B*y{qy}zH3LYV zLQ$f+e3b48EeeCt`|Royh3}QMu0`>>A!+kCj~|FG z1l)v#o11_dvw9R)V()o@#(A9ajdE?GDb*1v_AXc+N|-!&ji=Y zLu!!C!BJAK4Mt&OVflgr+wowrRc=m7T6%SRH182jiHv-z%(HB_{VPZI&6_e8A&e0C zMqLbw7)@KU?7y4V4p}Xw@2U|2jbqz(Wm;WlTYhpe9B|MnBQ)jAuuxI^n7Mu;?^^)~ z+m5=z7mHx4eiLBFoqq6vHw~RxeL-r>u=IyCBtYOi?x8e!I-mWzEZIx<8fj>W$l7f@ zSJHDcbS3?Wn3yRA#?$vr-5|dsS<|!V`!iO63cSwW-=AA4rUnM*`*_~b%0ycoAAu79 zm(O;wN7KOT!|(m$+1=Jk!cQyL3tQ%+$P!x;l4^pOsoO}B$VkjCzhyvHQg%OIYJ2%C&0n!`uX$cX>>jCoWzWb zG8b>vk}pnwe}`>sZvMsFk!$Jck>oWUBs$+7rOl|fRP>$&Wzc5*aLfW*o6c5iO_!=O zPlfV9`q%2yB~0t1;w-tBPxqhHY}CR)wkv2$tNArfUo9B-9Ril7+UkrL$EocB0*B2+ zJVB#w^wr5$o=&}uR?Wvax%;Z)E6<+3zNs6@LMA}o1Z8JawXU&h*C40yT7weKcX&>> z&6Ajv*Qx`M4=X!6-&`=>I0^1>`wRPT$JxPyJPYnEZP7odB%{yAfW6F9?0z;jnd~&& z;G@=+_hveI=r@*n#NhB_)ou8dA4Bl`P3rqY0p+vG4+Dt|wMuW^z>p|CeL)cWm9J#; zbD3CDAjVP?^)zlP7pcPBPR!ud}5-jK`&$7%pd%wG%S>!2FPvaC66g&o_N6ripo#cI;FaX$i^=1S4u3L=i$Vw3w zxXdM$9u^PR5SACJ=b(~@PXyf?seTPTYtk=%ESX=Vpn-@cJycEq$Veu#>xJX9Xc#Vc z_?ZrUSQ_#IctpXKzLOY0la!&$HeqPb0}+Vk+z7l|s`0z87Po%|ia$|1i%1snYU;m{ z)cWSIHEaZY!S&(pVn|;WaCtsPFA0RJJ4)TOP_Lg=6sQi_)k_AW25m69vdSzuXKD!= zHn4}OKp@0hk6M5ize{oYUPSXSY3S=y00u6ES|XU}`3%7(D3Bh?XQ5Xf)@_N5M3>&P=^HlY`cppaDJp#ji*-AEAKD5N;Tt-%f!{YtUZ~NEBjI7?r z`yCOvVYRZ}5m})2F^OEM9uevOGhTEABQ(gCKsu?A=_cy_$;EQip7W4bWGlD|GFL9W zEnn-fpXz9hF$q67G;_=1$^CVAmy0)onN85T&oR3FMNqMDzs}JKQ+gi4xu5+F0<|!) z;2uF5V62R@WUcSB~o6#?l_m=gqqCvp~Farz9xWB=u!Fhl- z0p=xFxjQT)%UCJr#4fh0Re~ryZAQ7yr1$r zQ--c*9N})(t*efBs%uBOT;jp{?nwSZb9uzFVJS2(kCvI4IW@&;AYYC7XvQpS1<FKUT?q$8sgOc#4jSAKvRdx1ca%%i zv|8Prs*^Imzkm1comSQRX9R58fscBJi~WV|nd+j(aSJs$sb|LRF|g;OS=n#X6gL(7 z{=V~s8i`eJVnYl1^`4tpZB-__yE6Y)(A)CT(b0L#a#B)KT)~~^s4)-47OX6^d6B+O zWbNFYsU8J}LMG^l%+Yq4tDf#vl8nOfaQLD(C@PP53%$pwfu`Wj0$2rojB zb?mw=J|0)(?)0rJ9nN4X4=TW8UwjG<2086NU>iU*w7wg{fsD>>{Rg%}$_s?(=*bhj z^>(ud)B^1=9@@!dp#*~N)%_GT1Iyn2VS*KrazvjROSE&|bAI7JoR8#En$J?CfY*+* z#hQI1@T-??cjr2fBUQBD3xXoZ7T>2I6&3Xp0K~|IU9mDVGn0kgUPwn0!3haP++bV$ z4qiv+PxBxkG}8^th&tP_Er~pqW5z?7Z}`gP%Wt{s8&P*ey@_%a|Y4e)=^BBA@l;x{5RMI%?M* zX-4h2puoW82#&UnMu)A#jU=mb%fk4i2>lpmr~Z*IN9+sj zft4P;#utGLMC?S1A6r1Bvlc?>-4TRJ_^aAX35`OS2*mFS7j$C@a$6M@?EBjjCA#O& zX?b|?gMxxwum2Q5KmMXZBjZ=DHnn^EZuVlW=8#+ee&xgM<#EjRczMjLlh+)~0lMrm zk{-4JJ@BjtC*$;)uaw}*4cdX&C0IdPPGZGpa z592Q#eWfwZQ3VD!-j7sM<&DYFP=hUJNBs&{GxU zI_^xO>oqyPva{m^>27N64>#c26e6CXpzfsxbu>7hVL&EDl{7Hg>fsU?6tt4zviJ(% zP3rUA=HVv9eVltgxXD6=d`0%I0Z&FSkPV}I#28|!-W03IW-7m+$N7oF**@es{Y6-u z&{k7oe{mHJg(NVr{~1xe8QCPpy9iB>_StP-APoY>A+Dx|M`ZG2aIj}6jV~@UGkUy8 zX>GC9gOQCb2&m*dy}VX-chNL8r${bzh)f=ZA8O1&K$ro9Q%M6-&CYvRL`29yr3E$V zHC*Wm`nc5g64L#!QH6J7v)|nwe_tj*OWkYo%lLTR^qVquX?@i9s=qJam3c6;skfis z9gcs=!m8y$!Kszpd7F6+R1AT^!88Db1$Fp%OSv-Rqhc><07$s}#>U13=b9QDJ7F*c z;GKuZ$4`IXyTqU4a42!Yi$R-wZ?%+^n4pu zT{~*@yuCdh34MtdQFF`PBD(E0ZFk!I<>Opk*|7D7eWT)%$?Ln@R1zhanwGN-@$h>3 zD`P8~C*<(8HkQS=>&Nl|y&he&|H$bw?k)b{I`*;?Q9yvfUcpS2LX$CT=EhkUjV(%7x0STj?~n z>2Cv1b6?2f%MZ~BZgP5@TLJwdF33`#Y6fLUr?PWl}7hF^fWwjRB2X6%z08`Z1 zE>QsOnyg$&>f8wsQUQPv{G?6+;xq)n62wklrG!X@N+~Ky4lsQvN0+45A_feItPRTg zCq?Uwe-Eo@y)@<#p*#`TwJ!bEZSvaimp6mh1nrOiP3Q-TT&LBc6fUH4{YBS5LszcK z8gL$@7_-L>KV%zyiQno!+HFYpsNiv(ySGyKV7#-Z2qh9Gh2*POaFgYF5kM(M49d

WKC39SZ9Cy^E9{9H@KVR!4cyeG{0H!uyttdIPbFM z&gWEjmF8k7bUd;nTnKfOz5NQYIr+Q@7DhwrTtP%opo#nK`ld{y8Vpg*W?*e%9#nv) zUXAm;Zf~}_%)r*{uluUHXbo5+PKua5Fo~b(85zVTku!61TnX|pAR?@;ub-+(kfj?0 z2=96XRYto@rqfkEs>i3Lh+XeLUaf092|TcYg!L*A+N%zgx@UssQQiB-3E8^TVYy9$ zRoZ5Fkx*8x?+~E1>@EE>{tGsT9|`>;oM^I_$no9rNSv(w2H8g#FHg4IzbO;74s23x zX(?I=Cgr5#*uT-1_b=YHn#mV(1HDf9a8tNV+X7Z4&l5{;9r(ttXY6xd-vm`XIe54h zp%IOdb%oj1Yh&fC8-w!5zrQGb7*2xlE# zu@Ns1Q8Z#LMLlREr)IxPQ@#Ml!2626H?oApmGECGM;a&;J1#CR)fr?40A}hH=#b{0 z6*S(6O_0rui+t^tgmrJukC9LQF|N9r2f+jl>42=~ZI(#Tqj=(GaH-Px5GB=gbZpx} zI}e}=i=HQM2q-BluK4bo^4*SML?!@d92^~udju*n1f9a6y+E zkl|8N&wW;K7PRVD}V*zK2vpQR~K2EQa1cGDCRA7>fcvZ2n zSQ;7{llfXK8l{F)6$TYYsk$Y`0-yUHVHY9$E$SOKr5<~_m=y7X^5-Zf{}Wl!9^`uq zOpLZZ`a6Y=`*;1U3uBM+1EYkm@)7+bL`Tf(304c6kK~7+DVk3zob&R2JoqW)p~3Ghpu6BaDhUJ9tZrWn_)})q z-a=#M^mZ4D8=Kwk&eH(oW8^MN?246$WR7Cg7JN%5SiYbR2GYGS`>mS~VWMXWndlZC z!0Dc<%|3z`!l)V2V*xvU0Ok1SxAZfsr_YJUV~u6sBS3^L{6vL3kCVpr>1=ZypUc-r z(zw4gUbZPM@a?-IszWR)Ug3Wd;SLGI{nJpmnx+9VJx)lY6g{KL+^Y9G0kV6+v}bk> z7U#D$g~E#=VC>%!u#c~iF(Tn?1A7xC0*)X`W5>5 zghd!^KH~Chv7%U6!w}gASvn+ITJ0m=?6G5(c;~MxVnGVMO|{BF(MHo=xMdsL0E~T` z?fsqLReOZvpUie5^;%r|KeDz$c@-09Mmn>$V$9>lIoa%iYPSxgJnQ;mK>Oo-*NDo8 zYSRTMS@KpvK->6sNI)o&D4*6n3j|cyqbze%i>IU($W?Z&cC*!3R0@kpY}J`Lq;g8J zq+%j~G?Q8;9X>}Z)any2kid!QQcEXm64A^_8A~1_mlC!3H#OBi0a#Qg+^^n#IO}w> zUkg*{pTp&azi2Gmw>^!7)@*#jUjA7--Sq^LG3O|~#hU@;F=qDts%R))iRBV;K%_hkqfIv>qOm|M?Sfswul+nE8zv<|MAz7!-W(Rm6 zotjS}eOmbha3_SNj2MtJ5eT5ul~4lh$ui_{dA2*iG6q0vj0r5C+DpkakLi`6h!5>3 z5lQjztdUuxGi1zdv z4;@yz#1kbMdW!isHZ~Nd1PsG$0MH|4kB`Ev7Rxko=v))TF-4${!TY`PyZTM&ec;$v z0YikA$(Pi)FNdJ^Eqw{h!m%1uN~{5DFrci3_)rk;x#~{;`SjDcq;HO7ZFD@H*)hfB z{qT7m@|mMwOcaiyu;%)}`VSVCzFHX^S0N#)j)`Z$$anu}hiVBhGcc?hS_WX)C~hxr zJ`IxBEBb!fY&SZbfZ9jU&_!*2Rh*nGch;rq)$%K_sge z85ABCWd5-wHa>&xR*wvOdlN`go|OcIhar5^*P^+d=UKQO*$6j8qWV52U2!BHNQ85T zryVxCaQzLD=EXN+4oPhhW(JMoYq1`RHk(wNyGd^S&>CStI{Sia%09JVReFObL?j%N zgF)ePs2b{8uG{Od002IKC^OHLxB_g$nkNVr*NciB0JKQ*4Yw0zqQA$bxbzvT*P-lN zuOC+lQwNbyvvA6ldaOwWY95sS4L|Yu!gclQpLRkg*inQuLm*Nnc#M@TWiZuXNai1j zOKPeqgd&1>u7SY1Cih4@zoa-hR{D>i{twkDLYl%?OI~Ai=uw1U+uE?qy?}nr7z<6z zynrNzS&flVvuiOpfH%7CaAme@q3y$xNzNmqNA?|Wgz<n;GNH~7m1Btrn__QmvL@9?PFs^5n$g{nu z>Iaicy%=`RAGq~@_=775Wah5LCy;2yT&V{nVS!pTf8nt+^z_VIViEF;#Y@4hR9S(M z)xoVKhKQIOpE;6qNbaB#6M8k}OO>>?{fh5W_3)y;?&K4`R*;KOZ~gvT-$1rNH76ro z0$c_fDaI_+4kitFu;Rvdj~1Do(ztdDN5?4Yni=_SpD1HaM zt#FW;nVCX`sa?>1l^XyIBq8S8emrte+gTkx${hp(fNzHWA=PPhH8nBdx%4<;uUr&? z^#82)n@`V- zG3w)j42cSWCqSm5z2xHLB&DYIWs|U>Nx)&VV`KyqAP*Hs*8u9qCL}}x6<_vOEQtDr z9>frIrX)j65J5`}Am97|TU2&aP7%_*^0CgxzU2(&cF=keo0Syk0F%h$Iv@48hFtvo{8ThFPe6YV#8Gs-|Nq$%l*fh< z(*4TNkm?b(9!wXgUgQVm5FksVHkYtKr%AH%UNg#nl5uGT1%=kR#>U1RfXp)|$nR{# z*YQU}izh77UwROH`t&JTz`-={(<$h_z`{jAMiz5+=4}FPTad@5nI^|SV}<~8{_*3H z55xoXE69(5fQ}4`&~#259Iw>xF{=(Q`+K*d0dnA6Zs{o!KwI=)yH) z_?A55VSRm;_fcOLZBsd{E4X^K?CoEyom@qZY*oKbVu$tu8Q3n-&2j+d-zZ!Hf?hIm z@^?jw8Pfpk{2K+D2|%Zi)o{9iJ2?kFF6b~+AX55I9$VFF7u^xm=dUp+m4$jwG`rKW4laY4`1X}H9N0y4A`$ZL$o z^5r%aB}@hq*$~4d%oqZ(Krk0ah9ZL{{Xaci{x?-Z9~gW6Yh4!fR;maG{Nqaht@H6# ZbZ5@y)mwEZ9ncU8k&{xCEEhNQ{~uXNuzdgk literal 14803 zcmbWeby!qg7%w_VcXx+?^w5oTO1Cr!l7ggkNSBBb(%mJEbW2J}cQ;7)U3}-Pgn+s&hQQC;)Z-UJxtQR1*T;czbkUdLR zat6sH!i-d*)Sq|Y@p1qETn3jF;Zk+u$mgkBSXt#XG!R8cN1H@-^z^{Pz`%q>MZv}P zak)mw$o723Vx9W?mxhZAmnLo}eRFDRs<6CV#>j{Q2M1^6=*S@(?M+YwGPV8}g}j*- zFTQRwXmblH)NY3D{3s0-!pMr=N9iXVGcZ79VQEPtAkZkAmi0#jSEA-iS{izlE*fRV z3#HU+=!+L0+cyU~HV3e=z66EiiiOm>!${F!fBg7SDXm;h4N5JADlIJy@yDih^B?@~ z?vXM0-Ml-D9OB>EDPf*(`B75|X-vZ0oEFkPsOXb*@{g&>YB9Z2nT=6gvpS#OL?hpgEzR4mD z^aOMF-28kSpTAQSV03D*h>4M9Wo0!r;mo&3ZW!5O{FQn`m3n2Joao$+Qey`OOpt;l zq@<)=NN8zkVb#>sil=|M9d=flc7YC#>F<}jznuR7fnbE@+*+43Hr*<$JzS&0u$dQ+N>)~= zs01wSnc_iU*rF>dIau^-3cCcCeeYN~IXP9ew1VB2eF)iGugwzYdwY6J59ewHeRwC| zM`N)z{`y5LAV92BZXj-IO3lQ?+_CDoa=01%+3<&#(w1Hy-~ZBBPwDKaGAAKvokY=`s>B>bt)O6 z{H}Z7wkRbS3X`zL(S<26K^KNaM0B<-Jz=pOQ6!bE`h2;)_4N4k$yL|L$S7uVQnOUQ zws`Qcq@-kZeH|8Za&aO3?%lhB{lymHkIv4LGBPrW^PXN_IpyWbIrmdpV70l>B@UX} zmfSoImY0_o78f^J&7Q9JVUv?be>&Tu=H;y$JQNWYR#j7j$*hIAYA5;st;PwK2&k#y zZodCFXz1`(PY;=ok1skli|cu=#u?BD*%K!697@Z|=H%u|U3fm+ULwGPU5>!HtzU8zUuk z4UM?3U$Z?IXK_XMZJA|s{EcF*{-0re_4ip@a(617}|TCu1T{PG^jpzpP58h z%*MxL?JK1Q1GYMc#o{Yr7i*XNSnnOqP%QG_x#}|)nNEt6|2c;+=*qU7&y z=iG>pg2u*VYe57S>J6idiwh5jEhXFKWD!4b%>!JD+nsi}_NdVaA6ti;b_rK+QY$j~$d^WVKc$-j-C zTN4DPDF1}We?PR#o2MS$)FjyOC~#q^0)p@thvJHgP*p}aoyfS|vgbmYii(RB4kuGZ zu%F+ANBnMip8apMw6snxFL^=bgV-)T3$t%&Hl{aoGRqs;1f5ptjQfIZJg0Fw^mSnnR1m_uj)zenl* zmuEUl-yr(O7Rh9otz*e40oym@_7Phlr?M>)rejT-ArUl5PxOwUM@%!9_V?$`$5!36 zyfO3jAJzj9DL9@R5)qEiWY5&IMCujyi56FnuX?55S*pg5Tjr7?ybb+^G~^m10QY#w z;cPCddz0Q#vhr=^+*L~M8~y}8v}uwGQ{zX`=Bqoe_X&LFcFR>^n9;J{a>Dni_b!4m z&Z^qlAx15p$%5$B;^yu8;R`UkGy&{iIYK3~_j?$id{>lR-*I1!%uG&_Ow!8y3D82o>V*)zhhsVE?BrN87N<^f*Bp804>SEXU6MiXX zDe0lzPAN7CgD)WE_u0f}^`h=1sQJm=aof5~5F9HjYuH{{+a1G}Tl0@@kPo1`wB8FR zQ0PZU_Jn>R`VpqGCMvGvh}J+>UZgRBgh(d5^F&2)?uV5@?;Y`Qb62(6iNM0&011D< zIlWA(hjv?q6-^wdYiZ#d7#f;dSs}hQ+jALufsc#Y(f$SWdIs4OTn~DWhf{y#wdEe4 z;SysloB;+Qau%{2(wo<5xfm%w`KSeGi37?5lAiu}cL#S{`y{+W%gf6 zR2m|K?Yb{kO7}EVjfJ3dz6!y<%evX=`iyefc4# zFexcKr$_d=2R5{~X{?kn9-(`bmDRhmhXix$PM?wpMKEWbe>#Do4Dk^_>r;#&6~P3d z0IUWLsC6itvr?(vQdB)=b99+botf4e6KHI*;LGZ5=JTGv-^%(MQUf-kEWo6RdFsAL z^k!RTyleBh<6T%-SXo&y`SzfPBFF{JLfs_fps|M=_%YzPyQ!!tSM!aN5kAqlF$zSB zC^^!A)B&=bgZj)&NJq!sFWhrYO(i9Zv%Fysy9sj-)>|G6;W$&**iX?tZ&pIeUcH4{ zb(3{}i0V_(@#VZ0MMXs|N^%KgbB=nh>8y~|y7$c|vSe2{Y6t3k<=;$0#G;6}fea=C zTd!t}D=ABGb%Odfi7kwSgM%mEHsHEgnn~3T*UH_CV!M77BujJsh7z70+!VUjUrNhp z%_zQ>FEy#ShyF7IOrnuDXcM8z ze6E5b0|$tfU4tE{;mZtWck%Pshm>W`L$=f9)GRDYn6rA6O!NpYg*#a4^|@x|<1$iS zdymrebBZV`f$iG<^1JbsFW;f|UXqt+G&eU3pYL1wNUH9^E^~67&Pa>O)4)w_%M+flpj!fo4i+7%&{v9xuiV z^32t5=~9`6rpt(;paw0T_^^l=lg@5`1A6_ZsOr%PtVGW7o#N?7<94V>?DdLA?&CLfb-6g@t6_FE43^9M%_eGSIoR{oYjuj@c zOS%`Am|xceYdQ64t94&7Pu-x+mwa_~Ri=N0bBxbeb{2nD*X2F7uhJv7MEak}T`X4q z)70nAToSm(DAzRH{kS-o$WF`17}m?N3t}TVw|oi*t2s%l#{nwZdG>^{BHk)_z$R|O zdN`}fYZlxR!{hs#GmwgMnpq}r?v{2Zj(q2WKP(ZDxM zt^cY^-O18Iziqv#qL01sLW>iPyB>{uTByE1n5`y)ntcxc`IAgjsm3Me5`ouX=5->U zw#|ikNt_!Sy7Yx~@jI*0^q&4W+Bb>cf-}vs^K`h3d$MH-ItDuiCyTIPcU~R@*UPj8 zPjec>Li#N@4VpdTdcCJ0__(ZOjQsxhbKcF@?UDJ)!9;(UX^lHT7IFQsb==^xi}T{e z3o1Ny((Jn|vutXF5WcIna7;5Xj+w&kZsC(g?|X(VF%ReE6#kFrNEF^D;vn8*7bd-b z4?+{x3<`&9yQW5gL=@FKRTO?Qj5>q4z_A?J6hZRH*F93O3i;hXOW$2z&G@AV;+skN zQT}r`aBTGNzz%wo79kB<5nz1g8~lYKSGGeW>TMn=*&%rqpI=3hMir8#68Cc`U!pjx zE1IB^aogy}9zseBz>M)V+0$fsfggYOoDk6zA;ovm|5rHJn)!WhxtG%Kc?YC7(|D^H zu6}-(qx<86HUkpomZJrnH6OQoE@8fF0d5}N-IwX2YQkMTLQZRN5|WY~res58!nh^M zrJYpU;%Vjc<1}2JzFS1Gz5a(ZMv?ZY7)!!o6$QR!XD;apj?H+->|ivcm6Xr`KX9Uc zu84H|0n5UT(?ih1O*gUE-(EUV!0`HL%^hrVG{dM1U5t1r7VqAz#>rB^{Fy^OIrWV- z*BHsavzx6}gYI^^6Sn1aWO-$36exErFH^R7UZ}ICVDrF|vO92_YH!6xgp)M#G`|1J zYigJ2Fd>gKSnc=ZtCYmCn*Oe;ql%I6J8iw-w3J@57!$pXU0W;pZ`k+_7yyZ}Xum4t zT|Znc3*q6C#-Qh;r-o#d-&Cb^0*l-%fH$7%H*;dp<-l2N>3;MP(vTgV0t%o zNZ&ue3D&cenh$nyN(qRk_F%ck0eFcTE4Fk~Q*QpEH?tv&DDO?4%*cOFcBhkiVNLLb zY_;#2{`m33@!y7gcTbP&tteRi26AMV2FvCfLoBclwJv7I(Jf)fQBYTs?tTmk;jlPb zSvLVUd5{`jOsNHdOaTo|HZpcCPBbSM1zyPNj)H}-vV*kY!3uPY_ z{mb{wY_sv+;oB6XkEonrP~>>+=RSi+I%j9~_U(ff0*{holl~A@UNIKFdYz=0#1zxG zX*J-nIGZRySs_Q)d3iWr<*?Y~&~s6F(H=aGTGr(?PE-kbk2d2qsklEjUCj=b$PlNP zM(>21@s@A%?(Qy;h$lA~)23P<loJiF+=*Q~mi89gKPjXMUcXf14T4TY^|9ppvfJWqha&iKms=Rs~-5~Qn zp9*_tW@B!)Nr!{ndUqrwe6`>#*n|D=a|7+KoPyF)Sp0Zi{SDTY`G0TH5TKdlLT*&T z!p-wn3P)@6|Hf#h4?3g)p3Frrl0czIwA$ki3*hGCGw%vP_W<+9HKu3%AC;Ee#h33F znP%f52CqWPqRRjv{m~Oi7}T}u-?iF4HKldobw>%YS+>11KVp5y=dkcR)wBKl7T*fY z&(4VFf7_*I-hciLkV;!bq@Tg-HG4pb%uS-QT3e0&@9&QEvO!VN) zLv!PG^9+B-T9{NLUQqsbf4`!w_xf~w_4v3V(Zj{BEGOMrxdMiX{_k#YH0eyOEurtN zydUoSv4`52ouOQGL?OhmTpEo6@pC)CbR-El(}O(qoS<5MPHGrw4oowITALbC2IbWD z!}0X(|b3d(qn`IYYsS^^c6mI5|N{ zNJ%@0f+8bP!Oj9mxS}gD0^5^C_hr+S_5dOQ^(WZR8qnRa2zgQ;!`mO5s@}D=wf#dw zpXFSRA5!=BY&3OrXxZ5@k&%%d-QD@rLbeJ33?L>Zo@sE(5nm&mBFw6=H4f^o0}1ln z&i))B;UN1D!DuhJJoSU67E-e+qNU(qqB>hb&+V-j5QVO;)4uiC98N~0 zJP6pzpB}H;pys`3(Xy%2+91rmv0KJiSaG3RBIBx#KICkFr7&TJ`3sKd5p9Ga3ZhU9!1=z6b ze}n9&fOxbOk*neTXY&tht^BQ(xGe}N(OC6Z0L!bg7xB3=F#+Nj0;ym_37*b?-#%&* z`3``p*Ktee?nFV3ahor~yb1u)ByUNqPfp#yVsz_V=yEJR!OMQg9;Rvgn4p!B-k<5k(;-M-~$P49ey;-yz)0=Jd`6mDv~kFxQyyZ z)IA$pHGDj{wh|p7!vZD_EXc~<-rQm8@Wry^gai{MBR+8YZT!rDeW$ z#_Ye%L8wVPMCk{&e92+czV0bB3WLc0n~xq26aC@&)lMv6d^8t|IQD!=^FF4smh|ln ziT+BkJ)%xSy_PrFgxEF)2+3^zrVkq>^+z$z*J5X%u`zcs0$FoDAv~2j*gCh zZX24+0d)lSdrS=eE91rjQxZVJO}l7C#$FBIe3+>@xAu(e)9t%=;Nz+8;wzbL4OFN* zQzcLYaM{@V6h8j9#bU%X{@b^2!F&;9`EWP7%xiR40Uz6Eu3q#F+N-Omh`N4uV{_%| zK3{i=hXT*c%xn&L+IQ={(fgeFQZ%N*6cv&Smwoep{K|*p)JSPviURh7?aJYq}u{rYCxsUsp5QwwC7H z+4=QY2f1$bJuKLY;n>^mIqifRe-|_B=;#2R&(+Hd6BAQYv&QK1dtzeO;Gkm3p}n2m z%Km;N$OhJilDQ0fOG`_ek0*!#we7z22@xm}L}I?pb)ggXU^_rW#iXI;?s$J!^Bj#= zna_hW{$Lh3Si3d(h@teV(j=H0o4$15;h`PqBOY6l@UA;IBop$)dsY!-qKL8*eO&1U z1xa4Me7U$J%F9c@tW*9O?CS&HnXz$kb6lhH^XZ$P9`A}tJNC>HzI`*9EL1b>z1@gc zS=reMOEmJ7zY7}cIQ_Z~_{;pIP8(IInz_wW(KYopVwze1H-!ytKGP54_&+QZq^{h$ zBwH3uM>mRh2>{t$^O#}ucA`L~F8~%%vqFf}wY6k9k=-~ZHI)o#2Go3fgp|J5SvKUZ zP(wq*W4)v8?N9&%*Hb>U?s9H{?=+>>IfES;Y&yMPUn-)_8MJbVZQT1aAd;51#rAL& z*s?iCxUX)(Z?$x${KosS*!8|dFW-caiDaf?dT=&40)hVPfYwd86b-NQmQtNr4@$l& zV^dlNZy$5wpq{$=*QK?^sj2CLUr!YB^72Y-iS8uFiN?OB@$_%ZfE>3Nu_&7UoI?z) z(_(5bs4lFEG+B*Dj3`Yaxgbk@C|K%mO3pjwHA{Z6%}nB0S%@M`ieU!Z!qGiX9OGsH zDeFQWO5wu{Ld5I^ST*`m4E$)%-?FzC6p7Ze4NACg>mA?|2M;Unfv|AB7Rg@ed12$> zLF6a8vy9>=S?I-h%kTbFlvJBzHXi&$dg%vXx7+f?Q608`fdRU#5ZhW{3--}2TQtuuQ&^{e7sys16z7{N2h5U(2=I z$rU$hGWH=-^<{ZrqGaD4iTivQcU?dB-c}eRCE?7R%hIFCU;=AlVPQMy664*|t28DZvxAqSA-F>BbZgTbWX;C{9QW zr|U0ifi02<=fI@Ce>lj>4}rW-tA7dfxMiO!!^G&p{uVaFrq4j^d5&=O%yh5!qo;qH z##3(~zexysnF-LiALWg^aMdPqv+zCsx5c&1`&;F7U!(1Z#Z$uPk@{ov_Z5gMvzTB$9L z?$3cq-^mlFhViGBjd{&NKnsPAtu*RKo^6k=UrZb8;NxiAp8ks^S`mOqdMmkGTD&Su zDx|hjnv$V=_lN9Y^oa!X!Ywqk#SNDC!$Jc#fLghCbR>eF=Zs69`>ZT)>xMRh;UJOl1q3&borY61`ff$Wh}Rb zol2RiLIIx=3ir`mCAyB|9v*(3C5Rt`S2z48+R0%xd-Li#vmkTqRosA4W#9axUeKh6UNuAhs`((q{Yyh$wgA=eMV7}PBc$%J|9=SzTK}-(z#XMW^T4KWP!V)_36OH|H(dk)!8uuS0rUf1A zhCV-Qt9n9Sk)HgS#x%7!jtl!2y;rIH^!Nb%^v|MHx9UZSPDS0GsG$3HRZ(eax7g!p ztOL5Ry`7aq#;|x{XbAy*OD!To-W7_;=c<3DMbFd)=;xE+jSufbL`696%*K7o zlOK;_!qERo_&ZO-;t)?v!f3?drGHMIx03k%jeMydK$W$g$heG*Sde2#Fn#|)rT8#7 zJlxavc)<>0C1MZ)dS{*eJcs>UjdGtO4$!j_=U5K9lC6RgGgH^;GV=pBE1 z34qGb-L+5IJAli4dK*;Xt}9E0^&CY&Sb8%wmng+RcAPGamk@{@r$ zglpax(-TVR8wLQh`ADh&5tn7utjH-=WDmc~4%X$tY!Jw?R>ov0dq8H6jg4)(H`!!A z_?=bP$A;%pxC&8Om_eh!TUtK&`tdk((G+&QTUJcK8d;sGGM7_S>;h~w)OAmjj-I~QvWqJiNZw>(zGQ&C zdS*;RFezhlbLmPA8h=ha!Lj__QcOARZRK zUMqdMmCenOjqTkT%W>KoA_k@tSlhp)9bwqCCmW(ZM95~)4tOZXn}6># zH8r2pP9h;U76LW{BnF24>t!#$4&Qs9(dFm8J3GTmEFN9ee)wEp35+5&L8KZLb34OS z#%qkmy_VDjvW>cpHS}0MEEbgB@KggdO3~)QMoX4iAb!)Vhoj{>$oKT-Mu%M_EA{hX5ub)zDNfE)K=yKwBMmB|FK4K?-M5)txa%x5gb6p1PK6s(K&Xmm}g^!p^L zGIoDP#^o^WgoC^m7l(PgUdPxTDC1~Pa z)5mA!=OX~K!pg^I2rUWRKY!{!7z3(yHJWcp6+a&^<<$q?1tn#TZ%+rxbxsD)jSGvK z19oeZ7FuTY8Cf-dDaVx}K?D~xss! z&sZ<;XD}IBE`OzU_(O2q{1_VTu633!qWS(L5{!QSvo2&8(c?D>Wy*!oZUoAoy zqV8BZp0N^>bBw7Ct`eBDalbI|$tAj` zJB!j>W3M=0gI5@ylv(SxH{4kuH698!k0T_+LukE9l@jg3H6Re0+GH5&=kti~eHWaQ z-&p%JP;U5(!)}J-8R3JN*Yi>5zw@xfg?dma0GW7YLhnJW6c&>b5vCUzo4sdf_ zUwd3%UsC~(+2-aZ1Te6UAO!4fa;<%A(7oR=W67u_i=SXHKb);Kscaqb>+Nnox!E0! zYkG+f#OL+0u@7SgeDs;qW}WM&fLlSzQVpZ}MK$D_5vkQ$QaRv`Z{*|!FYYGp`0*p4 ziUV!9KVm>Iy(mC4XrbMR+gFF(NlsAyx7x8ofePN&vrOJKx2OvZaj`naKKP@;XHhnO zQP3$BtI3wDgX&SU=R^~1!pl#O?!ZEa1gSLXc!5U3jWjzui-eA@#TpwK*)>;ddvT+- z`nyAA?I6;s>Xj8Py*D2WAWU|7d{xxObpw+=m{)EN5TvH~U;u&TO`3Xa-W3A;(a{mu zae9F;x4O9rFPeb~UVt`|l9he--yI#LEi=4 zJAGDEkY0-U2$VRV%${azwVwQximsA-o6)@hf{$4f094;RzGG8Y8=WM<$k z#9!3sc4hL_L##H%3XXq}pHP@whFp{`-wUzr$I>ew-Rh`8Vxl+Q=JnX%Qo zRxb|5T#{X63HR1!=jJ%WrN&J^Ee+%Fa(qO(d%vz?w|()^ZOe`P*OX)E_ode1t&y}r z3g2tf+B+M+Bq6s06IcW^6E-%sgN&zphU)ryt0-vi`l)8TxR$myr)b6~k+~9090x`0 zblH0IzHi2IT#Bsj_4XE-Wn%C#-0 zC&vI??uxlyzTOvW-brhIGLmw3I%<_~YGAC=fO6oN(@MN8#hY|LK+1pBNBM=B=hxM< zgtM`=mEGD(ks}kWzQ})ha-xckt_u8qp!cVa9UL5Ph5D7FkcooM*wgx6R|F6>TXKl< z-@@qQiMUhkjmT}_ung-*P}_!@=vJ6sn_p6T0A?~@Ir9GZL4>@E3%7!Tf@Z~Zg>mdF zjr^68idOYScBh(kn#o~ob68=HfuLB*wrjMG)u1ZowikVhwrUGAXvqk{n;@(5*e|Q+ z04X%&wVfRc$nL@Z{drXE$$;m;L|z_QCYsZm%T6>g*1fL|gW99%oE!8<6eau9%&xzp zNch)RRhxSRX!=}8fY<(SCZ4PY3)W9jds-?N$VWlztS9I%m%NUPj9_<8CMG9yZw(po zes*_Dxg9MC&?o#RhPWMQqtD0b{N1gyH%CA~_!%qqnDhz{rI*MYpcfWi-UMX07regr z9>BH=KC}D%^`+g})-Zy=Qu#qkRienrCM?T8zqo_xW@FcwX?*AC`t>~wUXNiD-UGj` znSel8&+y+whFe55>9hE_jRqrMzkY2ZKX&MNAICg1i~tpKD+#f0)9%GsF14Jtl`hrG z|0CE2(hK^;!P*gdLt%TyJC+m?Yq{q-;?HE&$3t4u z;4146y&?3@LPiK7L!`;q8`lg#Tx%~4)AR7%I6J|&Sq_@Py4j^QN-jiOw(^U8O1q(y zSdip+%DM?d4TnV}tL|4xTiE+cP^s1;a)E=%5$lI(zpl5GD=yimx}D8_SxPG}q>DeL%c}e`E%+CAl++qhzVfJ%AxL?t!)(5Dlf!>J?W znE)_i>EoFBfGl*#*M!$njDr-I7-zh#xr}BnP#ky{q<~iY%;J1r zK|XQge!vRGcW0_(_oMV3zQgzWa3KBVEf%Q8O7~OR>rC6sq6u2hKeR{5?;~Lti^?U_ zzDADcTRIAmU8BICxZ}9|sw6t|P)t#%y6QDL7+#KCj`R(s9mO7d93z_bqN8c1#oEP1 zRzXyWUbik$FbIx^XNY4T=-VBi;QzEw_hWOIXx^P=O>!Ohb1=mi#LCQmoLvPUCD-hY zS<&w~LKM{ zUX3B3>?s=8xIP_^Z7^^p#&E+xIzs}^MvlP6z}4Y18GkFt!r|iT<_^400P}&}0o4Yw zmXowPwuJ)m-j97%)^Zu3y4ZB{Fcz~*h#SJ7?4dZ;-ZFuDB3&W%eKC^6BE$)Oxt@>- z5>^c#vTnJ;q=ml4m@tYgkfOX+ATx%{cCT!?$6O8=Fr;u2#dfv0^wF4RZ@=Kk}G z5;C8RFPHov(^Hy&nuJ@@p@b9@z%#u>;U2(q+;(22tqf+!Kz&=xtX-fwDTrYA<&QUI zA1SQJFx0Wxvhl@cHo-UUFjaP~&WV6fqnH=h0?6&#ZfzNPjT9eLqGVN^^~c`D1kBEV z#NOD0xW}MfN-%tTnk^Ok^;2!Cbb0wV#S7GQobOov5#eg5bKLtM=8bngYQ@Vu1S3JM zfR(PB*P7?YZ|1NgWZi4WPX>VJ z>S4PM&wkq!T}RQp>k0=#8$paRC~b^^iJl8v;>mi#uXDql?{mOsYi z*l8?v;GL+ZQmKFKIU!9#bVT^~tVfA%&B7$zs$XZQ!A$}ZfaR!@3+(iSj_7c=4+bCt z>ggN@#Y8qZcn1bQo}x4BjcFsOzP1Jzn= zW|q`V8UfxE&#I^I%L_u9MVX%5&|mY${lW69R4&<%61nrhoQ%K6F6AP=j57WTT+~P} z!+kXVaY>kM`5;1ykDL9y{1&w~%{Z)8R|GQ}M)5ol=KZ%BJUAnR zAo~#PiNHkPU0B%mLY!kJG4#VQB_>RQ!D^4)qIbvGUl$ErEbVA85QN{NCIe?*_nP&baTrRZ zDLAd7o%e7rQR|mpRg+;HR-nd!lZ_4@u&HHbIou}v+K<08HT2$>S_IU^GpcU*_SjA8 zz`>8YNHYS6Xn2H#IptX}Ca(iZ{-1zPPxk8){BanDm80CFM~9h~&Vx0u*E!(=0x>Lz0k%g{9{dSzY+? zni2`6SPGQ#H9p>Jk{AWJ$mbf2{ivN*dBS?k@bIoz;#D3gH~zKhuWoI?QSsIDEqRzT z1O_q3#m5IBQh2m~mNhq+91FnW0h7S9uwDo!ux0hwB6j{|55%8bNM=Dp?&#}aBf$M| zkj%=;KZQ3xfGbae+p)gwOY;o|`0S73VUx6U*)b0rUkG%{N0 z3rIs#lbev0vw)HP+Soe`YJDU$G&qP-x;H{yT?EGQ{2UXtd*UPOwLebQTgU@6{ z&}N|uplC@lqW?Q*6c-og=;9*f=m@_xx*Q%BW_EJJ30_h%1m=(Te;EWMvakSElGfYb z-~U=u^V{CV|B6{4z<5#UJID0Dini}5xtN(tB!}vprv6XmRwe@-^h-_q{{=rXGZlu? zQo^-L{+dq}UwCYOR{~A!{QVpF5Rh%=lX43RXxZ45FFc2T@aWb%kOQ=Tx>r)MmTZ*{ z^T*RKzq&dQ9+kl4Pp0^@3qn_yh=5gp(lqT73N%USDKGc+y67@CO9KP4zvYG{v!Pww zn#RTzyTy31YkEG-2?+#lkN2N0&ql?n(MfqLFZQRaGyEQt4S|VIK5v|@{^I=H0+6U_ zt?m!Et^^bmzAC_nPy|&7a~gfa*W^VA$rH1`NGLWNP-A0b;N;=f*4LkV zH?+29wza-KL2YT30i0jp^hFfe=-OsEd=GN?zEEyKp*wmC=T)db5 zt}m8S7Ua%b!zrONGc#=US0^X0rKREh|9X)d8Ku>KvL-zPyq6X$;2#4jD%ca)3?qOh z;zC^g^Cu2S*-6L=!JtI3{0}{{cp4m<>ihOe;{+UvDmqGZdHMg9CnrqP?t1KOY&wBG zzoH4dl+E(L1e+vpHlRpEY<#6gh7fe{#iU0Huf5rE%T@XwkN&n#+jc=sP0hq+RIO1H zIPk+6@Vb3%Z4I1o2dtK;L3l!x{9@8A2YGENF^0WVOM<7y|{mFs_PiW=NB z+y7hU8Cz1Wx+9SyybDhENU?+XEO8zP&b$2YFDDijhPcUL6R;#ES2@sAhV_8)7c@*w gc>RAplX&lkH1uVHb86`UoYI8IODjv2OPB=yAJ81C)Bpeg diff --git a/docs/images/chapters/circles/08ca09aacb271735e063e7e8d941a195.png b/docs/images/chapters/circles/08ca09aacb271735e063e7e8d941a195.png index 7de43a50bb839ed28f31f4d9ceba166cf58da22e..9b0228288b8f0651fb896cce7c691954031154e4 100644 GIT binary patch literal 11969 zcmeIYWmMGP7d|?Gh@dozG!ETT0+PZYA)!c0cY}m5G$IHnHKcThG}7JOAUSj^&44uA z!}s^{uK$~R-`p2>u~-b}^O&CWMWb#~vnv{K(+>^LxjduW7jHHcnC+o3oI5s;C*PdQwc#5Usi&KiaNBjHV^U?cD8k&2UB({(u z>H*2qYjl6x#fA+=poR?=K3D^NeXMG)$dy+uMZnk`f65{n6bg0Z!3+utx(B=ajP}kq zC`eXZ+hD^AmS&rM_a(hIS)XH-E#_)Q-lupt) zPQ;G~hld-rtZHfUk27#?Zn?SD7h;2OaR?m3LmtV>ecnY5PWg2!i^r!;*OTH=`LU`o zYy2cqEigk*Qy?RL5FBilWc-=XYc1&9Nh!>YukI%UbS!^5ru_Elh?YrciQLA<48|Ez)1|GR*Eb%r$^uwhQTUToFC~&+Zusg| zva06aYk4OA1RH;U21CN=& z?@uLxansYOY-w&*Qm+BT_=gG$? zNRX^=rWzt&5PNPIBx%|VZ(%&;V znK7&o;1HCtap16Z~{(6XRJpU4fT`z<0X^y9OJqi;e|mh-b$^Cvr&Lqo7OSu|++CaRTGzh!6Uxj9}mhc%wN zakpd~=f?0#>Q~&s)uzjV34>H`EA;h??ZFr$uM3IArje?mUHMK7ptF~W0*Ni~OIxQ~ z>BXjv!b~}0WK6A;QHBs|Yi9hac4^z)!s+>xY`18BhE_vkx4gq#CmGIKZ)X(R} zU*r914`0J?y_`NF{3qQuxc;|jdFuB6>F0S4}6(us4nMqK`2u{c@FS6Qh_HyK8Y8Fa=Z zP9t{FqvMEdZmxB=x^#j6(-J(@K~BiY@mm_*?7HmvTNi(RCN(gaaC?=TQ{7>YL%_0t zpaaru+=R$QE!|uWHe7g}zN&SsDu^2U7A`>Z;D!BHht(yTTi>qGoIghBRSe!AIww zVm^9XvD8iEB|(RH)N3IOZVM*dTGY0W7w*>g96iP+ zWeLq!GyJQn+3+++SwQzm^Yx^Q8WAH7L3MxcXY&;9=4pXBg9{lxTdR6jhs2G+!6z)$ z3syPOObWg`Hb^zOAQp)^n``%S<|a>}LM`Jx>M2EEBGz~B%sy-vlnneV0_Q%Vj}M{hqjFd{i_^^!^70!xd-sW&gbw_OWO~%zjpIGw)55b z&FSc568gJx6-Hw|aAw3-*_E1y&xCoL+Ko@8zrsL14o=GV@!6Uqv#ZB_!s#!t;G4j2 ztHF?+#Ty}84KK6QpUsbHlS<-X!3J(ryufz0Wh}djEfU-^7;EbAaRar;^p5gz2Qfhb zvAjb~jbQn^P4QdHq1TLxW6|pO-uuEbqyl9rpF=i=0*1u}ocf`NENuZN%k$8&M81nHZ?;+q^?JR_L{|u*EEddS8{j zy`9L_;_cFBESorb@3+2!x_Z=ePT}#zAIi}G{w~u+$XZm`;TEXJQ}b(ACn%Dyt^h@J zzEyU{`rg+=>u_MuGq~P!{O62SwNy7U7v9KJ$MKGgj2@2+eP)I-o?A-%6fY-bONq*U zigRnC-;*`*<%w(D)`yZ|jJw;=d3bcRwBRxLI_=?;7Mh!-!V5m?G-+q^iw%l$*Ad(5 zct+8>W~l~7#`}8ie>(mhAggAIWpUE!qQ(pL@ilbNl*Mr|?oSuL?kMIq9+;5<+f~Ri zeS>`5D@Q`B^vi`q$E)&u?bH0HssciXuiAE>kJi&cG>G5FB%Lmn4SVM+*lC^`Uo@4g zEYQphb!?l^NM!p}+(f?MH8t1EIcbU8fejSDT_SMZ_m~t#H#$Qn00Ke!aJA}+sX8;!@rp;!F6vJg{MX?76fc0 zv@dL?AyYaSCCzppM!xk}QsIZAhfSljEZN=Gxg`DYl@|&Z_XZ~1Tkd=cc;xR^UNqk1WwiPtv1mh!NqL9hco_5-#o;OC~u7$8^V zSDDXANV~Odm*9@DM)7-Uz4j~;lC~=_&Bg?}@my;ST|)7DC<;1-{MG;`imR=ow|FeBKB>c7@_HLgXHp}nQR2aOyFT`BLU6^*}DVp}T z9dRNI4$hbUAqQ78zT0TftymcTXF;g>9=2Xev7W0T`JXHje#-%*2B&sst2DM6&=rfx zw$&e33sa>WEEJ22xgQEz6-j8bPX+}J)}nG!-E@-fR_k|xLkOr8FTS)-AKom2bhW;+ z#uq({@7J$^p}Hu>SM6W#RKzahFZ0=ILe|TMHt_X4-pou@L9hi*#Mn~-JaWG0-#4Ku z%rXJp+)YEIzfQiD_j=eg?5aQ$(COb6XtdYww}DtyPej=5Xapzep_~oh)xN>Co}A`n zFk+EFM0^~7QLQZrzh2m0Xr%v9tmRE zLBV_K`+2b(Wy8M&u$^z{9r{@K3B2Jy-z)k!5ejiewqy9_F^}z0wQ%vKe6HX_PQV^f zVPEqb`dJc9RQ%kPhfy-Dv}b~BKkLz+{N8)z>0){#W{ImZ)9}wLJC^iaki&IHxZ=Q;hcZCP$tKF&asog_-2&<~W=CwfN`PP>zw%`Xi5#w5=5S90X zpZKmH`)z6TKyc&0cY8*_>z5@pQ*$PSW#ISEwPYW$3R`PnO$80WP<|J^ z`KvYvwPiqarwdqtLchdY>~4WQlFC{RKr8n*$|gPw&x@K_5xYtv7IVnwbY@jTHs;4hOq& z1Kg5f?|qYYzTl^sDeAg0q+}m3>)gzkmzPUXogGLavjxAQubB*s6_n#2vIoc3+>*mcS|0M7um~pU^-S!{~QNsJ$&_B9miymQcgC8Y@q6BG7%V*+R&(&EBmD z;oyr)%aogcicQ?g0~`Wh z4gCP`Yg+D6uxrE2g9j3xBK7z6lx~iv!0ffSdY7CQ2CCx^1zmezi9e+ruTUV{e6ZHSt`0LiLj@W2{w9Eo~%L!a0jKH>fw% zJGxQ*wsG#>KJ3b;u#%5A=ag=%xJMrpMrgZdc%%o`;*S3OnIyg=Z6vN^S5IQy zfF$P=Hr+0&7DL$py^zY5GP4`5z|;aE-#1uH9Jvz!#F+ z5=1HN8b*P{&ocSLM}$8rnjbAD+SWg^ZXlx-qZx?D@VcA@3ZN&UUPd6B_?tNst^fB> z-xCcRd*r`M>!-9#_}VM>g^NpC?y{cm@F;rb za?U@)vM=-lJNV7OicIbS2*`Yq?7FW2!NE~wxs zg2iQteS2N9E8o_qjh1nO?|>E%$eaDiPHBN@;RDv8x-Yw4;R!x)oMvj}CenIfKgm>-oe@U$-$up?sFmfIfM0s#pEWK3l1OXGzm z;r_1Z^g=W2Y2RT_17y~EEG8o6)Yxp|2>hANa^8g->h6I zE7m+z8VpQxFJC+EI3#tDyBni{khZrMRIwdhKv5lBe+{YAjf`ZQmqf;S+Au^$iA4=p zU_HZ|TXf;moYK&sqLeN0$_&xQVD5a^kx4k2b$xUM>b3H*ebeXGFTq@`r{A|u3l9Bw zQbO!IrlqeY5&S|dXoI;m#`0?ZNbQ438kv1BHPxkO5D)f>uOXW^bIQ-50|Z=wRPQBWlz8{GEDXpO!M&!-TO zEH|qvAYh4ZYfld26Gc*b=nP$>{j!Wghf&HcFYCphSR+mc8Ih*`N$9W+^nk$riQ+*- zv^`kW5v;o3^rb5loUUV1TRR5GR#ww4_tzan)jjPk^^4MA7M zhRx!N)$?!nZ8axMdZri#?1JIlJ4(^|Cs${gyYiBK^JtNg=yGyEqIYokTXhu<)U(OX zIVxIYHJe1o%YZ-uD$#;JcyK34lh@m#gk9DSV-jj~iSwg1 zO1W}fY1vwFiG0;E-2iw{n-uV&u+k(-Ct_uS#KIsFndN7>vI_J}`^;`7NE05OEHZ}Cp8oYF$p(=cdWOpL3+B0-BuUbdpE#5Nt zUK0ynGFu)T5b1bAV3LyE`Zu$ejRd0Z$4dje!;hS#r3-Xwc`)ujVjlHbU4Zls_|tl$ z6AHz*NnIi02(Vm=HIkky;=0rVkUo9o^P_b%%{ph6aPb?#{pMTm;AZ=&V%_lD{iGzy z9EJF{XqNXqVo%HhM6o_j>AE_k1+i4OF-yR%vJN65`?!d;x5$EqH&+v^i-pz!44YpH zKQ!X3T_b|jx$X~bjb{3s*7^O+?dPm79NCYLi|dYfCbcz|-G8|Gce3cseBCEyH(kSF zUAOg^fRq`M&PKGzm~0kyZ1RA>=-*ZR$fsv#%@)_2;^k(83FA#-0s?V_bgvIaWoW2w zQERd8MH zxz5yZd6;{!EwA8MGO+`eE|f&7DQxa?;VQ z!VQu<8Q&66L%6xQDaAa`&bFJbj^4~x*}n1i7K>ri$Wbr)&~Uj|-+hY&0K#4aN+}~V zGx+;=wj&|mbJQkjouGd7w5Z3wvy-;MLUup5cEh}GHrE5%wY7-l%}e2Vcy{(7)V?7k zrLHBb^puq6`w}Q%8K>V_9XbeH;YTdeEne6K|rJ3$=?Z)i>=JuhCLJG?l9Ax z>9SleyY4WmtgNhqjYRt=+Wjn^-WMlR`qy4RdAmF=EK}_Vek_G*$dhv&bs-x)s_jqr z=3h~i(DUh;gjZWlu+n_&GoHv-%MF=o^l)7yGfy(iZfaA_SM3$}xYR-vy1Bp`!DEWk zc<~BG|9iIDe)!^Kn`|}{3uGjH4Q>@SGe^-=Bnsw2(sG=O97_w3;Bc>|Gb-+u*~$<+lk$gW$GU+ ze6~Ighg3)6;&c%Sf?9O5=&1BJ6>gSbDShwY5=T>Hwm&R-cyn=~k#MSAK!B8pc#zV> zxmURYx_52WkQS~{D*hH6>=zzAISUcekhoO)NpT#Z`$8E_v>}@dEQQmW%=*-0V<1O? z>Nzh#k{n)3&=}AKcr23r=wvJU5)_2XsV{+5jU_vgUuvJa%L?oGPC`pq>|#4zI}UQG z6=`7?P{jPt!^<|hmwMIz_fD0)KsPO^laFs!)^xEXtbv(98jWbw`92g{WwVHP7jy1H zf~-p9-NE5u%K8^*;H$iEZT4W0hWP<{fITDiQ|$j@luB`u8_Af39*o|^W!FzWW}tY9 z6PzxZ#WX>cU8Qb75WE*4Xlb9p34&lc)*xgv3$ii7!^ltyMK0$!AOoLu4ulYKFk8|4 zzkoZ|PqEKUupRs8RMUHiIF9VS^jecUwj&b z&W5GS{d61iYHXU-Lw84kxjhBYH?XsZKyTt&J7d`$ z;+O1XsLeaY1-WOuq3t`_k(-<@TXXqi1Oky02t-lYxrqz3m#0$kJVOP_t3r+dMVNv! zP;&_*3v2gg)pp2(v=>SQa!Og%{Vz@o+wX(IBPg***V?7|pauOUoIwKe@{?Z-8;A)| zyDG=*|K(d*#xj6o^asFkHhB7&4Mbz8-8!muGc^&vfO2tUWo3Xf9tv%(ax2%m z%zNJr62;YI2-?;H)m^$75L7doAw7~AxXXH%B94{kqQhX4#>$lseBBx~t)^0=czB8m}YFv9zovZ^G)2%6-Y0@D0LnDTK(hjIe!c%jkcW}U0*4I!3_vNMa)%;g?kv!3X$*P*|3 zzws|@1?xqpdh>kO7r(I(lM^fw3pM}rpGKv_;t*)&$(V^RUKM>9D&SIBb#ujenC4OZ4z1liL$WF&ca|< zsFNwPi+T^&|BA8T(16bv%av1iw-{kC%lA}dTBq}m11SEUNf~S?ls^YMjBXEH-mO;j z4MM-?n?(;$?;Pc7-QXa>8S4@NBfqUK-j%0(g{ERK@bqk3e~?-W3T=*w!{TayP&e(b zR+G>ISmg^jzrljYodps%-Zs_RR0GB$M(``BY)=>QheeF%y%y4$TBcWyb9`+SM_sYXKxJRx}IScZnwr<*;@zpu#RM6`8uCn{4CGyeCCTmbLp zOeQECoCY0>#J7m~5*i!i{r)&`YhgB;Wo>s=~!Lo?m#nd+KXPyS-g*Tq;7R|dH^lj8#remG3PGIAz#MgJCO z2p;gUAD%FW>R!CLPpbEzdj9@I{!H;}n55}2^$!s;^|)8lV+VNh)=R)wSoy}cW*I#)To5s_BYBRWIEM*eqVJ>6?UI+KfRQX*sS z5IEYdkk9(wpIsNgsRc!M0Y)!eJi+XpiV;qPL`PSf_>G}h5gNtpm zePI@EOxd*ADt>_{cvM!v6JdAkRwqiMCW~w}h{^z*dAJs3R==sCUwJoc$M2t|PD~;< zHM{cAgyJ^V&@(uEo1w`>vVC)W4?gL?BhAF z3e?Szr*`uIk%cx_!np?$F!jAm>xrgD$Ye`ZOYDkS_NVY&c1i)Uki^c=spD`mseP9S zSm{Wi5h>`_xAh=^$1*EcgY5{rlUy8g3qbUp5#TY1q&E0s4^IBJdY#_cev6H;^N`R+ zd-A;)_l!UZ)z|me>pszWbUgTVc~Y($c28DxlukchU?FXw!qS9Y%a03SK|;luDO>5} z_S@=PW1W0?fFyT2u>_vV^%9P(Y6Xn)x5wFg&z}K($& zGAiuab~2Ty3;mt)!i0o2{dc{)~nxTe+US%%Z{Ci>>CKF;Vns@WqX(XM#B|04ZMxvz=xcG!%O$^ zK{5o_bZ?g=c5kLi)jIx1h$(|r-WPo+HNn<#44!#?k&_6>L!xuBGYCLbv)`n@#zjuR zzoFC5k@NWmJX}8NPxo{KjFyWu6k&a6R8H|PLwYu49=+)E^hhDuKqA(c`| zH-oANGBmI5BL5*hW)R2jY&^lAMVeSHKpDxJXjS6aHIZ!$e6F&!)mYWSAxM1b;!nmq zu-Dkz3()i6N4>ot2l<~n<+w~$Dxeu@hf^Qy)qkA-bkNIm2d-C{_7Cj$feT?8H&fh& z0<71^UWN_Tt4VFcy}N@H9QVV_Dd`s8|MC58$4lLSb&_P^#DenTx;~rkk=*dzZ?Gie zNp`2W^Q-fbrp8^XwWpXtQ5AN!l55}CXnEfYP<->PeFaLQW%1k3If}L#bWDIWQv|If zM;-lKfvG+(Yz3to{wn+UP3e8Pz0pAa5{{UZNXmOh=M=w^B=p=QD1MCSFmg_(W}6cb zU{3M%ZS4SYS5hly^t@o%nYYfVsHhHyccx2yJei1yG-}IKqC!5FBm7M~|L5q4`SPH> zgI#|JA>`CE#r1v~&@N}vw^zKuUYC(mb(J%bqUz;4N`PKO@GPwL9$cXgN_>8HyZ`8V zuYCMGyxKO=QtGk=Biru&V_>J4tCl!yDk2BI*b!{S&o#1xYGmX8j1MyGa5Wby`4Pf4 z0}*Sp*2@j3rMhj|t2JU`dei>dVtM+HT3R1Z=R17<#n<*<%{kd>&@-d~0(d;_vt@$Q zF|!UlUCV+b7$)rUqp=W=W}~!?4#WcJyYr(voU-Qdb@{W^J6f2CB|cpIg|Nz^CvO7^ zuc-qDwh1AjMX<8sP`mf+$j+9~h+LKKkrHn94}Ha=;X4%=Ou(d*nJt!;2NwRP&LyCv z$N~*MtN?c(zoB}%U_B|F1K<}#&(E84<#I!d z+nby~w-{EyOw*32Y1)l7H~$%O*)H1P{HLM!`Hm`J(5rIgn)4RknZ;yN|6%Li!S8&3 z+w4=^4a4lCbaa$;NX*vkp)r2uwXV7rESyKT$r{2_`e+&`u~v?9C5y9r6xK=%oPuy$G}eRah3NZ zA8(<#E?7Phl$JL>wX0dh1A*n~i$}5S+}rjm*f<1S!QXhSp^IKv{8m3OPqB#yykSrV zM3gZnsr@Vo5wT(alc#(=S0TZ7jShL9Zp4(SK6Z8zHj^PWwLr@Z2%}yiR_DD(sLjOw z&Rm7hv~~=f2r0A`Q(SV;9dxkronGXc54Z{~HbRyG^VC;yoX_tXHNt>cB1iAbXGx&fnG3EEOfKykS~CtlTG`{UR}Qp zvA7-O)hdnIp;sSCPfw|Caa;aC1?i-8FU5_;DCwI0y6UyHp&_VxCEgscU$s(rn@y;l9?9Q{j=7tw4o-l<|{v-xIM$y)H$9C zgh>I&pT^7NiIIJ6XHpT#4i0;InOpfxMh@_M%F61oe=y74ctQUKiwOM|sAB`l#?$GebK!@t7Uhzoqfm3K6}OMx)M7?`uoy%E~Tq^4$ZV?VQgX0+^F9g z-;05tbVkznF44~;LydfZ^Sh2`oRfT zOprirj1DlDf$hV!hLN^*`BqrCcYPr2m#qNpa=q2aSXrPfDvwn|}QPNXn zJ)BbLaI=t|`OL|^F5dB_`!QTob*_5bO_<8tn72I zg<;X^ttrrmb#rZ!mG$>xW2%!S1l~cnL%$3g8=L;jK`UL`{me=vbaVgzZx{Z5)2@8(BhrK9T9?KB R2YS*Wa?(n$Vo3wP{|ja0@K68% literal 12174 zcmcI~cQjmI|L$lJBO+QvX9y7?MDM}~AqXMr=+S%cBs$T85WNecg^V&<5WUyvMj3TR z@9iGn_ujkSwSM<^*Zar4$;y(o&$iFmdw=TlJmGJZ<%q!4U=RpI^jaRO1_EKl-u}CL z2N(%x7U%KYRzH`~@%)8M0@LO~)-M*{pzPDHc_@cROp+0}DSaEUasfoDl*cCC3B) zk1LA>fk0lW284ylzx^-6d#fnaxr{GaL>SHG>2eHc?7M24(__d0c~U|G`4H!pwzl@z z4{csk1FdCCvBDV5{f?)x;n#-0AJ<)~i&qzoJBCw%Q=%*YI2}xYm6DcT^d9cPcKxR-a~NtW&Ejr>7NE{C|1ff7yyxPNce&_^_<9viLB6Mn_42 zSB|w)!L3qjT zGlYn^et(Y;NOMae2uL75xGC2l^pPMV@9u72SxN3B1ou7i^jth?%a%dyrO^sGfL`zx zWqh;5{_b^@*EPgN@zz^wjJ!iaeEW#0E!Hnx;Vy)RcpRxS3IvYZNR}-kO_t%-UQfAVed3q`!$>3 zkA27D)RhjsKUB$_=KA%_w>vRPKH+UZSk=`o&JTU1k)IzbEXjTt*kkAUdLJy!WaH?0 zK2Gywy>NfY-rJTN5f`WUBPy|Dk2%H`hp0?;2PJGny45qc%^&;!$n5$u*KW7;e5%iJ_ou6I^JS z{#9W{6s{tMWzs1f&pK4acDRWr?jBdp;OrkP=BY4MBVVou~6 zkfpfnGqnJD)3YnHe&ao3GHL@q%U0tSQ0?;l1MH`-e8gOK>?G_O$Hm~TC1;3>iv{Fn zNwy*-87rlMO+hmhB98?@uAQ{8WwtbUa{D(;p*k7y zO~9qpGnBYK*R0yw`Zc*YAlg4rj)l{=a29|C=b%W6SW1+4%(HL$=#9ztO5#F+b2T`s z7@LfPzlErJ1gXTcqINv~fVG-vb*clx$+BIozg!=(RRwFY1d%hUxyr?)Y!nf41vn3U zoYiwe_4CfC?;r;CeC18kJz3vm*ue3d&kE|~3Umdl^k4Wloe7c^LGZ$GcecunT5%5s zPQID%JDHjC z)sBCt_YS|i;>~dlN&+w3j(`BQ;Dpwx+CUZ+_VMtjh~vpc|1DFdRmOs>_V)AZ%IH=H zoGgS2I~$Haee{B*K87T2|4)0ex4gs|uGwHnFkyf5)pE>@%WPxU;CHtA1f-}>ayu2T zd@_Gz>wIH(s!*J93|E+UBb(PYs-H#TlG~BN|IKy3pz~q<)-D^{43{Ep``XFH)kmKk zU9VHChW)Er_vifj9>4Wm54X(W3-6H@_X_=+KYp&+`Rj-|!VwynBU-Qzo+u4uf47($ z)91)rGb=qYoI6mYvRuSguXWf#lb7Kf{SBThIOwyk_n3XSGS^eN;gIy17Kl!i^ag`~ zmyqHQsj})Hce9}eR+Aq&Wq3!AIS>AV#R+NOT%(q&2IU(zSUt`ylO^azMr<@$zpjJZ zBabosqV^&8c;pJob!1hOxZovUEQT9Jr*<06;m!s=m7ZL5sTeYQ)c}vzj+J#LVw^;3 z@A_4lo3BchZ4-MU!+8eNUAOv(A}lR;x2sSqANKAk8F~Ks%E|55AdF+VXRIVDZ!uPa z&ugAC(J%?azqw@VOUX}F6qnXyv`fmgS2jgY4ps-j8g-CN*^~K%KF8nca>d2x#MWHA z1Gd|YCK-VS zzCe^|LG{BqzC3srP%EoSBDM=k^DgsT6ZDrDD>6Ym-K(#6MIKEeF^GPV*|8!s!K~U( z*zH8u7Y6HdGl!n1m(DQPumix0>l}~JHA-tea|l~a7N#svxSL{9%Em_t-^D=UH-*9^ zY*rI}eDqsm-Onn@wTuuDz`NwD_b-sca{cgbO#}X@^s!8tG%lWD+P`NbiUnj#%P})$ z^2JhEq*3>2v!)X@Q#hjet@{oW<@tn9^bTfW6`(uB9q#L~=m?g(;8=@;%o`~nC~Ox4 z<_pP}=MzEmd7Qv37Ba>T26#JjCl5ei* z+*{7pQs)9m`CT5qul87Ha_$&f`ZH9=Yx-fTOrFIwI-2&i(+OtR|Hd3+lp-|j_L>xg z7dE=O`c1Uw=0viqi#zEGVmJ1yWGdu(%b=}52}Sfy`V+c%g=LIv?zhjkZ|XEWpO!kJ zy1((X(WY{eEFWd8!%v-@HB9|#BzFluAeogL&BsMg&Z#FGFp;L)mS3Vrq9IAQ#na(b!vR0*Jwtx#)-$A@Bn> z?Vq?I#Da20#E%RS)FkGQcspDq8g`|fH>Jz<0Ru7N3tlCUq-^VoGh=9ImC5^|$i;K= zl86orxC??>_!ZU?Zg~?pS1vVP;o|rgEGmrmXv=)w5avqJaGZj+!c{Vwsxicxb--yj zdg!uccKZrNl~BQX#;s|NqHhVfIQY|LU8CfT0*wdJQpxPl5BZFt|a`%QZxtuic9GgAXxZZ8i|+dgKb2>M36ug_~>U(MO2Y)%LL z7kfDRa|~nX;Uy&D;C!TNjw~U3Fu4WOZPHKjgwzonE8p7{@H-I6aKr6ABfFBId^d*F zJfmAMW})@O{WG`Vq5_m%3Sv3Q02um9&Gr1{wt3z!dDfYE*PsVUCql6dss6_4eiSY4rrjeRU{cp0+lThMwd;`D)rT|V zVbEZ3Vd>Mw=5=WS*l6HvNl5E(uF;P-3Qwe3`*%WzbN42TObW`ND+y0&=pTb$-8X5g zI|@tpkb6}8CG%ZC&`cn4Yc$=;Kn4-d`PM_>M8h_m03i1H3w|x-cYAlan<=B4oF-Ln zf#kEN7dxTt%ZxGT&w?r!DIirLJaG(u{&cz%18q0;(P9K_c=EMB42T66Y%l`$V_k0i zkZAXwx%Y{S<*?MNn9Gbg6=goO0l4QC&8_1#1N#s1FbwN62yYTNZ7Ld1p9zK48fT%H0LH1(mm;Pf_bbH91D-!Qp; z*&`5P;Pb~tBq3H+g&FHehH}$>WShU4<=?}`@ac2k`3M?g{29ORrJbLcdy6R<1&_1_n+;zY+&m?thdAZLK1;@j2z=e1 z>8x@^bL?JUAaA_rb3g=%z88!2TYXsJI((GEMQ^#1@$;1Fj+N?y8~vL!wwRdmKm>Tr z3ofq_d#_x{$%TWZV%P}kcn4RuMDIASge;_FY8So0lYb(1Ed=E4Iro_|X!|fkz*ghq z(o3AbBR}unjH+mt{8l1o)HqsA+U;USdAQA4a;Y)RAhD$0+?#UO>#ecBl+>yju;b*) z(rhU1>JN@%27+O>qsi^5er0aWZMQ7kL*q=wx^7t<@IHI_E-})i1chh~|`JrWvndkl$Q zX}2tKa@}a1OoA7lKKK6#dksucpP)2osTypJSD6O(9+$%^7TrxJ_yxwXCjBY|t(Fsw zo%aF^Hr&=qet}2LR{_L>WvH$P=xr72!aQ$88iAm+Y8=`sCjyveHV~9R{z!1(_FZI& zNXO63_6$v!G%GVo7DI5X6rGqwnb=s$`xRW}d0UFu!s6rj57)bs#5<$?+bdn)zqe)6Z&RDpQN;Okf zCtQQ%H$`1d`2OoN;h7mG)a~7`i`dS)$;sa~nyWZ5cm}oJmL5^iWUrj0=d5Qn7 z@^z;*+9iEh4km|<1ZY^HYHB0wlEa+hJ{*5xm&%+wKwBSW9p4uO6(QT2cIu7VE` zfB(F^FU^JAGQtGy4M?QgUp&h{(6#@a%oBZeQX!sEQ1NuJw>J=IHo>4IF$~c0ZD2_%)ms9GjWSPv~`0@ZHyY zCNWGacUMSxv|zion)iCmWoT)TT`F4hEy(|>MEFFZJvZmtu$}vK?`NdY=?>LfEeiy> zoN55A{k4z9c#jJW`5t4r?KqLTV?68j<|wVc|C}3pbPh;(-rxr8K5vNtlW_3ax;kmM z;X8O7FW4j@pMYaY8GKD2c8{&;iOC~L0;1`r4hSWG=vg92JKqK*BnuGPHB$IU03QZT zXl0g?V7Pci%;Mj{HqdbWXMwOhP*k<& zR--$5?ab9PPw$3@`}iNpk`HD8%genD0K!9>Fi;T!rU`k4DE*{4)GR3Z%yX*2%_9wa zrL-o?R@Y)~8Zdp5Y|E0TRzm?hqq?m*vT%+{Wvmz8@G^;ZUUK?u$Mv7i-B&f3Hz#fZ z+p`2zR1RmWz(Es;2Fu-bMfWt}q#_YJHk!tXH8`dNw6*&R@^7kP;qqE_b)j_>dwV`N zsm^;h%d#_xTB{ow>*J#CtmXr>5iTy_32mI>!;M32vmwdpq8mO}@$JoLm1wmedv+>^ zDw4q~rfS|N_xn8xv-WcA#RLzbCvS+IX_kN4_7W?RT3+-E4%Ip5cQ|hzwE_Ua5#krGR@3b_mkAilI-QrsS@Yse!!B7HC2aqF0Fb-#3S~g< z^N~W|THB0B$%KcR$$U<;(F%|rU96l94o9ruP~NM*qEW3UjpNMix>)CE3=RVNt&TOOhIE6I1xp zv2QY1)n^@z?~%jiHIjLa>nG`XGH;C_4~M|La!NuyT@?LHu;wNUzxba(XYzy61-)|Y zr*W4%65q4P5d|ATlRX+D|XdA@$_yD$^6IuT|>_!1$%P+0x z8vaf)b#pQ8MnQtS1_`@#J+t7qwF15qZ25auc5{3=Op+#A>0|x33mnrcFQKhx&Xo0G zIP7A29c#<-I*NZQe!D$3TDt}fA0`SVx3;A_4T!Wgc)ALo;CG!L4P^%cr>2~>>rmeC zFzU&rRXX}b>%bM(PouzZt)z|*f%cw zk9<|OL$q`avYhAx_6Q~>?h(ypvT;8dFF@d!UbtDz&$>}8If9Apd#u$48^T?A-0{Q8 zq}pEuhq}|*dQkrnb;ToX1dyP(*0$x%iN`nIr5~Z?J57PFq5un~&nvB$Mo+IWOj{!& zDk7?2Qd;N*;=*(@1bFWe%=0d&sEAhFX~>PjXcfZ|+xG^rhJ;obh5Q1wp>w261c zpM$v=_h4dp9-FZ~GgU^9qAr(~J+gXtwLVN%FfzA$fMu7_D&p~AbF2~;T$>Ev>?tCj zC^9M-?Tt%MryGeSEVLRgR#jJbXjKxsa z%ZA?QCqe2~4

iPvlSorR! zD2D?IM(Nag1Sr>Lbnbe+3o)JiOwIAni2TYW+Rf^NflHd3)4V%dg}+g(Ei5P?^%*MaH+bT|vCZ0p;Q(McxKfbpiFecc60}Rq0a!@$ecXCq7l9l2n Gq5lKT>ee>^ diff --git a/docs/images/chapters/curvature/392624cedf7c78aed6d4c6065a014b42.png b/docs/images/chapters/curvature/392624cedf7c78aed6d4c6065a014b42.png index 2f2eead74d87614b62e1a8c4b509e45ee3bd4f7e..9e36ed1162088f0da463af327ad47d02d1e219da 100644 GIT binary patch literal 105439 zcmXt91yEbT)}}y{uiPw22;z`tO241WiIeEA_KB@XfN@0Hh8 zngjts0wE(VqUHfS>+)=4)YQKCTWO{-Z==!UwrOHsO6@YkMMflCq)N;K8%cz5(M(R} zq>K{yN#N5&06DCII4YB#xWs4Wz(f)fGH6IrVcwaGZvAtE@v{mu^_W`_*HD!XJ)cR~ z*;)I-s7J230EhW_g|cx=<%ZYh-nZ=BV5m&+@0H*mhNri?(#KJ!S%^ycVx$EKx@xiG>Gr8 zr__y~FNbh4wlx-{n-h8h`-d?3Z;VWO2l|9R-P#K?Zk$g(XxqO3$ur(9-xt0`@5`eP zE|V`2!7Hy#2Tmf@fLz`4SBJ-*UP|Vl#+#wrgi3@-^{QV!e9k@0%{%{^qu@Cbd?>*@ z<%95X10vu$`(Y+5I@uK_`>M`7P;k14d>$IFCfxf9i9Xx=ke@9#x*eE<8g~T-cGWN1 zBn1DWKpeh4&c*)wZTWg(*@G@@Imqen-ucw2D08SRMe7@C1GJM00d^v%T15lmNqiHD z&G5SndLLoJ7yGPtrPus>{D!9LJj=x?+d7eZ7F^v44Xz=`4C1&Awxs^*bKPw$XhD2|BFWM3F# zf>u9r{T(+DKX+=8I2)*x1k589m)>tM>_nVKbx8|eG7)%2IGNJ_X@S9e%hc%pQ)S8; zwOD8bS4@2&(5YGN}m6fgiTQv~BNR%JxdG{1fxLLpV`y$>F^>-eE5$Gg%}k^(e< z3CxT(wy_Rf8&DwqW~n)J)h{}7%p}TPfQc^^uG+bU79(N~IB`)!hI!Wq1TZXMFK~`{ zHmN%;AjT2|xOYNzKX7=gpNjXw_5bJoc~_Th^E&hYCY=a6b_2dzRn&~XhlIxxMbD7I zDyKVkE-Js^+$!|UBjE%+tBJnGvsPv9;BF+Aic|I&FMC_Gi&ZW~thQKYf?o|}dK|XJ z?O;V*W>Z(2Hke^-n*@|&>dnaR&CbHTlPBs9H-5S+-R`;!DHS3#`0!fl_Wd{iTIjHL zuU{sTa@mh8wZK;uAA^Ees;IflutaL$ZZ!p;AB{${EY0IsBw)E_!8$52sVQHD`NXm> zG#*&@Y26f#SUR(z_B0|9Jw{k+ad$v&OVe3TF$oA)eIzfu5}^7BQKRmEcM}3c* zr1CWjg#@VHajt&i4^{@IKHUw-Nsmyy(Ym@ih`gKX)8JR#f1}_hA!yeQm&<|M9+ur3 zsy>SU)OANEQGhqx(V)+0DOUWY9+~i{S>nx6Pd}?LXDe-Kw*3!pl@9`qOi!&90}Dhv9!=P4SWEi;-^^CuW~oosi5a~5cF`^s%JMx zb0|6iiG6u{ky7`7=zBaq$8uLKt-Y~?R|j1E#c9a@834bwl*2MxL{5GVp>nd?S#E7t0hcCiCV_tYD5kqtPsQ8+i^%K7Cl^h_-Tx4TPG>rH6oqlf4F;AQ>8 z51d*zbnuy9hBT#fo}fE~_AYlw{1neU{*!X;Np@-5-Zm@s9?p=<=%VDJT=>*evj}<5 z!!`Iy*5OT;5swB0Qn7u3R+5Lwa>W%#6C$a641W$xJpL#?4ioY)+b~fyB_Yc>lctOi zjm;8m$NzXQBD>{306YxCYTLPKXGU0@#bgL6=ksQ;dSj?j?7C)>?s5#?+3F859kV(w zg;eyPT_As~gRHqvG%5#8%QuCsfMcM+o`UBA*qh8G+~BeQ(_ zb|Xf(ClbT;EwE!xCG?^@djmbkD>uuPm#|{~L_2G<->lOOU+}N?TnyJv=hNU-a;d{d zZs=c7rf>JZxzNU|eOR~O`Isg~dq?ewg8D1a3{LB39&Vk^{08Q@1s%{0$S9v_&?Kd# z%AQzwv5riK;ZP(&Vh8#-xKUby@Z`qvjJ;FoW#8V@wdO;I@^s2)a6l|sAD?&S%&y`?3_e) zoyw4(y##iIo}fHNRlf+3BShT?7@%ApTt@1T_kpjSCM6!94|9qtAfeeKJWIy5IC(pWAI#~`vCcHA#LeaetOWnzF_?O2rY*A#xFBF4#J!`pQy%Id;o-l{2sZq< z$uqBaWD!8R^hD8fM$&Wq{5r#&KOsEcSXYOt*8PUmwwL&~+r=Ebcdl8aoZmN2)sVSm zQT*I!$7I*kRJcDvV|5d%r!R$?% zAE_}q#u(X74ViGn4g$*)RdPuMim}92iRh_bdrbdx@~h95%7c-)JlOWFcXH2r)6L#$l|BP!W=tG{_Wik$^sT-=G{%=+vcX zn__EQR97)5HzPiGKTxAVSV&C1o`YUYrJc&7d6Ic%!=ybAfSbN1(`u8-i5lFh_nUCa zedOJTv{Fh>wqdSH-}vz_@mtG(!wp*!@>0)({@oNP>%^ zofH7enW^R-?ipqvzN<05r+x!C8NWBm6&V@r-NuC%ed=u*5H zlZ_?id6Ue;TyIn5g1CoTe`5P-!0@PGPf%N6?QA8X0H`e<8E-(%Xe&=!slZ|$@hzrc zWVC^82Pg08^tp}TLZ0gaRN7Xu(~4p*;6pSGpQY}BK5pOw94XIulG5gv5i<6;5$9$c zf`!Skk0sC!d)hVfIq{wRf?zB`LPu7uFOtC{ud+iQ@tW~4c+>tteFHUfGuLTwAb?l; z+rv6+6kCXj>d;9h(20BsTw%{g8q#2UrP0fhPaMIOMb5;3F@(WUeAzN8b5i(J&i zS$?LmCL~MeH=vb0dsVLHhT~TIB<0i1A&fv$bCeyv;G3n#rT4a_V$fm_QHwqCzvbpc zZipV$?1r86y~s2FPZ=!v9tag3=xoTr!!3(JP1`-a#nSqDc*;_?(s_Ja<nS_SJ09O?lq~t0q zOLV2oGhpzL8yx8oz≺k3mb30>;7MPPD-Pm)F#+R#w})+*O$U&*T1?zX!OJQ|jK} z{=gjJU!gwCHfJXK7m*h7gg?EukdUdN;WEn7%u6MzV;Z7AejWKc zfPFef1YJL=QFi3mmBJ)%F;rIl9!_Cq&ezceRwT#Lm1d@;StSlwgG|Ws=E!?1YU8DV z%{+zG{5gCmzwp#S-o?dRXQ42!l>xe}SO#zlJ=ZG6&z#EFQ^Ng1S-+wU)U9GpRITiF z&Xs#=_s6@F{}RU#GU)@k@xWQ;wpHrhzmVzUeRzR)cLojtqK4=z!}qTSH;E#X1K^_b zV=G+O;k9NqK8qNYu*)mMTxWWklOw4{qowkh*CP)Ni*=?8`Hh!=dGv45(10|x!m+bGKJQfJ^2L`g`ua%0afD`V4Qa!7rg1I~IBZgD%W3=UR+tj>(6Dmus%t29~73zVC z1zsi->Ws%L>u-ip&QbRgP7sHt%#%6)29D0BLCu~_gifI4jYNaH)`I~^27NP@HRh&9iatR9e(p{S8 zd!f1j?V1y^l%h>9!I-(YG$}ipk>Dg8TAVn6qxhDF-A#H_$5h-P z*h2(X*~jXq{aQcbY1|&fgq(lui8(PW!4_x{-yvB80_Cx)mYG(7DE z_S(AqbekIg%4y8W;)j@xmG=Xp2i3p57Zi2~^*k!bB_ur}@(H=r_REPUn%;M^BH!wN z5z!Dh+gg7t$XoD4<5c+Sf)GMohKVGC)E*(%<@9pTT{yLH&wZ{PKIF9Gj2bIztCS<_ z+OgL6576joq6x<4`ajOd(Y*#{;@fB}hwD>8s(kQ|X1yC** zvK-=w3BlNqcR>GXAZ%N418J52>A0(|oUVr`mfD~DXKh-|D^JbZxR=Aufrua!LKITa zi}^e9;h`btf8in`2K7V#wRdnljr!8uMJ<(ib7B&HvDBsc%U%^gQx2WAVhdv%I`qk; zzP^UX7a@xnT<%42#~kmkRY*l-i2%;_!UW~Ak zQD%APNd+5C3P=-GccYGa%C-gL88m#qzCW|fxi2FVAHts|?TLQ{_Xlbp7lq(J7Yn-7 zN3nl~T5_j;{S$H(t^n&SvberS&bd^vFII#e=1F(txS1&LiF-6|3Cu+WG1x z?SEJgDnx8Lk8SY;Y0Y0Dw%50~G6US&XpVV1v1nHEr#KX=GnQdmDXchBy5u!CAX9U0 zL&@n3YGV~7x|zO{fZ$HhaK580^j+&&*-}v|RM*VTon(b9&2f%A*hz$|nj3^G-3G@K zCV`iQ6kc(|bwm{wl;?_p$7l4uP>>rtC>C}v1I{sAe%U96{?dr~_@8fQue#lTZ@x0} zTO#m3@|S7uX%0F!dyz-R1XaSt(dpAo`yT+@d%vK95wo;-?u+$|xez|5E#93Pm{DN( z638Ev1k;o_gC3P3BhD80C^MEZw|6AS<4LPOj*17GHZG=5n!mv0ol2++Fjys>bLAGv zE)X=AdF-d1->tfcLXW{zp3b;1-hD>Xo6_{4lAWGjDOku!WN%(bj?XvHohy@t;+-w- z0A^1KY7|v|788xt0l}LN@8k!~9=oU!fP!*BC?PNYo>;d%w^+QBuMr}O`@JIjPlj*& z?;Ib<-k~G8bqXhT$$cXs1GI&&1#dI{$JYqwIkoKzRRAA{gH=HSuS&}J8h;3GmOZum5_F+j43-LcF1I@( zPwIYLgo7re?w0(;hMyI~Gde2!jvk^X5`#`fl~mcy1%}*9<<||24tIYb`;z~doc%@6 z#cFc859j=-hO?gC;4@RmnS;*Er$!bsN~pB;6j^nuR8F60n&v=3ZQu|w^le9#Xj)6s zy_Xy><$l`%FymCFhN-=?)g}w8zN2JG6A`zOIL*Jq<|7jKyh>-|=hHXGCED_Ck(Oi_ zvvkvxzKjngNjzQXI!q1xGyIE*F6cW&sqdWmQ$z5xeTY2xf6KK}(L#b|&o%Jd2Z-~| z!z)bpT>|BTSJk@*`GLd|UzJ@x2qUquv2k&6d8}as>381_>{(t$;(BlWG6AJx%{l^n zNkPtB8v?DphW-T7+n>FS7zR`=khH2hW}lH#bsgf4(L zUwAaSP1+ttLTWH)YT}cw0n?&~NZn1h%88<0&i?(WueYZM z-?h`#S`%su+b>|$^&^0<8JST#8_VYv>gwbDvz}5IwgGgKSopWXYK#3=&^^WX^%PfR zQ=~S2hmr13w4l>70hIejaPY>Ly^V)$Y5F&Rbb;Y%ok!dDKsP+-5w2V$qWI5!F>cvu z#AqX#N1n@giaJ>c-DhNOJ0n%q7hby_(xZ&KVM{r&DL(}Ox=(f)*218uM)`5xPswQ- zFrq;cb)C`iHaN3B=dpZQ&UU~2MN~TX)F{9hBXX-X*-n}dHP84%%K8jrOvg0~pt<^G zB#?g+yltuTyKjL#*(mf?mmg~}U%(88xc^7J$mCwY_6u`=ihXc-|J96CUvXAEgCwOZ zR}C{MWLb(3DstwL2Q2Q4x~H#-c)6WaX&vonHuu~yb91A5(yb=l;M_NiiQ3nj|58@0 zENwXjY2l(lGskA#H6!I16r}`Tef^KE=$P+Erb}5onK-z$+ob>`dnIe+)U*jeS)`d7 za-+}feYmJf{SyjLqN}E{T_Pe{I)y%OG#P9@s1WeS+Cf?9&0A z!FfFDZs)$^`q^N=J`shxEYhxtra0@8 zoc;N(vVgrDV@-Nk(X|tnn1upVRKXD^;u{v9cS2Fb2$K>qGn)~(f^0z)`r6b+CAfmd*S$CKgw9+YDgaC$PH}mMmoWLT{Hgv%Py%><$2zLLu7>Q`7>)8 z&lP-g3G-V;0Wu6BzU}+V28WdW>2SKnczqqW^B&KA&#NkM)A#Ma*ozgF8=0V6Oj-u4 zN00=xv{P{`)2(KTm4A%C(ehAANTt|XrHGEm?4}%BS?U4lOazf62x*@_s~zq{b@Qs;I5lFo1B{-yN+hG>3|_t8E_0zo-L$Bw5+ zU>0?2-H-nD%=GH1dW*LG)kr!zK=&(Orl)Q)0L6iyP&}_6c1lfhtkSpIT_U)NR8VI= zgYUcdftD7Kv~ThaePW3QOcVdF#Qa7T@8AM7sC4yy?c4Qo>btOVe}#8)CI;hAtB}OP zZ|meB_P=`SNwcRX2ZilaFeOB#RB~YG0S_pU0l2q|J4v#S;9^pOJ7;MwC=LbTLPrIs zA!nTuB6G>bU5iQ`A<-9#xfZK=v#hmR!03}5$z*kFm9EyHS-@4zSBs`r*5^d|8DYgElQ*5+#oe;#&Y)`|6z2wxxem9~x6;|0As&1M|Sf5Yb+kh zS66{pHJY8agnh$kgi;cKP4pmmNx9VKAeJU6D<`YfFE^a_9yW-HyWAAD?+Pcx9*cSA zWKSoD+vA#jH;AvVala|E95Hkfu8UeEX@xi4X~L1GY8zRh2Wb;XLHsr9B%paSErsah zcQ#q%Db1f|57~v-;HP~n0Lu(lW~E!^erEsOyg0CWC@TueD#HUcW^7j~~Krc9&YsHQc&XOWo$(iOy zj9H)JB|El0ZUDY}s>_&Eo+lR%E)Nco@Ce@^^a$^l*za!o{@tQ-bP;-;JvN9fysn5YxKX8$qumXoR57IY^ zk!#t2FES!d$#gq~^z&qxjF!iys>#5j;&}?G79>2KdE8W9px@DN92WrQUJ0i%!)oRn z#Vrv>;erj*iIWbM#TGPsi%i{5?S&R;1po`8rLKVxSFp*~!0YG-I9qi-1`h6VC>EFQ zGM;-v)XCpB`azyOT;DhmM@SL3x%`4wU_S2Fi1{x1Hhpp4_txtQ;h8-dY^;V`XI*>m zm8rOYGkUGxF&n36W_3eCz0n*u6c|@oA1oFsdadoW5VZ)|<+8WO57c>_5p=VFwH$zIH*RoZyP$XZE+2**4G!sY0ER2ix@T+cmKwEY zD9T*xNe8g{hjOJYeU;U@@zhd&wUy4`SIFJ3u`)eGQ=^mo;<_Ps>=W2Cv)ul|_D^4i zhDdpS-s) z!yg=Ug-@ZDT3kx5wti_lz!q2TP}@l!ax(qt_w;kaafS0sb6p!V`^K%tQ}t$I%i2E# ziXsM`fEvf1(!^%0uG+x~s>G-V0eF52JLdX2Acg~+`l^_$iK4pbS7n%SWP3yD33)b8rJ7rtELy^;XfPU7@xVU$L(r5HK7f`!z zEf+5*#)#1m$iw*_nK#AFmnmXvKfH}cz*Cmu@7c*9Yy;$bh;b`fPF8g?PNsNF_^4yw z+1f_shF_%gnWfSvnN)nTrBxZ%V+$>UH;MWRJ%h0=^&F+7b^?3MaUp|VL9HKlfz19y zhwcV^N9F&xJ4EPKe>rKu4CbIDPdpWJijwrt@(~%=H8u<*C>q~Y8cqzVw^>@A$i(&W zxrd`l+oInV>IocV4KFpJg-H+lFPeE{oKdm zA}qgaikD_`ap%m)IDz2jVo%Q1t>*5!^Pw19_vp4^Bhoz%heQl+&yfD7TJx+W12rWiNAT}=JQc?MFPq)Sy#$UXB^M!xlu}X3` z+dg!$3pq3lMga4_HNaCOTx#ZP!>(20t|okU?Om?0mE}_doNt3;MXGdRo!kE@cWc*F!W4@$=H|0Vx>F zjr?4if02!PFG=&>Odg3m_B3WN53RgY$CXMdL32E=4J)86Ra%F9{;aVbF}t8Wb5lEM#+Ei<#S5+;&RK;x33# zi{H=903Xcu{2b!jjJ^&1Ko%4GUpvGjW&PFeEi|4Wd1*dS3EzlTC72>?GyqrF0BZhXeugg24Gml%BWY`hdo@Yh$-nOp(_ac z$g1JbyGD|PAVZV$a(dx6#R;GDYi+>9Abfu&3g^u32bT_G<3yk&)ZvFGVT{moA&k(% zyGq=k|1ba!Y+oitMTk0&uXIyj z;=`WfZ1QY8P#&GA82&T$WF*nyGm4Uv9wCD@MqZU>JYg{y5O~3LAk!?|v!54`mSM|Y zLGXx+tu%1z$!h3MrLu@w{ot{a;ChLU&Sayt0@$MM@T8qkWfzoGTX85D0MaTJ_)BKz zWV;+se&xC1giZuafRm|bosZJu4gs3A?jZ&AZowCx8vZnsPX)F=OdCC}meMI;z|Na8 zZ~=VHvU|;Ub3Z8ATDK4Je?fdrL)hkt`)Q@rLAPzukZ)@q8{r-5r!Q})uU);8{*u`+ z3i-xu5fYp8pb?XiW@A>#V873O>~xv^P9Wo$pGz2j0Mg~n46S0r$6_r-aI>U96aqZ>SHcDkGWmKqKeVPj_Y$KI)s1{)yC zc~Bky@6J2A2_)?`-BNO{K%*d@&o^@)BGi|k zhhD(JQtxxfWF4N7*BNcQ|I**#7A%ax#>SpJ=_Nk71~YQ)ZxYOVxR)>FcJGZnlqlp7 zI@MU!8d1BTIOpp@Y*hvHK>v5>iSxfoiW_2n$8N)_q$1j~x!=pG%G$)>mPb_9Kgrpb z-W5lKC5^3n(Iji~?j?yEs z%ZxcETLSRhr4=v8bC;lmOJ{j>?L+>V9}htI?^n4A=K{I=rGl6fTdfvdZ}0!HrhsiL zZ$>9|DDYT{&(XFc7Mw<&)5a#Ec3f%vl2n-BVIL$avL{dLX*B6#b~wf{xJo<6xzQKT zyU3u0X|NT}UjNpt113;>8aIn;+nqXYQ*#baVjABknAB|26|CMTvOH*PN=Ki=s5Z^r zH;YqyW|%nSt|@RGB~O<>F2=+%_co~kmE(p>I;SEJLF=M>S^n;mkL%oF^%va^iQG>j zD)jbd0Kf$QLfg@5$vl0ycdqv5p;6X&G}? z2R5PuZr$zoO@rWujc6P^0vm1lZX-*|OuxA!C*_DQT{-_|aWPT0nQ%N+(*b0s0IQso zn4~)u!v+@1>p=S8W$+wvHjA%Vh=slEAn@=l{_-;8M`7%tl(IO6gt^PlgoFg*6Otet zW*M~ieZ%*Vg@)P!;dtWH?OEZsWbjOOX zH7Z5zL6^^>(rb~(kW*&UdP$gzL2E@7$8PPTi?VSrvF#??iDvMdvuWJ&fthe{NBywY zI;Z;i@F{4E4#P7%+GpYuxdJHDn3zYnk%>7O0;QhtgkF%wAgGeOzcYWMeWI{yXF7L08{ zUF@Zpu@)epLfwKLvV*Z~Ax{HEW_63tNeIy-%S&viW)YJAvnMuz&fLjYWg08_*&r`+ z3d>>3`6*Zio~g2syUJ-5B3muxAG2M-WJJf+C`dF30yMRf5<&;TQ;1$hB^+8=>#G{qnzyxzBM)xr_>depGyr>3l}qZhHSZA&%*q&85tJH z@h;oY4lzHFnA%PLpOWaXEw4Oo8f-Z5XkdcfXV1rua{c3qA&^~l&>_)0`nAr`C*TaJWaXgqH_wlhVe@lLlxapY%nUvY9}X+oPqHu#qm4Ai$Kgn zzK9KMy8sq|kd9RG-A!T7j;)-gj;&-D_tHBE(7E&CUw{U!Wh0Ji33X#8Q5^+Am!p*-G$s9Gjpca%se8D2(0Hio1 zIj3lU`_1-MS{JqCyqgC(f zv=yxKJl6DD90)YTC9=$N_VMsaU3xb0L7(=qK<6;HumO&9mawtPk$-$=eJhAv&r{Tj zW;*%g+T>i01ww#DippaH9uak` zpd>9WnBt?f-GY$_%dhE}=aByKi9t>;@W|letH=ed0P#6RDVZXvqNG8^$E7(u4jF&S zx;+?4+2K|C;nG!hTv%!_6^lKX+tjkYhL==%dS1ptaqP`x`dpK?sKe9Tyakr1?2=ny z>$lN5{bAijf^(5p=qj6b557)V(X;t&UToY3vx0Dzdz9B~r|ul^AiH7X`pZ@Ck}x>v zK+{G2$(Qt#=FhA`qa~FeLTg3zr!S5ZFOzp2js&CvR9}jKVHR7d1|5E~I>+7e1xlVQ z6-YX^j&h^X8#M%q>(qeaF%o(_ugtLMZ(^*Eg5{1HcC*B-7`D~k*KLh|HRoZpwQTt3 z-fsjv$dR7*4T=88BZZ=ek&F%26G?W&$AM$?#mg0uhmRF&tWN|CSLf(N-P8=&p*iT3SLmfRl$j#1|*6%L11ney}O%?O4a3kfG^t#;0U6e9DW3xMIOmPwt zvbsvRfmO$-(Zcr17@02G2g%KM&eI}2vD95|fnUonBU<{-7KvUa`y_`#DS7()`yu{Z zY$$l#pxU)6d~46%c*3IUzH)z(yJZJ~K81J6J40v+mbYbI<@Fd*1{x<$jf#gR3R=&h z6^oBCt`6z7wpBYh)~5t#sRW?B`I#&}m zYZ<8S%)rUgSr}Hl5}*DAwrqXMiI3!;)s|}~JGT0z zLo1u9_c(UlL~FRX_bN;&Wyr?)b*4lJkSaYp!GMXcO_^-K!7bqf?g;lnOc||&YtLnE zT-0qWw(IU%;li4BR8{N>Sj#iYV@Ek-ORe3aO>MvQjYKcV)YmkZhPd46yx=)Wyc~<~ z@)~pyko!W1y7k|XS^#0j#b)jrod?pMj-U!Q7%mOxq3`T0^q|Lo$&Mk2@550|yqO!5 z8}%mwuWb*Zu3~ck@+7Os`&<*_yi901%n*rfKEtOhqo4V&?Lwp@-f@j+jqAbOMB`s% z9@KWb=o+zOI<~k3CXaP=TmPX)EH_0HjZMm$@{k>X`n_%6?F7?Gm}4A5Swl>f$7o)O z7gFJ*ufS71em1Kack{!qV!+|$(Bp=FFTQ`oyUji82SQX#!1m@F2k_(luS2h|_`*L{ zi(g);8csXn7pi5Y&RW7Ma>un9G^Cu`=Dm5~sj)r@>Z%dU6d{JkI;1Z$Hbcp|xv zhigmF@N0@hw@XlKd$|xjuJ_owf)NO4jNuOP@51lyxRPdixcu&QX;AatwYfi%`%008 zat`y|ah4+Gmt7vmgKn_hP;43PsKN*Irm>%8#-z_Jgl6}vrLbgc>C@Gr{lNpd^XoI_ zo^-J?|O32uGs=ki#O zBAQ%Mk`@V#9%Q&gU^__bSC;JEHV`7CU){x_l)?O48(z4uN)NLVRmc6gaFu8&%Fd{M z=XcThpB04p{T5BsM6_2|1WhN`hB~8|dD>7$`hbwbXTf((#3w|PB zHT?so-8;o)n9tI7L&(vDl~yUh&LD52W0q(jb7`RVSPGbF%&$kv1R33trT>EC6oN~A z=)DRFPfU@f5%G~l&qh~?ngz=b0#m$*(-X-D@A0JiL2Om(nL{CUKCKM8kKvKC9lDj= zkHT+07R0HG-=A(i`6zKhfLYmgPx$u>jk30Qbe6`q=V;YEs1DdlZ-w8xqqy)?HsB!yppyR`}g&)=k+X; zB<&Jh;zfGEg|_zk=mL5Rlu5tE zFxL5~UGhO^e_n27Lomkf?Gxl#hr<(9JJJ2_IAnS&!QzgOD1eN0F%4ZwMWoRydiqtl6Gya9ZAJ)sgZ;F!F zVT+8ECrL4A`Sklu#^F_eeQzDvb6RkAxw^4fie6cw@%7^v|_SL73W4+pbXE%gxu?wc^6A8-#$;p4Ggo@|GGanO?Fu;$T^%oRJeH~zt#EtO$2h~B{eN+MVwMc z#NnkEsshJbf+nDTk;TE3}4^bG5Vj`YcjURt3C&0X#ye1MG zOo}ySDfQ|TuY<7i-7qWb%ziM(BBp3^6G<8U&OVy9HkPiAx$@9MUx6aP+yGXolL8bg zKM5S4nt8KJ{EW>U-##EGYNZ20sMN6qGm0XvjZ5ov>KXnE=-#^aB9HwUiAzUMZBlRi zT7s}KxA~@4P8ZX36WxiK&H-~LbC0qgz!f`EQP0bb%i_T|WBnwcUT@k%zq(<-bF5sZ z*q%tDNHe4(ax=~Mj(07M0jp-XJ|1b%+nPQ~utet3c&2Cn+I%D+S z968vAhio!Du;#nPNfYZ}fW$qY^!JD4JGZ1t&a(Xev2Lvo~@I;j` zfy7hO$4I>RVBpnZ6%u_2krc|`>fYy-F}gfPIk{)^STd=)sW9h#EG-n06FQ{@}6sc8Fsc$&_>m2LA&()ngGJYC5jqMT5DvT`RP(Z zI_4GR)D!}+&{TZdTJj_)*Q{Sn1{z4%r6Ba3D8VFPMng-!ho2vE1iqj99sMlt(w>U2 zLU6KLsLjntt_%1s9k&QJC$pj2j#}m>dmNi+gO42~sn`KI@1n#!B=Ys|VOWL9>p<^3;d%LK!&1eUID65% zxe0&SBeL|Sg4j>mgcg>s6#hqvB(>S=^9tOPiUzq7s&z5w(MQmC5XRAdUFxCsqr{kH zK#g|=HbdVgil_qP7ZpB1mUW_O``mFG(=V46*JnJ zz3O46>E7rKvHjhLH$xuV2lJAyu+$N_&*N#kowtPlIjkD{A9&<=WET-@#}xv6dPy!N zCO^j*7lEoiq0ldbmR-AD**QqDa_Wo(N}5>qoNtW!@f>Wuxb#CKb63<=wXXyx-bWhn zcxs4C1udO;5%R>qofK>ZbEu{j7I94^YGgFsc2}G_y+#5z2FE&{52NEr#q6xID3CjH zYc$l$^mNxavCFN-3kxn>%tCDPP5h?Y%GmG0Ud+G&3F73ZqXiMz*3zap*pYvmKqI`A zWC{AZYCs(1+#r8IvVwTsbT<;ADC~;OZZ8x@oQxFgiplS=>j;O1Q+ooBE>$5fpD0zL zJDC@}7cE%_gk6qMh3!m)C?FX(S}ewP-kv$fKb_M~nal^I9`fJYU zN7?9Dt|x98u+^tP>v7Xh4`j;C#6=7|Xwc7+rP44KU~}c4GAHAqLD#fxVeO$9Vlsz~ zUB-5?jf?HPpswP12`fupI5=^P-9Yvxjzc1?)3V3;Yv>>DJ^lG=^#6$fYseg!b$RKD ztMpgvxZPmm+143Zl&N#3f}c~w`KRZmi!Ea^9iLK^jv*kAO?P>POQ0a}=w&7%O*oEn z{gV}~UjROei}k^oYZNo5L-&z5$LzFLjw=Y?a^Q%|VvaqIQ^%%o7!;3r@vFBGZG%FB zzG=AXomuw(GCGc*dsCy%=YYB&XLfn=`>+y%_{j70wcRB9Y&nZ=4t7~9u*NOH` zjPt&(-@B!YBGmtT0aA@N-sy#y(#X|NFbL9hb}ZA*gZX#k{I7A-b2zlg?m7Kmqb3>d zbk?=l0nRl&yEwIfGpMU6Xe-gDoNQHZ<(YGw3KfxzWt{Uj`$0=H(IE=?{>68W1TcE@ zxz?Ls7IhX9W^{=I-RJvu@Ap_MqHD_<}c?QmRuUj zC30%otjZiLxqHMGQrjPP)*rHU%fP#M>DM>$*p1YbOp?Xy`iCTnjQ~ZTn+6k>C`UXK zXYo|iR2TP35Z=R)v5!uOcFb=F?!{n@p&`BbdSypEL`m5XgK~LT zkM@$7%hRbIy|48^km4g<{woMiXWaSqhE4HTa?i)8trw@A5yfDN)&KEyj^UL=&DxG_ zJDE5Wn-klZ*b_}`+Y?Nz*|BZgww>%4JGS%XeZOg~E2jUtMs!gWCJ_lv8Z-pZR5K z7H()nXQ%VJr2r14mgg92={f4xNBf;0m!9C&o|EMslhy`<-dnd250~4b863PUue@0B zp4766R11aeeAc6PD_;%wuZE#|G3^0>$P5CuK_3}m5r%45+ClPxzdorTXDHn8>nH9!P$1+{2xEwYlZHDS_T)H|l8NDba*}n>}aaE(MsP~Ajj8hUb z9fSn9Vz(%r9db*|o0K@*QLtg0a!wZ+q0N5Kc7Pxqu*s~r1UpnoYf&x9GSS~&#TEUTdTSkx1GxY26I2T`l@!2up(6ek?b1>22(C)Ckjf8m#f zViYgF|6mwvn0sS*EHJoohr0pB0huV@*QTo~_s&1-_S-TEPm~eU=gpV<`j;~M_-INTg ze{C7EQ2T9M=HlepDiS{wl@mb?j%ufle9mlNY{o^tO^1iF*jxrPZ&F)Gy^^24IKY6? zarQn_HQmH(Uwm)IdhE7>qoAm$Nk3uf^hoR$OuStc@S5wE#qk*JChU&TH<|YVm5+g` zZz=srR1Kpo#heaAmBY|w@l+K2PKaO~v(GSiHG4U>?SY-r#-bf+&Om8V3ahRpt#k}8 z^RJ}u7ZL?JgS!!Jh=x*yttCos)wq)CYH>OOt862E`L1x-xP+1)LbdNlGE(;?#r`X* zHFW=?3c0olFx!HPEe^LVI^R`?7W?{$YA)zw$lA$tAz4pp;I8MS{QKI<>C+8j4y_sc zsT@_<0^TnZFHjQZJW>SfFi1-B6`J+tJK&GH1_sq~_yMn0mVe2%*Vph+XfC&2p!@3J z)!Op^Fl}p?V)MZ=v!6snY5cBe0p-e^bxSVI){RAe*J;H0g4U-_X=Z1Ku1xv?W zcn>{pm7xSSnp4aP%i&!mA~ef({M`;A5VcZ#Atfs)Hxezq?1L?IE#|NtgDHT%8Izb} zCem!XNy7*0S1_X9ap6n}9J*>x3?nAP$4I@$biG0URkFPlBjebpe-$%lOIsSv^nq&d z-iS$6Q^>zDzG~Hw>FT=5(LHJoFntTP3$;BQSI#1fn7!i_i^`syhy)e)v7i{07^GD; zUnc>>L_7tXF{}7h=IH~x?~rV^`r{1%XIKd#62z#+(6!-@psPbd2f?@8|F0MNrO^9m z>}a5f+gJ9sZ10W(&f{Qo>wDo2?7;zRb4)bQJoO;*y+}^p%^z*_RR|xclGfzcS5Ra+ zA|9$LiuRhHzvsp76imKvMd2;^8Tz(Q7vid-Ka!(59g%8AG7IOT7}2)}GS8(J<(tmo z`odnWKQXgnpimo%LuLvXl_SIyipv(8#^$QWZsioQS?4$_4@1m_^wTkw#B4Ql4_#27 zF<2aw|0o~fh3Ga!vXk!*R!SR^-V!_{Gw-`P=xM#!)o$%h0KdA1*u&StG>$f_2Z`=K zj}K4+i?xpVY|0KZY+7yXX=9jpZ3tB}Z1`-fX%(A@nwGF>QizAFYi2@6%YHSTvtZG- zm5NC?N77sPH=gZD17-^tX_(@Eo2+N#VAXC zZ{*g$QB7Y6=$JwwchjZyw*8w}|5s-tzW8dB|0724{^@+%kIK>KMevu(!d~a)J8hQk zPMs|?4IK7jZiYZ`A%5(Ow4#<;K}7#e#qdbV&pu?F!DLm^=}~7boOH};F^qfwxP&qmj)aJRnldZM z_gz^}G%;&K z*(T$7iO98-r29Q43Ruh+XB0POqh;q8Tw(w$CRa=gN44zn^g2%?!~r&?99QEHy3Ty2 zN^27?lDpq}iD+4lS52n-Ae=E~)7E9p6$;*ygul(4RD74p>)9BYPS;hD5i#CRn!^Lb z0v*=B67os57FnlEU2C7C@M+*6f`Et?^Ns7aUw#?grl=cs&F}>I9Y;_bZr~3weWmJY zHX_`KhnwaoW5J{7vtnf1O z&~nRe8fG}CCe?_ByNQ5EK723%WS_P2WB62YDrHdLgaHLSO}@Oa6%`m8Y41#+an^R- z!e%KU4ZtDKg+<6Y_KVNNbEtKTBkmBFZ|`TfF`jC<@5x$u*`E6FtDec^B-{1g_TPI8 z`{Mfn8Wg)vrw&8);qn+w;%dIF{Bw^^CZcv&9f}zO>}e-v$Z6tSQs#R1O9-i0yU&c_ z)pmDqux;yKblH-KL2c5DHo7sEt!UV23poalHH_ypwK)cuhz`@iIttABr)Uo^3)Zfx zhzYK(s5#qU)7Mw0aex1OFJ1-Xr0z?TB(8urb_KX9tMQfnW2%%vrN90yFTw|*c&e5i zmdpt!;LwbyJHLYPj8!sX@{6WI0E&~ef;i6ISJJKLY!hAW6;N`X!zY;dYIF}5eF3r5 zBj-ExC{lOE?h3X<5>QJ>wUT=n(1~3UGd&>k=g>DupE&X#H70l8TCvevM2#D6G;zQ~ z!Uz8n6_#8hnX=#%rAKRQJN(83%_@`#&;(MSACSwWhbj=wTyiV10SQpQg2N zTSl$3o+X?K+a4D+Ka%@FXLcMoYxavM&EkWx=|oPgSS-vyaS3icc>F7lhd<7qH!ddE zCzSsR%AlL1Hvl4Z+P6e$TTO)OkDj-3pBca(q~zYqulohJ@&>b5h4Pb7s8|5L?{Tlu z)l_nvQ6{N7P>|Z`>;oqN;8X4=)ljAdr9G`@Mj&(ex6*Vmc=NDM9+s=iv_56XDx1z& zfi$rE3F!q6_T*wQOKV;|fhy3S^3ubxm{T{r&n$cK&O55BAY8pKtXu@JDtnInQdtO# zRML%y{(Rh*k2!}1rguLp64AU?yc5yvrFWeuf(;S0*D~mn=!{$w!EU|V`wvgQ-$WG- z%rmG&>)A?+Lx^eoLZqcco`h3%%3L^1>j9_QVA&&a{H{7D#X2XgwfX2;_L2J$p%R}gf z3iI>uzfnR6dO{*wcRlPzFXoU&T~bf&BZQemgcn~Rkg=tavxz{L zX7o=klkQd}$8mS>5_4=nXqslfEgbAX1fLE3WKv(L>SCrdGZ>m@la^O26rcPFb5t4= zQZF}S0u~C+0?ERb)L9~ru|C5e1OT#Q8Fu&+Ptc0fEi zd*rjDbuj(4ii#<$FD9RaAoy6Z_cQwHyVaj+3Y?2Rg9=%_#eyh*Q+z^>S}(?HjgFQ; zSDn}n6(uN%S_EI4|>xGcxRGQYI+evI39-s5`S z({GEWh%y@-efo}aJ6%Qe_IR5J@P6LpcK^EE!;%318ZF#CJc2oEQLMd%QGeL`Z9MT& z{JqH_f9271u(w)0uHRSc66o)X z{YhBk5^=A3=dapg5xy`U1`LZza59?P-(FEvw`l2;qb3}N2O~A&C9yQSKLa*wXc`tB zFodi&cGBeQhvp*fG(FT~P?Z0MkzEjU=jXv>U?T?bqo4ZDTGqbLD?uqQS|Pigt&G3B zMF~CkHnY6|xD&EFOU$f1M~|`oxovB|IE9?l)$aF(Yp2K?RG6^plr4*=i*PbNzHvNt^st9wfMN9zTE{~U>nIYu?I=k z{IBUOXoH8k$D5Sqj?CtcgY&OZ_Di#`A0et`{9V}}eNOS3=y8?4wu<4$d-_w8{fD2G zNH9a-w)ZQuqtUi8Nk^qKexmcInEQ3iM`wHAVLnWB0372X%{fBJM}^<$==nvt*uYsA?ZD#5$lb zaasdNF#8vn-RV5U7m5top;Mh)nLt#V@1(83Xf3E* zY)1TJc@{dQ4aX(cQIr~NC(=P}L@Q9Oo$-~R>Ffp6a5nxx%S!xna*Xk$Tc|m_4RM_Z zR&c*g+Ilo?VoZV_=j|WGmeYoH^{SM& zZF0Ds4~hRRvXI!`L3N&%L94)zNheo^FwbNw7j;uA6O$+1aV9g1*3xHJe9CHkm?1Y= zMK_O`%hzraeM}D9NZ(1MUY>Jq5t4EL~3{s!Fr9^i(_1&Ov;F_y%)&(>H6}WDgljz z#|w4Z^YGSlZF0M(6V=aZCjQpd@D{?Z^Y-1;H%b89UL0dNvLUJBp!9CiWZPV7)U|3Rg&{_N*^=CLT4!GvBTV^P(@$I~%@j z8@#pe^?t)EM9`exzGWnvgsqtSIVB+4vW4&08m7i@Kqe1%kNTqxioQF$6lq&_EABfv zX~zelgWm(E(-p^(zlfB9ibU=zG4Y#SUI8NUs7;$?QXvH(dCTBi?W89>rLf$l3-%Ql z(YJLozOB^&LQHmBFHd=_2Gv5nb2d6XuKmVQS)sUiptcU+@HdO;+=9HjQi@1I&oc$u zV|Rv9u2os9&6|5?@KE*#)2Z(#f}yC1$$w*liS1POl3X}ZUzU#xtj7OEgh4~d5 zTBz-JO!G;zEL1rdZvRS?6qNeD;eLIp*%8W^?lu~1m2^8`kK!@r?0y}4`D|_MwCeU^Te9-`sNt}0|$mEFn61p%;H@NuAkj&dg zOjNdN{>lgOlV#8snptY@*e@|m`JMOjf@OYrSJ5|n>r#5H5r^AAR)E!Y;XMCNcaIPJ zQI;CFC{XcS7nq?m*~jb;httM_(ennQ$BA;J2|dB^fJH%|y#Ghayb{#}S?i>~2!_Gk zxDLfxKfavtp3=pLG3lI(oKYl-YBXQ?YFL^K%({%p==T6-k<>SNC1xD3#d%Fs2w1Q9 zf?oswGSoH`1v5t#)NBasuV!Y|d|s0Mhi2ClE3!X$BRP_X%E(j4@xtO+6B>$eQgE71(-xnV4$RHi*6Va|Fwis1YJN=%v>BIRED47GZLGR57^3M z$hB&tALi2Mvz}R@B*_#AG;&t7*GS712yIYv+g+fB@+tTME|7UUvjYGAHIt&9LRCH& zKF<$%?~69g1GOG9SPS{=Z9#lhJm9Ui4f)OvXyk?YTb}ET1PIS@Cn8uDe*EL3-whdg zp(#Ijp!w!A`l*Vj?{Ry6b+P9#iIt&?xh>C4=gS(Uzw#r2y&XkUp)XV&oRa=qv{zr% z(}S|51xytc*pNF^28X_}Ymrsoq5LAkns3@wE-`{p1ViG-Eet+W`KeAvveEZ2ShK6EA=#zeq8bBqMl^!C(jUvm=szWYm z06_+6UiUK?r@zjW3S`nQcJ|l&H-mZ*BBX(QQez>95x%((QVl-Bcb2X6!0oTO;eu@B zAe+hK6|ssKPrdo4Mg8!o;_d~d{QrcA4Ssu*JF4wqui;Nr?k$f(rLPyHtn`rHOHqIp zjB1f|z7eJ+Ya`VR&Fz3o?lpc7FLI-|?iT~qcsOm6uZ@(~3kqphR{!$Gh?L+1coB88 z)5os%oOlXccy(8V3C#sf%&$`s%tGMACN$0E_ZE=8=uSp^^GqJP1eOE`0V4yhhNk9I zn|(Vvg2&Z9aYYr=ZAI~y_h0_kK23jaM(vJTzwl%Zf7q6fhc|J4nZR$dU5&uR!unT` z$xHD2D;UT)`}|o7T?D5)M;bb?#yK$xgx02p=JV(wo*ell<6?cft5p3>uKEL^!3{Be z4|LbiK;f5u!AogmBq=L+i+3Z)gUP!S%BE6TYpA+n#UIjnJ@gHPe4uvUYbV?WGX+H% z1+)$OP10dmBUuJ44(B&yi%Z9@(5)_qmz^<701!uwV zjrYrzK=Z|=>^}u!*KYCFf>9C0IHVA5zj1%e3VvtK9q7(G=M0ZwR`lRs08u@1=pU{o zyVarjtPiXrn#U{D&$mv-xT1rmA;IG;aDO?;4oW(!6vz`kDBUkF zx>}_^SO9FUH_<69RY*=P>B|wNHX{Jso`0YoXdAjOe<0!8a!UoyJDdV^B@Ij*cQ8VT zVeZXUfRHEX0t;{KdUBeo?hS-*X|byGrkb0nDD2|Qe6DWj(0Tj9__z;tL*>A<#FL$eRuLd)#tS2I{6uVV&VXV2a&Ge z?w&M5V`Gut6Qk2AlbrA>ymB1wuV!^wQG}$fLjvoQwHV^{r`7DPQ1`AQW?rv-+9t$h zVn2wlN37;^OltAGR{cR8X8i)DxW&Zn10MJ|X%6(+y?fgSWA$X`;+X z&1852_r_Jrx0zPcT96Io3M^HmXeuj5y6zcgvYREwG=d%APjw3^!bRVBH?@D;qyoX^ z0{J0!3g;3rZNgl{Qlz$^#8YjI?G%vg3#;KP=kv7vf#I(ZSi_qy zSaD~R_icSa!1K+H!KjFE{GsW)FCSq%I6ri)qTMa=v3Mk*eM^i9>ejdy_Yg_u*C6H# z9N}Z8Xy;Dzcw8XE?1@JGt*ZMXX*@3Uf3*PDZws+qS=^u3x0NZT*Z0JM0TF z{Q(0Of#14z!Rd=IdsP{sHK9B=Xj*QJqKY4c^4BIO&BMy|VNd>gFxkN(g{&GQ3I zLg3kKDw!CZH^GnuFVRe9q*+uL>m!k=CJm3EE63wfJ+bV-XzPqkWaRF1^bZ7>CnDYE zF|QWX{;`e9t$`e#PvDv#&F6PiMfSsjn9x)D;GI(b*>qTw2$dkH#{;f%DfH}8QAB`$ zG=M(TeXm_&=uPZ1?Eb&_(etN#)p^S%vaz**q2Nb75g&+8)N3St{_+%*q+^+VbjpSU zYYLAstYcg)^r8;_dD|+~W8e8eoWbv&61(k2=&ABYd9Y+NSDc&R3M*+GPbFl8ZBPr4 ztnps_&gM8-F1C?)>j2UE?kaTqDsWrgowbHH0K)Nv5s}+BmoXIJ8A}o893L6iIH>I? z8kx@kJe*ZTD$&F?T39&CwvfesQRoY*>+OAbNiyfTB{sTjj_3*_Xb9*=0og*?I;c(;y)Y0kt^3X|m6yZxuL5Vjwfqx{B zw-Uh_|6H#(eeQ@hI@AlS=8-+Y8N2&m>q)I-CM&b%_9LEFb+{$`449Nvl9l2n zhi7xJ1vkRfp+3@4MO%vy-_^j{oyVW|<&#ly`*IBP;4F{AP0Q3x*t=XC-F|zZ`MrIO zHU;v>bQu3*F^n4!H{>K5W5~vn10L5^5N|NyZz_!J{yuS^8L#Ql_aoC*50qEm@LjV!p#xlalW`ZGg{xp2uxQ zup`%QgzeAw|IIQ;I1|kr?)|;jX~rYs`opXvhp^-uZBQx8Y}buiVUOFUOxKy^tpRVR%&#J{`mp=H32E z_pOS;R&64oGD*v>`AEe*h(k_Q_Hm{ID;|Io?gRXZW)RepK|!HaBa>5=y<70$!rsHc zAQa-XO|;VWLD%?3MmOWf7MD|bZsW%39KzvRmm1;kK(j(vT}evM%EDwLN9u%zMP;pg zT5QWTas8>$dT41^6L2YsZ(TXH!XOp)l?9iUix4 z_3F9!IX(kEg$5D-mFoP&nMl#cb1XiLf$u+<#u6;r=9;Nq7B5=X#`EXbVREE`aT%T| zENa0p_U;|uF+%nwi_FKQ&J!l_3+j<;MI{Vm4~UT4Y>;ar^Ufw&=bAYNv-p`W1HL{8 z64Bn%nbH1_+&Nn>pSfM1uT}q(R|u31Irg|BzUh>5p?crU>Q&2aN>)h_8{-B6aGbFE zcH>@oyhe>yCBgDWRKW$oiV~(N%UiRCZ?hhKlIouJiQVH5M!`0bf?&hah{n4FCKhSW z*gJ{p?%60##-}ACvX8Bc4fKlvg8@NC4LRM^>o1JFibM#<4j{BDgg=PX;$=k@Y?5nr zu4KJF@CFIkgof<=qiqlS6PB$TSa;@ycpOV+`Mn$oSxs^nGYp!tg0HDLU8H%4F_fi~ za(}>aOl_~4Wm<&yzQZA3@#c2&SU`gf)&uRsT%gZbN0vl5of8VB_D}9gvcsXBF4Dtw zFW|J!WEt*$+?v%03x%FD@^ry*+XT+j?-oa}8Ot;^0S6HDy0v>Iaurybzr{}z#&-aT zfYK2YBU8Nvk^mOYP|3MMDn50!;Gdvo-mIPFz6O$?LlS}9>-ZU#B zbtMd+Q$jd!ad58RHMkEiJq}X|pHewZbOb2Ouj7}mX9)OZ{Ou|j_-nJ-$G_!4kIU5H z=zpTQznERv7Ck?YQD1P^j^{S$6fRyI;lnA96sk8y>x8Tt)}xRzs*VfEBQGAg6Y%$Q z2rQk0F4o@52c}1=_-keiJ{ksn?nTc=jz+BD;4A63`_Dkje#RpgCSE+9$2RHGVB0^Y0MO#Ur98VQV>Gm#j3l6+IhY@zE z4xSWRmpY^b>77&SD)V)DW=;m*bdz;~JxfEe&aK_x&kyiEKOQ4{+HSBHJg-vYq}v!~ z;J`}ortk_rrXGkhq?67`+Z?H$SCQ=aOQ;3L&O}1wbbUBL2%+V(>5XYS@*xW^w)c0$ zaj)r1wn#fi!r=y&o{zP_g=m6#oRwiJ=Dw4WXqQEvNEY~Fcbtj6s?$+WpF_Yqn^D8F zsn!N|m*-($SLH^zK&HLab0o0v5G7DqWMIXGaLFF{eZ;7ovK0!-u_j4%|d zb%@eJ5UnBP&DVDawNDUQwC71O|4KtZEqOtxYKX4Pp~X(cphtXx6^arm-;)lNKc==n zu;LnVB2j6kLRw2LS=e{s)92;Hzc+8;^6CP|_P98-XJ?7|5uUf)frsh#yCb8B1&*m< zsK!tq`{rAZDZ0F^X}g~b^vaefR?;?naphFPRo=El#ArE)5(;1iI_CDfEjFvDelsKzia^99}VNI=%vE~g7X2Wsp)3{_6SZnucY+$_>m3ky#u6ZAQA>Nlu& z?A75ABrN==dHfx!XF|MH0QdK!ZNuP0CKUS>OgN}1RQt;BFOiloICVDkk z)7|z!lq%R7Klo(%Hvy_wsd}J-NRvvFA6lS#xJ?QBFksAKh%}Jq%Q{q$=Id!c0WCc? zr9wCHP=Ut@gKF-*5mfQ6f0qwlhiyY)aSN~5WkY|g^25-igm&^aLA6U2q81;*A4AxZ(}!EmVK*;QQaL~{R2hG+k_Rx<%{f=$A-b*4AS#mv?FP| zwBr(8ZQ23<>$vqA7JFs%bJ%-SDNloHD8DAW2XTZR9NLnyT2IQke-&+S4~78C@&w~(L|23(<~qXWJx+R)1_UuZN|ak#F=hFX;ZsZG?VH`#F*?X zaE{IfMu+pbH}UU_iYjnsv-WOxWtL{Yqc3!Kz-E2*LVfxE+xv(eYjw~lpG5>mGFHMP z(llI`cV^Tdr&O54f6og37g%pm8r9-*<1YwHctJe)LqvY2_#QO2Le)HPG2hX$? zXC`ec+QBP99#BF{W{_$tEj+F-wPEs=r~OFuYSCtF%!ZmiT-m}-PpdbUYQ?mGyIE0ctZre>ohg#wmH3> z$4vzONUnDTQM@d8ElgPiW+c)l03Lwk0wqaA8N#1W7*P#9}zyprqcUcdv7k6VsJ z*M;?TwUVz180>nQG^XCd0xAouY|?S({OeokkY)=~%2obwTeL#%@%>_g*vfk;5ncgD^XWSG#;yNGaMo;0@U7_9s%JukOiF#nRiu~tMO(yEoj~SNp6JA~4M4Q2&*S zSPMAEKLz1~kuo~pcGTrt#0?ZRi@@`+MaS-m+Cxrd(L%Q@{)AM<;_$!aY}mTtYrFKr z6*!U6e2M-q(IQ#Iq#|bq;Rozla<;>;X_KSR#lT~a96WG99}MK&fcMDU&GlR(D(|k3 zI<;=r?Z42K|3?^0EPW=a}k$vNE zthVWMfNT|-9kn(Pqdyiygoet8aGH}&A@f7@qIarzNg5Efs2v%<5jV9lko(-!fPT}$ z8L(9!n_J2)rO3O5veM1vr&!Mmc`v=hP>NB*bribdOSJ&WOgB}HS{^8l;ND#vWzoG>~9KkD>`4cd* z0Gu)BX|;0vjbOgkS2aL!j}-Efd!#F1j3M?xf`&Wi-HC=yw)kq2RACKJVG&4xq_FUF`g-GO zZ}{wei`jUTS<4L3OO1dqb?a%>{3Q1OYA6-o=L>~-U;b#@9>Vc!9N|pE{|?vKSY0^c z;+OZZv|<3Sk+U%sg0;U^dh}9w%oTCZAEQoPi@XA8F!hS&wm{4OWop3(I|oLsE4B2` zT@bPaY25@YwUxnCSSc5CI{|Mhv!V~K3|yFsHJ|z2z>3tD^z=a~jH7QEs2`vypuE8w3auj+;ii;kk-vxg-}cpYhjG8KgE6|qEHF&h!e+~V>7cyHe52Wjl>|$k8{wkO z^mf1&pC*Noz^^w1RVtOy4lvc?zGZ&&TajC}0NTd-kVw0!)!2Ec)0qpF#+j)?*TxRZ z<<~cKh<2mG;xs$#i3;5_{7K$s^ICiIqk)9Mt%wYEdYo4pr>QD!8`8;t+cjxr*cJDo z>JiQoDTtuDCC#rt(9_X8BN+duU5^KkAdF7UvWv?Q9y#jOD@n>w`1=`p{Kue{GMa_# zZYc=*ysJ;6ZrVyo#ksc;aV)u?=-Vp^pQ$P@vXw*RI{G_uF5FcEH@YkrWR}+1eO$&Msm}ug^(m>OqX;$gEexV z@+0Z~_imw?H^ci3jKj%E*@Ll0`(D;<(srJ%(nL={{5^0a6mLWAG$ZqP7>EzN)^aUQ~oknxDc|e<&mf z=-woml09+uwMA0C=T(IoPPjxCL@2B!)%0^yhJ9#?wT4tmZXwQpA4KsVlV~AEA(_Ge@_XmR@F%9yGzs za=)Txj~|&_gYs70C(sp0^%@1oLMyr1(xZAzh;E$A!65InzO?}MN~dHx&LCGV-!u&6Zadv znR^%0DQ%1f;7_nQb*e@>Y|0y)wcsdB@*z8tIy4gHdldLXYTIgf`6UlL(u3m)TZVq> z`@y0~n%5$(O9JQk*^wG|gyMiKKWX2D0*m(m%kgQNKNarC%{4r+rUTqd14()Y$L zTj#)U(qetCeh7TfP5&z3lpRkUJ}Cx(I~g64enx73oWv$}U9C%ZJ$g50Q!|f?&TIKS z2mYK8mjOX>HgGVc0s|YshVR1y;=#GxMq996=7zzpe$yCdfyT*FF6LDVEF(?U4}57e z=}Ys~V1^ff5-NgO|8jw^9OI4xlX}|6x68s4uEv*~X7elkX)H*ki&lnbx9xmenk(05 z^~HFf7xq0bhTBNh{+%$BGj`4R#Ki9jnTD<%ryh&fWqa@Lyaw_8Z$w0`N_hV`kV}`X zOj~QXWuO|{XeX0J6Gnc)`11IX*$+N$c{5AcIZg(GWH3v(P4UqNeIc01j}Su!FHM$Pjw{owVa)#q*`uJJJ( z7`PXlWoSWG4jhnKeiXTcZS{e7 zE6nmY936gTP*Us<9=1~|~84!G>-b#u(`RseHDA@4)jl4lG|%@*!bksy=< zgcQa)po+4h@Nj8#0@_Dj`sKSbGX#rWj88d2Cy3+k}7L=#g>d^?~=D!ZsR#3gGJLIxM&|+;OcDj0&sI0M>!EiN)57#;>h_SUs}X`;0PpytX?se%AGfaxx7z9s|G$OhDgbOf!_C9 zD=otaaf0;tLReeR0S?lR>U{ z3fJ%6NRajP->xDnU9qi;uo>&I9nxeO{Nsd>{cVu8n^5H@5Rz;?M&Niu|9?A^e!15yCSy@3@rXk(@vRmx<`?}a zy@~x_D$hQDE?m!vWQw(k{Cr#uiUXH}OrQ)jkisJxPJP9SD^T7?kvK#KLpk0%Q4VY| zG$+#*>qB$yt{lXtmd{~EgIc@WUUJ(35S4d-7lPg6lg^4R+&vxhkoPW77_w;g4RlUWhI{6ayq32z2dw09N-pGEGn7jh{FdJIY(KR(=! zv*VK*uhI%C?v;ekzh~Ap!S3QaG?^7R14*Bpgz8D&Z#|g{{TAK8VYqEWb*FnL=kad| zZ+^8{VO(@UDXl`IGQ#O5nAc#yOY^d!a@&8g0r4ugE=JW_mK0po$-u>~Uu{&Zv+ctL z;x5xIKAS>xWR^!IP{?O^i3SSlg_eg=A|XGd zYPbjtICgM}ED9M13GnvEOWiCsir`QKD;z0tx*oXj;*5M6fgz`n* zz8wq3h0^g~!(rM2|>AL!;1_tx?SlgT4aUVS;CK3#|-1Q#W|gP`|qWPwH#I^nYFmSh<{PLVO@`CcIcE^ z=Zb&+w-uQNNE{wF3g8H>p&$AHq@?3Fwi`PHwV6A*)V8%Y{iC_ap&3RyZ@s{aeRbMY zrQZOz9bGhr@-)B+CU^+XB_XEIfaE}oDQMo#E_O0HRN1J1qEVeZFec85!q zl+&dk3KRLW>vD2zdm%f)fXH98Np3l}7T4>j*D~M$B$&||H^BorR07w#U4|%zD*V|{ z!+RaX!YRQN2yp>5kV+}o<}W+9*hhgz6n2*;=K$9STOw-R-Q!~rx2JU#{iDxeqXqjS zt+CaUK0_}d>S-SNdc|iEMMj68he}=sAOl!y<&@EB_cw4qnTR4i%2QOae#DNUiXZpP zedlhK?&FIGgFRPI#kKneq=<%2n{bE1(w`%!Dz*@UXJG&#=UNSivIq8O*B0?^at#=a9l(c^c&tUC(fZxQ+4TDigj9u7(?9!*G znP&DhYY}xL6}K1q*((1w5B^Q{KB*w0me^%Nn~0m?Du}0-z4FhCL#1DTi*F zOy=jRj&X8V7C9`3FoBi_-EDMOwLXCx4KwA_37wKBTcR4K&rMAGv0L9bTMSaV$!H=ZiPr`CG(@> zOf!8crx?tippUl)#vk*J$%M)c_03V9DrY|m;2dKNCK;wSk^QN_=C z?#{%%wU9642AWKsE$RZSpUojy)cUr_3tEb8w2!6J?UB_mACPg7Eb6$D)bhX?(`5CTD!s4?n_mt%Q;+y#0RPPafuWEh&zhy4dMmvZIeuYP zw!mUta&Vd*V$IC}bPp(cM>| z8iP;k5}}uV=I^F?|B>Gn0m8Xcnfk`+#KqmccMGqBA$05Mw5e|z7|?-x0E8dRlW8ZMGvd~n!MP&4f*Tj?S-ncgIA_hDwa^axP9PU{kB=;LPLt#@07_D@ARPx1btp zA|~4rlge->KUXC$H@Db+AKf(EWt@}sCG=HPxwMo)SG>}3zjL%A0K`pOt%pw^R2#f} zQ-O>17w^OBBmcW(+jZ*5?XaGSs$z=2hpc^RAA%9Bd~2G`6FeKX3#`u?zI=NNf7bMl zt|k2tc~kNc9iN+1wMwCaP-G}SYmhN%wJ}n z?!N3hyPxhK=u_Rdy1FXgo;KMrqHOf9t`L}`Xi$b_$3yI%l5*qHtC?&E>t`&Y!^z#o zi@JMb#hxN>Nqhf6lo$1=@GHp-f-n@b*XCcL{QpX6;DkeSDhm5HTL)oXt953c(39km zx5!d41c?v0o698)hcA5=1BnyhMg?naJGE^&_~jCiPE*@p0LtmK2`%!hOyGsm_2G=V zI$%AUpEmS`ZqCW{-=?;*5?35{JS>Qh@>oy%E(xFP#h?uuU~O{!h_}EAjm-ux=aooO zHN-`cVnqbRg;a-TuM*^V=fI^;=Rg*CHNa&}mAH%c0!bs&RfC<*lMM|V&1}QRN!7GY zy4{Z_`#-3$4MAuh+P)4G7b(ap-0SsLJ(4_1Q&=LUk%&4L(*{tC4W&!xH%QbD?Z%nF z%wsCsh4JPrq^*YYJ@mTjcg1@(nSf?SG6EF=3m*m(*EOxkwHhT{wQGA31){GUXa~Dor=@nBiGdbaS zQc0w6ks|(PfrfIlMl<2-Zs73Nw`r&gS``iJwb9r1;%*3VDO$+!my3re%NjXJ(~W={ zE0tDKuy9u+gBO2_SbNbcZ8L&0y?)nMYy?+_v>ldNO0OJ5;~i{dzovQ5u3yuSd$F#i z+UW0~RZ1GiH%A21`O9G?u2d2Iql{~+6pca90j?o!byQF`>~hW1LytrKHfvT($bpx( z1|6od&i2E~M^=uDA@Q*4{d1uVoVvYvG`pjociX&v zv}pFIi@dMKV0#=nzI3tr;r*oakua_L6#w4`jIT+>{G86gnfa{Iu`qFAE zgMAe=c|7RjQw}E#B~h_MX!cbWZ4j4^DXuuV3fge7{`Rd$hXkANB00AHyOKx&Q*VR* z^If^X;idehAzAC+WE4FApH)ANG-hIjXWI`d zpprVE`QYmq{C|mf6K-+phqYaPZxf*Z>RYP?Exy*o(31m*soKr<%k5{KxYmpMcPYA1*oJl&Wa7YOO2NKy8a*{( zA@nZ55S@$TH9U*dj!)AI>v&EH&6k-G{JOn0AkC)FIaUDyYW9|1H=TZ!=WkR04qtHG z-Mye9TI9(@q6sCk@w;u|RN)5M0qUEWD(Ue3^Xst;<7imRU zmXxSy$fqm=e-|0g=dvRCyS*C>RTS)eIP!7u?%JY0afafTOzK}>vDQF&#eo7Xbe!Fm z9q|!A1p$$ex?V>T3kynqwr0`zAbx24FI%dlPBYr7`j4;f>tB($+8=J)t`OFS4j02C z-XZrJ#KKHaeJ(O~+f5o{1q4`4T4SH@Wy4Yn1qDrXWnsS!{B6F$hAN+voLOdM zDR-qwJm@1cSl#|*16T4YbV4Vs+XY|w8wPc~dq|kf&Qu(d6nJ6dEZl9W46yeBc+n+|Blg0qlw1xlp~Qy_%!iFknN7IXDCgV&Q0cu|0JDK zWcH#iPw_(vM4&+ajItp`BcBMJyp&W9^ELFg%zVeIBGB-~MN$bKoS=|{`(rcft~CDro-iOJ4mK?8MU z4YI?P!TOJwOv8gS&O;icNAdOeTMd4xU-2fetYWkLjnD$DF_{OiN($O}Kp#h)lL0ro zL3ogoeYFg@PcwH3C%_tp4nqFu=YZ9(@}(s4%AD_h|K{7$TZB!Kr#KaE)7q#-0h_N; z|CCM@`7xGjj5IG`qwr{x&VQ@!t;v3sZ+(uH8XK!bYBkX8PB}^U3*Dn=+NdCS^l(x% zGH16AZ{LPdb^-r+H<2qWjGPzOX=tV;H{M4aDULBZGDj^KoB5mL%NoS9WdjIjD9n*87-T_Fa z{l-);craFacCItTPj_C6p3ejKh3;iCfBsiMzTwT+E{q`)ctLS$NBe);7~bnpA=1W& zn76}Gy32FWaLw?PJUGd+35Um}RK}xItds7CU92QK1rJKNV*YTjdDn{>#Ml&i4T^jC zTpY*S0{e@)1A!RCn;D*GBu>fvoch*Ny=jF|NR^a+RL8IdCAQ*z@_G4|HbDGTH=yh} zjI{AF^DtJwEz-n6C2|ommuQVi%zYR4rzw)t&afSxE-oqOE;DaxamOV|4zyPkEW+r} zXxF;?AH%aXU^qm#>DUSteyBTC!V5q&6Ot@;=4V}y6Hb8GgfL9KSahb#CB}Cr z;+Im=$-3+luVnyA}{R1DVZ zZ@#0$x)%;PQ9jXOWY79(xsm`i8kDva7Rs;+wgy3MhLRyDXw~(Cc(6N|)33vDER3r{wC;OCs_pl*9LQpWq!_Iq-IvDe~I0-m|mwSE&5I3F!TETmcJPjYp{F(op`z~HmwFBbTaX@3+ME7 zB`aM|7pL<`?i_yS&-FW2Y>9dSh!#t-aV|mSgDqS2 z#nyPAsl%OY#{jGR9ABsEmu&QX{-B-=MqEUEuta501k0gDqC4iBNQifqb0(#51T=V6 z*#~}tNQcBt{9pKB=PKeKC>{_1h{>IjoSA~UP7f<;P{KEGNO(yWzl+~TkXPCOLhjHR zHA-7K_wtBM#fMItAcp1fI%l%6l0MGVe5L(<>+T1wMhURztqBhwMrb4A?)P2 zJnV(5Zmy}SX6FU|7)KX%PdB~~R#vhb)7q(FtwDWL3^1nw?7V@^?CTde^8B{$Tq)G?o<`pZm(pG_|*%w~!+{XXN z;GXQ!TCs=uYs1pG24oCIpLPQ0Sofz;%b4C(GtXzekow!Vx2>N03l;cpp^nbvd+vST z|9_r-S8?3qxVaf{*VPR~4T#MaJ+$+$!U=rf^MWWlCh6FHD^Q@5P!D4ffDcZZ8ds*! zIaKI~p0Z?>{p%-XWUKoxj{T;c?p2_n(pv-+X)1WkTUK7*e7oHkFeHKh6MK(pb_6Zh zX2fi7=TqKNt|4#BH|jtmG0jj|eV?5An!r`ERsoQPJ0{~W=lqh%qfj^{I-fbp1S>R* zI^wN17738*1jO@=T(TK+T+BNPo|9CKGbRo2oWC%={8bL4_`xk!UHi!$HJ~()XkeXo zpjy&7tu8V?+XQ)6Cb}O-(N$qT6SnDy%gk^ zSxU9JQqp(F3N6YtRa;y2#oW>~ao0Xp>u$ZE)sc%+Tw?b0xO>uBKllf%+KPf4C%~!f z7LGhP%rE=!)2lZ{M;Z+-(o#}Cs-+;bx<5)Y|JYdiax$+obIh7=*_X~94VnR40N;s7 z;bf%c#dM^0#?4LT{xtIoiOyRa-lQhqz|Orl-kRV&;GAwNyq1a}?U1pb-pO3q()!do~E1I3-buLggJvkeATYUk{h1l#FC7w#yj? z4hg?VhrACJ9=wjo8@^p&=-Hg53zdnYSttZPDgCl98#by$87)ZoE2F}BcC-JX@o5?O zx6c?D_h>QoX56^^Nd(D3FAZ@!Z~2Wd?&yk4(xf#L-CQ_i$+_+ngca(9F!88yXfcS6 zDZZWUz(d5>8Bcn3^?gM%bjC zS@|~65?6w5aXHDknK6~BMB&b&h4zh5UT6=rvGI>Gs*5cOXHn~1p#!`z*X=IxvRp|V zJbQW2jx6W7SALj{BfbRl<1kgZ&IAN!c(GM7bZggM%-}Y;iV8ZR%R_dZA8iv-SmI^P z)i$s-#kA6HM(u+|Holz8{@^lcXZ0NDg(pe#W0$?!*IM?AF_1~3EWzEqUF4<;TT)XL zJ}^<$JF)=kr0bL!W|R*lDh&=A*3S4HW(R?2ikHPD4($~g;LTPobf`md$VKSX*Hl!G zqn4E$mJ0mFwFl50t(*Inj*!1m9^C9~((c3Aaq(+#Y1K^emkN^5g*_&x@%OG^yb=5E zcVzBd-;kL*Hpnsnl3_^&RuTu2h-vR77ksWQ9*0{ z9pW(nU~xmM0|6I8bC|7WKJDZB<9&qhhRc;`fcq=XCbiHdh-@|sp0h~wwYaRDk;hdf zs#yTX>K(u81UPP0%MB=1;8tn*JRlQt>QZN8vUuwS$vf!ukrJKDY%}#7YkC7y#j6y2 z4xH^cWUUrT8wm{bZhv2C`?NcOAXj1P55kyQ*cH4Nf+#c(?Z5BnFQtQbi~!52ZIl;)%n>AiEAwa zA`(Q;X(68vR_X*DLMT;-$2Iy30c*-~`!$h&bmmPrB9S+6FhuTQc3h=9_Hb@^3#{pt zTECD_w?aLNP4J_gbwr=hJ~}Ni|Jfy&nXj-dQ9UZhz7UhdrK|EqwYbZ0t0PZd+@y6o zBm7X?jq8SDx9$w^@}c1!qkA$UCYRGtdDuuXjIBiP7`ICHUY(G|$?rp&jI2TnH?~rA zbgN$-w7zYysQ!}b@*(RKzF0+qjrgyk>ieQ>8yW_gEE7~-bu4QA8r zr?}uLGZ|~A*ak!t;s>aYG{T5JNAHwpH5$%4Gvb}Ke-d}^5K*8KJ-%fjh#uN?%QYIk z833_YVIT820Q?JRaNbEJVYJjooGQOX|Lv`6fsvJctv$@@pxo`e;Kh( z0Zy@)TN8HkB-9J-mKd4uE145kHYL*%pNdZ#$`E`|w$#Px!CsLJTAzuE0&hQs4(KL!elZs#~a_A0a%(nZdE^RqNFjC2uMRm)UG z&})cJmBpfF=PiU$28MZo+Q{Vp$dq#@^b!s7(rf)4h;qQ2SY+-Txi$d>|6jg$w8^k{*3EGTP~4it&3j<*ZH!c zMs_uX-LfVV+N|@DDh!io!atRKzVZ3YjYG4c@jhxKLH#E`Bq#s6>t)RWaeQ z^Vi4oZ?12g{cqhEvF|76aB5zSjOK@DfoPvJNW{eQFmwuf-Nju@GJKJWs{ zg8^K5Cj5W+G`WFJVlAiRm&TivvoFVt&*+XjcZOb@o6I#jrPlYN_g9#%Xwrr2Q{xXR z9dn=f#SDV!#A1_zRkjj|Cado)G^n}J%`v(|VnR(9JZ!%HlON2DTgdWXYw-j5fG)?) zOV)T#OPA2U2?O;p#ATUmL^CqiYK$yS!y=ui-a(z}!*l;L|5)cB-1 zPyLC57rKsikmug+IWJf4hJWsKY}=|3nrY%DBq%bl;@`l}yS^2tum0(!&I^KAjy@7t z^+e9B4B%Xd-$)|6CT{L@IGkO@602vmU+)W|RwL_zSaT4Ov%7n&a)KFcO2hwMZ{$)U~weU7W|%Io57+5PmEo}s)RpUM0#Z*HdSpPGb-rMJ>@#9vB% z%|^LjRF0*b`W9YEp&j?3k4KEYft|G%A6YDq!eYCwK7lcvkJxUNom7~c9^Uex##ICn zQJr@?_4qpGEd7SW;8$hNfqEnxL4FB?9s~o=A3$uub#(j#n_AApR8a+kF3=iy`S=dd z(cC89V_s5gM||4_J{PSFKU~}hAg?;6a*-)P7#^-_;nQs{rO4AQ57XBSo?oM`vd@^cxU&&(D zP0d{E(RN)MPV9WZwOMN_o(L2fkxfy%>u-2>Z0qnUZ_UQxn6F0@R03e;#l$YGK;SWh zJOp8dY@oW5{6JWAep!uk_gg`8xpA4nXFwEk>RD!L?uyGE=Zk#<3E3!f^}5R+x}bAH zo}MKH5<5xqHuPritL+{1@AvTlAybGT=A7bA`FX@l)$;@NW3S}Z@_01e?20vKY5 zM=9&+jdX_^j!V1H)IAE`F`VbtYlyY;0cP#o;x2H*?9c137m?mCkn+WeAu|>k?T91h zx9wP?^vJmIb0Fc(Yu`nf`}$MMk85Dx$xAJ9aWfQ9Ma1X0xTCp5(m7>e)ABu807t@( zAu`kYb9d9`p|M@%8r^y=C8WIlh#O_xx1u0b!1wRR+lLZY^AQQRZ<_XDJ)tg+cYmoD zPDEPi$-u`qU1LSYE+6%V8kvy4r?DO{Y+6&V{N|szwI3JyaC`oWf7u?2 z|Jdpqcf|Y#bGC~D!>xW!%&lPLWE5QA+~#8C541R88(=zicIFz!dv15Dc@yXWf`%cW z7b+-hlPro;+377cD5^itHN39W3!;9sne%)kMB;jZC_fFKtfIQ8ssT$4%O5|R3hlr> zzuCtSI#k-1W#7JqR|MMj3;exi)7cqEGe{uD$AC^a)0)q9G{Kp&aMQEF;Aa1|E=_jE636ad* zPt%4s`1js?wono-f`(R*l0qHs{@~(Eg0s$j(F~t00F!ph*XGpJE!@5%aqCAt153c& zpR3&<&7*qQjl@1I=i{rhCW+hm!&wT9YkG^dMeX|Q^0+x!!(EFQpa%U2trq#&@4tsD zbzcy-V=HzGcFUz3P@CP&bOTn(Jhr0!xTzATTtzZcR|0N#;{hQDQhNzy0gWg*^q421 z4{efm;Q*n=@Am-za?#hq+XUb^-;YrtVRzIH52UkNf7nn46n4cB6$e8w0$mP)-|rnyI)5Gn0TrBkitfw9kQ2ja*iFK7Yx|z z<;BzKtdZ?{``$_NcD_%)V4MPU-j;=BSX2#eD!o&V#@|uTW`{qg^)^U?N6k(eZpUqj zyiyi2b|Vhixw0d57VjpFV!2VQV2BpK=hT{;QCdE3fjR0wyz%nQrP~4#?(3NlXM#4t z-hfx9dYs`6oEwapYHxQf)hwFjj`={`<5{HK#-q~LbZzPHXZOD0m-p*G9&0-zugQFg zu9rIjo_B52l3Ydq{Q#E~OX>eag1?*D964w(hzgkoLj5eoIVa#F(f*2Z=Wxc;vN`;; zs`gygi!I%MCXUa&be0+=N-rg5QW9K9ue-gNmuSXNNH^|UOB;28 z8m#wrtCtC#PAaAot0I9Bmyp-<{TW$P-s`7FpLOEY4if?Ylqvw#ne6P8S{6klgpr0K zYd$(}^!D$c&zxTt9WG=xewAr4Dng0cY9H^J?yQFq8rZY_D6EprhVB2w;B?OiR~v1` zN1+YCt78`RVGiQ{yAy&ii42FLQs5FEfEWO1v8yfne=WfAFu_zuO>}4;pdcxBrthKG zQe%MD>$uZ*9c;XG>aygGLc~}AImAs|_g&rZc>G86r}fCrjBW2GcDU;@NY*)Y@aK1W zj^41J%L%FIo|1-Rl`SK`u(_5(aPY?pR~GEl@PTS;96Me#Pqg7g=s9T3#oRHp=PUVVbPL9`nF(2^kay& z1J*J%1QMv@m^_f-6?QN1fVnNk1ePwg=7l4Tccj&~`oKY2DO$LRBf=#?=%TdDDYV-* z{Jb65gOCsvgq*6rV|nj{b|T#^<`vG-nYhX=pFI?(Gf(D7vPagfFljsOpWIMCN|$85-|hOE^ZhO2F$tdj0v!E)MrRU+@l7VpHjzl?dW8u! ztlL2|_l@^JV#6Fn0o_eRcqs!p<{8Izgb3}q=^7E}w?TzuBZ#=ysSMU1-xq=(Z;hG3 zVfHP;uNYyZUKi;?CI8hb;f!wq7Z))o{pQexB!HaqV*1pFOZp=P)`f&PHzWgWsEZ&y z#PIL;_f=MYP)P<4I)0qP-UtElPjnK#pz5v(a|?tjh%rACAAp)OFWF_ZL$5(;dW5S8 zhDag0Jnpu?>&X*p+l+YNQ1&owmr3TskBAU_GVwzEAo)^Boi!nVVtgV@kc>aQA3_N7R9 zA$8Vv{RXkeq-$%xAJRfSS-$e^Z;~N~jX(1|e~pzT9jl1GNr_TN>;kY0npIxzv6(-a zl)ctl8X5*#)IT|+r4-ni;QAa>O3!CB!QG$x7c%vP0o>l`PKYe*o7Hy`>?x~H$$rVF zx98f~0K+WyRX6{kwt0y~Rj&9M)*@5zu%N3RS%6aVCd^!th=v{ASNmm|>8-MCa8+_C zn;Ouc`~P;Rj18VuulNNk-p(LtNeJnx2U<*Yplsf0B)N+|6ySMt>C9 zX14Y_H+!54@wvywt>z?(WSY+970HI7*7XVb4i##{FPcWT0<}ypIVwdNXX%V%6pgV% zbma86|HsM~=c_s~PA$eg3He4oyX$^jCc!&PFE5|;C=!jV3xufXH(%X?=gp{^TQT1l zHxC&|H%i@fk>t$r=B~9av6Y-GSNnDeN=@u&jq&FNqhWz{69tRqqUEk(+%>502a7s_ z9fH`#)6)I*NuQ05+i`(Vhju@^e$?MP1^`uf>|SzZ7Ra&J{~rUM~i4kxo_rR?W2+s2vnRmq(({w!8Bs zJ`Mjq<-*IhQ#gqf#*jvn1APHV!?sByR+b2Z+~6G=d-@%3bVF7Y587cKpZgBH zN;`blE4IC5S9<%U3holmRLIhwKf~FbY5JKm^$4O|EX59|JK9;OL&H)h?w0em=a2_Z z%i1k-v^jlbI3CMr@jGMRMRaWy5C#_e&IN|u1fNgoKL;*(kl3KUQUM~+1~n)%=;1p@ryXKAUS!(^r#Eb^AT zixx~FhemDX0&wf6w|g1m=kcuv75*|5Kw&kLZAieNaFxZzCz~2HD$lm>FXf{x;~sU< z^Jl#R^wCBrZ*05)&R+dH2e$L&FYkptWA@1eM4w9Elf0OWZYtjpd)^PoHZr|tsfwJt zFmQ09DJE=gpHFwn+R6+Qjbm$hs)=2x)N8}62+m31@|xvBOkmN8Z6*0MOhfBNJsew3 zt@|bYVf|O<;0boh5enmOS4B7aub;OGzCETKTeS1cT{1BCY-OG`nHN@zUMBj+RAWCE zy>!`byj(HslX~6|!SMVqukyjyH8yH6uA;=#W6|>@v>z_%`ODFg!dd*z&{zkJ(~b0; zfuUwZL{ZDrkotP_?8!uSub}UR?J&x#x@yOTB^pVv>*E7{lxSjfG9YPI z(vllS5C*rnAE?Vm5{XnGZx#q}_$W?}v~!-;vM^T4H|dK344{d9SH_wv?mV}}HKS7l z?{MwtktGip>CHcUXWQsvpRGk$DjyE?gm0JvxzX&k2g3c0uvZj+CqmW%?cp0I;DdDd z_bfkoJNn*XexYqq>-k0wHgkPApBqvDO4>rc4<9wdM6)N;UNXI%KzKq%V4}^zBg*^+ zn3t7X(xi8VIQ`Q!JNH)=N>kdoQE)d>#SF1cYREOcL0|$`@9%g=iyL<{LZ&jvP4o$2 zWW#riv2rudasS%{$mCUlP*QP;T6%Sj$cK)FvQC2Q%H0R2;d>wdI2h5Y+}fg*CwMjl zDssQuU^P2!`nnznfm?AM(8=)F10)>Lig7V{w9`h7iQ@|7vPMi^t+9UKRw0P3EV3TV=R)w#`r1kvNQn`+2ng}CD*A6;u>=~N&zq! zF`)AMf$pf-SPZAE6Z)$a=Rxo-Cv5Y7fNU3WMtY^o2!hoBfsEs;jb~EGC;_SG(Wjg` z*%hTFjg?tOzf{=HyyXZTF+<)CgS_pGr(M5{IKu80K=Ag}{s@fW{uX33}9YX^G zxz?wqFmAx_tLvjM!;qkxZq9$_{%6J)g=q0UEXC zq`WA($dn75U)2{(B%uL#^EiId7`+!KLw<6SNY?adtD57HG-eQPnO~YVh={;nj4aan zCTdr;E_hU-SROGAWc2h@9%ffi{`(5L0>Iq-q@(H@V2I^ES?=b*K_o5zWF17#V_HIz zsX;4iqL%|nt>u|P%kdXXvbQz)DgMj*$6)5Zmr&+Tsnb9V1D{*6?Y9ld;+Hd9mbXM< zi9=UkY*S-JESB+5FX?x`^oB^^v9Rhe#)YsC76u-l;PhD2<{;gqmFYvv4)q^aJ(;qYVwbPPaRk>Wy(E_ua+WK0L*)bd{Z zanqGQ6~9r<%=V%x)OR3xR-h)e=y9SrV*5&!#`CeGSp7eSrZcBPdzx|#pT5bA{ug5^ z`^^WB_-T9UnDBck`nZD^+Z_CpKPVf3{u-?m%E5{M4YK2I6^hxU(tOtR`X1@sTacAZ zQZ@oIp<3Q=z#G74<#kE7A{v+kwjCc$E>uAe#7PVDhu~DJ*{dF!Db9;n@z~$D%Yh=U z|0R-AfMHdxsgLyTVD$Vo9?9KIs_{j0FftgOb zx##xEiR>)891W>QMF~!w18B4?b|D}U2pwNLY-jX&tfB>mCshG?Cfdmz`wrN>Na>&Q zgQzNNXN&G6y$Z=vT3E%y1-~1P(7I>svSZw2>TZ^R$^3oZDvD$s8=sK*7@@A|3;9;7 zU2Q=@*6i0Z4q}0Cn99po$LzdtuB+bM;~fqJ+}5lbQ8N^(^Mz-wF`mu{l&NY zk<2$@Y7vj%IXu(o!S{_efV!iMJ;Qc-e5)5GwTM|h{5y&t7SVcatv#a!Ynq8qS_Cx$ zJq0?PMR$g=iIs_Kp|P=b#R5fD8K|r(=({30-SK!X*whwI`Vq>Z3q34tl24j0FS#IL z-}RzGE3nVlyJ-E(E(7z=7rb|r?eq5FBjG2_2ModA%}$?uF7c#IOx6y2i2m!jqTaBx z7D&-SpE$TqjT;xv;bOApm#nov%qBweo?z(Id&bKuLH$?a3FWNd`WdWEo=+L3(0!>3 zDX2+Ce!Z&%m{8srk%z*nY6@f18fB z03J--5eNH>aT(0)6SJoR+X)BC5UMrI&G)ANhC<}stifWCoKq0EeeAB5SV)aO+0jC8 zpb|iL^O@59x^$u)dJS{q_*;Ne@)cd&DZ)b@bQ2j+k)EV-%I zYs8ul2%PXF3F!G)keT>OB?$^sHfx+qNFlIJy-T-Sle}FCy-biGBPTKRM27TqSK+5< z_goDzL>$<+&{eT%_?2hiqmEAlU(7z)a9J5n8aH|W5K3SDh7^tlO*}wjr+3bi)`9d) zH8W4OPY?ezL`a-=!u`eo2gBky0Qf4UR}yPyFQGWuQCM`KojY+9X1^#%MOD1$|H#US z@@;{$V2vj|gTiVQ>^Hmfe!dd_Rgtd7oiRF3jFLv5%p6jOq7sZ_erg7(+h3}_I z0*+Syc;P0yc8-shQF-^A8|kBRTJXSbyZ>9lw1X%wII26s6DGuFo= z3FDekazt_NHprAaJ?6Em`ejvO6(7|t6r7`T?sS6@&G~wA=TgNt7dYKu$P4cOTJJ|w z&(@d&_}TDk|AkphskD|cSk{rZns2_M=PG5#Y90KS0#4UGX;c24#qhi{eQGz3=l$rt zx9#TZUgbNSGTr&jC-na~#jK9579Us@6=&(?niF7l(IcONWtfhni^hPFV_0!FAkC+C zybKT2nEN=yKu?3{gp?4(_=PhJst^5I_(V^?26mYMwQ5?)P?z1drB!adP8;6D1)N!) zbhct9NpU-7#5GVOS;73de;nl@x_j*rmYCNmO_Iz#t$z2{F3Dej)E3wvXIhGUfGie= z$;wQf1+HHj$&!$sn<<2%j8*|chUFND(go*)2QArd@4<}&+DOQDoxbZ`45~9lG{I%d zk}_{+wpI3a2=H!@AG7(L0HHm)t%C&twtijgk+-0(2}v<~JRpz@psjuRJ?7_o%ze%H zJjQr_x{`bh8L$c7uN1YUH?A#It{w$1WJH3WCY(HEP{Yf0>S9Cd83+5ADj?akZq}c` zW5^pHR=(HaM`DH%_^I08>`WAO(qt_yzNdV3*@pz-XD#XGPI%Q+2O z*9pJadti9S=l!f!5&_@t;W+o7)B*HP2zadPo#sY&d#@fu-%nadRHCa$_Q-agP!(IP zcdgdAG!6N*Q%6h+6PDBsf1-rcjF~3Km#dvrVm}a$FFZ7V@8}$NL6|ClD3666MK5Hz zI_A^9)yhY|zphifhjyY}Unj}G^SnI0qwQYtdprIA|0Ang;hnpYgITR6t^!_rDMyMS zDiMIZAQW^pfAh*^pLZ)bKQ!+0VDisz7))_sBzfYPa(AED4r=IFoohovG)A&o96~}C z)AQ86hF7R1(Wv^Crl-00!}on6DsikEWk!iQn8AM(+DmPLUy?O*JDFN{YoYqv2%{WQ z+h2dFlL-3{wzp(x|OV!xD)pVh3$%GE9D{$f%Th=J*p;Z%oh5Zxee9l^QM^G2FQn zzP-sW=_g!PVw0OSb`qlj*~MXWWY25odt3RQ53{DECimeVHAp7V>mLjlS;60<_9cFs z)9v#;a&H^={p9nQiFwX1vfo2icpVcJM=|%jwI3wYlL4UlJY(d*nd74-YNWK_wa&&f zH$820t28UHUeIRjM^nh~2I6X9#+*L2P_LN4BXQ+>t@W-Z&s#V|eu48J6F6;1?{)yV zEyXTKpS0!NQVl$5y|m`FO;^Ue2d;m8U+x4C?r((pAYKQDZq9Y#jQuwXdKdmm=^j+N zb5J>|rBXE7Cp0uZDT<+%p^_yBqpj9eBHyp$U{hz0zh(8|Iz3akN#Bn{xA4{dl}1+4tR2r zF)l*k%({P(e>R-fR3_A9vCKD_h7n!0+!zz@wdBSBMd5UgymNPx>eFxe@lpJ-LG7wg zQ@5gjpuwx*Mi`(;&W9n_-&-16E1|kGf0!QlqHX)cO)$d`>j1S^igb;Av3nhxU2fB* zrWl1y+0;yXVQ;KGIYo`P+wM*)tJBJa>1dLN)W=%-Mdrg@nIgAI&<8S%UW3$7EDM>C*+ZngFSicJiPW$ zrHsu&z|@^@-187kVKs~f6E~;-wAa`5wf=f9tK)b_MNo z*gWodP&|I!jzOw_ZuLn6*mTab=Y8&9&e#@8mVL93Nb@?xjJ@iPk$&&xcP4lrGjdvU z623ry-3NoEt|nS&O+X?aFsIXqVbzeE#eNYtO(3XAZD^{c>PI162NYta)oi7#iq%G{ zDH}x88`3g6;);k_ljkiDW) zW%hU-jdzi~39Yy+^E=4>KN=Qh_^2c{YgK)7%F30EKMz~9FDIi4doJH!Vic>+8s2fK zmFG9%LSJy(SB{cK&KqJx?-6P>^y6Xc1(BKI=XxfgS{YeQa2?kg>h`Po8O5!$k;g+9 zSttCHnVy3C{nU{;B0pn`TJp;KDja4FqeXaken}!PVrpa*kKr{R=yfQon=`w-(-7Dg z=h6koCBu(Gy#U3`Y|=+bxl#ow{7Rr{3(SV?itmPrRlusRqMArft!8VG5-Ic~aZu6= z`RLENUK9P2QDJCDI>9SG2N`i%j2%pmX_}uMx0%(b_OqbCzVsg_q^@v$<4XId#54pmwYu#xqhPcg7Hk*xQsrHqodfQx5KUN+GFSsN1z zHDo<-wlC{@)yi={UNcK#kOwC!eZ@8Ysp(_Ce6@or7YW(>TsA{#np*Tn+d^Zm@MpAm z`ReTYc8`a3JKAgw1XTLX^3ORTc!h5`8`d{W^&O)yD_z!>enKbJjA{OMDLEc9r2Wd2o%`r_*#0(%Qwp!j8 zm?vnRu0D~2_c35(2lvzA_Kno>r*seP!YMB$bX|Iv*BZBrHMc>4^EGn80k=DF9y$zv zdYC_F%{DyS=U-}S%?ht6APx03O7!pPw2NQ#jsJ9ii25h|NOM>o%1ZWbGji?=4<>m| zIQRWef(3s&rdnQ+dapHB!=23(Y;P5e#HGj>9)T`q{PIA$b4}aajV_%cL1I6`wfy1l zj$@*kYm%H+N2Y}K*^lDYx-mCLo0W4pX3_)!ve@-t#MX#Y>|6(4^kKYS?L|sj0Po*J zgIQx+l1ZKy;CTt?cMFcJGg!YDd;7+Kvkwvf z#w*Q-Q{8i#f6%MC9i~wW@erk{r5FO}Wd8kG_I=W2&HV=^#!l;g4~vr6IsyvYR*gW- z!j0Ku7{v0 z{7n4?(bzPt4NOQH{YMeM;)`6ofg%P?h>3|LR3)*sjPti0rsX@<#Z*vndG2>4J{aQy zuN;{V85q@T97!PEkcOHsRUgP%%ITjP!=@~Q{P_bxV@>`X6>c}nxWcxd@xgGT|4F+$ zxpsg~m`AOLNBxWHN}|0n(b3tRT&wz^+j&D^Gn~G&eNSp2-=Bp;&mxFHhTq-xE~vn; zoxv-ASxrszGOQa!KYu{d8M+pRfQUi%Wyx0xRaVHh2cW?r#q}_1vjV@<0>1V_ot+duHvtv}G|gv|a4@KMrx7^!ABh zbkTnyxjf6_Kk;-G}@JP}NP*T&@o$>|)GKRw)Mzl5rmf~bhM%y;Bhc&E> zdVw9L1D7sP8Ke(550F=~t;79{jX~4HG!T>_)g|X%<20xd7VVM}M5h7-`h}Vi!^`Ql564nAC+qUxmSqL@oQ)nQ80^ z?afBdJ64NO0kIyNJl~=#n%$HPyxM4`5+DSDMrc#O3!8VUPOo**TX0}nnM;(63ztd& zbB5yeglPq273Jc_Sg^wjI5Kuf=^y@-;HS9JeVej5r|xzNzhVQ6w4*Kt7OMAQ3deh4|JMtK=5Qxd4-SPGga?S#4jA6x4DFz0zSQ#o)78~PczdtGfveWg(zAI|#8)Mqb(3pryH@>Cgf?;YdcT_5Hy{$nh0sywf3ayYc+1^he}+ zzo~_IZ?}v#TWgK6yBCpAG?XRLQ5U-+?s4s!p95)sNjJdKnPyjCz_SCoPe3^w$%mXJ z{uH+S4x_~MEw=q@nM5L&eU|uvN!EZP@y~E zenl{3s_Vc0?yF}X^N`Tvbl#SD@z5S5c}}jIZ+rItT7c0S`fC7meARqn0f2yidJSIq zPc)cPVi>QtqzSxQFmb<8XBA?Ad6Vf{rp$;Zg28yLWqv!Qd{Cu4ftwdSf9)_BP)IVa zv~d(GGB#r-yta2-&e6y3iTr*zfRy1*{PD2XJG-5GvSJ89bzUCZ&1>tSZuY|~m)f32 zS^XcVR#aZ&Il+?UBQu|BXqpR|+FoEUXHbsEKpT@IWzG4G|eFyln<`yL*8 zZ!QG!wh1AWiS*l#63ES8{Z+G+k!gM*VVtkxdyg9Xq0ca6hS}ee_dO(7Sz;e7TaIaoXAf?ich zxJ(nv8};9D+I(j9=YxDi@DlX(<*H{iNfy|VxaR=BKY|`-f3foRP%@5t@yyTdL=iA` z1%HS!wqN_R-O2X)dy6QFfl9|+2CElq&r#RG!jBAbN|{DY#4P*;P=SBZ9=+)RP0a>U z6A$_{RqRIU8KdC2@&r8xH9yPIf;ymD*s_K8(L za$0Jt>hUK<|oIVchtbrnPues$fr*d+PB{eP;A%)KQGX(!D72`iZ5 z1)}xo_V1r%2!m4^uf4QM*$-8%7Sk8t0d5ryt6V##{6moL)I4+WuFe)k{-Tva0xGcsZ)R5ffW4r=d>|NiZ~l`_nG%RL)A*Ds+~@Rt2)!yS33fItuO8_;$!Tfe?o@zGjY$A$m#o219cfF7M-JQdGr-tMLoXfQ`ZY@htvR9ou}1~p&~DmRhv$v`kGs%lVoV|95kw>y_HY=Q zLz8Y3gz|tQ0+%%+$Xe#k2$LQ0=28mD&V1ukP$?{Xa+a3``? zacXT~VnY9qUPu4e&BOqLz#Hc0IuI6#Kp<2fPbsZ>IP^jkOV|`9GmDq4=Sa}wx+`4) z8V_+j#dTdaZ7hdJ_VtQ5Br1(0G8~zRY_%)T!jI|ZvTLvbe$Y56eda_;#T<1OmrK8K zwh<%eA&L?-k6x8Q_NTQwgv@Oqq^xdWzD_=qOoz4pH0-2X`^Q6;ZAQSAH(xo<>{`H{oSK1n)>48Wn}-?^ekuPM85zzw0;IOz^y~p);DJ;OZL9m zN=N99VrLkUDvGgTfF~AHcX&kd#}%V8BXPb8?XC#`>=%Tg)q}lebk+(K-XN?C*~y!( zg{gJJn3{3YLJBhvWvK+>!`O}N?bIM4 zV7{{gC(DBH$8vC!D@i@9wd{5<9MQ2Zv_D)nsdTE2(;Eiob{zb1Fjw#N>mETo|6~rv zGQetkTG&U`*RhZdh2Zz~w^FK|>3a&zaLq%uy|=Lm%^&2_ZN7IyU;yS8tm zV`r!m?8lN7>#4iw2A^W`K?Dys>fI_LHzyNw12bGq?BWUsSS?fJ7>}>AjSL1|^7b`- zr`<{pbL!GzswKDUf}xiR$j)x$>v0u_>Fc!cuTAXuHY4xD=;V177bj^N!lk95Mh`|E zJ(qOcIS;Ny2(80mh}PR&Q&&#gbr(!H_w{cxsGBrR4PB$B8k$rt3fU}z#jfYw%=Chc zU!Zwv+7r>BYBtLAH>99IdRN;{s{^|;P7ebO60;_#ZgZSUD=5pRng^*+mwugwDuo3} zC1b4T8R}2`%&e&QtWu(Zk0z;?!5vnqtG1}yQEnSf<_r0p*3eK)il6#nGO81@i!z>9 zM9F^}zz5UEvjC7w^#9Aq?IUeH8c zk^@h_2$DGfWwsgV<|!{g!zhc!%*egrKE~O+;tl>adU}gTSDD81bt=pMR*aQ$18HW4 z#LCIi_iSOuH_0&RBk<^N{041HJthIh81}d!Nt$=GGa@u1cI6#z#aJ zh@)BV<_R>ha#W767I}QCM*H3lB|y49M5vISwUhbXdwiK;)kbD?Sx*nz)my{%xq)(E?q-vDvV2*snh!Nc@D zy9yqej%Zr3IfK`}H*gdS1Y^>3YIE-~v4y5i@lu0<5NE6f3_q@Nu!vO3f0Vezz-|+y zCteLbLAWVw1!q`^GNF+-_`@;Mk0r4wHOozq_IKd3KF$}~c~2xUz1Hwb!%L!*Tc z;G$zrupYNRr!!Q;&)A>Vsnz0jCL*<&478KluJM%-lCvJh?A_zS;eO>yu?qGN>1!3OTG z6T%D(xstj`bh%;UEm`?R}^O=tV||O2h;3shE=_vWo9Dvq!D=vo99&?h~*+OkoREb^DFF}6G=q7_-#c>922NWU-|>br*z@Tomd9MxoP&eOtKPKcR6Tivq+o=y&e-PK$T`&>}`@j`UKDd)! zJ^Cp@(~B&>a>I^!(>wd{!>%iAFFi5WX<`U#1A4SUMub3i6o{u7!_!C@R^rM^kFDoe z)4__g1DI-2B6@XJ!Q4R{n$L8Roj^iw!T9b5uCHkTlS=uUr(CcCoY>5;UR=CG)^-w9< z`<7|y6xh!24zBVA5WkGT;UpzU_4Qqs*{Zj$A2=YvpKE@@u&N!I$s@=eF3_kvv1$p9 zq(s%CG7=azBCr|55y9{wu_fJpFf9zC$j6CqzG?ZgR+-vvHh_ro7O_g{Uow4`Fa z%tWlU4RNP4C%f3%$eCYzAyy3hB>Y93yW0NAsNQ?Ey5wFb_$_H-bKiuOinD63R2jyA z=Y_3Zf-L&`DCc1O3tj4fwWO__>%xrfC(S&*ivYI0o!*+gtAUBeY?BdcgHjL>m6F-T zYxW@~E>t6CC5$v7os7n~*C!^6bK5bK!e{tdI&D57uYbuVi`n+Q=?HTEIvaHk z&U=^O5_*FyDGOzD?eiN;Gq=mVF77AgzaLUG-q_Gkr>zMnnxy!1WS&oAe?8qj^ht82?=$Z1RNM;7qOPyA&_A|NaukN!q#mMrjinQQWy`c%JveniKVxA)IH*KG8J`Lx4h2n|Dqca0s7 zm(6_4A_j5oJ2t4he93ij>VQO_q-NpTXKPsQXz;M8yqW3HHQuq%q&ZT;LmXLr62pK!r?V38t4KJO*BtkS4EFTZ(ZR!Ew$)HO+k*hMhl0V?5YHw}&# z*WUGh2cr8&!;i)Kq0G;JbRl0x%e@^tPn`b`!TDj&l~6SQHrLuR4-_-YX!R$Mm6A;eND&rD*Dx?)X3AV_i}Smd8u|e70Md9 z{C9C~juEz37s~d~hjwOMe%ynQBK5;uCr9aly1dxNK*jmsJ>B?xdo~+>YX;+)fhL1D zHev|9b+AIU%Q<#*`gb(ixo=`@TCHyl=ytoM^0Ji>E=G}-aXX8)g|I3%l0|Gag<_wQnok)&8za89cjhprL`tEDY6&FhFJtn_7>|IOcLZlvWGH; zg;mZ=Fj+RVz$szH_leDMT)^kSDr@WQmW|tb>Bnp6-~D7GSx&1=g|UMN|AE2*9}%c8 zNipceX>?v9zsilPV6`7K^CXc?wm_2nq@rffZ>c{G7+FYmA_f8U?|>QZ=gHK~pHoW_ zO`FH?xuAI3cwE;K-b3ZK!n6irf-SzH3RYPjVwf=t7AF+Cbu3FwB{TWP+x>++Cv=1kAA8qyxOT1b{!pXca5-a;!x!Jw$Z}+W9?oExnUQVl0ug) z@Iqh9|LU#dzZ18drL!BPtnKqYia^Zc7#`9_Lv?gChAoS;bP)tKzKo(xsGfcAY76Lp zUB!bMSUyf)@LVZs`q6&}2Jb~+OWg2>Mjpx|+Durtl*v-|-z(p8M(f;wtH30PIlX+c zbm1Y%Fm?1n7NX9_mC3g(^@DzT6{VD==16-byFi5oXzhpXPPnSbv${K9Ieoi3PE9(d zT#m5w!l*z}08^%jgo-ilX@*te@%Um6%VkV8LQSE0M~~vzR~^1k@Q1yzh|V)p2o#{5 zKbrUJ3x7l%P5a7A#<7oFs{yzuhOzvdYT+trGY+2Y*jqqgg$sn}x7(u4MAC)$% z$tE2trXlqvfYfuRHk)#agjCZpH9(_GGmf~DQ2-4HJfpAy_O_98Ze5bO-eq=&{Vnm< zT&+|x3^RQ&iv^)ydjUu5? zT#gjoxzw^DIS$~#vc{05#YlG2r?6I%7zOYGUYZ-vj-_9e$g`j)DUO`G+v@O@WJjIE z(3f;eP~u;gD+-O>uwCuFk5PWVCaL(A;>0pX;}XhQtRqK@I(I75s68by2$X2B{xv#S zl*p1)K4f6?Yf6F}2_ckI1m-|Am{80+-{)17lT~R@QUp5>RtzY}c=be0zzIUFsQ zcB~T>InF;vYr0V`tW?&I_U`noCpb&F*N56^%ijVawY@d|n_ku@{noe%7rArFaA%(I z-b=S|bil3+lu=!Wu0S4UeLbJ9(GoeQtM$2^`kOhcaP-C{TB(xw3cVVJQA_*a3W6gQ zqDc70DGDb1U{08~?DjRug81?q~l2wMj&vE5*)WLZOb-}A?Khhm7-kPRZ#>?-sW zw!sK{xQF9^ZjFStQfpjSSRr98i#Ar#cPwhmt34a4a@C>i-$aYXwCUX3ZB=vRhs1PWKHrYzz1e!Aqb#w{K1;7|xM?@ea-@ zFKL%;gCjCB>fuvfjY`lPTkJCvkQSZuh4@xeG0uKCc6CbJZ-;=m0&6jv-?@+>IImbH zMpZLpkAW`H3r!JIFY$5R7fFj%+_}_6{twk_7fL?RQRv-C7eg#MYo~WF_?`-C+kd;2 zmdqq68#RNX)}maR4hmx}5%+7`T8r76cjP$C6d3xlT1spfsUwU4D+&x__DM2j0Y?wI z8kIN${)!<&d}270gZl(7u3b!NgMWiRgZCaV)@&K{=5OKT>x?K%nqnwy%75@^nwJ^d z4qiY10nh#HFMSE}KbDgAAj%j_k7o};Re>*TeOha0IA}_ry%?8Gy1q=P#k}9lB1AxvhIV_zoE%h4ummK#Wt-)XdH8D)Q8RHd-ee7xNp4)FRJ~CBG`6tugbb zn&wg;-lhoH>3D`rrA%KJIGYN3J#y1ko|QZoQ+owL3w;`&m8QN@O6Q@R28&GKyGAb>~wn**C9V>>}X1j-=(W22_Rk)G%}N0;`?8q%GzS7ex`Q8 zu!b9OJ^(ymktz#_{(jTmVS%v?46g&EOg%SHOuB7D<#$&{1$S|Y-{mRk_6TL5Rbc2& zC)e{cYS3*|YU@k4wK+(6=1xS)iGvSRL~N6Q4z>@p6|>7`Iq`_fqMtrJUQ+?pW-Jc< zJV1HY@0U6#^D*` z!u2??9&oUIe7_Gr!nL=Oq{@#s&T%m6-0Q(Mj2zT`rHYun9`6iuY$*~K&~?cPp++F< zytcqP;gfgRYusdGVx=2c9sEtSpR8;>Q5c5+#Tm=Ah9875IYAbc?BcT_;2ZX%E7WqJ zF`?9tB9jmgD@axei3S79^!>M<@;f8Vec=hiS;+u#*sH7Pj4j&>B9u&#l6?!>5OO-H zXUlb=*%WY@dW{i|H3G(#k?g}ir`c<9o+wlC!ORZzw`l0rDK@2v&1U^+<6IKb?|{z! z<`f@$3=z+SmNKz8GI(U5;AY(fCX9TYM8m^D(%AL3JQ-v%zc$cJ54&_t+rujnVBsDu zri@l!w$PX3?eM;xi^oHAa!W5uEZVQkJJ6Y{fN~lnTY4|j>=iG!y@r*0#zGD=8cI3Sm{pfky`|4;pLO!yX z%nb!@9m4EAYouaGG8QX1cn{VN!&0u1P}z8D&ac^cIn7g5UUpg=3%Z&LX!@&qiNi2F zIU4ItF!n>cA5M5=s$Zfm&NLQQ>`v>r)2v{u3+i_|bF@QZE|t-o1E@uz#uqgKXeRUT z476W>R^B6=!0(%|h#bmDD*#(ZHi>GfsH-iN%4kZ56B@wRcen}_r^78<>zFhx@Ie7C zl&@8>7s@*_3NKxPPM|F)((!MniH^x}cLElto%V-Tybq#2>c3yag&Bzp{h??x*@dt~ zWdl{j+o6n?k!XvZ^5o{SS>lb1+}0@TKevUio8P%VIi$1IY#P1gSpHo)UfWAiOt*@2 z_3?gf%$Tmwo4v}{3(~mr%2k6p6uddZCz_w_`d=4uz{gS3bHx zbL~^lp8o#=^u}p#KX1%Ql(61W61~4+?eN0mfPQ_e9RFANNmx$a6SH|U^tS$WzxvGK zoc=yDb6b7U;EL0aHcY*&cTu&BSRG~ZkwZhEd}+N4wvrkG&CPz`Zwb{#M~i`_?r?e# z;V2eK7WFavji~-Mb*!O3ekA6GJ=e~ZE6!v7Qs^32w2w6@Wz=4}{>MF$edj0bw9p4d zOYR59;t12-pj904^EJ=su{;0Ymtwf$V#;-&Un5cp0gE10&9hXEBZ<*b%<=RCPq2Sb zxKgw*rgLTgZf_wHrjH0~W5q}YEq@6aTm69=&>6#PGHkqurvBt7%F;_RTCEQ@y7j zA+;PzOmc}?2pFFoB+O8At7{+;*o(_MLpU580sk(khc)ec#~R#}tT5F)E8~5m16Bk#NQ!U$k9YNUG(BTHmp*%D12`bMGxD z5@;7+l?FIW2Ld4Tk=o7!pM#&Ogs_0N&DaAtkbmhZMJe@qTLFBEak6#ocC_UFImVY; zbxFsCJp`S~CqEtQJBzlLSs+2QkuweAXS;D=t|YYqUVB9X+fV*GFT_=`g>lW=ijB?D zK?VG46$Xn0;}*oBY*|6v>-BLi4Sj9%wo{gjF=G8G>#44-MS}M1rFSoOtnY1M&{IBb zMg+ZqZ?UO2D*C55JV8|LV~pR`lTpOlT|x#Uy3^Kx|K+6$pwDLG!ddP4X@jLS+Gg_I zBif9E{3%+^RCiMFn6VS9&OBNVQO@?dhJ7q-ct1SIdS5;A1NUDq-HrobMtOXTaV%JW zLFbLZgoN~g9lJ!Y+aS4U*$x~QxuA7{7O_32hF}Mu`Y4w})@3wsr4}=hC(Rmv6bObV zM}*luAQ}L^&j({O#y`ld94H0=s{4c1DKY~CW=tcw*4N*6xW0xT36HW~C_}*mGwd^~O~H*vLig@{4{rarMs?URT^Dp9i(enkIR)y!qcyt2Qjw#z=!L90wti40 zpGq>;!CN009N}S{!PTv_A>kk8H_?@Jw?eF`8=j<7f$(vqFK|jjpyMFt`^nMAnnqMx zUR31xKx%m>0x@M0OAJNYXq(MuLSFHLNpCiFvs3m?gVqqq9e;=&5A-proLQOAge?j} z2nKbuw7s6??PB|%0Q^o%1;?&eu|4Ux_TzG7t_9-q_CAO`k{=#iSz`hClO_gwM+^v* zLs>6g>p##y0Cm*p@^}PNQ641m&Wbr1Vq;nujM1OY@rJ@8l`FlG1zOdot)A^#IJ;E* z&J)>f8Q_Ey)@$&KwCu3f&b3qY%&>3)#AsUE6rCadBh8R{qF4`(lHJ1q#co2g0>DoW z`e_0KNS1#Ty+8uc?15ega0jw`4+AiGqq{a9EDD14veOoX`gXcpVS6Nw{100eNu=Bj`Yn4X@-u$ z0+9F?yMQUc!av6fFrNJ2rdC@xZ@gY~J+@!7>T=h;2jAgTjD466Q)H4yICao!sOcvU z7ph~&-D-bVI|_xKd7GXXV4V)LY-t|OyHx?Abeb%@BXlwV{Af&?k43C(-GA9^k^4<; z9$dTcnX=|ZE{&6!qd2r}wy1*s@ch!9<)5TWgdv<_fsES3T)k;7)j~F? z0N0ViRW#0je#oWiAw`CsiMatv)@7%y%EWdp-$szS-Kj+vSvRi&R#*R3t8U2ED6;F) znbzWnE4nTft|wy+HWj7~4)cqFdEgXVlm&WN3F1U6d@RCp3Z{ zWGz0t-bmS1`-kM^yhWm_yx+r|6iUx0U=hQ*EP`W=I5{cu@}cf_eX9WZ&!@}Y(~=wG z&CrEC-lH7tpZb_t^Q!~y{Od-lv#oy6R&=YiZO|D6Ch+17G|*|A+im!K@9WA;x3%3K zDYBhulhAQDd7sb3dvBl5PmYK8u(yN3Csq4XNh}C&@pKma;0))8P+%yD!pEIK!rNw? zN$d0}v(c_2Ej<@G-hEh zz}zcB3p=rDPl1*~n0lY?f+jHBw=)VsYh$NY%--(5#lzn6390RTBwJ|0YZFhf0DIWO ztTLIPo9+Yvbx3+rDNkh*ju-YP`x1N@)|D%?qRC1Jig6zU!;}|4o%7Mx0|s%)rL;rJ zOobIdAPkNuOdIG>2+IO2I+6VhL+2jqWP62$hegY6+;Dew>!yY8@ifj7r$2IQx3&fz z?X0e}@%AUUmQa^9Yo(3c9O3IFIT{y2o$25xH0Z36)`@z}T!$x1Mj~GdE?u`LXxD8| zd8@Eh_{^)T&U|~osZ;ZL;ujw^n-&-fdde&2xE`fgQ2SH)S*oUzw{Qs?zu%Bpt0~!y zC7JnjB=>7t4cbSjNXhBuTV)@7^Dr&PQlEWuqyM$nWdTq2lSrt!1h-&o9$l-#esH zh|GGWh|ZK_`QnT=Px<~L_OevzbRU{LN17Hk2({p9?-0?;x*2g*=41d@K011`jt^Ej zf+2Q6fwSv+uV}yR#$51{UPF2gZ)3>sPkF37B?ro*Way-0Q8_)1St-||Eo-0^*{4R%+&nw0MaNL^7EKTd_{MQ3*(*Rj_(BJuz2HTktHXpnU;#7>OP7kp;I zzA00{1UkTV`H%O*-JPi&A5b+E<+^##e!wpJO07$-o*!sX*m6t+I9hqNm z6)*|7rk=5vb!v9UzcrOU+PTEk5Rhjk+Ih%YZCRTkQPT}MjGb!uiA5q~PO2SzSOZn) zqZu6=(#e-Z>asQ=BkNHE?ydb(R}D?ON33V#u!(QY=MaASmMyPq7&}wKYeDK;g7)P2 z*d;W9?DwTfK)xW%(p;;TW@Vn&ux=3XH;*X?fV;(6SChPj^1Gft@4=XSlt!O!5=_jtWq7hE>mAqf_F|RsO4`_~2NW5D==(l`zBOY1H2peh+wJcIb_~q5<)mgFw^)zepr0#}av~F=zluj z2`4rwVcBsQbX8ef_RzVN@?jY_>Zru`O{DqQPJ<*iRXie!E|)PD*;uQ-1VI^t<}fF^ zm#77Za*8-_J3c*PZr;tn;t@_3oKZeRY!qxtQNg}3JYXLDUQYg}60N`s3Qbj1r4#T; zV!-cAV*+8IrAvY(ySEG`SUxBa2uTi_Vk7xX78%V%P2)BiDJBDny@n;5qyP)^A{ZTN z4?? zdNzGk5OUuG=;g$T^kSP57*?{pL5oF(B2PnsS|Rrhqv5p|HG02 z!QiV#SP7@UFEu^)M{rk1ozsMei=CM5(7fgubJWi5l%W30t@?9HY3&VL?PbG3?l9j& z*ZNbz)lo#8K;I2|X*wh%5wE$Ujl9@gPtyt~I=vE|_0P)6EhI2>FXm7kKhhQ?a>V?= z+8f#gjc4`#pWDW)q_@5V(+kT<*K+?RdD2<7hx?zM&j+ppgUo_(qO3MBz+;kqutHF$ zbjZx`LkE#Bc{Uf_F`N~-h4Ytkpdl$@))^vU8Stt5LFmVW5z`AfjP>`f3AT&)V~p9z zQ)DO3&NRZ%T1Jr_(O{bnR^h)+yQ-$TSe8(E-TaUM&e_C$>}26&75J`1-Ctn8D#vRD z9xSDK=$Wb$og?QJRi5A3@~F`<%Iq+J$}(zq(F}BUNa3GMo|Y zvb4%VEoZX^f*&Dl^MLQzj?EO>)-Ac75d>{3h@VEwdJwMPq3A&zk%sGI*F|hX9Px{{ zVA>e?BI-|d)cbpet^PMVezI%oEV4?;K8&qhPc^aK!a1?l%sYc7GFM}#CY+|w>}V%% z%P_4%D?R0yn^N*uB32`v3;0k6~bK*jsPvFye($|32mC?Vux3pE&Wu z5aDFt(QP14QDv!pd3Paerq0qDUAW~5DRF6s_grFGI_lQrvW;B9RPEPO zSx!o?)oU`SOa?0O8lx52HXXEs0cx=0$325dm(_BiiTEeQuKWpvo|wOTe|4cRNWD&% zkoEkK1JC``3NK^dKRvRPA-D7|u0G@aXNWFnEou*xqy ziNw^A*dTEGcVnekB<1P93z)uQP5a#9>e#4$jWVije1hxS{|7C)&u?-$dOIRicO;5+ zeQN$-960)ftZSG?&33YuJ^P!7pA3iE1si>HL`me?tD$D zy>7#)in+ahlP)e$ z3Q6HWOspxLrirB(z`uW3DVaOu(H5RSW)GD>^&ko~-6@Ijv1P<NG$&~uU^;2=*K!u0XmfT9Z;oHg0P`@DEz z@b3%HQl&$*2F6_QR^@6PeHv4@zP_IMoGf5#2moakv=8@B(0x8%sDWXyR!w+%s(u92 zbQu{v@SAYrgx^r{e7#g1WCS`KwgYWh$<)N?Kes-CK2ir*Mv+~?YoZV0};LlCN!-UJ4GXhe)q2GdPfn8bOn-P*Hf6Ir8bZkXz1VZh* zB?~bV+V2k?Pq76}`Jr}^)ldn>)K&xrYj-(tLpd3g<;&fKux9&!f2P{%oB!e3Gt&D&&XmH5SZ?y@Mgm?$dB3J#n1C}*hqpv&Z*AEe|KI`#h|ip@46&ciO!t+<==f z-BHB0YR~^$Zn#iuGM*`W+n4y2k+$^ms8}*_TF5_$$!`Y$wMPWGM0w_=v2re&nVKmj z=q+lV(Uw+Pso;Q(_!v)TKZ$U=f+;0-4!PPHHyt`|xLD!BKx7hzzc$+avT(8QC)r=J`0PlhYef z8Svp0D24ujvNPLJPdo&)p1bScTUY;!tN5Zf9>oo{B4mPbYTJwSfO^~GZ;dhhm@ja| zU%U!Cn`T?_2;N^jb9g6R{x)ko17p=wO1c%1H8@)RuAMBNR#qbr;Wqei9Dr#Begtjx z^n*$dN0^3)PF^lOY`!_ZnWCP0gqBVAxF!g&ips=N23xR%fbmyc3%fmSPCHk~<{C>O zrZ#DHzWd#5oo7$ASo~D0G;qGjv)axH^<3Yep0Aaq=Sh6@X1Kik$5Hp&F#lu4I=53t zD_K|0B{m4qbAnzO`3-M-2w78)=YDmjg1f5^!Blm24YT^XmIj!0K=Hf=K~|qbRf^bj zv|^1cid1WC=@X}-5PY=DMV(u$q2HlXGjq|LogwH#MLE9tZ)j2ag+o|*N5BXc2poA% zn6ZhqUn>kG%x?+thjnlCzoqEI!1U}QI;;_A*LV7PZcapbf<|(v5g>$&C2Z+ zQ%rUUcdE50v>eHAYvGK6UEuQQk{MNaUw!gnD)Yn)`|n<|I`VZN(OM29KniQN_c1}r z%nn=L2f~F#Z^oWEcZXx(md~B#@CWSp3kTXb3SJLTTQ#_`1))JN9KVB`V_-FVv06FE z!(_BB8kRr<6MXMmXj5V6#u((ilUvjj#WDu1TiDVk2zf&|uy_wL5{$?m&2+>tw%nb` zbd=TJ^NP3D!ZF(Ben-HEv^`*j$@>wY z(*8H!IqAkBJV&P>9Xq0KNhR+W%_#vD`i8rWr*l0n3rBSg7SmrynD>Cq8A|(3E?aQ3 zLr7tKhMfRW1*N7Fl@EJbn$%(Edm$E=j_o$kgTxkTy)17Jt<0+<*7k2I2Dl4r#kl8H zcTO{i0u`bo2J>-8krJ5b-ajymVx<{7k!rQ=_Ym}K{x3baGh!3-!)trJ)(K~(Zh-L_ zFDdDe7G?_{g|vsqOVu$g_x<9`d8D0;m)D=~Au5W+6L!_lj#?(xoM2`p!aUKb#(!zJ zR*dplVNeqjBuOl@ji2HO(Pzi$Mu*IztBP$NFO9#Ulnz~?lzk&If)mZ@A^J-K)xL|Q zGw~>8BU&ir3WSJ#^qD8Go4@1-j;p)1tlNfnwI~WR^xBzDG5VtA4Vl6 zbiG%`(+nJx7SutkrR6umZ3v928qNlZui~tV0SSVr8g}u7Omyfr;{w~Vc!jCLTn$Rd;iU{NYKQ!8~HWY z2;1r=y&T2{OlH4~rEXuEnl%}F49;vSj|#UGJn7ajX=K}=K3*}`p?5w$a_IRT(!I~K zExjCI+c}NAf&4qU+V%Fkj&pwe7pF|S)aKAEiL!j&V%TN(n;9AFb74hp@4y(DUOIT| z;VDZ-tQ(}#dgcr^dvX9`*w_~-V!u%C_ssxAcbse$i}@kbzKy}Mk2k4u1N~lT+;MaJ zasn(-_~cZoJPKrMO^dWe1EPh)HZ!R$qkU8iHG4Q#EgPE?hY;8r85|iTa%+tWEI@I# zk>_qt3JN46xjDG<2~85OmL6gIm-jb1Ly;|^v}V|@g@7|`hBT!!u-kel@O)E19QG|_ z=e~9n)BY+Yz=2{8`g@3moFg>-nUdvm$R#v#!v382>=GzR?Z zG?kE6&fbwElItxOGZ2>Z7zxNswK^O`)nKcG_nIij4>c$3X8M6~WLss_-$r@zzK$Jr;h*eB z>Dbc4%RRlNnmlY|tEOID_bMZHp6i7bc}kBDE7uZ%L*X$zz$zuC7@};wB&+fTY@Ac1 z$q&p)|14jicv1$OUyZsep&eJgQutWu&b=@3-rs|zF%LpFq(%=X8%^h@P2ufPSHqTa zKZmx$A17ltYSxL5oV@XXAr;xWH53nx@``G3aJy4g(3~-hRNBn6D~@^dDta36e0!Wc zBTzhsERp~%`48(Q6(_wK*;r0iQ}bIM3-$#rH9EW_UP5$mJ91$|igz?H@JRr_T zhT)X}HZ2Ee4oQFykyR5Kve5gl&xWtl={sn@bEbb4Uno8API*1g$bO;zk1OV_&*#qU zc=BBtcKY_brmS^qx?+JTn5taJ7!JBXzUJc73I=0I+QYgLwu>|cxVFd8Q=4&5P{R0F zAe+jOzO?n1C5c0#!vnap&1fkQS~*9Ih+*x7eM zI7;8XkT(CWyOj#j&CDzio`Ry6R6gYf>-*@#ZMciA_m!DjE=$zuRFvk)MtXy{8{0QJZs znBFXXr$hoy7N9JIU)d*(%S0)(bAN#z%!D#2*+c zQpeTMYWTb`XGO8jZQ*YEy2;#Hz@Lh zSQGI(==l+xp>A{3F$w!%R?%N`pe=T*wvfl)sWv-d3Azvw2|@es|0)kc@^Z+7r5P>S zPibo>zh=c7|8nxvEw^-}k9RW0jyy?MQuzL!r`C2?bT^aDkUEPo2yPuT`3L!L4~Y?l zg7HTs*7sVE?w{JY2=^=ZROV*-%p`tHEbdrq=KI+U+r5e#PvE${YpL5KUEkd3D_5;)Q&i2 zSG{Uo!6S!_Sy)~O*t1TPtT-ML1eEO?v@ktj!Owm^E^>~|GS=<ti zqkw}+O_#-WPV#5Xq36}x^z}cXXXE#ko_BW||Ld?9oh!l5$FDoc&qKvebia#Y^<}RI zd9%3)$+B`L_MuFAx?i4`*M=B$W|=tR%A| zq%wfau{^j(;z~$>C$lbpMDSi3eb2WdQ?fZZKl63|R(%iiN0BAYjuO4-!zlpx&?tk0 zn(Bf%z2|J{mO~8Yy>XPoVTT&vy%&s~ZXhjQ*SAECRJ?~U_XOXR$PWKCZ*9Vz@t!J| z`qIz96 zgNJ$Ql>v`Mo6e5$27{^zbVe&=SZI#GGr&&CHpASnb(BlYR0Z$2=*1IvSD~i%lbw*i zudr)_S#-<~F?%r$&+`xSkK3YHLmhn~w5h2^2~gyR^WSIL@ZS$EGQlBncGWf*Y&aO; z=_rAG|Ad*U>T{DNez@UXUtRkBr3A<~CsJ{6R=nF2m3&KvfBkywJ}iDRxh!+SBp85Y-c*KCBCkd>%mdffPyrPqb3$@;F4&Saz~H`BA@ zX|%w&J5uy5vasU7BW8U%YnCQo7~l=QVCu)0BMSzd*9(5Q7J&Ul*9D7=ij zcHXm_fLK*Awr4k=;&DewQS(hz&;z(Y+OXa8Jm=+|x6!vl9loof^hw7V38FOQt;w^qMX(Hf&MS z)qWG>+nL8p$3}WK4HsboA?7}kq)iQY@iCrMOTfl6$By*=<2KjplA}~+5x0( zD7w%HNEgz4&kucqA9(a5{Pe&6r;vgmJ?1kD05Kj~CJA@8x*P`_3(k<*i@K#rF%(lwL3)*Hy9SFsWho;OS?c<@Vhx#I{9#7Lh2I42tTT+BqimY%;@7xOnOGVWhM&x_kQ4lAahF*U<+! z@zud*O;Xx;s{JhVgDPb3g8*+ZGob^vyg8+`@$U?woKeT#FpoR79xr9oPDfGHU_oz+ zpLI#tt_F`nQUk#uuG8R|jgNJ%QVOu|nf} z5^>&v>+Sct9kqAV-nUG9QOIo6C&>&NH#Dhy?UPVH17G;ZaP}kc!Y{yy_ru-SVCgip z??Q1!04808*+@Vl)I`(|0zAi2C_w0oeux77ABT&HYZU4^e*&qz08?ohns;FJJluK( z&U}FA(la<0J_I*kfwc=tJ}WIRlGePWK-0=;xc-s}A9xFvP6)IlQet&XtaNn!72IPN z1ftv|1|Mjn){)z8n_2D1ld%yQ_^3}8Ca~bMon>+hbNaV9{Ygf!oIt4ek1zJ)`xV9=rI^)P)@Ta%pMupTDIorX{9dv-SXwgD2FIkDsqYPg5EoIYTsOLn4?L>BKjo0yC4eHuJ%Ch-M+S zXX@tIhPeU#?qW-Su4yx!FcH(Hrq3i&+lT(0P#z!0Gs$U&$(a`dwdahEEif}rRf`b# z3C;N?4cG}G90jTilriaVqZm>aabKme+IZi)9SU*O9xKF#gU7Ehe{7@v9J z*VwF|B<-g0^E0eHMMy`xV3m**ZKmR#(JnZfVszYd0y6U?HRpZ-823Swr1k)WZ^a}Z zafvR2eNs|2y9!KAyDJdi0P7)$UjqA4h%cLOwDb=B{=UlgZffrcYZgm#(Q*vss6owuzpuNs8t=ijcm0U7@>!GE*3e*(&%t02;`^>ZNeY_A+mp{4pfeUI zNvHK(DyhLE2ox!u$%vV0w}dAg$H#Fz20iudnT&vq=Rzu@&v@7uSjuFf)6{vU(i*47 z1A(JtQlQ66_~Z&6ld%?)(pg9(Y#b`Q=e-wrSo$EdRFxNQ?eW>?F7v{3FS4_H z102ZBV!TRG2=btd?>P!G7>v|urxIGNn)s5I*|v%%6*B5%=RpHW2lnUp6*@pJjc>;S zIq9T2`Oa8BxJ&}tV;z_A^!U7HPJ1LqD4kIH+B{`#Z-Ae3No0~rU#n#FrFW-LPHT`G z`%p{@q*zAA)}Dw7O4_hD2Rt$KXZTs0L}=diH-3m0z9$t zY$P)<>_TaUVEb*{6OY2R=i$r;RWE1rVZ}pNk1d!M+Yxh9KViVmZ>gcQT7!mi8TW^79|b_RHbvDliqe9p1O zn|M!bPW&d=k3sx0SoZ;2dJSrTS^IoPet*Bs_HJtLNUrt{Y_1KJEL_-t zAN@{v>Q|1T&s($=nbG+RzI`fP(Qd_69}5l6h;uoN@aFEsMo~W_`W7ty$d6-L(C=!WN=?s2Arwn` z|D%Bx1q&tUwbfCTsscx|iNI1KsWu!3RyHzh9(#m8{wMw<|HWVWD?Iv%zrbwe!~F84 z0sreSJew2DGOm#g0BL8D@|JehM#|CivrzwE=|OV_&+NnXK5UQS`ao?>!f(LU zdbAyF?}BYQfX-dTHNW<&dhDs6fN39YJrAurdaknt=@Rz12kDZ0EOY6YKQWKBRc~f4 zOB=RyI|nJl7G&r0vnuqPM49>X)q8G@?`D46KUmz(LkRNoFaqPxDN3>y&1G-dk~-vI z3h6xb_H=A#Ps~ntU%;y0)Hz!S+!Sg8S!IE&s90A!!yXiChrXPsbHKr1%%^|-DSq)^ ze}+Hv6F8DM=Ou|G;Crwf7!-AmBf{{67IV^A;LJI;L4ZcBW{5T<2Y-ub>IBI_$9AF#6zGXZv zL9~#o%?pIiw+Qo`k-P=wf#4p7D2Q}^Q&ULPpL=u=2uV14?YX=y;XnYEZ`yS&jX|Ef zsD%<_ysDU27uCAK6(%Q)c2SVIH6@~)VkZYXe7aG-4#BWeyb1D#vVgZ&V@ z=<&fP-p_yeNBv!ER1 zs{1ovk(lF&d^3f-*qQx?UT<|tz-BatT2+1BTy|DpS^%ADMW0E#qyDx&2QxDT3Irxe zEs;<=4U_CvG0*k7CONGNt)?x8xT5nL7{yqr42^yDrRAEWaFe0b5813qW3!-Q{5heo z8gEzvH=YBFpu^;FGWMH~-c@;G{VvSqAE7m|ie z34Jj#AMVP>$C&zek<{+o{TN2d{!vz{9VQv2Z zK5>3&0ea@3Xfl+fERkNMv2_jDnWw4sn~OBHUPCgp_AYTzfM}G+@xt@Z^WrmK7pxDF*P$^3WmA5IBzVN$%ItM9b?BV=H%V@#1r!3{O-iLU#Bgd zhVS?=l*`0Plc44pmV(%LszUuPLnB`cO@BM@b1h2?Y_7AY7dP{b%rQ2MkE_+nI9?RX zjtn>n!v%nd4Ty;&!>q(c<`xqZ8;}z(yg06{umQ(f0HPpfQ9s8lv_)vlnwcau2V&RK zY&J6c%tDPpex$K#8tbuHEZ`Kz`fO4L_<+WqaRdpQIBUtb(+vb9ijMp`!w_l)__QOa zyD&5Rt;(l^Yq4_Q8h_KdG>`bbMe+i;yp1r!F6y?iM5$0t2udyC+PKJ z^cpsj)Xni?0A#`H*#b6q!9I^7q91|y6)`vQRj?`$`#=Y*Gr;S>dx7n*^Y8Fow*AhE zVjOM1=dF~5CoVvz>*FQi%nH1DOY*MeisWr4RwQA&xeF(k#pc}EQxx<;N0PO*n!>xb z8(L}F3E=L&9v_Uf&qf^{zHf~?w@-2D%~yHk(f1s#Mjcy$t$oR@S8H0mI_O9eH*jFD zVRQ1B0L;#Tl9gAM1bU*b>BueE-Pfws+OqBg)GCskHCj5Zz64u$ZPIChnPyYZS1bBE zXef=Xz65)F21peYPS9*XeHm&?u)iyCk2n)w96v^)>8m5pBuuS? znU)hhN8qB6hLndroXNzLHgV6TjFuh;^ZTCe=gqMrU+hdAYZWY;fStZcVoP9e1?n{O z3D_G#J-0|!D<*V*_h5=t$zfhF;4@+tK##Y;hzYQ#%be zeiN!Ep}&WB_aZ#@1MvL+3xDWuLhGiUOQa?LS~@ELV>{rbB`-^t^jNwm;8NKXkVzG> zraeveG6o1`$_vJ3*b><3?!d|g*nb-~-}C>o_vTTOWp|z5=e{lWH6t=|t*p$f>Z-1# zyVZ+YtzE4dKti+_25Im}7>x0lv20^>W(NBh{Kt6g!R8#nc;<|a4<4}?!2lA_f|ihk zwr*{$ZuL_8nwgbrL}p~{`&<0S@4k$zRtrg34Qi?1IeFscTO+I9yZ3(YclnLM&H=co z3A-}r5*U^qHEH9x=L)OryE2Fk{IMj-9SD2WZODCFIzjD11&+T4Zat4bbq4CIFmYrY z8Qs*ZO_-u)TEg0-3b6V-EZzhATRJ+rf)`D3{rU~mSHHlYY@6&?daSiP3JWKQ&Hf~B z{09-%dw25Ydu-M#GG@-_;oJt?vjkT+VJatM_gV{1&Pzej&W?^w_NC+Z zv`#As^;&2nx@IyA1H%sp!mt7Z0Wl*N)$Pf;VM{t$Zzur0_L+w?>T+&6XXEnHrE|(SQaGgfKWL zkUCVL^mXHwe8--T>Iab1eObqo?~oRjhdoz7Puzft3lmx6WuYjDtr9~m0mc*2Rn*LDaX)`&v zuqqAl%mm!r5f&$Eive=JC;(`qERAlfD}e-sQ~-v6p}C^8KezPy!*4jnNB;avEG#ye zE2Y^ii(ztZQUHt>2!PqE3fL)4z##3jRThvlHwD`jnf00~NKEv8U8HMTGn^BM6#Ah zLAND8iIn^_YmlFm5)JKr0X^vnsN51@6iWuU{S5aWeUR5Y_#m|%@PZCI)f;^2W`oKn zud=rOA2~fW%juaC>BUKwCZ?FEwRIvyLL#J7QOL$Hw)UZfNYkoaVQexn@!e$gdb2MH zYayGHii{K2CR5{tv|Nn9R3cA{>-bL%e%R~E)0uRjHH2JL1DYWe64K)jG{`Ay<|%HY z8bj%%LYZ3F2?c>(cVQwb6#u}JZF>V`##&3-a=HTpf{4CPPR|1Nw47~q1V`fVOatyp zLwzJ2Qqh5?2QxanA@T|-{mSwZEP*dIW*Qq%I0jcf3G?@fZ>G5gr``@<`7L<+zl8lY z@u^r*RKG3}%HjFZv=q{erX>_0nv&mP%-F;-!uG`T5-JA znQQAZ@dqLmzjHxU3+)~pn-${SGR22lsmX+7JSyvFkHN{K(CvH|$5a^xUa<(Xv+(#6 zS_-CRTIfN4z&J%M2w4>{Jr89V+8>4hUdtq~Y+bZdZ*wdGI;NI}b%!z|iIQ{8FqlA7 z5Qp#Q`tmg_NBSYh7)#V54k)cEa7^hFB3c>XSOUs?U-tRPB%5szI7){G1mi?EWq_eW zG{Sm40H!|IA4A`S0QEOY+ox*DhAS-)AuR z)lK69LYHh@abw+Jw&0PCD{ihEOchK{9Z&P=&t2pP9=@AmCSYsVB$M#TrC@j8pw|t^ zr+sQogVjwXMMG7O)eVEvgfA&>6@zrrCzXKxnqsprUYoe1Slcj|obbt}p(nFM}7JTBMR-c;lZ9`PjXR!n@)$94i-*QC4?kjnT({ySzErV>y#f%tq_AHi{j0y>TxCsNp_Az#wxRC%O>CoS6;ud3Arcbcf#7|mSgCSn6Nua>4br|JL4AVnZ zdw6k4@eB5vcsU1WFkn<~p$IU$F6~YmQ;lHX!!I~kqXBMB@2?FQ<-lS@P;297Z0vTQ zVavb_T&VZ)<54Vkh`ZlJMGX4&4sP1RW+dA@Fd80i+s9TeZncGS1B`7K*N9?Y8Q`{T zR5w8NeB1>aYsBp|*xkhPOoHk*es&(aUSZH35*rNYRVz3+c)Ob@-$Yfn@th)d zt<0eB5$iPR)v83I8CsQf%)lYo-GIPB?QP>t9>=b3F`PV0q_#tUq=+^4=r#5C+S|sl z;*56Iuu%8cnx`Q0!R_}eQORuXS}71K`P z?W|%sDMsZD93_ctgHBUCX_2tB$s?k37gfy(>wz9l!yHg_$L*w|4RnBcptZkuFc0lluz z%?*p036EAA*0xL*r#(8IfW;}FV%DeI`3^mQFIc{#*Rw1dzK^-Ih_zEj_4*oYgdj&9 zl9TCke4il!%J&#{JE*`QFfAGQ+XlmthtcZdM;rkTzK81e2y6?(8xagZ1wKK)j|vol z=cBxUU=W~O7lsgEV2nJJW#M{52?ywp2rNUmrCy)FwooGvJfFZ&s4lX<=ZOT|G*BIn zz)}R3LJcjHA24)B0-i$85x)Kt)uA5*(0HJRiojK39kLaHuFs>IQfRXu9e@q0E)0~P zP^h6oZ4L1=5>#Mp4Ds`z+5t2T{Jg?kAL3`_zQ(%3E5fnm>{f);$!gzFb$7u0kqFT_sBA+jE$Nw6M53M!t4`O1$%)NFc|(Un z6o5dE%V;)ge|Ny#(Fl=QVR~{C5Gz4t1Mr0%3S97p4yghdGa_fpX2G73@(j^Qh)xTe zTWBd>gEMLF6n)9)!R6U*R498{M1NiO1tGdmQT5 ziDlAac(;MV6yOrch?X@PF(6uk@-xIT8AwkMi5QTZl(2x}41?!7^8GSJB94;6mA#s1 zV6P2`W-^d2;5bMDOUr}g40Ov-nuiO0qS-8Dia0j(q8U<#3x!#UgDxyUXA??C;KpZ( zmga$xe8+SF#|dC!hKT9H#4J&B1QWB8SzDZ=)2R_F&I%hdIWLUSlpmapM zG^Hc5zjOrLo_u;;D9#fNdQhBW)Y&Iem_vC3ngf?7pZFq)tIzR=fAm#GgA6UNMgI5? zv9$02$L?NcZgz%J-r>q7yzV4?^sBHm2REOCdzawmR*!|L2!Rg5gD7K=rww}+fP=1|@j6+<@daBCM9XTga?=#Jpn zg7j~(nDl?;eK^=QzLuAHlIj^oO5298A|cb`8MuLu<=En(@igP(SUS$y z&^}S^0SZ$&ZA&C+u1KcwHCtqx4DoPGTZ1^payX?WYM})0K)z;<0~ZY4AB57mltF*g zCmJ|%ubd(O`ON^5hWuW2A!&-FZP$~2P671*mMrlTq)bRD*!Sqp4=L>y_=%tUM?C%~ zpX2e1Pt)u7dFR_6A$BB!zaJ4MXETC23EYaHPD5}jf^iChb=e+NBB%xE)qG;PC@Kj- z1&rh1p9gh6M4pYH?t{p!2x=OUr|ko8f~fuCWBr%S^ZYkhUXiPP7cRD?6JCJDX*jzq zDK0}-lucLHA(zlTn=;H6rP05oi+u9WNI+1`@GoV%*OVz5oP?{DW&d!z&8i-2~ z(RxjQO|Ag@dr+DY5L4dN5z{FNBpnjW#e;{>>ARAqR_V%rYSNY{>7vr110w-B#vz$nr7JUk3Hv~X7KABkaZ48k4P?FRN zEsIcyX|(5yWUpw!g&y1&hYJH(ib%>1ffz~mJeaqoHbP1r3;l4 zb{vKs`5g6R=G#e#WGg!-`v4*-NkxmKpk0AT5)~dFNyDHivaZ^i$khgVuo=_qv5fe1 zaz{kM=ETKA;SRKftxj+dG?16XwfTFr$0h^Awp=fo9!uFeTFTbeDQ1V0*0M7g`xjvP zD6BsZg+-{`fT*bG-J90mhQ%$_W`}hYBTU@Lk=e=*V_|OyZW3PcvpOZ2WCE_! zm${rTHw<1qEz+~8oJh9)K+ONouS&2_xuK+CnYg0pZ2k} zFX!aE4jIq|KQ}5;GtM#L`mXf7ejwh}(~EzX5Bhg#nVXZ2^YJHiAb{Q;f}3#gZwP~N zAoa4e|7D z*g6o=uvt?!v4sDQrG4X=1JcCrqprpSQgGtxSDp zir@RxAK`v@#3TRk5mKonK@E%ou!~65E0U-d>`9^;O~8>qP4yIN9)hc2oRz=<;~p(h zI}X7%GC`9AE(5REzkeStuYjDF!4k2>PXl6ryz3a;T=$5?BxvjEEs4cEIxRk)DZReF zQ)2eWakkdJOeFFK`~Z&V0D}FRfR^2=fSqJKAONScy zj}CzVqM4F-bjoE36v*d1T1_e1nMuopOJ#Q)I8dp;R7u#F?QLi^g@G9jwI@gWbt+{! zE}ijdHN{Vp%!v1OYHH>FT^JP!SsusgX44SIH=Gz;mhM^pP@lUb;& zL${-ghQ^^;m(YQ99-37tpq-i!&=QMF^mKV$zDptv)v|#*6adt!3K)vUwMVimeqYPc zvigk3+WRe$xjAte<0SLqL5`?2r_S8LU^>BTvuBB=VDn0!=5Uj(-Bmul2`f+i1$%oN zJT!k6Gw(7xRpL}+l1OBpd@2d$HWU-%#I>9y0H`yB3GKD%jU-%PtEIyOG~g-3$7yPX zsP1%J0ViAH%L!tp0Gec|&-cV@Q|Zft5{3_?boc;_y*OE0U(*+jbJv3e|fGIAIflR-lpYPN`3LBl8G$fts*q4(mh{tF443w|I z>={&N|8~z!n6%caN}Tf{Z%#+W=-#-%S2lbSS0Crm&A(#UQf$*$NJM-$vw5FpR|ibS;o63P8HF%5rxzuI zMJYJ5Bp#gT-(@89cY2A(;o-Ny<6jhsL>N90AZ@>H!)-4hP=UOIVXU)l%kw%K38;zM zx;%j?$gtCw_eg08oC*M6+D7XTs0xCyL=N4j7?%8QngVD>+B@XyHr>`B$Z+Cb;u1!DRt9{UqH8ql z42|>>M;1?0Z#q=#CP!wZP1q`r8J(?Nk)+K|!`8O!pPUe2Q*S^%D{bV?u7nIs7Deus zO)I9SM8;-Y0(3lAz}3WrNYr-9FgqjR0Xy3=36xGtTUxIR3zW|b=t-p|j9`09I<9mE zc6%E1q=c2P?!n{~6iTqSBLzk?IVf)l!xKvgd(&>{d*vi}pfIHwoL!MBWO5>POK33F zsl%Wr`?`)qPfOT%gTyjthhXFf}#UE$43 zuY#P%OsT{jP62;;f<$Tp>U{w@NgYa1w56l!c>-*<^?Pi3x*4)P#LqP%dO^BBs=52mwlHKHVKkrxVct z#!=F-m^wXYq=C%PCrBw>_)X6kj(h<$BeX;T-3jaC68e0lXZ2jhU?`?g9X%-6cHyY* z#74+D9T#RzxZuJAHe7Pyqy^PLq;w$>6gT9Y1m7Dm*?t%+YeBm#gLNwkJI_Pm4jIf} z{wSP$1l8SxwU5DFKLHzG6$km5cfpmvfXo~$ydEz6Dct{`36NO%oPeBBPp%DeJ$E1p zPrfGurf?g885Nesj)_(jz>3K=K(@O*`8>RB*cD*Y(ClW&;5Z4{2O@VHbmX(KC4Z;q z%1^j2SrkA61WrQM53@XB(7@pkf__s{*LqF4uR9c7Z5TSBW2J-W1eDjtfdi%`-(3d@ z(AgiyNQcMdk4Qn$sS~jJl+?oMO02Zet?7F-^uNo1X1mLvQsK*g_9*#lm-xtI{{pA$ z@#SCrNfyrjI19OT%9&r_J@0Mt_|6%A^&L@u<5Tc6Z-Ec~)qs0XMfl6-;p8kly9&!C z2?&@x1YoKyQ6U6@fR%|9eCY}tpN6YjQs{cO0TU@7&xb}wLIkdC!0{OYJEg3gUu(hr z#{>vX=jD2SAoZ4SYzv5KcjYxtCw$hcvd!~F4u1bhNf=GU?duE2c;H{S_al3(4{fh19cks8gM_5e94}um&Ni5$oU>vEE9@3S)VP!sU>K4 z#M2WuTeP=Fw5peGJI5bXsqa>4)h}`4^xNq7W0dy<*aU&3rp=TDxP+i*MuQe(LZoLq zWdS-_4RrSQ#z1Ft!yu8+GPW&gi}E=rSG1fg2ic4;Khx9WB(<3-0YKY3(CLV5Y&d{= z-5{0Hg+q5?dRF41%bO613Y*gH2LH4fL33SC69-qMsf1 z^!BohwIaGw@u;u2@4#rN$Yezt?+#@`xLTF|D3*cREl5vV%r6(2s1|w6o6geN18?|a zSlKRX7wXiSJ6x<@<6^bS)sH_$x7Xl}NAIPd?Xg^(=19DVbF#!#G=bR}!DL)!&U#u> zw$nXSFJ}b#=s|K?5Hq3f>Mvvo8T+;}uMUo0SX{gr#I5u zMtZ*k-cYYc1jGP=X$!b9Bf``G*b#Y&tf+hjfu>9~I})-nc6C;OrvY+!6rfd&2k}o0gQcd`gd%hw$*d!UWmh52K;!Dfw)7 zbgdi4(k?^c8HS`v+qSUtgOSV#nug4-_QH3~8rRRUwS3I6Wzb>kNM$n&IY2W<`ZEL{ zO7HUyt%s>+o<7btbz7`Ki$b@9ruNZjCjIa}<&VLP(qP66`-4E*WToY9Hu`>u6wOqU zu4WnIk_Tx^?zQjgG6sRL{&`FKj*2JI(fgrf&44u*<}KLtV8SFY1E~5iYrstpmQA?f z5x*zHFFXGO|Lzn2C!hS?PjUZ0yqCM(S7Dw2?=l#73b3*60`DBCvm!}zkBtbfMo_b$ zTKb5BbD1A43IwZjNs9X!B8SjAWp(Y%lXj z`Vp2`;>!8QICknC^oCI?d*aa{;J~9(D#-e531+qTW@=J?wsr($<+I|~2_vMl8J|i; zBy1%uW1A`o0NUP$wj&J8a0vTVVR!O5En}M&)~38ItW7Qt&8DK&62EyODN?mUQ9w(z z0#mavF)6G~G%8F_xdYv{L_miUwA=(_^H8grWHXQ}iU%hW7jRQv*L8U^u+tC*D4LK6 z>7Krp6BBkPofojvu7e#94j7!AW^>v#$W04t(+Q<<`j|*Ut4iu}?}VPX%}&cWXV5k< zCS;7%Yf2rxP~sNL$(VE0mvb9iB7=^mr5s0QCeG4{IQ{wzulsxV(%ylG2X7-;f?LnE zsSR#%b8mx{EtdsR^%7Wd3-wSJOHpe3;0+(VeY_{#4-TeYW0?KRza$ zd35xrskaA=LD2{SL_Gw|WPI3EA}@F@fY&MblKy-qfIAdyA~ST~iDmm4`ApBj-W356 z6UR}_br?njkna*6YsCccC_`UsI0if*FRt4kGdiG|oDPANz8%nvdXOj_^4%oIfF_iw zIfrCwwRL%^M>@so0OVu@&={tC=TT4lY??ypR<6oRK4_0;FnSH`uW1~}(}GA^fKXU8 zwAUER(GD{=d#m_!_s}ih;!3bW?)=yI=yQE0%h!1HxBe5NDntE0{37Y2Z((`br9b&! z^2Uc8dd?Akb}7ZizJQV4CQKwD)q%Y>6jQLp&(hAR|B| zU;Ax2=kpD?djY<34epo`5LC*@xuuMN4L=b1^-digSoB%1i$tx`kZUM}33}idJi8+M zs!j2!Mji3n+;>cSe6qUSK@IM|12)PAxs(uK+f{ho8HtEKx$rlCbzZj1^pteApZe@r z#-+@$Cr4?ZVVEMT80h!UM&uFQz6|~yTlPhDeIpqy~<6K>$y1RchfF8zwJ0N%- z)O`qpa}t<*$)2E>)$$6+`5s=*ojXUT)8WSO1|O(?02lXQ-=*Otu2rw`{`~t{W|>y) zB8$f!Vq;r8H^qXqPhkK-F)twlGgAUqb~NiUGX)!)BB`CoYo?|FiDZC16yQ@V%JnL{ za(t;IK*zBqz+h)vf?rZ8@y73KugydOcFQn50}~S-jk*RsIgzb(+9G4?_r!OTFF-b@ z!v>}W^i;N`rcE|43{E03W_daqv_zs>b~Yj4q_HmyP^KVKv&f9>t5-zwmdeB4rk1m1 zMMaX1C|O^Xzaf!^0dJS~zn&y($YtI}7*)6l3W39HFKb4N;A z9E^=}a(u4|&a_s`v_-9=Og&kh5KngCgOh@G%^((wFt-$A&M)%No9?526W%`g_c0?r z>tE~ANvyHA(PXze;Kr4&V6|+@A6mnR`@Cu99vV+}Sj5b3%)aKO_e z8!c1IX)li(NJr3crIU#my2zU;)IW^bnz#Y`{#d5g*Pfl$Xw24x(zT8+t(S*%R3R9O zTjI$H)PZyjTZ!br3?*!83~Fp8GnOU-paL|LgO;+%G0M|3i6H}Yh~*ImNO$Cc8iS(S zK7;Y6LcU9&WY*Es=L~}}FfxsCF+FwYIy#8xrC+qHspPU|Vb7F_& zYA;}NQd1&L1I^@g#$FthoIB_cM3Q4Kj+U?y3mKfyr!yzO#W2T#2C=NJm6L_tn`1`D zwB&E?w;-NnP%l$!4ydktfsN*fVCM=?R2Rj_fTT<7aEZa}UH||f07*naRJ0_lQq+N11S)lt`%emE^J8!P9`oD$O_oRm z?m7*ZuggrM?|~O!`!1d`HApcJ;+R#dD~yg|=&%k~7#pQ!X_le4X;#Jc$F*{NEgOEZ zA3}iQ>$U>f9)eV_@C~zR6 z!B|hXw=q*ZouNl3ZOUZSz7JVL+WU$tQtz%Oz-Pu5DRWK(t6g6tYpX6Cu`q8bm{hW_ z4wMYo4B)5%D?S`Ec(3t~`P}s5JpYOF{J~vS{*m<_shM*fj0Ip5sfE*(e;SnqZi*i# zcox(>z&S)QF#ostFnxC~uYjEI;bnDo6@W+EkKP7w!o>|-9&J6!FHHOb-PSs%k3Ynn z%TRC11Wq9%o}5NYig(cMay5(pId`60kcAMhs ziKn64G)SdoU#AY4q5z;yT@qsB84aW+Wxdys^;lYD@u`9+QTlCRbxd22pHjO28OYeE zu`Of2Xj;OWf+-1b>enEe)8nlcB&W3XpaXUi{Gq}d$>ei;OU96P5{4~ErVO%$B(qbe zx$^|nufy9LCKi0O6x*^5un-MG%>c7dr2&+;c< z`wga>H+b@oex6QyM5Xm%k`vR!CQmV6aHu5yDL?*5oxj|^n@{|kDF4^zBmDI1;m@9d zH{Aixu1gzJYD2XR`9wgqE#T!=749g(xeZvDfE#7FV+t;9!Rc8zzagI&HsQ`0xUwyN zo7Ed|&w?CRN{PH|QiGt$v@j~Atbn7*G&FlM`R?cx_=yyJ`KowrF0D)Wz)oGxX%7^p zA)sWt1}EmkvooC&HYXVqTgdi4yzZ1vYRgIDakDGezOV+1Q=&9@=;V+)j((@Lo-gxd zeh#WNNfB~A2m(B1U<849a%@{OJO+VnL4P3s=(Yj9!MG6UP)pkSI+!5PY`Ib|NOKF& zk!}JyLhuvr3(q&h|FgZ3z`{M>>1}wm?b0ATM5? zN<|{0i<1I?rl-dN1F@JKv%L-7j-pV6ei#uwA%JLaSLAKQNvW+9ofSrBdqV@5tS~tJ zj>youE%~t{Iti@~+2S~apg1KAPNOR2D$*0;&56awew?bVz}YkWNp?L~VUb*r9kOdqT6YAbjhezquJSQoD$VusOX zJ6ogs=V379{KfOE++3w`xd8!WZf00oTH^HS(>TtzH!VFJ4mp4RJS!_JH0qKvmO1}n zmNFivOBQmq!%U7E2PNFb*nFe)!+^(<_1oB+!_qbz{N|zc5d3KAnstZ)vLfU3u>&S2 ze>;;i(ANrq%wgbTACjvbcy=^vgSy?b)7zH6+tR?uNlKlYFmS+kHT%+$npFc0d^AfF zI0>maW5@IhH#+d;^oh{fTH(gvJf$1g_{@L!O;X#>aq&-{W4GF*(GA$BRw&-_gG}0g zOl|U~xG#T;>r+3>8{g`X$V`Hh;m#@etIP17SHrJ;3LbtlJoia?9r8($isj=1DDnyY znS}K!%%uXZ?7^uScxFvP?e;=BnwFUDX=dhNd+QK`lNR|}F$15y1gA>@S1NF74!&|7 z&dz^*YFec(zgOE5KHvqy+~ndS4L-9V@z67Q*lik2<;3@67$SY!-iJ4w($~#Fy)9~h zSX88H_Z|~@+U$gk#m_Fn(zN`yG#VlM_{wtIa`zdA*KTnB+2>g4wyDR2CeGH%EIBbw z$1^x~MEh$Dq2waY*nbm^2vuh@mZOb4=~O~+b1+%LxE%>)yNdCOuO)5@?W-}zVFU`z z7HR1+zz~rjD+AT{bYOu7GPXhJGrFx1Xz0@%Y9_}vBq7aE!sGw@}|h>1ko#_j%@-XYf23d<=&}hC`V(^*oP$Uj_<65YX**568Q~+up%gAAxDGalMq2d_l3RuoA zU=p~KMB@!m9)>@S6KP}FLq>x(ZVW3DU`3$0Z`1D^lA5MHIMIZRw|Z@1Y@8@`8WKzpjmdg450SV} zw<$5xv6Mb$LOe4E$0Q`^AdwT0)f43jC`oUNB%r=66UvdKPkT$!KI0jg_)Tf%r?W30 zC!!^5>FIHzZ9F66FS&+hdwj5y0mC-L^4fBs0gpS-W61<`D-y(FC&2AObV9&T;K{hJ zzpvE_3GjLfs|cnSVd2OEJqLndk@yVM&%rzI`~gNy=+`}ZiCyX!0}{q9>icc>D{%G7 z6Xg0+T=|pd@CGB6Cl;tit2~%FOZ)jg#YB#&Sb@Qu#X>!UQSnJzQOq!0VCX|&j3sPw z-2p2BD=PHj41MX4M;a7`=Q@Vyr$cW}z!(69QiU*O4aUGHWP75A*cmsF&hsz;@{kTV zOmOqiT6b=jvKgUVBRBvx2OuYa{r!DDzVUI|Zu_?Pwpy)HtyZ~o=@M^y+uJb0IuPGv z+27yi;~)Px?e>dZ4=!yQylt+AbC|LDwt>ypGdw})w}BVzW~q>!h`$u4^Og zPNUIa*zB?Y{L`%NwwSCw%V!@iGgn{d3xDxGf?l8PTAO;iOMdxHOW=nOoE4I(%k5j_}Mo``BxuS;<3de%U6g;yo;Gou(5(9>U&OpFS ztux-2PYSr1&cNjzST4e|YjC#YQ*DbpZMP-oHG2?ugblIu7|{>L*&2>HR#JkDZ6me- zm1NYH(1D}|HD};USK+P&xV9sHoSS70b{gXpwPHq+>;qp)eeKlYRf}+S-QeVraddR2 zr)6w$seAL_3D~Sks6e+b-)Sl*lDMO#Z|PNdWdZhUH9r32muVmL>9SPaAys$CrGA^Y zWhO}|O~Lsik)=7dfC|qSVBv-jjW3c9%jO_5I%I7QLI`x@cHqY`wEoIu3``7Z6Wu_E z5t!p-wJ}T;N04N-WeZky06{r`0#FOaEKx#1 z4JMaM{EMIYXZ*|G`k(pG|NSA1dllWm3;vH9RjO3E{KO^Rw(vG`Z_K?!Psz)Bc?INr z4=?F-nrgMmqP@uF;pG=SX3<#$AeYPW?svcYJ91y&hibJ6DowpUA_!dCZN=6Wq%#9l z06RMd*-RhH@+p^X;;}wX)TO>}qg07lv`@buWjII^i8kqWUA$2VJ6a|1oAldB47&wv z)85O2Sw&H1)GrdS3#t!Z3GZfukypcH67Rx*!KjR76!1m?eRmVv$fJT7t?mtMD}zm* z{mx}UZCUMz_Ah}tRwVT!Lee*j3W%y8m8e8S9-f0BgPy0qLVQs`PQ%$^1lAS#RB2A1zf%*lXXeOXvP0&*# zB5_5#lE8}$FqBQV;$S2^RK%fGRTxo^XeuUvX95WHSTOE0D9i7mk|vtfliPX>*VLYA zI{~9M#3r<5p)1>38+w0Sm^ITuSdI~xaTrv!_cJO?kQsxZVls7XiewL#veQ@zXkCT3 z-}P3yD`2OAh|BiZT8yUm=&l&pqcYWQjmGAXi&wuuu|LDLkDh1H=`oj?WlL@F=E>Kx z@x(Susd-YdIHNoqAI#D+0;bv-MpZ>5H-R-&V}`{;duh;s2Fp=)=%oiRz9uN!}^0Ci+jE>vk=FJk1K6)QFZb&-P z^70OkJn}S4OJyE=>@j}u2Y>LJzL&${kdJ-rV{|$lN;hwE-=mK*bwh$kc9xfU`jJN{ zFD>!dV~_E}r{YrM0=S*k`34~88%ftpQ)}v95HN7qljFY__|f1dl0Fm>?cC1lxC6Ns z$n_2b9y_WZ{6irFR%|Ra3!|eChY%bFG{X+fW)r_Rq*`;iQEykKJacPkKBcmNK;6R*iOHPw;x&Mi>Y7atq+gzoB|_zf*-!i z;nzM5|KktA|MNw7_yla#<^8*R0WN-32JHz)Vv)0PX!I4?IBYfGL`mw-q+$XPTwg#+ zy({2mzdHsyvpG0dgVQq{fSPItGO>VmU#A0`&>c!>K)ol#R8_M%yUj6+Gm#woazY8( zp5EW+8e|gkeN0ovz3qW`Xm*qfwhSgS@Z~G8 zJSE^K1Uq3&^u=57y3_K?{N!7|>Er*(@`B}{J?M77aWIj+#AA&LKN_3Bu{DF!)wR$Z zYs}_!h9X(hvbTd+>VR?kim#bn#?m!q$bccFt_?%8;T{CGw2OwJfsaF+rEGn3z4JE8JIT|+-bsf500B~Gmx4&E1vjr)&f{i^0}noI>gV!`JaCBm-+3L z-@zPMlu9?b@4iQwn$m6d&N5Fw{Rrjq5|3>@#=G2iecP=5_rvlE$oU>#78e((R;#=- z^-iu=uYa8{$HL-~QnsYZ~o@Dd|$%=#|g+}pjfa$m)d>R zJ>a@g`hDnjq1i|giHh88b6tvm9+?w=&g7)X)iyUo=4M!3y6re0`1N1q?Qef4ue$G5 zj7G;PPDTpKWVG=l9&qFTNvtV{V8 zr1L*QJSpJE7*WIT=1}GoD$-?8OJc>c zV0P&3r?6rcp6W2HIhalwH|R3#r0~^z>TJuRFDnr_tWUF(5spK65YD9Cm#gWwg z)IV{P0GNuLWRJ@vjY%R_l#WrMppMaeY75hX36|M+u9K#OZCdo~J+gyi4A;vTfsND8 zaA~uS(<;JlA1fAM?H6$OcW^p+tg6CG_K571sM<|vL^0|nk&=hIpJK4xLk(h3f!GW< zMugJr90SwfWPF)O5t`4#58wAN-J4*gWsJIixlg~eNB^=((cYrh7|?0K+U6zN-8fG@ z`#C1ONh*zPs;vsgv&XqUy2hJJuV>{`w>X|$z>eAkNmycz-K{pqoAb1u8!^#OqUSK z5F8FHu$)6lYKH)g+oRbXU^eS)UA+RgHt_aWxv^q!WaS3uuKgyxc9UdhohPbKF{7GX z-~R%;wIpvkAld@LX z7&QBe<9WEaEug2~l`&f;CSWG3!A(wso9PUktHas3u}rPjm3s#P3_V3gvppfGIS3s{ zYROu1oU9fms)f?E-PWNHf;3b+#sTOlrNuWhkra>?)l5^nC!nX?fY+XY^S2}^?fSOJ z*vd6wdgOZsIF{V+p%YR|=fteAMsK-G;-i23^?&W-{>t)AmTx(j$XVus+vC-vh)58k z5%J{2qGJG~5S%j5U?g1pfP;AFKu-wyp-L?fCMcAsh1EM1fGKTZfM&z3aU1LUhoYqc z(;)D)q}kU%Df~WRyLwpE5_&ToeXoILo1#j5n0;UBqUANSQ`78C){q$Jq#=x zYkDG`n=!?QGpE5%0y%ci$D0Y@rY`{JdH{DAaMgp8CS36Z0A2Fnqyd+JGYZZJTuNNQ zbSz4x8@%HkAHXyZwb^r*n3=i8M?d<0%H?Iwojb>U_ucoMeMr6smRCT|_waJ&%o#3U zzRXg4iGN=B=RDealp8}mQ{yc2Na_)e+DAzylbkql;_J>|U0vmue(9HZ?z!gNoH+KlQ7;@4vf`-~RvDd-E_! zsw(gQb0Z=%BloPOt9q+ms`r&{7FiWR(E&wNltmPfaT%Q%5EXS^L}bPpw}BaP_jSP; zXU5T22Wb=-cSnDMFo5hzFGcTNdsbFft`Yb5$GK6N-89h9Ev;}r&r?riMMU0=s?3at z?>*;y&sTqhZuU``G7UQ?TRn38B$URW&?Uc*8@IGVr>VP{1^Ld9@nPth-7+MYHLtBH zXiDm5hIe}8Iy|~&!pM#o-7{09Q!uha3==x?Ft!VFy;(db#rSUMT?n-)aZ&4?CpOlV zB6QD#$x$f_D9nb^Aiy&6y`~O1y_$3<3~ho~b~zp0DRw^!GZvnJ)-d`Q^e&YQnaUV+ zE`ZW*aNJYL%z%;2knMdbu_O#V0zE5UPkFpaWkln4P0$=mfpaXeLcqjEkJ!v*5GX2p zJiOi_&Cw(QeKh?E)b1{3_atdnG`i#ybi~mk9-f;gh*j|_2}+eIf}lh=7Emhgp<^n} zNO=cNeT0NgQ}njeSzW}Qts9Avz;EW+-h7yOY7IN@`8`hHk#}dXseB)c<45w?PaYty zQ>4hUx%>b}b)3qB-?^RHsRf{6TI5FB|B6Na8m62gP2Hf^n@8QN6N@F8J#!x2 z-Yn*_kaDcZ+~#6(ZWr0=0=m&mH4`x9u)DaG(COeezxz4cyd!wzm-mw+ORZL8XK4qE zQ%m^G*#F1N=e(S|zkUy^b1SLDE2RAtGZLK)#z#0Rww6a8*vwJQHS8Q3Vs0>t#*UD* zpCqF)gbhV*JcTZKB%22T9cM|1h* z-)F~;9Zz{ah>S>%x#_081Wtt0Y;yEXH}U<;FK7E$9p;-^!Mz;S+M+spMQ@%Ub#34J zoUjSX1Kmnd3!612Cnj(zb!w$uOuCQZ4~;VR;C)QQb~9(|0QcSU1HAFgbd21?efKL0 zgSRu;yoZhTAF#N&m3wZwoO*SNY|P=lzq^@iDob&qM6FSwYw;;~-}^iK{2c1JMJ!I- z!8em{U}bKYTu+kU)ZWNzj@!b0wUxYjmdBdDB>#GUfvaAo_|APWCl4!n;j_Pl%U%lq zaW^dQk(>G5_sE1yJ_)6ow0PP&Cih^DlGLwkLZTE4I%Nrxnl(|CWw8niyP#N+Novm# zOXYl0))kWQ=$J%u#w+56rj$fYI?TH1NKRUe0angx)#8kmWzs!nqAOCLqv%Svi5nBS zi6S_Wm>DTa#AfFh%o?OFQo+eMnm}qNI;{^(FB#If;+6W+?kThY> zL~O$-JvVGcxTkyLsOhMUj>Jn;FEf2kRI-d}6;)K16PR>14LHV8GbZCj<3!?Wx~_+o z*g>^S2ckxxjj-kT85zlxG^8BDoFkK2brZdrYLDcEGKn?Iq^xzga#(|U+Gv2;o)JVq zTH%Z+>6FIU-{r!9G6A7Rn|Md_Mn#$F}b{AO7%FtX{p6n{K)h&l8*J zcmCA}x%S$R5d|Pgf7@+9u+p&?S%4 z88Ii=u~kGzHV>0Uabfe*n#r0xPMI22JGHEnrT-4&eJkAtk5^Jg`MgwtQ+|@F9 znMkgc6`5W+57(xoz9!rwjZBjn)A+M0H8n%Sa^IKC!{}CVVoP^OopQP$>&p`k^t9Y& zGhK2|S)UTariq<$-jDhTW_wyjPVr2uX?rT~F|`X4F`x8|B-ODb9bMp+A)d~X?1G^( zbkFOe(FK)J=;()H1`>r?#IrE^Fm$c%r!oXIAw36n-&3P!^vIRY}%6(LtGs z+9dIs=HcQ-W_B*&M|c06MyJQ!qjxYj(Z_ggj9OTy$M50c@dr4z>v;bAx*u@H+%viL zx?4G+=Xf5Se3V(qnbc{}puz0aEH+JT;su?@@Y8?#Ij@=hYBqfBW{yAgcyM6LmM3m_ zuw{!BAnaRXvp*sulAbMFo^YRDvu&(YdbiDrU_CVDuk zXA_gdy<}1`IujMDWrt$1$Ygbphe~ytyGD3;`)a;dxsL~nvv@>>?8*-D!1mRAuY4bm zjL%`KTW5Z1oO|~y<$L1~vT=9;^~c8X-6jtdXX0#pfSqIO`1bd&m zOZndDgKQgI%zq8e;cxZest2(yx;OJadb?)Gy(b+8T*u}%c-7M`I<*|tw%*l+f zXR1J#U*fJ>3g4@<@j-_`yWBWXW2~0pyF;VwnV88BZr#R%L;d(;8@Xq25$k$3Gf_*E zR)ajUCx+(){LJ*Lz56LY;ZIDpTE+MMgBantH{R%|n2sEXF;OXKN}WxmF2W%m7sZ-N znnuqFQQa})eikAT7p+9Nv^?l#UC@-gM%tp>PLy7zTKhUJVIy@?YVIOG38&Ropo&Ctat_or&V+_86Y&`d#W5|dq$a0eM9rQ6W_n^R z9Y-QR+e5LA-WrN)+E^$?1}QE3%~T?S7P?{`y+pw#4NF~kG=xQtM0yq|*sduy%sIQ~ zNzU25vTb5w;z{4{=POp^Jjdzi=-}+L&*q_r9%A$6&5VzWJ5ixfn09B1#bVPpK<)hS zhd<<--~8rY>$KMU6-530{baM*13s>O$NB#Ezt3&A-8Sv+25|4a_j1M=XP~v_vdb9)2SdCAmi8uO;{t&p2- z1dr)mD6K#g>29g33B&2K1F5@-9CK69+-^*#BJ+WQ$s(%Bx~ZM=|8;y#a|*nS$&*Ji zgn|lqGo&U*hf!0ViYa$#biO6Pi?@}01J{ak8W!A*mcg!*I-2o?`_yTIbg_kZknaQ0~q|+&!JTwR3lJ#$-`U`l?iKo+e z04`Z@A>jz%fs=;D_JD9XnO#JQVaIVV$bvMdSi z=I+!hsghu+e?ND*CxC`!`HkG;oq?~XSjcAXi=9WW8stc47rzcqVD0ivWSc|$T)mu^ z9eo#Z&*9d|WBKz_?qFPX^V{-VF6eleho@#T6~s9zyOsYbE@8eu!o0*N-x)rVlRJGX zVT@l-EaI5XI`QHRwCmHKQEVDblgsvycIz;jz;PT@Sf@0lnRF)6%?jg_RYF~3vf9by zK^X-qOwYT#01tlCWw z8)venWm~hvNU4DeCKwHp1^!brJ`8SV(9 zQ(5{_BMer0=}wF=RLSAHqYRZC(rS>2$r|(g-_xi}ur~J_sA+NxQLEOd>O*JwT&vY+ z?k9R_$9r-jVxGOa{F;6Zk8ri2QJ$KqiX{EbLikga2)1s6!XhmeA z*Nmdxjw3UC4NN<_IqS6+D~ z*IaYW^twW!Kwn=UAN}Y@x!{5eo^p(xJ9qNxSHGHpfdRhpjc;()S!Yo!7D=bmXs!1W zVG9;4VEcBN-ixB$iG=(_bhS->Yb^~!(GpS56`)!R$>t=|Qm=}gbSwd-qS#5t6XJxH zHIJhRju)5ovvf|fZz^Sp{^(GILMjKvA!*%_%84a((^yLn4akjprU2u+#c?fNkcdz+ zZ+hm8L#_wBgckefER5HsEFqBjmvHtCU?j+E0z*BHs84QY7JgeixE|)$wp{^mx4x7u78ek z_;r&|ACYlgUqp?YgvO|>Q!&tW5lul=QgL)Br5R2_1X3a^&qIUbgRX&!LpToF100hD z6^F16ULHw3z>MFJZMp#}Cri2DnRUi!K&9KZ0R(N^%)Zmc1Ddv4oPD;r;NTL^Bx$%1 z&%|hC!7HTDT~O#MP{~b^ilv|cdIzjowi-4Vop~-kVI5f? zHs1o5ykZrPjYD`pT=2pp`Q-rYEWtTzAiM{b&wyj*!p0xNt4~-#7{ci7aKTH~()B=@ zNuB0(YhdEnaO&dqEa-rZ{|y(Mv65du2HT79`n53jD>!}mTCxe4{0&@i>N>IyLf(gy z7s12dhqF$Y$s^-=1}EV)Yhd6tWxd89VwS=oCesRaCDBfRmr-c-0G+(!E*!Ukkcm!~O8a z7rubqKY~}Cw3Pg#;-FSY!d;KSxktmco9evbtaVg=1*ffC!Mxp4U(=C>`**<0SHjnR z1s9)oB!joYYhSXQ#&6)1BS0(x3gYcT8;2Xb&x4igh zuD=Z~`LiRq?pAoy%U3h;BRJ=j6-@pdUh~3bO#KR8b=)$lcfc!Ou#Czb@ap51QT$nr zvrk<~lybLvTIq>IJklV$=uUF>*| z=R4o|&H)<}WgQP{HHqNFV$w$^Zk#a^CS@&VQdS?A+NzC6Vs9}NSx$$hs3$t_MzuE~ zQkR30v<25Qb&GDR%|K`(L7@`YHdmP*IL5IpiVi7L4-`e06ykmvy-qn2J5qb8A~6y( zbvQW%S*fp~!f>PkQt0f31uI0zBs0)GS45IhQm0cIm2-BsQy!xs zT?i?S%K10bArTzcGnUUq5l2xY^;k-t)2NA9sh1^!lQR8q01~;7%7k1CeA6c|=9>vE z(;KHgWm48mk3p1ivn|PM=umF@6FF#3$(&r=q_nw45H?MOralQcEy3fNmLg%$is~pw zqB=nZ+@wTu9AzRvQzBlPRk?n6Nr|*XQ5&sAw1idTzGl|Bre~uQ*90~58u3W@h`4cL zh$C6EC7zsw*{0;NuD30Xph1Z^3F|E}H7!$n;=qcPHQSK6ON%-Z?ZAkOIiqFK~^qaXPtE?o)_hGwD7!uSH0>^;K_|c z-tv~W0AfS}aOYWP5qOWE-wHgBJ6|P6B+0J6*5oU#=+Dn?{wAOQkc{U<_b2;4jtrA% z*$+pL-*d(L_|a{*?zJv@9CJk|fJ_qCqsNXsPI#?#F;`O$;2rOH2an!$E7hhkT?*wm zk}=7i0eM_)ifdQ3*;*HE_dTdK6{)z00Dx3XESXbrF^&QeCFQ!*(M-0jo2-fxTDdOc zHUjyZvI&@|nWZrk$eCK4Xqoq=?k3tcJ|)XyrB##TIS_X=ut*J~j$`Mf!AQ70R5jQanHgL-=x3G5Y+Ub3_Y}rCzU*9wQtM-plknkmPlgW$FnZF#mXNdq>cQVXg1oKzI z@?)WU4$N8zz4N6nPQE7~<^x(HHI-6JxJ-@9`8-t+@#83YTppA9pG3xpn53M)8x@GB zLu!+)+MTKqKc=4`j%Qkdluh58DW;o&)=F|5iL8hhF9l&!N&mvOhV3!aFDI62^~;Hw zv6R!6erD=>qUVU9(bM9^H7>lK5hYQYTC*ZzN;iP^b-oA>!j|B1d`(b6CiIN(aT5}? z(W(`t*-OwgjCeAlDB4GPK&YgjoY~Kbw~3vmndCEK$cX`w$f?W!ju zR&C>oD>l&IKlqf!A)orxrvP}{+up|b_&9_8{cO163bw6UMZ@=L_&(cKtzyF!S1{P$ z&*m)Lu?sKLe!{m;Lo$v8q>5C=~ecOB6SJ?vuRhU6)arnv|>)BVe{{*}}_@=$;li z&B_Gtee)~veV@1o^(N$e#YbK)DI70dnWff*Pk!=~eDm7N*t&HaQPcB=5k&h68sBRP zn^c>~iNsARCPD#Pq9xrXa#BVBq+@#vpUJuinUZn3&L&{8u4Gwz{I*Es6f2PNTQ<<* zq9)78aA3@|#>gi%<5RLOvWgxv%W*T=kn`JwPn*+PXPd#mXgPY0gCCb0fT@~@r?K*W z9M*EiVQrtLptCzf7_FKMsRW#~w8@+fIQv+!tv+#?{B3hOTcr_}KQ30UUQH&Gp+7Fh zM5_|3@Rg6Rd{!k^aYgqP^vConZUPer}!#i)0^5IO0!rNxDsEyFwMe)kKJ z*&mo_O!3Npd@KH2S4{UFQy0uby?!q0>^UUPUjQ-1ColOhmwokvY#ZAtUkX>tNhaZl zn_7Dfjs{Y$)Z&x_S(lOMjWnlL5}_FlMA+qB*xi(+QZTBGWpq>Obec*;(+maMLRjF! zrVth?+#L?|IU;~&D~b5@IWiLn1iHQkb`N~)BRSUMi*ZbrrKgng6UBoerbxaKrA7VpIA(K3FdoYc&mtsR31u0rtYR7g2&Bi;wh=uN#a*c|5++Fxbm#$GpDX>GBN z>E4gi!pMn3qhum2A!0{r1dN+Dsca75>b)X3P7V<*Zky0)OJxhqteOtYMC5d@93yJt zx)rrCvyLj(5;akjDS8dsg;1ODQEfs<*X3BXF%Aj?Gl}Vl;EBiLoPNyd+O7-pJsB1>AD!n1iR8Y!f-YdF+{NmU>f!&s2M) zCr}j9t#zrmJnl@j1Wh_Ff~Gqo;$?bW4LXvoby*)KYl^H7drC4^6y3=iAu}g0ON9|d zk<(fQ<`iVy;3}^A*4J1$dmcAldiAte0mWTgf6wQC`F$DCe2;@`e}tnv zbK=wT#hey8ivFwq%p8FR$Kk4PUdPJWi@EXgFPU@(IZ^0u@06^FW)qr_iKn>mlymsR z|N9J|{)>-KXUkF7t=gQJ5j)C}rq2pUd2)^}OGGNnnYx~emd2lz5UL7h#bB^0Vkoa* zM-%3_(i^8!NmOP6sm;p=Vv~j?ju;FqbK!R(tWd^L&4C9(V;v2w0v^)D z7sol>Kb>0#Ze?(A0k_=p#O6`4SnP4z_Pqu~**P0GZsfJEeeLvq`x-@k@Pi-ZV;}n% zVHonE4}FMRZh8E*XWvqWtFF3=l`B_rL< z>wfWzU+~5^zH$0B_8n)*k|o@I_uYG~TfTfb_uhN&UhAf&rnupT8@TDFn-2Q#f41Ha z<(qq67qTX+D3*j-Mj*uske)3w=2;(_6$dvS5=%_dtScHS#Gm0(n$*OTnnoC-IMzhD zisF!qn{N_&4#gow%4rffF^a1xt+ z943Y#IU}G}jZ++N5Gcjeh(nl>=S+-9?p1ZnBi#v2Ri!i%;`o}WaSylDps742Mnb%_ ztW%Q>nvTchsO(oA_h7O?9goszh(^;GcS&_aQ`IPqH3U|<0kv|Ra;Z)&aH)@J z+?1wK@(8EuG>B6gZ4!nKdK|oXKz%ZXF4b}3aVE#ggiQq%X_6Y0VgwU)nqi!Y(FQbK zn9w*mO|#_EC^rZj34-w&VGyG+9zau}rxdCeYULXBQXL({sgKv85knONXed-wq1^yg z_Ry6E&QyZRL=~r*pk8dC>n>^{gqq1A&q1|ZC75d9RpXQ=YQ*b4jbfEpBSCYbj$4Zn z*Ktl?bOyhl_&r-kA7i3Afkuw(*kGT&);pg-NskH`qjPGH=9j@Fl6)Q z%^Y>qQ49`^;KgEe7IHMR^Qm<#XVdB*_}#;J`PrQGqLbPE%T?U-%Ue0_#FGK2S1Z%6-KbR=D>gXglvDWh^^Y(z zwjO|~YJ&0caSDY36O&bn#Tvi)=?1R+>`h__P>(ZOl)Iy#QD=CpLdeA*p3xdX&0%!HrC6+^f;g3iB2$@QtdyWw ztU@Ek;FxAsB4BLNr&z4u1hQ^ks>25T{&M#M}}?MLC9q^4T$1 z!wFpWj4Ea)n+%)%<0`?9;RdnLVb`dlFVkeuY)hyF+XoxulP)`l%k*S4yGK2oz8XWt zB;(^1l1_rhb~i}ivTaZkb2U4MU3%v>**P4;?XNL7o}@HU!P7q5_tfY}#d&N`iJp|p zSTSUN0aQb?v{U0akV$B9+ik~@bX0(Pg~#!yc73zsf#W!wcG_ud+_;f#+qO+Rbmelb z^Go8vN73ayG$yCe@feB_Cd>Ty1NZXpZ+|xp4XRq>=HLE?Ti*S*pf$~A6QvY;Mn`%3 zzkiDVckz|<|J|DjBg=UdVPNufQ8LrxXtbkHL6hJ!_u~H5I+}qLeBS)qpK;53zY2k- z;WVa?ztO0R>44`@uQx!!pB?d1-tbRvqh5{jnH`CsyCx)Em24l2SdGSRh5$+kxo&DjCoV;++W z%fudVX)X@&?(=9Y35eb6(pc6cc6W^C;wIib9*qSJyoWu)&JZ=|;Nx@J8K<#v(-Xf( zN~!(6-}}1;09<+HmAvOY@7Zg8`{R4=xrdWZIti^cmtK15e#ef^+0nKQ8#eIW?|ye$ z;6&^9_uNxXIfaWazW8Z<=AY^J!}ERrfUFCj3x#%eyr(zbnl)?qhky77uDa@~y&@^k zWW0}k>|=-I_&)G~5Ad1Ke1;V(RveP?4i$fh@p|UA6_dB6+s<7MkU)pfSO$)pAZSR0 zC7y(QheSDINr-7k72=dsnP5CkCJSMgq-&)+p}reZ@dWW1P#JV9J3^CP5_wu|$F}2f896v$hjbd7nxe63IA?jHz2nNtCDNO%ihx__+kNG{jR0 zni+{O5d&QohqmT!hMJR%#veJlfVpMpAww?eg1at+{xOtYxgQhRZ5Z1=lSxNzs&1i_c{`5 zUdOCivl#jH4ZQF3x3R2G_C5Xd)9GGxB&k0?hJ`P>0KYEdEm^*b-HjCaJfK6~dQu1Z ze4b`@3IF`5PqXXJJDEEX^8K&>BLJ8F^*aGEA}RFr;s5PNdF%OSQkuA(bS5W_B0b3G zrAW+;`{eUJW5c`o`i8GNQEUI4eBNj2(&dco z*-1X1C!dCRGC@8GxRB5LbmgGjfP5bEX^15g%s}q{8&+Kj}RiVEJ zc9dZLOqjJt_C0M4WU_J2I{|ug30@s9(%qS7?f%pFpEiFA=Tkf06Vu(@&2h&axBrdn z5>y)j{>ZpQThb{gRoF2)%#vAii6x|fZC7!Wvp@R}3=Hkz8-M;D&RTl{Z@l3jx&HO< zU}2Y7-8)7&yYG1keT5!AddAzi;ONsOeC@b^Bi7PJVSBEmhV3d2a`va+$-wXqzVVJv zan^AX>50E#0exMw`MWoKfQw#wJ~*&sem^@#cM(q~S}z#Y_=F~XEp2Ll8d@fuQc4ZzSdpB=-)0-F=7~o4^`qFgf&i+iO zRXN0-G|8`&$JseZ4QL4I8OTIh2s<)Mf<{KBcSXSEFXTtyh z8w5#2K~z%G9LJ=SRi*BxW(*VB-PCHs@;Xl3tn;O|Cu(vZSxW~sv6~J}ijbRV?Oz`S zx6On=wHXvN>xf}MWCjqEbTubsQqrOc!{~WY4Nj?5ucJ*ZPP-XFl!At*WrmK{YeZ>k zQQBH$7@$M5Ph>36W;oDZw=+G4aU>1YKI54Bov6>IiqUEydI0RtI3x^1e*W{H^Sal) zPDD;3!PQq^%~9F6a?+cw=F9*7Uudnl^`R2K`0aKszwBggy88mDC)Ro|g8_8NH|`CY zGxl>%ed){jn}55NtAG0+47z>%<9~dCYu@rQ{_%#dGPdPXu6gUrxZvuq^AA^^%UjPs z6M_bfPz;Nr6g6m>5G4F0@BhpXIBDh>C%)(u-gM3gT5GQPVaWR6=e+ri=WzMA{#-uq z9@EAM2~V1?`>wR_07>Z@F|{p``9QnzfYj$Gd{g@qYDqC0Gtr8uW+$D1AydzmHYNm_ zIP97*sc?=o|I8#{a!R7IrS|5Z6;q#6m3@;j88@oWSB*0H{bi$A1%uh7R$1)LzXf3z&=A33%LwIS3y41m709$ zl5sh|B+Qq_b4kV${NP=G%h+U*Q$G2B82Q9;{Pg$tang0~nI1~L`_It-_(PgO$p374 zh;#q-8ZJ2MbQ!a09L}JXx)y3Pw$BgV{WZpFB~JPK^B6h$<>+tkBvhK8-~VgQ{inBc z(TmR$;S!kVHkHigfqbc4q_##OMgQ6q1^UMEtQbm-P7KtrhCEPkU?`Af5^+-N@nAfV z=XN3WH}hOsMomD|h`B*gL$^eOnCHM&4f7os&@dm^reHR(8z`8X9;F}!>;O(gj{6z1 zziYr%S6#(5*Id(PPpugn8{^)4@8#Tc&*g#(E|@lb3W9(_p>RO26`4nkjg9f*7r&VE z&p#in_4G9-8vEJC8V^`3&qtI@oOwy9$BCsNHA8++^~z5o4_&K7)Z`XG#|lVxL+^2r zoD)!33NufD}E) z{O)CCUljwdLvH$=l)3r$z|FKoQP^fG5Q!*pQPVA_H6xf*yd_F#wCaMU%>jaTk;LBC z(b0rpR9_SA)Al=FOHgTNZ&9;XEsptJX0P3+)c3Wg5sEq)^7kXo%a zec@WTa3KQ&1Jvtvl1abKQBBbdnq;$C6fjvX69fT!why$4oPZ#d@#21pfvwwUG#YsI zamtgEc%Fx=pjvN`$z%wdHBy-@TefT=n^2s5+!44=Tlzk7zXY&!)jBr+PK-$WREApQ zwB{+wl^U5$hML)Tbmtbjdc}w&s>w+hmorVPCI_Ter&1T;(k^nUO-UDxgh#sFX)Q7# zXfqs0$3%o=60KUC=&`u4DUzNvOb<*gPPHK-C2pn`SI-tVw57e!o0WQ(6Bo(0Hyk6v z&WzPV8-E&&bYjI-*txV*m zW(*0E#<8s&ikU&qSUr|P!mI+6II4BK;?7nAX1P+6lQp8L2+T3!Wu9kha~u&XF-O7( zWsS}#X$D$AqCEw~xFD^i|Bi#yJe3fu>BoTMkjVc*u$A&+QdDIhOBRLj}<)Be*(f_Dqz?OSL`87_(%T zOW|^=EgD@oR+h20Nu=aFS$0Ud2Z`COZ3&aUm4A_`C(SfS zowHhom6j3*ELwwo25uZ-Bhc1m@D{ZuL-Lyhb)SRirFVQwUe`!UGk&>BASKF4VBe?M z-oGexqS?krBVhBGZLFHeU)Ic5c2t6u?6Y@lAQdns10@1-4zTfTaR z4Y^{)qT6Lu8TZz6QPnBusmbT_VAmP+mxBuO569oC?vgGX7&*n@zRCaiu)MTf^Yzpd z!$)TEy(To`@vd7GKNfqnK6rgkL=GNeYy65~*nLy=i~Cl0my)0hb~LKzvSX8wanD=m zgAK3BxlKyXl$Z6hMkkU*>(#d!q}d{;$7(D{J+|f?$M1QCOwxbdT27nj>VRW|TK3qJ zGP`Paa$yc*tJ$~INAK`yXBAP-a%<~5k~*XfplM&3dtZ^HE>1!F1VCgRj%BH27x$ga z$ln^@%7pLmB+QCvlT4iIBpP{Q3x|$2FVYyh&LZ{c#y-+mcA2}O4#M1=8T&UCy){}g z@@T59qVGH`?y*n=x4)L~#EuQO?d?ySKLLts9uvm8rQR`G*GMeDTjOcHFF$nuMqM4Y zYUY^{f2aOC3CQM}i}zN9yBMAf<(B6zHfR{1+3+DaqT%Fx%27Z=uFl)d09C1r_JLR8 z>w|dx3==0M<9i!cX*M|?*lnGjh~yRD%lX8{3HvxmLdC+bdsY8axn?VRyZjQbcIpQ| z0mA2KjHp}&%z?J_Gt6?=KT?P)emp9!RT@NUJi@2~@1>$4zsYt=)SF#aJJ&lHRLXFE zj%y_IS)y;C_cU6mPo^AzcE_!^Gi#?OYkOnnUJBZ)xtga3X{X$4_yN9%y6<(%7&|K$ z=kBZ+sYo*}@JWic1ctd6Wsa+uX5M!JIPC*)1Uw6Hx@5XphNkeuS;+5vynFX{-9n~0 z6qDAY`!-JOFrUr!VDatL=1b$&e=clsKfDniY3W&Z^IR57kY0OIB_9r^-}(lUsdJOM zWoy5#>yRyCCfCrjYxATwW;qH??HF$DRCI;g?QYY0-u}th#)?btl1l-t)!0Q!TmJ&a z^H}@GZrw*Ud+;>4SiO%`hz_s$||q`I7Mk2@-g8i58MyueWjb z?|YN~J^yR(#xq#^%mg2+RrR@0pFr&0q$BSvpB}lb6P5*`De2bthf8m2XQEUS?s=W) ze*7LC{^)kR*k&8s!A6AFvHy>8O#6l;l{2Qzq)ZRv$D>#39pOI1X5>6-N+dZ1gx2O@ zPK=AzGzZjW^2*xm;qu%0L$8mh0MZ$g#J$p@rUsCxl@-eX+iK zANT6r4-c31C+@v;o6MhJXAmeU>rjC4#Q8z!{6-~Ng{kf7W=Py;zJ%7UyzZsZmsIs< z-5hyTJXytkOcu80#r|5>g2K7GbE4-MHOlJ;daiWNS~jQkoH*Y?wsV7(#=R`>7o_-V*5v?B_@`q^aMp*SVoM8xKb1uuwg+m3FH zLpyqG^??;&&kPS%lsPq0zNgxfVGvluZDMuNniV(lQUS|}_A_Dkwo`XftYUP2UAOZJ zb{{yjVy{zJ=VCABMJ@%mD_bi}Gbu(VL{(NQt>5pS;?_;p6&{KcQ`XHY$@nZE8(UNH zLIQ}5=cpT>5?90m-!R-9y(i5o^Qtnth}^%A5?c7Iz;{wEtAShn-Xlhz3fXpt+Tddz zLQJao@jUax8NFX3Cb^J3yd$gAriLl7XyL}wHwI_kGGj=*ES)UzoVJV~&uEIY%FPmm z;FXw|REpVObT`3lomBT-P~5F@Evj8GYuT3Db9;*S9$=%sqfS80%IW1eMBrA1R{hn|dpakf#N#)mT5%Mz&3fvWCbjb_+?BbPk zlsgdp#-ZN|q@e8$z-~?9aqPU^FVqV4N?*3*1`keFDq3act0OX{i|343bu!(Ua)3AG z)T}m01BNXh1UO6-nf&%qE(16&@(k=QARF@2Cg87(VX|4Irf=XF&0PN08 z>o5f_1A8_+u%cnomPFYIoZ~|wI|lI!ZSIY<3;A5?#k_|C%;a}7WvYT%i?KT5<@wHR ziXsB2JOlK#$nA11Gj`t=pPLn7~wIA}J*gk$2`F5UK5=t_9Pu{m%e*ZF|g z!8n;tB4I*Iq0zhGG`J-eS)ZGA-3Dnxt&%B%JEZ=!hzM7n1lj4WuX=&crKhU{B8+CVOWV=ldD0|d7#c3 zOcXrA9y9o5pXAWQ?mo4Kvdoy6`#GG0UMU*qaO4VW(l}YeNUGxzyVw=26TfV4-1o)+nh}qT@*|ph24qY>)|QG z8fwj?116N#7kai>nA(5gAL^m=G3d%LF$uG5g~b;97V zfCAcTI!0s^wUgR{8e#VTP!K)L$-r4IH^a+Dlu&0xbjGNO4qq6RIoVW11nBViH53)J zgMj%{OG{Y3y}_addL+@=NPz1TwRo-w7e7%QbU{-~;^xKrhf{hi5#hhVj$sgZAExZB ztgm~!pX?%=+V*4g)f_bs^_aL^)>{bO$SiRCQ{7w<)Ek2W+cC7naM*cVo~dD(C;bVc zGwocVPQijfv2#sC^DjGcDt#*nTxS{gLB7rhA$&<`4HTQOD0tna z@Il@*cfysxX$SUQdmm4YN$3q;Bhs!9rGDs(zCyUH<%2=!X) zE50+7k!W)SM;Fc|TIbPTpVwdl>Zud?hVT832Vy|-1h3^z9X{ir3FZWyi5m!iPFZZ0 zZavhe?SXFYjp`)K$3po3EBHMU$fcfQHjP=V(Fw)E=rOpn8QCsq<5D5{p&AO9X~5eWlV<7e^<}j$%W8Q(;EV9H;~;1&6SYH@k_( z&0qU(=LI*8Kjzq4O-=uH8KwKdJvO^kpQo1*QpgWgk1G4i2q(KO#?lLgl-%U?Wm=V&k zAdZMj-Tfx6&VQM5ooq-Z0HV$-ohe3A7~bshJZ}ERU-&_N-fPo&z#*yLylrhZx(ly_ z9{nYysOQ}vuH;;pV`BmZA|GgQ-TgE4W*4_m2FvwxL{uUu?iOuEmDpxir?`3UiLX+i zU#3sv6S0oX8GtR>RR*QDJ>V@Rm*MS|`r~x2W(aaa6(w3+5R<7DPkV}r}MiWLIo+p8(}t)2H*=d3R4Ja ziO3-4q2@~Hy|VtXUmU;f>Z0Z!1_ys){k+gee>t7KaJsPhWmw!i?;nj~SPYtnsB1EU zn1=cB;F&U;S1HOY6YwhNoxe1ftkST^Xt)yt;=e8U`;sC>G$EwH=BdeM{ zz`xk<9@V!7X51$No0NL&&TB7cwN(U`jG0!YGd?>u%~jvn+>0u8#T!sXmo?JeZS(>8E}8N4E+Y~ z=5|89s!9?~2=GPE0O9?rl`P{6_O=F7z;P&~8uDjnM!T$q^v*M+O36Qg+`XnnLVUw! zHH$Zp=ugxKi^k?!oD_6B)s}@g-@lcVRheD#B9C&d%mDLlM5*U=a)Nn9_=UeVNXl`Q zU7eW!vKBuH6CJVtp!)lT^V>hJWhZwiwUrm{mnZOBJTg+1xf#i|)0Tdt`%OJ9io^S-#VpPeezXN1q4UPgj;8<_O%t>&>+O0nf>JH}0m+ zWHHWJ|Irx`h;Q8EmI!=o%TDK=Nb`OvmRq?;rQB0^ATmWf84LpD9({Ntk{8Q9`fh$E zX6ECM)pfqU3R4G1v9+QUo_{+q;p(u;QP@b^tCTTi9-Mjg)t(p^VlDw~2M{R1MvOhQ zx{2X4z6wZtG2exk8x5-WjEB|&lXHuKKwt~)&hGNc@(SC4JZ61EvUWp`zymIzLG2yS z3&-feZgu6TD4xyIaaYgSqDNcZZS9U5AP{JvxMx`%$-U!~Hag5^bp-uVvb$*0O3(x$ z0(rxa4`)sr;s?WfzP61GJc)Ywt|IIrkqQNHSACtz?Xi~*6v@ezz7;&}fmLs^5>unj zG$eAhebw{_Z7uzhIIk)b!qDoRx5L(Zh&6El$_Xnf7Q-O$_wx;s~f;76JlEKZ*e?kbdh- z_`WN^l%RzWKwOZMd^<2kT_#{Xknx$Cy-Us{{U5uVz8^=uAGi4qi~wShXTaH~Y_@ir uMSnu>AWrrEBhOF4JYkq=yF+nzx0{~toICE_ z85zk={;a*%+Vg$qGv|s@QIf_${eTJs1A`$eBcTQZ0|$VCfyF~cg#LoxI%ERR8% zEdlfP@0Hh4k^}=o2_q{Zs^OV++~JW$tL=CCQe{9!KW;yPXp@p4$^H(Oo)|3z9_r)8JWF zbw*04)|9P}<*h*OxV@r>b}rwlprQV_SI+uX!==DD08>I$f+&2j;mBoEbWq=je!b;e zOPh9g69uTKH70Oyv&#>EHepB(SdkOD~$lh~j97OuE{-ApqQc1~<=c$gl8WT>s# zdgjnJPHTD}(t3j`OfL=g!FdhTajQ3UymxB6Kr-~Rz(Zbh;=;KN;~L_#{_l4i$D%)% zPlPJr1dcDHq)maiR_m*S+(wDC3?^F8A%6cD92Y>kc7f?R*RbJM6Ozq8KmPp3)3ROK zOW^D_bK@c|K<589#M+BX^xeCWGfoc?wnPfHP$Q%jn5j?Zv^qQ>{S8e@M^TlCU$8y# z0>ZW;yy%nVD}+@;bM=&Z;U+|>nBAAKk$RseFChUPmhE`LYc5GyqLcy0Th3>{EGDMe zJt++W1lwP4E}v4gdnE}mKzxXZA%Q+mqvs2QnBE@{p-1Ms4rIM=@%`*RC9O5gh4iY+ zM)3ZU3EAunr{&n0g=o6(j5LRocu}(vGQ-;9KQD*) zP@t0o=C~nsxh67Wt^K|4+V47?XEo0+)B1MO#KC;~y2M0TAe|F}+XQjvHLsAI=%ApW zo^Yw9q}(~`KW9nnhQ>qTuYxDR1v0TDVk@w0e`I}37ERjnx9>G0Gvy|22%?pm3%50- zQ0oiuvxh%|wC$P9F5AW5xEJpFHxhctuHpSD$)9xEyPg*EtW5I~XwO^5d%kp`q#2=( zdyTow$`cdWYyI+AQ}RFc(m8z;O8G!UL{#bVXN`_2W-E9rDsjs*{-)bKwEvp={9RdF zSq!0^k}^)FfVP~HVRF{w=Y6L7BncBhSU#sWU*n!+$AKG+JWdII=5YFiZs;)YHY4dh zebz-meO%8#&ts=p$GKR+b|vBQDs$MZbZqz8n~(u8ii#JpL?DabF$|53^qLDDdhoDd zuA)}a)x`j+OL>~m%BTTRYAz!7c1AigyRBIv>`S`xqJ&1pDeXZz6b77LpVGpQ-3OR2@>=uS9i`!Bf?nk{F6Ae0Od|AgOYb(Z@6fS%bn#FVjz8fF=s+xh=2@ z519&}1&Pw4^`(s28LSbt(oN2?MtR=+UPv>Xs2_IwPJ`j6Aj7ayPVX8UWX{CT%rA*F zutb>btn1BnqJbC8&PkInv3`%({y2M{m=<`3<*kK?c!YX8^=|GOC#llX#Khzr+xc87 zS0E{etBKXT*FD)#k&)MLIfI^`&z@3+hew-{?f&{<|E zR!Ij_Qn*t>&|Yrjq{NVz35|r_plf0r3d$C}&$|+_SYZ^@G6U z{>=tnZeVj4S!r zpQ^%@xeq4|-0CVAv*}1`67g6?GIUu+tqOVnj9(8`!i)sZ*(y{l|Ewdzwm+!j+rviR zVNu%KuI}qqh)WKX;{56+ROqjjRzw^(Y(BgqO6ja4uv9>T@M|{fL7#eH?Cf*E1bv0h z9}3yXq(CjwI!|_!zY*SlPMYh&SyFfLmZmXpRtlJZCE(Z;|C$gKT+wKpre(9G?|5Pi>s(r}j*s7}T9TbxC-^ZR~d!-)p+S?Sk zRGmLN;@(QXP@3?ZC8_@Xvc-%-G2rE5^8elue#n+vvf`gRQ_mqJW=nH(`0=m=Rf_eH z$+|!q%Gr;t@w|*(0AZH0`_9x?zWcI@9gIhCC+5Ka%O)(rA)XFL=>rFtbNk%%hk(4j ziy433E|MpK!ZA7kh)q8hUJ2*OvNIAl@MJaMbHC+)p8I<`$FdhmiR+TI^Ai7lOE~;w z`JnR_T>SHAP>)OR;cwpNm5=V+(6tE{@+6tlbFv;Uk2%8Bw9QoXo!XO9etvE{IRt4% z#6i;@Bn--X*AW}0jiXO8MAXW|pcW(2R#D`gg#(bnm6Z`~Wa0jj5SLR9fNsJ~4;n4D9vmV^h8bpbV9>WkDD)uf=Myoj%w13 zzQAZ3p%5B!O&kyr*_*=~zmOWZFXok)LW5!L>4YvmZo!F-MKa!Oqom^9P@N$@%?Uwt zGW5#1!8}qFSa==S?-i_EIgwtmGV4iY{R$5w$P256bC{8=6<#z>7iVHrm zAQh96o7@cLNQzW`m9DDOqZU}QWgWxx;ghT_Cy@XTA_TUQ&_L?`W*p1=0|^!y+WFdD zucmfS%`20EKR}}*Bqs&xA;MhmjgAXAi^&JOySq31o$owCBa>b16_wJDc^!Z57YbTv z%17T@FDLBnKY<*SRwls!t=tt)CLzAcw2VNW&};Lu_-vC>6(CgW5o5Lap-GR8mINAjNh+{E?NN%(N53XHXZ z&$#Kn2_53?JWMZOwd)jM8<)6|%}+F}LakE+wniU=yTxpMmm0)?8Ph83m=_#eaql-#5kqR;;eCjescelT&y~i&z=s)}!&R7ASLe?p`YY87|iv&!}AJ$;;MLai%PfVf<~e>SxyOW-}6W+H3-lOeTSv{$=S!*f6Yk~S+S!+wZKR)xpcySwFF+ra zY`^;bPq@w6@98CCmW_!8vm_3x-v}fh>Krr&Ie#7kn^1q2IffX))pnEM;+Ob;s5}m{ zo+IlYH0_q61_ZhtT%mN`wuma5y-(y{3{gpWwIBeg9>82vaY`l^qxX(=;#OFg-^s%Lz7bNp?OJoWMQiw= z)-70VA@DR|T36MLmx+T>{vvNyEowqv_NkYi*o7px~HlsiJ7%hAU4FDGgpPpZgd->N8_{o0RGbm{7 zEVp{zlYIXVijYE&a7{i-%WqPTL{2s&DcqoQwQWq6Nw#vWA(8BCz+fq&t0HoB20M#e|;% zVMgk+$A79~VjV|OA7q!r{sez68!5{zK6F(IISrnUOXJ~fSy6rry3(x&-lgOz&{$0>yl7?qbpcueKox;{VgbAes>qFG~l9 z7{~30KyJXJf7M1Iu0~l1l3Gf2YPnRNcZFF&+)2Xpw474Kega-{((DH&NyReWRwagm zcP%pyApFIsWJy-NdZ<|KI85Pi>|&s_Om4!>PBqzU+9`_rAxUsz-elJ+Z*_|Zm8%nw z#BMt;TJU~Z4-|AYpWp;M=wyU=bfUaI7!T!gpo}jd;!CHB0nvtbSf(=~c8~L{>`ZFQ zu>CK$2Fm~0T^N?pkIRd7+q=BBz*=BEslBB8uoYS5L%IPgOxhp7+`DFw#(Mbyn z&$@HtQ>`8`B^KDadMCtZ)fq^T;5NvJiql81@=%gcGt zXDaDbX5m#$uS!8AFsoXedQ-V18uoWoQF%RVCAo4vH-0XxsOae&*Z?k?aZ00i}5z zs-f90jnsWrO)mmr_gMCSjY;>Wrm zMr+3U=?onF5^m;ZA4M+*f5Rcg8!l_IaA$qFtN8d_Be>m~wwjpnjme&HS^!k1wpLu2 zp?P{p4x}K$9Z%?)@cgwa^vd?UoztXI%($bDbh|Rhy0eKMP=271+MPgx#OVPSq8PqZ z@`i9!BrN!UMh5-8^l31Shy1ES4xu@nqK3mL+{`v<#r8f3%(3WVAj|CPW=v|#$RGQG zh7mcIL?-6Dc2J?&T~tg{eM``>Q-ypH7fyvvO_8l@V`77WoYpk06-z+}B>{MBWl0F2 z%R~j&qmxiPmy;j_sG?C>|1^FPvE$~RM)^a|hAj9|$(k&9Ny+Qa#^m4tEkHEz=1N%l zhT_e3g3}9hUd7f;@k-7W*DF63=1L#hcs)x}1jnCo%nJ;7*lHxz5i&_dLq{j4I{xid zZv4pg+{=dU&Getp6Snh_IbBIi9e!Ihg*5D`u!3&5saS1x%E*OZ+hU+qd+bAn9{%6h zQl5^c*_y%0QQ;SF)wqJ@y4J3FE%4_SweL*BV?2l_2*`sRWqpjZJB_h&(Rx-&j)Nl` zp&aw4^&g>WcCq^Gl5~GgYLk6E`ZJEI`(c~ug%)w&#^i74mB7?^B5Xe6wzQh#;cQ!rSospZw3Dl zZ@#zZZ}hgk>I0x5k+~a9n4NjNyA>!KInE|8C@BJIvD%Y7+bX%PnfiLfE%N)i z3)PJT*@=zaCq#LBlpLbf=(K{)&NSk9fl;%QnvKGv%aE_%2gYdC$x~-o~E)Q_7 z-iSF~w?B=Z`M~R|PgP42=+2DEXbc6ptO)2i_p#HWQ?!du4iBS3^H_%CZU*!Aw+AMv zw;iWV1|zZmOJRRNdLSfcO-Gvr6k*XA!d{=WR{QD#(a+(AF2SuebaRa?^0j36u!+oW zgJO~?p3y`t3)x-<>-aQB-ku%*P%Pqy@uyR6UN5Q3KUURUSGoV?u-SzdQiD>+;zS=V zY{o?W;03VCxrxdVJjv1H;IA__BCxgc41>G{`|o}l-RV}zDBqP%(xX?*g!k8T3w)^G z=pK?0(FxPrSFaMKLoUS--FRoFcBP(vDq4cDhium{}9`q@BD1VT!{J@b8v@n_9^Xy7SfDL%X4 z1;**jRvK_0gQ!$Z;rIeg6cuAzxi*Os@}10>x#4De;B@K1LbA#TX#J9Lq-Q_K0j*S} z#`tY|D$KaEj~A1p`J&cu?p39W2jxb9Y)Q?42FhjY$j(Cdu5MIXbUTQ++7G!t$mcj1!1p z^L=CRP;g>*<$<{o z_JPRWUMEH~c>I-ruIBmjzk1HT;QlF;1_n6WHo)q=yq!KQ$uzrENyRVVtf-WeGUvou z*2DPao!hh8h6tl)_iT)axJ+9rc9Fua)e)#k4o7Xmp^W}#^!c@(SfsZDKq|F#{!oNK z-w#)VEX|*HmQmGvZf0v=18jrO%T7bfXan)toDD-pP-mg4&^i zRF`CJkp@|IOr*NT$U!NdJPw_`^`DWcFsR_W&^*rv1nvnSH93-9Yy9KCh{5T8Vvb*D zbLmGUZCu)^*L3ml(Pq&bB^)0;1EDHc9vKH`QyKkghR=&#L1g2n&j@g_Vr{8^F-V+C zmczp=lK@jej8Ua&Ia)bTvv;WZ^zH`Blw?34&9W~&I#O1mVYrp3vctGuevZf;n#c_A z7Z>Sa;cnE*^{#5|^&@_AAlT@yk=7t)7fcD4c)=W6iI~-t>LA8p@(4gSWtDsRjudRQ zUq#E8s$id)T*yqRGNI- z{gq6*%Eq9H8mrF*ulQ9m|1bE!)};iLC6PG$m%uZ{pwgO2ca%cSWpa?V0)a`lY8_qU zW)NjCbbDV2u&E>EQ!S%8tp$!=3Qw9ad|yO=1hI3en36B4*G$OH{;FAV7CeR{t8GUX zK1^VPj0*jO6YgeOu{ll}jU;y)X3oyIZwP6a^KAyK z{~Cu7ig(1)*T1=>>i;GlKl6p7u{t(*NV`CDtvLkRBbdT zS}9+aJ!{YuEu*8T-RWe&Dl}xlfbTl+kH9s={|E6AgTrhVJSxij>NMj%)YQhC{=@h> zLvi8|#=CLG1j<2t;Ff1tqWYTNd)rto=jZa%lqAA!OU;{M#+#vwBguUcTv?h47RF$# z;VTGkH$--8%0XPx|ktql%N_TKPPv8@wJn>9>En!RI_A+@x7-PyAs!%Y=C`$FV-%`2aK2?Cq{ z#ebs~_5!8^D-}RjFFv!7!91e9HtoPo-!I*@k`?2lEwE8hv1V&2dgYg7(&THM2AcF> z*su3|8wXN(dXTamJKPxoNlTWB(wmF&FgL}La!GmI_Y=A-mA11e`=6H7ayc1nm8?YH zr(4VBtlI?_h!I_CBxjyR&m1&b6~bCyVb6HTO^EC?ChvB9HMAptFRu@c*egnsGRhVV zAVS1TRVQ0}%tW~UG~-yGWyN33YrMZ?eb4BBU3zR5lWhFn>q~MEJQGhfwLtOP`Q4)}AYB(iXj_N-M73#bLjj7dRt2`lo@XFd9cq{~R+ z^&jZS60>H36p53fKp>nI#t{XxDJ;xGF___kS`gh3vtT4(%0RJfVo{|X--L+MReN%! z77p4EdVag)tkjGUx+n{O!!T^?3Oz8!ixE}e_e(M}zBqZ=c=cU`7Q*)XB=uzg$j4MZ)aOKh*Lq5E7(?$x-C0T+USwpV?&X&Auz{Fe>$zmIuOf=}Lb zvIMtqAXn3Qa$9H3b}f&^sLM9Tc2IKG6B=n?{-+XqZTYR3OeNS7az@+L3*l z$;?f=S8`Q*7LzELb0`IqZahZ0}~!vVQz1&%`6DeB5lkEWN%#M_5*&sZIfM9 zy?tyl9N@Jkj5i-ke=4!s+uzAb2f63nLPVnFF*j-;b%p678{r) zW$o$Te%`rRV!vT;+SQPIRLyibq2=Y7w`XOE-(|D^v)bKQI~)?sOfK5TW-WBE@Uhy| zjH#%UD!Ykhb_V1hJ@W#hwq?i^0EVn5kh)=rZL73W<%ggFhn7Yc-&u5x*;V)=3GgYj z$hew9&C{Cyx6p5uh5;K`#FkVq5k^^R-rQJm8?~brTR9Xk?0uFdj5MOoL)u=JbG zG}303Qy`lkLXHoCz!NF5z?`ouxCs55L8hv!hFS)>j9-JjX704IGBYFR%Nn5c;&RIM zYU=97dCWkZCs(J_~liD0KDW*#q{WIAvb_BkZduk9-gt#OA&tBZu`Nz zroQ|nWQpAR6ra1KW&HXNU~<6#x&i{;oStiZG4HDR$+ItfKKmYvpIZeWE;FCk5BA-v zQc<;xxqZ?d)p+|*5pNJphw1R8K?TemUAjMLFP(DCTeK&$8skis-KiCwc^W}XibqqW znIpgn3y+9+7xKuz7I~vz!_5>6H8_ml2I>ufApYp>Z84yM@~7^n7hv*XVnO2YG5A3m zODid4FqX}j5LRyk) z8211?ogLxaRzCeX6^s`>Dh&*es580>DD8T|VDLX)t`3=h6FTl7F#=KvI7av`JD4wu zR(!EQK!o|W0+feK8Io}nMO3TPE+0IQZ>NL8u#f8{Rn98q5Jn1VGB^@~(~exu_@JIL zw6lRT8P+j;?f>S|3}jx#$qCU0jQxhHAYdM$zFKC%hakP!@Kmdtb`w{*xoRZ$kB6v3y7*CPji$|DcNvF(*@$W>e>pMLs> zm-5^QPAsvxOiiN0ND@8M`ezqxyQA#F@A#E#Vi= zZ-&p;C3$z^sS;Yn%71*!Olnb7dFZ(Ep1;lVY{t96PiUvJ{d@yC^7F-N6#v z?`EM90+mumo$8B?o`LN@oNIvo=|wN6(%@SXBfmj26gqYQWzI+_Si#(>Gw2rOO_E6@ zhH;!~w3U+b2nQEOOxxWAf2ZuIn(8&>k>lj_r6@re(FNLRiL63_j7#`v>A3PYAS1y5onxqmhVTTC2tMT$0Ol()HTZwu*Ul&x2F@(-Z;gaWZTxd@~#rHpoZyR>jSo=`6KG{sC;3mXrI_Z zQ{7yur2ODL80eJs=veq=%yFL-REW3-GoxV-a$^7$L;G5)BALtq`N(1Vze&x{hw1>b znBC0AQ64YgU(GA$wKr0bfkS%^I}INI$dw zidebsMF!vVe}>9=YSJKlH@e*Tf)3QJ<#6kb!)p|q!kk!O2jzk4^%V!^;Lb#KsWN(L zB+P9a)ua%*!D*aX8v`9E2;MGMi%!rYmUE=T1FA>-QkBzD*B{hYl;SU{4N|)rA8WT? zmQH!c>GUm_la02tjasX*kZumFfz!=JfbdbdXR3)GYYLNBQzj#S`&!GMNk4RE#vOT) zcq$0aN^FZ<)g|LDGOVg&y|6O-qrFk0dWC7n4J-QFDgJM@-ithvp7l4n9-lYdG4hBs zLwNo017?oxuqp*Mv&ynG)kB5i&dm(a!kFII={T=XWdL`8$0UPBWqwNHPdwIH_hyN z`YXj!QwA9b?Ry@uunvu8B!ek5lHguKxgE5p)3js_7G~_VV^ZE!-7rLdEyMqZhxog~!N#gh`sG_Dbc-q1Jp!Em(=gG6)XRd>h9GJLOLxI$?B18w~ zqa?(((y}@YWYG-RbTxghO_fwIH<-5wo@bRwCxBM>QxH4=TDR3uFVM`de;EbNTwpV2 zlj~pc8&*u$pbVQF#B;QMAju|Q-@fgKDQxi3;f6^T)t^lZExF3WsN)HYS^`(FntuMST&z8*j|oJ zUXwBHz|^KEo{%Y$kcm^Y@)3c76kiPweT0`FAJt7e>`&t!qwe-^+5v z1kZ7-Nxxy*IVL<4TP0jfImfmvX6<>28vl7Q-5YXee+qRhG(Ggt#el+NQ1XgP?cR$RRM?c*Lfvl`BI~%|MHQ zlZC$oe7vDdN1Hx4We_*xu9F^O!eVCOlxE_ikod?#c9P(Q`E44Y^ke#mJCR_;;NfJT zvvwNqg>Ke~=stUZlavuyhu3;@o|6ER4?R1l?!4oZSzL^6>*}B&tL2S#U^8=m#|Lcp zudRl;vnfA}cI)cn;}<@CYe0$Gs`u4sVs1d`SfSh3>$j0G1520eY`L3dd=t7oFw}~J z|9*N{4|MGwnr>w=Bw@eq|FLN2$Fz5SPJhcp+Wj{9(= zQka+}^phFzBh3>u6ggC|dmxdLsOhJY4QZVIY&HIJF*A$`!y}uq!N=iVjq{36h(n7Fja)g$wzu+^ksh+uj5jUdHVWbE@* ziizSU2L{^E52zw(ZA^_`;%iapNGGz4( zswIKlvJz?f!0tlOFVF7MblsLNApN(1VI2QxY1^x|v8!+UfogIxKI)1}7x@P-*3|F7 zDAyi66rxo)BmTTj1SWS8y%0#lX_pd+^W{ZwY0E*cZE5t&x95~7^Uy$j=KYC=Odd>SN7j+b7mZuN15=E`mr+8>LKWtan>zKRcN_FsL5efQ!` z>>$c4E-7ortP7B!oR6zIshf`b*7WU(MNrDCtDUP@^RT7>qPni7-OA?JbD1E0YY8o) zvd^qvO6#z_X)ZKaO{jl{rYuCre5`fhlB}|t4fb^<4?za0{Rsk&`hbp4o;e0zY3o*U z)wwQb(!|{%)wd$t0Not3qwqZ2Oe+q{CFVKbkxjVR*vQlJi6DCTa@bM}U3l>#%e;#6 zoMPC8;khOzH{O#lH9QsY%?w|_5 zsY0oDrprMJ-glG-6o$NJd%W)>TPdt!!A#3x@Uu$Vg05qsyM`*l(hlRP4ogB00fd<- znW@{y>s=IqB`;UPW~P&iuCO;}erfyHSJr~NvVmdQdn-T)-5-sT>o0M^p}dlb#FaRj zj~?c&t<@{f4as?PPLH~%@Dt4=+tPS=+5U4iV8+>v_^6Jm^?mg?Xalf5o^m$6l&+#< zBMuK+_(1*U{#pdV7_=EGXmib$X&u(BJP#1$sKW}JF@;x#v!g2~atfNd83 z?6a#94mBB@pGq*29un0`4zwL`mJC-$MG%8eI+R_KQ!Qj#AF%|{OL>i=8TAA%ao6(v zyv{~;?C_8Y1`i2qYSo>u>WVGD{>+`p$BE|7H!%U_*`0*to6?~iSRVi{;?XiC3a4vS zO4AB-j0rY_qWzC-#`{!HR4|e;N^hd8ug%J-a<*|=ZjY!jq--Tz^Zu6^n(_5Q2Wmmr zD&H+CB5g)NOUw%~EqIrmBt(^Vg^YbQeh70U>o5!|N?)K0alsn7Lhb9yH%SOYVha#J<$REJuB+I{brVW7vt{gH0%q3}5mhmq|=m!tozG;93 zQd;_^ls>4W+h2!6237%akjk`J6`~Yn?W!5urNA4M+0ZdY(gLG0@soy zx12u=+Li3EQ6_{kMH7*`fwGgtyr5V2l72~~WxNceLMx@Cz|t=(*=)dXWhgULB=n$|IHmteh5*%?MORVD7)s(C++6?CE_vCF=&R0>kcI9B;$Tm zo4ZUjWF0@-1B#`gr2YArUS2jIBLX&5WT>JuY~i9#dlHlcKHmP4u=W0%8fF|~B)|Fv zJsTfcst~1A8;@C!fN*j$#pNJGotO|UQ$Iz%kn5cj?FlpIsLd?Qb!2p`5cQqMt_D^X zJ#P^9C&9Bu8iE!hyk{Jl4?M3Q`$sln_BkQ_hic`pUm${LHYDf}m%nrmJ0d^JP?foe#VcUmWJ0 zWN}!dBk0)no+D;hDb(z=8TD^|SBu}*pGRL~>ymZfi^A^YK(S@;=8n6tUscUz@{uDb zI75HsG)@y!@?$bBvu*d`B;Lgwblu7GKAjf-rywzl^uIQp{Mq=X!0h6Wd<2pKq{(!w zBecDMks|uRo1pgE3Y$yQG#Mp;*dvtV39wNYUKua~>OP9uP^TFgQT9tL&w%2xa>`XN z%BRonp)lKRHsO+#KYFz6&!A z#}_$(Fv`F-<4l@+wQ8=X_~7CeibJbNTlICgY#+n_6lGS7I?!yb#I76g88;_TA#%*Sv!|8#C>sipj@fZd3hsx8w!8MD|MC zfW8_pz}(p@W|seBo|W0ZE*I-`T+B^k^eU zp$fsSPd@iwQx|E~wvMD@Qq>SfYJ7vkgsSkeKLD}^*_9>0i3c7!A zJALj*+vty?S$=J&{!Xi29%**Uwe$yzXYBZHM9b?qxIGM|Iqq)wCavm+KfHZviQkN# zBgd;{@jnSMHs&11YCpv7-c31syIgvD+jH%yU*r+$cR(Wi<$MSK(jb9|5cRdn@6$+1 zbTTYo^pc)WaUBbQU&j{aC0^;99c$k(%Xdfq1d6Djo^G7iX=&%_PtUtf|KrrZ?;PWY z_Lbz5Nd^<-LrM{Da2^#!;3R?GbzyrW7HT!j+dZmBr8a8GxK=ikC|fn$} zj{M-g^ol)Q?9;6A8qIiN^BMO6ysf{*p6>{UW!@yyXHIC_)_*@+;U2Ld%%-|q#58U( z6<%-k%g{D>^lXaLc43{=eS66`b6N7tHuhiJx_|b#yMWl8?Gonr<=}eFq{G~IHK2!Nx-k7erV@+d89>!MIbJza<*yZ*Q`zlrP<54pO7>uWVbn$@4C!O2}P?<-{zb!arQMQWjvy{*d{~nhtCyXGE zC%KI~B9)5+%O;NWG?*hbt2E-0=z(&xX=Hj3ui6Y!2W|PJ-s(9Ah+}L`auyXaa}2aW zCpEb~Uks_-EdSJ;5&kCYlnFhX-thBOPAMSUqAsh795?B^ut9w0E$R1MxRLr>)*JpS zHKzs+yM`SDonHt|0>oOUsSjwBPJ=o zrFh$AMe;hB9ygnx!LHVv`Lp7GCNS>Tdw{OZEN~ZP1zZmDO8+tEJe&6NMd)7B|H!HH zH&UlF+DMh-`rlg;if7cWn=i(DaL^F8vDHebOr;s0w;0;-CiM?Yq(N&9ODvd+hRAV?$ZES! zqs9`-eFx>pu_~=3j^v2VLicrGPWZ zZv4VsCuJinEOydd{xk<3CZH0Gr4F=#cQib$un!wU*YFS%I8j@!dfYq?+BGVJ*sL0s zj{@2nJCJPu6@!@CsC1?Bi!jJJc(@=3Mdwlc8_Hb>Cky#$eb&gT?>f=s9cl%0!!{Y5 zt)G-Xx9z1ZG#{nu2yT`9;D3{Be<*nkym)(Corsk94XIay?Na1dA}aqC{e7E-hvqch z=&u{aE4|b6LV(NQCVuzN#e>fVXg$H3-Nlot;c8WRI_GQJM!juDuV&AiJIZE-hgX|y z>xikNNY%f-dvrYGzMfksn-&tBZ`Q26(?QpSTT-nlU4_U++3<07i?O;D+NCLht{YEF zcX+RlydT{J=<1UgsSEMuE$^b}9ZYI_RQ>L9H+Ka+UylAK;K7R`lAdq_S3veo@zKW{+`-<)8p$sofa2 zP#-Svp+!s-f_A{GOTKqJamQ{o+YuS_Jriy ztim#IvE-_aA{syrly$=niY;Ht3Sn~I2&^wXAp}kfP|kr*E5(0l&K@hsjK&DFFxs86 z(^lX*Vpq`TuX@R99gT_YaOrd!5qqOU=ZXc=a`Lf#lf?N+_1s?7$>7Fz%Am8+1~Uk* z1Aaiv`mP(CSBA?U?tcUJHNE@2?c|yHYys!@Ep1C3L*T=S4lCfg6oH&Dt1diqAw zanrYBRG1hg1#XQCx`mCyb6BqgtsGHD!Vx-e+GR$ySs++Xfmv1SeApG!kO}th4iSQDoB^{F$pc9Z3AR$kSS_e zA*E(VRjat3vLI7!bd5DF!*9kjD%3ELzmPZqjr-73Yzlfdy}B zO4u$rRR&~Cn!SctKZ~LG#LQU!mu-5v@U1PyBHpqdK8fBLl&e7n?X?nFs6wRay1mjl z^I%_fI;gvqB`osipr4+EmWm@@`}zG@B0x$Sz2<` z+!FudO6bqeAp;JlB$;_z%>jxtA1Xie-B_%!utH6-Hm$T_q9=Y-_^sVn?UAYW+{)cY zAOycTdIGl1UN;jPfuU5sLwQA2byE6gF?C_K9A;B|>LCx+=h>490rbuH5=EVS9jVmP z(xLs0r(sKu_=Hs*rbsg0y%Z3p!EWBuvAs@^E*+NMQlTU3?qKMzG2V8@xNFASv(Y?sGY z`i_h*E!SRtzWqML%Z+S!;sRI3Uu?TJ0S9{!##V$dyXAA-U<0ngNWAV!b{BcH_R@0S z;Dhy5b-{l!%?L|ycv2tAR<>vqc;gQgZ9SZ@F81#r{5#YhR}Y&7&fDRfbd z<7_3-!jf@NbyIuXm}%>&De7602=eopu!Z zTzItg6msYuoueT~vcDGc)Wsz~ThaLLnDDJAX2z30WKn4*ae{anHTFf0K9%M$>X*1* zz8Uer@xVg>wdilooV4lKsgBg}C96rxC-CK;`peXuSe#i@T@w{#CcY9Lk~e-u7L4?s z7%>?N-XwhO^4%_GbAHHNlf8ZYV%CCLXqW>U;}ND|B_z;eT6p z?ig@-6(%Hnh+48n#Wja%ztbH5?PidY^5aJQosjcyG`WXw*KqKdqhGvwlE#_WLIpJG zg!IldiNAYkq4LUny9RekRpvOA?d?9Y6pZmcE_g9nv90PONsRL(Od*n>_oN9a-jA}G ztpcH2Nd9eHsQoly6O~;#ohlE*7HDagka>D9;4)><;MB%H-YI@X3h-tqi8C6$PemnQ zf~WjAr;?O|&9I8B(Y&oc{r>t#25E~x!bTy`61p!3c$tyR#enmRh6z6^kM3%6~%W7|$TPCB-2+qUhB&5kM^+qUhbW81c!-0XesIghK>s)zb& z{u5)4G1gpNZ8MBwsYc zG{4}vg(pd;eI93WksFlQR7eP}ffJ<3n+c&GR-5xTc*K*F7I29=%jBIozC34e)6U?? zNUI0CCm%5QrS=o_&16kM5UjFOffsi532MdiDr|H(JBdG0Hq6ylJqqRdx$^qby7179 zGcaC3C3Q!E)~X7=l5QT&5gR6FMwL2_L{O7XWY|6Uc^UX8I}K7}8~i8x=NMrve^vbZ zi$3hfWbY$&$0-F}LMC2y$y%*zWlULuuqlOAed%wm3rM{WkQEALbmO+_m`!;aasmD2 zM%tbo)f8T6Z5qnfy7-mw%d|N};bXtxkL#C&TjLecXMa1aQ;$JMSI{|Me}EDaQ74P@ z^CU^nW{*K;f2%v;f5zBM#4pz9SOl+ch*%BG+o9~b$!21%|cqQkUlWLs@6HytD!Oq&i! z&=kR(Nii<4E!>)!n=kP6W}vN|kqe*4X-qDwC%wB5+kc8I z8?M!brVqy;^UvY@^|l`Y2B*?YUB1dz+7B&OU}%mZ%s~fmaoHBIZPr+zUMp}L##ciB z3Ek3|u~}z@E z%M>*6NqnuNZ>-!2=EviLHtMCKr#Y?UGO;iO4pN37WF(3CP#I}{Sr{h2$BE})?$B(=g{fe-G{X#G>BC%&Lwu7zi-8Ko`y#})_r+k2pAjP!fzrF=U`oxeQ( z1ntrz z;7kwpJ@Qf`v;mO%E|Bn()6zX8X>BE?M>H|Y8_zZDisR?MNXrvi%&w75dDtg%b|v^H zNKxzNPqi|h=8JFB={pi~pT2lkHiK@LfQ8nu&#ch3t~mmFU~m@jOLfVzzvObS*4X-7ty^{N-L{P7b?!|KlN z>${C?7&)!?lzz7m5OHw-fp{P5vm+j&x*7hzA`+K&qW`^Oeplu6R{!mN*Y^kC%P2)6 z=_DX4krW~OeLwN1hDKM42sYP!p0hkF_{74hMcM2^osz7ar=?3wLLz42fTmIfHid4h z$qh*DkZBo4@e=UE^|xH?@_ZY7YvfIs6Er@QiHvSFYUPLywQ=71vn!`wwJvGt3z<{? znM21+w1?`VKMiW$^1NHP>hBEymfS;TYAJa@%7B=)YH^+Hx5l2j&TJjj9?a`b$bkzo z*UKw+m*rCB!?7Xq4p5RwLp?cdEV59_FBbl%SGd3Jn;#CL4j|lomp$=AEp5a7Sa8Nu zG!s&v$D6`<(;Q5G!_&619Vz51&V+i=j4{=xC1%UTF{exdP*=@A-!m;UF-qO{PaWz` z0a*L}?u+-w$AwA-E3DGsOv)@RVrk7anta+>IBOS-U(>6YF~%3%A}j8pV`Ir?PGe%T znrvAiCE1D5XA`u!WM!$?KH3P-rDv~Nz(w!m8(lj|N_yPTPaLITl+@l$sFhx)pJ3K9 z1A4iHAA+v>_&pF8y*GLP{~trsQiX&+OvUJ|2@FOZz5Hm4<6pF&vnUBEu5gS?^h7PL|2!Qk*aYv%n!Cja>{BDTVRlsxb~O zm!5}%;`TBtDP!w9Ei>KiEbIfur78)wsn*=bJf{k6KI2G*RqzFp+d1H(Uy(z6j zi|d^KQ4}FG-Rym$r~7sFCu(6o6qb0^;AlpxIu)g37#mJ1tr>Ns)vwnst4*a^e}C)@ z{1ug!f=YoA@GF?=TXQ>+#pABJa3Vro#ZY6&+EmiQxiQ~Y@63a%#!a=Pf5+zC>L*IbHT6dXATT3oAaOu3-n%{03!Fi6Qrp+h7-%hy@1abA&6 z`S`DO`GDdb9L<#KsTP$%PDeIh|M8xYbT_{1z~|>zN+)|<^FH0#`&Q%9%9XG8R^sL4 zEGccg>Fy5_BuGf?odf9s!S!iHFmy06Lp*G7MF9nI(Vv4q=d)p<>gx?)ZeknkEB9TuDb7Y)OlnE`ym>B;&McZC2ut zRSl`MzA?`9nHXqb;;d#JgKSA`kqB?lWy0tPIcGqd(@;?>=TM#DMTe*>?O zVM3R9r%V(BTZNspf}qjzJUn8RT+%>cEuqG1y2#U0gu&Xd&0tA6keO`)fLsoVf7*V) zwKm1@5&c;qYHKd@mLt4(8vgi!urN;YYHqt85p1$C;J|d^e~iPCPqkAK(oXfwh0k^J z6mb~oUBgAeb1FSYJRx+tsfK~ROuSN~0gQZ=*AcYYY(3q4_<4^_f3Mgo)y@pt_ZAjZ z&#j`mTQL#n`wo#gTH&;{XT!cXb8aA6kPjknDmP^(Vr-ka|GFgYsq!q{(^RF_;RXuJQbI89f4}_-}Dg^ zqXz9!TlD-W&nx~#!mt5C_a{4Wx+{z0!kBwg0a5OKJ1dnX`G?h}GAPMjV#(h!Q;ne9 zOh+uk^w#AOq$!#(1UOjS$QG?4BkFGX{`ocR0m(pHnt7tFs~-+IwXK(tGDB9fAL(D+ zoqBOng|`#DRAV2>Z*Sz|8J#`L^dF>$?OURIku(q~el(la|G7o;q<(p&w5Hz-A9m?) zCjIIJrz~NewLuyVTyAyF5)NbI-S>4n{|njUj*FElUimXQ4G!u;W7=Y*b;}nbnOcRW ziZ;wEp{n%T?q}9DS1I1P-||PNoj{`&Srd^?@)-kBNq~8Ki>ie7EWd8`&tG*WWsClb z5seGh1~L*JgE_*^>-~{AY~Uv_%cat`Fc|6ujlC-*iVd%35)?f!x-nCcj202##%_V%aC|$LsQ8F(L)S|C7z*2 zA%?vjd;7$U5@xsG73&Db^8N?aLdGDAm^q4&<=|Z>hOt5Ep3BPHMFf*#t^D3NJZt?- zIf=SlQ{7RW-BBp&<4n)D;89!89rufWq3{d}Ncv3b$c4=FVD9j-9txMt%3z{+Q~m03 z1m@<>XlkZeOFV04Q#`aPM$5`TGbNgIF9mfK=Yq~aA{qlauykR`i<*tpX*v#$wxZtT zvwwX~4`WB)vxW(P0-5)tE$(3plTy`)J(yyespPbHG$ZxWJ68Cw2F8D&p=I|&(Y7Or z_6r2^I00dEe9F|y2Ac6`|JX-GmSohKVf0xlltS1z%X6u^d>`f3y;}rzHBHRc=mm5! z{cE^?Dcx{(Ejvbpa+=@CV==#~%r(9 z8(B#U7^s~5S1Uonq_P`9(`dVyG~p4I{*CxOKH5O@G4WTyM%lXhGTpINC{O`syY-|J^+R@rKW3FrG`kcaz8RFU1f`1dGwwQP>rl|L3U^JP7e;^>_VEZ^r zY!+TP+Cxgz)wW|s|E;7y)CKouMl2X0R%S7JQr<@r3Vi44yInfmDc!iZb zG6xC~sF!pi4%kmNC^?)z`z+F?It&U0*uxXk{(K|u3594jVQZh$!~ltCjRDmuqj z=nb`wMv$bdW0AFknXFD`VU|tvbn?X^*Dat3-jzK+acWR9B;$}uCC5sbo3hFou2<{{ z)k4KYpu{q~u$5rUH9rPs&HZvmuf8w4^L(Sh_so`bn1E6TUK;nI9t`fHYn;mZg*Kg5 zT(teP>v$z$U34`4ZK66Nr4WX8q}*lU|r$EkVEu=cfgw|gyOVMW~-^_ik( z5%>pROO;as2^1F!XT5F;&EvYi@zP4Gb8Pn4OWa+odq zXI~~)Su`*EZ&zk-L!BKbPkLcnffV>2H*+@pl znAvd_O2XY+t*kHwAeh+ty)X3cVAg{ymfcR4q0yr zu<;jz^dzYp%Sn$PP46ew1q3BN9q*$x63=RJ1*a3R+?nh1Z;`8>*B8c0UBZynatZp# z+DuRnC|%C)PY1j|+lcSEEpFSHdVfUb?;d5Ku+v7WnHg3NNTocCs9#jt_i+324?%9y zSBHJaLwQAAR~5!l0U|{<|E4BncUc|<=n~HNqV8^`X@?7^Q4-Du&c^DKw7rQZoZ}I*>h0_E?^4&U`*Qo4md+Hu%s;4n zzUtQp!advW7fb=iEcETRBR0?7C)@pEznd4@iY!9XeaLuLe6Z8S?We7W^!H`hFGxFd zyN(Ccj{Cvxfy0Zr_YwH&?Rs_qNHsKSW97Qd69)F=JvN`N?U5-;h*z%=;* zD)}k5JNNZ#GvdbG4|d^nkT2uEfgx_2$)g6w7noJy-7)4LCQ=Q$kT%{VyX0{3d88;w z7>4V^zH|;CSYspj(NBDes{=5gUvsxP2>QAvRs(X^xiA{q0z%>sNlUrtkLWWF%iRGH zxOxpCS6}gdEe`pz zEW~(Y-RH$CFgye9`5*Li%+xskGO&e4JKezxy(y-rlr<4q-XD}z_wB7oWGqicWK>H3 z*rFVO%mJ%=Qm8p)_%n52!v;1%25BZ!!OI%hRwv7ykd@AZy?ATbnlB`gdfxSGT8VA?-P9%o=igzrt8bA+uw#Mjp;A(^muk%MkG+1bh89wPSo`uPC$Ouh^ zbuFIXk3g<5)Z03b*ekt!rXiOPM+Th@M|1q>fmBjZoVigNiLAaCNrXssph0tB{$Gc= zIaDx`^_M&zzlO#EfG-rbKHIFoa_J*O-g5jDuBA3wrtbdQ-$)l?6){5_A6!+v_jZwyeO-$o& zHetpWo@LBU!C6;IT*nb@HDFNNW8q9+2-y3^_pG-04q0uu_EWM#=ZVLDdw)50)WR9G z`n{FiVF{o=G`0hp+Vb0xXuP2LznQW) z3}#2*KA!XCHdgBWj?0nhbct^K3j3$X&ik`6)rRvn3wlFM6)3+v^axQ@2$A{*DS&+; zW;Wm0f`v{?si(*Br1&Je_L^O&rNEE?cb65Jt(UXp9BD)Vfarn{}hFZ&F8Cl9C*=$&Djy8Pr%aFwzRYc1fYLsk+w9ZHM(u^tltFG%kwfWRE!_i7EV8A(P`C!9bUGP-C~Av?BB6ibv|F0I623_RYqkXr6` zS@Uv8V=~CtzW;-ZQxrb(tseiu?K|sy2up%&xVBvP^nb8?R}aYKx0SfL*APr&pIcOi z?VkTj9`3JL!52E!#N zJpJHGcw{oy%<_}f-4@zIg@CO4xeWVp-^mIXuGjUpfkmJf2(AWfgF1y`ZUWBVaQ~AK z5IT+XG26D<`=`I(m_u3@)bA;dt{3So=>> zaV~^QYd!kFff|t>lL!R336X6$5^FE10VU!DVJJ9ZBr=5`Kff1JrN8`~qwjybMMsv4 z(2oB!wWZtz9vFd>mBY`>MCYyzbB<|+L|tGoHm)CxdWSg`Q)2fi9mOhoqaRN4=*Sn= zjd>|>Hm{4E9q@k16UZx!{o^p>>)k&Hw-l(4I&fj z9`-K-7S`dVrllS7>wb}b!^-lv!PZ&56U_jvRqp`xuRi;`%csteVxFm^nC_?Xjsy5B ze3|te10)F}z5rWDI5)oDZEwAXh+o5(d%x{H*qb2Vx`wd^>epf3F9|>BeP6q~L7K~I zYmxN6#@L?^FheN+sPYcQB<9+4Ae!T7SP~rJq?zUYrc&oi^*nzVj!#~HYiYn-)cPhK;BfO z-)Bz$!aTsKwfEar>5eI-+5j5FV8eqO&~gFI=&whO+2#=IF-VyWgrmi?WV(n^BdvQN z@2^KDhZCV(c$$Hodk#*b0-%DlMkaN1yQagGIu^J#>Q83M!gxs$X(EHvO(LT*$~wt1 z+KM=If22ApQ+hO1506-MG){3Ed^AolKG?HnVt=q~P+W`uhn|%bu*?n5$1SD+? zsh{1ZfPDB+1$dNdOP{wWW|h(xSGxDLlOv}`24Iy)8vwp=nK(2x-gMT^djCiaeH!Oc6wh|EBmT(e>q)t` zxOML`y*8*K<2cQFpkXaB?fyUt|kQR2$>@?R&z{flM^P@2im&FL+MwqD(buFG8T!HSa2f>qA{E46Y*9DAtqx;9F8We&5 zDFIY0W$-rA{G;FzA4A7E@#oUGSIH}4L`EUO`;P5(ZhwtC$Yo|CR%(dIWK;9@2#Jk# zrBQp}(2#g~k@(AK1gWt@1%G2mw ztXocAL>#fQzvnOYfhlz56=AvOzn++1ji&~m$DjJ|CX~qiPyxGQs>{@6E{|Fsq}^ z5}hmHq=C^zu#YC#>ThCaTVzhNaD0w%n;wbxlNvofm{l1`1R1=0HAo{-`6p6fwo6!e z`hgd}NX*z4nHNDEO_k6I(Y2>1^3h*5Z*x#3_2uNvw}Wea<$D&9={~TAsIkui!Yf|r zqL)Y$4$B#5VbQg6f*VPb0SmgwLb1PN<+%S`Ns|8vZhJf@#kR?^sPSB*yX5rN3w?jbV>&Unl%J~!27pP9SGQCU+-V^A5|1yq>(z_&prcA}tdEv$I zoSZ?DxQj^Q(3*0jina6K{SIw{5I)V4O}GJz!O-HoMx!3p^04tk3{HBa0Vsp%@w|1%x(g2E>1T0PzFLRFs`nS z-zU#P50!*BB8Q&AgcTUd@0HrOz@ z&~)d@FId9(;yE(?t%ea#18msTi_A#`S#A0yL(dDMU4Om&70cVOUO zjnSal!}C)^P!pqa*8>F2uSjbIOaa9}=V9Au-Yb6N&z-COC_R_|O9`d4AwllmDlP=E0X>jj z_%9ky)Puvm^MsLARO`wANRG!$i!@t!4du1MA=im@)l|;t*nPI(D=I7RO637SN*Rc? zeP^c}|0W!41Czl7GMbDGJTdF`RfvjhwPxz&BNpKB-a>9S-CHxDRkxz=pEY{D4fW|^ zL2XWczN@D1Oa8QeqM4tckl9iJu?m~OQ{X%Pw1BT#qO>On{Wedb2Bc}c72<&q-8s7F-02Yv;fZ79y<+Ebs#Uo4)`^jum?_u z?KC&SBoM40K$<0+;tS{}p$Omk+JN`-;?`lLim&x0y{Ypy-@bKpv=Z}i)H?KbSZN8* zz4*=11f{LDo3u>cq9(>3Tnh28qJJ@-q{uDzcS9qybzml6Jq)*2VD_d$(8>de-Qx8h z%vfTmW7k1h`Uf=b4JwFeh+L?m!B{-w$iIljs>qf_R2(Ob0wh*N#KUX%L~@1d#hQ&R zkE_-#7nA+VZq)SKGgP0K6YP#V*PiS8-kBb4oQA&<4pdWNGbpDf4vLk~Gz`2+(CS3w zDEwdhUSx2=vG*OB(EL!W3FR&a_U~M7yKsE(6S&Cv`eG7{W-zEM!}vDS-h?3Y@?9*?Q^o$mLP|AoX;R5USK z>_X2@IHy&1Ei9>nS(Q@|6yZAU1a(o7jpE6xz~pQ$!sLh%;8A2Z`;p=(XYX>FCYZBd zXl>SZnzXxx62w4V6X8`RmpC}RX0e4+P>z|zHRYP5ku~d0M$sNjyqn3_h5)V^IM|)& zoaEh9OQO1(W&8C`w3#7uncy;m5>q2xAJE9`K$=)tznMNoE-gRRS9`DeHnEK0No+@t zJXUmBTyPeJ71BFLIO)R`FwAc+8h%sd4UiA;i1gH|a4f&X9&l=JFl^mUo%M_{x#MgQ zd(>cOgSgR0IoJd?jmlt;8)Ppb-Q5THhz#w?zI3~rQKXt4^+{-r?dQ7iXk>k`(Jsjr zd$nMB?mw(IC5X=W5#{)rNXI=hqRXo?sdK$qhP04IINKU1+EL2L6pmWclRCmi#)j0* zXCfqS7+egRvYgc}kF*BA!rw9~rgt@3=wQ=@oyY^c?%)@3N>q{Ma8PPs)4sv`^pfM?)tBCo1WQaiW zR_T0RMoA@)7GU{bt58aFVMk^bmh`9bzh*PANyD4FH9f*d!|}h0A%B43lcvEkOK;#= zQ>Fn+62Em8?`oH?`v6f0vFU)su*f%>@(REF^(nuE_-r4(A^k6Op0^duH;4a%pmeRT zvoYJMO3*RKJQ+3BEf(Nl1JL)YDB{ilS9O1u+7T4oZYCE5=D?C~ID+ts;m6Qn=`$RI zY~oqA6K~}qW%m0##jy)F=u_w}uEO*JN~V7JcW4U9?i|H8X4#mhZZCj+WdvDvwzV*7 zLdc2ED_2N*`l;Y&sAT+g&;{z->1I|oL6P;N{Y(;{*=0T1lJtVGR3Dts`H6R&fgY&^ zl$eq+g>{j$w3$F6R$8eN=3_670@avE+H4IhP*Z&NXQmdGj8VzRve0{V3QQ#N;esTq zTI2>xrpW(h8`zc9;-r@fHh~ z2aeIEjZ*!*N`9;jSlg*jT!dW)VXATCS>&!HHW|b5OqZ;sXm3>{%92|%#vJzJl4S9KUaAk5sQtJ6e zfMqrkVL^7b?R%zkwPAB{X|RbQh<`4mw@qb~$3!SQhPX%~5Wmn*X0S^~YFT-HA_)A% z!cu13TSd{-J{>dNPym}P8Cv1d4p!vTXfTqFuG|fuOT^(zxmuan@$M4J0q|B*|BZ3v zyKUFs)mR$@oJ<-1o3i$crMaT+&-amq&m*t(`r8B(+0}vbyBUebkuqfkhkHCTMZ^%Y z)gxjrZx{k*hY4y{4BT;oiEN1NctSFA&Ftj_O?_sj={>6?5%q)$Q&=?atn7Mbj?0qd zD)2%z0GYIKn(8dK{B@JxY!}l{=Wv_nq4fVU$0*l_05aLxQ}7?P%7~s(exNu0OUY9? zjFsqQF$C|3L{BuNPd*EB?4La|&~1TDC%h1F6^zIetRnEYr>i5!H~=l-=WtvN1AUMs zM3o#2)=JPXYYPuD`RQM8qZ@cE8M_O~6~OS8+tb}y6S$q+r60nLpD>t8L5DTNu`S|PS8zKQr@2!Dr~SmIdzJ<8E_qrQ1lwR zyLfXybEH>w6#_gojVwF&0;4C1K{NZeuL1?U9bb=J#3u3n8bhDyQ-8XdevCi3k{nHg zybs0fX6W#OX9EQC%p^%_>xXUrv~r}#hcjrXKOJ^)wm3yB%DdvH}$BS=N!fETSVHnS8J*^(1EHrbIP1%}8W;QN<~4TZ2`-Rx`VY70Pa zNQ6dBHk0iF&5#Pg)7nl(Yu(;jf04Sg?!s11Fq-netmAJX=bC9dENo`FkfhfinOUe7 z92IYb9ZF=9_-V}0Ihsvi**lTWzsMJ?5%%@Y3Ro9(WKH! z&UU4AXx(*ZMTmc7g%_<)jEaU@!dmbVI^%^663xB4j1m1wN6!h+dDb|EIG{Y`al<@t z5ODF!!DwYaQu1VOWj@Jbtx7R9q`s6m zsd1F+p9PPtovs&Pl2&^C;+aHd;eb!f=QudDg}G{I9*1O%yli)oqv?LBBCKw!wJ99q z+ti111}Qg@bXVl}100+%i7kyTvX*L0;R^XYkHr%_f%8K$Uh5N0h5N5Q4ag`nBy$4h z?wfDM6%~En=dJ!X6~+ynn%Y6>qJDHEou`V;<7j56Gs6KmRXVCmyd?g<;~~MrN%EHg zCIzy=eps1OkX%kN@A~1Vkk&>93*F(){!BxMxP`w7^aA6U`De@SH=`hk2O{O^2X^TC z;rtAWm;%NRvFh;=c|O9gkqs>NFXofS3(U7a|I>r4oHi_ULgx_vC2_d_#p!PK8gPFI zN3Jq3TRnF*dLTyUprlEDA7rG4z*Ps@8B=*qs*qwfG8zwSc(!QE^HD0l_xW~pv|p+W zZaBdpGQBULjunj<4Z?Y3r4)S@)pmNV!9`)7*(>AP4L#E)h%?SKWPRe3p%E+r9C-{C z8*g)Vf=h;uDGNbSg_~KpeXJpimR-XteKG$9Z;|rTmODLn?%nV(Fdm| zy&Uj(ds=zhF7_l7!W^y+X_n7pZkjv)asoNG5**RLUC``gS#||MIF|P;V8P!5`d>tF z!SC9py3YqDuTH(iE6kRy^eHW$^Wgr*Ip`;P4IArG=Ii6UQP*^o%dGmpWrfdd8T#z4 zHY__;oKe=rsB)EQ&u(Ia0e9Bq-b3GyG`2WVA1vpMHmQvnP30Jwlcow5FxxK*JGxC) zrP`Dfe}6f(tdQ32;Hfos)PvQyHXRlRSuXZBFB>Z&#wx^`xYA@rn#=<8;wwZJ*@VSe zZtr2}xqC&zOblEe7A z1mO3O6=YI6x<}~D`LRqCoqICQ$88w-((E@v|M^ip_py>)^!3l&-!O57NqTs?MbnHE z%Tg(0bwtvBz08?=)GkM7*1|v7@q%*uE&eqr&#xJ zfp4VjJ0G2G>2Ac1_O|Q30)n)D4>mAeOfZHE1u-y`sLVl5pu~|td6UPc?0PzEeLNW% zI1FymKzoIP$-c`JOC0A+PZOPDd$_Kom)1>;Od$cg##L(+CbOgA?qZH6u54kPZ>ZTNLk3~#c4ywRr_D_L5y?Jg?9WmouhPLQ{#ghzH%Gd~H}xxp1; z1Y)a){CSYB+2Us-Qys1Mw0;TxSkBbiWbYn(uP$?qIb2&Ofq6T6R*UF~RpM-W9g4)( zWG3deV|pRK`C2$syd0FWDn~8t8%G;d+y4~bW6Gc*&1;-`2sl-l%){=T+wwDI5r z1R3_hi4zJ1Mv&KMrmf+S+ zT!M_ZZOG>X_Yb&H#BTp%as3L?`T_yz;anl`iom14i3>6p5fM>y1dcFG+%lHJjI+WZ z#z>6y1higC_+VB!eS;*is5a8_3;U=oGN`x21v$;QAX`V^wF|+Wv zp_kDu zt=&$32cKgffZ}s{X+t*%giX)|Y7xgct3s0g*>;qi5|!R@&fFkf>l#PJSnFdeJLt*8Ni6V!je~NCK0)6s0-0Nz=<>7U~@yg}&Zu_s~UN z2oQF^Ct}S-cm(#GJ{S_A4o2MRp`Kk^&eO-q$yc7|al-q(GJc>j{ZZ14zsu=pBQa=N zhl4Z~)Q;WvU11?<+#f9QuN{h_oX5W6y&8D42`4`Qy7F=xA;Y#wnjHF^h)Xtquy9Ot z_z>LxQ(1Q=??B)!JcR_>op3JOh!mSqqlA>0kpt>9L;htkwj2cxY{b(3z14*b)zb*O z!J%~{vuaumf-WqzU2IteVr1x}r>$8Iie_t~_s&L;vK_6?VlwMu2GrM5JzGpAoGreh zIBQ=99*rzX$say|)e;pAm(>tpX+W?5I5G4E!OsGqGRi27M%i*z>sP!NP+RsUy|Jpz zgKMEOB7cDtZ$MsI@{bU9ToP)0ZHt$7e5x}B>LLFg7k}gG>agBdr0<^%$SVrAyz@=( zh0BsKvIACf#N+AYh_J3vG5oTmzOs@ec^i+Pf)H+2BOH6JXW@^k^8 z-Z`rH5Hj-PJXze2SwP_ z4o~2z-L#F;R<1m2RpyPu76l=@_TgRn`?O#nWQO|Sid9nR$PNz}z-KJ?CeIyM9 z7CL3UgkGq{w$EH0U6L$caU4SHcX18?1KOOraO`LEL3xDYVdY zQR&7$RGK7$C+toxfVpkx5k(84{-WRX;{L1K?3!ClgKnbDKQ!gY`1N$)B;0a^EjMdo4 zUe@FnV6W20khth2fpD15O+-(Z6aV)@D@Rdjz4O03{-IgE3>KPj8_?cFdrS-olsUSl zm_3r~(-z?xe>^mX;NVHerCPazI-3a8gR1eGa3&eD>VqJehkFGqG`m0fKR~nvVRB`>d%aOg~R7O?zE@^*k?&?+p9B8xbo0m*vtd>&iWCL-kwsmr-U@Mt;R2 zSdt0R5njH+K-}3`tDGyyEZXcci7Z}+6tM;qyzfA~k54<79{sRN@v2eAz?j$=A~ho5 z7+e#gYVWqfoHvS8_WAvYUtEj_s?opb_Tk-Xo&E<{5uIyaEFJG3@-LQI-%PmkNLqjw z%`L}88EyXSxKk&49;gVn0Y_wVR?1$v%$S>`zJ-P35|LKQ_+IS4ftzB{fr4D3f_8J) zlX*1vVf8#M6lp)XrzE&HxRm{?<5)ol4C?kAZG;~A*6a}_ ztwvH7HrO?-#Ft~ARK$(EWY8A%E;Q2Ozcd#Zo+!y=@HZopTToaFyZ0m;^EY~lWr^n2P*vSAxs(gL)T=O86XtGC}5Y>BQIH1muG2}#P}Be0@{ zzyhG)*s}-opDmKEfo~I>C|qI4`+32{(*h{tr7&xzKmN%l-D8A_!cB~Zpsg;HTq?>X z-}!h_$SI#B@wmdN0lOBa?&|22Qt#@cySA~P!!8G&Z*Bieh$;B*Kxkji<)$&Wi!B1} z(hdRqJ=56n!Mfo5M-cN}>Zb%54LP4lX_TSZvezYxyV0%$2t|`F=~iyb6wBhG&YiWt zJ||n6k@t9m$82J@OsC>e6uV!59=p%Xjdf4(u#xi$Uq}u*&A&kG1#T#)1p{; z;K;OsQY+uM2`;FsgBn?vnYt*}K|ddj@+~vfb0XD;sAixln?McJ00{bVrh0g9fzqPb zK}$Zmqa#RR#R}{3lvHq~-_TIHLY1<7{0w5a#|b_r{bh0<epYD~430)NK`xt$CN{QN%-KGBO8a z1MH*xHBTZ-9939Yshy(Bd65MwBL}v$CScy&Z4Vc@FY^hd<|k@{FU`!bf2pRf8GHO{ zss2RWybk;d`S7;Se{5?0Cec!BkzSq|Oqacz?}PH*J!4z$09(4CU=utv9nJg3b$-NyySGOeR9hPbkod29YfCMeRKXfeFmf^t zjIu-OCZ3>JNa_sI$&;@s4lYk4EEK|E6*&|emLP3?1KFH%{NAlKa^_K&#Xo`mcc`{Y zeKj&i2_+NbbJbauvDY3;^Tw;`>X{k6=%4@P0&Mg)aT?ISTGbXE;UlBX zq)C=ZV>DBdkYAoa-zer5h3INVstgwnQ$S;j%m$jSbaPYJJ2EgK^DDJzmKQ zUS+`jYAhEL8rMxF1+&k;KlPZK-AcUgUHLQ3Ne3$Mc0^ic*7m0&uo*pYZ~N-I196Ke zU)lRp0gCW9YkJ{dC>(erxl-SqfrmaSBJJ7Pss6{EjLObx%r+K->UhgXpvaOP?=Oxc z4rHjFO~=&CyJgwk)pLTm+-wWdiejOoz<-;q?htgi@~!S_uP(9leUps6l2>?%hB?~e z1ArAikYe@01I|u3?;*0LCRZ@CB`}d`)qi>$s7QSnnS;!#VX7fuD!LjLusQ~%zj#?= zsZwm`i3>H!1MPBQyE1n;N%U~KI%?EGbKB!bA7$n9Z0@W{F3ad_RvP0T7FgTLk0MYa zdRS?;mm#cmJp<(Df6STgzjXh@)qDI2w0nb@Q2&BC{O9@7-;RDj{}*&l_#bZyG}PxPJvFz2%U%Xx~CG(;6aQ)kXaVK) z)wB9fBBW2fkj??GKoyr1knF0=P(Sv2Z(do|^TLeit4@2CP&xLtH8>LI%YT@@6?~JX z{%x`JKOq&^=0JdP1dhLPAweH=Hd-@5YyZCF3k(R#XKvpU;oHnt;LBsYJH#@@B=f}T z$BOP`mbsY2UL~J=bG#kD3mOkm_;e=91sk0DC0q25Z ziQV64PrYBzsDcBNb$!`h6v???+V7&(&a?brl>k{8B1B~nzza9H9tMNY9IUg&7=MW6 zV%X^jIS61U-yL%4!N+toR`0P3&)<^?Uqh{y?3Q)Vn}UrSS6{U0}}|lef38o ztrWgNRt{Yt{KuRj5HP_>xVmTu=)QZ>Ybu14VP%qV8ym=+wR!5ZQJTj(lI(_ z$F^;&V`s&-eY5vI=iD*Y(?8Z@jryu)O)T7KgA^nuR*zk2-$n>5GLF4+aR3+R=L+_; zX;e8#3I&YWHvDt|(p-thtFb6J0G}zy)(?I?K^n%1;;_#6_JoL7*RxuGbcLFX^V~2~ z*;-$G#Hn%rO(AgKCLzm_GS?*xRfyTw}lkWqm zxPE*$z7!&=R3ex{d{i<&FXV0Pft4IXuC3{6eU*7dNd@Vk%P2kpVa65LZ?V+_2RX#W zxF0V_Q=kyl-KK|K+`NB+IkkZusCF{-`yxV zu$hTb$RD*Vu`EM`QI{N1{^Pc?B$Y*m9Bd@6`h}@B3?PjM_z<=*0-EKA49e$vr6vH`S>QaFzqEpH6ydonLP;k^O%?Dh5F1tz6l3e$WN5q zpqG@OuxUBNi%Wg!A=l_$9YsPTJ#t~e8jqrW3&k!F_YUD^7YSY|`t1)0IQlay3__8K@iH z(3;7qcCL#cDO|2S-#VaK^G_;;k2p7;F=KJcNH-8jnL4{9N)6PUj_!lesMPB)deK)P z8;r#Rof|!=Bbgz|xjMX{rdVH0_rXQ%WqsUkP0yZ+6fFu&{b;-ZBv zK&Dlz2U;&QR$8u1h*bVJVB14{hZ=Jdhps7o$)}7<+P9Dq9dt9mfk!CG&$ERRJTK! z4u|{TC2Id7u*c>|7ZMe_oXTNQMxnwpa2)if5hT~*OxsW;;ZnY$_m8g_-FPV|K=8rhl5`FE0+Y<@x{M1zGr}`PS3-osa$8l6LlR z0W?|562*SMXa%GuA$vbGqbxAUvQPFZWE!W^QLlr#X#Rc$aq}+?NZNchS2}V}J%zrJ zErG|F^EkP|RBwT_WGB3EOJeQw&Ljk6eV~P8c3XtrALwx~3^q+rE~r}<(wD2wY0eW* z6*^KX96;bugp^)Nkou(XhGYshk>58#VGfz|cEHX*{CVi^45Et_oA!u}s%ZF2@`GGK z96Q?p^=3YfT%lPR6p8~&Pc+9o27Hy@i!R4Ks#^13cZO9&p1vO{pkeJqPjB5fiNfvY5E05=?O`=C2@ScoS0N0 z5Nu>#RcJ|xDK?U7Imx)>g(a*FuE?J|r7HdtU%twjQ*xVT-JaVUEG2c~d5uCzqDAz~ z{44v=u@iMK^o#hW5r0+ne?o+o!dvoILx$REXZVlayzOwABCDCH*=HT{-O)z!Zi}&K zM{*R4M2Zx81X2W>0kjK&qk4b5Qo{abh|n=w(iEdIk3xWAoHmW>>Js$X~WZ6A+( zJTA}B5{Ou()TuP|7E$YnUI-{c${^M@qjRMsyHw|+pZ1DM(|z^OcoPtTV!~k77};0_F*aP@0^2*=dPVQ0?KN_ngYtRK+UwoI*~=s5_uao(atjX=m`7`(X|(21;@J_qpS~6h)oEb^ zuHr<~)y^o)x{|<+NU<#i#tFeVhJN!TC$*cok`&{+h=}xZGR0d5!%J5W{YP&PZ4iJm z=}6;WP)iM?dfH2QK9wUYb$t90bWNLb(5~4$yz!Zaid!!1Og49fLEiUQE ziNzNbNgBe_7U!ssaR2B|*?(2gdBl+i&SEklt^&@^@cY~a-QB(i0%FDY$5}$Uu-TH% zt52IlE3P@wwqidNJ1*P*-!q2tkgVJHL{5;{a_+%b&tJO8?)7(>q7Fd*D)7=etvnkB zn??_z62NK+stsur(K=pc1D?%R>)hQg&y*C#mEz+uAF_7yd|f#LTS zgZqA3aYn861S2Ue{ZvlxQcgBIhO${k?OK9=BQ5EmhA)_y*Y4hv>!sz^2;J9N1+?HB z%y8giNgWcwLYgM${^h+mu~suQ;XHn1gZ8fI(WL@5@iB_w7RT+ePC4k(I{_0cTwlV| zG{H}8cPL5 z(Rq>|T!s_Ja5=#bvfpy33Z&lA(ILVA99b$VuOcEztzw9OyBC$X^W7%slv$V*pB08P zqXs7xkz${d+$0u>ci7|dVM}mwe_11bCsdgk@|bZfmZ5|}tTHxes*y^xg|tz27((6J)sT=$A0Y&2bbE4$D|wx}^@x$3uT&di>J_egLW zy!?!D9p8JtBd#EqPH=9U0%VB7_X=}NMXKSz2fl1wGo6h0E2X)NM*gvx!a;)S&%|^! z?m0eMU!%WTR(kg7>^XMQR?bBME0bmZIPqFaVMeZ7a^Vu{ljWP#xHSQ0`vUlSyvW}u zeUgT3JyhdAU~~c$wG2_cEGTmK#l_sL){tn6f6~>RfBR`#+(c`NvopAxGsfCZ%pee* z(t7BjN!XHx%uNEV6+1&}w%p@~79%Xgn{}_G)f@_GGyN z?#KcMM?A00_B8{3A6|7~6x1MmDr+In!C!1jC+2@q#C@kSO_wO+yZ2~b6Y3lOQdWzA zTUMQf#T~zx2%SE$(xzMwi!6Vnb;&k9J^;lI3xQJW+y@tSjc%$aiOARTyHgW+?%plk zD^05@xN3ZCYpCsfb`rRnct!g2REJdmQl_bjkir~`&lv8T@*f7XK)?HGA$dmubi4kG zLD}ewnCOYMcOKa>hUqE3ZU`z54ufK*Bnh@fwDFJ@CU^n>S2;?Z8y+T0k2)O|9~)Q@ z4jKF@E5H+)d$)eNA4Luraq&J?%@y(O8i?WAlFFb+%y|+&CJ)9(!gY7GP0zYhmlCJ*g3vW88mjrBMk`gW}bu({6T8@dwTr$a^cr-Hfp2m++ueo1Tl*8 zJReyXdkWGHzFM=06QUD_juT=z-$K{F9757Q@h8RL9@)*ziM&8C{Gzf|2J#$>@x<{6 z_{lJQNlZ|4-gm>wKrCb5azv&8`VB5Bl!k*Gg&N*?zfRnB89*NU!Wm9 zPMd~>`Ng{atHY*$%NABsAtcFIQ2Z)9r;%wdArhx)>SZ@7!Z-Iy zHId?NEsJOg&z_-UDM&S35}(9hWod)Y?SvMEm0N`Mdy3|eTyy>Mm=BvML9Kdq-aW91 zIILvxuI;H0>1L(nlV0c+VO!Gn0(Ecug8&dGD3bwaho*=}ll^PkQh1;#k$F!m)z~xl ziGoW?w82iM9m?+fA0r(6wF|FGn0_$QOoOBhpTx4qxl2{=GMX1ldJPpE*642az$I}d zv7@;|9l~jxJy%`V3)%7bSrTtGKkqUYRE$FSgF1tyY0PQhQ+D+GOknus%m@8@KNGy~ zHoUFY0Pd1zTRt@Gi%4sb?VtJ-7^XzAWmNeJ`mv)bf+deYsk%+7{$O)XXn+gxY6t5i z>O$hmYy$zAR29+)t3WKy=mIL=GvVxu%83n5?<&`(7UzgF1baSm*5H8v5`VLgW2vmMp@GZ`+_#hLj|(X0sjeBI8p7#O9z}0o%2Tvw7a@~tzU=URKuF} zwvaxtM)2*@V=Awi8ngB~3d~aiMl_mmF+_XWQKa^RDl$N(W$BGcc5@i92z3Z$uPr^+50lZ~JZ7 zB%^x!oiUI7*1is1Y?Ef0OY+(awRKvByHp#-Q7M$Pc+%VEnwSUS4a0$tCFS0Z&=KaK z`-BYl%m3fMB~`S?3xDzTFs9Vy6C$di@5GENIPB5N&e(>T64L=l71KqPH_%DJ?WwK1 zE-I!M7$_3EI6!*Bx!B)b*rmvU;qvLJ^C=G-haW9ui{4>N73IH|TgPAY&e873(9i@PO4O)l+nlgxlQ>5KH&zgLFPFBp{O}wzv2dmyCBvR{v(6G`f_pb zQyQUL2!vQ$RB7)xWb@qA;xR3YNk(&`nSfY&PGBsf4nEF!JB^TA;~nK!7#o%2(Xsr< z%gR+?*5+n*SG@KJhSG0e2~)o27I{#g4ijIg2VU%8CtcuhtBKpLQr?h=FqMr+BZ33V z5RtN4BTHiG1E<~TT7ZP5Mlbn^2y+!=UiC!@ngQ1HiK7+XyYtvWWcdEJ^Y{{BR}+(2 z@AyK9_t?TleK!-DEMzKsZ*GX!YF{qK-X2qssyADp^VRulq;M>YlLcn?1v-Bdphb@M z!>%|#>VZ4C@hmYpMWCnBJI5d-35`Y3Fns+G*k<19ZvbOgVgYQU;ASfU2CJ%G@5YFL zh#a=jXs^}UA=R1eJ8fRc>qYA=!yLq}x{O3A$24xBd9>^Vd$^%-Ik2ua4ESZD z5Uq#~$IT;QH1Y(9UK6O}gD_-S%+C1%%fEcx`(cUw)f_k+m-Q|ZaOB&ay%!-1bNc`6tj{mJIuWVx}ST505Atvf&faJRrL-3g`Yglx{&OCilA~n{U_*Z% z^J7iLW#n21H0e?T%p7viOl~9oDSzJIlOiD#p#-*y+7ppz$J&#rfX<4 ze5e+9C2viPW20?zbVPbhTHqQuK+|-Tn-gMEmKxeC?5k5zG4Vpl8yEE-`hTkir@kmh zC>iy=mpTo+R-1WP!{YHJW+sEN@L7W)ZF$nwbbI_tj^TW$$IjBP=r6}2FI%@7Z%}La zaOL#GaAMY2>N7Hhq8yfd0p@-w2nL=90wuAw2AX>AL2199YM5VN}wmHz557b2~&Ojn*<)Ko6 z@oC$+2M`&Jn>hw3fIzZ4_BbBcq>q%8^!_5p(3&_3M(0g2*3q5cj$eF^ZB8u z`UTYApWX`b{^WVS4&OeV`ror?3@W>ffppR4YvZcZTxVIve{2t~@po(4zn2>~t6yL} zkrABA*$%IXics2)tk8ez$Oq_M#?NLE-yEjTIc8>4CvO;q@+=Lh!tpqj%h4+LIJRtn zMb{O#=x$o2kt7A5N&G&&(hH6=3Hcs5>Q8vOv&+6!Pd!Fbda zuI%PCFp##jF3jO>rP*m;B8sLX<&A-GdqoH?H0qaG)-iGGH@Ek7kYSno#BQh8X`14z zDW8dPULc~Ayrw400{7d;?!lNQrzq-b9_Sy zAoV8tx>@f@e_ryI2z+)_c^eT5_B;s@j$#2L2{1sJHMSCkp$|HC{|2GlyZlv}s}X3K zYmkUWLtt6skU9n9i49kDx9)!sl({s6S|!3eAhcEYE}f+ju*KIEoq&jW=OoYbgjy61 zTTS}qef@O6^6=bweITj0Gf#o(K>v&j9C*%#q7l5&o-D`>&3&c8#__J;vj5KvUmIp_ z>pLPdW>VJ0M*v;7m)56HUW=oJWY@r@gBN5YsL|Pv|>0El=FA&(p0-5NK zAhhQ%0_Gbn-%r>p{rD~S6jW4zwWd{6z`rsLdK_DyTNq!8Z=8H^gV}3kHpo5Pd=ea0 zjT0J3j?x4v`s)o*lri*ISPI`7zQ*e9gdIX^PR-x7GsGTs4WbA7Ga^WF4z%XaF6QF< zJa>GiQl!pWk|fpIqSmi;f5L2SJYnFtBbosQwG5e5M%X);8zYStu>&ZIR}wh$e3y~A z4IXgilE_*TUa=9d1qzz6p$x27bfR-J{0SugsVKq=PxPGl5Eg`w9K^$hupNLa7$_b+ z{XM$hvv<~$K3WvfSQO4f%}6`jck`7A|L8+29bI>e|K_MwE{2w&uuw=2F9_d1lV|B- z%(ivHJow2l&1oeypB(*`X5oIwUy%Q6|LN_FU}3WsTTI~VEITO+EZaHp6EGf-8MlyF zU}m@c;EK{!8b5o9AsAbbxV6DO^_J1rb&N#Gz-ijbm}TYxqlzUmZXv2|cFFd^S@PWP zNH2#qvOkA^J#UrB`>iHSBaaajPtL5N0OMlhV#@!_us!o48kpR(&Nm1b`1N-Eog@1C zZOm67^eD93b(`2Ljjvss^y0xQ3>oB-7N8i71YMaeHJtl@xd8jDj?v;VZT_sSlTmv| z>`cS;;*X_?Oo^{K!rz7r-8{CR>Ev%9(KQDImPDbZpmK~t5ornxp{k=GN}jofBaozS zyswG|y5Y$fF&vd&+pL?2MS*WjI6ZqH2bP4xe!F{VaqpVkoafA(mBt9RT8b|OU9WpWtKu4I+5y*Qo9Rw2{KaWF`qbzAhRub`4ntw;0C`)_ zos_11NGbm$-vFdsMe=)8x6#gGmzHMOOA-SQN*V+o(}j%1e^bQtiJura){1$%@@A^x z4-8H<^il_ouOMPjNDsZrd)v)<%_>q!51kgiWrRHHngrn^g|ufxNOVZ3v_{(B<9sLh z3ZpgsWwthVT)#aQ!#oBwAaBjScCkRf0odZ335L{WemtR~BmM{!mrGDxxC0WqP3t}$ z)~n3ttF!Q1n{_FAOL_J!pysOw7kc4jWgyUFFA>qspSaC2-$`|XCKq1MWlTsK-te$t z+?_{GWH+!V(zkE0;NA6?PcWdd94+fRlvC-gxuxMrYgywI%h1p<47^K?H-!!OaqR3C z4;+D-1ge7_AG!;TIFW`Yys5c!>AL-Z7|&lK z9FO*}ben#KW#bmOy`rD93N@*KUCxX!m-xiHyj7M~#QU{OzstKg*O`tj#jagcuURXa z@H!nScaSq&Fd~`YUfl+JXcr|!6`zE^o5@cl*yLgfe+h7XHYu6-KUokt2zP5X1cxX0 zn07y>sgASx2d0tUpYD+e!M+>dLjH=_lJ@a|lmOA37>%tnsl|7qA3E9t#@_iF8O84u zc~mEXDO-5TJ5ZjS*L@I5>BkkUA+|K z%_%v>`u2sVr`Qe3^8}xq_~N64KNI9+qLN#wv}hh>@3~*Gr%WGs`A~#;Pfr1Vp7+OE zd>Xzt(*M+KpxA{SU*~haA~r8TM20c-aK$)cMiOo!ySi79$cI&Zj64D&c-}XG zHPq}}P63lWXa!pUb_M&bZ~v<5uPQgE}$u*cr3tKy{v>9;JK_wr$|a z+SXRFPSs}}q`az^J3+u64g_RpC+rhpkYr7-02FY{*2%ANs;!Fj`*pwL0^Y-SI(<)u zO!P?aJS7(U%RFpQGy2Q-cNK}L_MRlQ%U!nXBqPP(2KvfmbIVJ{<(0_6-7$sToivA* zT{J0$8u4Nd#tW=W5?g6Qko(z;WJCP6@?G1tVtpH>g*Mi(^Bk?+MZ*P)A?Etg_5?TEITwPMEjnuy^bQ8O6q}^N_Dt@*STV8FsErgE9 zcVKfVR#a^95A|9cJ^_b!)vNh$>HMrLLXkH7g8Bnr$WH>_;CGe-f|d+lCk7iDc*|dw z)={5+x04tcZAMtrYHM#`&8|lIZ!JSiHV_wXw|qEN24NA${$69~^b=_bXXb<^saKxk zk(XsCafl|pjV|D0)!v&W?(r;)T4UhfTd569?OHkICIq08|Gxj}Y}e*naI&{&68xxM zGXGbJ`n@h-CNRsVB-+x4o=Nq)_B&Y0cnAgSebAn%hV0s zTh)@Y0ZwlXo2?+V^*A@NZ%y2pu+j?xwYfhnta(@=eeo>{=p7AaK2Ml9U`GmlY5)dz z6>|9cAFYVQ;4o`cn=+BK`wKo@7pTdoPv;1P=5qPMI2ESQz&2GmSQ9TttqI%to_N91 zNa-Y2{gMvX(;>E&_OZY(s_+&Umy=w_ogvaZ4yt2W*u?YMt|?`)3Q(|9@nkFXVb61g z3E>$MEfHX0*`c7uUo>?<+$DgUi;TC+LvNZ`io1galo%b2#h-41?%ero`Xb(5L6+Pf zetd3~?R6QuG<}%EZ}5X|`%>v`sIkqC)W87wyGnC-X6V~Cd_i(LKs_0bZxr6VTwrqy z>w7I>rYDadc34^)%K(kp)ft_TIXN?$rl48cpjju8`~u44FKg;}@?Gs@Mrw!vC&XwO zixgJ~A51)*r99xzP`cJLM`6#q>OLLI1JNs}jnEC!wV|J~wCxA-2EMftKuC#tl4UbCHnvSza(Pf@Y1cooiT#&@Ztv)w)ZHVPK$M@y(~z8JRoqA`IFTY1I&f^? zCa31bM*xIHuq`XO*Lp2ywspN}g7=4CcS&0MXgf`+rjhry>|I<_rH&3-Vyz8vGSRc2HTPmI&W#{onwkjIy8xX}2jj+^=hh}C-GaR)2~Tv&xu#+G(- zKJ%5CAdiC$;5Ocr!$?n$@Oo~L|%3XlTe|{{jCm%twPmQfBg`^-9 z6ExkiBLs+ufOff$t>2n1T{S>_mNs+gq0F*8dNV^fs|Nn!fyTnHGrEP@K^P_ad5!C( z-rIEZ4{o*#x=;T0#tSX7#KYc4&(5@$LqJf2@Ar_$s(oTyGy&lfpz!G7L41l71ncdU{4dc&pQj5S0St9fe0X`O5$Z^=}N5_!*?(DATGIrBMeQYyO!AX(w%H5t|b8 zy;XO!`B(qrFiyhtFoK_d(+Fno;8KlD&!*nTdM?KKzE~G~KDs7*jx1!RFeV3(j0Hbg zusQGW{|nK0$_YGmD!5mhv$hMoxFm{|jLYo<^Hi;7d>niPHle3{Y_IC@S6YwMmf@8F z*1HLmg#NY(F?R3l8X75MefXMZ<2jb+5oasb+6A+2@xvCQpP+K>RCE)s&)nK%N}AR- zzGn^nG351vd0RlH3*DKRz3XbvP1(PH_8Q_oL0JS6b9!Y_VVT>46BmVhDNn2g0v3yB zIU2iO+7Ds4n(W)0kFQ2s(4xVr<>e}1$R9KTKKMkhwy?wz8DO$)JRFc5WQh|G%Mfnu zsestT>gJi;pljIuxF24GF5%yPLzh2I%H@sBy%H``zg-<{;wyvM5N{$odBLlIwe>!# zB8U+fFSbw89vp1>7P(X4 zqwWKyf9-oC&%YH*-q(ourT4m?Oanf1_X0+s%HD@-SU0@mH zl&>EmYHp}VS&`a=<|brC^J4Ao#7vHgEv5x>V8Pkl2zQD&wLg>{ouX)Sm$yH?PW%W% z@GT7aIfCt7)-Fzu&$-*X4pt8HrC)un(HXeID&`559BKG)(W?lXLHPC%TqkmWzs~! z>wQ%=ex86i%+`kZZs*5WA&!UQQ2Zbr!q406*BwQTKT;lf8BWfkWwtfdt6MyoPNj$y z^5U|FjuCCv5B7&`bhYf-8vUzvG@w}xcTUCw#>KF@1Y;}Z-Y@2{3ZnyTnyNCo9VJJa zoPc~>dvO6pb@oFsZN43y`5j;yuK%OS!(v-KBM7UC62N_mm49MAG5Nq%=SWL>UK5vD zMeTJ>LKZ5nu3m7!K{k3OyJfqNjY*&EH*d5iKqG^Ig{K<|uLDy`+o_?uD!bD_8rpt- zwm0;5dG%9^$1izGiF=-A4~V(x4VNItZ*$4rm$IJWxUWMB!JEMbx1L9Vw6Blc-S=z5 zqWjyuX#b}cr8g=eh06rRGw9P-&eK(R)W!PesJ97%KmdfbCG<=<&gNn$Kv0l>urBaSWM5#luc)`v$TSS z2|&`BEUe@x*(SgMkDyuf?8K%HD)Bv3aKa#?A%I6VF{ywQ>E%a|f89bO$$`UDWe9nh zSl803b-d0aPIv!Qv$%cOa$omLRyOUab$DmrlvVRaT5aG@0h}yV=4dN6K0<3A%!2!; zK0Q4ZFJ1g@sStnOY407ZqV@FU1-oF=_Q5Beq7yz*5B!@n)UJ1!wh)N| z)%p5vQ;h4&DYKe7O)^U+V9~G{dOfI}KvdY(Z%w2Smv{PkH+aEk82c~auE$v?W615A z(ZQHieikusvmn$k%F=q8O7iWSiNjCdz_hm0paqVB<%@FuA&xX|&Tz6=Ry^2-(DzlP(jqFNNEA3ab3Awr;NTAbyZ^`fEk+l_+dX2=GNER>gV6XDbh-P zMUdeb_rZxHj(`$2t!ky(-*DNWL|51hP|e<@^5$0V7{c-1VVup4dPVSL7*kJ(bo16p zUsGRFJ6r)6PrKfeNPa?o_dWd#EuD**{{w^uUSVy&!!rvy^s|>>Io*1lQP_<9xjtrT zn*@#}X>z_prO=3%rltpdv^U@v@{Vvydk1|F18UFV&Wpg$>sUO}*_(u^GMO^D2>5Z? zj?;7FL3YodA%=%LGyLNMPg_$9 z_uSQ%!3NaKK&-|}T?}lP7J!)x#_>p5C)fA&p%i8(3^(XCM=Q$+^LQuOzI%0#9)Y8BVySxNXMlnRo!~raZ0nOS|HPOEX{o?|Q$jw6GVObB)Vqy}j zNb6sPN=JJ+`8w&&`gw%*F9(hJndtev8DfwT9g5grbGZWDC{VTxJwfhpMOT zj!hGVvSxD<#HNh_lA*^v^((4J*3vftfi2Fqi_e%Sh3nuXy%eUBy+fx)^1I@of_%jJ zJO*r7MU>-OL@(SHE%Is_{l!8$xZ+wMpG${gOY8va*iBJbBEzcYQ+EFN!E*DL$9 zH;3>cby%Vg?q5Xj2Ki)?Cjr1r=#I&g!O*J2RD6OEonZ5G zd~P`eM9z^}k<$6hZJ!L*SI6*y0K_8^@O2|e{l5Ga4lL+_kc=}4xUL!S^b6N9M*44a ztlhf75J#mrKR?YeDJzV!o8=LTbj!H2n`nh^O|zQqptOBXCzYVYf?hGBmkU#PG|BsS zve|nk3jV10hAM2qsW3pj2HX(>IB}g`PQeBSc|2Y zT_;}#)c~jZ+~(_bT6e_le`;}vI5qmUM<>Ax8_fAkF*Xy;#JuNq+xyHD;lVDT`Hyv{ zUS%N!=Ed>7y;nXEp;LKmp`QY}m;?u776voVb1WFlYo5~#hM(Iw%BWQ$keV>i$L#ff%`kV;F#2089MU^4|2~!C$BVTz_^79Y zB{kY{JByzgaz|%k&+A`DDp?6-HA&mBwqe0&^edTTE)T@jG_B{MD{cG3{k`W#;LnWE(28eZ#4Az&($Lp?Cij&7 z*y0bmcZL*DGNUAM87x@eNkn4gqzUcZxQFF^IPlp-4s3=srBDS=CMn`6h-Uu=d1bn_ zs22sw^gDnBuePc(9!FBXH4QEw9+mukt}3<8<<|I?1}AWMo%uIIB(&foN_pU$Z#p-R z^o8Lk`0Z0xU*|K};Z8X4t=kd$fBW&7qA~@{H{__0Ty&d9CfdV<@`I{TV<}(3ak!rD zV+A=Z)JSbR^_c{k9_|kWc&`DNP6k*Rcv(SM7N-b9a}19uAA{{Cg{JZN!KY@u#C*gkT2EyQ-2hJ^4CMAUr_G1D#MXT3aL3fzp zA~Oe8ViyV{6K@ylxi9}E-Mu&dF_!!O{o9+w4kLMa$6Hq6aBRoV9c$P^7uL2O$7<-T zVh;K~=-H_VTLji*mDKbRz`;^$l}d+cjEdktSU>n3O6=_7X61yYI$BL~Yh>NOQ|oC3 z7pQ{aK@Ra4>Y*{O*@>8i0vM|)Czl1Hl?U(tvPe($LtEG+4!l$!cD0u&{j@=GusiW? zBL-2MgjuNMK=qqRx+*TRJ*C03sa^cpHbxo~gQsVB!OF?meN<=N$nMZaTwJ^yUPn9V zEXKF+7?a}Ox#Doz-la3wXXdKQ=kw@es>^pSrEE>t@d(1rV{mv~N5(GGa`%r4)BEB@ zGOaYVbop$qOJ32K(2wbGy&Hzg>00OudMNOXZQsC~(1%32OiYInX(Q%C(7Gpl4{C|Q zYOpEv@WwAVx3uO(0Ph8LjTz*d>vtyxA9a?99hwOTh^-gnRGgAI39Ipx2`0!#R*o=EBz&Yq-ch{QCMG7 zsj^vj2MFd%)i>2<#^34lubVR6O_~vtMfoW(1WD7=HP+Sr>MPQysOX7~agBY2sosNy zHlLitsuNh%cBuh94rtRh0SBjFM;&3wDaw8vu2pFuC$uYIdHiFav}{M@TF1Z$w8#}0 z_SCm;_+ayx{6S*m5D-R{6A*2%VBX0Ik)Uznb^8#cg`Y$lMC?p@>_eVFnwJgontA)^ zMaD_lPL-}Rr%vzgpeNvvD0Qwd1Bu2Bam-MJj^tya>vH-{9>Xk{?*VJmW;lB`Nk~Mv z#sWCqY6s$<^^9%8ccrSpQVG1)gm~iTYm#uoby4+ZFY#_gY;ULPI+omYm%B7ERqKFu z-xTH-5bP(ntx{HIX9QcO0?#WGmmP97@9zoje7=Cx^bS#=GHAz-2v8-18(BHwsVTtg zpFz@zQnwz~`hwYTXSbjS{$s#cg~+lpjD3RkW>85g6I_A|GdR<=W*-r|f4h*AaWJLy z4yPa4&AZu0l0(Pmjm80beIV=KtY>K{0y@hE=UFaXSeCSEny}Xu2G6@|s=z4S*y-Ah z8de?D2pje$(bza>^eVgHBwWg~_N0bp1|tp*T3~%$!C^j{y44eeN4a?#KE_@{8-s>r zW4`TIl|FOXKw0Kd>&vm>!75fySZ=_6kMnDt;;lSV)^4vEq6eN*b$@s_=w0wr-z#KW zNiNmXAX#D0HCthma=-7ExYyV6T+g0^%62N;T+4?&zGkIj3xklNpio5K^pCeF3eZ0u z>k|EdZ!BwAi$+g*ud(W!9K&)7Vp@D!w~s<=U`A(3-iL=2iQf5e^{c!nRrE$ z-Rn#a+_o3>jwen*IO0dV6pVM>(tbma_B2>4+}p*z`pCrFbZ8)n;m`u@P_P)?5uBnntv_8 zcQRu(nFUeAu7ERE^FhOm;s)?ece|U@9X^I8yC>c7ck12X_r|#71!3BJCJjm&w-VBjX={gRVd!mG1`ZakY`xM#(R;wxs!ecC8d_!4FnT-^Nnb1b| zbWKL1PRsV;-2M|tw{vuwFJ+2fmI@dKyLeY^zra^7(p)esj{fTWg1Q`df8HL*;#1t& zksSMaSL{o!xS!sV{Ru>Yu7>^1-0^o z2%t3w&EqP~I-qcg6ZrYnHiPVUXqtA&==ZTa|5Du(^wRVN^6-k+T>XF6cD@1wK})7v zh6roz$6_sfxj0iiH{3PATIc2JKiQ|WTymVBNMbi23#}0hhs;r)Rz0wpB; zKoyq;tJRRSC}jNLPx0$G;YMo8(dYQf5WYx1Znjxn4<}3ZI6kN`)K=Bg%p|+ez|*NW z{qFfw1}GkFf^s@&qe?)o561L#pXxGj8iIw}lToVP!wrnlfjYR#lA~acqP2C`=}7+i z9)p{sq&Ux1{9++fJ~+*_LMOmnm+w#wfw6ibsiBiWcf?OWu{reNr-h+GrYM?|yJ(&b zdCk#$K|7}5#vdxn5{gqJ0bLSjOit!-aA z?N`sc8M%q4u!0uq`G-Uzbw-NIy>sMb8R)#>z+%=jb->C%>?C9ZpWvKi{RvWahR27FJ$ zdp2Q}9BwLcs?-ju%$h>u!6Sp5eU5uL!QMK!om*HX`oX$+-r{Ty{+>m-n zWYWIR0S406v7&<1J1-Qs0l|9JQe|lcYGbO57|gL`dVjP{?IZOvf|ADe=)GK`A5OH2 zspMCqS~;7!;_K4I9d=!|yPj?)lV)y5iTH=RkT&F%7QFc{gnK9qV|AsS@PqMc>f_p0 zXYa_VkdyF|X|hlg{OTy&T#E>0LKuH?P?f;&UNe3pp=rTksqwOMR$4twL0_$Y-G zocdo~Vuju0jVSdK3S-D+qh>ATNLf^FC;?nOJUwHcHeL3@h!MuEPsEd&J1sd0W*2ap zgU8U6{bo2HVyHe&@pa+*;e!}O|lyBA~gP6L6 z_~AxktZOeqs}zv4!DYwMNSM3;%2KXrWTIwT#hn7f%>{OW!i66esr=%{m!HavZPaLj z!wU!u4gEWQI;Q^lfKF`Afw1se;m!P!$BJ~e(pdE>G& zCiDtK(~$ujiiJ zAZ%`FURhozv2&QUbwFvszt7ZFdr#3}owh3)G26^f_k?vJ^5+ddIvRN)s=i^7j>3P4 z^vkW?BIOKoXn`N4a$YW3;;_4KujGEqcJx?AGW~Un6FLU5@DEpV%yeGTCT4!D6}Td+ z$*wRY1pAB=0ODJ_11`NKslw02v*TRU=yJzr6omb7hlJj*4{0i=uE<*^eSX<2YTr zX+c6=(T4Xv!ayYqQ19+$B38lfTmr@v_j zjMB41KQUCc@k-+<3MIEK3MpW9E8r`1!I%d{y7MfcPszbV*5ohFIq9{>@xm9!M0oQF z;kM`LZmBpk`=CrYOP5H935>i?9qZ;=ho%J|VawzF;;uUFW5}tex6|mraM#vQ_NDz4 zyDUA<)BL`=zTXLr+wH9`cO5Ra4ca8H$0`cHmc*P6;O@zH3#gBYg(NwA0-t+~lU4w` zb<%P*!Z2toaprz($O_7qVJSgsRp_T&n%zc*p3pb{e#CK{RjD!{;pBK&f|%`!o1O=3 zTtuSMLA;t|*=H9?tPfVS(l3p)G+$43g*@Hf@kDFxJUjFFV7VT0_?|8GuQagmy{2C3 z{)Y+WtZZ@Q#H(#b>gZ@c(x5@uP3iBps{%qF+y?5C5_-A33g%Sba%ocA&cL2$y!}`r z)N0deSu3hrFgh8cb9N0b(j=l>H?XKrXEPjk^`I~^DVIF4$}st4DVn7fSZYX7LGYtD z_iTvE(Wc^fT*p1>JfQ=SAc%o+Dq2|4*401(4v%IJ3-{RximKgjCekGMTIh%FobVVcu_$4ZDlDX!X{e$f49 zbC8x(XtdSN{u3xW8)}8#nVFtd3VIKz^#`gjf4AVRA^`#p?U*^b&AH-x)u8;*0a{#a zk=tdPR1%6!!c+M)3zkkcaQ1BgbNdIDqlv*Y6QJEw{FV|PzL8^u|27fFoU45VsZ)k!^z_3 z&TP2}@TZNFx%#;@Z=h|3%_gsU=mF!4#o>^VyKbQUR2X(82ZzE=Gy40~PvP;tSWF}) zxw^KB9M}5t>9)JOYwH=O z5YxiqH5py}q!x4Sv}$D6hU?Fo{shqmNPd1VTYN8UrUV0TOrkf8%IdyZw6a|C93UxB zP(*-TQKc*?g^pqmox(J^Z~;%6Nl_(tMw=8KZKMwx4)Vp7EvzJ=rOVy!S%gaUEDXma zdkE`r2u;BizoaR8Zh4@xYL{!#S+i!3iE+1blcXv>d5SqfPHsp3?Cv_=A3-7aovdX4 zzwBlIf%)9*9eM}HPoq^Q)_`1blgHMPJEPF;X-9lqhbv*>l3hW9oTo+Sp?((K*k|&v z{X%E9_?0z|wGGZWP9L0|{=wuAfj;Y4O^IYb;|Yt{Tu`R(daW|@l(ruT1uk5d+TY(2 z7s9norwmf?gHe9T*`zM-2SUKC_jX41N+XiDk&(|)ra$pq;t;tZ7z>-2`UNlirq}bo z`9>M#vi_Bc$;lI&+u8l9^R7MOBum_);av1gkzP9YOc;oY6Y{u*2uj;aM*b4I@_P!N zpL3SpSWnSY?>_j%dR8y0mEceCPAZdNT%-T)1S)M|$V2C@e773{%fU<6k^6cj@#?9F zUO3TLy%xDj`dy}v)+)^fnVqd1vDA|j`b_5q4dFYHA93tUvh7Bc-Jf|mCDv)SR~TN; zxowZ&QqRW@&9DDrEaa&iFI|og(x9F-!iA`)@6xtc`wLbj#mOI+iXPU_4kgIkMTR#BxRg5h+~@(QDH~uo$36*45L#tn8QKMOJfjT#2j8&3S9W+U{{`t+CkIM?b*& z)(Pm3@5U_6Yg)SMGE6_0N+EM|k8}IqdDb_4_=#puWh1j z)2S<^Ik^ankvl-esS;~|zRoej`3uV#qL#O&UFJnCMmG8Rf0M(~CVT79#inQDvd1+hdXp7C#f4kd^ zyTr+SeO#sgs9Ma?@z9x{mHjBG_czSJ)?0$ZXE2C(J;%NAY|g4=sA*{H z7Il3gaNc9%jy-6zr89a}iml^ERw>K=HShGf&fCMn$5k9FqY{B8Ck!QAZ-H&Q?7Y5I zf-?{)(C)+i^hk+pma(8+#8o}%4h5BuVSVvh##ixnX;g^;wq)vIm$dRq(Sj?MGEix6 z=VE>X0Qwt8`YTy}&PXE>w?^tz(Oo4_X3jF`^&*|D4778Eg^u>`J2KzS^yeOIdTxxb z&lB7Vl zYNEf*{KYLMXx!Gx1j?Z8lBGF*l$V1E%i10~ZkooK@ft#_+A&H3x8Y{awSWwksV3@o zq)BW0p_3I}QYTp{wXKO4gcXK#P90x^sfp?PtN{9grgr-~8G(RA6~m#0DfTL3IQX1R zVS97Y*-4pl-H$V_swxy3bNO4Y4cyhnSk<5OChlB#^>y&c$fj!klRZ(K8o$hZ)7VfJsY7HDj z`-c3NiTJSAag2O?!s#`u>$*IGH;P;_{%|TN9<+JZ4?K4z$0&9~FFi1H)K6E%lTKYn zS{6)ZyMY)bsi9g`bG8-YX7%@6&&88C+w+M~)tefDE$Rvu10q=5ad z6L$V{=Mi-gPOWsX!`i?yKG8?evVmIP(J)>#NH1`6GClWXOCaPCmgUeMX`(FOi`X2l zU}$Uy#7yB4lEW^3&U}A+?lUzc;ajuxForV?hF+fyY>RBcU@`KI;DYdemp&WD{dc#C zD2NCB%8fyTV!(+~ifZ*$56P+7o7HSM$mex86M*<`IwNfPwv&zCKNlbmYxj@y8X3$M z%2KALOAN&j>Ir7xs8<-iE1cwsvnOloVb|W!QJZ~+*|Jk5kBF!4VJrq?e(P|aK>$Fx zbJQ8*(^=H#CQkDSaqoloh)+(kJSTcb@GU2fvrxeYodA+f^QU+>Da? zTQ-Md?`d)YpPWKZq12@>ey6}_T?Uv+BO^=QJW|61->SN@VFnQ+%b!0l^^l?@b|)DH zhA}mXA=;ZT2ZOziJuI0cV-r!kupwB4K+)0p#A0G)$+-Dt_r1K((OGz5P2G!_)y7^E zfXv%!6KfEh@zGFJ) zE7X9;xv$WwG@;R(eXuJc$J@__Ip6E<^5yOB@;Q<$O}u|u{=Ft}rZ%@RCMe0932KG(hm{&bg+{<*XOA5Mf4tQxu>7ASYWDnvX5dz_m_T=E4s0NufZ z?dsKn^twO5{L2&Sk7ci5l|5&G9NykVcYrVDURQw#l2t%v8C=Y{+rXGW0HK27lNVs+ zC>8gcwsDl`XCE(V{wwdf0yj;BDq&Zx2#*59114;8tsH!%RQ z^wS-mj04=0g&!HTk6ln7ITM`aJCjqD6Y*nZ$EH%+Z`3a5Igl5$WRaoDNT6bM0Rt5i zlHU=;0}Gu86xRCP2WNTWHm6^LuWN_mgk0MSO;t>FhpUn-LpZ&YY;=?>6bxi_Yt zBRjnIWzyn(ADy!&DaUxjZ@-2&eW0~dH=g`*vka#4_uLc}mti}i5katmNI)Zgn4n0h zqM1jg%GHanxzS{3lqZcPQC7&PqtQl6=%QUDMg3t+rlM3LBTWRs0>SRk0*Q`oF!u1U zp!W3VEx~^ue(LlG&jf`4bppMwIz4^b?0V{ao8|rXR_|z*or3WZK46VP3?#P`OMtPJ zr35OVh?0#?IA0e-=?$RsS#WT?$vgk+efQV1e}tw|SPd)zQdke5PY~eOaL>a>4Br1; zQ0Kq#36vi*vyG>YZr(dw_L8GlXqI0--#u)g^0V(`QT48#C1WdQTdNWyU43!UJN1kc z0aUzi#;x%?T(x%XJ;trQoi;z+Q0FCh#@qZuh|S?h<6Q1ZBDQ1+JTV(GzF^Wc3`mt# z{NQ_I_Df;^CofdXvF$^ww-YbykysV0p32Q7Wx=L3C75XAYQ&NjKM8*ll9a-Eh6qAe zH1^B5j^Gzbg0Ll$@8S6v0mCGvWopra?y$i*{6DaLf*Em?m=Ydf$sSn@0~1k;Zk6Mm#pK~& z2NQNG_+}-4rJ0zdkenr!>`H_#V%8l!=n@ivh6|}G@lf}m&Nqh_)PG_lRb^`oejqn6 z**;k&Bjt$jQ8%eWyYDesoQYV+7#&|Gv6i+`3ru1;$KftUm1^H!pFL!%jkvv;a zn=&K9>S2L5rGa!WL!!e{c}#LZ@FH52=1BBc&!2xyQq*K@qY?iwy#2CFxcc15Udf{~ z9F?ae#FfQ8^E_nR5=cTkrl}4^sc_MZ|DEV~8Hh6xqK32+$DcHX1xb`RqCSz$mPO2* z#{*D-{JRU9hL%US(RS_pzGvX$dOQKZXET|7KJgQV5Kxa6=s4MQ^{JMPv!DwUi+Yie zjT4&yDn@Om64`x#cm_fUsYl7r{WfuJ*S`ytGL(@?wAX&;8sRKyn8z7%PVIibR8aNs zZ8>`JVqj$fWNSm#PZ4%mEV;e6;MtOCc?pzJ!?nYzl~aLEWg=$UJh4Q|WD4ohM)Kh2 z7HLLpAgQ5q1SkL|W2Q*Xo4PE&zJ)*YAUUX?qjbis^8188{}4D1uibcb`l+|D1hNcd zYOc7YOxjHri^q{u1@N(28#9HFDo-#-`eE*V)@(YRc-Yte!&z-|vX?3%*W)+gLR?vM ztZkXg>v*DTSfylQ8f^9YGqD)SDU-p_OpL;d^iP>n-t;q`4Du~x4A>k>i34Eg+<-c+~ZNm)?oKNCye#t>kwhab&MJMaXoM>HxKD&$9TVBd<4klhT=uy z1T)d7PSw4#o_^8!R^m+uo)UW1aTj*=lUh-wf7WYZv`cT@#CUCFp_Qc zjfRT{Nz6AmMR_Clr-rJ+Miz@~4H>|}=yIh(d(cq`xH&06gt+6=6B$#I{V5La+CkrR zs#AQeJrbo11R_Q%JvbF(6^c~`A-;fAoS>xM=c)v5xJR&&#xPuyF+f*#%X@i9rog{g zY*o@^QWt$~>Xyureqk~wJpZ<1j@n-Jh^KZi(;lT|Ai+!%sMzF&wR&E_xRxWYZ++{o8ab8>AXk|$M)E>XgNsW%-Nj@>m=uJ&zGpwWSx_o0q*VKtU|zw znYtY*QV3uL%CD34RFe~b6dH@wy1{FNaC8lg-Q`3aimh-J)ITh{vD_YQ&Ic0h+`_$Y zQ@;UUe$?9920@}f9H#_4_zY)2=51(QTmTZpA|0lUUJ_NJO$ebWAO@k0cvuxap zx=JIm%K|x~1>-b16dBWzA1CqmL(6eyHd;}C;6*c%;Ivu!3L=F{f3y8e$vhKb9WUi4 zS;6guazvS)F^n=r3>3{V>6@_=<4lCoG_ps^S;|9%C4L_dDyJ~+wxz+@gD_g zQS)JMP~t!B&#UOkdaWLcb_4!vN|V;mEW&C1gsTMp(gJfZ2t&7ov~cJ`!nZ8aZ@+_L za6C;9=1-s%tpl6H8`!-^=Vf#5w_|X$%9ghL>LIbZ0&!Rm8+8k@9J)(~pOeng<*sS) z*@H_C=H}6S^fD8iiky_!SXy!-F-pPYZ0r{$i?Rq>iCW;44VZniwkZ){2T#-FS3bO_g23fLVs6HiH`IAB*JIu8Y z`gdsc(S+y+C8EfkSbA9Iz@%k~s%yV%Zx((y7Q>9203}v5Bav$*^CQ8eVT0|zlP9m8 z-%DOMr}mc}*F;2}U!*FL7#;+A!RFx?u)wpMHtbvzHp08!Ctv>Ri=D%JAxM1*u{GeI zqMXKk;~-dyT04M8C+CY7RWfVF0>qrXD)y1c5lV1vx0L+dr7B2}=PCp;gV^$-s|0u` zafm8TsF;d^%oDUx8i@deCFZZJ?RDgx@$GKnXEc}u;oGe%&AP4{{H(r7%x-B6Pq`x*s4rb!tB`Dn9X_9!z^z;=%nJ6 z=EKsbs?)z$6$(=q?Kacjyo-uJ(}Q#96=Zhb{kxi*S^Ap9%Oz>dTZ$-3+!821fR~B$`wogH z7yE2uq0+jU&A#~%4~8Ta6Q40k?#=ZCI@%$Sc!|`m(YXP)3q)9Jek5{GJ!rC7PNRf+ z!(i1b{FjI!urP4~cV%@V>RNzYeLQ`YZxWce2(S=gA@rL_)TGYBSiOvYD7x0ev}@NU zhXFImPwbqe@A@EZZn?dB`tJ?vdr+Pnn9;!J9K~ut=U6)*wRuN9H$%o92r|t7mj%d$ zW(IW>fS*G(>RmX>t3~K1+WBN$|D8NzU!EXmj+yxe#vNMk(7`&QnWw+>GFT2W=LK_s z+2qp--$wT76xb5*^Dx~Q`SG&Pc6$8$pY5n=<|xxU?caW`oZhpgd$o(Hz=|cLHs=Rk zCgGQ_8Ltaxrh%vDZpP>7O~1IO&OHI@=H`)u`ekE^hN<;nyuZ-fAxS@=3s@bj3ay&$ zZ6?~l$m0tB`t-(;4;3z21wtNB?qx&6i1w`@47;ORJ=-&k?h^qr&0&>Zf_mcZ`(O|2 z+4oLhe}Pb6TKBO%V!OK>$|?0(-T#p~;(HVw!^($f*7n+`ysbi6VsBfmE|QMhgE^l- zUuPf*3?bZ`KZQcp7Z5wUhKd({;VCeZEUFE;IKv}fq6XgbC<{v0EF2BePrIC+V`*$b z{;ZoZRt)T?m;Ey&zL}j@HSQw2eN}FC?6z;Js2_^4t}f>`lcYDLAf^TGh;G5%!%w}E z;e+*S?e-{S&8SAy4dfB;*%N#>i>Vi>wFHxA8^{j%V+*R6(us(Zcp*&*JuYzR9 z2UoU%vOeLohE`A3oA~Coj#=EqqWKICPOGNXj{DxH+T!3V(npE4 z$u+{LjRMU`V4?5JiwdhaGyI?kE!Xc73ZEV@21vUcLFZRh3RpUga+9KlwD=>nnLUy~ zcvASz*%fqnGMIh%{6Hll!g_^fe0UEVl(C~_Hk?>vnKxLVr{ntWp}CYJ1FOfSbE+%g zDM@$o&!)mrp6USi+zso2Z_hbR?Gz&3@9a&=QD|gt4e9TV?zHpMYC|h)no3#79QI5` z8>GFM|( z(!qTJFOWJx>p6#(8YBS+zTL+_ZMNd=AVA`uHlbF2j=W)1!8Ep>QRK$m-zdj_tS3}i zp-=sM@jFMtG@G$0vk2eDp#%(>|8D({=45qt8K<2%iM(I8_e}I6w%*31t{D32H7%|} z&+rA#(%Bh?-j=^UZ{R!a3LP(U_A*!Z?ATjGG8j}tbU60S7E{f4a2vVHvd?zu*81!C zU3}VWZK!>9dE(RAfBF{wpil&i1%_NujVl!5mr zLj6m}&1L9MeeQ;`+@gjNTUV3W%eBQCi!-yk69GZL5vb7wpT|cegQxpF1_r^~EFujO zMazByng}WdsvEEQr)V>E)TXxl!iw zfJhc;F{O&=(aSxeO`4 zlm{4{Rj;LONOj8B`8*46mu)ymF3E#3Q^&Y4;4Ettdb{97UtZI%^>6l*C?tcnde8a` zYQllui=-4jWSO-eiLrucs`E3}6ZOSmO6(fSi^__GhjXZnsNWC&xhC+A1cEvEE$^Lo z3J`HQ#%&e=tIn`F`_W5W-{r`Y04^E!Tf?pKtS`15ZQfu5)z)=+fxwlrMkxq-QC*hW z8VuH9ep}9b%wSk`-38BqfR5LmE%;F>&#I*i=U7jM6p9`of9#oAp2nU5%VDpPzj22< zt;aZfcev(r9Dgdh$WFKZmqpHo(rTXKsD~K~EWPo##3R z$>=}7>OFao9Vq8ntE9WI_NsiyCPKlIEDJlq!4GP@ufmcK@sRpjnQfesMHrMGBW-Ex zi6N38pM{ZLH~OeZL+Q;1J7TGi=FG~YVRk)8;FF&8{cjs`tin3pc2~mt(c{AdGkv#0 z&;2q&?H|2TE9SsOWk7>6Zg#)_5d;)44xB>mPgcsX8rKn@=2(kfYyP)LnJj0I&m40J zD(#%yh{6s637YEmE&UemEXHyPvDKi=tT^&87D5hOMShdO8MAfOkw8-|Qg~LulMue+ zRYy}-f}fSD5WPked|S;$o{jb=ERGBAaiM(YEj?Ek|0hkz>jpS5mExqGkA+$ZOSxf- zpr3AX=Zlo3z!3hl{MN!feon0pMWdqDKM32wS+&jvw;aq!ZBffPMTieCJ}ns<+>w(A zXZ4++RbmfiFK`@>6sF@Cd3DHj6lU-<<*<4fVdFIE#1ZBlw{Gk>Lp+- zl(w*+Z+2OZrFRdfaV{KO_f(0GG(SIvpO?N)$q^g`M%P>sKj;0VyWa$l0OuRX4c-2~ zQcNo@b?F!ve%Z?+eFL^56squ#7MJvRv4}pqq5=v=+O6N);P_FTC{&CuBg`c%AqxGh z9Y-x0O>d(1JM^bJZb?2YAYy=+Z1ev8cOj3h7V&^W$Mb$a`b~mqegRRdDOow6KhaQehrjQMAcyoOvZhZMK}y zo;F}|WjyOuVw~)hM0d*$WAS^-GOD^v$K2hvHb(<`l!(}~U^qTuf^3(M)4u(@@R3!v$D!JY!-Q3UXW}0HVta9Tq|0zCW>~9>fxg#%k@meekc^D0-uROXYk;cNJoEWP3&GCZXByE_t)Fv_Pk zE|IF(nFGD`VZV+;`7$ZH)_H#}8^g~}aSN!AgjjU4(n>EZeFJ2>kr|&JG89@j@<;o1 zBNwoCAMGB*0H>P~3)s?%agEqB7)26}#zr_Sz28OLbm>4NalRYWsk0WNGoBBA^>_dfaQ4YxthL+BtN{PMBc#A(yCit>UY-SB|tZ=}JB6Rs2O+6Iv|-6VLP5EQvH0X8V@?94G%eIhjJjw?%A(<`B3$u?vcS}fr_zO_7Y9X zS^fHMK6O~vFC`?9O(xEQ4BlI|aYV|rNc<#Ym9CRZn$y+Q(=*sRJ!8JPkDgZ1p-q&< zfEiZeRVGXI0NjkGjUL_YFU^JM7u5qOVPbNYXoLRNL|Bgp<$)@@M{V5Ky7b)|i-8RP zrTm)gv`|=9Z^uK@G9^Kt@6}0Co74_=f_}e+Xu<8*PAVB5gCju=`W=3Zaz2OB4Y$~C zeK-E`;g7qfTh`@hmB7A5plf8=1cFc5>XEmlK7q>87N8%az*}N!tG`Sq-=_UUVSi^d zEiQY-@qll@E|jy7ba~~e#$3XVqP1EHm&fy+1LB90B-gHXmsDepep~3c1W7W+3jml@ z|3XC4X@4yG)p=3?s}?O8mZg6*>zcb86m4n!D5&CG#~B+3i#<}YW#h)#GG$rV4F@JY z&j8_YC@0qeU7#%0wm*h7i?KaYBrl&sRLE}P40VUcN6X4S+5B3lr}H6GU|-2_iLmx4 zo=xHQo^7p2iFkcd_LWmHeF)4o6NNVo{*m2a$pyIIh~H7WI`VeRh!_086?*HtqbLzW z6E7U2%$ea-X;Rafh$v5iTNY2pW=$*pmLm3byG1XD)o47P9l@klFF8kri8=pqo%x=~ z;ihRJXL=Y$KcZAo(5aW>Bd_Rz$YQ=(;f& z<@=`#X-*V(+I`0ex}3haSvSkSq`&AwQ0Q}?{e8E@;nuA~s^!*0LGY1OLHIWKz2_OEA4|r`B)M)6O>w|D0NVvJmYI|VHyTl<`2HweyzxYCBF^5n&XFlJI*^7U zU2cCg;|2b(&m&fN_*q?V7-PvTuIEQVKCD8S5{eaUV&~KoCaJx#R++k0->yX*C+Y^*$F5>jiQRD@8{%(XJH54b&1bTxfqqE-O*WDSlU!JBES z`d-fEQtz43IvjTMh$+kshx9y-`%o)QXM*URu@Os7pyWx`6bpLByY)&bX8R##bI z-&!`T0s1eT+g%t26U)h-n5i<_S$fcI;vqNzo>~v7T?ayo@w8PHlToMrfiLvfhWB&oD5PAlg|Tr1D+^mbl%Rdb}rYiD2LS*cc={3ASS2`nM=fwH zFQ-d%?&kLC1;wu~=%7*Cjq3k*_=2#-bA3~{v};8xAhzQ7 zwq)m?-}sHQ)Zs^sLnj5Tm0kbPhl%6jw3rVunzfV#OmodP5P;)R{li&(HOm29WBR!6 zHk0Zthp#dxs4FXh8NT+sllDcYg;b-YQ?k|*y`>dm6v57rTZK#4AiGJ`L;|S=v4A~j z6*#-l2ZH2jWDd^C=b7K$hX7r(WD+*n6pT0R`-tm*!Os11ARB(pQxci;iGtjDCD!@o zcV%Jf8ofg?-jB_|wIpAE+m&^}GDKF_&Xkd5Z|gF9MYl|ng>KYH7E2!-w49@+a&^$z z+Y>`M2fO1Iw$}z0$EtW9Zz~dKznoBze$d9cRlHe=Zmeuj?vfI72p#exVh=}g@?Bud za2>cbAE^~sH7d7vE#S;}m?8QHMGI=bHS8rLM6%dI6b=oFKt3jIxbDHr=tmz&!$=4+ zgjDrm`?j{X!Z2E(^MwRRknLyqKhbuqeAC@jk$7feB^0O>1*V&sLHOk=q_2}?Vrm(k zN`|T=(&1OXae;^Xd(29b{dy1^r91EA`_sxeP*bym=A?O=uM0LTR zyx%oHzQ@o0DhqCZy+hp<4xYvmQ~dfJaZubd_O>hq=+bcL>^%yCx!2#Ou1c3_8@<5W z@MrM(-+Umj{pQh((ovVmsv2;{V%BhKqV2RC))e&`q4Seu=!w!#m;TJtb6F##@kn}1 zV=T&j#n$Sw%Rt`cq>10F4#JYFA!gVlk&fyR@V6JWf9^6`g4P02bwNdUrE!Hj)Xqgx zU`N^3<}+LV?50;qPKk{)&lFcN-u}#=jkej!s+`@DEpv%m_Y`AJvUkA_mb@^@reFc9 zSc(d9Pe^)C*1(wn5+35x4a}QndX5wIJ^f7ZA zU{;Q~@Z{M z0&gI&P){Z?r$rY}#Dfna2enSm`^Q#5Y=GncjNX#7k1@!ky?m-6fWSrex-06XC?L4O_6E;Vtg|Y~cKw#(RJIU>NY>WN$Cj7JJ=cTWa>~fZ_f>E2n3} zz86XY7fR3!HwCu3VEfViBwT=767I#>d}QFoWr8N~4J_C9l1un$bi8wOe75{kdaC=a z<2ml?1EqqRxF;4sER76;j^dVvXmLhXy=d`-SPe(E5?1cxF_an?8E!msE|goVg&L${j3ueodSc1;bc z>UQ4DQUd)?v=2^?{%~nAcaoJh3d6)SlT}>Aoh<=>{hJh9OX%ONMwfO}&ssukl)ue_YcoW0QKL zpCjhk5Y!Ql9p}Y8R931BNgqhQV zUhKkRRz1a3o)2}WXVA=kveRcaOo1(z_HAVP9}VuuY^46ArH9&gkQbr<9oP|WVxR)M zTn6i9jt0iPlCleY%6jceyUR3HEG8}CuJ=7`m{!#m8kZRza^AliE#kCv@L2FvgF7|V z(J!hhN>Z}L!J@7h%(b8{RB?(2gJXD$28WxKPQ=})j+@3^EH>_urKiJjM@!nMjkH+h!$%sHwoSPD)B~?Ip3D{3{?WoW=Sr6DGQY z;TUiQU6$Q8(g01QDBzkyo%_lYtemSiL4MJybyBF6Ssdxx%s7hJ?x(LLKoy8~CnptT z`dB2dqO19KD6s=|H*$hA&o{AKJ-ZO5+GP!E`g)FUy{bGD2&kUwNzadYqK-hYsx#s- z#`3)0bq)ghs^kEwO2`y}j24#u9TstsZA!!F?h@In6Ay`V67c#W4>3l(gkW;g3T~>nmW6l8GgGf zH$1a^)Dv~LD$9Jkj&1Or3LFqRy>+uP>Psyf?;Cja8YDdadh(=HrM99a%lfdB2UVf& zKHLSWHhw+x5-NBg`zesBfgzTo(7##z&L>^K#ZxZLzOaoOh8L|@RQ6&_0>CAxVyA@`D_x2dagX-|q zYmF*GG#iB$g{s@wmn>G=`EFTHlADmq~;b*qy78ReZd6J{eN!Ju$ zvkuBLPE5&pv*1(>4S?}7vbU`n-cmE49nGgs8?!@|qi5GcJ{~f^XJnMJj`BmfXR&X2fw@*GbmU%hlx%&EG6TTETlC z1j=&IDZuiP-^o*lL%`*h(-zY+-WgFMob~S#P^fBCu$I=z=Lq*}aSbeFrBW;HQ=~;T zZ0Cjm+yF%u{mvJTH-1Nu2xunyrl%+Zh2_paNBgi-kXTngkiYiND^*GaXX^M#V3tbF zo=-dQk2o5ieY6ebbSSif^Wmjg4ftf9Dt803dS3JT~xtQd?P*IgcTB>X2lp4KZgyZ zl%I=af74a3^dX&6uhUM3g!(d)=v?t(P-;+3mn14AY`B~6N=?p+D}n@nNk$8zaPN5o z)nQ_e_03Ozcpx16DHLcq6!S{?3)r&yzt3!M;LX_ay)(64o(5<*fE)4zrrtBfh~F!1 z2nXmhtD$xhc@(K=4d|gs;fR@XcV`30n}Kujy;-^8b59@{2CBonaROWwmOGP2+Y@Vp zJt>Pu0y7h}txzq7#`WGnt-ES4$jW1)vQ>OA!Mno7-^sOph;-C$f*0Ug{1srplg`iP zEhC+ko@h}kVz`OMn6$c9iooP4T#XYhG3MU!C5P*G)!d__-1eaw!GNrenvKG&x=Cp(Q=(y5Pgt{!BWMk1t`E!W0qn3d zV1KB$**LB3n&pjt^1y0NTjE^u!EbCtv0PgED$HJ z4DcQ@$|GzAcRXMMCLSs$E(A_J?T61@Z9uitxZ+s)rCoEXb4o~q&568KW6RW`qx}72 zi7;;dpd5}E^xFm%=}bj~F+(I&NA*ZE`u19`M2Vw&BslYR;dhkCKY^9DgmCXUDwuo~ zJ*hQ4?9_5FuKs+i(p%org@s|PXvwifv_1W5k&Ysh#ZfbHE@<~XP)3X~0mjfw-=XHvo*nSpuAPM6B-PRrR;EN^*0d-FO2DV){E7dop|ywG!5rHc zcMGYiMoxyE`h;{b85Omr{`=M9<;&lY4eoeF=uE5Oq#0g}y*b{%_j(X1e{K5`7oFZy z0vVmfRqWDJBGL9zzYB00CR;=2mBPpeOuCtMY@W;*$OIQkt4rD#lBjtHYYCS&uPB<- zdENu{F5*c)R^iVSYtvj!Po(FO@Gr)%=l9l|teG(W#oNId;0}X0ii;F#lbco40v2ML`p0Rmrfcl zr{OJM+0zwjOVESxdEkag9c&s0nSm#{ z5M2!$ib=`Z=MT8%e-3);j*tI(t+{Pm5}t%6ep68Kh>92bt`6JG*`er}iWt37071m( z&nNk{4U&!c;IcSN;&!ZH0(uZ~|KosDiAU5o^r>NY_(e+U< zH@rU+7bnUhhD}0zobm-=!5%FqS(Z83e)v^{h5!vF7a8))36E1>u{-axdn;U&w4Z;= zJ@x6Uj(F$qC;nH&JHPP2URA^UNA4F!z*E2*KS`?cH5qAndldNQsX zWw4U1^~i2QFWzV^>V!17p;RE zl{{INu9_JJ(UyJgu4GRKGz%($OXuihqLw#|RmR5zV1cSUrk*4LGXbgyVP@h@;M&27 z)$dvXo`$refNj7xzxPpAed5Kx8Q0SC=;5UPF%|vwBjtTKK=OR}VYle=abWx0Sc^=nHL;;C&)!vX=32C>nl#KMf&I8|#?1^4s1Y zFVB!adxCcIa7_3SM4-0jN1$lS1%;N9ok zD5Aw=Wo6~bDQV8GLcqPMgU~2HGadZo7b>heK0CI>PZa#%UUkn~4J9Uw;?98=M2yl_ zeG@zc`HP_6$%xTR_vgKe+kO|Uee_H-;xVhW)!xaU;<(U9d3lZpWxmx37I7*7BU_+3 zk^{q?z}*fxj2odVV2xomix=@S(hI2xhC4MrJpJYb;?NoYyS~+MI7fDxU?VKlcap0Y zj1^rkfy1AG{HLi+HTlIaTVq6U%4%~$Dc%{(=X@iZ=SI=yP5)HluSDXZs;B2r!!O}B zuN|G$s{LR4|1w4N;}+ZD|nMdV5baYUy@OqT7OpzW~4vWs6$5XSK51mdQk z1y*!3=&=yq$J!=<=HI1Q-LQ{q7`XPr2s`6O_CFetDSBnZ?$0%!-=5v;34^nlja>=r zShqfCe6QAw3lNZV)h6z|;B;l3?0??Y%fUGi!N4H?KLD~oO~1zSo}XET-KI@wd^)%G z#R9DqmB^M)iS@a;uiNDu+}hLaY60#X==ddb+jC*H9I{*20^PeGJwagQxb*5xR;&6= zA2iiYHU&Eenvbtn4U01o`x2oLf=m+fnUMXu7D<*XlDRcIn)BzgaB!&iV!0wP(`;-0 zUMh(F2?J@5eP2Mwi$ww`O$`7A!~WFkdOxcb0iCgtr)9HHs~PZ8)FM>Ju=C{-v|F~) z-ts04t*u1N%98O3&9tbMNJ6Wj{=8UL34_^GkEhZw=y+VXc##i$*L(Q>|NL+A?SJ;K zvsZkSUwA#@Z~x<8=AZq-FY(#eU#E5BHj!`Q<-$-N9M8K4^jI40+I5lYba|r<##}ZT1 z9h*)&2K_OSYinQ_O!ePFU?ydNP=MH_z5zUG&ww9Gt7dPup)rO}-GN(OoZ1++`(M-F z;qi979dGw+VF>+(Ch(U(0YvbLpH*jk=|94x5A8j9V=@&Tn=V1WPM9j{ex?kAhNNh7 zHl|-!LCszRG9^jW;wg^m-6S#sJ#(ffzl7bZD~zCPK+gPszp3MG+ps_t0hGeB($ose z0y|z@shEb*u{(zXEYqQ$HY@qnS5;MR-J@WdxTyQ76YY zew?YV+tZm|$ZV$XQV?Jf3Wr#hRwiSr!Ddb8-W?YHnDKN0mTX%KYjauHoWTh~j=;^F z$pMXj&Ro)Ff(cwpo|u@e#k_>8`xt;S3dQQgOnsfP0a7tX)Q2Cz1Ob+r@!t5+B7k6~ z<(Le0F;VaU1eB)O=Ej6o2B+{Bb_?;J5I9 z|GWQ#AU5N%t#9F1e(M+M)Vm55Sb=EA#a$D4aXcUnVN<}yzhMe~UXu?eyb10pN!pwZ zF+J{yg={Uj3E`Gc@H#jTK=>M1Z&PHnWr7E&fE9u@Gm^I}U|m3qoZn>Iu_xyb*j^$g zCS|=0_pQliKAAx|Bk5WtColP}LxG>sM04G>;*{3VBM5cwc3sC+^8zlpw1|SchoZaF zNy*akY2DvG)SS4S)8mdEuv~JfwSmau^hSm!o_c}m@<~=IE@3DiO)&@Qgwod1Np;rE zdFcVuS+_cnN%-Wm9*0da(#4#veK^!P@t6QnA}*q0Fw(+NE+gY`&j(b%R&-I(*{An>_vWv)s9J zjmVD3%_6)~UT;o!D5RE(9~A|i#az63ut7Gh<7!H`5M$A*S^d|v4`p8zcAO_`-#Vz#U2${7|Y zhy;R$zS>H9&>U;OXG3iSm4vRfKUI5^044!5D~nMb~K%)g%q{Y(74 zeYx9ld&lj4!?YLpOb0#U%%E{ck6rq8SUL-z{^xM+{qWqsg{^nO_DhOxZtXyBSwJR{ z)r8-Psq?d7<_CC=kt_LA?F(j_frY*i_+!$#Vlv?;GB9YV-NrVoorBvi!I{ShuY3mQ z;``v%C0M_x!k>i|7dQa z-81*o*v3aQ3nWP!oAYg3%#P;@+=TM)gh8mtY}eACZ3zI)uhllpjpqnNIp#Y3NK!Wp z!xPi<2SXp%bs=NY>3X5Y#Iy^;Q0J8#7=$p6D4nnJ$ntqU_uNz5YTxBs&V7K7f9^L} zU)dmSr3ngCtOE~kOMwG+c9E=RSH;M-rzO77=^c z)%}YQ-2!U`=o+b88KP~lFF0`1AU!m{c@3Fm=+WTE>NXOfkvSpL-z695t+YF4?V6|KXD*;jmgnQ=1E9>e@nsZCg^etjX4H4Mf4s*_>2aUy!|)08Bn3?PJMA>eYKN z@rhC~X$HqrA!`e*Zq4iS45yN&S(bU+{FKMO9-nmuWHR%h0Y_-JF@z{9wx>RXa$2lS zZvwfPZuh59N@`FT2as}M*RVDz4>}X0QT7CoN^!W}(Q{1$c(DQJ3(%iIIRR-?ug5zk zj*qwF?RVV*U#su+8)8ZJuRuHt(;?pOHMsCTc=I#x?jMH6Ey=mCl!lfnC`2Gt6nl~` z-NV#0A#E6&ObPl;qI6OBizlFQ8%mqdybGm`1wiW6#qjiMhNTf}L$4-~(XHt^M?HC7 zEC+@i$d<+AY&`{y10fLMfrA z->buglk%DN25@RcYf)jO#k+Q2YfAZysXl-nAB8A?6EfhLC}#_dNk!*|XkQ3KVsUKH`L27U^@(gt0BSse7 zGb6E_h30ie#>qc_LFT=P0*z~y0X&|i{h?_ON#3Hg2`A_TlD1`S$z%mhqDF29@rJlcj$vt zHlCUj5WJ~0H2S%OuMh3g271I-_>$J;7c;MVt~rp)?*y*USm4k=Z=x6tybj42}1#* zc2B@%sigh;23(bkA^SBTf|VuF^xeLIN6PqeTvySU&9?lW<$_5qZm4r#s))6jOremE zux-QC+_P^wmDk5L1xDx6wWisZ%jx)LQ~g)9B2eVdEY(Ewpl#CUDplR~r@BtZfnG<~ zS*nP^2}5~xh*UV*iwV%Qn>Jwx`63MalC&jFZMA+!3qR{8vz*?12S5B5evBXgtAC5P zefwW#Sa~lWeznQp|J7&ssh|CM{@q7D!W#!WO!jK(=*u}pQHP;uxWU+Xa}q}XJ6Qw` z^o*n|6Bj#V+X6nFA(4&zINh<@%_qgqU>TX(SeRGFu!Di26~dXnSD{WPGdCXW1o705C7_K z@?ZV*FY;G^>fiBCcOGSW;{BAq>mTsn|1Uqy5B=z0=N*qez=Ic0bJ%fNDg^j}7Wb+J zjjgGK=F)r3g_mZpt{QfradX9#huE!Y0T6UdwHRXAQcHaAw`t!W3BZ|eTc_Z{1r2^v z6Ds4GRX)!I4LHuCFs5a(U}X%;;Wz}gMPM6XH1#D@@k7mt9oy8DGihxY<|qi&hHI$NQniu(^b74wO9O!-*M*nq%qQiFs@uE)aLLOg{^;rs(QrBKEbWpb)o}sHo?DT*z3SH z*lQ5&$iu?{SS7H#wg3yjRNf!%qB=RRg8hh+)vWWtzCJ2efER)BSL;jo*4&Ok&R2M| zEXmO>Y{0iY0Dt;@n(Jm05htq(AxI}AYrD3i5Q20ZZtlW*N$2m@1$035an_9PcfTPz z+_quA?ocgBVYG9osLE=|;la0m4`2A)Z&A*N?AG)FZ!E)MON%h86|qP4md;5ebiU^Z z_#D(VMy*tIe5)-0k@zaZJx7uFU~Y1Y~1agPFbznV5;9Ho#;!)qoJ8 z4nO7I19D6#LBxW!@yx2-)EqlBG1c?iwrw=kfswBnAd_;THPL`}r2~5-c%ui;x1ct` zy*Y-SuXLN^?Rfh}*a9C~w{`pKr(oQLE04qV&p>Anu6#l=vAZt-5hU|4Y75MaI$w0( z8h0R`5wHnnVr~Xa!{D?R;AReX<_yjelT#G1Nb7!-E-ye%uf9l7o8LbIHI+r!K<}Ra z`K2!kA&9co1)!NfKI%)R7E2tZupJ#UPg?8M^%^WoQPa69v=4~#%g{b#6wC0!bIZnQ zc}pyeClc_!Z-r8cD3K6Yn9IIE`flG}{2kTHF%$JUxno5FONqECmN~lzh&UPpW-}mK zNYLi8HQ#t_=0!N?KqoRl!nPK^5a0N99D7mwW?}*fz?{n>pdew=&KxtwOroP&H<4$& zKtA%&OatAXMJAxf_;kjR_WRKy%phs&xkiyf2t1oGZ9rhxgc{^sojVAjWIV0S0IG(q z>V&ZFK_i5j#qxve{Ea{Nlk5yH;{^^cJoh}8uDwLK5AGQ;I?k5-HqNq1PfP2qvM_*W z2$6>hc_tw8^>$ftU}|BPAvys2w5f&jFhnmaQ7yU*)|$>`0M<$15?GG`dtasR-&=G$ zmaBc0w?YQqa{=0&8Ln43md{+(0H3NhF| z6uaZkBJLi*=8}H%JLd5R&sVv9`y8*m`ZDi$$0O_<2(YX#!LIT3tX3tPJ8WnL%d=tc z(B|}p_U+ZAD_$-Our%9}pgES>SX&ZUI%t>#<-7n%ttODORDtasi&RozrqvQ_Q!Odd z`mhEo%aXFy>x%y`6a;2!HCS1Zt`%C0UX!Q6wc2 zED0dEMBoyMAQm9P?qb92?CiwO;k|x2hkNhwAHQ?^^(+99q5%pE?5Uc$_ucrq`@M7S z_k6?ea956tr$k~t>O(wjFu#;#!F-t4Kk_iO@_=^3r`CCi&)(Xi`|K4CE5E^oQ!Bh_ zVTI%!D=e?BV%NLJf(A^9%S`E{w(d|QZgE#4q|+&3T-pODq~x5=NWf3b5tb+u7m(5) z3lPDWO4deg8A#F|=59j1Q2y6;?YSun8S#w z7sD!X8xDtZY~;-;#B?nKB{OJUJpg)ofQp&0Gl2ORR&@eaAJR6w)P~m-pgMxQD{?nq zC)>Q*UTr^2E%b$LDV~H|UxV4R(A*Vy+5PW_Z~huQ`gftW4uv&IF&ngD_M9wDOPHBa zPlK8k#4^Ip6ix_$N#rm_9sS+P>TP)eKoL`u$U}1%7A`>fI-Gc58Z;1roqkg~vB?l+ z**Zl{NoN_1WkxBO$Wn#VG_GJQ@3G&Wf~9mxSedhLfX(L!7w&<^I+RXKWo=RTK(7H_ zQhwHMU1VqL&xkLlx_uNT;K$P3ynY>?{2Gt#ThtB*+-i=QJ#m)!%Fpta_j~LN@8m7_ z+H6b#&I^}b zpxf{AM&5{d5`t@BpM&7$vA}^f2sC>W?t!&1m9d4JlB5>ggmBd-yawh)P|pf;qh0{> z9H?y|EnrBkgK;mYr@*)$coE3{U_N#~tnF2h^Jmly&>2W<^b6}kq0i?f zCi>*O%pBd^6QJWcaBCmd7NkE}--Fq#A~az0K>RwcEyl>zIe6Qf?&K4XeT`G6+N>;O z*sF-;V^OE8MU;K5sr@!HP|SwxYml-yE8BBv@!Hf|!qm)^plHHg85ZZnPg5>yt#StF z2!NTLfl?vluqIxfVnG<3MpKR}7NiMOUu zgu)EeDloqwanXlmVRtfF37n}P%D#LFs{0b$lgbD{>9k~A&*Y(T2s2BNEl$BqN~fw0 zyO7F>G`!P*!kiT5ZXQbVX(|VYJCMpkEU5^FX%<&s%WEEdEsX;R=y6cJ#n;!H9DdS-TjE-rc?6^p`k{aA3G-H#< zLvv5OHG40}>t0Qci@=3H5mqUdlxrmNn1hyPZC)-}t7%Wo;WUvgRnkDIAf2MClO%)j zQPNtNEWpM$;q+@^{~DaQ7s|IZdvhqPO}8dLQ>akA7Mq7{Mo&J$*Q&6)eaPkVm{@*> z2sCIpA>(mx2&b0>Ai0i&5L9&fXv~AdCR{uNcdS9b_d^_0Wf%lAC0Jg9 zuROt#EX}s%7p0g!%I3KOXd;=~K!;XDaH-b(i3hG&>QM7hPZ_?O-X8-1QI@AD`*A+hDw&Kn})2nbn4rTNY$ae)fI{n>`#38I4ANqJvwl z*7x=1KknD_3|Pv+;(gHQnheJV3ESZ2uFXs~B$t3|n=~4D0m-<*=8nm1F(j9Ua>XQ*4oSon z!=cIUu0f%o$fO2r@0!fc$i8~RVmMS}Ga;^Pu)S?EQwm5X^=Wen001BWNkl8X2c1%h$0r3RXsuoIxIF2F= zEh=S2Iuqi=4EA?SO7j730_tUpe96G^`UIgxt*l68Lfp7PdCw$26X3?-@W3WC0xTDr zH51bbabt>5S=1{b$&AACOb+)TTLLFBpmt!An+>qy2EB^Isne%9c@i4CuOqkcpHVI! zuzP*Nlh1vV{k?B8M~Zl6$kO5r_oe4Z27NN65>^n>Z?wU+vHAmodJE-RINbq$tBo=Z zoK_FN<$|fO>m4Q;8^a2)4!ZbBFdc)^-0OgIb=+XS--EIULUwg@vW_Gn1ZXroX# zjXvXog=P8}wE;mwVVMD{)+Wq4*rO4Cvx8wMtiDgT+r=^_Q1b~(F7{-EUu$6~h1Hxe z$%Ac7P>n7@#=>n67J{w3MAgfL`8AwTmsYDq442X2E_P_6YWsw_6`W?7QGZN)G-6Pz;NlV-?qY-%M*V;w zHiy$FGaC5Bdu;}d8eTk0r?QQuT-4zXU}IEw31;uWsqZnKzZb8z&v0TAZ`T;KyTp{i zsIrG^Czu>;;+P5igH0?ej?t|#u#yA^o0w(-Ro=od6R3ktS(Z03tOV*{1H(?D4mL6E zB*ES~mYpKl-@vw0Ov>B1ri-BhM*R-siNZVFVRZ6!xQE+JvMZR|mpJfUD$o8dyZaTM z{>?w&pykmCo+f?nXSn}F8*}wx{`$of)m{wCfSGK_rA?b)+-Gy&;q~VxtnXRm;!vrD z42CvlIAML?W~CIeQvn-`a57>oPZduqb=fr)Jx zgh9aOfkig17!E?#cWo9+0o|TrYv1P7LO{Q-Sep%*D}?lW->=W#?d|(|J=>-g1lT9n zu*!!p9AaoTM;Qi&Qo?8)Ny-4F7%7FRLdN~RkRDd#g(`*^+XCX$FvO}@_-7_KPrLZ{Ot7DH@K=1yZ3A;?GP-vJ-{WKUL;ZRF|8YG# z7K{CfqS;!l7TN6gkLP;zdbhXSu~@%1T$I_U(F7KnaBUlwXC(r9rvfJyz%T|>I|-Ib zlFh$e7FH;^B9gL;7awF};}cw8Kf#@MEU>mL^0Y|+CzoXgtJ4>0S-cMi)e$FGJ>qdm zP+KU$+KL1YB!I^sN2<2Vz48>=K%`p|E@?eDG;mMVRHtsZqC2|*} z6oWSl?!1Kj_>(?`nK+fGJWURQ0hobHA`3wutkC7$9cNf5z-!D4q>51A8?wJwX21R- zk8O2mKKW(3!)@+cInBAn8B!+~SxXj41(p;ODuL@8Fq4Kth#Q!Yk3(yMiz8`i^)a3u z6TNV=h8K&`Pfdu|1If}Zni3e0OhGY(&=HwhEyT4vNknV5B$K;ni=-``hCFz) zdsOYoaaj}3wnffX@9E=AJja#UslAbW00#pY#poY)NZ`O)48GNahf`3V2z%sd$wOA_ zshvNb1$w3TFF&@|>l}cS5LV^pm*L#IVEt=CHf_;}eYWQ)XHQ_{!E&k)aMAydTjOvuklkU(jX(YPn?FI&QM4JlYTXb5e5 zV@xcYh3pKj2NdSP3ZS$k^#o>@;qs7JE+@x0Fo4EuRRBqrUdnPp2s+l zw9(6(qI57_4`qlN;mVfC%wjQ*c7M#h=Unl%&O$cpvR;9cOHu-1Fots{B-ke&6Nz`Z z0c%Te|M~Cq-MnJHAK9$*&E-SUl!w|YqqK)5RFaCPYuT8#jiD57Iw=neW04q-g8(-c zm(S^VBGM7dk#sfN5vD#e??*DTsKBOef~Dj*Wq=*xs}S3DwPZ~vp}CeYGs=+b2jNuS ztW1%PnWjk05>XEV7Gw=!Z#)y0O{n^i(O%U?2y3>mIIaQn22?|OOJf#lvwY~U{VZSk_?LO& zrEf7Bjd}ln@;>6L9%0SHx&YyZhj|x-w>+%-AiUvW+yiWSQtWff!#D$jy^we*h7kwG z;=c*6f$;u&*PalhExb8a3G-`EC}LBzT~_2mP4T z+8GU@S(9UmC8!<3!m>!*4tAwhP9g<%ed@u<=Al`I(!2ngdRe6FsT`EIMdlVuNdUp1 z3(1UJr`r^nTPh3Prb#vj>AWaFVk-i4+Pe65Di8HtStj#P-IDd(T#R6Fg1a9)feN9z z0g2fTH*VZwbJu43$*)n~{xb7+1}od=%)%1qQzhK42}oG5tHTE( z=~^MCqd`O2-_lcfkxVUVPm|SRCJaMV%oG`2h&Gspkk^J(v^ILAAZLh7K_T15p_oF) zYI@u_lw~&*S%QfeMhBtDDrRhWaR?72;PMbod2&AxsX;Y>6-Tt#>0`#-SKALqQz2&K z7G&>-#s-YLFn2dx`UISMv&h4q{TRIEe}MI`LS_+GAA;*&5Rem$VA#}B98WwdPC{g9 z(YEnGz|puTKAOouWH)Y7lF@R@vJWKYdDIapTBi(N5+gd^%ZN8FmXQ#GSO$h2`R>It z;>(GoY=u>kr@0Aj5+7;FS`!t85F`uYy-7YGK?CuuydS0HI)mmkSuIhRh7Lr&oUX)1 zo7GFOcnY?jgVISjxCV=N!tP~Qx*K+0faQB&_XSuu4+qyJ3VP!iSiT$fuhOtGRGyf)kC9utm-)pSe(|sHlW(=T+&IIB-t6#OUxmN$FnsBG0V&&6n9D<@ zC6cpAAmNjRG<>6N^3VbtG$lQ>KaywYbON5c1s7JP=NYB|pRTkJa~$!MItdwPK&Pl#QxDKl;ZWU|TcH3f%bWrX zOM|6}j+LIEWpio@YHU4m=4zJ77sg1E)s6ugLyrlDF-?>+4Ft`Z5F8szY$f9kKrh4? z07YHzt>??{r=VqMEd@DCSfEBAKAd`>6U=Pb@nKmfuT2yzTG+P~oVT&p1K#$bw=@5R zIez1dA1C--pZERb`$!d2s4^I{p!UFA(ZxR(K{nLv9P!*lZ~ z9-C58M_^avoP}9QV6$hSkPqqgpw*Viq%hC`W%wy;^5j4(QL8W5Gs z5;l;@g>+jIQjp3>EOe~`GxNgKRLe3Wm(4+WUxEaZx^QSigPuZ3Qq<;_Ad?H|G{l3G z%t$yvzbhdGu>`d10${QQXw{&!0GR?bG-%1>p|T_IBbkL-#gJ!`q&!Cq`yzqw+2SdV zrR3RZ(2?Vvn5?JrBAXwyW!+7HrPADS{tTnpBoF8AB9?}O>tj0OJ@yYb`0`H3#xtL% ze6YzIPTYf43|L*5YM%IOvX#d2MvKz>d)Iq%}P>|2dp|GZfF@J@c<%?S9!;yD1>2docB@Omt4F69a!Jg(D410FVjHd@L=XCbu#T z8n8XNe(abRr@1SD=H)(}m;?R3lGfUX64~526mL$aO3#clSiizIc0v-@zsjd~zsRU#uvtA|=KOnEp3e|2 z|2f|H?in75Em26D1c4;g7c;PANqA)3m2RirgP9CGbpswc3+rW=&q28LnjNeKpuxqoKgf**WKB;f5o0bl*PFfmb68a#v|53_@i&@^=@1)52* zQQd+3VH{ihINjmY=Q9{cOljo5G1QbLGLL~OrDT6przg-L$WX$n_)14eJMv^Y4rN^- z=a_~ZA0@2?`gqqup~2B8^5dv!G&X2Ijv+snY3lvR*kBo2x~AWoXNc@=sJ%ofO8`+L zfV2g11k_|qnZ!HrW#Vk4WP)tfh8;aVjDeaWSTJBcfKwK%2i*D2yZP_3MSl6W|1pE# z81j=p{XUB662=Hr0#siI2aI*n8hSl1!2-dOk6PEk1J8i@0B{+MRiLKrP%}u-z#9;- z^JCgx1v#(4<~gub3W(UdyH3fhRS-y8+Lc?7&j{%8{nzrL?c_zehRb2yVr^Oahus5MS&;1i{fdB`h^aBmY2nXilcV>rUy)6mNgK?R zV81MrEt#}T_#EmYprxX$FD!`ctfE0mDkaA?>Jm^;C&nyUoQ!5Kdi4xV4838oimIM*RQxZ+x zZwbqjNQ;CmTY_Xp(P>CfP`U_R4WLp5&FrKE#I!4r$jW-FB5Y40C&{X*f~CuR0);0u$p)0V*^VG9E@lhCo!KIA-|rltsETJmNlrVLJ0 zxJ`xA@PVQpM8+D>)JO6mirAW%DY2Uv8ngtmZ4`18@j23f=1_+nM68Z$2t#9{qfRy8 z87ov;&(ax8L6Cziql(x88t^DcnzA$v+2(5*n-9dn=;@LAAb_CklGVliXz= z9&+GH0C!kW)yxwb0L3-fN{XKQ72o`kJam{gbj$MKZO35mDVV)m;*qa@8qU8TqhE#1 z--C-k4LeUla|7=FAYA<{WS8K?8{zWrz-#_<0TQ>qEFdQs%e6tkfj<;bq(XUMu|0Vo z0G0+cb{zbHJ|`|L3<#j{2eOPnP{h)>QlIK*+qkEJjNayH;NvAFgusc5nLqGl+o%Vr z838$f@=^kBB97|M0zW{+}t~F!X+9#udkL}2qF`7u-9MjZEYf*%>1|U(u zK&Vp(A|{7$8YQiCBs8Wh6}pc$r}uR1sl+Y_C8S`eB#6KO#tI{n%0<#N#j%hDr|5NgB^AA7q%Y6D{pW$sEd@C!-6PR;4QEd^zeK3|F+ye77s4HMx z1p6rm^C1YI1LIEM22ju#vkk@>;91~druB4wOxvp<=SQGf7R(i7eQ^fvI3a$U(FkHz zoA%iOo%+?=&Iy7llTnoyTh};u*Q3O|1P6x_A*~G9EyGGAL#v34ZE+6v55Oo2&^f3; zyA6e$W@;+py_qe+_Kry=1q=Ed#}a={ttQgod>*oafSZMR@$&2)$a*0!3{JljC6rj2*04JS?%9a40csih4mFKNkQX;6wL(K|RMg8K% zWjq-7<+;K%rt5}Y_mU!m_eTbayr^F$I-IGhrG~LIG`Ar=YqNTKmO^coN8WZHy(;+Q zpTf#?*u31L-rDEcgKIojJK)-9KE-I#i4dS;Q;#0>VZ#Jrcn?M+0w=IBJ5 zqzijm(iYXoDaJ%=JqS>78=AfhWJyP)YdKc{$4CR3%J5hMn+7z4m$5npSI?kX(}cF7 zC)Yqt0Z-o1k~A$V7zY{@O=J-BOf6G0q3ui9K!|`8&F+x4WNDy_$Z2+GatzQI2pID9 zF}?!F5>R4l3CGbsbOAh_y5l1c3%wBfrpOw6M3zwpPMYvs2oGC$U$x-P3a%?yHlUA~ z-#kMG@H}$Pya7!mai$|=7V-Pp+B>q$EJO9GL?X}Lg;jYTx-nRKEo}UuJm{=E0#|+q z5=A)iCb;x5n7I>{AA(E21!o_H(jD;9ry+X+5+%6tIk@9}7%G6;jp=y)NI&o)~+-WWt%uGdwx-PXVcG;1@OE|rq?VN+i}CV(cAp@FPV`XX10-cz*h9y`uU z36OC$cmescj=NJJ5;SIv< zDPe2egnZ|Y3V}9st(;@vX67VR*J1U3Emte(WV0-cdK^}(3~F_r{MfIPx^p?>bac`7iAumKls}J{Pb*ywjer2_m9CuI?hRHIeaP`|B6u+zUEr|biqhLGZgm5Fg5FM%khT+%-OO(0A@_s_F>h6vVydMN*GWL zVab#)&nXkG`y}sA^LOmO#XtG{KjZVi`33HO|NY!Id;seX2(L?V&+tV7GT|j%3+EC9 zXD6u39>y9_K~lSuz&dadcp7*OaLnudv;_{_|Te6CW4g+-X16V}GDg{f(` z4Z0m+dQxeqRpppsY3k1@7GZB!*qC%iBxC)q6#tARpw$q@CR>1d#X|dt#e!&gTm_`Hbzp=%WlgfgZ5Oyb9&`eKLCVf*mS>@E zHPPpebVR4G2NBnhZ5c;LP&1XfX(q=o49(nV#wS{i^(gC_GK=RJ&{Na%Lp|FU0iS`A z_ZT1po(}FgTDnXbAzy$IiY)(t~H zoCGMn1O&u{8rYcjRKsuD8sr=YHpc*u?F!QYx~9wMb|xnRJcg;Sr9sdUYZI6K&M~GZ z0y%a@m>SS>wE-kcQ;_2x12mIpo0h4?(oo;ham^#`#nGT9o|7-FX=|CA(INf>~k!HuKAhGA+3 zCX$LN1Z5avmr)T2QMVswD!F$glRjc{^z|%jIyOv{0Fcm-*M>2b;04H}Td4b9)0p~l zqT-=rHSLdM`u&VUjFA#n#zer2gS_r&`+~Y0A$UO-_k00~fqn_hvtZq{ zVNMhQYNTaqFM{y^@NFa}y7YtpD7_Net03n`xIO>;^Yr_DZcT3S(dtL>nQCM%jV^Iz zbcK%;Kf)k4R@pM#khVS1Jp#cQ*_FLK^|UKomYOG04A6K*T|8L>9`LO`n_aN! zG|NJ%C$bWBlqgP%nz*m3!Cq-JiHu#NkWUYIgDZv2>B?KsHkLwa5q9gr_pblgZ!0=RJ)b*FN* z{EB=z;h0SAm``WemY+4c#&IU+@aFVOZ`tx*hh0eIn6xT1x?}2_UuUQ76YgK*Ylk*Q zx5t;B|Hmxf`5w~NzhSiWvplj~;l_zhwP9=AEFf3b0X|z`O(yC`Vx=Ky3-jqrMH?55{xASzzf0{}Fnnv{ymSk8s=A z*Z|0}BbhjE?E`^KC^xr&IeBIpXi0vgoJl99svIv3;n*MyP{0xg;2|LI!W5)4@fdHC=p$s$;2+C;Bj-hAJM!uN*BW5OIaFX^k zKp+A#o+(8^Z7nxA3L((@#!{OhVs(VI@u!2De~jHpS$c4c_AzBy;^uU)4m9gxF$F*a zB?GUo2k%HCcO+#)GePFDSw;EXD^pE@Pk3zK%a=+`kQQ0uI}qf}e=ZxsBDCGC9g}WV@G|?zdx85EQXGjs`ZM zy*T~bw`os~ia=46pf>IcU{bCx>S`Q|77d-|ObdyS9dX6HyQ+|5c6s>u)Y7v5Rt zi`57Dy^nhQzh8^<0MkG$zYCAR?>-A}ItR~h3#nA<35dxj74@$8Wj3pDegU4_g0&gA zS&`Q-?!rCG@WM7+T$SZrOK@cm&dtO6A>4mLwlAbY0X-Q3N3$6XcJ%9+EVKqv*U`16 z49_>N!MO#Iv(4uONX=%UJ2Z|$0uGvRetGKIsdvOj);nkc7uNv8Yy!0tj_Y8)GU>i==MR1dz!w=3E0XFPdO7q}a8s!C9yz z`Z=0NQcCylOqm>8`*{K-eRMQ7Sq6+0Mj!E$8irG8@hm$hIceCxoyrqvd zVb*}HQ2aL=0bw5K0BOXV*)kxnprT;K6lQ1Di@JnURI3FigalsDwqWELu$a*cQnY8w(tkjO!8O9BhxSt(eaDhc@MRAuR< zLkU;PO#R83d3o0Aw?(O9SmOI}Q_$Mg9_Tc*_aHr|gP^o6z>0^Ay5Qxs;%FkzXze{M zu}+E#$4kp|r$3Z!gPO(siFq;#P8+Kv7odI_-m&xuo|&Y%Go+W;r*lgYj`q3MJYcKU z;f1e1j@OUV@3rvbL*BgbTK2zo$l1&?`BWCQ6z60!kKHs$c@9Qf4-S4P9eI?TmN%xr zC)$?Pz~}fjSF<&GKr|vR&O`}A6v@;gCME(hjs{6+;1dN7On?kWJ{=WF%_8qkaY;Y;A7QzS;%_ne_~-tE*hR zcoEnA4~u~skH%GFr|bU9WNNp`)jaX3fM#$;A}Kr0-bCqSj+Um~=Gi$eSJP2iPJGJ3 z9QW)D_Vu=7tWLkKfs`*_B3B0vlu}d1CnmED%1cSINEA@ukB?>`hRUV4yUz9gbIe`8 z$`?2ODarlkx&HAdIjr|+_I%3CCZ#*xNy+{M&C<_vUtxnQGk=aZyu~GvDS?;go@JBI zUxA-{9sHxu!MolGFMdIn_DDgk2Zf{nisQ?zI-JNTt{e(z`Syk|IJKUX*{XJ+l%6hy zsgdQ;Yb`;GUf;8zxOND4F2l{TfTEjaSj-;-J5l&RSEO)(lKM3Hq_9QztirSFuvCO< zONv@|G((f{M2`06yWr-&0Gei37@3$W60`@-3L~^oknI;w!l?!Mv$R^#^!=;twszk= zjIZD1`DdSHtKX;T8GuEuS!UIaaWR?2b#=IaWrFKlTDuX;dr`V4Uq|~5X^BXGB?YRZDz#Y=<`kU*aVU8 z^P}@Uv`kI2O`d^D8X_YOLrL6>JUKaA6advg>deDV%Tf3?W}{ZIcHzxo@$%Fq3^pG6Jeh5qww?QhWz z+TcSbn_=}!t6Y5OBJM3$0Mv!=|3~DN-d+VcuQmXbQat>?Eij(86`* z#Xn)xvCTY|mBF;im}UY~m2lh&hG7v*W{D-*fRD)nuGho1#`u#0dwT)4HzXXVsa9>0 znLhp~PV-8fbiR)YY}Rk2$j=N3LxaX{ie#=$JU*dSjbphA$AeD8VKgv_rzC-G*puW< z$AMl)SRpqqOiwy5Y)-c+AjV6ElDbwDd0M|MfF`CvNva_0gSJTM9ZvvPBBN6=b?89z zK>VC;GNgYX&rz{UFfg!Zzz$=aJb98q98`FU)H2kcgZEx|C*w8@ z8v%o4g~sI&&)lNc=u+8NTzlyWiok9YKxzuMSm1iziWVB>) zqLsra`(*4m)>yMP<3NK0Lk6A1u}}fWkh%#+HFF%Dw03-(DUs9>Xo(~cM_!!pcArfo zPm8u4XL%-SdcKdAKNwP1;>f=f$<)-#ByAe>#E9hTQ_uv|>vbOAd7Q4_z3uy2tyZa4 zt6aNwjYl7S6wCUPxu)y&I*&j8INk1dyB=KIH+giij(41~`S$^vqxA@#@tsUg=%dVI z0RzW9Io9{u?o62+|CmHgGd1eFF+JZALNJt>jN@_AlYy4u=<7RiiHQasG@t_qj{9@Q zI!?N|t-nkO0XJSsm>A94wA*dQogww-pJub(XQuKrPkg<`Vs(co7)>>LFU9_O$3($$8qnB(+-OTHI*OdefY=-zNA>Ot~*VqkV&VYuJ zV@G-%F*%ytDlN?LH~-6z@OOXZf8k^Q;$t`uSoB8y+x~XdW|b;cu6_9$kFGsR;dO;S z8cGCUIr`KKpBTL>Ny;*OBmJ|4~m$FP5iJ+I4;6wA3tmmPVyL6LPo&>Mvwz#<5R{6 zQy416z~96O<2V%PjIN_Xm!z{wWAGxbog$w&O>6iPmg$j;pQRt(#Kb1!ongp2Dm2Nu zry1MZm^dWOReWy;LWcxPsQ5mH@-QbwNF3lA85pOq(hY2tfg3JPwu3Rwv0FA!u@;^` zN3}b_O$P*x9Nnu!3^!n0&N8w_grUKxnjrLi#>!$?aWG?Gx)!63O=`xJNvk2)$&hgw zAuP1wCRWgrHtnbOwKUsF!niF2h?9U|09G9QzLsAnpx-kv9mG#MlKmqMf~>flW51ksP7E9eED0A;yCRGzRU&I;fK`m5M05UnY zw6we-RMW(@Xn6#HOq8Xj`;L4z2I?7BM!#v8AQKg`pTT z=CgnBS^9pT_4Qdk^O*;@az(_}7cT7bp7(s4lP4=Y{`lj(>s{~qW51W<@tDtk_OtZ+ zeP-9!dEhgjVd;tlk?dZ$z_;J?9x5kK^7!MA^X?0ADUb);&gy&*kni-x!ke(IBL|T z-D;r*W6GDlO{di$w(~8%_3VHX<*R)AiN8<3)5Y(1d1~vQFqdCu{0T+tiJ zeH=^D;B!fRZc>hkSejCL>cgpZr(pv#83_d_rJ*%6$tLCfXoVBqk$|pB3ubZxdhT9< zr*6X81z}wxiP~Ha+5?fY#T}@&P3CeEFMWDWGc#C#=36KNDQ-npn;3}PMMm44isR@YzKN0 z=?L^_P~z*p*3|wyAMxKrKFA3IbkLJlq4rnW)-+wsFa%)P`pG{6K1%oHfxcJFnx=wI z3`m-&qy`_p5}DdiK_e7N^-w|Ghj~Z*H7*i2y`zH$%Au$LQijOUmQC!N2Hb7KHD7Aw zY${kYMbZ`lpJfwvLXod+DB^b}`FlVAKl5uhf1UBjVs>_&2OjtgOG~=X-o3!Lzx^I6 zm6JTa_c%Z0|I~jltN-KBUIjTn!fkDBje5P#2htzl#^}a(_;PG)K9K$Z0E>%@-}M^4 z@AJVAevnUo@{@n!`GcxW)++e|w3-sFJToT&0_`UJ(m(o({L=sWThuBBi!0LKR(4@=Ro>UZ zwgeKG7HnJ}v3$bA@kGv+FG|F7bx+s8Nl8?6JoO&DluvahB%XLHPGU@LA6)OXB+@>O zvWa0|z_2@vJ87^^;3md&4?PSgkLisV)RLHR-P}pe7WpJJ{+zMcm4f zpul$bd1m8hP{xE#P-ZoGKb22i!?Gk9>s6sZ+dk01vO~0rTao&QWNgBWG%2wTw6)8nZO4K%nJ(w_tv-8TETZlbpK2d-v8&RsMIxr%R;JnW2hJ){n%rN3$Mo~Z+XOtI59dY}?;DQW_int3H z6%fJ}AbZm3{q%O$s^|B|{ZyY5l0X6x6rRs@^;LChd8)e4={ohT`@X+-lbO!Q%Xlki zHKeOeNZOZDP_8NR2~!)B7V*-OmK3$NOkKB!c{3!8u{ zq*f-El4#9|OW>hxuxdVR+T9>_n&g~PF-^pgem2J~lE$BlyCv##>Td_#Gziet z>kzdgg35F2wK##=yM8j}qKkh0daQg=zK*Hynf|qcdiq+_G5yjoX=>(}Suq(<69F_k zrhdjma++N+8AvH;_QX-G&D4&LB9WZ!HJIf}@3)8&BZ7WSZI&YTL=EP9@RSdW9M}{{ z3R~mr#SUx?V1)}C12!fe#fi9dbUe-LUUx07+i=aE{Se*Vk8=C%e@}mZ4-Y-`5XT&I z%YtXRS0k3UY&WDl3MUdA2ccko2z2>=$y7IRMe9Ogylkxr*sym;|` z$GGK|TUfbr<@9S6hLw>Nw~#yTxPvRMxPmQPwlH(%OxCYo&$;KGyVv&NK9Pu|fwi~q zJ7%GvX>NvzafrnOzWcrF_{vv5OJCn+YPBjqyi-#w^>M`q-^RvA9^ih7 zkJc_2+6lSVD6!chfbva{N}AbTBnq}pVCw&Qdn z32AxKw&aaeCSknK$7vZwk7d9;1vgV>Vsnht%%iB3YZUuD;xnfx?s3t~$H|qcY)jMK zoI+P!fl5}QJb|#4qrE!zv)M{n)l%^P+fTepkkZZUtUX4zUP}MQ&3Hsf=_a<69%g>zXtvz*5D}duEvzt=dcAgw$#-x81giOE0~H?c29M*X3Z^h-B4m zx9ufxLfl%7Rkz*79hY9pwxJTo%{CTPdpW8dL{!J5s7=rN4G|o_x=$phTArdfHikbo zNibfdzw#(e1H<&+|7*q~?abT0mEYd}ZQ>K#$PfOOhc~)p_xy^Y{vA(MzR$es4j#Du zebg#t(h=DB(RCzZ3C6}In3x))W6|mGKmSa$c{V|&hlPoI`C8&adUHeMXD0b+@gm-E z(iR>pAHf@DL|8U6&KDoda>ZGSum29ZbFj1n{^Mux&NJcvyB`*}$<6%k`{g>8Pr`Uv zqAS}*B)i5@Qj495Nu$cS&9V(ZCNAP86PE~1Piws}-&IhoiU1m&5|OiM#GxrAf+pt4 zedJW7p0d^yMsiZ{SRX9w7E#ljmgNyw>Qu&xVG5g~B_rFkhK_2_5Ie1=4yQ8*J4ax4 ziM)Kk?$hqW*@ZNQihOqL|;DeB3wTe9%;$6(oP_{~!= zx82xF=K==D#U(A9gcFv+ENh8-SUD79qOH{;uS({`WFQui4|ugAZ)C-U?`pn@Xsfza zzBW^_pPa7k((Nt6DTQ6hcRq}d?!U4| zjofzI&2)9mBA07rY6AZD?=I(v-d?V|<|aOV%}4m;^;htrD?ZEeBhKQ>U%Q$cKDCxz z+r#;Qc+${7GP=;;;DdIx!y--Vzk}|ClWO-YB}SmHY(AbNX*pQ!zO^e{EHrPf7HPh#B3uN2Hm|(nL36|X?RB@=REs?5 zu1ra~+1M_55EjqE_-;tG*Ug(^*}Ah@Bn3g$B_4?rX^v7FiPF@R#G8PKm!zp3hQ^?^ zBTpp-r6FjV4+R%uxptyi7=8p=md&9!0J;L{S zh%jEBAW;c;qOh6n=EeN%fnTDU6a04gK4!#cGgcmYMqbr=9Cuy+UCy5I zTE6$m?{Qq)YMz{UlCDH26v}!^`-T7I#8;mP4s71s zcEJCgEn8&WGiz-2M`%RSwtA{_;s5|307*naRC)6Ow`nssJ|hz){`0fnY#u`oT!l<%@-?0S}EC+REQ!JcA-CAlh3kNlG1 zsS3-wvuymtYbjT&%;;Xo)BkiFv3QD1>nuk5zDBNN9w-lrU!m4^42jD3iByLf>N=P1 zd%i_2*1}-sEEY`tgudi4#G)QOY7<|LT+Gp34-@em9te(QP491bDp<(eh{y8uF8+6L zC8sw%%D5lr@yRY0H>DXI&7opd++dQiVvQX;b}&#)@~gfwgOB#}yTN1l%D@ITPR`!3WqfzKZ`I zc!J&KW-5Wp{6ru3jw~a$?S7sdS;|+o?PRQ)V8TzZFtv+&hI>hEy`Qba%lSgzQ|v0Y z@TlXH^Co#>tc~hZPqSh07{0uF1KUQH@rB)+*;Q&|>+g!VPQXYd!|cQ!9vPp<IK?6*Fy3YhX^12MspB+=3bR`DpFE!H? zDKJ?}6LU*E^cxpnJ8XZX%y>1!SGEnYeR2l>vu87pjV&SA`6wGl7BiH8obhUgq~1f} zX%|=d-0{eBezc#Ha=A<_cKF=YUJm1hwY3LhOf>GBI-62i3g#o8DAr^`BCYX=xJX1? z>gEAKJsMN@pf~Bkfw7l2tCX&%m`ytdnBqkriONJG60M1QvMewKV}V9hsP{tCM$m)~ zZDEwi*Y(J}LTYOqB^Yj&Hw zN8$C|vNu-byv%8CZstvIdJ~U4@(7zZZ=U`~$Y!(5pFf|aOP3OfL?{#r``!QEd+%k| ztXa=_oICEgV|xD7Q%@zA%T2$|X0xPHDNa1`ME?EX|NVfeanIz28*bp^AOAQCok(zW~wh7bO(oHbjC);G2p?|wWYlj#T zOpJ&rK`Lzg*)8H`as-+>rCzx>Ci_KVQo|fi!Dy|)Fkq*+#w4;Le$p*<$JuP!7!8=l zpnlc3>1Lp?jlFsi#&cn-5Jj>D#&;M~DdW7BY^gh|MN%-nUG|Tpqy{I^WHN`0+uEdQ z%1{Ku=C5W`hx=18K zECZg35X*x%A?;1Z^N?&!;>RHE9{jCv z$&nXP`Wc*e;yF|whD+xBIl(aCfs=yj4j+GB1s?FYG`-@A`E$hl!q(0 zp39V;z#EA$m>Hz4uZ>JP#}UKJ7+EqZ*reK6H4Y1j^(4$v3kk_8`A;j$%rs zSrU7Y`@L1vycA1Y9^oNggX_2~k8WmTu#^NvR>yX7f3T8HXMj_@J^Z3}97iUeCa;J2 zLG^UbJLN$Vu?XKBSixoIJ;b1&=eJ{Byrca|9v*LFvKC==(=P5FoXfoADD#q|e066p zuj^`|=tua;&}`1`Oz@kbCbS#p*nBfXdpekx_epydp4`(y_l#~v$I}#jk90asYQiO& z%+jP4s*)g`PLnBnc(Dx4DW70GPAZimTZs_p2)WDz-e`nGDoaj#Ox6-)GfASO9_LMTT%sAu1Fgv8dz>CgUP>Li^~z+~_;5bb}3RZB!_F!sxGv45g9f zJ`%YJg;1FO8-{tpR5;gI{YTuo0fBF79z6#gHKqX-V}KB-dd-buI*>HoXd07J90amo z$`egB;X~9#XU+aoh~Yr!uGZm6%9;+uNv*FY{c=iXeYab;j^61@v}a7Cd{^ppMl`fK z($IO9f^8b+nh4Kq2X+9Rrj{pjXxH?^{T=Xf!HS$$0MTfal`B{7m3DTpjE;`d)^^}0 z$#x6_GtM}J!NEZ~IywO8?Chk!zn@qP<}YlgP#B@5O)REEw1!GquEV}B(VeMcKqf1e z*2PJw|A{3eYLjg?^9AX(6V9h{^=M8yFUC-%39*z`j@VQuGEf+h2u>;o6GLJ)lg`7Q z&5&!8`O)3txR%UGufKSv-Ul$-E`4%J6Czdu<3JnF!0;C7H8$OHBVTWro^CZM(nj!1 zZIv&RSOzA?9lTECylc#&Ci;xvF(N0PtJ_CMk|NNwstcZ2Q{8#jjcaj)YckRQvWS22u5CnnJ(6@_3F zw8wPcNEMZ1MslF5^5v!?vb_^EA}edA4iGW(5#;GzN?vR8J*a*5?s-#_K!dKC@9gOm zGqcW&hX)QWv2=op1JQJZY6iS~3f%unn3RWXEEsC!m<;4(LaKvr!3*? z-?)|G@-Xw}4e;1w`+qPnckTcn5!vS=G$I+8H;);Q?ca!GVD4N1;+bYHyzot2uy@~uGKR^-2-bE9!E!{K>x3E zoPT^TnHW6yJ$TbOi`hB~6A!_U2x`!3kU+(elM(h z%@Se{PcorXtXmGd?}d|lj$mFhJotTBcg_OZx5D;OSa&3B{{@_~WC@uV?D-X}d-V~h zjgX7OsqZ6M7fWyjyB}np5!L4p_Sk zKK~2&(=&S6d?%dsnjS_rz;TNfkl6*Dxn7Q_%KhogCiwMJMP9uu!S^19Hy#T&{T%-6 z)ywGr9$c_?DLcLk=dD@9uDf9E>GSFP37m0a4+HnYt5^3hbRV3uY90d{;LO!M?D}z; zvrk{d;LqUPQx?$wBY4B7+;7zA3y-@6U z`*Q28w;r@HQPweia$+%wsDxJ16J?0UB#Pr1u@k1Px~85cbW;oKaI})NG2dJ~!*!av z)pKaZ;wm$auBV!XMg$RaQ}3G;;D{D;0!;dvYA_&prmzVchQ=H@pQ5Sb$#`-tAJr1M z$ss8jaiGb#r{#>G+GB*yfDt4SpihgTz}5ifIbtx-;h+Z0AJhZNsxEl)z>pSO=?Nf# z*h_05qsT_G43~!)7?{V583(v#56F9TI6prxC!Cl11T25p+72qVg3F6G&MEtGmcV{UY1Bn%G!*%&^E)TqZ@jUg~dm~?8UHbHOyWF%^mVO zotKDAYqvy&l#)oz*pS@hW?NuQC>ZVqqu+c`K_$6=}mQ6=> zq9SrY_R(cfX8UPjVT6xgHX^EQ>RpZSX%s+ImDHdb;e!!7(?W!X`3AA0P2Wn)F+%}_ ztLv|;=6f^TCakS*Fd$HkbvCB{wQ-$x6=Ucw#v0knV4C!6S?#>d|+1BF)KJH1qbJR8M zFSIyxUHkqE&{`9ZK&d7L@FkPho`^`@OSpfjrbtG`Wegx05liM|RHjgOJ4?r4;u*_Q zX1j!O)2#(!A~062*W`dK8!wx6B@tQ$W9}9j3S`Z)TtcRv12I=KQ4--ZG6^|jIM8Gq z(Xt7t(W&`$+vt`C8|k5m23zSg3{R@*<(YW>?-GlMK|qr+DhMs1TSGH~w3rKoeRM*n zwU$)Cz_@WqE5R{6^1(UpRbmbO=cme;N6wMVre_m4mKTTcday3D4sSotyBm#(WtFwR{_DT~i;FJ0XrJ}tO_q=Qchjmct+4DUm`fQNja~u zH=<{bgRVz)923dWfr;Q0>%wC?O>N37tIAD#D1zL$5gu9uPf)H$ax_L5m5orV08U-3 zJX6qUZK5=l`Zk!R+bS^IRb`tX6gf3HpD?=Pgu=xLA#|g-k?k9;qqS)e8b)v$#f|pa z%Rza%sb`}Dn=u{;#m=<&F*p0#k^NV+tsoUo(bKbs%PzZ}<;%Abi&crms;pSCjms_* zXSIVh4q35c1+L?;r>BS8FT0Fw%a>D)#i+(&Y+JE{+b_F}{+=F2M@RXK_x_79MN*u0 z+Nr$dEf=Ab;>UN~!koo2o}=KtAKi{piq|iXa`v(;ci(+Cu7aQ4^}j4wuz>fSp5@5d zXNEIEqjGiA-nC=%8x;Ry}iP@!;<^qLkv#j#Qj9SN`{-)BRZjetyd} zEMB~rH7lBV%Tcoal2wY8Gi1E$&Q(-uB3#<%EalES?&z>^`4Va?>Q0m?$xOGtd?tIXt!@B>i;Z3y?qhtt@GsrYYYei z9dONUH?a6!Cz8HsKI%_9!BeO=bfDI?qW+`}^|}`M!i_mBowI~H@Bg8Qj8Y)i?xY8C zla4l}#Y7;XaK@1fTuzC(fI^}*13qM35ha6K($jYPvQ8n5LA%08jyoOEad#qS1bYHW zR~yo%?~WFc)Nl6NreVGVk7-x|D@w~r#Z&Cr)5GnzU&gj=%c)jlRI4$zZCk&M5(r_*TC=`<&wd@`T^{O70l-Jgj;QFU;H8fcinXt7hZVb;r}B{pPNMFy@t%7l!*{K1)n(!>dU+4%Z+y6 zL$+1Y+)_;$pbBq|fnN>9RU$8>XbqM3TX zn`j!wcF1w#StyN)DS?*^B;_n4H%?(coZ199Z6dI$#?)Y{u_@>zBs)Z?XkCx$I4xRC zvMREyG%PnWZd{@-UP25B!qm0ee$v!JvEmm^s+bS>b+Hs_W;(hProX8u)W(dE&`6Xd zYI^mgB?9C)puBqIMOVPdB54TWu^ZCMRMa>cH;5DEHAH!8KtxYRZ3vGWsYh_aC{tr< zn})qp+QNuUxKG7IJ&Z64eAC9k2&0G>W$mi9+E41 zocEG0xwmZ1$mjo=k(fduAZ4R7e^eZlF;OzfnoJ9^iLx{|jz*zq3Gw4Mg^Cw?g(V%&NSjW!V+ z7xd>I2ggBOvJiRV1ynWH|KM}n_`|Po^LuY#QP%=enIvITQGLfyJw;6cWzAVDPUGsY zU(ecAXGsXz!8oWPR?!IsQxaYevPPK~G_-gy8A!CI)05sfF(nb6U8V`>o&Yk4_!-h7 zXtruG7#A@{pP#=~d|jRErK?G{B2YpnXNv35d10 z6sN$82h@THh0z+yRTKtXynKzob0`cbQeF+;iBK5y@e-PekqBOEg-FU}Y_LSKVZ9U4 zj1PG@Ej4P6Ltzk-Z9e5vl(Eq&H5Aoh2c0te4EUflr4f&G3shB!@u2`Gu9+P1@FuHN zmB;u{fS1xt6+F~Lm74OH7?$lyqaK)LNXA(FQeDW_-AYA1L&QBAWIoPeqv= zuTTzLsv{aVsi}^8=&1@d6=i(5Mi4mY0(dc>@^}P2UO`2oObkyF)D#pIPTa>IkKm70 zs0C5RhO1C>VNBy>HMMb<>Qt4W8e@F4OyEbT6nv;D^pwKsgwjNr%6J9sN2!cdpc+9H ze5fi^QK20lHR++JsyI_IisL1mT8v7eimte*u|QH9d>{*@$rAol6|WR!a;!|W5~o@y z5vj(gja6{V5u!TAISXIQrot0!8`;KqX&e-!vuS3}p3VIE^QQ&Q{yu;A+;b24e13Y} z_19ldwOZxLC!gfuhaYCDR3(v2uzSZgVqMGV=zR_GxZD>^PDzLDHD{ektaB;}lvdFbgPD3_*u6XoI*x#l)*yX`jC{nIySN%qsx(Lt?J znm(^<&&%;C$LGXTU&F$;eSt#3r8%Ew^R^un3I)m~Z~AxgD^Z3W&GHP{bc&XRCve8Ue}wjq891p9?!EV3-ddevXjD@u)Nq10dxpyd z6^G$5mqMX}@}m^1iga;|k?|OXLJ6u7`bRWfF`to%IE6wHRg-12lRm@avdpPP*fXrj zwfPL2ecd3+WJM8m#~CO@C=|-Xh_kc5jN`lP8CJ}Q*BBU;?V~Ejj{Yj1@34DVF)LkT zk69N}F}C+t$;DlE4ouRK(ew>@sF`I33JFFFMUrleZF{Q3aM>}SiMX0wgDxG38oP!f zIJGkUBMHXGi$qnNZF|ZzrJ`)@8>c1VGExYbl?B(=ENIp!C6S}Br(z>v-_$~hefVSi zetV^l=ejOuoplyZKKUd&cI=pTX-uc<*O$b`P3WmTRL3UK@dyfynw%nVUG9D2QM%sR zOs!akYQSfH{2e~}#!FbL9L1o9Ll9C7k}wv{P&-|pEWmr034UVq^4&7@eESY z1SiZ}!6)DJf9cM($!jf05tBE~w$%@8Y)yI9nVN@dRhR_SxSf*L4s0en0a&pph%D zypn6Kxn}xxCX*qRO0jzNYCiU{kI~uL$#=i|UGBd7ZsPGc?|8>MShHr$vwpT3@9#eS z=}+^o|N5`=^z=|H7AY2s&w8wUKF_`P-n-xNpYQ*N7mvpe%GKlLP-thzdw%16``h0> ztmDD{Rs0JX4-V}-{}|qn_L=GWg1PnCSxW(8;?lVolp7dVCCZd(f@&WmqA{YaFf|}{+SNS7RFqU3jE+J&4Y~7ILKqehyG6PXBhR%UMDo$|Pg>*W`NVG&UnnFjQHVpAROggfzl!izwMvw(v z6p>lp1&LgoVhZAkDAly2%aN2QPstr8>c&W9Vw95*O~j~~ZFB^f6o)pyA0kb0Djr1B zQK~8M5;20bh+P#CSGXxPK|C7A$;AmgNaSMZ1SHZ?f&|2qF-U<*%J03ZWfjM|tK`eL z4VZawpT9z(Ku=H4UUstK)3&CjCNBT)dziO)G4EcP;Psb&itm5vAGvPy7rgl&KF#dq zC!w|G=UW5r-!jeHr z_}kSeH^Q7*-R#=2oyP|>3z`Go_dk0W`tB9{=!f5D%MX9d4S)6qdgdf3mVI)$98JbX zK9|dp%ZT~MjvYH_dujtmt#}XVY@Soz@kw%LoWT#C&=e~Ht6KUw|DEf3|9k$1?|lDm zQms)&M@MPNCYh{4E+@?>BZ(NT9-ZGbXP1YVpjdt(S(lAZQq(DCb-+)zKudF+TpHY1l+Gsb-8kK?5&@dm z2HVD9P6y1|11(K)&Rh=ZOq2_bgN|H`^Wp_snsTg|_q;Ao`~95#pQfM0EXoOgfqGc>0% zT=&LzaK@se`QV*j;>|~&&RI*2k}-n-$Mw+H{**vF^iPhl{;vP#ZD0Bq?tagInUn_? zM>$6DNMyx}MgRp#m+eEl_|>Kdc=zY8pedc_6Yp5h87ogg|L-6Hn*PEd>uhj*IX9orvvlcF3WdTxCQ*O$H-E$3ci+wM@GuJ&EFcnzkVqsL z92|VsW9M?Y>GRvMV+RjD_#p3p|NCifZsxk{uH%d|&N$$);_*16qoezb|3dyhY__b$ z@<&RT`sT%?{wCcnJ#w;LknI+G=#J%(oeAmg8naJ;%nX>Z3c6Q8W+u!&O_t?*ptVzYF zWSBjJBLEaehDfGG-9#I*@1m|43p6)3v*_3}`Q4@+{L_`|x#go`L=y9$5{S9U>;L*H z9{lYF5*`cy;9+3JRMLc5MZv*x=AuzQ#VQLzfTAgaW-o`WcVWn!A zex~-R)0wW%Dap2RQv=R;QxR?dbK7r@+B zc*|-r5qN{?4S2!=ao?TQEPqeWe)C%nmP`u2IVY)VwLt7bO-)WdnPOmgaQXt_lZMyB1? zd@UEn#uO;m%m>h%$#dq4Q`zzV?&o7~`4=wvx%(ZjXTP|nAeKlxvp`yfIx=q2(KpK7anK~P7 z%nVWrMiq28xH}chap7r`!uAxdGsx)Qi{+JwlAC5PCc?x^K(bBx3MJbh z)dkTUwDgMD$;^SKB{JW!8WP<#@{6JKBuI3@jME@HA2Rcy8*_>xuZvLH!sl5rnP1el_LQynrLJ-k= z3n1gP*65Jtgx1l7ejqAp6kGWe?-P}6`&l-hWP~OJp~G4#(ol;N5wQ{)6hOVUMrv|INL75PvChQ9 z+8QxZ0O;+Kbqm_WGP*SlC-%s?3s=c;y4H*(H+ruZ*ZTf{f6kASeP(uf=9#@_ z_Dse$V~Si=jge#PmV?SOWY|I3=PS6D`t0H+eqgFkjTNn0cH z->PUB*?$fxWjM-acv_U$-{8HDrObUhppUNTK2-Qi%!ZbM!1cIT3nmzKKvp!_pK zGRe?^M_(SQq+B2M;E8A{o2CUy_eCko)TR?^&2v1+n@Go<`C&8rTWs&_~t7iAo}q?Lu`E2JetOhiBA-y7AR&g_3V6*EjHWNRd=QIs?6CpNKd zBH$4?A)5<7xbJC*8X=WlGD`Ika(Tup6;4;U^pewu4$t|;CqBi?MiW+IoZ(wHBf0pB zajHn)gHqTUEl-V+E^{t`)`_!xK~9nyQ9i*&qQhSan_S(_z_l2~flbNH$w~P*n%Q(B z{1s9FE4)iXK8^QQ8io!Wm%+UFQy%zBkLCYgCVag+5$K4r3?#`}@4z z$ub#J0tLJVDaP`xF@>;9^mJ*+|1UH}a%r@N6(GZ3uc1zvWN+B@prw^txNaxVabFH3 z5@khIf+ukuJo3q$PWCSZ3azk66$jUKc?2{YwC}|o+^`YtHue>zbRVe+h>PYF zv540S*jq!ltnb8uLw6+04sAnD`D{<_ydA~yULb)UayQ@Y7vZYQBwG`XJ~RYOMWFVNC6aF)aUyvjYZz0- zJhm~EE4F@-ZSEoAZy`?)SN8U+!+zADz9rocUQAhA%L7Bo`CqeW>snh0Ka=00y1z6a z{=wkxT*-p|eT;ztx3C_fOfcPmCTE8@R9ag>1JQt~mwzZf z-U?sY3z)s*^f@Mxe0xByE7iuM)tRGtoM|_pDHVk5hX!OVv~6--!Z0~@4Cv@d)>Q*_ zrz)RT@Ue17g6Zu{DXr6vJ#yeP2R0uo@>l9F5p?^YODom@?qXoZi->uGmCYphL-1ti-xq zA}2HuQyqpMx+Oxd9-Zt)r>CYm6|tiN58b7{oYhAtQD@dE=Bwmgu6V~#h?a3o6n;F* z=_yM-vF=h&QxBTCO4|U}HOq15R`y|=4dvR7k9DSGzkpf>lZ_U7w{(-JQTtCC;lFw3 zHSuGiwN|2x$o}2Gi_znX`EFrem3-Zkdrlu{hOeOakmc3apJey&Q*7fO3!DU%UT+JjH^$W{`4sK*wH;+lf(u+m{i%-BS9-ED9?ldTaY} zEQHVSW~T7ND~MI!GsuTpOQpY-etD9hXkHFX;|ch&v?MkLZow^ts!j6sQ!WuL@L(!$ zjHkO)%=0AiD=Ert?rH*krG%kp zVvQ=~t_0YXD)PeY1&t{KML+QVDr#k#480itm@-|%gQZBV3V6$cs_S-jcneV3LZd>1 z&C4R^^2>(>xc8LFcUOH)E%+GLFDU`dwFA?h$c{zvC|zfle-l+4$xt<;HRj3NNJZw8 zKBF>5+5|iCpvnOj)@$}%CJUjI>{<9YX5$3B2QK|j!EIYBdGfp&C>)wt2vD)7gkfr2 z_zkqBs$5k%C$DSGJT@A25GQ_r;YE=KJ{qz?iMS%<+9ojSJEg-~PhzR-#9uKll1#w_+nqe2Un5OsIj6uIW1)>!A;=x5Et7#o|uwyb!Zs$RA@ zT8;aL5Z@$)CvV);w_)#9!iM}mV+9I+_?Kv;K$F@_^w)a0ee{I^XTxbjqTsbJZgUjG z?g^()31>_~SZJQCWsEyKvu1AfqW!e^I50;iO3*vWThK9)BE7i8t@w=#M1#}>EwV3P z=;#F*HN?)?;FUOfPc7@OsB+PInX7KLH3NKVh*vaw5`GfW&(4QlEVfA$u{}6Yovr>> zebL~07+!X2KDy~O)t9?K^jwT%C`%wddegno%sQN+JRYbM$~liI?6PAXF+~+W3y9zL z0`5gn)(DFU-7^d#T+8~J;@V~qB)hMhu9o{Z6?nryVZ86#v^8`eD|~+ZN@pbVO2d~` zBx0&yz0(5|3p1V((Jl|XgE5f?^Dh(`xF=oJiZ-+dy-DoZk+{k4Tsaw)1*}bO>R71x zqsfP2#dIc>s$hvKw?ZoEfm_xx#noIwa1;U`y)W;ekbP@v{!Ihv!v>(rpfsyR{J2+= z;4Y}P?tvXp=DH8Yk)BXinUG^Cc`)Dn?;ryyRD+F3weYO&(Cmfi*BLdh2IR~txg^Tr z4riEcBlD|^0035MUr~z|EX@oQEGv(_0Y)L-1FH3Ub%kodA*o z0#{7BW^X!Y2UHmicEt=L&lGJ=1x@;)rkO*_nMX=L`TQqZ!lGY>?o)g~E1EU}7yvRc zyAjWbH=e-RQvgp2xM2)LV^Q(i_`m^!9`$z0WiSk@+KyTU(T}`FsR@9Dhm;~SGGASo z+trRCER-U!i({=0g(jStLu^eX?{@xv_(18u++%hJa$&L3mROOz&?+XuOY2f%h_8K< z>3$$+nOPr(ec@+^c+-N)VmV%oH%s1rsa)O)7E}NGY7C<|vOAw`>zibrp=JHQVO0u~ z)Yaj)e@0owC?_+I&tNRA64JJvv^5Koa+eKvBb`ebU=%~jdwTx`n!Lx?x=dTc!StW^ z1=yn=A;W9rQ1_3PV3Pt^A}pCw!{j*6(^^^blv8<>8(>Yv-<;G4Np89>Y7dsa$NL-e zyS+LBoC~Ur+*+hJbOKqptr~bz*%(Tnwfdxj;+iGrfogIfvs9RuG9l@bGDPJa3AjQQ^YwHMG!#FC7 z4B`m&?h~^hbZLese#)i%MXeRJ)cMRuO@m-=$WPB7qpm+Iga`3Rs4{viq z5tBnQCOCZ|8^b`T9wR#z`i(x&4J9Ten)s~G`P47jtmZXRnws-6n8=TH&jr#WM?DY= zFkQ37Rer0yfby~7UCQ2be*U!VwMKQbeW7Mhq5L+)w&K-qq<|oi6r^IbJ+rwfo_E%y z>NC2=@@Y=?vf!DCT>&tH1Q$xC|$H#rQ@NJkrI2@DAlDH;#S+u8n&5V8UO70 zAp9l?+EUCRqVWa5eXD<*agGua^#tZILQTlO5v1l|AQLp3Y(Ni9B8h>sH|}c%RT7? zR_XZuhjOREt|AL!CXW>LwIk>OdZfmU2NQYu%M6I~1an23wkJbKCCW@n=q?N4!U+}i z*#yuKGWib)Y^fJe0VUq_8s?Vd=wccVj38L)c+>B9j=F+Pet@uOn7}{gC`oF0?Z@hd zT-F>oM@XUBCHs~sPEQyNur?%1={c?MkIJwV{Uav1NjxUg_-q~Osk`1F=S@krc&$5w z^$i_r=kD-cRg&cvgHQH;KD)o=@!*|suLPjm=iXDJoGCZAX^z1Q)jOt0Fz=omZq`i_JeHFms?)g+ zp>DpbkUu7sKSsE9Y1uogAL`|JN7Oh^DPVUAhZfkPA35Ie>o$|h;&n#*qMID*H~jBc zb)@>7Ii-Tw-)|l*ZG_}4!U-ym<i@dO5Q7gU@V@8dQMs#?Z-dd(IU6FGDK zQZJ_|H^&;i(e!aTtAP!(H+RZ5LC*l&`cZP~>8KIG1d2vYc zzDmN5z%SEh`;X-V&WwyQzebVId5lICEzLrEH0!p~K?KkF2aj(p*)rzr@h%UhH}Y2J zQ8nL)SNsVREV_@k(mDdOJAyGCiiV7Pr!6ce z#_2zSxV}HW4j?oOEO}#n^Q6|P)z+o9rByrkC%Dd0=D5^KXata$koN!a)Vp2zmtQA@ zrjr1ya8^@zrC%nESmqFctXYqt|KR>#)sC41P|peIEqX$*|Mzp3BMSMnOI%*hhVTxc Mt7)WBrDh-bKQPgwiU0rr diff --git a/docs/images/chapters/curvature/4f2647446363ca5d93b11e414fd976df.png b/docs/images/chapters/curvature/4f2647446363ca5d93b11e414fd976df.png index 46869d2fabf63b9bfaf708c3a9b6aa5ce0fcaf21..58bd72b9048d5acdd2bdd9cd31faffccbcc39ca3 100644 GIT binary patch literal 51523 zcmb5Vby!qy{4Gj1(j_4b4bt5$J%ma*l$1&ijUX-E9YZPz5)#tg9fEY{fOJX2-RRfz zJLjH%?tOq~VDrH2_uZdZYkk%mstH!a#d?f|fPjFjtR%0EfPgH5fPhGZfd*V5sqZxb zUeI2vD#|0=KYV02eThdvphZxYmwDk1+imiIK`c^#-+veusnT_M861OwMCX@9pN>M{ zjFuXLMqm#0zX1(|U|>Y?ap&SRerpSSiK?afA|ePO0Gn4-S`QsxUm%E)oHh`jaHi6^ zzNo0kS_1BIUz;^Pfs;+E>^3&PH1cI{-zoD1_WUJAo#Vo}3#E0*hPvjrDdjKY_Lke} zvghNs&Qc2+|Hq5H*)F_k=40fuHT!zM{V&(s-t}4k$IFt^{&~JaR4z-{pU?hH-!Li# zE#g0)-NbB;f4?JEpm<);``=|^3$+}O;(tFR@O=FL@&~5xs?rul13drzJFREfW+hej zmZo)(e7&c3Wt2Pr?;Cr#P1}gu?-2F>fA94F@zy9*1BZTiBmK8j3YQn_5hO<5D0Ww% z;zsN5S@QqA>9a(W?@y#X(to};JxVNFb$v6n63cFK9qrHa>DK)3mWA=M74*(P$_a{V ziAYC+`v*+qTH~{HZyzxo2+e8~{I{{6{KH^Fv1cPO&G&>`#)w)N-RM$?rj*=e>$?9o zIl=8Io1-AJLBQe9v%__tmGg6HHpEylvS;{pgjHwkCPUgOW#a`BU;len5u}gB%jyot zIvfUA%P!u%(dnvd3!p_cLde9EAr!}hI+arIG3)%@@B}TH448VxugC7ED(-LT_^n$H zr4jLR1@-W9G1NL^>^@xb6r9V{^B<2C|GVFiFqnE`0jkClmIeg-%t(HoTv?q@(tY2` z3zLp}xQz!PhoMBmgTC>;^3x^Te~;CoQ!BQTwZe`ortI$^YVnJ~9jz+xiS;WC6LzF3>!=N0!fP2i>oDaGB|<|V$uXOI9pd8S?DUk|Nk*Z1eX&8}CYbE> zLq6K2f-hchDvD;Nn&INhSa1{@J|bG=W3vzb7jQsP6hz~az=<9Xwrs&h$1q=vOvY~` z6d^Dp9OEOZTGxAOz^z0CMwtft`sR%Xn-PXA^NiJ)#jZ`_q6u`{t{`C*eCU)8i* z9Qc0>69R`{vf6$z;tO_WP;kCm<>fDCDD(*NZL8(w-$4oElJe}WMkP{E$Q8y2muYRr>twrsQ9-jFnXw4x)Y6c09Mdolg5s6yAvo$3Z%5 ztIl|Hq&*gHRuV$P#rlCScJFlBz0_t%F39)s4mf4Fm`cjk8E09Hxi`ZJS&k>OVNpcB z3envI)8^cdz&qeyp08`hp1A7Q;E)>Q23XVD_@B`pc6Eh3TLCHM;kzcbWkWd7T~=%g4g=)HXP z_Ov6uC5X%$FLtMf;LHzaU(MI{2UG{v;QxwJ5GD<^uomPr3s(q5_hy*8WKGT_tv?C5 z2+g3f7Gr&<$NJ#}X=a>q623;Mca8I6O3rRxsm1JHtp*kJdJO2fo6_BE!+A&fur>x5 zx?;<7goQ^jQMn~av%n_4L^84L3YDd9-R7OR?IPlfWhm9P>w}f}`f601_~)0?wO+XA z*JZQ;dTL!C#6~1B8mPsO?KA#`2LhYpBGv}nv~OhHvj|P*nabd5w>KB7A41@871YA2 zyT|>GTPto$Sn7y-M3{vZEPDbYuW(kG4uVjI<2HdkW=GFWpUc=nDo78oz%CFUv2-0ig5K}6Q$&dI}p*nQTyY$}KC=VW^$4CNqY z*!O8Pr9u zM&#l`lRX^sWtQ!vT?K`x_CpG1H8mobkvgm&sY*G?JNu(Hc9u~HPoC}Q5eR3hnug;( z-f7SmJEEf2nCuCVwiVayapE$W#>rTDgcqV@P=m#0ol~}f=5z3G*Z7Z~T_XyL*r1Fo zD*S3YG2PSF-AHtXuO{$|N(jjzOPZlGdaKd{CHHgc zp&TRZ{CMFun1`4|{Yzlfh35@P@)+H=-N2LS*8kZwNwdYiw}R>7cNc5Lkh(a>IvnY8 zJ@ftq*4>H$PKNlt;nlN|RU%7Xax%R#VgVb8A{4{ zUN;H-f@lASF6cwcCudul)llTh_2x~47N@TT($z~F_C>!=un}c_NRqH`jCCu~Lu_Ri z+imR#rs@C1`n)%fh^l-eo}mfDAOHlhk*_qOPaT$N;50=hWj#`{RM?d#xKKWK+S*(U z1bN>eV2C11Nn}~FgSH#Ss;7_2tK&aEgXN+v{|;4M@>)tPrm(sqH;5DsrKN+@&&`Z1 z*t2*)CZm&h0VexWTla03)L~SY^;}o5Xhiq_0MrlgdNG7&F22&dkB~r^UB}_Di>z{J zEDvRfNv5(-P&L`Hw~LGq+J*J?j8wH>0S*&JB&^dHgoY;baMH}Wbx6?%Es9zPi7?uc zf{D-fJ0M8)eUUyJLM6UiZh+su`wE-Agzh_nTpgG1MHn!w6w@Xgt30iuPvJ6Bni0WY5dt8O%kPDW*w`v@K-Z~zfij!WM=Z^fVFFxp(cvhfo}5e`xk zxgVpGiK(f>0!J?880Cl=!tZ?aMk3qf!L262bpoZ|(3xG;Xmk5Cdfm7jzIC19b?t*a z1nz!W33m7Vuc#VMAra{w3cXatRdjvhz%c;b(0DYK1&6#t}YpZHIs&Shrm%?srM|9FshBIGCSl{ia$-jkJXRgEuob6qO*W3L@_CM`97&?t8-7+!%V! zTR9biuPj&)X)BhmQ_A|d4m{F`IP28kJ2y_3V2VYB;}X#ML3Zk7G6XqaI{o+|*csRn z6qMGQn1^`bbrkI8I7eSJFOnRwiJP9WhMtl0iIZaJk%&_MbFdIY%Ia}_!O2rO2|3ym z=T$e^C`!t<=jXVdod4nH8F7lr4r#Xq7v0SFzgQ>LQrrEd@W@v18FKuo(6TbDsH@d-MAtx6coq zrq{M+p*wA_20r;5>@k!k@97&*P02t|)i%(ODon_KxPTgcR2O~ljFVcrXh{FwLrTnk zZs_d(oObF95zmg4&xlZT`ZHn$uUoj!@}!exnCEygZeq<$`aNDCN7@k65fzo^nr;R; z7H!3PrHJJ9kA>R9px}!;%qLFm40N1o>)Mm6wQtCzY;S9yFoQohalLb&nLG5$mIIG) zBa&c)!r3jbP5TxOOIvm@SpIGMAo;jY+}{5F=3xh()j`v_<;}$rii|WqO2iYXOkh1p ziuU&!zpS(J_>7t2p4}958HQ)Um+|g0ZrY@j98PkrM7D_f?o;Wl2)pbrdo&mN zQLt9>3A0@>Hf2Z#)eX2J4X!=6x(JeL?YssPpM`1} zqGtkdn%aAB?+-m^u$$C?!!=rjrQm|D@SP&HnXee~%XuD!R=ye;ZuCWyFKRbW<%DfU zY+3LGd-LM;M#gk)h20o_1tKAAj@no5T_C|}ZcU>WeB4BS7lzNcdimLV{idgCS}Mg? zu#qb7#Fii-0QrS~i3&Pf(HlXH#4Ld=Sd(2lWS~eL9h?_!FTYR^Zl4?= zT9*mI!0xx(kh2ZB%j3p|b(vL7#7oreHt1@X6K4cjEq}GJT5Cz<^*aD88XGA-)mtst z=?!~RQBB0T%r?_`iAllcRHVC{`jPUiN%z0#j_eXMSU2aTxZ69eyy%j%F!EmUI2@O2 z;ys|rvYJqBis|)@OcAwEO}g=*9@0^gloU74%JeEt!|Vjc$)k1*YsMXaV?5-3jYPK>5lI$K%Rp8PC6B=0%=3Sq>%_-3t7kvzM}~ z+xP^5_Uw6GR>t?e$5k@Foc_6|f7oN?z7%y3#IpV>E4_QrenGxuX7ku*=1C(1{PslB zYcpNQp9ga7bB7sn(J^i3Z)MUKNG@p`MI5JVjUlw-b>UXJ;?e)79ZR zAabo}jY3=tC3*qV=l(4&-9s%&;8CiaSwHG|Vztqqm@r_05Y`2Hp08$_Z0Jgtsup!j z#MZwwEM~Nh0E2c@z})AF=Dm6jE>3&yY1SA;_R0#Wgs)pfT)HLKi3pxdza?ZZl_W`f zdH8(!((6#-;Pu9hPi&cg8RBHc;r4`=_x%E$P37q^m+;E+kFPqd7f+jR%n3+MS!CN! z3I*j#@Ae4yMb8Vs8quKvm{g>18d8P)4*Iu(@a~+KY*>)G`IM#+@H7b94@;RzdY&sh^6ukyapL)vK$kjbR*YW5mrgQ+uK30jkVQza^w;wlzn)?#%gl>s9l|~C7B$x0vK`u zmY1(k#7EEx41|L+Y&)ZyooUf`c@$8hvYh*DhKE8|nNR5)R+;znLR(g=jM^ohKaGR* z88k4CP*p4Re(#8i2|W)9!ekmT+_?lKA2T;R_JHRPOR#XV?2BW3QS?%$I0b-}PX@>G z=0=b;tB#UD#_0z86d{t>t{cZ5VB)lBrV+im;pv(g&Q5)Q|moVDMbr7_8Y@&@|^pEvYi{BUN|AvV2} zd8H4;f1^fTr^EoYHBx9_p<%CO)2m?*x)%8aVY!RUn5s|lc>-=w7|gWsyy;-084t_$yS<>0DB1;9I#qK8{_p8vmd~(@!}Z5fbdRuT+?977j$)OF^N*{z&96wRjcsO!h7P?p0nprUB=-mxyP2XJZ62xFA`jAT?YIigacK1RFj zN8xT%%_h~pG{sJH(ep}XpZaf)nI2K-)j76Pmg?6zf?BQLy8`?Emp;82QBbQguo#Ak zM2w)SjYtehUVD*`7a&Lq#&&=IdR&>+F0XDw99Dq&Dy3|%f5z*LKx0@Dvn zL2SbqUou~wX_n5)*q!f3<=q{6;R2*w5%W|drsY3!h}0+h>vLXIyOS4w{-%%J-QC_7 zTG0C%wqd?TA<^x0y^>qBaDY}8a~NNn)W=E7Xzng zf|Ihw{u;4%of(0E+r3*DxE!^Sp#jktb>#QWyN|JU8XlDm=nn) z-L~3eLBjiTz}T@Jli8**_y*hSDBO6xdd}{82RqS?pUu$--3+IP$_b}t$y@B+x&LzT zHjDAJkqTf+K7*rFfA z3nKyvKaa&YODKmQ>vEZ6%i)6US?J25S zY6Y6LX%DrV*U(v!PRZ-ZVtvT_?L36j^K+-eHuM%di&23o2))I1Z`L=JJi_jj9J7wl zXgsMFhZR{Sjm*LIzIgrbN`X@fWQLuEUo!hoMxn!|D1FN*%A$k(kIxAXLEE&Y%=pP$pf3 zN>`dN#8@_iG3)Y1FqUQeGv)4f|IBOjZazCn@a|--1LA7$mS)_1F%e5QkygBt8PPVl z&Jc1(dO%&BGZADd_xk&k1l|B?WOvUW_S!$kkgDdm?1!$+LOkt#EdX=uF9BBT#o6R6 z`32_4K4;$&#+}(a{EwZgX}Gh}_nbPn$1?>>b-s$qvuH`B29Y){l7qu7tR+ ze5Bl!(CVfEq&Fl!%R!T_Fp>z$8q~aKSf@+Pl46k-nN=<`%iJ zcTH>vz7*Xs&NB$HnFZH~$W!QzGiW;yAEeT*DX+=4KtKn*)FZ}KHFYX{J83v&H=w%( zp&60Th}job?0x!#=;QHYgQ4NYm9jE_S_htSWoV9(-lHM`9fJp)E2voC&>(Vi*f9g> zM-)=tH!)7U-Z$>Q7d`ExgjX?|j}yNE2`X~ETUq#mO>Sxt+Cxp{b=W4?vgCC#1alr! zMv>um-c8gsS^+xJCRxix*DLR85vNUuNqsBCGWVm`k3_dupW8R)FGuU|auAqtoAv`Na^goDa_E zkD%aVNmOxmbK^7X#sMy#+2-OOU(x)UtgE5gVb1x_O>CZYK#3=WyK~ERB{P{yK*+7#SzASVWge zNmz(=2_S z(KNk6iT;y2W{=Ep6rX(Hf5>N`NQj@$JAGpn7%=CZcO2jl2XLa--ANYu@_xg?CeZE= zi>{!>RmH|FnMI}bVO&0!1K&JbDu+eGD@P4r95|F1K+!^PX`oKP zCo_xss1^}cs_|@|QWVtXwjYxfb8Fi$PQPBq9T3RwRGAYJU$8+q3%Pm)+PiXrAvoGa zy%n0j!YAO9WQZTLPf$`tkXbg;B}N+lq2KbD=Ds7uhv`owL{OnqJL4--yAZ=OJvoV( z*q4zH-hOviOn82Cc@6WXy6HY_DLp36T#&6>TI|9dfzt? z=k)Z{VN?bkcdx03#k7A(}`jUt05SEM0}HZxeAkv z<#qaEP?1zJrJKswI|_gV^r?pHDKo6=trL#rVsP)L)iMm99m$tQ_tfTOxyOm`G3Uv1 zfW<;>thnZq%mg!g)?Sz1A`nWRHlk=qVI9n@xhzTj(I5nUelNHg104WQ@`6_C?kM`Q z4bK~)wWj%u!*Up=T%xmfJiz7CvEw%TmRYA1()U?8K1Ixp6T#=A{N62 zB|Pi?tM#zhfw<&J>Ij0vUe(ki>Xz;N^l6{_TcNF;=iQ{hWQ75p$^ik0`zPm%o?PSk z>T5eo89gHRS0hwe25oj?d&{yYj?{i@=&QxwA9H5Y(!I``z+WM3V2$00ZmXrqto_!V zn0aZ}92H6H2%X4`R{z{-5ehN!i;hc@B}bjTAw zJTG4ocAdr*f%5Q5=k2j695+a=BzkEeoC35O%%0$1^M9B zn2SPie9Hk6mO&}J>tc+OXrwMwmO!X5BIJ`EhC;|md~e8dh^NJCbe2SyqcLc!FE&{j z_-U7gRVJKpI&Ji;Tv2FfF8EN|E1H&$(J_a;%KB&ac5ZR-$WLWU+V}vH=#Khi#p_Sn ze@F>Euk9>9XXBylu<#lVpxW2hR87rTWeoujc5zI=`1Y6X<>Gnsc~F1~2`Vt;VPsNI z>y*6S?HKX7b3$NJOKGK8N*%#_;uZQO(@0Wn86|`GyFA(rFn=IA;=O6a6;rl^zD?r+ z2F7GZW5udj8q0yfaJlKu?~g(JLH&bXGc6M_@UqFFK`7?iNnDWaBb8AZ;<~Z`{?8TD z@?)V-*jEenlW9J=Ze4lA#SF``!LOXPkMreYi*0}A==dofeDrH>Ipkz}$mEJTr!i$_ z=>6?^<(nU%&5^A8dMA59QmLyS(Zb8V$j^4Y`!_?wU^TsiseJm4?mQaeM<@ud?3)CP zo7^R(nypYZq^@DYqI*dgNiV=CvQC;`TgNWJeUGc@4QAC@MZy(=p?(Oa<07K^%ytGK zmJoAyD_;87Z&(Ot87ml_IB#uI4&OG2NCNaK^o7;3Ri)osk*Q#%-)i|8+0Z=u$*`TD z#bR_D+N)l0*}QR(H}~P_0(oT8gQ+8dN8Qa$sM2c05-2xbaIvL71eTt)A0NmPP^wyx zS+qnFdugpdcY^SyZS_phHzO}bsoL`q@!L)_?dqi7aNjpSyXurTI%D5XcJe4&DhTsi zOEa{0;>pu?<)Fwslh@kX9K#Qd!KlbMt+pIm7zw3WlYefl+25b5Ih^N2eV(^KSi^jm ztOfFO(HcWh79wKa4N2`i`^B(&DWVU@_sH1#^}dPTH_atZLjvcl?Z-vP-&CPq;JqE56UC;Bk@+?)^JXYteM9H1?lqOOkqQSsviavgP&nZfEz8&|>JaFyM^dtKz7*%Qi~+551wmP2V> z?B5!BXQMwic4)<(2t6RVN0JvNn?vd6U)PA3)J%S#e|vqm7BF9PNMQ2;p-diAH*?Za z(2f_+eUu8HAZ}1OO&=3`Dm`^394Z(Q-; z=mg?_o9o+0V!PaygDGt=5ho*}+hP{_hZJn13#O8_x97qpfqZjY^I>t0y$kQhoCKV@Ju@q}r{|M9*w7Ib*bI@*!lx zQ!Mc$@B%lEr}c`N`Va6pfoI73dannN{6>KGH3FoZ;B54#Mmb=21rI-Dtwl>^KBQn< zo379d85ZjB?V?7lrwNkK{YXY&N&V?hcg#0B8A_}n9b@l8mX$0a(lV5fkvTPK>5WZa zsZ0c2*;Y8Jvqz6R*8K-bWPWZO9{mCveq6G9$zv)n*-yr0=4d}DSU^E>h; zM@mT~c8r*g@N?eJrx5|j6=TTb??ML5yr7mruXEB@2i=v6;bu00Z>!2;Kem<#NSg5M z2@2x0@hvkdG$P0oFh!IJI#xOmt`58Vg(Q}7oOdH-V(>2|3m}+M5(RZ8zmkfXT19nK zU$@*h=`zJa%3XCgILHW7&r{s&_@lpXe5(MklJwUXf(%?ur-aE6N6hPk7AfHC9&O>3 z)*WGYbKkZ)F!z34UfX6Tp7fg%konk+FR`pLJ(vLg?!6!&ReRb{cEJ=S>FlGDw_3IZ zFAfRc%ogVD`+mCkJ-t+ZuuGj%14XU@vHKGNg?6fqRpz5i3>+!MSOP?nYI)+W_6~Qu z4=ioy-N{CjOJB6U#j7x6jAB`|1g6KXdE`3H+7jr+65x$$^e&Qn#O`VSt{Ht(qig|r zT5V;nzSWMeO>r@*m=|nC??v;!TQZvj9>(AZ0J{potyprHEHxZ>2rVPey)I^{OJ$Y3 zvAJ1$;q?A%z#aMi`fc4T8g15y(|W8TFMtYjS>7)Q^my%`sHH*Kk|Cb#MfW$wwV-Xz z)`0h@hlZoR%o6-sa1%mgxXT&3fa>u-W!l9>ilKQO*gC#KJ5WWQEKFfC%1H9JhIB z4`r#yMU0%t`su-8NjlAHA4$-Fe_+Y6ntnOT*BLGh-OntqJ;Oz@U zdZezC&d9eyjPFD@CV!%ThCEGeV;TdA{(4zv3$@Yqs8gf<3sY&S@bWQe>7gvONdi5}%?dvHX1^jp& zi#M1VD`y@XPYQvnh4!1q-=d_-mbj{Fd_@TGO@LFStK>b@{0P$*>o(HWv+vmRHL`{$ zbd7JxrT}u`3iG@anEx)?aVE5+^2xvz#JKbN(~jMirA()^t@M0yg42sx4mBtI!};=J zC$0Ix4B-z$7oc~K13(~ck(qIFGTc3?9P?+Ouj3S~Vnt##sO+52MaNSspMUG~)imo= z?kA<)hJ3Q5e#f~e8tY%|Y@(c;HsQzSq!z23)j_ZX8p?pV4>I+L)@ndSPx zm72>gvy)7drnVTGo-Zt!$$T^}BWtOT5BS0&r4L^ z!gw6_H{dHr!G>w+e;F{z|1jX+yEBb|G6aBwesXz*VKgQ{R{|74Zu>>?I|Ke2xW~-^ z>-{jNVrI9()MDE)nXc-A+QQis!^xP__}B34Lu6*P`Qi*q^|2pgs050UGTUuOUcJsP zol3USD&2(8mMBVoQ8kR22zGkx;F5D>p;12M$=Zr)L}+B(b`*IA>=i_~Z{Vftn>ud5$|%9iPIpMY)l(68>&^pA0&6CxC8&O8*GZ9VqvDw7SUw zM|Hel7i!vCnO~!Tk~PqvcNfp|$Il5B}%}EzzSIA?ZZbIXZ|t$|9-ohE`u0 z0=aGfgq#UBP?)wzFDc_ju}*B248ja2E1q?O_IKBT&;lg(0L;3Yk3QC)=%t|3bC2q)XL=cC7f<`zP52d zxSDK>kCrh$=)R5l3vyf@=qFfYgSh#2&;EXbjaY|FORC`T6Ai$?mYsR!vBI3By0S?`WR_*1lXtux1U#lcm9??8 zse`}hyMe)gFNBn#Q{XFZh`TbAMYOp=;PA4qcmQ0P3(Ry7n7mUHN}XvF#S%*xE2y`Z z(2!>s4UUO1n+OY`90_TN9||}7?oBOz>gj&ufy?lBRyb_eu-BbxmD#tEY&>k}wjcl? zhX?{7M;QPn4G8(dPVnIk_2E~H$pbvs{^7lQV)6EdZs;T8_Z3d&UsnUo!-kKxlBI6o zVIa200<6#ZGHbc}62jlQ_5{6uZ!mT~Zqs`JY`#+wo@kpnV-Cl2-#{Yqmt>|H!!pf9 zs1| zbog6lC+#bNX|7@A>JZ#Gkj*>fnBFi1-e=zSlMA#zazkLVz9E_BSjTz8YMJj_=D7`9 z^G}qp3?{m?={M{z-Iu~b2q;6BLs7>}UP53jyPr$)Dhb)RH3?+aQ3=H|h8n&L!}s;r zn?$}j27X4zNcr5Q->JS<@Q?s@TRwuv zYX#eQInBD%)MUBjmhQus4A)hEkCt`6$)v%S-$KR4OYH3+CFv6Xw?cp*^2*s1kp!cp zmb(y^kcj&`z$MRm8`6Uc%++qiz~mm0GH{f-GGxlISL5U;UeL%fsAIb7%M&E1;TXkK z_*&GfG79)l6Z5RJ@N%$N`5H5WL$>H>C~O$cm{DRxgkDI}FFF$QewD}lZiI5f=`Zsb z&u(Z)pQ*H}d^f??e1Cf|Zdapt9h)#j4D_Q^<#?v@5y|Vr^rf3k4&(G8%gb32c;#LM zm*X9mJv>yA)Ra=7m$a-8rmDmqj^FF59I9_C4NEa6b!3EB3&#=!V_oQ<5WZzqr7Wl|*ovs-?MuKYob|00lcPh@X0t*o%Am=Rh3LThmkc7ESh-saPRme zBJ{05+3~nJ>hV|&WxmVbkva-RzU)8$w%Tt&;_W)H@WONQ^*dycbcMnc_FAVLmGG#Y zbcu2Qydxt4?GW0Ffz^6*ERcIH2HJr3YSLhc84<&0Ma#0){&B3aXfvR~yI^AU)*kw? zILdy-a*x?*w_K&k4%^)mn_}Lf%JU05-F9!L8d;w|!BzU2o`nxh0LxYS71r}s`(cS6 za~B&{%1F`wmA9w~Ls7nDzGr)7TSWp8zegL69^aoR&kSnD@*x4n3P9uPBhiJ(kO%pU z(>!PcOIK|g%ePs}{>JlzV{oJP?hPyQ-lR^8Lfqb`>9mYUy-pG$RbOkLevJE^iu))e z=IBFj;^&?(T8Sn|*04i|HiQ5Y=*!nRaY|uxe0tlbtLquUjz7kjD#o(nxq7`{)}Y%R zK#V|nMI>)(LwwZch{U;hdm!e#^SHXt2~8#(I%pNWabbU#Qbc3a<#Rt`h$HV; zOBZmfElJwIzLzA-KbmSfUL3Aj_KYh1#k1U;B48jq&bSYr_V@GF&r%A#z(6XfpAqiv ztyeRe|D@8Akc+CUpRVE(H)Fs7H9J-~hap!eQhmta2$?QC6|+)=(K?D;Hi;sbDkUgM zCu3_Dyja(tED7iew+Jct+gjr~0D~I^pCz}|FclyliU4)ehN+;JLMBS;&YkqBUhA)k zqNPbmKx$tpKDhC{JNFL6{??fEC5f}k(y-BKAuoebJ7x!MO*Pb@12emrl2&vp_cNbI zuZw9hi4_y)s{yh+^WuF7Emw@$HWhycci1fG#j#0f62t|=i)qw4^Yr0 z*0}v4Mpq8Gnw>8P63aKRK`~!nHnx3jyig;pub($yPD$blw#rJ(r1=v$-Bj0mHBWk; z_=v-^d?ceKeow!VimIRh&R?JavlMZFcyXU>4tdUbUN(=wC4I!RuDhcz`shbajPE~4 z-0n<MM_FCe} zRfd^)3`4crv(geB)3bLM<5?z#BXi#Ne|*Fm42P^~nFb^oq8#)=@EyoEf(yiy z*v4;YQwevYO229?qsice`lm&LDGc&eJEJ1ATQop9M9SOO%K2hW_Qfta$fXgbqO?e? zvHqoTwbF@UD38fC0>Mn|Q@UsnP1$Gt6SKA=laCnHOGg>6bzyo2Uose;pf9XzhjTyVPCda88e z%Th~VXd;=&>m|IhjSG(QLaGbN{0U+=W~JBh6s@lxyS%9>FZ_-|q8I)aN0!`loWkv? zW2su8xBV){*pln;?o;m^MX!28ac(PLda{fErUWDp44 zhw3{OzVLyPia+XLkH3*-Q%_Jbjqr0N;?wJ4imxCIcw5605a9SU9#M)GR>ru}z+_i; zuKUo2yV|N_z=Hl{_haeMMm=yfHA;K z@hiYsVS)A<7vL=WEA<1Mn?z!^9g0%ii8i7f0phoT!rVh?H?V#LNq+RVFN!{6DKP65 z*Rt$Y7b}1EQ-Z|_t8=8r9r_x)Bpi<0CntUE@;EVUHyMnaDJ&EaP{EWK>+9$G>dLXP zRU(V)8t;+o{1Yhz${$<$8f=OFHabxXvBHX`40Nb6vQ$~`cB$&jLJ^qk@$z(Abkpag zxc-L&*YKPJlJX7$z<}umd%f*DgXZl`Oniaqan^&IF&-?6xpcntJ2U-wXX1A8u|cVz z3YF6!kNCxM03{=%IbaZgIGbK=tqh{U#Cp6cdztSK*V23-rz{HvA3aW zVca&2nZ@@hhZ;=#t9-+VD^4#NeJ4wP&$YIlewpHqeuasa0Ui|z6EZ}9%%l$K5pT_h0bA}YXdqQsAtHit7*i4`cl?$ww&$nN)E zG@tI(-Aq+XKzvFPZ3%QQ8)`W6IZXnBcSdtQRVsFDe=LH40h2h+i}=k)I#ccsfE{|gbS`qa9zO#VhExepK?#Yp?_Ys-@-l+wp-Vyl z19S0GFEYpd(UbdFd*fU7w1)kTI^7%pB~Al14kcwDaNbmCgdG|C0yCp!YJ?MX7CgUpwbaF@!2?kYky{m*`EO;`zHt9f^iam zwLIag2x>ZJnD$#1Qp3Pg1QXhRopadB+dMCqt*(;}7@G)qJt~MkedejFqMiuJD)pUs^U?_sk=;o<9} z#Ha5aMNFBv-dZ7B5Kvq~2^kGp#mb)p2YMQWO8~nmFpi3V`SA0S5RYO?|H0Fz#|wZlY7a@DnnWPI){(yJ0!{PvkWZ=jh)d}YrRtg z6YZ;}(+AN!PNj>{DK1&oLN!BhYV23rgy z0S-KtPY(xAeSv;o_PPs;#$!E}pw77P#p-|+tWSCAj-aMdZ?nDbME|ONVA@6olGGNK zZaR)DuW7_}D^kaK`Sm5^cZ7L0SVBljn*59ByWIxZ8%9tcUXhui8T@nRidD1??A7)c zMXL-XfImR5lL47I1k`w^mn|GijE5+YiA}2{Ypfrr2lCyI+Zl)Ey{q%D6>ft1Xvm^y z=V5&-4zRkz8PKgi`=eCbl&eGJSn2PB}mn=x3C zKBz2%cat?D>k&?uGpie3_IExNUM_foN-`ErA%SX$VQccmFRnT*Sfk=ADpmAhFj~f* zh{TFYomZ9<#0?4;I0St(u*;B zqGzjU3QFIdb7-x9;X)niRpU^mq!8pQn6f@L^X5~lV7)3-g=GE8E`C)n+iZ~5ng8$* z(%BpKswp32(PU$95g_`f0Fc6q1rTWY=?~`%Gk~R|?(S+w@n#XI%IuY#H;+82eXxMp z7vKu~EvoZdr?qml#)xpnQ2`45ioCjdfyrV^2Uf|`m^8-*ng0rKhmPszu*jg@S`etS zLf>P^5eRCnkImMbFBwcErbNacd}y*WdHZ!dfNz-!R_ZUzyCDg+8e`p}VSc1hf>xVJ z<=Y%K&mQk5OEWID49Q|}CNsowdA={A`EluvMyEOZ^`G+)g3n#2P0@!X&LMWSE5PhT z`j3Ak9UrhqP`NF7zzYNhZabuIkjZXORTiB(p^*1G2PvKlk;T7U_LY6IFv7=3wc~VL z!5YIN5|Q=Ah^)G*`SLgXzqE%ZCBU$>X^(9-6)v!kK8iAaZyI-k*<@%|DbkB8p0glf zsfC4Atg{9hm>0NzvHy08Xh!s(7eLcmY0SWFmln4dlKomsWxBs~og<@LR594~RsU*z z_$rk^`YN#t*-D|+N^IUCJlRBgVERGRuy8V4?F@x|_&xK~;CsmB30zgA={O3Qkuk{w zsqogsraVjb6orCAYUk>-V%C| z=kg4h$&8Bb)t}Jn-<18|W|a5WOJ7Pl+Lj1VKHVbp7l{D#)!KvR>snxv4;)CFEsy|s z%5Ka3o!73}lQlpDUyZXaHSe*B)7Wofp0gJqh~FWU8GEOiA`L>FX$(o4_E3&U9{X-7 zCyU{TN+7icn|{Nua6zm(TA&QH4AoBU7{_gE*4~ZKFh*sUot^mW&5G5#gzM~qp`mR~ znQuGyeHIq3VB;2yjMvGvk}u8GMKLeG7w>U|w<FGAMSMWhCP-Nd~Oq6l4$w#4lOi zqNx#gZeCA6fvku+$UDfI4^bby5sUYC=g(P)pPhNUF6aAjS>b$3fMZN`Sc0c zNJXSdQBo6xrJMH8A zAs1J@ijwV$lE(z3oRHoPU1Q~=yN^`OQ#89}bNdXAhWPcF0O{=?#4;pQHr#h<&wr&d zb2X`Y>deX2p5LOexoYlM)-L4rJj!ntEQISxy;h3RaC7WE!wXh~Xt=HMMO~dSY ziWkE7+*$Sugdk@;F&@sM+u0WSQSIIr#0jCQd4q4sRI?$b+eGqpXpyr=CueiMkRZHWJU2rsvI9QCjsTPYL zA?w|!aPLmO->jY+t>E>VJKx}?tSCdq*>qi_|0Jo*T?*F7T#x5h5G71^y9*LO8?Ko=l7Om-6!H;@gsSl-l9E> zmZzG33oM?My4m*Dym$bFm7%;&n`#e$H`9F;({;uo`G+@5_Xw~C#o&1#20S&aPssXz zXgbI6xV!J`Pi)(^Z8v6{G-hMl=){d}-$^F6t;V)(H4Pf$nf(6O^Wwajm*4BmIeV}D zS?d_N(;Nln-HeKq)Zga%CR~xS9ojTSGJP=0e<*MJ$m@=q(PUKhFpF8busd`r%DMJ~ zxK+Mw`1lqlv!o4#6~L$cmo=`=7%@{Puxj~=TidPG50ChyBBuOPdTd?6V6XS?d%gxa zqglbMM~|mO9YDl9BWwOUIz>aZYT^>2H@;Q~&v~&YT}~KFq>E@u2zjqj2Wh|yd#9xk z{b4~o$z|OCW>|*ORhpmEt4~|6@W+ArUA0ea5YNTUAA5nB>G`J1juq!{5_;^Y#|UUI zmE@v1Oi}C)>0I~iH}PkeEPH_~9Spu9faY(NSI?i-i!0hp#d`^%8sx_@2^n5XYH}9z zb_t| zLpuX$J4%KhAbs4~gt6XM=h}d-*KyvoQ0GlbH)gO9>yZ7rwarEe(!d@r`6-UMs37fZ z$~bimK4+(W7z)SDO`PZN5B+`f%1`aGLRN>;O(6?g)g>3=j>m-2p)`CgG&E`Q(h z{khCq@^DIMUG(*8Qre|&sECe48xqz|v3dITDf)iZ{?a}2ey!)^I|ECp$!U-CRG7j@HLC7OLAg(ZJM=Gc8iZH}Ph6u`-&D*!GTNWPtL zi%j!!9s19jDplJ=5sGHjktwQR3j#c_w9{|OO%t3pEssB5xmNePdhz^=aCM~<&?ggc z+%|0j_7BB9-L`=>7ddLoPl&SfUzMXd7D*zVUsHkWZa(aH9A!`!Y*KTSrcMTe-y$`VRHrw+MG2SR8RADeL|E#-VMO08;NPeFXf>+1^1dQjoM zlN+Q=IP~A&RF=;bXEJ=BA6+nL?6mIz$|9<>V+LlYI zG%h($=aR_g78Q=EZ<0lL0HEDz3axVX|I54znwf|N*&xHPJZy`6f6fWL5xR&wR#;E) zhU>`9cFg1qC>Osu8lC$GBdmj)j>u%jk&JI)buHO!x2jP5ZFi)u(C=32qhIWqhF9N% zbo%Em(#h%RQ%c@vPbe!)dJIN*4#9A|(D?$_=Hp}`lFQUhE4T)3yB;E1?e-Uff`S5n z>>loKW!N8YYqs8p6-JZGJ@4-C*XKr5`V7}|x6>w@bH!g^VWqp5Dv@gCi`@-Xw*#bx(Q)zajo)Uxi86YSh^ObZmY16w~^BHN1iG%*~ z^m6zSwD=e65Essuq_P>*O$b234?_*71x#E?IRUBcE5PPcMyac?s_YgMCiVIG3o)Nx z>sxmL6N3!7w~5Ej;E6zfz-Vg+xQKe+V|x!m^FNW&cOJy)JfLCohU~6#5MG^y5wP^W zyT88yqmc1&{uidNt9fs$8Gois5$Czv?&cKRz0GfII}rHuYUSs0>?6hbf;E&|$+mMMSGnTKi~X|QtwmL`eBI5@M#FgXJh zlA5cGcQ`~CwP!v@sn@{FB0V1+WA6!6Hq?`)`|-ompiD}TB6{+RL@d&3yoxb)BZed+ ztSLJ?7$ZUST6Lak@wG|<@3XnN@+0ScwDUXP33)L?{|1$coH{P zX%Bvv3Pv{g$N8;|Sb3O&IEpH<9#~ZXWuvc^YM4)fm2Wa}`CxYIm;WOj@>?>*WpgcS zGvwju3N9eorvb_k5+rwu`bap#UoF=?YnM<_Y=faF9$f;hBp_%x%n6X1a_>SN4@vD{ z*kl_3!1znO-Jzh?UuNG5c---ey)B6N?OutFKCyKt<8_}%|NimeJ#_&dj9Ik+Ki|8-sM>QM+BBeVaF-LJKrCiVEU(svzB+CQmUXTFN4$)Y>*Ujc>7D7MUc zAD^<{dhT#6yinoENLTXlu5aSsnxhj;6dB<>(_vLWt=A|$eQ1r_dkmysaIc)WH@upJ zHSF+7YZ(s-ivSk{6fs%h>QGeJF);3{<8G)OgfwYsSd^n)eJonFWRu8iEdr=twm_fe z3L?}I5)?sp#{zKew#W;5C|bxa>swghMiae37hCy>7R6>bp2FtOv1)fO_BK(m$Ou*; zh`=Y;YaiKbpH2qWX3Q||*n|Iv78xyWa9l~6S4)=>+x?%-@v4@sG4N7g#uYk$e$ZaO z_0P)@^1P{H^9!6uyWFdvBxIqd|kC34;MKNQ*)!aB*j`^g}_p427z~H{pI;nU?+UIsG#@Kkb-;?1;W@cIhRAGg zv|i02&P=T^n78Xi=aH$R5vUmh4EL3B*Kr~Ulw3-zr(6qYd-@R|rMwHT6PD_LY6Ztm z@Cm$TY?={PDcW*xgj|Fmlc0^JxUf@fQeMacfQ}ch6a(v8e{aORXx`r5K6YPzeW84v z+MEZb_T1_-x2w@e5^jatjn~3T?m_H`z3NFL;(O-eE@R){oZ zu1QN_uusAhl;S0PC9ZZ#_A2?4aFvw@A4N&{>M&((SW%?Y&~jf2=ekX-^7jx1=Im_b2_b_no}SnyTv_Jxsu+xa^N2r%jWjE> zFx?6x4`Y!3fIS{NT>-K4g{@i}QRGjzz)-_1C$N2U@k8c$$V_Dd!kXAvwNZZMO4jeW zR9vS}OzlRqmrbW3`&!4)=oafu?m#+@#oylKpXiaT*kbtmiD-KNeG~0(0^i2({wU4s zv-5YMHhqUbLI~jl8%_f#8(^e$18m>g@Y;{D?Bw_53wUF9YzG@LNXG*wEt2F8u3f{= znjg-G!E;yeeJeWj{O>$MrqI&hBVhBsf0zAyx#k!~%Rt&^5V8Aq#IXA=l#Si_hT4K! zL6tY^YlQ^2%s6+g0p1yky*N5f#j&Q|lIhpcI*8E}gz&*^@{zhaW}P@03gUk2raD3v zXW^)jV$xW$Y$DA4KcyF`UvQPaJ0aKl!=U9Y@W9>0Y&}m}AWjc$yS8*W_-@{}U%rKx zrdTi9hR~r(binL~8*j7qBKA#PNm0XcsrN6Kh3_mtK=3r9l%Nj$hod$%9_--&p+sCl zB5z#c;&euS7Y8)yJC$NDI{z=4EVO{;1|SkZ42w5DaP!D*J zV5v5hA&hdG-N|~i``e^6wLm{)U$)V&f{F%w%=*9FgRA*~6xFx9xSOb-UBN2{{Cco2 zt!MR$Ub{4Dwe>Y|N=9UwoGiz_q>Y(sHHvz(rO`YJ!BEQ>PV)}?KCGJ5lyAI~u|>#7 zb;N`~XxoMI!ZtB)ACodQq1%-f<_E~hn0K$sGzj}YXH6rn{nfUyfG_@@5+AYsd(7{20h(ucf zm{C@%9F-6lDr^h3u+_nS{6!->G_#Igv|ZluCQXoo9$@DO+rbFP$c|%${_JeHPQS6; zdI#uaFMe$29$mU;WS6K>V+95w})s`O+l7Wcri*MU*b zld^)*D<#`XXU1fqqI?`H2CgNjk|3ut9Z3oAH>+SuVo<*Nyo)L7Me!{RyYKG}@@<~LS8aL(a>~xgq^3OQ4&$_g zR^;WEASU-V6-vgN#veGa4gE*ijN*c<;wAh@O(0;k26o|Gd{EITK5<5f2kM$M0kkFx zyjtYz8A1R5sy%+D0c$C9UGed5qu#%a{P*nI6iPy=LJFvf!q3D~Qz5=Aitb&G^u9b@6AI{kY}`b>U0iCOlXzFGV!7($6%rqnZ4f3jYwJGHYlO11-Cii4y z=T-X=SQRfQ7G@ZROqqa>=hy*cso=Y5FnDkosMwx@(Gtnh>b$|h4-5PjsnvD%-K;@p z>t*ZVrsU^y3Aoi2Gx=a?zkZa#I}jcOTj1FaWyPK`K1Ueyb9?OAh~P#-%L%;H=&>J{ zTDk{iyB|dAw;i0`hbF=H!&sPW=wyo`FH2RGBAb*|)R~OzKPSlzx@${mSmJXY)K|1l z8d=OEG2Ec4!V*RH6{h>?$>v%q0s|w)dbF%|D#DPb@G{+%8KguYka9aN`ut(cCCws%VCT<%X!mi+bmuPzGc;c7Qw zl=9)?MNpQOR;Sj7g^0B^{-Nud1^K9+T>v&5O^xqg$oE24AY($}_4osf{`ZO-Qp^aI z0_9;0r+xlk90h(rK~*H#%ZPM5@MadC|E)3X=E0q6*8E_q6XPT0vS&;omOJ znzo-(<405DcgW-o_V_)nss{9@i9X3c+hyU|ZA?U(kB_n%{1>Qf=rATBp*FdK9yJEk zN>BLbh6q9|wNc}uV>(`IU$Ig@KEAUEsUrYaJ6KT)2HF2vcyc1}i?DbLnGms&Vi4#V z$p9x$ts%bG)z%6$xSQB>BAD1yQjGlG(fA5bVMZP#Tz~<1V<_IZxgZEs>*K{I#Ujqg zOPe>_EX(6xF2c-XGhfcSy*kp`{@<3`=Fd(Q)*$_f|D4(z=u6H8-QaEF;6tRw03->U z<7d`Jz>*~wqF0JB5vPve0P7RH0~6HL^~Nh|{3zqofr(ody7l&EC<_(`yJD$8EVz72 zInr#T{aQ1TazUmx;WFY0B_t`Vh=B!pxg%(5rEH*E9J_>i2}Mdu04{Cl*NERRGYhs} zZ4xH1H(32^>A1;J8jKAWB0h;CaHzGG9~9YkY)sSZQ@2|;l;E714>E&x+?f6u%IrGz zo2QF)7IFscxrjGq2$9+JI~rq0LY7ge`bImJcpGcb7Xj3TS5!Sy`X|35x&tZ8GLmIs znp|5*NPU6f1*OQC>NwMJ{#BcC+v3%18^xO(zezw7@c}irZ%DKMQX=2-6(Byff&iB$ ze<%wsfY4!BX6@R%rwk&^dZKM)a&*Nnr<7}2M_HScW+JgK8v#N=afh(o)g*XBj*RWJ zBM~Vscl!9_PlccT2~+23+_vP;pJLNKOARAtoPGyXWVWnRfWk_WALyCm>Tgf?AK)!R)R4``vd7 zb6}**mo;7k;JYCQm6IZ+`#7k8E=tf@0X9-j`@klY;N%u zWt<%S3qfilTz21YtQSY6WbgeF`$8Uc!+jmh~|MtdqrPc(BZ_A4c zY@82oJIb{6ngVCQ^S_0k4Pjw??yVTS&K;Ql)$3rN^mOZL2#@Vjt;HnaYfD$mW{0Qi z@k94;$;WjevjVmEZYBzHmCpE1A_hV0jWmQ{Mk71#nt1QHMaU(=364z3D(U+&b77&eH zNfe|7CIl(;)RzogAvc=tc`oiXZ9A1tW`tPRGN$ds_WeVnLx~kh`*9$6S)M<>M0YJ zS4z=5V+?U1L5#(C#M)Ft0aBNdCP88_k_=M(IC#{}5X3W}4Etb?7H6)%`~b}e@TR5~K8^4Y38KLjE5mccmkZU4 z+1aLXtKQf(CII8c^)Y6}hmI!3XTOHAOOqd`Kfl2ss}Ze|aU4=*Sr1f3;WV$fTzfMc z)Mb;sKEieGcR21(@^z;yR%qS6KlJ43IkbgJ7W97Fg4X@7Hof})LvqRe&t#pq#(UDX z_3eILr^SI}*XdEo^Hd(K7#rN92DcF8D+@63slB-@MEwGLOMUE2kTO`*^8qUzsU89E#64HQrHcpuyE5g3C1~0{2 zQ+;(u(l5kd@Q;Oojme7(*SVEiot11|l>1AJQ(bOGpd#K?(I##)+=1)dKbu$a`omK* ze@DZB*tlAL^88-Q>hzL`xGUd!BZ4)~wPa^BBVckdjh(i!S7HxVy%0GiKm)cv9Umg4 zFfAhzpQVTEu#@x6`oOY+FB9g0k5ZlgV%3r|QBf^GiZV`5<;pco=H&LSdNUo5`y z2|RZ=9cyCQj4deI%&fTe!=_eFs7qNd^5yEhJ4>r`KK6M(>>?{U?hoz!gHp49l;wCN zMfQx|(9rO@OZHB2hKwC`Im8zCC;5*$KbZRXeRDYeG?&)x2I_Fp{Jqv>>picx)nDc5 zmv}Df#5E5kvT?Hs)|q}jZu_0y<{j^L`C1PZvKvUY<<>G+)7@|lpkoETMk7NMXMCT#DHo=BIwJ7nfUI3lh>br$0~E@W?+$vvP0==h}^umxKbITeaU}6 z^mkez#fi0g)xPK{Yw2-z;M9d4*0IQi6wD<`Fpvf*s5@*$)TU#t&;~37t)@19weGJY zz|RW3A`6?xq31U9{nU*udvnR)M+nx@>X4I5LL$nPB+8AJ z)ZhANsO#x#SU4N3+6N9Mln_S{sBIeN_pB_mT>>n`feIFCvGV`M{2Mr9(k>Qnnj}md zl+=NmCQA4@3STil1~|Ix;cdQqpQxHq3>z~D($%eush_bPVL{`8Q$k(Vk^nNl&npRto)4fW7_7dH4e&+gW(40h%Q9{=z` z9GBL{mBss~0oxr|tMAHcMOYi6Y?7dl0;VLTkbqO!dRYNtxs~)a@Et{Ct#M=HTHO-1OX@wEXkVQ6lOMw5-t`50h5J zU+qlT>dv-s>QB@sbVLZAPr?9M2(!KHeT5KCDX-;hOV*Z_mHZt0SRBw2qu5&Oe(zQD z$PHn^3Rm}r&?+m%M}dyzq_F3X#|90j+bdM7>1Lr$q=9G(<736^SfBXH;16u$V{}ViEvU&=90~S2PNU0MD$*BBGXXdVDef?{KsYdIX?0_3!BG`Ixu}0 z7@_Td2!PG?C)s$4``h6d?(aD;auf@t>y7x0aDjOncG$CTMuEQ_3sy_=y77WdRkEC{ zP&fXc=6BLuDQ5^)@EI1d_LmSsHA4bI6JjMJBcxbwXGIFqYD^un9Jyk@mDgFx%-TWm z#_5}glU>|GuQeYJ)B@Fo+BEtO!ibJD>I5!1_8^+HbT=*_@00`I5LG2=X#xLb6FvN= z7oVvt>h%DYx2EID@Ag~P1NWbI!_m>v*{6=GL!jGcDGIdVxcUwbY3_#VJB?-Uo2b>dcYWyh|URiYR}@WKm6?sQh?HZ#NJwN z{5tpC*z<7>AApDO=)l)%wm){U>!ah>emSOVVcK#l1k*_z<+}P%HtHQI3kK@V*?bAWX>J{$(vIt^ zLNEaK8EJ&J@yz%7h9^3FqG;6H!leEx_ObnOF#W#WbC4x;b+{1RVa?-ipJ}0|O}IW+ z-xSH6tQeIcK8`VC$_PQ58kiOvg%Egbn+jrV(2FSD7_n70(4wQXi{%tVMpfupnoyyo zb$1blUYX!6-ZayOO>bs;5@M0nxBg+Wx^x4*(x{)Ithn-@xrg6tD@|~CVO)m&D0$k< zjx2*o;Eg+hTI_UhH3Pq9SPoYI-9)n>m)GZja%wA`pByb!sD~P@V!KB<@#*0MpYXee ztL73Ff=<|v{ta#2)DScac)|uqLB{#`iySDP{*${&4#9HHO&!$-sVaDnFKbMyp7=R# zJwWi1+oE(HwVhi9OckP5*r9c)hG@xoM8`PgQ-jg5w&fXg!GA9w#yS;N^Q;lx=-J ztC&UwVugI0nMcx_3N3)?zrhF9;#5}@d$>c8H?xq_5vMv&a33G8_Ucmx)K(XrlvvX1 zOCc~}*Ed~@EA}b?-1?hIO&hVNc%|!hm?jTe=jdEXlM0QiEl?!3qE+r%8ap!0&x$FA zdDs6-wzB4>E1HB`%RQLV&J49~EEdsouPkE82{vUkfd77c6@pTBjkUA`I*j^KRrP|A zaV{7!DmyVwC&jSh^BS*#BbvMVgnbROhnE?NpbCAVrPno0QdNM;1aI;T# z+w;?zub3?$j7Kr8!Tm{L=?*zwrvou1avWe(HD0MB$eM4-PM2@l3Alzg6lR`KsGHZAUmjT+rI z#tfPVmcvM0#}Zk8UXyfe!f565=vRUMVj+vufmfS|J-57Ge=D3uZA~nY%cXxELwixX zs7>wBTCF8+p@yi;wtVJffDU!5w_Fv!Z9DH}W zWZO|)rQT^JEFURCc%H$0a#BnS5FeupP*)G1bUF=!+GL=}B6W`<-CEr7i=pb`R}3`N zqk_5AKC@<&aV}1;Vo8GtJuvp-6?*(%HVBxU!6Ap*yLHtEV%nu63qS56irs=6&!aIe z07)n}0XVvl#g#kZ7`h>ttG}j?#RLsCbYFjEcyLQan7lHs2=l84R?$pXWa6XVOtU z!{yf)A^ut-Y8Q>(@cbTszIWF16Z%MIYKe&7_?#n^zWYZg&W;7~l(28m!PO|G3^8)OVYe z1_yqSbiV(q5u4xiR*>N)Jlu<7I|46Q;~zhQ@;+XV{SLQCl>Tf@8w(GP;V|j&)u+$O zB&|eZB!Ecr60^)`X<;NPayAGRzMm+%N5k%YtxF(f8@)b-CuS0lqa!zKhh)JM!wp(3 z-`EIdoN4kHMyXm#UXE0mB;LmJpe4m6Kk#yy(1y`hTocoT6l36Ql7xExkwcuLp{<>p zN%bTm!D%%SGH-t>HGw&U%_B5?)DKzv%|sr$>nGuoS;0oCKZ0#_x!dP~_CYDEMcNBO zLmB~JpNTadB(@jimF@7TZ`e~mX%6Cos@y-S+U3U1q<@(Uk(X#LL@ODlHaxYti?K$d zu9}SG0duA6Cs>pHvhIMe0{)pzv*dIl+iu3X;@`T097D~IdUy+6eP~TaY5BpSDsdhY z@Q}mYCyl-~Y|Pq=33WW@iMCi^#(EZ}qS0F-&rh^iUb~f$HcrAK{&ZB{f~-AG=vK4@ zm2Bt4EO6t-y>i_JHBIbevD5xiw|w!;v_j`r1F3i?`*vre^YYiN=yDH_$$nOqrtsyK5U%>+3Hj{CqAO4@wKWdgHhcYxPSJr z<5}d2t2+d8jShMiKtUOygbkR8-?tR`h2MX$APk|+ppKu%fak;PEwV*h$30-{vg~~I zCiHj$hP)JO+M_>33vkL{6|+wLf&8PU|K;4ywApl zdwT&|hTb%v3lNS*W-%N~;%71JpNvLY%$D&my) zsbYaqN`YmKh#y%9BF;%NCjji}f>L8_%JhhQOxMQdapJiQp_Cv5n|SX+2CG0vBT8DE zDG;Scy>3{zbUWe%xWhv7MFU^k42$$i8;PcSt)?Wqs3*p#wnJmI`X4@EbN~79d$2LD zkZVpY3g58~mZni;CJ3jB3obdF*h~Mi6sm8O<=_*b&k5>2bn@Z5V-AZ5_L4#>CY|Vy zak_xuAmMLTPo?wd>vv~w#(PfdpM5FLKVAtqAqcz)VdsEQRL{laYXl#0(8#LX48yEq zMv+0R3xneGP?@WvV*?Jo7tti>xK+8sHIj2nK$leML4Y9?rj(8JFgAcAX+nU3?_tU(`ik@dCrG^*4lb6iS%Kb~v$ID~)bb(a( zr@`g%;7zEMR9J$V*kW^RF?(H1GH;oZuN@v|;rFQ`!Y_vJCiKj`mhF&b=?BNrDS|GI zSkiUS$}&ZyDn{Rei>%w)=n9?O2w}eo)|3F=5KO+QXyrTRnpM%EjNtlE6o#Nz%N_yg$~ z*~?7EVot$6!<#7LM^5>`O)KljlDpga&^);?UgU6rCHB2P8Ezhu>{ZG}!WS{(Cx+|? z8-8aoJHLb#wq*rK{WEF%VPV*7=5h|G-4Fm#OI1czA|O=v>-bzGiPL<8wz%?#3;#d# z=%aV4BrYRR>c!>AtOZn3ptl9&sOZXvI=%>;%Ac_AVy8V5d~Mibj#sm7Ve3p@8Y_*^ zO$2G4ms@83IwBcfw+U#*r`RlAZ$eFnV5o;ZTck5x`o3J>t3kgiTB?oat|Es z4p`@`J`7jh^PuVqEA&}db>CxXtP-59RVCX}fERt#T>+;q{l0QS|*T>n+mr-B_{(Xl1!nBx16EQNVm%KZy{s zms7hga!{+GK~|j&4p5}FRfY>=#`h#QQl#YXq0(mZgqcvu zEK2(*E4oG{rR7m;M5{WU!bJGVq)O`y-SyBf@cYSB|KPyVYAGT;V)o)3g|~sN6fbkI zq~7u=LB%LJP7BQyNXu{-Y zEQXAoc$7$igLO-*%`iyfI=ECSVv|#zSw});A`vr3DBTu*+wooni{eLJL>>{ zo05UK@y*fq{-wU9VQ4ywbqNk$lUey_HgV-btGIyAye;>=fJAI3^i6-c8(b| z=hx_Y7<};}>!(b1UTXtAxZm;-9?fFN3u~GV&-*a)dVk3WH;P@A@a10<{W5o6u~_ zVe2L~LF_z0J``sys0={mX(snXcr;Ll#qRYDBUSiLYB8Y;;9&^peuDzNHRbwBoknCDSXu9bSLwSl_~$ z2P@b50Z)g1cZFn!mySPQ95=Td38(Mwna6*C3qHHeb{wEb$7EkH9N0fD%m4g$S|DHt zok}L?!dNT{FSb^o30NwoR8ALu+PV;F{#qOQy=pg=*>C-K^@5XyonLXSVgtWCl2Yvr z&YC=ud3!q6_w3d{6JLCF^X%HV1o+u>*8Mx_IwiE9?a^tqwV5m%KM0Hh4JzlRGVwuC zW6RE_#)PczZUlTplnZuqtl!lh(|N*I&zM0rQhkHxC?-u={L)=LD@8dlQM>`pxNP>O z)4W#Sk(IUszd^C#_POZthjKF7&Cq{${>I8{zj_qJvA;Lrvoh82wt5CO^={c#`q1H zWv!;utnFJfXQ+&Z6xsR}*(mR3EUoQ+a_!FhTCa=HFZVI4%#>?H@RLsVQ>R+)9S>h~ zM%86CfG-&IK9GPr;yBOst*KXr^h&XOW)tqLMUqcIB09xJnL7RSm&0KvDgPDli!vx_ zR!vG>>2$>M>KVd;r{RA~RD#GPmE(5rYkH^hK#M(rPR`jfdKftwAj74%Oj0VdL8~~j zf#!Yw{bBY={PN5ahau5|%A{3C^UDC|(z|}p^^9KpW&rhj z*82>8!>!UErb+AiC8pc`D%P5tGJd%IM<^H1a`Wm6{2Xn4`nB*xaiR7;R;iCva17u8 z8rA0Z#zi;|S9HwPda;~-yO)o}`)>=1u03Aiw*jPZXnrr&a?Rlu7UY(gZRkVa#~9c# zt;gzc7ynmKoC~p4U|is=$Jf-*{pKQc>%tgEf_|1oqQ~Y?n~oiy$1uso_&4zNK0cwb zIM{;SU984}bokUre0gUL`^P3XAj)Qo4AM$pjV1JoUY?{rC0VCeHU(!aT&4(tw=|Jm z%}~EU7ty^vG);5nhC`+KuS>p#KRuC`K&bbBp=k-{gml;rWXz7yH>t}#88v=?#{C{T zJi%D#Bsek5!tRAni!OVyr>CJ*gb6bOJ>m=LJPfI-XQG3zs-3Tvkw4ETdf@x;$9%on z9!&V=mR1Q<9HwBguCi8*0 zCX^OLK4B6~`HmEeHVIbSwY3~?Q6U^>M%0yxJ-m4M_zC)%ff%b|?2sx3sSYU5xER;x zGcEBRP2#Pd6l3<^90XtYhQ(+)KJfg`8pLR_#fH9=rtws0a<+R2X>}R%RA|+tPQTba zbqO1+E0!FJ3W*}5DA4g)kk&~6paJF)S-SKI=m@&=5c5Tg_wdq%H$wITiCCtg`I>mJ zbUzRWFc-bKOe)lk=XpbL5YkDPd#^<*<5>i_t1bK_Q)`8Dr`iyhLCXF=n^_Wy1lGn@ z=j{^g=la31K7_w#69y(%W#GJJ1N|m1u#tG~ywK&26JcXKf`5@E+L|{-NUVm3P@H?q zM#I-Dq-R$)=dnK3E(CGFCDKykVC&mN=^`urbJ@(Qw!Dife9iNHXKP7z*adrPw;eTZ z_8D$pgpk(G#};!zCYiM0)Ju_;EgvQ#Uc1L)CgW++)Y}S^1C{y8R-YVUAgvwYmQ?_{DyU#|RjI$cmnFyNy&SE-0*QBJaRp>!u z;@aI3@!|}vz^#Jidx7&j1p}xv!s!-sm>>A(8Vp7 zxzEd$yMJb0F$j14QQLu_IPmQR#RyZr#6a6jYBEitoGDG9qyK?DQkJ}nV65azH%~1p z=`$aGm-Ydt&=pW|8$mVzxeSKgMU?ot07%@z&?KE)K*NRNCj@($xrK|XXCVYzRnENb zp7De_C|#eTL;nKkm$%Y16A(h!g=XXz%?`5AF*6u6rPQ)L3!f|vAW>G#Rq^;jZC8+f zpau?;GGW@CCP1-jX_U@#k5wkrZJ0BpUQDR}-jr;w*^v&TV!H@whHH%NF))Ma{Oz11 zbX1DHRe{=Z;CxWu*o!24jS6=P3$DNsJN8eA^tp+h7!B9p!kgIPqecXkY|m8PwfueV zs}L?LVS#7k=*{=(eI7R5!Lbykj{WJ3=ujkzG_2E1A+#On9a^tPzL383Bp)5YkK)xO zZxJgv9-4NM6nmlx<0evUEyceZG^S|2*?|HoC|_p%9WPu=v2o?TuZhQZVC zz0G~-Y_lKyk?{8N(BQZ&D%$Z85WD^6hz+{Y@3uoKc;}7^3)8_mqa!x_!kPh$&^d{U zas_W$WEoE8BUlk|1YPy{pX_R@TL6lekLm42A(hX}?EOp$4cl;BCZrW+py+mre%5Fh zGGXA8__r)HaXLIY({o;Tl%Px`XF*mW3@K(I6tfwdcE~WsYK=19Ysc9#RH)vQ98 z03&p~4w0eIJsF4QaOo!(E#Fk+T+&T;)d*o++%42s$ zya|Usn21G^`*j#n02n26CQOvgR~n70D^z1Td4n|eV}fk5Ud5dQ(qZGx{D*vM)PY(M z0?bm=?}hnwuH)L=pb@qZW4-A1M;WF*3VH+|1pAfe*NJ|%oGwriV3waWfR)rqECyir z9|vPY9ECr(<*n&z6I4zJ`CXe_d}zS$R(36H5g5jma64ncwh3pQww;I2V~y9FoMpus)(+w6Vc z+0Xj^!N+3O%o?GY?&_+#s;kaJohfFHlMm^D7Y-AEnM1tpPezKG-nS>fOR z!ZUVlf%`*B4TlFq=79O@RSF-wX#{Zw);>&n6Kg|t)+#5}K$v-zjcqvOjRs=EHx2=5 zWrn(owY-J&DwI)8q#T67A`MnTw;x2Ep-E^gg&)Rk>?o$sP7)Po;RSVm2h|8 zEjr$DIoI{NS~TXLIgJQ76f{mPDd=N%~lpGhP9&caCqd@1X|9aRBpL{2K@ zPS4)YQctGv9jb;J{1MPVcMD$7slmi|b+YYie>_d2$P$(AYNsE4DcyrUy^(egGgZg? zpAC^ev48w!6zp1wLjm4ogB3h%sY9R2n#K*MA5TUqG$sxt#QM?~8(bM<>d0^SO7P?V`DSb3sHsrEh*Qz=f-(hT&rwW6<7QS)jEjElpfIh-cX`R&T zUT*khUx>9j0z(<8uDp(XFDTNCv>SFhq1}!ewj?Yz#?4vT5dkF1f;Kuys-9S+87Pe@ zTmG$*^Q(9Y-PQup=%B*U&6(D&c-5zIKWOF!-u3BW$35JVCK`?&uIa+raPxNa22(OJk|p!^t*SxCa%`e(K`r14O= zee3?@L4>)vdY6f7p^om&97$Bgz?fV@BBk$H_4>_7de)kTO4G`;hl?L!K355PtZK!! zrz~01$+@o>QD|wjH^jzS`Fd-h2 z@7LrcDym;gB5zx6%K2~lN>@@Pmd zHlMSFBV}?;Lm4~0Z3xMY>Dza8c#>wHaV!Q6cv)oC$c+`>K?<-z$TG>Wp6xy{yd1U# zKOHE7^XRF{FtuNbV0GE~PGKUR$l zsHYfKKEJ06Wy-~_TtE;yMjH7mRa!-+mBubNDbc0=cfaX6PD(zlUFJ#TFCyxVLJqFC z1Dlxxk;g^*#-8InT=ll`+;vJlCI>@33V`+&#Etr3>CMjxMvO7@5DB|<=~pv+kr6hA zaBbsKcZBaPY}?eP14m=3>{Ky(MSZ+3d{!+%bBKV1rlRdF-}xQ!@kSjjs*QG%`DDWR zAE!#aKA;osW=58&7sY zBjYGi{I6M)cKT7z7S%URGcps+GE=J>Y{lYy*veYSi?YHTzlR>j_JZmNmcbJqemr9e zmHuPWc${XC)KlnrP_kJKc*_xG`Hr%{x;@^lzOyu1iDFPzs=_3 zwPqUI?|tCHlnX7uSYr|A6A%E%0|3u&dS<3K@z*brrgepeM>pc;c1n=lAx)}A_mCI; zM4cDgUMwAc+)Vu74E9fi34TL7LW7=J&nbSjI;x1Pn2sG2oe~-d zgF@2|m7YzKiooeQ^>K{w!p9~(>h8@qji87F<5m^lEzUhhzw4#~s;H<4iXlqVLdsxL z+26z2RVTXe9pJ`}M(2be(oUV}i~gGZ>!Y|r(koGBD$}FL)LaKO`N#pn{nDnly_{gx zKop!&(GI2$vz1BT3CQguc-w#Oc1vSHUnKXpbw)yLO*B|Nu{YcTarnI!QaqO^sDt}G z9n0UyoCJ=1V~o)vA$10gG4OrMX0VPD{N_;c`}?3Kf{K3Omog@e{Z+cWqFwc}A*v!t zgqWRSvDdB@}+20`REzq&am40x%a>uF99-PlA z7qRJqdH?2RXIn>klR_5QPc{OF zNEeI!Mm~0xcMNt&Ce@dwV3>P)SogS(rpejObC6M$<&q2%{J#(Mj zbo$8mXPo}eMIy-}r17vnMKm`NPbTR*{O5qELL#8!v(J)g1E3;UEPu7QN5{t2G*_i% zcsu1u7vC>`{DH`K<+zzHI9C;IAp(FldHDIq*H)O=`4;TI0BkuDMF7zfr6^w#Y+$m(X<0=x&gY_vB-&7T2v__f%mR%fLwPykWO`09$c&0xAjeRw#Z- z1#h!qRzfmPm$3qfi*X#?>%8|iJW5XkDrxkNQG?h+RJgl0N)IQcB1YEb;$=L}dzO#3 zaC?K*h59YBKja64$WyA)qP{5*G@I^G!wqdfNc0&q?9JYI&~knVEYT+e2W}8~Xkf|U zI67?2Qrwxw=)0r+vU}M;742Ker9A$w92y*9=&P?Y=zXk=(56~SwN`|TA zQK`2g%qQ5PeTW(MuQ^yze#KK=ij&U_0=e($`if(q!(d5<0W$o_!!yh03xMG!q?NeT zKR_TYnv5#&JDbs(-e8Wi#d`+6g>JXgky}D1%r}v6El7*9_LpGspdQ7pg zn~`&#zoOr{^4)^f8DTtD)iLeY)34#IfR_#HPwukg1$c+}3L zFi+BCK55rkV-1?C4-dzBs51)q35wJzkYqB1P9#JqhW2?-^|JSKW~_69-C708zS6|T zfGuo-cF0^S6d6lKh9dPBPRxyZ;Ama?Uv8ex|3ZmwlF8B{yJy5GKZf`iEgyoH%gD86 zw{L1PkDDhYWXG6kJt&ypzre_c5DYg%9r{wT#+u}cR#HB_^~2)vc4<9kR4BgIX82LW zN?9d`qVf)jF!wz#yPnEd#lG(bxw$IyART{AdngP z-8|=0ZEDQ7fwW%06~b3}n_nUX;b_#*YGPJy|8CZ*HS3QxYxNjXda-)Fx4WD3^{d&N zAz}%@-Og1Q*sr!{k_vw~`A#~{eSaS?>ArfNnoDIPu28}_?1egJexM+w2lZW4cjdf6 z)Kx)GgM44C_t6K@RF`V(IL0K^ruj-%B=u6hs6HvaGzf&jxpTc$I$O-gq1y5%qqkRP zae>(37@?poN3wnJsF4L38_u{^CnlcKMJ3rH^~n(#L614rvEROlEun2Z^h^%H`ILlQ zX2onlQ0<&{z~?{SRFMfyHTM!sgK z80;8p#A?rk5f{^_GAqG}&o>iHe>AS3deFfbtFoAAX7(1B{u|!%huk}uRJSer>aXy4 zjfEBU`h~JJ7%#i$&94Hn8_r-R+(a)N5;yB184sVq*S{Pp3X`i3GdwNx(`JjvcKV+# z-M^rVc7=%hiWfW@9FP!vHSs4#)z!7O*8A}zcJB0}>!OX5(+>RMVZQpb!Q8hIA)&?G zMsn}F%|F(xxu+*($ypHTAA%xed{~{HhTdRovdyz($($EjQ+!9kDE99nowGl0TY}>q zsf0{Val~JD6yBDaXWn6dKdpvBWB0MtIJBXPOF3ulTV}EG?c}S1-2TZ>97@wZxiP`4 zpi|^TxuHq4DIX2!r$E9wm-TYNSM)_)ICFklD2s4lx8J^zUsmvcn1bDYY|9%iZN_F( zW$%$a>Na-8tJuk8}<-+^)#l^+52W2{qtmO=4x;WdWw$)+KhKS43Yhcuc%m z(f4>P2Zr7^tMgPVmorjcuDqQwLk~hGw}0=F9`C|#1d63Q>SfdgN0^@Crd0I?!Gjrs z-0hNNqTD54P*vHpSYqrBO158rYvKTZ>v{x%>W$MVEFo1<2hxpwnc1ifowr4E)Aa=z zT~>%)Z@RxwV`4DkzI^7}0w`NVBpQn{lZSwbg;zDLv=W-7A6%4a8BBuBT&wN?)B1de zvz+mJssTY->ixCZDO7PTm+{!!wv&8^N`H?hf*(-6DFLy3|3Y;yqpGI#k>+Dws z#yh~ot{$R8x8&kO?QK4U#Jd*LpVu#9>7rDYbXtuG7Ye;?K>_?JjSW+?vxNl~Ll@3{ zFJBV#*FK)o?-CK4P*C%M&7R1F5(&S|@qAX!*AIxH-?Aw-FLZZO7by&?Q+`&%?)O z>-iHGP%8hFHiQQQWkZ#F+jAW7z*ztMX^N`*nqN^73qX3)fScFU4r&!V-!-rn;yKKvm*-{)&hFdI5L zD~=B-iP*tYt+boJSpp{Nf~)CS&+T@m(rxHg$*MeaU@PhK#N?s<>u_PnWm=3TrC@qv zYSZ4HHi0mC(0~km)W#g0XmNI|k1GovEN%)@52KwLmRR*tVug#9q9!P7297}*)CmsO zizs!sGG)woonkj%^NsNn396r;U)itrFY!|Yp#gK$=oowZ`(}rW_0iGF4P^H7>tIPA z@j=`E?W(ZRAiy!+IT&|{W#nLsI7_TaV6|@ggFqxApCS45MML!{`pA+Ej;0W|$+lvw3xdjB|O&H`>6Fa{89CMuekng<4 zJX}wDEV|h<&`KcsH9FSjcM**QX{d@ny;WgJUsZg#7pbRjixjtaD5$Sb{_+JM$l~em z{p(tly*PLGkF5@!$Sw1KLog=vn1J6Z6W>r&qHPU3xD*%7q?$VOH_|c$kyQ-I(xI;u zymUV2kPwo7=i7P(=i$tJ4fS+c<`cs+*%$K{a#c9-Wy{h{a zo=G`!F2ivtH?b|LCEtXsps9y1(TGYhZvxZ^OEJ;WByQzWZC3RZ!t;C_Z_8*C2hSdU z_v1t&WbGusn9XhQ$*tcwRw}^e<>d`;!&kjgO8T0cI;8BC`ox z2sB%DBwK0@T0wtE?V|dAG%w{Z(4W^AhE~+)+Pe4|^zwfI}TPQWt(M)PVGj8msloufVtl*4E1d zoogdB%!~MyQ-%&bE#rlXOOdnT@wX%^;ruNrM)r5FL8u>2|HO1`^vrrsxvS(8r|V2i z4YuB#SJ4}F8I5F*FE_i|*w{>Vr4A_tpPd@p19FVgk)R{OgZ1*XW*VHaE<)GZi3G{s zQ9@Sv!E|OH=6-cOrhYA|odb171MVyVfok_duQ7DYF1pR8T`!P;pd2mX>IRAPYkeGo zxe#cQ%E8LJPnq+{Yzt9KS*Ke~8F9;jc1Dmpku!K6q3c3%?6=&u1^O*B;cc3VS;R3k z#q~&4%-ECI6kFDZWRI5cMYmOqp*Vi?rWTeUEe1cqaXoGO&4%etx^FsfyRB9q2px}a z`2gowS7&cBQ!-Q6&|-FVWO0Aw@w(P3j-+<=s-dy?D>n+0izTO03fpu2N$EG6iNPx!Rbots@buBIa zz5OC4w9>{w%i6m7`fA|Xh}2`|#r5^{>gqVq3G8}x__d=WI}=`*V4kN5bs~?+409=SQ7`mdZGs-}s4x=2i6{E!?U+c97{Y+&S*0an@iB99X=!r4Cf4UY5&`G>oZ>8SuHFa<#70-g!swRYy~t%9+n zkjR8#&ypnQ``)CUx+0CK#6$%xgy^nODOXb(x|$?tX8Ctubnv~cCM2gxnl)?12F6zq zUl&45_Aa)%;2V=J(TS-2u3|bhebUB5WL)Wt3@g~la;i;hrCG&>+AkFvKO&O>g=G5g ztnhea!)IQaS1caMW6yki!7Txe8~^XBqNM99vPy74?0_Geg!&RUr^}$q z8(jLP8p!#^#O~~`4iop*tRP@SVtz;SF^Hfq(75?LLpTR(;nL6&s=D^&-(CQ^Yl{8n zF|aX@_Cx?*+o4Z%JwP4jyjPF{cLJ!0gq)8F$FT?8443KFM?K?h1I6NBhZINE?LDe- z8|?SySy@|0xC#07fEFLG!^6Wd8K+8&#FMW_cFqaRV&@f<=n~cTIXfD{%(({Nm<^E& zE#*asLaFB`tfY!FPP@R0ZM0-8Vk5g6W;CB*zhXwO<59ANqQ#*T(4)eYTG(sK4m&*e*UJp^;_B52}8Z5As_(s4h}~?iYPFff@zD8km$!S z-T>yYX-`1b#fsQ%ws%5qk0ZY0;r5`*FPBUc$MH%+6msaoy=WPe=vYW; zW!`6E%lL)SadDs5sI7+H;b&;KDv%k|g-59Blzw(DrCvyiUueu_k(!2<+=ulyt+#V0 zMUoPBcFX~b6A~eJQN!AJ7JM!OUDA5pB9&b7`EFE{jtZQi=88k1@H_LeU@yma;HKeP zu<}BEb+u>YoyAbUG<74mGB5JZ<&D?VQ@JINL@1H=2-@ku1tp%gf9j-nr9XH>f4b%HJ9UmM(s#I74$voF! zS01k#&nrCIPy>U4f~5TAR|7KlE564*lca@@TWjy<>BUY_+W+*jW6NL#JY5HjKkF&; zWo$j8-1(QC;h$TW3r{MYeQ{7j)Qc|?a7~c(Dz#ZayC}n3E?YaYa&p7huFcD#Syc&l zCt^H2?;bzv{?b5H-jyr%_Y=!z?`YR%nG#y;@+|IYeS>NswNyQ;PQ=0q1b$qZdlB!? zr>9_l+=o<-mz=qjXi~o8^iT@|CbY>c8?*BXe8b6$msu?)@bvtUft7o)?ZO@T;+EHw z={6=l{zz7QSzS&@qRAi6Bj6z8t6FZbAdnL{(BBQcNB`?A@{`tSQ&RlHJG>ev3^>HH;3&o69W8 zEx%B;gT3(0dw=3m|8_uYs5BanKn?=&MyWad$TH#{tLj7(X;Uxj?Lw6KZ2{+v9mV97 zLeI^@QQp@^*@X02a)r;5G0BIzF%dZh?b34D*2cAAiN5m$U0kY=ELy|{J;YB(siIG> z`Va8E9IdT)KX)8M@qjWvq_B|g@qYXX0x&Y3B}zFsDE6dh21z2N^X_54rQaQWj{{b9 zcQ|CVvmmAjAz|U+EhP<;CwIX99sSP|vy<+0VaNjR!OzV+W%a&)y77XpzBIO4=-!1V zrI1BRo4&aje~g)D?@<=E0}=}${cUd-yS=+pC{PF7bo}^^&U@@rH~&-Lk1o^@ysR+Z z*%u)syPUYq_F8}_3M3B!vaccJlhdpW6y8yDUFCze96YV3lx^isb%J- zFD<))a%Oywu+K&o0DgW+v8RLyqIyMX8t=??!6H>&~!})&=EM)E4Gm;oF*%y2_qmlxP%H0=P?;UQ0ygBkAWut zoOuH-`CE&7v9Hzh@heI=pW;LJyNmhN=pT4_n%^Rc#v`B~VB~g{*Ur5OkFTlI-mSaw z6pq%v9BeimuHC>w&hKlJjaumNhxo(IoVwkFX z^vy+|ONV{beCc@tL>~ZbwC;!=Gm6gwO+NtV@R{%}T5vsN&IM?0irxc_78_B_6)>CP z4s0oAZf@@~Jd>fZ1t4dAm%*3_C>u1X+yh$<{E!c ztWAD05{RHNJeKzaQuc8`rX}vrCHs|kluN6cSo=?dHynePN575pdiRJrht|gP*-dGq z{ycu|*kteU|B8wW{y~e+IaXFIAlQ)5grBBpEqTKUQ)$}=a&J!4+B4#Dd;(9@_&h}Gm8i#d~i7a9St;j%$z(lZ#z(lBnZh6bwwZ&W(<2%dS+T?G$-p06C+A`7;xUcAK5UlyCUTd2h;{f0h zgg`m+wonepW-2M7$n9<~j(_%@YhC}r-nKS*?fXpzqyk{AyA1KRRai>z6^VFQ*r^cmg&@5q60W1cEB}xdo5K|RG484 zHT*O#7ZLil3;hi&70-gt|HTQ$r_b7d-D6|DL`|j@USO~Cb${&Gp z$TI-jbyDVaygZGyjg84ltJl^Xv6;2KeGfr5Yvb~d_gV(M4Gv3m2Xx$#wr z7VFg>?ysfB(@5d=Q7yJRC1kjdva-#)h5PbGUXk2j{p09*-AM!npJJ^V7;l zvk^JNxSIiTN|5S%9ewk?#m2;KyGP`c&L3*v=ueO9mGVzB#ajzz+wAxzE3>gT=VAG5 zdT=P+DV2YCwEUnr_WP@Q+xyyJGxeh9tg0Up0x6y`L3mAPJ7C6wzvBkIGT>pq`FBa_pUcXCQ9yP= z1$Z9fI>vat@*XMP_whVioI=2yNs>kloB}<2k52VAiP}uDOPZHgUrju2)7mh7UxQ7o zg|0j1`aX5o=tm1-OoTqRltNj{zK@8dqg{P%ZH z>vmfH#R(^xRmm%mP#j@d!~oO4bcr9xvc2(9zYUC3#{dVUj{rr)r^zXzG5yxn8H>$w zVZk>>LR~TPYAU1l*U(7D=$EO4h@Y0I^xoJU&NZ0hvd$&oNL#Y9IDI#sDiY7d1QE)? z7kjAMOKo4WvS^omj=Q?Q(h-}tKM7KZ`8zJwMT+0miZeSk2B^Jr-<_rxzfm}9yY6%y z&y8J}S(mueLLl8YYZ%;e`5>fWk+jeah$%D!I--A`Pq3fxbR8#Q>?c4Lz5k6gwYr+q z@VaHL+WXQLxb_ii%PQj~_&my2XM+QD+%wbf{+yeeo}BE8q|XwdEc=>XJ1&+1NJ|pq zRGi?wd;!$LM*Cuj0sT@uQBl!+bw(m0qM?0We*P4AMT~kwGc#)7=`x+#kI%9+;bh(s z&wP#6b#$j^A?B^t^D#33tlHPt2RQr7pBm2-HAwoDWh;)rH1z$hoOrrgaP^ZAdtFu@ z1N#UnsgZ5gVnl}-W?!%h9mfr}bk&BF_<+Z%2YL~^C<=}YleCT6S8Z8IsSl%+q-Kef&l^5`k)MgAmY2sc(IHO_V3JX~j<9 z<(QrO6Cc|im4S^letttkh>~g^S51VIrh2l-(_B zxacgtp~BH=R}2{NB|DHjuH4Qxg1q$&0C@~x>5nk03&5!d3so}c44y2&BMlnn#>U1a z8Na=2tpSm$W1lM^{H8Ws#wK~TF;})uRTzB#usEB+<_09@1r3G{4&|gIB#;vVV`3tr z1Aw7wQeMxOI5-bn;h(qXvpWWk)fr#l;qGqLvrZ^Ri(#vQlvIccUE&9z^$WnTF1#<* zTmbsH&d%>D`2_^5?d!P!df5hxM+7@REr#_tQi`QMcvTGo3pR=?gILO z7e^NR&t}C^Yp;SJ^{TjK4|`4Z9iw`8@+*r-XS0qBx_60SySdlp4>Dbg{ZX@z4d#a- zGIn#}vyZLjaslG@hYz%NjncC#S~U&Abytt_j^xPYhf!q@GUkW1vy0)>cI0to22SOE z1G;)bXRF8NJAw4szEfH5vpRfHRCb4({ddw-Cfyu$1?q!z*pVR>0aa#K9vyeTLMj{% z*U4++wvTmuT#}?*fcg%I0WZVGvHiw!<)V;(^um#0NQq2X!C4+xW*JyF>w9e_cWwre zUr}^XhU)^WzQ0t@DmjE zt>4Jw8j`rSp|F^x>gBxnuWthYjxa5ZghhZRdpt!|KkD zTv0}mm6b&f&S?elWw2Jx&CJN?=`EQc#lJh&x-=t_23#SV$&#=36s4$O7?mg4p=I}Gpg?2K?q4;K^XD0FXg_=EqR=bPSDstyP6wh?0WF(2{ zz0cS_QvmA>S!zc2QV(v=kJNcW@B|9)?5=1t9SF47lvF{P^vC?jEN&YXZ`ZtCMX;Dr zR7*}}S^4bD39PBB#z%kb)vDQ9JjRA!h@exyogRm;gvGG06=b;*M-n+3pawWgF6&{h zLGzAJP(L8F0Xc0@H)>poqtkqVgFr*SJvy1%^FBNvCwZ2000bPR<>Zizo^I!0>Co>5 zw4Dc-jc$g{yieatjGqgvhvGB=g`}H_a4~6I8R@q@ODk7=(VPIIuB=|$?m6RVx}4Q( z21-Z*TS*#!Ld5@soV9~!Bl5jFHtHxQD6nO`s;i4>MzY}zmMGSx+BLM7cwAni%BY@Z3{mjg)DhiEzqx^^c$Ud2+D8wW8xlboGL$cu3EIFgw` zM8=I_GGuPyRX@Ya9g(do!w`EGsuAI+kD(fblt~D;L1Bn3ED-xOn>CUi6d^@m*ZbZ> z$#(!z^7~K+I^h+qaIpu#a6hk3nuN;z0ZY(Yq^wn4t=+Ut?1) zfr6e_H}Og*G)`Qsz$7|_SVmFfHF0UVe6l& zS;2X4Op%jLNvk`OG4JUxAS8^oT-w?ipYzREq4b@(VPc$o+!ybI8r{FH0S5-UOL0o| zkTU9YqK`+zb%snNE&wCXi7>um?oST-r@tF(nwkM6TN090_|=p-8hET1DDbnd!|6U) zQAt&6Q^YHC>JmVc1J3(JP8Ni1vgZs5F^Ikk#e-pAWQ~-dyqfDJ!eW(yhknuLQl(4# zh=n?eLVzX%;b|e*)ugWYWJQ5`K_MlP?Gb7FYoIKH6swM)&tp@8U0JF!$kLdc%Dw@P z9)_5PE1f>5Hc)W-{4VpipfS_F-*1iV=M``26z84VKItImo%JKoAg%(yb@6L}sdYf6 zXdFEAUW@cn_j%T<((lkN&btlJTl?yJS!%RJ4&cAzwAYjyRwj{2WZo7^P{BH!%h=~9 zuoOFOx*j({O55GJ!9QySa2;!kIFpad?z)CnT?D(K2t@*}#X%ODhlI1aTj% zIK{^C2P0=~!(4@Nm2pEOLuw~Rn+GZEFyT<61q77P?IlCUGfOdtpeZKjj-=0@T;>HA9QIxzw;7) zpOuCG1lRc`N5E}>#k3!WfNtz1r(Py-Jf0!RTN9p$LQ4hrEXkp#a5Ol9Vd~nnE@iGZ z`sWIphUBOezB@;$iUYpp%&Ywb*H|+)M85Q%XzH{0n3p|6`%0&`SrKKS&-Is+=dc4tv?KX3Xb~sKMYf#Gw$We4;_g(VZ@d3^kl0Z`Sc6V zgE;(^L^DzO#=JmdUlUnI%_r!~n6hMtviK=p%$=3_s1py#E?YZy4uylm+XDMtjI z7;`%(P)_1?4t8>%iDtx4H5^LdR2}r*S}-!XJnul6w@Jld^`eM4bGWEnWYS9Ad`~KS z5%cfJoie=rxd_$WW}*u#>U&T|VF3_UZ}vOtoiWd&AO;Mfh&h?t4^i;~oZxhw#Y97h z2uCb(dL53i7b-g988*4fr#M16Z}%@RsEo&y<);=CB!)(b`~-wQyVNHKNh^8~Sbyk3 zJgg>wYsvjdCfOKo2qD!p&AY?$*u`U?WmkYb9Q@Dm1|mJ~&YA!1cm*AYw6sJ7PGgY~hSTm47q~=Egq~DK`^Uutar6GLMI0P*gGk zS2Q{!<{2qyJ?h(=IH~b93(YBZ-(>%QQ}Q-^^PzcK1#0isi2iJi-5P-x-E}3PN!`9hvjtz^-}n2ugy;owOx{ zb8=?T6WF;Nvkk6#rt6{uM8Qep66yGQY(`SKcX!~t3_W@dyd9+cRu(RGp%k|~dIT5z zFLXWYY0%+U6P~Y$w)`B@sViNhJSekVrx7P}lG6lbc39lui%5HY7&K&MVxQT+3{hO} zBC1pcjn@*zYJ0_;Z)ru*ckBt6$56ziK_g90y*I^=c?l)>MHR`t7>*xeaX8j)15Cjd zOvihK@&LX#3F~@$#s=)3Ub38Ortss}hp~|*^0;rY@BV8M7;&y+F!r>sU?8tfDbTt; zf&$l;2_W4#P6Wb&r*@)*2t=qvQ1wYjMLb!Se0vpnt zkgn42*`HpbafUs}e)IZx`E1 z;SoaNeU;b>{`w6ck#n@>qC(z%ojZ%Ox@fHY&)3c)5evAtl}AXT}ggwtdc8!SEtxoH*CpNmj>AoAqHKt}a9ww?D3Jj2b=C;mj^246EQ>He7S(F@Pow+7~-T8vZ5M;2j2B^o_km4t zj4($5G7cOqM{fKA$83KRw!}+$6BW2t8XDgdIQv(D$0qRP`#x0qA|dF$(Y&UQG`=>y zg9K5`38Aj_Kx>pqu$z#IP^NzuIua6+jkR@Z1OM#hKS{?wca~k=McHtA*OJ;WRC@)H z5N@37&88+R9651fa$ij7Agd$=tv!knNml2(TQ@j()P$3CZ}qzZiSeiaaf4$S9o6BP ze_RlCQ3by3G>0K>mpU*TQGM1LQ=4m$uDhm{_qFtEj_4+D7V~@5W!D|vH*H23rW#4b zIsV!IJrP_s^Anx72a|R9`CNM5LS&0JvD zce%c%g%BIL1MLBU5d0}DY%5E`$3~(b8HNrdGD)ZhQhU#(xWdCtSlhZ=4ZKnduLtoA};f^*7 zI%0sDMMeqa@2aR!>CKm2qg*cjv-Ej3k;3q8S*^m4!(<#1a46)R$)(LSysL?+yW;kt z)3w}9(?j;sfBlRo#61c*WuWEHE-v02;EO#G)y3{7Nwn{ONbTWmIlqnn=k!*_OiNNy z`jk)KpP&D_|IeBa7d0-t0yzi}nX;urs9_v(%cYY0uYIPG*>X?JvN*=PkZ+lQjg05Z zqF+Z~-e0d*;cl%Rvj3lt6a7Or;uOk|pqdsB3#XuliZ;vdtcVY)%-9osZz=dUe(imA zXG0*Hmpy^YCkL)Qej5v)8x2bGM|f|jf5EQ&&o;&I8=Kl8LRvNWfmXScl1DL@OwLJW zZ#brfbN*`(BUIQy7r*8a)JrTW@H$J0B8e=$canffn5lpE@P!E)i#04ZHu0fXRSnkI zyRfe$!tegIrs?5rnzf*M4)GV2voyS0^Lc;2tl=5i+VBIa9wl^q_lTjNoGxW}!BDHiVl+*_q6*>(4pqzj-QFfzO3o`PI3z6oEV~{~{Ex2O%^pKw z#s3|^)@Yr!_l|AYQ{iSZ2R+5MxflrVV@3aWI)1F5;Eiq)04)=|rY21(jd^YPfjk+C z{Wly?n?Wg(Hqr#J{sEF&6HFh=` zxC8&Dc_)!97hAvVN#;DDI$#&7kc(|;VB6fmUrHJOYe3=S)9`9&nXVxh8w8ymaQ$$z zA@59OJv0Qx-rAJ^8{LR+Wd=Oi-m=EV#OlMq%IcG@jx38Y^|Ssp_U>-oc_B1^nvK3| zNJhwV6MJJ6a#)ICgCG_W$!GFm3K-qG@2q41T+sAId^w; zNdmRy!owJ#k>Gjl`EN96OWE@l_K{xx(~b;rISHgc0u2saXUX}Q!J8-ae|U0hgZhiA zXleYLT;v8P<$J{=lZ?AcNvP0@E%JXx8-kO28%w+RZ?2%6*#(F8nz9NHiT6d>Yl2Y?ggul_z^6IVN(Odd|osrOk7uZEaD{DKAX%gBV7afYtGdI4O#njLF*K{=K z-8T9*w$~|+Vyu|iVqvin@p;B3KurC@|9guS1N)#2B87{q3tH$=HL&`j3cMW|IVyEo z{u|}F+{^G@BXwRgAYl^h1MRJ@sOb{nflSk`l#5-48 z+kRrn z0d$v3bzh*nPyIF4|6{s7z|z;~?SDVHKZFxgM<@F5s+>qM8@a_k!}zK1}LJ)MK`0nOFBkK4e5}s z=ZDwj{oKd<{`3Cx9*l!+Y;f*8zwwFhFG5pYh49{kduV8Agpj9-+GuE)U^Fyzay%U1 z8OpjIGvEh~g{q1o+U?!fr^dV_G&CkOh@!lXCw#lnr{STs+wHCYdUXjZF(k=ZKp5+u zg1sWi_#`IEj%xPk~Hg2(}Qp zKrDq?))4qy(_WgFe?Xz_kPN9-2rCF&c)I15zC9!4u)aM{CF47J!EISo)o|}Y49>al z%>?}7pAl~ZjV(LrqP*kEtjV!I4;p#2hJC6)q8Wozu9*M(7yrF#F8$b~sz-}L>EGK7 zgJ!}Pj>w$5TVJh$QtY35z=M^0l;D4#j?9VX(pmlQHIahF|DWFQywRj4!v`Ao_s{j| zn-^A{Ou)s(#h+?u4BOO~{kPu#z4$i~-m}Hu{(o4F&T2F$@kC!X=)dJegE?Kg?(<1u zP5Ir+YOpKJR{U>6x{}Qb8Rfh(bxZ7BB_nw)o*M#j)QrnWmmS5tw5TkJ^bVs%1x)cUbI(ibgPdgN5>;Jd9YKaF@ zrN(CF6NAn>vaMgH6zU)C5-NAZFb0(^uj}7u4-RV*nWYLo`$`*2{k}s-{htKT&KCrHJ^u#71+W@jNcw)yMCr212t5}0vr@Ul zr>g0;=v*jo2AW0V4Yy*`8_Vh2FP8iJ|3!zO`|JAI>Yk(1_F?tlH9Z9+Nf3F>DlRb@ zR}T9?uMa!c7G~21gG1|m^PRUwOn*ty(p}Q&gW;~j8ts20H@PS>e`>3G#xeF7)=&nqY|C?E z)<3swSSOi0>PU!~D<2!f3Jg0Y$c?BKG6{t6Ow%|Q*-wI{>@fU^Mw6{jHDjJr&C&CF zf6vq~jVXILX&8086!GreyWhWmJ66*L&&CQ^)2l)4~8 z%zT$?l}OfGRu#u-Oc&nPSi5&0wVq);Dd?qS!Mi`#wO85y$ zi~rm2a!Rd)imEcItVTM9Gf`pDX9fmTxAZki8}Y6fSaKcen(>puE9j3?#;@mbGti1Z z3`Vhf@u(SID-wCP~NsJy%d+kXc?z4;k6*z19FX}@nIbbYeQ*@0*^y;+~52{bbVYT=s z#VhM(kKD9IOgMJcHiK~*NS<1we`kbj$fZ<=g5KE8YzaSvdMhvI3YEQB#N(63m)sVx zVT|~%fH)EYx$We5uOI+F?ux`fM2>8vP|bKNEumV-vQ;so>&Ydr*5Gm*CO1jH!%9aq zHAJmpNS!{ap17EDXmAIOMFG!j;vuKYf3XaWi-ZsMwv6pB)tK);em$XNbOwG+Y`H8N z0>)DyRd{S>A!;X5Q5H_K=(;dGJ4z7Up|E0E(QK<8N7%v|@12Q;8D06$E7V1y@x8YW zeXzd;R9s1EXr0Na<;zq{l5DEGAYbx(xE*I(JMq_fg47gI!MJxJMA4O>5NY(oO3y&M z!;Yc2aPkmwadReAp0Y)53~><5!x<(WVOYI_KFC>UNc490?N)&D-#GlBnQ-LowJW3V zOXpQy1~f}$p3CPIFd!9? zj)dySkLuBt@mLyi(K1O|j2)Q-0LN z?j#)pGkea?fK++yi%-g5dm2-j_BG$g;a6q0^=9`48Lt@uN%%oh<*J)S=lp>$=%cUI zWH))xm{Xl)TA~ZeN=Q0wsn3W$Dq~LIa9d;H3$x0#c4fkK@{OurdL#tP!5IWtf@T<(* zNBz>*BVhg@U^jlfHsOep7J>>e>%e=SA_t-l6h}ORVAg^HWKRiKJc7}sJcw87VdY`v z8CkX!x>Z{Fc3HS*j?C$TxIIU?#ys^Z+d0L0iwJDlr+=XUT~}k$n0=Lgo$F(V$wKqj zukiwRcb(kl-SljR;1CL>%%my4FzZzL0?ra4y-u14-!8Obwt?ZqF=e)lbFWWCP$a3S zDBZpWTqp@4>&R*4$jM{hm5NnU*vSox^Id!h<5e#pG#BQyV%o^9vhIg9Y?tU3A1i>$ z;ZeT_w5KnxBK0)A<=m`O8|*OY0IjuqCj3ZaHp`<72w-|VxbwL=Q=V%_gslG~{r&HK}3jQmKf z7Rp!&BdI};f(l7a)-naLwpv+q))|A<7tIt1hP2vvt~$&roZlr>3k$iZLTryDBOL5D z6PGa3eHfn!?(7(@9_jseG-WONl7h7dKD6}oxp{fE74x2|Qw~)IzPFc)dAbDxUdQ$) z^}7=8ugWk2ZBG|6E@k)8C-gQo_N7LM{K+iogID}$(UJb{G#UyR<++ti7OYq;bt(K(Abn+Q3>s>S_>RNb z*;xMG(P61(7PCUAstkGfeA|elGVT}<6tq?R6z6`?Fo)=j_Fj8yNnb(vla9Fx2Ro>bm2{^!r)n^*+mKh56-!>!;;PqkBDcYQYt9BA!L972+fa7Uvd zo8SXl=(NSO55dy*33;gcM#<_G`t#4>-?pm^5A9SECkkQSf~0G+$g39oN}a6XS#{&~+>6O_Mxy%G|!H##LXRh<$+EgNs0oyHF# z7c*Q5QC?#HQlTrhqJW{iS1Z8FfyM%H!f8qM%6y)CHse-H^6q)K^^W7WBUKvHVzH)L zrOCQipDS5H%1N*nJ9X+SYs`*~zjUPj#R|>Qh6!Wy^Mox?^*bk#s0Q&j574CL4R1PS zIi6irh%g&0*9vhZv68XI*NFGjkaoyvQ3)}^%|_QmFF{FC`NxF9NK?Uk~# zFc0X#Fr)c#o%}2`vhjq~WLp2{-f(>iuHGB7W@-1;4A8@&SZOfnBPh94_QqS@>DBY? zxEspoIPiKw21E0OkeTnvf}7x!>^?=PhMwa`!wRv-VV+$l?ym;z%}osrvE!1w@^+HO!z$W+K@u=i=rY@4JdL6>u@e zCkcw?qOXR@YK;bd$AcslX7;lR=pm~U`37W0sv_URZ@>5;HbU2o&i7P=EH|;3`Q9^i zQ#bhu3AxV)scOh1Rx}5Q<@ufJ8C~T!yAZ`0(%hTAl~hs;ZD1A}-M;lxH(`%}IdO;W z*3+Sk+^EBv{rF0EVn-m*-ROreyX8KnW=dwEHuawx(;=VAVYn(LX6!+%Q@T6U_FYPs zb*G^ksFS~l@ua)PsfIT8B>9^PMfXgds5R)vtHHYb&@%)xZ1{{cZV%2v1Aag&2|9kIB8y227ii z=yna~AwvBj_#=K-=i9%_ZhzLo0@`5LFRA$m=eut~l77E2VlVruq^3-9Wvw=Id3R0D z>B%IPu%4wZN7rpCmV$rEdv4woU8z0( z_dxpAEs+vnS}lsQy{5}@;JazN(R9&YQub!;rSD+~F$pV=SmB;~4^4KEeNt1_YeoG+ zFKD==Ug27HGnAaXA53-Ie(AM%<5Oy0I+Tn*Z^kX@eqP8}7$Ia;@zn0gQ5c9c%4P@vDM^qFBj5war0>C;-|Fl+11urXBwl;VF>R97hzh702SHr&JgsM zBgAJF%=Wv*70F;!m)=YD?qYOPl+(>S*Lv`*xO310CMzT5PrThAwyJ9Q%kVF19qg`k zcex=k6+hN-qMJ_3@OKmIAoS3|xqYLPTLVXR2Bi`TD1}ryd&<}Js zc|T+oNUA$3q;ry|KT-&YY06?|f!Nq_2b^L^4cm4vNSWU3`_!HuQ1^_7+7d{fxF4+% zH-`D>$q>^_5>I~A9O?&8hHSx;T0~7&fU(_|4OzY$}evZ4=2kD8t<`<5K2s6cgra%E04n^o+_D~i6akTVa6lx zffKp5;W{$QM$G*pmptwJ_E9*+CpW*tSQbi5&|LYU3IA`K@QuE7uafP^zCF)A8oQQl zVG}cMhXmB$XB&%0z}*M%0}#|=(q|NVvp$??sH+uj8SkM9dVv5DpGG@eUg@=W@m+Aq z(~^&{1&xi%ktxsN=zQPHRbmXYSQ+BWq+!fZq-g+lpo`2h->suw3fwh0JnGBv%i z27J3dHR7W6CwqRXwh`Z{C4$J_9Z34fsjX6_Nd(zI?VO#a;szCe>@mRtqvMW8wYrfC zW7cBrl5t}s7rZW8tD@(j^W=EH+SBS`K z4+@EU;n9rp$lcuUUWd0#>3V*l9MQQ-+z_Lua?#h#e36(*5eYi(hf*^cQtB0juWrurUdbW-2Cu$%$Afq{n@s2 z5=QS5STJk47RoYZ8X*1QnRU_%ov5Q(dnjR{5hAgjNRnWY=aF9W4k0C2?(JJXkpE%! z`GQi^%+61c=VkT|{^PSXr(%PBF=m^#;L7Zv@W9%~tr)Lb4D~6mk4K$ib?B3_yoaxCKDrja4Aj=By{g(yr0jx>_c5vtg^^SNrX0hk0*TBt4p1?(-GG z>?}W%^9}hy=9ZUPo!r^Nk;{Wx#Sx4O+q2!#WARk^Yhi)3c3_sCdb7aANdFUfQeqwL zgLt3!N{^|moD#UQ5P~Kg#qoO~v5e8)I9#$2RYg5`A8dNMs`cygp%uzH=lguf*XC{R zhcE9k)w}a19Q7qSmN0bH)Lh460Q!U)|kmy~s0f7c#t6{ljLgD~`rrXrKdV7MG-(@2yIdWwt{x7%k8? z>qo#1*Bpq`MOnb_!j(v3gUKZQF!gG+`V#U$WdDfSX*skqh7{r&Tf3utdb8|*rCWhNc zq0VWZr<;xHbR-`rr^H)Su;tVu7^n@H>{U;8Y9BDWz(-nLch_*^4VE8z|35-rlo1ixcNSf?;IuB|8MQ|ZNag(xA)m) z^Q}*D-6p4#ClvQvv3I1{o&aH6B7CG;lOv*l)YAodUvUXbiB!Pu9)#S*8l*j2`QT-a z!Ydn3IUCEK*;(f{mV~qi!&QY|3F1$KG^#a)FYP9_R+Y&#M%7Snm2rvnjiS=;Q}esF z;VZju(irS~UwK(E~5)oJm734O7F zm4g|EQ&)xfT!-Q^svt(Cat}O=n1-nnC_~CQ#%XFyK^u4s@fUBX-_6^*Qf06UaMxpQ z+8c8LMWyeg4}f22;ujk}_`$2zd${s#KU5a)w+u<%I+0Rz3v|=N2B>_xrrHj=8TY8F zxFRFrdaF+uFG$1|JXyL2B%@TFyhCoNO(|M`hV}S9r;=J?&bc1J|5#|vjh|uC zgGhZgX6qT@M7&_384JsXKddzOuKtQ zru~Ks<)KE!YP?f6>jNTi-@~fs*-Z=%=5f;g9;aQW&9{Ch^B#){Q+Da)uYWL~%sTMp zaI{_NI$XBBzb^*1iJn>M z7gNVG94ok#;F6YHNYY0U2ddKrAHE8i#_+wXZKu$TogpTF(eK&S^Y9qhYap7ew0@YQ zTV{SL)x`=sfcGzxjWx8qCt|W_7h0n3m}r#{z*-6>C|?TXo|UkxH&8cwk3H44-iDv3 zpT!5o;OITs4LPj0+Ew2Od9iV_?D=K6sA5Y@>ONP1kgjRWpKL05#}Ro~<{zi_Z&%NG z%Uu83I-i|t>YZvjQM)yF$14$)u#Yu!~a9sH~SL^JrK$JBqh1 ze9v#sM~Cx+8vO@cM&J07fa@*30`sDW=jX=J{_k2MWw!=0H4PL*GkD*<(w$ZHR)=0N0>FV*q%$Vki#+wU-TFQM!R!dZCP3Tv!3z!~ z*N{qJQ%%_y16@754sgK(I=77WbC5XGfLn(myI}dLGJU4#q`p_C!RyIR2!1)@A&5OX zqZ9vLPwF_Fc~)!6RnQj0(H{ff!nR z5MHh{Pc#$iCfdbi@3|3MMEECMr_kB`$yyZu0>KLH?5V`C$b@d5*I za1uYkfm9rEz*+lt&7%KZ_aLMbjW&|x=a(}z1qFK#ri&oGP?-(C)ZSVENr-eEQ$vG$ zZ|1|D0nPmOdT()cVbTBx2gehjd(*&%o$ZIoUPTUjO&NP=aenX|%UHl_2_>ZaC}@2T zEeWJybA55l60onOOS>qx*=7+daj_CYB&5;{(1iy$Cbh zLd#*d`^~<6-}e*P?EMtaaS6FB{6;{7zsf=y0jk@YAsi zo#DSNmeG;`h?@1%cA^yk7_9xu%f|UsQI+?M0I2QZ&1rLXmBWl@OW2NVYqAK_ z?D_(?}G2FEe!>2kCH}KZ3}>; z(wBV|JJnSUbjW`GD28~-X~wUpfpCJq3;B>6_I!{i`vFxAc}h|@z3G_6?!NQZYaesQ z!7}q*a%E>`FJ-1BdBhf(Bs7s-d(5-gsd(&S-6rfgGTEmx!=cbdC&G$_^fB(27AoB?k7z=<} zUZEZhD-SdFQgy2xbXFVg)z#IB-<+)=&X%L_+vEANVCZ@>vsUM$UwdxS0k>?bJcSyR zXY9fJXpwf7*pKvy9jCxp7VfzqODl#Q_mrX+g4ZcR^Vti~Goy4vvM7h7+GH_V(B3QL zK{Yv-tM*(lUQTcZwF|!Epp9)nEOifD{bSORRP)Lv>tkUDW_yET87g0mO(#zVh5$3} zX68Svu9_d0!ISA|ebDk?xs%?x1)YV*4h;o|5uq|Y7|wmOhW$?}jJ$t+ZLZN3#jHFU zlC1DOcS+%XCFtTSe4mPYx}y_a{1gd$RqaUY=jV4vLvGLFZqb&$WjHMZeRgF1?Bxfj z9RDP7X`}OqJT4TF)S+8NH7L;!s&XhfhATx3TCYC(XH~YU;+bR=bbw z3_f;+Q}dB{hDkyO7O* z+0E5Cw^40;7^A-@>r7!uN!%7d5Km1UimaWTomcltLh0T%99Va|0|HRO%9m6nncpK% zd=H{beJ7fu1e>%#sFxPe=DSFQk^lJ`0U@CPbXoD{xKPVEYBrluD(NYFlrwW-8Y4%wMNWa_)Kf+;V7U(}< zuY;Pi5<|VC>BzO^Zy{_`O}eSHDNKAAc>>((kPQX2S>946J+g%Ox@!@t9*>A6NRoPd zYY(?N6RNs=CKDe1Z3HYjSck`4D6WLi?taL;BQ7(OGi;=_i34W6c#!|YDLnj{Ht6+X zrY#{x{-RUVncd{tuyW?ck4r`UKZ>BBQ}3FK?=CExC6aka=F-t#UQM;4k#y+I=7*b$>?5Z?W=SG0(wPz4xP7y1DWRIn1i=a-iBBNO;T)ju0 z=)iab~ zVT;4zAgyXy2+d*V=z;yr`IbRQDqEBi)!{yz#3wO@htMcmhclM_Q^eSd9L8);y!@je z3pxpO@!Jm36M-1(IXtjbWWC5qnP2$)YcE{5%AEt#iXiI%E&`< zENw|U`7fN3t|{{(%>I9X6>T#;w1adYOF9GlK2@p9T3?wz{Tg9(Bz&sXJnW%zmBAe* zTv+wg)1%JgZ3?HGOg-zICr-TZN)6Yg{v6L2A@^EWVmAgASIddzw96i&)M53;&RiW! z`wM3qh`aVX>Ny^QwVFTC2Me5ru$%Zw6i&kF+V59HiTl{lDjh$CL$)u-r9$*gO&BOz z@i#>zS)^G}!{Yb;?8ju0B$TVPZS;;vv}VLh!>POFt`WBQEMA((ocC(D0J!3?1+$LZ zh?WkB1)^vH->Vfyl;PsVWIxW%v#(^fSplRqFE7u2sWr3Oz0A~vUc$7+Y{?ai9}ZL% zV1rw*sdB4@$de>m6L#hwyFNM?R1dMj_&U9llbazpjH&)RS&6{>vlcX`du!mgm8y!R zhjBeyNd*FKq~(4J1o=p98oeyVj~A{)FQ{J37e-xDbPFzyeQ)<`PW)*1r{Q#oT<$f? zX5U_+5c_TS5!WYk-51bgF{W=OC37|}Y}rkS^^Bm{T{Jf*Z*Nbm_t+q90ksgSV}oAQ3ZL+OkCX7GDi6<21i&r z!p(d!ySi~cJH)_*r0ux1#zZ@p5`OO}`V7}td_n47$Nr1Ht;_kpvcyTbR80EaoTI?E zS4ZHDPw&LHOWQ>q-YA{>(Oh!Yh!GMIjgE~~%s5G9ZBA^t$z0A1yYxNIE1h$yzaE0| zHY_#2zFqGg03VWggN_l1kJYtcr_@+(Tteey*I7H#qoYKFN&CpxXz3A;^jzP(EvkE+ z&E(<~11%flso58Dd2F90&XsS7Wgm@4CYv*3pKu?U^MlpB+YkCyikkAFf;9w)yA4<7)7m~x^pY4q1 zvjRurdp=_OZ7;Z#8YsNc0D}XPi_O!6a7L@a6Q05jB0e^^Bfft`CKKq^Yb%1VS)J9* zQ{(QdKhMxK`rtiS^%2|Yt$KAq-s7p<$w9miUzG+unLRgOB2qt-3&!&y(lS#*NGc7T zYT_zKXPEd}vCHl=y`EfqhsO-wTbMJ~WMRL9H`Wb8=51K&ve&Lz0Jl*)A>iywQB)eT zSvKDDrk0kLzPdum26Vz=r8ZyREZPpQZ8I+RvlL8*m%g4ddhQAdfgqikb5-G~)ltqw z_cb@&%ZAi-LJ|xmHX%B$_3Xjyi)+leMY%Oq>Dof34=BAg-SF*dpKc_MSLA2p<=1}- zf)C6o4!QLP<5`GRB*7&8${pxLzDxXA$L7ub#Et5F^IcEk5q2E!`nEFcZbNl8)YWra-(6uV41Sa%|RRtU{kjXZ2Ehi-I8ZDCH_%SPCjtSfFm?f zj;DtD70g9O?@dT?i|Rtq&Jq+Mbf0t++{=t0@C?>pc(E-!v?5)d+G6NjL74A(NgwWn z!gQj?IZ~A1l@FNJ4b<#3`csw8XiQ@#95QWt5#yDG$3-U4d_Z^g>RbM8C6Dh0>WSu? zN~!y5%mZO{+i8+qLWme{as4aiMB}x`2b_fknDr$;h6B7DfPpc`ERR=cD8U&CO-(X3 zgXx`LQ%x>>hBHLpJg_x!1uV)8D}RAsMan&pFQ za;J9;x{+Q*b>X=KgLYy{;ko&$5Uz08xYIK_17t9N+#tAyy-K_O)sqg*A8dIDk`NnX ztOSwxe)+Qg!AZ~M5go3>Livr^S>)jl-Nuz=8kW5eaKoxAc~9TYlpF7VYrUn^@Ja`{ zhE?LJX!QS=`9g?{Cp)2luwnW zH8-J05^n~c?gtr?r0;ERpV*DFxiUo?ZKXLFJB(*)+Si}fF_~^`Pvz@=ns*N_Vh!dX zVDqmtfDR?#o=Y{9QZe|;75}TwBLUcbvOPoZK5c%--leX8k6B(n?GwKXa&|lnb*TeI zfUUGKMy{Vo#RXiU0?fPcv)7PZZ!Ol+QHsdQ&5?h4a)!0z96$va)fGKLPEoKGZ05+| z5yG>2X<;BpSo1S}+KDPoQ`D~w=kint@eZ`C_m^|!?F8SSb;Bp>p$1gJH`Ib7E*9T4 z6B!q&!}26pc^u7&3^|@j!uM4B4T?B}yNNq!ImJ)&e*~uAe?~KncDl%h@$?TZhYkXJ z`%%(^_dYeRRp-5(4NXx#U}O}{XRlH0a=U_9&(!a@K;m}fRfYHgJPNTi*$wDHbCCT< zQ-q$HbZ7+0-kk_L9mNn&^ws(0VVjpB@r)7S2P~uykMGflr6%!-CuqBU#s=HbMjeuf zb@tj#J&NprM%-hL-{hqoq#>0;Lu;r%b-c&9#DUW@mMD<}<7Spc;V>C(?rXk35t zGPC(|x4vS3xgSwB+3it+$m&z&d&*_dq%zx{E@CHt#KNJG7`55%qOL=BVD3vQzZ2Rp z7xV?b8# z&^PV-L)ubl^A#6!K9a5IO-y=ty3LaigCF>1Oo6E;AwB_w4%t6WMD98;Eo$JSz`OG! z6ztd6RF{<)vDo7kjuk9Quwgg=-D72MJb9bWEKJs#UIK6-@3x!ya8D)+R@{&_;gG#W zaDkgZ&hu1UK3%~sSe!#`>YEKY)Z@^anwy^+CUVXT#>uR#=y{J5k{}MQGKU(tV0%by z)QUh0U`Vj+xQfyX>i*GSoFVytEVSn!FZxlRX8?o#wXo82%D#~a9wMzG=Ho)Q;g?ip z6+n)j$x}+)5h1~$GzmA(ylj67SpeeHZkP9W0gwS27zyX;k;i=)g7QLoZxscMyLD$f zV+43~1sqhFN(~G)H-(9FKXrBe6563~8b1A6$BKx(2O^ol=sG(S+=G2z+#^pH*hWr& zRt3;~?(jo(eZ}Ro@ysJ_jtH6q1yYq)&kT#FBeI>wKP3*HhzF4&GQ}qzkxKicDWQNu z^6O2^KjdcE;9YFybo*U~AaV^l)8M(ct>ga~u1fw}66eegS9{@J>uFMlJuF5py=)~H zvSeI+dv{^dLDP;zzz|o5{j>}WH(!eVjxrVp+HMKNl`bA<=@wG?W`J(V>iKE4Lv;cylK(*$Vetw}>&qjI$I#w4)>99r99@I& z1(Qf+Bg?~1g;@nr~-BM$eQ*PobJUgW!P6;(mYl?rBctJ+|rj} zaE@PzN}pm3RIt_}r!8KEs14$OX96*EaQ689^MqL@r)PpA(o$GzBqDxylOgR?;3Y^q z2d{6}v)~8RFBq?nx6q8$NrLLfOLy3l9V#8p_{uVQCLxJ_hQyV-h{=gobwSW)Um{WA zIpF|w$uKAqy<2)1u=IQMfcAYiTLf_DD-q*>@gFKjX(fJ9Y;gqpJ6rycwXbd;XFr9f zxRl_f&z4q(u0=7cCw5?yS1N5M`&d_h(@pToRG_^ly*x1P@KLv7Y?*3ZqK$);R54%0 zB6q7ebCBssG)?@UsSeUZKaLM&or&Qk&nC;)>-D#CkQEZ+;AYKWZNymdq`gOn`e(Fb zoNopRnjz1y89dP@OXdm}6aLWXIU;!y{xn6 z=1WPSvMVIAmgINcG1Hu=Cj7UO$rZe}_*_#`amT)PFH|i^A0ZOyt{ABf;yt1y>0Xi& z%j2!#Gh1xF{7x9*{A;sI*$Zm_&y&&W6h2io=T;DD*>I z-@n%gMN38t05b+*fE*YCl$#?YZ1@fl%#i{mp6pI_N;P)>IqOj3Hz)#h;4d)QA<)fA zK<#Q%-uaJXu@_ZM;`|fxcE((;y>RtW-N#xUj4$Ja zwa$SyYFJvm*bm3K_@!1A@au932Mxdrj!*UF5OgkV&qrO3M4dny`GyN6FTaWm*8Q$> zp*~Zl&sH|R5i8fHvdx=uwg4r#R~7Tl{CGj@F6k!PHu`mcR5fGA=f6x|4rhrq13K-U zlKz29@2+Id;RBz88HO5Kvz5K1-I{D{(k%e|sm${zzE5?-^6;ZUF0bDSYtz1|wU1-& zjfW|oQF!O>6bLc}vzZ2GMFa+I2G+{FFRzvCdQUC9`OthNjQ7nO{WikLd6zid7t5bk zQef8JR#R$K7**+5#H+@d@?2C}qfU`4V-xqmDAt!;qnjywJyLN>iaHgKb~RGxNOejq z{}Fi#OlP=(p(4VpId&6C`>RQtn}2|%_*&zhA7A#>DhLcHmR^9ae^*3z(13tSSXeNT zIcp^bDpYF2u0hk@Q9_)5bhGIhVH5Q9g^}Cftw5zyra1RG?@ZbYI*J4B-4B+>=C5XH zL&?Ao*E-Ac2`$^ne#N$4)I?b+-GgJrNuc%-;zu?Prj&88r|x zaLq(~HDcEU>b@yrXWib@pq_8YoEsL3l*z`ctMBN|CsOTCM?qf8Ber>QG_N^3PSUm!KjZ1ym*z%Y!%(h&xY4b3xB3%T~ZpBm^?~b9whft)@?7`t6myj@KP^sy^qCPTiZr_7ghtcPqG&cT`7QI=2e48!op;6~+~ z3%65L)Yv@Nln&0$L~0mBJX(RXpg z|A+ckfG39!LYDz&Vm)rQoLfA+5 zF$4#xXn4!1Kv4=_l97Bc?kE;{&8LS^73Ax!-#+SMCeqMR<$f4sPwYxzC0Q>u$I1$` zt5I3bd(#vsg2NZ9R^qqwJ+;ro+Mt=uL!cMvF=u<>E)Ynv`lw%EDnr8 z`Yr3vPUZB>cek^;3@C$pl%_dUMr$y8lY29Ej_z=1^1(AvA>Y*mmDq!%1_{755(lX8 z44|6b$^8Od(RWEt)78}HR{lR%Q8oYA;(1lB7>p%z@ncZ?t#HkPH<#?$g$2=~gDsKF z{u_!f-z&~4CgNYgnUbkQxhv^}!@_z$)LA)o8-PnDC~DYHA^MHH5GO5@*)e|Yg839b z2?Ts#qH<0cS2S^JH^GZe3hI=uMonv+>@bF9??x)qiQy^38V>jesnhH4-rEGof~hss zX!zME2tP&HrV_jc*zki2e65(QJmZ8eeTT zl4VqwR2f*`abp0K-Z_YEr`J=)FvAVgREN)A9{T zy>gqr1>awidKIn{Lb*kCtOe$ueV9YA@fpACF>#yqleb{;#(#QL53OT=+WJ$TRI0PQ z63$jOFlE(>xyFz-0Sxm51{o2}&EKaO!Xt~}i!$=f9y~uS)xauvaR35VU9IZ z1vT8A%KZ9;@L7y6k+*keBP~6Jd1BNmjZ0PpxmS~k$HK@bL!ET;)Nelme?~UwaYvy2 zqoZ^csHn9F6_aXouLzj@x8W0-)3^X`+@0e;B)&or8XDSl_aOjWYuQu3aLju*cmnu& zO`W@UlRradyc?YL#u6`2 z9p^PUEtET7yx>($cCCIYjHh^Y))byAp5R`$z~FTf=@7t0VK!)RHeu|EslNZ`B+89f zycj@Pp3Ij!HH?72bT`{}$LB-=s4K9(!gwnVFd`8;F(+isDR?#hiT>3*qvqnJG5lbV z3vs^dW9Rvqoa4G!Z2&CUj;UGjbYE^53nE3w5T9FZky@Eo^_Ibp6Y1*E{Rh*L;>%?WB5D%48W=hZ}vc|FRGQptT)KD!h@aJcFs=<%W?>YXJ) za+J$(P|(no#@jf>SHyK^TM9(L1a+%*=`xg5?$1d$PSIF?l=cw~d+1J+m*vf=A$$E> zUp7&&>w6Ea<;BIt-NZ$ziT5w6X5hjO-!m4*pTFPw$5Ea?hMzYt{%oR*OIm&Q(dLQ4 zxReG}XsDN$aly0ia#JDdJ_hN};^lwrEYLC}?+0e#3yKLzaSW-l>tlc%9}wd6Z2bCw z`+(k=$JUFKyC{XeLbv0YN1V9C#627qb&WEP;I9+AicjfSNJpW}PHGxrIu6W%(V$&T z8im_ z5Kw?Dsjtt_2soj*`N_4ols-Xs*2k85I4v}68D+e@q!X%0@q3%esYkOJ>t1r5Ro6EY zx8NS&vH@Q$qfCRBAu1kxlSk$wV1f4yczL=nZ0rP2^ZF68AMTm0x-~JO#<#Q`+OQ&G z7#PUpDG&&_ebvfR@-t>99AvZxO=}1PG&D49&3`SiRvgpGgagk^p=8|ir;_LLV|~Q4 z&OaA!JEG_T8-E}}EDrFrVSs}l19TnBkVyEOKqfr#gAVdnfFWZDv;A>56iOS_)JF>9 zTQY3+O1{0=U6VrWDmfN2@i!+7G%O;i~i(6X4VlAF{k|{Pj9v> z6zOIx&7w|x{7+?%+v(HFs`^-8dBx}TJP{5NwG!3(iMq8mPloo~df&doR2N?RT(4@B zUy@TLh^;s8_xHLA+D z)h}~&}}{i0Ou>`g@E!~m7BDZc~{V_-^>Rog#VZD)Yh2gmSi7A zH@fv1-Syr2rn!O8F7OyxXvjkxeMf9i70o+pTq)T)N}K?6Gk zdbz9Sb?Sd-0gQ)^tUQMKc7tC!u>Df|BYy0}-g%C|5W!JW_EktfZ@nWBT{C?ELkayT zWJ^?obh+;^6{|LIV0zVQxHTT5fV}rbnq;2=A?BM;u&N@R zT!(U827=c45fLQ z|M{AaNBz!_|3O<9dg)`)frv9&XD4l%$kJboA?~B3GJJ$Bb%gt?Li^M9_tE=}%Ekwn+F}@dMRMx8)eGEc}!XmLFsu}1hK%nd-rl6ZgP|rTL zd`WTEZtwcFkJzF_3NjQ4mPHL&(<>>7?!8-(MyE%1)QW8*~#BGrEeA21Ml)? zjZirtK=y7(dZjC-(6A;J7y{BPnS4#8_gcrka+7dWoLyLE(jeLvFG`7eCVu$^9VMfla*l@p-893l#P(*d|xbRue$tewq zh6;otV5q}2zJN5q(}~;N1^O(*@X=X8^92`?%$XT5aYfX0vctA*L3VQyn6_QF<@7N^ zvLytv{ONZyYg^8Ej9R>Vl>$Ujl}6rLADOo6l!nNk>Wum29)zXEn!{!qzr|ilQ5EFx zRbrIVQF-GJbvnd`c+%06l_m1Ebl@)!5xis#H&ckK6?)gcyzBa*Rp$UF?5j!ss=;cg z*(g@p4CDo=a+3FFop_SqAoGvYJSAd8tBJs264&fUUdzs!Z?WZt@TbwL9WZ;zXr_v|HlG`Wj-*yBtrB9Jb zzsB-byh@tSIq)#n2Q#`onlFvjC3l3I&N}%7q01eE`w@@lm;tgW|4lu!uN9YZU@g^j z2n1+hfGa82F;%~K?X+;c>jB$qJauWUNg097HPU-58K%itvYY4kV^b0+@Wv5hlF~kT zqCWk*Rx>yqFQb@&R8XD+MG?j9%C4SF68Kp@ZSGe}rr1JDUxxsUys{E z<-yG}!Bu=@XH*NbJVy$DdTHj!;^J`M2QCCK7-r`<-SdBF`pT%bnl4;|yA%sfaVNOD zwYU{85Zv9}-J!U?cyX6v#ob+tLxJLMH+=WrtTiitvvTI_+554v_j3}edIOXwWR5p> zk*u_sL;I?|rOc9=MmBo$ zoDyS9b#^Y$3#V34W$63H0Mn-)#vu59a@h7L<9;c^T172VoR(vLI+=GSklfP!dWM4> zx^(kQX#7!LlW7!;AgULiZ4?7Th}dV-Ca<*qET6)hXdj}i*l);tWg1IkH6M&2fB?H$ z5MdC)$Et;>e^_1*KD`)xAAk3LURyTrex*}d2Uu}+{02B`w0l$MRJ~JM+K=H&LFf;$ zYG!5|Y+CVR2IzA4polChg^=-(NM(k^!Metut^(k>lMscZh%Xt_jCL_iJCiqevYp#e zjSaVVU#M>XVBWYG{~DW)kwR5C!WCN-p+Z+y(q5g{Crmk{`yokZdlariXE)Ukkx+|5 zqXj{|7yGq73NN21<+s3ow+j5|Ue@rMbYe}%QZFz1Fs|n*8Oy4)kN1G|*A@{TD;rgq z!6?Mh#~f$1U~{w$mBfD}au+25LPCAso!vz%ah{faa+L<%aYOz$L|u&!0x521#`iFH z+syVkBf6VcsUlAeybu@#GHZN*C~As)p^&}r?CjG&%Kg!cq(D%uep=z*Nz`&5lB!Ez z|4OoB(+{Y+0WW6=qDOuBy`Sc|4~OqKt^4MGw<$PU|Fh%yJ^OM}Vq)R@!pFyX4h@Yn z7&|H=@^g^UbXm7c54~6+W8`iYW?B>>QkSRk6`?|mV??AhTyPCATUji$B*QW@_b~HZ zaLak}?e+3ht!0FYVs^tJugNOMd8&4*OCra&>-pKYYYM(O*Bd=UT?d$@C9Xxd_9HvX zO}9r;sYXjW4H-IQwjFfT8w8bNB75cJ#FyC9R?SUC7d6gAc=lxuRpRdU1`$*B8-P0_ zCbjwpTz`=obcif28=jU=6+y0zY=C?dy(dp6LHgKNk^ivJY@&=*|FkE>MY~jWBjCo1 zH^GIM6r@ipx;Scad(?NA*trxEZ5$0YaF&#ow!ObSd*1v;$nd;jQ|}rctaF^ys!xg5 zwEyos2tP@-Sd`#&rIQGBbbOMW3mon#`tdKW8tLx2m^L>T2-t(9oKd1$y{biq zcy<&Q^1DpP^M)*1LJLNU_Fi!p*@0p8$d*v@%uhAAh$iTNy5B8 z)DnF+20rw!!NFRoRVzO*B@xLlC47vsdgs%B!Wcrsjl#$wJt%+93YO>0EUD!Sq&@`K zN7&#%N2nuuL0<%vnfaw8-L2&gZ|eQ(dwU3sIY$O)GzTycWl@i%)|;##n&-OW?W<nkiF2E^L2HA1e!I4PJmJ^#mZY+jww;a!?G_(mcL(Jn zi(703zB^lYzu%A-dI3o^ z_wg~(v>I6;!PM{u=e_|@W3YjEu%I%B0T013qx=BG3_LNvaVSTh5Xs_IeI2h#ZjITwJ2KDw4~Qjl6G=>^8S1fl$Ma49Y_kc$z%p_ zA5+~F8Pd=mN)Zp!upOeTb~3{^!K)M1Z>EKX)-Nj%^%Tk&HX{ZTfTNyNX&=}ALUugl z31qYUKWMeh|Je;)=z*MI>u`PKn;xHF{{^FgR7p!(2^nVOkN_XVczG~H$DdD$Sfm1H zXb7K=O8w{eJUfR3gmTmoHiKnnX%Pfam8519AYxiI_8=<`Z-^=VJ+@|`{zJVp+zbWa z3*`SSGIRt*yo75;e_DDjk-t{gD{P4gYe8LiNo#ziMwrDo|{!;0X_>(SslrYt=K zb&Pj{(4X~?^D%iNhZgsTY^c5e^O~T;T1&=w5ug7x&0@8-MA3@)3LZmZP|pz4kKgr3 zs`Sdp4SMi%6kkf9LbaJj{LUiG@Z{)*#Hf=oVbcfLN9$DRU-=Eh2aLPOR6`ao#%&t%=PGFh|n?V(JXv~~)(Lk@>KVZ`ciuxAxn`snO zi#J+k5kgKdVDXsV_UTO-4;6+{Q;LsUj6ZDV2IPzzL@mT#_pNf5nT-7CYksyG>-o?B z6{lPktOrb3plsk$DuQV-Xq7p|G}0)4VcrhR4GS2J9d4uD4s6-5Yp`BoA?pedsMB@9?!ULqjY)D4(fLc3-o_wO400oXMXIq**I}a_!Yo6tMd@ao*c_jjy8Kg!aeW^D z7J@tVeQ3P<<{Vx3!<`yJk+z4@yLIJ*eDY8#eQ#KWhT546LzJQ&|hb- zMs@!0es-%R9JKF=jJ9Npgdtm_>ahmh=zi|JY*dH@hW zS9DZrDkjXDnk>=RUZS94*D55GlgZu8%p{i|G7~v85YuztFHfEoScX!C$COe=Yt+xa zI#8t={zjTnK8g>JptA9UYT#p|ijXPT#9=(%7H)H##NU@*Oa4#)=O@NRaI2=>{9{x{ z+9pn;Px@qEl75D5=EA@iZ&f3?oSI&Xy!8u8bf)uCEtb60E&G;EJMWFV_A{Zgv!+`~ z+fL|(B;e*a*^*rr!C!AuD(+gzD3x<>#MDYC9%KZ!-lL*`jNtg*`mZ1o-WMQh>-=-ulT~VI})Pzn>#O~AQ!gK~3onG$;0I(gyve)*>^Gmkr+#k@ z+j`4cV&i@A2+lXE`yxnsh@JNUyI}Q0CJx;Q1s4juq+a&)wO9{T+9|NAC46@&zqr{-&8w@mtCAY#%Vf}hAM z1+meEphuWY!VRZ7L3IRI4e7<6KyAPPoCAewyWtxz)vT(&E#!;i&mcJu%1j7Uv6C3- zY_cUlxlLQEowy-8&1cNx69aYG61fOQ#pF>fA>~-{N%OVy*7tICBgg?O=y3m?*z-uN zK6RwcdhsHaW3g^8DLV4TWFSQbj{Q-iVuZ_xZNpD1P5N-0Q`adQJS@!4vV&s{^FDa4!8Z&asN?cZ(TU8fbp!EL)^TH^uN32_0fP0Bzl zc-@ni65+KJSYF=Vbqu%>1D0#tajLx?{YI)4B^u!xx{JmtbwwJocQO>cN4#;BYt2v@ z9;kZwH4DAfJfWA7(p2%%^p>`8+C+VRaFnZANj)sKCjVf$#?ho1uwM9*p?wq%+ z>Hps{@uoH}+i3NTqTxAzAt2+m_JpOD(qYTJXi*0Yf5kEV3pVawSQoWZyhsTVbQnCD30&3M0b#{@AVSz)9=om#R-dc) z`dw=SnkLX#YO#dgTW|Ek?V;ZA-#5Q{7?|aZ${S5MRjpqdX@a<#@k(K4a)L_fBvXAD z(5D`HgCc8*j>am3c)~Z>7ITJ{+?dZ z!xOD`d)Yz$rPNp3c&N~(VNL>eVFuU>-qc#It{`c0beqosmni|{eF=9q&UOCH#fPWC z7fKxJ@HzLSgmk<#)wT^j7BPo40vKEydl0#?ea6r6I=5GTpDkL#SR)|3ZnMqjys7I@ zR-ZuUNxpE87yOz)ciH!Dp}w^fcD_=Tej_&h_ot=nVC-b+Y3WQ7q@U%yF$J56!N*%jX3ya@_USMHVms22sEl{YxD8TPN==BZ z6vkzTUy264Gm~QN2C9<$bme2o@(`SJW}>xGTo3I4a7`ST?4<8!-Y^n{!#HQo-WO~J zHOcJ{k{!Q2yFo@9UIogF0$!9v-qS9W)at-Kj9`-APHGM};AIai;w^;EKIMf$wo6ob zGR%NEEAdX$j98F2Y(0$OYApz^Z}(42z8*PHyKss}N~0OXJRXZ3*k+t?j6Jft?)c^Q z%kRJIO#Ml{N9s{Y-5k>}`a~72P{xeKBB6X>7o)EiIth@%`x&es^(CG{FGwLRqK z`25y)+WHPiHJj6`HREFSp5!xu5C4jaPVs|>XorPR$%-KJ^HTK^oP5Y1>XaByTMN#8 zSHuz>W>1l6HlhozU;jbpw01{8-QXJ;h27euY?fqT;U3UJ%t-RKzvjAvq2haH$tUOA z&>|JiHHgjMa z*+#neB%LSnh>J~D6s@D+nY56E57$%ak?qH=I7*;*vMEo)W*5q3C8sA1Et3Q3_Q+D3 zxRoCtza4q;wbh$h+R)9LlrWHDP0wDLZq(;1;~L`BA(_c(t)#3^qi_M#Rfw**5?h)C zc`Hxgq;m4cF0vW^f6tO@dR7kg!aYukBfYa}Ki28Clrvo>RtU|35DMnIfMU{(7jcKB zO~mVQd{EKqB0WGyfrw)N92tX}(edI?lt)l|62F54KZ_RfBJ?pDz5yk78wx=ahq4+IRB4ooHc{Fn9(pdPHv6Fac#51gPfj6OZ^9DNSw?1&DKqS` z!35`$=8#n=Ao<`L3MhtWAVHE&k+R(+IGVDV#{tO9S!;nY{+U&m%xX!GS-Ght1nisA zE_Ojv&{y7-*>-Cel$ZH<$vZt z)ollaZ%gk_1|P_bhj?hU;u0hP#t70ELAvNzi1j<-m;aNR-P=}4RrSTWAVQ@=PEj#Wm}e)-)rFDfRM{XNh(6Bb zvgeRd6p;U0-_qPme;jc~B}h`;$rawKU)j(~>?+u(8Y;($WR`8>U>=s(^pk@i>#d6s z*6_E2&9D$T%qFS7i&-hJ1BCtT^$?SFv1t#CjG>XJZeS*8^Z6m8`hS?0M|-b53_P(P zB7AD!rZ@QpD^t``kQ>ZT&+kOG$kL0w^ZJQD7G#rT2)-KPnHetNB^DPy_oI^YpjAu; z_A(WD?FaJnz!{$uB&CJFOCqD6e>#vMNXQGtC1M+;wrC;Wpp=(mqDP%Q1%n;+G>a@( z0ARxdjIp{n7?cKz#AsSr8%`qF&R}R#JTtgI=`=Xb>OZ;MNHUBLbA?XVJ249VBNh&G zr^YtlWEZQyKj)n^#whh3Id$v~?Om_D(iJt#k{KXXs8t|m;neeYu_o)qw);&Xd0t># ze}g9LyM{G+0HYB#z{H71o56rxP&tnPWD*gWInlud-)DUcfJ9>ntGuzP@=*MQDa$3b zR--?rHfBS``m|uxW%nH6*+v2g=I*h->&`|&&zaJE|F`vUp!pu#(s_M`ih2#Rj*xv4-RqTLvd z$v>Gqosl^L?pGhpc3#)Z6qAxU-~ISR<@dJC-&P7^6GletpuFJUfW(ldH6!)ut^jCVKc7w|~0#s4{=5`@9X9U-7w47E$N_w<7zrV5W<%Ml<= z8`7Dc7ZwOclpmg&aicgDa7l-WKQr6v@)b<4sJ*Ie;5*#e`V)0;NBzSK+I89{oP3UQ zpw9aV3|2H*S;*d7E}Dhzm@a`?_$l}@WDNaQnectaGgqIeu=y?wHhv&Qffqg6B`(!`Z`zN} zkB8NxYAf0n=Y$I2K#>9x1h|N=yJzN(ix21lI95Vl_A8*Byj<)QX+A4jxBYBgkR*

S$&y5_D-+CP~^u(B7UG@NFK;4gI;eTND9;woQ zBsS82^8SAPl8 zX(RR~^Gq) zG_=&D&#jxGU~F`2<*8;9(~kLh-|HR@DWGP;0}8H9nQA4xk432*b7|Xn3%oOSkG=|7 z&sga&6*qlWLM#&3HJ^^N2%G+ z9KtA`z<*0_(>j@W3#W?Jbf=2kXlh2d3RDzOGzPP?jhyoP`8??5R4GKOqx|SAMFE8c zq}n3YvGAYR0DMg}RR2K-jG`SaK$WS7H!>PhYhnc?NB*u0B2JF#Nfbn-`_c|F__0zR zg3@40AgGF!UVaocgEv`t&CbgJ!@#V?LC*8kKh?r>_CZfDMQM-_uKsM(#+9Z@3fSR9 zRTS-nz5Q>v6I7r>6=AwPEd**tx;qc}jjKE_TGByIKG|<2fxJ7DzGo^_QVmzu@~ z5S*_Zmqk|SFWQ5FY3lAqO%B9w@G1r}4#DbiDzRsk3eRav{pw)brcbA4ZRrk&a_np&YG~B};~K;?B?5qzl7HlA z#Hs!l4E~P^#;AGtNLF}d6IJu=?wd`QH+Ek4zjJ~PNN9cMzJkw@1ldj@*(==#zTf)v zw}MSkyl&aw-=M;D@|7vzWL8zCaFw0Dv)y7DLEN(!p_r0B;&of5b zTeCg;w1xBH>TRW7XXZEKUEXWH7#ms9uVdJ2hc-Xel*~Ou<>suEt0Nh8wOGwfM0=dsHyL9OH`GrRkJQ3MiL0n<&PG#IZLq1Hb6EH{umN$C@>0Sy_$8+MqREkEszJ%%B z(L$FeDZ*2@I`X||YdQJ+&%;J!C^PETJZ~&D+Tb_ZIbWI!|7!}qz>7U)@{evCRzZgv z#rAvt5|}PlcsQ_K%q+&6--SnE;Kk>y`SHkNYU2`N+KYi~CaCE++mI%EQG$|ytP(dT zl;oc8a)YJQJ1gQQ3(X4vTtf2zKU;cK10-zRptv$Kb3NT(1)nQr`;vEUI_-e$NS5SK zk_0*>X5x}^lfCOyK9)nZc%4{y+Swq@A)fqdK#sMjIB8^T(x;2mT5?;_iJ`LJf2&SB zxBj|c?Vi6!xZ8L$+IuaXJ41LSM~FY`3g7<{zXOtA^u7?dX(w2I^jAai`h(}2I&0aQ zoGFx1a??~SheQf;Ep;5n`s_5k_nf~j3|ZRiO^DdPBiX+VBRuTf+Zj)NzgMQikqXi% zAGOiqq?ojNOnNcDF_Y{)cLgvC3&RT>ZhH{KrAVF=m%ZPt{11MZpqP$4LTC=YiadRY z3dkn&lP1k}e^rM1@~`a6tn4+3a`GvDKoaQ@lbkax69OBfmv;q2Of1pK@M07PbOf@O zb|!uRc!dxO4MMsLpcZR?xf$;2MZG~C>B{~5YFM+9yUuyT@}vJn8h5s{_w)vJ?3rmY zs-vm{Dv@_Ic7)GYZtr?nK_YK>O9y36(8nfBx9bk~?QYCqcf$Yjt6kf>iYyV!US|9Y znXva9bH}w(_3lnItFY$)d7;X(7oaj5N4iUI=J5+H@Dm*x#7R4@p(ctxyr-r9E4DrE zg|B@u<_z&@{+~^*B_>ez-}6KXNGfoa#!7nVwQ^%tmNO?An_V)My1rTY3NMZfY!^3q z!sdg9-ZO@y^D>g zJEuA=eK#Nv&_*tMoK~k>3MisxQeDBnNCtrm6xQP$k8!&Py;&Y%?B15UFXHnanLg~g z5hv|Fdz54a8WloSBC-Fkba@u3(tiZ>z5P3XLoaT}D0=?}0b4JkG`o^8{9dHr&NaKo zmrc{Is$MwH-ztb6liz(QTQ<6bcQO7IE!afhOk5W|l%BW|FNiwXjdHa;4pOq8sN+pz zn|$i+p<75S_v`}k%Oa&a=AdU8YiCZB%dQn1i$a+grSF%FTNE2BTUr?6;7&NN;op@M zGnZkvw_3o)H*ZPxVhm4f@l1>+oE5+4$e!>K0Y( z3$+o;3+$k*73|bt;O7UVK!qxD`Dkn`i$5n1i`9sj`d;6`OSN}Ksmqq(mJVuR5-nOmXNHB1E3Bx=Og_1I1!4M>WWzX3k zw0f=5%#3Z97!g1&*T7x?%RK$a9YD1|cGts+D4%1$%fGr&Ngbn{Pclj@(<7dLvdWtp zBUDh4d598#zHbu!C`_A<`M@7uQgyP|_=B6{LaEkcF;tWYe5u52dwuJM4G>}8VAUnt z)vAx{s`j(zEve;Fk%LfrUv#+^%9oOBe{+MZggJ4Fm?CtQKV3*@EYXf4#Mo!-3E6wm zj>X+whTD50K-^pe5c1{aIPZIshLrKl*JJg7eh`JA>;0q*0Y!5Or@~>?0 z6;l$%UP1i#c@tSY$s;Dq-TopX&&NuSdv|%Nt&d}s`gK7Kn$@P?-!WHQE7)d{%J%zf z$w^NFatnm(@`Qyk%7F}6!Q!0|ybKq6v4nL4?3|>N>Bht=4KtEHKW*s=of#7qP+$U! zTL-spvO@^8oC}nK2?_ym74xGOaazv>0c6rMXCsrE;77Sk4a=Izi;v$*Xp@?ZO$%<5 zvA_;r5JeeU@n+4kddFkgJGm?W|L_^EVJ}HNb54W*LKi(rd8Lw0kNJVck~))uu_N98 zZb&9Wdx@8r?`w>|V)P{8);dT+DbRD04Ub%wlxyev(LeAtVy+ z7(xsI#ov-o2Io{+%`Nn*36#e83PKyrZ1DE2gU4X!xqi-{(jl~Ljr88&Y`qVjG07`o z5oMW7lr(s3pTEM}ueDy@Bm-@QQOrufet?p=$i~s;{ns__pPM*%l$^|n>fzrmUEgHW z(+Y#6uwjCsLfw=bY;&A{HM zc%hS$Sk!O#^c0kqxtdbO*`zcMGIO$XragMt;fOuK|6;j;phiWO@v%$SZYHz~SE`Q0^j|2pDJ5OOtGa?kgUx>xOSH`vUfV+G@N$vwF=$ zxBhEC`Os4g%@)h$rRGWSygftY7UrVBu-^-s1``M(4`R(WD^6E*w{svG8k+R<^nj(V zRO!OscV}ylmn7~NQ|>-_^9nR48dV1|k>4u+5s-ITwFRWfl31-u^Ali0VY`POkNl8a zO;zYk^aOB4mVvmNO7>>^^~$s6uorvE`|>Gm6zky2>BQJ)9c9M`!R@d@jw9uK`lkWe zyHZCGwND0vcsDmqdX2b&9!xIH&98cWOELagu)}|jbtjcgu$c5F8jN;*@(%%E2^GzG zMP87>Py}492rUaIJamXgktya#Yt9qGhX}^6?LsKVO~T4E#SAW8@ag;Vi&N|F`JdwR zKnVfE1?T zp+4A%A0&t*;}&)6EY`rY5BJ3+)UkYz-_YEpQ`oH4n2|)z2=CC)MYVxPf+GIskG+my z3?wM4`67%+RAP@Wz$~+5jkcc2dLz3-U%}RMOc8+xK8@;Z!-ck~?WnXjZpbXv)0_v0 z7nbO|(yWMj&NydX3}ZvQxyN>iL4dY}$IOH=l${_nz(+yDX|8c8Ct$|lP;h|E;KN%u zy`(FtjT?5+JHMOx?JQFl5)0_jtngIgs?&^t(6YTR1iErd#MiJ(T-24y^E02gzsEF| zIJ?`ZIay@_!s6#U>yCL#3EmweU%Hrb!uJqJFikD3tzaw=fD59+6^+bHYMa=1w7ZCy z9D#l)Q2!>hdu#SKol+Zp+yoTS(F{j^Hr#sb~T*cI0bQ z3$L&u?4kDRYrt&UGlA!k5bDg4NA`aSQ8zu?dG&RD_A)WnGLN-d9iH16Q5@Y753Zu*_ zHyI6%>~#LA)U*7MQEUH;w?5R9ubA&RSUf7o4BiDTrzwY!e^}VTvVUt0iaXb}jQr zcsUpUY20}*G9ZDKDbgQ0x=>ue(pLJ9H3-rT7jH)Rl&^t9HVRWYM z-Hbm@o6Ffv#mHFIr}3cbPi)-V9I8ROF#OIPBjLNcc*nD*rBO*M9z|Ifu}Ka({l`(w zD~PpT<|8}+l1Z`+86O+tA!JdV%?f*U$CYe~>gifbJc)qYMJS0#LE}A0uD-KUrzwgf zHHJ;={PSiqp69BKnB8m!EOE|K8{aBd)gq3ij{2{B(;W(I#)bY2W=fklS*`Ye3KDTs z7JPb4|Cw5$S#lIw?^3ao%msQ3uQnEJ1*QjU-;cOI(O)laj zBs+wZAIFaxs^7z|6W53uqCAval4HA@_{*|(il+WT2IU5ANNG#EvLAktAI!06QMTmC z4V|eJ?Ii!RJ9FMsvK4avZtt;~wS6H}YZCmW?@GqdkXpJBkqylxJH2tn4zaAy_UhCrqQSF1)}WPX3%mtDYc^EW|Wxd%CmM2Zgu`V{kI0@4jysp7*Vcw}PY< zL3#?awCQahT1yp%@C9)hDan^m&otEaQU_DXHG8q-9x;J$#*!IU;74|&^C_DaJ5ne? z(0D#KAsvc$;rC^A%Ur$AfjF{tMelobyA{tE>HPHd+QXifU|nrLiy#$P z1xTZSgKZiqvuD_(5&coZ{4lnKi#d}iZE#g=`8MDSKZz0!Yb9l=K{AskFg0k6#JbT? z092>|fjQE_gaOZ~wb6$^K(g+A>`E zeu44cGbDV+##HeP^{5^S20}!R?0kG9ADM}&Q7{PT4qKh~S5x;Ja;Ep6jp~3WG#ZYR z7fw$dG(wE2;F88{Tk=I#g!YxCET4 zDq31cP$|KlKy|0_x=*No*K-~-Q!Cqmk5S`V)Zf8YJJnFhI&!PZUp|?y;~gMKc4^pH zaZ5LF`>GHi>a6;JyFMQd+OFh9TAU~uEEhup3n=X*x{!iZmcONR;Rq!%#SGNNa~Yb7 z!#8@|yRXHXi$0L1#EDHgEx-B#1rX3TsmZCiV4u%Js%pY1?|Vzes9H+&F+pmqqbI#= z5vFi^l1b1uFU=sHg={kiWD*B#cHY-rXe)Ty?W5owEsflzm`Nn3CX+4ygworboAqw;S zVPtCSRd}7y)35o?!Rn=kv=#}4&d#f12~dqK zaTOPvg;D>B#kP8D{4syV_8|ceS4{Pg5Z50})CTsM&4qdZ-)ab>vAd?7ySTUS7n$b_&BNS6BeVGz7@_@2gJPq1yLvHb(!U-!a z>1fVqeB5Rz(Q`yUevfFX-k1KzwTq?jDx<)w6TVHi=tESQIi7mP@SCp_v=u_Cxo4N_ z)_g3@qgae62-bblK(rvxg9|mLU==vw*{+`YEmt$gWiU7}P^lmEd!e3_86Y(A6P`~h zq9iou_&W;}P40k2>g~IRwEfA9ymAe^fDm*Bu)FXClls9&qf&o_oN zL5TLiXT$v!kiX=WwXCzp49@f2Y2cZ=(^$%XUOpnf%*J9Wb;dFkbRU}VZ4Mx09yw`z z-un5AdVH1cCFvTP+EAcqfQ*GOW>BtFvV0*WZG*f#F8MG34OWuSO#N@Yo$i(9Jsi|Z zGdx#U>m^ra{(2noM828_2dtJp`4x|;i`8FFK>{LT-977q3Y~d8j)2w*Q)?X&O-=SV zt$-Sk4&j9sfBf)#QU~_66H~)n9teL}CLlgkzM4Rkqo)$k85WSA$xU_CB$>&5C=riV zomzQo5gcC5jmpRFsTnft5yaElJ=T5xD<|vy)$C(yL9$>Vl=lf9nlB-QTFMP{{K%yK z=iZ%#!qOU0P*Cu(?G$3-;@J85M@`ed!}q+r2x167kZ<`PeU9_Jwu>X<>s`dP<}-E) zx{1P~c?uh0YOg+i$&rVyXB*#L4PwVm7R=0TogW!laAAp0tu!^(sb5TR>s{q_lt~9N z3#r9T?nd(UE;qV%P5P7@Q-8NZ0)1_T_ISc>hgh+&L23pjwlV)z*ON4uUPM|#q%F_I zPMkSGb}A?bt~t5sq&@Y`!*D6$Qa7$%!~kPg1NhWnE?{)utJHGJqAEZw28V$ZRc7zu zi~ID^U=7_W7aeK%+gFcMZY{UCy@`}k@P^#B`8!ni8-R}ZN4O_#oXS+>YaOzSlOG`= zwe~^0)rpy{0$EU2x|wOxD|FOO<_sHi3JLbHhInvRYij9$cfNz3lXaX3Q%wbmlO7^V zM{}+0Oh_Zv`R_`6F6+XT?CFsF`g~`r1O7Oh)ye=9R!)W_6dYJ+zs5P8gd^NktNk`m z6H%&tSjKD8m>McdILoj%GR{w(y=lG3gFCY8InTU|;AfAcX`H%1ERCwI;brX#-osdX zV7eeEe-S}HW`2r60 zR=3`ckmMRLgsEY^8Z3r3e&7RiZ&mM1NL*rp6uVdU7XNp*(S7zMeQ#S$%WKZi4GT^S z?kuIFKSp&ADf%62n>8gueV`AL^KrgZcwlYo?arXRla^@e?v8k986tN_-vmB68ni-nPZ0Wl|l@tkOdaC^xp^_Yvl=s2yx8bx1@sCEM^6i{mtz z3glr*wFVg54H-`LGTc@NYkew@k@cbFWxy;cm;V_Kk`Ovmzx?J+CCNXCg0;l|wbC<{ zrc9UZ6RS}ZYU%|cY074688XaHq%!D zTW{MHvrRuvk1-XoBe>Yl$P080ok6^iwmm(DvS0sg1Do;}&H;5^jq`_d`|Df#*=vU~ z@D`e3F<4tryp3G~b7JRTN91CO=I*}^0uce}1a0ZPf|_aGcO8l$vcj`FV-HddyC{t| z*?AO;1G73`>>pX5j|he=HR$l7B^@m%N&3)j=6i9(+z1rGVP;iQl`BPEpy=pm5qDI| zT9DJ+!_4RG7u1P%nM$4iat+8ojVb2}1r_VfudDP7ri+A$w8GQ6XD?ouQJ|=CuF}-a zLf@)}ma$t*k`6WPHkM*+r#p5WOK54J7S@Yy$o^3x>*d@n%jg4|5p(%|N#zI!Rq|Rr zk%#4oDvXYX>k2*`5|HyeOcIoE% zx#n`Nt=f(6DekTohx4?Bp#`@L0esl3+gNlGyC{98Ecj=DMios_xT!UL=GU=%whAuj zB|Ht53pOiP2V&IjSSwi~CuW#^1Yqtb`F&5-N^Fo0a$oc1OYhD3M-Y1Jl#)6Q7q+(s z8}?;WFEkKfia{9=wgxrSPuDC)AJjqX zO{hqj)`I@;^X|v{#}@xr_?Gtz+VS!63k3fQ1j)h~+E?7X+3s$&G9dxNSm6SUb}c61 z;(t^dcYaN0-yw#Z)8)G0@4ws}Au0bkPOsk0%}bCQ?Ai`a0+ zZ6ol^Xb9t1YE7*sI(}l5212fqzG41SoKL1V1(D)(0;=uBl~AK>^@tti#+o9QRe-hm zwW3sn#i=&fV~?tze@&(A%}9QFf<`1xEN6yo)sVchLzar5l}yq&O3Q-(|5^acktT2Q zAe2rlop5U^W5=U8ILW@(FqBU8ah%BhUGe=EAIC&F*j7biFX(hjadv>VRDYau?0h6) z>X(~Y%)0O3_=5^3y+JCYH45{wZ?}##SVN(R4jqv6X`DW5>GFvmk-I$}IRC19AHSws zqIl!XaP$$Jz3smE;2mXLJ$o-UY39PyvQ&C+r^P1;Xsvo_y~{WJV1 zsz1Xqo_GR*ft}3%WuThCJ5Y+AgS3Qi;(N9UfW)CPgpv)~e1m~roDvLzCEt>>v#wNt zq`@#P*d)YobF@i{V(ny0D$~D0BKF0$5LF7P4`)N(uzofP)&;5@pZ{S8YljWy^D>Y1 z;-E5z&&HhA9Jrokgq>=-0tqR9{sn<#f7Oec7~({87rXWNuDK`pm>|60+@39ScxNQi!z7nu)a@CKa{AblnDes$4JzOx(#8hjUWk}DbNMMNI!O-LOomqQM z(W1K8*$UNi3xog4toYhoUX9A^avDoftiNEt$$_K_TC{-!T%A}&fGvSfz;$~|{2h}M zR@l}cRXa<%N^w3Sz+ALOxCTdbHYn!vsX~8<3ig4}7whmgAE@2tTHb0vAN)9ix{MB! zHgGSqr7JSBM&Y77&oAub{xA&RBMGISdqbAj&jN6(C!-6w1rQzTlAbdfg4))2_Js0mo(FGCm}zaBZETQ{!fA4PW0~R(c_xp(D;Jh7ko|u!@Rd5f!L~n>KdKcG1hR z$XpmHL|$1$rlM~w<{xT;*4BVdq)r#*!i1#B=$gO@5YXbxbS{UDI&|$W;B@n`u0u_~nSb6O&5v!>d3n=ma>$E&TVfYS2WIYQm}x89gC78r>O-XMXVx-Y%*M{dBlNC2?2`d*Dh>R6 zqJe*F=@sBK@<~?5(R*q+wXCdlkGXZYOZoE$7qQK&JGGLBE>domgaFe(FL_W*oUMx% zaDJZ4t+%dUgQRrIDnJF{iHWcD@5td?g{g%Uoc39J>OD0EZK?Gk>YBrRunLh=U}N`u z?`Dl*RjhB%VbYS`=x`4r+1ucd$J*NFU@Xt(7k~fvxeEyoH);Gm!C*+!7vwx8lvV?A z@O)UkOx#O}lt2&*#!9;@V;UqybEnya4`8OY6HpK;;}2W($b$B<@$FJ&S2LI+{KuyRUn7&1zzw$7 zO;P~3i~ZIiLvIAz!kYO4=fe=$gd*Ii7rQ+;$Ji-tWWa8;H&rPJFB0y(d(@y?D0MUN z0=2_Zso}tNfql}jZ(itSgUME}Fo5sJv0|6(ug_-!Di?%?t5>4?p+X@6(td?RYI#x|2vH!CR0Ev8(}#CqAg=tig<~$ zC~l`iWxl7EV&fJ9iqlXlQ|^JeC{D3D1dr>@g1<0QV|&N#w3?9$+a1Iw zCy*4ai}~toSf;+(DSGDqBq;RzqvGerN&f5bdEMt`tIquX;>(E1>rbR7%}P_7IjN2V z_emooqn!*Di|Y)$OJ2Pf@z&ka-u{pRP7g#$QnP13pymIm?k%I@YM%H{TmlIYg1ZhH zTm}YrcbDKKxCM82m*Bxeg1fs8GDt!o!JR;G4Yrr(_sDDWhg8YZMI&htqjZlT(b>s$We=xFHZZta~63QCLyesmJ=B zj--|2Z0ByF+c%M~du?LzI)jf>Soq<~{d0&ykq@7D_^_?<8S(N(l?R7_fPA@Tp(Yc2 zA0=?Zz4qNKz2SGE(awV*Sd1pqMavy|p!S!+bi z3d`f35xwXb&E+4>g{R6;lYla()>6zIDg#`ek~z3=6xJ>~TNKW>ZQep=eL1{>XnJJd zvBQTXtET4y%Su!kg%k*~C>jvqx$U33C`wphH3`2qweeAt&{>(|vYQWhiYaLs-xhu;iI75 zIFr`nc|OzMj%HuC*QP3K_P#l}3BG#KH>A^Rz7jl+#uD%v;=+}4){a1Xx$>HF*ir5$ zhm^K8_4g<-1g^d0$^39C7vQF>ItTmN@2%a{T7T&%`6U_BhXd zsnI#8oXnEPk9_JX^};p-14%`tk`Wxl(?O*R#GW1KVnqz$C~?W-!9J|c{22Dd67--S zs3jp}=|go}ZEgm44Z`bX=K=^v!bO4pylrASGoz9YwVEaY_JT+yXXf_+>iFY^!s6tK8G?^Mj(uy>RCDd>dq)M`iR4Gvq z5rswP=uKW<^FQew2zngmG;{)_qOLCCCdZXYfN-Jx{?GMu+Dyed z?+Y94YGa{yw`=_o3r|}ckHgIs56IZ|l91a{H!7^pu{mF9KG9T9g9L*)3ypqzD)}=t zI4fhOG1Rr8T(QqzXGU9?2E@8PXoWz?FcMY4NmXL|UJi#;5A@a>1SOr%`A8%}{be@M zO=s#ZS;D{^`utrzU9U#I6WzP23Q7V2wbYChU8t+24LK&>fE}f{Wc1jf>Th~25{s>| z_59OxPJM=ol9xw)&Wuy0u&r)H*n2DFvM_2D2!BuqZm0Gvd4DG85Y8 zXWu1u61HL{gU?KVdVDOz!8d|a2_VbTmiTGAw&&P$+T53ecYNSc;Etm?9z@WE#k?F| zOGWBkaNbqxc^ops-b2aw=J-BzF&M36DoyZD`MMv8BUVlgd`9DFxU$%%4W1YxmCLv(iEW(ZP0Cl3$Saq^*B zI|VS>n#V5Y6S7QfH=-kA)bmcO6m<&CO?{O}y{< zvsuS&lVGd6{1)|e2G(%qk(jzrBxzn|J>nJIo%)lC_NmlB-?@o4^v!imPO5c=B<$MN zc35o5xv-^UL;}g_-r{^LYg&4k_FR}n-^KRzBoNL1u(kLtxGu)LfYs6lvyON8gr$yG zrsaO|vIPd>G*hq-iiWd&Mr~zBTZB=Ek3v1VObD~QM_hO`L32g)5=P2`%xeUO37_kJ z;+=oCH(}a^+D`rUYV`&ot-5?WeHC`IB698X@N3B6dMe-d15h@z`zio!l0|cK+zg0e zVhNpN15h%f8zVdNBqPg73yU!wk zaY&oYT~|rJ#`1|Vt*uoto@2<;!EgxFHPKpcHHN=#(eIcrSWB8YW)f~5 z*CFFFJ)>plGv&}5X()*=Gw5rv;Je}K+NM;ox*nv{N5Zp@yup4bGda0empO5ZHu;U2#oxw>qeNcF6p|{#J%Sa!CMN~ z^}vXXj9O>h5i1X0!N?h5B;d6S2Fc2oA}%Us9Be8ovv|htn1O+L@WW#BMpCa7gbx;r zv#{w&`w}Oym6JRomyVw0+cA+=Uh-wlDWl$Yvdx$jWW)KMI=Z7LtvpVUYJ(HJ6mt5f zg5*5gc_B^rnwvG`M9dCq=~q*-!}LobL{tV#!BJ^Nx$xYU@b{jX!8yY7$JQ_=juF)^|7rAF}`{j>nqHg_MEzh?U0 zaqn4lz&29X9T~whWlSofoOuC(jhLyCZK^bwq_Oe>p^nsT5 zb-p4nsL!zF%%Ea?%+76ztHg3NTvRNCw^TEjfSO>5y7|)E(hq!cJe9**IsDE8(u$VS zn4ETkeUdCnj@1qg8WOcDPRW~>rC^X1nbbwPmmU3)r&`C?nUo{J6d0mi_s}lE&m3q? z6RgE3Ozs;Cu2j04-8n2OD~q@Sgs02)`q^*QlDJ~LewCL2cZAHX{XxxFOUgm&apSWK zWee#`Lm(0nxm#JmpUsZnx105p?d@5~$jD?$28tL80eMXTO#tu%=bL;`eFCn{U6v#; zm$+oL{GJoALL#Q$^c1$?>oae&bFdAncSn8IEwGRk)eA3GhEj~Q+J_*6gNl8@jF!HH zF8RZXfuqu@YjXAU+EtTEEu4)7zU2ojldG}PCCouoCGbUf$`!iIba-G;Vz@XcbrGE% zpT`+Jd&f-AkRF6jt-fp}uJKZ1a6AD(Cb4N*Nve{n>Wm-0q7bY#kM$8j%)@o)!ZkS& z?%IJ0CI-~jL@WGI0Uus@3@pj@m0nB4G!MVOvkV?`y&4;Y(wb^?2~quwYu6{nGrXjK zEu)aWM9>d+r^%SIp@5`mT=hEns#D(&WHF+8y zhx5c=&jW`tGw49J+_2I=D3XkUS%Z0hHj~h0QvJ2G7yZuai;NN?CprnDObrb+8ULD$ zha}Pn50S~`0)*`zUUES~)$iT8s7-!B7iV0tyl!>Az2aE};|1Hz!b3w+F&{(%gvGGm zdM#usbE1>kuC|h~b5Ir1vLvEmgx2Z>I+qyH9!n2xQimLkS#xBh3=a@w8jkAsF@sBQ z_7sXI~GO*!-eY%Tdpgl=r)JuZ8L0xgE9!44gnmEs|Bb(C{%NFH0Rf zoMvrUdZ>!#skL`QY4lRe8AyJHK5sG3sU7`ks1hG1rFQT!odmb{ z#Gy#`WZWKCi@lB1+PAVy-JZ)ShXF8}Z=%b5ZvW=M;3jZKlIjSg9Wz_Q(DDk>^imSGm0gZ_CIoO0AjALbrIIO}O&qT{m} z`=*{L_|ka-7kuw;uUuDq;H3cxYp*LnNLHw~YxspKKSY{ejZ!Ont#8hi|9aLr{d>c- zs#Zd7zUio_P;a1&7E=(B3n)UI7Suftu3SzHRq3dlL(f08m&>Sh21(*xt#4LZM6Po- z^5f!SWQO9(0mlH67n?!E(S8EFuX$p8Jl6eldyJ4{tBLXL-nF@NFmD)Rl-!STfs>LB z)N+uS2*5*$^?G^U+rWn&9OsMJx}O%_pAJz0N})igE*fyz;V3WQRqHR{@wK=1-YQW7 zp){~_IeH2gSY|n&*3(Q3cf^kkjwIvLO^Glm8~n8Dp8r}TXQSMf4db2$d`9o6yquob zCA~XHp=G8)PDpTT%Lcc~8l(a3OT9XXrEvNQtrIYh$W3I5lvNPu4+3kI#H#3fJKKGJ zhT@}uf*fX8>tWqTPqfq=vorjV(h3t9#u9wY3qO2Q^ZW5GRwCjB$B%%-$y!=6j9(*j zzs;u?-8tW<(asqH3ec<#4Grubhlan4G>wJ#=aPN=QB{O)XpPLxF^e%*HTb6b`zV!U zELlg3xd@K^7_}7WsB%*uvNfS#{5JK*9@(ZPu9C*y^v*Z0S{td25BS#$<9v+Z^+3LG z9<_{Hf1K1EjN%e8ba6aeWuc3{G(S7|eKw4byi%0VPSy4<7!5PC#&}CGn`UH*oL@z4 zSS_Y#>})@thjB(Y3_!bE;m`RCxSnCcv*r;#XE{A-Ced{$HmqEhwfeLwhBxD?XS>-n zZEm^s%%N=!cAd?(2z`Il9nZFOMhAg0S1w3M=-QT7R%)GnV*E%g1K*=RmZBmoS3vnj z`1A&{P+^D{R~UJ6a>v6Ghlb}4u^OsxV+7m`Yo|XgQLp-rUxYLFUBa3b{ z3pc_sDEsU}$Tq>4{*#vWE(N%#RRJF77pUhR?21mH?Es zXpNguHuw`-5Aiba_Jh}XFtMnTtPKFY)~J_BZ2(WF(7)DUi=rQpTwm+N;!-EfJa-k+ z+eqw>z&ZnDGr2fjDm-iM1BVR^uu_;{^Bo^*7?%^i5y@D)6e7$qa}hEejh-Yl3#maj zUqR)+8kP=`GtauFqW2qhJaYCjH~NeG0%&yor+){(!-;jHP*-8kx#cB?eELj@{8E=&3)Wc zo3(k=@Mw!sLD9unpj=7P*_<^|DtEd1Dm&IaP9C*Uyh_j70cLcH1a-$3sfN>#W>%MK zjCFNOK3w2)G4SZ}Pf5d5E2$pBYCHOPT;;s6_3IWF-sVfIrCtk2#wa|O)!@P4+2X2- zP807gw1mWOmo;)Ks3#& zJ#M?6E^>#-ci%r@__&tf+&k!KVrOSJb96Z38hH7Yj06u{{~;@j_yvXVc=G1Uya{2S zaX)+zKBDTIH!wi@_VAt0(P;`0)&MxJn=Y;E{xZTI75BhtNkT%xf-_U@HqhcD^eS7v z!dF;@F#NhJVDTQP2uG8+Db#*FFFK%l$M63dQ(|V%&H1+WGKw+4x8wd+aX-*q#x!G;KyDbR-aB;-bC7PrdVLrg7e)$Nnmwn}h+jIY4IVAE&k)r8~+pw26T zyY*4$^b1Ho8+m)!qAJDmTQ-s9Td6tv7d2aR6!jG);fhL#Vp@n8`UGk}b>&P)Q9WfO z5GN)kxOsTWb=8b4 z$y?f>STl$+)|0ix$VyFbNXqqAHs4CYVG7=^)p1Qr5e;8|WU$$r-Tt{(#Jr$ld6d(bcylKY9Vxpb3J9DLn071lIYT znKx%kS)Gn}-G9DqonJz;?yEdH6}SY30kLcrt0&yJuo_#h<>&*Ukm2_$HxlR=uzX_yp zfX;4cY=j@ie>bQ9=k^xBeC|vYNPHy|kOUON+fm)QS{xIHQJbf(HmSxxlmvXt2Qp7T znDlqgP{9F*j?-t3h%A+QvRYDjnzt-m7`yxAmN!!fL(O`>q?zS@aF70Yl_{>u#24e^ zY^KJA6GNu6?8M@n!sertQ`wvAG{(j6Oh0la-+yL!v8X`O%Q1tgD<*F6DR5#k-H?pE$O&WR-`_JV#{`rlGEG=0 z%%sElYg|#AY(>`Al#O@K^?*NAL;|piBoMjd9CmK~M6B`J=e@62b`L3v!i1Z4j;t+R ziJ6+z^}b7L7~$bPH$TwxMEV8g^pI$`a?~L%Wd?o>>%||0>d;Ba*V)J9dR+h10yxBxWN2&`sykqtFkVpUs~~Bu zW_UYObF~I=ZkZ4F&v@I3x3;XeKIE+jFf*l9(+TmhJ=ZDSUapr`L>ZSwdY-qR@0^)~ z@Q)eAL~1>%+C8EU^LQh2HG>GNl^!DzhUSZ4s*1f<&A);C<> zO~)XYe2##~Bj1W55SX&Jw+F!0NzI+V?Pbv8Prr9)jx$69G#$YffaA2#XiIS8OK!`$ zw6x^Aib%0K3d-Mns*76oIVveB3Df$b{Zz5>L@~@;RrxiT87-TA$i_2E#FjDUygNl@ zV`HN%km9~643ujZ&+ctRHebG?I$vIrj1CQ{`ewk;?u3>Giw>0mIU zgqnkV^D^hf3c5wR_+T`+$n<$1D2-&NAD7_oModoG*?A>33WE-763((BA>QWnsag>nZHDDTe7|{?1nW zuM`l)BizS?0!c1~%Y$v5?1!{-S%UX|-Rqf^!WTjbT$8wVF#r7ToB8W2e119Xf>4Cn ziG;^(AMT&VFP>ZO#eDX($8cMEEaU#FIi+eOrkY~uGa=7pYxBvtA<$ytrb7tr1D_+g z&fJ{3hK7WaQru*?niUVoOtC>wWYU(|X3&B%E%CJ0S6O}@K)2zAO<6W5jWVsY8RRf$ z?fU}@t_WVkz! z$&-u|&2?H-byxZ0Vpmc-6pEy7%nJ&fRt;6DM~#s9bSimGie~BzXH;;$fl0mZn@hFJ zn_Rnq*UO33B*O`0k&$e3YVtH<5SobJYP($vsgg_XzvnMMkzK_-X)w zR6Ousewsd0@ku836tFWu9TT16-Svptt)`BSD?mykZEn~Dw3h$SGWY9-k}?`UU6ze+ zKkCZ9ymTk`T!j}tFTiqM?Fo{r29poR00l=LE&zMwWWv!urKzcDJ@j|BLt7M&?VQt! z-_^xF-}5Me|FIc{isk;BNhw_ff-c}Y}Mr$iS(%e zM*O*h#a;`CE!Ho$$tK1cGo&=VR5LxNI}L)qtrd#-!x0Bb7_U4Kp_>wI#)6Y5m~xB_ z*Lrq}>E4sx`9(&BLX!Mwx2KZVg!4nUDdEh@iVAP`)@pAE+YMI^(}G>oJnp9x?qy~U zhd+$&3}B)7jwzomY?2 zo95`@VGEblrcX+=QxZtDSKIs(%0e&m4nOOCFunWRn0iSdjj;WE#$ba=@5-5ZA(v?s zJCmvy1gbGZOBDZO{Q~dfkCgGrUbJnxk1Hi#vLL&FV%$Cc_8<0_VlVoBVgISX^j`*q z`mo$}M_27|#rka&e4CLEGi)H=GDQ-D~+*t@rHSD(n&-47Sl zfOpk7!73;%4KGp1sI&RT^fF4)S$8w(ZaCl$6+lmUj$SQvc)6ZDkgQw}$-U9FwXykB zQSq0IDxmjt=d7?1|JO~pCn`#{*Mu!S*$RL|5%PaH%jW`)&B7;9D#w4mcasMqW8fbH zPx#8$rs?oHM;R$90jSST|57g7Vcd6Wubhup#Q$8wR&2kki@WSUg<`E z2r@Lm1Yo4Ue%UPVH^JMX1*h=-iR6l!mBL0MR@ul^I~f`pG6sY(Vxy2{nb3>$fRj5C z-Rkk-Y;Yoh|Q6P~&0n z_*kYB40fe3H+?HUdWos0xT}z_+#Lqa!*nw?`8d{Rma(R{+iU!B@ZwZ6e?F^9dv;h_ zYLZA4Xf8ltO@U69LVJ2dIYKz`+S=P&e|`SaaxrHBKRP3+Ch6$N24KN9vuvx@re!E{ z)hmGD3wgku^TTz* z1|@%Vc%7BfW_4XuIz~BsT8kK9~MrfU%Pv7<>2i6KCo znx7w`xa+0puoy4roGsMYcn68&%Uu7%iELg-44W90E1@lVX&{$iL6w`#qP#kwY}oXx z-9KwyiS@9*cCwMhEi>yHtT6fu%xiuU7h|qaVpWSH*5_A_WrXh37Y&B5@|{Q}!X{<3 zTMsH=REf7g_Sa z!UaVkQrgM~m%ZS(7}WMg8lk+L!T-ejmDYwoM`*k^MB%Ny&}5dMs=v$+L{GbhU0p9V zpp~VDLni?LBNhnzI`u#2J1*txEN@vlgXrt)w``>w%mbR?sVnUT1wk=vT^~AquIijO zq@QH0M7shn&d=efKbzsq{rsum=XamF!3<^2^yHmR({Z3e>nqB`3QlCMPVFuu;CgYB zyK<8&chY3&MQ7@Ceo@JeHt({fK$0|YQkwmxs?7$flWt9&S2#`|y{~YQhBVx=Y~-NW zK$!_&(Na!#Ur1vKf1UeNn3XM(HHWoIrkX`}F%);r7$YSXCAwcMw^h`RhAbmw3*rTSeG(W7PQK9BXH9~uLbM-!KGvkoS1IdpIAtQOq}}VaeC7lRijde z8;N%tSd3}Ai&zb+XfYFNts>wJnnsRxJztcTxlH6aC=X^;Qp>-R zLVX1AmHIKz!ALh<)94k*m{iTre0`xH(R{3!Sefw z#FNfyAT1?@JZg~)cHJh$V}2mOC5a&22Pdj`KjDx7DqL`BX{nrTf6So9tF5=daEZqP z#p!d&oQ&%9QZj8JBs+_;K6SZFA|~(KoN87$5BXf0e9AII`*s_bi9@}+Rpgkd{f zkM`QWzdYm~dhtGBz0v2Ss5uttetbVZ%;p{rZ&K9_=X!thAt5)W^}!HFpmwRNouj+? zVGeU*My;8Ee{}ytns{g@!%pwzj!s zSlP&qhTHr1HScEQ&M%uZw6q3)G}TdheCE zd3aDQ5m%p>K7bnZ?K>Nr^`~o& zZ|;BTzNI*p4mpZ_EnKAR z;nk0uGK+3EJ_EUS9X+`Tcg!C5*QQ-JYa-@KW7Ns?YB@M}o6BT(K(pc@k(Fiex6LB1 zN+SN{5kqa5@mQA>d5Exn!)8(^y0>0qxgyu7hkw>Y&L$E9qUyI8T9&RjSqE$7S&3y4<~a-j@aB)-+$sCvZGe~W3toouJ1Kw0msF3kzK zUBaHqL;H+#;yM!jalwAj#KcF|w3hCht!PMjPKvS56r<#8KwP8R1VNJQBhF905sJ%0OPR$eh*%~enT?}-XRq}%Y->CSY{}B zAJHT{Ei?xl8;pPfur+pR+WCfc;YLrJ}R%D_^KBq;Q3j5?SAa>`8e z682{^BZ4*$YlID!GM_}*T_HNPhs!JDDM5bC-sa+_4}<4;f}TFH+<|2>glPe9@gf?zO*WlvPJ&Qm^XLSfL#p7q8?vwBs^(U z`k$uc1JFAHxS5n@AWwPNb?fl7l`8zJE+N~oEAz>#05JI+84ABtJ6;7PrROJ7&8K#3 z?0bY`akBA{p-O{ha<1jkyE|q4xUv%~rH(Zj}JXU@cNH zqxoG%=c92X=VdS{tG#1_rX_p%nd0rmfL!Ibi2liFf#;Y8pHRI`8oYv`hYh+CB1aX4 z`4SaH`L0B{xjkky+sNN!yC)f(Ud!;L=ZqX(>F&T`?<}$b2)4U3kw;gK<=VMvO{T$- z5ws}DV8hokz(Jun33F?})}8cd<9R>sAh!K$E%na6r?zJ*&xo+QEry+Z0U-aJ}`QNkdc=xAVi%aYb`K*752$GJOXIhNvzsE{5Y! zUu`^p@uJTE{*RgY-oU^>@rVBYeqX>@mR40&&A+-;AEMF_`d@e2gr4n)-y(dHjq1;` z?$PCCc{7+7vXuM{jpnu98F6fVB(JBgH6`|P7rM`|#Jpe^s;JvOjb%7e_*xVP@53r{ zMfMSnx)3AbcDPg6QKNEyE7#`Y2+2p zUu0S^1O6N*^|Jd%#?Q6LPd~uz7c%RG+M<%Gws1noqhD#0#wUES1ZY0&jHB7ab;v~O ziYMq63NZT#o?(*KOxb-zeT#$=XAv$=oATAfB7PF3w8Zv7cUw*5b^%66M)rh{kog_3 zDc}t$Dl0#h*RB8NDXR9$GTo)vd8FyH7C0HHMFlY1b$#4&W7d*mCD;D27!aWmmKMA#pUjL`z< z;DXYcZn|IqdI}JXUku?0f65SS{`yiaY)X?%lfc2tr$+NqDwEuf30>)VjV)U-E+-Oq z=4R(kSDs9sPx$vrPrO&+4t}urhCZ+rmxZ0jU_93y*Ogm`kDGL4KgGsY>fy)pyKn+O zYldU|fNe7$-7B!8oFz5Rv)SX;p7&Y~Zm>Tp*(LwxyK>#X6&`F1xN?5|^ICY)#t7=i z=Zk}&Li#Q6vmzZq1vsJF_YNbm5pp{AIQFgk-k~<3iNDtTV;_1}0a`MO4W@B^ShOsP zA9*Zoj0XGxTMvW}n>%d0b+4(WC&Et~I&rB>w7ZaC;R=ZvC2kdQkfe+=z>%BN)6yaG z@@o&W4mwM`p6yB6feTa#_%kbG=^Wo2PY&DNlbk35I+=sUw&`;7Ed=nj+{-L+1mRRq zbhPWs=5C?N6CgAfgqvl3tp51M9P>%Gx&?sf0ijmTwEPvpm7B?TaURsbqkQ}8TADra zudgp&5M~EzW;xjfU-_QY1A3hgU(?cB!R$EHZ=c~GmyEMMTg>prvK5NWa0(GaRfAAKde!p@57dN$LQ z27xVz8-RF=i5k+skBRn8zPJ1a@TX|l5%0ufQ9-78FDL)jc*W*MhB=Vk|)Wwkx6xb!p$*LtSKe>HN?Pg;b3y)}T ziD0cmTkqf=R?`#WhciveY-xjGRBBxdx@AN4w_npj$%CI;RfY4po z!LNl^V^xku91@u(jwVJ4h`=l%@V*Sf2|^=TGzUr3qvz^w@)yBZ(|WQ?uMY+fx>BRi zWD?jaRG%z4Bj=%28#0I^6+hB%NPcS}EK&U(^d6xMnsJ~U5IWIP2Z+&!du&rX3)r5R zI5PMN&}heZ;uCEzSeaxngMY3esbqwe2>WLog(jr^B*^LiQ61EX(u80e2rk>baz`nm zmU#j_zmoEDjt#mDOT?l|3p5ntq^C41W3Z!U)Zv1w@~!k-Td{OEGc~92h-r_@cTL#9 zRK^~;A4&T%7)($UW z6}9nH462Q^gCZdUW+(iK0%k7^NUhomZB?Zc31@bKx_}SULMK-Bw>~}NIwOQRARYa!9^M(#t~eN4{W`iyuieU z-ykR4sN?Ozm51}sT;-s4^1aFt4s zZ6S9WP2j2020zi3h(-v+u>Tz4*W&lBA9eOTRhmayhx1v`JX;CH7feevoAR_98t*oF z;XAUDSAt#SRg?anCW^W7^(MCtvPkOw=?OVgJ*^d)$&8}1<&X&hF%zTNfLcC~ztTMAMUr5_*~)4miqfR#f_yOU#-2H-i>}= zn)_O7fF+lc*;Ug#Bdd6*W*>SnJHkd+*U;_Y0_lcaxqcuT1qEO)@cXmsecF#R_4Uf= zavnd0xf}!N3;&s2U=HIen``Za$mABcao5ZwaR$1#g9kBVUriM19XjglR3Oa5A7%`5 zbh*7=ZzP^jVGFWKa8 zmXMU))Qocu25IvR`jg;rqvU+aI{M?j&dHN=UBJy&w);_gu6~2ceQHAPtxEEJCEWJs z9<_LBu7AA6==2N&5Zc}uh>XB2ic*-d%RGUbl>9hu(Zd#Sz$q9$!;+&b8g_V*=I~*s z74P&%@8aI02y-{ihm#ZM>J|B#Nva2u>Ws!`Fs8$*Yv5ph@;!& z$`c`v{evQ0W5)iJ7sWeP5Fd%b~>le(~nn!iJcW^m0h-wrV?VF)+&Z*m(ya6nnM=Kyg1x8 zFnDfa^rYu32yQ=ZVubN%V*RB{9vmFHs+^>lCcK4k{0>T>H@gTULK#P3u>b}GDBS8F z1I>`pLU{?l>1rm49h_Q56+GyjyZ~Io%6jvXdnoOTSM5R|>Te=lEh-{XDKI+sze^<~ zi{xWvs~z@R&F#%6= zIEB7`6Wx3jk9-WTGN7C@+RL0)L5JBL4*~A?oR*@|WlJ=J#k!EgZa@=9S}U z-0)JL+-~J;j74>{OC%?i%Cyql{NLy?%u3KB`%~(Xng`Khw_wKin)UD{o|{;v=}OsR z#k)NF&u3z#5qUMaZ>+Z@nMK%e5V8lIuV06M;)UPg6P51{W-7zK4`uwHmzW(k z<6|5CDu)8fOVY@&xJ%Z@6uCl9d=i%A@f`lh zHJ6_xlqj5tf}(lB&b>52;_nwDBejQrx|^h8N3RLNT&}Sy#gYRfqR;Gc@45Y<^B^T9 zEpBRZ|FNw5G{^p%a1(u=SicbvYH&UPDewR(()=Mon)P?BS$_|!-zKI>&v{5AmmfEj zDEhxaynYbg+_!}P0Eatq49{rI^siUmTRSoCfxB0)D$OLd4F9)X9lp}_CWy2#xYka< zh;W7c^QNt>Gb};FZ7W@rU}89f8Np~5Vzqx(!w%mUJ4RT_oATds+K9fZ(LwiQq=1mZ zQ$L@cUZ22wfKc;5E9@JJle!Mu@d{w`|DJ^Sd!xl7r=YT>#>O`WhE1Pj3JYvuO8)b> z%PA>TtO}0zRe?FHE)9$bYpB6c34H%=Z!70TxYHaWc->g-5gk+aM6Qk!SY2MHa=rZ5 zQbuv4robq=$Iq%PCJ=@)v2{m2kEkh&{M!vF8#ksn0>%tY)4D`)8;54tGze1QiQO{g z{|y!x?BDRFqA(5_cF#2zMPg8wq@6k&`){WVFiJXk7Amf)^1^Ya`eCkSX9`Mh#Y+AA z_1EowM|2>sSh{@tnzBK3#X1{e;Xhh#-wybb{tCQFom zha-n@t}jn*Kx^V+m2iI?HP*s~I%B$ga{RcL$(p)Xks|Q#ZYGh25pL2P=O%JQ_Hazv z^|ZJ&p2o%(=17kN5#I@`_b)5|wiBLY6;k^9SqV}2m-u$Q&*2(PBqEdoQ#PF??CeK!HUYp{|U6$HO zD=QhXS|pcUU2`uJQK*c4`)Bnx=RYF-O5KxU`te#+()s15BeCGZ9<|ijC}OO?J%~n` z2u6Qv@><|gEQ%u1kn24^IkuPp$E?hkoPvLcx^z;q7oFIeTQqYaFwXW{1zH~X8<%el zH1j`W|8*XZAe}`Hu`a+T>}jpUJNZ~(BmF{6tpW7+$stn!(JQ0Z*}+B(a*i9BliRS< zG*OetpwB+1e>>COyShr**f`7gRqp@pLjUro%2JTu|Fr6rrS`u= z0fgtm{`o#qYF2pvbS^LeTK^j1{|_Gcf9V#%G%{3FRN$IIZWG^}Cs4cbe_LGMb!g!H hJ7xdBT0HoO6%c|R_Hpo65ghO%C#5V|C2kV@e*xaUQI!Ay diff --git a/docs/images/chapters/curvefitting/134baa1043d0849f31a1943d6d5bc607.svg b/docs/images/chapters/curvefitting/134baa1043d0849f31a1943d6d5bc607.svg deleted file mode 100644 index 89d9ceb4..00000000 --- a/docs/images/chapters/curvefitting/134baa1043d0849f31a1943d6d5bc607.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/curveintersection/b155682162a5b6da6d40c7b531164a7e.png b/docs/images/chapters/curveintersection/b155682162a5b6da6d40c7b531164a7e.png index bb895904d95fe01a571c5199fb4aa67a1d5f0052..ce57728de9df50bf096614dc134bd06481501ffb 100644 GIT binary patch literal 32404 zcmZU*1wd74w?B-cq6mT#iV^}6f`oLVqyp02DP4zd6afJN0TGdsk~oBbbSX%elypgh zboYO4XYTvn`+YOx45M()-uro;wSKiuki4uUF7|C~3=9n1r%%KbF)%QBF)+>%T)qUq zLsZeM5C6GjC@m?5afbZ;sX8MH1LH2nQ?bX&uJM1y+_aToMhO*P{p0#q(n|OP5uYlo*RtTC}@cjTJ^O zlIZ(teAqgwzp8fOoylZ7&f5I+R6SNbZ_<_!W{u$5fz@qs1;Xr4nVFaUzRO38qQ!=z zcGCpA%x#zYvP@2iMP7e=t4;OFj@ViS@2mO)+3GJ}=~kY4^d#boT&5J$$wWm^ze2uz zg}?eJm=ykld@UJ4C5il&$mJ=8wC3x7|5(J8@ILC_ZzUH;zST}iDMQvNF5NkS#L{Uv z`;WSd`x<Hg8#}w(dOe=w%5d?hpdLy6yZcJF*q^s^=BsIFNu)*$=z4^vOUImFkEuPr zSgot)e&RCj7U;G$r=YI>L|y$>eIOpo9=YC04QAEp;qq*7M{vz}xdYuz7TtT))P7hL zo^SDl_ITa?gLPA2amX*mrEs8z2yCucM#UC0s9vp9V4b}PxD<~*@EB5g7K3d^^&!Q*W`}x^@diu$o0X;f)c7j~hHkS#n zrpfPqE$!_y+l#%(OLPexy>XcDyiP&U)txHYZILgybkKT~`6n^|jX`|??m)G!iFL8n zP-L!FDQbQ4F~;HETDWfh{OQq}>3nB`=lXXn&!d&nuNM8mPorp~LyAl4m+R{4_7_sa z*Ecq%USN?=PMn=`iw58%!&cm0n@H+R;Fo>@J9u(ZA#gl|EN<@Q)?6$UMw#!Cn9%Q_(^%Vv#h2$`A(iNy>7bFc|{6t?;Pvx&^Na^ zgK4DSfAKs(NraO>laRQiRbushVj{_D`B%ipkF9ShMfc%)%g;`aT3TDPcaSB|uRYaF zUsmZ;m6F1Sm7<}ddinbG<-^?-Y4{Bc;j?3QOss2Z@B*%GZc}S(ddxM9^RhYz8QK1F zf!D}hzj<>{Kp>ufwao9$8ypN;nGfG~S4K#Ly*;BR7~xbFQ;Ub;i)6~AcE3B&%;>dW z>`7ATEY}s^E&-43YZ-%tq)hkyF*2b$pbUgPyJ8_AO zieeNzE1yp^Y6vFE(JqgiPxJ`mvL3dZ@Y;(=NC-`#mq`?e&CH|^BjcBlm&aeLJrlxU zsoAGW^4cA8T1zeTU{>fy?$64~YOy*=$m^r^NJ)2%mj{MssZ9s0LAl-3`3%{F-``$f z&JM@&IWYvMygO3rr<(uW5Tc%|8b%@9*%*2!C_EfXzcZe9)N5~SfYgv4szAB%)Hyt( zVohGpeO7aG^K=K=moHzw4ho{=;wow1ccc_M$+XIoG$HreRv>qqiD1KP;2yQZ92y$h zUg}40f3K#(yo57}&aa_jX1)eH5_X3RxzDk^(8;#aZEhQx7-pTzSFfVCkDHsHxbLlo z%4Sf*+8~E&Wu%mVj4brKzW!95{|6abUC#~ik6hNRZ%EiNcu+H!i=KB>ta@q7spOTq z?<`?=txVJkPxzb(Ch$3(*N57XE_$Ef#*M+R<^&HPK7@l6y}X#67wr8xj?3C`wl#YH z;NYS6u`9giGZcyq>l!J2LR3_I`~%TnukcgI&EB;(vT8Ss&0XgBv0GSFG*sclcyMq) zlsLV%)^)NsA@}6TYle-d9*HBm;snFu^-wCB_cx|Bs$2_(lQShwad`r(!bL8dMkJEr z85C;{6`EXqUKFdW#c&73Z8z#n>icUvmxPe#^@rnIT(;xwuw~OEB`;5L)YR1AR(yl{ zKINX)ynzpUug;3gVbndoT*INjh~;ElVOb%}gv`iu!Bw;OqiEnY`!=T1%v-tmBCf=X zZcu6lsIKC8G9)E>QXYpb8;R0I|_CK8{2)V@W z?wG3|Uin^dB+8*n7HztF_Sht>diW!!KeU8* zHSgcQmvD7;T^xW_t~2AfK}6J$t%AlFtoGpU?(Psp_uxcGc8Dqb>z)ol55)O-%&k?>WrYuUYg`{sX~q{_ftz@I5)& z>#gpLQ(eC5L?}L0JfLed{MG#1Y&!bn_*kCMKR!NQGod$I zg%R2-+OzAQ?V>E^juQ9=1?kDNXD00*9`^IGaB;mao$!u>?PtZMl`bxIMO@*Qu(tp- z7NZ}puYa439>%_U^i3Zc8hf&S|eBW8ikOIrsi$hD7d1_*RIjh(gq9;s&Bhj4^#f)EgEY7GgL&vX+a3j zh|@+gwVPG{8-}Ns7aZq@a6kNT{q=2a=0{qGA`u3WJiD)i^H{H0$yLIP(gG$5)Yo;PZ$K2Qy< z_W9E%-|=dXB7^!%aMV;+EU}$BK5|}VVPSdR{vp|!UgR=2m(x-oo6l8Bv1f}omoP8+ zO;uohltGGjyz|t@5>qHjP#P~@yvXtN+3b7VJ7(_3JQ${0rO($sJ&q3vG8o9y0f?2> zD9D$Gs*OI_pAJW$MOH?lNCw&~P&s^ zf8}WO*G$x&?&LqvE_)kiQ;}7-R(aT!np~h(}xO-tVo3u9dflC014QX&k_@*=H9t~|Gp=H#HyWNTC(@;q;SaizB=9- z>PP3}-r>T@Em{3+$D~_T);Uc4_w{i@MwkTNCMI5cREm42m5_(#t24h2w-wYvx|Ws} z%KP_&BB&*A+`g@uF;IcpDMoYS5&iwSI+b!Axv&=F^PlWBHPf3o4%YTi4i3vSpAQwa zwcD%QwiX8RDWE;&$5_74sVRXqpndjHId#fXq%1^9L-=jnycyGZFY~6GFdlYtI%zs%zj!Owi|`l_VsIlsk%n^ zDMD6fyMntJo>|VfM6`R$<4@w-hVq?n{k|Qh-99~#<5^sfpYj-9HvZeUM}>ujJQQTl zX7X!QUE2>97J7^X#i6YMtTup!NcY$EVkD_~4@VtfAPK+o133KIKd8iXsahb?)^QNS@y z72g8zQw-z@#f$Oo`QyiwkDblg+uK96bP%Eq2qmCHv2hW{efRCqpES05aEDHI##098 z^z_tm;b-#b!Q$fLzr6q>CeTpl084*wbmBd09ulz2(QdY;+&-6zG5kJo z>jFD{N&qUNYd+IUEmHWlMA5N3Eh&$CZaw=f6RVgP3WZ$}Py&G3gEqZWMRR_*DS?BT zsH&6Q5zd+kleYKQNl2aobh8+h(23|Dc$E<$r{|B~dAGj*m3-0&5ibTTn!si=^}kJ{ ze(I3|VO@PaU^Y{zJ5W;9TvriT3dM_aWCGa1-na(~p1?YW$KGo0`Ym+76hLKwxmC}# z;NQkayDKV{1?WG61%7>f%8hSGlv1~r{c!XM62$`0S198j3$b1o@v=FpEr*^7ltBPE z)SuDvsGOV!XC-FB;zaZ^3sw7@INK|F}-LsSm99(qaqpu39tNS$s(~ z?{R1y(&=q!@w@!rT_Lpd4EYlUN~8;;)E~U>iEcyM$$KWP!T+|C z%2C=6#3AW2*B#-a0rqm(j#strrO;zT-J=o`N+cI_S3VF>Q(*ICVrG^N$iMOz5vi1o z3GotT$;!!2)OZb**;AA8J71Xm9A`tt#1sl|Q0Ta*hyexqd(!DP-B(Sf|MZQ)F9zRc ziN;U&r|~{=+jd6MNHbXf?IO^zOI!B}y-(a_Xq9q3{QAExG zGQ>9h*+jsh_2K3FD;j~&n?W-!?sQvBk8#}_a|sF#u2?PL<>iH|+yJsl097`3xr|N2 z>seVGdVb{!rI?d>S=GKTzKG!JgwD6w#P}r~b`of&fN68G%nACE|;lqu^i)>}9Of7>!<7%_w#JJo)>M5d#pXQ8vJs*quzce zNm6AvBs@F=pd<2NS1Z?a9df|M^P-Q5Df=C@85&&{`dO`c`=+Tbyo1Rt(wBf(msD?d)|JlSk(Rs z4LUy*>Ssg9&8==pigKN91_8gAm>8|u8K6?CJ}9H zY>a3!HwXv>pT7<-YF@Gwu?Age(25IsmPi>d$W~C%Zt^xXG?*1@meQlzpR*U4c0KRQ zRDdhYCJU2>vMllpc}nr$rP0aijp!@M45bq|eKw`3y^Hh=42t~kEENg;#Y2ce$&iC* z-FK}iUFEBr+1cqy|21h23^wg&m#1*}LN8Tr&s>I|U27>AS%F$z=^0UE(nbt83_zU{ zKVhl;?0Rvz!+h&Hr^0+yRn=&Un)>Cb;WRqQ%R6o|5lZUnna=dlQcs^w0f#j_-rrbo zw3%>I&4ycN;(B}O+o1v8Cx_Q>-c;lBw1%@kFeaeDI9@~|WkR;Os>m9~!XtIAB=h_J$lJSS zIQd=C-}68XNYWzk_e0|yPK6jmJLK^zjUl!yK%NV2I6VpqG(1pL7Vf>Ms3<6*cwK^f zq_;yQR_p|V;S?)14sXR3hg+>1o}ZuJhNHpnFoz=@ z!)#Lb>dJ)+7c$Ahnvlji#Bn=Rx@&_#qzns}CIqh;i&iAFVI?F!(hCJ`wYj51g`d>w ze|Kc$2cwW5#JS`rJp&ueqf)^58S;i9O@zFUA3vU%naL!(*7lx>nvE?I}cmgV` zMOw3cBnUdc@^qx3xquXx-O&9LSyL_G;eXHQGvv9(jM!eXC>mPd-KAw^l`qC3_&&ixeJ&<)VYlm}2*nM$Q^cWj_ohmQyVura#>4sJ zG~l>B>mq(x!BR+9GshV@6(M=h-a;A_)BT>&Mk%qZzS#xj2?d73qVF|SZa&V&?61Xc zThBp~fZ{#l6z;dEv9z!NEbzVzt^uSu5xU zP{_7(9{h6x{bbu3M9@(0y7m8m?7+aZcz$Ii6a+d01B02lIX>n|ak#)cvV|g8D zlarH+ZMud){W#kK-S5WVZJY~G=+&!NWipC0EAdFSELlP~EIj;}`S$PXu1Ei^M-Rni_4V)i z`}?csItiaT^=GTx3YEBVg_ zdA#|73V0wsIl1n9(l=Ta{}&BAIg=^)Ap0YtqLW7Cb-?$974a9(=4LM^UkV?>A^Zf=w$rY@ zVbRbBVILvrfi$$VpJnqX`zYfAgVFsa@GLk;Yxw?dkrCmBel@wyLX$SoCDY`Rgq(V6 z4x4lTvwVS3IoaIiu1{yr9Rg#Q}I=SXo(FKw29Yt*;JH)el_g z(?MSN0a7e?%Df_sN=Kvv(SYsWw71J-lqCI!ly7I^;)%~f&4Y)HtEK}dQ>=b^KDXYg z2u&7%j~fY?gGP`S)M!~RR-pIoiGpzosS&=H?bJ?2~x)CiC04 zIB9QhA&}wd)pFYMyUjpBYJyGuK1O_WNHlY7scQ zacw58P?)ZGJU=d0&9~eACB6?uyN;iRx0STHTtNo(W+>I5=X_Cr0Ags4s%5!$%D^&I z@cqNX-^8!3{`Ez0U?;64v zC59KZE~Zje6>C;ibL_jiO{Ml4_PKF>*n8547tVkJ*au8~x-Ux!K+|W{6`P^Li%~6N zhK4EZbveawo9p0_Y#goCZh>F&_Wk=t5N5y~;IJ9}u)c2h($7yiB$jb+U4hUa)@XBU z3nXqTkUxNQUne9Kb9CeY;Wr)n;`W~*cm^LcU1Rc|s_LPAj!v~^9I?al29_Ca+nEs( z;_hQz(r;~};o$hHYm{jTLV$LaYXsOeP+A_w%7EYshs4|48%_<*B~kN-4xCE5%&O?< zmoLwQ^8>ZJL7x?$lvDuR3g;d_b0_tLBC3ir}Rzgw2QM;&I_No^V9O>*D1dKuY70Sz=29Sx0te$8QU>9{*u zW@Kj(1Vhks0zv1u2MYtXR{2Kb9pu+yKl&zgLujEjjYm85!FAC1LA;vXoA8;3r#?$@ z1`oyW#fVL1sBX18APc=IQCU$H0QsO5fFOvN6tLC$uC*CCT??DaQr+uY=5*1zhEF^g z$#va}*U|kdcie4SrI+ch+at<7-JMhpLjT+x1!@JtUkv(oj$KP+&v7ocT<8U z&WPPxi-&n*Y^V1AczO#N(bY`^zJDxH=QS&;Zf{MlFrz&*(gPYKqI9N~1c!vga@&$a z8Bc|U?!+!ogv|-INFs19F#CjmmRM_nFVK)89)eJl1ie}+5O0C?vwH5^G3!<}mspQ{ zfNmC?!U_xu6x0`&YFo9%>j%K`%6C?;Yq5xuUW$WCr>*9z+vI;9nx*p7f)u+>t10F2 z**e<6PpMf+FLXOjE@%3sd}e@+dS29N6n{xFV(jsAUCN`bkMh1;l-^%2Z#GD~yNF#J zt!XY8vNCZ;85$byZ}-Ys_Wc6o8tkll_wGSuTh766<8@uXo8)uK%kTVWvhj}X3_{<2 z{|EwRKQP2;Nd*p66S>D5fJfxik`wZORLhQym&->Jg|dL}94^GIvBfR#6fQ?S$z z2t2VyJ@twJr+NA6`UWj$#-kP1y!(|ur6g=9Gd6=S?{3hteS4JA_+p`2+j2GQaYp*H z*A&(w$7SmJvx7A3l0}vdO0!Jl;6 z^eztpu(`XqYm5nVQPup4Qd~zP-2Ia$ zPwqZ=kRu~g9JB^nw$aW~e_C4FPjC%@SPFu7ixJO7DBF=F37}fzi~c!Ka`l0SfwcPW zyB}7J+a#)CBBsKM1w09ON?Y(l8xR6OfM21{nKw3e>ouijXhDpOA)px`diHrt%w?LI z3ac)|g2~zRQ=jq9UlmT4h_s;+5>G7|1{N{}i1_EoJ=Y&2$U9XcR5_$+h|l>?b0CFx zGsx(z3*D(C4-GHCZYdifwiQvn_fs;k#ZalZxm?+Jx-`eL5IgrPHWx|`ZHHsiW2r=B zP~*O8MIyUI!(J(C+|AXObVo$T^1g>ct>Y!XbAI3L#xSK-Cs4$IYwdWX<{!WNIsncn zVrBH5BIThwgR zR{;SPGM+nQy5PnteU_OLK0Cmh@Z7>3A0JQq%mrS$7LRw;eT-bKB$J9JRe7m!Q-QB?>Wr!GqLU8>r!nkx3G&mMR}61C8KxJ3|=B5rB4B2hTzMqx7rw z3*UvtKRj@zI$k~-hFuZ%d=_aLqOU2&s9tVt{(Q#G`DhtmnCck!UQ)}H_6vWT>)-dw z21ltBghx*^ym#s_*K4t&mO)wn7Y2U*Wlstw^-W#Z7hqZ;R&xOeO##=)mJ0A_)GU7G zkN~BlVP+1aOY)3_)h!lPjFw1EhGYZ~X@_H1g1L5*Z{4X1>bZP5s$`Pk5)Vx~AB9oa zz8tce`Xpn~lx&Gw%LgUwK8(V|t z=cNF)x|d$2?wjlwAh}tttX#WRZHH^4H=`{`hs7-XffTCZ$9Y(rf0YHxo%`EC1az(k zvoQzDdS|UVE@KU*NnVYg9{VYeh$%B^=DvsjeIKLe9Sfd&gl+AK>dIT?XmA6*1>lZ- z8EM1A`Z7{%HTbi*y`XMKnI?uOWG1o=T9=;JPV_M46$Q)3XDcmPcn7Ndm2!H9U5nBj zTiidnD;_fWz2Wd3{)~hBQ%Jy9HWc%Kl08nzP23{z-cszy*_wtzL~n}tRp>BJ9zQ-O zky!UwG5Vg??cv zbNCA}KM(3*I9wF;Be{Xs`WlKPuh+SM7M0%=ey&@m`R6WQnGLGkPS9p7YTAy+w=L;d z=P9}$RgcTPzMNnCo?jCDC}!;^r8Z*=U@{fwo0pCt1`wE2mm!-YCw%e_c7!6iRG!^* zD0d-9;s`1LyTAXMB+*$MN3Qntht^s;?fPQWQ|qV9Z^>L2w7OziY8au6g(h9Sx3)C4!_k_M2s`bpXFk%VfcRBL$b^$W@60)hUxS3?Zj{) zsi{|OCKnlwPxvIFqDjdqWVo(Z(${kE&2NV?*2O%KeEfv@&*=1F1WFGH+A<{8wuRl3sjqmLI|PwVb=PNKJd@6H+g;fwVzRi(ZY=+0Bh zyqB*s^P{m5aA8DGz5@#NL`n)FO!>bG)4;8t1*wmQmG!Nh(194aRCX40#|P?Pe0d$_ zzSvF(71w8R|Kr&Z5I7_T5pdtQ<8j09d%$;nYOdXT1avm{Xy*j}5|Nl8r&3Jas<cj8KX4 zx(daF_^TGzUw3LY@<64eq%?PR(E(S2?)<+*gy`WK*9|HN2=PI{$q3YDpa;qS+tfDBvR?8MJ;;ep|XIsp0?hn@G$5Ee|WuuDm4S9mmkNa1fOjTB5ow; z3}k~~6aiU)Fh%6sxAN)E+z>}$?-CHz)+RkWo)?}?{&A(kWvvqqTO&m6-gNw8{e{N6 za)nY@n4(%&&efHNFX38(tR#G~v9W@G(8w;1k2 zWT7qAY0;OevFK1&4;W6UlA+&QkT72iBjDKGN`CfQSU92Lm1u;&uRw&>-i)^8)VVT& z_<1O0w6NN+b#)4Ut28j{RQ!mut!aUsP^4p8e`X*2(LD|N+!}aDmwS@Aa=0C#OhmJXHkDyDqx5adI$-#kP8Kag` zAOzlpg!F=fg8IXXDHp(Di}pZ4+fdV~vwLYe)_8R8s4#ILhGpR*>=MsD#t=loGp z2!zDK2LU%WFq=Mo`69n!wb-3JiaNXy8G@fJkQJOS_J7LVN|r`~c; z`cG(E5YqOSV9{gUO=t!cp|tP+1H2a)R=%fB(UW{fN&E0V=5!NovyI(1QKe-aZUtQT$mz0p1l9=Tvt!vaLSf3NC(0$Vxe zJ_-WG0IHuY+LeHAuBfd118|%Kc;pzYaUDoZ5-?PG!&6m!)vnpXQ(}nI~)##w#(!5Q=Ttgyg<^#?VwP@t+YNbDzF*jOLYAmq>j&T5}~(M__8WM z={trs2Eh_cES9SE^GGOWe<~1~c{Wi62I9>_8U%iWqVHORxzO3M15i;jc=Wj=x4~as ztKLf_NXf1NKd&usQ!PUKcZC2_VpHjV7SZKst%%Q$FyZ)!80?XoIr!BLk@CB=_WaBa zxyEH!z|Uj)Ui`Aa3oTFsm<2xlZ!`QIdFsG&P-v(LB%h$y#X}CE$aUlQZR;hvcxiP1 zCU~s{c2jjIbWnL}Rh0mj&1f@pl+VueDc7Db+v_C02Ml|-Db0hru)E@ywRQCtf3YuK z4S%@m)jMb8!QnV^N(qRZWf5)(@(D&Byy zBye`TWmatCH9|fc#=rUq-W=j>*<@0=UYjj+5K0Y!;4eu0PGYE_<;^sxy1gaVKQpg? zwZsCOBCu<|yWIS4wTJn>0w1N01iKIB?T%Sq#k%T6P(8n>Xa8a{MZDmR;S%sb10jL{ zMIVVIK`Mm=VhM*{;tEJ85J?onjzP|Lf5m6?Sbjzcx8s65WB?=}g&W_atHN7UjflOl zNgi4cb7T}Rx9)j3c@T$OgbY_k$(j_@WKc@}j8(=!LuIhVW>^ZBNHqYoRt34{ZIHlL z>|W7ln6$MyppZO}$cu{<3GbURFiktPnY~{vo84UMveh^c+gG#hRh|0ymk3WC3X4Ux z%=D@F#djzE%^Yrk3djzHQ?tZM<4_RY-wLucug|Ffl0t(1OUJ^J-%60Bv*9gRT#a;Z z`1YuXh{tgJb(m|ik-QSfsj?$#js0tTOS|AC#{Viom$Sv>Xch-UP%SDp_I?Ok*(d(B zswn@fHz82|T?%Xf3zFTkx{#pTv;rCP(|ad_PD@lp^ZY!n@@>P#7Ls&%gxB>a*BbIS^gTiz!NZj%z%e4E<jGslHKAG$My_8dX{~Y1YC)Y-xl;sWgWu@p(?MQqyO@AZb{%oq_V-;s%O&Yn zOuND^H~=@k4_igM31XT^9u*?c7J-V^jB0r^a29tL#@_@yK?QlB=3&bz)%16Ez6s%A zTXLH|F{t~F<{R5>R^eN z#iOt+=l$av;}6*V*F^i?LI4T`UVy*saE`z|{<-LgU~ph0thVChq(#2wTv$OBDp3z1-!{Npeb;6yKpq7IH4k-L4baa$gxXD4_{q2#y_&hfJueIF zGjTemowWQz(@IA@usOZ0!OK3FKtV|hgL2z~vA1d1)vy&@W>m#H{~Nf)I0hXVaXJ8O zfaVagoJau~V(7O+)gCT7feaQ4uqB@ZtFZ{Q*Eu`gt6dx}M)$sdL%gI3Q#QbguLR{! zh!$n8FO##B|JW&)hSiu|Xb@0zz+M;=c!N>BTG4dJ5%FL;Z>ld-@+lBR^o-1vjDBuz zX>Qhsod^m9*krK)+8`n&us76BCmH zhk02<+{z*g&ZH4nKoB1yn<17yN6vBCK`ZnF>(#fgKY1M~W7rQ65?;Tcv(phljfrDC z*II?AMr5_#xGLXVG~9npTscaM@P7mr$bJEpZ-i=7U*mPOFjC3_iMf6YG+3m6hKsMF zRfIV0#UXnE#C#Jn{Ae<%bk%&_4w!XNTbU4?(j*kW-$4^{lZ6&MG0l1Df3IVH(5Lf9 z!+FM1ZoGyjw0pQ}#JQUImQs0`6wp>x@}cN#Ie*=zX*LAHMe1M8t^*4Kn|}-R`*q+1 zMNpg|Rq}H&Z-UG-9h}PH5^F|A>hdp;zN-Ag`U#TK*U0#OK*@k@+4r)3iatsj1XT1` zz|4(-`BvB$moDzl=3I$tkjP-p#Y>6YIdy6xkn6Ofj_NYU zh$u)0z*rK*w~fK}j_0v|3C`MkA^l%pn-^gjAZ881DVxI;dkR%rSlM;P&UT;r0Vg=4ECyRz`1=1m_W7yrxZF=v?zHxMRvr z(XlReTUYbVxb94|VvkLONeG10Nnz~Db$7@V32wudgp8B{$EmHM-LnuZnQ?^DKiS~2wVI>--ytmEzUhB| zd2v?Asd&iB)kDt%_pnu9{Trs$@YfcSsbD@3NU=c9%7H7WIo?DTy&M=R?C8EnEmWBQ zXj8w3wIQ0Y=Q0@|B1NyWtD)hE;f6w>d48Q@3(e86kj4%fy5m)pFqxcnOF}+1LnW zHGw3rtPXT4&@Y}v{}od0tzpK+*whrpngABm;U+XQ&A5iej(5~&S9hB_9k_X1pPBAh zJf=!HZ;U>#MzD}dJzzoCr*^vLj8y^h7lX@mj6fQ1Tl9(-m@v>Bo&O9fRD50D$%AQt z`wt#G5X__9*nye7ezaO|wa1=i<2+7}?hbNZK}$h3k-6}sSEF_Es}UFgc^Cxyq+9BF zy=gHv1b#tC_>Ro|ZFICa=$XJtM(vx(qIX%#hP$RUS2P|F$7njOB9aVP=`g(q26L}_ z`5X)<0v|3Wr(2ln82hv-B10lBb{D_p2Qks86IHwixt zM9?{W9u4T!XVquDmFMy4uRYytezT^q8mwQt{dEI2Ha7J#J4)_xSEJ}*5ZfS^gb>o{ zeXw@BAfV3cb;tqiSjm5b*YB&E7xXLxXqv@`2dbYLR=}!%4aQe5z!i(q|1~9H9@)xN zY6Ur+X^eQmk&o3LSoW*hI55m%*!-4q*4wx(;y03qi9>>=4};7?}7L^BMyai)AKK$(VXjzR0a9 z#b)1axtaML(fX+Wk^>64H{chLU#@~cg9@|VC+Dkp?;s%Zch`fq3<#P>4`c_R8H%&R zunzGs4Ct|1o*^+$@$aYwEsR>|UsF(!>|V@Sa8yl631hRHB>l+28}^TJBO6vTB?YmF zXf<5}$c6W}(sbS}|KTJqA;DKor{r3q9g-ku%G{l+!TMcjN6XC2ci}E|7H=WO3-ksO z?_pHs$59Z^CPvyaG7ts6DH{yTK=nmCd0WrVyhC!a`yddOg9V#Aa-$QTHG5~Gpw`-e z3ZXv?@g^Z^X`{HKyE`AoScNr|WzT^URZAG$!ZisaDt3VAZDSZ&8^k=&3DWW6P(Bc^ z3dTkuZ8j>FT>XJU)}%Mf9zQg6X+sh#M z`Ds+!L{wn@a9c;mEGW5?fba4ZnC*$7qTFX?#jhNbwJ#8ee{ESr#J?}Z9tU{8Y zqD7R9*r4IUxZuKCy1KfIylD}JT6?@1GHl>V8kn*^4gS^!qNOGcPuYQ`*yMoyrLE=x zbHk@zMdQ6UvQONu{=}4`NTTO zjPz#FfSmirTe zY;R|$`i3QZv+{+3}8aDox(6v1~V|v0h3@jF7;hW?S2A92$cWqZA3Okd?*Ny0a7Ds z{LhdjT)>Zkff#T;-gVM+n{k+&V*KxMn!^+OfQ~qBB&q>gxgJ_u@h0eSQl zV7U*ra$$z!0fpxmUpyi8Z||9mcztS!AO!-0=(TkQF(A0izyw=e*V<;;IY;o`GK>@{C!i0;{7rdhy#)vhHP5f(xe`B4#uv@p+1+0^I3aA(n0&Py!L#8pmb+nN zx8}NRhYnmNCSDvN0dfK3e*Q{P`|`T+*rgS{Z2;98wcdh=I|Xej%gMt@Wd<~WXkZdG zr-zmqrA*mf`mB9z##4Z^fP~-8u=zl8<3=4Mf<$1LbZTmf@aD}_dTjUtSP8r9x;`Wb z83p#B5kLRtdu~=Y{O=^`r=9^o!*URwM?O4Zu+qiW-wm%8kLw0HFXQK{jw~m4rW<-$T|yR0Z_pCZ(9N z^rx&Zgmy}YOr1cXh3Xf1znYi%aYny;`;rCMeUt?YCucM?Y(K=xh4mh|Z~e<8k^-XJ zZ!%o2gNxo75)bGJ>Z{LB&~h?La|f0eN6XpqgpXO_ZR@SLwjlJ?3#`NP^fVFt8?X@B z@_|oS=2s)rDGXj6#%xpb^MN2+Kq5{hthcEt&@|C40Lp#Oz<;Ng7`$>3!20~TYPpUn zBR#!7+$A!kH3}cGklmo50!cXoIQ5|C`s(KIeMs_yc(uN~t(%dYz6<@a>(UKt(G#%+;REE(bS<*1X3eA_`&Yk5O z(&B33;PRXGqy^2~xcRAfQQ&hkjCtCXyP2)d&9n2qx>coPn6;2?X52T8?v0Gl`s7TH zo2ZIOt~f2P{O8+UaoLV&LM{L{E*6zeV>)MuC!oBU$b1WnaH?z?773x`$l}OwC0^M>hZzP$~C*CV2Fx2$n`nT$FQ6!N7o0 zB2{69b@W$vj%RQ%nWN3`+i<@sR<0Pe&u6FC-%c30q(AoiP05}H#A1Izz~y?BrVGTy z{~Z?ATvV)ui33OsA|kE*O!IZ*GYc~0B6HdjNqu2@*rsgi5}eN_Xv)if_oXOl^>MI-n|yZ5jS+yNaYwfe~>|0*7x;5^c@dpWOtWEjz9^R!OdY#8WJ~sg)m)mkzx2?6c z4&5xwuF&17XKl#6f4 z4`=) zF`=o+xB5O?6`ib#HsFdwqGpxGPsck)b!ITWf%aD5x?w1jz}KHn&~t?w!b`V+5jn@$ zITwREuZv5e1-cf@*S&x}D4HxkSj}6srwK!k$_y%j1G4cvFObjS7}-k@^eB<$2T4*# zx7VV$xwEqo=7330N%KA_W&wZkhhbuf%(bV^I}^3i{7_+FV5oQ?BtqJ2RXCYpfxG(V zs=rKo7bRcTb*Y)?tZw1v3G}=o{$B6LoXcHerDsIo-*nNmIy-Thomzk3zrhm6KK*Rn;L*w1rlyh0vUtEsZD7`7ONGFk_7;tsz zeH?|%=t~1l4~KnO%Mz+bzpF+2G3bKhQ&Om4PD?EMoqp@#&TB^IfI;JzgT^A-_76Yj z>4j%T-he+PBqAzjdybKKAB8aqeep-KXrR2R>fEkFMBZp|(HZ?UID0^lr*0%!)e2-~ z919DJFNO`+B`^cmO7o{sGuOsV|F<~}4UMer=M_uuEZZ03Rt;aCtSy|23Vv0_Df6Ok zn6^Z8<0;o?Cx6ZwOEyjX1xfKR9qSMrG3VQ7KB<{{KEL{A4r^a0F14KV1*kCvv%t-5 zZClL;&9;CX+hUx@A)5!aV!r#1KXMYOsOpU7rz;y9CE)2AKzFFpGW$gLPI}OYD{hQ! zyT6xNuSP~(mW3nkLD}dhaS5x-Ys9w~o}%n9aGo<}YvNc1{&vX0!Hl@6E&Ys^J%jtZ z)5U`C`o7&}jti3|b#oGeOswCJ-LJFa80}T>>N+SpIIzR#HUx6Tn|CGN8Swx%xdESE zVFU!%t7Z{p>N!4~D3NMDfmcVgqEqco1SsG;Bq|mQy3hm53MwiW(-i?$F08wi?|V58 zB*NP);u_o8v68zlP{LeaMb0y3{E!Cqauf|R6%D|m%*&Q`Gwv6b(ed~rrHtO>ii(fH z@w)PdK@mg3FH5iDI!e-G1c+b!J#T?ag_)*n-`jCh^$up%tBHZAhQ>#xB_fMW69?vD z_-)2)fzGnO%s9wYFTxbZOU+yfYsDLeEZSd=5{J{$c8&P+>HEs7b+MRk~= zWLB#?%v5Qg2vt~ik#R4NSKvJXo(dY<21KTI?#H~Xf$532A!H*N!Dz@zJcMLR{w?SE zr{k)VUxVIBUtd}ayt+i3_Nq77Bhj$nA^w(Um8ReKH+QpsFb0h6q;>ghMR_ynV}F(l z?C<>Y5Uu{^rFkzO>4?S%IgQuUAG>16$KD3TXtm!X(dX3d&qyg=42$0CmfO=c4sPyB zD;nf;my&C0$YJ<9UwYU<8G0{lY5{lII_r1R0$pYto8Al5vp0H%zPx_)&ce}*t%vGD zqG93C-Khh=0C&Hj0W=O@cGrX=?Ed8&-7PK6I>p$t_wTc|8N)yHa!((xW~*dAlMN6_ z{!)qeR6~OVB=k_oxZT7+GPBLIs2>dtlyr1hkCso#Vd@cLWg^f;1NTaa;4mXNbS3}l z|7-5O!>R7$_~Ap+A$xVoJj%F{a2p5NBPA)RBwIsP2uW7fAtWtLl7y_3nNf12l1kY` zRzgA|CHlR--M{r*&viY2KlkOjFIUO=j?Z|%U+cpX$rb*{V-%(HGpy;m92~w*AK~KU zYyg)Pg%`qK=BTkt@@rdb>+pFUDhz_asZJyQB`Cyh8EzOQ;QYyH7kq?A_}=_W;i&G{ zzu?C-jh>}5Vad0(ZnE+Eh3S3wHBNsl1qgWiHC484S$w7O z@w5`pwYn#N*oS=kT19;C&d;A=EvJ0)+rg?o_dRg!Jw)Yk4_tma`TtJX>v9<+Yr~i$0?*8N7an>;7N#x4a0Qw#|gnorW!$|sq zGqi#=!Qx}W`{ss4H2nx1+|1nkxPkRqn2%^!ZJ;n7Xqn~vnrSzb3(Q*8MTj#Y^lzjE)LDg27w#N)_rBnFQewb=L z9V;CINulJzh;z`hzn7Dv4Bs9F6)U&~L zfn*7ShD!-9Z0{Z-GzKb;U_lKMu->7u>~xSX)u&N-cFJFt8o6CIB~PPKY`yx^cS40G z4hx)pp@t=4fbs3HKXLMRb+ou_Y7lF^(%Xr-YKP{t7)fLLt{Ua2M?D$=Z@rWg)pX z_H>(j1I7(uHUCg@v-X;@D=UD0Vx_3Oqw@~2g8ajNmAhAOS5;LdGO~#3%${;}2zZ1K zlr#_hyq|JkH193S&1HiVtJ>jGq90r3)jLj_8ziZUIoG?m8a%G%tH;GKI}~l(+vZ~I zPYM@b_|EFFR!?1BC@U*#^3NZr+a$EK)?rcb3f~Bn87LVI1nQ#QnP0zx37oC5t&N45 z3AXvg`x>VvNfFz!&^U&q53V7z1TD>V6rY%KAZW%=_o(mrcki+i9>3CJAnxdq9z5V3 z1+DK?Y76y|Q*!$C=~W|qKk{;OeP1w3duG@fim1tHQRI@u0<`#hGJ>RP8rZ{?!%y`J za~@XWh;kCQ+0W@y=Tlk>WvmEnu5h_4k)EKUX8N0K7SS$tH9tRpUS|4#m#g5GCQWwq zu)EjQ)j3gEkurB0my52o7?y6*{T?oo2t=1djmcB9N61)D2*DzPlSHwOkIw1cwzIak zpR`Wdy=&KLxY$5O)Wz)5{E8i{4*EG8O^tyyFQvZjvA1uY=>KPgZ%{VZEG}i20d3ER zxK|9HzD)tI)U+N7E^qE5m)!1(sL&5DtDmN{*XDv*7z8L|U~axT;;j7bQqDt7&!6Kt zM&jDAHAJ+J+yYziUe~T*a-pnJ9d#$%{uR)if}O4P?rHh21YNzU`eW;8#p2g0Ww^`; zaF=Uiq`t_gr{#{oKy^aZBP)>BPkIYWIH;p5sz=W^zOh({_lsYgu9Tr;%oc>P|MTmg zgS0(a&e{S+ndO}lqem^dR@}1Sw1UveJ z2c1jBi%a;PBZgw$;v*{(=ur1(S=A$xa<#P^xqwQVn z#ip8s`B`B(6;Sq^$RU#6aieqeN57eZ&``5RKLdln_t*EMsU=$@kdZN$OFl$lB?v3f zA&uRi!Vzmzho+u*WyGiZPIjxMZqyn&iY3R9yz+@?OI?LcKXYl?r_qXS30>?l(dj0U z_jb`45}G~EK?28C327Y|uCN8%wG4$TfsYz`d2Q?&2#UeP)7{9ClA5|I#jI6VwdK8> zygZ`a;#P?TMO>QY55wXdJTfvuMs}ZeqC=*g;SFkPoX?iWPQPus$TbZw18EKh9uFj+ zB!}y0VypxN^ACo&<*d^;IcpB*f)k>6f(ByNIH|N;;6H*XKr}5shN0vl=xKD(#~d#{ zeINL>Hy#R81R9o~XY&<(DOHG;YN7LjQDrnuUPb@RFpuUAYt<*F$Cqy0RXf1*q(1G& zdDe}5QNO95X1hB=pY%v(SB3g@miY`AqIZ>mF-MwPI|EafJ%)(osR`}}y`hYNtM<#b zZEmX2OOiYmzwt~NR5H}M{o(hHj>X0Ag2xg>3g?GO5!P}I;EUhaGW|1FQ`>vxOJ9vJP#FV%|B~ys$M?Ist`D5* zOM;uR*e`4_R5q2CoNU5{yK$F1P0DmN+I6=jqk8L`Kw5RH=L7w*#v;kagQXdFcF_$o z()zbWtqwXc>{BX}ZQmna7U`=*-el3`?FG5Hbu&Ya%Ev!#*D!5+ z`1sG#6ru75r)iULI!GK`$(Wbo05cj#(K{|w)5h4rS98*>Zb_c>dTk5S)yyaN4%_W4 zm(YDH&a~NnzI(s*aK$cI>H5MJy{dnW=Mn2d&%pFc$n$)$!)b+tAU%RG0d+f2KtR+J zeH8xgA#U!2_baKJXAn3 zC8sB_XE{wX*r6~){A|e))pb<9pPw#BTCUX%p%En`@%y8=bwAoqu?QpW5B)6>x@MQJ z-4ICQBJ%t0V*sNLe;~>}W97UYM(Nr_Zj>Kamex%?QN!=)M7h=QswnNozoixxYSv7- zSIU>x?%+ow@ciI|>tZ=AZeaX|AMk7UfE_V{2kw-xUBZ+pk%sg=SjQoFOp1_9;?I2^ zSty1(9Oq%E>T8v;F|yMJ5Um@H1Ob9!<`*|k^XQYf}}!tH{nq`JI`l<5wDV*jMl zax^NVy}T-iLFQpc-YkiMXY^E%7FZx=`>&T!_l4jnHtOcJ_YRjS)9l|zmxLMcJScIL z*f(AFe4zfujn{WAuEzp&6|vNr6L{Iwgz@wc+!X!d#f#U)z>dzqGU5tKA>ZrwPw0M~ zkS0|zS=M8tsW)SB(OhK6O>rQy%+w}`ew)8!(8Zo&OC^ZnTO}nICw*uyTC$7K_LySP_E`B0v+kGx zidbm5^KYatsK1obXiQBLc_|fbv9->RgKPK4s&B!nyE9z(d8E`Y-Ojk(MJnx!ZjI9~ zGa6uV5F(;WND>0yR6;;Ji4p;o@M9UuO9i+nQFhweo-v4GoeR3LH?INiCuKNlxlP)4 z1qA%9`+-rf@T)wQd{XL5hTS`zEyvkdf8SoZY#4_I0{qEK*d|n zrsHaXY<`=QwZ9T$^HuF(@9KQM1$FVpG`J9kMp7N05gV?6fB^9@ZfKw20x3~FW;{=j zpcLd0f{x{7(*cQ*aP3K9XVep@FM*ZI#2pjP3vJFbQ3DNKZp6DKIcFqbFcpT!(id- z&9P8Ii0xY=w{fEhdSfiDgLThsmk`-xij|h|#Imrweb4t)=c2i-=6Ul@uogCE%hs!RaW~=9x|nlmsb3%F2Hw<{(hyPrU+KRYqxmu4p0v|7hQ@=h!~dL2jB{1NQi z8q6P-@2QUw5P7HXgx(jqY6ZXQVwq7>)H02ztK*510BHVEN0pwQIe%Bgo7_hz58(up zOc_nH!4w_@ZLSZ&2&AlI#j|{a>=(@|qWstf9-hu%s#veZB9o^oySMzwKGq`pj;pkN z!#Q)(!~0Y}p1tDqnk#dN9mBb=`@-=|=*dLE>nx!8n?|>CYUvz55LoC`J_5ihUz~+L zi3=^an5gR?48oHBi{NcfeW{MF-~&FoOOe_qVfdB)YIz{%{aWAr-bH%R{fyN@QTw&iI1W1i^o1E{j(fe$|cy$T4v-Y zT>3g%A~*V9{$EYkz7xGiOL2zvM_OF!rN|;gbV&%X7eiBs%O`y3^_W%DP&EaAIG+id z-IJ-XAKF!VJ+O3PgCIOi<&AYp9FYDcLe1@pEzNBEcK=09wwdU!D7$1vwKKEr2Djo; z%K1pxMo*hn3drQG+trfa7?&flW5??grUkF#a!B3+{G$`scJ*uqjBKhfW zXTE1iW;df5DTe+DowqA26GSj`54Extc=oX$&+cSoFu_hM2%*r|d5W#A?KFe{MD)o5 zf^k3JjSk9sT{#T;1Vhv6!w8?{Z9BsdgShRHXt$nAxrFQv79DXYJLy*19%ptrLF8dl z-OL%**0Jps<>53zT8*&@S26+@ov{hnfFmX9W;lL@P@j9!M@q)W-O*J$A?eXRtn9yj zRT7|nfwEnN4-@JY{q5U1ptPuRTDnCdiAfP>1TbEjW=03SOsd?Vt=%Kz7oUv@SZln2 zM?%do9&ceG%OSzFWBl*db@{#jVfjQMVZ(!8n*VsKULoejb1BKhcalT42n1G($jixrivHN{AcExyDn^9^ zWgKrd&vL-SHFja8>Wl?`3y~R7qnjGUj2f@W@F3s8dG0M2?apK#a=%g4$B~fq*yH}5 zQp#IP>Iz~06%-N{rG?ttV*dZQ~4A-Uv5y#0Fl__Q$o+86^8asDpVIddh z2m`EcP^IygV%J6`uqTQ|M~)mBf@VFihh^~FH$TG92$-4xzzCOP)yU>(pR1dO`A%R5 zAWQI+D5`~qyRdNVm9O1vza`hSv1#OPUa9GKSLYntgcp%hFI#ioty5B@iWG0V+UPnh z?GjZ-%4{iazDb0GS@w0zU%P(Y83EhX>GCrtALzszMpg--zR}o^_N3_^SI11_r)pkP64u36CVnakc zN|WoT>h)%VCc^L{BCYK^xoRo&X%t_nb>yShdKv}8h-MO)> zJuCCQ>t>DC)@Mq@$Z6uVo31Jc?J9Pt2y#e3_wfjXQhRJ3Psm z0_bQJhP7LI!qB}1^s#WoumXxDa7P42roZ=MKHDI3sZ3YUZ}|B1&|?~kA?pdfhx`ufd&wJ6*!UW~?gOda7ftTw^KtQxA{nW;9|J9pSF$D{6} z&&tt+o9+cn1IuAz2ecmDYlyf#(fl8LFXG3hqvS8>aQ5fz0rwA8Foh|@{uTmMi%8kj z{XK02@uWD66vM+)M1f6cFz6k)ZVA44yGPj!sON~Fh+9Ci;h-?yw(X4hv0rR6>V4% z=rE-~9W!D0m0w$5^r=5qnh|Giq_MFuCyHtdL0Hg$C;?)uJzknT2GCS`dE)W%Y6P8m z(#f|1j(h{$0k_$MCAEiLiC_x|34r2B_Ygo7zn7-M8$rU|Cy2?bDk~*Zo{t!nlV?mr zUP~3W+bdF|X+)7!VhDhk_pz#*@7Lge&qt1!v7a}R`}9o7Qn>z>=@G{CbN%w0MgiLT*|hH^V%vnBi1 zVEX~P@|MzE@h+$S@wTE&#|VL?6-&Il5{Cj5wJoJ`B&FtB=~x4oj>Q@{FM|DnmTS9|Te z-es2TqLbpu<rY~Nh4Wwi;sI0;Yl*w!7+~?}`$m!iF8XF9)@T-eSJlkw7ZF2(oWS}uJV&*W z_C4k7bch0^fbwa0r&?xP?90O|3e_fB4wgtLz7;?5Eatvd$xkg5JhioJbMla|!-HqT zz#%H9i0Y1Z^4kW$OHtqoQ5&pZzkUUBRmS(U4CKHcECvOWx3~jHswjY?LpM+Kjm?Cj z2f`613}6|)Upkd@tmSZHtjRj_6e7j+ks{f-E~#=D5I-&WyR1hgV;9ECK$=7@ivcg+ z9aC8>d6^FyBEZWk*?Jb#_YPYbKVPydW4`LZ{p$KE7&@F(R`Xn^oDHptG27}}&B+?^ z%*hmE$-RIVD9EpW?o>!^AnjtZT6g}LB<@MouDKL2G%L4QA_5uzHoN!k86xeC$ebeB z7|9eevSS#iHk5Hd3V68ZV~1?H#dUj~gTTHJ2nlnRRo6fT0H@#RXJln%p8%Jg#w#VL zkhN>q294)tBTkJKBv7^sHM$~JUqT;7Vl06+O;R6}*Cpi?6g0F?l_Os`&X0{MvPSZP zgt4^2whBG~d8JzG-Ah<+|9C_ zS-*u@PLPM+^^eZs>zqlftH(K|TPK=x-e3Q?iBA&v9o3l~zar=U0NG(LJJ6qfXlV(} z0t}=i5@3*>*4ELe8ixp68n*x(BN}E7sJGVR1P!Re$=uvq4jvq~NImR*^eCt`CZH*t z2@R3Z{t%@Nwgh68K(FpQH@*k2RTT$y=t-ri9_4cg`#CaSa~v7BpU&95doM;zE7Q!= zU`s75IPc}E6<54@n=6*X2!7@!^LzyEr$0wBTg&7@8pW&X_pP#c$kkl{J;i%kL3bN%e8`v!hh@BiVcqy*BvZnB5jal zGG=CqW@dWuhnIBdKF5)gIVHFJ`&|?fxm=SFcOG_jDcb(5(~phGfRxzdDZGAqEZd~W zx0V@Tm(#d#Q?{k2FDHVUTb#fTjSiy*?I zSP7&Z{Xebdy4DP5xs)Q~R_9tzx`DTo%2lYv@PCj$&Po(giI8sp)Q7|^V%H*+Ph{8k zKOPlKv2}laySp=5o4?w1c1O6L_m^G9WGbCmtlBa0(sj=YnLhFT;X)r<|M@skw4dW*1;^jrcfkV`6Brl1 z4SZ}cp~fL=zBjPP%ad+G9=Ixe9NFLBFI%&V=&)~AT6;xPl8@P7!egp>#lgc5L`8j9 zI=sGkTaum*8k_jH2wH+A*@ZayNBe|gk25k(KRrVoO%1)I&i8zIK3fi4zICKw%@8`N_B z>rn$T_2!2w=42{kZB~-jhL!JksvVP7r*L5Aa*5`*xY-4Nt1W}``c}t8711c{Iwjfz zd1l0R=_o8;_V%KParou0SR2y!wPZ=>`1k6NU)83Wypp_=%oe@!$SMKzVYYQEm@!;d zn_JR3att}8rFGRY>Hi9GU9nx6Eu9~Fi;JLrIsA~iVZ)vJ#YzG3(vIR9Y_r6fNQB-E zAtBLk-}X(lHfO)-5zQ?LSaW#G>!$KIJ@Z{;-)uO322n^K(&jKEzqzF9mLg(UL+d-6 zTHoe2z_grm@7_}sH`Ej}@WgS@FqRofMo;fNph%pT(361^e6}Ac1A=8*eLlANls2&` z%Fy)!C%a>S5V^wiOjsp*2Jmz4AOfzadtEbH&rPP@(UwZG1+YrxV+V0ku~tx%;RQ4r zd@AG&zOaB%5qCa$4gt5Vm(4M=Uj*V*C$Wc}6uY+1N9&lx>#8Ih4?`0eT{0{_I`Zk0 zq?(!_B34z;rt!Qx0=Ad}fhNce{BEA7u3!(a&GSfP#Q23UP06jimR!%xQSXH70o`hX zfdE>dlMA7^MqZT%p&{N)QY);x7gVbtyKk#9C&>ljhkco91xfmcY49S*o zVV)oYT>%FGD#vH-*~l6f%^dx65#w!!r*F?{>*48oz2zBhkOD4HE$oyP@T41&2XdiO z-?@p4$~c-TTtYFcq3Om1`^7cg)C(5_6u`qCL!3bK_qmOungNi1XEVnCwx~Leg03?( zTlH?hlSX#lP!SW9F@9vMN;3L;7rI@@3+&>IRkvQ@d3WM*Yq^t?8HuIDy~J|BKE-G? zx6c6+la++15K>T-Tn|OLsL|Qb3KD4?&bYq6_pXf3fCgnL%B{=IvA#-@9_&{!Nsz^G zdO>`-f=NdOs|SMMdZwXYHd4$M<~KC!U$W|t1k2AT`I-HO_%I#K-aRPb0b;VFlBpQE z433P>^K^GWKmY-_Oct4teb3an7y*o2;a10#&iL#jM(uOCSyfXTAoeR*yrNNH1IJ0H_oKJ2ej`Q<9pp5I2Q|7;QO%7v@1lWXhJ%RXvjjz6pns4K8)jLdiDUAs z6d}SXaX?7Q`SZ-E0LzTN@An!S{j<=hXDD~+fhgB#Z_uQ=x;it-zACyi1JS|=)BcDT z3E2%6jMBM5z%56#1)i$>Fa4ZTlIhU93)=4>h%<0-5cAivr+2iut6?D|=n?p>Rsn9z zXN*ovf)b@lN;1P8ZJGMxI8|4@XQH{Z$uy+U6Jo4)h2I(f0JtQ^=>MK&UdMfN*3jcn z%^XzevfyrYQzf&r`*p>O9(@B5JKhbv?-ee!kxLefyId5RNF3)lo|HT}YSu2be$ysy z^wh>d6%+RY)pa-ZcUOfiX`%m%1+ptiqg^>ZQ{3f1_xQ0*V1DYcsFYNJkq;|J4ixg) z=Wf_VA(?|XV8lMye_{S)?wVv;CfaWz1&ToU{=UsCGaU{$7Ws15(0xEm0zXHb@wV+! zHjd?D4R>9xQ24hevf-QjeMMX1b6IQ)gwSFf)V50}{Lm80%Di|A+&vjSI~Ge1@Wsqc zWZy!joAd<=z8V^Qf#S(++jjd_oBJ*#y#!W7-*<;Ej|l$(WdGm5Skt!AjQpLhSCC`+ zWJWT(q(p#p|M4L`h5PgQ5xsBo zNcup6N!-%nmH^si9!GyvKp{vFEV;c7gNijl5DQ%IHu-xgj=sSIr^}T8$X&SC<7$0) z&tfrly5n7yoUBns^$XQCI8R<>>1w%H^k=S{YCbgUp* z-+S)epJ`R9Tk7rZ?zdkgdb$~?*7RE=ZmO#4#mSvmmIG@GSH7t8Gi$gJ7G}9Q?!C!b zeTj;n%5L}wN=Zpc#g|@eWJ2a0ed`_Fu{G5>DMdJ6;VL?yAVlxMTRq zETYT)>i-WRB#FouhUsCwoE&@AiK(w;Hxa`s40-RClP4?9!#Ab9RbRV&R(sxP!?+fQ zxoip2SC>T3N=#a^C9fo;<&l8+NN>42hKLf~G}i`nBw; z<_nEZ72O0N__HO!+)*H>r7GSGP zB$%G+2oMDc97MN3^7*~sOGqpMkj8xoB(?fRO6qxIw7CenmXQ^doMW>;I#1a3TF27DyS1Zuj@@p*?o`r!j%Ne0XdB%4 z3c)p)MZz#@vr65e!Lgj zA%f`zlF0z$_9!W7qGLAhq#70n`KI#W$hmu&W9z*ynrPpyxJsY3bIuq*A;RFE6Xb zVn%JPLlqh)(L(zt%K@D~)sjr3A|gnb{b;ePyk-Z2?@H2f{O|e=lCSlOWE5t`!G%RHh&gcVo5%yAMu3~}?oP~K zp&G^NqX-HAvrH69Bnh-4U|d*8o*y0wk_hUK*==JYfH_~N%YpF_9NmYq6(j->#leRk zW@dWFpT*CXmd4Zka~4g+MXu7gZ#7NJ-n}9R7z#S#P9#ijCGvDnIG@P;p=AUuv z?bxshlntyRO0$zc7FXEH+(}0v4VB>0^dVjmkwkEch>z~v=tFXPd&{7+)MH<2rL30` z%jxzQ34m~sh{7RGMFRi=S67AK2s<>g8_RnmySr_xW^w^C3Ttp-;jD2d0vxWWc?h9` zt{a=m#IYwxF8MrH>X9r{IEsl_QyS)-t7W4oEa4j8H*wm6n)_pr}Oug#+`*ixR2QEC|F-Bs4I-btwU}+r47*{kiF1d;X z0pbVUkjD2~Q;kFBB4ZMs2gtXRfC%C=12j+hdnYVpUY9q^kC|(SJo$uc+T{3#IL&cj zO_4)T#Qp49#>rD(HnOy&#?PKDTk)20l6d_*jvwcAG^0gt%Di6j`GJ4A&VMh6WN|}= z__s4FnQI*00Ff#il5n{DU+dEd%1tZ(e*2og)3*8gs6Tfiq~?S;Bm7Sop)ohu zrg`FxZ#b5o<$3B=GEJf~_Apega;RMWa(C4?-vBlDhNFh({H5F9z#)( z%J|W!x6BL5k%9ANX@3@n4xF;eZ9U{g%%W&oLTnyyc}D!~f2UM(62JHFw~RdWL&WXi zlI2k0|GhQCYIZ@*|9+CE$j}6>4)N#zKlsr&PSzEPrkO^oJXF`ErrqeyoX^pjazkZp zxPh;-B$T-LbG3Z`FJ0K~Q>|j9qdJpV7*k6vh`SwBSXXHJ|9RWr{)py!f95zM-HT)9 ONn1^J7#A8kMg9-JTa-Nj literal 32754 zcmZU*1z45awm*!GN{C9Apwc1TprlGmNTalLcc?T7h@^Cf#FB0hkS+n~mKKoi`o^p0 zf6u+o_w3_7ZdmcIcg{J+ug3D1laau@L39HR4Gr_zQ_&Y_XqR}<&@S9TzY0IWt!#Y- z|94eKQbH8%9Qh}yCNm5T?H<}Q(I<+IajO#!u1dS72b)6^Tq_35_6>5%1e8hlO5bCI z4rEZ4zbMg`&u;&bh^1le|vM9qJn49W^$jZmS=ril`gY?IgPBnz5NkU&@NB0R+_*Y+0pHUy@l?0 z_4zts^j?Bbl3us+Q}Wl}85E7A<45lOeZ(DMbP6vctX(E+uUp7pZ#CR`8hZcV-#6%9 z%ppI5pH~Om<@x(-VRRIwSnz)yn)COeH=_9*%!4}S2KSuZ>ejns?(g}Ojtuu`!cz@< z=}6z+5_XI|THUppWtl7&^5d$_9PNzfi<&z9E(S$}LE^K0do@kt6*d*N%eS+$ zv*Y-kKGl0(D{?<|#$(b@*_)i6!ZPTLW^vtWqvp_RT(;fX z>R{d;F-)|YZ^za0INhaNQ#@n-_L$;0)UeX_;w_}Thv74G8V4vd<}R~V*lxm3ZGHTq!Y z?5QdSFYgX%qxMToO!|k0qEw3v3g9}?GGyZ08@zAkxEe55Z(Lj%LS?!g@9q4Vhz%y? z8$2STGkGf8g?vKGnTxcvv}u`{Z2IkYBbhYYH)mTu3%GE49FI+*MRQwGdYsIXFOL)% z){No`qsw#U^%{LQVxQdkc5>o+@zNzt``54=!uF~&1w8Ag3rlNyLxj#+wa<@UUc7kG z@nFW6-n`m^!I}1_$vHPKbK>TulW|7rwGtt;or5*jVXU|Ty6VKBWGG(#`e)C#1 z$R-MQaZS03C@S6+Iy<;ucQViH^gBBfO{Lff2f3z$g;ayB+Lj#SuCPkEdR*jL3D=(cRZ zn^se1n2nWgPB=_iZA?FjlcAn%4$3KLwQBlWB;t!zpx+T0uwpt@8_%NK{K)OdHo;zVw@+p`jRrtX_fm%#%BD^e^)*VfAj`zAY>z zg`w?!_|ULBE_-tr)}GaQ$4Ez4S3U=oX&3QaX`aBhQGRnj^uEMxT-+<)-d?!0G+x1G zGMGElFDsO8GL%1VKW0Y4YwNdOcb@R^}Q~?r_r>d-sW&)XOTi;)~>3F}o z?l-Dtk7#CQ#&vJRFfBd3x5VUSsriK3%5na3%>c6dvy+GNv%AzSY9i9o9^`9OG&BX` zkHF3naNfz{&X5Qp!=0+Lp1;9oznX>TamXJ_e9m6nTCvGn)hV6}G17 z@Ym!gW;ZCjvSM@d7xWqMZnm|x!8^*!$)UY%(@TLZ@x7@jd1wVUSPor&1D43Z;xF#Jv@wHt;8P!W~ z5fVN-KRaC>LTSe@9v>Vey~w6>X8%*OKN)by3kUwv8CUfms4%2zd6Dj?M>yo<+ADO% zB{gNSZ(2)CVl&yz$GK~d|5ODuEcjC=Ov7Jir|Uhx?e^@ipz1OtqnK=$`@GA`%biI7;$va9GRSoC%I9Qh^+U>*|D3B|@kkJP-{%NE28$Lvb+TGJFY3 z^jMoKdP8tn4b6a3fJduAr(h$*X>%?Ht7~0YVW_qk_&(kMBS&f%8;lzjpE7{rjG9>TzHjxw`Tb5)uyN%O~2j zd>~Y?GKUiQ?D=!+k;Mo9+H2xn9xY+iP(MmrsMCOrrIH}SSBjd2F7$hrXac5tB!MQz zg$ox>e&N^M&ip!en0_5Z02w=adF0% z{OYUjIG&x=k+8Wry-M5V1`|}xV3v(V#e8IdjYKd>TL3P-#vaf0zc+cE`gqRKzpw9w za=r#~#qf@Cd=B|KxEx_^qOjkWCu{h4Q2DOXZavBY4u2iBRQlQy?k3)N9wOx`!8!fZwd3Wg=(P8%A z+wV8{G=PM;zpZ{EDQd-v|NE$?*x1v1ljNJ~MNlAaV z#HaM-nR{&E=fx;(chx+Bmml{u#uDmZS*nix{p8Z~RyNk3!omQHQEaWMHZ!V;`i9?S z=OiYI4Z@BZjjlcyGoOPR zD0a%aHM!6?;HJUIpJz)TJg%rGr%mbkdlDNEk{VmcO4}R11)}!!aJW;q$jG{f3v|^_ zMp<2@i?3>*Z|eq=oQp3%^2$;$n8ab-4yGaX$!heE6PD3re8G!Bp~eBvXZb8Ox3KVu z^o~fZLZ&f4Rh0rY8cI#mtS3PL_Q%Gcx&0B(iZe1Hp*x1dB$~&@phmgDL}tEm$OnV4A&$p}8<0lICi={C znan#SqqBV}w;H+tyP$}rDZMf`%RRbgfBo9n-oDSq8Oc~=7&chNrR;ejr~a{;28p^s z4;4Kpo`;8r^Zp-A-T##S^B{fYY(=HB@Q&8jS3k4lMa9Kg4LS)}Sy{i7l<0Th9qlZO zJbj98Wo5NGT9VVvS>>=U^Ww#enkkDnZ)gMY7%3?z-tDc84INi6P&E#DaanVd6jF!X zN@*jJk&_I%@0G%z-`cyf`+<<{D!eo7=ch4(@k;*wxzD8@ERPtr1^83kyZ07Au-~WvDW^V6khbLmK+q}i&kP-kKyCM zfx;l+{7`UM3AMM*-P$N8CY4JB2WKOo6pv$GYhhpcFT>Rv&= zKyHxDf|UL0uzX$~bM3)beDG;{afmd)QK5atSSGCiz1v%YZ_|sHAmZduh}-M?4FoLK?VCX3q9_bv1uO zVK3u*>M;8FRl4wjJTM?>bg5$62f(O%{L}Mpi z7VeiGi;9ZYIPKYsb$4eC~IKt)ccjnNO`9m13=}eaThmw-+ z6&4lcRA(sX_y+{&OjbLQ@H^gm`{RckD;wLpm2De3d~q(VDf6;zV2Fy#cG=?#$s)dJ zt?liV%K>`c-rhhhS$ankF%2ivFp$r>wh%`Lb*X&7zVju^(aq|z`jP496KDS5!3vFt}U@!ef|A}>_&3& z>-x)m=?I+#lrnTIP!YH_V{+*lR``rb!|YRLgwoHDR;RQA$i!5`dD9+FyEFKjQ3iL1qc`WD?X;+Z&_M)Lx_Eax2|b)KGYAQV=oM;P6U#e0&%1 zqt84xRPWxs3o@UYfSv!Z&80hu$p&Yl{ParHR15X4-=UI90;c!ex1Ztv!V;;^tYVSB zam^*tUp$iyh6DT{3bqHTqGM-{ha5N}1wVfrkXbs?=S82OXjj3W8(y(O#%XAH;9q7jYtL=m$!4kgJ_2f#UZaByJvxjqI$ z^O^lkpR_N^Y3q1*Z?Dt9##5Qw_N8$NQI!(B_$}#F<#!^+BouekIf#M?f|_H=q7xI> z`>t>1oAxZ*{<7pA=d6txrDtT+ZwkOYIU1_VaM}N3j;}3QjE|4s0$PS@fzB0we}B9D z+6Bz@q1Z7t0t_Y-^=EVuH>%*+P{cf4Kn@RzFn{5Vh!g@fQCc+wta^TZ$u!U5m^-8> z=jh8Ick$;_xS7wl11MW&EMFCkgkz;n0;~8}6rL>0jW_$@+}GQdOXdb8NWMZ8eI|k3 z7}tKe4M$jBOMVVbg#j;JBrEFYPeo3%(eJscML(8ppZm@#>+NzRvj>TL4j3yz+w$`5 z&q5gU8#3)VSd^c}cAnI0^vPepe*FGj6!4t5ib{0fO!-5?TB`5|Abd68!kcgsqNt!e zyR*r~cKImhr03>JEzA9$v*dm%C8f9YBN@~U?PbgH$)H%@Nr0@+eIKY@poM#zpQM!i zRTAX26#iQ37K5VX{^jvnYQvSmyniR-NxH};L4+@gQL9?T2jy>&IbfU)*T(+26F!ua zmcm?kWF+jOykHWZhRGV2&w_5ebP>;&E%~2APX#dDzQNJEnno>wK668jFrUin7r%RX z%z$yzt4DH*ifOhq9PR&51hTo9CNk4X2sup&1YGuV32Hg@+rwkc$>>BymTh^<$dBl} zEIoKjh;njo)4|oAG4;aL4z18KF)8zg8|Nz1OGYxhZt%Je6q=4SxFu|0NYrNAD;Xt> zzMrX-^K~>GF5;n+Sg0;L(L`Y@s9l&)Fg}p-DKE=a`{dSA#e`8a=ujf>5kwS=R8yJ| zX6ts}oQQ~sih--208mmqeCVB-=&lA&_yh+!M&z*J($?_9t`BmlxRCfg{WUHy=1cWmDKm z1pUiC;OT%WQfz7R0x-1iTv90(8uq_ggiHaq8Xs)DnD&+yi2xD` z(J-XEv~L}BJvQSc=8eY+YfBprdiMG?1)MiPhCyGyJ`&Tpr4&TK@_u+&WrSamJsLT3 z8c&d_5!o`eOFI!ZYssxrX-f;#Ast#Zqr00t@)krQ@cLf#$cbz3i;RdEaPeIS=`c!b z9CrHZ0toSxJ*F>f@W@q^HJO+SRhZ>^v_OgZku4P37j3wu8tJ>0&f1?Lgw-z z0mRAdn$h9~s4#3qLCxuF^i5M!Kz|D(utb1sd{E=yNymd4m@?1@zjj`m|CE?0`|4Fv z2h-}8JzHKm=0P17uFF;qSt5z%vd`wjuOu7BsD zGmfV}ohu~eP?4;*)NxY>Ziw;R`ubD~vZnEHsp*_m@JhYLPD-I+VPXB0L&a)&4621* z&=Y=?m<-`nWLO4m4GaO`h{?O4iRADmg(k)zEF`%wRn)D#;0=$0& z0MVIo@ZNYL^0s~B@T^|mgWv4o?H@TP?#^(;zPl(YAz?G$9$`=+^MBwfR|kieHI!X> zRhxy|6rw9LIsI}vB^R92MQ~CEHV>-G+XZ-fdwaX~cI@EUJHH14DMWnOC69%jpxuX? zbi=VS5^yacLh6$+a)h-tG4HXm7OI*{{ZDnTpEtX?PuVzhnOz(1Vlk5-qTx2d(Nz%C z$diuc)Tp#=OJB>($UsD8gRU4QE7F-$=*bj0s1lv7Q1ChOLQpay1p^tta*CfrHwfRK zvt1nyEiPt{D$!I?d8(sB*`FbOpMjxMf9*!ky~ZIm=9i1e$GMv{^uj1~##or9o*s{k zhS{2ul9GmjA+5OhE;)H3RmsZ#F7i{j$((^HDU_fNAQA)@R}{cb*zW*~8ybYsT;1F@ zx3?R6dO})TpQfaw_yq(AqoVvjaw!P9nt`Cw`I*i4O@BsjrYw1S4-c19Q+uweO2|33OZ@|E5VUw1UoDhG zX8+9ky7s0-YA+3;bB!IB-K$qmx!`!^`GqKZ4$OR8uuM_P3CGG5Qz4ako@d{_YTwCu z4X^4_Rlon8ucN4S!Zf9vAK6Mq1IDhl#;`oDwnXe-1~&=)BWE_nWahoEDbnGB#5E6i z(-yR+XKUsYmAEBpU7py2O978!f#I0M3A)8Vx9v_4=l}sz)Y+LA%hxk2i$+*PlV;ERngy7F>!KjgebS?~T<%Ga!F zh9lY9)6)WQWqmy)Bm`SfP!NzzfI#bx_$5n$smQx?x&dKpU3jk^9XGPr39uNXrqC5pN0&aqT)vO5l4x>Gdq#Qk#we^HQdV8q3uAj&{Y z>Y+fsROKM~oF^#xRQYVV3(j4;l+6p4FZAqQm8jE}jL;U0!oPb>ejcn8dmSvHbN;@* z**lZE`@xr{;S@Qa71Y;~Kt2O^hbz04$9fJO5Y>FV!{n3n9}3>EWq~BN%ryFfHE@|T zHt5epRX@n<2&;A5pJ>S|n^co;We;zhxkpXyyS>89ZY0~A}ZpKAHB&8~qy04I06`EI`H|)HM5DY>afh&W(3zbhiY7g|A}!mz^C6l{rh>Kqt@C-*(2Kcuf!M?G zGOU4JdU&yIbeyx78{Bq9kBOIY%^PGFN6GMThg{ye786mrVt(75oOUgW_&~`fq)ERp zMR-_@`KYgcP9!UY-wW4n`+zcY%aY?Q*gH1U-`lqprxgFb|ZTJekDpC6k z3ioVFNXyVrS(B~}7Z;W`?26-XtY_sa3f9jg!=M-+vl15T;l@T_rD~C!Gnz8tMuy6p zHm42UUb9Fu;S)O9Z69a7N@RPP$aehpZ`sdXGHOp~ec86G(BeH^mkE}}j2HO)eE`2K zHYdbIM-zZQI+ayAn5!z{=*SIz%R>f}}WMMPx2CAx+Mq81V{*(!);q=N< z$%$>A%JQS%;p)Z@^OC#*z4b2>+9EfXufZ!hRHP>+5E{R_pU{NxNgHj+hJrauc16?p z?Ozg^_q0{ZA1lt_LS+MKDW2DArr`<*C!Ri{2!;s{zdPx?lDB{!2E_rcx_RpS%mLI6 zV6d~>Jwp4@{%5C0yQ$iB?jS4qU<;jmgnAdgnBanl@uk0$sX=rqH64KlRxPFLZ6`re6!LkAd?A%O zH!lH-OxVktJn`yMs13By*qGAF{0%a#mvaY4N4Ia?dJKhAUS59J`2N4XzHzjI0?e&R zGzBpAU`6}9J(5wKfZy>EbKMCmT=e#^M<=Mcoxqfz%E%bZGuZSOHL$t1k0Drlbjw4j(E2^L>_(i2ah zyE7E|%u%@+-;G;_nzmJlQokE7nrgA&-+1o7!SREO^5Y}zp3|~cy&tSu(vMzBGLm%B zO>1afKTWxKUCzic-jaM2T4;d4E%W99k#ddRgeIM2Xt=4pD!p1Q!K6q@bbtaT-NQ^z zKMRF%=G`^2Z+fgs{@Jvf=@}VY;KF;uUbBKKunne%GKM}~-G~|8Lk}{rd>*i{gaT#= z{rK@42oZ+^D0iiKpAJ$F?UsyB>&2I|y;WtZ##d?@%>kBn?Vb z%kP04)iQI^y0cx>;V#Ofs}WmhwV)~X76%Jq@my?P03Od!dv07e<1gdvi0H@U;jY$-h$<7fBlrmep9Q`tz+N{e(0{#!YBldJBuiQwy|e@Ie@1(lqN-S{qs03;2~> zT?2UgQ)DD!W?DF(eNoEs3k;lvB4PK;wIzg{8g4;H6!Z2{YRDF7ut4M{_O?xoU1a$; zFC(64$&l78h`mTGV|w~knLjFh>?Q`rTzZt&uN+jZ4wzL&C2Ek%cr{jf2C9#nQqG?V zW)}NZ3LTxo|A_1Z&bIvXr;&fj{#g9Dbq6DElaXWFK6#r&+d`StF_yIfnd z5&VUco#S|23ULW=(bD>ZQv+Up3`q;*4IF2@@uS%c zv5?Pa;c-k{cQk;i-pfcU1Vo~otMUZ!mlFgZc-&9Q`zUCc9z3|5r`$186w08+9vmKy zHEc>`%85x!OA3hw62l&GAkd=Ep9l2K1gQPwiYos!cf0Lj!!b0pxhPfjofufh)vD+Y z>?l-k{aj;fG`fy@+nlR*yie7ruMX?du$`c<3FFDg&Hd?3vF;$rr;WRrs7%?!{O#)h z>JwjZpJ5_}NXvQop_coh{-26PGEVdHR!E`bG|jDnD}ppN=9-<`YzCdviRUMoaN)mC z4!6-vCtdkC)puvEbH@5~#adkZXBxy9y*-+{{W4U;dLpPoZzx|AbO_O~D@3+)vFZ!r zRX=oe&*x1@%3XsvFNpsM@=wxDV$L&*(2uy`?0xMASKs&AzzPi}1_j;+y$$?l+C9Yh zWN1cHwE`6t6@Ws2mSc-pS+PJVn{A^`d;t2b{7_d%$1E7`i(?jb^N@O&g@lMHuB@8H z!k?;jDyZ||9D{<29)jYUl9ZG* zuyA{Z^ki49)BYN#Z_&t}ZNV@pNe}IuwLBq|bV34BuS(pV(Q&Ebx=_BcHwR~Z91K|3 z;xFoZ&E;^MM3VQu%vKE@uGQD3D%J;saDzXgb0QR86IT&p`xrh!l@&WR7SM zi~vUIw3O^Smyz~~Ou%^q&QF2y0Nr4odZsP6xh$1R6{^zunZh%siVEb{i$dqB*8Urg zbl>!TgbrcU@W@cdRVa53Ef)4>DSr5HJ=5fC;$7|tz&$6qA~e}A1IOKaC}()dCc+f@ zUE{1lYySm@4oSkauV}k&UIA?9OYUA?Kg0*|Ba6CI&gA6eT$KW#^26dy6GWh%7lJ*S z;Iu#pSOy|0%DicK`mCQ@ySg4yQGwqw`qNP=Q)UE>N%Pj@axDOCv=5?wYTMe{=5#-7 zaPcCrn~PLK-?p0Y{1ErzrUdIYi!uB70L9%fhFFc9EhkhSk+>=#Q0Kf$hpIc{Mb#eB$qF7mfWi_L`^g!|jm}7+ey&zc z4EzQ}DK*8v7W|``dun+#qx&Ule0QiwE@##nolWq=*3JjS34q~;+r$J<}CO(B_psT*x68w*3LUi^>F6t zYPkyI|IM|sslGZ74`7i5Lhk${BSS7xigZd5Li3e>-Jnk$g4D8>x=e2{+8#;| zsoD5%kmb`^GtEVP-FD@?8WiKA+wcZ|dQ`fzCO%Q7F@EhmPmJngnQZVDMcKBNC^Iu6 z2HJt`1poS_V{Kzo@PAYZ=aTrau*RYu!45Eo5M>hdnl8@G;W4|F!63*GY_`**K1M{O z+j28P^Z;rK0T5Py|2CbYod>t?ybO&l4=_xWrlg_@eESv+Vkw#{Bp=|ZEY@?a0jMZu z?QYnut)RI%o*gdI`)CaA9RVC#8Y!ZNL=F;d8Pame8|5H#n!|xO-cw~|yt>mhiyV?! zTVDFJaN>WylVzLX94+1Ar)F?NK56`-%wvf{U!Oq@>DX|yjFATM*8Mavv~Xqu;Qab& z)h%a<=omAcOBVP>KUnVZ`gule)08NYn^9DIO5pR@?apN>(t+vL2q;YDe-y`3tu=E< zhe3yq0pkNI3lVIVnf9;h>5vuN&e1*%0L5Jwk(V>YfK5W3_v$rzz|T+OS^u{F)jcR% z@VGmW?OE*J{ZGyTSK9)@9%JTJLC`J#)E=9^fB*iss~aE{5R$@x>u1O&b_1UJqM7Qt ziB*1f-U$qjzSfZSvtMv9m7rh(9GOc)FNyhpiS!GRV87srV*d#baheO2co0=CXVQtB z*Mp^UwZ4r>&z<)#YHL14?&=SCMc?Ob=Ma}M@^jH;~R*!>5@~5)0i$E2BL5#{qzdTP105*iv^gzRhXolvp zWz0l{%_6t!!6V=YOyatYzIXlxEwtRO|Nig+gGx3*3atEi)Z*M6#GU{FFu`*a|HNFWV1W7EJ) zsWXUYFSXqwpgaYTdz{E4A9^v-L&nwBHT4F`TWB}|DXHUdxDmVz$Ke&Q6uVk7&N?+-RUW^2QAh&=)#Ga*7f4RKY)fz&O5R^m0UASvVF68&AV@9p_Ogw{!D2;lJMtsw$26oeMLck4p*tf+V6Pd* zKff>l^)gSb>AU}(LjZqY7~W}(tw65j75Iga5-Wf=;nhW+cD{aoDwN>-JDcu*{KYH( z&pAw;3d{W6n|cipQ)+SX-hktSsN&o~U@#8XIe*g>4}TqC5p18a{q#xmTk;3jtD;wM z@LvnxcbQ}-ju)3cZ+(oVKR0k_gOkuQ8yGk0Y>ikx>ytHa0FdN$)M?eaex^=vd=G)l zLXeb{owhT)o_JnmdG!r#(&OAMUE~cOs7O76#~)}FG8D(MN+}@8(f~A=zSjL@v!$)O zdjZ;Xp2}`+R#w*F(3I06DR}INSiQ5{&!F-ZEvxd-=x`HKH+o=Wp6r#?T!f_sl`FFC|mkE%} zq1O_cM;C)Dwhm>qWUvhc)h^YbOij;hUI3mSUUjrd#^84~7!*rp-YF3<`DClPD zCdnts@vt&LZt(#DuX^cg$0s^MHY}sruNM~Hn@hsS$-gj4heiT4t_u{q}MqKf@H z2>Re}E`fsJeeK50o6n$A!BXLFTOvs?=r17DQ9@@0j|QR@Ds}EP{JRP&kT(DhgoKje z{zr|Jnh}>xI@7bTusrvr)R40FKi-0)R~^!L!SibU=m3%PxcTcoW&q7z(cLoEH%Tzo z?=lIX-w4FqYt(W0B(J9oi+GVtb52pq@!P-Tk`ET?a}ZwWgCT+eQCv?5gl>GlBWu}0 z>hKjZN+Tb7Fu}A#(#Ei%N75#*ID|$;wSZcnHXj!;_!zkYPrs3p+7y-dfFhHF)>aXRHg45qzsNVru z7AlA{ONCK+-W)nQ?C}V0oOuN&2<%}!1UCvDkMtO8d{K40*7opEM=U^68Ep^D*ROvR zQ4^FH2kYba)dl${JAC2(s=d4w3=71r?0bTl?y%exJE{dv5*#*< z(-mz*YeH(}`}gNT>9)M_NM@dd+Y*Z8y{i}mKaxx;$BuUg_mKEHC^w=S8gY`NIm|Ch z9S*nj_tvL+s-3nq2J`vqrijPbV!N~iULvqtr|AZy7@k^MT82el#qSbHf3Z0iCR+#) z`%mQx6O{9m7%bAHasrvTyPuRy>$BwhIlr|=(BHyIwzW+u{F8hZ*Um`UuB}|Qt)~Xs z0IKVTiF$l?0q6xXnId`0Mb5j%3GPR4z{%6z6bNi;fanxJk=GzN@gFUvg5d|cWffka z>}Za*3q%yjk@FA9df`0pVBq#br^~WZG3-cgC8~Jw&-#Az9>&%6Q{NUk^8Fu0Ey>gS3L@0kD zh$}o;MiGzn)trh91t=e0N3HIL3GJyZYSCQ=kR<#3==H^df&yIB;^N|VCtFWa|MEe9 z;+d$53IT*O^v>OvvF#;{OJX1vQDi>B3-vG@Q99r@iw6;<+tM^NJ&1C|lkEY-pt_0` zjJvS4Pb@B7U=4?k&c@#B7&2M2K(%;4 z;8fxbzlvnFA#e@oOL|$(>*h12O^t*Y=tY&v2Kut2N|inby#fqqZ>L{HvtBXa&uQYS zyqlwGo-z?Clb0g>HnjNtZE&iB=0e3}z@Pvd?)AoWJ?^$YgrTl!InG=`Vy$57eTko4 zS&^)BKRq}*URUSUke5Yro8;#QzJ0HM#ts-Gk}>~Q>wP8j_uhtlTkDCi3f!_CP=)pJ zVn-Zkquk}++1|*l{aLM?N7*SMD_)TMYd&A+qT5q?8i*bAJ|%mJnE8c9y_BF>Y;0~m z$r;!}V%TDVcm*H;L-kQxsQ_EE41>)|x5i~ZqqsO65jvjxzF>wpRT8u;+m%7&76hJr z-RGyj%_YgSX_*iSZR;65&HNOh6O>smnz~VZ1OGiPYq3Z=dvyvy=T$7~LxUV-5jM&x z4gYJ<0WUw;uRBMQFsd-Zfn)-<$E*HS2ExMYmb5eH@V&@Z{6NGJ@R!@knSX(RMBVx6 zI{(fO5v1bkfwpKm^09Men>tpgky=Yzn8$}`JY@a+Bd-HNZ&jFv;skjM(G}GZ0t-(-K>t4p+4Wo4YC6b zWCKVQ9ci;P&mlU%0iT?ERys@G6O@^M&0Z*NF$Q6<-U{D|MuQtgF$-Ia&J`?q1#~H{hsKC z+i5&0Oh)K85M-_aAsov8cWAVqtBrZuwDW{x^YQ zWJGflpJ@)1Gh_j#r>C!$aJU{=f~e^6nzKXUjzijuYyw`}EOpu6T6OLM(7c7H!k#V5 ziQeIk?+(yF%C4}mFp+KT-0l5^>lLDoA>g#kg{zh4xS?N_ZZyvuy-LtV!>JoZVPa0- zKJ_?qr__n0j2Uug*O1&bl4n^Pi-O#~&F`PIk<8k`YI)`xlMm|a>yaJExoxBCc@^^) z7NVd$}e|A{Mq&=@`sF_*b$o-z^y+kiG zmb#a&Vod)?7D3WfBH0U|%^Pmbwc!$r+~|Rr!;ye!O9FYOmOl++8+pt$orz!cdfRS zb4ujJx?vR#p;Wd(%d~54xB|0P;NW6{((!^foDS)3V9f$TQsBMY`xJU9AoO%KrZ)`G znY%QB-Mh+6PtTi#kFH|8A%9er$Qt>NQ6Wvgt zouHiR&&p!Y+w!V@ef1>hue?$&<=QaxFa zH$PGhIhSwM@p%3dB6dGOIJN?2}uO zbBkP{BE2tc_SyKqtkCP%uOXM}mFRxt4=Jn<;<_cXM0^^UuV5=ee%~56eufWy%##o@ z0dA{bUVX38z!Psr#5)jlk!|vxxK z&0C~>H>TG15v@pnl^SfK|M%_lTmKWk1HBt`efh<&pE7pl0O$RwJ9p=_m>dtc6k}T+ zF1Qag>n=V%J?V{DE{hB>f_6tOmCF-hoz64<>Tcy_TN*kAtzG^`F|$108uUfhzJWej z6Hl+$V(iRRGd!M-{HJRw`{oKx(~f^>eI_ew*cHQJ`}?OqFz{fQY8$91Hql+23y%ea zFji)fnJd-&t&+GzO&T^Ex5J;f6e}e23hW>Ta(!~W+@iAxNQ2PkHe5nB=m@AS4a5+K zc1Om*QRXMDZzt`*lmuIR%Dl#ja%P@5VHYforBCJ*h3IAa-owDV0yX=28_eJnrjWkoU|>C zYto_M;rR^C^^F@h&>nr3Lgv`Y%ei-)Yv>zsBao>TFk}Ykqu3*$Bn?*B7(yx+{km}~ zz?n;@wt3OpP7lW?D2nTe-8jP`$C31;V?Kp-G**MlL9X1VEh+t*>BH|dV^19~nt{?# zc!A!1eSHD5M0ciuyCXo(^BQ(On;<0n ziKrk1{PX8es#Dh$>!Lk)7&7+>4mxV6#DbqnNGK87pd7Ht`Q5lU61YTeyWe-=Y~O$M z=&SrDhoViXA!lr_5j&Gv9j*QPWEXX!YeRbk{dH_?i%=6JO^mF^i@EY-#9*Y^zfA$t zvP28d-01 zK};l0z$FsEKooSDp&ZPr2^mYvM@U*V*aFs~8j39>V*Ejq{z;HX|I`I@C|wdC!Za~E z+j-J%qsK5h<~058nxCJak);dV5TG*fv4lmx^BnC@iI0_1UnsZD=JUl0dPq<216boW zs9kvyEu>V7mHmaeazY`Ud`geV^{M@`OoN}=z z@!Is>7*}eq;`->IktrmlcL&PDwKVG0p_*--Z1=E1l+M-rE~Hf zGtKu=H;9&_ShuP8t{z#Zw|*G;BR6-HVo=MA`H+*=n%`ST4A|f_c%l?w)B;5o4<;Fz z=8UP_cw7PEB$A1eeKql+O|QX}bI7GTI%jpVhB&P4JESCtCd3ahl^h+{u1Gl^9MZJ} zG}F`2OqbQ2G9q$_B9j#^G_5v0*4ITNU&rPbM1OV}J8Yl=D{QatrE0#lw$XvCquHeo zaWzymJby${^3ZV$`1E)AWRp}vi=4I%pc%HpqJ8GIGxqE$8(Yg#{30&&rmhz;VPOVF z@45uZrf^OJ02pHn=wJ3!#z_ErGEBZ`pJkOo36pRIUs zcqsq-q3*KB$)*GrEW9m&R?Cd-*6F;7=n>J0MfDEONn4(EYkW4s8d>FqOrr!9t!{2&c5S%E6-cWJ@xtgf<2o!8Eb` zvf=+K3)04!uCkGSm+?|`do6q5*5y9ijsKSg0P`R4c=uw|_{pnY9Sk)N17^a)1>5&rLWX9W z44)yD4%X{)NFxFEGqp{zACN0)Cv#b45j`H_Zj0Lc(vq~_DlUESX}pOp6tXqVj*IbeB z2LKY;Z^+dynN3ti06y9B6*{R;uc^$xSMmMSQ6W^fY;3|+Wso(3K`M2 z>>Te_4Tt@t{;UG$wl5A4Rn5c4=V2_Z8UPpuldZpiqGbRglqP%slSF{$anqemkP}=w znVa#$U%hvYlV^ymRBgjuPK$$}_sAsBrJ)Yj0@K!?~a5V%HY z49wZ|Wk?ePX0L#dZ_SuR06-Kiw_T-4h*zok9#wd5X%xa}>tAJsI}cQ)#$OK`ujMJ{ z+~r|HaO;LOS((JOk|l#gOx%T>mLw$`h^_5W%`3a!8G55ghf>w-`fZuA{fjbFj7%Mg zLxPC9^+N>P3L;w-Y0@v5`Q>1ss2^r3OZeZ}U1F42%6*vPRGXl+m875@h zMd8JZ2AEQY6i3zax`mdOR*=Xdv>;?+8JVr9O$ZAU(bIc?2HuLEzCH)d-qyg)#D7Cf zm!>^xxRGoyFK=avZ)Q4`T~v7ZMxP|}=CE!E3yhT-)M+UI{t5!u258wO-uP}4cPvPf zde$e|5|DK*SQegdaa4LA-I-ZDOVkhffhMEH_-T@a3wXj;r?v;QITP=+W)lRW^M9dmc_Ah_`H$V09<43v8 zN&pG0te=oMX!uHq>7(;SNX0x^R?PX4A#NhZ_HSZyY%kl;euRc;{ZXyBz z)JI^#c5kY>M#8|jlTa%$ju`qdv!~eQri*+j2n4eMRhpNYgwr!InahQ;Ic~gyu`Bvo zt^YOf%PNskIwbMmku5OG5vOKy$Q^O zz%PXB_TsTwU|+G*Sj)~#0A70)W>tc=d-&x62?+|anr0Ws=;4EJx`e;1+Oh9|y-m$g zhVC16Lgqq;OBd(Z(v#F_(2}$}DJB(jvNWnw@DV_inwnY@oeDTf&`USY$EvnBGTK+M zr>7XC`>3S%aVrb%w&%aSi+kDoBHLpzdfY2k4&sN)Z#$BLJ!uX^^7v2WaKtW%(a2u8 zT)?0Ah{i7um+?m-vl+}N_cOlfU+q}OmdcQP(Z|`=CEaaEmYDozYcg!ly0v26t;Yqv zzd{3aLKu{{K2oxNnb%f8z@l>*MhQPaD`~#ZNwA}&!8D{V77XELdwWhOdrnant#1yi zdc%&F@EKIz!UV(XU*G-F-XgQiXvkN8tk|fjtFH;3L4$kUn<55y_ZyU<;I*-`OXXI? zG4qXD$NOf7MI)xo(JZ=;fYr9@v3?-oxvlMSYUjJ?*Q6trz)uL9_9|LHwwjqUU+^{=E(T&>uDy5({OSH z{q|9^RQ{G<*K;>q-%F@+GdTHADj69qzd#RwHYlc3Z z1quaNty4!OHP}XbWKmBaa64|?drQp5SjtISvOfyLexQ%Qz)BqjOrl)FB6<5FTU5`9$H09DC&BR#5C7)4P4r=} zgdYR5uPEgcY3U%zJt|9y(7#`C5Y2K4dw zS{eZW#FGWm=Nt6E$9vdhq^?ekgrXSKOEF;O?T!xq2v>MH!B-j0vz7jl)6|p}lVM(x zHn=;$uB+&Ex1MsZ`@R3EqhHw3tZ>-Yf-Du&YtE_YuGdwZH8?s#>orQ4OCq6AWiLiN z_s%M0?QCloc6~$Zc=Zze@0X+`3pM@CPc zweOV3qpf?@vO0sX_jdBj;#V@)!;%WcFw?vF)KZFq2n{U1ooV!aVY?^yl#Pc|Nl_6E zt$BUyT0&6GHMq9s^AY(h@P*5K3pxgZt}#G@8!=IIKs#U(!dq^~jh!4A7=Zb;TiE;? zZxveBx}nTFk564pG7mF{5FQx3#-$3&ZX|Fc?1|i)WmGM^jL5>k z;hhDFrrQsy?fAWgUr%5+5Y8$oE2ls&0HkC;cC3^$Q0{)p3&x0w3} zl%cK3(kHwQ3eXXS6%_9L4*PV=<*d4E=qEMR#q!N}>DwtLE)xkP@HJT13RFxa zgHF|83n94!#Kb;Z^f=4xndRXVv>@$y^+?5{%d*)K%4-8?&=~I?PnYhyEcZPGW9Mfj z%KLuH?$0UTnoW&@msssrRXz}~Tm+!}24#JHnKV!NCX051ErFXgXyEXLN#7r? z+^zXshRcnIbK#pC1}DF{s!M7!vVJ2bXXdcDBEaNUJ-aatUqrTwHK; z=Ka8FPR+7)$eoaYg$4-o|7-5cqp95gf43nSw@s2v*+Md9p2?IH6)MRPQc;EsDHTy| z^OPo}NTd)_35m=b6^Rg$dB{=iqNLJ&J)Q6Ockli4{&&x^)>-GQvY-8YruX|beeUZ- z*1Fc9VLBBQvL|TI^A|53=~>R#acyu#c_%9`AGqir%%u_iEB2J;p=>0NB;(% z1aS(Dg>#~dzK8s@ou5qFi_iQraCB!jf*%?y z-@8dfTs)BwIT4{5Rx&$3o$<@rj3f5C?U+!4IT|7{nW^UwaOjnfv3Wzy< zSY6nzeau>k`b}!t^6=+OMe1cLVzE13iM;$gTJqg+!=<qmUD zw>jNGcwGCe?VBSSfbo13uMJG2F~M}b`SPd5DvXciZCjm3OI}Gdh{bbl+xqDB zppMcVm1`3F8CKiRJ1mb0Z(nr$DB3Z3LYBg=3@IJ_B)rhHG{G;Wf}{&{svmO8*qX-! z(oaWqb$3gusjYxgd^t&7T^(5joFs~v7*Cqjgdf-8hi|znoDxAdO327GEgJW9r;;GK zikX<0@TJGc#MB118$<#1Ux9KqSLMPWtgIKFkB2G+KtL8n1S%k)VSDTg|7X9xaLH`D zHui0l|M*R(5pmQQppj8qjPt*~LE0!D8MS;QX(nLq{tv}zCN9{$oOGqMo-_)+BzJ(4n#--m_;lkT0}%vQ4zT&fmv2k^48|`vjG)Oa0`y2 z*lI#fz-%H1Nj2?7mH|6zV5(po^1@uSza83=RQl+VgoMP+9fdI>A|gcPt`CrOd;AOk zjemYU>=zNE9M5tZIi;wmNMUF7bxE4Ex@_6}Qi^R^c>Cx`h`W4eRi}^yyPMH_whOA>SaZ`o1YIG^9oVnz&H837eIhBKzHkjF86nGcD@nxxUdYk?*$0! zn10FgG$xG;(2<AsGPpZnQGDC)TJ)@kD1%(m{!>jt-~6t8@gAL-B%EA*};25fNTn7Gsj7%_?U~g z%I(NiF&`ZRHA@HC!vv4AIt=6+KQnq?zMIFjyt(DYiw2xCg)U{E&K>%(Lo!)HS63L~ z0h&XhITjlLOE=|XPEH=vMcoybu8p@B78VkzPblTNNt#%8fD@@}*VYezR}T8Hi}0ZZ z&lHyZbNs~Gs-2FGUcam65M!qY6JTv~=mKUbv~}2PuM+P#CLm@8i{-)N<}4?ZQ*w5a zCYSA_M;vr86U0huX{4)dW2sELO7HTmn{P{a?Bn*Rd6*I0TXp(G@jfqQA!~St$D+dl zAbbzzX$AllotrD;-T(OIG-KgGwSW$I6r8k|<~7Yuz?EZ$1VCWsLOOsqCYY#beD zF5Asn-NXmG;{aD@TcM=c@#4N(-W(h2LdnDCJg-j4nxrVBv!j*%m{yO9oGPA!S&leZ{tPN@2?u?@40*L z-Xq^FHJX<#EspwWYjsaL9q~22bgg?*JI>aF*4wUzjHO)_P7yST_i2qx_D*ASj~L+v zjZIA>prz}<+855=-SXb4@;DD*vQLZ|Bm^+$?P~398-k-ATYOf(rA4u|l7BVO>XHY< z>s%zBzN?oa@TNIS(>m^yhxG@Z3+%mGm$JK7zsl+U(N%ba6c;DD;~E+>wL{#n95X%? z6ca*WF>(RwTM#EVL3|mUx?xwa30|4CB4G~pMNc^+syl8aYp)sSpDY>U*A&#jH*$&i z!KJJ$MSAw8+3i;8nb6$VB78M0R>9vQc8aII663dhSR_u3o2eTGnltky3MUcn)z(_wDl<((aMZj zbNu?D|Lcr%?J?PV_`G5IcwN%-A`@c|_2GPz1Y@efPyKNod&jZhzS$#!V7ifkW8~wb zX!v7nMQ*)YVQg%y_$s-83Q-gVI+RuwJ%|R!d0W zIM?dh@RqGl_EDKb5Sv9r#`lV^hj7?Y?IiclFBw)f_DRxrdIXS+WCbMU<+(7!30X-R zIqHg>oZN-f)ZXkKu+CfrV_;@R^E&s011A95V74TKZ$fM7>jt9&w!EL0Nl_-+G)?iS z^D)0otl!*f+8uFi+Rr|3w?{bb;Ek4J<|~StcMI{$xakU*<9#w3(nRCna3WG4iC8#j z>+q352wyx0#7y_sFU$q#2-#8HW^!eJ7)0#hSy(jYg2ni#16fY5wG3WLXN2Pumr;}A z`hP}^SebHt=zC-pRj;e)z;e!B^HN~7O^kNl#s&qS(*t{Ujg?!&R!yij;aXq@;CVn- z)Q(6lg;f>FN)I1C^oNY_TQE=3Lt>($k*Zm-L%+6}5p%*T(-LGN{h0Q>*PTvvk6K%| zVoyEDyeuTADRSuhOhB_nS69xuF$aTlw2YI>lNGn#V7j7sgnPbYuPABAOI_HrG;#MF zXGGwt!uf;xH?}V?br^8t-Vxi5h$SY}1obd?okeEKzS*Dqlh#$cc*J{Pd$6%NiwYy^ z%rg))^fv{27z?9G-=|XBLY_e#tbz$&JIeSc(d)FHA!GgfOH z5g_Cj!#LR*K>M8o@sex$B^W)`p5lFtzy4U(9tmt7R82c7z~k+3DR67|xiM=Mntpei$J}ZcDSl%q{(-p|&Io2>W8>|5 zelB#RD4^ykNgO`RDAhuY67bEZiysOW<~n!4Fx@|?V({blx1guMiq-=8tM3y&J&I)t zA>S(Cyvl$=-|@S6MDeaMcXk#7kx=5>6SKFkMmNA|$6~0qs(5L6P!e`|=;+*AE=q5I zEDt#U_0LMD5LPy;D3cHi z6Km2%Q|X0@d_JFDQ@w9;x*z5?>qvVU$D2j_g&m*enc%#q7|{DFxaKN4US4A55@N(% zc6O>_iJJ))5@eiz=Oh}qwJIun5GMLY=NDjIGU0tJ&^Im-3*l(CQ$S$i1b;5RtJPLc za?J&0jzl|6lRPzL{E{Q+v0BgF8g?iU-X?5dZf=oAZd`l+L9>c*0veK`fW4~?CtjvU zLA=5Sy(o8(^9(dJ*$*D@L!a~E+ew_u-KE-Ack}X$F)atr{PbaEWmV>9pet;Ma8W-8u8?`ChfK>3^s+)<1tt@Gt=EXpH;$*a_|41md}}=ZX3t$Sl!;GoBkEh82uA;2 z5Skht=+e!WjB+E4)bN5Kaaa(jbnO9E6*2zoty{PHK8X$kZIOz^lN{Uc?*pHpKm#Ao zfg{E5X9>qNdHf=hHl?j@IhpA1HVY^1Tv6v=vL$Bc1HsfZa7hsa zarXmSqJ|+tMAMH9Jo|^A*cghJEq(o@Qj}u?I2gzFZxokljL6zsXzE|cEc5W_Rb3f1 zob^=68nw9V*X(#&$KlJxXL0SeeRJ|oVn)>D1tC|s1nA|h$zmRW4p zQHZ<^(ldSNPA7xKeiFLP4t7BJzR$xV)WH_tkZq&nuaKe~5y0Wbg|2oB!%aE;rMANx z)`Z&choZ)X4VZr|P(JH_8VxwVHEMY&XM|Fypd`DS*TMZBFx(>NnyVa!`nMF zJi&0~!I{%=%))mbwX-YIj*mYJ`?cT?o(Q1r+XNdXxLF#&n8p&}Cf!BXE@a$vdL63d z#yQSkr>Rd~KDz&0#|CR!{hk@BSEs7jMaMS zZ=UhE|Bdcxh=~{kN(zrm@6(fN@#gP)Zx)o73qkx>@bVM#$N$^WJ8CT{C@D2P-|#FR zL-aNwfwE&$MMEI^UT;lpvv`IIb)m{Gkt<>ALhFDRkDOSY!M*0g*3+N8g4CMmv{#Xt zTD5rFjvfJhB6`Oasuw@V6A_=&cQTviXL|6uZ-ZX|JALr`hZ&fRktaLdJ@A~1lkckolsJnakjDje)T^aNp67hbiyYj)rB zwi!)w(2@DZdz+bVeD9t>$n9n(l2S?>l;Z+uy@Ti9?D3y^MOARA7S+!u+@^BOKl5F&Z|?>e!ib(eh5qXG>vN#^5F;Yw z_}Dv(SGqwys7)G$4c_3L2mg7TWwnwU-(8}R!wh+&fL0Ehrwv6J+C)+Ge&jz-@n-C(IQqF}* zVj1j2Hs7>WrNl*6NuzWaTP_aqPPh3qTaWNIPm}2Aiuv~e3E~^~`GlL*&~wU_i__N! zF1K512+3j?oq)z92Br{J1me}WCm9ebF(G)XrthDcg`u?^ij617w#jeUKu0~1oRRVI zfqS}@46#^vt03o4EGwXc(0E0mgvZ40%P>BT&fki-t%I6E!N}thpErc*+=0;6}OE&qECJ;0j-O{%dV~# zCseRNM3!HQ2~7`LDIn1MH0lftdd)43eYN?_NiBe$O*O4Yg%V)vaHW0+}`diqE?n)I-Rgk@qn+#Q6rUqO`<)ATEx|| z|5TwqA9Ji_ufsZ=*wfd?d8wsqd_IFeQ;)uaos~%4Qrh=48q}*B;?{#94m&;1Yp~+e zp4+y2wVJ*D{6fUVO*}eWH4hCx-djKj7IG!w4%&x^GWPF-r#F;4`DW6ey`51EYZo;j zi?nUzDzuvkdZ`?;4Qwk&>}|+vCKIJT3`dC&FqemZ9T)^?NrZ@he)i|*Ghe-2klK*6 z5>p9^QWmthRXq7{|aqf8yW<{z}k%#ZSL*f{Jh&d=XC3{2J*r!g(Wb3##Ae zNn6;{H^oFm^K(H!J>-pt(tISTv+LZ_*(LFc?i}+xe@kk2PRFQssJy!QTJz0YSAQCt zNYMY}BDdT+xzhkTV+`L+CI!!{UGgJeAB>li+B8fKDy!vF{dB z|Lxd`ngQo_%-Us|$D zuC%;2ry*_8j+>oqgo@Zq{I$*tPfgSA+{RSd9%oh`>lwc+n;5}DruD=8QWh*jwU@^3 zHMMR%E6l4M3ER@^oGqyc$v{R_9B`Snz-Ea}1RGW|Qj4a3k0K2fS&$Y{f(2;2A!?Te zBy-B3XkRfMn@lvA-0X^q)lfv4p@L;)?xUt&7<6@4@@H$6rfIu8GK=<%p}Z6}XdRam zRVdz^V;|mdlVlbf5nr?^x%4o`{rrW=cOi#V(}~w?yHf;vE?Y!&a^D74{d`}i*eh4g zb$I4T-igBdi$v%LVG{2aD~EwHg1F#>EDTPzRBeAr1R;fEbvMxI zdcS*#$$@AmP@YUAxJOMP@vqEQl?w6n%g=mu z;omC8K#CCFWA};%o6DB$41HUsHTCr@D*E~qbW=t08VlKWOobgZ*%C_fUye({-?sLq z9Sg4^yQm@e)Vydz!{(tHLnT{V(KYqlMp+%%Mcd*~Q#l-CGmDF`AYYPDHJVC(xiVy# z{6p=+7JnryOFTLlj1d&j5mqv$u-5^XJBNMv4+|+?M62yoIuV~kp-^7kai16;C)~^3 z#s9%cOcVr91n&duO^%vOooP#+Qdiz4@>{_|&}yFBJyrHB2U^BSiY?MOxqnag(EZ{{^N z+4%Xi&9-=Po~|C2PfuAMab9j-k|Mb`k;cc3DsL$2zDj>_!D|lW+N+$7Y~$PK!|BG} zH`&(@pJo60oh7BaFPIe*vdj5pDaaa}_=$6K$4CU6hVBST<8!DQhk#K6zA%RG9MxQM zYHBpt7-aiW)z#;6Pv^=CaQ?!EWseMM3?|SwYdgD0c)WO^H#ZLo(tr>q5)z!Ok`g|c zg^)`GVoP`lUdY#H1ZekZY5`j&xz^00_yS~?kdSbhD5(^mrCWWlP>%DF({T3{dgDlryVJMQ=7 zpVNU;6u)s!zgc05;9-+9Cq?$1+!~+5I#eNj$or+Hk&m)*q1LT!N5gH8n#FaKP%aoT zt;t$qp5qm8v$}lw+hc|`YTl{i8sB3R6U~nuQ-{!;^sfBn=;HiLIo$bxCH&D!!NYAN zbBXwZcz#6H;j7W!Y8ndq6M3yeGc4BruJ9p3=m+z@Q9G9`)gOZAz%yoTqkk+fmxqqej^d zyrs45g&pYq*e|{1W=is!^?ZCXMd)nplB|tZjkRy&TW|AyY21vp=lp(2t)Uk`A8wzl z%?!)-jAf@tTGwn2U$au(yS7g<>QF#x{dhnJuF062>1qEM2!4v`UHPV*hrrgw5Cfa4 zdXN(@>pZkF#4aLUm9ER+S-WD@+KTu3x%CpWQ}d@}uWX%~UsQVkbG~vuRqM2z@3TOC zDgG^G1MQcC?${Lg)Xp->Dpj4z7^}3)oHeQ5O-vHsMvdQXW7@Bpo~3rpRsH?KlL<1p zdf>&TBUVD&yR_C)Wl1<52k(yWpCY+B+#20>A=+dA!P`cg5+_mjX?pC#5ZmnBc^=*p z79tQk2I}JsPSFuQW*Q$m<@HAI3>np&8F#QrcME*Xnx4`{uf65H&|~5`B4IYNB-lfv zO!(`FM)wWaJsx5j3R@yuW!5!*Y2TJAI+=9kXn?gX^Vp=CkgUHju?U01s|Y5LII!TH ztVbQ3gyRT7@M>a$RNBG5Nq!Ud>GjF%+5xPoDqPJ%VjLt2dw8kW*~6VeaFM~@#9@*Luts5EjMrA~D=5cgipbr-sy^@*6jdvUe3BbTRo4c7)` zv6cRU|2_lE-?0>a{&Vn^51}is`xgSLKW0y(Ts@p=$xh8FJK*EMe$r`<%VV7G7O>*p z^_HyuH@%1PL5Ql@42+W4Dv`?*K7E0ms&Jr1nnroOY9;4N{_{T{?^(rgx~CA}DH9cX zKy@{w0*OS1$Nyk3>pR`}Ozd+*m8i4}gSH~^7xIqbY*iEE^yljq)KR}wuk=3jyBI+Z z=I6@+J2`83lVWnJ@uDR&^VA1l}xvr@-vdo#51I|?|SL!s3V^6~MJ$guJgr-#t z5!5+p+9noBrR~I1IUXYISt%#Q*3rm+w5i8a{ne3DvccXgYwh`=DzP1RI*kkMPMoZ| z)3A9oNVFJuJAVu-bTXhJ>#={I4pMmh)aA4SQPC@<4;I@MEKAy4X|$q84wMvb$N;(C zb_|#22o`vC)+Ny)YEvpdrB-brbKCtJXCVAM^>r z-g?l3Gz19=?l71az^Q2BGe+ppA(#vvKSkcHV(?d%{n@D zS~)tdU5c0GXSr3p2%XxSlXBSQ9FkzCiX{Q_$3fB=OIbj+!iod!G}rZqDQocX6^Tm zS&uRux6a!|u!ar$ReUSP1##f3iSbiu5egMj@cvI*V-6fbVe$4>6RA&xzV3d}|3*Dm zJ!yZ7J2nuea2|Pm1gC5jWyDr~-?%?SY^Ofz!J9FpqE z;2RN#N82U03Fir1lBwhvN}UPt=Fj!Km@rgY;|r3~&U&Uy;9*g|yT67kTeb{c#oeN! zTmwaRYSnR|jI^jIccj?^BxZLjnc2z%F#L$ftf~DTMeF2xGL*Q+7)L;rhEWceSC~XA z;sbN~ojItX$8T5gu$X!(CT?&Wup$T9eGMTf$!*(sAOQD+&sABBOr_lh^0O?>DiQ`P zmA?-p{6_aKkIWp`oQJN!i1|HvECc#iDBD+pD6qfp%A^r3L7VGC4s%SD_^IeGSZX_z zGzaYCOt)o+a?LD_-)v&ZKwc_0>^n7(4yoXls@!@i?fxz}7l~#CIWVe!Kgi(`n|fZ5 z1^cbNqAvAM@b&k6MX&C2ewSP$LdX6ce^`+TJ&%L{=fD4*lHPwoK$IlzvosxOzh~@~ zRpMlRUS2JMm>hmB)wgho^4be(3?gWzG(iS4A$>4f_;zHU`A=C)9`#G3X=veC^LE7o z8@l{=)W1Q{>xMfOE^NL1?;D0t4VnO>8utOZ^9bv`pM^VW8__op#?MUVMBX1_2tx#=%f~g_l-)DcSfaE!{iGF5XoTu zC;QD$ZP3#bf!-H^E0~U0WX`^JSz$NXB@JhM?IgE1t%@2R286Wz4?Q{N`}4xf<{6;{ta`IxA4ct7QlwIO5?BE~*~TIFPl=k+VMYkNlgfk_A+*gHKdZBse_{ zPM-D)ZG#&Ds(A_2@lZz6OEM}C+BCObJOBE1un@OCelA~aS}X5GZf%v!+9d}CRO_*F zn(_x2i|A-;0oJzs-M;mDdShu4VGCcU+n7g&H*MdvX&K4YWyVm*b?Qqf3wLkN6JC<_ z?%m^UuR;a^!@{Bw_E)0>se{x^MdeX_?#QB+Zr35V_tgwi_?IsMCnC+GH|G@<5fTg^ zNh|j0OHj7kEv@!HFHfuzXwpEkB{8FN*(SDo~xf= zV8Y>CgSMR%{(O7{n2V6Mp9FE3g6L2a&Rmz+)7{NRlHE}D>~qPjk*4nolRDJuDd;F_ zu>5bF(vzk(e0TEPwcpi`#QJCh`Nt(>N*L4*`$;5g7oVE^`h#4!`^y9aI+knc>29iN zy86DVY06N>OReLLw2*r6c6UO;r=GOYyj^mL!id`UeAw^0;Bu^(7Nw={$UjbO_OMmWsB z+YYEBoW5+4tjc1X3}wXEKxRg09$9{|4c-2nqHL?pZ-R?m5-Hhc*`I9}EQKV5NHJ#T zsHTy!fdxF#zfWK^n za#*O!;Evw=3@rr0{eoM!ZJ}fDjms%+UX9c{iCgs1%8Ch_M_Z?iOg?_ksJOay?)#l8 zHFky!#-IC;8jcYhJNg%}{74jtSCQ8Szp&u=_;@#O7jQ^)--s^dM1<5oA19W;WJoG0 zwJny~6*S@=x>$H)`LThEWyoLI3YiV$Stnd^QIvD{R@iMF{)E7~wAaH!q3i<|sb9rA za{5Jwrl*i;en#x4gr=q-uD3!!(}(&UjSpYdo$MDiI59gN^080v&p!qkplVQ&VOa4t zq&4F{3}$rVQo(XiR>bAExI07=q3}R*0)IsXu5drMT-!%sK5;PC-7tKeLY+#bLj>a6v>3s#87v zc5lu41FFU{m)!7eh;%x?xhb1;_wN7EUX~di`a#!4?ko73Db)$wgm@oWKm>mZk&maZ z??=XI|8cT*Q;089$%?P^n_842RZ6NM?P%vT zcA2C#JEA&zO9eHBe<0z^ymR?cd+kp#FC}y{E5m;i&y0cu;0TM~==F!Ym4mbnotQj^ z_&P!iFrzcj-96ED_1)ha6GMmRHxU-?{QUfTE+Pr$82y>$eQj&J9UUD@5VVvYSNng6 zEp?Fa=o$J8mrj-`eb7vQi5AGY=|yYngMBZH{}otX|H|gv@W8LS$d~l(Qy9mUD{HfQ zxbS1MSBLZG_h+nW-bP7+ZSU`um`)y|gznW2d#D15VLWb**-M{mI7IpHR&nQlB%LFe zy>$lP-*t`?tLMsAX+qTb`?s6`Cph!>I!CV)m*eFmJ*{p={sj|R?8%QhyXR4#a52ht zoj^87d`jm1j zoN+WP!RO`PhiLf*giP*6F zl?O8wUaBI|7$LT2>iV&UJ+IK3}=iwsB{nT^w#V z83MEn3H!DA)9?xrfGZ+TlfYBC;t>eV;!klUm+E-53?uMrYl*2TD^nm@Ix%>19pGde zoaE30_g_~Gpvj+E!yl8gXR@e@p}^1?z5emnnJA#_MDQ-^*?&6AX_WWh5cP6=Aq@D0 zqMf?<6qk5X&mG>FBKteT@w|l}M`9zs-gxW9){zqNRZC>&ZCe-|`*d^HX`Y{&sqjjY zZ<|I$jb5k~ad+2+!7q#bZ`cF96|a{y?o*4dR=9rRzzgV)W>UR9*^K|*IwoqmiR_=& zhQqg{hx^tBuF0dChXA*B7dd;0UE+Bu@%Il!yWef5Ds`S54*A)3WFTno!>Q+WKhhI) zuu+3+Ytxkxxn@kvGa7~dSwnbR%D4O90A=10zuxrqBLmi-81yk+x8f2zMS$`N65C*kjPYOnVks}%lwdmOc) zz&hFV@AqJ5$n6^Y_d!jba<0_*`^#yRVv|j1F^T{G|JNT4Nib&twfX82@Qt0k+ebTs z)iX}S!;k|>-gR9=;*l+mGB!!_nMrnR%e-qX3sijU5ld&Vx_H8*!T;0CX#K-ddctFP Uv0#%w3IA-_wB4-8#P!_&0{xyuZvX%Q diff --git a/docs/images/chapters/decasteljau/df92f529841f39decf9ad62b0967855a.png b/docs/images/chapters/decasteljau/df92f529841f39decf9ad62b0967855a.png index fc22756e7653a2edb304f4c38a42d0534ae6487a..eae2378212f6c30633128a3df05530fa2b180771 100644 GIT binary patch literal 10266 zcmb_?c{r7A*Y`%o$UM&!l8l+BVuw(M5K?U0Ec2Loo~6kg84@8wkuqmY2xTTxGEZCP zX@6_q_wzi*cYoje9>@E~d$hy8uIs$6^E%hL*82T^YsKo`P^TbcAwwV#6xTG6`UnIb zE$%-F5geg^_47UaBC^y{Mn|>PY?($#5Lp<1K($>-~EjZyiSNWcAH(JsUITf zBbf!SK8&v%&UpM1nHG{67NTUY##eW{sNi0bo${>!*KN1k!=JUJIlZJsy`;HvH831; zOhjUfQB)6kKC_>S?%qE}X`9)&3fs>0^z7GX7tY5@%m>WLKjmzi?%U7qBVmd{<%E@$ z2^-(OUE*G>DZtU5@B5OzPkShSeY&Y{{PN!i7eoyIlT)<#>w%?h`POG(T7M0ta#uvCXs2uQshXHD3tqS| z@bxP`f=SYW?ko*yKwu!s-o6V%GNuae#cqG^9`f#8j`4{JVs;lhyUVJo1TXZ@x3sp- z%+9tAkc?f$UG%Wa>Qis;Rdsb@=Sf#pqhZ>jB3ELBum1?Q_!}mgE+;nSAhv;l0X{L+ z+ze7~IbBoJxCal;7#O@q&41Z{sHw>Hgp?aT$$l2|Ur69^Zhk&NK|z5TWh!pI1EIym zf{4pT!|VOu&`K-Y*&Gs9)`XODBcpoa+(_bs_}9tFWtx4jN<9e>Rqx&{S!>s-;buie za;oVhb5k)FKUDMZY9bL>%-G=O&G0^DtXUATx=`6IKWD5X&m#>H^fyT`pIvTU9VS*0yU%I^5?|4tQi92Od# zm{8?;)miH`wz2&>wAxF$^mX0L^6hZ1A`J3yJ z99#LmbUK7SYm-;>^r++F<1ycKlu?_rUA{kK`Kj30$ko-=BQY5n84puaqu;%giWFmK zW255WIHRuKi7Ic{8Fyq-4pPw9*C%3EcKdnC($ccOzdvWg*v?LXhK8npWF(q^_zah{ zw9n{#L%_Z`JlJ)ng;10{e=Fnl?V7x@vNLtwtNM$(>Y2kNdU-Ej5=BNv3J42#e6RDN zW@3u@;4yD`e6U^d=8fON_Kh1ijNiL*f9~#f`_(})`N4yDZf?%i+dIXju%@Cy*VpOF z6}&T?N(w9QkvyT{;o)D==gyr=lXl@?in^k&|FvRyd17|MpVRP&%vBoDJ^0*dM0Xa!!Tw2P5nDR{NFx3>ioQEzW=>(7qP zPF_Vtc3M8|hnzRP6H`+e9UL7EPfFp*<<&Jc`BhbP?d|RG?orXv#9`s#m2SV(5?G}l z4DBv0ErnHAS8L9nd95T$9$xJ@!XqqvhMJmMGbt9yz zibT|JYk|64kNzBOi#*u&`_d2z7NomztuWo^oDV`Hz}#hIm@*@}ycf2^-l zB8)372?GKGX0NG^Rp#VyS=rjEa#3*!2#_>~6Q5C4QPDLrk`lN^dJ6x3uhoJiLf6FP z)mNcnPI$y=)n(4c#zu!B<-oE|aWraOCMug1HS|KQH&c!__ zc=?h?N{RvTFg=|VAs{5wmUlJ&X=7@hBZ7;cpEx%+S1?F!2~SJg0B4MYMSimeKYCb~ zii(PmrY2=rmzmYo*LugQ6PS8T2kU5qe~sq+ty{Nv1qI3Gzhs4HWn~$yXQiY>6=2S;8hF zEzP*Uzb|N>-@6y3sjb}>MtGWvf#IdQg6SMp{>yvV=(1(bCH=y^aSJ!JDW(zAT1|RNPmO|(cANl^B;*|l}D}&GDCzOFV=Wjfy%(GG<|Jr}hnI?wb zn0=zNWQC1ZD)Qj(G=>`xp@%tsbiGEQAp|cVRaL|5PxS8Ib*?y6-ioy0izq@_9RK=~B~L;|rlO%iVpM+D z33sZXq~!FMFJH_kRSRKqiHQbSS0A7G-{eTJO=cF?iuO2EFo#e{G`%f)kPZ%3lvHgtWRchMf%?F*}8e&05zEikkF!V`&=?OJ6 zvuqJs7R5NE9dk{$;N*&rxo13OH`@}PV;K2s?D^?e6WAeR($}wF8)gKht0d{(d{B4~ z+bO?ZpRa%1I;vDI)(#d4PfScSUAIkmo|+=7cF*EiRWP}wvom@pn2Ae?&KysZPpz=& z#BK5Gxl59g11FTc&C!Z{h;y{zg)951qoc;!S#rrsoN6JIKJExjKB8;M^X8_erUH+c zNR=kL@C4GCz!B1f;mx!1=gs#*VcEO_R)atL7=)ZFT__KxV_^7sxZX5xjlJ2Jf4dcl zj$7hJ{qot4aLQxcd|2`ii1p}uwLgh&9g84o4J5b` zc39#8j#Y`7%8U$_XG(!A*4EZl*l@eV6j_D5qLZ2VdG&3|7P9dwo1Yv;*ql0%h>Hdi zVpj~XTxK~Lv9USx<|HF~Z+gT&a8wLppEA)jZUqx+zaYNn?=LTWx1QvcL9rh8=AERZ zpBU+EzGHLsxWe;Gc=BPn+ew`q{^UAJ`)7xgW3J3-VdzEQSc&T&=a-bkuQgJIx27uc zg*`7RAP+w-1&+7uZaJ*xgKR> z^K!$_?<6vfrV$p`_y(VNI{AkW9~KrCniJPpKRNWOT=gMDG*7g!KFZV37(XWybR^T# z(lR>&hO;cTv&qYaXL@>i=BOBjHc20`D=s{0;!D1UX^O(4UjH%mZDmarF;}~Nn{^e- z#g}Nq#{vNhA7NE(OQCu8*e{L4s@fXWyCC^K=va=8jcs9{(ei5%q9pgk(y^dae^EC1 zJLTQCH8q`Jyq>uUT1(eGPe)@ndIQzs9^q3t%p^0EU_F{xBmz+xX}lj5atkOjfZ+LA&oM@i#2aOKxyp1q3$p^Cu5>|7UwV01NmC2ny=^-Rpe7 zuA!-^a{W3*W)njTI>2mO9-5kD@0=#08XJ}FjaQL^CCQRt zL4Fma-fh`gop_XyQ9Q67@-TF-sxM*)Vz`W~ENZ4DGEK^f#mvl%i;j$&jtpiT25)2b z!JQBsnXJ6GudnlH|4%e~s>WGYS6AKp1uU%N&+*P9v5c-I#X0=#At-en~6)aIW* zhCw1KYAuD<4{l`lR>BKeSa9JJ5JWc&4-G{EEcmrrGtIGM>g2RqVM5p{I$D7ukaC$4 zQs#rK2eIfyK>@1k@i|I%9WAZXD=RC_qU0HwnaLak2oVvH!QtWJBa`bmFzV;;KQJ~% zil8Ektf;IEg~I_`29Qk-Me=;wzeXWJRfMT1p@`k}=~R(G7FJej z8k(>-Z^R?l8R_UER#xtjoWegvNO)e6NI_ug0jF~IP+y4)SV-Xf`Hyhx4S)zad3pHG z&d#vUHa0g^bakn~M#PMaj52a^EnQvqZF914b9Go-xVzioj?-kV*XiWsWErm|x-#>o zGr77XoSdAeh=_OqI6%xC{=+Y27Eqk-`dAxw2NeJ=MgJX(?v#t0miGW7{{IL;NlB@w zv^1iykRLW1@xKwmEavbRAP=y@;Gn_2kbr~)ErO1NAmrIIwuq>xrKSAk1OvhLS2qiB z=<;Z@TZDv!ai~ix)3M;pK@WlNU#QUkIr!7B8DGstN0Oj*e z;?@E}7cb)W+rx)cz_flsylsQMgF>Uj&uOJ_2?-^9GkGeEd9_EjVDUV*mWP)YSYN1w z?H5-?y70lNsRs@IyOtlr2#d;&6o0C5R>jPW?E!~kobvIa&fd>> zk&S5OBRq(KsycleDfAr=bo{ju1R8um2s8k-eGDf~cDDJIpPkJSurp@si)m-X?Nv4f zU#j)`KVK>WWjZvts4Nb)7DwxRMD%ao)YH`sSy-@}{ry|f)90<&%(pDIX2fGsgs-2U zCD>)NPCCE`(1y_6dL|$!xIt1*<5qNpsOx)&hiF*j(f4k=;^N{J0sHPxQ&QX&g^V^9 zZ=A8<>N9ASB;zqh{`2Qg!1#7-N&!rO z`G5Fn_>Ewx79R1cTflm!r7`6P-H5Z{V_U)+HHH&Yx+~H(JnFI6?e%6a`daT)b0k*tC(J}F84{{0X2$06rSN=jNU zl-GxKbf_IIn`2cjUq+0miVFbjm+@SnQcL1Uyk#Cp4?D_?^4djD$27E&k&*9e^+<)w zw9-Cp>C1(>^~ToC-`$v_KYi$lcTR&H`lJ^wTqwO$$KW_pcGcZo{0y6n#U-DBA{$G@ z(b3V5<>mN<1oE=zJX-VMW68%)o?u3rP7DhQvq?pr#xLPaS$HF>#r$NGF@G;U%7Kpv zm)zcaF4EvATv>EWF^paruRdMpv))l<|Lr};13fn{ul8(3e+ygN=v#O16MA`hnNiv! z0d-LUan{h#Xoi#h-@o^6yBb#pV}o#u%NexCUEk2~0Y<;6s%m?AJySIJBmjI7f27=e z-cLhQueU8k0nI~^+@zJcXg4Djg7(cF2eLXQ!d`F@R+}mz$_89Jd-A5Zf;%=UN`#AiS8`G>L0Usfj^HD=`op!l(Kptoh40k1ZM!0L zd)q8NynoLnC&%Kyvm&;4U5mWc;@ivXzH1-mMSa%H%A#9YaX=b}BOX7-2X3_O>RZz@ zQ%YRQ$3z&2rWgP@#O>th{?YMzu=JR)Wz%R;Q3`9NVzgXi!q^5$kz|131G4C^O1P1P*$+xQLwW_XRH`b^UujcAyq%)<52BPKVVnNIulDwQ&6DmxBt(FWGBur9)HycB?9y~B1xSnhj?3VS+pTJ! zKYu2JO!Ii^u*2KSYX%qYVFkB#vXS!K+}s;~{(QSXlkLE#cJJ$pYkupjhrQC%7N4J< z=R0)rf4Kx-+N7rg!Z$bFvl@tpv=g2+9(K%u&zv0X``s65;Q-mG)a(NdiyvT-v>0%`XL4Seg@I_DEV)P!N0x<@1{!d@3AR$ZI%i;; z-#yd0G!ifhtVq^&VnZ-i30@drK#$+OJaZ84kj~D+#U&*siCTolw0Yf)mD7Q{>^PSg z9UH6U$XrOx&(A*+;gEv((6&(QvI1I>nAehl5t?RA$mGqbTjh7#LC7(Vza~jbn_tpc z3EE-R&lLR`u;3v> zYpf;r0+dhoc!7@$ax{sZob1u8kJ&zNFMJ*p2e1MKx|M1B@^nhhYHMM~)**pKfTD$D zY+En>|+T@3PH*%$^Egh4)nXc?)^_ie#1BYmT-xv_F}b=ARq zdU#mv&XD$G1S##tLVl9N(Kgd1iJ3+H<0uQyw5tOPJV=p90gLu$a-4Gaxm1z!BlGRF*AsvdMshuuVLrl`isB;ualUNM&` z1qks)fft9geQ|UzC@9EbJF&W=qJM%>mfHF1)vGvdYp zJX%4W8mxWbz^72)JZ@va%aV`8;Nsl8y|*X76*{XR;jlPBps(qFtuyUu}3%Qc@24 zf2L`jTUj9o(gbp-Nx`32HY$jtVddSg{{54PW_Hg_^zusIb8(yC(Fv&{>`NxI3+L`(0o32cWc?# zgB}h1%udm$Ou3v7W0#&Ek(tTL!^1#-S++XiF2&-Xs!mL z3$JIsbmKI!tcN?OPr9f`(2DcGVcVXx(6_dI?h4|0E+0RB1m)N0Kn@fnRSK~0>M8|d zZ?W+Cdp3pviQwR1nCSVypm?$hh|`pT1JX&R0h9J zOYe+VJNlmN&r2>@+(;PF$YcPE4~kc~teh6951E`a^NUMUudlCn-CesA*${#|IShk< z)%zY{I^7`XmPI4C+Io7(r>3UJQfoi~V_95hl}C4wTiMvSE3)OOgE?Cu=0&jsvl`HV zj09oy2S}A$XT_2PwD>@LV_9DRYns8>!)Y_h=!Mvt8fig6K^T!iI`H{(@&&zHx1u2r zr5v@Agld)q|U*-((dax zTzUoSVM_b6w4g9BPwokD)L}UMSHuQKM$m&rdUp2qp~=bg7mm%xSe`fj8W4}Zitb4G z_{O^1b*_gwvR&5FG%6^X=6QTM$dtnuO{++t5YRmQ`t}D7vVlrcTE3jxTGJDW=MXbP z9xmghv3PiR+#7pCNJ>g7--()YM%;Kkp*A>fqqur{K9i zS!^MXV{gdF%cI@2$sx28{Z|z^QdS7V1Cx_NO`#v>9P_B&e&om5HXR+>dSNC z^jg!f&Qg7Y9vgG)1Pv*O1P_21W#lr6Nk~Y5HZcQpc6E3EdE67>z$bL!;zgX8GASr1 z8a#i)SEGRHvlVC=)|Ndc+SO6BE(( zMdqDa6CKL1w{M8Dv=T;|ty*^Rh0f8YY`e_5y)HHIdoWR6Uj0|$4(T&FIXPQru{bP> zO1{j6K#1}_Yd!KiBL>BrAP~c?c^;7CO?Pds8V?p|KWF8Q$8P|15ENIg3G z$h$Kenbr*|+;Kd-#98tNYHCDKr1F&hM17Wzk4Ho#>B+8?TE2QxiVGgpT8)3t&l5vz z9@IMf>2A##$;QTockk3{swEu)0v|OvvoD>afifK$FimS&d2z7{=w@HPeZxZlz-EH7 z3Y1Hs@Pg#Sj)V%&8Fsm|O=?06l(jZLD!~yD2=A5AP+;X( zGAm=;L8{*rsh&R+S8o>W(dhPmcXFqiWC0@@wP-4|ygRK9VjVs{KDkZ6 zK>`%-JaN?GG@~%`{(V{J$@igib2g&n1pNH```&)1PoIt)nVX;I<>gg9@c{iA^blZp zv(smd_4UI*m}vd@5%1cyYnq+hk2HS#_`%g=GgW`zf=Ml?w;AnHkBK^}jfo)vrPA)$ z_2#@a0zur0>jm)tK>$tya4Q!G>uxujCZC+8y?tzR^JS2q@O6(}qJVXSs=eR$JYbVF zJLtgP_x}Cdk`gVB?%`nusLa_pI~y@Y|EFS?Vve@8C5}SjRzj*mIua^$UI{nDpFE+3 z@~3Z;ed6QTQ0CgDgoFenh|v(1aAnXRAMfNqb3j^V<^?lBvH!a`x9KP+FF!CbaiPiP zhnQ4hbHJ}3KSFkPyr)LAp#Z@Io(mA;!55ek z`ulaEf=7AK>I7^T7x&BqFoXy$NlAJ!F)>lVdco#8C=@pYO~mD?Xen7)=H~EpO;9B) z(P+HOxRvJ&KB_!Ac6WEDVqqz6)v-6l-Q1>S-ugk#+qX~js)>aL8kUwEkN|a{7P~o> zSFE9>)%vwSt2$e6vd)JRSiq;xpIbm)tBz}DTyMfEf%xVIENEq_A#-`gHOiTS)uJ9) z`t1BXWyVE2S{?dh3=J)9rNu{lkd!KEk}gKExvWn$Kmrao5M;+RrOT1;{}GH=?`9*Q zV`aTy9<;#v9BVW5G%=CFdAhN8*^l(Hyank%Ms>C1+V{Fn+;^z;zaASO&&$sbx8XCR z4532XqS2}nw9t%!bOi2DKV3oe*WW2ADbT3^c+n3L!6cun8w06Q^Is!Ss8CRnOy^vo zGa`Zg+!JCbq2--}Yo?~|{_|)HeiX_sy>%NwB1Q-r@l);h90=~0AtAC&h%2c3URh%n6;^f)1j9`pdrHO=eERP zUx`q5KkVv)#dNWq7cZc4&xtvOkJkfP*E`@K4i-kG%5!~65w@d;BKa3w3xw8FZH}C^ ztq8lzd{1&qOUXg2g*}Dz^t=RenjqHFPfJUyEV{@C=A(BLjphY4v43hxSlPzLCMqh5 z=+dQ2mj#n~6cpH?`p))1AB&qXQMz^%AOfh_H21C&W&84+UR$F_gw)Ep&@rn2+;RGM z*0|fN3*aJ#MN>;F`1)D;H05A^1%($>WY+N6*m%gpW63``iAxcoPys57F1dx9gzDe9 zp{YCU!~MRBk=`CZDz;~m3ucWFl4Dp7q#Zug8VM?P_O5<(gd(CJ^OM}aPQWrCLQqei zw~FE`CU#WdY)bK0LNFIG^1Fd`Jf7dgwdD+QF(_-$UA?Neu*x1JQB`v*CNJ;PmH9Kt zj~;~rEBsEOe?2xf7J}?Ixih%SfBY*4jA%W~YB(b)*|@OwPR*xA(!n9%5ggdw{;zYu zKjXae`uy|YsQCE!qxpD75fvc@K*jk*MRvw!)^3O&{kf{Ho}LCQw*M*VBpNlb;-+wQa^rG&(}^6U#fOg{i!bbf(oIE6 z8xCee5I9Oup7??VdAPapaMcRe9|(635BkR4ei>XjO<$h|pg&Z7o_37YdR>4Pg9^tH z;p8CVxS-%F`$Q^9*}uy=0wN-v(3{dfIY|!qV|MoJ^bHpjDtBZ2=qSL-%8FcG`0tR7 zg*wrFU2#DuBtW$ZiZ0OU@$`X{#k1O4xl+?=GI+P}!ov9&H-P6b$NVV~OA=gLP4j;` zDo*7bXcq#Pq=~ia7mGqwpjwIUP1Tz$dR@GD5m&Qn6~)DC-!J_#7z{>3N2e2L9tTTj zclT-Nji6-5YT$OJx@rj-NKUKc)iLq$sz_wS*JUIR_1}TG#C89LzzygTfh@+YtPc9` z*1`a9#{dyM#0*|a%A_mu=v$@6p-@{%NYg;zIv*gF<9e5%kPt#l$wqYQlt5yuXu{F+ zljrp9cp;xZYbJ0i-!17=lb}`E61a4U4zAzW-6ck>OgC{N9zJ_U{-Y*c3tM`HiwZ)( zi~pzd0cmD78RHhEPe+XSSCAdDrG;(|@#3A3a}johP(RSq(*s_|2Fai*F8S}K1_#Cb ze>OEZuthsl+*F7wocCpBX2$76ot(ivnhJ&qbRuG6)TbGQE*l%C838>~jmd8B==hw< z5@oTw_Pw+%7hmY#O<(_;t!008xB+N}zBg#(z(9W-`W(kakAF6>op4Is5!PWaRI1I{ zWRh848E=V+;<9~GFFeVz;Nle^Ej=`ilY44=&clSTFHDMmGm`Eu%LO5sOY4LaVS--l zPl7Mi-!gnr@3Cvd7EUf)G4jjaAk0R*wEEYe+jB01<+_!FI#cvDGCcCpdJnc*=k1=~ z9~n~|X~U*sHN|c2Iaob=_V~)_K>2m<#QE0ViWn+s1+6UjseP-}o2rR~htr!%Qx(PG zUGO@Pn8TY+Ly3qZC)rq<$jIlRj0|4$pV#!#L?sc8c>yr+QSCDzt7$DEe5Ugm#4)u_=9*E^;tBZKqz%uH%U z1w(&-KZ3ZpxHvjHOP`Cl$xZv}1BVN|K&Z18(EybQicVNOhFcmMS3C^86 zH}U&-i<<9Jsx%M7l^u3A^eijRNp*{~qKlT)H9_bK>%>DX+^xOZPwbs}(hSm!qe^D^No)(ok5 zw{PD@Jk#XmrBkSEo%K_wCgy%bAf8Q4(ry_U0$}tXUS+11MkztVgS2#&wvM@lX>bN@ zot>Q(%3gGRo)^6@c6D`q>FTNSJ zd9SJ}(GQ|_B=N%5a(a5qbq6bA{<7zG_x5m*4CNy{dipB~e~Jst8~d_75M+O6HJ`ss zI;CN2zSl0=RajPDUgj5z2=;~4Ei5jDvPuL#_^fF(+2kYQu)=(SnwXk~rqF&_M=3#+ z&IBFom79CT)wR9g*-y%~OVh49$epBVdWjlcjJF6j%@96UTE>e(90E;{w@ZhpPFe-7m2FGD;OQ zqP6+VB;)QxiG(q}Z6zYL6t|PxOthr86LlUJo_!-jh~<2#!DTYxQ%fcY3YfHiSL*{N$&f^ zKY#wD3Xs*VaYUUve_o6wt+4R4&Fr_{l9KYVnc3&g zXo%*M2U=Qwo3q{1qm?#ay1V_NqtD0%Gk8VuYT559=8~tV`g?jJr%s)kNN^Zye0xz* zF!aI0hs0;1zeT_QsD3FmEiK~RyFQG4r5;6u!YSm1ot=*o=OXRm8%oKBrJu#8>FEcI zTn^lRjgG!MsH&<;O-+@4QU{)sANBtI)Xum^+ur7^nzlBPl#~=1nnCIl&jx1yq z=k0}$>x(SR%&PkO6b=p!n6jLl%PlP}6VqW*a^~jd(IenEYMPn^@ZFck{86i1DAeiP z++6KC*BhRM(b;Ex=L_;Rcpj|qMt+@`o6F)ifHbBxywCRV;X~M{)*mT2wDcv9xw|76 z<9qmoglXyNvJaKi)zzuk*ustub{x1bER^~A`ksU5S`Oqi&UD42jGDZa+srNY#-lRI zq&L>;0s;cs-4yTMrJ|)x$1c$FJA1(ctYzMRV%QANDBU;Zclq<|@#DwbOa-tx=MDrQQL1gDWswPlTQQ18>kU1AIQP`%zcdqy2%#3Nx-c0;Z+4FexK}ku8;mT;` zP_+Z!=SmR~k!0{g-2*fleNIT|!4sSe#b(;>$5Yvh9`Fo@g)?(v^>J~Gmme0LGH~0b z<6STL80KgjClma>YMX6;&cj@BWacvpGGH(X)1p*G^7GAp1~n81L4IByg;{rEE2bm* zq3%hF_0H;eCgh_ttScV=t;VPD@#h~O`AM3)Tx|P$`J~zrrH`~xU-`oQZf`15_4ScZ9U>cdpdJ^7u-zRsb}C3?czvhf;cwtIPzH>%x5~>r*uoP=JYn7 zn>?1cwB&vL<_)>~kzbvlpkPcV-%hqntGBJL))6IS_q4kGrtABr(qCi*mpvE``p=Z9 zb$hQRqJEtie=+~TIC;^e_M|oZhbT=G)Asgu1w`-afX5u^wvzuH%kdL#Zf;N6bCbWY z>m?GSE^?URrd14HVr}|Zs+UQ z3Vw<~i$uOvS=Y!tAtE7Z`kY_)N$sS0Xe=Mq`Z+t>tF*LKvrPcKqnLR^F3^YpQ_EL! z++47pvPYtdLh&pgut6HQCM=w}%_iqb*`?(f7|?j%;iLm1kDWfKQaQ}rlhRTN;o&5> z6M0}Tw-B^G(HzALSKUfJFi>KGO(_33;yb)0LDM8!_e}~Twh&9DS+XrvR9vj4r`OrB zm+1;6IbFAJTZ&UD#&crAaKU4Z%1wZSu{RxiAh&(IGk33@&?Ou9JnuVUk4v@YOsCmD zI@80_C%{p1{HEYFeeSx3?#VO+cf~6V38Hgy%h`7zcFq6Oaa(~=bJoc%ZCIRUC+nDS zWRs1^L#?}|MfK#(3m;`0CFIwMG#QQoV*#fy-usy(YMG|!y?oDrPR}b>uC&hgW=13> zjpy!VR(%VbS{K%Ln5nm?%da*k$YK1n&vtx#4DUnW$JeFR;2-!kZJ!M+X}j&iz+5M2 zfWtx;`9GBTr?h@69vtXUKCm1MZKTW23WtOQ3*66A)`5BT1@mgO;w9ec)2E;2r++`< zYk9YZxl7UT3T5&1X;Yy|o2r39SksJjdVl()&a@EXIdnJpYb{rE^#IfS{5%mQCE?9Q zDSwN!#5=;6e-->L{9UEvv$U$*NgbA*&828d$F@C|-M;;aK?9H5PEI#+PuMv_oH>7K zXlR^5M09LQWH>C+JaOeML#Lk4mcKms^W4$V5sVfixDac5U$?llAX|0WGB7R(K?Vi} zVnpqD;{CWr zftfBT5j;ge(Dd~yF?GPH*w|Qv^#f<=Nm^jlEEEh*ixXE8yXvL?G1s&%gV|e7qT-8tbU1xtaj}|%@0pVAcu|qtL7$@ z@#)tvL_|dVyZBn}+@ivD-&V4HWgV)@Eqn9kO~2l#>yVFz>fFS}JWkxA4m342Kj~fP zi@JN7g(W0C{Q^Q+XID~ICe0aX^sCBFyTA}fIY}~C&HYs+^SKBjWMXE9AXaI>qWbga z&#&MIFAoooVBqtdxVSh~H8rpOeP_foB!v7FF3#o4mkmu!nqS(ZgTN_j$Ay@wsb2$D znA}~T_74eJ-l}=?2Cu!NV{&CBZ}fdslut|ygX`gX8xcAA11m6tUwk~%K;EN~$?dSI zrKRAaBL0f%Y8jp=o*=D*W&T*D%14}$jX!=AR$Ir~KtbL;Sglu6RmI}ickkcB>!$z| zhu{Mo?H3z+_Sv&%@5=f51_#qVe#8g-YHnfS1?zuENSK_O!iD7ktv9u`5lTu*&W)7w zwXJ@Cv7T+#^p)tf)8>rP&!Q&`0;Xg&wPN~m6ciM4ckTofiKM5evr9^{9_*~Hswxck z_4OHo!PYi6dv|wJQc@I@lv)7Yg~!DB6Vq_o4%aztnqvOEeM=1`%Wh|R;SogW)kjs{ z9npN#3;mz51tBt$28saSK|Bu+57=yTe?J`@c|mFEZ5f$2!@VtNw48>fW@=3hGk+|Z zdNwUSKK_@^&X%rtVIvERfW|k3&Ew;2US3`dqPAIyMX;4HX#S8Jn1Zxn94VMjHW_LB1Ou%>76!v>um4Zpz6K7#kbgZaggTGGfrHe1)H> zoX})lf8-0!Z?d~K$#CP<1#}o&7>A6n_=eP+uP;GqH(;b&7pO0(;IP@xeq+(Ab*4MR zr{CgB&h)O0P0C$dT3Xss!ps+f=9_5)KiQiQCgCR7APtwYhLyF2MI&IzLWk8ct5!RJ z%TJzM)V07x;0+_<;-0<`WH@`}g#Dy(TP4arvG&v@~8fJ#PomPfj$xsc_l3 zIylb|48U+{q@1!_>O_2HtTyNkA*C?{1B0MB)nN8^Ol9Q_xe!JlzWPIIVq#*f%K{Ms z7c^4exw&o;LTre%p2G6Ku!4eXOQV&MiyCCo_V(9JzkR^6s|#FQCYbgp0R0%zD+vtZ zj@RD4eY=zkTRg9mt z^5vwZamUSS@Z zWV4omo6qj9BZfQ;?d{CT925}Hn5$d92FcAPA@Cc}8RO%_J*?ILto%~!_|7&pJL~6m z@F&Hf-W{8(uj*Aa*xl7h>17Uu;KGvS-G$Hjs#;oYcA}y-lTG+FhZ`M`VQ+&qjwHhw zA^U3=JtoE)CP%vv*X7R=DvnpO&r=9#_9d6Oc@sH4TJ?aHmD4jbP4+f0ySvk2>phBm zKa3^FgBeiR8ybN$1DDBXjxzj`WoFE-@4oj$ugaFVva(Vyv+0wB%b$r3zWSE;S9I5Y zN2|0tY1ldPbQxk1>_fcwMfLB3bVdLT6CV(Ld5Rj4=a9ZrBib9Pe=rX4{UW$oQ>+BbXfg zS$<(}KO@Q#34AuxZgGGEQaqSaFSD0;TZv2wfDX#+8}qaAIu@uRdYNgZK;wb%(1BwM zj;Ru2S?#^}>yyotQZ3o{Gb@xc@fKnklEcHPDk>`{rB3!J2eZvF``e4!)%I5;50(T< zdw1`2?;RYRzjO&6LA<+MK3<_LAK=VM@N}%k3F}NGr2Qw{RUFw_Sq(0$wSIKGnn2WT z5<h^frNkdaEa8{XwBnf$HU*>U-ibR z_&GELZQ;vlIqYc<<{Qk4vWy8C-iXH;VF`|_EcBfLeMH1&hD`G0&?b$^@G;4^4>#Um z9yP_Xb??gG*u< zrbdSX14)*KOG%-YW-{%Nc|*P#$TtXslbQPcHPTt)`uIQ$BB!H-G{-Yl+srB}#R(GY z>+9=fN)H#7m&-#eU@r$(khgBbj6^Xv_y-XiirAP01qHpFY9Yp`#8Nl&KmI~>S-SvP zUtf3Dy?sxD<~(*U(I!Xst79P#i%lEH#uAH}6$VW^V{ibGL7JVzIQ9vkq@UZEP_l}D z`MM5Ra(yws-gu@n_5*X0xfNhUs82EaroUCYTNGMbT9UJ}{39z12`QKx-d(y6kT-M3 zgl1VpRMZ&K-|kv7Wy+#YoNb&*dQMKx8U*3Io7FQnH;JB}o@*v(vRanNEzn&|CT3?d zHw|tk6Xcv`5_c>Zn?)aQbnvb1>}29Um74*5<(8=_XP#bVYOx8d`(@t6YAHSnN9LAJ>n98+#+pz zL-@L|@EH~H^;4^L=(uyh!#m@INC9G)30>V*QVPzSLF?DKo^AG{G!x>S#3G%Sn0C6+ z8mGw87_gkNt&s3_K0Zo!clVsu^P@#&W#}5G%>uhc%>m<$LLAGNC&z9f(e%Z?wg=>E zl9Sm#NV+kOyC38Z*qHQV9FF$w#E*BUb!M%Dlh_QVLRiG;0THx-wp4x8oBMn{ET8Y5swi#Z-DT&BGLvc(l{}8u=cXX%= zps_dQJS;8%uF2XuX|hQ-p@uTDw93)o@lm*7j1n+b(081~hh0?3pMypI z7o3QLrNKh%nv#+d88E1i+oRe((m14~q%2QvleK~luBxLmoXdpF3;LOXe;2%%l+K++ z1N?0o{ZP)vWKDDmS5{Vb0+ex`RYh`bR>tekV&GUn>nUflX{k_(zfNvpVWBfVcw2%d zvB%yj=i|qZWA1dJFFN1_&qXLK-Zz5P-<8>zEH5po>gs;>>-6V{5wxHVWs}P5m+TXm zL_MED|C#S4jEahK{u0RW8o&rt2)WNV82IwWy;nX=ai;BPdz@73<0pG`JHU zfAh-&$rC84SW&S1;pwZO$~1O#ke<1!_kC^~R3PNg+UnWUr=Qxl@mX0}xvyOLSjs6b z;C|@%LEK3_ZE&L$$8tl!uzFg}#srHa0l_?#c-f^o&e!T97$Ex^sv%<#{gnlNR^3g5 z(ynf*sfSw&{fcQkI-rRyC7gr%>Qy-?{bqY=;z#Js zogKfM>d4bXA(l7C$Y^9}0gtB)3n_Pz}Yal$q7>O#T;kZvGw84Y+YWHf&JMgbZ8kvlJD z@vkr?r_mL{0{Z*6B&lvAO>feZ{lWSd>r=N&yFmxEz0*HBgfc--Kx={7Om1sg?Suwo zH>Y&JSu}(q6|NT|Y&AL|B9N*Aek$+oE=fz`(>|MRCM~$JHae7<=u84^W=e!NRRwC% z3mcne5DY(VIDve~=;Ua%lzt~N{QY~hosMkx;nC5{y^SYP-D-I6&2x_a?t5TBQ?!3F z)~MN+LrTg3V*?tOjzv6Ed~~$@xUwUt#45`q=HM>5B}q(TP8*L8!6S4+w$G!xz!xwW z438NOY{EPj0t-^NtD9T)>5lU=Gcyd=3=ylbnxFGG7GlG#;E7_+TjtE7w*HH@{;cGz z;tWvWS4$F{In#BAXV5dhf8XSc{E^TiKDD^``PHt-wvs}5X;%oeHHPXvqz31yB+gzl z^lTsJ$Veofj{w@6l9NM*m0p)e$}5%q<{CY5@KiN4yg?kI7=bX|QW&puJ557_XJ%#w z7~Lj;;s+iR6B8)G6M(x|$~G%;6a@?)2A-j%^#;zye*Qcj3Wb91fka>?abVkXacB@g zo=*b`hai=D--Zvh8rH^E)24oqY12>F9{m{s)5zbwOCq`Vo90bN*i<2t_4njk;a?xO z8#t8|6&XJ{=GEPs7h_GBDaZo+3-s(NXcb3alFnX4clY)dH(PVBS7H@UqhF+aHHeBz zxAM4>|NKh_hc>B`!!*~<*%@9$`$d9h_GzM1SUfh_pkr0@V$8TYKmK?i2!5; z5Z}1@4A=JVQrv9ywgU!$rKF&a=#i41j(P9fmj4J38Kn<;X|(b9OHFM=Vq$1k7Wcld zaZ6`c7YB6!d(_>@iHVyMGDkc{eNT5dp3cH4q_RRLU^m+`mkJUP2JSRXA_+kdTs!UCv~K*l#gLUA}Uqf5l^E^ae<|+-&6~ zCEW3);JsMdAbGTP|8T4KUTqJ|0ua_>X8}~ceEa6BaG|N^qaq55vU1_@37Nct0ueJa zGm_j}mnI*kSy_7}!_SWp^0UkFWDJAXO$1t*APue6nMt?G%7aO8adF>e@(z#U>SH@} zXU<@z0qag|4;sirmzMsjUY2-QL^%&YPGkiH$WHF0s%qF+YtxYNP=)tMKS(LNEvR&4I_Y50-<>BLgB| zQ&ZEsm)Ms7ay2v?ubPyFq+LjW;7Kbhd%r`jT?|T?T<_TeSt%VIoofy&dP~2m%XJUR z6Pk-JLFXkZD;oKXWE8~ zhU751_dQ3O&8t_^AZ1u?8sId$*Dp-{u%O_Ocx+o0^>+plnm2>`7%}W@O|A zq7M(J9@kC(!s3z=Z|RieUgkU+_Ldg!8F1&hJcSFXg%1B{rR)WMSUh2H|&1Nv|-G|`3tBsxLys0o4s1%*jAt4!oq@~ zm9j_F91T1CXK`_!>*mdRA*h2ebio0{qP4XZtF2_HB!xmtz5$RUG#8{(!WU_`L%|+8 zRSv7H&_qxJ_&CZ*OTrP|0Urc^(0xQ;L4`u=1}{Ab#x zR?jZ!MX`oD0U4p_&Nry9lhLz7{nN=2+JE+UA22%reIC%Qj~pbv8iSt6aq%MY{CvI) z=cXK}HISoI>_fhK;So~7fP!lca6JGk(*wD>j*GeF*FeibPU0gs3Sbn$czMU5CG2fkfov5ftdzNV5wQP z{j&NW#VTq>#=w>q*}j1R?6?CBr_$UOqa8|DM@Mwa82b>@J9mg*zkV&~V)7{QnZy)x zHWxQ*>$GJJD4Zx5j!=^%_&(nawtQe^#^8$;o#5?a^=q9`SPkC7BAZ`X3uXEAsp#(? z$J-0W?pr_PJ3&Afa;`TeCnpci5HN(CHrUhiy3%IW7%8X}GBqoW6+Z VEsJ8>I2iLo?%YoeZ(_viDAxvry5Nsc5(AP|(A8Y=n-1Q9*q?<6VQ zVJw=dhkvBj*VI)IM}(gjO$CVv1hQ6BMbW@7V+C{HOY1J@iS;$xgwMRsc@tmE@^U*U z-L6)=WMD9oYhPHGY_{}L6}!M$I^(tYecKqFyihKJ32^FL(lcljSh3FgMrTy&$BeWV z;VJBnwdE7#^F6xmdEq&C;v8#VQ_qg44!*m|H+YjR=3dtPm(2Mk235W3uDFnVOrQFQpx^L5(;C;pD04d2%0GrP9RHP|)+x zgEnR9!T~ z;6cJ6=i~f!wxgqimXi~mnwm;F%(J$cgh5D}8yXr42njiUoz>=}*U69$o0+jr?X?TT z#M`q!e|oclEz^L@-QdXM((m<+4Mk&Pmd|#tzs8np$fLEoOL)$lsk%3HImJarM&^9i z-flhK%YD}u^dXZY`Qj{mCIky7(3=}BneWK!!KGdE}Jwfv2iUpLFkxrK|aronxd^sU!octQdl3k9k3M16nyKb{V{ zf4|CMknfvwZ2)T0x@vmhMjtgHO6 zXX8J**$xo1%lpyj>FGs3dgRgYYxW^m@D&FKK?c<}9CkEw8@2iQqS$jS#Iy&l2;-`5 zXvkPxT%1vRv^yKOfN8vh-|bSE8EwJW3^p^KGdg|Y0$qH3ym6IvWMPWB(2v_jMx?r< zp)TDo`%C8!14hfuI^xf$uTOaPaSI4^SKU9t9~PFDM&t@;=<52u{a`cs&i^DOrK*zB zu`ItCvVt_O;9c%lTF=AJo4swddo7cxp?&RI$CpgG{1zpBvM!UPat$lsecsw3` zpOuw0RmPLA)aV?-^4I6*(NCVJ!h+bt0JcXs;?W!O2FAt854W* zw-)+49Hv&r>yq^H)CA9{qYc0T*a-`cU~K@8XPfhV=dy!-$9T+t;pnQRzPx*-AB!VL zkc8%rVw2!w>uj3IzQ06gfc)J74Fe;i-0EsNMn=Z-A>!@r?Uk;Wx9TI;(xqI!=Bl3R zVjx4z%+6X#AoWa4$Pr@5C(&_nA-1UaosEo`4;ixFb_zx( z^UvPi1r4NNRLs?tsTMAm@!HO>Im(%nF4&bC zcOweiURG37i(2-HcI*_>E2^^T;t>!y3H~?TeD9`4};1O*+gQ#NAI;JP1!x(a1C%0?|hIA%XQ>4+es z7xP*EMvA+_OW)P6@EjE=b$M^$OhKd{r-b#bdbgZU;i%Vky-IyqD0c`ISyaWC^M{W~ z=dhy^@sHeWe1NO_sc!a|J&9F=47PWvfQ1mg11foE64)4dTV%v(t+c4-9U`i%me9PMRHn`e>JV-D;+tHh1$ZDM#O@xpLOl3jm$WQ5c&a_}l>6$0jEcIt&2(#+{dlbi zpvl=6*Lb{VB~I+T66ttI&8+VP=UaeafOwq!CIx;tSi0Flr=P7+<|UP`z2QY^{9CKh zXXR;1%KYvmBqRhQ(}myhZU5xQX0<$8wY;j4<2BL`6AkZuP8Hu>$cwXlJvKc*{}4dw z%<0otBoBuR%+0c|X)92V1WMqfdpE*&CIhJ)B0@9>1VE(n{(#o<@J}|y0|>qblXQxMIS~;yGS~#m7gAPb>9wBL|P6W$MsiYuNESbyxC*{-F-2F-?KY<8}ACdJ_5X z)G6g(rJ$$o%~s$#qxSg9&MLh;8Ov1*YU0fmT z{r&yBoua|qu)?ras-J+gG|OOtZVHAp-qHO(S$w@&re>(5xL8S5HBucr7oe}pFnQ@A zZpbK(-;!(Q_s>rdHdkVqL@Mv^`w-SccFb{8ab12qFrfb|Ie8}iUUNK#D!zDwcv(3= zKmWze3iJBlL)lk4nL>nf*`&%x^w_r7x{TY_Ddv;?Q}_(BUt0E&8!5Ye!otGhF)$TC zcJ*x@eNHiltdNy9G9hg{~^+dJMyE-Zr9~PGVE?<9|<=y@@QM7 z4vF3Q$dDoJe(V(Uc@=l}D#G{&R{9;&Z4UsMX}P%Krl&38*bbyHyakV|=>GJnW3nkQ zXK>NBPC1{Ig8q_U4pI;EU*`N5QFNoX`TqX;H=DQ|<%d+=eSIllt5;%L#ebGx_pQ&y zDa^DS9g5-)*FrwIKGe9u!Z-X5zlV)}{8)vMJQ*3nU_&$pdTKkCd`XxqRns0)-Be6oz*9Hy=hjTl0}NKx^^fX&%gF$S!GPa$QXIM&b7z&p+UE$ zm?eZ-y*HLAii!gqYG{>e@Y)+|1Wfzv*|TW?XM!Ubn$vIeHm)_tf;xF~Z;BI)|JUs( zP037IZzVM~Qbe<)7PVcCiIo`3i&5;Ix?Dn#54R~C{n0fRW`f_jxw&o-6QHKOD?R6R z2=gZ;$*5I%Q5VQa!bxX`if#ZmBw@&lzRvX|y3cjfzOnB=_JEW!0fR0U({i*QE$0BL zj(_?zbZg6NY~|@k00GkdowXE9l#sYqaZzdM`x>Xw>*nTYaCJAq+J0a1kqQb5&STXi zdFpW==Cec2|7Gs({3xKHypSV&kW=c!$IqWtHh=Zfa&U+=G7kShRvMfuCkpHXau5%} z2SJE(i-=Gv?2q-?*5wQR6V&SY?%cm)XJ-xU6CE4dQBu*o)Ze1(GIS7tD!Ms3Kd^`O z<_CQF`SYh33x$BNuvfr41!#*Lc_`$16%y$s_xCgCn5pmtfD9UsG3%8scn=T6XZ0?Ij?fSZ#uxUNi zBgLHxb&{Y(3)UD_!?Lrp^IfCg5aN6-?}P0fml_q(rZ!Q_U>JqFBvib3{No$VB86N;F=F;j&obfmmEf6 zV;c_@4v3`EDLXqmjT?Q+mXi%)2HA5>R8&71dpOuBKn+gW3F76cbj~*H_n5cI@9=NmG+2A=3$ww!7Z<3uu z*6S%Is@UPYmbJC@byHK*wJvHFF$5@ouzCrO@9K?LT8eINVg#<>&+^Ni?9&`#uwNx! zKHT>n8r{QTa|Fn%tE;~j=yFl8ULbC6UOi%klf;5{8QE{p*!S@CBor&5p@^}v!mM9v ze4+dOnjlBpJ37po9XanxrPDGo5kz-J#zE_8?mbY!8*_=eJUl$@?$nL}V%m6J+28}| z@COevYhM-=G&&!j_PMR1Lj35_qxHI$!(Jy;!ZzIRaI7InC540a1%*s$Byf*xVRqD@-~>UoJ~jXpmh0XyB&wdi{)~s>Ei*MfyVtBtu_S=GoS1!H1qCkUC<;Y? zf0^d}RXJNbyOAT3r*`i-lUsJGTN;EV2$ACDn~ zEt)6pM9iZ5?IYSOLn|soMPKp-AYhQ(+Pep1ukViTC%!vD#qK8THe2-K#c2qi*>6=g z?5k2lk8=IC)T1&Seq%DC&XZkFIo(j98q2gWTETU2c*wRjk?$(!vwQ^#K2XXa1A!{F z2^m=O>J8~KXu3-OO}kVrzS3K-N$>A%#bXle78s~25Q8&m&nAm~7YEKdf2$+{S?ssB zSYSAsy$~RqS^kBY+!2R+RZw69l7k>OftxHZJU1B5PQyB^X2c+Y#inOEV~r~K4jzUS1eMYi`6?#3{fG=#8Mxyb^t*>2^r(br*F9n&Hjw&f z8!3BjQiqi7?FG~xv$Q$sAj8_*mEJfGlM`rpFdGk$t!<#K|mb z{tOQDA-?B~$KI9$1)CHFSI9xtvQZkvgSJq>$B2lCb@bynggRebH;|0`@;CA2i3a-P zq!d@Qv>q=fBT-G}aTyux1d(9&)+-*Plh*~4C8VZCgEWBXlejl&>Ld3gAy?qMuN%e5 zlUMZh6V&S@a-twoG6@V9~}=jpfb+M&nJfHZiD*GVA$eDVV4711tw5$2|g5ao4!(N z)@bA-mj*aSD9j!`BJZkw|6b)W>qXK`Io}WAB&T?NM!hzOOI1K57J|%!f_<%jCMT9z zG?$>&fZDxeIiQSr0Lv*?jN1J9{j+-9X@G@~?g_gL9q^ODxpN&(6-|5)tx$knC^v1G zN%d*kTPk6Z^%5AJF(kqh^A2EfsbW^)pSGPw%dIwMIw5r38e0S#-F}1*{$T@ngr-Am zCr{P6V}4&$k|u*TwS{ljQULUKgy1*1##RbHh7uiT-w_;G^r|!#rl3o*Of9?)g)$8b z%ffEofB}a>Fgw&8UB!ka?Bh%#CJ|Eyzb+AY6^2#b?c9r9yM>CyP6(A~@*_dfb^h z5A*YR3r-WQ9vWteSseq}^9b~WCI%(DQ-P$SqEhqot6%mhWTK#_EGjPccuWJ`5B70t zcIi-C0eEIC}4g!*F8=FUq9HR2Mc(uPK<3Fg5ni@ zw>5C>BsH~qK)~Bi{w)1huU>V-j)*`POvc6HjXiDi-qJZ@V&c=lQXNs0p3D2c((k$L zteTnoy>e%NZl3|_A3~l6SduX7+9YQ2wIHGRQB)*yov8oeF(3u{3HZx9u<-=c?lxU$ znI{3OjR80zse6-;p@UYiaQroA7QxG>nV|48_A09t(H3Ub);wRb z71p+Qna%x(N^VxD0Z@E;Gk5T3v(luj`^y)f@1U?_AodIl3|SHw{qvTIlZ7o6$C8Mah&?2{MrCq4W$l3th z#eG*zoiT5R9kQW&@as!vEVTA`WM#i&4-?-)!|pxGo~#EDHuv6Cv+E9M1#UsXo^+RZ zQg>VER}=cRNlYW5XaDRfDnP?F3$E{={y;U&-CG4sl;DH<-BMbj>4=ahtgmN* zm~s89ks{i>dknfx41)|C81{iz`+12vnX(PeC7KnZavUcn8mTvORbvUxuGsl*Af}~} z2+=8S^MRruX@qd)FusLN#V$lGeH! z&50glPZAUoYX7r2-w%s)N%@{hh6q(rQIY0I0YU(XI*v#dwLlM-+|09R366W^?)V!f z2A{JEo4sWKIpz6X%6rM6U=}n&LSkYjx}+Qo4xPS9=cI134wXAu_qT7QghDIOw}AzV z?|z?IS}MH!z2+38%(_Xoiln%f#EbK15}^@kv$wVQ?XCAY=p=F8NgsMIu4V8Zs$eka z`FSl!xnIAfUNz!)Jv=wJpDJf+v(fHF(_3Fg~*I2Gkw;9(s8a8o; zZ2r=Qp1ZKgo0Exr+U?MoCh^tSvSfc=7;RqR8>lR-+rZXFgAy`5p6u8uY1u;$7dXeV zE=kK9np{c5)Qm$kf F{vXMyEkXbQ literal 7085 zcmb_h2T)YomId{Zqkv=(Bq<<`lA{tukt9(hO3p}>jASG}lq5-lWSSt75ooYQ1Vmzk zWF>!fkEujP=Q<8bgaG_?vqj)!)!_Q9<5p zVj{*gTI5oQaUQyIeCE}5eMA1?J9ZZ%(ml%t=w9z?+`Q{}&Eu=Ywc;o)4oYs8J3@Gx zNlKi#J2W8JRU{Ca3UQh=hv$ z;xqUrhfs~YLl90#N!X5HV`rn|QsB6N3XKf?qOGfYE!WxRbzE0}KRI5=ZBNGhCk(#M zg#x_1AxI>$lJ&Ds+yf+M-Js;wPPK8*ifg+3za7jpl_HVif1B}dZrZ@TcYlth(t>;g zx5wxH$UIABnYFsKn%m#5_-~&7@7wx!H%Xfot9(13yw)ylYJlwafv-Dj#b{fy{~DHm zecrJoCkID9g3Qq)bE3`}H9bAO#hJ?$Rq%2W$HT$H!-I#Hmr4uW(?e_$aKMSv)*W?5 zRzezhWyp#p{w@&_kwY983#;?~PQE6(#di;3@sUD7QPB()PFhq}<`Ca=S<=-)&XEg& zpfE5nh>VTxN-+t**a%V5(M8&iRtWoKm1`=SnqJSkc{k|sx`yNxge*>#@Sxc>CR_= z5RbocwWACtti7tIrzhpNPXQY!XGj~;)6oeFJXtj%ASb6C=dC3}g^p{-vq&Qx#t`05 zpRx%F4cT@6%BOtJ4?`?JK|ft;&x=ycQ`Kut`Wui;5s;&24OK z?D9Bw?i^npow~aElDmW^|DTs(p|mV4Eag4{C&E-(+^s_2y01n*TU&GZ*5X??&;Val zUy1ZCY*(1Pj~g1&e$?VqTW-jK-=3(#t*NbjK~eKdxuIz3JhEk%;hobYQLIu3^8G0T z8yj1)+r4DI_LsbRI8u%1Jmq+nWVgrYI7M6AtIf^LE@!9yrRnH6W_exROUZl&&vJ5d zM4GQ@QN^iB1SfH;v-9u}w70j51svB8H#9d(L6Hp&4+}G7Dkv#+3=UGY<&`t8mX?>- z`VD)*P@SEfrt=?SQ$?M#h8xNa>)0<}#=*tK9c%O`%_+T7`z2kKYpe;sEOH{(0J*+0 zS?_v&u`d&quaZo_$0eQFqM>)@Uj9=pO!At#VXkdG1$~Tgn;&7)9>=q$0S zUvD*M+LtXG-_~|3hEDLDsMEJfroEk=u31zBHkqNmy6ra!->7HD?eBZT7U9#0Mqd#T z(e*Uspit%_`6B0i)9oggTqJ~n@qA5VsMylWrhk(nIx*E zj8kR$48?*iqnw<}$NOm6C*^j~?-rO%>cr zp`fK5wdlp>v9PdkQF(b6Tnjz%%iaBeI51XH3jeY3d8DMc__m*))YrkmS7i?juM%-<|^ygH|oIvOu_He?W)KofeZ|}=o zTzF7UIAznuC$kZ(<4kD^%F3eSaR@dxT!~Dw=d=RvVmv*6PLY=DKF^f&%6<3l((>~1 zSc|XdBO{}U-Hym{O4N5>z_X&#(we6g;#ZQUPd2=g(JazF)Kyhg$%%a{!=;HC88J_{ z7XYPv(XTGVf)~fO%*RNLB*QeWozEKlDBhZt%B}2?%T!TgoTAIO*V=C_L07E^QL8;JO@-k(1Sv? z*ZKJul=3Jc`jD7^J{x4h&GOw4#}+3!i&PsjV22m8PNC9{plq9-X607TdNGNYsGYHZ zrTC;v5`m{uQ@%f?Q+us2e}zaRpjH-fP%zE_!bxax|#d)4V%eQ;0^j{DCkW_o&h?CylayB}nTyWzoe zIyyRH?n?=4o{`$I#FgrT9xK|a=|Y5iKc|(=&2#^8*=&m%Rv_lRnX@Jm$#kwz{GS4B zMhG1~KK^j+TLTqCEFQ03MW|X~(&WX&6+u-8@5M~7O3(F&p#7j)QjC36Jnt!#pb=HK zk)!f?18v>igg!0C07PWLX6C3E`kM(^S!BTLP;os@5g*wQs#&I=Qw8OmWw}ic@ z(Sazo2KGbHk<|MiiKt~taQ^e^V#@ntPM_aP!MNw~OxMP$Al6TK{PIksAAg8xpKS8N zl}e|&n0O*~gUt>vsXG)>ZvXJm{Ot6kdP0+iG-!c;kctvBlg~r6c5s7^#8bL&V3nQ( zZ)j+!=$$P64o*?y``t1<1B0ll$LJ2o?4!M(dPadaR$bE{^j8=)eaTZMzkr7~*B*io z6Vp6#b3=6}ace$$6azT%$w^s*I{j{#SoiMVONsytokydV79FSv+CvN0jzpZX=ItllpwPbC@)rAE-QPDas)}=^at#{gL zYGk_`lg|TA4$Zp%D6a3GZ|(<$asQ4P2bh zRyVh{wpdFcO^Of8Wkl|$pHFzLN zU0hv*o0=qZb8``RMslFPfEYp+vjdGi?{TE%^4w8Txg8K7 zGg4vnd}hWJ!(V(6)fo<&;&ZmF$B-N)9w(KTV!dp3 za-ih_T)s_C3fg?R3`cee_5CV@huFl#L^Vr_-m1pHQ<6i)#)bzAr}OCe`PxzAq{%ot zLDejCAUPfZ0d(<$LhKGP>j?_Lv~=Q6ID{Bt*L1ae*ifDQ{pufmt*fi65oR41+*e0P zK@NWaJ}c`JFll811fsiPK@d@=aNp%4g7OD4@Pt=OOKTH)gUf<~lr>hpaz;j(KaS_R zo@?srh7T61%NGu+ti7hVDn-!T;8P~@l@*CukCFpep?~Ar+&?|DdmZugU{nr z(9;tJ1_p+PhPHujXt^H3)Mn`15(fXlYGhy`s88t#U)wQLL>u&)qRvxq5579OO-~+* zWi+?6{K`H%j`-Qm_8e4KZb=Ez!}3ch0wzit8VQh7y|W$pSJfY?3(}FOldy)g2gS>t z(t(yxx_vviSMeRW%svh}o)!JI;2!0rOCww@NsSI;mAH5zxaaZ76x`k4u6b(rLv)6R zX%rO|%URMuIGcft`Leo$rN$VgD=RB2Ga#sU>hZ~0E?>V+hDCIr-#kDx(<}CucZf+z zu}amlZWh6GLp5NxCfjVLEf^$?xPt(S9keuT+ zSwNY)rCxDa-5{g4 zHk4${5*MK3f;E_ZdU3Vd@CjYU!rPpw;ZnWCtgN`DB^#{5C|Ol^Z|M{2?f&$sBUkbH zmr^|e3oEORo}PO;C&dj}^4Hrwi8zi6J$fI&%qZe;%hk0ifwWMC`9myY1f!_aM<*1U zYj6G^&|BpUF2l#6^-z9@UQqSn%*VtlW3B#@eVLN@AZ9S@8oX(}R@Btg@W?FuAFD;xg5g87AopqXi`YUrWi9D8el$Av+mBrl=e0ltH}DZqyb`X14jv(cM@Ru2lm+6*%Pv}<00Ejp%<*4SOyAq zY#SY=?d$8)p#B3ese0ly-m_=VX4cpFn_|f#1~WtwK=rsqf0&T@Bzlr-Dlc=k9na#o4f`S1G(MGSRdiG|hG=fKVZ&GwQrES2j3 zUF6_^c-HHAGntW(Raf7LVG_SsP*9-%K0q3JB8$4AWfyY&?wvSR*|?IDYpQ8Cx)!(z zKW&h^#7LW}XUoI_5u@{Uuq&0OKxDU%K@0jnS`Pzf#5Bxc26B zaKob%;P6CfiHM1aDqZK5Na$~b5=W7OjWKL@@}{DK49b;&fFNOv2KUiT&4TIU1=0PZ z>9aUJ6MvfQz|#i6D9iD;w#g|^g&9bFcGtQuYZ8V>5dpuFJ2^Q)Qb_ymzL<;?f?m~o zHjK8?|H!ka>ctCcKBGozIJs02M}{~iarP?tlK4R;n!9RhA3&W_Km!a7YmY1B00ci5 zJSDoDRxzDq6KH{eU&K8|DCnQT=o65VlB#r^5W$ouk}{$22ujTMlPY;c?we=rCU!nKb9j&2VjY6Um zqs|X3yX#k55Q1*86;IC2cHYq{#yI?g{Ic|nj3(D6W)*q)_&V1xHNRF(&KP#4i5J$_ zu}LLgI?OI3V{9HV^zc!Ths(*)UL6q;DDKEef}TCFle$1!&;0xu54e%1hctE^SBIhp zM{Z(ik%OCi{$y`D0;?YK@&v8>n0gw3o9gfT=I*v9MulucB%mQ2D7Z%}CN6FUa(1lJ zEab;=mb9;ire^TmoVkUywW0K0Uc8K)g2IK<%?Mec5<^g@85vBes&AAQPxs!RnE`Ns z%UB~3u)S~}a?kYp$HZ8X;OLl`(9%*~92ef6xDsKv{1MZZ-Qtv_q&)B&u7Uxk16EI2 zALLJeb>EX8pJ+#*R^q5M@t`};+AKW;+UeqDtoo~Mm6mjK+*+#e0k)kVP=!h z&NWpos)|Pq<{sZ1L{JVQdEf*NyrSY(O$`ELlm6h~K)=PuBQJZjP(6FZh@_aIP&{)i zj#<)WeO&naj~~K5dEY=VY<3U}0KcBsMeq1X%N%^C0=}MwFxN>e0;mJjV-a-;yq;Hk z>MI@K4~}krs7Jq1%ZvtXG%h6?aO@2mYUQ6`$L~zUdfr&yH?9(&uN(cB@r(Vw+KogD zQ$4}L)&9@Gv*+N+);9J5=;26>mzC7+NZbr36wlwb@foVTYWZ@^egU@ZU}7%0(T zJ3*h{MZ*W+VhQ^gqtJ=^Bvent;^kilo}GfEcgWk$%F5cGm_8j@s;{YW^L;yNE2vt8 zn*FFi1_N;j2nc|0KL?%?C>ULPw)oN8Yrnkx3!0mcqwEnDtk`>ifRBI02X?~Ne3#rC zFU5%R5%IC-~EGeXqn?jWoQvZ%dTilmvgf`#HybdXbWHMON0trbW_gBd4N5 zP5(Z-AsyE>Hz3KfM-A1(9*|L$HUsRxHzpYq6BFsAyb;%pn|X5N*(cyOEo0-jQaxlY zcrAcYu}VBYe*E~@+pFm0bp86HdeZ}E!z*ckKW1O^Rd}vllQT18Tj1&Elu0}^kBkV``=t%IOqR)>aP0%6u!f%$?7@olR%P6DOPw~BcH gRP`^DM4p`ws!JE7o?K0V_is2VifRfKa;DGz1I{IoF8}}l diff --git a/docs/images/chapters/extended/2d17acb381ebdd28f0ff43be00d723c4.png b/docs/images/chapters/extended/2d17acb381ebdd28f0ff43be00d723c4.png index 6613240aad3c78ad310b76e632fd13c269169a94..df3d4ea0e0dc7bbf4eec13615b1f331cea7e6e02 100644 GIT binary patch literal 10991 zcmaKS2RPR6+xIQ9_ue8Q^A=ed*+N2+>`{>!*?aF*euXlVkWKc=h>BEXWs}5h>$b;x ze*68O=Q-ZvJ>IA8j_T`sU)On^^K*XAD@IpGos5{77==QSUDHs}N1Td!gN!l9CdW zkCl1qY}M>ocb;Oc&oh%eb2#1Uj*E?LajlWBUP0-Pva(gXCVLa(I%Cy{Qjc*N&*Ybe z|HKX}UERP}JHs7m8g!_B&W$KJ`JFW9`l642>_=&rDdmOUhU35}n|RUD-2D8r)G=LI zs;f%a!YRwb-8-r@Jd?cb8XVQ<4p~0z!U!Txom_N7R7W@^uWs6eGV+{Hdo35u*+t}jW`~`a zMk(XmYdIPLWySg3z6N_4|7wweQjx*S=4RFh4;~N_5zUrgW7kcioI|ykCxwQdu@h$wkGREk zVS!RDz7U0)Xl6KRg8?ziZhc5o7TVcYQl|$GZ7fvx`l&c_urswZjLvbVu*=P6JOGex+jMm zkKE4D5f4SmQTE$2y#n3qcm4Wx+<_F`XcJ@OmB2#Ramj&Oh3rF*Mtn=61lr?WOHc3V z1Vkp+j%4|Y69^9F`H~Y76Nl;-shDMw=Bb%gHE5e<_BA?M_|bpbkE-k;zKeIhT8;H@ZEYn#eL56$eaCWXAfHcAa3%EzqULk*^6dIWv>qOA zKNriDu3u**#V3M~z5QHrb{jD~*2vfA=H_<4)Ar^KvBPjl`!gXE6e=brW@CF>Sw#hJ zXKm8KtgNw-IX5?VxQ&5ppW@f_k=gg^r7N}sGl7hT8zX{j6IB>R;!jZ3EgF8Bu#3d!i{T0rgLm{RkCtur| zdh|sus-gm2a;N%rC#ENfLtRY`{~yf@sNngvOfQ_^@81pX-n-ZCC}Fd^Hp$1wSFt~) zs;a7?r4^=`zg*E@X4dp@WmFXPuDHB4_JyR&T&CC5(%=qmZCjB0ATuX9F%eNa%?;U* zI0liIRaMj;9v-OR+0G~vmxA4$o!W-AN}HbK)YOp$5AeC*U=(Kh7XPg%w2app>%ib( zJEO9vXYJ=7zAfxSLqll_L4m0rSwSa07`N@kFCb^+Jj)ab#NNTXo*sg0*RE-3YIanA zktYcX3X(lssctE|ZDK;wU^+E>xtr14e^d46&!6s1<~BCxkB*Pq8Q}m`4e{mWBB7z7 z8zXHeHN7&c<;^Ops=2kb!&#r*2@xN?cE6g5>v4dBX66@uba#dSzMG(VGvnz&`RR{U z-;j`yul6-x?ZhQ0vVR=x&Bck7BerAc)kq3veC|y3< z==gYdJC;>N$Owdho{d?v-Wvfd@dXQtdTg<0d zuiIxKS&o+}L5-LO$vome#43?Ic@00TJ&z}Gjr?!MlcPbLBg67i`a8zFV(A=d%Oc1D zv6}gz<>lqQ%QcJ6SUYl~H|p4olEIzr^Ia$SRqSlPDCh8MjGabAY;OFm1J7T)Sklz_ zo8iAsOlbS@lp@(v>CvNVEFMG{Y)V8%Mg~zk-$bT}$UwyZG`8av9U?)FT#Db{x0RJ~ zBT1P$yjVj2u4(t>-;AB3Uu5^?@2dfKC0=~{`W01V@X1R6c}at#$bcVm1heP_B)??& zH|6E(c6NNzPK5Zda)5z_MG}P3_GF|boPR(a;;MeV^>EKw84VjQ<$>~UroDxNrNJ>- z?}~d~EqUPIu?sETmihl9A z{?Q`&%e*{2Lqq&b9p1u|y+ZXU>F_`8tK)T4w6rQWC2Iff@O^mL`g@_Ce;p?uq`p7x zL&dj6egD|O4zx&c=DdE54_5xKA1hwJPPUrvu7t?;)k^0!k}W?tO3eSLad1x?BtnvT zqw73=Xjs^T#%>)%eY*GVUATLR3C@n#m0X{XHo12JmXkuhT3A)}q@xQTHeywYlCDL3 z1XVpo8le>O`?p=I2^A~y6Wv?4A`aJ@lFELLA_l+}wrGtfVD}DQm%xibN{eZO+u7ae zAO+Zh8rcF2w+1U&^L8@|n>B`yjvB#A+io5?qBG65H)E)7+GD zImC0w8AQxg3=C)|_kZAm)GH6yCZxG9T-cn9%t3^Sg3LgiVvg9GqX;MNT32hjQAf{w(s!#%|kbW=EF} zsNP+<$YAaOi_y@~m}#P`L!4GJHZ~URvvt?o$7gdg)&%i47G9lRc z^Kx|%Bq9d)U%ff<@$)OWyNgRDVz7@MkQEZ+>XCW*HV#XZ1 zr%!ey`<`EpTQ{5_h9hQX$Okab&dxGQyKzsK*3*H6GW%P-BAdVSlC~xz$041)V$1s$ z?b5oB-?sN&WY78iDx$xBFm}?nJoWRkvKg=x;#RAy(r$QA6#eUFpMka>1IfaGPRvS(MSl zfw7PD_3#Mxek4^Ozv&2LJI)4n4rE%?=YJ~Yv&C&#W;*7QZ6z*dcb!y-Cz3voNr(sE z2@^DKj#&%v$zdM3jKAj_^Un2|FSST7gOkEL)VVpSy=Xy;IMv{Ktb+?x)UfKzS^SvP z3+OO^skg19G4-YFBW3JAhUULEM(3MjFh73rgyWg;s9TeMO=pTA+BD7f-y(wIPsPYM zim6f)!%(7z%dg)zy~UCvhGs2ub`Ho^bP`u~LU-ETPxf|A_KiEYNv+8B>)hs?poO0+ zBmZVqpwQ;rIVzJ@3A^W1)DwOdJNAy^Tzq!Xy~|7Wein~1F8q`#HSAr<#C?|Bu2eiU zmwmdQy?OA=i?hXr`*{NCXSJ(Nck3OhR!%V5SL4zZUGV)?pK&l+JieFo?lzu=ahve* zGP;Wu%^MJyy4zAe<;Nt5__CxOhN+JnK%OhTMQm5ylmNFH-PR_2R;Yb{~L9Cwq zTsb+3QIVReBcS;lwzY2YYi+?-np??uGMVHtvt=xU=fl^t^RU0sQT0N3>55Kh<$Ouy zd@;p*NyYqXR`a)P_cdBn33sRzm!0;-)RxJDbi!MDc5Qlp2}Q>c&H9PqlM&N!8TpEJ zX43agshFiOHQv7-)j}eM8|tq@xP?_pP9hAEH)ovXRZC7{I1`|?DjI$^7A^B`ocf%+ zc?)~s=*tnz=4=$l+(N2_=dqf%6fw=EneX^w&Qq6-PwW!vT}KF0(m4&dx(O9TyYcre zed{?Q_kHh(#26=j%WPwI-I!5~XOSeQkiF`6p-jz6=^!B;$IJb-U%TrFfcNwMEmH9t zG&Wt7qH|FP);k(O;hG1hvok_MLVGh|6dS`PF4#j`A1-2=Icgai<(A>90@v*_sycDC z!Z-gAZc$l~Nr_PQay5HXE4Lqa>L zYBf$QkKCK*WXaLeAI|twN`7Z(W1LH}oKB3on2;dTjz@%AnP{MS^s7%1Qu}l%sWB~2 z7mpZjnFHxw!)4jEw`rhF78~3E^rKfCq5cP_tWm{KQ0F#gJNNz!-q?4(rwjSOkUx{? z*5}A!1z)^FUrzpaEY<1 zQ3x9;_>N${^F1S*OnnxE4VtmSbzZH+sPc@wyu5#%*l)eGdLn4{s;VyW(GrVGNQ8<; z&dgYTSa9ol{~o*7Ibdzg*EAS9FfK$HC#7DrM@ zV)r*DW7TueuDJi*UV220Jk`7a?Z>Cy-p9|LMJ6XFh7E%+&1F=bO=WhECf;a2Po=MN z)|=ukrnv!di~Px+&B|DH)ViT8eniZ>C9E$kE$!93Z|6#?s4qs5TdAm?Ma|rI7w;D| z3N~^3uMV{RU5x8&BKhJ!LX?lL9_bQgF>!Q^Cs_Mi#?RJ_n0LAa9h!tk0lG@;$w5=q zlQ0Myg}!=uW_!w)#9S>8+Y-Iy7T@f4cE2}u-q2lqCsQinV3xwtF`0rr?oUQba)0aI z97*AjJ#-3bAXn_-QKQ?i5 zjsjq1v-+bZG=tk0owr{5t|oz#sZ7_sN!tkH7uocAI3`$*%7#>0S{lH8*J!29zH`q! z(&JjLvsRR`v#MfOdVGD$Ju>8N`ij0#4(#mYPlezTkL$?x0#7C5g;I)%XEl+Uwmv>S zA3l9rZoAL+)_qBzmY%+`#9T{@#N2!KtnBI0_Mc`;cD*!0ZRp8%R>s00zF1*BC2MQ0 zSNa#1f4coFH;#B-FU81B4WI8_?>U2p_`iQ#NY=~;)bU%t{C}F~lJbkR#Nm;Vo<`6$ zH+DL_fB!!D<6Zaw3aT7)Sd?MyxpNB0#>>QZ@b&c%{rzgd5lpw~tsX6i1Sx83lm7aW zz5c}amCNseimMZi9wh2X?C$|$>jBbkc(iCeR&|e%knk!Xd14Y0JqEgn(8cWaD(99U2rdP1dd3x6dmm zaM<=|gyfP!td3JLFxZ;$aC76`kln-+HgAsp^hvYO0D4qPYHI72%qhcU)Vu#Q z*GT&-RW^T%+=ivGWp*AzVxj)sicl@-c(_;j`lDDJw3e;*ZJ;QctPXFjUKTtvNn%05|WZAo;-Py z{Lepl?@nm!YiepFcF*|^j*b#SCywrhHfO~=XLy>d=0A zI{V}gZ;5$JJz*V^p>W8qX}D^o2jyOUlPZT_CX!Mu6%25r@W7d5P_})Fl3He;MI2z-5tU{8(m0&EB4WYkT`j__c8# z^2GTPKVc_7GN62^asb6HUBfPwcU7Kp@1;6SDJUyJQmYnhqJBIQy>B9)^x;iDT0t3^=a(fVub8<^HhPr2dGl%M1f+#GSc{8_DvJ$GT7c%0 zaG8?z#0(2HjbOBb5)W3Yzl>(d`I#CfiCP`|)z-1Erq}C2ShpT+X&tSfo``^hR5(w{ z#4}(1bdaS}&{=h_H#s>um`=d(kMD&px|c6sCck(Qx3c00<#Mpd7)&j^^kCuB)6qr` z0WL1C6>}g*#E@E78F1)H${{~-A%Jq^CE7Kl)=xzzOEzX{u)wA>lFW=pY|jH~CoyZFLKD;QzMVX;Z)_Z^&JjQh{FU0( ztHGb|oh}BQ5R8;t^c=3YDdZ^5twA}ev>)O>+W!)?=4Qi2MNga}Zx@uL0Vb<;o{Z&G zZVW2)R8>6-bYO+;7xw3)qc{r-3wygkr(Qs;MTT~f>86dwcr>bW3ki|+ri!fZ^bhV} zS~j~`#3ORTAbf!bO&7Brs1BJ#MO+hs)(kmP7y*UxI#s%;P2^Bqxc&^VivCW}~Q*_=M3$>mcIIv1RMA`Htz6YB;JU!msEi!a+5-Kxqi5;u9 z_43g^yR`f+1Q+TJ0T?}gbG~PLlH8Rt;#zV4ZZv`lL`6k)hGd~zxZjtCBO z(Pp&Pi6xOkA#r!gj{+K=UriSV2-p<}90LO6kjHa#*4pWp&I0#YsdKpK7_``*sjR6P zTa!pxd)p2Rf=M`z21iEXJ5Dt-DJm*XmhpaHMcQpyPx4XN*Kw>Wzf+E@?00!t8Tov7 zLiI^7>Ff1ufn~F{KsEuxl6QNX^Oc|oc99n(K7L4JnRKPk^$$ChtzC`r^S1>oPQ1p| zRaF^T+%be<>fzTcKw-~-3^X=2KER4iBz&{$s&yJixXtGMUoRvUOE0WF(2a_z-i?-W z`1#>+VNp>oN17qfl15FQLQ2ZYm0s)SscW7zCuE<{4-}H3#o$E&PEH(!qm8hh>5BUj zFE$aR4W~3}bia7AlzkGj{!BxG(ZFiJGwB0lR?gJk%trMty8B*`sA=Pu$z{_TPNZuoo4yN6APQjF31RwW1ygEJtd@quBP?YD}_Napw*b zEpY_aR1Ky*SkCcUe0QQ%_Qxk3uB^`l?}P&IwS-KyceMRw6Vj$N@S&ksQr&jf%={s} z0mpt7zR&LSlge?T{@YR!(}$U#TWG)_zSJ;zI`i>ahcD`x-6Ea zs;)lYvOCoYVZL#4xIPaDd|w`j6_DK-wIL)W{aIRmaNwIE>q8xMI6jS3SM3Zbs->l+ z(YFt1GATIa+#aMs?)Vow=M3n^C|{4#wSwu5il4fT;>?*3LvE~czO)xFUL0);3QGF& zd=pR1w!aIk?fR>a$^T@35s7vlos8%^)%Ry}qu7`0Y$@v+0uCkmvlWtnf6jgTmKOAu zCgZ`+9`Hx6#G-BLpwCjtcYXlf@No7F@W%Xq=)AS|>EqGH(*h%@^RZjR&nV0i?NsJd zZpbV8`}-rCa)SUG*RF-A<+Z~>AI^0VIZigxwYRqe8ocW0DEQM&8k9L$5d6T-?p*?l zR*SK*G2!^iU9=oSeYWL)qAvnbD@Ou<%`y!b$YjmK&==AnWaNbJd$SX@Id5 z_1}94n2Gb;xtz{@`ff%bk8=%+bM=dI4EQ-8YoOhJY5)jQ;z_ZZYW6Xzv{sGkNhRKV ztuQBV%FN@$YeMI%TGz`*`fOL6IYB_dS9eZ|97)TNc`F~zDq%1X)G((5x&RtrJA3=( zDrZZ}b1=-HqNWbbRgN0SHyK#R_a`?Z-wh0F9A`~cdLwDwMc9-&wB+u2tU$lDL)^gr z8OLFEK-w}aG!zkP``PT1lP90=-Ic$Dde)mb9k9#xS|>9W7W*;Q`si3s?#&1X!jhAu zW<5yxD5zM8w&d)RNX+S?vPvx7v$a=JC39?08&r(z4AYFBFi zMg3b_W8w8zQX{YV`^y5d|0(ZQLTnZxH7%tfMZ}$f=i2IOMAt~{#vu(qyx*S@Gktop zi_{wkmoqiS2@AccHZWdGlk<~+RwS~kO9jy~7uOlT-Lqo7|rIAL0YM^r9d2x)`lb>c~k#{qK z3QBt+EIF8%m_B^|?D26ojPk}e5t~O4WB}pAU%ldv8ZIs^jW87^upDawMM4*a@R)9H zZY5qeu_|1oY%;VW%^PPBauAdD3<&XApn%j24F17>qe&>~Uu8g{h|`d}00{{jySoH7 zeazm$_e5U9TrIbii1Q4~<%#``2+>HytT2~?!Cou2W_j5j8hHYdwxdf3Jr7cF^YF|; zDyiBxojyxKpVQF51gr%FA+BWuh4NqQ&nYEUe=3=ntf-@FMNLH&^5zY{b{ZN7wekJt z4NeJ(VPu_xU z`izU>=H*42E!G}RY3Z@X45i$w&{QZmU*HT&$?4#}e3=d=cEjlX|DL*8jtRshBt(E2 z(nM`Gru^2e_BLjrRjkmNI6B^)wi$SJ4J8<*3ALK+wRU=FPR=>)Oc^=?3U)ly&;LOC z4QN^ZdkjZlpF+eWjz`Dyf6d?Qx8GccUOIIIe6iB*8yB<<#y8$bNJ`Fw(xg0=X+2ke zL|`!^Hp}0OhOPcUPZOf_ZL>BA5aVQ%*F1osCC9Y>Ar3TTqjfGEfbx-tpq`;$w>XtFc4dXNe{64VR_bWpMKhN83Br(5?cqZ- z&yB2jK>Poa_e4~(Nq}jZTUtunpFDjU0c|tR;+O0mEs++yA_IFl+O6>Y2SiIS31m^> zqUug$i-Ry0yi~_$N6!)Phst@PJ_I})yp9a!)$aGAz43I4hc5Xo24C1gswD)f>%pG~6U{y} zL_|bs;t#01lQ?c94YGzqXNZ&2%XsO={MN_ED+?E~&eN?42$Kgy2V`JXh;pR-UcC8Q z>atCd9lBS%*nuE&L)PRF)ivfl4+FFt6dq#`Az zk?eBK@HjVzFl`v8V{!B7jhK$Rl)wE=6cQM?{BzJuckP-fXGtPHo*SkY1OiUQ%#2MZ zTRuUgb??l0ty8tU1s@E*V29M4AXRp22sxl5)O~#$J(|D5V5ib)T%xO|=e&djeYNcu zPy1%dO-EWD_Clv-U3dq(uf%&JtBW_=UuixRY)KmMFwfT4zq3EsIoUl^Wf=7aL^89(D_nF9YA8%vA~qY#cJJDYR{Blz%jNzE$eDjjizXrmG z5f&Am$$Z3*>EZ0OGz5ix^X9|><~Y;_v#+#Dz%}5_X$3RK&|wL4sEb6_m0%ts#t0%; z7IAZpQ8=p!EZPC{+=;A~`H0{9O2;s)BIVD_7IEL-x<^Y#=i$>zxmsDz9zn$PRw}U? zUu$uZgozZtv2^-9e7nak$Rud|YFBF_!vF>grwUWb4c5#*gF>(&ln|JDh9xE42y#iB zfdO37cb{`F&(BA!O*DkTrIf;H440Y3L4CHF>xwBX zF2+T%%KWj6I`_avJoe%5uiU7sw{8txt?j)6Gscy#66PPkRHz(X(sj=}5nQCGuiw9m zpKBqCsD~?PJN-E0qocnEC<=THz64+@ou_A|di~->3*?oJ#>f5s9k9z00MDujH>r4v zJM*<_*9)J1TrTO|sR81({5&!Pq9pim_Y&$|N5lonxa)A=Dek&aMb4|vs!Z@2P62`9 z)93DA2lBP_jEsJ}vw={olpGusMFzwT&E&;~3+IK)@Tp^P;N$9g*{Ft}!_9{cuoX2m zbz_MIOv5ib3|sa-6G9xH!&~kLOZ8>gwwcEHAk+7}1_c7EJ=tqO~i||kwvwA0m zEEuu^OB@OVUWf%fU0uuV>hY0Tss6^m@g2|Gi~m7RoBdbTN3n7_>SmO4XZ|0M6e0@K z0!qy4pQ))Si4i%^6-Gdzjm^TTy;PKxmd8zSd4*C9AsKvqWf*OM0A8!D>-Uhsl=UC& zZ$Ezg_}jeaZX|NmVqD$oqOWlnnM1h8S{)el z*gH7*t+w&PRG2iU-HB(fOrR3=7SpMar&i{{$1Pdt^4Bp}3b}1%D!CVG9 z4VgOEJ{|t2pQcpd=1ZRMfI_0I{e@VXac{(P)g`Yny`_e>pM(#AY zT|hWAxOh;o{8gGp3E$}ceV$8VVsK-?9uTr{kuuy3qD+O$1mX2~pqQ=;FrViXhTBkt zu0-JY(OQsEluSbh?Q8m`q%Da!UtJNGriSAhe7W#9kmjht4F6f6>lD`)eYLUssp>Ph Q!HT-3s-se-WEuK@0PE)79{>OV literal 10944 zcmZvCbwE{H)Bd6J5Ylxd6cOp}4i!XD1SADS5D-whK}l&51!+MMDUk;0zLZF+G$>ut z{mu5i@Av!u_<8Z-*=O&))|xf*%rnodaBVGhG%+JF3WY*nyQ*>vg~Fmj{u1KD6`I_c zO8A9up`osVIz#@;sLhE-p;%GZR1|eRQdY;k4Rl>k2{!hO6R{X@t26?h(y$A$hutfm z=6<{I>c{8L39`xMhOfLgKgYb6FV|bX{ODcJw@bw)BF^Jn*MuTp*lyuZX!BiBxlH`L zfpz3D{ckrvf6_=+VToqa6yfgq9T~YE(s`fX&YYg!y-Di_7)-MOX?KP$_w>q&eec@` znO?e%5`$gU-$E|S^%7DnN_?Nx9-d9Ed17sy)5C~-TRG@Fvz75pFlE(|YdML9Gbpw$ zWj{cz-HmQc!CWbfCS_mdgX@FT%k;ZknuNvJ!JQ`5oIwVGO>=z2R7&{@=1=H?iP-d< zW}Y==lOX@1_~WTPP$7vz?tUId=1l6r!om_+Ku}6C!&P{h-q>NlBCS4lRnOp(i<_I6 zUE(75>2lLtzkXfY#00IX zaYYNOa%DPtYUFF8U^($ng6vfEI)IT~J_Rl}pyHlm< z($dmaL_4v;Qal;Aa-FspbhxOA_Yd{hq@?J`Wwgmqm7c=Ps;XQ4Un!n``682@oh>nl zAx;%$Yr(#F@uD^t^~P4s^Y-&+(odghuu<5sQrJ*18~?rF<6ES}_d2Vz^yR8sC^ZEU~9yQfi>Sz80TXMHye4ORRjT^C@N)!~@<}M6L z_B^PD=4Q(VS!XM6-dy!i$smJ!E2G7z0OxsHg}l`xs}FitTB@8$P$7}LThlkqClApPmijLi-?S@EL+0$&2jfhER=^#+VstvuCA_$!q#NR$H%cL zDUsFH)pF~P=Q=1g--vW-Z|!f+o)j`+FwtP&JUOPnL!S$W8%e`ezKN#Yrx6l%XBBoP z`18k9O<9?6Y-~)xea$%DQj-iL6PBHQp?S-s#?8DhO}6sMFXGef9C`E37Z*|6^VX58 zy!K<2l<1Nv{@z8&(EI8Mslx3W^?jGrFTH*0-PbLG%P{PJa@^FAAachb>l zLyi_D@QSO~t~LJZdaZNkj@ACi_~hjI*|sQU4-biyloZWEGXsOLj11Q1t+6rV=GmYi zTn`VA?W0pO{}a*P6iFO;dHMXiCHOINhc@X}az`t|s6&OBt%H#M%J;))V&bMGVe9T= z+(_`~`zRH5%C~RdQqa%@CtT5QYHh{4qF*RtH~#7s?f(A$wr6c&Az5y2?r5>4di}Dv zm{?O^A9;XW10y42B;SayhluSU7c6F{rm(1p|4njo&(b8dR5t{QVF4|6V8hBUSs5AD z_)f1{x5Mp+E$!`=J4-`XH8oeDk$5fBroAvNLQHVpXlGs>&uNM$5;? zH(Kp_MMy{pHcnpt{7w0j(mUxmNK7Kj_NhBK3CB1|P1`5It9Q=p0fkQYb6{2P)4aUf4bH8d!|bdas#$sSnLHZsyMe?pFY z(ls)Q(8*HkC$-x{l9ZBuk*TVU4cE^{PGksrfBg9It$^TD*j^HVY_3V&o^2~0C!R;PArPk24Ayb@Vug#mV}G;#JS8n{n;TP#3i1T^l`ETH zmC1`?*N}{dMNBmSD-on>7WnYtLqiY|)u=;o{J);j?K0bjuB@z7V5QhLeEmCMDC@5i z2@fm|DEJeAYmp-7JZxfOcvH4N@>}uUt?8U}KP7;$p9M~| zFq;SalV_))v9T0ynzR4jIz7`G0XS|^gVsWlE$M4Mlcwxz@nBzMR#oJwTA`mm3lYa3 zfqCZSxVgEfR#(x0BjrZZB4EI^+2ez)u&5|2*nD_?POdjyJ|_2e&gXIbd05>y@-oAy znHh_gmX=Ro9^^_4ybhOuK$$)6f3L(}x@qyF^r3x)IUS6-sII7p_5NmhuN#va!YZil z3o$G$FT;8}-LCw5r8iAB;_~hH0ev}URL);>RKkcZX{9vq`)!hx|NO)RE2r=O{yn_J zx{sZWEpSzD1-}PNW9LH}B_1B0$3`PT=D_0GQJn}uKIT8y=ljyz0ld4tZoa|Mivep^ zh3e(CZqBxEj#|bPk3Tno;F-f%_ogV`yT_^)gXw{ZAnxYPEbT@L!+`78D4Q!?CFf2c zmOtLW3GPIJCwyxLXrnlH4%^YuvD~Ow8s@bfo17eRur*&Y9%AxVfGu0Y+gqBLge30e z%NBTLIJ}Z?((j1(y;kaV=hvhZuwf?A7}@9Wp3mbU335fK+32-jz#_&kncvzPMNV5Q|Z?e6q66_vtgR68>$u*?`* zO#k3QT8&Z4wIc1hQYM8$HKS5Zik9h5{#<`Yro}@1p+;|T0fip8bJM%RG$$HsV8?6y zho=4`{8YmeAwLd79bSE=S71%D<=xevTwGiYhdavztFkVnUo(dkbImgbg|rzl%kXPq zVbdVjDr0cWmwDkwh*U5>YN};k5x65t#Rmfu;F-!?W=p|4RG8b`F2X^j=a|*qZdYg+ zGf`W>CGrbAP754F0e^yt}E&rJDZDHfFm0i)juS@imQk&B}Q;l~ULQ7)Z_lg4utl7xd z*cC_YWF$xqLfYMb{Ir)5euKZZ#$m5l6dz-%h>iLZ9Zl9YCV7vDfe*4|n(yvi`^f~X zxKQyQqNG1W1B{y$GKbcO-EDVjPEw^5x<}6J_2bC)FNP*sX_ z`E6Nx6%~HFua>~CXxulewn{pyq=YISM<)M!%Y>-p#=%`faPqM67kRF+d9F4(5BtA; zKQZGq)Rh|3S)N*I303I`Z3!nQ7d($)$7-t#pCU%fot}~$&z?V1@Yv@n7q=V*G)Q@8 z9`H(w*YJ+vVA)c}i%CVhkujb>b4JH=?YQkdT*s;E}U&|3H^@L~OHMxF9A{NU4iVl>W8!SfF6SSbwna(vjh*HWg~ zXzyi*P0r(HNWQ#&QSNYo4JX_0sM{AE7LhsbPujjKS&&DQB=H}*DWn`Fv*OgS$E+Wq}mPISpDm|+vi(-@b zF(kDwv8>$u^KQTj2 z3ahyXw;702&YY+<+~#SZm8M04pCJF zgELtAY#0|hwuuoujKxuEuYjy9i^gld)(<)&3XcSI5+y!~C6MX=tlpCo*`SzgG!Vbk zP3Y8$Q>Y|S?D~E%gqWBZ(riR)1YODaKcCdiXYSe3*d7nQZ7Xx5VN5WDZKas`9dzv+ zuab|w`y(Q{54zm;jievv>V$`o@n)M^@DMW) zIa(jd?wN-Jr3KDyes+3dUoj)Vr758WxpkpG3#kSRJ}S95@L{<42~cCCX(a zqH<@)&Kn0l$Ga8vRy8!C7V^*Vm#Wljqh0 znVrGg9t)YFqYmxUgbCL#cBkSD57VjZ)E+I@tv1}N3*{aDrBn<0hKq~Kz3Hal2elOi zhs&oY#e02yKfFW;c-oh%s;V;Iy~Fz)OM$DaOK+kk=JtPTRszHIrJ+0t$U3`g<95SW z^^JBf(MCWXe>jq!&WmtCby0Eg+0}~4ws5LTDAYVHPk`R&o==q?S4$P%f6H2ptp3rS z8&$V&-{zfQy3L({EKje%FgPF}KyfK@j69gAT|`W5rgm?ldElme6o+~=D<2#{Zr15ud)X_-N!Z7#L?aDg%y{lYmSb`FlH;qHI_QPS0YIV~B}E_}S$Ik^1) zsc_A;deY>D?+63Nfcik|``~B!qh`Rn%@(Sg2Bnx{h|K89$Fuko{%5}V##QIOfBzm9 z6Vv|M%rE?+#w%7yNrrRbD)MRfhq&*rjRtnJNXPE|=>l&GOipI{-J5FngnFg%83FKx z`yRalk^TNBH)`u1&#I}YsAObkeBU)9hM_8w1@Up{cYEgJvg1<;-I+h^uG8^2zg zz0J=jWfHR&@MaN`ut_26LYU~meERG^30D?YR)Vy-T58toNOE;;4cZ0}geGgqgf55@Q!s4{U$HSZMP7;1FQ~=Qy4XcWa3#8<*cdw8V5}@YffzXCJPr_$k z0MaEpJ3GLp{FHwEA&p+X0aj+5GY5Mdp^lDDNHdLw)`zmvQjklm8&1T;#2D3H93CBI zWo9<6xH5zULKUH`j5Faei{`oe8F$imFSYYcU0t1xgF~I;I5_w5FMj{V5HyoO`7*AE zIbCok#W&kB3C}+l>W=?tfOy1XSV~C8Z>jpkp#a3Ok&6pRtZ_sG;-ier6-q1{fzG)-aMZo!YH`XJnVp%C` zA>2V*(YljeqvL}HP+wlx1!&+K>a*sfe5$$zm6OAWii+Zx$_olo zE-o&De0BBaO@y2B6!uR{M2k;&G=H+T`uO0+4fM>+%wa*9?~z+&b@fauomr`mWk*J) zY`mh&d@l>LtQYmc!2!LzpUiNfNkp+_yHck1)VtTFPb(_Kp^C=$EsV-Z^CE_O62%@d zqP7q9-W@&G(1_?CVF+m~fAn)>rKGQ^y`3ONc9$Ngnzh0EqwJ5lt~KX1E}T1_p{sAG zKi%O+3EUO_Pch#4=sV;{0I*NieHcc*K0#{EXBkyh458%b+n`$6?BIyGxAnWn$;~aK zrbc>gtm>Kk@icj&fI0SX{$1e<^%Nj@)IMMo-Yw9uv&-vrv;@mO2LJJ!oSks3QCBxK zEUB!dPZoP5RNWiLp}v9mt=ZWj9Td8UtD&T_#8gbS_V&om^`=UPfwD4>}Id{iobtHk7LDW`2JtG}QGH++Xn(|!u7 zQN$%-O#=R>J}{T8zWNl><|$B+fczx6+9Xie`eqhV5fE`Lko;&!0bU^sArWk*JEv)lRZuk<^}zZJxWI8AJ5=Y6XU&zk0&VZn*IfKp+UW0E!)*F-wXjkMev?v$2vtn zKRLicw51A9ng5v|4Dzx!K5=obGk)}or?9F2iEFiLa4J;b>U;EYw{x`254RWLo2>wA ziK5)x#+~I6LCZE$NVn~6sbuo%(aepoTbsSo>#XeTPgmW-g1I!S$E4Nfz{F)s*NspT zu1nPC&Yg=ityEvLpvSLqUvCqi^m!)hy=!n%Mdc=O%>1dWtgNG}s~M`_xWvS;5i|dB zfv9(7CnnV{ME(8!AyOck77jrs@Hkqn=pP&ma-a021K1a-!su5ZCAIUY-|cpzHE5n~=~s$GU?D^=8S=p57113}i!poMAfX zDeXWzh6p6P{>anR2)5GO`aJ|XpYBq~{W_C?{DYX{E-rinq?u2KDm|Zo{UFUmK zJNo+6K||xzO6hRZXi*DsO*D{(`v5FI6c)ZWOXLH&0iax2K>;Oqx^J1h=0>%P?h;s( zj0qF9RLK_wUp9snLPRnB+53GCs&sGx+@z!=gECv19(H+6be@2qSWJgPd)R0bzyE>8 z@tS*m^N%vSyUT%(0AY~rq7PA2Q^W5Wk%~k|2t2g6Z}C6fI!{eUx0j?IbOU^+ttFf) zEN>(xj5mobpwRn$D^xj zX7zq7h-bXhc)fAF)n^Ih18~#6LN$bbzB#BskiEQRerxlGU6J{~`x~R5tw_Y|M+Fvy zzmW%n%*YHiAmkp8-$~XROL;|aQXWQ=`lq0&%E53y!YDaD*^TfWPoABef5mSW{pe?T z@!^dDg+&)BYHI3xlXboy3JO|ZT-1nt^-7IRsgjPoq@U{0zuaAy~gVL^;LwG-vPnaq5s|+hF&FQdl*_?!yw4_Aaj2j)(2ToD_t&XwmoKc<$Fn@kQH&7&!21AIXUDJTQc`I zMnyECD+5{uaj;S#w=2z)pUyfHcgawMbfK1YKYfBalw)7NCdk!&esZ+8=Uv30E&%jE zGhL1aVnJC;i_C5HNAT#VL3g4c&Opx1^V>{X;H1;AT0|oNsX`ufYYt#)L_|cO-BLk6 z0DtGCbHjW<01O+bl-Y1+=Z?MO7#Uh5Bt60H)!Q#6%yzZI?7Eoez(_Tdgx*bh_ei{e z5n9^XkPb~o1`3gk3QcNQ|eb_=YpuW#KkGBVPE`ekX!R`<@GX|OZY z2|>4VfoS1kd^Qx39s`sXpCX(kg1qe+IY$aja8Prl^V`HP92jk42hyGmndX7&_j`k` zfq?^t@ZPoWPLpLqIqfXnp{Ak9{QUXc8Nh@Nh?7wDf$R^>8yw)F6vV;wf)+uUA=lqM zy_FkeSkSimvw|(Yla+#yf`TFd8W52T!uZ#&T{Eb0D_~2sm-QrjEF*@FUyCoe^N~Bg z%lNh#4vKN`{`;?AwSz)JcD=wcGCqDJCI!twTcJQND=Vvc!}-#sONgw2m&F+< z|HRpul7$6*^X5(H75eOryQ8*4ugRX#TQ1E{`k%={0-au7j<}TOfi09oiqh8C4+ZE0 z&5Rs;3>Y5{DkwCRgP*_e)p;7WX0jN;t#o5&_jpaT;qvd_zeC_&M5oPd7a)Dmn-UHU zqOm$(CTLC|{0_JTbV_&-`GJ=g4LNIS)1!8m_HcQ9!U_imrx^NE-ETxmpjpwJsT_)s z_IuwyTWxG?AOabJU5a2BXm?5Y9*UGZj6Y9FVRn||&RTg~k+76B52cK%we_V}Ja;YP zy-*buBf7H5F#$W<+bY+u6Z4tXMFNR8ZC-2#d3|HUNdl@2S%#L`O>LkO5RV)J0u&H! zTs%B2pxkcsh)=S@6-ZfMZOcO5fZY@2e5DXlT_Vp@pl3H~G`;Du4_Jvp{ome~Vc9=U zC7a-%Kz!Sl`a)u-?X?SgD8p4iT}H~L*5&fQ#GN4E^tA*R7h(NDkE6D(ZqUn@)GF%g zmIr@+c>w58(9wnVXQ^8*4;L8J`AC9p=`ifQGEvLKUwev#IXjU!+Msp?#^xMQ1jx^D>32CB@=OvoPm)1`4{ zC=~HS-HK z$JNdWkv`_roSB`~xqaKxBP}T@JeY)T1`?SFbNu1Kf%(K2PXv5V4@c@}nu19-{`_hB z^Q)^EXxTB8HnsSVlWrLsM*U48fwY6)$hM^8vh4;t3;%>R@NV^9elCCVns4p= z_g^r*tlC@~WOA6I(OGe3J?* z#njhKWgOFbX)ZwNSV&e#Bvd&QfXJ>6h0iIV9@Eo{Q0PFzb+6#z4f!Am;@;rSdrpw| za5cb<|M}-1ZEfuUgwOyBQPb2UMGDL6dJ;I$5GJsAjfa+llM`Q1P!P$Nyu7N=E->}{ zMdRV+g>(@-=Hk==(?=Z=C~9V$%)iB7y&!}eVOXFBd>d&_jF@_(!Ka~Tp{l7FY0~%n zW{39U1%buR&Q2mofMJo5E$e=J(#BsrD52T!F`w!_H8&TOm`IN-9SZD}$XgPo%Fb}P5SEa(95{e2@Bx?kon0Wxh^c&AbnTScow$Z=#l>Z#zbiX9>G4y zAdeTnT}+Yp#|#Y(5t5Uu!<=DE%|;gFI_M7*zZb5)pSg>1IS~kva=|DI)H7Tu(*Nh0{ zpT((T%y(X81dYP!6pGKkc9epg(7{9+3tntOW7>&M0aYU;-)Byhf=FpW6IopMViOk^ zZ|<{+2}W)U9rv!)hE@m2h#OEI0I?w-o7MJ#Y>dXz!ocm-$-@avstnn3I zF3hF5LIKK*aoE)n63W610S0^+stIqq!Y(m$IhfFhOEaW+?T7^WkK_JpgkJ?evZIgH=_w7G14`hU8C6B&b7`1#fE-eu5Am9mOo4zB!Ci*&YFe#a_O0$~8WuQu%n z{Cai`S}I;_?a;Cgp*Gw2A>E7s-1rcT0+hM&`}d@OuULq~n;l@4ZsBkN1LE^ucdcpI8ox*- zmuAPOC+jD{e#e{Y;H;T_C&3Ha9lOZo*4a(cM&vsnyQIyBi>FHEisnkpYJKJ><82UPbGrKaS)dw{HnU1gGc`6g#$JC%5w4>ANp%d`_sqBYNGN%0P5<9-mtl_bLqdVO!C^i-! zlwr1u7h5xTwUIXDwBHCxa8MA?rm3Yn^7fObS8PPw`^sLv!q570jFoTVjhmT?M(_uM zDZ)s&bl+ZFflwcIzSQr;Yh|p83Vf>4_ec!Idg)TReyq!Y4Q@wgI35c4B>G{!{oigY zSTelT0;O;#A*R!+$YkIwzo6jJRj*m%OyGW1&JE?|Zrl&;{+) zuIdyFchSWL$WLx5mQoq z#hd;@V?xi&>9f4?pqYU1br5i%pbsf6EtTsUGjfL0KOyO^Bdd3R>h^$6*Tomz9VqLPyMDvwR}crY`d+3QYd$b2p-NrFzc+PhnD@CHOQkeDsEt>sU) zsi-iRiM)Dm{~I{i*rdx)jiwS26EE$Z_#G}@;^g7!*6!-*iQ3%Tv-ShIUs&96Vx{5% zF&N;x)buANURHnK#+ffweoi9j5JL_lJuG@OO4`QJUsYG&5-Cz%?MN@iD9x-VS3 zC|KbzE;P7U^uXJ@X1Z8~-5hkOBwz-9cBiY3{+6eZITOojT{6$JDf5Vm(nQeli^*6= z`dALYvAHKcq4o9hAaV__dymyPz)?mx(b6g!5fM=gS=Mhdj|u!tS@U0}B35L_&6RI{ zE#0Lzh{A#jn)DRO(A^9gMGkMYZ~_vri2*JV78Aj7hp-#a*H^NC`rq8j7L?dz+086w u9pilKmMknSq!2kB33otm{y#6<_Yd%&%$l1b`3Xl-QP)(pR0@^uJ^dfy;nko3 diff --git a/docs/images/chapters/extended/37948bde4bf0d25bde85f172bf55b9fb.png b/docs/images/chapters/extended/37948bde4bf0d25bde85f172bf55b9fb.png index a714c4c5dd07086f32acd175b6773f5a706870f6..b73290e377d96f2a04697ab70027299ba3b426cc 100644 GIT binary patch literal 9203 zcmb_ibySq!n;*Jc=>|tc!2zT}!T|(_6o+mEK~O-trAJz7h7@UOk&u=lMM0&NkVd*e za_@Y1_v~-?oc(L}h=+6T`@Z+y_r1^aeBuG6qoqbg&Pon}K&aI3!}TB#JO?d1-)w$hhOI-cruban%W#H+F!oU@95e8_#0t%ORwmQOx#O+A9d}=ukv%fc~TyP zyjpQf?62tWSUw_A=54B~%9$E8&yHs)I#~}i&l!6yx4h@GwS0cjOQ=L^Nf@h2;N|6I zGwpa~)JA@^rqWldd(;2&BH%l#l{^?ZEwy(?)6&&OS}DWdTB2za1(>05e)0%1I#kGi zAHo=yGj_J-hDWi-e;NPvr}W*s-^i)kj5^lOm9Mvl^H6)>2@I4nX1W74;3Ep_4smM8OHIQJl z*h9yG)kQ52YgSABY#?Q2b2Fw#ZeeBP+A?bud^leTvN_GINid$ zo7!ha$mHb~T)a|QM!0Ujh+5CA(^V`{ZGTkGTHqlw835~^U0w~#&UV}1Pv&cEV)OO= z%;V+A#@=m>rk$X3#)gW;rPfe$2-e1Krx3KXc&z-np@#7CN(#Nl6+xEB;!taB+Tem0 z9v)u&?j1hl`0rn(5`A7#QTmFC3bP8c8G+xPS=5rN7o>4NPgLcWc+|AD?f3UwKSe3h z?7O7potR%eJM-^6G4vHdM19F`Zf3_|FvTS$4z*?ldP)U)v%8D^QPI)&6cvMY4;)w7 zboET)U$b&(IIiXXxMLZdb5m|Xi*(PUYb85-WpjxV({P<#)do$Ad5J7lQB@^&eEL*Z zhc;|@*s6DMiwr_aakHz;$o`G~w5qSK%+g>^7UZXkh%qX!LsKIh0m?8@Q|wW>4< z=>DN{RoZcyU?B)wjMB4b!eFMv9ImjV=?uKAfsc!|{b`}D2aeBgS!734RsHJy`n4-z zJh#DY?9QmJUfw-j5BUYHN2G^tN>yG%QfdiB?4t}E&bU{6Q(R06Nl2Gzew1)$=uc)s z`UzpGkZhDiP=AlBk3eISEZ=xP5dnee&gB{Fx0;N$@Foc=Y*;U4I*ADd1p!rSQ5B37 z@;kpoMOnWdnRe^ZBi22MC&Za6YPQ5dlN64}**X}6_qMj-$hevTSP$;9x1`B`{rY?t zlSL<5#6|~+B!=h+e5TwSI!mvttW@#6BWb=w>+xx-$tw|qL0z721vU5{2(GTKeyg$9 z?(FWKD@ZdBNlRlI$dnG9sHt(?Fdmechr0O?*nRy>3=z0O-qJH(V^0&uE*+wM_iyKV z&a&=h(XWu};j-d^j5rZMbP3uj_tm*GcIDvif1%`Gi- zSFesNd5=|Es99U{P6u5EKos5G@5sr^)3LM1fOlb=Oif843RlJpSAMy^zAoXhHPsZ! zB4*b| zKoRX99EOaioC+(=!Jw#^n68;w`4RKoyLUlohPDO$dhTrQvv@TuER0)3ql1$`p<8#zq-JA|k7^;~lhwGb=GQJ1IC-?Ii+y ze3;j!scEA-84(dtaakE51bo)|Nv*B9xwl?3Ha&4xThlGhbvIKqj*pL-CsUJ>LLG(* zm1qh1`T0LgT>kzPdc4`RWgL><5@0uwi6Ww4WIp#{fi5Jxd-u-e&yQ;hOG|UZB@e;N zd~>V?nW=<8P%_>&kNKXbmSjDC{!DSZT+S)oHBO|)KUD4amk8P3!LQX+JM^A6_m7g=sh^!)jAT@v>$3D-GzenCNPn7W1rIplYF z{WShJcjVjWzZyN(s$6E3A?*eh0YnffkJT$nUo%J`;|;ES*?ybS#UqYRGp!tu{Y?T0 z2{n8BLSItsCgy4TYfW&39V4X%u77tQn>KsX$z)NVpB|}sdrJ*Fio3md;kvt^zP~m` zGv5>6GhS^2PL!tPm&DJ5wJ{ODL#MO-A?@wmUD1-!2r9NJ@0~fVtjEJ6Be)d3uc@g5 zqAK=hqy<$?w&rkC;JM7R$%gN%U-&@6=<4dO%(N+NPJRr%7fC&Py49xE7I+SO{P_wZ z{SH~Y{2^N9L`iFcOV{nkB{<)-(Kyr3=+{uYAB1`6KUtb?76;;X!pOjn#nflwj zZSUg~^K#@o$O$R81=0$$rih|!U-GA;WlCTVwtD5d_nl#gUS1gn&aWUo6VlUPfltQ= zz>b3PRy(M!2oQ;10Nb(K+2$zhjv)EwuHE^PfCSt~e*OCOFX`eGl`V&?3vM4D1*JTo zWS~ch5EMq-k&xh(m1R3U+_>lIDFLE2s`lwv-J}iPS+yIB(d`H_4^%Zl;YLYW8QjrP z@NnZ735cLC7%`HkV-@0mTghJ&W5Z&8Oo+rE4)B9R-8mSum@mo+>-(&+&i%a4jUk*%B=!@Q@f0t7~XDzFWJQeBBo@eY(MJLqpjtFm{3df_ze*fgzS&;@hDK1X7>gaB31J~ z&2>drdu|wqCaw406+x_rjT01hfBWN(OJ0x$XXod1TwL+hHa*JFq@7k;l1^B)?h-3Z zJOWwTb#Z=L4PXRd5I+=Zz49&BPL^|mI06+5I|?ol*_>&M4>&!v8mlm4mU7{iaQ+#l z6iSRaZ?I)!h9W=ffpwnTWdIfG(;KDG?VQUq67a5!(h?KY5vY`^h2WBER#sNr8FR>b z;>rRgog>p&a7t9JI08iZ_VIi?TgzslG-*gME(oq&BYOGrrT3qLG?>e8#l91sU+v6l@(iVQr|H#!BW|JBB&a}Y^5h(eQG1WR^HRkE><0ZH#8t1 zDA*ClA*-XO7cyq5XgCz?V4s|pW& z@p9BXQJfm>sS4^$>)Gyu^~sM6Vs>BJ?Q;Sme*d;C`NhlD4HJ2N#S5L@udQ$nc#I!S zZf@=|$no#>&n-$RWZ%nEG7#i9V~0&~C0fX&4#!k(8dP)^>*nP@L#>ly7w}BjU>h+Y zdDef{Qds!!5aDnjmQAvwt4n3Iu6IDn2|d0FHkl>w$H*q(qJJQdn=bZ5S&+I3zPfrjDv9c?Z;ZrRd#GC&Wd zEPooMq@;|0aA4To-L1015_fiX-k+kErp=&Y@Z*GSyy6Nx#sdqs4{0mJ2A$1wxx9Fh z)OgJeCcyxW69rG#*4BC-O?jCR8~aEo>xoDsJ{SuLn>GY@babRDJy1|k5J=GgdE)t( zXDCj>^5@m%!JM9S@ux7ih5P=07t^-4w-dm0me6o@A0KHP{y%&;Mm&dl4wUt+?8v zzc98;0O>I8fc1MZCW~w27ZurDoc(nk&;c(j{`m3Z`v?Rf#DAxY3E)#xCkLJQA2&2W zPq_=N*A&QQgj-}iVg$(ZZ%eA`RZ}z*)b&~lX&qlJtHr{r3tK!gvgod-T zJC==(fnh$u25vjm$PWD)grTSnC!Tp|IJ2J)7~0*F6~30!ho^!!Z&v#rID$j~#B0wf ztSr?u?|b{WxTP!`MD(m$FReA8i~5RRsn@~kqvpKGXmX#aO zkqu2vY_;>)v}=KZfq-u4GzcUVeI06w?|AfmcUtYXgOh9Qg9oFhfUQ(L?q_U1Km2t( zlyhP0abC^2CT~1gQrdOpRpA@-h8UZhyukyvUXgx3Wa-ge$H8p*sQCCTM+&GBz!-3R z*Q7oD2$dAw+(hpl|0eL?{zT#%8rI9|A&nrW;iLp;EMS&mIAL>*cZzpb_(f#X7mmbd ziMWB}3Jn17e1d}SzFz$A+>-$E{{8zmIs)J)AuFrAKE9(vd4J`5-jwsns?&5c3mF-i zfUxl9YP_O5Yb^2ewsQ#_w@yyJvcENBv6>ncWA`IZ7?9Bfz{git9a~NTapM*iroOni za5nc_OfdjZ_`_`wUh`i(6D{W>3hIH$vmc*(d!L-*ahAO?H!_;|VX;@Y6mM5NaB{Gw zYi#_=`Bxpwfsf{Q-Im>5w_6si?4UlX0O}IyMg42e0k$^bC7-kzOhroCKKb#5DXFb%@jJz85b{&CEcM4LOEP0NAVYSVJ~% zcTf-?HM!0cCu=+|bq?XSJXjsMuc3keEjiHK+?jJx}%>aA@*1T^vx8vf&L{OnyH>6$gvF@14N29ktNPqd!AM zi^9eb!}qpizF=$g{193;w&)KZ?f`DW4G5y*(73X@yL*xEcLS0qti>s-5*N4v1!%-4 zz}r~(ZMg{f9wCver%1rUb3c602hqvoE|WDqW&ys7)l{R0&?VKI5iTX#u!lujWS~qd zKY3D|kdktZG=!U7&Rb`^$|`jFoYHy7K#i6VcYnuPWt;Hl<&9qK!H=P#@REl`%(Sc|;esGlKsb#+x%QhEvI>iGA$%$6WHu0v_Qy1(&@X>GJz>{uw42jJ++ zSY=LMOSxe+KF|fVD|{jkA3ogquEu2r0E_tf0^j>bc4l?1AQQ%$yl&rNjh!|*WD+)w z3a4O7uOSKrv`xsj84yMu0HuJLt^BI5REgKLC9{(k+c`Ms0vp@zzuku3gCVzY{AD3Y z8ze3l?0)Yvj5q>fxjtUK5m;)orIji55}=pme0S_y(`-7Un>CSr;C|P|&sQT&+ZN9L z?txh9@Y`zX5utzYXGg;zDJe%rcJ`g4+%gR%B|Kz>DJtkb^qh&^++uJT@e+Pq7!6MPGOGX?&fugAi zh*cC&(uCm4*8IA02>5Y(S2R<#!w|H0QDSf55wEVh57qd1`gSvrbS!f$Q?(sI3~*&bzNd*i%)owt&h@ zxy}&+o^#8`u;yK9DfZX%wpbufDVbr4tDaH(Hs4`n^~(G=q9e$@T_CJYYM#&By~4<7J@NiAZd^J#vJaFz*TZ$Ayd2&q<>ch#37MHO zni&$GzJArjO>MsebI4ZDOiHr(b1zyIV55i#UC{ZOg){|v1ur_1?3=q^01}ycC6ciV zajoV>9+un z0&0m_#=UsBoYqLZ58Ih;PoD-j9>6T`gAs#*XW_l8>YUI~xg@|h6}kbl7CIr-ZjcA#gPoq(OtwG5M!3B!zFLC>FK5MOUW1% zPweTR-eFyzY2yM$MzgIv?Sx<{IYUW~dL|HJ#l-^C2Fk-k$g5DjLH-Nyg>XVwJx7oo#2k z(MItv*2hHasa09m6UA(+Lb;L2^Ksveo@p0gq=9CCtvw zwky$IxL+oN_#&Tl4CK1pO&)sGqewE#X-ub)krA`FW3rJWgEZ?$)x4M0PupJwe6&Ke zv<95MEi0oms4!L1(Mjk#NKZ*lRxBF&G&3{vJ5bNx99UQ;<@Q&fwtFu0UC-(KySFR? z@LG2juN??`5&vVC(9`ib)=Y4=dZRTb{p{^>VyK|=0|QDn3B(6qc-#QAqmx|>HJBwE ziAFPjU!{|l$slD=oWyhrrjP^HN=imXSHMX(p@WVBfd^z8D34(XC_}CoFTRriqBB$7 zA20?u93J)UDxG)@28P$!)N%pr6%cStxqQ5ib`Pz1M-g+S29?%8PVsuEz=LDMQOshf zwbfOfM~}imibs#L&`s#0A%zyJSFT>=;pY#dP-8%SSqjWUWT~G4Cm|2mCcIuMa0w=8 z;Q+tzjT^J)*Wh-RFRbv<5=qB^u!4k+@8zIN)Qmvg802sDSYsryKz(V4Vmp)V>1SF4 zny>|wQN?>sqKM!MaNX}nnA2}vM9f2Fp}|Q4(I}eKP*_L|NP}wmj50jSrp5~2JrL>& zXg??*cHMDN>}i^j>h@r&CNJ&bk;14I8e|)>li%dj)b|!YuS^*a@=y|*&8lgj^L{Tb zKHhgN-`IQ4xn}f2KEn{CCeCUBbX>%H`#1F(SqKDZkG#028m#;modH8dMXl^2ta(MVrovDnVyHzoX+Tbhnt0vVff`fn5ks*+;Y&6K zRzJQYteKdYhR4QC8gBCTSQ=R2e^5vopPHIlu-~Lj4D@7zZhYlG+};j0E>C@PllRo# z%j(Cp7)oAVUf0Crrjh~taB#DO5O(%P7;x_4;oX-jSMcP5#t-mQQ&X$Dx>k3B9db3%FuXZH%q1ky;(8}iaI~CUk`#Cg z2=F>@xPlMT#nZ9y9n%RoA20%5*WKWiou>t#wY9}OSBKd&Ha4oU+p1$>7Sm~E1zH+| zX^q!-o919vyW5C49NRUhG*9`DA^bDflA*SsAVBZF{uVrh5BpGG#f4H2_Yf3tgH#ZWvA?W|8S37WT`8txHwq!q2_?L zwzfva#i;^|ZA+3zH1S>}8AR@2_@Rr3$Kq5K9O`xqV-c|?G&MDKIXia6J>)>m1%3AH z8MA~Fa~!9$(gIcY8_=y3z*;i+(;D%f=8ibxEY0N9)Hz^e zgmaT_A00*NmB4Ln`GH6(c$V1vP6Pn~4iKN17y}T45J;952?VEW)HwSg z0`4sBkA6~X4AjUQU==745izHxzsjb`J~Uuuj1a4_o2}DB?eDt-=P$=z08T7*hE@mf z`7mK&*`~W`i}!f7rM71HUTA+t zr1T$mZ(i}*5kFM1P$W&#yLW;RMN6yPPe+9E-W#BzGyXyKGWLnHQpK1N)cs}BzBLLw zu_9^;npF2IgrD+l42PCM|COOB8(V?*>kBYx0oXhyCpuyFHhqFo2ExDM&`Ic@@@RU% zej%#1!!7XMeo!^Ec_Yn)%1!#c{a43>X|;`s8WPZDS$g`O12msh?Cc8pd!W8(Y-Ats za%VR-Vnqw+9c~)Dl@g^t*xW4k@=|lP<^O@KfMfK?K<$C6sMJoh0ly1{LU~`FAK^hn zF=ZoMklxRqI}&*A0U^I%A<4$UL8Yy&O-Mpw1B%YZsII^j9T3xghqoY>drM#M-@mU2 z(%dNUF1-6SS;)6(7bOZ4xeZa|zqE`z0K@9+>Z+X_KCA-#l0!*J$vL~_#l}x@oS%zx zoj@M$ob3%_r=~b$y*BSiAiN zCiE`?B#1~0^70N-v6*i6@eL(}fcZdo&hEp+#enRVB~ZbT;LJ>R1V)r-s7Sj9=y}S+ zTz$dW`FWe|1KfoS2`X?w4IpO!A-GycNRg<|enCBM2T%#LgmB%W;KGH2-JPyWvogi3 z-<0<-=6ZVg1$q;XCZNp|c)B6^B}1~ZAhoFIdUJC#sDmZ-aiFo&;w%38^=sq0K3h@H zSF^-MPpgiQprN6GINpaB&STVN7qmnv$zASMXxMDapH$F~0cn+= zpPyPy4N&CC9aa#%tMXVnLcIR8+dXepqPORwZ`V0Jee$FoNTs;P{i(IXx}_t<<>glo zl2h(eQYKM%x_e=zQ-~u}Q$s>SO{Q2fKr<2a;6UT7)+stR7S~!96%_@tpW=;!n-$=f z5X-*T*H%VKnLrP1;$0mAiUkbSL^#0O#wIZ}H3IZH0Q1XV3;=y{9FLlxr}tpF{X%9N z9m#&TLQ9}{l=n;h27g;$(se!|q4LEvKVU3?lME<5Dl@642JMC5q$GyclhvD$cAzS& zo_q%`-oJK<=;`QQ;%p|+&EX2z*L{!$R5d=tbA9}~-evns0^kq5kIG3erua|ip{X?O zg@MqSWShp;DQH!NS5?7s8yg!Ro|l_7Ap!J=er!umXSD)6O&J&fX;1&G4eFOIdmngX zY{HcAo;8?1j88wQ(>|x5qUr+7C?O%?TS@x=^Z-;`T&ntNqY$6^+4&lz`e3kjs*j7T zG32{WdnX;?6ehw7w}hmnjb6{w%fGURs=oTO8TZ0#t6r8Tp*K+k5i>mWFwQ*NZNb0+ zCxZk<5HdV+?N=3MDv}zhcuPBBf5KdXB%iqBzk0g*k4}OiWR!?4;Rz&KQ^*IP(2iK4 z-SzY(C6h}$8p@wFncqs}vq8JRu0cdx$Cpp&s$qDGv~{dChIvvMI_k}kt_zP^X+5_; z{@9DdIzboLeBsg4StQ4uMpXpa4kPGM0iV&xMBaSK{X0Ya$9cW3fR_4k25H_1bta$B zS+L>^wt-2J=25alJ}WfZ68?t7P2X|y`ESib_$61TgGYHzLK77H6%RyRMGIb`Y#H)z D2E~s3 literal 9201 zcmb_?bySpJyY>*$-5?!zsA|`PTZDx}4cO&wlp4pZmV<>$*0v8tMv!cyxFW2!v2c5v~P+ppm2g;b4Ig z%97Pa@E?|yiUJ&Ri~94nwKxR=VTLHdWpsSA_ZC8Qbi8h)4mM#@(&rK}3G6ubUB5(M zb7UH&F)uflq~3LQ+HQvC$uFFl*JnAZWVqO_|#``({9r}230T)4Q#Bj>WYYH;d!{xbJ6*gS(72YuHCk5qQ`Qoi|6d;xDZ7moGaxvbY2?;zFW*#6`#5131Yp}Byy*^Q;_wH7OwH#`ZP%N9 z(6f6E-$oUZzDat(&+a#3JtSS2Vuk3cp+9iJwYR5on`C}limUN8)FPF@;pgaW*=rU9 zj&Q9h?~qEgK@j1WCz*HN7K{-D71H{L(@qoG&bLew)FJ3ICGqChiX&!?lSr6Ebj&`f z@1W5?V(|#*vqeiMM0X&>Qt_?Dk#4L)2Nx3HYFLK7=e>F(aVZ&PDZ4IDH_Y8Q>m~-< zCrG+k?2ry&mh8dK#F0aUchg1TCSzo%Lf1Dp3u|jB>gwtsp`oD=xV*e<@wm)8Ew9hs zHO$NYg|yRjL*MRZ=DP|Mj!Y!n1X1AZl!Xe-+(krP%I>i#II9t&V_e308R_w<#?Tj) zmf{&28h)s%a;<6o^ocArHB~s|MqEoE3PM_XuZ9LmY*$yn!FJhaQgZF2urPE21k;J9ufU3OD=lTB8qS_|==&=4cC7$5 z!LCu3HF_&6B#DT$k`gY&V!Cq7Ijku4$HV?g(?UL1Gw4m#0V~n{LXwr@9%*w9M%>@{ zOf_iNVLg`)jgL|LdF%DDu2Fbm!%kV*cFP+UPR(fZO8bFbiT7$*QOnB~a0UJ4-5+jcmvVBt zkkAiLo0VfNWhdmqn3uaE@n>r6h&?<#HTCqO-oJ-NL`Sc^#AcLMR>p%c^YRjp7vF}) zQ-_SZ%)|5T`~GZ60W2bNd+u169%Rv{uqN-|7&8TF2dHF-QoSfy>_jhNfbN}4>(5M!NSo{F;LX;seD2SBY zlL4PzJsF~85uN!4ipm( zg5YK%df{&?-mPYcH*Y8+A|ov~hBChGoa`lyeg7VYki5XDuCCT7(Rp=!WqO=*sh+{# z`>9H=OioWPZ8BJ748Y89o-pROjg2&%>e(Hg;n?msSAGIQLb4>73G^alDNwbpLWE@8 z^71m#&JTb4iq!UYDJ?Cna>-gfZS8Qw8e7Yiu1Md5PHZ%YcCmWz_jf9!#%chu<-B@nVFrM50y1MlGZIwj;lI;*?s;3NfM8{1?K<+b zwSE6{l$=u?uA!k3J@W9{d%V02bgJN?DV$IHu$clwcXyq{Z?BH;adA1;RGC#Z>>7g= zZk+C`gU5^cMECc`Q2#9mZfY>g_7eyN+6Jv}|%{fbIT)!Cm_p-*;5wsS7u7Z)3! za&vKYBl)b$PY2NqraaI6r@GkK*)c(OADo`{{iv}^&dQ4a@Zlai6l(RoNa^6@#OvaC zcxVW&sYy!3YeoRk!>FpPB)Yk|(QBBVo?iQ5m)E<}9nI%5FYGj1i%XaIUd>)dM`yWd zGq?NZ`jYhS-S|{4ebiK-n$m)pg3m0S<{L@r60ua$dE}g&9s-=6%i*04dc*9GLU9vWhKoR31(GU@J|Q%v4(ocW?l%7!@}FQ z%sTq|K8{Ovc|lZng`7f77K7VJt5gj}w|43zEoVh7X8nJXQd38hvdLTg`j&r>n_JG? zTj=!n5&{ha%i?JL3qY6%`9gQ!kaC~9{i*|7Gjzu^LQXW`mBQ>s=rOUe*Zypc&wTdb zUG&>z_%P|f&dJHf#)h%7GR0)h*>8n>weFPY*VEf;*wsBai2w5COH|b!_CAqepV=aZ;O2{YC$D$x(zR_Ld-dR<}qt#n3_U6xdusTk0 zBVB!cYfy$W)z;WL4=8PikT1`VH=jULnuc|Pe0pK|j5R(T6KQE_RP+-56ib1}gaEk@ z2iOoOO!>C;B5ib{arL`3BWPl#Q>kd<0xc&~#s|m8JMSK` z@PTb;FVG(U`lY-I%4DY5iwjJb{N_yr>ZOA?CrJ`|z@l*L#&ZsSCH!!b2QjaA`Vkft zg(ZG6p>0rWPg*5AkJBCDFJLj7QCj-IX|A47%zHOfudEmJf`o_&%;IbV1&MChp|k`8 ztWE)Bq4)lLY{o|_{pApCt{ckoF_ zNSM0}U)5M68(FHcRmH{uo(El?t^ha;@3iOu=yU)&$zR`8eh+Q}^2f_DXxahDVcXpw zHQ2PYv^Y389pED>KplgF@uQKx%u%uU9_yhc5}BEquQD=X6A}pKJx3)B8r@jr5sZ5u z*ODDAkd2{>G_bnpG5em89I?am%{(?CA*#VN?(UD4-SojG9)twsoD*3X?@wVFsx$S@ zAYo@96?*ZMTzWA02ZpFv@?U#k85v+1^G)Pn8EGdnw5&Gx)1j%-3XVvJIwJyF{+=UN zj{CK>wH5&AW*Xhu03L%z$!l@2-1++CN##q&j#JszkttYWeEiN4GP-c&@#DwTOd(f% z>mDLESDR97EG+p7Cq4U!lk5gYS5Nv5LoC(Ls{n)z+Jhy&e*JpzU4_>Y6h`2BPIH7M zE4tlX*j^lfTNIvO{{7t_&)XKmnd=+Im7&>iu$1ZIHib4`3Az$02>>?1o_(dTv9YQ3 zBjnK3=TqHfo+N*t^csNWpFb}EMXEP-(8L%OOQ|JgxqVN+(nQAJUoMJvjFzK z4OovsTc8L4+nB2Mo6MbZttz*8nEN!?S7~Ksp8``*#3Q?rW$*;o<3(Cl1wC3%Xn_t7=>Q9h)uR(9JF-#RT@1p2@S~ni>*LYdg3^IsZU9tkno#tq#?fZWdLI2}LYP~(8O z5D$RH)^IL%({GUnAOrTW93SIu!DSq^Ff=0d|m?rdc+KG&IGKkhmJ7 zcVRn6B7n-Z{aOg5<>A%wXhT3#>cn+5IS zP$2iE7$7kFZ_H(aL$;3?L#__@>z7HMKYwo8yiJzHjAPL1FE~|c8ZYQLWk5*AYB#Y5 z8{2vTI z2ThH#Nr2_3yo5*t7Qw-B$I{Xg&@sF}{z0FO!0KOpvDu@Naar*G>uoGpXT<&K)2HT3 z$1T#7FA;L_R4jnD-+k=T^NQsO`&Re|v1SJ7gH$~2_o^@R@|X%z$B>1&;kuc+2JZ|b z)5Fc>Yx(H_A5c_Nt2?LIP2)Cxm7VQ$?su|1iK3rzad9x8eUqU~L43$Zw-s4HQR#s= zib+iDUG0sZn*^0r?=Q!K6E?Y5?*QfGv>whBT)4>j(=(j)_^?>&8dv!pS($#-hW^iW z5C1=yA1|D@{K`6X*n$2S!GyFl9(|9;90EXMhh8#tkm0 z1OCS~y^ zYz0$jYI-_8Z_7cLg;19B)LiZJl#egAv<8JBdp9LzW#p*900n_7DJd!I;ql3BZ&q+{ zQ*5@O5WLx&(c3G;!OwJa(LbMHj~;0&s;g7s;p5AdaAHkax8_cH&a5it=jVU37ZCQ^ zrjSphdv@{&l!otiN%1~`P+8vb@$nOHZ|~^-GLSD98c7wIbcuZq!(hjjV+C@1X}TnX zpw(7;?->Oiec_CWjddU3iz!;t=;!L4cf5m)WyZ+Cr*IP<>Q){I z3SMq6rrSS)Td{DP7^0)ToE@&YAFV6VB+4r=_Ho$Q*;y|%dl}T&5(0RB=Mt^>f{2hD zs7XNfRsbzZ5>FThIznoy${g`j@;pLD)@3l?68%MR+1ftPULUP`@gADDwG3nU!}t4E zEMLER61zrFs;#>v?~7VYssPEO8u zuWtGPpt#k{M@y}SC+luR#N-g)U-{AE+j_+?H>4_n2 z^gD(I1N>hU&Mr!jXWJOgiZ0Fzrn`UtKA2;|(Vs)rhM$%(@IVGo>&C}KoEe~BwA@_m zpp?}khR^}>0lK?)?*vST?!^wK^IGrC*7^Q!{X6d+cRgF@=zh9uSZ+{_ju)xY;+Lu@ zuwaH!Uxt$4fq@#DnlA%>w@!0RH|`FAcjCT2K#JHGsLBWg0=ISm;*Z*gR)Z{ilNoW!R7OwhA>nH_ z(-vRS+nc}bsE!)!`2-WUtH=xLZf0iY%48$}T13eCpgGKSQG5g)dSO@huS7c1jMXFIZa9TBTl5}{t~*_6?v zrly7>5*wp=u6{B!qzh0bB&5OIbH!n48`K`7rl#f<9dZEgOBcLsvUx&v6?8(Ur8b7D zwu{n<3K9{HFI_#Uwq*EWp*~AN)V!^K$S(f=<^``-UFj1HE!a@t3u#q3$v6-~mi!s0 zf-W&;@i2NF72EooJx$BnnoY0LBsLPCx@%D7yE%2TcFI*dkzJlri{G(rPk+BB8IvX_ zJ2AOku9AtFSrUL*>;3sA5Uuq;{~Eml%-L$T_Bs6vj!)t1mgvHps;gZ3l@?2FK}d;! zda%N#2s2}hdN!270BCO-uQ`#ctE*DL^E#00e|DycN2P9C{_CCX~^CW}dC0lWh{+}3kyx!$%&u>6Ryvl08TE$OitPAeoCUd%F z!raq*$n?g0E!7GPKy5QRxJ?0_Tg5SS?XBQDlF7qsfp%rB* z1+Mns0#?lo*s(rBWt}IRN3|$ZsRbxB)OuR|Rat#jUs$65xlvsODu!O%mt_3C+D@G{ zS<~HOf|SKvrNDz_84%|9iwNfAV)w7Gt-XvM;4}qd6KN8QJVQcS|M~`bc?kmW;|sp> zjb@4;$PsZy*0!b0*b9jM5ymE>?`U*i+s)ingxCM;2WJ3fh$KdUtHkHN3Xf(CA0QWY zMnaO>C;4ayP;d!VMJ|7|Ch@~7CL&iDWta3=#{q0$tijbhdV=$aU7Z1%1l9%WFyZdw z5th1G@kH%t+~t{!9uW}{lgQ!kKQp|@`mB?6wN3pzSEdwwXvJCKz=20vm7KVd^1I(fWLMq0_nvO^QcC$gW zx3@PYAt53?eK1EfS=+Pew!mLB3nf-&V$m zrw>Lc`Ph<81CNI%z3^$CB++idz@sK7B*ZjNDe!nh4fqS#-thG%6H(P>(G-K-$@ek8 zsMXuCoD_hjmwoaiIVA-nM}7La1n?^XF`o@Gu@f@~rx13Pn`Dk1-Znst4-5{5s~7Lo z<}%~p5E0eyG>-VUfm4+cqtSP+o}uQpWOJTg&tNpdXUgl$JWSAvN3q98N21r|WA^L# zUt{Tees*IvN~8!se%vrV3M~5%I;Cd6Vg`;I<&f;`>~8^+8PDhe4zBXMPwCC9M-N>{ zp@67~=2U`VfZeYKf3{9gCfm8k!_Y#3!|Qzk0Swn9Id?#29;#Yv9)GmR!0IsS3}PDyt=1}mYin!s`pB)*&OcBU%~BeIv3SrofY-H_(F53py_3`K z&!iaf@VQ7Og-5ZMf3~{81tbmlDM^b;^C0cPIYW4(DH2j|KJoMSZv>E6xT35cAuhDg z`bDK(Fw)#C8r#h3;rg!;df>TbJ$6+FD-Q4+grSu$ZDV3#71Y;L0fU^(OK!bwUP!gP zHN_xbqhoqHeGLjHVn!V~fpVFeL}@IbWJ$=#$+_pu0uSX{&Ns4X#C<#?E|thM()ZiaS^jm?%wRpE_bMZWElra6Xf0XdOl?Z%;8O90O!sWd?#)hbaQwN*$1amr^qG6i}0g4$Z0)ykN8#9TD)H*e@+ z$4dsy<`bb94-^Bo49+O69FBQ1UzlWYg5mYy+KP2@+^he+x3|5RR zcZO@~=&*y%{j2G)iZcgydhDZaoq9T*)=|L>8;f&z44K&_D_!C4O*VQ4N(+gAtO z#3-g1a?>)gN78f$L4fM=psnhePszbeH>W@itk1c&%bA%m0=EOmSJWDLMk@fBQ3O8R zrh|2I^w&Q7pio<)1%$l|taG2Q%PtQf5pfB;WRjL*Nr$ zj117Venj*an=a5yid-TGe)_#Yo}HhwNJ=vHBbYXip>T=yG0T6bwZ?-~;h>9JR38*T z5T^hu%Qqmv8Z=Q(ui?7IO9vXx(cKrz=|+!j_3y`Dam*4yviB|e`6lr>#zo<%1t^? zep?$O&^}rXD(kf%G|I}#5J;iVUf8^o%iq5qHMHO}I)s9n`me1coTqoKy9m@Kgk)Am zBjSc$xFV0E;Pl@r-&k9lZM!okAHW`wR20o6U=<<(5(v=ObS*5HOG`^NNEs3jYV0b4 z-?KizgOE^AOzh?6{0syq#q_$PYp$LJ!9;OP@1H8v6kMkp598elMoRuq)$SgBGQ)sU z?QPqv+qfVYIqKNv=H@1d*v|bs_!^>Cn3*lDW9xf~S~*0YsUJ%FsS;dWUBRq+aLNSh z|MTdHKcjqD3Y;iFApd>h^!3es)QOYBv;U>xLnkvUoF+l^f#XBA_ZL1v3m%%P9C=^+ zeaGnN@6%&;&kbhl$p<#&Mv>*|=JI|o1g@^HffwX(Oq;@OT+b{b^5dw687D0yi9sw% z>gEsu2?gxX>g-@;VDxSCD;Wp`BNO$z0HX@Pj|V3IeC5>kWLY)vUVs3$1mqAY=$JOy z3>KlPuHKg^=mV6=xy zm)}(5VRhm&5Sjuf9Ho^PPC>zRO;1x}U=zXB)CfTVkWx`$K>&a-@bnc;?_mX>;Jv9YGr`Rmk~L1+L_qWYw6EkYLx>k6 z1w{m~IK1c5fyJe7=GS}Vsi=rkY2Hq>>}_Cx&kPy(@}(2-H$I>&fHpR)5AgGoDI8nQ z@mZ7s+9O(UaBvVD`OHVwaBR~?DRHho349gO9g{}XMnfy+{Sf%DvrdCwzQ{azf&uJh z=O1anTL%v6#Pl?qgaiZVDAG@Z`xN+J-dSa25R_oS!^0z?ql@e6k_`l`5b|wf zE&~L=5CAZ6R_g;jwav|4±=3e1eekelN?hAGf(nR5L$Xf=be;it>USO)p1v*igw zwtd3d8)tPRBf4Dw?Gm9mfVgu(>V;zl3&EV7eKjmTuC75B-}L@dzcVn#E2_5eS9gE3 za5ZtZm#z46VSKyp;fK((uc2g?sCJ$c8u;lkwnO%#3sxQ8r4rT1>`I|Si-N%Cz+0a1 zFYD%AaT*zw)j7Hvkyp)bKr1U>cBh!67-guip7pV!n~O<7?|EO0m!%M)LV^$K1x zt&1XFdWZ&*g+J1_lETJ8_$O*pOx2u60t6{8N6CW+Ty?}ZRbkega7cHN0#JLMC3hPv goHibv!f%;I2*f(`J1*{k-{FBM$*IFDWi7(~11M0Avj6}9 diff --git a/docs/images/chapters/extremities/fbfe9464c9653f5efcd04411e683faf9.png b/docs/images/chapters/extremities/fbfe9464c9653f5efcd04411e683faf9.png index 5311becd5e31ae4daa91ee85f15374da4a84a438..190e457d5efff0690db1397c1e3777341ba95b57 100644 GIT binary patch literal 23484 zcmZ^L1z1(>*6pT4x*H`0k?xk3E)@|arCUUh?k*8QK#&r!=}t*ON))6^LP_bqbNT(} zoc})eT+TVqcRn_I@3r1{zH^Q-=9nu=S6h`3pAH{|LJ?k9Q_@GFFeFeYG;&;Q_zh)k zml^y6+d@NC33ZPA=Ve1-G75DObzSM2p;zYWq_>II;W^IcNJGKwzccGDRU7KDK3| zVA%NikICP^8*A{{A1JfYfzNsgSJ2hfMIuSoFu(XE0~ZgER1&+m`|R|1>u0Lz+@GI4 z3aWZnvNVL5``IZqZf9%2HzwVba$mp=Ct{2yq!)eZJXu@qwUs-Nw9J^wXWFs)qwH`o zKbW7NKd_*nU_f9veu?Z$=Y#9*`rntxs3er~zmbuVnK?LIKz*#LQZ+HjY>q0r=w8F{ z;a&rG`3v;|1zTHN3AWrPzb5O%jLU5=c@R|0*C#zduVnn^GqnR08fib4lyslAiO9+_iHeE_ z-_P0isUqp?>XLVNcMo#9FG&0L?c2|+RC>Hw8s!-M?1qL%3OE;8xZ{{)9=}wKAW~CT zZ=I}rG+6u4A%}HP%KqEyumX+Lxwa_s<)MNLsOBF(Xk)1P`bI>rT)6=2w%C-1eE|=z zC5~Qf^W^YBx$PjjR=O~$ZW}@3OL(rk~nY-}-Pb2QH-OPL*>K-=h(1=~P?UEDi2wrLw7! zhRXG6_r`g%%>z~wPE$++h=_<#foW+p6>A}p9y-T+YiL%}8rPHz>FQ15B8By>tvU8q ze_$l3g??x}!hihuv4ScK%hk3AcuCF{5R`Ow-{0P@cAZ5>&CGs^NE36(YL1FjqlcKw z&p_fuMbdQ!<$w5-8kX5(WvF04jN{79x0eg8l!Wbu*m77YX#3ff(%nP8ayV~8YDY&$HQ9}1q$w0|wY5>oM0$9aDeyxh zB3LCQ>GX>!;Ji~&QMtOi2gk%*ARr(JjaP#MmR|-dK=~E+nuLrjxV*evx}TGs{Sp@! z9-Q0FKYxPZl*r>nMMg$?(8n<3TIgP;5cmGWHuUwYUY-htv~+x~^%pf%^Vh<@YR`94 zXNR45oZ6Z4(~W)-xkxr(L-As(8xmBxOrKRXM=4b~Uoo?xM)UK-+S#T2SQZmlwU0ks zL>jfT<1x|bcW1d8-l8OlT)RM9TN?_ME+U05)6nbtA*TKL!NGi<_ZS(HP7YyL4GG|$ zP-gcjPp3ph?5&!frKcP7K8Nd!yv*nKTob#-MwM3zN8CiM*|Wk{Z-CmK1`cZ}P&4>>t|KM$!I8SP6u zO;-e*`yU(~J-gdQyt%)xs-_ls@t6ZrBR3yxc61J`A-{mYHH%#G;1JIFElX-EE49R= zOD*%AytqVJv29A6kQkc3eG{496A%!Q#vg7V7E=FU!> zj*bpgV1}3rI;t;IBEs&wH{pG-`~A+YW_jnZZ2O&Aq+8sxPA@F3|!mE-pL2M?M_jXd^ez4o4s*FenIZ zVq#)uaS>ZiPL772{!@Sdhk>=ZxjFdKr%Mtk>S44()@E=)8>GOBD89=N*7o0ypB5hX)>s9o90V#w9k=T0GsiP_8BSvTx~_mOjI zw*{hM@LP2y?mOF9TZfN)FlCjIk?F_nuah8Rmd(85*w7FsX>*emf@R2qq5Bed!P~by z%PT9C#qVCcV6?EX7^rr$L^Xdj|1<#qo6zW;Dzy9h3YUl|T#ZfP>nklZ6yLpi`3NFL z_l@6Zs2Y>3JD3d(T774zi;xcX_V(H(Rp6spoS@CO~dzwks7uUzf z$8oG25B0~f%xS!Gf}k^=QL#|_g}QG&a;($i;aGEVrF>@L3#TO^LFwq|#L|lqad2=P z@|yM}J$dqEiL8w3sxe%jq)tXpDxc-(M{`u*=H{laUp-v8jKs>y%4mfHH3XeO{@;%r zi`<8u(|1#GXJBC94>&v7?@PSN@L;J=d2OsBrb0_z9tD{nLrSVneuvh_fmIwT z`eN~a-?!e476?#c>7L5XDN)DKi(UT6iAJGlY2~wJn-8~tB*>jo9ISA@jAW*vrc`y0 zROb>BDl__vc#`^zOEJYCaQINDa%1_QQX177uGN`C=&Iv~%{9aXAc<=5j{h00*Y2+` z^S3?(Dn<-9(gVUMc>4T#H^3pdoXhy>Z;K;;6H?(oYSo`pLy(jn>Fzoq4Q+$BqhqFD zLVS8j7i@iOECHNNT*eS2ptiN12N%A7uWR_8AT0yqg%n0IN3ow6Tmb;I3f8D6At<81+`T2e$AF@uHe|~E8y;Xr5GQ?DkXSlY%KG}2M6uHmY zpFOE?B;ak_Z4fA=+sM0^)?J-Aq+nzbLv3ac{8!s31Lq0Bstl9 zXYX|b^4in>t@+MDz&!5bf8VBd{W|O(ys(4zXno>Z9G!@6fqIJjI4AO%Gk>>cFoIFm zm!{$IF87FEDQrvF!oq_4bL=E!Q|KG~PwB**C*)(Ou6};lgj{9NKfAH<=+`ip$NB`V z#TL$IqUQt$>vbpn%u?8IO^NgKDNYVBknE!XxeMY8eo$uJPkeP4Wgb&ZO-*e~bI39+ z{~$N26W!RBJH^DrL@o89Vc;wVnBbnyS##z(suz%TcXuO2#M=7RfE)4&p(0Mkd)Wqs z=qM&>FVuw#7lw=U=0A;pdy6L>khNcfffI;zA@E*9sp&25ZP{n)Xb-mcJs}{ALiP3a z%`7a?Aq~Ou!w)JkGxK;P@r9ItHb;+Kt3eqDzm?kFw3AbifMuXw-k-|rYN*v!REc=q zX@x^`ri2TVocEs=mIx?_DS+Smb>A6_s?|y2Yp@T&P60BRLpK_{wLZ*47)-XDbFX!(7cB z9LN5QDfkCV%*~%zH!K}~6L-K4>EW?+aO6JT{iV=(5kg~#pu)jMhbV9r4@dIRYFYrr z(qvL*p-!32&7MOcSy>;lVE@y=m9If(r$Mi@f=F5a$euM9*rSk3Sv}ldf*ph(cauf} zNOi5BKAAf?#l4QB$B3g3@XC10)7*u7@axU$9Osh%3t2x@6sFe>?m^Qz1`p34Szb(t z8M4QJ7Vy$Vo$zZ+cs(eP5TE^4gci#fgHdLar;{zKJl^qLbYC{h#InQc{6~eu)Y7!y z2^!$pqELi^-0j+ZiR?iw8#_A|F|#b-xQLe$JNE@|C&@)UQ!lWelmNmuHem{2R8v#a zJ-}c1s;T#0A@^b|C%ny200~tYpS@5uHZtLeMkSUYB^Ivo97FCf?@M9_6gYq1oDD}3 z%Jt0ZYV78wYu8WrxX-tyPVxL$kg7Awm7*RR8frFP>7<}K`_o>DeG8C30HjaN&GO5a zk&C0dBrHs^zP`>cDoP~@tS|RTWK2xRyLayf1dzoO?q~_Y*=naY#bv}mAxjb!7pMHr zxFR(E8oW85)WXsdE@*aPfz5*gg`AiLIFcXB%EgtKv@c@s;U~$`vZ*CP*W#MIw&j*D>gPZG#uYQB4+g@uI!dQ&mTW$>qc7Rg{bq(qWz|aZaA^->dj1<*k$&6l&+d zGXLG;&p1dmVgV>r_h=*6{8PAs>fa>%2%rIwvaic{IPS|mR)SqW|FDSsCrF2pAY~)N z+u7OqDZvy(0(lgiIRReZCit?%sI%>Rpom6zk{`=re@3&^oH!p-F*Z? zB2d^fJUl$H=V$N|dFj7lxSG(~k1`w~n?4Fjm84B~kzAJA2TMe3MCMiE)iteHG;pszl(u2L}-gG&^A?K zpwYUJA3_om5(WgK<6u!QJymB1$t4VOL{45}GxBc})j@z8=jYmE2EKk(1ZX;#f1TVH z@&fDyRomR7_vr6$lha7kbltvHF#HtF|&KEB3rT6xt z?T=D+NUNU*nPBb25awfH!ICcK!t3$W!V=DNBn~lC^Ux64jW;(bCGqjt0PTVr(E-I} z7Si0_+PLMw%WKNVa)S+0{B~p?EwEndqfJdg^}-Sr5fTcCh`^DSmUhq9P*ZCLl(99R zAZG#Eg6pG4ssLpNO7Cj`5Gz{2g;oADUiE~72@6jzC9oPUR6rd=+)V=Kb^D||ptRJ05 zDzEbI@9$>U)(8R7%-KMI90iV!8cBQq!Mti*P9muu^%0IBusfQ^JIkOxadY0jeOuiB zMBH<2EDZL>{N!Nk*B#8tG_1ZlAo&BYwCKz2zG8!d1x&I1l~x84NV0xMFX5Z&BLIaC zD`cp;p&%zPKTSyq16F4N6h}cdNAQ(?@n#QSlNnr&?L@Tz0;K8b>2oX_Qgd=Hoo#2I z&&Zu05(7VST^_gu>4`bpZ;$q9eT|A%JLFzUQa=w&RsoIdrmOA&IdH2r;G64Wd z1&=XrSp13N4%7AP*8yzL^xmM1)#UgD9DWP(FEC`4-@IB^h)RrNVIe+!{Yu`_(z3a| z9a31x>Fet&ehrP3^e3_SeIJ&|(uYMCPd`C=}@9F~&iv?FuKPpV`>p1Jx30-wxAKf3|9d zs1p=qd-tOqJHVG_pPzEkNP1juE6D7-O@Ez@4AcvY?Zq$EfO^A;nd2ex<1@yRu% zF}Jb;I!>LFSA@lPiv7Cu<*q#)0U;|8EsZADf0d$Fs4qiB!r}*oQKb z_oegB*qQKdX(337<{h!L$PU#${Oz?eD+L)M`fi=4`NmWOz_bti{wH4L4kOyjMgWca z{EyeffaEqK75%b(-hUHW3%gIl!oqSaJL8qd3L)MX+-q(!yVxC*M=9C=zH=AID!<#D zlE>N@<>}Ebxnz9D(8x&VOiRe@%1RUx32bbbKs^>CsY#RKyFPzz{r>JwiTh7=SgDlt zyHB+ZaSQaLdMK$kPS82oFw+FFsy(lsJAAl@w!L`&{In%x z)iXm3L;Zf^D^Pols+<{(D;=Xi&77~A_P;Ve13`yJ#3&i6BYTJ`Vn0kG8Ep)u`f=yC zH@5(|5+VmTPZv)~S=qcb6z^a$!=?OsXaeS3XLCz`8zJX9*!E?y*dQj)AcS0=zy01)RuI*n?SE z02)E7$N{}!;QE77+S#&k5ZD(OL)2$L@=g=Hj}L)c=ojNNRfof4VpOlN)P`raX?BsH z1oY&C`2?{Kw6tht;S%)O%{mCo&U`P5AB0SPha%@N@;)phgC0%>2^H1NU<}BO zT2&4sygR?XT>upU8Ua#$?^Sl5_4aCJ%J@(L5TusGr!>&lCxa_3vFauhmx604w1T2- zKjOdbH7z@{`r~8`n*cRdg>!a@Q*`6XaZS2NP-v)_)9ag9M>}ZoOjxUHXcuBuJ9VB2-`w48pYKf2@Pvbtuh)3GcPExk zTIcfS`be40j}IfS zv6PhWpUX%}9&a711iUsx!zTR9)P8s{Z=w7ZVtRQn4@tnW%yLYHI@!EuTfogasvm7I zWy$!wy1W0L3dHB!fP%t8q`of?7binBW5&kCg~jVi?*7JTI!oWr%4}6}f4qlD#L#sn zBuaI#ki3dVaCSx?8Pk(>du>h5&o}VKo4_Pi4T_T6W!9sm_beR@_M z6_l%T#BWlgZHnwq8Km{<)p35A%i4U2;-97ZglP$SEks~iK3 zpF4ABpv?FJsY6thSQ@CT0?_J#<8&hu)*iia+8ZE zHw$+nlKP#-LIEymrtt^GM39l%6&)OSnN+)?{mo43HIV2f`B+&o5M3THYFNBNehSFB zp~VJ;xsc!B&Xo=OVqGAN1fE^J6JBE3`KuSFmd9Of>GbS)qsX9;XmxE3dDpkvM;r05 zL*`H-!~@Ri5w&62wCVgz3b+$;B;f8q;L6|L+n7#3I$Mk(MGAngyBlLLhjW^){epxs;u!Dkv0OBKUoT1xyDjgB+ z$Nc_%c8{m-5Z@kSkWB`wTudX|PxLP8b7Ej%U}9nA0PI9gqrU!=0YD!f+E!L{RJ`f>^47Ie>u(=u?^v zSsiTU*>3llOz2 zcfOxSDX~=SYEj;T(1F*md(#9k@bK_xv;>?Vo6^wI;xQ`opR6fJ zs)9_8WN=~IL?E6X@z7zo`PL{1(y8VPNlL-zn#*d?-2EIy&S~c87z=n4-l&Vj2?IzC zaA(R8a7dga13G%8`5awLOw9LmcSKEFn@W-ldcd1=?H7;;Pv+1flZ>7*T1eE^yG(%y z&8lv~Mt~JTLryz3;z-`I-o>|{X%3C0-43jjQ*?+ z5&!}M;mDxP&dw@y=HM?vPl86~u`ueQxHt`@nuwg7oaW-l&mBOEFD0pM{;7T5({FfIX+hd3p zVG>aJ=!9+IW$re}Fs^H3B0KAo^w+Lk1MQDgk}O34I`?l)o0#AlI)E{S#w${W0&qc~ zDZqVsHnJROw8DYXmje0@$O|%hmcw~Nm@?5yC^UF204c$3prn))(1~ zDyfEVZW4mn(gq3R>FNM8?PdUOsWE*7#5$ahwQ>= zr4zlR$1(~o3!n$7l8VY45crORt$6{bF=6-+2|-cO4%jDB;HvF0)QZK1_>58InV?hw zL4ej45QKcsq@*N-+;qeUtKNEAmvE_JuNaQ@6hB4ZfcT40yXRcHH)-# z=Dyu>ntBSic}p-B!da0>L?k%i=y9y6Jpwj;4855W$>jDFfyzWTysq4O-v&XiS7Cnx zxlF{80DN2p0>s?bHuCD7${Dy~h0a`e4zehI=LvCyiOhGzDL#H&JCLLnBjPfZxzDW| zw%v4oN__9$z50;@nVm2qdBbOrdV;&U)Od``)s`cysfeJ9Cxe4>I^z3Vu?& zZs?@}K{jkDT#@JkVO$1m#I0AA^s$f9mESzFi9C131N{_ z)UiMl7Kf%;dwZD6+=Y)m`_AH$lDW{bfdrSU59$>o&(*5}=B=2$8Da?+j>wX<^)FMT z$@)p6qobSI*>QO=#C0SwTt&_^gR~b3AU|aQSW00^WaEcP}obcQ(#e@aT7;sb50BZ$h3#75lofQKDT0y1d%f>;<@!@rT$D&AI zn^ll(^5)yXyGocll=#b6C_ctS2JS~w7Ln5ar%z2cyE*$wO+(`oH1Qx%3n=L5xIHk3 zr_oxw0qM75V8bG*qIXK76Ru!nSTGjB!RT`R=onB+(z9pb_Zl8k+D-rZ zoC<7hZMtb_r%>N-IpDJ14``XT4Jm7syc|Qnxb|h6Qzbj_0jG@`4|{7ERHp(^Uh=Ld zb8PPHP)J699|Z4Aq8k1=`^A)h*Z42Sdt_#^mXV#wt@BKB-g8G z+F9D@9B=_OAvnY=;DIoBrAaCoeREFh0^Kj*y9kT}LW6_C{XK|r`{?ikZ|~5Es?q*B z3e_19^WqWQB6DM7<9+JNw3E|QQZlmK$}=lLT2NgOSk*c;b=g>-TZfnDg($>b!lBP|#Ym#^=>7pV_izPKRCZv5O&VRWFOauPo9(a%) z6CbYvu;5R*&(5-5))KF2BY~=_Dx?4w9-g<&Z#VW>$)enwK%$u%?=QXIgJeBv|C8#4 z>7ptg(?$kDI^jT&&y9~*XCWiKgsOrp3kW~`J+sRpufM;;mr-nx?r(U~5PV&RJcumm z)#ddXHX$MMTklO;Ak|GD*CiCaoootVhJ4rx1X8cY{eI58?Nx2*mzT~CmzXof-OwTb zBC6*M&}lq!j^^QE3g~iy`|Db62!?h%V}{Ee0|NsuZ|@xa0`?MFboG*9O=+yU?9{SjknCooRYeV&8#cI3X$u51W`N7J7gd&@T@2 zwUJ$gWPbp91mwhZ{awZM0`Ns4$_IcQEWnD()cn!cpDN~awg%>5d~90s@5&c6d~JeP z!TG&oG_b^|!k!XL3Sg{3=UN9Zm?<+z5Wr)uB4yKYs-6}ErXjyn_n@P_^e8-GaLW6k$-2P}HenW$3q!jdP)8R1K1LZfwmzV!aY4;qpq&LzI*|N0FD1`g{XV!KegNFU*FI$_`$S^ z*!fNx+^PAu*Eax?lS3*{|3+=c#LCM07uG&fU1iM*~d#NPpZtx2}CqOJ3P)`t= znKN&Q%j5nz3DPHI0XNV)^IyH1+wPZB1tx{apS>@nsx)OUaj!HG08c=~b2!+4VmB!& z`5^c}!pkvj@&`RrDe8zz6WnSDC;|V2cCnfgGjb(J2!huFQAnnxb)$sg$W@L0fGq)8 z55cw|lPNyWX*={@auYk?ef{lSc;6MtPrXx`%U$`(f`Wql-hZr+7V`_qhh&o4VJ$3$ z4AAPI?Uy@m1HE(GWP={SFJNc0k@t7+drlVo+d(pK;WPqd3=;=O5fCC6NKk>$OF|5S z)v#DTRiJkP4+Dtl?II_?mygB%0?(=Y2|N4(VGq+}PC81p?cLpY4&UFlKs82iq3_-g z8^D=Yzkxpm(f^<+-V7NKh5EuI_UY?%Hp~1CY%&*imUVZAGx{x4LqYBXLHFD$Pzw=~ z0&S-y-s2uzGRpZ|V=E-jZa5k3~r&Iqbw?@1gJTX=;iIBr|f@m`^N)1odDPKu9x) z9O#^!oCpVj_6=07POvikOY8=Y#Jw%{3j}PafoZ7Uu)%R%e2CZPZMUAP%U{{~eoQ{urZqfU_DgMoXHLX&1*2}8-y5qwn$KmjBN z&HY#q5Ruz42^k~YtUKILOKrFvDatB?F=SL6Sgb}5jSPrv>Km^O5mh5J-r&+!o~y_c z^#U-Hp+Im&tXYT-2DC~Ua2A?X!YXn*@87@2#l^jxnuH7Bzc*X%yg?|RvJ-&EH$g?c z|9Em(Tpj?6d23Qb23T#;mR3(B6(xrr42@88w)f*=mJi*aNs*| z92^{C86=)KWG{<{DFnf}{vUOO8iKHsr4V!K-ux^0T%$5;LTHYhOzi>D8f^-Y^W2!a z`zo2hP#6LV5eq;$y*)#s_Z-p3z^yy8v=jkwGza(T9HoCzHs{g&wvy7X?{$w{4Br`3o}HZ;HTrr-6;^R_N2;;4 z7H0cLLxG-!zfTjgAp+_+^ZR#?g{mr`C;~W03P8!Bp$4qr1tdli=$CsR98z*~-$+og#(P{BV~fuandfl|)}mliqi0#TRO{LhAL31`5*o=jAfLff`y@k`!-M7X2K6c2PF>V z%1&JXVG|lGM?AQ4K@_k&J2^C}cD*Efc3_z^PojtQO8>R1WhY^mzSSnp zN)Rli+2Nm`X<{BsV?Qc-UZ4#X=aKIa?2??`5&q>f5>bJp`m=apir4e(V4fHbA2Y%* zfvg}Mc09&}VX${=!17512L~UxT!F^l1`?M*%~hayz06m zXlOXIeC`kseTm&yj0+)k))+=pWohM z9(FV)?{VOb03|_lsZVIZe=xMH&IszK4P^5|AYIs7CV7B%n1WCZ?KG+tw|6;{ZTt9Z%RbkVFlKu4Q84 zq+#6d*Vf|6iLHI(pXGlXdAU#c`2_;+6^Jo&hw#|BykFa?~PE%P;UH)i^~Cze76gQ}q%^ z(;kd^W{}m9ZXm3GXU|lHyn+HcLS~RIIye?g1eMH=$f7XGqVfm8@=F1RY5Vb#fzyV% zjxQkWY`=BOi}wX-s|XZ=b^872ol^tV0LXTTf`M2wfRitTe06a6vqeWor@)));0Sk> z!0A4ON$4Y{lv3;RDo|8dEL1J?{M@?lf%iRw2JhA3LA;~zqb{!$v}98)vw?mTak_M% zsvq%a;SWU%y@L(S+#$e^NQ>3&{LBxL8=+7)gYRbc=g%9vw#2bu78Lf?FIb5;?t&;r3bZItJ}BdNFW)FUoqNDv&x zn$OOu-}3f=QwId1Eu=FK+M?^jhm;_$-ay^>eY3sbMyhVby>Amf?)dcGZj8&(mS>%2 z`d=}5Pfae8MUhm>61)<9sD6Sj}*RYq<( zPft%o?YUR)6~yd+fOftga1K6V$(P`^t}nhT|7c^X9Z?ZVEL9yxKL({z*r(y&U3lsV z*I8Op0^aq6LzOdAaAveZeCvS+5N2*v-=*r1COL1L{ECv2n+j?>`BGp7tG3tT1P2P= zz`>wi+dnR$u`6q^}f_P$_x5sm3%NFfOQGu*$cYndg&LIljLpz-o)F5=#>V zPUVyp0n&yyP>VqJ?*yn$%D@ndKtp&5u_c2YlgC5)1yr^m&?o2jrvo~n zay%^fb0Hh`h7K*en%dgA4^3x7hhg$5>g@g}TdIH!2%*d@Pu5jIFQF(H`hKp5r3V+h zzHfc5oAcE+-&rSc`p)#d=J!0ZT}a6WTEvfJtz?y-D!eI*O{KEXc{|!$ivjiZBD5Xo zM_k(8EZka_s^&AUtHc(ZcC@s2!aw~?`DZQbBI1|<4Kxox526E`+1g(AV0)Z^zuP2h z6PE$ABhB62#fA)EPbE&wTEWJ*RF#aQ2YjzV=(F5swOa1V^>! z9NoRb`VQ0clQ4s%LNoR6AVT;{EEqVrb0-hjmALQ0$CnbJV0ZcGy=_gvEXx2XX5mOo z$uI{@H6V`RLM9E%&SuV%3jo1(h$Uk}Y&JXNd_g-{j|SpFj_L&MeEe*=JfEERUG3B3 zvE6qgvy1cLfd?ox7Phnp_lYt`#OF&p?|1FokJL_pl*IE!v3L`N$t@rN$fyAf8}WiR z*Nv{KVY{o=CwG1l0yGcV9#9@TA=g4pzL8`3a~68>&^XA0Y$*AF_%rB8@zBOa?)3nA zVt+ePY%IBgv-9&I;Ad!=o@Rcq*c%k1vBmeCb#g~>gL&rBC*ttMa9!)mm;5pVt_n4% zKjtqJ3z1ePO3}yKHeUSbmtR`MQKA-yf-b)i3I20RlBmHfVpbmbh@Br2MReiQ^>$~KnNeA2@wUwAG#SX#SaXP3$9=5xK|>-O(?#Wxti zqLnnRlJqJtE9c(2!=F_5E`y)dl`C{efsZP{1PdE7FoZB(a&mHy-`@$rrDn1?C&Oz} z9SB1BOPEbDDz(A`2Y+b168p;=%6$>vtM&Hn*MEW5ei`W?8P$6UfB<=K)>%I;I=U6K zP*Nr)B0yS*sgcjLkwTJ+RIkR(-5s%6AblP1Vw!=^0qkbvl2mqh$>|(4J3TLb`+wi~ zd?v?M@>D*H8wZ7GH}CanKNGMIWxdKNy?_7mJ!@(i_hN>WH-;SrDXBAy^Vm`M_Z&kI z?g{Axn}8+|#FAN{iil!ha`)~QV4^&z%#uxAMc|}`qgL`@NgHXG0JQ#j-_BTjK87I& zZiW*q)d-OivhJ1F)g^&e5sW~^oqwnP1$+`|lz>4m3K?Ak^<qM%gRQf<4Q0J&U444$)OI&I7)2ndlSJq&oAAR^~TG4tx{dFM5S^n^19g%#!Z-^+I5275g$X(c^ZBY{Eu9#*1*c?+cZf+WOY_LEsf9)5UzI9qP9y2#u{wUpln7Ip+exrkVhTe!<4~C+UHV7>(EuuR1Wy>YN zCLz;Xw%=Y;B90#@{SNIu&-T5+Jb+9rBA!72_z_iAVnFFV>=)EipLRIYtMLY2LpM*^Y{0Me|^_6`DXjqw=0Ny z5!50jFE4Q<96+7Qy%vJ4pvuZJ5(>4X-nZ=hol)CPTik696O=DRXbcSx&jsc|SJR>8Z3~S$0Ok?ki<$)v z+zPP)HpVAag&qqkwh1dWhcQzI|scFn5AYxdJk=FN?_9jQ#>iC zXyFYFG9a|*fsJLqsjxbLIZcYX>p7Pg7vAzXH7+sVM{gAsY>=WzsHr1BGJ?3Ahp{W@ z52}D=0C59D7Y-99ZsvF*BsJeD_W@i>0*AkvGTT||j+_;X(0&E3GA0w7OmiDI`7D7A zLJQ%~Tfs2Gy{hHZ4n2G{Vq#*XBMqLK0ZAY2ZF3d9HuJhnU? z&=P3Zx*zVzAQed7bC#T$IbnI(RvHw1AXf5VcSjl-AQHDi=K|!jAQ%q>T5Im;NRaU6 z)|Q>S;FSOVmZT_E-7`MbCkjE})rLPtI!;h7Nurd%2n%BrSD_}(1d@H@N1^C%BhLj` zS-}G_IDS}{$^NQ4OaY)!s$b(5xm?|?UJJVI{M6Gdl0-uS5@6?>zR6p-a59k*UYOrB z(aU2Yn zL1chbf5G<@vXs}^i3EHnm)*e!qHzIod43x1)9p>;2^MT@D{0Q zX?>}B_xw36Gz`q`?4rQx$PxBSPr(+-#GoyGCO%*rYhq~xD|L*00z zbz;YY6#=x85-nvu`|jGRlu?bE93dvWeERd5EU$!>ybOxpxY`MBqB%Kkr;oaL^WrDX`>9jM!A`~;nF1c<{CyTBMzbRr>vcI75&zSAaQ+2}SJ z4l(|V=<-Y$dEgoB3!K>tMwr0NMGR2u zVF$NhKtKQh1>91Y0vx&vP`w@GRx_Agg(;tT$bp~;atBxlTwu_rub`5E^NF-wG6^#I zYsRdqX=>$Jp1^;b+Xt|c-)My=2a&_r(!l$>QlL|kfd@G_A%PgQCPcnHI6lsULE*@R z1XYmlV23cJq@*Be63M2kfC$47i!KOZC7?1Q4BGg_+GE zG|M-!Uq(9M^n$6Q1}pmDe7x?wsHjgWVSZya6*nYMKcJd$*ARhhJ8f{sW#KTy( zyq0gnB+B!m64qJ-ct7F>X|!H%jeh0ex;aVWP|;EQ{$}i09*4EqokZNYnW({c*V^N_ zaPo@Y5=E8;XwoXnpP&)UEIsUFSbFZg_!I9L)8-Q)*MH2_e(c=c)=v|g8E^;uXR zz5KDC21!;kT{sH-9bmjgz+QiULS!T^SfRbizinu!9&1QfLf&51V2BKt- zSA|}`d2}MiU@)HtEa0wvmyohDx}#5e9S?S zkzF!dTU+MfjPu=W!C{c}h=e>D9TU@V_gSP`-QZgxi3-RzAC2D`z56cpcoI5YGG+W6 z$x&cj%FW9|?gP~PKPFrwot+Udulfg!;7^j01{bdN|M>9(k;h>szzqg3n*C-xTwVEj zQ{VKt2O_^a+F6l$eFq$o?BMWD5u5S~nVYjfEJp1Xvo%Xud7x)a@oFjMYdIt(KB#ie z3UaG4LA)~HkP?LJ@3#c-v{2P_dexY58|horP4xI6+B8Sy@}7uiwrL!oY!gfhg^_8y<@Q;^Aat3$@AidnWNFkwOSSAccGx;Bw!x@PA;pn`Hi2;Mc#rvahIZy}K zq0~iu3hMbe2Q~@{Dyk=`sj8lyqArbx9PstXQ#k?`7ObFC{wvnt_U&%KsxWbA+T-{V z=JJH_h={Bq5Kk63<;;NM>eyrk2 zr=+A%P*TFDHv^6KZ-Fo~0!u|%;|I782RvL{GkC0vG_(U2mX^u@7=z2V13HSX z;tZ0k8IcpW#!R~G?BKg;!h_6d3~$`fGcnP0cIE})oP(Pi{}Vqk$s8>KXN7f6Rdr3eN0TF_ z?D_1z`;5Ljr1hJvYfZ0e>6m;;&Ssfsn9oVi4peD`i`KVwWbu6VTQgi*D;yq<<4Z8c z?`a0iZ)|MfNdu0Z>kl{+g0WA;qX7$jdx(mbc4bXpSxIT#2t2wgV6~7tUC}w*8+YmM z>6zbeI+wXX#q$ylI}DqW)6;7pe*}hz)0r|y+SPF!5K-6nrW!W;q&7dCj~&soGTymO zTYp-4e71)DV#4HLmUwpZ(IshyN|z7x?u%a@;JiFNrgzqe$V>c+VrFJW=BmC}TS8)D zka_%cV!{L7G2PCL__)`rhdDWFlHkHQ-}+2c98TB z5C0he^Ehy==QWQ-`}3*%W6Ap6@4G+kTUypF zJ(P@Fzl^ARNJ9JY=+r&{%rCRUT>Ga(`j3uhu8F&~%8Yxhr5{TqcTc`26o#-4`$nid zbP2u{Bc?sL-C_$ftF=`DNlze#LMehn^v(X;^D!tMr#?R^yZW2_2|i7q%hc~Yxux-h z=(Q^G{-3h!^9jFI0%4it-I2I6va+>T8@j-yNC#nc6bl> z31j>;XpgRvFRH%P)bJ(#Nb2|vPiO+U*%}7Do~EWg09pw4mDUgbyH~)@)C;K=nHl{+ z+pq97Nw#TE`rg{PoKr>Pcbfi(+olsr`SV-bAx`5ntyqtaJLciRMg|E?o73lnVPPZ8 z^wOh#4zbzUQh^(zXb%?U9p4^TXE4e9+VlDSuC`~$83J;anubQr%*-sA@d`ILu2aW@ zxkn3I5K!JTK{#pRZY)24{zUQ&=ssWx(f}p|0tvf>#9^iE{<9*Td65k-=rh9BRQ@ z3or<_7YcY=dpkTdL;;+qFQgvrc@{%{VqswnR?xnJ4-cwxuClBeuK{5P3k&NZI-DI= zF0Lrvqbq-(5Y%u>?FX`=|>34nka(}ip47fQWi9w6wnw3?4l?FgFI6y)a?qhh62jHy15dD2PPEgPE z4Gr6%vf<(5{|U{3gXogVjjX}k4Sb0;U_$V;qciwgP#dp-Y1iA^`>jdMlk{{tM^XB4 zbAGdC)ZHe4#gNdXAo)S5X|X6tOi%xXC;fk)9Vo;fE-!e%7}$S_^mfR@;|0;kn0h&G zSSHb1(M@oG1i|ShVv_EvbQ*`@Hx@9cLluF6+I}55Y!()G>h~r=S{XAdg}-GREpaj> zu^Uo@`6Bg4uaBpt2qk5FY8Ct9L|wwGrh+?}AISx* z8BcmDV?k-tnFKqzZdXlz7$~Pz<1(EqkoywJYyBEesd97Sxk{5wFHuuo^N){vX~P&u?cDGs=FCMiO2*el z*F3qeT;by8&VBhZ5M(RM5+iik*z@QzYG z<>dvn5TqYNfbogH==JN|>gqHo9N;L^1nDE(9PCUA05BWwzJ_T|oYV{Ud?GWyFU-#F z9#7ccJg?iBNeBp?YMAW&U2U>C(iHsVX4T}ITRp>kEbbU#40ih9;}Oju%=n`6@L`2b zIH!XfQJYdJ`~NOsJ|GYk!OlwY7NGk&&M%p z+kb~cj`uhAKOZ}ovJAN$U}0lVeXr2t{+k84tm8(uI^MjPMH7eV6VS@6K`I#@9yTlc z0b7NJfo=Bs?)K#8-HPG=o&M_U4qcEVJEUK3qGhZE-C$ZUx&5Cyu09&de2Lwy%*V@QTW4tF7Hze6glDw35-6a`@GQwonZr<`TB|~1zY@L*mysgzl*O9?xG$m(f zCy6L;p<0T%pP$;(***9A%WZHkJMk`%=p-1@!!ZMM z4fS-D;^JZhV`F`Db3S-Hbtu0Ntr7?XgvI(;6MY2fdZSpgv4ed@7@_ku@?mR2!+#1* z6#^7sA#MFo#vBdzhyQw(!^vq8?AuBQf$1VEkN#c9TJ$)cC&EZc7pN&)vVBQL zB`W6Url#kl(%s677t5|6QUvyQJFwDk)k62HD|~$)ohT(PKba`XoNw%1JT(mx%QEV< z5ymKD!3RN7{opCNDDe`!p||!~H<&AAt zoSB({XyYuh&HxZ#qF;(rY1zu-3f|A|68eZX-3#k?T>gH_R+Mc{i&xT~Hf(Hc#3n}z zhP^#OJlEXRvml{jFIK9Bma5OmxntakWM6vt(AT(R3Uun6yx2X%&-dz7v2+&J1Jd;@ z_ae)BM)M~vUJpq?{4Mx0$U90#MmVd2<>a)K0-Ca8T_xtT1V?W!k+44tm%9%pB_;XC zgr!`b=pN??w-s#2ekg(SF6u;DRi&s}TQU$hJ2jtsbd(F$4BM5NtnA{&A38cZ{G$$& zPd(v4>;x7!b7+Wh>z&n&+nes*{Wkwr#DTB_GI`G$*Deio3A`%%)=v8|NIjVpb^Yk? z3w&KfgLAWioRKox^Oqu-gH8;I+4B^NtzV9mOKw0|!s3%a7HQCOHa>?9)4x1BD~sX7 zwT#~qw|$YyHGTT^^Y|@tWaifYY-_&2V;wUN`ZLFDbP0@#>l%P z$81SRe5Phwfo%ckCpLtlP-4~#y^lSrE(;AzJ9ou~d+9>xCmieM@A9aKIT=~16)Ui+ zA`D;!gN0MfI5+(PO|Ck zX>c0z=dw&C3)P`8T~C?>{UKqFNSNqa8=E{}B63DA<#T;~#Gu7tkVjh<3uQyTl1_9w z%Dkx+xvpp7er7qEYm^EFg1;plg_*geV62z@%_lhq>y+1aw~=Mo*GF2*D# zFzt061S?WBOitzIYPpK{LmUz^#}6Le;W}!Rj8hJYV_o^) zB^N@`QSC>J0@hn?ZOX6CfM^yx>{+d$(F`h0c{5-laK+5JcdtrqV(_nu=D%lGvswUg zxgG{+RlU)!dlUjPH6YcZ!BBD;aQ!DHCH*>BHg*TReq=w-fr9OuXm%lrRlFV~V-WBS zz@KP$PC|uFH$Op~!aS=dZNa_i+s}Bg5)Hca$aoBFIhhc4LuIEx)X=aS>My=KJ`eETcz`YiZdoKlwi9-OJ0i3SIm{ zH#fJ6H%@98OFKT^JF3q_huu=*qJV5dX|sz9ku4C+EQCh8A{0Ko=W2U&awLq(e*eea zruq!?ThufL#5m!XxNNV?R1 z{QiADPxD!FqIk~vEllbO?LvK2Ds-<&TgjS*VcP)fp8QBJWWkd(!nVqj3+?Q?%XIBd zQP9~r(e_cPqcZUI3Q#K7?HgXkdo7{S>FM?L_4?NCZf?s~ukH?FU7KL$-x;=elD*@S uzb1K6`+Wjx&j0m481&Eo#W8;Jy(dfW@5;+(I}yiCar);@d*RlDKl~T9?GCH} literal 23872 zcmZ_$bzD|kyFCuyh%|yUlF|x@beDu6rIa8oC?y@z-5?+$g3={jf&xk-EfP}Fok}Aq zyw`G{bN+e1Kc8oR+*@JYbIp0hxW*WhhzDv4S8%CtQ7F`vdx~-zC=|K~3WY|3jR}83 z_PNspe#10XR**woApgs#&x=Q)uA}bB-O+MS`8DOHt7&n8xh?js{K*Yj(#us3w63ZT z%(s`6RmOC~D zINUVMr40(_zaK_Gi$dk8r4sXF5fwSFOr`;`;IJ?zVPVR>t;-k~b91d>mvWfo1)h!I*Lxk3i{8zht(BA5HqemJYO4)aB_JyJczU?iIxs+l znq6EB_B!0eNeQ2B>+ipMa&jW?*HG}VL@!^vh}`qfFPsQDMroNr7xBb=tTbgsb!C^ctIqw>ty8w$ zi^?-#$%0n?KXmiGiGoP0bk;{4Yd}})TcyGmQd$zg!!-tl=mA=QOrW*5q zdfwbBYH^W!_xm8ZPMfxmMXs=jw~qxvS&@Z|>swxSHoCsPJ~InTb6eX|>V(~R6^@Yo zwA{oe+ah5Wq5{!0<#2uyh;og4WqlEhi9O1O4pp`jpSJh*ihk+J<(Rv3F$}8IvBCOY z)z?c9laRzECWa}*(q*cpipnT3G75bCx5NbG3Ghy>@TNp`bTNN^Pc*vWOAx8w`}cr< z%Q;vd#zQ%5O-ta@OB&4b%jJ~*-`_nYYY<_8BSuz3m`s!9)Y;W_M@0pHbEYvC_P=w< zDe2w2mg#A_&CSjJA3sP$86kJ$+D+qSweDJ3S%pVN%9xv*SFC7iYR0{L7kPB#?y@ya zOHNLHiG>*wc+dtG(0!ea4J(Y0{_ewv>jcsjeg13$W6@;o%c@j9jMANd~i={eo6-}V`9TM6YKl%&ng zS?wlk7+egHS`qz~H7p;0bGjZ6WjVljX?)zn-p5#9jPRD890WQENf9pZ(4(fNea}1GNXF8 zkzE<+aohy<9TSd%o0nI{=>rJ??QX~4Q@hUIVFkNV$;}zJ zj!fBe_ua2J?d|Oj&dy#I(Gp&Nnp#`4R~;e7#U&&jKYonn=H`|n?08AKc{r}6Y*fPc zjB@I;YtYA!0-pP;0UfdQjjk`CtY+SeXO$PYpDy`#+Q-n)kcf;d_jNk5e{ua#krM*a z(&*sp(ag*i{6<^fhSU2Q1R?+A=ilC0j9^kxQR#=`NIcs4v))VS^Yh<&`|c2=N9U(o zjw|13y$^?>DFw)?_hHx$VncphZ&ekd<*Y6?InqNa}e`t@s85rd$hV7_sq zq=BJfW>Jy;{5b`efq!bM;lM&?XJ=MpwfzhoMC_Ktc^s31!_K_CuD-t6c$JmqkN3U$ z6Tc>^38n93=j7&M;Ne9bZa%ZH2#cb)&3;639U@G_7K!FN_wN(Z(a~*hZ#Nz7EIc+b z@!whK;Ha|q^c3a1b!*1+9R8D%_X+k$@uR)zzmOP@ZEde)WK?+WE_R7kbI>fC+u6my zxzxECb8v8od;L+#JVSb3VNC0AIQtDLDf;p8aa?h7EMoc6twA&#o z0=vf-tE`7{3F*Xdc%Ci9N@W>0`i|9kI1l_4-uhANY!MtBe4USvgn@x!b*SKn{)9@J zc&#O6ssYz+J-nhEA9`_j|2Mbv+0UquJkh!ag#x#`rza>TW}Zn3I(?*BM+}+7pMwK= zB_-U?E^8Y6$>XqXk-f!#PG%Bs8^=%!*)V9^y8W0Q-7Z>l(lEFI>GZna6>Mz$1_kj`NL$dE7p9`h^W z(UOXaispfAg&8k=V`Jl^lN0iy!v>#=qXoim?*x?pcT*wTF%n#B-hKKc=zFrpmb;SQ zX_}T0heS}m-2L=Rj6~a8_~qU0bJlWXS|iX8AhOJs7Eu?+gtM#I!onz@6I5-jk`(^` zytLUrh>{cjw8uZ5EnuY&uOWsW>tXSOlgE%3+7Hy!zI#zzAhj>cFC+~8(W5dti>GJ_ z+%{>*LY9`6D*mA!4F(7$_Hdnlz=w2yr^7#>xu_#Fv?=4x|9(_&Z%A5NT0hUfEuR@R zUZ_Z>cEZO;X=-LTTbP`<lP~Ctn zEdSE(?(X&YF5xdo{foOqC+UrCYip}3DyFmU=<=bmk|9BfY`n%{*ySleIn6`QwY9bW z(NP_}`q2W-`bbVWB?!)UOziBq5}y0{`wOe#ZKmOE^BO|06`8pO1hmHk9>pFM%1T7Y zY3!e7M;;dzrYSOO=-I*+6s9dYiKwLr2hjT-We&!+dBHo+LNCSjQIv*<=B#lUe!eUu zBt%&!`9+TffS49qU%J$7s?xhiLa;=IhfB{i_!KSbBAey^-)1Q#rlrOBogZUep%yHD zwE){$h%)-Or|gE>s=^P#kxj|T%WLiLf3Ql4Jcs=jtQ~4x(TD%OT2KV}(=E5XW!16q z@s`QSy6%`R$dXHG5?&8k_=1rdUr_Y-^e`K;MB7sZzMgFR*H6Cx`IKge?5h+MVdS^; zwALJd!l9$e7Lo5s_|+N5v^rVqyoMJR7KR=W(3GpZb?$4*Snes{u+&|&!ez|mzPUM7 zEogzfDbm}+!&e0Y0@~>4L|qgsK04xG34oZ|7@au>4BKBoa+LyWf77skh?(`0&ndb1 zXaUK~bYmNeRNT7DjM1#|UC62^si~QSgmhMgk(X-0#lQUK7RS-)U~Y}isi2Cgs(Ph) z@8IF@Y1*;Ii>VngKf3ZtbQxK+3d->C8*I z5NBz%6L;KDD<>v zNqSB~f>FKMtoH1;tx+onECQN2vO8B8iLS@bB^vwQIozDWW2Jei6kyR8X`h&o>#pkM zb)6w|b_)jwL&t7+4{x&v4fniQr)-Q1snR!DSTG(wd??fLuW-|Te1;y+I`_bw^<^N)<(5#P+`C^}BJ?UM`w{OL>wXCYR4bvU{fOA$wohVs2qXI{axR`hib8FTbf14Weo)Z{x5K@M^XpifqbJ>;hQq{jjoxL zxnS67{=oortnKuRGlf;k{{PqLP~a@_RwWc3KU8K+FJw3A4-YQz);7GaC0xP;Yqlz` zs}rXYak>l$mqZj70XPcK4FLXR6cur}kRm~l;Bc~U1=GS!7qr|D$YzEitNHZl)BhaP zwYj;u)tSb|ejZdg{B0u4PR-NPM*ow>bAWC82L}V=9>sOCN9vTM2S+?EO=l?#EcoLbXs_l&v-q|L= zQC#bK!V8%FzakyY7T%wO300;(2*}YJ3CWSQQ^{tz_uwq=!l_}g;C%! z04$f+rU*7tmyrqV{zfAMpU=#!o$Qct6&A#+q@wcV-1Be zMUBbxUt<91J6-PuBTHaFKtR7bJSF^c9Mn$bM6OKdmG3eNH;IsS9c`2~ z&d$#>{+FiBXZt_y!x2bTeT2nnFd~b)N>3k4D0w9P-!TNfRLVtQKY&wMy%F*`{xI}H z0DNFlh{;e*7G#%lNmnI^3xwJa`y2u-{lC4+?}p9&HU1GV-S?!RUtK9T9yo*ot-}6x z;t06{EDLBpd7gyylEDE1Xz=k-zx9t!nOr0pHvSF4tVj`Y-d()MZ#zy30|u-=Ax4)o zRF{*>RW?HI5&*4S+N6Jw9YIWXc6Ro^^UBHy3$)nX;TbS#=MMjSMwRo58mCd6?kXkj zoPz)(Mq7KkQ$Erft#V*%n!kT1Kmt-h9m0qaqoOMlHzF3x)YQLDJTo3rEaB((R52dY zuh^MpH&hf>QF80ZbOWd|w5X`4UtMu(aRt5J6OH0F01Fe_D=89qL&Y~;cwzA&08gcj zjp@IChrL&)gr0G8womFJI6IpO|H_q+7@2kRzn-&`BG)Bo-T&a%IAbLVtFLN75aCXI zcI9Y)agE941@9cpdl&{^o?ih33K1;uGA=G4Cx=d>8BzPq9pD=;sZs3uYay&6XW#5-- zTOG)rDNKhQ08)n6YrS%L53 z$;`|=nvc329HWjjC?$l;sE$?L z$AVwN!onudeUaRYiD?JWTVeB)*WJr28zwZE^f+_?Ze_hX`+GR!ezJe_oxn3zm)A}R zpo-+sDV+txPcDk7O#}rzKmUf!OarZ&np%>eEfwq(vkNE6zo9`=)MeElm>OyJgn1Zi zbgS)Xd3kxcE&H#-K%dHoA|{H{u=)7`Bi--37EUl)m0-YYad~+@R_eUz6^n9kj7)+G zSKc?bv8+S8-FdSeOZmQ@J|5FFsD0OR1%hOiIxCJPY(!$eI@?S1;YI$tv{ zra!LxD==+*=HH~%)CdUZC0}&Kv$aC(K`nyJ%I@ER;@N96R*pt8?5!#Uk(T0LbTj_CjROy}t6$ONh{Fh#PEbK5^H((iDRs(NwdZ$cuuy=(7XE9la=HZwx)gf@PXUw zkBzRLo+(tN{(-p_(Xze${jUHpDxf>Opyat^>~lZ{r*++h6UG=*AhH81EBSu?`V}0L z#V=}Fp_@tlfQ1+b0CvK26Xe56-5{S`)`#w8nyqha$g(FC9&D6YO2jGVRE>APdi5%B z?MGg=ezhH?q<2lfqt+WBKj9TaU;r{1$dVU!TEsU0mV}0fhnJZR#e%G}udjnp>dgP; z%d45$HxibwD6z52`}#_%rAtObQbEP+zIpQ|jmOD6HwOhwR2xeqzPj`+6|YzqZE-mZYRp*qV|2YLOEegmfdtC3-XkH3yAH1%_Q`gl zWqe$3^@dU#B;FN@+eUIQGN~iTGW&rD3^iEsZ1>y0z2zIWUmfm>Z)%TTns95rG`nwg zUG#HCfrnw9TGPQs(~rEew2c}1mx}%hvAFMONe$wtxlK?I;?Oz+M4XqeGcjeS+zy3M zHG~}cvo#ajCwZy@pW1=8E@v<7xqk!b1HdBeVNw@oB$XC@?elF>!}c*3Fc+0I`VNP` zcEyEJBp@7H`hHnP!J~@#;X`k4|91jW_Vgs^a(6F%{_LNu-%OU__Xv0$AU=g1W>LsS zmo=VKK|N+)!D9ka!g|e=$u4enjzoEAq&oowVXx)YFO%ixU z{^qtZfrW)d=ECdLAoTt$42O+}&FN*y0!2p;_)xlalt;5cI|3)G1)DHo$5;&wuG!K6{q9bpSYD$zyjB=X;t&6fnPt zkO2VIvlm*x-0Et$-bZWXm}lWXg@GB7k(C{;1c^mKA;ZI2n*G+%o@=To8Xa9*x|g#l zik9}e6n+2TrK*oJGqb3s2A`?dvKrV29_J9B&cSV$=*etuFYVr?xUcmn1`+KgH2 zHoflzj&^%)7ybU0J-nw)l%NTT<$18q0{hQ|j*h;+_5+KHi%UkKfbv#J;N2zFj@U*Y zmX6l!qpSQ_Sl0zVnbf??He2i9<-hEC$dy!_@wDgAt}r`0D1wA-zB_>_^?s!S4h|ONoJ9SVoO( z3UM4F3mxc8cS-q`NZ2kNZlZ0BkWD6Q3QyXOa9-!HGD(`~&y;>h?yyPYa8xd{^G?V< zro5aRVGI$82v~hwL&GWD21`g{C8t^%;Gw?0zNr};qfr1z{qF{Jy6xRvfGrgO-87Gn zP-4&Prkak9UN@Ab)6cH!IowwJ@?6?e6v}S%p^^P7+-H7gj?GuAT{m#wabP^s)ir?- zJDLll-)^HYHtBlBQeIJ^?(PSrX$0s+WG*|P|N8(J;_E`x`-mH$Fslpx;1kAyqV^o| z@lQSnZhU+h4sV*=1n!|@y*cI9)6=6Ad(KHEWcLE%kof%k{Ae{hw%ZVDd*NM3m`^r8 z1K}v~XmbZ0Q)w7cJ3Bk>N81eJ4d<+J;nN%0v63xNLlG{}2J{Brw<@cOa&r50Qs+lZ zPzZye-DAVI!gO=VVSDxtoNw5V1>%Ib=;$kuqA!8KhbqOy$EW%4+X(L6yI13V%!`Rn zdEMnTdoybozY!4RzQ;=x`&%={t2gAV??J1)bLWobXeliK)S$0lU zKDPL>{SW1(fDvtP(o*pAaJ;NoSWP|+tJb!ni*w0D}fH4@Ln=LldGBUEn_bep|+0(thpH2YPBXa>ev~^fGG6?Heu1DaQCVzF*B@=S63(r!!%`WefszmL z_ea&#)R>$d?IKV4U z_{5;jy<*i7ppo57!z~lWLKWal+FvrrB1wzP$TA(7){B60YJj;OSb!$IV<0n}X28_} zZEtRCtM6Vy6U4p>Ocx+3BskqBxS5`r$#8PG6$^p{0}~Uvq@<+#@osK^MLsd0$t!?p zT3%e^yGl*X;X+f$ijR-q4-CU(y%&Xuh{)=6y;vWM)^EooJx4@eEo8rK-0&5kUWMm@ zHS}5>&`v-Cp#iBm6GjyP)L4^KQz0?(3Ex^<(R&hacZ?eQMdOgLW`IfqLZEK$FYplWytFEqY_SdiC{k7Nk zJdo_wE_yfz@plB%LU4?%_E!+}sReDCpk5-ojE0VhO&c*ZS>w3S4@zu`s4K!$L9iC` zLt(|E;6e(kiK!{03r#}voOKQnGcnGe&8esgvkn;ll|YNZyM6mM9CF4#@+Fg9tLWh^ z2+}!N3XBX<{f>&g5pAYG*`bSq3Tp76fyk5i0(o0PO2Z(BG5diOT) z9j5RW(Dy}bobty{Pq!M0u3hUQ@qdR*M2M>ZN|gzyH=sTeixO+|fcyoiDa%aE08kfB z5K&(>;CB(-auc-WZ*S2dMKTxiX#_>Ieuo4Ho9wR+a0>|mCzHSk^bC?LzzYxL+#~Vc z?P7!B_Cg+Te|$U%?5_U4ds`&grS}Zl+~(gXfz3AkHCCbf(VDEdxHvdQ&N@e@ z%t%JzdLgMo49y+rACl)sW|`UfoJL)l)*QfFK;@Bz%DBJ34<|0SG*oGEA4DWH&G#yI zG&ElII|5~NusXm5lMERe8X9bwRo$y>RwpN?t0W{rK%~KDqr>Tg23lTS9ropm*v!|) zR)CYp0KyfS`+k0Y9_cDjwjl}}0AAoqJ@|G8)xw#ohZz!zRfh)v?YEYbsu3q&A zh}Z-^i&bIvXdR~^X;4!UaEicx0Rb}D^3L8~d24I-Qlt7vpVL1Kw{EGfCb2I;=^m@J zAZ$2U=RnYEs)%!FWsy2hX=$nMhbK78-&3QYY-cXKd^cMgTo@0|4Tv8^KOhy2>|RKq zAxn&p4{8k~Wa8k!dl(V1{o;jOYK<_eNvgE8X1s_ZO<0r>6a;iEf<-b~6z^Mn1j)hp zZ(?hE3ktkO7PM9qP&bkAfvOV$a0iDrB0OJ-M$(%SdMKPTg=nPSEPN0JY+G?Lr!E_F z#$k6td7~rxxQ9po_nx4?(bTK;PAtX5#4q2zjR2T}O{>H5Yo?JN;^b&C{yxmtORp0X z55G5(pkYLXmg;@Pg)yuZKn5Hf`wQ4Xu2SR17+??{0}AOMoS$%$-_FaK|4$AlB0~G^ zmXh&tZ?~V3@e=A~Qj%`MfYuutp-oNFX_tqfW}a?u&&eX?8Zr*DkL=0g#{qz* z(ir7FA0zr(tLU&UWisNB%6HFKt&v*5n zY<&+k;0iO~Qi#=-m_0A}(ob~HL_G23=boOPR`?p8$1RtIMMQ9D6-rj0SXjWk#a>k* z$Ed#E{HV;uskUw~HuHs#cBYZLyRcD%cLa3%n=Up_mpIP;svo8s%daOK%vf0=P9HLk zhe!)ZVT_+v26C?F4QDHH4~z72(Mqg}L?EH34+L7AzC4)edtq2 zsxc!s8>8;0Mbx=Z8o$w_J{B055|Tm{c~-us%4%tmLg6giCnOfYxFF0h)zH+e0MQ?x zS`Z)PVVVE<@!%(HY&5%v9X>)ri|ZRqI>z*eQ&WzeGxYaUQcP@Y@P&k`-9{!Mk3k%c zq>GZv$|@U8*gpd}HBh221Ufy$E1f%ZfBlXJIm;U}90l(|S;)%D>fYzmZ}1k8JlmE9 zNSmi}N~C`9;72ZN)j?I1MB)K2o`7UA!vjVQo!Z)3=<&p7XJ^|`_3>!K__e&f#lepi z0z;N8%*3n0)={c4kqD+n$QQr_Oyc4+FgLaUQ~QF5>23xr0qjX!x`Y<S9}dB zG1ZNIBiLeMxOsRGl}g!w$pU^`1;%w?p@rIcMr&{n!N^1TM5d;uK(z-#k-Jg1wK3!M z_mpKA2yqUW?Pe}oJ$sQ4Nfxqa#`~3BkE?eIIM<~ z6ONpm9BkUyq43gDF3-cwXqdN50WeYu+0m35eqnXt9J~(9Pz#bv?mMRc)GA@?=*l7- zPDx2RDBB$XnXV@&Wx}X7T%^-F^>?^f8Yp#2VTbFcW@d3oN#XzJ{mFvV0n#RQVyPJh!rHG7TDdl?>`V$*>4$ z5ML0CI>9r(r&lhH`}}5CR?!fpW$4Jxw44Emt9Pv$Aa6=*UDMTG8 zVmP#UdM5FjAqb@o{N&l$ULV-45C@PWu;SpSGiwgSumq+kmuB?aH+&!`@&TvT{waG@ z$@(fjUcK7RP~7*d=G&F{H*d(HZj?LBwPbN9u?%~IVp33nq=eer+q?VV zUAQ72zs=ZHz>^Bn(ujF)c5UsZ3$unSxQh^y2tA`ZjO)kI@FsQCVBvnk{<%0yK6)-tzKqKW#1KWwPNIS8*e! zk-mxSDv%wxhE_LKiyoE$ryb-ZBm(hx7uo?d3E+1F5x$_d9gKDmz4KWd>hKoZvuJvs zY>ub(;7o75c=3W8d|w|LDH#Wa2?z*uE6uOGS53y0l9DPFy0f%AHx~d)p9jMz@;3uTG!0c&L{>i=ADa&8gXLm(>*4f#rM* zF#^fc0W(8LjOOocB{dfJKfkUK&qE|xe zvZ46Ra2io`jnByW!OSuL%Mz?qK*_Co#qKOO5J%*EHLEXMruRDw0Ij;-IUVMY#}J?h z-;a-96DdNJ5dHS;HKcA{<1@bwc&(wK0U1zM^!msehmWq@{$PAdE_sg;Ywd6lLq&c) zqrABS%v(A4;)7!}Z|!6|uqdgjw!uMeAB-Bej*U?Q3jPZ35@lE32jm>$f2wgbg9Ow0 znSeN05UEnBOn*dPjBMw}eW^S8`llr;w+%lB!{CY}UH+rW_k*#@a_^eKdke891GWAQ zmm%(36;=pzvv$Gum)S=bcuzm`($qMhKtCeCK~^9N8VNX5T8D;60RpYEI}e6{PDR=`aPd=Wca^)dGR2i`Ch z^nZ4JmLlSe31J$Rno8RqL)&vO#sA>$T};H73wdDr&zL)aZR13`zf2{Bj)@OSxJAXa z_Gx)I2C$G(N_g^HkNTcoR^0mI8T13%U{(Zvc=+oQ5Xg{t2STR8X-SD7iX|7+W`wXB z|7075bO<1eLcw7RBypSeZ-K8+ptzrQlj2kHV}k{VLNulw>T)Df&?G!b8_#x?2dIP) zJ5$SPIxP6nqel>-3M}u7jo>3g{r&y3FvG(0KYRq6AuB8E(RxwYRT>(AHbR+9>HK5)!%;ErIkJ9;dD47g=JzdKiXPE10Eu8( zg_0^*)Vod0Y}etEp`IH*zR zJQ`!>cGIhV7DUJV?CI9ld+{ZLi{mp2<#NR(?)V1D7thVWQMubGNv@ZsbX*mFo72Ro zJViFP!N&{PSg?Ca!z#a;@&L@6ottBD5iiVu{xT1=NgK4|p1+c>*h5vnHv&dQ4Aam7 zvw_Azh%10L>{@%s+srgbxVRjd`OtK*{zqYq$L3c&s;Hznc#;zfV_nyK2oH6zbLg`ejqBTX6}5 zNZoo*ASCkD=Y0!LJn{Vev;>T`hl*Z=JdWsY$!fs(YCfMOg;y^!iSI73Slqwom)82N z86Rb`Y!VtqLErNA>t)!gw!mw68HuG~3zf00Ri!t@?|L>BR*=yddIkSY|E1j59%fP^+~8%&7dK>uZd(>7m5ut7>VlG@za!e@hCxE#1sF3N20UvG z2yXoj^ty4dzt4SNP!+U{?0he!9+)OwgR79`+ z@@3)6RzvIWiRud1jfcUZp{9TuR)1Eq{bNxSwjAca;uJrs#Dw!$A`>RdRA0#Nu54G8 zfAaKcb7y$^c6|@bC>3B)hjNVx?VVhdfJhsiF>bY_DDt@W*d0uirhRLpHySJ}YqLet zrhRcc1F9yhxSjbqPSQ#hx=rn_V2rBqNG(5-{r3vkW7f#LorWO@|4e6 zlRTP5=QOIL1m+tNHFtnALsVW$(BfW1M~B2H+@S2OupVZIOCZ-dIbR&k1B4maO z@3U-dZ4p;TL)?W95RV8I2UjkBP1b4_wC06}i(#j{sbSFA33NQjuo?R zUiXu^WTJuqBnX8xe!}QmM-1fJ60)+g`+wFD!D0dW`dE!46R>AIHe=jK?XG=gK&eY9 zox{=s*~_%&Xg<)u>Z)R5)0%0wLgara<^jxx zj%76U)77yGCWOa=FvWXw#}Zs;(hd&XK-B;MWmc@bCzqoF!y*{*+X6B0keTOOnnbct zkmg&7hI7ghyOFC_iMFeCocBWwC7#cv_grN7j-F6fSzLv|1DL09AW!CIzXp|iT4La_ z;k}R$B6pl~tZDcyin@OAkR9d>F{BpK0 zwKo5P@t;Z?ab@?1Hu@E{l*?+%^0Dz|uhh$YBmnM}+uO!2COsMZRE~klIeQy8uKLVu zwmiVWb1s;=aatN=U_|$($D*g4xR!l)ic0%XW;h815h&S+y90Dggb_plHfSFBLm#?4 z01v{MFp7!kFHzB;9G*YF3(oBd(>B~3+V)E9WT5%Ga^YTjaHZ` z3>E1R@>zWU3MgTq%vdUuDK3)Q!<6~WBhJ=N5~e6stF(PD;us7%j!z|;7Hlp?j&?W=|UW;8F%-vKZ%Fw>~|_sP)bOsNIMeI&9LDvEwxLwZJg zAap*t$5!{Z9*vaKDLFOm$^`kR{A_J37{=RyA_7VR12G80EfdS(LNX`^z&Sj+HJyEf z9l^P0yLd2fh5L^%@_l9Rh}^xzpl2Vuwg%K!t*VXXOwJ9Tpc_0+5*W&-_!WU<1s z7@n!6rp~&xZPlOt-8YT>)mCV=L@> z23$x*cpZd96Ls5s0UB`2;GhP)r5yX{VtzN3Py+b4%s|gU=wXD-MF0Y*4q8#CE@_fJ zk)552Km@_PhO2Ob3NY|xWM?~FfO-4!%EAIV7|!6@3m$y^*A99=PMC0cY|TI2Nq%eF z{juy2-`>4%o6B?ZnQU2Bquha(#zf<(lD*U5;2`+_4o0F^3JMAoRaNDIKltpnWyJqj8YHNe)>g;k-KQ?U$H{=F z7z!_VlbVu((3>e@ZUji%LHOTBKR-ad=wO)*TvED!zXO;BN3g&nwl}bGbois8UnM5a zQa5vn2!Gw&*@+Esxb5QnPzv$RVG|IvW6^sf4zv(ZzM8>?pz~!niOGhNwWW z9>XfbAV0H!1b^BmZ>?OQ3exL8BPARJ^D-;rt_Smcjw1@CYo|*!` z#xJCW0V9$7>e-}L2vognP`?q!2)LcgAGczHz4a2UsEhQ*NC}6J+8?+(fS8KGE^!3a z6X7o4PFWVf0mNsOm&nXmcs))r_`c-vWz~25&A)zD?!ZBt0c>S=J4nmthg)%Q=gvPY z3=1;)WpZ*9yeM4uBL_?3zYznu8w*$xT2Cmv7Yz-~gFH3bAwc4bhop{<(hl6)XCBQI zD16EX+h>UP5N!T)5aWt)wHz!I(U6_s`^W}v2e2JZC?S0eAh$hqy({mBSlL)vkx5^% zQ?;qOV1|*h`qsbmM*B1(?5um-qh~qfj(c z$j=1;OEn!_;sY~;I5<|ohzW@s3Tz#i4G`(&u+n0HOMs~w6YJgr zc+We1X%Yz@N7l{2+&6_vee0)K1aYQaI zE|JlD-{BH6L@O|1JNs*zAa+RW>r;a(AGx6q-;J1H5iTVE@he4UDjt=W;fjhNJRn*x zaGfZiED(AgN^y|X|x zB=K7je>HVPj!*DQa2Qe;Ca1guK&spMVQEAu%Q0WjYBPWMBufZYa#xc{Ag~$c|(JGN^(qwS92V z2D*@l7mL+<%q#v|4d|?DoHbcyPceSY+;(BtaQ^Y6JcqBhhNFo3A{6KHRm#$A=}aDZdh^y^}hT`KgC+B;V*&o{_XXMU;UbS5`FltOd z+JM*rSKylwdlQH~GGJ+ky7sbX2^6tl;NamrXE!%vKq5umI9fb9D092Hb_&!4@^|35 z*uMBZ<)<~|+R(AX)~JBBI4UpQc6s7*H2wE&v%9u=*-yIbt!iIAeP=H9tJw4NglAd7 zMk(89{D`_3rukESuRluT?~l!<9Ar6SyYe$`k}gVgv2Jn&vW6#_bpAEtV!ClI9f#&E z@qLba#&~P%RIIHtxvz}JtX3MA zAt7O!$MQ8G@4hA5H*{_J9*P0#GE99Fu9!FUUOom66mdn3jwULWDVk&UK6yH(^f0vK z$=z9e#(Kt`y*)EvjoklUD18Ef2r$>)kmh~FvBYIiBP}C?lsd?{AgF;nfWTsCL|X7} z*CEt?%HcChO9$VNi`E^i_tU;TE33cj;UNMJDN~r2QDD@Jg^KG4R`Sh;Q@a)9QuYTp z{ZHOSaPdWNzcT&r3z!pnCkBO{|*1cA?;owLi!UrPO5yAQf>7=W!`&&3eSwBPB(%<7LkLd>9mlM_!1 zmF$5|m^~iVx$j(w70E9vbNjRj)pQ50lcCCePM-rTRR;AAMwBPe%~7zja;SOwzV`5M zcER9!0WicSAdmpMAXp1qU@+^18|j9rAhPB4FUu1M>$y%oQFK@KN%jDrY`02hUw=T3&>qGYCltT}xINbq z1Iz+lk^20Fj8SAEtV2LPECNw__mzXZj_8kCatc zCrU_2fJ!XPrUJlZ`D+9T3j8QyQ6;d15~W*=j9g~CZ{1{6U#dA<*OVw26a$46MCUin!3nekHKgD~lU=T!% zvLIb712kO8O*Cq;tLF1L**E7a`nZ1rF5hJkGA&^7sJfH&B2rRP0BGE1$db?e<=7R^w2>!4rp}V`gs2v{{|QGbCMcCnsky zJ!QgczOkh@4Pf96$&j zkN{K)z7C}VOOY|dNw{u22EymcbiO!9`#+={QWpl-CF#}gjXao zwec+Xc<02kV+Yb(BW^F&Px#O8@m$8dsG97kl=mR%ghxlS2nxQrsreo;Y{UJgZ;@mi zFpIU=2%3G=yT)ze@BBs(0`5CF)FL5&(|k{?AZW&_?dbs?Ki!#c14W4#$|n@0+Lbgf zxC9F9-xm-6T;;jSz6{6iby~66kE15jb$|RQ*T!Ly=iXzVhbyMWWmbX+VQfwkzvlTT zxLXMuyT--4hg|r1@v(q?!`cnsoSp}9KQ4hs11a3!%=uv0U7E0OWPw8lV=~-KA zx4q)vTJbzw>s@z|hl0)mWHa>c62SX8rOD%8z{Fyk@ z{~_S5Ya|=JWg#X%x}28etFe>{pMIL$$QmjzQtaL z935jvNBlDlbsefdez%6vE_A;m7oqn?$%e@|?v$)>(S}B*JcIzc&d3}8><{U4K&m8$g@sifp_!~}xi2LsPYr)|3)^=|Tw*g!Nu~x1^U}HV z%IyBy@~QQ+Wc$P4co(OQ+)u)@$v~`A?ieY>u(uzp#Dd%W-^`EFF9PhQXl}us^*d!o zREa+@R)u!-`>SR}fDVB&3wHw|pjQDU#QjzU%BKl8r9o<8cz9>Y@zzDC&ncIxxJji5 zrd`w|L1yMQTq*SysXLm_{VO#1iqDl@{5U83*~Io-OsO^KGHJg}#FbJ-R*AOr}!#c(l&|s%M zR%n4U;7;y^DWyggFRgU{+;!PKT1tvmpz%-mXbpB*xXa_|=@_($Xg_&-D?OvQPlJ_g z-upx}Qq$fSP{SKa+svJ-zLQdjy?=?Rkh!q@e9beyizlS2Yw_rk0^av)OXO{lwDl~Ksal+{`l(U%Z$K(>)I4%clr!h39w;@6z|`6 zBNqpyk_Co{_^#k#YZmam-M{~Lg&ZvFai#Yg51#WtlmUu=nV#MUjT(M5&~>;28s8-c zwH!03`G0m~k{ljd4nIf)s35pHqu+s4xRa_FPp*ZWDjAR84vuvA_aJ&8b%v z4nq}f<P9*bYcb9i%hz#CgNzT4(ej$8|J@3WMHna<@Xl3x0mg!!pbuG(r34qyn*01U+w> zeGUVrHZ7OnqP#4-)20JVN@F&Hh@*agpZWg}Ow@{M10plLgr4kYUuB=3EvBd0)fH1} zC6rY~%E&}p#>6x?##7@?baH6la9=(qyo!fRWc~?1RmD#mcq$|rlwoEep+r!CNI<3k zX=4tqGyoT0kmp({J1-0CIWK>+YMAL*>2AsMI;4)B=to6IX@98|`9w`sDy|@x6J5|5 zNB;i_EjoYZmH>(`RnwM}8xKrQ&Q55n?{b(L*Aw>neC)CEeaih?gU{TmddEh#0_`oK zgM%1Hk=Y~PQN!vt)!~YGT8o{o|1aFo*VpI4`9GMUvlo{z>`~=)zP-T)>i=IO*B%Y! zx`zjQcb0W(x2Fl4&FG9%a;r3QzwBxz+NhLEw53vPmoBcQ3s%?-o6XeLSIlpRT# z1|_>B$&hiWJ&GcXOC`l|iSxX)&pPX@bJp7Pmp{Jm^?l#_zQ6bRJ-_Go_|-nUMIC(; z7HVN{cPP~NSl0sE^wPhrSNCkqih5y4*EVT;63h%;-rS6)K}F@dp(tajs0*i_>+t*c z3_yUVPq$xk$Ru!+>|e0NpgeaQ^Ko*aiDLF^`z0;3iG5zqf6GldeB9H1>`ij~7k8Z& z%fY6LPVbZ{T7;WkIvv@QX)j?Pd@-1j)|r>C-W!iEFNCJ6_(ExEYfFTkc3Wd3Y3_lO z*eQs#^Y7frLYh2R@%=EIArzvF@C=?2c0_&~4TvFL zI;O%a4MHlQ@D6ynynA{=pDHuATrf{zUd+ofZC4i;G}UzYn7CFHsuUGVfy5=tb#-%) zse#6D`p*No-MLCaJul>?q%1`1n1(UVg5FlsYuia2QrL2MjmQAGA{+T0EQ^p}>=8=c z%fk(ji)J`1`_)yCbjKa-k6MdOPxad|v0&SQ@_z@2UVvFd+Y%7A$R#*j?!CRMd8w(v z>kSfSqiownhVAbz$ojB9kyAK1IaNU{4bAgzG)=Z2dwYAkzA}WDvev8of^A4X035z& zC*N|gv*VO>nuYZ*h6?8X9L89UCVIqQ+>7k--dt$0)6QYon#Y38h74t+qhnrK*;!Ec zyM)pRd^nuWd!dX3q7@)pn++`thK3+PU3Q36Gv>C4zdtpa=Eh}b zFZtv8)40Adaf6mIKPfi$i}~{u;^N{EQm;pTItua{XdCkEM4TFbTpx?;jZl>l1PuQl z98TKY=H8a7vX!TGE>$QLmUN0x(5SAi?uNG&>hyR(K@D)0CqWc|%}gaT9J~<`XzaER z4Ah6H?j^*-cCD8)tMt2Xc`*zIGM-MC|KLjpv`3d zjFcUxDR^7wbU9t2C%L`HyD~kD7e_@o0XA3$V+L55p-wKmM}vz7 z0xcQsGMa4Q?QuFQON@#>HjOV7l^{=vNmT7##?qx<##9}uzG*Y{P61*%fz`3nXVXX| zf}zYMLDIZ-7R+hnKk!qSXkK9$D*j$NN&^8KJ1km)7b3fVJkz!na+-x zPZS>SH5gHJVKEw>!A?gi)IH0we|zNW2`_>u4rp(VscL*Re`|Wd zl`Fq>bX3+K`1MFan`+niw!v&YGUtdi>{%$Sw|Ekb0?%TJ|t-b4g1`2Uy3?IoYH?As; zwmv+3^y+Ot^3X$A=yrNoeoV+6$p1*x_w9Kc=vsI@ki!I&~OO9a>b42)i*HE07=bAzJhW|df5Dr4V-sl!>D2;EGuR7s($Aav^hpNmAT)X7Z1Vhg z6!Zd%6+8R#;xY5-12Ekb`B+?;IL(#78ZWx(Cb~b#uiA4}QJ486S_(DzM+!IWmFm zwFnkX-$677%RM(MpItM7G3K&Uv- zTTHml=!r#^J+A}0o!#6@Q0?)xZ*JLxA_>A?a6!uPxSSjY6hGi%x}m7-0;hGL)c{4o zAlO%?fQz&ks~4C_#%uVnlolVy7X-*%i&-l!BIi8QgQbspl z9H>=#L^l;BTow)_#rgC9zG~HJtU4zzuLjMTf7<%{`wJ$mtUj8kQbnxuOBuli6B||h zd}+%ZI?-Vic0gzp<_5%ip}J zTUmL(u8xZ=9bOpK=q8RpTNx;r-5?!-#}$ZxcM**y{V>gFTGLm;VimFt%osyxpp5|m zbt9r(TKr=#RvylUA0-)>u^Nnq1q}bJCxI1>&>MfV#y@Q&D{6Y=ygRxnLGTAk)LFSB z65n{v58MQGYEV2%x!y)<5q%d<2_|vF1~8eOeI^Hu)NsyM?J3^k=GJ?~nVqXZZn?oi zw8?#LfJugIa0G3~rZtONnwzho(;8kR6x*@X4L}$$7hARKE+xHNkD>81nK^&wZjN2~ zBeB>As}P=?nH?w&BaYBdd~hOOU$uLKH@TnV@v1=iw2w09lUJ9Hf@_#HftiQ<#4Fp)@1vv4v z7(=?&)>){E>y6G#_1YXPeSU*mn{bEFlzt+ diff --git a/docs/images/chapters/extremities/fd68347a917c9b703ff8005287ac6ca4.png b/docs/images/chapters/extremities/fd68347a917c9b703ff8005287ac6ca4.png index 8c7e30d1b3b9781dacbb53247d9c6bef3f7a58a4..a7818ea315eb4920394cde2051c79471df0b0c52 100644 GIT binary patch literal 20577 zcmc)yX*`wd|2_^c2^o@^44I-xrZQ$ImLZu!DaqVGL`dc#nWrR8$edJ?C@J$08f1(j z6hg_Ikl}w^eLmmc@B8}S5AOTfZSVcsduy$0UDtWOhvPWj$GM`64Rtotanq4Vq|FC) z$tENcr4orm!MceWzhQsbWP^WE+a1y&lh%p9GAiNjOb$P~pTTbY=wm@+m#?DY372Q$;*9Zn zeCozIl9m~n8z@AUR%;TEQXM~dH}EYn^M*se3~OKqy*a~E>0f8I};86R;~Y4Um4x!w2e z$lIG27N>37WOTK>Ci|j3sIT*0xNspW`##$rki~q~=t1%!d@bI6?E5vP< z&}`Y3VVI^&3ikc|<7x&7dE+2^?D+9~ zXB%wO$BuMgA$5k%$0uJ6@^t_GGnMW2>vQc>A?f@ z6Q5W~txs$o(WP#xzME&&ClAfXy!<1g{f5QxUiY){J&ziFALW(#Zsy)gBIV@d z+;?UWl2!TDTS{{^NRkTA;`Wl8*)i9=(8{*=xr?-{Z0%J+O&e!t31t@Mae>K0kqoRv zJm`glgha;0QIYz}eOT!^CD~FVBR!e_h>*jhu;Y%os(uTDO^L$BJK4rQ6`v{M*RQRu z-FrPY@^nk8n6gIH1#69*iq0~rueFrqoaD)Ys)Uype`Z-#_`dM_GwD})@|D-_mbK_?cBMkx73w} z#IRMC?8YtBex&av(^gR=UQV*UzCK-LfQXRzu78hhZZ0fD{8ZH9qQuWU9ae()Y2SrU zCw_DbpEL{p`_=z_Q8dC=$I#F)O8!Q(gE(I*k?Hf~jH|@~e7|NniF{;<0e@>;K?NMF=+ zeDGhW@&CTAm#cS<6?c{#5oIPGW+uMdOY^oq^{!pJk~hEe#fz3rKQuMzo_ZFAC43Xp z(e?jcOFLv)qT1W{?;ED7*5k2VoT+tor%qA%`S~d-Db;j#nz?BZtG-$%8lh{@-qAtS zv~us6Z*hM++S{+*xDo!dyD(cPVcUI+XAd&25CwD@VVcN2{Qdj4)=bT&ckjsVPdD4y z^|}50NJ9z^-^`zNX)5MPY0#BWnk{w+8oVWp*|%^G6CV~-P?=5+PiwGE`msHIJXR$Hs) z79X0k)YtnjzJ5;JO%68#sB`eQ5M`R8&=G7H8~9!D(NHgwp4hWey!W6z8!0-xVxR z#@%PQs}>Ml7qMlibRbaE#DwMP)2GulH2fJy9vz3w52l7)A6GM#DmNuH?mkd_c-Ys_iB&(mM%zkOA|H#tR zH7uU)7w_#6_nAK2RjuoKXntY1iMhV9aR{Y~6nxJh#jfXxE$Y@N59f@3MQioBd-v{* zwPn*V@~g#CZ{BAA=DOxo?BO!^Q}bhOlFup+Zq9F|rXis-hJSf=smi0OqC!>N{ITV5 z0D6xG79rD2T;}4B&P#WLuyZT(9Y)eJOFz>C4}Vy~f6HJyS*omvP;{vgjbON@FIxBP zT=&(hSA((FLc4cwE-NcTwozGHT1q=KF?oA?XIMOQ8sTp2>|~RalytjrL9A;IMShE) z-(Q7EQ}N{KKYw(Ljg1rkG&{)NN=eboH{;lQBy}v$FsIP6hYtJR;UClb`xc6zkT9s@ui*-Z2lX}p#y z8nm&Qu@l&}YuzhXFStrw`qJYbOR_~4wuclv)R;iO)%whW2>GQxZdcSKe?IQqX*Mm< zr2P35*N^rU?Jqn=84;W7uM_WJwQ4_o+L|7?EOuAL=W=?`+E!%Yed{WFayE*TN@AK!+7r&>tO1t)}JhHB;w0x$ap`n-N+LEdi85>LPxMx5gu=2}STU&eN z&3z+x509CZB@ZGarL4-UV>qP*ZoZxQ)L3sXVsU4x`0^{Fopmqm4QyCThUo_ILjkvm6y=FAzQ z7XALwRYw$yu~xw=VPTm-+hd=f?ILn%i`wFTPyb}Dy=RSU_R1Q>L`Sp8K9+21Zm!`E zS_#>AX_Ck>@99Ab6I0Wfu{Pl*2U)ZG#uTWH5y{Eys7`ciw{G1MFv|4Kv%0Gh60wE< z8{R-=&_+B>QuEcjcjNo|EP*8 zx$gqQ)T5Ipq7QjqXG`I2rwdCnomDTh9WAhU^8C5v+SyNY zqt#hjK>?ws&c|~Py|(>UA2k(l2Q>`MY-@H0|29p@3iQ%3&|Bwb1Y1mqy3#bEWXIWWBie zeQGqp;mnyKAWlW!`JLzc%cEM-_SyHB`?z^|ecvl)vNcR21Q%NQQ?=eekgwlAj-UXd zK-0#@$8*U#Y~tnReRC)sG%YY!?v` zl#yX3sKCcZ$3t=V)wsfLi<01!6dqhb@1o!d0$Q5vWLx}II`G6})Uav9oq>DrCKPOK zDq-P_pmmcf&Kn$9{K)pba)l>fdeX|s$S4M`D5j>SO1lm2#}dx+QkmQa9{-9$5g8Lh zG)h}n*Dai!obx}sw}1eEdt^D$?3JO&$_|N*UFp4XV?^?JzKIA}3=3Mm#0#l?`*^w^ zpRCJ0bdzb@ws^E9;I&LJyv4<#gu|uhzqltHB9qN-y?dv>GOE5_gY1qZ+V7j|Vs7s4 zz8=EtRh%5*28{V(|7kgTU7e(*ZqH9uLF#sPb^?jIt*fhN*D=I2#Q8Bz{oEpoE zs`V8Jrw&1K7&qg;GSkSV;w_z;mS&rkpPSlMq8U_R*AVUI;Sq|}A`RlPUbv6YfsjdG znzVE{b*iSWj`G*%XQ8Oj3A0Shhs#ufv5+G5pzUoHewQ8{FTU4Al|j0{T5%QkMn1?v zw=-HJvl_^w=+cx`s(R2if(+W&*wj3&-yhgPHshcR+rTMlS)C}Xx-8!YrDGGSw;2t`}!(_E{EYY!oUlYlaph8WqZ_CrZu^Ecw*wQ zWXi+Kzkl|;ytwPywQJI6TLlo9ugJR;9`dB{iYHx@WSds|kgl$yug|bB`GwxY!(;we zFDEu^q*9WBnY4ZT_CcUtMW5O2*zDFak2CLX$!PvIZS^!p<`PRfKlz2ny^Oc0sAv&^ z8aFs-<2@-s^zr1Yh7W!}lN?j=SSYXEv8gyMPH@7E8-!s@GX^NP9_HgRIG%SJ! z?YA&d14h;g!9l{JWb8}X9r7sxjia{^8FK%4v8eCD1T8;5Kfv5U&%FgA$$D-1raL@; zbkZI_eq7OOf-5U4YkIhu)ql2m$5_`xPPAT0Wn`lx1vVP_YxnlXZe;muu>AqW9cC#{ znlm+$Ph|%GX$D!qYCk#GE$QIkU;_w$OYR(pM3)MG5`!k1f8gqGb6#+7G+Yfbd84zl zGr6%{2_;vf(5j;Kp}7=Mf&MQ1s=Fm)KR7rTytevxC_6#zTXZ-Y-xxBt2Qm2Eb%1fl zjvW%mo{$z67P9OsI8%5x7%zlvh)zO1u|e%7U;{#sxZx||;aNfJYm384M&6#WkxeAz?!=c@oLJ6~SFiQ~ zh8aIRy1T7N?}|Zui1-H$3Xu!A)5|02uh6BYD*wiFPHcbn;K)M`9$vYty4&3@U5dcl zwyF6?0{q#xW*ivn%xA4!{6>H5#EF=AT{2nxMsji(7`x3>f5li&k#ObunqOMV%XgxU z`|sOj247RWG|Bt?{FmtZdb0o8(r9FS{P*j7g;Zxi9&C`RTzgJ4ARVP`Yd7w7?dJu< zh)7FI8}mkH8)6#?u#uIW-Bxfy!MVSD>yZa$%zO{^dxpovj!3c@^I{|I8{=)~zQ4cv zc7&*a8z?CUs^pcK#~%^cZ0>`4BWSzZ{Mg~ezkjpOe|gEJ>bsYhk58*nSSZV6+0Ub+ z_>7VBmzPIIt8z{!W$!SnFy4Q#-1j#f(LYd3h#nfdO_dvn18T;%i!PB2!DOKWC`)a* zhd5JJd{}B~YMvAo)$|nEw*jxT%|u4t=R)+qx-|8gC`j8>eaEys=D1UM`EnnJZO{`^ zP~dE9YwLM>ihg2Zf{BTVWNvQ$EKY~~`91WB{{H^y+1c9Wq#ZIrYyO>u*8G4V5d{SW z#Cpu%K$9f464it}+LBgu>Mcid>hlZ(tj4B`Qf$U7Py>5jU6LYJTh!o=4RnsPA0OTR z+$fU8;{BVh=lN!7*$v?g6tQu0Q&nD@6}=dmnkmpCEG>gQMuEmdjMS;jyd~fNJl)vQ z!3tE|fMPZH{d>5K{c8e~i}OIbcq1w$wFMei|-A>v2+D^|BTyO&(QhZ+~vt#e-9nC2nEMB~LVFRt}$8kQ7W|ry? zg`OYEg7~@Q=)9Nb-;0hm@>2u_Wt=($>S8n8q~OZ0INkpgH+@BvlWu2zdAUk`eZ9@< z@^6532M^60F=poKhM4NUG7(qLg$asyUJ5q4j58mq?xa=*%&n8EvvsOhmZ~F}0UGv? zwp5RIEWJKcF8KX77t~2Q)>W_`feNKo{_Rob?{f{#A+Wa@oInBpAfTONo zzXGjd2sp<3_YZb%Ij1WGD1G~u9)wF$KIY{tL}p&~Kwc=Gm!Cb8!wyCuPOkgvpIlzA z@f~^iPZ&)hIXIAfe1g7MJeq389w{_if?r;wh}s_XH->Yr>b_sVKDpnk2J!z9e&h=Y zVQ7A*=EF;X{Xkw-t-^8v%kyHtmE?Z-R*Zf~*YWmNjE#$vK`SmnvRnQ(OH24&QhYZt zG4YL4$GZOA74tnC%9rjPW)sd{aedi;<0cIQKjTI!iVGK(dTEKmC~^GhMl@UWUjd?J z#`3Fg11;A;Cm+v-RP@NIf?G^XYx{;hIA?NF8~U z**cV~%M_v1BkAdLzOVlN8F0?oJ6yL#z5ZjtiJ8KnReI~dB@t}c6|{Fp6&mJ5H6QS1 zdFLL2sMwM%*Lq-Jg-!xVp;W=O z`@!#iE@yW$atd-Y4$Km%PK|`BgH+L0Xsv$h)^s6NqVe3mSozOm>NR4 zt~{@9vwtmU^c_WedvIeyi27QStM9}_uw%#URMiE_v%N00l*TMszBej9l@mu#b^SFI6Ld&M=TA?m(*0*? zi421t9E~zSdJ^EbCVn=^`uD_C#5z4I+x2x7fgoB(fp!11j|Fq1QdNOTqk4_6#|(XkJyMi zV&>vnY}p_|CSR9QQre1NMxYnz>Ta4a^L}6KUQ|SJPhaed2i>_VqDC$2u7y^v{p$ad&n&!`It z9hsS*C&zH?e&RW9LKGbIXHAq~twtds7VS)90xI-W2JE9|YBne;D-+?D!vQ#03*GDqfPE~t93AlDx3Z{H z843pRYCta*fZ2`2jA!WeX5yKlc4Z0U9RkhpZzcWTFI^>D=># z0+W{dh8Rv}CZ_7_E^^KswC))Yyl==(RE}?LPVB9>MH`K=s|Dax&;Zyaq^q+P$T?G> z@K5mA1cS}ij;qbmiXp~PZAFohdIpdsm6f?ql%8*x{#wV(##W#2C~h(-x&uI`YJF|r z`GLxlZci(r15Et-^>ui7V~4DjL2oNQBxaZSV;hj$vmN&~drkDzBwdJV${?N=krJAo z&aa)YEe><4n0P=zyMqT09xL~j`O#hY;YdekhNXcaXbvH%g1(kt2*T{Fy}jLSb@?Kp zWI!$3{rrX?Adv;Q(~0NKHF(6YQ&U@9U0vG}BO|YXi6{{`^oYr6a`#;94som7DXCYkY;-t%`rFaT-4aG8;?2E}O!xfm`>CsU zs5aJVPb<^r#H&|Hm}4N83=!kn#>PwpGKV0RN7iYIjARHwBwZIXb%R1p#2*5H3{8TJ zG~94i-QE#6%zJTKP(xGmFqE~n#ChT~Q5Z=3-^JWg;7;wf60;Wk{((!sP>qkEbCa*8 zUKsD+D#k`fbPF0r-s)>2hX%iWv%5Mfz|72^hf@FK*)y&E`$?kuw}KJ)7-W*EC!vrY zD?QH_zL|$mW!-EkHj8;A%`Iy#{Oaq~I`8rK)S5j92M1I+m&^}&fBe=~lvY+(Au1QS z^vZ9m5EK&EMqd+^Iw!Z?;qCm+#YHt zbn|UuQj%#_+@aoETweUwZQ)69$wrDo=Lx3#cwA8(o_Z^4z|2M+hTDV7Q2cge^CApLp%qPR=>{XYxy3Z5aJO#smoUq@;w6 zui56{n`QF8S1Z)_j7HR?#T-7S(CWALX8qymt#ko70MGF}E`0qh<}7ZQX9?O@C79%s z@n}a2t>PeBaDEv6eLmo7h%&G5A0bir48KU2T|nYfG&lzGrlP*5Bza~C0H zoO*lXzF9uYkB>ixrh^vtKHr-;v;?5{ZS z(K_T!^N2gKR7i7zj}e+Mk-_sC(0`=;7MzK}49d)_F-3BY4yr{2D)bP5ZBMz+9u*Z8 z)W7TYUnxvYO^Lby;ZZwOoySJ+nb?%G>Hp6om;(rCNVFNoJDIq+A|Y5*m^SwJSN8m+ zduVTK8eSdm;d$Zj~-jITkeuT_oES`)7bjrMVU2kOHC}nwo_2G+ZtMjHP ztxr6qEH)FfFfhz)0C<9Ye{dBLGGWD5JhFI3x99XbLO=5TGr5}>*=GQhVc5o=>NmGs zyvs4cT~2BL8Z#e6Jdi53zDsl84X#8)45hEHoJsuf3KVCbjwSK~>5E3NQDA-eG~}b5 z$!{)GKahxi&;c{c*STUe#b}86c92s2i$#RmgP}nSiHp-!R#xg788xDdXrXKIu|T

#Bq9IfrBq1+W5PNdw*k14FyZoMy!g~`MHH_#V~;2rmH!v{X=Z)!YN1+yLS_# z2sALWe3RUFK4wRBx}c&)peEK045V*nWKN_$e(add$eUzB1wQ)dSk0-{sQbqXjdsJ9 zvKPRtnp)>=er_`bY0H)^VE9j)8h>`?hhD#a{c^9cPCOaf^z?XMI!9mkgCn~pdP}zu zT>+yS((pRfIRtP1Q?{ctP1YG10J*sja>aKTrdDHK_XKSlIv8#9ezHRo+%C*4ESGVU zg#1GnrU4ArZqk^*-$=l3C<~bPo$b2$-RyFZ(o%&D8#YXX&(`+zET%Q?!>kDd zicGB-4pK1Ih=R1TvO*f>|3u5k_&Vc2D2ara+CU-gmAQvZpe<7haI}&vVr>2Nef*r)VO^l4+Yo*9$ zn3K?FW@Zvi1KOI6y}ht9OY+sv@xws&)t^6eV`?3yrsgy2;Edq}8Z+tYKcnb>PD@v4d$?2LioAUOu`S6O@1nFG!;evd7siGO{E% zF_9UmAC3ZL4;+fI12bP#WJxu2G{Tqxz1;@s;_T)b`}qzpX?lKM!u`#if`Z*B;xU$T z1v?KMH~@7&u&fxkf*z8^eRSBebIE29IEhA&p%t4?h#8QnJsJohzYI4gCEmU*rp(Ap zd+p}U2)w`s`?`siwg#Q7Y_@>W7xA*V57kQELA=k;q5p{#2>(l#XD`LPf&(|c_1_QKIcFvHlbd#Ewv!qk|OTi$sm zRy5Hfg%{&@Y{oa#{U5yAKlzm0Hx2}?tMb4z!Z-ctOuFr0b;zx~jx6vs5v5O%XvRP> zGUxjgF*{Ei9UWJEwp(TT&dr;ghIdsQpP1i>kBiefdv@myiQ`$<9KE2z210HKHOa3B zK!9b80v||JB?GH+?+~CwVW{tDM5AvUeKiVn+)0BFFg5J#vJM3Dc#XVaM^zxWj0#Oe zq;4ZIVIjDwzrR1jtZHl%6GUR0`bcKXZVwzg#>K;{nzbO>`0gD8>Y$>>_t=*a?czw_ z>EFK_@O%J;+p*Tn$p_H4WI!YcmVqeE8e&R}-pNKs#1aNYufH|K$kboLLE zf((xDVKcg`QiI@<4p{cYzqCEEJmL`;V%tp*)(~BUQ&^y#5@YXMDn7CpP8HeKQ6pX= zuEE~+VQA}hU%n71=?<467i38Yw3z_*Tx$L=ut9KtKq9SIUz^!N zSR1h2c)G)4!y_ZM2=hopjf2B>YssZSoNqpwjBu& z${X{=t~4T5rn}6o*_SQ<;g`-MM)_tBO5_ut|M>Ds0SFdOsVLW~)f!?Lr5i3xO=h9KggLJQyni$Xv~u)>~&#r*d$ zz$G4 zD+J0byo*us8ex$M2~x*C9f6|NZ5=d{#q2AornU_lbOUr|>2n_??grUc@BKVX3eL;R z>(@Wt3q2MTC=RO<6R(k@u$_f=d44=peSL+ML{LS-E^*nEnY{(-FJ>%+U<9>Ct1+{* z{`KqH#zq7m!^y$l{iOM^Z^@v(&YikSd9+mRNAKR5#Xz$I4He&8HGczM% z)n;%)WXBF+G*sY$5ym9lFlQGR8Lx?BP{4;~N8SQA(4bv_1c+aQe1NGw9yAQ9Kyp;w7(o%v4*UtnZi|gk03n1$VwjTm6K*(vqOo7S2R->ReGvEh!Ea7qm+F|F5SRnWV26P%&5Fd{zZrm@PoCN_At>*FA&Kw~On2%NZD`Aj9@D$8B zEIqfkYiQizg8$lvi5h% zxA++CR%(bx@6y%z|AU!0`66@+bMo@+5%AJ7G6yh|Dm2p7V)0ry36;`wGW0Dm%E&Y1 z^PKEE?#U)padBbdc!^6d2MEaJ+qbtO;vy(%7$f7g%*4?o>*}LeWf01k9JO6wu!7KF z>*y$^Or6|;#g;~gnV%ZqH@&AHf@+5#XwWBb?kjOLCFV2v`T6&cJiKwAOXSA2YnO?s z5Qa-x*-Ybdmo8m;@#+<4AEx99Gc%3+^!j8od1N(Ic(L#AZik|qdmNDp0X^BRydfD* z=f6)b{ya*^B8-fTs3MF^&l~0B<-=dSde!nsqZq;xF_-!H@guIzoESgWTPj5iRJggt zu}g+;-oAwoXJb6C(q*{*?rvsGzSUZoTwY%8_N|^4DU>J^{Tss}P0=JG=jVR{Fv>GAPp5Ue~IT>zd;1nFjHpp_ItsYe`93ue}T5T_KPP?myLO zKbf2>ny5=8HLt2K;j~RbY$y|>cF>#yN*I{SOWMj(-};^IB9 z9BJtY$|^#5X5tIc6@hR=z(qoAj*5!P8sg%eJmmRSs%qsj;#9)&1rwqi!R&c`TC@>9 zIBY}H+qbNIBYflI<2_{_J46jrcVZO|pgLS1+e8*OEwGRSq7*i*>H7i_-*LS_da&e^U9>;Sm`?Mz#40adXCrEE`XHd9{w=?j}5F$LVF9>}m~PqBNq zX&B)MG)h;KmX|+z{N%;eH86N`_YjR-R{K^!G3*V|=OK9L0T0+dwk*300t>No_RiTH z6PS3|VC?JJd`fofqeqX-&gFGO&J+ihSly6W*Bvq0|W3yKT|qPQhLV$9rE~ z5C)|xcuE}+skcK|INQc?2|kVj%Uf6OmqRKiIwDY}fSA}pPu`09fG95$@nnF4S|STj zFys|;I4nAb2_+cuAf9ewY5`0`ctq6xPVC84ie&f%F?gnVN4ia2_-5F2nDXip`yW_5 z69)!{?v(&m7!{v=>4j9A8fKFo-`^bo$-^kvJH*I@r@+x20XO*LNA#XEEt}pK!S6xX zGILG+BJX<=b~VUGHmEEpx}o^^F;DK(jv|of{)56Aqu=_hiL)g!eWD9idWpXZ|)@ltS+GOH)4Rm#L6m+ zpw}ZH8bMj;=rq8BP(&#+j~K*bY~J_f3)J3S4lBfLJ(PMgi!AKXTjit3mjMFoAa`K@ z*-%OU$vY6}i1b4>WW^0c;s*AwU4O>9!|?3Na5aQWf(hX|nc9saLPEX;;NOHii6@D` z7%C=S)9f@VHzfCl!m71s>@C=lh39rzX^ZY*0MPwHiEI)}_r)*?CJNVfmV&>@sA3e3 z7!7F0aZ?2a1p)X1g6i#tj7sbm1kR(6Eosru+7g>cjw3sK!A|2O_g@9q#R6nwhq-x@ zFnr!gbU;J_(P&V{^hAV(DIu4i?Rg?tx%^|-KMx3*T5nehFT_w_3@2x?_gLmkLZX4E zPWv@{>f$Y+63qPkQWGjivf5u>n%ed{c55hbup5;eiy_|1!^4B|zp$X-0pTIy5Q3`j zd}|ZLc(Mp&)L1w^2&3(mZ7NKB5n~mP!S{g%Yd?J00;do&R(M~^Kc3dvbp07t3zwll zPj}*v!R_yL`uK=eU0oeuMOZ<>+>|SQks06DAHurFmoz}B>lzlUf^XJbKR!)$6L1ntOeef3$gJ80@cGkblWMXgqEZX>A z+~W52@_#M?2WQ4d8OS(q?7_u!4GbC})ooX%V-8*s$;!^kB2?5zCtuw-L{9qH_F0t1 zc>Ve_08z_I|E&PGPCTKw{c;>CAvypdx8pbo9+n{$<{nU&L^$yv-(;BN>OthM(0rJZ zinc3^?jUd{o$`ZV58Bp%GX!aEa0e5t3@|Y(`zpnorlxAJT1*tEG2KB&gUV;N zz2N+R64b@?BLum@$;rnzK4TAmNPAD)sK7o8kXk&p(_ueyj^LGqdt-6-trP*QZv5jY zwysBvcy=9B1sG8YT;W1-fE7aQQbji*!~t++!nUoYW>o+46wb&r`0C}F6SvbPyvUlZqfvG+I8KahLG zEL}XGhE%mOORu&(mMxSnJp23Uwx8n`s)ok}+9in9``u-}xcsxQ2-^-T;sJbsDJ;u1 zm#OHPczI*-v60A8I}7!bo14;~dI1z92cB3NSYP8r=!iemL0vfrq33d5-VRs+MU?5r z^FhIE+p@F>9LK}kuB6=d-$WKrQaXN0PRhEH0Sqd+3+dBd!Yr&UAR}|Q;N}T|I#0q+ z84-aI(KS<{gFD8_Nt7hSi+FETUT&@+tc~D0rUz9~jhO)~zJExEuh0$4Zw4^RO4vxS zm_<%q;+>Cf|46FMxNvGQh5n&{sc8T`?(~a&IDG~sL@});_wSp13m;zJbDm8F`&bN&sk(? zZrQ6u8XOgob(Q!z4vK3Ge~I4dH-CxF)9< z$|>Q=!Jr~=D9bzzEf5mNgPrWKDSP8Jn>KB##s=y6ukOpjSn;1_AdXv(Ai4*H+VT^^ z$y)9CRm~Vx!i-UmqW#8F7-bj%0$h63Ru)S*18)z(-m&1ITgn3e<@7+*(om)DKGgTs z%Dcmxv10ns#Kgq7&`LGX6lmGk5puzcfWPttbizS`_?%6E;hzJjnkR3vWSYw_>Xc8v3Y#7kNU;P_ul1SK^CYM}0J&&SOK$9ed0 z7h03)z0rYJmlC=gk*O7@U0mYOIE57zk9o4S%MyY{&cP&Ud@7E8VZ()KiSt9{UO zC6$3xscjb~J`ol`yhlKpj+XFI6C@M2u8lz^Qru);nzK8+6kjJN==i=1-!@*?JC%R; z&-}P45yb}iD6%DQb@&6WU?yY-)d%55!om_qJUAqd1)~rmJ#%D~d;f2amDPVB+zv>Q z9Fu(Rq<_y3+-VOv9Hg0Mc(BZ_Ss=m<7#uihD=(@anA?Aovq*!cd@f;-&* zmX04~lNkQdb4cumrZA?MLjmpbmV)cH8+5?0yw`SE_dSHvD~LxS2?+)qBmo+vwf#RssJFfph8Me%^m6Z8RVd?J zS`0t}=Lsn}Id&LJM7EjtT1ovZy8Ha&u3u#r$lG6op369O?u1l4(;J|A5oc2&Pk)dK_k@TwiF3M9-Kn2WGSCnPjteg}8Qz^waLhiOmWm)uu_tPG*zMa65MI&&r) zy^32GZmOI#*bo^2IMCJ(c(R#fB24}}@~{cj+^Bli9jg0RU_vMk?5}XrarBK7nd~r) zbqvGV#37^!c9@Phf`al*hDoQ8N_SpedElQ+vIEU0`imrh4r%~59EIeZivOB;Kw1cg zZPt+Szw;~D^i5l~5XUHi0xUG8?hq|>2>&$PJ+=y`H&N*|{{`1gNJ{7yZ@WHv_7H*; zIv&(*tFhFv-M($Zw{G7iv~x(2@BB*NzI_`C#uHAIK*v%9`49@|i;F*RjO{ej!ygdF zEZnhE=Y6PCi67fykd5H}rW0WYaAem!1lh36Z*On^?yf2iTKHv+Ok}*?7>2L}C*tAF z`qP>sng{>?f8i^b9$Lg!V#p1TFkGOL%5=%qI6Q>U4}sqPv_{Z(ONHbA$~~b{xU0X? z-oVPZZdDId30xlE(O2PT?#bqx0LaevDC@O{TV%hHDgV6G zmoKy6zlUWL2Za0G5sPq_+Cgk5I6gsc4-^s~iTrY>*noarX8HIlZj$`c7J4b6Gl8>7 zDATQefQ6gz4*(Tj<6Pb@PE~Qvoj98T?*;1GrEL$76$t_0P~~7P+XImnw`5PzVJeNK zA`x9%VZc8W*lBtb<6*J!lSx=?%5&$=L1ZqUBm8GLPbVfoq~&>t|Acu0^;Ano1_}6?);G@3%Y-`~8qA>^5(gj( z5*o}w-%rEf>jG+OYiVkjlbJxP#FZ`K22|6tZfcH=HU>e_x zw+GkeJvvfg!45?nmG&|brsLF5$%yKUu|)anRBGZF9|D04(;8-JMD1Wl7r@u8=O3+tNtR2%Sk18NjDjX2({rPFo}lE7<3D`YSloCHDHoWJ}(-a`2`zq$;% zACE^TCSnOG7zQ?p7lp8bqFH)HeJ}g*nSgezM-t3+ceGn$ItI~SnDB%@p3S93!M!E# z!hO#uol!wSfw*q?|Hlc=M^V1GW<8Fg5kn33K65xXa0G&L-*d@ID;%vp{2PGU>}X14 zNlbpe7@@?99J(Q(qGAc2d=1C+kXE*RFD?)#0?@PCclquEpi{p#2iG?oJRE&z&|M)i z(CHIZ>;x({&d7$r`n};pI;FUptaBuRt08p}J}zjwFq|B}-Q%!u?C7Hm8gBzO|7>I*J zGZ+~{*3IdjulzU9Ot^gRd*?&*1e~&FKJ3W;ygIU8Ku!Dp6-5S3c6zb$| zQF=ZOrgt0(!kEa4?K89`#anX@vLc}2 zVKI`gFY=3dfM24kpy_~6>SRVTxx-RQKj%|RpJ~K5~2n~8?zq7 zP1E33%cLRX?hd278Td|uMU0Y=CU`C1O_=u480{g4MA`)Zk4lmVEeD4K_h%Yo zP=+66ar9#-mPqLSu4g7CtKiHfD(9X#F}GDbHr*es4^ClIH5i5I#u zJZ-+<<_6b%NLm`7q;;i)GO=Xi85mD-0IqP!*j<4-IM2Zfktj7amBmGZ(EVOC`GSeP z@L!T`NmoyRl%D*`>fjS_ymP8n9?_8u?|~er*MEJn);Bw|);RNr6wkWWAs==_rg|bS zJlgo>FD<{!{A&X0ua1aQ$$idnnP|8{zVJ!=tjGa+li8Sd<8vnVDd)o`?nz{Sj^G#;XZIQvWBMH?L4b!e0nct+>NvE63+TQiJyVUaM>bqwDQJ)qM40H^@ z%iHoJn>Ln9aL{wajAZh`K|#A|)2(L;^2%10mb}Ty$tBO9e}nke>|kNMBzE^(nMwIy zb#ra{KjJ&dJFM7)n8z51U}W^1Cp;YU|bxzmvL~`YL~w z|C$Pl2&J(d(r#RSZ zDR=KCD!m8HN!tSE4`+8UCq}saOyq_1g&Ns_<4{|9c{|H3n3*qFmz8CIprKBUxog|s zza_zAU1NKUjqNq87b4ye=RBcuXy*wFnGP-t{6N!>`Q-c^=gbw9feDz0Vs~q+s0!{p zyHYo>UQ_7maxr#ZWhrOKm+uwd`Ti?C$3_=ejoz&mz010LcN;olBk1<=GWXoJ@yI5& zvVg=njC!8}yGUBTu<<0S-|J2E9ukbw&feSclwV$D2ZYDm`C6rLUXepkSMPG^K=`mj zWK>AE^`-N#T7N52Qc~gw5l%Z-!vaX`#=V+?yoVHE#n<65eDL6b_s@dTSp1&suy(|J z-;K-LIJkeP1h@!MQpS&;>wT^ioqm<6(WE^p%BE<)$K2SpHv3-5VYovjtSeJt{n(lo z9jTwx%c(>{#^yYD5RH+-47wo?b}6!Hg^n?Ne63aCu;AWurUM$6=NA6<(qCUH9k3az zw8mT)Gs*}Y>)}aFj$GL$?j8)0);}OXOiJoZ>H`6H?$56Y}tR}1dof0OJ{dCD?h)HvGLVJ$A52J zd}}>8uzqjKdVNDRFDH%PAMVuak?*cxfdc{puHU@b(bc6j{p?)*jTg8)kk6mXpVuNYEvK*R_lzDT*Xs@pq|9k3Sv8oJmqnhrGq#PD z_&n+hf5)AeHL3L9r?1rqcP}=4r!FtQ?=vovSNUg`q~vL&r^n6V?fw392IAu43_h+% zQRc+3re86sd3Sy3tgWrje{xR!Qkw8HF|n`s*o!koZmP1^ivKQLcze38yZa^xQoLV) zsVVF2+qa2I>9&%kx2-A(w$)6j>DbFMK+a^cvg`r-v^ScAVgq z6@Lu*UVzr5t^{GF$z;p8-hmyB&hkVcXlYTE*4Bxo+qn2cgofMtICB3*02LKg`NHRO zsD7mEZc$=p@BF!hHu&8;ec~G_8tlHNUFy#)FYgn3jr$|rfABy^Oe_jcN2Lbt8@F#C zJbzv?G&Gb~*)tN8fWei_esEPd*V7 zIaRkH7nj|HQRIB-+{asX&HzIoHU_`lfP)d9a_mZ)?@sT*_yJ!bL1|iL{3D&_t6 zl7h#N;~^-{zS&h6TR*$7@EUV`4EJe=dVYDqF=piDwa_G=)Lj4c)LZYVia&&cgwuLh zm)H{{yH!+Dq0Kq+7ZVTEWS1j!CS24)Sz*_%7))jp&pE;t2u@Jb(UDYLT}^9`5n2Qa z0vvbJPhH&Hu7Zut{&-mU6k}iFm6MaFaR~OP`5QuDWInEMHvH~tms4(je$@T@JFANn zi44XQ@{i;6E9#FDh8#GmMRJ3Z1(#SIHUbjXu7i5`ywN9HMjg6)=yUJaPO(i} zB4h?QNjrCirU!-ZJ|Q6O=6`~GkF3?g09#8uM3HzlJ{*l3g-y&b0AFRJtEX36R~LHi z8Vd;u5gEP&5|G$NZKHCpS7|_%tvx-unCznv^GAU@GE`&ZRamMXJ&FZ&d;jG?^t+HB zKb)#Bob1Ixh8B+xsx+d;KM=zxvsL+F7x8_swSaRPmN}djy(4As9lQg?(D2bkmVN?G<0`gdNb1> z@aH6yep-6^l*551Po7j+o$oL^a>VwTMUeTUsM!>;29H8-*ynCK0fh?YTj5@W-8_Rxz~2d z{j5t^&|Tc0X=g2wMNSGoC!0x%b*0O?>9?@Gy@4 ze&73YR@$DE>sA!bE5&b9RnFd^Q?Co4Ro~D+ff9EpFi;Ku$Gguf7U%Z1`3y5-6ujf4 zk1@Hv)#ISw-?6!mM{zE8CO|{OrjfC6?JASbg%anju_qPBZdzA6QEl0xlm7PR*XoeA zg~F-_P0Y82*p-SQ@B27#fAar3UAJYkk9FcFH=MekotqQey_<^g#0d!shQQ7(D=T}} z)in8x(_1%8@ptGh~N<2U^B6^V^!GDlsAMWamH*oGExv_pE$P( z7l@#u;w|(6wZ%co%GHGvB=?BQy`ElPPPUtweKM#nNEY?#&~q9YZKC5IXdD=rQP^HD zb7zm@CV=&`_p?_&1l1-PDL?=8%WdwS{maii1hO$$6N zE-p~Au*PkE48d3{V4g-GuEqkDokFtjracZG9>2rOe0#SQM+{X=!)$B(V=uvyZ@}m8 ztyMaDczV$+BDwTC@Hxbl4I48k`8Jak{@f> HT>gImZ*691 literal 20784 zcmcJ%c|6ta`Zm0zRD{S>8A66op^RmSWynmXB$3KcA`}@yB|{-egE3R4Ohr8^Orvq#^3ueg`~j?ypZ z?o2uNCZ&do@{7fY7P~3t zrR|9m*0&Fe2mi8+`c<5nR8gTs>Hd&+KxgBHP*219NFExBRunIL+=bAc$u>v58k?Fd zMyl`C``{lcesYY4g+oHgkM_F4m;3_cC-2$6epO5#&;9o|En+9!L$&*agoO{kye2d= z9xNBL9@zD~O&m6cu8;`mWuC!+N z4abl;`G2lbx6P0frYKl6^qr;STb9xPv4oO|)= z73Cu+4eRuV?Jm|WxlKE|^>mhDszejfkStl)1X}l#;UP(*J%3dlen$-1b zeD&&laBy%7-7zIxbu3%tM?_F!BG1Xr^3{|_XMY|W7AZOObb|A%(5up?M?HQv|9 zVHsmK`dVFA`H`<*gINSMvlKa4oVfny4vsiDggk#vW#i!JO%7OmI$BV*O_7zwY17|V zRFQkcCvDQZB}s#I4~dFuViYU^J*z;+bF6f$NffUVg|c_=-kJUw!{wR$0{XT8 z`SjfW*Q;5KPfu8?rTcngImnL_J3@*4q?>*0UrT-yZElBKB|lcd|M!d59FboCTU&36)mb6387tJHjHGC~SxJd!ZhBnHz`)@4 z7&GtgYUBa8v38l6KYyORdZlBeM%Ho++poQXo}RwYwM)5wU_d~si1+t*e+G;Eawdvn zbMH2vDS3vKAv?cyRq-wS`PC<Y19q z^&{Jm!os9SXCLsoySuyfJX_IxDhvz%H*2#$Iz7rkv2Dxu!mF3QG%+yzDmvb_ixT+! z`SZ9dn&gpx;Y?+#z7(Jh3;V$S-DC8|>D&I~{n`rTF9NN%6--r@M+=iv=YFb(Ce_(Q zlFa zS-O#^+rppI-#(WzXBsAJy1Pp!?LK>+j&VcxSZSAL>B$ZX#i1!lW4tp!fWPO$V19#? zq^KJHYZ+yO+)3;2@<^I9LupZk_VrBfEANPLN!n7*{TUW%mLiADCiLC=P#d)rqway% z@gEE?@sdHJV$VB6mD=C3$+pw0z!F>|(vt0A zs8Z@X{XL_}r8#xQ@A0mJ)~@YjRqs#6bEi6pve9oA7Va&mT71@=rgaKOr~m8MJt;AYWwf-qd|Fhh?XQvdZl0dce~*2Z@}96wPfs^}q$DZ1HqWZ$*`HsdEZp2z?%dgQ z&!*yWBbV+CiKjVcxf4H|RMO0H%`^5Vo%>#X_NPwr=aSx|>v%p3nXqj80CD`u6QaYr!EtC7&tP!8eIL{r$_8DqaQ5IvqV4jCJYx z@wGu)Pfy~YMxJ?|m19f#YO7*L8g=y=`s9F-)-5}oTeGGnJ1cy3m%jJ)U1%>kUgb77 zUbPhYF4M54u8z*W?#iC&?;Xh-1py6GQc@h>-PON6W~kEE)y0>Tltc*xYUr67eiysQ zL(*hewQz`Mq*{uqi3uy~x^<@(=g(S~d$VY0Xn4=hNFkmlKOAuFshIA(keMlRFHhWN z*<suS%Dvav*yvu5eXR|op)K?rZs9x`FwsqO;=~CG1#7BqW!3IcaXWpzbk4jLIik+2@!Tywh5`FL$bD3u z@TNTXy%VA+p@9eoC z>F|cz^ZS?C;LoG2*%xq@QtWCjY3u5Wm)hOOr{COmP%>cAk9)WG+CzSmE3U=FH0+j> zktvxfb8>PzdUqI0!jdhBeLJ1x;>^m*YU|__W>wkhSkKJ2jCHl`Q6a_EIP+X$%AEIK@<&+M z?Aw=Vz_Zh}lRILA%(K4ws8fHYoIZa3oQ7Y|48*%mv>z|XURZEaUYxP#R`guOqv%PY ztl?2)qrA<~z1iZA`fu9v`|8!Tet(8D3m-nb^yJB|bMup0l%w&tS1@qczPQToy|ggh zj3B&ssEU{JYIob`&l^WSdy8cVSDz$bheL_V(oDfn^V+R7CXJ8b9j^R6mi&m zXAkYn%uIK~_9`k8r1R$F`zQv7aYGrWckI}a{5dl#E9=>qu(0sQ$ES~%t=C=L601|> z^zQE9@9d$nrG@bTsbh_|q-7}AW%Z|688|2b^ECYf94g9wT&oyYBw1-hAxG!Fe_Qsp z`r}8gy1KgU9a7@ry4lHo4|qq-Zm6YH2QzT$aMFv;-mQE2a&={8rQ?Tt>&bydX%g7E zlkL}Nn-)1T?)};y+uL(?C+Cdp-@o4u0j85{ajig+aqr$FheHL52Gi>_gkySoYC>MT zc~b`ncQq!4w#=I8wjN!$31Oqyw}L5tE-D#W$vV*f>cpb z>OWV-$YeA>?eh70SHEb$a9$W+|q(qW0KNFb93{d z_7d^bv^27a&D(>DZ=uQ~ipkyg_a8~Wd$$)^VrDQI5&4uB5%aXAg|*~(+oiIyv$yLq zu&`g^gqoTf8q^I8m{1k3-o70+HRbu@tNX26x2oIQHy}Q<4pk|C`}WP&!C~dfl_B)w z4@faQiL79JD>^9dkwC0I(`t)m85vac4E>0tBwk>LtJkib&pS|hxU14%Qc_a2E&m|P z`t_@K?b_A6{X(&kv$NQ?LzP!9UR=I;^XBflh>aAAudnaUlV7+PH_Bd&iCK5T#l`OE zQ8S;YRQir5D;wjd$3-mf-p$(K%vQPhXOvTU>-O!;+1c6JIy!17k7#W|{O5l2Z`rc? z&!0bz?ZsOTly+pKaaBc&)s=moF}E z8J~apl+>qwqljF+eLtt9Ys<>YK0a_@CK&%f>B$uQ|CXF9Z$d)C@=E#^S@$yTfTabw zH%UsDA2>Ai_SRmOl#|mUdz51)PJ)ZvfS#V7jsMlXF@cXB1#EvfHAy3E;5YmZzwPes zreGBZ&)tZRKhSGtPMn=L(U-I;6&_4RG&@9cDU)>Y%3YDe#9e}E+ zLLpoB-BqQhEc`K>wYD@n;jaBPr@i(1?m_g2A1i$23=-sn5K;EnY&S2jkRY0+2SQ&- z;xtHBj(GG)o@DjN$Vij}M!Gdzfwy-!JZo&!E2GuPsQ%ue`jXKs)~$(u-@bj19zCkA ztHdBdvQ1Y$_yO z+IN06WD!X8ob1&gm>0{fj*TD>L$bo%yLVI5RyzPX*i`tWSQOZZii)b9I3aOntUb8Q zWAGXJTtG3+%N$#&^+4h>9GsjXLqm>a?P3g-gGx(Ft^AGZghOmb%<{9(Hp^`fSopsT!Bx-rK*!tZGdQ&G7K>41j2%@AOH_{rgi;T)AYN*)|SxUe(E% zn4AoXieg4jb?Vzkse9ICtSG1xzkX?;k+XGi*{TS1oJ~_^9f|d|Lm4EEDHh=Jjjad5 zBh9wSDoiCD_iPtIr{(VQCZRQlFgu~AU-N+8Ic#h=Mb)g2uN^OQ3L1;DphT84H7 zEb(lVJ4ppeA_=7AZZU5va>6~!3X8AjY{LtUT6f=!^S31O}W)(t~=t~>DX#M5WN43o)A^DA+zdsebhsfwFaRQR5p z8GDAuBPZ6j=UMOuk9VhK6%?}`GpsPh36LW@ov`~XE2>Utc(_`cX2kU=`p_hcs&oFb z@?v)~GefY6DR?vhk6Lo3XlUu~=-%8$I%4_Qh)`{YuLA?UxQnMi>u*6I;cz>BIjnTT z-81k$2YuERL_MkJBxTVtafb#5Qd)D&bXOOg52ZdRDXE>A?341Far^eAY#j=gX1uJ+ z#3zU3N%b`X0$Yt*zL15)0nqgHEUuH06Z+Wi$bWP@+k`ZdAlU+Br0Dqg%TF(?#5%5@ zzj#wy_jX*|;kS48kfY7Sqj=G^YJua-nKN9nC%8Z>Y1Cs6#9lQt)L==-qkE2b@@$kn zQH|b0Skb(s@MZI|981wV(qO}i(E3rew9*{L<Gh3 zf|rT)y^In02vP-wa}rxDC@EPu^=5>c?hAlfUQy8lJ_86#P*il)@87?v5#V!jdDsDgt=itfx_ii zw&wF}WDsSm9W1t~2+Yt8I^IU{UuelJd}&N8Q(>ffl%0d)F#4m=$Vg2T?;^)$0a-b! zjNK>c?kD4cI~)SbJaRt_&37%a%Wa8{4vc0gJkBY3(t5J#N<`3wl|hQ918YKp_CK&G z50F{U#bw)F9W>POM2;+^q4Ml2U|cmLDRT|_4wC+kA*<>lq~`;$YH|7lAb4zsT@C_a8OlBnDv zXw2+>fBS@vZ!g%_>j8ZCEEfKuQ%~FES=A1T1?AwZz-fs7B;r!r9TF! z^o{jCBm1M0Ux|MjOy-o^ksQGEv3>E$!9z>4>*^nHhKsSOzN}&nOP`s1@bCo9?{8Zc zTaVXA-W)I3EGsK}OC>R>th|cES)#Rp1uY+*SdUgs^4z@lqbE<= zzI4Zka0bNhIP3mtOjENdxk^u8{7`AR$L9xwxemM@kK^dG8q{daz;j>3Vci6PK++%9 zW2P=vJN?Io$W(z%f@$9#IvBE^IaB>QzBZuh?9VR4G97MtieADhZYBC#w=4Vt*xful zYLLe#L7((IXi}~L+1x!kn)?IHQt|$iC%2uO=~)Sin@j!Dny28CQ!+Cxe(u?Pt=Dyb z67MdRrP-erCr^GouI!ud?k1`%A}q||s;lSs-N!^$LFdj%1h2-iW5NQ0f~RIC?AGup za|7lE6I8c}HUHc~?`)&)jd@>vgQJx%CnqS^t_2#Wg%3(3E1K=rKz6CFqnnwi=`Cx> zHD_1h-SvBR;dmgR3LP`=m2C$fZ`!W3TZ@;0`2fBToqdu_8y7!qdk;9d|$BlAOG3 zLV}jI`$cr`QP~hi@x>#fuVQ=GU5RFq^#4;e_aVn@aZ%<4YA#rm=V-7d-&-8ALzy8& z*Nnb2i?Pu2+FHBG-dckA%?oTI&}oN8E}#Fw;P&IlCzrefoJ(g}_Py&3ULM%{lYL^m zB%6+|mSNy}Laxq@%bg{;-p1#*?0c28okp82w*0W=yL=X-V-=h-GH>3ziG6%mGttIX zPjAh$ni|q(=_SaspmLk%`rXSs`|I11S7F)d(HkvwS>d39XegRjxcHCwnMB^ZCm~_| zXzlcPS3SbOth9Vfb{?|#GD;+`lGqVHUR~{dUml-k0F|`o+&N`oy<646)~}wry1z>K z4`Lqi(bqrvrOabxPfrinpy$uS*7XbzZvlyJyoG^*p)pQ|fj~`C!@*chjl9>zvI>+$ z$HoR{Wr-#zcn~5e=;qCvL)y-l^_g~hqy*-8|L9*$63y#)InpMOkV-en9NfAJA)88F1EfPerS^#xYN+0V`W!OL<+aECMC zL$DpRC%!z2ey0+uEqE(F-hQC*_Do-tillR^=oOw_9G#t=srD=^?;oLUIt(b}cIHeS z07Tihb3((^m^4P5b~ALLKn9mkd}}~e9enJ`!h^E%ig0@6vjg$KEB{v$v`m6lvk{$zkpjE8N(#4Y1k#+6F?VkC|M*z1z>g^YQ8!oJMfRjEe z2i?-e;9$bektgo%?G?agM4M=DX7yW~9VC6#UA^1q!K2!v)a;&I7j3Qp%}-5D?QQ(HG^`u;wF zMm#i9gKE0<_U+r2_m8YT+3`3S?NAz6+EhWO-4TxNo=(`;SD?JluV8QkXLj-8#p+L= z_Im>A$AYEg;MiszitX;ZyuGp-hvIFq)4t}S(9j$`u)yGXH4vF#moI|d1h5K=%wW&y z=x6)i`FytIKtLco?++ba_R(H7nu1>JI@u0@p-F7=v({Eqd|>JZ{%CDUs5-?bKC#l! z(9DeF7Q6)~N+keta00UbSY;b1+?!C%?!KjgDR>OO{ezz8hpkH@f3n z(nskFQp}?*PP+e>tT4G(;J0%7W${@H3T5jRt2-fQndSTCZ9x_fwG|w??dtgm+cXFE zgz(_E;^KPouLI*@p`J-7((+ORhio7&o{}-|E(8b zbem%2S(=ydJoaB6otmjynp3&LCAIA7)2EJ~?q`sorGz`EY_%!pf|8=Dt-Xq1Y^@fb zf-n6fw@?a4oAad^4Z_se4!5Npyk+F@SBwBLw2f^d99WHHO7s@|Ai2QSe0X{MDingc zni^VCY?b{LHKTYLw{PFx9384%zW|_YS>Yp7IsZd<>f5In-LAH)SWO*{_1Y5J_uP^X z7@njNwF%v{dv|$b(~NM1d&|3bQfORB^1zO7uc!Z^~H>;tbSsH4d$mecc+pM8#jR z=5w0`a>Rnwf*{b6JG7oP-b8C}->1_dFAeFSaY`CFHa#=5X3t1q?yN_^& zc3oN*t+_0r21=BDcbK&~bvNDwiY zAB{E&)#3S>Nv_@Al1RrjbXtYjGl(&SE`HklmWAdtZTE@(HG*L*S2k<9NuAlF+Ch=m z;jue*jLyMf(Ea%QkB04Tr%!M7FZUX^DgX810NDHUmoGC!aw{ym<85I7FDVBHUXVyR zR)kxm3Y`7E?&~#|rjrF($3Qw1LV~tfC^6MVBXZXuF`VRo+rNWm}P%Dyp_^`l-fROl=?9yV zwa`ldkxp6X5$&QX$XJx^E4kT!c*UbsLCZPIViilvGUsyo3-=Fw#`@K(D#)9Bt9=HF> z6#&bL@>oyUkL^|S*V^RPvSe$=2Tr=bDuGNBzQgO z7FwwergO5nAl}ej9=)5`QGRwEwnuD1_^tqo3q*l)Cw;-q5^4~cbQy(rw>M?u#*LJ~ zyu7U~EiE}4pFE2riViFxC8HJ}zBb(K!Lx%^ONC`dMs^!ZbpHLKqCH2BY+kl(+0~?^ z$Q1PucrK_L`b)dcspBqylryh0ckY$3Is3T%vb>_Anpms=KYvf~-24248v&KJKiDcP zOgHwqwEp7ib#Fm>Q&bZbPrHHAq0}-OosBRyGc(<>b!%ST-SFDl+NVuT%s?sj7u2by z7o6Yc1mlh%tmdv_W*(FtqE4A025;?jVkIQO`&^3$PmMjqoZKm(gBCJ8l3+SGbSxtt zfE4&_qw*1^`gkF<{mp3vYfg7?7(N^BSu`qs(j9HA-(*EnvKy?r;$&4J^Z5o&M zhK4Q&>qRgWk!Vqs)wkWej0bie8EQ@=ya7b6E8~@O2HB-YH@!F7AJaEBMr1JLM6*Xt zX!Wmv9*kAy1;0TgEWq;X`Jp#rVlF_Zv3}%6^WnpXNA6$g2bz+Z?|iFJ&@s0V|Fn3& z%EI*L+Z1)KosL&o1J*U+IrvD4uf5Ef20l!T2W_m_T`)%=Ich-n;~_L)T|li&hnVcX z7D6T#*#7IQ*z%%GI^+E@_AV}r2M!!~USDtVs0kUy?faLqmUBn<0xYH*7Na2L9yKO)55q8OnlLaPtt_=LlY>r5l$&F zR}h)reGzgKujRV#+_|Ib$qhz>NIwqA{+9?zf^IwOKxzJvw&@D7cx{h^WS zfUMvRW`BGoa06wLh+_GExk0Aw{odZ*-KYZS)N{Bs1&IWhYf+$yBm0-);TK4CpBnqj z30XEUIhk)Gj{uJ+Y*9otB~-y@<_7P$`j|6&KY~@?R|h;Xg!~O z{(RuQdUnRGTU-|}T@u*5IlVE$5Czj86!2>BKNCNH*0f~k8t0mCCDmBN-NQr5wR1Nt zHiDpokZe)~qgk`2;<+iLBSqHt{PC{v=U|f{umrjX$1jhHhUBSK(-la(H9#?u(9f+z zn(U~LyKz}Uuz|ZT@}4{_D{n>psvnt>!r%VTwr#s})U>mo5_t1wo387c9DQUF z(xK1H&DFeqt&gK1kV6NG&Tpz&i%aF)Zcf2XEz6L=p5KzPpP1-TCzsQY+leN?^p0j6 zH=I>RjvUz}B$Oe1X`PO6j2+xhglGg4Cyau}vgIibXjG;D++n1AXe#89r^12Bh;&Cj z!`fQ4;k|Z@V6-;kXC-*o`?%0un6;2U&*$cfqc|2V|D~B z{C-5PAzTA#Nt&azYyaCr^g*1t^{)O#oJ~ScW_a-C?km`xItm?~UqoajLVfFWS?W)C zYO-^Z#7;Kqrbqhx8d(8WlaTmOkS>77K$uPu*8G;N%Z5(BpZfRUzoMfPJ7j&UHSqs* z=-8ay#L3CYwQ*w~It~(JIR_0t&E_0%C5%}_L@n0k)j?-dPh}YB%s%wh#--~j&vers zId)6{#s&yFel^teI)kL$VaOddcvN@;Qu~Y;bE1cxg}!ujl(^HJC$iaaO9oNE#xqo@ z>9km*HTYD$q}}WBU|5;v9YB(hFAGn05F1Y6)8;)gh%;YbL0MVzT!U2#3JOG^mOT23 zOQ?=c2+um&RU?Qolvz|73d4|@AQqqjBLw@=EcWplG}Xz}^%>@CXk z6YBH7+71yh7vQem?4EWWY+vxy34$&HO|q6JWU>DF^W^}6b%4)CMn)8zWnNv+YNRzO z7!1%Vrg#j#N$tC8E7RHOgUpked@g4y{N#ujKR*phz0!yLgAInOL$5#o@`ZcPo;?Id zqP>X#oT{m<)$o)^v@(@1r`Q`eOkfBW`R$l9!>N6LxN#9=6b#87eC*h6$;3El;raDA$G!M2JE)ar{`Ag2;?LcH+bp zVm$z)c;q*Ge@I@j2?9OBB>2ID9e>pVes+1cZVGu?U^iL$DQs*Mwo2lt84JUDcq)Mo%2$h|qN;^N}yh;4zzp@im>Pr+jUMQT^Cf%78t z!7i`w&vEF8z5~4CtgRhm@94M+ryS5Nlxs0o>Fkf?{QUeA)6-h$G;!4yoyEmP(*9DZ z%L%0Y^2HiyIVT-9lUuvIwh8OUT?}F73y)M!H$w@KI`fT`e7jHgkNT--va`1IQ&eFO zl{pz(0`jrCvonb^mi6y?lKlRcN$7Yi1F9wKRQ+lJ3bFs76N}gzBfLjACM+`Tfla`| z>0-xbHP1CB*DiBzKRh$hGxX_!2oPgRLBV!K)@&)ehVvzk&C9^rsO#v^Z`kR$NlY^X zxt{@Op&r#qRFOXVgFEa_pj&kKLM~mR*HbrDLi}qR8rG-pjVT1vUdg{Mu_(0|;1^zSuh5z|jGvVD`Fz@_A~b39I$N{#)HYq1?XdZC?& zkH(4>E69p~iY`3%dV{RIsnfGKZpy@uAJ4+piQF*~rPg_8L%B)2bB6-mj0Gpf0Uqe! zeebo!ukfkxl45XIKc1@9ycb^Oe7L3v{RUiw%3kwVHDYyIo1z~HNfzA!2w+@~wEYrR z#jApXg5JJc_h(d~Ws8VGLidPsI{Vbrlq7sdL_%E2#J!eBYY@)f+k`iU2sZV-x1bT8FqlGD01aPqEy9nNB&4apOp%s*ZM?Y)+LvY$LwC;zaW3m zfRo@?WQuMfUJ95!kx!_5_5=;dTSe8gz9v!~7F__f2nHvF2rC{lp-?D2c>o|}orpm< z%zCUm8Ld$K+=kxr1GuRps1o;?nqHikaD(x#b10%X62>7CfZ(5qQc?avHJCWb2?Muk zl`1-qwx1-Wn+DUwh7iRsQjbj$yU-_WUoc3 z1_X&U!887v2BAf6i|{>PzI>t9^KU&2$Nw7i^*6uYij7@?o<4(d-ct#GB7&A-4TDQd zy-G&S+5n1KAdG_#fu2R(B4ai?2RHjVl+41=NXC6Wx0{XJSG_O=0n+hwYD)6J%!OE0ePMc|Wio>>xCs0q}k5zWORl zpJcgEC@BKM!kUhe$>;tM^V58CBs_Q*FI=G2jN}djsgm_|j4kxKkdV-dnoEr8$XldH z{-xNcR(dY~1T(9Ty%G2s(pP{6ekQ$^CHgy10|R^!MQ%N62piL0aCcn0&PZHh=8J?) z0XRTWHSfPLoyLn9cnLI+Xow`sus%67)U$1(r)Cm35c2dsoWu-8RD;Te-vrf^bI3!e!jMl=Oh?C@V6hQpe z4hJUqWrIoLuB+4Z+}tAN+}8h}RyIYmz60JNAGk%*>|u{lYCs7jodBu?eVO$(MJD4| zb~d(6z+XN-K7=>is-V59g!DZf9Tl3Kn}`(%3c)Le9%P!W@WsImM1$rOnK4U;P-lR_(WRCDV#ic^3kb2Iuv{}6BSum4zMheGBRd+{jruurR3!5!9z=d zNRBo9Dx4FlCtQR5F@AE!1;1Wtk`4Mi)jj+4f5)WIFME6Yp6}oDuMzkCs}CUVi8B%~ zA~o$Q(8;PS6Cj+pV)+f z0-*qUY6Lf}oAkjD#Lgm!zDCY20y&m1e<_g6fF8=$)fMQ&sSGXk;XhNugeW0B3?Qah zEzFmXOd3SsQhNIAnVJ)=MMhFc$jaWnK4Lk|Xr&E=?Cp@?yH(U z07Hivog+Ah_9%ws+s>N~pu4=K;2}KlCh;HVkh-apv&uon8LwAy(kfe(y}on9Zk1N- zgGH0ch7Vz4Vxl^nT^yap*3+nKq~L;sw!NUfK}#@>TT4qzJHrG{MA5us*n6a`SaVoL zv+nxOe!WRH6bAlniu6vm5i`JdjKs)G02K*kCu$5WF?;JA2<(On<^_zD#KL2O_iHal z^bm;-xyE~<8$~!>2ws+8bPECqDD?F>PD8PddjCxdJ^6r6j2H=Rv{cb%HzALck;bfUZa*~e#RS6^8z8rVUSxrva5HW`3u~^I ztjt5?$ba&G-Md@DD=9eXvC>D!pu@rR3Lr$B&mo}5Az#tXpqtjv;%5}R*&qc&HPm1` ztm|7S{D_$LojDMlC#vBu!%$t0nFv3B5Sg%XVu#}%MglvKNU)(HJa=By6TSeOA2&BQ zkx_<*omvm5r_wvM2E6=F(0&nb!_0<;xjDy3TS39|BiOm#3K%$XzE@&206WKcH9ns7 zZDbl@1{s|cHL(b1_rMXu{knR37cp$2lD!GTc&&gRKwCZ74@AiZ->D%MbOIsBM0U_S zk#M#o)>XXo`iCWf-*r}}$xufJ!zzi__K~Mw8Ad&iZUfA^fRfTdPi_-+Bo8uJKzxV@ zpvwUDtQ*?;(a)p;8Ha|2oy{2RD)S&SSu9Q^=;-YLoXF9?8GBcn5FVhV{BV@u)XhZ= z5=%R!&qVP)d$!`o^%685-Doi_+>s&#ezgG0wB}kcV_FU^Q!5q7o!mO|Z4KxluqClA zFNEtqeZmc`*WSA~`-ca-nq@;27M5YqrNZ6B1_m>lpd$V18wm+(R8&;_=Eh1v2(9A@ z(f+#g{(sfV{a*(=pLj7t=MmG^fS=c<@%3w#wQB?a&0(sa-z+3#*ZVvSvZ@+tGU=pP zHXzk68SKv;vF-!#xq~GY)A{@6&U14v=4#HhliC64}yp z7MlWs&+hQy)MsR`M~=eFL0Ua%LK<2r8%P^RZk@0WRu4~XL8wKve8h1DDUn-D*`ujR zckkXk(_5%9+6y?B{3xeT6*P^Fqle^8X^~W3Wig{+k1Sj6LN`pN8~@Rr5@KUBTA$_~Fcgb@24}RCloY07 z&N@|qXMi+l?qhx0w*rFGh*i{d&!(0&=woXEUqIo%9M-iOA)_C#QIfuDTQODf^?$tR z#-F;*i{>W5{sl`h11IOrn;-UxvK1B=-)ns|@`t3_)}kb;fU%W}fC-PkzrWsX=@lRq zYNxvbm}zNg!9j%qFN6~pi-5rScR%+II!$@XP)N>0Y>NGIF@+6P#(wrd_%xcC!P@(FN;#Au7{Wkpv* zxK+!?o?Gtak&|J;!Rv z)Xs9|P0jm?Q(vA!EP_8FvG{BO&OcBNmg?E~uiVm*C*HSrgn3vsukRfk`qM^2vF z1WCUB?c256wrwL3*mDkC6f#pm%HKU5BT_ezMvkYwiwSdm6kk0Qd%{d3x(pFVC28!4 zLBOcR3^_V8Q8w!p;h~{J0mP?-7{KoTHJ?CdmfyS`I(9n$qMsxnm9|=*pEf z>FMdAk!lDWDV)#MASO?Amhbmu%ehFqlBpYcQwxl;g|VOyc?*1Vik?G(P`S(>MO9WR zZ!d&0?^B1|Rs^_(VW#j%we4#$%S#-3Jv}|}lY>mr&U4a7aj~(nkw4W-4fZ-Dw{z!n zT*FzO9n0zihKBSwkOmI*C|rLIdn6Z9Na5jHdgw}8EE&q( z1{vpz=u>d_O~f_*Dps7h>M(^s*b0{7oR>5dFRy8O&P|xa)x@VWir~5}YruDqaWODc zB8v12O@{&&f(r{JJBMPj!i)1?PX7L_iGiQ8woWile`g(Y0zsjHwVD7^92TiYA+~3z za>S!!EgbjvO+}bOfwF{XO`f}rSCirCR8C@C4+~oj_q~K-xWN;@S!n{d@L3oZCE`Iu z+G&zMeSB{0I{F$A{nMvU%l;0}MS>sU7Zalg-LLaX9PbXP4c6by+f5)o?V_h(l~NlX};0RGt-XD+Lt zI=k-!8IDLyWJD_;u; z)iHv%y>W7fq$F|6){VL&K*XSdumE;`EOv?+61Ccj9u50MT9kjXX0*BFzb%PxFtVb< zWBvXYQu3+(7c8D5t=XOjh`9E1K!2xG3)P#j#4g$OyVBP=6~HyvO=1YVX9xzX!Xwj8 zV{nR4xXYI>zk_{59d%-aT5b>axyrL1xX}`~@6sKY)doC@U|0{dw6)`yy*={h@0KE& z$zS(*N5gVNqP4G^$A`9FHWjM{{4GLOkT~#A&yzdv$56B2% zDtHejdZBS>5wqHLkWq9|SVT~e7FGBXkMx9$&-=LS16dD8Xkr< zrW|AunKvVbfbO0i!i#)p9ud|Oh`xi|EDuaM{1CXO53|WEw-Eo@KlOh6R3CeKA;ry& z@@*^#hEx$lYJH0~p|B8YgbzmcB=tYG_%Xr+&In9rq)x@rKt20Ad2D97ON>5`Px)eh zvtT#=nXucm)6`QUVxN&wAPO`wMW<%jS7@gWNUY_Cb1cvz174r_ z>Mt?zh!K+SUD&@nL{Tv@HE>omcN!Nw@G(O|i$D3QTc^6}@7}G};tOj?vm^Kt4tug{ z;AMuFCpx+#!`mG{ei^8f5(rKJnN%i+dc=< zmgxUvstF$apARo<%O}oOz5Y*QShLZ!JP2Y0kyokdXC<~?UV5v`1NjcrUNt(XPu*+Z zOD4{$yn$i~?U9W6fN#^aI6#K8O&)?`>_Z8wBg_m^Ggk0T^KIdkXc?@~w~**Ukm3w5 z)YjLNK7+7K=s;22TJ|cw?|)QQMg}$ztddevPciwg0eKx3d3U^#V0^OXM8lM)D(Ys& zajUDV7vqZ-01?P28R??H4&&;&098Hlu9bVBNg@xE+c0>Sjg5qQNh}3~kphC};2~cz zQLGA>Luf_fq5rFbg=h~BTgUW$`*sBth*;iXWk99CYj;|oIU|c3TnWrq{`1W)geQw< z;lJ@7(Uz;07B{+=Srb#3K#P%K((b3I6H>O-6bRb}NUMMN5|6tF_HH>IusAD%*dl|q z|11eDNOEX1v^Y97kj2pVuqbJW{QLA`1W`~~*#=j@bycDLT|i3ztE17gDAlM6$MCDD zRGI8T+qV$lJI%N6bVdEC`Lq3l@~>Fd>{-Wh(BlJj+iI!i_rhXl24C=T9#a``E@^Mg zKIoNUdD*P-#9sa8k5&&Bn_u6(aa)Ec%Mxqj%} zY%u6h369H#Uo&x)YrUtJ7DGe%6uj4dJ7Zb2%*4cp*GnF;udREb%p>CA1BaBCZT$VcqXE(lS|i&=uBrq!!QP+qzvI`lvuj{r2X@UK6y#%>IJAfJ6B2?dwn?mk z?pisQKPZ*Cj$_-!5ogPXYn+cqq>Ai#=Xl`J+V;;YV}HBtI@P;8BO~KP$KzPEE)U+Z zL@Up@MAMQXSD03dK93#C#P%FcO5pzZ(8V^`K$C8>@b7c?QkNaJPsv(!|3S(lE+(Gy z?{gy_zKlt^@9tK{oQ67hAK#jx{AWn;o_X%4;m{YfGrtViLYah;L1s@;4;l?HH~pNO zg(WE}@!J`DMJtOfTk2Yi7qfCN%y{>mMK4`JKmC2r;A`n!pIch!lIA355#}!&8qx|2 zuOc7bN{kGhUtPufay@v#u&^*xW`pSlWQe7bqn8KmGq0{qn;3o^9Q;cqzhOmU+qeA^ zb?2sobu_D=2~{op-iSFpjDAZYk&`DrH!&)(64O_S@G-(japVjs>jOoWSB!7uTOH^b z&~M7i$XJrfGzgB{(05j!@p=Gn#moy+3d)#SMYD=D%-1jVc^)ztb?{b&z~k)RfEhJ=tW6q2i8L;bijxG;GFS~ zUdP7v97f7F==MILod4^?md?qae)YiR4Y$4T)j~;oej3Sefxrp3RjUH4RYkYU-MES| zt+@sbPR@;&S^N9B7W}R@d=*epN!H>$OKlqqzoL5XP(KBJjBD^1w7h*=jhg$K?Ano; zj10yTG=W#v*pE8Dj+<-h>$9t&YwCOXvTnd$Ek1s@T+I=ToTUT@%)$=N1_IKOV(x)3 zv##akHQ2j%_4dk{)W5g*!p-^k1$J@EgEma63zh!Xi=Vq@vI>plc5$pvnLpPkD^|yDJu# zln^+(VdKVGkOZg7tUsQR1@)?y7V7|}F$|^KJ+wM~cpL9u1I0avmlo^w4W+pK=p)UNP)ot(% zPfkuwqk#okOYP&9TzPhO$MDrCHr>-21{krUysWKVE}y6rxl%rHv)8@CUN`wmJ7lnq zEG+9ORomUsw69QYrKP`>6{(i>pa3@$8ak6-4LY->y}iBQjlRCVjlajC&_VaIOY=(Q zlUB9+b{~qIwy64B@YJ_R?%a9(?c291V;@^v`)&LiH~I-BVINR?tfQhFelV*{wGaL& zcA`MMJ6EkR_iE5=ckRY-Z;s2g$BW4R(>07|=IW1%+P*n4Dcf^^)H}uD-40d!|4NUm9TuZo#+faFO?kb;5d-JC&75Lb^BB^6Tla9eiFq1Cj^GY3P3nWg$;4}PuRAuD?W-&C?@XdKNV4*Rw2 z7sIfN>5-*u|8%I#Xbz-GQ(0pU$M zVjkTdC+hnGYQzM-HfWh9%ew=1%Z-eU_rRWr8AUL29UaQx)j)B3Pj`4J1T4n6Ml zL?(?Ijs=@seX`U8kjd*aOgFnL?F9tVdad1mOPHFf9uxMwH#0KeR^3x`^#+~C;NsZ; zmQ(MvBRTP1B60WcOGiC~JMyzV+sf_>%*uCia>C*3&B@I@i*I9~Y4W{NKGmX&ZZr|& zsAU!=Pdo-A=Vtp~#+2b}05DtXW7~*JB>DgTlrsNOfYu#jw_thH4<6)cxVe2Wx1dr$ zLgEVK4lD>65FZ;GBckh0Ik_jFX?^$i9;D9SG)&ZWcc=TU7*Zo@o|9|Wqvq&Z70@`m zvmO8ONjfXf-z|V;^3KxniSuZvQ1y4WgKuh2JA`TWg0X|YPP@i`A4~L#9~U10t^ted z-|ZWA3q5iqaO({eW9$)jeKWc!!rz3{JVmhu9HX&)9VgQ`Hvh|nn$D1{aI?R^`k_nR z&q%%vb7OD$Eh{!WbPZX-aOZvtR|$=Mv9K&hc=)EBY~hz9QtuWl3iR&Lh~z{s>IR~k zrs|yN$##WXe$-1Xa8gnFs^)(DaEG;KRN36bL=7k!lL!<-!G0V&g9q>)N(s?>X+iai zX-d#lEC?7o$8)cPX(hAI!;Ez31k=-_BX{~d*77l16m)L48X39rFu(qL$LN!)Q17>% zGAvRWAk=X9!5fx+*3r@N#uzvaMm^scJu8DzQ`49 z*tS)o@qytqpTd*xPpsc6Bz7E@`pwsRh`QTxvljKzK^#K}Up=VL-;cN8m?~fz9eU`B zE5kyGcMX#xYkF*9;J({0E>QW>;f)Z-WvN%vcN<*m^YY#PVe!GX$KswN?p5l?Tc zssPbtm1q0cVKU~=mF%xA8uL;3wBO?)`!aWb9%z($xH2qCj{a6Sy*f@=XR84kvVukzP dKIH4tFJ+cf-xuER!B-$s_G%bV^VMw6|1Tr>)^z{? diff --git a/docs/images/chapters/flattening/0e0e4a2ee46bd89bcfde9f75cfe43292.png b/docs/images/chapters/flattening/0e0e4a2ee46bd89bcfde9f75cfe43292.png index 47fe5413542a1e94b763636778b80cc5053d2883..2b5dd24f628d182f8228692edd1c655d9926290a 100644 GIT binary patch literal 10622 zcmZ{K2RxPk`}dKOk-b+U9VJBmW5r;0$f{ z+!r_ym?|mAp-zy0A8ImVP^b$iMY&rKU6a-(-E!gz{kP95doyHq@A zInEFxTIvL4wC4F6$@r8B_m!)wt1rZ~k9j#|tgRd!y%e$VSz21!#9|xjj>hXAwMMdf z#cOU9TD~cD5VH6gc{EPY$hODlv#*pS`r?#|ib`-~WXqdNN<7I_(JJ=21qC7%x5MPq z1uAWN)9z&DUtPF_%eu*YOGI!;$ji7mdLvKl z9Tq8f8dX(QojPwRZ(M&0lYw@|gGZx9kJ~yrzWNhUkNxbqAw7AVZ&(%h@gt`}l{2Zv zC*_7uDv1;f3|Yo}{QMLtUOS8&92}d!(xb}4QhP05zk3%-E`3Omt`OA{XY6gopV`;< zfXA?s#?8$wUi1Y=EpGX0*5FfDH@DeQiamHS&35Ns6b! z%322mxUv~=M}p_)6Ij&Z34I$1%I@8}=N}Xl0^vexvgPl z5-Pav@F!ndR@U7%bHSrsTU%RsGe!@-vGXyRnVD_f-6F>w%P-3|*yaU{)aDX2mBX(c z%!Tnag)?6F&&jc_B7c+mEa)6#8lPNbWMs0@b1SRf6B|oI5fKq+&ygx7f|goaTl)xC zT3U)DFN#b_Ny*C2rqs?ipiGyK=;Ir1_W0c;5)c&?HB#eIJ}S0!{LDjnk~Iaho3^$k zGVQ4m&LZ|8C5FZZ8x@Dmz51TBJaF1$Ln_y(hAAN-0rwbQ`o&|{Hgki`bMuzDg~j~$ zWC=?fn^l^eH7AD2%C4?1oD-LFyzkc_^7-SV15qzrNrUIuI^X5Xga+pFz3Zw6A?`LfTbz~+iiJyg5~99JM3IDZ&Y^G z{Pi-!aWlOdci~7Dv9#`xHtu-6kaKyT^~<0^vkMDoQ!}&e4)25UX<{|CZq@0t$wFS9u`J&b163*;r229d8$y_I*@r930fF-kS>}&__Spb@};?R=|3QXLfm+`^opD zE4ZV`4>CGb;Gl;eU7`sFvt2YdS=D~X#C&$mz#ge6cjFgi^@4LFX2E(H9 z%R6^_I`OwriPR^>7&H+*J-wIegTm$$BF65@xpsg4XT+x>8b1bScfG0 zK^9E7C89ao7R{Z$vbR@K>%1{lKSGPQlJ2l~(I} zk_s^9U%q^aFGtfF=(A|fKMICFX`YU=3Z>2s>s;Efj^4qKGy4k)}RT7_6lgF$M-n^m8)hi`t7O_2zQck=< znxCJ)xxL+7>viyFOJH{Z(B!rvPwER*kE8kH$BYv$s;XpZ zX=&=3nuE59hE>k9KiXqZzNOZ~aPm~5$n*7HEPhg1^j)f{sVUdx>`c40F=2x`Z`P-! zOpyN>1Wf&FC2C?~Vj7;Qy_XEZKTGDQm#=<#W_C99qylP)p+KeM5>;AyI=jD&y?s%` zh+ZCDMn(oylBJTaqQ@P7R+^idF7Wc^E3{70(ov_Kw{v!uzk8Q(eY!4cFh_fSd)T-& zM(yV4n)A<}Kh-bOti=+sZ;n|PJnDKUEMMMlDP-s1;71{Q@=s?Z0r~lJFyDu>f7@uSIXV}R@N{$hl!h~qEy zfx*Glva+F>C2CYjNy!YXeb@b;aT3oi{=AYVV3R@n>WOiaWdoa$$j#2oDQw;mt4lai9+O#7XgCy!Aw84nt| zySslRsWh9)4v;L+#k95Ez5MpxQIJRXdnuIf+pF5A_Ov_hXj7OAGWt{8RZ+R)=_!et znQ8Dt`I6A{A`M3AZ3rICvz_>+q_CAe8z(@mm5o+7WZn##Z;iSrC|IP>I*tsa(dDaG zzkYc!uWaPAzx6X-!iD!uOw8>1`qWa1LMBq&v)glZi$6Rl-DpfnOblxdqsN!Y461dS zkay|&SXM>{xI&&SZ=AXM3DVFq<-?<+=y&g){)r418_Lx+(_+Wh3mb)kKC`kizH;Q! z6ialUFWv)>4mCGFzn#f;bnV);yl&It$Fc*r0eI{kA8bTS8Dc~L)JXm55}B@VZ9Pjy zcF2H-oY2zJ8saL~c}5#Xbf5ffT-g;GR`dLStlP1ljU6r;$P#U6eyT)cx%qsAYb>zNm>B4|*7WtvF+!)_E(k`{cjG zb0e69zx8G7kRrj>flIf+XPydK{s!SR0_dC>py*0 zcuPWvLmd-5-5jktZ29+h*LP;|x%v63@#1HBqHC;eZ2Suf_-gkTn7yQ_GaeA3V9{XA zpVGDd?RuxWo*pqIigV02lG>IZ9xbU$l3p6f-iJAGu;&;})y*(Xq+MFi1ljD2gkT6_Ajit%7hFj5d z&BOUf!-KEI{5@t>SF@P4MwU7}>2Xp1qqa7E+TV2bjHgMZ34A-B8H#~DpNqOm5Efw-_50&j8$AvSCe4n|IgkMIh z<`Aq56EulD8Lz6(i!OlOyBmt#4OjRllMkEE%!Q^u{vjy)y{HgNumZB6*w9qBBZyya9gr(&$e! zRdVL_F4JB?P(>woCmi7EPmmi6pN-#u+}toeKHir-at=0#V&_}0C5Ng! z>hRC<3lBfn_S63K*|YlMw22%!IXTpyzT+vGlbf2F7P?|hC9zG^=)8i5+}q=$*T3U+ z0Em#kf1mOg8|5>TVDbJXBxXg7=vkYq?-wXaF#sm2|MV%L>8Wt^M!uJDx15}WcQ&S@ z0Pe!1C&C);CGYF+7jRiOP!;L94SkWG5lxx`UUJHAp_7r2UG%vkA2dF7yV3f=11fXD z1n9CZ588YeiP9tq7qXN6loP9^hc--MM58r%`sJoTEH*bc3mv&{M3e`%=UbKPtxi-> zFMdxhhEq}3sEe$F$XFW(vNZ&(`Z$#`O@Hesx8*+@Eea_2yVPhZ>%7vJp|povDG+!* zC8?Atc%d-<^(1cdD|vK$d>hxcqCdS!JFk@x6U$9chT#I^az}1_z;xzuJL{8S@$n1* zXZ!lH&oS5}yq=_#HGv$vw>KYkE%}>oV=U2VGEB*r%1YCP&V*Ri$1>2(s$yh_4!1gD zRn6*WSqjayfjLkxGKMrZ%Eo-UDtu688L*||9mU2h~n`Rda-MM+5(`-%x@%3T;Ok-ok@QL;>OO#ExZw)Xah$to90 zMn)Jm0v6hC!138>oU4x$D>HJF;> zuTNC?!Oa5$110+TS~;m&Irkqv^v}w=?CRm6fBE)G%=`C|Vh#)S!^6W6@{oY9W)41$ ze)~3HY)o%&qBf$?td;-s7Y3BNfVbopf-Z^A2wuNV$;{kAN356mc%stj>W$}9NaKhL z5_BpW8ykmY9~gwYtqgxH&mxS}yT($gw?{|I{^ZDgkA^-SidaBEK&j-RuAz}R%=+#f|IrA>d1b5^0UezQ*G27r5F+v2kGU$@g|TOf zAVBdy$PlcI>?SXO`6@8BFVX^X*vggS`3l%%@k;d>(V^ktuLEKxPCg?K9!d6xZRtED`4On?x!ef!o3^;~7F{Neu6P3f}3V?|0RTY7HQskejXfKX-q(;+a_ z*1yQ@6ED`;-(TD+rr&|x;hf4$@ebXc_F=R5p49MBF$O0k{T**SbXYm}A@C{~DD0rN zF&=7*jz^~0Y_YiSiGsq9t`u^VeC9<-yBQR~J{+&^^-E|phlGUw8y(OJzQWoAjYAzk1j?FI}1ing&dVP>6|?o0j;pcCL)B$pt1ugPvb=uZrqrkP{iQcmRZz zlf_Fls{aqv9pKZ*eh8pYu4?<9lG6CKj;^i%1KuD5UIing_YDRTEz#}fI3Oc@=`V)z z^n-r$;H0Q|@8FR&o{_){d?CTeMFmR^;Q<296ZbQA&%sH%E{VO%{aGQ2q?B;=NXfkE6jlLJsz9-Tr| zmCMHZefHFw9Lf(hH6Jg34!4l4PD|s!LedDmq!b&hc5`2DqRz*dkX_E*y^`Uk=a&(H zik$mU<%lBW`>otsjlDqCsCo=gKCPlcS4dcxh>MFW9XoF8Hp+CjkLRX^c>Ih~I!WjL0TvPIj=T!+@8Y56k z317>}$T+juouXi1z!J-&+m|ld0)aBiCB07s=(B%bi*%cnF2=hDSaebn3!-j;G_kj& z<`dd_-a}bh0|&K-f|)rC6aa&2SIYJE^*3wd{{`G{SXcG*sKcaz1_N@?DN9y6QH$>^ zFomPPoCOokA&A|KpXRJ(jqA$dqJ{mLy{~qmbcA>S(=cINIBC<(|uZDBVYuE6v>VyS6*9;4!87P1p%x=oh zOpwf=p?92|oC@Y^ur17KkQRsc2L}d%L#Vkzi;9YD=PiYp1VG=yF6R`yD>2r1*K8tF#?H>}Iv*jz@4TvWWGxP)EI2AEsJvW+n3x#n$P!XlC;dP}BRVN5 zJXykJhkNLqur+796)=#I5-W{zhX^29kB$%b;|0tJjC_uq6n9j-f|}({u^t3h?@oCj z5!&DcWjgFm7#+L`?|C&=zE1KH@6DS=KA#_vLgB15%mn2KgTc7YygXaz_WMb?mra{{ zVf>$0und3*6AaVJ&eZHqeGMdKE_B&2>P(g>Pbn^Cg}~A&v7}5&N-DG-=Dpe1FAw+B zZ?6o)+`OEdGnJEW)pf-^|DJ4!#X*7v7C2qBcU7l(T< zM41?(_@8t#tWMQ1`w@^X2iZIu27YTd-$DvJ-3$PFtZGna zy=h;>!om|zAf0V(ZTuq&O*Fuc2+7GU*q2(Y3u*$prB7HO!M_Bgm(kCD9TU%x3N;b{ z0I)kVs9^*L!SSCnG6DiVGBe+>sYhx8lplv!FXm>`zFSZTK*S8Ys1Rw==ufmaSl}IS zmd&$JTOctpQH;%szRY&=ugL1N-(d_Yf!UH{R*;UecMm42EvKM16cUjVRTVtlg)@Al z{5cyEO-a%|=gnHeF^=K`B+k-jR}m-+F@I4+gueJmSL0{RoIhPQ*;_dU1+5=$`Q^^J z=3Oy*@&v7wtMd}1T+@-lN0a%_^IAK^O!P~84B05J{08bp!NS5qFmNzD z?wYBZ+=wXE=1-oybG-KLT~wSyeOXaeKa0ab@ZFc0VrxHt{6HXWL(1`=%=5+r=oK^i z=8qy?avvcHbXJK;fAQQGeD69(iUyIR#i3C;ZJW%Gq zlf9^uLPv2<44$s?*!PQl<2N}TSXx6SpIi^$9ZqDO^#auTGk;Vvup3J*QiTM~C zOBZvnHS!str_D$qoF^|TWzBTNQ?ZQdv@&-ccZt9vrYRn*(j}#=T(%In?SDE z+!=pv2I?rt4$ThTQWi@k?zyR{c#xo)4mRq@p)hcw(Q_4x@2_Y7zDR1IWvt9IsUXJ< zEs}S47yq6j9rd^)b~ZigW@AePGbIg;`O<*$4pO+UT_d%!vI51VMPj4c4{*6s<{hoZ zr(q$be<-8%YM5d=-G;&|M!Tokbj%k!rW=jbFcBi`4)`e7x5xZ3?Le0Z&lmUhHi? zN|ea^c+BM{P~oshR#aSkUqhqcwn8s2P%jT*YM`?H$qUbaFKot#;Pk`o0-xro8V{ty zrhupSCO5o;qAx1!=#$aX(t=w-jwr(5Wtc&(O1hQO$zdhX`?M6MS@zkpYqla&Ef|kh z^$DGGD25H8^S^_>2c^lUvN?N0J7VA+&1`H;TO6dnRO+kigFr;&TOIxI=^URZw9u0A)jqn}SRCDh zzzk4i&A@;lBqm@xK-F_GDKC?wm7Je^85ej=v? z;JIGj(2TWx1xrWlwXfsP>n7KzcY-USM*tM?P%JrOL;0_N35B3pa&871lb+Bb3kIa- zD=yL>PMO7k|CxW&)t2LFPKDWAd~>dZYuCOvIb$7@Z@k&8W0~ICnx9KgxL0f73OErUeu%FieZZC z>J$@`lc{!ECe6XLge)wC^a&&e=DhGJnD(NU%xo^AF&`gIpRn@P{Gmef8NHD!p0Kd+ zT}8!PrluU{eGXrE9&HcD-+0CZ12Ma@66STVgJ)S2Ft*b{@%CN-}2w!*ngg@Nlm7%=b_V$9`zJ2@XsUN)hN|}&dS+{rKNgwbYP{Q--0!E{Q zg3dzZf)y#=taR@lgb}{1tnBN=L=teH`~w0~p+pzd;az*Vk^M7zS#}j+SzH*P?HKkQX9~ftk0iMp}-wns_K^jwY-|aaqEX73&{8gV3`e| z|L`FDbxKNQL*JGc76{n=YiIEC^5VC)?2EV199I)xeVAVg zqc{aCKfcc9zv1$H4JnqrIYb($)GKoZWY_0?{_QK;0^R48Y!+sz3{Ne2r6 zP|hqa4hJbH3i0b|3WS7*`-4wL9|JrWh0@A5Ao@_MNku_{f{1%D-|`3yKn5{;PDx40 z*YWYe8A@?;8$KrzpVU&wK5Rc+g5(}AY)vOAB}IAud?ct|D4G8TkOvm+0CGWj*coQ0 zCS`yg#qt?PLPq6)sRnRQurWqMOY5W;3raR*h;Rl0S)co*m6c3T^uO56o^oD0v0gAj zp@_JUy#QkY>a z(Q$BbodvP%6^O!5y5C1^ZrW9OByrt;K58gn)iFBC2y_!?R8~pX0kD7e=g)wI>lSmp z3T(|_WL!(;jvg#tG3|>5L9n(ER@@BO`b}Ur6uNIe{Zdu6q870F%v+B`0|4jD%#1G) zHTnV%&pE#8%~Q4$UtR*oI@~F*8;qg{fA1?eM1LP>BCr&wj7Gi+xNZG#o(}+cjxe&q zLjKP8Qp_=E6+t&ut+bcH%}wVjv9hp60JbBxL;wjrQgDTYgiHX+@bH+I>nOW}F+%R` z?Tu@E096Q)UXd(GNk#Q)wAkVzC#SzlU>k267{dP94)t%WY5}1mlpA&^$lbYf1}wMG zrT$E_Z~t8J!kuF&I)~`x&LrN1T7d#;a$gxPd?R+QP@v*|? zKwP?IMM^btAoN6KM_|%5RpGqbX)-dhwH6YU1i>>9De+<8>19TS!RRZnD6vxB?;tbNMOUJO3t&Hv` zmpk0=_Q~HE>V}hwJZOBGR@);MKDpR>Q^9)VZsM?Z)TgUFfLCQ6Kfc6kR2__9Mabs3 zMG}a|i1~oKREJlKsDT~GfwTag-1d9`=tdZ)6|K`}&aeaXgkg6rXd#LAG84euhAD$o z2se8HTN$bIk!~fqABBk@5u0>npLC*$q)&TqD!?uU$0@L({X~s|JHfM=6m+VlNZ~V| z1|gTnFoOQ+fhRi%MM|0Xj>2K$IQ=Ou^YXTU-4vaYLJrjeq;0dY603OrN3SyaS2RFK zeVY!OWfZ+gY+k6|Ygw6gt)wGC!Rnrc4FY%7)Z)ymUkjfv_zv1bZ%>a&>HfC1i@*f!SQN7AQ$`>UDn&(^iL%yqTwqwd=-W`c>S6}R9xJ?`ZX=M4v7s|*`bJLe;cY+ z5?wY>YTZ6X($D=4z|nzvdYLta&k?xZj~_oA-D@fIdHT!*Qt_0~F-s&$t&1eWH*OT^ z*EPYI*p{pf6)*>n^kZfJ3u($~YSN-+0Nn@3rplO@Tu4ezKET=Kl7Wbx5s6RP#mMIf z)(m3ze){xj(!NxWv{WwwbRLL&@DHl$$XdA(UHrO5^=qx1J6c+FPpz$A9F&nqvXb0> zfrYBDhoy~|mS!VvE^q7Xl(VtPH7r$!Jw{`{Y^!_l$t8UuBhkiJ@2h~eYM%L3kq<(GMQ2D&#)$ys=kQPHJ+5%lhWE-;{TN&e}%ot2w=8pR@UBi)d_ z*XpT#J;tmb4+ZM#Q9^z*Viw7W7+C`OMWNn4{n+cI$<2nUukqN$T@taL)iE|E*I4u& zv=n-?o`hxScAh6nFK9E*D0A?Bp`OD_0tT zX!~anp}fzdUcrz; zv3`7q zh9pY}2%fns&+NZTkgb!f|IMriNP>!=>bt_n7bKf%S*}d@a@fd&)OsjapfXuL9YOy7 z6gZ3wD{0uhOOq{2kTcV~e*ne>5gKv{K8lXdDBRcA*EU6i1)O-O z0}0B3p!T8i@W~cC6Z|^j_?RQ08GtB1_`I%86-5CuL~u-%A2k|$aX}|dWK-VtIiSmw z+1c5+l-{uem6OMo6%?|d(%srSI3V%y@rh1Myb4NaaraH?jq2Z#y`nK(YIJ^Q$?0HI z4-F$Dqx?K*u1}wSzI*T9@*uV-o*!IkGs(C2F119l@d04-VBLrinWBJ&QuDpw(PPS4a68;&1+XQdcyG@Y zgerY;(wdSIA;r1E$N!&ALEz)QIUPnssVJ+Zt<9yJj2|1^#-JM67mw3Lsio|)?P>E8!qbip#Yx3%U0Zu5pjsO4v literal 10637 zcmZvC1yqz>*Y*%1DILi~j)gtVlCfHX)ADN@p)NSa6rDj?lbg9u7WHxd#e zjpV<_=Xt;Q*SA~<>)iK!VxN7keeG+XNNr6O3Q}fL1Oh>ELsjV(0)a=1`ynQR7W%g{ zmGFxQt*)YkIKlmUU6-4PKyV^%C|%RTJYOC6G1A*UKK*;Qh*puGD1?;QX!;&!@OiD=*=qwJbZlUi_>Ux?snSa`m+>qi)aBoYvllx+U?P4k(48EX9! zd#mVs48w@?SZ{BW2{&E*jE-zybgn0oE0pSHUMdv|bFDR$s)>x3CzQ%u8(Mz{r=Yp1 zi1&z_WfZ?gQ|V^uBB_Z9Uj%v#YxZqiUHV!d&n_2ru-3rw;lqc@lj8%ot;L?e&+ksO zbuJKGTSz9cia9_s3N;*xJbChjGfA!3eL<7osOakCWooqRyYy$zJeRs}85-uUyW85@ z8W|fmwzZu$sc=@-)1wRebs{4zBSV0=D{@v%UA-9HN@`k*p;inhZ!C8jrDA3#g-;wu zN=anAR)WI9PDwbAMGTkN7MPvnz6xxbX^nPU9uWA&;os&zx;y+bT7#z?f@mjaVv01%*`OuT=)N-5nWpIVmnJ z{V|}+n-qh@{+g)kAjE$-vTZGLwxpuMw!&r7CI&e$G?b^0{8=3v5<&nIX2(q7dCjzt zV419pjg40xo@QTU@Lrp!0pwI39)`+5AcjUqGiz&~n{-V~up1Vd&n3fy*gC&m zREBmC66n4I6vRl$_mXrX%TG^;_=>oH7?PKl|h#KlHq zPS2{VrGJTBk&!Vf-6glny0Njr`B;^Tk#SH!3-)qO>i!Q^|MgZz-pv(EZF-yn z5~;Ji`F%T8e@;#H)z4dz{qHA^|5SS}r)poi`|H;)$D{r2v4z5hV`;F>TB)P0ZJWT! z&ZrB&h{(sjlS;N(ukrYNoas8cx@^(Ke~s}?W-YI?!a;DM>y>R|V`JyYNP1XkXi;r# zt#gv3xOmc^y**(>=ip!pxcpn$(}>i90=|Q(5K1w2LvwR>a6Cd<(lE#G`T81{=50@FIarElMXyZMnN0qq>b2F=*iz~s;^MjxWpcC7T>|3hmX~ZNXy!LC$ z%Nfeb$}U~L+_?3#Z+dAdKbL$v1bUTD=$QNb`SVp>UD~nn4k}iEa>Yjs1c(o2wC5Ba z@Z=W+a6MKP_WosBVxQ41M~m{%HZqFH$>E7m%1#w`;H#~RwzPX2l%0(~F`RHS2Hr~5J4Y5etf8LbM5aN`}@9#pl8qa zQPcBY*?raXs9Y(_SZ7VOD3x_gT<$XBYyA&L;A95qzChbVL|ojj%~B>Xa?dqOOHNi+R@d2Zj@~}Q%vxE*`}gJd zoXV7ad^m9p)gzRs3@SW4JYFkSSq#;+YuCm$imlqHSXs#p4Gjs2h-PZm>Io2KLbV0? z`AwyEeVjCrRA#36Gm53Q8_0eVKBP`u1giE zgg{VIQtBBR7MKYlcs9-Ra*wxb%%%Loj&WL+_SMy0}y9ao6c#~8$G+GJ@>+{`4 zmzL}+D=Qa0Sj}``mXsgN>+;=^BxGcY%F3rIz~5r>@-F%B4BL(!(cElmZlBsOXcQGiX6=sU*zhl04#DZO0k_BPCw#!lb1*A?Cg9=P!K!+^;wnY$f;AOxWvTh zc6N4#K3bE}(9()KKYQ^a*7j9E`R+x0k1pu_ZC>8g{>yuO($dVE^IvC^_2r7+78K}? zt~EC|6F`(;-bFq*_V0Y9(4Z@ObCiLOZfdn+a)9@NXPz}mm18T%8f|A6c}^wn_e*sP z=@)EL_X(4elLJN+=xI^=uO%fdZ_>kGzHiIQ=p|hyR*Eems$Qc}szUk(2B`BijZMMF zXYs&|f0ji>kz#y(mt-hHfF&iESNm?;=Q^*?#{u5dDG4NggHzwD4xuF*-HV6oF;dW7 zI#bH>GJETCHo(QF;o-AMNlCRW%U_0OdRU8qeQn>ew!YAUMrPf*eEaroa>0g&aVjZN zp<5`wN8;cV0l8>lT&HA%g%-JI9L&1YII|cRZ)kesrV|F!@qN;F3OCZj+dSjAU1iKm z^#F`aFn@i<;h%Wc&M zTa4IG&Aa~hRisWV4oly-N&WLizsSE0$&vQKGTy(0ofd*Fw2ZaS-N*R&WPJSiWZCw~ z%a>;m0GY}1x_vU~1j#@!*$&s+IKa^Ry~;M= zYx8|%a{)H$ta@!nijdXk1TNkRyOUijwYJe`7_rm*t(lnH32NC)5{{`~ zobH_&lIXWNr$n}OJ{LkCaSy&+`_KRC^Y8H8)|y>7>%(?z#lY0%AWl_!E;GR3e_h7) zXx|fq!Dv4!WA7g3$~sL?PtPRnkyqa*jKS^sHG6wro}1}eYhvUtlj>@#5kp- z&cZ3$*KD2A=?JBws6X6L*8uopCv$iRL)Z&rf45(JW%;u=8(z5t5Tz9RDsZ0!LB-Dg zVL>B?02bXJ5g*?MoO%02cn|n~1S#+*MK3Q2bKjpAzzM~0Q%2poMe`;zb9!+xZ%r?T zNRF8&D|%qS;5-ixw~UO*U@!=oc4kRSL^kI?(q*qxDYcS5iv0> zPcEnn`|UWsAJEd$I*sT|7cWrg6ZzMUY;+6^Z{EDY?~*^@fA_AmhU}{Qr}B&rR+Kbu za&MHQO9jvWc(52ly^#wk!t_*Q2!P0dH}Cbx+dQF%KR3G+RPNqAe>1pWy_QPk(zJLX##TZ^e>7$O0E0&QRKON z0Q2cHtljWN>}{Twl?&(3PlYj>2i4Wp)$1bdp>Lb_RaIwC{&YQXo$ovgi`)X3=^Qb% z2nb6^pd#mxHw9aO+qS%G#bSF)V`MYXsLp{0CkGagH*}^V6(`@O62fE0GKDialH2Pc z-?)+L5Wa8$l&`S%&6=JD5j`zBdFR)7G(a*Avt{TLuf>N)4g;PKl{qlh|Cwijac|+q z{nnGIg2eo`j5#ZM9Xh6ij#paR+WP9A1cLo&N^ae{6>MB$lL4?RAt6!Qk|kw1+6~y# z^=@W$yoQmCfjDG&=8*ltzqAQ~nl<~3~#K0^6 zw!wPvt;k>bZ;-?`|5VY_h%o-#-u@7BupEsv-1He>`V2F(v&EQoG?=y*yeH9nbwvF6 z6*qjL`n@+`M?3hcT^$mvJ}m_b6%`d&PVZumFwzbVc4TDa6}OpCc*_T@&9k~(*bv^( zpwNOY#x=o*H=^U?bK4pTLZe8K#()0&xx~k(sHgW7o|56ZRZ~^{1ny#Ktz1EL8vlj$ zIj)or9pHBx8~6JM2SboV7cXAK`1zd^7EY+3F3LgiCe3tbC|3Ia`3OC*=xM|R>G6gs zabL5tvdZmpaB$EuF^PWighu0)JR4-WzF)e3<7j=Ve1P=W*ccdO_%#|m_2a&jh=@pc ze?Q^wM-f5}nedF(Rz#2k8o6QnO{uI<=sXR{{I?hC0(7OZ%{Nq4ucA=FFdCmnSIZ}Q z@cXxKN#69{>Kv)2pHNmV(z|5&R{K*;9&t&zbC{pqWo8x@9X-9QH4loWFG+kzk45dn zIiUwV-n=$pQIQX8?$41TIK>1gl1Y^-)&%HDLvyn65MRE0ArcV}DFp^aM@QFvx5xKt ziWJd@`g(T6R8EF=<&m-L$e_5PNx5U=?&O1bj=(*<-7)w0`aYzL!Pl=}yS>f~L5^eY zt-0=tWm?`1_+9+Ies*=Fj0~t5Fl!*k*#-ymd<{Df6tMosdl4$`pO`VSu$AED{d@c9 z&)Mp{gKyW@@eq(J?@HS$N=;0(sB35_`TNV>zkgrYW3d~rD}Fx6NpV+$y#KvPMFLO| ziw8&a2v_GF$4mMi^|x*tPmiq(BZBTUhrf-<*1jZc`|bI$da_DSPmgpNc4uueK9)st zW^aA=HX}{1LS8CU>d44#z@wS3l9Qp3PQH8hPTRsFo+U6))uN7h4iE2MS($Rddx0+m z1dr=|#q*C2Iq68Lsi#7FZZ11D?@$#J0qob5VZidl=SuWD+2D( zmdtuDp`%*SkbS)Wrtj-3J>kE36X(>;k0=N2S$ZG96e0=?3%Laa&o~Tz2va@Z1SF~0 zWm5idZ#F(6Dk@m}lH;=0N5`{ItgIfhvXHg+X6Nl}hh=7-%h7uEp*+6*_~?L|nz~Uy z6svcQs1dsLDNeKJgT%s3r*Z+ zE;w1m+S>X>bsr^%JS)1GX<%T$e)sXSXQR1ihebt2-F8-tBb2yn*Rd?AVs=k7M<@N( z>Rer2Kea^CQnRo?h7z6^`*UtCti-mqjb9nY7;Oef;u8t=w@~WM?rm-ack{Y^)RbcXg{% zd`e16+t~Q{@cL|iT)|XPtD*CE|6Dj6^{s znkpMKS9Eq3WZ*Xny7^3k>->3;NDvLj`+V3qHZn-Ne*F0H`?e~Alb@d)$UxnMysRuM z5{bO$=%}Qjp;2hg*wfvO`_e7t!V=m*PtS6!(rvu_WHw$w*l9$}aiTVD{vuK@ykA44 z3=cki^q>2zst%}vg+KO@QY`=;`& zprB^1dlre={BMqULHGSe+yPHdkLt_SvT6z?87jI%u(Glu?YUuRXV(}-&*utWbNIWN zk_m}cYM7&CV~h3KoOkQcXw^*t6e`oQl9iQBNJ`3f<1njmT4Q_b&>0+zUkkF-*ak>+ za26hd6p@}|A?B*5fzboNe)^Q|xs3N2+$pK{HPG}rM~^3p12_|_4m}$_X6-{Wl$z;zCDiw zZ49KO0#nHg7fz`s^EZO}8Lu48K;fE$o!wa-O_g%zX=-XJ^4YX0;b^%>o|n4yFkha# zyu6%u(8A6e)-O`cCKYD@t{XvHRS9cR)j|GzQy2gFt!}Qc+l-Rq zP!S&Y<2)@S2z%<1)Z)0zFOF3znB&3W<3SSt&!66)H*!A&C`|*otF0XS?J6$-E(ujV6+iQOH~ggSmneUY(iM>o**nodTwimezZL>vTeVgAvRFw!dozDD8E2_4d**Ffj0V9`7@+0Uf}3sOIx4 z8D8O60_Jnc(h16Ajm_q_a^i_O7DaMPaCQPpHwh;oL!XNa+sGev!(u<;ci98q^gf=8 z7f*L7oXDQxkdJx95oiQb7<-xrayk<}&a>{FUwHKZi*v-57A4vJ#ms%pN54DIN4Kwz zRzP=+>KxSasRxLGQ+<5evVB7Qfr3b})@5dtLl zoV+}Hb2vpa?kxTIOl_Z$CY*SQ2Ly&Iu2TpgbJbtZp@}DJ`(Yr!C+2%=G-EMN@5{>KgbW<^JeM176Z0`O zFI^?8s%n^%j07QjyfseDf7L^m_;No2Y7@&+irdKoCgI>3+&nx)l=6F=5>6wb4M(dk*Zp)2;6%cX110>u zxrtkxu=~75N!8PH`Dgu+J>DGD^?2vx?dNZirLHR>rDmp>q`_rO=7{Pt>v_ zVq=xjXl@h=g+m|Nogpg^h0v{tT{L%+8t9$fAfo=<1d7x@Fz_ZTix9AQ!TGtp?_f;T z@7>);z<7Z{FbJ8);3nQI(3vuRIzAXF-0vu##urGHBQPI%N! z0$~YQ1$3Zge|s4ynv3OReL$thPZ|Jkpnc7@w*wg1*xu_FL{qu9I3P*L$#K;9@ZC|- ze*JgiB8s5*pBT#VO$bL$p|=AHrfp|WXJ|)6(q5dIOR>0)Lg6zA+$O9tjdL0;zk`iA zYZ)`$o2?H24;5PxpE(l_1qFNpg7tClNxXYC8cTbbQPpb=C&zw-r%yMJkF&)q9A5#g z5`h2)ohsqP3^FQYLD8UrB7%(g!UGB|kEcEWfpTCOC`Y=2eCjy-@f0wSk>^mLDjrr_ zFLht+<|-{Mo%*xzYT*LACN7gOeSAtE8fH?4HV!&U!sb1(Ap z@+Lfc77a6f~tw01pBz`XE z$H4uVxFcCW5p)d1r;ZLXK%UjjDz28N+Z{u!EG#U9r%r9>%am*I%$m~z;XWlKBqY1t zulu>Ht8J{xotl+324eL6=3NkwV+2gfEvK8pAT#f==O7^=!Nq)DK0aJd{!1jcygYgN z=;yCraX?dlbyYZz_x>>=MqVV6^z9fPPCIBD{{^Bv?z|U4X_HmnpXp?OuwiOp;Sun2 zqUC}2%|IjFjt_m_UsUx4c~8gGlsru%+-(-?CK>l185)saP*C;8z|4$-k&*HE%o8fw zyRNP0|KYJ?kOGPT;{G1CedRWo1?$Fj` z_085pAS<%@%FJ(G%Y>7!;!;zYpbmt~w{&wf@DO@>dQ9KXUk_ajX{1lIed$9qJUl$M z0U3vmu5JhfnEP5bkV;o6oQ6^cxP_{^dUG4cgBXAzG9ZLTwHPFn>FL8Rs3%+g z*_cBFQ7Rl#DJv_xy1Oev{E~2)Ab;(2|24?dUO`=iT-eqro7>TGaoAd)%}Q%e%4gMq z6YIY=(#0LRftH^lW8j3U0(|~^d%JZ{qen>}fgq8<-33^m{T}CuHRui|5=bZuN)TY!y#@BZB0Sq*D~1Ce-93;Uxz(d}Tz>AAuKc9`tmMu$*U6Hpc+ zL&HIKtDOrRsLn80_3_ly)F*l%iVQfH#4#|OFa?>_Imj#$d2Tkr#%;gg<+|Phj70@l zDKQDjHC0t&!1Ac5D4e2oxcTZN6wWm^o+F-wmgnjvx+NKXR7}(`ePSG>R+UY?khwrO zp9Z_)P7_p=pc)?q+E-$FXL`ahdZ0Dy9LOewgoK=&oCpwnxC4*>*n+x;9xZn&Dlfkd z!$Iei4<08yp;*S435Ey;Vttt&OaV zF&R=DF5pOz-QgZzk7uXg{oC3n3;^QL);yuJ$Vc-?K{5+TBRu!Q>>AgtouFH<-%mTI*fJx|DeYm7-#+lGeRiDx|oc26<5VY}g*Afr{%0 zCkOS2nSbu-d3+Feo4=)78;P_uYa>f~`gE!@O?0T-iG^Q40PLwY8y3+Bx)mHp{z~T~ zp}H-ajm=HBpS>5Dg_Ouae>``din#J&gqV2*6DcnbPY{-jTJdUw8BaL;5s-mKH;qgR ziCJOEp$b0**z@_@H!Pfqa2$2GbLS3bZ~D<47ni~{y+{%mg$IN(@Edvayd(LJJe^Eo zP=`UGuF`^8_++Bd;UjHiVUHeVW@O-joOi`%<9z-85+BUOH#RCV&4EJ*6f8nMzJFUt zrn@m*y-+reP=o0$Mn#eQd<+&?JVMPwxuPdBXC1{hDfDA{I%7?*9RURx5+NolTs2Wg zTRW(>R`ybo8mOg2z&NmQh2viD_`V$o_c$?sIR(?&+FI#4jj#Day0rttr6_NX3)N=x z34zGFpQ)f!$SidC`l8E|EgATOJufu@PkA;>gOQ&0jig*Gh2UM>_peTV>05_ z7@0+F$l3i?knr*+P(ed7myS>%HzaX3bY_ap2Ua9a7d5;;g(dO3KO_0g@?iG>TIXEm!!>Ad6Q7w zk6#xT7xyhd0+9##CLkXuRL)*UmavMFT(AuE2Pv$WxOm39PPVqVQ<;^OtH;qs{uH9N z^UZ@5%#f71FMJa(U?7F^g7l(YZ9-w85RS&(=51;?IWoLNcTzKWOYcYQ%uI$+882I4 z;P*?syeL)G2=;gzOXEcuJQTV90VYG-VSp2kE%;QW&n6FoQ$V0pm&-0!+lX60fDWN; zU?B1O&k|S*RoX3Y{&`@40ovwsyvF$qN9qd-3po=KMU#QUshH@lp;u>QxzAlMS+0YUDcoz z%a`>f8jv%fq#YkmiLl&RaWDFK%j;m`i(2GxNf+M#SA18IS5{GR7-#-l4sq_6MmelMNgomW*wHjuMrXb^h@9V9luN{)OEy)> z9d>lM*~Ne1LfGm=-Lnf9FP4Dw8E9*N`f$Fz0CU2mlld(}V8F&yI**c^7$MeWm z8V;ND0mS_fA0HpPu}O%HSg0yh8O6oLbv8GL$o=_|2uXf)0xyeIq^_=RRTuEGf_Lxk z0_n1fm*3CBrh@d{iN)ranCiEwLy?kP7%~RHIFJ1mAVkDuWX({Gi2ME9HFlJeS&SH> z2BO=mu=(pv&sQmNNDiML)W6*STL`@3{p&e4_51hlQ2Ojys)Vy6@(HvdDCIW_KoQ~` zFE8n9P zjQbd&Dw*a``#%*s(;Bava|vg{QmpqAk^3XbGZkNBA&(-s)>cb2LHKa*a#QkQp=%8g^G~I t&#|tM`u7AzNxH5>HST|1L!3FmTig|`xW}Q_2lp!xHn%o_qYg=RM~;-&yPX<6Dl)wH{~hXYV_&>$(T0t8W{dJoO14H1G2MA&-@^)dRejRm!JIlRvIKDPB`+zFaX|ZEMMY_WUd!@lc=x)!#fc z6qB*I9aO+ISD510J!dtY6mYO{yORy8*ki7^_ul#eo{ruv6sw~4+7wS0EzF8i6QByE zq{r2zqocoCm~CKxD$-qJscUFyIWCcyGo+M!f3G|vU@St;*vcyNCV8yXmO2SS$Ebhj zaBpyaQhfVstWb1J44)gjn)H=hCMM#n&Z&-Ko?tDWY6C1no%cXth*T)YZ z7#*uNm{-@<&RR6ny;KS?wl>La5)~1ts|z`PO;0cQGPfV2ZBLG}u5Oh3NUHxXWnp1q z^Y`x!1=xD|E-fvsI-ykGM`2vv6xMAoJNtib%=NBJB{}jbDRC1K5tTg{JEkYXR9RVx zw(U7>VQoE;A8;V5lPn*SU%g9(!{Iy^N_xxJi|_g9Y;CDoFp&@wb?jWdYiViusG_36 z{it-wo{5c3ghTehMOp9lx1&o-`F?&)&p1`w+)4@@M=pFG9IP0s{?nHHb5X&6-;-BP z&Y80wMrrAt+#avWf&w(QK9pL;$S8Ta-#V3&|MD}t-nC##hT!@zS`SYXTg9ta zua2!wws7t`4*qPwF@)TFf6Hx}GVsCjsB&6anSRt)ivgvI^77f-j3w*NQW?V6yRvV1 zA8Mb>T4;~l&~L|k8&NU%4L;*i6bTLqp%fDnTkwx|t@?O7-*|w>+1Yt^+1tgXcyq*U zVQzL-q^rBz?QO0&#?a94!(MMsk4-@I;Np8xp1X5r?v`SfS61k+-?&ln>&DzssbUs+ za%y$zrNM=k_%jTZRaGU5xX;W-t3@ep-3)@5*RNj(4^?;^-`iU1b#QTZMv91t$nVR^ z%IY__wER4*$Q#H#8ZLLdaU*uKgq~lWA~7*h#oj)*z+=fc8T(_?6HLL_lW!spC!(C| z$z`akt2-gzRJzfYqWGSg*GN%uclu3T{F%GK<>m6cN=io$9}e{I%lJs$ITp`?2ciaPLu8mUWmCw}Lx{f{F_HGC4V9Wuj>= zXJ~h+d{fbNItuMNt^4`QmrHmBkGwYYeVu%T%PKq-4qFwpD4W zVN_M#8^~d=Kb%#HoDd5X6clv7hK53>r|%`FO1K_;5>t4KvEyi^Y-04I7=391zQ8xoRpN**q~$0B^8xpZK+C*ZU=>2bBV>q3Ds<<11_)M z^uMb8*i}PMe9lVs7jaWiV{Ssze~n<5Nq2atas6eI>i} z;Ben7E_jO(~puXtn?6tsV(c|xVKx_<52!^+C@zp8vb4!J+qB7LqE zGCppmrl$7FV`&)aCM*_88GTYxQj$kThNYyWq|CjQ+`!-{EGFgU7|7;%WGq6d+qvo(C^=Wt{Kih4J8wjl$1>?>tp%)HCp)8in6`^ z8LH#QGaPOc4029bRFoKDVHuKESm+bh%g$cGD>}F0zBeFGc>4#M()f7qxuL8NUE~N| zBP09hl?Jsz91ajlZt2f8&>@ z#IDts)3#3{M#sjmUvITSv9<{Q_D##M(vvhsnb{CA%5X#-(@W&oaQ+5>5F=Lj46Jb20~OfB&%Z?(S~S!<|382`wdqu2t6E?{4aUxJUo3 zrG@(Nq|<9|=wmcn0jp&mOI*i@iO<^gp}V@eMw1+?KW{D!u53?4@?W^XH8wWpIh__J z-vz}*^~#lqHuJKI3VuGmW5&kDoW?#-#?cQ(FAAVgrGtkDTO<4^6q(3#tv{v9%MZRM z$MlUhcg)kE6Y?jnL6K!-Co!E;jo@MaVqc-aM)4KTTk^Y1+uK z`Y^%^;PhZVa$~=Zn`LEXrknHqWu7a1zP@*=qwgPVX66o8`IygkWrMHlN^E;y-Fi=i zfHMsY&iJfPrxmCA?QLTBHWxTcMW1WIk>`eP>FNe*#-fzzNKW7R@V0&rO6PK`%*ef< z^x`cwb#*=zYD!s2{!A#PTzV;SW_g|YXzj!8l_u^&^Cp_pcS}$7=9@Idow`wA+bafk zuN$3!^x+o}AmQcZHJV-O$-S#9@_eC`@=;)hc7oZ?>ZJEah{Vs`-As>o?$y7~}fi!Qe~*zPNYqUhSL63j<>35wEJNxhg!C z8t3wBo|!j2!~RIu)GxI=iPCvVk%+M>^k3rkqaz`D$|BylKJ%0QPgC%FD}(SywWxMZ`US&MPa+ia=YpagdOa*{G_Hly-FJLO)kU#qvr@ zGG1&6u6LSf9NXVnqoAZDM)NAY@YkA=BaFF~01mDsYB+Vwv&@!uis?CiwLxuEtB4$kJY;l{qzk`}Sk6xRSeu%}O!pt;a+o^{;X_?LR`1 z(6-!npZ<`iiAhPx<0Ma%cN0NfO*uUv`&iE3|I?>W+lv)T0HVmQ&AMgOkRE3q4Ph~^ zpP$&ocZHPf1ZyC;(XiV0i>J!h`r4n7RSs#sy0IWvvA zySjoQp?DP)IpyT!JZH1B=r}oZImvnNpT%e}X42r@r@*z_`)lomZgaO5WHp?fCH%K4 zMsS{F79ydPCokm?-SC4*$BEbMBk$k8|MuppQrDu;HQ{>z!p<)oyQ3K@ncFQShn-!HF+Xp6^W+P8j^pURKU)8Sgj1T9>;^`1; zC-Cl>8~#xrK1gow?(#wTf~-5cyD=v;>(uPbilX`U5>|eA7L=Zusgiq(9|DU|lB0N4 z)s&bGxau2}@jh^f9;c}A@DndD__6)Wyi!lVxt(t=ulds{)T0Y;#IvHiv$C@(0e&?l znE2xWikYuZcY1Dg-==3``r<^V(1UT}5tvUjMojnQvYVNkJFQP&Tc7KVG?|oV4aG%7nbqoXm)wcS$vJIdVKMqW z@f;rFNZPKi!+}HXdkdTqnv9$oXgTl zR`K?hhxiHQRSo5lmuCw&+8gQo=xFkkBh`+)H}Lu6LR~byhoUiRX7~qD6*nZ1xza;<|rvCHFR_ua}0{p9m4Eb z>(yv&Kh<{``UC$uZCFM}PRrGBc(5mUwE5VMyZ*OKsjR6@PLhnH*}#YU(<3!XIxl63 z0KdPlKEXdYT7Wg1BhGA;_gv;7WB}BL@nnzTF$shKTNovS4;xyl2Lh(hu3sWOD{CUr zIwiWCv*?^VJ8Dzj)FvD%vf1LVPi5X4A^`lKd=4Vl9zyo>R=`SbSovJ^HrnzVV{mBb zI8@f@?)M3B)8wLlv{O3K4fDz0zfEEB_FC#u_@$@JnnN`Ga!Hg7mpRc=p*$4&pPU6R z9IW4%A1FC~_54aOtRzOuVg3uf*^r-6R8&mU6QO5hv{-+U%7K>BadVThYEOv(5VOBr zPYai=fn`DHQy3rxQ3(lQaNg3vyzAMou4n5S8iqqYtuGGkJuCqWl|%J4`xbY~b1KoA zP?N>(s4-i*;{Q9yTVgAESFG!tPl6cjt*4DuM5%h+#wq%2LNt}v3o+8*pV+QVwQm1R zJq(tz?ctG@zBTLKOmvbH1!(GRZtf*bRmuyu{KC~GZA?TRgd3TY(I zlYh2XCV+H^S+%|hAz%>LCKgsa&|!-p3(X@M>_$Y51*~0gt-3vb(J9Uw%9Q8P!B(MF zE8F?4UoLO6v*|PPN>aq<`qdSMJ{i({GIZ!*A2g2l^!8TCxUSr(`tT_=i(B>Ov7C-Y zpCiGDGkH#@+!Q0!gJgz1i6I;70I<*_k$l%Zdc3N0tnY4wAKK;UejvFN=fn8Y)$okm zXUfS14@ zll(=F#YoBnBwg1z3g<))_$Vs|MRkZ1X;};Y`t{4zzy1OACje#b+{XEBU zLTI;HJ-gw`lYkSzSzQk`Bg^0B zBB8*%)e{L03o{?C^t$KZpbquP*~LX6rPguI>20@-eL(KuovT+&l9@zrjY4zQuW-** z{$opJK3g)(d=we5Pn;xeU-$E>lKo8reJip2!C3%7(+IqrpjZHiRL;Fce9y~5CEvJ; zP;sg}u#1BqX$L<#E)^JM>(vbnaUyj84Fjtfx=EC$3u@aB2G_fB;o)i+45n9SKn04~ z@@k9t1e5_;r*RSjJIubzHyiiW#Rb?bxbTMqdU>|{y?(J3wN=~8>@B~K4nvQwXWM*m2u>dGvFc*IH($Tk9m$NxXz5Rb%+cAZP*XiZ!_A znD5Rmb8Zj06ONi;Ih(`2dyIl(E3wY(f?l z6`?`=IBi_X3S`7QDykJdh$MM)IkFKT$mpLxS!)|a#AL5>a!!2y{237_=Q72EM$<({ zM-TX^sH#RMCsQY4LIKVp&cq%U!r%j;9&;X6GZ;b$HZ-U;G&RlnzWoV;N>p-kE_Ez)Ttsb*pf=)z?{!TrEfqsUj==aoBgf8o z`0J!zFuI%NW(UHT_TK$0wHi0XnQd4Oi9UTeg zmj8UwrevVQsN`@pq2{m1v@-xo7KxIQQd(warnc0G&yr$VPR{q=hl|5iq9i0FgfNvi zk(u&Uc#}Uo^D~PyM1>bvm)bo$JIPNeg+ReExz;_DoaT+;pTR<|)tk-ykvykPWw*3Y zikVb%1d&loeERrt#Gf5>uZQ{h`Cm)IpGQYOv`+P77QLOmnzq2R##V-Q;+t^m6eraHdF+di~S`4 zcVE1C5uuZE{(f71UmWB)wl0_w67CDF(ta%uZv}SKYYxDEw@>!OJsM8=;A?Nx$DlMS zx-SfT*)VL*!z*Gyu6eMtY8hryBXH&n_1Uv$`^ud7L9S`sxZJ#mKe)yM^|`@3QZW?v z83XA{YHKX#nfja00MVVMzMq7H#})g1bujpS#KpOB{SU-3tDLjIix)2lWC*R%oM#|_ z38m$Jrr@zST61)eFGyp}1l%hqE5)1YoOK&8B{J?7P4{*_?2jrRkMldfV`ZIxKNh4B zI;Bjv+@B>dz#GAIG|GL+HH(faFFVFY5*9L;~v#-`{uwpC0@iV0%> z1k(8)Gc^M@H@Byc&zpjRj+ihhQ8+nR3j6sH>Gh8)0;isig!cDi%pBRLJ2)5 zuAGTa)8ITtJ(4Q|VwHWLQQ0;jCgzdMi0_SA>Bj=rE9m>lnufZ%rZcKf^C~Mj09VG_ zoh0S6<@;(^3>0!4G{);3cl73Z^P2$2Z0nxp(_lnlu~^|#rwV5h8@3jQQ~@j+drdNB z-{mc$iV4n(WsSU>bR|g|JKtCM`g}!LbBadwJEu7?N}i~86KlA*LHo0;+5$}_~JDEep(JiP!8yab5!aHl;$>)&7NURz&p=txs_ zg184SHxl5d5XCVG35lraXzcn~JEtpGd(5lyx{9B)aMYHIyOkl3ok0dy@ZlVCKWtJp zzmrBi=K5|^m%eZgmrKcVkxwGLJDAcmxm{rDn1)(t;X8C;sWsAtNPw^2;3i&w5Mw45 zsAeJio{FKi-0EM?jqv&{m~rg{F()E{*J1_bXrdf*)CyF?PYC2S7JW;+hGJ}4ULFmk z&2QM$$n-guZv!(9CLH@Jl=`2Kj%oP#9F)7wp*UpmpubU=ezsADZ5ms%JS?Hg@z6nR=*LTC~d!w8nXn>}6 z{KfT=%_)SZ&Zt5jitYL0cF!i!=;$1I45Fp=?^QDkpGcW1{Di_Ts2$J4%S&|O!UZdZ zHn1rjJ^e$Vjm4Zn%_(0=h^@OcKcOsPYfxJWg!|g;P(B28T3T9}|3SI?5b~~JP%J2# z-@I=HBogBTP-h8NO}-M0ll#YzPJL>4#8lE>(>GwxVa@Uq8w?RR4Gw7r)D&J5q+Wtk z)XdIdk3A0X@zc_xf(?+k*e+_;**OauO( zlUT>rT*>_Y*A`RQSghH%b2#u{v{#=69>4mJwTET=E>CAuWVQ0y`QD5w2xrxWbb-(PvtDz4Ls8I@afRQI@7=K!BFY)AD3E?^uS z#PW=*p7a-6v+L!93goxD{ux+iCzRV_pj|nCgg5MVCCIl&&f%yJT)W@j7Kc&>Op@Jd zF}%63&lmrpO+ZXo7`?5?5iE#IZ4IOBkci!e`a`zVo*#y5ux7zZIRDm&i;H7DU)yjM zr8Zvt6g|^lEc7TSh&n~{`@g<(3<09N#5S3i7|`xa@xh!qDZy$B%@GV8su>XO|&O`2T?EPwhdc)&P!uXc>S~G<-fzT za;?PN{sJs9BNb_i|L*9p_IR&Y$)lDh!Ojh=W&4y3hAL zWHIr*_Qwa%(lrpXifQkjH-WBmYOa65l4g0-8o9T-3o^o+%*@8RWcq&Y3Yu8*R)P4h z7RzgEs+yX(zx%p#qH2R6Dd^%gINZ5Wv@iZKFm?-j%=JPnJ{9x0VT=y@1Nsd_j4THq zMji~8LGkzLvFVzs>VTs|m~8>Tj(a7_w>O6V`(4no46D3vKFWJMA66f%X?+eS=b&OC z4-EF=Q3eedTOwIC1XrEdux~+lOce|Db-b{NYi{}PduBL9FoE%db4xW{loS-BmNd~g z4l#^ts{p`dabVzt?yu2zGTPpy4BaUci%*bO%XBJ36S=kvSmEgs8+q&1djD>8f8LwA zug${JG7V}UFnB~DurfX&A#y0~NDp<3HzdT_-8QNoB{=Vojw8%dtw}K$u}YAibpJnyTKfz(PbNbUm+F`!v6U%+SpP^?>x%r6^;v- zz$2sC?Y+H;ff8F_9`WUZd`b0aB0A`=aMf2Z7lABO_VW4!Fr=R&lyU%Ks<&RP5sTl? zbVHpb$Kt~4(K}b>Fyj~ec42@MJ}Tfw-19mujVMxaleNo63YbBm%O4$2FE8b6Jw9P! zing}4>mM~Nx>-?(f4De#ggPhJ<{a}^!u=>bEMrP(x5=~9Yl&d#Bj>Ch~eL~7iH5jR&UW)fPLWN zQs1s6Sd>Y#xPd?rhY)@Xz<|`{Vhr|imRtNfDNIYH;;wSc|CR{I&!I3v@bC;UFgS^1 zU?}w7ut5Z3P_|pNf2l5zAzfj{=Jxg_UGs*=^beez0PBlOoI9tG`$kt7hC2^HkV4eH zyZOH0UMHW9j=)!FhzaH&zeE~^zWYmaxqu<_fO+dwR-3T7*qaS=s$6qaX3x z;zXE!dBRlYw<<1F-~cc^sx&4WRg4yLprJ_U&Gp5fN&x^Qg$O&HZ~xWkQ@A>(X6Z})GExtz~9iqh(4T%d8a3Wc_FD;arNFkVNe*xW@n%L zpG4QJ?WG*Wi4$F9nxb%4lYoPpJ~Iq#Y)|Ir?V2ql7FH0i;0F?*j!SN3Bi){0&0Ygl z1>j{H@CmT>6VN9<3<{k(MG5qapx+WidW5Y1zHH{zR2l>j8e#-t+;_1hE2g&gA`D*5 zI$kNmAh4=bXD+Bm#;)&GZjH~D14KvCk^t@m_FK@CD_HRjEp2Vm|D?AY!~Yw-Rqy@L zotmxRzA`Jf?}y4{gZWWROpN?Nym|tF5_5P9iu(vyOXCYE+tIl>Tjw);HR_+7Jj9CFT!4 zE@+`HTBrv{`&9)x~) zCuT!kUSyQI5K_|~rL3`^Xkb^9qIF!qac$hM=&?5B*&O`Z&Qv(7gw?0%(2+hO<<7hK z@AoJ8X`j)OH=s{WP`!LNL>Al98d}W;(Fc+!5BU{;*h}w$leC?adVDsVs7Ftp+|bg} zveXqL+^A&UkkW0|>~c0XGU7E*(vLmTAKY7~V}dCwhr$tIsuV<&?D070;4tV{I2*f~ zDg{hHKtTIlTv};uWDj`_FT&W^xH@2}Dm`m3(dv84obL)#BN4a1qsSyn-LbRz9}#Dg zmHnmOiCU-Bn$&w%RmPu+i$f|&t9zE*3&-r=^e(we{rdImN1cnA<7^Y%NP(7w{*zK& z4naZE^5$duGSlV=W;ws#S0{7bJ#idtA9^d1Xxz<-${Xp@p6%yH8{~9!Mc!8>C477a zpK+F!mh`Kik&8P{d>k`vFrKxU`ygo66+zAW_QMC%_0Ky}i?;pJqc76VG<*%ZwXde=zkc(vShKI184AnKe2(TEEVj z$=%((BJgiOkCkxK)$ufLtl7^KF_+mU@$Qo1Qe_zB1MT85?4WV2s&Q+u+WUTfUub20 zDhB3mklEK59{66K?k)VtQc}cc8y*?yM3X9%7`RN=Iw|I9hrfGAHB@D*)x1$WW-nyX zfnVvdqM4;#)N#BuJ-w-?XB0}z0Q1_MVAJ7w9~K!&$?+iS2ct{XdtSYWi3$DmX^Dfq zJ#sJB;7#<>ATLm;NtPQ~N%7ZGHoKb8{X)>L&z3fSW+U&gD`Yi+}bLYnzaGP zHq4OQFMqzuL4Nx-?8S>09Y!wA*1PphO-;&ApFU0Xs+FHTW^B#cDfaaDtQlkeyZ<@E zX)Hk$@#Xm%yJm*$oq($&y}hLmX7-8xU=pwvyYZ9#eF}xk-8ahz6P1>oDl9Ld^z`)4 z$DOVWexCCWBJ3R;jxv&~oqH>gXwm6a84;n;{PBS_DGo6ERCSeg^E?{0;aJeJIOciD z@?!VZ(45|e<@lJc|7CBlU+q(^VYoq+P4q6;}el%?Jef;6V4riPhva&lss zF2Cy8*xA{uuVbe_J8xXA8k&{rMn3!?5dQWp+5F-n$Myb))>TKBjauU-pU1%=Ap~IC zg4r%}SQ?Kb@Kn6EQS<6(9aBvsY=T6=1F96P+FKWlpvIVT=`7ztFBI4X5voYFc+^O!hm@PQSaL2m5U9Nf{$%x z4krHHpw|tK%$qkM_4V~hm7Ixa5`A_`0yfu;asEFmD~sRX^GbRw6J7736h8W&Y~zy% z+}G43O`ks6G9rY;%O~4w5K(4(Q7D2bSSBYYd%k~f?@yJ$f^c$j*1tFt4me1iGyneX z4kW0j=T^P%LAdx8%-K0QEUF0tSsf{e9k>D!^?TAbv$G&Gz`iTTE^~GD8Ke9Q{goY` zWTodUiuiw)mgd3L({BxsTOQ1e1%NTX+@JcWNC!-%sIE?kOH3aOpbhibn4VckU7f`F z5%1UU@6A|6tI3#|k9KzpQHqbl-t)HJZ4FRAJw0U=71db@h-H*|ou!1bmoLc8W#!`| zYH4YC#7!mXx&H3!QfFsplCU))_Kh3l)YKsh3k%{G&+P5n!L&lpey{>$?&s9&b>d)W z7o83b4rb-&*Rb`M`T>HcoZ7^eIUpbaKr$ScgfT2FjS!eXLeQ=764$w3*Wz2CzY1nBrh)ax@m>2!j^qMBu=+n zY&X{Xp9m~0F2=>hy^3a#u$XIk>C~bSO0XgIQch`}oSr`7-W%Nc;k<_vbuKI|rwg}; z=!9_w75>U*ygW8|zA@f!ezCs3?s~NT6a<5VoaS6>U|?`?aNEp`iOU5p5pB>;>tAL$ z|6|94i&P1hg`$Ft_OW8UBZ0u(uXu6E$>AAt{tMkKfy&CtIN<)&jI)S7BNKO;3hwDq zEnVWT%TZ=#L!xE6IKe7xz@OFXpxO?K_@6b?RPdhX;H1Wl= z`0J9jwKYBw5e@_g2j^~)PHoXsFkhAJpvBs7-tN)SePd(Bb~L)CmY9%m*pY#ZENFk< zJ$=hM=`DK1b9t!LNaFIiu62x%6!M7Y{ps&Fq!li6^4#y$`;tZQG6T-hTswKeK#KmuS$Fg+4 zzIp4Bj!tfklbc(mjZvO<(OH?HC74?1`Q$JoGbX2{q`g+O!+WYtK)5`f(Z{yzLC>pc3)U%cwuX@&;pAU;mP34up*c7RBo{`?W zJQ$1wi766f`S7SHB0w9a9G{q)3dFD|2v}HH1RD#`fC(1>-=G3SDZZ{e=ZJ(+l6Ab) zsK5aCWB*{fjQ63yUyxaqeSGS74-cs*VFR*S1t=b&{mKYik1l=GvyIN4FLv4Q1lWybIu)&Ab=W zl$GPgOhoxv+GHMCh#q~E4i3^j1CLI~0taj(=Dk;&H`&(Uh@OQgg4l6e=FkCX!1)>% z6b4(sg#G@0dwa+&3{O2=AM5>vQsf7ZZc0-ypFGjle% zkAGaj5vuO(JzR+CjqRC6Rdw}tz*gO^FMX9CJg`4{^VU-c(PO3kJc<6(%l$uUo0C;| zAdjU4yvp#rj8|7Dm#e&6I!stJ$9hMat4Ol%wJ{-+ohBMa2@Z7FroEK55Sd#^sm@ zH#UR75M~4mEwTcmKup*$UWK;fhkywx=_aM#y6sas1}(?a*-I z3NKWVJm9UEl#T$$lCzGxjoH(`h> zssu2-yIx$raJV)i`SM&8mz3FY3l_(OOj2#TErXaY26%XjMnL(|BQjwTk)Z|;TL+2o zO(Jye4aXXEb&h7bbjW0ttz~Bfwa-DngqVvDhdpU%IJx6I!4r?kdN+>i>yuW?i{mYD zp+0+g^u!1uo>$~?A^MjX4Im5-*fsvknaE?%}c5Co+ghuLW9} zp#=p68fKw!FI~UG-GWI*V$K1Htnk^haGa>X0^5>Zld440ai_gwNE#uNT~tI+tXJLz zVs&q=AkzZqnsH|eDOwZ}rOGyZD-0lXM@PrUrY1%j0pq)|%yK0&Q1`neIM4# zLQat%9UbY{I?@4Ti#a@ePQ)ZVI;FwRW$vDYA#c3Aw?aZf3JVJ%kmqZ|H^GGqRm=)P zz#P0aypp_(-Q0w-b8_aFmO{W*w+#*sV!9%=lL~cs0JF>b{8`(<*T8`8`}gmdT!u)O zrUtYFccTKWL%%YRjs^*(^7)#!{B5n8)92>^qq|^}7DV5V8gG z*LLXzZ1uvl;GSaOO`>)adQ3mN>wn{l=YmZ}hGFK5rzNV0PDfAA@>@LT>Gj2q8{qatjFLYv z?>{yVzr)Q9fiO0DAb3?%#6DtmZ!@CZTbE9kQ(Zvn>L7U%PEAE+-W5eh3pa-hWXRoO zVPOH$m8OqE=QdP{wF;Q+NZ&lV3EB4S>FVsXyNY4T&p$igDBs)f6n{9U}Z|=G# z|14dXl0r`_gXh=_fxFkLK!PG8BL{ZhNjvj-INQ7cnk z+5tmTK_G5FqwHgsmi}U5Ym(>=Skb@v-09EO>&`tfD-UT#7-0zZ?Ck97$}A~wW4Usn z3b(AW?C<#ntdIh1OoM`f!oNwVZ8V2U%rm+4e7Ldt)8e+8nwrnykV>URCt*i- zHyZo`$7Y{{MzSY-BO@or0;hq~h-aC+BkV{#6;XQ6sIq!m8?@wL=%=*jr`ZB?QYPtO zOv06SnQLJJgtDipY=~~TX*8QsC@Tp5H@L*@DPj&&qkv{f4{`^SNq^WV zmh#=V+Wgf}r8E~l3LtE6+{gvF^!>f6me$H8ht&we4}1LQPa2?|QfSn~ym7!kz$@MW z|918E{@Hj1=BPH^d>0>|eRg0OE0^y}{?yM;8h{izfM9A~T~%QkR7*;;-;tt?P5!qd z4M350e|ZdMwX`r}?moyzr87`=ba33LeNAO?#IIk*m5!6bL!+Zs6&b0isf9&Fb_b1S z0cZDtIu2j#j=jYe{Pvqwc3GJwKAW>DkdhT2urBh@QEJE@LL(o zqybm_cCl_L1%L(EPZ8XW%8a6;zWuJXfUEO+ump@0!Tp)gyTr}UKW3N+NVbq=m-*TL z5(h#Brq^AVY;w9U%RM!eWad%fFLX^tgS1Wft-P%_e`b7Ice+Y z2;{@gwUGkiG7+0TS}JO4J9!N6I6qvQX!ftmZ^eNCl1ffPgZlnmy`!@exzG^~tJ5;; z#}H>`xv4+rTTk`$P@E4F%|Q;vXf~Vv6ze^4vgCG0;lN`&m0{`8wbE&+!i_r z`d0uf4wRyDe1`9NPU~0P7L1GxY3Kava%BL(G!(_=B#s0hA{v2ZIWh?ZE_K<(=RNPYpFcKeV|;3A*Ckyg3gNxZlt4n3yiW8pSd@19zIL zFF{=01L+5Ytn2IcxdjCr+}wp4rOS;{{DFsKI3Melf!V}&F8ryy^_m`7OWg@t^5MC; zP=K_Z78kJi_{nx0NLMV2%dU3!*cdekd!{nMzFk)XZEXr;6BEwz;$F;-GBHI52ji3& z)31$@6GV=a9JY` zl*KWePRyPP@dB{$5sbdf8IO04W7yOf|LLoC1 znGF&h+-XYma*^e4#VPAOb|YjQ$%IW;6tq?alcxYx0h>C?M-Rg^NdS%xgg%~Wz?rp! zL`csk&>plXXPTLq7~0l$H;PuMeM*DvGfyPyTfN(&>)F0_&K&!*4If>_=dqpWTwmy3 z-op=k%?{s|fNgPkc93Qfhp&rI`Fl1mhQCR(T?g22Sjz+Hw*&*4EwzI;z6!Ntk2I7`h|l0mTmRkp-I%bt-iL0(^;jXM*@X?&hwtO&)>WemXrH~#nhJaVjd^#J634habw;PF;r zhK%oA4ieKTclcR`Pe5=_QBLm4YipT;K>3(!hYvY5&EK{TS^Z2W>ZfFVFDhI z>6X!X9988LTM?v1?+$cXk}>XTkxE4A z5DS%e{S68DdokznPv>uNtda|#D-5-GHpE`Mnc8DI>6j$inoysg-<5p<_@m2XBcsS} zX8*qI3(n8p;ta6<9jJWZb<_QmMqs&s>Oe49RZT*FD1rIeLndH`kB;i_=~uK_TtMBa z#F=1;%xYbh7UGK-eS}8VT{R*fM@?M&EhEop3FL1zyrxVX4DL`C0$H755sw|5wGg!{@Q zsnUBEFBcOFnk zXa$M7EhrNhJ^1r$Qvi`mtJVjOIoH-3iJc@K|Ib_~lhE)xJ5Wso;tvQx8}koHi|;bw zq%87Lz6Z<6Sx=P*`UTK9tEZ|9WhRY=TT7mieC9lMd$xd+NFbR)~3-4fMP~wv*h;!iLZU1Vpbgix^ zWflayqIA4!qnGh_;0{#$c=J18NUvVK;^yNM_2>;ti=08J9jGn<0Z7+OEQ1WswLL*% z|6))}OHQ^sJa3XiK;=;Q36FCgWPSYDwi?$VE5a6^p%fh6*eF9q7%~J-%s)xh56B}3 z9B_*q7W2hqg1AFQ7(!7|F^XA@k&!Vf5L}4D;^L~iV7fQxU^V{-6pf9IIeO)$J%fXF z|86Dt(Gu6uG|CpAs1$;E;@}t_SIr1cYf**bpf6a}kc0--6$7qjhL(%=r9$oiK%&`J@ z45<6t&R`U`gTuq%f`XNodez-hcepgu!jw=XNT5@s>*&NCY4`W{pVy+9V=-rxa>RHn zj933xCLE-RdpbHgzX2ZnYmilV!0YXMJjU7Q04PE9q0uMyMu?`hXS*WOj&(HmU zGUTm}=y8+ONTQ`zlQnOHn68s0_t^c@Q}dm&m2?3C<=ww|B;e2yD>}$*7pM?AxaYYh zHEer(n^yjXPb%2s7?QZPG#*6IUpdn`ia|seV&yq9+W}e$oJUXq{iwrJIw8z{u?!kP zcR|lE#Nu-#_A0Yl-$N$$5lQQ$Re-CY4iK$0h~?1p!7LzL#(Ucw;4mZz&@YTZatg4e z^p`NZ3};lXZgI59n85m-_v(A?w*Jw z`AQEIp>ck+j3SfBJkv%{YQ1qCFSc$z?v{F)RL!W?CqkJW<8cnhx<2u_3c|YaHZ!x_ z7ElsE zVF5?NhOBRFJpP{rA0u_E#x!{ZFlFARq(lH)5a-Vd2OsB6Qqx9U$f=epzJNw5uC*(C zczBqMnz~bjid`EnK2gY#zzmyE*K>>nJM&%JpHf#BT2d<(!oDazisaER2# z)US3W$f&OW5~nSDh8ObxTXAo0P1kYGH1Ll)?DW~^S5Myl2e^!ne#wFwq&IxK4nq(* zox=D>H~yut)!)1s;^0zQk`TMEqSb#1X>lV486c!n4kqL8gF^+#>ErL8lWV5%qSBp{P)X_W&ipGsQc@}&s#A+4 zb?%F!H?gs#ib_hJ?~j! zc;(rbE~95VV{>i~s>^iXQoZ{l!!8_gXXhI{q;w2?C_p_I__tYAQ}Z?`tx*FZ0fNrJ zxWWBC@S5m;)q=+C!a~)R9d-_m`;Q;TJt`i{R*7X!IdY!)!s&OkZe6hrlB=?+Y8aU! zJ|r7lDt2yeRoR0+5q2&v6@0eCaXvmiDv(AVu(Xz5otVpA6t5g=wTUNuv!a!LUbyU@ zC~UqgR=eeIN(DdSCo@hv8+xG=>iP{0KC12*LK>(>B3j>h=}1RgwpE?Vn6L->2v zK<&vB%6ICV;A@B=;Ce2my0i=#iLkOlfcf7Qh@}j3KksF{13tq56_Hwl7OySC%QdaQ z;TOjT2TE#c1VR4DEZ!gpD+k97%dV)e;1do|C@C|TEqcNQv?KEB{HXrwdRRllacl-NqJgA+#(&ZJ_d=nxiUKlGZAbaJs%jqOb3lACH5y3u_5h!8 zJy~8}UXrL?{PKW?wRPU|pq$@f8=x>@f!7C_xDFYTZrC2neL^cpN;_m`I1X?DF-~r& zHK_2oRGB_{^mbs0pFnOfwAys`@-V9VWBuR~gHhF8|*Z_l3V$^^f21 Vk=n9@(kld_c3)GuOws)H{{i^r{FMLz diff --git a/docs/images/chapters/graduatedoffset/17bf62e05a1fc3387b0c210f2decff45.png b/docs/images/chapters/graduatedoffset/17bf62e05a1fc3387b0c210f2decff45.png index 69a495e83feae1d0538c5ff5964b35be902ad32d..318a9deb1559bc9944fe7432a5d59bba96577f7d 100644 GIT binary patch literal 14415 zcma*O1yEdF(=H0(4TN9`kl+v;g1ZF>1Pzvf;1=B72TOvx3Uhjt`qgG%!Ar^{r^9uW!&B z3K}X|-#2XJdKA8asHiyPVzN^4ugLX9*PR<<)f7zMic*ou)lKXkItwHuB(S*~ba*TK%6#N41ZU-cXRCg3kc-r=eO-T>Ri-KnRl62E2^sYwELm;4G%ZVOS*ajBToN4vTZQ% zhpA4B`8ApXNVz&FBLj?}q1jVmGJ-un-#^#n#sT~+Q$QLVn(!*Uz0C(j7&97t)Ua#^e~1YD>!U7Ni|+qbLNBL3uh0wz1+q6P%>IxH*kXe^Rh5(2cNPu~oWiNE?(PJc5+o!ST!)8752axnf=$Kb5__Q?mv=-3Dal)b zSr9$>t~n0dCQ4!>BbZ3J#f;+bwqJN(d_+hZotf%)s^jMC>?jLWiV4pIiM@FK#4tP| zkpnr`A=9F^FpZ+IXN`%S#&<74e>OehCR+O=`_V}4#Y{`-vm(TQACH&Wp!)@ zeUe4E01L*(#(q~(Ia?t+pxN17J>K#@3qI7)(4b>sY1v^D2VEbzvPkkA*x(#)GSg38k#mxb zj`NkSy^kht)U;YpL95>Jy9fxE)Z;`%r9`VQuI1k7;E|Lu-^`RNF+?XWD(ct5VfUsz z2Ke@$H?RG!myW%9R9kMR5Uupx`uT8o69Tg`r%Zd7_Qg&aOwkh=c*mjaFIQ6b%-!rOxc5-rh zV!2`{ZFCLAFvR8y*$l@Ti7B4{tl_G?{)iXLsKemUVMOyXPJFtGo zvkmF5U%$Gp`r#oko1c%%u)xVW7HnFEe!oP^xastXW_>Fy4Jn}BKV(Z%lG@HYTn`>g@oX4W&N1_IC2*KvVqoAYtdlEso%W~M zj0fY(uyDW~2XiK=*>!bl%8j;CBVtt4as~e1irU2r{k~mSs}!k?dEeU6=hi}_FXqbm zN<=^zRK3YYaKLdA)SsRGL#$U9Wg)3x;KQJ!poz4iTeRSNKR)WteqddVJ&l)YIey%TYYmkaq$1&p8VPLVDxZA!c)E=V}* zGuIDXA+IJ)1RECq++!^E>pQChN1n}D6aj~+SB-}Bd-{P~?L z@9O%*-tcoJV&J~?M_lyuFnfe)5_J8^Y%xzps33-6Zk&YnHW&i@O8*|IEah_`r}jmQ zD7PxkT+VpjzLpIj2R^?*@L=a*1ZbpliY}K=w6jo6Yv}Z!-=HhmM=&+*B@cGX^s|A! z1_RbP2LA|tAtIYUHdS_2^@IC;_e1P|`ko~bOC?xhv5gQ8q~~f-Zq_d8_Qm&jIftaO zM{Oq9Litm3;!7*(jW0*;jKb)|#M2#4-)XfoNkCV{pYnMwpTd1ZyaI-AvH&o8=5rY$_BhqfDm(eWHZoyt6j*> zH)1Kp^YIlBb_4mWK4;*e6y{4UeunI!-&1{ zvJRqF?2yywPrKa2Jw=^Mk6F9e{h#u#8XnX%y@?AmRL2TSs(v3QIV4ZdW|7LJ* zYZ!~ZKiRA6Ly#&kY_kh7zMd=|Sqd^1v!lhnokhO4wPZ)n%dx|^-p%>3f)$niwX@BJ zUIra`qJRr1no^HKY;j*`i^||PUq+2KsqlLk^IAgbHc1a#jzrVBg27oQ!?58?N2@yRYXy*%O$!AeP^alUuY1NzJRRy_W~m7Z0cdm3Buck=H17rWu2 z!1*2=z^qOdi%F~UHPGw#-0_0P1+EkE^hUCVyaGe3qT(;=-&RLF>;0;eq+RgD?rBuX zOu9nQf$F9swifmgAP{UFPM=aMW){EQsM2gct0iga-@5=F~_wJ;87!Ik=1w|tB7 z9ADaT*Nw)|`P9I)V3;;esfzn|fJ%6{pcSR>xFRtrld!l*)?S`TL;RUQHVw5qJ>hmrCZrdo_Rn z9cqwbHoZli7F=lo(JLy*V14~e?pY=6wD89otBWA&@SWz$BPn%#rL-LAsreK*u~B7( zYin-|hi`nuOm+g}zgwjrKBW@udUlLyMU4WKge-h3>s=W$;*P-+heAFb4&k6V<$?v8 z7r+CyF9Z=AFlQ+fb?qA#;7YEF?W=UZV-8HS(g)la>h@xdbMO)4x22=d4mptqp$wSJ3$w5+qO)F(2-PKS z&~sC3l{Y&=({hiW-~vg|nnnVWReJeW`@sCJ0SfaZI>2m1%{)t?_ik=k_>*CpydH?j zj|hHm84u8MYBEiiVz!ow8muS)*ND0abS5#V>K!~`daNg#6C`!?xGR4b_r*Q0R?ZD9eDoLj;53y^|6}t^BS-Y!xrSnNN!9O zC!JV^`2`L4l-TyCR;czz6IS$3WF<25Ry^p(o(Dg6t0N%pf?HtiQc)OiW(l8Id1xaI zuWj~8H!0Yw33^t*e|k`9MQkFA`Unc<=RSFT`t^a5B3?+>FD`bnS#z1sXSzTHe^?yk}8e zVvhP(e@pi15&b4?s%<+m4U}QJ-+n1^<5k*+6-1mjowEuqGgml5$`=liE`Nr_y8g!r z*OjFNoNT$bAf%+q{a7of1gh92fjyKu+pSYM)5C zrg5Ic6ndjoUcfCPzFIXfa-Gaz^WHPx|J=Ks%#ow!FXogL4%9a&P*WS_;2J72#WQDK z1^*#E6>ap5ryKCU&OFVX$!FfOxB2HjQ8r?o#S<-uhq0#L8S?-i-6NnKlQ1=+CJ46X z=$>T!YHtjP6CDtzmVN5^s8efquR>wc>Aa~6#B4Mws^F;9>%+!oW{xgj7#;6gR<~DR?I)(EAkpUvGa{&+&*A(Kk>XwfYA5G0~%HVsIy(q+}{iXRa%2T zg+#7`bDI=AatePyOZ4G9fB*g%@aCwzU?`ZU27KIG7y0`;`u-MPWGMZ;a`WT2Kww2_ zHuH5Q&p{I`^elC;j1#6)RuCZM2xO@7aZbiy4#tc4Hg%LY)h8A5Hl*JQIXQ(py!P=I zQIChTa-$&?u=M>wafJ9THUt{)kn)(lBWW6wr0e&3zB=o%0Mo zWh1q8l!CW!kPeq*d)W*o^Y8YTm$fBjwyksCdFucTKI_tNY(?H1d&>1=234-oELe|B z?aT;oY!aEDC}ItndvsE_HwldS=7@+G0HogH6)iMMv1LaJ^W1By`QrLLSS`u2^okXN zY7P1%b9KA)iACHVtv*I@y5+B2C~DyWXX!5im}8@(KbNc}LAVta6~M_jHzVT(Ovf9B zby{#LaG$(v`-jLo&)U6n9m>oICJ{IGDR8bynIPu7d+)6ZM^iqGhWbhq?;IRt<>Vme zPPmP&s}#Nia2IWD5&$Kj63~1IYglh9Po06d%e+s(=~#&#+r+Z380aD~IQtHZ%kZ7` zI*>MwEe4{a(#vmSw-lyM8-Q$68EBhl1Vv2m4d=f;1S zH2>HN#tcVTI(ix!^mFj!nO}jDphWgV@rqMRq0c@bIz!rMEl|8*9+iND;1ZMc(<~7c z9KHy@n82NxN5EExHtp?B{7sE?WF=%*CObbnRuMH!5G%G4_cGoxz9g?T*NIOD*oULba^>EI?T`4YrUNg)g zam(+~$GIP^52OTVh`~uaHqSqb_;m>L%b&(c{++I~TUgSYvT(jRz5Rl#uGkJfZk}5tz4$mlf;?>>M1FLR+h6 zZiwA$d#x<|k?V!-$wxRAkwDjzGj?Z8%k+ta|)Ix#8*Y z4!|x6F6z$(ijx=S<-0moS0ZTe6|YaU@RZItzOS%ccPIgohKvcg@2m1*=Vxr14q2H6 zM6ZY8C*y?@vdD6`x0%b0sE}nbcQ?K%EI}~`@t0niZ(GKsfQ#qvz-{BognT6w`8{(> z%V+58)A=CBLj)HmXJnkq>3v8}PEH7EVR3PGQBf#>y*VGwbM2JtXh>#`T|C9b#&Q`# zb5p>X9L;U>E z@S+p>Pw-B5b5plc+>giwA4=n-2L=XEnyhaxcJUtEo(F&0iQ_9uPU?99sJ0f8qMs( z0q{>zT^-d|IBj}-oZS>_?dyTF!0zrYUGVQAW3)!7L}q$cmainKoA2FGqfIY=c6N4E zRTaYMt(U;`5_Hfr{o_Hse)EsOc@FLG?~MzZ(eK+%y~!D@LI>@`*ia!BQXpIag&iQW zX%bN?Ax_DX+&q`|7oy zsO3sC6u>PVRy?n?%flw0%sTj~AuNWNgkN!X;rVqEFg%RegM zs`;rR9)?A>;a?ij!OnaSQ@^U78#Al^zEQZqR}8zOb5DGPWOF>e>xxsei1$ZAg%tMg z!z%z3dD-jE3fM@l?^jB*=D!8d?reu7FudYdyxt56^H{eDA#jgtfKGQ*1IRKk5SG`F zW;@IbL!5XgzFo4G)wpx_JkXwhL0_yeln~_d5a*5A*4}PhQp}(87-{Bq?MF0iVB&6Wd@J``JbnEmt37fy(dlefVSSbqxnE@XDw?&x_@j9CVc6jR=T#Zu&BRUcE5Y_FoQNUG(0}7xNw=2@rTQ! zun65@mwIit;8zw=9gK`G9_B1>txb>ZA!`YG5ouz68oY=e;h;IrDO*5wqLS4Iaw5Pg z9-jmV-~a0G|N3mt5@CempHa`C5jr#iOg>4nE*V(nu8)2Y54~N}KRR6>el7nR2Jp^G zFx*@Ryqle8UaVQmytfV#eSPI+{_e?T*X5?Ow&SVm=mI(4!O=lRn%S)3r1ss(OYp46 z--S?yWR`vF5_VJwzNv+U!THvZ$4MK?qWu8V{dJEp4bN36gSI#LUarbCK9!A3eC(t! zE~)N(tG$qv%JW~iw7^cA-gaYm{r>#EdQeT6`rwota3*?<5pG?4hI#R;gswmEI0!I}!Q9-;F~2w}0H1)GH01S8LXzG;rsZe&y0#qIjbq#_ zezN{6Cj8bNo8~Pq_=3&`27eES!x`1eeww6u{;8_^rBqL(Wz{rZ6-K6dkJPj9CG;fs z?MX?af_p9sJ?tE`?LbEtmrC1JQ2_LBBR5I$$Ms7kzGEuv7m_O!^sBhQO?jA6 zx}SI=4CyL5(y2XV#C>IVa>@@NWy%T)k|wWNq#AC2XR$)w8dEyEU+Yy=K0b|Yny;1Y zDr8Re>z-POH}5jMLiTxHohJCS=8hx3(B0g~S&)Lqi{i_(r+F)jl>mv%Rf&P`;f}m~ znkz`HRt{53pFgI&go#`V+Q;L<{t%_c2dj|jV5MjFNFL5H7f2X_VoIz{Zk8}uCdR9i zsAEUcWI#OWa$=J0hOdB#a43oA{Bpqtl9V*C^7OExq^$f-Wg#-7EZvL&Yu;vwL%CVsX=0%5PF0>VcA2!rWtHf~SOg<=vzb0=cH z-Hp?L$zsbeRRD(~e%xNi9Ud}iSMBlloXQzn?uf!f;d!?mX@&muofZG={BewI?R25u z;wcg8$m#T!XMvSB@Hs~vVh_MQcC5x?)H3z$c$LCvl*H3 zV?_7Id$$^^1qmxFCWoy-Jzz_I;HeDGO8MPhGQ3{Y+mgU7iN$w|i^=KCRxrxTD0Hk` znAOBjR|FP$89ocU7PU3_lajX3)uY+0b9%k_$o!c5c!wL$kVovq zA|F76pJ9g&rvQME%dhi-KUkoY&&fkAPD!fqbqly-`ph1FlB(>08N*G2@n{_@A+}4z z=x2qWUlfSDOwKuu{nYX8;d)aJH$DeK6XG>vt#W<@aWJxzY>%+wOPE(#pGn z1rdCIR#FcT7Z`Ts9onIk*!kyBA#2BF^v@r%(k9G%2~bJqPoBC$X~UJ^#A-7lM}x#U z!WwTc4XYgFvD_C9WCz(eR5phy<{Pjo;iCqF%(`8^86#r$vuhuvf>}WVl2nAt)`AeK zGFk)#yz6H!!1Q`T8G<6`8*~w4v&i*+)DKOCBBviV&d0BB8LA#nv1IF>?9R6Rd89}b z|4K>nFkxbyOKe3#a<)v;Y|A!V=3H)P7P(LLzS?hzIiIW*NbRr&IfhGd4w~^dXv18V zIkNX`Z;@mN>DI(X+?qs${bR0E%cjfsM<#8qc^KDeIL>KVGDpyD&Zz=eT8|vKLmP09 z@F4+0)c2`A!aG|_9l{-}d_Cg?PZARIfl^qyTGXfnGo zUEtM6f+EI-s2-vkdy8e)ii=Y&to9|af6j_^n!}TI_u>P!#PAbiV$8|o9p%n$(Zrow=}oUHIBZS!Nzds_DwR45yVf_LODUWH%-I zgcytTrCnp)Bq)JjLea-YKlQon`-9HLrsbvpWdmzN^I=~7X0C^K*9YAELiae^F4jGE zd-yjFHU?`mPYI!?5ap@eOXR#t{vF4=gm{Y_&H=I7;8e*E8%9sNx(+*oM6xXEeO1w$ zrd&ZS7azvZ?^x^qIEd-*b`w=3d{1m^vElEUIVSC?Vnjk|L^?35X3Jy^*;Vgq;yfQ{ z@_xie8MZVW>BEsFIaQ~@T@3HCb*iC%_3{O;JaPK|sHCG;6(q%hY2*8_@kZ{SNPcH* zb+?;>y}2f@aCNBKl2ErvQUV`E2yW~^#)z|LCkpMOUuc#7QOXU)R^i25*A$Os#|C)t zZBX^pK=*ls!jbV;b!3ii!n$nU0IZR@Yo*HAIm$|@iELYwHZeL^-^$AlVzc5^)y5~p zko2EpbIPHE$)dlEJj7P9FTl2`R^4t5h^iCP4GN@PbFSDEd2EIX`B?{OPoJl=@*h-< z+7IXGKPOEL1ToXf>qegP?G7^bQ;2t2pCYJiOJ8y}i^dfOCH0i%@S5bzN6`5v=;6`K zID|_Tsmtk?u5^o@2BhWtY~(S#+;cKJ)M~A#)y+bS#Yd zuk3!X5@Xw?Hj0<&QM|k<{XySCeo)93clOV3PNzn*DtMa07(2576gU%nH-5EEB00Bk zA;VAHlb~|uc9eXdS9BrwMwMFDLx?N@lO!WOJ^e|y2}tHXCRy&ok)F=b1NbOZzueGq zM0gpo@9PK+8ss<{hmNfI6LV0GhaFf{Ismr2@ri?S5EjD3MaV?Sk;}^N@YNebq4n32 zCuUr!52-HR@=r+04=m)R9;?wvBSLX&H=(U)0k(~I`n|){^U(Mg5_{AWgJJw=5+N_+N1xP+RDWFpiPfrxrep{3?#^kn2|Fu4WmtV?g>pmy zLY2l9atlu_~E6H@~LM+TVk62trGmk1K_#5j2q~pf1>AhvdLK1#cG|MENARf z6RodI2BIS%BI*K8d~6%Q=;Xxn6H+}5_*Wjf$iumK>z-`-$8yXG^tjpEErHxX1hVF_ zyn!1H66GLr@Hhs(^6=QV0i0KQPItTuhaJmKqzWIze)WAW@9G_VtDaRL7=ak`BeI0V zd(CiY;h(K{tgF)s93{5RTItfZY#0z96s$G-`n;>DbESO+)HgfcCBv#csQaj-gKm-K zp_?-;k6cPQvIG}896{!?Abyrl-@A0ZCVJhk4)wZgX(9!xYh9vj#vwN2>sa(v{DOd3 zcpaC2k&Da7YkUM?3wHGUcsP9l?Jq=EpXZcMax(&^EB$Z?UP; zf7rOz&?cKptQVj+=44P(f8|t|%&fLYij|n5DgS1D(|EKQRbN#7Wo2 zM?49i+}HtHS;*!>kKY9+30m`~@JgfYIaqUeGChH8b8YAa2zG9ySWWnqeVndu=fRg2z@_1sWw-B`Qi59 zyToGDm$o7GEDir*ut6}ghqdBQk{h(; zu=Miuh6IziXTOYu9}usQmK=tme2VB%5nt$??o(1##(<4H?u;;2?7zJW4$(6(beLCH zOjg7|rXfh(a*jJ{`Z_OK^FW!VW@f9a0c12hJaGjDAFb;*-`Ciz3^yJ(4*&!#{DCiM z0}od#BjedJH-jc6Bh49v#r_$5^|(WQ)H@hs&AE^oD3sHZ-*@Eg2NQebIUcoMIt57x zQG0aCFYZA>)d1|X8{1TYkcSOBOOnGn!acjm)J~iIPWDG+c7_~~*T_e9>-bRT$Qu#S ziy*Uj*#vgKOvwnvBgXCYrhwUvSuxo0x=hDyIb%6vI%hR*<=h~x@Qt*+gM)gt`MaJ7 zG7|ua`E>o)XL54#1mM8gZ4D9tq_xf<9NON#zSSVwl)oXm_Y{dTTl{qWt*5{J{gI7_ z62&DYQ4oMLu?M_Rx9bP%$-G4=o5eJvxrK##x&BLpmYX4)crJ_g3W9K4Wo6|i0HQK9 zH0(Sp1Tl4`8Fk9Jh1TU?b!(Fr#tlJE3 zUIqyH&Wc#9ZYk^kqz@8?8GnYMkh&ritH>A`QCC-2$IH?b6cn7?J$USEuC$w1XN;L5 zV??4wl(N{~>ZodH^gknH?(PB#LN`>$uq#ru_l!RP)EB@mxh-c~0WcrI_5QTS%*x8> z;TGP}oMa|F(rN%-v%XrZz8SkFLUn{v?KtC{-LZgS)23o^efLr6 z>_<(j?B2~aueM1FbOv)Nj7>}&u9jRJHhSO6L;3xNjFCPt zFckh5{m@55v)0;Zy*ms*3++b)uK=oCYl@(|EAYbU-NSv4_dUue5q8|=H)hA(bNO5x zsC-Whz0z!pxA6ZCXx`~Ix}NF-EXv_q_`boJr?#t&-kKyejDjw5?9Ko$?mf zJ{EBzRSKzTX#OSmfO?A)y?9zP>;6DQWro>jOVdg{yG2a^H1-mEAQ=+#-kTMx@);e)qBhJmj9{X=pe8nu|8RT%&SN`_Hz zDSkZ*jEU)4odMcvVw?_`{=gYnSt47fUj1_0{o-1C;Irn)KUZm@;9Vx0@+krEdd4Fu zo?{--07dfTpmIb}ON-G$#`l|WE1!4UzOYSR_kb%v$)?Y{mbCm(Ax}LGo z(b!uD{HR{zDfkDFXnau-BQTyJrrO2!z_)R_lQh0kXRP(# z04lL}S&>F{aD>opAQB4Z$*e(y9(U56f*-b(w+6Ysc)Q^njp9Fc>)jwAP7=IY>?k+r znFTn*!q;tBw=-yfCu6f)>)iLu6GbP&rcq4;6oJ={0*6+<$rLsUOhADZKdYc1*zJ7F zt^81fcNs>{QXeyAPciR*w)F+9$@w+5nR59L$}kxVTsr z${!1mPubnh3?2FVj>VL;wJ}(KV5nCF-E2v%e*FU`AtCX&UJaO>1`-M5?STE)opu8O zu8F*w`?1d9Is`lh8Hxh;2}4xXSHS5>Hq8XZD(++8^gamf^R@~|Qcgo-9xn8qJcB~d z9z(oBM=H>j97tpMHtI-mTVn%kMML~oF0gNmM=VsGQxtmagId4CVe1<^RVy7ZL>T$;{E85Bb(3;Fp}#IYZfvz zSg4|te5JAXG+C-l$1)!8&^!U;@*XhKKX6eBeQX-(5*ygJ72aT=$tZvs{9p)+4D|O6 z!jA!Nm;;muhSx5z$VFYcdg=IA%FMMoJMDL92}4_ewyY6C1b>Lj(gllpdI~nW90L*+ z1hyT36y1HXI|0Cmi*TW)!mpP<53@N4!}Nxj0(%7MvzM!jXJ!l#^9l>s0Vc8KY#BMt zzMud4_MrhQ{s|b?ad};$EXX(XR<)})iW2ZL(M5P)DX?kQ3LtRNK9P1YEu{(_I=emv zSuGw_4Hn#xJG3ARfDML@=2MKS=Zv#;>@oe%H@ zfRC*=`EGe<3cz<%BMi$CR%3>>M6MjinLV%p9YuHhP2AqSr;upS=><%CXOx_fkg#md z65;-`wk7+&t3sw8c{{?N3CVHHV&5xu@+vDKiHU^C^4sDOn#C$bDhz4TNkFJNI4v7W%uS4f6DlQ*qB*23}%Df;S$bpLHeN$0EZXM5V9NqnmlrppXn^C=D zKVwG69Ec16f48?slJ)g!#akHINNe0R{Parro9&2AsFR#TM`# zxZPvP&Ed2+iFimvM5GQBlZ<>iM@N{*Rp>Ef2We>y7M3>8KQ#cw3}+Zt2+qT&Cg%44r_j4qh{qHpEpTKNJsJZpI>Z)R#r95_N_ z1{jQPGrdOxXcxqe5wSpk_7kWTnJ&?GA6s9ju`;+h+f*x7`T3|`=h>3-e?(x&p2ukW z&s|=6rn=7x6!Td9IgYR>{QsW2)jGXqd8TVx?2Ln6ii2DM_w$@W zG_qky{I~vs1|J<0d<&!y#X+zBUny95PN2N2&ObOL#0=mD%vYHqEV>`VWD_}egRkgQ z*bISdbnJeYw>=HbmAn$&?&8bk~s63Li|rOL)Czl>)U5s>3662 zTm!cJrunkElzxAv^!SMbR=INfM08*$4vK*axO@aM&R@;U@)b2T3+aM0?t$1s)$1aY z3`5;yturXN?Z-D7+4!{Q)X$e`pmI&U#S3=Uii9NwccYelE|l_z1BM?1?64m(11R^v zz@D5pK0AvKKjnkU8@2nQesyqgP^(&F-BC89rm|38_8^TYrE literal 14364 zcma)jbx>SQ(=Q1L5D1JXBd-{LF{wxao>ZeUVy&6#|di0 z#;#T`r=7AX)hN9xDX($%ul6d|so1ZXwXuPgqt(>9T8R|lMFwb>Q^ZR^5MCL9rN0~w z9Nzo*`*-Di1+lCof0Z*IK0p1dOr3n5%nsRHI=S{4;WY>bfpPpCd5CvE^Z94&<0&gn z;$BjeW{j`&C={}rl3-to3o0|8?cb762fRlqr05XG3i^&^M1dkI9m<0Jf*d5Cgc5*- zLm~G<1mzVrFeWGn1y=-1RMe<$Azw++5*Ttv`;e~Sz(EL>lY2df7U7kXQG};;*<}^| z9)+Cz|9ifmu>W=Vq{-4V5&RfkCOl3~P7ZEv6Sm~Da>pWF@kaRI zl&LBzVydaBX)^x6ZC#$4lF-n=Lq|tv)Nf9TkB?{FUz>%q#a6G&#AP=hG-OUmnHzg{ z7A2RYg#5e0gz9kFmx_cW_>-Di#pk{8B0f=U6JSm;d3kiM)TqJ1PF>e2_RXD{92HNc zVgca6c}i{Md#n?FM7OA&-I1pIbr}wPdJay`XDk-ONT29Oew~zlBHbRGL-=T7vN3l! zpM(n^kR(TqFg_uL8Gdn5fSW<|VG<*x^CXp!D5~dsYkd*Tew0K@ov>}M>80a<+S$Tg z!@yqF$GeC93Psi;lUwR4vpRHFgUw0RgU@`Q`O*#0dlxd|0fBSJU3?;v;%|Ijl9-ev z4G1DVJ$Q3-GUg_2c=$O&TwL69FLaAk@n~(l+RPP8_~u}_;#^plU9D}9Jkiq9A{h2Y zm!5is7`(ppnmU|y;zYouQ6R4P*1NGbZv!Q{FMbL4}UusJ$KRzZJR8| zDD*wng5k6nkEdX{UN`}A{oV26Jrz}OzHDNp!-^0g5z*fM-b*mW2hLzHgj!ieMO031 zu;{4HZb4FCe|dEyiijKI={_fdNwj<>77Lf#`u=*^JLYJi4w9VQ7vpz9`HoSO<&f9^ z@xb8g(=(HN6y6k_5a z5KmyI6y~@nDFc_6_1o@*37t|X#nXAx!&A{KuogWw+GyXrH~Gs(M;A6cJdE2~>w2J0 zPfuTeeh(87=^ixN0Ey)n6pSw~V_YDQO1Hmwl%mpyCaydlyD$1)FBRG@jE#vnJ6Eop z7=ixLBqdsDSV#EC7MijMmGsk0Nl{T&e!kh?(-cmd`t1}OkG=fFdbd3@UC8JCOM!%h zgfjgW!iesU{i#x|ddIb2I}fYK!uZ6x2_P#+ra4PH9c}F-e)pgMojz}z_Ovm8e~JKf z{%5m?$iZ~r$A`ceho&PgK7Rg&!^(j_Y;XRi6I>hDXe-vov0tnx3O{nx)YRk^7OHA! z^gPH?5>s9s%}HzgLhNaEITO6o!@XhE>K#fec?% zS?=rRIqZ+=T&3BfV<&oGOo)EPoVzWeV6Ei(#)ee-b!SK$T4^M^H3ZLGDU-k!xtCHH z@O_~QmK^gp#X*&!`oqJsS{BBFofD^$+eK8u66=}bLa%Otd^#Ts9S-6LROOS3 z^_v?}PK44=Uzs^-7^4-|yJN%NN0P}z(VHfIdn@x+MY==0V|$-kKlNRJwnZiiE;&7& z@tbUZfQ;WnK1s8BzLt}xMe(_c-Ix~e{mUDt7vn)7N|{JI6}UkKzM;(@4f^{BBwon7 z;raJ1aEd6)OyL z*HrD%VqD~dpz|GW6~X_Eice8Dev^`AqpeH&ru8ZRzJ2>@fQf=m8Uzst2I#v%^u=dvxl80~9Oi)ZZaU+VK+bA78AWy3J{W0$M zXBtJ(a-hDDp((L6F3w$T6A3`->CIV6!SCgoTffU`RyBA5#3cVfW>6uo z0zkkEqj4KM3f|WD|M@B1Crjxi?0~F<5~8nz+voIBWQIT%E>w`GD2Ms_E9In6N?Z~A zOi@mAobU-^=txxen252i{pjO=hvjAU%7I!NQ*cd_@Q*&=8&~yQni&Ddoj3!TNal9E zfROPgR4x9An7kRjr(W5Q06Fp%J|}4|lDtM|o7B|ge^{J6LuX#Nn~vKr$~GElD{*#&0p_w_FK_?pZZ+Wd`dLoQ zE2i)B^4@)NX{4W`9Dx`sl+PYQeIfQ6xNS5^S!2jnve^HERgPCpL3#t3B&g zbBp5HxYaV5av&702y8+zm=;hp`4s9^Lh3?fF@PMwA7Z<%1eTL`Uu=RX;kMvZ%{!WJ zg=nx!dRcMJbGAg=$zd1V+ z0LhljM)QWEt#kD5UVJY?1IBe$#|#GTef(QTjs6d-#tt>OL|s-2nVs^^ScSYt1ufXT z`p=CqHIN$aprERF z_>5s+LSE3K4$ZYWWpi-z%YL$)+xR;E^ze$HA&EV%q!b#&c{^Jr<8GIlSh#KCUp@Op zR)?F2dqbxH?Ub)%`rBtOue=(u}3u$|~eb40M&br$83Llbmr6Wzt^**9?Akc77F>IT)&(n6nNGM?l8Dcoq^5|d!W zv&wMveSImKxFliCVl;m4DW19h1(;k0|2h>>s5^F7cnJK8Mb38Eo-sN)RF_U59{7~CZtoT(EZV)u-Disj}qL2!?@UZ(0U0SS) zYDN+6eg>!t$M?HVkdtxkix@lEIwS5fscz6Nn5|*S?scz?0Uh|me*jR(`C?20M|K> zN66Ikh31B#br3_+Y22|Lws4y%oU`@hfO^J7d)GJu!=$9cUPsTeynPx#V9nToz&QFC zjQFr-t)e?Ox{pth1lsblk9?^un8dR~oAst7;r43j>mAj#(}M$f@Bg=^bDKg?0e2NC z0tfx*gq6xg`D1x6+!(|Umgi*mJ?Wm+*wyg-3CFaBLAXpl5V{H!-+`aiki`IpAa=R8 z6m#(o)qwp@0*KF+$@%rys6>uxJ9FzdBne+;5!ELnR=r58kY%ghI_G0AWR&P9c1DQg z|2F5!ymD1oMx177;90zN{~>p;rj*&P>F?)!033S~EurNzR}#o{W?)buhA(R#hsIix~0m-xNG=^y-?#pCXJWh*ZM zT8J(EGwHMvUrxFS0M`}qvSa_N?|f2E9^jCnF#TL=CP}KM^3Elx)QTS+)o5mQBqwmD z`sQuLvemIN@o1Wkl&jKTefu3PgW>8U?$<9cS@PSxoeT;eRqSnnSzNV6mT&_KQ;&e? ziWw3qN*w^Mp9P!Z)Adgwm3~CtZYQJBBE)%=;*sgdskxz`yt&E+$>)G+2CWWVBR!oRXd#rGi{DxuB?hjE%4vEEgUBE9 z)LyuSL-1DVqG^mU^BZQ6C%IJM#L6tJ@b}$l6ZYeKw#`cx5|u86rRN+eWwvY$eyX?(N$4~2ysXVh4FQe=V0M?}elCTFu7^O+U zSNc#v3~mKzOc{bY%1qr8*~?-yC7G{!&%!x?BPMk8;fmGyhiM-vqTr-WJ}s~Q)oAg^ zj?qa)U2-zsQet2T;Kw4phe`2BGC#+fA|@p}Qm1+OZ14l`AFE^>k7BxZW8Eb@CfK0#RMo33_7AD&fZnV*-(KqyY{`%Tz~KenVx7 zCso+H(h$17!WG6l(-3}PBoMWkE=7}F9-#S=I)YQas}fi8wHViP(viw>c@XzR;ml2= z-nGQyqzf|f`^no_KpA?mOr+DFPoBf~V{3yRcH_-NqXL{YAv4 z?x;6)sF~2Q0o}8Cno!A05`pLOFWBnLK6~38+TS6*l~{pHF{Qxg%MXNdu%0&Re&U18 zOnCU<+MTa4mFG;Vz?7*YYXF$@ zpXC#d^;&ou)8y(Ybkv*f?9q6UNKV;K#w{wzHVk`_2U9$%%G|~PRbugzc3gE2Nq#s9 zrZu&_vvu~eY+rfL4^HZObXcfkC z6B>{TCv-=gHWJey-s!-UfMaK2k|qs9WF|A;PnW3NPN>#N*IXi{rW#my_;cWLs=S1& z%X9h6_>*xPjz66H;GZ$xqxm*qp&3{Whdy|Hz>Tu15=XgzvHvuJ(jP?ShjUwezUjtk z9TvsV`zmoj0)TRchK2zA1q7yGWsQA!@R5;~9kb*xS?Xr*v2&X^>s9a1q)mW}GQl+c zTO?0;nVa4=;$)q@@m^Y@Sxtl+gSJHIx*`xW&>3APXpAdsN~@{8l}%)ds;7(}E?>gGPUhK}Gj3au{ykhP z6s~$v>7E9)x<&J{LsG4J*LPU_zDKS@_djS=XGp_-vc(^0xy06K)|m(JuvKD_LB8?V zi18g7aH}<0TKaxA-!due{$+02rWi zeN|Ol1Q#ybk6Wj&?_2vfjE29VBXDqWWeojHcS1I>0(q+A(6QN(@G4MGrnL4a*gpK;FcZE?-pLq!A zdLPwTltHsUU*^w@K7o>j;qGo7LM>bjprMds;|GB53=sz1!IoI;@3M7Xlv%Gf821`6kc$@Mkbi`pNC^7EUke0WJDBz7;o z%!i3@GwR^D6o5|z#=J%CUiBA?y1J~bcmfkTYElu?v^Qc`JKihGQaHugjr~dZeo&2?W z?8LWC*lc_7r8bu{Z;wE$Kig**JGgv22Oqa2a*hc0rtDo zYJR#qVAQ^Vdg8``qIll1&(Dzg{1q`DOcDW@W+v%S2b69bae99g5i@%+Pux|p7x9^8Tn#T1UPxHueRW63tJ z3cCfav$M@38Lmn1la}7AqlLb~!A&XQJ8ZOfjNhk($jQF~u;%ti=7`IPpp=5b@UIk? zk@D3Lh98jYa>mx{h$Fu#PQInXJq{-$KYD8d9h`arbs_!-5T0CXNW4b{x=B$e3jtyr zsKWc&zNGdcc`(j5PjQfj=@KvRX)2_#dzcsIE!0?st5IUjLD=8lkEE!v`MT)x@-jAq z$zkqENl#DD)cnd}F)blMLP{!R#f}Qxv2zZJ#f=z7^J9Dr#p(wls zdpd#beu!-6XHG|}I_WUZl%02o!O1Mhy}3PCvf4tY$WTyG!2ux8&F%gst*I}Qlb_}n z7W#&U^2^KR-B{=nWGX8wJswVbkz>9iSyOvLZgQLhw|M&c`U=Zvz`t5* zVw!_Asqq#3sho>ae$@TqYpSo`4XK=eRY=n#H~hSAVF;Pb=QpPRjhN8N6|d8bVy}xn z;mTkeX;p5(*Xy+9 zsnV;8?cO6J& zOgY7;H#03aLPe&K8-8Cp)o<{0hIia@tshOy+kxqKAvUP7y*k<+AoW)- zabVNCAfo~em3!HPQm59Iz(?SsKU^5*cehtqTwYEkD45>Z$d}=B*4MrhsqV5W)Qo5_ z(${BnvfR>q*3W=Y=4JU-XU2b+`>8g_{2;bH1(;*4AXTzWuBok8G^TUNQwv-v#8j0V z6|@DXN18g4BobZ~dFawWg$2LIV;DF5 zSf}xUHa4b&ka}5m&A@GN(qbd$fkO-N=Q}Cn6CY19m+)3uk@=YidzwF#9iX7b#zxIL zdlG)PL&LEg34!xLb`Ul7dVMmaNlsLmv7c-3UE1Z?T#ND zFum6Jn~4I!;_{= z1(kG17zljmS(%aag0uTt?y5GG!!B*9^=OL72XBWKg6>Uqh(Nf*L3w&DVgT$uC4gx>HLSc;- zL@Z>LDo^n+jWRD{dm-U@F;8_0Cw-WDEuwT>4Ca-fq^%qdTESvW61~4$D9h=@C@%$J z)Hese^VImnL`S-xrR|&$ow~?~j`{ zx3`_|kLuX%=Bp=a@3S^!-+XK^qQ2Bqk0z<61UvRM@6L4(m|Xtt4)hL9%{o}|8XHkf zWon8z>h(en6pv1iYAfvZN`hwI%f_Rg9{6&FDSE#F5^>in{+S}VI}jK-*F6n z&lCuK5hbT~jY%k4LA<&--GHYtfxHtv1H~C&?u=x1i{!iiBfD@8mm!;4a1R^U@5B0KQz@LX8{&t%w=>Zp&+^zneQZ z8JSm5S`cr%xi1l!xYX96xY-cJ8ORAJ8nFBa}ItAY4^3#2Uqm z@46`+)hCKq_u~$w-R;M2loKi!l@33qj7m(-mors_PT$Emw~|WN1+%}W?+`yGjy0^8 z*KQ=jSz}XTc;f+jCH6DK>)J0%U=zx%-mE|)@6alDFU-Zh6|H7En7OaPpZUVONsgGE zKZ%<>_zeMK`FrJwGrSU=<<*>Z7)aPv-!L6-I5ZRk@^L5hg&0|BkL^L*xxQ{BA#fFF zd7p=afPmKxuLYhlyzoU$|y zl3u=hDJ)fElik{;KZ??(l0NTFYL2#ucy}@q{L)@+_aGAAb;8yw@RQxHQRB?x(_670 z-RJoyZOr`Zf?dYZ<>DJ69>$i_eX3zQa+!{Aw{Y7if6fLKhoVAaudrkJr9hgmPFZ(N zriiVoVj(3Px!X)K>^SwT7F?wmZ6m>Zs^IW2LEmf^V+TQ*2?N;2l~!gtO)oU9L`IS9 zixbTG^|N11L!mEplQQXT+SI{k759J0iX4{2CE6e5lHWT}+N>RhhqCPL@2^L3FK0D2 z-Ua!J_w+9k(D|q@bjv4hL2@P>yDST7xe}=6cK-fdZaZDcpUEabF65v7A(=7*OU~F& z-_&b}FLr-INtkZiJJJW|{9NeB)@Pt9K10?@w`cc<`r|Y;#gtk+5<*qw#L?W>U_`#C zV%OOJWE0`5nYplep2gfZRIp|4S$PTZ&v&N1`B3+EaA)W8q%y~xBxbliJcva=G!_J0 z!u5!~-9K!LU2f|7eQ^M3n*xmDA9;*T!;cAgpx7fg`uPo#Wx;KGXJsm?k$2hpdMw5>hl`Ym%_=l zK^iGS8%_KM8_23mW+ExD)EDXC<7K93pT8J(WOkx&i4=#;G*};>q?D{4g(-I1VxkC| z76v|vy_O(&LLbD~t>HS|`)YA|C!p&$Oh5iwb+Z-S=yg;o3-1y!b_iwjw$q}M8hM*X z?pHMP0_8aKRg;Kx=K08Y3o2*twdehiv)jx-vNOp6jN>#-1*`r~!&j|D_Elj&{ryCH zEy#@R`NX#$WRspcGmVo#C*g^6TiHJcM1eE5`jgA1bu9*dNQaQz@HTO*QVJ($GMy)qU(*`PiUD78GFba6=?SY zeQfC6g=!!!;W)IJd1in&=hwc!sc%RPf7RDjqQ=;thLMcvvREI|cs2B3HA8}5t+h#= z5SPvp_`35APPJBL=Y&n2kpus?OHKA^$`BTk*^paDL?>R)y!iByywnp4qyEot3`pi~ zo9xA&&zCC@dNIx}>KA}|RWqoB#rcLVeVjI9@Q?}w05`PI*)CeqwAjp?fvp@MJlozj2LOOk|MUI-;x6ADUK3-uhz|z=8?L_0m)<*l zxV1LRV=gaWz@w&EtKXYQB1w2}J1EWmJ^us~DOeP&F8WIA6;&XoE-6vVRjy{5-t1CK zeTxDk7p1l<%uM?ws>=E~I9j%o<@Uf5qB05^;5Q=(>!op8T+iw@E$m&RmV%lnxUtby zI2ZiB<{xiKU`~ZacAA+^cR9G+4rYd7g|U#stgbFDl+@IMyN!9*4uI>|XIIy;(b3=Q z-4V#h3Q|ooav(Zp1@mrsi88O6=%zN_WLXP~?^$XtK%8o)PUEU)e)TzpRyv$gnA~la z!zGF@TCVdG8#+uIhO@7hg$QJZfl+{CAmPH>b0S2=*(gfNO7Nve9G!op;@9zunH!VQ zT9$ufVenOkPB=2?FWxwdQ-iMUeD$l;A<5jP*r@HON!|kl+OZ#w`D-$$AK(Q`%G%of zHqaKX{T6eIp^LCn0dE5W*kzL-u_c#NuNp+D|p3K+Y@ehi%J` zGLQ5!B88V7dzWKoE>4SwJ(CN;OB+;_3bN7qQa~ul>NotEMCQz6_x8dBqqPnDy7_g1 z?x()V!EV~#c5y*+NAkzL>B&jMsyUuRp+V+Gzpuz6tJl`)H2{Sk?=PQ)GfI@{)J2~U z^E-5T4~dsFNAmccf8@<;5jiIKVx0i-Dp&kUWtSooGasKNl0j_2iIzYItV5BKj&>$-}V z4K*U(Rx8B)$`Q?Uri(IEu|)CgQj+-X=KL;;Uk!{y(P#n-C3x`S^u zbg}yr$?(=qrWhN|aC#&7=59Sm`X#IU<4xe%>@$&<81EY1tU>>1;d)WiW9lM_= ztep(qJe|CKE!9w~eeI3oW!|7F)L;Kpd=6FYpaJtV3lWntquyz-9&pmXCgf~*RT00B zgA~|j$q`3u!~X@UKd_N)(YsUaUr(^nsy&XUJ2)tn(UlrU=u}wRQ5?)6hlHdSS{{$)__0!&!w_d9VI(v2giJG z&r`5ni@5ZoJ!(EoDB+4kka%gly;gWSmQC>Mp{G^eh=OP)(T;x$*2Rb&mx<6XGC-n{ zH3r63)}ZYAph@1ha(B~pVMF~t)R-q9(F%dg!K@#-;1C@xZB}7nNGZC8%c!U0Y<(CZ zrv}kaBtrnc1B#^n{-KO9WWi4o^yaFv7%@NF?9=ssylj5FTF5@sRsu?XfEuD^rygXS zxAZTy-WHpx<*mDV?aj`~-d~Yol=FuV9FjN8eq4Ve2v)tI9rwXs$5ko`Jb;WG_4t(q zw|?{T;+wDF;xy*>8RcVDF}yjocQ2%PFm5LbW!OUeFju^d|c(~WJNKyCUnbc zV9s(`Ia1aIapS9-A38T@1I)o41Ia{;jNYF$caJW z#-YnZRj;&SOISfvIesc*z1LN)d^kCmSO~R<{Cpnes!${!6NcJi*nNn!JvV6+33Mu8 z!MJB?DY&W>SHpn0F;dRB(pfrnHCA44wie_BOXu9WcOxC@0ta+AZHlC}hCk{|7E< zEJ^%Yh4#(iTvf=7zIW8(qOP{K_H3;k5uC%5HkeRDLjzFF+pK?LIP@QM5qo_+z} ztSU!{X0I-}ue@Asxn4H8zdABCH+Md1xyB$3+uYi!@V&pv5>8c8ElKDy+Y22a#6OpL zX&z&i7)0wPSpfPMgQR6Rcj#QpzPq~v3Q3XgV4>${XY~MzJJ;gP4J0J;o14=B7$nf( zabIIFoj>vL&=y_|cU-@LA#}Tm+or!!*Y^0~deH^5yu5tTbw=;Ao7?k8)qCV=L^D&< zQJ})`<^-rHoI8=g$^p;Xou|jgbKLX_8XioS`<$n)Y*zG=Zb1DX8H8cCBXsoi>kj_+ zL<|}~LHzuw>HO|@Eh6LGt*7Yw`}?~G2PVgh4S}U`s;a714i3m!xw$4RlYD%9(h5K7 zY?9uKr~X2NgomR7;zJ^XYE^x+`PuLd(0;4GX&mhA5Qd-i0lj4YX_-$Kg@^Ms5dg3W zX9CWQ(9?~{;gzzzCcn6=D<^|`+3L@xLj8w2g3+E?T3HzZG+3-Ip^9#z zG_f?C_7DM3+ra9^f_+oe@UQ|?Q<*=5aIomtZexN6cPlG8x2AM-BCfxHQpAzLZhCDO zYNMKt8*Tpb{tkt;0)>t@%h#>F*KIIggbcqsD{C7Yu%coiVB9pBJ~&?h=)-YBo;!U@ zQ535u@t00dgY6ufef`?E_TF4QTfW-^F{wn2GR?1Zw%b<=4i*cwc9YZ7Dw>*qnmx|} zaS_#+tE<0D<#r%d{|~lDyL(NI63cklA^tBE_hkX|MR2>}Do}~K4tzSSYbsrK6l(1j zdNN@TJMD(WMMXea>sB>q;9kso@d!Y^?Bqg1ek_`43N+qVhim5}!aGCUEAY*ltFovv zY=2%^r6?$&ixo5`A(8Ytf65xU@VH5r3E=DY;e6z_Pti|QMg|oCjWz&}{}Fl(D^Lx1 zFKz@39kWCV>^^c+H82<%PUrXBjFk$FiLn??;~mTp^jQm_E&Y*;6i4$k)-tRwNWi`6-BHE3w$ka{#yNp0>w@HuRr(UVsx~ zO&uCc;f(M+-&R&vrw)(^wWg3WC2Z8Ue6I=Ulkh3PovOb6Fd%pyz!sECyQLV!@lJ`z zg0kCzqS)b|dQF;beu8aJmr$UL9d1(%E<1NMko5PF?x;;mOt($QWviKhXKQ^plXW{@ z1iQGfQ&CX?yr*hanTHvMeLQJC$3W1~bDhG!Hnk1*aRK>D$hN>=xLHT5byyjGdc03& zH9-Q7uWf)98gg=@!F!thmx{L0lZt8EX-37x#h=vG*(`>M<7KFt+)vB_ZFzXQZX-*Z z_HTcQ_Z=|HX}z~WS)nYv5uy$H(pVi)C=gbPX|z|dl0jhnzwRwcw8zXK6pL^a2#ZU} zrKs2mGbma!$Pqw;Bh#DpEq5P-KheK+*&f6`2oCy1-eH;sh(2)Q4#-rgyPYh{`}($M zGJ}H9v8gw{i)Q2NhSCCauZaeMUtj?*IKKC5qKE_rM83tQ#FD({(312R1hZ(nFC}D6=wSNr#iIK0gFEa(p_6f(`dkt+3Ka;UgYG4Ks|(r zG4S4NAdMc{-*V7|`8&1`T5L%yCwxeGgJU#)6 zb9gA7oty{VP6BAanY_fo#ogQ6%l`EXa(9=8ovErc`s$;K zipp=y*6sHh0_cl#toQ_Dsx8l^3|zLyvX)Cy@l#^^D&9cFCkn_7K;q&=fwPLt?}mI-IM98yEgK#TKAa5z=~2_ZII8NJ8g%a{h>hP_ zA5Et<5b$ab0V!2L`VV)ufMy1~2GR^vvOltB4(J#USC_p@J&qeV1+Etfj~fcZfqi<7 zgcyOxOzIllcK~f0&XI^fB7<#nx2<*K8ElSbi!HjZ_y*Pueg|1~7VrUJ~Zh5RIQblm4yJ|f#k#TS48AIpic((&iyRY z3dr;t-cw3NO9KrSg+48a$QaTIY@KobooGcTsnS4kmxfMr>k*kzf8{h`8hAZ>(KtPj%;?4YYxejptu0GwL z6joNA#UK4A?78dZ0~(4l&e^KB0r?5sxaO;Xfnz>K_=yL|#_^a9MMq-*{Tz>820$0r z8e{4I(pmA)*llTMMpdc-?J=(S<#lQOj!Bp2khieZiYryC#(HY?<6le8|Fl&2JJ?Qc zPzqz%1d#8Cd5vkGDEi+)E@n+)(1nn`J-P$#8M}=M{8-2Yc+oh8LI}`m1FEU5#hB(* zldqZ+xN&IFXa1m7DQ<0P3D71*=KINOBZ~XMtIaq&^L?5bpuHR4i0tuw7^57!dn`Gk z`PwKM7$SyJxiil=73^&|2!WD=N*i?zjq_14%=EJwB)F+p$^AE#Nb-R;fV?o%c<}$X zk?o(xv+Q3BM{#Z5zfl9;M@o|4^4i%89E0fq{&JS*N`ab>|Ev3L9fqjPxmWwW*77UR Q?uP)DP!RtqY8dc;0Ak*0dH?_b diff --git a/docs/images/chapters/graduatedoffset/8cc724d5343c65685d88c92b2d069a2a.png b/docs/images/chapters/graduatedoffset/8cc724d5343c65685d88c92b2d069a2a.png index 24212fb3d5da8d4c0135bb199f21f4319cba608f..2012eb6852781bc318e797e0cd36c89382c3e730 100644 GIT binary patch literal 13404 zcma*O1yo!?v?WRm55a>ItRaw~!7aF?aSI;Y-8~8J(n#Y8?(P~a1P|`kcw=c?o4LGs zZ~lL4{`@tI#iHq|dv3|8Q>XUc6|Sr(^AeK`69on3rJSs!DhkRIg2!KURNxz;VptRK z2i5qKj3mm#u?;bNpd8cdZF8SC$nq0UG>O+9{oa&(<_s;?fA!}R>uy5o;=%cov8AM>)N zAl@dW2I@Z;1v5u~a7)7Fgo}AE;4dgZ;^^bl7z8wu%Cb;&n(yM>lIVDZ_#pAHA93t> z;?IMF!f|Pt=)pm_Vn1sZ59O=S>dZ?9sTi2Ppq1u%{LN%ehvKLG zXpw4Y%M)=)0iRaLO=qKuQx$3tDV8?Y#2-VB_@=};WnMdVIVd{E^9FOYmky2NSMKD= z0)keju7s6C3@-j;eD()a1*pUEk}ZJ^8qw%2h6$r1C`nDC_( zS$ZQ{Nl`s>-jU;dt1Am~N=C3Lo4m=*&gp<2R60?I2)-Rm1dIslMzxovZ);{td%0`Y zhj_oO=a>$oM&q%c!c}9(Six_ASwJDI6d(GA12=H^xz&TiIjF|e@mvNj!RJ1rni)8Q?YNxLqmk&F>9rvBHz-mW&d-;9;zPm<8|a7JhsINwIKR zX4^?%S0R8GNPWN{;9$L45+Pf)+PTXz7hoANeWz%S4CIo{j3?*Lhqs>AT~>6}nTKzr~)rh=!f0cbU`Z!^6{?$2Ew zIahLrL(>$*K8A!#Kq;+geqM5;gNJh`pYt@!m~*G#J2o0A5k81E$II3L{xB{+5(COW zeeIP!2n*RE#?8ii4YWWID5p&wn=eA@K0AM~WM~%(o~V2s&%ygbL)}w9q)Cr8hFVln zLE)uzF5cJ8P1A^o2&VV%8F<_ez^IArcz6l9av8u!O|scV=$Q`TD=hEInwFeo?N0K!6n418d=BeMd=Jg*Iht-g}SX?shct{DUTU@S6^#TfOon#L~{ zL!XU{bgqmB>i-NjYBE)Cyg78&xfaV){mfmVF*5rhQjpj-PtWE zTl*QlLmL8hOyu?;EFgPm*=0<)OuIHD-M;1Rn#(5O&P2d+kO^=K;qCB5Y3%E%F&wj4 zx%@;6k`)jX^s;z(Vz6{g*12#Fs+=XaJnKtHOaA@+&Ng``-?5+D<$5^BxnO69!)6;? zr}JJ)s(dNnK~etG{$I-_%MbRzho%Q)CLvLM&a`zZw~c1ym|y@>D4o~Ae6hg}WjK{9 zG$`n~vy01oho4V&|HbjDl93TjaBwgMpW_cLeaJxX2=8i?^yz>QL%@y@@@m6&p)S0o zrR8gYH0QVyJlay&Cmk}`DXIiS9l|A{_~QN3Fa@f+Zi(D`KS4pkVj-^xHMeMzHg=j$ zsHYu1S!`gNIwb?dB`h)q-e_Q9!RT{w&}GYWdyk!|*dIZ}A}c4icfPRfv8rHTKn1k& z$jSG3rM=9sld9d|_;T)KYiPd7iB2Rt_O+pJJmBw_ma7E{>0yzP8}Tw!V&zA0=$YGS z&nr=|iwl>M@VWX&QPI8aFue7tR{jf^HvFO?&O?HjSEv(3LFWo*7=CcDxj6u#5cP^6 zE{&|aqvgaZjOU>X#4iuo<4?`AG)8qZuKhh zN<->@mr3k+Vq1S(`ReZ}+j=)B8$%8bNU_;fjK(E9hYnfMh-hq zRO*c;k}wU*(&>`VZoo~Lj7?15vasZqmDwuqaDVuqtgb$_0nK3&x>^ggov+CsGm*}X z@9mW&l-k3o{KU<}<2>}fQCU}a=;CluyVhJGdxwyaP}$Is+A*C0EUDZyLZ#U0ngwi1 ze+=VCk~-hf2s?-urB5Hil4PW$UdY5!(iw?^^0f>kwqZ&T0s;wxh~)5ar#!}9{Cv!N zm%X855vGW4zsu$R!^YLo5{=Ruq2$EGpk@31{qvhck-q-^TANu`z?HQdY)QS_{eQVU z1=d+e-~O5Ex#-Ibui}`f=C|)|&xP=7xR(PP#Pel5DTE{X_j7F#swdlqJ-z$(AXb62 zw$10xu4D-t-N%Dg6M1q|Gc%k%7q%!zE&HtU^71Vbxs@hcTU!T9%>+sEQeIyCf4SEk z_a^cwMEx^rYB+(o^=EikvUK{w3+E#co4vfeo-}HAOJawQ+Y@1IdzGoreyONy^r242BiLn&xW=GxCRvD1)W!)rTj-mnUXz}OVUwFpJwcI<#h@VkTz4$6xl zuS|rfUa1u-=5I!*))a#KA#swelQPD25nzQ6sR(OWw{1BDnIRgG#ceb5Z&dI&4hrQf z(}@+iVe(k<+WAqzY0E?YDTURnKZbI1bMv`tj1XC}_mf$!>{{o$J?nST?iEd?adNX} zQ98Gy*`i7k5)!d2u?Y$I<|Ao+b$gTH*>O_s^NNA<-&i#jDinL@Gf|^kF6mq2Emviy z7TbMzD?5<_W5Snu%J7#3ewRxwuNs$}v9gh8h0iL0#Ej!j=!)DH){dojVdkEP;`X4f zc}@h?{6k7-+ak=lxd|4*0_CKFp1#oRP9X$$L-Zr< zNXSd_mR9QA!@a_6sxy9kP>zyFY-}yttP8L)V2W_XlLvI-x^@4#gCIP{HegBr5mw~o z;bhNu5Y8)>JLhRUz7gUEszRSmCjRyfif$=GI-jg-Ev_rZh-b*bd?$|ucAHl~y&Z6QHE z)y?kMCmARz85N3RY^)1Na3_jK{iDD0)BG2teC6f|ZZ)qR`G9BZsJR@k5Nh1 z)6lvd@Ecz6Z4Qc^CL^~rc{$H(NL_m?2d5{^og6gE#6B`~aL2}KX;vDcGNv>44X-3K7VmG3&6zZxE z%Jw#*(QM3Ly?k|%B@@uICwO3A@=`OcOwqZHjCeaQp#-Co=CEtV-qE?COI#x`xb|HC zK}tpj(H)2-zFY^S@T3+L-i*}3PGrDl!9txS1wX@*7-YL$rXGAbiM%2V_O1;qQDkAQ zjLn&n+%^T_aSdxXub0RjuT6Pz748pbh&XK92EIs=?@N>m2MdX-? z9}hEn~aF!Yi(bjWpz%qjqf77 zjQGHOs~kp0XLz27I!F(Il^L@6Eo%?+}OPxwyfo16R zdud?49ScT>5o82J8mKW^>0v6RP>sal;T%&K%*EGHMrAvPpwH~Wh-cbxKU42vgrR>rJ6HP_-L z1;5FUni)xVZ=u>qNOQxxK!@_f7C7#=c=m8*e+TF&w2LAoiKUEk_^K;b*3k7l%(p>H zrHFfuZ_-NYb*Z_%JLeuAyVCCsjiNGcr_Hi&5=n;{#f|YPNgwB@2xkmZh7Mq+_#0bV z#zmWquW3+QjMDI4f7%KV5?2Rfb0JT)VH4jwktKjh4#1=lgA_qe4mt_7>ZIw`-vLFO zO!(&Kr1-XB&%uI%DmWawA}v$myR`p{?^6B=M%L_f;Kh||#?XLs!sf;@I)_8MBK5BJ z^}SYH+CssyJZs}7X5OSkc0FFAUtio2MrMp?O74zV>WtOZB;q0C=sdd`kvxAOh7+=IvOXz-JD6&%}_ZBHT_))z)x@1*6oE<-?rAf@r;dhpjNKt9}at?RT<>h1oegQa@bFC%4VZqwk zA4xLcvct7^YA8-bu;fGh9_Q6G@vAf@qP`ZA1mo@I$&qm0U5b>iDV(wV#AQXW%Xa> zNso-+AyD~6L@K~Z5OVQ>ZcemwlDnLu)D2-EC0Gc52XwzXDWBl8tJFfAbaKdTXNqhY z<8lTbCq5mRs4^w(b$0GyyB~P@j(}6%$H7xrT{E^Daf4H4&z8R@01EC}paE8Xn_ph$ zCG!G@j-H;dVBk6Lri#e>q$c(jReXJV@OOF=wUgTo9#Ve9T&zm53H0*xp&NewB7Yf( zp#P z{%%l?w4P`sla(^+nIceVBF+Pb{C$BZaFCj0GFcZ)iKxEq%{EIX+gGIB1YLt${)?_2YDeNe+ znTJUb(<@4|sq@$N1P*>vgKc}oORe_IecCE{c~QrG-+!lb5P*Jm7zt3^+#fqRFK#3* zT{`5A7=G9k0QMxh_aJOV-biOvDEs%Jr*jzw$d1m`&i3Vxu89%puXp~PPiy70R=utk z8l()Mzw6n zit*C;Kt{pq({#6t;@YZ{lJL9jZ`Y=If0#{S1U)zeDMNjTjsD`|;=PVK1x3YzqN1>g z2~}g0(2H0>e0wNPYL4S$=uyo2ax~^IXDbhn8{9o)#o;0S1CH7V`Hnp$*1-!yFyCz` zv#>qn;$$ls-M{p*y<43OTF9(`(!HI67QD*91WWC>nk zr;LHW_Lta+mq-*o>5coI*9YHJuKfm^lB~&T7sJ9YoE>4YL%UuKzaN$#+<9&0!NpHQ z$5h{U#j(j=Twea-OQ@Ml?3?C4xt?=ID_tHS-%yWLzZAoVn_0Yo9#_6J1Eg?rZfNT1 z?wE>=TcCi}6XwN{46X(3M95&!)!dIIx|6H611pi<~pgM%Ui>Bg|H56 zelRQ2q{3^h6gW7bBeJlP`4>-X{XhS=f74hI!7t~u6@PMWvrjgyz3@0=;hiwZVb zMppFJ4Sh1&Ym8JO&eKn(H+A$j{Yp)dYifcNI8XJTxmP{;(PHXj9k7EcsN7v`uwVKR zi7ami&$kh!cmJ57;b`@v;NlyYDS9g((PhN3K_Y$Y8__K)5Q9*i6v5o6y%Jj&h=A4p z;;r+37Vo`Lnx&IF{$m{b?!(#%H#GzT8Q(uA{7vXQ;Wly1A6y;0eB9pli6GoY4@Yx( z6$#!EUHx)ovREZkil$LHYPtjMlGN6WSY;qYTkP2Q?INU6A2`RvEt-1IV7}`eV}sG3 zojny<<}J}-&id{EGlIK?yXgKbd8$aeTz`3^J;i4)KV`1TX+t@k{oUhv1b`bzt8Yds zj|-e;A2@ba1H=yA@b(If+iy#@rrMlLgtQaev`+TsCb6Av5CnWV%VOP|bRHEgE#4_=v25)ASe5%$i`4T6D$1!* zZ-eY^&k$m9v(dWPDgmX zuD4QrDA|~q#XZ);`{U_kJ~ulvMC4RiO@cbz9s!Qnj+k=|i|G;#03`BvG;E=^uML0P zQMxD%kMQW9RW}K>>8^Hd=6T9bpk3DOsJ?U;R?4`c22+v08vfj$k(fg>i*H;NBFxLk z!;=gE1%rcwEwP!xKFN&Pz6pXyO}S&IWMpJa%<$`2eHB6~-+y8aw)0P2^9_xSX(Mu6 z(WaHpGzx$G_%IzWDkFu_qM-|?YlO4b^Z*TaHYH+2&Ddls;^{) zEL_)b$o7EFFI@3a4CF5K66g+#YRQ zEIK^)5rv+C0YDEB0E*Ho{|!?!Q{*jXVS}(DSQY<;;g$Ta!uo{_3d;}dY=XhIhkwo2 zcoxou{Na_JgltMl0f9oK&CKGay~%Grw|R2hUL5Xofk6ssl6H3QYc0l5*D6|%asX^g zP!?ryXlTCKg^@Wa;^d^lUP49&4FEbecE@v{H1-)%?zX6qM;YDK2l(RYF7ppkEfW7r zE7#wTC24k6g~uho{Pz~K@`L}MTJB0G-4#{)Azfyy50QUWpPI6hJoPKhVTzboS}L<$ zYEnUX3i%s8EX;Nk*e zo9jQezYpyc0KU>~wAVOlzg#i$TTO1fU_K7ma_^g;A3wfjE%#WlR5K7>%pswu_$spD zeBR)`m5zo~Yrgz9DW6$xC(C_flMPT&62-j6MnHQ`0;%slUhn3{-gUSi%yHy5HGSqt z9|7K5+HQ@s;}@0Ro~>t#UtM}O@tWB-V7eE`Eu=~c{#)Lpr6oxofRMJsgJd)O`S?{ zV--EyOZ|g2Z>V`=G#ZP6so`zEknpi?nQRIN9c^3GZcTLLYqssJ6vN-^j#rd5okl&V zJhpS}2@1QDlS-n(Wv8V9w=A$)#)Mkq>0b`0X$J<@Ws1q0`9CR5<1%=7tG0Tno1D9` z{q=($Pq&q5dGs6WTxIb51!tC`-%5 z@h`r-@`K->2Cj#GliH5%^Hi6RMN=B->7t+kfB>+xvB@niw)!B#y6a2z3W%!Ti&vK< zZLJ@;9EC-KFX4i3!E$fd-irLV!Ng>n>gY0rudWxLIIt`5z9ktUHdc?8hf2!GxJ+wk zA&y#)V}AF)&|fx0hJjep&q|ZeZbe@3&6&Q@Cg?e<`_&NUcq=623uC}>&oa`TUpBQ0 zX`o5L{aNtHrv9+`g(Cw%q5wbf?$HzLO+WIv@bEg_{5ysZuTo>84y9Ithv40vtTe-nD$*hY$rA$ zV3E=koH}!SOFn|5lc|`zWpSl1hCI38o)^4A7O&QR@Buc+%unypo>KkBM{0GRMtnwkn^hEd!7IC2}*0sTS2S}*zd4jpU1MFo8Zq5az|n6}y! z1Yoe6C5Ez8SB2egT!-Y!5LT5BmhDe>x<5G?}_GY2sK`>cW!HV6s zpiGvL#we|Q{2M%zzeS+~u)aus-z&#ot>_`*=f_!Qu!WaJ1h@WeHqPFBKteK@p`L^0 zDKNcP^DnoS&FQo^*!GVu*}*m@9}#5D!m%-%i*I`(fst+SJJJ%jz`U^0M8gUEg1* zYiI;7gJCf}XpU~X18o-}BC95jP5c&hDp}JHTs|*?v$Jln`oiazQdciEVsK5QWV`8E zS!3pFEkc3E77oYM{smX&L|k;D!~^HgInytS=A2G&WPodD@k-!|)pmzfHf4ALCZ4-^ zgW6<{RAe{2Sn(El`A;CzRB!U~<%F6y7Dn~>GkXwH=adFHui2HH>VZ!uvDwFmSHu4H z?OTlO)ZPejAQ66h2h+M;y8L}`qHIUfS?;2A{#D85k7;7mn=>UTK@oP zI$Um4E}qU~JMg*dA15Z@0$GM@UACjz)aF9oGa`R&v)C8DseX|ACLYC2vmEC>kigJ( zf3eu!fJtfR7I2GoP2Pdx7xlWBcM-_93G*-f^4HH=uKoM4d^sLaxAPTAi-GPmh?9X* zj^rqqli{^ZXLjJvYXMnNS}J@i=H~92s-2UFtP1ZWs|C5T(Ri6(#=e+V(2u(o_`Gc= zottHP%R8!30795&Wh{4WA-eWTQ1R4g$BIWmpG$RwkoF*02rSMInKP-MxHwugU2U2; z-dC)Qtnn|{%vnxRvE2hr4-jh;ZF~JB3F3&>ReCIHCDI+qEp@2 ztTpv6&lBI|Zq@aW40@p@!sYbDMfK?-t$J8B^QO_$pXqW-XA3NwWoN70PM$>Y5u5^r zU7wm&;FML^W44XO*{y*-g%e7*M%vazh4imp0b2a|Z-^wwboBT(!=x)20~Sw}8STvd%lV0o69(JOKhp74Z`& zRLZ70;mHZKGrZSzVD}mk=-ioGGFZ@EY%1o1kcje{`$cpIXxQ!S_pvYyFjK@o?zyk6 z&bxEcpXDx2JEcdt`n!genPs2c-W20tHDg$Jiaq_OGenc4Tx|@yi?=Pn#bdDTHeK%y z&TU|Hx7HjHzEe>YZK?UixYuzl=n1Gm;j*9rR+qyUZD&gwlp zKBn8SepoJM4oqsx^EAGhyV|~9m~FNe$KY~prkj_fp;epNRMk+(6{!jT{TB zPVcJ$XeV@{zv|j$MsvkdxLXU$fnuBo6ED1*87yt01dEZlm19ob^YEKJ1qoUqs{LYE zuFa-deEA=dRG*&((ZXfdf_{5o%J5_WtH8LTX^u^)wb%(6q#!D+ZAqcOUVZ&|Bbp zPzi&@4MeJ~2)x@hZR*sZ3A>lq!|8TJdHoD(Ac^hZUA4JjjHo3r?Benw-e2RGHKxDt z?-!@5Ds^;bJ9o1+6m-&d$;ekW96L*moqmb^CORtx!}iO~pPCV#KA>2K~YN06K5paz$(-}?pI9HQ{k-H(tkgtnd5CLVhPlYz+{9Sjv+gi@w3f;& z3M$U4n?*=R2A1{mP5Dc=*^54avd21Oik@8H6N?8qLG_j(S8H8?V#O8$*G{uZ17^Ua zJKr4ys+rjo46Jm$Of?g`_NDpVQuUGm-9W2<$K9k@B+?!pMb$4X@}hlN%H8<5n!&-9 z!ep!W1!o1I%LzVrkgqbw*DX5 z1^JsqD@nr>VL@yjP6Cbg(Rn{4`mEV7Ess2Cp9HVdGy}%&|?GNoQ{1JWFYkkYz5g z041v`H%I%5&yS4{SQ7;=OEblzwEnA#J-xrNVysCfjz=ZC)3uPVt$>V|Jfdomx!04B z!|165xxsN|IzaHrhVPStWU!~f(~7hz!v7%eX(S!+h(Hplz-!=sX2z~9avcYQd8LtD z+)Rgr#zGNA?4$=^80(F?)EYX+Q=~A)Y!js247sS1um)8jlo&I|nw7t-nkP@7bp_xR>MIiOP?+a_*E2 zGm4yJ$1nvUwqiq;MP3l$`VFL2T8uNe0{>QsD#uuabPs;!86J6boiwIH zmYS2vXKV*}YS=D2sU@hIS%7RSVOCEhcy6dWZ*)b`-}qrj43h?x_`>Gbg-q@^yHWRZ zk*l?50RA#xnl-f1WmD@$&CAy=`L=`V*v82<$Ek$f+q;orsiP7&)~)NY(awB59IKx- z7+UWMP33nnuk;3$ib82mUI5>YRoV8iU z9}TD#pUd@zVz`dbuXF|kRBoggrd|;?xibEBFN{BH${V-L!8KWdqCReKLWv}P4;TLW z{!?ga+wn+0S8m2;AH&6U3ofmr2?z|xAiPKDHETX(3<*u{@b7JV3>>y^1mhZ`{{};^ zfb^6)J!E-${N94daj))UW50NrP9(1J6;Sx`IJFm1T|B3sxm+nz+>*7cw|N2$*BF|3 zCjPt!iU}FAz%R{(g+)Yk?QuNA%~xG-H63^wJNr#QK!hGxtYxHXMo-%<@NG6YtNC=E zD9aT8({r|hA^4SL1`2c^O<+Bmu;JGJobet-vQ*uZO)r%x;*}Hvk0}4BHu2G_XxCp?dwT^H0D|f~MkX-xQ$!q@_;xaO#W)7sv<=!xG;Haz6 zGgzq|-2XUPUZm>lL&d+gsTQhRTaUU5uGWRbb~Ea0V- zpzwYw7%?UXPt-MgGarPHMj(af)C%7ur@E;n3)Wji;NE82&&)msaS|BHm0lwv~%MpipS_n#$Cnr=$+J#8`S(v@-}HB6kn!(tVB){jW8Olf+*{#Jm# zGiZY*lTq(1zDgi0{JnV+hno^JGEW6Bz~BHyrLzM_ggCjzaB7H$Y=oMLYfE?vJ07hi z@Epo8n|}Dbek7}O@c14L4G(22pdc3GsOa7t4E|K*EwSvr@UWwy7%v%V{UYI zSl{@Ef8pX`{?V=M%{2#HlKqR0>EwH^O@I4#zBKTLv}iw?eGTG(^BD^_Q`}xvx(Y1q z!cN%@KRfE6bGUA>f>F#jS3W|FL^0r&QydWXh0)L!xyOe{L8qTIb4O}$4{O#m%}f&5 z@r>twpOIDFU!Ni?2@X4wi@8KU-8ZwH?#(ZTtM>iJtI9aqfkau}Ii>damyM~{OhO_% zsi7-cam$|3qzIsorY&D-I@m@TW^GB`43}QAM_ULI$4-VT^U8&V@!0oB77l<6GDE8O zgmb2@Ny;>S+9%-R*)4@&{EAv4x3ckhh6dSfs`75fHMaci2Gxsid}^m73{t|O+~#76 za9D!`;rd7Kv&%(f%Zh_hH8-4K#*BaHjA=rQ^g=Oy4d116z#n3C^utzsIj2VOXpN(9 z*;BJ>Gv<1AY&*j1QblmwT!O+rZ-{{LjTz%8)WSX?{#x@94XxH>Fe1vGoWsgaO~NJU3S-nML* zi8x0TAM_DGKLOT2teWu2TGz9$29+|EvufJf1CL@w05mb<3P!g98m-6ry4BlobiVI@136rNqC$xokmMgS=!df|Tr zm8$*dMeZ+b@`{Smm9vvNpa0}RkQOf+iHY4iJ&NpbQXVU<$WD`AE@yOOi8*~4A!~!L zylQN-=0?-3JVLAtwVzFV0zh1+O|tn|pQGkbVV{ej&D!9msis$uR4xlZZffFsSO?@o8`e%ef6>#UoW49owufH62=Np8cD6f); zw14dy**jG6(aqhx#$t?W*!+Pj#7%(mk(*>-$e*>jT=8Mk*4DONX>0CB0I>H-^O#?Z zNq*b-cL=!E zH3+zOL!t7&GD=HN_ktG}hMb&#`*(92=N+%@u=Gmj8Q>rPo#2SbiAzgc5Y7+*fqox1 zKZ@l5ZLn(PhUm?B?A7|K0Q6r#ZsEBe{QB>nDBP)Y*e_oB*_ky$BAIVd)y z(*X7=;7S6(rziX>=>TX2a4%Qay48w=9Hu3J&Rp`pJuK6yFS&(|?Q8(lEUaPCej`Tk z=nYV{bvb5s|0SHE6u0Y*5hoa!$^VpY{@4c<8&q0s4JIugzhz^yxI9{>2Z2-#3~*c( zSoHv{s7x%?*8d_USXGPF`7%e3DwF_ir=Z7?WWK6f0M&m=IqB{-f_GDb2H4*lc4TA< zdbmG0ivnU;p}T)Q!Yj3I`CZv`uh4J z=549e+e4|mWdX96(*5=waU`eR!iW5d3j06xcl8cUc`H|?yhr`1*ji#@Pag5r%eT>@ zx(&88qM`wF*$@8@(!XQ`Fj0JY=>*Tod2Dx&{G%EQv;76=vsrq#fb8)9bRp*db-m=j xk`drC&1-E6xyRczuM@o|M*jN^8pOjB{4dyp9H+{pz;z)MIVnZS3h^)B{|At>tJ?qo literal 13395 zcmb8W1yCJP*Ck2}OMnD-f?a~UJHdmyyK8WV;I0>UcXuavaCevB61ceCX}*7^YNqCY z_1-Irg6i&b`kZ~XuDv=`PDTs~5f>2#1_nt&Tv!1H<_+5GFFY)81*2rU8TbQh@Iy=( z2KxGw-BARFf%ys}AuOomntrTx^#KLOg$?(KILOsIpUocTYLLt>L z3}1u-0s}%nlh7#9CH+9S{e?-~x0{FP#L+QkLdwa>X=O`uou4_Irgpu{CidTqB9jXE ztWW&^=>`9Fz5+&*LWL6!`h{(k5FO{rL%G{ub?;?2LbKs|~glvh&Kr3w_kgHf$-Z zy-^kUz@kSy{*L)kZ`I$a97#93^95(U?(ajYV?Bx;%kV{h%n~C1Xu<4M#g;c;LfLLXh2>=no9cRB z#yh`DL6@tbs?AOTwGSK6HeuQ;u%D`^`=b?uYo|)AQo0KGG zZceFD;he6C&4&siv3UQC7}=)5dB08B&PK%C8Q+`APe(rgImIWEn~Q6)#eo7jIB&r& zyx*_iV5=__tz5eeGjEDRhW1m{pC?*)J4`@Fz&;ka*I8oXRN~op$1{I4OoR)YrLPm5fMjER$IW4c810VNul4 z8Tb3}aiFGQix_1^#7I8dSeE>pRRP1oG%b)U`2@ z)Okhza+%Z>@nDKLS16r!HK=4 z=<8FvtX?3JlV2Y=@;P!56%<^r+6^93aKifg#>K^XZ`=lA>FAxUw(#nnFV^_qL-q|> za4BhNjTb7l{RwT&&GpApS&fX1YwUJJH8eCH+Qk^KcXNUX4Q@`B%o>(6H7n~Tw3}?S zCtCXZ`vIk@+qfAP<}YC{yz!pkrcQPXC~{yIpgcnVn_XyhyWY4ULaS_rB{TvY+6+-MoEVYP8m@ zx^>%)6}!8x@)?6n@K2<3I=BzCrF)%u-L8qbxUjD@S{qJe^78!q{f-AR%FHw+8-b)+ z7hF0>ZT4IZ#?c+tza~n`4n2MhVL?qth`zkEkIF3WDVU1Y)C#j3;4zUwi2EL2P$0M@C<2EiKp`W&CwWrnT_@R9eO_rzIABFEvyBoxvfd6rI#ilLt??}=+ zV1|+P4v$Ccs(?^<>gc(Vi;abuf1=bMveJf!i?DJctH+AbMRF=D&#vag)N!n}*U;nX zzP23*g@t44&zk@F!Kx=YnX?c<1Jj^E>##q5>ePlxRW>InD!L&yLQ6|)wOk)sq1lA& zzG_etlaTO1z{CV(Qr99|@$lfj?gRC*>AW|I5GB=Yw9@}OoX|1Onj*!!Z(?FHy|lDs z>p?Chtag{-O_gPvy;mi@TftbfJ4G_>Fbx=;z&Uj^;1Ww^GReBkkn6Ctr49%1f73GW+bl}atngNyol77b0!p`oE} zE4EuaDbc)Gj!)24_&Tty?P=In?jo6v(Fm25^G%I@ZpHmjYu87EY@<#M^u>VjC(!*fmAs6! zIWuY;4;656aou~29&b*c{sr^zM*BP&R(-`exEoHOUaB?zc)RXFAI(8(!>_ z(9OomN;*`uM;rM0xw*jJt<4}j8@=1JH7(DJ{--SzpWjAI*NBmgjchjU7ozK)5NVT% z%rVkvo*oQUEx{i@eoVAbtCVlLbUhM)z@dOuB`wvN);XUp^JS^d|HAk_@BLe5?ncoJ zY?xxVql8)%eOh`JmMf>LD=Q)*BD-HWWo9%GfrGC5^Nsok=EL~|bIAJ;#le^YRd9}C zo~qcu>5t;KzA)_TuA~!uk5+2Oh57k<4I05tXGrX;zh`9mBQ@5V=p`zp@^$`Pb44vu z`m~4rVizCklBO`!f!>a9*<|H@i%&|rbxgj|azn?ES5W9V$tCf7c#nHZmx^z*&1z8# z-u}aO5E^~j_0EapiI1cr^C~3-V&B-z`LrvsJaRgE@BF5vtX9;CPr|2(wxpy)>u2|e za;=s)O`Dd$*;ys7H9p)Ex*O-_Df5$EvZH$`orhm-R_P6TN6}<85Ky@V3 zd6=+_90t`G0e)@6%wGsN4SR5#vIs6V^be4fx*wK0 zDBP4u0{PqC8l&Z;cdQ%|E6=!@gL_FGr@6DMI1nD0c|*(J^D<7&MRS=0c^Zu0%hy4u zoY>sQNGYfR8*^7&um*d(TlRhw127b(-S?{#8x|`Y1tZlr@u(seW`v}?a3lZD4F_Q( zAV$^`HC|jyM1#4gLr0ou4;moa--J#X2>yBgUOmm3-|JgFHMYo&3M@2=u*(|%=8SaP z8_Gzk&lKub&WHSb`Ti!=ChUbZy50Dhe%qlk$1xds;1or{(m}z{DR-eSCJTSUQxUNS ze84wA1fMG}8<8MDCIu={gt92U9D{)%Kja5h4d`8ivbbUEWDV-|9@5g$&t1T*iOgbwA-mDw+Mb39)CAy&w%(C$T8f{VJT#A0 zsBuMjeD?!IRX8@mhkAa9=|}eRb4!u@#d6806YU6|7}LvJd&2(y!$EX1EL_DF2X7O? zaF!6zP4JqpazEKTov_$54LV=_GmB$Lp+PTRn-cDyX+YoLh%`Sgy0I$Yj6Y5jGH zp{A-6*!`QNrC#6QD71gD>KA2heQ8%p3rZ+$y)5X+J~6{4)MIOe_Gs5=CVD9|Xq&EH zSYnn|_!Tt{_484Z1qK63yjU{qwHKFKCi~=ZOx^x@{8P{s41cDsTS7P@MR5&_Y}%$M zTGTB>=VrC&UhfKY1 z*`&AybGf_}xBJM+7dN$Wte<@)R_SdoH)7f*^%)7P>)qKg#H7N@*fl0+a4})hZRH6D zCY`c09=P}^ku2;hCz?E!^JIDB;;R#t-~ z9-h3;xuL;9?)y7e3S(uq%i8T^ZstIGt@Q4=xDr%CeKLJi-yoJAC3&-_;~%rUx1VOZ zoR|x#Rf9Wt--zT49PJ7R`=e{q{HP;k7cNI8A|g26+%YVAMIKxpss3jWX zk(%lePo9c?d(Vp61j>;J8*TkmrsS1b=_u3G_S)?ZeJ?y{QFWBsxN9l11r7D(_QWIxTI^opSR;Y zT-x?Wa;`>C1r=fnVj|_vOh(`;Wt$$ttu~Z9_q0sL#K1s0G!MT&WiMI1;i$81iM%RE z3Mhy)<*RC0`R=BPMTMc294l_}^iPOtwJVxbxgvhS-4*(idro?dhK8-Soh&j?WI#WX zCUUoneo6zVvqihbi*`}S4sI2%k97^Wo#RuW_qzELsRWH|E{_<a>IIr_94rJPYlxfQPA<5iqR1mu;PMF-U`;*BGk+l7|;I zB!iTBJ1-kSj{w$(zm0!_)F|U`9X#_BKSET&E@P?!1%6+GB;{FB^=4#Fr~FKjU<3)yYR7l~vBcY?y+zsI!>cSTTN5TCCb$_t`& z&mS9pzp$RrD4SBQwv?3A-*F|BeG%Ir4O0h?ckK?7J;y|i_U-HAVJ$eTPyL7SGO~no zk?nnpY4Q3@n6LBMFe^IWc7Zjxl5n8z$05+}hPHNmvzjkNWqV;yKrMY{R^jiH$Xe4@ zp0WKTJWs=TQD-|QNRcN)Q7-WXWVoi!rKL?lP1&_CwrkyQmFg^C$Dw9ls~(=ZM0`&-E!MVk^ML^wRu0yL45ii^lTsq z9t(g6iytr$AV9d=iaobgNAIViB`7o4Pq@|vV!oSjXV?&h7v*0qyMR{HZTPD?H+p`Y zr;{+s^zPHA^mJ^r#<;E>u3)>xbKeUaCktx?ABwZIU-3lxc!jpyS`!m@=`A{?wyxr{ ztPDxaIv9I%9m%)ime?Ha-LoINL4U%PVH(>(U4Hx-i5qMo0X${-}IA( zpU$aD*Z%&^Goi-0;Hvy1D~7gL7AqjPl?Au#x-|w1cv4H%y-Tp|xL{tQSZi+8mp}rGGZ6YLdOiKvFF2Sau*@e67w#SH4@YGj z88Q~0tS1M{f`LVEe;H^q9-}QNZhq6-AJ4E_ZLV3skgRt}Mo!B8mcID#wQ|oylAS-o z9osU7!|M&7Rbwt9th#!PFI~inc=KDG7kdj7dbZ9%g%RadhZ+2^OSE1UT`8+xWH3 zD+9tbsOrI0-Dop} z!twE$At2RRH>OqwMRA)I9oN9l#bt|7@DW{PYb=T>)7CBe&`$ly^#OEwjkAfP(*cQf z2bX1vRxRYAY}G!lvJ!#8;KPH1Yyr*Rj1x896DgS7NWbFa|UCyU)r z_Yg+hNB{(?wOpWs0gxM@AVpGFS1;FWib_mO?H8@+v&Z;&eW{@U_6i!m0zH zfUOrb7_*IAKGq4IevvdHAxo)`_wB!OtE+~>`Vapy-yf142JwF?UOC@WRj&GZ+)bgL zs4}V;ZeuKX6rsO>w(S{20Me*PbUL2bXTbgzLc58%Lu$|m`X#Vj=>y_7%wrSm+`W5PddBu7h_C|Bc%Oik#)B%8aZcB_C zG#)#0<H}Ei&HJI0|IVBIVp+gLWt;Z+(=P2o7Kg-*x3{c9=fqq8)aG(tZQ;I58@~a8Cf`>z#qktD%W7U6-JVa(PrmYVX6Dv!(d#d6pY8e@`V3(F zyPV-<|EX{9>T-u>z(AUrmN#dS!tG*HwUTgjWI+hvR+-e{zG@XJOiWA<_^)^zpmSDh ztr#46SJq=LG2?gT#p9osPYjl@=HU-)zzOXRDLKc&t+HrkdVl! zsHj+|;q|zw2G9@hfBr<(_J1W|rl+UPtgS!Dz)AbOVNrTF$C5VooiMWO8{#103;i~v zknwc6D$p}?=}=$R5Dp?u*^3+}DTcQ3a;NG!D?g<8(e`gRXOehKh#jNFO0G75QM5mv zwfT3N98MPi?&`|3!|Tzxvh|26)BUJKN0pUYwbBUiI2`!MjaExN3934r+c!u`b^nE> zd@517zS-Dx;P}>h4Mjb&Y964&B4X8S;TYs7SKfZ^hUqGK;fCtmXm+OhaHjp}lu1?D zn+myLw$QIN=tq3Tvflds{#Dg-vux?M6Upx0*?Cx)W@c$=I9IC7E31c(495?>pP;0t zrywQu4+}#)>waMeFryC!w>F&0Ca;F2F z7u0NB>6X0o+lTGIM?U?oDMcA?py7Y1U6l!L`*)K24K_XHF7A3ioyce%lBYp zMxU1_Z{0fc8BpibU;0~5-pM;;|L)(%Rb6}skLnfk-f(|EDa`=ld!Ejd>+1 zA4Hdx#FlRG3hd!o}ooc7R;U7Pl(KAgJo#|P)6q-c$D zni00RY=jP9^WUPPOA9A)df2H8nLfYSq60!-=@jjBH&th2m+K z#;%sJW!^S8M`v|NXV9d^G35_-ooht>+~3-`81tL+Z2Ot9Iv-NZ2=Y;+@3@*S?wYG1 zpe%A)I39iAcS(2AOC2H7nsiAI{gc8*%#o4Xq(Qov}9NO@nyNWJ*S{ZaS;*Qdj+7yFvnx7e?s<1y42n9p{qyE1XV0rAFJk-5Zpg~O1`(IN$-kuX3Hz9{MoWR&P zP?sd>@NHTG2|A#z@g|@QmrI?r!2u}kBkeIW)0UR%)G|_ti~cY-`qu zmqc^j4!cR2g}hU3?D|v)-P==yyXND6%;iI=6LTrwHC0tY%$Vu)Yh5o4EG#WaZ9sN| zVjc+1T1LoB(+y*X1>I)`zUKeDiN-ZU*X zf8~wk82hb?8%g&uE&3&tNcSg!;#KWIU9#4IQ|eKQ2w2o`BrK=c=i+{5#T|K4)xLB@ zT24MW5tUyLN%r?(Gzv7cN>)=s;<8@eqU3X|rB*N5EN`e7_}TS?cd&nb|H#90CxB;~ zG&0juyGkWm-kum@hcO)2_G_F#hS!EPwA89)-gF+zJ8~ZpycEXCafZGMBUKb zV;Z(%UFCAlz)OI4f4ypB&f&tp2kB~$AnxMD+>aUDSAkW$}oN!RSVKSlE zi!f0P3~}8d1TpY2hhm82m=8jkoV=h-AQ_*S0XlXO^Po4D?UT)T7*w5!^rTvWlg(>q zF{b~Pts1L4?^E2#$;o+cR~?eaOdMOcqQ|f$A6t=$-ewRM?p=B4Xg6h@aIlV7R(elV zIkG_pTbo?XZXy>?Z>pPQk0d(G%GgaTelQy=zDDlxS4dV;+HKjpk-QQ};A z;lLra6N%M}DL67+IX|w-bEdb~&RFJl8mI=3QPlEscgBB5JPntrP+4;Sq*+G%eb(hFG zSJ%>!`^wyT&XEMzs7r(GWyWDvWT{tNAoh{BFO> zP$9%p4@!H5=s_LQLQ?3!gADG>p)0qi#MTlUWvIGxIZIn0{uW!^klb%iIcKz;sr>)u zv4le4k-$@_z^*QWv9YeSr7M7^Xk}K;Nv7IXqT{gfC{)RX(#3>>hRjCEZt6ARo72g5 z_pa}CI>zSC2{{lEC+0hcR3W>i!Xc2*KMcPAP^y|#@rliy?XpB-fTjol!4w8`5>yen zDEUhEPuJ&GH+&BXr84N<9nI{sY2v22J{gRPQQ|3@xKPx#k%A>S@)m7>>3pHB93gQ1 zPDk8VM8YODej?FsyU|%aTc?aq-jc;&Zr_|@aNwb=aA-Af(pXEf(^K8v^%+gOF9;#v zT!rG`dRp5fu1~$cBkIu~k${dKSiuj5b{oz=50v2{@*-E!;vpelrl1Nvtl&j?Z3#~q z!Xt-yYxC2a7|f#6y4hb8n}*TX*g#z-detc;1^Pk&+DetM;&u_ieK>So zv*zrH2%yYAhTZ6pg}*2XK}Lh0kTQ6@2%C|$A7_+HKCyaErj95eb(bXurG==yb7pXJ zbE|J^a(KG4lLF0DA2}(j#@z&3x_fo45DClsz1Lb>)XM6d-#*@M?uhl7=%6~9-;4+N zTRg`ieYn;84nyQgKMy8}fz~4LuDc{d$ilSf-l%$?8UNZs;o1?|NcNG#_{Pz#gPRHE zGPd+=MK|Sj{<`aY_}vOV_L7uAbV1}|{;rPT+sU)bILrl?neqiqlxix=m+BR&J|hea zeWtjSzREq&cA?ZM;BNksKc_R8|7_esx=R7Cy8P+*Z17Y53y9(A^&!PR>7g^H(NjQ= zsj%xiup(cwpRs?;gvlYbinM&r<(EeT^=3$%*znEm59jxu_AgF|?mlwKR{!B}5I#jW zs0~YHz&VpLP<23iaHy0jw;?Lo4+y2$d`m&yc|X=L-Mu&J6Vv@8FZ*2ivdWPn;|2C&wJ!1_1jlh_uHYdc;nyM&-mpC6tKocwO348D@M0t| zu(GU`H^(1gdH?*9S=I5TV8Xixh_!APDS68>#326=T|lD&93mp(6Tl$&ZfsRlT z!|D;j=e+kG9pfjBxwpmD_m7R_(|lC`7co|go@)LV!6;DU54=1-0wkDI#feP4-#hl# z_~%FVU-oaWJWoHRBTrEdbgF`4AKVHuXTxXR<-OA0WsQ@aogYz&<_TUrx_92dCFo>S zVMb>-YTtSP7LqcNCjyz8pq_7ZbxqtV)zo}5y-s=WCs~Cwyu!oA1aRm85ul)P@Pb{> z*E_tT072l%ZJuNZ&;mTafL9o~KO9R@Q**@>R3nMDa?BY8| z;OCq)hSz*&6_}T5M!qj-#pTgSl;^jpZod8cLMxC`aymCy4K0rpBu1^amcY-kJZ;Rzn)V>2+j>aT4z7FOf=YvawI3}AJ8vOep z8;pSXCsR`>U?SiXxbBurCTgUA&jLoEEUFtaln|O9`gx0$xc5C@$DG7$+E4do1lVV@ zImWxU-LkNi8QcNV@QkQv{|N+EP2A#PBZV{SViAZej79IrL89Lo4)}eDiG{`g!&G%b zS~P>KY`i2sJ$a%Q67|HhWSI|4R4;6s$b==nZwOOG@ru26fHJL*p&JN=Jh1A)mAU*) zP`zArn7AnTfVGK`EFmKU%Ax)xN+c`kha5soIrK|%?zTjR#ajoUpAOWm2Xu`2%HEQx zxGB_dP`wG6DYko%K{1;tzP@`~c>&}VXafPrU}omf=p=QsV=mZleS@WT53<1$!k zesgw4q8s=nPwKxdGDYpWy%{VvtaohDlnoxDPzBYba(H^TFuiJ(XL5Rq1b)%*z_fCx zU|IJT?=w1uO2)5+bL@RgL`FubI|knw=Fo~U3ysrY#=cTJPkZCsaOr`akLPrzLQ*PV z5-F8(weHyHv3fO>+Y{Ckr1#{2732Zk8yp-QGKW?AI%-gY|V z;I-aaV<83wNT7iL${X33BNg5TsRUN|*bb>oQq;^0-?}uKflxAw{@dz*Jx&}?*VqQ# zXBTKy(u03%B3$YMl0So#lW8CEj@*y+0cC7^)!XdG#O8Rf*G@65)7xV#YnA#fOr-u@b)v=v58sgmb{Ey%Q z$b8;>+l3SB^x2KUWZJ7I@Rz^>9bqHYyjTq*cZH0dk}~pPwbgmIr;!WKra1!BrMHEd ztnh6c%$cz)z50jl{7>KyBD!B0%kUZoqv!_$qKIZC`Qyf8IVHpaVEatlf&ej1F$b`$}R?;%Gf7?h*-+FEf0ABIQ=B=eLi&`Igd{ z8DGNYse=`CC&bkz8|4BH2(bXONugeYt}ICB_tyH%E~~jp2iqisgc~IAKrkXLW!3c^ zfx%W>ZDj*(oUd02Kv;X;$lsSKtzJqT9hGyx`Ciz>eU`UW2o4p@67gdTw^`A@TK;4; z-ul0;BM2PUPZ(PJAT@xpZYp-HY1NC-Q7#SvAsS#lsqF^y z{DZ=s?%&-OG@;n~zH{kIH01FRfv$djjRI~PlXuj?R3K-1HBAnz;IL@1W3U!nfb z;(&_+dGm~{b_m`LQJBX$7dBC!JIh(;k( z?R=^cA-dzr22@;ZhrNBr*Oy0SRojkSL**sdAVQNxBT@K2V{0WniqnH>jye9{O9Ix#HM z;eU05MCiLBmN#6O=^8dBa+7pnvkMDNXwpE!N-0u09Hu(d0S7;J zoZAqbkkrz~Fz4B$1Kh)MBOW{gcC-LZAn91o2bweyySqkQTwG?<-}H`Z44*p-t(RF~ zfF457+Mf7*yMpXdq)LZ`Pa5+nIQ#BrFYMdYPL!s%xJX1ci!cJ--rilTPu~2hx?b1C z4R#!Cz}Vd~I$}|S>0k+f5RHITIbCR>&U0MqNeJl$hQiUJXF zsligUi**9K*u>6`5$Gv?TA7dtfdCT7owHDHvLS%=-@0-INMQPp^A%PbtqB?I8xbwp zzxw)wfp#PJCN_9K(<5rGO!d~Y;+y1z55WA};oi7M{%>0o*LOS7b=E6vGMPL>6;12I z*9*EEwrk9=r7iB)7MZTwq18_j0q~)yBxQ5BM?dmvGY5C%k?}dQ+png73s8B?T&Sr@eO@VT9(jqU_)=|Z z)}Zk!N{2S4cx3AT5I;q;c8EOxy#bD5cXwB>ZmC+024MZZ(PK(sNE>@MZ}q=59=w!M zKfyAljajX?r`UQOYrggw-_4JXj^@|ZF<%k=`b^4lXzS2JOimuy*x1N7c;2#4*a>|; zzxB)jup3KD>YrRR4b(HTSkT`GeQl^66&G9%MdIS`v3p{ diff --git a/docs/images/chapters/inflections/726ece45630c43be14589c51f1606bd7.png b/docs/images/chapters/inflections/726ece45630c43be14589c51f1606bd7.png index 0b4c961ec9892e3ac0dadc8d50d52185e9447f5a..b31abebb01c58774e52d7fb6578bc39d99e9f720 100644 GIT binary patch literal 11076 zcmZvC2|QHo|Mu9IEQOGLnIwb|vKz8w6p|#{NS1iWF42%dLbik?TV;=uJ&YxiY-LII zEwY5{+syl&=lT7f|NH;EZ=I$&=ghg!y?mGJzOHYi(Is6*dMP-Sr)`2jkg<~f z(Ne>ew0x@gvgG8798JTm7rEW5tV!rmA{(wpycp zWPfDtu4F8RllmAZKYxTC@{#@ELnGA!-OaCs)@Co=eHD3%&fyk}RSS~uZwfK7qG(f@ z!srBX$Js=1N6m~vt#(6TlE3|GsrH)7>~*r0Pa{K6QBfI*vgH*P zh8l^^RD~g})|Pygo4115n;6T6qGQzwoHuXY6twB??EJ{>|mvUJf_WDa~%Z4(2^F0DL6ip?MBQkZ2LX&pIvyO@iut&Xh=zNwb z`@~e)x*$1lGp`$I_+0fKyL=j>ST21xvwAF_BlGiU)=RCC@a2Vq3UksGsxpZ^5;;;ForJ0#BI%QHHGz5ab&Eimb_&3$HMhdx1 zLN93H@bOwGb@?qRilf&Vh&UUq zG&8HWyuI1BnwW7wdApp8J?iAly+42c^z`*n=9_YD{OG>FC2MMCmR(-n-_W^zA!G0I zOke&L^IG3C4Ac}{+}toEhS;{y<_PBLnf64@>wR8Cbh)v*wmjXh3CNO-tl^}+ub zeVASAA!m5w@6p=cfdOj#zWNh$3F>BP&J!mhs}5E8n3Ix{zE`^Q%e#%xBH}t`dwzKz z8x&{{YeG_OZfs28Lnx~Df7iaf-CvM7$Vg40!gskvlbq7#Gj7D{-*B%xH=}m(70dWet3r0c z4P#@nVza2QBOKwQj*A1udBw$>G;|0#Il0W79C#5kBO@aMk<Wru0|7EiNveb6Y{5k4@yX-EHE-m3Jaf%$w|o-rbs?pQoi|wNZ~KY2KJ<3`GR% z^4+Bj-e$;w6CzgW`ndwiuOlGVx8 zHM(FFdDkfAB_gyjc<#kL$xLjELZBMkE0f53zpGCbbbqf8QvO}HZTZW5P^8K)~wvo3WDv zH{3567<9aML}D^93z_u&!bALz1V`3pk=o7q~7Jr%+~&sR1!AzDl-Sq%tezf zjs&iE7WD}sE#RjQZyo3%pIkiYIIT^aC7pV~bt@KqOFj)#Ic#iZ)@osh%GSil7&H~D zH?hN7JbU)+A-kXu@|Y;vp%J|pTdJ9FI#sjy#z5~cl#nJS!8X1~y@?ur`1mM1 z5jGJzmd-ulH9O8$Y^oORH7B|Ul+v6G{s(Jqlb+S98{aOM{pjWTJTW2AVt2Msn1?Qs zJw3Hl;@GsV1M2W3@`O=Jsi-mOMA$}OCQ}-tDbyCLon2YUVUR4D-rtS17o$&9e5=|Z zVj-D)L@AxgvpUdBJvlC46g`E%K>TECQOrI&J6pzQheKb$un51|(|w&yU`db=MQ6A8 zzGSArxx3umRxt0yN`a|4Un~IVW5|1L-2#jRI ztvrt9y`r*ywL0rq_rPgxQ3(nASJnjwK}K~mMY=&E-}EX7@%i&C5lB`g1*T?unUhxV zE;bQls!h$tI$IGN&1N{pX-| z$FlEDTU!Ku*n`E_N2ikGF$qgBfnUeRW2dIs_l#a9Vk7xlij z=Sl1LLzlr2&V*|xTx!MX7i`7iz77vZ4ZAm_FFG7$5m;_U_V4SyemA_#$;YR2_wM;Y zLqVj5MTEZ6MlAZ3NxndGM|^sEr_U8yy0{8Vg5;s_mCdcKud<$cCKo%T^>hDe-mP|7 z{!N;uUfg^7MDIm@x(oTHb{$X8SSp0Z2{_9toH{i@;1fFF-aTPuaSz2{hN>BZt2#Ny z($mvREsDv8A#6i7gLngJ?@0plRf9D)D(U<+m>WCMfuJeOKTGdo*1KmR?XBiZH7WE0hwbmipc0UfFQW+d;v zE+hfSkZZqkGp1s(2f};%`)5^DmY9#eHIH6f^Qf|vgyZ|NFQigTU?2|#1%;@n=<0U* zmox?Du&}V5`!~4hUq6>`cA&5hh-^YeI+a@>@MkJ{m}yxuii+qU6nT8u9&4`hUz?Js zl)SIcEwWO#%|}>QC$gS7a|U4Fbj|r9MV|#W0%3*7ClpUhO6_ZS3T6$-YZx#iMkZBSQIR&H8S>o`Yb^HDw&8bI|CvUb_!}0Fnwn6s)NIF( zXO))r@LvRUS@c+4AS)(18gDN{hWOd}4L4fyzHk3WpIVY>oHu65TWRfwONnOgH_Xad zo%31m5y-Y-9M*V%OMfSnpK0!o7Y;*1L#x(Wxjhay2lp$f7`XN}OG1c*mCWt6KY#9w zKRpBae~oyPuuxbcEh9r-Hnc<>a+TK-iq{ZJ5>i*^-`(B))X}j{l!(_mP&Yub00a|K zP~fn8%gGW+DHJbyMoOKRiH3)XrpQbjJ^=wsupa?f^2I?mT1ObZ=;-R6m63S>D-s$XtU*1jAYv4;5a51nD+mzLecIgj8${J5kUQI#&=D)j;pZoPE|GfG^vwCJp z33CJ!@1N}u?VQR=zdc02&l3|}%5}-|YAkf`wpZFR4u>*2%`-5Z;4XlcdV7tdA3tug zd%J^`PwRvQ6oNjv%5%d3{ew>l_9&u3az+NTT1M>i=RCAwsE67@5P*Wa=d>i)g_V`L zhlYj-!^8DKL4hlDw*muIvg3b%L8qjo(C*X?FEjbaU^r9IBKEg#VZ6QB+$y=jLs(f^ zONW=?C&A5=tCb4Shzegf(J?7KoiWSk;d!s08b}sFn9=I$YSGiDHSgb7qd#)Q*whpU z!SBhFCzL#5&Zku3V?wk2;ROxciaxdrJ{CY5x^ zaY6DvaMY?DFZ$|E zKeO3pbaj_f;EuHK@((K59yq?ZORawR>$1n*hBFas3%I!XOy$S}$BZC$g;8%-wS(2> z;070dKG||l&d3zeCuwPm{Oa$#=DMm{W0BJ>qJVMx3$INdZjVPaPd>~ao6EpH?&`X< zG**|kxpng0*y^hEyRtHh`i(0TcDF^r#)Sa<3LpZ>1dq}hnVPc62)s0L?%{B#h*kI+ zAd*N&3jF30um$bt(7JMk6*fplb$}Ifz|@M1At78&R|+e48EmqZ#*MU2uR#D{sR#li z+$kaMNukkb+uc92(p)hBg0;Q8WJ#c#UoATE)MawS*6-1%tSrIzm6fNuwbc{bafkcc zxtlxjq$Ftm&Yn(N=$M_=PrzWF_2`^VhtOUphP!6cyMy&{tD}*$K4ug9n~Vx0N7qhL z;t9za;t-FkR_gcf5DxC<_1}(;jz%C}n&e{@>of#bnaeVkLYdX@u(1rS5dV8?)LC`r z`GqLs`ukU58{48h(BXPo->_!ZAHxzfO(&{j*a#^Fxc_`~(sWN$<`Co@^v& zd(^pKU8XVCtVntZMT1q0xnPUO*0j{aO>8|t>xi}7;XjXzV9FI$`TI8 ziM;z;%gvDWDDd245HT5=cPN+ zUItFUq3lFXPR=na)DtJUqm0bc%vY}pN(lxAF?{^^QM5Yn#JU=R^UHE&D6wvZ${KY* z*v2cbAg;9;OxzeJuirJCSgx!acB?y{d6I?2;dU%#v-J13H)-TuzXu-|H$&HU+RJ3s z)^fvk>j-_~sf<7tRaI4^!v3AH?fQtuM$MVIIZ~ir_SF(g(&?+eek3t}TGzzK!Tu#@ zxF!}$e1Cb&tJ)G!1?e!C&{R)Di$JdBQM@BCac*3C|G@)tBInU@#iRMA^r>ieX|bZexSr^G zex4$^BkZ#JAsb6%i@N{2tSmBKUfzi{e1}lerqmqr*>46qDdqo zwA~(MWyNb&V$Hi+v+_)7s)IJ3Rx#+uJu|_?yVw^fBOB%?Pste>$2`QZU&lwXj>$4- zT!{Lh&PiQ0%f4cCSfjKek685&)=v+{S;fS}z;<;tqP;Pch71v_@`o0&($H))eA~q( ztJkU2wvqe;VQ=xD%__IND+2N1&u8(SwODL)QIW*Kv^qf;ky%$aidC3ft{mU0ocw3& zN*d-s@VT_kot^2}v-0xne>O)(EI-Y9@+f(W-WqV+G^~JK1lahIopSe<`M$w3>_^_c zdk2iuVUE^`@J14>?MzdkiRMf{SOoscc?ET_*gz1!c<~}>llajG-4seA7`S*Qn%FrW zd|I4>*vuTd`u^Ve%J;Hpqt@2e)%2A>nzmSe z=3VXfl)+#hrF2D)DN4AJX_h>-vzC@=a<>iD*T;aF{oI{pWT^w;fbd6(D3e#_EbrF& z(ThXq)TuUX>%`(sz%yD`uU^&m6I46gxMD8B?)u~X>7Ucfi;Krlc*<(n&S$6^%59+EP`y{)CjH23-w*Xj-h35hA~Ik9Swu36uuDlaczdPBCD zcB$ATX$l`wd-klya!21cVJ&y}^S(>-8it*>hg|at#$C-BsgL{W7Wa&&hy|KZ~ z5p@(ZB7XGyl2{ySV&uH<4+E4%!~MgJoGjM){%bbrTbn7(Eeh#-J8L({AGA2y2_0#R z9gmLSzt0SNRmfjgG(_h&0}u~g1isA}0f9x&xyjj~AtFEIT2XcB0%Xh29UVg_uwJ7+ zWB22*`wF(33ob!CiRhHhVDlx$c%Q}KSoM1O?220Modd6l2zAQ)1h0v{d>8lUfe~|~ zes$`4I*-ny19|hf40ntRxeQMuCO#*LR}l&OjdSaVywL!$Ay0f83tRP3)M24-m(Oc= zt87!@pm}I}tK3QA1~rdzxN~kHIUZR0&h0RiwLbm)qIfQyx#0Iz_AR2`xZ3kSug;^s2tg3mR6V2yGz_ zI2IPDOI@zfz-2&2pKejXW~leak4NCZ$AbOd8N0bj+RL1H%yXU`p%}C$S5R1JY-JVm z=FOW-E{KhJd3nL8EIj-e_`BV6yEF)u{iQM|XXi#dUZW_2`&@Fz@%7Y5ciJE^wco+^ zZ&LQrIFumdP3S3IwPkDspVEo_THHlP}Ey2 z@jf6(6~fLm!MR6mxYAu$SC?iZK=pH0Ph z?#a^f2RO@$708N?y*V}^fDAY+kIy~!b#aprUyFOm($(}bJG)8!5&P<;^cU*b`}jWR zak+WGQG2+1Zgd?0l3W=gU>}sW~To*yn6^xUNp|>ecr@mx(y#3is`< zg1S3`Q$N!L*O=?lNa}*RBc_+TA@cHkp3Wo)axGC)I z?LAO>O{Y)wTus5T;`NnoIiF|eD#w`)+OU*r*O?K|FI5Tq^BG~W>X3cNUwG>36=VuN zb^QSVqSp71D<+cC$(f9`Q8pxD@+~{AFK4?Ru>^{D2H{?XVr*8U=!q>EJ5mVE0+15s5vm;`~|aD8`oDWCd; zC!{mMQ+u1+6zaSBpAJKnY`QaO21;lMdnFllnFcCA(HoMoB&!adox-uCNqJDVAcEx(4 zSGT19Xw=!Y(UpLV6_-kioSd;X)14OwbHh5HB(kOT<4Q3mmJD(V_ptR!`No@@gF6df zPV~LGF~=A-iNRM=YI>Z#Gkeh}RU*CXp1#+v%xbUhSWLa+7z?U)ivH zI7M6@Cr7;0(lURVG#m8H`s#5YSOr~LrRS%Cvq?_VZgu84lXTrnY!j`qhsS7+YYGuhh@sgR%ORA>a| zDkf&^*=4m~K~@q>|LsIv^7*8sF~RLL1q$~%su3R!&_MEqbyss@xa%+z>l2}4r)Gy- z$J#^jZE@;-fsTd)k* zgw*6=(w&cZJU8Q??D;`#NhU^xZ?)5%+HHNa60vG(7O?UYU11xtO(V#Y=>Zh=FRd!4 z*V6hNB)Zx#rAIm-fr-?UrCVK3f&if!U%h(HpjZg$0T*GnE+{VU(i(Xm6fOWbXVnu^ z;_OkF1jMCWuhue_;5`Gq09cn`hqG&y($=KLf;jiaxgP3dV3~C0r z`S`*$1ZV?@Gvl$3AKSUQGMqnu{w(&=J07Y2oB%{ISz25`pc=p3Tg8dPD~r2b7g4Hl zs=doO0j5&NHs@tdkpVJ`Q{EjqFW9rglHf+_o?E3Zn@FDx>MN`*`IkPj4rvD1e%-!( z+u`k19FiR_nimgDT*oIeFfmbY<{wR1AU)+!X2|{bQ^)>!$^giBUgH!b9_=WC8I%@8 zhQUd2|LqIb*4Cc}L3i_3;+^r_U6OwkN0}q7k0|V?kJM)Qa}E?H5d0qg{CzaxB5C9` zwdYBFIZk@5#M9E`^FV@Ik=mV~7yYxou2rvt`S)%d#;)9g#9}>qM9*Y#LC?-OrSuCh zB#;u8f;SnnGMvxrqD+n*7CNLtN8H@pKH)PBcoMoq(dP27r}ph)45QO`WD}7{b34YB zbnybT%pOM90o0gQcsp$BcLLQOJ$j_*bX+h}*9HI1a^>qQ10X6iSFRTJo0r;>75;?| zpMDi!+8r?ve5?8e&@~_Ku0I$9k;v%AjSdU$#99A*zW(*B4xXn#7ve0D7KQyLSUPfr_Z+ih3I1?VmMcaxz3A}pP*hS<=h7u6T`-|F=q4rb!0q zh^eWm;J-kM0GXAi{9)XX=0BI4nwo%V7~r`mU9aSug1-2vtLtapRooXQdRBHuX6Bo}N21_TB&RzzZF;tU!cZ`u`OYL%Vv!G zDMqNTuuukoNS%Kqe_;vLS48(g;=?+Z)nH}kc;$6Uznu{u&ESvtQx{S z#nNxohX)|e5G?Y8d`FUZ$s7x#I|A|KES3-mlhW`-5cv)IIRU^)R7$GtKPp)BQ#ySp zl8^vx7^Wc;kfvu1p2fq0Jw5gLTO6ERK5OTam&DEmoLl9Pd$A;T4>|6J1g18pjr zI=O=?{Sr;JWtUGP?Ia~!4?H~ij}S)Et_KZ;aI9=19$cK98bGIPCdWnVVpwuP$^S%J zz=Sm?#_x)Ypo(R|8W6X!#E9D8H7-B4UpB9qs1|U%f%V|lRRtanm??My__3*QZZ}%x zMk!uW!mj%Ts^ldeSQPN@tZ%;{uC{|Sc+9V?auqb{l{rCNEx68r!=NA`4QCSl;=W%x)cl%J7B$*Xnz+eP{TLEVr>=eX# zc{{KOgGiv3ak)c|#S@}bG>B9Q$zhKmG>qLsHQva)N>ck`kb5dlonY#trO*o)smM#q^zY zuO?CJ6PrryXqR>kwZz(3`}Jy1QkY(CIwExk2Ax0+^L(ogCYhF!eB@3uqj8XR0Us$X z#1G`CC5q;oigDSX^k~^cj7{=Q!Ovmn_Vo0B-vIE1q=c_fQeX=n4?)gSAuR4C3c~{Y zDMk9Fh`qf4kh=7~W=Th)U@<&BJxL)Bq&s~jXm5UbIm}4(6j~WP$$$SAO02eJculynf^Qc#RRzJVMD#Mmr!m8#8sp&q@wxRe4KfPvj?b_d6 zxy>dbOQK7R%*{y(14%$l;0kMJs6Y5)AYon_4Dx8g2I(o2{8Ff&A<3IxbEH$lOV&3l zhTWz~v!+U)=)AGDWilr&cr05wwUv0#O(ZRi+?A9#TB8*7Gz6Kv{QSw96S$YKa?W5a z7O&&uXpK{FjnWLD@%*luW+Kh!#Y}0sX&4%)Ibd$2>3@Qst08M|`sF<}Qtj2_BqyEL z3GxV@wB>+$85SQOpQH3)mJ0E4FLf~Fcqh5+vNBe)!gIFwKpUy4sUgYHuV0I>v9o*j zp->4ntr3w>fYqLO3g~{L0<-DY({0ADO~?@8V4Az1ld6i0CV)9_dv?xco*^twTBg2x z2D&pQV4Qzvd%at0;2TB#H##`3UuHH_A*x)wD)!w~9tx?RF8*XmcrT-d1`k6_Je_Wf zS3P@dBVeF~(oO9?sZ2_*G@6;UY={AolG}?L&D~gV0b;i7s|cigEGUcdK;rCAy*-(_ zaQ0iBuaBZ?v$RK$jtno!BU}eNcFx%zp{FNlIwfy(J`UPMjOf}}EsrY<$e-#U#jWJL zfD_o%D?#T(Ys`utr11xOb#%_ z+{+oQO<_m=r2wMX&!GNg2xR_B2l2-A+O_E@sd~i!%>(ajx z1W62Bura)OQy4}(x#qvxVMAcy4_qZf*n+;v$;H(~Vg))TDfHenFc&00+myNA!-W9r zn5y6Z-MqU#mt|H67>zs}1Kt39eMn7>5-sI}xRt|+Q06-u^SznNm(_NWATm=R=70Zg z1*SDVIXMDKt4f!ljt^MDBl`jWiU_McTFkqZCFXC*T?XI8Y}}#Kr=?FU1logL$~yXE z6bHpam3AK`UN-s*_zEbrn}Yc?uL~$|FoTH%X9wvCH2`R%)^A1U#*H*AcWj&1|0o1a zps-4?(~MqC_`A-^m|s@V6++LIyuudMds~^s#hv4W@fjJh0MzXr9V0%N+Hq1V`!3ConH%AO1uE|7Ky7Jmr11U|B#E(`{asEBGn2!8i7R*rrr+!yVUdl zU;hchG3zg`QruA%rZbPV0pIoCBt_`|T`EdsW=>?wnCo#7hO$_M{)J20Z&0=m{tr9# BLGJ(n literal 11254 zcmZvC2RPRK+rH|_D6$D5duH$1gUlPr-ZO;AEL*v)gb-S}U3 z&-4C%?|U5oqdKaO@AvcljBA|Nd0t`n)fDiqQC`Eqz`#>dlzV`IafulH2OAScNV69z z;4e%I6$Lqr3-qt|HCeG37`HK$oLy(#V>o{QfgEQ+_VjQF>&M$2 zS6*1fbVLv^A6oGxBAEH>q=PB=IuLw`J}mr+KB-bWw|7&vl0}I-l3aTalXtbpH$23p zc#?XWo12eZa@0Rbu&H-mFWjN`@pU#eO|u}4-hm-*ROhZJ0!d)ONW?6IeNQm~n~sFe zoE~`}i3}rkEG;dMtLje0iHM0=Dy%Ko?;ne$)RvZN4%8tZq?Rnf zqz7BGQ5{w?vxO`759!!X*fCg#NJ_-~1g~mbwJkFkDKHk#3*sA>DttrkwbYTIl)%ct z(PYJ^F>Trtdq@4jjBe!qg8!25lEnB9<8Zv)D|v>crKN^~0&H0#X7i7knV11rrm>1F zs1K@1Ij}16G2V}^v6R?ZynM}rL)_b-Q`6#oe`s&CB83AP!-&Yd9j8pX&K}CcXimgT zL_~za8p`$COU+M9D!{0R@Dd%2?(FVf(j%@5R>U6+p%1fCMNwi{_a?6v3v^?^jS%^tveo|Mbk2kXWJ7DyDRDn*oUU#g_#Nn+^S>(#wj zJyTO!?}POZrKPbc;!lZaY5S5zT}6CP1k!wVa0UhjXSUYCH{1i z@y?x|6+Uq>F-3Lt!(&utCiBbia5gqJ=V3N}e&T>vuW)fO1N$C=133s$VFo^@7WXLf zJM)q{n!Ccs$7k#0bolKSp~Ux|L?k%FSeZS!@T?F2(`om&9E!;XhK3s1T9(&{0$};? zl9GJI4%?#{I8CY(uxNc=KkSOX^!nYq&YLgoa|5l$4Cs`ii$jQoF{* zCMQc|y2w3IOhC~0Zb!o{lz8r17gTMEMXhM-=(vnOd>io%)e=swX=*C{%k%VLV=>;) zalVDz)GgvI2B+cISJPEbEd?k&mlJLDD&1BS*7~QXdm^dDjjlB8@9%ROe6j6b3Jqg1 z=Tq9--`9BXps^#CW&E2fi>IfjNnd_W&JG7+vwBxeSw+P-3d>ZV0(BOQM`i2ixZFzR zW57QVCN@_len)cb%lO7b9Ktt>rm6(uDuo}`(jw*6g&`_g>_gU(9#wN{jV zb8{?jtWv!h&+lep^@@lnHxCal$-0t~m`TT~$K6|+Mn>J#Pp9U+B))5ttAihG)m=!< z`NsXy>yq%?VKgjvY_S!H)O$+kL&Qv1keByrCCxw8eQm^ZV+7Z_bi99UzdKdZyL&4` zC)aZ^LKv039NNmi%1N+K_#7PdX_xwifHC_|=OzYrLyWUSXF&tt-k`gQjY*8>=?m86T?Ahj1{px4M zrVk}fex^_P9!?ho`O6^Qv?nCow6L&{mvrXaR1($iq>_-@DZDr-l!}OY#@^>3H2D2{ zqqeTG;4)c{)%mfK=Ld~cH`f;q5{nG7tNUIML0fMz%CP?3$QarxF2(oe)!R_B7#^IM z;$GVAAhj&Z`Np7^Z}#U-z6~9X9wdp`gMLa7Vy=Mxm;`^oLCG>2`RL36;g?E z{GBye?6vgQ2tkM;F2Z}5SfoSdVH(OR;iLjntd643`3H*qm({Ep1O zfB!yK>7M)XN`9(DA>xXd7_)CtVZi6Z;3{(-DI-#th$-rL0AF4~_73Ja$X*pg6i6UyKaF_WwQIA>4>WBuMjNO!MN6uFFw9#BjOz0bd7 z$o!$Gwv|>;P|(PGqpYDIn1D-D2qE*xw?|JuU+34_+CzL|!UvgF3CI(8XJ~jhIyE&? zJ{`wH8~G&B#hhsUOySLYI7e+2WdxokJA!!)j)#buoT_x;&BVk+dvEUz^fbgH=0rE(q_uT*W7E>q<&X$Mq7O_CO^imZ3c)e^ zBJ^58#tFztnAgSdI$=&@PX5&BsDhFb4vTuK)k<%QIXUCpO`o_&d-Nnhs$HKlGJ+(% z50Jy~e13lZXDxOxl%MPCxE}8CDKhW+;Dz(N=f5Ff{a(#VD|s}SX{LKG)SP*rrs(;U zKO>3s*75ENIxX<>@-nWmEU9knFh3TyDcO+z2;o-Qv1;QhdM{DTOBm}`Q~p-f5Tm_n zOmwb=zZP6IV4fHAw-Ug^!~68<69E+!)p|~-Gg#G|xHt`6T?JiTatufwKbDp>w6x^h z+}vK<|4dUNW}cUVm!wRIXl@4ZRL78V!QqJb9P!Af4{SVBQMrES&YfgQ@AxrGcg5)7a6CBoSNEI zjFnze7u1m*Zt&$Bv{KzWOI?9Rlf~DMd2rhH1yf-i{ey#xUU@pXVG$8bLPGKF?TT(< z)Lrp+c2Nsgu*??ae{Q%-Y;J{QrMi51;IWw;Fmc@!wq(wi*xsGDoSBp=lmE3cwX2JR zg(d4nAeR4&2A=Rzo4%O%c(Ue+NMpqiC6UOeF$W>FWRZq}bcO8CpWj0S-rRH)po}mK zw>LAxi0zP5SH~aBR821|#IG??x3HiO3kzdpV&e5ZaRU&fw;gW!`Ufsi){M!~ejyV= zMp0E2592yH&6DdyEM*g&jX_u%W6Z;#MP;v8)eog9TvW_=F_#>bW5uDoh!U|o+N z--%x7_4Jtz=xC0oMFl)cOoVlP^UcW0ihcJ^8NiW2zR-PI{kfx%4`pR>k&!nrz7Ou3 z^r@;BUb-oi$wz@SO4vI*++7Igeqd_a2dNOEKaNLSYmas5cCP8!%a<>&eRZ7f6t3Nl z;H^1F9q)E03HLqjKmL_lSS#@8XPT;14UqN8irP{|zzbaU6oM`C+>8{ljCR++&aXJ=&85zSajO0MOmm*7gZCKV`(t2zf6ruOdB8xDg&6p1U$K zm)Ya4(_7I$ho-iJ$m#AktUTgHDv^G|I)Lw{n1DE_UYfg1+1_0)2z)-CFAq3aq}S&3CP7G z?D}lYtY<4JPsggB7S6iLv2$?fmpi5$64xH9KHHK!>QH~5i5U|+KOZvhJ-eIjQ)Zy* z;<8hbW%$(H{hOc0Zv#g>*9)I7dFwqFAB6cvp|@wBKpEWZnt!T+RthO}qX?!U)v{AxlTpz9Px zWo4P%;YiWsWV#`UC%+zu)H%V))zzh;slCm=8fyx$YhJrvgZEIsg z+b#bkH@LwOSbW73b22P^NArdkvWPT4vVh%{l_#j9a^9QQ`t8cP6ZzXLDNYZ!Y`{$l z+CpwnTR#yMVkBrEJ zv#Lv<2?$TS5ku{QhEP7oOK(X?Nw-gDFH-aJIIms5F6wjir zsZ~4cC~a+R(znJQs3ly3f=sc!@vjQ z4`Bq<)IB-cxe&;Iw;VGjg`~hO^}QtJsP@=4_1+Mq@%l+Oy7+IU!)HIjIpeddI#Ot= zZfUt#v`8iF_yZzAzzPP%daC!P`NjDe>TLEx61;uNebm^f;>q(68m~}}!s?g{RNuvq zXayQ`IEwY@s<%sBZ+DLtqp-LUnT_WRwEigGaI!lx=>td9%;{=6*Ka5F$b+M|VLbVI zLn7z}I11iSYfDRXOw58^zLmAL7jAfTCBb`C&ZwD^l9KtxL|L+wFJ+C%mQae{!TLA} z85t2N>5_ZhJ2=QU;3T7qvLQFV)tZ)A;lT4WQTXTMOpJ^KP%J+11$mS{+o zMD6ZIU_h+}HB!OWj-3Io$vB9cF8%^Wzr3m^We0c&*(52f&2V=^NOU2&@$%0@_p{xS8?XH2Pr8b5l`G zEwzG&Lc;&YfC4REmt&6^>||oX04Rl;)WO-=t;+6S)j!0J`VI~}x~=)uCAN{1uF=ua z&8@952?-(&=L+)j!E0l4(q)CUFMl05kJV0F8V8p#u6yWx#a-?Re1%(aU?Ad4`9PC9 zB{m*2(9i@6|Ao%=i(TpF^RBx3JLMvcKPG2`XV1wxIP)mJ%z$rJX zh1*8tOL@6#3#o3A$A&@qN{X~}S`{I!?_Y)0m+#zpvQ-nwAruJYKf+42R<;#Rw#@3ako1>@ghH`J|nXt$*w1F-IoCVc5; zE52{kPr0(BT$gc0MMcrU{K-b2*pX4jm50?!fbDfh>qSck;3;cCuTv>@_Lo2ySXf$y zg@-c;36aHiG=F(C82a)hJqt@n7kAy%@H>d_^z7_*LnqZIzjBS7TSy(RSe>I#^a7L? zC8Zkr`Fp0$SHFG{aG=DC)-{^Q2E{VTm1Q11@@#c%wAm1NC5 zEUKF~Zw@-!==Vw&)}4FRqISB>R}e`;5#SQOW7~bw7s)+I!l!juc`X+6a}9wB2xAqN(* zZADQvs98?o>An2gJsJ%Sp2w7w>RaQZSf>XKENMI&3u0-TZ%97Yox9uG28^1r)aK&_ zBS-ZRt&$p7v}Yju=G%JmoGdLwce;+WZ8=7|in_bIGpkFw6YH#H@V?&Jkzi+c0Zeg& z&pzI~IWZ!gVuKyIdkQGKva+((m$dOwwZCZVsaJT5jI3;+OY9Lnz1eT6Z?di8m|ouI z69Ni{SZ13>N5_5KkH5O@eYRY0immZBUs6`UxTK|J*4O2Vym{9l&oT*VU#PuY8v6Rj z(*%*XnzSNC{dW|T^K=BUZgBo+z{3A9y7mn$z1`y3IQU&zS>$NoqH-Uxx=5sWSvm33(~aK|P5~W%`JZuA??&>a$s#%H znwrNa`qx5CcRg~loh3%6rn(R7E_~@37>uvHg5s^foq)k4sNc@_-5G`5E+v9_u;Rde zc-5`bk;ZqU?Dy%ctowe(NSp2X9*c-uJK;_4>N#6a-07mMZxpG1lsh|I)TblmKRp-m zh@?)t2PD1DJr8s5(sQb|XP8IGH*NVHikJI8l-+o1`lz`F`M)YTt^ea|mMNkfAW0Dt#8_vd#EZ}2&pP*bW(!|!|d+#SXjDq_-y z?P}DL4!AzTZ`LgbGn+_A zZvGF^n{--ObMS65{=E0$AONZ<|BJJ{x@Rr*-}b%%xhCqmT=aPItNFvnQx(xHsn0G4 zHq-+~Tk(jTc6{n9ddIXD>fnQ;(`^Mo9-g_wnCE5%BxH@Ii`)JI~RywdG7AP<%_A%|Lod(?OYuV4UNAQ6tDn!xfte)KkL{pwv7DS91bm@NKetp zfw4B@VK;ZplljY(5v{}Q#1CNR9-dfW+IrU|iX+(c+ z`hI0aMS}QdMbrM+WHo-iP8ZbP(iZv>~?<{dUwgZ z{~2rUu=3-_`6c!)$b{Q}Z!9BIm{jTqmCeJ0p63?Q>@Pvw1StD)!1nTeQM$`2jdDtd zQ;T|gLL&DU7U!R16`59lkt70gEH)7t7V|NEevJ31+&^7R`0a?3-(c~T@knZ^8Y9@0 zb(7F`J)xl7@T*FX z8-?HpcOMiKBq#T3#tgTtk7G&6^Lq~tK~m4MCacIlJePFlC^n(;4=2ZfSMI|_YFSiY z1M3K4!;ne=&mJfK+<$Vq(R3q9iu4&@gi5@(g{({rlIhS@Z ze(Fqj`Sxdiv!=vhJ0?Dt_NdWu_JK9(^5?9q05FPXyf_T|QW!RlwpCWhNb3DBg!c{_ ztF{`0u1B8sjX6ZNAUSl5#DrO}q)`3N{!|!o*J%8w&~x5xWV|a8+_ASBS{=NYakx?}I&-R2)8WW;!Dyqxe=Izir*?Q@<^TaDd}SJ!{1 zQ9|Gvd1frCuT1aldAaxr?7Hn)h%)Msa{d{5cMOp=;*_NNTIWa2}3 zu>RciVgjZhVKyhaP1E$(ZNBdV%1_=&rHndNM^iagJpa^S$=n;SvDJrQ0xtNfRS?-#}%>FLqYW=7Toro#OHClP#Z> z2fEfgah=YmJ)|6Ft+;LW7bm*(cJ}6{9tvwrY&b%WZf=i0ew>>UeRlkCt1So{1Gw&O zuiWbqnC$HDONcBifbh7@#PsloVc3jw$YN^UN^ir`TUi9IP(&;5`Q=E*KJl#pwUsgC zql0yoJ5)|>LH85DE`W~Ndb+2g_#`4x>wJm3_Rgk?s(YPjp)0C|jlPjkK-N3@W`~97fi9}u zyfisJp8Yc^#NHG5O1XRY@Vw9F+Q9b<$X?ZL5woZ1cE;}9JqW;4^hB2vEiK<_{A@jY zb;9a|S@ARexXA74B@3XROnjIsjMr4o-=rzN%!5tv@u@uS6 z{FRDS!gw{(6CEFht$^O10A)>ZSc}Dn4g({jMOc9TOjT9V(a|#)uoC=dAU8K0bm^!*rG(HY=t-bzp=tUAj{(2;r$wenJ@H zE0Q0FI4FXQ*+pLu$pQcLydJx}Yz_SBtavUC;V&zEy`V082M0zjF2^zZeJ@FVn_jIk z_Cgc}#`QFGFTj_`55GM#B-r|t4g$0*s%vYdpu7#P(J36&)YJQ{M=iUwtL*t8HP+3Q zWMp7qzL-N(Q9Vp!(5-s%7z!`auGVj zn$MA8FC>DL9zM)Kh9wZecdek-Rabv12?gQ3j0bpYC)k9MFgyf2C%S?y{-WHhMQ{G` zc3{KY$hGX8oERQL&0v@WG}DZS^qHKnG7wR!S`q)@_t|6{<>=r#K~flf{5RDf70)>d zk3&ubYP-+|v)laqgZp}wA{IH(l4hjrg_V`>hK968?V;FEb#yE+q?q4B-+(&!zcJsxa;TV_k@F~`L7rrjdoBb z5~X~_=ElOLxViDm45)t?;P5)bu3XMho}l7l=ufd^GJ|yhcmw-mQ*D&HB$R{)Tf7zv z+QoR2972?c`5Lgs_i~4csHs)Gy{py={A3VJ=urm;tJYT(^%*U`>~ZYvWe}&}?&qey zxfXDU@|n3oN-&6qPvhg`{}JUH3StO| z)nF;0|3ZNv>u67xDS#8}{A+6~3kY*y-9V^w2xf#ck98Jaj7N5e2nq?sq@<|+YyoFJ-fd{jkzOrS4iS0{D-l6d3}+%=SZKYsiG zL8J+K2CBJ!Uk~{cn;7C)rjwn44+4q^P2+>tT4fX^dpZ|UR{M>JK0UK<17HD7y zpTI__B>ee5C>k9d9Um$yA^Ku7B2YS-noU51X6NU}Ng|LK^nu3DJ908JgJDxz?4*-N4vSA%*rQW1|~K(rAE2XaCkZdnh-!7 z0PcWMSXdZ62cu$FGd?5*=th9v5+yn0k79z`cZln>bajcTsNjH+H{}}9PtYXwFfTI6 z;L=aulK#2x1$l-M0Y5{x;vBl;>&;ihxI*dR3XKD3wSyf!w^ch$ktJtgadGs|pSD1` zw(a@F?*EBDh0tEpaT6+o4ipic#A_akbYR(r?8=}UNEQxP12|agQhlg4c#)vJrii)* z!}@@=2s9?8lX(e4pfdHQh_`{>O}d_gP3Qs#{2uNHWIo=fn^S11?=^!tA0`O6Acs)+ zkTzbyiRBdmjs$2h6bkj92LK_1;e6D|{WEhU%nTdvn#(0HQA{{$2O66%@hGY8IO)#| zjOGjJKwP~9A)c!awez1bVivXc1Geq$?bx7mVuNo;i9u5Vgl4&n2Un}uWsrVi#Y#j%2J=JHYQ^C=q7Pq%P-$KUP;sZM_et+WtCHm<+=1< zHPs!F@gN!+f3_Ff(7hvYE}-Nwg_e=fe^EW=Dc0kxKHaPW&WLZ%i|m$|oZ3B5POn`T zt@RQNe(^^+ohY-<2%<-lYx;G6%FDwAb*qgF?YfqyW^K~q)X~qecY?TbH|i<`2`D;>1?@4a)yzu&;Snz7 zrc0xFgsERd7803037~}#0(T#zUn)}Z@Vfi%@<8E0f8oG=U9IMZJBB>n?}Bu4AI?;J z(sYg0`Ab8K(^E;EY%NDq*xvtc4lA=GMK_0ain;wHxK73iXgBENM|yMzQ`AE-Pma=m zn#DK{T!}0MG^G&GAEghk803%erdjP;AH=;WD*S|ON)j?>WxXvWl?o)FSBmA%gnePL z&#}v>W!3cN$;oxu|GPEJvNu`O0dx>OeRQjp_u*znPR>iT0=&OA>diUYVVL-D ztJq7_E9CF6X$8QiwW(rkU2AJ=rD-D0jpxfrP7|hn#}7|yeNWe&YL7F|plbC9Z{b_8 zhd!EhNMST-hNadKHNaZW3=Ej}_4azC=o%W@_x1Ep7f(T!^)@&EFzr}AdCA+yr&^MO zqswpdarroOpjnnrZkTlC<>r>!SX-Z)3HO9fKmD%`iT_JuNLg%)iqgvjkRvLppNP+j#0S(-^laM#^3!xAng;=m zkBrOcr`T7AaXug{!1}EYH>Wv`E8X<;|96)d6}R)R%}r})HY;c_b9y*3I$CNwK+hzd zNx%RQ=U<9Ed$QORwLtEePlRT?HB3y@yu8FnNl7IR#w;-$b!~(k#_qbh@(a&AOJD1R zPr3P~rKI$rwOD8~Qs6B8^5qRQXp7t9iqFYJUK#8e+OP~zP2r$TB2Eyd`>Hf@RD~VQ3S~?Wn^es9hJ)|}QXxk)| zVFgT%{Q(hk|>alN2@nkQO@ U8Nx6Hjl392_tfMHWXxXtKWq>XmjD0& diff --git a/docs/images/chapters/intersections/7196d3dec75d53f5df9d9c832ac3c493.png b/docs/images/chapters/intersections/7196d3dec75d53f5df9d9c832ac3c493.png index 31294c1a08c7813c2af0f9fe429a6ca302cfde6e..a9b641fdf9c9890e292f76fd2675f9442d7a8c2b 100644 GIT binary patch literal 16525 zcmaL92V9T)|2LkkVKj&k+A5_*r9lb}?X5vmlBV`>N?Ov=(h{Xf+C@V&G!$(lY45!? z?&o#R`F-!-@BhE=`{C$FpU?HVuJ?Gop6ks|QU1yvGCDF65|TaAS0$B6NVZTDzjyD% z-yBHmEyDkH8pvLeB-tc>Ju6KKBOy6KA}x7Q)iHMBr<0EA!s`2(Cbq3xw{jIrka77v zH+U-hRF-t-sX^x%ZIfHKmo{iR=DQoZ-v0IdW!lYR%hck(JYPOLF?ctW{OPG8nwU=~ z9-aAU<*M-OBaKTzPC8=iC}UiNsI*IU3C*O{M**v9Y?jSvf{3pP|>! zkwLG`U*pn`@Nk*rZYs(dNfFAd(W;iWQM)})hy<{)c%B&c{y%+{pp1@=rZ{xweuhSt z`f{g6mY#9{rw}7kQ*j3^!@)G^J9jeMi#62Ra-ujj-;hMhJ(OcHtnl2?q%)~CHO80l z!J6Hm!8DM^W%16=J#-H;GB}Q$z4J^uj73vhdv-9!k|DA_gvoEFug1{QvZ>OK+G+JH zv$w=wF)`QuI&F$$w`|*a!`N8rSDBmB@^oT#hmNkU(?Xj@G@ogsZQ<_`k;I-dpUsV> z^3iPB`dlj^4-bz(>wucCPgvhi{(NWF`R1f-1m~9bRaH`BT^Z@Ougi(8&#T~f3)!iI z#!5;%lU}}L2&?VPF(wz^T;t>6;aS`GGo@2x^;fQG z)7BmO-Alhb66geEHe@Wm=lMA=jD`dU-+%S$^p66ImSwv4{k37=@K$MQY3(7>o~fy( zp#+cD8^w7d0o)ooGfM&k(Xp`&t!XMkwu2ONgP&EnGpYyZ=Dxm~?Y{2(M22}|V}mVRX0RdpW1d+{=Q{`Ptdd!> zABC1fV_D@ajEvi^U%$>IZP1#k^oUNtUwmWPt4Vw_cB$JXt5P^3B7##p&o}VMIRg`u zmyh+A9TJOehjJWe49?!|+Wp>Ts??{rxVTz`nu^N*(fPYwMGQZjr+Y58XXz`vxE!dN zVQ zUN4}ya^;G4!5yvs+#R>Rysxh(x3#q$ZGR=JsJOqN zpy1JQ@p!jCiy?7wx<*~S)sGH;jN&nzURW^R+*p??7#nGt`ugHZ$$YacvsjJoP^0zI zq%H=ihWPwa=Y==9xi2vJnDWM0p^t4D8s9MG0S8YjXK5Tt@z~rbS?&#xi{{;dX~$}m zl9!kF>%aqVxz+M=Pg+`<)U4y`ufjZ$gpG4*X{yY#oI2efK0N=_)U^8Z%acDVb3;N7 zlSlXN-8(q9^W8^ZNC?Xd4R|g*1cE+avs(hTVcS;H?aRcq&3N4R2Om_XQ z3+E_V8H^c~P=1O<@OyGnK8)qst>z@s$;nBbclOufQo_Rz5Q`bRhq>BK`|e#Xl@$48 ztz45k*lIOF$Hi@RqUF^4hdMho#WvS$85kL}b8>dU0ZWT+nTmf7jll z&rCi)<Eeh`b)PCKm7RbgzN3-$#X>@c*M8wVu8S1YN30j@NdXNZBVyLO}K6>*Q z_V)If`sHPxgi)-V<_5nOSX_U6ROmy3$3~jo)QB}%zU42y(k^^kt02BEiIL(T)N<`g z(JgfjCkF8F;k53Bc7DB^u|-9K3r)0KIz3+d#BL1Kg}0`wk>AItQd3f1l#wA*Qc|*; z?26t&e(;1eRVY(hlho(9^BQixkufp#RROfMpFUkkQ&rATKOLBLz&{}1$?5mNfK-Zn+;D3eb$oohqmz?aL+)~sg89$40vN~V z4pZHu9F$M1y*xjLP_xPhhllT=iH#9N-FW=?@hFFWR@o~1-Me@5?*61ZckZ0?;<(o6 zki+`wy^@j=$DuffH+gwUW-TvUGqu8a46ARs?m5B15rFGVua0HW`psmDB=(n^nK>Qg z8k(BQ+9>pJenixdZz&4L^_)kTpPsdlad#JcaV4ld$|axCOqwgw{9UTAhF{;ccoD7D z*?R7Kk0Ya^*zjF(!EKQheYy=UU-FVWf*mt37(^-WPY zK&e1OO__DIq@h7>WT8Vp^0iL!^GHWyYt$68N?%H4d;3M(zLBRXW#Z*6DGHHKU7S=S zenqB4A9iWFdggR{d;0?)pCiV1G&D5qr@CLYw4sVWo3j; z$?!GhP`<>(#79Skay?uRK90$U0+-`3;J>wooS8O*yr(^VOlz22YuMg_pdyYhOD8Kzu&@ml9o>k5qSA8aVqK!o_e_{&<@Xdgqsj2Ce+qc?L3h!#Y6dk*XrKAs_N>8{YIt|5+=5r?k+YqoVB&t{Tu6x z4<0nI2z@(z=sSn0=4ThTEi{=n z#j6#Ph(!qc10(mibl+}&pJQA zS^}41Cuhm@*vRB$BOVOp-V|u#fw#BW(LeJeEsoC40A2x^xj+<;=|#!jn8Gft9bpuEJM}s>_9#C`blXjcXxL>!wgmS$?nqno(eD7Oul=gstYz=yY`s? zof!X`9@<|exxcQWtLrE>cvn|fl!y~Awt-x{$V%~;sN*z=sp$|ymUL)k&2f*058mXm zy4<3o$IQ*mnWQDKUv;WKeAs&a{CPBNAF44Cw}E7-fP*5A(-&esb5dJ3O76FNl8}%! zBkli`Mmz&$lh35##{BTt5@4a^*QVy?vg+yw$-E`{eyiN1|FbluL7Sh_@pb5=fmvm| z--%&1Ztft!n%nuAUS8W~W@hZhJ1I%@_4RdhbXt8_l1I9V`A(iZ`N@cPVOtfwsIw%R zxu2h(5_gI!dt24u$UU-}Z68J1D4~;?n0$8&@A~-@2vfd>Lxibiqw#@f?t1s$hP zpkoencQ^YQFzgntYijbC%nc&{{8S(Sc)4e2$R{xHn%~EG36!Q!iPB4xKV?Oz7RJBj zF0Za`-4XQH=}FvOQ#3aAS0HH{sn2W7 z)XLS&HrN4>ARMG0+sQ|{{aE-wEb0BAprF<17tdd5x04yc$(|`@&+Wf&Ov+&dL!+&pe;p$Us8>&X`K?z-~(t2M+UxlpKcWR5e z^y*2uy7%tYy!hoJ;H~=R;jxzR^A~m2ERGF5S0)*5AKLLY@36uy)4ndX>sN`1xp?{V zWpBM&r?q~1+Pkj}tKTVoQc)ur*EDa8y#CCoPa`|uIs1dnDf!s@2euzGGBQ#fj+YJ< zc^HsFpH5DCUqr2{O63$&y~nFJBK&_pWpGGHqS%(}lt!Pvv`$=5zqIrFy^CiK5f8fy zFkVwjORdv!&&2uWG#B0Ru}!yI#`?cvBVEouOVq6~%`|iDjm5gVz!j-+i$}<=hemw; zNhNKJi|xs`J=ur6S#Jb7m2D@+Do(XOzV>^*xmJq`(QG9}>?gb6 zhzOJ5!G@|=+WGzi^^w$v4@(2RkjOp1FLEuGm^@X?z=aDW*zEnFE29zy{(moce6R}e zk5^TKxQCd_;`m7h2G5t*V(9K1b^Cjto)Q;NKn1#U=gz0kpWDpuNQXw9wV?a@^{b?m zlv267TWR&V?L={p?Y9v+d-g0Evr1MK6L!%lm6uk)vur~4I*=X_uzBQ2eH^W#J30PHF3z?0|BMgDx}pph=I zwfFYp6Hl(&*zj_fOnd4XoT4GyxsU!4aGcB1XmyN+@$~6E`}XZa{n<@Hk#sq*?oW7TNi?}<$U5*Mzgcrq z*BC1cOLL*6{_4_HQ@UD4p8Ilf8r9zoDJynkxo;otx%F2=HAM4W z>3jRKptLD~)%m?Y*tbys(9qoQS4L2wTA+9(_HgZ*03fv=KVmRIXDq(+d-a<>a_nt> zbK=QI+<10wE{AP>HTv?-y+=QJ?jQ$a5Ky$QK1n>;M-Pa$X8ijVr7la3Gg1>Vlu}o&@ZIS=vvu3H!@_o{Tlo0siCH>2`Kvt0%gal> zK3>pz>+$2qfqVUZe0=`G&#lcg_t*VK7t)@HJ}PXt8$Vqf|E7{mPD&Jw-lEe0cGCSK-@W_9nKQw@zPo{6I0X6n z{$4h0TwL;oiMx02zR~jXTC9juPQ%hO)g7oD5{G{l+xkT6yKAlIGD%~m1HtV<5f<9} zl5!|DY_}&qiWh6BMJJs2p1-#E!}92P`^@hLss3KuOjz4MOqd&KJw5uevN9$nrs;&u zH6Ku_!{atxI3iUFuRVVq26><-+1QEy8hM?!?qoBO%8?bauBXeXr%G+iT{l zwxLqk6m#kBln#x{fd>yBkeqRxzQJgGi-GBM?SPF1Evjb;J2f}P3zR%O&!ze9i1Z6q zWxYTVY2HXCvqwzQW;^OLb>FE!6?1zlzhY!!A_Jrn0U(mZQx5&X?9bxF=#s#zInDz! zD}TeMDhLO|Eh+oJ%8V-=7&Yg<(s^$6Iziq^)=A45`Q%Au;w2wDI&<{c?K^k&PE35R z4Py-`i3|(7l%-eN2^RSA;T}eI_R-?8LtaHs8F$lx^#uYig@M8F-R%-wFn8N*)98!7 zT&yE+Wp$>zyL+^Qikg~q?_RF#JAKuG0h~+Izb|p=6urmt*DQAzh3LZ3T+oAUOwa(` zclOz)P4s8)bY1|_lF6L=Zk{zVwT*38wkgBzDo8k692^|VR!2GRom5Iw?RtBU9Oyc5 z49przhf=?Qw?qYQM0V{|na)kiCv*b0zvLLp0=!)BUq>Ab4Gon*)dWx&!p>;YtvN(g zO%Y~xc7IC7%hMnCiPL;@DRN&IKCPX{mhU4C!DM;(MUa1qiG{_rNG@F&&g*L-&mpgj z)_At3DtX1mKJQ2z>&R+?CZSn)_ZT57KnEi(OzeYlfHfZbaZUU2&Wih6w)vMhtjrpt zM#<=EG+Ui_`yDTBefoL5vzIAhG8i_F7t6LMmGs&B)R$xN|t) zyo2lBF+*bi+z$@k&mtdt6~ikEacRa~PEIZ@(%s_DovSi32GH50=ncQ(u~97&&fDjj zd``Vy$OYikvpmz+5O<#TK7^)RyAkD*(o$(TIU`I!re5jOH*el3ckr+PbEPg4H;lGI zN>1LQd3APfZUt0mq%|$9dLc%@yt;g2fg4j=ANLWvBDAvph|(zf;dbSzDbSk-Tz+ znvPBe6)53dBZ|4Tl@(Fcw(@3nVeZnsOG-pA$e-}3zP`R+-hS?dJmd)7a!;&o+ljrl z57@a`S$)95K<}>Hy2S>KNIUBf^s@ki=ge5Fs%A3DR4BSOwzi~u_EbT`9lNKZqOyWl zjC@J-+QOG9jJ2m|{6E^>X_b>(2=(LV+j}pde(bD!a~qf>HS~XUzd-H$+u=f&RaI%w z9?o8{R|{ifkq8fqh!Y0xvF9lS^Vu0JpZQjvG(8&$O2VwW{}xwzWzxfbU5=;Y*N^_tNCh;m>t zEv6-C(LA@loalBMIyTspkflrdKTmS3 zU8A8+RDSnAJ7k+|LxTG%Dek_sGF`s%=3_-ObjUeq(K*fww@VL%c6I5<#|l2WVwa6wAPbdLVB(on+pg5QGk+7?oDP;`;jz?)-ck4vJ-i(#fDu z(h_M{^LPeIn{PMs0F%F;;;GE*g^v$#s1xdpcHv!%V4l6lge##UQc%4UMgu$Jv}MZ{ z!1RazG|N4_)xb%`E=#Yw_r;|XROpNuUO=d502W6wgGt7J6z)TfXlSV1%o<=86RLo+ zQ3=co@8NHb8aIaRpab5$LLryK8z3ckb9RGO_pjC>T+NQK~BY zM62wY2H~wfJpmI}5QzewMUYj@Zeg+Jxsn7zyK$&_e zA6JK22n_v*7GnmTjJtF`RH-^E9aV+B0cG6E!NK(Fb1CS(Rct~SM9B7Ri!u}!K-X1$ z$!WQI&~*I|a2(<>c<^;r(5i<7CIzJn^Uf2Ca@)NE{QW=F*9T0M{b@|~W;%a9X3vqc zXPjn@UJNLP<1P-VYHA+B`Y=V`cXV+%A`l6UqGxdMGKzfQ(<^Gsdu_LSfxh1Ct@PWz zYZsyQzZeV+38}!mtbnExuVEoRhnS3LPZO zVhICHv)G0i>M#GT^7b4-DA|RUzp86$YH$%J;vziIS}^Hyl9Jm&U#sPJu?*yR0g#MmFjBfgVIvb#>U1x%U|8M^bdmNh`9;>!*Zfq6DuQ5 zU$?Tc>CNxZT8yT`(ni}KB))M2gM8;(P8DwS z$n&Tu!EhNVj6N#EWrxWx%z{zXg_hfbJ9C(6(PwS!S@`*%04At+(#!lqmVW>K==0#= zL)1Q%lYO@@g)+ed+nm89v?NJW8>XmOfYP%}qF3(l?q9Un={B=tZ+3Q6nkxPitgPl; z4&c6J_vb7=4kS?Zsyuuas74{-;l?)J+sRCJn3|aII?vz2K)>=eFzEeTt;0<{EiPv6 z&I*ZId`jOa&)?MiY&nfxHt<8v;7sy>R7&o_*SiwlJ4NA8imAXV8C zKcBg@F&f7%9V%}qVy;`P(>VS6BO$?n_AoOtUP^Hsm??3@m@@M5g<%ih7APp2Gh+3!$RNG-W~xTBV@c^GWWQP7JV;^yXl^@CG= ze#l{{@fa{j_3FY{{@e8ArZvfZom+d))JPf|8xuAhc*Pb$aiiT%boARoOiij+%3m&? zUsqH0CnqB_`0#*qpso-222Ifi*l_rj{+Q2k_fMF!1uAl~)h7T_kw66gLpzRyu(7bP zh>FGm%bkIWNH}eTc9@-g^4quQ;BUFbzsf{~A|m=VwQb7XTqI1ISib#7a%9j$eT78R z7h_qbRDKXx0+rC8(*c4HLsp!q%fWw)x7ueHy`scD-2FPM`Utt|I27S~M_E}}V`F0l zJvKy00G~cKH3b3LjxF9?H(r??$O8uZE`Iw%XQdkp3&W{Xejw(1=|un$$+qpJxB&!& zn(p(j9BC79@WHh>ZG)e}_eOh0M$%>;4AbhnRuozHy(uVAFMny)Er&<)UlW0+my?qN z8q0k4th!<2)84Df72o1M^o_tc`8_qI06iX1fR&Sz0-9>|<74&B$+B9-HUoTO;X~)q zVawNM1B!$`Z^x1}f_cEq#PlG5mixIXdz#XVonVuRFxf_p;k(siaTEiI!*sgH70D<~ z1>H>Tm%RS>@Nn!^DjFKqegWxF8IfrjuE>lf7BHU5Z@H!`lkcXMYs5E{M#CwuS^K|p zUE$T!(?dOg@V~lH;kAqJc5ABX>pvprEW7&&ul3BC{Tv(|aR#JssjDilf zx@HiQG&HYQ9g($0Q2u6XSreu!b{!9nkAIw>f6q@4^a5|%rCoEH3+pCGY}FTDa#RDw zz^FV;7#Vzja1Dd0?otu=wJ3sH0dr(ydlGV9;oYAUye3+Y4)J&bn$9+PY=mWIazb!x zb>aN}C_EuS0iYSYr57I5`}X#N2J3iMXfDKOKv+C;<2^}fC5;`czx0kz_4R#N-D~qL z3Q^I~Gws(czm~c<96YTpDIu{n=A7lAH?7{8Po1c_l{Gc+d239$9Qr>|l0ZfvxZv=A z#8b=>1&y5G_^)ztSnbKxA+<)}V+*GjpY~p;sQodk=R~@D_j@St9-DuD4QT71p-_^e zxSFO)MNi*|QOGVVG;N6Du^Vaj1{gs5BpfKnIQ+x?BEVM?5)%G2VISs>@~xqk*ojGm^Ok?Pp7YZx|61>09^xlktH{tNJKqpJZ`@#F_7AQp+K=GB7a%gcrn28h<`J!-t1B9aHBx$((R^y#K(6~(Vbn>iWKi)ez0{70F z{vqiAqkpY!|G)s{v16aH^IMmEOBTPIYwJfmdi?luqGHd$fN_tUY$V!0$g^Ki5UYqt zN$VvZ73+~^W~3J6IC&PJ$+6&!0a>7K0Q!VsrR=eg-a3cY!9%0L7KaAfOArP|Hx4OAw0#Uy0;s zhgeeNFh!3mbIe5f`%h!lKHPKUUqy3q7eSbNhomL)5gF&`_%7udp#H$!&zj z$2Y>Xt_1KvL2QFD-*eM+@1qKnP?!aiS+W@^j$& zpIOf5yY6gh`MD$zoYT$tJRmbOlg#2fkq&wH?mVhW^YwD|OQ%G>3g7&&JuZfc^kt$p zP1r16$$^APPHyfV;me8G$S4^E_QkTx4Cbk4MEQh1U<;sIw%kh|=w|L+mnK6?tvf~e zRSk@!*kiM-qWsLcyIt=;e)K{U;xn$J@p_yb040v7{9oW3LYPhO_<6TtIo*iI>1~iN z*OKpW`X!&eYcQ#Imfplem=m!IPQ8*S>%Qv4caG|UCg(yIAwnbs?s#+au!-bzx!u@mxMnNz**`*UGeEDSv$J#@eqL9;YyqP9SDz|l$vE4LULG0$j$M*2V^F* zRh62U92Gcmgg1kV#)F_vZ(oo&*PKo?@%UE1B@hqs%*e=)rfEW-%(?qh*ucPm$TMh` zIA+%>u+UJ{7zOxcV|^52Vsjy8ZD5=8f~JZ68Rvs2nbh_c7})?Q4K*!dHMP+n$%H1r z$<8vj^MHPa=!Xb-UClUkSfb6urm9kPl9Fe;tgI}t00B)Z5o4-Pk`6lr*ZS8E-d&x0 zW4GyTufFiUfCS3vd#GYiv(9KwVyi;?_k~WR3>z41{WD}pIp?!EirEyiG0&9KikuhN z(KJVYTEl}~xKto;ZJ0OTMUp~_C5#7RuAa+;00>%VuIWD1@WK2@{Y#5E6jb-+-XB$+ zQ`50@j`LNa? zLr$T!RTZpfW`16E(^97-$TzKv(cl))vk~?IjHsFQrC*HV+(WP4St_Y4XiQNbh%#(^ zIKHR#mbSHzJ)Q~D9U&nRn6I9@{83~iEm{Tj(4_~a&E`Cig>e&!iflz;wmv;I6W?=?2B& z&H;xPOumFhjCW>l)6nQJ6*GS+oPGPoj&rhQ@nLZvY4tt+3X7ozLFKpXF3|}}y(%lq z2rCP!nl(s^`{ud>kmz5^X-UwKP_QFV!k=KjsDBUZ^0Xewj{+duwmo9;qqfU3ryVr8icM{^8{9_IuU`9P85ShFj}}7?#cT*q4p?~z1cL7D z9bcri?v1vdvYGN{l7=}N93Aba!^J8fZ~%)9`5ONhFBoCu5a}S~HPiQI(dJLUjD`gLdRP0tn~_ zls+OK_|zQtX&(02+^7A`S%;3H8wCY=5)-&i^2F-`qqT^OpECqP=%o7>KY43k5I!ns zbswug$E<~g1R{IpLJ?HiWaZkfV%u*;){K||?2oHI#}4*jnh}5qM?%fe#H1R7O+=o= z*XOCR@CjVcsb8iuG|}drR!xm|HJraIGcN8JqH%(4HoXr=A66U3Hjv-l=7C)RqVyaj z)!FH-j`4VObVNioN-=so$1g}s6^SwqWE(Vw#>UWR8KB=>4x~%#g9r?3LWa}Q!a}$I zWf1|P(C|Z_KCMF!B(egVOCFmk?&Y{_SBaKYbM>!Z2?%b@z%2YdG*p8+h!j;|L!`NZ3?ji^AupmcUO>K)nnmu5 zskSyzx`kYqt-K$FM@3zQk1zs6k2=+SyG$NxGk6-;%+q)8T>QkVBHvl}?K~!I_r9*q zU$4{|TfU8I<^b6az}r437LN||Jp@#20q8@ZHNP-_{912s@2xM-{#qS^z6Z?uFgKTPs3~Dr4+LF7A)(*>{YmZ? zi1jY7tXOYuxVgD=G80CHnA_?ZgGygs0r$oE`F~ygesE6h!Ag*X`&<@HA+)Yinyn zKe_LjaiFPpd|ao02>$YZDk=%MM7DzsgsuPoBHTGfe(_@A&oLy2&fFkTc`0`jtVFm5 z;vu_Dk#efzZz3b+%wp;6{C4$=a-t%!?P0tl;V0re@7pBy=aFUE%0pyqf?n&sn{81L zpFz`}`F!pdCP92SHdj$#WPNWZIy^6y%WM5|^izl6`T-K`)(p#(EVYb?>adGgtjug| zyAcb5w|51~90si)q1*ocexRA1&z?nzjJ+ z5&{iz7u&78Jli380HTYCl|hIm>`HXdp3%`d@aG~31dsw+oeiEVUX#{_%KV*6d(sgB zCS_-5=a3F)<;n1PWRaIlMt$WX4JJv$dT)lRk-WC%8aNZ*orl}HimXH^0D+OQF|%|i zQ(L;4Y+ZeQ7`&fkh+}Dya9$!aU%gr=D@BV`h6|dEm~N!Go43SC*s=0Yc?=&iNe}#4 zf*)Gi3e-G;4jB1FCK-x84ILf(I?zJ|q8CYbf4-H*T}O1gOo@)bXbfeNZicOy1oc*y zAb`p$D*5yo4DR}J%-xu|B=pqb)A3Bvi;EElo09zf!o$@<3;;E_Q9Q1Z!II6zXz8C)B`^&TK{!xba0rMKHIaIhG>q{BO*7 zu6%Pn2qm1)pmI0FcS*P`BLL8Tov>J4H`kZYcXt0BprxYnlw)~;VD;z#7PXa~UF(mt zasOl<+F$F|gJ~=r)YvZD;0(prZeS_paAulu?fu5}r_YpdUwSz-P{UU*Atl~yk2;>^ZWPo zj%We%gBAP%PidJ9Fxy0U82{4K*Kiq=MgBGki5{qu@%EhZAGJWr6^qHGmp)Qh_Z>(~_96Uul z#HGYuZxC5H1+{+EtiF#Paca==UrZ&Fa_^V=b~AJ1(YXO7b}Ejh1^E@4Gzjn)ojzgqaL)MT%|@sl+`f0EeOf zftdk*pI~G2#op{48>`1y!+|?S#NuJt_a{f{u`kU%*pZk+Rh9Gz9@z)fLqwsO`1xtT zLy5x$FieS15UvbdxaFVdv>y)C6u+h*l`v-fH%M~*Cx8Rt!C|Rul{()`3#xWqo`!76 zL!5P~tds;jxPX?7l@R9`YjS{wifXH(!CtLz0pY96yFM!uuw=OOGTgr`_#N-uJSkG`Wg; z^V;o`LPCcj1|jI=3lA;2(TApy7LbmXICsITp9)a|#;6x6SlV@VG6`d5=q?E8^dQ^{ zZM&{DOD$}+ftKn)odivosc%qF&Cj24;4yr*gL0+S?e=BCrKP2MH3zm|qO3}yDPLba zLbzOzA_*JqnW5Qv{i|IxtWms5a{NX16Lgm4>t_?(77t@!GQMOd2vq;dz7NdDMST ziF`qj3=U0YOA{?n!93A zRuYGg(4El655s4BnOq+jEKM1dVh4Ve-k3vuOh_A)G2@Mb%u)BZ%F z3y`oMJ4sSf(s$E1{{-&bSZyFc`u}UD8IC_;KV~sRZZ?POj;1$PNv5e1rWE)Do}yRA zn_k4c>FhKLpg72Kgnd8@e>*Dkk_1*g-fOVK>>qJ{%OqttzS- zZ7|zYvF+jBV|C}JN61e7uO$#_8W`D8(vjOAF!e3Z$LC9GzUUE z#aqtW3rjGc46AMRPF|k#wz09Pv{-kBw8Mm~;`{d(umr6TTOeMvIB&7;O-)&k`jMMYV1d>JZ+l8|T-wQ2+fJG9|)$~W%@jyHOH?gREbc)Pb&6UAGYmgjG%@@TeEIL}N ze*w}4K$u@(U==Va0J3`PVAS?7PL@z69#k>{WWb&w0!^~!i~~&hza2hA}^8u{Sr?xf!vy)`uO_!{RWQijo*?ef$9Ddz+L7mk=V01 zaoKfc;Dn9sX1A?h;Ju#qS1%i)C?)D6q=`n1K6l94oL&9fyPby>|H(%pJr^)NZKZT| z`wRA1?s>wbLc^bfBpDVPOL4(!nnQ|AI5W9Vf)Z${1P%gu_m;Kn5h7W+);k zK2Q&M>}5QksSE0VLX`i0QPIO>t|PP^yYGWaYh1ni@&o7j^G7UP-jBL$$lff0S4D(| zkOl{*-o_qI7A~VU>A2dcfP|O~d+CofIrZsyCTZcoo7zAAtT~$Cw4?Syz!}wjweW_} z885-I=q$1}ERN|_t&Wdo&>IN&_sEF6d~H`z;tJV|FqItVZ?l7U*neD)aJ($q5c1!n zBb;x|Nm-g|Yf0~)ywqrD5~0!S#+NHnQrbkTRZ#To*$Ec8s2fF0^~2yf{cs?KYk0JI z;NB$am%9@(5%zUthF8f6ReqF0P*p2?d6UKQ{of1AgOZK+MnnkZS$ZUV%e{oZuZH1; zW&cD|Tudw;WR0LtBwx~0SW!!?tgLPTRF4iInJ5cgx>Y;@R_}l8TpTTV+Dq8H2_Fvb zdD=^yzr+ql*f*=i)bPm_T_V^D1)WeyO9k9tKYHhlad;F7r2%{s$H=}cyQ5M~&(F7Z z8XDIjIUdTS1U+YAVm?`PegYsD z;%~U)&u`y~KOK-O*3nBkq+8~C_`g_QR%@4k&8gRUf>1n&GbK%3lS83-Op*2Zx~ivj ztrb=$g4UZYJ;o!O4=g1iNnQK(tCDuMfmCtt$a!(3(K3er|8xye98RjKkt|cB2a}Lv zI(6=x214Hv)v(VMt!kxh?ChSPNB(ILgv1n@pPz4kgu|>`&ub!DrQ}op7yFL; zuS>^l`)iosh*rV~myD50)DHEJQ^Prr!+a)Zoacv4#q?T{NchKp<);@j^M_Bpva$kE zmlvoSpp4w(A#^Vibc8J% z-L|K?%-KZz-Wt_Xxw^XE_k8R{V%ivc6F4oo`u}33-ur&mi-!6zFMs}n;I`nuhA%Rz z8o~%QfW!x7C*2b-1`~@{=HER#sIV&>BQBC628y?@M^r6Rqq`roeH_A4kbL zOPu%+EgSfeT7Ia1@bSaNV@@k=J3BizT~_lkKzftk?mx7DKJZ-8E&rc0Eh2ONj|%=h z({k@IpLshcc>P}pz{^`xTcbBL8Kx$4vj6)DK`q*y`dJ+V^WOs(eaH@GcT* MDS64)5;r{m7ho9*d;kCd literal 16795 zcmaic2Uw47|93Va(jt+RN<*5GloAao+Dno4*4|mALGf#+p{1p%p_E8_OGSxjYH2ST z@8@zq&+~tu<9(0!eV#-2eJ9uNy3X@^e!rh}@w=iRy_=k#oQ#ZYx2%km5*gVBD&p7n zt@sX2THh=DW2^opX(_UG;{VUeQi93Ij*!VpNvJqSPkeRJQSJD>G`;8UvyJ5J_wFB| zI8Nc)XA$_!cQ!=JjN%&igObpNPywm}Pg7cov`?)p%>2B!&QWi9a75D1KR#M%m(cM@mW*-!gZe9yq~z_sDit$||Y7o4xn`-@bb5T)TG5!9mr5o-Z~w zmf^ImREKFT{;}EG+Pb(?wWxeZxa!-%YvHF*Rd^5o* zUq>r5veLU16%@i;7k_aZ8W_lsSY+|Pw!W%>SH#e{Q`n60Yy0TI#&GJhjDInldu@E0We=;6R*2l+(6dxZiVBX1foRc%7ZScT> z18ucWnDt(jU$}4~*;cRIvuv}>x#Css=CAeja$#<&W8o{~99k*DPMbE}Q!2hhV)&Z9Fe0-$&u}^k=l{*3%#hYswy&A>WSC|7{+K(vB%*=3Dynp}xUVgs7VR27+ zdwYT7$B#QZJFArFsH@X)b8}mLEm1Z6b;_!TJ@j=(1}BLn*K_r^(pYHYYHR%YydF;N z@}&CC~Fe-3QuGgv)*?wljXS_S>9H$l~+ufPkGRPMjG3oO{pu&9cwk zyBk?AM^)14xxbTU^pG+zV8M#Y$B2YR@*C%wd^q~+>uYHvBcsqSW+o=zM@Q?QFiSiC zo;$AO7g$|2L91QnDro*GEA-G>a|G|b%TI-J4I9F2ml=g^Zl@^5sYOW3K0Re^G&|I^ zfB*iIu5%__+}z8f8QJv{PPt>BvTqt0vHpqV(#UzHQc7`1q&44E+1=gU#@4o?zMlLx z@6BqP-*cuz%`pMt;dHld-Qv>9ms#Lvm(|mY!(F$vwLNC>tEiCFDzuQjdiAPzgR-(R zFF$|(%wU84&(`?1-tdNSp78oIjk?|~&+2y{a;zoyp9er5Axqo=2*l$6x7%*>{c=PFyRw9~pT-&vW@$Qp7|z~Q+a(^= zCH}6L#(Q?`*wGj#9@q5r)W@uAXS!S`x(k@ko{gECvv-{6zHsgw*=wgs)>Eg7bY8g3 z4&54UKi*wvxl@!U%5JD}L#j=&kIz<~vblX-U0r!`rWa6S9F1iIlN--nyy(NV*xs(9 z`KxL&{2H;IX%=yX7F~A>3IzLWf`%u{*Q>j_BFo$sadXUHQqsHAczTOS^ed*#m3~ws z?P*lcpFiJqfbV`y5M#LZ@KDp!`}gl(H!!##8+&-eh7CPsZgkFT<>6R1vf1I5l=BZL zgioLTkg0v@S5L9>kHRAB-XR=^hlkFD5#^F3+rc;+bH>TW(XqO|-XGsOd81vbnjr4-cQEXJ%$bM>*{Xe!^fO`A6D+qX}SyP=_>KeOmt zA5PuM%4~~(Y1{qIOzqjZ5#=rs#W=BK6g7_RURqioKffIZg)FY4ng{(mgT&WlP`>Cm z1_lO?V`H1~4UG}o+xPEpe)#ZVdgIsD)*PK*G-+Im-R#Y)b-`@Ow$@)u1dEG{qt7@6 zmanfpWmAm(J~DEAQLICH_nU~2)uWe4Qn z4bRfjj(_R={5gWrbLmcMD!Yi&uf~y&DV#lyUrQW5cX#)Vja^bv39%F4plMIGQbnyTl_eik2p2bGDAa-t{( zEt8&}KHs8?^!c-fQDel_ABDtEqcluEKL3!R&ZAk1Rd_0V>nH^Uh5cCPqu8@VkDzD3z!);im4 zi&fHuYN5idUx!5O*+bnPP#lU7^ANdKV*jq9V&kPtmxfy6^sM6O_wAM?S(D%W?M%Ve zr^@pU#bx@f^mO){-|3Ql1!f)TJ@fISQ!_J1d3b_~?S?TNw0=b?%l3M$dlcA)56@;O z{F+%4mHQQZw68AXM4zAj(LO)tqp^~Gg7qI6B=%A(Qd8lp;Zmc2lfm-h)KIrhH>3wChe z`kQoL{iEAE9*2k5VBXZ$*H=`skikX2TGwwWDr?(5H-o}Il}{qVq}(d`#> z_$lw3ZQr?*nUiy;^2Z1A(M!j|!hV>T(LPJlyq#<)OvUEu=T~>0kB@Kro;~UgB@S}L zP54L$9p2EI-76~~*q}IwA3ay@$#!S>qm&S9Ld14S!KCEm4c9GnT;fj(d8(LwtxkB? zw1E?=QvZ-`_{blvf_Ps`Rm$xDeDhPfDkq6Vl99=&?paar9%Et2-xjOrt@$y!!_X^( zN-=r}*UY!>J$&2z6RWa@#+s4IjdyZg4exw)Tc@7}znrKLG9 zUj~;g%+DvINnw)RFgI_g3ZUCy@bPw7^iA3$Cr(hXv9ZZtzI@Bkk@duhES-a5Vq$Zn z9hYeNPoB)Fx3sjdn3|o<=rCsf`_YAkLe0(1KSxGZi^S{5MMWL^e&kY#mk(CfXeVZ8 zw_vw8CvMs7p zywEPn>Jizt=Wu01Ljc-hXSQAo>DZ<89>>NA-tn)mi`3#C{mf_G-hl>M`S1Y$(pYx6 z@56^7tCLnnMi*-W5210#<0OSzc@5UnbQhSjxlN3XasFAH`thFk+k2($zGbP zZEfAl!;{TXB*@Flbo#Vr)(R)Hg5Bo$F!G%{FP+sJxxnH2z010MwfgOyO)l%Ji?{6U zU)cU3?V$Yv5(Da!q1fq{Xqii@8-d2;9J(}U<{gPksmU!4q7^hhjqze%cvmH?NV zw(ffD#>jCLo7t`;At8~LvRGDYo?UkF+_~H1^9dIMWMpKn&;9(6_xe<1WMuNMfsmC& zU_J?FXJKI6o;M!a3vXYMoIJ+I$A5mxW(06BZV2Ok_4>7Xu_fRFdZ25j*tT0uQhUwP za&o4;TY^uRV!IooPb)8!B%BdK;}2cT96{$1qDPNjV)A6~%DIQT~TZ`}2>HM`tKgPT#$9se_w!cPKk6{%y;aEeR1vh)2-^ zObd-^r{8&1o>*gYaw%R3;yer9GMI_}`?Jf-%a6?Ze%2Fxq#z_Njweu_ATegi&z{{* zNH}`y)HdF2FMQehI^^W!$VEg%$OF`sJObE;ZqrOlm1rlW`bf!4rLvS~!kMxo?UJspB|7T|IF){GQnXt2c&uH1Y(uMC4rSSyrWExW7l$Zm zO`@#0lTuO|6zoPN<&0R0etEV0vAZYZ61MI34f8`io~>`|$7y!iXO3}&=T&aekVy3| z5##ey+|fT|?UyIH@$m{*xVC!ilgIRG#x5#S#Pbn9aMk#q3nwMDVkt$;TPokuep%bP zKYvY+?ezOjk-5X4>HF0R9m1q9R0{t6GbdQZotJEn^&PMhFRhq=HauIqaL`C{Qo%ZP z&UMG%1tLX9%L8MKC2^lRbqd4nUU|8A=4HVHZL_RoMm`HXrGh$if6GxuL7}>>O=)BT zbN&0oL=#$z;Y5K&{EB17d7=etr3NqucVwvBEsSg4y?b{w$)+Pjm=E8-G9(y`br0jw zYX+x#<-Tm28E2ko@%QM_(9k@Ij(!NRhg~CobnVKOw+F3D-z7>12|N9YmV8~o89?mm zc`zk(BGoZox|!OR7USu@Dgz@U+qIQtkuq(SgMTk|jD|cSrOe*y@9!@oCzo)uCh-1)2fJ%zSLIB~(zUg<#;2!(-apvG zZ#y7!%;>pDpH<+I_H*|3`8`zwK=F-{{131)LZ2l!5*L0{t1XXdq)}*bz_=yW5X8&j zapdUVrPxbD5maSoFS$!Kza3 z-Fq(l#I@0rXF7;hkv@Xe-cL`jCel(1*fu>k$8z%IUJ!keW(5bLO3hrCrkDVgQm4++ z(b2hkcO{ix>JGN`rE|rLwQQz0-N0-rXHMGUOTy;G?#m+eWVi6NN?B%t7QCsF5 zaYDnuJP(Vx%V18`)z#hjJvTBiIGAhJ!2$TE8WSrARJDEP)gJjxjo)Q{Qsz6F^WLoA zWp8iqjG^$MZJ+us;sNOVw*KBA_AD~h&!yz_mJ?fE`0G?Dim$Se*LnFVO6=7&V0*lU zdd2$cY{k&SCF9NPj&I^!FEd1LuL#iD#^HKhkN>;Eq2Yw;zn2;zR}pBjJJISa^6Jgw zi;j+kERzg5cU$*Y#RR>#4LjD9JhZNzx@pV3owcH|enCNd$WXc}SA9k$&O>jBr5Wpw z4mld@tNz8Al9CdvE7xn(^P=k`sXviImj#}RmcM*?&}27I)G+`H&;hXEjK?bg!fbA_ z3Kz;u?d4O)gSB36rQv-S{(h# z(B(F;T=hBb_PU3cEC$r%w=#0mPaPehP!R}L3bCjOwI){<_w;E4XLe~tQLkqM2s9WU ztB8nJ;l$l7J2}3pMOJ=gkVrizBeNZ_s_N_47@+A5Q=jy_c=fw{ZF{4DKt2CVoPrM1 z7{;ybJdzwPcPJhQV(R@K@euRGao4F8m-k71Nw^I;24F4=l5NuUh`6GFeDl1WJ4|md?AWj^`VbR)GxBAIy!7HJ_eE$47 zM7umE=W%p21Lha{>T^6?@;4q`UA3TL~L=Ej4 zDv(|AVnKfXNdW<_A{{-w@}4|&YNeR(`T6;^Ej%82CAbmrZ;((wz(2H^!3yu4ww4+4 zI35*VXdn-1IMu%Q_n$BKT$L+x8tdc+I^g(Pm~OmTeYm!=t&Iek1LA;5M>-9fsQqYr zTE#p@eE^Fr;G7R7^Tm+vLSU&-s|+q&a-+@EcL^OG9bvnnT~G*4EUqznt-OO~!twQu zYKB@*vE2i3?HiVs*G9E`Lqk=)rMYN$`S|pEUfm*kPW<@?d4QeG(WlSL%kM&)(1$3H zXMU2zLdn3;2=##gMISzWGRDsn=t<(<*1Ez>PIRxxNdJTjT2DN)ZE7T_>H}0jTu7n~J`uN&=7Jt9i?>Ma|Ek2)QRfw*IT%68uUp=y? zN5>P*REhb4ce?4|z`zmoyRS>-?%WB%E#vk${T?ikE4Q$41azF)iK1RUC|KT=bBl|{0PEy_ z9TC1C8REaq;J-0p7{uH)Zg{hR)l{0?bfAhx5Bd{M z5zdk&HSNn>Vk^t~2M4R%Cd+C-9%;k`Z)a!o;J~PxG*(Mun#|YV*|bfg*!CDCIC37A zk&zLdVq0&>uj~R=H<Cdemep9W=lS ztoM#xyDA~HAGgH)_@JDw9nU?u3C*!!xUa7Q;?efM^I875C80}00}Qz1Q2?C(tdK>v zFUIK`6V?G8{CMVLvS<+pe{}H<6Z_mL0_baTaB%cteOQkky^B-aVKR_iiq<3Sw!kI6 zwv^b`ZL*6P0uuwn!~Dk03>f^=i;Febh_s0sEr3olieW>KUX1f?jyi zWyWBrG4egOQa#1e*udbV^*2%C#Eel^fZmfmTH7w&2;m});AxK2N&+>6;2 z;szfxo&SkX%Df4P8ApY}hzaoZy$%ryD8E)!hMF6tg~AB9B}~rh`j*^*LqH(pR)2M_ z!yzpBXuAoSzUWVtbrI)`aF6BaU+ z3YBubElHMufY{G8uApExMg5yM6Cax&lQiuUkiH@!E-p?qEA+}2Z8erg#>TT>oqBU` z)=-2Fwm$Y_rjketJancSqiK42`s%Ws+u5^cM|#V}dCFJvJs zw!Sr6Pds!8npZvQQ4=5;KT}(28u*O**RNmWJ;lMOo(~1)>Z&R42{1uyQBKYk2aerg zsIm&kv(s_wD-TdO5Q|k~D7@BJPrZ5bhERsn6_O>0+F%DfC@m`^?j}HTV@tpn+#0Wd zfC{b=MO@>BZW%qE3A1*9KH(~zq@<+t%)k!Z8u1gc?5vWR;6b?{MsR?t1{=chXITUV zRXa?dKm`Ln(RgWeIi@M7Ey7PB^?mQoL!~AXGu=L61&spMm(ckBX|;Y4R|JR8?(ls={nN z7#1dJQc{x5+KP)-zRA^;3H4(lBJ?~wJeXLjF>;Aa{(ez}Opf{n6uPnRfGne)o^FZ9 z!Q1-}9<&N%R4gqJgw_mMC$Ty33eJ_we1|$T!0MVB=}ZM?wA-bDU`0TdI}x|uQ1~gB zo1Ng|didf6mpr4GL0`C@{(-}RMiLvMwrtVxdB8v2z{BgZ^!VY!T@~-&kFlh{w1|$0 z0TSr@`61CvRfY^^n*!vWh_h~mct#Z|-m7*f&5h4`>w-;asYBfKUOv0+$)>n*?|}np zbMAP28ylNv=qr%%Q@>J8q577Go?6@7x>ix;lw(r?1m^yjHAqRC5ZcH;RrffQE@3zuGKU2uOu0owinBaZy0tBcDGV`DbA%=mbwKgvp7g8+!T z7{#K2C~am2WdCkG(zDDpE;A5lCoGTUguBaJl^A1(;TzJt*3yJ{Qdhk{D8tX?R+d5 zECSxVPb3!ArE;&jL14eC4<9HYf~>8~Dwa4+Drjizht7TjMvAJHc9+|+UOgN`Z{C`R z2i|`xa~oXiAuJRrsm-uXZLOKvANpzD7&H|VoRL|0h$8&Ymo6ii?x&N6cl8-of2V~jy zzlF_onhc>EFA3f&1ZgR`wkRQkKK&>gt_g zX?}o-&9QN=G>a^P{QS&fV!BxxQ|@{J0M9`|!!KLJZesCu&|uplSXo)&Zj_(2($b=b zHN}KA%3q({Vj@w1{&OCja0Q-Rw3r9IrKKf~Mm?r6aH-LIpWSj#yIbGBg=u!5u9?pf zMmjP88;@|Zkl{y$0Esj-H1^Wbu{qt~=uDEOYHV!$oNpRDz8#VQ+AmM(L1^o>u&wW7 z7$;tQl&*H-?u@TFtHTj-@i@@2Dd4_a(2l>k&t3*4eE=I7s2Ot5l{C8~ApOTbfBxk7 zblP)8m>84eT`vc}alUaR{YPl+Xl8w}GsfA1 z`uwrP!Q8uuvmIlTe2VZfqlG1KQh`y7QT>B!!yznwU`sdk^(%mhQWE6XYrw1Y^z<-& zAx0|)n9_1m(bLz%X})D=_a2I7$H?~3y;VuH{(pSoTJgdWq~+0l7#Ii_=kQtA5HyvU zrc=GCb2?Dhzqt$_f?QS%lOH9{(Qx6g)32T%fKvmBL1GE$A?XUzsdw2~J5&mTYC&W* z3d~eP-%qcv{jr85wzT$VGGkolxdpN)rK<~H5E6jKluRGf-Jb00yZsyNI20utx$Id$ z;OP1V%19wgdJkkPPSGZF$&x#cqka^0q6Ob!uPw1rPjj4=jzHbP@CSYKN;JJqDG3H&<#t?Vft_X8T0Y6`PTFgxqIDpvOQ{8+}> zw!#$_xH=Q!(BzzdYR%*5`qGl zyT@FHZ(I7tin`p5i(@4EG4TAu{k)PlTIyB->d<5M;6mhnY}v0~2|t&Qyo=xh&a2Xx zBnfMU4U(0U(?_R*)2!5ST=nU{m9tx{=^ZA+x4FMa8+7FvCZCY!I9d+R9P~&M7;V(T zixFzmKwYrrE1Mey&ox!5TKZu%ztAf12Q!$#a;Up}(dj7XQKiHL|CFj=XVxa2B$Stz zH#$c<4)AVcZ%Q^BiH*kQKr*uMR3Q_~~VcdE!S2?=_8-ZV?bC*6FfPO))t z$c^Ylz@mY`T7##}V0e>lhLHvHjslvV=hBaD-iIA`rhhsTAY11mW%f}zsnB)q=Q~2W zfldKIJO32 z4hSa%5nRRWUHT(*SCoq{i!6?Pz_QlCoUK>ofmN6B)Z2IO<~|kV%TZwYB@7Qi00{5i z!xY=MUtgXXB%nEvg-(1f4HFN~9w#TKK!(#57;q7%?VfA(CeWH?>(T|2W_^W)g;j)u zeACNt7E5QF&&*Bs>fPA#3p1Nb?+pVZBV%&&A#LS5ckZ|Yy+Q_z#3++U-o)TtqRbN@ zdtufW%ba?RX@m3fV0fgc3jwzVQT7Qg52+$*XioOFfAaFujZT10h3xe0~|!O~q|43)UbG1rr2vWabJY zdiy~O8AR-Tk(S^CA%^*yz3}ni_WH!aDya_aMdpP7S~Q9Wge(TNaJGt0uht`_?mwzj z>dfgYnDyIt?Dzq4CF(MBA5nrltFIzgr8duVvW~V%OC5mZQH`2eMu;lnl=b2D-=E6S z`m5Ko5(hT)Zlb)+dvcH3r0ydeJTxJWvZmGXE`F%=(+dmF@#*M8+h9?LKJ=w+4_5M| zLS&Bc^*K2F0mr{Dzt2EOF1Mf{b84^N*WbSE0f>H}9~vx8{jm5}8VU6}2Ut)y*KNyg zx81&5_{a{`+BGpx3<{iBcU|@lYEprELRrX0d~{GpowMbpYUg-GKT%lxici#_x{ z4o)8IP|)C}JOXEg9_57iOhdluM;kjk&O&)R24IS&byz_VEcKWT}-d-7S_pSchH;CPi;P_uJ8wO$(89E zMqs5W6v+%kHlUHWFvB%pJ8}c-)|+aW*idW9S5#CWNYQs=?fe_L75|Oo29w^yK6Ymo zg*dgdlg?k<2KYPzu{*RS%2Yn;;Hd}Y4Lm%x+}|5w^z=I#$Wc8nUcA8U%y3*m(hH4K z666ydfk3&4Wj+ZGCOm#3%Z2PhJPtwMfj5$m*z+SqUFWvn4jy51+{8)NuMRFr>iF7MW@TY+#Bl9N## z+y+kv2M38ygHG8JdzKjv7c}t3rY6%*S%;vR6B08TNj-UR0$U3uJ)<{ma9=>?6PG8U zS+nfxl(&Djt4Jy-?duN=x>+=6HBL1yLDfp8b+<5Eh+m|cEVg%~zP9!pqJ+?}S!9oL zat5rdxMN-zP6MOM!6W_9)>ctlyZz?PesVTb4lXX1GiP4ce{3i%6$PK=htiy%SM3ER zO_kc8`&Ni~Kij?H^b(3lA1u~)_qIXtfBgC@u#2rFqnLXHRPn1SDwXI^0QEct3UILF z+-G*AD8wAahl7^9behz~ZaGhV-vC+T7}Od-I6mYo3z7BuwASfUQq8%xmTKXxNkki_CHITNB2TX1)m}rZGmFnrm5x*qW%BT;qm6(UOXG-9^@l z*t_6)wppiNy@Od#P2#HzR_n+#G&P@ugxp8-L%o0g@&%b(MXZK18a7b;tE0*pp%{nL zNVXl%-?zduYUJ#0}+Fp%+|Y8BGPo| z1gf)Cd+CpQ(^7%eL7@5Y@#FiFG(paXYB?V**v>RL=JnW|?!25ySTYD3=he^?(Mr&R zJewbq3%|Zz!np?bBrqRDu$M+n^lI6Z#p`(K zwii#HP$DGw9buTGe0+H-PQM{cLQOi4Da={AKPf(50s;ZrIKf&E9cn^%+71sC8325v z6($4Dj3KXsgTn!I_&AS+$e!W*Ajf%$G`n~2ZjL>>^DRa!xI70!vQRT67jmYN)X66sz`mba+oadqU68? z1OuOhh7!5yOQSu-FY}$J`?Ln_R)0H~!k<#YeiE!m7BP`Fqz02vK1lM*Lf~ox+fD+$ zmcu8tAqbH?K5l_eI}UZ09x;JAkkU&qyTTEYZ>Y-7=%sV;9UdG!=Hlwg!4=Drwz#H7 zt2p5=3z(1uX+Ws~ur}%Riy@y7Oz1xsphRu4=T6VEeK}Y zl6I|p1QyJ`Lx;HCCVu=NJkihPUghJiehB=6B;Q1VxzE(G!P&eAy!Fy~TAwgnpnxl- zQ>h!mR@W?ZEf|ki{ZCTt9nP7-M~aKm()`!o-lm^u@uOmk;5ViK*fszN7xZV^K7Ut4 zRY;bU1Vup>+Ga@E*7{XhMaA${ZSXx?N5?R$o>xrASXt$6=ex`rAz3x%Lw_g}Cy`=IaQrX3x{BbX1XhS*OroL;z#>peq!86f=512Ko!)QWh+zs49f3#?p@6b< ziYowl**Up3ZKJ5Ht^MiEoAI;-SQ!!ggYduX0nRV}!RkE1l?+=EMW6|Tws-XCQ345r z+93Ty3tL#%Wst_Zap?`HFGTWx`Cc2kc1o_cvir-*qPFw0{l$OcQ&u{6^A9djkPmT!CCPb9p20DlEl?+k>q$@3?TMNhx&Hcvp z>u(ADG>YotUV6T(^66>#DZpbNEPrdF^d96$8bF&s*%{G{Zs2$m8CB6}ITQOzYj?rq z^Uz8|EWI0^&>F#mJ{d1HdbV_NN%=9Wx+_9P~TyY9?3i<{vg zA(i;3{U@)JVS;4YD?JDH2ZHvpK_Mh0B;@4gzWeqq7Pl8r(MGnsv_uATPKw4jL5p6= z(4(OtHONkhgHPAOf(z#gGV@E=V8tmm?$92HH-OMz5CI7XkRf0QWDL`;yyS|R3~tKo zA=u$^R#sMNey`0g#Vi)(Im79q2x1hk1_}c2tv}nOpNo^+|MRB|!WAZvV+ftLK9tiP zA%;AwRlqUYq7ZLaMe1NW0^Q*TxU>r&B7B~smP>tFS7vYfBX(9*#2DVz)%hukElUXv z`z5pwIx|V6?o5bKLA~B3Uf{tU>w2v^Bo`i=bY{9BswcBI1impj;X_(>3!v<4;7gC@ zPo!7x40V6=SXiHuxUM0BQg<+Ah(L-64~t>wAs+k5Z)f3Tea0z)fj@^#LdCS?-Dl(p z5h1`DH7J)QXd1aMUn)C@BvXs)rJ@k!yEa3O2apWP z9JNo+B&x~$Q=qz@hpL(eM5#(_N0l>Ee6zssUi3gmypKeSGM}rEeLQ-PcjQT*p;{d`|swqjfRsG&b-{5VD zef{3|>CFYIk`Qp+vXge|=Lcqh#+23{pLcuj=V58Tn8?lwv9K}b4CZxlGEcEyG=ObTa$AY0zTA9ZX}>ppMW#di5Y!=z0pq?(KLkele*9o^y~lA* zwclG#OixDiL(a5mj@?^UHa31Z=mbA4|ESy)(9I>%p3l2~N{NAiVSXI(b0R>0;-8zTLX&WwRxp4SDp z0`4&krIygp2$lB!ex7z4L0^4@{hDFfbiKO896-mT<{)x6T30|AX-RX#Y!lUnFTzrH>Nj{(cyu2FV^6-I)5oLCH^-eE{4rHdSb!r}U+^`DnT ziuUs)RgYToLX9=X;6!dC0`JYG&kF=dHqivpfo3LRVCt7Xrq2lf8>-~e%)1Zl<;P6A zDD{v|$$CC3!r8@Tn(!|W1VaB-jR|~q6o24%ntIg$41gDU<*`H>!(r_CsQqR^CLHf+ zcc#JgQ&k54oz)flEQPxXzA$X>sG1Zoj#6WyEpyx;qO1RE3@-bwuO<={WB ze;gbvJ#D5f|39G)=r-5&^%F)z?r%*RI9E9uNJqqPTUvtPlPNU-bIZ!h51B+~XJMhgUwnK5 z56gI4N0;}s@###}gZ`h@Xr}Z3rk8f9*GHY7LE9$MZx9n+qM+y(ue=DdGxB$y=h1~M zjW}^dx5%9{PK}&8DIR||lOgbG7X7kh+}fcpZBBqqNGR;admmo?N?$FSeG;jh;PKn8 z&kUHGn9_u~>0QcrTo;_ZL8bd$9(oVrJRIE87x2+xJ<>NhI9Uz> z=+v_wX1}dt*U!1q=1#Rtv#>U3@sGPz; z<0^7-$6s^ThZXjO@hraY7CDSpI;#T@X`3kxg$guqatwIJbgV4Nj zT3_?1tFC^o;%Qj^9PB%9Vtsj-k$_REg~acrLHWtkfF44Ew<3V9e4Lf_d3^-lh$W?^ znHERTczDWjv!+F^%Qzi&yF~m-i@e$f{hlfE>C;OncBR*P6hwM4!&ys93v^7Ocn}dE zCFnXV$UAccARJow;yUqSHc)M7p(?{H{Ny}C&>JNSix>IO`AxCys@08kbx-mO3L2_1 zO}s)oHwn$%VJlh_tIg~sk^K~GCG(n0A!0%|Kw|ZyF2sM}J9&~wts@H8l}E^@gvuEseZR7%8S5W?Am|wT-K_R9Phtg-AEE`2C;*0ytcp}95=Ua-aG{ly`ezQ>{P4bUiuP@4_><= z`7ZOcyZ?ImX7WET-_%<@g`SEyYr##tA)f~gf#VCu^AKh#s!iIzcUj$&7+e2J&u@`= ze)3Kxua#lOL)IXAE9;Bo!uFu6ey3^Rz|)q?=Cz-!;%q#82-&+xvfhrALn?K|&?Jdqm~pdD6=-Iv&x%N=S1`m6ik` o<+54g%67!>{{7W=eb27%r9veUnn&=rEs)7xP>{-!yngq80r@atX#fBK diff --git a/docs/images/chapters/intersections/9b70fb7b03f082882515e55c0a1eacff.png b/docs/images/chapters/intersections/9b70fb7b03f082882515e55c0a1eacff.png index b1954e3011338b9ec0119fc081e52fae5ce3f7e5..4f26d74a2940b4a4f180ee14fcbe64fe27f3ab47 100644 GIT binary patch literal 15088 zcmZ{L1z1#T*Y*HPiAaN}C?QHS64D@D(x8M$D2+4@(%m4T(nu*GB`qx_Qi3!hsf09$ zH2>P?obP?V^Ig|})Jx8|XJ+sHtY@vecBq<)+<8J;LKF&hUO`@31BJpONB-dB!Yh<{ zv(@kyu8ESIH0l)j^}7CT913*}r67Gr3zM=s>1Lqycmi)@ssulT{f%$vXS3K0{55y0 zJGXG<34_MN6V-oaWpY_CWCrEt4jsnl{0veGOA2Ac4hxf?Phz>5bnZ3F6C&b6ZuUQi z!RS&VZtg>SPw}hbeRGLUMTbH-hN({JPE)so(e2mHW~kvyNl96{z94MN!ceHz&MwKj z+q=2dd?U-iP(;sKJ`~9dC!i1D4Gbfom*&h(Ad#H+``3>#xz*J)>-B%=o_wp$*r~Mp zR*iM_>Q(%}k7j>{;zcI>_#Y_iyt|J@9E?L8>`%$2!&~!EGrfx^J;^n`-%Qi?3?AN` z{{=CLbTLm<-Lv9rOibA-UVKvzR@aNo8Y`?v6m3)o&?q$x4Qz9B^LO`4u{{rVnm1-z zN->+WW_=l6X~D!-`~m~d>J;joMHxlRd{q?>tf>*i#>T$3RFQwMz)V4ic9@k;NJyA2 z>l3qf)>qoHIXCWs)@|?*e985&DBh%>>@%xTtcd$AQGb8`-qChn+r)6ugK$3U;gHh? zlQu%`_9mK^8|fZ?-|O9#z71!MRY4AsfLON#>4G&OZ~b-W@j zVIl5@>T_=v}9RXS+Sq_c-CGZ1$QDPHMPy{&6_tLKYo-Pe<+-n+tL{Qvh^zuFK_%Z--MqZ zc0yueCm*Rxh`Esw3dPOCHC|yIytrtQkd!nF_0tx^C}(73bgkjp%q9kNawM20 z;^vpgV?MjH^p%^NTS8VAx6*bZi1WdR>Du*%mMyi;Igs9dq=zZ&&rAi|co3 zRZ@il;VQHAtDc}v>NWB`wPRnt44R*R1plE>)drmKrFi7ziW~=R!vn>1bi>uesYbO8TX<0u}kV+{j=ZL>3zY8iHWW0cE=)Tv9aK!1D;HNFz!x3 zkJh{2mS!L|bYJCzdy=8~R%Nzsxud}I2^JRC-fGp9k-0hExpU|0eh=%DRSorSZ_agC zZ63b+jIll7my6Ai3jWyKjI}yZ)d3YpYhX6j;CYFFfMD@U1`aWuNXWZ)?5L(DU!49N zC2EvUbo7ORfr06@HDa;!=!q&jMi!R5Ui8I_7iq-3>Z+gf{`}dKZ13XIcFC^J_x5)T zPEk=&wc9o`?s;114|p_}E`@Kun|fmV;>C-E+jeyC-n}b?iRVG1qB7!J2~ONJ&B1S-CdhJ z$HT)D|MDd^W=NNc~Nvwrm2v#@uJmo*~;0^%^!i z=Jfba3@k@`!ru3~O1PafYm>E5N-eM)U&_o?>kijyU83B66;|2Lwi3+${23NqFsX80 zPEKyqtEQ$#z-^1^yVs#@sN1kzRV6Ig^zQ0}yn+Jh6u-PxF|&%Rt83Yms)fbPk1Z|9 z93p&GnAoVi$!LwnQ&wT&OOv%v@ivxSi$q!1XqWJ_CD;WCe$uZ>dJFV`RBGU$LS44pxVEeKrkl#7x!G&Jxp*_jQFMm64(cY$qzPG|fL8FNw;=zJGr!Mbw>Y zu00Y5b@Y4W;rZTLue2<==pcW8oDZ(6Vav-_d;9ywdmA(MuCDFLb`1nDD~K-fzq=!% z(1=G$N=n0RgkoS|_yV^?wrXfz@Y&9yS)al$Gz-BQY|E!P7(J5paIOZUDX+uA?oCA{ zr7`M<9{sa(jmNwC{N5)X4iT{X6~7hpx$#pL=U8nGmZujp%&Ma^Fl2L-)YN4TbM2)O z5kxhu)}inhzuVR$m^#UkBjX}JHjjsT%W5!ZuU{BT?I>O8ox882wx!0h&mv40j9y}v z8)*IhI81JHkls*Mhxeqq?8>X&dK8L??pO$S=YYa2C<@D03pQC(yxISLB zTsqU zwl`Ex3s5Mf?GtX>sRwOiomY`VUVr#Nd)500jjYwDPx823P+moIT)mukPw($Z4K zkBNhRdIlGFb}&zKHjjKfw$gz|F2wJ#nVHG*V4i@-9{0iSYQeIzmY=4MPWya)E}Gu9 z(;x2wx7QIyT2{8wVpz~tDl3SEt}PBLy|3d_3?tgKov4P>Eu&~*LQCXB4}JCvsL$yi zKhE0O*?DZXQRl13{QwX&K&Y*)ogn5Z^scZF>MLW}c@SF^j;6%M!J#L?fByXWwISt` zzfN86rijR878VxyyLV-3U00LDUq`++k(58TGFeNluC6Yxq!ecz{DvQ13yCKqBO|$d zIik$GFO1rN@iY@&qbM?{!S?p{Zf$ReetTzVDfjxbBq@0w*wb)YfgJjkMo{-~TDIzF z`fLI>H#hsIPvzw0@$2gAwNsS)mI%bf#k&)Eqc^h#%E=a;Wp-V6ho zc45!;p!bD&9`4d}aGbv)L?EO!GK;9pgB+jVt35Qa*=(3MD;wt#X=Xkn;VT{0Mqz_-asxDwec zK_i7^Z3BY{0&?aisJJ_K?!aZw^-T&b`cmt$ql?wQ`igoK1hSXr?Oxvu1H>~3vu$0Q_# z01^_K*!K|I^ALl%aQ5lbr#bVPL|V%7$@ZJGGH1@5xkgWq#m~<_TX05(G*lYCPeMU~ z;Qf2{=DG7t$A`slo|RX56RP%!8=3zh+T7d(SkNEFMzQR3U5|^0ha6fbGPVs24B`jO z0EZ|lD%zhmTyNO_iSzBl(0PU!I%x(`gtj*F=5cq(6Ilr_5db#rqU};dk2T+x9MJl zrlzJxn;lm-wmw9-|M|eXc)R~=f#2JO$0(n|!opVm%&aVj)?PkjX>YHal`D zMst*2(Wcnlxr3F=ZH&#s%}wIXyp)ic$uJB%Xm?{qDz8zTc2RD#ocX&i4ju^wMF7mO z^TWf#x7yqRtvvZ&7n+^TWKjQ%lIvl^8Pv|&)YLtlRJ3Q4tUCBAY0*6cek{T4%HcK zEL79CZw5y~^$iUTt)_8taSj{P&A1b*D=S7OCIgF=)+0r}T|dI`c!r+>pGXmK$m~tX zZjajB*zhSXE*@M^Y74NaOTKOQab$%0aCa2}DcM4AUcW{GLjLsWQ&ZqM>ZXNlda_V8 z0|O$!5X$LdR~Tg@nyicKnKhpP8}ctJ;{k@D@B9tc-q_syp7|wHA}GPO&o9_6P3l0a zfFk5>Z*426gcDUx5g-*56dbL6$^{UeRuqU0ZZw@}tdKb?vviiI*AWeXrAz!aP44UU zBa2>q_vio-SG%s#!j};%z@xZJ!0-%K4nJhXxEkd0RaK8rGkkaSFgRS zf=iekEj67rA)}-`i@INjLHpdhcRjYUP_KgUs^<==&%G-HD1`her-@udL1PPL(R}mz z-8&*!Wn5Hf2TB6AuEB!`a0)+tuG78+Bm%W52c%K#XhZ6enOWxh_hfbr`^=XwUslu6 z34&&ca0l2y$Svo#8ocSe_Dy#vPib|gC3s_N>!V%cG4pmEoAvO!W}t)m&M(yU^uqeH zOH5L{E0N<~a>R!&@cHgx@ifO&^v|(^_kv}*$ zCuxG}Sl(0M%2PiA9$%IYiU<9a(LGmNPK(6T#5}_S0?r_WdvP$2{M)y0B~RB4otFk= zhL*b8}KmM8f1*=|Iwu1->`4wsx|7;>E*6Ec1h#*e~cz#;?ys)Aj<2eW~)Y zZ2>m}wc4`M8l6sMUxjggk(>M@wllueoF7an;UZnAhNdS}KRmB#84heFHKe*)0GQYP z7x#^E%>CzaW!bph23XI+8C5G29 ze~j1w z*i1}J^1sVv2Zj_(0l2S&XaFQz#>IuN;b^N1_wWKOFFvs3Hc&ApriZrRZn8gYs2e*P zCuC(lBk@aEIXNypJsNl*W$=Zo2-yNv4RoFj*@d$Npb{x65&~_@F-%Y#unj7&q6-WR zw1BF}lG^q3@Bj$xv)*u6)LZ=mwgo*9SMnPw@%gEhK4A)%t}6xS@dCDLp4+tFUC~9V zA0}r{!~IIAR1l;9P~QD{?`ux|Ei@006l~V~{Cp%&I2Ix zM+*!XCKAzbqoAr#J}1YA;Z-BwgwXBw2RoLa88&_SLef;AGBG=AlEb;2Ly((b!Q$qc zz>oA1fWo5{*3cigA@grzxdT~O) zK@nL8)zsj5NbAvH`IWgl`St}NA)(RtkI$iumIiW)g@P2W@LJthn&s0Lfu599Z)+b16C`tq^AK_(Uxn+#5t}YFq6$Ss3Z}`AaYJQGTgxxsBI6uzJpcpb{ zJrB)&bo4$lsw*oi@z0+xbPB&#SXU=fU0cftN^7U2Brgsn`T2}RVol5VSFadBIdb1! zY1X%^4|Mr8JcDdJ?RNtM8lbCu?CZPW^s_t0>5VjHZbB~0W0wTQte=WZtgM946ZsoJ z^Pu`;JE-#?Qjzxk>lX$s<-vek!pb>N#b@cFgF!CJx>sy?i<)$LNb{wS4@yf*%V?oD zb;m+G{LAGMRt*q4YoD&qf~W!d&l?Q!R}fvZV9hl8YV+_T(1;8GqzP_z6Cl2@s_)N~ zI7{IJBLU^KwzYMNRI#XY?eNMeY|@E&gn-;xbzm8uxm^(Ybbsry2go*GhKAZ+J@g7~ z_Q&g@D6TBjuVT1$i_Fr}vZTB`UNDXQX4=tQzipIXu2it4()|3q5x@k@&LZj0?nHl} z`JKP1=Jz(|pm#_C^bS8q&=1&tUOH7sSszh1VCcv?I^IGMl0qEwkZ!OJ9U{1?itV2f zp;P6cu4CA`)0X{Wu(YN8Ix8z{iT9}&vNl&eHn3dRzJ&r~=&Vn5|FhOSF>wjF;zy7c zf6%4pUSMU`i2d{2L;C<0WCQ+u1R#M3zyKyFG*%4jFjz3yH#Y2HS-V8V)VZ{jX)`|G zR;Iju`!<3vBw&X7`TJk$W@T1wYirxs*$HTD6gTTl$t!UUIe2Zdb>Z$WZrf0V@{N_@ z!4}!dS^D*BuH&k=mW4hXm1vT=1V?FWX$8RGqH8=o5r(rs1H8y7uYJ@>2Gf^;pI>K9 zH8AAa*YWg9ewT#GL4f7(&7n}_)6>oVitb;Xw}18^Odj+@5M^J-^M@XMNixZdgipRs zf?x7vQs^5fk1&sY%gV|I#((~Ym$9?6^PYT;Do<&rBDY=RwvBXJ=rsoQ8&U5{J@y@Q z3@?HvPYP-$u&~!>TPL_9S(t5eTFP@^X*YUqPlX8Uy>R2hM!+F5`9b5GnVroXtPFh? zQ`LAY#fedSGIOj=sh>Q1BIQSH%=`QM^2*B64h{~fs~%!k;dj-ONx$0K+Sco5maV&xd-vc$B-~v_ zPEKM3WiIxAe%My~Z(_Ey8+u^rg8pA*lJ$KP+x(1In0o82=Vjt1DKBka2^|s6V^`O+_yT^aY+w<+Nir`9X`5OckmA2;@;8um zoGM>{oF=f{zx4O#kxM1+x3#ujgJ}&M)Z)3E^=t)ho8~&^83@T5$M@cxq(qyUnN4r+ zmkd?4+4kFpc8pe*QLgKszLX_jsMa#8gP~{C5_EoVxuAGst#&=1w-F2<#H~?KP+0u> zHoALrKJ{vwW;8~q@VuGzVZItB5pzfN<76z|&h%JwS#QH{896!A;di=)ckeKRE+lJW zlBqcUgwT{Incw!Ks`zn0y4Qh%3BCH8f72SRq;u+ag`CXbmfZn|%xI*SxH;8-#$p;@ zDvF*+Fvy!@%1)SqsBiodt_BPl%i+q0ONIIXG!yDqve7|YB76K zICv(}!09pp_b*?#aAEJ@pmHUf zavFF8!hM;Q6MXu_4wZitAomA@!8G;s6lw#*Ez zDvgqYVsK37!GoToqoeyU`#~!3^z?)Q+s4Yqme|nXl~s!3{Zzc0$4B+{^=qGtmo6zq zjKyxhkTu-|FShx+_o=7r&Z0WV(uGMF7{E5L!)-ptv%JgCSNw_DlH42=Pm`2A|7T^4 z93TjC#r5@`rZVrEr>Hp%@IfquvqYf)KC1&U%FkyFtx_C_y?5EI)_I8xx<{$a*e$>z zibG2%6cwjI2*3&^W@bsbw?hxCvd?RbQk((m3<8-|R$J@m&xAmP0SIBE5KjyE0~IwH z3K4E#w>8b!x3)}dZ*L>y&!Exk!Lk8U8XKb0zh*IqWol-CcKL7AY))Y1b>U>GuOXToZapOFhDjgVo?H``5 z_X3=_`3jSSNmSW<6#llvl@y=dRrPAb)m+ieEE z5nd=1Z6cBdKyPN?5<=&^*gh;R|kq%hu zw*6D~;k;%g5Lx9^Axc1VF?(Vp^ctFv9<$xNX~bjJZS?r@W5>`D3YX}o8`F2%e>jeT zB95HVKKuikjg4&=u08+w54aLgAH~X00R5mg8Pqt+#PsCW1JVKs?Y#K$4mq3@DBq~M zr1Q~TKTP^FWfFKTWAne#>q*S?oybPg4lNd{=!0S(1m@DUTembgtIW56WaDyiabeuu zO~2Q<<*Ni+5Cl@k-SNkr=}i^3bv%jP*>QenP(y}zG2=TiplKK%>@0!jBr7c)fPA*} zOj|2R2FHhgI`+30C@16fIzNAww72I57h+~{F?ZRy^=}cr1h^8oC937&8io!w)>T5F zC`*8?OT~WQhh8cuSLjM9K&I^MZ179p3@JS)8KB9bhy!GfaC5NdLvJN%=7Fq&_zR$A zwZcpS8F~}|DJgU{0Q3XjYNhGpHI$uQT>i19S+WFQgf#Gojq6=`#cHqKQh|7O2?rT|5x$3*q#nYizs6N@W(sZcDb&EHyOPYm;SxG}n5py?_rM;Odj zoN2Rm-;5V(uQOvRt*r4r6+;38upGJ0EMend(1S-8_53*rEJuORl>ywF6$ue^qO^dA zbMx~x%*qJ0LER{BI3NUxR<78vUMrQ@=boV3f7k$AO?IT36}q&%JODx%pgqNpy^{eO~=QOYd$Z(=acze9(w=lHD{en&RW*OMr`6A1@oP zvefnn9N6k-zw-wR@be46Otu*>C-^m7sF|$W39*eRsFKS9_7b*f%rTWxOKsU^Oq&pr zh@UL~2EZ&JC}>vvbU%7L0{i<17dCKLgF#<#Ui<>~$#Z`B>pZy8z@MnO47FswI+nWp z(s5oLrv%uNuW}ZJ1QFm;=wY=$k%amzS7&8m_QyT{5pFD>Bhw~O7>AWnG>BBWLX@at zv+7guH0Zv6(fe4E<)8$*Xh8QLE-{L zSA?{Xvr&0`E{o{mE$O+mWgx!IpFb;d3{UoSb>aLk?}L~=pN$!M=5TIIE3{EWRsatp zb6HCbpfwN;L`MgCPRm6o0HtP(?k|sNg<>O@<R~h>N%& zxxDx(8N$TM#k?#MD*#LhiQ0f*uAPJz-616{eFmgqt?%Yi@}^pb%7bo9o3I7Y#vjk* zaMJ5d!FD7iAwkha<3SgH@#BuX6C8y4eOciM1b`F~H?otH6V@*Jf7v5iFY)LH)W$ED z++|^9m9((P)r)L;05}CmftHRkNMwXyfUxVnKjY=)1&H!FRm|gFIo-kNDp0XsCMB^} zVcvi0>3Iy6vnj|Lj^SgmM}&lg2_XN5M?}OciCC(Mgx#jXA4WOCO8@(+Nx^B$3zwbKAkJPYW;=3BM;(keN_xe6BM<@d}E z&d$6byA{W3um6L5q0&)+d=>4|Npy4XqOr6ZnM>7fO5515zfnoe->LLISh@kD7swP9 z;L^Vi%hN`y|4wQJ2WfGls(4_EP0)TC1u)VdctC~j)8OYXH402Jmt$JN2xAjFSY*tp zOpJ~7gE`0mTm$4!McZT&H#i16_X(2&3^=e5(e~34ITbKq?gF+k0i_2pV*0@Nb!pPU zqJ&&4CI>hnL{-g+B=S1iLjy44w;6p+#ZyJfz*fP=?6W0D_Ugrp8Q_@Hjx~yw#h|-` z{tdCEJ%EZY2AJu?-n?v+9%BpzVZXolp*W(v&e;A3E#0-%!Df=C8+@N|)1AoyaIgmR zV=!defp10>J*Wj9uQ9VqnyGB1tBm?lZ^#8UY)p)=dX_9pT_J!#1}?4(z^OI%-`NIy zN*+Iv9eJm~PC-aHt^$l5CKN8LL}2Cih*9okQ zgP;Te7S{4eXa=UxxIvYHbVQBh1cS6T9Eb{Q9Ae?%X#4$Z1Z~oaA6O9sZvZLovXC=A zLXz(b2jnK{QIpF=Gd0-HV1usyrNPrZu(c1o2e7am_BM>`j&~;rE&WMcGKSlv`x!&~ z(#4BQ3Sw7QY=GRGLOs$5I%Z3bUz1MFbwOiM2*>WhxPpLG!ne7teGmH@19rV~{5a;j zvF;v{gE47pHv^D4bLc9Sgb{tV(*%9oVWIcReLcO;my6pWL!qXtOW@?REgNm!vriZ( ziKxu5D6pekOmb%rkw~EK<_^*M^EpZ4S+8G96p`arFntH76ydj>aW}fI)1eT24h_<3 zCt*0nPFtH2garbig&>q-fkyQPa~G0?s*|&M$F{_;6JlDuPxf!%646|%!k|cKXz+a{ zd>|q=%)%5T=v7khPmfG6Ukp&$3RtKZBrlm4EW*5Coh;c@y*o9;tgL=-{>OR`QKb6CAbrJB3mwxTKMU#^bk^5^Tnoj%fBi}h4F?g7 z3%0LGPgrQ7u^u%J0V}=(Hi*wcjW?(T>bJ0wVw8Od9e@JT3k4-~6Hz4Rb- z(BG=r7bVL8`PU9@(!(*WAOl$q9bN|$Uk-LEL~&4}qV+lzcSHEUR-0gd;*SOB(gATM zt9~>j&DsFskU)-)_=-?@{sYPKR9Px=-C-s#7y;oz52LS&gOmFO3H;~<-934wwd5-se0;=$;LdvUG!6Og7>&K3fT*B&V?CFJ&68U@c1q<3~zP?(NvVJ|b8_Ml>KenC-Cxr)0S=C^WBsFopsJ*c`Vm z=BrUvS0{xQcmWA4qzGybQxI74@P0@rcV==hHa3paJOi>J%qlz_JYs1zwdZvMOxbtU z4h4;pO)M-zAb29$wHsui%gVbRe;W?~;49H0#EPA)?9 z0~$(8Tbs;9OR~(Na+AFR$T31gu|GL-i?}M@Fhi`0>c$Zh+eHgZOMD`q{o1XHD0+zP zzn~PtKM=-7j-t8sC}f^`^fOi+&h*XK4~s6;JBlu*56YQ1FdTUvzer`K*Hf|%QvmvZ zJ(@`jXZoT~nD_0aTMrVn6R|=+cRwdteZ$w5-5=BXbYporyJ5gvj4|L-;4lUG>zJ6< zdK8WPXpJaJZ#jshra#|=!GUSB)~89$M^WWR_x`8c%m5s&ytU?vSEcpcm`jcj`IgpeHv-D~9P!!HNE9Ysf>17(CR(?~@p`4* zRFtUuuF=^0$7FvX^OiX8xO;Ws5TaOS!=Wa>jv!=Z~ zH$AOp?iK+w3W+vEpaCixPkwMI6f=k3HB3o9J!CoyLIKg~@wzvXhYPKGy7&bPWLvW6 zrP0!?R@(uCQiv3O1WJvIi;FqhD{b|J^ki^xL-T;j(AB?r93-tJw1E~MTUa{mjxgN!I_j~ znVaSLebRr2yw=v%Hgh2rV=BVPXursdJV6|Wb@%HYsH#STxXj&7qhD=L|Fys0xVgD` z)|HCOu>0BX<+dm~(ec@B?LX;w2nQ6&#Nq{ z)*s@eUk*S5W*{o+Jn#gwP5`rU4GQR&^7SguL)X#qaTCyrXKg{Hkc47}xUyCZE9|Yg z1)s@AZ*kX^QQy{9>7B*?tX^}ZPM~a{hjM)1ZNFp(UiQ1O%AwyL5K_||!O%`(CI5pd z#Jvn=P3B{VnJU2zfoAt2k(u7subnFHAL39mv+5{qo5qHZ}D; z1U&$Cl7hTo<2ZG|7&?WSI}b(0%6M0DoFbBq8Kh`H0pU~}AF~zWk|rBZ-hnIpMq$h(sXOF79&e4&Ao^boCaI6d(cG*iG=|7G%j z=6_qhZrmQkHIRjc6J~qMPC)NNOC1NQ3KDt$+v)Up6H{C>q|At}iWxj{04p*=`01L2oE#n?@0~4LddFn@l{-Kq+ z@_$(~>e;ifbOp!3T=i;D^4K{zmPpa`j+t8~?|}-hOjKO}Ljq>EzsC5xiwZW@*5M$& zvE|$aq@cLqBkC}F25C2te?fdXpa~swGXU7Y9tVk4+iD`fg7HNUmiy1{i$@FMr&!S) zEk0N{kD^8fjv$F(4*@I@&jShw=$}8Zd`f=%{ws~NKKp)_{0>LPGpImG@|0!m1w3LzaVZ*xt5&V8!MGj_A z;viTqa&&x*_B?QY%+Jvej~F2F4oJ-aYnS9r8wgVXtpWUUZXh)XE7tAd?;ISYfar1y zL>-qr$t53U$D3$uYb5;>AAHQqQt{`z`vpLvSyjdaF)2(V$g+Upz-NDr zeA#_M7Uq8&RUyptNM{3#lnpsjqIpAQM<*wQR73LhV^7cB?=zI`VAp_4B~R*xwY2l|hSdfXO%p4Q*?7xY{~ z)N4rLpdfnHys+)?YnTJ5PWT85a{%vPzKO>DgWG8YMc63zoFRQ~G&VcVbs5o1B+}+`T zPmk9qCCk;r!xW5{qzv}0g{izeh_M^+QAXeMt~88^5`MsyLGiBhU%PPw55*nQQJ^Vr zwWy_IaxFS|?i>}o8Vvc^NQ?fjDdv8^_82)-4dhW0P1G~X>uwlM`B(0apVg4KY00PF zC(c*9yv5i+u2577uwW2{=d>&rd?;TM&|6XxMiJvZ{&;IsT-mb1*;}UF+G@ynw@-nm z{gIVaYVWIAbrH7?Yske8+Y#M4ok=zRp7$ViBITq-sO3W!NkZi-uP6nV#JPd)ZX)RO zeYUp7L!TTo;P1A>N`7+}!H2LXDJY9&Jel9Dx*NT2dAs+7tMy8Wzg%M!R=rrDjPS_x%xuG;M1fR8|7$;{1_pIu;PW`>M4b>pDU1o$<^mX>2itB7kq11hmJ z5HP?5XqROz8d_QtXljwfH=*P4!V_3s0Q(r3?lt^BW&TJv6rBtAaQ9>ys0{@rW!;eA|oN=&eU!k zVGN0t`!$X^m`lMe2!ky(>!DRn=06X{1UEnblMSuTzY~vEn}XyFs#?mAn3gyl0BryD zo);1j^~DdAgpI6Nr)?5p46JQqd)q%aI2fZ386^n$azFk|fqWi}i<@k0Qrg-t47hX& zV(?g*Z!HfOUIL$GdVc=Q^bnELUx4LaU37T(ISA1kffAL*&I&rCX-CvwYK+s*zo@=} zycJMDNV;2sI2X7nKH0lB^W?b15YR5 i$DD(Q21si5y!#d;zVt@_j(|s4Q3^6D(nXR+e*X`_gL5bV literal 15234 zcmZ{LcRZH+|Nl*qz4wfChmcK(5+Ng^$aWLi6xpNP_Daf(6qyl`9l~wP9wD2uM?w_8 z*LBYMe$MCf$M10-=W*&j_kE4`^?JWv&vk`sYpIZvu#lioC~`GbrJE=e9xd`O5g~j+ zpEp|x{~^Svt0n^6C8Rn}Oeo!^hbvQk>^pa~6y;GWX&oEm#-67bESVHXS+2?()^K{43ssjS{ z-8tN~-1AHK{OV0jvkxcs_6{a(_NzBm>WWrwW0*o7QQ6tq^?vm6uy{yMg3I)zyHPUh zPP;ojHf9)iR&k4$1E*W-smMsHNMuV+PpcTFsx88W4LkWie`Kbnq6!QQMA_zidT_8~ zYh3AaWz2R{NlEE6H@9N0-(A5yQbr-7v>IbeZ1ao@_UkR{*^14Gn>bw4qHF6rA#1Nhv9!+1c4$ac5Kv zySJKtEQChQyE zI4*f#R#qmU|Bp9( zLey3A%j<=l7fn@FmFMm%Gl#rySR~_x*`F&T_!afuKS|&^==eRna&GGCp4kaZc>i8B zPEu&yGy2=D+qXl~WId2?dLHcXy?_5+*t(0!cc)Kff3N6zaY*Y%x{m zM?^tEnAnNiSsI{bVhU?$xNK!(GuM}?;^N`a`r_h);(LAER(J2dg2N11K7aO%3LY*z zzl`Eys;*N(F|6Z`We?X3_kaD`5{kSeCs%y8li~NyzzyGD$G?B{@CpdfCAEe=T%QPs zD^gTZIi;JU6Ghb^we8S~$(fdOg;20~3u&8}1_$}=i4 z^M48kM$O`$J;PR3N@rzdH8eD!#B4q*-MmRNI5ZTOmKJ?{v{!FlVn9SfqPVyI)pf3g zJc>mkzrCdX_(*nea4?0<=jZaRi8?=4G#dSVxRCYT$Hn@oO4k!o&XdisSlVW0l=}Mm z(ibmoEo>|-1domyJWEK}Ug(>MVV5noY^9iASokzP&YGzbyC}OsSzn)$!m_h6@@{CO zvxX^RY5pCh>bzDD&U@Tb^(gQ}VX9 zwyK(%!QtV=&g0GDw0O10Bat`^9;)u>z+TRGpDHFMMp#@Ny2Z&2gG-k_qFeI|3YsP- zO>+aDi;9WOtgdz)G7fZilLs9AI%jq?*Am?XcUA20MYK*Xgo3>zGV$fh*k3=s%^m$Y z&PxLU!@5wT%$j<2$CKDGI-#qwf?^0!q3Vi;Yz9p73 zwyR5H^3U%Oa#j?oqobqNlAN5}b+LjFl zXk&kIO_D{%jdyl-_6gjh{yS4H85x$Q5ORBwfGb|U-+%x5p{=j4Vr`ASfB*jU_wQjI z8JSn4fRJqFTC1wUK=H6mE5e84S;!Yc=)3z{r-ao;{w?32Nk8I zK@lCtN8NfiZ{jxzCrP_<`~O;Cw6eCI-I{9+i-~EOZ;u~;?gQPd+~LdP{{EY=>qsWn zmzVk7*0+n)>1nS_OiZ}^IdIR-&*wgU8inlk(9lzC?Ch_yvYJ;m?g_z%*RNmi?eACg z_m`KGlk;qal99YP+}=(+kaMH8E$)ohcuK|iJt!QcPZnD=31gaxDyyp-akw|Q-vkR= zT(B#k_Fn}YA8l@K2G-TpHT*d0J@?Z0F$@AdeSIlg%Ykf7XmYyo8}qws<83`Xp=oI* z@h{xm-1c{!kM@mvO9bsFQZM*{M0O^-}NG2u=^e^b$x+T$W zqQfT=<)Tj2Hg|~)x8hj~YjEZ2)f1&QJq&7cf%O5om$Y6;0d&Xb)h+*3~j=574t+Ut}FV^8@-$1R$vT@>qkjZq9^a^)}GnqK02`@)0vMg0_I zl*{b$97zm&Q(R z5|yIT;|O9d%%D*#4QD_;CA*)*q@;Utj9MQi<5*4z(0kik?h+e`Rc~ z9uH-S-fd|OT0E~8Kztd8r7FqG&wm^lNos7IZLw=ZDr{lJf(m?dsFoTVvOo>^p*f?j zj^n|%&$uKJ%VO8LYpB56l$LxpXH5rf;66Nke6B*(`RuJff5=aBRUeHF8~vOn8y6S% zsJXcr9e5TgBK*wS?9gBW(lB_so2<}h-uNPvg~=U&Jw}X zSi!!2%{@9chFKf?u)mnLR_Xw6V^=|h7J4*8JTtT>($)+XsjD5u`@U) z=bVFs!@j9t|1BL6uB;OYf<}aJdBtBGY;(B9zS~l8t(C#tgnh=%#r0?-8qb6o?q+NT zPy!JpWh)HwU~exIeMMMCF6C?2Fe}4_mF_DEsxPB>N#GnrToMxWoe6@_%uAEzQ zv!eOBN!=gv_N_oy`X$y^uU;|9`&O0>#NC0_uvNCO;4(HbDe>8J(nt|k8(V4K!bax$oPt%2&*gB}@vT7#J~1)z_wkB%@Ui+Jo_cjHw9KwFS;m=}874`G zB>31!PeU~^0UDw0K=urrfl2n^Tlm<%j8I_dJ#^LS)zv6mB%@{DYvq0|mGwTBV!C(g zE$8|9A15XbX%gKV3)zBN;zMjzu>B-MhqQXTSuqQ3K_c*k&$uFpA(!rckcVrQurlC zwG516HS*oY*49(7h{!Cwb0^tgyr7`qs)K_-et!Oww6ClpBAJ`}TZi;x0s=>m3+!1m zG$W}P8BbACQU+HlzM?RAy$OhA24F*Ru`n5nxGKi+_N`kpIMOpG2{f->3lXU_zUxxe z(bgt>O}kgT*0YZl`cYe3+Y?o-*=sl~Z&_`voW@Ie_D`Qa6_=MM7}Q)at2>EOx)y_% zSkSe1F|i=3txW_KxVq{PzHnAcPa~|6Ur;a@cK7e!zo!=$(S7BtvPPw)DVJWHBY|Yf> zNscCAxT1z=Rw z-e?OZIllwCl9G~2pFN@N?d{F=ud+L9>p{Xk-f7k^CAC!2z({*)4^ui2@ViHj_uYu!Y0W55MOG;t_*hu2- z?G3+g{$M|1_|;Tf8wIq6iJAFRQ_~4R$i9xScBgrHiLQo_Jpvf`Zr|VYZE304h5;!l zX)_K-+?7}aOC;@gARfbZ@v%{b(<_x&4)&NPVY2t7Fdsgh09<|M{CQG0~A*CX*uFt%bRHc?c?Wb#)b%5PV!Ab!>JSpfC*oa_N@&iK+*iKYuR#@!g!h z!gk4<&f3~KEG(>{*6Ro0o)w0Q!-E6N{^pFd`!XFqK0ZuzLikph8tdWiS`*MPmzhQa zGEVsyiJhUX>!T$>9&dw!PWT@zof{n;-CR(IYQ#1-E5RaiUA)LlNJzLQQ(od}Wo0#8 zG39^N+gk>nI-vU)dB1~(p&|O+-CZUCN!9MOM;Ox^Hz+wcIQ9X)0U-ZsCH=9g>Y{*v zfFXN-z_-urWK6<=dk-`;$TsTFX{MZHXHUCb(c0QNxBj(8SXA^eEWwMkv|nS=zWc%t z*T#Zl6^<&~cPV2{KG@MaI6C5?P7snc0MaH{glla32#6W;=`lfAR1`Kj`A+Jg7+KJZ z7i_?#&Rn|0j$nR-h1|IlzP)`Pd7!%48jlJK1}cuLPjT?FvRl#-_G5B2;gUhcg{L{T8mHyIa}Ljz8go8O5bbmtcsDj*m-SOu%8b zV%EOaNL^P~|ND|Y!1=q9L{30MYx>@oNwEH7=D7+iSh~C)32b+$Uci{7fX0%^QS1KdOi6?V_ob_S3XFE4LkObi8-U6fip`V*W1-Rd31Dn;St6b_uB8Rf>LCqs`@iMT$ z5HjY+K+(-Rp7Ft*O|#3o&MF~vUVwoVR<;EQ$n^Ytvqv4wt2kL0vQVf1-&;pZETLA( z;lV=NCb}9*p{%J%zCKkSi|ktGsk&ODBp@s5vOgc@y?fXCn1Iv;B$l|Ap}s!6(59u{ z5;MP_^nfq9HcMnplC|ZMV_W6SHxFj*fB)H=sx0tkyrLh}+uMuc$yot7M7OjN^7j0JDVV+)ae z{+08!o?G}^TU(>=?~?B9>`YHj2R?a1HI{J+#;Yk>LSs8C;Mclq?LqRuR+SnRf zuJHJS+NK~`90iP#Rjd$4cT78M0y8JG+}BIII`k6s#RuA4aS|{H+9WtF_%*iffwDwJ zPaoWwc%dn4_~av^`Kdg-2CJ>~6SfL4lm`r1c%TM~B_mI5`Y$d+s+ zBn}iqWN_N?XgFu5zmn7kr;XM-gC|Cb&D)M+E(i)bO4RXyMpQ=-mz5Tu1&D49-Q}nb zwJ_#OcgNT`>b!o@^!7q$@5o3is49q_v$H%z`uzFxBsnip>%tU*g*zR&TKOTMZJD1i zcNvg|?oM?dd$nNu$B&uy^_a3U@hFZs11F02`}|TNmNVctH4I@aROrLiQM?Nm zE+7qqSr4=m%F8~&?JNMemoap)<-m1$(CAqh47qRLHiL*X z{q0+*=|}IawHhGm_wL;b?6$689+{Y^ZQnS1_AI`FvsQ1!CNNsupTj-Ch#_y(6EPE& zuD*vL*8u=~wx`}Ju)r?+@H7ukR`&!wNRRz_6a&zio}HGHV}pwb2@WPe8lt`Z3d6Z! z$1m>*l9Q7Gr%~ChJ$d?+hL?|Ti$3leY-L875opG#6$}xtUU7j&)Ko#4+TBuSJD?2; zk)xxdlCrYIFFBYrpa)9%)t`Tb=_hb6Z_;Bz+Q9Sk`YtQOf}qK8NlRC@D=Iu@Q}Abl z=TzzcN2VIqJ8zfu!?%v+hK7vxGM_6CycmRx!=M_>x98hcuU~I$3?|)L{USE`4G#~` zcO}Y=UQeSMrQ@DNG6#DanUCpmUik26-@hSPn*x_|k8ex!QdN?N$qnMI%|108}Vk zmRh`0ikUrc`uDlM=izKLHlcaKuuP$Vh_F~}28g|YIuN!v#-0OWWKmJkDCpJ&!PvG_ z{OJ3aY1p$H5!iw}GiZ@y%%Ue{AFeh4{E(q5_#djH{DLozkRL6o`V{#Iz-&)XX@sH! zcv>C%P|&1&(lfwE+~zZz#tUg`d3ky6u4k&W^t8;Umnd{CT&s+YtL_tUadFlDydtZs z6u9@T+WZqf6B82vGF7Eynmd>68?IlO+c{{B<@{mVJ6IW#2enE_&eH4?_Y`oO_-S}Z z=We7o2jZP5_S~9fXtggJ0oh^;l(rct7{R*sn6`7?DQ6yek~N(^bLOgfXkPIRqvqtP z(5>~!kC7jJ_7v~jVLpHRZQzFw?m4r5-+>HC`~Q|MGOHIF9E~rZbtlBp`pH#yQ=hoA zBp+VfSUdxH@(8f5@1f2N&1&qlj>#Hp5U`A82!I^1e(!L;$iA zTX&}r(9`!Fl~np2N`mtG2#Y;wxO(T2j>r<~&YG|zrBGrdXw=+1JSRVY{!BznY`e{J zU8lXPs~K7WH3!G!(q|*?@!TY$D{Q*kZ}`rhB>+h6ceslyv1}D~nZ8^?SM&R~uSTjQ z`6&vvdt1++D!FQlP+uN9qk&r(D$qAC{nGaNb4Y9~CEzdI@NKwlVB6Dkb3q`^MHaGi z%jPCsmY3BSp95Hi$N-=agnIp0wD|Tp4dW+ZWMmWxfBS^P5vV5$(HqGsmAkcLxO>>M z*M4r!Ty>bLV@IwfS>JGi> zGHBmZ_MBm}x?z~txo}(@%q%)Otithc+VKkK#P1Pii9#lgudar={8-ck+|lvpIlHYw zl>j4-ux^7NI5J1n`0(^YUO-S# zc8BQw`G-jlo0~!@5r5}~{I8R+OO-)I;TiKcHRS*cCI$ci#SJ2!h)i7^Na81fF68Fs z-h2PCvXTjh!?ksEJdTg21)yLUJun21sQ?roxLGm(KR@-~n$$2neE5*2=N0g6At51X zDM6{Jsmh}6Twqq3&Pvi<2<8t0d7*%XZNuo@uP&ERSnM@?#3 zja|H6-W{}_c3rdF|9;r;wvmjPnHlINte@6(HMJ{X?Jx)!pa6~M31kfR_6CB;3=NPi zRnqaP=b_R=_sXPN-(NH+s$Y`t0iiY%FvdBbDY+SN7MY|#rb#V1Pt5gsY(hm)nLo@DRBbKQc6z zG~zX`v#SeRU$4N%#)g#kWM^Sk+!Wk~>;9$%&_u)*QT!mdwh=q4CYgR6Op9Qa=|SIa z6&Q>oSlDoc=%|! zBkR-ncnfRm;U&-4pJQj+uhlxSlUrjj_^6~;7fisN-SFj=qUqa+VYPH5C4>_lbMl| zb4p!Z-MHo%=0b^)ja9T*HDAPB~SJ*Ax zwX9mdpOl}XsH;n>rKKep^Z1g-Iy@0JPX$i6TbS{VVo;@n(Z8fKb8Wp~ot5J5XZ$OM_`ux{GRu^sCFAzZ-T@8nle z{p!u5E(Wv-du95gYW@~+PmRY$25c@6j$m_@d;Uhe9rFg%WRHl5NbPYtFhgfHWnW)8 z7&WBen7uXpaGj@~$b3y-_$3IU%%$rS_bv#>>1N5=xp=6(6jo*tWP(JVNCjG|;s!tj{} z_@*FE=o%Sy!aS}>LKfZEM_t6;&W=e!;={wVcfy7ddjL&^!7b8ob1PGwb=R|$6H4p^ z(JF0gcNdg)omZakIo~x^Tr3n36AOYp0c>-4*aN>Ij7$B}{s|*M*+aIYv3I_4AX=vycw_`aDOg8{kqctAy%5=Hz zUlYKcM1i^sJ-Yk$9y1&gBqvXWP%PGbu`3yB!XUoW9?k&xOixb_fN!J~#z{#>qgd;|F3}H>T3KXteG(5G*k$<1VOKWttmwAZ;f3|2Yu3U|LIcN5kuP=jAOK z#v2Y57g6T+_T$TiF93G>qK4scc%B%RZ(44Sk>v-*~j2^Ltx;}*b)`N>0u0hnO{NX z_nS9w7Tb>q!Jw36&N!zDw9VzggQnNYk-{G9#9C`b;U7LoKrvHLgq~T?w$g%$ z3CdQ3b*a&dRbE>l^Xcj7#U&-50MllrQRtk;TtLGJ-k*=Z0c3<5e$fHTRbt)E;_$Wl zR`$V}fBb+tTt3l~KnWu)DoNNvF|m4xkDtGBZz>@6R>AG|!SkmP&kc0&7xI2AU#r#T z11>Eo@@${8gK^F)D2M~^_iWz_y`jNDb1*VDx3?c>W@eTR%v=O42un^)Pk*+r;!j-B zRc@RTh|n+n5BR_)IUBy1m5t2d-AL1EUE{NY{l9dNOtyJy=xb;pCap zwGVbf{OI%N)vdPVdDc9EtXW%se$fs-e>jh+5jK2pzMVE3(7eX|`|nj}Kb*p?cyIBH zk)G?)zNs5*=mgI?lbaz>(F*K}e^c4K|Nm^2 zzP`S{Z51w)B)y1m=+l7h7M##qu%1*9|A0!{GNnRm!h=nc^$>!(xoc-Fx|EV7>Fx%E z{QjQaH&CdAMMUDlTN_|dCP_N7K(gR;oc;r7OnIJZFvpe|7VwRx>*6&=2_$_^XOOV-t z8$Lf+wgPt;xc?`Jv)F30#9h2;R&!(6h9)#5q_Lyp`f~g}Ja_2SC$Yi7S72^!kXW3-jhyZ2w4h}wpODj`a#wjJ(&>$g~BzeDTJy{dy z$y#D47%;yeC8TZVg{%i7?PWU8w;pDg4FWLj=_%p1Ri3cha3JNu5>9JZBZ^lf8`E8M;4*7CBu3H?eqZkrFw{% zoE!(jM+OKXTjikrg7icIC|Tgnd!^Ym+I}5TT@9;+v}QPls)ok14%7#LuU#({I2RWe z!Cj9BXr3izm7UbqAsY_j2qjopd2dPILsx^~1k%u9g0-C75UMne zR?@S|Pg3Bo#H`V93yJD$ zpxS$Qir&6eg3i=@TM(q>M+F7v%N@rFZES49%hbXu+pR?6b~I)oegx3+3hd8SM<^qN zVIrO>sKH=kaXsaZ$0_xTgQhTJD-QVrPdx7AlZj1l*ejqC zSVC3?9GCC#jALwQk*_hm6`uxcrh2 z)SR5LV7YM%2#_6sZ(|=yWgEYQ;a*Q}2J5T%{d<~ms$J+<0;aW0fDoCaT`~;DPAlDR za}%N7)0l-846aSs*nv}K&Hh94CA1Oo^1A%?XGaA4Z3I&l7L5`8Q zAB7MpfX-gGKT8S+5)bE!2JWIGr-su75LBZbtK8Yqp(LYNc4-Sq-2k+mvsnB-fb0GJxmaZeO4Pr$ zeO${~C5|>_5$aZrSvc$`XeL%$|Bh$Fv~cDE9X3aL98-aX#3Xo>r%erwHgZXewZ{4!oKJryQJIM2Wrbd)7zMDjB|$@uNcK z%a<<@k6G`XGXk;a;)6x=+2-)gt*s>&3M?;o)p>MUn%ax__!E|va}PONg~<@keB#83 zlLQ3rxsHl-$rJSPpFi{XgZvm85&ri;M0TAwr7#zMn~4aXj8yzSNIh%*Q9l2UK*;T_ zXxCtQWkCB#^E&4W^R`CD>`Oj<^!aeUkev%!H@ zqHc2+vE9&^Ex((@zO{*FiKYikF> z79^U8wP1q&dlF0{n2oe*{@prq5jkyJ^BD5cVF+sw13E_3Trn`mL7SkjoB~%etipMU zo1dQqAf~8K9a6n^a#eNRct=w421x}4Fbf`~y#uoqiH88J&VUbM#EEr0@0)3`z;=A| z=0lY;Y`dvjqwug@zV~v07H~yZSJ&8<=Jn=$T@q@;hGy!cPeu2i<>loRve0_3A821+ zU%)Zs0HR#8Rl^!V!NNE?Mgz7hzZjp?xv=eI`M>!C8aP(3URGKlFVM`!{(c&uUoNh$ zGgGfn4HXUg#AbM8+U%y$0oH$L|H&M9)ig43*I$$f7a}WBNx+UCm9$R5T&cVKxGg$)EqnB3=#Dn+QrB4By({^Xj(+5MT&*L z#LuO5x!ipmaE*wXIG7K8nN%$N)CsDxbe+B9<0vX_8j@QYCqMI~lei3CoaT2uuFxb1 zlt8`YGGd z;neYC7;t=iYSZ5Uei&3Zv4ea+lBM~NI0`JNp!Ke|PncfMFRCC>f35dPjoER_EvD?*6KvpmSsq zFgG~G{&b9tmJo-Oez2&K^Mgz2CLQKQdnn74nlA_+h@k*GLRd`f1PYPG!07jF6*qm3 zLhL=^+OLyrf5Vuyq* zz>eoVf4&LAal!z9b_0m9s*f7!X;Dmap2A?q#K0K9!1-#|Qc_YNjrncd3}TpjH5rKy zmX?=E{D!sx-y?1~O4x3Y7gEN=X?p-)MBo-6V>JulRngiyySJ5w+mS?H1B5Ij_xgK| z^SA+Wh^fBWd3oVb``he5M_J%jFx@E<+B!4G*TDi07D9JVRiK@(NQ8l0j~aRVO$mvDPVT3|3ui3aBq8iyJkdq zdAVdqXD1X$gNLl8lT%E<(T|sqF_>Kuvt6B|LgKT5&>t*7FxX#6G($4fsD{Pv)DqyK zZY#slBc}#xUs{9m2i92wIDwB~0)txeaF&%z={?lG_~h%=uQiMfhsn)=A5MifJpv;g zg@n#QmQ(Wby69|_3Hf1wrkT(rbiI6!eti=b7H$By4dKy~g~Oq15~fSy7Yw7v>wj;n z!;)RgI&VqM?Q|U~C92rIp(7uTbQ;pI5K^~&zu(uRa2y4JnMN?)D;z;nMXJ23yZcQX zKNpGm`bgj<*V&Uo#+89^);g)5Ff(Sr0Z5XtXK+r9i=&3_H|Ah;yC)Hb^_}Q*BSFKn zb8{sS`U5CO;>#J6;XwmoZ;1S;iXt7eN)Y{zDuBUY5LLj1^NLvb;m%SJNFHi$)Kn9p z@y|mBAJ&~vD@`Vt`pmVmYUdxx&0yjo(Z$mqxUAek6L|M~MXFz=AZus%_x$hkM8F$)>uIT%K>*W;cDsXIjw|0l}l`EG-5I8-d0 zwbiHW!_!M#S$^}`NdiXYWC-Xi{jw%s?uy<5zYc{0e-=!g;9OmvEcN(E%=^D%ezP!Q z@oj#RGz||YcoYCId^5V0f<9lfMMa`OFQ2`ws_D?)p$?_5t7?gYYA{=4FFpF7baW-JUu3VnJBtK2C%#5 z(eEl%42|Msd^_gp^XI{^ZfC^B=|Ia$eg?Ru?9@vkKzFF|&W?4Z+TOYf@0F@3-RyI%bZ z4aU|hFhxPLQj?%mqpZ_WHCBYK<_5wEC(vavB6#OA+~1V3RYt4Wgc_Gh6?Piqc(MuTb~ zvKJi89fZxy0oaxMhV}%kp$-hWOvVe-!B z%IaFcXjt)JZWk#%d?*Q_V8h;`oQujY+_K)#{U9MmJv%aq4E0do%+`=MWBtob6z=-$ z+H2@~;_2<^Unz3y$g;4vjzLrivMnvpL zk%alFUFJv~Ny6Sfef#^VV#~TGm`Ym|YjQU2H99(25`hvTLJzC1?oM|XU1bj|EiDS- zWLlvQ5}yBf1MjgJZ;iR9W)2L{aR!KU`h3=E5S^knSd_(IEiK>Q&-~T9#0Sj|WxjnJ zJX^smrv0vQDp>E`>&H{pT__Y8o3yq$nNT8PB9<(SQiHs^R=3QZB{p^m|6$fI{UnJQ z8{c{du{yJI=2A4)96p+D)8qzd-&bmLgDxeR%kHP$d<@egkL6|0oT3|K zf%}UVw$A)Nb4trhTZ)dg?4vAsPHCxQlvGYI0B8#W;pp7jui3U#jk!_aW3{2It<3>clLP*iROHxew`22>8(g{!szK3mI$+E31;4!BBs8pXVhovB`+a@pzBeUc z%C=#Q}@^$f_cT1Fb{_~p2|L>2_;59I2 zi;=*yFx7cQ6e;STwMy{8u+8**8mUro5la;Z`9a`y@WKIA+`dAI*Ee>nu)C@7QY}hN MSxc!%(fslM1KS*B^Z)<= diff --git a/docs/images/chapters/intersections/b3f61036d8dc9888a6a64a1171583dd1.png b/docs/images/chapters/intersections/b3f61036d8dc9888a6a64a1171583dd1.png index d763288817b79b6c55732b0160354273cee76889..49eeef913ead32ec1d8c4ae16dbd1c08d720adfc 100644 GIT binary patch literal 3173 zcmb`Kc~p{H_s2mkv7Cs?5-CljQN=X0;-Z+ABw2COO3f*+GY+XG6*=C@0Vl2rrHM)! zIj2~bieYA@zL*nbiDowu9IzBM%zU5j`u+3!<6ZAst#>`^yFP2}bDn4IbN1Q$>?iqz z(-Dm|x@$lnkOslQjsyaMVLt^~4OljCMjr!M?c;dF4z%*~VKtS~K_GQqf*t-;Oy1N` zSim_sqWz12wt#uW33RnYJ-6=fx2>L_cj<1p({}SpR`TE1Lnl7+c0{VZTgI-|9 zh~}9wvkVGq_U#x5jc5FtvS^pV@&~6J6#|{5&y>3dT&GS_YQ*KA5Zu8ijiL241uiV`^zV*-FFPE&`M{=sHi*pdGjDuy#F@M-7 zd^W53ZoF22OTEmr8EwN6{d$ImD$cj32g;@W(#gp_*l}#(g$6zO!?@>1Ll_Pulic1H zGtJ{jR29^wP2WKr^Erbek<7oUtI7bT>sHhN( zjCl2RDEa(}X8L{|@b9=~972uL)kgJ>Pfrg#OH_TQ0PWK!`n7PhPQdwp@3oSH6AXw- zqkXCjgf~gK9@o+dq3`#aPrgRSDW)k4=h zdv6G}dq)ys&6i3oU0{-#NFHy*2NtLdOHxBdND~!M)Z%IF3Ignr-ysGI6+%VB_$rZmdk0EM#*$)sm~_~G{sGC@zKNK$11?w64uiQB zhznvphRS5!p9CMuvkZ%~*=|oSj*FbD?(cuJW7K(@dXkzHQzLJvxRUkbtI&E}z>D?F z?h_Wgd7Ykci@RXgXq#lW?9)}PS=2BP}b2H z_OFh`W|``822fZ^)_Ma~#rc8#)E?}8!?1!(*^(RFn~Go5YTpF!5_ zv=9$Qc?vI9Om4a*rby2`beerwanf|HfdSDg$fMr5WAKXeo{ehRe|-$t6sN{l`DG>l z-`NCvp+0~+ut19YZh5ZZt#6W!mJWVs&K_78IH)BKeBM&-hL8P?64kl9i5}iWR(diZw$nhPUYZM+!f=+ zv)~J@8j9h?n7*dz0gFr4b$O?HnXs@r_t%-t~0pwwH_vvr|_QrB9^k3*+T42dIYg@p#>x7GOk z{CwTbo8OGKr5X{Mrt)`fOjSw#1cBLw2Eio~N%Ca^Lm5|h-9bY`Bd{%HgD&9T%a;)R zy7Lw=y97)&)&iE4j>#qhxQ@vV0RR}UTEHyUlaap#paM1xIy~tndR3m<#i^<~o|u?e zm$kl-+0thR-wcNr)7a`=DyR}1Tq2e3rPioV7c*I*Sa`ghIUZrfdFE>Y>r^AF{Qib` z0yYTovo*(i)j>N^pD1^?&S{=u_bx6zt`B)vL|vMfNJJrbs7SHA` zzpxS!UNtJj$C{TX7r%Vc3AK*=gW4JCzePYq8{eg|T(IWPjAAee4J_t}(jY{NH;+dK z#>6l%34qI_&?4I8AlFe@Svd(Zmgbl3LocA|9f-QAk3ScH+G#OV3!UH{N}VfYvemcm z3s`@fa5S8hc(|C!29Z^~Sxn?607XpXdH}^tq&ivUWFZp?0RR|*DJCrDb3L+3C4fF) z9T@VU9X#GV0A*%jVZnYM#>>Yjm4>T^q=oD3c7K7WHe*`KO$JIzP^AOy*s(92&m+w9 z`(|#Ly0@CcZbXn0tqPe|Hw85P^Z-Jqs4myIpO+W+~w%C6FA5{I&w#Gf>pjU9-Pp8{bbq;cI zV`iogC|Gl~oBoC^lQDklK!U?@9mUMf1dNi(5p(9VnHe-$Bb*xIuO^Gn}OS)`4vPk-fS)=e3_L96WjN0kivX z+zY}n^FkVkmb3WN$j=)82lSW!KxfI1^xzuD@_2J`lm8D6kkEq=?W{AFbRT)%mL1r8 zbk+2Z1**n(uSYn_r17$(K@5TF`vT1tpSckIU?x6&UiKG~D*NKG93kkCew)@Z{Mn4X zyUR{K zlBj&2aJZ=cgVOe&F})Y)QT}K@xnb3yh%}~6o~k%@&#N?6oJ=*%wGXFn&*%Aqm=DNK&5P)`G1Y_=*nxo Wi}olAHyZeF01@n+>?&=&68;VC$-bQc literal 3194 zcmbtXX;2es6HYKdB3FP#lmJ1@g0P$_kz0Z>FiVsyf~X)jL4j}x$Px%(2r4Ql0hWX_ z%B@_=;YC0Z78Sywa)^S8+z~lKRyGKHqi)sK_hbL;*3?r^b-!=T+cVSsbWaNDke$qC zmCX+si#OEQv(|jR->w5InTTkG{dasy6!)TAc(FSyC)r)T zxtkusyv6<%@aEIv&Ouacji8XL-4R~jxEQYASuerGXrybZX_c^>(o(H&$`K5HW)6g! zSC*NdclNa7K;kbgW2ahN&qpn_`O7R%?ddIiao#sO%G<|u+8>&dX^{2v5>+zz8p&vF zqQ5&r)lLwapT}nRl{jSAUw=x@_m1Uu`FgOX4v1ujW?nVj-eD1W`Xk+aus`W7*RExC zqNRUr{LI`a*R3<#P%e@N?qyCMpyt-;y|_{KKz2rX-~3=tyy*6sFVn2`aLuxy!Rk|U z!wo8`s?yXH8y~zm5mR9>Qn~6Hb#tzu!#O)-fcAN$?9n5fjEu~w`BCohZU0N%1y-o# zU1x_b#j!H$Vg%%TYua(QJ0UKvuG!SsSQ8TyrRCd+Ll-G5dnfb!BcY)+zjt)J5LDvKSCIbQ{!LK{J?!^k)E1YK1sw zd9Cg31Cqb&XE9XW(0Flg1!x>-+OY8OLLrVNuO;6WKtmE30DkeMr1yRnUe%F?-0}vB z+TlqPX346Wa;#$im}UnQ^!-%RbiK5+l#`uY9A@=wO5o9w6$9;g5IMtKeG}1h@1|?b z898;!qr#>VFOQ|kTfCI*jn;56Zw23`_cHOCM4z-W>jf!qCNh{t3St1QU~E-LEd+Qv zNs0l`7muNjY7!p?(7fnIXE6mL4wZojX+4bAQ}U*ZgP~<<5=mR!(9U$DM{2i(XO$TU zOJ9CcJ5sA0?uO3X&+-DEQD~(B0`ura{k5rCi<|Dq!4x1AQZrisZbeUL>aoMzH1{Tip`yu}mtgW)t zm<8azv+h&AT5YoJqvh<~h8_YH6e{bfUx+B@gDIIW{hIGT$`d$M^Y42k)RvUAM{e)B zG}!ci#UrPFtca2;8+fc#L^-t8(@+pn(7{G@>2=o6&Q$Lgt#6ut7Oq-FiM5TIndD@P z7b3NcT)^5`yLsPb=MQiu<2_`{e-fsY%nzp3gBD&;j_RjGe{}&XVAmCj>_02z-wI07 z=}7m8JvckdRHr$ss4$z?&8J<&)AqT`I{T0b$6Gu^%ZFz=4Qmo8!9*EHYaKOf)sMqs zgxyfx@bL?g>t9ke(u*r}{72*6%%crz{ga{IV$Nepa>YK2f7HB<%8nG(Z}1WFt+4X8 z8!k>KM&>*nONwLXg$Y*MZCQ}4=}*n0wv!Ge)skHz;iF0&3|x_;z#jmTlAe!5J9K6{ z$jCUwKDgMC1{^$yt1%#1@}Rm6l-&BLo6AqNpWpTe7uP?xE8}W(!x)I#AJ|>O-{_%X zl6nxuH_%SYN#h5cWdFR)3${%h8$m4&q zb*wBz_YKnG^Z@gwnR%7jkB5UsmJ(ghCD-zBCuKa<1}7A!GWk8agN6MQw<1&uR(4MN zU%MPx;mia-l~gi)#xOu#uto6aRACUAJXu;}mgd{oUCSLU@A)8mbTaw02R7Gbz9ykW zGQQhC% z>nFLP%@y6<-JER3j~uoJNvc5}ugHU)<)pUB@nFqGscoRud@Z#N1rq2)^I#?iP=IU> zBJdoRskcXwmCIt0938K8=b7zC0mvK#k=Kyk*oYmrZ|x;|jSbYLr=`i7V6hThy_E}O z`i*d`SPGf0yHG44=p)u*>gC^DB-1bU5=gQBPI>Hi z*t}FPos(KC5)@cJhi$`*{IU{048LK!8@PK9dwTYz&Ud5s&ww38MMW^g*A-53H=%Mv zR-bvE%rvsBz*Y4U73kU~e>W>!&9pgs(@O(HAcUIibK+xNwY0=w3#Rx96nq7q&gNuzOR`TY5n z)MQ(nzP>)Wg%y+alw3D_FQTjIBKi*!UV4T)tVvIV^qXI{W32JL*(W5E%NvNiCDp{8#pNcIQE1O!C-5+Px=3aT`kG9XA(x0(fe=kqx} zKjKomPru-|i!ohNufa6791DgWRSnQm55tPR*~aseua|#lkppb&xPmTAg(z{@Z+xm# z^UR|zV#S;j$-8>K)TQ_J{X&P8LwIdxR_a1t?>=b$Qh0lmrU4F$%lOv>lQIn?T4+RU7@&;G_Iz9&XgU747GfdB%55UV_bYeOIyl<0qWIADdk zV5tuLhhw3p42RsH|711iAt4YphzeXz*XP6bY@nv@@ip$=DF^1a%6GY;WoohEY<3%{ zM4ygus_(e%?w=)5@63?S5e8?Ng0a7(2%Xen)sT{44lL3xJUE(xWm zaz|f1BqTRrO)E+KIWaogHa7O83agT3vpmCkY}tu~JSNm!9-d={B1__>gTlE8BJuxU zmlhbxz;#M6zN7b=Gy4DeCS@rN7#wP!^$OkgmlUaJX+M7cj00KN*dU0Ie;E=2nf&!j zWhUDPJSvX1fk8}vKVo!Z!uwZ=E?>4TUs7gf;@+NXU1Ou#Xm%a?Gg*3tNl8gBBO`IB zsD0T)y(kC>S+OB!=WjWM4s3a;YLhZD60Eq;lqay z&Rz}<4&~ss;^N|-&OfV$#)}Qf85!e7Mzqs!UWpGU9_wqMH)+)1iQkjJ+)?kbs`=!} zEy&*KY0n1<4>EG{;XmfWkB`56`SLm~EvBL2L41$vix)~hKH@VoGYx?!OK}42-@g-r z-5A$oio0QIX=!Qlr7+Y+L`1Mdp*T)XPLq|kq(VYM4M(($jD-IF{+Ac?H<86o7x6JQ zfnv@x(FX_aTQl`Kd9t!|D`eSM0zqT_aj$f~+?P&d{pj9tfbAqLt>MX$J?Pjpx$iR9iPVsZNlvENnyw{{qq#fMkeHZA`t92{ z&z&@LdA;oAZ=9f3>g((COG+Y(i|;zmw-7TjGKNP+TAXbCLW#TOj2GUrYzRIA4YA0e z66brWKu>!FVYRb|o7+p(Px1~9h27J6a$z{^($dUNo;<-LCXUc9*z#gE7DV)N%rdg9 z4rN9?D~@bzlw@FFaQ*X*nuz%UmB&i|R}g2|i!J?v>1~_Qd@bJFx8*c7$slHbzNLUH z*d-!%nwFQBfBp6?P8uTe^=mT7HYJJez`KYzS^*AlD4=%fWCFxNu2t<|LAtuS_6`n4 z>TM(O4Z<%vKGxUM2L%O{)z(IR{K!c_K(MgA9slEpW|3hniMqOa4#`^VT@eu~dwctI z^W@}YOo-3b@dQSdX^P#{+c$5**VgPHZL68VBoJ|r6{5HIj0qr(4-OCA)62RqBmJ93>3Z6``idvX-x_Bu#q+lGg=N}s(Kb83&J=0EpL8J?VsHmI~= z2Zajib=02qTBP64Mr-%{c(cxJSt*sDG&ndI8wbY%+~(}+bl&Y~O)KDV;Qrpp3A*NC z-aAH8du>=s>qBJ6pblKOCIz?VnhoFh1+Ke{eR<+@(8sfXww$h*|vqBU}7rKkM%*#lfEre;;3dF9htqo3-TB@n5?;Oey zQ3~O|>y~;@BGul}vADE^`Sj`2nU>j>EKNVZ2k+j$7Yx1*;Ns!w=;=}3&?)mHqD+Rn zEV}JpHLHGVPvTT|y*zU>YJ5SQD1AalK|w)AM~9b&fG!&RXr_Z&Q3c{Bl7Q@d^!RaC zACiZhj4UKSpBFrG2gh6SDmb`xw0|aSOjIi5#Fm4ERp!;`sP5$CB%6>Bg+S}2kfW2+ z&&9>W5)u;PK6`BL?(Sp^46($FVxBUV7?<_#^lA4Z@sHz5us%x{ zl$0nvf6hn3B88`{tbE6$iH=ds@yA6yK{8H9xaY`(U}8oI7pT;|E?UzC(1hDSDwlzn zU}Iy`om2I;(Lr;yf^iW3IRN~?u?JlLUCl2p?y9i?%}23M#iL!*tY4?g6vM0-5h*>pLDnRl!f(9r7IC7 zKYMpb!98@tYW6=}`j*1)bH1MEw)TU4?wdY-WK0BwM_bi>&<1lbe^v$<2KW^s(ren6Le5!tBVtGoxr~3l!?M0x@bs zzF2ShN45gGq5gK4SXy2dbX$UlF(>w8b#x-={J;jicV@nR4Z|(>-Zgo}c%Q_u#_=o2 zXve+fWM4BBiq{fcBxhNb3*3ZM_7Zjt9=3gGXo$B5diWJ4NZ!o^8++p5z^|>NQ|^E2 z{EAufExWw*5-@C`g~NbP(O zql}Eq+1kfMnBS2dl2B4emWc9hZb18}rt@4=jMH>YHrR=qn;S~Z>D}nvq+8~w8Ea>- zgW;K(SLpEm)R650U7^hP(1r-$%tlRbyS@_&wHab>nl9;00-8o>YHF%t^L7FvaV$Kh zo)mTR#Mqem#n$9YG@PNYENq(qt{hQ02`kUV_}TKn_kc^xW$u-P$I3$6ODv=9^jGZ` z?VS|3h>I@`4GbWHasUs-oTkD7*c2FY@LSZFTl8UlzS^Jlo;Uyb`LWOCe$RgJ&1o=b zvZ)Q+6+$+Y&~YKNsi%N;O|}I3uRi22ReHqd7ZvlYxVtNkR?Em}M#66WKm0EwsMTs} z3dCR;fXSmrkJ?5HboPJQW}5%VlGCn*>mf>E1w}=08yo1x=4PYgQ@#`c@4Z7q%4h~K zJL_L;Jpj&UYZ?QwaHAvcK;2pP@8vKfsviey(XkH>BB3b$Mcm>r3ypHz( zTO>JB%vIx}Iwk-aZ~3djdbUJU+-ypgnwmN~F%cIM04j*?foFavGkNu9?xC65`};0R zuC8TI63CP6!M!~%hm4BujRVHHaxw%kZtwbH*I_2pPtVvm{&;gjfeahKdE!$}s|v^W z@7`r?GL!`pL+G!K3e(lqQOnB9*R>TkuJsd-cdR!T79hB| zxalvNPHQ^H#;5_?Ie6(#l}{+L!&!^;4m!OyWbiOr9zh+qT%S8QPS+5-H0|SDAEdUT zjDmrbIGw0mcM+^_lIEq65C{d(&TTl!F><@&yZ3Hw@J&!>XXjDvt>&U<)dXp2X~|5D zJb?X3NYL-LT-=)rJT;D&ledaJXNE5KBt{{TlvFPWI*H>HLINR7RwS;kCD__zAZ{H? zS!FtHeSP>ZU%s3@X$`u10NT#}pyVzD0!5%gK(Fce`IGYEWE%@&WoxT&FBSadHbKd9T!R~lQEfbRjP?#H@E$3^+Cf3&6<3$Ex!ylD7Wn@|=c%laO zg;2+L8Z%5Z+8P=gE&GzW(bO+CHdfQvm^ka|v1=F(ar?+8wd#_!l$4YREL;-L=@L!e zg{>_jNbkUaA{>s5E%(sTv8ek&D4VB0cgNLq)tEt@D|=2(&iRWG9M`LJucm`<{60%D zJbUeg0(;byR&S3o-v7fL7w%E^ml!ujGE2T#02MdZFdxr&KTDIh!I2UhV{@WZ8KiQ^ z$&rVrXE+;Py4)D+)W+G_SyoX|(Y#leE8sRwCsPhS>z`7|=;?3U3&77|y+w;rEPi-m zVq&~}e4(YKrQN$T!QR|}l!C|p^F7sau{{C;p{1n-?5V1z=6TbTkraKe&KDJh4xp6BWUMSm})KqjM zmseAZ>mD-oS-i!5`*y+MJcj3AWisqVfP0~U8ZRs_bAps23}wRuWStyzz+5+3Cy-6# z*m93lRVhq-|4?fU*4{rISJTj-T^Y(0xIEu1%^9V{0!qH+n1}`nzPUMGOd0g|4{dy= zahU3Eu17~CvAl{(%-q%3!~~&bPXe{7nwn^kbH40lY0J4P9w{j;NCJ2P z56!rxbl@iamKZL?rXQ7ktY>?)HVp1Qa?T$w24oxHc;k(=)eaLDKs+f0a@zp5G8cRk z2&k#rW>^4xjS>gXW5l}|o?{7BBs?9@wBrK+wjwy~eUA{`5=0Pr$P zaBU@Nu_Qsdqi>yEUAMr#x-?|3ta>|8P2lH6faINxyDi_@*yF5!z%WA6gl$jG3KKp_ zdTmhEI88?myuTmk?yqF^g@^@;ew9o}dyh>}kldy9O04Hm?-IQfVrzRF z%^W_+1Wx^k61uIZs2Cj=*J0?=90B}*f@yXk@vta-v%uO1tvA<_CV|IKicFdrbn?_w zKxTY|siFYcT>^R0Q*2oK%1vCQK$oxSe4|jm$wv?!1}{mtlI0s1JIzGN@ zvfT2MbJQatC1OfaQc@yT8781?x}}Qbnvcf~Q6lyf89qz+1%~X&Z?rnF*LT~!`yD?=uYVZsusT0l7x&)zFz7-q@&p{~BX#xew|5QATwKtnk}T|p^6cjt92gKi z$yWTRS4hsvn%H`Cwfo@~!ZUTa1KTdRNk&rAFjgUMDU4L6tv89Y9JmSK+Cucn;eEgR zkXrisc5j7a2nh&)9Ewr2-~c zo7Rcu9|#0dBq{5?f3IRp1&{sRY20mS<{KSllJ=tl?OeacF>SP%EbWd;$L~3BiuYe6 zy?4x32h)AFt0&AXIoDZflT?tGE>Q(TTyu@y0>J1%=O6mP?31HnZVz#l6cu$|UtR2v zs0pASsShcIK=?xUaajXGcrtw2^GnINg`OsZq%V?XkwVWpm4@aLhDQTa~ky%=MWiTuCXGh6_I4X4P)>?agiIzPiOH?HB-- z%LREE$yKBr9+pDp7JR#0bFIO$=)~XPW=|4@rNJEmV7eOU84Jj7GquKkY00$|^b7y`XxNls%$<0Z$e40l z3R`Py^Y(CjfW9cL1ZArjgb)IJR$N$^a&d7{NrEQnlTaTFC?+6Ta<5f8qW`mB-ZTn3g2ktCD~LtHXIwugp>!hH73!bW*2pipLLWQIz>`H{Uz z3Rz5NZU;7O3$e7d<%oDMHo#ULQxxw64$jjUOK)qM*mxtJMVbMS!aDaq1X5B`M;4_C>VnC?82=5Vm3F~^LSMCupl z<^axVVq%gr%E=*2Kq+WTVG2Al3=E9deO4TvO7IZ+h!~V`jDTqi5zznS4IDWnfO&`b zBu<#tyWaw)2Ztvd&fL$3=zOOVaJ=yZs42bNl4v|wC5+q>>Sk%l2@E2b`|n5OlAIis z1e6bN4AVgo`uO^~E!uG;rOnRs2HH~GoJ_j_kfEvP=MZXy4SPe~fJVv`U>AZ^5>h^F zP3yexJkyr1^%1@k9o;X4IP&G$2VGhGOw|fvLk~S~LY|#$PdDGQeh3e=eQS<+6b#)* zJD#{KK5Y+;$qfW5iILa+TqUKYQ6PZoo3&(G^C%}2BrsFO2#79rg-E(GPL56Mar2Rh zw9e}4>YNfWMR>5#Ur=&O3kxCH3QZ{3Lo#VXvba)gbH?q_qyKULjYir0>+}q{rij{CYxqj z;83NIlV88e?RxK!CE3u>LZt_|SVZi|boo-S!4b>E5h63@RHQ%&02%b?2X)e)$~qWE zRcrotK&y|Bk10IiR)6mZA*z~@lK}O>AyarMTK)R>%D@0IF!TPi0vV`%u(PuRObkUs z!}oA-`FnB1l#h40SOD4rv)Z>%!3u(&ajZ4q6xIJ~_P-B|f|bGa>H53Cj0J2cOg{T{ znJ~w9%M*vT6J3{`nff!};O>p;Xm~j@3R+<6@;xc-5$id$mL5k|WOU7~K;F%(NClj^ z0?wv3Nu6X}u?8C!LHYm|2wZlovNae+!w?q>_$Cn0x80t?-}WqZ{Ju`>%*e>#y?ZyM zjvw4>-yG93U1jUfpDq|k+(JC0zb!>$V*tXm|vAq@gil`Y`!g6x08~w+q8A;^i!~`ucmVqv{(kg-?cG6WYe<+YHZj2TFXDLM1 z#jE=jeVD^UiM?@GM)djMm-bvsXnA}*F%uKh*&ahubP?ZwBR~wx-SG-?axX{kk{b{e z5>RGc?Vs@0_b_Kl@M|RSo1?O8wP)zNPfs5CxOZmN+W8$I(h%EFBKA_x9SI6 z@ZZ0GKLU{Xt8xRDCKLETmydezg9Il4Yn!1j9cUT}<^vDa)k!k%vyUL3MBziEP9`nO zDl3RyA}`NN89Xm{)`Y1%UpPArWQO5LeLeBxNur@-sutR0sTZM~|Dbv*kAQqWdX2 zewSoru>n=LTv*zYj7DxNelM)gz|^vwYx3n45a>~64Ft~8!paKH{~k@vawMr(q9FPI zco1Ob0D%AifIvQJ;{!ClybK7Bit=vBLq#`$jcXN-_4f8YR8YW@ij9Tc7NsVg3!FBs zFBJ$%c-83?fw{iE4yX!!sb+Ll*|{JRxT_~@x9@a@Zci_!Ws(|JGt@j6l{!2ucfEMw zLq!7sAp5Xu6+W6i7Nw%1f)6osf6j+9)7)I2`dE#SMdk7KficI%;6_I-dZGqvs?*=} zmf8u@GcY~+*{|x3pMG1eL`0lWn;!z3C8r-f22RP zyM&Uxud!W((cItJ>k~)5e>Ks3KfU1hoUM#h5^*%yV>K6vhQ&2Cy)m!04rVX5!T1U2 zs7Qwj+(Nx#c^jM0Ts_c+)Ul{O&}-450mBb_H#c5lyoZ%xSnrsbG#SW3s_Mjbd8z(0 zO1e^0RV4#7)O*es7s7_@Yq-RM{QC6^kANW5=-x!#+*UX~B^teb{QNBDTLPuWlfM6X z{rwLhxl9l<&`$t!R0u_T;9-&dys5RbTP>^{Eidl&oxQa!N|nmsi0621u&i zfc$Nf(xwA6ls7c{uTE7vpe4gSqq>l?GMLn1zp$&TYg>l2f0a6~6o7A_p|f=PfCcJ$ z6Z{T*0@xW@z&2(LJR%2vA`WD-&W-ci_wVH%tAp1cWFSA!cfb0r7z}N&KEs0iAJZXr zJWSwLNwVg#io0fY2PQI0MqXWi1;GJyFXZ+6_i;dmp-l%dF+B&?f6t9%hyEk; z+hniz69>0}nu6=s(4ranzpil>DbmmdTuTkn6iEC?#9UszLtP20N$YA5+DI#18y{A4`_ER zyH>va(^Iq>Y-(=a2Tq&!cuYb94g`!#8Y{0CJEF*GXl&|`Tq6ZOWrOk0e6`4Np1^ozi=W=jhnSXtfPmL;-i$7nio64d z+CDvPxVv0x+(;w&Vk;ysFK@0o`snL46FFR2*?TUC&d}Yxf~$wSfwlDi@Qx1ONS9gsvN`)-U1hNr5BA;1vsqilREaMBXg)e*s#U BG;sg` literal 8600 zcmch7byQT{+x8IBJ(P5#fH;6kNC=LE4yjU7Ln3@Ihj0+Q0* zlHZ=^dEd2u@A}?bd@cBo!1Rd{9CC&Im(^650D*8oRN(T@JkmC&y`Q~vy(HW@CL%Mc6%>8Tkwu(~ ziWeFj&RJ}>w{BOQ{3ZEkbYgQ+Z)l@3F4czN$vLuf`mXNu4`KZ;MqMz2wgnN&2v$uF zy4ziS&iM3oVib4-+mvS7<^%*DmKk-8#>*Zt`>=uKztZov#Rl9nwvXe9136monW71} zm16GvDdj`TjGD=ueH3pUPIs8D=hf#f>GUeG@;^jSK)G@*VTdqgS~4gX%{5GL82lTI zlS}h|HgT@6tWdQzXNQEb#P$wPuBN_!KfAhm^Vb`_*hAfc=Cxf>)c<RbBwl97y86b=oAR!%ci*R`;y~Eg*oer;f*76w|@&F9Q^$|CL&_D zt2-r}lwk`8awAa=IoHaCPdLUUF3zB$qT;qo9Ad@?x75(m3OqV;GiX{vMKiNMeD#Xs zwG{u#YqcA<$lE%iW9QaJcEJlDy?RwNycrRQR4^wNnzVwySt~J8>JvDJJf_wc;#>m4-xL; zCr_L<#zoR(eVMGQnIh?L1=iGv2L`QcXfHW9e$C2~f>0gyKf5V{2qRFT!Z&F4xlTyQ z21g<}t*xzpOiU2_`}<4pwBUus#u7qk>FAKV^X!g)zL%XvrApj)$9{r968Ka_Jy_D! zP&4OC;^Q?AnV#w|soo|e?EXnMW;(xq+a-*Eg9!JNQB%Z#+!f31#vR<8+}z!bssZdO z$;->@*4VQ)UmU&In5>~JDk}1reM6X~RRpu_Weg6UYFQg0q+&B7*BMi*t#?z%|JZ?T zpwS`~7Z(pkp_rw&wx%1Xsj0L5(-RVcQpBuUJ39#l5=oR|SXSK|^h*_ee5B9M&*R>` z!>`t#US^!0o*u7t;#PS0kTbdGFxkL;D;16YFr!oS42)*XCj{ns3Cc z`e#=LbFl#+B_#zWVcCm*bB&aKc2J2uq^8FG>tWRN=TXe`H0$o}E`fu-jkR^_&-^Fx zZ{G$Ivind$${ptvN?z1ZaB^~vH@M#(Ezt|g^uL@~IkMd?4LF#5@m6s%q8 zi<%>;{sPt1o!w6Mh2pa2)`Qid!{dgjy6w}$^}*56P|zbMD+|Kzcfu)qaWEvi+|$G2 z>>)F!sjnXa8gY1ed23zQ=|NvJx!8z^ z2&0-I^$h9jkMn7Sg@qN5cC0W4gJ%1h&5kd0q z-MiUNc7M2z4#TfszkWt`amGhIf|K0csJBEp-`*NPxx9|PwaddMF}XVl9L(M*4AK7^YDu=fq}Tv`@Lcw$J6fAQ0UP1QH5FOd`C1hTEdCT+Qx>QAR#uE z($vgs_ZuNQ$L-scAflzgN%7h&)-FCg8yy`zs&8z7VuCmNr{|phe5dp9_Qqf^c(-od z(kLF;+1qQ)jg^h~@ZrXd#K*;rV`lwnk{^nS22m>7+VsCY7RWcI>Z)|&7$@1FQ0&~W zvEzSrHbc$I3PKme*r`i$xB1F-g(jC7aQUM7?%lg+8SgLO{3IbgUiXX#G(4hM@6%^z zXM?#t($l+6LP|RfI{{P9h0l8tH)N|a$II+?2jKBw!c0TlhO+X=$DRepY32M z%{pUPHEz&i&oDSs@o{lKmVN37geI^tU}zXk${@T#DDLay1GD%}8ypfMpF21U0$alG zT;lxV;xHMu&>5Rte<;7Yw$|!0AcF!UKn&L#85zB_wPpF)j2+MyhZzd~!sh+xlkVWd zl4hd;)&d#40+~$d*49=kHny0er-eE7^-P|go?Qb2ox#LVDrV+LFykAvBvr?qUNug= z!6h*jBC)TA3OM=r2zrulqW}cxOjN^vgD{v7-ysyuk_aW zX+={LD$1^_rbRsrT$!hl`%}`7)VXl0nzZZp?{L0oZD6l-~GNO=@Rn=do5)82A4DcXdv;V59}9Le*()_@>Y4YGJL%E?0)%>8lc>CMJ-yxzeub-46|CrSWcg%>e*i z!si4AAo1zvni{j!pZOkVn>DDW$JZz5K_6IWK+((|D;)=9jyZUFuT@$0iB9fz-Gd=)19-u3vIk$5 z`(OF_98Fjek(0LrTo$qWb*G9w*%lAeMxy83GEA#qO7DHN@eQnQ_A`O_q@)h2Y*|J= z?UtY51wU6`V^fy{Yn`!C|({2UV~4D4WR&}o;`auJECLUe*Skq-#&bU z2!sDc(nN|eJw1KA!i>;u^H)gzljQjzO4$HF!s>5TicwtW`yy>3rE4{i(SEl5l@c%z zpUc0?Xc#hZ_*Q(na~P}wg){Wc+Rma@*Tq4^|~ z%C3UK!sFBHS0d;Fz{DMz4(YK||GS(?DRBwsp|grs#Ctjx&?E*y7;I8QKtd-AMUB5P zbvXR~{rk?*qyx9GFa=1;KAET&D%0%i6~EKd02O@olU&Wc?C;9zYB*SJ{GK39x8mMe zS$Sc32;}zdGTG!iUurw@dDs6eE}9G#s4AA)dEYgfR1Tu?N()j^F)?tukTB)2 zlEF9(JYNe5pq|Ky_}iLOG4&P`g-(i~fyF0{=x4~o+7Y67R7+o9--=;MgZs8393Fh> z3c$#+KaH53o&Ccz0c>1Cvb2jqMxm6GljGszQz+FwF8cH556A|iR@93bcJ2=1+E4%C zvF;zfCLsr{tCdw%A8Be*W8viH(*ot6go7Gnl88lTNg610ZfPmhB-uphSTCk8qHnxp zJSw@#Sh4&GY-)NMV6n-8@VH@v8>g_a+Lc#N(yj4&SHbT+Jr$^MI38R{@$)NP9fMPR z;Y*96U{wIh1km|%>k$FXB<4HXVjLWhkdP3}CppIF&!6j7m|pX{ST|lc{(2?DE+YR$ z1*U0cmRsaN2ZkQdWQ&VY7dJQhJ4~vXnTB8e4AGJ<1#!vA{6v8qU4NYAi$_QT zIf!cPr)dlTeL1q)c(?W2zNA1gmwM6pZ&v>pZO2;reRnmWhZwM$MRSdu4g{IwI{9Bw zfWe%cAWPRe|9Sj%dw*|F!O+mK%ffomW^GDO%B|+xPxGQ*VZh_xK^gfIvjS0RCY^ zpw{oA{_^5tb8^Pb?C#a+*jEx`zrXo7GvnytZDqGw|^hug~i2S8Sf*qy5@>3#u)HdOLw=* zK|LfcJw4k0az7K?Y91FiEnb(_TFs58lEIP^%D})d<$rZ?V5u--gEdJ z1&A&)84Z_`$so-0WDeyt-$wk})m3k=+;ZgCuZS@tZ*qs`zkGmH=h>r_Md`^C?px7J zMjp@jvjdoH8m;7YhSQTnTbk|5W~yx%hOO@!`Cqsez2JPC)()}+Uj=WoylA1I&Zk-9 zt5XIbn|mDW!#L#82faMP0s@r>*^hDuU+yf+U0nx#7KKc5jCG|E0h7s6Nq%X;XaL64 zjEr56vck7GRM23Bb1Ex;@>$VjVwt$_ar!sHH`J1%Dy?np?bienzb{$RxVyQTg8kO5 zwobgnLBe0B@*r`i{CY!5eT%E?Chy^2yS8g%!iC2}3eDGg^;9feP*AYKecPz-85jP zmHKZ(!;1!ZlC+eR2=Ceb3V!#FHj0sRwJfshqsnC&=9d-C_Ea4d>}*<_(j7=QrtQ5I z&DiLu>2iNMz`l^-QYt-}e_Q?cy7_fKzoaHK3`cNyq9LP_2`9g*ZLm}sKro-HApG7p zQ=68f!qQUmn>TM_*$eH&}vgqePI_*Cg)1oF>ZI^9@e~t7ADmjz*xX^-m~LPgklv3n zm!X;vC?HD;cyI__X?XWF2QAY~C9pk!%TF8M=mONK&TC6!2956T_+?`ulwG zO%>l^6$glU(8R^Xr7(OSKb$4{d1!dP70vSE;?8vmIL15?iL1MHblTCOXi)2jv%l0Q zc2*I#kunmhG9VUymnm8=)@8Aax;Op)SYCGw+&t~y3+m}o7NZL+GG4Trnwpj}1Ykve z!Ek#0i-rKB1PQMwl+dVoy(>|VN=RYKvU}bacr(Pa=2M496PoaDWG>54b-25pLm+VTb zFeeB1&6_ujBCShm>0KWuroMid>*x9O@uPx?NmiF3b$nk)x1$w}af4el)?B^Ez}K>4 z`OnALD;`n2RI-FQ=>*+UZI_+IE;3tDMCd+Z9w>yCfq?-sLu$!;sbo--6|wO3^>q>Z zDG4#xM~k(#_Yk3?3AOq8*VNS181GuYyMCYB8F7!9uVvPXW~R{-Dl02%xe?aoH1Qfu z=EhesUaU$zc%^&6RDKcP^JnMq@R7Vc&NG3LfI|RufREelIhAhu00+LLEyNPOs8-wkEkdOVC;DPJfo|x0WGW0 z(1FLa1AX)l`dPQ6v;pek=34)FKR>@_M#Dq6?}RGInFBm-PEPVxR)rxAT+U=}bbNSyJeI&Yk2MS-{&9tvkLOkb zCNQRzm0FUZfi&ULk|vhvZUopIJ~sxw3z-BD5c!~nAy&}aBN!_BWY zL=jOk3FDvzSXVTzkIZNOt7USMS={0GyAOg~XO_uGT*l_%-Gc+Ti;EbrM|jsFy8I^W z(2QPu6;)MY9Z}aYPT=uzuCHGc6@0?6v620|2t~ZVLp(S*cxnd(8rN^1eN{w;v|ej} z*hXw$8P23}6A)ckvqN2aH`p&VGn4%eQ(r*mmv-z)N~p}3Hd=y&W~!8E+yFG4cO#-p zcET0S_+CXCxUmL+XtsU|+ek1(5&v-Kt5C=`!IlxGQQgyG;dpp>nudnAoOIwvom&|E z6)^kxw(wfF&G&7r)D!F|B5b1c_VmPQ2@srs4$exz(Ni)B5)u-OI%P(QwP3pQR7l)| z<|f%2H;4cWrg4Xx8?*2sw$fC5wwsRUfd#5n12@T-e|v(LebCN_P)qG0EP>?|gzV=dzkrKO4WbX=+JFdUd! z2N~8909po{Qt2`HHC8kO@9&ZVMf3LGK7ZS{{f_Sgvm~puR2AEa2D8Wjz%d9UUSOhe zV-t3_-Wowkh<5QKTU(*E_4NwhGdH3@h`PEuptHd2lNU~YGR+nL+zbtQ`SRrhd;3p* z^m*FF*C4=A$OWLBENrL%GOdq~Po8Z;*GEa8|1b|2jWyq-NJ`c&w_by^s6Xb7pcjVQ zIl`P>T;kuoYg4}gSClwMgNhdMugS?d_d@};$y(|+`T34NS7-BvILklN#@bDV?dp^P z7lST8KVPfniFq|xEntD-K&mRM=JJLP|JEg_q4XvuAOWB-@Z^i4`BTv0pE_Lg1%Al( z?KeaFMqFz>dbxuw!*@+UQ4%gwdv_R$|81j_%IqK9D_brHjAaKLYTzuc1qRI-Yg_Kf zxXS-4;yI6PZ!PzL82IEx;hzFMc%pG$?jcg2cIH<%@@FH8o&Gif1f1}06LMf{fqXP& zD?{DY|+{NEsRquC7H`Cz4Jr(5VCj@#e7`V%TQpd1JnwDEq zn%zx&%nMfa(U}Z2dD|QYE}YCQB0>#BNX<6CUNehcsyMgJzU5-`xu|te(A?HmOmHxM z?lBsRJ3wchjsg4qFXu=QJ*uWej0BQpc(maUrx#eX-2zcU!a0dwH__VaBzEp9%+KZyRh6ZLJoIL=< z?SN{Mv@FD+GpI;nNb1Oc`)|dr_0eQgHzTUO5v6KlBb9FTm)KpIGK-EoIU0un>Mi7WWY~`j<8&r3ag!BS|usmeCy*l{w zXCRPZ>T-;>C6<7ZSq@~}T6#ru7>VQx1de8nuXc?;u&}*-1#kk$neSoEHz22L9p`Xm zPiFBTD~FrCl@>i3Iy$KFaor1X@zT=L)30Y|XA1zPPCDf-2OatVJT0VHWr!@lf*!Vt zK&g57`4@@`GuEna0k>=5jFu$i+on5B)D*Mx@li$tW%P&;C%1* zKf6yifQ(Qliu=ie1qF3Ds7u3hK-!AG!N>>aAqge^zncDAXu{zgUHDLkubaDG&eS=7=l89HmDpr_Ttkfv;LXR353pKPi|5{vTbDT!8=p diff --git a/docs/images/chapters/introduction/8d158a13e9a86969b99c64057644cbc6.png b/docs/images/chapters/introduction/8d158a13e9a86969b99c64057644cbc6.png index 06300409b5c9b7af31cb8a74c6cfb29dcf551354..ba7e5b85afb8876a6bc762b0c998a0395a9ea9fa 100644 GIT binary patch literal 9924 zcmaia1yoku*7bvQcXtR1N=hmqA>~5~D4l||fP{d6AT1)I(j_9@QqqkIFDfbBNQ2V( zuk*h5-v55%8{_|nI1ZfWJm>7Q*V=Q=HP?>N(oiNQU?4!DP{gV#iaID11|{+z4+qXr z=gro@0mt&TvLfmn`Ipg<6Nf^vp;Q&+biGs7Cw%YedYs{I?Xz8IWy_fN(6Z5@B&Tn$ z&rVGKC}(Pan`NBe;a2t-&Fa)R1{rlPGs~A@m=l!e~Npb|PUJZ$fiRqO|*u59C%T&N3;Ql`@55*_{ ziS_@#D{|Shf)Qe}pPUV*F>7m%53Vu1e*0FQ4C~|5=ke!{43N&rFcGbDR&TGPVxY%U%b=v#ZU5JB?>_E*Kt{opt`<`J*Wrqf~oabBs zc7cq7T9<={^j%U?Qg3hXgvzXjBGu16g|@3cGBPsX>?<$v>%R*)7vZkH%&*T-?c~y@ zBWZejWm`C``-0ki6D@1&o|{fie*_d9986+kg&MwBsb9Q!k)5BvcrMAYW4zWqHYsV8 zy^M&dgg z6O7lrHXN(^w&%JqPzCx$@_v3d2nY$Cj@&U0atVj5q@&)ve$6H-N`3$SeeS9Ky*(ve zT`IpPPm&wiKmerdPa&&!q zaS^A|khsz?TGD;7`Ri9g_r)GN_ojF6IC6FKl}hk~0%=XXG4S#6uWsBmG$arZ5WovU zrE1u^F?xA<9UL4?pYBh&RB5)km|nod)YjFt{>#Gf**-eF&{~jgde(BXM5n)wAc)QU z?Bp;()Pd$~uezzS#%zCMLfgO~A~KR7E?#r@$=c6~E;@VS{j!CNmbbla z|L7=mq9(xp{pXL6Wjh|KIV(mca(dc)E?f0>Yg=1vN(xaTI&^F6AYHC zp6x)LNx0{cVkM$+F)0KWOca>)2S~^N7U(JHd_90Mjo9gj14h(`E{X}-*;shLLSd; z)%pCXQolD|fRB$9%aa{#)J;!M&k6?O{k$ey9c}HWSBC;Sq>hFlq;;jwCDIliTt1(Mk81DiWDt;*8Ve<6 zlqwYKgDB|UxnrfBAi;~g|Bla-sNjJVd|3HvqN1Wilz^Fqog#5kfS7!Gckf;wZ2f-K zB7w{x!j{_ud70dzq7XK`6ER$4%(uJu`go;MNj5e%#0rLAHVn=&lmay959OVo###oX z8pOm)D(YVn5D;HwwX$snV3Rj6pl|wI`_w|-I;6Z@I4yDfUuP4yaSAJ};4zu+-(#hv zrCC<9<0B*Z^++o8GcoepR?j>sa(w;zb%s$j!0A61{QERq_|M(!d4gL4U%jH}UT`Zj zip2`W*S@~|NDBV?i>?u#JD?FSU>4fc)HLFxg46aaHpprgpttyJr^aZ{Khq?T4U#v z(^$iRRFfEo=5m(R%HS6B=D~uyOba^|R72II(G(R3()5_C@p-Uo{ zE-Wkvq0vB@2I29N9}tQ#7o^Mh{(y{%ip!JlRU1U^roX?8^QUEQ|2V4)2us-ZrFHa* zPhIh3eANygH`lr^33_jt!`QjtY1rO2KZN0!h$YvYC}K;-#B>Gj(=KXm5VPC!Xr>h# zXd>fwJb(1w0z;`s4bBxH_QW4QevmORj5^8!h!kTGnl6$huJ0`O(}_7Iz)RD>Q_-@g zJCySB@(_Y@cnh0vtGg0j8%!>ZRY3gKq5V0UoPvTcIn9iKM1|WTD0wpU3*e1^AlG(( z`C^&y5y=1?NRT)lO^z#9&};f!(Qqczed#ZtUs7q`J*qltRmj6ic1}*rEae!QI^xg& z`BN#H;$MHhtE!?gFfgE#bbCKiq{0G0p#J&ur`9E^t=(PfI_kf_GCStd)N!yq*Wuj8 z51D1SJ>fMagzkaavVRK0Drq{4{Ps<&%&3fzi1HoH`|NCsonPO9k)uY6MzA6NgfEyR zgYYT2!zV%~AT)^?!>hnjErMmJ;lTUSLn7=~uM*42$-Vq+?c~ot99Y!$_w1qpcbb{K z#w~_y-H1n|>U-GjTg?ThlU*8H-mAv7J9iYrSdtEA!dSmmyO`~)4r{LIuOSH)(I+V*8SGx_b#@Ki;OE~WK^{R2*`&5X8IFrOf6kJjb54sBCDKIGL z0?^U)kdX^aU0sJD`-M{9eP?vK40CaDv1PSoH68c-PJTgw>(;a!oHc>LFtj~OnT$8{ zkB|$-VP|8z`L9!^Iyw|Kf2CaZWb}-&kk?r6NffE^I~2CX8-w+gGP<`uCgi?|uBq9& z+hTZv|Em9oeWirY&ZUjoC6+QHS3Dw0H^b|mYd0S~y7tLyli!n%L{8tp;BrD8&Xw4# z+`__kv-6|7^?&}lt}hVu8i)p!Am)^G)v&?{S^g>IlYgaN7xF*x#=^xlGKq3%9-6sN zBFPMzP*e7lid1^%6V(q#blVpOhI&a-mhIBKM2YVzGW1Qg4cI{ zqvIvJg17z;40!o##b!+}NM#OCK*Qv9)M1c9Ib~K|B)Nn3M9fmUYx+zSv@UEpTKf8- z>toe(4f~Una8-v0r5i!Am{aTg=QwgYx=ZjnQMTOkTYxg90q6dHC)-_?)wAD#SxGiw zCAYVc%05q%_Klo~RIo~Tsc?LDycwIE90jZgbK?!8x;HpDc(2y&BF&Y1C~e#(Wmb`s+T zAL_ikJRk*^olp7&A^HWkG&D#cfu@(1cyz*kxC1yiPBk(E#6vdZv*ms(Js5Kw1{>P z$NBs!Ztg-JyY`pd+@|w~+w@ApICyaT^-GjC$Bp6E(*t>TEUWqOefQSn9zT`@a=+-w zrx9Yo;d*#@hzc}De9Le0<<`6F^Z}z3h!#Tq z%KrYbMtzfU)1=O`=#I}TUJws2Uk1W75;0s;S@TSJvAVlDY_Y*mx*JgD6tdgH^KmZ? zt-s$CWIkxCvg|!}#|4BC%ug4@(P7ATEWhb!iq-f0(7*@?MXlZxK6Tw)(WByf^DAKS z`DdrdH)^QBm>A7E-?v}3tEx>+O<81)esO}5{8{Ky>G+cs!G^C)5&4k7i@nL)yTfLf zC}EqPn?(+P1g7jwOk!Q1DX%V&0WR;t*5us8KOErdy|!0YtL zgL-pvAvD*Aj~}TZJRLB;sNQ68&MQ}l+-jLRc&T5}UX`~{!ONGvp|w8qwDlz!mw@3j z%Qpn@X=Y|-@5;i`lGW!|7rj@Ds-KtGU1K;nJ{BJ>&dA94?y$VPti}5VegM7%e>(I_ z2`Ba(q?eSqM0w~J8HOAjcvI2PRGs4vHZ!`^Kh0DKlRC}fZtryErq~9UMarR(rT|%l z%P-&gU_M#@cndVSswzHGX))1-zqU=Yt179k7KcRpR_QQovsq-?z))e^7aAWQzb*Pk zZP2lN-!$OVgN_L2icuBO<;$0Q_k{G6$yPQt2Fx^*^Eu z^w+NkctL?M32~L`gLe7ZQlK&w&}ekQan%CI&F!A2S2R+)eS2nM0b7nN zqkp+)Qj_StEM-;uQco~7*7ZCj=MFKs_MI;v!Q#p%gg&_u�ntz!-8eg# zThH%*o0Po1&0 z1f>9|@2f)?E6$Tu`^oXteEK)}^F{7tg67w-p$NQVg*4FPkx+(x4PoQG-K0n=;G&Z%z5&^C1 zF79#U({z4T_i(xyVkV4%h2tqm?+Zu(9!> z64f7u2}y)`aPbHLN!JPt0wuD1h5g|0ZBwjAsj zWnqcAYo+*tMLH6sjd@2TbuWzo4Hkw#!w$s}@Uxwqj)0h$zK%nN^GIpSyY-6l@~u&a zl9_1n$!T$MicY~Y`BS=BsOiQZFQp=RU*FK6?tHwQlRgXi)Zx3?Fq|i6x#{NSR{G#O z$BpxoKfO3Zu^$D|qnR^wu^>v23Tfh#hlA@etPQS<7cEwYij47lFGxyC9xTb8M}TD@ zsIrD?Ye&n+32|s^YbVGC_(ySSGH&c2t(Lgs%Mqr^JQ=0qs-Eyn?#=0oTNgr)Y8e?t zfmXEKoNVY6;-KWwxOVLD8BPG$AZ0i6c;1|Z|A|DJ?}j7-DwN!4vN~pA%C`5|J`Ix+ z+r^-z;d^gKKY8%6_^t`Ph!E_LOT9k)Hesgb=gZGuy=;#h7_bjg$BVEZJ`kpBnIIYZVLTE9E3Di zn!zI|%747`OB-mJ+~m`vGVS;ms$^J&W=$+e+ShG1Mqyy$gilaZ2KSd>_#Xbg&m`kV z2MA3kVng)yih-r^Uf<{$f=l7ycvj4pnMwv$^z`*DA()pgUv5bfb(o%;>!0=4kv1*H zFzT`!4s?=Ing<`o{+d!d%_%i4tyT9+_D(Z0<)g!&?EWO}d6@>YRa^6f+K zS{z5_oVuQ}N6*7WdHMKiA7~P;xP0vF=rCwdp->IJ^GwyZEBxNqi3#T660^;3eWl_c zSMN3XOCvA})mE}`_1m>Lg|LW9E5*XW5;Iw#kKeS5p`bOdn0%r-J3AYxa!OX$tW)(P zq30$ZCNHyj9$pE?(Kg_n)M=n7)Cig~hggVy0t0G0zUk!F+S^=A3EhN&81kQWUIMzj zZ}hAb9cN(9yKq_k0&hjpL<|Lw{FN>n1HzOZ+zkNflhNU)l8Omn2Le*in?2dptME{4 z9{}Y|#45uCVxuMf{KU3wY+2W*fdzaWe$S2Zh%OdiA~0i^u-B&7q|gk-oWedk58_tC z!<{z9YY)0)&z=IRUhkC-5@&rBZN!uHv0s3jN| zyajqhvrYORJB6JMA3{9^yv3}hrY7hzc>|qx!vNlj@~2+HM=>SEFLwsgfg(;$Hv_t$ zxP1YLGdV+}{N?M{HoyZHYg=1Retr@V1EgQRd}()%l1iL!Q!EtXs2a$=O$$E6bO^0! z^Y4YP+pUBG&4cX<-Rr$SD{RZ``nj#|-=As0Wo`c|&_o2)UD+5%tjeCt#}SyoBETT; zrBtu2fb;!;1QA;j)5iIm=MVB5H*S>r?Ao8e;%!kNm43#P?nkjd>04rgHNWsR-W4EpZM2dQcouP*fLUhA zZ0}h!GXEBrD;egzew~pay3<>rt*fi+;HA?Y%1`gbl-_4`TMPH3Ni)JqZjYqq+xlsn zZt+)&WP*h2yZMdm6A;D+fCaIyUIjrxKknMa3^SZPcw6M9Rr%wDe+k@P8ce7z!29g} z{8>&{_w}~f+)$e6Tz61V&=#=RYaA z#PVDp9bBofNojGao(LUtt|LxNOgvbt=qvR<^?A-DNfF1b-U`HheT}|-7mSD?D2ZoB z%B-MHZiOKTd9L028)F;&YPSli*2&k#4y9bwM%|-(Da-b0K>vDB@iI^lxO)&20 z=mbU43e27y{^^E9gPaZmF`oe(%&GJo!K&@qC;`h~-!eU2NyFZEeXw|ixYr|h8%wlb z4iu2vmzFrPF7Bp;0nS-0_9RxFc!y0dEQEle3Z6PN5at&bXLHj1<)O+T0XG(b$P7~ib^DK8L$A10+3zILRDahUZn&$hZSgh6Qw7vTFb;8RhFq|qRVUHhDy(USXdDn6 zHU6hkAc%$>ry#^_;HN@5{`#JU+w>`$WO&)3`O8*UQ^sCZXZGur-R1tSH7B6ko}=@7 zU%L1fe2g)xUZ@WT?rv{`Z4gpkKw-OMi+0X>bmv0u@hGBxJ7C@c5lBVGs*7cZtQubA zc8Qv=j_hb8>ln~diPYP|TBLr>XDnnQ!;C@j-#T>U@U`t1r2ofo*LCK-^4shg**=+f_l;@xKR3gD@n{YOg1)F@*BZY- zvP<*EHWrA`a9HtK_4Q0J&s_kVuE*=ubD#Cyf%a&JC!a=b~OEbfyt7ZQ<7|qu!AY)~X0Tp9(b}u|72u+67l0w`o zttdUVsWo`>HV)Xpv6%r1dEJ6J;INOi174E z`s_11Fjni%1Hyi5I+%3yJqz3VoKs#}uX6Lz;o(f<@kZNne@<+CJZ`E6E(%gilq6?! zw6frHvK)B&@=#rYq_XKtBaT)I?&(4MZFkTjg9G+JW$<$uGWtyJoc{jp213{JU~2{z z;sve!z{3BC|GBDvWIquoOsC&}*xYECg~hvidx`BU8SE=Z-qDDc`99UZTrra>Ht?IG@oqLPy55IdzLk!r|iNN=J0U#Qx< zx)k{1lTU`Yd3bVuA{8xk=(tZ#Pou^4zJB@g@Z`Xi>AEKoGV#EX2)@o;I})<$yqRTp zV-0#M7m#XZrY-FHA}WF@;e9KYpdcCKU3g4P*9Z0Fo=E|A_ROEv3X_wQ3D@i>xYd)M zL9STa*f0d&2k8_Z9^MQgW-MYx+6@2n0px!W>Qbu4zHh}sSz1{|$Fo*(#D9%$L3#>^ z|0yCu3EeP-n>SIN(TrgbK&!S;VzE(i2KC-T0Q5qjTF?oC7*CG(sO$7NKD)07 z+T}V77k8RRL`Aj5$ex8GjMG7D_RN@H+>}7F?dYT8dW)b?OGgJbz^FVB2&|Q8DAGCL z!N3dUu8mr7@~$0^eW1k;9BA`^4}dU3^dh0tNGV2TWhF8TFt&%lowql%sTETrEG(ZM z9m?~*AplZQ1PA>E^YzeZM)7yRFV*92gux*xG;Qq!kU4mGc!D@zVHOZ6H8a~0rByJ9 zygw}2CVGw>wt~V!svZ-}w>|C)U#Jm16UOQf9ocxKB+|}OvVa))rF_6t)6l>Pv~llF ziSDIU7LKnNsB>E&Xgojj1whKm&BbdABkck*6td}|E6{%)zp^c5CC&drH|!kkSqcNK zp7h}}Ha2GHrDYIvr0aaa90kap1ro`!TFsgo3S#>{KXwk^7D4ie+CL#R4=H~VhDKl| zTsP~c>7(UXxkN-6VdvDS)t4p>$oCXVtmgjyJ8Sw83>FD0_xcazWzSDUz-QkAPHhEv zL>{+97z>?`wP#ULx3erxyJA?oKF9G$nuT)(TUdr!3n#jA%yP!uuw)u4y~gnfHU~H% zdja|m<+->+BQPIG6{i0QQ6VCt#BGC9G;peb^ew%JO$PK)WKXs*_tt&}gWtuNE&vL= z^HAQJ^2BJK%V&2b1aM7JS($N(kQw~D)!~vZ=y3rSJO@Ub$Ul^yAE)Ei$n|g3di3by z_MyCT3`+;ZPQomFG}>Zzer1Vpa7uZ`kKJA%6rbL;kqinQd2ZwwX(8_`_N_~+~b=!NjwyrKKKVPl<)I(p|bWtnp*Ki3d zXmLri?g&A2Zkuc4OLy}Gt(;ju_O6ceE|qIey|m>We?oHq2%=Re(7m+w5{0s0o`Dd9 zOYp&Tolo#-)M|?ePZ8i!C!&1y5fTEh9b9lYVL?P$&Im^*INpY%G#q^w|Dv7!6Cwq@ z5hk_%B|>nPY3%Imdea2jMQFz0hHbCFmoT?G^2=OVu_sxr76SV@I<)O*m4~CstMMgl zGw}2jyT(BLNQxOXUC}2K3>2)Qf;T=sE_L*fD-1mJyk$kj+lz!I{}0=C*dUA(nsshI zZ+mEetnR9)s>jgP+VV?E5~N%oCjGO-!+GhKIOxtVHH}ds56SB=IP&t-1wKAn z)+scozyW2c=;|tjGM{QRV!&2yNMfK}8xto$fdw*^xE}ceX$rv{LVps4ivDmnYRvvx z0|x5v9SxKJ>}U}5A|CniLnql^0y5y>m+zB_#E+46J@e7G#3@J zc|(o)l^;BKP;vU_&z~~u&lL5$Lq@2;(o#WhA0PQlEptHb&UFF;g6YjomlJpB9KTk( z2tUHY!byXoddl2!^ydX;?^<+I^$PUaWn@@FLPGRU9j$}dKKgAB2TG@Tma(jQC^kT& z^!TN7})U8TG9=B{ZA0As21gj9t;j^6-wEb+nrl1I=eSX#1W z(wM4rxEpuOBJ~IlBxtYc-8@}{lFtls`b7z{pa5f&l41yAW0R3-?3hd%`fu;EYk~&p zlGfDL$`$D=6zO9Mo}+UBVR_%=e|G>=5{l8A&?}5Oe)INiD`@T@HoBz?qWAM(pt%vw z1K&Opn)&q|QlJDc@$%YNoDV@iOG^8{w@I<30iBQ~7HS$eYlaP`sySWKl3)6#O!5C1N zNV)M?xV*gFu#hwEvzDITeZQCWc~U5(+e$`eKGA}2^JDqZqVpGIZ_`!pmAss1?d^(l z>DuB)y$9)e!{E>Y4040{MgZfM_wo6}h8Og=QCK-_>QMF<6inDqVOxO1(h4CoD&)iq z0w?LxjnkceeP)5SXS?@GC|ALJXTNkwP{VUC8Z$*8Td*AeUQ7Sltc6_NevYYXDsp{pipLPPrJ+=BX($%U Hn+N?L7(fSw literal 9857 zcmaia1yohvy8Qu_?gnWQ#Gs@kL_$C$M3Jt;!JtKHkWOh3L0UkhJES`Vkq$+qJ0%1p zeBbtW$Gi6*Z#*3i9GkV*T6=vp=loXCJyiu_0tNyU3PpTJQSLqpg++<{!^43m)Y&s1 z;RlY%T?IMRALMUp-G^utiWPN7PFlk$aecy7Lu39FXDcAsQ-H%;pF^&RK%E|6?w*8t z&?Oh;^=ES$6-p}ICBd<8GNr1Y=HDk&i|4sb*zCHnxOKR)^Lxfb8RU2n9E zoJ$iQ8+%>R#}|uXshe2WF5L{OS-ODb^M)3ch4za_FqGdLry^FKT>)D5d;kwMg>vsESJN zH+EmarQ~ua(i**+RZt+Wsv0)7?!fJjA4$VwR8NG$BT@B^iAn3}&CjEP|6`-(78e~K zS)^g&$jqik^7T+CTG~D7>6zzM!xlY$_EuK;?IJ=2d3i&tUv$FvLh_USa}!s}NMBlB zk(aa(7Z)G<`qixLU2*=MPcC^MHeDoLf63|S(3=X_I62MbxP>Z@u%M=vmd2)&iRVwu zAB2CnA*Z`E*&v0QY7M&c^y$;1M^9yagj5sb&Z3q^3sbrn2>h@gWGXM0F^jWva>}o! z7u4`82bLk{X=4*UW5UA0#VaT6CAG3@rEX*s?2XMG6-}F#(TYnEboTW0^s>d;_bN&( zEUc~$ZfaU0ArH$D+b|;#HEkqYP*CbduwbrnRr7;}gOH0u<+kE&G;Br&ER4uidzD zgH23~Vew0nth;;twvi+JA)@0$xx2eBjpQeMTfZ(K&{t_=G(Y?LHP!rNOH0c%ocG45 z$M0n>MsWusYisK`DUX}-p$x+dLX$7|nLIOZm6Vlfgtb4cw3@7cX}dPYPb`Hs41vUC zjKTPLdZNyrJ!||W@y61RI`M_iZ`1oLsHv%++1i@zxT&j?o0ymsJ?@EjsN}tN4LL;5 zOw<)_ZuP_Jg~4>gIu|r-blrM&W#!fO_Vdu+S^;2jG4rQSJ!4{G+6x`+?0l6uE%v_m zrz?;%FuZc!Szu#lPn%eWcvsigCrah^)Hp}kXQwgH*XMa~;8c6O-MwzDcmKXOOl@|| ztbvXrTUH$B!zU)bjYhxQwv~~==2S}#=}8c|QW#st#m2@I zulViVyJTnQwFT_QX=rHJIXL8I_frWV<`lr&Mn>X|_?=d=KSaEFlj$abX(#G56$qif zE`PY*vp!y__cD5Ld-IA`ft;!;iN3!6v(sZ|x7~h)u2DfkJiJ@ap7B*XZiZ1`(PX`O zvwp&AaA06-cTjD0_r_K{eqkx}o3zptd- zyVN%})@k#iQOFn~+wQ))I+mWEUfr<8r?C}VV(k%(B-5(kpHn&$4+9+}ZpE^4at4-I zFfSVU*aoB@K8)EG?D!7BY1eomy5T>6-cwiiVwQ9XobZ>1bh+$_-WGxML($hwSQgZ0t_w953?Ut0G3*!*VryS>rjy^lCoSBnjC1%-vv z^74qz;1Jyc)32fqP1O!vd%iRAA$avr0F0bFeUH3HNXw(|Sy)&IC&Pifu)V##HRVJ43U+S- z3wWW^m|>KxH!G`%2(1qeF+~lvPwQV75wzP|d1$x!gGp0UGrwYEC_|N8CQb7Ji+!2m@Jq=8v(RdF~U1~5=KR6(pN{V=K1gENr0Oe#%^kl!4f>`zXw zulEQke*>!wG%gQ*ASWYZg@Xx&7rd@N`C`1(_s*me-3O!h!CeDOpY#eSZ+M|)>`1Gs zs=&e`iAhOq)@`ZafE&|upQ7$WL);hXKRH615sXKD7xz=>B)Py)+mx29X@_)UD_bk2d_hz|@7lGD&|I-Hk>H8cE` zIc05a`HxRdV985ho*_~9V^M$N^Eh|!p2k1*PN9fo2q0lQw~QU4>A5*m6yMPW#saKvm&ZCb9brs(goG`g zSZC}C+Ti@bXFrGO^K<|@MDjP7>JB)@;Vdv!~oox^GiH?p|WMW8eZk9n{*LZ~$Wes&{ z6CeMx|G=_4JLGXu-bcMDxG4Rd9oYq6{7XN7$(x&BPD@XB_Qg&uE+)5WI1x_vICU-% zo%D0+X>EK2h&nS9$2yh=4KCDjg$q}c;9MRRfz$k)G4fz<;G%|>)&(vug*6A_W-ytt znOV@E<3%YrMNK+RoQ>`IZj|R#28T8`m6f~o`+KaS-1w-7l$6PseEJMX{l>6)?OG>( z+Mo@S<1*s!`FfQo6wmruziuP1!DHPQ!;onnb-$U;{*c`4LwfU)gZSf)>WT`Q_yR0F z9i8maq|zUC_}+cXM|~X?{`YwK`N35qN! z{N{JE0Vos~r;3b^{W$L=LtehE$p8oHR)hPA#t77Z`4Ep*Hn?LFRskTow`TQKWX_!v= zbIkR)H<{RidQ=|5m5humRl=Zu!CUwC?Q;M;#lG8^nPorc$gZgoS5i_E{Gmy97J(=k z8E0ACwy==%JcCP;mB0KKdT+aq)Y-4n{BYY*xoTKTck-tir7~j@XZOzEABXX#8y#W- zScLCU7ggfTmkdEcK~9@>`(%ua-?@+M|AO?@t;?7%U%p)8;qm|Qf$P<)S7jueckkZ) zSW|;J+@9~cz+o?RS4nB(&`McW_RJ5L)oUS4;xyq>r%nMsD3@6Z-`aWo{_O(ro!4nL zR3suY((CQ@CqQ~=#2q(UBQGcv*{_bM5>lY>{vvsFKxKGf;4J{{Ik~yyw6p<85j058 z$svGm^V;rIGU$OMCo!%d(zms4oZVgbT$ zE$zKYNJ~pAvE6xLtlBCv15>&x5x1;%fU=AFY1<6Pgz-+gUxsqL@aQB z%wVPpBae+S9cNN1gskiSMjLFN|M{HaRlN!#2L}g460n}E57jbyNsVOIyDuhZ)N{Z~3@5hXDaB*RJlO>VW`|4cx z9gf}=bnx}3Ln?-Nnt|#U7!uM7r8citQCCL?A8J3)Ds*QA8}L5&*u6)O9{EkwIIh+e zRQPlYzANaR?Fh$ARNDuJhX<6Fmj2kkTaAfZY8uYfDt6p7M82rm7hk*du}Xr+jdRkC z;+wT+09V)kY(LL#Z=P6EQnJ;rz{1AELtN!@3M$C1LCDoiAQqn_F0z=@FT1)w8MdgN*iSf~4FBo%X(_BTzY6f*vrbFYv=} zR;R~1YDg&o`0=R5r#q`0@Se$!TIayf&_@w;SpGr&8_XBB=c3iv`S^-@^k3KDdwT-+ zGXA~0yfpYhjn>}cSQqEhg>Z?q{pFnu529#S0DTC}`Uy3)d2o;tAZ06fh|rJIzIU=& z+1b(>8X8;RR2vQvYye?k5975DG$J2w63uCj|+u!#V~ZTvD1ECcx}-ypG}I&WlNocJXF}D1|UaX(y+TT|ut9BGP0HZQ=3cOQN)~acW5zTdk37u9lu& z@WWiqcfej!+<$X#)NI}?AE#eALLmER2jVd}b%a>ww+AkzH$+8IfES<-7*)hpPe<3Xx&IU?|qmmWIacfGt_Rr~&)3hJ0 zZqIr4*+(?!fqs7W>XOn@St~264;w9Gm#7Hx^eRap=mG(j-h0pXMEK>E|`&Nn! z=g*~I;sXxTg(b2IQAt!$TP@2!QlZ@lyMfBQOu}S6N`?=>gW@cxR zRZM=ZeFg4r{=q1n-Im@yu1sISba$!$@+dj^_+8#QUnH(u0!T#yJy!{ENts)@Qrx_8 zS=`8sUhYwIiN@on6Hz`~4ICO7VbG62HOi7o`re7=L=WdY)UI*Jzth`qRv)f%=MKK1 z>v}oKH3^mW?x)Rjh8Q`b!z*L4hA#Lo7Ctuxu($_8-HZD^eG23fag4+pFK`fF2FwA# z?3#n#P#e)r-iF2OQB%Hjz5-7Xz$?v^`hCfGApW}rHAtO@lC1i$~TUyGwxQHQA)#~V9 zK?{k-z=&zTpm!nU%SejTE;pyHu>9)%B;_RyZ|xQ~K)cWxu`W<$BBP?bTU+Jeyw3DW z>=3nwF?G}TXAE)w;p+TP*6ZBTCl!2_ltBeenOo8H{yb+9n?+-2s3tOt<9>IGIoSs4{* zuASZ82!DA^t2$mY8p6Hv^Cwc2k)Q>cDbJvW22@8#22{Z7Z?EP8U{eTZZUf?F3*3$9 zyfBg4%|nivnHl97F;aWxI{zTx3iU6K?mc+mbNt&aL*xCF+k`X9GbNveQ&id0v0MbA zlr2W7*lN7OVN(c0(%jrkMnmII$)(E5&5aM|r~t|l^UGazUq3(l-^3rfy1EbtM|k1> zq=y0|P8k^)GG^wm_jhBPo0{G|*jLMxEks8pB_X*C5!kTq4keSF9n8ADc9%#mp!-)T3vOL9=)`xDV)8yOinxw@tsHMoNg zHCpZd=+Q+)c<}bd20eHdP!pUNgzz;)u*79{Lfw6HwJBBFCuD|!gdRTs(|mU<7V6H! zJNe!jZ?OqD9$rKz??^x%i9b0j`ViJ=525$r$Fu{gHmLUf{8<^{iynWDSU~!?2Q><+ ztf-h+^N%0QBO@bA{pkTE{`x!$u-?k*s_o8#3P^C;Iyy})EoWXH{kjcUmyDi1=o}%< zC7(N!!HW@x{T6Y8MwUIXi@7g*=6`pi8l_bjwtxQg3J+JyT{Jg02dqyjO@>$pK*;y1 z7Yc_vKz80iaW1N;@CQv80M$oDN_(-o0<7Gg(7bC`uZCEM-j&T+Twc!pX+cfkCx0Y7 zw{Ou2Vvr>z3>pWL^g1S{X}tY>m3`G^qrnC}50r`mr_`V0>+)fYK}VYnMx`6}YvFs^ zh1Wqo8{>4T8yJ!JXc^VQp`9!0Q{pegc|#c+wYxe>Iy^j#1Sr49shj0wJrn2{_bySz z2|Rjrb9_KYSvf>g!9Z?7u@KGvSM(-LvA(H2=E`<#b%M4)6Y3G_DXwQA58N!>SR)k? zuY$US0Y(!i=_(9qOS8~0^yh4cEZJEoy3kO`$jD&j<0Dg5RRydo`fLj2nK_v8#zRAW zf5*tzTcM8>N~A7AR;a!1c|bFHA2i_%AIzrJ|KAZN7g zt&XD6=>Ckrbl1wVjte}koSc$oU)#fx1Hv9Ylh zCx(HagPMo|wcBZ&J{J*p*}atPwtWXdpNQOl^LR@Z1jH%uPz6iNtH>Ili<#Hn8UT}f z?JV}DL$YIIW9uU9_y_Le^x28jFWduBU%Jq+Zg-zENI9a(51KWX;ILxCDJdxUoVQIa zf7I$YtFC~XfUhBp!0Bkc(spHtTU1;e10R4`%N-tR`g|Pv)Ya9Tb~|_r04z3&ot>Qw z06t_Tw_jhMbUng_LnWu67{K3r(`iSiA1ARl^Z;zk8v`E%k8O5l5)pFVk2 zR*J}n(0RRjMNlp@F)>l>@#j={58y8X%9AF~M{-~P$X8ra(!4QIBVPOR#XV6`QC0~F z#_8$lk1vme5hX7qgvjk=Z?wx8+7*7$(Swo}D+8pFReT@$mWh7-`qgEeotxY8we(43 za&j1GAT%IQOW(Uk2CbU5qZ+d1#;eU47eSIgd+r?DrAv6AslZ1ms;PAVh&~66!w@=t zY;as(Id<}zhwSMl4(NfX(9lo;ZAhY#2?@bq=9vk{Nk1S@dyZ-1IaH;t*%-;Bb1_~$Mz2*Kv2b$7cLnU zl`jMeGl{P9{mPNVblXK4800PIO)9FYRLr*4C*`((*n=vwgREz5&9g`t*<>6?$A+jfa&oYM zEa+`miM(?Do_`$Q(!qmbK4oR?ofE(Ia2C?H%6}opwQIx)A~x-SqRF7?Hr*1C4wcZP zQi;A&ajH8U=nn-!l0%>nmIixQyHdXeAQrrF{i7 z{ozAu$X~VxTWB|ka9EqEosa)T&yF89Hny>qRnMW!5Sobc{)`*jz_}J*4df^EJvQzP z9?H}P+b{^RXpAz%^Y#R+$_^c{52{HrASxh6BVrPxB9+{Y zxI?Br@eaT>9pFgOMM8Q$e|&nzhx@c*ZU=mT zq;42>dV+493ZG4Yhv%7;1oQnJLk#m-g666O3pv45yMjr)tw)6RJRn-XuG!?Zn{Gar z2<^U}6F;ZRKtm}7bfEFrWn6W1Dg*Zf?vt!8p7`zhpk*GX&ljC81O+lB@p8m=BOwR~ z3@9lmM8NHvySoYRXHy9a3ma^O5fQfde!kxw6!`56Ep!lNK|uwrL{HKPs0>H}0I<+d zCLW8B_`|en35PQO)?l}H3?ic_0(TY@3K|X$4O)Eh-){Zlk%k!nsQ0YkA#JK=I2va_ zgrK{|@>ESntDWNe&1_mg``J*`@k5wEGfUsjCgrqWbYBEb23zrd`)1i4?3>Rca@pMMR-Dm8g~-GbN1 z`uA5*UaK=iWYGqCdW0&Yp85atMT1byj{@yN+|a7rH7EqOdg;m)pUO(2sw!p#jWrPc z|F0Ps4`fglRaigWd10}Nqa1ICjp{2jl8OFQOD!&2zr$wE@K z7@nMH$ON6aYrRmvihUP4dBJYGZ@vc-80+1)#JM>mKmyrk=+0Zb-S6p1Ij-zQ{kc8++4n{pf`x_Iv zZUyh&NdtfL=X2c9pDylTwy1mzcT?cP!eC}W&VKhGV;axG(h}ZV<&l-DbcJw4X5Lxo zk%fF+GY;2O(m#HryERx&LQ6Z0G1MJFI`}cYdb+yJ6BA6`vAi8Ylw9BPb^dkFV7DIH zg$7q)3h3Q-bwGP2PQqD0*ycxG|26pw|BYW-+uG1TY^@Am6vjhcJAf|gYg+H&;j34# z$|))aKDJR_{~3uBAXIHEIdXl}1vE-pINMA&qST%gKz!M5hGg-o27rgo!Uov{1nwi> zSEA}Fl$(?D+9OtxLq`EsLcpM;yQM=wL_`5;q)P$mkX8{95EPJ< zkd~A@*LSVY59WO56XO}<9`}7eGeS#4k(iK%5QRb!D=T4iP$(QR z6bkDc0Y3bMvbO6s{0rY)RS|rK8^3f4TF*n|Kwx4VZbBA7kbdr*tr6OiF{hU8ZDzZgphTr2) z?$OG>-Xm6VbC_Yo!uv=6j}LrJKHW_b;f6uJr^#v}HJ#5=;I`zj-}APOT`reIqtTgF zpO?^K;mDt^b7fu3@FO(3^xeOHR8JU1ck+|poZL=-UsU+Ndy&(dZ=*}mheY^6s+ym+ zVGtWJ+qg0TrKP2+Bkb@XYn~dFoIsVF+Led+L0IB1Ki&O16i-cWm7FhkADkm2BNL12 zAN(JD)cQcgqNCaaSrlWKL`0}zWSJaj4i19#_4WC0Z`dgaLk7qqU`XG2 zsKhWl*Kgj8cra4cVkv0ydm}QE@cQ-Z6T{^eO6uzJ4i06Q(qD0P2`Y*%lSJW@ct$+YFW9q12KDPiJIgki!aV zwl&VU8+ioe;r;&o8v+*Jwm!SrYGT?#?)Zq7PwZLvyy5!~AL1T=KI=St$Lv!I_r;5w zayM_j7$^_#YAxfdp+LsKf1U^xu(mc@$k1PIacD6(I4E-Q9GPd8>*CjpDmUC;zw|Rp z>{zXBR@-V_7qesJF$|%;-8nP&-dqS|5?q`JJtJXY=55=nJ94hwMv|>=&Au<+xw?O` z!Nfa|sd;Eg2oJ9KkI2?5m|7l{4P&3#&%dC&1JSs}= z-o07MMR2;YH7F}=N!96+tW+Br~zsz>SPgn1{427S5Z2W zdKk(u1OaO-qqqwJs_cF*L-wmzXda&BCv)}N-)*X=W3Hz6S*~@l#NO`9l;53`I$k)S zmBsL03>=pvk9$K$a>~bNI1y5cbp{MwVyziR)N7ODQ;Jy7moGXfX%7$4sp;vyH}`wl z&d{NDw@2)0hf14n`1nXv7(SRz+wzhO3=E_>K3FVmGT)r^Jsh=LIB$&)9KpFRz~aOEBo z6BFvny^qx^8-!=hkTEj)clTnn+ctWoV*hN+nQnPPYJflRFFuN|zl za-}-2t5}womveA)+kX0RF_Tt_0_#GlYIv72U3Y$`dq9^m&w^Uphp(wr@?^ad8$@iH zsgD#HeK!kw&kzw|jE(8l*!o#nBogO#_ZCZOql%-_rr#$FhUYy9M}1#4FU8DoQbqkN zxT5m?T+9rqShw9<5%1XN^YfN7KYwPpul_&>3KaJKuC?wSh2ng_aMPmlGF=UpVw+u<+qHgd|aa330 zK-$0m%~rVUXU{%IG>s6Xp9|6Qot=#SM5yOi-FXAX6}@N(DPO4JYg3fpZox(}5Nhqu zTpA7v0yU~To*gI8bZdEYp+7t#qUH7752%SJb454GGX+{-sVDPW59H}LnV_v{(U>mL zXL1;g*EeuED7c+&Aak2^5cF1@{PyO)#Y|V6lBQ;uLRO9Wo*_* zNV)Uo8@wQ{zdBfutVY)O0nY#ymQLqmI@k+#`229A7A&%BDP;;6jRK(ylco2N`0Y~G zFHCSYjp(u>AzL$xhNYZR(vXDW1OJAh2hOAJ92Bd;JFX1Alal6fCF(A}%IXw2W*5d0 zX=EH7o`Q}Hkn32m#>~(q_uwaw;pgbEx>9s^_uRXJD`WY!g}Clj9U(x8)E}uPrD4C{ z8E&pTUsb4NC%`@Dovuf^_4^p?pW|GHLwiRtnMBK&!m67PNl z1cYuA;mb|Wb<%H)9uvCvLBhe{fc2Lwj3y|1I$2L=Wl zV4YI#xrQ5EOA1eU1&{tIDTYFpJ?GV{(48z;-6BXicZ;0DyTrESHPVr{BI|oyPfw25DW%^bgM$EWC^l=pX>nfgwmba@2@feKl1;)GUuZylk=-&>N!dD0UkYCTu z%*|z~Ci3cZI?y8ZH9s$p0Px}8huc)jEwi5trn*AJA2R7$aL>T5bc1_MfnDp~Mv4|57PyX+*M~!4y>%;f&Ra}<8 zQ!2*LJ!B%VOITHSn|qzp_^YtCHyS%{`WGCVi*;i$nVD#)-qUKtALs%AyBg-y*VBo= zBq6k`mWPNQS<>3wYYwA*NFH}CGAfFUhUPsVJs%+KJW41sHB^#hVm)`tSpswiA=t&l z#KL1^6U9SsnZcS9-tSG7pZoMOCq@Rt@X4}LCTk)S2dcbouEyl6WVfI_Lxc(Z>*<4wd70o~IhokCb7#+d_#}oEk9nQ;u zY)Xj&rbLTpm^5!K{$vII^$TnmY)wtg{P)Vj0qyXvpol!i?nxIRv{UGqU1wjGVqc*F zUw&3KJ%ZF9CnkoG(2G0Vgg-wsk)`2f|o>r!VdqvG}J6bAY>kRK& zw;+uRyF#Fk>qr4{vVyZ=Gjo?Kb&_%*Ra zDq>CdFT%cbcRy6fl7R}D6EpQ}hBJYXGvQb5DH`Fi{gkIl z4tXGxQmW?r?WZa^AHK8wcGJ7Z6)2_$Nv^{G*pG-#R12mZWy(S(bWUUhjW)>RbD}so zp@}bEL}20IE77+;CB$xWh>7EZSQCcD?}0YS;n%<=UU6xRhGRII z2ua9JF-!!a9&1c{JF9$<(Awd&T!>+ij2s@m0a!U768WjZ%eON0x|nz3luv^1-aWQj z_f?W;1(ttaUrizjb=31EUdB;9Bka=~n;~AJ3zso4;AmvQ=_EWJ%R`1L4ut=&X>D47ipAjz6Tb49&D_C;P#Momcb*3CLW)y-VNTgvoz$g+ovq1Ba1WRP;PF zU?;D;M_Qz{wmjf{=j*aVs0nRXV20veyy)(!f^&G6?TmPySR~~5KPz>s z{N`;0`4-hk={|W0D-G22yOr4Q231a?Z699aF!$13g}!we86H05ouE@^GfkWP{thp4 zD~co-fe9(M-+U|c-mxR17r*|C!!A~yq)!MS5^i5#9|0lZ6QpjGsl36%A{3)&Jg8K~ z6u-3}zlLNvGcz+CMMv+Y2N5M^J1b+=TVHRTB%YOeW$8|o2BiBr+p${8ol)o50Rai* zK-la8LLGr%dU{f@-t;^)ETm`V{1GdM zDK^m2354XGot>>j-_C+8W+X(&?TO}xYg=0hCj;GJ!=fJ`?7E}F&WzUF9zQ;!W>I() zV8oj!9;pab$%XQ&`%2qmX1TvvrR|PFE-GO_0D9TGckWF5@Lo+25A8&%7cR8-A3v(W z>dsJvabynMkqHjnnI_dn0}czbUz$qbbk zGskzAc>f6Pjb(2#xj%9a_hJ8Dc#^}9i%d%GIav{1V&gs=#CzMzA9Av8Ntfv4r|TH| zpC7x;FP+<8v($z>$=UnGix=kw%rS31ODEx<_`mgDy1e_55;6;1*njn8;qopkX86y-ExW%>V7BGAQ!Uao=`@=T`cHTb5HW97$&{TQh_43fGqA@kv>=B z(Vh*`x9KRZi^7iczJ|2mvNC!Bsnkdw)7R&cE`01<0J!Jw{`PV;w3mH3s?;zJIv5A_ zw|dVjI=Z@l1I^Zz-dq&PEt}o!X%iV?3{c?|&0!D^yYIZ~l{$W;ouzPc+Mf?HXSQ)L z8GgvOhUo!80Zk|u^mhP$V%yMR^MTMlTQmn?{X5{Dg8E`7cE$J40!ZD?+tNr8eHhQKi8P_G5I0~UP(`Lq4AULaK}fJ7XHG(nr*?i|El&Am zaJa;`-#xOaaXAL zOdhePdSEv;Ho}s4)W6PhD}-!N=eGRraAzE{I{NqRd&xNMY*^*uyrR;7(*cC^6Lt!Clp*ZReQnQ3toOgF~4X0pwVI+`l98wyy0?;Eek2jQzzS_4w2qxo(~2Bdj8F}W5m8G`q3jI7zd>W(<`?*z!)9sB4fd~>}upsR^s<& zC4tT#W~Bx8lPACtWd>rf@2$Nl@*ZR#Z*Wuj9ISW#4PvAPfPlJbB`ce!aPk0LndWyP zX9VnjNWdEa1r3zg4wqqxjQdXo-X3nhxDI7l+_Pm_)FM4e&xkce^J^=tiSY z%}FHQq$f!!xol=Rr**Pl{fk>3`^#_3o#!-Q3Oq3ok7Y2>QO8dD@aX8j1!S610b{NA znRAz(nXK|hAl?~%zXOj?yjF5cIo*qtR8;2uuhe%BHhTXyNp~t)#iB4<$#Hkh3_d_8X6EsbH+ABoqOkV0gLRIFT7=4>j-rOR4DBug=to)(!4xrd3n4T`m3S9AL5>Jwy4F2 z-lR22$Yu@FwGx8Do_AY!LLjd!kwcn-xQ1q-WvV@D9fDnnK4!nFaP1lc~HS8CwCkT zRC|>q7Y|SSmmF1~rumvX!~7c}l5|W=VuFMHo?l@<1N1vHa|Yx4_wTb~e(L|Sbvw}4 z51P@Cnb5((s9Tsi`2zF&9Eet67v*;MoG`lzQ+nyiS5BDuQG}UafA^P|kFO(_C)=3o z?aWq=|6F`aB!8N;ruhGZssE|}T7_tyhAu#wm0g0=n0Wt*%{h58I03D?nvx-HZJ7M& zimRaE5Xq1xxjMAsy)7(jm_1jzH5|gKF0bwKi*9Ap>sJ4dy{$zvL7Me2se_E0b8dQ# z0E}{*{6AGdR&C1oP3lI*ktf``;$ZaK`9h+C-BaDWfj!Jq&$OTu3?F^gv(kLldpqNk zk$ZRVx5R^$G57J1KFc=eUP*S>r*6t8@F~BeosIc|la^Uod?6t$)1f%Kc9h{hCtk?-RI^E8nMZZCpxF zAqbr4TTOej?AYMHbKf~!^tyhuj~u<-1#xn$q_-1`S13U>I0#>#9)B#Ho}5QzD{5la zR`w|)<(0w!Zn9ikr)f#&_*@fiO(|c+cb(xPT`UkvhV7dULk&!QDW#4Nt~B93CFqc9j+xJ$%bQE-?L8vy?;6<8P5@1P3MH{SF=7(CULxEJ8)fpA_4Er+UMZ)&x>12s=T_lZbhA9^dW_$Gqve?dOej+WUWuJ*}xQw z?VOe|@p&R?<1ci-s}wk(jgA=Qr#{oRshnq1QI1BSnlOIu^KEE2qZ$ z|B*Lx*Y86QPH?8qr8n&0;6T-{3W{r6dpk2PF9|?m_1VUH`WI?ncQv)NKI@Hm{#(un z{Muwi{ijpv=wWa$?j!oEUs;#QJaJlPo9%N3hn@B$G!|_fRl}()?0*F(^A%B0(a}8u z`u;ZLf_jdciQ85}flvMBGdBQv#f^~Kk}O>{4%lZ*4OL{UJ{v2!DJAN9hp0q$q}SR73978k|d#kaHQ&ioRuwi)wQhw z%HE;>Rx8h6o<_q^JjldKgf7FQXpt4*`= z-GHHE@SGvzHmo2xKA4kA_E1c+-d?mHRQYTx>ve&#L!1YoMSiHs2OseYCFwsQh)x#GoB#BVz&Yw zwB~)%B2hgnS><9>LhRXtpN;d3WWl%f`h9T@HTZ5gR#(ux?F^P(X}#*0=W8>sszF6H zIbM3e`03~D?TrrN^P&#v1AU`E9^#zh_ujVkIX>L^Y+3CPF>qr>X$DlkVd$5foSYms zdSaQRq;TE#b42X-|6mcn{l-B-h#-wimrhG>*k9xWRmQ39rxy`E4#-+ek?2h?-#2OO z@!h27&IM7McwqS-Of!~#*RZwa21@Q$%Hoty@3npI*-tM=3WgMCcsPG?i)vd1|FD%; ztR%noI3r#mgu#qv*C8Qm@%1qMuzLxiGQF!2Z_@Y5sQo8j80V_pL0e=uP$(Ma`KzK!>!->;SJ()?tmlo$#xvt6mc(KMk8svww4Au?6b16vX9f6 z2pk=49FHWlbahQfIVhU%rw#JSPArMt0z6FW>grlHvfRUv1dD{MYBSt?G+fL=Lk9De+ZyxTHTMu$RIxismE1aRly3+hz@M$Gg zRXIb$qyfiyY$`F40HD=UCB4^rM8>6g6Q==ix2MYlt-a?eO|Py#UuaM&trSj^=1;OT zb_Sz>4GNKQt*HOXe>5x}FB6cf;^N}4UZ{mO`tD<+IJvk0%DGQWv;-OWrmC?8F%d+! zR;X9v4DcC#{+{reir0uBS|JFUj7+^=_TSe@$T*7fTe}Ha?p(fLM|fH~TR&vU9LM^B z@aQ*I3Qn;^&Icu?y|hd_>oUD0hJYZ)wA8m)=_xKt2)lCr-q-+UCIac7ExbFG=oo1f*BG&Dk?a3Phmtc;)gRuvIp2sUa>>R9(U zmg`@dSt5lP@lIdYo!1?4siE4>LeqF|l%W6#Vh42OFayVeC}pOVFM)bu0X5FJ*2T)G zr#RmYbsK8(c_HiTQKr4@X+D7ncZL9b9}gCx%5%bD;m}J0c%Iv;AJoXlH$ru<&K!Hz6qLKJB6F)y#NPVv-X>gV?{6FfPJ)N+U#lVWH4q47BkDdWB^coiS7m&%i`6DRunktbWmr zmJs|rW0=GYt+nIpFTqR+bT}+QWKL$xScR<*jqwr))(z6#J(&0gZl$N!l8AYo3w529 z#TmF}RLIx2xK$10n1JPHCP+Qcsekd5K~Gu_KN@cE7NMqbkmGhT>blK8k!;`C;(xqv z3VC6&o!S%wzKv;%K&FX>+%yGqb2QA%bu6LVh*?;e>I@aPw3nAS3~ciHEx=ap%CC?a zL!%5dyWO_)>({T>K#R6m_>u#Xh`TZg+MRWnPBcqo@=+rdJbS{bt>3y z9vDst5L1FCi*ep&ansmyy=k4#cy5*beH|QFEdb9IoxtY32_`GTlxcyrL zVMP=a@JSdX!s}LQ&qPu4h3b!N5*L~J{b2=~pv>owliNiehoxbGk1wvA@;_WOEgdd< z+I35CN!77YU%Fl+x^7BbdU9m*X`F5X+w&qfqD8vzGt&ct4c{Ug<_h?k&;%Jrgn(%) zBc5%t+Ba-bfv=?u8a|W63Tb_Ps*YI3*ug@B>8cUCcySVAoTwE2YLD1t;EBv&x7K!c zntrW=rg8)99oHa)JLK`__mB?ec<1X5)TUSmWq-Q7lR*xHX(Luy`~!6M_Zt(y z(pES!6$U0$2#T{|O>Vx0-cEjpv>*@wA^yj{{(y;$h*8K-C_g^9{EGa_%lOFI7 zh`}jY_<^jYrR6k(2+tNm>jAwp_vBGe1#BWWWJ81`IAQte(7DDBOe3F(-?YNW=7p-t zi|Cg3Jz65i!B@b~3QFn^?%t>=;6otd8uaTMwb?H&U%t%myXR75Tu0?T>KHcQmK2eRcqMUus%x{N1fF-eurL2(uRV#FXxH^Zsra-U+A==x~{;Ip9%4!8f zcm3uU)nVW;n>@Cpxt(~cePWYuzPSe!M@zo2`||e?=(Xt^8&jnGc``YsfcFC7YA7J-v|jC~CRfe&a3Al|3)oYf>vos|wJ~$=31BCf9YU z+gwj7CNS7YePUBChPjh4Z&oG7U++Axm)OJV15@%*-Ga&|eAM_`~U#DQtIs>q9)*#(iEsj6Sd_D`BicGk%50xiC25R=;s*Z3V9U=!qSC)aF+6T8 z`{Y@D9)s#1M(Tj5J6MnPZ>#s*SObnlg+9zE-IzkGcx<}kCMan4-qfh~PFevX`{94= z>jMmu$3~Ax-1Fxgt`{@!Ip2dp?7m0UWvOdUI?|yXD(Y!Mgzgc=5Xpo~QJ-h(W!5vD zT3wC({{6eKWLR#wVX?>2pSkt4ky_MY%3tCI>mV{&5o^ zVn)9DHSBlq-j#W6Stn|wp{-Rvt*B0NSEkV^6h>43rXILqyPI?4EWieLx6{*yA~OQn zE2^N;I_zy)nRJ;t0GwBQaTyOVS=ambaES8;;<`fWUw4t=VqJAxa`1wIv$C;GH9y2z z_G8fku>$E~Zj_tj-TRahu|D0YN*|GT)2Na!B~~e?6U}xt zqLe_d^78w12W9A4#+cY^dj!t2ALF=#20o=qMj?$kP($sJltv6m5`kPhISwI8(S+=r z@6D)jMtsDh>Mk4FcJT^GNepA_ov`!mhSM2~@J(O;cwq(m{0eb$Io{t7M09-yJjX|9R|;xi{?H3W64L z3gVZ&prBxFvtJX;X)BM#$@_%H!eM3Pb2sIRG@?x-x2lE2+X@x06nxaDyHpr@+dbW$ zonAzi_3fOpukz}ThV|X)*y%n+#&%Fe>b+Rz0CV6$R+R@Y%h6V;zZsNi9mP!AE=j3} z295{2j@(Aor2hLOQ`b z$@QtQHqUTbX&#|>uO74EVk}uX{f&vow>AbEXF#Mw%m?Q*B_ znTZ4glZQR`;>Dk+rieOXBV8+U_JGPek)JwacYgJ8{x(AY5{WsOv=#Ha^p(Y^xNU5G zi~8b&%JtO3?mN;dclZnNXkV*rI(T$fMwpbDn`%?KQR*{ca+;0T=}D=xw@)7N<8xxy zPbZ(PK?6~z83I^x{K7#_*Nce=Re{U`XlFDlD;~RMYFpaTCVMGEab}_O$bz-bs^hdZ z7@#~rVq5b+-tmV@rZ8>!G}lp-4ngQ&i*5#izjF%IzIR-u^5OsrawMJmx9L|r zdDl7XL7?2-X*!OEPU*b=(MB4=bGEP$z6Cp&?T`8ki(iZ7fG9ye0;Vj>>t5F+i$RUv zH;t^CSY&CP$h)1ACpcAC?*s@A{Z&rk+DGB5zY7;1=az=N8#)C-j4fSy<=B( zEdSsEkKe(%A|mO7zo{jhjHS%74}J0CMJ4*^`GQe^VpT4x;rL`&oDkLlSN05HLe)Rp z6ML95hb=q4{SpUM5dI~PpX2g*8I!%^ka{`q1y$(0Q@vNjfQ-Y}lvu|_&fz44Xbbet zV=J}b4aTPxv1`js^NVco-a$Bo-QOShJQ?|gx_`=R+`Y?M^(-=s))aTgLnk9EegPNPA zyPXge1aNm33jSl@R{Re4%s`z*9MkF7f8*bFGF_?VlfQN5`g?&2XO76|dvt~TQ*U3> zwBhCViG61DBmZ$=TlG!boLtM6Ld%4UNv#+&C+(k&!FxMGtloX4VyH12{NB5OW}!Sj zM7FNf|Bwqgd|?b;1T8HsPksW`^A1pQxm7>g0pzOl;w~Jo)LsOZHtxN*Sm|N2s@~MK z`S%rpl=9S;j$hGT<+CVM?Vf9Ub+co*NQVUkEf5VIAYuNi^=sqVsjeXlC83YZkPAa| zah);s-J(Qr1qf$n0qnOHzx@#jUZ#qLKj6V~od3+cG*(BA7*QZELXAa4e`HI*1!UE~ zaZc>Rx$T^4GPEEk&aD#x2(6)brnwolg zF3rC~Q|d4nTHAEMdG2Gb)C@jrGDU`!?|I&yVcwbff}UHOY=s`Bt9)jnIP&f7*44sr z`Zsi~`H{LFY%>>JTT8@5uaTLBF1ZU z;9&sIc5_EGt$_QAQE*5IGbB@(7;*sxv7BKP6skO|Z^0fy1B}?E=p?-;z*_VU3a^%d zL0C)BDGO-fZqzujf;`hX*q5hgWH z-|rm~5P(7)`$9s1m(U8P)|_k z=5q#jo-|i?x}@46hFq|vS26Utw0|4!SIs$)Uh)KQLC5X*$uhLhJgA*lmUZXh`=L`# zjEuW~O#WOmTtuvt@q9P(dQDKI~ z3cv*0)h@8Ptp1<}Xvu+i89+xpZOY?$E|E%%IErY-(5ci*t)%iuAijI%y{^tN!awz08qyE6_B4jCzF0Nis34i0vRxc~bGQcE5z6ATO% zb4qT*V5$K|AE|DMW8woAFou>%L z_sqjpCE|0?=mAgZKYQvDvNlaGPLGq#+JkBW<6_ zt&(~XQ0(j1-i>3a;h`)ArKaMP{pCE1@xoMXkX2GClzpl{$ZJo|mN{!?Dm&0v?NV!K zWJCv{>5#U4Eu}=%x7}CFsY^iP3P^60lp(+be0+J88vJ@lVg>Km+H|axIarf+*ILgS zUnstbPN_DMYj2eEFk<${}IX(mxd;| z;+bW$WXEH~AFX^R1CG+nF?{%@*2CqRyQ2CgK{Ou{qS8sV_uCqS^9~&rhRy-O4cQ!V z1en{HuV3%dHK5BHw2n(k`hFvQcXyYGnHdYxC%9ERSYjox3kM^SJ`4*09sBYF>s6izMW{A zeg309)QT=n0ci?(moIsw#d_=LknoZS-_vn(*6|UqU|HcKl~h!$`;dD_w2Ec`qoBJb z1Fr@!Sr$hJf8bsj>X)KFs6uNHm1f&Z--gBltbe{ws9vrbA*H1~1EPKUQO80+KtM<3 z2yI(OM`l@BnS4z|mv1);H~)Y%`?uT{z+?mRK0OH7 z-`sYtB4a=MKSRHZ#2-i#7CHLFfd6kInM=W=R_zi96&!=Oa>W9Q&ajqoG zV1^*Nx*BTq;?B|2M}r%56E3=;_#b>1gxuRu-fZ=`#`*r{>Nw*|ao2Y?r46Bo?G3?s z&_;X)vl9`6#BOBO17TVO0}5Dlpip4Mi~>TY3&e5Qy7lVOQKPoDwu;HZi)eRk|G3PS z9%dRKnvv=aZYcc-td}%=I#1VXK3NKoRYLIyAC0|5DVC)WEvulQU=gT{A0MH9<<=iUE;%n~NecJF5MINq6(7}% zn2!KUpe&6v$We%p1T6JfUZNVt4&B^K4V}Sf9#ItF2I#wIt`DLM+X6_e@ep8Uef{x< z%}sPI{x&OSrKLDBo+}zypUoKd=#1}S^Y42yPZOH(xb|Klc|BYrvONS^G6RMad+Fez^n1uSC z?u-7ZFHeHpaLsq5c~OX5<_PS(b9oY)BA}$+{TeJJ0|^(v&-B+@_K4fY*vka{o0r-& z`vyJ9(UlcF{%}{hfhR>|v96rhYbJib4_3bd}IzM)wH>JS! z>c3CzTZv4Gx&-nwJ3EV**aiK)E2A}V9^L^_CUB>V6m&Qx`h@V&=cQ_)1dy(h z#`$!>K;8I(FRW>lnhZ;&4R=p2%W}w=m(2LoJ~?`5GIs`BN~)-(Au=nC;2yH$>9>;1AvA4~_In4Np%ozusG`AMths(&w z=(6~g7{DcxTLvdHI1lUi%p%U2{_+bV$oxu5O3i>C;*yi2ii)_ojcX~~>geHf&P#ak zBOtb`x|^EJxcw*fCHyy(?Fv`GjRbcokw@C-)hjV)+L%KDm*93XlxB-(1}hlIkch?% zH&>8bg?B$ZlOe?=I?o?~cb3dJkx2US`?CP0UC#Drf@EW)PXAh!s4NPN<~7O=-sNge z4Gu3sB51I@%TZQMjR^cRG-pzmmTY}M6+qw-^1{Cqd=`+#TcK9M483a%o_GJp9#vK-(zW?&8$gx7zaTu=z>q5oNTdBP z(gp*dv(ND<_A&}Z{t)?IfGowcT|Y|-xf&J@H)ghJ{>`*wM|o1`H!1Z^fRzx)Z0JT) ziUgS<18783UsBn1sCXV;^3eBmrMpvt3SNSW>ln^Ip46qDWk6PleIG85D0+H|ZLf~U zLf)GOP@_<4((t*|REmg3Fz@E7_pKQV?~(_a|6$T~ITLtolKrSB=A!Icu;!`g0>GOu zuW-ewz&cLPTpD^EIQnQI&aOX|+$Zs49V=>F91YGiq?1M}&&2d}5UeQDVR9LiV35Az z$&)~6N#2c*oBqnf%ybM;8)x5GB6<28m*VTdBnVr<`IK-DS!A;U8ToX==V^u$wY*!< zbNFAZE5UJmK9rai2!QaSq9Rqo)R~z(aKnW3;lqa~#r)NOn=2Uev&iFUElg#9Jd%&W z2-2uqv{+nz;Q);+YPf=GHR>bC!J%|ABHl)&a&bBip_^K^g40Sjg|HL+wrKIK z4>O)@AXPcW_p~Q1m6#oJ6&Nm9bU<=2g9|F)7=(7{6vE|ZW@RDi064-yNL?KaDO1{(UG8JX9e=iwTkf9Emc+;%F8Lgs!x- z$qOepTVO653EiX_+RU;W`%pENo8ABXTua~s5R%3ibT$S{P#!DP z3j8N6c>cZ!upkIY3)Yo1Z^ztd?Hac9G4k@tNUj-fr@jOx6}ZEaUe@vGB_5@g(Qb#Z z_-95#lWfvHp)x*Z&38>94DUKSPyLsJ47>@T)5-ta+~DNS?IiIS?jeH>{c%dlqfQ2I zA`mBr>)iRC@fs6?LW%4!B-yv4PG|0n&P0WmOie{Sy*Bv}ap4*0h82S$(<2bD!FU)>qlER$2Kxti& zo{;h8=y}QM<*}rqJH`3L3h}SX70&JVy4qVzGgb<{VjHPgRR7LaeN_HRz^&5e`}yTe zRK%Gsxyh=>Lo)SsyMQ|XI(dR+^;HwGzP9z4M-fz%gZTZ&Y=YboZ!2;39<{^+d=O4 zpAU_jzb=0Nn z@3cL{B(@;@+n7Y+l}u6+1=)p%))5lKG)`<{i64O0zq#Zq4&V}y4btPlUCQP>%;17fiPVA+n&! zrh@r`gibVIXpxnaU=4h_{CD5~eS@uK{1*{^>05b;&r)G8j5{A_)-{-=e|Tu77mr4- z=RfO;|NNMx)k*v%E;bhSLsUaUgHg4^!(YFw!F#L*_e35_2jr+GJ^>Q)WN0~K-|&1x zKcq|1us{@aJj6@GvFm1eg& z$`PNpIEnvS{5A{XuvQE@E=*kXJW7!z6s&KJ^PIWoBlX57>prKxExV!Pl)lW0P^~Yy?dP>W>waJvuk&Pa)@z1J%6&Y$ zL`X@H=5DRkCI^;;ao}L~guH08m&~XINYIzh`^-1)cr8;Y9!WD^_fX{M zNq)^9R_?Pd4}TJK{BcL%n>GW+exHLA76*bo(E8s0B&%Hy0${mA`{qz>LaMitaQpu$N{iN+nnBb*`L!63Gw9#T?_czm^ne34+ zcd@nS_*T|B802BoY49*{-o&WCr*M8@A)}-Oxk2f;HX-wWcA)S0zM9g=mXS?LDovWH8Cr{Ogg88z=gLyx zwbr|z8Kb{YMXou!KkwLP7m5hYLSIvUCDV{0BsAAb4kk&R%1Zl#~=EvIY>??=BoodwV`O`GP`1eS?E|U`qFuH#1{KEZ%rutk_4>`5bI%*X zFP<5E(wWPwcrhj=C1tp|e{4+M*H@C7&*Y+zP(qd>^_b8E5nRy>$x=LyXgc_LT#7XuDw*dyoj%cJAW?7uLf*Ab-gGVarr7aBS%p55ASE2-ncattp^6q=ns z9}33$pcW=DiDYfJMdo@;3z&dIlAfNPf}+B4;mael*h?IiQS|%OV(cIOT(hljq2wj= zbOC?#Y{SkG&Y49l{+`tv4SUmVY^><%80+pamztz577h*$)a}87)pEoARqs*j<+%id z#-_z#Z+dZ;@Mv0*YYi6}`+$*a2PZ!H%h0aug!uT@9M!F*9}Q38BsneUNJ0va&``y6 zTo`;(XVpdKi3<+GW^K{IEk-#5qOM7PAlX>I$bCeA&dS^z;Du9x8B{?C(#JP{SR^wj*GB zS?y9{rH-ES+^YIKIx68c{{OUb{jpGoVf=KPEko&6b2K7kE3V1U<<~fLDq~ufM&^_& zx{!qJ5Yf4F87kdrE)=7jEX|f9by1@We>A@(;d1h$BbQA2yjuVH{`39k-S>Up=Y8Mj zeZJ4<^LcEI(yXBT0F*9Pmbt_g+8L$A*I8%wj(%dAfyuzeaRZYfd+_=2uO(^W=j#RC zQ-sF@k?E**06d&CuQEJ>YG_&bb<#=DOtQcqP(U1MIARu_H`K;EN_f&hTI3i8j+u6; zQ%DnyWn}h_uf^sG*%BL)5l7)ABPnlXkezu2xfo_~E)G$^9SHN0VC+PAx{93Nj9P7J zA}SV(S^wOPc}lwuDM00x{H>QKg`(i?R8oiMyjm@&Z=--(cu%8As_e6cdM&?fxHWNp zH{&DnZr}nc`6@fe*Bl}_FuJ~TnSr$DR*#Bm_D8f1cp7$Il=b*kg^cmE5h>73YiF}x z?+dCa;W!3{h4nvsr%x<7{xdxd{E304i=iW~U`tFxHMHZyIgCBV9dW2bo-V`O^!E>t z4V6HO0Ro;MX}H=lIwDUi0rjHgbgI9ACi15n8;`dvw6P(nj2~*6xsgaFZ?nz7&5~`4 z^(3pD@WqGX96#V464J+~8j_MS$Ym>{8-v5c^}J(a4=`GncR}FL>Kd+r0cXo*BZ@Wbz1uA)Qd7p@ktYF9(uVgc3_HC9^$#OR~Iz+Ggb#n@0$ zl4)LPgMnJDemU#m3EEagZS9PV9VGQsB8`x>#>k<nHgIjW6Q*JMHPm zb#d3RlJ&5zmQ{fV-N@V0p#oDUPtkTi-`nVb&p%;nl0=NmjPXIJxY>#(fZrpJIM{J= zv(fvT(f_0D)P(-!zRFa14cdOh&7eM9JM~qFR_^v&xfX#-;>6(GN{EnbFwG-4ip>vh c)oa%V_432}WsjT10AeW=mR}H4<`b3u7tXYA00000 literal 21273 zcma&O1z1(<_b$3fk?uxdAt6dicPvmTDFFeIQjwAd0YSP;P*6anq(nk#kS>uD5EP`9 z5NYYSW8(jJo^#Lt+;h*~>v=XC=9=++Bi`|jcgzTFtt%vijD#o@isaf=j4ld=BZ)#` zQ4!$7Z)j>d&EY@zmg-k9s1xMh*Y&xHC=?s&8b(pyD{cA5LodBM-|^R^eZHK(wTREB zjjMiB&EW#Ij!qsyULHsHYYFUFKmN~|!n_jE;rkJp5<$(}Y=N(Z*~-G%2poBmSd8Oy z*zk;N2Dxs1Nwc3nz{z{=)|Ju`22KqyT|PG}bZ(Xc{wWVurIW;*{`2v% z3Muj{iai?If4`bDG@2tvev@$9B0>4|Ait3R!$CU6MPF4}m+y(C%6hw=-a(vKAe%yl zwwH(;oJGg=Lb);D2jf4Y0wSqB1cZbeqcIF|>-2s8GxtRp7XQCZiTrhdoiZ6vNETYaY=oBeFbV2Ts*uCtB%;K8XCdIMbD&E;Lb9|qEtwA zsGdH3YO(pNC$FHOW@$brCx?fRPsPJSER=*Pw7*|FLiju!YC8t&CK}Llx8;pys(Gmq zIYmOy++55;|1>>n>TnO2S;C1sGHSNu;pW|FM$t$PDP(CW%@dS|aBtqcSz>f4JX$8E zcEy{CFeGc+lSTTRY8YupiABqM9{!cUmM}67PEM6(`-o1%q@*PCpP!zh9+z5mpk-uC zXQIP9=}P(bXvEL4*#=&|*NZ~w>W$pKF=hHjTO$5N%{f$jV&dS7e2RoPUVeU6Utd;@ z=VfQ~-YhG^kJNN@>glu;d-N5YhE~YE(9qDJ9jv0y{E;MYh)eK|1 z6a=4PDco27$JImk;dB-zE}}WZwjpc;F*T$1Vm@2HqC31k(q5--qiqP|J~#G#9ar(O zQpTE%FS3-sbZyIxZOXsfnc0Y?QczRRl_Dm&XIc09HCxitry1GjN&k$o7H__I@}w8_ zcxr0u?))-eUuEgK2xCl#B~`g-)|I&SXod&P)4t>=bCH$SJRR5by^-O*JwGd2?=lgg z1Q~ZeELH>$4u`*Z?K%(VsP|b77~DH}B%P+kxxG?<2GyUhmn&r@!Av^tIr)h1nU!la zi*!X-mOf^VVS0L6!g+@6<%^fW4Tl%ccEqvT3`T`SFTbPK=!$D^>i#cBdT%qe(|e8R&B&a(3tFKBbb9QnprDxUc@zs{}PWD=|#E z2bKSIlX=6o#H@16_(9WO9!eyZT6dXVx+zSTDlQ)~Gb2GCqe{#q_IBd3i@{r?fDGlZ zuYU6W$G(@0KM*t=ZV(-BS2kE2?yjTw-<{;!x!w8x;r_z6_uSwt$SJcuYDFUFvCyC4b#k=x!|zdeCkq<&O|%#sBfR@(plL6^TwfQv zZ6^2&m_Ol_xN|C6T3UZ>uIlSkzpGu5I)DDWLXPh6!S1bE4|XxJitU3KCh@O+vNru} zxM$9Yd!6u^RG9Cq{0NVZZoS+m6O`&$v*)N$cLtm_V#v5I4H&! zFFZWF(KS!<=bu8=!89l`z5X2?pDOE2Bknuv%<$Ya^) zbGme8Q%CZ%XJ}zz%Eq?lf{(Im0@q!M&Yo>5xKYUHYmdR;e(LUKdMZy*POO7#y_`&* zkea5&?J0u?0j#{mx7fnOxqjxWV|XEw|6lFQ2(_<3`Hmd!(Vn3>&p}$NM7k;~#s_BF#cyM@x{&!2m( zx6%Du`Pez~1nyB>rXj}IxspzH7d^e)!VA=dzQ=p>=lWXGTVzbotY}-tcU|o0>m_3j z+WAIvPBn}PsZ~*s4g=Mx+2^%MDXP1;2(;}aZpEmOCZsZxo{3_H7_~uo$%Wq%S(DVB z_9DiJP6sWW;6M&Z%-ox$C0<DT3cd#5i9s zA?=H8xe|G>>Mh^BA-C%mfB^n_aV4I8p*|+!ZRJ-)bDbQQ)*ItkQfqq2DZJAG{-kR4 zdg#6vUXDRwh=I=^+dg|S4j!IPEAhMV-7v4a@2j_uW@Gls4P;1mX1QIWek=wC61jwT zY9e1HJFefk1NSf&`svdr)kjwqFhk_E>cN4cu%h{Ob<6>mTu{iKsNcE6c|{1WCzea( zf;?WT1{W1}P*9Fm%SGhxI9K3|RLlI1QVN)r%g;3g@yj}~U`eY(kkvlxkxN9#0E^?& z*M<&bB10i7Xes=hoV>AqA)-?@a1f7^4lyb4HS+K8(vinORv)q{=B4>8vIh$H?_W@A zHkv#As;-$CX}T7-J0oUp78eej!{X%RB)F4+1L!Bg1YBD333$c(^)zPm#^WbXD1*UX zP{;|VPq%Z*NwF&dZblD+osv>Z_J5vX1TjQ76ifiFmRoqM86B4Q+?1aV&S)V>pRE-V z?wb;>UR&r4$5c1{bjhfyqMuGk(9NBpL4s0Zmvvy1`D4@GN(u_mS3T2#YV0^?U|m>{ zyCWqH)y;V#MN`~3!Ju>bX;Km!-odB8N;^44>P{mt7>bs0KSM{*m5>TiB$4BHYZvf4$Fe%qg^-*`y-0%500lGgpj-n=K|Qi)vd+B z@TjOnSZZXO8G2vJM#Zv7tEJ;p>{apB;;5GEp6H!-_iv0B)|+;B;A*&qtYgkEg_!oO;^^?h#tC6o)A}9 zCT%_i_E^0**Oj>Ny}S_C90Bc`b1v1kmG{r1ZTTta`Z@<(k-eRAsRq;+^!r_{){=f5 zVH9&F*QdU|D}WQQv9bRdW}Bb$flW@@*cB?B&7X5~Y-lt#evsmSj+T0~FH+;Sh(7fh zUR-HU0}ga0i>x=btoK$tJX;H_L4=D7l3WwUcpXS_ZIPdRjOKn7lG4z`5efi$oQ@Uc7s{UjQ}8;ZtnC`FlxlWJL zfTK*~;M!7d3*-REINTY{B9O;}5Qe@kNJNTzABh|*U7tT=EG;u(LpWhWV$MgRV_D^U z9gqaVRmj4Q#B4`$6^b*Y%nFkqALVGZUA$0aa1mG0W~lA%`)^s|_aO4-cs+dhuz%4_ zn}^y^c6;OC%FZucYk=mirV~=^2`2M2UmB$VM+hHhBfu8EeEH^_2m}rtj>E&vK5!>x1%<~$ z`Q!;?7vF!k(7%|G)O~84qc3HB`m?T((G%mtUK_(+Gfq!61JY##P?YA*D3o}yCr=+oc?=&QLg00)vzVQD9O4SsTW2*;IJWb-P+ zt0ZpmB_T&aT(%f_cdH7}3oPd8fuXY8X7%&aU5 z8X8V_Ub7fJz5&tYvC6oEeyW5qwM*E|e`O?+!^b!OR8CrU#7Wef^gVV{o&E7G)*F%U5Gusmv60OVY~b}ix5lKFKoH)BZtU?UNs=id8oC9d;Y z7#s=e+DIjUvlN9{n7i8vIq*U}?LkecrC zE5H4-IQYciw&)G#kQ>;URp($oJ~+=JRe;8lh%YvH9lo`-wG*p1xU5e3_g=${!v~VA(_YySUpE;Y;hYaq=_2QXf1!y+ynm z2AIY2Q_EZF7HDX_<@p05h+GiZ4GXSfBiM!@lm+>E>}b|s)GE{__BIdZyqVFqA78v+ zL{7dl?ivSYCjM`==>($y2L}#pY)JRdnqQ9u8vK3t_G*{_B9XqL8qUc69!efV$gAZ8 z*}!+RNBpXmma3Ce$?2*7fh7yK_urNWWUGO}sFR$4Efuuxj92}kcj{lS{64O<58CWr z5Y@T0QTR4oZ0$%9vvRY+PxgxQi*QyxilflM#n$aVW6s@tmH*sBr>^Q8_xHxX5||-h z(lDIczUz3ZvCJ3TdF6m|8F;WfQO^WHEIc;$*{M~u(PH73Q)2OI{e&1WR^xuL#^X<< zR=O1I^mTypTTCXEj%{Fg{HM`tgV(OVsZ$3*QBb}J-u+CUKe&c~(RbjbYb|8*r;bpC zp}@O=Z5b_fx}IDF*dKRih-f8yU=tSzeRft^J3;&BKgGVht&XGTvo^Q z>_9!h#?Bs~w~!i8BrrfD<~SZMy;`Stng-J^kG!^GyW$+)Dck*wujSS87B^h|8eE+} z|Lxn>ceiQ<-4?V@uX)$ATHBg}95PDyxViB2wZ9{`t(Hei!PFBG&)I;>yuRJwCrD4M zo2!J8TlUSNp}?}un-f0$c7^v}+w9z2pRKf7q&y$sOcirz2HqrFD=6G|T+)@qQZf=E zxba6{Nu(r!o45%#fwI1Lk3&F!OnlP!{i!iuCi2i9OAr6*`r8H!>GNle!;Nl%D#)MN zT5b0rcwpOJr!?U4vSFnVon&w&k-}SKo7E5ne+y&v3x`rgF<)OTJ+g1+;uVLN!?taP zw;zREmp{^9(hu(ple~p?xo9hxeYx^AcjmA6MwfQy60bY0O({ZbyU*N}A@%^u{Dpz6 zGpF$!K%SN&0 z-CI>oSPTpdhaZ9gn!8tN>+`+7+z0M0PEAW|30s~b;hYJc1Gs$X?xT$^?y7^8S_*o4 zo>M#0dTN-V_nMR!?g{ z+;mt3nsF_f-8ONVpRPBh+ChY(y2Dy{8tl3i7!BCA2Ruk#{B9?Z^glC?+0m7jZG^6-8tI-d-dwowN7IQg|)pYd<0+f$g`(;NWo%f+M*Jk zJPFLsKL4Miv);VHS5M||^FQ8X{TE4{A|!MSVgiIrqafuN-ntcY^Zi|m+4dOl!(>Q9 z4(%I1y9bJI%bWTi0j5G6(fTG08hePj4bti7m$FR2b1=8vXGiX($zE(0p1B1Gi+^CV zXR-NyB3`A(Of?ef==e<=5BD|`5)#znbpw;ogw>myos56HcMB~Kam$= z*5>LQp0{WA==;e7(^sXAXm6GrrT+?8d+N1xaW0hg-f+LufqNtLbRH1%yq%)>=@0(PnLbN z_nM6n7j*b80@PdnAmpZ>)GExp!n2`yY@K(6Pk(c#+<1LR@A~x7&SaQsR)Q2uLx-rO zT=U)c>=!Q*Ay{8U1rMWx(bguH@>mXUZN+%Z{4TNoCcL#UfCIbx4~uYj7f@v0;6Oi4 z2W$KI?fpFtQBfLLd)?eGTqT|BSad_ zsff`ceIo9N2~)3iPP=}!F+h41<7~<-_Fy7fCri~CbLMgP&*p~XzXBR5f>^M-y1B|2 zIW^pp;{M=WZ5%f>M+Qwy&#X)JTE8;cG+_BF_V#aw7C$gta;Y;lFrd9eON74us{Pi3 z29l5#orBugFtV(cQmfIm<@Y=*U+}`nSXI^3B7?tqhn3rnDDfxH#`zy+fjc}bf|~uU zX*DHE&|}$XWkWCQG8d6`{B#X@J@Qw+3eJ%Vr)eb~YQn=m5Bw9Qo{EJqk$F}Bqsm8$ z8wwh$HKv`|+>g9b3P4KQz4`T&kdUy%ya~H*t(gddVP9$KZ)zY<6vpyIXk>h6Q(53d=jhjeYf2a9h zH2iOpbztg(02___tk%cWROW8f9v-j%9qCIyZPZM?z`H#~88H5dQR0@SoyqqngFlCz zA{zf%0Q82t(#j>8ytf=U{Jl!f+GK0OXSM0>`|0ZWm&h_+``=XO4araOnB_@v@v(#O zv7}hcMHnrFpKDXng>84ljqFuD=}pH%H8nN$^nM$;-7uJY-Ks5;#;C^a0&v;JqVh4S zIEv)H`+r@<&+Q!S_<1A~g+=nl8{P~ut0W1VE~?IJ4?>}`oVitFUT`>vPGmIyYFsC} zQB|%dt)<6N5cK0`^UPDbwf63lKsElb0WYk;#)`Sh*mJUxBW6A%wH|+F!X8sjx8s;8 zPzT5#oi(bZ)#k-|PTX%}>a~!OQ({b(>A5;F>eDA%^?N9PvZ&jc_bKv!sK?+H!wi$t zPUhf>MztOeQSx-2F)Y%wCr7LP>~_roJ_dy}0(EQHT-RSl1H!8`_rQGYSaM=7%*Nh( zDyseI=K>W&F#pIEE6g_gTcc8Jl%V6dPt3kk<0OEC?YZQvdsP(PG}4?H!!jw;(X~Ox zI;p>`B`KfQ8;U;$%1@k}Nzve{;B2rcH9~M)-SPhKQ!XMM`ctuVyHhykJ&cW(dw=f< zKPO(T_dG+(r>o4Bn31PWXP+XB*Tp1`mXc;dX_K?Y;c%$Fmf$DF55lVV*?D*Q7?5Fn zQj*(^r|Bw?VP;1ti@cb9Z?s_sKBM%=Mk{LSWwK8P5=x#5ZxE<$4>mxcE$@7 z<#i(^j0Z?tY*8v*DljiIk+w(a@u>bRv@RQ|PBxAHEZ~kknM{>=DEuTbQ3&Yhe7*aF zRqK!GjLMgmu+N|{I{2?;@yX7`0Uh09E@&!4c5lP_(sOA#^>{|jMM!ni_ z;+O8cwDB|f_6*cnUeh6Tj7Rine*+6w^V3w0R zZDyscn?3)=opP_N+o9|8iAnT8&=>L?z3I7#*_6XMsx-gh$4J`*O7eEdemPn>J+oJ1 zTg%?RcizZ8rw}%Iq?MnP@dn>!QjR)HHKutW>q-J}IFXTr;hEvkBs2B=;SLG@vD%Fj zQU$POm)~po|Gg`ce=UxTm^j$1xC1I*AsmigUPjYz{I8<@V zQQd_ovSjP&eI5x<{NG=FW3h1Xucp5kkp_C?rO%4wAy|$eeU)>>kkQ;#=%_-^6ZQLR zk@AfjH2Yf&apU7oRi=iXs`|XI)sv%c7Y^H==j0sLlD7M#)^CM74|zU&C;@_0SYe=3B|OkrM~Nr1du z1!_m@mRne34x?}ElOmw}i1&-HS ztx^Onu+5C?3!2Du7c)0DToAO8BJVF(*rHL+bc;+Q8!IZeS1`*pg9D9q>aUMJfJD)jQDR!iap`= zFJk?W`!nKk)-&)^0P>_Dy=EdkIqwj8f=t~xnB$k7Vpw@4TyT=tOuxc7_6cjt#^ctk`KN1RL` zP)D=3Pxk2|qoPWE_t3$?3v25gjR_nIPJ(NVI1RBgRTugANUE!=d-z?x3=Rt2>t+6& zCP{_Btax_i7tC2@Wt7GB+s1#c>OT^4b#)zR@IS$~sz2Hu14hp4aJ?h-*c!-jJ?aFj zvZ>?^TY6Eo^D6jgqpao$C91a`+>XDg@r$0g{$#rNev~-JG&I;l6|#}-4fk?Rseyp7 zbBuAoOk4H6lpQbnaP}CGc#DmQ<*(1r0hk(^CkzG_6c(1=ALK&VfFHG<9e_on;E`G^ zOQ*WHUtz#`N}Q&fzmwo_Y1llMY5%XrS8r~;OUL`}+o>9Fr2P3K(G$L#nlfAc$^>L& z9ekDJf#AG(+b_CP&kpU3cHd?q#vcckU>9s2>-9v+ScO9i4w;Phs_RY2X3#4|ij~fv zv41qKFc1Bh=p{3(ExmM@6vsR?-dj;!J#@ z04d@ndIKN7s2`k`_skfSKJcDb#VyO4z`fTaDq&S-uHE4=Zu9tiaOn3Z5XO*(bbfxm z*L=F?sLq4H9Q9<&flQUrQmZpt-t|L^{SN?azW(w>_wCL1_`p20LCjwNYVu&%kRs~9Qtyg;WYD)g<-cn&Pg6n;u7Hp7~j<3UsVkYbC?rwoakAM0!1UL^W&>I~6 zxEjP0>9v@X4Dy+P20#Xsca~Ct@)WZBdI@RMvPgR*NwH+o>RGt59VpR6@?qD6E(m>4 z$JKX_7FiY$RLs?VEl!liRjqB+GJ#En??w<1Af2DUYNug>jzpjH+qHlybPDL5XFPh7 zAFMh^4;QcdBW1DE(ccmEbkgCVqB*jn?H>hJKQ|SwR%-?}Iw8i{|h>JorjIKcH^ab~(mvDML zw5W|^G$%6fps%WWXTnG5&Zi{&>nE~6T_e@Uo$hDO#d};5E^`Dg{r2x`ypl`DBi+ln z*q0MdcA4!>{kx|34p7*Ybr(39ANd< zx9?>PB=jLiKg>u4&>G{9c7E(m1(7**uy|fgC%uIem8QD?he6zbw}mx6EzNiMouxa4 z3dMuEb$l+ZwD5(61%pptcW&LB8h7dDD`#RL->a$Vt%?w~VG`wa&FhSQIjxGDSEcD@ z1&3gms;sE@;(k?dX7bGB=e}$%EdGdaaOG<6j;B6h-<#`1${rwf=NTccRyh8MEdID( zupOQHCe8W)XwGKKQga|x`BHNHCB4?zUiu&Jea8;#WF2znSJ8W;Nuz&r+9_gznz}#m z=6>=QMkbF}0`~%x{n~z1%mEjQfmC|#;amdk=PwP1yR&ihqPEmT%&aZw+;g;1c(?tJ zL@OM|LMQ5dMqWA(>USp#bU@WBIa!|Ix4+1ZL}<7f-u=J7x!t8Oj)kkHk#D4K8pP}h z;*3m-=vtINKEmV5t{_tdf;ul@byMF&*k{{bBVFq0(G%70P;zr=z6b_d>%+~kj6^r1 zxx@CKmmNKpM*{+I&LGhja<{#~+wpPJG+7_o$_ej_5Qb&b@=v~jp5<3hY8Y7L8Q3TA z0(7N_8$9ogZ_Ifgq$L;!u0p`rt^R;5-X4s`bi2pTjmdm_mL}y+`0d-bDli_|v>bn9 zh^wHjf`}xx7G%iUik^Qm`O^M!puMNP8?dzE%u+t-o={F`OsTsKgxWwHn)&c;E9$);TlRA3@iE#@Zu zqn90DWPP@sPb#!?s=_eyh%_xZO@EEG(7q@^6XvU@UcreF>fYWpS1k7mmzHcbUWl9p z>CkAkD>oV2j)hv0fyr3@gYf@!NiVtAA!`FXqnUeq4EM&xQ{|ToB!%<-A1CMqfcn&X zqoD{z{yH=(#do`&h|AwTExzi+8oq`bPVQ+tDC+g**u1l?rNVa~<3}^;I(VXIeHoO> zZO5sZKL082;x1P;aoOLx@Ams^PnOV8mq5c|AOei;-@k7JY-{DnwzGrnr3T4+<5iG& z6^chHuiD?Pgat*`YIpP1NwknvJGiG#qUc{!LQe@({=BNIRa$`Hn0bV=eDbhHLQ3j6 zCkJ0RTm!4e#=$Y+s){?9i4s^FwThd5b#llJ)z&3!6pPL3v$Z1f2F4|4+ga3}6O-!LPuK3!CKDW*8V77XH6UIfkJNS) zRFK0IRGepHL(xX%gsje(fCD22*C%&C{ClMM%E|Fjj#h+BYjG5_XJ3`GA+Wa2pdR#? z698#f6ZD4%K0fk6qTji6IhCO&qtze~eJVWce0 z;S?OkM-vmR(6#~@pi!CiSy+b8K_5Erc`d;*hAin}a$=zFgo8I&<+KLc^m^UL>?s8ueRR%t)6Ow7xddB`8a z{XyJj{YSMfU;(88Rk~rrZ_E>6KUE3fH7LQRJ_6+4KwB zZ@R`HU%e11;gLZ&`KtCYsVX0SX(K!9RFT8AT&?u}27iA9`#kx@jt9gpz<&lXZ+cm8 zk%|YCz1SUndt0|7>*{3QnpBd4jFVp2n&4M&`fvAF8ZyVbv%B55j|2UV_RF7(@{CnJ z=oov@5aalxW*X9fs+-$~p$inEd7OPYYf0z+5RXD;gNuteSd*wJKBWm$=cULquyo4x zAc(IDy6VRFvHuwozgt2{S^&?1SeD&gstBCnV@Q@2Lzo#(p&Fs966dp0eFpRx?XcZM)vOP2Qh^t&cAd`zS&IMB@7qGwo#gp?Xj^T& zH1+8Vii|W;XYkDZ1J2@DDw+7wab0+hCmoxlpV_{GfknhDjjc#(%rW`q^MN9e43ZLs zMw$c#lbHS5Z@+7I?{Xm3ntuB4845XAH0bkt0ia>upB5{**Y~pR!~MavorV(@q-BW66`|{B8}>)4~k~;PdJ7d#9y9I?uJlB zT(>(Dg;Z)l3JUDDU8n(qxrO_cZy?SY(y691tgY$-^>csmxXW+prmIh16Gk14g_#2? zCZ&FnK+Qlc6hAhmtZbSZvkGpquWp=}NxPA}&xaSWXn ztK-8xzx5p-WT`;f=eIk3MrQN#W$^Y0!`_XIuhZo!u|xO9y^IwUP(U-r0geC4Yir=j zIy6z$J_;$hf@TaFx>H$ka!P)Fa_w=ay3GTX*{jFbmRv8$XmTb`ze&+v<`bL_bES*_ z%8o0Qvs!e60*m?vQ73<4VA~3&l3eylmeF|p&PHX$c|k#Pq_+sd!*CPFYCNDSRT1v? zzUXs^9+29R>!P{2&p@cWBnNyHt@zv{mDtKO=Rr!SMG$-jXn!Un>|8U%?R%gA{lxvN z#SB_VOyZ8MFGTIg$;ls|Vf7{V9CtpGq*i>ujlpg7f%J2hu$uTji;~lf63s_hlgMB} zj+sCl-E3|cV zu%>3X@d37jxWh7Cue|P0bm}v7Viyu2i5W!*-rwy(;5VWuoqCh%xKVS#3qaW9qa7Bc zFT)IC83Y5FJ^V?twerJcX()W@r^CGNhirgbnIy){ykZ<3){1=|su+g42l~U)lnJ-* zs5q*Se0u8v0W^V;B~Y71nq#j1yP64dQTHYNXl6;u>#~2Tkrl4n_;j&qvsb$Q-EE{y z1V(%YD7Geme$I$Q6bUiL#xI8Q6X-U)67&oVLZO?VRZy^0@CpR3nIAq-1G1tQvp4#a zi;G0q_jgIQSAIl6=WM2^a>AC%gwJ#F3m&3S1=$W#Qq@6fuL7hm=)3C#id^+)$z9gX zP*Woa3Ha_aj}ErmSdZ7vK&8_m@RcIrq--=D8cHFDY8Pv8&#=|IN{FbfvU0i`zE+I@ z$fa$>H)V966XMBhS0|Jd{?hAbPKEbV2Sb`|u>b);Q=foLJIhrY^+n1?f-|Ph*u+8I zdwV$oxVdvkhZnGL^!zLTb8)iU^yhrn6hGLFXqDALuVa1K@HX^1NJc?}>R9_n3{fUS zhL8n%-1=d{&ZbR*704-o8!);=J2&cDQtHAHd$t3GI{N~5{C*vk^9^SN)lpJW0q@~r zedo?JRDoNcE;|~xAbgtBl*00lS~@@j9Y_P~#fwOj2lX@%05C>IMo|+DZ{HF_j#g4t z#rHejyGKn!li`1IOfOpN{4`IS6pgiWnjF`_N>@?9bXipnSi z*#zP1c=LVW8^1=C#-C(t4@#DdAxq_$mWXf^$#m+=x@soyb+rWe8nT8Yp51ez2`od79}wFGtw0`fE*-yyp*9uz`E zr?#&sxJMgDEBS8(LBc^krc!+}6esrkBnvI+Y>E?cU#M&p|BYKEcY7GO{*2K91;RsD zgacwA6?ABLjTz~w zclP#%4%?~#bu&%75~d*!+0(P0L^JE+u$>K7M>A21+>~mciINN zsO8PM1TX0WiDIsQ5ky0&?p1Gk7LkyZ(-WYUiUT$5-6qmp9yF#GsM0iF ze)pJe#!HiQ#Rqqh@K~mgWtI#CbpRMo_M10o;J_e~Afm2EmNDcHSz#sb^`>E=-WHqs z-w1GhjzbDR#pSlVgs}@5B*%4)9pD8uJq94a0$ipvY6?&zQa3}{z7Ac%+3{*uK^T)E zBfA1h@-1)KBsESD?AcmD&)KNRVDyiNDa1?=YJy74X7ZyfNCF1(^|*Wa+bOh zGNo`oh6LE&K;DbI4qBt>{0sVops#>+?zFBdaGj^b%2f|mr?zmfF?W|>#{ulr{Mw)~ z&j8J(9fol*e^1Dj`L(qo`yrGUNP7Y#-T0`r%|3Z6K**h+Ker-82|$gqGGSma*hECA z0|ElH^YuZ|Q7#0s&o;{dHUb`A-e%~7XXx_|RJ-1Tnu26~9nv!-W&e|dNzgiF>0f_( z7=0xwtvi)?Zq#L|{#W!7G3wBs1 zmJ_Iz8>E|$v9aN4ZfVIF`E4TYj*c^`_ZEHo)s3LJEJ{8;rek2W(tK(1gpmJ_O`Spv zWx$KVRNpZ5;Nj`u9?s@zmtk7bq*8Mzl{k${|F8ic=XDT>f4!pm2eQWvK zX2B~^;%B^mje=?~@9o>*p1aEK?(Tn9hiye+D~D`FL1NwyB?2x!euc-95Q{b;nCxzIs9y*nvc8Pe(a`*&b;cPq80}Mp17inIu8!$p<7Xdm+#h{ zP5!NjQh4pm%_Sp91WkiOQ(dfd-2ecsme}rsWi48mzvd?3#$H>uz>M#~OwaVWSy>6r zwEZ77%$!M89bROp*8uW)E?#T_6M@bkqDe?fqJ{#S1{v-FnUU=C7bfkoEK!KK!mRSP zjE)I8OdC*flUhQX5cSx3c0w66tf)rLNF+ACR%_gLrF*T@NpYti{r&a&Vq#}R=jP}$ z@^Zd&EUB6>$YoZ1h3*EO#(izA zrYM0nKG2hbmaPQPbb*$GGPR2Rf1ll<%dNe&cl*}T!4Xt3VRLiVFw12Q%n#3n3s_F; zTjSCzRXEW9M%WobA|fo**|TRMtGwsF&5O1TjE^Um*_%uB0#S?|R3dE2UE~z66#-I5 zFEBS{gNMQt~PxzlaIt|l4+pT4Hx&;9U16jusMa$QW5VT-C z2%mM%rzpJhy)~-Y%{F*(=xQ_p=3CD7J3BNfY}nx+xAN_)v|3c!G5P`(oS5UL*+ea6 zWo04TAsz%INlHqVOTWXz$DgVB{Z3g)30t4{M!HIb23HGkF(r_Nk)RfIACxcHXO*IJkXL13sxkkFI>ABa&^wFIq3xr(N} z{Uwjpi9g$R%9x_rPg-ilP8^7O#PBQI`#kKx!4RH$e3ocq%u@Nd@U!rxwu<0R&+^Vs zQqb}N?F=lX=w2y|LBOyW4D!%HPYtOANtq=cgZzPAQj!5lj}wn}Zf&hjB3gMw5wf5RYgL$+ibes#Q>icDq=g-B(h zS>?|Gr$5@@FONw1Ame8MbT_K=5_)T3m*o-3BKw?#dZ!+l4drY8aa^RK67bvuB8!dn!OE1=VdGm;dd;SOgLJU465-2YlZ zKZgb~ z0-IyI0)tfU0#Kl^8e;$c`X^QjW{3^eX`K3&Pm0IWP<;B>&cLzxeb? z4JP4GbY!n<_I{<7`W<-#XgT}mdwC1cASD2)pi-#Y7D^tO3ePk;a0h{o(8CRDXsGyO zq3Ylppcx25!V}OV6t8?Y&Z5lWDqd3oXN-5jW0SDoEo2W#r189^L<9`VDtA8+X>%AC ztN3KmGUAaDpU>Pcow>r8QN2Di3i)W0(63^SrkF|B+bB{B0VF3@`xgO7uqd$ zL@z&l5D5+qHM{&qdXGHef!@Wr1^PUomspnJ0{rf}exT?4lkFyGo1w>>t@Sj#j(&_X z^XpG{6)kmW^_Anru@N8~zp$`yiOWwlplo1}6;J$!8x?4%bCBaxmLjzx2RSoG(Nm!8 z>`=RyVR(_EXoO8n`%F-crbrQHNW)$|+}JGDXU9f_F*`e(4Nd@(Bt#AbIQxxOdL#@L zBI_bP;l)9WA2gItnLwx4c_*0^Z>ae~B+t8#qu2IU>@T>fwYA;n36P@yc8xs~zl4w7 zmce14CCi}Dj_6$Nk5~xyRsiG>;SOYLC>)XO1ZryabY6;wLLti|T9NA1I^K)HTYu*H z6|&+vmcE=_w2&%4y_}Va(wwA2O=tBJwHzg2RY7Zg_xSFg zF!pg#S65fiq8S$%a5&gm?SPKgl$#V%i2yN@33HX&JXH$u;Y(aNk zx@gccUlQh!QHY_>A|1fz(_hZ+si6hg7i9(VUI20(oY806Cb>hlsD|NVjkig6vPkBy->+%`-n zePJ@4FGMT9N`iz_`wWNlh10A%N48r$*qt}L^gXb{v2u2IeRj0+0qfn*DOgzpMhDWd zYvel0)MdA&FTnK1Vh+`p%2&0h7^T=eb?)r2#i9S~#2zoLq?tyJZU#urV3tB0Y3w3u zbeK$7t=lA}Cd5NXEAvJa)S$}k${)k?&bd4%rodvv;M&B~Q-qf>GHvh{ji%ow4bceY z38EEJYYrb zo$EON|1sj^2S&dEsm2&ND8CfxrBB|M^9tmzkO9 z<;ezCz^yHybPNOi?0IhPEFfT@F?YK9%YgXGe?$ZpoIF@Pwt7)BBLBGIU^DGlBIj`q z_1pKo8uaoxOcl0Y?s#ZbPg=VsJa_o@dq#6RY=@hk}*h$Uh<~6=}n>bkxv~XRPs!_&Z+a7<=yuTP&`6~6g2W%<j*e^`9b?lv5}{`g9UNK^k>UrjPvleTU=Q}%=&oo4 z_g|JNSK5D2FZZ^;z+xbx^9hE4@L`=EXC`jDu@X^?a<6dG%a^2;VFOs@nveT;!4+RE1!Owi6Nl2m6 z0p|;V^KoCJgjr6I86ky#fB)l}8Yx7Fln~GXid;TEzSFw{&|!3^-~6d!(0c#*n%vZ+ zi)Q}W*SelzY?A6`I8INt8KO+80z>G)d&37eNby5OMP@|>Epl(bFZ^)q0G1l$lQXCH zHyq8yDZ&tXPb56`81;Ssus{E4$pWqs3FE@>Ar%HQQ2MNYwdob%i>GV^z#_nho{33W zU7g6*&dzzN5i2_SjmAo@ZLe^>&zBpsFnG0hbPP;P#D<3AGs}39KRK_m(%sb+@$W=sKANAbC89qr z(tPQgTnKv|&*ADFvgH;db-D~zd>I{0>|wt!5ImrUz{A5EeSeQkN=i!G#3Tv^+W=uS zIaanTx}h&$zI^%(;3_)%3E{)@Sb`E?E|e;`G{EyR9sF+`|hSSZf9RT+GbG7Apcr?I}Th8=|^bu z_??^$j9Bx%=eo1Qt+Qzo?bT%NWcIP+gG?7H+ivLUNFIl#+Kv-B$&VlcoWa#mLc#mE zP62M{$k6iOySkB?1Ul5L)fs*B%4%w#>i(pxKM|Pt(ef2xqAM!IU<+qpyyo%bY6nOb zrro4;Oid3lJx*x3$y!`R1wnd^(L-8}muv)^w5iW3pKGdpyw6GtD(A0lk z*d4o!lj3eV!8^LEn1PaKagIBuu=&g;l#dA-L*7q@e-;u5l>Wrrm#*?9NLjV;QJb3Nl~0BQYO*d+c>p+m#p2k-oMv$0hk8XT2~~vY=wUsKWD~ z^T{}MW~kf|OVt6Y9*8opdb6RC5oAgOIHk_MzOa^QWTah$QS;Ot-{@T^5xWsqo-Y+0 z8|Gvu@wm51M`UN;{7XTp?w7_)x(rJ|ju2JO3y?}ui*qGj5MJBb*~zM|W`GCY1wm7M za`G7fy`Z*&5o|9231>B=v#wQNrcmlz8zNk6_`6@CR2ZtljaH{*=b2jvpiW6|cg3%?MjIY z49{(K`F%posc+~J;j1x*t05tZ#jfj12YVQ0?D=a>)u!;FXJAlNR>t8st$GaV@Rqyp zn-<-g|1V>~p(GLu%0!$yGlJg?F2WEr6tu)T)C1qYM}GfqoCbr}(4c`)OL&_~Xq)9f zr_;auBB|e+{^bCh<35hf?@i@ggXT;sb5WbuhMRrp(-XR_Nh_#_vJoVdd%+t<`oDY$ zgtH;=3A&dEu>#5%AaMVMz7SKfdY;$e+7iTlo%)FzeJ5LrS^63*p_cs$|Jf62kGLY-tw9`yH#wP+iM7VS7Mn1fb{&~7zP7}P$aU=qHklULrl4p zLk$$*V-bplu;76zum+~94R<&!^h{=>bJC9E)nWR7GA_>#o$P78~)J#wf zDqC4`AYP))6O8m-`eb*Q$?kJxrdXrCh}wdyCvfHXNQ8&3C~EjvoX*>Yh`6s<+~^BI9?Ud)T9{PVv#2Ih!J^VjfDmKkgXrCX3|^YKuUZ?rIkpC zxirPBF|r>SG$T=xlTdlk8>!RMhA|sIgVGJ&yb=YH;H~ZL%+gW{VE3BtY>|*l`-hcv`U-{-uhB`BWrQg4G^z?!X3V7kg1TE7} zuYoE8S{9yb7b*imDXGZbUg9)alfa6K{7oMh*9S9X{x*RltGK%<^{P()$OYX*;fgBC_(b-oF zLoM)~LGx331i5!;uEEulqLDBfW6?`e6gQs`D6*eHK}1X2T{1y6*7_g%r)a$KdUG9f zHn4oI%ej+168*o#lq&DPynuv3**m|=k3}WlTR0O$kFeeCUZZ zGw)d!T&rGY2-MM6B;GCUDW%BDazr)Gheh>I95Vs!Fq0-p!Abe`Np(VpG*)Jg!jtm= z5C{TMsDT+lWm`Me8<$leUMzILc6K@EQ zV_eVk{xq^0npxCmDSnB;T#qid_E6KIzE>YO6yz(%3I+WyhYomkmb#Fcni?TKKEC<; zhXWHYMSqJG8v0O&si%fZ8Tz5;YHsHfPs7hxC}0|z2Iumnj%xHR(jDz zMNW-lsasu(>TsgF<;n^blIHIvdRtl%^LzBox$}MZ?Ai1G?|Giz>np@dgm<6k=H%>F zDqo(RnW;G_X>eKPy%rG3;PcGxc+;cUK_VH~PzX($KL z?(2L1mMe+mgUR%duYv^-LKWfo@Rx%CNHZ(wj4qUJ{Wd-Q7Ricp{c#;g%Jk@ThHf8r zZYm59vdE{o>$V(1JPp1<>5$xyrzi%mLFg!e7GRgW6i=R-S6}+-p}j2D5`qd$IF_y9 zR+qZY>@aDVB_)r8_ocF=-$UKv+^rl058#EYRuvi`tVID76b_=fY~pq|WvqGS(H_samu0`+ZolVUpZS0Be95LulozdVAH zvp$Cr&Ml9{5&Dc+uZ68Att%=j)-ah|4;FH2bmD5&Qxx}l2w##W64`=%)z6T)o+(8m zbYbSX8cpEeMgXFq_=%y}qZ*EqHKn>xT0_hV^Cc%dZHXTd6d+V+e%|56{YjJW9{IO( zpf&K8o!w>G!$4OF$H^(5mF#|z@qbXD>+Q-<6ki=YK20P2}q_{f% z)HVts271Kj(94~j0}~>AjsMp)do{cVIAa$XwZ?wd?1lXv}J5XOuwBj z0227xf84IuC?+p#T(~b|svnK-hL8OD(tpk4z4TLV V<7~}sR||AqnhE%UJQX)0^EX)1P*VT^ diff --git a/docs/images/chapters/molding/2bd215d1db191b52a89a94727b6aa5ce.png b/docs/images/chapters/molding/2bd215d1db191b52a89a94727b6aa5ce.png index 85adfa93d1b97e6fa7c9b1b695b2bbf61b1d6157..260066f34add1d02d033d5454141e65968ee4076 100644 GIT binary patch literal 18875 zcmbun2{e^`_dk4)%8*PUQygG4bwAJZd)B+w`>y}`|IU3{ZpU?fukX0`XMgr*pRhY>3g-xE2~jB2IVDAm1`35I zibCN~oIMSnT&Qd}f*+@iRTMC&6Xd_tn#@=fiV3BJk=Alc`Z?jLujTyb%zDrGtBa;I zylBnt!W+Lxlr` z{iW-?*{{gir!L^1rlDEz**|*tioKox%a_-V3tPvTwB6#17abXEdqgLL?7*q8!-R0D<6_hX&28F^ zpDs5j3FV|_DPU6eRkHicQv1xr#CuJyF$)O^8F843!;&mFYvv5%MBIXL4BSo1m;XDS z|KB7U%!2AvyDpqef+8-eMyr*YLOJkbT1u>E=?!Kp2cp-3WuFI(#=^CIFj` zfzj;RYf?HoxEhC+xhsmMEypgwpUzFQ*iu}lW*0{eNe}! z`CfIJfdTGGNBzC&-Me>Y=H?B386A5EleYe!cV;&`L(gZjBkow^RA?l8f$dB~XnJ+)H zDLkBhNzI^waT0ZQcBT<9J7wBE9`XA6u5oWFSyo=&1u-5p`D?{&He{5k&e8szoo`*D zvU62WqXYy6gIPJyB(lu!<&xgoH*yQL$b;(@uiv zOzR#sc_%}^u3pnUkYqvQ`ibQ*v?W?)eH_{7(xnh*?2(gfP_2BSnwSJY$pyPqeMQI8o^=>)~;(v(uw= z|IPL`Ey}OK=%CG9zpEL2P9B7o^sA=2c z$pJ0h%>&{G*589>6*{jvZQm{b{cUAEH1v{8_sb^c_>;XC%B%y(RNgc*Gpqk(J$6S+ zE6BF)i0am@Tl_S{G-6JiZ`@`Gp3({uPE@%{`c^kh*Sk)7hdzBu`z5I&zl-hd!S6hO z!vX~tm)~rQH+&97OuG_-j}CV6P$47?;VcTzn3$PywibKZwwL?IysQ_ylV6EBUq(#@ zp1)b2rJlhXJ~l8ARJB%12g3({fa+cFq~g&Fie^zr&&oP$Zf@>>g2xtqM<|{yv&$vP zsyMyjSJ2o>urjYgfPC}KA8tMBT3bHKv6YAp&50efiHIe6{;9<$wOp?;A1_U$S773G(k zdu7~nt)_hWeIjq=kG{%Yshq4V&Zn0ok&}&?tQFjAx+J9U@a)wq8sx*vFMq_O)cpC8 zQ@&PvklMeDn)3Si(Y&91+Zc=3?|y>zg0dw@I=tp8D8ze|`T6UD1=z?8r|2Svov@Z0`DSrpoq#Q_)NCkvJm5 z1F%gV4(A(;e6pr{D$fFYG$uCIerNFB?E6HU##HG*-pau-WGcsf4pv2+mPjvOzT7)9 zau$V+h>(?+KYjW<kk`GA+vGQ*AOixOnOx>Im@PK`spZ1SsPx#z0-Nh&%R&n zTcXI3OOnT5mY7k&_z&o*)ZVZsTp2EC>%Nup=;TvDFW+?v;mA6I+s6k6Pw7RAHD_ZK zo7h9qF&!q2X|kbr^z_0RZn)H&2{caI)msTY8Yz4TlP~7r=otO{dGEU|jGWv<_>4x# zN+yH606B85t?F8uZ^&as+_pOD}gM;#l(D@ObFm@g)zviBqx0Rs_}EI z?0jToRZE)fZ==bDo{hmLE7SHB{N^*Ks`thx?O=|+%L1Cx#4o9^%+CNmoKVm{WX{Z!La`LDQRLlA!6`v zipt6Xh6T2}5~w@6x*c+7IA@MZvh6)FX`J;9q)zOn9_b zBwC59{1~o2Zu!6$H0Mf`g@hWex&9iDI96h%S{;lz^^*Re%(^AhlBm#TLQLPj<-)6L zj|pw6R)yZae-Hh5%OyyAPJs4S=$Qo7_kM=^fHha1&bCF(0z#pl$nNOulyPwpVpo0T z_wAdW*YWj}P~g zyVtTa-!#IJ3D3>V74kjqcZk7&AHPY ztx`>`Ay{6!3cVPc-kC&(7M)rTK>#fLc2iR0&cg<7zbY2JD(w{g8~8gwOW!{@l0VKaiqqx0UDnCv{bUF~HsHg}r8bbvSqIOn>o0gqsTIpXD(9nzYn;Ak8IAQSVB=6rs;@J= z9}GU88qg{H5Eioj79<^bn&6NCJD7B;PhHb-dW%In=R%6lAFlB5@b#nJ$=S-4{HC9y zAET3#&s~f+V+%K{CBdNT=Qn2DVlT9*QoZ`g`?l4LzgXc4d5Qj;mz-an-^Z1dlo-t? z>=qig5F&83^92{Oi0fM^CrPt0{NR{n(y!mTl>>eyt+0^v{wJ%3sfIxN#bj4BA79i; z3x!-^H!()s%4*Pt59c!9Mn4+r2pzQRmc8ZDq!J9qkP3)Ov{yFT|;>rXIw3}bv1jj+3SNs6A~Q{#e96yV?5va5_uR{Y{b-w zpB!$*xF6r^9GSZ#wO4#c%2S;2yvy46NPx#bzt_+q<^^Ct_Zk^ql4Sx`@y{Ux7Uf4R zWgWrrlAXxn3d#0YM=@IUw?CA5Hr&KVO3W#~yUm`n`c^`KScJ{_&(X256pgo~3|iPj z4`)_naN_$%MzC;tDJdzXGW!9-cD%a&L@jIpX)G77$Wz(wk%Z~8QR;sKyhqrBV zk1$apS#wFx>Rozv5c@CSKZ-YyB!B$G$E zMP9~^QmB`oHQ^Zrlpc3QH{kZo{ZnP%J-EMnu1P#)ln}#?YH}8KqzUbRyi-3fLQmoa zP)z5e>A8f21pddJl;X#Gv(;iUtoHu&FiCAIEghzG&xA_qr++iC7>1@Poz9~48VXVD z#5GWXsCFncR|ckn1kmP$+B~9g z%Hntp!r!UH8zUIwhAS@^f*$*@Q$-zNcMNnn>#1M9e7P9!TYLOxW#i9Efn$>znt*Uf z>h0awiC~sFu>i9{Wd09@Dr^I8zJRe*bdr2HUeb zE;NRTig3Lp-=J#!SJhg2bu~Tmci7aQxy&)-~3Lp`!nW-@6h&~lm9Oov^ z2wM(ufQ33&+zvO|Ha6NWIrev-ZDLnIdiQb6Xzq$d{LwUkO@L$v>qbE7>AbUq1``asIc5 z`k+}ubO=>5hU;oNSfC1fj%ZE-T@@=Kx;UQuA#xE^%-q~Wwzd20-~lO0wxUG}JIKO} z_$a4{swSSnfvrCHRRzH?DHX3imOH;3y|l=dUQg!0^tE$!C%w2YfMtgV+n;53VYiH# zJlQUb4Q!)r^~ULcg)uXdC;R8~t2aASQ{tQbZ>hZ^9F)PSHLG#_2$m0Vj(^wPAMZ@a z+&E@HaGD?byq{LVRbVS!EgzgT;MSZyI!+oUEiy>s&$(1hWCr3%6h51CV~-_({n7vt?8a)@WaGTU!+JlQc5u!< zJ~uTqG{nBJ>Fm4L7+Q@M;{!&L($F9`?Rn$v!?WL=q^Wu$NmF3?ACf87w$Yx+Ar0}m zAFi|LWlxm|r1MjCmo~5L%pidvn8MyFS6i>W=};O8j}oK(=OUa4Mo4&eujuM)Rh(qT z3+5XLdeO{OD|OE_1t6a6^PXKN#dY`Bgif{FRZ2qqH$H#N>mn4(Pfm=g>sx1wlhP=< z@uGDnV?sH`Nj$MNK4$+*%*vCv?KXm+(F}J4BOR^zWdn#Q(!;625 zlB8K%g@jQY`}+EmnWerAH-IhnFu#uaGRfqIz8P{?l8{#a{PDPAP)Ug#xB}lJOC-#? zb9sL_K#|j!4qH-OB`#uO)w*m-ciOq1MMCY2Xd4SAfXwo*!CCn7If%RXwMWx1~Szx~^TyGuswQ{NL@RabtxI&tm~=f<2GtBb}5HUuLa z!v+TeNg4?{PZQHHIVrQ^%m{xqB1U+L|@^onpvW4hu{=k4u3c z{K7H}=#6x_i1i-dV+vpJ4v4QqBY{teqH*)Vxn*<_FJ-ST8ylP3bTIYvxVR7q3)29b z!RBG*(qP2uXLbLm7k;AJtH5P{D+PneDd!_sXOdIq8Lf2YX7pHMCZ^#>dG7v_1MC{C z7RIVfJ4tg1>z(G;ZDC{6I@1zSWIZPIB}tUxQQNZ%Fa=837ygRenQehz6HJq=iyoO3 z<8XQK65OIZ8?CKEOGmfSOmW)?zyxvzY-)D2(=^dxu|vX3xEgw#sUJUCjRxQoQ5pRl zDcS?V1$c=dwJ^mJv^AW?ep9;N$<{LAhLI~@t7KaQ)4~p^I!aAezm+23#}1&_4B#z5 z05CK%scVy*T*$~FG}S4$V}`J4y^)B|2%H}>aOeQU|1%x`sx8pydz~AK&ez7n}ju-*oKr!MKM-&MVs0&{p|$2;KT{-SGtr z!FxB*oA_tY)Hy<9>s_PlZ!Zeaj9w=r#_M==x=ae<0z23DoD#Js=IYu~}FR4+|4 z{VYp@t^V69&_f;`9t4JW^YPbL31i~h{1wNN82LyB1k$>V7bob@mkQ%b^0KoDq9lIP ze6pG7tP*C;;`PO2AUT7aUKrTc`IWZTRx}F>Uf`MYzde=e74LJ_UphlVUk?D+e(>(S zX#l*7cgfR*?3lh@%^38T7>rc>nr&}A`RmT+C|$ZrVVslI*eC0~!2+LP2ntCKcGniK zH^5W8Syon7itK0QUv(`+7z)!y^3z6&cniGP)j#U3x>$Rl#b&8^9{2 z_i~Z#PtF07vKlGGff>oN(@rhSCRIpQz(E~p9r5?^lU-aO{ujAiG53swnsEL0ty|}S zyTeDht<~%zQ4A2pOnBL)*^MNP`z-v-$00KFlnv<&lAdm|nOa6$H+#x&5b z(>`95LWjq#adC0!@801nFeq%e&Q4ATtQ_``+#s37Zk#=SvCgqAD+yp9CN~gJSRc@h z{I=i4RTG8I#j@YpTqd~~Jwg*HkC34#eh8XWXx&LLmA|Ya7(9Cf(jQclye9ClbD&is zvs7BQ9xo@`o%EG7{qoA6#C{Y|`h(d$r_nph1NE zoKra0U(iOLroGk1{o|GD`-u1PEv>DMW&&t~m(Q*#=q9cyMJY@A{=A*}{z$;^UYO|j z<^KA{;QBWbQYXjtDJR59H;*nNhF^SVKogj0OZ+aRTN!Tt-@IIq7)Ig%AGO=Cx*=0A z8Qu{~jnC@xR;Lv&KE!BI7mEpR2mF)3jHY7@U_B!{K!{U~r zsw$-zPoJ39@KcIg+H>#Nn6&fA#Te3`Kja~xFBpyxbzGRL+5IjH4-Fx{h;~xO+}!Zg z_}%04`})$3hY9Afk?D*ovG(~8`Sze&{~~X4ik&5M?siTF9F{mRU@U7NPfMX1@8f+iop%mCwiB=XI9=&v5nXRg~Yq81Kd|;mM~p zp1Yqewyo^A?CE*hBg_?F?Ox+z4;}I9gv32gRGv^t1!*wFYvQE~87*O^iM`p_kkRSJ zxJS;4L)rF^dqjjpFh)EZ3hwTr4h{}>U^HfuUB5pJluaL)SD}UUB2d;eA?N5^j!W0- zP8cJo`C3noCQnKaY5@=%IJC6@j(+{*O+&T&7AncJTsO17VPz-1sHks^ek`KD%=ST& zxck?&a(}z6u0cq3xRp;x-XzWKL&b8WaDTx>=az9;B|QGO7C<^;VlcA($9rnAmxaSd zj>ZSs9tg}$(fN+cUP)>hw&t?>E%?4|%>~5k{HiddKc5Y-kw>G3XK{p4XLAZ7Kl z;`HJ-OaFH&!QIak-8N`1n1~4bW&R%2`8ncJ%*YZmJ2FLD>R;L{49TKb{10&vQbb6E z)k8WlDIqa2qM-H<;nANR%dN#4j~$u2cL(=FNB%`TKWq#lSd6-O(@SJ1@BW$Bsv!{8 z;Gur3tX#DjFW>4}-%HOEsjt8%{QC9l;l7)}Cx9j5yW@@!JTr%o+>(~X1DZ=pkG3}> zl!=Qh_?=cD6-PoYc6fby|C3gt_1;1*f#lkSD4TDbFd2mRbJ^m{AUZo-5U!J!md1eu zM6gKbm$m7WnMs2nk2f3fs^_hg@qrbc>AfQByz~pR)Va-TkrWW<<7T zCN2ujJ01$jZoa>j=yLPf&rneVdwcuEHImX=_SlYZk$9*SuT27<{m&0h|Ga)+0VE^> z1r6QjFPNB`;-Y3|{nayM-VmyftXTINe=ZU;aTT^BT)YjWV|vM_ixt`W=!7hmT(;GN zT_73}KMLus=H4_}=r1P|^p}-`gN2KW0O$#7iGuzgaFql?K3OF~i(nci2a z;<5dg`S{L1@ZPzbD7i9GCH%(ww=VQ4L-XolS7PW*?>|nWWo?&y>EWcvTUc1cKidBQ zk4B<$@cf;pL$fHp`n)ogS=4%cMa0S^Pz zKmt_1F4I8yjEcEMIBpV>kyi!)Y?VHLv$oYE*%p3*d;ObbL7MB@cV-FU=#h7{B_di+ zLO#z1YcP&(se*S9Ic#E}|FthYAezf2=>ZHT;*EjJ3rmcOWyRl7d)8niWopU}G&aST z{-7E#9k`$4c$#36V7u>2=RMCz)DE_AG+A@BFCGoCT}g;;J0052m&>;3wcc>X?ay*1 zlDaHxVD2Eah{Z$g@9(dw=|-GF5D)yJ2*?>f)sHOUTG2PV=2RmX&RD3ntISghrsrd~ z^@@FtJdnjimc8B|4_S7zfbVlizIT3po|=c4lnWt`AU8I>=6iA@rGfcJI%Umi`5X$D zc!eb`nQk*@BU@({2M@OTR8yG83(@*20P8{y*6J=@@;$s-{P82wW)`%WHE8ja<|^)3 zneF-C*^SC_a`>j*Nnv$ItF|=aZiT(eq+xGlcx7+b&~hZQtPonoHU%eV((!+ILchDq zMChlO#}-P$&#?~uUl|TBeXtDapKVxr;g+V!9Fwb<#Rf)7=FuaTn|o7u@DZWtJNXQA z$BgcjYp+$Yk&!zG7%(+~8rea8W|%skqe^Q^A>+l^OMcNi9 zyNFE1OU0aeBAP9=p5vGmin8r99RtasCZKyYAJkr61v2TWF&;czoSgNrZ=wvUT%_N{ zd|zbwp{V#ir?j)H>*XBoDTuOLAS<;o6T$asbc$5n=$*1tN(6fq(PWY!bLKtqf$-;X zPtvihgL)2-NE8PJg-(?VPo?V`FMz>-v&*)9S0A;emY37ml-+oddG0dcn9MgGgx5{K zz~Orj3ndryU6NR}+A!@p0eyU11Z=+N$tF^{3HJ=bWTeAOo2`UqyKk&D13kd+v~&kn zhFn!KF==kjgq4jg4M_rJp8N23H;X#3+SKlw!nB3ch`q#xw;Cj10lw z^_3zyP2TFLFnX;^KV`T<1&$`FH=C5;^4$F=i}Ul-ki3%d@bJjG&0vR*kDryB+YCsF z#pyDJjAJ#FyRd_NMDjiGj_jdCvwv6#lL^(EqcwQkA-b`LeBjUV3e6=g>?!I^rOAql z3QjJr4R^URySwV@{=09skaG7%@}Ob-cm+X>fq)xe)aZ5`oIQ1NbW+Cnr|SJgeJ7fU-zTwbf3bdfB1}0 zPIUGK2dDlIcdV(_Zc>B7rYeW-YPl!zwmNfNks9W?JWtWsDw2{%GLJVescz#*$E|hr z^ic2Q4Is=W7Z(@EUSeP_VNgiar_~Z)w`+fvX%#(?|RI@A*g&3}3eXf4> z0S0qS=EwujYXJLQ2uWLHlXC|Mh+34=%7zd({9YXk7O9z}Cpm-gZip)(ARs_2&IYB! zMUNkYBBNLGxeH;0o*(G03UEd?e66uwO}kS(qsd$tS60T3(lHqc7x@HX(fHZ-?gC72Jt)mdTkY31U z7TaIv>)Sr>prxgC_I2xHP+XwedT`(_wI*xliDQ+Smn5fJsw(b2KV11A7UWR8xG(p)tTLMG|) z+mQVWBf!`z_F^Dq$OdcRbaC2H1qnhA4;Lj7Ap(q*GnW9IAYe6cqIOjiGMNrx&}ASd z5@2AMRObXGk#3*TDv7Xw>=!?l>0 z7|^C%cDfwzw$&wU5C2#M1@FQIS$O}p4^_R{`ZIP0*Z>*Si+GgyP?j%|sHpIF@!;PNzoml!&z-UG3P2knqnv6k%#2gE5OKzKrK35BXQ=8fwq26TNquztuJ z4?#5{B2WC^W{-#fJYY6->Cx(h@p{E>8^+@hBqLOUzLsJ(Bc8}Qmon!2j_q|$oTY$ zYH)BcAjua(qxSCZg<25|I-W8wq-7amQ1ZFMw!~b;UU~Zm^yp}c%D7y;QbJKt(Q)d~ z_VdsHCne$B+#I+~o#$J=+kFZkLioV6$evyw-Qj0=RWHfx6M@5VxIEFH76l>)8#Ez( znq>z7*LBDg7(RET2s6+T6s@n|vIKQd@bJpHW%u*+A}DaZz}a7W3wg9DQj5oLJ#z4P zqs~lZG(FO_M*pBqv*-Zx`%oy+zDsa8=x`jPLF#f#?d14~IsP-m!en2!AZ(<9M0iM& zN)5S^Otn1zMZeRD*WWJYUDJ(b%#?Mrx}=CPVAJB!{bY5~=U{BoL0@Jrv+!_XZebzB z2?rN6G6+Zkxf=K6KS%uU+f=YEEuv$J3#}a>5(Oe2H<73OvyjdA>Wi?_y%z2}FBQWz&gp@w z$Id5_Oq?{JMz_U)As1a@VVm%r*Epum^X5f%j@ zj6AR;s_dF~X4XJ|1-@F^Nx+qzBc}M@VF?H4M$e~_zFTxH*%lab>YsX>qfb(pIXKP$ zYa<47BMr+U7Es5jW6qlLa9zIKcushn z^u-7!TK*rC`{|cwYG&jx59S=rgrFbWV~R?!y>cEoZ_J5-QTQxE?-2->R#>u3D* zN!b|auW)H4d%>%Zc)lkCfBuK_as5mJ1FsuS^C^CIvfu)v!5FZ6^{XzAK*N9&p+wp7 zZ!#A5wJLjaYwPKs&BSxH*-H5b#MyuVggS3yX&Fd-_Uq=7PC1I8S*ufmm+K|NtN52KwKZf2{m}phX|31it zuOK&q1U0x+QP{=@|31e?S0lB*!_5M4Gx*>3&!35O#M2cy!eRPh1LJ){`9stJ#}m#F zE_oETnMKg=o{=^0d3{D(sl|DUXpvLJaX=ZI?&|6SS)7#A3|TZbA?ZKxzv^>Q?a}Z1 zh;skh#oZDnka6~_;$A%{ZCH@1mLJ@af{d1bE?bq$Y9plCMR=JBnfe4oJ32Z5BFTUv zIsOrYnA2HZU0s~^aBQ=DrUpp(Qd?)nKE%ZL8#xw3&+owPa5frV-ZQ-Dmy6AMX_i2b zLdMblq7+;oZ=WJ%!&h8^u4WTgX0+ zXeH%hdo)Y?moLP3v_ba(PWVF?2u=+WpBNY4Ap3kn6hp>+r*sa&caUqF`4ywlQYN?l zhg6t#C5~vpjqM0}@!|!_6cqL8+U?P)chkQ8Ji(`kZzBS9W_H)jHb5EEK!Mf!IzCPe z1q<8T+sL))77l*-w{>$qe2~%8qn&f;^L9cLrsn^0?E7gaU7!&VML;3FHC!9f>Q7(BKq+eQrei&PI+;FFqv3b zSn4m5MRO)5{rqV)Ff^22Sy|b?mY1KukdB8rg;EUHZklmw`f2C?_mmh8G8s ztNS^>i%Sx*59nVC3y>hZ%NsHwj_2tq$Z(Da z$zwZC0qwA~v^2#gPnm`Riz{VgW0Oh61bSNqPyvx5g&@kSa~E`~KA$U<8JLfSqK=QH zAR0KWq@*PM1~#@g04A;aPrjYQ{ay1G^OuOh`<@k>_!Ooh6I0By<}bwm_WLqwkCcJG0=Afl7} zm~xcch0TIA}i$@OqDAgC5g|Rb!X>|qE@b0`piLwFa5+KY}86# zRaNzDF*;m4{e;JwpXT}N*I|$haCXb>GM(+Tzznee+X+6~t+4Z~*4K^1ej9j=;4`<4 zmz;{2NmvTNnJ&M^N2l#UWeK2$xgCpbZXfpM6*u&gAR#Ml8`d)}8)LgV5zk4ll-_Ys z!b1RJ+;7u?gP?1;{PHpew*3ew?D+MP3#-KACAw#KWOyIN%VMmzO5@gNxYgGc)Bk$8 zYbIyG%lVAXj@y)F-Sb1IPieG2Udp~VYqB+8Q11bH2&n&?wnX$mAUbF64CgFXT-e>d zd(U8Sx-}Ih^qQeJTl!}xv#?JO^d}%wSMbVaqmL)Wk1|O}0z&qm1EdGBg<|u61)wq* zbgP~W>y-`xQNmzcfEk>;yf(Wu?=%SFBU(bR-#B7%YK%)n`o@nt6|bRg+L)_ ziy$D|$NbA_I4oc2MB&K8DZe5LpY(J|;j~tZyAt9&zi_tMca2KRzAT2fmQKei#TT4sPE7Is5s1zVC zxvZ7QS8GK6?s?NMQ#gpV<^^$(BuoGFNjo(c`}S7MEt1~fFG`_g1S-H9nz#!g68-eM z;tuzmU8MxzAFS7qHqY|+nj4U65d}|*+adHX47V@mXmEiLA4K?>xn$8j8r!rqd1@mD zk|Z)F*QB^4hnGmn_S5t9(zPx>fI18qd_dqn>!m&QB-dnngMi9Xk-`lIn@$nd@J7b#_x{$!r&1f*bveO)Dl-s+DUP) zv&1keJqY;H-7WtQozqhn2O+`rC*5aXAzo6WgcdxHY3RzKrQSPSs>OGSBot(&nR$9K z2?^nlW+i%cAzDBQQ0%{b(8m6-$M2sVV28!z{>3Ekx4n!`ze}j254C#juGWJwkQA8$ z`-^KT$9gNq#sS47DS7E{ROV^J@D>7;87OwaeIkc3-s(s0tcJx--tE`>3fV)DWf8=a zMq`v}JC^^G<3lA?@pwT>+5aONT@*H)#e~;u%hr- zd5NP^i;C2zaD2S^__P1S z35ykDSq(5}2VtAQ=;$1&`l5l3N^x;9D94#W9}^RYHUCdc)fM7%^k*741dv2w&@U`{ z)4y4TLK;Xq4@0pwM;d*JHEKI;Ew zJQ`B98%Q0kfaVM;t(O&MY2I~ZJ-aW{qutx@bxS1*QC*tKyd(1fHuq!t8)}3Q-4xA6r-LY z{}7H45UOy`@V{vaIEMNiLD<8%f7l6aE``7@M9RUckp~md&RyiVsD@-jh~M=R1SR~# zVuHK_*v+X8H>P+6{Q?Dy0qbAS!hd4|5)IS%OcY{VX!7JCJ>yHSB4c96%O`Ar(n#w^ zlhgUC{W7nJfDak(}TdX`YrHmpOX#zHJ~?WHBcIA_p$>bW~V7d#3Nn z@$=8~innjKLH!dXwm`vbm=Qqi$E!?`5v(}`L{VPpTxZH6LYN?Voqy=I{}d$4w9-Lx z;P={O6ac>25IUPB2k{5|jI&CDZzgP_MwTxCw*Qew0)}uqf?A9YO`Zd``*o87RO|@P z0?iH-7`jgrAds0r0?B!Szp0pWBVobMT!H-0fO<*bqWl|-{8U{wtUUgvUMMuhOZ6BE zM~wT&Qd|1|P_6gfME?kfaeAD)72$%fkW5UO~ZBi?35C6oC%% zy#N=ky(R_#PXW&*=So(_@qU48i0UGVhjdA_;<6Uu_G&-fN?*Byvu>72l9R@Zl;HU2h2EqLF^)$ zoCok%(Xe3zpRg}M*%%M(pmeG}FhBaBoX3WTH~u|=!B|YLZ7$Wd_Fkj;7lP=`$wD9R zUJ14f5cVNFxO&7M`xA&`?qq9?6hB4?_Gs+8CBb#K8j%Sr;H}Z*ML;;nZne(z@6aI( z;sCn#FW`a&7P6i>yocUQ0k@MbM99&Fm|$PR3gWuzX?aSYKiSY&m|nB6D9Uckx-1Rw zwxFOOLi?kKBn{8xe8$U(a9j>+_bt=EL3Q`#OA+TEFOWN^66sM$(#^1-E#zFapep8A zWhk2&&u-l0!av;!Bx%~pZ~f+6ohUoU2MYnBXwaQ2Lvil_8wB?fVuWo8PzM@)Vsg#s zu&ZoJ@{;crppMg9*cZ=0p@1ytne_DZ5T+k<<+}9wk4^aTp%F?H(I@XGga~fb9TP%4 z4c%l_wsP>{2S3)_+&uNjP>?yD7qVLT@AN~$<^P$d9VIo)(0cUVT4vH?nYr`GX(fVO z5!8$@B4jC)uT1ZFkU=u^f9mIDG8HlCJ_)_ZN9;t`D4hPzhS2oTSm8oblwHrjjbNBz zCpV5WQr_MYH~%cLG`F-!7YsMeJ2b4iG?@t)gZ)PmRgmvwa!SCklIu=vo4Gi2(vmrM z9rgce9J-a#WIXNYgGs)y;pzZc9U-d`KPX&ey{Bbl@{rksUV;!M(qC}tw^|bRt_oI+Z6=W(VnLMRhg_@OS96zq zh6V;GW0J6n<48$iD@ z^*7|5lTHoNto|&)`155)_fh}K>Q~Xm<*jRsv-sD*f+1Q$Gp~)R@(=AkSuEg_n)Hz6 zm(TE*&sbiU0t#YeWW+C7M1STVm0&u%1Vc?{=Qr{9>=H+ZMaK*DyeaV;G<-B0$8|L~ z&YBmWpAtE~pK!DM$x!`l^J+y$=XgiH3?t+FyIICp*dLf<)#BFxFj>F?}JpV{~r-qvIB_^~>Ck3wk871ZMp zOM*gQh1$6!F=yrX=07qXb6pj9D_Hm7!8uuYTl&L8*%r|G5VAqj+CCvmKHgs zug%RtjnjogLqjBUTR4T1Hf8sZj*fn;t;LV6-uyX_;?&ev@jDXTTMUW~?@PdbeH$6MBb4}ZY&Gt&?&iS}KG8E) z3EyM={{DXVHU)Wk%1C)OAm&w|GAsuADnfreFLvc~%;%jI>NC7NJlPo;vZGeoc#t(b2x~ z@g&{{wYkvZ;Ly;I&+&ZUzJKRkjx9ze=0@5aWykD>nX~g1C<|S6pNoEy zQVpe)F^}<3D9VUc3*>=T!7=a~s$QEDzKaQdo#)HhWgYweDh}B;C?z>HOumd!!2biN CsqI_< literal 18910 zcmbt+1yogQxAq1l1nEv;lS(7qu~Aw<1Vlo?mnZJ`k2}U43-gm|`pZQE8FY8_;Az~y#p-?0m>KJ_#>Vz~3 zg+oP%55J+Q@3w>&d@HSs7}PQHPfk-|5(>qJ(!eMicxQZ_@-uyS^FVR4FN1*IRtkF` z_gq%LVTWn~*5pCq*O%!x>qZQlr;KK%943aQ?v_b-t))LnH<`IBIoSI=DdVT6*=5-&Yy?Sk+LixetPxyy%=%U5n)YtVqm{(= z1%pZ1NT$|Qn|E}V_s7T@x4AajV=c@OLn#+#@vMiJ&`V=}R+iK)`Z)L(i^X;`O}co% zEBtf>>yT0)KN>{o%xL5%ky=6;gZyxwb^7Z=>iT1gYhsOroBIWH1C7HL$!&0_4r*OF~%5S`qQjTf7Vez zbt!+qn_mL%s%PY0$B5kif4zn_?ub7R!x{Gf`7r;}jh=jR3V&nezGjs_usYDn09$#9 zaf_dpn3^c;;sO<{{!X%f$*@Yv@GHOE0sGKa%gwPCIF4wZOmD`RPV1^jSYrKycf)^H z`af>z|Fnw#r?dHA-iWu?_B=mr)O9+hm)~?Dp=Q6JxV>Yb6=*2^mGFp?e zS5w3>7dTGihjiiF%5_@HeeM{v7h{dm0m~S=YFd8bFAMp16>4WaaWVR~)g}q)=}~!k z=%JAjT-2_w{QSZ~!o!Crt4##l!jVfpwsS{xYdw4RtkUnN%h2d(jd#=Y=ai2gJ(BYK z!T(ezJsgWAyeGV?zii) zQw?6$%kR?~^AV$6R0g|lWRsMn<4!t7C%D^^?mCUv@Vj|}Z)NGt!tQ7DR7qD^O3Jqh zDJc;N92bdF1t=ob)YR}s%B~ZjQpFu<^YZdeegN0uC3iR_$;!5UzpYIL)iP36^UB`s z@z^&arzu}PR8CG#OG+_Cg8HMB6zl$+8>xzlOgk%f$ba$*3DL<2pefUoin)-b-C|FWs z%V*E}fHH0F9V>1Or|qK{T55*gBH?QZS6o5|KU2zZp(%4;&_z}5y!*ZPrKDnWxLnAx zx_~${akQ$3{R=wfU07qI9J7okA#9rJwQCIPbt|l)5fS;WGen1*s(pqtqlK67P%W0i zd!lCO?JI&P8g61DN9`h~m#y<|xPo*lLz=sUde+m&e$?s5BDB@jiUg?8a9hje_fOim zbufWlcx=skl&?09C<(||JF22NgolT%cK0SkZp~xf`+W%#-YI;+WuXPp7?I=9!hCr zy6HMgmrKpUJp$I2<=*tBU-x7zM@p~leSOgvsvDwCX^MKrRK{B>p4-^^;6nO|cpQ@_ zxKknGSV{WxXds^`ZoTlQ%1r;M|Z!DiQwAxbwWNNsgIW3XnYm+Qn#?Z zKUeiSi`Vai@9Mg`E=N9aG~S<|q+B*O#*U2{^IN=TrsC0c+WG$cx#;0MU2{uM5)U{} zXq>_U;lRLvK&P8brg6RJS;KPE$ouzk$DAZ;P0MLFSVXK&8djLctghOhW|gCF@Y=ZG z;P8y@=xJm@FJ6oNt2zAu4#ClDm+h)gpbUtl>+y<3#NGzeXEJW5^WPvLBkOVv++jL% zQyX(sRGhrHY|-)|vter^z%M@C=j2uZmGG@?o5O&8k4X^)62_RPz4Z4kBd7XX{mPZw z^Bt#{<-E!5`m)rltvO5u>B32v!V9wl8NDXmFGjh_M_;gxqIalf94x%lwLaA_P~$3{ z73lkuWFa&>-0I6vv2#|!i@V=U_>C)b_E&-q7tR>hd2mruQdZs?OD`<*MG50;iXH@Pa(s|vrDeUL8AKBVSS-U{Z6v5mz?$v(IHJ>dd!W!$$egM*yOTZl`{7Uwj_3y5*3wHoTgn9d+Ny(SnT+Sf!2eP zm@&WE8L5{+LLRZq7&9GlZk@CT7K$+#0PMR9? z(?Xa-b93XLw|%eV>nlS^Ma7=dpvVB}paqui{$=PaSdQ36-qX>wfeH&nW}<*K_~5FW)l(+HUrc_9c~g{d<9P z?AT!fcI^_}8YcFo%Y7fZ0;#6u;}flf ztgPwzCa~__jc?OUvd>CNWDl=l3sw&z)qO%}uG;cZA768@^{Vd|Fb*-#HNiyK=Ukok zHm9YaaI+SWzB_+X_sH)|EbAU6o?V~id_m;ioA=pPzrS2v{q*8{Z}jQ~ClB@Q^5!c^ zkdw&M04QIJ&-F{3Ru{(5&oGMSv09#)-$BwR_j&tm}n#2(1 z!{g>TqkHJLLgyy+wf0we!{z~BP-}rk5;-+Br{~+e4(n)+DsD!;YEt};^YwcR6p0Gl_KaI=<4n+;?nR*>|5Wch7= z?IfWWcX(*=L8}8SD$?9hM+Y_$pM){AwpL0^OboH}fP-D+F*kqB9J8{svj+w#^tO{J zMhE=*apQfe2r8H-%NrLZ?6bYN?D*x&mvR4qt@$_JiwPIWX=uWRh7A0-dw5VL66akT zxA0m0zKTrxtYp%Q*^@(xIDrzFmJK}g_TQU}Km{W={PFdZ`OzY#&AzEvk4@A#2Fpfc zK*aEtsr2%>jMXQ^DjsAfR18eRFJJ_!uWGf46Pe1Sg_W%-w3f5T#;2xY)6Uyj&E2nn z5N44(n9O|5+B*N8RTW_-LBa&Yy$25-xE<_%UXD26K%@JcgMv7@x!as17RIWqr92i1 zec{U+sms6#R0^lrNh40g$^W2NI9TO~>m(s2R)wfiIQ^@eBW^Z?Od(ihz{Be2!CHlC z?&aBGBa82t2N(tS7f-p*8J}RV;gw9&GjY9u)r=gwEa3a)C84eLf;cgqpwm_}#>(x5 z31zX~{Lp}F<()x%r|Cw9Q>RXO&)y?&JO2HXLqWlOb%;x+c-SF#*kPin-(JjjH{BJW zomZWy;NsGfxGQzPXCb$U2*u!2o%WV{C-lHuoBV!S^(1m(BO_ZEdy=*mlP*oW2}Vn9 z?-(`$QgfPk)oZ==c-yf)YLG!^|8Q^T;N5L)jO6VXk?J#*meHq)R6}}W6UQ$sCNoi# z8swxX5PEfZSzD`OddEw+IK+IJ-@WZ`Dy4iZAH`Z*I62SIZ7k&(GF26OT$#QEV8quVPKXx#oIjE%u%`j?v0&w3@O zMqBV~^UWOcFCQV%vV|{$|o0p%z1N^bdegF;TL$r4mt8UMdhDD8jC_gLq z__B=?CbKuTwx6w_Ba}V5A&&y@W$Z{|eZ7qT!T!qLW;>bt@_Pok-F{W$8W)xe(e&(- z5jjjy)K#q6Um4uZvhIq#G4aY-%70H7$OyrAqL;^Ca28&~P67q2=u(J}A02-&q_{LVQxIahdY} z#a`4tws<+Eu0c=fdWJ5~>-}XWutgHchPA(cOjNnesc>>}iCb8UF$QeuVbzt)(y(Gp z&5f=1Rq{_Qkr?it9=0@ii9@06kR5=JrZhKlj2!&jkzx@T3HEgc-1JSrG9g}t2p zNf}B43F)C5ET4F1ZO`%XRo#3?i+EN-Lc*UgHy3Ebje6hn}T(DOC=`=SofMD3KITbxH2tBR(S!u1NK&p4_{4 zmeU@mPx(l@&z~Hru!t)yaW}@vHBt+C(5P8YoRlw&2^LeRhQYAVO3TwLW=v>}oeKh0pPY=F1Lc#2 zEMmdjknHWf(e%@M^nwa<2ogm0m#$u=k#L#B_NGf(Z_c)Qe9JP7jJc9 z3L@XEci`op3KQoJy>+$r$#vfP2z$9-IM{9>?-$9Ybsl(phNZ|si+C)mH~tuRRKIj- zr4e&~Z1H4HxeN7d?9NJ`^}A#NB$gsYA`PD*fS(Vl*}>Sudy+h8ooHI74_C!!Jwou@ zb4^&sG&q%C7ja2TOM3&*+ImB0VF>{kDU*=U+~<$l);iRoK*?3>1_xpT;&EE6muM;(7TFLy%y1JxU zkXsj5Rv_fJaa8p*SM?;YU4kIWS~wVqWSri|;=*a#spsuT%>(yuV${^`e9tFO2=_<2!cU@z`|s4TnBLLqjyX5W_HYv8MsJZXYilB|({fHz^)$c% zSzE^+TPJbgOpo!lu)s;mdaZYS87=`du+>3zY4*p2YlNxbN}U7-y|@;sPE|D~^h>e$& zByx-BwQY}Ok*)Oo?y&o1m>Bs`SMd-yZ0KXGn0vh`?=$C)w36?HdS^u%BXIOu zCJFq}N#L6*n z(=QZH^=^0yNl0R>W?id0oFuH@JfJ~v9AbM;<1bI;Ux?ZMw8u&6z^ z2B3ZHb%1O-lS9%+Cy92es@gkMg*+qo82gqya*K)x3om7N7GEj6eED)PB-GuFnKQ-3 z#ooIE7b$3IM_dG_-}Zg>w1VV-1b7NMx(Fk4A3P`^*EUnF_K!oh>RJ1&s+4jsdy6Sl z_gV0uM^}tYC$NK$?(`QwQjZ~tP;Ori@1EK76L6Qw+=J|038C>cvlKaG0#y%>=a&2N zQoPkBW!JBNVE{r%Fpj9eTQG&#$H%A4g<9BoTnrM?-MutM*MMmVM@WTA6oxC*v`eY4 zuU}8yH@Gj771TLs@4ek9wD$|M=(|KhyLfd7s;uyfO#Sky$Fb!5#3I9q2#Pp5+`oM8 zk#Ut39w6#4-AK&+cf*Y>77|3OSg16thM!#p;M!Rc^xIK8OO8P_OGlZLS40Q%c*x{y zcS(_2Nbf!NDNig32~Zz-!91NN+rNL#Si?=b`}(^0DjM%Y(cx~c!qUdwb>h%DKRC9i zxVROmoPPiMxnR_MKoj`mMYeMBVlyp9g?s_*VR){i_^piTGf55+m5A3j*yC?2sBjXB zj3&N)t9iKJ7t{%XW)q_Od=F1{sOFoSfect`-o#i{ zRi#p>Mt!+Mj^ft5Ck~whqua8B{QRzR>PH@;b74U%+MMG0MO;99Y&7nW(h8V_W*;A} zNCh0coGJ5VcfXOe5dJ|-{D>b|1|-S#WyvFnVt0LNV&kze`fOqwY_1U_f44w|TBYW_ zt*r#Yt2;(S2}0^+$zR8-iII|U=5>Jk-sTOkQT5e_>bQGM#WZk~bVBA%ck)pp65=|V znwphR0YFJ=Io;@^uBAlq!Ay=H7p6!jZ*h0fJ`{=S?BiZ z8P5JN2|YRM{R!W22FBY71=!~PO15g@;3>E#FWx+yR?1p$_t^@yXx*rufc%{KtbR-;!|1aik*x5G=;K%cw7;Ou~vC;X=%rcK82|Ki*fVt+MX(64SR))b$fX5El#Tzv=fZx6x`NOm`3&r2(aR6fNlVpOl3^?T4VTNC>m+&&##$~ffqIeK zd4C=uLv^#{B4=kUbFbu~I395G#DlN$0$c@R+FPqu%Xs|GeOl#1e~sq@SEiFe{y zq7lRfE^ouqiA%ZGr2!rMDFXw4eUkzZm3@j^`A%_9@v~=bKtAbNSj^0S2j@m`<=0Oy zQYD-jfo67LkyWiTt9OIxmMvl^A7E@qtDmc44EJHn%8b8?{Y(6O+ zLx^(JxcKSYU1V**M0Ly@A9=mckiG(EB{Q=KMBvxSIw~Nb@m}G6EtD#tZvNDsrIXi+DucPVAy>VJwTM^)XoW+~tj{fMwoqpn?(5Mnu z3VV|)wHVQnyxy#pz~@FJR05VkN+GYAfQX_Z-rOrt`h;=w{LOhg(QVo;rB}93AKLjO z%VoUE=xprvQ7^ir1iTfH2X%T%bT@5mEa%$7yuVg;xvgY+q51i>VdZk=d3m_Nq@6&$ ze=J_|Mo3#1Be$R#f;W>qY>u9-|_u*F8*5OXyf>-k|78_B) zJ6y*DpScSwE6Gun)h>>}I7G7Cbt;>9u!7ER&sB}MY-;*G23MvBlG)PRM`QJO_>%dr z-M`Tuj<}plZWs?o z`$XOm$_)=G=DjtG;|Iuz0e#*9Wj17#Rs2jBkH-=Wyp^;seROim$Xlvo2$KcP#^?K_ zG`tXuYVNhCmaIpe;a7ZlkfrcU0To_A z^tY6`QKan-PjAH8t5vr%$55#G-)PqZUIJ^<(~)7-L*V01yZO z8?2-<6(a_h?(3#RrkziDb$r3$ik>rv$2&)njD|uz)D3~HMsddxB(oqTvO6#i#F?ae z4Ke{0ub{pz0w-+B{fVAQ3~IkEt;y@|y4r*ah9F(T*#2JxzifJV7ocY1?0_F>eVu%~ zs0-03ZSEEayyq|O^(ofe=)k&G+V(cA&NMgod9YKO7#L7P;<*P2D6i-h1=)L(woKpp z*xS zQfHhM!X3g@I0ltC2H7;yL-(haIg~R|nYyFPLF5Qu;Nju1(X=jtKyC}Q32qM@V z$o9|S#t5^kz_aL4XA;mHiE1sikp)@NcmE`NxAqpm1@VH$*j_(zMQ8bCOnZ@ z^kTZ!kE6PG-fdppon}$g($w^1UvqA7m&r$I8XB5Zv0Icw4&pz)O?1iL;FK{t@D+ke z^s=js=a`yWfr_=nWi)-GpzeH|M(?4#n=wpSpq1)S9V z_N@xS2Ddp}C=W*bd7cU&`xJN{v53DZaC{VG8wpoEdKQ+L?d|RIr&G4? zAs6M>*Vn(>n!2-+X#}tyUyAuSfXz4`XT5OqIB2|-+2tNC$6;An+DHyJbI<5ATnF^7kT>E?8_wBf)? z(pDhR9#~;-XaL4%1%4y1=Kon82k>BJKSs^17dfl?~>fs$#dt<{h^#$1^_cwd2d+( zH_CqQ91&8ftyMIWq69{;65q;4c76ShCk@~K%`LMYrq>R9$duh$PItB1-(I2@wIK|4 z^l}d;BiJ)Nqo)N&3TD^#`Dni#C;#*5j3ILh!qHd14>>5Rk|*DHoe7Gb`MvH1#wH}} zFvOcX$i7Wj@r{oWh;HcXEiNuvc3>&W%+CX3iIAkdiZ!o@8BwdAGGC_}em-3o9vXV- zJ9gu`NPd3)^_aAzq!0vBd^aB-cLLms6kSJ9(xBPdTluWXH8d(UaWYLJJ(9w_&Xr%W z7n?JzA=1`3bH!|ivlBWGOdmS~Z+&{rsoXC)`!9prL^0uWN}SI@&& zjH%^gweB1slIU8%g&z6!t!clF8&FLO>{slFX zn0y2+h&iF4ed5;#U-%@XIHk>y|4rRMf)~^ARQA>FMZjG9QEaU93z#9DOYAxH;NsQW3Cwaq>0N>KP4gI9@3Hnx>JBufh#XIl5F!PH&URTm9*k# zV$KshwQKU+aa8c|M@{n^r1c5J6S9$v6_V0t&z_YEI+Am6aA=f&IMdVH`=%|780pPG zxpy665Juoap<4AP{1;PaJ0zmA|AD07{4)=HVes2WD8Ce8W`=r~WzsQAYhtp@`bK%_mpjP%Kw)CUJ zBqLR)b6%tz+hxZd1W5kcXg-6KS}slB4Uu)qz&cUFE|ayvCeIBaK-gsm&>}^qrsjkG z&*T~xOzkZN&fj_|8}D9Bq?*6G%KBcwHByeN-$6q|!O((t0 zhh!mlefTd^GhqL9SE)UMtm^>{pYYAM)ZI4E1-l$$`nL>Ajqf3a4RCK|uZ>m@L4MUp z15}LnH}a`7QKJOxB`+C68o~~22IAx76r4>2Tf{NkrsXxWv3u&>H&yIZInDGdxJnGm+8|TKXJ(Q- z()7)aE0R~ZRzAWhE`BwUi9AEri`2FGw**iVLk=OTm@FD5#!#zM>(MqZ$2QH>SoE#h za`g9J%2DBQ0x25c7umm02Oa;K z5wYqZ1XHApzLcH;bq?m@MGhXGu9SA)VaJN5@P`knwbMk^VEb?{{1eXn`lueXMv8I( zxlGRVHggE;ulAsN5%`-9q{-N?N_HxfC?UuHH7i;`Ku1(-1c5?`)?$tB={8=XifXB2~g9}%w>8_Q^?SuTlX7#7-aAALwWr9N36@y zIYq@Y#|NX$2)D-jNE6mfB=vW+qT82AGKpeQG%f2YvOgio$VM4#;GhAen0dhWt4MEh z(mt3FS4>(alN7p6Qh|rH6%jw^!95gzf4+pIZ1OW_gup#;G|J*RM2*8f`o8Ed_I%v_ zS;g>{cw+rp?v2r6>5<4{Y@CW%$UmV9ExXe8 z*@L}s5?iqDTjaw^nlRg8qE%|hm*um}z;9MhJym~)N`?j9j2?3x9mu~J<&si;wZt$N zYK=+3$K7!qVGmr>ddQ5{D4rR5G}xmR(Lfo#<*Q&55lNU(qpDV^Atvk}RGZj3ZP7eeoxYDEy z>86G1hG9)8D#%n42)VgazPxv}B5v9gexI}mxgjCgPo7en6Xk_84wE4L(8!au(045Bkg^{E{LWkls!`bF~oT>Z6 zgoE3h9_XOujcObCM&KRWp@f3vcUPR6{uhWzF{>1MglunnAS{*lk56`C@OQLkNJ(=Jc}CKpaxVBWe(f&}OadA?f=ip+Jb=?cOw!%cLm~5ssqy&e z7m_k+KcNx`PnJ*vC(bK*EmU_2->!TlzlI%~PC_0TtT>uDYVJo8r+6eK$;9RnqQ80) z#U>y?0v)vxD6Y;1TzXIiGra3bR2suE7+nPXqzEDGYGB;@$qck zdl;)_Bb``nZiLQt17=&f6S8fe2%vxEPHuEx0?sPUGqp&inU<`ITT0EFmVbUrQJ)MF zx?@B+CeS^Gv05?GA>Eweas(c#a3M>|9ap!I3b3aFOA5A;!~9Gr8i$+vQH)^=o+sIl zIOpofBFE5YNFO=8eGb%Y^?hC3iMY*<9-zm|F)RsqO?L?Ox;t$Jy7FqL4OhX)Y+{DZUPi20RrlAlvPzv01blmTd(V~v!+b=`6e7c zcPv7d<5Py9X}M&6%{@CO3XRALM-69Zfmbe5Z@D9ih!3ClFenDfiJm_{=O*PZI^7yV zkN{on0jjx3b?<*ls{G*!AltF0^>0tuqa(NO2tbq}hQwFvET%`dx8yqiU+95n;oDf% z38?KidyFxQvaRRvJRP{+>dC zPETSr5cC-e0gRxf_}i4aq~p!*fvZF5hm$u+EO^09i)#AzLvH_MoDp@BMp>hli-^DJl(9XmJ2d zVcRgsw^bo#OwLdco~Sj9a7WWWPicNsCsy-G8z6O1Ps}~qg902l zg72D6=2`MPB8daFT0U;MT z3(M4!x6Xs8C>#`iNN(@XQ(HbD`tP9sr*mWG=TlsBwV7cob@*Q*zyicOVr z&+D3}z>{gVD91na>UsQbsjlhsC1}e4uj0qgO0nMzfck-4Cgkx zcBtK?;yO8rZ1U{?_TYrZ}AF)PW#GoBKI^)j7f#6e$IaaXi092%I?*wX;xEIoBG55G30>Pt8$yXj! zzjaazH5~q~gyEFQD&+zdkGh7&Tw^G#U~mz16YMfFjIeIA*gK%8J!x<^V1^f?S!8K>E@g#&a$48VI6U-+DFhsTP}@BBCH`t= zYNzj5)t&;BNn&<(9Af%Xd}vCuq!TDmT&(RTRwpJ>f>h*PI|D@m{hP46 zfoVurnS2_iiag7bO>5n7K#a%=vlW<6i>Fy+sE3D#c?AT%ZTh`?`?mHoiUJ4q4|O~) zduX0C)v7~)B~%)9(9qMP@cp^|Evyuan+{#g|0?%c0yh8%^ygnB6S7gmSaJO2O}hRk zChTvssh@pIsfl47d1Rt*Y8nla(>@+G>RHS$rutjpM#EK2pZWOs2Fk8W3c=x7Q31}> z1m5c&W^g z17I>BC3Q_r2T5}n4!Iu^GQ~PNqad^(lGWS^*z+=P$1i`)f9h(y0rtmtD{7=OAwaC+ zvy^cG&kY^OpUSqFdl&!ZE(X1n%E0vg-I35FnK$y8$AE`Y$K;rA6NY;hH zBca)Tq5N$Jc$oj#WwdY!_Vj%Gc(ml4N1a zGkg>7pl9fMZz|>v+Z~Kb}T!mLec2fABLR)m`X{tMM!Q%+mc3et@oSxnp=I2B}})Kt*Rq*5+Nz+4`oAmhFu*}6f_Um`S?nv)}Y25 za;PQ~MWaO?9QTPP{g5(6`7xs%q8h3@tg6(l>Vy&$6ER6iiHXQ?(9qD(Bl83UObqajn?V4m5*9hF#T;xi$_b@m|1N#%7kl;SJ#!MPfd`%v>q+;cP$f8xG8Xa-mBlHuQ}GmYPxFhbWy z)?WUZoJLMD6)jsGxkb6MNuxS_PQ9Hx0xnwKBp3Xsz2?)33gxToZFmh zcdIkW4g|mrk95ZFZ|@n<;pX;WnC+yX$579LxEH2A3`>S17%1>aHN>UC^)HtUgH*@$ zdmq?ng6u!V<|}idMNx(fXa6)6%uqp-N#20{q|Eyatcc0;~w#3mi(^Japym+_{7MCjc=$Q@hr*6y6}=P{%C61JT#x z5?svBe;RxZKEde>7ejv!MobbiGpFY71X`wkm1UqL+V9t=4TEgU_u*vSDC0li`aD6`;l zO5yY9Tl&zpt0Bu1WsG5EnVi@HdB_9EP?jGHGe?_M6=_v+g@ud<$ ze1VgTo4TYFclzTTHT?bM;P*oD{{NAsP>llGm`tjFQWHK3sE?v%m!4A;39^BvTKJ^Q zSm4}FWzZz-s=o%uhA6gt{NRvsViQqMkJkkSq0S0KL>1N7s0hakl zN&Z);78kmE!c$<^0eAWQ=ZzkltpE26(p z|C>zu9Swv{oKy-i7_?`A00YoE00t42nm#T>%A*FGGK=OwX!A4zshObu2BAaD^DzQ{ zc2ayeG5c_l2N|mZmJhCY)&p&mR_MdCC-uP0Pbf$LL48CKdHFA|6Qs*|_^!XDtRS+8 z>al}{y*DR7irTZ32?TfsN%{K4!k}CQLe+A%<(_|c99Bk+J&FB7p|XY<$7x0yhxew> zFJM@_V386|^tu^R>>$`mq__|jc`-TZX-XR~IYMY@V zy*5#fP_lyIvO98F(7j}Knd!lHq?A(Vt%waL>ea-wLXjhYT#*Aq?v=*;E@n>1 zn0syf7K~2Z1KUPscR(X%lCn@20wvqCf9JuY%5OULpirm$kiQF{=44NPj;X9jbfW*# zsts2CrIl((zmN~7==q}USqjT|Cb5Ju7$C2hrlF~qTgbh0^z23zPaY184 zY@CI+#A{>!0y=L5{76LjCjaY0kBWaj4QaBcr$_4USNgtp?0LFZ{?HQn&j;FeJiut6 z<=Q!;fD=J_X5L2{V@BPN_;X>QDvZcL7mf~CLCPwHa5j{vf#h~0xO3y&?_X@9^*k_h ziSWL8w3~m32r)(yV54~8H=I^b-k7@O7L@D#0X%@eq8sH|uGa*y|NZ&DIR|%WMgS*^ zrnu5=vr1cf;a1bNKNnDoGJGu5ZjCi-Kq%B9TqGVM4MC%FQ)M5YI(V!E?03n~oPl*r zex~E+$f6zFNfDedJUrgBXaBI6R!Ju;$0~0?8U&Mx`1AQYqqo>naKgk9*UO}{CT#8A z^0HhQffG^TG;#7k!5GY>?^@Z|3%j^KpYK6=66@!u_AQa?(RHJ=8*byj+InY{r)!3GRA*hsh zcz_u8_Pk+;M!C$8I1^0~iK($9dElU<9(@{lX3DD1UQ;n7>5>bP(qv#n39FEff}cG* zhfKGF`V4FU#sEO=j7;`JIe^Si!MnMGoalsFuI=$e@sbd&5gy-IUU=yQN|1y#df~inPNS>E*XbrE&J!oTx7+ipK+j>6aJ1an@$W(DcHN}(18ohvfBkm+?4>ox5|0E(qQt|ab!kZIhw zDn3G)M@qE!m)g+{B$JoNzDXX|wYhYaU9Z7K!QhvF8Y2WNA0KI39s0PdQLnoW_dn5# ztsd8SN#^PlxHESJJsv;#QRbGGHZQHS{6lmvtunvCW6Z2WKi$RU|<*4N3flHCtvi-%cB z$-ehDX8YvM_X+V!oMv;}@-}U1{#_oqv|-2K<6%7;fs zf!89r0}eL}`Pay}C}SGyD3nGHe*OC1>X5ukH%iH-g%^oYNxm6iWH%`0JhiXct;PMq zPk8TQXCq~7irjWGN8fXP7$UsBgM&hOT%FzWUflA2{b@iRt=t)Cgzi1JlaAo42!xiFPZ?bP_?b{PZP5wnD&1KoiN%(RG9o z?bl|W;98P3&s41R9UF>`=h(kgrlPXm{`FhY&lqLM09&yRTX8B(W!tyu@v~RRz?lCj{Ot)o za>AnO>Z?`8PI@CPEx|B$^%O)7VJhtKo{;l$qjnkqct&RCm*3=Yx^*=)B4uP`h6V<5 zp#aR5XGWD!!}d!0SS&(|^%>j;z@x=TyR^GXfV}*dwa){yTMG}244>838vFhD+EaVs zLkyGT(8rI7pV4 z=TFboVEHCBE@qv_anaF5@RvQlxhAM%fEr2WoT|Hh5%fkxCvjzNFGt^9pR6l7ZfQ~a z@~hmeVR2_)e*$EaDWhvwRaHaK3E0!^-Kp2~^m3s+m+^rBzPj?U-lG3iW8)$a!wtQB z;lw%=`0I!Bxb$-l4h}e#@mlwM?ZKfEqY7%mlbjl@mTo;>Q(aafVeL1*@9wG}c+lCo zxafj)!*W1-ISewL^R_NymCCNujr&Jb@C7I-%=9Ir7a@HsS`9?*Mn)+BA|H(l^6?E0 z4LvLnQqz7KK@brUan-`20v^1ixcKbwYxrvfcp=JWjoy0J*4A3KoeS>E>$o8N6Vh88 zgYaOUl$1m~9v%_#{ph!!^UQ1c;gO-C0(-G%pFUkv%h!XZs^DATBnCm)dIzR-@cI_s zLzd4(C#ZknB1{wI;o+&a?jn^XLzZVu!|UPX21$v8ghap|WQ1+^8>AA4rl!iaw%qrB z2f~=Ec8Z``Vga73mMrv?pB5GlPflj|CXIjn>h#4AQd5?kPu}$OG}rlv@NoDe6uC=s zPHNaE5^leqZ@m#Eb-9gYz#h!VAic`T;ZY8mp3U0$y75cs-E+ zl_F9jMP4uFb|W@5%S#P^Rtbe-e!`ZZ-u=9E`EsLcpM;yQM=wL_`5;q)P$mkX8{95EPJ< zkd~A@*LSVY59WO56XO}<9`}7eGeS#4k(iK%5QRb!D=T4iP$(QR z6bkDc0Y3bMvbO6s{0rY)RS|rK8^3f4TF*n|Kwx4VZbBA7kbdr*tr6OiF{hU8ZDzZgphTr2) z?$OG>-Xm6VbC_Yo!uv=6j}LrJKHW_b;f6uJr^#v}HJ#5=;I`zj-}APOT`reIqtTgF zpO?^K;mDt^b7fu3@FO(3^xeOHR8JU1ck+|poZL=-UsU+Ndy&(dZ=*}mheY^6s+ym+ zVGtWJ+qg0TrKP2+Bkb@XYn~dFoIsVF+Led+L0IB1Ki&O16i-cWm7FhkADkm2BNL12 zAN(JD)cQcgqNCaaSrlWKL`0}zWSJaj4i19#_4WC0Z`dgaLk7qqU`XG2 zsKhWl*Kgj8cra4cVkv0ydm}QE@cQ-Z6T{^eO6uzJ4i06Q(qD0P2`Y*%lSJW@ct$+YFW9q12KDPiJIgki!aV zwl&VU8+ioe;r;&o8v+*Jwm!SrYGT?#?)Zq7PwZLvyy5!~AL1T=KI=St$Lv!I_r;5w zayM_j7$^_#YAxfdp+LsKf1U^xu(mc@$k1PIacD6(I4E-Q9GPd8>*CjpDmUC;zw|Rp z>{zXBR@-V_7qesJF$|%;-8nP&-dqS|5?q`JJtJXY=55=nJ94hwMv|>=&Au<+xw?O` z!Nfa|sd;Eg2oJ9KkI2?5m|7l{4P&3#&%dC&1JSs}= z-o07MMR2;YH7F}=N!96+tW+Br~zsz>SPgn1{427S5Z2W zdKk(u1OaO-qqqwJs_cF*L-wmzXda&BCv)}N-)*X=W3Hz6S*~@l#NO`9l;53`I$k)S zmBsL03>=pvk9$K$a>~bNI1y5cbp{MwVyziR)N7ODQ;Jy7moGXfX%7$4sp;vyH}`wl z&d{NDw@2)0hf14n`1nXv7(SRz+wzhO3=E_>K3FVmGT)r^Jsh=LIB$&)9KpFRz~aOEBo z6BFvny^qx^8-!=hkTEj)clTnn+ctWoV*hN+nQnPPYJflRFFuN|zl za-}-2t5}womveA)+kX0RF_Tt_0_#GlYIv72U3Y$`dq9^m&w^Uphp(wr@?^ad8$@iH zsgD#HeK!kw&kzw|jE(8l*!o#nBogO#_ZCZOql%-_rr#$FhUYy9M}1#4FU8DoQbqkN zxT5m?T+9rqShw9<5%1XN^YfN7KYwPpul_&>3KaJKuC?wSh2ng_aMPmlGF=UpVw+u<+qHgd|aa330 zK-$0m%~rVUXU{%IG>s6Xp9|6Qot=#SM5yOi-FXAX6}@N(DPO4JYg3fpZox(}5Nhqu zTpA7v0yU~To*gI8bZdEYp+7t#qUH7752%SJb454GGX+{-sVDPW59H}LnV_v{(U>mL zXL1;g*EeuED7c+&Aak2^5cF1@{PyO)#Y|V6lBQ;uLRO9Wo*_* zNV)Uo8@wQ{zdBfutVY)O0nY#ymQLqmI@k+#`229A7A&%BDP;;6jRK(ylco2N`0Y~G zFHCSYjp(u>AzL$xhNYZR(vXDW1OJAh2hOAJ92Bd;JFX1Alal6fCF(A}%IXw2W*5d0 zX=EH7o`Q}Hkn32m#>~(q_uwaw;pgbEx>9s^_uRXJD`WY!g}Clj9U(x8)E}uPrD4C{ z8E&pTUsb4NC%`@Dovuf^_4^p?pW|GHLwiRtnMBK&!m67PNl z1cYuA;mb|Wb<%H)9uvCvLBhe{fc2Lwj3y|1I$2L=Wl zV4YI#xrQ5EOA1eU1&{tIDTYFpJ?GV{(48z;-6BXicZ;0DyTrESHPVr{BI|oyPfw25DW%^bgM$EWC^l=pX>nfgwmba@2@feKl1;)GUuZylk=-&>N!dD0UkYCTu z%*|z~Ci3cZI?y8ZH9s$p0Px}8huc)jEwi5trn*AJA2R7$aL>T5bc1_MfnDp~Mv4|57PyX+*M~!4y>%;f&Ra}<8 zQ!2*LJ!B%VOITHSn|qzp_^YtCHyS%{`WGCVi*;i$nVD#)-qUKtALs%AyBg-y*VBo= zBq6k`mWPNQS<>3wYYwA*NFH}CGAfFUhUPsVJs%+KJW41sHB^#hVm)`tSpswiA=t&l z#KL1^6U9SsnZcS9-tSG7pZoMOCq@Rt@X4}LCTk)S2dcbouEyl6WVfI_Lxc(Z>*<4wd70o~IhokCb7#+d_#}oEk9nQ;u zY)Xj&rbLTpm^5!K{$vII^$TnmY)wtg{P)Vj0qyXvpol!i?nxIRv{UGqU1wjGVqc*F zUw&3KJ%ZF9CnkoG(2G0Vgg-wsk)`2f|o>r!VdqvG}J6bAY>kRK& zw;+uRyF#Fk>qr4{vVyZ=Gjo?Kb&_%*Ra zDq>CdFT%cbcRy6fl7R}D6EpQ}hBJYXGvQb5DH`Fi{gkIl z4tXGxQmW?r?WZa^AHK8wcGJ7Z6)2_$Nv^{G*pG-#R12mZWy(S(bWUUhjW)>RbD}so zp@}bEL}20IE77+;CB$xWh>7EZSQCcD?}0YS;n%<=UU6xRhGRII z2ua9JF-!!a9&1c{JF9$<(Awd&T!>+ij2s@m0a!U768WjZ%eON0x|nz3luv^1-aWQj z_f?W;1(ttaUrizjb=31EUdB;9Bka=~n;~AJ3zso4;AmvQ=_EWJ%R`1L4ut=&X>D47ipAjz6Tb49&D_C;P#Momcb*3CLW)y-VNTgvoz$g+ovq1Ba1WRP;PF zU?;D;M_Qz{wmjf{=j*aVs0nRXV20veyy)(!f^&G6?TmPySR~~5KPz>s z{N`;0`4-hk={|W0D-G22yOr4Q231a?Z699aF!$13g}!we86H05ouE@^GfkWP{thp4 zD~co-fe9(M-+U|c-mxR17r*|C!!A~yq)!MS5^i5#9|0lZ6QpjGsl36%A{3)&Jg8K~ z6u-3}zlLNvGcz+CMMv+Y2N5M^J1b+=TVHRTB%YOeW$8|o2BiBr+p${8ol)o50Rai* zK-la8LLGr%dU{f@-t;^)ETm`V{1GdM zDK^m2354XGot>>j-_C+8W+X(&?TO}xYg=0hCj;GJ!=fJ`?7E}F&WzUF9zQ;!W>I() zV8oj!9;pab$%XQ&`%2qmX1TvvrR|PFE-GO_0D9TGckWF5@Lo+25A8&%7cR8-A3v(W z>dsJvabynMkqHjnnI_dn0}czbUz$qbbk zGskzAc>f6Pjb(2#xj%9a_hJ8Dc#^}9i%d%GIav{1V&gs=#CzMzA9Av8Ntfv4r|TH| zpC7x;FP+<8v($z>$=UnGix=kw%rS31ODEx<_`mgDy1e_55;6;1*njn8;qopkX86y-ExW%>V7BGAQ!Uao=`@=T`cHTb5HW97$&{TQh_43fGqA@kv>=B z(Vh*`x9KRZi^7iczJ|2mvNC!Bsnkdw)7R&cE`01<0J!Jw{`PV;w3mH3s?;zJIv5A_ zw|dVjI=Z@l1I^Zz-dq&PEt}o!X%iV?3{c?|&0!D^yYIZ~l{$W;ouzPc+Mf?HXSQ)L z8GgvOhUo!80Zk|u^mhP$V%yMR^MTMlTQmn?{X5{Dg8E`7cE$J40!ZD?+tNr8eHhQKi8P_G5I0~UP(`Lq4AULaK}fJ7XHG(nr*?i|El&Am zaJa;`-#xOaaXAL zOdhePdSEv;Ho}s4)W6PhD}-!N=eGRraAzE{I{NqRd&xNMY*^*uyrR;7(*cC^6Lt!Clp*ZReQnQ3toOgF~4X0pwVI+`l98wyy0?;Eek2jQzzS_4w2qxo(~2Bdj8F}W5m8G`q3jI7zd>W(<`?*z!)9sB4fd~>}upsR^s<& zC4tT#W~Bx8lPACtWd>rf@2$Nl@*ZR#Z*Wuj9ISW#4PvAPfPlJbB`ce!aPk0LndWyP zX9VnjNWdEa1r3zg4wqqxjQdXo-X3nhxDI7l+_Pm_)FM4e&xkce^J^=tiSY z%}FHQq$f!!xol=Rr**Pl{fk>3`^#_3o#!-Q3Oq3ok7Y2>QO8dD@aX8j1!S610b{NA znRAz(nXK|hAl?~%zXOj?yjF5cIo*qtR8;2uuhe%BHhTXyNp~t)#iB4<$#Hkh3_d_8X6EsbH+ABoqOkV0gLRIFT7=4>j-rOR4DBug=to)(!4xrd3n4T`m3S9AL5>Jwy4F2 z-lR22$Yu@FwGx8Do_AY!LLjd!kwcn-xQ1q-WvV@D9fDnnK4!nFaP1lc~HS8CwCkT zRC|>q7Y|SSmmF1~rumvX!~7c}l5|W=VuFMHo?l@<1N1vHa|Yx4_wTb~e(L|Sbvw}4 z51P@Cnb5((s9Tsi`2zF&9Eet67v*;MoG`lzQ+nyiS5BDuQG}UafA^P|kFO(_C)=3o z?aWq=|6F`aB!8N;ruhGZssE|}T7_tyhAu#wm0g0=n0Wt*%{h58I03D?nvx-HZJ7M& zimRaE5Xq1xxjMAsy)7(jm_1jzH5|gKF0bwKi*9Ap>sJ4dy{$zvL7Me2se_E0b8dQ# z0E}{*{6AGdR&C1oP3lI*ktf``;$ZaK`9h+C-BaDWfj!Jq&$OTu3?F^gv(kLldpqNk zk$ZRVx5R^$G57J1KFc=eUP*S>r*6t8@F~BeosIc|la^Uod?6t$)1f%Kc9h{hCtk?-RI^E8nMZZCpxF zAqbr4TTOej?AYMHbKf~!^tyhuj~u<-1#xn$q_-1`S13U>I0#>#9)B#Ho}5QzD{5la zR`w|)<(0w!Zn9ikr)f#&_*@fiO(|c+cb(xPT`UkvhV7dULk&!QDW#4Nt~B93CFqc9j+xJ$%bQE-?L8vy?;6<8P5@1P3MH{SF=7(CULxEJ8)fpA_4Er+UMZ)&x>12s=T_lZbhA9^dW_$Gqve?dOej+WUWuJ*}xQw z?VOe|@p&R?<1ci-s}wk(jgA=Qr#{oRshnq1QI1BSnlOIu^KEE2qZ$ z|B*Lx*Y86QPH?8qr8n&0;6T-{3W{r6dpk2PF9|?m_1VUH`WI?ncQv)NKI@Hm{#(un z{Muwi{ijpv=wWa$?j!oEUs;#QJaJlPo9%N3hn@B$G!|_fRl}()?0*F(^A%B0(a}8u z`u;ZLf_jdciQ85}flvMBGdBQv#f^~Kk}O>{4%lZ*4OL{UJ{v2!DJAN9hp0q$q}SR73978k|d#kaHQ&ioRuwi)wQhw z%HE;>Rx8h6o<_q^JjldKgf7FQXpt4*`= z-GHHE@SGvzHmo2xKA4kA_E1c+-d?mHRQYTx>ve&#L!1YoMSiHs2OseYCFwsQh)x#GoB#BVz&Yw zwB~)%B2hgnS><9>LhRXtpN;d3WWl%f`h9T@HTZ5gR#(ux?F^P(X}#*0=W8>sszF6H zIbM3e`03~D?TrrN^P&#v1AU`E9^#zh_ujVkIX>L^Y+3CPF>qr>X$DlkVd$5foSYms zdSaQRq;TE#b42X-|6mcn{l-B-h#-wimrhG>*k9xWRmQ39rxy`E4#-+ek?2h?-#2OO z@!h27&IM7McwqS-Of!~#*RZwa21@Q$%Hoty@3npI*-tM=3WgMCcsPG?i)vd1|FD%; ztR%noI3r#mgu#qv*C8Qm@%1qMuzLxiGQF!2Z_@Y5sQo8j80V_pL0e=uP$(Ma`KzK!>!->;SJ()?tmlo$#xvt6mc(KMk8svww4Au?6b16vX9f6 z2pk=49FHWlbahQfIVhU%rw#JSPArMt0z6FW>grlHvfRUv1dD{MYBSt?G+fL=Lk9De+ZyxTHTMu$RIxismE1aRly3+hz@M$Gg zRXIb$qyfiyY$`F40HD=UCB4^rM8>6g6Q==ix2MYlt-a?eO|Py#UuaM&trSj^=1;OT zb_Sz>4GNKQt*HOXe>5x}FB6cf;^N}4UZ{mO`tD<+IJvk0%DGQWv;-OWrmC?8F%d+! zR;X9v4DcC#{+{reir0uBS|JFUj7+^=_TSe@$T*7fTe}Ha?p(fLM|fH~TR&vU9LM^B z@aQ*I3Qn;^&Icu?y|hd_>oUD0hJYZ)wA8m)=_xKt2)lCr-q-+UCIac7ExbFG=oo1f*BG&Dk?a3Phmtc;)gRuvIp2sUa>>R9(U zmg`@dSt5lP@lIdYo!1?4siE4>LeqF|l%W6#Vh42OFayVeC}pOVFM)bu0X5FJ*2T)G zr#RmYbsK8(c_HiTQKr4@X+D7ncZL9b9}gCx%5%bD;m}J0c%Iv;AJoXlH$ru<&K!Hz6qLKJB6F)y#NPVv-X>gV?{6FfPJ)N+U#lVWH4q47BkDdWB^coiS7m&%i`6DRunktbWmr zmJs|rW0=GYt+nIpFTqR+bT}+QWKL$xScR<*jqwr))(z6#J(&0gZl$N!l8AYo3w529 z#TmF}RLIx2xK$10n1JPHCP+Qcsekd5K~Gu_KN@cE7NMqbkmGhT>blK8k!;`C;(xqv z3VC6&o!S%wzKv;%K&FX>+%yGqb2QA%bu6LVh*?;e>I@aPw3nAS3~ciHEx=ap%CC?a zL!%5dyWO_)>({T>K#R6m_>u#Xh`TZg+MRWnPBcqo@=+rdJbS{bt>3y z9vDst5L1FCi*ep&ansmyy=k4#cy5*beH|QFEdb9IoxtY32_`GTlxcyrL zVMP=a@JSdX!s}LQ&qPu4h3b!N5*L~J{b2=~pv>owliNiehoxbGk1wvA@;_WOEgdd< z+I35CN!77YU%Fl+x^7BbdU9m*X`F5X+w&qfqD8vzGt&ct4c{Ug<_h?k&;%Jrgn(%) zBc5%t+Ba-bfv=?u8a|W63Tb_Ps*YI3*ug@B>8cUCcySVAoTwE2YLD1t;EBv&x7K!c zntrW=rg8)99oHa)JLK`__mB?ec<1X5)TUSmWq-Q7lR*xHX(Luy`~!6M_Zt(y z(pES!6$U0$2#T{|O>Vx0-cEjpv>*@wA^yj{{(y;$h*8K-C_g^9{EGa_%lOFI7 zh`}jY_<^jYrR6k(2+tNm>jAwp_vBGe1#BWWWJ81`IAQte(7DDBOe3F(-?YNW=7p-t zi|Cg3Jz65i!B@b~3QFn^?%t>=;6otd8uaTMwb?H&U%t%myXR75Tu0?T>KHcQmK2eRcqMUus%x{N1fF-eurL2(uRV#FXxH^Zsra-U+A==x~{;Ip9%4!8f zcm3uU)nVW;n>@Cpxt(~cePWYuzPSe!M@zo2`||e?=(Xt^8&jnGc``YsfcFC7YA7J-v|jC~CRfe&a3Al|3)oYf>vos|wJ~$=31BCf9YU z+gwj7CNS7YePUBChPjh4Z&oG7U++Axm)OJV15@%*-Ga&|eAM_`~U#DQtIs>q9)*#(iEsj6Sd_D`BicGk%50xiC25R=;s*Z3V9U=!qSC)aF+6T8 z`{Y@D9)s#1M(Tj5J6MnPZ>#s*SObnlg+9zE-IzkGcx<}kCMan4-qfh~PFevX`{94= z>jMmu$3~Ax-1Fxgt`{@!Ip2dp?7m0UWvOdUI?|yXD(Y!Mgzgc=5Xpo~QJ-h(W!5vD zT3wC({{6eKWLR#wVX?>2pSkt4ky_MY%3tCI>mV{&5o^ zVn)9DHSBlq-j#W6Stn|wp{-Rvt*B0NSEkV^6h>43rXILqyPI?4EWieLx6{*yA~OQn zE2^N;I_zy)nRJ;t0GwBQaTyOVS=ambaES8;;<`fWUw4t=VqJAxa`1wIv$C;GH9y2z z_G8fku>$E~Zj_tj-TRahu|D0YN*|GT)2Na!B~~e?6U}xt zqLe_d^78w12W9A4#+cY^dj!t2ALF=#20o=qMj?$kP($sJltv6m5`kPhISwI8(S+=r z@6D)jMtsDh>Mk4FcJT^GNepA_ov`!mhSM2~@J(O;cwq(m{0eb$Io{t7M09-yJjX|9R|;xi{?H3W64L z3gVZ&prBxFvtJX;X)BM#$@_%H!eM3Pb2sIRG@?x-x2lE2+X@x06nxaDyHpr@+dbW$ zonAzi_3fOpukz}ThV|X)*y%n+#&%Fe>b+Rz0CV6$R+R@Y%h6V;zZsNi9mP!AE=j3} z295{2j@(Aor2hLOQ`b z$@QtQHqUTbX&#|>uO74EVk}uX{f&vow>AbEXF#Mw%m?Q*B_ znTZ4glZQR`;>Dk+rieOXBV8+U_JGPek)JwacYgJ8{x(AY5{WsOv=#Ha^p(Y^xNU5G zi~8b&%JtO3?mN;dclZnNXkV*rI(T$fMwpbDn`%?KQR*{ca+;0T=}D=xw@)7N<8xxy zPbZ(PK?6~z83I^x{K7#_*Nce=Re{U`XlFDlD;~RMYFpaTCVMGEab}_O$bz-bs^hdZ z7@#~rVq5b+-tmV@rZ8>!G}lp-4ngQ&i*5#izjF%IzIR-u^5OsrawMJmx9L|r zdDl7XL7?2-X*!OEPU*b=(MB4=bGEP$z6Cp&?T`8ki(iZ7fG9ye0;Vj>>t5F+i$RUv zH;t^CSY&CP$h)1ACpcAC?*s@A{Z&rk+DGB5zY7;1=az=N8#)C-j4fSy<=B( zEdSsEkKe(%A|mO7zo{jhjHS%74}J0CMJ4*^`GQe^VpT4x;rL`&oDkLlSN05HLe)Rp z6ML95hb=q4{SpUM5dI~PpX2g*8I!%^ka{`q1y$(0Q@vNjfQ-Y}lvu|_&fz44Xbbet zV=J}b4aTPxv1`js^NVco-a$Bo-QOShJQ?|gx_`=R+`Y?M^(-=s))aTgLnk9EegPNPA zyPXge1aNm33jSl@R{Re4%s`z*9MkF7f8*bFGF_?VlfQN5`g?&2XO76|dvt~TQ*U3> zwBhCViG61DBmZ$=TlG!boLtM6Ld%4UNv#+&C+(k&!FxMGtloX4VyH12{NB5OW}!Sj zM7FNf|Bwqgd|?b;1T8HsPksW`^A1pQxm7>g0pzOl;w~Jo)LsOZHtxN*Sm|N2s@~MK z`S%rpl=9S;j$hGT<+CVM?Vf9Ub+co*NQVUkEf5VIAYuNi^=sqVsjeXlC83YZkPAa| zah);s-J(Qr1qf$n0qnOHzx@#jUZ#qLKj6V~od3+cG*(BA7*QZELXAa4e`HI*1!UE~ zaZc>Rx$T^4GPEEk&aD#x2(6)brnwolg zF3rC~Q|d4nTHAEMdG2Gb)C@jrGDU`!?|I&yVcwbff}UHOY=s`Bt9)jnIP&f7*44sr z`Zsi~`H{LFY%>>JTT8@5uaTLBF1ZU z;9&sIc5_EGt$_QAQE*5IGbB@(7;*sxv7BKP6skO|Z^0fy1B}?E=p?-;z*_VU3a^%d zL0C)BDGO-fZqzujf;`hX*q5hgWH z-|rm~5P(7)`$9s1m(U8P)|_k z=5q#jo-|i?x}@46hFq|vS26Utw0|4!SIs$)Uh)KQLC5X*$uhLhJgA*lmUZXh`=L`# zjEuW~O#WOmTtuvt@q9P(dQDKI~ z3cv*0)h@8Ptp1<}Xvu+i89+xpZOY?$E|E%%IErY-(5ci*t)%iuAijI%y{^tN!awz08qyE6_B4jCzF0Nis34i0vRxc~bGQcE5z6ATO% zb4qT*V5$K|AE|DMW8woAFou>%L z_sqjpCE|0?=mAgZKYQvDvNlaGPLGq#+JkBW<6_ zt&(~XQ0(j1-i>3a;h`)ArKaMP{pCE1@xoMXkX2GClzpl{$ZJo|mN{!?Dm&0v?NV!K zWJCv{>5#U4Eu}=%x7}CFsY^iP3P^60lp(+be0+J88vJ@lVg>Km+H|axIarf+*ILgS zUnstbPN_DMYj2eEFk<${}IX(mxd;| z;+bW$WXEH~AFX^R1CG+nF?{%@*2CqRyQ2CgK{Ou{qS8sV_uCqS^9~&rhRy-O4cQ!V z1en{HuV3%dHK5BHw2n(k`hFvQcXyYGnHdYxC%9ERSYjox3kM^SJ`4*09sBYF>s6izMW{A zeg309)QT=n0ci?(moIsw#d_=LknoZS-_vn(*6|UqU|HcKl~h!$`;dD_w2Ec`qoBJb z1Fr@!Sr$hJf8bsj>X)KFs6uNHm1f&Z--gBltbe{ws9vrbA*H1~1EPKUQO80+KtM<3 z2yI(OM`l@BnS4z|mv1);H~)Y%`?uT{z+?mRK0OH7 z-`sYtB4a=MKSRHZ#2-i#7CHLFfd6kInM=W=R_zi96&!=Oa>W9Q&ajqoG zV1^*Nx*BTq;?B|2M}r%56E3=;_#b>1gxuRu-fZ=`#`*r{>Nw*|ao2Y?r46Bo?G3?s z&_;X)vl9`6#BOBO17TVO0}5Dlpip4Mi~>TY3&e5Qy7lVOQKPoDwu;HZi)eRk|G3PS z9%dRKnvv=aZYcc-td}%=I#1VXK3NKoRYLIyAC0|5DVC)WEvulQU=gT{A0MH9<<=iUE;%n~NecJF5MINq6(7}% zn2!KUpe&6v$We%p1T6JfUZNVt4&B^K4V}Sf9#ItF2I#wIt`DLM+X6_e@ep8Uef{x< z%}sPI{x&OSrKLDBo+}zypUoKd=#1}S^Y42yPZOH(xb|Klc|BYrvONS^G6RMad+Fez^n1uSC z?u-7ZFHeHpaLsq5c~OX5<_PS(b9oY)BA}$+{TeJJ0|^(v&-B+@_K4fY*vka{o0r-& z`vyJ9(UlcF{%}{hfhR>|v96rhYbJib4_3bd}IzM)wH>JS! z>c3CzTZv4Gx&-nwJ3EV**aiK)E2A}V9^L^_CUB>V6m&Qx`h@V&=cQ_)1dy(h z#`$!>K;8I(FRW>lnhZ;&4R=p2%W}w=m(2LoJ~?`5GIs`BN~)-(Au=nC;2yH$>9>;1AvA4~_In4Np%ozusG`AMths(&w z=(6~g7{DcxTLvdHI1lUi%p%U2{_+bV$oxu5O3i>C;*yi2ii)_ojcX~~>geHf&P#ak zBOtb`x|^EJxcw*fCHyy(?Fv`GjRbcokw@C-)hjV)+L%KDm*93XlxB-(1}hlIkch?% zH&>8bg?B$ZlOe?=I?o?~cb3dJkx2US`?CP0UC#Drf@EW)PXAh!s4NPN<~7O=-sNge z4Gu3sB51I@%TZQMjR^cRG-pzmmTY}M6+qw-^1{Cqd=`+#TcK9M483a%o_GJp9#vK-(zW?&8$gx7zaTu=z>q5oNTdBP z(gp*dv(ND<_A&}Z{t)?IfGowcT|Y|-xf&J@H)ghJ{>`*wM|o1`H!1Z^fRzx)Z0JT) ziUgS<18783UsBn1sCXV;^3eBmrMpvt3SNSW>ln^Ip46qDWk6PleIG85D0+H|ZLf~U zLf)GOP@_<4((t*|REmg3Fz@E7_pKQV?~(_a|6$T~ITLtolKrSB=A!Icu;!`g0>GOu zuW-ewz&cLPTpD^EIQnQI&aOX|+$Zs49V=>F91YGiq?1M}&&2d}5UeQDVR9LiV35Az z$&)~6N#2c*oBqnf%ybM;8)x5GB6<28m*VTdBnVr<`IK-DS!A;U8ToX==V^u$wY*!< zbNFAZE5UJmK9rai2!QaSq9Rqo)R~z(aKnW3;lqa~#r)NOn=2Uev&iFUElg#9Jd%&W z2-2uqv{+nz;Q);+YPf=GHR>bC!J%|ABHl)&a&bBip_^K^g40Sjg|HL+wrKIK z4>O)@AXPcW_p~Q1m6#oJ6&Nm9bU<=2g9|F)7=(7{6vE|ZW@RDi064-yNL?KaDO1{(UG8JX9e=iwTkf9Emc+;%F8Lgs!x- z$qOepTVO653EiX_+RU;W`%pENo8ABXTua~s5R%3ibT$S{P#!DP z3j8N6c>cZ!upkIY3)Yo1Z^ztd?Hac9G4k@tNUj-fr@jOx6}ZEaUe@vGB_5@g(Qb#Z z_-95#lWfvHp)x*Z&38>94DUKSPyLsJ47>@T)5-ta+~DNS?IiIS?jeH>{c%dlqfQ2I zA`mBr>)iRC@fs6?LW%4!B-yv4PG|0n&P0WmOie{Sy*Bv}ap4*0h82S$(<2bD!FU)>qlER$2Kxti& zo{;h8=y}QM<*}rqJH`3L3h}SX70&JVy4qVzGgb<{VjHPgRR7LaeN_HRz^&5e`}yTe zRK%Gsxyh=>Lo)SsyMQ|XI(dR+^;HwGzP9z4M-fz%gZTZ&Y=YboZ!2;39<{^+d=O4 zpAU_jzb=0Nn z@3cL{B(@;@+n7Y+l}u6+1=)p%))5lKG)`<{i64O0zq#Zq4&V}y4btPlUCQP>%;17fiPVA+n&! zrh@r`gibVIXpxnaU=4h_{CD5~eS@uK{1*{^>05b;&r)G8j5{A_)-{-=e|Tu77mr4- z=RfO;|NNMx)k*v%E;bhSLsUaUgHg4^!(YFw!F#L*_e35_2jr+GJ^>Q)WN0~K-|&1x zKcq|1us{@aJj6@GvFm1eg& z$`PNpIEnvS{5A{XuvQE@E=*kXJW7!z6s&KJ^PIWoBlX57>prKxExV!Pl)lW0P^~Yy?dP>W>waJvuk&Pa)@z1J%6&Y$ zL`X@H=5DRkCI^;;ao}L~guH08m&~XINYIzh`^-1)cr8;Y9!WD^_fX{M zNq)^9R_?Pd4}TJK{BcL%n>GW+exHLA76*bo(E8s0B&%Hy0${mA`{qz>LaMitaQpu$N{iN+nnBb*`L!63Gw9#T?_czm^ne34+ zcd@nS_*T|B802BoY49*{-o&WCr*M8@A)}-Oxk2f;HX-wWcA)S0zM9g=mXS?LDovWH8Cr{Ogg88z=gLyx zwbr|z8Kb{YMXou!KkwLP7m5hYLSIvUCDV{0BsAAb4kk&R%1Zl#~=EvIY>??=BoodwV`O`GP`1eS?E|U`qFuH#1{KEZ%rutk_4>`5bI%*X zFP<5E(wWPwcrhj=C1tp|e{4+M*H@C7&*Y+zP(qd>^_b8E5nRy>$x=LyXgc_LT#7XuDw*dyoj%cJAW?7uLf*Ab-gGVarr7aBS%p55ASE2-ncattp^6q=ns z9}33$pcW=DiDYfJMdo@;3z&dIlAfNPf}+B4;mael*h?IiQS|%OV(cIOT(hljq2wj= zbOC?#Y{SkG&Y49l{+`tv4SUmVY^><%80+pamztz577h*$)a}87)pEoARqs*j<+%id z#-_z#Z+dZ;@Mv0*YYi6}`+$*a2PZ!H%h0aug!uT@9M!F*9}Q38BsneUNJ0va&``y6 zTo`;(XVpdKi3<+GW^K{IEk-#5qOM7PAlX>I$bCeA&dS^z;Du9x8B{?C(#JP{SR^wj*GB zS?y9{rH-ES+^YIKIx68c{{OUb{jpGoVf=KPEko&6b2K7kE3V1U<<~fLDq~ufM&^_& zx{!qJ5Yf4F87kdrE)=7jEX|f9by1@We>A@(;d1h$BbQA2yjuVH{`39k-S>Up=Y8Mj zeZJ4<^LcEI(yXBT0F*9Pmbt_g+8L$A*I8%wj(%dAfyuzeaRZYfd+_=2uO(^W=j#RC zQ-sF@k?E**06d&CuQEJ>YG_&bb<#=DOtQcqP(U1MIARu_H`K;EN_f&hTI3i8j+u6; zQ%DnyWn}h_uf^sG*%BL)5l7)ABPnlXkezu2xfo_~E)G$^9SHN0VC+PAx{93Nj9P7J zA}SV(S^wOPc}lwuDM00x{H>QKg`(i?R8oiMyjm@&Z=--(cu%8As_e6cdM&?fxHWNp zH{&DnZr}nc`6@fe*Bl}_FuJ~TnSr$DR*#Bm_D8f1cp7$Il=b*kg^cmE5h>73YiF}x z?+dCa;W!3{h4nvsr%x<7{xdxd{E304i=iW~U`tFxHMHZyIgCBV9dW2bo-V`O^!E>t z4V6HO0Ro;MX}H=lIwDUi0rjHgbgI9ACi15n8;`dvw6P(nj2~*6xsgaFZ?nz7&5~`4 z^(3pD@WqGX96#V464J+~8j_MS$Ym>{8-v5c^}J(a4=`GncR}FL>Kd+r0cXo*BZ@Wbz1uA)Qd7p@ktYF9(uVgc3_HC9^$#OR~Iz+Ggb#n@0$ zl4)LPgMnJDemU#m3EEagZS9PV9VGQsB8`x>#>k<nHgIjW6Q*JMHPm zb#d3RlJ&5zmQ{fV-N@V0p#oDUPtkTi-`nVb&p%;nl0=NmjPXIJxY>#(fZrpJIM{J= zv(fvT(f_0D)P(-!zRFa14cdOh&7eM9JM~qFR_^v&xfX#-;>6(GN{EnbFwG-4ip>vh c)oa%V_432}WsjT10AeW=mR}H4<`b3u7tXYA00000 literal 21273 zcma&O1z1(<_b$3fk?uxdAt6dicPvmTDFFeIQjwAd0YSP;P*6anq(nk#kS>uD5EP`9 z5NYYSW8(jJo^#Lt+;h*~>v=XC=9=++Bi`|jcgzTFtt%vijD#o@isaf=j4ld=BZ)#` zQ4!$7Z)j>d&EY@zmg-k9s1xMh*Y&xHC=?s&8b(pyD{cA5LodBM-|^R^eZHK(wTREB zjjMiB&EW#Ij!qsyULHsHYYFUFKmN~|!n_jE;rkJp5<$(}Y=N(Z*~-G%2poBmSd8Oy z*zk;N2Dxs1Nwc3nz{z{=)|Ju`22KqyT|PG}bZ(Xc{wWVurIW;*{`2v% z3Muj{iai?If4`bDG@2tvev@$9B0>4|Ait3R!$CU6MPF4}m+y(C%6hw=-a(vKAe%yl zwwH(;oJGg=Lb);D2jf4Y0wSqB1cZbeqcIF|>-2s8GxtRp7XQCZiTrhdoiZ6vNETYaY=oBeFbV2Ts*uCtB%;K8XCdIMbD&E;Lb9|qEtwA zsGdH3YO(pNC$FHOW@$brCx?fRPsPJSER=*Pw7*|FLiju!YC8t&CK}Llx8;pys(Gmq zIYmOy++55;|1>>n>TnO2S;C1sGHSNu;pW|FM$t$PDP(CW%@dS|aBtqcSz>f4JX$8E zcEy{CFeGc+lSTTRY8YupiABqM9{!cUmM}67PEM6(`-o1%q@*PCpP!zh9+z5mpk-uC zXQIP9=}P(bXvEL4*#=&|*NZ~w>W$pKF=hHjTO$5N%{f$jV&dS7e2RoPUVeU6Utd;@ z=VfQ~-YhG^kJNN@>glu;d-N5YhE~YE(9qDJ9jv0y{E;MYh)eK|1 z6a=4PDco27$JImk;dB-zE}}WZwjpc;F*T$1Vm@2HqC31k(q5--qiqP|J~#G#9ar(O zQpTE%FS3-sbZyIxZOXsfnc0Y?QczRRl_Dm&XIc09HCxitry1GjN&k$o7H__I@}w8_ zcxr0u?))-eUuEgK2xCl#B~`g-)|I&SXod&P)4t>=bCH$SJRR5by^-O*JwGd2?=lgg z1Q~ZeELH>$4u`*Z?K%(VsP|b77~DH}B%P+kxxG?<2GyUhmn&r@!Av^tIr)h1nU!la zi*!X-mOf^VVS0L6!g+@6<%^fW4Tl%ccEqvT3`T`SFTbPK=!$D^>i#cBdT%qe(|e8R&B&a(3tFKBbb9QnprDxUc@zs{}PWD=|#E z2bKSIlX=6o#H@16_(9WO9!eyZT6dXVx+zSTDlQ)~Gb2GCqe{#q_IBd3i@{r?fDGlZ zuYU6W$G(@0KM*t=ZV(-BS2kE2?yjTw-<{;!x!w8x;r_z6_uSwt$SJcuYDFUFvCyC4b#k=x!|zdeCkq<&O|%#sBfR@(plL6^TwfQv zZ6^2&m_Ol_xN|C6T3UZ>uIlSkzpGu5I)DDWLXPh6!S1bE4|XxJitU3KCh@O+vNru} zxM$9Yd!6u^RG9Cq{0NVZZoS+m6O`&$v*)N$cLtm_V#v5I4H&! zFFZWF(KS!<=bu8=!89l`z5X2?pDOE2Bknuv%<$Ya^) zbGme8Q%CZ%XJ}zz%Eq?lf{(Im0@q!M&Yo>5xKYUHYmdR;e(LUKdMZy*POO7#y_`&* zkea5&?J0u?0j#{mx7fnOxqjxWV|XEw|6lFQ2(_<3`Hmd!(Vn3>&p}$NM7k;~#s_BF#cyM@x{&!2m( zx6%Du`Pez~1nyB>rXj}IxspzH7d^e)!VA=dzQ=p>=lWXGTVzbotY}-tcU|o0>m_3j z+WAIvPBn}PsZ~*s4g=Mx+2^%MDXP1;2(;}aZpEmOCZsZxo{3_H7_~uo$%Wq%S(DVB z_9DiJP6sWW;6M&Z%-ox$C0<DT3cd#5i9s zA?=H8xe|G>>Mh^BA-C%mfB^n_aV4I8p*|+!ZRJ-)bDbQQ)*ItkQfqq2DZJAG{-kR4 zdg#6vUXDRwh=I=^+dg|S4j!IPEAhMV-7v4a@2j_uW@Gls4P;1mX1QIWek=wC61jwT zY9e1HJFefk1NSf&`svdr)kjwqFhk_E>cN4cu%h{Ob<6>mTu{iKsNcE6c|{1WCzea( zf;?WT1{W1}P*9Fm%SGhxI9K3|RLlI1QVN)r%g;3g@yj}~U`eY(kkvlxkxN9#0E^?& z*M<&bB10i7Xes=hoV>AqA)-?@a1f7^4lyb4HS+K8(vinORv)q{=B4>8vIh$H?_W@A zHkv#As;-$CX}T7-J0oUp78eej!{X%RB)F4+1L!Bg1YBD333$c(^)zPm#^WbXD1*UX zP{;|VPq%Z*NwF&dZblD+osv>Z_J5vX1TjQ76ifiFmRoqM86B4Q+?1aV&S)V>pRE-V z?wb;>UR&r4$5c1{bjhfyqMuGk(9NBpL4s0Zmvvy1`D4@GN(u_mS3T2#YV0^?U|m>{ zyCWqH)y;V#MN`~3!Ju>bX;Km!-odB8N;^44>P{mt7>bs0KSM{*m5>TiB$4BHYZvf4$Fe%qg^-*`y-0%500lGgpj-n=K|Qi)vd+B z@TjOnSZZXO8G2vJM#Zv7tEJ;p>{apB;;5GEp6H!-_iv0B)|+;B;A*&qtYgkEg_!oO;^^?h#tC6o)A}9 zCT%_i_E^0**Oj>Ny}S_C90Bc`b1v1kmG{r1ZTTta`Z@<(k-eRAsRq;+^!r_{){=f5 zVH9&F*QdU|D}WQQv9bRdW}Bb$flW@@*cB?B&7X5~Y-lt#evsmSj+T0~FH+;Sh(7fh zUR-HU0}ga0i>x=btoK$tJX;H_L4=D7l3WwUcpXS_ZIPdRjOKn7lG4z`5efi$oQ@Uc7s{UjQ}8;ZtnC`FlxlWJL zfTK*~;M!7d3*-REINTY{B9O;}5Qe@kNJNTzABh|*U7tT=EG;u(LpWhWV$MgRV_D^U z9gqaVRmj4Q#B4`$6^b*Y%nFkqALVGZUA$0aa1mG0W~lA%`)^s|_aO4-cs+dhuz%4_ zn}^y^c6;OC%FZucYk=mirV~=^2`2M2UmB$VM+hHhBfu8EeEH^_2m}rtj>E&vK5!>x1%<~$ z`Q!;?7vF!k(7%|G)O~84qc3HB`m?T((G%mtUK_(+Gfq!61JY##P?YA*D3o}yCr=+oc?=&QLg00)vzVQD9O4SsTW2*;IJWb-P+ zt0ZpmB_T&aT(%f_cdH7}3oPd8fuXY8X7%&aU5 z8X8V_Ub7fJz5&tYvC6oEeyW5qwM*E|e`O?+!^b!OR8CrU#7Wef^gVV{o&E7G)*F%U5Gusmv60OVY~b}ix5lKFKoH)BZtU?UNs=id8oC9d;Y z7#s=e+DIjUvlN9{n7i8vIq*U}?LkecrC zE5H4-IQYciw&)G#kQ>;URp($oJ~+=JRe;8lh%YvH9lo`-wG*p1xU5e3_g=${!v~VA(_YySUpE;Y;hYaq=_2QXf1!y+ynm z2AIY2Q_EZF7HDX_<@p05h+GiZ4GXSfBiM!@lm+>E>}b|s)GE{__BIdZyqVFqA78v+ zL{7dl?ivSYCjM`==>($y2L}#pY)JRdnqQ9u8vK3t_G*{_B9XqL8qUc69!efV$gAZ8 z*}!+RNBpXmma3Ce$?2*7fh7yK_urNWWUGO}sFR$4Efuuxj92}kcj{lS{64O<58CWr z5Y@T0QTR4oZ0$%9vvRY+PxgxQi*QyxilflM#n$aVW6s@tmH*sBr>^Q8_xHxX5||-h z(lDIczUz3ZvCJ3TdF6m|8F;WfQO^WHEIc;$*{M~u(PH73Q)2OI{e&1WR^xuL#^X<< zR=O1I^mTypTTCXEj%{Fg{HM`tgV(OVsZ$3*QBb}J-u+CUKe&c~(RbjbYb|8*r;bpC zp}@O=Z5b_fx}IDF*dKRih-f8yU=tSzeRft^J3;&BKgGVht&XGTvo^Q z>_9!h#?Bs~w~!i8BrrfD<~SZMy;`Stng-J^kG!^GyW$+)Dck*wujSS87B^h|8eE+} z|Lxn>ceiQ<-4?V@uX)$ATHBg}95PDyxViB2wZ9{`t(Hei!PFBG&)I;>yuRJwCrD4M zo2!J8TlUSNp}?}un-f0$c7^v}+w9z2pRKf7q&y$sOcirz2HqrFD=6G|T+)@qQZf=E zxba6{Nu(r!o45%#fwI1Lk3&F!OnlP!{i!iuCi2i9OAr6*`r8H!>GNle!;Nl%D#)MN zT5b0rcwpOJr!?U4vSFnVon&w&k-}SKo7E5ne+y&v3x`rgF<)OTJ+g1+;uVLN!?taP zw;zREmp{^9(hu(ple~p?xo9hxeYx^AcjmA6MwfQy60bY0O({ZbyU*N}A@%^u{Dpz6 zGpF$!K%SN&0 z-CI>oSPTpdhaZ9gn!8tN>+`+7+z0M0PEAW|30s~b;hYJc1Gs$X?xT$^?y7^8S_*o4 zo>M#0dTN-V_nMR!?g{ z+;mt3nsF_f-8ONVpRPBh+ChY(y2Dy{8tl3i7!BCA2Ruk#{B9?Z^glC?+0m7jZG^6-8tI-d-dwowN7IQg|)pYd<0+f$g`(;NWo%f+M*Jk zJPFLsKL4Miv);VHS5M||^FQ8X{TE4{A|!MSVgiIrqafuN-ntcY^Zi|m+4dOl!(>Q9 z4(%I1y9bJI%bWTi0j5G6(fTG08hePj4bti7m$FR2b1=8vXGiX($zE(0p1B1Gi+^CV zXR-NyB3`A(Of?ef==e<=5BD|`5)#znbpw;ogw>myos56HcMB~Kam$= z*5>LQp0{WA==;e7(^sXAXm6GrrT+?8d+N1xaW0hg-f+LufqNtLbRH1%yq%)>=@0(PnLbN z_nM6n7j*b80@PdnAmpZ>)GExp!n2`yY@K(6Pk(c#+<1LR@A~x7&SaQsR)Q2uLx-rO zT=U)c>=!Q*Ay{8U1rMWx(bguH@>mXUZN+%Z{4TNoCcL#UfCIbx4~uYj7f@v0;6Oi4 z2W$KI?fpFtQBfLLd)?eGTqT|BSad_ zsff`ceIo9N2~)3iPP=}!F+h41<7~<-_Fy7fCri~CbLMgP&*p~XzXBR5f>^M-y1B|2 zIW^pp;{M=WZ5%f>M+Qwy&#X)JTE8;cG+_BF_V#aw7C$gta;Y;lFrd9eON74us{Pi3 z29l5#orBugFtV(cQmfIm<@Y=*U+}`nSXI^3B7?tqhn3rnDDfxH#`zy+fjc}bf|~uU zX*DHE&|}$XWkWCQG8d6`{B#X@J@Qw+3eJ%Vr)eb~YQn=m5Bw9Qo{EJqk$F}Bqsm8$ z8wwh$HKv`|+>g9b3P4KQz4`T&kdUy%ya~H*t(gddVP9$KZ)zY<6vpyIXk>h6Q(53d=jhjeYf2a9h zH2iOpbztg(02___tk%cWROW8f9v-j%9qCIyZPZM?z`H#~88H5dQR0@SoyqqngFlCz zA{zf%0Q82t(#j>8ytf=U{Jl!f+GK0OXSM0>`|0ZWm&h_+``=XO4araOnB_@v@v(#O zv7}hcMHnrFpKDXng>84ljqFuD=}pH%H8nN$^nM$;-7uJY-Ks5;#;C^a0&v;JqVh4S zIEv)H`+r@<&+Q!S_<1A~g+=nl8{P~ut0W1VE~?IJ4?>}`oVitFUT`>vPGmIyYFsC} zQB|%dt)<6N5cK0`^UPDbwf63lKsElb0WYk;#)`Sh*mJUxBW6A%wH|+F!X8sjx8s;8 zPzT5#oi(bZ)#k-|PTX%}>a~!OQ({b(>A5;F>eDA%^?N9PvZ&jc_bKv!sK?+H!wi$t zPUhf>MztOeQSx-2F)Y%wCr7LP>~_roJ_dy}0(EQHT-RSl1H!8`_rQGYSaM=7%*Nh( zDyseI=K>W&F#pIEE6g_gTcc8Jl%V6dPt3kk<0OEC?YZQvdsP(PG}4?H!!jw;(X~Ox zI;p>`B`KfQ8;U;$%1@k}Nzve{;B2rcH9~M)-SPhKQ!XMM`ctuVyHhykJ&cW(dw=f< zKPO(T_dG+(r>o4Bn31PWXP+XB*Tp1`mXc;dX_K?Y;c%$Fmf$DF55lVV*?D*Q7?5Fn zQj*(^r|Bw?VP;1ti@cb9Z?s_sKBM%=Mk{LSWwK8P5=x#5ZxE<$4>mxcE$@7 z<#i(^j0Z?tY*8v*DljiIk+w(a@u>bRv@RQ|PBxAHEZ~kknM{>=DEuTbQ3&Yhe7*aF zRqK!GjLMgmu+N|{I{2?;@yX7`0Uh09E@&!4c5lP_(sOA#^>{|jMM!ni_ z;+O8cwDB|f_6*cnUeh6Tj7Rine*+6w^V3w0R zZDyscn?3)=opP_N+o9|8iAnT8&=>L?z3I7#*_6XMsx-gh$4J`*O7eEdemPn>J+oJ1 zTg%?RcizZ8rw}%Iq?MnP@dn>!QjR)HHKutW>q-J}IFXTr;hEvkBs2B=;SLG@vD%Fj zQU$POm)~po|Gg`ce=UxTm^j$1xC1I*AsmigUPjYz{I8<@V zQQd_ovSjP&eI5x<{NG=FW3h1Xucp5kkp_C?rO%4wAy|$eeU)>>kkQ;#=%_-^6ZQLR zk@AfjH2Yf&apU7oRi=iXs`|XI)sv%c7Y^H==j0sLlD7M#)^CM74|zU&C;@_0SYe=3B|OkrM~Nr1du z1!_m@mRne34x?}ElOmw}i1&-HS ztx^Onu+5C?3!2Du7c)0DToAO8BJVF(*rHL+bc;+Q8!IZeS1`*pg9D9q>aUMJfJD)jQDR!iap`= zFJk?W`!nKk)-&)^0P>_Dy=EdkIqwj8f=t~xnB$k7Vpw@4TyT=tOuxc7_6cjt#^ctk`KN1RL` zP)D=3Pxk2|qoPWE_t3$?3v25gjR_nIPJ(NVI1RBgRTugANUE!=d-z?x3=Rt2>t+6& zCP{_Btax_i7tC2@Wt7GB+s1#c>OT^4b#)zR@IS$~sz2Hu14hp4aJ?h-*c!-jJ?aFj zvZ>?^TY6Eo^D6jgqpao$C91a`+>XDg@r$0g{$#rNev~-JG&I;l6|#}-4fk?Rseyp7 zbBuAoOk4H6lpQbnaP}CGc#DmQ<*(1r0hk(^CkzG_6c(1=ALK&VfFHG<9e_on;E`G^ zOQ*WHUtz#`N}Q&fzmwo_Y1llMY5%XrS8r~;OUL`}+o>9Fr2P3K(G$L#nlfAc$^>L& z9ekDJf#AG(+b_CP&kpU3cHd?q#vcckU>9s2>-9v+ScO9i4w;Phs_RY2X3#4|ij~fv zv41qKFc1Bh=p{3(ExmM@6vsR?-dj;!J#@ z04d@ndIKN7s2`k`_skfSKJcDb#VyO4z`fTaDq&S-uHE4=Zu9tiaOn3Z5XO*(bbfxm z*L=F?sLq4H9Q9<&flQUrQmZpt-t|L^{SN?azW(w>_wCL1_`p20LCjwNYVu&%kRs~9Qtyg;WYD)g<-cn&Pg6n;u7Hp7~j<3UsVkYbC?rwoakAM0!1UL^W&>I~6 zxEjP0>9v@X4Dy+P20#Xsca~Ct@)WZBdI@RMvPgR*NwH+o>RGt59VpR6@?qD6E(m>4 z$JKX_7FiY$RLs?VEl!liRjqB+GJ#En??w<1Af2DUYNug>jzpjH+qHlybPDL5XFPh7 zAFMh^4;QcdBW1DE(ccmEbkgCVqB*jn?H>hJKQ|SwR%-?}Iw8i{|h>JorjIKcH^ab~(mvDML zw5W|^G$%6fps%WWXTnG5&Zi{&>nE~6T_e@Uo$hDO#d};5E^`Dg{r2x`ypl`DBi+ln z*q0MdcA4!>{kx|34p7*Ybr(39ANd< zx9?>PB=jLiKg>u4&>G{9c7E(m1(7**uy|fgC%uIem8QD?he6zbw}mx6EzNiMouxa4 z3dMuEb$l+ZwD5(61%pptcW&LB8h7dDD`#RL->a$Vt%?w~VG`wa&FhSQIjxGDSEcD@ z1&3gms;sE@;(k?dX7bGB=e}$%EdGdaaOG<6j;B6h-<#`1${rwf=NTccRyh8MEdID( zupOQHCe8W)XwGKKQga|x`BHNHCB4?zUiu&Jea8;#WF2znSJ8W;Nuz&r+9_gznz}#m z=6>=QMkbF}0`~%x{n~z1%mEjQfmC|#;amdk=PwP1yR&ihqPEmT%&aZw+;g;1c(?tJ zL@OM|LMQ5dMqWA(>USp#bU@WBIa!|Ix4+1ZL}<7f-u=J7x!t8Oj)kkHk#D4K8pP}h z;*3m-=vtINKEmV5t{_tdf;ul@byMF&*k{{bBVFq0(G%70P;zr=z6b_d>%+~kj6^r1 zxx@CKmmNKpM*{+I&LGhja<{#~+wpPJG+7_o$_ej_5Qb&b@=v~jp5<3hY8Y7L8Q3TA z0(7N_8$9ogZ_Ifgq$L;!u0p`rt^R;5-X4s`bi2pTjmdm_mL}y+`0d-bDli_|v>bn9 zh^wHjf`}xx7G%iUik^Qm`O^M!puMNP8?dzE%u+t-o={F`OsTsKgxWwHn)&c;E9$);TlRA3@iE#@Zu zqn90DWPP@sPb#!?s=_eyh%_xZO@EEG(7q@^6XvU@UcreF>fYWpS1k7mmzHcbUWl9p z>CkAkD>oV2j)hv0fyr3@gYf@!NiVtAA!`FXqnUeq4EM&xQ{|ToB!%<-A1CMqfcn&X zqoD{z{yH=(#do`&h|AwTExzi+8oq`bPVQ+tDC+g**u1l?rNVa~<3}^;I(VXIeHoO> zZO5sZKL082;x1P;aoOLx@Ams^PnOV8mq5c|AOei;-@k7JY-{DnwzGrnr3T4+<5iG& z6^chHuiD?Pgat*`YIpP1NwknvJGiG#qUc{!LQe@({=BNIRa$`Hn0bV=eDbhHLQ3j6 zCkJ0RTm!4e#=$Y+s){?9i4s^FwThd5b#llJ)z&3!6pPL3v$Z1f2F4|4+ga3}6O-!LPuK3!CKDW*8V77XH6UIfkJNS) zRFK0IRGepHL(xX%gsje(fCD22*C%&C{ClMM%E|Fjj#h+BYjG5_XJ3`GA+Wa2pdR#? z698#f6ZD4%K0fk6qTji6IhCO&qtze~eJVWce0 z;S?OkM-vmR(6#~@pi!CiSy+b8K_5Erc`d;*hAin}a$=zFgo8I&<+KLc^m^UL>?s8ueRR%t)6Ow7xddB`8a z{XyJj{YSMfU;(88Rk~rrZ_E>6KUE3fH7LQRJ_6+4KwB zZ@R`HU%e11;gLZ&`KtCYsVX0SX(K!9RFT8AT&?u}27iA9`#kx@jt9gpz<&lXZ+cm8 zk%|YCz1SUndt0|7>*{3QnpBd4jFVp2n&4M&`fvAF8ZyVbv%B55j|2UV_RF7(@{CnJ z=oov@5aalxW*X9fs+-$~p$inEd7OPYYf0z+5RXD;gNuteSd*wJKBWm$=cULquyo4x zAc(IDy6VRFvHuwozgt2{S^&?1SeD&gstBCnV@Q@2Lzo#(p&Fs966dp0eFpRx?XcZM)vOP2Qh^t&cAd`zS&IMB@7qGwo#gp?Xj^T& zH1+8Vii|W;XYkDZ1J2@DDw+7wab0+hCmoxlpV_{GfknhDjjc#(%rW`q^MN9e43ZLs zMw$c#lbHS5Z@+7I?{Xm3ntuB4845XAH0bkt0ia>upB5{**Y~pR!~MavorV(@q-BW66`|{B8}>)4~k~;PdJ7d#9y9I?uJlB zT(>(Dg;Z)l3JUDDU8n(qxrO_cZy?SY(y691tgY$-^>csmxXW+prmIh16Gk14g_#2? zCZ&FnK+Qlc6hAhmtZbSZvkGpquWp=}NxPA}&xaSWXn ztK-8xzx5p-WT`;f=eIk3MrQN#W$^Y0!`_XIuhZo!u|xO9y^IwUP(U-r0geC4Yir=j zIy6z$J_;$hf@TaFx>H$ka!P)Fa_w=ay3GTX*{jFbmRv8$XmTb`ze&+v<`bL_bES*_ z%8o0Qvs!e60*m?vQ73<4VA~3&l3eylmeF|p&PHX$c|k#Pq_+sd!*CPFYCNDSRT1v? zzUXs^9+29R>!P{2&p@cWBnNyHt@zv{mDtKO=Rr!SMG$-jXn!Un>|8U%?R%gA{lxvN z#SB_VOyZ8MFGTIg$;ls|Vf7{V9CtpGq*i>ujlpg7f%J2hu$uTji;~lf63s_hlgMB} zj+sCl-E3|cV zu%>3X@d37jxWh7Cue|P0bm}v7Viyu2i5W!*-rwy(;5VWuoqCh%xKVS#3qaW9qa7Bc zFT)IC83Y5FJ^V?twerJcX()W@r^CGNhirgbnIy){ykZ<3){1=|su+g42l~U)lnJ-* zs5q*Se0u8v0W^V;B~Y71nq#j1yP64dQTHYNXl6;u>#~2Tkrl4n_;j&qvsb$Q-EE{y z1V(%YD7Geme$I$Q6bUiL#xI8Q6X-U)67&oVLZO?VRZy^0@CpR3nIAq-1G1tQvp4#a zi;G0q_jgIQSAIl6=WM2^a>AC%gwJ#F3m&3S1=$W#Qq@6fuL7hm=)3C#id^+)$z9gX zP*Woa3Ha_aj}ErmSdZ7vK&8_m@RcIrq--=D8cHFDY8Pv8&#=|IN{FbfvU0i`zE+I@ z$fa$>H)V966XMBhS0|Jd{?hAbPKEbV2Sb`|u>b);Q=foLJIhrY^+n1?f-|Ph*u+8I zdwV$oxVdvkhZnGL^!zLTb8)iU^yhrn6hGLFXqDALuVa1K@HX^1NJc?}>R9_n3{fUS zhL8n%-1=d{&ZbR*704-o8!);=J2&cDQtHAHd$t3GI{N~5{C*vk^9^SN)lpJW0q@~r zedo?JRDoNcE;|~xAbgtBl*00lS~@@j9Y_P~#fwOj2lX@%05C>IMo|+DZ{HF_j#g4t z#rHejyGKn!li`1IOfOpN{4`IS6pgiWnjF`_N>@?9bXipnSi z*#zP1c=LVW8^1=C#-C(t4@#DdAxq_$mWXf^$#m+=x@soyb+rWe8nT8Yp51ez2`od79}wFGtw0`fE*-yyp*9uz`E zr?#&sxJMgDEBS8(LBc^krc!+}6esrkBnvI+Y>E?cU#M&p|BYKEcY7GO{*2K91;RsD zgacwA6?ABLjTz~w zclP#%4%?~#bu&%75~d*!+0(P0L^JE+u$>K7M>A21+>~mciINN zsO8PM1TX0WiDIsQ5ky0&?p1Gk7LkyZ(-WYUiUT$5-6qmp9yF#GsM0iF ze)pJe#!HiQ#Rqqh@K~mgWtI#CbpRMo_M10o;J_e~Afm2EmNDcHSz#sb^`>E=-WHqs z-w1GhjzbDR#pSlVgs}@5B*%4)9pD8uJq94a0$ipvY6?&zQa3}{z7Ac%+3{*uK^T)E zBfA1h@-1)KBsESD?AcmD&)KNRVDyiNDa1?=YJy74X7ZyfNCF1(^|*Wa+bOh zGNo`oh6LE&K;DbI4qBt>{0sVops#>+?zFBdaGj^b%2f|mr?zmfF?W|>#{ulr{Mw)~ z&j8J(9fol*e^1Dj`L(qo`yrGUNP7Y#-T0`r%|3Z6K**h+Ker-82|$gqGGSma*hECA z0|ElH^YuZ|Q7#0s&o;{dHUb`A-e%~7XXx_|RJ-1Tnu26~9nv!-W&e|dNzgiF>0f_( z7=0xwtvi)?Zq#L|{#W!7G3wBs1 zmJ_Iz8>E|$v9aN4ZfVIF`E4TYj*c^`_ZEHo)s3LJEJ{8;rek2W(tK(1gpmJ_O`Spv zWx$KVRNpZ5;Nj`u9?s@zmtk7bq*8Mzl{k${|F8ic=XDT>f4!pm2eQWvK zX2B~^;%B^mje=?~@9o>*p1aEK?(Tn9hiye+D~D`FL1NwyB?2x!euc-95Q{b;nCxzIs9y*nvc8Pe(a`*&b;cPq80}Mp17inIu8!$p<7Xdm+#h{ zP5!NjQh4pm%_Sp91WkiOQ(dfd-2ecsme}rsWi48mzvd?3#$H>uz>M#~OwaVWSy>6r zwEZ77%$!M89bROp*8uW)E?#T_6M@bkqDe?fqJ{#S1{v-FnUU=C7bfkoEK!KK!mRSP zjE)I8OdC*flUhQX5cSx3c0w66tf)rLNF+ACR%_gLrF*T@NpYti{r&a&Vq#}R=jP}$ z@^Zd&EUB6>$YoZ1h3*EO#(izA zrYM0nKG2hbmaPQPbb*$GGPR2Rf1ll<%dNe&cl*}T!4Xt3VRLiVFw12Q%n#3n3s_F; zTjSCzRXEW9M%WobA|fo**|TRMtGwsF&5O1TjE^Um*_%uB0#S?|R3dE2UE~z66#-I5 zFEBS{gNMQt~PxzlaIt|l4+pT4Hx&;9U16jusMa$QW5VT-C z2%mM%rzpJhy)~-Y%{F*(=xQ_p=3CD7J3BNfY}nx+xAN_)v|3c!G5P`(oS5UL*+ea6 zWo04TAsz%INlHqVOTWXz$DgVB{Z3g)30t4{M!HIb23HGkF(r_Nk)RfIACxcHXO*IJkXL13sxkkFI>ABa&^wFIq3xr(N} z{Uwjpi9g$R%9x_rPg-ilP8^7O#PBQI`#kKx!4RH$e3ocq%u@Nd@U!rxwu<0R&+^Vs zQqb}N?F=lX=w2y|LBOyW4D!%HPYtOANtq=cgZzPAQj!5lj}wn}Zf&hjB3gMw5wf5RYgL$+ibes#Q>icDq=g-B(h zS>?|Gr$5@@FONw1Ame8MbT_K=5_)T3m*o-3BKw?#dZ!+l4drY8aa^RK67bvuB8!dn!OE1=VdGm;dd;SOgLJU465-2YlZ zKZgb~ z0-IyI0)tfU0#Kl^8e;$c`X^QjW{3^eX`K3&Pm0IWP<;B>&cLzxeb? z4JP4GbY!n<_I{<7`W<-#XgT}mdwC1cASD2)pi-#Y7D^tO3ePk;a0h{o(8CRDXsGyO zq3Ylppcx25!V}OV6t8?Y&Z5lWDqd3oXN-5jW0SDoEo2W#r189^L<9`VDtA8+X>%AC ztN3KmGUAaDpU>Pcow>r8QN2Di3i)W0(63^SrkF|B+bB{B0VF3@`xgO7uqd$ zL@z&l5D5+qHM{&qdXGHef!@Wr1^PUomspnJ0{rf}exT?4lkFyGo1w>>t@Sj#j(&_X z^XpG{6)kmW^_Anru@N8~zp$`yiOWwlplo1}6;J$!8x?4%bCBaxmLjzx2RSoG(Nm!8 z>`=RyVR(_EXoO8n`%F-crbrQHNW)$|+}JGDXU9f_F*`e(4Nd@(Bt#AbIQxxOdL#@L zBI_bP;l)9WA2gItnLwx4c_*0^Z>ae~B+t8#qu2IU>@T>fwYA;n36P@yc8xs~zl4w7 zmce14CCi}Dj_6$Nk5~xyRsiG>;SOYLC>)XO1ZryabY6;wLLti|T9NA1I^K)HTYu*H z6|&+vmcE=_w2&%4y_}Va(wwA2O=tBJwHzg2RY7Zg_xSFg zF!pg#S65fiq8S$%a5&gm?SPKgl$#V%i2yN@33HX&JXH$u;Y(aNk zx@gccUlQh!QHY_>A|1fz(_hZ+si6hg7i9(VUI20(oY806Cb>hlsD|NVjkig6vPkBy->+%`-n zePJ@4FGMT9N`iz_`wWNlh10A%N48r$*qt}L^gXb{v2u2IeRj0+0qfn*DOgzpMhDWd zYvel0)MdA&FTnK1Vh+`p%2&0h7^T=eb?)r2#i9S~#2zoLq?tyJZU#urV3tB0Y3w3u zbeK$7t=lA}Cd5NXEAvJa)S$}k${)k?&bd4%rodvv;M&B~Q-qf>GHvh{ji%ow4bceY z38EEJYYrb zo$EON|1sj^2S&dEsm2&ND8CfxrBB|M^9tmzkO9 z<;ezCz^yHybPNOi?0IhPEFfT@F?YK9%YgXGe?$ZpoIF@Pwt7)BBLBGIU^DGlBIj`q z_1pKo8uaoxOcl0Y?s#ZbPg=VsJa_o@dq#6RY=@hk}*h$Uh<~6=}n>bkxv~XRPs!_&Z+a7<=yuTP&`6~6g2W%<j*e^`9b?lv5}{`g9UNK^k>UrjPvleTU=Q}%=&oo4 z_g|JNSK5D2FZZ^;z+xbx^9hE4@L`=EXC`jDu@X^?a<6dG%a^2;VFOs@nveT;!4+RE1!Owi6Nl2m6 z0p|;V^KoCJgjr6I86ky#fB)l}8Yx7Fln~GXid;TEzSFw{&|!3^-~6d!(0c#*n%vZ+ zi)Q}W*SelzY?A6`I8INt8KO+80z>G)d&37eNby5OMP@|>Epl(bFZ^)q0G1l$lQXCH zHyq8yDZ&tXPb56`81;Ssus{E4$pWqs3FE@>Ar%HQQ2MNYwdob%i>GV^z#_nho{33W zU7g6*&dzzN5i2_SjmAo@ZLe^>&zBpsFnG0hbPP;P#D<3AGs}39KRK_m(%sb+@$W=sKANAbC89qr z(tPQgTnKv|&*ADFvgH;db-D~zd>I{0>|wt!5ImrUz{A5EeSeQkN=i!G#3Tv^+W=uS zIaanTx}h&$zI^%(;3_)%3E{)@Sb`E?E|e;`G{EyR9sF+`|hSSZf9RT+GbG7Apcr?I}Th8=|^bu z_??^$j9Bx%=eo1Qt+Qzo?bT%NWcIP+gG?7H+ivLUNFIl#+Kv-B$&VlcoWa#mLc#mE zP62M{$k6iOySkB?1Ul5L)fs*B%4%w#>i(pxKM|Pt(ef2xqAM!IU<+qpyyo%bY6nOb zrro4;Oid3lJx*x3$y!`R1wnd^(L-8}muv)^w5iW3pKGdpyw6GtD(A0lk z*d4o!lj3eV!8^LEn1PaKagIBuu=&g;l#dA-L*7q@e-;u5l>Wrrm#*?9NLjV;QJb3Nl~0BQYO*d+c>p+m#p2k-oMv$0hk8XT2~~vY=wUsKWD~ z^T{}MW~kf|OVt6Y9*8opdb6RC5oAgOIHk_MzOa^QWTah$QS;Ot-{@T^5xWsqo-Y+0 z8|Gvu@wm51M`UN;{7XTp?w7_)x(rJ|ju2JO3y?}ui*qGj5MJBb*~zM|W`GCY1wm7M za`G7fy`Z*&5o|9231>B=v#wQNrcmlz8zNk6_`6@CR2ZtljaH{*=b2jvpiW6|cg3%?MjIY z49{(K`F%posc+~J;j1x*t05tZ#jfj12YVQ0?D=a>)u!;FXJAlNR>t8st$GaV@Rqyp zn-<-g|1V>~p(GLu%0!$yGlJg?F2WEr6tu)T)C1qYM}GfqoCbr}(4c`)OL&_~Xq)9f zr_;auBB|e+{^bCh<35hf?@i@ggXT;sb5WbuhMRrp(-XR_Nh_#_vJoVdd%+t<`oDY$ zgtH;=3A&dEu>#5%AaMVMz7SKfdY;$e+7iTlo%)FzeJ5LrS^63*p_cs$|Jf62kGLY-tw9`yH#wP+iM7VS7Mn1fb{&~7zP7}P$aU=qHklULrl4p zLk$$*V-bplu;76zum+~94R<&!^h{=>bJC9E)nWR7GA_>#o$P78~)J#wf zDqC4`AYP))6O8m-`eb*Q$?kJxrdXrCh}wdyCvfHXNQ8&3C~EjvoX*>Yh`6s<+~^BI9?Ud)T9{PVv#2Ih!J^VjfDmKkgXrCX3|^YKuUZ?rIkpC zxirPBF|r>SG$T=xlTdlk8>!RMhA|sIgVGJ&yb=YH;H~ZL%+gW{VE3BtY>|*l`-hcv`U-{-uhB`BWrQg4G^z?!X3V7kg1TE7} zuYoE8S{9yb7b*imDXGZbUg9)alfa6K{7oMh*9S9X{x*RltGK%<^{P()$OYX*;fgBC_(b-oF zLoM)~LGx331i5!;uEEulqLDBfW6?`e6gQs`D6*eHK}1X2T{1y6*7_g%r)a$KdUG9f zHn4oI%ej+168*o#lq&DPynuv3**m|=k3}WlTR0O$kFeeCUZZ zGw)d!T&rGY2-MM6B;GCUDW%BDazr)Gheh>I95Vs!Fq0-p!Abe`Np(VpG*)Jg!jtm= z5C{TMsDT+lWm`Me8<$leUMzILc6K@EQ zV_eVk{xq^0npxCmDSnB;T#qid_E6KIzE>YO6yz(%3I+WyhYomkmb#Fcni?TKKEC<; zhXWHYMSqJG8v0O&si%fZ8Tz5;YHsHfPs7hxC}0|z2Iumnj%xHR(jDz zMNW-lsasu(>TsgF<;n^blIHIvdRtl%^LzBox$}MZ?Ai1G?|Giz>np@dgm<6k=H%>F zDqo(RnW;G_X>eKPy%rG3;PcGxc+;cUK_VH~PzX($KL z?(2L1mMe+mgUR%duYv^-LKWfo@Rx%CNHZ(wj4qUJ{Wd-Q7Ricp{c#;g%Jk@ThHf8r zZYm59vdE{o>$V(1JPp1<>5$xyrzi%mLFg!e7GRgW6i=R-S6}+-p}j2D5`qd$IF_y9 zR+qZY>@aDVB_)r8_ocF=-$UKv+^rl058#EYRuvi`tVID76b_=fY~pq|WvqGS(H_samu0`+ZolVUpZS0Be95LulozdVAH zvp$Cr&Ml9{5&Dc+uZ68Att%=j)-ah|4;FH2bmD5&Qxx}l2w##W64`=%)z6T)o+(8m zbYbSX8cpEeMgXFq_=%y}qZ*EqHKn>xT0_hV^Cc%dZHXTd6d+V+e%|56{YjJW9{IO( zpf&K8o!w>G!$4OF$H^(5mF#|z@qbXD>+Q-<6ki=YK20P2}q_{f% z)HVts271Kj(94~j0}~>AjsMp)do{cVIAa$XwZ?wd?1lXv}J5XOuwBj z0227xf84IuC?+p#T(~b|svnK-hL8OD(tpk4z4TLV V<7~}sR||AqnhE%UJQX)0^EX)1P*VT^ diff --git a/docs/images/chapters/offsetting/03b8e0849e7c8ba64d8c076f47fe2ec7.png b/docs/images/chapters/offsetting/03b8e0849e7c8ba64d8c076f47fe2ec7.png index ec35fd139cfbc7794b7f0d5714f8a22352ec6dda..f959fdde5be294328382a39d463ca7748beb51f1 100644 GIT binary patch literal 13212 zcmbWe1yEFB*fzYt0*f>&-QfaCE!{04jYvv&cO%^$3aqrCh%`t@NG)B0bT>$cbp6LS zzW1B?|9|G2ZZSvBvgcJik=Tp*I25YJTRL=Tix7j3gtM6iaHvHmDqD~BqEjJvRs zLCA3GPgFW}X64c?^wP`cXC;g_$Qb8S&tD14KfKHBRqGoQYCEHuz1e*ibacc31xp1} z#wR8+aC496yj~35PZ6sjeS0Cj(CXvh5I-;3M>EUl-(fxK2&adHX5tY+L+KhN&7=vy zqzcNA&=u4eEGdGwbpLK*Gr_=8E+NQpItWMKkX3EFNPYxe^pB{aMHj!Yj^L{+A148r zzeUnMdWM+(76G^e{r_;Vi^&PQ%U?_x=^erG#h&~ zKR-|wu7WBC2`2^vBbpfZvJ4M2bVb3C+U1f%T@$rTA6-@!WIf7liskih5}=HX?d>3F zN)ZzsKMBiov)Z)8M9HL)%YSw=<6vAYPie0#2*Ef&(DZ_W_Jl#>$xY{ZZvtRo~Evc*|aiJ3@k z-fM0?3;&29KvE%Xz`PVN8e!_>YX4G?+aU^sm>>fu(r-^;yW8nIuO8Jskdv z5r;!TIXb^{)gt{aPd`9IXZZ4Vc=lLM6c1- zwrt4M?m4z%#&eQ5bUr>lE`+j;vzmbOgub|*7TtKKhvx0qfTp@U4|YumNVbEz*riRMn)1JKXMxyrDaUxgQDZ&&bHOx zVZ>l{pqHofEx#uA~VdRMXGniwuU(*?`g_N7C zW@08Ebsj?2bh9cTE-hQ4yUjWd<{JvMF4jM1BsMKjj0_*hD=Xu?di9E?(O6y{^WtbZ zX>Ym`4Ky$?@N0ZLw5W)w<)nww=VI2{kfyY}yt~4%83d9~r0N*>@ZzXegqJtGI4_Wb zSuGd2v%>;vY}pzXSVo?14*oOW!rHp|oJQjl?d;i1?r=}Nr-6Z<>^IGRHw#!#gu=bA zr!X}IMV3obJ+Y@F42O|Axz}XujzH_f=XLux*ePU`;^ksFs!{8`NnJL)O)c^c4s7)f zv!P(p2+OwH#Uln05z5=kqqd`~lGnvm?%S$1b@~~-{LV}9V*VHHt1c_Q3S_>Pl>BfG zxV<=#k(C7pUUkt*2Rh$6e6u(5Kj5?BrD!=D5pik_;^m=Vu(AlpQqUotPq-e?0VU=g z1%LYe=hwtU_~0~!ONl7~20gC%DGpPwm8szyA{xD%s7~~1$O&C42OLf);m()-& zN@dIG6Fe$Gu1$DY2Pi@sMT(%O+nzy8Ow4D$x;J0D^kg-P)(n`a*G6esS#cE=91Xq7 zX41+rMx%H1XlM@h-e*>Sre1lDi0@izRZD|Asv%!88On@=@hP8cR=c?} zbZ&$!`+eMhM*v3QY&k~F_1k52n6GynFn4uz<;ohanm^#kP}9)R5cJw7^AM)ycUpjy zYm2P&9Rj{Y6o_ZPCXn^Mr0l2zgP*7CO+Wd(Z40C(DaB>|@d8d#O|S@msSuCd`>jSx z>%)kQlvml#X)rerPl0x+hQ9t2&`hhpKx}Mm&r9=f=ETIrf_{HlEV#*^KYtFY3hz4K zn_k}<&SK%@lz;b5O!M_?LMW7HlMj@yS+d~u6O{>u1b^3uF~Fee>h0;gX1B>I-4-WR z`sKQBzk_2Zh>G4fEh5Xw0IWSbJG+g81G3zp!K8M!%4Uq=DF;U= z9|I=tBKKP-3Tf?EOjB$s?YqxT6cm$Ht(<_fT<^S`=)OH7RiN%;xpzKo+*>;;zBk5Z zZealij<~$1Cm%OA_d-)-edKnPSUHJ}jftl$H+Q^lOXjji-sPE>hev*M^X*TByoLrL z;2^5L=7TWoMIVk|(u%k72M@hxanR3=J-1}d%vfp4EhGz~O*@#Jm{>u^K(97AgZjI3 znb+r=y*X+8_u&Bt_Qvk>G%@{D->rFTHiuT|)OGvPd1PeP$Z1zPw@s@}`ct2=u(C>- zn$p+Q){c$aB1<@LTyN|o%szD;AD?Rdd(__|M8ikq-&+WY{{|02CO#fCJX}UU*%?um zmZR%t_4Zmcnz;i5(qsL=Q)x z#V1rF6x8sQTKqRs+K)ztm}sEWmn5ZV%$%NU3&VH|nz*Ykx)rbXsIjZd=rsQR%=rP1 zQ6yM(se-&IPxq`G9JL<%#?D@ryW53dyL%YY4n1*kQ{oF(dz^@K7r;=@EN`FULtJ$n znuSNjSO|BcHRq_{he;AEDu+f#VrYxVmVlj>z^XZ@K>G7x2=>zCHamv!%V8`gh%B=F z>V8u^;HKC}kP5ZF0+EDIV;DgYr8i;lgN8>Cve4K;r3>1l(bi=6gCWTTlM0vXy6OMk zP|Mlud#3vkFOkmsTS&PWiVt#upjIqRt^dYuZ5$Vj=6iN_+*yJ8ek+KT1|w|t5`UFb zPA*s}Be;e;ePhD~l#sALGgp_I^Eo*ajO8T)g(gJai@}~%L$r;tVw&-O|KUp1)o|Ac z+jSR5PCE3z*6ul?$-NQH=2JHuTd+lrowZ)w=}V-O!q)tk+1gJ#S9Yj*7`;SF zt!*37gsHJleXmR}K1z33u`|KK+p6_L3>sz}ipt{Ojm;kH52_R#bFez|1IAU&QeSnE z-d`;uoXC{%S}x@1*>Vp-Ct`=r)f3fc-F zB542kJSFOfSgbyswj0Z==wIq4{4sg8o|z-zfitm%+y2M2)=eoHgkeVx{oMo46lo^W zQ>f$^p-B&cFgiF~^3{{UkIJf6+u7p7Mse{L%TF4dYu$^c>RkY@LS$|c&Tfl=DWm?T zTVH+%!m+AT`2jh z5XZa^Dxd)BFFdOR;Ow@7Zf8|56CY%xq499fnilbU*M}A!tG4;z8VxSU130GaX zFiZ7uSw3Lc)ZzLZi7g8&FLv0uuk2hKQHjrnMCie<9=y{{PEaM7Z;xdO=qRbc!os>B zJzX+vd*h#XgXIRD<_gNP($>W^laq@(yLKT)K@?uiv(|X8y>;q;aiM;g5MqFXCWTz{`op2jT{H^=TAskF(w2%jFAydS`Gt@g&7q? z{Cs|g*}uXrlZ1qXk->~_eB7^UrP2n^t04LLRTq$d{idm>`fO8U{5o#jK)h$;+*K$K<1k*a)#jMBkZ!fJ%Zld%$k73Cb_R(+npYfZQt zHySUU8XOTU;++}I@>Eh4PhEq4ap|)ZYD`R)52dMH1DbI_N9eDqN_+>movQA5+J?aR zn)k|g_hNxIYpKdG2ImfxXXR~CDeOy!&=c`t-03V#$&WI)si%=;E&y0#F z&Hw6or&cLChcU+1sJF>1_ehu*6a`h)rec-!fX|1e?kTOo^*PA1MIc57*AIM&bc<%n3c*#t^6`j&BN*9rse*HlmH9yd=vDT79+wlZo;Dt;CD z&G{bNwa39+`**1y`HskCfkzSTXT&REZ-T-axd`R_pP;@ln3?KXDqDD~9mK6}*a@k` zqJU#ry!r`^3Okt5Wm8tz2n_Td_QHOsK8FSrICnohYgVVo+g?U#4D0Hs_b@P+)piOL zud*W+(h39h$HtIpo-&)&R=edE6g=1m@TEi+;I%CIfR)zVj+bKJMdK0cSqo+*bVq*w z&K1-*{@Fn%5<Zn)CarsF-B=g%8!6v-GQ@ ztwS#U9xHn9!>UEH(x29RH~aB#?7u}!x>cx8^Hrx3!J z&bDi(vL9p}fy97PLRwRnEwFmcmnkalt7!WUi@}J8SIt#&6iuoJC8n8G3sG7}hZL1W zAXQga*Cq9|UEMFcx_^mYI=eygjvV>9q>yr}sSTvjCFBEm_^i-hIyhJjqhh7uT~Md- z);nC|QQgh9tWjhuBKmHM{i4TV4*9$qDm7ZxjWnyy|MCR`mWFjO<)wxHrf_%;1Js8< z*jwnDBrQ~itTyjG-bhmu=aL}|>zbg%!Qf|PePLce@E~muUkehr!|=+&l`wvoR)~A5 zSk^Z~X3sP^iN3MXe(_6&?IbaiXGmWes0B>95x=Z#yVU9=B5>xnn+W2htfhZ&MKs+bf8g~|qf#OpWZRw~6b zftd$Uuajr9&8N&hX_umaCc7ptv0|WECNg!^6w68vG$VNuNyc|;q@P?f{lfae*%G7t zjNCl;-pgu!9IJT~%*Ur|%+=3J=#I;o_PtBpAimJ@eoPTL}_^>H)P10hmQ zE6S%2WXtG|j7tvpI-+T{gd#QHTIvN#Px$z2?hhxlR@PpwOt7wqEV~YBFnwO;oLx6} zo`MhgUS3Ey^g2Z?blWN73DG3@{{89B?0$7@x<6ZMT`zMS+vsDK;_K_%H#pd}IhbLXDDzp;_a<7o~=d_;^N!z?LrmH?Ye2BFuB4Gk)xKgV8hm>;4fc% zWM!*PKP_?p=Kbx^u+m;0Bxj?*^3(Y5QwHhc$6q%NbKj=X1f00!?Oow7GucaltRwH~6P)tY1#&$y0L_?}rfA*{9i6=11 zCyKhfEzx4QFbYa*Y7%x>Y$6V&GXds@;{X!M#X<9$I%{C0-f{jPV3OhGyo?tyIY6ZU zZaBX?`cdyuoTO){cll`mT0srS;#Xo~O{eU;ckQXq*sQ>ix0CkLZaz6#P#T)V(^I$i z85v2m_KuF-^UiGv*15h`1A9%)%>`P|k3?V)R3qO_Vk$uw5a_kUe}WAWcKphg%mMhSTG&hsqpD zFuM44sa~TCy|lD6@bZawcL5pO%0`A*p~0j^J}7Ep=TCuG!uW^Bh0QudE;@Pt&~hVH z+uu2FAI@qY_~4aC%5O)pi+mvflsu>W>w9Nt$;Rfqs5+IXZbJL^Q&$4n84%%s3u3J| zZz5`~M^#yYlm($)I?2M$jt<&iYNdSn^5wcTD(HU$6%K_sFfcKzUDxFRh+^@ey0He0 zOa1`K^)RMU@2a7JBKz{k=Btl?e%06cWMkBTuvLg{84LXUeTo=Cf& z?dN*KU~?jiK{7M05Ba}%Cx8N79tXD?Q!L1cKu)fuK>*#t`Ed4Zc8H0c9k-;U1VEbF zM)TG-er)G)#H-CSg1FbKv}W#jq53>u6zVSx^!~EVNdQE=rw& zk#+KWYk4(cyzRI$nJl4wJFgaX1Yb;6TKWo{*K|<&?VHO+A0a?9HQ~%I*V&VL9n5VG z(ku6m243)h9GL%(P@8mNEFUgRkiYRMU5DPe{Ati2+bd?13Z5h_sVts_thxsX^Tfo( z$w*0|RJ{GMHaI^-5f{PkoX&aa2(RMwD8DLcqL5$-#E82mp0sP=A+)tK^VCsPfY7d= z?R@pzn^O5PfyL|TDXE(1yk6r(akyf#ZrYN4+4!ihhJ%l0JyP`%i;^EiR7NSIfS(uF z4}hv;3kz&bH_z<&UU+=>sXjV%9dyC0T)N?Vm`*L$ph{bBrc^0vAQkjaFWLC^=Dm{4 z6>eUfB7klJ`9dT%DaX}~wv#zGfD}6TuD_5_Y|x84TD~C&YW86sfWGH3USY?>w14Ka zM`AFgOt>sV@UxRsk> zbaj=qe6~4BfYQZjMnpjoJ6Wo;es_KL(75>Crp|%VXO??;aXly9E_^`x*!Z;J<`Aap zAq->s@#DeynF2vtGGf@OW09h22>|Jim;8_ZZEgJbILDa$lrCHGRNdWPMW3%>M_>F_ zxK(ZTjtgHdpZZoBBjmz<-T2KYDbWLW{~Cc;@oFnj_E5DI)>xjP?G~{3H>Y#AelrS@Nn~7Sey}h{8kLX1x z?mGh&67hbiQTy}Pv_{T%5X_7~Y;un9@D~XQlJc1QZ{93~htv72=)m>b0>yxQ1;2Um zaCCmH2bZ7m!8GEuHuQ$Y#jhf%;Z2HGSQYnKpse(7K1=ONEez? z(!8RL@VhhLWA1xhw4UNY>*pE58Bz4@N#;ZYf({wH9G9S)p*Tbt74}Bu1?!25i+=e9 zKgXW_v1SL}Ll5)#FG*?XXqvV-#W!z=D~wuqy-1~;fzY=~T2gZ7xJ*sindJOPQu*{i z`ujqZhyX4wTOA@ioIog$6lx7~H8jjoJ2K}-b95~05Ok&r4W7ZYMdSJA=I zSnaW*w`Mx>#pZP(H3916C(BJEjI5$0S(<}al$bL#7u|7^zC!jc$^=9VCT5yiS{rG~ zX4_@qm)gfb7!3Mz#h7dSJ*U!?nV%Ax8j5Q25U{)uMi-#ajzp3okQivU{TEV*$smW@ zr$M>ZPlU!;?-H<4UH%Xyr>-^c2!|fSQJ~P^gWYT)_a4j9>{*vv{yqP$c^$W|eCbxH z1Z#KxQ@;J8&k@AI7v19gQdcp-vHi$gV|k0O_-0}r z!u7Z$<<)LENpDD z?(RHO6^45$Ig|p~n`GAsqc+qX)^qkg-rZ->NgL+wcfKOR!O@V)(tf3CIhI*$C;v|f z^HHl39?_xf%9gzkclCRs+P$O?=yb$a}jz(%PDJ@>u}zc5V0vj;W?1`Q1QHK(@1dQi z9?2o-^41;Q;I|N`Yg3bMT@sZ>$RSL&98zkRZ7>MmMe z{eKdD{6OH2t`Xbsfk0B=Ca3pY28z43*zqc#^yx?s5z4Z{+9y`e-@OayM%wTcsxBuc zuztRaM-A2=Zri7niY6P2jj3fm+1c5-g4ZHGHyp>yweilx*MOfbr^Z|Hyy)%oKl(@~ z8*kH%oK3p|Lgsp}pZ{!GxOL*fOGl_Svn!sfPO08F!4X9&nBGkG1*|(_y16J)|Ma4r|XqApOK1x#>v1CtN*IgICW6r18oU z5J~=bm8jVmoza?|P4)Nka{cC9q;Etuw1dlY)kdwyd)hAC%K>smUH#f<`_EaV4R3Np zRF@OCYqbCr(2m6P=d#$fk03wt*}EKkq_Cf2YuC!~5&r4kI;{KQZ8Q=*{h8eQyv5P6 zy)(jvPl0vnFO$E;DAfOx@fN6ku5Qm|bHEYVd-g{W2c#mC!;-kYGfx0swxA;lmrV}O zEc(JBtD8yeedx2CHlvtUjU7>~x3Di&Y`%InHg@Mg6dn#V+BuiBw4~3-MS@IRXMS1! zj0-^T)eD!CsU6pLvY^|h+d}TRdKwxX%_ajm(a@I=v3rVbn#6{e* zJpK2kV8Yf+w-Bzl#% z`#pri=NzhBYqnYcbL)MzZ%<69c3AzRI$vdq2BZ+N|JmhmCyIe>ynNeb&zC(Nfw>bS zhEL~jq2_QQ+ctz?EnNmuq>=prvj&DxYqeFxlqnb=F&YJBNGNLl*XVnH z7nj{LMg|6TeSKj|nCrSa^O8;;a?ZC~_rrI{QK{M|)304Y|J|?D^1yF>E!g*Lrp@}M zOj}!9mAu{ScYaUPdo#9}*pAptU)>DNc!Nlh2WmPgu+XQcX*0T0kMnkx65dBO=D#(x zw0MDMsCqoF*n6W;<;5`OhBqkc?CGAqFmznMoe6ygSWKwDc$H*S{;w&UyV z{bgI*-^M(KFjq@A`d91j)or!px6Sha>KQk%A^nJ#OczR(X!!L?Qxi2jqMGyZv(jR- z)nb#U&+S120OH~G!6F(Trz4Deo46Qz95BlI9)+4hX}MgquUieq}f; z4Z)fwD9VdUP(~%^l~lzwRh3N#ag75qEv2FJ@ITx!GBS)Nwf@uY+u|E;{GJRD=>;HZ zA6V-YHEmKh^R;0D_Tp7^J~Zw7_wTI!L{SlP9aaMF6x1`^y=JS{Npwu{U%M^pjqhIb z?LJBCZ8zg_BbycL>{3*QtkD2Iw2F(wdX4q^k zS*E`~ur71R%L=v8XjeA8yTgn!lFsGkc+jHkp)~8mE`%WyOCRn! znI{3*LO!x;~FDr0Oo!-+U94d>7z(Iflq{{a9 zmlJcsQ!ozBlGNLYp_loKg&!T||m^(^$98}oSN+3GiDR}b0e z%7ypYa+s0@Cf7n>crV~&WaUr`YjN~FKPZ+(D~H*BPx`&njP7z{=-3C}eTr}iU|aE(7?GzY7^!d!K=x-9ZhnP-5r zX~{u=F0F}r=>Dk?k?Hd~%*GAo>Y&9RIb@_&oTnTh`cH}nSREb-unx}F%+RjAbz*E2 zE?_;lEfg_0g()Pv^YZ>TKe(jZnGs24jUytnZ1AH$} z%#S!lU9z}}LUuDVJVq?|YMtKN+g+v}mI7LBloi1=1!ap`DBw92q?LWH1#%F`LdIZG z-X~pF372v?OfXFy2_WM9>vl##NEl){n7-N>fvu3p9zNwW?_a2znyNdm{-51DJ9Wi$ zX|6r?rpmhj5HsbeL2!p@q8SnGD2CuXDjS&Im5Ck(n&(=rzTF*SO4oZ`K)uESqo@Bq zLBGYDN(7~OPytGyu5%O|3TGJt(f!YG>a(qC0H%OrBHYbT7Yud-jIdu?P%`!r;;LP1fgd!EkU{Cum6>&L~`XO=}SQ~IT<-dy9kgW|xp>N;zqhn9S#fzGn zC`f$K-YaeBrr$C|jlrck4MUUD(B%Du1+u{O=(5IWUJSFdWe(A$gQ9hpeT+#fD|jQ|%CnE(>PmaO(H(yhv8=m8ytNu1Y^pyb9qcA;#fgT&H zCO>3x2~$Nj%c$JN!h>eR!-=GosLe*`h2V;> z%tR_%?qGz3gbLd43tDfJ0&s-;`#^qz^%HbJk|qRq-THmS(m82k98*@g9c_8a$VuhWY3|X6TpVaE-kCa zjnzSA#`t>9oai-#QVC2z1tuY)_g4MM?|f8G75Qs2Dr(D-oPk$#45*>fQU=|!>Wuh* zK~_ro6b3IhHczUpt9A2T93&SvX(o(sh`~}oVUGI0^qYr>j=vr&TPpdDWIH>-LpBy` z(gaemMco?!cD9h>fBp2oB%D2w<|rWks(04q+SujVaXme94zo4rF}cxFe!`U5`ncgy zT}FESv9m<7$qah%l)+ij`*sjl(8K()I%k&SJNr1q1w6DNIspkk#fTjsR@F0@E*RAeJTw~QxP|`Q((lzKNPtHU; z0vE>T=1jN|J(PJ+Frb7ZNsGrJsXkPRGp^Ie1?lP;Ca9B|*3t7*YHR5%dviz;;4|35 zV`5Q16}IlZvutd*Q+QMIMU?qq$zJArof5%o23l(Bh-N+$-0bOl(f#h8$owCAk!xm0 z`ziuI2rbHr3h5w5i!$*?w19VR*;o9AQTE~qc8L2nNQ#S&zQg@ri3mVDJEMa#R%v_O z;QPyqy!G{UYcZ?ppFu!Eb5g_n#}cpo8&hNynsAQ8R+?e5w;Brv0Dw2giCHzjO&m-T z_gnsvCKPbRb+JFoQ+n$^xYX+JdUtgaypTl)WAKHe%pBS~AMMjv{wJ#;)utI!$k`e` zvSy~s$Y`1nFKg`zPyG3p=;p)!uo{zgeE%es4NHF5AiA{v!&{W#xjFs!>FErdocI}> z61ogEWo0;HV`F^-1D#Q{fr8~ZEiIye2y}IKvJ@07up9vWr(U$S-Y(0{qw}GMf&sn; zDS9w&6O1p;Wxm+NVqsmIPL`YhKO9D7?|*R^fG%XEZadc>&(_vv;!>02>=~4a&r?!;ZaMgs3Z*DX8$jeBSIP0wP8H`KUBxvu1;HL3=~kXRkmJG z@KWF|`EFB>(gdKIJpf#l?M#}(uaD4D;;*G>nwnr_H_GrZTmEpGWIYloyFXKn1}GP8 zoSmbLDx>Hi3d+h<_{I2H41VW;R!lJ9>KLsz=s1jaWm3l&9W*gHxq5%O{7>x(NF)U| z1%abtW^P^_&{|P}XVBmjK?m`9xZdtfa-Ofp1sEBROINH;W(6Z<4`DhEyuP9Dj2wyA z>Nhu{NwK*2`0~EKLVz3+rLl6u(9rNd#E_+cf)dcco*R*)IGUQ8MEy-dSK>F`!8A0=yxip? z(S(G8J3x)qLfyDJrAQs)cf^=@RAiT)$nI`a509s?wEj&^MFMckaVNIVZb|X+fpd`m zUFM=wFD7k3@X@;?|8=h+D;{MSbei>r)Uo6#50F;U@jwON=_iJR>yahjPM7a%YiaqHv(<%a`bOc6RydEb0K`5d4hPX`zvDy3)A!aIqOkM#BL?>7um( zEX~xx!6D$!YYmN%Lir?VSX#m|fZ;zUt`YNmN(i=Q;)H<)PxAyuBX?}-hf;=}GJLch zmRiW=lW3?PUD{3uUici;O&^ME1K{h-4|isOggG9Qr$qt%D4*o5_s5SP zfFXsE2basR3P+zLExuXsPyh2X~?ZcL}#I2s> zp%Ot_0pkDj(~!R!EzK{y(j2;Y$Dj literal 13267 zcmbWe1yEd3vo1PFU~n1S-Gei@W`bn`0fGki1b25G2p$3iha@DyAq4lqfqLx^}K@UOa%dxqJ|9UaZ@9{xA)Ft~2RXJLj0@G~ulSS^n)z$o7#T z_1n{NdomU#T#UCrUGu`wq9tW-`u;HWhl+H|ij0aFFOS9~9Z7h|Ub>=&gUnR2n84@@ z9ga{|8JUILn5IW_tO3>n4F>~@K_@7bM154UY>moA_-Nj|r4liw5G@-ArlM;YGLi|* zpPWgI+4+bR63*11@NbY(6%GzxK}X;y5*sHcBqZRpK#aiQnIZiP&r8H<5@O@y*S)O( zVg07AbH!GKz$_*qW^g#(|E9?QgUP=z-UD{NATs!;m4PzAbLjtHUjg_G9n6whH6Qi^ zx^nH+VT+NP8XAty#KJPhdn)@~gc+QeBTsa8j@Hwo@S3~eC3;^H2F0@i{jC6dJ0;>A z!K~2xVDmu=CbmeE_;`%oUQjNfL4#n`{;Al%wHj0j2g|(rp%Bi{ajlCw^+)&wkDN{J z!@W5$g|MTeLzheyN(mH0LL3lcYU)|9YYf1}N1@e11tl>!sxAVj z2sTcPtSoe>BS<6PHtCr<`iA>TF9C;OHuRDA;(aKvH@2O&R)o4Gp$2>k>bS`te&&>o zEXIuQC>T&xc|RT-1GPNJH8ViEE1o`mbSQ?x1LQXlM4;^)+M-(uf^cx)3VM~cgi~{1 zB0DX*7!2abSBbM%AOw!k-|G1+S5F@de_5uq>fFAfrw8ETqqRzy9MKaXFpw?Qj_~OF zU=D`6JIB64J&gV7MZZhMhG=ADrZ^S8f#F6Im3=!q zT)%&7A5Z27V?pO1V{1NnLIH?5{^Qaf(bB>oDk^F|^-UMe_%Y*_g(N8nmm{>Gl^(Uj zI*ovHVEi03tV~*1SRo+=*}J%4gIdn@cFCCCENi1y%Inq9Kt|=$`Oi+NfshXzITD1B z%Wco9QaR4Z#GqZ~UDI*{)no<5l)d zf>}NbqjOvJ-q^Y*YV->m+}_+cLkGuKTVMnr zz-e;rXn-TfX|K=AO5&i9=ZjTn`cuOOyFW^^@EN!JTfYULWkciE>#Wzokf^T2n?q?l zQ2)q?w;yBna5Of9FR1$b4I0S9{%H0` z!ntxnqIXEPfH$V1!mC%brqV}*40Cs84VNw<2`a(U(cKwj{X-)?J`dmUUF7O;A6hR> zWsbMWxeQaW5|kspeBs*I^jaU_Tvl{*}7u zxVX8qP)$uuG`#v1k?Nm-IP9#p>a%UFnAxv(numV;{MoWEk$!CdeDZMBRpMyDW3;fK z0HprOeXU>Ys!>;07c%;oj}P|zdBWiN#7N{(X0?#_yYXEE{{StL8*u|-Of;i%3d((2 z*IQ%G-~Jm^`nFHN2$+SfTw66ey-)dwg3?g=0`a<&EWtZ<_HW<5z0lE#SXf{<_$_uv zB*?*mHUGmYTc~fNBMf7fLYl`Kzp@ zMG|~oYPR~dB(Ez^HsI)FP(ZV>ul;_%eQjI*L2@bZ^f?(h`SUWH6OWLyV#en(KKs+< zsN~>;Ro*IT7S}_SCWr>lII$ z+srplzF|a^@nPGk$|r3Lz(w|sj*)t$xv3iMuB%F96cj9;?|o6blO-uxS(3+IUS8+O z?QE!%((Uk}hKR|-N(9jhef^kUzg`~vuEPV|IC*$H@l0TqQ~nA^ugpMb^Lk@j?idOA z^5qK%WNm9(R8cXq>8}Qll@=C7!eEtlT3N%{hDBK~Nifmq>FLMb)P1R}RMB}QU|s#p z9A~sU+2-+kfux>j@;7BBA7bA{Hj2Ttc+&rEw!5(#WkhG_#>Q@;O{-~AuC$Nk!~I>Y z>EHeRCvV>HfSB0XAIZwfPC9ydd$0Zas^fb$u2QJ|waR5#?xmidhF^tId7iJcx;icv zHnvGc29TynTJsS7J6!v}+5#JWa(`nCZ{(^31<6+Y_@T(4ADWC(I?EDgky1e<@M>3A zaFIN1%=;r3s1_I~;~x-^l9D3iR{rA$h1u0l{i6uRTTps>dabD@91ipG5f>H~9-Ew8 z*Uo&&L;k~YF5E894=aXpd*?QG z7D>qop38g=2K`DFAt4+bYF53(j&tnHl zE8F7!25w8EhFyY1o~qh5nHm}*f^IHWcT4hn@dR2zhqJ{soBMt^&2!{b2xsAsn7&TS zZ+2O3Lu~JPN|l*3PjC8vourw|uCl8}7B*(-9&Byb5?9t6e!CBBt>|GTxq1{8t8R#! zD}-hD3a+Xd*YP=(&5=({*|)!8Cxl^YYRcZpsRsyKP=kguEdzt{(8kt}w2EvvN8ld^ zv+jX3&g$Sh|Camf{THQEIlwGR9W634vX!gt{65*s1&mfx`QwEITJg1v z_FMK}Z^ZI~uU`Y><~J9ACQSu^GjME*y)T{Dy??V~t;hT0M{rSKi8l)vT}oPd{MRqh z)rFT|UmAS3C&7JB)btnU3F}p!RF0{}`$S!tn!TWBOO5q@-1zyV^i#TFz9M0CCVU<-kEFI}?%L8bU3Ug6n9S z9{%9#&aSOXN+_mAMVjUpISvlI=JqJ#H~HDr7_X>cd-Bf=YbBWtcgeN2wJ{gEK1R92 zPHZoIA!n}wcQ4$$J6oZeDXp+z>Qe?_eFsV+eq^Dcvq;3xFDzlCft4YxJLw6lUW-Q& zM&EvA(Trf?ADcI~r`^v|(v;CzI8@=`;Fxv2Y7pd1Vyf08y=%Lk!6q@X`Zhj7V!`PFB{AED=YPrV$OT=IjFguPHQI@wo1k%}0Q|@I^vTFCoRb1`<_O^F%!} zFoKhCa!&9=neT%q<fNoV=ov5wHTbhG8>+SCO4jFkQZ)Mfd6CXN&`R;v99*ej1jg;_2 zfL&=lU|iL2I*Gs^wC(c2^u@N)a{ASvhodo4q_P?>@01lL8WokfT>&=}ZQ2P*{NO-o_ z8jXZm;uB#|@`Jj&-Cw?}BCaYKnzGm<)UQY=xE>QTZh3>e5Ee7_wPc?l{Y3jUlRTT? zWPipGzpXwa%1=kzKqh$V+wLo5GV#`IQy?`8Wp&}@FNN0)L_Gwt;tG2H+!1$a;*&yP z%divk>ZYJqOtc>BlEegpVo|WIiW@^yCNn9spGoE?Rf6N?_#y#SLUL4;kPsJ_tBVf_ zNoS&@A|?WXDVQ}&o2k^fG&alhH;Zv}60i?=pnnuHVkhpH)6}lyyqGm`RnQANJ6BS7S?1dl zm;t!0hMkuD1FYY=MiP-`PepH2eNn-`1=IbS8OmqM_j)Pa&XV~Delt_Rm^mSv`E%{N!%#VH6CeM-EHPoHb_JRGJmX9TSo z6)ahl&iuEmO&XHPx6sgk{|?*-JmOZ0(SEZ-Lm2E!z#hML)UEQ~!or`pJDX5YJT?*)QspL=phHj0+3$0Cm`&)RPMP)7q^Z&Vn8nIM zOBdNvidR#?uTU6j)rk(?-6dw(R=aLvEM%w(t`gTFn!LUJ1ST+vfA@}VA;{;gic4~v zf2&ueIkJuZ8SUd0)aXvX``7$!lsGVW#!KgHDUO~#C? zjpj&{ORNx*ySp8GQ$q-K6o48K`hKwZJ=IerHu%0;-801FY~A_x=C4t;?N{7($)YC) zU+|UF$U{!AqGq)}@Z>g8$3#T6c}nSRuMMH8so}(p*l0%}k$5e`g5;Rf85y!Zj(^m!U(w;cEn+z2YZ?F^Or1DdsA#U*zru@s5qGQKX=y>=EG<2k$i|7? zaO132FNqI@-P4j$`PyX?5@Cc~r=c4f@}&sBkUm;+W6g89Vti{e$RyoLwrfL;nnRQ} z*xA(hW{VOLVHiCo1M-B3g%*kQi0UR6G>J(l%;nk1x&M2L{ETe{H=*sQX2cWMmCG5; z)$`Zjnd1vpPA_an`BAfjic(xL*mcC({rWYUqnqPr971yvMp%M z4T3$bPQCvws=yyZj2{)DIMPB43CY;N7#asQDejDg)y{=_1uwe1*mrfW`HT3Q;}FUa z!hHzdT{B2ikjmiochga&ru~_KnR5h~OOcx+jAsCQmhkS@Z%bg|JGUTObkqGHYA*eu z(?}OBxb6CZS(&-Z9}*L>8TI`~&5x*fifY1HvXm4)#p5(QPqj8kcEVFfr6*xE!Vl6$ zeFqdRv)BA1L_gN@aT#G+#y>(RR|vgt%7r4-*&!*9@VErnz{wvoQN=M|OAPorzUq|C zO`mMu6G4BSoAqiM5n4Fx;kFoEOT2SWxw9|V+j`;T^k{88jGQ-=4Pi4!TFX+iEHQ(s zjrl76HUX`!?x3OF8WHl>>Bp<1zvX||;14#p>!|`3x?L;}{3{wjF;6)$-dQBHKQnrB z5#9MSrw3_SUC`FXkeizeS?!sfHL$j^Q7j%;`1;b{QG}RK$Ay~4QaZIK{%+a0rE@ls zQH&8;GwXO;bmj)*hP~^sfiC~l9_}~K^4HP|GbLYrvc<-Mfr0Vm>({UW>ksa&m6b$6 zK|%JfUw2;onJ;X5vgFG||JlRBJoHNS*2K!nWA?a|mT_%%rNr0qIH-YFQUQmr0)V3i zGYah;tP&jR9WB~OSayz$-~d{&6!2H}ZDOL5!&H%f1I_dG=b+H8Dy0Ql4*~b*jey3^ zM0)AZ2QQFog1LAR5goge2Rr%ix7u$m5^0ZHdzU+$>(_h8mtjVcs_svRY~R21_(PMN z{)m=6YgoF4BJKLx9}r$N{5v}T;*JbD%W+?6(SinOL3yv$r`Y{`0@Pr@Ef2N{NXNo} zq)wF@hJ9;qFt_g}q9Izrqu%@yXOb8RCJ7wP5fl0I=Rl&?Poy-Xkge7)X2x=2X^9Zz z>wqffR$4=1(Js*h<0Wy*C9d}+Jt>()^jb(!M#p3&^O+sWGXs^BfP;f;!wp16_l)>` zj}3P4yT~cr@81PU)HJaV8)L3ZMm$<5ASJcB4&x)JOm7Ei&{pzdz02~u2RsyG1Z5_? zBmW-xW|)HSXj0b63S}{=m`P93(sC_?jJcjeaD(9zWVr9~Sio?F57~x9Q3SMHuhKyv zt|42Ig+FN0hM8UBop}+ueS4fto#)ijm~x z#0I&V4F$buka)~p&=Dkbi3X~vH7W6$cZ4p4JlrkucXA#qGi9ffpe;7TESS2yyxjMo zy66N2hj2pz{+~(#kdSpf1Geo;H}Wj zjbYQTQkN8!x3fd2=JgEz{%e>gxLURpiKthWGW^YikXu5Pf}p&A_(Tlb;ca zFBt=m(DwHB{&|oy&ljDXoZQ3kq6{h`BBBKV8lOJ{R}DVy;hD3yw8y#EX67bqq=Bq1 z($+^I&y?n7sm6X`zI|J@4nG;*nxIKoo>5XwlbqS&N>_jP?j2mR;(J})Dp1Gf&l(Oz zcA6`=pPVJ`__s1h`y3+g?{3hl&TfBY&%Jw7M+4OS9wrxWp*q^yf`Wpvc?{Iwz6F}y z&D-vE$&0o`=Vs9!_q=sqUK%17PZDL$5A51QSARBlQ7mOuILFF{T0t}IMI3HHRdQtH5lz0=N5l4DS&bYCp?#~LKYtcX?sBvR zpHJx@UyX(^RJNS_d}s39#)ch;1?;qeNC0K9INcidFwN)928ylm;(u@gg_hRg_PB4i zitITmqj%DPw(Bk_B(+7Rpe4ulFR4ebon?*Nx^j+`7p`}6P;X!0 z)u35#3abi7+kJdH3h(UV!gBEaV+aWqpvx{lh=PECKvhFSSw{zw=eI=-$|NmxQ})Q2 z`FnQt2MXYmcCOc=V{?9(MDn-fZOd=xQB@@jH`z?~OzgmEI&MixYG7w51x%5^GiM@a3*eimiHtVU<8!!S-T3&pL?G&@>{5gZpFhpr zw&tr|DW@YYdTG5H9MiKY9<{64E?r)ZE|BOM5}xsUsc%%BK~e7`Bj=7J-B9PZ$!eG= zDM`49oKuUTmX^@I7AG4UW(*$SbSsLRs7 zYn&h2VW!_1X$y8d;i@&Yh8YetC2A;eew-QroxbNY3>UE*L7hNNt)R8lWDYae7cK{$ zAtI)~s?ShS_oSrR1QI@pHr39v9k!>WkwnK=fnn10Ug!gy`Bm-J*^O>(jeX|4c@9_f z7aOuG?PtElDUyci>gMapOj5t)FWI-md3JoI%n!nz~%4jcDcBY7G-H- zq>dv#iyvu55wQqNXg+c;FE8t=F9?-1;&zEVf&dH0#$J>e)#29F)wQ*y&*KwRFUgha zUGaCHUmM8Z?@pSnNKI zt^HIJPMp_{%jv+;Vu({dxJee6xbX=PGpSd(Nvx!K`JGEYVz1xCg5W_%e*TgK} zy?EwONq)`kLjo=HSO>3?HA)xGp-*}f4;1(@R}03=vmvWLr47hhLAtNy5*I{8t1i$s zH_57*VF0DpA5O!SYpNtA5ZUSPG)9U;(ula0?my4r9NxRbMe8z zF_Lj+iKFh%5hx{P$@fU>qxP`0XBnBv@6uC}iHrLFTB27KYnYhiYW{WGC8br`#*^Q4 zk&;)6HBOAFIoxttP;=ldn5m`5xl#q=5EBa-*4h&Mo}UlWQ(KtnTNs8w0xCLe6)&~%`tM{#t!HnBhFSc zGWxldN*Wmn&ddbB`sgmoY>8>U1NGyQSA z`k9mIJ{ea{VS^2pO<|SRi#;<{X@ePadT(~MmuES_^29HM+Iy z5SiM#@`E$Zig0C7%g&dIiWWm#=c;0}zA0^%*eupsLrJ(3B|*DT8z^bkjLD1+W93*H zVmsV1pR15rB*fYWFCBjCrkvWpqV`&IOwoi~#xlV~E!TG8mN5~!d#B#gbWInWj&L3& zZLVb{X+tx{!Ak+YjW-!g$Z&fk=8+>uwJ8xyrIyIlfF9a1?}bHV>$GM^2Es&RPDm(= zO_KUdq8RrDiA>8v($nASY(LD-hq4P%_s6c}=)2!ZVSY*_KOuo7v+xMH)u;Lf(os=? zDEL+a9xj{Ydh z@_XwX6kwQ`=}A(xB$2Abx&O&OWRSdHp1IxAzsSKUZny6j7ShcJNQ0p`N*`l2;GuK8 z2ztWl4=E*?vzXFv&KTEgUprh;KQd_G7M>7~2mmWt$7;iEOW6+p+TnbZ*~XeVH&eE? z7M@^{V0eldHnp5w8JWrX-%}2^LXduy`+OmN4nE@hTB$ihGyICNWLj}Jf*$^@rbJ(+ zZVyGQ>3beo(Ly=g8181Kx0+w?VlpeicsWM0khZmcYrp%!4w4QLsv6{hTw*0S|b|X1i4O{W7BfIyuWjbz&KP2L#eSLm#fJs>QmM`X$q*{8A z9J6lTE4b6_?qrY{jGY5acX!kr8yJV_jZl`c8XuT1wRVwccrvJnB5fcY+1<@&6R}PAck*sH(1#r~pMJzo!}1>iXwIKzXz>*SWz6PFuCujz zb>NQB_gX5Cf{-#W3uyMLGm$~@c>np<;q2hgpGqMi^8Cj3D&F2KZ+o+7RRU_{?odql z(#CY4L9@FL#vt8o#YpUBU@)aq?y%cS6{ppM&Bn3yl=W-jggfe9-eJxnoE- zI1dDq!g(!AONE$JiPSPRCnqKrjQ#z0<0zZAyB|sl*HYpdiiI6^_o~apIHjajeLH>MOML!Xk9%y5^J**mgtu+P6V{>^jJH-{WOjN49WI5 z)+@L$z`S4LmXM>%sg1IO5zPKV)bzPDPPzM9D&o^IXR$CYYr5RZh=>gV<}KMG%CrG# zNJn!PGHn-1#3Gin&e+z2ZczS`9X=EQ6X2--3zx}N3zU|%#rS1Gm&B&A&8s-%0Ss7K zO{S^KW&uNs>sTKnJ;)$wh^P->%RE*bp{&3mG@0!GTBYrG2seJ5%(TS}sG3ib480LS zm=rRKADw5tO-uziP%wTDZ|P_J1bEr^A%{b3-cOU}LSN?f(doZYAwDJ6ibwQxaa=a4 z^Lg2E9E%1J(keUJvT;c&o9<*jAqD}9p|pd6xrjMi(&nO&K=8LJs{Rh{Ka0+G78c?k zo{_++SlhZp+|lk3(I5#Z`M^L6V{oqmQnNSdtHWhgx(wxyNbD9-%% z*N~BXd7}TU!BrtDQtDpxr+02Ojo%!>J45DS10*m`Xh5pm7O0*Fu{dXvFkcSF({eYW zUwFl&I0Ru6;22R~JKY{#9?zSzc-oM&PF5-H7mH)b!0ac4X29-dVzNljDDGU(#1~nd zIW`i>#6eu1>FEjL)I{GNsUEen8vY;#@JaIPQc-smNypzV1m6VE9tY3#-J1aNF<{&P zAl9)ts1Pjo!k}M{kt*}*xUjHr%wAMPkJOB24=Es>SS0hBTaR;VJ%e7w?e&+*5InY6 zzfi88kL`@A&#V+vH>@xGLLLrAxp{fBg~G}^3wokIvXDLPHGl3&%9HxOsB`vk`4K5B znN?i~fv`Aje@sEaaJb*>kse}UvHCV0X_zw5U~m7VtPWEB-rHEzK7~9>=JJtI`Jq5E zQ#hSg9KvXL=}0pVQzOSbfBALWlw&$_KNVLiwTd`E>Q&XY5w9KxAF>I_9!TMP+YkU4 z$E5PzZP;eepa5xWmEYVQO+b5edpf!vhRqP_sO?pNFC6Hx+R8;lAKqgL+Xc>cz87Du zxz7*z0d-Vbhwz-pXn$zjmPR_#!kvA?HTT0`a&>lE0}bNHh8`F{HXY$;p2#cqY8Ey+ zy&I~SbV&l9*=c|@;}(xN&0NQ+)g|Rs>awUEbqbw%837RRR$*trp_p^Bv9UpIN6NcF z64=8g`YtL598O%H$wih2ToO(T_-*V(`F=1H@_Nv`1OgxV9QK%_u#hjkTgUc;8r|pE zh#sk-*Iwq9Ys2U`EMStlOy8Sc`APU%)tJHCc?oE&TO6Xq)G{2b1gu*+go9>O=xyxr z`Jici^ndJN99HG0iw7@Nh&$n9afs3^q6A@ev@y}x+LEc@zeXjZXK1`}J{mw*oR zj@eA~pde^u(|GiTcL30Kmhe8Hg1RipxIKoGBrSyzB9#{mRTd4+ViySun`sFp83XbZ z{wo1DVe^3iW>*?AcW=jdZob7&gpf`mR3LB_4>D84l1frU*S_Am7~J~xD_?VBhO9Vq zfVHOM1uSR?ZEGvlRV+<{M~0kr%UPoogSJCV>gOnfqAJ;V{D84BX$23Wxa$|-I38Ro z>Q%cWz}B4}HWo`~m)rVLwnRexZtKIf?^2-g@eaYh1@iD@E+C=wtLkn3eO#|ksuEQy zWI63&L_dpOuJ>&m$ZV;^X^l;6K;_W))V5PXD(1ejqF@n6afs;OIah|C5N7;2bWXV6 zE({dEf$vs!#SF$_@fSkGSy~Ebe4Lu+MOd+l2#J1M|E^R;QkY|f!ekZ$(w@5{9?__m zkFaE`98?(ZiA<7Ibu~cO`jro+k?dRCz}XgR zW|~=>O7$LwpUFSESOCd-d-Z(h%ab7;Hdc<7+iW03_2S*_Ft*_<9u?i)!Pz)#nYS>7 zh_ZYv(HCU)=;RZ=xQbytr#pR~nijTX8him(Qc_iiV@ak@O7J4ml{owp@=t_ttDYvl z9Yf(+nzGQ+(n0{Zm)1U!POG4(EAi;4aXY5v=CGM6P0ZXPg_PH$ZP>!=ZE~$0v$b_c z%KP474nH2|ob(f-8+;(=YG z_HcYWhz_}s&}j0zOk|JH14ktxVsmq|;kQJ1#l1%$zLZvJ7=L1rEshd8@cTC0_5O@H z_>j`m6X<@#GQrqhA(pssef|vNjBY;|{GTZxCT@g_>Fjk)fS@hz(}x!j`C=aOc_lSA zb!B3i%uCA}+hH2d*PHguZfhe%-jt0w^j;Na+jeP8u*3}{8*`X0@##r+WOOtO_h-yM zCY*n7%{2|GK$)UyKr*8rKp-Yvx;%VpoHy?T=lzAGMua|Q=M0@Z{}Pv($*HOV2fhD3 zJ=r~K>Q;?~VUl9+7?7Voo=nWd{>a`rZ=OsjJQ{2y_~tv`F7KuXEe!zyy_-D`*hDcmaSW zptriu$I;VjUpy@oPt2d ztBDCvIYs~Qrq!rrHPi24N}7@VanE}(aCj(y&i_v{$o54!7QYGldwzhG$t03xaE}8B zNF@hsUs;f(V9^^o)&-#ag3q6^Y<0*$9q1Hcy8x5R%#6{<$fzpl#`}v5NyK;2j#ZG9 zG#bzrs1cs-f6o+7HV~(#3U&Z5Pj+hq*JI1-mDbCpZh+*e{oSantU}ozG8o-zY7l#R zxfGwbVEpUXFLQw7miOny#N-?{DD0LUIP?tjaXOTc9tnE@NNwfmN%qyr33cEx%3NBA z#tfxZa)eroiLTU+%Er9=0nLPS036XHLkUGp=2VKb@1lA+(iwhIfByUdg-+BjDaFQr z|4p1t@X1A}q)e~${9YN8WJA>m&cZTFx7KLgO}KaRWZ$W?;A;yQFOGB{accON-vH72#MNbdE6P=7NF*`du z#B9*IhMN(AK!891KhJ6Y2eX(MIlvGcYu@SAGcid>=havGqz$wRS7xiN_?@J(25c=X znBTm4({i0h94U?&m5D@xI5i>*e!Sm|^$XYj3e(jkFDfoJlpClp@%4HLZfOdUf5Q+& zCfoQw?Fb>s|GOPg#{`Bj0u@y!Ymy8)2q0aG$Hm3HFf&UE4@WYp1lWsWg6hBC{^1+VP`?bCm5ZfT(lxIBDnY`pOHzZ(+U_u^qp zTd^BG)~?CABzD%bB$R#L+>48Oc0D}*)pVfKmvJ+o#>Zmgxr_(UGi-Ar}WKw079%E(1+mF zxnXO@vMo^A3f9h;hB2ZuAk?}b2_7eX>JGRJayoZ`)CkjS}12P-z;55iZA(gb5>>m51#%7+g}kkQU-n|C{*xrsnLV8?d+4l z^G0vf(dz3Yozf}(N=@ZVX7P=S{gH$2EJxE8Uy|j!Lm#_VO2s2Kdn6msd5Qej0x*>i z;Xup6^Drb*^o`gE0$x6e{r5V=|Hm6A|F73W`g?nOF?|9$NHGPzRFNx#{=ywH>1wY5=?Ff!-?UF)a!Dw}0IYB(R2_ahTBr+@W*oi?hhZ*WsZ-Gi`o zeILOQmkkZ>*}zL!IZDm=ntI&p%T?Z2QuldWr?j34=MtAC2cD4wgl1fw9Td10dunih zziSj48cLc|K(MmYTR9W2woli>%+9_RuS_aPXl&B)^*07Far6jvl2B#k%6@lWp9y!G zrS-7s`Bs%rIq<{Z)|o@LRd&(h>5pD+yaB{(Fonbwh(VObCPw#8yiN#%gKOtzLLn&N zKDZ0Mkzp)p`@eGm^~>Ho63e!*eNm-{&PSzxS6b2()3}#i{O*U6@g`tS+IL3M=;-O= z@@Fy4fV&}MKuTf>*|RwN?bYCB&APRg$Mw@F)~vLgq!uJbEpNHTf4Dh0xoLFW>!{Y=A#;81n2(OmC@({wX1;vyUX8&xDjtE|4Bh-q@HNRp=6J zcPy*m@3~ZJTG}gF;^gkj5=*FN;o~oyw7{bfez`eb_9o`RkL-3~GUw;FU}C1WJ=-`w z$>MhoYrU_vxbLK|Z6mg_m+OR1^L;BBl z!WLH?tv&(vrPg+NkRn>J)7P*;?JpmTZm`unyM0F_VSIf(@516DU@o4RvtWR0IBC8> zdAW^#ncF3X$dCF%Ee!s~W_B6M!b1D{AN!|9;7x)1@8|SU9Ik$vs>-A_1+YFH(L4aR z)xvH<;`x_NOC5xe*6uEB&KfHZPhGN)jEqd( z{AIKGO0)a&Y>EQ)#N=eib0@qglOvW`*Ve$7%Jzp72`49nV$Jts3ay-M$s;s|WP*&u zTCyMPk5O=_SA0AzrhjQ*;o!LV*8AQxt-d zF|8Q{s9=s~P{Qhywc6R++pzNfTGp7r@ zq1oQPHMh8}7xB~CQ&UFfKSPw1`}@Zo4-UsMT_m(RZv}}%qx;C^PvCyRmxz<$88SNl zp+t-t&me?~DLX&Et>(>8`?lt*IcBIoLYxOp?7I{a#uGr7fL%4xygh=NdsVE+ACsg{v~B1|IOcCJ2fF!9mZQXd}caW!#f;RDoZ> z$j^lRkqD~$ow!4UE!6g)i4~OC*z-$iw0|C+B=c1*+rCkSs*N7AeOO=#CyL_5+Fnh6 z76clIJQB?NUJ_;CbRv(FR;<5QNsK4g+c+PC=KvZGl9r zyS;-h9(}^qfj>1AG1-FJ#Ek|Xx;un<4$P*D##G29^g1jm=1D&r`#?VLTuT4@S8C}@ z16`STJr2CC#m;{woI4Uufdod!)9ihV*iQlfbHd_d>wecRur@Z#bRIC=eO!V>>&sbL z>j!wlpe0L73m&J({ih3QB@fXsT81}s_$VJ8x;|P!LToIX#Kotu z+`pT6{cA^WfF13F*n2T&pgWR{5#cDJ7FHWk@PG;_Rs2r*h4b*ZkI4Q1NlwN|WPy$L{Lz`U@Bqi0WQN^-;W3(|dy^7Wj9Pd7%fm zR#O}Hh@nHYFLtzr90<%TO5q9KJgc`jtXHz$l623327@m$9?f|zsdN3{J!gzDSj~Uw z%}CyWftc~9V8M>EjPwdt_taeI=i4{J*Q`tq-Ko1rl4w4VR%1XH@`VxSu0zX++Hvb0 zTz@Sq^!iz=l6h^FX-2N;<4i#PnD2i;{<*x@n&tV_fwyk&!+R*CP8v%i=1vFZY*5}|6iI=i|;R$yl~ z{RiS7-h6ple%BJ(9+QUc=S`ml1W9g=4CH15vk>QXI`!?k-+%h+8B(^h^h?4jo4FIvg_J z+0XlR7b&S@QpyP$I3+;!h1Z%JeKuiU^s`5E_a-`+6%0P-l=)}Q%>^!d)=Y)GKKazN zphY~&ncIkeW()c229|MjT=syRRaS3-Q%7F??uN<&jKt&?F!nEYx1<8;Fli-&vtN%q zRHoW1w7czF*rR_G+3=JG;RS$-G*_Jqmbrl9{A${$fCl z!Rx?cEu*;Z?*^wGNvvVXdPD$y($;*F6P--&F9i6|-8J2;-^bD$#9|G$2RCJNl3kIiK=1zrmFkAf$SnT_L zGKeuTD`nl$8mORP4?3UI{5|e@op=oql zya=^{P0&fzr)I(oh6e^5oLRV<$0ESKK~tL|i;GeTyrQ}&OJUGOiA}C;NTo2HB`Cqv z6~u>Gk+3ms3paFXXv8Gp#6-s2O8{>eR?%gs8aE>`vu{|3?d=<-HJgi&rV4DHvOyEQ-r<0G2ykWD_o_SDnIKs z;FYhXwIZ0nOvBDZUy6@r$(UEJL3q)<<=$m3d={(*-HqHzOO!Y3Mfc@W@ae?!3(J^u zB+d$(jii^O=+>E|YU|4*k>MaQ2fPX{I*b;HUfeGtu6WLyT|y4W0Y8oDRImNXe&q`L zdgE%v|Jo&1)MHF&&1Hya6*95PL2>0nP5EA=Y`hr0WZ=#GeAi0PSzlIP29WIc8CqDP zduF{{8IoH~><7lgj1x~6s;Rw}&V7vM)IUa1P$Y@rxFc)mw5{8lOUPu_a*aT%e^1wk zSX!#`nSR*x*~^IX>S`mN(|pt-+IGsq8ml<)`|G;}+=H7~LZOg)*?|I=>6x%8_j&A4 zlQ(A6xl*1@tg*px_DWsf-x^aJ|3)-{u)!Le$ap5?e(!@dA8lW1IHxgZR)#Mq{|mD0 zeMw+ICa*KSce-nYew-q5@qIEgcw#mJS^M`d*S_<)xgT7y%*^^2cK$4>@rj>aViRr& zd!-cPe-X^gP>(JnoC=rMN%Sa4$2Icvx~U8cAGc^Rn!LW<3&{R61Gl|xtVvEFQxFuF zD#SSKrO@bNt)IdBg+Vs#{$ApVI{FO?JXv3K_)C_Eq0f_--w(@g8-d#EQc?r6B05+2 zlWNBl%ns~01QAe_nQ0?=xp()=JZ99bIk~XHFNB0B+O^2h`E(Qw??KhP8Ft$rS4*&j zi*LCcC%Mr8Z?9F~l7SE&)C-9dsEHIyfb{3&i07#iARFUNI@@mNNOek5nv=j`WMmk- z$0zTm=}%kDupcl{m89I&c%6mgRw}3X(e#^B{disBP*{vP1XvoGDn;CW}YMeRQymPk7E=YY=)brQgRUNGPtF)c5vwrAKDmOqy!VP;;oabA6 z71~37evQL?srYx|$ljsm(JaZ-M!G!y@r_b}_KZ?|S~+`?DqH-z@$azOS8{UXX}`;q zOvOG5I}PkRIb8qMQJ}`e34g0NDJv`6ZowIsk)f!qoieq1!F6=>?XplC@19=f#BJM7 zE)3F}kk@nGGc;s0k@rI};{?@=x~s$3pF&2`i<0RGWS_YuQmv?{h=q-9R6Xw@WXqk# zmNt@*lvL$$^%VmX^QUlm`L~9^SW`3bST0MwZBzHp`x|$Q4!f=JCcPYfw;sg11(OeE z)Dk}%biByV&uxjTwgTq;C-jYk^B0k&zjiCGILr8BnpX!IqB8I=Pr! z`1tW-U~EG2MLQBpjOORCj{WX6r#Q{egA2F@q(7Spo0gU2-sX6nMWIx7tGZKptANC?xafFVXBQUu$xZMPx3|faTAYebsz!xU)1rqm%`4AdDERdpZYz1#ax@r*lUw#3%Exn$#wWKpcOidD8i|@_oJO zKn%u2Fn=om005|h#qhAHgN{zyEGG!`8jXZRWb39~>%?$&Vn^N(e}(xVqj%P{d~M^m z_;dc6nK8w&o-fDVtmk%^_QGYn{w!cXDSMZ8VSf*wMeTqpvFpEdgqqdX*43$NU+R|C zc_X%0!jp6_E1zK2b+lJHAHwuE8`0NQdEVG^+bnD~b)bmKf%in+Z_=j0ZqJPt{^%5) zj*vS(x7)a+d;PV}Zgn);=aK?IYsQ2h7@?q2xrhK`#;eLjDfC@j%B*`_Kj2h>Xi%HsVGp-x zd|ce#>>aM~GvVaCI-@U4(kOVNj88x|ti}CS^S?oCIXT~xl5jW`0l=|RAfr`;;uY_= zHJB#XljEzDe9JxC`7|3#OER)`P|5mKk$QG zmHO*sB_ZoAkXB-wx|f48%5)M+FXp z($4HH$EGj%Ld5m-2#=1A>X(lVIFg=e>Cm~NN=xy_$BBiXZ%w$Z=h`=;MTBXI?Cd?$ z33+ey9SRp`Oly60TVWy#$%^)v%R~kuB{!aR6rpjLW1@- zRU-C}?lqXPrqw4NuHtEi~>EpL-7+G$++r(j^^qEr zT7gtpm^iICqd=*xQ)qxsAxvSTrl(tI)6tD|n&Y$)k?UffqcD=8_- zfGwpb9M^;$@`sa_8KFD{T>~BAFMke6w3_Opx zP>2P23uhh5u5AVhgiZCInZ+f@;hoE8ot#32-*zo z)i(<`29~Yu?X$ntW9;tdTgNBPZv1zKRBaRjKJ=?8wVK!c4^TunL%_AlYzW7|eLGQX z)%U6K){9B!b!DXhue;i%|L6K|dnOCDHUv#BN8~nWSBi;>B=%)S!0ue`@=HZ=vwWFH z+f@0e0;C}WrFv}ukYy3Dr!k&L;c$Cul*D2<8twaN5*HWep`V+5mRlv1CmpeQI7u8M zbg)=uzu9~Iv-gXo6=wCbM2I9uuIDkjj{u>8&)v3|zM=Dz`l#NMkky6nM}vc8V)y36DJ8td5QOTyq%1?}=&8xkX6JAc+ilzRoD0 zxyJy&o}YH^4G1K|5ZLVOZ)Z6vKO?ZT(nNMeScJ!xd(CA-Q#YDusurd450_nwm($2{ zhPO2F33>)R>#|EqB3xI!lh4l`G)q7Cl<7BxMi9Eax2xhgE$GdC%vmD{L}73E^-~I& zj$O@3==4gpR^Lq4*_r&g`D@}{l%}e^V}afT$_WvWyKR2iziF@=Magke_r8%#rT$0_ z8d3B980^m;pXuZhwj*R9r56}&n zoelpKLZ>cCZmXZ+YRRHm%$j}cc5?q&fba|ZzNIY=mkl0XPN~fpN(_bmrluX;dV9l3 zmGMrUoMH-1i&rZyE^{cZfM|AJCs5hE%xlnjoIg6-T|IJf1a51^`$iub4%YDbJh|cF zxM|#z5Bqsa?;`-uKg7YcU0Jb%ajM71$@k`+7xqda%fw?N7v_X0MwFX<;o`G+9QQub zoQisje@KysKmhSF~^g?FH-lsB<5M%oJ)eqpV%0I_ji9idR9Mn z;)hlgPg-t><>cg+`N#vdM;KlPv9 z**1ElL~9dt&aq3^Eo1r3>ZTk zFAGCydV3MG6qIk4A+258Q;9sZ#SB~i%~=>E`HckHEPs757K9!cD5|bb zlbvlWZOjfH2`$#@5z5ZaQM6IV6ud-3CNl#`j&q`B>3PITTj8PZyrlQDlE}DmMaIO_ z7@eDr6MiOptMBRAv}1!hC%OgHr@WljV6?)+Zj1k@u+Nr|owq*x31-{1bM?5@FyhKe z(5G-u_qRbEZRe%?+V8=Pn=XLc8A5{JKXRp_vPLd!(8T6LMB-zycJB;ayT{(19~hGh z87?$>R#x%q)(hT(w_Xq>O7j`3Zl+^b#+4Wb2G0n}1*jUfGx_u2Gl=DI%{X^dkINKS zC8VjXjSyOk*rD*}+V2QLQROqk*m^#tE;)Z>e3(~*HMG&wjcSj2dE?ut;HLcnayY2- zi)mNJ?$cR#X|*h2j)5F_u5x$)z51eV<@>|2RnK~8ZUY{%m&zN*(S+L%5J7- z6i|kAIa=s>Lc&2y%y_S=y1c7#5X`14U^qV*;HpeM!X4gJEu}n>KhtWy;wG;ikC0=~ zKpsKPiS>LM!phGtxtgDVWyD=nB*76kzGnRN@OXF0y`%g@?>crnV-qQf52`Ly(W{bd z1|t@|ks98L;v=`W2auT=Ob~~gphcHOsA)RZhzw1+swZD?HQSL>qabB3%2R1nIHq^% zajR~(EDF(g7ViNmwYZbfA7|J=s-I=x0rnWuX-$84y_q9JTbEm^DESKsv72)=*_}b` zqi7m>1D0>uk?T~!HWRgDcO+NO#K-s%n_|}btaP%k4>nR0~YxFaE z8A?Gt>$JeY(7or|v#$H0L^RC7mpf@@PvKuXAH4}K%&Acr0yqQ-iQPmQO@RExoyy~> zVzD&h1CYNpEY#gKAD}X3Tq`Rm<9U(9lTD73MVKCZid`r^#RIU~rz>>4Pf@a;5@ZXK zyU*P>?+RdP+|P}ov0A>rW)qBt*JHj(DU`9YX4*dO$-*GU?+dsvJokQPG12yrDn$sFqNjXfm6)#X}QZbr*L z1E_|c?QP8IpAY#|W!p=?CqL;uq)0)OSxq+Ry=L%AMbeenC?JA(pi#nx-S@`5HIU1{ z6jc`EIX(}e$YJ%oswL#HFK43EFBiM!{bQ;7xcdjK8xCD!%v@gW=li;>`U1m!+XDTB zzNn}yZi%5PEWh5HP0Z7wq@4W1Rf#|_nNYYtq}Vuf3vP=hCSiHrLalpHMy;amtMC|* zba^TY%kuIjyf=Dd`~)dxq8UKcQLtGTm>MlH)w|>b{Vc;1qLBND0p<%sH$g6Gw3{%D z>IX7scxzcsi_R{o@vtqf?%CFK@M7-SMy5Qw_n=*c7Q_N@mO-1IvT~Vy2Mr>5BbLUZ}*|d;ImCB9&)e@Vc?##;L~cv-$!q%v6odO>t$Y8 zEqR*KM3)Wr9Iz~6Q*|hbF;iQ2;&NM`nnx4_^@(YhUFlqG(e9M+;cc}#G4cEtpl0w3 zKQT4}-^Yr&>xhEgR83tfIA!ZXjlAo!`83k~acx!pZv91A>G>@Uc+$8cVR7K;rGUe| zXRhD(4=1W(7SYf943P-~E&2OZ>n}64CukVNN^pkR%k$zEvXm}qCR=dR3-l4DgIL%U zcv6p}3UC<|kr^~;LbUrv#M`NCfqM_WLU`%0^)kbGGINXBXD={rHyR~}2& z+yL8zj)}?a>R@_#C~+}^i{q=QSvCV>;9Ddw2eiqp#7#@L-(e*WYP4pDos;>H_fCfY}DFX6!`<(O#D{?b9;B;9$4qzUyjf zs_4TzHD9X?B^n@`?6a*?AJ^$vnQ2LIO@fWVPVH%|cm7}l1!-i rC8T+P^wrQ&gs zlmDSsKx3Nsj^X1wA($+5+ogZsqnPJS_7+$|&b}T)erIv|OC)$}>5W=4Rcw_TA-bIF zGmNJm^T9Q)ad`=;tqV7uVKYfa@_R_SB8c(lZoSL-2i&32O>6e}ls3PFL_^bS^+h(x z_bsU5%BOCU4-K{4<7{6)yoDe($a2NU#bE$6r0r3M=jq4t`g$D~!vn_cxH}BpJ{ziV z#uc~C_YMvYww^P?78?-0i=BPjC&r=RfR!KS3tcU){Bai;vZM0t(>poG>Ks>DseBK` zh@Hj~c*#UEoe>{4*mJ>h;pFTMH}HEKWiJxuR!DO4N$Ogc#7E9r1CIwLEoJ-GzyBB* zhUAvvcqYE{)S#Lf!ad{7=pIRx=&r#VUOLCBoCN*2m=*MH((jhZ$+eqNqkDua0;%cf zylXdDTrrTI2$FZKoNY|DqxO#HbKI%$dGdfLvtTo^tK}IdCyAHYp$9^M`*N$+p3yNZBH_CDNL3^zk?PK9`)T{7IKP^8ILEBJUke&L6OQ7OtWX8FgQuj&`%F=MnhaF_@T9ZqB#0 z0}B%>-_h}X*3TmG^?Z^3to#*q?)KIV;2=S=Aqjj#_bt>M3QjX$t*kcVm4%H4;;14B zct(fTQSH1WCc>j}onNrAH@vWY2}sah{;aZD?z8{Z`V~5os&sq%&WEd&Ox1uHg?n@? zq=xf_zz`J(JN&0)b$23jH;E&>(Ii9|Xkp=IdP-owoNoVoD;lV_PqS8_-9-idFLfVOm3Ad-K*;jw|omyJ&yG+?*mo~rr1ky8V z<;(~mVV7Bw_L04`(6%(HSucw{8hr0qF13VSFE-%)zBWHvVo zhJ}ZN!@_cY{O}tcRcgH|wKL*Q8{c=jWxTzG1YD2*K6o(Us3!9JKL4}t8-Ln8|P@B6AIDOf6|EG=&)b^ZUM3aDpCy|*qLtPR6pLr_-Jr> z9iB`5hgLS>eVLPQ3Badk534VkC*CKXdiCG@fh%=zbZj^*YeLxHO?Fsrbk;7_92^?b zY4PBKvA4IcsjJ&`X_EW!cfj`DJ7Y!Ptx$wvkhwe`h@G~u99KFnj{J2=2x{>Sv*>AY z^FF|dJ7)as*GKFb0ceu2dA$;ZJ?JeG4y*l!P0uE@~Pd+wOu-| zeNJ@k(Due`+cOuc;3*8u-Q69)7OV1o@c|fVP30UAg*;F7L1H4NuM(eid*E}-wLEA4 zD4!r{9~?@3n?Ay^Ve5;=#cqqIW}{yC%WxcOIHCX`2~tKzWB^DFyXM;VKm?;K^2A9B zqN+9@>osL(b3XXwb5(4}X4Lt{*u-SM*_~tcwwLD?z;LuHzPut$`WS*jWPAo>L4AGI zrUMwzq#fqJ!(UB(rvM49@@r!9Zew6jhrVmaPx*}icKe*$KlRJx74;Ueq3~}QQmK>J za9P59$5P)iXocArR_lBiNTAE*mkbho98>_Vx!3zCdH zuq_iX8^=F>O@H0q8i*$p@<`A!^iDkSeR*KisR}$L(8v$RW$86N$hfwvTXaAAJo7dVd`#%%Knk8{(rp9r+T3IIKw zYjC6iEW~=EmciM5E0KW9!nY@w)DQ?CitY;naHr$xPt0y$&h`@=VL|v&-aG_!LQnBU+uW@kQoX~WsAs}=4n5KgoLB2s>b#g z$SQ{nqw}YGa(-*dE>lVg9icWj8BDAf5YSLHh*@_9ZCK^x=IS&$(a9$>_W)#2HMP!I z3YocTb5wxr$f)0#Kqa4a*~H+a>%_0!{9!Z`n{K_Kft>i;K89XnpwAL-Yz)_o62S?o zBBp#wDn0g!kZ`{*M;#C2-R@DGb)>ONb))r8Fb6L(MZpvmV~XwUS*Y^FXOuQ( z(D{+_#AgNu2fGjqg$N&-GgN9SN!gZ=aSHEyQ z?zWTdo0g#>0$e0$G~K)NNojgB3zjpn%^*<3*l#-0j)JuC(IhGp%wv|gIMwN>}=uR%; z+8^MO^vfC+q3zq0Q2UzRp$pi=6Ww3 z#^{`!?sG~iDJ;_9U|@12OpesWw_YvQ%S9^4fn_Z%`WFCor&FNJux%9ZGkiXeyjNFO z01-SHY9-A_36X^nY7UBj{V{9|Op5U2Vm0GKq}2V!>)Gc`6YALn#7dWd`Ye zqkX(M+eccCGuY-*+&KHz81Z$61jsF3T`?|D*M$ZH+e68iKcoCrZt(^c9#eD=gP0AP z`vFC~J$7xA|MUq9sKXI>w8Q{>P%~a1fUYH6p||iZ`T4ixw#!4 zZ|zsycZqzSFS6pZvc@KU$`6c;T*kl(7bNr`m^zQfe62E7M7`hFSi0sw}Jyxffq z3=9}u|D7+20=hY9SO$?6Pb(LVt%b^wm9af9}k zfh^l6fWy?N3VdW8ARLafKp$xXkB&p@uEY(pSn?2Lf#2YIjt+u6GFh;4@wwg-LAkJq zsrjk~2BUxr%s#y?;LxEH!E=zuZpXz1wc8nzwUf8~lUr8Uc~dS9!2 z{yaEeZ4R~bc~eMtn}6Qzx4Rd&C~EUJ@6A(ayh3@P@=CKU%LKf8q<*=tZ* zlO(W4|L}V-ophwYG|~XCp)8vNKvBDW`!tPorEoYlG6FUAzYO}L0czi=z`cNBp*dO3 zFWhrwVEvD&Wggi;bbt&O06Z}Pp7bIEL)(C%AL0u=$X@=THEr6o#PA#T*Fl+YSaq2! zPV1d-fy&p!t``}XNo+KfKA8g)ks`NeE`s<0qy4$f6qxCIaHi#xeo&}HDXznRwb)3? z%Ok<~-cM%zr^ROHFnQAzIx$rp4mnD`LE-<0v?zd^yZR=75fxd0RiS-M1Wv|0(>*xr>909d=X!gkVXlw$=Z0h(zcpO` zU2cN$8yp;rB|I(!s9o8o-+|$$r{7={^54N4Sh4?v`STCS-~tyJB?Q*av;8w8PLXmM zK>VDWo`idhQle=suw1Cf>}oSO$+!Zt6`)}XP7G)=06L3?houJx@5c&9hOPg$x)hpR zC(jCe5c)l=N((0j0#U=D3cDc(bS&>azy+i^A4Xj@UTeCXtc(B>=Ec)5R^@&HZ2$=_ zg)ZyE8^B_8cXxl`7t~c&W00Du3VDae$T1*5v9Zb>9 zl;{NHTUrC%HSLwlzZ@B0_y1iCU%GZ)eFx>AqagqdB`hq_fWovtK$HM`ES+09Ym%CO zmT}@00)$q`ZdpCXWc>X2ij$32{L#h?$q{AL{_k z5@-9qgs<_3GZLs#LAxQAuHDA})dd20TA&N$esY}ieRN+}ABpd9@OuJiar~r^gAmXy zEDG8Gl$BBciNpcZuzHrh@%ZQkbiDw%+Q0fcpfT+B<^Ck>!JZS~Lirm}<4%;}+kvz) zGgG)3x4>OaUOv1onO+Dgertz@FO9X1yQ9ax6jiH@ZTFsflblC;O zf4>||6?M(b|2rCGVzSZZrw54K5v)M8X}Z}PnJW}w3xt$rbo^ic=e|NHBmBSG1NVSN z8!kX8>9G)dL<@Ih0ah)Y5S_DW+U(qvgsQ| zInWF2cV1gb5(q>KQkIj}^vF8S_VoCwm2=urSH#Mg>>GXrCljHogo)OIKDPKPFXmUh z@6zm~pZy#6c$LE#3DqT$RpNpErWlcVj`^)(EyAz)`3$riV_#4amFdyY=-3#)Yo_RF ze1B^5r~H9wUZ$ePRQD(`@d0Q0pWoVr_CG)8@zclg(+EUq+;v3cV7S1*z;U1HS4#1h zN)rv2z2nq&eB@31A(y}hKJO;1dFfDtNU?Ybf@mA1pfb3KB!jA`LF@48Xwta)w6HHR z%)yA#j)9{gw5S{r$jHcut7xwgrEh8IEvNQpZ1}soyBo99{zl&+J50G&~1WkF;)pd0vQ&Nbn_{dR%0;;RI7@3(F1O-WW((Qg^3he{8ghXFn zUgrJ!MJSSe5;mvBN~Fn3bZ~G`?R8}%r=Xzr9-{=AhoHOaw@qLrhg2C60|SCOi@xZp zInTz~M$hlVtBBZ7a?)LNqiOFEK#gAf7(_&ed79tyfdw-KK7DQr|J)X6T0y*JvrZ_( zP0;O7T7{yP1)8>zD#Y73^|!2QlCiL0y7Us?+^jTyx=&5a(npHyg^%nF+TLa@H+tbQ zHLH>;nacn1!)6(a(3+1_2cZNP*bnD#tAH@%_3NcT&x?R%eKz}!?6z-{LRC}K(}*Az zRuNESFVD6wuuMS#0w|!rzu!slcSdHk6Tc%m9WUKu(1|@Q2R3QzoSQMNcSXyRC)L;$ zg(w(mUKkN+-MmqowAlGd&W4LLYjqk@q3tX#yq~ygzHeP;y5~Y)I-@;36A2f=?OjhU zl8uNT394KC`!y~}ON~ls30k9#k*J1JribNZK-CMHuCvr9Q&#pWvSj8b5^_aP7zs?kl`5N&;lPpgmrO`S2?-M^As{9W%gLd3aV<8v zzg#qtoSI2S2chENxVMi9@~fB4(5rHUk6=;4{eXLen}v#Hk;Gu+1E=4L3heH@kK~A7 z=LC?3Ca5!=*S*_@N`%a&PQ~f(Xf`O|>({R){OrUS0`|*7HVA#Lu7FerhW?Ia8l1SU zV5XyO2##hTu^Wy;KN6uE58+>DuxgwtNgPW$?&({q?es?^1>QcYlUdsU@VZ>i? z^ifnq(XP-7ZEO@O*R4gSqN0)=cRE{_-rm`vySFyCv@HDj6F)sY zeZKzFQo_XJ-GqPtYMb}u%uMLo(}}yscA}z!xZBQf@_e0b>Y>pfEsJ&R8QYICUDog! zR(UzZgq)LR1#@#oflrGe(IlLk8nWWe-rgV(cjDw=mN15{t}ZGnD(U&{&!0SuOiaQ# zG$Jm8=Pwo`DRlbzAa>c3>|>wn1xprAPDOovN^1W{Cw9YDk3ma`=UX!nyD>X)G}S^Q z16AhNuOa{beI{uS%?AwFuV<#rF6C~^reGPCDov9oB^B;fAf9i1b z(gC=poubZjbW$Qe6G2i&Ml`0TreUqCPRHZ|FK6NNO@87<78X*DT#h5JWhF{9zT=XR zpnxLcJ_InP6k+J#`D^y0xh0ps_ zE2WNKxGUNPR}T+X?PXWm_lTbKoG|Wq3>+Lq9i2I>s%#Ngbf$~lts0saQZ{NY)ReyH zz_>UQXVrp&g5L`ZXaRjvVZ7jgE0uC>w&%+d$&J*G$DPsv)?^hqIXNN!^B%yz-eTqF zOL>dtO~EM?qO#81dNH}d%i-_6R@jc_F#YyB(60;+ZnH-8fr*L9>eDAzv9&gDK>@q@ zfQB4j;+r*h8t?ZUDBV49ZGMC2Y})0Waa6!k`_*Ox*z88_K7*EiPlraLL1jU-LZ62e zMC$A6i2C~aIQ&t9VV)zbh*6+i*b~3BG{Jb5-K`@omH*sRXvLXqz{2EK6%Uj&6LgnI zW=8FP+IF)!Ut<=xiz9JM+uYoI7936!#==Wb&?>g~@gp58YY>Yt8HnQNeE4h5woCjU zKYrj65Cmsrkhk7$M#oLe4^2!czJHJR?QLe*@lwOqJOBHs(gA8wH_V%bSi7v>{X0dw z?p`JD5GE(dM>F`(AF;Cg0*;NIiT?z8h}E9t(h1y|V2y0*X15YUt z^IXk^$Z3S#+b^<0oT$WSS~&<=TU&8oi*571k)bDRd1Baq4@x-`Wha&z-WN2Y!;)+M zjZs54^ojuv6e-8k3TV$~o`fU>Tc6f93h<3N2a;)cOIz@KqjZnhWkinuaH>C`((}r7 z@9_$8bWI(D@c?Iiy8a0)kdO{C2ZzvfKN%9G)6EN^>#_T+*na_nBkVrhXy*FTPh3HR zn2E7w6TLQ}uqc{?bD9%L`^^L!cI0G1oz$jk9d`nmNVp*ThA+oniC{ylEVo8_@T2&-bhheyrIg&&sMo($RVN9V{cAHfNyDqdPEsFQcSS> zR8hMxxXBj|{0t%G2%tVy5X=C>COx(!bO`$F_@x*f$Nh?P(4;*iTD`&GHO4XXObm9& z3Jc(#B$(QvKlfzWMZF$Pga3VZ>8!Er%oN1hFrkm-6C zKf0{|+dzH?4eH;t@5U{ni1PBwx8wpT6cRo?5p#Hf@_1dk zY`zMg^384xHI_4Khg8WH1?MEDzHEXRIX-s@QM)#q_gR*GhB<6g0q(6u@799O z^345$T6qt?fne#wx_DC0ifXG57zL(*h$nzti9=AB&IvC6rBlM>Ku~JA=?C1NJ78=a z?c|xFy<-n1$HK+$e&>J=+~PyM$}5cQOe2TOz!k8j$@WVRNYc(+#QeZ0#!!6nVIAi$ zz_Iz{v`f&6WfkU#ZZfqFn0x>MBD**fOww-W6_&u|gag*68Gvp{2Cc91o(BTD3Mje| zIaYM)*$*Y0#v#>KvHzTcgqj2A8eQBBG-!1?HlY*IuV5O;DJR0+Tjj)mhyCu{$Bh(P z6#+8Ljk_~}q}6BeGp{gyHcG9T)w_ia)^@R(PI%g&*0vW(Z9?G48ma0OaPNO&8)3$60MkifZnIA zf|5To^qM({*>0fDfUT>@>40Q6mrio(lhOx~#Oqij*!6!XN^%x38JVzXc*)`oYri2) zF?aPp{=t;%~_)dSC`=h$ZhD1dJ|+HlX|8< zF7}^`P#kz`(6mc`rQ%VDgy&khH@v%2fho=aE+X;Vwl%ozDOMe^pH>RL_ua3SrIJl1 z*iOwZuWF7hJx+ct%sLZK^yR)vp%8~bhU^AM=y*iZ;>iv(_Y#wOQ zO5`{^eVILRNTAdmeTSO|^Ex7Pm2gwS&=v%EJ@4T}LjK&`?M~1e$VDPK7#6*J;DZDE z_&u?mG#5t_l!<)UCT}ip&8rY$3XSNF>7?g1oIu&<8L?G(jt?^eyjvWn&M=|YEhdS2 z4G%8Bf6r~j(ZzL@OFQ@z;CKJKjXbz2uMxwT^1_Ar&{o5$n)jHYGKdwk!}2&FJ+GIz zv~@eHx+k*(p^pgK%k!x39l6TM$3m6_sY{XBft;GuOYb$8cfIYyR?JNM$u(=yz-zYy z@~ld@r8S(}6x(v%_>mU_UwrmL*!#WM^7pTy_mw+lOMU;e2 zZlgW~KrF2(wsZ5^#9|}7<0o`WBA65At9w-=R##J zRVj`=Om(;kS3@acBAK!gc7|;Ghi0!RoSb}IS9p4@9d7OnVYck?>g`)86?9ts^C`{4 zr97&=&X|agcaiX;sy_cUY{Q6!(b>(SqO+N{@<&Rtng&_+O`)vafNAQ4MA0Y~`9iQJt(MT+_Hh-b1SGpY z?Tz4Iw7*q%bFgh^l$RIn#FzKoG@b-k4pDIx8ZlZ z<8j6I)w%4JyB@C<3G6e6Wob{UTD}I>R9aLWK)tNUn*}oVX@^NiLj7nGKx8Ya+ zfMCTZB@I`Wu)Kh#sRie}tLA$6EhY||SR`Y@Dbq;Qm8v)bV@S3-%UR9uBgIJR#W}qa zgEW*!3lo!zV1Lh8Hlu*PUeln=B3`%l(RAV75mHgE^Cfqp!x>sb_I{zB1mi=_CL3n; zx+fIuZ@?KiYW}lEFR;1!)E4`B9Ysp2^X4=>k2d3vEOTw1*OTtsP*#Exx zVWw+#=|Djk%;pRID_L*rmnqIJJs=QGo+zq z0u_Aq*)t;+RbP-P_ijfvGQ#iGI6ui(q~gSvnd8AH_HKgHbG6^smh2e_oW6u1ryLzt zKDL5q&;XYRJ+C}Pug-P<%ws3fO4h{o+d&McNJC(UEwOsL_SvtUV*Q?l91EkFn6S~( z>DLUjUr=XN;r(JTNGP|{%DhLRx7Tj{#9OLYl@ZFcRZL?f`%LyHDYpEq?X zt5-{Nzm4Odqz}pM9U=@5Lwp!tMPALx)1SyYd3kxM>*=9qtGV|avL>SgKd!E>K)udS z-)8;>sg_L0^Gn}10)iE1?BZ|4SJ{d+`?K!1(V1g*!!Pwqx3=Eg=HRX35)>9rOKF`4 z@EiLyvL~3UWf2ZI;`B=+|zg5RO3F(b0{*sOdgb7JK&mv9f-@buCQj>uQ2@rPK)l>*O zi-;o#^m~4O@A_I5z!4VOAdtOwo3E^&sBD&SOoomImYstRY-Wo;>*NhY)A2eXMTcU2 zf=(bqN7KH3u;C*INsY27;5n~29_3!iRqC4@QtnPzCK;Y|f+1%!#i{C9=cw=qx5%Av z)r0{#)Cb8iKziq@{SKG*?k@!`BjW7fpp(U~xDsCg&U~U2i=%p5Rj&g zP7qkV4h;#(tE$2=u(nP!{3v>4*<9%$O-4;EadO&8wPs{xMas&is~eci#w-vmqW#e9 zn(EZ_fGa?J6h!ZMbm(~Jm6@q)FlU19v!;%aLSX3b9}p%yp082!z0-+&$OAwn`dYjV z6>g-+K0u_v}iT=OGnR(zeCed84Wg}q|Q zUh7dlofy5(dd6sfKw9F?)YWQT)_-12VKp=qsLK(q%c1zjo&(!{1>M3+gb*F+^XJcDGs`XRoc}H^ zs3zp;6Suh25)DuH_BwxPl>tCYIKS;|USXlxpfECxZ5_Pjgo-gA{F~G$+9a$QDrMgP z;$Le1MnG9NdFmIwT^gE3)v>t0rN8-yiB`g|n`qaUiYKY2w8CPc!ak?T+SazGtg1aq z$L$Hd%R1L&Xwzc;*CoRCSTlv~YG1;h^$E5M zLcAHrlG$`j=PC`<{2eUG*8*uF@4uXPWJ~ljxrn4}9L2RM`(mo-QDAxhc%*V~ly6@u z?OHn=&p%KiYpxP7*$|=lBNJTBb?7O_(r$EtM@AM280yAmU$m8-T`OI-1b{iikTX&y zlgjdPOl5-ko9o&M30Y51;mys>v)iqBv-8cq9>2(>6mwScX1n0Y;bk$?GnAcG5#M> zl2t+*55;S5Z#`I&hv{;^S6MO{fV4fksP{+^;RG_VREqIAThHK6I$r!JCH;Kl#^&cC zm)F;EzHjfLk8LOMkmL)0uD8V8C$BY<1`(5TEFJfbMSdD&+)TD zd83new$+8vnDZU15qqz$>~ILN$+V%hWV!y&!q5A;&uA7F7Poi7lHpc*Ahox)eW?fn zk6bspd!HT!|JHxd&ny*0+$;K9SJL>$mGXhtgp3(7=-KUtsLDhB; zq+QmB!P&aLryE`ZBK6- zKaEQUd&lC7e)?qRu{9vd8ix{tyx%`g(Kfon+tl4<*%(8!}=y%g(4k zM*spLy(DJv`*+OEa@?>M|9r%RXfH9FLnW*+uH@q*7Dp{EvVDa?!m*xinL~D+=J4@4 zPu`VWa?~FQ4qxyn6s|g(xa>zjD8@?a&1qpm`f|(|@~=ASG{K1-S)d7YeYzIUYdLNV zK;0g9JE>IC8g+VDg<~=Pm5Z@Q)r5~cW>lvs@ey+LFl@2k)lP?mjkVqw?rg=Ky*eXj zT&1xwL;o)(LtSmzDCLv#ZDL zMmQaf3He>XKMEJJnfb&QRJjxKL+4&Ab2i_@!v&k5a&~R0lYcv>Nt58f1 z=rc8X6UEK>jMY4NU?iofN+Kv;7W$c2g~{=-f}#C2Dm*JI>)C!uu1v|4NlzHoe6uSX zV_{5H)usD7frYRek$vE3-*=+e~k725U{Xpu{6?<1oCh$Fu_L}8uH<14Xkbe-R+Ge zxIHv_K~Gl=YdLAgKte*2ovpzfU~|n~XGVL2N3*B*S<740aRa{Pd$Nka(7I}sR0VXX z1wGW+wE_!yeVC|*A()>(Sxbf@OHU3(dE+HThKa(PK1$%%SYOGPzW^h=MH zQ7+`s$1UgkH)&7G63&h|t%gSmQ`r?U@ukTw?T{8*Kng*C6v$&4qN?T&owf(@C;?1d zSXkI&Gm<$$B+hOg=VN>SyALM;5WaS6Xl5)~A9HSX`auk)z%rRw;^$|vL(RScgn_sk z-LZ0ZC?m5Pm)WGv!j7wZZMbT`3lHytnwMV13p^`oTR)yxwl^@ayX;a0qPV#9nr zF^GRJQeY>fV}9$PvB|`Ga6j!IN@A|Dnxu!>avVQnZn8g?qVT{P{+we_K?QrUdM_Kd#sE`CW{CpZ%pcu$(T?J|$>tf?mB!(jJ+k!3A2F zsN=H%SMrqxEny7dII`ZGh_fgt3>Gu25vi|(xdJ6#0w$vjUZ%p5>R9FEmCm;fY7@A9 zA8QebG1LuSckSlBNldkreV>SAVq%vyweYHJp!iLLu}!@MyU==>ZW9v(3y6z=?7 zAU%B7;7F%bj=B1?gB0p<~TiNA@XeV>0a01t&?1k9k@O2d(~1*71=gJ!FY zCgdkRo-dJC2RqB1WQ&qqG4;&Dr~VUWyoF)@dejXXbfMiG%?Vur|6K-=Q2ry%-_Vm>~#nIFzC_KqF8cS&;P z?FRMPhkPhaCkLr4?2lVMBULZtjgM@(wE3Dn^2k_y!06=g$qjV7qf3$H5Aykwl;bDRVhH+ z+ng(NflMnuhqeV%$|9%U8>L>JFX^xrLV7gXulj_)T`P* ziid`URB348T%MgBKr59d6~betz!jJKVbkUVSG(4pN!=nYk;rp}tgP$;-|3%XJDb}P z&Oog)_K29vcJFSh+0C*yOZkFVrm|g!C$B)_lwRVQO-)zYe+R)++gWq*XV$zU;dM+*M88{EftdXRQ*Oe)n~tyL1?XRn`uPh^z^3O3ieGnzLc)F$YQ%|M_V7JkU#V=DSbE7dH@B3YiiM2~n?cgf+4K*DLI`x81CGcj zoZe05(j}{Yb_5z3;$M2@X4&Y#NL74E8wU}jm|>A*Xq~e=r89yYKIydkyo8Vs`0+2d zjbtzFOrK=}A@Ba$I3V%BM@_!>LNy>u#pQlYrJQ(CGOh$2HHOnM{d1NfmX!EeZnn}u z)Kr>2HE~3Dxu_~We$|y4l8G!0a`V%i(3<=3yP>Yzu@XCV+%F{UtO=4`ONH>4Is{Qf z$2^JxOhB=ynyV&y_7;wMjW=D1D$xm#phqRT?_b#gQ*y-FcKnBP|K~q3lC3(?Ypbbf z(TkX4$$lZTS69W;HE6 zsbx;pxd$(|+pEdv1-f4}!ZDrmLZgrlyvgVkY4)~r9Tv#=mu_S(<*Bz+ROMd*%-wX+j>!?fG%n%|)=GcmjFp`G0-6gYTj=#?ansb5z~$X~1|fdmqhRJ>HXVr5VkpLf zDkXQZqhJ!iMERPM0wg#qXU!_bK`O;EKxU;~Wq8ceDYj_H`$0hgSq=r~=Mxm6zvCN} z5xPu_u=8;&t%;TTs_rRakV)vemRD(wi%NN9ixy+cU(FKs&b}uUmpPKrs}5|qDao<7 zH06D-mx!gsMri_H0TA_p4e^p*^Yp_k(JhPqSnvO$99i)RZYEsg+QjgK0@yu?XG6`Sqsw<1z7o9XXJ}GNzYs*y4ZAIOoKbPJ! zuvZJMmhouk_d|l23v#5IIjA0PP6sL)Rz+Ytjr=rDSwGCj6iTGJ69HlEFZZ_BhY;{` zQgUB(o18S{)wZJObR5adR3Z<`Am9o!6PnZ#qK=@pOFTGl-MH?*Jd2K!V-@^OU0+K&{ ziU<`ierV7qN3rqak7zfYGpZRG86XhKFBE$&d{JVEqm(x&US2$GbX2w6;IK(_32o|I zO`7sstg>VwWkkd`i5`mV_@@z}w5Y!Y4uKXP^1+CR%U4%f=VBV-Hl#neufE9-(&3Qw z=-==M{cS^Csbfn(X8Y~7Zf0fk!(?*bSo@}{=)EjCA*QDbCj8bW-q)`|X?-IJ*PRTg z8C|o(r871ms#j;9343E>?FzTDC{J}a_ADLw3B0T8CZ~lqEi*>ng18vJZ6=yqBINpy ze~T_L{wBqUPshhkn*Uu8o8Jyfq4`NX;UfZ|_XXEDJiX&d^eiSEmtA_74R!Qa%s5iy zXn_?r8kc-hU=92bF^` z)i?|{>e8@ zZX5ojY9qx8n&lP#yp0MVP7!33tRXEG{T2|?H&c~WC@LyqRlcBZfhSTlCJf-DLlV+! z@kn>Q$XgU=D2Wi)|1i~K=qjTnB&_Qmx1jxi8(Ams`pKNSLiU-ts3#_`C}DFvK@<+i z?_T}hckuxDN9Qm|Wg|`Sz)>ap?Z@JtyvF!$ng=7RfpW$A0|n?Qr+UMPfoz4X5?t}>B5KMg~NMvy?yw!R~T>1NAC3U zbmKgUAok#;_-BiHGd?D!7a$2ao^Q(Y2?(qvsO2Um29}qzBO)SVQefgo%9U1D_5)5a zyth~GSABf}fS@tI%jr#IP)tZmi|Fi>0tk-bP)A2D;@5g101VQ#&pBdKC*s9MpNQ)J zu(SpO<@#Qe4UdfA0Z1hPwiu6P0B-lcP+IM+vJ>}V*hJ<){RD&l)`6CGuYgF#PE6Vu zz##zmjeR+7~gGn|(*oYKkKNp(@4sutT}c6i zGz~04^bqyx)~uOPrSx8l>-Ns#H|4$U3O!zo1_qsBSYL+@5IpqJO*}jr zxBk{UA56bmcAYcuIBD8FJ0GCg!tuXF_%k(iF8OjH2@n7&@bU2-ua9&|vzCPpVJZf| zKELPY!hzi>t_!iUqGxAkgFwX^`&|6R8j%qZ5vNG>^Z|X*3z+Y;h~|dvB;HvvQ?Rv6u-aM$Y`K0#-6`o?%=Za`)ZQ9gCKE}odlER%H`EK3GcjSM7 zv7R6#3=jxLRs*f0NZtl+@#<9&)6MbH5uqccu_B((~&jhpo1o zRcA`G_(V*@gWQaaC_p*Zbvo{eKd6b767|4EA~@6D&)3oA4lpJdcqoe_9s;l_HMO+pUcbh|AZ7>1jcXAZ=jZ1_UYC2g zKtcCk1gvSY*0Q!h#i6yUK3Bh)(8?a_?L|y1;(II++}Az($Hlb|pV^e!#wbS7!IjiCxbvBin7E+28W)^|rlqANDZ=u%m{HS&U#t4>|muF#TcWGz!Q$j6M6iGXzK#bl%G*LnLV(7)WD$Pbk*%&RRBzZR>+z(2Xg1S&ChX9+uVS&maa zZZk_dm6Lcq95>|8*l;_nim^kgK|u4MlI)E-Zi_N6Pa2tvdu`_3raTl~T6)wtC|Ea0 z-;v83Q?N+LKW9ciGT!oU(G*rkS$1EyYI#SlxQobu?2pN6yXgp2TJh|+PTqPBk!Mze zs`}<zmRc+Vg=;-Lx#$E6LnOgc_*vKg|1k3e^EfFbj<%k?B)BgDAaTNqG zJi)=!5+CV_5rAHfDgX7A>_B`isI;#;X4u?R948=M(~;{zkTHZQ?iFG6?;{Lu=|NR+ zr?J`o(NQk|Y{2l&%^4jdd7$V39;dFaf27IOLV+6zaWHQrW6H$hN;lT}KF`%_W0CnD zXev)E-6*TQvjj6G|D;whUf+W=%J8KFM_3sfOF7mfvu{uWQT7&aD=_N<&{|rc?-d#v znkMLwbJ)L$xcL_vad=1va5AUoP;Aw(5n1E8tSQAja6#36Pxr6vho-yRguc|?MjN`j z3y$YX#sdUt*l7ceD?mYeAMY;kL;_#G4*&HIAU&L&IgS1DIJnux^3tS?1PDfg1Vluk zL~Po5rKPBV#t;z^v9PiA0I>g|(xFkKBVBh-53^fA0`4MmP?`oH1-zWYP}J@zTHbVe zutgiLv?{$hm^z^LFx2CSQx5keXtcM7^i!L-LX&yH<=>(OcnKB5OBl-qN32JXJ^%XfOQ>#l`!xk<>Epk^ye_} zdI6@^Io$G3;2!-DzZM_Nh6)n;2yWm*lcNWW3WFA60|Nsg=f9Y+zC^>S7e>SKXAHnU zarr;{s9uC>eChoPJc=1`%R(`I?S_NYmP6y?{mV_xHFgU>(SWrcDR>AZd)p1I`l5(_ z+tk5gb0h)t*+w|cKLbv=TOxC?Uz~57x}Tp!k;W_F`hF~cT@^5kfkyz0wvZC*amqEK zr=xL$$16!bd->CKGXv}@ebZl8GQl?_Q6)bU{ZIg2U?VLmJUKZzfRP|&iMnG0eJ!BB zA!}$z4V}n2p0a)d;4^6$Y4^1kKR-V#djSyErgD;h+lR&E)tKaeaqWS5i2cFg;esm} zKg1_SvHsXEqeA(7J3!NIdM7cfp_LqykwFF^>A_0xbP9g_KnD!) zg9m?7w8sw$DcqbW-?rZ$-ie3K8G2&@-1@zf6J!vjkh)87;(QH+QOGe56-HVR`hJ{k z1IVcx?dULc`?D}8uZA1iJ0ZYdbHhhIR`hG^LjpM_GK>)TKa({%4g12nJ1#Q2>rO|S zd&&q12-!azcVhWpDaT)W90FVTPbp+=P+E~Kab&6Sp#=om*x1;zuPxE9V)Fu+=cpt~ zZ{H$;mOXdLl!|BRPTv%3!~PGFQc)SMCMS+Yq7bxq`AFgU)AIsa;%x>a%EC7_H39u$ z1urk=dVBGRtSsWe9Vqa#{kPuuP+K6WP@N?qDT%inHvN&9n|tcvxA&WlkdTn}zoRKa zevd++lrCF;%x{H-2tdUk154q`b%W#L;^=l8buSL9I;abTA?}Q0Kq?O4CL&iWz)^)@ z44U)6rd`lO;jOJz>=rblvPIp^cE?vdE+;jLieKU3;|I-@>BOg*{d;~mH3PbDal2I= z&myZn7jOCa_|^qG?qbbGFr{(Bj)}ycE~f^^#}$Bh$-~QA^J%G`F1JbdB2kgrsnv3h zs$Bfb-{EAo8rwOJ-FC_@WkAugo>=x1b0!YSVih(3huHo6^bqR4=BqQ`=G{>3|KbOv zY8XL7=HyrNdwW*-b#<=uHwTA@q2=Y}6pQW`pG17`T?Wp_#>QkcH2#$CmQ_`e01u|p z114=`e0*9RZ&utfbgB%gv}%is$F;{6D4}GOmC>lZkGO$!*mQcU@c*465dr2~COpZ?J(oT;2pf__ z_W{E!0~ZsLu&!pzkMEkSczAdl)2@l>xV?=3?N)0KT&%RXn*r@RkHdJsqfV3EBB>;+tld|ybz&>Rzt_g^D8Uu zHOJOoo^QHQXnfYEul~qL0`Dr|uo~li(ll_cPYUfiM1vZ`te)qqPCx*V@SY+xexD8$Q0P~w5WqCEZN*Uvz{|CxR8`J;* diff --git a/docs/images/chapters/pointcurves/067a3df30e32708fc0d13f8eb78c0b05.png b/docs/images/chapters/pointcurves/067a3df30e32708fc0d13f8eb78c0b05.png index 8a60b8a2fbe271d2bd21080e49a7c318f68f418f..4b069261edec62dff2684218789ea6468307d12b 100644 GIT binary patch literal 7135 zcmdscc{mi_`}T-zP1e-dhOvK;#=d08p0b7P+n`8xk;cA{Y#~{bNtTkeY-KkLNm)a7 zLr7#B`|=*&_xsoH`fcyO?}ckzXPM_Y=RD_m?)yHm26|c-Y1wEY5XeRCo9c!T2pRL) zA2lUtVR<`O2RSTPS1Yw8jDgP5ccQV>MF+mIjfUT44#`dDXkY>$txJk(lDQ3 zq>%o?LrLx;PNM?XC^DwFYn3xcaRq9h!&(*rW;&pIvfT5tIr2ULssf4f8o1)x( z{`>jiXjOI9|H;UI7zD2XUIesMo%#I?Gx6)+uKb5l|GB^ayFncjU?Fzh_g?!y{Ymme%WPYh`cUy2YK^?YXy(wrUDwwXv~zvADIlnK!uXIfswc zGi7%ZNq+N&@A-=t7Vs4p#bgoU7m6Fk#>PcS?D3k;;Su7_ecVz~QZwZt$0B;KlrDZ4 zb7l1*{l4ouQO5`|u&{WVDe1%`CPue1QcielRP-z@ElOW$pTah9LnCHGeJv7YN5&!N z2{AV}cXn~9QYoK`f1#}BdYMeQp~Pt89pRr!WT!C~rUSPwLS`N}A9DV9Zyo`Tj!#ffEx+Gv(pgtqyA>SM!ybR0 z$8AZUoEK-R+3+!Y?T30k=_Eeww&XS*4uxki!HDDJojqedXhOKEz=y*lvm{bZHGG`8O)tZ-q37p$IIIuVQ1#PIBna1@^;uFi8S_;*ogi75Ui=Po@ zWVG_Y4u5^M%Oqx17i#_aG4HOS_%{JP{8DQ&wUHdgPkpBYSs9U$0aMq~B92x%$bEb) zvdyc7K)Q2iW?jPmD1M?JHNjLX@5htgBknb)-f(ijRNQ-2XmbCWm0Sos;LE$ZV?-@J zE%xR9{yrLPlaQF0ri~5!=-ju~kTyQ6p=EK^{9m`CCMZn>g7}1mY0DdT*rMqp7N}m(4ypGYXemtMn(Hs=VK%b7!|?6`tRO3D7ZIx)hmi zgP~y-)WiSi%{F*k!|--C49epD0Lo0ocHK|ZCSX;1Xm}XizW9n~VhFn(h5((~pwp;i z*#BgIaRbk+Z#P(|D;F$8AA5WZzSbKipvR-61i$@4G)%9kJxlswpOtCE2BQj?$O>f^ zoC5)R#m1lP!Gi~8weFmW9P;NvJT!`M_Q9U=ua!D{1oWK$j8?)V9I2~5eDKJ0+4xNFr`84|a8R zwN#+nMZYgw@>}&!tGLC{LG-iN^7FDj7sk}@yA6#bExui3=EdnX*YzmV?i>|CLEWhQ?< z!NQvs%6z52+*aA~-aUkZ0vpJU8c&|o4{w3>N==3OA|x`Qse<~N#>SVty}k2BR+3Xv z?4=8@8vm?O1}D8o9yKIbC*h^i-`}5s3XsvukqhIz?q?R%X2)J;TwE@$V`Xe-$9-AM z3ftMK`QEbj+i$gzzXStZQybC?JvB9zA?ZZN!^^ud=JZOXXwX;w1x|IR6Y1pz%nlam z6Y#Oj8{b1tm|0j@CKLY(8~eY0fr&%U34n|(?$8N!9jzcs7qiC9?Cs#Z_e>;Bg@;6- z%(6`E>~Zxzi#&pYR8HS>tCywSa4E%iybjeGfmKOzDmr}6wdQzKz|PEkcdAUYUg`6V zTv`%sJN_c+?=;ue;&&|&u1}vnh5xQ@xN{rTf%Gy$AP^bKAxd;`MJy-|vpC!D2Dli| z;lP$ad9C?KVsmrT^Vio*_^@5^vr?FVyO^9D>{{bC)6Uvt+`zy9d<0$G7lG7KykTYr ztE#H1a2uDgY7TBRt%eB?u|&tlVB_QQ@6A6{JQx%h`tf7u6I?Cor$dJSv!x|xotFwM z>@u#Qt#%BCse&$VFSg^``S|$}GBTI5Wj&%3ba!0an5*$1n&gb=%>t77XyoPPVMsd& zu=~gR;!ZJ1@ekQ}qU26!wN_(cp#U&L3DUiky}kV(8D-F&gE=C!sqG&e%q=RP%GbD! z-yb$Ik(w$daX^{rWk3ar=Te4S*1EG`PTxpES#}A?4bM~o-SAesiv6PafH>S-6HEZE zCzehGr>LQ#5@ua{eO26eg1$YH;qv8ZFJdQpcWp8#*e&|$A%Hm$ZhrprX=!P7f%{cz zW#0|Ff}qUjAVQaD9UUF*{!TUeHM5R$<`(-}Lz%0vNl7uPxEo$xmH9SYib zMn!rWHa59xrh>Z;NfsjjNr|+_(8nYt(SLh+eR1in89K>gk$}X}fi3XdUb>^DpKsJ~ zXQcrqFx1+W%zMYyHVJH<0!G-{zF3Ycn#Pwqy(c*%V>DlLC@29prwGQqt#lyF5x7Dk z$NsQ+e33BbNJ6W=9B`JRrloC56EWpLrNy)vcyWNa59Yg)8-F|hy$MZg0yf1ow6w;z z5li3V1zipk2l|=07g-n?!=j@prKF`>M@B~a2ba@o3?8f;`Kfr^h}Z5Xe)AY&3}~q# z(xB|PG{JRdVrCY19Yt#CYY6QWv*^Dye4A(Z7B1Jv5@5I!x#=e|=H#na1rmJzb znJ@FXKEscdg=m@j>?LM(UNS!ZntNE@|CB{URAoT3e8i#SDGLNrY7z$!q*~D+?)B?) zAeK4+Ds+A4hJ@K6VR^$YR}uCXED5Hjro6no4*&ZR|It)U^1lOHgXBZT9suw_MaN=(?n3nq~%zqg!=~A+d1(VhJ3+>y&Yax((q~0#2P4cW5K?sD- zbuzx68LsB6&KIm`0x3wBol7zF{B=`F?A5hStQM1j;ULyWgD}gIeP;h=O60=no`><# zBbxlqB6~b9jmbz;h$19M`zEWG(4$}S^j*m0kEw6WT?WoVvwDNGSQ7nAc&G2rs>eGpCBP02c z61CwI+}mLawGtZP0w-7By54$0u_3P6V5Ly#x{1*G&1Xp!++G@gn@>+Bw`6wj07!9q zynD6T9gqsZLFT>X>9~~QC7r&RUhZ&i9sH_>gmu$O}b(C@5pG~&ml{-f7Cd3*a2A0J;{ zLj#B8{jV)yiz zv{{M~bQ>2rs29G>!=rSfwRojXxUAN?K}oWc$*v6C_kNz9?$DQ~Vpi|X4;n=sBqSu( zrvmVZnd1Tkv}0ujA>jO-)VC zZmV^h&JN%{4K6DLDN0B%)z{ZULN_h0|aKYu|%(bm8qRQYI@zWDun{Lbola(cQG=?M5)>Ai4^LphiOAjAS{bnoi9z~?52O>Wt3O6ayJT4!`}_*PtZJnCMp z-3y4m*}MeEz17v#pnUvZ_s3uqpp<73(wW4SqT1F)bPWRjVO(1z{s0xblGFySy_Rycgsw$?FlM}IN7|jjB zYP^TC^SNUV)YaNHWM!E7IlYNuLVhw^iz$so!eyC>^Xk%JXEeo}V}XgjW#))ph;ap<~X$H;z1&5xQ1p1!++;?~3aTZ@b^ z7?zBj68L64=hwN9)PaYWr?>G2JC^`X3kXSKu4WO8=UW_F)#S^UFLqLFAQsNR&}{5o z0EQ43v{NcVPJS4*;9bYRTR{VW*5W9kQZmdV zDLIHMVahc-Lt^J3fgrhoFm36>sd0J-O_})u*w$8RS?l?G=pRw@syE-udl|gA0Ah3k ziNtrsvZ#9r59m{xnO_~*T6&&rY{P6b~+inY{5Xk)t}OrE?=G-Hnoi!bFEpa zhViT=q33W}Sy>hLdM-QOzuz`JJ?;MY1yWs+svS=b2+~!{nwexRxQdQWymeVCb4hLt zdM*syb1Fcd0Ry5b?*(muAfN$yKa)pnfgqtuU$lW27S~biD!83~wgaMO6)lBunVOl{ zyDsg>4F*6iy1KhtL6%_T;Fws!G*sSN1;bmw;wrp|n!peHnU=7|cO4A4vI=mf#3DHFAzfTbYtzL6@n)!u=*^Vo=>W+@{ zI7@o)Nd8Ae4;z1Qrw+`<*{y&w8#at9CN{i@B))(xs}WM^~tm8UFnFb3Fo5{a;)U z%cm{(Xo-RL!i9|?BYockv^kW~`%-jl+?mh@#>Z>&UcPEEgmV&6i{wZ^&z&o?X?*(U zk83=uBqgBX5msWg4To(E2&Y!0m%ZC~7O1W8%*;%aV+-JK4j?|-!y_ZS*RQkRDqt#a zI(WB0UgZxuSlaP4FO`DW0(RFlEG*azjf@bf-De%}c6=;A-RQ+{FN@2{IvbvBA?;cq zxn}(7zyK3JKmXp&xQDp=gq-5;gil}XEunc7qc;NR1A{B1(dvg++$ZY7q}1$ zs9IFbp*P+rU8lN8%SExXrD^yk8lmbQhPHP~NJtp1_2Bvvd_d#HTy*0{2sfqkkpI`t zRcLaFUB+lEP$>a(@mw-c;?YV}{WJK1k@<3@b2N4RKhd$AR&}0apZ)$ur`tm8Pgc$2 z;T*BZs8~Uh?ox3bich{PmzyRVsfMeou{U$D8xC_Qn?3@We9P1>dHszU~oV*q}rq>?p5aZK{>;mY%D_eunf`EPxzv>=>Z(&jna^N z#7%#!dO)1M-;9mK-GMT0Izv-qw%e0B_SQaIbu|Pc_m`c-5n@>gRf_0XmPVVT0WSS^ zkEI*xT06Cvvb{EbXP;{1$BEX+?SC?!?@_BoE-67%15S2oo6-M;+_>R~N`c=d4AE#C z_-(LCTg&_YzNB=#Z4?M#ieDBgnJBIUng8|69!v)(^@8v%M?QiuFi<(4Aj`loz;)O( zNw!+{a<4q#dlI`W9l!xKZ<~N31TO9>(uZfU6-l$uZ6nr>7eXe3^4&a=dezEps ze5;XP{U_~l$aV5_=M3)NJ^!y)l8#qBh@?Cz7__3A>iv=jty4+uqh&+j+DA1MIDDWu z+5aAT`(p7-*%Eve$W&TdYC^QIv}9yr3jg>~3d*I#fkva9A3RV2P-rly5xlVQtaO7{ z7}p?N3F*zHyZS|gvp_sD`TkIf?QHbDd-rbY>D9!4J{{@k=zxS`vD7&rV;B|cRQCK! zgT(=5S!T&SZtyaP8*%NF1o)`v>8oeY?3!tq002*amJ^%m! literal 7488 zcmb_>XH-*Bw{0jQy~RT4HB`lfB2ADM5JN|#C<38LlOi1iBvh#)O^^}Q$vJ25wdY=I&bgxXbTsIwS*alq2;Ch`H3JBQ z1bY5MNdeAWE+sUAKNQw?HPj$y=ij+4Z<8Sq_MJOw%0|9fE5H4WUhU1UZ4As@5C%+N~dTpF#!{_B>KucfsULhXi?-YDTXVqo=ILSN{Zfus; zeL)HyLw*$}mCvm1ruv=xvm$LxJFUbXgY9c8{Z#RRUB5R&o}Xm~_xw=Ai3Ptu^3zG_ zU+g3fl`BV8ucz%Tl%8~M$Z+C4D>e#GemAzXJRA#kQ;5fJWIbtqhQ~(z)n|e7GCxP~ z@bdik2P*kP{|52}YZ;T4lD2*1TC|E(Zr={6;_&>hhd3!%{9#ODjHP0t>PY_&9Y|S5 zYf@H_^q&tKisTgcjMAjUVWY5?=>N>Y3~QAdKly0Lyf1qKkqVZOCJX+>sHmyjfd79V zY$lqOyKiFxamp;^D40ImHDM zNPa=V_o=D)`TjIMDXGgeSH#*sr-CCvP7ey(*IO4)3iNALGggZqBSPc|GY1$X6 z%dZq&`!8CrL9!q>7qT#zgFlbvcMYytd!-}TD=mSkfVEqJJ0tf~SG3_!5fPDZ(K$ID2x&!RczSJlO=zy!ixoSG5Gk`kVpdIcLE-r*qGiN{}5R8)*gNYL>0m4R%JxfWW81_@I<**27vl!QR|nWJrUem4+p>1B}(J4>ULYlF~< ziHRdGilIllGTEO#$q#e3rZY;jy21-ae0BNJo!f={f#%!S;)4PxNt8Q!gNVW0a!7V2 zD=P;lr!EjX_p{R=G4py_WCra-!xIevMhoBB3l4Q=0P^P}*!uH@41$e_FmoB*+1G5b z0J%ji9$)$K@1L#F8-B}el3PZ3*Teq?N0&Mh)_U^sC|vuLPxZBb&+jlpHfW) zo_JyXnutrmLGfRXXxy7xFH;z{>Fh^kQDAsOa&ow?TpQ?!nS7wh>g4Vgpy-{YkPt&5 zRiJV`EKQa)JUrY=-Ss+viJplsrNF(*C7M~G*j_#Cy3gw@_A@9K1r-%l)QcBgeaV9E zgqiD4(b*I^W5Q8VT1;$z`vXMgUioH7ph4u3NMx0qID-Z1Rz>iZz`OUZx$AZu;=#Sb z8v{1!2S-PIf`S**#2=yD-4T~Ul^wE^LRC_(KuMeZidA&N?BRU;NF?+7`<1x#UP>V! z*U(2RUVmv(*`slIc(fTA)IofT#}5&E=l^ z^M@B~JNbGSWn#}-A`>D;pC=NRKBnj4f!y5O{Pyiz$d}+VHWb@%FDu(XNusJX+Ie7b zkU_jCyz!n|*#&s+E z`uWYK+_zA6b{0F`ZD^Za{r$n6R)dV^~Vapr#WCuQ)h3usYbB z|K?cNJ2=>IDBa0bqbBfLKX0}tUFL?5k5BVH;YCzTOs9QLkln%a?Wr3gl9Mb@!S&>v zWAew3ACJ{OhB`Vravn^!=6AyCB#mODlQp%pI-g&lsRC>F;FI@C_23$3O7c#Y{$16s zbeW{FF%vP<599%Rt35q0t{&BC-#0P{n`=JK1sVQ?Hpvs7BJ~)z;QdPxv+6 z@zo8Fz#`mL3N#W>ppF8R)<+;^L44}I3sKa`fOUutjEON}9r zJ`W44w&`OA__^q_Itlw>5=*@kj&NrJUnY%iJgxf#8cAtsdOrnJ_(u%-fhJg&4<9~M zffp$%^-JglsrQVvNj3CF-+*6-3kY0rcX!|4>=zF>sok*jQn&p&#_d4+M#i*qAtUC+ zENj*8u_4x~t{xkt)o8^-N(KfEGqdrkWmSx1o*70B6CjB5?Pr|l3qyDA7-hjmLsoE{ zqahse8In{lcTkmPN65=6o>5(<<|8kCf2y{MAoY9 z#DoNfE7#1h&!3al*45EXJUdpf(WK+DZ8}HULQi^Q6+j5Tpdct)mScN8B{E95?jBoP z+q{;RmgyWMrwFRY4*|cf!<)nna)qEy7T7mSQ7dWs?uxNgJTfvYL&e%%p=1|Pfzz=u zTb5`_NaE|)F<^P~K7JgBAK4qRb752Y*}}A_j!6vNlB?! zyd>I+6gKG6;0fC^TE2o&xmEb~EphwbP`zTT#etH7p_S1x12~I*ElDMkLha?{Rj>F; z5h1c0iOA#RXQq1j@}=AAq^vpDXAC$QhRCxJfWrx(u&azJz;t3`RuJ%EQ2?Tl5J;xB zHJcc!$D0MZXc!o%P+`&HxYWgR31vG6KJzv`oyF+hy6l-5t>_Apy6mMDWOzXS3h|c$iE+e z6bTck*&!LDDd>w`Ta_+Q9t|~=`r_L;&A}`FokRQCGewZy#n$GI*Si>c`}@O;M0)zg z+i2Sl#e+08DZ!>m%*vWv3wrwd{SqsmygVBu*GQ!N*{b%Q%V9xY9*MlqDmi2z^JXPp zDtIr-*4~~^f{*5Y=JO zIx;fS`}Hg8&Ye4Ldr8^Z@!Q)^K>@jY`_L(O0+88rbd6PGWc;^0#c;@-0kimx!-wT)iRu3*+V=ctSsn}xJ%(f|FUw+zK+neq{z!@(1 z`Q@rVxxLVpk>tv3`ONum?i74EuApV)rnJbr(O;FAg8V8WMRv$~P@;HDOpM!~EhW{r zLwUQ_!l$(^Y}VGDc?PB2TRvTXp115j51*XbaeXqC%^G{6`GgwcR2l=!=WWai%05u)!9 zfj5zpE8hGla~(a7?yu(3(e8A3STJu7p)_q~#3v-60CCW{?5y5_=b8Ztr?KdX3eHfM z_gg>57dbMVAX?dFN4xP3|E=qor#PLzv~QxelT}G)inU(+^B8TkutRfdQEsYdc0`R! z-( z>9hES>;QVZ8dy%yF0A5FTfDGg@wsd=ew5YfyagkwBrX&-9NAD7}6KPtx=cX1O-iawGH0d7U*gGh<|9bNYo6Ny)FN8FHKif};OyZ(~Dw&vg8ELsI{|p3tRR z78;}4Q;m%>>-&EC2K)6XBK$6-WYSh6()aXoG0|6yJ^;6vt)&?U&n3E_Xh;& z0Li)Se`|bVA}J^5iC;tET~Gv!L@pPjGe~EY`HxRdX8-)L^ZSaWf>v2|Q2;I4{|iri zIt`bXpI9u*|8%gmfLR%@^F5ePvy#qqUz@rGKsUJfFfTu!clqQGH*4)mqp>@z?%rtC z&QP1R(QU}-+S;gTeM`WeYbD;a;Ah9QY6R9&sUH(+!6_50+mp^HB_*|UD6Y6ZBc7X< zb`^476CVYiNM-vk3cp_{Gp+g4froM)g4BDij{0{YH=KTorK91)KYywK)5Z7GO^|U} z<8X`B0(VnQT()FZS6B5!Qc}lU#;QobajMHw9Y={#UBMUa_>yms0qyJSl|jAHr&dJ4W>=`>XZBqtwznS*w?RFV__1wpOL(=^XS5^ zkUky~7NVfN8o3pkVkWzoV_~(9dlBHO7${G4gZ@^D%keTkM+0+uemyGTC-g$TN)q6PKbe+77M#q*j zU^G8!6fU9FiWX%(Qz9Y||K8Te>ok7a$=YC<`PXb%RQaF@nal9wS8^+_N>?NT2BCVe zpo9Z3DKQdJ)6keW)T6*MRdZC3OlUuHYJWXomsK)~`7Fc`yScxIeJ@==)rwrmRB~G! z7XG{A4>t3!L%qYIrY`I|X|i#7?ev+2P)1#0K4fPjh4ak>GcxI(mL-a^OJJI>w<(0? zR-jHO--Z0Leu(l@$L9u41g0fgE`~Rohu6N0X$SUfz8Fw$u}lv zQh1uRqO45q_HBrYii&fPP!M50_5=j7cSt)1@$BnKYQ8~f;M~Fk=V+$tN3m3T0%9^kzIC2G6lUW>jJD^!#u*{RBbd zT2}R<2RLhFVnP|M_Poh+ftu^n5*jOE{Ez`dum(ny`^vaf^WozAMCxPCG8!3#PnQH@Y#`Lo>3U`lUu9%VX7KFc>Ud28ezRr--!xN&| z{1%z+wxI*%QwhTXK?hu%bIVIsRu&i%OQX^c?Rgh><0H=j!3q@nod=C>Bt1PnCLe65 zc=`Caj*hb>jDO2uQqDykDY#HlQqm3*ZJwSGP?2^mB=aqkP!K0p5nOjDjF#fkrAxQp zk68D_Tu#X1oxZOYrDla@e+7I^;N3C7V9^UX!L5S~D+pdm0~Y8-qlXXUIOM&#MMY_Z zg@s!bBboA&z#!A1f;uw;84~*kgTYfv_kEGPHlnf7Vc(W^#%k;tKx(h{nAc*7L%n}w zuh#n$JjZS6=e6aD1_r~oI_h?I@N?hli!lnjd;w#PN5i4e`q@v2!+g&vEj zM%0%Hng8f{n;&L+-Kzu zz=X1<8^hsH=hNfh;OI4NCf_mOdhp#X?q2-D7;yt>PwwUItx{@40OUsVjTGA|MsU{n zEfp2DxW(7RE({u_sjW>H^Wp^)BTLlW!j-O>f8i$K41B;aLcr`C>sY~>HtKspp>{Uy zqGHrgKTJv%BL;~EVFP6~>?N~+3Q-XAhAGH+98r#P2G#@_}J{R(Br#qF;6S8}ES)h=M^z7~DH^-qcer084v^7H#>xY?7 zcx{bYNfjp7bKY;2)!)5y2d2jYSQ8MUBFl}@%TyRrOCZ9Hc}O?|g(_QD6|c0dE8r5? zB7q|U3Av|me0+@Byb-<3iXqLdudg4)&u3>lI68OlyAvS_Q5sLCb4R@nke+|g%Tq0E zR#sMHRaRuE1kSk4O^@SQhF}Q9#@?P`!ioDs-RvWhUJ@HSyUwAZ8axqzCtbns!~6np z;0ue3%}yH}M?@eD*B_mN7!ZKG@+HRIbOQ{;8PE#3H}r#wM`A#BCnhI{4Jw@lc3VKd zYZ5c4LK>$0X)>iaq0FS3v~9mf zyexZk)RrFd{x5LEpL%=q0?!)wJk^0mQ-I$@uBkKrN=H|Y7PNs(dQ!yLGldH{T4~tW zop`b*myIgu_*G+%2s(V)j{zca|8GG;?so@EN!DwFYlpHzLX=wzgBpMtg4)T*&hEU^ zocEz&S}A^EoOoxzuBNGtO>X@4D_;E4ipxVKB_(v03=`NYb#-;Am21nVp0>6EfMZ%) zTbFTO^$P@@pI`pKao!#(xb6;?uWU8v>6zb;!aI7UMz=TxJ>Y>iBmZ+9r$m5l&*0Gz z0p3NL|4iTMzwK7J(V2i&o>RNWd20?Ei|=RuA3J;hL(}j7ajB6)4D|V=EM+-C8$$&L hnhgI}+i&BH#FL9R;_*bi4rqx&?x^diRj61#|6jHb3JL%K diff --git a/docs/images/chapters/pointcurves/43875f6ad588bfd04cdb65b591a62052.png b/docs/images/chapters/pointcurves/43875f6ad588bfd04cdb65b591a62052.png index ad8889fc822550f65c1a981beaae2b7e7e4f7c70..9aa4f5ef19211a8c3704a1f08dd6ba601322ab7e 100644 GIT binary patch literal 10685 zcmc(FWmKC@v@R`QDNv+EiWH$dYOmG$Z2x8LL)sT)7R*H z;GA{ptD9VsypeI~A?>U`yWMs;dM{$u1iXmEw6)3W>+6F&EV5;Ui@@g*C2$X;tPD83 zT#HUsc~@3=y+m_|{gv4F-dN!Komi&?W;P}kA#LDWdWFE3VKM)uaY$87%{g&->}?>s zuvrjNDJv%CLn@~!k(UvmI}h$S9p6Ryf0F%g75+~f$tuM#CHru+{-e9M=Ns(O|5v>K zQ}zGFToa7amtwCt(P#eIH3fs6QF9z@S^OQFlLbxQ*POHV%k;doUm-^KGg`~D(H!@; zvaOxdHR`3!?3lCoG+P__iWx#qfoj_$$GxaU-NAvJjGqkC`>3L=T{#|neD7lQj$tGf zn2#@GFsXZB8SZ2{xVFEGJu~wg%n+NNie+huM&I4*eAqSD_O;L&;AoW&YBQ%2)gv7@A~idVCp1yszC9hwP1Z zzS~l){t7BD=PpwX`n8?4%_S;Yd~;=JYI{GYDl03?FDSTti?DvBBRksnr?W^c1LB+U z{m8-h;o2O;swZa$<7E{VIyyh@;Ctgw(iLLty0VA4PHU9I!S{F*ZfoGv9YqD1BczHf z`^k`5tZr1Y&)#Rs(qv$PV&od^-NjfpLVX94ENv$eZ=HA1lY_;o?TI-2Ge9?!LG+c^e zPympzNJN9a9%r@e6+pJgMg^kF^%Ukxryc zXL6}OH2LN%dmEf(sM*%CP!{$!j>{Nh>*Rg=k= zUe8F!M-&OMv9Wddg|{*Q;3vr|7ooeq)Y8>261PR>qq z*T=nQXGcFgX3I^sHXPhn#p+SZ!+(y~9aWFw;Mvu6b*3L2*cV~@e?~P*3RSY7w(5HX zDUFQWoh*B8n3)zKbKIX|5fX2jnN{2EYkBXZJiHdTpWfYFnLqlrX*DNkE2b49;Z{u3 z#oZ9+Er-=3BmIN<_{VhNbQm1YLLrng^jo5Ck?>gzyUIsktVN?~l-0mskn04(j6Tvi}jJ(<>f5YLMi)G>VOLTSoZN6TebY7 z+FUx?)2-vxzF10#drUcxZLV3Xohp>dr3v5tIJ<+_;tY+ zi(WxaZ{U1S!EPqP{PwaSGaBEIlK(@+tbY7U#_V$O;G^4Yah&7J?QTtj%}sKi@3{Dy z2-bLzk!sjX`#rC+hsP_ATV81~@zkqsQmy$p0OxpRWu@WOMt{b96CtUzG$uP{yh zuboH)6A>UM78@J;&$DN%+uPEgKD`8f{LX^(et7>LGz=e(!y?pbGeRdW)=)!e~pW_!Jllo)X8NaF(D6OuqcjZWi^Lt;hhm)}8e7wJzDlwqJ zCnZ(dUy!_gf)3Y}dAF82l8Lq0o|U95ELn4b`g}DZZ`EL;3RwN--MjAz2}IR3HBb6R zMnJ4|0i2ry9A5LB`xzPP2V-YipLMt_M<_2w;lr~fP>#OA@ZW{+>J~Y%2tW)DH)ob7 z>8$6n?p@9y}Uoxw_Yc-$o#;cD{9#mKnvcy`7oq@>B-?FSo% zFNhB4|1=FIdzdb8i}Be*E4iK+ZtwK95HqLny1aWI@w$Ug*s~BBr&6OJFP~e!&KZ}o06CN_Hf@H5pLWVPiTFCqI)hs)W0U_=SLC}=m6dn+Ac4ngMh;$AvK>{@A+Z% z^;(~`9;t+z$L<-0zx^WGW$5kZ;Af9XXJlkqtO?m2PG_FV z?D5MYE(V7PNV?t&*4n<{<<%Na?W!i@YOMBwWG+Uo4_u&F(nYA4<>i+$_Qs2s>36-( z>DUaTuqTPWiG{PA=}KE){+7-_8o*F#cgZ`&Lx=ZjF^-9mA1#7k>uz z5OmSzKv#~(*iU?**WcuR&cwr{32kEP+U-IcF`A@!-RZ#Uayy``E6HRR%JMV%8|B2#YF z#kK^ZqJ#MI%q2Nt6ZFC7cyBhkE-C3OLoktnjO-upv)Bbz%cQ{m^$Z~_Vb5PvMW4R) z{{`@aSXm#P>Fq|&EjEQh7u`Dl#k8H<@Cornlc|wcTuf3_`|@B|>ctCBrOW{_X6ELQ zKmZR&HihWd-JQqI3@%8CMI{Ce2N%Dv~T0CZ9%U z6qhBR?65zB+Y^cI=hxa+8yM!PeYW2}kVk(LLD5@&vS~S2kp(~u3+(S&Tl?W`&WVBB zxplp*VGUrQLOsXdDoJ_Wj7Qm5p;4lGeE=p31IM#B8JL02{-U?FnOy9L(nfGFy;G+w z+bu+ED^+7ZslbDNGKjM34aBYZ z8m}+fmms4$0O*e5Zt#sR{Eq<$Wg3w?;4x^E)wGtdDfUML?~b2H#I#-Sk5pL7##Uzx z?N|-C*6JsWkKt{$s;4$=uLqJTIF8WaQzeFDeNj+=D3}d@o6~wNpo&r=q62{S_}nn^ zWM+D_r(Bw)KApskZ+{Gq77dPh86dK%txLc(z{$mB{LmRx%I_K$P-9kvjQYsg=B8p~ zVkTzz!!D#GIe7t@z=TD-*{D%I)bitvRNFmo5eJ7=gR#SZ#KSKj=rdY#-VXvvUKcUE z#=&2BR_?Y~YQF6;K`Uy1gQXq*Zjm@SaUBj9CQ{W&a&fZ9V6(A#yYIv-l)WaJOn*Z0 z>Ith(XSM04BGs?>%)3M5DqNjfM@k)BlDM`YtK}$d)~^JD1tb^NYE%Q4g0VK-M-B-LDODN z4$y-v{X2Q>urBpw6s`e0Vkmf6tSlN7jGJw#RtMp*a5^tI+P%G*KtNgZ2jxiF6WN=5 zsoEaM_5JivO5EaNKOORY9|S}*1iHFWi_>m>=SoteSL+q zQ{$UPN2`V$!FBxj9Sf~C%3wSIq+DO*4 zL#gObp8WVTB6|D=i(z?qwf%gpSl3a1%w+?{)h z?}*){Wj1AIZoZDJqVt+5&35XIvTDdBO#+hBIg+)=f}qrd5rms~Ixl}CBDqoD(JWBp z;5eAOtCo8=GkRJ6nI?_U0Q9|zuyk*}I@`+Ga>N326@mwnTQ6OvZS!~lS>By4={v7I z^5r@ptEAxluvicH-YLwfJg+jsNZK~b4yomJRk>ih6L}mc>nTVnmSm zf5Ip{jVO)c5&}I;=%-2wU|^8sN=e+^YtKS%i}7!POS=M^XfKOXtx>w;bik+@dkM(<1UpL$)vl$>90 zT_WfdqNwN!%KsjSuog2UyE$uPqWb&W6uZUrqQNhja2M-0UF5I@EG<2~6nQK}71q{iMc|az{$*l&>TOtl zJqB+Q>>gioO_eQDu;MQ4+DvV@&CFhws5OWTz5gh*)!OgPDA`pYNE5TV3 z7lOrZjhv3(s+ShRNYm6ZrbB1`b^WYB;th|L!`1|Yq%}D?79k-tVLFtfOzuQ&AzSq) zl(oF4n^AS}6QG^VAc>tZKc`eF!cikGYXUn_&UMd;l|eEUgqv{DwUOobZ+vX;>co}% z#|PZg2Kam;TeZkeFQ=xaUr|w`UIje|#opgPu15}Y1s2%wfKKLq^S$}kXpTg{tqIt~ zIusO?L&Hr}uT%sh$8jD2-WiSAO*E0ReFlLXEH$bWu6PA_w+m;(>+zlKRxn6O>k#bNC_M@x zC(~Y{(L^GsBWy}=ZZ5`YFX> zTLXQP!$@Z49z5d@s34Qgu8<2Gg-P{XO0nlA`Xw}x|V7aJ)?)}eI1=ngnTebJK)5QU@#3em!3_z|UZnhOq1IGSohit}qz3?@05Eh%7IjHu& zHcvbR@3O7ks0*X*YVC#9OsU7AHNa+dudL93!C=_IlI#W2Aq(Qh28Y8pHa7n3?a4bi zaiV?v_;GG-j_9gI#OLPE{(fn1RgXEb%<<9OjJn83y#!=L#kb^&!1OUSHHHm zKcbHUHem>RU6|Znpa5C+FTXKaSXr;nCY7IjkB%--{VC`9_!(P%aa+O0#${VE@GY>N z7tOv6LLvC7563=cG*5dOuE;7W;hEv%W)BPxGjMRk{{Ah2flDqigXB|KULP7BuBxh{ z1%tPj*I%3pm0`C#2Ss;kR_+cwfme)06h(%dEKZoZ*w5lIJ3HI3u>M_cg*?ZjbT_Hl zK^(RXT3598Br&E!!A9mk6_q+T1 zs+t<(u^%5>jyhjZiuzI-7#P^t*icho0FLrL9p;Om5p9zho&5HF!=heKKo+XRtH&%N z&NZu@f<&4bjCTd_WciK!I6(frxjV(tt^Dyz&;#7^UbpcBFf%^aTDa58P?R~@pW`^ zL3XyKEA^~Yp;^=J`oni279KiugqWSBvaD>Ao6>%?=)<4$Sp)OQe8tAsM^W~+A))E% z2jxa8Uo_V*gsT_NA3VSwC4{1)qIY+9)b4f&7>b(TNYL|a=eeCHg|H_#8ZR&JKNuJy z7arQNq;EZ#V+Ena4xXK@0{xbjb6)O37aV}=`5}-R$91WmNQ&n>51GNZ6fk>@lM%?F zP%w)}HqFH#dQD=GeOTJBO=$+YoxEI+-as-^B9Xm7AKW z{>7ppnQiy;`_r*_ds@ErS`e_asfq2=+M0_dOd@s@8L+NqssQXoo715Lv`?EAt%?K$ z5`2b2y2*A|<&A)$LMvAPx?@qOU+Y0320p}&%5?4MJ;`qrRAG`h(S^80g^`e?%+C{F zy)G;LfSq&CUIbO_%u4}6!?yEf+Dk0WIYm>~9GCAKO|T1=#C;U;&X@zRsaj3>Th9Vp zcKpGLW?aJc72)@uHbKU+m;KHRoN)uDgG|uTmzJIwEdF%`;3PNEE)7&_u6Ej!^c9;~ z@HB#tdFFcjbCuj>L51h5{&AZoGAT1V-q9w$p57vIvy|A31c1`=I{3x+zS+*J)jcO}2sXXj6u-8wr6G4$rt{m}RHe+vtF;BL4MaL< zxD;Qg$*^33q&pOpXSS4+Wd5NltK#RhtSq2)vm)TRrG_Li3y0hNLblLoiUnS zRDdkULLIe9AZ+lWBz~ezP>ra-_dZcdqonv`;l_m0Xl*;B`{o=7XjO4sUBAfC#sL(V z-*|!MdkwF#=l`iF`uiBJ9Ouc;ylnQ`=}_8W%$?v5RE-|-0^%XXNVy`eD0WAOHw9*j zES+5)z0xQDq{kWXWrKC?=f-u z(Kf|&3;UOI0aB=}53aX$Es5XG z(nL_itEO!$O)XRv2O^m@Racg&M|&cHr15e|Xt&hF?MM8f)K zjF$Z*y%Hi%$WKzoXAv?&MB<4f&=j_)=qlB$Aa z;B^$dV~c@!j=x@aWL8Xu=cMJ7Af83ZJ%nmwc- zHa1|Dr}K6F_-N&3>~sPJqcZkZ`*OR!MF>col-}J~d+X^HE!3C&XF`Vc;nT~OJ5C%* zO!7w@owEM_h%dpp1JLG6S2fI7q0-( zfH8A8XHqXOh4F#b+o?*(&Q7h2r$V^!!&`kb6JNc$tFlQ2GVxCLuHVblQjy=Xe7}Vf zP9OB_kaa~($E8sbh#hLz6`q|<0$##i$0hU!7)D`_11$}Ja* z;B63>N9RBKR&z1CAkMU7M+G{C4{jcfc6$6C1Zlh)2#Np(T2*&H8vbAT;iLkDud-px z7#%L1zen)QMy6~{Uf@1+&Q1MUhRHgjrLmY!OY1#`;A7*lW@^*W3s(T9{mn&?ef#t< z07@Ou^<$H)#7Y}tO){&d`kIWq7*N(#+xdG4hlFKJ(b#&jC$C|CFlu@Y0R-B{zE@v0 zoVUL)u75n_C}V$8s7C=(@E;Wk5OAJ4kUBK!wd#KsO+EaEhUQM+M*{`L3Huv_^=r|7 z56{*2fvnjLSdXnSge3!W{C*!fxx;u^}cstNU zOx#_v;x}6Sxu%j?QwasbiLwmF+TpEIgyS5xRgbeH8M6q*l1w7v4r}0c8 zapv+P7pRES17-dSR2!jZ9WM5c>me!~0)zHz5NgHG0Dzo!JmSw9W}X#Pq*nEK zB#yi$)X@pZdg8?i`es|@=2n_E7NOxYGdM70a%Ek?u{4`Z&1EifQH61STdFENDKgZw zVqMX3yxvS$o#m4c<9ilIRdPc}`TBci9S_a^d@ewYzAk%_D(|&o4SFZJOC6()% zgFeulK}JrZGmkxuCqNJJ=X#qK{Jhzf8ODh9s;kQN!CU{~!3*kh&45~V7YEEx?#!zs z=lSMy>++R=5S>n%m_kBFRj_mn8X+5CVdy z@u07X#0_ifO&+%d4Qtk`2Wuz4)J%GdjrvDM%q9v|N^kexU)U~6*Nw%Q$%oBDy+{PL5R& z47qQOR4FemDt(0y8|;Rg^$;?H!_PR;v=CmTDq9Wi^?nq9;D-?!VYwB@b@yf~jW%J& zZwdQ0z+sNy$tp*;snJIZbt1;5CVX!J9@fmGu2)dtWDtv& zTwB9kBmJRc!|4KUe+jVq`z7?+l`8P4UN6^xfkT)49%R4WV1sSfBvNlPOXikyV)FW| ztRcng>>ZcBZ)!3r8-OmJ7U;xcA`i>&?)W*KRKyQgz?|0{-zLpRUQ$yl*{BjIX89xu zi$3%LXOi=tJ1@(;z1@p+`0|I+dpqL6Pk>W&vO{X&zM+Afl3?>O@^5tMyyK-lUP=-Y z4ku%@SOmiHWR*WHZqNR<0DrBpiwhU;i_aotqSXGE?e`@3nwra)I2%qb+~#chCJUWG zEdmxV@9*w44z#_1QBS?OShV5ygO3K|gAKP7kd+SEnw;qMM0@Ay0`Yc)ps}XUTqO2$ zA2#Q-mandiAKKhwHE+&3#>ZnZmY2DYF3h&8jOhysNIgf2IDkFdvUR_dB}pM&sxOJP z5!;2Aq*RKdcvM>TrY|0H zr6+Cve7Fb-dUe(L!m6Q!EIpYeJ2s0(6t&pf(;fcV{lPmsKq3K8;7JL%0^|84kW?rI z1)to1dnSaeXQja*>r7tnjfbbo258q+*40B1iUDpm>r%@PC-Qa$KRHe#`F1#sgDEFo zV}4HE2@MK*Hk`&|ys%YOV&YSg zEC0Nwyq&2zLlHPh!x^&cGi9kl_SE9dZrcFs@czACm?Z7UDb{@)`*;93kJYC#pj4{O z&dgHw@_pMM2`A2)0!8C(`*xKwW9c4TDme+Qlm`l)mzpE+&D=kRRsa!N&8m|kVD$9X ze#x+J;-{7XKy1O^`3HtRVdvt?UH=_g8X3Aeb%Elan;YYO_kCLsM(@&tL-56T+OBHq zXMWz%x~9RgZh;av8Gp>qoEGo<&z(_M?(lR$;tgK(0Ptf7PK0KGlDEqpze-gV4h?^d zD*~ocrv~|1Uv+TZRyiKHN#NAS%)+$Y<2b{&&( zN}xz2U-f+!MK@K;$3fp3G5xjFM4ur#8f-a(gZWEq_mj6EeTQF^>oB$Y!kx-n&nMWJW^ba&nTR7EW^2cJy?boJSRbe6D+UYaeB^*b`gHjwdPU$RnId%B1m@ z{f+-m>!yGiwGTn(YZT-G=u0r^^C{XSkG2lcVk0Q$id%<%(73?ni|J` z4~FOCl8@zSFcx5f8T{_)|00?cQ-wo%bu4I8BfRcLo74k;|LvJ{%x zE4ylH7AG=GOY-pzG&ilP*Rv=A`JAb}vH%~SuAW}#=)hY(K2lCj7eD{NbiUBg&{e1c zPJWeekOG0T)Azytj%O~p znL|Sdw%E?~)j2I?tOrq1?v0Iq4z_x@TzAI`?DSf6YJXUjK(&_K=AnPRJ_fGFvvWMz zs;^hSIL)ojqVatuB(;2(lV@_z82BkLFchd&BLD&QaWW~Gv_!N{{jgT%MJhl literal 11356 zcmb_?Wl)?!kTy|5f+V;T2<{qKG{I$qLvVL@x8M+*Ad3bIy12W$ySoHf7I)e2&E22R zuIheVZLwQS_4M@gbkDRrVe+yPXm1JMA|WB6NlA(-A|XA)`uF$dH86rxxLFVUyf%=L z5Jf`#`}}Unk4HlKh$JQYRoN}=XvNe^dFJlwbS>8v6H_uPn7T#ER^g-cuTs%)W%KMe zZ$mNcu*tLgbrqr|zAAqq525%drs4mdoZ@}BUo}!z;QNoSggqecI;a~jB_)MjOynBc zY)6sVs~j*)cO9TymTQ_hQ~OT8zL=O$qobo!xtoRcyRm0PnCfh)CtM7utE-3OP;3nV zL${INH(5UjhxWw+-?YM=qHnU^pkR>)Qc!{e--pNiUxSctjra-VVo!}ApMd`Z!`=&j zz@!#_BPN;rh8&w*pW=ULaPoyJppX4$me00~sc5h~Pd--kMcMCMn3w9bf|sG1xw*M9 zaT&7GCe(Bw5T#|!-)arcSD99Mnazz$^YQ~7&sgyAn(39>ho-RP$jFN3%H&!;TNf;p zlGJ8oh@W~G^E>xeDOfq_V;dkI;@j%1HjW?fK`y(fAvP(1 zxED#|&~w^Eq~_Du=;+Mz@op*=ViKC}AtMhD2QF*H#l_X*;7N3jq*78-$Hn~Wo+zO8 zL3_8QWkEmZ=ulu|{Y}8t_2Bk?sz9A10|ot$i{3{w&LBL7;qbsPx}lMsiy{@+(+-{s zDNp+<5qEi2Wp-XZjoe&8$2y+w_7plv!E6b^?Cfkb2|X7V4jx_-g|AL=vSMXr70CC3 zb}x!}W4Tbt>h_3RP^~qMADW1sKtE9^_HME5h5{-P)qS_oj%o6>v%8xzov*Jf(?|Ly ziUii+f`&{)^xSfKeWv|O>;3{(o5!Vc;n3=6l(_ZMWP!Ejy6*{pH= z!UfjbWW7bs2PbEmwx?$;LTeFW`a^>WC2Ote+=)y%S@Bo`e9i{_!;l+Sc!>ox+lmq# zp}Yvyd|)?xZ8=v)!w!PLeSA=l=c2hjiLY;OD#9we)l{yoO{YuLBc8K!aNIt&A&f_M zM>sLwtefNEiC-KI4V~|8Z66l96lVY0P78}|C^yGaO5w_Btm8iXY^A4Grqc~y=vrjb z#6Q_h<$>t=VfS=&{J;~BGFrI1>^@osMfa6Yipo1mzX!Or`rZ}8H| z-5r+35u%fkQGGOr!C+~boycUU2zK&;!K8FRGht0j?BdS9F4@=`__K_OQp@4Y7@W0OB5_Fvl^=m-h{-`Qm<++5cwww zhlepC0zS`Lbtc2X!P(KMtf@ctjO=mIebF>dedrGY0_vP*?;8jRERtwQDfA8W_yiwf zYMGfI4pVJ!^tL6VU`rBfr&|+h0|*?k2{TS7?;V^@*JG2TVMxh*V*)h}efjB8U6v7Y zebBRhzWFcAAwj3vNqF(QBU>$5A|OHnsWtnp@&5gB#r^p7ybCX5=QeyK#r zz7GmVl`k==cbJbBxlC%Uq~_N3o~LkbxQV`e)bZ68&W*T>!0$NqzQ-rU?;kSkV;^42 z>{iO3W**9xO8%8|Ih8+EE{xCvqgFO11gKqe(}4e&RKFZamLc;5~3U(`TiA-Z(ri zb15_#nwg4c`Ce+YB|*l9*vs5kX<)Cozb@8J?1{xHH@3U@tML5!^Zx$+kKEizT_G6f zYd8!A$?+nY!DygLOT4JFqlk`b_(XunN@K;zimUlbA(mqZn&HJ{o<))|n~{m)Ys20M zaxSifqeHvAWj)nB5- zv-7y0Q3hQmHl`)3ZL%@<_G0?1BW)m~+p&FC9XMmuXqkoHTJFheEiap?v`QKQqT3P{zA+!deN~Kw&dkiH(~@$Hj7DMC{ic3!QC9G&AJlX{Z++2@^5l=?> zf7mB`f-Zd@eyEy8CQwH|0V=!b!jKwj91c2Mk{oLkV_N&zs_q`sCi^7Pd9-9?V!Uwx z8*a^$v9GQ}*1Kw9FfxrM)6Ks7W%E@JWy^T2W$5pZ7a-p~bOx8B?AaD^S!ve&>nh=` z0rvcAX5HY=*6m9~O^An(gm<{_@o;;8YUg7DKZEY)6NFsbwG&nTVX zs9(RbIv+fa9+BGExOph|<;18yyUkLZXkb(E(pxLU2$K=~vb9(vB`N#)f}z}vFfw>w z*JprlOi1oJNoTXH=jRObfK@*zK91Kxl#C*t<`7Z$3ue6%#*-|WZ=${MI47NG6J>De zU#Jz@B0XGFf4{aJNXGL8O0QC&Qd2Zp_63{uqUE|kyXkX_j&HgO9i3C^=Wj=GV89Hd z9-jD6@sS*Nzw|}P@ei;6{+P|Ow()>N)aql4i9@fo(Z{5@;7R+!Q}gl8@o6#-^=Lt< zvl}Uc2ox8I!!Wtma;@fYi0du%s0{E+36!$GItp0VOBQGXe|^4gW{c>5ZK>U6cIUJL zHo3C4xP7gVXQ0VRsP0WD9U|yGZ(}SrwooZ5)v}efnv|0AnXUEgK&nP%AJR85W%uP|bmGSgizEYO#xi8sM9!U3 zX|d{$eA8GA2J5j2w(}e{_Lc;LEu8$4t$n#^8w!s5(MPPB($Z*`aHV2FLsAwh@*jV` zer3GBG0cGQ;;`n7Nf7i_l^`hho$m5R>g}V^FlB!0K7kwB9%8X-sP=YudhohJINWC$ z&)7L;nETAsRFg~8a}Za%c}dfy-DcY%;ZLGaSm7__HSV?Uu?2{jNN@q5pz<_UDVYE| zrS+EUFF6WLggH4v3VdRpEo3&%-OrQH56dVsc=KH7#5){1jh?*BX5JEV$~R%=T%DXr z2&(Yv}V>OE~tH^ zsBFCY&(IEMe{jB5)#{KBlZlfjxP>=~F4uP~L=pk>xpqmYGkwupJid}&BjFC7*cHkw zcg z&`uHS8moFE_F^^LeX)P!H=lxfzS#|f2bZ5Ih4r4E9DfV*)i~WN0d}4IU84C@nG6Ix z@&GqQSsTKzLK<_!A48dQG;+HkA|d?^p5KoYQAhc^jIM3Us{(dMW=gv$dY0dGP356P zxn$th$*0gA)KDQ|4`gy?|%s z{PVH2WZh&+ny%6G_t?2ERJe4z;`pm@rt-Q=+Kau!_w+0+PrM3rbXwt|fIA1Ux3Q|% z$@*5=-}26{pP5oQKj<)CRqPVe%Q7>g7HKBUkdp&eEx(IL&7kTR&P`lrbX$cf(G z!9FMNZrqME3tK66A$V=9Dxk;3ZM(Txgkw(h`7gH~TEDVKdt%aZvlF^@n#53wQn=RU zFHvc6)Fc~gDh_Ul1y0k3QWhFvWQ~6UVX$g(p3ht<>umJ zi*H#pc}wt>@j@(S9EL>qW}5vEclCWlN*GgT_{YmNc3%7%oBIk-@K!@TXGfVLR-94} z$kqDOT%2!yt=xM!`iPyg3jEO^)VFQ5#(Tp#pYOnbeyU@MX7u3fjEp5Yo8th(Hocv3Bs+WQpYEh5!|Cgi{80xeT|<#@17H zhY3{>+m*cecq5mqGjAzuA;*PoXOzElyZu@9)Af{7rnnpV3r)hC&fT3Gt6hWVJ9!Ags!-NO~tCP%!?qn1)~ z0yedfa>_J*?@XXKaidv1aFfE}%GEVHlkk1G6JAH@Ez^C$*<%s5PV(8IdjRv}< z%^Yc9Zb|}n?-cYI3E}YX3^~^CSXTsAw^P$y0bBa|J9m83Ccw;H)SLTa?Vy2u_=-PP zURrwcK}jwveDw0~xL)rgMA|(>I=)B+WWO!h5Mj*-76LPCAqff96tjZv7Nn{aM`Xd? zs*1@E|3MhASOX%#%K4kz+k4Tqy(xwgPcyyJm9u4 zqNILZq^!{NHh$aA)}@l`NxmYw#mNv)kaU;pA9av)!rf&IM;asoqsDAT~st<`p@I5bBkPMFqLE-uNH$9 zFAFa1owng4sZ<_oL_2%Xs;aKmdpngxW)-*?ybFe#U9(+|4X>5cU`JAN zQVLnpU93l}1A}p-)Ncn7CNzQ^>LyfqxxjOJ{shj`E6|dY{5_W|RT36yd~vpO0bfZy zoGUBVuG|Fkfh()fe5#lnclE48R#u5F_6bGzWLM^Eo|w^%>$6l9aG5leH^2ER#S1A7 zU?i01AfjtUkShh))5>>UIGVbrP(sgWRC=4fzyq}-QE8WmDBv4Tm; zG>RTT&8%SB1!_4=gpoM*G~5b>vPiC*ve_d;e+(`^FGsA3X=uEDAwJg8)AJ)?ZMbuF zHN*GmHY7(PdUIprtGxVsRaI3+jp|T5U9VUH&#RNpcPzAb)=^pAG-d+>-WSJJm6f4U zQOu`28qh@RrCR&D3v3%fF9`ib5rr4A=F!4RS`Fn}EvTji&R7 z^i-#UGX>M}blgHT-~-iWY9YzVmXzYYW9y^>HwVk;cY>Nko}QkZgg}^^RiarJ#jM>} z?u+nwx;tdq9Zi=pGo#Iy%lxB|BW?yR?+H}Rr&=Q6{7mRZD-Sz5D%1=Q)*<0_`fbn? z&S^dwc)jWwJw7h4im$G#K823u>G`NAL7T1L72M39%R!`iO!d7u%ja6v>2UVwM_-9< zTgI3W;^7BI=fO;|`D!x=$nAKY56UfO=>LEr&)a{|DHW(!QM=lBb& zVX%*NF`!&GiiHMO^GUX!=^L8!Bo##0t&wC6*=qSy0%!b@;sL3tM5U#r^-WD2J`Zly z8jlVGQ%b%=@<3??27~pFj*90@@UzNFW!S#0=4J9-<{cQ07pmaiAPb)P+l%}y#O@>? z=i=}D2TcYBVk>Kuw{I!f*(=Apvd{O%OfUXUba!{N8up_8kxhSVCnWi$ZEw2G*EgW0 zG>-iU5|*CsvnEQrIT>l^^)8KJ|6nBoI)}kFi15ufxW7Ig<2!B^(bM~gwACAF*d6+T z5wI2DboCsMi|;yxX>@0bR3JjWf~%{mau;5PIWy)9zW0o6eQE7Iy|MaSjT7zA#mCHh z5BG}OAr^wkCT=cq6wam%CXh$06cxC_T#={{0`faDO{Gl1MxHi zLqj4t6XSKZx+$NnrInSj&CSjId}`TjmU#eMJC065?7af25V*KNbj37t9{H_fCn7%K z@e}!3c?8UgSjIlLWgQg`UulXU2E~<>FapAaRaH4c4!@@WK9x^H>vbBJ7yd2WO;7dt zbtx^gk;`PIv5ZEo)ouH^#&g7OTxgw6ur~`=X5D%MD1w5DsL% zPrRnai~5^8uS`~32H#gY9y>fb5kC^7_zvwGiWY_fSTuZO#v zYTHTuI^H2P;RfzUghuvJcn%@o0?@1jUb z9(TM5;ZSDlwAK3sgfpARWtosr#i7guo<$W(uKQD>&|Q8$y0P~Iu9ur)MarDVt*q25 zFL2vO`;b#Z-+Xz@$oVtVSK5Ea;pJp#XF^nW=FZA3d_*5<(0X!hKn}e;PC*BU5&kw+X&n^BMkfdTH)cG zdy7OgX4_um)%g|hEc-&Hs_D{%Fc#0+Qd=5m>%S{%tbpB)bJrbu8(z|=COBI`*-Kzx#w3y?B0)OvdMqvw7a;4CxHF(F&>HA3cBrSv^1s6Z7(zqX0EB}ayGJihpS z^~ux#VlpH`?%spWR|UGdN77(n7dVN<{iWCDUg{uBw0O$l<38h3sl-i5G=Aj#Gh8rhRUzh@Rn3h?1i;Kk|= zdR~pj8KM}r5!nbX(JyOFm(iMaH&?4OZM(z#o!tur)<=tXd9vrb7T*oorZdv<9YeSI z8hnKkpd}~fQ{qR8gX!*pC^1Cu{VuaaFsWbX&ceE}g(=rEm^9sbpI^zePkXlG#t>n@ zjE%KO(zkC^u^AO|E#;qNsO2|v*0`d{Fu4+9SnxjQ2`Cckig;ayZykdqN(>VI5={!w zzu)EHFkixD)}7^+VL!M07!gKpLZgwMSirY;`#6b`lNpBnD?PRY4v2hrIXA=H^KqCO zTwW6-o4!6yPP>i!Jp19-+^Q6C$)xx88UbQOsT7;RU=g-C;y?!Zeiw@xsl}tCHxdy1 z&jl>ET;PL9hMLM?C)b8S~k80GKy0L9yNjb zG`74{yJtqG#uXsy;21KWf5|Ib8SCkZU#+$Nu?{31UrpVFCOS}(Yf57j+vGZhYr3W` zzX+lmtrX)Pg!09XwM3IfKieO^%9LzxKyV5fsF#0mrrpJ-xczLk3Kn{&vFQg29*|+_ z_(gX|Jq7RKKKU?MQCFE~Y5WM=PLLM!jTr`sB26kOssU}Pke>~XnGI}h|JL_;)SIF% z_xm*5OYQ68&;_0NMD%Bul;T#n?ZzKHBV)SyHx+uaVKPz#8Abqd*Lc?Ni306Nv4PC= zK&r=Y)^gr>+%hP!UO!#o+W}!%fF9Ptt;kkcTU=&-#ZV^o`|J3*>V2$e(@_DvA;)Hy z(aU+~E!$C=Ud&X;7q>}!;u5R4s7K%T3T zY@5sj>ZrFrSwnO-9T#awsxw&0bz{qwB+ej4AYJVqmj!nhQx-|tZ_m-N{7|rd+Z*EZ zskTxxBS$Z(an)cI8$0un{~(JZ-RYl&0%U~e=vYERZkH-RQO0eLq^6jSDDmnDEX86skkNcq|2eDSYfiJ@7zu+3Go`(`QlZ#zI8>7{Jlvq|)tGcUW@mRFvUsHb z&d?>^vv5TQe2OU5D@+QETaPQNjMMX+#Q!9Esdu3KxH>SNb!Yd1Dvpd|X$xb^V5@(e zaHZgIGv7sBeCMF&C*youX3ODhp?3HI<0r6}_vMeX9=wwQ8o+5UNYP`zTRJ7Yr|WF{ z^UY9eC{g(c>+;VN5)$gGe^`J;6`)dON}&Jz?kB?M<{3KK>vxX_r97Avd5&iHE}bme zn_$tG?iP54FBxgTFxYxdwB9PeS|fJJ@Sh|X$j-IeiuY}9pd`w^6EC&z8H!^-fPJBc z%4yEQpMHnVmlrLvxJ>>k#TR(bHR}C2K1f8S2TIaeY1PM6moNr1X%_)?y{^hm(VQQ~ zLl;A7%-X*#RB9L$#q@^=be9?7-3RfF1Xx{x2@DIB4pg&L^`#?E+Lwj0BbbO*KAN)MZPfcdZeu@eb`9|q9npdeTWN9yr8)dK$=#~g z;5%{Hxw_%=5IQ|(4}>}^iBPB;?Gw$jmu0>R zUeQF3or95%0DDI}JrlS3#k^#oujk)-kM8fkz7?%MNm*0eP^K4JKhMK1DqE7+&#`X zv5dc4vaP>@p)@qy=l8PR895yCIw^l>=5pZ|HlX#l zvyZ~((jG%@B$qL4#?3toUkCsinrj)aP~G65#AXDGY_5)v9_SN|>P%8xvgAqok9qSN ztFZ9SI(?D<>>kQ>4xr3Jw-7ZpBX{Mj?C-P~Ze`r?_i9cr$Yw==g=4FHAA5e+t?lm9 ztK96)vrV3F%W_M{r}pvFF)*lh_r_g~i_H;0)YDeTMtiK-f1_LS5_LVpO0~T3HtP~OUuEPAXU~P z-re&vkV#W~DoX@tN|pef_pd!-L%F{bk+J+njCodb=&;%gw=BLDc&FCqdJC~{O(RCJ6k zrd(*&xjBx1#AVb7+-iBvF2ik2A(y{(5J{BQ<`6V@cbVh*1*r@J-%sfW5$NdVT+YCD zRr!z#(p+-kB$&bvvCYoDF}kr6A0oL%6~6)x{ucX*OU&mIvC$PW)#~jTj6pUUrj*a) z3@6@)K&nwuDGhd{kxFSOPZq|6rXC9O-9W4{6^?rj{JexT$jGfg*+4Y_S*gy|c9Mg` z9crtapbIe%U#^eE)oya&qz*QWjm4yV{b!)Re>|ZYxr8DDq?^bfL~{^#&Y;@B{fJQO1{NF>}j>~m8Pes|EgC!IZLx&bUqD~ zh^x_e{%8imVLnyQW{XC6F>gPT)n}pJH#9V4c-!ngyjmY?TKs+QoJsS9abw|wp?6#I znOmi);%Hj2@AU2I=jCtTh5tmMi6ig_v^vG>&ANCvhx9W zp0W7~V(pr6$0Qb|j-FQtdX;JBnF6s=-LVUnHcGZn0dl}j1T(#>1oH-KH_el2DU@nJQ2sr_8j2QIF{1W82B>E zYQaAz0ViD|in2hHyA`pG6&D{j&0MjZmzP(uSzP8O$QonP(V-Rkb1SK=jhcchK?iZ| zYX+1JnH8`_I=lOo(cT)!Dr2)rNuiRk`ssFc7Ob}LwFLOTc|4O*R8pK~R$v2)fL7X# zzdk(PD?uGiRm*h8EL%LWJRf=2#3ZInmU;dIv9z{}Pq*|I2fx|uLjWGr=2e5E6)Fn3 z(-ie8023=sU^ut>iTe_bSn|kZWSY4~*`~L<+;~U&Gvbz}X>-f`?{>GJY39yUjb$NY zWH`|N4@0pl`yRa^^w@2;esy~eyE|ZTv05F*%Ms72siiLWaDKj2oh1@Ydc$R1@O;n~;z`a+@Ooay|KI zDwn=lcD}a&w3l(lvlOX#Ys@+<#`%#m1JbHpFi+e0N~u#>2C;wBY4coLWgI)X|wP(1N@rC9QY&*xcO-YRUN|m_VQ9 zeiM)Z0Uk<9Dk^&h=BU6_ZYeOBlAJCqRZ2=rR(ALQ!1`sJUFUz#6!gL+1|PM50lhOI zrvwj2Nc_ZgvYha@d4*sXNWUqdWfl{&-_P`|$HwYfu$+?wd8hoka8hLFkQx{uS1*1A zz3O-dz3{w%14u*%yY``;S7mQ;PR=fJjiDMBg}Rv@){F90+I&_50>?t%_AS2#p2o9Q zyp)x_aoYEnOJc5O*d#yOz%!=OMSJ&O;W7pEf&ic?Y59L3Na@%!7#;sVz|;R0IP|~9 zt^RM*J=)&_k_hM-1NhNiB_n?I-TzU<=>LsEz3{^UV6`6Yga6>S`ThS7KK4n7_hg&y UCo@h202fA*5|b4z71s0rU+9@r@Bjb+ diff --git a/docs/images/chapters/pointcurves/75f7b5b31e98444e13f17e5c3e5b7322.png b/docs/images/chapters/pointcurves/75f7b5b31e98444e13f17e5c3e5b7322.png index 58155c5c0031f3557212adfce19db7c53a50f70f..ab8b792712c5b2f78c2b828c5cd9489b90f9209e 100644 GIT binary patch literal 12654 zcmcJ$WmH^26DAA^52u{!m?(Sr8x53?IFYkN4 zZ+HLg{@4REoO5q?Rdsb$chyq`SV=(=1CMP4w0gHH94>XTtH-sjIs z@FKMIIUMia?f8t3qOQq}STz^#Y2F_=JkR38%W&0Kk$!(N6ICvQETQxO$7jQ&gG|@ERz4<*dHwKL}!ElkI4St3?;1s zud=h&q>?TwFr@Eh+Aoa<9ub z_LITI!3cp?Gb(rGD9E3T)<@%p!XMMpGA8#YzLrm)UHwX~WOjY^ zDjXG^lJa8!(YtpDH#e(0$FX5Oq9au-PM2O~Jv{{Spa@j74ja(@B$Y}Gsp&l+9sYX< zkxJ_F0tC+SX=utXWn~WA$>3jy^Y2#jk(JAJ=pRepqW)lURM80f9z?HEbsI)Ddvqfw=ydM$q1$tW z(ixWu)v(q~zEh*)6bR(|$T{Ncvzw6GpU>PP-gUHuTcI%<5e$9}aoK!> zrtcY2$!$kj*O)5#fos!#Ojr^6^mux8g(QO|oAE{b@lN1!-umT35G@g(%w#%uw&ij( z6FG0iNda-+8SsBfX#abp0#Q>&WV0MmP93qy6w1tX-Fy3iY>W&U>ujwXTyVTNkU=H8 zd$pNhDzZMzL4+ZQgk^RWPgQU<+c8~ja_b3Y+T*n9HrV^Ol>(cp{y8F(LQ5yoDQOhPeBPSi($TC>uuqsG*Qkms$UqGF+J85gO({sjSF;<0AM=FM;QTARsW zN7Y|%k2kuSbH!*!N7NzW%JA4A)QQ4^ld%dtFgVRA=9ApvYT9st7oXCAKO-!X$u8XvtWM=_t zU{H|#lgHoJunlk&anxmW(S2jSUy<{%*k`D^C_#f-E^uEH{-gjFRz+{t3mPWn$WP@!~^H zNGKsA@$DKfnwo+SFIEmhWnW_-elgL z?*wdkTGnaTZdvGVLf z9a2>pdY z-*z8!(?A3BT@*rOuOxPLFZFJpZzF;eYzY&+A-1^!f4F=QO2_MlVJ-zANMG=Eme^ z6DO4)-wN8gG1Vd{5RQemYUn0(EfL~CCzv^w!bAlzlMlTPzU#SV&_-+#U;KZY_5B0oRRiZ&ahjTf{!w%B5@>C;zkRERtEP(FFNe_#|L*s}UU~ zL1%pUC!&GB9z@b9(wWeGJvZ>?=l6_3#)5vEYR8Z~y3P+=-=@}k7#8cJ{iVWufYYrh zZA?h)@ph+>T92NPQuD^cbDX6+76*dOFzMfwE8g2%W3()=Z9m^V{QY$jQy)w^RoDL1 z*yGvvg3KSC(|Z0}KaD@5pFSzDD9Cw2h~A9VeB!0^_DL+6->N@zTbT>5f_&$ul}Ww8bwI&TBpGZoL$w7x0rv5w+# ztydh{lOv%1)_^@AKgH$!&@E0_!pXSUu>TP+DH(Un#s@B7f@(YqeE$G#<{Y0JgPMs> z&&O!)x`#UA<{YunES{_^3W<}3FtY8q<(2_KZw;7J1`vhzyb3}V3&R{QQmnJD@T($& zEAhOuT1hT$WIb+`jYY!nxLfu%HSIu+XYstZ>bJdHVI|=85j(T_>By(u*gPNZ#*R2_9)T(l7hTL z2Zqxej)(>B%koS;yo$ps3ENL)5eej+nr7cU`i>G$y7IM>@~Xg8h_hrWp*lkgum9nC zWbg+0Lo0U8N4wWQHk_O9j%*UhbD;doQGpOR28ro3ob{*M*SKy-d@}tC)p2UWL%zPdGI^D^@<>_x-u`^(*&d(~ytwZtuTc_cRb`5XxvdIZ zs2bF|B;`{pL-X-%|1L*WMSGKZ1|PZZpS8L)w$Wa{NuZE49Ki0WTuzkidTVe&g~x`v zxKq%`p2lT@WNIx8C)NCr_3D3Qf=lDXlcjW+9s0{)3fIjkUIQ`2meXCb0J@x)mqxLIefQeH>cB! z5OaKF0sLDc`V|mN(U<&4XWXVD4vNht^FLll8L)3HaiPWRsq6YSLexJl?w8#6iTsW(-j(g7o@PvQS$8hU4cJ!CP^>`by?t9b zK&Feg&2%d+QH`&mz*-aF6%XuDoS!zP`Fq^I*oB%-szZaezu3g{mE+%Qu z6(E{UoL`5BSUpQ;nr92q$mG)UUf$ofS#moc5F9LH2zt}$f#r2uQZ>K3m^;(tgA~nV z%|Pa+%ize+fhfYllt3Uoq_0k+kMtbaGVWXdK+s>cJ@S>q_4i~U&FE!kK~L=$wG+&*@BU#tqF;nV4b=IPSQ z#gK;7ax@+b0ZPyiA+qn}%&JLEc@29UW`Bsw9)mIJ032B>a6v0Rs(cGGh)l>U(zW)M zig+(-u&C9(>*=j=ndc~!RgRLq48X9MLfRN&HnE#mgQ*a#k+FW-1K`33Mn zMpk_Lyq$MyxG2N`>DJf;4bkfD``QeIwdzpS)a{4LT*fBOVp7z#h6ff!kD^g~r}EE^ z7rri6Qp6U`Sb^;pk_A6-DeXMnmvUs~`>*IikjtQhNrhoNTyL<=KSg@-Vlp!`d;O~% zQ)D-z(Tn(`3zBPwmIrl<7Y-=}Gi-11koxbEKQ*|PPMXqma&}TuBgbxjvzSwX$-E~A zX2;lxg;voEXFp7<70OogL*FF0?z1#lZK+J{(9?ZB{Ic}f0Mm46%b3dY@zE0JeRt){ z=^U}tB#~NX5X(z00;7Ac$=#2Pg_|*&0K1aeBPJV5RxeFt>N?V;kgK=tTFeWXRHsqE zJ9k3QJZ{Tod^yEJmw5eCznP1ffs&Ch5-YmN`;GMnt)YLYkui{5%*Oz)&;tfMTz#mu zSi}3HKO~EHu97TKN=5!L;1CTDV6Vsf%zbjblKYjtu+)JE*t?qf-tMY}5h4&U?_g>Fi9b=qv%WbfsFV zFF1+1RAiZ?dL4if6d`&|0^_o&JUsamLug*izarl=b_S$)xltl;((i< z@WanmLSQi1Wq17h^TUoX*HUk@=eZ4E7W?Vx$>;3sN*6-6wMC=#LqKCpM$cs4Qg^5P z5fKqI3=H3%9&Qg7o2JY*{O@2zTpxz!fE@jG0*Zhju+vn;f?kR7@y~52RgI0QCeX|E zn@gy{*D6E!g$8@%cH`IN^8 z_Q53j(qnl#@qHLLw_osR)@@mqTwmWXdBaD?VzL0ms*Wh3#6_d5L0dinBtPi)bnO3} zZs2ztf=R+xWesJTZ?GQ=BlDB8vZ8BkZ9Q0SNrz1pox_TChn?jc24UZ_cE_3`663^xc!bb1nXtY6Cx#s;Z%3=TGpR2Uw4ZA^E^bE8Q3@dDUA zy)aGpEl3E1QCZSQxP2zdHR$;=G3&HP0^nZ{lD3@*lxsw8{XV+UD8)^ZaqWXP5ouy3 z_~e5#33(B$o15D7Q*pA{CoF)JaNsi1^YUmZeJ9I1X6}!9O6CfXSRxM$3|Mdwy?XV^ zj$dC*?L9ja^R`|bv?=gYDJ~hlPj<-{Q2j6c0^roY7DwFJ%5_?2K|yE8Yy{aH-Ssqs z)?IyZ-lRBV?Z$TV8MLXP6xSjO{mhJin#y?s%$d)GwY-(u$N*`8PP;>a6% z9p17Tr;qbq0tx!D$G+D_wOVvO1Un8Z+#gHF`?H1x_7Lurh89Gns!A8r^HoF~SQ-a7 zeW(DxRZobE7Y!~dn##ti(JBXm(v8@2&ss3%udgF;0(NJ~fEU48P1CfjDXM+=I$U}- zIhsK}KVWXTA0bbf*(4TOAoBl22GQ4@li;(lqYe=Ux*wzm2A@G9PI8@Ola`370vSDFZc|cH+jKd9o8j7w z!&@k8S{v1PxiXftY;W4jPJ|v@Z?8BSf>uml9T?csY7a!2_cdqXXqQDs;RZ>)&PQDt zIUnC(B%OoYK#`tHa}y-|;mW1-xRh5)#?>|a6e(SEeAw1u8Z=UNJxyH*IiAi^3K|g9 z2H^5tn){x~-?Si7xw%OoBgLo6SD0tr!V<_)Z^^d(U^EHgY@G0_h!Mp*hqSy7!@^l9 zj+->q?UI3!?_Jf(a#Rv3xdmmF^Z2LM+M*$2(&&ZgM4Um_E3ekYV|1BXmye4SCbqvL zCo<(nO&FV6?|orIMRzI3mq={;x4q!-#3C&h{CMB-$-0du_rLK_P2{H2DXv6ImLMO= zVJL9|+$?5Mbx1v!p5IOl<+z?5bh{>czs;|TQ5(}E;d1)H5 z0i8JihBfE`?eP9cQam^0l<$01JlZq_%zz`|TWzw>o>XsS1SV+f#K-UEo!rI)E?&Ss zmzc^ZpwtHH*xR#jJt8uiEdP+@ef6@OzkrQeW$CZUBsOfF81Y>nmCs%&ZoXe0M;sy5 zrIveoP@U(Y+Z{C4H+T86*>t*P?CU4=_x!BU{=(v~{k;r+?X=G$hTix=2N1YWBFN@T zRJm6IGIK^3AxDe8;_(p)o0E@q|Hr7^-6A4_Fn{YKt&tliU-SoM+GDE8j=*pE%0P9H zvlD>^w%k%1lCPxJnEexodd<<6oC)K}xNTNX6*3R%Th81>{vbn95?X!NcCuyd{ohAK zpq+}D*ZlaptG_fS6A}5}31*Dx;uD^*8NfLqU-5t;B{YZ$uu||QoSUd=!XYI5^FB5@E-%YM#R-_mUn|N^ ztPs`xb#;*=jo}Z3{4|bhg1w&m|4wZ<7b8{GjJH#!BX!vl9sFBI{>63RX63?)j~iI1!vG-g z+eVEkhxcY2U2KX8jgh7V6MWYAFSpc@qtfcWYbfuh%DnaU96f%mYEhU>w|a&C@Q^y> z&xq%>9z;S?wJ5?_w&FPOABU7}!-or-9?mX=)g(>BQ{yJN14CU)R_6SU_CB5ZlRf@p zHA~`_0o}tFOLXljP^B*Z`o|@r#et@xCKcLjIb}ieUVOhOyW)ZXhKR(KIa+b`vON%^ zS0~qPT?cbD?(2rl|HYOZ>d)QG+0*5GYP~osHP28-XY|JBXHZN2GhyCnvc((rNzWPz zq(Y4fRsYhZgc*SxS0+V;7H!2l7V`A8$nED7V&LtIUrU6#l4DL^K>~t1Zhmdx0}r~k zo!uzTaAjro6@mBHX^L-ayVegeC{@Gf@OF4)flU^={Csp@0W764<`2{MWk+e{BYf7RnvRAWZoTz_qqyRH7qQBr`hDiTFIr{f$2FmFf$pSkk({^{FmF7 z?%KKrHTqy+7`QR9@Y;_fgVdc&Hf4e)f2MyBUK2rdWNye$2J|Ma!heXA$ILIYwePch-gU@3`0cBR%DP zpFD|>5Ls6@V(o8#4erd&cI*bUMYHSU zFXBCR7Wh3*rd8XwP=)bh_D|rQv%xN-JkI5XNLd5JHOgtYbSf;*Tnf?Va|QCn++QS( zR19$3FdX?gS;{{qSmeUF&Zo(|?P{;al7*`O%F}w9W)BJZ{G`pi`eBt89S;syC~Lgl zk3*7a@8nEr0;DF}`1(I&7TBBC+nz(zOF#dkagzc1Lpg_j^IcMJU_^gie8UrW5 zZpm<%z~k1G#_{k7HrjuFwza;Ig32=Qc)E@q^j*ZH_e6Uvi&7yKKtI5LsN?ND^zTaE zN5~Ajl@3ibeZ!hXt=2vzVMBO$U=ez5o}39!?;lSeVEt)ld_8_;Yt!HW9+@{I_A98Se8-Xj)og_f-Nl%5S|7Ed#@&QMxruTQ%1W{ z&T6Ej=mhNFySp(3ri(FG4YSQddY#8gd7)^ifv)>9=}M9kqW4!anv@lJuZ$m?c?;Oz z=ADAH_VpMSzFzj=H~%$ytphEmD0Z`bxlu=kDL}I+-P^3)Az|ZO)f99eK>~@Ocl9ln zE{lzh!#_EPewoZC1P|b{vKB_VUmof4RWtHw!y>xXn+4TPBR%WDn7s>K^^bVK3 z*H)fG!8oV={Ynrb$ILzL;r;0>!3*DW`KR+idJYn#a7K*t=wLV*p@mBO{p4oCc_*<#Lbs4E(zEbrMe(@W z#1mp-{&P&<5Xxl1+u?hAJeC z%^5UZUh48?y#{3TbRaiIHQkr4WP7WSNV*J3(r0e%Prwo!er>>%<0}3`FCD+-@}G{t zSwBSvP{gaZU3gWvBZq`Z+}{pVXsS-t>Ad4vXk+p9;r#tQuD+%UksP(^z#I#B?jYOjo$;fMu zKEKc(S16yP?GW|+Z`c}G&iS+?*=eli@@7colPrnBt^4%9O>XUXSb}O0H^<}U z!jx!WGu7y7oSmhC!b$3U9ZEtW#{m;)HddI;87ML2aDJ$`)zMd{Ts3EX;23Y zerUADM=He?bA(}I2-_`6xs3euIq5`0#fRxHr=RocxZ8AT=|-KNtr)|80q``+R)e!8 z0unNvZti{pO9=xrGhT+E?1U?~9zG$uPn|JUWh|-t$i^Ku8dYe-wcM@Sh!q@?;Nlc$ zV=3ZK?Db3p?k)zj;;RijlO-om(gf6srwsS?u?b^#e^gwuM?%dla{vW*@v<~vyq?G z-q&^&_9&|K?b-k0pjTeSHvX|R&y%2@dZJ^E4BzHmIDZAb<|{vvB$+RN%tk6a8K|g2 zii>IPF89M^3XS)sOFX>1l9#9{=oEn>>Gp7%*#BSD1rmK_Z_nI+u-&B=egmH?!U^{! z#CG=&Tvawhc=Me0qXv>uhHJmo3k(`RyD4BwDrRp#aoKt}_P*=*u4*k}UnFWXzTH$XKh*=xkTh>LMqHt_N`;r=bZq9ZG)p z_HEARtqRHQG&PY@jq=$Bdol{i7)n;wxHt+)r`2)tCjtK~yz1&&B98Dh`__RN-}{^0 z{Yt{in+=AQnh*?rw;xVN#OEo@I~5RDLViQI^JM0oKKXPWuRD&{KwAXzLY>JXj^bNN zvWs-Kbh%-#4Ge8eVsV>Ctt&HA$CHy>>nFVXtHb;m`9P1g&JcmS-TdypzJ$2A4#3fH zJ&-FTz1lVMO@tr2({Vr!IaO~iq=?4tL~ljz|GM_=FQK7E)*$1sJib(zNQM@FKYUAL|IeE*ZZQ9SHXa({IS>^6KB!F~SO&1@qm@h$Gxo&m%*LTL2HwKIpTvbav?$hmyR~lK<@X*G5uLvn5 zqdwj{rIhbq0x&RUQNehbzyD8xD*}4|+n!NGex#s*&X7D~Cv+;xM(g)4stWAV65(ET(iE=ndQ%%t=R3>#}~klXs}i5T=!Tkm-rOS%%}D{kUdWm6fG88 z1<6{^Q0HJUnG|#PuREis6ZDnixz^M|&vx(NHRw|U>e?aE`xE({YXxhXz{xAIO#wJhld-^RvEeb90nqy z-K>R?5wIA-iG`v!t?l&1l9}IKz%F1@VL8ac|7R&eaB#wxL zP5d@sek}H_u<&$iYq2_A?bA~#BOVPcxn^ab6^Ahg(ZI+M1wC`@#e>f%K3;qYG(lNK zG$0IF{_(+rm{9us#v_Y>H~xlHFud{ph~LknT!Y6;xv!-8leQ!h4y>s9w$cI!Dw}}- z3IP%EbSuR`rNK@Qa8)=dIXOyNT5t#&ezIV_!&>KuUIIQB8leA2$oG!LupKTwKHlz4 z!sR~H`FhEnef_r%X@zbel2)?^tFp2(m%o3EL}zyh0=J!9QExQt;%o>iVAt#duGRZ1 zwa}LRhnDu6*|#6wr=Jjcc?$}POgIQ=XlOgExHd{uz86t)AR_NJUojV+w|pY_3Kw2nSlxgplt^2+VJQP+-J+) z$4N^fHXKB0oYr`@3-vu#b(6q32bX7GqvZ?StFp-qi0hMWMlnqL)vo z;C9(n0xo(!JDFbNfu@5P4C^mqWViz>s^1jyrM@3Qc@T3;XreME+A1skCQVC@Tcz9hO z$pcZWl#EQLizlgK7$%8bleyE$YDdi~FYtOUYZ&lSxbWy`)1A@Gi`|K!Bzn#Ggap;A zdEnI6DAVfr!mag3yHi~4FR>WPUp6O{8Lz22eFA+qtuS^0fed_okMA)g{Mhf_b(q8o zN<2S3eyy{@^}aqb1+G%SHNvKV!x5_I&T2DTRpU$yjf{!e25#_=LPDfiSXiV!Cn7l) zCrGQnBg3=J0ZK~BADNjWSBDE<8y(2e@EBGn3uQNVcFutiTl3yqMQ?6xy(1;ncjH9+8J#Frrg(UKWKAC-=5;~{>i7(dADr3l!aLR+IQ=#z zu7;N1Gx%>hvN)jPvJ(Jy=P|sJ3MVYywE5&|kGk^gD z1$zH-{y5UAP7tcR2IeNwn~SRgWs*kOb6RQ^}&=v7tM^t-iHN;&ZmtuhTvb9@Jd zq9S=07Newtx6Hb+!C)G0y=rhKtm&pi5GW%NSSdn}#>nX;I>i%*fZON(&zYrezL)(t z&Zums_~O4yz43>cv$%KM)Y<5cvYPH9f2n_mUKrvwP#Jx)GmHfmcqx?{Zate1$RZEO z64e69L4i^leWIz>;?2;M``$Fmq&FUH9na$EtO?0M^e}D!=5R`~#D49p~0!crhL}i0Q}-;8=_~`TE}%T%9uF0s3FF#)V(2kjN|t z>3*kO1h{sb(gCc~Rw|=*2!JRr1`WWVMWun6qJZ#)fNXHEKoIfwU@%SsAZ%QJ(U|ns zd!VkRR_`K$gDKY8J)jEvv2RK)wSc z+ufX$a>Tmep2600PoSt#04T}=8-#fPfbmM@4KD;0JGwO>lshP61C-!^ z1Smm)L4%$a5j6Tit{d6<%~kUDm(DsVEG_=1B6_ZB!OKtIg{-tC5#|886m15CV}dU& za#n7IXxhHQ%s;<`09{pYXCyice(J6uj+8O0K`6<8Ck0D$HY%wNA4M%%LvprYNfk2T cdzb!?HZa&SQ5GP4+2@b)7Ee?x21a}Coi@Uo7OK^e(cZUGMf(3`*?(QzZ7kBRQUENjP ze|J~Z7H?nm^z`)1wEVg^{IjAI8Zsd=3=9mKjI_8449pws*AF5*@CLVJvjO-AZ!9Mz z4)gN*m(yAl4+HZNMn+sj%`@Y8#luil{iP4;F-fH$)#O2EhBHW8jgo)>3#-hC%sh^O zf#XQd{8_KKnATq6&-a|S_lP=hu*yHbhEXK2AtN%4zhTO>Vo!g3roEKHET9}1;i>eW zPVqiRQxIOx@IJZeFViSZoOrW}nxrl{{i5d(ZH)mn&#oh{sF53t|Ii>zLbr={EwgdEk&HVqBob>mrr2B6e z{+rvDE$cRo5|Hu#mCgT`kmy>k`nmjy3#_`^lZ!yfsAdAL$gF>4XS)KDG1$MZrh&IW zqoks(^IgAGqG(Q4TRnKxRuU9X&hH;7St3pz|4l{4@iT@pCZ@Q!6!ugt9cFmAxZ8=m zv}i!VshlG-h5WMg#70*^QDJPAFaHBOoAlTija+3>zcsZ+3DxLW%a*&skegNg0@b}vS?zQl`pN} z`k3Hu?Imu$5JedZ?caG3nyZ+xhThy%YNA=x)bO75krA)9XGc;>X*HW5{F^Guj~kE< zk1TK)%eo{G%6MPD7M-lKl2Xsm0D}o)-oCs(F7Ul2Pyb_YHk8VKOPhEZ5wZ1C$k)@P zA43O34dLXlrK1qa3cK~ZUvt+j??XRb8QMK<->dh1xHTLhMDC5uRr2H!t89)=C-Y6% zClw^t$Y1B7kk8P&zP~E-d9Z7{yV88-B9w`Wv5;=}YjN#DqjEc+KlaN<=BcBy1Ert+{HfcI8i^qYBOXqK|leQtGpO;-=bPMl;NYJ~$qh zbBb&Z`pj6{6L5Z4YJW~^S!wpVf%e_mz4+Lx5}O*+y91SZq=_@Z0{4X0w|ud{JLd`yx2#!|NbPcKiN8hErkMttB&vX6f=G_acmfbft@f^ zt>_%kN##p-M|hpisoMSS>9Q+zB>0_DFTXck9uLh~Ssm%shGk?BV_{=?`uAhV7Z$1{ zgY-fWvH5Fs7#V%{$@%InMeaKS9~(`PP}=W8@D#sj&k@v}EGFI!B|#5cypHCZlaq3D z88bP;`BpVRJYm34K49xm-y<2#7G<;DY=mmxccF90$i_7>CeMI)Q!D2x%uc${*$=m3 z2yn1EpA#7hndNwtf@CO3g#9w0TYXbC7D~M_WMjZ1YeK@A{8$z$F>64&_l^15AG^!; zt@kBzZbT(py;TlNT$0!Sy4Lwmw+n@Q!QU%|;;@Y-5&fU8@H6?`j0fVVj~449sO6Hz zb0Pd#ot?CULzEYxWGR=6Hnhx5-xPnN1*;6)60q~6BuVPKa~*efE!>_ot0eDCh^KIL z1|eaOTi2zGu~f6AkEju%dLkgyd}NCHn(`r>!PQ?xA?^42!>QMCMRSAqfm+u7WUy?N zkOUTmJRvzH#d zUX@-`&eytL4^&CsRu)H}hmZSrC4V{>6P%j5&&i9k@6v}r~5}@`cU|?UH}In*47qOz0_t@l*7-@^zn(?paSv`vlX=X zyxNvI!P4@VQz89)HU#Z9BuE+(w}dTdNW`{Uq6&7U?d_NDeN)QiAX%tl1Sb5qREo}^ z?;VUGoUIf3*kDIQndKFdq>zSzhXeZ;nn4Vs)7{I)$C=1lQo;zVZ%ngtcRH0lW!T)^ z_nDWhT&tsZrJ1g>_#-22*cLCZ79$?k=B9PDO<4$sl=C%oa5T%TGli9yfnn>xZYk_| zv2ORJ@p5lWrVpD5@qj^)L`8LDlwUcDuqXMGlQw~NV~hmhXJQ^YT54Z zE^X~v>&1l7)0X)3`%7{*j)dghF@*I(1kz%!RgysXi8eF2!CIYFo$8=O;TP0Tza_h7 zm4r(Co*(Z%L`j+}W*s%$90$CmptwEgBTI#zbzL4z1lFzjh)w=B0*=ie@s#Plo3*YW zq_gv(@fwSfu8ucvU=d)|Uid#3usfAC#LbfXagGu2yVAZPL?iNhh}zp5HtH$Ue)-@x zmMezpwT=1q=?Mw_J*YnCE9u_ooAE{kqK7ta59)YU-xIyK-BGi@cJ)=acYBThc9IJ# z2ART<=9})fu^V3eJRgpAM4Wl;M4xEOI6iAxm3cAl9cgdo=M%8I(%R|P8akMR$iIDi z%!=($;`eZVytMiQT$5wnKif|ws;NG6klVXky=!Y_X~U2Hq zWj{8dJ4n|7@Z0gJi9NbA41T!O#Osvl39vun%xBijA7f7t-1xQ`!U#}Ny~XF zB;rBWo^$p^d*$`#a^kSug4Z7k_eP=9=)%Cs)5XO8tx0tHuV|zB)bKv)dJosO54R@zN^)CEMcGMB#L1w}ZR7UGw=sh75qL&bP^>FYVO;7j*mt;_vv3yJ zD^--GYu+(34(%hOyeB?(-boI?Sz3#eg!Pq2R=WQzUvv?eCHecPP;JC-JdD<`Na219iEyBT%pTdRbwa^ z4)!`}JowbOZ_e)>_vW~hGlO|f*&58d%^7^}byzdTV*v)+oEetLInia2fNW;};v4*p zUT$CadaJ{5C_I|V9W*Ar(va^{7Y(Gp&#`IF+E4ggU3Lwf;&g8=+{}^D)N4`HW)4O{ zJfSAQwd<<2vel`NPT4jEgxATM3HWOr+~&I>{PGQx^7*PHx-D4f$(F!RmPH5t0RrW{ zBB*TZ_Ix?1lOi?h>j*5PV=QY6yuh-DK1}pO1rBXS^CUiNBPlkhTcm^i&lym0{zYl} zSFbxNwLdUhTzwI0fAIsfs*{Bc3J#vJk0C9XOq)2I z$rV+S?IUXC{%3lJhkEa!ygIokJES9!oK^Wiwt{wAN+PI|8v$jokXfQ3@mN8_#+bpCyP`V7?=_L5IXQv;YZ?5LVShh zX6Ypb`FzZ~;2o`QW4fY??K160b+zuX({>hM{+`yZZtGw=)%6Ypkqw>ZFWtALP*r*! z4aO5@cP?vTVg0rpJbizt1fU4|6B3Z{GvXIPR6;5@YBcA@krvoJ@Rvo7b2NBAT<*8C zj;CO+f7Bm@pQ|Gx^}dNR7eN|Ax+Hc!ly>hJTTp1yrZ``Wv^)TE_yflR>1M%+=*lg}y z0cns=2teL9ogIN-)3E8p`5<}t<4z9KM0FJqGd_Q!lalaAQjG=g)ewtHG6nAqYW&@W z(L6Z0IpD&<2WMtw;XO`5#7Ij z?zdIss?_B8)U(?ypEiHtkUWYKx!8n<;DPVBeCxrCG+=I1kcXz~3TA`A7%(@^qf;i_&@W61H zJhc8U^~M{`>X|G{olfL|1D1B#wC4CO6cv83y1Ib5 zCs$8c-u>vSO$9Vez0%Fih`q8a7%&AmEi=|l)*MHA)#3*##DiDvqJ3aCm-9r>67 zeJ4_xw!=nRs3!8gT%Ip~Kez%+u8+QP*%efD{_8hmi|gckWCGSL-d8|1iB6UT2;%$% z9_arX1PgGGZF@Cp;Ug^tKc@2$9#D`An7eZI_~l141J9kq!eg7Tif~IyR)-HA0BObQ zWE6LU27)>Q-}cmXZI6>%E=-!;^f)DO<@(_OG7!t|94nyIrFD_ku66?i^Q@MOYJfJf z=7;47j{^gFZ!liJ9zkB9rpfGR)Cy^qJH z`N*epK(8lx@kw}CgGU?0b>SS8QcUZt2)JF;P^V+?z#vw9<<-Y{kq0o^LSbS;#88hr zJ`tHxbjnvZOfN6{=EC6CTiWefWGIc8?XHY~FqHbeBeZbgc-u>4O7r7aZi|ear*-Ep_d($1F|J zh>r#oPc4}hLLMK+%I3C~O7TN}(VJeicntsZTRQ+}9r}&XF~O0M)h{Z|$1>aQx)n$> za9vbm2kS(=9rx~U2|o;x(I%KbbHG!=$x*?;U+)qD^t*D~zq2o%!o!=7+0t5A(Ye!u zaR2s)E?vdRd|-@xNJW|a0UL2=U6(i=2%Sg{z7~D%_wAIDw2TZnHFa>K`+>pzMkv{$ znV!WiWpX~lO=NHYfO|bJUlHlH8H6hqi4{#jcajaNUG@~y+ebHhBJ6~I)?CC6^`wSR zv8Q%#mqrl_e!;03P`g+I#fRr7Ys0sh4dLN2YJEOL-KuPV-no4}bUO9+$2r1p9$2*)Js<96At5W7 z%yZ3-DaC3t7Br&y)HF2g#@&b-pQ1G~qGp3vhNU_FU6i?)gs(j|+D%Is&g z8E^G)a!z_}tEx|J2_*RPQ9{G(J+4~94Y6-ly;rOCb=Rl~bY_1OKV9q}UG! zN)5|Lk~h2krWiPCzA@2;_{Qv&y#xGy8PY73Ul77;Sar)@)np!n69 z6LS5YhaY=oWG7oys;1z7LjvL}zqsHoDd8&Tu(N#>iu~a3mKp7G47QrcYjVps^Z*uc zKQuB#`9}u3WT-eX>&>< zma#sYRh`eJssM@J6a!aEkfHDazg-!Cj!)Mi*p@_9W^lL*RGvBdC}xR!v{_Zo_KtY> zc)m<88+3HI?9_6XeD->J=o=wfbuj5&(p2k?C*kM1bvY?s<1Sz5{DwtISeQ}KLBAa$ zD|?86lpMX+bSVD|;oH2{d=svnS>aH*nBxrenYkoq+3=_8cA~%E$)z6=Tb5!GB?ooz zCx1_k(u?U>=d^I!iTwbe#d=`<4oK08RK$${NFU~2`3XB zwP;g@Lp_ZZWT4hB9c{iUnj5isk1a@N8jiIMqCliOT7gZozu4YBZ8lxZp*!z4iUe{hUr94DnK~lW4-1i8y}yusK|uc_o|7(kj5Tj zmXZRO-`ooCe2(tF020E`(RmgY$w7kqfHlwjF}pCYP>@fk4H3{u`^*ZnRX?4$9lyon%5XsX51N8NDuK#OA|OW`K_8 zBq3rQ-r;9Cf_E->+Dc>Eo++HI-_g;1bz*l>Wj6sgg1O8f4@2sM z(e|KtKp%9W2H-+sPrO=M(F0LqAMjru*|C`y6IL{2(PrYy)bSXewz>fOYxhj}XVcHf zc`tK@`*~*%2xuL%njLoQ{Pz@{>FFDsdQsD(y~`Q`j*B8JiGJqlx?;#SvS-CS3a2U- z)y&o*l$D-&JvzZe-mEe>QFp@Ol_tI`l%$wJrkEbRg0xpKrlP`&$XrAj%cjy`AdH91mTGLX}~Vx2@&Y&UPK!4@iHD@oNS4GaEh1i<$Ha9 zc!`RfH?rcv?HP$pooFlQHhS{9ac&Hi5!y9|s{#{~69cr{&Wph|JZfsuRfC)?_~}<) zR&|<__SXEGMfwbu;mVctidUL*cmupcWnP|VBIY$JlbWzLutZj6^PXmi85xtC65#Kg zcja|p;U@v`{R^2laM7Y<7l|%yflL^7y$9b2Y7CBef@DOM+V84ssC2U1%$<+&pH3h0 z(7iH#cx62P$L`9sY+a3cby)6DG~KI}*N#xk*yNv9Obu4af753eKt`mWJyF}~n>w8Mo4*~erj8_d zck$m5B2e~QwQRv9cKSBblI68aK>BYbJmD0rf`lUnK z&IfU`w}xN$=xMB#nbVug^`D9$!4i-)%^)U+pNT5+^l9oR*hKQ`9=qzg`ybDexjtWc611|V^WkPi zN-OmJ{sFZVARV##%>(~%aH#R3 z1HlWLq`U9A=BxH$BtyvQ2*8pCguaaiauGBW)*rpmWzk@0GwJ=R6J>O6`YHmYlBPs# zhy2nf2Vy4*u-%cC;>dyH8yFaD-Pc|Kb&cxnbrDlG0*fp2*{9Twd@%$=!8N0`A-Rsn zY!C2n-?ts~W6%6|Z*KtB20cA9Hz$!Flv`u*N0(ftRr{9Wb;W;Fax(?k)=)jkUbPCF zA(J^Q^;Sy7U`^N3>Ex%bG*aUv+AP%D@H6211<8Xfw!31c@-39$|JVnh(mV*rab#KO zk2I<|I$oLDR_f2Z_^r69Xj19rHMYmSGlLJ9p>Kw8-udjxudcPEX8J#he4hH5L0hst z-sl|H`EGxTBSFLHjqAUxNC06pGo`ho2(Wc0n78F}xw(tBLk~Az;3U2*ya2e#69As%a#)p5 zW`^)MuEX4{`L{##`+(Y;-e|G%ev$mQhttp&fmc2h$?68oLK=D^5|k`Mqz_x@_X4*c zm6rVw!l%|8rSUmRX9njC{1w#IafJcrrc<6o=KnZhU%4l3Ji3h>xz)F&s;Aew><3uj z+>!8Gx>idkAaXvL5JBS$%XN;1w$m63d?eJFb7Cprm~B zLGVlMioqPFc*Y6CLmE^e=&o3A*euHc_t2-O|$XnHFc+d5CG*_hv?jGI~}iVcb+ndahm%})L`^X zLlta8yxL+pU95~s$Pp&|d>c48D7)?KH3G}{Hw=@qxU8)9A0LSnhLH+BP&yz)lg~hO zXhS05lZM3peR0@LRQ@dK>^|I_TD0hl zyl~lx0u@LO!p5!p!#mG=S2fu&`-ClNHiBB@OQUWVmfv0vcr9ZujP`e9{>979F?xaD zuG&w3n(Zm7Bp=KI#a?k&4%^jX_JM&iStTilx;$=;7fAgK%dx+=gW5D`--Mew3ntw|W^a51klncwiXVDrj#CjdX#+S)3hz~mae7RB=x z$maB?I7}Fwl!})w{2w+AMe>l$QcEgOsfZ5xsi5rXwXtQ_?k%Y|lPl=ma0;b4oB4O9 zUdoy_!YPab+z&h^9oL1eyu7^q#X1H^v%A?y8fON-$7n}lGnteA9AYNFhxzGhD{#pK zFAruSyiXcKFW(`|yX>Muu(9@qzHpEUUZ9%ChYugw_6ocRbRnP`E{||ubS^g@$t%$S zob~-zT+SHs6o~8Wp*V!wrtQ2RV8RPbW29Y zcIbrt?nu}%XLigOnA-Kg^^Soj+xMDa1xU8VLf;efxrSAl^zmJzHIPrdn~jbxD1;tu z4{RS?UCJx&hDM^MiHpPh(Ea?k%VRZs)bOt%8}$5cY24uM))S(3me|L8mNy4Ah*(s{; z1C?0y+(Ns*{AW`1T`*9zTq9q@z*k94^GJs7%5$YJ(x_{H!q#U4uD1;eZ{C)q(_rYdQ z@n_35;r*a&md}~;iO=B`JSWTf^)-l|{<}?OE8bTFAmi!^MUPKQTU@`-fL`qFL%TD= zTL(;0J+IW=SG-xcuTMsbwQCTez!<(XDaoj&O(RNpbn|PVk?1{9h)OQd4bcBhLT>D{ zS+2Z;djdqdEOz^U>$czD$Sph%=~ znhJoNVGfqG-WTWn^HqU&F+ziy-q&YS7TK&;o8me*N7@%H;d@KhcfV!;)H?_`9eJW+ zteRe<&y1m?369ljL*GuH`^Jd;Om7-RPfk_?;p8+Yi*!RJxmnV9oG6|iZu-{N$P#5G zN;S$4T-});`7uH7lM>=OrJ6l1c1u(lGr-?iSz5)NEorx;qBif9^TUwPE4?b}WqUZw zb!4>4w*bs%;kPNPLD_AFKxyLjryCLO$%pco2D%c0gVJRJJ$zPH*4w+g#$`z3sG{CT zY?y-{aZI^x-$cM$`wu)n_I8JJ)hqRf)A$nKqTAwnanXd{{8Y$)aq~@mnZbwEF2{Lu z_>GMFF|0x~V&bR)@)UHBZA8I(`U9hvMYlIQF_AHwGh#Sj%F1_2;n8Cr%*6zeO(b&A zwPyQpYriTGQuuSyeE;(`iO?{1d=W+>Uk2LW9-FWzK9b$0V|*HE@u2XUvvZ2Os=iwe zhKMbxn7HLecBQd&DWGX5vi(X!!~Y!VORF7UwOk1RZq@?E!{gS`*!ya;m8H#4YppF~ zcyza>jIl|{>$_ilRGFK6e9$bcggBOEs2vUL66R7M-<{{e;atu6asA!yoB^*P7|g>K zNA%_L<6SS2%)vyL`PIV3q@%t$=rDt)DN^`6g-@cKKbl8re0EJ}XuNm{*F-erJYR9|TfpzD_OG zUpyTxl&w?;7q!2TlCiMdb6SbE+pp}$r^ybzC&}0}uj}%@{>{r2wKh3*$4L|VwkEYh zSvgW#!3;_ngW)sqORcr>yYRVemElg)EqGdAKf@YJWQ}x1nE|&O;m%}6bH28>ww6Qa z`PSqWagM-a0-BFb5m8EYwT@SslRZ|w5Wdo>{v3xsHswYTL`kmtzxe*KUy=3m6T+rZ zGzI#Te4st}Wjb}pje4W~v6MD-)~;?8@_Nz$WaL%s;XwLU78#kR)#zQiMuq7}G>QDn zwXQ4^-l7$UYh*O2#Lb4!^+?({nOVH2Z;9G;)ZD^lJw}+P#b1~}D%z1pIQw!^MU9Ps zE;jb^cp<(|-z)gF`*04Ixpmj->e20IB+$!Ji+WwU&|Giz6&-*stg|$&yBs{*U!M6H zG%HMdBME%2W;AVAniBiS{4>UY$m~~NQbK~`)72ci@4ai81_KbY2}(s1ms}A?^mL+e zFMk!yU5}L_B?o~*_V+DUeQzLa?Cb=>egbA@W&%%_Q^hmZA@517o4IMof$m4ypDMAg zH!D;tzTEhHMK~n9qW1?grbqJ~&ylDafHx2Sj{Kbw&p1(b^YJkZ9Wlqs5lk2bUndzD zQ^_JEORcZLFc{I)l}Ln|!h{DVq+_Jtly_pa7#!@MvgXy*mZ=yBBDOp{ReIRBT&%U} z8@X8)Tj~h3S}d7TS1K)K)R;sWq%G!{WZ;^&Pmi@QH3i!(a=h;2TAQ!eVR`-jY&nr! zGV@MpWg7((6I!eO7=ZR`H8_%g8WCpUa@rIJDQ5Je3qSt-`sQ5#l8 ztJJNJ9v+r|ettOBIn*TeITP{meQB>ByuUi+J(g^i{0?@aBrn&UIt{1U_mD~E7BeEJ zsdv~@`@&aLUEMF_6}LI5a=AB}J9Su6_Dj@)YfeY`momL_ziEyd7BPBrCW(Ul z)KBqR8!5G$lOPVqr5KgKa3IlvElIqLK^LQ|AJf48(wb$|6#@mIgC7YA-A4;GUjGV{ zJx^P2V5Aik%z#45jt)-}=2#S@oHPA6+j!3V;l6l-9mzU{_UDiAzdW`#pUz zHZgev#PM3?8bY`0?}VN%RZp4@nF9g?ZK7oq6rSeT<0~M|Y!0gesPONAMshSFF7QE| zcUhCi>1dnpL&J`b=gV0aIx*1jnC*8TezMXW9~UPoD~oJ1S3xZ(xMJ4{YeSNlkT88! z0<@j$mqX9%z8t2R+|35UnQ256Lbqo6eyNw1#m2_YuCE)~I6D4z8_yA$1p*`OH>EK# zF*9pxYxMC0Wqy8svv4SQ(^kW=lu|ldo10}$PENDUr)^8tbvoszKmu02b=tT`YLm`o zR~8-~J`1Q(*Tvbntl9mz8Zb7CizTWwHhOv)btZk$v+Ne5^OsNh;gr4Dhw|N??nFx1 z7=NDwoSlcRTqE97EZ5uXdM@U-O-`yX;>jr>`~mt(-;SO-xdMeVSy@?WY3Yw3(C#l2 z?Q)H1pl1FFuy{WPOo4h&-h`!fbiGcwshOFwva*P=vGK^ZyO|j^U>nH6;2%JRY3cY$ zmi4taeO{t)p))If?@X2c!@K}$VfUc{^V2#49Ndh6J9I9i31C=jyHGuA0x+u67X!sZ zxEOtWeJkgmMrOjnn6@Y4GSmnWXs%(1Wpx_jjt)TGGM!ajd*0&bu#ZsM zJcKesY-$h+f-I`|FWlNQO^#nEY3pg-uw@@AmA4=bA}pn9Qgg+B)0izsRkA;fIc-`Ne6WJ}7$(vIE}}u%+yE|nh817dDFKjcx*5e?d0J6tQoaj^ z;xCCJzEE|F-)iD}h!A0rg~W6|bMGTn$OMHuN@yMgfDn^`qQotNl$!|las(oFF5Z7n zF~Cy@WpfVFiBwSd*Q&wJHsg01jSEJQao8Km&hDTl#K_P84^?=>KMdhM&U(^HKV5CytA6iQVjfd?iD{1lb?S0z!0< zilb|B7nW8#=S+W90p;7}ASY@K(9fstzc~!q;w(T(V`w*E&2ovPY1?hU1cv^IyRB_R zdVNSk#Kxy-%0U_hes6(-4e+v{=Yvec*I;luK@V)7w1YG`;UG(lCB~U=~l27Iy_UDhd%BN$vkQLap6YrR(bCn%sfCR%AX7FfSXIg~WZFPXt=F`=<`>4&}AWnJpztBCufjAB6vw XBk-w0@P>dN;lRj9D2i8z8V39iWbj6C diff --git a/docs/images/chapters/pointcurves/eab6ea46fa93030e03ec0ef7deb571dc.png b/docs/images/chapters/pointcurves/eab6ea46fa93030e03ec0ef7deb571dc.png index b26b54e02a3825e4e4d1f8b33ce527e471e00583..4755cc1ae051a4e460d7d7706fd0d133d0fa0431 100644 GIT binary patch literal 7453 zcmchccTiN#n#P9@B0wn2)Hxp#T9^xbY?yyK;%zl?!lvHQX$YJ)yNY=dv(D~C#|H73fEBudU^ z;G>9*^the$#K#SXoj&Y{r^fQe$tT9cC)-nuTu`W+O(y9pQbGzg@^BkDsGNX$j52OC z9vNPz04p0SEf_DI6-W71h_5b~OtnSE| zNb7&|lKy2|G9^MJ(hpI26P@1xyV>u-VZP5e#eXuhjRS$ z=KnGq|FTi3K~z+fvW7*x*vn~Z z5;weDUnC#uvHg5e3Y>HpKP!WMx^jhO;?U$bTOlVL)aY+`6l6x;?HheC-_^*o1{C z)Y2bf95glRn48bq&J`B2QBqMoE%|8z7t|Q|+$^YGs(^uk!OLg#F$e;|!^0B`xe{x; z+>=zh_vp*ekRlix#CoAEI8)FLdwFHWbL}hDyW--dT-bc)<$il|8n>3|`0&@S+kHiE z^E$+>KPZIW9OWMliY`1U8#7J6Pl?Zw%#fU670gtWmaio5EAF>Hp9mh3E7rBy8LyaZ z@??iBqR!?q51K^vjE!TGWp8L}YHFOEoSrBs+>!aSN&)FmVhOgdml5ZOMaf}A$&KvA z4T*7c?)bLNbLR(z$Hn3Qnyx4C($dPLrx$?jnFpQC!9KToQJIuzcIbWR`YO86XJi!| zRjii5#?FpuX=#~G@72tB|Nea^S68To1=I5K^5WWBR7?!+K--&f_kH7>-Tf-jaS)%T zt4m8SdCeLY_vTvtZ)bdFQGqmO4X!UsO1S%ylzUu|mf_7KVf?oX@P}q7YHrOuIa^yc zeI6=4%TF=O%eJ$h0w|t7eR{qaK|)GTKfN6CW_&}OVxzxIeDpmC3_`dFS!ZXb;lTbv zd-6cQhPJNmU%~S1x|YgMJ0p}bZ?=PGTVW(L{FXh-z43>$-ovcY(o9H)K{+2EALND0 zL}f23@XX`pD!`K2&x?>^`)@o7F6*vHy!2u02n>W)P8Uh*O_U?IdT*&t(c=1gw3h63 z-Jk@|YyAAsn*f+-1{{JA=ZMV6pm7+?j9$2%Fg@>x37+{;z1DBnWK`r9YUhPVNJ0K} zVuHr~(0}|XIjbL_ z6@W{nj_Pb~%-oF0d&z*0GmO9{>I&`eS)6V577Or~RlOwX5*|*JlQSlj@9h;(FP&*_ zZnpWBDM%~gN}I$a89i|MQzt7k!TQ9y=gy^|zB$;U@C4ltJp=sh31sYwi;JBR_gV}f z-JdDV2{t1p?^;F2rLEu@Z+k8+Eg823i7Rq)zFkzt-mY3-GJQ^!M4532t)Q^WOtJWE zFT}ygX|p?B|D*`xkE|;qbdG(twA`^&>^`yg=gPlTIoH2sMS$hAx3|Ch@F8Ilmu6?; zSGaju>B>QWY_+2t@!Qbb8)KvcQFwTG_0DDDrd}}1aP+weoG1C1D^T>+bY~u(I3WxM zQ_qv4^YrwbY!8(UsPj*$ zmGJ3)3Re7PCbQ|w&+keW9KYi`=A@;b^o-7Uqp+%o@MsLHg!3w4ASfASLl}h}hoEwD z;bSct(o|_2_0xKu>$faPrpeumUaIVh2~WVF{q|;KXByq@z?`QHu$)1~u4+i~!Z=ID ze7p>cMy!9n*C?&1sK3bLb`GTOj?8%j!gYNx8`0Dx7IeAW*yO#TiS#war<&J!zzlBZ zsdIHt*4POKp1S^;tihDMz4DIVk*4Eme4O~}18$!HYX%twMa0@b26`^Vd^WX_FiufC z9r$6=^otH~_eY5qOSy5)U&Y16lWoD0^b8DU78x7*J+_%C13LF8&j;N@Fx}nVgKn?S zzB@f~s~pxQi?EU7`!YC4RQ2gMzOAiowdabWzL8O5E0KJ$3oX}bt-TN(m&TjO)m3}f z={me;&z_06^=of2M&mVq(kW83nXY@TtfnTfq7pkeBd9MC4eQXYN=Zq9kdTnL&b3g* zQ}c$a7S{%t&Fi=@MB|;%%SRKw1cPjFn-K^+{Y5NW)}{TN0UgwuNKO3NSs-NZ? zTV|?H2G=L(#l1Ml$;ngl@`#H@CfDGo>unOS_v&LQxC^!NWhNWlt;&Xf`{=t9QGpL~ z7Rn+-=Oi~bUd?b0;(YlwOy}w;?}M4ibYT1aEbStxpCmLU;{Gd!AYh=7z6O# zaqk{*zi2*J5>^e?u-v?vJR~G!Zd}GFw6zNO9&K;x$Qq`AdnWP9P<^F zm2t<*O@6LZ;#7E!ErHM4#Ix&9iZbU@^7HdU?mwdcN64Btd))yO4d%7GpCxp5E4bOT z6cA9}-Lx+Pjzf&AtE-P%ZTCt``=frm;hU(#iN+hz zySlm(2>5N!DCW`O+*D*(RuTOeR1alURq}=f50*R8*nbK-?Wqls+=A*crYV`3xM0t( zuC1A*-@(RS&USBmnVd`_B_$;QKX$rW;}5N|d%#4{9a(w@Oc_7wm|QbY3NKgF(WK33 zgl%+uyzu>d>=Mn~?E?cABM_<13ZUka(a<3DN;FM)Grrg#oSru5l`+zQ9*8KoHG4ok z)7D$b;Z_JFM_*DcCTCYlq6{fRJ+RFrt!Eb(r)$6ZO|ZSa{cDaAf$k6ivM1nz^2%S@ z-hhP|5}2jw!447r=CJVz=N&hgAy-mL%1BU|W}7^_=QTWy(XEJBWEeB}tr}p$j9>$P zR+L8&E9mIxl;lzF5dn}C(Y;5xIZ?$D7Z-rcNG*uf%84DD9meWH_xldg8y_aJg2j&x8x-Ga7kvDPzqYn^aB#5cg(`V|!u3dm z9>ZU}=g{w6UaQ>OFu9SNR^41rP1SJ<{r$HM-Ire5D{e|49DF30!XqN0Mgw(G+04xE z`?c;%em$H1k%SG+4|>t=oxXsGS6*PVhrkccMGr=ny%J(^!;&5Mues9zomsT{nOOSdWn>`k z>%CvR`g4Sfhlh^xz1e2W#lx#=wG2K5YwLo+ll}2+kGFN$ib$+9ourLxNY*!aZlN>p zto@VBuLSt8et0ZRAlK-}S)8%b$5L%G$noznvpA?m7b`QfcragP;aCt3m2kzPP8zEq z49NM#PL=1@4^Djp16II$m^j3)BHh6*K8k6a%8&wecSW1}K4&J>NWyK!kAgmHpU+0N z$MaUgqm|RqijM{d`7laF9@!3`kk1fEjW)DvDoxf{E{zR%oE+w`1EMJ}q63^RXoQfhRQtne6G`jsHcfW8Oo zql9^*B*+{FA(qUgBeKMlbdb;}94;lC#zs0wf6Lzw3N785@b^U@wX+O>M5&jmWTTo% z9;ydlwj8&~jQ{Pq(ns{^@aw>Neg@aHzLn!6)+58LVn?=&FQnzib*3fyb{)bY^P6SI zJQm$=$eZGr<%Ig!fh_gRN6&SUzCV8c^xXW;YToKcq*z1- z2wm>f;^PSW*I@?vaDeQ-M2hwcWt6&Rl&9)35~a1ZD$f<M1%zou15A$*tTb=;)^QWKQ?y;(;dobUN*Vi;wSYxG|3kOds~%7*YUS0}yt6 zTLdk3B`m9=#I3I}^woSnS9~wb-(R7qe%*o%y@0ciTCL6Y&!80yK!%m(jl76#vm@eQ zqZvn`ti|2k#N3&i*xumoFtNw8b8~NXKTMZj4u{@Q2nq_uBqzHgZ}OxA4~8W77yoSJ z_rJu%@z@^y;NaWU-Te>A28g6G1v&H#Sk9K zr!=})3JnepDgYFSe`*`w*QXeC*h}NTQ_+al4Eyc3%xPR2a{Bs|B5u=IiHex8vwqnm z#My2ifjmoo#)#7+G`>80R$Sb8Pa&sU|O3v(O7iG8Be?3w~3F}jL_iuTKg z2(S>qH5}RNV+v5=4XZ2@(H%m<&gKJ27(`#xeY&r0W<^qTcxX^l?=rEB#^kjOE3kgE z%X*)EeSHP!6HCj==pVtuqe zcz(HXn^e_)Lr+FV*5tFLi)<+~tOz%)h_0&Q{|gf{``Qx*LvX01NK50aQoPA~(BiX2 z4)%EjsL_6n0rbGFaQp6D|9N0Q-r3nX!(}jEmW6^#{Rsf^w)16zAi)0T$3L}RrKDJI z#Y*jdVR3k^I<^gmJd{o=p-WVZjEXXWQ7O;Qe7fl;`Omp^aNa9IU<-^@|1hzL7E!zkFb7e1u7Z>kWC$k+B=UvbklG>iu8 zZ9V@f!0;?xITZ`AQT?~^Qw1d@JD_S~5)&1If}{wsG12nu`*&>)E-u@?vZI6QT6*>s zL>G{$v9Yn~bbAc+^yM;`9cuij$cqbjPk(=Rf5}NdEgKsf8qol~U!7EiMgge)xCH_n zhlJU>BMjsG7jk%UV*>%u1+4qynfGHWuU+*mEZlZJmT%ae_|Zs_npO$JTk@t{)4@jI zSX*B=CGO6@Ip&sZ_)&ufQ(Z7sfMb_l8(G#!V3_LbkGeYj^r@7OOodD1^K38O zhYhi)f&Sp(94dgnSyI!|$jHfUPv*YAm9LO_@LE+pUj{XB8T9V)qO%E;dTJz)!^cMk zxS>7;XspZgSpP>C&-fPu+1yK zW4<6l2|?d{_L0c72V6NS$rN1L`*(Hab%%H=FuE9};dQ+M8j^*zosnHFikTdE&MaDy z_g4Qh+X@1zcc0=8%!jky!_!q zlUl6c%hUP*$!O(%0rk(%R8k`W!QTbX4W#qRKY0?SF(U|-K%kYRs|Y(VsepO`9!9gq zhB7UiMK=(pMH4$UcKyWuaJcQ(kK)}nhD=sA?CwZ&)o0H*0Uii~DM!b^pxG)No2?*t z1psi{{drWTRDe)wYO1M4GOXN>BpOd#S|!!v^p^?HK-M=`f6_DDWp}^iDc60)jF7uL z_E586K#S~7>kgpTbU;)tT<^hnHCc((>|KUo%p!d2u2M337a6f4c`ju2D%VdD7Xv7D@MZ&i;SZ%F9d-_KAg+5R- z$ru=19+%t1h+%`qQfXfz!IYur8tFK!sIE>3iH<2kJ0TBvd3!fniQ?81JhSUIX1l^_|9q%u&oTRLj%&ZZT;q(fyB90E0ipfwxRzO960M?ynK9$ii+4>U0nkE zjSoE3Tom8-^@tnhmV%hDH!`iD2AfFH(^IfdBP$x-u6U!Lww z3F9Ut^nKzS5ELuq-!R^d0-N~#@84dtpMo&fo-*SS6E{2*%^nmB-jf>2lOZz5zxyr3 zj%@i-X}YG^-kp%bS;x+fJ)Pew3COCadt0%-L`0dAFg~>ZylT-LSFTsG_(9*M?}2xU znDN}w#q;OSAz_?q=`k-PrtrEWO^^F!Z^%G#20lE_F&xqXxXq(#mDK@tDp4mPEktq? zcH8dK`YapRXETh%3JMByKVHUzJw49Z4~Z_FwT?EqaFE|dW4yE6i(DfCtbjjvzQxK}C~=fl;x~r-1vslzOfN74UhfBpR{Bu}^_d&miLZL{%047%r{OMFGq` zO8h&(HMBNyM3*0%p)~oNI|S6$Q?<{)S&kBbKX4x4dA4Wq5J&-#Q6Uh3W@o=JbQ*y# z6i-e;f!34dR_)A2RhH=5Iv;#yG2p@#mW~kDN*h>9;lKl)46xXeo;7b~wST^uJ%LNR z=3*KiCn}e3Or8MUT@!MBK|LWzOL#ufeiMa03TQK{wthL*Ui{__{lLc7&l0=qixW_~ zfx?U)aq0nTS<;7a==5x2X8Q{`&2pXlKDGYN9<^6y1d86T-+O8Dxxh_6zq~XX*C`tV zhX5U~9~hBRQ#;_%nN>liF8xiauEE)qc1h{jwn#+5R^xcu4mv3U*)lgTQcGev_+RH+ z|MEoVzaN7A+eY>hxtg`U#Kvre;~vDu=%X1yGdX+V&z!cZ0M73J43*Ee#?q-5}D9#FA2rNQojPEg>!4-2x)r4bt7Pe9xZm z%s+Gf^L=w>{`p~<9o~KJXP>y^x~@B1Rap)fiwX+@f#530ORGa5D8$G=3^eeBG=I4U zyrG%Bl#_{+^nY@RQFlvDj+XLd7tbqj;m)#dKfqLt0!XTjFl1@YBw{W+^l9YmTAIhhpj zbL>e81IvKG%Ue^Wj{!Ibp|zrP8*X?$>&Je+^Fe1Pw|>Wo=W=G{Q+7`8JIw<1ir%ej z=9bcKm!*1)#R<*Sl$ELJ$mv9%>I_|P3DPi2OXIge85kKTn57@zZx4nCn}0_|CmBo;47xra_&H^Fu~)ypv*5VUL_^G?+_658i~`vhO6%^7ASa}t025+8 zA;F>+@gzVCLIJO}wKcG#FryMVO-+)U+uPkou`hzqmQFTD&h}@j9}TrYGn6;ogl219 zbK4IgkzELPNy&TjzUPJwH%I;Q^70ZcF5F!oDT5*(@-y=DldI>G@H@^++t{$;prJ<5 z2#3B(=34?Av2t*bH8P^*bC`vcRgndu<5z_#T)$hT1UKcrrQ&}@%x5~XOoYQQf$ydOAiv$Y`%W1I%MdakS)W!9}9UkPOoep8* z=vb!Xgt+k<9UosdzPln{|CJatQ|*{lQ$wL&?eHK#c_#L4Ou?8<-q@Cd%1MXlQ{9sD zQ^e(vf13rj=|r(%OdqLLC7=0lrp#oiDJs@rij?x%M9=Lyh2`k0w{dPpN*6Y_hh(QA5W zhzBy}ujfk4#C$%!Se#;fgx-y`5GWG^Ufha#+4>!Vj8pth9G=WW8Ef8c zHk2yrc|PZX0tpNZ>=_)C)zTu9jid@mNuidOlUwmp&nv@9S`s?xUF||b;;(+LI=%6hw?TyHp2AP{X3#DoCP>y`|ZXOrbU!MQ^0S@Q}{az7LiS zUR+#Uz_YXDvPw&T&lm63o)OQDcvGp6jHfe9RG^DBWF&E*0H)Tzq3?ndd3U(W2f=!EJ$|)l`GS#Kbh2_9aAg z;p-wfW^+gI0aS#iM`iLPY!OTwGlb2M|MQysTImD=UP&nnS9G+$rm- z&2haJ2}16yS|vsy68WSHyYnB2AIJ!}<$oHX<7|t?#HEOo4GO^~U%oos?&<4$pMN8w z$NWkvUl!fGH1rWEem59WQ(6jT*RR4k- z=SqkKOr9RfYLL(MTvLQ3S8!FGDiW=?xAzGfTl;(?jF5~hwD)ckj*I2_&rVu{P@H@)3lDc&?)F{4Am-n{(xOrmP`XVm#Mf>BC6_8`1PE?48=QBhrc zbM;(*e^LHQ=Eonv!fmR59a%v71&ZdJF>dH=A?d)v&W>YkZH24jkn;Yr1igXX4n0B-^@*$_oqc^{jc(K(t8<|>6#F=tLN$l`RmHpAs$W409 zacP7|3bcwOeSJl_xw%#K^|uea_47J#$Cg}Pp_1ENQ?>_-BuQqpWXePk(9lGlZcnUC zSK87HZnA57$`yIy5FcShiyUE)aTvhuXR0zvOG`&wr^EUpLoK5;@0%yYOkm*ScSmqG z!DnX;EZ)3vzcQd*G|KBh&lkKU{Aem^!~}&B^nc})d-H}#?0V-p4?n+$7srHVhZ!(c zWBnqppb#jV^+Hw_ZEJg*&QnZE1}7BLiz#38L#^uuBR4nE_C!fouf9xG)Ly?HTREf77olhm_UZdOoWUY--a&Q`qeF3OBCIHs~U{ba}sUs_ta zb%{KKjZ*O_;;8jstqB#kiJIsFf7+`-xAPyZ_qpVLw=)@7QNd$XIqkP?d@X8|^}Z5g zaiJvp^}O?X5E@w++INGkL9yrnw&!DH0&IK-LCpdd)*Yc8+$#+{ji^4kT19>d?? zzh}**qq}>hsCwSTlyeTdJva{=#L)||CHDBV2`+UFjl-2_aZo3hW*nPkOia?TVSkMP zs5HdD#C!pAY80*LmXiT<+>5N?nmp~j2vf-u$Vje=oSGV;jg5`L9BDO3;@$xP0g%U! zAJen2bdMQ2!@znRlu|_~-wvhj9HoVVmODocAr?4c}$o8vQPFHOzr}M)EV9TkW z3t95!i$ZCQ`?D4t`9pCi??rUJkBK2*IGrwt(@VGFLJgQ{Fn)^!0;5t43=EBa%4PiK z2?VFACMG6nscJh#;6#;e?Cd)B{qJDxI;D_i#Ayj3Ie8dBaW!@N4pMttcrn*M?0+(z z*I_~q#=a{mVU9>;oy;|JgFMka_)U0MxP+9P+#H~>C+zGUzcZvzAe)<;NOY&fETyAE zLB_5V0P0ijmT^8;*5AQsD{K3ZO5LcZL6H28L+WH)0gq}jQ$5~$udy=rOG*T|GdR}f z@?;I;NLw0Bdh3_!rC@Qr!Miin!gfnmS+! zp^+K7PDV0c99UUN$_IJ<>e-Wm;(e$2NFnS#rPCUxYUQ45qF#@?#0-ZqUfp4YnkH}L zz@+<9&hY~WUifYd_O|bxA>k@3Z&Fecb<~U zx|K;8Zx@|EQqj$16<1bF9|YB6{zey|=6pV8uBbzR)fm3NqcY|kNi`) zR~pal_#i~|s`)J|OF`dkww0UvNY#dTq$4XMY3O2@05sAuyJv}a9HyMxes`>$x!V6N zPaH-BKy9$I1%*+vczL_%({o!ZhgR*ctiwkmS-ckNnD1}*>Z8AxQL2aEXFw=Y1o7nM zS+`8Kio!lx`!Dl^)qNHaArM+?^zuqq5A0ZKswb{DD(NmC65DoCiI-*!OiQCPg6;B# zxmM~ua}%Ojb*UZU|8U7>0e_eiq|O|oc@Iaemw@oE0Bf!093=m~fQD|wG&38=M8fda zXj~gQ8rWO5S|32}-R)%3xi6VPjvpq*wHy)lc!s8=|~5^(SxCXY*Xs zy2G!|7Vt7@Rr58G$n+3YhAeQyAP@gNX*fA?0|1on%a<=TE^Al-sg7;f8W&r){DzP+ z2KoDSgvv5>%V>zoTGclP`O-UCNRCPYr}1uV8kzX>=MMofF%oXp)Ci;qx&P>Qg-J4f zW#H~P)D>;#u?e;rwW_S`HUPCJllI4tACllFx8!?v_DMgl@IzmDrU0%d>2c+TLCDXo z?CoVh%>)6s3~~rflv3W%R#IUeH-E+$8qT4%_o?JAaAClNFuo|RAxG@rYUH9gBOGWr zUh|vOXyGOggdS~qOFyt(DpFZ~#>pw?=~+`yzZ6fwJZcO46nV414q8?F_68F4?ZxK= zZJiz3pQ&gV8AWw;NNanvKXV=k_AdI7ftf*7DJmZTi+CTTIe+Q+xznt*(ssfGN`Xv> zAltd9D=1(xGBT#Co-E|L^ZPf@PH<|w3Cy2aTX=MVax|3h{jD*_!M-zu?jd20RmgJg zP+#>l#5DpJ#U)&>d0~)*goK`hqqzSa<5jLIt0`(sY;0gRE$mm5Sh@8Gb8tuqgMdIy zf7N-lbB;ebhe69ydl(@R(R)B8dn}^psDJhJI892xHvfD-yJAsJakA=1*1&6T!yS|Y zRMiNfl$)iE2a~L>#F_6aC2X@cniiCA4%+eaii>6K?AWvaM4@`C-Rj>MuojOc9U}H8 z=$V<(kdy#YYY+u;J-X2fKsQcSBZha1ii%JG3$nAbP4?#-1^jLV#IF8k;qgYwX8GFW zPHbn4Y&tD>gyY#ay1KkXwNxradsJa_T@os{iJGp~r>3!Dh+PU+D1p587k`K;DJc_p ztcd`sJgjcM#VxlSkTf%66uVeNeaymQ-e0Vg^0X02bFb|$0l0-g3dTMF1hxcFdxE&X z7&!4A_nDj*yEA|T{}$r~8Moq2Nch<^TtJ4m$BWvNcx?|h#P29^ad9cQ%FXD+IBgR5Vr&JJhmeK_x2S3#76xkYB0vBFo(J^E zy)|5}Ckwc&Vvt}xK*Mi3$J}KhXU_VOiSzX7Q_r2U?j3t$keP1Q`I}Lk9Q%8!R8jWk zy3$fp_kLU6A#S0Gd-nBHeimkx^E0ADmgO%PooC7ayj$4tnf4c1+^t>+9k%K?U+~9= zfTOYiu;BFcv}b6j8-SwV(TM2iZf#@V5RF`A)A8lhnlW#_|1(EyIutbAmH*UD&52?A}B$vCdLug4h~%zQejZjw)@hu zvdbalMlC=dp_%4Zg4Y3{5dP^CQNvLm8(z1*o}Q|<_C!HIY_G+|Vh}zQsDP03)Yjv} zZn_zAjWqo#>UO!%l@kA3HnF=KR~>dT0L88|t&^{t(F4xsVfbl2N8ydHJTfygW4?Wh zL`BClIa=)nx62ifHevh^uK=6ZsekwE)2C1LvGN3zlv8Cz0}5kfV}q%E-bPxE8olzuT?$O$eC1Pj{A);ydJjeZ$t#iA#Q3cB}cHYhAg@*OxA&qXF06kqJ)fNCWbV^ME+a1~a z`uk}dzB}64Nt>Exly1)#mM;Rdnzy!heDNW5F$q&mm@gSu*kiBbQMAY(Q`GD-^KPM= zst(}D018_y6r(S0MO%KJFA8c_51q^iBG}0#K@0~5wg70%0>~Xv_tLYcE%Lg=s>}P`O?ak{%M!(Q&R)^6y?s{`cb< zzzO`?dT=jjG%*F+TXFB$goC$a0!OigycK(Ez^r*`M26E;j-CX5wZ2*>+`dl7~ zId)p1I(5oP{li4v+WIL{7JJhl-@35SJmEeq%eYa%exO}-VF9m*O&G9DE{C;za{O;b ze2VZF@NnJy!-Ul9fINVKg|+n>_hGz3cgY060@UrVYOeTo8-X)zdvJr1pP$&z^77cEq^;7^r68!Z zcNCziyMXsS;o#`{`KCX8V!K{6jo}Y~_eY9FMMW0?%tj>hJ1%#U8%s+_w4e^Xg`d!X znUpm&h^rhI@Di0mzrXw{b`Z(~gWss4SAJypLk?_NfeTO8xwC<64YeIFOjXn}Sl6CXE zMQkd4Rt|{`syB{d+5nr(;^pTz2O1lwXI)2Y{la#3qo%J`b7SEwdu3aDfJ>IKvataN z#iPJMV>hTFP~RL%i`-fEP!rX=#Q2`lST2hS^%*mze%F9729T{Hg&2dcU({ z_c=b^3~_S_rHaHm4~I(6qP7RmP8aD{$0R0V0-jc=SBZr{K>;>M<;9|W%4uhkYju6y z45UpA{|nPu``Beq878TYS73j;2F-pn^&b0O=`Vs}&i=g23%uv`h4(;PkCM^hBdkTe z8qEe@2L!ZZ^DP;W%@WT3W?^_w!avk8Klx51ZS_U0x%)@`3T3zdvob&Dt6$Gb4!bmvapXA0JOfJv7kCzu*(J2vMP6IHR~A){|clSRiv}Q z7TFA%G(!(CiOc@-YY=80y1G|eL7+Mvg;9xjb4!} z&%nj%THc6n5%{@Me6T7EfMtG84s)SS**zdwhiMkK7#9wBo=v?y9G0V7TAjkaIQu)H z`eQ8N>*rwSES$zBWp8gG2>r8XuQVrmWsufLWKsWa7xaWsOi&;@lVzyd#S0zpS1m-i zwiQxD;v>J`y)Bcf2YHb8K|45Z>37Knz)`uW44CQ<48p?vUOpxQeT?}f9k2zU>{#8E*yFB_ z7wLO$eDgT?Erq}IMcb$q{EyTN+u9_NJqFM9#J225Pu=l*}% zE(N~#%N*u}!j6F+HL=FS!pl2`&~tF$Kz2la{^UkVm-+emkN{9cQ9Sv12#_ZRhz1EM z=@LLFT|h5`zT45cIgIuXZe6@cPA<+7gruwI?-=)6aDyYG7Z9KTUif~%GHAfE9VB(= zo4(I}zMMSJRp6h4dnjyIvX94I_q|(NTQ9oB5yNYOk&%%-Jv|bZmN~bRf4;vgbK6wE z-nQSUzNLX}<>wv%)@%-t48SBHpSuW@u-tC5jT7suARBvwd?yvi-h}`5T#Yj-V4ZFP zNQo6-_m#E~?B6z(6k*W@cgHy({k1G0ayw4+o!{2NAQLHkBz{8?V$ZH-NJt#6!XuDMMUTx0&Dq+>_sBe z!N2tx|K0LGcJcmo4X_rXSb|y9cw}QTcyxv5|LZRPu@*2nJQ#E*3%k7wS+f7H*5@4^ Y{l5PhNm$`M&=7_w$S6w}OPaj@FG2T~WB>pF diff --git a/docs/images/chapters/pointvectors/b8285282b8a0ce28fc2cc0392e9b607a.png b/docs/images/chapters/pointvectors/b8285282b8a0ce28fc2cc0392e9b607a.png index 817d0c452b1d91f47c3f826489ecfe7699f1cae6..2594a68369c9a7f779740fba16be17665415340d 100644 GIT binary patch literal 12267 zcmai4bzGF)x*b3|r8{K=q(&MkX%GoPq+?Ji=^8pEhEh=h>5xw8&H<5<91L0-1O#dD z?)koRzjMz0=l(2w=bioTckgFE&wAEcF}m7nq(qEF5D0`+LtXg+1cF6{`9pvQuFw=O zHG+Te;P=#&A%8HxvfB!iAP{zlhVoqlMEcIGznOv88UEg=E0a|tRFkME{5hi_vT5DH zn)h9{t_i!vgJpc&m|vw+0ulmF0v;2APEhu@R;zsQiVNkQ!II+lJWpz)mvOHNRnpmE ziMX=~t{wMFQjv%V_oSzKJYDteuxRy8l{!eb6=ogrUG>l8vZ9X+C3}~awz$4tG*}>= zgP=}aGB%W9GPnFbKHhD@3I3}mGZrBEmu-9<%EbQ9gx*Qe?^3Zqqw{(XJDIo$%9)HOSsQOBY$gH+q+zr~ud`p>O= z&Db;Y`OozgkS_PJYp~JPU8nrPUBjBUGAD;YUNN?7>+1w+{d2aklw~bV zu--EXYd?P`Y;HQpAlM@lN5$g5fgkW7jlXxvt>N%W*mM5XLek+7?g-mnxJnKWf57#f zo!ORaJ3HPy{BdAf&D-10di(pCga%k|aC0w$6;zG6k6F(+Mlm8mK|#b`C!4SDEw^K5 zb`w*Uv9FuaEj~CelN%_>SxgL%tV#C3QzJ^20n`SaN&JY3wafdRs~xw)B+;NS+dS$hBjWN)J| zi<6rh*JpR`$?rAn&*g$Vtf&c7CntdfR#{|ovvmAuOLH@)j0`g?E9-2gY5g;X*0#24 zzu%jJiCHqflrR{qsGBmGwx$IP z9z6=Trbqbt7QdbN_U)Tvz)lR+%@-LNvpa6V!NHBq%_Z&0n(xh#dU|@-IXO8yg-ISf zcp%#Mc5tN;ECMGNS4-Q|(_bqkJK)FbR8$j%$8%I5+z&&;!*TlJ7{XyTeO7<2F752? zyV9jR`~qj&{EruQcjs4Nk=A>2{=3$*&0Z#!HLv_hK7cKhzJ|fi$Yod`1rC!pUNpLJp5&H zGJdE+$ijjZ1W8Pm6@2A;@oP}1WXHtB!~j)r;Q0C+C*;>J_mk7p>dR@6IpCTT1^1OK zEHXEf{nth^AR#_;zWANHYJ#7_*eM9W?4~O%6K7_OAf4@j=MV@kE^dm13o8UwsP+E+ z`@Mq$B~?{Ch^Oyx0?Qo(DQ39LWg9roy%C0GMxj+z!uk36oha1%Z}y3=4ET23 z9c~9*AYLc?n0E*7FR6gpV?ip+n#KiG*U&uvP;g{t>Ky5`Gi3%ooo~eXK_gN=f6h^= z4!ZO!&-%D-16G;-_Ty-aO5rOVjG?`qN}`s)tT%jW@Zm@*E5c3*rHs>LmLOs~l3iR( z*kIza__;@k$MNCNDf3f#57>ziyPJB1Oi=j!jNR-1cP<|5MLE*>ufal}w5& zN?EXLb#s^}$#gAUN3k{so*U}!y_R#PWT$j2mF1+xL^fso&nF+gm7ts#Sx$;Vs)+_GQW0zz1gz8!xD*>_%zSfzG_>75XCXXSxMKaZlg#4c()dSWa%w& z^7g_a3YyNka1lr7&J4$==WHHw@lq2>6oI<1 zF{{ZUJy&n<$c}J%L%(w6Y+;H7Y&A(D=IcBpy_%J8_Iy4Ejua}eheSo;U&9V{V9xsW z$0w*vzR|2wXH3_kdROu%N<&Dg?Ue%L_>xP{!Xkk~DY7WRS$gzv^02ko){iomoDV8nR8oT7BfNKX zG~FC;#NYRtSO3M!mpP0?WpZl=@_$Zs8m7W1WAaVQMnq)AWL-8>Gb7f+wLSc@%Pn&S zrKK6`>gpstR!KtIM^Arz)6!8q#2m0jzuzU=sNU&mS`7`2lW+4^k$QQmIhGwk(6j^l zp}&Sm(DGw{sqV=Kg$O_XND&bg#U&t6jz-x<7;*AJO=J*z+328{=41;$!ua@jNw3Y< z*G0G@vWimd+%C_&G7>dkgGhLpo*pObf3$FQd9L~?jv^=*o{NfSlBlf^+K5R^B+t#w z1+Q|q@J??D2==AEcqTu#R2~ozIpyV~{fTV0QnwUOVv#X;&1K*V3hWASIH$Y2yRNbE zfV+3Z9u*B0sivl;Fua!_98SV0O5m|Nbl1d$QCeEs=u_O@Fg_dPcIt~QQN-QPlasyc z(m2y5+cVWTJyGQSBBL@ky3CQ*RrV9mYxwwF{QPn6w+Sq38*qYxoM%2fVT$@hu+-kEYdTx?P4>~nq2pjO^_?l z$s)UX^Cq*jXY^EKgsFj6GgzpC!c*=*ZVG~`#{-niQtsD$etx$W6JONUAng1#&r+j` zQjV8HFR%CX^x#xiSF7K@uj222XKO!ZW9+5IBLgjIm}j7dOM!L<5ji|2U9Ol&Zfj(ibt#`!H;>5GetXByqO znAd-#9fJZBW?b_$r@o%{_iypWfF_!{x~!CKa*Fw}mjhe{>&ZoFl5U)MH@PEO!Rc4pl1+1p!G^7K>(IX}c_K9Ekr`TK&jtVK5bS9I|i_vX^_^8V%Vyklz} z?qcGrSCldVCsp4%OZ)m%)8&IR^U6-1BMG(Ug*Jw!rf3KW3Dxi2Q}Xbr893JQ)YsR~ zYp!#&CFxN>T*>t9%#jA1a_j0sCa2t9a2ywIj(*M#8k6PQ+VsVR z7U;2X+S;DzWX7FcU50imTA_%nKC`3Hwe|{{L{))z_AZYVY`E}K)5vJ3TqnSi($PhA zcPoPY<@yePY-nPFiY@4bLz(mCChVnp#LdhA%4N8X+s<>kkT?>Ru@?g&9%~`*?_s&P zEKKsU^Ahg;!VU|A5ND8TWvynfofD=Z)o&<*Z62SP*av{Hdte}DB0$~emSLHKx;nv! z4auvHbd3498P_0Ej2MM|N4=M z;6a+OH8v>VFVoUu^Yd?nFIQ)uu-M>}mvn@u z%^Os&8HmMRIIKV`H7qWUq(Yw~pQ)4|x;hE=pM@p<>sJ%AM%U0S1A>iizSx+i-(Z-! zI+X_xD35mM6G4koF~OmX`umefQij{t^b54^`uX`aq~=GsqBfB+O-J`8pGQUI<>X)$ zYG*7-)0FPb`q18Cj~6q1q&3uJIi6ZxE`TXXwxg`yJA$*eJo&%wFLyGzKz4ROesKBw zuH2FSetZDHKYsj}J;B0dO}|hXj4S>OvRxJ=Xgyk1k&@yMa)^Ne_1CXoT`x}DD)g(> z&yRRjn@2K-K?<>bi0XZC#+?DmTn1+fuhUmyu&oF#0`%nUh z^Ma)d$0O|qNd3jzw~+}6WEB(D^p*iRDSu?15TR33@Jrue4fgw1KC`CBOA~iebafT( z?(S~X?%x&@CldB9BR<-3>|Gy|^|7$WD+_x#95N-oTUWV%XpKY+CJXhKJS?T4q{Nn& zm!Dg))!vu;W9Ftz$+$W0$o~$j)Jm*S>g|k>o`h8=q_?-%>RY{&XR}Snd=QS@ImyYM zQiTXi$WUL4zYc`i+2IBVSVI;+s>{H`>l93-jT~%a8B+o7LBBU2mhwOWA^|lOv6kUO z!OR@j6^`$bLAmP%ngV%wHUNjh^YVnecaX{>L|EmSE5mS9xoF-|gyt3MQu5$w`BN zi~atpy1Erpgth9%VUTKAV%pdwYAG z;6Y8=&(VhECePC(T$UDlqU@ZUR2&^y-n|RzY4ts5)E?e8Qh`;!1$nu*SDE>aH%Wu@ zEC`D^Y4;x=^SEBkgE=mA{J9h=dopm%v@AEpXdE1dy~D#^0FaR7rVY7ub#Jyj5gOS~ z>~EQG<;#P2ed`UXm!gu=;_`Ah4nCP{29c18vU2CZK=GiBI5w85kEg;(7rufyi&|q}Wwjtdgq zOyl)z{Y(qv09f^OkcO?Zv$Ihv0`@>(pCfs&NjvHhXpQ#)GPN1al*NF+_LC*z*U8B? zMs_?eUSeIW$#?3=V{xfMzBS;~((B&BOyMioX166}5>^>{DJdzl!tOib;^HY%9?&ar zyUow-}jo`y}VQu6|u+1$3K7nKIA36%6pr#(H$4D zLbHIP3w>nyez7Xi$P<*0;p-P1G@Cc)pFqT8V?0ty4z*8IXkAJa?w`-0}VIssoESEC}%xMhinh!m}?*u8h> zhQ0Wp=ox}Cla=T%HKAz|O(T8oM$a%4*nP`y9G08QNlioJmVq1y_#;cfOk!a8bs^c3 zI_rYcdqzw`Vk_Wok`~m)YppNtcu=Ur>SS*bZ>J}H&c(9rY_n8dN2mX5og=A^I%(FN zi+&P>NrI@_B}-P^$)Rs3wLB5dn3??RI7(+XITk^C2gfkWx3alcgxYfXeu^*_3WfSO zFmP8>lZZH#cW>h|QgX$886(q={^Rb-jVmZKYuw(uuSO-%)? z)|xa+n=nbZ#2f+#1Ypnhml~M&wvkr4}B)x*JmN{y;C;|4in3Co4X=BadK zLid;Zxo+Kx-7+}*;I(B4zJq`N{{0u4?*aSOiHeoy5Olu&;R7{#v#HrVu=tPAkHEr|fWH~WK+KUg5j5*qExsO9a;?WxV z)8o$P#GSJ(9rCLi8!^Z8!E90FY<+J><@@)7FNuxXnzvMY*ZDUSf~Kd1UTVGvj9+A0 z8k1@~BkC;UJQLst3{LO3JYoWb&%)N0qw5ez;mb{HnM$C;q>sD{>MO0FS+sy@J-eWw zc3p-Rp85?WO(`XzuE7lu8X*QH^!;axQ6X7bZ!_ZPvG^^?dQn*Yi4(2Ysj7>(s2n$w z^WVSk`n$4$&QJy5wbEt!q@<+%?;fLzU}sO(9T!ti_Xk!H@}D}%(aXETQu^J@2Exmw#o|6I;jB$5Nn z0gCx-y^ptd=jIf@&_hxp>{qWsdX-V+Z|L8ksFSwPl?N4;?yEx#7%~zx-96wH>ni*k z1C00Q&vKNInbyh?OP-eBP~ShP)<{QI-}Uk$#l{Kk3dee>zQ3}F)!2A()mB)z6wJab z+K=`g96CDk4&Uu)abM8{G(^#$USds+2qhIDcuJ?);7Z(>BT^F$drM9rJ0AXTc!qh)1sFs2$6Tt7=^Pod2O3{&Ve7n@=mUQ!0 z`|bc}vt19@$DFz7oSvDyc*pp7uZ~CL=}>3A)3;^;eIM8L&MND*{Vx2PT3TlA)G~&j zKf9e1M^i=c#l*x;1}(3`BAcaWgh}2Cnj-<3F;kUaS6zeSy{&7o&Z`d*k#}DaiM8*X zYjg{jP=ju4Y@7goLHE%ke86d)o%w;5gko{ArrPehuvN&?-u#5Vp4O*NA)|8o>0gfx z)gcSN(U$!$IFtZ!X^UK;IS>7k5S_I} znXj189rGqySSM)xh3w2$%>A*j5b|X3t#zsaQEffnPs#od)i(X`?T;^!9+hg2T6m<4 z5mFwj>;eL$;PDfuNnG)*Z8U6OWDwbsW+G5EHP96;|Lpl>+)H94zS`z6jz$Qz?qLg> zHGbY%SH3~n)zcGRC-dYsLxi_Cg@Ivi>hqL$zuaF9j?GP8^DSJ1-hUTCOeeUo&=n4j z?GqGc&UnX=#Hd2_+Y2$M~NK=c(d-2BzRi#y<9Iq;78?UYCz~}tj@@4lUAJ3^6{;1IML#rEj&Q3Wck^1QO@Sa) zoPnev31WC2`}0_gitrY^TeTw}py*^l^KMXQ4F)G_YC>^|l|O#eH5(X-oMsTArkk&d z@0my+0V&08dj|Hs+3VgPI{o=x&XnJ|bwH6}G|9A)zx>Eds(TehZHO< zWcw0rcloA|U^XEjpQcHK-pqO|Ac!uM(#*`vG-`C^sCS-4g1WkSyo9#UH&ap?dS}d* zN$6`!2KM&%yFzjB5B(R}{uDI^MW%?=2tXr`?g0+Ic>i|L&70Xp>-jkw_OuKGL9put zA#A~-b9UVscWCGaHh4HVunKjuRHaBg>!qKXYLVhzPq@-z30pDk>_HXxs-V2Byy? zKQFoO3kO;dnCM+?Z8Ff1Q{Dashp(T#BBX*^O_v$Fy1D7euEK4jjKlF5Y`?UOTmB)E zxvbq^I_M3>?smptVMU5PI2UgMTC`3HerE{kj-W zN1il}*}nBeW^rjL3@}chvgb54rEYmf6Q-=Jxji#aP;tM&d7-e-WEoDsqjWyHzA;la9n`(V@8iw0>xY7B& zNx?Kt@A%0^SXdYZJ$*FL@jNn!V4O9#pwPCgJ^8B~db$qybNeIPuKz^gL@-K>IBc^X z|3=HcKY@i_EtT<~I7iDhE@>Tr)N{Hfr$}0R0eXi^M)H z>b^9btCDD8r~>SpQI*x*1~j99ypPYq$Lo^!-(4#l%cF@MA&Ket4th%M(j7qvAegGN zv*4>&uMo+WmSZGFxHZzli`!R{aM)gDFqUb9!Fd)8TAb12Q2CTVUu`Qof}sn3b=XR89@G3XpH za)F^CEr<--)Q7MQr@(<`ug>o0psGT(kcR^orw4i_CdjZbY(R|g!_wg6rXBgY-FRQV zbe24XYR4T~1tc&*AmpzxUI2}?36}a@QME>Eaa4Xo5Ev-|aqhG{|KV-bO?Gw&Q08XZ z1MgrSm2w}pQEd6Zh=7`UhLaoBt8qR2zu+HGd~l$ta(GCr%t>`qh2*J=OX80o zkGH=!Q9WoT`&5>yf*Iu3P6IOS$#9kq7xwU*e|Y&WFTP&LQ$4L(z}HBJ9M9p1JLliw zPH9+ zo-KZ{%|g*x281G!M6Z+QL37S%SV^Wo`!X@5<4cY0zv-adn{#wjPdaV@S^`kZ8@JON z+u8^LL6LXxj2Q0d7>jvh(Bd5L&Vyzt7yqxDe|&m#%kJ5;?W0yR!NSvA+P?oJheW#1 zF>{Va!6l`nfLNcChodz6UGifbz5ajlLmbjeUa*bU_8UNT*4Eah@Z7BbIy2Mxkf-;+ zr46^g&JnA*uLn->--MBpXXh4t+&3_$rR9!xmRvjl0NrzQtVc&jDS#eqZwX+?wG0YX z;KRknrjFHKFVLa2{+xX`MZ}g2G~Y!7&Ugq}9K&y|=+)JyGvHxeT{(lx;^@jAw;sHi zs}n90!?KWoG0x6N^zT=PVb=5*!KtgO3)K5%X_k09ylt+y1RxJrqx+Fzd^pZqe=D?cZBy@_(xZ=nV?7R(@#uEL+sx++?vnK*^!u_~h;x zM0!1gaQOH+AyrLMN=md%f5PH)xhaTT>BT27>vBs){#S*CH!&^PKfKUKE-+Hr@x5qN zs*LaLfPetkg-$GRb`$q106!72&iAjfV{gw34&q+bUl%_O50e57T@@$`KinC4M#Sc5 z0MkJch-t*ZL{$GO9Y(VMy82(V%M{`vXzLZvy8mJ6w{A}-rO;L)0rR4632H4Fko{&_ ze1y=&MgkUpjjIHt>zt~p7f-Q0A|7E{>Rt-Ymjp2}F@U>J1F06hVZ<^=jhTV?HC$W{ zVPOiejjMn;?McIQ*4=Np@4d|~Dk4x3?dZ!#h+rb>k>VK;_}FjVB6dVqJg=##(K}WV zv7*^#W?=y?p_#2nAYA|a`Lk@tNY(Se2M8@>X$k&gb(NEsccAE$ z@VO$;5{)KDD~0wEZts*XDd8l5>j+ckylZXE0bT>w;;+fsP!K&gYWC#4Eh3_4X*ud8 zO`AxEikMEOuweit0@UFbZ{DE1_%K&*+;Y5@iGWk1?J0=Gg0BKzzkQpVogK1ez|Xjb z`qBN|6141+fC>XwqMvUDG;`&h>oJjnkJ4O%l<TDkj)z1^+XJ8u5?B4m5d)6OcY> z)DAZaZSq*lEh!;d8_imiritzK1p>v{@|#Dkm=Cc)*Kyz3JeD{P*(J3;eB{_^mo(+lb1o7;t>*x(t;|f;fFWOdp{5F zNqR9j3wl4R5(5G($WW#%6UJc!+9hGHO@oRFm_JnNHjtmXB<2Ic*@7>wPfbk$9Yje@ z?M^^IJ}7v-m=6Faub;0Njr_mF`B3IqqSg<0sb2>NW1l}mc1;`h4s(0`{m$jGQ`X7+3ws2ET_J|duaNd{d=qKi|+^5~L4!iRLS zk+Xz3nVKG7`}$S*=jR;3Z1u%#Jr3;@Z?T8(C-L3YzwDz}&)lP7-N z+190}c|YU5sm8s-M$_2%iZ9yt&Nv4gWSOZ&eI`<&mmZ@?(Sg?4H7_!A%`>~_BjEtW2cQ3 zBO_=1$E{V`ETylj`=U?sZ;w{8kR=`@;B2E%T|=V_0KoXmJWd%{;F+i7ojaw@)^|O* ztUR~H8QfD0Km(AcmY7>np}l-iQBgraP97B%rPi?aR=^a`#l@w}qRnJq!9!$WJsKca zc2Gng6jK8SPY*y$Amc=vw|H~N%Udo7fnFdcB_%}=ZBx__#CA0`HGGgT+?IL?A)Oc| zs-%Q3;OcZO(C<%(FKfMtM=Q{jTF-yB7}Y!_1+waQn=&9?=e>WA3mzkC925{|`--Lt zbTf$4mE`L{NdXRznJ-^3N7KT>0-(9BTa~T3)h{5Oz{Dl&I9)c+=IR+?CfL98t>K}B zEHLtQA8t&*JXW4)%_ZqkyY$)BnC3@OB9U>0Zz$JT@-QfQA;%a`t;G3OIxE z3kq`E+A_90hr$LbhTU9UF^rDz(;<3D^P^9|mDM#r-!W*Tm_q?F>d< zTk!=oRUlq4Ss4bq4p~@TjRdGcR8-VXrW=gu(s3!iI-C{-Od!Yfg`l4V_NOvdO3H&b zpiqL1Tl%X|cKf#qLEGEg86WilEwhM74X9NhJG{)wO8m>`6-%s^$Vv*qa3!6gIAU|e zhO*ig7Odcw>(tcwUkh4WnSrh4S%(-LPexX@+I3O!*=#dSd3iaQr6cst@J6ZqL;;78 zkk0QxS%*m;kk-3LN68r&7$6~_#B)KRctF&cX>ehKWM^kX4pxV8AQ<=x*&DgD3(;jR zwu_}=Vj@;>nuug5kf<@)51))#_wx9zsFml9H=16vucr_zD=R}) z);%e}GuXD&_UGzcOGV{**Xa}*Edt8y5Qyk==6m`)alT6znJ)xdBPdM{aU5iC1~x_5Bs295L{kdRDw>?wBThdCl#E-o)&Kob)z z&BsDfUVvkveuEPDUwS4clJZEJFt%M;9$QA@YuMkLSK^hTGF6SgEJH%9drBy1ep6js z?wYP>*+>rqbL^+oz9&!SW?Fq|Z%|r(T=9M}xWoSP1b(op{`!qC<}Tn?T)*4Ic>Xy@ zKDT_h{@a0*lena(omsL|ImRe_nB5AgZ4C&ScKS5fhGjIbWs@`<@*Y7w`eDhQ4d(_0 zPo~Xeg&dl$W6ouFKTZzkEwCToLl&mn^V7CHe8QoE2`gd90~C4IP1;BcxJQH z?Jt&MpI6mv$%zh(RIupjJvXm^n4FOje{zCY<}SWjzsjtSarT1AUtN(`xj$jW#@X#2 ztENdvC?_CkjeT?LU}5*q!f|S!131)Ge!m?8j=tDJL`QGpg?wvp=@}fn_d9Uyw6}Rg zik9G7iO@<4>3z4EcOK7PF;rpv(hp<|Z5icr<+6-?M{QOrZU22M`qq49z2w=5)b!y& zo{D3{x`x!m3C8W^wMCo%fESpr0q6vqbDf@^06>fQN?Et1YaeQAT4XE0Xnb$Hc~w81 z_LB4*B5>RMn7zI(0S6B!6k5=KJj;afg<_?N#50f{#7uc*>XeP9+k`4WfE*0WXn4Tp zVX|BWTT20MYH0r4Lo4awo719(1{zXXpet%aOZ9(}Brp_jxi2mw$T|R71~(q<->R zSgtast&kWg@@rVyGCFoQNcB&OaP;=VZESqK2ICG7Jq?#Wzx;n}>flW&3<=)7dq;0> u+exVyEiTF&1ELMtuL(?y|Lt=>2~P4oF?;YH@(Z~4AsQ;$$`y)MVgCd5OL&?9 literal 12354 zcmZ|01z40@+crFel!G80N{DohG|~?*m-H6g4oicQN zYxdsn_w48W{(n0j9vknu?{%+r#d)3QX_&gI0wDnn0R#deR8oXLg+MS!(ZBIgH;E-$duZ;S?9=<-=-iU{{7?>pzxZ4rP>& zpn)>x=O7hjhKU1sJq@BJgd|}?EbgQ7^nyh$>KHIdcy_3=fR4MCi7BDKC7Jqk7L}zx z*S#*GrY7%eD#j+R0Y2B20Y^!1uY0Zat7(i2s9F9-Po5ZKDjW{ti1yNo3awX~AJoOe z74{=J(-OZtbbb5w>qoryY&WvOlf}>aDtL@=Y7Uk_Vr_L8V_>O&I6h+_xfb|oVTOq; zp79NGQdvCCd?c}~PV?s4n$Zt~Az3__g!r?BQDI@>*5+ovZ&;dt-<4-nA$8&=UFH?zBQ^kVT)?UYpJ<3eO zXS8DDE|O(`a^?Ai{^BI*L3dwJz|x6xsu0jY~^QrHzaj zFtM;4)`l3J-@g4ajcaLT%9`^c?52qC`fyrre( z{f?AL8{XK&1ZFG+gL@EuY}yAD3dO+BPikpt`S`e~pdcVVK3;w|0_3dNQOixrDyIBco>j8(jkfJG%*Nq!bjvAj`78eIwWVZriTRTYta2Kb)NNEV%&MiYjr{sIYT! z5zKv(oI8A;Wg;6I&23OeWigPlel5nxh;ev$=yv)$;asZT(%PC9r8LmgX!o-MpPZcB z-oc^yaC36Mn>8I}H_FP#N8H@pT>ABEOGgxKTw>y9=Q)0h0R|o()$C(A2~t^SE+HWb z7z}o4*n8fVTV4HPp@zuS)ip)ZTX!8sLP8Q78_UO*_SL($r>E3*M(o3%=YiR3H+ zIN39vShD)M78&J*-mZV@*3heH60B9PqZtodJG-AwlK2{d;bEBl{rzlgY-{NUQ=Mb0 zUL;v~T|mdf&)b)~ioI}h1$%`V~3kHne^{6@dkUH4>D}-d zq~E7r4V;>prm-?^WP!zi^erqbEX&1;tbo%{n4({+j-7oO~|y z>0wT5=QM78eZ5uxsg0G@M*8{-UET0yj2P)P&zDnEfpu>xL4^FKPm2npL-4vCTTgyy4&9s|IObYXnT^cW5;2RnmMwy_C_(Dt2 zj|T6D9v&V#J&|k4E}On9D|u=h0ypZSqi@NNQ>VLaRET&FF_XQ!m399>BXc&@Z@+Ytoga|VW%i_i9!DISmITW~vsNwp`^J3F*mG7+$|K$1_ zIR}f{T9)C1*!p@25>nE*xHyb5ox<{^cMG*%rw=-!??${RH_xmnG>b<^ zXth+*hU6N}^5^G`B58$hogeQ`t~-RCfjZCUbLFM={CPIW!1{V_;)q{mh)WNUtB$E@ z1{q*Q#b?ZqxD6ZbFf%i=adM8HmW6Hf^!5%hG6(`jo`Nowyx<1x7oqkf?8hbru8g_<{lyQ`(~t-#?BS1u z3=H1_PTZ+YoYX|$WN;|>`1ndtp+OUugEefm2(a13T=-4OdG z)3*q3TwS@&qbxIWajKm0b;@z5_;J{HaI}lwU3o9Un-nduMicQrUA~ixpm0bE`1Vck z>C>mNadCd!dX><#n2&2ApocTvzmJ{neMGIRr+1r%CcLc;en$%7wf7<-k(v)thZ^+1 z(O5S-LwIqLjXIa_T8_X^0H=hiq%6mJ*;5{k%uZc!Y#FMdy-qk(4 zl9F^UUc3M>(H9H~9s62x`@wm;#|nlAqtC?S(=j{`R-V*doozlicTAkCupG5N+<3OU zx@rn=Bu%I3L#c3Kq&gei!Xj^WC9lAZ#XB)u4XpGd&&%*y*WIkDD&(dXo<^7wC8_sj z0`xu^LEOiW!6_+sCe+mMhjHS0S@cVIH996IDH$0VyZZW;#mHsTdOk(qKR;f|XX4?x z1u9XM{Q~CXXX32)-4vwWI|)!TTifWdF)dK0=NDQ-z%GQ!vgoOxstq5E1>%$PSq$7U zv#@YVC14~-bcPMDf`bJZD8Hnnv)XC3N;QbMTBQ<6CEIAm;N;}=DKC#nSeOz_M(F!M ztx{beOLWrmuo^;=ra%OcXEQ`JH%4 zKGoDjVHUaEXWQt39Dfia0UFeJoubs*Kb`VlI)o1FB~HfRgCZ~F}lsFG1o zIHraxogC>E^AGv3JXe8&S0MhcI$k^qvyzi*`ub`_#u5%gukI&UU27|rhK2@l7<^4| zq&iD~bNfTI-Aq*+-P0=p^j+!=A83Kz0aisoKv1RH&URM>^5Z9FNkfkd%h}+k@USw} zzw5gC%a{EwV)8lVq&nYkoh*|U^TVJl&r~9F3?IA)5P*V5KNOUy?7X~?P3K5??T@H9 zVSWq8)G*_D7iVX_F9W`U&fuhj)6?nA`;g5p9EGM-IC_;fevOS%EB(oxE@C((IYA+x zHMi}zbZ9RwKTxibvZ_L24IGmm#3x}Cyq?{(U;6xra$mbp#}9<8Nk=60-k(1swHHwK z*|{s6x<3Sb-ZhJUv)-?93g36@uTYtSjs2&@-;2W|2@BkGhHAgz-iz^p(0jMJumrc8 zfp0wsoOIvh3@vrWz`W0#ryG6fNy4y6!XOkB*AcLy^ZWYoumqJDK`0e zEj5Z(xCNvJ7;X9Z`2(V&2=An(?zboMH)SjMmXx4A8rLklb7b?fdKI8B0f4g%FoYl^25R{by$)&d-ZAoa2F?sTT@NNFPj>#Zh(Cr$$FQFKUA+qkO*G5 z9lS8WbLffkXm@VVg~Ov0K*NL8!OkQ>>r%IU6CFJw9n;0lo6Em3LT+K;5@E)(DqpLb zQF+5)zrlBV6L=Mrl<<3cdO!l~iC>G`A90nwNp@ywMy-#OQ;ZDeN8kf=1Ax>c&@vwI z^S_e3#mqr)6YH_EUmKi3l1`_FEF*Ik5?O~-Z-h^beb!u9T@9I@)-TqC>3d1W#zLm+ z*zexzU2;f`GvGTjk-a2&@vrvb!V%m*^3)?&7DD9?%S42EO-xKEDk(J=zpNdanCNg3 zOA+-k5KD?HZGDYi$NQUjcunuwHA`K04T?2~t3Og09a?d=bznN@VR%MXf+Z}kFD@-P z0C4*L<3~_52JVR7yR4XO!K`w?LJEX!Mn;B6b5T43aI2q-mwUpz?(5eFC#R>D&d^65 zjjiPL^f85nTsA*SWex&^hw_fk_ZW$MP6Eg|I8M5{*PBr5Bz!8koCUwn5k9E(OUK)T zuPvIs@b5nR;FdHp={B1Cq!r-j{dP)2qy8j8E;E)KwOwH;dYv-aGo_sdZ{Kg<2n$Y! zp*rl@b$#GvJr=*jwd4{SgQ7jasF+RoNZUH=@E!itLn&{>JH2=NEs=n2+X zMF$&t55~X*R%SEx0F?Do1V`Z;FajwmD*DF65E*!G3#ueOxPe~BM{XClqZdYt@cBQl z=7k#<#59*ID%t}l67nrBPWAzvAY*RMGWj|G)#gNzi1Qk;*u#gzBP0F*PIw)^q@MNh zVUMOyNQ;6d@<~^;f#(+SQm0i_i2#5<-x73FKQHU+pw#sp&l+7FH=1OPr&!|^7104ql>=#@U5@0yFq7r1SJ3(1YrKhzH#I8p-s9%jPFfK-;E7J z16r^;A)6n*pmv$oDq=T!pZgL?VIxlb#_PfyeEcvqKf%3T>Lw*MA3y{NTE94(Y}yzf!H}1e6R?9TL~iu8y+332IsY}s zXEnwKmy=@zcnPde0dR`e`=q3#FwY|!uk-Ei=;6c5+k5|XHQoA->r9QyWzZ)zYirJl zLU*G5{U)bD3{_R%MyXQcfC}WvIhOV0+r^D%8Q;hOa6=03vg3NJ-d*2ru6=8n!mkmf zk(!9?IS5osmyCLnEJR3C|6_JG+WlZ9^UD_kOh5B$U9~jvFf80#86Y^X1;gO}%gbhP zI2_G=wMM^u@q+#M>*w-v@}*8;CZP$8t1I8{$MdA&E9UrPHQ(S25JyK}N<-|HRzH9% zQo-Yf$>PBlWv?Y%ph)+|pEq^KOWX1o+40ssF)&E$gFX*7RyUlBl)6|DD=cKm&bjg0 zI{wPi>fM#pSyRccpT6vx5cls(fX(qxXIvkh>F@kyXV`MG@05NfDfW~-o5i3&0BX==#9)pPS?07@18LOoYI=3oG`z* z=ntqvjh&atP^Nm<<$@9r8h+=;IeB|_br|BO6`JxdU#9x!=2O$q(i@N5x0Gt`kU#6} zUQcNJ^%FBXdW3dXtLlYU?Ch*T6s|^zK|Kv<1OY5{Q$Kzfo&NbfZ@HA?&Khy zvZ+tSy|$?O1r!%|%Bl+V5D?SGAYxuy+6Mb}K5}$?aCJ7>X!Wt;9tFh+{Z=opGOw$A z%*230=)QoF9hjAcEm(e4zCrI0DB|cCulV*Y6L5S7s|rl_1Dp8=745=^Zelf@tDaf3 z;*1mEL%?Xim}rqkjVficwzl>W?7OUbQ{p8Neo%`r(g!EO z@bcWyoB6mQHahxkU%C&jLUd$E!RFqQ-isGtK))$sKMxrQgWFepS7^ygHv#2|#!Ny- zTblw6TO|d_wGjy9`LM#}Wwh&@H>IDSAB~Te4gDRi%PM)ws|$Ucz%HxtM+ zD=-pmZEY%1$NSdqLc+q2UfbR>ctY65)dsYWXU{1;#KeXK=RNt&`=lKlV1O~O51d2J7xfIqaNBqs~Fc!FNL1lQg=IuLE`Qkp;YFyivl*48IZ375Y1_8Sot-2RI9{o(hh5ez;)>mKeV zd(Z~t+9Kjk7`CD+j^riT$qKDe2Xy?%6LjPe@x%9<<12ZWold3vk=pVM$|4t7N8|FJP$?894e)ddS5s#P(?~Wvw1U}VAnCZ|%kBW6#mWpebzs{T1V3YUmT+J;}}O&c@C@a$m5$LPNz(e0>e}rhnsPi{EJK!3SYA6_vKrgVhk%>POI|W(v|L z=gd)2QI3>)hbYCNA3ySytit;fvEF~cu#uX#>LHtFXTor9{~LP)O^6801(WNR4#gC_*l0M}k)$!_3BzuECR;&$ac(-l^12yb#$PEucFS4e= z43u9qU$*Vr&QytsVer<;^iUf?y<5lhvdUz6adA=o=~I6IIEytI#Y|$`1B(us6{7FR zfbwEuYI>WNHllE**5OAt{*SVN9F^aAw{TdVj2L)H7+YIMeEM`BAP6h*jW{ZtfQ#42 zqH_l~adc@EaLe-6q2e>e!KR~AtB?P{ChW4I1p)y*tSc!gfnmL(k6}=y!b55xUqc!p zS@`rRs{HAQKLk8o) zd9LDoD88^G7vX#3rZ19e9g6wD1_J{F`QR17(aub-3kN&`T8vynxwp{!aHZZo3z1q| zag*l)%*kZ&%SV=$?0}x&hE-9Q_N>%@O*2%!=|)Xm>i(PD+53#WMxbs}jUIu}?}&(r zQ2?7FE-sGaeP3$YEh&q-bgP3Od<|WgHcJ|L%1b8xIN3uQ&Iq&{PYDatytQk>pT)b` ztXfF#1GOj^z^r>fEaQA$JXU)V9AVz{n0oA0G$7a>)YbWVpF`plzjq4CDkt!?&(ykV z6h_Ht1Zv_uQ_)&au$klWUS`=eAN&}t$3%0RAPA7Uk5pA}ZRX7c>VfxvU&MR5>yr~R zW#*)hXPjPA!YeB)3y8RGz#hqgTclMsEN91j8Z6)68!~=srh`6pP0fItlsxvS1jb>_ zWbt{2%C^&dfGBSI`AtkU5*h)hcF*yGB&O zo6cQ))Nq)_7;l`SuhgZI$IYm;NNMI)vcIYOpn!(3EB;VJ8M(`MgOXVCF*qv&i@gHldvzeWC6*g57$B0=B52gak@J zZkIW&s@6=yq&*B(&d!*@P>k*(j*gCdK(biuq~FcVHl@@qx%J_^i@*e#m6bI#H5CSQ z8IW04Bj+X~lf3L~Y|>Aj^k-OIp>GzmB~JIix$xwB)dHS(w=Dm)|v7u*?5UjOR-W2zfAvE~h1-%N1x^z2QoH5vYO zlh@Pp)&r)WWYpC7pkn&_V}i$XjBJ`bu=fUq`@qQ97-QDgVk0Rrv00a`P1)yysc;rS zn_sZRr`N&)sFaEd0icgS$H&J(nrdZCwwukLy}QjMKMz8@15CLL#CYJdcX2Sdv^nzh z$4ni5$M9{Wh=y2zFWs3L8PNGl?H6Qi<{It->!+Qkz6@Hc062EMM<$T(K2;O39qdh@%xF#yIp zq(IL{MBg2!DlC);te{%!%LvMH7H9KA>fu!TJX{V;C@+PS(jEln(Fi|(1 z9&E8HIk$k0ev^XBSMh$9pqz(_T?9VoU?=`xG=uOQHU%$EvA!{@XqHpCdB6YvrK;c< z^*k(v;%Yh!5CGEY5fDr)!59S*piIQRd*=`00VDt*<3jIv=&9)Vr123xgt`5PimIut zMaN@iLVIBO>Jo81<-eFF0%}>_h%nW3Ru!l!z`MYNV1It}2AnU}+8da0j-hNMXs7>l zzoBtC(j!!m0R`#AIdj`uUy3McT3VWj+aAN@=U7%$F<>PE^EqI!+kg)31dMjIYLGM$ zdXf+QvbDV(heU$OG*ZUN7Ec)U_3vweF1geZ1v=$2BBMjoF*iRKpIw1KjWBO@aKp|qZ{OUC60 zqnQulH+zSNa*B$0fW-p)y0$I{>mE32J=u@1QVpQ`{BL%{WK4vYWp!{e`wItNwn7Xb z08Kvi_4mgmC4~b@Ufk;>7ErtttL@8s3n566vb0Ts(75D(mt~l43T6DkYCoT+P^$|~ z8+Kb;TSh|z9he)J#K^PT++o!Yt#@ItV*NZBmH%k15x|H9_FpW8BHrh^md*q8a27MonW7c0h0q&xA!A= zYlHW>x}IJyV66j2O&l+4Xb+X~SkasaP%^istK)fGb?yy~`vGxE-M}CkXjM!5OP#Tb z9%bfH+#a5tSp~gdoT8|C97GI}*3u$tINx~)$oPBr?)d{MIZbsyg@%?kE+9=ilF7=-s!FwU-m)I)?BcQj=n-Ik3bKAi zey`H5^AF0IOHECo?P%M+=!+~MCV-PlZZL{jZ1vxgtYd1l%%}&9{U2M^&aB+>NtWWAy;_M>czeuvI z(W6?a5eA5*_0d%0ku#{yx2twOXuofcTT1&)z^|c4t%HNxbacgzcaktTa}KFXZ{51} z+QtT6A44QRQ^n|->|tjI@m&hQwMX({AcY9=@G)9id@~q)J4ba9WHF0`6;|UMg)eI< z0M#ar9-+eJ_>FQZ5Q$5dpValk!)0Tn-@Z_ek@i*A(11WnLh&-{?u5qRBCe>x{4+hn zc)D%3zwt?_$>$1dt(zS@nJe`cE#tow0Z|FbEaXIl67XpNXM*}e;L<955pW*QnytqM z|E?Z-j)5UkW=^=#5e*zv+$?Ho(5hOj#%qba<3_AGZ(RMlP;oUi?4)F6L&NX4K*MMP z@p%*2<*>1_(E}W0@BhQG_4L zh`mVY{LV{2mZ3Rf;AqMOX$Qhn5^${q5VzwQSG2bR%*CxhO52<)Nx)A4f+HY(L94(8 zcyVja^>rUW+c~h-)>bqM;l3QhO8{8=e+E9+9sAVi=Amv!5AdRJpmTu<3Pg_E#R-fm z*bhkMKo!9StrwY`%sG7xq=`Kg>OI2PHX5khkdc+*#6)WO==afHz<&q_#5=l)gV8iF zFtDqQrs^Zw>;g6b8QgFb{Cd3Lp1YL8O6kYei~5Ew~!5`IWwo3tKSUn zPiE54fZ>qf(}NE=xoN4p)BT-U7s2kt4&V>gcr>T7#M=l3-qBhC>3a)Hy}SV!$XONB zznr#qst(UST%0#R=4T}mWPT+P6b*vXN;ZgG{9Fqed2dc*?Ty0)JjfknC3d-GF=`UF z(^Sl^w03sucXxoNtaadp;cSW6KH8Kgi0T)J4zI!X%TQQ3$CM$;%HqNBZSflex?@Sh z*_R1Nvv{%c=+9`sYC~d&Y3GDVe-bHD{8Tcq#TGAU1`n?)M$lT-XzNFfp*c-&TtdR6 z)P1PBctX`~@!Q$hJbtLfgaddLKNxPuE7OU;_L9#Yy{5af9-CyqavAXE*AaunC7GAk z#`{45oIM~AK|Oe4vZ~vltz8nFR0D-e;K5i+w<7y{3QeEbicVmIHw(T~QeGZk4Zxby za@Xw@P+Ltwb!fdK>KMq*$!Q9j_WPZOj?0RXVReZe;VUG){3$Owc^ghlt~bri&Fx)XI~C}?1Hk~! z9`$*k5PjlnKZgFgk~6Eja7hwSMki=59&K}BpP05A;!}%d6bl=$G>^NTA8ngTUhK7A zm0)yToUZ4kFEO&u{k z%tYHzEyYH4Zp#J9l5Q+30srVf_V)knLH~c8@Fm&V0U9niT#SKiEv8!fLKdO??7*j; l?N$?kws!xQf&Q9^n#alPC|N5K{G$MflAJ2MSk}n@{{Z~U_Q3!E diff --git a/docs/images/chapters/pointvectors/d0b09bb338bd42d4164ced871d1f77ba.png b/docs/images/chapters/pointvectors/d0b09bb338bd42d4164ced871d1f77ba.png index 403dac6d81e81d68bd9f3664fbbc51a8be0abd81..c020bd8af446bc92f8912b3b265d0d7cd53eaf52 100644 GIT binary patch literal 9740 zcmc(_byQT*`!5Uv0*Z)$A~7Ic(%pijl!SmZNOyNjNJ|eOtsvdq1A?@4gX9213PZ=; ze1E@t?|R>L*Si0_YrTub#F;bu?Ahmeo=-eG{JoOQlSd?vP*6~w$jM5oqM)D>-2Y=e z1V^41t~P+DhsN?Uk|=lgf7z`C2`DJ9P~;@V)jhuLEqc4CZ#4EE9jpsB4hH|kU=!yg z2o4RFCs833ulW9m_Wc{GXF0DUUc~$&E8Mb_duD?dDZ`kH!KlSF7h*9oDk=9`{LK%V zCn;|ppi-t{IZ0Z)(sZKi6ZOZDl%O+;-%S-sfmdt1h!pV_yUO@_J#&?D(^jseLWm(W zF*%ttjo6!=WK~t(krNpyf`2;bL?V$zJd)sTA>0$Wi%UQmy!9nG98cmoKAi;C2U#c< zr8=b%Rq*?e|JT8s;?mL&6MorJSS!9Rr=m&*Dk|byT2L$cWG^tfr-n%5&@+jj5->@u zoJZJTR3;jQV7#KQ?EhaKOhwZGBZJ5LsTaKa|M7BSrKGQd>8&j+vMVa^B7aFpNnxNE z85u!`%#4kVv9PfIG=%KLu|I=}4`+!fYiPWfoSZDJOG!#n(a?}qQF->MH-=VN_`0cf zXXj11ySn`3{c@vmb8~-9O_ftrd_qgxDytk@&~R>slAAlsZUhq@+&q|Ra%EO4P_lIn zQB_^-y1l7zEj7T<#vLA!9ms4wUwJQN>Osr+m|trVv~Uv9;?`ASomL=!KGKWt6ubd zove?t;3QAG?B~6AglnDF^*bpX4|npGD@QN`$wTqzAuhEMl_ZgFgF|4ROl)1S%-gqUXb*7=|8@jM zL4=Z>;lq{g2l{kMgKOR4>GLP)=NCL2MiS&GbP0W%0=)K13T|$^i0N|o^Q*?G7SCfS zza!OCk&~~By&%es!mPX2#(v04%E;6QhmBg;@M!$dEY~L!@gl6gc>S2jZ06d6pYz}A zoSYmSGX9F83#Ck9IV&qB%%|k&hKGMU$;AA=k=iu`rs#Wx_mX=LZq_X<=;kY?KkP_m z(RnH8Vo}_bl$0cVxs_*PZmy=OiD`HYO)>CGJwNxTs;=H%Y_8W^rY7E^M}E(I0tu11 zySkdqH5GOF=Atz*&B(k7;)ymjUh zQOfEOcBJCq;5fg!niqEKje*$AR{a=qtDcSOG_-Kn@$(a1>xq7=LI^H9M|RgopLMuh zkX45+C(}zqM;D)%XmUrYv8%4HZ+B1zN=2hjYkB*v_61z*t`N*e-1D##8PQjxIf_6e zViPDa(JN=V*j(AQ>!~Px+j2m-R7nlt$@JC*#^f?&GXkcH%a;VbU!L9 zRgTWi?8n=(vtQBj@@il|dGbN@`mhT(w(#fA_Yx8kyMFCXrc}c3OigLid2LsJ$5G?F zWDg+bGBe~R(OIn9UG_(h{J{nvxr64(nE&gd;EjvNmv!Pi*`2*${;EVOFm@T`qpsfE z%rJc-$`Any!gwsJq(sNTQMqUUwuiUdNMwqcL-gYY(=YM03i}sK~_!P=E zo-bamuXM}PaOXEu``GU&(#^ebuWp`!g~jwwwm8aQrbznM)>iT#3ws@1T{2=~;w=}! z)zwuLL_ilvC{-=3-(M-;g1pK$AIS_34XyLW3HW8ay|AYjHmyC~Sh9^6A>F58-$eOzW_^Z z$vTO#qL)7xKM_i)jE#*YUL3>cZ~RPgwN{dun3(weuKA^;q{u{lpgld(ytZ?}rCA#t z9q%7KQW~!=(4){5458#rPK&}xoA>NlU++s$_{JZ{)_xuT_3Kx=4C{-38^WhEpE8>I z8$6DvsAXbkL_{)l>{}wO_x!IJS*xJ?bI7IBUP# z`*Z5rv=l|mZl&H_FZlfVa}zVOD%*KpUYnV}C#OT{e6d+KyXL~@e?PlTCx{ZP#Fw1~ z7na_c5)cr09<-imllMIMgFr@6h%Kcm*2h9_=*IIEtv#WGvr{D>$G1NL0F>X_CCIM> zQJ&Tn_SAWIhQnoOh=GW0OpJGG^Q~qYDbJ&1*2$*lMIchr8uUG8ykI9@I1kZ0Jw4ag z*Wc^vDol+1!13OA-mpo@mhz{*-1U;%9U42Tgii-!6eJ~IoWEfe#*lnf95hn@-@js zamrhaTUYPqm(^_&XL0HmA7=UE=E)RmY0Xd>Q8h_MUYmdIQ*>~)T40U1v(d>k)i15=vZ8!jM&*>c>Oxx{lLcU!0)Ty&4eG3o7eg? zL=d-q3uV0^F3s{swjAj&OBq}W(FG>ivcu!zwO%n%g?7KXUJrh$gQcmkZqX|ozkdXK zb1PsCgcGSV+4p2kOlZ7XGebi|HReADok&VcyA{_l#mAS_3*yrO7+az0=STKAbW5O&|dc z6ZefXm!qFwORKMUI`&td70SnYjhxs7d^v7mh14ep%UJ+@b`S_q^s?dI;SRemab9M- zjt3LSLK_%l+@XT1pzs6+238mbA_Xqw5^C(?9Ufp45UuRujw1@o>-{3;O2hi;ltY)V zAD-?8dhe^0+QNAH$SO!GdBV0K?hE2BF5JFXr?uWoiZU~`1F;5of?sNC1j)U&RO_uU zS}y4Pn%$j`_Ub=aWjK7`#4JhJndJ&AOx`O76RP8lGS3Qu2$kfW7AS1%w^mfJH+YO* z@?WJGz##4}?V0bsXI>)Ld3(hda=K0)Nd*q2qiI!jU-C<`~50?m1I^-Fm z=Aj5}Pzx*625H|D0}--lvZ^U5FSnPcFTu0=H^c7Oi>p>!YS5kuhz6^Ei>$3Ji@2X3 z`qf3H-;GykI-T&>-r#V07idwba+YXfVq%5mI9&{x04(6QtXaz?Q%@D=*0>B}6Z`2} zozZGIBtHh=q$jBB?aL11iiZOR5=t>8PW0*o1&9MWI=V|?8J`=}OvL_(VLpknHjsm(k2g%?jG3bUKM}Rdnqv3e-mob2+!Kx z4k-mF;!cWOGTV+U-42$9((T(AA^CbId-Y$TSf4Sw z=4$^Ql#g|^gppV2XOw86GBPp|5-ybt7&-mu8Xq&Xbf*}gMOXt?TB1`wym^%SDI|p5 zuLb!QN-Iz15cHw=6)SF*?ylt@9%5#{pfy!uZx z@LFfl(2x+IV?z6uyGKLwcw;#*pxURcr^!7Kc!P_2E6_H+?HA4n9F6Yg` zZ7YH%)1?+q6H`-5r~Tp&A3ktcO)#A9FAn^WPib)3#`pB}{#a!~n40<_>g7tX^!a6w za*97;FiO)iy2{E*zvZuF45hvh91$Yld8w4M6I{fY6?adPQjGvfL z*C^G-=m0VV8yA<3iYmyRRqX0(dU|A^3Cx5uE+xgh8uB7CB)u$E^JLDpZl~q?-@y0p z-yJ|0mH9nZuLN`fq}fcBC|f`-HvySXTztGONu9!%;cX{On>TMz?$sVMbL8(L+03Su zLf)&FI1fSEMz**6r%&tq-?*9$rAf)kVxjy;S6rR$(a_Q!&3uZDOH1pQXKR%NFaen3 zJ*!uQRL}Y>f*IV;-+5C(qeQE7XR^5YMCAQjOH0di>?jlf0NSD)1TcsTVvbi>_#2~I zzQeShyW_Yb-wGTamX$>zm=Oc`G`Ziok=MzV(eVsK$c^3a`kZ%u$jo^_V*29ZB8rqR zChPX3V0XTL#0;V{aI4!~+A0_x9&R(Azjt>Fl9~6agPAFFrkbmm(?07@th zF$@q-?Yazp$DnC_?`S}|>b)1hS;HniqObcK}ZQ z8{%7bX)Gh@i6VZhu1*NZAd4o07X?KwSg0uCJj?muEu65ECgk2c5d{UsyLa#A7U6Tn zm{KZQ6^5vNeSO#0a52o1F1WR|bzxzl<=LZNa;r&4bOQqexBZ2_g+^zcOV`cYRifRK zqt3I|vl_chJU8bVLUcgL;E`-?ub7znHxc!VZlphc{J5CxRM z#Ka_feEUmeBwESm$1<_xi(|dUPjy^7=2Q@mVvk6pr8UZQ`}&iZ+m}7g*`tU*>1nII zxOuVvrc%w*v$3_ADI%A_eE6Hn=3r_bH!ts;&&^RUg(e`v+PAnhm#TzP;3ue1QBx=V z{w+(($ET@6D0Q)(pq#idkTNe^)(Y-7Ak*J}{9pvQqPcOSubi@DYR1$*#K+u!w6-&W z2$)FlS}cWEu-|`i&)&b7jNhSS(>e?nq6$<~Vo_0%(xh&SN2RK^_T1j-2Te`pn@gmc zQih=Ev1@Jn1rci^^FeC?gQ8-F@ZpHRMk2-SdzIlV``THE|IGJH^J(X;gm-6)?i2<- zMnPPA6USsiF66U3H;MJV_jstNsG9T!$n9I6{bnxj_#vMm>@ZCB zk`LGKl&ivd<};qQg9Q|aLiF3s)!DpH%mc%@8cUm*iq1Gii|I5TYmbWY@kH057|7uz z0`>alz`=7y5#DlL<+?xB=={Mx9nZ?q0{Svq`VxugU264)c#@>J?GP$t;UcD|$2KBg zQBWYFEM#I~Ri;IC%2>7hK5u%yJfog#ojh_uBH2wNT!;3r_N|5B@&)J^op-lvB?fG2 zERRyw57~(YT^I~}kx81v2LFl;#MKF3Yh|l;rrXyI&z|=)kNH#S#&mVju(N-qxY?4~ zovrR2EpTGAZ^LeEoO+D?;a7J@o3Gxk&oh|gH_rWqc{^7p=amciivKMpYwRcW6a}JJ zx?k1&m@sk4em6S9&rj#=GaLmKGejP%6~NMX0H>cSeFpRVhELbl*0$h0BC@;GI!|2S zT;#qCm~EdVC?RwSfb%Yyp4vdU3+A!bh)X=F2xTFVKpXZUl8)$ncPO{s}L%E2fI?E|8Dw$+ofvSOicgqER4&X+bRif&fya6F(jJCjSmES z%f)6oNc+`;17*OO?d=w1|CM|MW$J4}o-09!14%(xUs7E6@Hu)PalUd^%r9erb*PcQ zj5E$o8{s9950ukP^!(JY*#4fdH*&`Fi-|4I7PyrG$ctn%?0N_lAkY&UXUAG$wamUWm>VflfE6Pp5(;Jn)7>vzC0%7VVIQ!!i_s?d`6}a8^=|2dBH0xb%K^zLRrw!OVYNE9BaQLc@(jBy(HFjfhV+KRd)6A(hH!+#m&LWtU2YYH;5P5 zE0^3)2^=!wx`E%bw%>fvW616>Z~M%|WMSybv)&rJWIVf0OXm)^??pus_b6JTC7vyf z=v3F#B%q)e0ig~YsfTY0j&saMC@SoiMJ3+8<*`h`+lSqgiD^W6^C#e&0^77G<9x>p zJl~6$^0v!}tNrHkd1R9@z`}{CsXBCQfU9c2IkB^21+K}ylP%BOy=hiF?HoQ&_UBfk zpr$5KkaQe_l_L{3-J{7HhH*k2NS;|wW9EVhk!qt(HHfC#BF=ULjk?JENGOFf2SWRjc*1R?Q&bXwZahrFrSZ*8Olmx_%F4=OfLODed@j)-2MrLU4kslcv9T}z z8L)i&Vr;ybgD4PhX~-5P86DT%75b!Wdb)m+7Z8!qU%x)WX)lhmDCr*BwI==jSY0yC z?RW75nMXk>8MZ*DF!55Z1=MlC$;98k$3qbp7eCn={fwmWK^YOjBqp9>PS~5bRFJJ0 zJ`=Rhbk(T6I;h`2UI}}zmNzwz5Bqd(xfn3q@XzB0P_QN!9}uVvjEqLWj3N_o3PweH z$o{D(>VQ7_;h*hSf@z^T_Ox4Kw_Q!8Rrt3^HpKCn|1B2|#3}%v3NtxBTmk-W zNK=zwb24}BFAMH}fRDJHzIw^kb4eRq(PwMt+#x_69)hR0reXA8u3YYQaK+Hxe}X>* zudY!K>S9iF)aE1L z*wuhz`Y_ur$w(eB?4I*HJ1V>k47&x@2@B_%rg%*+g8 zs1VgaY z_r6ypFH+w&6nHRAQ+KxW4r}G59R;1HG(H>-w_YTqY)P9B!`70)hi`7yU3mf9SXu0b zyI3RjV{QN7xU!g{sp$^_9fn}|@fY?+L|HQfD{I|s=fuQB_fM5Pt!6itjg1%m11u$a z&B?&1>MM97@$)w|?4CY|e||ViCim^XjVM}HAkDy+>4b#lQ-=s)d5x1b(U3+^{boU{ z2ec{P7w?5VrLe2kzh8?Hs8$n8d5yye9|9RNxVzMB zyE)ad19ofj9x!FMke9sHn_F++hWl(hf5|s&*1A(v#48{Ah;k0jVdR9AT`iubQIeH? z#m^7j*|ADXPe(mGI-=y~P7IbI4|IEw*O&`yN{+#Uv67FPJB5_SbwB(Zm?7lu1kZ2> z$Q|3>F6O@ENiXESpUC`_Qhl96lkSffjib7{`oQV!%~e8r`usmb9mBuKtbpedE-^BC zsc(9z!_m;tXk~*P@M0em5QE#xE6l1eaUTVvp5#PyuO7Q%=U~L*BrsEl+lrLX1v}ux z;t+fUAH$D-v!l+}3n~kvHoI9)yrLBT)*BjL$`k%M=nbm|I8V>kwsf_yI?;Bf0yUwp z8||quN(acuj^PZy5*>GyHNcrX7yNK%Vs=ChJJ4G0Zm$&5xUmDg4??C5ddrTR*6yQ6 zdBBU@I~@&_eR6h`wp$X`{*Z9IqT?<0MmI%cfB&szigiz3$_e;zL5Bo2kJJg+CvJe z3o0QuAr;l%8IDpN+F4Oit_T35*qH4C%4C9S?ke!OIr)Zy34+pS$>b;BSK|4 z5)F*8X?2y<_o}6|e#y8mo>u+Cq|>%4OS3!X(e?32HGtU5lkMZ1;fSEvUk~V&w7jjK zl2zP()HL67`B?kNu0gz+k2?7Azch9ZlIV3fTunt~l=HC;@Q0o}c`_)cWH!Ih=&+6j z3}FF*;LD5740j0y9AF!lXc_tL?9J7>1mgt|gDk>oPzyb#<9@ysEhNUZhAr;z zv~G!?L)KmBCW^+C6YG;kRMz;9bI8ka3Gzn2%#taZZBmw|Qw7f-@35m<*nInK@8_I1 zHG~J-^Yn#&|1RlwjM0dKf>C?F7r?l30d};O0d-Ww8V#Ke>wCh7&S~+7QKRfy8j+8+ zGSwogfZ})SEH*a&M4Rs|Rz0231L0c_8Y0%XpU;i zL3MfE%K}SEDE=(G232|M5q82rzCw6MFQ(|_R?U>+GZ>5K;=1gfrG>@cquw*eYY$@k z)(^3{3;<%#s%I4x6mYTV(aUZQ}`UB60z!9cpHAAHZg-m5l`{R0L3ya*B*JOr0;J!_UF$VGc!6K z9-b2>MoLkUjP%GzRDOczzuK?~&WCNx5L2ZJ8G@5uAx`J)Hgo9yH^r`|?WtTV?RTVJ z$g%T9tQ<)B(}bX)2i9K+iMVbjHIHmg9O_@`(eHNz4&rrOpjw z|0x!GUVdJ!3k^$(bL`2 zgHJ-z|FJqXD&(=ibAm3~)=YUXkpg4m-&ze{tIfFnXK(rlSQp>(^Q){TnNVB=^@`a7 zKsVsw6O!OQ6Dl^ga>eM*&zJ;cWD1g!=m#ywN@1=FbOb9yo{Wy{IMWt(aKUtZR|SEy8AUA<}u(iRwdIgO2}^-Ernw)1tU6*jY+ z!`}qPOEG5jwwK@TG@6T)-*IEU`fulI{r5BA_As!q$*2&*y1I183(n5Y>OFVsZ0BcM zJe||`^r4qHdn3j1Sy>|uj%)7;LkcpzicGzY zjj2G>{IpGvl1u`&UiH6iE0n8*Yv99n+VIxTX}o{mQTeIY<9sDd#|i@zlK>xI0vI_% zf)TkD7Nb<4Uo}^ySTH^cTJgzj2E!-Yp;sSMFbR!96C-dX!JZ6oL(#Q_gYUzaIM~*B2c>nLakX*nDN3=S3Jff8JHgTc>J@N0EOrQfL^Z(epbnbtu XPV8P+lpO-LccI8hDM^+|7zX_hp>rM& literal 9800 zcmc(FWn5I>yDu0xbcrY_H6S_CDL9n0NOwuM^pN7vEeb;;AV`bS&43`%lG2TUz#!dc z@%#Voi+j#_b>Ce0FwCskduH!vJ?nYCu_B(U$`j&I;bCE65h^^Bd4Yw6O@93c!2u%_ zg^RV|AC9?_ybRXW^>21lK_V8`Lo5ZEr<&gBTQk1ingt55IjybtD4dv!yL6tgqBAnM@~OqaFU3NPL&xC5yPUPJv!}@z zBBdbe1lZDKaFpTqUy`BGxGA`a-6>~B2jV`8s;hnj&WwJdYhD4AF*$5_f-tI>-v|Lf)`PrEuhG4)2g{wEa-&p?;N{3(GeR&df&}g^ii=m_@1{1m zNbtC!AW>ML`2X+gBC&LYq#({?BO@}du8*zz(?67z5d{qnK0iI$iEA^JDjuz`4cSP5 zk)od#6&F8ZW6N%CZf@Vjr{s27K;Mp~m+a3|NDPmTZZs=hwSo}1YbuR0m@y)6@(T!5 zcx-A@QBkpRG^CuL^D8Qy-H1;=C)6mQV}4u_8614$%a<>r{_saUM@xeY$Hx}O<1MCR zoZgo{l#K~aAgE~U$fzh=8=Lu&!u!SnCv^Jy=Y5hKkJi;}ZA~)0&G8i!wC++SXAcor zbP*jdlTWnJpYE^AB1Dx_Ztyfigy#OBg)Epr8qmz|uMcA_uguIcJe#bnJWfpZ62qrl zP0TXsOyX+4&xaFm$fw&vw||PzqF)hxgNz^{8XWtjuYJ$)K%hV^Enuw1nKsLRY}dM~ zipV~-{d=w%15ImZMK;#)nLW(}*;;}c6uuvy!}cyYSwmY}4n&bvHQn?Wv+&zpl#F~} z?;E8*XJ&G|3tj7ylXc1U(e8;kZ0VvNk&TVwnLeuz==?BGXAzrR{TX8Na&kC<=S%bt zBoO>q!}piFZhCL!oJtEfTJ$_ke|k zWpyg)(t_)cPiIj|!t4*L4gdTJS5;LF+id*%gQIb` zllpwiIgp8k<(v5I$BR0iM$C<59usg;VQ4hm*l$;M)@Ln4bw}q^8988&AEzeCwUVgT zoL^QpNJIDF!Gqo`$rcUo2}WMtx8P1=X~pF9^itMw)$BPwSodSztPd-wszN_UnU$U$ zZZUCl_g@FA`D6fuj!9g+v1e(j+A;Ujr*IVN{^~&100@Yfk;P2CN8|oarD@M8N7~pL zNw1+C*=Sm^dqK6e!jX}YoI<)4>MANRr9AHuF9u3CtE#KnM>PJM!?}#%^W+Jw;zXFQw`!} zqjvA}^P#fpcRv;tDGsf>XV|SDAS}DzvMA0q`@bI<(TrLX6ckJy5TJ^Q)Gn?%4p>Qy zz_1Gl2rRFzzA!dc_45_N;=>i6&IV&T`kBiyRtc92m7k zos#K9#VTnkmCS=rOuTk}vbVoEKHyjK`E&fQuhx5iz7D)iNElk#1(ScB-Yo$apNfhY zMeFxa$=|;}gDRt1f4t}T)$A`uEU`~QK|w*=Us_VapJkw>WqF{Hij2TeSz1~W&_AKc z9eO4$eN!|ewjdJ`*`3Jh*25*4_kkgCscTI`pQ^tjf~z-+{+S%kt=l4XAC@}X%8W3W zcB5v8kx%NHnwqF~W+9>E>!cZPxvY|o*A>f5NOv>{T%hUvGheA?P@-{5c zmOnsIbHqqd7nO6qUl94X?M~#YlP^(oI}zlOuX?zIN{%G2_9TBqQgHNkO}3O*mn$W6 z>q%T*PaDX*NaJ#y?dCJ5l_}>IfpJ?6Ev=8rDZH{^<|5lnpI@E~1hPlwxC?tNc0{bY zfs`(DnE2ebc=levG^9)YZU~=s|8sqPyY1=d=;-Abq1o)h!ouOrs>xXzG0&tlr!W4R zMY$o<2#(~vzcv?b#3jwE8>7YI$MbI*&$b(ijK#o$N=>C|=+EG@CUS1M5WKw1SflRF zH7i^@Jh&_SAulhl9ipKWXUD|EROh{GUa$wAoNPDj(2PB4*yu~K*bqw*Qp-RH%||1C zS~E_^nwzD|%gEf44kcLstG@N_k%srx#gXlmfv zTR##nqhDKCTk~qavNme)GwgI zQ_M9WX#Yh{clgtWKd+!5t@#nho0h8?ccq0`at$!?YXBc>!BygxY0Xd?&|W$57TysEfQU6JFMI6Cqha}YS?JA zQ4jjob~E)X0zZ_{4V@Md^(K#b7!~Wer;R)Xx5`eN-`Cd%Zq#kD12%e)y)))@Z>r_w^P(uy<=7$5VkWc2=VZ_)p9*O6lN_e1WB&eq7q_@5|veXEFRn0dD!Kz-#13hzv$(W$U6QP}Hrz#vi_67E{g2wtc9c-U@yG~D zmsN;{Jue^M(B^?*vtP>Cm^RiqKh-`$B4ryxd`#IAlypz023@Rx@_tzJz-&_0+grH7 zb8GQK7~SIF2kC6K>D3*z?O59Fitdpr*`$!AXDDwhd3Tg3Tu~A4>}aRUhW`bCdcFJW zE`RRdCn$4Db-95{F0>p?q#^%v^B})}LN>~DMG+;?Gpa(S`p|3<$8g9D!PM86U*{@8 zl_Kh2Z<~JJMV~gdLt?Bl7UY3#Z;v9n4+~meeu+bH-*om_Hw7O4P2I|81CbPRP}M(r z1s@Lw0--Kzs@%>K*)KMc4Q~ZZGoBBO2`C$lHdiA&Jj}#T+0vg7IyYjE3%Y)7Fa{{p zmfZy&ygu9=&md7=Kvq}~0N|l3mfAYZ(d~e}#GqiQYJ9$%b)lk5Go5_Am{R;Nr13AF zVa-4(hMNq^j#(xe&S5-W;bC|(o0Y@(dukk;(&yauc!AjL8(!@F)Kdjq*aJW{>@d5(kJA-0(s=b4IO;FR)cz zH%rUv%3ScWAmnNrFsqucC0t09L_EdC)RJ<9=A>_{MFf9Xfb4F0?uRBO&B;I09Zpzu|CQ?f~ zRqi9+2gW!8a_23W2NHvExyNmIeDV6&dwBR=4&$#C)lWZ|&>cFW${FT|4&VC(2p^u6 zU%m3MM^di4q5wWtH#7tQ&iYev@lEB1p`)ES`ZsMs+QnO*DkfX7t=PbGrua{6C|Fx; z;+%Iof6$-iGaCK?8Z4S&Yu?}JlYb!^@|&Tdefb}>ZuDc5gxOKwhGFLg9e(b;FTO`k z?Ck7$?IWryEBSp*JN5XN$MVC{_qFQ7J7eBVISUcGIbU_iL&PR%p^h1)mMWIoCT{g7 z{H7c6ppS23_)IooRpZuKZB0k1&>3#s`5^q4*6nct$T^9lR?GrXZ4ib6TP}BKwT^1* zW^3mt9rmCN|GjqUEvemogyY%%#sLDW+KKSx0{Y=tiS0wC+>gTrq*{{&M1E#uXGE1E?aaqmG$r-(|VMkUK;Wu;~^39Fx207MnYvHD| zt+;E9lKiL`V*mG1nhti&j3>24*XkD^H#~a(TR75dDb+>QJKd~>rNw&_b6F#9aR^W6 zm+>x+!iSadwzqU^4e%ZY*)j}cm23-4{rrgraochXmhMd#`D&0=k0{E|4{g}4kGgq> zXb_-ez#6jrJ%h%L8=@Si>@vP;24&IP@T=CwW2`*GBHB!|y>@1qZ$m4N^S|Y7ycn`| z|IIH-fVdgutgrl5FyddDFg8)^$|M~|BoLUDm8GVyPX)5=J^_yB_B5`Pr6o&BN=ipZ z2h8X1%WIY}ibqT=pF5OYw@oiQRk0aoP@IyUuKe`r>!y|@CjgbHLAV(WuxK8!@d(hFr%r4LohRCaNJMZAvajY}Wcm>4aIHR`h1fvC8WHV`ml-LksuF=k`m!zf_BDyX0xaU)yLWd1g?zwoL(-GPwW(R0pWnI2iYKei zNmru{T4F?rK}~(jK4<0CJYbW_I6;`u~gEk7- z93ZR^0)k4!WwZZTcvKYMbrhU0_iA02da2bNLDS61VKg^4*R8Z6webD&p+{iMuJhYB zfIml;POGVSUqVDP%7)=`0K_s#Ojyf73N#!vH8dRPF(sdr`7-vQY}wqRJ#F{`S^{le8=AVw}XXVk9Y zAi&YA)L26!@zrK&4LhRA82hf&(AmkOn$8Z?)OZHMt_H9&34_gpm#lilZ{NNhw5tLz zE2n2XBrMGB^8AFT3lAUvfq>mzK<}BEm~ICB2e#iqor>n<=K4;{*$WgG6{U+(3Bh3T zGYwvR&a>iVWMp7UOlPO-{tyvCG0uT*`eMl6kHV0+t+Za#Z2*+j`}KWN2gK+TEHEoA ztq06n&*{Nt-vtX6K)x@ntu;(c)BqG;>WbTOJu4Y4`Si)<^h+p!I3>nSsb9YE{`m1j zHzQHH1E4DC;@0HKNHxTaCoDt(fe`gQ+TQwn`TJnIg`;IgeJCiG--U#HU?uTp=8nW{m&<_BS~spmGlzlP!r z&va|3wS|+8iRC!vj7uA57HW(|Fu8|(e&62yvLl={UEJp_pbYXRKel?)?%3M0bKjt+ zxSb}{o^^SvY;sWp!a+j^ef|110kvQ>;NIZ4(h_4rlK|UEb+}f|EZ!ZWiC-s-$R%NM zWvuPSJ=LuG(Ub9(`4CFwxL`+J_WZN+^uJju;ZCkU=6(RW%5B{E9q`EvSCQn?Q~Tj3 zIy=qwlj3R+lZAJ%AshZ^I*C-ZCnp^blHH2D=KPZZxeg8v21|6QOEMtYi{pg2UxsyonQueisMh zGosu`AvZhJ#PtAR#N`HEAs8!FLUIEuGT*laSpliWO$ZO+v7WgsipU$*{k5O8;I*N{ zcR0O=F{dJKtDl`_M8O)7fBu~0;`G46$cg)^!W282iUpq`ek>XBXk!`e}=#98kG=HWTdj2EuA-( zN<&u{+?y#mx`$58vqYmN@wt*cN2S4+!_L!#H!}?kY~i!ytnicrmDv!CJ2!43DU+Pb z>Ar1Tn!@?zS^hypy==jobehzFUHtoE_o|(}o=qWL1d@>orlt>Z-n8Xfk=0bAht=8x zuWJ0)gd~n1`eCro|0ZJ=qrRA@Vnn&q0z~ab*y0dHiOF{o5?qdEB-nTzQ0vvKJOlh6 zbs;>9UtmCMIYgy&M4Ib|9tgFlf6$$4#_V?GvrHf{jNJI-YN-l_&rQL5nZBCrpGJSA zGNU(Z_nqCRaug!dj_6Bt8Kh;AwWP_R5v9Il4L`f++rLH4XZzP7F;v`FC0pBk5m~JtL%OipO<`3@3g6?sed(i@$gygJG{1$zr4Io!Z3uB zYue4SmeEH%JT zE@?2=gUpWW_TAOxRl730dC+l%2~&=aMm~LG+I9mJh~!F8AJzoTZj{X=3jbfl|My&r z!~2ze|x@x1ZXvpZ~GW$7X{=_-gRTb#@eJNJ-m(=J1GvBe%8|HCn7g zPD*NaYNkw3Vu;%nC;ftK+BNKvrlOM3Pi;^b4MHO1dzqWpp~LJ7tZHyYyu5;TwSXI0 z0UH}hxC<|wD4_Eqd=a>~k5yQn1kgCO(B>;Y_J2i+maj8H)tW$0 zMk3LB?upM^CZs-i)#}O@5hFJ&Gr!QT#lwYE!vnU*J)Qes^Qx=0N28`*lm}j20yWOd z?_?#pqqFm{HZzCq&rCzIgm1rV4;L=K0O7|vJqxd_3xL#lyivnYB0tBZ-KzlC{Gck?dLJMR|Jv zce(O$FLZRmhIExQG-&1H=vFGa=rjsH!jHM*Ammk6WX?17Rl{6HAlD3PoVL`B7?7a% zf8tB5EiCL_-0hsy*R3Xaa!%5pwKF7u#AwlxL+yH*=^mTky1cxkjXgO#hA{S?esN1IU}6quzUcLJyw!34>)B8j6EB z*a$H$U+asZW?*RBXOw-t)MYzcml)I@W;Vn)jFy)UOAWT#w?Xma2EQ5gb%T3j$KjG6 z2Y5F%nY~4!z5^YxyRYwKPEIiB7_0pmQx!V;xv!tTv5Aaf`K{B!OIuw0CWmo2{#VT` zFBg={j8Wze9mVy$n}L1-<~@zh+JKCXt{)wO-{AGahD{k|dP)Y@U4HBjolj5cfJ4r_ zHBmvEN6h!&x~r~ciyIOVAtXsYhN%6Q(B0QJ^LZGMr{dhot(Pes2Gyu$B+16!_kWfm zJiK3=S6SR#&78q6T0RuldoZ&Dehi5tB-i<)z={0aZ_$1$y@9oYB*&hP7TQ0Z2vSm@ z#^h!-hkj`iz$^g9J-$yMR5CvCsm0T1$`jYU=7bE#tzx!H8!Tl zJvsfJ(svhK>@)Hu#^@TI(t~cIk*~b&at>stdt%nqJVuKXe|{-%%}kj$J(qo4dS@zA zne#51y&N|Y2s4*|lUx4PI4qvf(bLDN7Hn2p$8rm!jNH#vJUKssAZ?8=GxZBBJO90HVF?)N`^mk=eOY}@ve)-xvmX2oOGKlU)K1mcz zvLXu$3;Sw6&K*ifEhi;~g^i7U4Sj%?StuGV4>4I2$WqQ;QeJ$0^G<49L0;ZxfX3Wd zOPZ!$LKsT+iqxQk2g>FK1w;^r`hiWfj1?LF0+lf(bn7~GgyY6TaKFnL!E1wcLa%4z zFxNPQBzEP!8KW;a&JSoArYG8K3ts6!!0+nTqMRk6#yHy z*?Pra2F^{gK->pd`;KWl4wsl04rWbKsJ7M;?oA(KzQnDaSqrrU7S>MF5#6gX zrj#+A-%-XA6+1HQZ;>T+!*#_QD>oVVyV6rw?wnUw_gcaiu?yUYSM#mdfUz_OR^-fk zZQS)cpr8@A*?1}9O>%y2`uq8PVUnC2!$V`Fi~JLEXj@wwji`J04GKOA%`#8e!8@QMXXzn>bQ->;%*}N7ELkUpb|?kXqn8tU15E7?Ed_hE0g~InrUx z#Ia%Yq4XEwi1^XXkl#32c!}BB52Ns*>*#g{p_-+(?fs7s2ET%v_e_7hdvxoTd#_rC z>bR|vI)lv7i6*MVs38fY4d8dU01)u=U(?#L61}C3eB6ayzYRfFFf<@}mA*b)z~`}{ z2|TvmYb9^>l<-Fg$;_&T*^PYv(4B^zCjb&ZQ&aQua03|ViAUOS=TA^n!ZUZfVRZpe zP)gPs5rh+Ee#Z-8hV||dXFDx(qbj6m4(|i1&qgAFm*T*&u$(cXU3`j#b$j=EFF?Hq z;4KYC*Z6m@67j?*?dBIylI-?=!LWaUiX?F2j`|M1;!{__KO_YErW1>#iIWFlhgn4A zL944-pWO>fR2XH{PW({fyE2R`D>W|X-WkPdj7a&t6BsTo?#thUc)sLGAdU)_5F*an-)^=jn7dn)^nG#xMPLU*ZO6Xgdqj8w$A3g2z=J}0$$U+6P za+%Nm5Ukp{C1p%I1r$#SFmh58xTz#1g`XyxDX`r$Bcd`(3Ml|GML~btTc2=YFS2NjESmuXq7k`N;k_;Xm&9 z;OSZWSSjYHr5#Y;!^srBJI4ML=ys!XC!`B)7V*EDTeyfvq_1(ngR7)_djqG*?*?ls z-*KslKh=6c#=_Io@mXj#D9t%E>xw2b(D+FR#5HEXt2Oh9KUL7w-e1vM3e?~{r-7cE z*_`4u)sB2yVvVM+Z$*%7fvX%DO<7%*Nq(>Aud-j&T8NJq{O+zbzY-*l+?1cH!h>)^ z|9wrbdl+KI=tDEBqi5JZEhhIL@oOK^q{51rOnp}k!No@!zkata_10szRpnol z&xGF({2#?CT}jcrXYpIQg33p|f?U0VrRcWSq;FA$ii(VY^1#W!_s$Y(NOq73*kVCM+~rprELr!A^*h*kFH9J5fMYIq{WM?==CQ!`ij;hph|P z0Qk8}Ny%q3_%MmfV61R#vdX@@yL$mR#hkjIgYM9Ix0=rgj@hZE3nzGNjurkDBSS+Y zW9;ng9Q+IA$Lt*(G>K?M^BiUy^?c5@YTi#z8(ISB{Ud=%{#|~4KKDS=dVbnFpixIj zrU^QxzED^HFfua2=6|~XxTUPI(J&$^YV*ssj*0(AW>!`XpeO}_tIh!^lKtv(=c>$a z_ZuY`)V9wQag&RPh^RQZXqXBlE?o0D@Cz8!r!Lj*xejCv$ISZYqQXK3QPKM0UH`Mg zA+W*VnyflHa#3;{K7>MXf+i+(i;7ChvtH=wbpvLq*Mtx-J=OdKg!E-QT~%=#TdHm;?ortTTN=0Zj4K`&v{K86FPtn9U{;jEAYW{YQKH?Q-o& z_ayVUfykxBS-yN3pOK;B?=SJ|*RN=L$!3jG5Y6qTBV*7u=03FK^?g)I_TH{r>&<8z zv(whpguPmRCNP&zV)Pb#x7%d2b!3a^JN0{g% zY9bgWdi#&>yWhI&-nH&pcje8p*3A2!_nfo$dG@oPy(eB@SCf*Ai3|dPP(FO1ZU})8 z*ns~+q{QHp#Lk&f@Q2tzTT>l!arv*Xtt<@!;eb3;zh?}8yEzx+Wr+zB#chARLK+EC ze?ZqTDy8x8eZ#<{17EF>M!+wgfQ1bm-bZJrZ*?V}Dq7F>^!q+5d+NBxW4*g;ldCiH zelu5hme*b0{q3iv_o9go4;v*7+WP9ajb7h2uM^7VXn4hu5mGycKflv*>jB~x#Af!& zVYu_1;*TsW*HFrA=fiRx??^d`kcAuaszv(J)Kz3Njv>!dHXB?++FjYR23CYZ3>tg) zf3nTEt?VPIQ0nSVl*Be%T)E=7YqV1@5t(s)&J^HN8AmJ+R~;i%ls0vU0UjUstn^6^ zZ>0^3j=Fli5pquAtq$+l*qHqkdA;K0a5fcq2MWLC>!UeBh!c77|0Y2Y|(~*(FepRUcP+E!9t~m z@jspEIAcyI;mQ!PPCy_KIYyKoqVFyjF5J`8^J-)*73{~rx9E!Cn|49Z%iDWNs7P7;h1 z)YNJ)n4iDD{qOH35J<-YH9wiUmDsxBK+fi|7dNpDHJBv>BjY{Zw9eg?0os&a@a8hb z`{EEgFsO!g6CWaMhYofGfdJ?(_MP={ZaEBVxAZvxuLw+FUL_ ze8O~ebRZCVVGAdhn=UKV(W8i?z)CP>*|f1jh!|agc=|+G0$_SGFk4#vw2S z%~n)hU0q z!JCHoWn|SP{Q3iZ(tigcRwaEu{uJ26ajaN_fq@|sY0*T)nLDH!c`p-HKK7sa$e79J z-RDY$K>qFRNjquRRBxcu5eOQUG^%SKu1y?c~yer*% zi}xu|^b!MO8t)7!C@A`n_A-4dKI4Xz>K@BKz?W4g>6RZtnH@bnYc^*X{H>vH%4tPY zbMO3T(c6T^0-7$eoROhQBp)PjKHb_Li!mErRSfVAI|gDA>y{VVnIiT(zrH4WI0~f` zkwp_?*Wxl~J0D?UEyZz@VBDn~X*ZUIo~$9)CvI+0B2vUQK5u8Lshb?IRj*5*KDc#F zi~@?W@}=ZTjT^Tn8EEMSCgr@ztfsy`V&&@{0$d_qo$x4> zjg^(vtkxk~qs?s=lRL6ToA?z%KooWFQ$sl^QSk?V-L9lq+NAI!?!x5j+*`|X1!;a5c<38SBnw-q~&Qe58sH&8sS-wt>(5*ggWVl9)a5fZ|n zPhfD3W#=_d4p?nP202P{V9Y$bsW#Uh()`wHqVNNmKA~~C_O}n=Lw;eO%?F6)r@oB^ zWz$GZjLy$f6_FL*%1o`jdusy8SVJxA7LT5(Tv?}pj_wJ4S^OBvm$(}MOaDzwY=gXC zZ^P~}`D04J+Xvq#or`pq@bW1N18vMy@uR^lWs^-MtBqruiItvBSIve(RY&kktw@m!*-IppHkF9 z*rnVBCJF(X=c)*#fBE$_kw6S5BMBzB+7g(!vW_g#q3_E^@qDnl0r#IPKR4jU9Y3K8 zx5gN3X1ke;yv%4a6-a-j!&|f=55RGHW+ruD1-WJU?qscfe>%1QJzs|>aB+eN2(ZDp z*3s2k(tanV-8eFZlivNxXDOK8FVm78dgMWrf+uT-*>1qE_V)HB(#w377-31zJVQ}0 zLX~2xt3}k`8GzHkzxi!B$gEU{n3Tu?U&=gtzSrm)f>Nq{n&MKO{*X5f9P}rio|+L6 z9elci{^Xo=>n%zR2*OCxgoSc2Bmo@z2v|G(%xfc0l{FM(# zG8l}{(@2#J1j#bRzon+9r$OL+rCXM&TXu8Sqv}XRw26;M zOuWb*zdFRg%6bLz?AZb-PhnvgDr~p09Fx%*SlyvNce>YT7!s{Y@Y|0G#JjnLzx8rB{4i}&w#1IA+79xnco~HL&ULY4!lCDy_{;PNMb-7H_ z%8jN4Aj<9Rcu)4Z6NJVpKFAk&Ta|TUS?6>W0ISrdjPG;z9+j*(mM`dydohTot&`t` z&FhFjz{?NFE$+h(EJeo6A^>+NW%ZF$D?yX@wV(hK+d_leB_;}CI?_VcEp#_UZBTy# z0JS=>!|oL}8&X`5nTaO6_jjTL`@<8E- z73$}=4+e~(-d9RemSlmZJJ_hf(zLoQJp6@M3o?>IjihCTedl28-*~LF(e_6K$OY0UWDXsDv1nJ5FraBKd}SIh#TO3A_TOz{W|bP(yJ5>`$OF>4JF z*3FZLO*nShtBkvbU*$qx0}sFP%>TeQ0r^1>m;1xx3OtV*EI;ztkTP-*pRW)|6)li* zbM-^!2XFXl4#~9SO9rhH3JA2^*wyEC<)mcECI9d2ya%`*Y(klni-BEKMPZ; zXXcv6HswpBRwk<~O^sRzvXG1D;5PM1JFLH);3O`?672aU;qjBV-t!Gr3evignMrFN zg}|&-nsr$Yui@=CryIbrLH-UZqR|)nM$KixKx}hUP&658mgWj^YFRwoS&~-NG;qSL ze8_yV$N;x~wwgdJ2z=dZ&L6W_)ijR`UV3?(;Y|heXVl9l_TYxWZzWDE0QWz-P9Ssm zrXYoF*2%%msFxqYDNrINPw_8q0E7Pg?9H9bs$hc0pN1cGt~^%XbHw`B-UlZmdrdLy zSZddQnevOhh$y!ODVUF&F_HkHT0~<~WdXzAV&>3XETVZeWbyq*aYh$NUXx9XY3Z9XT2pz>bn{di2DVdXib8 zO6)T3)5e%T#6}z(C*K;{)F&3R-9(=0k+Q#tvvc5*4w?ZO6WG1d2K$L@L$}Gw&XkB# z$63GeN7BlH34~p7T$ux^23r0Hk41*&*=nc(iQdr&Dt)84xq{#2X37gb?dv0TsQu~0 zEICuLI@mDf=jVszQ$D(CSWX96nrnD?P=~)_f*D0t)|32$uIhbw7;aG7=M}4Vzw`7* zduTF?TwS5e^mLXF2a*_CLq62_rZN=~xYUyb!lk-5V&MAp%+RA7;o8RTv4!q$ueV(>71Pw;6 zb@-alLcTWWBtpA(O|tkla7IbrijV!t#!*k#hV$UROEQJrM6adtnS6XjmD-Ko;BwZe z%l7yB51OiQ(0nXm2UFW$p|JDuG>3Lnc?B#wvZqKw}h1MkAX^sgjeIk`` zejbiG{!@|BIPl-c-e~RK$;|KR*}zq7Ds zPT$^tZa5XeS|JHut#FB2eYTtWUgwhRU2rT_V6a*EQqtkk zNQ6hwOafO_O{sTXk_9=)#~y~n)RJcFkfA58n*;7O-tL(`?Y5k4c)4A&?K_m0&&8>8 zk$Y^r61+F$vanX`4Z8N<3pbVg_#*J-J`b4u9zu%wM zEQ403L(ia*!j_HL?)dAU_a-fcpL+7j{jq*wHkh-&y43Su|E(Ldy6zTPUrszeB)jl3 z(A3f&Cc;gjqG3PP^ISYoN1{GH;bKt_k47y^|}2ZjrlvpO2FoH^jbm0{CD?m$)hug z%~!gZU3yK&YF&|8~fEXVTJB`m|wC)S5K#Y?t#AHA${= zDzB2EJXm%F5}sRnG+!A}I;j+{xfu^p{b1#M{}C|nJ$wlxEs482L@Q{RazJh17zf1iG!9SS91-C%5RR{_V_8**s?)&o)q!`a+rc zbUYqOXEk~*B;(gcW@P?cSsuGrL?dhVDiDZ6?{k22ZUOf`JPV%<#Sb+m2NN_mKW7%| zs{kV#KFT5R3Pg5(@?Fz?s~o`=ul{8cLm9oip4pj+0=gaN#7j73mx@wBfsoIZNq7f4 znEjW|C=&Mw_sJa|QSkmqx!cRRYppex@+!@nqidZaA02zCE{S2|YVad)4${lU8#|;7 ztO0%C-fXx>s{FdYc|@io^hn})C${p;YbM+9^2E1?o!|yKP6skR$&OoX_FG5NzD!}( zr*rSzoQT52hRpbO?HMGyRLhOp@N$n+bYf>W^>lPNuEiVwjh%C;hB;!YtuLYNl~xJJ zm7YZ?zj*b|2Ik%s`qW$Y9QB_g@sy!B?|gn1W?yLI+c8O2rRfw~FDW`^1(t8%T&l#3$_jY;VUF^WX9Q z_2S)`KJ4S$Tg`yiB)Kwf63#!8hWPww4cxZr)oeo>dNGNk z4)99%<))Tt;620FW85ZYW)eX%%%T+f^LI5Qi>y6nY&ca^4y=Y(_0WxJlRi;{l#}KI zkL;ph(_8>^e%QV_oz-i*y%PeBey%*{X~f%3v|H{pVNE9Aq_ah>!>o^z+m4M*5*Qu40>Wtv6$-*?YRXUyNI4msYt8 z9PZ+D>Y&gX%7m}eP|tdR$s9~c4k!DYjqESYlNx1$Vk`x|VzrZc(KB|cS(0lkJA`QA zrrGC?y~!ZY8qf_~5$NOq?}Z=S^N^YS_+Mf%B`8ikY?UIj^Py;5(IOZCL}`K`IuoiF5~*!hs{ zhWBk2uyGq1@;6$EG)dpmgqh|#Oi)UHU4c5$@XBI9CBv-R!&ecI#W^ z^vvT&3pzc&^Y!cPvF?ScMUAdbD>>4=ZicWMA^U$j!HER7z2%}k5{(iyk4H2i=$YWT z-Y~UhW{=MVKu9msD+p*2^i5D-wiOYX_S0koyQb1GxgdR&v+$_}tT|8=e-!$&HBfYa zdtuwpbk!-}%8DNc3_uJ7U^+61CWVFfIy^vuM5sVq75nW|rcgnWKnT;#zx(_0c4AxW zwcGTcyeLqVb+4q<>h&tqLB6^jwjEL%aS`5|LYk8@LSiDttqpV|Qoc!KQ}|ghLkyy* z=+di1J0g(9*#i-JuM)*^GLdpjNLiy%vg>Xug9h6OD2bKwpDRw*v5hi#QqD_;D}z?O z-+FSe-7DL1^oW9zaxvQ_zqZj)1c16cDd$~raJRIi#-bJpl3MT7EmHNKO;$rNa8%qo z_A}CmGfV_XT>V)={%lt7ob*-Ek6pvyqss^z%Uw7P#yT)>e6C#U7FQszOy_SMC&^^E zurq_ig#LiIa_QotJp~mNSGEqgeu0_~5RxVXzn5I{Y=_$Q;~P}G8mUJ%&CSKhD}q1j z7~sNy7O*{^6h%TUKbMZYLb$6&uf~O> zGNPmxq+9ORRs`HXnmQBpg~{z_{_f1~`nimxrZZqr)hWwL2Lnopx5an;X{sE|LssZzQn!bZWjiV4W% zpbXIsG6R!ptKTi>FjvuZ!Y}p_*33_G_Kv;OLV&_@ReWgl&gFEqMcbDBWRX1$C}U7f zB=?e5Tl!z7`<;#dO2!Dr_B7vu9J)>DEv@EwbQb;$*Z|OPUvQqTsUk`zOj+VBsKM7O zc!_nNH{*9K8=rN?U*~(f)l*3UrQv#R_p$xggCD^R@SA-Us2_I##0&db_kK;hnYZ-~ zwj0EePDoc0$*Y|EMyMOz3UR*rfOx(K=q_hxXLrI+KLa#0=I@KF*r=0Tp4qIDvOJ1YI>hLmrtdrWoeknQmBUz}O@L|Fm-(_WGg%=d-z=#Dez)U91MnXGYKcp<8xp3li5yMS# zW~NC_d-!azjGpxn&>bimVDM32uaQYpoD4$bX58;_zA}1~kOHkfzdnC@vD_i)VU(!R zeJZdU&aN7?}CFid#;Tz@48hK;2pO(6(!Z zoM}2-PWDp@irMNP2{xlIwzdlwg7`+q9Uq36)V^cv`k61?mX{hZ~u{GJlqU4G3b zR#tRids?S(lTIGb7QP=wAN_pRy5y5(|>dA@3BSYu51Ke z_OMS=>G}SCKv}DUoTn}O{&iK3bS+MB(IOXgmw&@4|pHbP!TP6c+7D4?Ef{l3fXJ`24#8we;2b; zTrPI)u~n){$_*(U1~IV+zI;eH4vO_HfQFx8nAr~pXFWL}bY!=?9a#D4mNB4GjggJ| z$V`R58-9A{_~aySlSr=EOvu5{j}8zBpskzLImM4#ThL@CJ)P~hDxM%cx(oK2(6X@) z!@xqs21jR)m5%W7QNJ_bV9UJGOsA3Dp7zlE&~kj4f64>Yclh-3yL*zDHzT}#UV>t| zi1nAzQ5w+CfJR5IMTwwkw^0ci&G#E=%XIFZeX?R}^ybEWAxGy(Pj>XEW^vE$ep0XH z*X)WB_mAKhJ)F2!gW4qnGGJ_+Wg~@QSBc?mQ0nGppl|>R-*|v(Fe`NXE|d7QVmQm; zG9NmK5L?|^>E1#6ZT#BeM~d#~=M>$m{TnMs_JWUebg&bqwqK#tckSG}mbLyL zHTWON{Wo&Oal@TrPF}SF5fMicxT6v1!nq7?YJ{)@uP`rM?Ve#QMZ&pyfYg7*?Z8<) z`YC!XHk#(}AmSqM*-Rtl4t%ynG}5ZfSim0aX1|kZ`FgG;%Z8b<0+4sbdT*TLJxzvS;`Vgi+>kZ=bg` zgrGX4rlv-tYG-E$0Z8zO*^eL@zE;Rlkm=$k+^rWOfd8DAM{v5X>2s~>2EW$TqEX}U zpDdIc$NxTkwdx4Xk2nu{d&irQ{o=4Ub!2UG+{U^tGz$sERg7I*mgNQAZ=5SSZr2s3kAnze4 zxy?+KtMI$@xmVGvo`}=l0|;s8=@PqVd1CDiKpX$xtp?s#y z27QuJ6p2w6XXuTZ0iPtPrrkV0&16(e*h%*+u?!SAwNTh3zSm|PM7&Wx(K+`MGPpc z(2?zijY`#XS>bg*Nww~}w}U!E{@kh?5Ai&skKB=qSM8mPeS!~wZei2d6@~4sZ>IVE$&fMhE_e*?|Zi(YEXjO+> zdL4{%2?hF{5gmu**pYLJ!=Js8j=g_x7mibaW(S*yGk?_KuS7s?L19TUe9tt(zi#)Z z4pN|_X=#b6Zi-Zu^MF-KYaFy{fJTW$(5hjEf{TNNXUXIs5KS+uM>@{-ewQH@P899% z&RFG(>+2i!RU!E>fpo9qe@|HygX#Em-=dMQ3NJvWFOM0lX2SRlmxH*SgKE1bDtWFv zG#JrIL(QsthnylK4oU0|JVH*^q(Q?J(9tgsKgb^99}i^M zgDmUw+5XH%brTlfPu&HCROUzHKwJTR9o5aNOp`ydNTAy{R&Natk{|!lz>?cS&W{RK zb`+?U){~Ze|F6oCT&nl!q4#Xq!5m2Bjnx8(9!oI31w|2;;kTXHk}hI0GBKcnUf9vW z9ylMg_WWRmyof9P^}-2#K1E=j$uuz7IjQ5H(7N3-%IN1pn8UB4ga>mZpa${pq-_7{s{Cs=V*o@V#-VODz~wo<6VThtR~Ue}wP+y3&?xPAObK^5Ti6 zB{RO&!#=}}1%UlskanTbl|jKROe0oM4)x`4a!$J^Uf4+X6eXsfZ|TJC|9B`jCMV47 z&{qI+m1};<2UJeIi+_6FATwD69`d1u^3gb0ip-I{NGQgi^%9mFpD}{^ZVV{<+kx&E za?(F!PE#~DR1d4)(v}g574m--+)jdOM7pm|ZaDHC&LsGCRFj*F%UUoov5Fq%5s z8`hCFF5Zym<>eJqQc6xurCs)Eeu|)n$SWv-hB7(f(EvZcL1w3c6TTON1H=2A-9N6f zul5cmID06oRg_n!m@oUt-MV%aAVjx=c%@nGm*V+m{Mi$ykg81A!)Z*UUIf$qaw1X! z*q++ioaRo}uH4lo^$Hox){8DlR^eMfyz%&)l8)vl?ZwR@Bzgh5LFu0YrE2EK}!+<|3O;~=jaDVI4 z-FJRzyi7_es;|Ee&N3jOAC8Y&s+WKdfNFcLjQg){W(#3lz8rVT){yhSAN$p!)9s5Z z7bNF`;X8@UqZ2%y6H+HFB^$?1qeAqF{Pc-w1#9@A^_JvNr4%a>*&|PZbk%a>=hYi& z>{HL^L1F~1$jYsWxw$Byi<=20-w+I@N$LT9~>!JcJuAy`L0CfI=?uY`tXyozY z0m< zlRGIFhbfTd^&_xG_(;gS(aFs*bqhh90NIOIw6~&2seW1iOmbPaR$BOEZr(+MJE~5W z3BQ=QeibPnW@g&1-AjQ~jqBeZI&qm9awKjRZp&RX73k17I-ONMJeWJMpAB4yb!m$z zH7qBBxmU51wB&&9Ep;x?Y?-ao^V;XFl}$iHM)P+GM(jnqA3577A^@@!rxvoG2q%

NfJ$YNj=?X7eo9yt=lc!fiOx-2K9*nPvE`T-3>S2RrEZ z$d&c%$uN=!%fK1sjnKH@u|wiogr14Y3l6UuzY z!-{0Xg!@`E`+&53O*#F^6DC}VD8Wn}Q2xK6a9V?^$55`}@BM!SDN_a-QabD~wr`_J4I)#pNf9%bUSN<#m zm6p9|?W%hif6##V(dwNxTnX2{n;?X_qr}#orbigEKL|&Uy+bdhKu44W^-P88?V-3O z_L)f-*UayhWT4joXaVT#g~`%}%S?Fve*N}N>xM|S;?EA{F#k~ym8c`k*lyH@pTXRX zb6tU1`<(0j5BB;}xqmjgGY5x+?4069Oi>*j9TC%K=^orjlXBxPpatyGTpEe=_D8!b z6fGjZnrS_YI0l3Vg)nA2e(MY8?sgav?D>gcV}omYWmB%M?EZRwFM9+FweqRhWk3tv zuP^utr1_ebi~52yp$s8&iJ*v*nYdN%7if^})CgdHy7rc}=9S$?o$M3LG!iUaD1j(Z znW?bBwB;EJg&#dWDl|U~Rf*`Jn8%k({Yg^iO0x_dPF4Kmv z4PiFr0di7GrVGKpRrA~ z_t%6Q!nTn;tLeMzZdRSbQSQ zDSbpdl1=d6F+Ko}?BqlQ^h$sq3IN3z{S?2XbK7)5+V_Ozp0u3>U>h)zj|Y&UyW8j0 zZ>3Lu_9BB1V<^!^dP8Rnpd-1qwzfqRHPh(cMICXd_HTb*SApP*kV(7yjLhA$f8*6rC5qv2ks7g4Q?uBAu`7!1j6!RR!p`2fn#hE1 z`4)7QKFZ}y87rSyQ@#ia#YM5J*P6W*N|D8%Q-`gq0d!3}Fl|ZV|6f;RaZL?%k$&HA z5d&5$-__|Oyi({QIp^Jjw+YIpT`XH|O^rT&?XnB%|ArM(gZ|upgM1*K<;QB)7EC*w zmy7HE-T4Ex+M)#dH(&SP^0U^w%jQzVSaTi5#Lix`UsNwZNA?S}kB?6eIK6|dZ933s2Qv2F zW6)o>WQM#)inYDCarmzck?TQ7#ZDX?vhnQiVyykYy?bX>3&@S%tKVYI&*g0kJ`jNE zT#N5s?q=jO%Z=qm9F$~6lCZxlHYPm(T@ab;X`|u@d=>X^e3XhsrA;}*IsYIt{3*d1 zM_pX$Q;_dQgOC9Y6luXPtHh*LvYwACW@{P zN06^iDb!44J67lm%Sg=7)M2JKaepev*U3|_4AOc?F=+^-`?M(hP-#qo%ep5J#0j73 zlq(-Xo6!ejngkl=`XLE~p+^Hg^)dZPEbbbl><@Ft3{>; zqse!Dxr*=m>}4#AHcyNfH2)XnDzasWJ@r|w9{?RRhPJ-iIR-&vVWt+GcaS_#>Bu#K=1X5iPiuH z`b?R>u0bcO)61-AXr*6o7*T>pM?N8sB`-XAd>@)Wl8%gzb6q}0hs?M)4IM=YdOuGB z?N7#Pj3T$WxfH~38H;h5m;F$d&U5k{Y=JE|)qng2t!+thaS%{NLIxGi46N7w=Vui8 zrqhl{#Q)b%FnZ7miI?cV*|ZP?5?HUr=i_sn#k{|cEhu$^{*;&D01#4GHhElfD0 literal 13157 zcmZ|02UJsQ7X=tV0@6XGDpe4r_g+Mbf*>NjN|D~A2I+>b(u?#aAVne|y^C~05do!! z8X*D!gwUCf_s^^~GizqKp1yVO&AqwjJKsL%?7b7CqpeO(!bAdrK*%3IQqhG#@GQW8 ze8f<2P}31WBJYCR)q{XYNH z)(xn96FIf>bk^kt!#Gt4qPs-Z=5S?H6fdEp=oFsM?h7tCTdUdcwUe) z1jiRkrJR2);|X-hqcNp4Eqtq5zP|5pvrfM5=Uuy6 z2>!RbBRe z@x{>pzt8@E-tfPJ>we7yUtwi>B<}6U+>jTihg;Dq927Y4kZ#mLqpe7cdl1HS;tkF# z@AOvT_@{1p&cY@knEdw7vggUH0l00|fB@N(lanu=YlAg219r7(ELODv&s|HnzAJ?;`Q`Ca*z$Sy|cYGyd&+z2dxvI)>ct;$cJsueXy*8d+?`a7f{qT>*uodcOmPPX2P3_J0b82bawXzO zLI1qT=_qDlnCnMJCMXojoi1Ge^{dejH<65Mw4qI!76zaH`@RxX%4~^`P&0MT4({&i z?(T?UpK`tXFO5xAd@ByG7GvSQC`PaA1r_)b#MkiCI@Gz7Gv@5IiC@G?uc}np^!@Le z0b3EUU{O_=KbxCxN*wi1)l)T!A^K1OczSE+$bV0w%c{`UN7k+Uc7cYHnRP>)&Kkt_ zgzi>RN?K!0zR$(E+1g(F{P}b8?D%i?EaaegH;P=s^X^^5f9LZ(WSb8#)3d=eD%{D? zlrn0;#4M?xphjxAA9bv&y1gWOsF?uiQHfKliW@2t)BTyD_2!t*Z8#jxz|7qJqgZo% zYDxm_>pfZG2;DoDL!qP7=eo3}qfCWiLqcA%%?k?_eHgtBZzy~J_y^B^=+Mq@R6TpY_v657Im21N4FMt%7f0dk>ud6}WizO~ z$&5lvOG_vGjSf%vi?JxBeCn=z*^|CQr&4)A3s&yjrgi0>G~w4TuiMDvsEIKVy8ZOk zzd0uTi;L9%_{5{4nZkno8rT6sA|lJbxBiG^h{?#LJ@|w_hL=a#kIvb?Ir+wTpr8PcS<+4A%iJQ;93TTbjbrHYb2hRrYC+>dgDT2~@)i`*Rs?ME&=Nup{M+69KO}()BWuNjlRplY?0jytW|mlacu>Wt;A^G}s5(1Wpwop( z*fq%$A6>)ZDbwq?xug{>^2k!97Se$PGeEXHX=-J2b-HFHX&We1pUq>~5?^ODg<0nsHy=D%X2hV`@~Ko(now5upX ziIWjj8@YJOHj6t?{q)WmoMR*EtVC&*-p0^1>9v`&x*3Re$qEaHg@xgXiHjGMmJ$Vr zAOoGeG>YBkj0p0NZbrAx5zeENB^Mg)2{JZ!Pdr3NLk^HH-$hzoZAqWH+vfLEGfx8H zp-I^q#(9)uV)=0tZp9p%;OmNxLfrwtb#_92Y>1`tn?fNzxOgW|dl3i+TmEY4o9Uf& zQD%nLgP7^47JmU4dG`mo&tI?L!r=3p@^YUhUpWwcrqp8SHArHXj0X};;3*v?qZYSx zutc6qvhY8sS1Eet?dQj-Guboq-9q)@=^DAjw!3Z?xM?~syaD8BeZ7Q$4WCAiNu4j8Mo&mVA#Q5Q07AB4(u3(f zHKT6@b>Ir(d1#k?z4vuXB>I~kgnzFDxs$!gs!xE3pFxeW!n5|MV)SR$&oJteoFaeSxwzH;M( zT;aF+i)wk^*_DIp#>3%vro#b7zZLsN%_e+qgQXvz9dFuwSMF3VDN(|iK2%s+d*l84 z_nz?NXtOBGBZYxGRK#NZMs6ObXj8=2?#Y)D`P?R7I2Ar8C+A#Tm9jF=qteoJrj#!t zn9`a>LCN7I_m2zTFh%v!G_5yo+yI-OI70dA!X`78;Es?JMk6cf^jaOZP8q~B_T*Dg zJ<-nEP`WV4uoptnkM39-&)>;HrkPb7z79}^#HcXM*1Lkz$p(4hY|BdpDYO|M=qp(; zjobUQ1!auDM=m;_G^ z-swTmwjKFHN+cN~!$2pHo692b=F_YCc$mI^PtD6AIm2wf=K)Q(sNS0QVX;DK0_G$T zDW6SRNdCsORwFt$pv2WJXLRVe1k%x!AptBf#o!zCm$^yE9OvKNt#sol*aeT$*&sRz+t9(#I-v`Nl zk&GwfC|6Lp8xi>FEPl;Z;*pvpY8z>PK;T5?p<- zOLjV%9s-(=*eQRcCxLUD%7``g^^<#DE#4ctWMpKt=dug7oSrQ3X&W*E2-)iXv2??} zS$m+gb+yrWhfPyhzdxBp{J0NW`Vbof*+`Qh21IcE)^2&Z*^@3TmnX%4AM9U?70eZ? ztFPzCLXP;$;Hm%bG4IUk(oC>F6J8FbA&uRLuQmD+BqqDadm)9$h}pSkcYkDPZ96;- zqtaryefxpbtz6ubGJyAHjUHBesG*E-2PFOjns_?eCmR`dY8kSaX~T!{omUt>Q=PzG(nUTc+oud*YDJa{UUAgH8Qtf2Zpg11s9D zinx?dZs2QtBQwwcN9_*URWYU>6bjWkN*V#_CN>$@;<%3opXMAE0e5Uj0e7r4Z=rMV zI6rJ1y$;3e46z|#;V74YBGf-{T4+tYV1S-o{Lml(<-Dg@GkrZMt`HKN^pM*|^=XU) zJ6Kj8Gr4D*4)i?yFg(WAY!fwG%BzpQy{lWMH_jQd{$p;AO!ZkT%dx`BZ7MR)NP@(x z%zd^25=z45SLj=ML4l)I9pZjlsZFl`o=Ph~FJS+j6au=vK2!I$I@Erkf?JZdEfLrl zwAx&}J{0Z+x^Q~PH!CbR^Rra&rk#AEkknMB%*;$GiYWT*QPO^D#Au(1;4|_6>m;Ed z1i*R`0)hC5_WY@-XAYu$^}V+%q0h@Ce?g=-_4TJ-$CO8OVesDai$iKg0zgPX9w+54 zjzNZNQw6E{@kh<}l>#Ei!}ac8@9aD}I$AA}T+6*Txv&uTyj0t2zS)QC;Au2ZZ||N? z%3x%>+wqBivAhMVoY&_+S$j|JozC9CkM0XwU+0D>38X(%S0{oD2dtY2n>M)h08D!| zp(UC6Q_-A=oBPbD<|AILu0_ZE-od_ZK&t|5ovBG>NlzqyPnQ7l@9;1!5Kf-p^D-~K z&Vr*t7-n9D?Pb#WjZg9%ObB2D^U?7hi`#upulOBZ*^TmP`bO<8krlDH6IE_-;fmSb z?mgZ|YX-NE-feil`cH%dBLp*dTUTgJjq}yYl63Fgm?#~3ttG`rujkQW!QuSh*;@-5 zn3fXIP}+EV(}GIG)uE=p29?RSP_re6B8HH1d-l~`73KFO+~%Ua&?Bh_%^nP6V#g0u z-hwY6+%5DpDj}rGdvfLp7=5Gy4qc29&u9h8*=k7O(><|B2cy4N+2A@84%!-abGuk5 z9AI2&IyFB4hMw}9t)R)oOnv5UnS($QiLZ{+f#W1N2S_EaIK=bjY}xbD!Qh-lK}D&y zcb`1TEa-XXC}_U8eJAag*_n0kcyC7N1CNACw>_EW-ynH5Z-s3(s?#8cfNZG-gGmkg zAHF9o3k5-03YR&1xh$DQdZu&{-k18H5XfwUdta7&yYaX)GAI?MOZe;F@rrQ zpL^!%wM0)MtpF0He)aQ|_T8R~nYIjRTb`B_gB-V>u@(LR0e4!D27Lw_Ey;s>e(HOK zE_034ph(=@sY3A7tR`qwuPW)?haxnQ*&22S@5PhL%YQZd@Q!3lKYzs>7W$VLeMplU zXetn7wW4-rOb1u^OFPE;cxp_L;MXX=QH<6#6DkdG9w8! zcxAc%90RI0D#KZIF~wfv_k@MOT}3A5(p=WJJz%Po`rSKxs3Gm;Du6^HVcSjUghzM2 zygt~hv9$cP<1G`*r5UPH=-bZfTi1ty+gByR?G}s($_<@_T<4o__QkVW>M?h*X;P>Z zzrZ#p-1dUmi##nEqib4nyJNj=v2OXNOvUGI7e?84cq!)|)BD1Tii+h_7jFI3C1(cm z>@_7ZfB-|p?22}AunKc0Na%cv-%8!Ig{ik}7zE;AX7u2{3ut*oA(O1T_)1ToQZDaG zfmWSvKCRrY@yi~aC%a?EOz=+~*%i29~#()1_Nn~ImCdSSh+dLe| zx!{z}1bH!zM=PSuQ0GdfT%@MN2Vvpoq60Ci?BR;YE&2;Z(%^8~FOuv$JnO zXVKO5;Og@qxRU1{MG=G0#3oBlk(8Ckg1loywC=^ebcN!3zGTLqwlsK>d$x)6&;-&DW)xZQS%_S3q#YaaH?mOe1p>hVpMHS$i&S2YI{yvbC&#C zs7~~OMY$yi-iL}|h!j1z1IQes*U3m)iXm2!xzVsqYWOtQ`h8KzNGgciP#Ql23roH2 zWn@O-c$wB>%;t4)S`3-PX^}S?W$M;MDA;jUv*yZ6GWLsya&Q$XQGOSY6I@%2y)<`{ zS0Hwa{Tg>@CfeP4W-8V*bmiscr94-O-P(`sREow`8zfC3D=Su?oM<}d{N`HE&iM%u zl(|g?|Frx|^n7L}eEf?X2a>o{FBS4e!KSg9!F-nUaxJ*P>pa|=y?UivQXZX2hjxRu zp=MRrwX}NNgh~^UmO=C$^*>)Uuf)k>H!BbvhQ)7Or1(lRi*@xKbNlr{ zh=r~$QO5#y;6-;NIo;NzshQaQ5aL%anI~nS2+|ahWgjPlhXtT?-Mm`8V$%C08IrR# z#v|VOGrcIEu;Vw9S;%iYxQ99ZC&dH?pG_jx*X@Q5H~9gi_DiNFS<)Apte#Dpp~XHQ z86-piVeu&`ybNX-QmIQ@8FR(KGLaT`J{dq~Rm1z2e{nwVSH@Hj}c$o;pb)=E+ zoS^XO2c(mJc&G8__a;Gbg+YG^@KEo(ZPhp9v9@SYW*5q5O!+0WQy%PfUHzMYyX!7^ z80F%xs7meNlr;NjeZRhy>)E6b?6#g0w)qt(`m%o8-~0QCtIS*MhE6lQ0+Sza+o>&p zu#GA6PC;;KOrTV^ZPm|W-&&r1tH=F2eR-t)#3PK*pB_HVPRgGx)KvHL#Z0X|<<%W# zKhY9BgP?9J>65ko^j1K{CFs7H7P@L~2ed}!2X}Bbi|s+Mr3iYLkex`Vam~`E z#2yV4K`>l2M?d4im!r)I@Z0zk)nEJ;-z{Z6y6(veL)VZU2fbkVyWB1z&V{kqVR53F z8yv))@vy*)6*@I_b!v(z^xvr2k(8Mxue-S+7yfg8i$Q5$B_{&6uHz9?_Z(oLdUan# zSLP3QBBqH(&3I2r^9wt3Dwj+*McUMe&#K!cVZYyfuzADuke7`FI>+S6w`|-p2 z6!(x=dh_5EZuBSkBlBe&_6q0qRM+N=y!PKkX6WhnT%4cYE<3<~e%EMqUrr%{oI+FsgTDSN zbdKLly_d1@dcS0{)lrfxxWK?E0}pFa*7Y^$)E5j?puLuvBGOOgTznEKo0)hj1g_A= zzODupH?|O!X6n}UR= z7pTD4UPjxUJ_qk|F7n)L36)GE9Yv4G?tC8&x$t73V7>Wwf1u{!{hqeK0nm9$+%Fx5 zzyd&Pa3iK@oM$0$U#ql5M0!t)o3VXW4vKiccYH!(0fCJMEWpuuAuO4sz5=+@(r@%U zE6fjUg`o)7)%50{CjzK*(2G@?)R6$1x6c{Tw(e~Rc@lxJQh~)1N=J4%-(Wv``Q8^v zZJnQ8_(gNxXWIM<#5@I>TGns`u}R7z1JC)chJx7?`c{h^5L_@3eR;K_)_S%#r0w@I zsn8)>1r|;v4c>}5#8{Ncsi57|;qu}fb}(s}_-dS+Bto57IB-Cl%!0Z9tqEV_rCWgs z=>i}TTz=Q|Ou>aUsmVkF2ZFCoZ)mQ~50phPJB!|y@_-=P4w-0T!Y9l#N`wM$=XyTz zdDG=gpp`7%G4Z&VG_kPV8?_4YBg>jeR3^9H_T34f~M^5;4f zq3vfJpxD98<NE#x1F9GrNr2;S$2I_;->-Hrk{Nf*ccNX3Q)~YX# z_`d2L4&1-g09h*Is4i30HMiL3pXltxY4=Xgr94oO;KSNH>G@QD)uO!)GRqnocth;o z@!sJgzg|Tx6G;Y^@lR)4TsgkYclCG~TJ^29IV;3qq8U%L008+5t%ng1 z-IAqf*Nf=u-N!zu!G||!3_IGA&ZpetP9Eq4)q&sseYN`5uWAB7SfWRAyqbCU@5?oP zU@n`mp1eriO6W;pbUN=fVOM?jCFC-=kFu99qI|H1pdtBVk2Tg{z#85#g=vYXcbg|s zedclU&#|Y_mnq@8!>1%iTcsdSARModBJk+T#5~VqAs$6;?(FUB z{(#IS+4J|eGFU!F=k-yB96iid3}G2nIIXO^?NzOnCGmhbti4^Kv_&@wz#ney2~qjA zr)L#!0xc{)tTn@SIdTKGr2}y-i-eF>Pw7g_4i-R()8WQ)4tEygA4X9CcCc@wD&-DM zVt~xg&W$IwqH!>OJBFH{Is@NgjQAqKP{j@?RF?Ra?=e@*(3IbgVhQ(!cYzA=`{+^G z`P8IOTIm2l)d3CR1t5U{L97NO4&T4O!oU%u*$4E`X$graZgqn4 zlfA7mfRA_l(7-KHMUN?si7#z*=rbWNm;_|zyhi1dpAWdTY=P1AOV_uBrbR$BV!x3J zq5vgsRBQKsV*cLaJNWP39~WR!7j=qHzM<+8noN*pigc{atIH z_-`!uU043ulIB+`C_>f9?u)_0hXnPmvyq0DL6acPPxRMtd8M=y%CPlJRj~c=^4VGP z0{Wjnwbw8nU$}Zr_sc`~#*LD!{PJ?liD9LDceHh&+*{A5sZgiW6OU&326NCNA>SN^ zLzZ9VH%UWO@3q^0o;ly2$#5>d3OJZPf!b5;GR`)unvEXXA z!Hw)b<(jwnF8!*u(e^8{%^uPOqku_hEn~SonC}{Ww=&l{A`Ko27#^?V^eh=#-+up1 zHn|4mqnAT;(|J00fIZ}$y%n=b3HM9%lxA$cwWc=;iqh())(>(5p?5sW1%pphOTeO- z-$`REM^sdhg3Z(2-3_*5nwhp6c^Qv?ibs4VF$C`SY?za(%=`S+ssBU1CJ zkB?i)=ANi+BMW%UYSfBtumbq%Er-9qSix6(EV>gOWIgInzN^gsVt^)%{!<{?g6a$G zjQwP}KIjxwW_oLVhnY(^zLL$VC`Y>A;UGv~3@E1!F)(~Iy%sSr9w#5Dz<_*9jYu`I z=(op*3M62c7D!h2_m=gI*%8rH&o9L<7w5(+u)_P7FJTh-WsZzcs5Jv?9!sU&2f7Qx zXC-Qt>$r04>CO03YKnbBI!`FP0sptl8oFG6D3!N!&?_om9_L_;5Pc}{5!0V8{8!Xc zaysv!GF!k3?q5cT{rxlXg|iUV3>icNRyeQ3DrPZ0o09l!@e)q;y;5S3y8bB3;c);t zd8fG_UcAEQ$c2BSyz3~-(c*!?(2OM?)IDRES?H~_AyUJB1QIWYZ`O$Av1q|P%OB>d zSm)81RyF24$_SI(9|bgV7qhAd@8-wMl5X_AO-$hne0;MCrQ&`GaVu0wR>E7g>VmfU zjqNAa*tCJxvh{yj=>_bBdFOoCzqOr`?RbGG0wtk6p0k@2Ox(+_!ujV81#ZN1>O~wf zwd3SJF7C6uIB?ROsg32@a%A$jki-5d+WwmtHKQ39XU&uP?fds@X@ZtwvJ*yxh+1`R5b& z_4G$-`q68f<=!S2p~O&UOkCTCiWg6O+yet|0ria-3jg!>uZp^Q)P#BF4R`~Y_Pm~~ zKh4n`I`kqKCA)~X-Oe1{_m=hVP2zq^^K{O|nCnlpv&ti=Mr=hKPz!JM*s_p?K;<|N zWZ^ZaRI;%t!rE9}NR{b_sj`t;;Wo)yER zJ2a2Dldn)jj45LIA2OY0cesvbcF1$WP&&B|_4 zJaxrS9wmsRejulC^Sj&J+&QQTK)&^CW?sz)K7@;l;G;l=O4Ej_j}MBNg<+ZrLt39# z5NVc|Oh<6Vgo8Qv5aveQ!e(3gK;=0UUr9kRuD5r6fK9u&L?U|kbpF)`bcU!sK%*^} zyZzJvxX#bdfv$fwjAzQVTjWg)Xe1yDok~gH#S?_=-R=C`fFFPxMAdP6zui4vMRq&T zIyeeT9;M}l$SKtAHkABcYyhG2x@UIrFkgl0%681 zfl1I{I8`TQFyw3_Z&1ImFAmSGrPO-+Ct?4^X7erb(uofStcRx+il)m)yDPuGI^HOB zGZuNIFDW4r2@17?SnT$;+mW+_

bj%Re#|&0Aj3z>cgKjV>>CI+2p5v%%sY_S2Ce zrx6EBF20*l$&2pi_{l09VJm&{D-(GaU+c(|d5d-L{lnoNKBKKPt}WrV{~`oKTQP_j zaZo$YV0MVwk5o~|=rBac?W3c<97L8u`4bZ~EmiY#Ztx$ocNJY$!Q%s24hvhz0BRhpwo6 z&w26}52iadDYCGiiiJ6uRJz94xR9rhRIn9dK3#1;n!D+ped})cm4m|4lJ!j&+v|V; zXPK+5iY{PFPqugF?^V1M#ik-44U1LFzp7Ptww4)V3r92S(}o7vBhXMMx><)^8*#2TsL)4W8=md(q46sj%Hoj6L5WXz?xP z%L(1$5*?zkbCt?SDR7Cz>YW>*(2AGR+Bq`fO&agJOie{H$Up%uz{oT=H|IN=%!dnf zqWCp(UsF~7DVZ5dnCoA+S0rdZ)qP4!Ai>kS*MBE!XgT|!o zgXyQJCUkDk)MwdOr&~n%doIP9?BSh27z0ugSdG}+++5?KK+JWxAC!F+Da1yQ{}mN< z;)tTWr25rJ4ck@CCn=4e{tAYvmEZPx5mlI@KkABa05ASj`?+T6nwIP~bi+JiR=f(d zk?ajQS!)pJ4%zR@N@*vPT~~V292qY3ibV^zWL?v#8?%hjL!ivO>5ifV0{qhaPOxM> zAW)iG^p+0OGRW9eZtVGE=xcNiXlJVu??B!0yuQx?H~p z93Q4rZYB^Vxfpn(<75(%@5toVc3=>Qg|lXR=_ifeO&37#JATxC7T&!r>yT0c6w?2P ze_}mji)rf{ivLBNFZ#Yba;|E3n6Xn8jc!V-!Zd>ZSUX$ls(}WGL4lj|q6BoEktcI( zVL{nr7FSwgJ%!oua{G1P_kc3@G4IQvgyW#y2}Scw>su@ee5FyUMV+FfkQU$HvmHG& z%BZ`F$Je*pa2BR&tiN68O0{#oHF=@Orl+HTW@8OHV%Xy8g5fTWj(#d@B%o!o0APV} z=TmuXo%8h6s)!}8Fw9-ZungInzxbN;=&ya4zm&3{^yv*y<9`*S@D^O z%Wz>I1>>cYNL$^m=?)6H-SF&cEaLJ$I_LFd5|<{npEKU*N1Al=22i;}LKML^js(F6 zq>0VeOkLwn`zi1}(6u$ZX<{}JAR?xIAk=?SWlGEr)N7CP6Q8dQ4K@dxli3%B(A*}d zQuLqqUi~#x_($m>*Ovt^MZ(Wdea?C;ZNFIed=tGjXV(*Z^HJ$Ua>(}1j^OEZ2LMVS zH?5o;U>Fz~yFhJ-jEYJeMUMj&b!0@F&$ONlWM;z)0%q259lRs=vuUEv(851DDRK}Z z3B39YQ$&1*ZW%2gT7aZ5J~uZk<3_l-KU}wuUi%@U1wUloQFA(%;4@wZL#dVA*%v;^ zGZfK{Mog7jicI?Sj|<~53t)RNaN^<48?|w9A=`yrRdE3RS7sa_8iTBl1F--)ueY~% zKG58OqlC|Y`qQ_I>+F~sQN>~T;D`jSYx^mbwb}VR3jGj@<_ogVnav98I*sN<$L`eaf-HAbTP?%3AY7{3t;T8tIgqWD?@8aVhPKb#~N&<%;BRM&_QM32G*RNj#n(58d z)RggoOjFA1`+P`BQI%4~sc2ZB(-Pd@X826Vp8Esqszp!Qaa!$n$sFyK)q!N-Is)mr z5@_+$6q~Ckcz2;>V&bs-pJFgo0PC6Mk=r7R)T_=10_kuohJITSa5bomnI-IbqriRu zd{5dJS4v41k|Xhv(uCN@iH(7H5UEFZ-b6-1!@J3`r{_m`m(r2j+bk*^?;YGNBYS(* zREo45TwGM7Ze@@Sf{T^dq!?3_`^!uttjc-3LO$R*T^LXNaBybg7N_ z!>UqFi))gOeme+6V z$#A<(;TX5>b(63kEq)FE!Vw{OcIx@E)>tz=a_V6KFbx9HM#}q7!pb=*6dtWs1VRbC zTGd9Cg@?fJ=5hGfSnhZ=8M1odvHbn|T=41ES`fZ?N&6dm$@NhDM;x*AuSgG<2qB?` zJQHp-43VMbBigq*I$+3YGOhhfQr7e>#^j1xTGfn0t4v0Z9iv);;FpIDk64tNu!?7+ zf$I+f*F`BuPH!cbY+#Q!xAO>)sQqIgDiQ1JU->x#72F+9-(rzuHbx`gG|1z)l)z;cX6VFquzsKV>{**7xsZS-*QCDP&_Vt#Zg7&CP{P$3s~hy;!-uP$CY%b`Pp=Xk z*YA^n)+ASt>#tE}qxv3lM)V~B_wMd3`TGPo!;Z{4)8jP$8j2@+6(BbMvu@2*jdRJ# zE(e{uq8dU>1$NGt>BH=EGd|pI-ffgy$ALIcRc1`~t|N=%VSsnlE&Ew`DlaGZ4%cyc zZdC8WxZQs0X4dw)ZWT?d5WEy5P+O`}a(` z28fDMV5mV6p^eNBQ1H%xMbfh=_nWTHNG9Vj23CYqda3`varNrg2TrZKz08fr>sh-I zf`J8Bk7x&WhC)Q~i5&wJN<{PuWKp;>JFp0t%CLUm5CSXfN#&dltreH+ef^j_fEw+K>IOVfM2?Hw-eujssjh}pjb zjlBnq0pCIEqYB)^tsyR>GP$32Sri2@jk{8y^?-EjGLyEVehH%lFYo~;MCdmnVJ|T( zi4@9sGSbpxEt0{vRGR?mSGCJ;5r`7-8PLRB^ES_ua15#fFAm(w7XRw>LjLc)B3;l7 zk$zr`nx}X5jThl8iLvK0i9@?|(Ett$RKuMh?8vT|fkj05&!dSW$2JZ>e3trz*2u z@7g(-3R+tN%9KWT%tK1NQ{0fOI~R{QjG2kaW8+70VMzKDZuamG2hUu*199_yE2u6w z$k0y9%LB_5(9Zb{Tm^5pk5%x*q&!HOn2ds=5P_f!Q+nY^`Erk9nsDc7F#g|P?CK32 zXT{mq4fZ<9?&c=kJ2?Wr1zxp(UBxrv<$ufNaf|=s)O69$tYoQsJxpN{E?+Bda{QUe}x2IJF z?9k?I)_$&0k>RjE?CfcR^vkRl526_*Z8Vdgkht#Kv0v=6@@`J|zE{qcl{^=2SiJc% z0Ephz7*Ky-0guWnKiTVQgpHX7_VMpnwj}NY2^>kP4F#G*_)`PF8?| zd*1|8JoCaCJ4nFPf<5iX%iSOAPjY0!{(RmUM$q}A*ptE}LPlmala%8w(8qxR9VDLB z+=$O#TiIFl*4YM0;N3gR&<#1X?~J+8Y1od>GgvsYe;n!_eBF>!t0!y~e-t-TTpKIoiLqNa+Tc$KNrvzj{RuzHE1%v$RWA?j!MTA5qv zW@{pz4@UR1TgY$DiLD!FgozPZ8Njb8V>QZUyXR2~HS|>}&Js1CZp#fb_Jt^}v zg=xFvG8kaj$gzFJhdxN#eomUCmWtTGjjn)M*9PRNeTB=+cNsxyeO66!!eMGb>MMs7 zsfQ(a6al-<246r3;1#&|JnTv7x*Tqd%#gg&fWzYQEA)l7ipOKpv}Vt5hk!c=LY%ybUDK diff --git a/docs/images/chapters/pointvectors3d/f4a2fa1e0204c890b2bff07228ba678d.png b/docs/images/chapters/pointvectors3d/f4a2fa1e0204c890b2bff07228ba678d.png index 1b0e876e32f54867e912dbd51b6e3041d93198bc..70a69df75eff3a818f1a141958e865672259e404 100644 GIT binary patch literal 12694 zcmaKT2Q-}D*X}4mf*=vS_aJKYULtz*5{yowM^E&qNr>Rb=rxEof~ccS^avu#L@CzNhW8_p_h<%qs)^hvYXHZa^RqaxG1DBM1Z!3VsDi zh`^OM$k}h;50Qh;Lv_gI)lXqZc^U-52GLSiF$vDa&WHF|yM{^Oc4ivPv2bpyjM{p^ zS)<&e4C$uZ369G?I?w69=q?;^JKut8%jzz`sPTp0e`{9`j@r~IvM&21ETWc1^F+@0 z2G(igzMVt7bW#rC!1dS50-bgt=I--mAu_IG+xhF~%SW05&$+Si(fr}4MKzO%)KPa5|8Ey{ji?`3D9B_cB&_jFwK+-W>6bJ#Ow`l5 zgj~I?BhWLsl!${PM>P0Vf=B-%H%C2vG%R7zg$(-ty|@4b+O%(CnS)Mp0kqb z98~ck>gs#;xEI!n1p}zAi-5B&^Dcad$3iDFh0>XbtgI~4Ha*oyEE4mqw(CMRE-udJ zSMT+?u)}$aCJ*}2T$$*pDs$A7ev!e)za;Kp@^h2W3X{*-67GWT5p{I!EWBE`wh=pu zN1=*Lvi^gc+mx-^t=1ZnIS=YAVg!dRT763=Z1B4m_XTZGqVn>Inq2s%rn|zIe}wtc zbu2nVWr&E0F+vh8VT=mk@`akR68gmf;3ly0=K3Mm={-7k#73G==bAdreq%~V!0(# zo`LD{3|`BAbDjO&kQ4Y&CJ%sNuxm~A?7Q1uGQ745kpI(k>Fg)`D%WsqUsOisI>e&h z=~bRhZkiI)P1N7@irF`tHPcYpJPrPIRjyme#YHy!sP1;G);qS?)wt8&f7gc(en7=* zDg1mkGceJmL@-0@Yp>MtRZ0FH98BMrQ!yo_gs@3H>TG9=sX3+hS+jceE(T;$;yXGzYL*&pY;Ao8y@qz490+|` z-|JUuS*2>E6M$1lIpAdZ(uUYlG})+c*iV${kq||bH;T(-)wQ^)k2Y-m+Z(}+So~YB z2w(ScOjlb*TL?CNIPh&{is4$U?~_p{uRHnorlc>cVn!9z-#7o7#3NAsVU^?i(Z`-9(~)Z z<+X`%n;oCgBT=5X5$QoF{IGq6#tRJaHF6dL%pCMQK5i&z(>6KooVZ%ATxUUe#ywj-zH~t*U%gS;@k|oqHCcp&8 z`BCv!jW)PZ^xM%dZAtU~%zV^|mgK2|c+SchGNp+OC~Z`HPNN=fw2?UvcHRCT2H*|Z zobdihdg?UClDakZ`l|T>u4yaKzwPD+H`=|z~fI;x@S#nrDw**XYdcGE`4S+CjgRo{L^ za)=iR^ue=gs}&<`Ab4n)u(q<2j)ldm3e4rX^VXQ!kgy7MlFc{gnU>vt1;)Y2w_r%m z6@yJp!BL7Vd%A(edS2aojFFm#Bt%C3Q=o5YNu#bDrR^=b>6D z-xm`c2T9?tLbI*{sUJV-Ybk>EtZ_HkMIK+k5?y{*onIMJ_i4{K8 z-%rf_jOk&NztcYBK?D3!lrn@-%1fP(+P6*T<{!PWrM`V5wP$UUi-!cq?Z#o8ytotG z%}0dZtAYMiW;J3>Djt5SntasM{sA_uyKydWgToZzcJ8$8itPhaNGne{Ss};!#|C_S z8yglkA>W*t73brPD(*M_8QhWb`nXFJ=>BVe$p%i2{^FcC_ZH5;O6}Y>$)v_Ht-`CV^V-Gf+s<vV6#ALafEzs9^ z)LIL1=IN`Mn)(L>6qS?^#HtppbxI&2w%uz9TCD@OFlKzzlsIA$d3kw90VVbfw!G2L z(#4-gFX|KoW|#NtS`NO}PB=h&zFf$Jul@Xu-2lFdvC_D8H?h*CNlMf(uI$b?5N|95 z;|%x+E7Y=ODTeQ#HvvDT; z*C0Q>*IhGGRvl0um~@<|^^DmGjVNk6-t#O!0KUF{M)23JH1PP2Dr1959(j*#`P1Kx zxYT*fh;6_9M4}2QHQMdJs~+xXOwHb#82=nnCIuYU(2y3{&6~cqu{5ry3FfI>)z+=w z{ra9uHI_f^#^I!aPyXo8cLUP>>NYO~Qnx2M7qZil(C1)0CQOrL=-sELhh?30*9G9B zx0mp6(39ZoX2`o|Z9biDSZ@&Ku6){V=JROJY30lWrq7G}Y&HXlrtaw(_pIXT>gwo| zwNj}%eoGK08&fBD_8W#u(tyWhjZi`>4%M76~9r9`D}752w}R_W;JTbG^@38#dJ z1S>r0G*Wx?=+V^T{&86{0SAYxXs|+myNYDURCWad!Fc=jZ9elXB^nZ($qf!EJKT%J z*1FcpN>d$j#R3W&8yizhJCZFLHb@RSbtKwJj23d;+}v!n-)HFdB}lOX9hCX@%9-Qf z7InpLIDA1V%2vE+l%qfMJFX~)I_dSmd~i%u#j>hT6f+nTat9XoGks?=wQ0zR*5_aW zt6$QAH-MigHq3QbIIE8W7?%hq$d(KWqlEy+U0BIKDRU6L@&PtzNd`?8HLF=&SCHtP zOL)@ht5!GI0bb|;48IMomau7jtKHuvP3~tJn>v%DV+l!o&tCrBDFCu;FHca$#J1=U z1e^ZXQ|~MM4Om#8g<^M47PH(t(o&UX${&aFQR9n* zm6=3DC;^ZtzV4o(t*pojqC3^_iTNw+R5I$}X>M!3QFQ!2!~G+5q2U)s1dwXW1|suX zXx`p=c$cBuBGIP$fg|ok0-RC&7WD&5kzfP@kutox){Z2C6uqlck)@h>gT$~mRzDWk zN{x|lpI-d4I_#%7kUQU<%fFYPIK)jA!pzA`}R{3+9TpZwN+fJYC|sUFH4}}* zP^F7AR5Bln!%B5v0IF0z#2b0fNx4Wy%pnKNr`Y$Vw*n`$`$zNA+3_B!qA-)>OIQiC zcn!>6-MOMDpGp|w@#7;Nt&p_^?s>gO`!khmhJL_^yK}X9w33p7r96H_?yZlF$rYfe zZ$?YaBxENdF?efmC*^Y~6vR=W-GJ}YVkorR!eD9d*`&^-E?7U2AZ`K%0SQQVnO=_1 z_FQXrKI1FMYokLQO2dCqb8B#MiV=ZF+H01b?Gqt-sH1leV zOIhwLPadI``Cxt!eM zJ(pOc%A=aOKl^(|A?=F6YYeUG&RKX*+9x59=;&zafGv}evGsN^COcf4^=guY-7_Qt zcq7eJlZj7a##|0xF{^Hp92k*c8@O+;6(9BD;VHX0&$B#`o;*AGyO~`eJlwylY;B_A z#7tB>Jc5pIPLXL(m-I$ko44X+qI;rF_itH(-Sjp4x>WPYzK19=F_A52Wco~0vEZ5Z zwQ?BMe_B`ejz2z~;K`NoD}yJK#zz?q1#=en+2h(QR@bB>B}$39`J{5?pVz~aQ@A)o zWwy8<%M;bU8Oa~Fdr*M~@oL-Tv2G8pTZeDX|0f1c~KsZIuhmr8_a z5ZHNwAzj536f~r!D49GyN%4;cqv83#&7VjutlS`xIo_T0Fd>_l-xrxg(N8$t$f8lD zmDg&gIMK>%k$O3j68>rAB3|{^@N;K%OoTy(W6I)l9imF|BwMm^2X5Nrz3|rDHmq?e zIaxi@Aw}S4!WMEdpu$aRTD6Gs{fi(j+isBGfHNC6G&B^a2u)2)RZ|>GB`~Z3gD`zw zFJc##g}hqx0L*lpi&Q2)UPUHM&a}jimRq2q>HEPh#Ai zD7c^wntS$eo>floC49huZ-re8q-;#B|d12-8z zPoccLyrY@1@HkzbXfzt#rqo%#P|2Vse^Xb#Y4orMeh?gn`zqo4j@fv~1+ki8<#Tql z`=F7EtbRq=gp{b3bVMS(Itz(#9;UfEDy0wfA8l-i(3&3I+oUxVaKN>_WA+PneWOtV z5MUn&s5Db=WrkHyoVlyY_d&E9Cb4gyu>os8>)pjX@Rr%iUoHy5ro}B(&6Z6NjdHF` z3XEEVF+d!n|JK!s0#g-iyQgJjD9G?eAmIrQc>XJG`-q?fq(2wUEEl-A3bSo0l!&q* z8(eJ(;~MuMgTk@|g0XzSLx6SvcjyaF1c_suOp>2r31yZtztk+ zyGr`KmM+hv(sRVMef?efnMY2Wylfs3v|I{<)>n5tnUt=3$+$^L5BAJqTT?QGHP@(u z&;~_CLg=wMur`tdh=I2z{LWe-iUp#5g0N)`S#?M>$~3pV&TMFX&mIN03Eu6az2`1Y zDPq=Cht>fpaC++&B5>ZK>XSQ4h>V)b z`7c63s9vf29Q?M_nI$KTG)UNoy4iwTGTlFsN*lY0%s*exlYa5{!zwner(ZZL6#b{{ zm31-_bIWfk2yb1RJhIMa1>|{xIgicmGDx^}^h>6yV}TdBRr-C76Iu?AA4t&-e$Pa& z7et`{G;t4+*gjYqk#rN5_Z#8HZl2s)@ui1Ip$D^bb8~kCs-~ghWB>eYe=g1varW%@ zOat-w6aR7SJYS?P51G~%)zr1ll=ame?|WCa*H|h1MZ7e70@_OE)S&FTo?5vhoURQR znd8Ld=09H>V^4@7v(4TE6~^U0V+w~B-_aiZAA+v`dglJqDuEGTVWc=}P-Ac2f*|*v zMCNuIMyUPocb#w9ChwBe6DwSDHWPA3Dwy7O@{{{)WUp9T?)h#sq;*f1btIN+iN8A950}SpVF-BPDReM+D6O$fbeI z_Ky%zts2c^X?tIHVBe?M=H)Nx0!>LT@Rm0UeKJV}2N87bH!DgzNm~%$_yRsHXg&Z| zQv%?DHS!wGiYZvUNh24?oQ<0Z7Tv*U4!2ej%lfn*&|~lTcNoWAYmbwCpFvd%$9%i* z>6Yf#{O9!i2O>x~|M?pItZmNeQ7WxyFtZ(+`p-Swl)34hB_F!)s;m{5@ z_Zk#+5<}x3kLd##j~mIDwcq;=wip~|UjC)W9`IB#szoxVOHJaXf!Lq1(Zm$L=6AxI zxv?XXwjure-2h4WYo^w3955g;JssHArIQc6;wYvk!Z7hKWJTZ3u#fch0i|Mwjh}ag zPdNZ<3qD?jrTk0ld0vK-*eAYlmPv^H^2_V&`O=kre|WW(HNJSj=$`Im+20v<>T`s% ze+XTWM;4p>+ErQcB7^2^o_uCp&YV+>Z_b|eLX0#iFjb27m`~7ku0TQSl~N}Ol`b5czeojBtIv(j>@mwUZ6qdmbm;3Ic-0 z{$vo|P6mT6;9C;ZIfD16%=5OtBzA=AaF>UuVc&ws9qKNqYGKi|5cHpn=0)664FnH9xbX1O(gbBJzyEj-9)MdvM; zjXFt$P02j)@9c#5`;$D@TO;1#;^Dh*YzsG1@v9anG>uqh1cz^~fRp06MG6&Ou*mV8 zsx;l$*%=rwc?bqZt;N54Cg}@+gycn2Tb<;r!r@$W$-`t-L$_I(uQ#J0-^P)9vxDJa zS^gQ0q;x=@2MIKOPHNO1&bZfeuX*m*hS4uKd^|_-fH>iqS4j70hw-C{CYBl#HxXuO z9|#DTJu55kCv39V>2hz{vgB(8+Tnf{np7$UTh)ML0eGqAg>VFx*m0VJ?*6FHuoGS9 zWGES$=l17zzfl0lujupO^L>=4q$HTT7v&An&>j}oJ-9e5XgGZui6o%E))n#Ky8a);DQZ#n!JZSQmdC8j}jNf&!UkjM= z4q)OEm-Jrty1GY8d#eT*VLd48w&$4V6?g`3NlWWz!JF|Yf*$~8whe(;Kckb%Z#8=q za(aO6Ar)?qJl#f&r5VzE28wd%j2)g8YjQ_r9T;8`g50Uq)Nt#5EJC&NXx0*dY z+@8O3FQLcvU_Fc<#@Cg53$6P<`YEA~IS($Qe5R zVb?ZJ*zy%}R{ae*+AMdEEKW zO`g$ZPo+Z+s=@eR^qrMoj0KY28BqEt6XMR;2cpmFg2^0ma;S zq9Ah7^A_5oYJNm&fy@T=Z0#$3IijAeE8@jNaoA-(8!Z@ClnjI0C!=pSrKb6_O3*#= z6o~LiQo2)F!sszZUa-`HqmR=bG{=tPu z3xvX?rB@Mp27(N=LPeQJr{=G3}%u(Octn*W?g zYTiF>V!lG_zFW1fN7O{tpbOLPV_j5|k($E5StelQ^>d}8{AH8Cr>KY>5XKm}O(Us= zP;u?jRE(>1CfCr1LjdL&GCNg0-yq%V`u?FetA!@jST$c&IRJ6-qccnnT=DcIw$V-H z`aRoBF($v4IAK#^$!6D1pTkZy6<131=FXe`B$&okI=d4*AXny9jRiL{N_u?!7_{9S zfH4Cl9})oVRc&nxu13p+|Ku@b)puFSq@qJ4_@0y33orPNckjjdXrE5O!%}3>oO-g*oEyXkg8Xk?CI3VpoiF>wo1|ENzpFN-PRuA z(o$>v8hgitk*GHIe$B2JfnJ(3eaOLYO5*9bN1d>OKb1rs0DEyDa3N zclu)rD^JVCORJxF1CMbuo*W(nCf>8Y*9woA`Z=-BCzgf3rC`eN1dHj}JeUi-g7Tz4 zZ{us0M~#j`xGGLk8(j{^+(D2jZfLkYKR*vxoc^t;DjQUn8`fD&nd_JL6Fj{tD67|# zWLJ12TLyi$D>!Gto29VaZH*CW7|ad1*a&eIzdr`N)ypdLS`JK9O*feK$j;Tu(!Oe- z&mP`6bi_>8T=|)+6BYBWkT1#etvWUyiO|e;i4l4zGtv-t8kj2^04ic%wNQBM#U!x| zvrMmkQbY%(d|fPEvuyTaHZkoR988$7X^kqw%OYdbdhqFJ=Dia;(GEwZm9kiHj!r7q z6+m7YN(1LwyY2}gypkcrPxBV6+7L_E{>Bb%;+R!*Ea^3uFj4+U?kBO0)XPBM@6yjX zsPQud?Vm?&4J_Iaj=Fpt``PFYcCxUt(zEs{TH2JF(}R~c35VSf!G|q`xKT%#OMS4G z6w-Yg_?vxL4B+Z#fZZ))CU@EvGUL~~w8|)dM6-67UHzH0l<3(f_b!mkElG79j}*T& zM~zsLsPMg}g0DF$7Epbv_Xzct;e}<#B4^q32!1^)4;hC}!)@LG2z?PZ5G&H-zt&Z* z7UUpml61ocD=FlRJ&S&*S+H&1?znKpc{Fz)W5AXpz&h_mg)h!!q!R~5sl zLL0lQLUkuzlbBH>X{`kArGLQ3y$n2R9dQ`x@y+_|5`1*lkKzcFVk~_N5 z`%W7i;kKf>4S2+Nnp_}zdhz1LM?g50BkuDOc{As>RRkW3;mRX!Fe#FSE|GQ2E9C5B$kjr^TAv4Ij_AJL&gSS^9egK zy{=4^Y7%Et@W!%)A$U%RZA3u#PTV- zEi1eh+VL~>VD4aVeHGKQeeiwrKB(P-A|YV9s4^QL6i%F5k?4?1e811;Vk0!N3F{(M z{6kfue|o+{5OVlD4ZKXQ<-`{}x^3Rl-x;qhF-B{CJFXoen3!E428^#f^1_;EN~^2%kgiXJ|km4)( z+U7EG#&mCN@BspwKr0ngr8kbhU6cN0V^HDrHN~_4?(Ij4Pjx9~>#qebU9(%(z}Py9 zeogUmwk$Rg)97haJikybJZ|8Ih95h7pxfyUn+56Lmkm3KX>#2niBFGpb`a3A_ zf6C$-2;N8dtmk|_OZYhRh-+ul-j=iB;wBl~98{bMZys|}M+L8C4C!wv{I&^gGURNYO>rsLvE8Ym*P(#mK z&(0Z*GsR@uRymB*QIS$b7YlYO3dVGIU0}K#K%NM+0QK9q*-&?3!$29JSw3MLU|p+A zQAO+CzGq-svetS;4vpl^CvljSEE&1Yo0vrkC7*Zudg626Z8Dx)E6b2|)Fr8l#lNF7 zkb*^?#0FL8I+nlYpUcF1Ro(B)5_e^bxgz5Lkh(H{>2 zXkyO>{uHDfF}o@W5o3Xk&sw6PzH~U#(J$Jl<}L--O+pXhESn6G^_z^`uQpe|x{#2v z``{K;auvfEb|NlM0YgLAeotfKJ{bjtnjtkPOWlI}@ z%{BSIrRkf4i_*NY_wo6bJ}dFFG)!qvhQyvwxbX823m1FgcDp&jiGan2O8978oXf$- zed|`=Yf7gxV+MTGIP~F#;*~f9eSuUk2-G3X~x1JZ{>Pft$d5noZ-WN6D9*Otc? zjh-&e?wG?N1t-)#8}AGU!WQw?VN)-LK;6M-WsnN2&T@AYK?}BNwrtzU+{?RyFA~t6 zf|9Q6q!JGsu?$1qi#2R9&rk|11yE`OJw=MMRa<=9bAB9g1f>0Z)$P*qW6eIWh2Gg{**&wo`)Py>=)ka6VmVohNs zsc=NtAFah6R-z_Pf{Q(+sEhE~wsp%@UwXh$4}AMZ2{PN*eGF3iz!L<}T$m_rqv7w5E5lW3(DOD&c+jL)1~xONjVvegW_FU}jdS76@%X zjiwhE?G=AlZBd7Rg$ifU2TgH@R=DFtDH~T@TT(F}h)mxCnB7tE)qDn0I~>#%-J2Dr zlAH)BxRl2wUp`^#+WV%4mR97YBjbP#>v>Wt+`k_a)O9*99kgw23>AhAFM+aE3pn~h z)-91h0^u>&LIv{dmP2$1Z3h}kLZ6km+vvTVvjm^ezZx>Dy>7uHv)0Zci~CLPXP;>! z4?Ah4!(ObNcDMjt*Us~?5W%N9rltV&1K|VsSa*ifzb*zfFj@?%c0g|7SYN&5N`2yQ zSnIeQSN1fLl9`0aav^{en4<}QEF3Nb%Cw`F<>idKUnL-_>a_)GE}_gfSr3*UU2wge zstO3*)VU$)4fl)2qutotRH?J*1so+M2_ddeIGk$d1q$)eao}zCtO514<>h5#@3r0_ zQ`0Mz)g1vdyS~?VEE?VJWC+=`C8WG1beyH6S3QCkQ1KFcW7jIhEHDyF;x%Pi)h5pX z`v8SrCrb@;a}@c&_P5Vm?CN^u%#Q*64m|X;y%8z75T5Td4c*A-NO+0=A>PVqv)$0G z#mT#ij;sWuc?61=$4ucnht*ZaGv~*nLJ}q=K-9v^D=I1)wX_8EGO_e3q>w9Evb~*8 z^$GCtq;Ve3moXw+^?n`KMZ-SFA4`Tj6mKplnleHSF#Si`BF{1aC~Lv3Jh?8`5s%t) zZSsh7R5!08UWYYNOr+o``{i!IV%m3CT<`6E#a5*Y(Lz2nqy2SuP>luplNd_ru#@{h zAp_J-LNGNU7+?}sCL_WWS!Lzq#%E?;`AEwL5LAOC3~=Ke_Qi<0e_qH6x^|f=!ZD)n zC;A;z?}q_hPY#-^{Ew(_t#4~$N(xnCa&ieMMuW;>W^Qiv@v^CDA2V28mL?uO#T1pT!mlO1z#rXcVgzJzbd@YTG2KT;(jag< z)X@P-Y74qnRmtB87z{^X#z4mfXqKbFhl($Q<A%gxm#&@_sb#5s`DQ84HC;>-fHHbAln2cBprJ6arvB-D0*XK zBNC1oT;Buk0gw+K^H_4=KYL%@K0;E2L%NSINg)dvY z=H;|{R=z_LLos-wV^E~8U`mVM$^VpdNJ?>yRF?t+W?`FsQO{7suR?-nd0;9ZCu6>s zDU5LJ2|QZ+Q3ah?yF^}g6QkoPLKnmQIwP8VTgfcFYFUZfQUr$+G)lmnTdhY^vC=Xf zzcoDlN>`07+fPPdKILvPn1w}xmc31uGh3#p1P!*ZP2z6PW<7-kY~S-H)C+8u6zV3E zv|Y%hi#O`W#DuuF-Sp2cz(R^jD1BCl==BRq z(5}!EJ}ROZRSO1rNfvB8Gz=i34XT^fKT1aDQ8?@lOPTt5guB+#5c7V&C3 zVQvp|ZrNTg!dCBwA2by74JchatF}SVBrBfq0ggV45q-H)lI#HFHpV-(2Xy1AKugHT zh_FZW2g)hVW38FnA-JREWXFZhFiKV6>jQpto(nd?=i_s3cD@WkJ1!hNAAdsKYVFnk zPfn}G=Z1oyv!iFBs{*zvKqs=@b$P*t#e+e4G!PU81BDId3wm`nhO{;#HcJNnHuN(H0c~+xh8B` zPY8shvFpDfH3t{5ig6#Tg&cvn6rhZUKnMa{7F`?5(ic$8AXvCm^jZ2mTld_L7aDd4 zKm|Jb=ZI1a%6;|l6jUbBxP%}8ua6ERE`$E=?=LmIE*%1+14YT6(LA}`m9$62<>jth z=Tj`4Vx~fSo=yl2-waHrRXhb_`LU4A4f(@47?3ke?416Ab$QHSCjI=d;swWKO(BF%co)sbe*D);mIJ-_{)-mI6gJi zm*gTp%6c>2{iNpUR8vq-PtV5h-??tW1;iI;V^QAW6VkHhy8}@^()fj)Db5`rn@?i! zb#>jB>++hDe1~le44Hx_&&*0393D!2>Z)M`vNx}>I-_XHZl&|vxL1!LL34Oaazuak zWu_Q84z$GD zeG3YFVw;VYDxf`DkI-LgD7g3QV3q(B4AUQnB4SnX_9rWW;((84JskUD){11-F@$tH zfEmEYE2dJOrS>tVblT1xcXA#cquLR?!G5g`?$@hjf@E6icMt`->T-mdp3`YU5N29B zvSqn%Cyh6Wb8ZJQmVH3;8EF4ghNc8i2?>u4XNiQp;Z!NcKP7=mKlk(hQR#a!wpZ`o z`hPV1E8lPQ#Qj-gppq%|ypr_a{rB3;i?!?=0K3>X+F~As0f8=6N{5Gn--sSIR#hVp z<_ZU2Fc>)a?e*Wk6_DjPfRMvB6a=&pt53p#2>5?~QQ+Vb@23upZOPx}r{Ieg5UmIL K>Tp%Nm;VcTh6tVj literal 12730 zcmaKT1yt1C*Y40N9ZGkIG}4`dfHX*hQc5E^(ukBaDBTDM0z*i*w8Y3u4Bb7%AkuXY z@Auui*7v`6U4^A~HW7wxyb~BINe&Z*FUG5(L5kQBjoBgJ$l{zxOaf_&-J+_!{0PLO0OHWz{B8F){i2 zwu*LEht)+OEfP5*mQ-$(AwJBaz9TBFkuIDqYBy@)HKn@!u)2zjKvJ$ z`I#|en27=}`RI3brBq$t$&`6P-Q$$Y4pW{tGf~z5^$BymfftH|I6;h}MN#pQ75J}F zbf=o4H9s~laYE03e)uG5V9QpMfQ>0XRArrqGPZ9uLaCsrm2cL{*D-ES$ljMri0J=2 z!2fep*gS3Sx?1s===(n`kXM%{d-p=+g-;_Q9(65wHtKPNx&&U%x_>ww%efK!xm-Cc z#hf3?2Q_nk_3G8e-pQm38)dzicX1PKM-NyD^gr-Adtlh=Bb1(z zK^)h&DXH`iM2ca0T!HEGv%9}2KHSuuPkk)@IZM21AyJGHF+n(^N@3SAs=htur|`p^ z)=S1HUlr>()rvSpCNzj)m1XD`Gt+nxU?RuJlCh}mbMrH? z*9j*q5VzmDo;k>EWZ%h7bE|y?p_S|QXJkkrcDPER4h0=uSl_CgrlzKm3|Y-IjRz#X zxXfKG()4>evMZ9Jg&+aG$uB7xZORiq@!f!#B=Yn1oE`^;Z1@NX??lPEDZNJbQf*q1 z9R)RY7iciLun_BsoRmxEG(jY46f?@9TgO66+EU;7ixP(*lzQ#cT}7i87R;fo6aEq; zkt`m39^k2`zo_=L9~%$o0p5kDl2{pV?IS28#QnWhj{{33ITwj%;U-f9BYbR$GJdX2 z79AZu#AW4ZXUAVS4;nm&xTnmOy+nkQ@Vtgm=>A5KoZ{+^y@7#FJ+j}_HE zv#vF^xSMS`e}TsO>E-C`EN%|`&4q(?&2*aoh6k$5N zwlR-gb@)cNaLi8jQSjgrt4QUa3s)v=*ubMw-3hlnC{GF&1G>RjkoYZkjOkG9zuU&f zUssTknkwtyfE@Djc-g=1W;x6U4T;Gd82PW~>hYNU<`_LIhe+B&o-YAk(3c9l@H=xgaeEGQ^QO;7*%>kH@TQM7S2#D_To`MJH3%q*AlCV*x`jM2ewO^P6?Tkk`lw< zQ(?!crBs{fZ4d0uSESWq*u2wyCI1ePvh;UvOQ%HeLnp)6GR)`VAt_wzfB&5V?r%@p zD#M|}*-@LPl>AMpnf~tN{Y#+UNqS-s=qWT0RE=(hr-Ua@+> zdmRU-9h(keiwDs+Rjn6x@EAmR@S6%TtrDAPXQQ8~1J*!AmlLF;6qtZgw~ju zI+5$^@w^l>F5LZniO;czzo=0cCM=(wf|m9fQ$okx5zWl(T-Zc6@u=eGY28CXno1R# z>WpH2%6Db;VtrbqGt2fvyl?j0HIgb8j5_ps}x9(-DTG{L+OZ!jmREQInTm z9!yXH){vBSY-80{U?MM5I%nAjDR8l{(u*Rg>5o372vEOejBx`KFvEfs*+ukN3e1j{ zTeP=TAoY5&#UK^93om->oD&p;W{#JwJxN=}#>OZpDdkv5A(G5*pB{FMM2L)gd2D<5oE#lv=-3O}NYTk^ z2Vsjhd{KS7cGr@+B~%`bnVGrY`svor4th}7rKV7g03~F;Sb<{*8;0Pt73PCdlfnq! zxSOHu(}mrQ?ST8AtT4khN~Q)^NjytQluH#Kh2|4*(jgRw)4%DoLopqO2$yK!fEE`O z5k>WYBfq=477B!$Yiarp5D0w*^74v z+#2zWjVtC&ai6;@wNy*`p0S<^mAJG>&-IOVn%GN-Kke_g9lIAQ4uLco;0PWH3G5sC zwyt1PpNf=df>whGohB``W+o=oDg4PS@CxH*r9W!V*^=HmIbn4yl^cL2@OPaVFWBxp zr7T0o-qs`5Vdr@F7`opuU_6r*H`?5&i*JN?cH{;(6}TLE`_32al&-EWm?k$j_dVO9hXM(n|-zv)j9pOVKP=79% z6!U8G5tZ!;{6qkur~i!Wk;#ddhlXT9Y4V=BdP)DL`_}HRqPu&|)DjEJ+{l0E4uY7-A*Qksc<{D*%$WxlkUSG^_(WcDOC zr#~J}UdqbOR^oc3z(52XkN&SsT$E3%JG|q;r^d`4Kk3Q$^I-_(>l#TYuX#(v_O{*G zb-+h?#<3l?M$?L25499Fj+E(}+jQ%~LMlG;aK; zOIqdgf)T#_ittmo`Z^ks$VeFy~w1q7lPK-)R* zy(sm@9Vsst?L#h6|9c?ErQk;~`-7sQBIrjSj3w`>(C+P@+hQV>KCYS=EI z{mP-}ILT3;pLDjW+!`jHc?!RAklQs?-Ac zJvypgJ>u}0+7QWFO&)@<`E`Hc3#Wmlwe{DvK4?-*e*hR~R1or8i=FJz+=%-`sXp)) ztTMfB7zx*4S_P%D(mA1v2NtdtlNA*eSjsKcT#t15{33S&n4?t3_;dbG)MLEz@^TQ+ zaI-Z!w0RR+K{w`?<%KDdLSMgscj$Brf%mmiN9sT92d`d+p;8IB-+W^L$%J-#0Y9Tis6Q znG4yap9IDspzDYoT(qzQYN4q~IFcz=v*f_>AYKE5^ChM_(t^LvK+K#z#(jfI4WB2! zv?l6t^2O^$+hHoxMmL+zIA7Ra_YiBh;-nR-DWWJ6i)bCw1>ER=w*Th$1s<_cCrE*n zSaM}QFIIDj*L{)50?ZE^07_#NzY^*XcwP&h>$GSphn2e`MvLq8+WH@WH!^m=}VHz9K2iR80Ur=6E6m>6oc?cz6ECW-Dr(q(z`F%Q#^FiFTJI zs_7$flk7S30)Rx|2Vlo%^yK#-=^G*i8H#$?*f2E;8*Euj7cpjY$W-NPj0L}=;b-)# zrx-z}GuV=fdJf<$`IK5d<;#ueM~2kq-QXclkyO~1AydrH&^GZ^qYsSNghm`zz}nx97c`q z(5NreEf6BS4#<|LqM`@!Uw6XV zG|n#?3xNZabT40v`t#xm^3Ffe;pC6rgD~X40noyLJd$!R|6FcTzp~O3HPz_;#^_pC z|CP^7)I-j629+M;cwE=Pj}?_KIgAQDL6QhRLH1j2>4#s(vj z!#k0$Y9E-lFUDnOXMY;RhY&q@=R-N_#t-^I~#E^dmjz*uhp;xAaLXP5elZg`H;f4l~U|`Y_J?GH2EFo?yciN=j3uUi}ut zKlf`ZYfa*;azOB0?HII|UYx6|%jPjGM0~qDv2>Z0VcVK}nCS5w^9g?UGelf^*Ve2j znvrUW>|$TXxWtIQM0Fyyj*1i0{H=z8!>)6l|1?&tX)qrKe|+()u~l(>89-l0O@z&s z4@G46E`cS81gOhv_ZdFqi;@^>k(K@g=7#f)49+L1*v*;HKjronSwzwrhj#{67c5f z>oP+$7Lwzm>&nu9tv<$4@V1vqt63-HHFC0CG~nv2dIoiA46R*tGywxmTF6a1HWZg3V^QkVb%iu)%Psa9agw-^09!iC?R@guZUgB z)$k#eZ{Tz-e|~WeI`?g2KtgdjBOY}ud8X>vVCS9ykG6$GDu6tFYHF%_HgTPc{3Q{0 zP$!ZP7Yjy-W5)9=*`{Tk+okh6tj%8DK|DPpSnpFYDd{jc-qaqYmRcTg_6x1P@2<|? zk-eZO^^vqBui-ddVk`xIBV3#8QWYQa`=5V4)H)zm#!GaIe!O`>yh$dIZv>vEwd+m2 z>KPUaF%p@^Tdtg{Pf_k7y$=?~Uz9Msh@-4cw$((Clggvzetmr%(YFe~icq1H)h8p4 zo*0h^2sW%>>b+HYD%apty$ij$&o=AW%-{3wU&52VAYG2FHuOi7J`H&d&z8D#I=$Tk z4ZyOy2Yq;j#Ia!^pJxitD))PhtF+iZ4rhq0)*{RWYR&$4*OJa15IyF+W}TpiJ>+>A zhN;{Yr)~kfLL9&}zwI0sKcL>F0DgC8;``{_Ty)JFc=}8Q90wD<>{Msf5+&X(Z-d>@ zwuBXw;nFrep5YkwO6w|qIe=h10^GSOtu(Ys*f<1YQs>xtcGmFZQe=pn0PcLDG5#Dp z&SM{;IrNdFn*&h>dUL+1?qZS{v|;;Bi_N7@oB7UM?KH<;GJH`*Z@5u<$&`?K!S@l)AEJ}auTDGb8mZDD_V6Ya$dbKR!?0j>R9P-}*T~fDHc(0_iVf2S( z&!)B}zYAwX>a;l#XOR)%`Jg0hS9U{j+!If2^a-2@z}<2G^q& zfVs_K$zYP}g)Pv?!GU zEgF}5XZ-3Hd^pXa%c1G*8Vn3=!gFK6JNO|8TQw}$7_j3?r&-~zV|FdT8wdq*uj(Q^ z-}vK4B!_f5FsqZcEAe_Q4n;DX%MKy^Pp02+zb|*C`0bfCHuTMz59;(o&!4>VW-7Rnp@u%k|_}GEp3!s1(rc0Ky+-Z@kLW- zF2%LmnwoRhQmu+u=XX0mhWs32+tgS;_PJY+)-PVns4@IeiXI*XMGSf73FHn9ve$;7u42=ZlvSyZ8y;eksqF|^l{%& zhXA}z5f*1X0gi=7^7`*s!$I5S3p6Yv9kA=zRYd2 zIg$mkQqTmAxD6cOSl-YBs)TilmPfi|{8*(l8aZ7ipST7^J+|U)BvQUuM*Gzmh4G4! zh9E50lPMvmU#ReA>s`W&>_pu+qMiKXZTSBT+cp+ZiV=bqUoc_Rz^gN;kT|!U+g|-i zY5!46U(Kb|mWJ1rk{R0Rz^G2@vOeyO=ABH1*V@_&SWqFUga#VAxOC0*(olCV^ ztjA^@w1c7K?pwbZuU5$0gKUqqJL%%sCBBZ$4W?uUcRJ*${=y4tR&EIrO%SCaSX~=P z^76K^UCU7A_ZtAo3+}@QF%MfHu5hW-rS0OJu$7>!!PbRo6c)#s5>Z-orvv3-Ay0i9 zZbLh8bPkA$o>Kkqs9&70sriSHn&t0yjM*59&Q?an9;8Q8h3i7j{{Q-pmPH*smChXTi!NiX&q3 z<%4psmcLa}kQ(=nq7`+8@U~w)ttOaJ%sacs7{+T9+3A4xP8+GV-)>U2@S(oE93ME} zs=u#>+|I7?EUM{35a^Z{IyT;JJ9mBnc)o54ljEsod;ODQtab8O{|(?9mbM5=np4NQ0V!S5`26? z0|P3}&2Cw;vLAT~;a`odP?M?o6z<-f&583(aw9r`r~?4%4YDpF>ON-oR!U#5F8AS3 zEmeInQVt@{Qc`^al2E`_ll#Xe!9N?9wK~n#E#t_Xp#@wcEeguZou>En9S#NNK1B4j z#(yRIb)7eW>}uQ-+q*M;A!ptMqT(eD%}ow_O>HueyK|q>JO(VGiZ2cUO-B@OTf}MG zX=~9BgA%tHwPG}Vv)~Qesz#skOGE@BTz?Nt1Y2i-U*>4S0k{kRCIIU@<)x@4CeK(^ z44aw_2H<&ts?I_^sTqSuYbl_cG@zxWha&*1BJBMBXDbU zfT!rXLx>0d2fmOXKj}DtIrHo55o>hfW5F0Wu79_*0i7~E|E2n)yrpKixM)@+S1&Pa zKv4ZNRjH9Eaa0c1I6hKVZ15*RTkhtxLGdhn?vx}uik?xN3r#pmHoQW@(W+vkew zuS+_uLqALzS5iqyNi|&VH@_)LY~6Em=>h@u#ctM-({wc!T=EEy$Fv~`u#>a%Szx{$ z}7$Ggx9Vt*Q2hfsnmJ1CmAg3d8MX?a_g^A>FhbM z04mDiKTyer?V=3S)e(Gl{>vv!^tr*{UtLQ=18yQcd5tH#(;*`Hn2 z1kXIFa_CjX9ql|@hYuV^_1(Z+Q3o=bPl^hG%TJmaJ0^l)9iIJE$W{{9{S|#u=Q#Z# z@TxDcdn8MOaj<&z!k|=29hW(6DDNNL!r+TG>0M{I-5tsj$dfX|S$W4mbu5>t%4_u9 z^LEZ_!&nSdgFtcWj+_K>OS~ema7K+SxQNX~ANTFf@rA3rFUWxjs)n>An!OKt?;y9g z$gkeU7Ci!N*nQEAJg>X-h0~xS^~T{X_uA2fOMqrfXlL__%wa7Px;Px*(%xO&e;D-f zMN!jm!nZ>{Xv&iu^1MN9jlJEu#<9_iuMSkXcCRtfCvrH{|Lw>nFmCScE%N7F5|Cnh z`@cN@ysr}a%JNXFSIV9Vg12D9Ul(MZmRHB3BJbkDtC1t^UD_sLh6`KOpF@Y_b`JVavGZF{^EzGfMAjZmTT$ZK}LxSpW%1T zw0+9tNP3^REie9_X~02=@s*JotNufBHf4!P5UmWGy}0k}h}yZK^Y4kL*~ilSh=cO{ zh`_;WcGgh&EAyoHXBH)@;H56t7sn0I!{GkNAr!ll zA<%{q1flQogZhCL#^t6>m}O$VWWRLdUjCJ2UPCYaB#+f4z4{*aS|S(OZ+0o=f76H8 z)80!RnDTY{$R}!XQ=6;|<&$s;7fGAE0hv9g^<0qANTa>2xIYyXCASnKLf6B2D*ZsO!FqxMtF+X7^j<)~((tXf zv_;UZG4WiqcuD>@6tu;VQ6)8uph7>_ZHABevOfuTcErx4_1+CSMw zFYGk6gYp>vd$P4D>6zVz;xT|jE;Fuu0NC}SS|{w6PdGmhPD^AkreL(4t1s2fG8xuj zL$5uT6zI|OwGR~-6TTg+z5$+HFPmKj*92OH0f7)(-Dv?p$qOXsfF?zq`+=m@-E<3` z=J4fLEh1*K<_-6oBN+@rDa^m87@Cs&F;c#^#rpl#sW#f|5fl1p(PwDOrw9bCuYcYL zexVlYr;(tF1ZZ)<1LYSKbnWwSM|7H?H9MDPiDUU)jCG*+P~?GVbDrcxaVgSMgPWu- zr5s9aV`Hgl!uo`anZvfD-L2#$&v{b1P$)i7^}BPY01)&YEOLQ*T z!|6iFE!mCMZ;b`4#qM{cYa1{zE<0G@uJp;AEH8(o+pS;WX`sR75>iOGM9w zsOY?&*tASO2)w;K`yDQ=m$Ho9M|Sz77cnM02YKI})21O%sJ`4#O9Ko<+MZ2KulV)D zq964X`dDq3W=jqh7z;;qOGA{oty1^M<_BRXM-ze!Ij03$CP z>*J06WF+Ybt&MsCK6_yYx4XTa^Yb^H{?7vTa@y(l%rO~@WU61E2hoN%*8lmC?f>%g zLnh)Si`$1#q{*<8MRJd7`72j1(o}i0-#>^}4yPHsRyOv@c}!)!z3b=a8K?_&p5PG@ zcC8-f0y3#gt^ibt2jk)JidEPbi(9I*5E-K5xdT6(hSP+myBL)#jcX1<1H?i?uk6>$(vo0A- z=SHlCYcJxKmmi&<(8V>pry#)K=I+H_TseyCdn0o5k_>s7Jh%?R%}G z(w6J~oEWH>fY&rhd!WTaQ>+~c-<~`nZ6dfrg?G?jCm6C`?0gRz+b^Mz@I=dR2gcEr zhPN>u-nuXYPX-K*sB^v(%-d;14fhIoLdxx?a#K9&y!n2h! zZODw}^|x52XMesK6h-YdHU|>Oo^|&3M_8qtr$-7cJhQIJ-)Nk39kLB}@nepl zl>o$ZP|cfQ!hqkcc_3$rg6X^0d?QrT!A@k6huW)ht@W=fM&^^;#eO~g9>4@SjJ>PQ zsz5$xA)%n5`M7wX3}{%haGnn0116TWqZP;U^|RS_EV}oY4aYAjVQo#jt!AbQiQ~wk z^7SL8h?r>-weqhZ)Kh?96BFmR#@vrM!Y0EX`nGy;9RLpGCxSuQO8`oqi$puSo^bKH znQ^d`-tWdlAp-Gt7qWys(d#mXc-1Ism-SRXRuliMIkuv0KkMNID2cB|HKxWQ~etQkAr2db(gLW$d~s}g{#R=o&OOMqU7A}Zh38n^B~kt z@~~TzL8asRbn#&3{vW*=c~8^ggYs9x*e>@QGpFFP$%D7|GvOHFgi-Z`frgHVeqT8? zPIHyh?-p4dx|+0|ljY5>$Z{OUBgev~Jft`Xy`YE`JURIiAU(7Cq51?q=;o;BI|00Y zbaq~74ix=t=uOaYMzF;V$+yn!RnI-Zf{VJX-2)qbxo|pbSZhBqBv|`GY3|3WbNs%3 ziy6=}P!qe(G#UxNLX9L1My40!e_rsan(7Ejjq_dqG&R0a(ag2=inh^Xn3A-4L_thH@-;}uh^i9sMP1Fr(hYjodqUFp7$ zM?e7hXB_WEe-V)TnR-zteq|sbR;E7pn}7@F&7`%PSph3kK)fUGT544q>wfE<_r4h} z^|&ZJw5y~23b)d?zV_OL#|&{;lXH!Cy5I|AErX>5W(%YzA|lB>ktAOt-~H(+bW;qe*|bUi$Gm8 zRQ~nGB8a*VVR*mR-5Pw3IV#LsKSf5~zY}%80nv;+ABCbIl?`qEgP8XlGr!{vjf>c|0#-G? zia2+DdT8EqW=TXS$?ywb=4x4RzoCF~ePmszHjoW)?3&HmN8KPmAU0qfuTh*ZU=5T`vGg*rz@UI?1qkf`ya1R795>LZycPTeen5^< zX4**N@9+QC&Q1P(Jjs`& z?YI9LK`sMEQ)Z9r62{rUsR+;X$D%J4r#sIajHBoa>~?D7Fq+J-0%(H>%n-0)!FYzj znn2aWkGC@(ynb}_*3%eq$}R6(R~7+391uv9r-ds?NAD+1FG&Z_jQ8!6*zY>Hz3Bss zMxCuH#}9Ati@o>USnai9z0XcYcILMr2Iswj7P>WNmZ&LE=Sju&g@98kP>!2fkO3t$ zS5Z8D{RANXi|Q%0cd@p{d^3{S85i)E8L~1X`fdf&x%GJRmvpfBm|-Wbtidx`^Jknl z%r)dVQp&jlvc|X!$c*bJlGG9I18&>Y_p7`*?>%Jrj(Y@R3n*!o+M zgy>^47Mf&41GT@73zrSEzyeOUF3-b#z?A)RK|wRi4&0M5X9Fb+N~DlXof%N0^M zgcK$S4Sf+lykO@E*v_dGNWp?!vF`Vckk1^7(FD^ykoVc;4bImk$TZJ<<9m$A20|z* zs-J+7;XcUu4^dRpR&MUxjZKYFrc5B+9oz(*L8Vn6!R}O5K`A^hd)yzzQ9@(xb5K8e zd-giI{kW_D<|HR~8)cBU;UC0<5zO|SP+Ih{M>5{-%u|vmf#};)<#$Uk8FUNXZOzeCx@Y6;g6axfk20n!-T~M&S24=w$?s`4 zxaRr?jEXI9{?FK!r*c0pQpcJI;>gNK{ED=K*AJP~FMe&=q} zaZdtnfk!v^)2J9U6}}2$J}4THh>MF?x~;2*RfLF2L+Jy~Ubi644t^>SN_~tAyjner zSrq+9-@2JY1ihq1+n?}ZzA+yh>x$YdKTe}4cBKS%Qu$?fbT!&xf)*l)^K}6C%G^hq5#Y6 zFmo}p89-7Om{|!N=y3?W;aQO{SU!)#2+Fy=+Js&m_Ki+Vgoe}k+!FyB6HmCw7Jtvd z?b>6R$PDMEfg#N#)Bk~vAG%?oIK1vx4l*zg=57w>W~Cv3?>1{)y|gErTUd~;S?&iSC8mUa z&(6l*R-YDS`1!T}G>(e`oLs1u;(BxS{zqLMpy*QakR-8q3#J6nzuHx~Wv~M}$)gu! z^jEo4UMK*nstRo<#>YPb8OY}=Lyj*WDv*fQFI2RP;CMy-&iKG}KAf_OTON@(;JUoq z6`-p&`O~O(>tWWnwZ#sE*vm)fwkKL_r>>@`o831Zv9y*Rmp^ek)Oh`ICA=I{&(G@H zj=3EHFiCjyOLu#?HqL+7Y6|d1ADE}z{MR`xI)OysH;`SETlf)&6VhYz)oZlet>^a7 zATPWnhe{dfrGb?{O~bq6(}CsPOs>kuin;F{Bc#r*HbWwMSU@IWL?dZ9D^N%0%N68d z{F6D(+TlW6aou?Lf-lw%=)6HvZ$pcCHez#`=CHcjL4V7S8p}bqxFA7Ew1aB?7eooi zo&_I!Y-Oa9nhV}aw-d&+xqrRE&IO24Rfc7|dsGh7GIejvee1qZ={-Cf@m8c2p}^G0 zLJv4N?KU!1jL|@NzIgHCm8&aJB1;|&MhHY;rW+=Y9n&;Uj&d*vC5hxT30?oPSK34! z`!BJHt?23SKU))*F zA{}KT6qIm7wXfdkGLDJ3&&G0(+-|ALSH1Fi#|!S&>lSvEY*_{*LbnxZIAp_dxMG<4 z3x=qbCxo*bDzXQ+h|ozaSAGiSC;h&c=(m0-aVEI!>s0Ma6lX?_g%_`q$0sbT|ENH& zfMGP(^F%kUY`Ocvg9pVza{F-PYl|md7%p=#en@*TaXTI!nH&x`hbj(paTXtQaCzW= zJ2-!bl{9pN&Jc$=G%SC4WAE9sXVe0Z@onr?|G8F~r*pDAju~bMT#$mCEtptOGnrV< zK-Ns|f9YT)77tt*{tIuS6PNg3Z`W-soVAwv=8cw)PH<`}t+T6Z`?qi87Co^c4GrSP z4<6LnX;^l3b#d_U1RL;Rb@0vaxR+N}2G`UGOGrv0CleDBIj&z<^zo5=#81O>{rbX> z9~d`o+?bf0d_I$*p3A2tf9DRa@9|Hn-k*P({nl1j5Zir&9pSY3^@?0+3FI9@{E zy8C-DH#<9fuH`0vP;4yW-#`5~+>f@I<>ci>SJj0iB_(@%du1#wEnEDymX~EbJVY6! zyu&dsQ?_<=1mzOCFor}#78Mo6r>7HyE-Z8tXgqrCnN@0sUtG+ZEOa+GKHghLPmhJ2 z9lNaMTx@)UYkjiH=E;*M{s{^AQd0VS&kPxgi&MAv3h5c7(xL(b6X9HImCbm$c^6*k zl&*ogIw2P~_b}&HXGlJkCdc*8d&l=gsKreyewy=;@W{Hl-ntz_8}jKTyVFdm`bf!z zrFFtojXgSQApK34fEDHZ;amxG((ce0o6O^P*yiSX@@xWf_wUmf8X1Xh>fJQ2pw`vZ zMfqoCEp6{Np>e;zje-ZKU0CzN>`!9E-j-19GQSx%!&O>UMe+0J&-k1Coq}CVchf>{ zdC>icUpm~Jb9}dfKTws4j{5QI*YoL;%^W^0*}Hcy!}W*Zo}TsN{ub8OwvML__7i|`m5gy!V1yh=*yn4V6nVtA`vIJr+9 z8eqV4+scZ4a&nUC*l6i?<0O^ zH@A-mv#=}iM%OksZ#y{&fF%!)j09F!->R&t%5G|+Cyl^({P=NsdAWZ|3e7vUH(280 z;`n}aOS|-wV`F+he*czdNswVl$a??&G9@J?wWOyYtR<$Akbkm<0E|WfEFCx}6`N$l$?qi;D0F2?>*LT3|^? zNSvJCbGDaH7Z(YUWB&8!&%E_&H$C@u z?{?UXO$W#fVG}SDjm1TgRuQM120N`YJKe>|}3cdv~{E{rks-l@;83_wFUG z9vrxxow$T6uu#*uSd10xsJ?j}rInjHG;9_YMs&tZj+L9=LtR>0ii3woTRp3pA?X#m zxM)7z;H6_wV0f$pex}mh$tZWjQEvn)2xhy9o`< z>oB5tCd`rQ|w9uebUB#St*G&VMx zK7JgEj!kF^tEYk;$$KIBQ8%t#;gjbLtWO16xds~$tAG6bxiDSlRO|Gckn*}=u*5f zbcLSIXnMFsSf3m^&2REPs1N?}<5BXPH&=Oh@GtgFF1jX}qf3`|kGFSr+Fo3{%S0KK zkeV84-+YXvT}Zw*Ue>cZ`u_gueOwGQFfeRv!Yi$X_wsebe%#b?#x4#NBwrvP(a@yLY zU?1hyqg=ruA;XdcJl4;ib;U`ahxaGlxXQ;zOebs?oSvR;D0Gb!cX4r%&$5ra*rX*R zPko;}Hw)aUV%Vx$Q)j0F8y8ok$H`uyVU?J;cuH-p2pb({Lv&rBWFGv%u zwX*Qz+J9VYVmESIpUBpZ2X(agzO1=DmR`KvtOF}e#1UxM%e7f6gfF_5Iwb!J)M`Bx8M3F2@QorNhBS!@#rK?GbKP=qWmKe;09Yh1`JbpuP~`STB0;-dsu^lU{~cOU{w9seN136W!E zV>4f9520Ou@no_+Ky{Ww26bw8YO zTUc5O3J%7o+A3$AY#SILgx5uOT*Fp#)afDi!*o9;5DT#KSbn`0@n$*J@oCb241T9m zbLC!#cHVo#N@9m^rT7J`w&%gCVA(?kN^zGdy~?Bwuf=o-@Kqt!_+AY}y=k&uv3P+Z)3 zIOnGk7b^(5NuwJ+$MyB8DBQj+$CAKNna+$I*%lnZB|9=b9gW-{v_p7YoRy0#Yt9*> z5RZ=BK^8CO#z|=(brxr1Mam+hwT+CTt{YXG!fZ`6xRn_a#k17+PfSpI9d2%TIrrMP zoV^=z4W;JrWMnJCh)`y2KiHV60PC}zsn_kGCw)-vvZiM{QGSWh_b9u4F!olZ%zrKJ2NDmcft$#FA?{f6xrAGO)saq1he+{yq;d5&4-BQ&dc%dPE zMh+7zFJ&4@T?#&wnS}){)AxiEv+wR6zp9R8+4!r`cV&F< zjWiO4jNZI?lQJ}<$tEg#f9Xt1^6Y(5TH5E}&`^Gs1m%TidoF7Cn?yfrS$@%rPzb!y ze1z84*5>>nu>g4q974kG(gzLr=iRrOaL9DWY!mx`PE^@Yfxx?+(h1jf@VnMD|}8};VU=-&c>K#%t&>xew*qHu&27ZIuYk( z;+gL%24=Rl(eh!0%#aBd+S9Gw%UZTgF8IjssluX%n2bz*u8O&sR>+#%+}zw@ed1=` zxcqRK%%kuo1t3N%LN+k@bAw}@ zOCh_Gc;k<&ps?@~L@Y39ou?GXWbEqxgTI$Lh@`)19tQtife#p&gFO*s?5`$Eg*8EP zUZh>#KqIHD&nh3H+1Lndu0uixflC=}$#;RLAWm+y{6)jU*Uor1Z$f_&cxCX$$DOvG zKU)12uHOu!>4XUfGbHa7xys63f~XUpoUDBJ`U{e_-@7{(=W`eG?(QOg|Nfm{TFTw7 zhK+V`V9~~hudV%er_9eoj`#P4mVWHd{-_GqXS_%n8Oo}Xg+ZVII zO7r=hdwX8NT<8S1b`w4$zM@fW&bXKMFU_d1ef=TbdxAXMT_r4ZY0b{`t5fQj3nN~7<9l|(Gf%mU!2NWjOV-`x9x8Q?s0^Y!ZoXWiE5@Wf!7XejT` z*wR`;{oZ;l74o9bf8hKY%+we>zh;O?qhx76sn{!`{Oc3Vb@Nwgoj(`f&-wSxpY~S1 zUP3YJF(*IzySl%7r~4(bImL+mx~YnaN_;{>z@YR++SzSft>NF9ezDQf_^)2Q!WS*I z=MDa0G4fWQ=#*W}T0vRax20lI#)g-mp{}lNbd&-BL3njsH~k8|gy#Vcif`1J>zBdj zfSb_^zkXo}3)jf5L2SPw=8WTawq^oKe!eFyIJ$B;S81ut?~I#}UMv`VEurU|z&CH` z*woX)S68hSluTOF#qaGb5ASv*y;CpgN)wGUvz-=LH>Q4SNXCXDrlt;u{1P6ob*`Q9 zobueKY{y-q@nF&>S=fFyUI_FZ24nz@$G_gB+9eU^HNgkvJ9pah?j{kFlJYN$`1$$0 zdi`1|^YZycXv;;*=|e9mDv7Z4%U9Ljw|DmT;#|6PY5UI~*x>E35Uoo)V7%y5R7qxs zf}kGH5B)BvDk>@<3A*}Ch≫$2~kcx)VeLF*!L#fa&~Y!qovx-Y5IC zn+&se=KL-sS65ew>FG;8RpV$)Wy^=bZuTdVI+DrW>4Wjq;4dM?_MQm%fB2|aSyDhi z0HwL?!_CFzurhKDqV9(gcB;&zRT}js#AhxnEFf{!%-&wwR?_kp8r@l;wuVjVuL5q6 zpMJ;vCgcyVr4jRriis`W@H@VIzIAaPx29*1OJ#;gOju&u$;yZ|JshnGaIFp-lR}^& z9cS>k_gGZb)5Ss{P_%V-hav-#V`2z^*SL7PR8g&XCkqM*Er6)SxqNx4ohlM%K z;4yPuCL-#A+=rPms?JXFx)?nhOkH4RCS!LTsN1oKRj$A{5(zE zEw*%?xQUdKGSun!fa6SPpEJKIzzifQE;gtLgsYmVIXLiv7TfJ}dhQ&e9{LFsUHOw? zmb{D`s^QgSI2Z<%Pe`=y-Gk@7CPu@Sbi<^(NcVlMT@dkDiD4BUeE(HXkhpA|tfQ-o z%*&YTHP5H8J&|M&I8~nux60Q#zB_mC%4=#8OQysvUjtbKS=Mx|W6+x3t9MQU{F0LN z{jaYt)f31GKK}l}vbT@+ubK0tN`Cv>cki$uJp&tEgxMGD?(e@0MpIB&xKK}#x2%)= z=<(yl{c)3g`71f&N8VC@&(0E3Qa-^QZ@yN^({v`70Dz)5=UwaBGl_ zjJP-=*uxlLr;tv4r7gGR;^1fnKZETfh+;nJY;Q;Vny;Y*p<%p)L)EICRa%<7l~?R7#T!IY>1 z^UW+q!UkeLSn4I3dSWvtZ4uuV#fYYbU|78byHJ%#x+&^NjL|%tcwpH{L%Wx24dv{ z6H`-=#@n1Zg;$BGvXbI{{`}e0*7h?zpF=n3P1pd$QC;wOtR;jm2L{v)@gs;OOnB~0 zbtt?a_kU(%6O@vY(!TZni;btYWKlk_Iu99}R<8SyL+1ib#P2YjtbF<}75N?S#W~sh zjvb7p=-XQQ<@59N@(&-hpYTRM^1Cro|B~eC)2E<4Ft-18zHDle43=ZY3z4g>tp`#O zzC?&iLvv#QVHc>O)w(R*->M)QPJG=7;}|;@VJf0R5#CXR|CkO}-x7eH2_Eum3x6 zEcnjWj87mlugU%Moz&_%T;*^vXBXZlYRTK~5-5N7@P7Stzs51~*Tg<=-@|jYcLL`` zZ9B|1%J%Fv|Ci%$52N>*bl)@m+41=4k`PCPcka&8SW|NtesH^2`JV?!3TZqNt>V{rZLp zAt3yaW_gA1W_L&4G$|ii0F{8jMkZlOeycuwWM@YxEiEnKb$}196#VYjK1K%S)G=`? z0Oe+nAHN%QlI8Arn-mg)35GWQjJRDiLILga<;$g0)GzeHB4n7LPJ%JG zgfBrQZ+Ua`7dR`CuQM|-rKP-+0$SK{JwY-|LlOBYBsrWhF)@IJ3|(s0WGcl}QKoy# z!$)U_b9KlL_zO&@xb($4PQxZFTi_wZ#l_o4I~KKdb%hT!?(4*@K9z5ZjKqtk6~Y4? zQi)Yuw2v6Sl~ouRy;;v{Uup&n;xsBNEX0KuV!qOCUobDEA)^6BJLPucsGEBk=e?ZfSaEM$6E!4;(;r zEszO!clZCCc<8kuL{3HJ1b*b|CWw7O>YPjzKbq&+KJg zo#;E&6ge;U&Mf$m#p<_0!v^0^{owadt>fcVZdi;>9GN`tt8TtAZF@~5s8X7$SOTkY+6{BJU zj1HtRSDrp6>Bn=?k&%(|N1sSbjkHq7$}XM$5rO;+mh>t&_gZ5Aql1GCU~u&e2@oa09P1ED4JynruUxrOYZpS{Ny65; z>%=a7@mIjw#s&%RW}{ zJv|-P#;!wV5NMX%>^IookH{0)n@Ac+O+!Oa+XN`LhtV%OIhooU)=JDrn0Z(Ozk>-_ z6QkeXo%q8#i|H>ed9ENo>v$| z17oomqJ@R&u5>h{5#&#og6DmE?43eI2rO=q=t?li5F8&T)Fx|RE>3u$LMGY6!ymv21{T*3_T1CNY z-NUBWS3k+t-w|`2*r0ahjTS5swbItp!$Q?Mw3V5}?-i9+cZFQ`J?kdy?&v@d2nayn z_Gt`hPEJP{5yF)j7#QSZ*WO&-pgO_4Q)Yc%h_}*YSy+$m$rsvkg4-B%&@MncjPwE! z5TH=mLmopra@hjy9riid*=DT|AO5X5ob@7lBW!jum{85l&A-MNuW;i#;CBdEsk8K|(_lVN_!`ED(vy4iz0ousvXnTnKC+#y@u-QVdl5 zH|Hm_x3}Lu_U3tD7(H*w936Av-TS(t;tIDOPKEQ+n;`HTo0=#}ZJs`j0Ls188h{Qt z#)!gNFh1A~E>f%j{A%{}De#cl*;z$bt6zs!#Ru;}PX%_TayJo8UX#6W54>tmCVk>y#R9vzu(OjZHZ zyg^Dz8rsOCghK~D80tznlnrX8@PZPH`y@&$}2vPu04LO%%Z>i?Lf9^J(ZZ>T1 z?dcpnqGs;v1JDEUYzH73M-|mH9VJ*8vZt%{xDnOt`#l8lT@==@;OA1CB(wWE27U87Pfkpo#08R+5D{opp+E|$2+^bwX@=xAGG+=1HDmJ zR`!QFzTDKZ7mTAt#_>bF3x13N2^(mNuMS}fR1ccHY2d2}AU04|IYYNXiCO&_!f>-Z zv#yA{;v?ug{nfRzG)XT?r8tHzU~?SGSXtxngucCGjEr$msT;DiK$}1QIR_u-r|etu z-5_2zf)dAgPD00O9HBX@n$F)7i`%vP6WsxVc@G zl478spg8K0KEDJKBBYuustcS>&nwP*NZ==ME4)^@5+#>fdiW;^ouZJPGAd17EELG-1HsYE>d~*K9cfgB&wDKZK0t*>7U zB9dwag$t#VVJwXdi8gFTZ0BlUXaQCU(tyOM+E!LwJvOv~ z)fm|%udf>+Is!Zz?cVi2eSEzyQANDv`Dz?>0If9^UqMtIT1RQO_LkYneU=G{_-jBQ`PI z@%C<%#l|&lg{f!Y0Duhaoa6DA)R66M2hZrx*#3cm+W?}GBOso+s+66!vcJIO;fujd zBtLs3h}rbn&D2wX?&_MmICqsMN zJqO<=6?t)>_-w{_kTi-v1%In?mvD2o>3udY?i=WyKq_=HY%do8iCAMsWN#(x@gGUP ze0XCko1+7gM!$>GxrCQ5gP<~OcK-MH=s&B5?;eNC4tkV%Pf2sj>7+joa#7)=bUK1Gkz_V*`UN z$=Pmbxu|`A2$kUQ$VfR`Tb_Y5(IBWwWmQ({pr?~6f*LXh)Fk5c3*X$r0*Zi`0Ww+6 z^ms3_wUY5=5``>Het~-iYcK~X93CFt3iwjf$f*B${v!@3%TCnTGqSL-lt2DL3&maS zpZW7z@laZQL|jzNdCCta0#(enZ{M~;9|I7}@Te$xs95yRyr8P8tYm_!EF5%jc1F^D z5TZKGZSjE$H_6%LHGgvrMq!{`k~*44hnJOSd&vw%KJXFbTs#o%WaZ@0K=MHY5>guh zPgv-z{52|2!I7M;!<1VS)jB)NprxgytfCSKZ4(9t)XnE7Hb^eX<6@KSd%7X)eYQUi z1TzTELTNlFCkNDHncf3D(w~~i9R6QqE}^iou@w{*7offXNn?lCR#O;w>74TN_z*h+ zC2;9b&W=FD3G^U|fM0`40MyNLeFm2p_F==*-dF#g4a#_7cTgHBnK=T(g)%QAq@{3Q zc?XAb!-n4{Dm;`|g@h=i&yTLY)#e;4Gl>K0{Ndzd4wDu^NCc@M2?xkwE~E{kw8BHlt_?9nP_@(N+@$XxVY#R-GXX$qsK0v{j5YKEDb!>ocAsP zG%5hv4u*EVw*Gz_g}^-EK~OdZoMcLkCBZ-hX0?5A5SpK#-#usYKXrfrs+L>GBg59d ze?MYmL>rn~U@hCg{2|9fKs)qqVMfs6KPE7!&U^bd3+N_vJ_XJG6oVlBIaAtXu$Y6jBG^f7avvu4%XFOOgi;G@7o-)cUCqy{DoB{Q_Mlnzo4iH_M+ZV zrqN znHyc2=tq>AS}rl=>nNxKCAJ4^CKmu##RMt$OA%P*p5eh1q z(ptd)M-WvDniGYBZ3ZPTNWua^GU<)u7 zBvwaXA0800@{b=ofU)qou21;3?IT-n+{DlF=PqIX;i0b9O}n9+&qaxJbuAV)9)v){ zC^Wjk637nAC;j)n&GC;PJ_IHvj&f_{SWv&afs-NSGkNF{Dj9LmgKHTeOQLgcU*U0VA$Q ze&bsqF;tS~sfRzO6r_t$NHEwdYzmtNMMd>99&Sr?c1EB}t1)pD;N&Vro%+@&ctEq= zpTk3JRIp))`BHLBvZ=(S+kAmB2?VbVc@^LKutuBVL-S|)@e(->hEH2kvkUiT>BE?V z3Jd1nyTekhJjl1b5~E8JnOXJE8(IKa0gHe-Xi#r&W`*+#`ox5Dd#`dVE?H`Yqs1*b zdHbNArTw#q5r3AY$b?oAzyZ?X`EhHQD5U>5`1nxtA1@sA`fpu@Y5Vzjb;d}!S4T$& z|41xzPzf4k;7+0|8jP#MqGUp~4GkU8%L)LwptKaBDTRfc;EKb5Up78(zo*~0KW2oq zMQU;=Kv|<@YQ+i^(jRL}2miNr`~R^AKTF6I7uxf;erZY;DeW5+ zDus#u?>E~2m*#z&R#Ir;w`ql1$^Ue_t_y5u&Y!Vc`JD6cS07M!6x9_<r z_TD${8{>}q-uoT@IEQog*=y~!=bH12xr3AxB=N6OUB$q_z?YU1Q^vr!K!W^_jR~KS z=FU{X3#Ngrq!`9I@-MwECk6xK7KXH#i0aGt>yvJ(&-TvFw-|i8?q9s0dGYrBt5O7k zj>&=XNl!Fi78z?ydQdYdd*c(7cN7fO1;?9A+Qb-l93?l`#a%LBBw*yF(=!Y7k@9v1mg&QW3C4y^gY&Y@I{n*z ztk^gtbbcaIiP&^_O_^n7s)kF{ntnvq5&jjG1LS06{UwHyzFvm840|7RR`VEg`2Ii6 zBibZ}txxp-HROLgwBW)6n3KjOwH0Z>T+>K*hh+NyJxBEilRnj7+;&jy?d@}Ob4pL2 zzRu3Rng+|p$!HW0p2Z!Hh%(I85{xy(~ejb z?HZ>bddYBBlP-$v@xif;$uI4loy{G2(dp@CM|%RI$NN0h&YfX9i;_{XoKe&~HpQ-+ zIw<+%4^Ce+1w8+-(mUQK;qnj9?KY!lfzrt>O<;s=!!rrMe@^_gM))?MNZ||^!Zl*tafg#@@8UU zG{sfRt50)PuV24jrss+dc)zfe@#zz`&zcpFg#{B8O+wOW($lPdUgwqTISMIFLqkJ) zg0Tq+JYRI2ow>5Jvt#4ql~q;!Ac|hpIAu64c1awp_TP<}(AFg;CjPselU7_=S#;dn z(b1uII+@G2cmMuQQb?jB+n7HsgJ~GEn$J zz{<+1xU`f*oe&1q9Lz5$xWmE{u{vC&qNL;_lbv_0h*p~>y(}aoIeAmVJr;|jh&0jN3Y}ks+yYqTor~Ir{y>A-chfQm2({RYNcgn-oF3f!QR=Fr~nLU zHeD}>k(%nJu4zH_h<|Ip{z7Bp>q}ev``kygf#qp(aLUN!WEl&K`_Q$#f*|qdPgMxlkur0V&Uj{e zM#jL%NTQ~uCMW)R3poh;KRY}0%*=j{9>D>v6wyGW;Q#YWRo!FzOc^xL;@cE1`gI5;{|3b-)AI?Ydc zY_?2I=@cbDpMJ}hO<0+<(Kr9vUHc9ecITks{Kev`!NSRb)hBmi6#q~@espI7 zOR3c;yNs-?w7mR$N8AI~Nd;lOMxQS>ulmb{ArzgQIKQ~8&m8To`n`S24&%Ie^Cp>A zVeineE;jahbWF^>hYt&nlVf5?e6a{W>NdRMA0BunFOM%RE&Z*vm6(LYH$VS@N{+%^ zCZ+<79J+*uFJ?{-cVLw1Qd7l?7cUlE?udD z?<%KXjWd$dcSk|Of?HtR9u;Tc!IVct7&Q-aUGx!;nglj$pV-*glv>KBFDVT9 zSo=w2{a!6+9wb5oj z9Fo7Qy60nStCt~}7rgiq4^?w##4w`$o+AC+MXO7y_eq~Ved;WEr<-q`k(qfugr#0@ zu_xuxpTLqVQ)7Z&QoE%6OpiltJHaQXk7hb{QLWq6pbo7Iw;I{I5=XE1K>cN zO%+nU`Mz+!_M1O-=mEEwt3@KuqY6W_O zdC!nILJqW zQdH{_Cf=|7qI!d#_k6V{+nr((5-a7!F_&d#@5sF}3{XPKPSyGmyT@&VG;k&vTTY3^`wae3~Knlinp zi5=GQ2!peCzQ{Le{rdGayxm#JFZyiMPAqhOvLjqq@geur^3wb}+j{ez*Zn5VANWKv zm2+QiwUA58$r-_(2&EMm3QW0b_ywYH6jCJ)4o>ZEzbt>9d^)#Mp-$b{LWI(oQ6eXU z6y%E}0oRA8M|+0NZ}F?Dt0_xV`jjUoCPs?&FL@p9>TgU`Ls2!WD}p_C?bYjPd-pCf`42fcJ1yk86xQ};`kJ-1wSQ+qSfmsc8E)Tx6CF(qKj?hjk7^Q|z6B78 zNxc|@hlfY#Ik#FC4@XEFi;N`#vQHtP?H=N z7x!!{!kg7~Lp#-Nfs&n_y=<%qeMKf8pTx`e4n4i^VyZ{D_sQ(duJ?Qc-(OsHWOrPU zLzXGkYhME=*1fEQwU0;Y;ftDYxGc4e=f~dX5yu-}dXXbJ-L~gyv@TJ(WA)xFk9%zH zjLg``*m&z?r?>RwmVxM#Ct(xWtfqh*jNo28g6M!?vMg8B!j4rd$1H#(13~f|whupg z)O~%7i|nEHcYqW3L~fT=*iH-8AFVuq`>{M$p09RXw!&r1rKy~&I!5s9gN}&E>jgNn zk4fXRn2NG;rpir-s{zrjzyFRY_I zpfiJnOf7)TT)B_)9LpI&u*BGGASx|QDydQ%0% z`YOz3-S^kpA^E)A8+_j0+dJ&u(0VaDqgu~GZiC_+YCVmh`>vv*8NYx4hru^&m-Vsi zgJd*P*;an$hTaey`(cgy)D&pghmRjW14w{5MPz5Yoc9*u$HqN)z6~+U>UHDR^SHMG<@zUp7(;O}Z29E^jzozgV7yF7qo zeff9X=BM5`bji*A{n4&dZUF#P*>hjh)ZO)1eRDZoMx|vcy6#*2 zx6D2>UXT5PA!pmknwIVPPK_F;-1d`x@$p|XGfmNVWz{RJC|g=uG&d98zsEDHSbQTWY?_-YJmdrfW_EJg6YB@hSJ}x!yN#MYl$Blgp?=0gMHBNp4l8|6 zHfMHrme1>m6Y^`ftrE7NxR{vT%FlanAIqt9dB|-zYvs&zrq->GYJDcv5s@Z8BEz%i&*uiZXLKR-WK!12wTjA9&@v9LI-Mp(J6Mt;nV)KnA2j{ZtaPiK1k zrTT|53=E(Xt1*;-{v^=gN^P&%ys zaQjcydBnuWw+`k%cbktpk4_wU7aIDo!TO7WogKfL8kc;Jf4m91{zbcKuLFD7{{8*^ z6#Bw%uI%6fYfvqWtoL&WYP?+X%s7E*lf0OK;}s zB|?7z^siiO3d9Yiu+Oj=U$16OZ?FE*FLz%KG6^L_fQX2Qk&%(zLKh_l63m{Sp2?m# zYUbwm(!BO5p`yeaU-6Uv3`wdlQyi(GP)-}Z=1igs3X-|VWA*wi8;68C1f5q@(XF-< zRZ+U$r*Qxax)ZH&`m6wJ3EjAH|I>cYDnpvhCmGf%|}dCHMSqk`A#DsNE2;xyuW@j z&*J?Q4%povZa9mBXh{4?(s;zW%6d3`eul72SR0tS-xMHC3U;GFg`)4Q;wIE6fU{_= zN_Nv8nry9WnWGC`NzFrOt?qp;a=ljlXdv$ZaV}vIz%=QSnHd?e1q1|uF2&i1+rp0_ zVrV%`x|C}0>F`W|m3tmcy2?~$@RZ`kfWQOC&A03I<;f*;2Ko72EtsFr9dPp{z>+wdC)mCo@aAxA^K`6r%V!C0 zJo?_1#6dk#lzj~PYp?nqk_LgrB|-T7S*%VcG(4|0DaEt`vSw~>4xqGRYS`<3-Nzh| zYu^^|z_m?zpL;>1mXwypaOvSMLsj^Dc-ZdYEB;Q;)>aYZgH;`SeS$8O9J0W5Uo+)# z2ne|IvCiQWh15swe||T^as`r+5EC~6tdF&MVrg;>1v)Y4i#Wo?sk&zL`$zUSpR**+^b#-Ch5YEW=Km_S--Fm#t?Pq2WD+KVD zaq_KN*KIjS7faKybk^8?Gbbk}klIvg@X?9#iMy-AivJD=96bwc8UlkfhA^h1L&j#J z>UCvhC9HPeOl@6VwaYqHTwI(?W&a)c9s&md!M)zuK<@UAj;o4_ilyoueRHNuP(_yq z^Alj83<{7za7dnf@9k|H7*K&hltO8eD5M}31U>duAsVj_0B^r~^=hn5mNe=n3d9$u z&8Zj2fpSpU0SwDj_924-t%0qKgjx#-6ppJy%-Q2+(b54y5?L8gkN_^jYKgJ|Nd=l6 z<4ss19Uu+qs-mJoT|)ylHc@a%MFp}}z}}r6gt0HPg3u7(66uEl5}urpoSZxU2#f)> zsqC!&W9`42+?{J`YAX5kX;i_5d`rP{&F)^|^H3O1F--{I{6K^ExkQAroR_>JDDJzt z&jrZe65b4tiHT8j%(#2}C8qpq9<>zc9T4?qoK!$c>>M3uAwd9tx(dqz@&CywM6@hW zVDU{z2(FKG@0FMulkG{@>8PY6RR}2Lf+Ya;GovE&~H-MO`)z5r{Kd zYI=d9%(OS{l^cc-gQ5fMa@R1@OfL4}YCU#+<>Qj)X@+f7G%8=v5IN%R#G*J0{NzwP zEjBkCt>!2%w@nKLv+oI9@HZoHOus#PSO8@P3@DjL5WTyV^&Uw5rXT#F~3L zagrdRQx2n2Gg&HFx47kI^=8R(aW&1-qW&=;P_@>)*L^Cvi}lHJ9~5BCox{tzqp&W+avO8MelFVPIp60!=7! zjztmMFh2zsiqy#1Sn{qmYddSMF+_~Hc`g$BRsH?SYwPQAHd#+mw5ag#a8$#sP$Fg# zwA#C$ZH;?u%dGZxcImaX?}0J?3tkXBRaI5LM@c)XtE(d!L`NqTwq8bJZS5l@N>^-# zj}n~{F`Sluyng?l1`?zw8y_DZ{3)``oooa}H$EX@px(>lp-ydJFKt@ss+Qt&Fq@J@ z>q0~Ez*T_BCvfTQp-n9(8vLbO8Q>e1+X&WMU$dNP$S+pggKRdxmT^=EA(Z z;I%cY(wE|e?Eyh?aik#GN2$>EA-S!u-17Nn0OSA-QD@Y%RwrB zNbaXkw*&eicm0LDj?A1@a__N*u)ZG0T}app5wY3U5mJ9n4*of&wi~#`+FY+k> z5Ec_k)We4l0iBnE4~75&JwymC)^F}IsDqF;PQ(DkFa)3wR4O9OQo}X^lkVhfCu(<& zaaKxhD++KIK>oZBYt@aIh8h}48iFGSl7MR<`5&fB2C2F~b5-JOvNgK+k1@Nq9d_2CnLd29zxT{C6Kc(q2|e?p|yD*3#k+Sp&3)LV%J>Rv;?)`THlW zw|#YGKW4q~@?@)RwA_;X@bC};bVJ)aJEJ9rSUU^dDu5-;NcCqnHhQY;XGf~+vygpW zUQ~Y+4Cku>Gs5WsX5am$%X$EEPyWtE0BA)nU%sQe+ZaUQM~8DUGL;6bXq24xix-GW zNX>1fgdo2%{btM?Zgbb5+{6Ojm1CC+4i4sZT)+bzDmFG2LGSdld-~G}3Z?+{QPx)JT6ve>o_lm^AI0lT)7R3+$MN0Eap|`i?f#Yx2fg3{o{?;EgPq*>@WG0n$}U#czcY~fhDeE%i{WK9d6TXZcge9 zcogmp27T9uvE$qqG&W)35MR7k=(wPLQ;UbZgGxtd)@oE^Ny^tzWcX}X|FdPInwr{0 zC1r#0UqTk;@{=RN``c~qu=k8cKO5Hq@fH&oUp^s<%PqWif3K@3RTGT@QPa3H{`bbl zlo7h7T4~_V=MO7X&Xelaufn;@F|F3))_YQHxp-T<*Y~-tef41h@oqdyEVmryZV&+` z4FVS6|1QWn|MY+YfC{bMNBgJuKJI=ffg1)m??TnVlq16H7k>ZFaGn$oA+z6IdKyl{ z*I$e2|D*YLa{!{^?B!@y}-soC?&?y;uic7+(qMP33~*~tYEP`V_W3kwU8%-$S8$l>LFk%R#i zmHO)Vq<@%)T(WJPJjnZ$OLFmDA0|H4w*rk)M3@05*>=~Ej}9;6*VsCub|15)@7%T_lCoINVAuDJzK~5k_UKEK8pFvw(_K4}3-Ck2r9Rmk56p=?YK1I<^J~?UZ-=jYKiM&+Kon_nF~`4gbHBVE$a4IQudn z4w=igZjTJS$cEl4vXXDs-e7;mZR+bUv3UptV;Vb!$5%oY(%g6kX6|z(rF0N9enM%t zfYWY_k&%*SgJ%3=ZgFw-=QN;uE|rgRePE4!8a-W>K`1%chROF-KAPD~)#j>%0Ws*7 zWvTlR6dYWWuVd&al6HodQSXkTM#&!4~q zvk#EMq2s~Su}})OHsyX(0|^ju&48osKyIb3e$tn0JyMJSwJEQI7ie1nGi#BH z=sR*4ALZYD+~K+o{CucD;|UN1@Vz89l*%OFb#1&d0)#}#eA9_^XH1;3Ox6ph4?A4e zMxr8w&k|wjOFa&3=v^5#V_tl5DFH7~=nf_qW-{3wAYBhY<)v^3yZzFLXR zB)|LKO7@_THJwyCX#g@82s|rT3`Lt;`;c!=Bz?cGdcRW&tYO3HR9OJIBY56RxlQ@sz)0VTW9 zKM&R+0hP%|xX&3tytm>pR+Z}xT59rQnzzmlX!jOSFGB$m{RG)c2g%{$qV!Arr7hL-m5moMR;y*=ZzlSA-4D63`luR@1I zqOi9R;8D0jGD%5E@O>O(6?lEL_fzH3 z(NY$Zb`=r0RES+9auHQ#Whk3>8u03Pz0z$qjA!ZRaxcqiUhmpyDIn?DyG(I2e)hW_zJU%bZtJmcfX3ZG z_yg+20e(UD-2D93gk$ekCrd-I`2Z^A&4=`KpYPwlbJO1W@%G=Ysho0`&3O#k*FrO) z@Eu;>M3BOX$;nN>J@hZ3ee^OCQN}@<75e+@5?XlOcUsXJyv6>HzL&?A%zIKS&M~Ik zSDB|gcjVFMkxb0YZeT`Ia+(prZd!yA2Ydee2c6(MR5@z-8g=emz}OJY7|M)#&go{Y z1Uo34^v>o&W4lq(%w7>n7;th?Rn_aj3H$;A93edDo$nz{V?lrZ*g-=MWF`x>$g(~9 zXie+0?;nP;B7}k^vT_AP`{pMf(egX#!M+Eii^P#c#EovqFX?-R%bD|VLL|BgX!*Zc z7S*)^c5^DE6HylCW3PVl@+B|ynAGkKX}Q<2y*l~3Wi?qts->lc2z5Y3Lw{ud7>3tG zaDn&t_dS8x65qi@>2G6|zy1u~a*CqcuYGx&5HePfU>;K|vTX(FIs|syTrIy1F0Xiy}nK90`MKFLm zEGsLUbzTcvAu!`~wfxok1<*Gj)8OFY{j5$cgjq?+$lT`Uj!#Ua04{;(yf^^O)dF>E zUq-=QvV$fQNXy*Z++b-EQBVYeE;uD((Bs|vHgDKu$cL*8LN*$NtbhPXUtEQQr9R(l z^b(xMpJouF5E24-OvpJTLMgs}kD`aY9ywI>;V1M5Ic4EWR##QA+RupjdQXGjrjRDo z-IpZ^##|0ybYQf}U~~3^&}%XXSPUg3lnSVP=*uKTEfia*JcBMnR$Q!n?0uy=(8t-= z*Jt*FC?|aOar*^_JclK{(bsy%dLMb^zxMSfB9Z_f@IoO-&9HXVDT;^O=@(L+bK?Ex`?$OF=+ z@Mw)0c-x|TnA}FjrH@Qnm_B6S=IJg zf`r1J??L@oCPhIvF$)l5sMeWzYS&b>&-2BKRA2mlxoj^n8gZM0LuIwHVYvzVN8o0& zyR4~W0ROA9`+X7V{Dr10KoWiv6WWgi1)&yv?d%M(nXKun^Kh+kUX=iXys7EJ&c#KF z|6@HtcjIcbfZN@>zf)5c3X52vu?H-}pVs+>g$rZlmM38Lz!iZ$`n{tAAb~IFRvEN!-e6E~^ertdi*Iyvc2c7k7Z(FS_X4eNqSh@2vX6y$b(dH|*=(h#eo6ArEq z6s*R+z9@wh{@-hBT_CGeou3`AhDSohmxfJz?_Rr}u&v?0hS;vP{$Gj7RJfl@C$aIm zK7RTn)g(p;Q~$9Sn+@1lj@23p%tQ)$s5A0EKa* z?m!205<5FP7`bG_z!5YV1L4h^P;j4WXqXr4h@IY%+eMDVcr@;25M=Bt$uPw|g#k_8 zM4)}gG^9hK`q#i(TSranaay?!{X6u$yj3kn1C>+IDJ3N()i{34A3$(8DA3xk?3=T= z-trZH)K#nBWz)|BlboEK-(@YZG>#)fA!9FmS%1=3-=(>@xVW*iGv8m}CwN4{WQ|2$ zN0Fr0RW}p6(r-CD}QlVNTD#b{HlV zY3;iFXZ{;QlO;{T3Ti2dzj$9mOQyML9|33|rH8Xl4h~H_l|ke?T^#RUF3|-;Lvm|> zbmqesylmD~$7ybuNcU%H^<)FwdT@mL3SV&HlAksD;k5G#BeZbGQQ?WlA3!hLj*zaQeauK)t2s;sU-~IN>aC3hnJ(`>*`r z_mR=jqR#R@%+O|EGr?{@arW|MPPZ|F=Ue8?Vygi4OV1xq~Q9`62cF0z63ZfBfL@ ad1&-HeG^xyQxa}MQnLXMG19`{P(&xHy(vLhf-6L*YQaGJ?gEYZFfYx#vA8ICv<|0 z?@8t?kAvV5S>D1aS5pDA_r|}%W3q1k`dXAD*g5e3IZ1P`%t&VN^hX+np4g5r-HShmhbJZ`8h)@2NT;Z5U>w z{^L;ptCPcoi7-j{IoB_p(%vcUg;hjPo=hO+2&QMm5gsI}=PKU2cj5i}_h0aMVwAJ9 zvwubgyMltkT^pNLiA+zvix+QbYSLb2rrh+BQR1Y{(Ynyo)I`I~+$EWANHc6Sp^N?a z@#Dn)YJJ$vH(oP+2{*Z4-sCPcua6-jXS};SR1!s(66REY$awqq?brjXSrJ^O@bIA! zKR!OzOVvnM+@mci@*^7B)4X;G|7hQK&pB|H!8Abmbt%&PS4=^6BK%wFVc{zGz zsr>lmQm;KG7SO{@ma56jPZ=2*xW~e;UkdvAr$1F!_y712(c7zTU})GlI?Af6TX-i;Q}+rR0Rd%4cX#vR zK(2*_1py?qfss+u@GvtBLlfEh{N|hH!*#air11=qxzXw;)czKZnYFcP@%u9;oVt|G z(^2|S5X-ftzSH6VSXP!tuA_iv4NItvR6{>s?_H{S$8P@J5E?>~58VP!?==;(+H z-_}NqnwXqiU&_ffZ|6JO+P!aQ$9sy5Xl1-U7N+R>NLpGNJODBY)$`9gt>I_! zhYvv*o}k)V85UO7$%zRR79;+wp4NgcLWybFNK3YmXzXPKd+MDor_XeM<4@a(BSa2L1tjJyoitP*fIp3QWDR;cvN7(zJ zw6v+#`|p>Lk*LH(#>?X3Q!As@_V)JIU0g(d{rdI#yTjz-A}KpN`_A5;<<}fFG#c$6 z7S_&n+_<(iZgqLicjNO_;vyn%Z*TRLcqB8-&CU4|ZVHwgN8PlUwJKt`csD@8(B!ye zE~})3wzahtHeKu1(x9f98I_8Piifv17at$`gI%Apg2IWKni}PUQqIelPr+lZ{dH=1 zd4cQ{!>+V@5B#ySfAu;nomXu{LBT~>8WD@8V1ii9{O@`1-XSUcaJ(*BJ6Aokw3M!( zpa4xlp`@WfqoAlbxwO>1I5jVpi3D2B-kz_kyL&SCPqUQJ>bYz#!V1^=I#$~&q%Q6* zYbBcj7@p>t&iFIAtTJ0$TOECU?R|2`QGtOc7F<6ak+&Z0w99c`xproneg6(b$HTA%SO!;j2g>q5|n4`h9E<^kh}g*2#%>1o0UCDEz9qaE8`19wHNwqsA7K`QM=MO9^yE60WTDtejwcxO@=Fw5#mcM-}nVFe<0s=wo z!DWmdTk`eZe^(}<8bRFJ>5pU5(pVV<4N!2zv#V@n_Tywkkb0cpg1^gggk}DKe!SHzq?)n_aiIo_T%x z5$WW*rK6)1uIw((r$#tFY^(OL+Om~QeU6f7aKp&h*kuWSF;&uy?8lEENCIFm6cC8X z<>emfj4$&S5^h@C*>MR7kk8J}hR4LTMGDuUt4xks9x)F0Nxdw)+ffte^;H_`r-<&w zix(w4|D2Cum!raBLT^MeT1>Y^JG;18uzt>wkdm_cQEFpQe*a9$t5+`D%ZAItW#r7! z#J3U!8zHKbGczrcZ`HCz&u|YGnl{IAsXR|g3QJCAsyRN|PZqHvUNW$4Y;7eH&@XK1 zN)XV5K+yB5-`)K)8y+6s0B30F=$a(6)Ure;pPLn_0HA1TYxjTo;=i-wah{u7o+p0G zeDYS5fP@57V`Jmc$9wdmqN2}VzC3yL>QzDWI+l`>65_NB&d|v%w&HQ$zpSXc{aH7n z;wglLg{j}Yd$;iW_YEH(X$XvnZ4VQ^q{C^X{wX^|O!dy4FayE1lq-%jw6u+q_}9V3 zjDq*?-`^9?^vs8|0+}zwI2kywV4wpI7 zGBSo{tHdHGsB``VH)Ev8(<2`i8Lv}-3I^5g7iAAOwb!=Z$Yu1CF$EqR_|zQy>CP)C zXw^6O4f&q0zhRpsAS$Y-r)N;>b)|0am%ukXerh(!T-nc0cF7=6LHIGn>wf#S^>%hH z2%VamT6k2HHETrF3zzA(NkDQbNbwTieRm}#rMumU7gK$xI=}Uc#mm1;OthTsN?2H2 zl=t$IWQf4R_mto-FE6j+N3%DFb$d~pJ=Q3%TDCB1c4Z?Y<`*wsTzax%f{zQ2Q+uQ3 zv(`vb={$L2%xmc)&Zvxxl$6xco<-NlDQwL{$Q??l`7`vNY!In z7M7@-ob#s`872><{?c3eYDOa4`+-}^wNyb1*;D6K}^y<}FJkBsE zC50Iu=MmO5!=Ybj(zv&^6!-kO|J5-MLPo~Tmgs}nkPxDy&0IMS4i18)!J_sA0eu6N zDF*&{^E4T6MhM1G)#LfsXOC$%%01RCmq)7T;hco`w&0RPjr{S}2?_UQrn;Reaux|^ z)X~uqZkmPws!@9YAxZJQ0iKT)6Qo`n-__Rp#W`M+}yM8r1I!RpOez_0AV4@RZmlZ?=DGTun({-QtlM+?9KH@ zTf$N7a_Md?oOj_%?SNUWG*m1S*Ka6DOaep_l2A>2$yBuU*1|7$ z&A{FpDsw!&y)t@B_yIbX2b1g%*k}p6*mk$P#4te(h~Dwfde_o~DB`D2kB=&%KmTqL z|8Rjm;but1m>2oks}GetJS3JRT7pBu!mOZVcq|oEyw-YW&@RIKVTe$#Xz-u)1Bf2; zn5GoUq7}`%BfEPtIy!m?z&c!xMb*0@#+gmuyMayChsmLgd9`l8uraA)!;@M&g*VC* z4u$CG>Eq(!0w8~5?bB(t{jcmMU3>H9jq0siijN+N`r<>G5@6|G#K*@aCfYLYXC}L1 z?JqOm|5;863$(dt-V_YChm0$J@KYQp^dv4ZaRXDK&SS?AU zieN=|7jrXS4=maXt8=2EW_|&B4qF}WAEov(0t&B#13$!?gYi{MtP&>GKv3Dzl2c1d z3p4j4LJe0lGGem1z0D~pnF8~pU}R;D9vd?Q@&1Ar^xGIg^2!B70t>XHpaM}4mW zthzs5#|n71SYI3PQ(fZwh|x^acTd$x-~_089#w7HN<6+Y0AzP3Hd1VH~r>U_@* zxj0}@>CCaj6zps}*%Tt-yLWksshOt(kwBFm>zs3a8GVvu9I05jr?Nt3 zHRv?;cFC?{bSVZ3DlHQe`t<43cun+VRAeOi2|_|5?;E&t-7^Qf8}KX^tU)xrtt~AR zaEP;_h@9xksjfZ?eRF}LlutT$GyG1 zYwPRsKyCl^ah2;L9Z;Q4Nn(y{+qoX>4rGKuJCkT*f||NIRRXEVJxxu`zym*4`2;mz zUl|>Kh7%`F7^rYqf0N$m5d*~lRNRi$M)jmF4{#?aZ@Ip=!qw;_l#<0Cq)eozq2FWt zj&|pdy5~tnY=E1#B1N>Fi4KO7mnQ(a2N8<;bV2*A1~w`&G?aM3Rno)Ds}vSC@0ElL zCy*#+W@ZBwk>tBwIXO8e0o7dVf|H5jRD+9)FTH&EvN*=_QsYZC(6~Hz(tF@3l*K&% zSOff&##kU97#f}dR%O_JuEjnuFwpsM&kZj^!4odJ`1^M&fa?14hd#Jjc>hUa;;FXi zv%m~&SX<8p^2M9`?GjN@Q59EI*s+GtWS3M_gari=`t8obx=?V2%iN88_H5$!Z<&R4 zpW%UcP~Spxp5ETsm6gfc{;Z2Nb-uz&OG~t;PHldh-Yw7Ng)}kgAtxhiN|Smrv9N&l zhpS57zvtiC-7Sfc5AWaI+f&HYD2}n9k$)MXkm+AhaTOxNlm<%&;@zPA5_+{;>0kQC z%*+gzO=IZ_v>T8-#0-hQMvX9inGq$(K-D)|kT@ce5lSbFjg9lZJK!~mTlYopIXVV~ zg^|FRm!>Bt8+WBdvIG9wd>tNs20&6ABaaqaSy|EHF+l_%6vuzAo0|*W-gd`}5Z|y- zK=)OB`UJ!z$Sq2qGz$ZvYKLI=AjPRGnB1W(r7Jl33M8%9)i5|>R##WYSxQk+qk5hW%1KskZkXH4+$lFK zXnB4*1=P^+utNU#(wKFf8lm{YLZP5?m2)lMehdvU)vVG800pj-xs4XF1Kq2`!zFK{ z&_M`v1}YiON}8dlJ0~}ngprXEnM~@NH{7774ck5V%~lEW_3{z&7qqI6_6EwEay8Y{ zvV@5_IusV>mvppp(ATe8aBIt{mWWE1X<|@O9C4~1aj~${R-c!Q$+h2N0)WKHJ2oco zvb9wY(tw~`8h=~3O6V6762iwX7#v$9Y-wd>*v8dj&t^vMOPUMGMEoQI$Pm=WPZ-K?43Q~)oPZdIMy+iq%Zj%wIhpQZ)(N#Y_$w%wnZP9*e3Mn)$={Au%`z*72plz+6cx3_)#bq}@< z2MP$1K`4O;Oor<0Pc*f zhV|@=E-o)yLjj))F0PU+`s>dS*$QUKM-su$V*8#(1&o#@Nc~umey2lLVn6}vQ~qzw zsjl>NFeWM8(!rWB5DeAfXWjN?nx3EC>Smc}{EI6uFE2%QFlc_ovFj;BV@VM@{2^NJ zko(v|0&&E5Ldku3t}Cai3hPFpkgUm{u-Ow#4iX*~!VNc@ApkB0i6bp7t)Z>m4BK8=-_(@y&Ye4E zzHeHudzkYeW?@dmRN?#gWU{idI4dS48XM7j zU(f64=|Kd-`*rp7n#RX7+$e(3Sr`DSJWRMDXvm?K2>O0J{-0_FwJ`;%8n|8-u(fg7 z$Pk>2kq8iE)zqXSvwd4r!_p_aFW6!!FQ{$tk`#54h-ebj0TdC-%Y#McD&VRtF%i>a zu)(3BSvfhu$eiB7Z~bv@X(p&GKa=tc3MgTQ;A$v1IxhL2UPIrza|aCJbY-aFd3orf zx;_5k;bAsI@@5;ncfao;|zdyXT4*302at`7`pkk3&ulLZF~*3Wirgr#%@p zwF<~Ng!J(RM={r$#pdf5aY{=A5#^xO!Gdq5$P=HLl@$ojuKXE55%OhXWu;fL>p7q| z>Ks2mS(1<`q{(y5`1p9PN;__URJS#71s$H29Mey(q(Ew=mzWT;5!XEnPS9qnNy~Y# z8o;{<4haEPXB{X&f9uvQFfur#q+Y>oB$Cq7qVw|jz*H>#*T*Oi@Hn0)B(wrZq;_|8 zHIWO~G)T4~j0mVRSY_(wXK!L|K7K3)-WBbcGoAlpOE?_vl3gzg_`rbTXgg6cF%dZY zXM2hY1`5)n5E2IV^Jiq(Ye2#ng-kkNSkiI~aF35h7k|4bU=)BP0I>m~-Ug$>N06XX zQ&VfdE5;Eb5}qdEa^hB!5JOL)!yV3%85E_-prv)i zN?g0f!O3~z%$YL))1vp`H<%j!@$vLad|aFhq6GdxfjDJkSpTk$1GRY0QT_B-_Uphv z3Fu;k;I7%w(b1g%BjXciB3esNk8xyVga)xgy243qsfTh6>wK;u%AzMtDrrF2V{U^j z^YWdY4Om`8YvPP6RwADMos`R#xvL`4_r+J(%F4=R;R^?%i+fYADR(1Tr8;L?<3XF4 zm>3=%?Yk`3INl4FM}sYF0a962O%1Vrmn4EI-d9wd1}9w=gPGaii7a$qN{}S1WNqDu(fF4O%y%i zopEmy5DQpNxpSyTRj~cku{u~(!`$arE<0-$;Qu2W-zp^~#TgFokJVt^ zPW569Z?Z?lk^TE2&2=+7PN)hqWlU03`c=SCO)SDD~O-d8^~2!yy>ZTJ+44 zgtC=51Q2Qv=vsZGv?^jW!0H5Pi*V0rrUb(TQWPdbmK8cb;^IzYQBqGMRoU2)TkTDgI=oTpcj!g+Lgu=61KF4Ny7|Qp z-}!Oo_37$eFF4-ea4k^XOOR6rC4p`%b3Jvi^yUdAHrMsm=N@8bchpZJ?9jU%H#3C}hchXyq4)^0yQ|~9P4pRiDW2LS4HWv{? zr|#g-EuTM~yqkt>f5Ksn52o3V8^Fhk2qK}@iaQ`q=ydY%^yIj5g#k$v3?}E!S$fpM zQ|Zf>FQ2<|<@>Ubs@TbdPI_Q3(0U_$Jl7)+De*hZa!$yk(my!(WDi!1r25G!Fa#Uc zh@^$oP85_fy&^M8Ff$(fET066qXj~=7+|do*^cBh`1+6x?SkdBOr&16qT!H=EY}{L z1j-y98=HIk{bwcEI3T}4cm;qk*N9{CPscJ{=Zu>We!5HkXJez}@w_^i9Egw60&LAd zg^R@z*qsWyZ*Xa;C}QOC)Y=IQE?_%pz}25#Dyk37&E+-ooMm8<^+~RK_Du;SANST1IAlYtURfj)FF!lg7~4cp8q{GZn)$OiZE}1oX~{iP6{A*5b0`ED#UZ)|N*P zXAm$oWyvDxiY}@@`qb&i+G?Mbbgg-2aWTTQ+CB62YZRCuieO}z`EKdIU!)vI+u%Gr%CxNjj`81Skj#`Y_WfCZ4k7n(-Cxs@~p zX(HkF>l786ba+F<^_G^FbCQzG;2z+t$kMgmDY?6gmzI`ln3=^aEZ8Ia{dxE7CPj;? zhdz&F8WosvdGFr?^$|c#ymG1w_`iD2lWS0>klHcOoBAB|1#v3!Ev;jl~h3_VSK1bXZg1A{y;$)Z8U^K?KkIRyoyPU{E#+VUEy@uUX> zq!P>w=m|m?*TUou2*4u@^i7q<*gMMSe0SIiT_KPH&NaN&Of7dkyI*?UJwhoEW$#ZhqN>c!iV5XckkW1w~5KV5e~N>T-pj{kr?(?{RPPm?mS53 zifSjWY#G9ePfi3V9z%77R9}_@!NYS2H%lyApS2rdi@bxW+|0mpflao)(I=M;sVtdR zc>Lo!tTT8*7It=59d?27XquWvi;RvYyU$Ua5Hb%<(~6+7v(xEG4#=-RmRT`2YJ%t6 znD1j>?~{5S6GQbfDQOyt@nPQ!V3iz6-RlTrgjs|BZ%*%ZZ5}j#N?D)>O{Po3 zyTmE&uad5T+SJJZ4t?_wnJ$@y#$$1wPtZyyvmqLLQ@ znLZYqvFmN9Dx|`my_)su6EifH=pw9@{=U=8*VNWV)B$3(fsV|>;glfBAR6Gb1n+Ll zPeV5*sJ{LK~XUjHH!y4kSk7p0ok9w ze`f&q^H5j-NECE@HFR`>_x8LFAW?wLA|4C0TS2h^DW(Em2F^NLVPT>3qep?@&VjK( zZfyLMU^V~a$5W^YJ(l-fhQcyWHJ5!PWr^h@WQ+bp^EcFDZaf@j9c%KF9rGFWx+6RIZiFwoMv z```g3B=b>`fS}E=;?jHS#o4yO6#ZXz2ym8}K%Qq?Agb@(%N_%n7k{v3`6<$L~J86~_sWMuE; zG(RI-1ulfh3@ZigP4J(O#B` z{!xpSPk|4Cw8z!SH%Vq*Gvcxt8uVQ?(5HA^Tuk%i@F3l5*_53fg|t`wg&zm6ZTxW? zyE^V;wPrLiP<^fDM_|L?-$r4(l2JC9G23@U!o=_+dY+!100oMY%a1oq+7Md!=jkFQ z@F$!K#R%9^J@$_FSHbvQZ>hJkvkQY)a~k^l?l7L_xzqfN~HJ z3I%#{kFFhld#4S(j2_9X&d1KqG$a4=T*f$(p@aQaJ*`XfnISn9)!VOkq3|O13Yw!y z84SdKzm=rJ6HKG7fu&K90I%-9om_LEHc$w6UjOys8n_+((v!W}N!R{=7yri*{--Cs v8|Yvf-VI>?sccAKRfHe^x7WNPj!ACpX5m80HJafqHk8^8b)`auyMg}&)pgjg literal 9904 zcmc(Fc{tVW_wJT?$Sg7swFzZRW*M?mk`UX_HV-8v^E?w}OA<0ANirvy=ggwW5Hb%5 z88Z)Oefpl?`Qvxa`R`oUx!&Gu+k1cZGpx0qweI^~Po%aM<|G9x1qy{asc}_J2ZbVF zK>jBqflrunrpn=k#Ns+e4RwtC_o^m44u#@IX{agdd8RD?^fLMGc0{^57--TYD)2C@ zg(`wvkS_s^#`4BioPRy{enQ`|T#?GSc3}h zZ$FY+&6$$3H<^w9sWKJ=-lhmn2idy%zECt*Zf{%ffgw2P>^?7_thJ;7UshLJ5T z>Zm|H!LaOXentx7y!`yr{E{?YUS0wM0)gt83kMR#SgJ6Upuj*>czajG*MSp-9d8N> zs3WL3R9sv(&IjO(jUNvW8$EyVVyd{6GbApKAv1JtvTCJP)xlxeAz-YxH+XZ?^?5=9 zpOn-KW$}$~U%v((?r&4xJHPr?bJJUanq3+dtxg!P9CmnP|5lz532J9#$Fj1?)$^jy zchrgcCkVApV-2B&&(2NqybD<1pVlio^c<{H6&p0sTjm z%$v+#M@OF=9QY16-_UV!k+^_6U_uq$?bBNtVUEf*;6R~DOR2BkIN3b+MVA1@#Kcrq zO~gBTOECWHz0zP~%Ib3rrA8-FR~@chVHAeqnVk1RCNlCzROW&RA z`L!P$D{%?@s$LaxsMX&=_pgOkp7Puf&z?rbT^s*PgK^L%EO)+dz5VBB+34H81v5*_ zCPPl$)NF`MN!D+}!xZ`YBfEv_nL$|L7QgMEq@}~N1rp!y8SL!o^YHNz?r$xJl$Kt+ z@8BSJTv}2Rf-9WgtR|^FI3TgW`S~dX{>cx+yETzkpNmVQja19yTh-CkW%&L3_t&AJ zN8#Zll9w)J=H%p*2)|5DHotq9JoYM)oLp6Rn3KId<@M{=%PT7W{#so4>#b*CpjtSn zn%nPwO3&V2=!rUCfoUCeq}mAx6k}s!X=&-pys`D)zR~IAFluROJ$UfoS9W^1*F>Mu z#Eqn@SvueQ`x`okDbH3ulN-Z0o+U+5amX@C*mhUXO7WzL4Y4gRFKg%aL-<|3apR<( zAd{ew(5+jy2-w)z+GZbmdU~48w8cJfaS6%I6=LV$5c6C$gQdCcj+DR=JA3wQTW4p8 ziT`ZD>6vv_RDBbT(i0{jeOH;Dfq_%~{r&F>3o)vyg!k^<)4p>j`uFd9DRLgv_exo1 zQ>n$1sM6BXR@c{qO)f-~m0g;7C?hN^?3mQrddzhw=G2p>@DoRA;!oq(oGc#OIr}a* zSM`Z1RfpSbS&cq>sPZG{M+B%gD-VAKr>Fv4O;mNY{LI|k>!KpstgI{+aVs(yPyGo? z5Gj{0t;>9Om?k6;~>f@h7Lqqj_ zeGIU$J7d|!#dHvR+Btgf-o2}D2qC?tuODtdSYQqYVYzsZ@dP2^7xVfE8T5Djm*;Y2oFYCip&ZZDMKk zNl}pw96^HewLs>_&3;EO!Sw_L1Z9Ux4)^b?-MU2=6&3YhbwU{o(l|WKs;sO`MozAL z^=jLl>(|;K>qI?%JbBVQpBQ)N*I>VfJqJ5GJ1qx?Un_&#&cVmOjvK?n!%ySlLUgmW zCN~%QzS>D5Kew|JP*G8tn4P^iv8$t_^I-Pcpop7^iOC1~ot+(Sd3kmQv}Ryop;H<4`@h@U z#@gC}2L}hnoL%&hM95S!56^7)LV1~;oo#Dtt6%8Lc0M}r3JnY6k(FhIpZ;2R7T4Y_ zPqekQ9n38i)!y|qMPFL<+~02JPoJ^E-P~-grewarefBKz-Me>L#4Lz70}h-u)8tQw zkkZUuQRL~#Ei7zpkK=cqs3&+HA8+3KM&nU*^c8JwT9Ng?Zn&ERDjsux`%~xE<`N@w zD&3b^;mpUrd`TfBB-E8CMtJ>&a8Om1+@c-YdUc}VQDmgDrY6<<_wOG}eI~Eo|5Ih> z>k2Sx3ov3o-9HUQAuz$z|Fol{W1@sD^;@mf4%sw~$-kUMN(ZZisEJ7aG_#bM zF>wirwZDU=O=hBITT4F@#jPoX&!0DJ<*jo$+;dBjc0N(%u@W8~ee#qrHgIjawFK{W zQw2*IB3qDM&U}hAPquqtse91c?edKrJ&UIBlaS4^)t+LFO-(C1j!(W)?sYg=ju_|W-sg(WTGFtuguK|r7HQ7le&t^ z2{;N{e||px^~+9>K8QY2?GYZ|GFIc0p)K(A>C?b3$(JDUmf+FSdT52G@$q3YE_1J5 zzdj-7@khyZ7&ik_L$c^$D6FI$>p+fbu5-8E#SyYt3XxUZL@2J^P!82n8zOzmT2{* zq2iFJ&zhPqCCTO7moEGG$jyC8ju;&^8Hn|aNlS8Vu#>a^uMXAv$>IEWt!4Qes3My! z@^Ce*t$2>6C?+9VX6A_OH18;Uj9XC03^gko+uGh@VP;kq@yI6!B%d!{4b8~W&F1Ci zM&;_~tKe{)M{6+wn_ep{x@saW{_|7Kk3Hv-Y$yKw!QV)hx)n={3QSIBlX4gig;=HO zk>8-YfU9nkrC-rb8l74CQGzXhz%w;H9h;aqlzAxjXn{@Ajuwelu#m7xIjwC^nl(qH zfnDjr+(bjjP?0$yVu&YPN@)dIJa?j^qnnG&n-HH(g)!smEn^f~Wa&k;lTbsY_RO`r zb8KhMo^`7zIn8s4P03&W*|TRwt_wPnQc}IyY{9rWGbqPHpB&f}_m?CD>3jKbINZ?3 zyA03b;)*?1a0AYX5{u$Lz*LaBio7;#ps*F;Rf*b~nh1?Q-ETNL+%NXqcZbKsN5prx zMx2UmdpKdfZ*o#^ZkpV=Q|$Ot7B)y)<-Tme5plXiI4%Ng5z`_eJF3TAajd-h;Oi^ILYKaUtC6<5r!ZG?)g%qEz&l=LV* z_S@f@>NuMJ(r@_VrB3|Uj*-K%s-<=BPM(}dkqOr*647z6o2bp>ksBlB%z_9W*yJZ=JoH~6v z5_aOdGZ~SWxhWKPKCY`v6L9BEY^|@%qClwG?)t2>-<~8SAw$fO?EVp&)(s z827`7IFX5*H?3v&+5U5}y~OfHfu_{`p+Jb*^78T`yh@t`lsPy|w$Df{Z@kxh`0(Ka zC@Pg*woO$`}gl7zP>+NPN$w3 z>&D)lsG0O8J6kook4I22%8gOw&pAOs3Zwi}**DW|IKm{*y6E=yb|f(h3v139^P;;N z8&5oD7Lm&FaM!e$O_gyWad2>e3}?#`_SBJ*h9&~aq_?;1qCkphr1Q~u3UNd{=JHt8W9+D6KiF428xfiiAk62wRG}YO*kxT z*-D3z_;_074X3qJd2vljO+Fi7Tg@vr?hjIB8M|*SbbvN5Iv+cky?d?G!A%rTbs!6jz z7-Mq1rdy&zYHAb!vj&HR+@%=``sg&SsF|*q>PDp^E#bGf*?s3%*wv@juLl>Me}6xx znJg6z=9ZRl7?wn8sdr79+jAV972=Jio}L&*{=`H^00uj|yRgxa|9r=;;LR8t8=GfI z+gQ=ob`{<}cYA`PGf85hA_4P|&?p^3n1M7$DGa8leViF|<;|Lxs^q&3Y zD(mUvQv%^tw+4_%E4wM{dIH#)PB~r!3Jl~N%>E17I5d4EEep#d0Fxi^y?nvK3@Vy; z0E&G0@B!QU3!ZILC}xzk}_%V{48qW1?cxJe{0SlM8k8hr1uM`9UHu=~^uR6|)t% zJY9|V6A>}3rBYI=s~Ssp^6>`J-P$Y<2->99x57ngnP3DX7*C<7s0eNopQx^>ah_}> z9uTX%LTmk6KX<|EqQwi?J+PbxS@p8jT20_oP{_!VG+Q8a3AHXHWPbQSj~v$LK2v^4 zA3uJq@9w6yC{BdvAxXgsho`1;0FK3CD)`3|G=S=b({OdmlJImk;j1pj*gCETe(9|8yFZ!Uc6Y0cfJWa#A*tFxp;EsmoHyV zOieW+!%54@WmZ%qxjjKv0d!F}iMC{JYD&erkr)^}KR-V>!n~!uJ#vh7PUZS_%5bXF z|9*!+$Bs9J;A7a>($W$Z!xMTM*kb0lw*4NiIkzOcQX@fn71Yp=AIiD?AMnlw)fX62 zwFdK->Bz*~K7IP6hxW_pTaewBv2Pw+R z)|L-A)58jO@Z1vydT~)l93=;3LXUlrL8~SJQbJG#GQW|PZOqd>SYkPW=Li`Iql8}r zgS)DsVa^fv)Zb%g)x05}cuZcRy{89r=MIPPNfP<>X6DuI_KMjpN*dSOeVj(Na6Y4ecyV+2Yn$ z6x6lLTa*nAJJ!RJsLN(%6M;5-R@~%n=U38bt3kGnI1C_IH7M5x!VU&|I^u3o<$qLr;`Wo1=ZfZFFCgxj1of3FPD3HD|22A)3clBD(g)C-=9GLaB&& ze3vK-w*yOyiY|j*vP^J&qst!$f2r}(L1q~7+S=MGDk=($kMCC)r)Kz{5$Ku*I!{;_ z@zV0L&*4Rqsr}i%o$M1W`%8a4Dj1V#bv>4cFOH=dKa?0th=je}v$r>dBb=0LP`Lj4 z_e25NYfTU#I*`SF9~e;KjV*#x4I3aOCH?;Wdl6olOyc6ji=fSzaZIPdQJSoH`I6P! zrja;DFRC*^qy%{YGB8m8Jj>BI?v6L_-$x;K$a(*MYSZE8QRBY&X?agk|HHoq$f2Ku z!F@X%OgvF9sBvE8)qUwRDH8(dS^O+$6#mY(#%t!uK^Tdf0Pt((pKp;dbDRDu( z!hCGt;pMjeuU}C~n&F`Rd>S|8yXJW9A_laZ0p*4=Hx5%SbO zAvyW6^K?rw{u^JgVBExn+04ugxb$kBu3-E>44#Q0x{U@jq6)PcQZ9n6*fPI80;(1o z9{#U13o_8q&|E!353=nCq^iq!{9!*sPcAPn|DUQ(j?zq#rnBu%F-InhOG~>@ZhJNm z)oBF+1Dr5RbyilGlqN4aqr99YKR+MxfjLLkAdxZp)TvXTp)>>WM({`v z!pMh)unkM^-T51DP;;ZvB&{)A7^rWEU5(&XD@8g;379HO8~Lj9b9*~1QL}z~Jm4Sl zeDUmAFziBFMy43w%O8A!0re;*rnPL`|MAwA`@k)-h2q7*Ld(H|JMghBM^+EO;!O2u zL`u<0m2Cz9px_P`DX*%cyf^);fXs$wsLE@-oxj++^Ard{6`hCX`+P{0`6bx`(%+p= z=i}j-#0PB8DP#KqS?nFiOTV|?nV zX=AU30!63L(9m#Q3MEqkl8+EuyKlT&={HKO=&etTmg8DmT1YT4oYrt^d}O^h1we78 zv)XIyw$WWs$>( z6k>p0U}_VKi$di;=U1BFKqh~2-Z*?_#>%APVHgCAqswdu)BTZ8Cj<6J#<7^ml=k>Y z=vLTtr$qi;1U&@!&_JPSjA4mQLtkH<8)J8pNfN3a;fl|m(GPpRT&HCTp#+KNe1i9M~xZ0O5Uyye=ui5x+7j?~OjThu7kODYq6gFvVMtQ1> z{13fBL$o~DSpz?_#hquEXlWP%Z4q>)U^hmK7Ce^{9^S^u|MHfIyTDM4eF9XD8T8pJ zpkpsef2QjH{@rYOxYTp|rw7le$A+y72IxCb-MlyLy3&=ng!J@4-_-_Egr2T9kaC&*`kH|)G9y?NAZ!%wP&-3E`Q_5M z8XS86ok>zq08S5;IdK8{UY$#Jdinj_hKJ=T>k}wor7LyEii$@&mj4omswURmT?!Ej zAY{4dBSYO%2rQ+*)6#NVpa``H6;=mt%=eVwRW-G}ykzwAjlv*$N6z#-tYIq=1T*aN z!{Y0Ev#vV>RX~J?3fBdCL`2Fnspx&v!@m&q1CdrkdeXgrp95u}cb^4a3T9PuD>g9-TPy%)k)X@FQ*{btSihHp9Fjef^Zf$@7PB4kJ55|i zfgqA~&-HQ^DUdbxa~Azf-Sz>YG&Dc|{0k9NJg`hgK)Ze~Tiq02wjO&WT)5LnDN-GH zTEfO`vN4SPOjXcUAk!<*Dfz(40f&pG89F(@*iXFH$wB7*+}dh6kY|`TEo!?376Tov z;n~^iMV@OGRyH;tK&RoBkYE4~tZre!JJ*%eBugzqWx)|B^7ckD2{SWuo>3_UNUIIc z&*@?@yz=po0g&?m;sByrFG^P{_h>rOv$95s)b5-Bay?Y;%m*4b@@((GKsZ1jUKyFw zroL;$Ozog$9dsXVkH_R2m%V=fp5pPT^RIKG$x*LBI)T3%XFC&$-4^vn1C9^#_|~Z7 z+c|)?RR0|?21brCF<}SE5@aj$Uqi)~kYF%sYD92YXma#AK+F0j7I)#18n66DmjoFF zMdSGKkq7M0;>QPTbR9qOE*GHWlWbBGAW*bax-9~=18uGFn?#%s85Py%?o>G*`bbfy zG5O&)!^b8xD&;grm8BC2SzvN?vI$UjUS=!~UXY%k&6k(N@Er)KkQ)j9 z6$op z{YEM)1HYCUs1=}uMD-}{Q&FB0u7}Qsxvg!hY}yZd2Zv@50N3C!*MndNURD8RWD$!F z>j$XeG8sn6{1oE*yzwWq&yVOl57t@{y`PFx zKG99H;eYQ=U_pcn4i3g*B-pBj08i6^UqDO*xI~Dm-Tx!!>FuqeUlD{{4lqRS0%T~D z?VYcB^@^K_n7F}Alp7SE2pX;y43{4f7Z(?jkQ+nESO5N_}ARK_PPOa(edqNw=^AdFbdl0-&6@hyA%+~O*q`mHi(hNj56{eD5Z`sK5Q7*tH@ zpZ<7g=u1Hp8MGNb1qBWO3@mc)!q9RPb6>)?S$!brah++SGpTaV?;J@@Nugz8x)>Pg z2#vbSpkjlD9-ELrinLQL^`I*kLbCS9!?KXv`TV0NKjlZ>*KI%jzW5iBl_gnE>X9&Q zU0TKz7qFlvoTg(IsQV_ZXNe=#f}s^LKG5O&lka2p1~f%uQ&Xd%-W-6y1_T&c&B5WJ z7_g`(tQRSCaz2i)b3n_SSp*lF)eu;W!JEz8RBD*~eOppSZhBOehDSt@VlY1s&KR`X zpzxLnMMp;q2nwno?YhU0W0R990bhaDdCr_6;^E=(+cRNRRKfw@fw&_F`7F2e>-cWv z%hqH;189WXNtT4$%Lx0T7A=*VE2fq{C$;e!9B48rHSvd)F7#W&avaI+UneAG+;%NB0Q( zJ$pLnUon@mMkEL;U%5go$$BHb@U&U!M@AldR*kJLzVp$_9U}HGXA&rIX{iLAd0k8bRF* zNnExp_P&!-gi*e_vN8emj#oE&lq{i55AF>F9tfRX0L~M0b0n88T`I4vY=q`FfYI>e zWaIe@^By`tDfj~SJ31WCJxgw1U!MSVR?yURW`2GeYWUK_G@wjwZWn>oBX=W!he3xO z5yqit1smmuYh{N(uNVLiSjF)iHRa>uJIm%;PO}t1tlw%v6NLaJ=r<_$Irro})TM%@ z(Hfua^JNRJl~os>r_ndcV=x#Z5)yYD=lJ*Up_P@gK!IL^?)Brx54(H!g7tI5)H9XA z%FsB2Y;I<2OKNO9(tQl|w-Hj$*x1;E-Q|W+CWBJe+qJdQZoe6kJPXG*U;i=W(e7i` zXmyi1e-6MD7R4+e)c~ao1Xc8n&N+2FJJJ?}`v9Zj`@^O7sUjPD`}3`XUL2&xRVAln z2TpRq`n`i`R@_%_wOZ7$-d!+g(OD z_63nX=Em*LK`)=JrG+h{iM}eC>L0l-aW zxx59DU<+6oevW=rqpZ_I%`1tsE8hcHjoll@OV?eRRp1iLnr9C9L`fTI+ z>Zh%6HQvc)4c)g_;XcGh8R>j+C{fDcG}pIZzoXnKDJ3f_r|xS~;rY*?pl_ai_+URm zB_l93H5G}=ZV3WMcC3QyFIs@7D?1+l)e&E&-#z$v+iv4u!@DE?!ph>*RPXM<#mByL zqJF zA{}KT6qIm7wXfdkGLDJ3&&G0(+-|ALSH1Fi#|!S&>lSvEY*_{*LbnxZIAp_dxMG<4 z3x=qbCxo*bDzXQ+h|ozaSAGiSC;h&c=(m0-aVEI!>s0Ma6lX?_g%_`q$0sbT|ENH& zfMGP(^F%kUY`Ocvg9pVza{F-PYl|md7%p=#en@*TaXTI!nH&x`hbj(paTXtQaCzW= zJ2-!bl{9pN&Jc$=G%SC4WAE9sXVe0Z@onr?|G8F~r*pDAju~bMT#$mCEtptOGnrV< zK-Ns|f9YT)77tt*{tIuS6PNg3Z`W-soVAwv=8cw)PH<`}t+T6Z`?qi87Co^c4GrSP z4<6LnX;^l3b#d_U1RL;Rb@0vaxR+N}2G`UGOGrv0CleDBIj&z<^zo5=#81O>{rbX> z9~d`o+?bf0d_I$*p3A2tf9DRa@9|Hn-k*P({nl1j5Zir&9pSY3^@?0+3FI9@{E zy8C-DH#<9fuH`0vP;4yW-#`5~+>f@I<>ci>SJj0iB_(@%du1#wEnEDymX~EbJVY6! zyu&dsQ?_<=1mzOCFor}#78Mo6r>7HyE-Z8tXgqrCnN@0sUtG+ZEOa+GKHghLPmhJ2 z9lNaMTx@)UYkjiH=E;*M{s{^AQd0VS&kPxgi&MAv3h5c7(xL(b6X9HImCbm$c^6*k zl&*ogIw2P~_b}&HXGlJkCdc*8d&l=gsKreyewy=;@W{Hl-ntz_8}jKTyVFdm`bf!z zrFFtojXgSQApK34fEDHZ;amxG((ce0o6O^P*yiSX@@xWf_wUmf8X1Xh>fJQ2pw`vZ zMfqoCEp6{Np>e;zje-ZKU0CzN>`!9E-j-19GQSx%!&O>UMe+0J&-k1Coq}CVchf>{ zdC>icUpm~Jb9}dfKTws4j{5QI*YoL;%^W^0*}Hcy!}W*Zo}TsN{ub8OwvML__7i|`m5gy!V1yh=*yn4V6nVtA`vIJr+9 z8eqV4+scZ4a&nUC*l6i?<0O^ zH@A-mv#=}iM%OksZ#y{&fF%!)j09F!->R&t%5G|+Cyl^({P=NsdAWZ|3e7vUH(280 z;`n}aOS|-wV`F+he*czdNswVl$a??&G9@J?wWOyYtR<$Akbkm<0E|WfEFCx}6`N$l$?qi;D0F2?>*LT3|^? zNSvJCbGDaH7Z(YUWB&8!&%E_&H$C@u z?{?UXO$W#fVG}SDjm1TgRuQM120N`YJKe>|}3cdv~{E{rks-l@;83_wFUG z9vrxxow$T6uu#*uSd10xsJ?j}rInjHG;9_YMs&tZj+L9=LtR>0ii3woTRp3pA?X#m zxM)7z;H6_wV0f$pex}mh$tZWjQEvn)2xhy9o`< z>oB5tCd`rQ|w9uebUB#St*G&VMx zK7JgEj!kF^tEYk;$$KIBQ8%t#;gjbLtWO16xds~$tAG6bxiDSlRO|Gckn*}=u*5f zbcLSIXnMFsSf3m^&2REPs1N?}<5BXPH&=Oh@GtgFF1jX}qf3`|kGFSr+Fo3{%S0KK zkeV84-+YXvT}Zw*Ue>cZ`u_gueOwGQFfeRv!Yi$X_wsebe%#b?#x4#NBwrvP(a@yLY zU?1hyqg=ruA;XdcJl4;ib;U`ahxaGlxXQ;zOebs?oSvR;D0Gb!cX4r%&$5ra*rX*R zPko;}Hw)aUV%Vx$Q)j0F8y8ok$H`uyVU?J;cuH-p2pb({Lv&rBWFGv%u zwX*Qz+J9VYVmESIpUBpZ2X(agzO1=DmR`KvtOF}e#1UxM%e7f6gfF_5Iwb!J)M`Bx8M3F2@QorNhBS!@#rK?GbKP=qWmKe;09Yh1`JbpuP~`STB0;-dsu^lU{~cOU{w9seN136W!E zV>4f9520Ou@no_+Ky{Ww26bw8YO zTUc5O3J%7o+A3$AY#SILgx5uOT*Fp#)afDi!*o9;5DT#KSbn`0@n$*J@oCb241T9m zbLC!#cHVo#N@9m^rT7J`w&%gCVA(?kN^zGdy~?Bwuf=o-@Kqt!_+AY}y=k&uv3P+Z)3 zIOnGk7b^(5NuwJ+$MyB8DBQj+$CAKNna+$I*%lnZB|9=b9gW-{v_p7YoRy0#Yt9*> z5RZ=BK^8CO#z|=(brxr1Mam+hwT+CTt{YXG!fZ`6xRn_a#k17+PfSpI9d2%TIrrMP zoV^=z4W;JrWMnJCh)`y2KiHV60PC}zsn_kGCw)-vvZiM{QGSWh_b9u4F!olZ%zrKJ2NDmcft$#FA?{f6xrAGO)saq1he+{yq;d5&4-BQ&dc%dPE zMh+7zFJ&4@T?#&wnS}){)AxiEv+wR6zp9R8+4!r`cV&F< zjWiO4jNZI?lQJ}<$tEg#f9Xt1^6Y(5TH5E}&`^Gs1m%TidoF7Cn?yfrS$@%rPzb!y ze1z84*5>>nu>g4q974kG(gzLr=iRrOaL9DWY!mx`PE^@Yfxx?+(h1jf@VnMD|}8};VU=-&c>K#%t&>xew*qHu&27ZIuYk( z;+gL%24=Rl(eh!0%#aBd+S9Gw%UZTgF8IjssluX%n2bz*u8O&sR>+#%+}zw@ed1=` zxcqRK%%kuo1t3N%LN+k@bAw}@ zOCh_Gc;k<&ps?@~L@Y39ou?GXWbEqxgTI$Lh@`)19tQtife#p&gFO*s?5`$Eg*8EP zUZh>#KqIHD&nh3H+1Lndu0uixflC=}$#;RLAWm+y{6)jU*Uor1Z$f_&cxCX$$DOvG zKU)12uHOu!>4XUfGbHa7xys63f~XUpoUDBJ`U{e_-@7{(=W`eG?(QOg|Nfm{TFTw7 zhK+V`V9~~hudV%er_9eoj`#P4mVWHd{-_GqXS_%n8Oo}Xg+ZVII zO7r=hdwX8NT<8S1b`w4$zM@fW&bXKMFU_d1ef=TbdxAXMT_r4ZY0b{`t5fQj3nN~7<9l|(Gf%mU!2NWjOV-`x9x8Q?s0^Y!ZoXWiE5@Wf!7XejT` z*wR`;{oZ;l74o9bf8hKY%+we>zh;O?qhx76sn{!`{Oc3Vb@Nwgoj(`f&-wSxpY~S1 zUP3YJF(*IzySl%7r~4(bImL+mx~YnaN_;{>z@YR++SzSft>NF9ezDQf_^)2Q!WS*I z=MDa0G4fWQ=#*W}T0vRax20lI#)g-mp{}lNbd&-BL3njsH~k8|gy#Vcif`1J>zBdj zfSb_^zkXo}3)jf5L2SPw=8WTawq^oKe!eFyIJ$B;S81ut?~I#}UMv`VEurU|z&CH` z*woX)S68hSluTOF#qaGb5ASv*y;CpgN)wGUvz-=LH>Q4SNXCXDrlt;u{1P6ob*`Q9 zobueKY{y-q@nF&>S=fFyUI_FZ24nz@$G_gB+9eU^HNgkvJ9pah?j{kFlJYN$`1$$0 zdi`1|^YZycXv;;*=|e9mDv7Z4%U9Ljw|DmT;#|6PY5UI~*x>E35Uoo)V7%y5R7qxs zf}kGH5B)BvDk>@<3A*}Ch≫$2~kcx)VeLF*!L#fa&~Y!qovx-Y5IC zn+&se=KL-sS65ew>FG;8RpV$)Wy^=bZuTdVI+DrW>4Wjq;4dM?_MQm%fB2|aSyDhi z0HwL?!_CFzurhKDqV9(gcB;&zRT}js#AhxnEFf{!%-&wwR?_kp8r@l;wuVjVuL5q6 zpMJ;vCgcyVr4jRriis`W@H@VIzIAaPx29*1OJ#;gOju&u$;yZ|JshnGaIFp-lR}^& z9cS>k_gGZb)5Ss{P_%V-hav-#V`2z^*SL7PR8g&XCkqM*Er6)SxqNx4ohlM%K z;4yPuCL-#A+=rPms?JXFx)?nhOkH4RCS!LTsN1oKRj$A{5(zE zEw*%?xQUdKGSun!fa6SPpEJKIzzifQE;gtLgsYmVIXLiv7TfJ}dhQ&e9{LFsUHOw? zmb{D`s^QgSI2Z<%Pe`=y-Gk@7CPu@Sbi<^(NcVlMT@dkDiD4BUeE(HXkhpA|tfQ-o z%*&YTHP5H8J&|M&I8~nux60Q#zB_mC%4=#8OQysvUjtbKS=Mx|W6+x3t9MQU{F0LN z{jaYt)f31GKK}l}vbT@+ubK0tN`Cv>cki$uJp&tEgxMGD?(e@0MpIB&xKK}#x2%)= z=<(yl{c)3g`71f&N8VC@&(0E3Qa-^QZ@yN^({v`70Dz)5=UwaBGl_ zjJP-=*uxlLr;tv4r7gGR;^1fnKZETfh+;nJY;Q;Vny;Y*p<%p)L)EICRa%<7l~?R7#T!IY>1 z^UW+q!UkeLSn4I3dSWvtZ4uuV#fYYbU|78byHJ%#x+&^NjL|%tcwpH{L%Wx24dv{ z6H`-=#@n1Zg;$BGvXbI{{`}e0*7h?zpF=n3P1pd$QC;wOtR;jm2L{v)@gs;OOnB~0 zbtt?a_kU(%6O@vY(!TZni;btYWKlk_Iu99}R<8SyL+1ib#P2YjtbF<}75N?S#W~sh zjvb7p=-XQQ<@59N@(&-hpYTRM^1Cro|B~eC)2E<4Ft-18zHDle43=ZY3z4g>tp`#O zzC?&iLvv#QVHc>O)w(R*->M)QPJG=7;}|;@VJf0R5#CXR|CkO}-x7eH2_Eum3x6 zEcnjWj87mlugU%Moz&_%T;*^vXBXZlYRTK~5-5N7@P7Stzs51~*Tg<=-@|jYcLL`` zZ9B|1%J%Fv|Ci%$52N>*bl)@m+41=4k`PCPcka&8SW|NtesH^2`JV?!3TZqNt>V{rZLp zAt3yaW_gA1W_L&4G$|ii0F{8jMkZlOeycuwWM@YxEiEnKb$}196#VYjK1K%S)G=`? z0Oe+nAHN%QlI8Arn-mg)35GWQjJRDiLILga<;$g0)GzeHB4n7LPJ%JG zgfBrQZ+Ua`7dR`CuQM|-rKP-+0$SK{JwY-|LlOBYBsrWhF)@IJ3|(s0WGcl}QKoy# z!$)U_b9KlL_zO&@xb($4PQxZFTi_wZ#l_o4I~KKdb%hT!?(4*@K9z5ZjKqtk6~Y4? zQi)Yuw2v6Sl~ouRy;;v{Uup&n;xsBNEX0KuV!qOCUobDEA)^6BJLPucsGEBk=e?ZfSaEM$6E!4;(;r zEszO!clZCCc<8kuL{3HJ1b*b|CWw7O>YPjzKbq&+KJg zo#;E&6ge;U&Mf$m#p<_0!v^0^{owadt>fcVZdi;>9GN`tt8TtAZF@~5s8X7$SOTkY+6{BJU zj1HtRSDrp6>Bn=?k&%(|N1sSbjkHq7$}XM$5rO;+mh>t&_gZ5Aql1GCU~u&e2@oa09P1ED4JynruUxrOYZpS{Ny65; z>%=a7@mIjw#s&%RW}{ zJv|-P#;!wV5NMX%>^IookH{0)n@Ac+O+!Oa+XN`LhtV%OIhooU)=JDrn0Z(Ozk>-_ z6QkeXo%q8#i|H>ed9ENo>v$| z17oomqJ@R&u5>h{5#&#og6DmE?43eI2rO=q=t?li5F8&T)Fx|RE>3u$LMGY6!ymv21{T*3_T1CNY z-NUBWS3k+t-w|`2*r0ahjTS5swbItp!$Q?Mw3V5}?-i9+cZFQ`J?kdy?&v@d2nayn z_Gt`hPEJP{5yF)j7#QSZ*WO&-pgO_4Q)Yc%h_}*YSy+$m$rsvkg4-B%&@MncjPwE! z5TH=mLmopra@hjy9riid*=DT|AO5X5ob@7lBW!jum{85l&A-MNuW;i#;CBdEsk8K|(_lVN_!`ED(vy4iz0ousvXnTnKC+#y@u-QVdl5 zH|Hm_x3}Lu_U3tD7(H*w936Av-TS(t;tIDOPKEQ+n;`HTo0=#}ZJs`j0Ls188h{Qt z#)!gNFh1A~E>f%j{A%{}De#cl*;z$bt6zs!#Ru;}PX%_TayJo8UX#6W54>tmCVk>y#R9vzu(OjZHZ zyg^Dz8rsOCghK~D80tznlnrX8@PZPH`y@&$}2vPu04LO%%Z>i?Lf9^J(ZZ>T1 z?dcpnqGs;v1JDEUYzH73M-|mH9VJ*8vZt%{xDnOt`#l8lT@==@;OA1CB(wWE27U87Pfkpo#08R+5D{opp+E|$2+^bwX@=xAGG+=1HDmJ zR`!QFzTDKZ7mTAt#_>bF3x13N2^(mNuMS}fR1ccHY2d2}AU04|IYYNXiCO&_!f>-Z zv#yA{;v?ug{nfRzG)XT?r8tHzU~?SGSXtxngucCGjEr$msT;DiK$}1QIR_u-r|etu z-5_2zf)dAgPD00O9HBX@n$F)7i`%vP6WsxVc@G zl478spg8K0KEDJKBBYuustcS>&nwP*NZ==ME4)^@5+#>fdiW;^ouZJPGAd17EELG-1HsYE>d~*K9cfgB&wDKZK0t*>7U zB9dwag$t#VVJwXdi8gFTZ0BlUXaQCU(tyOM+E!LwJvOv~ z)fm|%udf>+Is!Zz?cVi2eSEzyQANDv`Dz?>0If9^UqMtIT1RQO_LkYneU=G{_-jBQ`PI z@%C<%#l|&lg{f!Y0Duhaoa6DA)R66M2hZrx*#3cm+W?}GBOso+s+66!vcJIO;fujd zBtLs3h}rbn&D2wX?&_MmICqsMN zJqO<=6?t)>_-w{_kTi-v1%In?mvD2o>3udY?i=WyKq_=HY%do8iCAMsWN#(x@gGUP ze0XCko1+7gM!$>GxrCQ5gP<~OcK-MH=s&B5?;eNC4tkV%Pf2sj>7+joa#7)=bUK1Gkz_V*`UN z$=Pmbxu|`A2$kUQ$VfR`Tb_Y5(IBWwWmQ({pr?~6f*LXh)Fk5c3*X$r0*Zi`0Ww+6 z^ms3_wUY5=5``>Het~-iYcK~X93CFt3iwjf$f*B${v!@3%TCnTGqSL-lt2DL3&maS zpZW7z@laZQL|jzNdCCta0#(enZ{M~;9|I7}@Te$xs95yRyr8P8tYm_!EF5%jc1F^D z5TZKGZSjE$H_6%LHGgvrMq!{`k~*44hnJOSd&vw%KJXFbTs#o%WaZ@0K=MHY5>guh zPgv-z{52|2!I7M;!<1VS)jB)NprxgytfCSKZ4(9t)XnE7Hb^eX<6@KSd%7X)eYQUi z1TzTELTNlFCkNDHncf3D(w~~i9R6QqE}^iou@w{*7offXNn?lCR#O;w>74TN_z*h+ zC2;9b&W=FD3G^U|fM0`40MyNLeFm2p_F==*-dF#g4a#_7cTgHBnK=T(g)%QAq@{3Q zc?XAb!-n4{Dm;`|g@h=i&yTLY)#e;4Gl>K0{Ndzd4wDu^NCc@M2?xkwE~E{kw8BHlt_?9nP_@(N+@$XxVY#R-GXX$qsK0v{j5YKEDb!>ocAsP zG%5hv4u*EVw*Gz_g}^-EK~OdZoMcLkCBZ-hX0?5A5SpK#-#usYKXrfrs+L>GBg59d ze?MYmL>rn~U@hCg{2|9fKs)qqVMfs6KPE7!&U^bd3+N_vJ_XJG6oVlBIaAtXu$Y6jBG^f7avvu4%XFOOgi;G@7o-)cUCqy{DoB{Q_Mlnzo4iH_M+ZV zrqN znHyc2=tq>AS}rl=>nNxKCAJ4^CKmu##RMt$OA%P*p5eh1q z(ptd)M-WvDniGYBZ3ZPTNWua^GU<)u7 zBvwaXA0800@{b=ofU)qou21;3?IT-n+{DlF=PqIX;i0b9O}n9+&qaxJbuAV)9)v){ zC^Wjk637nAC;j)n&GC;PJ_IHvj&f_{SWv&afs-NSGkNF{Dj9LmgKHTeOQLgcU*U0VA$Q ze&bsqF;tS~sfRzO6r_t$NHEwdYzmtNMMd>99&Sr?c1EB}t1)pD;N&Vro%+@&ctEq= zpTk3JRIp))`BHLBvZ=(S+kAmB2?VbVc@^LKutuBVL-S|)@e(->hEH2kvkUiT>BE?V z3Jd1nyTekhJjl1b5~E8JnOXJE8(IKa0gHe-Xi#r&W`*+#`ox5Dd#`dVE?H`Yqs1*b zdHbNArTw#q5r3AY$b?oAzyZ?X`EhHQD5U>5`1nxtA1@sA`fpu@Y5Vzjb;d}!S4T$& z|41xzPzf4k;7+0|8jP#MqGUp~4GkU8%L)LwptKaBDTRfc;EKb5Up78(zo*~0KW2oq zMQU;=Kv|<@YQ+i^(jRL}2miNr`~R^AKTF6I7uxf;erZY;DeW5+ zDus#u?>E~2m*#z&R#Ir;w`ql1$^Ue_t_y5u&Y!Vc`JD6cS07M!6x9_<r z_TD${8{>}q-uoT@IEQog*=y~!=bH12xr3AxB=N6OUB$q_z?YU1Q^vr!K!W^_jR~KS z=FU{X3#Ngrq!`9I@-MwECk6xK7KXH#i0aGt>yvJ(&-TvFw-|i8?q9s0dGYrBt5O7k zj>&=XNl!Fi78z?ydQdYdd*c(7cN7fO1;?9A+Qb-l93?l`#a%LBBw*yF(=!Y7k@9v1mg&QW3C4y^gY&Y@I{n*z ztk^gtbbcaIiP&^_O_^n7s)kF{ntnvq5&jjG1LS06{UwHyzFvm840|7RR`VEg`2Ii6 zBibZ}txxp-HROLgwBW)6n3KjOwH0Z>T+>K*hh+NyJxBEilRnj7+;&jy?d@}Ob4pL2 zzRu3Rng+|p$!HW0p2Z!Hh%(I85{xy(~ejb z?HZ>bddYBBlP-$v@xif;$uI4loy{G2(dp@CM|%RI$NN0h&YfX9i;_{XoKe&~HpQ-+ zIw<+%4^Ce+1w8+-(mUQK;qnj9?KY!lfzrt>O<;s=!!rrMe@^_gM))?MNZ||^!Zl*tafg#@@8UU zG{sfRt50)PuV24jrss+dc)zfe@#zz`&zcpFg#{B8O+wOW($lPdUgwqTISMIFLqkJ) zg0Tq+JYRI2ow>5Jvt#4ql~q;!Ac|hpIAu64c1awp_TP<}(AFg;CjPselU7_=S#;dn z(b1uII+@G2cmMuQQb?jB+n7HsgJ~GEn$J zz{<+1xU`f*oe&1q9Lz5$xWmE{u{vC&qNL;_lbv_0h*p~>y(}aoIeAmVJr;|jh&0jN3Y}ks+yYqTor~Ir{y>A-chfQm2({RYNcgn-oF3f!QR=Fr~nLU zHeD}>k(%nJu4zH_h<|Ip{z7Bp>q}ev``kygf#qp(aLUN!WEl&K`_Q$#f*|qdPgMxlkur0V&Uj{e zM#jL%NTQ~uCMW)R3poh;KRY}0%*=j{9>D>v6wyGW;Q#YWRo!FzOc^xL;@cE1`gI5;{|3b-)AI?Ydc zY_?2I=@cbDpMJ}hO<0+<(Kr9vUHc9ecITks{Kev`!NSRb)hBmi6#q~@espI7 zOR3c;yNs-?w7mR$N8AI~Nd;lOMxQS>ulmb{ArzgQIKQ~8&m8To`n`S24&%Ie^Cp>A zVeineE;jahbWF^>hYt&nlVf5?e6a{W>NdRMA0BunFOM%RE&Z*vm6(LYH$VS@N{+%^ zCZ+<79J+*uFJ?{-cVLw1Qd7l?7cUlE?udD z?<%KXjWd$dcSk|Of?HtR9u;Tc!IVct7&Q-aUGx!;nglj$pV-*glv>KBFDVT9 zSo=w2{a!6+9wb5oj z9Fo7Qy60nStCt~}7rgiq4^?w##4w`$o+AC+MXO7y_eq~Ved;WEr<-q`k(qfugr#0@ zu_xuxpTLqVQ)7Z&QoE%6OpiltJHaQXk7hb{QLWq6pbo7Iw;I{I5=XE1K>cN zO%+nU`Mz+!_M1O-=mEEwt3@KuqY6W_O zdC!nILJqW zQdH{_Cf=|7qI!d#_k6V{+nr((5-a7!F_&d#@5sF}3{XPKPSyGmyT@&VG;k&vTTY3^`wae3~Knlinp zi5=GQ2!peCzQ{Le{rdGayxm#JFZyiMPAqhOvLjqq@geur^3wb}+j{ez*Zn5VANWKv zm2+QiwUA58$r-_(2&EMm3QW0b_ywYH6jCJ)4o>ZEzbt>9d^)#Mp-$b{LWI(oQ6eXU z6y%E}0oRA8M|+0NZ}F?Dt0_xV`jjUoCPs?&FL@p9>TgU`Ls2!WD}p_C?bYjPd-pCf`42fcJ1yk86xQ};`kJ-1wSQ+qSfmsc8E)Tx6CF(qKj?hjk7^Q|z6B78 zNxc|@hlfY#Ik#FC4@XEFi;N`#vQHtP?H=N z7x!!{!kg7~Lp#-Nfs&n_y=<%qeMKf8pTx`e4n4i^VyZ{D_sQ(duJ?Qc-(OsHWOrPU zLzXGkYhME=*1fEQwU0;Y;ftDYxGc4e=f~dX5yu-}dXXbJ-L~gyv@TJ(WA)xFk9%zH zjLg``*m&z?r?>RwmVxM#Ct(xWtfqh*jNo28g6M!?vMg8B!j4rd$1H#(13~f|whupg z)O~%7i|nEHcYqW3L~fT=*iH-8AFVuq`>{M$p09RXw!&r1rKy~&I!5s9gN}&E>jgNn zk4fXRn2NG;rpir-s{zrjzyFRY_I zpfiJnOf7)TT)B_)9LpI&u*BGGASx|QDydQ%0% z`YOz3-S^kpA^E)A8+_j0+dJ&u(0VaDqgu~GZiC_+YCVmh`>vv*8NYx4hru^&m-Vsi zgJd*P*;an$hTaey`(cgy)D&pghmRjW14w{5MPz5Yoc9*u$HqN)z6~+U>UHDR^SHMG<@zUp7(;O}Z29E^jzozgV7yF7qo zeff9X=BM5`bji*A{n4&dZUF#P*>hjh)ZO)1eRDZoMx|vcy6#*2 zx6D2>UXT5PA!pmknwIVPPK_F;-1d`x@$p|XGfmNVWz{RJC|g=uG&d98zsEDHSbQTWY?_-YJmdrfW_EJg6YB@hSJ}x!yN#MYl$Blgp?=0gMHBNp4l8|6 zHfMHrme1>m6Y^`ftrE7NxR{vT%FlanAIqt9dB|-zYvs&zrq->GYJDcv5s@Z8BEz%i&*uiZXLKR-WK!12wTjA9&@v9LI-Mp(J6Mt;nV)KnA2j{ZtaPiK1k zrTT|53=E(Xt1*;-{v^=gN^P&%ys zaQjcydBnuWw+`k%cbktpk4_wU7aIDo!TO7WogKfL8kc;Jf4m91{zbcKuLFD7{{8*^ z6#Bw%uI%6fYfvqWtoL&WYP?+X%s7E*lf0OK;}s zB|?7z^siiO3d9Yiu+Oj=U$16OZ?FE*FLz%KG6^L_fQX2Qk&%(zLKh_l63m{Sp2?m# zYUbwm(!BO5p`yeaU-6Uv3`wdlQyi(GP)-}Z=1igs3X-|VWA*wi8;68C1f5q@(XF-< zRZ+U$r*Qxax)ZH&`m6wJ3EjAH|I>cYDnpvhCmGf%|}dCHMSqk`A#DsNE2;xyuW@j z&*J?Q4%povZa9mBXh{4?(s;zW%6d3`eul72SR0tS-xMHC3U;GFg`)4Q;wIE6fU{_= zN_Nv8nry9WnWGC`NzFrOt?qp;a=ljlXdv$ZaV}vIz%=QSnHd?e1q1|uF2&i1+rp0_ zVrV%`x|C}0>F`W|m3tmcy2?~$@RZ`kfWQOC&A03I<;f*;2Ko72EtsFr9dPp{z>+wdC)mCo@aAxA^K`6r%V!C0 zJo?_1#6dk#lzj~PYp?nqk_LgrB|-T7S*%VcG(4|0DaEt`vSw~>4xqGRYS`<3-Nzh| zYu^^|z_m?zpL;>1mXwypaOvSMLsj^Dc-ZdYEB;Q;)>aYZgH;`SeS$8O9J0W5Uo+)# z2ne|IvCiQWh15swe||T^as`r+5EC~6tdF&MVrg;>1v)Y4i#Wo?sk&zL`$zUSpR**+^b#-Ch5YEW=Km_S--Fm#t?Pq2WD+KVD zaq_KN*KIjS7faKybk^8?Gbbk}klIvg@X?9#iMy-AivJD=96bwc8UlkfhA^h1L&j#J z>UCvhC9HPeOl@6VwaYqHTwI(?W&a)c9s&md!M)zuK<@UAj;o4_ilyoueRHNuP(_yq z^Alj83<{7za7dnf@9k|H7*K&hltO8eD5M}31U>duAsVj_0B^r~^=hn5mNe=n3d9$u z&8Zj2fpSpU0SwDj_924-t%0qKgjx#-6ppJy%-Q2+(b54y5?L8gkN_^jYKgJ|Nd=l6 z<4ss19Uu+qs-mJoT|)ylHc@a%MFp}}z}}r6gt0HPg3u7(66uEl5}urpoSZxU2#f)> zsqC!&W9`42+?{J`YAX5kX;i_5d`rP{&F)^|^H3O1F--{I{6K^ExkQAroR_>JDDJzt z&jrZe65b4tiHT8j%(#2}C8qpq9<>zc9T4?qoK!$c>>M3uAwd9tx(dqz@&CywM6@hW zVDU{z2(FKG@0FMulkG{@>8PY6RR}2Lf+Ya;GovE&~H-MO`)z5r{Kd zYI=d9%(OS{l^cc-gQ5fMa@R1@OfL4}YCU#+<>Qj)X@+f7G%8=v5IN%R#G*J0{NzwP zEjBkCt>!2%w@nKLv+oI9@HZoHOus#PSO8@P3@DjL5WTyV^&Uw5rXT#F~3L zagrdRQx2n2Gg&HFx47kI^=8R(aW&1-qW&=;P_@>)*L^Cvi}lHJ9~5BCox{tzqp&W+avO8MelFVPIp60!=7! zjztmMFh2zsiqy#1Sn{qmYddSMF+_~Hc`g$BRsH?SYwPQAHd#+mw5ag#a8$#sP$Fg# zwA#C$ZH;?u%dGZxcImaX?}0J?3tkXBRaI5LM@c)XtE(d!L`NqTwq8bJZS5l@N>^-# zj}n~{F`Sluyng?l1`?zw8y_DZ{3)``oooa}H$EX@px(>lp-ydJFKt@ss+Qt&Fq@J@ z>q0~Ez*T_BCvfTQp-n9(8vLbO8Q>e1+X&WMU$dNP$S+pggKRdxmT^=EA(Z z;I%cY(wE|e?Eyh?aik#GN2$>EA-S!u-17Nn0OSA-QD@Y%RwrB zNbaXkw*&eicm0LDj?A1@a__N*u)ZG0T}app5wY3U5mJ9n4*of&wi~#`+FY+k> z5Ec_k)We4l0iBnE4~75&JwymC)^F}IsDqF;PQ(DkFa)3wR4O9OQo}X^lkVhfCu(<& zaaKxhD++KIK>oZBYt@aIh8h}48iFGSl7MR<`5&fB2C2F~b5-JOvNgK+k1@Nq9d_2CnLd29zxT{C6Kc(q2|e?p|yD*3#k+Sp&3)LV%J>Rv;?)`THlW zw|#YGKW4q~@?@)RwA_;X@bC};bVJ)aJEJ9rSUU^dDu5-;NcCqnHhQY;XGf~+vygpW zUQ~Y+4Cku>Gs5WsX5am$%X$EEPyWtE0BA)nU%sQe+ZaUQM~8DUGL;6bXq24xix-GW zNX>1fgdo2%{btM?Zgbb5+{6Ojm1CC+4i4sZT)+bzDmFG2LGSdld-~G}3Z?+{QPx)JT6ve>o_lm^AI0lT)7R3+$MN0Eap|`i?f#Yx2fg3{o{?;EgPq*>@WG0n$}U#czcY~fhDeE%i{WK9d6TXZcge9 zcogmp27T9uvE$qqG&W)35MR7k=(wPLQ;UbZgGxtd)@oE^Ny^tzWcX}X|FdPInwr{0 zC1r#0UqTk;@{=RN``c~qu=k8cKO5Hq@fH&oUp^s<%PqWif3K@3RTGT@QPa3H{`bbl zlo7h7T4~_V=MO7X&Xelaufn;@F|F3))_YQHxp-T<*Y~-tef41h@oqdyEVmryZV&+` z4FVS6|1QWn|MY+YfC{bMNBgJuKJI=ffg1)m??TnVlq16H7k>ZFaGn$oA+z6IdKyl{ z*I$e2|D*YLa{!{^?B!@y}-soC?&?y;uic7+(qMP33~*~tYEP`V_W3kwU8%-$S8$l>LFk%R#i zmHO)Vq<@%)T(WJPJjnZ$OLFmDA0|H4w*rk)M3@05*>=~Ej}9;6*VsCub|15)@7%T_lCoINVAuDJzK~5k_UKEK8pFvw(_K4}3-Ck2r9Rmk56p=?YK1I<^J~?UZ-=jYKiM&+Kon_nF~`4gbHBVE$a4IQudn z4w=igZjTJS$cEl4vXXDs-e7;mZR+bUv3UptV;Vb!$5%oY(%g6kX6|z(rF0N9enM%t zfYWY_k&%*SgJ%3=ZgFw-=QN;uE|rgRePE4!8a-W>K`1%chROF-KAPD~)#j>%0Ws*7 zWvTlR6dYWWuVd&al6HodQSXkTM#&!4~q zvk#EMq2s~Su}})OHsyX(0|^ju&48osKyIb3e$tn0JyMJSwJEQI7ie1nGi#BH z=sR*4ALZYD+~K+o{CucD;|UN1@Vz89l*%OFb#1&d0)#}#eA9_^XH1;3Ox6ph4?A4e zMxr8w&k|wjOFa&3=v^5#V_tl5DFH7~=nf_qW-{3wAYBhY<)v^3yZzFLXR zB)|LKO7@_THJwyCX#g@82s|rT3`Lt;`;c!=Bz?cGdcRW&tYO3HR9OJIBY56RxlQ@sz)0VTW9 zKM&R+0hP%|xX&3tytm>pR+Z}xT59rQnzzmlX!jOSFGB$m{RG)c2g%{$qV!Arr7hL-m5moMR;y*=ZzlSA-4D63`luR@1I zqOi9R;8D0jGD%5E@O>O(6?lEL_fzH3 z(NY$Zb`=r0RES+9auHQ#Whk3>8u03Pz0z$qjA!ZRaxcqiUhmpyDIn?DyG(I2e)hW_zJU%bZtJmcfX3ZG z_yg+20e(UD-2D93gk$ekCrd-I`2Z^A&4=`KpYPwlbJO1W@%G=Ysho0`&3O#k*FrO) z@Eu;>M3BOX$;nN>J@hZ3ee^OCQN}@<75e+@5?XlOcUsXJyv6>HzL&?A%zIKS&M~Ik zSDB|gcjVFMkxb0YZeT`Ia+(prZd!yA2Ydee2c6(MR5@z-8g=emz}OJY7|M)#&go{Y z1Uo34^v>o&W4lq(%w7>n7;th?Rn_aj3H$;A93edDo$nz{V?lrZ*g-=MWF`x>$g(~9 zXie+0?;nP;B7}k^vT_AP`{pMf(egX#!M+Eii^P#c#EovqFX?-R%bD|VLL|BgX!*Zc z7S*)^c5^DE6HylCW3PVl@+B|ynAGkKX}Q<2y*l~3Wi?qts->lc2z5Y3Lw{ud7>3tG zaDn&t_dS8x65qi@>2G6|zy1u~a*CqcuYGx&5HePfU>;K|vTX(FIs|syTrIy1F0Xiy}nK90`MKFLm zEGsLUbzTcvAu!`~wfxok1<*Gj)8OFY{j5$cgjq?+$lT`Uj!#Ua04{;(yf^^O)dF>E zUq-=QvV$fQNXy*Z++b-EQBVYeE;uD((Bs|vHgDKu$cL*8LN*$NtbhPXUtEQQr9R(l z^b(xMpJouF5E24-OvpJTLMgs}kD`aY9ywI>;V1M5Ic4EWR##QA+RupjdQXGjrjRDo z-IpZ^##|0ybYQf}U~~3^&}%XXSPUg3lnSVP=*uKTEfia*JcBMnR$Q!n?0uy=(8t-= z*Jt*FC?|aOar*^_JclK{(bsy%dLMb^zxMSfB9Z_f@IoO-&9HXVDT;^O=@(L+bK?Ex`?$OF=+ z@Mw)0c-x|TnA}FjrH@Qnm_B6S=IJg zf`r1J??L@oCPhIvF$)l5sMeWzYS&b>&-2BKRA2mlxoj^n8gZM0LuIwHVYvzVN8o0& zyR4~W0ROA9`+X7V{Dr10KoWiv6WWgi1)&yv?d%M(nXKun^Kh+kUX=iXys7EJ&c#KF z|6@HtcjIcbfZN@>zf)5c3X52vu?H-}pVs+>g$rZlmM38Lz!iZ$`n{tAAb~IFRvEN!-e6E~^ertdi*Iyvc2c7k7Z(FS_X4eNqSh@2vX6y$b(dH|*=(h#eo6ArEq z6s*R+z9@wh{@-hBT_CGeou3`AhDSohmxfJz?_Rr}u&v?0hS;vP{$Gj7RJfl@C$aIm zK7RTn)g(p;Q~$9Sn+@1lj@23p%tQ)$s5A0EKa* z?m!205<5FP7`bG_z!5YV1L4h^P;j4WXqXr4h@IY%+eMDVcr@;25M=Bt$uPw|g#k_8 zM4)}gG^9hK`q#i(TSranaay?!{X6u$yj3kn1C>+IDJ3N()i{34A3$(8DA3xk?3=T= z-trZH)K#nBWz)|BlboEK-(@YZG>#)fA!9FmS%1=3-=(>@xVW*iGv8m}CwN4{WQ|2$ zN0Fr0RW}p6(r-CD}QlVNTD#b{HlV zY3;iFXZ{;QlO;{T3Ti2dzj$9mOQyML9|33|rH8Xl4h~H_l|ke?T^#RUF3|-;Lvm|> zbmqesylmD~$7ybuNcU%H^<)FwdT@mL3SV&HlAksD;k5G#BeZbGQQ?WlA3!hLj*zaQeauK)t2s;sU-~IN>aC3hnJ(`>*`r z_mR=jqR#R@%+O|EGr?{@arW|MPPZ|F=Ue8?Vygi4OV1xq~Q9`62cF0z63ZfBfL@ ad1&-HeG^xyQxa}MQnLXMG19`{P(&xHy(vLhf-6L*YQaGJ?gEYZFfYx#vA8ICv<|0 z?@8t?kAvV5S>D1aS5pDA_r|}%W3q1k`dXAD*g5e3IZ1P`%t&VN^hX+np4g5r-HShmhbJZ`8h)@2NT;Z5U>w z{^L;ptCPcoi7-j{IoB_p(%vcUg;hjPo=hO+2&QMm5gsI}=PKU2cj5i}_h0aMVwAJ9 zvwubgyMltkT^pNLiA+zvix+QbYSLb2rrh+BQR1Y{(Ynyo)I`I~+$EWANHc6Sp^N?a z@#Dn)YJJ$vH(oP+2{*Z4-sCPcua6-jXS};SR1!s(66REY$awqq?brjXSrJ^O@bIA! zKR!OzOVvnM+@mci@*^7B)4X;G|7hQK&pB|H!8Abmbt%&PS4=^6BK%wFVc{zGz zsr>lmQm;KG7SO{@ma56jPZ=2*xW~e;UkdvAr$1F!_y712(c7zTU})GlI?Af6TX-i;Q}+rR0Rd%4cX#vR zK(2*_1py?qfss+u@GvtBLlfEh{N|hH!*#air11=qxzXw;)czKZnYFcP@%u9;oVt|G z(^2|S5X-ftzSH6VSXP!tuA_iv4NItvR6{>s?_H{S$8P@J5E?>~58VP!?==;(+H z-_}NqnwXqiU&_ffZ|6JO+P!aQ$9sy5Xl1-U7N+R>NLpGNJODBY)$`9gt>I_! zhYvv*o}k)V85UO7$%zRR79;+wp4NgcLWybFNK3YmXzXPKd+MDor_XeM<4@a(BSa2L1tjJyoitP*fIp3QWDR;cvN7(zJ zw6v+#`|p>Lk*LH(#>?X3Q!As@_V)JIU0g(d{rdI#yTjz-A}KpN`_A5;<<}fFG#c$6 z7S_&n+_<(iZgqLicjNO_;vyn%Z*TRLcqB8-&CU4|ZVHwgN8PlUwJKt`csD@8(B!ye zE~})3wzahtHeKu1(x9f98I_8Piifv17at$`gI%Apg2IWKni}PUQqIelPr+lZ{dH=1 zd4cQ{!>+V@5B#ySfAu;nomXu{LBT~>8WD@8V1ii9{O@`1-XSUcaJ(*BJ6Aokw3M!( zpa4xlp`@WfqoAlbxwO>1I5jVpi3D2B-kz_kyL&SCPqUQJ>bYz#!V1^=I#$~&q%Q6* zYbBcj7@p>t&iFIAtTJ0$TOECU?R|2`QGtOc7F<6ak+&Z0w99c`xproneg6(b$HTA%SO!;j2g>q5|n4`h9E<^kh}g*2#%>1o0UCDEz9qaE8`19wHNwqsA7K`QM=MO9^yE60WTDtejwcxO@=Fw5#mcM-}nVFe<0s=wo z!DWmdTk`eZe^(}<8bRFJ>5pU5(pVV<4N!2zv#V@n_Tywkkb0cpg1^gggk}DKe!SHzq?)n_aiIo_T%x z5$WW*rK6)1uIw((r$#tFY^(OL+Om~QeU6f7aKp&h*kuWSF;&uy?8lEENCIFm6cC8X z<>emfj4$&S5^h@C*>MR7kk8J}hR4LTMGDuUt4xks9x)F0Nxdw)+ffte^;H_`r-<&w zix(w4|D2Cum!raBLT^MeT1>Y^JG;18uzt>wkdm_cQEFpQe*a9$t5+`D%ZAItW#r7! z#J3U!8zHKbGczrcZ`HCz&u|YGnl{IAsXR|g3QJCAsyRN|PZqHvUNW$4Y;7eH&@XK1 zN)XV5K+yB5-`)K)8y+6s0B30F=$a(6)Ure;pPLn_0HA1TYxjTo;=i-wah{u7o+p0G zeDYS5fP@57V`Jmc$9wdmqN2}VzC3yL>QzDWI+l`>65_NB&d|v%w&HQ$zpSXc{aH7n z;wglLg{j}Yd$;iW_YEH(X$XvnZ4VQ^q{C^X{wX^|O!dy4FayE1lq-%jw6u+q_}9V3 zjDq*?-`^9?^vs8|0+}zwI2kywV4wpI7 zGBSo{tHdHGsB``VH)Ev8(<2`i8Lv}-3I^5g7iAAOwb!=Z$Yu1CF$EqR_|zQy>CP)C zXw^6O4f&q0zhRpsAS$Y-r)N;>b)|0am%ukXerh(!T-nc0cF7=6LHIGn>wf#S^>%hH z2%VamT6k2HHETrF3zzA(NkDQbNbwTieRm}#rMumU7gK$xI=}Uc#mm1;OthTsN?2H2 zl=t$IWQf4R_mto-FE6j+N3%DFb$d~pJ=Q3%TDCB1c4Z?Y<`*wsTzax%f{zQ2Q+uQ3 zv(`vb={$L2%xmc)&Zvxxl$6xco<-NlDQwL{$Q??l`7`vNY!In z7M7@-ob#s`872><{?c3eYDOa4`+-}^wNyb1*;D6K}^y<}FJkBsE zC50Iu=MmO5!=Ybj(zv&^6!-kO|J5-MLPo~Tmgs}nkPxDy&0IMS4i18)!J_sA0eu6N zDF*&{^E4T6MhM1G)#LfsXOC$%%01RCmq)7T;hco`w&0RPjr{S}2?_UQrn;Reaux|^ z)X~uqZkmPws!@9YAxZJQ0iKT)6Qo`n-__Rp#W`M+}yM8r1I!RpOez_0AV4@RZmlZ?=DGTun({-QtlM+?9KH@ zTf$N7a_Md?oOj_%?SNUWG*m1S*Ka6DOaep_l2A>2$yBuU*1|7$ z&A{FpDsw!&y)t@B_yIbX2b1g%*k}p6*mk$P#4te(h~Dwfde_o~DB`D2kB=&%KmTqL z|8Rjm;but1m>2oks}GetJS3JRT7pBu!mOZVcq|oEyw-YW&@RIKVTe$#Xz-u)1Bf2; zn5GoUq7}`%BfEPtIy!m?z&c!xMb*0@#+gmuyMayChsmLgd9`l8uraA)!;@M&g*VC* z4u$CG>Eq(!0w8~5?bB(t{jcmMU3>H9jq0siijN+N`r<>G5@6|G#K*@aCfYLYXC}L1 z?JqOm|5;863$(dt-V_YChm0$J@KYQp^dv4ZaRXDK&SS?AU zieN=|7jrXS4=maXt8=2EW_|&B4qF}WAEov(0t&B#13$!?gYi{MtP&>GKv3Dzl2c1d z3p4j4LJe0lGGem1z0D~pnF8~pU}R;D9vd?Q@&1Ar^xGIg^2!B70t>XHpaM}4mW zthzs5#|n71SYI3PQ(fZwh|x^acTd$x-~_089#w7HN<6+Y0AzP3Hd1VH~r>U_@* zxj0}@>CCaj6zps}*%Tt-yLWksshOt(kwBFm>zs3a8GVvu9I05jr?Nt3 zHRv?;cFC?{bSVZ3DlHQe`t<43cun+VRAeOi2|_|5?;E&t-7^Qf8}KX^tU)xrtt~AR zaEP;_h@9xksjfZ?eRF}LlutT$GyG1 zYwPRsKyCl^ah2;L9Z;Q4Nn(y{+qoX>4rGKuJCkT*f||NIRRXEVJxxu`zym*4`2;mz zUl|>Kh7%`F7^rYqf0N$m5d*~lRNRi$M)jmF4{#?aZ@Ip=!qw;_l#<0Cq)eozq2FWt zj&|pdy5~tnY=E1#B1N>Fi4KO7mnQ(a2N8<;bV2*A1~w`&G?aM3Rno)Ds}vSC@0ElL zCy*#+W@ZBwk>tBwIXO8e0o7dVf|H5jRD+9)FTH&EvN*=_QsYZC(6~Hz(tF@3l*K&% zSOff&##kU97#f}dR%O_JuEjnuFwpsM&kZj^!4odJ`1^M&fa?14hd#Jjc>hUa;;FXi zv%m~&SX<8p^2M9`?GjN@Q59EI*s+GtWS3M_gari=`t8obx=?V2%iN88_H5$!Z<&R4 zpW%UcP~Spxp5ETsm6gfc{;Z2Nb-uz&OG~t;PHldh-Yw7Ng)}kgAtxhiN|Smrv9N&l zhpS57zvtiC-7Sfc5AWaI+f&HYD2}n9k$)MXkm+AhaTOxNlm<%&;@zPA5_+{;>0kQC z%*+gzO=IZ_v>T8-#0-hQMvX9inGq$(K-D)|kT@ce5lSbFjg9lZJK!~mTlYopIXVV~ zg^|FRm!>Bt8+WBdvIG9wd>tNs20&6ABaaqaSy|EHF+l_%6vuzAo0|*W-gd`}5Z|y- zK=)OB`UJ!z$Sq2qGz$ZvYKLI=AjPRGnB1W(r7Jl33M8%9)i5|>R##WYSxQk+qk5hW%1KskZkXH4+$lFK zXnB4*1=P^+utNU#(wKFf8lm{YLZP5?m2)lMehdvU)vVG800pj-xs4XF1Kq2`!zFK{ z&_M`v1}YiON}8dlJ0~}ngprXEnM~@NH{7774ck5V%~lEW_3{z&7qqI6_6EwEay8Y{ zvV@5_IusV>mvppp(ATe8aBIt{mWWE1X<|@O9C4~1aj~${R-c!Q$+h2N0)WKHJ2oco zvb9wY(tw~`8h=~3O6V6762iwX7#v$9Y-wd>*v8dj&t^vMOPUMGMEoQI$Pm=WPZ-K?43Q~)oPZdIMy+iq%Zj%wIhpQZ)(N#Y_$w%wnZP9*e3Mn)$={Au%`z*72plz+6cx3_)#bq}@< z2MP$1K`4O;Oor<0Pc*f zhV|@=E-o)yLjj))F0PU+`s>dS*$QUKM-su$V*8#(1&o#@Nc~umey2lLVn6}vQ~qzw zsjl>NFeWM8(!rWB5DeAfXWjN?nx3EC>Smc}{EI6uFE2%QFlc_ovFj;BV@VM@{2^NJ zko(v|0&&E5Ldku3t}Cai3hPFpkgUm{u-Ow#4iX*~!VNc@ApkB0i6bp7t)Z>m4BK8=-_(@y&Ye4E zzHeHudzkYeW?@dmRN?#gWU{idI4dS48XM7j zU(f64=|Kd-`*rp7n#RX7+$e(3Sr`DSJWRMDXvm?K2>O0J{-0_FwJ`;%8n|8-u(fg7 z$Pk>2kq8iE)zqXSvwd4r!_p_aFW6!!FQ{$tk`#54h-ebj0TdC-%Y#McD&VRtF%i>a zu)(3BSvfhu$eiB7Z~bv@X(p&GKa=tc3MgTQ;A$v1IxhL2UPIrza|aCJbY-aFd3orf zx;_5k;bAsI@@5;ncfao;|zdyXT4*302at`7`pkk3&ulLZF~*3Wirgr#%@p zwF<~Ng!J(RM={r$#pdf5aY{=A5#^xO!Gdq5$P=HLl@$ojuKXE55%OhXWu;fL>p7q| z>Ks2mS(1<`q{(y5`1p9PN;__URJS#71s$H29Mey(q(Ew=mzWT;5!XEnPS9qnNy~Y# z8o;{<4haEPXB{X&f9uvQFfur#q+Y>oB$Cq7qVw|jz*H>#*T*Oi@Hn0)B(wrZq;_|8 zHIWO~G)T4~j0mVRSY_(wXK!L|K7K3)-WBbcGoAlpOE?_vl3gzg_`rbTXgg6cF%dZY zXM2hY1`5)n5E2IV^Jiq(Ye2#ng-kkNSkiI~aF35h7k|4bU=)BP0I>m~-Ug$>N06XX zQ&VfdE5;Eb5}qdEa^hB!5JOL)!yV3%85E_-prv)i zN?g0f!O3~z%$YL))1vp`H<%j!@$vLad|aFhq6GdxfjDJkSpTk$1GRY0QT_B-_Uphv z3Fu;k;I7%w(b1g%BjXciB3esNk8xyVga)xgy243qsfTh6>wK;u%AzMtDrrF2V{U^j z^YWdY4Om`8YvPP6RwADMos`R#xvL`4_r+J(%F4=R;R^?%i+fYADR(1Tr8;L?<3XF4 zm>3=%?Yk`3INl4FM}sYF0a962O%1Vrmn4EI-d9wd1}9w=gPGaii7a$qN{}S1WNqDu(fF4O%y%i zopEmy5DQpNxpSyTRj~cku{u~(!`$arE<0-$;Qu2W-zp^~#TgFokJVt^ zPW569Z?Z?lk^TE2&2=+7PN)hqWlU03`c=SCO)SDD~O-d8^~2!yy>ZTJ+44 zgtC=51Q2Qv=vsZGv?^jW!0H5Pi*V0rrUb(TQWPdbmK8cb;^IzYQBqGMRoU2)TkTDgI=oTpcj!g+Lgu=61KF4Ny7|Qp z-}!Oo_37$eFF4-ea4k^XOOR6rC4p`%b3Jvi^yUdAHrMsm=N@8bchpZJ?9jU%H#3C}hchXyq4)^0yQ|~9P4pRiDW2LS4HWv{? zr|#g-EuTM~yqkt>f5Ksn52o3V8^Fhk2qK}@iaQ`q=ydY%^yIj5g#k$v3?}E!S$fpM zQ|Zf>FQ2<|<@>Ubs@TbdPI_Q3(0U_$Jl7)+De*hZa!$yk(my!(WDi!1r25G!Fa#Uc zh@^$oP85_fy&^M8Ff$(fET066qXj~=7+|do*^cBh`1+6x?SkdBOr&16qT!H=EY}{L z1j-y98=HIk{bwcEI3T}4cm;qk*N9{CPscJ{=Zu>We!5HkXJez}@w_^i9Egw60&LAd zg^R@z*qsWyZ*Xa;C}QOC)Y=IQE?_%pz}25#Dyk37&E+-ooMm8<^+~RK_Du;SANST1IAlYtURfj)FF!lg7~4cp8q{GZn)$OiZE}1oX~{iP6{A*5b0`ED#UZ)|N*P zXAm$oWyvDxiY}@@`qb&i+G?Mbbgg-2aWTTQ+CB62YZRCuieO}z`EKdIU!)vI+u%Gr%CxNjj`81Skj#`Y_WfCZ4k7n(-Cxs@~p zX(HkF>l786ba+F<^_G^FbCQzG;2z+t$kMgmDY?6gmzI`ln3=^aEZ8Ia{dxE7CPj;? zhdz&F8WosvdGFr?^$|c#ymG1w_`iD2lWS0>klHcOoBAB|1#v3!Ev;jl~h3_VSK1bXZg1A{y;$)Z8U^K?KkIRyoyPU{E#+VUEy@uUX> zq!P>w=m|m?*TUou2*4u@^i7q<*gMMSe0SIiT_KPH&NaN&Of7dkyI*?UJwhoEW$#ZhqN>c!iV5XckkW1w~5KV5e~N>T-pj{kr?(?{RPPm?mS53 zifSjWY#G9ePfi3V9z%77R9}_@!NYS2H%lyApS2rdi@bxW+|0mpflao)(I=M;sVtdR zc>Lo!tTT8*7It=59d?27XquWvi;RvYyU$Ua5Hb%<(~6+7v(xEG4#=-RmRT`2YJ%t6 znD1j>?~{5S6GQbfDQOyt@nPQ!V3iz6-RlTrgjs|BZ%*%ZZ5}j#N?D)>O{Po3 zyTmE&uad5T+SJJZ4t?_wnJ$@y#$$1wPtZyyvmqLLQ@ znLZYqvFmN9Dx|`my_)su6EifH=pw9@{=U=8*VNWV)B$3(fsV|>;glfBAR6Gb1n+Ll zPeV5*sJ{LK~XUjHH!y4kSk7p0ok9w ze`f&q^H5j-NECE@HFR`>_x8LFAW?wLA|4C0TS2h^DW(Em2F^NLVPT>3qep?@&VjK( zZfyLMU^V~a$5W^YJ(l-fhQcyWHJ5!PWr^h@WQ+bp^EcFDZaf@j9c%KF9rGFWx+6RIZiFwoMv z```g3B=b>`fS}E=;?jHS#o4yO6#ZXz2ym8}K%Qq?Agb@(%N_%n7k{v3`6<$L~J86~_sWMuE; zG(RI-1ulfh3@ZigP4J(O#B` z{!xpSPk|4Cw8z!SH%Vq*Gvcxt8uVQ?(5HA^Tuk%i@F3l5*_53fg|t`wg&zm6ZTxW? zyE^V;wPrLiP<^fDM_|L?-$r4(l2JC9G23@U!o=_+dY+!100oMY%a1oq+7Md!=jkFQ z@F$!K#R%9^J@$_FSHbvQZ>hJkvkQY)a~k^l?l7L_xzqfN~HJ z3I%#{kFFhld#4S(j2_9X&d1KqG$a4=T*f$(p@aQaJ*`XfnISn9)!VOkq3|O13Yw!y z84SdKzm=rJ6HKG7fu&K90I%-9om_LEHc$w6UjOys8n_+((v!W}N!R{=7yri*{--Cs v8|Yvf-VI>?sccAKRfHe^x7WNPj!ACpX5m80HJafqHk8^8b)`auyMg}&)pgjg literal 9904 zcmc(Fc{tVW_wJT?$Sg7swFzZRW*M?mk`UX_HV-8v^E?w}OA<0ANirvy=ggwW5Hb%5 z88Z)Oefpl?`Qvxa`R`oUx!&Gu+k1cZGpx0qweI^~Po%aM<|G9x1qy{asc}_J2ZbVF zK>jBqflrunrpn=k#Ns+e4RwtC_o^m44u#@IX{agdd8RD?^fLMGc0{^57--TYD)2C@ zg(`wvkS_s^#`4BioPRy{enQ`|T#?GSc3}h zZ$FY+&6$$3H<^w9sWKJ=-lhmn2idy%zECt*Zf{%ffgw2P>^?7_thJ;7UshLJ5T z>Zm|H!LaOXentx7y!`yr{E{?YUS0wM0)gt83kMR#SgJ6Upuj*>czajG*MSp-9d8N> zs3WL3R9sv(&IjO(jUNvW8$EyVVyd{6GbApKAv1JtvTCJP)xlxeAz-YxH+XZ?^?5=9 zpOn-KW$}$~U%v((?r&4xJHPr?bJJUanq3+dtxg!P9CmnP|5lz532J9#$Fj1?)$^jy zchrgcCkVApV-2B&&(2NqybD<1pVlio^c<{H6&p0sTjm z%$v+#M@OF=9QY16-_UV!k+^_6U_uq$?bBNtVUEf*;6R~DOR2BkIN3b+MVA1@#Kcrq zO~gBTOECWHz0zP~%Ib3rrA8-FR~@chVHAeqnVk1RCNlCzROW&RA z`L!P$D{%?@s$LaxsMX&=_pgOkp7Puf&z?rbT^s*PgK^L%EO)+dz5VBB+34H81v5*_ zCPPl$)NF`MN!D+}!xZ`YBfEv_nL$|L7QgMEq@}~N1rp!y8SL!o^YHNz?r$xJl$Kt+ z@8BSJTv}2Rf-9WgtR|^FI3TgW`S~dX{>cx+yETzkpNmVQja19yTh-CkW%&L3_t&AJ zN8#Zll9w)J=H%p*2)|5DHotq9JoYM)oLp6Rn3KId<@M{=%PT7W{#so4>#b*CpjtSn zn%nPwO3&V2=!rUCfoUCeq}mAx6k}s!X=&-pys`D)zR~IAFluROJ$UfoS9W^1*F>Mu z#Eqn@SvueQ`x`okDbH3ulN-Z0o+U+5amX@C*mhUXO7WzL4Y4gRFKg%aL-<|3apR<( zAd{ew(5+jy2-w)z+GZbmdU~48w8cJfaS6%I6=LV$5c6C$gQdCcj+DR=JA3wQTW4p8 ziT`ZD>6vv_RDBbT(i0{jeOH;Dfq_%~{r&F>3o)vyg!k^<)4p>j`uFd9DRLgv_exo1 zQ>n$1sM6BXR@c{qO)f-~m0g;7C?hN^?3mQrddzhw=G2p>@DoRA;!oq(oGc#OIr}a* zSM`Z1RfpSbS&cq>sPZG{M+B%gD-VAKr>Fv4O;mNY{LI|k>!KpstgI{+aVs(yPyGo? z5Gj{0t;>9Om?k6;~>f@h7Lqqj_ zeGIU$J7d|!#dHvR+Btgf-o2}D2qC?tuODtdSYQqYVYzsZ@dP2^7xVfE8T5Djm*;Y2oFYCip&ZZDMKk zNl}pw96^HewLs>_&3;EO!Sw_L1Z9Ux4)^b?-MU2=6&3YhbwU{o(l|WKs;sO`MozAL z^=jLl>(|;K>qI?%JbBVQpBQ)N*I>VfJqJ5GJ1qx?Un_&#&cVmOjvK?n!%ySlLUgmW zCN~%QzS>D5Kew|JP*G8tn4P^iv8$t_^I-Pcpop7^iOC1~ot+(Sd3kmQv}Ryop;H<4`@h@U z#@gC}2L}hnoL%&hM95S!56^7)LV1~;oo#Dtt6%8Lc0M}r3JnY6k(FhIpZ;2R7T4Y_ zPqekQ9n38i)!y|qMPFL<+~02JPoJ^E-P~-grewarefBKz-Me>L#4Lz70}h-u)8tQw zkkZUuQRL~#Ei7zpkK=cqs3&+HA8+3KM&nU*^c8JwT9Ng?Zn&ERDjsux`%~xE<`N@w zD&3b^;mpUrd`TfBB-E8CMtJ>&a8Om1+@c-YdUc}VQDmgDrY6<<_wOG}eI~Eo|5Ih> z>k2Sx3ov3o-9HUQAuz$z|Fol{W1@sD^;@mf4%sw~$-kUMN(ZZisEJ7aG_#bM zF>wirwZDU=O=hBITT4F@#jPoX&!0DJ<*jo$+;dBjc0N(%u@W8~ee#qrHgIjawFK{W zQw2*IB3qDM&U}hAPquqtse91c?edKrJ&UIBlaS4^)t+LFO-(C1j!(W)?sYg=ju_|W-sg(WTGFtuguK|r7HQ7le&t^ z2{;N{e||px^~+9>K8QY2?GYZ|GFIc0p)K(A>C?b3$(JDUmf+FSdT52G@$q3YE_1J5 zzdj-7@khyZ7&ik_L$c^$D6FI$>p+fbu5-8E#SyYt3XxUZL@2J^P!82n8zOzmT2{* zq2iFJ&zhPqCCTO7moEGG$jyC8ju;&^8Hn|aNlS8Vu#>a^uMXAv$>IEWt!4Qes3My! z@^Ce*t$2>6C?+9VX6A_OH18;Uj9XC03^gko+uGh@VP;kq@yI6!B%d!{4b8~W&F1Ci zM&;_~tKe{)M{6+wn_ep{x@saW{_|7Kk3Hv-Y$yKw!QV)hx)n={3QSIBlX4gig;=HO zk>8-YfU9nkrC-rb8l74CQGzXhz%w;H9h;aqlzAxjXn{@Ajuwelu#m7xIjwC^nl(qH zfnDjr+(bjjP?0$yVu&YPN@)dIJa?j^qnnG&n-HH(g)!smEn^f~Wa&k;lTbsY_RO`r zb8KhMo^`7zIn8s4P03&W*|TRwt_wPnQc}IyY{9rWGbqPHpB&f}_m?CD>3jKbINZ?3 zyA03b;)*?1a0AYX5{u$Lz*LaBio7;#ps*F;Rf*b~nh1?Q-ETNL+%NXqcZbKsN5prx zMx2UmdpKdfZ*o#^ZkpV=Q|$Ot7B)y)<-Tme5plXiI4%Ng5z`_eJF3TAajd-h;Oi^ILYKaUtC6<5r!ZG?)g%qEz&l=LV* z_S@f@>NuMJ(r@_VrB3|Uj*-K%s-<=BPM(}dkqOr*647z6o2bp>ksBlB%z_9W*yJZ=JoH~6v z5_aOdGZ~SWxhWKPKCY`v6L9BEY^|@%qClwG?)t2>-<~8SAw$fO?EVp&)(s z827`7IFX5*H?3v&+5U5}y~OfHfu_{`p+Jb*^78T`yh@t`lsPy|w$Df{Z@kxh`0(Ka zC@Pg*woO$`}gl7zP>+NPN$w3 z>&D)lsG0O8J6kook4I22%8gOw&pAOs3Zwi}**DW|IKm{*y6E=yb|f(h3v139^P;;N z8&5oD7Lm&FaM!e$O_gyWad2>e3}?#`_SBJ*h9&~aq_?;1qCkphr1Q~u3UNd{=JHt8W9+D6KiF428xfiiAk62wRG}YO*kxT z*-D3z_;_074X3qJd2vljO+Fi7Tg@vr?hjIB8M|*SbbvN5Iv+cky?d?G!A%rTbs!6jz z7-Mq1rdy&zYHAb!vj&HR+@%=``sg&SsF|*q>PDp^E#bGf*?s3%*wv@juLl>Me}6xx znJg6z=9ZRl7?wn8sdr79+jAV972=Jio}L&*{=`H^00uj|yRgxa|9r=;;LR8t8=GfI z+gQ=ob`{<}cYA`PGf85hA_4P|&?p^3n1M7$DGa8leViF|<;|Lxs^q&3Y zD(mUvQv%^tw+4_%E4wM{dIH#)PB~r!3Jl~N%>E17I5d4EEep#d0Fxi^y?nvK3@Vy; z0E&G0@B!QU3!ZILC}xzk}_%V{48qW1?cxJe{0SlM8k8hr1uM`9UHu=~^uR6|)t% zJY9|V6A>}3rBYI=s~Ssp^6>`J-P$Y<2->99x57ngnP3DX7*C<7s0eNopQx^>ah_}> z9uTX%LTmk6KX<|EqQwi?J+PbxS@p8jT20_oP{_!VG+Q8a3AHXHWPbQSj~v$LK2v^4 zA3uJq@9w6yC{BdvAxXgsho`1;0FK3CD)`3|G=S=b({OdmlJImk;j1pj*gCETe(9|8yFZ!Uc6Y0cfJWa#A*tFxp;EsmoHyV zOieW+!%54@WmZ%qxjjKv0d!F}iMC{JYD&erkr)^}KR-V>!n~!uJ#vh7PUZS_%5bXF z|9*!+$Bs9J;A7a>($W$Z!xMTM*kb0lw*4NiIkzOcQX@fn71Yp=AIiD?AMnlw)fX62 zwFdK->Bz*~K7IP6hxW_pTaewBv2Pw+R z)|L-A)58jO@Z1vydT~)l93=;3LXUlrL8~SJQbJG#GQW|PZOqd>SYkPW=Li`Iql8}r zgS)DsVa^fv)Zb%g)x05}cuZcRy{89r=MIPPNfP<>X6DuI_KMjpN*dSOeVj(Na6Y4ecyV+2Yn$ z6x6lLTa*nAJJ!RJsLN(%6M;5-R@~%n=U38bt3kGnI1C_IH7M5x!VU&|I^u3o<$qLr;`Wo1=ZfZFFCgxj1of3FPD3HD|22A)3clBD(g)C-=9GLaB&& ze3vK-w*yOyiY|j*vP^J&qst!$f2r}(L1q~7+S=MGDk=($kMCC)r)Kz{5$Ku*I!{;_ z@zV0L&*4Rqsr}i%o$M1W`%8a4Dj1V#bv>4cFOH=dKa?0th=je}v$r>dBb=0LP`Lj4 z_e25NYfTU#I*`SF9~e;KjV*#x4I3aOCH?;Wdl6olOyc6ji=fSzaZIPdQJSoH`I6P! zrja;DFRC*^qy%{YGB8m8Jj>BI?v6L_-$x;K$a(*MYSZE8QRBY&X?agk|HHoq$f2Ku z!F@X%OgvF9sBvE8)qUwRDH8(dS^O+$6#mY(#%t!uK^Tdf0Pt((pKp;dbDRDu( z!hCGt;pMjeuU}C~n&F`Rd>S|8yXJW9A_laZ0p*4=Hx5%SbO zAvyW6^K?rw{u^JgVBExn+04ugxb$kBu3-E>44#Q0x{U@jq6)PcQZ9n6*fPI80;(1o z9{#U13o_8q&|E!353=nCq^iq!{9!*sPcAPn|DUQ(j?zq#rnBu%F-InhOG~>@ZhJNm z)oBF+1Dr5RbyilGlqN4aqr99YKR+MxfjLLkAdxZp)TvXTp)>>WM({`v z!pMh)unkM^-T51DP;;ZvB&{)A7^rWEU5(&XD@8g;379HO8~Lj9b9*~1QL}z~Jm4Sl zeDUmAFziBFMy43w%O8A!0re;*rnPL`|MAwA`@k)-h2q7*Ld(H|JMghBM^+EO;!O2u zL`u<0m2Cz9px_P`DX*%cyf^);fXs$wsLE@-oxj++^Ard{6`hCX`+P{0`6bx`(%+p= z=i}j-#0PB8DP#KqS?nFiOTV|?nV zX=AU30!63L(9m#Q3MEqkl8+EuyKlT&={HKO=&etTmg8DmT1YT4oYrt^d}O^h1we78 zv)XIyw$WWs$>( z6k>p0U}_VKi$di;=U1BFKqh~2-Z*?_#>%APVHgCAqswdu)BTZ8Cj<6J#<7^ml=k>Y z=vLTtr$qi;1U&@!&_JPSjA4mQLtkH<8)J8pNfN3a;fl|m(GPpRT&HCTp#+KNe1i9M~xZ0O5Uyye=ui5x+7j?~OjThu7kODYq6gFvVMtQ1> z{13fBL$o~DSpz?_#hquEXlWP%Z4q>)U^hmK7Ce^{9^S^u|MHfIyTDM4eF9XD8T8pJ zpkpsef2QjH{@rYOxYTp|rw7le$A+y72IxCb-MlyLy3&=ng!J@4-_-_Egr2T9kaC&*`kH|)G9y?NAZ!%wP&-3E`Q_5M z8XS86ok>zq08S5;IdK8{UY$#Jdinj_hKJ=T>k}wor7LyEii$@&mj4omswURmT?!Ej zAY{4dBSYO%2rQ+*)6#NVpa``H6;=mt%=eVwRW-G}ykzwAjlv*$N6z#-tYIq=1T*aN z!{Y0Ev#vV>RX~J?3fBdCL`2Fnspx&v!@m&q1CdrkdeXgrp95u}cb^4a3T9PuD>g9-TPy%)k)X@FQ*{btSihHp9Fjef^Zf$@7PB4kJ55|i zfgqA~&-HQ^DUdbxa~Azf-Sz>YG&Dc|{0k9NJg`hgK)Ze~Tiq02wjO&WT)5LnDN-GH zTEfO`vN4SPOjXcUAk!<*Dfz(40f&pG89F(@*iXFH$wB7*+}dh6kY|`TEo!?376Tov z;n~^iMV@OGRyH;tK&RoBkYE4~tZre!JJ*%eBugzqWx)|B^7ckD2{SWuo>3_UNUIIc z&*@?@yz=po0g&?m;sByrFG^P{_h>rOv$95s)b5-Bay?Y;%m*4b@@((GKsZ1jUKyFw zroL;$Ozog$9dsXVkH_R2m%V=fp5pPT^RIKG$x*LBI)T3%XFC&$-4^vn1C9^#_|~Z7 z+c|)?RR0|?21brCF<}SE5@aj$Uqi)~kYF%sYD92YXma#AK+F0j7I)#18n66DmjoFF zMdSGKkq7M0;>QPTbR9qOE*GHWlWbBGAW*bax-9~=18uGFn?#%s85Py%?o>G*`bbfy zG5O&)!^b8xD&;grm8BC2SzvN?vI$UjUS=!~UXY%k&6k(N@Er)KkQ)j9 z6$op z{YEM)1HYCUs1=}uMD-}{Q&FB0u7}Qsxvg!hY}yZd2Zv@50N3C!*MndNURD8RWD$!F z>j$XeG8sn6{1oE*yzwWq&yVOl57t@{y`PFx zKG99H;eYQ=U_pcn4i3g*B-pBj08i6^UqDO*xI~Dm-Tx!!>FuqeUlD{{4lqRS0%T~D z?VYcB^@^K_n7F}Alp7SE2pX;y43{4f7Z(?jkQ+nESO5N_}ARK_PPOa(edqNw=^AdFbdl0-&6@hyA%+~O*q`mHi(hNj56{eD5Z`sK5Q7*tH@ zpZ<7g=u1Hp8MGNb1qBWO3@mc)!q9RPb6>)?S$!brah++SGpTaV?;J@@Nugz8x)>Pg z2#vbSpkjlD9-ELrinLQL^`I*kLbCS9!?KXv`TV0NKjlZ>*KI%jzW5iBl_gnE>X9&Q zU0TKz7qFlvoTg(IsQV_ZXNe=#f}s^LKG5O&lka2p1~f%uQ&Xd%-W-6y1_T&c&B5WJ z7_g`(tQRSCaz2i)b3n_SSp*lF)eu;W!JEz8RBD*~eOppSZhBOehDSt@VlY1s&KR`X zpzxLnMMp;q2nwno?YhU0W0R990bhaDdCr_6;^E=(+cRNRRKfw@fw&_F`7F2e>-cWv z%hqH;189WXNtT4$%Lx0T7A=*VE2fq{C$;e!9B48rHSvd)F7#W&avaI+UneAG+;%NB0Q( zJ$pLnUon@mMkEL;U%5go$$BHb@U&U!M@AldR*kJLzVp$_9U}HGXA&rIX{iLAd0k8bRF* zNnExp_P&!-gi*e_vN8emj#oE&lq{i55AF>F9tfRX0L~M0b0n88T`I4vY=q`FfYI>e zWaIe@^By`tDfj~SJ31WCJxgw1U!MSVR?yURW`2GeYWUK_G@wjwZWn>oBX=W!he3xO z5yqit1smmuYh{N(uNVLiSjF)iHRa>uJIm%;PO}t1tlw%v6NLaJ=r<_$Irro})TM%@ z(Hfua^JNRJl~os>r_ndcV=x#Z5)yYD=lJ*Up_P@gK!IL^?)Brx54(H!g7tI5)H9XA z%FsB2Y;I<2OKNO9(tQl|w-Hj$*x1;E-Q|W+CWBJe+qJdQZoe6kJPXG*U;i=W(e7i` zXmyi1e-6MD7R4+e)c~ao1Xc8n&N+2FJJJ?}`v9Zj`@^O7sUjPD`}3`XUL2&xRVAln z2TpRq`n`i`R@_%_wOZ7$-d!+g(OD z_63nX=Em*LK`)=JrG+h{iM}eC>L0l-aW zxx59DU<+6oevW=rqpZ_I%`1tsE8hcHjoll@OV?eRRp1iLnr9C9L`fTI+ z>Zh%6HQvc)4c)g_;XcGh8R>j+C{fDcG}pIZzoXnKDJ3f_r|xS~;rY*?pl_ai_+URm zB_l93H5G}=ZV3WMcC3QyFIs@7D?1+l)e&E&-#z$v+iv4u!@DE?!ph>*RPXM<#mByL zqJ&-HeG^xyQxa}MQnLXMG19`{P(&xHy(vLhf-6L*YQaGJ?gEYZFfYx#vA8ICv<|0 z?@8t?kAvV5S>D1aS5pDA_r|}%W3q1k`dXAD*g5e3IZ1P`%t&VN^hX+np4g5r-HShmhbJZ`8h)@2NT;Z5U>w z{^L;ptCPcoi7-j{IoB_p(%vcUg;hjPo=hO+2&QMm5gsI}=PKU2cj5i}_h0aMVwAJ9 zvwubgyMltkT^pNLiA+zvix+QbYSLb2rrh+BQR1Y{(Ynyo)I`I~+$EWANHc6Sp^N?a z@#Dn)YJJ$vH(oP+2{*Z4-sCPcua6-jXS};SR1!s(66REY$awqq?brjXSrJ^O@bIA! zKR!OzOVvnM+@mci@*^7B)4X;G|7hQK&pB|H!8Abmbt%&PS4=^6BK%wFVc{zGz zsr>lmQm;KG7SO{@ma56jPZ=2*xW~e;UkdvAr$1F!_y712(c7zTU})GlI?Af6TX-i;Q}+rR0Rd%4cX#vR zK(2*_1py?qfss+u@GvtBLlfEh{N|hH!*#air11=qxzXw;)czKZnYFcP@%u9;oVt|G z(^2|S5X-ftzSH6VSXP!tuA_iv4NItvR6{>s?_H{S$8P@J5E?>~58VP!?==;(+H z-_}NqnwXqiU&_ffZ|6JO+P!aQ$9sy5Xl1-U7N+R>NLpGNJODBY)$`9gt>I_! zhYvv*o}k)V85UO7$%zRR79;+wp4NgcLWybFNK3YmXzXPKd+MDor_XeM<4@a(BSa2L1tjJyoitP*fIp3QWDR;cvN7(zJ zw6v+#`|p>Lk*LH(#>?X3Q!As@_V)JIU0g(d{rdI#yTjz-A}KpN`_A5;<<}fFG#c$6 z7S_&n+_<(iZgqLicjNO_;vyn%Z*TRLcqB8-&CU4|ZVHwgN8PlUwJKt`csD@8(B!ye zE~})3wzahtHeKu1(x9f98I_8Piifv17at$`gI%Apg2IWKni}PUQqIelPr+lZ{dH=1 zd4cQ{!>+V@5B#ySfAu;nomXu{LBT~>8WD@8V1ii9{O@`1-XSUcaJ(*BJ6Aokw3M!( zpa4xlp`@WfqoAlbxwO>1I5jVpi3D2B-kz_kyL&SCPqUQJ>bYz#!V1^=I#$~&q%Q6* zYbBcj7@p>t&iFIAtTJ0$TOECU?R|2`QGtOc7F<6ak+&Z0w99c`xproneg6(b$HTA%SO!;j2g>q5|n4`h9E<^kh}g*2#%>1o0UCDEz9qaE8`19wHNwqsA7K`QM=MO9^yE60WTDtejwcxO@=Fw5#mcM-}nVFe<0s=wo z!DWmdTk`eZe^(}<8bRFJ>5pU5(pVV<4N!2zv#V@n_Tywkkb0cpg1^gggk}DKe!SHzq?)n_aiIo_T%x z5$WW*rK6)1uIw((r$#tFY^(OL+Om~QeU6f7aKp&h*kuWSF;&uy?8lEENCIFm6cC8X z<>emfj4$&S5^h@C*>MR7kk8J}hR4LTMGDuUt4xks9x)F0Nxdw)+ffte^;H_`r-<&w zix(w4|D2Cum!raBLT^MeT1>Y^JG;18uzt>wkdm_cQEFpQe*a9$t5+`D%ZAItW#r7! z#J3U!8zHKbGczrcZ`HCz&u|YGnl{IAsXR|g3QJCAsyRN|PZqHvUNW$4Y;7eH&@XK1 zN)XV5K+yB5-`)K)8y+6s0B30F=$a(6)Ure;pPLn_0HA1TYxjTo;=i-wah{u7o+p0G zeDYS5fP@57V`Jmc$9wdmqN2}VzC3yL>QzDWI+l`>65_NB&d|v%w&HQ$zpSXc{aH7n z;wglLg{j}Yd$;iW_YEH(X$XvnZ4VQ^q{C^X{wX^|O!dy4FayE1lq-%jw6u+q_}9V3 zjDq*?-`^9?^vs8|0+}zwI2kywV4wpI7 zGBSo{tHdHGsB``VH)Ev8(<2`i8Lv}-3I^5g7iAAOwb!=Z$Yu1CF$EqR_|zQy>CP)C zXw^6O4f&q0zhRpsAS$Y-r)N;>b)|0am%ukXerh(!T-nc0cF7=6LHIGn>wf#S^>%hH z2%VamT6k2HHETrF3zzA(NkDQbNbwTieRm}#rMumU7gK$xI=}Uc#mm1;OthTsN?2H2 zl=t$IWQf4R_mto-FE6j+N3%DFb$d~pJ=Q3%TDCB1c4Z?Y<`*wsTzax%f{zQ2Q+uQ3 zv(`vb={$L2%xmc)&Zvxxl$6xco<-NlDQwL{$Q??l`7`vNY!In z7M7@-ob#s`872><{?c3eYDOa4`+-}^wNyb1*;D6K}^y<}FJkBsE zC50Iu=MmO5!=Ybj(zv&^6!-kO|J5-MLPo~Tmgs}nkPxDy&0IMS4i18)!J_sA0eu6N zDF*&{^E4T6MhM1G)#LfsXOC$%%01RCmq)7T;hco`w&0RPjr{S}2?_UQrn;Reaux|^ z)X~uqZkmPws!@9YAxZJQ0iKT)6Qo`n-__Rp#W`M+}yM8r1I!RpOez_0AV4@RZmlZ?=DGTun({-QtlM+?9KH@ zTf$N7a_Md?oOj_%?SNUWG*m1S*Ka6DOaep_l2A>2$yBuU*1|7$ z&A{FpDsw!&y)t@B_yIbX2b1g%*k}p6*mk$P#4te(h~Dwfde_o~DB`D2kB=&%KmTqL z|8Rjm;but1m>2oks}GetJS3JRT7pBu!mOZVcq|oEyw-YW&@RIKVTe$#Xz-u)1Bf2; zn5GoUq7}`%BfEPtIy!m?z&c!xMb*0@#+gmuyMayChsmLgd9`l8uraA)!;@M&g*VC* z4u$CG>Eq(!0w8~5?bB(t{jcmMU3>H9jq0siijN+N`r<>G5@6|G#K*@aCfYLYXC}L1 z?JqOm|5;863$(dt-V_YChm0$J@KYQp^dv4ZaRXDK&SS?AU zieN=|7jrXS4=maXt8=2EW_|&B4qF}WAEov(0t&B#13$!?gYi{MtP&>GKv3Dzl2c1d z3p4j4LJe0lGGem1z0D~pnF8~pU}R;D9vd?Q@&1Ar^xGIg^2!B70t>XHpaM}4mW zthzs5#|n71SYI3PQ(fZwh|x^acTd$x-~_089#w7HN<6+Y0AzP3Hd1VH~r>U_@* zxj0}@>CCaj6zps}*%Tt-yLWksshOt(kwBFm>zs3a8GVvu9I05jr?Nt3 zHRv?;cFC?{bSVZ3DlHQe`t<43cun+VRAeOi2|_|5?;E&t-7^Qf8}KX^tU)xrtt~AR zaEP;_h@9xksjfZ?eRF}LlutT$GyG1 zYwPRsKyCl^ah2;L9Z;Q4Nn(y{+qoX>4rGKuJCkT*f||NIRRXEVJxxu`zym*4`2;mz zUl|>Kh7%`F7^rYqf0N$m5d*~lRNRi$M)jmF4{#?aZ@Ip=!qw;_l#<0Cq)eozq2FWt zj&|pdy5~tnY=E1#B1N>Fi4KO7mnQ(a2N8<;bV2*A1~w`&G?aM3Rno)Ds}vSC@0ElL zCy*#+W@ZBwk>tBwIXO8e0o7dVf|H5jRD+9)FTH&EvN*=_QsYZC(6~Hz(tF@3l*K&% zSOff&##kU97#f}dR%O_JuEjnuFwpsM&kZj^!4odJ`1^M&fa?14hd#Jjc>hUa;;FXi zv%m~&SX<8p^2M9`?GjN@Q59EI*s+GtWS3M_gari=`t8obx=?V2%iN88_H5$!Z<&R4 zpW%UcP~Spxp5ETsm6gfc{;Z2Nb-uz&OG~t;PHldh-Yw7Ng)}kgAtxhiN|Smrv9N&l zhpS57zvtiC-7Sfc5AWaI+f&HYD2}n9k$)MXkm+AhaTOxNlm<%&;@zPA5_+{;>0kQC z%*+gzO=IZ_v>T8-#0-hQMvX9inGq$(K-D)|kT@ce5lSbFjg9lZJK!~mTlYopIXVV~ zg^|FRm!>Bt8+WBdvIG9wd>tNs20&6ABaaqaSy|EHF+l_%6vuzAo0|*W-gd`}5Z|y- zK=)OB`UJ!z$Sq2qGz$ZvYKLI=AjPRGnB1W(r7Jl33M8%9)i5|>R##WYSxQk+qk5hW%1KskZkXH4+$lFK zXnB4*1=P^+utNU#(wKFf8lm{YLZP5?m2)lMehdvU)vVG800pj-xs4XF1Kq2`!zFK{ z&_M`v1}YiON}8dlJ0~}ngprXEnM~@NH{7774ck5V%~lEW_3{z&7qqI6_6EwEay8Y{ zvV@5_IusV>mvppp(ATe8aBIt{mWWE1X<|@O9C4~1aj~${R-c!Q$+h2N0)WKHJ2oco zvb9wY(tw~`8h=~3O6V6762iwX7#v$9Y-wd>*v8dj&t^vMOPUMGMEoQI$Pm=WPZ-K?43Q~)oPZdIMy+iq%Zj%wIhpQZ)(N#Y_$w%wnZP9*e3Mn)$={Au%`z*72plz+6cx3_)#bq}@< z2MP$1K`4O;Oor<0Pc*f zhV|@=E-o)yLjj))F0PU+`s>dS*$QUKM-su$V*8#(1&o#@Nc~umey2lLVn6}vQ~qzw zsjl>NFeWM8(!rWB5DeAfXWjN?nx3EC>Smc}{EI6uFE2%QFlc_ovFj;BV@VM@{2^NJ zko(v|0&&E5Ldku3t}Cai3hPFpkgUm{u-Ow#4iX*~!VNc@ApkB0i6bp7t)Z>m4BK8=-_(@y&Ye4E zzHeHudzkYeW?@dmRN?#gWU{idI4dS48XM7j zU(f64=|Kd-`*rp7n#RX7+$e(3Sr`DSJWRMDXvm?K2>O0J{-0_FwJ`;%8n|8-u(fg7 z$Pk>2kq8iE)zqXSvwd4r!_p_aFW6!!FQ{$tk`#54h-ebj0TdC-%Y#McD&VRtF%i>a zu)(3BSvfhu$eiB7Z~bv@X(p&GKa=tc3MgTQ;A$v1IxhL2UPIrza|aCJbY-aFd3orf zx;_5k;bAsI@@5;ncfao;|zdyXT4*302at`7`pkk3&ulLZF~*3Wirgr#%@p zwF<~Ng!J(RM={r$#pdf5aY{=A5#^xO!Gdq5$P=HLl@$ojuKXE55%OhXWu;fL>p7q| z>Ks2mS(1<`q{(y5`1p9PN;__URJS#71s$H29Mey(q(Ew=mzWT;5!XEnPS9qnNy~Y# z8o;{<4haEPXB{X&f9uvQFfur#q+Y>oB$Cq7qVw|jz*H>#*T*Oi@Hn0)B(wrZq;_|8 zHIWO~G)T4~j0mVRSY_(wXK!L|K7K3)-WBbcGoAlpOE?_vl3gzg_`rbTXgg6cF%dZY zXM2hY1`5)n5E2IV^Jiq(Ye2#ng-kkNSkiI~aF35h7k|4bU=)BP0I>m~-Ug$>N06XX zQ&VfdE5;Eb5}qdEa^hB!5JOL)!yV3%85E_-prv)i zN?g0f!O3~z%$YL))1vp`H<%j!@$vLad|aFhq6GdxfjDJkSpTk$1GRY0QT_B-_Uphv z3Fu;k;I7%w(b1g%BjXciB3esNk8xyVga)xgy243qsfTh6>wK;u%AzMtDrrF2V{U^j z^YWdY4Om`8YvPP6RwADMos`R#xvL`4_r+J(%F4=R;R^?%i+fYADR(1Tr8;L?<3XF4 zm>3=%?Yk`3INl4FM}sYF0a962O%1Vrmn4EI-d9wd1}9w=gPGaii7a$qN{}S1WNqDu(fF4O%y%i zopEmy5DQpNxpSyTRj~cku{u~(!`$arE<0-$;Qu2W-zp^~#TgFokJVt^ zPW569Z?Z?lk^TE2&2=+7PN)hqWlU03`c=SCO)SDD~O-d8^~2!yy>ZTJ+44 zgtC=51Q2Qv=vsZGv?^jW!0H5Pi*V0rrUb(TQWPdbmK8cb;^IzYQBqGMRoU2)TkTDgI=oTpcj!g+Lgu=61KF4Ny7|Qp z-}!Oo_37$eFF4-ea4k^XOOR6rC4p`%b3Jvi^yUdAHrMsm=N@8bchpZJ?9jU%H#3C}hchXyq4)^0yQ|~9P4pRiDW2LS4HWv{? zr|#g-EuTM~yqkt>f5Ksn52o3V8^Fhk2qK}@iaQ`q=ydY%^yIj5g#k$v3?}E!S$fpM zQ|Zf>FQ2<|<@>Ubs@TbdPI_Q3(0U_$Jl7)+De*hZa!$yk(my!(WDi!1r25G!Fa#Uc zh@^$oP85_fy&^M8Ff$(fET066qXj~=7+|do*^cBh`1+6x?SkdBOr&16qT!H=EY}{L z1j-y98=HIk{bwcEI3T}4cm;qk*N9{CPscJ{=Zu>We!5HkXJez}@w_^i9Egw60&LAd zg^R@z*qsWyZ*Xa;C}QOC)Y=IQE?_%pz}25#Dyk37&E+-ooMm8<^+~RK_Du;SANST1IAlYtURfj)FF!lg7~4cp8q{GZn)$OiZE}1oX~{iP6{A*5b0`ED#UZ)|N*P zXAm$oWyvDxiY}@@`qb&i+G?Mbbgg-2aWTTQ+CB62YZRCuieO}z`EKdIU!)vI+u%Gr%CxNjj`81Skj#`Y_WfCZ4k7n(-Cxs@~p zX(HkF>l786ba+F<^_G^FbCQzG;2z+t$kMgmDY?6gmzI`ln3=^aEZ8Ia{dxE7CPj;? zhdz&F8WosvdGFr?^$|c#ymG1w_`iD2lWS0>klHcOoBAB|1#v3!Ev;jl~h3_VSK1bXZg1A{y;$)Z8U^K?KkIRyoyPU{E#+VUEy@uUX> zq!P>w=m|m?*TUou2*4u@^i7q<*gMMSe0SIiT_KPH&NaN&Of7dkyI*?UJwhoEW$#ZhqN>c!iV5XckkW1w~5KV5e~N>T-pj{kr?(?{RPPm?mS53 zifSjWY#G9ePfi3V9z%77R9}_@!NYS2H%lyApS2rdi@bxW+|0mpflao)(I=M;sVtdR zc>Lo!tTT8*7It=59d?27XquWvi;RvYyU$Ua5Hb%<(~6+7v(xEG4#=-RmRT`2YJ%t6 znD1j>?~{5S6GQbfDQOyt@nPQ!V3iz6-RlTrgjs|BZ%*%ZZ5}j#N?D)>O{Po3 zyTmE&uad5T+SJJZ4t?_wnJ$@y#$$1wPtZyyvmqLLQ@ znLZYqvFmN9Dx|`my_)su6EifH=pw9@{=U=8*VNWV)B$3(fsV|>;glfBAR6Gb1n+Ll zPeV5*sJ{LK~XUjHH!y4kSk7p0ok9w ze`f&q^H5j-NECE@HFR`>_x8LFAW?wLA|4C0TS2h^DW(Em2F^NLVPT>3qep?@&VjK( zZfyLMU^V~a$5W^YJ(l-fhQcyWHJ5!PWr^h@WQ+bp^EcFDZaf@j9c%KF9rGFWx+6RIZiFwoMv z```g3B=b>`fS}E=;?jHS#o4yO6#ZXz2ym8}K%Qq?Agb@(%N_%n7k{v3`6<$L~J86~_sWMuE; zG(RI-1ulfh3@ZigP4J(O#B` z{!xpSPk|4Cw8z!SH%Vq*Gvcxt8uVQ?(5HA^Tuk%i@F3l5*_53fg|t`wg&zm6ZTxW? zyE^V;wPrLiP<^fDM_|L?-$r4(l2JC9G23@U!o=_+dY+!100oMY%a1oq+7Md!=jkFQ z@F$!K#R%9^J@$_FSHbvQZ>hJkvkQY)a~k^l?l7L_xzqfN~HJ z3I%#{kFFhld#4S(j2_9X&d1KqG$a4=T*f$(p@aQaJ*`XfnISn9)!VOkq3|O13Yw!y z84SdKzm=rJ6HKG7fu&K90I%-9om_LEHc$w6UjOys8n_+((v!W}N!R{=7yri*{--Cs v8|Yvf-VI>?sccAKRfHe^x7WNPj!ACpX5m80HJafqHk8^8b)`auyMg}&)pgjg literal 9904 zcmc(Fc{tVW_wJT?$Sg7swFzZRW*M?mk`UX_HV-8v^E?w}OA<0ANirvy=ggwW5Hb%5 z88Z)Oefpl?`Qvxa`R`oUx!&Gu+k1cZGpx0qweI^~Po%aM<|G9x1qy{asc}_J2ZbVF zK>jBqflrunrpn=k#Ns+e4RwtC_o^m44u#@IX{agdd8RD?^fLMGc0{^57--TYD)2C@ zg(`wvkS_s^#`4BioPRy{enQ`|T#?GSc3}h zZ$FY+&6$$3H<^w9sWKJ=-lhmn2idy%zECt*Zf{%ffgw2P>^?7_thJ;7UshLJ5T z>Zm|H!LaOXentx7y!`yr{E{?YUS0wM0)gt83kMR#SgJ6Upuj*>czajG*MSp-9d8N> zs3WL3R9sv(&IjO(jUNvW8$EyVVyd{6GbApKAv1JtvTCJP)xlxeAz-YxH+XZ?^?5=9 zpOn-KW$}$~U%v((?r&4xJHPr?bJJUanq3+dtxg!P9CmnP|5lz532J9#$Fj1?)$^jy zchrgcCkVApV-2B&&(2NqybD<1pVlio^c<{H6&p0sTjm z%$v+#M@OF=9QY16-_UV!k+^_6U_uq$?bBNtVUEf*;6R~DOR2BkIN3b+MVA1@#Kcrq zO~gBTOECWHz0zP~%Ib3rrA8-FR~@chVHAeqnVk1RCNlCzROW&RA z`L!P$D{%?@s$LaxsMX&=_pgOkp7Puf&z?rbT^s*PgK^L%EO)+dz5VBB+34H81v5*_ zCPPl$)NF`MN!D+}!xZ`YBfEv_nL$|L7QgMEq@}~N1rp!y8SL!o^YHNz?r$xJl$Kt+ z@8BSJTv}2Rf-9WgtR|^FI3TgW`S~dX{>cx+yETzkpNmVQja19yTh-CkW%&L3_t&AJ zN8#Zll9w)J=H%p*2)|5DHotq9JoYM)oLp6Rn3KId<@M{=%PT7W{#so4>#b*CpjtSn zn%nPwO3&V2=!rUCfoUCeq}mAx6k}s!X=&-pys`D)zR~IAFluROJ$UfoS9W^1*F>Mu z#Eqn@SvueQ`x`okDbH3ulN-Z0o+U+5amX@C*mhUXO7WzL4Y4gRFKg%aL-<|3apR<( zAd{ew(5+jy2-w)z+GZbmdU~48w8cJfaS6%I6=LV$5c6C$gQdCcj+DR=JA3wQTW4p8 ziT`ZD>6vv_RDBbT(i0{jeOH;Dfq_%~{r&F>3o)vyg!k^<)4p>j`uFd9DRLgv_exo1 zQ>n$1sM6BXR@c{qO)f-~m0g;7C?hN^?3mQrddzhw=G2p>@DoRA;!oq(oGc#OIr}a* zSM`Z1RfpSbS&cq>sPZG{M+B%gD-VAKr>Fv4O;mNY{LI|k>!KpstgI{+aVs(yPyGo? z5Gj{0t;>9Om?k6;~>f@h7Lqqj_ zeGIU$J7d|!#dHvR+Btgf-o2}D2qC?tuODtdSYQqYVYzsZ@dP2^7xVfE8T5Djm*;Y2oFYCip&ZZDMKk zNl}pw96^HewLs>_&3;EO!Sw_L1Z9Ux4)^b?-MU2=6&3YhbwU{o(l|WKs;sO`MozAL z^=jLl>(|;K>qI?%JbBVQpBQ)N*I>VfJqJ5GJ1qx?Un_&#&cVmOjvK?n!%ySlLUgmW zCN~%QzS>D5Kew|JP*G8tn4P^iv8$t_^I-Pcpop7^iOC1~ot+(Sd3kmQv}Ryop;H<4`@h@U z#@gC}2L}hnoL%&hM95S!56^7)LV1~;oo#Dtt6%8Lc0M}r3JnY6k(FhIpZ;2R7T4Y_ zPqekQ9n38i)!y|qMPFL<+~02JPoJ^E-P~-grewarefBKz-Me>L#4Lz70}h-u)8tQw zkkZUuQRL~#Ei7zpkK=cqs3&+HA8+3KM&nU*^c8JwT9Ng?Zn&ERDjsux`%~xE<`N@w zD&3b^;mpUrd`TfBB-E8CMtJ>&a8Om1+@c-YdUc}VQDmgDrY6<<_wOG}eI~Eo|5Ih> z>k2Sx3ov3o-9HUQAuz$z|Fol{W1@sD^;@mf4%sw~$-kUMN(ZZisEJ7aG_#bM zF>wirwZDU=O=hBITT4F@#jPoX&!0DJ<*jo$+;dBjc0N(%u@W8~ee#qrHgIjawFK{W zQw2*IB3qDM&U}hAPquqtse91c?edKrJ&UIBlaS4^)t+LFO-(C1j!(W)?sYg=ju_|W-sg(WTGFtuguK|r7HQ7le&t^ z2{;N{e||px^~+9>K8QY2?GYZ|GFIc0p)K(A>C?b3$(JDUmf+FSdT52G@$q3YE_1J5 zzdj-7@khyZ7&ik_L$c^$D6FI$>p+fbu5-8E#SyYt3XxUZL@2J^P!82n8zOzmT2{* zq2iFJ&zhPqCCTO7moEGG$jyC8ju;&^8Hn|aNlS8Vu#>a^uMXAv$>IEWt!4Qes3My! z@^Ce*t$2>6C?+9VX6A_OH18;Uj9XC03^gko+uGh@VP;kq@yI6!B%d!{4b8~W&F1Ci zM&;_~tKe{)M{6+wn_ep{x@saW{_|7Kk3Hv-Y$yKw!QV)hx)n={3QSIBlX4gig;=HO zk>8-YfU9nkrC-rb8l74CQGzXhz%w;H9h;aqlzAxjXn{@Ajuwelu#m7xIjwC^nl(qH zfnDjr+(bjjP?0$yVu&YPN@)dIJa?j^qnnG&n-HH(g)!smEn^f~Wa&k;lTbsY_RO`r zb8KhMo^`7zIn8s4P03&W*|TRwt_wPnQc}IyY{9rWGbqPHpB&f}_m?CD>3jKbINZ?3 zyA03b;)*?1a0AYX5{u$Lz*LaBio7;#ps*F;Rf*b~nh1?Q-ETNL+%NXqcZbKsN5prx zMx2UmdpKdfZ*o#^ZkpV=Q|$Ot7B)y)<-Tme5plXiI4%Ng5z`_eJF3TAajd-h;Oi^ILYKaUtC6<5r!ZG?)g%qEz&l=LV* z_S@f@>NuMJ(r@_VrB3|Uj*-K%s-<=BPM(}dkqOr*647z6o2bp>ksBlB%z_9W*yJZ=JoH~6v z5_aOdGZ~SWxhWKPKCY`v6L9BEY^|@%qClwG?)t2>-<~8SAw$fO?EVp&)(s z827`7IFX5*H?3v&+5U5}y~OfHfu_{`p+Jb*^78T`yh@t`lsPy|w$Df{Z@kxh`0(Ka zC@Pg*woO$`}gl7zP>+NPN$w3 z>&D)lsG0O8J6kook4I22%8gOw&pAOs3Zwi}**DW|IKm{*y6E=yb|f(h3v139^P;;N z8&5oD7Lm&FaM!e$O_gyWad2>e3}?#`_SBJ*h9&~aq_?;1qCkphr1Q~u3UNd{=JHt8W9+D6KiF428xfiiAk62wRG}YO*kxT z*-D3z_;_074X3qJd2vljO+Fi7Tg@vr?hjIB8M|*SbbvN5Iv+cky?d?G!A%rTbs!6jz z7-Mq1rdy&zYHAb!vj&HR+@%=``sg&SsF|*q>PDp^E#bGf*?s3%*wv@juLl>Me}6xx znJg6z=9ZRl7?wn8sdr79+jAV972=Jio}L&*{=`H^00uj|yRgxa|9r=;;LR8t8=GfI z+gQ=ob`{<}cYA`PGf85hA_4P|&?p^3n1M7$DGa8leViF|<;|Lxs^q&3Y zD(mUvQv%^tw+4_%E4wM{dIH#)PB~r!3Jl~N%>E17I5d4EEep#d0Fxi^y?nvK3@Vy; z0E&G0@B!QU3!ZILC}xzk}_%V{48qW1?cxJe{0SlM8k8hr1uM`9UHu=~^uR6|)t% zJY9|V6A>}3rBYI=s~Ssp^6>`J-P$Y<2->99x57ngnP3DX7*C<7s0eNopQx^>ah_}> z9uTX%LTmk6KX<|EqQwi?J+PbxS@p8jT20_oP{_!VG+Q8a3AHXHWPbQSj~v$LK2v^4 zA3uJq@9w6yC{BdvAxXgsho`1;0FK3CD)`3|G=S=b({OdmlJImk;j1pj*gCETe(9|8yFZ!Uc6Y0cfJWa#A*tFxp;EsmoHyV zOieW+!%54@WmZ%qxjjKv0d!F}iMC{JYD&erkr)^}KR-V>!n~!uJ#vh7PUZS_%5bXF z|9*!+$Bs9J;A7a>($W$Z!xMTM*kb0lw*4NiIkzOcQX@fn71Yp=AIiD?AMnlw)fX62 zwFdK->Bz*~K7IP6hxW_pTaewBv2Pw+R z)|L-A)58jO@Z1vydT~)l93=;3LXUlrL8~SJQbJG#GQW|PZOqd>SYkPW=Li`Iql8}r zgS)DsVa^fv)Zb%g)x05}cuZcRy{89r=MIPPNfP<>X6DuI_KMjpN*dSOeVj(Na6Y4ecyV+2Yn$ z6x6lLTa*nAJJ!RJsLN(%6M;5-R@~%n=U38bt3kGnI1C_IH7M5x!VU&|I^u3o<$qLr;`Wo1=ZfZFFCgxj1of3FPD3HD|22A)3clBD(g)C-=9GLaB&& ze3vK-w*yOyiY|j*vP^J&qst!$f2r}(L1q~7+S=MGDk=($kMCC)r)Kz{5$Ku*I!{;_ z@zV0L&*4Rqsr}i%o$M1W`%8a4Dj1V#bv>4cFOH=dKa?0th=je}v$r>dBb=0LP`Lj4 z_e25NYfTU#I*`SF9~e;KjV*#x4I3aOCH?;Wdl6olOyc6ji=fSzaZIPdQJSoH`I6P! zrja;DFRC*^qy%{YGB8m8Jj>BI?v6L_-$x;K$a(*MYSZE8QRBY&X?agk|HHoq$f2Ku z!F@X%OgvF9sBvE8)qUwRDH8(dS^O+$6#mY(#%t!uK^Tdf0Pt((pKp;dbDRDu( z!hCGt;pMjeuU}C~n&F`Rd>S|8yXJW9A_laZ0p*4=Hx5%SbO zAvyW6^K?rw{u^JgVBExn+04ugxb$kBu3-E>44#Q0x{U@jq6)PcQZ9n6*fPI80;(1o z9{#U13o_8q&|E!353=nCq^iq!{9!*sPcAPn|DUQ(j?zq#rnBu%F-InhOG~>@ZhJNm z)oBF+1Dr5RbyilGlqN4aqr99YKR+MxfjLLkAdxZp)TvXTp)>>WM({`v z!pMh)unkM^-T51DP;;ZvB&{)A7^rWEU5(&XD@8g;379HO8~Lj9b9*~1QL}z~Jm4Sl zeDUmAFziBFMy43w%O8A!0re;*rnPL`|MAwA`@k)-h2q7*Ld(H|JMghBM^+EO;!O2u zL`u<0m2Cz9px_P`DX*%cyf^);fXs$wsLE@-oxj++^Ard{6`hCX`+P{0`6bx`(%+p= z=i}j-#0PB8DP#KqS?nFiOTV|?nV zX=AU30!63L(9m#Q3MEqkl8+EuyKlT&={HKO=&etTmg8DmT1YT4oYrt^d}O^h1we78 zv)XIyw$WWs$>( z6k>p0U}_VKi$di;=U1BFKqh~2-Z*?_#>%APVHgCAqswdu)BTZ8Cj<6J#<7^ml=k>Y z=vLTtr$qi;1U&@!&_JPSjA4mQLtkH<8)J8pNfN3a;fl|m(GPpRT&HCTp#+KNe1i9M~xZ0O5Uyye=ui5x+7j?~OjThu7kODYq6gFvVMtQ1> z{13fBL$o~DSpz?_#hquEXlWP%Z4q>)U^hmK7Ce^{9^S^u|MHfIyTDM4eF9XD8T8pJ zpkpsef2QjH{@rYOxYTp|rw7le$A+y72IxCb-MlyLy3&=ng!J@4-_-_Egr2T9kaC&*`kH|)G9y?NAZ!%wP&-3E`Q_5M z8XS86ok>zq08S5;IdK8{UY$#Jdinj_hKJ=T>k}wor7LyEii$@&mj4omswURmT?!Ej zAY{4dBSYO%2rQ+*)6#NVpa``H6;=mt%=eVwRW-G}ykzwAjlv*$N6z#-tYIq=1T*aN z!{Y0Ev#vV>RX~J?3fBdCL`2Fnspx&v!@m&q1CdrkdeXgrp95u}cb^4a3T9PuD>g9-TPy%)k)X@FQ*{btSihHp9Fjef^Zf$@7PB4kJ55|i zfgqA~&-HQ^DUdbxa~Azf-Sz>YG&Dc|{0k9NJg`hgK)Ze~Tiq02wjO&WT)5LnDN-GH zTEfO`vN4SPOjXcUAk!<*Dfz(40f&pG89F(@*iXFH$wB7*+}dh6kY|`TEo!?376Tov z;n~^iMV@OGRyH;tK&RoBkYE4~tZre!JJ*%eBugzqWx)|B^7ckD2{SWuo>3_UNUIIc z&*@?@yz=po0g&?m;sByrFG^P{_h>rOv$95s)b5-Bay?Y;%m*4b@@((GKsZ1jUKyFw zroL;$Ozog$9dsXVkH_R2m%V=fp5pPT^RIKG$x*LBI)T3%XFC&$-4^vn1C9^#_|~Z7 z+c|)?RR0|?21brCF<}SE5@aj$Uqi)~kYF%sYD92YXma#AK+F0j7I)#18n66DmjoFF zMdSGKkq7M0;>QPTbR9qOE*GHWlWbBGAW*bax-9~=18uGFn?#%s85Py%?o>G*`bbfy zG5O&)!^b8xD&;grm8BC2SzvN?vI$UjUS=!~UXY%k&6k(N@Er)KkQ)j9 z6$op z{YEM)1HYCUs1=}uMD-}{Q&FB0u7}Qsxvg!hY}yZd2Zv@50N3C!*MndNURD8RWD$!F z>j$XeG8sn6{1oE*yzwWq&yVOl57t@{y`PFx zKG99H;eYQ=U_pcn4i3g*B-pBj08i6^UqDO*xI~Dm-Tx!!>FuqeUlD{{4lqRS0%T~D z?VYcB^@^K_n7F}Alp7SE2pX;y43{4f7Z(?jkQ+nESO5N_}ARK_PPOa(edqNw=^AdFbdl0-&6@hyA%+~O*q`mHi(hNj56{eD5Z`sK5Q7*tH@ zpZ<7g=u1Hp8MGNb1qBWO3@mc)!q9RPb6>)?S$!brah++SGpTaV?;J@@Nugz8x)>Pg z2#vbSpkjlD9-ELrinLQL^`I*kLbCS9!?KXv`TV0NKjlZ>*KI%jzW5iBl_gnE>X9&Q zU0TKz7qFlvoTg(IsQV_ZXNe=#f}s^LKG5O&lka2p1~f%uQ&Xd%-W-6y1_T&c&B5WJ z7_g`(tQRSCaz2i)b3n_SSp*lF)eu;W!JEz8RBD*~eOppSZhBOehDSt@VlY1s&KR`X zpzxLnMMp;q2nwno?YhU0W0R990bhaDdCr_6;^E=(+cRNRRKfw@fw&_F`7F2e>-cWv z%hqH;189WXNtT4$%Lx0T7A=*VE2fq{C$;e!9B48rHSvd)F7#W&avaI+UneAG+;%NB0Q( zJ$pLnUon@mMkEL;U%5go$$BHb@U&U!M@AldR*kJLzVp$_9U}HGXA&rIX{iLAd0k8bRF* zNnExp_P&!-gi*e_vN8emj#oE&lq{i55AF>F9tfRX0L~M0b0n88T`I4vY=q`FfYI>e zWaIe@^By`tDfj~SJ31WCJxgw1U!MSVR?yURW`2GeYWUK_G@wjwZWn>oBX=W!he3xO z5yqit1smmuYh{N(uNVLiSjF)iHRa>uJIm%;PO}t1tlw%v6NLaJ=r<_$Irro})TM%@ z(Hfua^JNRJl~os>r_ndcV=x#Z5)yYD=lJ*Up_P@gK!IL^?)Brx54(H!g7tI5)H9XA z%FsB2Y;I<2OKNO9(tQl|w-Hj$*x1;E-Q|W+CWBJe+qJdQZoe6kJPXG*U;i=W(e7i` zXmyi1e-6MD7R4+e)c~ao1Xc8n&N+2FJJJ?}`v9Zj`@^O7sUjPD`}3`XUL2&xRVAln z2TpRq`n`i`R@_%_wOZ7$-d!+g(OD z_63nX=Em*LK`)=JrG+h{iM}eC>L0l-aW zxx59DU<+6oevW=rqpZ_I%`1tsE8hcHjoll@OV?eRRp1iLnr9C9L`fTI+ z>Zh%6HQvc)4c)g_;XcGh8R>j+C{fDcG}pIZzoXnKDJ3f_r|xS~;rY*?pl_ai_+URm zB_l93H5G}=ZV3WMcC3QyFIs@7D?1+l)e&E&-#z$v+iv4u!@DE?!ph>*RPXM<#mByL zqJ&-HeG^xyQxa}MQnLXMG19`{P(&xHy(vLhf-6L*YQaGJ?gEYZFfYx#vA8ICv<|0 z?@8t?kAvV5S>D1aS5pDA_r|}%W3q1k`dXAD*g5e3IZ1P`%t&VN^hX+np4g5r-HShmhbJZ`8h)@2NT;Z5U>w z{^L;ptCPcoi7-j{IoB_p(%vcUg;hjPo=hO+2&QMm5gsI}=PKU2cj5i}_h0aMVwAJ9 zvwubgyMltkT^pNLiA+zvix+QbYSLb2rrh+BQR1Y{(Ynyo)I`I~+$EWANHc6Sp^N?a z@#Dn)YJJ$vH(oP+2{*Z4-sCPcua6-jXS};SR1!s(66REY$awqq?brjXSrJ^O@bIA! zKR!OzOVvnM+@mci@*^7B)4X;G|7hQK&pB|H!8Abmbt%&PS4=^6BK%wFVc{zGz zsr>lmQm;KG7SO{@ma56jPZ=2*xW~e;UkdvAr$1F!_y712(c7zTU})GlI?Af6TX-i;Q}+rR0Rd%4cX#vR zK(2*_1py?qfss+u@GvtBLlfEh{N|hH!*#air11=qxzXw;)czKZnYFcP@%u9;oVt|G z(^2|S5X-ftzSH6VSXP!tuA_iv4NItvR6{>s?_H{S$8P@J5E?>~58VP!?==;(+H z-_}NqnwXqiU&_ffZ|6JO+P!aQ$9sy5Xl1-U7N+R>NLpGNJODBY)$`9gt>I_! zhYvv*o}k)V85UO7$%zRR79;+wp4NgcLWybFNK3YmXzXPKd+MDor_XeM<4@a(BSa2L1tjJyoitP*fIp3QWDR;cvN7(zJ zw6v+#`|p>Lk*LH(#>?X3Q!As@_V)JIU0g(d{rdI#yTjz-A}KpN`_A5;<<}fFG#c$6 z7S_&n+_<(iZgqLicjNO_;vyn%Z*TRLcqB8-&CU4|ZVHwgN8PlUwJKt`csD@8(B!ye zE~})3wzahtHeKu1(x9f98I_8Piifv17at$`gI%Apg2IWKni}PUQqIelPr+lZ{dH=1 zd4cQ{!>+V@5B#ySfAu;nomXu{LBT~>8WD@8V1ii9{O@`1-XSUcaJ(*BJ6Aokw3M!( zpa4xlp`@WfqoAlbxwO>1I5jVpi3D2B-kz_kyL&SCPqUQJ>bYz#!V1^=I#$~&q%Q6* zYbBcj7@p>t&iFIAtTJ0$TOECU?R|2`QGtOc7F<6ak+&Z0w99c`xproneg6(b$HTA%SO!;j2g>q5|n4`h9E<^kh}g*2#%>1o0UCDEz9qaE8`19wHNwqsA7K`QM=MO9^yE60WTDtejwcxO@=Fw5#mcM-}nVFe<0s=wo z!DWmdTk`eZe^(}<8bRFJ>5pU5(pVV<4N!2zv#V@n_Tywkkb0cpg1^gggk}DKe!SHzq?)n_aiIo_T%x z5$WW*rK6)1uIw((r$#tFY^(OL+Om~QeU6f7aKp&h*kuWSF;&uy?8lEENCIFm6cC8X z<>emfj4$&S5^h@C*>MR7kk8J}hR4LTMGDuUt4xks9x)F0Nxdw)+ffte^;H_`r-<&w zix(w4|D2Cum!raBLT^MeT1>Y^JG;18uzt>wkdm_cQEFpQe*a9$t5+`D%ZAItW#r7! z#J3U!8zHKbGczrcZ`HCz&u|YGnl{IAsXR|g3QJCAsyRN|PZqHvUNW$4Y;7eH&@XK1 zN)XV5K+yB5-`)K)8y+6s0B30F=$a(6)Ure;pPLn_0HA1TYxjTo;=i-wah{u7o+p0G zeDYS5fP@57V`Jmc$9wdmqN2}VzC3yL>QzDWI+l`>65_NB&d|v%w&HQ$zpSXc{aH7n z;wglLg{j}Yd$;iW_YEH(X$XvnZ4VQ^q{C^X{wX^|O!dy4FayE1lq-%jw6u+q_}9V3 zjDq*?-`^9?^vs8|0+}zwI2kywV4wpI7 zGBSo{tHdHGsB``VH)Ev8(<2`i8Lv}-3I^5g7iAAOwb!=Z$Yu1CF$EqR_|zQy>CP)C zXw^6O4f&q0zhRpsAS$Y-r)N;>b)|0am%ukXerh(!T-nc0cF7=6LHIGn>wf#S^>%hH z2%VamT6k2HHETrF3zzA(NkDQbNbwTieRm}#rMumU7gK$xI=}Uc#mm1;OthTsN?2H2 zl=t$IWQf4R_mto-FE6j+N3%DFb$d~pJ=Q3%TDCB1c4Z?Y<`*wsTzax%f{zQ2Q+uQ3 zv(`vb={$L2%xmc)&Zvxxl$6xco<-NlDQwL{$Q??l`7`vNY!In z7M7@-ob#s`872><{?c3eYDOa4`+-}^wNyb1*;D6K}^y<}FJkBsE zC50Iu=MmO5!=Ybj(zv&^6!-kO|J5-MLPo~Tmgs}nkPxDy&0IMS4i18)!J_sA0eu6N zDF*&{^E4T6MhM1G)#LfsXOC$%%01RCmq)7T;hco`w&0RPjr{S}2?_UQrn;Reaux|^ z)X~uqZkmPws!@9YAxZJQ0iKT)6Qo`n-__Rp#W`M+}yM8r1I!RpOez_0AV4@RZmlZ?=DGTun({-QtlM+?9KH@ zTf$N7a_Md?oOj_%?SNUWG*m1S*Ka6DOaep_l2A>2$yBuU*1|7$ z&A{FpDsw!&y)t@B_yIbX2b1g%*k}p6*mk$P#4te(h~Dwfde_o~DB`D2kB=&%KmTqL z|8Rjm;but1m>2oks}GetJS3JRT7pBu!mOZVcq|oEyw-YW&@RIKVTe$#Xz-u)1Bf2; zn5GoUq7}`%BfEPtIy!m?z&c!xMb*0@#+gmuyMayChsmLgd9`l8uraA)!;@M&g*VC* z4u$CG>Eq(!0w8~5?bB(t{jcmMU3>H9jq0siijN+N`r<>G5@6|G#K*@aCfYLYXC}L1 z?JqOm|5;863$(dt-V_YChm0$J@KYQp^dv4ZaRXDK&SS?AU zieN=|7jrXS4=maXt8=2EW_|&B4qF}WAEov(0t&B#13$!?gYi{MtP&>GKv3Dzl2c1d z3p4j4LJe0lGGem1z0D~pnF8~pU}R;D9vd?Q@&1Ar^xGIg^2!B70t>XHpaM}4mW zthzs5#|n71SYI3PQ(fZwh|x^acTd$x-~_089#w7HN<6+Y0AzP3Hd1VH~r>U_@* zxj0}@>CCaj6zps}*%Tt-yLWksshOt(kwBFm>zs3a8GVvu9I05jr?Nt3 zHRv?;cFC?{bSVZ3DlHQe`t<43cun+VRAeOi2|_|5?;E&t-7^Qf8}KX^tU)xrtt~AR zaEP;_h@9xksjfZ?eRF}LlutT$GyG1 zYwPRsKyCl^ah2;L9Z;Q4Nn(y{+qoX>4rGKuJCkT*f||NIRRXEVJxxu`zym*4`2;mz zUl|>Kh7%`F7^rYqf0N$m5d*~lRNRi$M)jmF4{#?aZ@Ip=!qw;_l#<0Cq)eozq2FWt zj&|pdy5~tnY=E1#B1N>Fi4KO7mnQ(a2N8<;bV2*A1~w`&G?aM3Rno)Ds}vSC@0ElL zCy*#+W@ZBwk>tBwIXO8e0o7dVf|H5jRD+9)FTH&EvN*=_QsYZC(6~Hz(tF@3l*K&% zSOff&##kU97#f}dR%O_JuEjnuFwpsM&kZj^!4odJ`1^M&fa?14hd#Jjc>hUa;;FXi zv%m~&SX<8p^2M9`?GjN@Q59EI*s+GtWS3M_gari=`t8obx=?V2%iN88_H5$!Z<&R4 zpW%UcP~Spxp5ETsm6gfc{;Z2Nb-uz&OG~t;PHldh-Yw7Ng)}kgAtxhiN|Smrv9N&l zhpS57zvtiC-7Sfc5AWaI+f&HYD2}n9k$)MXkm+AhaTOxNlm<%&;@zPA5_+{;>0kQC z%*+gzO=IZ_v>T8-#0-hQMvX9inGq$(K-D)|kT@ce5lSbFjg9lZJK!~mTlYopIXVV~ zg^|FRm!>Bt8+WBdvIG9wd>tNs20&6ABaaqaSy|EHF+l_%6vuzAo0|*W-gd`}5Z|y- zK=)OB`UJ!z$Sq2qGz$ZvYKLI=AjPRGnB1W(r7Jl33M8%9)i5|>R##WYSxQk+qk5hW%1KskZkXH4+$lFK zXnB4*1=P^+utNU#(wKFf8lm{YLZP5?m2)lMehdvU)vVG800pj-xs4XF1Kq2`!zFK{ z&_M`v1}YiON}8dlJ0~}ngprXEnM~@NH{7774ck5V%~lEW_3{z&7qqI6_6EwEay8Y{ zvV@5_IusV>mvppp(ATe8aBIt{mWWE1X<|@O9C4~1aj~${R-c!Q$+h2N0)WKHJ2oco zvb9wY(tw~`8h=~3O6V6762iwX7#v$9Y-wd>*v8dj&t^vMOPUMGMEoQI$Pm=WPZ-K?43Q~)oPZdIMy+iq%Zj%wIhpQZ)(N#Y_$w%wnZP9*e3Mn)$={Au%`z*72plz+6cx3_)#bq}@< z2MP$1K`4O;Oor<0Pc*f zhV|@=E-o)yLjj))F0PU+`s>dS*$QUKM-su$V*8#(1&o#@Nc~umey2lLVn6}vQ~qzw zsjl>NFeWM8(!rWB5DeAfXWjN?nx3EC>Smc}{EI6uFE2%QFlc_ovFj;BV@VM@{2^NJ zko(v|0&&E5Ldku3t}Cai3hPFpkgUm{u-Ow#4iX*~!VNc@ApkB0i6bp7t)Z>m4BK8=-_(@y&Ye4E zzHeHudzkYeW?@dmRN?#gWU{idI4dS48XM7j zU(f64=|Kd-`*rp7n#RX7+$e(3Sr`DSJWRMDXvm?K2>O0J{-0_FwJ`;%8n|8-u(fg7 z$Pk>2kq8iE)zqXSvwd4r!_p_aFW6!!FQ{$tk`#54h-ebj0TdC-%Y#McD&VRtF%i>a zu)(3BSvfhu$eiB7Z~bv@X(p&GKa=tc3MgTQ;A$v1IxhL2UPIrza|aCJbY-aFd3orf zx;_5k;bAsI@@5;ncfao;|zdyXT4*302at`7`pkk3&ulLZF~*3Wirgr#%@p zwF<~Ng!J(RM={r$#pdf5aY{=A5#^xO!Gdq5$P=HLl@$ojuKXE55%OhXWu;fL>p7q| z>Ks2mS(1<`q{(y5`1p9PN;__URJS#71s$H29Mey(q(Ew=mzWT;5!XEnPS9qnNy~Y# z8o;{<4haEPXB{X&f9uvQFfur#q+Y>oB$Cq7qVw|jz*H>#*T*Oi@Hn0)B(wrZq;_|8 zHIWO~G)T4~j0mVRSY_(wXK!L|K7K3)-WBbcGoAlpOE?_vl3gzg_`rbTXgg6cF%dZY zXM2hY1`5)n5E2IV^Jiq(Ye2#ng-kkNSkiI~aF35h7k|4bU=)BP0I>m~-Ug$>N06XX zQ&VfdE5;Eb5}qdEa^hB!5JOL)!yV3%85E_-prv)i zN?g0f!O3~z%$YL))1vp`H<%j!@$vLad|aFhq6GdxfjDJkSpTk$1GRY0QT_B-_Uphv z3Fu;k;I7%w(b1g%BjXciB3esNk8xyVga)xgy243qsfTh6>wK;u%AzMtDrrF2V{U^j z^YWdY4Om`8YvPP6RwADMos`R#xvL`4_r+J(%F4=R;R^?%i+fYADR(1Tr8;L?<3XF4 zm>3=%?Yk`3INl4FM}sYF0a962O%1Vrmn4EI-d9wd1}9w=gPGaii7a$qN{}S1WNqDu(fF4O%y%i zopEmy5DQpNxpSyTRj~cku{u~(!`$arE<0-$;Qu2W-zp^~#TgFokJVt^ zPW569Z?Z?lk^TE2&2=+7PN)hqWlU03`c=SCO)SDD~O-d8^~2!yy>ZTJ+44 zgtC=51Q2Qv=vsZGv?^jW!0H5Pi*V0rrUb(TQWPdbmK8cb;^IzYQBqGMRoU2)TkTDgI=oTpcj!g+Lgu=61KF4Ny7|Qp z-}!Oo_37$eFF4-ea4k^XOOR6rC4p`%b3Jvi^yUdAHrMsm=N@8bchpZJ?9jU%H#3C}hchXyq4)^0yQ|~9P4pRiDW2LS4HWv{? zr|#g-EuTM~yqkt>f5Ksn52o3V8^Fhk2qK}@iaQ`q=ydY%^yIj5g#k$v3?}E!S$fpM zQ|Zf>FQ2<|<@>Ubs@TbdPI_Q3(0U_$Jl7)+De*hZa!$yk(my!(WDi!1r25G!Fa#Uc zh@^$oP85_fy&^M8Ff$(fET066qXj~=7+|do*^cBh`1+6x?SkdBOr&16qT!H=EY}{L z1j-y98=HIk{bwcEI3T}4cm;qk*N9{CPscJ{=Zu>We!5HkXJez}@w_^i9Egw60&LAd zg^R@z*qsWyZ*Xa;C}QOC)Y=IQE?_%pz}25#Dyk37&E+-ooMm8<^+~RK_Du;SANST1IAlYtURfj)FF!lg7~4cp8q{GZn)$OiZE}1oX~{iP6{A*5b0`ED#UZ)|N*P zXAm$oWyvDxiY}@@`qb&i+G?Mbbgg-2aWTTQ+CB62YZRCuieO}z`EKdIU!)vI+u%Gr%CxNjj`81Skj#`Y_WfCZ4k7n(-Cxs@~p zX(HkF>l786ba+F<^_G^FbCQzG;2z+t$kMgmDY?6gmzI`ln3=^aEZ8Ia{dxE7CPj;? zhdz&F8WosvdGFr?^$|c#ymG1w_`iD2lWS0>klHcOoBAB|1#v3!Ev;jl~h3_VSK1bXZg1A{y;$)Z8U^K?KkIRyoyPU{E#+VUEy@uUX> zq!P>w=m|m?*TUou2*4u@^i7q<*gMMSe0SIiT_KPH&NaN&Of7dkyI*?UJwhoEW$#ZhqN>c!iV5XckkW1w~5KV5e~N>T-pj{kr?(?{RPPm?mS53 zifSjWY#G9ePfi3V9z%77R9}_@!NYS2H%lyApS2rdi@bxW+|0mpflao)(I=M;sVtdR zc>Lo!tTT8*7It=59d?27XquWvi;RvYyU$Ua5Hb%<(~6+7v(xEG4#=-RmRT`2YJ%t6 znD1j>?~{5S6GQbfDQOyt@nPQ!V3iz6-RlTrgjs|BZ%*%ZZ5}j#N?D)>O{Po3 zyTmE&uad5T+SJJZ4t?_wnJ$@y#$$1wPtZyyvmqLLQ@ znLZYqvFmN9Dx|`my_)su6EifH=pw9@{=U=8*VNWV)B$3(fsV|>;glfBAR6Gb1n+Ll zPeV5*sJ{LK~XUjHH!y4kSk7p0ok9w ze`f&q^H5j-NECE@HFR`>_x8LFAW?wLA|4C0TS2h^DW(Em2F^NLVPT>3qep?@&VjK( zZfyLMU^V~a$5W^YJ(l-fhQcyWHJ5!PWr^h@WQ+bp^EcFDZaf@j9c%KF9rGFWx+6RIZiFwoMv z```g3B=b>`fS}E=;?jHS#o4yO6#ZXz2ym8}K%Qq?Agb@(%N_%n7k{v3`6<$L~J86~_sWMuE; zG(RI-1ulfh3@ZigP4J(O#B` z{!xpSPk|4Cw8z!SH%Vq*Gvcxt8uVQ?(5HA^Tuk%i@F3l5*_53fg|t`wg&zm6ZTxW? zyE^V;wPrLiP<^fDM_|L?-$r4(l2JC9G23@U!o=_+dY+!100oMY%a1oq+7Md!=jkFQ z@F$!K#R%9^J@$_FSHbvQZ>hJkvkQY)a~k^l?l7L_xzqfN~HJ z3I%#{kFFhld#4S(j2_9X&d1KqG$a4=T*f$(p@aQaJ*`XfnISn9)!VOkq3|O13Yw!y z84SdKzm=rJ6HKG7fu&K90I%-9om_LEHc$w6UjOys8n_+((v!W}N!R{=7yri*{--Cs v8|Yvf-VI>?sccAKRfHe^x7WNPj!ACpX5m80HJafqHk8^8b)`auyMg}&)pgjg literal 9904 zcmc(Fc{tVW_wJT?$Sg7swFzZRW*M?mk`UX_HV-8v^E?w}OA<0ANirvy=ggwW5Hb%5 z88Z)Oefpl?`Qvxa`R`oUx!&Gu+k1cZGpx0qweI^~Po%aM<|G9x1qy{asc}_J2ZbVF zK>jBqflrunrpn=k#Ns+e4RwtC_o^m44u#@IX{agdd8RD?^fLMGc0{^57--TYD)2C@ zg(`wvkS_s^#`4BioPRy{enQ`|T#?GSc3}h zZ$FY+&6$$3H<^w9sWKJ=-lhmn2idy%zECt*Zf{%ffgw2P>^?7_thJ;7UshLJ5T z>Zm|H!LaOXentx7y!`yr{E{?YUS0wM0)gt83kMR#SgJ6Upuj*>czajG*MSp-9d8N> zs3WL3R9sv(&IjO(jUNvW8$EyVVyd{6GbApKAv1JtvTCJP)xlxeAz-YxH+XZ?^?5=9 zpOn-KW$}$~U%v((?r&4xJHPr?bJJUanq3+dtxg!P9CmnP|5lz532J9#$Fj1?)$^jy zchrgcCkVApV-2B&&(2NqybD<1pVlio^c<{H6&p0sTjm z%$v+#M@OF=9QY16-_UV!k+^_6U_uq$?bBNtVUEf*;6R~DOR2BkIN3b+MVA1@#Kcrq zO~gBTOECWHz0zP~%Ib3rrA8-FR~@chVHAeqnVk1RCNlCzROW&RA z`L!P$D{%?@s$LaxsMX&=_pgOkp7Puf&z?rbT^s*PgK^L%EO)+dz5VBB+34H81v5*_ zCPPl$)NF`MN!D+}!xZ`YBfEv_nL$|L7QgMEq@}~N1rp!y8SL!o^YHNz?r$xJl$Kt+ z@8BSJTv}2Rf-9WgtR|^FI3TgW`S~dX{>cx+yETzkpNmVQja19yTh-CkW%&L3_t&AJ zN8#Zll9w)J=H%p*2)|5DHotq9JoYM)oLp6Rn3KId<@M{=%PT7W{#so4>#b*CpjtSn zn%nPwO3&V2=!rUCfoUCeq}mAx6k}s!X=&-pys`D)zR~IAFluROJ$UfoS9W^1*F>Mu z#Eqn@SvueQ`x`okDbH3ulN-Z0o+U+5amX@C*mhUXO7WzL4Y4gRFKg%aL-<|3apR<( zAd{ew(5+jy2-w)z+GZbmdU~48w8cJfaS6%I6=LV$5c6C$gQdCcj+DR=JA3wQTW4p8 ziT`ZD>6vv_RDBbT(i0{jeOH;Dfq_%~{r&F>3o)vyg!k^<)4p>j`uFd9DRLgv_exo1 zQ>n$1sM6BXR@c{qO)f-~m0g;7C?hN^?3mQrddzhw=G2p>@DoRA;!oq(oGc#OIr}a* zSM`Z1RfpSbS&cq>sPZG{M+B%gD-VAKr>Fv4O;mNY{LI|k>!KpstgI{+aVs(yPyGo? z5Gj{0t;>9Om?k6;~>f@h7Lqqj_ zeGIU$J7d|!#dHvR+Btgf-o2}D2qC?tuODtdSYQqYVYzsZ@dP2^7xVfE8T5Djm*;Y2oFYCip&ZZDMKk zNl}pw96^HewLs>_&3;EO!Sw_L1Z9Ux4)^b?-MU2=6&3YhbwU{o(l|WKs;sO`MozAL z^=jLl>(|;K>qI?%JbBVQpBQ)N*I>VfJqJ5GJ1qx?Un_&#&cVmOjvK?n!%ySlLUgmW zCN~%QzS>D5Kew|JP*G8tn4P^iv8$t_^I-Pcpop7^iOC1~ot+(Sd3kmQv}Ryop;H<4`@h@U z#@gC}2L}hnoL%&hM95S!56^7)LV1~;oo#Dtt6%8Lc0M}r3JnY6k(FhIpZ;2R7T4Y_ zPqekQ9n38i)!y|qMPFL<+~02JPoJ^E-P~-grewarefBKz-Me>L#4Lz70}h-u)8tQw zkkZUuQRL~#Ei7zpkK=cqs3&+HA8+3KM&nU*^c8JwT9Ng?Zn&ERDjsux`%~xE<`N@w zD&3b^;mpUrd`TfBB-E8CMtJ>&a8Om1+@c-YdUc}VQDmgDrY6<<_wOG}eI~Eo|5Ih> z>k2Sx3ov3o-9HUQAuz$z|Fol{W1@sD^;@mf4%sw~$-kUMN(ZZisEJ7aG_#bM zF>wirwZDU=O=hBITT4F@#jPoX&!0DJ<*jo$+;dBjc0N(%u@W8~ee#qrHgIjawFK{W zQw2*IB3qDM&U}hAPquqtse91c?edKrJ&UIBlaS4^)t+LFO-(C1j!(W)?sYg=ju_|W-sg(WTGFtuguK|r7HQ7le&t^ z2{;N{e||px^~+9>K8QY2?GYZ|GFIc0p)K(A>C?b3$(JDUmf+FSdT52G@$q3YE_1J5 zzdj-7@khyZ7&ik_L$c^$D6FI$>p+fbu5-8E#SyYt3XxUZL@2J^P!82n8zOzmT2{* zq2iFJ&zhPqCCTO7moEGG$jyC8ju;&^8Hn|aNlS8Vu#>a^uMXAv$>IEWt!4Qes3My! z@^Ce*t$2>6C?+9VX6A_OH18;Uj9XC03^gko+uGh@VP;kq@yI6!B%d!{4b8~W&F1Ci zM&;_~tKe{)M{6+wn_ep{x@saW{_|7Kk3Hv-Y$yKw!QV)hx)n={3QSIBlX4gig;=HO zk>8-YfU9nkrC-rb8l74CQGzXhz%w;H9h;aqlzAxjXn{@Ajuwelu#m7xIjwC^nl(qH zfnDjr+(bjjP?0$yVu&YPN@)dIJa?j^qnnG&n-HH(g)!smEn^f~Wa&k;lTbsY_RO`r zb8KhMo^`7zIn8s4P03&W*|TRwt_wPnQc}IyY{9rWGbqPHpB&f}_m?CD>3jKbINZ?3 zyA03b;)*?1a0AYX5{u$Lz*LaBio7;#ps*F;Rf*b~nh1?Q-ETNL+%NXqcZbKsN5prx zMx2UmdpKdfZ*o#^ZkpV=Q|$Ot7B)y)<-Tme5plXiI4%Ng5z`_eJF3TAajd-h;Oi^ILYKaUtC6<5r!ZG?)g%qEz&l=LV* z_S@f@>NuMJ(r@_VrB3|Uj*-K%s-<=BPM(}dkqOr*647z6o2bp>ksBlB%z_9W*yJZ=JoH~6v z5_aOdGZ~SWxhWKPKCY`v6L9BEY^|@%qClwG?)t2>-<~8SAw$fO?EVp&)(s z827`7IFX5*H?3v&+5U5}y~OfHfu_{`p+Jb*^78T`yh@t`lsPy|w$Df{Z@kxh`0(Ka zC@Pg*woO$`}gl7zP>+NPN$w3 z>&D)lsG0O8J6kook4I22%8gOw&pAOs3Zwi}**DW|IKm{*y6E=yb|f(h3v139^P;;N z8&5oD7Lm&FaM!e$O_gyWad2>e3}?#`_SBJ*h9&~aq_?;1qCkphr1Q~u3UNd{=JHt8W9+D6KiF428xfiiAk62wRG}YO*kxT z*-D3z_;_074X3qJd2vljO+Fi7Tg@vr?hjIB8M|*SbbvN5Iv+cky?d?G!A%rTbs!6jz z7-Mq1rdy&zYHAb!vj&HR+@%=``sg&SsF|*q>PDp^E#bGf*?s3%*wv@juLl>Me}6xx znJg6z=9ZRl7?wn8sdr79+jAV972=Jio}L&*{=`H^00uj|yRgxa|9r=;;LR8t8=GfI z+gQ=ob`{<}cYA`PGf85hA_4P|&?p^3n1M7$DGa8leViF|<;|Lxs^q&3Y zD(mUvQv%^tw+4_%E4wM{dIH#)PB~r!3Jl~N%>E17I5d4EEep#d0Fxi^y?nvK3@Vy; z0E&G0@B!QU3!ZILC}xzk}_%V{48qW1?cxJe{0SlM8k8hr1uM`9UHu=~^uR6|)t% zJY9|V6A>}3rBYI=s~Ssp^6>`J-P$Y<2->99x57ngnP3DX7*C<7s0eNopQx^>ah_}> z9uTX%LTmk6KX<|EqQwi?J+PbxS@p8jT20_oP{_!VG+Q8a3AHXHWPbQSj~v$LK2v^4 zA3uJq@9w6yC{BdvAxXgsho`1;0FK3CD)`3|G=S=b({OdmlJImk;j1pj*gCETe(9|8yFZ!Uc6Y0cfJWa#A*tFxp;EsmoHyV zOieW+!%54@WmZ%qxjjKv0d!F}iMC{JYD&erkr)^}KR-V>!n~!uJ#vh7PUZS_%5bXF z|9*!+$Bs9J;A7a>($W$Z!xMTM*kb0lw*4NiIkzOcQX@fn71Yp=AIiD?AMnlw)fX62 zwFdK->Bz*~K7IP6hxW_pTaewBv2Pw+R z)|L-A)58jO@Z1vydT~)l93=;3LXUlrL8~SJQbJG#GQW|PZOqd>SYkPW=Li`Iql8}r zgS)DsVa^fv)Zb%g)x05}cuZcRy{89r=MIPPNfP<>X6DuI_KMjpN*dSOeVj(Na6Y4ecyV+2Yn$ z6x6lLTa*nAJJ!RJsLN(%6M;5-R@~%n=U38bt3kGnI1C_IH7M5x!VU&|I^u3o<$qLr;`Wo1=ZfZFFCgxj1of3FPD3HD|22A)3clBD(g)C-=9GLaB&& ze3vK-w*yOyiY|j*vP^J&qst!$f2r}(L1q~7+S=MGDk=($kMCC)r)Kz{5$Ku*I!{;_ z@zV0L&*4Rqsr}i%o$M1W`%8a4Dj1V#bv>4cFOH=dKa?0th=je}v$r>dBb=0LP`Lj4 z_e25NYfTU#I*`SF9~e;KjV*#x4I3aOCH?;Wdl6olOyc6ji=fSzaZIPdQJSoH`I6P! zrja;DFRC*^qy%{YGB8m8Jj>BI?v6L_-$x;K$a(*MYSZE8QRBY&X?agk|HHoq$f2Ku z!F@X%OgvF9sBvE8)qUwRDH8(dS^O+$6#mY(#%t!uK^Tdf0Pt((pKp;dbDRDu( z!hCGt;pMjeuU}C~n&F`Rd>S|8yXJW9A_laZ0p*4=Hx5%SbO zAvyW6^K?rw{u^JgVBExn+04ugxb$kBu3-E>44#Q0x{U@jq6)PcQZ9n6*fPI80;(1o z9{#U13o_8q&|E!353=nCq^iq!{9!*sPcAPn|DUQ(j?zq#rnBu%F-InhOG~>@ZhJNm z)oBF+1Dr5RbyilGlqN4aqr99YKR+MxfjLLkAdxZp)TvXTp)>>WM({`v z!pMh)unkM^-T51DP;;ZvB&{)A7^rWEU5(&XD@8g;379HO8~Lj9b9*~1QL}z~Jm4Sl zeDUmAFziBFMy43w%O8A!0re;*rnPL`|MAwA`@k)-h2q7*Ld(H|JMghBM^+EO;!O2u zL`u<0m2Cz9px_P`DX*%cyf^);fXs$wsLE@-oxj++^Ard{6`hCX`+P{0`6bx`(%+p= z=i}j-#0PB8DP#KqS?nFiOTV|?nV zX=AU30!63L(9m#Q3MEqkl8+EuyKlT&={HKO=&etTmg8DmT1YT4oYrt^d}O^h1we78 zv)XIyw$WWs$>( z6k>p0U}_VKi$di;=U1BFKqh~2-Z*?_#>%APVHgCAqswdu)BTZ8Cj<6J#<7^ml=k>Y z=vLTtr$qi;1U&@!&_JPSjA4mQLtkH<8)J8pNfN3a;fl|m(GPpRT&HCTp#+KNe1i9M~xZ0O5Uyye=ui5x+7j?~OjThu7kODYq6gFvVMtQ1> z{13fBL$o~DSpz?_#hquEXlWP%Z4q>)U^hmK7Ce^{9^S^u|MHfIyTDM4eF9XD8T8pJ zpkpsef2QjH{@rYOxYTp|rw7le$A+y72IxCb-MlyLy3&=ng!J@4-_-_Egr2T9kaC&*`kH|)G9y?NAZ!%wP&-3E`Q_5M z8XS86ok>zq08S5;IdK8{UY$#Jdinj_hKJ=T>k}wor7LyEii$@&mj4omswURmT?!Ej zAY{4dBSYO%2rQ+*)6#NVpa``H6;=mt%=eVwRW-G}ykzwAjlv*$N6z#-tYIq=1T*aN z!{Y0Ev#vV>RX~J?3fBdCL`2Fnspx&v!@m&q1CdrkdeXgrp95u}cb^4a3T9PuD>g9-TPy%)k)X@FQ*{btSihHp9Fjef^Zf$@7PB4kJ55|i zfgqA~&-HQ^DUdbxa~Azf-Sz>YG&Dc|{0k9NJg`hgK)Ze~Tiq02wjO&WT)5LnDN-GH zTEfO`vN4SPOjXcUAk!<*Dfz(40f&pG89F(@*iXFH$wB7*+}dh6kY|`TEo!?376Tov z;n~^iMV@OGRyH;tK&RoBkYE4~tZre!JJ*%eBugzqWx)|B^7ckD2{SWuo>3_UNUIIc z&*@?@yz=po0g&?m;sByrFG^P{_h>rOv$95s)b5-Bay?Y;%m*4b@@((GKsZ1jUKyFw zroL;$Ozog$9dsXVkH_R2m%V=fp5pPT^RIKG$x*LBI)T3%XFC&$-4^vn1C9^#_|~Z7 z+c|)?RR0|?21brCF<}SE5@aj$Uqi)~kYF%sYD92YXma#AK+F0j7I)#18n66DmjoFF zMdSGKkq7M0;>QPTbR9qOE*GHWlWbBGAW*bax-9~=18uGFn?#%s85Py%?o>G*`bbfy zG5O&)!^b8xD&;grm8BC2SzvN?vI$UjUS=!~UXY%k&6k(N@Er)KkQ)j9 z6$op z{YEM)1HYCUs1=}uMD-}{Q&FB0u7}Qsxvg!hY}yZd2Zv@50N3C!*MndNURD8RWD$!F z>j$XeG8sn6{1oE*yzwWq&yVOl57t@{y`PFx zKG99H;eYQ=U_pcn4i3g*B-pBj08i6^UqDO*xI~Dm-Tx!!>FuqeUlD{{4lqRS0%T~D z?VYcB^@^K_n7F}Alp7SE2pX;y43{4f7Z(?jkQ+nESO5N_}ARK_PPOa(edqNw=^AdFbdl0-&6@hyA%+~O*q`mHi(hNj56{eD5Z`sK5Q7*tH@ zpZ<7g=u1Hp8MGNb1qBWO3@mc)!q9RPb6>)?S$!brah++SGpTaV?;J@@Nugz8x)>Pg z2#vbSpkjlD9-ELrinLQL^`I*kLbCS9!?KXv`TV0NKjlZ>*KI%jzW5iBl_gnE>X9&Q zU0TKz7qFlvoTg(IsQV_ZXNe=#f}s^LKG5O&lka2p1~f%uQ&Xd%-W-6y1_T&c&B5WJ z7_g`(tQRSCaz2i)b3n_SSp*lF)eu;W!JEz8RBD*~eOppSZhBOehDSt@VlY1s&KR`X zpzxLnMMp;q2nwno?YhU0W0R990bhaDdCr_6;^E=(+cRNRRKfw@fw&_F`7F2e>-cWv z%hqH;189WXNtT4$%Lx0T7A=*VE2fq{C$;e!9B48rHSvd)F7#W&avaI+UneAG+;%NB0Q( zJ$pLnUon@mMkEL;U%5go$$BHb@U&U!M@AldR*kJLzVp$_9U}HGXA&rIX{iLAd0k8bRF* zNnExp_P&!-gi*e_vN8emj#oE&lq{i55AF>F9tfRX0L~M0b0n88T`I4vY=q`FfYI>e zWaIe@^By`tDfj~SJ31WCJxgw1U!MSVR?yURW`2GeYWUK_G@wjwZWn>oBX=W!he3xO z5yqit1smmuYh{N(uNVLiSjF)iHRa>uJIm%;PO}t1tlw%v6NLaJ=r<_$Irro})TM%@ z(Hfua^JNRJl~os>r_ndcV=x#Z5)yYD=lJ*Up_P@gK!IL^?)Brx54(H!g7tI5)H9XA z%FsB2Y;I<2OKNO9(tQl|w-Hj$*x1;E-Q|W+CWBJe+qJdQZoe6kJPXG*U;i=W(e7i` zXmyi1e-6MD7R4+e)c~ao1Xc8n&N+2FJJJ?}`v9Zj`@^O7sUjPD`}3`XUL2&xRVAln z2TpRq`n`i`R@_%_wOZ7$-d!+g(OD z_63nX=Em*LK`)=JrG+h{iM}eC>L0l-aW zxx59DU<+6oevW=rqpZ_I%`1tsE8hcHjoll@OV?eRRp1iLnr9C9L`fTI+ z>Zh%6HQvc)4c)g_;XcGh8R>j+C{fDcG}pIZzoXnKDJ3f_r|xS~;rY*?pl_ai_+URm zB_l93H5G}=ZV3WMcC3QyFIs@7D?1+l)e&E&-#z$v+iv4u!@DE?!ph>*RPXM<#mByL zqJF zA{}KT6qIm7wXfdkGLDJ3&&G0(+-|ALSH1Fi#|!S&>lSvEY*_{*LbnxZIAp_dxMG<4 z3x=qbCxo*bDzXQ+h|ozaSAGiSC;h&c=(m0-aVEI!>s0Ma6lX?_g%_`q$0sbT|ENH& zfMGP(^F%kUY`Ocvg9pVza{F-PYl|md7%p=#en@*TaXTI!nH&x`hbj(paTXtQaCzW= zJ2-!bl{9pN&Jc$=G%SC4WAE9sXVe0Z@onr?|G8F~r*pDAju~bMT#$mCEtptOGnrV< zK-Ns|f9YT)77tt*{tIuS6PNg3Z`W-soVAwv=8cw)PH<`}t+T6Z`?qi87Co^c4GrSP z4<6LnX;^l3b#d_U1RL;Rb@0vaxR+N}2G`UGOGrv0CleDBIj&z<^zo5=#81O>{rbX> z9~d`o+?bf0d_I$*p3A2tf9DRa@9|Hn-k*P({nl1j5Zir&9pSY3^@?0+3FI9@{E zy8C-DH#<9fuH`0vP;4yW-#`5~+>f@I<>ci>SJj0iB_(@%du1#wEnEDymX~EbJVY6! zyu&dsQ?_<=1mzOCFor}#78Mo6r>7HyE-Z8tXgqrCnN@0sUtG+ZEOa+GKHghLPmhJ2 z9lNaMTx@)UYkjiH=E;*M{s{^AQd0VS&kPxgi&MAv3h5c7(xL(b6X9HImCbm$c^6*k zl&*ogIw2P~_b}&HXGlJkCdc*8d&l=gsKreyewy=;@W{Hl-ntz_8}jKTyVFdm`bf!z zrFFtojXgSQApK34fEDHZ;amxG((ce0o6O^P*yiSX@@xWf_wUmf8X1Xh>fJQ2pw`vZ zMfqoCEp6{Np>e;zje-ZKU0CzN>`!9E-j-19GQSx%!&O>UMe+0J&-k1Coq}CVchf>{ zdC>icUpm~Jb9}dfKTws4j{5QI*YoL;%^W^0*}Hcy!}W*Zo}TsN{ub8OwvML__7i|`m5gy!V1yh=*yn4V6nVtA`vIJr+9 z8eqV4+scZ4a&nUC*l6i?<0O^ zH@A-mv#=}iM%OksZ#y{&fF%!)j09F!->R&t%5G|+Cyl^({P=NsdAWZ|3e7vUH(280 z;`n}aOS|-wV`F+he*czdNswVl$a??&G9@J?wWOyYtR<$Akbkm<0E|WfEFCx}6`N$l$?qi;D0F2?>*LT3|^? zNSvJCbGDaH7Z(YUWB&8!&%E_&H$C@u z?{?UXO$W#fVG}SDjm1TgRuQM120N`YJKe>|}3cdv~{E{rks-l@;83_wFUG z9vrxxow$T6uu#*uSd10xsJ?j}rInjHG;9_YMs&tZj+L9=LtR>0ii3woTRp3pA?X#m zxM)7z;H6_wV0f$pex}mh$tZWjQEvn)2xhy9o`< z>oB5tCd`rQ|w9uebUB#St*G&VMx zK7JgEj!kF^tEYk;$$KIBQ8%t#;gjbLtWO16xds~$tAG6bxiDSlRO|Gckn*}=u*5f zbcLSIXnMFsSf3m^&2REPs1N?}<5BXPH&=Oh@GtgFF1jX}qf3`|kGFSr+Fo3{%S0KK zkeV84-+YXvT}Zw*Ue>cZ`u_gueOwGQFfeRv!Yi$X_wsebe%#b?#x4#NBwrvP(a@yLY zU?1hyqg=ruA;XdcJl4;ib;U`ahxaGlxXQ;zOebs?oSvR;D0Gb!cX4r%&$5ra*rX*R zPko;}Hw)aUV%Vx$Q)j0F8y8ok$H`uyVU?J;cuH-p2pb({Lv&rBWFGv%u zwX*Qz+J9VYVmESIpUBpZ2X(agzO1=DmR`KvtOF}e#1UxM%e7f6gfF_5Iwb!J)M`Bx8M3F2@QorNhBS!@#rK?GbKP=qWmKe;09Yh1`JbpuP~`STB0;-dsu^lU{~cOU{w9seN136W!E zV>4f9520Ou@no_+Ky{Ww26bw8YO zTUc5O3J%7o+A3$AY#SILgx5uOT*Fp#)afDi!*o9;5DT#KSbn`0@n$*J@oCb241T9m zbLC!#cHVo#N@9m^rT7J`w&%gCVA(?kN^zGdy~?Bwuf=o-@Kqt!_+AY}y=k&uv3P+Z)3 zIOnGk7b^(5NuwJ+$MyB8DBQj+$CAKNna+$I*%lnZB|9=b9gW-{v_p7YoRy0#Yt9*> z5RZ=BK^8CO#z|=(brxr1Mam+hwT+CTt{YXG!fZ`6xRn_a#k17+PfSpI9d2%TIrrMP zoV^=z4W;JrWMnJCh)`y2KiHV60PC}zsn_kGCw)-vvZiM{QGSWh_b9u4F!olZ%zrKJ2NDmcft$#FA?{f6xrAGO)saq1he+{yq;d5&4-BQ&dc%dPE zMh+7zFJ&4@T?#&wnS}){)AxiEv+wR6zp9R8+4!r`cV&F< zjWiO4jNZI?lQJ}<$tEg#f9Xt1^6Y(5TH5E}&`^Gs1m%TidoF7Cn?yfrS$@%rPzb!y ze1z84*5>>nu>g4q974kG(gzLr=iRrOaL9DWY!mx`PE^@Yfxx?+(h1jf@VnMD|}8};VU=-&c>K#%t&>xew*qHu&27ZIuYk( z;+gL%24=Rl(eh!0%#aBd+S9Gw%UZTgF8IjssluX%n2bz*u8O&sR>+#%+}zw@ed1=` zxcqRK%%kuo1t3N%LN+k@bAw}@ zOCh_Gc;k<&ps?@~L@Y39ou?GXWbEqxgTI$Lh@`)19tQtife#p&gFO*s?5`$Eg*8EP zUZh>#KqIHD&nh3H+1Lndu0uixflC=}$#;RLAWm+y{6)jU*Uor1Z$f_&cxCX$$DOvG zKU)12uHOu!>4XUfGbHa7xys63f~XUpoUDBJ`U{e_-@7{(=W`eG?(QOg|Nfm{TFTw7 zhK+V`V9~~hudV%er_9eoj`#P4mVWHd{-_GqXS_%n8Oo}Xg+ZVII zO7r=hdwX8NT<8S1b`w4$zM@fW&bXKMFU_d1ef=TbdxAXMT_r4ZY0b{`t5fQj3nN~7<9l|(Gf%mU!2NWjOV-`x9x8Q?s0^Y!ZoXWiE5@Wf!7XejT` z*wR`;{oZ;l74o9bf8hKY%+we>zh;O?qhx76sn{!`{Oc3Vb@Nwgoj(`f&-wSxpY~S1 zUP3YJF(*IzySl%7r~4(bImL+mx~YnaN_;{>z@YR++SzSft>NF9ezDQf_^)2Q!WS*I z=MDa0G4fWQ=#*W}T0vRax20lI#)g-mp{}lNbd&-BL3njsH~k8|gy#Vcif`1J>zBdj zfSb_^zkXo}3)jf5L2SPw=8WTawq^oKe!eFyIJ$B;S81ut?~I#}UMv`VEurU|z&CH` z*woX)S68hSluTOF#qaGb5ASv*y;CpgN)wGUvz-=LH>Q4SNXCXDrlt;u{1P6ob*`Q9 zobueKY{y-q@nF&>S=fFyUI_FZ24nz@$G_gB+9eU^HNgkvJ9pah?j{kFlJYN$`1$$0 zdi`1|^YZycXv;;*=|e9mDv7Z4%U9Ljw|DmT;#|6PY5UI~*x>E35Uoo)V7%y5R7qxs zf}kGH5B)BvDk>@<3A*}Ch≫$2~kcx)VeLF*!L#fa&~Y!qovx-Y5IC zn+&se=KL-sS65ew>FG;8RpV$)Wy^=bZuTdVI+DrW>4Wjq;4dM?_MQm%fB2|aSyDhi z0HwL?!_CFzurhKDqV9(gcB;&zRT}js#AhxnEFf{!%-&wwR?_kp8r@l;wuVjVuL5q6 zpMJ;vCgcyVr4jRriis`W@H@VIzIAaPx29*1OJ#;gOju&u$;yZ|JshnGaIFp-lR}^& z9cS>k_gGZb)5Ss{P_%V-hav-#V`2z^*SL7PR8g&XCkqM*Er6)SxqNx4ohlM%K z;4yPuCL-#A+=rPms?JXFx)?nhOkH4RCS!LTsN1oKRj$A{5(zE zEw*%?xQUdKGSun!fa6SPpEJKIzzifQE;gtLgsYmVIXLiv7TfJ}dhQ&e9{LFsUHOw? zmb{D`s^QgSI2Z<%Pe`=y-Gk@7CPu@Sbi<^(NcVlMT@dkDiD4BUeE(HXkhpA|tfQ-o z%*&YTHP5H8J&|M&I8~nux60Q#zB_mC%4=#8OQysvUjtbKS=Mx|W6+x3t9MQU{F0LN z{jaYt)f31GKK}l}vbT@+ubK0tN`Cv>cki$uJp&tEgxMGD?(e@0MpIB&xKK}#x2%)= z=<(yl{c)3g`71f&N8VC@&(0E3Qa-^QZ@yN^({v`70Dz)5=UwaBGl_ zjJP-=*uxlLr;tv4r7gGR;^1fnKZETfh+;nJY;Q;Vny;Y*p<%p)L)EICRa%<7l~?R7#T!IY>1 z^UW+q!UkeLSn4I3dSWvtZ4uuV#fYYbU|78byHJ%#x+&^NjL|%tcwpH{L%Wx24dv{ z6H`-=#@n1Zg;$BGvXbI{{`}e0*7h?zpF=n3P1pd$QC;wOtR;jm2L{v)@gs;OOnB~0 zbtt?a_kU(%6O@vY(!TZni;btYWKlk_Iu99}R<8SyL+1ib#P2YjtbF<}75N?S#W~sh zjvb7p=-XQQ<@59N@(&-hpYTRM^1Cro|B~eC)2E<4Ft-18zHDle43=ZY3z4g>tp`#O zzC?&iLvv#QVHc>O)w(R*->M)QPJG=7;}|;@VJf0R5#CXR|CkO}-x7eH2_Eum3x6 zEcnjWj87mlugU%Moz&_%T;*^vXBXZlYRTK~5-5N7@P7Stzs51~*Tg<=-@|jYcLL`` zZ9B|1%J%Fv|Ci%$52N>*bl)@m+41=4k`PCPcka&8SW|NtesH^2`JV?!3TZqNt>V{rZLp zAt3yaW_gA1W_L&4G$|ii0F{8jMkZlOeycuwWM@YxEiEnKb$}196#VYjK1K%S)G=`? z0Oe+nAHN%QlI8Arn-mg)35GWQjJRDiLILga<;$g0)GzeHB4n7LPJ%JG zgfBrQZ+Ua`7dR`CuQM|-rKP-+0$SK{JwY-|LlOBYBsrWhF)@IJ3|(s0WGcl}QKoy# z!$)U_b9KlL_zO&@xb($4PQxZFTi_wZ#l_o4I~KKdb%hT!?(4*@K9z5ZjKqtk6~Y4? zQi)Yuw2v6Sl~ouRy;;v{Uup&n;xsBNEX0KuV!qOCUobDEA)^6BJLPucsGEBk=e?ZfSaEM$6E!4;(;r zEszO!clZCCc<8kuL{3HJ1b*b|CWw7O>YPjzKbq&+KJg zo#;E&6ge;U&Mf$m#p<_0!v^0^{owadt>fcVZdi;>9GN`tt8TtAZF@~5s8X7$SOTkY+6{BJU zj1HtRSDrp6>Bn=?k&%(|N1sSbjkHq7$}XM$5rO;+mh>t&_gZ5Aql1GCU~u&e2@oa09P1ED4JynruUxrOYZpS{Ny65; z>%=a7@mIjw#s&%RW}{ zJv|-P#;!wV5NMX%>^IookH{0)n@Ac+O+!Oa+XN`LhtV%OIhooU)=JDrn0Z(Ozk>-_ z6QkeXo%q8#i|H>ed9ENo>v$| z17oomqJ@R&u5>h{5#&#og6DmE?43eI2rO=q=t?li5F8&T)Fx|RE>3u$LMGY6!ymv21{T*3_T1CNY z-NUBWS3k+t-w|`2*r0ahjTS5swbItp!$Q?Mw3V5}?-i9+cZFQ`J?kdy?&v@d2nayn z_Gt`hPEJP{5yF)j7#QSZ*WO&-pgO_4Q)Yc%h_}*YSy+$m$rsvkg4-B%&@MncjPwE! z5TH=mLmopra@hjy9riid*=DT|AO5X5ob@7lBW!jum{85l&A-MNuW;i#;CBdEsk8K|(_lVN_!`ED(vy4iz0ousvXnTnKC+#y@u-QVdl5 zH|Hm_x3}Lu_U3tD7(H*w936Av-TS(t;tIDOPKEQ+n;`HTo0=#}ZJs`j0Ls188h{Qt z#)!gNFh1A~E>f%j{A%{}De#cl*;z$bt6zs!#Ru;}PX%_TayJo8UX#6W54>tmCVk>y#R9vzu(OjZHZ zyg^Dz8rsOCghK~D80tznlnrX8@PZPH`y@&$}2vPu04LO%%Z>i?Lf9^J(ZZ>T1 z?dcpnqGs;v1JDEUYzH73M-|mH9VJ*8vZt%{xDnOt`#l8lT@==@;OA1CB(wWE27U87Pfkpo#08R+5D{opp+E|$2+^bwX@=xAGG+=1HDmJ zR`!QFzTDKZ7mTAt#_>bF3x13N2^(mNuMS}fR1ccHY2d2}AU04|IYYNXiCO&_!f>-Z zv#yA{;v?ug{nfRzG)XT?r8tHzU~?SGSXtxngucCGjEr$msT;DiK$}1QIR_u-r|etu z-5_2zf)dAgPD00O9HBX@n$F)7i`%vP6WsxVc@G zl478spg8K0KEDJKBBYuustcS>&nwP*NZ==ME4)^@5+#>fdiW;^ouZJPGAd17EELG-1HsYE>d~*K9cfgB&wDKZK0t*>7U zB9dwag$t#VVJwXdi8gFTZ0BlUXaQCU(tyOM+E!LwJvOv~ z)fm|%udf>+Is!Zz?cVi2eSEzyQANDv`Dz?>0If9^UqMtIT1RQO_LkYneU=G{_-jBQ`PI z@%C<%#l|&lg{f!Y0Duhaoa6DA)R66M2hZrx*#3cm+W?}GBOso+s+66!vcJIO;fujd zBtLs3h}rbn&D2wX?&_MmICqsMN zJqO<=6?t)>_-w{_kTi-v1%In?mvD2o>3udY?i=WyKq_=HY%do8iCAMsWN#(x@gGUP ze0XCko1+7gM!$>GxrCQ5gP<~OcK-MH=s&B5?;eNC4tkV%Pf2sj>7+joa#7)=bUK1Gkz_V*`UN z$=Pmbxu|`A2$kUQ$VfR`Tb_Y5(IBWwWmQ({pr?~6f*LXh)Fk5c3*X$r0*Zi`0Ww+6 z^ms3_wUY5=5``>Het~-iYcK~X93CFt3iwjf$f*B${v!@3%TCnTGqSL-lt2DL3&maS zpZW7z@laZQL|jzNdCCta0#(enZ{M~;9|I7}@Te$xs95yRyr8P8tYm_!EF5%jc1F^D z5TZKGZSjE$H_6%LHGgvrMq!{`k~*44hnJOSd&vw%KJXFbTs#o%WaZ@0K=MHY5>guh zPgv-z{52|2!I7M;!<1VS)jB)NprxgytfCSKZ4(9t)XnE7Hb^eX<6@KSd%7X)eYQUi z1TzTELTNlFCkNDHncf3D(w~~i9R6QqE}^iou@w{*7offXNn?lCR#O;w>74TN_z*h+ zC2;9b&W=FD3G^U|fM0`40MyNLeFm2p_F==*-dF#g4a#_7cTgHBnK=T(g)%QAq@{3Q zc?XAb!-n4{Dm;`|g@h=i&yTLY)#e;4Gl>K0{Ndzd4wDu^NCc@M2?xkwE~E{kw8BHlt_?9nP_@(N+@$XxVY#R-GXX$qsK0v{j5YKEDb!>ocAsP zG%5hv4u*EVw*Gz_g}^-EK~OdZoMcLkCBZ-hX0?5A5SpK#-#usYKXrfrs+L>GBg59d ze?MYmL>rn~U@hCg{2|9fKs)qqVMfs6KPE7!&U^bd3+N_vJ_XJG6oVlBIaAtXu$Y6jBG^f7avvu4%XFOOgi;G@7o-)cUCqy{DoB{Q_Mlnzo4iH_M+ZV zrqN znHyc2=tq>AS}rl=>nNxKCAJ4^CKmu##RMt$OA%P*p5eh1q z(ptd)M-WvDniGYBZ3ZPTNWua^GU<)u7 zBvwaXA0800@{b=ofU)qou21;3?IT-n+{DlF=PqIX;i0b9O}n9+&qaxJbuAV)9)v){ zC^Wjk637nAC;j)n&GC;PJ_IHvj&f_{SWv&afs-NSGkNF{Dj9LmgKHTeOQLgcU*U0VA$Q ze&bsqF;tS~sfRzO6r_t$NHEwdYzmtNMMd>99&Sr?c1EB}t1)pD;N&Vro%+@&ctEq= zpTk3JRIp))`BHLBvZ=(S+kAmB2?VbVc@^LKutuBVL-S|)@e(->hEH2kvkUiT>BE?V z3Jd1nyTekhJjl1b5~E8JnOXJE8(IKa0gHe-Xi#r&W`*+#`ox5Dd#`dVE?H`Yqs1*b zdHbNArTw#q5r3AY$b?oAzyZ?X`EhHQD5U>5`1nxtA1@sA`fpu@Y5Vzjb;d}!S4T$& z|41xzPzf4k;7+0|8jP#MqGUp~4GkU8%L)LwptKaBDTRfc;EKb5Up78(zo*~0KW2oq zMQU;=Kv|<@YQ+i^(jRL}2miNr`~R^AKTF6I7uxf;erZY;DeW5+ zDus#u?>E~2m*#z&R#Ir;w`ql1$^Ue_t_y5u&Y!Vc`JD6cS07M!6x9_<r z_TD${8{>}q-uoT@IEQog*=y~!=bH12xr3AxB=N6OUB$q_z?YU1Q^vr!K!W^_jR~KS z=FU{X3#Ngrq!`9I@-MwECk6xK7KXH#i0aGt>yvJ(&-TvFw-|i8?q9s0dGYrBt5O7k zj>&=XNl!Fi78z?ydQdYdd*c(7cN7fO1;?9A+Qb-l93?l`#a%LBBw*yF(=!Y7k@9v1mg&QW3C4y^gY&Y@I{n*z ztk^gtbbcaIiP&^_O_^n7s)kF{ntnvq5&jjG1LS06{UwHyzFvm840|7RR`VEg`2Ii6 zBibZ}txxp-HROLgwBW)6n3KjOwH0Z>T+>K*hh+NyJxBEilRnj7+;&jy?d@}Ob4pL2 zzRu3Rng+|p$!HW0p2Z!Hh%(I85{xy(~ejb z?HZ>bddYBBlP-$v@xif;$uI4loy{G2(dp@CM|%RI$NN0h&YfX9i;_{XoKe&~HpQ-+ zIw<+%4^Ce+1w8+-(mUQK;qnj9?KY!lfzrt>O<;s=!!rrMe@^_gM))?MNZ||^!Zl*tafg#@@8UU zG{sfRt50)PuV24jrss+dc)zfe@#zz`&zcpFg#{B8O+wOW($lPdUgwqTISMIFLqkJ) zg0Tq+JYRI2ow>5Jvt#4ql~q;!Ac|hpIAu64c1awp_TP<}(AFg;CjPselU7_=S#;dn z(b1uII+@G2cmMuQQb?jB+n7HsgJ~GEn$J zz{<+1xU`f*oe&1q9Lz5$xWmE{u{vC&qNL;_lbv_0h*p~>y(}aoIeAmVJr;|jh&0jN3Y}ks+yYqTor~Ir{y>A-chfQm2({RYNcgn-oF3f!QR=Fr~nLU zHeD}>k(%nJu4zH_h<|Ip{z7Bp>q}ev``kygf#qp(aLUN!WEl&K`_Q$#f*|qdPgMxlkur0V&Uj{e zM#jL%NTQ~uCMW)R3poh;KRY}0%*=j{9>D>v6wyGW;Q#YWRo!FzOc^xL;@cE1`gI5;{|3b-)AI?Ydc zY_?2I=@cbDpMJ}hO<0+<(Kr9vUHc9ecITks{Kev`!NSRb)hBmi6#q~@espI7 zOR3c;yNs-?w7mR$N8AI~Nd;lOMxQS>ulmb{ArzgQIKQ~8&m8To`n`S24&%Ie^Cp>A zVeineE;jahbWF^>hYt&nlVf5?e6a{W>NdRMA0BunFOM%RE&Z*vm6(LYH$VS@N{+%^ zCZ+<79J+*uFJ?{-cVLw1Qd7l?7cUlE?udD z?<%KXjWd$dcSk|Of?HtR9u;Tc!IVct7&Q-aUGx!;nglj$pV-*glv>KBFDVT9 zSo=w2{a!6+9wb5oj z9Fo7Qy60nStCt~}7rgiq4^?w##4w`$o+AC+MXO7y_eq~Ved;WEr<-q`k(qfugr#0@ zu_xuxpTLqVQ)7Z&QoE%6OpiltJHaQXk7hb{QLWq6pbo7Iw;I{I5=XE1K>cN zO%+nU`Mz+!_M1O-=mEEwt3@KuqY6W_O zdC!nILJqW zQdH{_Cf=|7qI!d#_k6V{+nr((5-a7!F_&d#@5sF}3{XPKPSyGmyT@&VG;k&vTTY3^`wae3~Knlinp zi5=GQ2!peCzQ{Le{rdGayxm#JFZyiMPAqhOvLjqq@geur^3wb}+j{ez*Zn5VANWKv zm2+QiwUA58$r-_(2&EMm3QW0b_ywYH6jCJ)4o>ZEzbt>9d^)#Mp-$b{LWI(oQ6eXU z6y%E}0oRA8M|+0NZ}F?Dt0_xV`jjUoCPs?&FL@p9>TgU`Ls2!WD}p_C?bYjPd-pCf`42fcJ1yk86xQ};`kJ-1wSQ+qSfmsc8E)Tx6CF(qKj?hjk7^Q|z6B78 zNxc|@hlfY#Ik#FC4@XEFi;N`#vQHtP?H=N z7x!!{!kg7~Lp#-Nfs&n_y=<%qeMKf8pTx`e4n4i^VyZ{D_sQ(duJ?Qc-(OsHWOrPU zLzXGkYhME=*1fEQwU0;Y;ftDYxGc4e=f~dX5yu-}dXXbJ-L~gyv@TJ(WA)xFk9%zH zjLg``*m&z?r?>RwmVxM#Ct(xWtfqh*jNo28g6M!?vMg8B!j4rd$1H#(13~f|whupg z)O~%7i|nEHcYqW3L~fT=*iH-8AFVuq`>{M$p09RXw!&r1rKy~&I!5s9gN}&E>jgNn zk4fXRn2NG;rpir-s{zrjzyFRY_I zpfiJnOf7)TT)B_)9LpI&u*BGGASx|QDydQ%0% z`YOz3-S^kpA^E)A8+_j0+dJ&u(0VaDqgu~GZiC_+YCVmh`>vv*8NYx4hru^&m-Vsi zgJd*P*;an$hTaey`(cgy)D&pghmRjW14w{5MPz5Yoc9*u$HqN)z6~+U>UHDR^SHMG<@zUp7(;O}Z29E^jzozgV7yF7qo zeff9X=BM5`bji*A{n4&dZUF#P*>hjh)ZO)1eRDZoMx|vcy6#*2 zx6D2>UXT5PA!pmknwIVPPK_F;-1d`x@$p|XGfmNVWz{RJC|g=uG&d98zsEDHSbQTWY?_-YJmdrfW_EJg6YB@hSJ}x!yN#MYl$Blgp?=0gMHBNp4l8|6 zHfMHrme1>m6Y^`ftrE7NxR{vT%FlanAIqt9dB|-zYvs&zrq->GYJDcv5s@Z8BEz%i&*uiZXLKR-WK!12wTjA9&@v9LI-Mp(J6Mt;nV)KnA2j{ZtaPiK1k zrTT|53=E(Xt1*;-{v^=gN^P&%ys zaQjcydBnuWw+`k%cbktpk4_wU7aIDo!TO7WogKfL8kc;Jf4m91{zbcKuLFD7{{8*^ z6#Bw%uI%6fYfvqWtoL&WYP?+X%s7E*lf0OK;}s zB|?7z^siiO3d9Yiu+Oj=U$16OZ?FE*FLz%KG6^L_fQX2Qk&%(zLKh_l63m{Sp2?m# zYUbwm(!BO5p`yeaU-6Uv3`wdlQyi(GP)-}Z=1igs3X-|VWA*wi8;68C1f5q@(XF-< zRZ+U$r*Qxax)ZH&`m6wJ3EjAH|I>cYDnpvhCmGf%|}dCHMSqk`A#DsNE2;xyuW@j z&*J?Q4%povZa9mBXh{4?(s;zW%6d3`eul72SR0tS-xMHC3U;GFg`)4Q;wIE6fU{_= zN_Nv8nry9WnWGC`NzFrOt?qp;a=ljlXdv$ZaV}vIz%=QSnHd?e1q1|uF2&i1+rp0_ zVrV%`x|C}0>F`W|m3tmcy2?~$@RZ`kfWQOC&A03I<;f*;2Ko72EtsFr9dPp{z>+wdC)mCo@aAxA^K`6r%V!C0 zJo?_1#6dk#lzj~PYp?nqk_LgrB|-T7S*%VcG(4|0DaEt`vSw~>4xqGRYS`<3-Nzh| zYu^^|z_m?zpL;>1mXwypaOvSMLsj^Dc-ZdYEB;Q;)>aYZgH;`SeS$8O9J0W5Uo+)# z2ne|IvCiQWh15swe||T^as`r+5EC~6tdF&MVrg;>1v)Y4i#Wo?sk&zL`$zUSpR**+^b#-Ch5YEW=Km_S--Fm#t?Pq2WD+KVD zaq_KN*KIjS7faKybk^8?Gbbk}klIvg@X?9#iMy-AivJD=96bwc8UlkfhA^h1L&j#J z>UCvhC9HPeOl@6VwaYqHTwI(?W&a)c9s&md!M)zuK<@UAj;o4_ilyoueRHNuP(_yq z^Alj83<{7za7dnf@9k|H7*K&hltO8eD5M}31U>duAsVj_0B^r~^=hn5mNe=n3d9$u z&8Zj2fpSpU0SwDj_924-t%0qKgjx#-6ppJy%-Q2+(b54y5?L8gkN_^jYKgJ|Nd=l6 z<4ss19Uu+qs-mJoT|)ylHc@a%MFp}}z}}r6gt0HPg3u7(66uEl5}urpoSZxU2#f)> zsqC!&W9`42+?{J`YAX5kX;i_5d`rP{&F)^|^H3O1F--{I{6K^ExkQAroR_>JDDJzt z&jrZe65b4tiHT8j%(#2}C8qpq9<>zc9T4?qoK!$c>>M3uAwd9tx(dqz@&CywM6@hW zVDU{z2(FKG@0FMulkG{@>8PY6RR}2Lf+Ya;GovE&~H-MO`)z5r{Kd zYI=d9%(OS{l^cc-gQ5fMa@R1@OfL4}YCU#+<>Qj)X@+f7G%8=v5IN%R#G*J0{NzwP zEjBkCt>!2%w@nKLv+oI9@HZoHOus#PSO8@P3@DjL5WTyV^&Uw5rXT#F~3L zagrdRQx2n2Gg&HFx47kI^=8R(aW&1-qW&=;P_@>)*L^Cvi}lHJ9~5BCox{tzqp&W+avO8MelFVPIp60!=7! zjztmMFh2zsiqy#1Sn{qmYddSMF+_~Hc`g$BRsH?SYwPQAHd#+mw5ag#a8$#sP$Fg# zwA#C$ZH;?u%dGZxcImaX?}0J?3tkXBRaI5LM@c)XtE(d!L`NqTwq8bJZS5l@N>^-# zj}n~{F`Sluyng?l1`?zw8y_DZ{3)``oooa}H$EX@px(>lp-ydJFKt@ss+Qt&Fq@J@ z>q0~Ez*T_BCvfTQp-n9(8vLbO8Q>e1+X&WMU$dNP$S+pggKRdxmT^=EA(Z z;I%cY(wE|e?Eyh?aik#GN2$>EA-S!u-17Nn0OSA-QD@Y%RwrB zNbaXkw*&eicm0LDj?A1@a__N*u)ZG0T}app5wY3U5mJ9n4*of&wi~#`+FY+k> z5Ec_k)We4l0iBnE4~75&JwymC)^F}IsDqF;PQ(DkFa)3wR4O9OQo}X^lkVhfCu(<& zaaKxhD++KIK>oZBYt@aIh8h}48iFGSl7MR<`5&fB2C2F~b5-JOvNgK+k1@Nq9d_2CnLd29zxT{C6Kc(q2|e?p|yD*3#k+Sp&3)LV%J>Rv;?)`THlW zw|#YGKW4q~@?@)RwA_;X@bC};bVJ)aJEJ9rSUU^dDu5-;NcCqnHhQY;XGf~+vygpW zUQ~Y+4Cku>Gs5WsX5am$%X$EEPyWtE0BA)nU%sQe+ZaUQM~8DUGL;6bXq24xix-GW zNX>1fgdo2%{btM?Zgbb5+{6Ojm1CC+4i4sZT)+bzDmFG2LGSdld-~G}3Z?+{QPx)JT6ve>o_lm^AI0lT)7R3+$MN0Eap|`i?f#Yx2fg3{o{?;EgPq*>@WG0n$}U#czcY~fhDeE%i{WK9d6TXZcge9 zcogmp27T9uvE$qqG&W)35MR7k=(wPLQ;UbZgGxtd)@oE^Ny^tzWcX}X|FdPInwr{0 zC1r#0UqTk;@{=RN``c~qu=k8cKO5Hq@fH&oUp^s<%PqWif3K@3RTGT@QPa3H{`bbl zlo7h7T4~_V=MO7X&Xelaufn;@F|F3))_YQHxp-T<*Y~-tef41h@oqdyEVmryZV&+` z4FVS6|1QWn|MY+YfC{bMNBgJuKJI=ffg1)m??TnVlq16H7k>ZFaGn$oA+z6IdKyl{ z*I$e2|D*YLa{!{^?B!@y}-soC?&?y;uic7+(qMP33~*~tYEP`V_W3kwU8%-$S8$l>LFk%R#i zmHO)Vq<@%)T(WJPJjnZ$OLFmDA0|H4w*rk)M3@05*>=~Ej}9;6*VsCub|15)@7%T_lCoINVAuDJzK~5k_UKEK8pFvw(_K4}3-Ck2r9Rmk56p=?YK1I<^J~?UZ-=jYKiM&+Kon_nF~`4gbHBVE$a4IQudn z4w=igZjTJS$cEl4vXXDs-e7;mZR+bUv3UptV;Vb!$5%oY(%g6kX6|z(rF0N9enM%t zfYWY_k&%*SgJ%3=ZgFw-=QN;uE|rgRePE4!8a-W>K`1%chROF-KAPD~)#j>%0Ws*7 zWvTlR6dYWWuVd&al6HodQSXkTM#&!4~q zvk#EMq2s~Su}})OHsyX(0|^ju&48osKyIb3e$tn0JyMJSwJEQI7ie1nGi#BH z=sR*4ALZYD+~K+o{CucD;|UN1@Vz89l*%OFb#1&d0)#}#eA9_^XH1;3Ox6ph4?A4e zMxr8w&k|wjOFa&3=v^5#V_tl5DFH7~=nf_qW-{3wAYBhY<)v^3yZzFLXR zB)|LKO7@_THJwyCX#g@82s|rT3`Lt;`;c!=Bz?cGdcRW&tYO3HR9OJIBY56RxlQ@sz)0VTW9 zKM&R+0hP%|xX&3tytm>pR+Z}xT59rQnzzmlX!jOSFGB$m{RG)c2g%{$qV!Arr7hL-m5moMR;y*=ZzlSA-4D63`luR@1I zqOi9R;8D0jGD%5E@O>O(6?lEL_fzH3 z(NY$Zb`=r0RES+9auHQ#Whk3>8u03Pz0z$qjA!ZRaxcqiUhmpyDIn?DyG(I2e)hW_zJU%bZtJmcfX3ZG z_yg+20e(UD-2D93gk$ekCrd-I`2Z^A&4=`KpYPwlbJO1W@%G=Ysho0`&3O#k*FrO) z@Eu;>M3BOX$;nN>J@hZ3ee^OCQN}@<75e+@5?XlOcUsXJyv6>HzL&?A%zIKS&M~Ik zSDB|gcjVFMkxb0YZeT`Ia+(prZd!yA2Ydee2c6(MR5@z-8g=emz}OJY7|M)#&go{Y z1Uo34^v>o&W4lq(%w7>n7;th?Rn_aj3H$;A93edDo$nz{V?lrZ*g-=MWF`x>$g(~9 zXie+0?;nP;B7}k^vT_AP`{pMf(egX#!M+Eii^P#c#EovqFX?-R%bD|VLL|BgX!*Zc z7S*)^c5^DE6HylCW3PVl@+B|ynAGkKX}Q<2y*l~3Wi?qts->lc2z5Y3Lw{ud7>3tG zaDn&t_dS8x65qi@>2G6|zy1u~a*CqcuYGx&5HePfU>;K|vTX(FIs|syTrIy1F0Xiy}nK90`MKFLm zEGsLUbzTcvAu!`~wfxok1<*Gj)8OFY{j5$cgjq?+$lT`Uj!#Ua04{;(yf^^O)dF>E zUq-=QvV$fQNXy*Z++b-EQBVYeE;uD((Bs|vHgDKu$cL*8LN*$NtbhPXUtEQQr9R(l z^b(xMpJouF5E24-OvpJTLMgs}kD`aY9ywI>;V1M5Ic4EWR##QA+RupjdQXGjrjRDo z-IpZ^##|0ybYQf}U~~3^&}%XXSPUg3lnSVP=*uKTEfia*JcBMnR$Q!n?0uy=(8t-= z*Jt*FC?|aOar*^_JclK{(bsy%dLMb^zxMSfB9Z_f@IoO-&9HXVDT;^O=@(L+bK?Ex`?$OF=+ z@Mw)0c-x|TnA}FjrH@Qnm_B6S=IJg zf`r1J??L@oCPhIvF$)l5sMeWzYS&b>&-2BKRA2mlxoj^n8gZM0LuIwHVYvzVN8o0& zyR4~W0ROA9`+X7V{Dr10KoWiv6WWgi1)&yv?d%M(nXKun^Kh+kUX=iXys7EJ&c#KF z|6@HtcjIcbfZN@>zf)5c3X52vu?H-}pVs+>g$rZlmM38Lz!iZ$`n{tAAb~IFRvEN!-e6E~^ertdi*Iyvc2c7k7Z(FS_X4eNqSh@2vX6y$b(dH|*=(h#eo6ArEq z6s*R+z9@wh{@-hBT_CGeou3`AhDSohmxfJz?_Rr}u&v?0hS;vP{$Gj7RJfl@C$aIm zK7RTn)g(p;Q~$9Sn+@1lj@23p%tQ)$s5A0EKa* z?m!205<5FP7`bG_z!5YV1L4h^P;j4WXqXr4h@IY%+eMDVcr@;25M=Bt$uPw|g#k_8 zM4)}gG^9hK`q#i(TSranaay?!{X6u$yj3kn1C>+IDJ3N()i{34A3$(8DA3xk?3=T= z-trZH)K#nBWz)|BlboEK-(@YZG>#)fA!9FmS%1=3-=(>@xVW*iGv8m}CwN4{WQ|2$ zN0Fr0RW}p6(r-CD}QlVNTD#b{HlV zY3;iFXZ{;QlO;{T3Ti2dzj$9mOQyML9|33|rH8Xl4h~H_l|ke?T^#RUF3|-;Lvm|> zbmqesylmD~$7ybuNcU%H^<)FwdT@mL3SV&HlAksD;k5G#BeZbGQQ?WlA3!hLj*zaQeauK)t2s;sU-~IN>aC3hnJ(`>*`r z_mR=jqR#R@%+O|EGr?{@arW|MPPZ|F=Ue8?Vygi4OV1xq~Q9`62cF0z63ZfBfL@ ad1F zA{}KT6qIm7wXfdkGLDJ3&&G0(+-|ALSH1Fi#|!S&>lSvEY*_{*LbnxZIAp_dxMG<4 z3x=qbCxo*bDzXQ+h|ozaSAGiSC;h&c=(m0-aVEI!>s0Ma6lX?_g%_`q$0sbT|ENH& zfMGP(^F%kUY`Ocvg9pVza{F-PYl|md7%p=#en@*TaXTI!nH&x`hbj(paTXtQaCzW= zJ2-!bl{9pN&Jc$=G%SC4WAE9sXVe0Z@onr?|G8F~r*pDAju~bMT#$mCEtptOGnrV< zK-Ns|f9YT)77tt*{tIuS6PNg3Z`W-soVAwv=8cw)PH<`}t+T6Z`?qi87Co^c4GrSP z4<6LnX;^l3b#d_U1RL;Rb@0vaxR+N}2G`UGOGrv0CleDBIj&z<^zo5=#81O>{rbX> z9~d`o+?bf0d_I$*p3A2tf9DRa@9|Hn-k*P({nl1j5Zir&9pSY3^@?0+3FI9@{E zy8C-DH#<9fuH`0vP;4yW-#`5~+>f@I<>ci>SJj0iB_(@%du1#wEnEDymX~EbJVY6! zyu&dsQ?_<=1mzOCFor}#78Mo6r>7HyE-Z8tXgqrCnN@0sUtG+ZEOa+GKHghLPmhJ2 z9lNaMTx@)UYkjiH=E;*M{s{^AQd0VS&kPxgi&MAv3h5c7(xL(b6X9HImCbm$c^6*k zl&*ogIw2P~_b}&HXGlJkCdc*8d&l=gsKreyewy=;@W{Hl-ntz_8}jKTyVFdm`bf!z zrFFtojXgSQApK34fEDHZ;amxG((ce0o6O^P*yiSX@@xWf_wUmf8X1Xh>fJQ2pw`vZ zMfqoCEp6{Np>e;zje-ZKU0CzN>`!9E-j-19GQSx%!&O>UMe+0J&-k1Coq}CVchf>{ zdC>icUpm~Jb9}dfKTws4j{5QI*YoL;%^W^0*}Hcy!}W*Zo}TsN{ub8OwvML__7i|`m5gy!V1yh=*yn4V6nVtA`vIJr+9 z8eqV4+scZ4a&nUC*l6i?<0O^ zH@A-mv#=}iM%OksZ#y{&fF%!)j09F!->R&t%5G|+Cyl^({P=NsdAWZ|3e7vUH(280 z;`n}aOS|-wV`F+he*czdNswVl$a??&G9@J?wWOyYtR<$Akbkm<0E|WfEFCx}6`N$l$?qi;D0F2?>*LT3|^? zNSvJCbGDaH7Z(YUWB&8!&%E_&H$C@u z?{?UXO$W#fVG}SDjm1TgRuQM120N`YJKe>|}3cdv~{E{rks-l@;83_wFUG z9vrxxow$T6uu#*uSd10xsJ?j}rInjHG;9_YMs&tZj+L9=LtR>0ii3woTRp3pA?X#m zxM)7z;H6_wV0f$pex}mh$tZWjQEvn)2xhy9o`< z>oB5tCd`rQ|w9uebUB#St*G&VMx zK7JgEj!kF^tEYk;$$KIBQ8%t#;gjbLtWO16xds~$tAG6bxiDSlRO|Gckn*}=u*5f zbcLSIXnMFsSf3m^&2REPs1N?}<5BXPH&=Oh@GtgFF1jX}qf3`|kGFSr+Fo3{%S0KK zkeV84-+YXvT}Zw*Ue>cZ`u_gueOwGQFfeRv!Yi$X_wsebe%#b?#x4#NBwrvP(a@yLY zU?1hyqg=ruA;XdcJl4;ib;U`ahxaGlxXQ;zOebs?oSvR;D0Gb!cX4r%&$5ra*rX*R zPko;}Hw)aUV%Vx$Q)j0F8y8ok$H`uyVU?J;cuH-p2pb({Lv&rBWFGv%u zwX*Qz+J9VYVmESIpUBpZ2X(agzO1=DmR`KvtOF}e#1UxM%e7f6gfF_5Iwb!J)M`Bx8M3F2@QorNhBS!@#rK?GbKP=qWmKe;09Yh1`JbpuP~`STB0;-dsu^lU{~cOU{w9seN136W!E zV>4f9520Ou@no_+Ky{Ww26bw8YO zTUc5O3J%7o+A3$AY#SILgx5uOT*Fp#)afDi!*o9;5DT#KSbn`0@n$*J@oCb241T9m zbLC!#cHVo#N@9m^rT7J`w&%gCVA(?kN^zGdy~?Bwuf=o-@Kqt!_+AY}y=k&uv3P+Z)3 zIOnGk7b^(5NuwJ+$MyB8DBQj+$CAKNna+$I*%lnZB|9=b9gW-{v_p7YoRy0#Yt9*> z5RZ=BK^8CO#z|=(brxr1Mam+hwT+CTt{YXG!fZ`6xRn_a#k17+PfSpI9d2%TIrrMP zoV^=z4W;JrWMnJCh)`y2KiHV60PC}zsn_kGCw)-vvZiM{QGSWh_b9u4F!olZ%zrKJ2NDmcft$#FA?{f6xrAGO)saq1he+{yq;d5&4-BQ&dc%dPE zMh+7zFJ&4@T?#&wnS}){)AxiEv+wR6zp9R8+4!r`cV&F< zjWiO4jNZI?lQJ}<$tEg#f9Xt1^6Y(5TH5E}&`^Gs1m%TidoF7Cn?yfrS$@%rPzb!y ze1z84*5>>nu>g4q974kG(gzLr=iRrOaL9DWY!mx`PE^@Yfxx?+(h1jf@VnMD|}8};VU=-&c>K#%t&>xew*qHu&27ZIuYk( z;+gL%24=Rl(eh!0%#aBd+S9Gw%UZTgF8IjssluX%n2bz*u8O&sR>+#%+}zw@ed1=` zxcqRK%%kuo1t3N%LN+k@bAw}@ zOCh_Gc;k<&ps?@~L@Y39ou?GXWbEqxgTI$Lh@`)19tQtife#p&gFO*s?5`$Eg*8EP zUZh>#KqIHD&nh3H+1Lndu0uixflC=}$#;RLAWm+y{6)jU*Uor1Z$f_&cxCX$$DOvG zKU)12uHOu!>4XUfGbHa7xys63f~XUpoUDBJ`U{e_-@7{(=W`eG?(QOg|Nfm{TFTw7 zhK+V`V9~~hudV%er_9eoj`#P4mVWHd{-_GqXS_%n8Oo}Xg+ZVII zO7r=hdwX8NT<8S1b`w4$zM@fW&bXKMFU_d1ef=TbdxAXMT_r4ZY0b{`t5fQj3nN~7<9l|(Gf%mU!2NWjOV-`x9x8Q?s0^Y!ZoXWiE5@Wf!7XejT` z*wR`;{oZ;l74o9bf8hKY%+we>zh;O?qhx76sn{!`{Oc3Vb@Nwgoj(`f&-wSxpY~S1 zUP3YJF(*IzySl%7r~4(bImL+mx~YnaN_;{>z@YR++SzSft>NF9ezDQf_^)2Q!WS*I z=MDa0G4fWQ=#*W}T0vRax20lI#)g-mp{}lNbd&-BL3njsH~k8|gy#Vcif`1J>zBdj zfSb_^zkXo}3)jf5L2SPw=8WTawq^oKe!eFyIJ$B;S81ut?~I#}UMv`VEurU|z&CH` z*woX)S68hSluTOF#qaGb5ASv*y;CpgN)wGUvz-=LH>Q4SNXCXDrlt;u{1P6ob*`Q9 zobueKY{y-q@nF&>S=fFyUI_FZ24nz@$G_gB+9eU^HNgkvJ9pah?j{kFlJYN$`1$$0 zdi`1|^YZycXv;;*=|e9mDv7Z4%U9Ljw|DmT;#|6PY5UI~*x>E35Uoo)V7%y5R7qxs zf}kGH5B)BvDk>@<3A*}Ch≫$2~kcx)VeLF*!L#fa&~Y!qovx-Y5IC zn+&se=KL-sS65ew>FG;8RpV$)Wy^=bZuTdVI+DrW>4Wjq;4dM?_MQm%fB2|aSyDhi z0HwL?!_CFzurhKDqV9(gcB;&zRT}js#AhxnEFf{!%-&wwR?_kp8r@l;wuVjVuL5q6 zpMJ;vCgcyVr4jRriis`W@H@VIzIAaPx29*1OJ#;gOju&u$;yZ|JshnGaIFp-lR}^& z9cS>k_gGZb)5Ss{P_%V-hav-#V`2z^*SL7PR8g&XCkqM*Er6)SxqNx4ohlM%K z;4yPuCL-#A+=rPms?JXFx)?nhOkH4RCS!LTsN1oKRj$A{5(zE zEw*%?xQUdKGSun!fa6SPpEJKIzzifQE;gtLgsYmVIXLiv7TfJ}dhQ&e9{LFsUHOw? zmb{D`s^QgSI2Z<%Pe`=y-Gk@7CPu@Sbi<^(NcVlMT@dkDiD4BUeE(HXkhpA|tfQ-o z%*&YTHP5H8J&|M&I8~nux60Q#zB_mC%4=#8OQysvUjtbKS=Mx|W6+x3t9MQU{F0LN z{jaYt)f31GKK}l}vbT@+ubK0tN`Cv>cki$uJp&tEgxMGD?(e@0MpIB&xKK}#x2%)= z=<(yl{c)3g`71f&N8VC@&(0E3Qa-^QZ@yN^({v`70Dz)5=UwaBGl_ zjJP-=*uxlLr;tv4r7gGR;^1fnKZETfh+;nJY;Q;Vny;Y*p<%p)L)EICRa%<7l~?R7#T!IY>1 z^UW+q!UkeLSn4I3dSWvtZ4uuV#fYYbU|78byHJ%#x+&^NjL|%tcwpH{L%Wx24dv{ z6H`-=#@n1Zg;$BGvXbI{{`}e0*7h?zpF=n3P1pd$QC;wOtR;jm2L{v)@gs;OOnB~0 zbtt?a_kU(%6O@vY(!TZni;btYWKlk_Iu99}R<8SyL+1ib#P2YjtbF<}75N?S#W~sh zjvb7p=-XQQ<@59N@(&-hpYTRM^1Cro|B~eC)2E<4Ft-18zHDle43=ZY3z4g>tp`#O zzC?&iLvv#QVHc>O)w(R*->M)QPJG=7;}|;@VJf0R5#CXR|CkO}-x7eH2_Eum3x6 zEcnjWj87mlugU%Moz&_%T;*^vXBXZlYRTK~5-5N7@P7Stzs51~*Tg<=-@|jYcLL`` zZ9B|1%J%Fv|Ci%$52N>*bl)@m+41=4k`PCPcka&8SW|NtesH^2`JV?!3TZqNt>V{rZLp zAt3yaW_gA1W_L&4G$|ii0F{8jMkZlOeycuwWM@YxEiEnKb$}196#VYjK1K%S)G=`? z0Oe+nAHN%QlI8Arn-mg)35GWQjJRDiLILga<;$g0)GzeHB4n7LPJ%JG zgfBrQZ+Ua`7dR`CuQM|-rKP-+0$SK{JwY-|LlOBYBsrWhF)@IJ3|(s0WGcl}QKoy# z!$)U_b9KlL_zO&@xb($4PQxZFTi_wZ#l_o4I~KKdb%hT!?(4*@K9z5ZjKqtk6~Y4? zQi)Yuw2v6Sl~ouRy;;v{Uup&n;xsBNEX0KuV!qOCUobDEA)^6BJLPucsGEBk=e?ZfSaEM$6E!4;(;r zEszO!clZCCc<8kuL{3HJ1b*b|CWw7O>YPjzKbq&+KJg zo#;E&6ge;U&Mf$m#p<_0!v^0^{owadt>fcVZdi;>9GN`tt8TtAZF@~5s8X7$SOTkY+6{BJU zj1HtRSDrp6>Bn=?k&%(|N1sSbjkHq7$}XM$5rO;+mh>t&_gZ5Aql1GCU~u&e2@oa09P1ED4JynruUxrOYZpS{Ny65; z>%=a7@mIjw#s&%RW}{ zJv|-P#;!wV5NMX%>^IookH{0)n@Ac+O+!Oa+XN`LhtV%OIhooU)=JDrn0Z(Ozk>-_ z6QkeXo%q8#i|H>ed9ENo>v$| z17oomqJ@R&u5>h{5#&#og6DmE?43eI2rO=q=t?li5F8&T)Fx|RE>3u$LMGY6!ymv21{T*3_T1CNY z-NUBWS3k+t-w|`2*r0ahjTS5swbItp!$Q?Mw3V5}?-i9+cZFQ`J?kdy?&v@d2nayn z_Gt`hPEJP{5yF)j7#QSZ*WO&-pgO_4Q)Yc%h_}*YSy+$m$rsvkg4-B%&@MncjPwE! z5TH=mLmopra@hjy9riid*=DT|AO5X5ob@7lBW!jum{85l&A-MNuW;i#;CBdEsk8K|(_lVN_!`ED(vy4iz0ousvXnTnKC+#y@u-QVdl5 zH|Hm_x3}Lu_U3tD7(H*w936Av-TS(t;tIDOPKEQ+n;`HTo0=#}ZJs`j0Ls188h{Qt z#)!gNFh1A~E>f%j{A%{}De#cl*;z$bt6zs!#Ru;}PX%_TayJo8UX#6W54>tmCVk>y#R9vzu(OjZHZ zyg^Dz8rsOCghK~D80tznlnrX8@PZPH`y@&$}2vPu04LO%%Z>i?Lf9^J(ZZ>T1 z?dcpnqGs;v1JDEUYzH73M-|mH9VJ*8vZt%{xDnOt`#l8lT@==@;OA1CB(wWE27U87Pfkpo#08R+5D{opp+E|$2+^bwX@=xAGG+=1HDmJ zR`!QFzTDKZ7mTAt#_>bF3x13N2^(mNuMS}fR1ccHY2d2}AU04|IYYNXiCO&_!f>-Z zv#yA{;v?ug{nfRzG)XT?r8tHzU~?SGSXtxngucCGjEr$msT;DiK$}1QIR_u-r|etu z-5_2zf)dAgPD00O9HBX@n$F)7i`%vP6WsxVc@G zl478spg8K0KEDJKBBYuustcS>&nwP*NZ==ME4)^@5+#>fdiW;^ouZJPGAd17EELG-1HsYE>d~*K9cfgB&wDKZK0t*>7U zB9dwag$t#VVJwXdi8gFTZ0BlUXaQCU(tyOM+E!LwJvOv~ z)fm|%udf>+Is!Zz?cVi2eSEzyQANDv`Dz?>0If9^UqMtIT1RQO_LkYneU=G{_-jBQ`PI z@%C<%#l|&lg{f!Y0Duhaoa6DA)R66M2hZrx*#3cm+W?}GBOso+s+66!vcJIO;fujd zBtLs3h}rbn&D2wX?&_MmICqsMN zJqO<=6?t)>_-w{_kTi-v1%In?mvD2o>3udY?i=WyKq_=HY%do8iCAMsWN#(x@gGUP ze0XCko1+7gM!$>GxrCQ5gP<~OcK-MH=s&B5?;eNC4tkV%Pf2sj>7+joa#7)=bUK1Gkz_V*`UN z$=Pmbxu|`A2$kUQ$VfR`Tb_Y5(IBWwWmQ({pr?~6f*LXh)Fk5c3*X$r0*Zi`0Ww+6 z^ms3_wUY5=5``>Het~-iYcK~X93CFt3iwjf$f*B${v!@3%TCnTGqSL-lt2DL3&maS zpZW7z@laZQL|jzNdCCta0#(enZ{M~;9|I7}@Te$xs95yRyr8P8tYm_!EF5%jc1F^D z5TZKGZSjE$H_6%LHGgvrMq!{`k~*44hnJOSd&vw%KJXFbTs#o%WaZ@0K=MHY5>guh zPgv-z{52|2!I7M;!<1VS)jB)NprxgytfCSKZ4(9t)XnE7Hb^eX<6@KSd%7X)eYQUi z1TzTELTNlFCkNDHncf3D(w~~i9R6QqE}^iou@w{*7offXNn?lCR#O;w>74TN_z*h+ zC2;9b&W=FD3G^U|fM0`40MyNLeFm2p_F==*-dF#g4a#_7cTgHBnK=T(g)%QAq@{3Q zc?XAb!-n4{Dm;`|g@h=i&yTLY)#e;4Gl>K0{Ndzd4wDu^NCc@M2?xkwE~E{kw8BHlt_?9nP_@(N+@$XxVY#R-GXX$qsK0v{j5YKEDb!>ocAsP zG%5hv4u*EVw*Gz_g}^-EK~OdZoMcLkCBZ-hX0?5A5SpK#-#usYKXrfrs+L>GBg59d ze?MYmL>rn~U@hCg{2|9fKs)qqVMfs6KPE7!&U^bd3+N_vJ_XJG6oVlBIaAtXu$Y6jBG^f7avvu4%XFOOgi;G@7o-)cUCqy{DoB{Q_Mlnzo4iH_M+ZV zrqN znHyc2=tq>AS}rl=>nNxKCAJ4^CKmu##RMt$OA%P*p5eh1q z(ptd)M-WvDniGYBZ3ZPTNWua^GU<)u7 zBvwaXA0800@{b=ofU)qou21;3?IT-n+{DlF=PqIX;i0b9O}n9+&qaxJbuAV)9)v){ zC^Wjk637nAC;j)n&GC;PJ_IHvj&f_{SWv&afs-NSGkNF{Dj9LmgKHTeOQLgcU*U0VA$Q ze&bsqF;tS~sfRzO6r_t$NHEwdYzmtNMMd>99&Sr?c1EB}t1)pD;N&Vro%+@&ctEq= zpTk3JRIp))`BHLBvZ=(S+kAmB2?VbVc@^LKutuBVL-S|)@e(->hEH2kvkUiT>BE?V z3Jd1nyTekhJjl1b5~E8JnOXJE8(IKa0gHe-Xi#r&W`*+#`ox5Dd#`dVE?H`Yqs1*b zdHbNArTw#q5r3AY$b?oAzyZ?X`EhHQD5U>5`1nxtA1@sA`fpu@Y5Vzjb;d}!S4T$& z|41xzPzf4k;7+0|8jP#MqGUp~4GkU8%L)LwptKaBDTRfc;EKb5Up78(zo*~0KW2oq zMQU;=Kv|<@YQ+i^(jRL}2miNr`~R^AKTF6I7uxf;erZY;DeW5+ zDus#u?>E~2m*#z&R#Ir;w`ql1$^Ue_t_y5u&Y!Vc`JD6cS07M!6x9_<r z_TD${8{>}q-uoT@IEQog*=y~!=bH12xr3AxB=N6OUB$q_z?YU1Q^vr!K!W^_jR~KS z=FU{X3#Ngrq!`9I@-MwECk6xK7KXH#i0aGt>yvJ(&-TvFw-|i8?q9s0dGYrBt5O7k zj>&=XNl!Fi78z?ydQdYdd*c(7cN7fO1;?9A+Qb-l93?l`#a%LBBw*yF(=!Y7k@9v1mg&QW3C4y^gY&Y@I{n*z ztk^gtbbcaIiP&^_O_^n7s)kF{ntnvq5&jjG1LS06{UwHyzFvm840|7RR`VEg`2Ii6 zBibZ}txxp-HROLgwBW)6n3KjOwH0Z>T+>K*hh+NyJxBEilRnj7+;&jy?d@}Ob4pL2 zzRu3Rng+|p$!HW0p2Z!Hh%(I85{xy(~ejb z?HZ>bddYBBlP-$v@xif;$uI4loy{G2(dp@CM|%RI$NN0h&YfX9i;_{XoKe&~HpQ-+ zIw<+%4^Ce+1w8+-(mUQK;qnj9?KY!lfzrt>O<;s=!!rrMe@^_gM))?MNZ||^!Zl*tafg#@@8UU zG{sfRt50)PuV24jrss+dc)zfe@#zz`&zcpFg#{B8O+wOW($lPdUgwqTISMIFLqkJ) zg0Tq+JYRI2ow>5Jvt#4ql~q;!Ac|hpIAu64c1awp_TP<}(AFg;CjPselU7_=S#;dn z(b1uII+@G2cmMuQQb?jB+n7HsgJ~GEn$J zz{<+1xU`f*oe&1q9Lz5$xWmE{u{vC&qNL;_lbv_0h*p~>y(}aoIeAmVJr;|jh&0jN3Y}ks+yYqTor~Ir{y>A-chfQm2({RYNcgn-oF3f!QR=Fr~nLU zHeD}>k(%nJu4zH_h<|Ip{z7Bp>q}ev``kygf#qp(aLUN!WEl&K`_Q$#f*|qdPgMxlkur0V&Uj{e zM#jL%NTQ~uCMW)R3poh;KRY}0%*=j{9>D>v6wyGW;Q#YWRo!FzOc^xL;@cE1`gI5;{|3b-)AI?Ydc zY_?2I=@cbDpMJ}hO<0+<(Kr9vUHc9ecITks{Kev`!NSRb)hBmi6#q~@espI7 zOR3c;yNs-?w7mR$N8AI~Nd;lOMxQS>ulmb{ArzgQIKQ~8&m8To`n`S24&%Ie^Cp>A zVeineE;jahbWF^>hYt&nlVf5?e6a{W>NdRMA0BunFOM%RE&Z*vm6(LYH$VS@N{+%^ zCZ+<79J+*uFJ?{-cVLw1Qd7l?7cUlE?udD z?<%KXjWd$dcSk|Of?HtR9u;Tc!IVct7&Q-aUGx!;nglj$pV-*glv>KBFDVT9 zSo=w2{a!6+9wb5oj z9Fo7Qy60nStCt~}7rgiq4^?w##4w`$o+AC+MXO7y_eq~Ved;WEr<-q`k(qfugr#0@ zu_xuxpTLqVQ)7Z&QoE%6OpiltJHaQXk7hb{QLWq6pbo7Iw;I{I5=XE1K>cN zO%+nU`Mz+!_M1O-=mEEwt3@KuqY6W_O zdC!nILJqW zQdH{_Cf=|7qI!d#_k6V{+nr((5-a7!F_&d#@5sF}3{XPKPSyGmyT@&VG;k&vTTY3^`wae3~Knlinp zi5=GQ2!peCzQ{Le{rdGayxm#JFZyiMPAqhOvLjqq@geur^3wb}+j{ez*Zn5VANWKv zm2+QiwUA58$r-_(2&EMm3QW0b_ywYH6jCJ)4o>ZEzbt>9d^)#Mp-$b{LWI(oQ6eXU z6y%E}0oRA8M|+0NZ}F?Dt0_xV`jjUoCPs?&FL@p9>TgU`Ls2!WD}p_C?bYjPd-pCf`42fcJ1yk86xQ};`kJ-1wSQ+qSfmsc8E)Tx6CF(qKj?hjk7^Q|z6B78 zNxc|@hlfY#Ik#FC4@XEFi;N`#vQHtP?H=N z7x!!{!kg7~Lp#-Nfs&n_y=<%qeMKf8pTx`e4n4i^VyZ{D_sQ(duJ?Qc-(OsHWOrPU zLzXGkYhME=*1fEQwU0;Y;ftDYxGc4e=f~dX5yu-}dXXbJ-L~gyv@TJ(WA)xFk9%zH zjLg``*m&z?r?>RwmVxM#Ct(xWtfqh*jNo28g6M!?vMg8B!j4rd$1H#(13~f|whupg z)O~%7i|nEHcYqW3L~fT=*iH-8AFVuq`>{M$p09RXw!&r1rKy~&I!5s9gN}&E>jgNn zk4fXRn2NG;rpir-s{zrjzyFRY_I zpfiJnOf7)TT)B_)9LpI&u*BGGASx|QDydQ%0% z`YOz3-S^kpA^E)A8+_j0+dJ&u(0VaDqgu~GZiC_+YCVmh`>vv*8NYx4hru^&m-Vsi zgJd*P*;an$hTaey`(cgy)D&pghmRjW14w{5MPz5Yoc9*u$HqN)z6~+U>UHDR^SHMG<@zUp7(;O}Z29E^jzozgV7yF7qo zeff9X=BM5`bji*A{n4&dZUF#P*>hjh)ZO)1eRDZoMx|vcy6#*2 zx6D2>UXT5PA!pmknwIVPPK_F;-1d`x@$p|XGfmNVWz{RJC|g=uG&d98zsEDHSbQTWY?_-YJmdrfW_EJg6YB@hSJ}x!yN#MYl$Blgp?=0gMHBNp4l8|6 zHfMHrme1>m6Y^`ftrE7NxR{vT%FlanAIqt9dB|-zYvs&zrq->GYJDcv5s@Z8BEz%i&*uiZXLKR-WK!12wTjA9&@v9LI-Mp(J6Mt;nV)KnA2j{ZtaPiK1k zrTT|53=E(Xt1*;-{v^=gN^P&%ys zaQjcydBnuWw+`k%cbktpk4_wU7aIDo!TO7WogKfL8kc;Jf4m91{zbcKuLFD7{{8*^ z6#Bw%uI%6fYfvqWtoL&WYP?+X%s7E*lf0OK;}s zB|?7z^siiO3d9Yiu+Oj=U$16OZ?FE*FLz%KG6^L_fQX2Qk&%(zLKh_l63m{Sp2?m# zYUbwm(!BO5p`yeaU-6Uv3`wdlQyi(GP)-}Z=1igs3X-|VWA*wi8;68C1f5q@(XF-< zRZ+U$r*Qxax)ZH&`m6wJ3EjAH|I>cYDnpvhCmGf%|}dCHMSqk`A#DsNE2;xyuW@j z&*J?Q4%povZa9mBXh{4?(s;zW%6d3`eul72SR0tS-xMHC3U;GFg`)4Q;wIE6fU{_= zN_Nv8nry9WnWGC`NzFrOt?qp;a=ljlXdv$ZaV}vIz%=QSnHd?e1q1|uF2&i1+rp0_ zVrV%`x|C}0>F`W|m3tmcy2?~$@RZ`kfWQOC&A03I<;f*;2Ko72EtsFr9dPp{z>+wdC)mCo@aAxA^K`6r%V!C0 zJo?_1#6dk#lzj~PYp?nqk_LgrB|-T7S*%VcG(4|0DaEt`vSw~>4xqGRYS`<3-Nzh| zYu^^|z_m?zpL;>1mXwypaOvSMLsj^Dc-ZdYEB;Q;)>aYZgH;`SeS$8O9J0W5Uo+)# z2ne|IvCiQWh15swe||T^as`r+5EC~6tdF&MVrg;>1v)Y4i#Wo?sk&zL`$zUSpR**+^b#-Ch5YEW=Km_S--Fm#t?Pq2WD+KVD zaq_KN*KIjS7faKybk^8?Gbbk}klIvg@X?9#iMy-AivJD=96bwc8UlkfhA^h1L&j#J z>UCvhC9HPeOl@6VwaYqHTwI(?W&a)c9s&md!M)zuK<@UAj;o4_ilyoueRHNuP(_yq z^Alj83<{7za7dnf@9k|H7*K&hltO8eD5M}31U>duAsVj_0B^r~^=hn5mNe=n3d9$u z&8Zj2fpSpU0SwDj_924-t%0qKgjx#-6ppJy%-Q2+(b54y5?L8gkN_^jYKgJ|Nd=l6 z<4ss19Uu+qs-mJoT|)ylHc@a%MFp}}z}}r6gt0HPg3u7(66uEl5}urpoSZxU2#f)> zsqC!&W9`42+?{J`YAX5kX;i_5d`rP{&F)^|^H3O1F--{I{6K^ExkQAroR_>JDDJzt z&jrZe65b4tiHT8j%(#2}C8qpq9<>zc9T4?qoK!$c>>M3uAwd9tx(dqz@&CywM6@hW zVDU{z2(FKG@0FMulkG{@>8PY6RR}2Lf+Ya;GovE&~H-MO`)z5r{Kd zYI=d9%(OS{l^cc-gQ5fMa@R1@OfL4}YCU#+<>Qj)X@+f7G%8=v5IN%R#G*J0{NzwP zEjBkCt>!2%w@nKLv+oI9@HZoHOus#PSO8@P3@DjL5WTyV^&Uw5rXT#F~3L zagrdRQx2n2Gg&HFx47kI^=8R(aW&1-qW&=;P_@>)*L^Cvi}lHJ9~5BCox{tzqp&W+avO8MelFVPIp60!=7! zjztmMFh2zsiqy#1Sn{qmYddSMF+_~Hc`g$BRsH?SYwPQAHd#+mw5ag#a8$#sP$Fg# zwA#C$ZH;?u%dGZxcImaX?}0J?3tkXBRaI5LM@c)XtE(d!L`NqTwq8bJZS5l@N>^-# zj}n~{F`Sluyng?l1`?zw8y_DZ{3)``oooa}H$EX@px(>lp-ydJFKt@ss+Qt&Fq@J@ z>q0~Ez*T_BCvfTQp-n9(8vLbO8Q>e1+X&WMU$dNP$S+pggKRdxmT^=EA(Z z;I%cY(wE|e?Eyh?aik#GN2$>EA-S!u-17Nn0OSA-QD@Y%RwrB zNbaXkw*&eicm0LDj?A1@a__N*u)ZG0T}app5wY3U5mJ9n4*of&wi~#`+FY+k> z5Ec_k)We4l0iBnE4~75&JwymC)^F}IsDqF;PQ(DkFa)3wR4O9OQo}X^lkVhfCu(<& zaaKxhD++KIK>oZBYt@aIh8h}48iFGSl7MR<`5&fB2C2F~b5-JOvNgK+k1@Nq9d_2CnLd29zxT{C6Kc(q2|e?p|yD*3#k+Sp&3)LV%J>Rv;?)`THlW zw|#YGKW4q~@?@)RwA_;X@bC};bVJ)aJEJ9rSUU^dDu5-;NcCqnHhQY;XGf~+vygpW zUQ~Y+4Cku>Gs5WsX5am$%X$EEPyWtE0BA)nU%sQe+ZaUQM~8DUGL;6bXq24xix-GW zNX>1fgdo2%{btM?Zgbb5+{6Ojm1CC+4i4sZT)+bzDmFG2LGSdld-~G}3Z?+{QPx)JT6ve>o_lm^AI0lT)7R3+$MN0Eap|`i?f#Yx2fg3{o{?;EgPq*>@WG0n$}U#czcY~fhDeE%i{WK9d6TXZcge9 zcogmp27T9uvE$qqG&W)35MR7k=(wPLQ;UbZgGxtd)@oE^Ny^tzWcX}X|FdPInwr{0 zC1r#0UqTk;@{=RN``c~qu=k8cKO5Hq@fH&oUp^s<%PqWif3K@3RTGT@QPa3H{`bbl zlo7h7T4~_V=MO7X&Xelaufn;@F|F3))_YQHxp-T<*Y~-tef41h@oqdyEVmryZV&+` z4FVS6|1QWn|MY+YfC{bMNBgJuKJI=ffg1)m??TnVlq16H7k>ZFaGn$oA+z6IdKyl{ z*I$e2|D*YLa{!{^?B!@y}-soC?&?y;uic7+(qMP33~*~tYEP`V_W3kwU8%-$S8$l>LFk%R#i zmHO)Vq<@%)T(WJPJjnZ$OLFmDA0|H4w*rk)M3@05*>=~Ej}9;6*VsCub|15)@7%T_lCoINVAuDJzK~5k_UKEK8pFvw(_K4}3-Ck2r9Rmk56p=?YK1I<^J~?UZ-=jYKiM&+Kon_nF~`4gbHBVE$a4IQudn z4w=igZjTJS$cEl4vXXDs-e7;mZR+bUv3UptV;Vb!$5%oY(%g6kX6|z(rF0N9enM%t zfYWY_k&%*SgJ%3=ZgFw-=QN;uE|rgRePE4!8a-W>K`1%chROF-KAPD~)#j>%0Ws*7 zWvTlR6dYWWuVd&al6HodQSXkTM#&!4~q zvk#EMq2s~Su}})OHsyX(0|^ju&48osKyIb3e$tn0JyMJSwJEQI7ie1nGi#BH z=sR*4ALZYD+~K+o{CucD;|UN1@Vz89l*%OFb#1&d0)#}#eA9_^XH1;3Ox6ph4?A4e zMxr8w&k|wjOFa&3=v^5#V_tl5DFH7~=nf_qW-{3wAYBhY<)v^3yZzFLXR zB)|LKO7@_THJwyCX#g@82s|rT3`Lt;`;c!=Bz?cGdcRW&tYO3HR9OJIBY56RxlQ@sz)0VTW9 zKM&R+0hP%|xX&3tytm>pR+Z}xT59rQnzzmlX!jOSFGB$m{RG)c2g%{$qV!Arr7hL-m5moMR;y*=ZzlSA-4D63`luR@1I zqOi9R;8D0jGD%5E@O>O(6?lEL_fzH3 z(NY$Zb`=r0RES+9auHQ#Whk3>8u03Pz0z$qjA!ZRaxcqiUhmpyDIn?DyG(I2e)hW_zJU%bZtJmcfX3ZG z_yg+20e(UD-2D93gk$ekCrd-I`2Z^A&4=`KpYPwlbJO1W@%G=Ysho0`&3O#k*FrO) z@Eu;>M3BOX$;nN>J@hZ3ee^OCQN}@<75e+@5?XlOcUsXJyv6>HzL&?A%zIKS&M~Ik zSDB|gcjVFMkxb0YZeT`Ia+(prZd!yA2Ydee2c6(MR5@z-8g=emz}OJY7|M)#&go{Y z1Uo34^v>o&W4lq(%w7>n7;th?Rn_aj3H$;A93edDo$nz{V?lrZ*g-=MWF`x>$g(~9 zXie+0?;nP;B7}k^vT_AP`{pMf(egX#!M+Eii^P#c#EovqFX?-R%bD|VLL|BgX!*Zc z7S*)^c5^DE6HylCW3PVl@+B|ynAGkKX}Q<2y*l~3Wi?qts->lc2z5Y3Lw{ud7>3tG zaDn&t_dS8x65qi@>2G6|zy1u~a*CqcuYGx&5HePfU>;K|vTX(FIs|syTrIy1F0Xiy}nK90`MKFLm zEGsLUbzTcvAu!`~wfxok1<*Gj)8OFY{j5$cgjq?+$lT`Uj!#Ua04{;(yf^^O)dF>E zUq-=QvV$fQNXy*Z++b-EQBVYeE;uD((Bs|vHgDKu$cL*8LN*$NtbhPXUtEQQr9R(l z^b(xMpJouF5E24-OvpJTLMgs}kD`aY9ywI>;V1M5Ic4EWR##QA+RupjdQXGjrjRDo z-IpZ^##|0ybYQf}U~~3^&}%XXSPUg3lnSVP=*uKTEfia*JcBMnR$Q!n?0uy=(8t-= z*Jt*FC?|aOar*^_JclK{(bsy%dLMb^zxMSfB9Z_f@IoO-&9HXVDT;^O=@(L+bK?Ex`?$OF=+ z@Mw)0c-x|TnA}FjrH@Qnm_B6S=IJg zf`r1J??L@oCPhIvF$)l5sMeWzYS&b>&-2BKRA2mlxoj^n8gZM0LuIwHVYvzVN8o0& zyR4~W0ROA9`+X7V{Dr10KoWiv6WWgi1)&yv?d%M(nXKun^Kh+kUX=iXys7EJ&c#KF z|6@HtcjIcbfZN@>zf)5c3X52vu?H-}pVs+>g$rZlmM38Lz!iZ$`n{tAAb~IFRvEN!-e6E~^ertdi*Iyvc2c7k7Z(FS_X4eNqSh@2vX6y$b(dH|*=(h#eo6ArEq z6s*R+z9@wh{@-hBT_CGeou3`AhDSohmxfJz?_Rr}u&v?0hS;vP{$Gj7RJfl@C$aIm zK7RTn)g(p;Q~$9Sn+@1lj@23p%tQ)$s5A0EKa* z?m!205<5FP7`bG_z!5YV1L4h^P;j4WXqXr4h@IY%+eMDVcr@;25M=Bt$uPw|g#k_8 zM4)}gG^9hK`q#i(TSranaay?!{X6u$yj3kn1C>+IDJ3N()i{34A3$(8DA3xk?3=T= z-trZH)K#nBWz)|BlboEK-(@YZG>#)fA!9FmS%1=3-=(>@xVW*iGv8m}CwN4{WQ|2$ zN0Fr0RW}p6(r-CD}QlVNTD#b{HlV zY3;iFXZ{;QlO;{T3Ti2dzj$9mOQyML9|33|rH8Xl4h~H_l|ke?T^#RUF3|-;Lvm|> zbmqesylmD~$7ybuNcU%H^<)FwdT@mL3SV&HlAksD;k5G#BeZbGQQ?WlA3!hLj*zaQeauK)t2s;sU-~IN>aC3hnJ(`>*`r z_mR=jqR#R@%+O|EGr?{@arW|MPPZ|F=Ue8?Vygi4OV1xq~Q9`62cF0z63ZfBfL@ ad1iH;91J-6bL2-JoY|C?HyOFHP>8o-XZd`AJ9;UP#_Qpn#4y@MF<2g8uo?s41D6-!TA&X z@0o$r2T{l)?BA@mf;b4|B}78>ow94%KGe-qdHVY5uUzb)QS48u zGrG9Fr8J_V`eMX7_;bl^xdOj}lkGBt$3=Q8+2N393_G7LR0L7IOx2NB=L7lyrkUJA z+(n*%8FS+z>XE4*XKI-gX)Npm*yq2db%UM+V~QZz@glzzL854V1t*F`;jc#J3;sL~ zMZl!+$JC91yzealc9atAN)q-zMD5?D6hUg@#@Mov#qk0 z_CUg8GNTrM>0l5i5q9uL2$l%aD(?TgEBt@?OGP0}usc@7*7?t~%PpK13nX{H`2Y7V z{(WAKi2&HRndeBsMg=xFx^)GA=NVBbnj@grCSw@_@$vEUT3V>Wo}T_Rf`kEj@Te zk~%`dpVZ&%gs-H%gzEcnFUMlHrna$>X0+ZPvE;fQ!TJJAKvGp78LQLzXgOluGQW{K z;=9yIwsj3OU@dPOKTq&HGcxw6)Gnu6(KWaEeHi{In$@I zCfs{75vF#fm;$f`0atn*=q4dy-ZBs$p9d$erZ!9I^&%%3EFXByxuQoapE@sf^&r(X z)Kv$2Y@J|;fm8FmM`gTRgZ1aJh825napo5h;?*_7*AKU6oMVHdOo%U;n96Be82rI0 z?gqCU9rB6;nLF+y5A-ZQkNG@AEUvzExf0VBI@DCeUYl$4;EJO;*y#MCTXyJY%FaY1 zyM!$4?ChLcQRRG~$!WdxV|Tiw<=%bNae&l?#cYzQyzOFv%*Sft-Uvl1n9RtbiD&uK zX55kY@2&zh`>CIL4#OLJ*H`36(OfTs@H4drX~}JjC%PZ?6(2aXJ^%LoId`|WA4=w` z@LgJ>{Nsr3$cxz2^dSlSoR(&P<}UC)@VVK!`}gnPUu!o;Q%fa6qQ8CnAT9lzkdUxc zqvn}!_WPh0q@*&MnnX4o_neRtwF;UyZzAI3aa7ARr;DNf{s`UO-TaRWeA^+k8J|Ye zcu08c^R$-bR`p_sM8S$~t`aEF${sQ=Sh6%3dwAk`e%&$G;6RKvt)lIGy{XN2!N5c$ z*gE|XF|}AyUY=iB8GUne!>PCUi6uHSQ%*!gq-6WBd5>vwatM67g$}te*KD0O8W7E+{=Gz$7re1PFY2z&Xnu*>u@$9UT|ks17z#HA=vz< zKiSf~A6uyp&ZRwA1E~Em(#cqc-f&8yljUuc_?-5rJ!ZG(hYmBfgbOX7dx%Go$%u;X zeQvUPOB5lU!JpjbacO`2`<=PDxo^ILumeNfj~_oc$DpllY!Ji;7o~%Ea$5O7%N(1-x~Hi7G@r#v*~vuDN#t#YyP?qs;h=Yc;EomA4b(WRzevn)}+C-AN1T%~NfnyRX} ztgLL<#Md0UC7;K8{=4IzjDeh-oZzUa?LK@xgx_i%H@HXbw@hL9%v)`j%N8{Um6a?v z7YD;@ePPLO4aHlwjXr-?P*t5N@(%m`5>7N&))bw9CX7@m*U!8py|M7)?vUj5QAb1d zL53OI*tm%90KnTLc{kVdGr;O#zJzI1YTJiU_Z zpX?wy28OlYU4GhL=hGxY-mPMM92^`3&!54LvToj?G5(okH|pSj17hdeYi;WQqSPl! zhEui53Dc3fB@`Sqp@N$DpBDw!573C|n3+x1dP838G{>Kg@{O%PFTluW)z-%U{3#7F z9!crTdgrH!5M|7nI^sayw`yBywa5vUHj&KKfT)TJu3>+;{%pC9@Dx(7agAGk_X00s zR$iWeyVtGro)gMrG~`vU8O~F4EA*i^^{Q}&pA=vB7rM4z?=miHc4UkRJ*LG_eqi;u zuzZF=CT(Dl)6udrG*@LvPe=EAlrKGM0Zpjto<%LB=;6Z#`kMg<8HTf>-Fcc%q=5nZA z-n`3T-FB|Jw!W?&YAz2(mC_h1MWoTz{NmapC1epYriN?hN&QqHe=rf=qv--d@;p%6gM&(H7Hq3|;m zdU9f7Pr>yKuTM5JcCtQ2CnWUch=nBy`3OGVc04YAub1uESyph^9{oID{n^OK$Z@Am z>{D#YACYOCtghAke(>L6&d{MaKACtcXPT%LLWshuX?M3~nv$mG5F)u}XF;a-P%su{ zxT5;t)D~^>@8#CqMP0jFk{-IoJIj9e@z1ffpc#B5*t{Y~V@#8qg=+)ECLw zO8?1-AcrsmFN!ngKE`j}a;pbJXybfVkCvA=MelA6|FuT7D8nuC1RR)^@wa^1@u^nv zT9(S{`=S{r1a0TMf%*MdWjvSYN7dD=ASU+pM-UV8*i&KfA4A;S+yurwD1N<%4@Pn| z4hqY~qg_cY5rCtl*OW4sRt|b1UX#w}ax%nNA+N3db7L^h?e?HSLQ-WG^}gCWZIHs>q_WQv>L28zkk#1Hm_rK*pfau zJw?XAz?kzQ7D2#9ewj0%kWC z>+;qUNjWGAg})8h#L&@Q$Hzv=Ao`(SH;;jYp~cJh~Y*RtyH;`$;-Hv0)kCG2Yx6}-X)Q3*zH#8%_v z1oituanE+A!yd0QA6Ik2m`-*qkSH|cAg`K#ZrLlDRe5k8am=NIA=<&Npl)&bI`*W0 z2cU+L>d(WKnq{xmD}AMyn(`KPrI03K5ilprIKQP9g8(Qf?gy2~k(DwpO$?h{3z-#D zRMV33Y;9k>vHdenp<1pzm?7Yi#9@IqtLM{}M}(CVBmu$Nb~H|&sfQPrgMKb7Dq01x zb_2F#wo+}jVm^??=&Xxeg@9`e&BKf9nv;m9CmxfvGg~z*x~&G_yY7`WByrolg2iP! zJ3E0zWC~GKFwsh#l`Y(g3Ql_>V~TO}Nq~zs@74{o{(GjO44AkKl4OK7XbLX>z^gsD1sgs&|2vKc;2VCT3`u z+tEVp&+UhRiOZ{|a|DjZf@A469R{)OYFO{#t7^&T>0*_QjSUEx{8tbOh{Bi1nR9d_ z=azJnOs_6D4><=XHC5R`~Ew9YD)5(V_xUMnFISJ3D)=>yidbJ+qFeSrf2j*20mvFsOEiOm>mYhnK%vHm629v<@Nj>d%;P}g_Y7_A$mg*I0fRzWMEs?$ zdAq)@0khd<(u)sM4x!WCa!NVx?cH_(`098alN5tb^Y&jN_QIbd!p0&M(?5Cj8xw}3sDk{<+jHT}F?|0tYL#DWs zI{e})Q8FD1@U6Z+g~qwYQo5^4iQ)5SPXLw`6&J5}x89ljz62I{2FAR{VN(!Tq(}u* zgJ4)259${oS0}^FU=%;6dchzu9UWa(Yiru|>GtU$ZRU8I`tQy!PnK+Z&>-Mpj7QT3 z*Q10+*~a+Sop6_63pb_>LbyUQV0krD-YaIDn0pJ&vfsRu3X`4-Q6h zVwS=aVUNN3h;LS>d81=Xq0;}n{HNsMvD@d*pEES^Assi<6|NwVXn^>4)OJY*>A2fu zcDvar*qN=+OSNhR7&)9wAf>eQb*XB}_Frg=%PEM)ET4yP#tbzyG~yEyoGwmA2M?T2 zw}l=F+jPggE}^t$68=aOFN8g_x(#SKR2PdJMt|yGkH5!IP*(2k@Yqr_sa}P3O18#>P4rw;q&&1y|*;Et}zaA|W@UPa*mz8jMualFWI3Tn^tQ9Ix}f zItmI30k0z+WcVBN_H4%k$Li`T|K*ayK}G|>w=i&i&BMu=n0dEB>v`OT2>GX0xR^66 z8`bLY7BC!$JWwVdGd|rKDb?%9_@?Jg=tsd~UW7{Fo`?|5RBTxrhlxAC^Hmlgy3OK} zasyD{fn`M_mjQz-ivsO#IMs#&=m3Lny50RW_(bk=XIJ7dV$25WL}QSG(svC#hK7b* z*FtF3>n#Z-BqRjQl#sAIjZjL$7+!NK8t`#>J(c0A!+)dMo5th7>$D6ePukenxL`Dy z&xHw`fs&FkS82Ph&$_{3#q07dfZpL+>qk%6e`HAmpf*AmNSe9rry0FZ?Z zSM{(q(iic9sqCqpQMW8j^Q)Wcw%l{oSi>a(?9uz?F7Gzx&JtkNEK(ZjY1%Wx8!Ca8I7u?tK4t)50hc457FbZicNfT8jP^=pScVxr~hn zuH{eXyVDPcp_s3He<1ZrMabeie@ksFoC$d`y)|!=`5D|XX-e}85r1^TFRn#POSQM! z2bZIa3=DJ4P9-L}IhZ2sN-_{TZW7P?^I1U0xb7wl3=G7@#l5$-E+12}iuZ_lWFTm} z-b&SWfjU($Y9mdAGQfZQPgYZZoro8eDorCzgDoZ|wz0i!$WG*Sk&=MNhfh;-`+nNI z?xYKm{1`yetACyV=M1jbx}nBwism&Qsu>ny6SqBS8+#ul&GurOaX?g-bEDF3UEI3$ zuXtv-=-76O{$qbUF9as z)|p+KD@O^qC6_ITBjs~uaX&NlaS=np%K9qk%f7-Q7eNQ|Q?1h2^z^jT^*K%o>v4Xe zC&I}=!^&iXjox5Aqra)zIAZixv4;uHmBqWis|JhaC4K}l2nty59%s^36ciM~x>W$v zn4k5C6CP$UiNhut!{jhj6hAw;jxxV_^Le#9fdBq%N_Z+>U0t03WWtP{D&vyJ zOP_0vjt$P?*~Iwx=(M!`YaV9isH7wUR#sN^8e^H3D1pDd*guomfw z%g8u9vJSRZRZ;2VJ8JC#p(13^sC&n8&5<{?J^o;65PbUH$*HcRg&tmcJW9A$^9@L6 zmP?H>_6~SpDK_x-|CiH&585M|_pS~7l|&bq*I(thqiEf+Vf1R4xx?tX zW(*#4Ui@!VT3Q;O(9uF2s_z@EdKRZ$Wg6+E@!hIan|4?*2RG>kaG2XxvPttp;8ifx znwCKliuzPU=ah`0W?p)si=9c=24JZ%m0JvM_F0pjUO@6$*ci@-GbcoiLr6=Jx!U4JIku1Ym#4=)7@6l) zAUA>-+}gN%xbED7jG@+di8_HkMpo?f4Rh5pU4>$aa5UJ7a_%??Ej}^P{%;@tr7eb7 z7KkxC>F}hQjU}_@NQg`yCr?KPHD5TzBpt{)V(G)?Rinljuv*Ur@6VLHZ>BQcM^RBx zEnIq6cl>f}L8@l;x^VuZqB-t-F+3Bn&LUR=M**Hi8~^wrbf)o zshe9^0BDuny|q@@H@G=(253Nez#@Rs|9UNqd11fj$orCSROr#u`=Tb(jjRPkL@j}5 zl_q0klT%ZC_uh`%K2dpjRFaaC&G7+T>fU-HZ2|!o9nb0YWyUyG@b^D@EXQ7atq@R| z5sk7erJ&&ry}J72kl`}IF?$RUsyztv2H`?qJTI$$)Nd`OgQ^h)30w2cGQPPj=XN^y#=c^NgSI`MCRh-Esfkc4#4& z5VsTxuTQ$k* zu1PL7Y~S=a=e5a3&Ghxj#yK84+VR?z;x{13eFh~MboQ@oGl*-T7jWjnB@)Q{1c|f- z5nhI?^4~AzAaYC^EHIYURW?+U`J%?58TAi1^0uh5wE2wJhF^vnP0Tk?GR`vRBfW%ocb=G6+bmRe)k8z-;kdZ+Hunsm2fRk0N&>aEI7KrP9 z)d08W6$_v?#!&h}t0B>Dy+4~yGK|n48Jh;c`FQ6=y!PB7$Yip|OHu{>Du;KmWNoTF zW4F|HtzAQ->l_45G0$ZpC^d6Ohy1~ILbGYV&PRxb{32l=+AV9u30QU~?eXYyxsi&H zQlKlZv70Z%qS{?R$2p*9WVA^5vij3Yb|8{;{bsMC05HoO7Bisvk|*2Nzw;kX{c@wU zq^yjZlM_!&f&U9eO8)oHC?zI^Y`$5TTe&!g=LJKvyLCeCDu5AGG|ZV(o9;}Ul)Rzs zwe^n8u++&}0o-F)ZfeQaS3-*sbcEcRC9K>;!zXd@6?H-3Q;wx=#?BC+s* zV%%q_FjTVR(Id0DNffAdfuG&Tb9N(|-cg;?o9@gqI;xg3F*2$xJmCSfpZnz@)F5uZ zQoje*&2nxbZEL@uEh;ZRl)ooT9oa(0>j}ar$jb5s6wq7GE8DTR)YQ~)4Rm5<%DnJ9 zR}7^N^jwP{g@#95!Tc2pih~YJJDKP3TqOXp|tNKay1H-8;DVLFiBaFH#=V z-b;>CGVGk3Crb`vZlhN19TREO`rD)FZxZ`5ukS#Uf(KAgfCf5WBGIcg#hS{OdGdRN zYvnAh#rxj9Kb&xVnj>UXVW0qHN+-Yao`Z4ac4zg4CN;{ueo;FyRTM;vpPtThqAQK2 z_Hk&Y3qO{KrC(rUHv#$rG)>eRFPA;;p`9ZS5mxf|TOnS&fa2)9WT=x1iTx2l|4ww|4*+ zYSz6q|3xP$shL=9y{vW)YY8aLzG|gZQa4dBbjq6Y&z*AIw?e`2gX>o>ACl}UqqAfx z`faOYILo(Y(g1ZP1}kghc(n)ABIaO&tPi)Rp8Zz~fTr5cuiMlAWOKiS)o|Rr@fBM# zBWv8?%m-(!V+dFi!ZvN?Qn@J?7xHd~U0f9dJ zwH;Gz0}J|;NRK|LxxML|Hh$JC{<}qGL15UFofUwG#5Y>~fZyFYIz)K$O)GoJVBNGW zVI56%E$5X4*B^qk7qhF(^zENqnt0R=^`(t(iLZIh7i!Hi%<_aQ0iV=)>6R_J_=^Yr z?*v0BN==*8`B+5rN2a51(msZ-SElE#PF&GAzqJhCp9B%I8QWeR>%-8Bj|+9-oQ^g6 zEJyW$DK2rSoT8!y1-<6*G*8i8tKN$z;Z*`e{1ER#^OrqYl$?s`!((D&3foA%h)U|# zVxX64qAWdYLvk4wMl)lz-k4hyJbfMF4)*$uV}+#4lzAPDZ@xY1$i$))ccho9Sm2;# zd|hIbH)%l2BQ6z4ks{?zyjW^HA6F{N6=Y7_#9FeKi$MW#{}r+^nH}WTT_nk(T_&p>^5>Uuo1xfuqR#k zWjDj)P=BS}EA}r=CAT(7p|z${L+)fKprnnppUe{P-T#D5>rj`4EDea|VcUtB@kYHR9!!Uk0@3yXxDsPC8s;C9H|Hn3ogTu>B} z*b~?PM8_W>PKm{xSxi3*jWGCR@S3*(*Uk57$341XA5jx11VNKC_h6@X!~R zZpXb(xI2}n=1@g^f^FDT?n6ZBPCKqufqKQxiC2(nh^9Kq)(@-H#eNYA|IvgLRn7&^ zil5DyzSP9-3SLGATxe%|@iRHB5@AkId9bpvX*4?)b+k;rN%8C7ibbg>E$cp1{UH@$ zxa3RjC+<5Cjwkt37M$}1eJ;l8d;N%rh|q<0G3yf3jT}+ZW~8Hs?#pSuV3eSrbBCl~ zYQ+?OtLMhOsPR)y4L zHq@;A;kxdm->Af$cqQ~->e)<8m{*_0P}M8Q4{3ezY@5^_t1FmH{j)5mKc8kzq+|Wu z`(dUi0*RKHBl72!sdIYiItVtxcvYbKSbMlV+LrUV!3M+-`jvm!0X5^D85b4fa|AFs zDyoRvv@R=OF|nF|UE;r=Rn-UOabB;fSK>%R$%~v<*KQqD0CV70Nk-1Fa(_RiHg1^$ z$_j^v+m%OqQ0LTuHfwj!6N{k&vFdJjMg^=@DvO?G4h{XRE1S;(ez9l%>&HMFg7S1m zuQ|%`Gp?!|-?`A;8<0hcOK}V+Q=MzRv2)wHl%{z_HQrw=h6D$j##Aq6YEm)27JQ?j zuga?Sxxmq*x?A{@d99L-ODrMz!BIIF_=%f$Pc~dqJrBc|_RO z-nxpXw&vjYFK+j~8=>C;K|!ZKD5H)`^zK7J2R|k?b#dJbDc&gHi@`rP_#CdIR~>=b z=@}+-96je;I-TT{_1^`^a%A?b`46e!ZoDGH*+E`(b4sw*EZPhIOe?I@4k&}_l2-JB#f~XWScC~Aw*C#cWXqb)Xy&wxRp%))w}O+~g1=Mk zAO?c zKVgv2&zjeo7q+gZ_@w#*maw_QRde5u_unenL=Jl1ri~`ob(rM_w zl=sb10^#2M>u!E!YjXS7{LL27XNQhcK88HESs2NP!|ZX{bRsFXqGEm%xj_a#iH`29 zo7Ei#8D-1{(3*H-)AMO0Wj1-G%b`*Vpn!M_*Z~8_-SC&c|rw!)&3Ja>t| zxK_arm(c-umxrL9f~jZ5WZDKA%yz$KHs)bo53kDW3T9u0+ABcyH^K>a8@Ir+s1U15~?L5-#?} zSl^}Tx`Vm>yPEs6wliL)LtJuOp0=+#W@S&?RNvc&w^rnA7c%?0zJ;NJfS;8@uS(6h zJXA%VKb944ug9W1E9hM4V{`kU4pu5~>8)*zdtVa@w*#z-cbjI9PZo zCsuT@C_XyD^+t%e()Pa`xsG!$_OyD5g6elhfc8W&0wdz;)RI$=l8CBCN_29@v zoeAiwa9whI5?zNS0j`J>TwV=j^wtge+Q2K-`?Z{T6n&Jesi(eejnX)kl(oQ6BI<(EsU>3i^^%-bEQ^OmCiHy3_g5MKwYbQ zr+2yVZQnWoaPMo+5CP^?CrC3HpXAeASMj>gdpP-acV6N1U9ZEt{m2rDZ|-;JmLdTT zv!1WNPa!y&KU6}rEcd8_p2A z`uzm(cPWR>{?X^}Rh|$&OJA|*3?9#KhfIceyacPGh{E}?XW!QDPI87Pq!d_8RW9)$ zZ{l!yp(tL@d7n}5NrXi1+9x!FrB3*a)PrGnUR&{cZ&F_d2{2 z#r1KEZWx1l4aftzZjJmt(QPjq^Ov*~G#gSEp`BUnkeu;sZFR8l%&NL{91D1klr`EC z49``&dyb{5MRJ*9huPwI6lZ^Ie^&8qI<(3$G)QJQQ;a&Nz+s@TWbb_u7uP91Pgty~p)zKD;rrd*~XGR0(vy+e0k> z8NgD-f)=$%WV@&@NgELrEgUcjv1FyAUw^TTDXnaeDLl?M3ix&|gHq_hosf{gf7SVf z|7P6J==|t?$o$N@Sn03g(XG5C14Gx8qnR7j36(FWJL{AvX>9naO+3fnj_MOiD;oqu zhi^HoNI-dyR``3>kyoSg6CBV^ZB0rse+Nq83Btc5HkBuve~H#|*q=*9V5DS`nqH{R zIC$si?skRwzTnOj7F<(}cZd`QrEK-ulL}9(VWR*}aCAW|F46eYtJskNGLPpVX#+*2 zCjYWl1?UTgdZ`E~^-4yVm!cP!qBHGKqP`IQq@V-g!51@Y4ziUv>bx>^Y=p+A9+23W zCa*dJ-H(FFpe^x7UGMA97e^~&#VREW+MSqK%hkctfNB7RXCCoa53Kic*Dy#y^(PdP z*9^&_OXE;eN|=@{Rh?Ruu7-9U+B>7+4TKp^H>$iu->f@iFH7aQ$*RWp_1f#}qgI5`DH zOulIrxT;NkqfaFSl-lbjcGf`QhmFQrTz26chx;rSRO3~jy5JvfW;^Vc-L{=>?kbg9 z=5PDNl#I-OgPZ{f>9_<;%3*Igo_|nI!`v#f=t~7X$QNe^{d3Xlg(%S$kG@J@m^eMejf?=7v~^pg9TihiCR+RS!Q$S!@ix)t7+4>x(*QFkUvLPt{K*wA`PY**_-SIWPkqPT@>uGQ}|0PKa z{4bs*Wp$l+t#k<09(&%-E0N;B=EVc!ziWMHNaY6(q)sopYFaz&wF0nPtp%oD%wfi_^78R908EL`)Tp`lrkmCWu9j*ShN zve&zh#+8mMe+U?f;N%YnX%}#J$h=23`_G1NXA?W_4{UI7ajO96uU&|YsP}|+?4}S& zOHv6X5&Z~{TK$?={{0;3*CQFIt?A3^SYGR5N8bZmLEBW3HOeqhhd!=k>Njx4Iv>Bp zuh9BXv1R~~Jv$mx!=|-orTwou2dLV$^2BVCOIhA#oyc7Gq;OcwwR`a~w_h0q`F(If z!ay|??w$`*f6CV|scdz;ri=XB6eelmrk(cUCA-nLQ97n$GQLxi!&-H8&lsj-$QgUM z$64}VroFv1SyWre8sp7eHzux%!8(hKSljlB8@J$q`&K$m4r2Pxy^4-e&;$$b`gH8# z;!7T;uEM!6?^GT6zD}lPem(cA&%Dilp;h97Ci2Ty$3@n!I@Z()A>T^1W?#9oqq^`C zR#}rmf+VFR#>H4U8DZoNVZ5a<`P#eOc_3yR!ItCi0<`X*>5^|9EmM*}ZwH{wAQ-){ zYKC?mjWFoug-b*0mLiTf3kM4w4-0%9CTVsv#DK@Chrcc_4{A6mE3jR!N$@Eg?Mk{q z-~iA_FxCUaxp2?{2So9KIRKj~swn!MfV+sF(fl(m(*{veH0&x2qpCu^ocI4@2Eg

rfy?8?m4`&2*6Uq??qFJpa`#NG5vIrdSX~eo6i^N+ngCr$ zr=+Q}%8KGIr~sdS4c~ry}+i{2oW0x0a!xz2(@!mBF@%K^YL9X zJCU<&8j`Rx-#g=jg}NoDSsm!?U`^w{y^UcDp@*w=&_wjUY+1S9JoozEpgS;EinCTr zB*>#Iq3~JYFtSYb$nGD0uW8_gC(5c6cxta33@Qeiz>_o*l&z8VkJe1+>F7bS12Ie5 z;OAU^T13#H;;MQ*7FKnWLvVz-wBY12%F47g0%ky;>_!xp$pIZ2?{UktK0Qd$*q=0%g zB(QUlrKO|z%d)%9kBTeoE(xcOy49GN=Y#@*7n2|Jx*Knj^l$09;1Ztl7OLAVw}KDcPHt zy`Tl#_X%1|+BcG(25$y~^R}W4zzgMk#)w5S*iG%uxp#j_8WR)q4#?5M27ryEa^5n- zkqiiJtQPAh#>amrdjmhwD`w`SQY*fHHBzsLIV}lbQWEQi+}X!Ql!AYS{HqM3RzNX` zgNIkVmEXkm!ZAjBnK|U4UmfJ*wc&M+IrmRoE&nAiHQcDbrlNd!Dfvwh6ZCvwO#_!@ zH(Cf_!Ce*D`2fQh<8qM`~43k#>;{% z<8(JS<{QIkdt#`7)#gwFbjBI@2!OiF5V!Od!ISBr{V3xvStmWeW?p!On!Yf6Ho(LJ2^c&e@#*}70^W8K4C87Hs;4t?ep9w>9KGcw z^lyM#LE}6 zYU{0yQYjT#PCI0znE*FQO@BZq9TKZM!``vUZ#C z!Ey}frNEhB+QQ_#ygc7My~Xo-730%6R}?w1CHjyrLqI z9a~Mg{rxLSbxCvCtsPG4IU?Ms3;YRu_|smvD)OtX+W2Lfb)jKQ&0uoHfe!0kMN_a! z#T;FzG#adfb4MjGxGLlE1&C?QOifMgA_tA&a|=Lz%a0&+i+SqHfLyW!;vKmuzmIh@B>pTm*R(+u8sQwHP7c{?SyB2P(=RpA#yum$!gg za)O+;WD5omrzH5~%>DM3beZoGG)gD@F1IN!?m(r=CQ-D-Q(D``gRRn);AqG`WYm zn)p#P)bZ+OfoxzqG?$QQ8OTd_XTo`6S*fT4*)eB&UFQl293ZsL|4Drry=6Hxzqxrp zHg%`1t^G+^bYEfw?!uBnIfmK$9iYEN55K?^E-oux1I*rkQNb24V&7&;t#zO!Qrlb7 zAo>;-!sh}Xot_RgCjK8O#^C=*F(kA`-qp0GTL|4Z$v>~`k_0DI)&G03Xmc8-clWiB zZpAnj_kbAaKHS?f(oXERaB-qNHjsFOo}~A@JB*?H$w>FIpq1o*ffJY(7mh%{o&D|G zw?OGyX*@zSW(PXHRJ4!keU;_qk$=|*8d?as|06MbhDPK#;|yZG=G_yRDiqe{{18d9 zvyj>o_0C~CPoi<*hyU+MQ=lk;3B2>RAjGAh3={E0FsM=oa*_?eO@j#+kpxO9n6LEt zv+AMYCXXqa4DU@`*w^85ZAj^fJrGj^o00`v%;w~GDIhM>3XX_)`Wx(rPJ^oB1&fNe zpyV2}0}j?V3~GV^h-d=>xa>leeGyP>{uSc6$)Z0!-Lj+N!Vi)SdS3uTa^Po$jJT;)$RY`W97g5;u}kU#@<%la?CWBc z7OT1^Wl#2DN{G7kVeo+6@%1QbwG`_0%RY*lgJNy#62ZHg_4A*amf3)D&%<)1y=}_p ztjoO(C^o?9euK2R*rE35x#zeoJY%K0=Di2LLi;x6F# z!F_48I0KQIgS|cf_x#h7lR5BkYPxRJ-*lWcN(O)cV9I($)&@~ijgyq$)i(72cssv< z!E#>Hkth7_2MnRS?uB}bi`t6izZOUYEymQ`bJ~CghaP;Xq{$Y@E$pF~RX7+J7{o6u zXpX^TsY3uK=Yqya`E*sm<2(%w4MPwHxvKNQ!gnefntI-u17}8{N{*@qpVZXn(uJ}D zC-7&WgnB~;Xet#0Lqo0f#e_XC0M!x@Gu_&i4*L50>s@_tzV9pm&38ErF^A^R4M{P# z<%JV-$pRp8*=kcDRR4K6k#kY7Cf&H8LkK9wZEDglmNOlYYgdG90Qrg}iVVg22uulT zk+{$4{;LH51A%3=?~nHdh_U?u?Fjs+D4oC(;sMm_Ni6!^8e{*0G8zt!>6-XPAoBxa zl}Z4i|Mg}z0wKiM!YG(GmSkygJLP@rAV8cQEHf6gi2OG zy!fwtaLnWE|EThTJ@PdSb^M(JDu@vfRmSFJweXmAe_{yTja&0&OL}olAVVQ1V2{GE z7ZA|8>hI50;o#u7dV7OPB3a1CJ4{94czwz9^pQ!U+P~$nIV=R5)_rTW!DeOjY-bA2 z6~J9^m>^KfnpkbaT9&{%Y%yH{;N0z*%Pyc(`l6G1v-Z8yc`JB2Ia7f%448tg|7IWA za?jOU5d#_{gIS?KXhs{c0E|LwYpe63S=iK$c>V6xg*9o#=$cBtXK(cHLB&OZvx@B#CG)Os2@+2UeDEg%48 z`|NUK`1I*h*s=&jBaR19s$p?7If^kf2gv@KRrtU!!tmZGf+l^rYzM}di`#8050KZ0 zG18P@M~;JUxU7`{4&1A`XqH^qYXu?WeKLnd;@ARTI@bZz`}RN*1dK7D-LxE1F&tp9 zdAMDaZE!L675G+DH>>JJCj9GBIS7u5zdSEp~!ibn;xKXew z_W4sx5gXhYu-12=Ee#zl6rzlqo14JZk1w6%C;)VXArm@M_6+c0yZv!Pgc9Q-U4wD7 zJO6Mx;82hYHo;`}5M@)pU7+90R}mL~$sn(*D{F3E3_BOV-PR4**x2i!qcG7ul)y;HWlaKc^o8k_vQFFKk6Gwm=1ifWD%pU|0j-aAPjzu7xvF_%xz~c@O2!qK$i?LYhtH+ z)gGIiI@gFpabD(Zyah35!W0Pu?C$KUc5MtATTl5Ly9TW}gYlizSeRWNZ zfQO*$tmET*>#UOGV)LBTbd(m0(mN`ig%zD01z>IbnZgkUFvVESg)UgLPJ61bQWqwl z;dH37Dl;`QijAZz5-!v%E)F3pi4s#fI+JMWvTYkV=o=USUXi!S($5K4UjDwJJt(SZ z13tb=%X!vBCY=T2Hz}H_PptgYl?}h@f!~5xEEL}~6NTOirvRmAI8b8(3k)Vj=cYZZ zz=ZkI0K>JwFYcTf^bGber({T-g$lV>pca5V4m>EnWx;DG0KPZ9OhO4uS8jUDb_c5k zg`x=s^>_6!(>jR8-^?v5N5D~oMqak!BZv&H2X(VBOA>J59+0V7Ibr68ykrnjA!p7(E52bS8M+jPz$kf` zZy98g!f_aB3Y_j&Kyo6RDd;uoZW*94`OO~%+&j{sw)zND%RcMHNUEJXiYc>*V#eqgeWGAicE|RyN=J_*)e2UB*=≷9t;*jWC4`fk6T zAv6OoK*M%1#_-(%daVd9of*A%&`vnGpt3ld|)~QLs z)9cpd%(2^j4e-v3YY#dcQ3eJEm?Z~hY6860PmW8ICjyQLD!v~@{Liq0CBJtA!@zMv z$Nk94`|Owc>_jjP0C=MW+*E;F$JRqEW}!_TMIzc+G^R3~NGAZ#z!T?xbK~xV5dg34 z;J^mn0T2bqCeUH1_%K#vQANqC4M)vP_8j>g?n@EKiM${J940olACJS**+&P*ejm@*8lxBGOp5`4R8~_5o8dupr(G;GhVDnyc~g;r1;s z4Z_}O(dN!{OP)<#2v7CPM864*T#y9DjvWK1+cbCp$Z5ypJqAG8{Xni64qgXx+16Y> zM1h=C>ij(Hldpg1Zxl=jE14ez0?gbY740`DfU&!{l&2F0l>_YkCQ^y7cg|xFA;0S8 zFmqL3g=embh}%B>7z}QVnU(cL$1U^-Mium|+Fp5Zv;Rov1$k?>yw49qaIPfdgU7y) ze+IaO|6v=@asc44wzhT*)*_e)_TYs0_)h?_%{AIl0>6P*xuMY`!^itjw^Jc$h3>MJfz$+QL!0Q{p;r~x-Umi_m`@j2aQ)ZixdCpi;FJ&IK zAxg&Hl8`A$#>yPFc^=a1HKh=lGF8S1+nf|dNQO;i*fPXM#^2}pe%Cr{o%7#W=d88= zU=Po<@8`Zh*Y&xsYp09JkMqTt%2YQ0MV$yc`9N5-pC8CWw};;A0Fy&lqG{-gKF6lp z1ki>2>AO({jz2UxC#dwQL#>@fg!K2K!RW02eDt>q=)uvM>tMTE1D7fWAKiOVQNfJH z6iCa+tN|pjhhizH;@W4zcwKZpe5mTkr<|oF*B-|+sL2w($v=ZoEqQ_$(*a6uKhT9& z1YDh%l>1E+CoDBRqEk2iOYz=MMp>Jtn6BsIi01vv?0}k}c&02&opYQ3|Cm6)yzmK| zCN9X9IS^oGs)FfbI>&cX3p}(yBMy3ayBUSfG+z0}BiP2`DXb?RfmIvxlB*zIzi}fO zRK0l(4P&HP%W}?zg@u^d*tNcG8i(e^MHeVvc*e%Yz%sC$%PUAChp5GOh7~;tpih^z zCV;@S!e3?ciL=Jr#V*dhfp2k{qJf9^ewN#W*fxij6wTLN<+DAT&2)rC036$TID)@n z_7yy_T?RqYWfO4*y%c-f4$`P<6yitNr8!tCF`AVl4p6(Gj2@D!9b>WzAj%^pJ3IdM zYk5F*q{cnIAq&@-+2J{1DO<_{03at-rR72np0aZK{IGB6=(wA}=g%1#nLN`ics}pN z3q~M4)@PlMq@xmVP$i@G5Td8zj{MeB8i`k93E*&>hTw&6hh()VKpi%9ET#&aGF^#1FjHottyhIoFyM9oiVza;BxyR#9-y!(j*7AZfFAFqO7^jU6h>r{JQ@Wzky z8}?v%1g3Sg&Om9jOM4Vd)( zkfxx(Vun$~>*I}cJm!rP-TM|H{oX8BlyjINxI!0H@nCF7gwTNPK>Z5)F}5UA|gVwM7G8dL{m&SsxYakGxX7wEjGa0kTMmE%tPVt0lTRWkb6-bKwc!B z>nZb_4n2aVKD0E|p1V-U2ItWQK{;N4L}tLYx4D%vFAY@g8W{c^fLXAp_Y?;BBj^A7 zJBDBTSwE(MaX_-2X2bjTD(xte2@wzAe@p1P%+C2Uw=sZX$wvVmrDYPiRdPr49Hikz z@k?1lKmLGdVf6Lwqo`#EsW7+yb%KMh*_5uplzmSpM=~ z1`;C%014QQ8>d>%230%uvNgd71V1-R%A-fK3krkNU=%<}4m+S)(S;A>0?7#pj4c}# zF7IK+YQ!^P(=J4AY~0t;)opV&>}m@qRRwOzj5hCb@Tt}_>p7eeQLuy={v&C4e!SML z9qPZlL95JQ$%xfG3?8^>f_*i5!(f2hG&37 zX#*SQE}8U6F!j-+YCuZh9R`Xyuei8pN}JyQB7I=S&*pqRBEV>1gyOc|r%@9+DSETS7FT z5d)OU;!0lRKd{_ElVcVnmMEnk0ko~q>C=N#?O}+?xPALSRGY`Nv*z=nqN1KGD?2Zy z?Nw?$-W&Yx9g?UgYWQ3xxsCjn@J2v$Pha2q&?7MeIN?~>gkSmWuQfo=r@eRMoSm#Fu7_nZvP<}OSCI^Z$~ek=(k2qVW_m(A5~L|8V% zCf15H_6MZ>?bUZ#px*TTiD zsqPc`jU8f*)*T5CXd3wfBF0^a~JT zWo1Q&)wHJ~3ZRCDhpnT11NXsrXa{o$YSF6om`c)(=Olt-l>oH@D>wJT3XwVu*5MX7 zeQls`VE&B)KGdNnMfgDA$}#d;Ixn704V-oBre*++L%f@*+H`y6_)+B>vZ)yvqd(}q zGYx6Vh8LCiYg~q<6Sz+2Yb$~uO7!H(^6~CuL5-xwMrCwlzH$SJw4(PSyX#rr*OukC z-n7}`Z7Nra;P9CEmD$m|_d||~*U$IdCoEsn0t+0~X}`8LKw$|}6?!{BtODeqHK5rp z2XL?s&_@eDql!$w?C(7|LhZv+Ad%|*u=q?(?mI`Vm&@;pel*;h%de^$_>1fX?Jzkm zM*RyH2)rIm7-aO-Ef!=NNQWcS$}@^SOI$BszHAKIgEX{Ou<4B3F`_FvTpPO@OVrB$ zaT8WOs_*@b=H}*Wu;>$PGZM{4z+nGAb&(_kO-d z?@$%aoK`3z32V=4sclRt2JqrT2Ar)JQf|7K*vLrF-CY)~B_S(k~~ zDL$WaK>gEYm2Z_LorX64HZx;G?NVqBxDD-7x1fCctaPmf#atW}DU?`At9eQ~^CgGOxOx134 z?pI$}gu+Q}-&BPhfYm;djWG;&^m_mfB?2uK(NXLLp##F=;xwpT#=T{KGbMxO+vsS_ z%#1yX!X;&8^&pT7Xs$?!Cw>%1#SLk5EMH#FOD%!~tYYP;6 z{GWD`Tp*tgHS7XbGle4uLmnT@K;f3lWXxRRlaIiH1$4#K z%q$G%>=_`q21sNeaJ6BD>4JO7`4)eJL23l7qSHeq=88c(@_=j-fuY=6x~0+1bnY7- z>OB#dq@Ay}zNl07dOMc1&^|OWnHgrISsA;0vDqn`^02`qdOX#EHE;evPSs*^#kOM3 zQ_*vAND4Yb5VA%EbcKm@8rmKJcUJu1v-1uc^Mm>Z23i*{_TiJDeU@KZ`fk13R&E-m zF&Jt%D=R)&86!ZPYzK{eboI*0%Az>f9Xwj(k}Y&Z&PlcQGWTIsnMTxeE zsf%mJAuspr3Nb=H5=T_p@%-oteSAgVWj?Ua$HRBk1OU3;lSl!A0AS3qf`ZA!_GnRA zS!N*mVv>@gE)?P}UuK6sFROsS@}D>DMEvIks_87UxIyaNirzC;s1IvuYM|HF`lB!1 z^!oL9pibMJ<@0N6IRlynV5EU(yz6A$w=e+dN8w*Budn@A6`W+myL2giDtNVpxr7io z_YBVS(s#UiT{}H(E~ym5Z7!;=E&$?*i_yPm!ZGFLt^J^#d zK>a%lHPn_-dd1u8I04~ZRvE3M%+~c3w-&PihOjocPkku+Sdx>%ghlk@M~Dx?sV|+4 zZi&Xp&ktL;Hie3KV?SZ?uk@Vu$>5YoOa#m@nCo=!Le(gL4BbkjB5!0x3NserQq!R< z(%Hok3%m423k9s%wAt^voVYUiHm8yH4}!sS z)v(cJdHyqU766t`INOMZga?<98(d|gy*Y#7l{e^-6u(w{T%~ke=hdNBj-2LKD>pJO z-^59N94lBtF)ld7O<9oRXf5{r40oxaReoAe|M21ThV^0`=2Pzw6~EE&*|x4IWmY7& zHgcckcM=9<#AHI8_$DqU%Z-_q*hnX3dW#Wz$~`6#mopq|84oC0_;(pkJkuJFKv0S= zyu^N&yGy6cli*D8(a}`W;V`@+3iBzE%9~4s!>-s_ZWoR`*!;7-_6d(%{RvyVRp<9@ z=f@>?^y1qWR`Re+ijRhw3$2p$Z*Ix>nmIExC=3Sn4Y;0vrA;UhR!t}xFmv;2T}oMf z1gy%B_t8y-#sy5lPMSN6O<7Zp_oq$XSn5# z3-d@c1PD`CH*%`v|4lFXV~?zPwL3((S^fkrl4#YO5@wOu8CS1vc&%ul4Ozp$-fg=M zj*!_YJC2~QrfIXuUHirL+zIp$r!^mR3^Pb^4gVLQ8-Ad6UHHl+6f|wqJh1rBo*H|8 zaNBsDxrdzQPHS@jv4rRBkdz|V*kP{Cv~pU|eQj6v1OLrK&)18L!A*V`$fQMbKQBO` z7YDXt&B7TLF{WT21Kg|rP_CL!D521XEIjLq>MP#enc8mqpP|s1TXHMjoJR z4QkOV!g`*HDo*J|MhOV$#`db^OjnUQ8awh z1qEDd6TTX2jm8(^SP?9LVdsm2<>bo&?NU@cJcj5}VNPqzrF0zA%dd&Eq7yaHbYYMv zy8pwgWMcN;D>C1S(jR^9jQbQ}@vq!MO76HBxNIrAIoxhJX(%3U6@2`K>iO&ZwKtUL zAN8p$&oZk;Pb9^xdR}Gn#rh}*kg(2^E_G$MZo3mw=A9AznHMYNt(t$ftYH2A_g~J- zqtdH$jT-HtG7JT2D;?09R#^&F{m3v zE>mp&zG+bLmxsH8wj#iQVEU&)fr44C{#BA?1K9%sMZQIlmH<&kj6I7)n zMrW2o>KUJ0HlkRzCAp0nFdrEzuP}Dy59_};O%_`^!HkKV@Gp$o9dZ&hz7??J(~nXX z{la{b&eE--?E+eWpM$vmsT!D>e{AsF@04wr}?!z=p{&=jV5Z!L|Ei(DC)!T)L)n{ya+9Sb%wR#0^kH zcN_=MJ1Z@R>Sz#c6L8KMXWe+11=SPgYmI03;mgvt82?lAHX_HsDcY#$*Hf)LlEjIU z&x4tDx%c!b%*GckM4S5=S$#cD#~B|H<9U3^t@k*UZMu~=A2S`ZZwwtKdPrfOFMpP# zm}As)>Fj8sPvvLM`)BIIs62C$F890`(V<$ipDERospHU%8elx_NjT zcEqju;UNt%IbwBMC)vW@aW@*+O?Z&qOlUL=wx(8xLMRTQJ~ropFpAnd9W(u!E%BCx=EC zFSY~BfcF5fToOobb9$9mT1pSZ;Do!1*9;Z3*LpE<7)YhUjYi1B64F&fEB{^XzC>j)`$p;!tgx1!UCkdhMju~Rln!}U_0 z{yRRV3|R_W?76!a$sWLK9+mx6=UQ3`Qs;>}iLa-GE2t>4bJYqTB7%1^TbKtHnv zW%?}eI;&KX*ot)`v~8JDzjURym;nK(frOenBR+3VDl79qF%G%u69TNJtJ(kMn9Tkz z>51)ci&&*I=^KJa238%TRIQ}{cOkB9*dB;l>nAJA<8DBU;Yr0xKxxOu#sV~_oZe~u ze84ijxN=-f306A-w5bKNsVBD zKGl$naQW)ht*K*i0TJhgg8u=&3Kk{E$>v6E>&DlBXoLg6<<@VV7TQjKAtGL7X?>$dq2k3}x@qTY}+jNV84I#H1*bEdP$rrCy73 z_pQ@|z6GgxjNt2ceI7on2u=>9=z;NrHlOyOnL{%`Za4MaSF1bKvd5;Jvz*neifjWZ zTLsy4AMcHC{YKfz8!htRDQRK^`*T!|!S0?QV<6}cUeBJ=ludU4+7-b=)2`iCPDo0M z&ZB9K2APJGXA6*CBO?;;ApcIQa%2`{Co4YJA!qFt=NACQbKLFFegmEPw^09fc}X}p zXfHq!FsD`|AI-=;b1f|Q4Fhd_TpVPXPu&cJzUv%i(z)e>Nk}4_bPK7JXl}ZgE^7zM za+HCB&_C5(2u-#HQRGPQ`cR7wi-NrT#{H{bNOo%BZ9;m9JjVeH1^j&n;`&fs%SQ(n zcXv(@nQ(xj1oMaN&>UJ>H0T`B>$qrQoGSJ(E$uk4{-P0Bii%i!T;)hS#5cl^B)swY z#RgkAs+?m4VeK8HZG<+*NCH~8b7PbKmJ~o;T>cv%mjxa`?37p4E!r z;dZn|jmzIH!=ywQl5xPvldm^#TAWfO>FVkE&|(czJ76->(bCe2-Ur>wHgryOt*joy z?E^%I-P_uN4XW)o16fy4Sm^CHSrgz-Am@#I zcJMmX2=qT_6QE@!XT0&)0tE>Mf$Knt_<_+4Qy-icNE-JitpK6G2nmr09*}haZrY`} zX}KzWPLL5RvRDm$4~P~!f?7=g8Z}KGhx)InuNIg{i0t+m_1W9@dI5)nmoZen;Xf7p zZ{E4(5l9~&1yv$se|{?4!Uc7kTj&fqfb_kmzn{$Kwb6-k9CqSql>q$LAivfpu-&XN z*pCXQ)BF<7x1gwKH;_xks$ z`yl{O20YYay6=q3KYILQNz ZY43z<&opYh;Wtna16?DX3Y=}&{{SD_cY^=` literal 25618 zcmbrmbwE|?w>7-!Mnbv;0YN&Y-83o!(%n+h-5rujD&0sU-7Qkm-QAs%@3YUjzxUoR z{`ubToUI-?Ywfkx6LXF^#+WNiL0%dQofI7cfndqXNGL%da51o7sL0?azOB3i;16WO zH_{T2N7%m^&H1qq2sK1j;j#rr(PNv-9F6aJpBhF+6I#S3CN*cI6tvYP zf802WJtazVb0WuaYO_O@NKsVAaa50!l#)=cg8w<^_jxW<=Kci|Zoy~*AtRm6DE(ks z$`X~i;hW^IdFHVx^XYr54Fj@&9MbTU=iMfgJJ~T2sFDUvL%R+2ba(NYqWiK>3xt^1 zn)=snG&8ylcNBxbU;q87bNrSAgeDLNI-3Q7qSAD+JrzTx3A_%12fyRAD!~5NDiJz0 zIN%Nn2YwILWAw#=qMl&%7;|#?=>~#x0`K`-tW8G8#{;*ww|#Yz{=HgsAI8yX0Y>=$ zI0XruNaLOP|Lr>eM<;DaFu?A+FHC+z(M_2$UV7uN3Z;_)F6_qtkG}B#&q*oU|BT+N zkstpV^wz#Ll>N_vF}g<8ns)os9Lj2H+4c1#kPHRJn1qDj`FU;BpbqM(7>Q?Ksf9E&2*f2O8g&GsDg01=T}TtA4CV~YK(&bA_VOTL#7x2@m(=5G zb8Wr%DMZZq((Uc{a|BLKZbnvCDFXwVY6+S2EDiO*z;dIWsCF~%tgL5n{Rs1aVa75;Su)eMa$?i(3BFxxVyXP>?{V}x|>G=Ih!MI)79pbO+C zX1U(Hr8>Tt(pyoUi6(ZtcM1&)do2}Am@og_#a-ULfe-@UfbEN;r+^U{Lp}QK6aVQO zf`rz4O3Lr|bcYRI@RO}Am`apHXK9{z+0xO-kf%@M{D)i|`O}A7GBPrrLWWy?9_7mn zYUk`8@2{>F-(4H*j{i8e!{S0S0K>H-K1e9YB{ZN*h-n* z+J)QomDXOxZR{u4(+X+ZF)Z4c^zT;X>Wztfe6NQsu`dKY#wr zQ2G%QA0H&_ap_ox9MWliH8eO^KQl7}<_UZHR?a`5p&WHbl7$zAn}yIUf-vQ7LoJ2> zPA^5o9#u6Kw|3s0`Qc%le0_5h`s|CNdy4{NS63LmqPBKK1SXd0(BXA1b7DV-_nm8* zrC5#CjQ=Z#O;^g+znjUfdnL6o-@b(%%+(s*obS}xuE6ttnhgZ&z}UpZpeKqVZH>!x zKul2)*Sda#FfJ}``Qi3R+Re=^WY2LjYkPnGO(Ltoe4U;8Ph&KC`IPNFEmjTE!#(W?U^M<;78Az!|x1acVtTK*v_5)l{ow6%=(Df*Yh#E_<@ChPQ~ zu^k*TvbmM(?rO(MAajTLckIAYQcp{*YiX& zf`sdr!#o9Qjm=V22$eT+zImSy?pGfh51t(NXNF75jRbw{N>J z*qVID$CV-5=kuqh_Ga9qU0q%04}-C~l5l}Btm+i4K`^Fs93enq-&e*1XbjzU$1LFuqL z(0=##EK{|pogH)D&HwxN@3T(gI~VR{Q=@wqU}!4xbJq{%zkesiq!JN2=6iGV6b#F3 zosfRgfqa)-pu(FV;xzIAM%LpAHN926x!ScvJ@5AGleG%{c6czqA^QRm=3{v);cP8! z(-lT=b(%;Jp=h`%A|G8{%3y2?^hl|Iloax&t3?lp@5SDf(dNLng2Fm?b6}t{MWB|^DawXUba^`e2UN)Ol(>Gm9J8UtB@nyvk6$7 zQ4G6FeKVu*Xf){{AqXi#Zg}aQr)Y9=a_p|h`tuG$!gV`&$wFM|(7)ak*eHiBx7Pc! z)y&izaHTgd;^kVW?C*{qAJ{&B{@k{9BD4{-dn>)+Ce4C_r)VY5z|EqnJv^5&HSyY} z+5=un(~vvP5YDxvHBU=Z6GUms{l#Xo>gwtwe!FLj-nXA}^7B!y=o`X%)(p-z2jSRi z88F~#Kg|)geUH)41j%M%A2Xv;0DMODoie%)1C`nWUYLcK1x{Q=tXIo3d7ij;l#ohrA6 zRSeo+1?ezPbYkm|e^T0`C=;F`iLR}Zg226>Ompsp~V8cutDJ671Ebe+(zlB3}B-~F$v);$(8IucAg;(OG$)QE3rdxJ2CHq5iz&|k> zDv(Ve#EOEZgnPrB6_ZZwn>2)R&C`0b%UrD6Ov=H*pGGCID=Pak51y)JJ1v+ZY zahF`sGxIM~i^xqk5OxxW?RD?{Qek z)0@F{4jOZ=++QvATprBXfk9rI`IsK`mVWQd9S7f= zgor%)b*aK0gx~~DQZh1F*oh#2#W}449{V*M9lT@wl!vQ5W>z;mHy;m;fF|jFA-pXM zq9@n}$0t2huy~WiX@ZW4EM8tDiCT||ghSY%q-*2yeNw@3{WosYX&>|UR2da4?XIt{ z>#K#~tRjJ9Z1c?zUY5BOpkVTacl)apsTw>!+$V8;#HN4u9Z|kO?RyNWG#nk~ObRBq zDN5nX!2&*w#(_Nw?7sf~ll{t}XCUcOd7a}L8XEfRJcoupVPtS8-j11?+YAJoU+K4G zRSZV1?-Wg$3xE0C!E~J+?Za96<1)D6p6!wB0C0*7-*LAgN7pK|P3&L-d5a(GkVW+> zh_D5vrBV(KToAEL1!@)+EC6Y6(1R9A)33Noo@p1^Z(LZ9@P%C5s(ttn@G;$s5M*pB z?`s-EBO~!b?|;y@qWrn;#PG21>gp;sSK#jMt^&ZZAOpQkVuxdjFYLFU%`=}MW-+Uo z&ctyNeP>xW-RdKnz@ill3-g&Fl<;8jn>gy-gK=Em1?PW7>Jc9jdFo85F?i*SvDm`?UFD|gW##D9E?#o>Ws0y z4!^OTuD!C>sL@d+tHq{o{wOprCE>c>@9gYAeBs^Tu2Z?!ua~pMbRm%KrB^_a0s&@tNEaik;zpuM(5;Uv!=9{ zc!gsp0Y1M8whVRWg^ba;Kn9M9jU6*5k(87Kspw2-@&68ut32-Cur;q&3+|r{iAsBW zi;P?F(7#W^EK$X1oA8>vO!sKl1qkGGcsRm|sw)WD005x?ET&#%_T?k)bDF?ZM&D~$ z#;1m>U12f3#=mil^=mA6Rq0E+`nuwi2)Hmv{4sxTz}8{=9?=e1aqQOf{6lLbT&Bw< zwTn=3aReB~6TMq`lC~8^Df}6Xxkt$m!V@81H9ao^(n#cJ(FcA)R&CTa9>SFe7qGd7 zZnm=_zIJQB3=U?i{al$Lb^blE??uA8x+Gp|*Tswu;NiiU;WVQtGZCO;VA!p84e`4d zl4}>C`C57!xB0^T{_`hTjQZI9VWl(V%lGfQZYeP_%jO2Q{N!}>^zgP$7{Z`bnXWSb z(C9ms(79sfa+sg)J-fmaPQ-SJm)>NGsHoGQ(Aq70N^N2eCf z;nBP!eS|Rxfa&)6w=>SH6S2%X0!;*2-^i(9WV;?k}~9H7b$!CQAY&o}oEAq0+pT zdNtyqu;|uIVyB21C!)7#xX>hSS@y{OgGh&?hhiPhRVT*1kcNa6N5~Cm>xfT92 zTZQ)O)vJR%czozHGAwYVNwaQWV)OE%k-fsAB1HfdpNQVEPpi-b2I5%wUH6(fv>iG^ zX^ed^7t0M04x6uZ3nq5;)q-&Fpwo{BQ*WxIFjntVTFB&Cb(=!PL$D2|N_E;x(P`+q z%ISTlt#~~F4*(L`sW6f@WDylAB2$RJ=Ts;d*nB*;{DW5Xq4 zprtR!#0ftMw`ex^wuL+JeWw zDw>d#R8@nks<>xACE~2AOryjcqDuS{N}WMQOGm+tUa(n!)%1I%jjYoXfVL_xes(nyMJ1$ZH!u~TftJSy538jX zile48>_xXN3fO-y`8*02t6G*9NyEN8=HE=DZEtK?>qc34@T#2S*)%b3Aw~W4r!F1Gv#56TwE1ix6W_VMJR|^wcCvM3qtX7+M(ds3WzGaIHnYjz3g>pQn zSDUmO#^{h`&r7#E_8k;6PHs3IamnfD`v|BIa4RtU1|tDXOrDb_e{X}!h=}?HDw&wn zR4hL+2t?*f_!mZk%%*nVp~78uF_It<&uKwW*7EjD%CbKmUBF%%5ag;0N^SyOJ_-Q@@+*R+qs$bj)ESAY3GsGQY!ACc%6SBv2QIo{|D zfs|_qNJaH}dy~YT2@4)jD%`iozkih~Z(DD$)G6DrtRY_QUS8f9q%@gUn30TLe`6Xo*M-s$M}L+N?xjVPUQ-K|~6gn&`=6 zVbOfW?V?1i$ByrvTr`g}#Y5OZk`uZZ|3J)T`Zlo#q(y}(eL&`9w6utrO))eNRR$Mw zF78>b7Ol=TgR=hGw34J{Q^Ll=0)4yNqev*c8=cEY`a|qs{#h^<1-er>AvBLlq-gXR z&*$g|PPjkmqS4RaBrJcoEPd^iimjLOqq~};ltdjMdx6x_eh;(T`P-w`XpquCS%*~z z(5CQ^o1pjUy6pPa)~&WTbrlO6FE1fTQfcO+!Yph!1x;-UQwtj2iRo0rSiXQ3);5unC}#pl6|xhMEulllG2|>z$8#GJ z)P*U4FjBG88hf~igWEwif z)C$&Sf2H#3V6nIJSa{A-?Ai>r3Qq`J>VC zA0O<_w-q6vct+}EFPP-j9~z2xbsuaDefKWp^5=b7)4jV9abBlr8&A>R=^E#^;<2fI zW6rqv_{{S1;YYsA5acJ65$WDHFTjlWO&B|@GL%q0kJt26(RotY`FD=A&|iuFItqJ> z{~9}vFb>{#N-}prL!7{>U017HY9EvO_!DMVBw%^wYCauAy+G8CkdGAq4)wPg;%PH; z5bHR)JJ^jB?E?wkuLon%`2BZ&c<@2ZNJh9?yo!-gkB~<-PkpOv$KU2l{Gv1}QM_$!SuCDV<=roNwaUstfMV<(?*95-6@6WtFs@kH5>R`y z*1zjc{0Uo|f~_ag0xg*C#g~`(0aG_Kwq==LdyPvkC@aG&>^7vVwCi!ZD?exU4vVq= zql4&9`C85V%w33czG}$h`uJx#j3KStR3)x zFare)ca!^7Giwc^9@eS5H(*w;)a4kXLW|?>mKSFjk ziGRnNX(GtP`G$PCS1#a5vIB8${h7$=_(&*TITzOoj72pcqerKv`PIu_H*VPO*34`h z>Z*Mn8_w}F9UWcqN@Ns3T`qTKYZ8&-xh?q8istLuiB}1>$B>!r&qc@&-;iV_ zBMx3#${9QRR(P?XQlttzKRs}JGa8yDH47DgCV74+GeuAc5`uixdAiXLSbfaaQkPf3 z1|WTxd*8dmBp!(Nhv5aM2G^4;h9pD}*fQnvrM6|kDH%a`V$e|$!bU+*;hg_Y8A6zh zaLZg(V9krPnMDJo#>prz#{p{w-C5h0K(1p2w8N^Q!PtcQygX!(Fpo~0dTpj>f`SeQJ9aIxJqL61+=s#E_pz}Ar^3C(MZNODw5-+MsE^8RFnc5B`mEqqBVa)16oSy>q@ z(_#=>Pb`fvT8c}hPe`yVv<9{r67`{Zq#q{Yzs=FWJc6*`UPh03-OjE=)9(Qtt&e*UEItF^3h7 zFP2Ty3339P9_9XSAWQ6m-!M9tvWEJ*?a_k6O zdf{}863XbC8klR%+USky$W=hH#|$}So_B`PGq;pYfI~0qblZ)T-D zgq(DuWsF!!7Uygs^t`a=pu2_`i5_5RvW#qBA;-3M;tKRrNT#~$ZSX6Nm^B|Hqg{+!DalZFyQUxbhT z1#$<$@;~Djb50@W%kRz_F%*|opi=mgRG{$fi0TZtI=_(62Kg!x#6k=<^pxIzG?*cmubNH`#YJgB8YN6b0Bkk~y)ZU!2tEo^rxDLB z$m6iue;`$mi2lR&R4oHBGAzul<jwQE0y!vD^8QP8AXWhgnu_Fl_}K3>6v-~hP*EDes)7@L)^#~mEs-bR$Y8tLrj z)5I@T3!~Zmdu8xqYL0r-K8F6_mQn<{?Rh0a>u!kxn%nQM7}S^}Pnvs;JN9QP?WS~G zR)83S>Fng*)yr;HE|AF~sPNHFtZPGzW%D)5rg-0z!K)=T_eOjZEW-JWgz_^YS~ODZ zwY9H9wtXwNJ0w`vhewaRH~^IyGhfSuVa+eTUuf0?fIEE;lYh#pBSW18nKAdkG$E+( zF}kpjN$B)9t$Kwa($C6D@ltP{c!qdp+jjk5ubR=Eq?og`u2SW`hS3~sq8})=6x%@i(D4AFiGONGlpNq%7gc(1FD za~+@FAkX)BMO17&uSdRjF~voY@}R>oS(aq@@h>tPUoyi=Pu8Vkh3ACOiGJ{9pYfgC zr?ON6FOx{o-e+FOK8Z=I5`(cQ9_3!LHnLyCVCk0Q860+6iFj^1p^|JdAhSSS=gYU~ zN;|CrH+*3ll&z#9bFeXpF77vJ7Bq7x@wg4+Ps*RZ(V~7RlbEVf=ipbxZsa)kt~lW* zH}2{EQX0`Qf`K)ym)SjWVxQymr7Wk+wq9EZ5PV4@UIM{bWWg|P&y8Zh8Rd`DPYr`C z<@K|uKC9E%8j=7b69XQth(ID|WSr~TZ&~dYD2WI%NJ88f&}Mkaj;)nKGn!&}S()b* zNgI@e(UCg}gpuHBNCO*xNzqJ?NuWGl5nwE%0Y%#NCshzZ zHLv|)&NN6br9?b@5cex=WTH2ZIE<}zH zUm97th|h_EgDoH0mB+07V|pYFLraLB?RPa0qIO@NM;#XIGw|PAc$F`l@R%E8Un|@DsFTZEJsc?-=UhpD){fZccdO-Nrqd}J zC&|P)!KEbx@PN+0oYl82nnXRb=26AQo}prwfoKhsfGA8lMKcIIR|AK}wwX?q(9})5 zEJ=9n_}Fl>1-A(K{wKpKGFkwGk72P8PrN69-9~2vWOWGhWy#3rqP1s5J4f<{6J|9l z+qv}Y@+hT5(lnam_j=`r6xIuKS$ZuRD`5{m%!V!8S6ERYAjbd+UV(^u$D#N(+`-ua zvuo3M*s9|7^RR(DAHX3c-}YqQDIq*~l)|AYK3gv!P)qq^RUxRD%~w6=QkbsoPZK7G z>ZZ-j!{f%w94A8u81tR*xiEfSh5$WMEp`DOUcgo2hC0Co!Xa4m%~D@TF)QO4q|4i2 z@sLBp3@JI+;;!o#eZR_;Fo&r`R4oYZ9GM0IA{r0vpm&_i`x84gjU3{)7$+(dJY@AN zH9!)muhfC~gIiT<`u>BAg7Xiwt?%!^e5}lrm;W)71iln=bTp})uIr}lpTLi1^ebX# zV^*3H#B44BT5I|2Nfb2f$qfmsP|t0w z+O~J4zjC(S^5du_hpQ+T~RseRPyU z4Sa^F+~+x4HLTM*mj7P-B_S;XL%#F{#>G`xSwGiYDu3C8;ahj|E)lw4=vTM7r_f5F zODBW&D-CtcY$Q9yFO91+Q-kPW+sd2iG_L64mx+Cj7HmG8ug3@M)=?^1sD3^G^dDqA z(J2GZ2``aitRwu&?QXv^+o42i29U@Hzy1tnQC9A_n4U4a`DmHRJoU2ZpTYr}`zKVp zI;;OjHlVtI@9Y;$sUWMxt9a{-V)6YQ@mxcYC{24u3uH9XDiHA92r8qxrHZyXEgCe- zefTS2B4O}-cB%UNy_{_EqWc~Tj1a7-;Qmb}12)m`&?xlV)NI*nBR$AT1=8vw#^=Z} zs#HwdAKg3)mxkd;P)ed7_8b5hAQ@MD^WgCST3~P+q3!Jz5au+>!A(6%4aB}g!lAnA7$s$q`5NTX_r7tAfZGDg zbi77A1LyP$9X`OlllgO+b+PIye95-gP}+y3v`?X`zm@^!*sm0j@HZ`@0K>A@!EH%z zKn1=`yB?_nGE7`jr%K^X)%Ay$`R$i*JCzi2#n+1~yzgk*>(l#>FY+5tLWuNw*dkee zfBqtM&h#iIpph@F8WDHVC>49$4z_x~uKFM$8Nf^ucgNevW{nz|AiFBji516$(DYWk zenR@myjt_6_X?8G<2=;Zm@Z?~Oj0G5na=}0SN#AtwL2I&7E7FQjv9@o7=yZHKPcC{ z;+aTS>+pOsfiAcDu)g#2#s*K2vd}5PMuY)dn3^(g+lggnVbOtgpQUKYz6OHJZ%I$> zqNqF}YVRA3q8_pM@qEdYz;goFplMQr=l<4L(DFlp516)68m&U+s?5-WpEEbk7qd?T zVy>U9HNyM;#sF9h57ORz4f{e(kO}>`A4{idHLm{?YEx|_A@>%|THLO+AJ*m<10QIk}v5uulp1Gz8<*oxXP zD)lyD#2g^lLtGZ!P@o1=$v5?}>J#6J85_^%s`Pxms!O5+R-ker>|rEx%8)+L3l33d zsAIH7-JkPe#L#6Gn%c5p8l=|KpPBHvqekx*Ja5-$0%ll+-__my*Hbq|B2;T^O<~OH zl&+#rg&EqA%&P7^%4=j>{L=8sk~TBa3vM0V2#h0v+J)lHEGQ|ekpWROz&=MI6S+yl z_u7x2kglAGjjd<3JN%@NxhzAk)q6oC9bu|!iS7rg=oFc4+Z((h8D|)NK{(GY6~W7z zCK79y#^_%Jsj7x(Y^r8qWpboJW5p?KL=8+!3h)`P!R#Kut`>54cSjk&A#WH@$?|$8 z*HoGBo6u|C(R=NDX zu)7>;17+N&4LA_sRjsRJe3M5FVQc4Blj7+=oD&U?NI9{c_u8w4od-8V>qJz`w(SsWj+?XeXF zGGjJ7yXqwK6uJtgFQ^~t!>Y_;0(mFB=A5dD5`cv1o#iu^=sH(?{7V-+Z5;3^2%QaZ z{T>+TN`JiD(rqS;?tcxcM)MKHYuUL7AW6jFc8lli_p6Ut!KWk81z#JgLDduS@BMqD zK&7dnNDi|w0j0#ZUEfytMO4RQ&PC@kph9u9-N?JzAs2)wyDzcLSg?OzYj=g!h2o#k zY)H!wad)!Atg)J!nlM9GK+oQsbuMxd!p5L#Xr1<5G!&wdWc7OO2?GLoDc1^9E~*@y z$5CuKLWn^Ub4fD-j>s!dW+$m}i#-LEo24cA9us3u68g7Eka)R?l~+TiL6Pp;#q!`! zSx&p?nibTRO=5ss?OUsj4%d)>uT|KhBBOB<)xja(}`5eD>^{HjvJZ1wrz+upI zozay>LlijwEyJC>*nmMGzx5uTw~}PHPj3BaAv`V?rcqW_o^{dXq6}>;lpDfcmayZ_`(#DR-jmrW6A@<@%K2|+QkfC!~xf!lA)(8^LF+D-E{#l(! z>`&SQ3Ag|n0YRP7U;E>I=OG94o@m;Qb=mziL89*iZ#^|`5a`Si=02&qdS&5d;ed%F zzC2tAPZe~588d_55|Sd!7>W>0J(mzC`qZfi3LLWncRLGby&wO9S{Y%&&3%F0>QlXi z!g>f0;sytp^X6CO9EanUJVfl%g+`9#8+Muxkz`8BAWbq3ITro=$q6&U^(V50D3Q2V z`T@VCwR|{`pI>0>nYCS23#jk^(O2uY`4@%N(MM5O1_7R9p*Hu2(>!g<&rz7N0%TR^ z4;Z6Q@n)fGoG{jL>G8p(>3sCf@87>sot+xu@?!Mgx(fl=EBD7j@duZ#o%o=F@{Dny z+{@5V60bmE^{c^|RJJF2N=s7TfjBsjCQRnA^{3luN{48F_gdn&gOg?CZa(x~C0crV ziFk7_>AuBZ02&n*^|vcooKex9P9ovRTIw~|gzNp8JqP;gb}2x2U)Gqo)-b-c{M&E#?6NH`R zGeu2>|LOMedAx5{lNE)zeMJ7$UCxc>zR6TzJpN4^eZK4Dt_6sj3X&5LSC6}A1nzPs zIu9`iqgehST(Gc$(A<^GOB0uWwNvP2*69DvXS9MMLn>%GGk z)euPjSeQwFg28acYp`C^_Dnv0%&x1e>p6<%gIVI_Qw4&E^gO>1=y)8`sn+K80N!g^ zESGR%Fte6Q!%Z9m6keg}C(<4Gd$R-Devo=voAw`_di9eY18EB@JofeE0BT)*hnNF3 z1Q_z<8;w%$Z_+u8kb$Aswx6Xo1Mu-zOFp8kIt}AHZizlGM337LrmM{Hts4(`$@y&| zfYlfCe2$j+y2AqwQMQT(8ZOPB6YC)zWpHqdOBcb^YyiBsKhhy>FVh*>I%gL9{YVq zf`&ODhzYLODFDlZoE#J!%c@-mgjryqM?}UB2>`C7^t~Nh{@K*qo&0q8e;n;FXA>Y+ zbQ;;t{!4HbNjxe7PoNLyz$<-xDlVxV2O$09HY1F=mFLkia6)vMJ=#G=09zvT zFQ|l9!kk!e)I@Mt1EDlPa*at$oI2oi6rcdcKp*B+7k^2C)ua=Mdiz=Dnu~g`hnf(UyuDLLGoa5>*M`V z-Q&aUy?c5`q+pmzmU7_F+29QN3eFjd0&`viD=AMZy>BsX1|}Fe`8wA?*1Wj5IEb>V z*9Io5)gU5WF>~`55Pjg|yaoxWjUBVbdf_uLg~@8do2e@rSCG_(|j}oj}b`8{x%%P}{xPhSW@g zV3gbJ%>s4DzilXQ_>Hn!^gliIcf1hR|MtftLtv{GC~FeH+sMhuS#G@`FtjH~Cgk$y zGAED+NQBG%8P~g$9yrMVn{WVuNesX;IF)^CKW0&#=DCTE^eJ^j1Fz_H#ZTRSEuWwA z(F4RU5OGQC&^*f7d^5IKVt#zcUj@qVL&;mVJvLwoRv2!`>$3FH9BTITFt#!I)zjl5 zkN`8J0Qz3L;;_Z_??xhr+v$3_hWR29>w7~LC6$nQS0p?WzjwRiR@LH0Rr=acvjIff z90q`r;MtKo)=tXZ&NeB>X^z7uYk+?~#@_jGK5TQ-ROoWr2<}DnCqNAZGxPw*(sEKB zNh|MZOmc=VlsGB650nsqLRr3|PDG>0^mqDXHudu5jji)00BWwFqoCY&r6ZK$>Ns9N zE>P>XhgOlkLv7-Mag4ijvU6fzARG&Lx{?N z_P)i$FMj3pboR0{?yOi`rdPXTD$BNKiz;;B759gEXZ3zory&Z%o4ojQ@p~)XCyGU@ zMt|@-k5}4Q3d{FT@^O2g^ofxXHlPw1UXS+^#!kx;u>`cMUCq4w?`8DX!n+MayH}Z8 z*tPAiPc=zt;^lByT2F=lD(Pk*RtkYd2Xp$({4~*3Tdv&k_9$vHgZ!`)bE-rWHBW|h zKuDJy8<$5!MFme-bOR%V0@#mW&>*Ecm>Ch$*4C_saVTY(60(1yeK_-r&!(Vaq!jLPhBF_0r6SdZts|5U3;+9HWwV=l>qZC`3 zbyAGS0Gc?rU7lJCNsfu}1*$oq<8eng%i`s>Cv$z9^ody*dQx<=ZhJM3>-QjMn?Q*>nnw~0{*T) zX~N-XxRm;xSQrSXgB=!@-Jc0yF$9-LTCn93xjrTuD60I&8)&d^q@&aSOX1SCXqvL) zs!1xJm!p4pxbS=23|O)3w0-U`UKcwpRce;NL{Z@08jp|vz0^qbg5fCYaz0+ z+KnHjZGEw%i}wx;#8_}@0a3dQ#4cdB)GRORaK|Fj5#&B17pT%XwyivKoqpxW2+07n z`ujaXbRb+I1M_p>XY~1cE0WPDDwH8>@R6Q9S*vrH;VfWg`Y%tIzFOjSXeu1tH0a2A>xqr(Xr5wbRTU%KP-X)DG12z&xEFwt#{HJc0XXw`=lAssJRR zw63kcM@f$!olGs6!O;gTShLK@g~6ntMwW0UN@&+_d$_-a86;q;wj?zUEYHJy$irsr zjdjG#eAZokkz^c2b;Km3SnKw$0MK7{`n>zUeaL)Ep7=257LY9i7}8wf0e^(DeDh^| zCUFc_YXVgp)`i1PjG5Y!*J#oZ&4n32D;?#Em%d=%&!XPwTrza@9_&b9(q%OwI8Rea+sTlk|XezN?aGLsWq;*0*)YrFb+r(mH!W7>pk5F98 zSy@@xNTABNgr-g?2Z+teoZGC5IISC&t)RU-u z?!L^lcnO{XwuBw#&8r0Fv%KQshziZdhsKETa3f&46)0H(>`aI8$B(sw&VPGcmfQRW zzyfb5)^q8%D$2-c`_Kxu20Ji2=<3aVySZ`S1pOZ$K^xc9PU{nqYu_JbWmoPGz`EH^ zF5vk4?D}%f<|E7xn3tL9Hv@d4v2k%->=Aoj^UWT7pfB2CxeZPV)<{Ps6Pd6VlA0%MEns7Fp38q=?~}m`f9X>XhG2Rjg9Kt z+DAd}}K)U38%VIkC9VuQe2xe-A(Ua%ZEeSi$*T)9H z3_qm)UR#?SIK{jKrrzZ$0jCfS=-TaM`^c0L<=_+ogkHd#FE{L!?%9v_{Ko=5&wDZp zWX()vCO8NfiGxun13kU78*Fym*GdNd3#I9#^`rYm$+WW`i3J6l|D1+b)y+7Ubo)4z zW&)~#-~8lJzHw9d%DPuhlkmW?Ws?M!CRZ|_&EfG)zi3NuZ|_^6)q^pr1n40Ta1E)G zWr8LRkyN5-uk6=Dw<<(InEH#6ZbVq3xpbJsc0`pv^l`w{A!H5oro9E$eAw6q8xU`T zUIJ6FmJxizqM}S6ZkHb2qnMw5#_kbxaQn8XNxj&|oje$!z<93d&G>Nt_XDM1kC?ws z81)Qj`LJ%jP?aMj$dig7_5zd131i`{U);^UhjV zqDgezFex4=$yi=Q2Yat9=k$40<=V|F1RdRH2qrQ zgwr3%l5{=g)tpH9j~AeqVQ%6A*2D;dMh6}MK2c>xd@$12w67hy;A!IpsIQTzS$ApU zq^5yPzd2sx!-q9BUo1gC>ruqUruReIhML8fzVSqi%{pnSMJBMpYi%eKU@prqXgaK% zI~WkYn#V6corfUad4v5Y|I7O5VQYUwP1hhz_lE7)u5FFr;hTHdIBWufB!7!TyOn6} zer)dowgZ!Z(H9vK`{hgf{(OC`0deq5g%L_@TH4}@>s;Au8xv>erXtSAlV~4y;9t$0 z@<8iu74_*eTA+5%DV1PjVUYwY!bvZP<6jFpQj>PD!SML_(_v0No5lHqfdtU54ukx( zkLu`fwYOBG0Ph+0#l1-DiAZ!J1a^0rsZj8x%u7Z_vDbT=_^>$v_Ew$c?UDgdy#wEW zM!#hR97qArDlzteOQjoFSQ_=!0&vOz6b*QI=Ct8FobGO4mzgE=@;&gz z0X51!fdN`O%yj_lh`G&h4h-d>lO9LSA%c%rR|=nzDEv{-^j?x9gf$I5?eFjBuw8Bg z_a7r4pO}~ptUm}L^ANw7QH*l5YB6<~i>Pw$1x+R_{Pi>g3#xk&!Y*T<(;s?}OGBG9 z)WIeM6&|`X59-b`z@gV*gDcn$3A3wHD1^Q6El0bBmqE;{YIfYg1ONEzU_k}0x3i;d zdjN_-6;Z}}-F1OoP5~RvG9>V#hrnkbzAZGm7=bAnCM^k##zSh9qia(q5F@5P^$%IF z4jD8R2QHX@2wYfWRgP?IOiBu7Sb4bJQ_Ge&m-y<{bbq- zA*&MWP6jhpHLR6&xT8~-M@vsY7b@tzhV!+&6NfISS#=e%zLZ$Td|9E6Svf-vT;=c( zSaV0~!v$N@!_8h=YG$<*Gjf+qfHv1Cnfe5LSQm-9qM{-=1`UiE9Hi;~a*p_wJst3D zVnWzw$;$;(6;Nj`wBqZ^cOMJ6;CsZbJw;rFC?kVnG6aEl-T7!SGFs&3w?jnU(ND5| zO=`?K+$m(Zb$hoqd}Q2F9hmbl;EBjID^cB~*=wP9OC~JNpx4mK4@53eTx{> z8iy_U@yW?mV4nKqqAkvXqiUoEdA%M9jsiwPMF2}-N2!3m1JIRXv@@E!eCD%9z^8t& zYzI7XU!^Mu8o|8&>41l|b%UlC#70s&c#=q|y8|1a&JXW}gTT+6hZOe$?xzXFaK5p` zoS!?UfZzul`nYDGsgqd9mEF4a{t|m=DfGysb8^e zJv`0I&IT=VPfAqxTueJ`8}yjit~XM_QUIs7f9J6jSU_v&H4F%}k!+G9iMvsB^ZGzp z3O_O1j)RaHXR^8*JTY^$#Np-_scs;cIfjesSYIC_mTSHbK{&DkRc(Vs3i zfGv5ju<_tPGbvJ-g#XT1zK2(2@GsG4@L-SvVS_|w(}XJ>O)2T%-~d*x`aYPTHc(QO zg9gvCwNLWM1$?uNz7h~;fZnVz=$M$$`Q~jAK$Ew>4ncS)jUhUlaWLO3rt^)8u9Mos#O}~a)2_xIe449 zdwU!o|B&DP9rF1FuIed}-)US4qJ_nNx zOv$k@cZPac29c!<6b;-Mv26hsJOpM90gYQ`9{8)3pay|)AFk_Pq+zZ#r`tQAoNyQ# zbh*$<wR z0BZ<33{cbwAo|yG6LmYlnfI(cyS7=~yJ8WcVCBJ{p#aJLRlMBA<6Ps}c0u2*&+?H5 zr|fUz^q0ZU>5!EeG0k71-iVugk-VmaQ+z6pp_xm`n81t95EoFA%cP`;ElFL1s+{Rd z5V$S@cktzze#}o)T*RNfyfa+Abl)~aT1Dx(dSl+crH>t;JM*S+h;rE4bV&ae)3hZr z)I1ab>q*nBMt5&s92=xzGvG}=0lUNy*ipfTLGT6*z%GTlKiW~>Q4uu8c)dMB$BFW* z+6DWp4dBf$U%tSFcdzbj#)RxR3cx%%pfdv*yl&#~^qDa>&#Q-(Sa&sF-;dOPz$*#F zfky*0eWh8WyA1;dF$hpAFzrvh)|vteh0ZHY2|N-YbWNRK^NhTs#|K1dAQ_)^h9)5F zQPEUs!8=a?06ji4N-ur~f*v88F6<2tuztfdYiuW4*{{c-GaMlVo1y~%Z4S?C+u8~1 z7{nIVqMqpjvD%58wAb{3W>Zg<7VCm-+}vIPKdQoq4`aJ$7S*I5^4`9dkO-9iU)5cA zIFuCB|u;+*?F_x<_2U-NufNFoAI+#opp;X$T$ZAP{EkX2W;U&Eyk zX#_Zh(p?JPLqdv*wfzc@URBi9Go18ZmX$a@E=3%_LaZ;!`P)}_#a-tKGqjZe=1RQt zN!QDZW1I95i?M&BPEllP8fTIo3(XUV$(LX^tsp}weH0M%kKW|fON$4(?oz$}Ktl7@ zH{M3zLG<+X(K1*+|0}lt)lRtpuMo%xajnjpMjsT6a`z#p=DrhzrOeWuYK~9}nEv&7 zi3L3K5vXoLz~^}wHw+aJI=jv37)}O(U-7rE`+#-xunAlX&U#H@)C*jU_bfxZ)VPG+t7=0;#oV$o}J+;;tx3c16 zdh+`CBHaEcFK1@Pq`teZ4YebMJ$KG}zi=EZI!7GoeLO&;Mk;c2aG=K|-n=NlU~YbZ zVJGd${Jfi+ySoc)S*T$p%z~2sra5e*lii#Oz;&xWEDAriGbt zN?1rJZ|qJ*sjsH|!kZs9@aHHQ_^2^mAEVef4102vZk@g{Zci}q+Su55J|y$#V}^2& zdwNYDJJg(LovJ7cJ*J_df$lLB1&$_`I-v-_#3ZHHo3=I_+!&&wqF5-@X@ME@F2D(T|Nl9X2Vl3bgpaLf!Kd6q%9KQSBE%Ekk z3g@OOZ%85QnV4KNc={r&nx0#Z0y39=mvU9S;MN+c_2p#{P)!7q`N_+d1FpXL5B$dK zu6*rDpW9li2<*Wxjlhf@IggFs!c#y z`Bq;4At9NrNYVW29?mdeQeiq%_F=_oQeWxrHrS~JC?^8NA66BP?SOd_U%q_FQIoaW z-#9t!6q5wTe{#$}***ARV57wu8z=3Rx&B?C4=T!CyGcEi5x8&WSAHoSuOi*pEC0^P zrb|@;GdQHk0BgkuEBM?}A<+wk78U)*6Q!GflOwtV(qC&2j_Qq{_D}n)TN3-`ac8JY z%BjTAXJutRYvqoXy`HIf(*3c=k0(c#e?iX>^iqv42SF_Z+`D6ff~P`_K&HCIe=1bT zSKDp}Ttixy!COu03=9mMU7vXiJ#p$gD<}FYoHU?-4&J=T;lmwEvOcOCo8rfgu>#;= zfn}ZfROkv5z-~*TN}JU(6n30L>Z=P#g0p~MHzD@QE=!2d~HBEvPZ z+4SnEAeX)F9lqA)s2wC}!PwEb`=bXMqjLN*E!lNw;rGm(d73}$swnfV z+JozlDQ3GrcO@B01=zi6Y*ChF7D6rzV~=8>SE+AzHK;b331^h0WDeoiiJ1|VY!hTv zZkw8N@`R2#=p}&j7_&Pk*k10?phApvm@#N7wW_!YSLQ89G$%syk=1x}RD&h)%im+m zHj*5umJX?vC`@+)K*Pq@bf<&*X^oGge5>@*Mq-w($=Hg$8vOZfxF_-xarp$(qk9=l zmu>8?H4Kn3MAf3}kkwT6e2iqK!-av%Dz1HZj-`D0vH$}TG%;Nq;b3R~DzokvL|U;a zv#y%eOMe!pr)?!ipTjWo1G)u-` zyM}5gGGxwQ?YAL|yt};YgEb*!y0@pNolyh|wgeExn_Si% zDuy9PzzLh>Ti{m$VEKa{8x|+FB;MlBJ2|gDsj<+7bV6wAKtn zIRI5IT`9uwF1QqZxbAgNZmpFzwqCtz%W5IYB{cAM7$5f3F?;l?^g%<}S3MBV&T3T23 z{?O9Wf}ET;=!FDS)J_IW>1P5_ZGWLpfYab-O3z=3mbQ&AqRNJwh>;7O1&S0B?0N?o z+msy?z9$8#f20Xf1I`v;bU|UE%S>AW6K)z-(%(>ihqjH&5X(SLBG|>4czA9$e6Od4 zni14lXk#@wzOzodXQGJKmkhK_xj}<(gUbFks0527|9a?|4my#Mom(0*wBZ2*?Qw_&BPYk|B ziQg2E@aibSzH<`pOzCpY8a#Wc{wUt@d8=gpq@=KXOAxJf>aUF@3|VY^rMiqEOZ4aA zfBr*hK{;8b`ZJCbws-wEw=k(>dN&x_BODZrSkiqDc^ThmO)S<7bxIY@Wn{uo)2?gV z2Vi~}5COELT1m)qF}E8|E-}!h=x8gn)|FT2{uTGW-&pvE1m)cW?)tdI?jsf3MT#<- zJPJ-tDf~8gcvuYDd)u9@RN z->cb0L#f9LYsGq01pV>{4%3qf#bakrW-#TYymfl7!8)LlYscF3rw&WE$2b|b<2ke_tBCAf^UdrFjgN8o+?RO+D*-Y#G}bea0!ry9-L z$U>I+4d`_6AugXz{*W>-J(*XuqhG+28x@0f`WAEj??=Pt4!%u&GB7G)Dd}MBPB3UU z4W8e-%|TP6L)*5eysx$T7PD0F47|(V_Nm%y?$DY-gWBVlFWJ(iojSnRm}eGXeBlR} z0!8Jc&P=bPBPn2or2#TU<{mOa8X7z{HYHs>1X>K2+`LMm@u{SE3VuMeGGL8ieD$E% zR|SriDv+*#6(9u_#6`V_T`EIY|IF-ek;yz_hlf&3G&p(oH0&EVl0ZhWZ@xU*|M8I5 z-H`obl+-nXy_za$#m0b5L@(o`R73Fx_XDIgLD)>HVU8L9_+0ejCr|lVScA~w8WK`N zj6VEFGoG>j(Ol?&A6<$`C4I)ig0AvJQC`b-R&yccFz=D`_j)j==Tx;1KBT+ZVd9Zs z@BU`r@o4&}DdYWv$|F+7-z@-L12i5O7_Vwh>PrMAbF6b!L_|cjUvpXZ9@uX@*D@EO z)Wc6L+_Yl&(_rj#H@W$AJFG6tkaUa6X36St7D8xW9Y1N6Lw{r~+^X9xmZ%|axXfeN zljmu~^_tF1z9Y%reUH2>>VEC#G%P94LVis5f_U}G;fhgwWOs+yT`kI|drERUzmB>Y zN9D_LuUr}Fk7;hI%6adGgAOTR$x{8hbswnHM$kt<0A^oXMk<_##fv2+J3Hy9WnoKi z?-sj_+6?i92cpk-m_N-ZH9GSItmkX_!zC5k)d8cc`_58+TPYa{Ogw33s7$4?#J9d7qn-HO~ zmi$P%@1r1g$yeViG^?XFv16^~KrZfZoc$!5ZS4u^h+`fu7b7gGkGjc;&&xsS6~*tJ zlVg*TOlwNFs)th*)cn4*|KE9X^Wx*ojeUAm6oz?Dx^K+?|QQ!RWL159H?$X=Vb621R8DY>t#KG;Rv! zDO~2$E)+d)b>w0G4@>z~X&SE8R)C290`6(^)?*x@X<}Mu{rtPT?VOXX)ZCB zLE+may?S#R4j8V>b=WXbKd#h%284ubi$g%&L}q7A*vDZmxc+J?niougVfA}xcZGcN z4Zz8orGqZepbPtDJCqDUw^nV{Ici=EZS^1hB_w)Vd(w528BXUc!*g{!# z5f3h*Z0H=8fm$h4G9;{Zz5P~@?FuU&-)-=;ooY%AfLRx;tCbp#mCj^H-hgVD831Fb z=nhR1m^-e=pA${}ps}(cY)!|)f(2n)KrrKVfp3>cz%=guEYfI)Cy`s-`D^!mG%$fM zgwls~3Yv}Co3Y+g^m#+Egt=PM%fSR8Oa?DjA7^Q%9~Nw^$oMSN_KJEG`Gzmm@y^8$ z+~Ql5uqdC#lgvPY8- z1k3)|xTCG@&5R5#jAn4Ke6ULP-k)O<5-Pj4T|1+lt5*fv?_WD6)uDCIY_@M&mkje? zBI=(HW_`N#xWB*0Ygg6ethww(0S*q1Yv9k^ne)hrpactsY&_FQLzpy&q_uO5$?t!ONeKR-W@HT8Ja)(*25M$Dws`y-e1Z4h zvg0vIyMOJkr*CQV=AEuT%zxt|5 z4=~Hu@ZhN!_*$U!!FC|HcH*0K%+{^9RX%eO0}5P10*aA8J+C(mlsP{$JIg5CNU}xUun{hN9|T^|4v5+Q0@l(?I=oJNA!;Vw8fu(JA_jOq+T9 zhqm(e=hzbuJs&ZGgg9t&KuJqXESUACAA01#h<@*N;7tq$`viSQ02wenU`KSv7(18& zX^kM-)IRCtPUx7l2k=0E^=Lqm1p;&qqW5Z=OaX@(qgGL2?e}d}2r?D8skIDk` z_WcrTv0Y8r_@{?pok7MxQfIB-ISf+r0H(Odj9(u2g90Y#SMUzX>MeXS7eP-?k4)p> zg8U{Ec(ik~KEo3=<&nO3NAeeYC_ZWjZN$EO_=@29hGiPzckomEs86V!T9~uiT1n9v ziFRo9wexusv012RY|IWc{R{tUYspyQDB&PcqEW7r1p!t zwX_oJ%*@OzP|Uu5rlzKxK$+F=Mc6fg(v)i23-; zh1Qm`#rR|~F8&S&I2Xv}T{Yp^v(Z@IF3TjcoC3a*gMUC^TVH?6h^=%~rf%8is=THX zP#NOIH&!i~+PPzj`1x2S=2SyB5C|SOc|T@lW*%LaLjyf}1eFLhR3;{N z9in3r5xGc%h1=7<4I?Zfk}jk9fBf_?{s{cp0OR2QhcCSw(4Ym(dOr+WC17H_4I^0LNobhI1K z#n4S-i@J2$NfBGmsjEq}6*r3N25C!|E{mCpR>up}(>=JR`oO_pxT>aumne~}t%d3> zxzP5>8UIH6%N%Lx{uk@VyIWPIxYEn9t?1>@}DkvV_+%DQMITISA8zUUVOoe7pBf#=gI5BF~QOzkcZdd8PlaHwUhi!5ZP~ze_Lw z&UnXzfaOCDKTCL7Xj?c%8xava$xTkp|E@RtLaxD;`$D|7_8dj8Dzjd6SSyd1@Mzw% zO`i|{<1O`)&n)5HJ5FDJf3|DaaO>*o0#a0dW~Q!VeNMXH;484VHW}3(Mb0K6aiu^z zyQ?FHa&C1swlxfY`vIeyzObO1qJ)Hm(I;=7i5eI3C-Nc7*};rv0h_q)?(PS?@wEbU zY2sIk4TLK)pyw~S~TB;mL3LHsuzekD)h=>&R^e#0tG`OyOzvj8} z9d~eWux#A1Y;p}kQZ%D@C1u<(vez2Bogd!X5A3t9$bJ19OEH%M>j{cA@mH=H`Z!ab z*_J53>wJceIu@6&VFRn)d+n*V>>&=sTXx{yl@im&r(Ub0gj@G##sB%^ zc(Uda9M;?{r~{Vgmf-+) ztC~r3(Toh)z1ijMt*vLzpNF2D1Q0|h9vmJjuqHCg$S~Y1G0Ek3)1l(fQqI+pFE(V8 zm1UgoMt6Tp6FTHwNrdR=un=Is)T4iT<~&uOo0EgOb?X*#6Hcmk@82ux>Rx1JWBaG% zWm!bll6cIL0%mTm)knhV_hO2fs&-iYp9~o-J82{=ZG|cP48V-6gng z-pn1erMPm1(8a|itLPf)jt9N*-Mg1b@UW5Fa;c}sMn}ub%eMqFZSU+To0!mthK1?s z>4kpza{I@ccNQ0WJf|kT!J|jl;#BkQZ?5XN?f*4EY@Gmbrvicw|_kBVyV?=Q9cO?mO++*a&xLUfQG13^3gyLazGIxQ3& z9l3FPDB-l!Z{H3@qsfVgi6JiLT}ugSC6Da9C`j6KK71e%&4>?OxIu`6g@cdJCLnMT zFYH3|$rdp@z|5mM(xVtJRJ=kPKZ%8f1*f`k;USijo!yNuU%zHlA0|~R$yG^P(&FXk z=Ti{iM90MB22yg{o8(-MQRd|4euFz9s%q1+6>Uxn@ukQ~#mdYa68eTTt=Ae7GrH5_ z;Q07c=aO8>;FU7zDw~=~*cL2ua&l!=Rav)Z;*RnwKUd!9-DqoXxAS;-Q#aN1oq#&>|F*UR-hy6O-SF^m4kWLLhzJBb zf0`!pHKTW6b$36o(S$cjhcjH0BMiQQ7vv=t78XQM|8IVVS|9rdP$cZ_?R9i?Ldwec z;Q=cqqNH;9=c0S9l^~j}1g|J6DIM}giI9?#)_SkeA!X@)jq@CoJx)%}mRC|f5BX^+ zE?o+RMTbX5D)OeZPflKSucJ3LHMJcpnF@5W6Hb3y#U8JU4X-L=(tN$~$|kDImm_}t zvhC>T$g%8B`1PZlP15t%#;|r9ozKA0@p0bAkC(WTi7sBes2ynWN-B?kE@Z@B1#)Ae z%3*m&K@XxaPYsPZIoz&wox>`5@G0`(z}NHlFkw$m&jI@UfLwa}n-Ed~6&0}@PVHd< z3~g}aoHuT?#$OdH^I11tSYB2tG+>2-F5|zaY;Vt{r>{R)ZrM$({h*<-5mE@1uNU+B zH7gDd4zsv;>S#q4D&p2v2U}Zt*pxSVh5C3g#>HJvu<>0FHp~|m7ZnPI<|1fJ<&BN$ zO#K!wd#+D2=oRkd&J+yV8kKg=%rKpuo&6dv98$_2vadq+u`gQ=&8C*{aOHbG1iyTZ z^VC3da4-qIIK|4!O0%Dd!aP=Sb~Y2FG;%bsxd%r_AI}PQ&7MrBh$EIxx2e7?!U*Kz#tXr zx4ks?CHdyB@rvtsWc1NZfyWjtAs5g;o&LALv9G1SpFAxs?en*9 zIH+Hql9}ReSblzfsUi;4P#hx37@p=TMX?GAQOe57?wgycQOh~hms?5=N_5=nMrB@9W)$(@elA0q_l z;)|Wfbem_E)-F?13kILAnx!d(;Ryr<1%2C?YfoEmI!+i|@gxsy9JIB9nvy2v^Ov6^ zDJJPCvsh-LvP-aHJPw1gfbf<4(`-S~KHKsH`}3DCiq_T~kYlcx4#Gb(%}mgO{D12& zzPe*!!8-kTm4{x^li=F5YqUbvB%;ngk?!4a^t-f(nMk5KySk=kxCw7JWQfj# z2Y1xyr9;}nRiJaKsHhN-l6rkUCG!5n0I8&s!uw$3?0DVsWqjmI5=1{ zQ~aX;-k(;!=9AE~(__o=kB_3`;ySX=cCMQ=1zbTl&N6^`!jb@hg6(}zODa)>Kr~;Nt2^jP7XK-a-yC`K)-6J~{_OH{w~xfW z1O(Rip`miuIYn;cN&=g{Oa((j>X}Cp@sQ)+vgMe3{{CzY#U*pI#P!yu(@Vv!2lCzu)$dzjg|XC!$Axz`lx;k5A3f@k6`gF9@5` zXIWVZiEL_^gU$J01G(7HM3XBkg&ilWNxKsm+N&JK28!_wHF9 zZY_!jpGm*u{wEX~1+?BzOV`Yc$7Lt!`bM*ii>;wC91h>_9WPlgDs8fWrja)F$*1Cb zR>kSQYxUljFJDpt$AtN^x{3zGsKaiJ6&yTr=pisUv#U zZKZpqSy-Oz+l9A2Z23`cXh8>z@wb%YU<6+%?5m=;Fp=VQ!p{1;W)k* z{?;ndNa6k5)6)tC&2_fr0zU~hKq}9*i4nVf9NpB}X<}EUC^8i#CG0LV6ZtExD6yFU zO5}Vnow0u8&jC*%5u2Z4+V=dKM&@2ryfil(8#XpRSqFe!7J$U}?nP<6l9X7?n55!B zxqEn+3)3a0q~Pc30AzgvIM8DFoo=d-4O!_uJ%hV<&7o|iN_ugfoh-|38{*qZu_h8E zUDs9(43zQy^WzDG`oY10JP-tCiigL0YvR5e%z$N)g8VO8*KJO$YDA|2g?=c^m8>pfX)fA8e-AC80*QV!I4*g!9ZF5J=8 z#mdeezAd@b_lg*R#p&_>kd&zN-%c8{4ycR#P=67IhH^bQaLbepq$|2tx>Q#mrJH(k z8$b#d7t!3@9BN|eQsK<@l+VnZ+4&Jo-z%x^<9XY*=8g{Kg@pxXXs8TQ-r)d~p!AD6 z{s_s*VQ%{BPY^-)mx1#QWcSspU+5UX_`JF#uJ~hi&-}GsA#=)C zKR4q`qTTu%hHH3a@hlvOVgqCP<5c9 zNyPmG&7H?GDAzObJwRCE-hYgW-in2bbJftN0fPWA={miR9JtS4lL>uAljoBWF}>Dd zhDD1fkLm&D(8~qU552uJUumXTi}5ypZCJ#T)FW=@`=HwK%F?&A_CrhCe}B-{-R1z4 zjf`GA1o*br@wIRb=gS-rs(>i;3SSHb!8ykOn%(-Sr9^P==Wn|Ny~2ePzRFIE)|vA& zpFcnAl7@l^=h7$RG$D0=udYg%`9jw6L8s8BiK9$e7XA0$$+uakJ)~$S%gnJkw6nh1 zWruIHI<-$pZSw8N~ey{d?0W$aK6Y9 zZ__}8gQaymRc?tNe>-~8^^m`Oa&5p?ICGr`P?@;j7W>DKACc=c)dr*vkBzl=cjH6+ zW$yVyQQcV?!Ur6exu-#3W+JP3P~U|a$W`*)?4*%cM?T@1n%Zj(2ne#Q?^xD-AzI%g zQ<>nt$q|VWkc}tuEr&xdKKZP_*3Oni7~a?Jm`Mr^_UF_OVpkk(^aM%5-oNK!XJ?ll zxi(u|xIWt&_G_|cJ#grz&GB~L-n+)@PW_M0$3K*oMk%xUy=S&7hTfH9)1Q^OuR)M) zBC2`NV6qD3-1l$YvhPX;tj8)976Y3x&G^-{-jh(~@y=|%i{b(1ITH2=?K(`jhXS<# zGy?)+;`s*n*@C+{(A84?%&yo*0yi;U8V+FlK%X|YwlMkUp`lnrl$e($yqX8}$*6tj zKN{-lv&F%1Nj>xL0|ThA)|S;{exLQ}{K=!}`|gGOb|$`y9%CzTo_3qbwbxRJDP;@9 zQGh|?2L^OZ9hk$(c`1=tu_&)qDaEV2R$aqGiQzWkfXHoB`l*pOwz|S=4mV`g{9t(V zn{|a!905@4Wt-kh7y@|Hfa@|aBnFIEFzAbuj5LY|9p^K$)DYfLbrdGd6d~rcO<$SA zJ+zoO9yc;7<~~cw!(&uoztn17vH1XxCDY@Ge{MBytoveYXPI$vK#Bsva;BsvaAQwN zhUIE2SiSIPi(QO9<&FcOOWo~+?Bgz$z|uNfW8m2vVvJ=(QQ zV3he{?2lGj)AdJJ*bQ;K1+=Ca6eQcuPol4@+q$zlhCWjv0f8LAjJwMK@_p=Wwxj9#HKno*|A zeepK%sD-5^40-TDSo>*bJfP3o^l1N;pLIe=pKNH5_Vx9ZXQr41aZ%OL(J{k0*EKz{ z=Xkvt&7u@J8+^VWoNK_9JGNTxG%0$rAABwbh@jQS52%dA#*nrg9E_Xg_HutWwEm5) zuY7rsEkNg_ru~)~0svz-rNYCDjiz7j#MQjx4Ew7uJ7!jT) z;5sWGMMP<4W(J5f@7=rTb-n0}@2k5A`7h9VCB8Ohr^*pa!>0;dE-U!V@6Fq{7{CB< z21=%;j5GN%Z{XaKntq9r@{l*8HrHvgnngk)Z9A;@m9-UX%2&6gZ&w0#2Gyd`=hlKJX$rGs>Q&HKg6m;Y%_@Hs3L=%`AOBz5bSot>SK?POk& z>HtkP$8EBay;cGOstjE(@%i&MzZoqdLBCAUiXerca5QPyJ%9N!60lFqG$TjmWQ7em zXgz#3EK-4~AVDdss3;T~NPaUjtusk@^M*ZB$|nJy`KpB1Zz@Tjzj46dO^1Ebk60 zq!3Y-8^wWA@gg~SGl;AxC4~m+I{oiP7EC%BlPpQGVQfeU%5;6)yO93^Doqm)9$;(B zy-A}rRlxFDMh3Ex5dtJ(qzaAJ{~9P_)E$)J3pfG$V$TF(bQ6|nkDktyMU=jG+q zFsfiO2CR@AurNCZ$5s&aNB!&z$M>O-ye}%cNK5;9HL`kTs9?TsIjjAuOGE2@(hiX; z@v2Aw00L4`R>lEVdl;x)EMg;DjYqRKGBzf3cz75$V*e<=u>f#(H$bw^8|NqQWV)T# z`GOCLg#h}!t*=ineE;6w%U`=Kzc0jz5S{}dB!qOUyjF;F^R) zPQdl#+kW@u(gZB=3`o=Bk+)in%S7c0(JzVlypcV>F@}q>mf3#@4pU-HVV#2~pC~J%w=Y8`FON z{5ikpFC9=c;)a4^JV4OZFVN}?J{=1-2PF#?0t*JYta?plLHqMs8%}KvGgqRzE8MZ~ z_-toLyBsR&ix)3kr^Vc~ZW?}&2Ph?bb};`IEinOyGz2#*&W0A$S3nOs#KAn%c{Ez~ zC>ps_z561O?BN2X<8(vaO6K#(NIb|`xYd@))ag%A&2U8)m(}a*jr)v<&IaVmeS*)8 z*S?Al)XFX>JHS-~X1W*M>y)NwaBJOCQ6b8#3a+mFz#t@K{_1iiTY$v#xvy{O5lgCg zQqL}&lDG+c}ntieye0Ck-_W2qq$bHB(UUDmVc_Gcv zeWp8Jm5ny?325lr$ue1!QG~883`#{K?oGMe*RSJ)BUKx8DhW8YyV_xl*YQW$g_+>< zQ>Cll>AeiFbW%~ZJ~tkN?gL(ddAX7Js+goC9i(Iz&@}qr(@bsI2s0ZWA2(_al7V>s zdraTX4}gxWBJfOsilcdJZ%WTTQ?`1Evv1a z$^sSuIIhTk$^7d}p8ZCf?T7x*7>es@qGH!m-b$bcYuj~lbzCRb!R~SeX2We%76J&p z2C5k#ZxS$3;JHqhi2V(VF01AZ&Q?{ln8G2dAaG}22!sm-x z#~e;drDK){8*@OL+5!69uQcm$6m7SzP~b{-*D+t!Nv)Iz%VWqZ57HZy&`iz1$Gg8H zitpE8lJA-3ip9M$fowe5Uq=J4b^KXdz9Rl)1~_4+1rnE)>ho@B2wnn0!qyyxFbZnw z$eyI@5ZBdqGr+04K{pYwALax-zRJ$*jgv{^eh{AgI$clU*YMAfeZGfSs~A;ZQ2Gx$g5aVoa4cIFIQ*7Z(==14BHhZAkG4 zgpZhoP^PJfaKU7pY=|^%d^`t!Gp(Q%0ifaVM_*q`{It8jFiq?Sxmg={EDGvT1c*M@ zox{M<8XIF17Oo`aa=a=mXqOA_B?})PIrJbmH#bm75gPW*S_nj4@gl@j0}-q{hVnZ9 zqc2pF^)0QSlCpseB$)Q6Fbt=5pI{s~nnn5ciNCh_cSWXvjSFCBmx2BZaD)tSR~yKf zn{!c2?y{1|Sv8;S@&P;CS({8TCG(^J^q|rlbh^W+pwo zggZ`Y^V#E~i3;OlFu=&XS`9VC`)R*i00rh+`pIUh6_obGSFhrA^Hf>H#8Q&4^U=lQU5e?LtLc~fT=(8f9%MCS%T_!a z;&f0sX3aqdphmha_c4IiGQZTD(J(xx1xJJw4@BH(k0dp!b!BltyN=jb>HqN!q=+CS zPRA!2E2tD9A|dJ6p9w}^5wb?Sn}a_U{RlSD;3tNHakpBP|@+wSkX4I6F5 zAF5o+jo>$Ch5a?sL5Yun)(v$B2)rLL6)IKKiH_H#HUz-k~rqzFc59bVGSOc0RoJc|KY#F59pYqpiM;(ex<`!$s2SHD*GOW97u+kiy}u- zF5qh;kT0f~4{(4w!u~`dp(iMK2`sO+)zap^-)-PT*=)8E^W{JnKbQ?8f0mgUmty+3 z=Z4w~Pdvw~<^S628=ISM(P{xeGE(J&Wf2V*DOKQ^@C12`OuZYi|6o*(1(DSW@_+#7 zD^MG&Zkd4q+X)haK)``3z-!1_Bu2p4Xa(nN@=d>J`~Rkhdj8G!wzdzQXPSD!i;5~- zho{wqj8A&;q9XuCEaJu;o$vai=i}CrQ&NQ1o6lt_si-BxzYfV`a}alZ#!6?162<9tZZ!N|CKqr2!V6nEiz<* zgoak31U%;H)2ACSqTqgswWX0~I}D$^3pzH~O9)H{y|QNdF)?7c#W#4O;6BT9Q?mg# z;7b0Ii-n7-_go2urh{A>Tze$(2lLd(xy_pB!M+Jd>3$C&iC)|lMaiKlKUVSpfOXXQ zVfMKK3=2#L9Xs`nR)YiukX_f%F!FGF8ITeqTn8aPz1y#f8-el%*8<$$TN8Mw8 z>ez@nAd31xS8n?|+B^1JR|w#fGg`gXOlt#ujUU8GE5NM1us>4Df!beSl!Gm0k8#qJbs_Ljq%^W;ULZQDRBl9io0KKDudXZca#DU!daPwa2?{_O& zJVbB>NP+O?Oi3@IewlrG#LiNQr51s9F=#7Z{1qxveLWq}VYn*!)DyPk+XyYRyW1RA&(_(-=L9yah?9VLYspUkm$hDB~? zCw`eS{?wpUbwaxtD0v`-2=`raR}Plt&U=oFPC#Np219zRV6)xpX%38^wuR&OVyiGy zFvHNp`tzU^b#}#>roh1(7cSu;rekJLp z2(~D=J{RffPlf}F47mQ|u)nUXq(TxMV$gMa=?;LlonM34c6N4%Q-Pe*iuv8<{6kxI zF0L;(%%=S>QE^E7xrSp%o_xvn!p^$iJ06Uc~GmLzVJC|VGAmqmn zqmt%bYPXa@bq~0$`<}d?4s=)mw9k^0cLzF0Vd{w}0t;R+VFEA7SATQGG9C$D0ry~T zeVrT}+DeBpG7wyMV{0a#eu`Ixnmm_Qg!^S4hO<}&1)nE{)f~PawAHzL7q`yjuV`N4 z5{%A-An}C3n8@mwB!gTJ?F$mX!qReXZLQbmRznWI8wFUi&H{>VBJp{!Y$g!Vr4}XF z9bqtu8BU0kN^%5O^Dji~YZ#M&HI|o`ml<(8Gq9ta3Vg)jvrs6MfCwP7IE&BJ0GNaF z84h$D>im?q@isGzoH2u6($U!oy^t1V8;)THsSn@=(pp9x*w9Mmm>fD|upj8+ zPsk*(M{fxW3(M!~w9L$8@2+0{d*wGM;Dv7ds&%1R*K+X zzsFjNHNRj|9Ep`pfJ2BF|wyD5U=)Y{s5N32;lN17q2Cmi?)4A`+> zyEZtX4udt~g=AOxG1s;R3JG~#z&n$TMB zfJr?#I|OwiC9hxhm>c2;K~3OnFmkxwC(E@ThDXL%JBT$l`* zg?SAG146F?C!Zotl`ZISk%l&26?q`rH}M0XJ9>H+-4RJ0W^%y91lJj+gJ54GKwksn z9PD?(7-cMNZEf(*TEBiJhT!3C&PxKcI6N$$ug4-SoynMABS1uXT@0ib5++%CXwgm+ zOq<~3ziw!thiV-ORw*0>yZ}Bz&(q|fgzu%LrPHyxoNJ(4VFP%&mT*;=mIRsd0>TRu zM;#VNx-Jeo*Z(Xa&Ny1`j2245ujxi9a8JN0-T)*b@?d9SZS8bmbql7xa&s>LTZ7>V zkdy%kNAy^Q$}@*RNV8wMrQaf)U`_vO$~j6gH{^D{A}BjDqRSK!F~q>6yFMqfxd8%# z>CHSShRyz?FsK#+B|%z72E3Kom6dT}QYfo;ScX7LhsiA%1p#qWO)Cw|BzgK#6}oFm#gRG^7qz_zo>=Sft@VPeXTGUbFa4_OF{gC#;)1x^Xr6Zpg(Hd@Fs zB>I8-2sY*nMq9!ZE6=F-sozFpa@P(0`T96`kk*H=YOHpCa-I{>1jj6^3qPy`=Q3QH zS_}0HkV(ILrhd5Kk%mIt!Lft(ZYSI(L`3=BzPCU#epNW$C3(WJ`t37;j9!RNwL5q2 z3}J1AIS8SRQKA{Ww9>x(NUd|5@5%+Q6^4X55NHmx5Rg8WU|Jm7<(06Kx#GkNb@s=9 zuw3U~^jMS&$6KXqr_reiua1?f{bvY~WOiDn`^JqMw6X!>Fy?tj0mBRvxO|LakFZ>4 z)VpBCKmmVEeG-E)074jvgGoW8&?EVPK((OLqs4it+`q=nH zo(LsI80vW9%a>g~R~7_pd;bR}i~pTgmc)JuGt2OG0{kQy!~Zfpy?Kr*FmMUAGx?qe QGv%mTDq6}ViVs8o7uR~-e*gdg literal 11321 zcmb_?WmJ`GxAsCQ0qK?omx8niOO$RwKtMoB79rizDIp*NA|fCLhzLkach?39>5^8I zMmo=Y-gDj`=NspYaejPbv)N;>#S`~^&w0)3y5@}3R9Cu4#7Klfp)RT@%WI)fI8?~L zg!u3YP4-*`{0HAmRY@LohWwLSn;DNnv7uDtWpupWu1FA9UY)q4=rDI5xsmO78 zvl?Hver=LUJmr!Y<5L6*{AeygS8YJ}_yCRgEiO{yS!=tgi;D0vex5tS^V z!2kQ5R#F# zr^P-9RAfHuRJWQ(<0cUAiU8uXvj89hbPR4KOBC;PhuinaBy&6x)PaDtq;xS z=H?8B6jnD{<$i5$Zce}cvMZbef18DHT2>Yz7K?pxC!?n1V7UPvZ*6NU```fsCMHA; zT^t6tRLWF;$Chx1Jx-a8GIDM7k(L%ETK#RrK&D2zVYwqF;23jyvL0yO9>d&cEuxvN zQ!}W(FNdAu;H4$j&DDRBXIRd`#)g6&OcHZq5q0`O;<gDAn z%@HS^p*3|~aLsbG5Zm$3KTU}b10q*e>|5H}=HfLZpU88_+`C7?c%Z^-o5x)Ejifphc|)Ua4wGXb!QU%^V>cWYUlxnW%_zj z5)BQF{euG;bo|t&R-SsM?@XMAvNEBG|IU3wUlS7(R8Usd)mN`ye=IC~F+xG>E%+Ee zH;P`wnq2Z^TN_@b$zHVCLYV!^6#`;f-i9=}a8uC~Y^lNQ&!6~_^5;A~Ju50J8>gnG zcKhE`C@Ou)$&r(j3*j3iG(_%GGLo%*g_@QYZX%c7FP)|-T`*$a5QLMJn;QxzSWqC~ z^n`uz$idoLT1Dl;@$vD@`nr?nPNkNw$n0Z^lcT+Mv1z}M&!5qc*MBh2&dwsc)bQl| z?Bb$jB4_OI#@3eX-r7|E=%_WZY7>55as?{AKcb~Yp4Xs+kf|NJG1G|g-Q^w5GYnr? zuozyH5EK&=3k?rX|MG?C;NXCem^dUnoM8E@5oxsATO)IGP9h>AL7#2wPoF=}bthlf zFSBP*e}6lYQz^z2jwn}kAxxP|R$3ZW_4s%2$B!I3_wN_+cP{pIv_#sJ8oWwODk#A($dl%o2N=!Tg-jp;o;GI@E|fGg3!y? zx0qi^lQI%_s>(e?heshGKoZVx`K8f0c&q0_whFK{5Lh2Sel*$*w2V^@otZI(6oGB! zxOz3h1o}MtIF=%)K1(m#Ya zuD6QxpCdct?aeMAKz8xs#avZauWiF6QwZ%r+Z)tGcuh@B`ib)!$0sKc6;EEiq|DOc z!NtWbY2YVX@jU#Fnb&;y zkgRx&BxWAqDNQRoRfi`{hlhfmKBBNt5VnDnmzM+%ZfPh-uX?R>b9-Cf!h-8rR1|E6 zTJ@ue$fgz<2!yoHpYe^WJ;ETK9*Z+Czck1wDWPUyU?__%w(6DTh)ea|TPur|3Bhk{ zZzteMAd-}n+}PZFqR;!Zyu93j2-(i4)^qRF!`s>vtxFB4X=rB0OYJa^m&jq?;}R3Y zU%Vi1X=z~-6{R(Ad*1khU2f+0Z|`YwLyvE|y1E>&O~}(_)0UQ;Vztu*i`wcR0!1KA^`o-(lPtOyO&Pbgdkd+wl9qz5iR#aek4a-PE@Gr`mnwq9e&_uQF zNyH)vFD1pA$Zy<~p_*7|Gk~_RuwWAxX9U1+j-cS&7&VWb+Sxq^vDnZcJu$=*7#Ii` zQ(CF3r+_tUhhT;MVW*5Fr=hVdza`f7&p(A$y)4HY&14`af;HRURhPz! zGfmwA>9_w35!d?cG%a+-<4hj2JhHNy-`||yKmOahIqB9r>sf!qtQgA@y}$1Tn{fR3 zo%%a*K(w*X5+!zJ?pM`Tbh>17w6f);rGutc!(h=Z(`Toie!F8<9i5$xp%al>) zHJj4$;ksAjzCpXTc6OCri2`mrJ3BQcBL;hR>+oKa6)y7?ko%6S!XYktyk(*6J*mGihCua1_yPH zw+A$?Ub`lzrbhCpJL$>u=OmE)@=8k0_M!vdf<0xhW(2vq*C0c;eQKwSid632ZG}KB z3^?(rIoX}4aGpK)sQ0}CU8#FXIExRVVVS**j0_GEhAQ!B?D764#(Rsa%43y?m6a7R zsV{%*VVWSe8!C2cDhlwCf`X!3Ox4oL${haIYc`Co5Xi(^Nq@%qwrFz$s%wx|2-kpn zqBY(>e^C4g+npp-0S5zUp{b=6{OlRgu;utgBC4kbJ#6=~b(X$o5D#1vHOQlljEtne z<-v>8_$WF7v#o_r4^Ns%ykxbfc*L|be};0Bu3M3s_-!iOKG`0yc63w(0!Tzmj4jgF zPwZq0yTA$euq|bT4O0_SbYTj2m?*>crOFiQ^LBju)=Df;BP~Yu<4JY(Eo9rE#H)Rf zWTrx^HKaJzLE0lPCUWCO5q}&v5f#3p#7gp_Y>q)*8=UuxSFb`LvDvw~BR%}ef`Nej z?EMg%qgTKIs3aFo+B(}Do~D~)d`NML8hs83Qkjxq;q8#nd5qsy=M{R4GaHNd0tJ%2 zqN1-=_+q-dx`^#Y@)dM-X}=E)2)X@K1H=Zfsaz-tZ~J<#NS_yh$xm9`92^`;c?#Ad zjJMB@|6b+eQ@C@72$)CvCsHd)O3Gk9!?Kyf=dw)6CTH{?6td;9*dOLtgNgCL#NuX$w{NZ;o-0Rs)8cIzfIlU z-66a}?qrx%v26A$v0|Yjj()Wye((L0;gLJ$7Mus5FeiW=P@TrNH{Hk?8K1F8_+U=H z1yiG$mxi0)KbShR5KcytdcHjt!H|fUm`7USIgv3jZ86NZHa6Q?=ixm2em`ECuCLdg z=2gai$HBzN0sVub3pI9TZLNn(4S0-@`yZYC!^643Y;BJsh3SL{d5)U>xv2evUmrHd ztUlnN&YeFG5!qN)NIEw;&yU~z=wBgGYCyrj&}m<<2w@m-b~J6i&=EIUYDb56L2jzr zvwZ5MrnYv-B{9dKc??P#N#GDjm zWzQ9hcbvJoxitsl5!(}(ASp2PY z5t5KtFml=0*hIv}wgUjF$(V&tPEL}uu*3jEx%B@1`wG7UVc3|bP><7VYM9n*{*o?J zPDD3_5{4G^_aC-BD-lMlAUzx%(ag^t>zSYF zs?6RGiK5lUvn`eEx@a!FmT}$Rz;?g8MOW>y`jO2AC7etGm$$}Bof2m2*iKvqoHik0 zxt-+VzN%!^P!=*~ujuZMumW-SKP}s$CvLxbuzv=#;jjb*T^E!exxXm8>3=A4y5BBk zw!1P0m7K2Uy+o9R@9qr1D##xzW0UHLir02JdyRWG^D*NlKn0cubs9b_7uA`$f z`u6JRW4N&>5gjUu-Gig%B!69#zS2t(q8AQrNjP9^ii7QihXecgVuldfOAvHVp1>MIt ze$mlJ;W^c8;`{mg0}Xr%nK4tl_oJA3mx+a?=e9CxX3BF~0gGi9br^4{qLy`(nXij$*#$A(*!e z3A_)~BQ7Dql8(3`bn;?YrvP9W@Vp{mHm_d2Lb%yJEI`Pth49ghp^*_zI09DxZ3Vy?x1l82$=g?$ZYBt*xre7Z*A(0=CAt8!4q$k0z_M&ct z7{VbVBkO*9Gv1v$w1n}ln`6IlD^?|Q01L;Au)zU0Ekc`XjLDNtrhC42@auSr4pNg7s<(;eqR)neJi>aw{%ao#OIPwHl$@NLm6w;H6;%!2VA9%`CWp5iPo7rcmqd88 z^`JNKG=yF5DPdJ+7<{(PP*_@8`l8cfV|$zWQ$1A7=81_**YxtKKS{}kE_}CY(7k_O zCPPamO>^V>kJ_cj;ue;cNU5zUu_K$70u!q`p-bJ(fW?7*@J%G-W*9Nu1|$&icUE>; zw=qE>#%O0x&v1R-?(&&m9|9Gvt*_S9)ZqS(B~PogPa>?{f0K|9hI1aDaQHv}p2DZt z09+yosU*ye#EVpEdr8NT8ULdGuwoHdF}o7l+uNHO-I85tkwkc6jrc$yT!Eny!83U) z{`FxLP>TKi{l4>^k=pPU6kDtwHa6VBAt89nu^2kgke<{2bWo5%RyBg11!AYIDtFqs zQ;@=^_ET=|^QO>}2?Qn(1q78)9^l9%;J{KXMfBpYo_FTP;|@k!pKd?`OG!yJ9_D0a z1=ZKr!`A-I^>sZVsvvBcj^e| z#G!T|A_BA#qT@cC9bZA2g4RHCa{iCwFITKtk-llE0iFbYD%e{rYU<+|K+3Z6^A!~o z@IdcjMN@E4zj{BESLfmL)1i4;9KcK*SLKLf6B44({;VdWu1*S4IADL403p^$YAtU$ z&b!*7Wi?!aQgA#n%}EJuqRW5y5T}SbG=UE6hYAyLayWamUSD2)tIwEUP5-%Cu5OO0 zuR+9L7f;XjFZXl6dx-^Spjtw1bn){|UPUxh`vs3d92b*E#bfzO75sabdgd+!=;kCe zh29p7nO`~ll3Hz2pk&t@H1zv%jR*f@UnoB)sAUbRzIm5v5rMq2fy9$B@&!j6VO?F_ z>RM*{bb;zSoQb|ZRG4i4mwTh%P@3IyyQ!_+cjw%E@V6S0j(GG9F^R0YZLl zZPyoP_;<5>&XC#KWF_JjfpSrdVFJWfCp3^VHD&wqOGdNgI>K zqn@`Hz|TOOi%*|$LGv}YL{i1Qe;>0pRZXmgT*q_m8s6_HOeVl9@WSdJ1q*{%EltfJ zpdK=Ly3Mwwvgmj$$Oy!Ks@tC3E1+HN z_Elb9`Nxk%xF|1lKPq;p8>{wo0trN@s|49KXj_WJq3hOy3ecX>e7c7?SIFXj)cVc? zif0uRgv+J{fh9U*+u9j_r5~-@}7?*P{E-oIm`$QP;Brr70zkgCm z60xD=H>yD8=|&-G)7janUugOywG=710s_WCp{Wn^em%X=_G8%za9*ZL{6?Cp7luV2SoPsEh51|Cyr zNFH#3n~;}0N(G1vl@V#xoqIa__r$SqoELtKcFfMbr z@obzii*v0}pbp#C1J8U>)M$^WA)PCNg5-#?QGa&gguJgMc*aQ*zM}FR6p(&$Z#nI! zhuq0^qY6fP!iJgiM|Pky0UTu zRM+I~yDJmP*R3>lbV4&f1dQ)`#t46N856`-Itu$AJ_ZIypqL(~a@G<6vH=SOr#vit z%>J`rkrm%Bo3Hu;AefuMyxrWZKQrE_i#*{G5NNOSKMHWW{giVu@qC4I63Slb3P|yN zDC0@+umJ!rA@eqpES)bCr|&=CByAWNh##x!q|&&J15TNCrn=DY52J~}!RP!=c1HAz zAANI@RKm`o3}~@8W2kDJz@CDTorNAm&Wu0RQ8D-s-5;b_!8Jeq_KRt@j;W!wxn*jK zd2Ma2Z+S=U@L`2BJIGX**(N;n3DuGF!ItI9R5hJI{qYvFlm;giLE{!>kr99U%gd^W zr-A%Uq0f#juW{9O{*m-mH749H1$~r$jhB#?mX`DH@0Wpjd5|)The8hUKCC;=FGy2u zCjGdWHpe@B7nKU{?=>GS23ob_Q(#k!NkMy*KwShJDG5>7Tz)~ZK^`?arU9HP;pFF? z_hR^HdG&+-`RC5cM0xCki}GHVMC}`(&Vj<|sIF@>E0X4=z18Ax#LtRGO}JCdJW8#F z(%=OFYonJNg}5M+#~X5xikt5X++OWa1~uWLCQi_Nh_>EdouFGD&NDY4`0WKqykBs3 zC}2Ncg80>Yetcn(A)6Z;(qMV~jh~m=pE++o_7x9^p4u`S^K$8ki5#HB2Do%nd!;d7 z_lY!u9?J?Ta0}=iYssIT7}tlS(pa+0Sjir)`^<(-b6Uh}}f-}jdyLQCLqKV@WKr|bM74|2QMSGPep&4QF_?oU^m zU0t2}T#%QY{RCh)JY0$3emiG=LWzNc-NQeHtM~8Ucge`es44Z#*)6du(ystz?hKui znT%TBv|aJfwLm|dfomH%IyzR5U*qLn__MrxD_2r-BHgBkZv|W5nFpI!<}@7x;ep6& zc)Y`7V3D`?Cf$_P)fLs$)T+mwKdCmy2r*Daz$=W|Kkc+IGkzAWmEH2>Jb@rstD&Kx zpdz2Rt9+v2e^~b%+_hQIW=#N<6H_sq#*7}%dGH2>#fvh1*=1#Aoo54+sd@ zZ)QDhsN!-ZyLs~_f~p(k^(PmBg6^L$RnT)|N~$&DI4y<3!+GNdO%yGktgEZ=vTKL& zr_4-qa3}Y{9u%_t=M{l%)I28Ki`SaCJxfb^_pA4+y3fv^Fwonkn{%xg*9Ahjr~s); zU?nt_IZOx!9Pbl2_Qok*Mul}cL0pfT6N5j{;mwgf+w7wSq+10523;0+!&&-Otg*LLo<5cOTQ! zz3oyI(dr1CbT(IxL(MwJ&CQM2pc|P|C!LN;T0PySf~Kd(2N>rW6hL3Q-3!2T^~SEQ zuC-=dl&qW_!UmvK`TUX?>Cn&+^d}g=|7#8>3s+2U+?;QxprmZ_WsNCm$4wQiQBQOLkP?dw|^sE5%?H`>?i+T+#UYJ z^r9jPu&&*?_bx;H|H&!%=rmoIB%O8LT3wcx zRt_2-P_f*(9niEqZ)Rp$al{XaU~<7qM9M`!OQBY5~=psUb>g7 zUv%zH1|`@TrFWe|6$hBEnDrViPau9S;={_3Nre7a!vhR*Q{&fDH$7J96{8u$0G(zM z^h{)IZF%C^vU75pffR)!UECWthOnQFd5bQB)HoI3)7SAZ*>VEH0t=b zMqm>-8!oHkH^5H~|1I7Me&`0c7)V#zYcW9&L_9)b?SIrHKe(HxVAb%W`Sj#ybNcLL z8s$hT@xkX0H+=V?aLA7xtSY~qA-&Nuhf7Fn6KXsMSdU;1=Bh4}BYG3*!mih!u?PtX zA>|DgX-D^K1R!%B>C7NbxNtJ!YeIK|Ny3Nv?61^W2y}0Y`8R*CEOfUMVAoffCH<0? z!EjG}`Lb#8*Dp698IzNhZk52r#})kj%JsChn^&jnKluJkw!(nx_ANy$R`u@PMPAw{ z`gUQjq4rOob3Wj4*Hz21tzs4}P=f);gQ8@Jsi| z@>vWi!28C|4uV%;6N4Be0P_w4$_%&P@Pbq|gQ_*x~w1{5Rp8jxm)lTe)?arM$ z&d2*MpFVwJyL=fBmIwYfa;1>Rihgx~PK0`<9EeK`VTQWXLpx-?<7jUk5=G+0n>R!d zCb>qHFWr;(F3(jA8+(2SJ(o{2>gx*&hX8H!U9TmW`58O~Q7ZU&NfX)$sc-@yv~6}y z1a?mY3ebI-e5h>(T*-QFEF zC#(6)*>aR2SQ!QV5M*PK&}8vl5gHsEoEg%4*a~N}`CU~&2Ba3a;Yed1Ic?#5pSJ70hm>jG$vgZ18C>Go@8c{vJcT7i*Nc6J$94Opupj2B@l1v-Q} zJVVUkr!gA=2i6(|BXW7VoMK{hKsK&&bIX8}fs7N}zaIu|J$+tU>(cRQ`!>p`R+vnI zABe-SI~lUh&V_t0kU9*u@P=zfWb0n9y(nV&gPlf5NEmb{qiQ*#*(9)T&$d?*UE{$c zB0|jy7K5?W`q~;P$ZhEGrd3ulfCVLWc6=T9oemETR4NO)M5ZvV#IF8gD-B#cys;fA ze0FFzB0cf4GFs65OFwF>qawYh&)mrx@!&wC6{BChg*lhYaXZ7d&dx_ zLh!QF0$EOdy%dOa7!8Slorf6ld?zwPk$A|UokS?jMJdCd$O$qH zB%=5`(#6Ha8BCApMc1FR1=rvau)#J%b6<)-3R38I@hd^9Onh`OF?p{lFfRV}b?eZZ zH;jl^A>p&_JZ8fic0rCWs;sfGF)JtM=-bCUPNqkzUjk|?7$K10(qT&>Xpj@gbHq9H z4_k}K<_yV#TL7WOlkkkzVYT}mt)4qf4}48Z7c9E2l}!yw57r_rCwCqKfp2^~5?;P| zY{fMyCl8vf?O?mjEH1jYD;WA#HulhzSpNEQEqz*4|LwEa%r;iIyA;{()$U) z{)N%6ggcEcOi+AQdnqGZvgzD!#>hdF2~Y@zL*DJ`qu4=ZWFzU?`SLO}mQHctctu1+ zX1<9{w?JK^prOH=`6+c3%v`wTJ9PqxIT-B#2mq@eEF;5;65Q^*Ko5Dos2my^8VGH0 zQa;|GKn7y6Z`ci+9&9Z}L_|mf4ncp3mft9X=9=!PiJP6iKJV*k5<@;(;^&I#tH@Ll zq>*Ae1$2jK`Jq!LoV-%jSEN+FX0$q4S**=U)N?K@GTs#|J_1(1Aot~AWK!ky^wjgV z-L?s|0__vi0#`?5(npr>fky!E3U3TkQr04Grk`6wD|N>A##)J6LRtOa*rt}w(n%O? z^2NK@?gBP?7&0xVs@nIsWt$t!Q4pjEoPr}O|j4dJ0H6v(FBV!^k#3AG7 zcN^xz$SEmhZ|UJlV`XtiUjL4V5e$F-T8A&Lc74z-$~7z}1y#~?t)K=OO<7v9zM61H z?VUJ{Kf@hJ+%A^c-v{w&_aZEgZNW2))=VsP*lKjoO231y0rY=i1e;oM03hb zH#9OK0A0TS@;^aXsKB+{z%?JQjwiS?@I!m`e;X$4VNdClwIVM+>8=qSV}5C{ZaMp|420)gX!Kwxl@o`F~J8~XIY|DGAj zNr^+Ap#Nnw=OsWOFCa4FB5H0a`wQ+~YLhoY$BM7zOkaMdNl2uji7m&CeJ?Dm#6*>q zrIOU}i-Ec3uS#`RNeO-l{#=&yo{fK&v~Q_~HUw48BvAq@RshvM*12Zzo#~;~))nH* zCHTsBn;3)dHqU&T-RS$9N4!ltQc||k97n=%MUmj!HXKB<$KV~zgo$EKh|2D|8uu#) zAP_yT6N$c35%G!;Yh>(TS+5L4*?7D^sBg&E*vZ$h!B;9K3?YyBzppVr2@L2<*v^bh z4x!Mmd@X}Gv44MCK#3rwf`09)8x!)_+uMtzUTq}lWqle+8?~rT?ZLPtj-#4Ad)0sy z@Q`9PU7k5!v+lJ%>SRS%rYL?YF@7{(<6y!kjzFu)+Tk?pZ34M z_nMfR$_9%w-Qdg!?p9;F6b)W1)~YRi-(syZZc)8Dz}ngcMrTm1d9vWTB=S>b<#?r& z+hHB4va(WhWAXIK_bhoS<&3=g{*CP-zge{=9zK4mqtC#7^5f#`)|ygxmnLD~mdx8u zw+j4`t+V!{s{xzX!^R^)V^dQqR#r?3?{jR(@?KeMj~@(NY+Rf_rqKOyisP`2+z&y- z5DYTWGTmkf#P_0P=Y8RLgitE*dayypSta>F*S zW_f*{9v_I_n4;m};<6ib1Z|c@k?|3Pg@qCExiDQG%!Sm|)p0t~(bKnop;v`2ir4iq z%LZmS8PzTh_KS#as;=*IG$kz`fhBKJqWPq-gHeiTKbl#C>zY`H&`tFT=X z94pgnbzH5E99JK#d7S$*U18J>7K>@##?{e>tovh?sr*@m@6CW-SM${(V-VJAQtSQC z>pi_E4}})_vRY=iXJ}Zow6v?hJP9CO35tUBoVJThx$nOJF&&0Urhs1F38~ec~oBOHp$-LtTa?$|p zps8c>0pc8G)+mxp+<4c~71xta(Ig54)=!BcaZC5QI67?UNNCHG%-anY7gxf_$f)R^ zH)6cY_;uL+zLov%1btj`GUK4>8PAQ&1ija4*sHFNVASrnQ`*!vEjOld>FKOrnYCVS z4yVr9#Es*XZFo-nxxYD+aB->8sImH(yj2S?{B39xLvL&5gcsL3)uJb&*e!GS8%Nyu z*I)RJ*&`ob@K`N0yob(2G;bVXLJ(Wot^>?|LE)G4{fsvzWzgu3(~zDcHz^f6d&L6j zuwZL5RbGkX3v+#`zixwI+)NqIx zU0SxylMF5y_TVq3LqCFnd6HL|4vAV=SR~*4c=r;GgmXXT%(n4>vsk+x=jYF#ZhIxQ z8R|tRf8eMPA7Wb8ZmJlJhQiWebNd8t+sPD@_2-VsMVSan>ydW zZZ%)dT2cuJ=oo+c6d^_7&5^-}*XVM9(;0$Eh>VQ98ZXC!O%U;OoST`9$3dJfQP%$c z`h@eWT7zkITu}%gm4M0Y?1AX5#TShxw+bxL&?H?C83>kEjg@)J-}J}xS8?(2f4m-Q za+9h#^xGhrlNR07G5mivZWoqU$1(@&4SxF2c*2>l3+e6ENR@a}*=0|Uq z#>SNAG@=(yys6&=JYDjrd6=JuT|)CMSA&qOcyNj)!@qpNd%mG|-mO(`A+Upp5CEG(nA7!5$PC{j*tZdQIiQg?T^ zQqkm!LL9i1&a|SBa+fNhKd9Prs>7zX_O@ZnENINED^o0l&HL7g2<xe|@_970-_b+gX|>(_@=*Z-!9WQ0ni}5w zkJAV)MvP^DsO9|%=CHVES%`5Z--s9Cdj}zSx`dhMI#Agcc#oKTH~%Qhq#q|X zfdJM95}Ww`?;-ol%%_&A(NQ}Q2N2e?rq!3Wd!nQE^h=A(AAi)BKI7vo6??cEpE8X4 z15%6l96t=3-Q{4EHFn)!J-AufN8R2}dX#&_apxX<5nE?BScHUxf3C$VOjU!iL*ZD4 z6fs>H#PNb>Zv{vxC}K1oE{Z42I7ZDlQprIm0~U=_2~9#IjGBNE5k*N({3W1bJ+e#g z1wU|7kXQ}JoqE7Uh#eQ2^AU)Ny zNhSSnaZpfE4LFD^9Jk(`Z`2i?@#!B@!*Z!DTI*3t|=0 zMSy(sIzT7-nv7k=SbXxGG>e7kwJ}nBe7usDR#1#kT1djHh?Ntwlc{ z%ZS)7UxI+WeCE@k3a26)44|djoyf3)GsPpM(nX?MTVuA7r4#yoYHbHAqO+22b z*gIQosjM|eV)=JaV2h|(H)dx%Pwb~kFr^q(xGrqdm9)6S_N46ogN4AvF8uc7Sx2gZ z#y?99Z1)4SaWl8ItB(&dwX_NQYsXbydvy4ZBZnaXYd3(;v&jL zlSG1E+@CQiqW}C+xqk3DKbYg1H^)}S!zJyb%)-L*A>e@|t5VIl^#W(~FDk-TrbAkh z^5Sp2_Cf?kMCFGSI`hC3O6PR%BebfqUfxF;FX!~@4IQ?!A3Xdj_g69T*T_Or4iNA# z%Jv-#aT-D?hfq2^FF~QzdKoT|BG!Xxi)Xe0m!`wJ_o2We?f;iS;volNg>ETfVE30giQ~ZvrYDV0wt-hU|GnFeCL~m}{pdoppDG00LHoZuz=nQO6%~K&-mIIi zR%n-p?E+plUUh;#lM>#S9yiWPunb%dT)qE~+A{PW>sYIIlate5vEtBl%P>1wGnP#= zlp3^Iq~q~M+trKS<`913?}@l{6HI5#InfNy1wX{3zM*OR8su( z`w#)$BxyekbQ(}Z6kQpV#+bvQitjbF_LEJ;I^>{&1sn4m>?qV&tjJmTg~cgH^)3Cv zsd*sKcdcG{d|bzhvx=X>q$MPOLG}IJ%QrYUyn0GZ1m)mqfm3@MVA0~`S9dMI(i0Tn z@4_p5fcy8n$w{zPH0kLh(D$RGGdcW#z5@RXVs9lED*e?X`?*CF0ZK5{e1Z5sr#J;R#E`h6 z5Ez=3GQJfk*{^_%-LLvl3)R~~vu-?;2v|kH`iWbHN9q05 zx6&hnXkbeMOBj@6`N=VXwM%p6g;4W|)9Af-M*s8rbw{dX7l0N#nzE^ZIIvAa8yEJ# zl8lQ=N+e88)u9itdLI_h1)K#BB}$jVi2+z_9RJ(5lYRX=*kE&>MUZB_s9&(jpWL;} z6A@P~9a{@j=q@M|r%}|ANB7v={LB{>n(0)p9ZN%>5m=&bDa|_U7D+gCLL(zOFrt{V z;Yzu7<0Z)AD~Ff!MgjvmLUL6dW|UJ0=+KlbIEc+Sekfr6(kzjdkwKmbejam@#gm2w zo2QT}s;NorB}hRd6P|z1;>2ItuZTwq)7B3*Rr+Q65~ia!v&YpSsw>04>0{Ml(}ArU z*bCIej&ZA@6{it1aYlEN=^v$`3F1o(ZKn#ML-HbrI^xPDWt=LwNpAcpO3o+8G5@d7 z?P>?M@9zq>=w)MX1kGDQvKLhDzZKHN70B}ntGl4@hPu2`d84=VD%3ojPS@$zH!h&d zA1r~Pj71}32q3%?y1Jq(YTd}^hm(x+=Hhrz;9LIZ)s1uDzR(3o+x@M&t<^`BNBh5j zjF(#;BEkrcqE(q3C9B4W5quc4MKC z7jI|edyj0bf)k&B)0rwOZonFqA3-C7Vg{!}l@&cr!+H*#uraSgF&@uNN}&rUc$^N6 z49-6=Af6|uc?Gpj_vo*rQX-Ujg{c7ElnLv#S-@p9$wNa>dns!wM)M92hwc?X$_a^o zYhV`JB&}Zb&m8MSl?7_kYhG1msJ;AIfP*Xu86fBqt9tuH! z1Rgr{!o2fob~JJhHn1%;(4|;*;sxny=RdcK7=VL=?#(kwcm-0S@G-FvsMX~BKZY%T z(xl4@X~VI3;Ma{m zeTLo&Bp5LAtrh=YLuXFwHWX1&`NOj%6Av#F-;t)ry+)509^3tL_i&H}6D_4sQvCJ7 zVl74R1JC;sfu<;NycY`Zsk4*-Gm0B02MYpY$D1lGEiLTDFZ5a* zkH2aFu2~b=gr9KtMb`4cyes5;y}Pu6EE@2*2m#J42sbIlwo}7@sIa&r+3LdmPlQ$f>~=p!nNf9k%CVQ+K9HiW>o3^6Tl5gdz?TtY7~Q zG!|C^gZV;wkAd48i2)u2HOW_CpTEe`evvh)Q>DzJGfm`|R_YS#lot+@{@jd+NzKlK zP(Kg6i_T92P0Adqo$#y4dz3q3U_dh z!l`sZ{)fHuX$y^PF6Z6AbXK9JY}^bHHJNCdC}WaV=Qqk9+s2BcjrCW;AOqZ#MoYw1}>P6B41 z4|0lpGTkyl;6j=uobv@(%6N|M4`%1rvAHW}+~RmmyS&u9g$(Ql=lNia_+X6Hoi?@9 z^z_Jp6I)RT4$abt5)%al04D&*-Uimzjtm3p3>EvO7GVGd7E)qjaCwCw3QuAXdeB(> z=i$uBoCd2ulK{@=0}#ze7exX~2pw329?}Hwe4orJ^vV3dmX;T7*}Q+7tSHc0(CM%c z0wmhjFPbF&69G-7E%=iL5I}hhs%`U>+z~~NR^9p&0PlGh*wvXml$7r~e`bTAc9C35 zS)0LC`DeT%Rd(XP_>hX3tyQ9)lamv)-x1mC7DqY)=ETg*oM9;ob(^f)i{DYb-a(Ly9C30(ys@f*2~B4IzN900(jwo*cyRfR;UUp2!GH4SV?At}pxB|yICUOqyXR!iZ} zRm+fE8I%Exk9{P?KvZMEpRi!Lk`-?+viy>v%8;RIpOX+8P;0;E0+RaCyca?Mq42>9 zd_umMV4(_^&m-2}{yykk0|2Nxy{tE9AP>4%AS=uVnH(`OG1!JUCLfrTn#R~(=->i- z_c}|N^$a%8&B0pjLRY>W%!rZ_h9QyR-=;zz{DJYGCEwf^4fXBE4HQ_2Kr5sGfEa*| zs2hKgMfYKBSzHjhU+>brZs|+E^(d$4gPYx^Eggj{*lekRtf+g9Fuz7_y8!wNO4{UM z+oKL2n`O^Au#QA9EbkGEYCxC7dA5KNCUU;2#F?CjfEaYMJpPu5&54K$f{vr5pLJwk z2%?;5@MIV))PTSL^}eJpt>63sP^TK)1tyjTIiAKIP6<43^Rnacgo+OuVCmeX1{jr8 z-BI3@*71WSV+KnG88m$Z$d%RA)s9pYY>5g7OKOjw+gKyHL7~PPC&P{3IGl+m7=WXP zX)-y`Z6Z_ZS7_+75&gTEy%j6p%r1)%I%Bl>xhPg4y@)v#>@a6 z)A{e#<1U0M6AUy3?pujq^DKt8`u5(3`g@~*IHU%khkeA1gNiwdF$rK+?@$!o>+S!{ z{c6b*v&yD{9=N0cxa)X|!Yhzj5t6>3`pCqBsjW-l5MQ?p!A7vX@((PZm@?c<`~V_F zelJys;g2yhaSspPf7eDQlhMh_!)t*|cTLZ>q-}C)Bp}}NvcCsh_jPJ7c*t}u01o)j zz@Aj1nCKB63%9U%JD-kfNb_9nYH#g5+kAl&Dp&&3=TVY?D-dmOgr+oMGyVa_%m86z z0m3w&{GF>a{*4)Hu-vnrneqRGs!}&mBv88o+Giko?))Zfeaf>1nu{siWPe+~K>hNO z#NNFu(k@R_T)DXYu&rlAu!ltME0~OpSOHkaF{j6joJk7_Sy|aWmO+ry4ZA^;G6#p( zf5yj6dgGycE^)pmuyn#rELl-F_$PF~3!bL%`ZyK$w}h7f#TTCS{-xji z=Yg?t?R;ebEz+^LZCo<+G!6ej`sNp!`>kfz<8jYJ6mU5p^TT;wCLjZE$iRL_m38?= zi#V=Onewd&{-IhZT{yGv72G=EMf-Lpfa{-HD>Z|7szF%!Wo4NCo9Xd1E3Hottbpf+ zDVvHoS?gUx6uO3qc$G!8Ob9C_j}bYrwi^jX(0|b5Y1q0gCYP@z$7Hbw;~?v+4HK_ulGalC@xB?>t-bOxh#AVmqN zJ%4HrY0?d@GhRQo%lmTxsx`yUOGi>VI%Fs)D9bL5fzfA+o<}PyArw8klSS+v=VlXs z|BA-btZ!`@Z4IYhuVDHjQNsINiC$-{Zy<$@<(jJ?y01Ms0DMd&`Af_dX%dQZ1g8r2r~k}>4o!&iXbGA!1t zMseI44j$k6lE8v3k-uh&_@~<9zl_Swg~c<^?gz>Y)~~~ecr)~*S0}j|DvZ_|HenGVgU)ZKE9}X3NdnWe zD26oFm1?(K8Vxv;;}R2T`P4yABnWHB*Y|9Q%YMOmDzmV#aME5BsjIuYzl9uI0wbh1 z1Xumlv!8qap^DMQtK-QnOZKA=S0NlWS`X3J*KS=Fu0awQ&YqK#lYN7OK`Qi5!+UN1 za220M$dM@7d>_3A(}gIC)JsGa6cmcqE)N$XVBnthLGK>zU5&n-{uMqdzLhtO&TWK6 zJ7~|Q0iWk%=G+&4ouZE8u%yn|?+f>D#Y{OR^*Hf&G~F40OEN~+s@9s;&aF7q$EK%K z#Jhdawnx|;{2pjo)=c2+>^xm#O#%@1g^bD>JD=;}KmO3<_m^1hgwT$%8fDSI_4euc ziGPm2ev70R8L{Kicv8e8BxdMC(e&$*(drJ$@?D$1VPoQ=Hq8S}R#EP^ z7eT`B-{U(tIEZpO4{kU>J5VHCHeG;;j7?3Anw_29JOcDI8P}568(q-d`&d+mt-OAW z>ThdF%*Mosa8kj06c}opFmJ=8S%=l?wJc(0EJ!`yWyNXqxA>#9ynHufi~w4?N|f$L zp=S`ZA=&5|SVa9e+_&>KTo|O>ewCH4HA>WxC}V&82+zu*0zCJ?;(R4vVOg1!s3`3D z?dV`SD-WADfbVjwF>C~0!ACvD~50H zUP5~bE{)BTQ&E!Ck&@IHWV}IuI_uP5?s1X_!`WyXGp45N+yKp4}h1wst;T zrMoNH^+Z0OE;oStAC!Xj8K8TJs*q*VwQ{mjH{(y=EXw7Y7HT+Bfs22={@_|}b#ofq zWm%*rT2j#$W(N{L%fH3`oh!mF=Jb*!x>4k`v=F~tLGSXrhCL-!wV)BfJ2b8ZCtSYi zB|h(4j@NJAWUBNp728f0zOMuQtqS9Q9Bt2o>LI-kMU&qIy?G_0h!JGsMK9+u$Aa}8 zJ(aH>Dzm=xIu9~`ivEs~X!o3W`_ zqW{8x=M^0HKIgr~rB0<$g&qz!0S|U*Qxhq~?qqke1SRF=i7u;TnQoLKg%{Sw#s>Qn z=Hl8(MsDu2X3wjy)zf|>Tb*#2{LmHvtX8d!goua;QWXtKdW}oB&sjYJdo!bpR*Eh~ zOHtLzpL-{}0vIY3_3j;prkmR$x0NGS(E=nN@ox|DrE7*y*uxgpu#n2%zh?zr-rXI& z<>1hcO`-Z|VnPc-W#*V!K=g_ow_PVX*D`3LPB*!~A;H;IuVZx4cln?-)*BnT4o~1M zPKFJM{!RidjBrGkD_*5_y`LKCI9~M^={0Dj;g(Grn6!wAiRN&w6AM0peZ_mbctuSh ziu7?RG-rzB9e?uBvzT7EL|OdN>N(!rh6du=MGwZ$XwL(F$|XH>S^x?6uU21&)f=yt zSQIjp>;KgPFptY`CC#k`_pb45?Nu}}Ue3r4Vz@oP!u?4AIhS(RDYCw=HQeu#s8G1r zyeZ+{CM7j_gU-(@lOOcZ`>T2?L_zid+puT-e~ENm zeZVn7V~3VTXj5m%9Di<2dHKz>zGL*4DH1DwnRuo5?}M|lviu12%orE_;SlYH-cBu- zHXe3>4k6sId^`Xc`lWF)ouIfjG^P7lkoLJF#SK}{)lqQLlr`_C z6@Lw1^4=}NX=Rh1*g<2O?rTpGC(anX<EmR3qV zef|Bt1NM?D^r$GbPLS3vJd&?ku|@T_M>E;JzsqN4=;%G$tga7ovOZdrmX+-QJ^RiS zZ#T8~??s9xpEWCv{`o0)GHn#I0=m(iIWh^NDP+oSDaFMds=hj*_)lK=^!@bTxSr_w zPmTkXyjnkMD+ZDqq4+-YlauDUsi}b}KxR0X9sA6OpA|~!{=jWhvvz$ovVbP7*wJ=Q z6Yb=(e5J@LEMkCt*s>8Kk~-oeV|HAt8D!VE02Yxgc?jaSzHYp}zAl`pqs}Q{ zVL?vg-x7UIM0?Ml-ni4JMd1|`cz3zFkZ9|AqkSnS8qvDa0fJa&OAEP6BN<9CEa-^1 zyStBCJRK=ywd=PPf2^`QUKaWC^=pS=g)kBf=y?1Wbb5{b*Bt7VbabLRIz+o%E5;IO zkX`5P5oP7%Dy-)?fyf~(k132Ll9|tS?R)DZy4GVYOfzZ(o588c_4v4=(f)BnN2+F5 zl{EQ&nRVOGn;oT|vqz3@&Z6~p_5`@eC>)NHlTikIqSOSGz?YY|&vh<_Ttmv&HwY`P zyOZKlCaMd(K%$%Gc?hr}NMk)n#>j|zcXt=cXie>56B34iWj8i4sjinz)GX-%sKf4c zZ3oHpjpd)c_7YT6Q+x616(UHlba6^T5u1m+s$7UPpl}BA87eH!mpEQp#cE=7N$Gk^9ZmslmR>IHY?+q=p6N zBvdSh2L0J58p<>|R+qv>t9IM1q0W&l!BB(1Ti2cWTpt;Ab%Hxu$}WNj*s2+;{GuX2 ztw?~v$%&T?gi(*n{jeMwmJh@rAqK7%pebLPlLP<8bg&IbCZa+2xnO`=SUj$6>uT)) zQ+x2g|V5b(N%G?^S(m(*igD49{HZU471f^+aNO8>A6iRPqy3^=Y?FZ zeDY`D;vmF`GC8n7Kuu3VqO7GAQCG)pY;61jB&vytiN|jI+jF(7Ut}NYV4ZKz&FSz^ zK;`AS({wObyVgTu+qD)W6g_9t7=?~3LDqlJJ}~gr=)@5Aft>aBCoR>7vUaZb>Kq=; z-~KR?5!2Q_Lz>7wdkTV&dZx&Ks+2TrIm3F>&FW*HRvDqQ|1n+4riNaRMQBV6 zP?rc~Ozo|l-Po;mA%m_z9sz*^ZZ`|4qC&=Z#FdqmlZHKka>0Jd`)UOg6@;ZNx4W^n zkupw#AFW-OyiV|&9#;*GH!izWeIMWWc#SiCBeAO6TD*%*`yN)C^oaX|MPVV*r1u@h zHy!fB5HWUu#&orut93IlB`W@_?*ts;IgYOpYSK7Fveooee0cNnoZC^$Si394mR1fn zo-A~mUt@i`lA}AO>N?* z^0)2?LZg+AV2xTEG9cTrCx1hO!kpb5PN?fn#@c$!|4fGijq!1b@0-UCJjAcxqI(}e zEKIl9@gRjyuYwF=ij`ef*J|53EDMKlMf(~dw}Ew+aspfdF>TK}aqQ)TC@iRLk`;f% z=DuIcDmEr~&^bPO@(NHC^um-&VmbID3b&u64Z1d>eR~%j#;@CrUsD45EjT!M2(!C! z!3K0aQ@HH|5EQTgs|m;u&=iK#l^NcEpRP1Pt+81MzrVjfOQBUZgW|Lv!4n+g z*QRJ>RUbcRP}oZ9apc_GMX?ZW&R;R2kvd=#+^^Ic407B~@Y3v;S+;fEJDKs2g*5Q= z?@1MO`>V)NSK{K@nfquK{ao`F)!(Ck^XHcZ+JG#EQSR8*Sy|2UWNWKmYwP)I6bHDy z@d`Z$gNi<6|6BnZCIVCyRn@UP#dJ=-Rbb$tE@rZ8$OkwLO7muq3lulnw~Y(YS~m3v zAgxa3v=()EO=27FCGR=!b^6#?w;94^v{kCJ{b9Osl3f(({x)ujl8=jU&7v2+WtDz8Z>-=(`;&R*Ue|4OP zFk(2})~2pv)VkiWmN50YPMal#*BJ%qaFpoqwwGFkKvgo!WM{=m0VbDr);?)t}N)hxXPIxIVs;f&bc(abnNrQJ%vp*m8oNFOm?$tX#%1J*Ui&;CE z(&Nc%gYg4(Kw!UFm!qOk|JF@I+v=&l=g;SaTRBuZ13#+^JS%0%_X{hRBf>rM*VgPz z3|@XVp{@P6O!|gr`S_ZxP-@>s3Oh3O4rBQ$Mle9?=~(I=@f+R@$7xn%%*EfZcKg%p zR`Yex)dXGhZ?8>IfV6H_pk;%c3;YhI;~O#(3Z@8HW6&p0N{wd$^gOryDm<9k^3@$2 z@28~}?>eCEvAMJZb+SoPMk!GJfYG}cD0Mu>$5>%?mZk;(QmQ}lR3DR~#|MaIC9Gw1e0I7IfN zOvICA>KMb@jRaWUkcar=upO#i5Hq%fo_K1kW&(=JT8O`Y{|*!Y-KvTL!R$so`aozV zEhlFw__!}%0~{8XK@$|!u_-B}$5k{F4Q*q47ceECft~ZtyKkjl_X<7w10&jiHI|uh z_Mt3aeZWk+7=#+Snr_O4A9{9saMm@VRH`0rRZOJ>mVU zx_4+=8x9ZVT<-F(20OPHW@`AzPXvp*Mepl_&erz%rNPdN@!V%l7z#xgo@i%k`)IY`;izM-k7os&jMSgwY_oE$lo#&n9<8hTw{O9xo-Mr{XR$()yM&0 z^vHY0iIC#QT655LFU0>j#OIEvG?|z7pY^nUV|LK>xiNu~M<=V@TVAYyItDteHfSz- zbGBtPoWe8YIq&q|+gkwG)Qcb_iP?p;AyY9>ker^K#eJK;aftk%Gz-W{0kAWk?@b+n z4RB)V-|C(1#HSN*fJeOA?^?zt%r_&QZf(8aD|G4 z`X|Vi2hHYnTo-~>^ArsB{+9Ih_0>T>;#6B}?U;L&pF1~Q+y`uyEnK!`PpeE5YH`_^ zRrkt&AY*2f@^;2AO}`9O`Z-qus!03t#m^@KZsxs_}B7x%#+4d@Qbw55y zmece;sH*+CI*C4aJXJ50i0^sunU&3OpH|61FG<1sgRR)lpwVbs4g<_}7k{-*wzFeg+@k02VaLvIb})ou^a0D<@{M(K z?nf*!AqA#H(|yPKa{CdR5C4{vYZs`>bgcac_Igbac9a;oOJcHAGd$*)CxbPnr_a=q z6PQ1_y*i2kdTrvsXXsGv0CWLp<-RU&CfoKvjmBi)YbKE2fgM2ORm>g>1=23izbtQE zl#T?1Ajo)NO4K9f=jSmuZ5bfciTe<%QalY1n&m&a_l4yl9k1LR z_A$8I4X*nwZS}qtN{hm0&Mnfp$flr{XKt<1`2Z>d-TCK@0vIuvbBseE&#dDXN|akw zQM0g+vmmC$`?(DwhG}2Kg=7q-%%(P6*~Zl!-@MSPuWy;z zV!wY6J3l}78wNP*db}Knxe5@X2c*$HX>VwDr)Ta|2I&sbhXfqi96F3i=c{l7KgWC) zWJYIwIKh7B=ICC8Puloq?mdNUrFlQ&T6i?NKRx6F77kpU;ElxajORR5-ujK-{WXyPMg*f0a>kezr~xKr zGr4X%D+085${jX-XOuk$BE936Un@kB#_sRsMMPkn8itd~@kEz_kg=QZs3{Oo%=)7l zB7`qgvmBjkSi7qOJ+m7!5J{8d#j|An&*2dS?;3+hY@4tte69^fgQIyxq9pSegUV=b zZxHOtTVV36ShVAlKXWh>5@l>teu^RC!L#>^S%_DkCudFbIYh9!S@Pg-US|E~cG9OX zVy-E208GHRl_tP>k4g#&rf72ZkethMs;(TK8~O0Ly<%w5Ir-t$_@?ZarQOs1p%^ zJ~aJd|I;~p*ZPBKoRGibYm2rHz)FPjBHB z!SuW?$U%fb4lP@TCg>}sECz5Gi} zQh(qpvqpmxy?TizLbYZTSZns>?&q3+cAV_ENi)67l?Ti?vVfC43xokFVt{BSlx4$J zmw7i zr%ufSvR2S^0GvryM#hVb2lepqa3xSr>{^0j81;@g`#RI0q6SxVpG#(dJwD$V#}ZAc z;y;HnhyYQR8;@GV!18IPUI9V9!RdN`-9gPQq!V@ykl8D9mOLPUHICUn4Hs&Z6x4DbX)_xrd@ zRk3~RY^|E>apv%J^g@9P^di+xPWohIg-yDVff>Ppkk}2nRo8>gc-sW6v_Pc-`k_!_ zEhmQt)ONo;`f6!;QgIVy5lYlit7b?YixiP5fszz8p?_j?6}Hj!2-=x~9`>NzDMl=G z1?NPz(tXL_d>Rg#u`(4CB^^*fAY8yFfVy3wfM+NOXi(lCj ze?t9tWawVgIOMs6yKPl5$BE^sypS*IK`z3TD5!XwxJWPKwo>hDcpycwn#<=Cpf~S~ zh-ZDy3h&1n{XVO>@R^>iM#z3s8XY+wIjaCKA;33^!`7!q)O3CX+WU0w=OC{Kp$Jqa zfGPlF5nLj}xvTWLlR{-xm9nmGba8QU>=u3jbmBTX3o9eWz<$4h8So$s7rrn0XLKMp z{l@Fe04fClJ9V4u%5W@dzwuy|Z_4I299F=IAgFRyG*cK*=W#!Nc<2Jw zx_!84%8Y}J2o3Oi3qG?yKsEmH<8up3O9M6{B_*ZN!;y8+aU)_kgom`*)UH7f>ZI^E z9vA67pn#S@J78l`qa*`cyzjDda@Iv#(84Lz`vY~}6L={rV>)}0mNE@?l^3s{&C(;+ zFh76$izrDxIXZ;m4$f=AYf`dmAW;5Q6t{+LXDB&6QL<%dT{M-l+=`i8Kw^9bXj7B9 zy#F{m-&MzVb36BNJ+>g!O;Xkr5JK(_|8#$cN=Cp9QBtw)D*Rc-NyrT;_nk9!qd~@~ ztqpub|400N+&DO}fcr`z1CaE9XnVN3l2TXyi;#>$V6~NFj+a+hxa*Zm$Qe(QnUw{L zkB<)&fgrULwx(Ct`ExyOYCI2VEsH0lmo6&szHZ#%gmW)1UAXJG5+k*)4*MZpDT+aDa#x=?oLAkp-E z!V&<@yTf4xZ031unR3wYqlE02Q~>4Hnc>ok(e=zc;z=Xu-fn!jhdjAI>AXwks2DOz zzNIZ|z9fd8E|5=dv8Y-ad(fOeWUUkmUINS(v?eN0&4Ws^0sOsSWMpI-v49`vG1qLF z1*9WL&F7pHKpF;EE+uvKXQ>Jy&?8;U%q8#F>;Qx0b$`-(ePHVw5r+GUikTT5$}a-J z43MV~Flz?&C%nq8t0MyKEYP?RDY^z{inf?gMDWo3-;hZ8C@aT+EjQl5VJq2w7qaeU z-82=aKa2BI#s7n@M$jqff9>^>hs=!&95QIqV73Ta_iU<4=9o{^Rnp-G7w`&tiec0L znT^-)>V{c!KnH?9XwKH^=_FTv*CsOPzRdN_^U0e?n;)K{>=DWFunzw>0axeTUXs;J zB?6(26ZT6qih%E~L;j*CzKTvz^FQzJmyYz?Dx|Zsr)ndw{a;Hn$qz>fOv`Rg6p_DLw~twlBDwtat+#qqApQN$X6|RsXG>0un^8h zIKb&Mle&-)2sVOaWQ1|jko*ccZqu61!^=!PO)3;sCna&2S0`TFYQCoU{UXnA**@V2 zTFJ36rR(WSB_i0QQy4j@QagC)d;r0Ef^i<)TB27R!GY|D@cuKBG#@^murb+?G^pKld&L~v4 zd4qL!Qhr5#`0xP{x=HJ*1@@Tv~J0 z)~hJx*t>gk0|8V0PizYb{##<4xV$N<&z8xi``^l$k&^&J8=Xu_l#(fRME6{xq`0*G zH|A5|lTRga!@99HpX1AL2M_(FTF<%tq<1OY*nrbPakEXXN{Wp>rw?VWVCGqC!s7v! zf`~yX4B)f)@0&9LRTUKw(}Px54S|~3aW%EMS6*FRT^JM}33_yE2(vsqeBd0VC}?j3 zHv=IBj$u8ZzbrMuzn210@E*&K+pj;t!1LaITZ=O4PqvZYQy zdJV;??S)3JmkKGN(1ZzOIYj(!c4yLBQa2@ft>m>!K747ukM4kbFa*W>Hx6?gP`6-) zboztr_60jT_COMAFSzAK>muh85U^%O63UPzR9*cf`c{g$*MjGdk~<3H*Nhmm{QVPQ z+vIfUjN|5VL%|2_@~vrpQ`AVzA+g_|(uH@df=mXvX;^$<8dgP0qCr+wxX+@-!ENB+ z9-EXSqFjhkpnU9DA3M!%poOCVXdh3@Vn90p+FoZ=lx4Ig2PlGipBDU_E&nbz3N*&X zC4@}mh^wimZ2!{3k(>USf_)05fea32tAS|I+D8bGoZu{pl$RGj;7yhvz4p6xUXM7PfSes5vU;DUZB8ed%ulMMjcGYq@jgj zRBwBlYNhfSA$L<%wTBXA0OIFrZF|8PF8k|a13WyuU&M>ez&qi{>n+T8{`>^pvR-I@ zX!s{b1~Pn~{IS(X03@2=@F+Vt{0ig%FFDF;BW@4^B@~RE-VHA%847!-i0{jd;kOg; zLXO88o(yN=X^%bl>o#vm><0!dW4{tNq~Gua2|uiZ)SYIKo;7%Er)w=R+`hLTSLtvD z&ewytkPN4AA7__@)_5CdRJBZ`R2R1@8O!A}8Ey@2Tm~qZD6`3YT7?Y?F%bGS^M|+S zBwty}q++>kWlvu3uT63{*|PHMQVd zkzSpvSTYRgzFWQ3yzO*Oo#|BYDS@d#H3LCd>f9fHxN#T^#bSzj!!5bR&Vhw*qp$9y;XHM27H+D~6d>vG;n*P}((610Bp8Zbl zqj!Sncm6Fq0Yay>HZHzV+dejJUUQsW`4&8P0odd$Y`VL{=Hmr7xIc>Te`)^lnEQ4j z-D2@f_9!KaNe9*;9%qzV2KOzdaR@HoODj1l8J%nrPy$SFYxkMmGQD~!jJtZ3(_CXb zux-y>IH^B;$*fc`NyKJ`mHW7i_>J4gOX(qAqoZa$GIsIDbgQorU^ZmuY26mD`J1j= z6MZr2F;~8w5RuDzB6*J@dR`rNgDh&S)%OXt2LAkp=2LZZKeLNLnmTxP7g&;;0CchdDF&;{M@Dv8M(@$%KvCr$;p(R%RXzlQX(OO50Mn8 zj)CaurAwjmdfve%WlcdjU)0XYxRF4K(c$;_Fxpy9c!&PW@dggGJs4Cee%JRgdS)RYAc>Jh{`YEM@Z_#Xr@HJ#e;JS3mc_< znTFeITAfu|Lok{6%AmMNB0C!!BZTY?B6>Sc#Pjg$YjsyWDZ156a_vFb5Gq}!U$mJS z;W1)9l>|5apL~3|q;VSigCTVY$ERVjYnoa3$0Sz=f zP;>=>S{)Ev;Lvyg$QYf80h=Y9J@!j?Q3(mil7PX2E)%rP`{9Gp(PHy-i#H$igg>yM z7eYc5&?IPkEZZKG;A!8>3f0ryUw+zB(CR$DM4v* zkZ73T7ggTo2#bm7J#L3%^RnAB&tj43DA*-d zoBG?nI34==8QQt}xFgUJ$gTRt4=kM_UG$y zT^HS7Z4M=e%v62qG_C57el8=l^!10JP~4BD;81Whb$AbjQ?S(!@My{m`nF%}es2c( z3ERZyYAG@zn`Dj?B#noi8cEG9h#6ig9`4&6dj9rR&s@%QpH|m>Mhl71#k_YFn3!3x z{F2rXqhT#KTzg&jCi^w-d#o6WuI>uv%rmMSR7|Lz6(}+^jJ1P04?C^N-_RLgkxP3* zWMpKZ8yugIV8YBjng~5-1+Xt502XjShU2;N+hEjZS2Qvm;7oTwiB_%Q+2+uJi#EX4 z%l(=0rcKZBh9f|?c6?(uYG^w2^c1~$Y<_yYy9RA%|BZp9wcVmJ44Fac`y0-c4miR~ z1A}y|d}rzWU-e2rs-Kb}4ytJW=G-|!nt2EIfGeSEEYldj&NVSmw~@>I@H-T(o%#yj z7y=(mPC;O*C5ndvKGgte$Eg~KE`YLmaAKs(e(l6DZ9rKxY%jhP4syOfgA2+>;3a^` z1|0;@Tu>^?HsYEQhJNY+lz{}CJ)7sHRk7~F(_G`!bpEiX&hp5Qw|ksbe}AwW#Wc6o zu~GgE;&2N&l*TTQF4SG+q{&K&r4pC(=$gTo!3_v6eDHr+GLeg;%;3UWI3a5ywvv;G zf^%R#FH1|0P{>K!E2C_eJR8f#xi)}S{P9iN-Qc7Cw`sqS!b6$^OR0#@7nn6 zFNGwKuCM(}((PyuL_9f(@okzN&wNIM0?VWm{Ih7|C2joRa-V1R=sXouSwJulyZL|h zYW9-&G@hW>(}F7?tzG9gF5ug#^K^NH#HBwq~qhZW3zEtB9CdmE@`cIUGnDg@btu%4JyxeN5tR@8B1#Z74*vsk^C)rHRwADg}5}bCSNkLgLCnBU^w0DZ0BD#suO`=*U~{4m-ht>&nt0i9{ zmxI~hK*ayJY4Dj51GVA&kJoF?RfnE}>Ai18_ya+3SnCSIWs|R(Rn6kjH%$05vn$4w zC^0Rwwp~wWoP>&shJTRz7t2!{lyFbiJ+8+I6h0JBSFJ9Itu5wvzM?btChNm#JwWye za>KMGxn0!l1irRq9W9Vg$0durj9pfY4y7TpA9mKii*50m9XI5_{qVy z#@FDA3ij!msH#sR9UBgbR}m*}{Np^6AIIkAB8G<*z;gvWFaMFFKzj}A4|%+-ZEa^m zn+GXeo!ts-%fo@K5g?;RottOJTg=16VHNpL;DO!ESzbNNEGQkmp|}o4$zL>QwY8m_jF@CTyg? zSa0s2ay(Q!jnHt~kv4^@SjaA##(PMw7ik;KC+Lp}mEbSM~KE3mWV>hXi}JxFONM2SqimL|f; z6A%&GM|NY>@zwXI25)=v)8{6u1ln41zYDL((>3~ zjN7euKe|H)kN(*ArOLW=;py6i+}inuRoV(83)_=v*>xv^q*Rn~MJ+K$j^yciz|e3@ z<&dIC>j6apeMNyCG(u7Y^t$HUy2V>zi-t(J z=(Uc2zv=l!-q0!G@dn7Em3uKtyiB8JQ{y${wRLq&52@NTtl+V0angaR$8v9u=E()X zATX~%FtO7y7BffEUu@ju=e5~Wf~vLrn;~T;yauhs+g=&BUkp_>!jaH=DMaTo2^SNf ziFO6KenRs|&%h4}Q zxBVA2c#q`s)UR~)#-FoAyf>RKh`^*dy3WWftzK^J&oK9@OvxAg@avE>D>~0q9V4)y6%%a$;zNv3u zU=#FM7|Ad{7~khz<-xG%M!M9(US*x-{Py`nt3KCWs}wB^V~aTkHdJx-(!aPg$E-f- zGz$F^n8nxms+T+mFptApD;47|UzW)3%6;(QLN6mbopT$5Q#;MZ46+8>f6Pbia~`U6 zyY*=ZAOBbX@zT)4XT+F;6*i;f;+=@uPxBg6Se6496+6Jz!rbfxg@rHqj%c3!qzI{6 zIMGBwX!TQDTw$Ssjp*5qNgEA;KlIjL(+}UvVYJ%2DCDwN7 z8^$)oIra_MtxIA##TBWT=Q`v};faM1SU>$G7_+7b>&3;H+`erg?t5k(YD2i8&=@)X zp%sQY)Lob^ljt>-go-|`Ow&BXHmwwB#x^~EMLrgkB8h&U9O33GLCHAdO9h^6f5p)t zd*&{mjn|2NebXpEM3~FIbw3e1ktI?raGdBq<9XfXGGH9Krq*RV)bsSW7gb_CEzLi zNO2T`-ExrY@f1#@_+=K~BPy(eUZ3Xf7h;K>xr|chFj@|yH2m8}9&>59_I3;$3;E=B ztFxjU(%D^-*G*h}e9+YnoYaEGsfGRKSAXZ!f&Vfz{0}OnHA7vaKzjE3jK+gTxX=8h z?cY&|a+q&|hRvg?KvdD=-l;n;5)X-niLn8GknR-rnk7|lfP)VQG@$bI?UW3@jY{wB z7y%{;+Ng_eod;7qEcSBgei1`WEDszld#!`z@NOS%vR8e2vqZtly6(L&gzjChGVXF^ z?UAQDRpd79U^b)|9Mrs`ks>a*r@!tQ$uN%>H@%??uW7P5%KvMO3C*kl*&xUuBe*p?JA03zFPUTa zGQB&`b&CN$Nhsnv-x;zG?yT!=VJvXRhcb~rZxk->ArPIz_F$ar!FYlmEx>xBIyAEN zUDnqJ)5p8*z}*?|%S|?tsQ`${vyd0rbJEAY>n(53+eCw{+D07P3Ql&)JX$Qd=3iun zSCY_!qqCECu;q%Sv<+((UV^DLP#K3~F))9ek>Ipr{{MJWa`;myp~gaRDkcz19IHM3 z<)N%@7!>_^Q2{@`pN2+kRz&{_xUtLOB-}`p5nc@pIcbN0O6HRnp%U?yl7ZJ^aP`C+ zv;!w@5yTNIGs`8m6*J|?rRp13*y2^wBVfNoMU^&*_TgkVCP!D%Z^A_jJYRG{1FP=0 zp{|_#{Ee%tt6%!7>Laa7f15o?^TQfhQK$`FuaJ5D~)mG#h=aO8{t z@if08ZuhL}8XD8{%+|2TX+gGrKl?@#9Lb{*x#br?Uc2vRw$#N=DUZfJw)e(7bJLh%yx z-ptS@XKG7#sM_1x2Yh;K7y8;VTMxp3*j@D$kL+^-z+n%{@iifmX)GI-{iU_-$n?dLS7daV%Q_+L7+}j z5Pgh9Vne>1rA5s{GC(hNU&x?ok`%l=SbXdoH7{QR!ZK6EFvv_!Eb(wdpa`0Siv zL74Md7EgUF@is5;qXXY!zS`Y)1+15h*}1J7?DZ@I3;R65cL|@laX6igO7dGKiV%HC3d47xY7H~ z)M=UN!D>ONw2SJHwb;Q%sibWz|m z&gf)OuZ;&^PHT6)((dYY^8788EV)Hd5>=Ep<0}6F-UTC3W)pTad#ry=iu&<9wp zZ5s9JX)4<3kQ8F|GooY>YT(~-pzBq8xc{?_yZSk|(d_c&*a6wPdwauu)VF{yR8AiJqwekwSN+pFCL1~$=t%e(=rBOv3DX6=j`c#*0S1NsVIqI6N8o3 zAugIAiH(;K;nW{EAxLKG!%ZF!Ox`6yrP}aRUG@9g9Ija{{Oly8dAIh_J z;$Pz6k7-Mm#p8DOCw+aWl;|})-5|-fA=vSdMB>FMO+hR+JV$Q2^N>&or6?)d_z2zh zVSs8Zl2M3%NL1+kLJlLdGCJX{7PQz~>>Cuac#bis5b=t61R`Wo5a7+v2WZV-E7(j( zD%oo-Q9}HCkNx?=y2pN|P8KOK{9BJL-m#8^GCwnH|JJ&K2*}|r-uvCe@;NUHO(aCI zLr)U90t^GjYZ)e+(fIlm}1>JmdN$z4F7 zvy%X|d2l_A&X{x~)<5M|nqMs)7&yV!#K@Nxh^5`XFC2|O);o~fBbU=7XOgWOPp2zE zTC1~v5@Vj?Fa${FnTZX&(l*oc+Pb{K4E{E*Zip`8=}($#y`mqK_5Twljcsjgup8*+ zYkVP)L2){_Yz>h7@pILS`N@>oH{Q&s(ZEA23A$%^Xx*6eFp~UYp>_K!{%tv0u77Kq zZ?Lka6pE6|&U8_<7M>UQ#eT7iupE@4{*(+^0LmqqZj@72#=q;qcece9uKxOc=W4cn zsNb`Fq-qU&YE&fxX{!KvXE*)?GNt6vj%Xy=ktDOL<@iRlOz-S{Wip__PEny$`muU? zQ)gc#;_BOPy0nQ?!=(t*4VLQ}j73(&Gj?NtBJu5*O|i6?#WzHMA*3PDS7!(i(7dW?9$mIRP?*D9s=l&FQmUnVuU_ z52qIF(uKJhTz0`=%-uwa1?+TN>WY)xOh^DXA?>gL|PT(uYysW(ZD#evs_l!t*Xm%xfn8-J{3 zcH90JJ7w{~$+m{{P9zRn{+f2z=T{sy6CzS3+PVZ~#DlpYNh?@aw}rlv^(Z!~<3k-o zYvA$1*lb8n$nwpz_~0q2fiHG;5;-G{_KQq5cVzclnF{Q3LZhHrKx5+MR{ELe@02Eb z4vk;?=!A%eQ{mL(H722pfJ}eG7D=vfX)UV=UC}D?;w+&yFHXOLu(phwoddu&^keS! zo1cQCL6vew3M31Id0p)eLSn1t{{0G4k8;-O8Aj=tiy$jZl7I{YzdIt^ibt*v6^YF- zxzLQXGynK;9*Ofo6v^1w_?P`KBU-KI{3wi6951J literal 24540 zcma&O1z43)*EV?Q?js=GAd*U_G>Fnt(%s!H-K7Xfmvl*YNvVJ|NSAbX&py82`_Ihv z&&)N5D4d7q*?aA^*1p%h?tLPZ6lAc_Nzfq>2$r0zqzVLrzz>1I5uhT2pFC+EFaZA` z8_CN^LLOm%vfBzjLLgKSIY}`!kMz9-Pan0(d(nq!lM54ukHXYwXlMyii12vm-%5*E zBDE)`a@F5YRCZ`bJ$Egxm)b2%Vt$gWU5v_REA?CocOsXCIgkZei$-1?)gQv@$5);KBF@W97;Xm%7C4eTRVvy&z) z9QJVoCFGZceM}Tq4N(7kO(aAL8+JL~`Ll0PWFg|zyXPZ9r@PZ(9O zFEKW*@O_qxiz_=npN^$Eoav^{Ui8N4J-+QV?m2hcperbm3k)+9?3$DRUny34~(R=zRX<$cRtLC1rPS?S!AGy)Cc zw(<)0w8Iz83oY)(^No(OZ{FAlx@`_;5DU7ojE#*Q1ySBXTHN;G9ZKGAu@u6tx;k1J z7#ix$m5TCm5iWO}#l~|LLr?5AO2}7jkQ4Bs45)S4RvEP6k++_%NIE$^rI$<0k04-t zZQpTUPbZgHYuJOl9f-lU#iM6lEqH71ZE9*dHdY{C@R3mw`N`%G@6snkv3PKy*+?d9 z+u4|WBAwh=6ni_9uIEOQZ4>+Z`m&$%ioC{sn3ku1hsO41YYIXzh+b>xc@_X`Fj)Sn zwY1%=(_|6raNmE5iEOz&R**PkwtLh(by-=nyK~^X$KrPDvC`>r{KWppr)OiIo~e34 zX?1Mn%#WLIez~7}3cQaszm0BQu(Rw5#TtvJmT3O_8h}rw{#cFP(mG_Ug_Yc{TWNmuv{29Y91@3iwkFR}fMdxZQKdPyz z(KYF?S#}1%jr|=;3;J>I5+`&LezqKh^O~NXK5^+V`St79`6!e=O3k+iEpx+j?-Ism zy{8vR?OUm1D1^0N5Mz#YhhPYPx27z-cXhoKaU=BZ3Pdt*>&uL+QxI4ij3TOtm1Fw9 znIK~Wwv4c6+BOxdm9RNeBgZ0)#~-^X42@gG@nkOi@BKD2$d*23&GgN=8t zquIwXr-+x(o<$(&v@Y9v+(&wJdmm4S=GW0N#|HvPv&D_IRgYcI=eRd}SQvK6%}(jj zwU1b@a=hPPiP2v3tV^w0z@zkPebqV)~2VZn{`(`$OW zwsUpZw_u5O9n$qGWn5id!wy|97V9oQX5AqQx*vSMIp3{z*&eIs-XD8(irxNk=zt-{F4YK!P9&rl=0BC$5seph8V=ZgsUsJ$klQX#ipcW+w+P4S#{T` zN8%SIsOl9uXqD4Zz}b)_!mx#{6~ObY1^s#>p9nWofb_S3T2CS{C#zuYFXFNl&t=>f zjqU!bIqGs)@K6@aUd79+`Rr_55rsRk#(qh}@>dF{inzG=+(=8*_#y~uI>=bdVf)-g z8@RdLc%h;w?PbMcMM(*>T!Bj6+f+fv)!=-U;{5XW4fSoqM%A(yge>uAUcSl40#Qp| z54E*i+r@>Mz_k(ySv0A(U7Fd}TwL!nGCH^~?EOyD6e?NOs(J9cJWf_ab`Px0{SPWw zxA)G0v%6k+T0J(QG8Fbxc)tPOqN1+;`RqG^qKj8M1y`lyS(|JFeMTLgZd*moL;tkC zKP$|uR7xjRw6s!doi~ST{?=MgvsrHa&PnVth^O#9QwFv$%#Y7s{!Ck~UCxpEri+Z^ zlV}_Q zkMc$TpZ#k`RdXE|wnZstb^rg`$Vy}xUqPdQ!~OXAQ(0FRD{Tl_F45pcs*;jY03NOM ztGrQ30V&wV*w{-GRuuNj?B!IYK2BKA=*sQew{I=1th(Koe8ZksG{Re2TH^hP2Loej zuO6V@fdgLPx;xdi;Y9ts6hBXJxAW>~S&BDb6HN3w$PZ4xylW_sRvz1=$ zeQ)7_4hbluV~zfDwZGGUm_!hg-P%gVmW(kwE5qr1VZ@a5P@nY#x(B;cnU*sp2}xwN z`=Q>aq$DYh_h5{XM7KNdt1eAmwm+O1LpnpozZ!ek351#!O7qnW7$itcAiR6`4idm= ztnSX_p#3b#FZyW>30Rg_rl$1&?6>2}(-qgonC39O@$vVVCJsdQ=Z`*&-G*%cNW8&6 zSGy<9uH3m4RkJpbfM36Uu*D7jD%){jfmz$$%E`Ohc&*J2x^O!rhA7fya zkayxCgo{XB6jZKmt*JQd5lv)$OeeQUw<(ax6sg1nHH=Y&+SjY`BA^?nJf6uG zic3OI)dVuI1u`n;5X#k`zhO|FQTPRZHR|{c0|8hA64I#nH1Dc$V-ivn2n6Iw@nL@L z3f>uy*99+Zsv}mD5Ca;}!a^E|J4wxl^`KN|N)<>MJn$4^V>ygerde#PY(QuLhE#b+;TV&e^dr`d$IBS;!{HdRW}5y?1(Gm*jP z4!mvd0$1|5hkq_7j-u>89n*&uZ;2kt$MnpNyNz^aK6Bd`G=HXcF1447_mRElXc zq`!lpOpKuAmxN-&u6IDax>q_*B?Crq0M%EI+8v-3J;S2eiGUd3xt33a&{U`Z+ln z&V*T}{+vr&K?C960Jg2b00N1ZU-C=Rr49Ab6h8T1cU#_o*%!fD9>Ybz7nl}(F9UB7 zR(^Dg9XZp1fz`Zc{k?tt1{?fLmXJ;86q;cprKdsUHI$$ze5 zjr@HYiiS7nlA8lJSZoOlVgL?gsKo%1I^4K34pn`CBZ|hUF1S(uMW2ixyFZSQxUOxBY2l?Y0zmFd=%$r|KX7I(HMxwxs46p(d zB5VQK&*uY&V z@Dbrang7onYS8?MVbgjsNhTQDwOKvDPY$~xo(`7k=+fwXMRj99I!wUKC4O~(Y)s5) z;*hTzApqv;)wz2vUaszJ3LLZd;9R0o#R>M`jn0JEbcF>yDW8b9)pD`Dtt5q)-CAHP zW>SMLl2YqK0U~bJdl_;P6$#R+nh^|F{C7HdJ{-reSzA7S2qNYRSDh398KwvYWCCSA zO(fAb0x&NZN(#dHh2u!v_kC7%-6k(W_P+y)q?&NVsG928v+d|9lB2+=1BZIxX!}V|E^PnbOyE9BLnR)`kR}9z$ z1lsB`bqH(`#&aTri202?fFo)Hy9ZLq;e%ZQmN4n56_)k~$s;YD9aLIrC2QSzjb-oX>?fJpSMDZKr6#?@gX zN=%%I2gV&@V}Eh>Z;`IOE_sQ49;!fwm;@_;1cRcSoSgsM#@fj{VHl)Z6XvBFQsW5@VTm#vphCen&XFOil!*_&-6C5rv4R(AlbA%G#5 z_%DnI6KLDJ;v<6;cre%!iU>Zuf?$UQjes4WC6_LzYeT9c z0eqvpH6Ivo-{Nj`aTZ0J@%opNrw|2dAnrt$dMS4qqd6$6rabH#NUiEfu*34^_{y`g zG~4x>jwO}XYejMuxCGi{Crfe^mW8Eg&=>m3=i+&@iMO9U14+%S$VGtMrTHLi6o$vY z9+eLc0X#l*Oool3qN*AsP)LOeQ2y6S4LhL&TFHr~kJvajPx&H2X7f24>`&n`wF2({ zthG+kg+wm#pWPYc3N!n+&6iL4fPX7v8jNV@8}${81J8K_HYyjAupYP=X@oj$@@=(n zrcmxiL}D&xx)S~|Imw*c^;7nQS#}%*j2jt&q!f_7x43xHSj$vf!kI)SXBGxtIzd5} z(s~^2PvJ0q4W7&o0`E@(T?8!Y0w0BIvCS^5VbC9Qnw8 zTh%EQt0gFegd1rF{loyFml4@v7Qy85c<)2{?G)p;C@1+A z-ALI&{EuMueMih{x(qlRMuPGMUy+Q%8s*aYFn&C0ZaI>6+Z#9l-DP^6;6s+QwbLggd~bA(i@c3vhwo(aeC~u z45q*T<~H2%5V2d)K|N;qU5Jn$It{Xrq_QNSVgHkbF0QD;;ns;8mjUFr>tDRLu(ScQ z)5Qlj1@JY}8dxgVfp2V8($NaPaS@V<K?vp6rl5c3v0cDXVCLRl329EmxgSphI(O zA>ZEK{?Egh#AF~#^w6#4__&|a z`1WESR|;>fKd%35dt6HhuNRJ!Y9)U51KPDtU?#}u7pNYW{DJdzBgQm^jV`G8f6`{*{XBgk@eb`3y)h2>T`<<4q>}ZMM7%TyZ zU;YtY6w@=r>x-j!0;WwC3l7TegFaN54Bjo_Ebu8;}0} z4Pe0U4u5}}rFQaZU3YTGGT<7qGnOFhlu<&kc)dEj_<`tWTK2lti(OR4m6AlMM#@?ab+A_ML+_$Yl!} zA-?CVh7sgZs|~GrjWm%ofDws|iZboJKJskNN^^4FAvg%MN z>~l|0#}Gy#X=!O0#@+u-vtKPKtq|AM)txaWP93}T{((WvrP^t(a&%fo1u-kLi^ullq?m8T z$y5^T_Js0fksWAWPz|v80p@UQe7pz$a7>=P_wM@Can>|h3@YXGaJ>mgYjSn#xprSc zYzm=pdA24xmQA|cfl_yLhnqX%lpjw3h;-Q%S&07lF3ID1*1Vf=eBaIZVTd>Hv50cH z7wi4;hRiMmvc{ekvC_2dcr64ggl{TTyj<3w&y zi;Igv2g>5sa~FSXMD($>?A?4glUI0Ru<*Hh^bbyI>NlVT*h?8V9n~RpYd+s*0Oz7@ zt7*i;!vnK*x94U?SmEI#JtAE;l`{pzz(d_hhncDH!S%@^@QQ9fyT!=1;3>40Cf#P zu>jUL{GRb=)VWt@^ZLouyd9zy=@gl%6#KXIaiR}=0PAx>R9p)`H^gniqZlL*z$n1~ zPa2lv#-mLSkb9V9d>kytJobyxq;`#PilX;ic1;_Ekj^r#nxm!12O@pn^Cw_Xp!MN6 zNT2+wayo1yT@vbvT}dg!qM-AOJ`aG}AQ|TBgEsdnqm3U`-*?OO#mTD&BAoOpiFT*T zgMMy^tDDkGSL#N!A^;|mF%-MELaXMtl3MWj?zHj2d=uH|wcgveSTL_t|G89lSl+PM z-FCSc)v)M=L8R~FICpvOB@7x_G1bQX;<$)shfDewP2s~&Kt4g zKcI4#g*e?TK(7~!iirzOuA87kKX~`=LLl6==eIT2N%oxBtM{xwl9H2q0MNhPQvRja zO8mN`W8WOf`;z9c6Rxc3P=pIgWPG^T#`OGoltKMymkkfXD@{UDkJSjqD$ovvrHaig z|Moo($x#D+UyO@Po4)BRKTJrY!ygySk+=O-HX>?EdP%)VNKoRGFVKI;g7tB;<;V+Z z-gWowH{)u3e6T$$SGeX4a+YfRqnYphqP5yctch3v&1^bP#1G&y78VvUhe5y|pZlUG zBQrB|mG7;aiHSoA`Z-p{eq_eyB=YGNBlo@yg6fXh*obS(WfcTTXvtdLxyQXdyYb+u ze9jwG-)qkcd9T>0dl`Pf*+T}(%JxoDejQ@uOc}3f0gHpEG%dF=tJCIm{`-b6L&4`{ zkkLlANO7fNX5k}budF7UjvkGn{O(z$r0)V=)XTAFk_7>DD+!E#QK=e_xCNVG?d}w} z^f}6&LYdg!KIt+4;Z`cW6Y0)x_`8q)ScIE5%&nYWXRDV63Amg0PfScKH}6*Tr1Lu; z?d*HX{#Og|TLMQ6H`q>;vRpLo{8CW?#s}8F;XP-|TEfX1Ap$k!pHV_aczg~fVWHvlev)|wCrBQ8>?w(7*CaHU9I5w&(sH! zAjtq|5ZJtH`@e`B3olpAkQlDg`}a|+ ztE<_B`dciE!x@5(L)_EL)%_GbAc+MHm)GG&Zd@wb2^A#bn98(B*E~J-KYqTq^gdSq z(G^MMWsTq)ji^oC>bV-vd%l^y&WUbJ1y$OeV0PTjOfOQaJ&Ga%C~-n+w2?@ITWH%7 zd7r38YiI}Sd*pZF9kder$-kea`;ZraI(}zo$AE(vwwseX|24Q+=`j~drY%vM;WWpF zWLH&j;1^5lP~pA;Ew^tqH4=Kdc0*sd8`ok5kHDt9a`KZM$2n9oLDlsR@j%;wou}d+ z=VoJLZ840pcjm+W@oH%K%*AmSB*1ap{QG&uO!@bK;C;sT94zDEble#3xfg)XyWsib z(ET^`b!x!ao(VJ+pEg5X&B+^4=&dSVKOhy!;8zyLqPY@%F5!o z1gmZ{6sDK}xhoDdmxB2Pa|ep}v?EX`{GeH7S}|pNHbUonD)Y{_<+eEOg!v?#^;_nk z*CL6P9FJU8*cA#bCqbBTq_tW*iQxNfBNH=(5CTv9IUzdnj4swCv$v+zf=khz?qpDR8Y*yCB~U-VPW4!`^2dk4ScsF!-yTqKUy z8_E3}6hK{p#FqR#|E(y(2MA~r1`MU~uAPpE%IIa@)$K;m*xvGbAJ0iC8vX1ijZaPe zaNW^L)ZuxgmtXkRg>Se>xw^`7Gi5S{uz^UmX`2x*Kvk8u!`dx-Y8#G@YNV)tfQ5GA zwTQbjPi4m`Mf=<%>!My$Zhxg|2VLtGDO)#yN+hJDE5{z$YRoz6r3q4IuOQcJA!vGRIE@*{f{rs9--M4m}T~UDz0HW;H%C~4x=7J>q z98s+EGwXW;jw3kZNLx~ox#%U)YiRpi15t=nZwj9&vtNq^ULPhU;r2vs=DVYnIcZV` zqI`*@&f9V5(l0x9>@Vb1=zbRX(BIGB5odgi<`SuyF4tW;=(lF?2q)5ec%FzQ?5!}k z;bgej>Ip247nP1$J+}`aW-wDE2T9D4hYw1cc)6lSDe{1baMFW6<#?0H)~l2HMPq>to=|; zJL96ve|1Z^Kl=Z+oPD+(tB7eZ7eV;RERFA*lwCn_mpTJ5mS=m=+ zW(=Ur0%;^ZD=RLau0jDH6M&DqD@R~>fC4YK>QJ!BI>C?~Z}%J4a>6Kf3NRzQl1DcC z$2!>iG2}CY)C>EAFo<&N+uK&V`n zNVnX$5F=s6_zA6ZnG1m>kn$ZNQQw)om>G@zC&ZpZBFD>n9E0xKD91H!jq|O#^7};n zhm+*S%}j5vXFXS-%|r%N1lVB+=jZ33dMC=C_nvcd}dZQGE2wqqIWr5f={1mbGuDw zEgwgusHb3OL_Wtv-tA)aL&*@qi(8zlp(*g~Fagk@?UU}E(MU5>5RV}5~>IwnBej+7DmEJ%_OyJXAGU9$}@ zC)0n_D>3`F@A1(^3HjSP4=?kotE=~$Qdd`v8W#?*kzzrv18IdO!|Dxah>OX~V}Qye zNYCO`BHi6ZLr1UZ!!eG{pW0{n?`Fbj3U;p0gidayG%pvKf3>=?@z^c~V<6+eLuzOC z8uu>*IXq^NJWdiqKre*kFji}!=vyO40aXwNmXglV6+#h*ppvmOt*BPqGo)=>Z?S85 z{PrSF+INM0;bkU-_rgaP{!=SGmrK1T)1P}%pz~v=d&%j7EKc16o2NO|)p$H<1LQ!EN5mBOPrcewMEyldN=l9ZWTj$dEl;kYLywEh zOY*piSTbB|#yR>p_c!k@&FO69RzGdUrG@kH@!p@)?Y|VSw*jo;L?s zz!gHM&u!VV!qICG822flg>m`X%1R@c zu%2P(Z_r#17TcxZ1>Oznbz&|Hr9xUPAX-r>on&$F8rX0GK+YHlal|U~rzR$XODo$+ zfDY^msK~rp;YpS0Y7ZPwH@+$!*F0Wgv^rgJjHwD##e*N03?F8u%ltRFb!$3AEDJ@ON%v>}) z%&y^g*}`RO-lEa*+Ai=|i{YJ#SDKhRnl*cq?uS&7&P|XflbmxhfOogu5SO964C!>P zyd(enm4QX?>^)soC)$_ZwVwKqti#jy*}1v!l>T>|prpe@ zM)<0j6`P)35k@h0==d=;H6k!Dux9b?xrb2I+&*xa72RoA--6F=kND)|1Q!)C&}#_b z=GZ3Z%~c?r`)=E?F!l2(e#pU@u$`alBSPcu6!&kd=EWKJkq7j$hv_+w{qyOZHTJ>m zRACVxE#tl|GJ;Ic+mC(!*c8%~b^$t&;MH+&%S=ox|GjHfTn z4935{br8s~u(Pw9TVt59RP8KQrc{PDjhf>C2u~*J*IM3*)<>Wy^kL&?^4dp5kuFeS z{syLYWut;~#m3na5skKb0)|rGQk=2x_;NSVQmo(4zlZVYy0vkm+1mTrZF}M1ey;|j zw&t=@*4QisuW)Lh*6vX!WxSFFdOJJ8Vg24rL_eWOFcOYPa49>EU6$~LJPo^DR63EK z2i*1bHH>^3zDiXn>;aFD@I3vy62X|+$qqnNy|A5ENpUe+YU+GCyC&e)7!}fA`T2=p zB1<&%-`@MpyH4xQ)L9d?-<`;_?YzQUJC4%SR}T6i@#N|tZGo7}2^u*<8jVpCbJasR zGQakDo; zicx>sc@G!$jpsT~e;ZkDSM|SDBo|VN_mb!kJRxWRw6B>7iG8dR+zd zO+}MdCg$d}Ko1Md-+28uO+qAW!7gmnV%-Qq?0@vOj*gsMUk<_+NJ&W@C*Nrp0-@Q} zuKweX%?zg1UlgKKWgc}smm)sa{CYYaPy8Eq)wQZcZ(Imo_s0fl_gXP_I~nhS3jvUy zhCO+%zPvS>M=aoCG2qF=%S%g7j}#FRf!DEU5xsft1(Vse>BY^@&)2vmzFz}cMQj*Q z*YmqJ)o~u;Zu_ENg(wz4h06JONzJGyv{e0~rH#+Rjfx71UN4=1ay=B?NZR+(9h(ceOuV|{o$DoezK8@^?M_NJ0`ie;+FMjB^)?Se7-c8 zU#@H`Tb#Evxer<%;V{X#5TEP&BK@H<`y5=VFTO(LZhT$tJ+I@~`{W9+wXt>9({5Wj zl{!uMu*uoA9!qbz2+V`siaP5#E^6=wo1Okdb0-WbkBzr<$iqX}+s^O_S+#??t)~q3 zW~$rLsYhos+S#QnKcjug8txy6VlaFhCg1h7_PZf>q~{Ik*2=DFn6l2Z(2jd-m7PZ1nN zNdpt5$Adp#>-e5edOY4=O0c6+RgaR8)el?W?g6U!aIBc3#8Wai4P^E#npF_k)_Y5@T&76GUq0gPQrABO z#JP_BtVyd~Z@X;Ltb0|_sZm4z5Cty{KU9FRtof$`eMnWKv1lL#M>|Wc&(UUSh%)R zT6u6{cvU0!<_+d`&Q<%@rlzMr=o4I>Ru6SOk1RE&&;{nKL_fWJ-WbA(T%%PA^C)!u(pcWnYASJu;uy}!M5 zpJ4$K#=GM<|L*{lN4Fd_W5|DISrL(NQ3!F?M~8fyOH3VS?;Eji!Lkzz&?C)yPRrlw zw^vdl4$sj_r5^d4?C>_77p|`%vsou!1@}W?@Z3LxU)z1#p0;h&wfK1 z5xyj|Yg|EAd-txII>-4d??r(HeIe_!XDqL}I!}*oIu?5~cthHWU$fxTqb;RofS%p= z-!5!yuc~u+=kWP~d**L*@tL{i@zGb`5}i@oiFNAh>BgQv`VooVY?n9QGm7ri`cuQr zcx)k-?2=Sj|16Xp8SeW1jdHm_hT^+%oN(xSK#|WgDpE&m)5zc=a`Eth)-_Pbpamgg z?Icb9T6a+ZOq}w&cOhkEWmTI`%Jsh($lTr zpYS{fl4Do{#nN)SZaz1pe@zTr1e@EJFOi@EP{1IA6crV97AY61b5zgT%rrXE`rWNj zu8qo2_V&e)&i!ru{OQy3KmzUN$qS}fpi}X2?2cx)`XxJB?5L$oh+Od0t4b(k& z5Ajz$HzW#=bs01Zpo| z-;6pK@zk2N^g%<<3$3I15$R`^HdrUvR6?fEs&&F-LQ&#py?ob&~d5v=F1DD(xq!6vo{$N+tr|t z06WAX&)ydN-oEY0`v$xHv&QzBzZb5`qPpdvB2ma%bzuv6u^^eFL`6%BU{>=K2FO8a zk(J`N;np%bKaW{B9&vW&vc9q5%EX5x)(I-tDv<_lRAsRI3LEF6Pr{$%+~+y9^vnhL>_RQXqAjOvEFNc$cgGR^+1p07 zT%R?CM98cha|bu1^Y>z} zvNOv6{(1XtJ=yf#6{E)mPE`C%h5pL9SGDi$1@?U+*e7yoYoVZ>lEPz4f*h9xQPma@ z(Z9^X9-qV<-Nb|%6BHdN$~(&cZ6erYd;uR2P;$%4 zFbNnHz6YZbz}yLBxxM)&0u5c)CqRntu;4oF+N?6T+8b4E`jfEr@!=M)ZSer)zE}VH zTcRw-0Rs>Bq4@SY6UC5#5s_OKn@!Z@@yi0x-s0Q*S)ch?jcfdcu)k+6BSAbheC3m=%Oi%t24EHR;QForq;u`{=YqGFjH7kDup!mt_2!RT# zZ=*oGGP2Dw$A~HbM|2@N>A6OBQw5=OMGKth{irJk9fR9eGHl!%8ynzAp#S@ky2rD zHzdHvhn*Se>yswqcR~ay-WEx`_Vo51=V9I(Uvu{27rzS=*Ptahfc2hN7D1-UoS@tz}chC)srHe6R>yN zxaE%RNUP@}ubrqEj!7>S$uQYQP(Q=_`I)V2HCexRh<714Yo3QvwB^`~X=HRlmml;x zN^I@aw9}R7>6Tk->DEcl!%t>~q5Vb$tY{%G#6CEB78hB#8LU+^V7}$eWuoU~kS+i! zw$*+A%DzzU(wxa%$KEonfw$Z%X~NzP`;!`LyOkZ%c6Mw(e*92UR}bB-Xu$?yvFFAU z58p|@xsmfslz@Pp81Zd)F@1g$qsHELQUV!BCnqOPfc9?Kw@(CHJ^(_|#GLRXF@K{y z3IW-YnMpB0!JD&AH0;r}7$PhTbj*C7C!ljI>E`AZSGYXbtXKL(vk_ObjRY6bU{7;~ zh&8aftgOcn=-~jNH#s>u)9m^@mW)3T0R`It^!Wh+Z|g3m-2skc&Sm88-hsYZkl;Zh z0x)I|=$ilIe^Xz;4r_&sZ-1Ae^bP+0U9t=mM4(HZ?BABrjzl(yX5MPieCu`74y7-a z!&^wE(aee7f)su=7rLHMJ@S}uTpMoTxE(8QOS2GcC&c*W{)qlHPDCD2PdHgm zlQ#w1O#S!24y#L)cAYn_*46$aZ22Z8;%Lju?e?JY2rzA9y~tn)k=~3UPP<2U{raaW zB7TtOcBd=DdRDg1Hj#tiG;|!106YZ8W&lP31%P4AEYNm`*4Ea>Uo=LT6mp~t%W1%Z z9$-^IWWino_$B*d0`1z19<6Z|or)0!@DY$8I8YDaU9C|jDCuEq7}3Cb>aZiP;N)jF z$OuY612~!|)4)vz@`9wJqoeHgf16$4X#B?oARMQFESeW8+@z zT5{s2`Nc)h*2aPz)GrHmLi{#}c4Y$Srq{f$q>*>h;jPFixc*5jvpbaI$bxt(3iC6t z1Q~4PbHy@LJ3lZ8f+PV_2b1j7D9WcJiNM^+cs__G}U z3KSqfOa?ZLpL+DHV;S8C0*nj2&VbmmwEQq+CIO_}AkYoDNLpB~#M>8zy#tz@KLnvr zKTCpvLD;_PbNcJ$RrmJy4}Z--onQnoe2Rc8Ge{OLeLj;0>6!XqF9kJTmFKA$>@@Mf z&z~QkD6$dqufz}y#tv@Zsz3|qq&B{lcO|NBK|1^2cYMDs)yVVd=lPT_N;b~c1Wi_G zMQxf*H5?kRzPFa0){FYptc|+=S$l!gj0~SWD<-euj!Y{;`|PDfg7I&0as<_Fniz6| zdiv4)(G97j$q-PZrj}=I(2=_svRyS>$=4^x^onFqoOay61AI%&X$7rlcj94COb-WK zL746MWqY{o;4{N}A{9M7?Bt&zdF^?c^R>i&cZZa_dWP)Ss3;ElH@-}6&^&PD4Nz^= zAYMmi@qf3;4Kw&d;y--~&d<-U?;f#h$ZKz>cwK<+)kXn6waW>$%X#vVFhGI;y&kvL z2Z46@quHxt0Mr81OM7wrju74m9SY57k7l2pPxt(Jimb42%?0o~YG45C`}$4;j$m$) zUPizj2brPR1X6+kXBy&>x(B9H0<}=P5C7lcj3}6Tq*T3pXLMp=o1owL=KFC~%e?08 zu{1cHSa>yeI?`?Mx;Gv}+Q2{#4Ve(DbNPc5(~pte`?~ z|KDl9$%{#H^mKK*RUmIb(WSz~IdM_JId~(W-D6^Bm+|!Ezc^gN0g1d?BGkag`Q^dm!Xby> zswC)LK_IQ751d?UeQ<_s4uD$t$Hf8X^#F1x&&?1MAZR|ym_@53?gX^uiSzBE#4f;C zj}Mm}j^l56SAION#4KjzWY{^*>Wg;lNf<@QbksHQ-V7SCvU?iK1Hq`2A7dX5pY|RuYr6)iIMJA&MPo` zstoNxRw}upa5#@U8Yl#Rk~#mJT~cmgP+=~yEQ*0D2njocsZ4lyN(z~|ElHO`b)ytM zfGuk_!d3T)Ko~Z8H`r#D9r%FEfy4L41ytyba~|b+y!%M-a3Bp+adP4TdGOk?0XS1X zQ)8|;rvbzcutSC5JoCKQZg~oq})kfR0DB?UL50l0wGiNwN8x)t8tX96Wl33jcGtxu?9UOf`={OB zna}ZCJDgeR6YN5Qz-5s1mR6TP_!bHIpeD)0W)~OZA;`bxaQ&mEBVfrYB zbgSX-U-uLlf4@m(Ci@opH;3@bSH17wYnoRw2@~j`#+B@d0k{UdVeD=6VCyM{uQNETMOiq zS7QYaIyaK_vB8@wz$%b$KiMR}n=}C5i<62XG`zbyh7}|UU%rsI=*T4kUS(AM7kG~Y zEYOpFrL2qx7-UXB`rPekbaZ!qKp07xn(`8k)X)z(h+xcSdya7D7hX`{P^lFZpzUk5 zQO7a!%@pJBNLJkAZO0=_!_$4@PVm;hW53SNFPFQ+8BY<~tEY>;Kp78&#zpwhDpVTK zkUT=7TPNuj6&3mtT=SbhgHDdmudJkp0^j@cpckg41ITXKT(k`dKy}>_Uk#@CRx|FK zrTgc5LT*s>7-Y9{ zo;=L^z&7!hZ7B#4`>=Wqlxv=0_ZFaH4J%eHgkgDF>1e)PKdvs+&D#bb17p!?#D(c% z`{F1mKc|ddUCu>f;H1xmg-FSDKQ1laqsY8l@<#B_5;h;<_x9WWmS)2f@Bu=jx&C#i zw#^N?|AxF|MdbHq=AcRSLcRRt;>clJ`7CUguWkZsd~pw>#q}4dR`-*JJE}4TmZlW% znYEuYtwItbyNDAP=nxf#9~RRWZe_Z^L3tVCO?cBTcPY)fPZkcvPgYJvIz)UUFctVL z&-%uXgWui#p@09w{Vg~s zy?AOSTU%eBTTl>k`Rg67J)s#Z^%u-CdRm6_LJvTF5|fdA*LG%I9BK7m^ySB=E90Jq zhk)#lG3`1!JBy|ajZ!2S=dyj#pr#DoT>%yJ-;v(EbDXnnSOFvy3q=SZ3vh(baNBPt zD_223jUPs%f+PBC_WrkX{4(nDo1;F5-b>F#7#zQ)O$~x$&qr&NkHoP0$8G;l$mh?W zU)x7m#WO`?nptzHtz#fz=6~CYF|R7zrGt(Jh3DjsDw*BUov4@(Z{2OWKVXcsdJ+#8 zf86rev}L((P&6JbNH#n-O-^@)3Z%YSt((^C0exWYGWuB?Lhx<`^xE&=A1@sGxvP)F zQ^A{=gQ>q|;9f5S%r=Z0$o)<;ui;Lq=cFetQ;z^~NOgK@$6?1LB7UQqbpr()2yTRA z{ObaPIiBX`=4G%`d||lrdyK{rf@vx#{H_djzU3MJ%`yV?Co%{Q#-`zAk+{!p~T)6t2-8`}* ze*w}6TyU1*<+snc@49VD=p}1<^FztBK0AUQN?uzV`OvYM4VoSQA@^5HBf!4(GHBu= zT6{r%I}Sur;U+K8`W=5SEW`k9tx!+i?#(B?Q_&~jXmljdnuq6xgYp3SzTT#bsQm~D zF-Qz{)cULC>dUNa%++H)w}bhJsi~=`_PsKEWMpI*R}G`KVQ>AB!}KG6=~3)V4iQ3~ z6&)ZOyp|6doMB(AL5LRj9$hHtfGb zG;R6i|53@6$3xk7eUU7UEoAKb*vm56B4Wrk*^8p=TglRptwk{gAw*fS3`tX2Lqw4^ zTO_I6_O%pa3oUo=x%52G=Y8JizCZ8#&gb_Zb6wY5zu)XmtPUJmEglCxTs`f3Y zso`mEZf;f4B06_JX*@q0m|9(}+Tx;Azy9{5-D58ff%Ya)#86xJUn+lnvR>pDXyW_d zz9lB`t0BH$WpOOp$J#r!f;`${;enRQ<{FJ<5uKXl=;!WMvBl0$JUurMeJ3;=XS=C<9A&)))@fnD|I3zJ%V}jg@ELD+{Rm(;(@C zE6I7QAf%p1KY6YRDoM*RIzM(G8lwvLF4_8l-IY)ce0ch~(8kzhopaBBel_unDSnuI zBq!7E)P-tejW8dud%2Y6TU z!h9_2pZn^VttB~rEv(R!R1b(;>akUx_@AR&T+^&6C>aCZPPi}+RWDWaZQaYRPpV~UFbi2_yDlMo?r+QFZLm&J;)_J z`B;>N3F;WL;S~~UyYOsP5IKAyV4GW;_7s^D#;GZM!5j&meP0UwMLY_QruG*5!3(6Qhl!p^a?Rh?LqJ&rO%) zZch!nH%AQ|s#Kd+uzGt6 zE<8rgFIU#h&?4sCmfYQ9vPo4@USWf0J7jG6LRPl4_Qc|psM2w?16U5mog!Unyeqan zhv&Z3jJ3fSseq-1jmeUyi(JLQB~8>M2!Z{^z^x_#D)b z{l?LIa6+*7tn{BY)_q$5FOj)FrlnvE5)ZZZ)x{FAaFI+I+{D?`l>J#Dr$aGaXJE$;38eh5WDpM zyFZ}mssN`)aEwu;w@#4bCRp9)LPiXsP$(x5)I-s{_vUjcO9f7vs0d}fL$&V;*Vi;F~Ov?87}-Mwk5mMX6?=t)5Oak9?UF%8%Fwcoigs^Pt`r+?Ev zZl|eId3`TVY>o_9ZFi#m9_AYr#G62jt0gkW#<<6BpDj-&7&3L^rJHO+Qs!jcK6w2p zFC{A8%iGvS7kPuej3>f4YoQJ)U+-dISccj+w0_5rLQHjVxmoy6_9X4EXkctX{R=(E z09|QdJ7w!rJMAkySU@41qa6;RTV~2OGzPpdo+4mk!tiYo12KmB(YNp>JsQg6G&Jc$ z8Qi1BGY;Wjs%Wkt@fP>Ukw?x78IE?QhDH=h;sfVu-*BtB_U`ocz*+(Drhwo6pWevK z*PYU87B}WRnEdr~<*P}=xj|tpPAVsa7T|}cz7z>>5EYSJY5i503|8;+>ramtVc@hg z%!+(A)C56wEk!V=?~!wPDIjP%J8#7Dl+<t#9W_CS{1GP&uP7KUtO1;dNzu$|CLfIRWlqf~Ld8)2DV9Kz5b26Uu8A0Jz*91(T zYJv0}6jXoE!JI6fZ~Y`MzAXuko*gW(?A4OITnddf-l;`h`6S4T_WG7*0w!Oq^>$CD z0y?lLSvrn!CoL?0#~O5qOSzpA)h?aBbm-tL-}9ee{K}ytjcKgFbyyYAy<12}!HhWb zPDwu;_NQC_HB46wtn92KOJ1pg#bYw6GOK3Xhhjb7ZC{__j?&<^`cD8n^wrptEIo-- z-az2i%cO~rSa@~UL$p|%6_FCe1jLJUG=!j_{6~*iPJjO}1WhJiWipoZaRaaZ79icy z!?%Ighvm8TgDdS*k=~h#_|@#j^DrgID8;6}5N~;NGFPU705Uoj$T*ii|5FpZoW=3u zF~CWL5nfXA?3>To(R$voAG2!91;RfWFZ>a3gkDF6125Sxew~%xW@0b%yy z4k49wB~zfi0n|Af@Q9$90PcbgzmuPFBAitPz%6nLSkZ~$t5l^ib#@($U_;%Gt9?`3aFxr`QXk=Y~P+OkqtiSNw6#PGc!*f^Ek*xX!!!|r;!hq4Q$q4EcC zSs1dk0}bJ7Ub0IW7*S5xCz*(alVewlyn#&O z%+1*dj{8}K-N!}A<@2XF07vYUj~r2+BTMBN&V7^^O?+QGl#d+pbt z9EYXGKM7runUM~ArRBYsUT57{$76yg$$8dma4BFWE~Ot15S(s zS77=1L)Gq~Q{f7KZno>>s2IIjSQkU1V7&-(R8PDHGAD9D0o^vJDO1_|?IKc0vsZ|) z9%dcED!ncg_{^`TyPk+bVJ})ps>4VD;h@ep};27rT=NlJCo&x)af@hiU zu&TUc9idxC4xx0q0hm`ADq7`s578m=T~~(|?i88p<5ZI6!}{E|VBI*sIxpyJ#kzj5 z72}?B|Ao}@Vd>vD*h|djtAMG3=BNX%8k`}(4*!$9ZSQKRX(pH2Y2DMx*Eo;z9oZ!? zUlC>K*d&?GM(qA|bsPNZ&_7D@l4kybrO7p*5=rx?V}+av(s|4^7P_YapF?MwcWg=o zOe=GBF{C1tCbB9D6F0`R7HdLho^PQmmJFU?{_|&OOevG!UF5Ad@YuvJjYPREo$YJ2 z5>2}c&|&GLCX>OMh_a@r_VmoVoD&WdRUHK}d<+V@15X~l?fTSc9iXi{6t3B4Jkj|9F9GY^OT zh?gA8UKnf(%PfLO!mZN2&`JhDY76OcXbT6(i6iUYj=JAAxDxtnFe5kDXJm}>mukZFD zKG<6IQ08@0&Y&Qdmx_z%WWguC#<&|a#O8M^9#xmA=j7LG>k%i*-S(t-_1XHL65&W| zKA~6dkWdsQVayn<#|V7KO?lh760B(1IC_NA#hR?ld<*&FUlOH>8~ep%OCK8>Pdh-0YaVUpbh6P=yI7nva4>v zP96E)VB~uF?681q5Q#)Zb#)RHPl0O)9O`V;$GP@5h(WYh?v|M7)@xv{mC!q2TrXH@ zS^Dy}-?u)mV(LG9&YOKi0SdZwDE#PnfB-|aFAp7Z@Kaj+ti@!_<-Dz|n)i{HlbJp; zxTt=@@+Q>9)fO=uOgC{NN29R1EWD5cS^=OMuqNuxDN%6+{ga#@2#%frGfMvl3XzhL zBq(kxXvqa(4HO{bBIjVQz%;K=D6OU+>QDLItuevbn(fv0s)&HqQ5(9Js45fl?T$;T z@++H()J=je@+fW|E*<%lU<{S8%1<(&zgBCY>${yU$KvB5I1 zlx?sTqqvZdpaq$WUnp?(-9md?o!auJIbL6RDs(SP z>|p}H>g_#BA~SA>5m@|#wPUfxaj`GnV`z~`Wekwy|%ZDi>o=pG!jMd@F9o*NlMYMbWR8Fhf(8Q;`d7TsjCcjPFa z-&*!nX+{!Sb~Z~kiwlWWjf3F4WP(2OchqtAEKWHi3ZP(d(GW~7X9A`jr;W;T*edPN z?T`|mPL0Q#_-S3c*l^F1#xUufZ1IN7_!WYfY)n;QAUd$P=nj(>(AM7vzoDT4+D}OB zQ!vC8SOl%yZjLm|4}}kl)s$}XwlPUXUvSaycoK>u&;zSusblWlErFP55fA(@zU?SQ z90HZ!)(H$1Nl%%6XyXq7VvjVv82Vtb7bQphy2JcWrHZr$%#f# zP;hhQ*7NI5it3KSF(lc-|IH9rl&1S0LTc;p4Dr))$c-KkB>f?gYD@?wF26R-ZAunj zh4AB6uI|aAr%S369m%0jHb!~Myex%B zFK#$O{`(kpfZY;Qg?JO6cLvx)%A!AYyAAJP3M`K+4cX+A=5iGkNSd?tuc4FmaArM|Oj7ka5aJBY*zuxsHqh=@l$g4s7JYEe_ zL))L`EYwAy)cSYcPhp6s<=dMel=da|Ug%3E4m|qI%l1?V7%7z8O}*i2$P!6?NvIVX z%8Ffp@h>Pi5Vf_!OATrpuBR*gopoqH6YW6Fc`&n@ctP7z>(^^-F+LspHK+X357tyk zUPM;~jd-ObFWSBP92jZro_@s}rG$?znWn^7of+Fn12o-9v&ImaPg?-L8{;(7nbsF1 z7{9mGf9&G^@i@7dL5g*#z4a8rn@^_ z{<-eu=Xd1%dEx4_-a7wGTxY|0E*#EA6ka_i*S}?f(R_T6zP>WH)Q(87BDMlU8b(e0 zQjv?kb?FeTqp6TDxO<7eXtdspaO%dQ#`KGFs`4* N*x)FxM9=B!e*tEQ#$Esb diff --git a/docs/images/chapters/tightbounds/9ee5abc64b3fba71e284c70539279d74.png b/docs/images/chapters/tightbounds/9ee5abc64b3fba71e284c70539279d74.png index 6fd045eecbcd2f08b16e417a83eb3d9b5ac3c4a2..983187adb3eb3c132a035c718e3ac8ebcba3531d 100644 GIT binary patch literal 12958 zcma*ObyU?+w=RsJf}kMXAV_zEfS`1Dw{&+miqb8Oo9<0_cb9Z`H;8n7i`V;|bM77Y zk2@R=bnjoRHP?)1KF=gbPDb<%A|4_P49pvGF(CyQm}lrue-U1SPcZY>YQc|}`jVnT zFpp1v(wcH(U|`uG(=BmmHtu+YW9wf zw&h10E2dG>y@d3TN^TCDllj<9618z+Fj8isBo?W-b&$98r#5noXU}GjukSx!A`b{N z*dF!YUb|lHv^8xvj%+V&9DYc>G#RA|&d~E88&lBI)(-nHL4_u}CeC{sysdZ8XJlkl zz%B&-8)V{h8Y)8}fMEANTFU=zL8Kw6zub2+bP|6WvP1+DC6YcNgmgr4aDlQI3#ve# zfTDszM&8FXVT92e(@;VgT{$`5`1p8p>^N8O_*qq6x51~YqJ!QFVE(^7@c+CtGaLV% zKRpd2$h)q&d$}2=tDB#KIzKaIco;%Ix%^9`J!nD z2Zz3)A#!^9u#*$p#-=9v34sBy9U*jRvDVP~A<=g6gUjjV-_HwtOnwa7!!WkH3hx4t zh5y_hKeL>#4$I5SQ*jC)^`a2Kp^jBr`^jZ&4H5s`BNZ2+=^r4A7aqRJ#T~c49@AZG zcRX5mh8r9b^1=BaAv@Vp1#Fb5{RP9xI}{djBSgP=*x58E{WOs;2(|^?cF4%FuZbFQ zeFq1nB-tkQH<(2OMSjY)!|}FO@D4~*+GIpa)I0aDSiM8Qc>{fRbhCan{EHQ)*7-2W z2+xuvJUcsE#Kv4pPA+Y)iyYEfap8)HSY}=JocgmAbz(cGr{et$amcT3*d|}o70eaf zPx5R&1euk2L)91eAL{Gt*Y@_zCxWW0DY83bTCjGhg2Zx=;&;0ZAHhd zkI>hoJl7*TfqVOk08Xdx6s~n(pm5fzy6af>M#|r`b=q^tT&2SAP3;P-Z%Ye53^nz7 ze!-0miP6HEyMNi7W5DtaXIW_0u85SZ1QJ(JAw!ZkmnnF%T65Z>5 zyuG;@m5?Czv$hnZ0ei(@TP3zn=P;)UKi@zE=Pl4F&4inET$PUGBbg% zz4(-QZPSAENlgu>tgOtmN<*bcqwC^u{*4X~HFXg1!arwc;ClN_3Fo7QSFQLqjcwVI z@oMFUePPWR_$+_##o0vz@yu1MUz4N3*x1+*x$lt3q;P$9cjx}_;RB50-h?=R%2Hon zkl$;vBvYMLOvoxO40x2dr>AFA!n0@3T;1LGe1=Cxa=mUZZc$XSd!x$+KOz4H+lqv|iu>D^I=}VtWc7A33>%LBpHh0l0&;t7XH(!~XvM zZ_&`&$7T7$8X9iLd;`NI!Hl$Cdv)IIx6;g*Dezm3KX#R88df@%}bV zOGVYXxJW20-#aqG=^7d$=>&h8o9eA`oRSoa&CkzYUQuClb)6&#eCevgqEQ4(-E*)< znp?JTa&1NLN|ieD6SA!@l9nEw!D#jPl)J7xMviRiu>UJ`*R>!@yhf1*=R+!L>h#*$ z+9^WGMQ6)}TKUuob4cdadwoHWqx@F;!^#Q@3QWgyq`w_|to4a-ph-|Eiz^@0)rvkY zZS=+L&Q%pYjAoIfr>B#Vk(GMgItvI2KC5*R8f0HMcx0w8?a$Nd`~S|{w5;y$5A+bL zE~`)SnsfzY-c(!8mVYlTEv4`o?gm-5D9L(wh|*KQ=Z4e!>obia6(C7@3L>_kz zH6ca64vbJTi?0(01``?Tm$p?dHMi5WH&KYO?mV!MJM*01PF`fTppKRjDolG?OK>pa zBv$xTSVGf=ZBSV#IdO;Jc2N`=`j^WI=mJj0)UTj$N{FZWUG|XEpaeX$)T90bXM>zi znu@YAW^e{uVUol9>KSHQi&hEVhVYs`GxS0&)l=3Tubk@k7OTVQ#f2J_g{ruVQ;j%|5 z^^)(%fd4kg62H&J%Ey3O(#~>C@H_Mwtq^NI^L96mu&}Vmy}^QlQ#?y)ceuJZMf*`g zxxJz)NLjPhFjW_*6`%}hGdpeO6t}kvR4gR!>~I_Fu90^${W!i}vYfUK1zS~3YS7Sd zz{JAZ~xw*QvWgKTQX=EBUo8vrKAbtd%YHFH_ zuQ)I=QlrN5Y=18y){xW^Qb)u2L+uanBn@t3@Dc$Hvf)%_=oE+Rm{q*Jo0S(5CzL7-E~rF@;*>Uqg=Vti4jeME zic}9L!m4`+n>b9M|DDIf*z*=_dP}VUY}(tK4}4iLf}{2%+H=2P zv%6(`{l|#gseIv=ykaPnAP9>dnc^-s9$fbIYkLf@IRA}06ueeO2AOl&vm!frG>`XB zpk2hOla2{Srj#>b{uV-V+WdBRbZl~cu~wb_d0$qvL`D!rBWX%L!;r389}Wl$D8xDy zUlL^dI@eQD?Rv*DM^mdFDnLtThR;IoO}YGGPB9rE}*@B(YxEE8LpFBA2Aq_frhq%Tiu zs!25RRvk`dZC0oeO~N&_C-8kZP7O|>MXE~S=H6uC^OrB}@2>1Xh(dF`a=E&cazG^e zyZ@->f|MP`^bmY;f_@p6HUQiC3J~6GoF$tIk(&@&-+tpTPItojoWc;QN)O)F0viPn zPqLt|}^6e0+Ry);}P{_Ax@y>cF+cQiDnnSjqZygn-lVBaW zGQZZU=dD@y*^A2l8PdL53;6Tj-SGwY{UYNANEPs+C_eNr3e~{R2U1)prnJBtL8$i4 z2yW(32+MQ1{UK}exa7F5lHp0IPWr&v9nG#s4iR6Yrs7pO#rZ7pR#75H?$dyQ=FJ&| z--k$5Rdue?OmSbUYT9>bdV{si0Jh|3qkySf9?ddgFJ39yh{*a7O>!4`uH=C}Hp4Le zv2E3lk>sC0g<8Dt_kL}Iyf=qe8msh2^+Cc_oy*Q@J7PzbQI({i>c z=iFvj`2YE1FO$k=RX(f=yoUewv|X~w8_fZnbULC(;lhN|TIZ{~yE`eCKa+FY3pgN@ zOqo3r9L5tm0#MM=&@$Y$G!q;Q=)@^9x*Q#>`L+xsIlBzQ2|i@Fvy{a?8OM-RLx^+R z7!Lmcq?Z^FAfO7q#$oz2SzFDr-K{qEj5cWqvbC_#yE~qHdU25>7=T>RRhj+o)}*c_ z@RgW^1ft-N*Yfl_i~V6ukv#;Xh8N&;XvxXHw|#>zL>tg~a%&b;=`lPSPIhHwWxI`D z)R)KvGiz!WO5pT}9E4(KI?T=m*$qxsLlOfSkcG!*Zy)+0i3m@~TU z4vK(h@NOYwf{?Dc5Xdk2R)+f~^6n{x~2*B(Xx z@#_US3t%$Yisl8nC2;*y>sY|%*XQ+jMGF&3>gs)vj~=!A#SNi65Q!z?3qH6C`04@>w=Aa8GlzVfLv|-CpX*(YWxz|q zxcS1l=BTV(ui^vRDmC}iKGw0CZ63I#CU4EuJJGJLu8I&ACOe$n`EA?DZK#Sj`>6~< zF6+#~*VgpGaZ$Fnx8r@RHs0(m61YEPw2UH)QIVkGJoma>bXoAeJq#<>)*mU%uJQ+; zi-Cn&E!M}D8T2fjBusJr$1CR%tEwl*LhUXN)Q2G-2MApjMpu}NoQA{7$Pz1AitMV` z3_NcSYlK8Z)^=u2N`x%b=!?j~SENNosmVD7d3k!ESAwdQWwaR_*!{on`M}U@-+Hp`k&9514x_me|bM-5Bl+-vu5!^DY)24*2G5iw{2v z@5!g8%PjkGM|%_fT=28D%3>M~74ZQrcsG?x3eC5l~(9mFTIIUB}=gesh->C6~ zs0zG_IK{E+L|w-+a5}7ra|Kl=Gc6w0bx2MU8JX=?j=+MdjI68 zhbc>!_&GxLk1Bab9|&}IO!?9D4c}=pq{O@dtbBUPFu6+j1Trz)%z>xS7d*f zUH`zq_1)Qs#bT`3rU)U}9DJ6&L8Ejs$7qF#?7eo7lJ&&GKk9XVrZg;0WLC0@PN)`~#L8^WeguIv!tx-seFW|6b9nf5u#jD2jbDt-S6_?0>dW;`VDlFe{J=g=6#66y7`UYt zwz%vI?axcL0auZnEUBS^2cpW=!$Y$)c53tP`pmQLNRoy_wKoNyHOf$bbB%WWdb!}y zNM()A`h!>a=VZ?D%|Tokh8HjGsaT?y;b3Q~t?&U~WFG)a72KX|(;U=+2EU-aU^E<- z2rvhxBzbHLt_E0_-ga9HivPfAOxo8e0bE>BA!o&5GGLz5CP&)HsA-*mh8DNcjTI&_G11sM z`@FBOZ#8U#lk2P(7i>6@cS|ijDK3Rb*GineJn*Vqq9j||+5RKnIQs}H_pOZ^8GykC^ zVbbME{ry!AvG(-Tyh`@+)?j~GS8}}%uT%xghQXn(liaxs3EKpFuF9fkvQV|}^m02c zhVl(0X8jliuzac&^OpO|%e$fDgNN+#ua3}Nmk!-3s%58qizvJ(RI;|r&F0YPX;q62 zkC&QYvO>xSq7ZtJm3+gKoXGqvKnOFASRTFc~dZ0-~xa zjh1#FA~MC7Eh&3g*pri!G~E{O^EG2*V-}AKOTN1k!Fvl_TwKrvlubQRMn&fu_Vz3f z4-Ww6#Y9Kzmkv3to3i2Ervlc76USsUL%6f6D=Id2@{SSuBk!X;#%sEjTbX5$-@azS zO(ANnz(y%Y@iuOMS<<{9if7c_f_NOQe#h2n9UqTrY~)2lNAEj5Gq`mandDgp9{~y| zB_(CA2~?b8X0tJ}g~i45k{8b>mQjpbeVIKsQhh)XNRU-!ID$#>SRuG8aE@M!CXK~MiZRsx zk~{osjc@vPr@nzUo>RNKt<7(9ARe9IZaK(vX=#b~?{C2rF1xRP|2hI*N*?0{V4Uu4 z-35c5>unGJ>g?&6c9)jNVDoXD+uPXQ-an>*LH?hF5>s_LmlHdN$wIB2kf`YL$$~qB zPL}Hg6$`L}nh`nYrIM?SD0M2Pr$`IY`EkF$8>kMEpeU5vjf?O#?JIKEJ7ukJ6XFsv zr{qH1`&Xz`OiWA!1OyOXy?PdUEH^Mb%;5?@n(*~2EU=P1#`n8B&&-i=Q?{#{8y1sc zf*h&DF+GBRbW-v}P6j1pC3Sn4xh4a)(a(v=2=X3<`yQ5_7t?0zk}YgoHys zsTGqjd)^gcag z;(VaS5L(;19Q@Fox_zJ6MT_O|gzh)yKyPDcY&_QDGpBWvHwH#%Mkr_rpLV8a^ zlCHusdhYP#>%T&dia%e@wSPwn3d&z9bX}}>f+66t^?ij-!G@FTa=Zjc_urYKC|>8i zasTyu#E5E?d&t4K0Ysz?-R3lFmaHT+LW-d367f^sC9#wFuTOAudlm_=JwXL zTfDVr>zxW&oc5G9?_VJOykpS*DXXm=%bXT^_+#vxu3xj{uQD*4)KYV3=xcH&COK+4 zI!n7o5CAkFH_}_{%F9QMfdWZROFNLKR_}UjlU%&u`O3&lS9Mca?!wibM!2~-I6B$F zk-%VXT1nUIJoZl)wr(DltUPr8xT5S7&M&=pCN;Iy*bNCom*j2Hd1XBH4H_p>3@z}u3F&6o@Mljb4C6yAge3QR;d9hl232eJ`<-Hvi-9m&S5pznaFB_fP`dly4Ll? z)U1@rMDX<&VaKoVmZkjp@;m4CR;AfD(Fh`2&}UlLS&2(ZzF(_BfIULcb4$)y96~q) ziH0nD7$*wkHo(&^t@JAX5+u=p+#)oi*bEerhu1c!r1|@=oKC9#?%f%kI_L~S7a|u~ zJ*kA_S#bW{YYy9vD8^zY1V}YN)tdm)_J)}>QzWdtE&cREG5L&L#|JC@;(e`V+M96M z<-4DszrD0OZn`X((Z2-^QajIaqadKj`b813wHQl_U4Kc_+y9J=gi&HF{GC)a$F8>@ zj`}2=ScIXP$=oW4Z<6ci!}B1RV+#Qi>!Zg-=fJ?gTIw|3x^}ItAGWS%6rb14Zwq-w zKF7OAvgp?klc7{qP}ryTh#ZnwM6#$IZx=k zThb9*Kky6jtsyY$6iu|aLr)-1;n%%;OM^g?XtDc!1_p+oad8FqZ#?8dhgEJgFjMD1 z3HC$8aIbZR?xX*4Kil45{eU~M(lAs^k!!V+mA7M{aq#1UAkL)XeuLA#h>A*F+jod& zW`)VfQ$gT%KIjE3)~LG~Za34?hO`KC|JV>Zy9-}`zKo0ODP%H7H$chdZNudqZ7z+~Wq>x@3vNXkmg;dX%FGNjO~4-$Cc94qB$ zAOu22=+mc972ScAod_b4QY`DZvfR6k+*`_gw+F1o!zpIf9BRo9+ry_f`z4uFh_ouu zG>^+87~UQSDyqhzc`WO6xe#r#7*N+hFEUeM65&l4^~9EuTAj89G#wa`q9@Kck+yYw zed;1Wp-&Z_GoS|Ajpv^}N~mrNgfi7LdS}PBr8IewEXq)kT+hyc zNOC*vW%<|;-zIbr7+t2=kb!7%B$x?&tBAo<3 zX;rC)Uy8sL`e~YZVPMcipJoB>R6$Q)6d)cg4oH=r2x7|j?**6RS2s7amksBC=P1Sz zz1NpuaM(JzN@A~Ua1I+HCwELOS*?Ced32*Q(0X}n-1hrLC?0#ksC4Vl;m!H}!D2%^ z;B6cLe{|G*YrW)ksht^b9N90u4|uLRpmzlY1)XET^#hfhlQX`ulC8a?<2-`)$3;@v z1P*G<>4nJc*+?f;t*tiIyS=Lie!As8Y~{izGy;F*Y?)8?cgN?u^AexU5uXRv!?`L} z0!r#23&uuYMGd^UWNU%jKLcD+zY6v=$^kGoetfvU1L-F#Cx^v$(VpOZ z&%I2G=r>(_dW@}0w$qQ9qT^LHdsuJ3i3m#EW~(0@!3TT)fp|t%%NeGlrb`_xI<>TA z)>`UfyD*$f9B|q!c*55P6|*8Z6-4czz;HS4yfx_`Iq^QaRZ>vj_Yw_zKW>vGljiVl zdn7g2X1Vze1X*#-w3Nfl+6@Y~Fa~o}kk%%RuGZsMbPw55n`J&^B3`lEi^JXN5@Pgf z>&?^v8G3GGQ)*#lvHR_q{vd^Aou2GH6a)>-lX2PnDWj`vwP}>p6^W zxd%N@Q^+@TST)w!If3=M`3smCiJ^gcXwrM*W4nfcq1EJ9C@7-x7{e(aI!am|f(Jy2 zqZBBQP2j$7Zr-gYnot!)MPdr4mzG9mWRQX$0!ZLplasO5i}gU!3<0A7m&-obaTEF0W2mImqN=3@-O9Ftcz4p(0KpGxNN&Jx z@Cp%;l!_|Q;_(K%Ehs2!iYpgxWp$p zS`_iWTH(TOD#bUC_mU+SdCDGv*q@&h&N;5Us-bkPzcYg{dJw_%Y(v(+5qrODs7>(t zgtCIf$w@;Yt$z|p!WpWrqGCMpxE|-ZluF>EUp;SGjcEmEHL#AWUv%oYTvSy(qIIgI z4L3C~5VEkKb@%J;gSz+F3iHG8!zZWN(;i|U!^Bmh!7 zySoc!=`sPo6xP`(6r$C@Kw$|IL-j6*rLw$YyCiP5i2n6CE5Rf@)y4U4#F>7LHYx|Cj%pW`wydHX(v2@RS>yA0Ipb0UH<=CR{jKZazsjw&}`?#>GbTBWIpYJ#+>|YaCa{cfTCRX4~W31viV*y|#iWvscXvB7Dz*Po@VD ztpTW;m75!rnu_uXg!p;K?yr=T*Um2w7wTdFZ3=qb1U54?Kt~X8I|d#%--ZK4`Uym& zaf>gQ?zk;ph6D%OIyz>7Gy>AnbKfWJ0u|LSg9>mGyAgz()&!%YqjXxea6TFgg>F0MuV z*K}KwMWtAJM9W@1ryN{1A$73!CsARys6AQ_$)J)-<@*I%$ekJ;C5-O?qdoOxOg2lr zGQYTm#l#Q*d7}qt)|Z#(VWFW}Ace4+48J8JLRO0Cc$)4gCN4vz+1QkPv_wQ+0**W9 z+83nDZ;T~A>MaISomuwptvd^jq=0XTiHW6w7(8U>f4p;m0U`H7X*m{9#mS&`b^p85 z2EdsBQ0q?q{xt+)zwlW=pl5oZ z=f2_iH}%2f;I(1Q%B3GyM$rIKvKBIc(j-1jUw9nV1=QAZfv9|~Vl_6l_ecw2~36}yaEUf6Hq<%j*cr`$c0eGSZQqYarU(f`Rzt6lVeFO@_(>PMGc7xQ}GwC$z zo&!4L#0~CpjU>}hWpZ@Dq+l^#!OJ2xDr_)FWVgTqc*vwTk|dPiLv%4mrF@(QVTD2+s`O+*jJULn zjLc0Z8a_Tqqj&A2nWAooRnve&{LPOz+kaC@znc)-A7+XyPVth9i|Z{0MrYdRZ%+{n z4xALscL6Ctn_t)TciLx>k}+%Qp~L_G^042gY=8GALqL(9Ok%9`ytW5)$5U1U0lf`$ z-HQ4mnzY^jtYe70FYA@#*>^Bh6&M^$N=q98CWTHwMNw2y=?3v?9UIg&^JJUAxJbh> zSm3L`f_vzaV5m*k`wUKa@_jqd z@vg47Q|lae(SQzUaI(?{l;qseL$)E3I%)eNc4H*=@BVK7r;fAZQZ%WCyIZp<{{GK> z-5q*0@F%fFSyU{7TsHeLpP&n1PeES!dpI8tJdee3=Qman&hprWQRHW`s8D^3+z-+v zSG74<~1X6Ykg~ClXQ$q8ZKYXJ zJwWW^y+%p2imGM%#e3Itc3Y$p%nz`d4||ftRKv@1lt;1j=cTpNO=DaDZTCA4k8bY- z5@%KV8|zwm-$N-&j4AQxAWSCb=19C-q*q^PqNSjNVOvN}JHP!v-^^^kAqENq-J%kS zInW5Y_J*-qMR)oU%bW`|f`ao@`=i)*M;N7%g>>z)%)S z%nt#T#{1#ggu{BF7qq+xZ{8SPovZ>~xs?XoK2PZrq7el0BP!6kgoOOPn}IRr)v7IV z0TG-Hh&WJIbA->n3++?%WnqpDVFw2W{yt``^m+82D%Q3Dqmu2_;w)yHz<(U?sp24v zWR_r^=_d4GHC8~;ejR-+kdaT87$|}Z>=k!`jkXtKNSd@Ea`xT)h;d_e^~-J!A}dh0 zzSCd`uMQ@%-T|=+>@3QV6#6y&T7W&^90iBT7X;&gCkZVDn86?vyhWN!38Xsv z4vhgLC6i#cppt?*`SV&pMGDN!;IioU)&upILwpNKo%m@)2y9pE0-J>%{JVH!QFjM) zgdm4Z&g+UDsZCJ!3C+-liDb&sIbC7{1vK)@Jq<96Mp#nEjT1U*GLLRvV`IvCPee>q z)Mqef1hedU_U6CEs=WvUCty;Q{3D^xoI&jRe_a&@Cg!C3N6=IOcnHjbfP&I9KR>TU z+yj_F)FK_lEu`50Zgymk2e*d4aXI85!T9;b2Zr(reQG`aA0xlE-P64CSL^1;kPsvoT-;xHyd3W{Sz!QC z&MG+%{rYWJO0n9&Vqkf>oVH{Q9jT;RCCmuP*^8h&$6Ix=m-S ztgMMZKOy$MB7u35hK7dT5EB!tl;{-kAAvz#p!vu@I{Z(yM}e_8oeM8amgvAj6(gyr$PuL|3+Y#PWQovgx*|I{2nO|xA!=jKuZ&GF^5nj65HPm{#}-V~qT zfYKTm9!|l@iT#NmQT1{k7Yj?o(vlA72v1X&ptJopU5qQ5Il414?X~n{t?{%QU#s3R z2v8{dG-6`i<8v_j z&inD>Isf+d_OygZ017b$DJiV5d|*h3MX8n3`oyGM3YhanB_#Cf7+J)5GUT_43RV&l z5V%TIro4Mg3}*&@=#e0<2$GufX(p7w`QCJ{g{M=L|#Dw%hi=36?hC7fWA0dlyr1t zQ_PVvW0R2tzTX}S%69MIU=|pzg^_~rCN4F3aMsJ(nVM3)di4s(5nvS8Z|_yWlT;VLiGTYU5d=>6|C@;L#Tan>ot>R(^|6&$2SaH8m*1-RKYs}V3^fQbJHJsQ@iz^m3nx1TKB-6!0pmXZ+r4)_QrK;Dyj=9( Sz%QJ@hzrXI6$|Kn{r><~7t-MX literal 13520 zcmbWe1yGdl`!0^6poBw-EG^yLDIwiTuXLw$!+G}e`ToEE z-^`ggGv|z>?(Dw%zR&aAb=}u}ZK#5rI64X;3IYNGx}=1N5(2^_Z1~@2NZ_4Ug)0r< zKO`e*aS?s-guh*n1={lm{%&}hGQ0cP@#9Co z4nndciuCU4Ha8C{Y|?;&FbeSRM+sI;A^d>X81Bz=KtwsRGzKx%S4G%HOS>(!--$N8XZ=V!m`FP zKO$Rv5X9H^=6TH{vB%1O>Q?rVb2X!>D?Ck2O)EP)Mbn{) zb(FclGPRLh{saXINAL7kw)Q0akHT*+)r~g z;{@v2zt{fW+>lq44mSVQplP6F9 z&fB#=GB7vKzf+!CAN;nx*eqsQaa= z9HSkhS>>Uwy^XTT?24?ctWG`eop<6_HaA_xZw4ON&W?`8(|rzLJh`gHY6*4K4GkKh zp`nVr&O0jWjoaJXVfFQ{35SjJx)~W1*kBF!ia9tmT?xwr(gV_yVP~DFOw{G=_e~>* zY3d_%9a0sniqL|BgjkBzt*wFrl_Hd0y(TBg=O`#jv9Ym5GL#@D=K@p}6fzqc8n#+E zGZCb^g3uFka&l-#RSQ+Zs;jFP4mGDYsiJv=ZdY{|-bM2ZY@S%9sjH|2`1|{h9RBHg zZ(~ZQX~GinaqX-Bw{LiwnwqXe^^i?DAt52RyR+%%sHnxq9+9uPHU<*tK1jZi0B7z@ zl245yDk@5^*OK&2&!cbv!p6pCIaBfd-cAb{gk)y3V|IB_>~mkf0~-_a_uGCoxHsU2aw($?%OaLmn{6Z@ zAaFWdXw^Dw{xS!f`W*yfY6HtIze?HidyT@q(tVL)301nA5LCdcNw2p|af%wnQ%5tgy~(=+qCu5oW9DN1#;F*%td z-hOclRCiLdafCqVh%d%l+#ztow2ZKg@^^v15b>6x5m`w|!|T&+kJjLUKI5ngO^}2X zf{r=uex)pML_z9%Q;$97F4k1CmFdlSMo;(yai%WkOjQ#-Pb(c(d#qZKbt$0FQdDGE zRtejooSKf^GeiDH2UaM@U80D z1PX4-ij+T-Ww5O*wPFWH$D;8@ml>8g=;g9Fz9D$xca@S6jmM2_n4IsRCJQSJY71KK zy{Rd4V;1Zja+JAbp(cp0-D0V(zA&LpFdE01klBdwZbz80FCGzJ;4WHMG}Jb-(a;6F zbg52Rr#G*fH}Q8KA`GX%HpD*epH{Q&Ebs`hbbf`4OK7vaH4!yy(`*yIo0MQeR-!jM zFg(~|^Y=0^Fc1mcV?BzG&bG9oAxcw{vU5iiQ&ivzsSp;QKW;(I$XIrUA9*arXC}CW z*!k>NW4vEu42QwHbk$z%-I~O+vR%mm`IS7!>ZzlF!T$b^pFe*Z91RQ%51S$&TI`J_ z9i8>CH5#yv7H5>J@(ahpzd773O$mh#tzq0}%HHvPQ(#xQ0XD3C@ZQT9pH_4Vzu42Mseogx3Mcr9)e)Rq^A z_$`jgdU|^4=PdbU(3^igD*`FdW4u_7mTR40cO)mC$8|`G6;sU=iEXFEF*}^)P-cL& z7qc2zb5f%N#VEv(co&UzQ|t{756|yn^-=0++VZ^+O<_h;1!ZuJsstspP-Y$3JCn3l zd$5LA1}gKTh1cvI^jWxh$v`CIbh&=-@bGZKH2tW~zi*oj{rfi8cA6v&WD9=$^of;p zTun(`X_A_Gd;jB&rcn_*WoP*EaJQ}7%*bCyYBkc7H#8*ePM5DPFB``0s^@z4+LWak zZmw^U^^0hvmfD!|{tBg9C`d>-c64<7`}gnI=M+?9x_OrQ(QN%R*P-Oi{I??S814Oz zA~s*Qv3RWKwUm^UNY!t!R&cxUdrTJWs{Sd2vn7_Hxx-sot7=cSgYj9-}Q?mwdT^lTcBCU z0a3)L*CG*)%OLY7g0{6U{Bo&lzNmipRtwxrbACY0jM<-qYdWdMF?%pq`r8mM)UyMR zX!+UVUv!t5$*VIHbd6B+x}%Y~jCT{{v7+_34t7VaX47q;lq@YTf4{phmfOFXJu(+? zxk<{orT87zG1KP~oaf}_DJduzaPIRB zDC(_pU`b>O-@T}f%D1~&V-ZhEBMWnj=-ramj~i}mOera$2S@^~Nk59zByJeE<5T^Y zB3~8D8Nepf2!=Ah3RM@u;$4?JpL%+Ff<5vTD6)OQj4;+oQ@3?u$tZImr7nbZzwBak zfvN&Cf#N#eM{$7@jJTh56H?~n`J{ZP>QcVs62(}D;o@(YE{=pM3P|Ww#-Q#Lk2;T zYSP7_r>9OJ7hCR_*~e$|4^~rr{eUXNw;_5U6CkN61fH_8Vgzhsacp8V0Gt`vLsHhf zX;$5KpxNz+4E;4H`!4OtIC7$8&2&@j6dAq7anaaN=47%k{4V@( zaNwZcc0<|BEf(8a70XjZafJ!2IKe@Iyq;azJs|78Ig*yGS#2iv$F10@mK{b z=%l8vbyQx^_qh`de$lhP$0C;z2zZkJk zYR%9gwg($a9sW;{kO=w9-oX!w0$b7gSCZKg{FAZo!u)@KAR*(<61~ZoTTM?4{Q^*HOT98-AR4Fgki?qMY{o0SqG+Ix10#*iG2N< z-fVUtvT^j^Cy-BZZsyV6obP8_PnRKutG78CGk{o9(Y9Qt45KbCfyLL`EebR_@0xwp zm{R)3#|m(@3t;i88X6K(QYZkz)JB=g{^M97R_?*BCBUzzV#viU{xorrkdV}d8Asi~ z@RI>BXk{>b?$E7yYh^iE>2>4cak{0r%gyxnF@vQg2RwpDM7DBk98VAhHJm}{t*x1O zob9L<<7XTZfHUXvG=^Vfv)YcQNC!eLp`_+|Ko%fJO`S%^r#?OpguRzBZ=i!Z#l^sp z`$be}zUd&bB~fQxay@W5Ln5q3)8F%-jtS4qPwyeMIc1~_+tfAv>%z3EBAC%f_|bnu zWL0bjZcau8ucq7YRkI<|bX+G;yM%xDCe_--XE8P-vz}M-<;&yqy;V8vf?c;iTDJiLy#fK?D=z@J?2doY9sR#rx1@@yKGSg5_?M52uQnhO5 zUA5LY$wyB;Cr+?E^3G}s!J)BQ$W|o%yj?quo7{&eV}HIl_&GiO4OF@<=Udd453BQ$ z1Tn0StppUNXlKstqLwi3-vVEPZRSy8VPWC3TYS~)9WiC>t(-WqwZqohhg)&Yg7xrT~d5 z!S+wnALsp86{SyY|1AGCMZ)VpoOTOCfEg&_@4*G(ARQZu?vYZT>!>`^A4kRGbMKje zeS<=6@Pr({mpi5WpVY@`#u9$z_r}Jmmfh?2Zx;p{0qYZFxo4&(a zIUGf`_EA~qLKRx?>uoAHZkLLtDTQhL65tA=?C%!49bSwujF25|EF7x)G{5z`5F50t zyzOt~`7@57vL%UXuP4ln5OW-rYY*EW(qqV-FZ^<1t!m*RLV-<3m5@?I%;OjUaL?|Z z+c1SPE!X7RevqwAi)6t`>lUk*o1JcrznXGq$n+I=SNZqjFa|T0qwebTGZ3H4JCnr} zR8(wkf0s}e;|65U`t6buVh4u*lxo$6b@(GIw|n35=|xi}XA_yyX@dhncL4#ecB+f0 zrV+$;ZjQgi;tMduQOE|SGFLDC`sOB^_pMtZy{2O?+OkN#NM3`S;ovX>JCkjhddcOb z8$y~Zcb&+TZ*7%Z;~HyXr3mIv<_kl!X#q%Ry~mjq2w_o>Tx!d6V+X8TddxU~btpd^ zyU%hMp#Ovk0^gq+c%>*p5yRg}E@Kp1m||S_+lnKF^V7;v@0;m1%7m1;I1?k$JXX~W z*p}hl^_kP&45px<;0Ic+AhP~$DT#G4LE40{!?v3!stix1n`9MRG-2=E65!mYczAfh z85tRM^8W75{?ylZfYVB`Yrjhz6=ZTpi=av$y-GESH64k31e&C$gQUTvXog5^ zCL@s~Y9$&U>uvuu00Z-E*vs&3_8`Ae<{GB$OA8I8%?jyt*P-R|fP;OoD0=O>2pLua zWxELQc)&gcdQ6|`7Q$4zruBVf!FgNERndG8z|zF;m%uB8;m1;7qt=9o-tL`N??JNj z+aK;z0s@|J+RXeo;g_e=I*QsMwr6o3ED67*)Jj^sN^^BrKW{7y+S;;6_rA7lv%EwH zk)u|ZCH85@Op{oW5;wiZ8Y*Ymi%rE+h zaNxFJe-#B8v)CfUL;e8pSY?W(3ux85gM$UvZbMGGvFISS5fUVjA%%Y`;!avoFcP|K zu>_9nX!_G7P*26IzPr;hGMXJv$MKw;6L`Dc>e6zFZ4@XKj=7@Gz+gK5dO9nrIUj{{ zK2u=e2@YM(ug+%LFkf=)tmr^OF7`L{N1Fr& z;|}E2Jalwxap0!OrPiU&kk$mz4&*4xnYh_dX&&}LvsN??h>awhY2 zJUaK=qu$Wy=zLHH8k(CSxw?ES`7)$`7BhtK@KAJhBru-6^&Y@sov-cc>O%DK@ky(k zF4Yz{FvvW&=Huhr�pPe0RJ)czU?_#?H=;jEt(sEH!^mn7dlPx33a6yBjC-0Fmxa!&S=gYlsd`w^s4uB>E?T;dB zy2%7Z;&4iDmHUZ==ang|m)FthsmTxDM+s3Zn*Sf$_eym9s@=!5Tt0;}5 zIpm5oNYz+v{7T9krq(ci=hMXIup&a_d>RP|DQtZF0xz9HxnM0hxfj42N*z7}Ht*>C z94Qk#(ZtMb$Ez%h8eo|?D71fXwwi#;mZ<&iTsxN~g~Ph%Y4cslRw!QzFP9fsBx7@_&GE*lw&ANyjx?z z&P_#iqG)K?pC!ifgZ9$cne=I?Pq%5u7s6I9LvD~eo&d_{_IkHmCY>Lu+w8((*!e_j z$fiU7T<~bghh(kUuC2IQLs)pqVW6&|@vEK^P2xaiXKXlyYqrImjgQN2 zqw-gN=)myLT19e~r+TXoS3|~o^~Rdqu7+vyBd^)a(6sApzKchbAh@ciNu-}7Xc3n8HgR%JR!Jnv$~ z!0e5j>B z&jO(z9}%nf_8qK$Asw_Y_LbJti$@U|{>Xjw1R05K(Y>uOh5SG$HZ*-88k2y<=*h>A zA75=`#a~z}yg00@H~y@z?irSPWKY-K1~G>j8~u2eR-U-i7efvT)kptc!k32x(Y)$) zdCab{G6z`|6<>35DBy4va2@>t&#|{=Dzo)A(^qh^*|WE#|#tg5}SDW`X;sPsAT;)sS_+@pqtKAFV`DN=8)zK z7SShU7r~OOzqQrTatUoqxp4&1&1+S828KK42StrJEYxZIBQUf7JV7&7(wBS z7Q9aaow37e`0FJ!X*-760@0|I)zwG%{B9MY+D2NO%}6$2NldM$6U4?MRugTDmyr+0 z`gN6&oM-c2p-{=-7sM80nST9?}*`!05a)cooou(u*FT%UD29Wh@qdrA~hN36@T7f1yhR zm^;oYMOrT0k)k-sPdyb z5fBh4(P_kzMMNhJU=HTtmpM!Pg0s;}8z;HVOU6M=u7oXrgyZ*wHFde=2da!It!|Uk zGN3Kch`1}{%0C?dJg3?0!Z4V~U_6l{37V^b$0%Fkf`IKxU!hUvQpERFeHT}?8*&Qj ziWoRa@~m0fSH(>Fk4U2XR_ zGMz(0@vjUsiA_rz2D20iNy+gta-+WV-JQQ3!Jll5A7kTB8xf@m^J3%SnM2kM0UTcW z`Ol)TspS4ibpl1vmm#Y}=5*Q&MNjC9%*Ok3_ISy@fB)`ULT)5g)=(9j;Mpzi$+If0 zq%W7NU2n@yP3`;pcWfh18n2%pukY|MFwy1dFZo8$k!UnJzGL{pd-M(g0mmJ#1<e}0>hDn!WxKeZ@?&3g_9WfuXf&10(EsQ9NZOH zB{|8_ks8PLAxKR_DO}M*Lo#|^XOk$X@^3bnv7votoB)%fXv(lQ&f{Z37&@YiaP}Yg zyS`2@EVG#hra)|sF@|`ZLe$@Q;fxaV$^7Zo;&VM90(A2Vu#QX;v!7vCP)tUKPcMsl zL#IMemG4skm8GCwtcANQWIgtWRN7k}GNhci78pO7x;_UI=IzZRPnY|bc^T3FI9l!1 zthIWD{DR0}Z>F;4V&U^EwPZxystxzd!xqd?eqmMmk@97?#_6D zS?hi-^{>sskgAK)l7y%EN6t>R%YA}G2-rkFUWQ= zN_sG{q4ko?4}Bm1a_0kOuWX)|$!CDU+TkL{P?j(rBUEWt653R%68~#jb*$nw%-qdZ(eAx-$!(c zqe!M@41k&D_FOF;;6Zai_GY(*HLNWO2lt_(um4Qpx>8kaxY&@zZNKNeFg0JjJfG*V zod$4@+iB}1Xjq8ed&I+GTyqFK zz&E}i;{E~cWb;hEqe+halvhjSur}OdH!;#k{EbO}<9jd(89RH;zzCn0=E^lju1Z1D zk0-2sOU)^Z4py5q_qQkVfS>aZ4n_t!$WX?UKZBURPujrX2l z*d_n*<0BvnV3XiyKD^M@)X)MfD-ydYDz3}Fnsl_r~KD@qY2pgtnTF93jE>_wfIo z{({j~pXSBIVcXi;a`W;|6eyA>bAGal7fs$^76)`fDUD5Zwp6arB8P#rYhAgFGiOy@ zb#=3K%T860+j2m4hasZzZlr0uhalkR5tq0vu^_H{OPv5Ui1oWP)9p-V+uyqJB0o_f zP|hCTa^9b7Ssa#2x#~Pa^=`eQ+c9!3UEMY+JCb_1OR~O+cr}`7{e#nD%okMO<=wr( z&@b4$(U!Zvaj6^GuGYWlZ_QRyUEmvx>Pma|lniz8%dqyNL5O>vPNUKrJdzU*`f;&X zy&TE)S3-JK?e4mv<(gJbQE_R__F~7AN1>e&p?m-&0ucY0Zwx_Oj(T2~02U2Yi_CEgd;!k}(iBzen3$N!rd`&6 ziCi>W$2bL{elpac7rFP_zh?08l${QU8- z&3bop(K|n%0?2vCt7B7y+uK_S85#C9@!W4JFV#^EW#)rU{IN% z%v{3)&oM0;d3pKki^FJtf8jtkpwKGU7lcRhWU)F8Ev*p%R5GA*9vf3wSy>4Q3tIto zb!K+dnBLdC}%SWrt!O4`60Vn8DuW}fbj#KgpuCzm!b1Z5kyV5irt z3P|H~Wi=lm0=GH6DsU>=|Fq#Tb%IiE!DlwS%z61d{X=B(WJwvb$^y}xd5N16A*116 z%!)`t4x_cc7$QEGh?<%jzMx-#(If#ZVQOlslMB-9?5x9DA10j81KlxVSg~W2$ampfrV* zmzN)d-i}sY@UG_gC!Rj_=kq{Peh+tx54oUahO^?Jg9jdo3TTV`i^r0}L1@dDN2_aVYw+CusTYtMk&TY) zCO73%FRWPS(VHO%icYPX6=V*7Hbij9Ojr%Fd-dk#B^nDHUaYh&~68v@0t!K zve?WpCo<|5c~PF7pPMcH@bmoJf!qSdjeCpj8GvXzfk8%DfOFK@{u$VxuRruC1u7}H z`+$iJ#s>`zd?4pp1AH}L_xG#(|0CjgvG8fQVP#-NJ3MlMg)ZT>;yXx_tP+kWfGr8E z07ULje+U$GcXyv_|Lo>QjfsglaDR7`0NBi@3imgQ545Yj(bMj>vq_)k>r@4<*06I6 z3M9aRH#%*{V&UTEH8eHR0tQ$aF7BZY%WCZi;h(*EDUV)Q#>)s^#N~s#tD8 zJc$&BX%K7I+oPhv)YX1tx#!i|L^Q0dvE$j|*6nJvG@J=Otu}|pzYCT72M3Ki1D@~A z)rSB6t>WqBwLPAV43rrOJvIw1JRpi*t>ot9Fne8}YPHulDYO~8&?@h)@m6UiH@pUr z>BWl|Aj~RzHosA#z$5o`8$PXSa6h30ZJ9Cff6s@zi@9$2>1T^Y9uQV@?z03Pr}!{{ z)QNA)JnwT=u43D*3)N*ULhyG=Ylxwg4AMuh&)ECQ6B*h~3}cU*6oDti{PY{$2VAu-_2K zpFoO4#?75X^4pEV{99Y!!yy2;Ise&j<=eAq zeYgicMa5N6Q4s-$=f3ql7)*BJTnud>D!-y;LuKLzA`3VrDP0ZC&BZ}7+@7f<2k3Be z1n(LUPQaF{9msETz;?lOH`&yv(6+Zk#a2XClC>}oq>=F=RXVXq!dzMUVzpAEjxSH2 zJbem00)xbD$*DQ@h7b*Bm{h)XkY#7)j^M^ki`^_7SA7DH!3&Z-wdbuKTU6tSl zz{xszYj2@7rTzXo9Pp;tL_{)RXi}ix-qt_onplly`0RUtqc7qU+$YD4gPR;t=3D2Q zO8lBcg#r8nAffKE)uiyOPhX{9fFr883D`__$O;50Ut|dMs8BzJTwLZtLlU|5B{ZlcDb;>p zZK6KkC2!YN>xta}ufTLFEa8BO-G1o>HGlGV~I+ zmI_P_;khzJLuwN>;k?@E00xyy;~meJ&(H$vr~~Ujr%hwR0Vf5fqVQG=7_daQc`D~5 zFu)X$s;l68Pmz#Tz<}r&tj-bW3ADgTNN7+3T_hEhiJ}OMKO5-0Hc>Sv?u9!4Qb0qj z3a#1{X72)%x$JCr@=(a*I1B*Yh#2-DS7@JAcxxio z1^g@CSO9$=&>K7hR1R~WCn?0cy$Z?ZIT%ShclP!o1IL0gnFy~@(d?uS;Z>o7VYFhu zC+RL+^fdHm1C_wYht!ZCE^Zp*&@b4#7s3#@d<&9j+?9|Sn4U57HJ#(lwR!Qeu&@Ao zMu6>@l{&R47#O@QD=YiYJGjCds8B8--68{bJ|PE^Fn+MV{(B*3u-6LO^$R=@c-zfU;mFrwSXhm`@)#3cSyYg}2VT0A!e$n?sJ*B$3eY4Irxq zaQUQq+Le%1Z8G42A-huy6BEfXUte|~t-iZs^7h8VXWv3-YlHr0;FZ!O9AiQzLg1Rh zw6?oD2p-A}=}7?q;SGE8!>+ZY?5E+Q)r-FyDMvNd-z2T9XhCwK^yKO>wc9~@W>^M1 zKsJpx0i-CvUw48ye-skF>mg1?lRM7+5^wsIZ%qw*cHQ)Eu&VFD6IVivjEo35xg4dZ zGu7tOZLVMxkzF&a!(F683kJcQTwK$cc;CQJ2qspx8wV^)%M3#xvY4aeK*(DDs*K)2 zRtkuS(TK|d)vD9%g08oQc;nc+7#sfkfPx^vh&*O?+R`H; z4yW~89Ip)dE`0HLEKl}z8VfridRH(;c{L}Yl8H&qf~zqwMnTZOs&UKd0~F=F?$yJL zsBmdgM#jJ=Pjc8yr6k|={%4590_ezphDf_$c>3kb7w{VZOppj~&*$wv^gmme5H!_o z^Mrz8zPi6J3p#1gIF{K*9<$VcnpYz=n{7z=d3tg}&cdQtJ_@EIFDIAw_GG|iK(caq zZN_S;M0T>lV7$;`+H&hi2JGbK;Q^X>U#Z@t61~>szklubr-8uEq(Al@ymy?|`z+gR z8k9xNdx(-Jn<56ZRXm=T%*4D-Z)s_ZI$73@RGH@Zcl9Ccmt(NZkf92oahX5jkWn?T8@6$bu z{_gD+|9syF*^CFpMAf|&w5>GY$zwgor>80)!pr!tx3UL-y1Xa?nx0de)9DKGXk`NV3tv-0;u*kl zbLHO{w0kNHyEK}d3iY%jg8p|Aw=K)b@p01o&vkX=4OL11;y>MXF178u7?OI9T2ipv z(jM*qiTi-af9$F$E1tgztdIs|7b!Ql2FLA*cb`&=j5gNSEguemf)Y#;sj1}COISB< z4%^F1G^@qRFX*uE0+_47fK)7+r1VZ}T(Po$c(~NbQyfq&l2TGd$As&nKo0w}t1Eks zMOjHHNx;h^VVV}Cz-QzTdwT diff --git a/docs/images/chapters/tightbounds/ed91133976018ec032d9115344debb36.png b/docs/images/chapters/tightbounds/ed91133976018ec032d9115344debb36.png index 7dd5b005115e94bc6744398b681a7f5069646c44..890e84c07590ee8b2a81622a590d5d4c7369b6dc 100644 GIT binary patch literal 12416 zcmaL7by$>9`z?&3D1v~1AfO;p5>i8_(nEK53=INOL#e1Rbit6S|7eZN4`tc+3M;I6wkD)RWsu&n|h|oU|vA{b| z^A>BsA1qTjX$g#5^uO`jiy9^IFdf^PUhJWQ4CINJl~^Kz zK#XWu{tH231nkw9-(MJcd9|#s-v$$8ta40V^7+7E^sTL}i7&J_p~87*bL(*N%P zxQM8~(EsmeJxjfhBj*!u+F5w2EO8&a+vbIc_h>%3Jo^5ihjvT&xeA^&{jc9uoDtfl$CPm;?-CL0g^*+u&gjW*sl*@jA!g4X@U;-ev4>g{D}O&Fki-Vl zre29?do5KO@%1RjeShnpwbEVetKQ<7HevnMvan2IIw<7HH?FUYA94l54?U1+_n)-G zcw9tJN1{_{Bfhsi>_DOCq|>Xa=#sd?!geZ5FW(R2>C+jIkvbr8tbPCG7YUf3@*={L zQEu`V$)qRFL$7!PiNez$E z@$<3->_alV(Noo56UZb(vA{S2-)Y=4oSi`#o(P!4WW0*H!;HIt6c~&! z=`}`Gawo9!t8X#y7AL4$J{md5<+#515U_=PDT7rwI?^~Nt9d~0=$z1YkMKv&neQTl z6eO{e3-;Ld;j8X+cx)ux8AvtI^wL`13_cVbojh7tRicoH z_^m*09b$sZtONC$vNfD1HPS5k+H>0P}E~f7()f}RJB{XlM~q!VxFZI z&W(V6$fC(4Q+S1F_{FR!Y^XptFaL%0rgQ#>g8eGTAYIp%qC1{Tg5%k@2Ob#Z&_&z| zdXz-X3>HsGY*C@5aOIDcvL&7@>iS6qO92SG@%MHzEBQpk`#kd~&&1rxei-@hmX)qz z$v1{88mr5wNIuL^u!*c@o&B$+_63SZLLKVDR?dSoybMBK*-T<+}Zz%WJ5Sc72GHYZXB_<6fDp|1$YPIi@@=7L!z9p@DISl)uX}v z>U5c;NyF;93+h&{#reD6As|2K!Nyzn1{-p5f)^s_7feC_eL)FcFvgv;7I49b)>Hpt z^OSt*@(Af*kusUMdr2+_mQ`P9pXJa1=6;kBd7Mdwpe>6n3NfeEG2alTZHt1v;<5-e z)c3XbjLjeom0!f&ra?(q<(MBWw*2}8$Cp+kv{dE?ClsQzPmHo2C>JU?RZfNxM*rzx zOJ6pI9y^}tOU*p3woSSp_r{dD)fvX13Y(Lr?eMSp$|%ip4cjqYAJ=2%$5O`+Xi^<5 z(5IZQeR3#R3Kk0SK*p1;iWYWukpaXQTkTz!q@n2IbF_SF`JJX12>M3569gj z^a$yy3Gn*UPM`Su%hi>4)8&q!s;X*tfB(Ige0&Led+%SpdiBGgzG{GGB8-bj<<0nd z2W^x-d83v2IG;ZqOFDzFPHbh+eATU#o10tSX?AXIbZY8APc#D=r}@Lu#)c`hjNGX&Vo*i0@~<(~Lxndto+jy}PEPJ5%@agmf!qMV zbyEjud1~yYKTAD^*>3ib>}j(UrePS(KEA$HyGo6T>BKO%@a$ZC2jYlzdUD{hVNi-^ z@lEvX(CzK*>?y?;;@<~7BYwSrdR^QS#IRD|Ae02yMKR!6IRr7Ybx<|h66tI)c5iq zuiNe%%J1foY9N`vtlyDwQ-`d2x4?EbXOHG1h(+QiBY8V98ec@+g7%=w$7gOyS{Ke zZhb)YBcRzsn_I>3scz3!we0|+&cx1%BiR9_ij?p}|QV=FO z<=+hI-D1h-k2Xg3kB?VkwG3OQFFU_NJ=1h*-c!f38NnvYEZnYdzzO|BtRy6-BYPLy zfvV9eJR}5HF-4$bp)KHiv!GZ;NoftN1DC^0!p-$1%$7?s(;sxNjTb$3axNr~9Ri;mY3LLPML`kPv2B~8l64`?wVvtc( zq^G9-baeCq>{`#jK#uLiPx)lNNDxFWeQ&DOYu$S8N0>Eqz13O=(~Y&c0^@!O2#AL( zFm@#Ky96Q-jHi`gSFWzFRb-#?7Durf)V2HLi^zbJxHVp6)*kp|#h0FoD<+tfZAuts zO`j1HH5Bas4CDeXv+LE!MAP})YjG#KYcpc;GM6Qizn#5LrX*B-`y<(wv72Y zq%jK_s47GUFEu~pbK8D)ette$Y?PLq{9G|jM8O~H@iTT~3=kRL?Wd~}UuYHFk3c1I z({OXg4x|WTvU?tw3xF+|;p+7&(yMl`o345s-6K)$z#Av1U+etU_xfTV9Ppt^Tdk<* zXzS4*Pww2gv*L^Ol8)|cLA#ZgGA<;YB`qf!lKOZyFI4ROZzWpdaI` zZhRu-n5ysRwn0Uu)v2G2Wlw+qaz7H1EaVxr+kBh)>z5dadhvRD86@+ah5IYu&Wn^TW0CjodU@ zpF{Nd1h9~yki>I!u~oc1SspMbdYc^gMvv3#_e(6i=iNosbDKx|IwkBC?_a-Y6fBF$ zP8{v;?>C?LY2bY_X<2C4s2dH};GaiB9RQz*R{san21<$0$HiJQ)G8ADxm$jwk-M)R zOS$=HE|t3uE}cv}=Bni?R@hF2fxSW>$)5)Gf66WKACq$#|Nj0Cmz*n*UZGcHB>nhI z7XFF!k6J|ln~l#W{IJ=EO3qC`wx?s!Ui93_uB#@Rb1Zvfj?b%hJhhuWamhHpeE!dxdzVAX z@|-E~hI}skf=uafLQ>K$040dPz?T97i6TB1&t5I=t=^9qxVf&yw5I2dV9p?w6EU%I zJp4#8C}B6>lu}c}*U-@LOvIbd?|RE<|Kw!cn`z>ST7sB+-63E2mN0Q!1WdJ!j~QN? zL6P&68y_RMLrhhbP_N34nvoF!PTJ~Vn$cCr{i{fQ*gZL&UTeieI-^ZtV)01uv33G~ zGDz@yyVcG)@L_Pk+dDc+E&50RP>^xk5Ff4$x}ViPvdp)}d08L^w#zidQ|47(uUdje z7V9+XK{ixrk8*YHH@eq6~i5_G?4p-69 z^;oTmZX}>IBA43`9)}qjO-&NB&JdH;exw7Dj<)k1bK(zrxW8XtrVemOm|EIfXe%_Rj~mHVg1BvKS2@gX^2Bpi z(MEOauR4Fkyho^g{=kOanx2vgpw+Qx@!CpBJXxvLJvM#c^eJ9>X0uZ5!0_-jquUEo zPft%Y6obMp^rrG z5f08|tqVhGY3XQzR(PIT?r^oEkvx5LdMZv{7yw`0qO6k-!}2rglDzV}WN>_O)s4;o5#DLQ7E--6#?WeqLT4z@J2r+72d~Z(=`T zKG;k&U6l=E`AF;+0n2#uCT>|PAQOVG^;6&E{9vWMs|(_~sZI%BPglMC4X?{*{Ybo0 zz<)btCErJPKDN@`FaGY`o!gt!W}R}_V|<}K41mxgAUY?!L!@AXcV$dOt?Au{;n1c` zP6gyxswbbv-b=@ZeXR8K^oG+puL`T7U;omyi|`PJqj6l(&V07eCK48HkBl86jWEN8 z4;zGX@vL+p$ARIUCJfy9onktx`rVNdY&PSsMUZI2d%Y?&V(h;a)BkNuSSdWE|d6Io+|WZYcaJ|E?X ze290OI?hSIPyz}T$D;EKgu8Zy)e{h7t_u%C6*5Ru2JMp&g0t+dF~UhAKKmV{EV>ne zAW0nl!eRTnRj|amj72jP{BwZ^)o$Pj9$BW4=Ire3aeX`y>{!uT9XU)3A98)}Ko1Cn z_vx&AR#q1Ad<8BZ9=(uIa$sPfjU(qjY+m^n7AcUJlRXbthYGaVZmxIz z)N;e*UE*lL6IPTVwazP#>fLwg1qJIzw>CC5vT}1>4wj+caLL3#&SWlHu?8b)IIo%z z_{(1Ro0gV9NW`SX5%-qqq8Q`gu~O8b2N(>dCvxAE&MWN@v~?^ptz?TVAFe zN=Tc9B#zB!01#B|1`iIfOUK<_61oxVlGp0>NbOrQ#U3TbQ(?yYyuKs^1TCO6vG+B+ zY6TcPX$J$#@H!hEUEUOVAMfLY@5#iHzI|lxv9_{e;)rFZLGiYyc)d!bz*XF74gQt* zAmUbBNdJf(^AIblCp6>_sgZXi$XWDCsUh+4T6yF<-c18Ps=FPZY|L#4rCob}ReCtB zSDbGH7j_gC6}`H;+P*OE>5KaN*BB5=>_hA&^&7^dH0**Gnpbm4Y}PXzOT)-x7`uPn zsKuw#yEfV~S$3QuAEu=xvl4J35&79eY2^VjY;Na8` zy_UKc)-v5W7R5Vr7m=%m?DQ#`=a8PuRcivmv&W^=?s$$61 zZs#o!P1@3nSV8&qr_qa7tJtve9NE(H79HiFis`+w`)`>=EDABHbetE-$jpQ)DKRiH z1%b0mBANwI=Vuwip4aEPNjp}`v}5kAyjU&EgLF;BWSg_URwh9D`Aj42O2 z{`O+_CvfuOUrQxtfM4puk$$-E}O7X$(wI@=hcbuNKVZR|{NRt8jbl86WpBr17Z_ zs_ba)x*XTq6p|MZ$7Vo z5AH|6S)E;67@3&LJoC)iiZ-n}fOc~*9owHM9E=7sKpq+cR(J|=UQU#auilkO54V~B zSR}tY*U10=@7r2Y$MHg)eYFaVXa=Pm>yey}(0kwiTQY@$<2_U~9hys4tY0ezM3Hyy%A|dz~Glij9rU#KonK77*IoCBPeKr6QQ*P1$+vfnm9ZG-EbM zeBV=jFRT>?sySihIu-?XV=AudgS4WVJN3IF$Md&06X|eW=Xki4H%HxDJex$RrUp} z`ZdOfYXfI1;TmQd%jYuOKu~Z6%N?*~-ScAejaHwA*O;M29*e@JH55I1-}8@vp2v9Q zjbupTfO#tKuqb$%nwnZomSwk@CGoq^gX~z}Cy;7qklROMr^ zWxL8559D8P%zBY2Zy*pg;gdgROrR|g4S&kv+&DP4#>B^~EqTMQdVRJOo;b>NaBgF; zOL`n^qd6*XucF6QV7+|kIssR_h7l4IOH5bUpM%mX#q*hXean5RweXx{xgyWoucy{S z8DHnbotXhql72)Pxi0DNVgtFeK3(njwOk-|TVSG=?NY=cv@up=sDP_hsKB~#D7gFp zwjoTcf}P77O(_h2e{r-yjx^$2=)LDFctJsRd?T|HsV)u_1;R!vZ$(G?rlzJ3yK^a^ z=H$Or{#LeE{nb11kdfkw=Ql@2=3JYTARXyj0C8UL;gC;>?s%spx$hr$l1rj681+n6RWaY$~%H1DXH_v=94XDrC1rx|D#Y7t4QX|%;;M< zVinQ=u~fU#<|){w)x0e?O-3cU#r$Q zvRP55F`JK<@1HnY5HdQ_lq$#SQpltS-FDd2uW8}wMS^ttOg@s`TizghGhTVvMg@0o8v*yQg!rVM%?p%yjHWNHB2 z##Z_gwp$iUg>#qhLf+@Ik1u_-4&*K=OAD8wL(>#Zsz+Q|4Cs1@zi14gdA>aCNdJDR zsV9;~dV5at_YX}2L&=#UCp)`6{>Of2PtvZuKdY*;3tyGYiM!x{Z{n#xCtt{n2&WX@ z+!VK6m1xCI&A%jOOZAQvOq$I*eLoW>!3L*~=OwDy_wX?_&7cf$)5hbMFbK_<&%O3& z+C7BhcfAqrP~*zwA3P|W6MsTJoQ*tMAM!7^9IRWBUF6U)`}@$Y#?pUI+)0jEyY(F& zp`2Y_=)bR*BiYEai;Lk}7t3lrXhV$^N5C+C>98$b^^9{cIusGKAQB&dyD*M6$NGvW zje{9dwGNC-M(b2&ME86_f#EhRV}@&{-6K0SVz|5Tkl}X`^27S?R(nlCqrZ2}f;)$q za-Ig~^xUn;FOy*eb=xotm~5k*NN3Nu2H4n_4tjb85HIhy_S~*|EG(Tw-X+ov9 zM0zPE9ZUSqeQyrpo7aVDt4&*fRn(2p!b`smH6owc(>e7hq_JK%%gy-RUb9Y%jjFX1 zi03e>WsurFn@z%2Z(t}b03|W~H6KFQ>tt^|Gkg;aQuZI^qm;k&>~H4j`!6`PESOcV z0G?Xx%+!L?@t%>9(QIewq$Pyyyivm2O{gHgFh?a*YFIx1Z{dUv)&pFa!GOSx`Vj*N zDt%OBzqIs>{w@$-RUY6y|6ORP@j|OhLwrp6g^KiOwp{#Z%O^|?P0cuQZaL>J`N8~d z&B^@C%^#Ziv5BMg$We%|u(#@|8ZliJSfcu`@}hJ{WjK=v^Zz;g$UqnTeM zKwntE9g%N2_9E1?q!d{e_uivsV+#l6E2qh?d-0*IDFC-#_b8*2KWsr8DaL43ZxS0|5D52+| z-~g1#X)T42z<>-)uByg+KDjqWA3Bq-!Hn7hGib^>sz#UfoHnDcb;$IwL1lIaPXPJqm=itOlTfq_3?qO_ zD1@ybXyGb<+DcIEbqr31hNV4JPKM$OPxuH}KjITkJ4opc^y+xRNUQ!E zDBU``x`@cgbpDxnl;LusM?jUJ?me#`Ewx--0Lt`pLAMZ~Sx;YiD!^KQ(Fna73jXEr z&4GT*0BX0}FD@o_H$~8WkL1sH3=9HBv=<-;P_Qt(v89;?4AsEiXLU$*ce>I>2z~(G zizr7r9`xG}y};XV$FvBwELN!jH?H5Q>l>nm6YJr;{8|#=ZfHn(0&B5(p>M9#2qkAoWAY~$M^*1UZpKAC5P{`9?(Z0Y#B9pL;|rD=o?x)d`Emn!dutL#}Jw; zX?9*e))$9@nrD~g0YigYo7Q@R(}RQfpa!WI`C(zKtg$)kb2^LOWgwKtU#1@FtRacn z5xgG$O?P5NA!UF8;kwXOMmspUFU~S1~xb^7w#a;u)ZBV>W>fZvdE|vqrB= zE)Y~((JSA`E^gXg#H6Mwg2I!Unpzis|Mo;J7T{ zI*iGG{``3_>|mAqn0$Uup%O+3v|TA+k*J<^OCpl8@&=Dd?kHk)VSWGUO=dN_>0zMF zpiov$4k00-Dvf_89bAPkdin}cHXzi0k{EiSu5mf^7jVx=AV}K?37ozJQH2CwQ`?df z<~itWrL+jNpYiTBIS`P1NqSt-NtTq)C!^1D(|j|4$U+|e`wzHM^LZf&QM4{8eL{P4 zzLsV?iqZlR{?l|FraA}IMK8kJgFd!cDKd{o=(o-Pz*Hyu#a&NnZ~j%!*Bp=QRg&kV z1HgDs*Txgw3!?-yz)sH1jmOZ?&~2rVoQn%wfwSvF z@_PSsWuUirn9II5@VO2K;};s-*gHwqz00j1ALzXIxwCM-5DSaAI6oiy{tgo?#yaRK zWG`KU+3ug7R$mIuyzwV8cAo9|p#B0e5c<>kTY+F|@hC)5xXrvpw`7kU9}*QQUjRYv z;0^-~+68pqKon_joW-c#d+=b|m2xMcwKgU@E8oLiL}QRWo_{RG zvjpU>!Ri8lLE*FSnb8bF9R!pE?vVU*B>JK^$l$W`N%8S(kq~~sM=J|s*Co>_W|I&= zh4Z;M0;jWd_;h7$O${_MY+khekdlSoWEJ)ryN@p-gL_m0aHUVN*Y7y2aaSeM^>|}yLaXVH}?E;4-S5wXkmXzB%(CvT! z6iiIgVO^vCplGML9}YsAoj9H+Itqf1iSxk5u(>R@J@vOdD6uS8P{UrY#bOqRVT;?y zfL=|)%d<0OW#zvyX7!&X*dEci=s=vt`o$%g5!Y8&-F7_TuvSC0S;#*?ql^pE zB%H`oL&JG4~OiL@LqJ{+dD(~TAB#e*JLZumPnFw>c@YDoygB$3#e=$b6h zfhsF6H{ij01E%+$RMDOuZhv8J0QEk+js!JgSy`FB0EsmE8iS%OJ$yh&TZ_;S$Yj}u z#ahLbWMpKlFXUX&Vals$O)mvhU)MafW?vuhxe{%bx{w}2^aCSTQSBj<1Q2|XV3$D) zH5<$}UjxU_&#%pB)2b3+uGTOY=J@BY`!`Fq6V7Zj&eH0M?>4vPdqG;<2jK{L|CdT> zdd|7;4N&3;C@BqB;JV?1akq!pqP~0QpHe0TU2m^}6G|%~o}!P{o4}Vm<;ct0wA3ToPxmY-2XfH&InE^m5vIob`3lIu3yjH4 zJY8Q9z?H+_N7f@VxA?}Asw^am838gLU?BRnU_~umbndk(e*?`2(0htbkVx%uRl%Us zBAh5TqQWSj+%8+1UfjGNA;y;^*A}Tek|;sA|0}rl#ve*IM%UHoN#|r`^*S%_Cx6u& z_?wT!zmWQ=ck}`T+6nv84sZMac^7<_$$XJ<o7YHFa7jb zsM8F1svb?(7U?q9M%4gM062-`d0oum==I3nqSwD(mwtt?vtXtngOUvBLDZb-oPZw1 z@3I;)5~e{(1LX8yIv?h3YsgT427Sp#5)6+PX&F-7ShlhiK_mPOlvBM5rS4eeJYrD9{7*vO)YGG!ea9LUT3Y>U-l!oQTCc3lMARdJhsS^=nwDhIH z20~pfLW76ezj@r-Cnn-|X6s$ne^Y~={2)-JxQdKYpP6=ZS%3;)&+3~EJ;v{iSIf()oGg4r#_KakWe?DRgY2K6ub`dhpQ+Kr2G{S zDe%GphDj={#Q_4)MFw1XYmEb1wNhy!K0l0_MW9e9UbyssZm2@ty9tCObaG+c$IW+6 zeg{-2(EO&OZY|}5f^7E-m4p~05^D9Dm@5ERfk(ig?gMC6=1mPIfn!C$>mdQTQ&YdU zW79!48Zk*xj*U2x$EAS!jg4QXc^@E9gPowg3M#ipDtL~K5B}N!cNVa?dDR3c*kxcV zbPo<%f=)JCFa>QP{A}Xm%`qAvYf=ebciZfa(ME!P1Q@rIo9N@q&wu)1aV1|;kIXg3 znS7^jD8Rj#-%CmeJ_1jOB1!6U<+PER*^X2q7SEW=AF*e;p?35b|8ZfycqB+*Aeh*i z?vErO?R??n#C7gd-R$zBEpW>7>jI(;+8m*Hli+PL@w4c#`zw16x$9ti9IPvm!(~gJ zux;u5#}XlVR%*MTDAn$j_v-Dx=A-#TicI$zDZnoGJJ-Yl%WS+MPz8l-6-^ouDrcO^ zAYvapF`L&cDcHHwzk;lT1eX?r=9#~g6m=S|YFofycYiOMGnIz7ZVk4k5q{TvLVJb1 zxh%F6LyS>YmRwC*MzAv=$i{~61u-u2+p93mf{*|Hyk;1%Bi+(>mzvb~7P%oMg;4EH zH+#0>ot-*5QkPntmPY04>+9yj&c~-c^66Q+;rsXO=*G?0uS46X|HNpP4dM3 zMu1JJrhiUQ^lA??Qz~>3Si2-7B{@JX3)&>woJXhsrlmi8-~x-2AW!dbqeb*T zo+)%|5^dGMV9~FM=!<77x0@{6OcpUN^rKQ86y6#;n|%Heh*yWd#qoiM1REG|0w(RG z7$t#VkByJd@aolFDyr%5AwMvL9k8SHZ?QF#BU#tDzU?KLG<<< z->fTa1*9?h*RKNsa|8A}uIpZ=FC4VvP>Rh)vtL&%Qk@6i~m@-f( zH7#xXWG<7$SMY?_p!EV=P)~0cQX!UjfLyQ~16Mc+!HXB6@$sYpsPDZ54zc~EE=&Rn z3Y-?;(fLF%FAMxpZ)Ig4z0s>OU+InQaP17Gc+di>+u$unU@iIadNN9Vy+1NE=*0r# zF;Gp_3p9&HLG4?h?s#)`&cw|^UH3IHNd%V_=%*!dX@pX z|B1-fId{67hJqaFY6 z*7^VAd^hPup)^48{)k$_llD0jZJp1Bg4X{3-*A6+`?!YtlTb1i5DhReppuFbKgEnc F|1T1S@WcQB literal 13010 zcma)j1yGf37cC|sN|z#tba$81DcyN!>5fAw(ka~_DBUF;(v5U?9~ur__u>2ge{Rj( zx#K9#;f*Kuv-VnR?-L~dN&F?sYZL?ogqM;MB8mtIkFXy8B0UG6;N-2;fFI8dq{T%L z?jQdAXv~R5KzN5BDe_U-HDzz!Q(f8V?*53jk5_Y%dIW4)sEEU@-1`WU6=XZN0oJh2OC< zF_~_pB)SRK)iDqZHVnBVEe>!y2$Za5QU&`EprD`}9v`!qDpg4E=TV*#7Ux{pSgeuCxxOMp}0)o(ml*RnPzc?&1zUcYiA@65(*H`lfDxEdSUwIA?0e>wS=!Hv+|*|rqt zH+9t9k#djov(ea#PDcn7F_WUPt z5E-U49ah2zA~%jL#ec5oe8#}!N8OGtO!_+UwjrC%nD{zZ)=CI@{XJgMR2Gvcc#9w3 zF@6Wt22+5%P{FMSW5WgK{YJ@6th~_lpI5Yw)HI|7r9yaj+M7eJR74(Ln+_osqli3lpU?c=Q5^H1wsVyeILBi?t@_yOdF#{bM#c*3@lOqaP~g>|c=`f04?aupx4_ zh{_}aCt`^%o~$beTRYyCkvt4NP6 zY=4HAyvO+k;dezU3F@-lK^J#BN%Gz-0f++*r$FI;V+OnSU7Luy%Jf#p)!1ZRs?=!~V-Q(+kgSZ+ z`^z#IA5A47kB3BV(gw3;Mm&Fz}bcfLCKO&DYn~y5)VZz7{p-fXDD(DZ_9jmZ*B#jh# zW%~sq^j!d9k=6)3I7=%2TOC_BTjrvSFpVd#OTWE!{PBc;yNCMI=lLjxhx@7DG)@)N z#82$y!tS`hDXOxTbOGZTmocW%+3uAwF`o|}4>jfEo$g7wzg3bco1owJH)=sqp*z)< z3Ux>2J#k*ia{W9XK0fbUJD_KN?^n;@64V>fs&_P5N4`gG(mOcF;i8ncX>@v5aHEE- z{5V#bTqPFTxmBhcq?k#kh)Yc$gWH=n>#n>7bAfs^s*Fds5kfF5#2I0UB!f$=0)|5K z$Rb!pPSQzHkoV^Oal;|dv!`;U^e%5AMR=T2z8bXzzpiisi&{nVS@bb7821zW7U;xi zpcpnTZd6K2e=Mz9jnfV;!YOS0r$&W|udnaY@^XgVCJbzcc{_Szf=HC?PrTFqH#QSx z;fxnP;;z2y)o~{~)y+}K$y(O+{rxg5tgPJLcYNOv(J0=%`|vY-9K@M&G6~d<_?j(9 zcdY)i%+J%tiBP9E(#;&VX7!&jkB)S>9X389_(=Ql>Op0EF|asiLk;HXGcDKy~U+4CYVrH_f3 z!&58$Ojfh_O%uluRa={I_37`FFE}ayX?2&LHLdanXdo9-h$?W|tvtOx+gt69B>k0> z!)iIjbaQ+AdHVRdY01T(YPnXy1z+N;IDrz;ki~tPt(~0$m7RRBi8RI$!yyydR49VC zaID$iR;1L_L6wzk`}6hUtDRw&tHk%+KR$j#;DW-b+5(6-w7a1(r1SdQdeRm(wK%LE z7Z8LbP8(}SK~YiBPy$we*<=old(p(tb2WB_TGfMF=W9&j2}VQ1A?AwMej#{_$ETBe ztWGfH=cxECza*ol$_x?dH7oU(Tm3w)_o^*tDudikHjcUk?;^i{M~6)m^~L5GAbISf zdtZ3GN=rLGgg1ywbmN!|%t#fe61cj%kLD>+{=C*{_Tty6bELlI5fDf#)~UxtNBTxmbRM3d&48Mwj}@b=EP z0OFd$W#`itfEEpo-+aAGEjWH*uh|W@{$}|4`3*ODx|@%)91PsB7c)05I8NXck(LU- z19M$x$g!EPWB>E#PfdOD-82OS#YpAn(UFqX)m7>HxoR7|o0}UNPmj&PR5EgM41D}< zu>JTfMiPFWbXvbd(F88O1Q0WLBWa!0mNcHd_Pl~W`tf-*8(d4U2Zos2g5HQPo+6xP ztAg#<7ta`-lq6+u&q67e_N%5Qp`n4#dcIC{vRG$(u~`t@oAL46^YLOm2@*s6HdGU) zZYy-hjJf6D+VYBuuzf*({#3iw&Z8d6bmc2o6BCNBU%w)-Sx$Wq4Gl%9D%P%T{qqgc zZ96L-2?^=l`}aPjrKR>mFHulvJgwXOU-tC&vdomAVkvwAxk6s3%32)9c+riFlJaC! zPH=f+!#|TsyWaUF9vNCOplffoT2fB#;cXz| zQkp!kF$oC^XR3&XMLE_QYw6=G0^!U6-SJgqe~U-P@4vCWi(7`Qs)*xTDH z-|}h0wo4Y`oAi#^^&GxD8O#3NacX%A4SN{@rZ!rv!^_9d&+UFjCn+geX*TTh=*jcH z&0xS2MFmiAJP$^um<*8itv*}WHC|k}Dk>_v?i5uctm0MAH+jAxP7e;Dw^?kGwYFx6 zrjT6$3j^PuPqt~g^#AeWhrKXBp*J#Ve2E}*AKvtyl=LAcwzs#lhCFx6y40)8aXxQ?Rqg ze))nhQ~6mKpe8jH)e-8%$N=9;l`pg4b)BXaAA4U2*+qNgQbWEym|{l%&;?*;Ntq)0X9bMyg%2& zd(a^K;R6By1rc%a(Hc9$PGf$iPbCaPy=iX5pFAZ;L{p`nR|;{UNIY4i^X39*iY(NghsIrw4? zKBX|554Y}SvSXPD)MY#O&7&qKCnuwzI8ML63HbGk3`B63&$Aau$jH?58+U$=nwstJ z%exJpys5MnRWnN9O5}8X;xebvzw&xAcMy5B*T}@A-0j4K!)iw4_Ue>?!xA5aD#gOW z0{i~J3?RwIbgSqAgkFz~oO+n--ZQtOOVb5#htlFO@~~u4Z1T~#Qy#5G2$p-@LZhOh zz~+&%utfc~qOUg?!)36Kq?UnTB_ol^?>IY(`v*mH_4JI$5y6z2N31MZ`0^E%mD@pj zQ?GZXcXM+KeajmMvgP)bj)cbL8f&mZLx&bT6R^jvD6qxyD)3BlN|^%tlMPt|L&NFH z&sYG<8{i|-@U7vteW92f<}ho=)smYkSXgQdDi)$C9dvjED|&aXD_GKO`)W4h(NDj{1|> zL!hUyyhEwVvh>@AzIw8#zvz%!lS@#|`t{5A zp!q(nr0FX7=Ej5KQ%d*E<%va%Vm9SuO(!c~8&Ex{o+ZB~o`6TROhG~{c-1%t=eiN8 ziJ6@LsE43wg;LYd7=W+^QLj{@%bzWiDAeqAo2ya~S6uueFQ|MtDzqbl6?=UgIzbf91tBH9f8|VmcF5}xa@1!xf~QmV0MQt; zAkm{OIC+(Lw{UyOcke*N2%XEk159TMoW}yyqD+u|Wm0)!!3ot;Ryqi#Hc_5eaAe#c zOnzo#J|C_Vg{uB$tbnGi()0Rk{H`*~(%YrB&ycSZi|>!e#$>Lza3eS@0(TdN+B!OJ zm+SG{!#@K=Psa@QpeG4k5jWk{9@0gx=X_=tbW7MDTnC(~6_n9()Zq}D6X!4s?Iw>z z99q@yadDj?j7{ADPD)!=49S093-sts-lt=oMp}Sc?wW-L;IP*Xjg3 z)NR=h51-vUh9RooGRqR4DG1XGS(QoItQHeBXY47-yZ(t+@H00z@N{PqEY!ev65+t` za0Zt*Ql_@v@XXRsR4i6?L&>Hr>zcV-JGg<6&|?e?3^wbz1Tdbz4Q>wg%?4F<*w_@T z8ku-1WNqz877RUV!Wg15d8}CgYSc<}$pL8?8&lTxzID2SZyROU3+c20V^&7y#=`yTuAS7Tj8f>p{K&@%TjLwcF9Cjo>MP(E-2SRC5= z?NPuK)XR;K!PXiB%HteeVQyvo6K_3R1LhD2$vnr(j$^@XG?Q_5<{bE$3GmV!sMl{E+(_tyWx=E5WKebG`R%L z|I*JeujB6^C#-aYNPhZc(i27Q2pf|GQA z3`|VLpAqlgu7g!#u>DFcLlVyQTH#130W4FQ{-4KnPCK2Piyp$MJWjw4Y9@mY|E7CQ z!D+K#1UT{9#zvewA1iA#(20ZTfun6OfZdiaMnB(Q;rFMf$WCV*~ zW?<&XL28Y=539@2A@p|gbL?y|2$-dVg9A278G2(HC^HGk*Jm$qnE=g5b6p8CN2KC@ z0d%ccSU^vfbUeb8y}NYnM+12g3@vLLh{#Sc=nySh$uM!MZ@bxkI1AAfAj+Tn5We#l129WmF4rCRr);Z5Bez!SusJ+cfXHOBBxP^36$$FzA76 zqcHuXW<%O}orYS?%=ON=7C$E=BVz@FZJprB@Z@AXfY*_cpX8~@gCai7<-I+tS|8O|^o*^~Px;lJPB5MI8d87%|4@CeM!Jn{S@ExVXeI=+F?yCV%|6%$)`hFST?FWeN~& zu?Qj^C@FOtB_H6sZKI>n!#~AW_pNz)XqcHJvt^UxyxtNM>s@UEq>9rOFLT)Z^gxeL zQP-*_b(P4i|0)lFi=tv;jO=dnu@%}~J6Zji9UJFjH}%}oDAPTn0DBzzYn> z!Uzj8$Oak2fl~#jjTiz}6Dn{lVkaCN9F%|oPvdbAuji#u=e(B>mO)wpTlDDgP#)#Q z3mj2}n0l83nzXbuifh)>sD1A_yHq$MPN zfUMG=!W~^&%7g%$EExhJ@^x|0iOuY9%LTA;=_GBP6793CES(jJIWs9u%{_dx!j zu8t4#9RNm542+E4Rt_ch(UFm@>580=Q(0ya#z7vgkcbVhcAzJB$Ec6SRX`5GYpk>u zk1;N)mLwn`=pD;bip|V?AAQ<%x$(vrOeHJg5zbws=k@R7KE`G$AS;98CB( z2U)XfXLG$ooGK>=-uLz8*7H0C4fOqrC=}lxcrOVHuaL+q0JT7Dr zP0ZBfuae*wEz{&u{Pz4M5(>)7#nGzc-pu$G2Us%0-KjET#N^?ptLP zLKVXEl7R9Pvmg@mN&(2L5~xzDj7)wLBsdIHr2RC&-2v>BPk{L-&jfw;@r`k>-p4ZqMqc$vA0ndA!adk-z<8_T$33 zjzLf?h~#n@6Y|TK>%gQ56;H<-w(Rt^@8&FZ!+4^dc9unyF3bE_)2B#X8JM=%nu;~7ZteZje~;BvjVheB7W%is3;+4XW$lQ@&2q3ci6n& zN1vsfSThiZBmq_~-4wGW%Bz10S=rzJRBb&kAt6zER_wSwptndy!ur+eZU(iD){4YoqCc>%)Qq`A)pf;+c4R(X#%MIiFIydwW;F zs{ZcSXN$Gx^h1_e8Ye9|&gr`Q^`GnFW&MLfLS97h9U~#WBFu#v^8PH;m9-wt#-1o1 zC@D%t_sV*V9RxoE)V%~t*Q}G?=$6t7<;J)4}7)PP6ehS z3d_?Qq6pxAzEMc_o3g4zcb)13+7iD$HevHir()gl-^{$E!y(~+vfI|mx`kSl2}%6Do8OKx40~MWO|FqoS*7?d%qy>^h4zQvPrS^ z#6v0P)hTXokFWzlCZ6C5X^|ENZn@^uuhjm1J6hG&Ry#Tc``dz@Q4X^aO6e0F6=x;b z<0W$Q=L>D?)23C{0}Pg?ynX7OB(Ijkw+rZm#`R&iL{$1s^p9mq__9j*;Rje0g-xwEV@`aR<@k1hx^Yw#adP~V(E^Bsq)xWF;jrdoH85zJ5u*gag1$+| zGfdA*tzb&w$FZW8JY}2^C{x{W-i-pG$SN>ehm*kqVxqpDmxP4GxjM@A*J9(zQ`wgrp+RZJWR6w$T!Z(FWl?C(5JZy&|k!qz3Dt|Zvj?&*S_}Z>!HKYhpPts zfqW{RFC=Sfd6q`&70z)o!{j~VQ8Zh|3s8}u|NE9GV)issS$u4}+ZuDQLgn<7KPQp< zH~*J!%;%%#AK=d9_e2pA?TBSU%Rz=@FpjlGMqHju)-Yq3*_*{U3{o**7R*O(iEd_W zGclj1wQB~EMOBcS+FcgB0KP<)D!r_?_QQcl4cYMKh1W%0e3^ZOTaj?|rgOANm=hQ& zC7>>~t`@IH4jt|>6Oe(|L|p)CrD;l&ZsOhRqUX^Cl58fpCM&@pdo1z&aL6aI6G&!t zY&%d}qkIEv_0f_Vajm70wzXdt?2_zz57QOHJm9k}qK}D)O zG7MBOmA$a5;72>;mhC14rLjGIeTKkZrqd{YDkmp5aCOnUN-J=36FOg8f9cV)$#>ZD zgzx_5Bw1)h{kRr)LA?OD~$>%@55+n3Q5v%yX$H z>!J2ywH}w{EVfG@$OQ!#Gn0&`4YN(F(n^eI_hOuTIeR(t#|q^2VB1Y;)(ZjVZn@DP zp98nnK5jJatWcxEKPL?u3^<3yNJJx8-$y*$rjKXlqiX`+ce182heF6mYu!#LfFBn1 zVl4k-Cf>6xvjj6Ixt0I8F9rxeB6NX6m#0>eCVO3~2KM{3Fn&vvGA9}G_;)j<9V&A> zX5Z_HrC{1;&z`k+bTC7F8dNv~0A=#W57NcHdQI`Wf$~W56U#S(c1&qc*b{W(zag9k@eNGDVhyltA(+248<$R)oz|=~v)1Yd9^2 z3uWSU+&Z$$v|P~!NmD^Eqe0Ph`nzBm%@a?w;-{%Jec6somh9!6|CAGM zLcC+$J%DM&cRHq|I-Nrb76THz?~kl*LvHtN&Vwwx1u z{r&BCrCTyRF=caKNkanoixb z@`MhsvR0vtGqd$Bj6iD^9SIO>P$Ytp^=HpQ9>=|f{V(-vvFe_op-qtI;oGBKJ`8V_ zm^(sE8^8!EwxJ1Qj^t&4{qT$ubbpAReaq+iok6Eg9!M)`EhnyrnMzN#{mP~ddijsN zkoC_Mz%bMr$rJ{bf>#^oKkvL=jDZ!C>HepXU3eTF;OqO8$ZZWnTFa3thkF3k1YMl) z?g9W?DXBGxnuO#>$jY^78Uov->^sZEu_w^|kgnfTz;^ z$MF_uE(FkOA$yk5u`^z-J5$8=pnQm*-T%03+=#uq>CZTyq&*!g+T2)QH=gLq_;dwx=KsDgSs zgTJH-ZO}o4E?d{pY8PPnA)53vReP43(h-dRxd1sqSJDzFoyg|LYT5(&v5AR2AP1Yx zROZcC%{DuwY5{+Qzb#wWWPft-4rvE z#hF!RUr*AK20Ba#EQD+j7ZYZq#XwZcN6({TUECf+m4-t$u7Gi2Hak%}ne_0RWSy5- zZBi?jq9(;$ccOprhB;*8b)y`~)`S z2e8S79mNZY+DnhM{B0m>b&)ukcmuJZlmv<%xhj3dx^*-UkBXkZ2?JpvwbgpO}p z-%}5VwBkAn%7ltCYTC=KfXZSH&m|4`P}PxNM&{oScP2})@V5x#!=!=vL_Ys2IzJQ7 z0xGux90?-VrAJjWW9$GEe*c<WcMXKuFtH1r#ErJ4;)z3-O#ggLd)vT z$p(9S-@j|QdG_^83XBNXkyGZCU0`iLc{rEHVV>hGY#~~GuacU-mn1+T^t}&~5Ylkn z+hq~Q*hlVd>(Axf1!avsl7;QF=d*kpYik4C`t|qk-$7|qBK9^bCr1e!s0zJ+Mq^tV zDn6}EjVtO~e5k2e6|45%-l-J7=lH1Xb4v`-P$RNc{H_Y8H1_&Wi%p)~pcD*pxyo4% z70TV+ok`;>_x>f1^z$C_c)3e#mk1bW+~@+44_A@n4B!Tn^<7lMOg`DC0a*hg*JS;` zUMPk`gjnnFDFBMI7c%oghg$%Sfby(rt<}emUxgQb%mJ%O7~met07#+_mSB!om^nZS z;Mo77sivi)YX_1{?D}s3aKsw7vYdEZgFf@f&;{wz6##alUFVqB8`}@1j!fh zoW{SF3F@yK)#LtUA}YmX3=9mN<5kcA1?a^(tDz z`|sWlaJ$a1Q9?V`5KV-NrT#bbbiSvEh?B15WsS9+?Ac>!AdL&D0UpL+n*q<3V-M41 z>r9DlFD@=N(W5VTNL8QxMnNEM1Sb`-B zo+>-4zEoY=ZJ-x|D>_j%Pj6hxz@3-^ggU(6q;2hPOVlFIlDsTF9?xZet_*fpBnskU zVcIAGKSZ}Ou2&Tk0LUyuKoA(SJJ0|I;4o_LgY>B31RE=>hk%i`>Z&^>DC0Dm(fhk& zE)I2d1pwALU5LgGj$<=ykV7Ekc0gR!c6oYu)Bvd`>t5#j@JNHIMR|vxctw9o zqb`7|StmL_Kc}Rn?aOW)i+T9HVJt*36^xPDId^qur>{t>nu?lw_!p%EpfaW6`DoFo zEck8saTO_Nm1^}3(;!QJ1aMj#+iq=bJyxWpHn}N_jh(j2-HL4b7z?OSq>9>y4mW1b zC%+}Icks^M9iuw!G9D6-2 zBLCFJH+poMl0``fFZ$Q*$iXBVdtrIdgjN4H#3#)E?RraEFBq8O5BlcgBOxa@SxUPK zDmbRD{$ZBCkY&N(1I;UXk*9K8gKu{!qa?CfQ-x59%93dav?S+p6-d1<*YRYpm>>3! zv~)(i?SE^A)3Q*Dt@9?glb8R#@Bgrw;`-yX86Jla{vCogeCS9t|8CjFe(1v`0PAY( z*MkSHnI0bXv-HY1mZ63zdAiW>ZFGY2-t|gu#;()~g=~`TDNH5#T)*Gz3_z1DK1aP+ zJ@KssS){rFu>Ov9j;*S8>}i8;Z*CCsWy-66TWGHmqycua%w$=&^pje4x=X-ZDfzH& zS$i#0_RRfWY6Ted_D7jkXI7ygsGqZf2s)$;SOe9wYBu+D36r@9yRq zf=gfHp^=Q$dgu;cu{5!m_=aviYDo=SYwfW#B2`N4CESqrahmLl(UH9QD~xsoitidu z-q|1~ZIzRXEzrngh%Y0B2@SZ~dOD$-%s^KGQ5n{gqHY0H%W^jxT94U>C!iaKXC#sc zzC=b@^qa9Q;}hUnpe9{5;}v>2zODvv_m%8}zR~e-4ip8QTM*x}uP;@nBOv5uK(tm_ zOl|;%syaP%3#ubhsnp=m5hgMC2mN_UQ5u~J`p_C$X|WZ4nR^183)s+Npp;-W`+FS( zX#ND}!{W349cOy9C1VOO+98EBl!>KVzN%^Y_-biM|5Lg^k-3#2IOGbK3jaRXyZ#B1 zKJP3+D|+2%sxV>xm81zvA({2aFXWz{p8Rq4Y*6e-Uojo`M9yH>c?I$(%gxY5=rZc& z;*C~n-MyBi$7<(GKu}Dpki@~%2E_m!TarvOBEG&zh)@Y(b%hU$b9-DcfpeT9=&kF= z&<9+QEAEF1XL_grB}hd1%P7f?945Lko4D=_~Pr#7)6Xc z!({&5hpo{ZG*Fi-u-$A<0)lXPch@-q!F5FvTLHA|p&k>pyLhM-Z3g)JgZe`^sBqH5 zgu&L8o4*j4$BJb7*L~vG>l=XRFezX3R*Z1la0kAg=()A*;XPpX%QQky10y0z%CiAxKvIO!u)aXYNW;0V zh9f|+mnneOa&=`z2w1SwwGO6PRV+QgDUyhWibnE zrGD2=02b&JTpdizu0E7gT>|fe3eimg^dkc98x+Rw)v5}noSOpJ+LY#-neQJSgR_kA zyfp2UEfd4f&kvQ5>pP%6pz;}%Np*2`H3En=gJbQ7P%YE|ZaT0I2}_P`Ph0-)e!rES zofjV<%Qgq<4-J0`9zNZIBIWEhw0BK+n!o9Ni$NugIiZtHbJ1czy}Ar;F&ZJ6JTt;B zkcK#v@@UEO{5R@M)Zmuy@UN^K1+F@DV2p>=;|isx-0{~R*y)+rBn{;K5aGCkFr)oS zX{pK$)a(%LPxefT+^KLou1D-17+42ZxMnr2uQm5AQ?27Rrk9)7u4`{Vog0ehs}JVV z>THPRHzWoo$n({fkKVDftE^i4{bkG%=zsK2gCkm6ef#nD;r8Ov@NKgvV;LT3;Cmj= z5Id`iq!6hE%EucfEgNeI)tHmo(A*pMJoz=Wyx>A{}F!xuM})84h}O2?1EEuEMTz~92QF>S=e7eK<|>! ze4SJ6LOza`>A;Fq&ZoH(_*z;HrJ9h?6psZiLD551Ll#(pp%d!=?Yf)k=*XID1|Bxz zJJ5;+TD2b9!9X1rB*hYW+W)g9k6ClHMR@=~qVw)l#>|jpU_RF^6ENWQ!7ZSpFM7C~ zoY$oP-Bq+);N2RQa|0ZcfL?25b+rwse_36_a_2qG*0wh3W>xGg_#)64%$`?{2pfZ` z(x6HL>WtYs_0A2a6IwXt^#AtPJ@gx`$Eqc8*<}vbJv={VYf!c!1L<30F znE3c+XFHR#-e`HI-lr*dKOs**Z~`T-2(3{}#ZsM~o(4$wqjfbXCx#Ke4$jQj)q(!Ljr4K1LCA~Rt*RZ60vp|Np~ z;9L5Alz1ce&%Zrt%B~)Ne4aI2!3whUCQJ0vfnIsi{PKn26Qp;b@Cc|cv%`i=Yip|% zafwO+F324XCxg5I>MKW*0CHbk_i4U|@UqA#|NXb#(90hO6osxX5$Mli^rWMs3k?iJ zvUhL@3Je4tX2BoFhKGkk@tJ)9+2+|rL7XST@ z#Q*#26!E++`&8g;qmcMgGXy8b>&+NaWc~A94ChnO3;W;iZT$1FmHQ{F`gxu6u8RHO QODYJGqMt;Hh4g*?2P_KS7XSbN diff --git a/docs/images/chapters/tracing/1cd7304fb8d044835bfbc305ca5e5d10.png b/docs/images/chapters/tracing/1cd7304fb8d044835bfbc305ca5e5d10.png index 3dbc9f0d75b12452e4d2610d69dc40de7561d331..8ec02d577e8ac460bf8cd19531e7219feebe5e34 100644 GIT binary patch literal 23810 zcmaI82RxU3`#1hALiWhWjEq9b$Q~IXiU`?+kjTv5dxRvbY{@1`_THNkvNzd#kNl(k?XKADQ?O`X}8#0$GD5y7S%;w$X4ZSAc@Hu)o!rrxvL)j z@1LH>nt21Cb~{fUHQ;vzc1u@BX9-G3h$nZ~zUFCwK<-5JM?iMr#oW#kn}7{MguymJFJxjU#w&VQLmLonzev(H0j zHXD64H8O2&?T6nD|8;B1SyxcORRr=v|J*`?GyFm>{fGH>A?FiELP2dgC$UtAAuQVa=|1X`E_0@>>;)~iG8SBIyjn?lsf?Xr5U zTd^t})|=lk$@_+hKi6;Y#|wP(rtw$KQ@=0?sX6M8)C^oM`)gy8_V&DI+ezp8{h3Pq zR+FB+5v^DymJ`jRWwyT;-=pPel{6o(*KpY3GBCj67@g6Ix&{W5vRy=tj*V?j)}eWk zL=;L$O7^8lV)`_{ny98Jc{Rp5C#6DZ)}QG$T>KJ;#wQSqI0cQ6h_q>ZoR%^a1C^|t zrEh7;Va33uR&4g`afU*|LNyr~S@gNBTeMtyy`IOZUzN*#>Z6am&q*T!19P$dUaM-h zO5N%4e#+w;R-4oJ`?FQ$7iEpmBV>2ljg9MlFZV9*a=4uyY|c2I?hX=BQ)`oL1_rib z7q-ze6fI0ndhU{l z=P@55Yzd>`b`++f?)dfTLFCDAf|vc7^j4G%tUVBQ5&Cw5G`Dri(OFnnN=r+x2&%D@ zUB7eZ%bRx*Sxzo4I5b}E_d+S}4vi8wI@Y z$laF%`s0;O5-kkyhpm=5pj9@URzyNuoB9fNldYbjqN0qmvmks6AsHD4A0M9=v8I22 zzskYUkpycL!<{>XRJ`UlY(7edQqqfv&?skp>$Mh4z&9xTb6e$Owr+ypYpc0-(xR@X z@y-kpSZ8Nve)(2<(T9x0m(2!pgI*5i$9+mnF}JW#%G12d$inh*#c_3IWpZZ5XL+z7 zCNoim)vu(4@6~AOh1$}Ys5YZal`LZACr@ywx6MW39{Ot(8PjudeI0TN3Vi7G2tR^I zg_U$~Z7gJE#at!({hQw2giP&2LDb~gsV2YESE{qa76O;O74h})DyRDygaQHrUy6$_ zZS3GpN3tjekIwg{$ynLhH6HCQ6H!uXPL}58<}xI0a{6a2-O{b3C@d^A{#d}j%R`B$ zmaiQ|bX_|3QzAWfPE2H^55(kD1A!RN0};u#Ih)4DM*YcJFJ__f&d!G`!^Jp|MVPW< z5);E9hDq0}k8h)9!uIQYFayKGWoq0{vEg>AzJ24N7IyqHqyhKh*E36ab?W4B`<$15 zu&;QTFH`gGW#S-g_)*Pare~O@QpvWX`ThGhzsFf+uXWdlfB;M`E-n=6>eZ_lU9rX{ zCINcxhnLoC&P5dxg*AqWf*|H4Am+Iw##d<{eOU1g?dk}j3T~q0c~!YxR=8sZJM*?< zPQA>AX&_%GjuRIr&X?rB z5!ecjr!LJ^`BPMcdwQaBHH)!Px*uAnRP6+H&O|G#stTWXknOCFkUz;#xF2&{2fNb* z`*Y^Lxw~yA1fXUfnP%Si*3IWp9DIC($-@2v+$xaecql_3cvRbzQB#I~Zrt5k64q+y zb%wv16%S7)6-cDqF?AIWKGsyeGGJJ>rM%u2$KAoMPpY|aTy_21owN1*zQByY0rXUh3jYdWFQKBX0xu}p(1;aMQlb*-0 zy2L?^pTzX4bP0lH|D4=o%F5_7-731!Oj972?wyyHS0|>IuB(${Q+g@q)rLYR>D z`U~{V(^oA=OTAwF{QRsh^*+iI1CO${uI_?rj+)!6gjh&9j*gDA=Z8auh8w%Pegy>_ z3i14#%43t0`a>flY0BwxLtmeF$dAMezQ#t~@i=pRL&b*=HHYhu6VZprNRscC6ZmD? zVP0sMl&(0p;cAQfmUjq)?n|@~QV+tx@J>jesjsgmBqtB}_U!}If)#ng_=E%^s6RCr zoki7$^HjK$JXm32VJOcclb(wxWZN2}s9=xTq?FKGPy{HP&bz!px&9m8ham9dKb zvh^e(2Yl4q*x0}?U+(OzjUAU+RG)0ge|tHIub`knaYr9D-}Nz|DTF-c#5a~mOG_(P zw~A8FZI_hc`gI}_5;RLo%M?S%9Z|Pv9Iv98+XTLupP%-HyBdFi?;jZcMx2#^M zxqOm8*_<;ihgU11o)kg2_gM8;9tXg2Cm@HkPEx1c&juS}!!5(R>L% zY&ewS_V#stAqWP60*vyM z38$^vSqW3l>a_Gu9AoC^=l6IhCx@M_nq&M}SAJn-<;~)vDUZeQl|0Sj`Z#VA-!NEf zm25|*g`4B~(kc>8FCSB+8n%Q5hK6=o3PRq+K?Mc}U)(r<>byls8X=2br5pac$RrKc z0_EBDkq-?OMk^BRFz$p|`%_~gU#G&~>{s@)!>u{g#wf27&Z~>H@VnkZ!xjU&0wzi! zd-ieH?MJxO{0P#Ga9YCkF3!U3o{Ax!+QbvG!T6M4OwN z7<0wU;^N{5GvQ))tf!^yR)&b!p60V#-MWq}ar%=#*S;G@HgMllk!aR#%hD)fHttE( znZ$k394lbU41n-iKQCNBt;2*{0IXiL<9T!|9DVs{_BudIy^tK)q5cXcC@iYO3Noju zzIfKa8~_{k;Oh26eALeJ;Ck6eOZyK<0KH|lCgOV*3W5%6ZyFn=w9D--14L_pGva*(R+%hPhL4_C$oNs`PeSmejh^*psB{B! zOa1TRVT0+$pueEpazgZ*`ETN&prF4{i;I$l)-kv!hbW4$U{;-gj*c!@r-F<{F|n~R zhy;#nFl>B7ON-2+DSR|W-@Fa>3s1mzau4&=8oIj`7AcG*Qc_Y5PWLAe_=!s^OmSWM zP3>P452sj#v5N%HoW_a92Z`!Uxv<@((K2o%dNww!V7Um;FmP+5*q-`^AuqmW@tuq} zES@k3d9i7+=Hjk%cfN6x6ax`pUjLf2BOJhtwSe}+b*r4G;QAzFU|*fX6td77J=N<= z4&IH9YD1mz!C1)q@QO9J#W`A+gPWT~Iji4jH}Uy{-7Q^7xvvD0-+%qF!&q8>T$8-ruV24zd3bbk>S_Gyc)1f?K_Iyu z)MVh(9JRM8EiKK!#YLQ2* zyESyMY|nMpgHA4+Cop2H|NIeNa;Gx-zRE>xzxrf?%dLWUp^I1^$ZN2yIrR@9pi6xa z$xOj*jQbb=OG`=3z12+d^Nqwu=|@VcFxZsEbLS8tb-G9FoO+=rNhU`v?~Xwu7P9Ps z&HUxki3Mq)vXBh3P|8vd&7z>X$VuIf*_OkfD=IF&KT+*wz1T~SoI8(`P0Xdig7BM9 zK2HKPmS};^XUkAC)n9I8#+)=rmSSRv>DW$^N%^s`yc~i@#d}5I_WD)yCpTYJ4eEK& zAng?7+d``;30X#B^WR@DS&V$YtHBFNT{s)}uDMQG?j>kDGNk~(PEJlp|blfGP2OPi_-PzPsZYSm5@I^O)zwkAFt6va!c-3ZU}pXk3vrM zySTVE<_z4MQF5haWuyY#y&6Nf&$FH3uwdcgHNk~!E*I1oY)sa@l}o4KHK!le}q6%i;X7LLO8nnKp=eJ*>_^g-|1~mg66VtHH`x3LB$Op*RJ+K-Pc3WHq z(+0j_24u+l?im|1!&}?#LUBQ20#MDyY#Rv=Bx}RwH+97?2YtgJCXCIH23R$6I>fTZ zQzC^>Fl-XuO&Y0?At^155g#A#8z!OF{_`gZQmGHO7XWwRR6;|%*q4S$OicXGXM8ry zP@p*41|Hxk8WeGf79!vXQVrTWJADCGnVXvcjkGBPrHIWyyPbgKmCdy>K-C}tqqur6P& zgX%N1LP!+FO~9qHVHJr@(_oI|^UrYLzP8a9O|7nm!ucbEk_VlaFeR^fAjJDd91TON z4TH#&Hs*z^aE^lij90m2YL|WOX~u;mS~-idVX5eKXWTsiaAo8nXP=e zqrKhxKl#>T7x)fZ5oce`Vl#xSc+47Q9959J{2*O%{le`vQQR&TvJgB`TQK zBV+(fB+Sj(F>on7VUG=mzr83J?T&Q?&WM+n*Ds8tW2U&Mh>a_pa~GG1EQCHnc5-^! z;Ooq<v4y97Af$D>*<{cG!{UxZxx+}ZAX-~{=1zX^(P z(Y}0~^O=Q_*S{dbFstPy7J}1o3Ft(9p-)>`aa?E9hz<@6oaRz{v*VWY(I5rjb@87>4ET)COg*-CT5}x+FBbwj+q_npX zvQF%pXscfdW>Sk&wA3O>DtZw*f=u12atIx{#)bwopqW~r3U1gdalvObz5|Woi~@!&@IR@bFwg@;1_clikwPhvo#(_h00JHs?>&4#Rm z-x5w2pqwQI=>a*ZmuY~P&=t@4IWsP)+vY?BeBSl33I~Gx{Cq1bD?&kRrL1>tMzJC; zya?fPezG72_*mlAt9)eX7Oy(c&DM{pVhd?i=XsFzU_?YfjR?`cmDu{h*VlIv3ZIl8 zHfe9AldS^ZBPk>4SK}ShyX_~FyZ4b!>ixTS7sbTHfGCZ+86@SM^(4ap_ztA!_^gnC z9VN#Ex07nKK3~O>In~9y@XveV<2#uH+Vi02q!zG2Ck{dfWG|tt{0?c2+KWH#CnZxB z%vSn_coAQ{DIjoF#AWY_pmf>n+S;q$JY|G#D&ZSLi_P8t3T)rC8#l0^Ba`yMz)K!n zrY8=180{4ph-&T$6$wB1iK_9-TTK5A@ksiAkQ+EQrA zN(mWQ_oooOuf{ws>d`6CoigY`Lt7t8#W(rqPg~qC^mo8KL3$+xYpBM5KO}LsD%{>J?=ns+6=?n!g_fP!Z@@8d$8j1mHKR zApael9j|jb@0cpweI%uVkw2zfTk|2Zm$Q=?*+lIsXHwX1GC_<8jcny~Z0HIDpjxLn zZO@llO?e~Bl{>tvh23uuxh;q6MlIZ`A4!lD1>Y;`=-WEMP&A*N9_=Ex_EmGx^5&#o z!lUjS$dX)dbqiYNI%1ScUg@Ny5yn-)d6{XAxQOJq4#>#9lx#Dmkd-Pa*+&{imX($DniWegaQxW>p6y@B@i4MuuGf@@DL>}v z!f#6d0r>0VoL~3DdXB6AA&?H7+W~1hR#gzZe>)tk??Y|$eC~Z|VpgoY$UnLOw<+cT z4;3Ng@WNeU5)vh#1!KNP%B4ec^j;wO3QXuFPlNS_URQ&{p=9f5`?K7t@l2((-mE7q z2rt2H+<6aZO?j1qFRu3X_FoDMIazCNhCO8`lVpgfZE9*VT2L$)lw@!cylnPh=>^Y+ z6LU}NmKx&NUOJ7Dmwx5M+j(aC0AqRV; z#Ij{$ssYLy(li?Fot&Hy_Vc12dmIr%{wDwTmjVq0pn&yKcX#)}T#Vi%a5Ks__tW-dXel#DuKw-*01E z;Cj+mTkH?#m$m@1JksPrcUuUhq#X}>(-S!Cz*QrQQf767sE@n*OB*N1FR2pHF2;+v zkbyiw4C%UTdT}ujq@ntzrb`hK5g@7{A5R=Foze6=SB1wW@21_AB9Nsr34uqr_bXc! z@CU2il~fxUb}~ayAsmlaOE%{}T2B8ftr7RnvFs}~)S-2M--BgvaYf!L6Gkl_LUs$O ztoe8}f>)}ms~3Ay(0E^sVj&3~PXFcUof#FLEOq%s3NkM!jo6zDNgl9OTxc|ccD0IP zr@m*W$1`vepmLcY5^NG^y?biuSY`NH1%L^zcF z);XKXql)(@%AN`Z@dAWVrre29zS*@Bug zO=N$b&#ND)q6?||g$7zq;MEmLsdzR75Trqdk!ZP=N?uxC&T#uSJ^;Og*+@mG3P|{= zED_V{igS~TMw%j=djY6qR@04_;XjS#m?Nj)&ZHF-@W(yQ-7bLN26 zlyeil&@O9eR*|qMBm-%OvHFBHN+L@|b7wR9+y9yQ)C-Ju=R4zkkmL+))lm6sHb@t5 zhLdh?LDvpVCWvUUD~`L)B8=}cw$Y+KCBl{VU{E+zvPNM87V``9_@1bo#ct#J>~C7$ zH_KQbYo>}qyYPNxck=$cbLHpE7GTA7fy>=!q4Ssw>i!_mbhO^#Hd&-B$&f0hsALUI9xjGyA{ zykjQ(xh4X;(07IbL1p*n$0cBG_fkQ5b!fY5eF6`Q$0+qJnTbQ;>KG+ zoPU1=9AhBRnIX4wc>$Hnz|4#R%~lKOY=~NI#510)<_f9p3^<)oKr|#YLiSf+q2pxl z-9v#|v{8piHwncZIi_4gB%#DDKMJPsddeka3T2VE?T%R1qyP_M_4+j_^0vpYOt3s5 zz|KOiE{>cyj6;!C2FkL-`dXDgzL@mTzM&Q%#>A&4 z6vQV|ir5<$^ZC+_2X(0s>e2<-N007*GHd^skJ;j1yEDIj{A0*S64yul}m%NyHQn8|C$ z)4x8gwL3i>mu91wr)JobD6%+^_Yv|(CN4;<2Jm-;q$pS)>m=16j23~i2JJ+9X@QCZ3{L&vlL5AZ-Tq-0+fb&^s%mZtVSU ztutI)&bZ5ZRRD0xh!{Oo<>GKQv0fkepA5;Kq+8IVmt4vSiuNOS-rCabjQ<7)pXx3r zRF#qc8v9r`QnYayGk4ChBbCZPQqgN;UI%^bWS&>Kj@Sw(`$n4Ogs8`NityJqiUqg#WS#zrm? zEDT#YWHZ@a$*G6BHwDhn_8^@22(M zfFf3&z~jXmY7ZFfP^n+HekzI~4Ag}S7nqaA-@b8J9~VJFkRd`AP}_YQd;4a{iij=7 z(a9+V8X^QC;W$j~ab&8{6Rs)v^<+vnLFcD(8{mfd@VCpLI*|xspXR^^fkJQP-~g~B zLR?rx1mT5T52ga3JtTs(rSV1kHCgxA(bY#M{*WNdkQI0a5DS6N;I)9%RsZu68i#hN zCzE`1vd8%u+ya+6B-&^WJx~UmuU)?`2`AYoMPclK=qzfup{Y8%S1Fh5BlTBj#P|c+ z(jCOU9Y)vZ-Np4{%MzvM`OOwB3L zue*p#amTadejE`dCMJ*NIJIT~12hmdAUQP6iYbA3XaFzJGZPa?G|pZ{McmGNE0<9lTU*5ZcuJcfNfIXr zIY|F2p&CQQo@_0?Z#pH0!E2zEGNUfSD?ldkf4A6F2YL7baBLVHBnK2sCWsveDL+)a z=U%1?YosV!opbKfSb&HrPYZM_4IWDQ3ED2RlnNZAmb-ALq`k*6w_&eMlL-e68q-f= zb9C@0I0&G1|MZE`*Uzu6tLss+@B9K(o}q6qNCpb@IP9))(d+5yD74U zi;Yzfigmb6^U|QhhFtZY2eGXwUA%a)H|Hq_>{IMDQX&lkk+XwiEuQQ?I)$%t5&<{) z_%}V{ggnkpeqt;fy|z1@r&4(&=Pi{EKN7z_F-o~^AaG02}ehMu#m6-z~Qk7ei1xSh=cmA3PNJwX5IF$ZAr1rxBUY0CpXf&Gl z9Z&}d-kAe7vL1YQpvQUvg#=QJ0yK*0W)01F2cmm2ruib?5-Fgw*?M&dJ6GYiD3hnS z|4Ov7&84@Cw~+xJ1bBfnkT42~5ShQ6QYOS??N0Rl9Z)=R*d<6-+<1pbWA zbmDV+450cy3k8h3!7li;V(xF^ump~d83~wW(>}#4@?ZrzsCFVszAK#;D(Kj zP5aJ-o^62xlLd>N^Icb+@>rgY*#5UB!0hU- zg76VlN92CJnvqXX%IRDCS3xuU4tI_#od<%>qCVc7RzA60QU`0Pt`kBP<1eXZ}m<~ zGdvpvR!5g|Fj?GZ;Y`?KmfUiUf0GVu#s+|I@ESZf%TR z;44aIMCwg2!WWGJc|@dd0pnYroHpT%G?G$`oo9dp&@ldX#lAf=y}LelKPY9 z9um|4(*guaAhHD*spRBqkz$`@jTAss zBBY@SMchY|bv_(A4_LTl;+D+Wk6eWE+18%f_oC2MCRZo@fK&TJn%$b~0P_}RzFtk(psowv{`!OrTnh9m_+V{|X&(9g8^KM0^HI1Sgv&q~ z7gSCwzlWLv0;xp7;B{{9pOdR#N&t<{AF90}pv_AIh?xZ3#TIt?Cfl&cmz9*1ngm4| zQopJIU;QwS>k?GKMmWX6C!T?_vKib3GaaA07S*%prB$79Xh}!?!i@I5{y6huIHcdZ z^;#w%j2B?hm$sWvpxl80SpoeeeYrRWf7;=Bx)&D#L)vMHm?w5!Wrq8N`D4 zeeyfKhEJBn%`3K9myLL9KqvJW3N zR=XWTM(a(NCr4yoJ=e`kfGYzKLk`%E)4#66GEGAs#lpwuHLF0HZ(d$9u;7ATiH-t} z5(_b3fXM`BQ~_9>ES37Uwyavg+Mf?_f`v$i!R0+QJKK0QB(QZFQ7#UGlfvzWJNit5y>&|;WQn;zi8if#t zumh7IH6mtfP~xGCBWX`qSQxzOTTjOmjai}dmohhBv3;a4Zb8ftapPQ0aa=XJvY;zb zgbGp;8p?dU@+!(|XHltgyITnMU904_va<5yH0jWj-=n&!m?-eL`+@vS=_U+0uLTH~ zLYp}`1=R;qWHPK#2HHWbdr8#=M-{)tS**gF@H zBm{2At%Cf@C9>~u;1&`$Uw}jimzhXq3yISe_>4mj_d*5u&DnWRMdfL>AQ(mtfI}dn zqqFQUY;0;m_`GJ=I{rQskp{a5sJOA5K0OO>H6EMV+}Rv8>$l#j8yb!v++I(s`+?sy zvzBjR%J0Exex@_@p%YU}fuy~?y*~Pu)*T=xz)uZsfUAgR2Ah3Bxa#XRd{r7~E3tQ; zNyKRfW&WQ|ETw6||ILX-S=j~RVoPZ=K{rx0^@jdQmdEYPbt(Q59j11r}XN{_u24cFaYPY;fk3j`u5i8MlCTM_vb zX@Ni7eaUP`%^FZ$e!?xBew#uyhfPY~$1pQjp4{s2<2i%XusOd<-fegZ>SH`{tJcgs zfZ4_VEE2NY+E^wFPsK7zUN5U5RIk{{2AbugJ0Ro{Gb_Xg0tBr5rhk(tf+@1iNWJ70 z&A8h>A`5(~^$-h`YZw^0T$;3vHF1}^j*VaXv35;x$a#X7XE;kk`YZiDs<-+>vsW65 zUXCIrxCys?niZTf@@*i>0>MfP|B`?G?!wE+_wRiXn*_+RhCY%~YBxa(gg&nUTzX&Z zS2Y1#3q%|-;!!;znDj_Vd)Dy)1NU>OJL!V4R7$r@vxpe!f+7aaT>p|1hpXwuuE5*2TUvo$-r#f;)M^GSJvPwta3#DsS|M%DZZ6NM zFb?gdnBstOVsD;ddr-R3q*BRyL{iG-_O!3Js@;o=a-{My>u;mc8qmvHtEHW~=ZK8JYW z(}});gf-K_Q6mM%89~%4%t#;nPj`s|K?-CqD6<+_T_yO2`d#?0_$~MTw460YtQY?!2-Jxh9 z&ZYjGr-}%j_dOCLp)BDqg1aaQ1?x)5e6l zrG>$ZKsiZ+$XNho4B>GR7l?#}#2xn&M|AY}D;##9xNDZY>QbBy2D-n{?vFCK;|-gF zNh?)-0^9SzlY?lc04lg4CqTG=eRB{pqo@$U6^hzW^D0dZROgp{MiQ zzoKRTt#|4_5j(NK-=!6H^gK9lq!zNr1s`t%7)d~{OmcdZ=IR{Wp-JcF=ZUF{iT=*N2Ul=OpLU%YsMP&!)^rCL7^x$jqj zXU5>D)$v`mfAWjjc^ewuNz>_)}NxQEzW0mm~H- zCZ?LHw)pVD;pM1`kza3gWlQjRouNn{z$=k zu=M85ER|q@ead42>3ya=2setzR0zWTD_s$5yaoI@<8gkh7cpZenDD@a$4F#=Bvlu3 zf)zOXc9wp1d`fzwd}s$Xb%lbTZt%pQ(AE~0RVDM99b-}QpjiZE745!9h~#8H2Z~M< z$K+OSdE3oDlzdF8>2jo_<*!35XV0SmrUDjg0H+Gf0Zobx8-1%eKcj;AB37Gt0iD|> zFn16Ht*7hZT#WiR(})L4N{7W@pl@!jn5yLdpn$^-(rxmNGtJscp!iY`)zpZ`oowIxC~dyh zC3V%v$dcwR$Z~+ApQhe@^)Lk*YV{&xJSYh_)G<=VDxFARPQ@GS2S|Aq;US73f+?nG zb`5VtZs6oB>`$a;S{?*FzA@l?b6|mDU77dCUTgI2*}X?6RR>hys^vMdznKy6-9W8x z*68_jdH`gg%Sg$}Vi|YFUI3Xj#jtKqU~5ol|9A-D_rbT z=Y`hOPKePFY$=S$1<=`N1qI3Xr085%EEOQHwDY+OPA6zLL{7f3 zG8UZC#hd+-o@o-hx3orY$HR(7_ItLnw7CmBU@1TUbbbxW)T!Ww9K>lkF6;`dvYOjn z#F(dE>BtG@&n7H{IzjJ z7?QWG@g8<@swhZ_LbVGmI9t?|XX;7mT~lZ)StZGK!;T@n#|k;zK0BT`PlXYZgSCqF zUPvhmWScxPpv(rS$ zew^2<)LM^+TW}erzOk*UJ~cuttkB)OscPcIjv4QNE5F#h2P1=;>zGYIBG20JulNt! z2acm)j_QMp;EnKWM9*2oUIkWDz1WPjJ5eMAq-sMTqTkAuwNel%y@-R?f8crH4NZ0m z{13VnoaficriutIaz6hp&^h9Gb!Nf|kC-atX!JOW>vqTHz9Qe|q68Vn9vym;LL#BO z6vas$COEh+#I6ciqzsmF-UT8EWDfKs_oJI&p^5Og+x}zGP7pCdeB`yLhhwt|Ez490 zN44ieW)iSeM*J!QV=PhtfS!PQCE57GGAow)6L0KslD_MqA+hM$8vAF0Y#a{0j^h&T z0t|(-=*3d3hvdn+h|bc)6(1_22hy%HA^`oQ>_Em=~`B5=m+-y;+xK2;rP`p1H(=g;hCHf!!RI_(K-Dtuk8gV+1wzrO1d{*LvqXQfKt z!dLwdvD>C6FaQ>Yf5vKj6UI7KDoQ@jjh^Qz(5Ynf>WOp|QGN%?Ib3z)S7ImdS;BNK#nF~|^ z97GsHGQN5~WTI=Jw}EL;=1e6WkX7 z{E_0|dJ&lhS*sI8~6tn>n#&l6um}c!oYr#l=>+X;hex>6;}9r3zqTKNt()>HbD)~|!^0e$AT-NJmWJ3J z%XYwR5g=(VXD=V{o@$vUx1H<0W4-+JXTfx2a8fdv*7z+?S1U) z@6Q038l1+>Qw;%|V-6E_zkZQIEtdiJ5Hi9GIrd7?`f8@hrGo{Y(vcS@k3Nt{k!`#o zjH8R>6@W=z6lz6f#WlaKEv@h6mnYREbQKhiL=e^piGXAxm>dC@3^H7ekB@I?{GWV8~RbkXyHTEO5%+7F`w1F^|qp}?wzFdLxX0tKT%?;^xG%$M)G7s@6M?DqUA zOR3!qXa2a_wW(3bQMrz~8S}fN@vDcRsiYW7MG zjC`9xYiIi7(`8UxLm)hmZX{NnQR0!MO#hC(>)NteDE^;t)z0b(`qyJlH3Tj*Lxlm-Q*H zD>D&dr0f|2pc}tDm!M_a^^|P)p*244Z#BTOl?3vF9)M<<^7x*&UqQw%Auqc8arrx0 zJ6}fWKmT7`k!EGPD;6l1rzpaP8WXB-=3j6W6NSaCtTwpQ?7jqC38of!`{etfdcnzG z9l=L*j&IhU8HNX8pBn}T$YWGv^aPp*yUA;R{s{_)={{uQ1eS+K)QNKopnjAz(gDC# ze?M$b)gAA2X&U>Vy+G7A-Se|k7<2yXKY8}-S&y4yTHif}h}pqA%t?R8X)RJ_lJYYC znbxUJ)KskU9^mJRam_V3-F@hs*E_iX0*&Mk_2o9UHQ&g4-metKP$)2mBmQs5n8?E* z5bTCLQNhlR2W)-@?dlj*0yd1()6?j``G|;=9w1=|yyfF86-2ruj(_km9+)KjA* zCLj@EW_V1A6(@@kC+mOj#WiNCrKZH+J; z`F;z~95CJg9-(al0uUwarZD&Wyh;vc6oGo%I;WVgHaT5-mNpE0M3XQ_acBrb zQ9!$&y^FsHBah-R3##A#{^r5XPesHU!I-J6!8_31jSuWLjG1tnRe;5GLC=pUND8K9 zePEaw3|#);M4jhgh6x6xLWBE=st!!tjg9Zc!Z}Y1q=9`^Dl&cq>EPl)&EA?=X4`QX z+JVr$YZ3=Hi&;NmJI@4&5j&!9JziH9s#LgA_PV*D?55+u0KR<4wNA4HAHl}5->-Zc zG!_U+oLqAXqklz#GDhzo00*iH^)yLz@DTP2;#y*z(2vHy5bo_%$bU9P4E#=Cz>GZ!GJb zQ^Kg}1r*S)Dc`@7!Hj|oD0HX%MZa;QkN_|x zQWh3@r=e;1)ch#ajT<*mRwm>u;HEMzx{8d0gO0@c;s+zJOoqT?fnwbZ$OH*nl!N=n z*GD@q7@3&pVR#9o` zGj;*_s*Bo|P zBJ=E7%Y;OsH)AMmkP$VQu9=;kodhb3a&sF7>%(AR!-aNxe?I`miQ%Woz^l#ZD}gXP zF@q5O0{CnI1Ah@?;ou-ZY#;Rv4eK3nTx|Fb4-eJ9=wAT!`JX$QQfq*x}aozR` zByBp!@`{SS-dn{Ru&9GEBqQv)GX-p9lU+37IYutSxcINK{}poRG*%rfn@FR$ZT zgdMuI;24V@uW$$K(lgj#-sYR-|Cp3SC#B?jBY1Ij^}eCubzBUz?3^4TT3URiE3g)r zRq|9s4|3Ad&;V*Q(3%vguav|)pwDymBy58jA2?I5=4NS72wR?21qN#WzRov0+6Aq; z_qBV3=sUZ+za1R0v9Sq3E(1$vXObA*lg|$$%2|mnlsauQ1D5d@$GmuV3Fi z+b!^TpaT=!iz$EX1mPrrIHo8z3Tsv8L3XGX=xF0ahD z>jU8dTRl$=$YQHW7=er_c00EJ^y!m6n36X3_M*%2;mfUB4=LfHEh?;0Zj)IB1v09t zq?%eip|Ejq63WA_Vcm&)sG#Fx zTDiQmBmqTH^l&!vndPIgzCJH7`t|^&_9yVGkf5Oc9f~hTZSRy`gSiZa%2N}{ULYaD z+fcr71%#{y+Qs0KM!R2?@Pr-EYR~|Gvk#AqSg(x=AmS}}N^9Xb0)rI}w@&0Xme3dO z6xOx9x#{)G?DcPV!-$2M88lRyuHmOsD9X}y@fB{Tu`QK_ooJrm3!VXorz0pJbdK8a zWTaV}L$tw~ilkp=GJtMc39$s$Rj{Nlb1ujd{i5rg@in>1Huv}yZ&DH)Z3Z*S(`GqNsS8oTTsQ(ed;`c zD?$JevWQ^BK%C3SSKCN}po+ALpaXirsrVayxB=9B&?I}oxdNpfv}uyv-Cf|*?S-M2{K5hhRn^FDL^$p`Q#C@n3jrKP3EPoDUE{mKRPQvAw&m^yHD+6=!y z)R(4EFlfsc5ZH`&W_KKjPzLa*rU+n&_x9}L;Ylb^ge5gJuEMh)Lcv)pyxac-&X4j_ zCI*JffX=Q#c}P!Bulz|#R3?Hi5S|P@@o}v!I8pis2WKGnBF-;xcZ0Pw1hVYI*gN7& z*-}zG{)m4OqC&gE9uJCcB&3zTOzT#s#>3-da0LF=%P{1M17K1JrijMoW(h)cMrLM) zYuD}}@18Y6_7?1PFc!R-qjxHY+;CbmeZUHI)G)gUKmWZ1{X4GX89_1;a#(8wHWi*P za~onSRcY$2q|X-Mx*NfS_laOC+!ThS>-6tI4u{%DAIt!Qwjct1%9n!}p8fMD2sHW0 z1+jAhU^f{5{Q(VIdwc%Zu%Ldrxw-ApBzk*ad>GG9#t*`o(_u{;_PH}&;K`D+f~uAl4hn|2 zB1?nppqW6Sppq%*$Pl8(K4i|w7ngF_+1b&D33vqX6!nM-r7qan+12Xb(QkXll&1O9 zl;lps6cmeJ{Nnh|wlM4+QOQ|ZmtjVz76kj=i3tL6 zLZ}DOPQjBK82>Ef3Palhl_h4CkIV^rE!3T*yVAoP;2uF{L<2 zv$C@47i+@o|3)zet+UVELvj|Gc?ZL_55CYKm|W?D;9+<+ZRRqR05I3QTyW02Mo zc?P-fLfn@DDHrw!^G7CM>+ab^AN$j=Yg6;GHikVJ&eG= zlMw<1vETf$Qhl*(2qc&=Zam?-i^WPO?Qf}Vj?-9mMgF&LZ9tjE3LZlmwZuJ zsG^{tfIx#!n(}P#w$_yf;fheCVjR_Y&6-o0nVIB8Woz^kbTrV}#2=(h?(Xgw#Qo;W z9@mZRdGqdF4U}I2j4L9z1X9*;t|WaJRO`sE>!n9X#q;*9BJ>)NF_Fn!l7i@J30jdW z+%s&KT3A@1;0O*4-Tb$?h0}c)x{hQ_cLC9g$B#QZ%h}r6HobUpir8vcTOYsWf%V@C zuYvwcKVN&SXjY*|Dx8gMLAG;Om9i&3VtaeLm$!Gq<;(Q{q^1(uJIB}&Ryg1R5noA< z31ON&@Vx)yd#43yW}dQCefsFai>B-J*N(Z7-gji>we_}4OH+49!hl{OUUUc#zfHc( zaJ7EIYf}&s#1Rn*OVrhsi~-GFXw`7wssT1epDgjrdhkFq&CEVIb2daFv~zmi!*ly~ z>$$n9-vtf%HRm+^bD%tk)pI${U@{xWpRX~8}iqtf}NClvPZLN9+_xY`)^UKO_p=pbon+KX| zP#8@$lqh}DxHQx_7{cgi^{FmxoO;ZW>2|sr^$_9*ol*Mm#D^fAaq+s5E8R0j7W$A zX{E@mOy-r@m!f0GQpdG_!8)du2wSVGgP67D9j$!Cy}=4VqCU{m&WN4}tAMw^jYo00 z2)n-1Ts-ira%Y`Gb`BG&pj4WG$|oi~NBPlPAC4Tg@$;Ky-JjSlEso<%bw;qqB(9KG z2;W3G@4@DD@${^{_wd6%5?<}xv7-kINf9hvy{EVb<8qV2w?}N#8}{G1b4L%pEGW4V z?2ZP@8H7K?kAg_vYG_C$m65jp1*ZvBz!jz3ZD#}mhGejHPs^?S#%fVs#?$5pcYZ%M zevqcohTM7vC5L!w20YCk@HAvjilVT~P(?iIdIL zfbvDd3(tn;;TfZJsRU|P)YVz#1IiHl5W=s~eGu(Oe^SA2e=M6{6mP&x9RXqob80Q7Tm#*#n`ds5iR0DnMzOM6{c= z{O$+CUN(%5j8-|N-9lRyOwgX7E$-#xljuLwxBS<+c5ZX$Jz*E7^3j(7!VCkIZ^>W~ z=_n?xs2I}jgtyws>K3D%;?lrsS-hFU&E_otG4fP`R!+h)Ev<7S+`;u9JG$rMK`&*M zlq>-(fmJKf@e$#=b{N&868%vI@~lp&4*%S_Q;!}!n(Zg-)UYBogKGIJ{Q)IL{Lp0c z)k3<-Dx&vKGmGxw@LNE)W$iF|7uSJ_iqXEaX6&cou zz4_y31pC4S`t*g!6v$2ZqYR@1ce}YMB1(>M>i3uQVpzR~*c@>EyuFtr9|e@OHMO@e?WtEkwb2uE}aFA@yFvEN}nsg`t z+s@;iBeU@a9QJP5&nmzrgd{y0cGJ^s(CilYRVIST`i3F{Qh@T;b8~-q&tbFKzvbtj z1`hy`LJ_MdLhdwX+S~6CdmxxO!gzdjvgaq$&V8)za{A9*^wb;|kk zu~-G;)aO!De@$!?j_@ECON;b7Vnbw#M0XM+HH(vp$;&g6&Z~%h|5bDqE+T}K2ANLp zbXDpdesEI>o`dRe=bJoX$Z_8u89B{sIhJc>h!~#k%(R^Rcv&mQ!V-EJys`azw7$uMe20|L6QI3SBsw4)#VmAz;{mKC5BBht{nDh zU@dYbtwvTOSMY9BIrt5e*$>Q1eq=!Rw*F$FEh_u3K1^eAJze^frzgMRK4GARBy?>REWV_9k@eeZ0 z#?x!<1|`7ZRR>D%huD+<4_~OQYyNzrzkQrJx$Z5od4b0yT`)DLTsL_z3mgMt{|}Rd BAZ7pn literal 23917 zcma&O2{@K**EW2mk~#A{v zgE>S)LpFaKwbUZMT4;X6$eY^981eb^YjdzPBSQldTzg`8~zvscApSB%Nd>=RbZ`CFkMu&WP@tbs8PJJI z{_`T;|F0J%4GkHkC{#iYQxt60rCqYbQ@aC;m7Xc;XkM7g+8{w5B=*MF&*pP;b5CXNi5Zqy-;ZLu5O1l2#l9gxx4#&M!}h(@B~1`8X$d-Zd~9h+z?GYq zuvnk0)Nrq3^4$Y+rD%2;V{g0l`F>0Q{L*e$!Qlo%a`LHSl%I;+p;lB<67<<}x|}Sb zH7lV*cFwS*AzAXSaV1Y{YwNK{7V%F9cMtY`(E~enoIN-+G&CTr{c2B7&r?}n?|uI; zd~g$&B+i2B0XfTqM~^7KeEr(k*?9?l(qTd}`N@{ZnD}@?N5>_C*8aj93=I56@;^G_ zE(x+FDyB%ey>^|`#>h8^(;d}-?ZpHSmby>wVE1sO%3%hE1Z?Z}*47iB6TDiYwgY@F zm-n{UFL4sb1(VS6p5W%j%`)O}(l;aV!Ef!p3c*4Wp1V=@KJQvia1`46dV#7g_;7_h z9V`zI;mXQNTU%QIJnM)E(wDmV4;mWe;Yx|wda!(kB}At*Q{O|l6~ooQ=cvr8p~yK3 zLBVudRFp0zCZ?^U!+UP!$rIY=&!3+V6QjRb?XLXi_m89M)=VxgE|22l9~hR}aGpCy zKI*%9Gp6&%(WCIi^pYVZDp~jS`NVW}V*B$A-dgU@c}+hpl32%KseF8VoHl>mjP2`H zy|w<{Oc0jbd1*uj{+@utD#Y>XI>AGawJ$0xJSQz9gZ?RN{wqA&ckd1p6BA>|z8mN1 zka&1_NP8})HwX$0#HwdM6}nC0y*{tIScAj8Xy6HL!@vgJ*<4c9(4gSe4Ohxi*}0nC zaRt-xF4V8WHSfaX$39d{OksX|+behtPPmT~aq5Sp4@{rL#$KBef(`dbJzKxbjtL^1 zlr5)ARCZfPxAt~t{4Izq%OtpsJ{!@2S2rVmelJgYo+?X{pOBOkk>|e zrwFmxapgC7jZa&`?84jv{9fGi?k!s{h*b zkeJ<<6RxZZ7pGh#X-PtigL->=uT;4}&cBqO(>tf`!}dbItYqOwZ75?Eg{rD5-;FXd z%!!0f&V)`>q3T@O-ud|9L$G0?Se3DH3WM$J4FO?c8s0edrBO#cy?EnE zQthG5$CI7V&lbA|Yb=V8@o_U0RowIRvL z$-5CBKe`?K_#r;%)0~&1m(NKbLCR+G?9bxjV)zGhH5p&Xernv*vHN~*Qf%ChXI2k0 zUDGVoo3T0!5t6Mih~0ar%n>59=^3Dzq`LJV|->NlwHPqtS3uVQ5CL}*Vk~FM%GjKDmExi zz=R!&!8@o!uc+mjmYFMfRsk@o5Dqx*9(T4sNvqN<9%Y9A`( z#8Bnk<>lpuRu2D%1BEy0g)C=2rC?IU)v+-|KROcxTn!u@9g{D3XmrV#j=Z>>7#I+M zah>fxboTlyoSGWGI|h%(tCL4hNw@NZ<}$)Yq@<+{PESAaknIBf#>NKY1@{8u&KKF) zjl;uxajj|6UV*P)pXW1p^St}VHEr$Rj>(W&!=R1~Ms;3*T=ntXfy3I-b2sv|PfFk0 zxLmXwpAI?pOQ|gbAq6WQs*-tMuGT*ffX>b(b+VBU%LQl;-)vUfT3uCzm6ZIG-TEn6 zGH7hf)Mu?nb@ofKc}C}TfGc5PVQ^b2MYc}^W>lR;n8^nU41zXC{p-7L$3A{MHa-1f zQ%s-0-`^i@mI4*#rIA`F);z9|ATz0RdNE?#{(QxT05Tq0SjW<^vOWwB4&F^0EHa56 zsl8|Z^IPek!AaT|=B-isHJ*ZR-n@D3xnc%SYCKMm!)IAx{|*3M{(}*kqvw(yJqnoq zn6yw!9K{ns4Qf(U93QxFBrePNzIjb-@52Rq3og7E*`qol7-p>OdNx z3HPZ}SpYi!yv@69WmPZo_FKe3y=fx>-4BI@Kg z_g#FKjJyT6lIC@qh-O&ug!a3PcZItkK$%L&<2y!1*e@mUH=}I|p>qpBQjOPXgxcX^ zi&NGQl8kB5;}R-q=NxzYHN>Q)rB&hcM@Ugo(P?kn?eC2x2J7Yhh+%!*&MEZFu+TEr z{C;vuO0nff<_{?s7$6i{rrKj;5);q3@|c~fc=o+wlE;!zhBc` z3@;^nq3h%>is|+rxqA90(=FQ!tD1{d+yZ?g6R*5hav1TvS2UFDQAtv zum%8pd-y^-&(zB5lq;D~MoT2iBy4R)Ssw{hG9R)^QWqB&E4HG{@L1{w`3+U}R#V7d z+;Qs7T#|Y(Z`MvD zKg&laA_VFhc9!a+Gn7IzjVfJ^4zzbV-l-pLni@X!^7Tq~V%MinuiB&^0B}{~4#Z&v z=a#PL^eSMz{ETY_0V2Bji8Z|b`7udfd33Z$G2^t0&#mX$X8_Rj*T|xFu-e#Ng~Bdn z&7a&5Ro~V(_=k@Y6Y~ZWj##8Hi`oQVNfH}h9)IV#yCsPJdR%r3EEn;&s1IZc0H^4a-WF5xy$%Pe9YaOT>JF<3RlRxT{|Y|ba` zcLQNpC>oOPQ!|CAx~L}<6$yyhf76+&sIjHIoL#XFIJu|f&NXfx9<#a6*{_R>8_Vp! zch~vei@gmW(2;fdTGOYv^3s1U7T_=i1%+bkX<=$=>h5~~18nPU3yTH-t`-ZTy0t9g z4r37c@1cIKY&}Y%E~>T_~576mtaqxu!6hQ&*vc`)tqU)VPP@@!?kzUj-JDMo(c-W%vzm zV{hlof|duRD3pQb`|^T3MtvC|ku^i1$mq8Ti2|fckBfT|$d<(V_ifNKTJT0iMKKDR z9r=_d7Y*#k7_6BBRP|!(9_;fMFG$#;@%Rr$fRG(_y=X3M{GyZ!H$b<11X_ZVoDvd@ zN#c$zP#*#BQbBEkKXJ**8qZooetiLDo1iq|+_c}Oq>zwMa1@sWq{`;!{WdFbxe}6+ z#B3o2PeMbFS$;|mgGZeXn~YJ!nsUTzhAmmn|DO-BR8;&ne&8OGrJnL~*Ny;vKs>av zr%s*1<9wL3+8?WNL&x%Nt~W>UR@0FWX>uHFQm*nA7Ti$wx>xY{HZq@5Q(~#Qa^jg! zsXwe}APe)BO-)%;R9BO-1$8{tD>S?XB|*Q=N3zUuGz#jYIRFs|a79(vE{C$8pBcMZ z%FB}M?<_z`N0Y=#tqoE6+F@8~aj2385|py3=?T38eMlsnt|C}0wS0&0QQoSHAyK!) zAROqfxwr`N>b~T5CF7k$9WSCb+G4oyk&(*437{g7tHX=(t=1WJzZb#PERMV>p;suh z1W!V>HERx6Y&}7M@$~WGhSWRp~=X%{$Q5l)x=Vz^C2HsA2 zALoz_hF*1lp>^nYoTbdQ#l!!~xQ{nXE)L}D7h~ZehXBU-kSfFC?(Y6n+>za;|K;t+ zKFpDX%#knJRHEKA+AY5h?}F>Q-p8rmYH7USU*D9w zTX-s_-2xar@4|-W=4}15x8MrPybTNK^YZd?4c_v+GOi^FQM{%1N4+{#XL~m(YvPsC zs1G(qt;jjP>_9!sI7!UjX{wEiRn~_QfDbN|g6+KRK*1otm6~j+A$hUB(d^deA>#`d zsPIyUeE)`AqiLBfR|w{xnKNz-b8r$a>e`PX~ouQI%6jEk`2-dY+>? zMaFC&lf+wr)smDvL=_WqNh2;RFh50sJgOAB1syW91_=oXu!Z>5oB?N@THQws6&3O1{v@kT z{^DtLNzm)d!zu%K?B4 z(8^LF&oHNqdj`E=vXmPMfHS4e_;b!H6Uqqn_0&&r%%mIM_FLIDfhRyl%fJAgN(XDP z{zD+g%z(pSl<^ip5F}e0A-_x^fqiF+mtJq${|G>mBMgs-2r4Qnf)zc|^ZJ$&tf3Nb zT&8ZmUWMOY73zLEO14f(E?dT~l8lf&X_I$$cAgz9O6wNv;08Vl;1Zpr27iJQkMQsw zf&P*%N{YbrE&oU`$y5j;?6xVT4BM(@qSCq(QJWT$%BdS1b;}*qZ~3gAv0ua05G*gc zwK}*XRk8*{y;3x+J|EvJ*z_*79c+0YMgtID#dW@q7?So>K@bj?tyVZNpQ3eh&f;Hh^taQRU$>5vYmlnSiALiubd@AQJ8|_L> z755n3Q^AR#z9`kt2NCyz+YrhSpC2eZ3wz;hZ@s@i>jif*1jwHXnmcbCAtVaQSLy!6HpN04I~QX}$T6v~nV+m-5Ri_8pgG(W(CpPV+_p?3+3|9xvA4{rtwOh$CZ_*r1DcM?UR# zsbS&B!t3XIRv*{(UKxeKGIIjv2k0dFHKsf8cl!?)=#qj~S*mLhr&Fy}>3G5N^&7}{C!bC(wp0_1|&xBIuDa*NVFCzu@qytS3GXqhjLE0xAS}Hh`lNh(eaFYfPPo!y^Sg62Ie_Fi2Alicak*U} z3bEJ-L=z7^tc~Q={bkzfE+%;titO{y`u%NE1qB5t=x=!4=KBY5SBtJ=sbb@@9zXO$1G+Ci4L1T3EYwW_cD~U9BbFv2LV(z}hxz0RijuH0PIq-uyn3|48!kFs(Cy=70vopFFt(tD@mSO%>2A+P%)jXXF zmmeg|;tqVSw3$N6ym4mmSl?9j$R*kn; zC5kl+B?%M&CjNUT(PB65#xX^|a1aH&M}@u#3vj*Au#{SgP%9B7IOx@xltL+*zrM9X z&y5@3$+TjvUH70{kdMIXIv1}z;6rK}8Y)Uk^r#6a85!~La@6^P?$!vP4`sMa?Nyz` z5L#WH-40zUsc>cJ%QFc8A+uk3CURfxeFAYzCCFdzq3wU|{?iZ`=*zPbkd&u)doK_< zy(Ec=jO^CpBao}5q8sbWyJp?{A`;5Vtenx!o6exRq1a0E+dhK;Gt?FJp}Z$Ih}?gy zOKd|TUtB1r`2R;(vJL#eT=s|aAeqlJ1mro2~;<>G5f5(7Re|F_t`mRBe{)%~5 zc0Et0X<(qNdqrU2+FuYz_BSgVFa4f8X1!p*5#YRZ?BMpR@3!IIprXyn0YRvdDjfzP zdTsVJ?s*O%r3pwaxUN*0b=eE|@om)Bu7=51Y9s0CY)ILfGo488WFAO({9*;b6b`&F zDO<>X7T|3}yTwexr@c2TZXf9Mj=$yz@Co(Qz{L=9aB##H#^TjZGo%fNB-FA9R>@tzpsm45M~_~`pCxr2wGZjFO#t=P(m>%GJBdHYHJTEzflFo#KZ(3s21F`MR5JYkfMKi0f2Er6g;Dt9V6gbZV0+ev&u)? z$89cnW!9v`!p;V+ER$jJd--udWelJUC(5JN%KyL*DMYQ&91KzvR56WFr$*&(us5Hf zhYLIC`0(L9gk%qt38>~VVOv{Urh~6;GKxE}W@crfNPS+;8c98qHll#9PJ{0Nk}A`G z59l)MfdZ!FWKXrHZOnRbtvRQQPUX zD))^=19Vxn1LN=WXQ!wz0Kqe|Knj)AnFdS&f)@r54;2#R18c4gsyt-8535VU779EY zZy{kQO;KjG)g!ol<|J0$RoIur0Dn9^J&D*t@ah*!ZTi8~a9_~81(B|3Z96tqKa2Lc zZ7ANpCl22ug1+a+FjES?BOa7lq*wwk&-coto*(5-OvA5ZGd->Od5{d*;R|{}(s%&3 z;M{a5Q;7G(2@Jr?_i$SuRYLJc;y}4d%%`6)a3^F=mgyCrVSXN0?tsz z>F)3EZvntv2C8Nf5;Z+M-8ii*)##Rxp?eio<0sfFg*Ua4 z{Y#LrwmuC>9BrH1A7(XgWO~%|f7$%L^tTgt=1au^5YMI3RIBt;;v=1nx)) z#I->|<+}erx$tcMy3R8@=}$5t&hP4jYVAW56N(@FJQwLrop_lr@$!p7lK)PJck|2L z7W7JN`bj{>5vhv*$0D-u^W2=bb{;Jzy!KxNrNK9JbuSs{=^ym<>7cF-LbUT>=O$x} z+7al2k@Mt7@0{x+HEvA!!8Za-sbNYQ`FXp8Gk-f`da3PEHQRHd7J^^1Hjcl$NgsN764)KJOP6S*2zzXS%eDLR`Vf@QR>;g~ zd+1G1PmYew4PhLjWZ`C_maXMv0{#9NDLtQjBy&g?R>^{&m6a8RCs3#F zfYEWX$Vpw@;KKJ$_K^+8G_1V)$##X!JT7kzyAbEFh za}+M%^Cch^SK#O7Am3Nq+kDmSjBgA4!J9U&zAHfVN>Xq3v|hv&ia27q{VgseC_%?g z!WM#ouGRe4P$eRKmgX_&=znC}I@8mD~w}?41R zhKT{r?h9nv7bPKkiHb>D^g}gR0eDEp*icQB6boSe76@)zkYHANHRT#xTUCKrAOtdw z9oY1`?fJ84gM{Gyz3JyFk5gVPt?$+X1bEMlN%d2Q3rizMIg-*2D?Gc96x^3#yPgGb(JiWXe-Qc?ZpvF4ml)K8k3nNf0Z zL<8ELb(MlfD;-iR5gFN3=+@xYTuvHTkD|dc1-%UkVzNXWh8VFie#hcYQWHWHRa{v?u%cvrX+wg3w!VC=m?yb zvKN3SWOX@om{6W4Ht{qQl|e*F@)4tpU&6~-is+=&6B~J2KR|{+E+?%054vzY9zDhUlj?5dn&>~WgwQK5iwc5AKKC%+H_G5_Tg>9Q|#yO3JMCse!AZ} zTs>QCP4)6-ElDwOeuj}Y*NZY+GnI2)H~B$dbpmrw#^kdQXaVQ|`+=<^3W`B4+1ya2 zFmMN?tl>l+fK{o-}I;9Ve^4`W0k@d5gUem}h6lV4C#VN(4& zZw|=#?J|O|p#?C-dflfW^--BsQ={VLZy_&W6vz5M1E zSCU@e>E!_|4y|Ge>unPL;nCoM(o;<>Eh=zNB%0TS@BVebY$`G{Zu+xi)D+*2gZF{- zF6cBS-}mY!>@*!b)E7<=z%WifzlnZGmZSl3m0p4_W(Dm@vu2W-^?@6}4aonax~unX zlP?tSmZzm}pP~cL$-B_w3{sE3cF(y=FdWDbyj)ZA%3)S)J#&}arIHn>grde5snc=m z;g+*krXRN3QGB<(TnjjjOI)0x_TJ`z?eL3^gw5^ZwLPoLGD@3`kT^hw@dsj%1}eBL zFzfdrbuv`e=;im{V~a02(C@06|N8wq%GHXXlo#877XwwM=D0YLx@iVTwM0xR^akppUXUO63LgMn`Mi3oMvwx&%5+#cCWhp{m*kzNnsZb zrBH|(-FqgIRq0Sv*&o|xAyjJ_C^$$eQ&?+U>GA^^%q&byJcp~+zX`S(UHWY9sQ>i+}&}lL*g#I^cSqTEpnDrMm(=od{{&A?osSiN`_? zoMK)AQiFShuU@_4QQJs{JCq!loU^fjTEQ4}IH!78@=`!{c+tA#pTIhaW3 zFunc@N){A`Wp-hlN)JqjDx8x)r4}>|FP_rh>+<97=sjO$dD>8K86x864b?G=Y2`xQ zv8hqgo|+cxFW>bjr8E+o&%XV5SzVnZp8sYz)Q)kmO{fAvI}RksduFiiOts^W69uUA zhN$x@`uOFAWw;^IJiOqa022|JSIQ0UfN^NLQ zn&b028a>Ln&d+rimK2PIawKVdaCDm0J>_sNp!)93zCu=B*C#@eSE3RW#9L=;S$T3`CrUY@nxxnDSjImE$`m>Y^9-j`}O*p{c*9>Ll zJS61W1n9S}u!z|;0KlN6qdN&4Lr}_<9Tp`4K0=J0-L{~&?wv!$KJg>ie(cj%(DB1##K>F3wCt}o=0T6JVKY!l;G8iv=icI8SSq}sE z{6D@<=x~SudV%giUl9e)BB*E8Uzm;q)dl7vWSr#i-xV}0cd+mN4I9PD)fJboSNIyt zeV}sD>GyvB&H}gGo9|#}#~06M5C}lDp|P=X>?g1csyVS_X0Yo!tfx$l(qJp+6NWEKAv6^ThwstS95KzTbLVBY| zer75z6;rr?8WY0A(#|^`EBznsiSlUiE5;d40<8<`;g92fF69EVX_xbZ9;g4!2$;-rl+)E6G3rUW3_p`FNS{^c)Z?#ZT5nJ!?{sF%g z7i_9XumTXVg`Q4R%xVg(_rPlHlgWwfV;tNy6jk3adJOOd6(}nENZV61WR+M3U5|=O z2Vx0{5Ru%GG% mLKX&#%t7M`8IX3FA|fPb))^~9tpFelrx5lmbEX~Fe*@7eTs6X`v6PGrlbT@)5WH4nTwG|hv|Ej_t&j(|Un?B2#=Bamg? zz%U{c5$IN+LAHJR)Dlcgg8(4%%j?MIhK7V7(F`uyfxcvIZH>T6brqH$PzCADR{w;L z7i#RSKY~xn6rN?#&1p!ltwr_w;gDJyM@H(prE$3F4hTqLt55IY-o1P&i1{z2MC4r; zPcMsQkj?Fn_2(d8-s(OiJ+#Tfe9n-h_%e?x>-KR!ItRrwXU+fxMUE^0pw%OL_xSku zABHbylQQtrN|D6f;w@wrF0v0ak#zLYuV!FjJuF?ldul>#KY&nprg5NI zcUEGJS^BX#0py+rAj&CeXo8?ho3}qcS;Tq=01I;RgEk)=MkDD_#revpGVsspjQPq$ zGpznI=y33t&3ZDGt%s{b(awTI4#Y@DdXa;@>d|T!&w5J_XcC@0yfNr}x98xhe~XcX z-X`=SbOBeQsDA%)rU8tOz7Eg{IJk@0s-eR|rW68nlv;Q)pa}P zQrfwYt&QFrTKhH30`<-B=NUaO2%{iycOsa?yY?bw?F|6;@vLasCMX$S#wcTSe{XKK z!f*o&ZE%>v9#m}25SsAQpcWgI3)hWoRl@Rat&-OAt)_ji5cH>*_gf}rVI^(gQPn)C zZM_1$Sr3@^u=RWJ6t|#0K${f>FfBpbuyfNaXmvWt(IwRn`^FHWN)@aae&9(}v`XL> zifb;eH+!n7>e)t8FTl3v&s1FQ^mBDCSH>O7D>5^e9UpK-gGz^&4)KrRN}n2?d|_Pc z?JeLx-**PA=TNdsj$AWLKvD`Y)_44ZK#BN>V5--uC1Ne07g2`(4CFh#MV$_}sh4!K z1}F7_W#n4_tUR}U8MYUI4DJ;(CM=lAJA zdcOO+Qet&sKvoZ>A)8saB`}E|LOO_ z({~%sZpZ*|1bYe=y3A4k0~xSHk{W$7Di>c`H-Gg((%&a>yZDQ?1gP=A znm08!D=+^PZS_V81w5MwFBjEqzwDkX{pfWMWPE6?zfPZx^!6%B`8Qt)80?H;IOuFJ zq+=)w%bWgwhB3&MC(2EI(T`l|_kUo;?z9hF8nDQ8_i_Z0q5ZpU?&8j-)8y)%W$Rvw zIwjSnmLr4Ll2$hi@6`&qvFkejayEYmyBQp^@oM&$Jbq7I^npFgDCxotya$k;Zz#d1 z;!;`D=8SqW=WVmU=6jk`8n}ADJxu-I2sxc$kaN~`}=Q-@~+Q^G9MlR zk~&d7(b^^aUNnn&yM<8f{<}NmaP9E|CQ;y#F@=N)+5A2XP6YyuK87-mOR`Fq;9LR& zbp48bq~zqy&>@@$KoRRFtbY3%Y(aPOeG+jWcdU>Z2P}L@lm&f>`q6#kDn_RAS6rsQ z2&~WQQA<%L;&sI(TSg{Z^o@;;@zK#U$Nu<3Pk##>)X-}*fmiEKv>4Okzbh2Qse-T} z;om~Rz@7MW^iQ=vpQa$dpnp9*uGZC&5alNU`NOHty zC|v(`Y-~zOBtQWcU$|XSaKWH$3)x!TcX%2HLnc6C>(~3qz&u=+xE>ZOb0t-p2~BH3 zYu8idW(zZHM;AuxcxkJDS+&XF#p?~cw4a9AZ-JHmY(G+1xc^p+;xAZS%92iK_<6Fc^ppV zndDs_$XXHQqkb%ZHFQo=v6|}RvIguykHm+@^*(9LN}QYrXbkme`7O<4TItzw{o!L* z;`cFm7fvypCglTL&613RN+k2qN+Sn%a*E|$mI^2>x5k@7fKfSv<9T{_*r>Us#SGAa zn!0+^F%F*q3N|Ssq(bu|oCb00xm3I9w7quc%U6TV&}+PsqWKaTwy#g^Z}y(^krBgS zW891*-uUjgf7qY8_Hc7OI8*($NUAiOgQEx-p5w3)f8H1N&MhOmHEdx0N|%#hRYg-x zt_uS{vy(iZWr30+U#K^kUsM-}TvSo*!%!bUfo=uMO%Ln~G(~jf*>YqszU>Kg<)O%s z4%ipyOj7aR=SO$h8|0^`VooN>`br@iRh{q7 z?~NlmcGwtl%a9%OZyk&uMtw-r$X=3$eKD)stJA3}_^kp>f1AH~5k6G{pkh|bfYyqwBFWbs4%?Qo_ zv%5+-9TMntmWMq?Z%w=plW>{B1IZUCsn)E&>~~`NRsH?2QInQ}Tl>qW1!$z=8^NQK zntU(3x|$ukaEssO__9it=DY$QT_XogaFi;6P&H4h(J$v~)D}WaXwDQ(d9^*_L|GD= zYAdS2IcZRZK*@Lqzzr-O0e}6oRj&7OX72tWFf(ch_Q0z!$y#7i&kg_qiZLlqDa~X= z&!I1VB8Lvjmo!T7`z9M+gb)HgwM21+;JxeD!GX7VUe4OCfNOc!oM>_nfgG zQ&VH$!YkPVVl| zkqDm$stmd&g;@!Uqc%FuO%ufA&c9T6{f1aVIW}7|x{{2o^;+2#{rF;Ycw<~wnAlU) zRXXBCbnIrsnQLptzGf2O%CJBLlQKkocy=*~6)ZiF*ttL_aegVW*f|uM6rO~^$mMZV zB`i>k@|oaNRIumIQ^F!Ef(x?BW62?Yv*HXn_&F4Tqd-PXG{Fc`i1oLUGce)N4K*Dm zk~46FX`oI*w}Q;tFqoriW`=zx?id9gF*CT2)7Qb$^(qbW<|&wGL4IkVKpM~Nv|Bve zNl+E8|6Sp;!-Un4X`07Z6BWz_4v-J`WGWohapM0fAWe5iT{`ljYDb(aw~065pKB+1 zMV*!U-Q&6~o^fzfOi)P@$8y&D|zz!%VhQ4GZ$lxM{Bf+afxXtvxN<^14r(yH}#^Yr#+HjUK6(KbCo^_eS+7!gR^$ zCEtClXB~NJbj-o;Q_ZIb;myyld;*pkgP}^K5)hT}>buH?6(-=&-6Db(R~d@>O=o?RtA`gfh3q-by)%t%rmzaGM~{vm!DhRkkHh$z5#pLVso{o3MJ8pwDJu(hEB z(QDJDY!{$EdubhLqnDfdV2vEC0(||k(s*Q{)o3+N<=-vl91F#% z#p{T_f|lbJV7(&ggv7)@9rcH|r~i{B@?3eU&Ji8=qrBDXSBw0Ko{4Xf!&Y{zS!p57 zDtXjMVNjVU?{Gg`ottAJuJ#{|ykIs-ZtkvBEKIFKDjc_+iMSDdqlj(NeBtoeAC__G zpAZoP9=7JOvF3I-PT@NB`ajb(|Fi>x{d68|+tV~*IPb`}C41<+1-n|%AMLVbUw=1vl=So)yxolC=Y^=9tGcIzBbzf zdK79_dkSxe1413DawCP766z-mxkp~D^YMamHUSO`OlsP%jfu3uGqZmXZ3`+YI$i-; z_JFCxH2e=IkLL9}e@_u(8oIiU!@&QGAzjG0Ji1VF5xMaX@k13ecKO>BoEv~dD-8{e z<2Ty@9z@Fd-69*lsm&ultHiQ1{Qobk6#%}g}@TXDg}n6Bp_M+m(>HnMks*x zNf2_Cfow*mcQSTq34dQSFWI#~yEX*FNUvf1nET|(Ti~dNu~~WyB&4}qlX_VYVXwlo z*lh)^`XShL6J4&u%v|CSU{Hjjps|PE9ES@Onp?ENphvN`FnC|NMGzUur?yhlg$@T3d z%S5Wg`n$Hav=APnV8viqr9CNuJO?fG7&@*4Mv*ZApu|FdWVKeu2?Y8@ii4us>S&w! z*Q1#IBSc|lM{WgJo3nhQdsP02lv`_{Lf!WEw>!sld-QVFB>O z(VyK@I{V$mKBe$x;)S2ZJnRG-w+_!0B(X*je14hENcm@Pd}aS?jQC-3;;xzGRHe4s zE?);j?YgNtGPZE26kNi>Z+jysDXGZmDL7mLlG0{Dx3n((W?Xa+EO@X}2r49vz}WXZ zJ3FJKgcb%i0L<%GIB}wJpxJeAn0iGm3;GH42TAPqu$9um&3!F$g@R1R~(u zH$#m4&6_C_cK!YRx8CEaqP|~*!5KfW9fD6i0%VNkOi3K*NyCrK&of@VdGnm?2ymS! zZ|&~-toS5Dc%!ZiwCt^)j{XskA5$MWa>Nw+6Lh3O-TwB%cZVo10W>@TX4Yc3VGQz) z{{H>DyRR=8EWrpMgIG6jGrPY1DidNnSp{GSATW?JPPoxwu@@zV9|^+=9<+3HVbS18 ziQeA!mR@;JGZFXs*|TTwi_Fc;x`&1yYRc`=`ft7Cfbl=#{Qhv@vBOuFnIkTR5P}qW5Q2aZ?W4Ka!Jo zpt~WH-dTJ*G5P9M@_<%r($PXN6+^a1w&Vi6vmG5B={?q_BFr#C(R|M`3Pwv!eAoML zdZBv;`sLTXQMfBYbtk9uD8s>113u;A%}~6xJUrA%aQGwEyK7;T*%TqGfA0X!84@Wne2GrO4mnUxa!U&xjFGO%)yG0Ys5*BBD94sYwyGo;>9dNCRNX$IZXQ^@SkMTbKX--SW=Tva;~a36~?&f?aW|P#v6s zstjG{#JU^}GHaVt%eV^fBnk|OkEeeU7uRT3d;+Ha!KZefITD!-L6+lj)5TUP`TEL$ zEK+szRbF0uhTOXk=Ba0APQn)g^vcf8hDe}TP{vJtARK~HJ)M82K zl#S(;6$MBcM-=2eJuCWj!-(YX-n|-^JGBZcdHwAZWDau8fBZr?_E!}aP}EOtgL zH#9j4h!QZkP_eN^LJ&=CtxQTl_g!Z9m7vmyLV*&N0*+STG0uj$+crAJNEO6O`0sol z^}KWh%Sn!^`ejHL1(xs)Pr_Y>e1Y0|D62Jio@-Q`9G^muPqkc0E zk0*jc)&xTbreaCO0FeKAs{uUR>S2e3s?ZyEtgRd54*qC!fgH-} zIn)h`5eA*o6ju@E3dg!&FyK(wt4e{6?~7fT-!NfJbu}3Kc{Gyo;Gs~;)^@A3Q&Kt% ziRjzpX8D5$g!+8+mlK7uHHRTKFb$^X*Y5aYGjG&9QNJDSx8mM%$xO;N6^38o;GmTc z!d>YZ8Ha$~Y6Y$NwN)28v@(jYWkGKM$JXOD^HlRyT9TuS?$x~;EO_Y<#X}h15f#lH zFHg@kYsu^P_MW!Z-2ZSmfWg@F(4RklfRO&>QVDyu?Z*&xj6Bf}p^vDnBB{={NcsfA@>vQhhxzlIQ1~r>009e}H+iO<(rV`D5?(bL90E9=KD>7n$ zesT$2Z-&H_1QTo==dnBX_R6GU5WoHUu&hiB8if@3_+uq z-|sTUiOuAPZD9&JmFGfX`J=Qn0XH@*HgxRKBq6ZA@EZ=Xd4Jyu3=QX#K0;=KR0i!* zNCLxw8x2*@FdF6sE-4G@LbU95E{fD&EdC2_!feM!`$3Me#qi@x;lKN7vWaPxJGuXlYUXUR|Z4qdOYZ zh!k!zCLxrVaooiwd-K=9#MKB-5CI^x6vGS@l0i6A({zbt5fPCwNOKcj(ts`zodPui zWE_69_KyQD^C-ddhge0^+GwlJ);YB?1;a5gkCKxg0#+3f6Uzi_g7l(Ww>U+ZBRyBO z!0V8gPfgW>san57v6Q5QdljYh=|m13nIM!1{($ixKaT#d3RDj{8lRg63xXi2mzNhh z=HNNpZl4vai`vxWi2IKNCdR>^jH`O3p{b+%o$?9BHGta0=95jP`4Eo z6$N1@z6y-wX-;h!8X!Ut1!5E^yFY&pgbEGsg|y!|I%*>Q+#V1+dL#DO<8}*A(A8rS z5<)>@|C`l_Ls{qJ|9549iJOVLu=DyrP$J&<$N=6P{A-zLsbZ*a~c-UFRQ zcEN-H<;$0*#rkM^qwR+aTnX&)7%tVn$#h$dKf(O#o7+g20)+@IZ=da(>FH_r>S2Dj z<1WrWfBuw!u~m&^iTeP5N=i#P`S~e;2J-~L_-_$u_PYxME0_joLr*$jQ$X;*%x)^( zfP`@C?bmHL7MyGButO{SD>_5$a+++d>H@5IgIO_>&V0YU=(jnFVI+GXTd0Y1jYxnt zOFM7Ty%F3Xje~=Ow^%GwcYX{@+1Uf9nwy&|@8u;4r@lq}pW(4_XUw@}uU0wZYJ)@h z1^fE@ZP!lIV!CJS*Gfh;{eIKz&d43ixmQk4TZWEI#e02VDe$`X;a2|$@t*w@}b|6q+<{(}G z`+grzU^6aS>NyoIQqey+Xb#@3Vg8xR{uj;ISB+$rV1+M`kdPoy9qM)hJMYNqVl994 z*fB1MKadr3i;7g8odv_g!yz|BLyN*lMhJyQYOR|Li=D5&ZtQgwfS81_=lIcb+4{%$ zA<|IR$Cv-l@Oy1oh^9wi4{#C@fN>t3-hyI?+z?tb#J(e89DL<}HzrXis|yJbn`#;w zDo}5jSy;3T4M_me1qKp8u7%7s4mOj}w6ru7RD>yjv;kwJC{a*Vl2J%bx&FVf-~%LC zfq)I?dqCnk&q%hsxtRc^swbV2`Tr^8+T)>2_xS75m2p|MoyfSI?rjpAFp?tK+N4^R zas8BAavj8sOD<)iK2b@w-0Fm~=u|?6QL?mz7RhDM5Sv0sZsB~N?4Hm5v**mmKVD-z zulIR=m+$2#zckdg9W+y5Qgptxa;vOxPHBa2$6IYE9 zV#MtayH#&(a_pTu8xR2q&H#)fLb(iu(WXr`>DrBIs`4i?4kX8P&J3O;?jdQpHL97v zs6@BNpm{`DM;LvO1FvzGP#=Z2oY=!;Dj?TGd0e-<6~z}6ypG^XkOBg`&uf#|u(PPX_@)hgTcWBg)oz61XzA_tdAs z%8D;cJ~|T@j18w}fIZEfuhVv`hF#5>60=t18XjZ$10ayaB| zvAdXWA?5`{|0?(*?BKZgcopBOB28@fS=f&O9V-yewRL$`VE0yr{6z)tNte7M1* zlMfd;q0sSvWQ;W)=3S7MEy`KF?~223|1!l;zNE$S7kTYO8KFRb5qTH(m44c!5ZnO&8NJB>nO^lR=k7NGB-Qk%L02!kMlG# zI%EZR4K(2@{xsDa#erZeX#vG0CKC3K{GO#NxFGO+lUGtw(pm7D>%T~M=p|=w*w7Hf zFl-*pl8y3|Vfp+TSVFsyu^)&D4-?t&gygK5%<1bx-0_RG=yGZlf`jb7yLGQ&<6i%V zBKG#3J3OL-YFLR3Sjz}?S?}6!Zl!al!HLe!qElUcea(HfwfFe?&x^;FScVPdMMs-l zxWdYB#wW4f@PPXgPZ@MGrv;wTHk*f7r6?c;o+tZfoam!h`{YUJ&RRid|E@xnsfSrf zs^>i{P8lCLUQWKD0->zYdrs`a8D^;+UW$rtU;;{JKQ5(6P66O9)aFmp(0Cs-HxEZo zFq{gKM~o_+`NYK76@*4H7!~~eLR94`0PYOz=K7Zb;~Z7l_#@kUWX1?Vf#H=h^476U z+0x3oQ=68>Sgc!?!#%5m3to0s9vD4`KrpD)Z5B2~Mk0cme;i5p(eZH;{4yjqk`&r6 zp@CvDq*kDtccHO{B%F3guewxqQA-g$dMya>ZKE+NSVf_ zrb{{kyJ?Jxu1GX_i|}m2(R>KQ7twd3n?-@Ff#F^dbvj~@Vo9r%tPXMH?7zB2BsF;C zEIBwvt9-fa$IzDn0Q?0MbJ~l>*A*+If#KRhO1)z1Rw=`qw!v zN1m;%sBA0nswBgZz>N~Y3}G`*UF)0k`?UU7kf$*)*bxP_qoX63`Wf5m_M{@v3ByIq zBE|bd36s!d3c)B2G`VYqehwtF@`%?qFKovB!Gi}5zt^XuLXJ>QcG);>K@rBRn%HbM?vTX*Gyhu{V^Rq&d>MaR(tdpYJKt&h(8<+gY8RpiRK{OHUwZNzmWy*` zw-vA_DCmZ%_!82EEQLcGA{-c?!4*XhczI2(d$S#nTUJ)A zZWuEAIS91882;jb@64aD+7{oQy^T#x0#*w%2_?Fy_vqp4s6EnhbEUjWC)W^A2AyoD zrsleJi(v#hdf`*{P5itI+Nq}-^LP=!$zwX_5dY$T`ib3k58h!JaYF_~M>Y-joT?B0 z_~Wa$N_Iea3Oz?;Q=d&1$7CJWH4 za9p3kAj=~fODIHaonOv_a3!sLtUwv3i8!N<9qaL^3;ERIg8Yy%5eV5Ec{;#<1=$`l z@uNu~x-oX+MQiH@f|rF}+^tnFxljy=Y`9EJnxddmJdN^;^r-@=?G)(=csvFV&|Fn@ zbvZctA?x1S8ofM?&R4Iv#K8OcpZ#`8fCKB_jbR}G$leQzEmx&|Wb4;BYrYTA!?S`Y zdjW;}taQ&SIfGh?H$7O4X-rXpFHUn-ZauabZBVmEbVkM+n-X@uWl2&cqgN2t0pfI+ zKQ~k7*n%1y>BsdHj9?PR#>OaVC}q0?lV{ha$RuL-77estXIk!Z_wq8qxV#k3pbmlL z|6NI!t8>Dn9ASmHZZ)81%#9GGBo($v8ces;5lJ9brmq=`0xu<{BYURd-PpPf8@^J) zkQtlzvnaM=JAs;(LyRHO)vInQNRAb!NXv#g9{?5*mFm$O>{uWLc6B2JooGm%ZT!nh zv9Dm-`FW!7{>CiZM}BWp#xqxffroo4dvk$L-y;mte1d4eP7Z=D`)6iBZncNtS#$G> z2<7&%5DXJNzejDymFhP_Uw!>dH{@ac{;vJnYDW>1O^mcYAUM*tU?yPOju+mSOWPmZ z*PD5933)$@Km3HKT8xZ(Ebty6HL~$c8fM3w+^)oMBtVpFwCWVTMa4tLq-6_P#=R7av$<* zun>H6H(|v9-EzEwo^_3Ta9UQ1GA3 znwm>gDpi7xi;VTX6Jh5%~GQ8q{{-ws+wQD!3I!CBd!Vei6bC9<3^HaX@ zQ&PreaVL7mU-$R*sR-Plml7Mv-!dW@=S@dPM>;YfTR1lJXv;``{P3{r&d~qYH8pmb z7Q4{&)`Y+O^z@vrzCOPPH#xJWF>|LI{jHb`r=((iNY?Nne zsZ87-nrHcvF5mhZXH`{|#oJi1`QJajjF<+6(=*$ZEHC{i^PHdD*5$wKi@#b~8^OaW z&(czvbNtCtQ{>K6F2qX4S|*{VgZ^wmltQ&n>==;C!KD|WLsML(tTS zUf$TKo00$P$3*SByV3@6RQ*Kj+K9oUe>Mn+vzQwGwtx;{|}iH)+% zEr!3ok2T^_YjeZ7Wgjhm8Ts(-tZaR};+3i@3InHv#tG(w`-yc9Z+mz`esQkN+;g&< zM)<_DOZE5UukeJ~tB|AgJ1)|$UcEo@h`#sl*`W`EZMnSO-$yedJX0*&V>d+Ai0|E7 zd5NC&eSxixg6AYDEG*1QJ;@}msHpmx`^e;*UAY$hZ*TYI%FD>;%f7hqW88D9n!SZB zlD5#cVT)tOqg7|0a-VqaQGH>e)M$hgk6vR}wn@?2qiNa)?n*hmf9CFx-+N@w?D%J{ zH0^tO-U@eF#!fjn9QSK8#Y&Fo6N?=7ASH$NmWX*sSlGev<0d16gXM4DXdZs8aoWbl z#!7vwf`v^bg7}EW8vG!!|oNXTWlu%DIJyhi9~_lxvq^_P#GKe5dN=Cj{L;jdm3DjE-J? zePhSEXWH77J1Tldj@%dAy*u00!pv+-1h2B$m|R{|6f@oWEmv*c-J!o1MU^o9cm1W# zw70i+qj+K^9X5q9@vN1WmX4`crwpgRF%reQ)4ZX)oJ1N{2z|8R=H|Bb#_K%K zDXZeAPc55L4jRNuHa?#C(^8gFo4wu_7D{Z~xUoBPQHZ=0wl73s<=J3v{MkIsK>0=? zUf!bQL&cjaDw&xLQc*v7`S}&`iWjQTNLC!p6npCTTaL7KNk!qZ?gD*O^<}H7FOj*T zH>nyi{S66`eRa2zQ&K3Sf)xI(shl?+tvey!ce^Fx^==_w5oKlbi^dd6LtDPJdV6Q5 z)sOMdR_fcxG&@^jHTbXJslDGXMxm(w{B}bp&#`0gMmDxEo?t3%lD@T`M_Tv+)+_e4 z#)HT8ISx@$?0Mups?bsLoIXnBCvS+_7j3V#d$;D@2(iPp`*JVm=*->^Z4^k}b=B9< z(Q&trukXm6qjn$Cbkz0quD==BMt&#izvM$vQ~SE~``4da5NwH^9c)~wrlywH{G^Ke zgKXa3yg+?&F#f~@u=U5*2c*8(uBD|lIh>ViD7W-;ZX{OPc?+43ug<&<+dd=1ZbA7F z|Mj*0idnV|@#rj+kB^B{gk=;vwrn^Z!_R2S3BgE(j7f|bZ}@$MMq~nUVL)q zbKujbvPnrvnI&uX$jWX`RtwLZ>@2XuCxxefWA(V_xnpPUMP9b4X-+X{`~y>e0go{>fYMOP<^2TbvO4kqIXf$ z)s5uFahG|8h1vNpC}fpTCZ0^+eYXMvXciY2MQz^3YSu0eC@=5y&q2#8b{{cxQsQK` zlXZ4>M)~%l9`Oj1yf=KKH?FU**U-=?M-PsQkB`80j69dU4oT=ldQE?$_Za^acK`l; zq1x!^Xp7~g-;*;lS#ED@dHqe7*Q-)Y4w0iQS~3mRtX-=&Y|F%KC!oT~>fqpjCSh9U z$D^X6LXC)c=u%Vp>62*&kL7k2`nBuUH5S?(CG9(N=1i)CF21Z2ar6Dj!OD7Lx^1tx zcq10u+fRtxB=!5}To5|r_wmoW@T@cW@RP$Rfy{hrTeoea;gNH@n3lG~FxM>nQQD^U zTv80HnYhzkw2qzA^;oTq8nHZmntJgf$?@|uHqzuwgYwG{A6WMd-BSq(35k_GC-B&@ zLvJWHZDL*)jX8voOQ1>}54*ssdhMmq(6%8tDKblf-uXU3>sQyd&G+ARb{VMWk&%(< zeiO5+y~Im=#flXhxVahB)zxWfX`>T5iqB7&1*b9FoII(Y={4NE6VHr9LciZtx;!g( z>U^Jq&-{kQ#zu{U2bqVHnWHw@eSEy9>%xSh&-CjZ?Rl0#?XJ}Ho3yglE27O}$}Q3JGm5xtzwsY5Vq7X*vm;GB}GJX zD^Ex?^nU#s$m=&x^ZonxzFgnQiq-S;y^(-k>)l3La}4LAD+pluwMAT1lyp(9A<ua*@ippkKXKy3+jvDT<;BUB*bV8KLt#{* z%}PEpYT=y0b{`%P^C2_wQ4OtNsk_Y1o%A~K3K^5nv;=~3b48>a+b^P24b--hGrN-` zl`3%qZxWs!pxil-8XksfG&^!0|lQS-7&sJs{WC=+M18b9`G$yOr&9@2( ztyPcQ!Y0qethhfloc@S+Ys#~cgP%TswnUd}Z#Oom-AWc}Lkp;?t>qIH-B7kTeFfWp zaHo*7mZOtX)QuZKSw^{r-a6!{@-JU_Gcqz{&h=l-Oy0m8MxWjXuQlqZTxB}Fg!tX;o8^vacWXV0ER({CQk zNEH*4l2StpjEajh9phkUXOF#o+Y+0%if8v~gCFbNQk`icbZ3T*d_3D-9xXkL+tKG&wZN4k*YmG5iUP0O)!l0t2Mz^#ukT4MO;lCSC#sfDBW!b(dg&r(S z6o43gb*r452{18Zd?y#{%9Se_wkU7_g=e{D|KE1QI7Foiafw;cYRAb0q@?f{7Z*=H z9st|E{}_t&pU(>{N&_H*x3WfsluOs8gs1_uQtwPO;_ zqqJSloqPG<@Lf^Ic2Q4nZ!$m)xK{2_#l#u36}t(pm$HOID+31POMJ4)k=(M&JmDH6VXd)lW1Vb( zW17lmKQ%XR^z-whUcFlN`_~VHxCm;x4gQAMO(rIsbc~#rh}-Wj51?Xae($iL%=#or z3Xq~-xEE_BDESgSPST3P4nI97fyOO)DhQ|h`+B4$?skawHE;3hCqR@N>| zhCH_+J&7%CznLPJYS|0l;kZS3{y*TtSwNozkaGq>n8#4F}oRIG?o z^5wkneN+WKdDQ?E>}UroP-pRl z9}nA^!zg^-7>7zktIcR-40--s9#E8-iz_P$C6u{6Mzg2CKLn464_L9fx;pR4lUmH< zV#O7bnw*WX#>U3?-A7twE_`>)iMXM`Kl$T_y1l(neS&gs63S?}@v+nKxA)}P?d)~$(;!#u-~*gOFRhQ04FV~$D!)k6 zUhK*zeJ6$Ppf!5rtaQ-1f%+D&T?0cO()bh=d9cmcWj15#ySclIfmbgs&NLLe4YB3?_E`eg{C@sl$=;4=Bx zX&X0fy3BJ$V*_VPEGac@S0uM=AaJ@I&zHpgt5KC!bHgo*L7EiCJ2^Rfumuv6lY>BY z&PhAkZ;*H>CMGUU;8E>(g<}JpScJh(PsOF2yLMabC6nJJ=jKL6D*ar8xfJ~5iL~KE z)7K6<8d-Lx)+?HSxAvqh{QOClm*+yaT#*VX%2Y~9O5^b1>LQ0W(X-z)KyB60{@jzI z!o!M+Bq62j-Me?#bicVdF9D&$!w=ZlJh+JFCvgK?PU_4jDp9-yG+p{NQ5`|iceoBI zAXvccitA{**y7KT+*l=F@6zEo#S5Dt)VwV5nz1;2TJVS*+1up*L#l=DB_|exo&wf& z_xICbFKF}6j_eSINS4<~wsZS8s@*2I@v?W@ z@Q1YZqhn+FHt#sQ26QOl5vi%$(HTPq2T$)dEk1HBA&a8=VZJ5>EkzVcmPpwmSwj(+ zdm}r$HFS}{WSeDFQa-gdJgjI6ORT^?kHMLtDH5 z`#Varpd{3u1pg&jqJ3T6qR6?PS81a>i>_?WX)WAKR7^}T2!qP->yvxufBhnMKc*jB ziU4zzRFnq4afzpxTI3c+R4AAyl_zY?#hVfW-rn9WZf+DSt0W7_oa}6jwo3+ChCNVf z0$aA@7BNI(6)#9}^YFN$>F4)uuv5o)*VfjS@tWq}qTspJ_%KK+2NxHgl$2rK9SWt> z22TkRX!)xszNudmWmkAYIc0O~{;QMH8y@`&?tFh1TDXT%3*TX_^L>ALV4~3h*R#ko zNm`kFK%qWi+Od5*{j*shzd(#c%7Ftt-^V`wxjCuavL!jdZAM;RUhF=#QJH29I|FMg zOUnQtK!W53wvJ);0G;HW?b8G+RdHZtw!2|u?#pKGx3ImWw3H|q2!VVl27j1DM7t$&z@c8lLSFc}Jv}75r%2oUqq+9oBl@b=Y zATN)2oGVtYbp7^+9oH*>A$p));)(0eh?KJb?w?k*>nBr z-)x0-4GJHM(U5@6;^faWfyuI#c!1k?4-JLwIsW*5Ufz+(pIGFoPoKCUiN?yG7rx7? zw=Ty~3uC3R%-^4X$BwL&7ePTQ<=lra0c2#Q5O-z=oa%M%?EIu9?rLRP4Dn&}hokt^ zkLc^CT>puxpm=ylL2K04OQ34pRrnm2K731+Ou9xOt3vnJEo1%H ztY5FbZy!m{ZSekeSt@5>UY;mvpN0lib08l?QVK%W3lviWn5ln#f&f|1Cth4hPZzut99;40Rl3=n z#Mm!u#a7nLc<39XX0QLCtB)^~oicaacd5N*T zj$IgyiE~Ykvlcq1C_uGlgPZ5}%-hRRE!K`llx+_qn=?Ve=SG ze4>_3x`Q@2)_fKN*Swqa@5Vv7SkXCbVwcxz*@Wk`VSFhH9Wv(Ob z!#q9vbn$6zp+g(f$0yF}hK8H4y7t`W(84P~Ho2r630RK-_h^PQ32>Ddj5?aI!RDh@v(%bXJ%rQ z9%su&X{H*#@Oi4oUUlbR@KIM+w-_DTHwnulLx@eLCr(U#|LDBSmxovK4CBLp%Me(u z$nnq50pv;zYe$BM$?%MROnj-eP4IotEOz3VTfp7BTm(c?_JpwQx%~S_W3IU}z+_@^ zvGJ%(N)IPs6N!)|H8oYSn;c5RIFER%HvnV14mE8jRI3bYJQoWan-Ew$E#6Axq>wY% zvM{Z7b$Jc&oH=r^p=B!9eJfBF9X2KX}6D_8bo*y+`R=I{&+4f%8m zFzd3tIkT4l{t2zxE%w*`OoDH1=Kco^4GnW1ihG%48Rq0+&}W;KKJ)R99uWI)xS1pO z?jWW4S`H+J?c2BGA#IdjYnM*y@i88VIQ! zHBn|Z2XPIwbFxvLw?*c0dit8WTOzew&Lsf>D_6HVn2`^&gHe-E$ht!cK&~?9$Kx~y zw$4S>^~K7#k_pcS9SPk_;!?c6P_VvG&L3Cjcw18GW zm-z&O7c|nXke{G8=jN-qxQHtKo;_-O>{ud9QIj9P34`YN|1oHyIn$iJm|H)D@Uq&- z$cSLTXxh!iKG|tY-i#hQ)2@PHY+SqQ`i&bSC+cAB^z`(AoIcn4l?)zI0SO(tk{a-I z>A()M{b|tquV24Tjy?He1t3eT^!siuR&nt^^bOis(AK*NrSo5}5SE3~(!6!*`1kyR zf*=6c$If5sKlW0onHDEcUYeix^nuENH;O_KcR$Iej$S zrTlVh)om#DD?~&@Qc_cUx=Jss@L6#mel(Qq>YICxM|)p(g`3ecFcA9cmByzsfAeb$ zXqVwy0{elXfZuBSQ=(W>#SR`iv|mMqrt@*>W!+@5#hG{-MMc@!*>9rK4HvTqxE_f6%>w`nqGx&O@Ajy<0BkLCNw^Ev=118oav)) zU*j72c#iqK*EqwPV!!qE(U3X|UuS!0eSDlc(K071Z7?-Acfik23DT{g@nOf@38vU;Mp*8Nm-yZ5SzFK^MhYNfD9S-Upaz|FhN4X zqai6wglq~{CC``OZY%ImuL4f4F?j;8Ki?g+KJRoRIrU+fU1QQJoBFsg$o1d`$V&nzEn6S&)oXDHK|NnWM?pCr{GK%gZzH$Zd>|k0!3XR`+@Od6a34xplT!{y1H z%RdD{BLbd@W^M!U2UH^6VPT&8ZvJf6*49-uHLbGiq8|Um&?2yy)5ph_jY$WHYuloh z0nNDOJs8k8@CqBY-cV!m>Z@0;x(q~YXX1_xx+|S?M4$hFZq!C;XOo`t09kx)7GT6G zK$@x1{dBqJWf6b|G3M{SAhb|2S-I{se4ZBz{amEfx&u1Pa}rRQ3E3Ow`GW@!Vy0o| z-~VHD!r!T=tX#2a(%yN|%$0kPp9S2GPg0T% z7K-8YM?QOwJ5o-UvBXJ$0Gf&yE^A;;d~`0WhO0aUMbkxkj7}593-A*}$Eb>C=Q^Oh zgI}P&GJ+?nJ)e&yEUdrp2TSiu8e0^) z@mkbl)VSn(r-LZaZhZWf+1XhOd;806W)~QpOXoHr5fXw<+}_b)IozB^0ye%{?ZXS(8|oSYn{ z9-+BMcWLB+3lcgm#IRMLFDFpbw>7o2RHIQfW*Y1wNBU-RAE?8q0C?q1K6~e`7474k)nlr6}+Zt<H;^Gj>*yQQ#@?qy{-QK%ue<)MV_ge>=gb8&((0_wD zx+_7Db{~6`Qu!0y-!ix#@ggwg4TtYats=p3v3`AhdtP{C=o!~O9$Plj8O5T8FxA9; zfR?V^ zW^vz{CJf5h<@d`80t3-NP?$B&WxqpU5*9jtc~O!v*i|Y|c-x3@{4u0mmKME$4wDY6 zs@ml|x+CbMNxJW3Q~migS_4T&xOqBqDL1DDk3(_`=ud&b8ju;mA&#B@gf`aHXj zKW0GwOp`~Q(i}?&aVXFb1j%3$5Hf5&pDxcqBF@&(6_GWBNFf9Wrxh`|B+%m1PtMpVPv%QZOAAF~F_i5GL+6kD*JtE^X@8UIM&dSAgcP)D&t0L}jiAZ&c z`Yn0_d>kH?(O>=Fia?;5v3`Pg?u(y7KnNjJ>Ka1vllK@cXE*l=07oG!C~(7#emE@y zN(dFJJ0f_M8S-U9wVxKggq#7=Z|Pr$ip7&_MCO9CAOdyy{QQI(VR|rd-2b+N7SV_R zG~FJ61Q!>V^w}2zzkeZKwM)Lh=%Xy=+zLE|)W{Zu;_Txd`}$|^Fjc&*6*5t_yv5AS z4Af5m3|5+>6rm1>RjP4=Z!QkPfrUBx@%DAr=|$n2w9qFS^SI68U4rlC1{6|RSQcQp3`_|8`u@?WL-5bK5v#L8I_`aE&9`G{Zt7ZE3}*hn-3b#N z-vAXbNAv-}ov4^t1<#>F+_B=TrE{{6@W&)+2vr~6wTpP+{O&!Lkl4t(QVO-{Tix}FCDh4$H>n~lp)Qt_&GA?r{B^)6? z#QIk6KJlz@QBIVch%CY)OAiR!W5Yl!Q%v8t-|U?fRV04yA$?D z-{*ztu=}&?+b~fL`AhTZw{+ER^XEMp-jHs(0zCUHwZu-X=LWLYW)_4bkT9V zQ1{SiVY*I)=DtsD#{kdK$Gwh#8p7r;t^&;vRJxPLHmool&J!fnAT;7=^JGi5WK+mvT_L z%k#Kbmn4Kz9p&Hpe#e~+Z(^Vu0Vsq3A7v>A(8^?D$qf*F_TOiye_Q>dkr5wSq(dTd4kf zNzq>dj<$#;CMM1gCP%`04kWf-!!ZPE4;`wrcMcOEC*8Rn8*72^8_b5ddqGb|KcKXz>lXxF)6gGKo59KXwc z>%c&ooS(9W)R^jHT?l*`BQ3C!j-fo|;0>Wq#QhM`Ic5@)Kw$)m+BdYa0SY@2;pirK z9pP7uvQsF{udZ@Ru2qlVu7GvYJvd1H;`i(Ds7AP1vc1+t|63kNLS4Iahk&2RcyXgc zA`)AB?$&uirhNbY2o|N;z8=C-jPkC579(>QSvCtbdPU6K+*e_K5r*sJ+*~z~$gZ&# zFoU8G;4eN%@)Epfvh}z#;jovSQ>TPK6x(_9ap5`Gdy9xO5xFOPL_%B!*4MwFnd?~F z@r(!+?B2b560Sqr&vs1zmTr6^Ry@wcX8AM;UNF~`E&5&6A=QR3r|q* zH%bYVmmQKX^`5H)sS&lTq(vZaJ;C7d8TB9QH^C)TIkVsn1Perr5Y;M;0~89>>;#`_ z`UrT5D_1k2TxbY$v^t_CU1GR~Hods809$?$e53v8xgF5U#GJbnpzqPY=m$zY2maUc zJc2?FC@WKjnyyGL@>^2B}wC0*eVzZQ2SFb|GicQ`y`TZc*jlHK1$+zX2 z*CH`x>ELh$W9};g*0r!GM9+_#R&YimT{Jj6yjHKSs)`nL5e|PPHq@$&gA_`t_mA=L zm!a#e$3H!d6}M%A9~6$UdYLCn#oBs0I_;JR@)JNb3S2$|L>pm_N zcOTYw>@1`wtQW+EdywOSgKLctg7IU=%>vqSggCC?w^(dAH}Rz`61ABG_#D{p@h?Nh zW2|%g=@Vq~&$f#+D_t;mvX4Ju!qDBHYgU@)_uHG05Jpck(sA9eC(>O$cD8_hgrG@W z3joUb#R46n2OtSYWRa_{uoEomz<~n|Imh($0w*V3P~QwNIw#-$MWMo;PQ(?na$j7A z0No8wi%?roZ9Nkckyo!WTm&{NmymU|u1YVCWqZaRfno=8YRK!9!FUjHGHX5>c<+#(5;z z`v?Yv!lEP|7%?yq6|k!Pph@_?2P`cU+hrxX@Fa+6@9aq1dVEv?H#ealZoswGAT}-w z)2B|_*nGvDRkO6*2JH)FLf96_>Zkkc@-~ol{Yo3LMe(wZ(gH2&Lg?O=HKczdxj%`! z2VHHR%o&}jhi+|PV++QXseJizCE<)f!l{l(mE>Q${e0x>x;j?31r@H5zJUSj&(GWi zg@tRdE#D^{SQq~0MLG1!1U1@9+z?YwPc)Y z*$TwA1wDTf-OL&>!Zl1xA!vRz-##>Zo4&~i9=_=z5%r%i5`+nIl3E5Z9kHW9KrppP z1Ro8+Wa*7aYX%`vN<#gt1E42QP;S|3<8NNRWP63{t6i!O4!^oH&`^XM&k464w4ZcAK-ynYK&Bkv71AN=I zsUQkWnN>wG84|b1d5CtjM85hkT&bBR<9X(L$b3?d;h{YTvezlQIv-D{Mag z*s%fv#=jbqx>I{>x^YKSeRIZ9NgF%$f1DcWGvC;{DHKk4d4BCh4(te5g&+ocnMZvO z-hU7gBZ=XMJbwc_p3~pL(vMN8>E76vFaXFvA1)w&`Mv4fT&$8!h0i_yW)yYLZD4Mk zDFPb@Agx(knBiKqadgyb*19p?2`>LPgd?I!M<(lO6DCe{^d%wyLMX@WuKFS2VHy&r z_DeyGq@|-H3kc8w32xfH68v!>nBA-#{uUCpS_o(uj>-sC)KDmQ9uNJs&(|H2i$nef0-W{zGtw~_Xr#e1XvC5kRkT#Bzu(^rTCuk@^ z0Rbwd!}Rwe=OhZuI{bII-`n5e11&}~{KKC^8M#Vxu#f2B1VBx*#&HeKTe**mqmvV> zOgmj&T=pRyobNSbQv_p%;*J1u5YP+@4y2T$eN{yqb#~N}h>blzHvgYcj{V&mRO;{p zSPyUveGpK|ooSL4PUsMzfrX3g*R2`uw;;2Xn4>!Gay#HMdr7n8U7(_P%r$h4p@}dbop)a={iMRv= z6$r!cPp&sryFK=F1_)fn*h3zu`^3%zZ6Lg+H~l#CW=SOY@KNUBlj61=aNDtR6P^DU z$gK}gTv=T7iFvo5La8(^^HYSTn&N&)4)2P?yRegpybJT@%{$-B_~0C!7bL4{)P@+c zY5B>`Yq@UFsh;3^1SX8!9|z|0(SyEdkM&^tM0%cE(OZ91P}ADF>0-L|f7iW%B2jwRV28SP9nH-VIgKdF8lH;dhW*njlY=tCg36Gf#%*Qz2D!5%Yy0RuE zKyR)o0w<6lK<0s`L5rfi@1#)pvx$1dB&mVchL6i9CBPV)dF#K*TxEoXIVumskVttW zvT&IvRn1Y0Q^r7e46OC9%DM>&8s>}|G*iNu+bSt(Fe=zkU6T@Zq@^hWJRErsW;V7| z_aARyj7mYkiBDfJ3`2v=5P#k$bsUANcua8`(nzV_P!&;@#qf0Kg=gHoc+F zG2~zwbIcANq{odA#0koBVb|QHx3#HpY7ZUlY$4GG?jCa+d`|=6zx~mvb)EtU7K#_0 zZd?IiQG+&v8>8o9T|oklJDm6218ME|8Ao?Gc9k3}zl0~7f{-qep&`;S@|vq0wbJ9R zogdSRlVL`)A`V%IZewPquSP5X-<^)}rhA0@t z`7TbD74b?$G6LFlNqJ(4oNMy`Omm=8UV3_ZAx)weopN+!K$}B`DU4?oiHlWF&-bRs z-(U!l+!N9w-YACQ0IAS`&m-uRwW?-T!~bh~XNhjdjT;jF8K5kvcz!aj*_qMMr2S8N z7oJ+U-=po!y4DgX(D3<%w>6iBQyw0Y$FTtdmqF;%kGn>t+V_I;Q2=O}d3iHGgQ!}pxG?Eg++$?aJ2 zJ6AcxRsj0q^wLeinF6tGDlvZ=VNp#g4~#`LhLi^nz5=J%Yim*{2AIr5mLMO5OSj+F zT2%-KRHakLJMmalEG)JHPrBetfMgneawf4?;0;wk&~8zZ#)H^=3N7w$R4lyR{Tekr zA7@Yl3kt;Hmu7u`x*eMJWt=Q4($Na9`yXJt3PF0c0-JX@pRiq3H8b*mMWUEvo_UOM z5@&W1lFe{8Q_Rp`o1t&gTV7J4@ap8=3IUta8iNTO@q9@*bZ~A7X@ofTXi|7uwHO412NZG`dfh*^hpiUeY(*&BQaar_%gU=$W(VGgSU|5W}x%Q`(h-7K|(i`}Ybxhv-D1L-D3={P$Q3HfD!S)r~`sU4>eUZbMH9g3O^dKZr1(i$;$ULULT3jF{D&DN~ z7zitIo(6N9h#CG*2fjrq5)ga10ZJW~S|}mZR!hSgJD3RdFD|STg@;_^;Ka=9i~}#` zB$C_sBD-3o(&|HX9|<81KAdO&C91|MGw-48#Mx7{MI83i?@&;%R^>blg9FI$eVcP& zmeEHt*=pn7y>GYO8c`8$+piWPOD3DT&eEr4qBC63V@SxA+P!)+WzW?9Xg$9%p!H+1MN>?L!XE zWG^$bq}tx&jAAG7ivkjf-wSY?LD74*90HNP)N0~;Si-*fFK7K^IzjHqsX|I!6j|}n zPfw4Yn(M&xa%j!oOe(MZs$FFLrfU}$u9*qVzy$#})l*#NkHofx+ge)EkT}Hq8D?kW z^}Wq?Cg`y+M=eJSpt|&ojAT8#RwRMoB}yoaCy1Kf3TuSOFyZWZOfw3fGnbIMQM##p zJa2EGBN-Y>Z;RQaBD5zkrsXj4!p)oKEg*Fx23piKMMxAU?>)#}|9|t~I6&Ti2DaFt0QB{_9}sGDNZz8wg)Q7ru)Az|E(@`9p2? zU~8$SX`dH$Fzgd91wJDtGa=Oul|p|cjI5fPn!a2FG>kZodIT>ZBuI1q{144sqtsEm zR|tx8NT1;btDz$y{J$FsJ^fR^_rl-6uKnkLs?iuXXal56CY({P0SC)mtj zF;hX}mp`OFq1oB_#5tE57MAhIw6)>NVMS??|k1a zbD>cVoFTO^eeS^s(y0nMzk!qUGP{|lN>vI)p9^M8E&fNGMkD(B@bp(B`$^)&<|pRw z=wfIb0R;u;f(f1ezmLW)%%tSpBG&tO&EC8G{yBZJi>!bc>|$oeZCmO@z( z>JTCFKoTR=KO%F9xaNL|N$A-29*j+{itWz$v!hbnAx-Mo2ojG4Uq@J4uyBz;H9 zo5%0`O*(!Vw0Pn7U*OCK4y5VT=_H@Pkb~58Wo@qS_YPYE3J`u4#Ku)GUc4ZZw8Y7S zC_NV46UR(UzJiCMbZV}Myb#nlEZlb_s0|47Qb-8SX5H^9A{YVE{H@Qm!F$1s5EGv9 zZfg=){ypKJj|{~8!UD~{g`({urL!zIcON64l_O_f!H=|tryST~ZLJy}7#g}7e!j5B z)w8hpE0HUzj!5(Xm?y+6eW6u^jkE$vY$&P?&-8${I>r4Tl~OGaD%Uo03_a7*^7ru# z>QP&-u9*CIlGR19Jm7S|%WE!l56fluZIywV6uQ!d_92O$y*%m2d;8YIhHoQcojmTm z`_Z9o_S;ELBtx@Jp|)3SW-=LL^1psB{hla!*&}-Qoj}`&kDS_IozcCQX5(S{SQByf)}YfEc2J^@M~>I2M=gd9;#Xd{y&~7 B|aw9$~X zX;L))&&xUYIrshj{~ovdaURF{bbYSteZ60=*K@rK*3;2opyQ+?kw^>&G${ro(kdYm ziHda{4gO|TOgU0rt28Faj{4cpQGm1pwCmo>dHS&lb|8&v9XmEM$WMe)9-EMwz zmcW@!tg8=DNzdiHOda$W>I)3cdbUTD5^|>_W}n&~I+ipSK{0(pioQsP#Q}Y>_h%cF z`$U6Yk>$!&izivvZ0=isl1AnG@9E=j;%B^7B!7Ng@=Tmj(n!@4X4BiZcQ4iK++2*j zk;wge&-)!NqdUwWXH|W7dthox{6*%oxyRAU`T6;J`FjtC&zb$_k7(L?yL}W2CEdYE zJ18hf^K`z10GUiKs;u+JzqZnFvEtthxc;C2F>s5js+GX(S7AzPv421S6=~nTeH5L2 zlsh^aR#sNo500HXcTOOxfto9h_&!b@r zBcvVY>+4HW@>w(+m!lX3+tr8JRQOWG$vRs;a3pRpbeCP-`j^LcfBEvoy1y#Gwm!vt ze)hfY)Tax*!8})uUUqkLe*5-K?((wN-=EKS>=566xRCnX(7W1ipI+34sY%a{WG0c7 zl(_kP<^{O9xh=bsx0~gY_U_#~7Z>Jl>);U5VR`9+j&x_-hRANQ?b|8(`ph?O+_1K> zxpH}VL1SFgNG(l0w5sCDY6*u{;hNf7o4$%G#=KlCEPhFsmp31b6@O6gpDupBhH($Y zi0A0hqrz--_`|kwW2*i8d$t;>oQ01cQ(oWNVPkI}%&Y8a!v+(!Lx+*tpy;4pZELM zuYKaNrXD-Q#TiM~j*drmc4>OMh+Yels#?qE*gb9PvNdF*@nO5MYwFY-?|^2Xi0dp$1BVYIWrj#&1E??shjLX>o7FK=%N z>k88F$ViIYSUa)slRti_hw~~&L`AJWc<{Ao_}Is1k~#?r0VWU3+w2rr{ECXCReq04 zlJ;ERxWmEtYA7vp@aXj5Yhe))eTRc%?i1bFDjTwm>ce<)mG3!rJe-U0H>zxHWm`?N zKFg3nXK+L0DV3GQbF*U|OwJ`!G>m*I+ugaz2CK0nHKqFd`{m9LZ>XxOdbM+fMSq|! zo14s1J5V1U85MOEHzBZX+gg2n{kB9qXJ@7zJ9eD1FJ1g@v~}xNEZRm*J;+UKc(`UtHI5-Hm#r*+2pOixL0*Q1kmKgPRuD(igW8++%8YBC-$Vq^}k zs;QyIgYUDk5m;Ur;>)-dz{$lm@Oo!C#mC1-dCL~E1hG~1PdPf;VQZL`cu1tCrAZun zxgq1-)Z}EvbGNZk0RiUV6m>)S{rmUJUHHcR(5gIaEfX(^gpJnpm|j;Oqx9KW(M4;2*^ht#PJflPc4Tc(1tmPdxi z#?-8=Rx;MIv$Na${xKmeEL_pH%{206tRH>ACKBmr`q4>~M^BrUT`zIVpQTPrOw>9m zumAI|Gtp3V7d(&47PPysKR#n`Z@*V4NB#7+L%)}ak9}VeaSLyF zc5X;dPtR9O_aVQMBKHt|W7_=X6?rQU-u|xsVuSN*nD|s`7G{Rl&}&ePZa+EwA=SCJ zj6Lz`J_^HCgV2T|#|{$d!_1R3j-4u-3pr{#6QA;sT;@iN`&*cqjcs~LF3x|6($zCC zAiJ}XSt1NV4|Ih-G`*g4M&Y6IGn0NZGTE3Yza$m!e77-^g0u~hzSLIY+aopo#Sc)7 zvhl@s4L7%XOn(XP)DP0+*OC@eS63(b{_vd0Rl1jOMv+uhz7QIeb#MEQ9THgRn!36W zo`4|gR3kfsA6c&9CwQB(kBzbN3|FMxUQ4QY^(v=@v$nQ2b;ksk{MjvRbOFM4tdS0H zZ%eMV@|s$c85zvEE`yU*QBhIKzJ;3Sydr<`{#{NPPa?F^@hAWNQ#F_C*5{kXNNQ?o zo!#9g<^#XKmtQL_^(cR?QBlngj>t+m_i($ryU(M=S$dv{ z6+aqr=T6QWjjdz`J13_b*fKVQuVQR91jtmvZE-{`UtU?>B_&0<^y$aCQ@p;XuAibD zp;Y(Q@J+aVNXyDv_vyujyt7PE*i4mg-#%CzijR*M&9fC|3z@cxIsEhtPua@yrDMmB z+n^UuPEB1Ld3NT^8M9K)qTPHkK~hb#Bkv0?&YU=Q;smisqnaakt&Z>BtfZ{m|K$s{ zq@<)~h5`P>#l`h~eS7y5x>4D$Q86}lc4Nhd|5RHJJATL&+j3Knu9num zp?3+d`7ZaQ?!O+J=!m*t-KHb_TR&K{TIzH+8~w)ZdY(rvkqxY*&12FsGuK|abZI@G z3OAZV{ruEG|KMQ2iMMxB4nI1PUs#C2wJ&4n>z6MTZ{FPRmSCe>bN1|6m#=R&KXz!# zscMTjd0-36DSLaH&o#kjXrKohPj?k={nlG<6&0>U;XKirrC)q@kZy%#UwS|FVe!SU zcb#oJ9v+R8vS$&uED5^0mZ{9;MnQo@VPRo3eMZI_dV2cpN1wB0>LldMas=$Va`Pt3 z{lkxF%F4=iD$7brvI+?aNv~K_(j~UA(2@lOrRb84Ryxm;}6* zKjOp5CmR!`m&_>XHqS474?~xyGe8O(z&tr2U8!M%lpUj%Ed32-Y6|q=H}sPPCvl9-KvZm-{Up% z3~hiz^27~2Y=c&_W5>3rEPP=-cJkzPYy*eA`E}`5Ee;hL%Qxnwiu@kJBaL?#(XagKD7WNzjWH&B=HnhT-j?_8@smlhiI;wi zANKI{q?^P0550Z&u3g=Y&2fqs9-@4+Hc~rkpLcVUaOvAK{^`Z0J@zv$*gDfmMMXtp z=PwqCbEV^TX0&LevFcaPetYr#b5#JnoJ*fC@w0){-X>u$UMMp1Dh2uZtr{L3B?}9$ zyDRHlH~2a>fIBc~IOFl-$CF(rlLX}CieK$mAEc>F+l{>(SAh|8jge=Sp63C#(bjAL zojsP8i4khd%%!bZvatT@YXfQFDlCe0%x%7Xx|?_v&I=nD7`)1~#4x7Xwrv}l%JfBP zic#LzH@B|}YJ^3v91jj^O(MGd)Z*C>&*|vttD92Prn(*Ts7P8`S{rzI8L`QWU%a4K z*vK5bbNlx7qzx=AsV%O6HxY@6mTBjf=LeXy??}<&y)<=nnCkB)vxJ0U+Zn7Be*OA& zpHP&Dv5|P}06@)P?r~DzRVI~+l8ZCI64%9#K5xxR-2JD`mv|avvymK>0}B`M?9mF} zx;&LH{ip#{DDarF#ck(E@%gyqUJjv+eZzhEm9tccdNj zow{}d{}=)q&a&VqNEK~u?7$Dz)z_HBO1@1@F!1p3jDLJ)XspJ{Tv}08wJO)-fnR3k zHaa@GnlLqrvEE@j+f%33X=rGS_m-Dejj;q%V-U)n{h}F=U>zJ}^tQHk4X~YGQc@Ce zS9GF|+P*EBjG6cDrKYE+SB>qyz25Mi_Fir68!furLS&YzH@7AAjf{-?BR5mhRgAjh z%*@RJNm5$|y?GSf{Vl{bjEvYYzl}Fv4a$nAkoWG}x9ao2z|=svY6#D@pv1J6I%?rk zheDtqt=_2DgS#Xo_S)K}=Gbjuj&W?izuA4f^9tZFx2lf?@yzREWQEX>HPB--8ed0h zZi$mQ&51&A{aiz9jGivqkz*{jMM$Wsqhk}wWP>6j^HTULF4kbmstXr>Zo0)7E3s?W z9>53O1Dfe+<9gy}N-tgT`q6dLW9-3GqGJ>cY+xQuk)ZOWT1~?*D7ac)UOuMG`S|fQ zJ9qA+sH;yONXEvrIC_+snVCvfR`%nwa~im}P|H#&p+k41_V$)}7hqtEhP$3ScQq!4 zorQ&ERFH*PUw+f3O^ZXSE0y^+)su~?D?vQhvh;IfiRY)4*|CE`H&H2jI<6sV+ra1A z>vA5G*50*b@+uN4+je1k(8bNI7T{XMojX-Mw}+Dzv*Y||E3HO2?}KTMY@?8hiakyr ziz8*znDtA!c3G2b?d&9+I(NmZUbZA|&Y)CUsN!`jV{F0yG&kLbpdhWei4U7TJULzc z;(OV3=kg^J;`2JCGD1gRxYxeV)OE#Vi{R420p0M`N)Ufo$XzY4R)!{(<0 z*&xC5v^L55WrRSRr8E^uRaF)1vi8!?Zx5~+Q7G(1PF*bMQ+suE7=gInu`5#M(tEg~ ze3pLow`3fA`R<+1i0kv`&)3n>RiP-vKE;OyX|{pXO`*HkpoMVYk=h>|+2+0_l_sx; zixs3|J-6J3wT#?ZKg__jv=bEA@jhO@er@P2M^4o$S@?Qa&U04SXK_ly!^5NR*W8$` z2H7Eh%fiA!2sWrEkCd$Jrh~V(SJv}gUY~7bJ~7qn{gVe%s_*-EreKNz6K!yo5I#sE z2?z*?^3+#WrnFr1YtL+LHQH`j!r@%DAhc@rYMWE1gdKQO1J%L$_o#yQ`V(uV!XtU5y=cgIAdo+(boNTid}{ci0OM{ElT_Pv+TI#m}FUw{BfS7m(U* zhjuP;?yELR)aXU(HDc*zXE$(g1kKK#2fUDzm$&&fJ2JoYv$r-(J$;JesD1zb{igee zCF7MZnPd+fI&_G6!v^21Ea8vkm#u0?f`W2RT3h=P9SXOqh7B!o{1w|Qwu!5&>*BeP ztXsGOX;8-E(ya#%cEl$nF!S()Vie`<+RPj|jj!Hy;tdD4iWdi}x;8=aLR0zW%P$Ad z-nS~}y?y((t;UuXj#L>TqHn&vE9*Mez9XY?>c@}3urS7(H*Xrn-3ZEZC%E4J>l^#A ziOim#7ZJBAyMbHbB7WxP)`QVyTH38P6kfIf74UDYMFu&}WC z#hJ#bzRJ}Ff-J%6vwt-OY3ZW6SgFCY@_6|XadBCx$0(GXtub0eVLNpdt|1Bv)cn`a z?@X63U$%ID{=m2gWlm=YC+oq32SJ8fJv20eDSqJG0w6qiK_>bNcIXl4NOJcFn|xr^XR@Ej}2Y zmDSa&F>$Vzm#d-x`bS2BvBDudp_+rO_asJoN)^H29p|>#h>3|253jDCnw6!w1`l6D z%r-(F09GXiCbpd3#QUybsoULBLYPBPEu_sA7Vo3$f;G_5($cQlCP-%KjSx_CL3{7~ z@F5LtH7Gc^Up=aoKl_l~s%V)_n>TNM+0c;Pwh*;Vx32d3CfCWny;q``nS*tY9N~%* zI#h*sGX=~OPr$ME}1CMPGS4VyM? zv))Z6&k5qyLbvgnYcu6g^-&@ae`v$y7Yq9!hbL_{U-6&=YbC&iu?KbO^UQoLt~>U6q=BPfi)Uo+?Ak#5~sT#E40ajoR=+pl1_{yO~?Yx~E2(!>F zjc?yFD7udm7}orW)8R4CrC)Q^F=D3qt^?|C5|xjPT$UgEJyXa#W`FiSvRRjxxw*L( zPo{lulIUpHhYth5fXikFwy57cB@E)XojJi&@qIvwFBb$_?2D$=Ki>7ouXX01Zb_Z3K}kOV%_QIZx4T8T4-fR0qMqY-HH}-#B&htm zTprv2=mPHi@x3=3;@MsY7G`@EfJCo_!Prxuo=X$Aa*c^E1p6~yUFyG^^4`&gW4loo zWFaA%!-o(1wQeD|Y(brZF%j6i|LfO4jQ3!k;B~%%QbOI`-GC_j9UOA3r5`+aAQ1bS znVmi0^Jm-2QGDagpS_p)z#jc_bH&c|mhFIy_4d?1=#_Q$W}1cgH7q9!E9;uv1C(_8 zlG4%w+giHyyh?H5ll;y0+W!Sxt@7{SHs0Smq~__Vpc5}AV0M&DJ|uaniJy3=N$5)$ znBwl-4mDEuB&f8U{V?CyXCr&C@z*gjs(X1U?X+XjPiud4QrGL}rwfxaGu7FKnONx1 zXmx3!TeogOD&DXtU}<7vGC4iH58N#zB!v3GAuRNB_lXe9EW4KU1D#KulR90qm|L>y zs5iKeOg_Sxw?2M6z~7&`(vcO5j2`UNTgJoZHF+iZ{Sf67>^=xH$ipMfA>W{aUO9r2WCuku-$1NRnig|j1$1g&^=L}cWZ_;~J0$0Gpa=w1Z{1uq*LH9rOn zl+3B@DfR8+v#X_-e&(eA`3i^?28M=2E3=drh-ncK7Dlb77_1%o`jt+>rZX5Y+r`*n3XxWt{;I>)q_Bkk4&mU0J-aT zmM@aAf%iz2NsM&l@qvN*kB^UU>QOxRb$yS|@6*M_#T8(ODJd!QGmWZ_N`aU(C*H*e zqp8GLCJ3NV6BOL4ZtXbIH!yG&WnVC$w>o7iG#JBxhLJmH^;$+Fk0p%8ILVWY(0?$F z^u2yhJSAkN#ib>p@{~Mhv(yLT?%W~s^OK+>7hhf}^G)&%(mdDI*=Y?O3ePLWx2WOQ zG>?Y%@zW>Q@lFXgHn!+;cU-LG*J!TCK&UKpGh7AMzvRv&q{OPXG&yV=2U1N+XcOQJk4vD11<$h!& z^NGg9P=F`fO20K=&xWCO(|_8b&c&;06w02bXFdTdt$SwiPQ*DUiUwfD7fpvy7fj#b zq1FWk1{(ABzPR9AQ(tcf+A}{t5FS&uiu@0HJf{;AG`txSV$k*LS7T#0o%vWS;LhEm z$8b-g0CE|+m%jJ-p%qC1IlqpdiC1u$PqR1 zSV2)yI*6xkuMSaKBM86`(}tiP@Nu@`o^~;tm3Mm+sf#-FfXBLdCtgK+h?OSD-^{)S5f$&>xA(pFOpa zq18hg9Z|T{-vb+h5MzIS@6~|%(=y}$6|t-E_^Iy4BDZ#0J#1wt__rftC>)I)#!7{S zhcgAP-|Q!0SC=(zKiQfuRa{ro)U<(sIv$h!fJ>8rbM3>6g<{)4%-1n7nT~P;YkFL~ zh!r##;W`x@#Cd(QSO158KM?H$^f>(uA{+)5t#CUz{$(#KGwyQHOSp>o!Q?Q^)lS4L>hh;bB} z-|EXNOR@x7Hm-*0gBmj{a)>dwn|z7!w%{uj4p|u%PZam0}*!|@^UPEi>1DGy)_&!B#?RN9W>xe#m;>3xf`oEh$d?E|724$8TOaQ6__l560wUCW5 zU>ve4W&R#8uzlpb=3NpT=0@JHRZ&sF6f$vG{Y1t4S4CYN4OpKWXKYZGKudEo)EV-- zc=@YMO?t$gL;~}QMSs%MevjI>#(yYwpWsBXC+FrK$@H|bv4KGgE%HoJy~@RDYW({! z+8NDy4!um*;MSw)z$E@HTPPU=a(_&jqZxL3OCsC1*8qn%L2!5zFAu3JUFh$gZuQ6A ziD)?H$6&!nG-jfgVP{fZUYuI{>&FDX5tx@~H`u~!b#)sZ9KGiIugwpqg-=2$OaY+xd7cdJqM2{*I+xLsI?*x`97o2+1W|(+ zAC&~6o0x)j7^mPS3a5Ayq+KCgm(tn5_JT_)gif3RtPV}V`;9#wXcF| zuZ*fAu-Dhdq;NE{&%o|&yuE1}OaxJ%=q|#Vx$!0iWu-J@gI?(MzJ?}hjlB#JJQAcA z_5i`>Q1GdTdoRn7@l!3RhGOsKl~VpW-z)Y^6{_y1P<&6^m$pRQ(x{ub< zbLqP>)QLXG%YMiC9oDt9w7@kLbYE|<8cm?{xto~GnU6uZEWTWmVy_Lv(!EmN$=&jHAtT}s8 zuKS5H)9DG@Q;wU`PJft*cwOCGILJ=i$53O^reoTWWTYJ1wVT9C@SJff-U@h$d98+$ zr%#{mA0NLlJbWr6(L%dbPuXXY1Fmlsgr{1R{mYjx*MU3-g@x7OoncGNX-C(kJE?Rv zyDB~|vhs?Hdg3m@&GU3M>7iw*=*u`v0>Yhqa^{n$X6f$gKn68jEIN^O?l0uO)6gL8 z2?S?uITsQv`E9UTgx$Ff)F3)FG&I0|l9qAm8*DJsS&RO7ht#=ksr%_MqSyJ^VfF?*Fnd{+;ndM))Re; zqMF-eo!dY9afhgv`*!f{^0EK9NXc4?OIy#eC^2g)WA0% zK5#8BFM9w~VEH-s2u0a6CMsi}IW7z^W%aXU{IUK?eXSx)(b2;tlRgK68Fdn-NhGtw zhmFnCii}GBPrr>M42F!az9$eT9~&IhY?Y3YOxr)!12}fPNg6TAY> zO#vj$X;s#soEFGy<2|KZs!Ox1e2X0w!7Xv{v{S%T=ZBIQHFX&4=Ek0QOjAO(>82O^ zd}3^9sC@Zm+!v2WpotjeW0U_FoshSN+aGYA@&QD>eB%_} z+DbzA78DTR%+6jW{sIhjFH97CDIum?7TdUirh#pXtt_C4R)PhF-@m_oyG8K^44J|2 zB7}=dBGf{_9XrSXW-r`#5MAi_>y0Fje%?qxS1$w%*e$U?cnR`_>6U{3o8?wj4m(HN*?77w4zq>70Ej zFBXi2qAK6NXVuZs*(E7?)krh|f3R^q=SJ5ccW{2ZlbO)OiHys|>0|dTOP?<;!auEU zX(?PB0-Z?1T{N|{_#u~K{`i#saw34Kd65J0vy1R7JIlNkk#Y!x^g^fjZdF$PfD_~n zBuEO*e0omT^%LrOaxy=}VnJAH7!*__h?Adz9paQP$pAS^G?FW@ualLsVT^ zlqB#df+~`dkLc=S4n2iUN0^-?t-M=G;4lzo%?gjvmXwt2f{^^U#FPD#p+VEv%SUQ@ zM_bmAUR{WHJ9n3sH;vmK@BQ0AoaI9ZNi)baNQCiJlDnnZ1hqM4LQu8iB(Z<0+)O|aPZ}-kk;_sms zL$V=!Uk*5f>({R*fOP-=+Qox;@VzVHa9LVf7JDu1ev@Z$1(W%7W70KuSBb~huQ&g= ztoY_hiNW)MadfZ^%*(GsiC&8X7`vRV8VCmlbi0fCFn|YrBd>G;@L(!@d<> z+6?*b287MeeU(>>e%}lV;@k|0m0*tGBRk4_!=>)lNdd^s>H-e|9@0VuBzytnYe?8?kgWA&Ri>p^&htxUm^0|C>c`D9@q zF8&(JlkT=}Z)ss6vSH}uqDMn1tsCAXC~gunxlbYyA8@&FfuI5}vo)jUAdzCFVYE9h z2O-&|bu3J_Z6vp#Tu7R(rOVHlR5P%nCVdUk#ikr5H0#p1^8vZI-$qu1}+_lWH6 zeg9|?oq~b_Q~*loZoD`~2nPXAp13bKzPRvR9SW_XbkzQf(YqNu$92FHz0RMO)uH4??!OAM!-bnX2=T~rl0)r|9rJI?HBCc$##SSRrVbMuixr3t3Q%rV zcJ`)}+Eplg$tV|)bXEd}=I8ICUb|LMR`zL?1nBjyUArn&c@+5NQoT=f~ms08O3)XB5WWs!^qdcUJ||EC-Qp`vJN&-U)9Pjq1bGP z`mVx`?c29+;NuHN8GIQWH12e5$8#>8=mkaMhn4cKR{;1=AX0PJ$#>k^@p8-hkDFC5xLi7-pA&+i?)(AE22)jm>u7ses63Sw@S-dGpAYU8CJ7Jua% zOgdek7-s6QyT^OG&&ZR>qSB~o!Xg9EH*An>(ewbh7m94m@%WWaw%QC%8P+bsCK#bla3tVp56JbL=H9+`{+Bt_>J76g#87}uO-0o$e0%5=xvAbhH^ zjyyYreV~d{O|B@uLm2Ttc_IlgM?f3niy7%}@UASOLKBe$vAmL1fGg=48C1fhpNUkW zUcxm|~WGF&zhm9lRoR1l@P!;xwLNStu3kN=Meyp8gW%*}0 z!Kn)b_1QU{Af4z;cuiKWjoW3j3L`fF20%q`Z@CVa{AMmLQ!Fdig|N#Z)E1l_d`0*A z(rfp4c(Ed2E(}ae;nQEU6dDsRjkbzF-l_uiCi);8yU5$O52FO|-9@)0PZF{>tXK+k zP-vbk49_J*2-Rc-7XF~2ED@ywkawIjdSzMaY1|pp_UTh;6+>QQujEazaG+8c;CTaZ520Dq4<%i;{PSJ2XMsSFzgUMb(wYkwE=KuZyPgqACQ`IYg9qepT%?Z6xkSIrVjkkz z=U{~wp#7SMAH`E*9;75EC$~#OEWjA^i;Lp`tRg~RZQq z`T8|I(hL=NwY6bwr=(v&FiPk?T;?SYID2h;+zCdm0OBBoKYwNQS|gcNF zwho^1FVn*~_tNp8psdurS7`{Ej^JRx)vK$aiO7`-`l)}2Bdw?wfmHLsUnbxv(QYsk zu0pcvLzqT%^HUH@sCXw5#^9d{EOrPl*Rz|pW8#&^w7iG`%X?1VROv_Yp+5i{v5JWdZYbC+ZLcB1&^ z(O#%Gj{%}JwXl56mHnSs4dF8Bi$LgwzsiJN5hv}q5k`}FneP>&tDy~&i3#1b3=Fj( zi2cx~;V!4a7>SdxqXE;ng7xpP@0kLvA$*waPZ6QPpoDrFoSI4|+$4Pwq;9z3*2A49 zqOQB({D5iQ{h6Fa;)>EDwDI0vJ}d7zcGyV*@KoK$%@!xCnOxl6>!8O-U?)N_DXMF< zKwN~d;jK@e+;-{&E(-F8w8!42=M>893REiJFJF!!(?ytgzG-QKcum#DN~>(`4AzK= ze*^9w^sQ|(8bZ7@tUb#?3UvaQT#uu^8&OBNOu)=o~_ zJ-+@O^eNH59;8{TaN*mH;$k_1FLrfx4S#we3+U|9U$v$n&0dM8V0G3byo<+=rQY0? zWx~+d50M)VYg4Z2&e_p6`jw^cmkHsvv(wDt+1WLit@&u^xp`$Frj{4iz?l<dCA0mOcCol6q+3@R8mm zSsVyJ$Q-9>DO5n?`O=4TIW0JFkiqMhZQsi`CjG_}YBCi6YEvukHKwMf zgm8#FZbJ7L!w%1q`gEGaqeYU=JxWAkW?a*F8m1xi-h5Oe64cDnT9Dd_N+wj5^_)`c z(lo*Yks)1PoY5^v%a)cADw-S*bOK6j>zNg$fk=e_h7BGB%%!l?74%{Uh+E0bJ4HE& zE-*i*kj1pE@~4HIdZ)9wm4z$H+|ym{NujagXx+z;+{7NBP^hP;r=gfoT8B&jjyowS zDNX+P;B1HE6w*EJsVR1L25ax^DtiBywkRMYbBl|LnK_#Z5&#kTXw5ckYHK6fQ|70n z$~p>ZRHbMD6^U{eke5F^E{r-*Mc@&zU*qiA-Ha zl449Gm&+EX55iFBhs|wUpDpX8J$eNk4bE8G2?q^i`;k1BkdO$S-Yo!=53_h}d!ALe zh~YiJOmKubzt989u!;ROO>7<0iPuk4qo&##efL@h&NU{wrz zRUnbAG5W+o;~!Be^(#L)HP#V!98&E#t3oC+o#S$%wRIHg(Lx6eD0SVB?TO}swz9ur z4~6oIFm{Ph9x8}Xh-~ZiH_Hf-^jz@Ik&Q^RTcSmu2sdOc1&af7pC-mb;NGyDiN z!v_swvw8j}vh(M2@iM2o`N6!9`LvHX7~B@o%+qBq&zeyHpR8d z16XVv0oq2yk#G>FeGH0pFVU|FWRr{oH**Ry_7A}!NP9FjgS6^hU0n8fdOp{x=WmgM zBq@Q^dOm#1g0yw}{%;tsv*Ny~^fQYvT$T(&jR=w9rKMyTPkp|XGmVskcHAy1!eSvD zVJUlqOwWBocFJ@2x5a2tCAZrN-_?bepFJQoK^W4x2TO4iM5z+0H8^Fm`+`G=p}d3E zo9u*t4U;(;A!g>bFvNB-l0y+npedMJjg=Aln#1#7;mD^F!AQFi`L2ivUuc8eI9kT21! zhtR|vb~4geK>HLeEjn?F;^abfI86=+H6YF%FEi^ONimepo<+i za6mL$7qe`$>=|x!27=TSqzQ-OQwDthr{byeh!288uoF3RKfhJ@U;sXtKAO$*PYCeT z_*p4(YF#Q$;t_EJG{M@iR5sjObWW!G{r`@->_zS)G9P~W;TBI@Z>%MKjC6q5iK_9+BVzZd#Ll0Fl6{hHXjK80rp7L0GfUAY2dS*#~Lkgg*iL<}{Zo}%8AM)| z&W%AM#~25eP=o@&^eF5=ZJ}JlQs6PdYXM>eVtKkpD1aV%) z6;tt}al4O)bn093>kG>_N8i1BWN2l;7DW~d6^V${zccQF^76cQa~6%p`rVIe9C-9f`c@LF|Bb}hlsJ`jwNzpwOqOCc9>K6wnVR)_* z+CnA^tU)XgJ_EQZv!d1-C+)2Gh+KDwFQNa5;Srxo;o#t~h8a9R{-_ZpM#aVId1XVc zJC0p2v$6T(*#?ju6V=3P=Q&0`6=iCvD9GZnwLeq3eKVMBcpV2i+@c*H(0?K&XTW#Q)Ke zv9Z%jo@-D1oBZ%o-h<2e+d%LEX22f?TMZk#ebS~M{(DMC49#p{j&wmJv;Zmy+ony| z@8G;+o7U+}cfgsHw6v=r^X%L3cy~_W01`cDBs{mKVkAQ-Ni7PewL zV0RD{?~prJaHYeCGTp)A@C8UlU&WU7)WV9k6+C%@%~*|wNO_GHNj*Vs;i zheuC!#YOZ1u_F_1HPK2m^T+n@ahyba^tc)*8IcQQouHk<1-IgcqRor!5SvD>0@0EI z@RzxG2*;uXpu)53%UIai?GsHK&mrdQUvUK%xT=3Xlt=`FOO}3om#Xdotg^z z)3f$o-5_x^6%e=y^-G-MGB@A(qlS<)fvP9*@sNm!QS$!;GI57QJBl;BkyW51d2@?+v3}Ab?lML@?ZSFUsxy9yE^8Uwsk z(Sh&Z!$E32d7|)6U&Nk%c>>Bfj!eV56NK(m6HTE=LGwb)vZT8-+rOsHpE6Z=asrz6 zTBNO0cNhzjQw>0AaJFhaP?wdnU7>_PbBU8fJx)du8JwQdOZ&C)CJ}_3hvYnVT1iG| z%kx+F{!cw3g-&*OYB#V`dc8X9N`xp`m821PQnewq{!4plh6j)_-EZ^2AKlj)P~! zEpHUJ2mZ$Q(Yu0|t)~)?`e%pE>9CaWpcX%#YUfh-v z0!79$MUOyMaEzA5Ae;B0qV~1UPt_|y>=bcI`|)} zTu~f3qq%P{I_7NaNhF$b;`ai4&+*bl<4gYmBt}E(Ec?VIt) zmSD>GIz))NiPKof634Wi#OrD#yyT`bZX`&wXg43XSd}wqhHhpBdncWM>Skrb}O8J>7SA!fW)xagDG(1 z>Ykpm#gle+cA8smZo235%2Mq2aR&$Uk+B#+n7xSi5IKG5s{NTsJ~@-iIHzLK|1rnG zeWYa#k_6I(DK*+mU}@Y%R=UhjJVRz?<~5KcBK<+UO)9GFo!sx?L&Fs^eRiz@2xFIX zPsw!BxwB{aqZ%wv?M^H;aD@&E2mV3JPf=DbuIrEqSa=l9S6~zq+RTTi&Z|k_Mw8Gq z%3XAhOMm+KkvN2gQ%Pm#mB{1*IR%B!*dz1bFR#$^@Yo!o{kwJPHKkH=CIRyo=Q^wi zIcq^}38ATG9PNg;I+d|y2pkukvsJ zAW;;)uvuvid~SMk-NVGMrTCc&SJsfo;8)H}@v=Iw-3Sehj4__eNgTrOuULniVb0_o z*voJ=^3RVL#4EVvw9F6&69i|1fdb%=eBpKY`};@haw#i%8y$WNxelw=F+7uf7j|ED zWhE8*FUc2^z5)lB79o*XEw~D*-`ImhI8NM)pEyR&Ng;k&0ASt9;!Jq5TNgYsLRb1& z?5dV+hco^m5ChQFrviRLZP>ucY2L5iy&L8;Au$3pV3Qawd&0D1qual*v?XNdo7ly& zukhlUpgCly-c>+rx(eQaGYadI&lee4Sa5&-{Mp*h?kb^8Ay-bg&OP2gkG?!sIDH}@ zNa|W_tZBfdDSr%e&H&_-7K^-e|jbqCJO_mpxeUO@x{eO z7{pL*I;IC+e+$UgHkQVK0{%7r$+UxfTic{xzzPj_Kc0rTE5eCHm-ELF5BR-s1el0J zR%oV>{8PK3BGh4auX)g5EZ-`8gIk^j2IQb>?1NX?ZxHCW^q5%4<&Mc7kuc`Dj2tL6 zHMP-^gRGyPyU~a9sRkw_@Q`q7h+ZvleI)|XPz7ls-!tnUDst}S#jr1S8`}utE~t6P zQX)ltFIdu%Cr->I4|l)SJcNU|(CT)o+|L*2M6fF0coR@65g5V&7~+s^Fb+mJj7>c4 zRW~rWX+4Ku&Y@q5P9c(X4`-QqeSS~KmQ2-g;lXMkX)foLDykpi52_+9u42yIP;$jO zx6f}ab4frEzX!FmjpOkVs#!+f3NELuj&>owd-rKJvDZ&}4;C)|e)8MqMBFyrwI6Mj zc*TVF^tit`s-5}1^UTxOD0yS!+J2XyRAKVxbG;2tGS~ZEIMSMhg`yfHHE7PXj-3B| zUZKss@EtE}visFaev@2dir3Nm9x6$Bx856Q6?8e-miKDpwmt9O{yS>Ub1Cl1l^5G% zU%zJO{b``>+guJugw3GgLT){Hk!#@JxAc-f8t@l~T*RC)*FollI84 zv6A??LTE`547Jpxmw~YEkEuyekqT4dNTe=3ZIZ8(un<1h6IMU)EGOq=BUMR$tt|ld z%)(+r6e;hUat517hLzR!*VAW*_VXZf@$>TZowp|_HE(-8idJH_>GljvZy)@hYpIYt ZKTM4I^e8Fem)?*L?AM_@+-L3g{{bDeQ33z} diff --git a/docs/images/chapters/weightcontrol/3d71e2b9373684eebcb0dc8563f70b18.png b/docs/images/chapters/weightcontrol/3d71e2b9373684eebcb0dc8563f70b18.png index 06300409b5c9b7af31cb8a74c6cfb29dcf551354..ba7e5b85afb8876a6bc762b0c998a0395a9ea9fa 100644 GIT binary patch literal 9924 zcmaia1yoku*7bvQcXtR1N=hmqA>~5~D4l||fP{d6AT1)I(j_9@QqqkIFDfbBNQ2V( zuk*h5-v55%8{_|nI1ZfWJm>7Q*V=Q=HP?>N(oiNQU?4!DP{gV#iaID11|{+z4+qXr z=gro@0mt&TvLfmn`Ipg<6Nf^vp;Q&+biGs7Cw%YedYs{I?Xz8IWy_fN(6Z5@B&Tn$ z&rVGKC}(Pan`NBe;a2t-&Fa)R1{rlPGs~A@m=l!e~Npb|PUJZ$fiRqO|*u59C%T&N3;Ql`@55*_{ ziS_@#D{|Shf)Qe}pPUV*F>7m%53Vu1e*0FQ4C~|5=ke!{43N&rFcGbDR&TGPVxY%U%b=v#ZU5JB?>_E*Kt{opt`<`J*Wrqf~oabBs zc7cq7T9<={^j%U?Qg3hXgvzXjBGu16g|@3cGBPsX>?<$v>%R*)7vZkH%&*T-?c~y@ zBWZejWm`C``-0ki6D@1&o|{fie*_d9986+kg&MwBsb9Q!k)5BvcrMAYW4zWqHYsV8 zy^M&dgg z6O7lrHXN(^w&%JqPzCx$@_v3d2nY$Cj@&U0atVj5q@&)ve$6H-N`3$SeeS9Ky*(ve zT`IpPPm&wiKmerdPa&&!q zaS^A|khsz?TGD;7`Ri9g_r)GN_ojF6IC6FKl}hk~0%=XXG4S#6uWsBmG$arZ5WovU zrE1u^F?xA<9UL4?pYBh&RB5)km|nod)YjFt{>#Gf**-eF&{~jgde(BXM5n)wAc)QU z?Bp;()Pd$~uezzS#%zCMLfgO~A~KR7E?#r@$=c6~E;@VS{j!CNmbbla z|L7=mq9(xp{pXL6Wjh|KIV(mca(dc)E?f0>Yg=1vN(xaTI&^F6AYHC zp6x)LNx0{cVkM$+F)0KWOca>)2S~^N7U(JHd_90Mjo9gj14h(`E{X}-*;shLLSd; z)%pCXQolD|fRB$9%aa{#)J;!M&k6?O{k$ey9c}HWSBC;Sq>hFlq;;jwCDIliTt1(Mk81DiWDt;*8Ve<6 zlqwYKgDB|UxnrfBAi;~g|Bla-sNjJVd|3HvqN1Wilz^Fqog#5kfS7!Gckf;wZ2f-K zB7w{x!j{_ud70dzq7XK`6ER$4%(uJu`go;MNj5e%#0rLAHVn=&lmay959OVo###oX z8pOm)D(YVn5D;HwwX$snV3Rj6pl|wI`_w|-I;6Z@I4yDfUuP4yaSAJ};4zu+-(#hv zrCC<9<0B*Z^++o8GcoepR?j>sa(w;zb%s$j!0A61{QERq_|M(!d4gL4U%jH}UT`Zj zip2`W*S@~|NDBV?i>?u#JD?FSU>4fc)HLFxg46aaHpprgpttyJr^aZ{Khq?T4U#v z(^$iRRFfEo=5m(R%HS6B=D~uyOba^|R72II(G(R3()5_C@p-Uo{ zE-Wkvq0vB@2I29N9}tQ#7o^Mh{(y{%ip!JlRU1U^roX?8^QUEQ|2V4)2us-ZrFHa* zPhIh3eANygH`lr^33_jt!`QjtY1rO2KZN0!h$YvYC}K;-#B>Gj(=KXm5VPC!Xr>h# zXd>fwJb(1w0z;`s4bBxH_QW4QevmORj5^8!h!kTGnl6$huJ0`O(}_7Iz)RD>Q_-@g zJCySB@(_Y@cnh0vtGg0j8%!>ZRY3gKq5V0UoPvTcIn9iKM1|WTD0wpU3*e1^AlG(( z`C^&y5y=1?NRT)lO^z#9&};f!(Qqczed#ZtUs7q`J*qltRmj6ic1}*rEae!QI^xg& z`BN#H;$MHhtE!?gFfgE#bbCKiq{0G0p#J&ur`9E^t=(PfI_kf_GCStd)N!yq*Wuj8 z51D1SJ>fMagzkaavVRK0Drq{4{Ps<&%&3fzi1HoH`|NCsonPO9k)uY6MzA6NgfEyR zgYYT2!zV%~AT)^?!>hnjErMmJ;lTUSLn7=~uM*42$-Vq+?c~ot99Y!$_w1qpcbb{K z#w~_y-H1n|>U-GjTg?ThlU*8H-mAv7J9iYrSdtEA!dSmmyO`~)4r{LIuOSH)(I+V*8SGx_b#@Ki;OE~WK^{R2*`&5X8IFrOf6kJjb54sBCDKIGL z0?^U)kdX^aU0sJD`-M{9eP?vK40CaDv1PSoH68c-PJTgw>(;a!oHc>LFtj~OnT$8{ zkB|$-VP|8z`L9!^Iyw|Kf2CaZWb}-&kk?r6NffE^I~2CX8-w+gGP<`uCgi?|uBq9& z+hTZv|Em9oeWirY&ZUjoC6+QHS3Dw0H^b|mYd0S~y7tLyli!n%L{8tp;BrD8&Xw4# z+`__kv-6|7^?&}lt}hVu8i)p!Am)^G)v&?{S^g>IlYgaN7xF*x#=^xlGKq3%9-6sN zBFPMzP*e7lid1^%6V(q#blVpOhI&a-mhIBKM2YVzGW1Qg4cI{ zqvIvJg17z;40!o##b!+}NM#OCK*Qv9)M1c9Ib~K|B)Nn3M9fmUYx+zSv@UEpTKf8- z>toe(4f~Una8-v0r5i!Am{aTg=QwgYx=ZjnQMTOkTYxg90q6dHC)-_?)wAD#SxGiw zCAYVc%05q%_Klo~RIo~Tsc?LDycwIE90jZgbK?!8x;HpDc(2y&BF&Y1C~e#(Wmb`s+T zAL_ikJRk*^olp7&A^HWkG&D#cfu@(1cyz*kxC1yiPBk(E#6vdZv*ms(Js5Kw1{>P z$NBs!Ztg-JyY`pd+@|w~+w@ApICyaT^-GjC$Bp6E(*t>TEUWqOefQSn9zT`@a=+-w zrx9Yo;d*#@hzc}De9Le0<<`6F^Z}z3h!#Tq z%KrYbMtzfU)1=O`=#I}TUJws2Uk1W75;0s;S@TSJvAVlDY_Y*mx*JgD6tdgH^KmZ? zt-s$CWIkxCvg|!}#|4BC%ug4@(P7ATEWhb!iq-f0(7*@?MXlZxK6Tw)(WByf^DAKS z`DdrdH)^QBm>A7E-?v}3tEx>+O<81)esO}5{8{Ky>G+cs!G^C)5&4k7i@nL)yTfLf zC}EqPn?(+P1g7jwOk!Q1DX%V&0WR;t*5us8KOErdy|!0YtL zgL-pvAvD*Aj~}TZJRLB;sNQ68&MQ}l+-jLRc&T5}UX`~{!ONGvp|w8qwDlz!mw@3j z%Qpn@X=Y|-@5;i`lGW!|7rj@Ds-KtGU1K;nJ{BJ>&dA94?y$VPti}5VegM7%e>(I_ z2`Ba(q?eSqM0w~J8HOAjcvI2PRGs4vHZ!`^Kh0DKlRC}fZtryErq~9UMarR(rT|%l z%P-&gU_M#@cndVSswzHGX))1-zqU=Yt179k7KcRpR_QQovsq-?z))e^7aAWQzb*Pk zZP2lN-!$OVgN_L2icuBO<;$0Q_k{G6$yPQt2Fx^*^Eu z^w+NkctL?M32~L`gLe7ZQlK&w&}ekQan%CI&F!A2S2R+)eS2nM0b7nN zqkp+)Qj_StEM-;uQco~7*7ZCj=MFKs_MI;v!Q#p%gg&_u�ntz!-8eg# zThH%*o0Po1&0 z1f>9|@2f)?E6$Tu`^oXteEK)}^F{7tg67w-p$NQVg*4FPkx+(x4PoQG-K0n=;G&Z%z5&^C1 zF79#U({z4T_i(xyVkV4%h2tqm?+Zu(9!> z64f7u2}y)`aPbHLN!JPt0wuD1h5g|0ZBwjAsj zWnqcAYo+*tMLH6sjd@2TbuWzo4Hkw#!w$s}@Uxwqj)0h$zK%nN^GIpSyY-6l@~u&a zl9_1n$!T$MicY~Y`BS=BsOiQZFQp=RU*FK6?tHwQlRgXi)Zx3?Fq|i6x#{NSR{G#O z$BpxoKfO3Zu^$D|qnR^wu^>v23Tfh#hlA@etPQS<7cEwYij47lFGxyC9xTb8M}TD@ zsIrD?Ye&n+32|s^YbVGC_(ySSGH&c2t(Lgs%Mqr^JQ=0qs-Eyn?#=0oTNgr)Y8e?t zfmXEKoNVY6;-KWwxOVLD8BPG$AZ0i6c;1|Z|A|DJ?}j7-DwN!4vN~pA%C`5|J`Ix+ z+r^-z;d^gKKY8%6_^t`Ph!E_LOT9k)Hesgb=gZGuy=;#h7_bjg$BVEZJ`kpBnIIYZVLTE9E3Di zn!zI|%747`OB-mJ+~m`vGVS;ms$^J&W=$+e+ShG1Mqyy$gilaZ2KSd>_#Xbg&m`kV z2MA3kVng)yih-r^Uf<{$f=l7ycvj4pnMwv$^z`*DA()pgUv5bfb(o%;>!0=4kv1*H zFzT`!4s?=Ing<`o{+d!d%_%i4tyT9+_D(Z0<)g!&?EWO}d6@>YRa^6f+K zS{z5_oVuQ}N6*7WdHMKiA7~P;xP0vF=rCwdp->IJ^GwyZEBxNqi3#T660^;3eWl_c zSMN3XOCvA})mE}`_1m>Lg|LW9E5*XW5;Iw#kKeS5p`bOdn0%r-J3AYxa!OX$tW)(P zq30$ZCNHyj9$pE?(Kg_n)M=n7)Cig~hggVy0t0G0zUk!F+S^=A3EhN&81kQWUIMzj zZ}hAb9cN(9yKq_k0&hjpL<|Lw{FN>n1HzOZ+zkNflhNU)l8Omn2Le*in?2dptME{4 z9{}Y|#45uCVxuMf{KU3wY+2W*fdzaWe$S2Zh%OdiA~0i^u-B&7q|gk-oWedk58_tC z!<{z9YY)0)&z=IRUhkC-5@&rBZN!uHv0s3jN| zyajqhvrYORJB6JMA3{9^yv3}hrY7hzc>|qx!vNlj@~2+HM=>SEFLwsgfg(;$Hv_t$ zxP1YLGdV+}{N?M{HoyZHYg=1Retr@V1EgQRd}()%l1iL!Q!EtXs2a$=O$$E6bO^0! z^Y4YP+pUBG&4cX<-Rr$SD{RZ``nj#|-=As0Wo`c|&_o2)UD+5%tjeCt#}SyoBETT; zrBtu2fb;!;1QA;j)5iIm=MVB5H*S>r?Ao8e;%!kNm43#P?nkjd>04rgHNWsR-W4EpZM2dQcouP*fLUhA zZ0}h!GXEBrD;egzew~pay3<>rt*fi+;HA?Y%1`gbl-_4`TMPH3Ni)JqZjYqq+xlsn zZt+)&WP*h2yZMdm6A;D+fCaIyUIjrxKknMa3^SZPcw6M9Rr%wDe+k@P8ce7z!29g} z{8>&{_w}~f+)$e6Tz61V&=#=RYaA z#PVDp9bBofNojGao(LUtt|LxNOgvbt=qvR<^?A-DNfF1b-U`HheT}|-7mSD?D2ZoB z%B-MHZiOKTd9L028)F;&YPSli*2&k#4y9bwM%|-(Da-b0K>vDB@iI^lxO)&20 z=mbU43e27y{^^E9gPaZmF`oe(%&GJo!K&@qC;`h~-!eU2NyFZEeXw|ixYr|h8%wlb z4iu2vmzFrPF7Bp;0nS-0_9RxFc!y0dEQEle3Z6PN5at&bXLHj1<)O+T0XG(b$P7~ib^DK8L$A10+3zILRDahUZn&$hZSgh6Qw7vTFb;8RhFq|qRVUHhDy(USXdDn6 zHU6hkAc%$>ry#^_;HN@5{`#JU+w>`$WO&)3`O8*UQ^sCZXZGur-R1tSH7B6ko}=@7 zU%L1fe2g)xUZ@WT?rv{`Z4gpkKw-OMi+0X>bmv0u@hGBxJ7C@c5lBVGs*7cZtQubA zc8Qv=j_hb8>ln~diPYP|TBLr>XDnnQ!;C@j-#T>U@U`t1r2ofo*LCK-^4shg**=+f_l;@xKR3gD@n{YOg1)F@*BZY- zvP<*EHWrA`a9HtK_4Q0J&s_kVuE*=ubD#Cyf%a&JC!a=b~OEbfyt7ZQ<7|qu!AY)~X0Tp9(b}u|72u+67l0w`o zttdUVsWo`>HV)Xpv6%r1dEJ6J;INOi174E z`s_11Fjni%1Hyi5I+%3yJqz3VoKs#}uX6Lz;o(f<@kZNne@<+CJZ`E6E(%gilq6?! zw6frHvK)B&@=#rYq_XKtBaT)I?&(4MZFkTjg9G+JW$<$uGWtyJoc{jp213{JU~2{z z;sve!z{3BC|GBDvWIquoOsC&}*xYECg~hvidx`BU8SE=Z-qDDc`99UZTrra>Ht?IG@oqLPy55IdzLk!r|iNN=J0U#Qx< zx)k{1lTU`Yd3bVuA{8xk=(tZ#Pou^4zJB@g@Z`Xi>AEKoGV#EX2)@o;I})<$yqRTp zV-0#M7m#XZrY-FHA}WF@;e9KYpdcCKU3g4P*9Z0Fo=E|A_ROEv3X_wQ3D@i>xYd)M zL9STa*f0d&2k8_Z9^MQgW-MYx+6@2n0px!W>Qbu4zHh}sSz1{|$Fo*(#D9%$L3#>^ z|0yCu3EeP-n>SIN(TrgbK&!S;VzE(i2KC-T0Q5qjTF?oC7*CG(sO$7NKD)07 z+T}V77k8RRL`Aj5$ex8GjMG7D_RN@H+>}7F?dYT8dW)b?OGgJbz^FVB2&|Q8DAGCL z!N3dUu8mr7@~$0^eW1k;9BA`^4}dU3^dh0tNGV2TWhF8TFt&%lowql%sTETrEG(ZM z9m?~*AplZQ1PA>E^YzeZM)7yRFV*92gux*xG;Qq!kU4mGc!D@zVHOZ6H8a~0rByJ9 zygw}2CVGw>wt~V!svZ-}w>|C)U#Jm16UOQf9ocxKB+|}OvVa))rF_6t)6l>Pv~llF ziSDIU7LKnNsB>E&Xgojj1whKm&BbdABkck*6td}|E6{%)zp^c5CC&drH|!kkSqcNK zp7h}}Ha2GHrDYIvr0aaa90kap1ro`!TFsgo3S#>{KXwk^7D4ie+CL#R4=H~VhDKl| zTsP~c>7(UXxkN-6VdvDS)t4p>$oCXVtmgjyJ8Sw83>FD0_xcazWzSDUz-QkAPHhEv zL>{+97z>?`wP#ULx3erxyJA?oKF9G$nuT)(TUdr!3n#jA%yP!uuw)u4y~gnfHU~H% zdja|m<+->+BQPIG6{i0QQ6VCt#BGC9G;peb^ew%JO$PK)WKXs*_tt&}gWtuNE&vL= z^HAQJ^2BJK%V&2b1aM7JS($N(kQw~D)!~vZ=y3rSJO@Ub$Ul^yAE)Ei$n|g3di3by z_MyCT3`+;ZPQomFG}>Zzer1Vpa7uZ`kKJA%6rbL;kqinQd2ZwwX(8_`_N_~+~b=!NjwyrKKKVPl<)I(p|bWtnp*Ki3d zXmLri?g&A2Zkuc4OLy}Gt(;ju_O6ceE|qIey|m>We?oHq2%=Re(7m+w5{0s0o`Dd9 zOYp&Tolo#-)M|?ePZ8i!C!&1y5fTEh9b9lYVL?P$&Im^*INpY%G#q^w|Dv7!6Cwq@ z5hk_%B|>nPY3%Imdea2jMQFz0hHbCFmoT?G^2=OVu_sxr76SV@I<)O*m4~CstMMgl zGw}2jyT(BLNQxOXUC}2K3>2)Qf;T=sE_L*fD-1mJyk$kj+lz!I{}0=C*dUA(nsshI zZ+mEetnR9)s>jgP+VV?E5~N%oCjGO-!+GhKIOxtVHH}ds56SB=IP&t-1wKAn z)+scozyW2c=;|tjGM{QRV!&2yNMfK}8xto$fdw*^xE}ceX$rv{LVps4ivDmnYRvvx z0|x5v9SxKJ>}U}5A|CniLnql^0y5y>m+zB_#E+46J@e7G#3@J zc|(o)l^;BKP;vU_&z~~u&lL5$Lq@2;(o#WhA0PQlEptHb&UFF;g6YjomlJpB9KTk( z2tUHY!byXoddl2!^ydX;?^<+I^$PUaWn@@FLPGRU9j$}dKKgAB2TG@Tma(jQC^kT& z^!TN7})U8TG9=B{ZA0As21gj9t;j^6-wEb+nrl1I=eSX#1W z(wM4rxEpuOBJ~IlBxtYc-8@}{lFtls`b7z{pa5f&l41yAW0R3-?3hd%`fu;EYk~&p zlGfDL$`$D=6zO9Mo}+UBVR_%=e|G>=5{l8A&?}5Oe)INiD`@T@HoBz?qWAM(pt%vw z1K&Opn)&q|QlJDc@$%YNoDV@iOG^8{w@I<30iBQ~7HS$eYlaP`sySWKl3)6#O!5C1N zNV)M?xV*gFu#hwEvzDITeZQCWc~U5(+e$`eKGA}2^JDqZqVpGIZ_`!pmAss1?d^(l z>DuB)y$9)e!{E>Y4040{MgZfM_wo6}h8Og=QCK-_>QMF<6inDqVOxO1(h4CoD&)iq z0w?LxjnkceeP)5SXS?@GC|ALJXTNkwP{VUC8Z$*8Td*AeUQ7Sltc6_NevYYXDsp{pipLPPrJ+=BX($%U Hn+N?L7(fSw literal 9857 zcmaia1yohvy8Qu_?gnWQ#Gs@kL_$C$M3Jt;!JtKHkWOh3L0UkhJES`Vkq$+qJ0%1p zeBbtW$Gi6*Z#*3i9GkV*T6=vp=loXCJyiu_0tNyU3PpTJQSLqpg++<{!^43m)Y&s1 z;RlY%T?IMRALMUp-G^utiWPN7PFlk$aecy7Lu39FXDcAsQ-H%;pF^&RK%E|6?w*8t z&?Oh;^=ES$6-p}ICBd<8GNr1Y=HDk&i|4sb*zCHnxOKR)^Lxfb8RU2n9E zoJ$iQ8+%>R#}|uXshe2WF5L{OS-ODb^M)3ch4za_FqGdLry^FKT>)D5d;kwMg>vsESJN zH+EmarQ~ua(i**+RZt+Wsv0)7?!fJjA4$VwR8NG$BT@B^iAn3}&CjEP|6`-(78e~K zS)^g&$jqik^7T+CTG~D7>6zzM!xlY$_EuK;?IJ=2d3i&tUv$FvLh_USa}!s}NMBlB zk(aa(7Z)G<`qixLU2*=MPcC^MHeDoLf63|S(3=X_I62MbxP>Z@u%M=vmd2)&iRVwu zAB2CnA*Z`E*&v0QY7M&c^y$;1M^9yagj5sb&Z3q^3sbrn2>h@gWGXM0F^jWva>}o! z7u4`82bLk{X=4*UW5UA0#VaT6CAG3@rEX*s?2XMG6-}F#(TYnEboTW0^s>d;_bN&( zEUc~$ZfaU0ArH$D+b|;#HEkqYP*CbduwbrnRr7;}gOH0u<+kE&G;Br&ER4uidzD zgH23~Vew0nth;;twvi+JA)@0$xx2eBjpQeMTfZ(K&{t_=G(Y?LHP!rNOH0c%ocG45 z$M0n>MsWusYisK`DUX}-p$x+dLX$7|nLIOZm6Vlfgtb4cw3@7cX}dPYPb`Hs41vUC zjKTPLdZNyrJ!||W@y61RI`M_iZ`1oLsHv%++1i@zxT&j?o0ymsJ?@EjsN}tN4LL;5 zOw<)_ZuP_Jg~4>gIu|r-blrM&W#!fO_Vdu+S^;2jG4rQSJ!4{G+6x`+?0l6uE%v_m zrz?;%FuZc!Szu#lPn%eWcvsigCrah^)Hp}kXQwgH*XMa~;8c6O-MwzDcmKXOOl@|| ztbvXrTUH$B!zU)bjYhxQwv~~==2S}#=}8c|QW#st#m2@I zulViVyJTnQwFT_QX=rHJIXL8I_frWV<`lr&Mn>X|_?=d=KSaEFlj$abX(#G56$qif zE`PY*vp!y__cD5Ld-IA`ft;!;iN3!6v(sZ|x7~h)u2DfkJiJ@ap7B*XZiZ1`(PX`O zvwp&AaA06-cTjD0_r_K{eqkx}o3zptd- zyVN%})@k#iQOFn~+wQ))I+mWEUfr<8r?C}VV(k%(B-5(kpHn&$4+9+}ZpE^4at4-I zFfSVU*aoB@K8)EG?D!7BY1eomy5T>6-cwiiVwQ9XobZ>1bh+$_-WGxML($hwSQgZ0t_w953?Ut0G3*!*VryS>rjy^lCoSBnjC1%-vv z^74qz;1Jyc)32fqP1O!vd%iRAA$avr0F0bFeUH3HNXw(|Sy)&IC&Pifu)V##HRVJ43U+S- z3wWW^m|>KxH!G`%2(1qeF+~lvPwQV75wzP|d1$x!gGp0UGrwYEC_|N8CQb7Ji+!2m@Jq=8v(RdF~U1~5=KR6(pN{V=K1gENr0Oe#%^kl!4f>`zXw zulEQke*>!wG%gQ*ASWYZg@Xx&7rd@N`C`1(_s*me-3O!h!CeDOpY#eSZ+M|)>`1Gs zs=&e`iAhOq)@`ZafE&|upQ7$WL);hXKRH615sXKD7xz=>B)Py)+mx29X@_)UD_bk2d_hz|@7lGD&|I-Hk>H8cE` zIc05a`HxRdV985ho*_~9V^M$N^Eh|!p2k1*PN9fo2q0lQw~QU4>A5*m6yMPW#saKvm&ZCb9brs(goG`g zSZC}C+Ti@bXFrGO^K<|@MDjP7>JB)@;Vdv!~oox^GiH?p|WMW8eZk9n{*LZ~$Wes&{ z6CeMx|G=_4JLGXu-bcMDxG4Rd9oYq6{7XN7$(x&BPD@XB_Qg&uE+)5WI1x_vICU-% zo%D0+X>EK2h&nS9$2yh=4KCDjg$q}c;9MRRfz$k)G4fz<;G%|>)&(vug*6A_W-ytt znOV@E<3%YrMNK+RoQ>`IZj|R#28T8`m6f~o`+KaS-1w-7l$6PseEJMX{l>6)?OG>( z+Mo@S<1*s!`FfQo6wmruziuP1!DHPQ!;onnb-$U;{*c`4LwfU)gZSf)>WT`Q_yR0F z9i8maq|zUC_}+cXM|~X?{`YwK`N35qN! z{N{JE0Vos~r;3b^{W$L=LtehE$p8oHR)hPA#t77Z`4Ep*Hn?LFRskTow`TQKWX_!v= zbIkR)H<{RidQ=|5m5humRl=Zu!CUwC?Q;M;#lG8^nPorc$gZgoS5i_E{Gmy97J(=k z8E0ACwy==%JcCP;mB0KKdT+aq)Y-4n{BYY*xoTKTck-tir7~j@XZOzEABXX#8y#W- zScLCU7ggfTmkdEcK~9@>`(%ua-?@+M|AO?@t;?7%U%p)8;qm|Qf$P<)S7jueckkZ) zSW|;J+@9~cz+o?RS4nB(&`McW_RJ5L)oUS4;xyq>r%nMsD3@6Z-`aWo{_O(ro!4nL zR3suY((CQ@CqQ~=#2q(UBQGcv*{_bM5>lY>{vvsFKxKGf;4J{{Ik~yyw6p<85j058 z$svGm^V;rIGU$OMCo!%d(zms4oZVgbT$ zE$zKYNJ~pAvE6xLtlBCv15>&x5x1;%fU=AFY1<6Pgz-+gUxsqL@aQB z%wVPpBae+S9cNN1gskiSMjLFN|M{HaRlN!#2L}g460n}E57jbyNsVOIyDuhZ)N{Z~3@5hXDaB*RJlO>VW`|4cx z9gf}=bnx}3Ln?-Nnt|#U7!uM7r8citQCCL?A8J3)Ds*QA8}L5&*u6)O9{EkwIIh+e zRQPlYzANaR?Fh$ARNDuJhX<6Fmj2kkTaAfZY8uYfDt6p7M82rm7hk*du}Xr+jdRkC z;+wT+09V)kY(LL#Z=P6EQnJ;rz{1AELtN!@3M$C1LCDoiAQqn_F0z=@FT1)w8MdgN*iSf~4FBo%X(_BTzY6f*vrbFYv=} zR;R~1YDg&o`0=R5r#q`0@Se$!TIayf&_@w;SpGr&8_XBB=c3iv`S^-@^k3KDdwT-+ zGXA~0yfpYhjn>}cSQqEhg>Z?q{pFnu529#S0DTC}`Uy3)d2o;tAZ06fh|rJIzIU=& z+1b(>8X8;RR2vQvYye?k5975DG$J2w63uCj|+u!#V~ZTvD1ECcx}-ypG}I&WlNocJXF}D1|UaX(y+TT|ut9BGP0HZQ=3cOQN)~acW5zTdk37u9lu& z@WWiqcfej!+<$X#)NI}?AE#eALLmER2jVd}b%a>ww+AkzH$+8IfES<-7*)hpPe<3Xx&IU?|qmmWIacfGt_Rr~&)3hJ0 zZqIr4*+(?!fqs7W>XOn@St~264;w9Gm#7Hx^eRap=mG(j-h0pXMEK>E|`&Nn! z=g*~I;sXxTg(b2IQAt!$TP@2!QlZ@lyMfBQOu}S6N`?=>gW@cxR zRZM=ZeFg4r{=q1n-Im@yu1sISba$!$@+dj^_+8#QUnH(u0!T#yJy!{ENts)@Qrx_8 zS=`8sUhYwIiN@on6Hz`~4ICO7VbG62HOi7o`re7=L=WdY)UI*Jzth`qRv)f%=MKK1 z>v}oKH3^mW?x)Rjh8Q`b!z*L4hA#Lo7Ctuxu($_8-HZD^eG23fag4+pFK`fF2FwA# z?3#n#P#e)r-iF2OQB%Hjz5-7Xz$?v^`hCfGApW}rHAtO@lC1i$~TUyGwxQHQA)#~V9 zK?{k-z=&zTpm!nU%SejTE;pyHu>9)%B;_RyZ|xQ~K)cWxu`W<$BBP?bTU+Jeyw3DW z>=3nwF?G}TXAE)w;p+TP*6ZBTCl!2_ltBeenOo8H{yb+9n?+-2s3tOt<9>IGIoSs4{* zuASZ82!DA^t2$mY8p6Hv^Cwc2k)Q>cDbJvW22@8#22{Z7Z?EP8U{eTZZUf?F3*3$9 zyfBg4%|nivnHl97F;aWxI{zTx3iU6K?mc+mbNt&aL*xCF+k`X9GbNveQ&id0v0MbA zlr2W7*lN7OVN(c0(%jrkMnmII$)(E5&5aM|r~t|l^UGazUq3(l-^3rfy1EbtM|k1> zq=y0|P8k^)GG^wm_jhBPo0{G|*jLMxEks8pB_X*C5!kTq4keSF9n8ADc9%#mp!-)T3vOL9=)`xDV)8yOinxw@tsHMoNg zHCpZd=+Q+)c<}bd20eHdP!pUNgzz;)u*79{Lfw6HwJBBFCuD|!gdRTs(|mU<7V6H! zJNe!jZ?OqD9$rKz??^x%i9b0j`ViJ=525$r$Fu{gHmLUf{8<^{iynWDSU~!?2Q><+ ztf-h+^N%0QBO@bA{pkTE{`x!$u-?k*s_o8#3P^C;Iyy})EoWXH{kjcUmyDi1=o}%< zC7(N!!HW@x{T6Y8MwUIXi@7g*=6`pi8l_bjwtxQg3J+JyT{Jg02dqyjO@>$pK*;y1 z7Yc_vKz80iaW1N;@CQv80M$oDN_(-o0<7Gg(7bC`uZCEM-j&T+Twc!pX+cfkCx0Y7 zw{Ou2Vvr>z3>pWL^g1S{X}tY>m3`G^qrnC}50r`mr_`V0>+)fYK}VYnMx`6}YvFs^ zh1Wqo8{>4T8yJ!JXc^VQp`9!0Q{pegc|#c+wYxe>Iy^j#1Sr49shj0wJrn2{_bySz z2|Rjrb9_KYSvf>g!9Z?7u@KGvSM(-LvA(H2=E`<#b%M4)6Y3G_DXwQA58N!>SR)k? zuY$US0Y(!i=_(9qOS8~0^yh4cEZJEoy3kO`$jD&j<0Dg5RRydo`fLj2nK_v8#zRAW zf5*tzTcM8>N~A7AR;a!1c|bFHA2i_%AIzrJ|KAZN7g zt&XD6=>Ckrbl1wVjte}koSc$oU)#fx1Hv9Ylh zCx(HagPMo|wcBZ&J{J*p*}atPwtWXdpNQOl^LR@Z1jH%uPz6iNtH>Ili<#Hn8UT}f z?JV}DL$YIIW9uU9_y_Le^x28jFWduBU%Jq+Zg-zENI9a(51KWX;ILxCDJdxUoVQIa zf7I$YtFC~XfUhBp!0Bkc(spHtTU1;e10R4`%N-tR`g|Pv)Ya9Tb~|_r04z3&ot>Qw z06t_Tw_jhMbUng_LnWu67{K3r(`iSiA1ARl^Z;zk8v`E%k8O5l5)pFVk2 zR*J}n(0RRjMNlp@F)>l>@#j={58y8X%9AF~M{-~P$X8ra(!4QIBVPOR#XV6`QC0~F z#_8$lk1vme5hX7qgvjk=Z?wx8+7*7$(Swo}D+8pFReT@$mWh7-`qgEeotxY8we(43 za&j1GAT%IQOW(Uk2CbU5qZ+d1#;eU47eSIgd+r?DrAv6AslZ1ms;PAVh&~66!w@=t zY;as(Id<}zhwSMl4(NfX(9lo;ZAhY#2?@bq=9vk{Nk1S@dyZ-1IaH;t*%-;Bb1_~$Mz2*Kv2b$7cLnU zl`jMeGl{P9{mPNVblXK4800PIO)9FYRLr*4C*`((*n=vwgREz5&9g`t*<>6?$A+jfa&oYM zEa+`miM(?Do_`$Q(!qmbK4oR?ofE(Ia2C?H%6}opwQIx)A~x-SqRF7?Hr*1C4wcZP zQi;A&ajH8U=nn-!l0%>nmIixQyHdXeAQrrF{i7 z{ozAu$X~VxTWB|ka9EqEosa)T&yF89Hny>qRnMW!5Sobc{)`*jz_}J*4df^EJvQzP z9?H}P+b{^RXpAz%^Y#R+$_^c{52{HrASxh6BVrPxB9+{Y zxI?Br@eaT>9pFgOMM8Q$e|&nzhx@c*ZU=mT zq;42>dV+493ZG4Yhv%7;1oQnJLk#m-g666O3pv45yMjr)tw)6RJRn-XuG!?Zn{Gar z2<^U}6F;ZRKtm}7bfEFrWn6W1Dg*Zf?vt!8p7`zhpk*GX&ljC81O+lB@p8m=BOwR~ z3@9lmM8NHvySoYRXHy9a3ma^O5fQfde!kxw6!`56Ep!lNK|uwrL{HKPs0>H}0I<+d zCLW8B_`|en35PQO)?l}H3?ic_0(TY@3K|X$4O)Eh-){Zlk%k!nsQ0YkA#JK=I2va_ zgrK{|@>ESntDWNe&1_mg``J*`@k5wEGfUsjCgrqWbYBEb23zrd`)1i4?3>Rca@pMMR-Dm8g~-GbN1 z`uA5*UaK=iWYGqCdW0&Yp85atMT1byj{@yN+|a7rH7EqOdg;m)pUO(2sw!p#jWrPc z|F0Ps4`fglRaigWd10}Nqa1ICjp{2jl8OFQOD!&2zr$wE@K z7@nMH$ON6aYrRmvihUP4dBJYGZ@vc-80+1)#JM>mKmyrk=+0Zb-S6p1Ij-zQ{kc8++4n{pf`x_Iv zZUyh&NdtfL=X2c9pDylTwy1mzcT?cP!eC}W&VKhGV;axG(h}ZV<&l-DbcJw4X5Lxo zk%fF+GY;2O(m#HryERx&LQ6Z0G1MJFI`}cYdb+yJ6BA6`vAi8Ylw9BPb^dkFV7DIH zg$7q)3h3Q-bwGP2PQqD0*ycxG|26pw|BYW-+uG1TY^@Am6vjhcJAf|gYg+H&;j34# z$|))aKDJR_{~3uBAXIHEIdXl}1vE-pINMA&qST%gKz!M5hGg-o27rgo!Uov{1nwi> zSEA}Fl$(?D+9OtxLq-yD2n394N>Wfz|AP@+ujI{WB2;>i>SD!?=QG{%9-B@9}k->xyXpedR3cXmd@7VF;m{$2!~AFZUb@F&r(q-;o!{_QXGe! zMV5{zqB1U}sd-6J(fgh7LP3j|*_=y6@<0sHagiO05Dd|O@1Miq@`AsDz8@eefd>6B zB6-($r8u#FA4X)OQ(6D7Pelpu2`54e$thI*glnCDOC;cmF_tGkcI|GN#_K@yC!IHE zf*{o1@u0FG%GJ@DwBSpYby}7+I-6${aiyaiwspMrx*botyD%CW+B?nyWQ1nsuD{dn zs3)CE)t2fSnwsrnW7xtE=h(ity;zX_rB>3z!$Uh4mtMbT2oOlVVrEadVJA*aMp|0L z57{K6&R|rCFSBk#+vq5kjg1YD=e1oFKG;=(j9cIk~RlvX&Rf z$jH-kbCd!CI>jeSHcHI!qxPwbnZb{bY(3=;TMCYc^Dpu6@QzPUJDS|is)!T6{@hH= zQ<8`rsLDdW^Cy50at2riE3*jA;TF$ zUDA#*S)|HB1P-Ccq&m@Mog#K^?7vbIL4o7lNRXH~h$iGYA0m&S$4%*_08u7(MX zhzG))6mhmS+_|(w`sdG|=~~-N#~VyaNrU6%_UQ`aka-@9UY$xbv=B-c;%eiswL)ki z0gWz4@8&k`cwDzKT$Y!YA*7_F=>l$8I?VV}KXt`wZ59knO@nL}8di2EirGJpN7qXp zOyg8%(b-5GDKA{TRc<=HT&uR6!g%>Itj2n7bwXWVwNBAysU=Ft>lV7<8T>AZ9+Bhs zRiDP-xwyDq#8Agc2Zo2g_?nO)LWQ+5XWQCsS=J1Xj*cE=tygCU^Kg3+pK^JB<6vTD zX4Dr;ez9r0T_@o4C;*mO?MN`Gc6hg@bHje5)yF4Y(4%H#SWedayi`y^PF`O5{rmSv zFwDk8L1I%=!w(l4J7;HK33^<9Rq)tIP&cr)j#AfigQKUXU$*19 zKJG*x$&)9nvYbNejV68u5pX?*JcCE^O-R7)=;&DQ`QQMjtV~$F6^Bn5(kWtQM#Ja4 zPxtVk|8$E_L`=N0wG|S9&%zMdSi@%AW3c@6HM#w)t9{c{`;bbe* zhhe@Z)#>TsYH9Q#DJ>28!-o&+TU%@W@w9dh4)Kj8*$4n%zs^_L#>WvHex>TbPBO&1eF-n%_!h+J0FzQp{_-=$(mMq9;} zysmVc8(~i`1C;2*L(qtGGdruQ-Z<`0W2*nuiSxLhu#ue^cKKjGIQm1@@Y5$GVc}Ny zz40sK8fQY^`{9qqcUcugD%{R-($9s3g>&=spF!B!+5c7;OJy+lhb6M|Li!#G-44!o z#^lH_SbGR6YirFlH8t4>87wLfTSL%E&3^p&vELDdJTx(pz}BPT>dMpTxPv7tBV+jZ z_{eU(zP`??t*u@E`RmuOeY#0aPUF$ai?@*SOrJ;1SuTk4DJ!?v*e?BPa5>Vmy|Xl8rl*f{E8TnwpN++Lv97LFU-Tej(Q~^yNl@1_ z)UL5I9L^NBwaYv|m`f_vZ~b<1Z;EkiGH34*_#(l&q|=g09ED)-%6?U<_WXl^B4%H3Vli7969?R}rc+o-9Yx zN_h{UBL6CDeQa1{8~N&*z>ButJwIKMyMIz-#jJkLt+hlqKn;p%NmEnP;(5cJ{D`+k zqcc+)n=w3C_!yzv&{FN%pY3a9c~N=ZEeZWhr;k6!Sp$u0E}sZz@lg*#g6(6Qrd{@@ z+c$?&Ith%e4(IEw78;!F=cr2~Iw+(@sv63e^jp%Q>!;i3Os}K1cv`&`FHJSN#Kgq( zgSLl;h6XJBPrc@W?Ck7hP+vQEaBz7~CzU-Knl4~s=SYkrPh16dkbZ)zSX@vLo1({I zyO4~F%ZpmA%xxnvH}}^=i63nG$n0#Ys)|Zq6|HUN_z?6>^aZHi-2C46b#fj?70C}L zy`&}bXa-3uB*mY-~tXj*z| z!2Xyra&aX#UrcIS&DVYYciKQA*wn5 z_wPda{*CZQu3bWj`mWwyDfhTsE{GgOT;ILk-O?)w2?^+_6m&lib#rwsUtKk-KayRj zv821ZQKPBtyK8*k85=#*b15Nqmz70C#p3MuwvB>Fn~-th>AWVwS2@F(QJf zhd@?AA<_F3xd)%cFmN z!%afL+goVu>;^oWnmcut$JHStGjlZ9fpY7)8t9GP;oI%edH!ifydqlZOarC1ob-WYU4VOE9QvMPy=#uasc4M zZ*FgYfqq#57NTEQZN^JA>DP_kVnvqR9r%R(UQ!j1h zSijLZDkwOZor8m8ves6= zhrqbe`M_v$a*~mcPm5YY;ecSHR8cPun}}$Lk&#gYRL~i)ATNz4bU*94ZQz=jns$Rq z{RbccCS9q8WZ8G`OeVEC^u?^lqwd!G=n*uXuU%#lJ5 zx?4Zp7-R%B4m6euP*@oRBrTp;do+}kFgUom&0%0*7=?uN%pSAELVkV7S77TgX)L#1 zYH8M+8_QQTFIFr4@<{UP)d098DVdr2&CVMK;2gLqDJeBy=`0Er&!HJdmTFbMyOsJC z+=5G8WTQ9|CZnQ)<6viJU}k2k8N4qw6;BF_`^%9Fq39oLwhXw{soz{wPYO@~Nl2@VEzlWf>sK^Ll z^wFy@4DjqCNC4hlUS19i72$YU#B{E_?O(`M@%UYBHN)Q3)%D%DHyX8OD&5uT;Xztn zzPms<#|~Vy>FMdJy5@!x`o>|ROgd*f*_y9HBT}={42mo#oB;q*ipzQK)%YdP_Jmt_%c}iFq5@dinFgB5+ zqP*NUC`clQ!@gbzkf>s-?s1Z8DZo1ak;UD&3lsn0isbM?;Y9zTm7yY8|4lKClnVO( z^OS)EI1KjxH-74hikjN5q9O+CUJ^XA#-zch%Z1L_EdowUS@!N``ROs=WfV>%4J|^II0I{K~~z zZ9e$d6Uq<4e`~7B4o-5elgCAIEeyov(FND!6CO;#yg)3+DD z@_6=^EfqQOA8+4=W3UgM3@a9|yKoYa>>5r~Xso_;N#@t=e9Dkcp4DojAHw$WcEvD2 zv8Lc(-p!Z7IGIs0Ic$0XInKm@<7n(TlE(V?`khh|Xd?}g)=48LzZJ@D)E9`EJXiJ# z7=E_poVlDV=To7?WCpYdKO!qC&cpE#CG%@gSU#be*tEhT;}b&KIw@;e@8|b})pBrQ z{UDz{ed_!32R>;KPHExcPj4?mR#p}$Ok?(eU@hO)2V!KuD5XyVDn!-0K5Vj!)+E;3 zU4fmd)SnixHq@6@Pbd(B&5uC}q*nprbUnjymaNzLpRRfBQE9fyV3Zbu? zIOtedl2Y_EG(v$?0_A>yN7*?#j=%pOFT?cjWf+_~NsoPM)6Sz&lonLwDR3yCMDdz} zT@q7-z4lC9&yjtN%lUOE_?q@R@J95;vbd$mFqQ#bSw3-fz5a{Z8zZ&a{>%m9TV@L* z8F@`8B&7Ixcbr~aN{Vni0b4Y5SBaTCut3yui26L8C|i}tn9?zWU&$;rqn0C5E(%lN zf)nL>MKEPbJV8{P$p;(4x_^!Eac&I3+T+e}yNvWBqnT-LUY>vN+DiK~?h^ecJa!I2 za*X7+WUL})d9;&dq|pfI^KEM*ddYgg@dywj;m$N4DR7a)$5KaLo-`?j^8di zR2}#&XirGWtn6`u;+^;?!S(T+khb~$R~bTwAqu1y2?}Glyu7>+-?iQxGOU+V9=cTQ z9VcrGCYzuZG|(ksh^l09ZeoFptQ?Taa%O7Y$dFFUKZgoY($Xr5OGYHW4f!80jZtm% zLVv{$r&SV>{1@~erv~?~WmQ$-L{9v{ETv8Q+x9H50FpE)At?#}{=UtuG}B=<>k5v*jC{oZF6(l zhkR%x0-*pP?oU^U9n5{00tbi9eq9>nCA;|A9!)N1VVZ29D7k+?06lnuSJZeB?OSJt zK*9L(1uI2ow8mQ7@o16QQ-~DE49{YEEuHEEJ57!Z>AN=U4Q*|g_LmC;Cu+DeC_Q|w z&BtOhu2JR|>miWq<-ZVDiE5G6$2*?$PZSV?dG4~+1f&*e{)~{bG7;pU1Q6%sHStD4@6NOnF^j&15ax`5pzQw>LDoRIr~CnBBjD zSO*Y~<7KwgPK{$@x*gyG#S41JU%!rTL<8&}g_sWqh>I6`P^vjwYa0V}!T~v9AoBoV z90Dpp{TMttN#v)oJQ0dG1HgZIyl$)HJOLQ+tFPyMJj;BNR;0mUs&v1w07uIJh?`M| z!hCGx;w^cbSQ6H6xRZubIgx?b$S_FFsgfYOo2?@n$`~dNix%?XNCDLtg^i8vdL@LE z)$MfMy*oevR4~Z6?(gsG3(VqMh6n8gc6NU_A->{6ZEUN)G6Ymc!Lr&MnA2sYa(-g3 zj@YyJRRd;$^uuO0L||fKvb@q6LPt*z1L5J}Y5Pxwy!iQbyv}X~7LAxs#kL_P2Hncq z8dssw>y9fJl@Pv%0Py7xwl_dNT-~3k92Fn^%ibA+-dp#1EwyPOu*$3R8dE=b zSVrn$2dUHCt~{%JTJd;%@?%>T8`Pa<2yNdua5)}aC4__oaBsRm|HNwNOsDX(1$EgJ zpG$X1Pqh}PQxO1G;pytCntKAw?|Ql(Hk9;4V4TeJnPSm(4YlcN(ZfDJB_#!D+W~Cw zva+&>&tX%pOG-+X_x2*v(~0)yYSAgBqE+hTK7FDW5D-w((_7rW9Gy@f)hb+7h~k@> zo?bp#>-`_wqxIUYe*HK1lKA`gSfZk$tX9*k3E#}$B(!W``5vxsmrj5AI6X7Fa`*$b zD@r&9JXrha9~}R8Y*a?@uMs#PZ;y6Fwgm+dA;)X`mIOTfIJ9buUv@e!2cGL&+l1@L zbtrwgz~SxvqWo1E;-`do_48om!dac?d7Yne*QfU$QK&fI53u7JcJ(|AA`O9z;d-^; z!Upu6pt1e^eI-pzqzVnjE9Xi;mJ~Ahf&k?{KUyLMoF$4`EK~@<>yE_V+T|W|3iH2P z&>@KDmz`f!TTm#Rn%uFuMJ+c0!H-@xxagk(ZW(L~x`l&&a1zHp6 zsl_g};$!D*pv&b9^ALp=b(NQwwp33G&BM{RK0LGI*~U0zc2K< z+p^z5^W`JuDSQ|l#r(`S-`1&gIF?p%KEbVJ`P};}tOh3UXCh-B9n*3rD|6Ff(hGjD zj`D!Yfm#K|nH=Y#Gr&QLP;Pz{yZxD&07*H{M+2C1NSWglA?Ey{Z!Px*(sB>4y4vjbE4kD+g)+3}K$X}Y%SVqU z75+Ki;^_Rd#X*{uMqGWq*#p0s!hJa+{9|&P#-RX3Xhp{^E7#MJhh4o+0G-8d(aXyV zps9=&=l)Or_>QG>?F0i8(+_G#jEz-}+!Xea-xg&dQ$*i_nmdqNe`jDem+01_!IygD{ZvetSzh6@LO}n_L#(yEnckMvgJ-#3M8j<>8d zH4%&0jrymTjkAZJUUmf|cfLHVC_H91eu*R_8ABJeqk8u43SYAOPVxAQqRyaGRWjf z@E?QQ#9^@wJI}QFwA%Y3vEazX)#RD`hla%2@=pvUUo$?p`}}$2tqDNjqvIk{Hj7PR zKnq%lU}^CMo&V+Aw&`h&y_}qCN#F>UDt#3Y<%7#A(Wd^U|HsD8E&u0%!OlCRfY?a| zWo9Hav>vug0*oLyB7-qobHa%JVI6PZ$URUpwZc za}OO8*2;YxR%EJz`a^0Fy}CHbi-nWkgJz7xlZ_fvGt20|=8fg@-TT}2yJD-*@zvGTu^NVGRVi<0Cq4j|$19zPC@6se z0Rck+Y}VG+drluHXD4>U)2qZuAj`M)?CC;Ek#)$A(^Yh&AMggeYu6%V;OsuvLLm2_ za|yKn+!)WD(tbR$%{ocG!2AbtB`d!%t#D=|yO$7$rCE9-u#^#0x4tnOTg1eS&<~li zdq)*m1wZgD6nIladsAh=IO1ouo_%9tY6=5^Ki>_h+jl)*3u;5QZiCzhnAd&GfT6S{ zB;q{|z^jeM$flQP0&5$0&Tqt7Qd-Jp&<06PP35`~=@38D(v9QTD|UwX{=#Alu97X~ zZkE)dAT=ii)@rbO9tD4VOm4)+wMnym=VUdPQ+&u91<|Y=Au`j*;+Pua1 zHn-1c60t=Ra!wh9~Y;0xo z@-%zmBF)Ee?@GI3?~Zu-@|Y1q)<+9r<1k+fNFi+-A6LnllR_McOcdzj26iZ@7%Z@^b!s(hFBdXuVhHJ*-wVJUah{to=3_uH}{$KN6-r zPVM?~Oe*;{g+Ka$@%b_MnB|1VX?bO(<;X`pB1$wO*aeK1WhY1xDRCr5Hesp<#d~FC zKcL74{tb()YT+ZMj~J za4#DR!h9HEh)Ca;6JjQL-EKa|2Pt?&8-tPg``=jr1*2>>N-tFRJ&Oo)BeoK5MNoWPRo`Q3UqGQQrv zVXs|<8i;WK#RK&YTd{F*Fc3r}r1tM30S1O6_co18f|!^uf~Ko2@muasWt%Q%E$i$z zUckb_rV^u*iwhTK7tk{^2ZPF8Sy|~=4xyx^bai!I9%N|^tgWqW;0`EIE`DwQ>&xqk z%QXSdt27oX;-aIjaYxnHTj6X~Uauf+lOqIc!p{EEeXMM!Xf3z>gxHiv%W~f4kO|#s zM`dLa^sPx+%7=Ml0UxCokdJGYA%ELR2<|_r^HkTW6%niIVv+}L=V$r|`J?9z2L|La z?K2cL?j&j*NQEh*)Yv?c0BKMee4d`Jp#3^xBBLfOXjDYP-b9#~n7HqNjDs^A4qDNjR2W2BP z5Jd)m;IDa|RZ>A_ZOhwQZ?0psnzu@U30272;4N2&t&*}0ud_l+Xo)Nw`&g#Q*JT+= zBzs$Y!`*M*cDn{2Y-bWtaNZ)aF#^}KW9TU@NC@|+6HbpspAPODB?8Cd9Y&-K1~IX& zt)`O>Sy)U=FM!U5hK7E?k_Tw66;2x*ZiL;IX$vbW6hLEv9~=#j$*NTPv+w1d$i?xn zX*Ww|-u$0Al$#mYCc%l90lmBfL;9-p?U5VSFpwqUtM;8|-GB;_9}WD;x5i@YOeyk$)C2sl*4nq`0%G8uv$n(kgyzm;rysdO}!F4Q}8U8Yv z_kpqr7)HxyHx~SuGSbp*U#Ml3fHGn_nAp!$gh(DNqAL*v%w6C_L&yL!xXpO=`t@ta zgIQdPI7BY43zkQ$hbd6|8gdF?3e=~^FBkoja?YoDQw;8Vt;ey{WfO!-+cJbnUNhQ3 ze4j1P#yL`bCH4PXCHmvvwuZs$t?#Q=eOeG&((hF6plLioOI*YIUl^* z{GArk>~YmGpH6Zs8%q9HT)C6rnXw#SCdxE4M;^~f`Y`Ld;qT+6r^WLUbQp=0z{CumA@R9TG$Ns=PJ#9uywwLV(OupuQ=<{ z0U*_w9wj&mjJ0r56ekgI^`(gSfhczlCTIzX7#J8tOg^l3BX{-m7=aFv%w_!w$o9wx z2qAdYGKeE+)-eNGnwnw!E{D*335WTguXwx=T<%ifaJpyPoEGpaz2!$&Bue!l-r~;V zo>u7l?p*Hl<8^Lem0DCrXkA_1_ka+5y<-OxDYfDaUi-EUfjDQFPerGv_Ew8cFDoi4 z2n5~VjE;^jA096Hmr?|NOy!Dr1!%BPCOY73pJ0qFErE4~7#JMlN05gODxfG zu`~(!105_30$=n^Pef469k>6&0GkX{{?(b$#=)*~K!qq%Qw7#hah9hS;n_TH$sgz5 zn526InqL)@!td^yhs3sjBRH3#<#A)UVpReK=zVusC~(&e$UATc5A1xkOGEsGL{ml_ z_NVJnb0nJ=KJp_ArG$KaP(QX&buro8A}^AqazEb-t27_Z1_mY5qMB&v?Z?JDO>6f0 z^%KgW{`4Rb+m>kHDOxbXXZ!pIeHXo(`&`8bRO73|1u<|9LH{Jj3R^xpidkMZxW2h* z@60(FiNZ0&3jVdFTEKt}3mLMOH=OGsBBE>v5F78DnRy8z!{R)ZK+CpA%?N!C56^WY zz7CAJ?a5N$A5HsmFL`2b4QEOLu?<2-MfLm{0-DNgyPF8{17-#x47v&_De~`2vbz)E z(Y2u1206K+vgY^9BLb>hSgQ0uax#%7=-H5X=g?0P(R{r&AOSXbl=ZJTIW`Da{XoND zPAVSVDyXWtw0Cfb%gkiqc8et9{ng)}Q*lj{%CjB!^GMiYxO|Rs;KSC0D#+^q)nH$bPew%0y6IiX{qm*k^GCmDLl>k7|-?2r)ECyDs!rEbHAoIbFi$5%@} zoeMck?4(ok zM#JEJ+}$du09=*a4ak^fMMW>txNX}ntvyay8*l07gcjSAh1T@1uh(L}wjjS3i>IXk zB+?h~Mxm$sGqc*y2eX`vn!lbaW(sxxdMat#wB=ff>Af?sunG1OKFo>760H&^4Nq+V zTH!Y#ukcT#%`p@DCE}hi8(7%_XvQTcA;oX6%AVKe3TFAPY+vGEFQvi?4C4LKKK?@vEj6{~@bR@#;<-x*L|wy;XFP3lgav}DSm*lX7$ds<(C$U4_vKW?Tnrgz z=<@7BasLVS!rOXrC&%i$6bhpXXeh0`rY5-X(?t$56&4cEU=4u_A|)mD!vTpLvd|9$ z(YddFo`xqVDN$dyXgPcqG+jqWs$fGy@aA$OEQ|sE1z8|LV%Sm7_iGVWfg+)}6q^9V zX%+G1od8?blLdlwJ_nWCRAwDrdEgh8XjUErC?c<^W;&j4rwVKYg-m|s9R z^9?6D9_YHJJ`9?N&6hN-e=cmz>|3!$Jn!*6vmZEVdM-zc5*-_R$EAElb|h0pF`qY1 z=dO41GeH2)i6RLk^^DKAM>JZ!o6ifE2l|GGdx14u?(^gg4c#949+zqAc(BX6sg zV8h5}Id3i~j9?1}=`C^!3emYNjAtXe5$UU26!Wq_tsGm%yC`&8V*RIXZ}v=2$pQ} zf>p6QS`K-8m#~P+!H?VOxoC{iJe=lj~VIlcb1;QLZoc6 zbe|x(w#|`Qt=_%}!b?C>lMKyb&CQl%#0%(MYoDFfnl0r71qyWi^6KgYq2>74#E%ye z(dFS;CAlXptypomG>W&vu5cqtL4mQidQGk{GOBB|cQiw&hKAk*Z$8GtXIo5%KZ^jk?l#Swk??p;K1gKeL_Y|L811SJF#?f26T(NXNO!bIos znigfAAC7owH&TzVcjY9g@t7DtEPLGZIx3>=#E;OVq1=gJQikaSGaD%wNKgrhiNVk? z-0nCM$Roi)N!Ng*JZ0_=E$g@s@d8-PCVs}N2#AE<_O;D9Y;vy%TzxS{`G96OG$a?w z(;fB4)MB+GV#=nIyH->vZOj__* zXhK_2To@0oU{tfwcS=XU8>>KK&d(Um|`A1{y`68i&xk-n! zsO`Bje2}~3#Pt>wAVJTOis=Cf){#nc#riW~m#mE`3I|0+p*kPT27#pNN2;1%^G2?% z3%I~qmCpan^L8g-3*+?pc8fiaXKxD;QI_YF-9|hEKzu4=P^MDQBqcVf&H2zc}CUe;`_DDg#^xP zbcAU)RUao2iZ9FH+=3T7-rhrZkb0fsY`J7zsC@Nv@T}Fr(8$og_K#qYO)zyvOaT#*%G2+iF|K%qytjorAU6{%1~;&*wLB^6c)S6a6rge}?VF zRmA>-uJYGQ4V&~RRInnFwh`$K{bB&ZQ&Ht=_IGxD4PfKY`RnD&#Qdn`AClsU2q*g7 zxmuz%4h<+hXq1$O!8=HXpH*5P(%B{K*DP_=Y5S=3!$+z+1YuojOA#2p2NWzf)9gO z@7T7O0I}FCg=f3L&Jpv<0VZo9f7=S#krn*B$4&bKC+Tx}H(VEIeFVt5exH9_80O-m z12cr|q;jpUk^^QzT@_zq^A8uhORD9x8*l$TGiG=@+s{R{XiNmGtov`G11P^lQx$E) z(|f*zWMt?@eA~7$=@RLEi`-QEy}@4X0(%vSyg_7vmI4sBWKIvfU=(71!_MFp(1;nr zn@pK?>U@DzsMG3AoR~QAtQ2@z1~;c0b>0triOjm;$;kve*jq^Cs5#krEr3%1ZOgB7 z z9Bufr4=naqI95w+RI)?ff;Soss={I=mkJsAPM_Y#mV+sh_J9gXxDIbXGi~w+NsRmWm4nZvMFZqSF4Yy=z8=OIPz_{jWLM-&OcpK~O+aKO-da0Ex~=#lIaPzavL+ zEM$nghP6A5R=BJA!6&P?LD|luTbAZ$9(46XIyDy3oK7bWTBtD)?XHgvk6i@>aD=)} zM%P@)JRy){&WmouM$BWj3+OOJc23ToE^q(u-xA*5E%WP`1O(IT>sh;prvJi5^hri($m0l`89av1U<{J~C<{ z7g!LHwTYz*MiP$CcRn!cHsHKg{|Ph*!L;!0L^FI4u7IX1HhN95WKc1&S6pX4d#>Ii zfeHX(u3QOdP^}4t={I$QI|bCaS!}L8E5XZB3T0`zhTWSi?F6tl8kLai<3>bl;Kwh} zS28s*d2VcM3C-yFTuj!lE5pYWK~`@WK*oa%etxFg3=RSkYG7-N4(2+5 zM*^JUPH?;DvM}GrAY~3qsj4Qt=ishpU0L4X**Q&Ab;=U^H;?pKoj-J@VPPL8B-ki1 z_6!}gM`&n|m76=Hxml1#@tYhR+9y>M&ZO;dPz2_i+^_+yk|@(_{tNOHfZT*^Gydb1 z!Dyv-62=}L9yoY-9dqM^F3(-onb)j0nKEi^rrvnn8FBuLqV0Yf{_{s+M<<&_mjZB+ zZ^AxkE-n|s6@296FtUk^B8GtTAQ3?hq!=(3P(M2*AIi+u1s;)y2o3xKJ@3t} zlA>s8@r+bqt&3AO!6$w!8hiKVl(y%TR+4=~+EeWT2yE7~Rhz+oDS)cd1=xGlFB{Q5 z+2CYZaHf{m*PjEmp}LeC92$`E@645nXZ^e?4tzhYz?BvFw(w!%{s8tuGgcSe%TGRk0bQnstxj8IQ zAUcwEXoO4{+TODO!VN|paGuJG>N3usJ*-(~k|VWKu(4iNt_GY=*~ zHki~*S$}_jyW?e%gpDr^^cB4ICTE+!`@qWsAv3L6*rLhx1Qy70;DjN7m4g=i{`BuM zR9KM&Y_K9CBB4=H)+e`!BQyil;lr|Tc@le+T}lK% zTdKlm(R)jV(YCNa48Q~!5{Q^IBG3IX&MpuPviIOh9v`oW1)YsIceE39f@64c*m$(s ze6{#YN$^$%2;Hs%-{!Rk2i~jl2V(plozfZoV!jSSBVFCy?Oo;{O2z~|=a@#C}>`OPGwpGMvJPxna|8tA>*JMCHy{q2fIKh&FJw?LS4l-B5X=`*^73l#XslO1 zQ(5xQqb0)5o_jS&{q5T~z9iXH2U;mGoz~{}Oryr?C5Xb$-Z$sa)Q73%;O2_{igzEkMVYz@WNP`u&-U670 zV~xcG!_E=J#GAl)AB-6e4Gu=Qxw%zzb%}$FJ+0z56U)hxxW@EsaPti#D^!csDnOdK z(8L3T`IS9_=f71{uyKH?iyc0TH;J-Bx7*Nt^z-v$nr`cEzDS9;#;xw4tE154<3?Y+^Y` zH}rckQBwy&XBohMG}SlV`1<7_oQBt{uBZ10_*EKudf(=!E)J;T!KiB0%r`zoIRZbdVD@&+*3C;pEQSWrLn9<$a`?~oLP!lV`&>Qz_JyFM(8(7_2 z)Vr~Xi9KLw;>VY3q!CUFFB5CfUF>d7P3y~?fFcY6{T%>KrH*d`od8Ut_2Tt%*)9rz zqhB%eu!%&TI;O;Y0vdXuPRGBoNdrsEV!ISjVY7Um9+L$T z$_>caTSgeu=2nm7>tx#i#tlY4L5YzxG|Z9&HHD_y6>%r{Yht1i7?-_%ygLFUcx3fb zg_!hT6D#_ph}+r1m7ArLZ3(TlfB^vTJPwM0ATda=zXd~xLU;Qdav6NMJv}||YQ0uE z;F|6Z8==q|3|pxXaFRFp*@u~JtgnABC_n^J_U~KE7y&tx{Ja1dfU@)!M9=xu#Sq!- zZokT|qw+!XIxjiQ48S9VMn;MQA_B524>Y(u;P51~8p2*4F37^6k^jzBtQRLHFs5T- z0#ib8zPpX;z?xG5UpjY^0s-QFby#{l$x0qbC#!ESb3j2WEKCYQbt0f2ECUm_q=XTm zi}HR-e2_FZ1hx-!>J$X=N}|h=K5?E?Clz#pC9*qNm6&hwA`~D|oLj-jRNq+reS5Z}+@Low|cVF}H@o_c&2H}Jiz{sYXJ-B&@qF>}t=Q@ps=wvg4;Rk3tqoAaO zCYTQF5S3q7|yu9wcCmg=AWy(x|$F)Vp9lHi)J59t&)5 ztTYusiWpJgiyK>6MS?m`Moun*T}&BBK&LxE4M;j7I(o0ml2-`Ov{t}%197c*{syfN z;AJC8g-JlcBH*<68csnIIR5tsIP4(N2c5hAhos)v&q{?VEdN)mzPgMKbU~*9Dt4Ei;IcI5aRb907B%qT&8vm52H&;O1fSj zlY4r3an>wC_X$Y;H#avL_4d>~(Zt$EUE84602Xcuh)CEPP3N0#1ng;#y0d#ll{#Sn zTroID)tHrQXVa~6BmX-KP`_?7U@Aw%{W6TZ%z~d$^o>H!6_W=j{}wIVQi#f=@|#~G zBT=9T2t;`aguMj-Wd$3^bTwcr+fekwESJppA`nRqGgPl{XyD_g?Z8+`f>k#eU~{#( zQ*csB%#mYe84+HEBNGZDkc;7yz?aAUv(F$y$U7#A1UH!#ovWOSAT z1HmxWrAJxMyab_|B7S=vAWW4k`YAQ z+}w2y4S*A=#urlqgVt&D3$T-nN3tbAQo}bVhZ2@gxTgz` z9U_%Cv4IXJpCmpFPHi%D_xv_{UtZ`e0KsF+H*lFAx~W% zk&TSAo{EuCj`{#((+md_nXaLj6@+8K_;Ba%-=#9vZ)*EAfxiJ5KFQ`nKw)9w`@#9v z_r7;Xia%QKE*t~&SzH7ZSR~Cnh5QjbIw-qD_>KbhhF9(^2(K^f5g=5gPY6FBFFz9Q z|6O-QVxubhLp^H`&<~g)ZkwMcntE)>26uXdFDix_gv%7693_=tBnrc*&cYs0A6sR9 z5LSVXwt}M)_w8GBZY~uVs4DG;L;KrAvDFoZJzeiWi$*F`;^_Ab^714m?p~YOwHx+0 z91i2Oo6e`bmpecqi>w%D3574{w1dIqLDTmDoB8`a9O)dbPTRw4I6CjH&SB(7e+Mb| zRbKL<0qR*bkktl;y$B=T*qI%oAKu~m^vn!785yM3cBuz2!=F17*rF)n^2{>t-id;C zm)g{%MA!9GxAAx*Nq@C^O0N@)<6&T6{LM~I)moqbe}D%aFt_K{FW}HELNEs@b6fS7 zTu8ITW{p=jr<7D_-u4{DD1Ge}yZ&e!dBgCc;~gf6jz@5W_R(_baHpZ!j}u{-!0q%z zKt-Wr>p-aH_6U>`MTt@qf&==ABnX4XC)QWoQ?+b)(mjB8a)+v(p+$T)zL7zEcXNGk zAwO;NV0(XgO9TE0F}~iY)(~j^R+`5ItYxvI#pW~t_Amcc>#(r=p|uX+Y#zsLr3hrA z@1G>lfV2TYHrQH&tySOgcpFIY1q`=RQB$up9<_7=YYQp_ zfW1*%xd&w!H0CZ~+y;PJp3d)r1{&{+E!!781m6VQvK`|%oOV^fFuwt~_`kI>eOS01 zBOOI#gwWH1pQ1}-6^rx>II5hEzdI-YDrPCk7{aV-vR5;c^bbrVbh8;<7`s zj9I?&X!me6CrF`3D_&QC(VHI-YbB14J>MvFFXKRpzaeCyD;6CHMkXK;mLg6QLmc@6 z6fC(1brj~V%-+fmn&q2mSp!;;!_!9Rhf@grSC8nR`*ak=Q`O)uDUPZ9x9;^HmhZNFtd z66tv#Qf47%2-#TB!Q+JaC=3$fC+Gk^SV=zs=M2HmgP28=`UF%UpZgO8=udgr+bdRi z{1lyvOzz!0Zwd^OQ+T<;W*fJF5{yxTsKtImBc#zaEW4FvLDm*+n9pGl10X%xuRnYX zL;i3{A;=6P*MU=WGKEk3*ZM|#nLt&YUFNj{#UFb#X2~N-RT=7gTJfyx?>J8msUZXA zrptm1lD&pmbr_ZEHOPeI;knRFMDC5`6UZc+si1{SV+Hq6Y?$K(~ zN|=I}K%6%X`kk=c+uIo_M6U-^0sj}63MEZJZ2A%tdmWQ(qd#UoV;Iq$R}7{$Hu8g> z7XeUo`{bmW^yPc7~%CGp@^0y|U z+i7DS(%?;9^(N!&L+jm#+Bavh7+GO~-?KkTXMZQ_gFNG43-=xo?M>xbWxvLq^temOi?BZ~rDyU!>~*34-!rnx>>4H?_gc z(_xsO=b_&)(#?_muul&L@=-uPgm_F(Vm=`LuifE`#POy;!7DG{G18~dYtwjy!vL%i z-`lM{+qMgx*N0X+_p#;3CUupj6q8B&UFU^k$NLVJ<)6lA>KYr-KPu!|J=X&47fjO{ zfdLYLp}6-v!^ywdUD)|8DX|D7I1vVSy*l(3s^7xHIjZ-iy_$zT5>Y?e)f3PR;eFyp zdowj&;o-wKTRId8Z}UgmCmPozhT8L6+pZ@nzyGw}J;oF1{ec^{KNCW@*!#Pv zjwB`}OSa>%J-^(u-ZZc1lIu&($VXP6N6!+SdMqH=a^HE1jJ%M0@bb*h4C%Qe{9kQ- z2RN1g|Mx-J>lm4dqmaF2R}RU{h)8xRdnB?W9J>=4S*e67Fd!xD?9P{P^X4zxcY+J)&l zaiBDCgFX)QCm7+PIn$w3dS8B5`@o}gc5w}BrD3P*<~Bs8n={B~0*%6a%{7mrUP(Lj zyd0R^+6imDe(5=3SKJ|C#A_-N)WJ@dvQ2`EM$s{?YNJt% zfLMGur@Qq7lgy;FaMhgr{`Bu(dc0IkKJQh%Uy|s~yNZ?;8rUtLic$Xhw&Bt4&H2Ly z?jaKHRTdY&<O!3HO4r}0uV+lKZVCG!S$mr|f z&JlRv2IRqH1aC&k6&-A!3^0uU zb3Su$Hh(?1j1j=TEm?X;qq@k&va(9O+zHX z{pR~}>evDF@#9o^hLN8?>tj__fBYsEc%YGuoM{qW3Ly|9gyXQdTgq~nTutF-ppw-; z|76h9!byHBIJkS#*WiMC%~jECC)R7mQ3cIfrOONVZP^4zW_Dw~`;)j41X*ZJc(L+g z=Fn0KbaT2EY78}p;VXo~-1b$oi7~6rC+7gQxKd%ySXN$6vejsbp}Sye$_oE*D(kq# z*0TQNM`&^JYEYWWwF$yJEyBDg2B~^ULKNWzZ=2QpHc_9E{?kPlJ(|}0DUQ-^g|)E8 zuAB)U&1AFl`!#~kT!~I$33C#lVPVoTS=;!5vNW4ViE0@;iiYF9PI87V)bK(2u8UIN| z^;p<`g1Mgg$qOcbzLVj=WWpl%VpAd_SxO##cNx-7b7S z{1{{*IH=6~bVV_7?~HU{3EAkU8S7Y{qeWG6mn#_M9K874Z6}SKq!Evs<s8E%sQ zCAbj#$zU4xDYvF)&pJP4aS zuh^{ym1L>Qgi>E$Ujt;Gkx;WEKrV^n+-gx8m{DGTai`G9RH8x+m0hRExciL-5X+%m zI~D9(4>z$dz7=Xhw~FiPI6fqaw57V#-Rd#@oL<>KUgb)LUoe|-ufAB{ppmWL7yc#T z5=K(&%dY>Hml*g6&-wVwb^g+dIR-R2!_SL*!@Rw35b_WAmLYpWXY06*R&Ze7^V zvxio@{o}`Gz`TdQ(hi>vJ%0~SwNMY;X<2ttAq_1U_adb0+%bWfFuWB%f3wo@;D8cF3a)SyH{R29|y6v;A0rL4@O0rtZZz^1tb2wn3$LWo$z+&A+7lMhYy1R z5Y{uM*IFd(;KVGgTR1M`1u(6knjl&^7%yg+1E9vSuThrX%&j=Dxgbx9| z_+<~SGG6K48Pr)dwJ>c;Dtf$s`E8*NAO-+VDTP22^ZT^2v<~J;^S9*?89FcxC9Q;d z*zHd*9;ZQdYG`Z>cCF8Uf2XaGtp86*=<=fB7rL7mV@PZ|7azii9&GjM0zp9!emL9p zP5NQ^RC{+@aTdho93OA9Y=~=*HF=qsYsmjGRfcAKd^{O+NIKi&Rg9u=Z11UKm z-5(yYM*M7i@tD0qR5I<+qw8k$8Ms_!K|M_lzqNDFGi2rEk*S1HvXpGn;(uIO{Bx-w z^HJB(2u?~$Qti^7A$2V3h=`31Gvrsjbt|vJfD0op-CZ#Q4WSxj_Y4%6gaiell%iB- zo)oxUd#a!3y7EQ#rX;H~N~2F#dJsklHcci@PE0wKq0n%Bpt5mnSOx-kk>Ms)=4#!Rmk|l^ru@nYZ{bG4U8_ zFv`WnMJZ0jmN?bP9gzrRX3|ucI{p2Xmo*U&m>=Y-?S2iRku&p7Qx4<+wD#MJi&*JT zbc3fL8Vf<++_vWR%nxZ(I&P%0|6M7tl1$_oF=aN^eGdVELWJSF;Ms2xtxa~-OA|l4s)cg>X9^TF}#syI( zbT5RFqkqI`%g93B9IWqq`gyPJ@MCS;eqPo5?@f1kBRMjXTv`^ZWBsI96Q&>_pR^EU zGv|i;XxpnD3%euq5BStnhd-D8<8|Wy*!$l2UCT2(kHN3S3gF|J+ZP{#HV@(dK!otL zjmKS0dhE|KBF-s5gk~&LZ}qPaiDR&nAAB;(g&`j z559_*@(kY%4jR9r-#Il7#4O@`*xQ_oqU&P#vmI7@CvwJjAhCWxNN_lGz)TYbHUQBV zz-m%#wT)Mq@zW|i<pjyfyTOo5H-Dm;j}~P_#z25o9>9Nf)(B zswwYxxc8E~YO{rmc8jp`aDg;&H7z1g*^M@E_r|f4bYD+Mcv&3;lOF60V-#wfeg^q- zV9H_>5*QGNWNY;nqJjezXo+1H2f`Zy#Tf>h?8Q<<2Lvn=m2C~y_8GFi($ixA=P0*X z_~8SK#0n&^7+W=LP`adS#Vo(%3Ry$ksPI%y&LI(snLYdL!PX@$3%BEhVRuv5^J||F zlf9-4;Y3|cWJ?U{TRfuvoQ~9oz$ZP3$c2$xj+>OrEr4j_F0Ng!(rKSYJV_|rY&p*I zmpba~+qt$}#$zhHJyQUWeeVrvX7Au~HwUh-6~O3{n>|mRjV&cFo0+j^_dW5Lky`N|0%tLe(T^gA_D9^beE8@D!u%*vRf~@z zssgh1yzFu9K1#o~3Ely58aeM<_-AqCsgYdQda8dr?lgU?;YOf$cWU)IA!P zhWgAiqz$5-h1n(~F&Y%Q$}*_4ttzFv`?IRMzCZk{+|5Jt>f_onk|Im}=Cls12DSWt z&NSZJ3&Pr>J3BkQwdN^up z>vs9h$GrUfCSB!UeRHXT*glC=XZPqfgD_ znk+^sJ2oH1q8A0|$7)VFnZ>cAVIS<+XwST3i;}iQp}r|8Q@b4AP)2>l2Tt@&?&=z| zzdNYc3Ho?B^6KD8=A4pFUoqADx+>TK_l=p0e2>rjD(dtwL?UMH%2DUVA6IlPb7THF z1VCr<)oYW@^25fIqe!u5zV_OyVaGA-C2t+ z(ce2i>SLNa>sRk`pT(5X&>4IuMX4W&kDKL?#U3$BFB_QP;T0FBgF|(xg#wq$omzMO zYnH1@1L>-H90mI%m6yxt>L9EG0D<&Yd-nZyeBv~sS*6}TPlkz)X($o&heJO><9hvg zalgHGJz53JQ@+5{#@kM`-s@YBV^(zHt0nm$^Z@Bmz=?JKbYb)L1W#WrsrOG345wbC zI)C3z-X5BvMWX=k0J56}q;dWM#rkEOU@>$ z1f?>eU%x5yY}K6RgKvX(y!OY6kL}&M5dO5~Egwb={PGX#@$p@7Mk)A2_YV!>vT2ke zsayi?YoBm#0j&)&zZ0g69$I66^XrABrRaL!-5qeXQ7f#CUOxlfC+hlznKZ|M*rY>I z7s0)G;-$TPv9#T7g_ocDh)|}AtfIF96uuQ?;DSVp18C^!@g7y1pX9k7y+xW~%UM?+n^)0>!v-7A%8nY^Tn^ZE z$OvV}KcK*$xx4NUT?416XvvNYJznKuhb?5Ose-uVTi0ssyq>XKAD#b69dp09iu>wV zX~k=WiV=o{6I>jmvFoa?(IPL#6Ymu)36h?l(>@!4tC)WAk_3VhgFq#K}ndhfc7_NNMfXSp6}fKAT+ z?*Ukmc(7S${FQZ^Dpo_>wPPo4*Y?AOFQdV@pJtll-X6zKa#>7PEym<-1q8!ew7nyC ztx*1N7?hf!VZxu{bRfx;4xrht%0mjs%)|oXUd(i=lV6eCYE?XVj|JM(Jc(s31bRj=mOuRMCYS97cb?(W z)GX`~rWP2Abr@T2F0MvGSRPDFP-GRsweum1A=+!c<)XJPH;FYICl*QFe{GJ2mX>I zOtr{+w_`Cly>yXE-;Pfd)A}~>L^MwDWpPDwXD1cX>9$9=?^U(Wy0_21b$Om=$PbXW zz(_<(q|EjA3=kh|1IkJ1loQip9Mi+$u&;vxBWir+m2uicbVRQ|grM6=4ehxJS;vF( znD@e!1bLkqDXY=c{-K>r0p`GGG`}~Eq|ApoLKR29%V_~#Gf2Jt!{yz6Z0o_1Dl`Me&<#XMJ{;lO?w&WdL;v3SlQWUs&8Qt2TYL- zC?>1C*Pn)#al3QenLv7XB#ZFLlZHPni9`M=pu*pBj1KbH4b~TZU8^Hj*wbKst`9-u zhyK22e%DCR#TAg9TagLVj?L-ZY3e;>4@<~X;#2y3zulW=nxQf@kQ5o}o&8NT{(7>h z=9jTCgjax_^Bp8hC3w8H=$jcis?*$kDW}P6^ieLK!@`znxkj9dFKfyEgz=?2`c5Aw z%c+uLEUf2L`sm-dxTBi7*FZg1kWP#quCBbr^HvrDj36;uEApV}ng2-DA@7*vuN8fT*&Cfbn8%eRH+ShDD+x6M432%Z50uspQ zA9Rt?;%9*q`U(x{VnyQC^^YcKi6N9P37_e7Hna)zo?P}l9a!RusOEoaAP#ZDiYgm#o;sF`nUem#@WbY z(HUPuB~&}rlW$(>e*Vo`U?xM~`!uCnfNXXvew9^}ua7EgGE9}~wkg5geYWm`&)jz- zX&8Q&QM0Lmsu51Gh3hY5z{7)) zkzoQ+;;TDySlb-9Wsp1e#YX6svvZhfq1xHAK^MhF51S^&R}V)#;~5k_QfV4fi3KP- z)men|;Bbzarz@UUR4_s+2r39zIiSS1nakJ0)%*^2_&?Ujpa{+ht0&KgzB6O2bg7L# zC{R<_rfA|9LZKL%kY@y(=L^c`KErrj=#2Xl^kDYg>8#VI9;lyMqe!q?em3hZV#g4s zR>cx0qV&0M!LP`fF4|$|SBr-bX_R<056y{%*vvR7uT%%!r~+D(NEIz(q97=2$DiNg z!D1CV+S}(#6JZ0Zvk=Hf<2@D%6Zsx#ERDZq0|lh?-HX55d)yDU^DIA_vy-B5+8Yf* zJ4ZS1M#ilV*EL@pZyhAybO|J~6{f4*@ux)A@8}8A=IxlInBsln8cydlyPV=my(iKm zMg&eFI{JB&`$b!o?d16EGtKJMmQ@rRkbU(Oa%`YU1&9Z?Wcasln{X0ZSl#Hs&Y1b+ zp8}wxYT-p{^q~{E%wE55uaDx|SyQqDk~dHE@>IMFtP-$QFLo};EXZr=Zc^QD%-+cF zYE71VNKZQxJOV@bq*5a)$g-*3U{tckO=g#w$>i=z2pT09YkRoZ&p#7j5EnhVdxgLx ztVvS#f6)fi=;6F^dot)1#Z_I9Wx;*=G#LtvBtO|LE*FQqFp1l`>c9TuMw2yB-92Gg z;eC@VsWTnpSN3np{@$x+T%&qm)my`VFl&Ks9ek6j?_0RYb9d74oW!_=FWSPfpwU8! zjKnfo&1>c?x9Qa*1-?@XiAAizUD9T)IA$+Oa?d81!cTh1hqec6}= zWM=&YP`~T46D8)U(tKWG36yCDKki(EAWki3SGDMeHeV9*xs{Xx#09S%52naw6*NWP zcQdFJyIU!;;ww&?Bw`=aCO~Mm_M7Gz3YGNfwe@l$j@nIYmM3YdqWGz6wrSLCAX}u^ zps#0p{x6JxlR)0ZlMQ>BQhbV8t!2vLQ{UeF+6STQZ}((8I|}z7py)KHuL{zULCdi3 z0DK>^_-|!@#IGUN>x9Gf2R+jkbACQLDP?EZ=9}Viws4p=OKTh`QYtSA zVJ*wp$~n)P5BVZ6U#TR+Svv?Q<*7Uf2r=TuUKRJDXYFGe>!*995egP3zqLk^+@hk& z^Ea@uos*CMH-NNQ1UnD+CCt+m`v{PpB8qwMb!$pi$#xAmQH8@D-2ZKq9CFTM?-aZ- zeTQeqe2;=3h>D^>m`q5h~yXK8B0s#Ud52cC~ zj1ha1inJ_wC?QCs+CO|K|AWdD6`o3asfPX4)9)E!UL3BGeI5+nP&gJQAIICcxATz9 zPjs&(tqy+u_5kzzec-#pH|2Cxo^rKsU(ijT+-ogG%4=&nO1&(q&E+cjAJM~H0&ya{sWRXQ!kQ*d(bLplasdP2J4v|k>vcFa$X36FiEdqdY!g8Zch&tVAaMr?=9?M~XMxMI~?(OfKW{xuq> z+D{ffunyG5)H3QX9aJ5Aqms{pFetdxCgtDjG_{An_j92Mg#4<) z^keAZ6ewgtQFAuWkTp;mQWX)z_TW_antv`h04BYrM+ILiNlTrWkDv*N1gl8icegcebZ+URNHM=x1SodgmzPMKc4 zs%J2EzcbLK;lrqk?}66##vFoj$9PxfR#qOXzqTd1y&@a*rP!0U?_N-9keiK!=xo?+ zzp7hApgS!{=Rpq>{DUU{=V+kT#Q?DL{+>oQU0q$>({%4AQwyGim31aDgPBFx@|Tv!c5=P;V#nLY}Fjae9eA9wL5oRp|w2J8y>)cqLlZRC$BJ# zS?{=2wc#Ncd{vz5Uh29Ofl#iy@9_XVUbmT?6Oh8-+@cl19?m`Ra2sq0B>-U8uDzo` zQF?v@4HDQ2TA=piitq4=IE#JZUbSFvr~7h!d_?n_^cht5m)s^_(6v@pRWI%^v5g<@ z6rbnXNtnV*h2`155uoSvz*`RTR;-x`>aky_6ZeSdQ7$yf`;qc9`GXlcx4Xyns|@?^iS|wyCHXsVcD& zT8Z#uR^7BrA8Xqty#0F4F$yQxi+Wh0ZU}9)%|@Aip}mBIV+sDX%8-d$x-eW#=Z$&M zW}|=(Df2jW>^%<0eh$JG%F4<(NeMxr`vAh;KqaoX}hdwTsw!5zc(738auA{gM5lRul#Js zM9pHcu~r*19YDw5)5$?r`Ff7>FEWtmbuapkcg5kf=3KRPuxTWT3iO2VOdI9LIdKd$ z2VJ!H+a60#sbD{yme>@dq;_dg-Kjqpmq=Z_&=VZBBErS@6rGY1XL4EvQ)b-Q*horF zejZ?6KR>^xhQml-jO~FO5l%)IGe|Ewz8n^-Z*#KlP2yS{Am6Ltm!0er`Rh^sK24&UVZ z^3RqJ4=n;F|IgvY>_CQ&$SO=kHT)Q~=n5slVP_2MwL!PlY|fV%%zYF7U*2flUg$aR2qbATO329yN0uuuo3X;EG;MnXe8| zP{!e5CRkiMd%-E}$uJPg-mEYM9XU}jXUPlNh=$XtY4h_p)wA_MSa14=dinD@_!`nt z*YA(5Mtg?hK5|2Se@v1!red?C3K2m=s>E)mO8`Q}Et$jifI6GsZz@awVYJ3mz>uF2 zqKF)kZ)HS9S;e`-<%%a^McuRv|HC6(l^=)9gc2}bAhAlrI^UQ}9Nt-A2~C|Dn9w|MZ$80SO%@lozA@f zhm`s=GT>ta!i|al(feILI`oe?0~uLVP6{pl%%6D#FPU=v`Tw0cTu?xGdU_)H!{COf z=;bQAsxvY3FA(_ey=(-a0a6;ld=fQ1POA#33+|0A$Tglc5J>v?v8 z3$G4C`ZEZw4|ubskqQc+yiia2?Fmr8g8_kh4j%n+qx11mNI`T!n8jG+PF}itHEQgQ zmr73WIpHMuo4`_t>pMk`i@T|*DdJ!P^6*GfqV<;~3+whSpoN6hR43b)Ur+1Xo~--* z(?Re`NsyNgE>{z|*TgV!-Ol`ykOdSe?EJ=NSC!M4+Z6Jo{O{q zQNu;^@$;A6LD-uAIS+CQ3Mf&clOII5_C4({`sx=Ro31}7YbwQJ2d$B5T#`r|(X^1t zo~=)YB9-&49JL7!^e?Qd8^_3^@xQG8d5LPFk;=24lp zEbtim{HN%t@ZQ$>Y{ctx=0#y5E!s025JGlg_iA?c>e@k~M;fOrs+N8O~53F z!hsl?*WAeH=GaF9dK@R>jH=lS-DTfTPnX)vY8a}j0pP<1dtdsv1(?dfVl@ew+ea;# zZ#W+wMS&6FJh*od&k1sOu%hI}U7RiNm0j^@Xx%cEzvs!O7v=%;rN=m5FpZk^y`%!Zn zi(_g|OacV5^qPGSD_J>>gl?U<5N&$imU@7W2qxiXX~*7WO-%}T?$15Yg7x+H!}5Cy0*%_}Kz*(#bv6M z%>@hE<)ju{@=|so> zfxw~k_4{}JzJUR9(?TQUO^e5x7c&75%P|>3qeOe;9IfS*E2_-Wj>*X5Lu}a7bp}tz z>gO>)@%Qt9Wz-{~o~E^2MAup-B)vv? zzYu(`U5YztZl7MB&k8av4AbISbiZjrdqiPxCcbJe!&lUI+i|)JIA?fHUl8oIY!IO> z!a#l$2)eA^g{m?}dP8IFR)eYaQPxeZ3%&7`ZcK6Gzsid|11_ z4=5=)sGsItKf;L=3Tg@pz}?>g_zA;!zqszF2AC@fIZ2JfgGTG%OF(HIg(i)Qher+k z7;$1c-riM6^4HSXn?dG7$P!@!RlxqC)zP*+n3Idm(+u>RDW;ku_CZ%Fo(sit;tc*B zVs~3gS510U}RvK7Fd+lyzerL zo-Snf zD=!Kj9KETC(DgI{4iT+pD-9tQKI?uuj7Lc#E(CUcZE6H$bQ0*>V6|tZcR+U62Fi8*EYHU1(fbaK|w%5K#&G$B^8iv5h>|z=|)kyK~lO)KuTIbNokNS={|G) z-uHXY_{JII#2;sG-J1bxt>?L)d(L@Z^SZ7{h_a##&TY!u2m}J>nXKeX1Okm8fj}X` zM2GK?)c-Jm|DhYn$w(ruk^iPQ<;EZocM;DdpQ^efZ%(`FsG6LM?hYcjE!)`N2j31Y zX9|v%d;KLOZ~0~Ma@JPVtElR{5Un@GW(+wAX6$A;b?Iw^eI)jS&mTP%Bi1ze;E0a@ zCNQDS+xx`iwE=pdVW4L1ObHoB3b$$TxkykO&x$~=Cto-xI!>@8{rLF!TdqQBCfbfH zm%l`VJw3M(cRxRO97U(gzRb`O#~dQTQZOYB>kzPtUw9!DDzk@4=Z`Ob@N|(r2w(g^ zU(2JA@WZ!|Uss4r;Ua&GN!R{W`GdrNevHZaP<83QU)uLPEQ}gAC@Ur~B_+j?Qi;fF z;+2ZlB%bkFO)0o=q9|@l#>1 z3%>8)zt63&hek)^Ckr~`wk&S_oS4uwYx1}}99#XBPh@0dgh1HUQ5zZ>@;R(VPx9hY z^ZESzsob$)+!dK&J^L+U(lSz{I;N#XELq6)<%(@YYXZ0F&jx4f+WPuT_&J~3L0(2f zwiy3Y!On8MNg|12dkJf6M{?uwDho1k35lT$35**U7*7)RBYg9h;$w$Rnt38jSf}VX zX%tes4a1qeFo$v#l6Va_MvH2lcW6aU=FsGmcop_&jj>UE#moeC>g*Vtc4i{f3SXfh zoHoZ%WMpLI;~zb>qH4bQhn1O`NnGN*H5q`0fty}iOD-47`og@TNJ>h|aQ|;Vx`lpP zj#66W>gwC<++5?GnWphN``nBMNlebC+BNSwp3Hjq!L{cHQUu3qtnVCcO_{E~BM=Y& z6d-&5y)u*K0h-vd1?=02^Y))Q8#_DrUVCnu*KIsJJPl1v=9J6)^XB@K!^I-4s@uBt z4o_>|P4A80m0K_f=z$*Y9hy zf}+67o9WeCfmSwr)E@u3A0YR{?RYz)H;zLBKIMA=7A78EP*@mNAP!Yv7#RnzR?T7D z{OV8^%b50b^ZG)*w)46dsgI68K;_@aG?j!r+( zZEtT+hfUl)Je0Du{7NrvWAmu7v9Z*s=g|f1b=+y}t zJQ~<8p?~}PLuJNYjAPm=8x?g|h38)j95M4|auAQF6vf5g4?AUdSEsY&q(w$rJ1W|5j@MmOPhNM`YH5~hywSV+^)c3zG+{?ZY7StU=# zt|JEPhQ0BT!BBTUTn|`UT9V!ixn*dMP(CBTyb-$Qc)dMQ5fsv4)JbmaA(CeGr}l%I zswz{*TNbZ2?W5LG?{yl$**TuClB;`wRe*RNlt zD!GbfdQEQmn{oBLYE^2@&5)c{Q#ezI7GCR_KICWQC?t);=1>@iA8w3Ad0t;0fBpJ3 z98x1xc(0v^WlV0G>&A^6c^=K%jw*YqY*7O1Ug`z0Pj&CKwY5z^x@fAYeKtxx+t}Eh zZwoyUd|~z{JF40M%);FM&JMj4L7h%a<=@niX%#AXvCM$D8XidyzHX z5k@uvIXm(GCf0a7r}20I78$X3$@;=$lIFTAi?ivUssqE>DZ9Hi)s7pl;GfQF%ZW^l zqu3+e5i1$pYo2LRp6PAvk2}9TcQzzc>e>eeBGuH@3opOcH^&5C(h9phKEFEOfhTV@ zSs4;R!5zpsdcmtkCYRJ~rC=XkuoHCF=jd*EQlIbm`;UM*`<;C49Ejy|$TiMR@&>!* z+f@8^**z!i4v#gLe&rVY$S*2VYrKX8uP7~j)VZ>cym8e|Ws>HrX4nDF&UhVmQ#jAy zi>9Wgmt|^k7rYQi9v&XVJKCC>{~aU0OMko=b86#7lm_A5fvqYmC@?xYI)b-bT3L~d z?C%j50o zh#4Zx8($k6llM+NzN|r2Hlz`7Ft~a1X5Yj_LJGfqU|oHELHI13Ik8Deg~94Fj4@?? z7oAt+kA`_>scqX>B zJ`yFam+z$-Eb0%`SM8xZTEJsq zMU}mz4R5(-Dpf0$#7biei^B1;;%1tv%ALe_f{E|Yin?pq`j6MY9Uo@o>k#7Ow-oC) z7avs|#;LJ?(J@YQe#`tgL%sVPVFQPn$KfUKZ8*iIx8=tqVk&UH9KGl7>;yOynmO zuggc)o4l=^S%M{$_lTrxiH5 zbVFe@L}2}_@SQ8w(F&BCzRAf%WTAqh2OpbIRmC&a=xQHht5q&SWxnR@S77x##_4-o zo7y`WwylPoeT#aWys+Fgr`eAlCEdq`#g9MBLB@UuxHFOFebs%R@RppMobgDml6vFL z@^XjQ`PLJ7m=kb@CvF}?jhy)L;oe00Q$OO7UqvHlm6ubakGtThUMi>Yzl&P`r`X7>BdzHO51ljFuV896`z3$o`Q(tM4n8UTvk8c6~6SM3FXV`t<43 zO(?yPlMsjlfECH(dQCDiGFU}K^w##*V>9O63eRimT>9BeFOR9Nihr^B)r&Abcu;Y5 zdCrZ6752tk{no8ptdAZ=!sy;%Qe~Qox=cH3EeAyS3HtB1DUYhi4kcp+EXom*?X053C30j)=~( zQ>Hm2Zk@y0OQ_f)QVoAqmoA|4BQLC}rB&|ZgK%1Czt!QsUQMAkrGB|3DkRi6v86w< zRh&;=;5Am?bE_aWgNOfVjN`DXqVjKj$KI(@QB6%vIJiQWS5~+?$$NWyjiK^}HZwc^ z@7unxxTq;6hFIJ`wS=XFrW7En@+r?Wx>k}kl;nsnKR)y>TdoYGe0AQL3GZD-)|xNe z*BE%Tul4lw`uh7Liu9W5yS=(PJGl^{y)on}&59c%N7mXS)LHB286x(7Xn!x)1bU7- z(D1}X{bD0B4ksZcEr$duPZ4zH4V9C}gF>5*lA(7Geg2^xHA0nbsr6e7$ zkFy_kPI9O~zf%U;OU-Y0BaU5{%fVTzBfbXBc#xs9e7n!g{V`)Wr+kV4NrTfCYE&1u zgR|^IO^SLnN$+qv0xBX1@GQ@q_hm<6@nk)>CiAIInPD>m#U7x`Ol5finjN2RQw&x?6O1*xi7(6tF z|8A;OB>o?os@wnX`>+2mFU`%#2?V$#aqNe0^!?ADKM2Ij{9i4?bpG*N$Vd1|4wJ6P z)Wp~rjfyJijVbZ%0B8(`HQR`z{?jKSd%sLPhdRpt+kWu>%l-WSv&#RcPfbotq=cG< zGmSSKd3k>NZ21;7u6SJd+sDEO`yUWK1)FwDm#L^bWbX}Tf8Rtz*Kwd60j}lzp37D* zPibRskA{%_@)GUXAK$14U?4)7ozA~9^>ai7j-{n#V#*69;_uzv&rEStghJK3hDN0J zt|y-p&|akAK#XU5*rWKpYQ{w1he(FHn9fXMQ$2BCx^lxJx=O@rO`*RypLix;S4Cc} z=#g?9wXh~ii+t9>?8xZ!KM0F_c_BI@e2*Tjqri9y54CE*~Ra=2%#NE;~U*(io+Lo$Piv8WU&i=5shhK zK-~X(pbl<^$a2)@O*(kdc z0-=p3gFSP4`$GCatx-KeEi`vjDNwtV3~_h8W$K538__q8qX14>d3i~=buKQUQm3uA zvR`6Dk3W8_r{}5S{yQKXxIG;f{ppiG0Bp3QSbU?e|8q4;7a#I~^aQ20;+TKY?U$KZ zKRE<$BOltfpU0;ed%T^{)Doc#m#x)}z-YWQKffq#ZHF2-H#A{OWX>Nld=z+@&k#ww zg@^dj*Js&VOdu|gxl9PLSlOk6Rmow@bjc+gw`3~ zxo^iuEqsJb6y%R8VN-5Gt?$%m=wonD!lf%}Czo(?;^pGz{tzD@d~o0(1l?3=b#-8g zHRrq8qnD@rchZayJ~1&dJ;%4`wy=z`6A}_+pFht;l30|+*lTNRF1vFm&{Wz06noS; zsmq?h&F^1{Z;VYrK>_p)Q!IymB=q1kf=-()0G9D7C_=2iouVs=oZd$m{HgUeFk!GD zZtHm-XuEiYf^gUvRWK2JnNQ5obnx+tR1!-iN8W5`=&pFMTY+6SfC*+;#bf%<&tJeo z668mj;;03ku#L*d;m@GxwFfLqbt%5|+OC@U@)rT==v)%PgGXJ)c%xAQw+sz&5OcrA zp2d%u;SIFkM?u6V8@i5{$nw3WeGq6X%4+#5P=~z>o(T>d=8$n6%$N4q%jhZkk?gO0m}SPvOS9mvz2y%<)eAUnyzH zJ5S0xwHN-*xVUY29BFO!AJ=Lt*>J|Z6uY`s^dE&RTNSz#uv6ID+e-jGdGF^(fRB$L zBa3^pnD4&&iUtD%Bb=O5d|9FfC>-d`1CcC9?paO;P`feig!n( zeY8m1_HbS8t7Zkd=;cAo@INh0;rf5j%kWlB-ZE3K>}DvaWQ;?!K+o$CE4RMlcUIZY zAQMHa7AjHak8c7&M2(9vF){JHu6<;L2m+}UA|%{wB!`Zb_3MPj7IPc|V#NXZlnfVJ z0;-xw+=P}x)Om$&nQ{nv{Bh+Vz2w1VpE+Xc&HtX*+EvQ9LejW%K{O*Of=bnu+DS&C zQp)#;bmz_;-{Q+Z@_*ak?Bw z{^VWq@WEMHe|LB5T&o|lbpQFIbFede$IQ$OnBZZb;!sL8SU+NJd%L(o=cVYKU%6K7 zW-j;Rpwvd$cLNnSx3qMlPA3w$mJIm&ye=C!H3kL-Bq<2T=Ht+5J-b_CQjy(V#&&_B z^aNh|u4t#-%kl2pHybUQ(5MRaf*8h7UxrXu&^XV#4^i|)z=CRByEI5}~IYyQ$-x#aG23Fo}ez`%?{{Bx3qz?0QF2$x0~GS|Yw;>+=EEUdln zSR4lZi9EsK;i`hularKCpXTi9nAq8+0W{tv4EbajtWV=@%FD;Nw!AzN@a;ZgOMGce%n} zS!xf^?dgWDNNWD?v25C%dY*@BN39rlO4+olOQCBm zhksrje&)Qq@Dg-bZ2>m)bB=zUU4Y}pXvEe`Q^Uksih$!wk1NQEzNx8(7lMHQH#a9L zpu18`6Df+%y)vz+s2~}!x3h~63X;iMuBxnj9UB{4r1@I>AQMohl}5Ag&0}mMp@+1K zntK{HgEN+GXE^fr39Fc53v~Vs$=q3c9JlJn{-1i2R$Z4(*Tax4oR=o&wEd_1>SPh` z2*_ui9tLQG8K6gl;AM>I#{Oe5(!&tWDJ>zJg!00K4H-1L;?oGk(+_%lL^1xhz29vG z{*s~wG2@ZA7;Nj4b8?U?rdAtZ&rNEz#-rDL80(3y-{LCf5cOSu--VamGa2WTZM}g1 z?ZoxQw@YHy+K9AqtjC@XaP|^byy<@*0L3*b__68F^rt`i`_+<3ulBm`5YW&>Kzy^% zFu!;r3@_y5LbE2Rab;|?8?H=^jf{xX`uqElWba{hpa1Gd zvAZp$xk0EDzk>^KziYePMnt>UkT;ONYcIAEUD)ifw{dcMb?T~M7s8pGZx;DqPY79& ze_61^`TP4Doc>$Zs4^!xJ6J=L>+L${6{5w1-WPqdj!UJN!J#hkR#(!B2R_1}e_@~*??+6_35s+B~e4tfJt55phKjPu-pCxh@=8gU)p-!@^DOTc_+A(wn6aI2Y3 zPVCqFyIFNQY@Xgi1I<*TG*ND{dwkplyuz9tkI{iN4%Mx-&qm>7g;%IT?G5Njrb4?@ z2}B6_m46knlMT)#19os`rHAQone-5UIRfBe|MO8C^zBGV2jK%Z_;C4SaxyjM4OE*t zog3)rl;uj6};bwk>tX*slJhOX)0X zHXeRS#3&Duf3eCKVZKo@eqMJ2U^(&Nj|H{YpXPqHl-y;> zx3m*UpuuQVLM0G-{O2uf&*0wME`RbQNOyz5WCfQmXZegP=OO197db+{$20DOmZ6j? z(slmvsE$$!v)??2pe9JyE&c6Js&zkRcRSk5=gX2fh}!O^la^wP6{mL@-}KPQXJCv< zyRm|Z;$0=7Yyb5rHugQo0*_!?#UEM-z|a*Dd~sQc#6M~3Iy&J1^>t#zm;ga$=Z@k9B zoeW1d0f*I}=uP+h<8ND`({%Urh|9}k!?8{m*9$y)hRfjaFd>|IR!!tvv?To5{XCzA zQa$JSi5V_NHih}$c?!05rp!tUGy0Qf<=8!I_PWHkzhE_|v-p>qOuV!(VVY*Om`Wrb z9+D#!p~@ol>J@20LBSjdK@N*eX=!&zSv3$!N=jc)gUA&dSnZbV=GJe~aj8hH(!Tb4 zAS2NF9reWNX{r%znMSzl2JN(cbR2VQKo%;t>kx&WNEEv_-?s+rl=?mVBkZlWPjbJ= zNDj-LGDQqp7&-7*9HgE;cEL>)o{qyeqJ7-W0608e)LZ0ZOw15>I$7j&Lu6b#+K>(Q zUG%x@y{lH0L7@PUEGUd@G3n4Vw{~|=Pfw9}7h-a5bv4hyx&VX&adGiUJ5E{dNs|xz z(vpozn5MK@)K~NRHPfpgek@>#O(2XpPy-?zDJFY=Pd?I45mL*xj1@WUQ%GhJI{BT@cj0=uH?>}Z`bTd z%6i->#J2P5G4}uR0;KCeV_P-YSns;e2>kW@>gxEaytS1T8y_|9GA_|i_ORr$BQF*I zh7cZ&@+>N+VpjyC>5tw|a~B`%8k@$kI0l`%Dj&RStMema=`nF})_mZ>FGw~^z2W=t z0m~i99z{t46(=SzaqvI)eRP(iH`F#&BnX! zmk_eXEBg=~`(2l;63iuVD#mkf)4uT>qL_*$Yw6y4F~2{ee5a~az~+Kgwd7!>{xDZl9zTXK0X(!Z!*6^maV)z#Hn z9U~o)vQknvUlnNuAj{XkrEZPNw*>W$8#!v;!Fyn6c$`s~ne+84HetvOX!KMmS=00K zFyXA;-P_A}F(j|iAT;($m$_}l`n3f8o0Dw^rSZupv^U#^Z%232UN%>7;9Z~Q-8=Hs z#&+glPo>%LzZt_#3{*l)`JW)ZiX+CCi?@|O>7gK2S66~tWH}`JaeTv^_l+wp+$Ct~ z+C8X+$itjxyvPQU6>YwHIg}U$OzUBeYo8c)k%E^0ZGybMj{(Hvjj@t05bZc$|G)>? z|K`)D4*K`$wtPj72fxYUE6lNM1N{_&d2a`qd|L-k^RD2PDh2Ey62Zoqtt zuCss*w|Z3fm9?Sa)8n!Dd?(ks$pIcbN`>=58JaRvT>^wp?_$dX*~naJ7nT{Nh!W`| z8cykN|5l@(|1KS_Ur%GC?2PFp$L%{MvZZ>Nc%ict zy$&*f_?G>d1VmzW*`CA8!aO*?M@Lz7?RuB(NOnRb>0w#pQ%}4!+k0Mudg6WlDuXS5f+=pE8_QSZgiA-5-Bs zG&ZgSD?l7drElAOy8X()>dy?n1z{>Gx0zIai#VxBenPd;%*Pew6Q@h!v?2-D6mB9z zQf6*#nUVOzBA{@!mes3N?AJYSFkJnay9YYO`qwQ8Hsks40Y85HXj%VV$Z7cfhMOH9 z-JA5HP|8kna^vT&euFEEbt|bV7ipDupbbUoBnQY(7t@=RGWLD=qx5Y$|2%kxW@gMw zlsX-jtz|#mK2-$*YsU$s!k>CaEF@R4QPMoKO6LJxlcep?M z*a$v5onm5QzEwL1{zjwJ0A(OmI1C&vz~lQDM?7f1LD81?_4n`JvT||~U@e$&+h7w& zmxC+ZEchrhEr#OQ-PS!sfnk|=$K~$Z+9%Wkl!L)G{?XN0-I2jXMQn}24Vkr!SlhcV zD00aY@a=AMT8=*|I(wm^5zHVLYXFqK%SwtfElr-SqgBU80`bX@8DW%A6`?W#UU|4Z zX7Avjiv4nKZEZsS#xSk&2V?)o&uC$B!HHDw|0J121V$H+SOh=F{Ci;bFC*O_e?Ma@ z^2f1ubKBI&OsjHtL&WVAqt^=`FL^IYtyC(WY`s!boAkUn!JCXYwL zFD;g|lwqEZ7%TQ@ta4`>_}Ucmsp2Nnx8qgA^lf2kg2y`hLB53I@-f9uk{}Bkn&Ih` zUePM<`h?|CA=3XaQhiP~inxP*HuM$G*5W$NML~%CnUp5E&|TD#SvQ=$y*;R?W?Kt{ z;s@Aef7xg3W<9|ekEM3EJ;1MwL^Ydk&J$w4*4I}tN z=2upNz;ShQdfEzZt&I~q3JJ9`OdgUdU_)itPquj-(1TVZl9H$pZq-uSBV82?|Ni}Z zWHTqWIa!58KtK>27KYqw9XI8|CP^gZ_i|k zd^qw;jWH%a^H5Zz2rQdv2A09}mkZ?3Kgc4g5bXD3jp z*^iaQEMkH!R@EOA6q+`!y30t?e=ztFO-C;1#0pq2-H{q%V{5AB-dS7J zUN{r}Jxy{hiS+B=Tc6X?NuKBT8;VL{30pkW=d>|m2h@ptDhg6?N>c!NK~Ll>9A*Qe-HsmJ}ZOfvIEwrT~aT0CI>XJxqc6q+FoEIomf3_GUY6Mz#E!d`B;q?iE z1BsMP%YS(I!Vdd3ip5ety!Xl{cJ2qrI;phaa!#(HO2Qe|w?IOLllKwe?RjhIqfpx^rO$*JcppQtE#HD%*^QR zC9?tnwXk3WZ67%Ns*#kmsU0F&khSIP)HN@^z&-x&Vtq-rSm@#Y{@l`=ki70F!uQ4K zx9|GL$2}v;$vAk&;S#dKq_A?2_+njINLD+HIYzG|pZglmXOCJpA}Xp2JZ-kXI+Bv{ zfpi=k87b}NCJ@W68wwIJi<9a`<&rDO!Y~VQ*9wzf+;or3;5D|G)GLa2#lnS5maHzX zc2Vy(YWrtNjQXy;^+V~E&Kf)Lm7VY)&}^O-pPN#udaH@asn+>XPU{{6!_Bc0eHjf6 zasc@)^EO~3LILMqhtR<$)iB}Kw;%b%pB8sVKf2viH0Dv`+MJHY^1UD&-ky2fjf|&~ zsT%9W5{t1R2@lCFq3w?owQeTV->$FRp$#RMVh0G;3i`!f8=h#odWjyT_cnCRC%{YF ztPL|D!wUM{`h`^s3m2gY8d@bM^+u7a3+G*LEBtL5c|>#JLjjy^6hS2!sWKJnG_;)+ zw~P^P;TI>SjP*@FF~;K3T53f_EFJYRIbZ!%{`L5NOt~3^WSgs(W8AQ*w>A>cXzt5nuEDiHBE&%OtVrS?du_QcuW_!#$xuszlL zN{qZB;`xp+`?yCX)D9rx$!~^0)nTI-{N06>M)&(dG302}K?W1qCd- zryd$wTBryx9s;9^BPb|XKO49z3AbL zW;N)@Qpxp??t&x7VyrkLHtVW-kyD|O7+~DTbN;gMMftXkz?2WD_>==YXa|X$%nkA*9y3| zSau(zdO9gk9qVX5e}4N-e_{aGSPG|Y2}2|`n!SWai*>{3Bm(CErSBeFXUR2_w~40R z=^B}B`Wml#-K!t37yipi{n6jP^U&jpNQMlqTx!o5cl3^1bx|CKVYe-lM2Ln_xAOB0 z!=K(KY@M9cvK8w9vnjNSyrCwnjfuYRv7CveB-$!E>xqxo**{J91X*FY>m=A?5|wZu z#;*2`rQ|U5=oGknn~q``qb7&#{zMMG5fG=Vcdm_EUsWW71z-2GV1e$$*KSsCM}vL@yNRO>6AOa;YHe=wu$S#Jh}H%0WMCxsv0(I^PE31GVFuq z4Zqot%U5elyH^1ah$|DwSNC9>9a3vG;K$-xBDV>=sPYeYUm0-w{d3*T27xvOulvb zdsEpnN6ID_^CsE0f%zGihL-mHFaJ9>J7PkpGcF!$cK`L1!svrI12;D1}S=PuLZKxjukJRT{t`+?6WD#%(BpB7-97KjZQ>_g~s?uT02h%ZM z<5u6QLSgp4KVi4VL<0Q=7u^Y{QeE@2*5&0~pQ8xi9t3E6P^!u#eD{qKMKwRgA96lxPvvDwFNPkW;6uAN_ZfKQcDSR#iFTtAe{ z%gcN&yXnIRV5T<&3Of4vmy@F@YXS-ivvgWQqr|*CS9$h0@P#vlQUJT#mm)}(nwn}Q zVxCwecqdD#|5c;Y`UHV)CMsP%e(QUSkb*yUC(XRbJ)fIki^9G5GvkVbfU1jw*d1Q3 zLj?I*3^mg`PLX^;do=paqc+Id>Q`}=(&E#%ON7j&5f zX%4?RDkk&!_9s1A+-bf>kL$gOK>E$G$k^LKJuWr>%>pgGZ{RxyNi@}8d4S#n+>f_? zIRimG^aG-5U;q~d1qFRA%J{g;uklkZ>B6<}F^-O(U|V$-S^?I>mZ}xsHWYG;3r#Zx zHcLT)^lhe^hz7KR{hQDCE0^SPo0F(|RIomuEv^>2`S-+PHv-4+_~9NXdLWMjWeq?E z;c_53S*+VYfB(K@VNjb{jF7*$p#@n0zL6LJ0|y2U4m_|{jmmAt_CWKOWc@8f-}{n; zmX@{_7$=}}gFbyCX4NdeVcobFn7IqAZGtvCCj+#R;|^A|Ft^O=J7_J zIurLwo!Ltcoq3o1Cr+%8oKB)A;$lFtmdqV3W`XQ6ZMjyMuX?a~5sMu-t8M-*JyTUv zV^-i`W`6qY*)6c4`F*%2^Zj?#nyHNPzq{F<++8|IpJf)4T|r?Xr`LrGt6n4d_V#wX zkSiCvUSlx$N*6%mECqenf$Y{0g?%4G4pMRDXuja8)nNx35qpx1>^ky!Nkb&Ic>jYe z`)t746)jQGcPL8bT2CCLOYM z(mA-Tb#-Ltp6kfX-~jq?!DEz70saiGhZ1H#%&j2a3T(+$96uJ91;$xFyH7~XXM=z) z^~cx#_Pw4(S(02-1Zol5-upJJ-Q~0QgDb`$#Xpx(BsT^x!%Kc;q|F@_6;&%)$idT< z0)}>omG)g6j}v6S^owfLWQ_hLbQGkY{7a|{-f*1fa&mjiar%_mmjWX>3V!m8FxhYq zmqx%pO7w~xuuy8Nvl!-!0J{*gI`3-{ASC7&7bP}=$Z3iOmq5|atgok}sG`u&E+>5{ zq$vAr;eK|g3g?&=d%e1&4D=}AS#{u5MgXr>$s0$$G4#8!Q3ysILbJRCb`Qt&@3FJv zg4u!lOi!ZMGXRLw(h5Z~j6<1HmEVGghU$L+L63@{E)pvEB)cw4 z*D=(Sb@F5Do=&3+L0w&)01cB5aNSx-a^Z(zJP@~oA{Ozcm8$CM#^5X%uXlWi1awI9 zW_7Y^4^zxPp2)tRp|b}lewbT8qS4j9N1V^K$oL(f?3+C2`aKY%mUMG&2*(=~xaT`V zVkrLj>4$ag8WcuzC_u?0eIJkuJhUPn!B$h%^AS&G-j0ai8!`P&lnZ|b1S1%pVf6r} z@gZjFbQH6=2(w-@5Qf|(4dY3y^D@kwFA_o^U}UB|Z^$u2Vh`r4xTjqbAAR8%uBLVo zV~jC8KiWbnac&2zzKSUVhU>rbRp;LF6~Mrg&gyJ${~ zkGFJo=1fpf&Ai=QTJi_`(#by$TWt^RWmAt@5d^~2bVWKYCt)(;_kUb?m+ino{Njs? z?_nqg;Rz(P&=iCx^x4Qm%KM5ROntjxgEmHT8DNcpX7F&5m#(vu+CHRg-i8kfqO#Cx z48C!W_4j1R)FtXw(`-{x8E{2xZ`K2<}O@w6&pSNk@o-8kD60THKrsU&OcZjU1lj zV)i$n*(s(9_2*Zru#PZ;*?4@LiqHUK>FK%F4=$y?%Wk zJhEjr^WxMZ9^@V(`YT(6Mw@+{afrYv4xlxj8X3_8EA~P~1r1?iV*^elA{t3L#-|(f zL0yZ6zw%zdtwYKi#cr@#?*8$?Nu}Bqjk|7*+rUCvSg845ZjA(ZpDvS0N=Y&6Sx;8p z0{*=N#N=I&<_#t*%@97|Y{>uw4<0c8=4Me+4!u82v0`_XU9d4;D;cgdUr4QAmfA`#$wUH|z){DKZbkm&nq|^lHwo zs>}WYa*OTxQI>lWl-ficYkzWv96GR!uYKKw)7x_Nt0useCloL(85$y)DZN8NObp5L z$mwq%1xyzS(MCaChcCc?psB50xRq8fWey9F{~+7$BBU12JF)+6ZuUWfYQpP#PJsexM|#R% zb0v&g5g4&D#pTU%m|;PJvxB4p5!EH(=H?c&&o3yLRa>iJ zHu<|yofAeDJPw9rfa|`2sI9A;hj}Lit!~9SaAE5r1!0=tQ^C7&uU$6YoV~lZ7w8qg zS9Ow1#73>}Z>~-^Jb~h8vob*Q6~G-NM8w~S|7!HgOb2NxJ3mJ=!rM?L-!*4L#|No$ zvQ;xXdGVEi?kv5=x&x*!04U{myQ1Sz`opGnY~1m_AOJ5FeYMq8PVF`cg8tq;0wN+m z=$pYrV=!K7ShSs19}njMIiHO|;UbZ^LmRREGicf&Hnl#>Oi(e}2epG7G8_S%7^7cSQQ?vW_?b{lV;T2_NhLA=um?6gLt&(PC4rv8FtO-y$}gR0-5bFT4T3<%+V1~V z0P|jcvT4sJW&k(k3p^^^=%{IfT`-iIXsL7;Kd-S&OIIEQ=+B(iGmRnOVY~sHrb4w0 zrWuu0irg>H>i?={U^>&Nd8Zq*)!x}@1Z)&=$lSQJLU}_&TAk#2?pr)iZIEYE|9}#v zYdbu^U6*FbJ3!vQ(ALHRk22B)nHAJVWW;jdMsN&%uOuNJOEa?&U{cIz^mQ}eM!;8V zhbUhD(XrfLbm)n18^TYa-W2pRz-WLZJQXBDF@h;eAZtfX>^?zoX2C83lki(J?Uosb zJagQyp)s!z056jCtw5T+(9$Z{Kk+9|)C)__O?A5sLyR0iwu&TyGo4?hhr!j=70HD6 z93#&d#MN&wxC5o1}#t)gOOvNees-Z zZsso1(&#fD`wE-0&36Dx*KhOEZM}zM-P0f(24Y#*yN~zwp1QcyDoO|4Yad_r>>K*4 zh(jbE%qY(gx3s*>OnW~bW^_!wx5>yBmicIySivG#pRw;;%L^iw9G1f7baJ7l#2; z!+pv8L_m(iKy4S42Tmw1ML{Mhp!33H&)$st&RT7af2C;7N4bNi>~V{*$Y|O7Zgt+l z_{N#X^g+GL^g=?^5Y@k(Rs8X}-`uXthV6nJ>x8l4d1wiNF1#loKnX=wtLtLBVJA8^ zHgMm=_Wl~J%G0%fbAq0s`K zSnzfPZj;ivjWJ3o0%-2(?fsCJHcQ%1Al~;04s<|yJQkzrFon#-!GSz5V2C$7Iy(AQ z*FbM?+ryH^_waDCAPfao+32z(;T@jOm6W;?c`TygG1lq4gZmzSGnxGZkP?!d1IN;G z_1j2%qnvPtIM4_$cH8j4iX;O>6*S6FOGshHmkCEGoh?N2J#42Tlq{qs5$UFYnQzQm zm(2!~A`8>bvp1nr+e?4oAO9%bwf}#40k{;Q6cB}VxLGqn!v*s~tq{@Lk1bi~+E1V^ zq9C$g=Fdza+c?m=OTkOVI|yDkWKws3WZ2!^Z7EPIM4E+JX5wV;+c-FYN!&5anQ?e{ z_>sxQ!`dWUG7muvf1K4Y@VKV@V=z-k4lBfavk&E-)n%I=r}FJ?h&XIbl7L3=rYhP0 zaaT`gr*xJ@Qvl3cCns0FV}fa7US3|L9TF^C+B7g~1FcEa0tLICoWsrx4Umaoz+Q!6 zr=%+P?LYjiT9uepRaH>h{d$62V1y8cUnL5&zNkD<@Ut@KkT>LxrWXkMeoj#-As$Ri z!4kKseW~*?@eD&R;W5tMlc^PZU)l4~oE5fv^z@Vc5nlfImu9^>c9&<4NGkw@8#|a5 z!a&+AWVJz5uD`5WnyQ%_L~kQ}h1oJv>-pdd&om>P2-L^m#wGOTC1TQ zxVX5Gq$mJALYNMGYHrSg^j&r|p(+ej|M<}Ty3}coes+wAg6lGG-H+jOup`AsQb?~E^4YHacG6e!J8@`MJqg6Zw4hGzn1~9u0V)7;9t-% z=fGf!G;bh(nSe;zY$o|6DL<0CN=`|!<=?Rj+g^Dl3>-j3R8R{}=z+MwQr1!7bID-Q zPd5>mX{%V@l6VkvJNnzeQk*qkFkxU}71$c~T7~QV!6S`zz{5B5c1V%KUIsNFub|-l z=g%Zpq1N7b0u0$1ym#bw^K(v8c@u-?l6rn*mIe`h;Lfg7zq@yonsKr-lVNjI zF7aWEsMl_#V^P2eHvOI07zUS!;=R--Y?cC;W^}ZVBRH*w7?;=_y?g>Wjf&E%gz_IO*)84|K=lYzGZ9C=Y_H{lDs<3MygL-f))QN${KJ79b~W{M(=g43(w8uHlwB9zu57 z>rFho~6D zyD#Tmj{NXY>S3h$-9IR35vk1quZlBG69F)F9yBfR&H9y;l+1)B2TCbSd-|NEzuWzE z=K12u?rl0TDl&d~Mwy<`-da-68>Dz0(O0tvK{rWUG?aZY)VUt5_QvgAPcW$~LrB!eFByP2ZS#0dP<&yGE zOHFe#d`ohsg^-v~JN&1HeAwJ*ZTshb@9??!*!HJt4Fk}K8Y-?P<2x5l&e!jyLl%mJy2YUeFcSA2|{t(fu?9T5l6g$G2&B)JK{QNtS68neOyxx}iWF;VP4Q194HiD(pW^6IIci zYW^gE&93`?_sksv&CU%AJ-^2B|D&((0H^Z(|36Yz#yJQ{#*q`mk#gp86cqe5iwj8JIU*&{LvAA7|A{pdS>|KD}}|E|jwr{g@&b3gZUzsGCbgl8XA z5@d4xDMcfuihd7BZ2hU(*4_PZ5z0TC2K(k;DzOhh?}Q{BNPvTszDl}`ho0}`u+LMJ zf8w(y$6q0_6ElO~B?VNgTlUWWgwg6?B+VeJ*v%83{UMB!>*;*|rN30&WQvy>_pYt& z3PL~!L`&Sy2#WaQlqAHr_~Op!JF$#tQD`1rD^;TW$5C-$nte5d_xGi%_+bYw=B`s6 zoVaVAv6Yo5y`+@YocW@Icd;mcn|qP|OV>9)bvdrIsU}4#84KQ#K8CvU>>ny>KcleR zr0@N?Wx+ao3ep|N3(=)TKO@fa)JYN^#XJ{`vXV56m=vOM_2{U}DdU`M&6!mUr!iw>sCSucREvv z)e$9%T0|;pGtU>af1Ptq#695Fexn-zfk4%%OEZ*txMI;7`93p3XKm+zl9SO54&4kf znfzn&E8BzYi0}^~vmrzO=1w-NQHhYY+vI`%TU+i~FJIo4iW3aYAV8UKZecM6XnRca zxht<5c4@LWWJ`}Y-L96YYS&5{jOubIXgR zshN_yEMf0nAwMeU7udK(mboYRx`-X=9-KVOxL`YRB14>kt$i*Ml~xr#)wO4E)Uc94 zA={v$VQ{m@b#G_QRSrFl8)np`(E!>`pT=2^uq$3$bdU8tjz3M|d;Hu@Akub=<|q-< z!QTBpKB{>0V~RPEW@J@$h4;uvrnd5y1fFVcUhDGN90?g1(3M{MqQLMel;Fv8_AGf} z-45SO$1~!~Z#!!<@tQ%rVMvcHYyFJrJi23dQ<1GJd?X5Qf{o` zjmzM6S@g3-4Grv0nMs9>0ximTH6!)y!K$g6r|>XKfr`DnLg`BJs`%Ws1ev~nU1eOB zSCi*zS$n|kyDp{sU(}dFD@hN&{w^x>a)Mwg=oi2~9$E5ohKV$@InoU zd6QF8bvbu|M$`7`)4L%q>(Cd9&{} z?igD-SUhvd_Z*#B4pl%%J%y9bN#<7(XoH~0@^N&Ft6}Ik)}ZgHfpqQ0w^pIv-pRAc z@Yo0EMkpq|P+dx&Bak&n*c#iJb7EVCtp~0=rN3oz=3>2D;ZZH73+I|madA(dI!ngH zLU{|ORug^)yQwV+Zd@6os?bffwtjU@9LF6?p26EVG~^`*b~|EX3fzoo*W0h(%of#< zS*2NN*w7+1wiPSY?>KX>kIeheB2h)#QKnPR9e!5d7~WU2tximcta0XY`tXICQKQn` zrM!#b4u`ebS8j`~`3m|?$YWJl0T~AVyu*i7Q!%1^0 zM?NiSuXM){p_uB8>3e(&cxCk-ruS1SH5z83P#h~nXnnlpCztEF<}>{sq*RTv5AIDi zFoGm|P{^>~&OEo=oFd`yq%I<+p4VlcYnq4MCt0}G+(r*{gKy%$}W40xLaJ~&<@c>0q-b~M0nVq)EFFN+|}TbJXzDPfz>OY*<|rq-jc)fcXMx^d}4Dk z^wbTzhQ7_wBHf3X-E+pmOdV9zD1=F*@QCt#@!e17Vu&#+JR%|(Xi$syPJ>*P9r_K> z(fDF1nV*^~C2wYyX(_Dh+g_XPNlY0>C4AvR)4*`h>>WRhP|Z$8i{Mqlm5aW<`Kms8 zQ{DYQrAF9-P!C_{ivtEt6R=axc^;@-UpcAJ93I4)`zTF^Y0jMjKm0I$dq%%t=dhw);@k{x z&j9!@^Ua%%?|JCx)9=g$a3CE4kDrxqszUEuTP+$sE92wMIu~N9sxGozbS7i--Qoap zx)lx_b`=vOVq#)I&UKLMQo#4!&0nAmmJa|* zGG1Bv0nRhy(%VXI9%z(YeMt+7gFf|O!s38je+=UtcOpQ5-4}~m#}^iCMw_1ySi&z9 zsLs#NJAj+ZXt+WTjd=ewN={z>2qQathVQ9dy?>xnZTe?t9f3aKqB>3m&_4-N+;>65~sL;ri!o*1Zh%m97e`k5JpMx%{K$(|Be z61WOgk6so|jpb~{`&h(Yiv$=+jG4;G$>~ip1!hnr&n8qorwN!HMX($ojYzo#r8Jgy z4kz3A8bQoaA{|4Q69x7Ao&fYwZfz4-;$R6!0Ux35tPQ@${zhNS?(b?EL^<#?J)|QF z7;eSzw@2cu1g&gs@o?tDBO{ed44Se7i}s&`$muGG7%=New@S_bMy@6xTMfx{afh_w zbSJB(VJ$|0BJe+R|D({OAwD#n+iMM7aSl<@?!ze&n-QOtoo(9tN^79({wXMSt7iwM z(Ku37g9NxuIAmf%qO1<;mpHR zjL`h3Q~Rw;pR}6~N4#iLp@dtkrF8}%HItj8;4F{(7 z%@CuS88DxrKwiAhljR|bH@`5_8^e?8O$$#l$XgaQrNwdht~}X(uG2W@pJiVm}t(1N!|BpAxKcgfn+Nz999Ur=Xz8mFJ=FJxPVaNDub?kEI zDsIcPr4Y#n>u#5_(RfQAv){Lqu)BSU+G=T9;YS|6;|=CUJr}k((5A4Kp|E!2xrd++ zm?-0$A0~lIQS+JR+v8v1NBQ#}p^(7gF_t~dD=o3Nw?1uH5%05xT8{hdt?+EE z{s~V%zlD{vtxe25y>{TxwWdtwgn>{L92KPzHRX6Fq|F1=+GdDCxx&Guo{<9azl`Bh z^#tWsu8sKE^b1+7sPd)n5BF*+$BSkKlY)e^lxrWp< zobdgfr}sN++0i@8obme(bn~y*K=Kd#J!_KmjW?_)hi zxvMK`Mw%_d5PXU?w4|Y$8=L0?jeYm!-z`_{+)?`#mgd^WZ%rQcb@rhHt35(k*u0|} za^pV%bCWrO1c$>xdNikVPC`NgAXSy8Cox3gz8dP`D$SdPEPMIy=XCc|M7Mm`IZ6^Q z*5B4}z>L3iY`Xv1Ae?UZ6$LEniP%7@tE%AUG z?EU%kQaypNhzFP3dhm%WDEI)RT+JIdu$tQVqXX`-H}%t-q7Cnku)3LBkk9k48?t5= z+B8bM-Y~N)VMDQFxyYn+W*e`M1uf~Y$!|T~uo>~&##3j@Fp~>K=vSY$#hd$qqM6G6 z=zH)bVijd-8t1YG_(OJ$_nwCb4kAc`7nhUiztqWT{cb)~-{OF~KMv{5hsp=D7393p zLAE=E@9GN3XVrULybOMXdNR!?NqHCC*D6(hxX*T#rhojmTMP>I?oJIgS7q;3(!s9e zgA!&@BZ6rC%<|^TTuJTlry+$$P{V6cHp{7VQVy%K8C>ov;`Ut<^;9w8}L>+(PUG(wWX5!?RvMD|l&?9W@?y#QpwiizmtZYgiO#UB2Z zt$>qHs>>LXtfG`mzh~Akh7bhDmD2D>OVn=ifu2r8TCiVajzC=)aDl!R$7BM zCnHLij8~=Du9QvTFoT8A`@U0HtPSa~*V48qH_${vD%WQ#8gFebQe2_yr&au;^ zEK7;H9x2UqOV4(=gb0ik7R9@a7=&6PqtwkRCIos|{*X(kRi5Mti=uMg!J9V$b)d*@ zHUcVtc32)U^-b(eqzbQYY`@`xD6q{ zZS7QlPN$ZnN#*#h-4#h#Y=KnSq5GQXTdz$!gwca?m{DUgH8b<-8AmnHRBtx+FKgtY zkHj$ra$1OI78l3t{y^hIE~C+GY*8g8VnAXoS1vRW;9%AjmofZhDtMmgPQXJn(@aEz zu)SI(*GVhcoPX}gVCPT3axv_^+KFJ#hzJ3xC_t}2=O{pa{J3K6Ibzfof0l|WJ`h7m zJUL4jecCdq_=BFj`KiV1_R3p->-gCwI?ejEviO?)5B%3DM}mp3G~6z1T^WtuxFyjOa)&CSi(Ar^r%ZRxS|XPj7F|0JCMzjISv zdmBKp$$@t6H9StwV%wZ7qFglbOmxfLkjsu7_qgd9m1N?bs7<51hOg&NN8X#e?pnN_ z(sc6}$*@!&ivQr&<6le3aC3|6D5@{znLJc5Z}PkeS*;vI$(hfoQ9y6kW0%6tV*3}? zW$q*y&Y$SmIi-OiV5T)}xU5l^s-Rz!Dr`9UF>}yVi--_7kC1 zP_8_J5`S~9RYLz2#x>%U*!!z^o>wBeoDxx!ES=vt?24KD&v=a+D-p-o{nPxx;^WFp zl-bS(NoX6^Q(qMtFHf(&Ja!%uGpD$#y}#`e)%Nf%#8kZSlz4n4+3a^QWfrj2C0*F)&7c8-} z7vo%ImrNRi$R!=dQEn968NODpxjbv@hRnQ%rc1sq=m8ZXxdCo^#Xhw?zNLC9RN`UL zKO+YZQ`I!DMa%-0?%o_Av~8@F!~+uy3kWt? zD{}!NPb!w9x^%$lMB+O3@9bSlYUnU~w|UJ423{kqLkZkDi&>a(+Eo#(LGXdseanzm zgOihCkE$3@4K-Z*x_bubzI@)+)RZr2@IJ@b-U=}0W9hK`*;^Uf<^3Jf-mr>^2&wefP^4s7g=FdfGcLk0|A$e)!+LQ07 zv?`o}XrG%D%*Zb09H-jyHapSxtL80Vw}z3G|HK zd0c#P6w%SqXhjUTmw#JjQ!7;N8}(fE@zvDLMWav*Xk?E7&6VGTFt)_b?$nxvGhx&h z8ol1Kyk@g%+^v7~zVlC`#m{4;r;5Dy$+ryB{Xbr7h&>*@vQy6KxJO`Js=r~xu&V6GrZ0%%0@lc_O-c@TqY{%j15^u>${N*jL(Pdxwn!YDjyC^>o} ziPDa}cYaTfB#Fw4oLg&@iQ$QwTWL;u|AhajflIO!!#OIsySM3Pq?=EF4Yj?kVWVQd z)O5B>BJEWN;e7zbEgf;ri<zvv7R^b~d%9AmvBfw4#zVtC+GKUohwo8rY^kiRzvgfD z6kIZ_T`2N*l|8T%Njin~`Vy=6t*0jlLKg~!#bQ0ho<9(-WlcKJBD(gLR-$Y3vU3LW z_XnMl&tEoezw31W5q)FN$^-RLow+lL|LGCG)_w$*2gfZG@f+FINY|nWA{1GmKX6uPVa5Kf zX~5=(>+_ZPxc6Y$Qs;5%eCQ)Rx@ig`3w4Q!_~5#>@A^?Zqm(2l@6v4+l<^o%KBt&{ zPOv=M|H7xME88FVKo=PNeC7v_vil;t!Th9B?zLFWAOlB}df`fWxme9bVf)&GWBmIL z-)fdAzU(WcSo}Wwp|3QqVktF1X2*?vK5|4z`U^L-OrWU&W#hZU2L|GD^&9Z;;;McA zN(pKuIC1UQB0zXB057pxJ$GacTixV|oEMS!hqtWuc5ugti=`bWq>fqJ;Be zWeuAYxbGVx8!HJ4g*H}hYC~7LUK*ViFhEPHh0-Yty}`Oo=CbNUR@9uF|JBHkzu}Mw z4axrzkNdyRp`(-M3*y&Ywk@$vf+M}O(1!f^^XH}J+GQV)B8;^Lvk=M`8BI##KE4?*_yrSf#Rwo2Ro8khI^yuZU~Eaou+u{&zSJy z!wDU}w+pe%w(S(6V^Z`|?U|new^>hUdOVHFS7?G73@x>@6-dYpsd~j_>qN-S$*L1P z$?!^kvH5uBC2|Q*)30;^yj|T21?aca+w9Hiw>Sk`=vL#mGKI@AcX=Y9TfE&~gyW>Y zszcOkk+l4+4BdQY(`VcIGOEv4miZy^iQMk8@lH0q`g-^Fc|eYoBDJ0=2VadmH59>A zSy^rTK*=(sdz!HG(L~7!#r~gW6qC&JR1&o2n6s8DAj2X~(^vTKD&aS>Uh91u9)1i4 zc?$H3P_#F8cIFScpN=RV_D&G8tNXNBY$jwFT=dIDC-VB_dyA~=V}kieUmz|n?u(_EVNDnzSLRLv z`gV$fIPO1{%0jhlA|aP9p)UqJi9m*aS?8=9VfmexJD3{L)OH8sOwgBTolh^>=<9Sv2#cGB{q zPmVCp8>U!Nw&qv68tLu4lbtyqG5qF9^`qXeO4*a8x`VdADzEI?L{54u(xJv%KU)jS z?(piV4Ht&AjT#!LLQ_lA$k5Xe8`~}nv|j1e4pA~n$`aFi_{esXq!+NBPqRq{Hx0%~ zdZ4Y>Wayz^5VT*|!9Ua@cx<051)dnGZh#+k_K$%?i z@a7gke7g=-GGK|YY6hH7B~Td#24)SFKO*Rn17kSjz42RD))?#aVQmBHO*>l3ntpwB zR2R-~8E`ZaiV7;$?PwmS&QJ#zTTr#FZl<%t0&htDh$ZG9mK60Fm}*+>gKGM z@z5zf7Fk>9KV)L#gvG)npWkixxanbb9!Dc9D~rlc8p6duW65NK?RV2}L)k{1l%qE@ zY0{)aNCt%xoqf*l?~dv(uh)9hUBb>MmFue)wSoIe`t~A!H0>;e$ac;eDBRFCu*6$jj|Y1|~=CZRS0tSu-PTW=L`J6VFCZIW+4zAx8g) zK7=6cph+w3LTpE&DK{}O0Zl~ht|x7dlEcoWu^uG0o)c;z0Ii6LC%)Yd7M1jPQ`F6L zO$qawt)RJFziX;RL;m{(1-svZ1XzXcM$$y?fA@zd?(KBuh(3+t_``!2E%|P{6K#c6 zu1I#bUsE>Ya_fW_XT#Q&Y7$bEL0Zsk+JBhBtQ`9bLX8JwNN_7=()2 z*>?AuVhI0%$8TKwPqPsv3}*a@*btoF9o*c?om}yeEfeBkhOyusG-JxMdT~@?cFHAA z@`*N(YnYr-sO1o4;_;c^f#oTJz*o536u0%KNoTx@o(7fg)A5NWVf=kufWziflkd`O z-eu_&fv9sHDH3$FgPE~86zE71L)NKB*QZY~>hf}8osL4<=nf|EA{JdNfH6!xw<#!c zy^5!9pLqUazv>`QZusDqamc1{Nq(;D*~_R=m1+l(X-M{`nlQxAc>wMjOa(a%Ff!;F z7)(KS)u$fPW`cuTYm)wo^|S=Gu>0+ zohqN15o{WXBHeIr4k5hgPGyu0df#NNnQjolz%(S&e}uS;SEtF(U1q;#i-Z5}75v7> z|1{MQ`s%lrsvKlT4642l1rj`4CVsa&Mw3@ejlN&*=6(CC^C~^-x2e5n+!bEs4O0{X ztR8|SDiU69mCDqZ=&LUqM|+kW`vbj-W?PdcI5qGOYH5yYadjnqb5OWS0nfhMFlPv1 z=geO~acjZoAS^qZo0s13vph$}sAC0m3h1vA-Mr^vws(u>R_cAFo3VuKY$fU4!n;=z zMTjU~8vmjXZKFnrdJFbt9I~=(;1n-!Xn6YRN8YTLm8PHKH`8x8;u8rw_n@%ag1%UK z|H5rm0*OCFn)|~t<_?>dL$gZ3MIhb;@C!iY*E5jX-93Z`XBv+XD`iHP1`&1?^R+q8 zouh=J2rO1vK;Q^sJz@z7!N>SB$8w&R^oonA;d|Jiv_BJErUz8-5TG}+3kiMxeinnH z%&~U*?_zw2TRn!)%=)4YRbX>z|DEHP$DKW~d6HWH8_WCm+W=$|cA39r;L@wrrykm7 z0`DY7!3=(JPagjV+7b90pigWBbX!qs7KF zwv=Yo)vmz|*=Uu9vq^|b?C)ctcbE|z4hg)5?3A!I#dbhIM58S& zXN5i!&Ts*q8bw6N=aeL`M8)0L9_U?%MqHrv6- z4cR7k9@j?;BakQg%RTphVQa|11j1`jbv%LcA3%~E+ciSgbO{3QaC1AQ9>>m09FAUv z#0gwu!z?`%gF0T;wkU|{isHms_ybOR1&1}duh2FLkIqdIw zz&0+#kQSkg?ft;FSkeqO*|4;>4!n_vfAByQ)km8# z3iqbfQfv|=+yMa4p5ReuzqjjZ;IYPwN)wZ28I%|addI0`3{G*0GG5}cIF1Zz80%)>Ns>RN?2 z&paHEf6a+mAOfKL9gVz+2JM^PGsa$!FdJqxLZLt#Z_?faVCdWb9_bnqa!?KR;qns) z`tPlz;t4-*AXjw^=CxJZx~B{!i9!^0pLJ`f6?e63@tpujx17|vR1bDDQ*kxb51sw-BN)af)2W#Z%>DW%MBqpcQ)oxw}#HCef{(Sx1syO?LtPjU1t)^#_pFBppd$J$ z!SGVp1_p|jhW>s=`0%ZX5Vnht2eaQLLJ9|yKE(V$9p7^SSqhHWkVlKbBc#a}mw3tN zVvn`ULe*+GqgnU!3j-zhs9g89KTArP9+j4sb^xK0YZb#EpmX|zQN?pV2F0&?aS(36 zn34_k7Pf>#{JjxHmZc>x5dR!+zX3aE(2-E_g2#-}!Hy1yanhP9?q3YYK^S;rRDUPN zj}tr3imWmLr9oG)rZl}hJ!@;%s~=kkGd07md*P|RJ9!l#v~mRP`6P&Y_+YOif>LJD zr+&yfbftZEoI^g3xDMT)1e^F4s1ZbXdE2}V936{b2gWagQw;R<1i0c1-Y_icE2u+y zV1k3gdNZ(SLNj<>nODKG#Z*pCE^4ek|Iene^llhWY&-s$8uuJ-vG--EIS@8t`e-#5 z_crT94SQoCxCokUl*OahNp0GX!L0k!#H}3Y?o?^ev#_2ZlM)@xvA@4rFVT9FFK(vM zT$l;3qtmZ$7N3w14&C2L*c!1j3O>B3rp5q7=|zc8pPmWJ7_d^Sy(7dI_mQ=pRC#CV zy?cCA>v1}(wXAr8>1_Sx>A8YCoc{Xy7)@WhD|D81P?JG;AXOhTZJ1)4_d*#3qDKrD%DZjO!tN}2CPKTAJcTwD}1 zOOf$nfDN*~)m+9FlL2i997kME#+C-P?ZaVKQdm?Fjz*3~s@95H+7&kah1|Z})>dAb_ zFlG!~EG(t+|5OLWmX5L4z1EK4?`x{99nTwPnj-?U#R@TzVtKOp$^CKboHcPKPbr>W z>Y#Q>70Cpa&fl|#wbJix{1u4&SLlyvE)Hh=D64yxc3}$c3zEje#)1nsr7xpBYhAtF zw!xb8I6QB`YTU7B?%0c@JZOnMC`(X)fNnnPpai{E!#P!fIrabXp=8FfO>k)UKap?~ z51%fKO7P}18ggw+w1Nq=;gB!t;A_;#D;OL-fWOGs^mQn~kyrlzZwdm*)K_C+T`3yi zc=-DIqACK|FzYlD#9Y|Cwtp{Y--oCd;S?TQ$gU9JE2|mcy#LoRejE@F9=oXcY+oi6 Qc9cb_D5~QNY>Q92YU327Ap0Rc&omJ&$;=?+0jX+#7Bq@}y0yBm}~f`rtY z$KSp0-S_@_pYZ9o&pvCfJ=cse#+-(zJyamTrNKoY5Clq!a_R^K8X4*r8w;*bt?8Pw{n5i!)mSR%E!DKaNDWSiD+nPNk~bX z*EJ2ICB7*fdiEF~=EIVck}x*0Cs{q7_aAK`2{|yNiEX8u{F$(&(~V?^{h8GGn6YJu zITMI6n6b%Kv3-d#aN!>8E0|=yOc>IG0|Slza|3YWWyV6yM}z!P78Vwn;=nEjSh3zL zYKRW*dh3fLP3)U~pec*xhepY3AZKZLqfD2Z%Jk<{`OT%JC1OU#sPuHEN6rCE;_@mD zSABwmvC|GbqA+4NHmp|G*JX>w#psH@e35zbgd#LFG&(NM|K;p=w@fjNMUu1L-TOow zu&$e%+vogz*^m+p(eHHRv|(t7^C##0B|UPjzn1qV&C`s#<8DY&!|D`8upiOsPadMN&VK0b8(A3iLyBqEx& z_ewIToSmJk>=%6}W|^_0jQ)6(f2V4=Jjaq#pp?!Nx|b)!boV=UR~K!%*6D}&Fb37` z?rsD^^e}FP-^LhoMb+4lTT9F+0{^@%9xf8a`L-?eSJ0tw$-;=(-`gH{{ z@LA|#{~(4XT^~DkwdtSL>B^6%b$=ubksgD7A@aj#Zy(^MVX${B#gHniZix>r&V1OK zt)Zl*re1RX&EvH8L)y!${^vGvebi|DqmM}#rPby0$-T-ccUS4X`UAw~F$evWWMr<~ zb=e88O22dKS3L%@35osKpGl6y_jth-b-_W};pa@XU?4vA(8k|y-vn+M*3(T+P9l6V zWP>s_av!SYj~*W{N1-9EKT2&K$dJt#w6Nb8$RQ&gRd)lG1_nB{Zw<}cyGarqt+EePNnHlz~H`CSg+M1eo zjwRxW_WxLWw4U*}u1GhCx$H1WNimj}m#6H~;$w*;TgiS7=RV3mTI)|ED%Xuz?THto zyF8q>ivhPvN}>s8lwx=!@rPVSMrQBgbeENvH>&g_Z~gHV*!z8O!1gG{sPJ%nom>+e?N2G@4f9dTcnWM`)jO_Hr4aDQaBa=mDw7XtETv9XlOs@>cr&3sow7G*>BC& z(ds*FcgC+0%4_%t`mob9+Yv@AzSMZz12Z7b;O(>1rW*p90v z@MVk&!%cS_SAvNv9~_GwdNEQ{c7Rep+EeE#|ZqzHJ`c)%t;3ul@J$-?|@ZuU%^|(=BY_N8Dmz@qKfK z>M{G!*CRd*b_cgp^$v1bWtDSPUX-X$&HfRd9dYYS?G3pH4{m;)j zQ_r@ZRXYYm&}+i0o;{;C&o&P?MlQ7_z++S296p^ECLkbq zSj3FYe1n_&zP&x)?!Mk%@QRw{aT6~}HargQwEN(OijN*CXlY%ep`o$q$jZtRb-@-9 z7M^@5&d*N*$)vNVN7lrI#b+(306Y^5fx5fMpw7lJ=1G_Kn zr*1nl)wcu%hYy`RJ;m_|316UN;TxrSUx=E9mylh5SUfSIGgf3M_4@aUQahDt5Fx#Z z_4rq_aN*qhhrFA@=YRh=RR!|53kV8Mo}KKQNfmH-%$e<$j~kfg<>l3G_oj#rH|>&VMvC8&$_F|vd4{tmTtaLb>~aS}SZt^LzPHtNJZ zC8vVc_V%T&>uIf@9wZwHlz-K^PdwKLH)UvYVZOQ6NsF8kCG&Xq?j12Tb%=z=VerUJ zU*agRIR+LMi;$4!#@FCb?f%3GLGwO3&!btFl&B|U{v;rX`7(BVU8gApO1m|*% zTi4%Z{tczn`9UnU1S__~=YoQRlM~W=9F~#q(a^CLV1pfNy~)NR$(aAW&kU{noSnd* zoiCo7mSmhcIsprxq->ofauDY+(3!*LfpaqPpwxJE$Qp=YO4B+uE8; zN~&f)(CvIRBf#^n%(<=84^A;*OOjJiET($wAj9Ca*Hl{i6jfAEZ>FZE#$*Ms z2*1uC>dP0dR~sW;ys1=J;&iH1rhGIJDej^&nd*7Pu1J%u$ zZeCRCOOuN7_s5uiA&kqxvhWK+LXpRBkDQW6zY%bOu@H?hAC;%UQgdiO#DXj=cm0m{ zg*X5U6WaI3-&+GH8gMz-;-!NlCT-*&KPFjPUOu<_O|h3H)PdV&Mr@+4sp$^~lMZj- zC47Q=JlWa$@4Ca<4{Gt_1wVwpcU9js@+8NY2cAnD1|`%K90dWM6{@My^6!O&UX)Tp zq@xM};$H6XU`mLNYtMhyk#YPr#16Trr{Ya2yz>$_I3vmiTmqWS-Q7(rupIyvtfqF~)PHa6C5QZljSsuH1+qaI?O>W207>C@w) z%fGy@pft!Hrbr)pjvGMM{iPO0Hp# z({p75^fB6X)h(q557?{mo4M;xcf^9o6^h1BBT#0ZMw`C&_U&7pVk5M}-FZ}^%Bzv> z#0YEig%V}-BZ0rd^rGUFVs9TzIbtO|G(245ys0H5B&0Y?FN39D>q>&!-|k$UZO57I zZ;EuGpxDP*@+h~0hrt*c7B)CC;{O)?>Rq~{%EAMmt9~KZ&3OEMR|J9&?|2^b0Z5zgF1pc6 zLSkpw<$wk5V2eCiPg{0rIDMAC3@#h2&tX{2RE_T&mYK-{SPq{?=<}$DkN^pZ=q)@k zzZC&&p5OcX$O+RZhkAumRXMqbH$_n%VH`DS)4tde#Bbb;(RXB-782a)+}FzRu7|0B zPRfgppP&EdbfsPX(bkqO;;B6KE_!{>t97M6rkk~|00Ge zWhDDjn5Q~8GP3>WU{;B73)b1MN8VOWSo4%S<7wN^>OHB)iwrwti6mOx3ja*>OI?P4 z{`{HnNPFYdc{H`ds!^bwn1sZ*l(~Z@zR&gVuiO%=Q7+Gu)%XM<%Xi~{qPB2M?Jh6Q zUDp!L?fWH;$}EfO>lvWPZK-+JEqXmGSFnskMC%9cYCL@!Qfl6RcUI4Ys5db4bKbqE;Z1bVU8M!#3Oevag6)T|FllEom)!OKX=$dG$2M5a9-xY=%R z$m-X;CNO;SheBOleM%ypKX7VF7n1uDQ4H68wywK0j;PyQX51X6m?9MTO+$~7VS-m^kZ%d zl2Ci=cP&C3)h3S{decWq{^blLV%(`9m@;P(xR5kAA?h-7y`J{wkQ#_MpWU2zrR{M( z81OWknJSWe?V{IWb!cbW5CxcIIB*KP86_noymbepFM5*#AaGxIevo=p zeJ=UC#JY2`k4E(p3POQWpxXWhKoN%4m&fVp2v$p z4|^oOn-paoC)e3dS7a0v;M3lIN#1(cn|ymPPvd&i-(TBlY0rg?YF_w&&nR}453ejtlLR>vToKAnQ$Z^N zt*b$dUX1|dkxOB`CTGXU%rfbwR`h`(J}E^WSvp)iJbtkGt!LjyR{yeny?3fA?n5_B zm-m|goqk}k=buq+o$s%102|wN+8ybiMoXa*?|ObxJ$Udyr`nMn(qDm?_3^Fx;_62)_8y z2;qyIvMzG|iDRUVJR-10t<*;AivuXZv+B*M2#VuBk~$~Xt9;o;_DFmYDS3P<>*aIR zH$>lz=L$Ny?IkZzInV`crHJV7k^^4y)h%rAVD!dMKjAGoSK-`R75MtBh9hD2gQBRy z3J=so#y1Bf7snk^EX>SkVz0OE>FcNb+;+?u-@K6!JnI9?ZuuzzVgB;vi;Yy8ls5x# z#yhTi>9UK)?FrlLZ2!ptlbvS465mdDYIdWOl0u6bE@&XVcl<~c1So&swv?N*7kQ5l z`RP4fY?nU@ai?9LT}Lb!UjC*0d$>Jqr8(&Ms)>RQxJ!3W4}|DeOnnr+*frp7u@U-# zI!!Bp3v|jXNKpI=LjjEj#iZ40Bsd(ltrxlN6aCE|H(uLW9w z5Dh_%(>ft~c$ANnwtY{WoALnFvmN2)(j(EJ+?8AoxQ#}TecJpbak-CV8P79efk$=Hq4K@yM5 zDAih9c7+^u;nlJ{A>5V$91>vNG`r- zK}=WF?Kd0j6BMID#ftOJ)~A0b98IRX3-v0CO*;tB7KywY!7jmh8o5h;M3RkVmczBL zS< zFDJO-eEl0Fctz;cWp*|j`L$~%{izbDnuk#`86#wV@_SXM@+Cc}6 z?JEr3_-P{UIB$^S;^Nw${5A`x6D0*=G+-fw@aGlfWRuS#kGw? zg<7p2NpXkXnCMro5EBz4fR_O?L8bI4@!!~gk9LV-yH*Z(#ZB>*bH0sIE-tl&74~s+ zaXB7W=jGwRa>jA2E9a6&Gfq0?RutfsVOJSk8*Rl7Sp$Is0g?ogGu<65>~j}SB%B}h zCe8jhM&ZVb%>jhFxgH+F}I%^fBlY3uGi-$ECRaw`cIQV}^ zQ017eZNC&uv|sfjviF=8xtjaZ%Bs05n=LmjvCTwULxarM*Vm{-2OwZL$gKEucZjnT zqS|j3)dH>X1L5=RY`^~)Kr&=D5-KW-WbKg9P?;AmH~>^35T}RR)}xW7UB5n%O*^*D z-z<1a@cu|qB@w<2A+}aWx!2-0`dbRWCjQ_u3e|#feyq|if8CFS z<+|Jl?h;nccSpv@WexT4HlMqgnzEAPV#2nF$;mO^D`UP_HamT%JCB|;Eb!G=j3c&- zx%}5;>2;9I20#OwUZ>Yn_g%80}1&{+j z@ES%!&O^80xzSY~j)x$UI^#2+rLgv~5EY&M{iZr=_?N}=E0ppa5xU&IvpNb=<)&AG zwqQL-U<#Hncb!TFbGf;R@;v+QzqDkcU+su#W@ZL7RvzTPff^UPgROZ+t8y9%_k(-6 z(Zu-SA<{sk2g|L;33uZcxD0CT<|XN>U*X);DzLc7Z*XLHze07mRk}T8&AaR3{IJDq zWo6T%To<2;@7~5s#mJuxw$*)!W!V6a0s;c=xa`oO&;}&XERUldDsfjnEPN`T8ycUO zcz6f_Mz72$?}T)(f*9TQe?Dv_S}J1$?--T{*u5 z34uz;0;)e_WLcRI8yj0X1hKlyAX>hDJF2Jv^}#->wl5sG5!CkbqH8=uH1@@r<5e5I znJNeLlamva5Nd17&G6>nv1*zmJ(0vgw8Bl7>aaTfc6qd+8!pKAC&ZEio!QjNP6M^{ z-$5`+^ZbniA`HLRnd|g$-kiEaFGYbPb?)ok_J;H62I)|Gx(ia(iDw~rcCCA{13${;b=i$_=eA>1)4N+^>CY$hbNF+!wH39QO56@9U=VpU z@68BlKUxLY+T;Yz-{`b!6w}CIEUOD=m%I z9!Y=G*~Jr=Ch^SoGb`Dq6T3S1qqUV4CShR;BJn+h5K&zK>+i6YJW}+Ak{N6{m@I!%jy-^WB zhf;$=;?CYw0X8;&ZLXz<81;bWK<&R%L?)1F2Y3c$LGQ~W?<|+yIo}ELzr2xbhdYtD zdLPt-gM%Bvw&16P1-j+k*Y?c38%QVe3k&bRdL^XF5vzUJAA9&*%KY8?_W>X@04vlh zzPCig#?HY+f1QaS6BW82(FR1wrNMJ!P-RPLAHE4PFwYq@;U{ zmqc~p(hp*24~klIwI^aVkx*I!F@)qN%q z2>)^P4(p6J9FeZhN&Dc2GDYL8!op7)4M;qTyUhU32CO~0Q>?77#&?2QM8E+=)^+`( zOEHAoQO)6r?({CW;aO4(l`OP&fK@2w-!^Fe{5VUI zCks>PWQwLpk4DmQ1)Iv0O3JHNv5sYi8R8UP|DLq;+mUBTT#M&JG=sh&Vgi&Lu{d@O z=OWO7x+lz1i~p|2-hWM1msOWQ$FB*IqmJ|OVaN0wSZSct{LzfZZr#h#{5ZW{)sk(| ze6-E4(AiQ$^%{>i4KG9E4ZW&orN|vG&#Ev=U^swy4mArfwFH8WOq%2yIvZDw%k1Lo zuymnT*u#%^Me9E0NC2Y%sJsNt9H*O0JtZ$b{X9$=@o}DJ@BtK^o}G;WuGt7Neb*+6 zckV?|Mg|&`hqB4NiVG#=tM$8QF)eLDP1&nESN_1`0ENv+RDht(9D&S|GzpL ziP<}m9T_o7)^ZP=juMN2_j-_2Rx6GG)ulqyJCC4>q(Enx&7{w=;glDrQh{b zSD6AHVg|VkH6x%JUd-{pPHDS71Qr=KL;}$?Ug?Z6aelkU2Yn2YYW$4KmmXwn4owly zE&JBfydwebhJX2dLIp$+bcPd~Y6Y4^d)B8(P6t09LkwFalf zl`+vw%vk9A&3j);Fu4C@=1*cx$xuVQeOvpG;_us@#@AEt^YyVX$<~8ORjWh_9`nWc z`1q_*7HLS(s;!{-$I(kfBS^R~^pHMi z;i{btbSP0QjVm;1t zOTUU3QS5o*SfE!)XgyZY6iRW^)kTBHks9)+2m>C6Y!D$u){RVVa@;=ASdg91mKiS_ zr7n;}swsDnJ9)T$@f7Bd&fDACn1Q?@)O2)$A)owf6@r@S_HU)QrLqW46JhRQlv|^^ ztKpx5E9}1srF{Gt5khwTuBNP0t{lP?ACQHB!J9u%5WR*K#l?Jl?P?kt%kQ{BU8T6G zRVJqX?n;uUE$v`@;2(DhnAiybg(P^8d=bmR!GTJ|h6-9az&+5@&tsHSs7!TgUHLyf zN{cj4dvgsjkfTBg4FuFA4zz2Xs}1KM(pY0FNF;i1aB(#^zD3WoFC*t0JF}98ejw;# zkKsLFl&B_Gpf6q!IuL{T+PqiM%3E9IR@T;7czKC{weh%&Pp=AC99d5XzgwH&C7}FI z6U#(Mw(P2@PutMOUUL9m$;;Whh{nwOVW8UMwd*9al5Bc_bQjLQHB(K7*aKZ|&Nh@8 zTJ|6McBU))3wiV)E!j@_LwmPIxS0eGggp5&QL@b-FX#`sS3Tmobgx zotbqMp&zoyzdr7P7{8^6#+I)y|N6BEs053``PX#|ul;LCw^+k_R=@g18ljrqY=Y+c z5=)RK!dGN=kVSEh6Xlt}h96a!DEbFEFehN2W7sKR{-D6cTFOA|9Hrm|Wgw7nY$om`*s?%%tk26;K-Mt2i zrmX2|6IP8Cw`1~#f6>(!(_7RV77zeU>E~=s<-jML<{N`_wrNWqsAiOw#)~hmfqD(>0(IoTn*xE%7zY&Rk^OkFvcK{ z3FPHYf5yygQP4vVd>EN=P*--ey&y}1)!h$%GM8EUF5H)s!$3*cm2>VrW@genoV|-d z3r*PoLfq@19qONr(v3x`!y`XR?DR#(2f{QMSJmbL7u?@warq73|c1uey5mGNJH z+%c|R1(hmfL~vd@Zj=s}!@m-wl3+}{Yes-&ctY}rzwvp4?PFqMWRlr~3A42FZ5^p2 zA|h(PW|$dY1b?$E1X2$lVM_x%-s3(}>}ftD5l_$=J;QF+Q-U5oG%?(|P$wYei$&H> z_2Y$Lj;OzOmU3V{m%aWAcLYxdOR4GfZ;h1+tr{g50WYAGRyNOaD9h` zMl{dM^P>t=7<^ml7k3a#05M}_bu|N0|HeB~@q}2v)V4N+Px3j;X|{$`QBVeSH=x9p-9;N@5pfq;mXj)j<=tS}25hqt@U{kt( zQ@Y&a$3h}Dit-L6NvQP*IY1FY_!zFPFh$EQ3pYM*mUZb_ecI2CZP&JuB`VtB+eJRH z@h0{LLR6GUG)3jk(rdnwog2Rf+m~lHYIc9Uy6cksZ+dinNPYbaZca|GuG&if%&RIa z8$IXziD$N2?NsaKj`7@JP z&H~M)k=78x^@`_!)PM!pSK`s5CTZZr;Qvb? zbKSf-dyb9G*u2s2;80oeT#09C`6daeiJW3E2@?5U=NlU(T*o{z*Y=#T>dr2|H-C!kp~34t^T-JyTKsDHz?#3mzqt6gZ}o0<2g z+B(-lHT$zY9nasc134@4VXi4CE-;rSTK+^`z1hUXWMZ}v9L)SR(6tVfR1?25{*0~U zsQYFwZ^l-&kN&pXRNBXr7d;n~(+(XS637$hI|)@PfI^3@$_YE5FY#X^n%6;Bf3MsI zc}6f%RW(Q5VXDMnABJHC+moa$5Vtru0=MVtY95He<3SyBn0mh%--L$C!BD#XRZ49C z!RbYL*>Yr*$I|}ho}brFvo8Md{{eD1c!tmTenFiBWo_Uespxe%wiZNuj3S2 z?MR21K*29;Xox>gOiZjub8z|j;thZM1OQt=8Rw|q34rL-rs4E3Gg z>6nz3HVj+{F#kZ#CJJc-m8)#oz63F+9n42ePd{s{hq3ro$_crVUxfQE$#trOJy{UV zCoYa0R8++wILZcYr5gWTmK-UkMN7tbo98gAZ}J5sM9LzG?-j1EF;Pq1;y) HBj5i8AvTPk literal 12830 zcmb7rbzGF+*6)CHmvkejNO!lOh=53mLwBdNNF&msl%xm--Q6XEq)3B+l%yabdDr~T zx#zv-{qO4dQRkW2`+4?WdwpXqA~n@j2=Qp~5C{a}-8)KJ2m~5A>JJVUymR&AVlDiI zWp+b1K3ZX|ein0D~kZ|C4De>ZiZT%J~2UteFEVNvjA zf^_+K)25HUWNlqts-Vhe#z{86*}=@QwoqoAw`@03NHE%PDb#R+e_-HoV92|mW0D6m zivW6 zdrSP1v#X-T#Kes6TmDc`aK)SeK8r(4tm5Fn^-k+!E|LW&V%T%;irmgzQ^Rtj*B!n( zzayT#{rwzm&c}`=A3w_9zfY*Ds>aJ+Yyd5fl{`c?SJF2R`bQ98CjMY}3X330`QhCvzpH%gqzIb7S zY&_=r^r?Qlaka@xOH(tjyIUDKic~bXPmA{kKd0%^*wme6#BRjp{!>ELJhHH`F#iTc z5~4P~9Yuv=(=V@tLL;aYxW2=SRjQHXVg9=ujjU~9)O}f6S|S2F4kB{Gs%Lkqh8A7U zO%bgJ8{>aHBbe!fP3Pk1eW=Y`PkneP?fKm-j?#`AvB?#O*XyVDuyylq+E?+)*Gh!f z(bGHl%wc0ES51wNxqiCW!J2T#h6<;B!6Vsc(2s9@wm}rZtUzn(nicdi2+_LE;ETa> zE!@Z0(9vg3PmdU5=;?~|PMgJ#)*wWnZBu^RXnz-*hPRnPi7g^9g3VWu8V~vu^>Xn_gYa$|)(td|Z@2-*E z3eh$D;Bdo&l?GmBO=e{RHtei<3A?|4w+-beew`9?`GE*0W^y|FyEdFlE#g3pj*d=3 zM;DcNT>~A#dFvLxX*)(}XsF4^ySpC>3mvRCj(;ybkBZvx*xZ_~K|>4=4}TsUTs<*o zsTU9Td3Ne~Etb&`x6c2D5G@V7e(vehOV^X>=or2^N?Iuvc9*z$v$uG^jjuX)Kj_GW z^|dO$?Z~y`<6~w8vY=Oi6b`tv!Q45IOTRt4$$E`cvtZ17ulG8)k^Ov)GZPs(d5r9{ zKf@}hTmDo~Q06o%(f>x27*XnbVAt8%nVp@D2*D=qo~p8+o}8+Ga?5jbB4}hpt8wqk zEy^CJnOagrd`il5$LVU<^D>fI`ca>GH`%**ZtgYAqF(rW`^j%VaiTZ*9<+Bxk{4zE z>vnyIvgJK9h{MYmGKu?DDe%Io8q)IMK%V~@)!^dgjf25rmVf>%ZETb}+#i&!fJHC$ zn)8WRKk%(E4+##|_wn&**z#@4TwY!-%TbIF2vcZlZ*Px2ovn2ZA*ANZR?I3kYWngv zHnqV&5{?wQHR#M~()lO)cZOg zXQyq0DJO~d1)n}`N-z_9II|^mE$sMq)tsu-C~S>5f`W8<5Ty0 z)v_;%245kXKUdpSK>>YabQDroLS|;H>r!`>q^FY;PrgQ0Y~+?u?`Mta!e&5@Mq{kf5uDl2tsOUp^r%6odtc(89RzWDjmBFerE21rUu z`ffKp*W@=DPDmFmyE9Xlc(5~{bvI8-#Hx5x@O`8c~XIX3E1Zk^RcNGDSIcYoEZ z*J#?u*>>AN@O_rg3QfZRc87k)Uz6=~ypb!Oy)W6lFVBy@>Xr%95{fuZg_fE3&{kZ@ zP>%@1r!pJNlonwa92_LZ3v16W+g}~**`94+p^S2*)gVFao@|sTvL^i4s+q6$jc(eX zBWuhcHjI3Q%pTvEbRU#Peb(AF>v4@!89cP-`N@-lt@;yDs>zMHL#x5euOdz}pHrKP z7jgWEN>YVHMHM+;dmc+E*x1}K@?B@Qu&~Jf_^~%=jub=o1_j=qt!X_QQSaOFCT90_ z{Qj&BQ%iaCxw*SD{J1K=d++H}X)FS2=1RK8#a$D#ZPi%zmx+n42fuXEC7!&nF2MmK&vwNnrn^GM%;3FA=znGR>m#wbvxctU2g}2UN%6gy`w|CMM;?? zR*omRV|28lPqe6xR^3pL-*Wiv?OKm@vVlzLz7Rc(HYQQ0nYF`5xIfVQ=&wlo{%PG? z?n_8a4EgkllTnHdC*TGz@6O=?S!*wTsbl1e7wz9ZK19u4RQ5=;yEJlbyk!KX$cLbHe3N=Zjl76`bu6mYC3_cZ)%NmWm z%ivDSC6TeQvBAo2kThz3fB{uiyz+_)*PS`(SeSns&QC~5iF3`m+&n3dtL+3mB=I|H zYD!jC9DjVJv0L*nj7G-evZ4dD+!0j=5HuCm8}m)2bc& z@VlBCDcnT01C4sRSkU64nYsfRMl!M`fPs;bkwWb_Fe)wbCXfi!ZF+7^g`J%E4s4Qz zHmx1_zDi0`C>ejqw{V3gSsV)vNjRJ`mRq-?z8&Gb&w2Y6Js=>!#>OVwq#YZ{U}#XWnkTf3KPsTqSK{Z*RXPV`6Te`+Hl5XOo5` zs_TxrdiS(*OAO>z7EaDDPH&JWDu*+JbIX0nr1bOz33n_;%aB{=zP>&_k90ad6csIB z(;2l9!okBsZ!J3Ub9Cg!BBFoZa=zs}QRjYr?&*;sz}19hoc-nJOuxwnpM8$EZ{I@f z)%$bl`M1nkrJ9>`J|_i;RP^)|aqeNpv9EPo#zVhyW%jKRxebEahdqKvxjZCJ~g|CDY3Re4lI3k8Sv`$>)*ay!F)ZC3tp%& z`d}O!95}fi-TLvv#4M&CQ!gHg#d7WT?c2{|V^#`jXon4 zaV=S1ED7*ssjP=laIR1}6_?m$tgn;u4o`>+MYze^VH>#L`limXASYu%{;;0{I4YVz z&?YAIYrd!=L?5kCGuIIWJ69TTzj*OtcP%$gi*JFP1+UbR1>V*uM%D#ioS)`;?aVpp z_+!pEq?!l95Y1xGE~J#BsnHOxQg2egu3>wwwBt|)U-P-q%hF)P+TW7=W+q z(XtP%+SvxQ1o^wPd6wXqDl{n8@>vh0+g??hDY9A|Vk{*jBpli}P?j9{w&d0pjCs5& zdl@M@=iM=}J)C;R<~QJ2(%j6LmzVea`E$OhCa)bvDJdxjnOMOVWA{V)sVlk#W9{?v zOmG6!$Cun7uhj;eOUPUtn->~Xz1XK1o1EjFT>3uu@!@+CDk|$u=YHxCt&d8V7w3+@ z)OI#^^mZ9b2WtEIEeF!YVFe~SA>WfSF*%<-`lwwW(;0?V(*)02rXLN_y!1JN?FKjZ z)hUMd+i=Ur3lU7aKvU>y>K^~T>au^YV+DtRpc5zoKmgH=*Qz}xDly{>E6-GAsUj6#@t2ayxTA-l7y`aGu4I7JGWl2u=EuJZO85fKjy+p)P1@OtwRvsv;R`D_I5~sWIP7aW(Pm1Laph7Y zbs&IEW6#d~rS|&;t)8tq8Gc7#W0UJax`%Bw$ki?Q{dYCX{MWaSR?jq2apP*+3GFjb za|G=)pAtQI@BjxN-~LF_E3uYOPB@WA!Hot3K`-q~F`9p0@Kn;PUln1n;z0vLZO_yJ zdkfYbTuf1D3$;epA*Lx`_PIFAoNfxLrSPFaZnS!~6WgU1m?SQu=ti?0saw$1a&f>b zbu{Bnd6d&1YF*za(1Yl;ti3)TaQaUQdHx(H>a=!=&>q31pIG~~obBN| z%{Rv+u&Jam*iYyh8pc2>XgI%ma_%{n+2Pyz>%jN;?_V0nTC=!&_xdv=shnHRMVkMv z&|PC^7teTR5l80-%rr&Piw8)cWaap`Z=pj&nhj@9!jGnP{ImrZU_I9^UL_~z0;ztO zueR=C{3&%;&8sjqWpQI8iv3>Fj}T%Rxu;K?hPFxQ69L>-GyMh@mrWktNP&+v2sy-> zc7&Vur|^IN@&(;o&BhBS_QtJS9jSuWyJ!1@UiG}@{f_vhmHi9UFj2tfOdWZJ^`LSb zqqM~AI0HKm!H__Nx!SS-Xvt8|h*I)8WUQd|fQNTt+Rg2>*N8vg_;{&*3J3@a-sS1& zx~rj~;^tQVK%iQ1rpgE%zA6f-BdE4k47SV^PR^A&q|A|lt!^{fESJ4f~-m*@RQ>0g~-gcwmQP9<; z%vVbdM!SOL;lqVVp6S}-d-w6mK*l||*t$OY{(-*!^V?CBlR|TbeyNc0#0WlWtGdys zt3x=<1)MrMOq8~LnqJ|abm`%-54;i**moBjB;%w#PqN zKiu67U=ms9hI#on>(aJ@(CalS;>pqAsaF?&jTLp=9^^ig7xW6NPq9L6KVcLA^qA5~z*Ngq;ML4K@}_Tp(qC;Ej}pn0P| z)~kv!89D6-S`GCmak3ZG6;@x%r!$*s-!y?tTw3}1#{2wWT)fQi(IZ@5UfzK-kHbNY zc*auJ1e_+nBPZdxtA5Bm)1B9PMs4;bP2s560T5MRHTJ6Vf+wLDcf)A#+){9HDV;q8 z;L6c00POCM%o@1(i~IdbQj&NQph?gEQk*QSpdk69M~`p_377keC65Hq)<#Mnk}xpD zbapCPSXlw%z~54@S^4t%mOh9gqi5$&l$4d->>lT6YN91MuZ_5@_NNKso0U<+`A}#W zajXOIPU1C=1+s1O^K(M_^!xr!w4uC(0hllVby2q%8L3mC%OWF_nQUIWuNZ(BQ=<DQk)Lkjlsrh-USiat zV_{(-66e7rhv7mu`a6@Xri!_eS(Lu6L-gub+K{gFr-nS)_%7}-Q0Lc#k&Rw>yS|*O zEWG$LdkQ4ajE4^&hJ5>`O)uq5GC4V!!fzhl@ig0HG0R((V&KYEmUUnI1ZG&Rzwqz7mKzt=Xhv%A}Dyx0)%^v8Ma ziYDi4r>8)4{yn4}M>Ky$vc2rd8XLWs3u0_+Ouxp70m5tCDOx(&htQ1%el_-5;yiMx z4zIq*kn|!#sTvT&&YKfuSAZl_{87cT*L!1QV>Tew{IO0czSv_BHvoG)lwA6O;2r&d!_i$((u~*Y z_R0CaoN2Ut$v7J?FEOl%lB+9!AOt@^K;yyp$0CsusjP{;@o8xXKS2E`oqY0K?VQ^>bgJA`A`s z7SX!WZ!uYJ2^Nyjl+AQqF_0!YQK!z`g^sPmS?1(~jEpUky69r8)GuFDQ2+~*rU==- z{OxXQKRP2&zEus|xYV%I+)*(o6LmEIj0pt?U*ynhYr0F(BDmbsxdqVz!L~C;3Etmb zR9`RVkjA>BkxnN&3mp;WaB_DK54&N2sDCnIm4?1Pb->?cs?DjYDDwH9IHouVKgm2~ zHzY?=a&iEFbOJ^xYWwkG;fAa_li9=7loV>N$JGS6$}w)groVxjt~4l`20$PY78d5W z_iLSM&%2kr3$mV#ojt6^JSBJQ?#{vj+DC0WwH6w4?~A{yzlT+ruuN&cMcIob*;30s%FwVd`P^YS)&#k0 zWI_B`MY{Mg%2&FnJtQ&=v#7FWn*69xSqF|znCERx?;W&Tlu-*yOQTMO@zOlESxFn4 zn#w?Vm`}|dz9}gy8@7tP<4&FAhqSW&`0?XJr7gLcnc2{L4WbJ~&VrWWxgGLvuVDW&Iab!JE zq$YUUZjh{KGg670mzY?N3+ozL>uAODrI~AA6u(>vwnu$Zvhx0qd4z@ClPGE8IP7YI z%jy-phy@IDf}wThhaU}MOGeRrrBl?gbsj#vM|eAz|GO5!{2XmeH+3L^p5q_irn2hX zF}s+1f&3B-U%q^C5s++{eJ3Mn`|>sw1l~70VJT_poM9X0#NMF_YmGbC$OveKaLjvR zyMW#ETYM&gI11tqPfMenY49QjEX&NyjK6bjcW*CPn=`DYMpVRkE*g{so0u3i>RT#I zPdzR*Snc$p_P`cS)R3{|QmbhyAKm*cuJ^Q)e!KXmb>Pvd8o`D|M8`x(V9s#mnQ_T8 z5)y-SztY)1^KyOEiVR&Oas={uS3D#bXkV!c2?=F`{7ZIJEXnq_sb@T9T)`Z{l;fa&2ip0_w5Yyn8(b*H{balmNNJZWoegEb_ zhc%&fdYTRl1@JpyD+A$3#KjRqgaQvhd3S&Qcr2xW{j2MBT+J>d^leiy^7}VK2|W4% zdhNAx7kpi7j!#ntpB?n6GW)*X1_LU*@^5(MSSCBYqXx*w%1V?*2IXT|yGfp&9N?hw zIkW|6%We8IUieL)Rw-LpNIBKab_^viJGBXnmbr71>!K@x|;54r-}=7#K56@>A6)0a43Zb4@md+_Tpc?`j%IQe6~-Qq=8)6OvO#sOG93b zm{H&P)bw_IH1=jT`VN_;8Lvs5*1)Dh33xSJB# zT$`9eJ*H;h(r4{10oDb2Pb@hF3llamqms5Z1=xczph6W`6SAwTzq4P73=(%=`7ymU{Q%ZO z1w*X4BJ&KHtIauUwAdc_n8zI`U&5O=Q5_M)<^bap)sA$ltlt&bD5F3N$AKq8d*fys zTO1ClR=upp(Z-dTUms0htA-gicrFY+yAT9Z>u_s20VvP_EKMlbr{S=c69JF4`U2M!m>|I*5(4;ys-#<522D9RNn4Zyt3E9SU zv5PH#MnbFiy3Yl3P50!VkG-t>P*TzZm{aCFF9UjFsf)_kn5~cGc^{1FWmNwyjjw%) z48JAk)gsgSFi)#$Bi|`ToA|U6DuRKESo;F zmkpn$HjnoDc-gqQ2?2@MrfW(}E}mffflXW+>ZyE5ZyPtxSPD5EXkx^qD>6R<<>jM- z2pCtjg!pFLRX>;Nr2yTD>es;~CA=sa36^-{{(xk>YToS9Ii3m1HaR~z2+=KgqKX>t z$dYJYII+K9EnVg*EqItR)+wGqiX*bcSq%;H7@_r&hDps5^s`whNJv*i7T_P9=^}zu8~h&CN~o z;X}4cNgmj;@3lYk7C?wB&R<@LRy)nA26d_vxX}QoB}(jDg~Qu?%=&MO z7H2EAPfoP3s@pP29O@JWxasxD1Z>02F>Hk=@`HzJ|tFMMB$oJ8cBi~xN{rpTyK}EI1ba{-8ZA8kz{~yN| zck9KdV(MM{^kyr!{gwNmfy`jt{}jet0STKf=8B%oYfL%C@IKs*qsaAF8j6Xh3ES^3 zM)7J9Kg2czC2VZWTPdlI56dKWAg~I|7!kOd^(^yNGR`f2P{Ed#b@$)@EjEd-%{JLX zyf<9Y*3)B?lspY`;#qJ(?L5?K=_NhYZ1mEJUe?;gjqF`RyHlHg{`^@E z{B7SqYi)_vN_y1%W*EJel|jLG{nCpOr*NEr0hxg5B;|8^0v)b|!xe9Qj_S`y8zEYh zCb*Zx<4G5e%h1xE z$HWtm2fh9L9@CG0&on6NzMkbDTv;jfW<*7AGqtxt8eiJPu^OA1g$0eBo&9*TVzAVr zm#ksyQB=jGy`3GDPCg8WSNy#mDm|AZZdjCzlErXtQLb+vdoUX~vc9P)AvGo=ul>~i zo%A5`_m+v;SOiiDP`s_enwXQ5Gt^Us4wXx;9(LmNvXjme-k&>b^gzliIcm-uk z*>$Zgq6{4Yff)s!kQXH@q#mh>Sy^#V5g2C5r{nY4vw(hr+At)RB?s!7J7`L^HzF~- zR5;kIWRa+4s)_H*UUlndn-Ey-j-i8F|h0Z^F!VyVf;AJ>sA9?hRFVoMy8w~#%BxqepboW|6T!F$SICEFfBxSqsCrg8ndsYwUQ10~Gnxov z+6w@rn<=$4vw&B>*pn8DunXIY*8vT-e*l^8W{)j6sPz!`LmmZgapoUl5z>Z2QeQ;5 zTEGRjs(Uc2CmNuf3+KbsvrzE(`a9{le_|TAU#-sUPpCaXtu>;hrRCmXt%tau)StjA zO841mu=L6T&VAECK;z=#8kVneJtK}$4gr$oX`ay~%d+*myua+4yX#6nO=nNfzU!TK zOkkL(IuH?qL?}o*$aMGPJHsr!aXG*ghOA#y=}c`d=Cw_)97EgH!-DJn9O!d_=d$km z&m-n!fHL}}CTKEeyNW}srBs{Z$$$#^3lIt{*RF-eDUv0*l$)R<*i>RW zbK_)QXvq8x9W|Fg$Kyy^U{6VDFx@))yN`nRzhd;JxCuC=d-AF(SmfEu7m0g&ZvIdT zO=nDY{*WALNuQLe$wwN+hVhzE5p##WXej-~A0KqRxy66W|HJ{$O@AEMmrJAhXqU zlFoBd$Yy^U*^A>`9T^-*tPD%>mPDB3Xl3!Kjc0p(D1Q_216o0j`;0H#o_Qrq>RCgz zlb%gEI@JAKHe53J3CIeRg^2h+j3Sus}CuU>rpudM52dX9RVH(42%x$ zX3!cl4qyTcV41kM_ypW%d3iZP>a)~Tb75Y2IT`r8-AUY!5eU%D-P33aii+G=QJ+8G zQA-u*`nW9c;dqL_k=Ol|D!ZMX9m&TYh zQ;0aedP~=2YpNBICRs5oCildnV6UW#M?Qiv4F5ZuY>nejA*(t(d`^fW+^G8jc7XMoIQtLF|rNV z|FIasFirHMezhs6+KEY5Hr>{?e$|)V|IXaIC{;*AWN{!v5>--#XOhUpZwuX!)Z8Le zhiuoE#O=;mNAM}tgP=q>5dedbl$2CGF~6RHfy;fWz3L=W{e8uBcu7gg1D+0=DQPV} zc!*?iltchk!xX_>n1i2+JFZ26oD}?Q37*S6%{qdYwQKZ7kc&h_6mo~NKYYM}dIx1s zok83^y$b7MYzLL2XdqrRmyW)RQ&X9(-!jD;z6-!Xd6$>B03-w|)|I_f?jFlK(80ly z*elT=k9^uZVIP;wSZYv238&x~%$VO$0ei!+`a+t=aA5BIx>wj5T3=ODvT*^rlezXM|eQqPK5@lf*Gfu3f*r1hN4NcoaK5 zZ;d!IIv!hTJ$MiVWpG!=Oy=m%76R|`^HCijA3l8eaiIUrvdA5BH8=&pvi~fQHwv1x zB2c|%C_@z>%?cim%N^wX$>COCm2EIzA2foIGBU=B62YpBnkZW$wx$oC0W%E3-4!+i+dCJM2FQtm-k+bH(xltF#mP6!=V913gdff1hfA z=EqJm*Y~HA9j8}E7+0lD-lV0GkdO?mHy2MVjVy1@QYmZ@0jH~4g`(N3R}>D@)r5d9 z$|Xik^$r>zwcmoB$FRM_wWD{(FP+!@2K%*i!VsjguC8u!!%|{r*ntb98kk@UJ3E6_ zkAh31KqY6Fm5~As-53$e(?XR}pe-X)YC~q%P53%9bD(_%hAGv5(P^^-u`jBH5y>WG z6cnAPLEb=LR4!6kQ87fvd@WpfV?iqA!pjnL&$0W&!2aH9oR*%$zj$(*-QGEn3y&9bBV=RCY6@AtO&W=9zr}ar(B;{ zS>u7If_3m>E%YH&pk4QV-o2Bjg>0F_?uqNk`S9WPoQOm{`k;&-m*6Nz95U;k6JJmhv(wTxPsj9)hE^H2z)|Zm>JS{@GIg@@mfkR*aRUVM=AcLoh^vLhpejhHEXB3e08zo6@LM+6#Tyu`TXfDdDV1irt! zkT7I7T|+ly*CpofeGQQ|m{cmB0_hbE5iEIZ60&m{L~u$LT49!wJ6`hGDbz@pa>ma1 zt*{8tO1mCT_w(z9MLl^1levtg-=}l`IB>&hIY;?rg z+1b?W%guihE+d?rsZ3KD6(#N5TCx7kl%OjGXhZ(h7K1C61Ffe)i!+tBLq63lH{R|4 z>8!IIMIeZ!|HMC{gY*sbs&ov~2!x1da!Vf0cn@FJ`}>mE8XATT^M1a+_#CIi5yiq; z+-*HlHR35yGkR-1m|4p}6PbHUPeI1xhL8XayndFIiPgKYe_~|=YdZvpaaxt@k5)7S zCYhPzeE?)qDyoRUKm>e!5tKY4@V|-=WYGOOJ$V8)$k*7bkxE5ISX`6Qv9Zbl0nJ5B z_zQ2TSKKXrIu;^*C=?^Uuq4_tC)$$G-Rk-BT7{J6QP+mQ$ZY1TR~DX`kIN?~rJg}o zgiqEg?9@l?7k3jm+Io84F9*DQMe);VGI)mjaeRx%O$eo-0zF=}bg|(@;lChBSorye zkN2S@i|W{bcBw^FeYDYMAAu;)D^XTcBf!JM;{ymNG;FBOC<5yxjRC*1FoKZxN|FNy z2Zz~owPSSvS7?-(L>3$Cr}&pI{UPZeWK`bjXu}{eo!5kto*3;5kB^0 zGBM{Wg&Mux;M`w_`le5xzHCa>SR|IkuKsJLEuR+0g+^=#z1{tN#V1dm6ga-u$Wr$A zmqPKm&`=Bnw_z7}frQzXgf@36W@q#fySuw`09$ju8{8n2iWT1g`oB9>oCs}#h4s_X zdENKbT|VS44DqF}FGhBCWyS2|&-&-yUfU{4wL5p@e0?Q==Yko+4=d(beYibq?^zfG zwKRvB+F^a>3X2M8H-&=J;z<#iQ{X>?>gpy<`Z)FGqutwY%*hG5^l{QKFo*Wd;H}w) zWJl-|s3nwZ(1U(Sl<2#6=Z=Yykr5X^e*zST`)Z$An-g9h{{C&6$Z_8c9ENCqtKk(; z!o4!*e;y{^DUb|H5+|y%-hq9%=ocImv&fv@513}P;e!H|12{P%(|qUgb$ib6Pql)v z`p%zQ6nGr}e8I_nP>!m3zo@)tBE)ClwRAUiMT7WDX#gkx!%1tAIBTgHq*a`3sGs?c ze|2aW$;0HbscWc#*7 diff --git a/docs/images/chapters/yforx/e469af5bf27a2c27d1dd6fc62a78ac27.png b/docs/images/chapters/yforx/e469af5bf27a2c27d1dd6fc62a78ac27.png index 8c5d0bfe7be418da060298427a45d4f0f7654ce4..e775b536fdace83266f19f5c4a46f46f0bba332b 100644 GIT binary patch literal 19403 zcmZ8p2RN5|`~OjiYzmp7C@CZ(MA-?Ige0?!sD$h&TUL^iY$Dku$zIVCk`Xd1Br9ZO z{6Ej>y#I6F_d2fYlz!tG_xHX(>-z*B*H)w5#J-6{BGDdESJoks$OT9wGGl*jtv$s*0%i6}NjuRQ-HyN0MKhXHsqt3`T}$hH4y|I4*@fg$>a53)4KBtEo{+n#HciR<#fF_#z>h(Z zlGiA;=CA+z9QLQ;#h=#uZeqr-{U|B6{PhAx8X-;mm4S~nQ?%?1;>XpIe#>U9@9@%{Vms&8lKI<+TS!_3C! zwP7uBxSCo#D{8*ZbAu~7a=n9Fw{9(9dB9-aX)P2;e9QkIO|8Kh+FhGB8NG1W@Z!ab zTN)2d*(fe{72fdmT^AP@XKHSqlA}x65}Wb+Aj!bcP^VJR?-qsiL(yHkb{*EyVLG?( zba_KaNXQ8zqtJKnG+jMBQZIHFb*Tq31a5GCezVlX0Oi&;_&tSk$iov&o zk*V*^7A=ysCwtE%vW{8FBAFIB7KRk!mErj zl!Ej={_?OL<1AbJHk|*g(XXI@yuN-@XP2e<@xYu7wxsLj zofXv`oew%YJofOGnK4bzDU!_-g0$K zP2KMIla*_9-rnAuSy^k}zNO~n<$aWynd#El_00ClWbeW3(@#}1E~TWmzlok(>+W$4 zJ2Y+FGkNCOg$hQ-HMG}ipyi-^(%|fn}}{Lx_D+`X~}%5zuLO5;vieV;SiQR%@x-?Xt?Fratxo< z;A$u3s>(S3VC(gqX>^akQtTH z+^w?CKg2sic@(%C!nrJlE;6s1$GUm)MDglXiO{gHv8D3F#Kf+Y^&}k~orKZ(Lo4#x z69oZFb2Ed@P2{}F>V?jEoXpJ3#l^){FJEqWdv}j-Vj_>x*|TJ%YrlV~oIAJuc2v}K zJ;&vbrF#_=cW#f+~GXlwV}TyLrPp+Jm$fJ)WSmDi@#}VO+P(3zwjmM z*!Ns9DXERMqI0>Uz2&k$XJ$@&^ws6J{qmrurq+y?*^;3h|6!y=)@4Gjbuw|H_B*%j zBNGJ?y1(wW-@Q_5G(7d~w@bV^wr(|hA9q07X@tSSaV+t7(xuH0)6*%`!a1u^dt`E5 zZQJg>ot_#{dX{)-YrtV;<@Mz2)=f{8PPU&tdHVFHehI()T~bo4>FMe0m8+v(dAamb z4^3K$Qy+2Nt(j0E74+>L#x+h{%kCGSjUiBnh zzju$7iHWH>&qRR~;Vf%Z>BWUlei_asOCs5}XWyu-R4^>GzJdDXCfP5|+N`IbASany zSdfuISVbtNMotdJO4u?mFpx-BJ4jVc3TreXA|iCX5@P3ECVFf>nxCPkk8ocxKFiZy zXp$&@jor)3i?%*Faf4#etstS@yYD}KER-WDW7{sQtfG>+NBfS(j`o~0fhST`%|@D& zi?98b^z`(+b@wiXqoZS3w?U=XK_4F<_POA^loayX09qwaPdOQviR|gCiaS2#laaQW zJne$qD1JBS{US+#^Sk?9A3l6I()~QI{$kgsPc|PXB?`=H15_*f1ie{4 zy2&I&MxJ=;u-)&Lp<&{i6FWCEguHAj4-5|rqZK%o5c=rR&g?VK>Bz{)t|~PeEq%|* zlfLZBmj2;`j??##nI}4|v-+}>6sY|eDy*+fkG0A=un!Il+?e=WYSEN%(D2nwhD)!{ z?d$yXX*!+)6F#Ax8h2bM?73DzKmeb(IJ1q74X2d~rHhb|P?D}tTY?=Xf113ps^a6LIU)cjw4-&p+I!U0)IvF==*|?`-OzYwen4EY_{#8mE`u{?W{9&N zSs%YuEc5Ho&^mM)PC3_GFS98J0(3H!j~od$?isw!pjdc#coTD^y5H24eOg-DHhzBJ zm+vUeOuW40h4<`fLFcSEiSLR}I(hQIGyF!(N+g9jP8xM%~+3|@sY{Q8W3YH4Zd6ae^Ttv;OYknm-4k{;i? zu;8jU$XHW!=J}=Sx;j4smbKjKKjw9f8~=FvspY^(OKP##ip*{?Dznxw4yoo86>0*j zDJ#jmsQ+?APfw4Mnp#OyljhCax2Hpnsym9&`1bS|J<=62w7~JOe;Cb-GtAA+WfL_e z68jIYFBjSu zs?#$=3FlUAu=if-70rHdaPU@QVw24ozeiEg(f;?0-KoBQ{kq?rxahB5zpe%zd3Gjd=jKg8 zFU@m=Z$B+4sQ&Ug3eEoE^q{d~=WV>lxZM4)P_&L2-g4o~w{L?FE>~24%skoh%vPU< zN1pS@kt3q*mP~d$xxe_hxw(rSzHa*U(Z54#UJVK^ zqGk?O_F0@6wrG2N3KhE1&d#o|OI}P)Cv)?T9Xs-$KUX__dh2JwKn8RBOFhLm%01@I z0fF-K^Q&-zSAKMgl`a2#D^jL6#xKwn_a^hamDNTQ6BE1MvKJHfQi{d5BO}SItcFW` z+Xx=Q=JV!_2B)+=sWy;4u(VW`Th8^m??!r@**YGtg=AF~s-&+q%H3!uU&hC2ckkX^ z_wwcR2YE40%s16auNAbpAB^qriU7)+w@J$2J3v!*zG2WYj+5h?w4habfb#?Ws-Fam`5elnpt?61y?(Q;{jjy4SS7>ISK^@(MSa(F2W#HS`M0Br7YcY?DfEK)afUNAHPV{jU4| zQg7xQBQx)20lrAefq?<_BS#Las#5fgG8}XO6fAjZOpryV(-^%=^OF&Y-m6>DhO2Q| zM4f*ayziC>8n5{dzGjZ9?JT*v8;E#OJ8*<7TEibrq8gWubX-qQ)fP)kW_WhAm8jOc z!l!+Fo7VIJ^cmN&oacoJfroR&+-?K8DDyE64h<2fUtGDmM6DD+llfbU=zV9m zMQaq8R+H0kNYWS^8^3!0zP@y_0{FPlknf{dCweE&{i$VRO-rg;WPE(1i;Qw0R)mSY zi%Z_0PvEs+qwtuYP{9$O(AD(|4W;o544jd_ZYq55?TJ&T9=3!$)UkQWzFR~@O-oDb zY}~hc55to~TRAu^FwBE)-mGch@iuxscVy?Hgv7tvFVi|yODe;gOsOl}dZoYJzBf+KPD!i5XX zm|wU7PgZT7yqGrj7?=3+<%?-w#WjOMYsN9F<5NGqc=XHD*$a0UP6p<_{5{j8F!25R zO*$?alI!e|F9WrKBOe}8*O`k=E?(h^$hzq3$KZVAp^os`lFe8Ps;;imz+mg=lWY1r zV&r!1+)4RoeteVNl49{b;ozCe>}mp0)gM0=9#O&Avj^EBTUnY@a&tQX+$e_rQoUuz zb+UxzB{@}9)fjOrI()Mq7UxVnT?->BTl(74e#{iTdA9f^4NAtFK0f4SbM0pg@=eL` z0R=9T#s;TPPsi)|(zuuA%e1$*Q_|7Z1F#-Hefl+S)J8&2fD)iG{62n+JJmlAhzZm2 zbE!)$YL5VQ(Yy8g2L`HL3`NdmH`e>f{}~p;O^FSEqvDEIUh;dfu{VLC+dV=r^{Y`1 zw2pT_7rVCP^y=I`P6aPc=mz`xcGj~AsKRtLFK{B6A%#Sy71&Qt~MVZAH&wI!Nf!XoAu04 zUcXaNa3g5J$fu`PuSiy4ApB}e z6>Y)4M$T!Gy8P{eg5aEE_r0`E9`IwhDC0V{4cxsScieIK{ixjua_a(7#-;GM09RMn zB$RaLQ!9F$2iem2^FC4c*%2%5j7(0&AW2<4J@bj47eo==-Fx;Xmg~r}*apMhnoj2j zvSgBzlV71r85*(^?ME1=t8^2;reW(=LXtZmshbQ)VHfZF^gZI z#4tvEJJY*F1s)vv&Ye5^Wx4nmFa+7eEknD8M>qkE^g&*ooc04*GU5thsI&hkEq-(sTu97tM58g&M!2YI(euZybpdn2x{Qj<;Lp7Z%oGwia9+)4Ox`?oX_CAJEFx7pXf) z1Z4QIUqwZQ?V$Vq)z#mZ;|@4dXlrYGX4FwzyfDh0zGj-HV-v*z#^?*QSI*WLoI%Gk zZFljanA6B{&(+0Ybcbq37jtV_h1WxqSw^+aa|6Gt&zJr9Vb=qc=u1JX<8g7^Vr_Yb zUdCyp`G@DCS7lgmhEAq6D-&;{{jVRV3901UyZZ(uPML?-%i$3n!HzA`#;4+B2DcLvBbWuXw{6?j@1J*` zjRMG)nDy*VU5GLVI&um%A$;tN% zNuJxc74(tz`#Unlpl>=SPl{dcl_Qoo#G|CXEf<<~>E4EhhKgFf=j4=eqJ;X3`5rq8 z&L@5K`yEwO1kw6!4D9TL@p-tlpr$8>`(a%qUG{pssSPf@g;1 zMQJ>=ZcV2Ryw9lUqyqRSCntyLee3pZA55p8@wc^`HOy*6Y3^v8PxY!ETUTHZ8Wcnc z&^iDTkMn2$YaQOdzg0Dy^V6fT5j1>Cl2N()7Jk*xD##YoXvpjzLk~OJJQV$-9#FW| z<^HW6076=O&MFM(A2~wh{G;=RKQ*glNz(^iI^Hjktq6>T!}L`NVcGhnui_gH#C+!; z1%XUfRu-!=a8fsjLh@plAC|PjJY5ZKGMYY3H__+_tq(-Zb!}ySQCm&WCP!3k)+{?I zY1=tVOTImO!X|ZJImXGkF#(|}T3cI3ZsvPACaPnE@gg6qx&~sx@UP4aYE;yQ1eq4U}J7c3fT@PGpm|-`2Nl zbbrnkfdDxS?Dy~KDk>^E%iP2tXJr9j%6)A;rtlKfRLr#Mx$5>x=ZG4co*uCvgFxjpx6_m^Fq{obiN z3ds{0R_6Pw*M)K)WCwg`0Gt?EM-}o&bl}E+3mlV9G_gi|8z3x+TGZ3b;&z-aupZdW z8$eVBo4ltq1s!J))R_`Sp0{Wig5lxeY*Lpv!HUVFHD0>NPz0N?WgC~XA9SC+jz0c$ zsHH|!}oJzX?(212zS5OZAJj)QK#_(5Fz>e+q>Cccg^PYMG>HM#Yez> z5>*zn--fAT`R7rz(x36+y?;4FVV3({o*%|E7(3haYhuIMFe!8bVdFBky?gf(LVoGu z!325nw*G4)-#hX$f!42ldBs>)U!S5BNY@xH7_9*c*PN`hQSayGrH;u_S3t(>mjcQ` z=qUq}lMzse%)!1qGg3ROOGTUhTdMKQuwdA0RD2zVf(0m2cCT*8M3hpH*Ug9sHB-}V zUQ45bLK=nFwY|2GK+jy=-T7o>*lB9fXN*x_f9ZT6VU{A!*VyCD6oXb5e|2A;uHMMg zf6v6L+BUDe9#{T`^H)m0e^CvU%jo{J(6;Jp{HY&Y4Kh>$l~t~^>nfy*jEn@Ax{;N& zyKj_@CiV>@t!XmJ}7X0}k7*#+wfa_>IU1m`x1?^6&wK<1Xeud-hd!hp*Df z1LQ;Bzi*O~lCtYA3TuCm-;e%6AiMVL)9a2MJNCW{==4wbPpJi?m|FckIfR!;!z|b> z`%8W3gc;wV!-o&=+`U`L!C_xt=1Ae?$`uw4NxLctLZYryrrO=^Fv!VQtwLeFdW7&B;47>=#*CcG}w7uHU$kUr1RPD->{+j&6`btOqbfj6Ms`%+H*lP z{a7O5*<>3&)5xkc9kG{CbZUYWA!b&y9^wvh;nK}F_!Lo7_rRjDpPx7jC2=bLrs);Im8L08r}<_-M|Py|f!QZZvj?e0>h%Ifb7PM#L=Ylf2hf zI}0o}8WuLcv2se`eP7A3$)DW3AZ9|Yiw2`RM$T;?hUWeX=E4eDP)N+~DU{S8v~5*u z>z#Rtu7pAR`M7}rP#!%YMFAP4R%(&gEnar$imau>ZwYSY@#{x*=vHw6Kq)})1qjvl zW9@rTwhvmL9$`<-&ZdEhJ0MG|_{tG-vMJ^c(HQiy4XRya(#YZ>Vw%G_fBF5tD(#t( zauuJ0Q?9}=7CZNfVR@>S-e;Evsd?anAEgmASvcM|&^&cc@Y%<)3Ka4O9%Kq$KQr8< z27EBlSJ~tu6PJZq3}9sWF6PED`Nj3X>Cz5gC;=YUKX*@nZpCMW20b-3RYmZ8jKS%6 zIybWS84lKJbfrIj{OBJW3&FxQgJA3a;`TDyF4QkINF2}`7}(f?$ktKt*sr7iPZl5v zul3Am-_{ka6eLQByb#|AY%At6VFdEXo+Ke4^Qya>gH^;>8TT<=JN}J}O!iJ;VPza6 zh+E{ndx28DuIoSvGP}?7M{Bsf&o{%g!VlzPLQ;0CLDv#<_@Yk8NoXhN4k^5UL1FEP z^yMqAQ~hKRRFi)h7Oi!%>68(bo~56s3f|H`IGFNF;&wxy{)>R|3p0e=dhY!B0l;8s z$8R(^vyE8xuyM?x28r(5=PnR%ypjj%opsyeEij}Eg7uF+n$$lFkH78FXIg+EX}G=c zxJGx9)dT~7ntgc{xv(%4!N|^zlS*R@-N6*B%FnKoS$IUL4ec3fpX;}-^m^N%ogy1oKRk+0Mlxp4fa>AxJ{fU zS6ua4hBkrYiZ-2TRW0jM)M4((GYNzRfv%Ac8u6q1IsHM8U+4BnDPqQpS-xWq z@S|eRH-EXQ@M8DfNvG3?tz!0T!=;?Ykq|CW{YD;7lD}fLi-vN5ipeIUZTs)<#n^ai18G_*of+HY2 z)V8;?l)KN>fR_;>$?Qmr5`#Z8Ynslk&tZj(B~GI{^J~wpMyZ&y5$Ag0?X`GupE(?ri!bepKm-O0#}5Ujrmm$n~nIBXKLH+Oas82?5Q z^;6B{X7!hMS=K<0n#mJw!|c0|O9mns48}LBzZPP>gCP!|zEE z-zM6|f8n4WORev%^!84B_|U8_i1Dg_QgASpGT#Yb@BN9PIRFiiu1kuvG&M~xT(}9Z zrWSM?{J{N%#C=)_R>a{&TnM>6H`c+wb7#<`le)Og`8s7Dg##TQKlX!&K+UMbKVBK{ zlISUOW5xm*>3JcJ+d$62!2zSG7Fs*H5E_BX1Idfc7<-nJ%EzVnrdQ_s_;>H7re+a# zEmTS=W+OGPeRg@UIl@|1D7vPlC312$8udL+of)fp+ z?qvw;m4TWKShVPQbR(ZjrHtK2cdTE(zN)&KEMDfSPHCw!hxfwD%1VLz+=aS^20*sJ z)}mKy&FsvI3y~0;fo`!7B5yJsR6Kd0iD)(-Kk6<+za-2Ah%)|YXWOKu+2D^{ZQY~D z%M>gM2i9}8MQx-li`Dp>EbqF*&}UmJh+K`Eo7qr(nCo!73p##&P!YAlS2PhK5n*o5 z2m49PbHScyC(TKRtv@~4(KlLt3wlaTO$`GVSD2Qt=GI8oRK}=uxAA8k3)(kEb_!|C za%g?+Z;2d)!7U9n%E8eQk%?0-tlh(0OFMMoTG=}~DlJu9bDe9`PhND3ODg~uK>VU# zwaKX(#YTM0HhfItVO9a$$^{s-)3x*pNooTzf6?z zjqdvbi-!K;;XpWqW)PVH&LR(U241warG{j0*H;lgsXNOw{>r8d6_{LH%m(di(Evy+zp&v#@3;f#4T_h-shfBqRMlm*rJZ zz_py+1YxYKx%oKJ;E7s?&JU7CseAFnlS`jvgfwVqGW>%Tldl^HdpXIKu)w({L844jA7>z+C_4N8%e8JC`woBK_61K;sy&)bPSz~@^{q$`FoQw99P z==3rB&Tr}}wDAR5l9raXes+y_W@_rt{KV%1t0wM>lPw(kqg!2ZXGZ!e6^x9G7?_!B zAsL&3s|G;oCkjDTgJv`GN|LM<+|YTUwblXaJoJ*N2{>iyXMP0hcQl zDL|VncL;PR`nd(qyB5Bg&FnaBXjlW`sMKY${(D>2*iE%~?@GBk*cX90IeS2{H{4Gw z8r-_XzW0M1Fh&i`zXD98vbELUs1n!o8Cy+2OClhG_4HKl4jcKu8Mf11olHO6ZM1Pv z<&>wFddq?_l!j6x(LRNr2rouQrx~Ox-$GtfK27q8pzT1rD zQf;&5-;}#zb;%Y|Tt#ckE~jF4Uagb3|EM)W9D`Q_kj5t(7an6@aCQL|Pk?cAb92-4 z=U-m1Q$ifU7Yl9x^WF9L_;Ubzt7g4{CG%3329Mdydu50I2D)31-Zh;{Q6WxBJ)=E_ zaP0wB>X1ZfhC|#w$Swog*xzrOmYxnP)EB}Ek>=e09oXkqxcK;K@3ndF>^lA?2L}fq6nIK%YPBd$-zX(_7)Z!{xHMqU=AHRwgi%AZq1NEc zI1}%cEdY26K=h|y-IR;?H&dPhGyd{&6E!u_hBfc+^Y;p54x{uE8pe28AQF7X6jtug zJ^4WXzcl$UVy(A^cMA*q;V^(q3+z7a1L~GhZ0`2o8c>ZZMY!0~GBQtMwz_e{820n` z=LNL~q4mt*H*gB4;-$sA>t03jPosfZEdIprm7Ic-G8_m^0NANyPJ$kpw1IEm0^lUI zxHC)qHGvM`LO(hot1}4Y0lsU@q*G8hqn#O&Za$b4{6Lr!6FJJI{}x3VIf2Peo;(T0 z2!&MnI8JvONhEVLDcw7_1Q>tlYQxzO(2OC0(%pT^7Yk{<2I{x{NHZ_O0hkd<&6#R&4BSN`k zS$6AXqd%n7u9u3g6{W#?6$M@C^5vGOb%^%awWWi{Ai@&BStQ`r&sJwJ8WkP^8qIO_U{HvhUA)aVmp@L8J~~iC_32c%mq|M9L~}tf6W;GTcWB_qdrS@Px`?dn;+qECSHphVCoU0S7=Ag?OV4P{(p?PD%olV zB|kpc65W1>A=h3nX1}GHHl%YR^MTjpe&7|pt-d`*!CM~Y=ZWLTt6XHtKribU+E{i# z0oA?3@ZR1q#=3-G5Y`pQBa!&gA7!}vzpj`y7nZ4#Bhseu`KmDJ(l1kHNXNfvDHIJ3 z2}%7uFtY96ExSwhAio5~+z2zQk5*IeFPOh!e!|B3wG$*-=Xu>90ub`j+^}{iG)4;9& zl|l=B(ys(1zk2(&7IZ5dLE2Vzsup(R3=7Y>9G#H0KZphE+q|dvO6QaF)X*UN zM(x8P{43f2{&kV)LlCl12wTyI*42zT55S@dOh$ zw|1+P@|bJ#>{5|~RALXziH^+xm_n$4F_ z9;jUT#Ze!+)t6c2Hf0Raj^lcW>0uJap#x*|*_kPHiAze#&H8_$iedYUg9qFyWW5E!gFyS^19PN z9*%8t1Ci?635714``}G@W)-bwTw4T`6OD@P$%u+a1e<87I8DFA@fz>t>5T#OJnv7n zDwE&^g}?yC)K4udi<@*ZIOS2r&Ra4;m=3be!muL9knq79*OQK&JOIScCgZeIUyJ#M z{ne{P`~~5(bg40||Fl=-;fHByrZs+41VJ4U-o>}mP(+iE4P#^3!H9jc(u1!gDS^Gn;O9PSfY*yg8J$2=Fm-Xn~a z2V4=fTEY5|A#kZ5m+Mh3P7@ff<@4PQdr<&HJO^Sz09F**CNC|~xJ1f6aW^jM7A61YPR>!G}jAqt6dK+H0MhG^uRX|a1FOJxn-tHanTFnK;h-S)%&*o zw|PvaabXag>zs$rGPE!~guNvX=dLMFN#B3-{~?7rZUwiW+sEt0Ow9|}(8Pm6LX-el z`bPIt<%sc?Sb`!F^rPq!oO!Ais<_Ppe`$}!z0G@TMG>dU@EBr~26NU-klT9V#0mCe zyql;Xsyl6{6(y8EM3Jt;c}#-7gychzs`M_+kK@SGy(XLz;NKV4w?`PDKX!I^6LDA1 zj1Sg@3(+I6Km5@o<78dvh@ynGTZOEbb?DQYFQZ%s5CEx7I_z6)9X^uu@0QE%zloX8 z3CD-5EdmFJu$g2|KgE&wPOe;4e=4`z%6!$(A{jY2LI7(3_SkTaiULuu2`La2I|$?; zDlR@GyF-yZ4Bz!SlQY1~9-JLL=2pnJpz!}r%Hjhk(R`DIcI~PLa3=N)pm{1hUA)e4 zHaS{C0QM&#tiY0dfkPGjcN;cV|I>yuj`K+qxH~Ja;wB=+W)NYlTYMk}Qw_xFr91q| zshRAJ<#YcpTS`uUJe*TTJ(Ti@rKJEUVE|HurpQgBVAY?($Qy+N!>Yx2(Dc%!aJQLn zX1HR|V+Lj8DO&c^F~$N(hq6m-;*h$ejOZhJY0ah_&96okp7%E~|1|nv7wQ6BC%)|< z;V%UfJ$(q8f$NKJqO_In+bbOP+J0^I8k)94%jOpTe>Y&2J?yx;#Ki`Zu^9dE5XleV zN`@LC=(ET>wqTtYhN>7NCBiEw$BB&-uU@_CmyNdqFjA%jX%1@3(kI*rLN5TzWN5!q zaaLMfJZj?Hk^TR)U?RNhnQ@X{>nnpiXf7}z0mTHzEkP!vd33cA!v!{L&N(BsHp&>O>Zl9M}UIc%$h@0X&L*IA)oGBTV(~4yfh`FHDq&hutf=@u6k4Q=m*rwZDcvr!=Z zRF7>$Q-n(h2$p!`+8(MWmUFYMFJg$!h(d;PFt|9wAM2$9U;0AW|OJuakcCRr0_e1=m(w9~M=kEX<;m zd^`y>%`Q+djT>nOBFO`|`3O2uHWylp7B3n%GNMs}#|Hn-oA&KuG&xjM7hdfwQ_xw-Y&J~)jq!$60Lw+dp+3s~{-wXb!<6_b-Z4l?98Iyu#W1rl3O zaQN+mA8Pbdg5%;iAyZKhQXlrcJ=)2e7L0L*b!vL?VxzOQqVX#P(2zAG^j2Fjn>xR; zR}hz{x@_~z2(Sf4IwC5s7<(h?FU6q5jc33gfDlnnBAE-+@$_eAwDz?fw3*r2b%-h4 zii-<{0gFvsQ@D9D5m7JmU={#|oRi(p%U@N$A%kZJj zD7DE+BO@AE3Sq`suWi|V|LI@Y0ocCKE_(`v-7zLpk7L}>$o?Xk+tj~ghM<~(^q z?Cr1%+>!PPSdQ^en}yD-nd;+Pw231e4x0IR*Rxk|-Vi940s=Q0X5F?c4poK#MW5)7 zQttlSo`>6KJ?q{jl)Z-9&WG+pWNLxH5gXf#Q=AGy%gfZo-ir0`plym>Cb=Qb;M%i` zwOj_yhsptml7a|@0KI>oqvKa}BFplpsuew+sk=WqH4t9J*p@g@xIk!?h!+-I?vp1~ z;kBlhFViDK2?Tv3PuE0Un8gCV_8$S%J*R&(~Gwl*C$Qfd+57Tj=TY{^TU$KSXZ z`Z44i5j#RYe*DGma2dQH$eb3Dsdtr@}3zgly2l%{=r6jkv6BjReB=-P#2Sr`14TU!~8<=Lxq7A|bNOe;U2_ggauU zE%X?@#8w87mE#DxL%lRZODm*}V<0oXs;@oH*oKy37qCM(Bs{zxzUlYRYUzcAg)QFK zA;Ud6|90xNU?MW*D=<#5Gc%jV`+YQUIN>v4BT6k2G{4@>=*Eml0z5-c5jHHOgovpM zN5Xx6q7fJF*0AX_(OrA@rXSlTLzrK!x*Ge zb<9%M5j$5fV0_HjU>XsoQW&=w6$m1l&)7|-{+5g(5j7bGzbCkDl-+rV(ZG;-ZpNPA2 zXzx9XuWzD>gxi+wej`biTLj>A!rb%G6v*;;_v#fTe5IXx_o{+FU_xSNT(_m5M(4~q z-k)EB8fN4UK1{uIv!SsuXQJ5U>}iXiX;oE+IORRrKqRs}vXQg+n7Bc6@4X>hgO{u( zo@Ce`qpYfm9q`k-w-{1tWB*zh+XVy$F)s5fuB=-(G}N+7a3je+PmMkHiAzrpA<>Kg*&ve3 zO1QyOfT0(R&M_BQXqnKmHnyr?&fX7wzfSC zJPMC`ZG)Xrc;>ibgeir#(VTIN=f~$#R@hWf_fi65<;-Y7PoVKKrQ7 zWf;kMO-z#)2GI4Gd)(OmGV9pSh2KIQ`Df31SN^E8EmjLrjz~ydUA7}KAU`%_K;B%y zHuEW{Onkd`QK8)uAuN;vyPQ=ZLy9*NAXM9#UQ?Sa%(~HGs1fy&Tz;_R-1*G%^r`8A zv-n+8+=l93iwv8O(VlEoTWd)=Tkcxf7^L5wI6h7t8QrEbj-C7dSMwgO8Yvi`oP6~l zK0b`_4u}vg0+~oa^+VRlIrIEO;~^@s<66cRIv(cMB@y(x^-b?7Geg57R~O`kpXFWa zBFt7MrVk%C$L>Ht6Z;6t3bp()iXxU4(rP;m=HAWJ*9&RJ+%m>v42GU6WbxO@d%@!O zod4_+BW6KS;xZWy3#6ZT&;x!oO~kGU90rbo@xqobz$yUK91)j8$-e8DSoSI>L^0oC zqQcM5xVT&S_Pm?8{`DjS0FZy4WH3KT4<^vr)n$&YIHc=v7K!b8cwPb25pRL|s18r4 zG^C_o0#4rtm&Z)jYOwfW-FJKxA#34(w(_YunSq*Bpz8H5dZu6|{Ag>t85E0>(b4&5 z&kxtnPIpO{G&kV+1cX`!Y=&i41JM?9ck1WQ8Z0lucuq^(PJ|iI)BP~sbwUuClDKEbW-QLIm_zKym+$e{eUBn z(MgP(XXQn5#iEnfv3eT~9+Rh*v>7IqIMgA+0>py}f^OetLVP_HE!^J8Nf|MmmWM}K zokl-wKrSTv>mkaPV~sM4&j5Ykw7>LT`)vY2cpZ2?iTg7{?k~#8fQk4h#ILbkaNvFe z7bj;k&Wcn8K~xM|)zCzNv?OB!YlkB52khT$g|39c?ieX&T4_-8zNE>M7nb@yWN85g zt?}{(YYHS&3Jhp4%#giIhEr+XlJa`p6F-J^?LR!eVoL?_Km(ZKVI>m)xai`K=GkgR z2^Cs?@e4C4LO%Nbg9pB5F(nF2$Fj;TF%qsu1S|95sSaD<-!{ljmbIG3;aY*MRi1ET zUP&Kc?TL{$=Ge$!$ursgT;$heUtV+(Hll>Il~PLulz8M3L{L;Tf5Li{*o+L61z0*h zwHVcbeQkK^0yI-%uM&|$>UA5AXRYT=dlB1}$kEzhJ9L~!&~HS%x8AT6y5QvcMi`1< zLLW=}{DCLodxDKpLLeYTT;lAk^j@RLBjdgnn~l^B$C4iFMiOumQ4_hnd}7np*)}$h%de(vs7zbD84@xUwscBgzqYnEwU<_Zk;rsre>81;(7b}~1iOS8nS{VZ zucge1Y3!=#M~*_yQos|l3Qwz8aWtvz4hjwo-1who6C6U+k3_hVB<^cMMyM1-`$$Qb zIvSxFF|x6B_Dc}Z0fQ9iDF7^z{Cbx(iD1J>Xob|e9HCJ;`P)(HT2E6`UbS>dZvz^A zII(XT)r(v%?cNK5TJuX|c|$P2udam$2OmCksLHWpkg*YNDJ_!iHBx<^zStg2s=`Jl zvkMnC0i-58WiHCdC~Gs{uWM-dc=V_Q_ODhGD)lZAk;@;i;&b0&4kWsj6)Ym)kdCmp zrm=BC*YGE=N-Tqd!Irhon;nz^EP0tLr{nR^fp6$Kjo~xrc!rn8q>!qy#8zwev)y=P zM>?E?NR5Y@kQ@veLzbJMYO)2G#UK`I2@?JeUgRt6YQawY0yAt`czIMCh~1PVx_kd} zd7UWDNw`LMv_N=iTI?aA=zY)*Eq;yvUi&rCx8s>;=(+{g&9Z6zF@6fFhYNY7?#0GA zIjHC72ps0Mf4^Dk>J`SDPMVmhFwhC97{k$lrArRG)wvy6uV71_Qhn-}^*{P#lyI4Zh1)}Ur zKYzD$<~w6O>g4{|IlR%*SNN?ov<(xTlSnxRH4Jt3^)-C^Ca_mTWMIzyioHGQ^xd|& zQ4hao4y0}DVUN8PL=0aTvFzKsV#GH*v(r6($mmfiTaF%eQ&UrMdAW*AJO#<_^5voF zX-2GapQpdOTU%T2y<7l>QcPw*gqGz_n`^K#IetjQzA;nLAj_x0b8}Z?GzAt`yjHd_UzZ-x6b!SLu3YLJiH&6_V8RO4;2(ns5WgD qmSoJ4EONkSs!JKf{QC zIUUv20^hJql;venC&<5PH965J)CH8H>~$@dS3kbG>uU|{ORsf{=*!)_PQgq_*85V0 zzel$2H6|l6D3>NhNZ>G1CruH~v4|=)@BZeRI^@dQG|}xAaOX^Se);yOM1V1k#dY2|nF}Nq4sWzM zQW$3n^aeKTR>FqxD7vd>!WboKQELaQb<6pLm>}1sv=tvbe@qUcrbLMRItL@OpM=_S zqaO(dhuP0sD<1jYeD9_)^1md$*Zpzf8-@Ujk_~wUn&$uS1p)s0yhCoKmx5(8`dcqE zR;`~UHqxn#U7RiiPrX0unWfhnC^*Oi8DnBP`LS+JpuCql8x>o=3 z2j1=5w>daDF*k3L(9_dzMiDW)T|Y`8!RSS7HNLK`WyB_;zh-UCQ^-|h`n1c7f&=yd z;~*o=q^rdE{l_AXizJPWjf>=$(6|xX2aJ-1O=IrTZd-0Ws~znUw#SDqK`qD|&)MY{ zp+CWeBERW!k+J81+Z@-gbhq6)j~h~hIt6Xz`&N;k?tkp=HWz2WY-XfSii;XF`Qa{4 zFC`L78J&YKogojE_+my$x0`kBevxe@hVd_7zC@wC3ir-)zDe1w#p68G-?)mJ+TU7! zlp_;lD6n~uF7p&C_2hot!p3bHv6uUj@NmQ7ri>dVxAtS7?!|H_C(hA9P{y{*f8Gld*38A0=ERPuG?nSsi$Y|4P)f;;5E&)9Zp?aoAR4bALbiyLayv zO-HapKkzkPSiW97f2S$YoKW3|oyy6`PfFXfCs&i5fL7qm2u+pqO5;?6FJ0Y4)f2wz z>guVfDU`ap`r-a^5z2e)y^ZO3g`KUvJ;v6S|M6t)k!=3a!A{fI*g?XAww4wOB_SbU zyV%2QaH1O&Y_+&UN$FYqWcA|9SfXbdmV{Y-v{_jx!&Og6n`Z{M4r2EUg*?T$x$_NH z0|OksA6>&Srd#$8j)|ds=Ih%FS5$H51M?s=6^~dK_&HeK;;5A>Xm6ds=Ly$gQC%d{NMe zg@_>$yJVO!S^SCI&6|;fN3~hCd8(;DVJWI6hzS?^4tz)&g=&5rE=^5&7Z>yE*Ls{E zDl!VxgUgJxCD=#uTON}-d8NeKc${5c8;1luC?gN>I5O`9t{?QnB%pb zpI=X*0*RUA9zW)N_3Bk*WF$&jx{lpsWwNereKJ!n>{(5XXX!#X4f-@{YG&ry__+Sp zBmSTfm$FGCxS4O?zAg4;;eT-ae)`FiCoV27^?iMBTQ-rH%O|_62 z+mUH<51t_QBYF3I#ZXjRTboA0g&&KU@p+bVoXOIcT$gV#Iw;f`Y@+5rZs!%-OiHQ5 z^G^pm!b76aB42PJwtu>4; zN%B1u1yoO_Jn?wdlOjp=RFB>bX&~E7ydQnkPz0@)NI_#7mRXmc%7O6jbGv)ua zoNBz(+8h)lIA_C*HugN4TUwZ}UM0VA3T1{I1^9ra!TUOkp1zstyVquzPY;*~iI4uvh z?rqM0W)OTxftrG~p%Ho%|2RbieOLJLxOYK;b>3&!>dE(Co|J#@S5#E|_f^oLEgv_e zMnn)-*VIJ4dNn?`{`7E`NyKf_Vs^+Eo4Bjx*Oeu0^p zEO>Z$#b#Z!C{^L&kf5Mz4<4|dK69pFcD;Uf{X+Do6p<$bf80iSXUa#+th}Eb+=}py zPk8(Gt#*U}RQkJ1Rim5VGcq!qYnW(1XYHJUE>`{FgIKn4^K~N=6O*+TY9oQWuQ+0q z^6!*lvB-x%qA$OhHxMsqMb+2eKlUZ1rbhCQyu2~TpU=-nkF_jc6TmXy-X^n28z|AY9lFEGz3l=sFdTo?mmbtc#nQ&||HbgoKg#!_l|QS~At<M~4!VEi{9DBYQc(}PV5W1`dNMP#xk04x> z?%lgA%*?00fB(+G#bqjb6*GKGTN{guE3ehyHUNaO=h2O_W2EF$4OGL^)6*n){!sYL z(eN_6jd|6zJrC$6{(t^DyDKsMqYV6K&YYQETKZ$7gAKPgu&+-;S68>p%~udQkvvvS z>)b8)r^0sn^w(?8X_D(sj+5k+ahPS5v7}@A>swo~?Ck7@KR7LG+Q`WTrB~X`VioWK zKHxdOIWyE&Syja%AV7ghC*h1yYD*w(3{MiaX&f0zSQdcW;h^c#QuB?BBoPr2xxm6A z6L<$+u)ps%svidZzriBgo_5VeqPe;G0y{hQ!sq0EV>7?SiLNAJV{>zPO;JCR4A#uV zcFTKE8-|B<*1)~cF^D_A&I4d0t1R~(ZdEf|HB<6F?FSDY6u*Dp27NIXfml4} zx3oTIPIzpneiQ;MxULYWIQ|13V=>D*H-$K#_UAtYnH+ zbWnV($IEY1uhYWAmX(v6iB^}oUSwDU{bb^2{D;nnP?u+cMS1!i&S6J;w0hN{)O?M- zy`&P1MXJ{4`!{a?7aa;Rkp;`LH}xuIk$tUtf{jWhvQfT7w)yibox{TC7U4+`Unq_l znmKAKAvw#3n~bZ|A77SJtt2*Y2o}3Qd$E1;Bv7y5vD4kwZ4XP?Ad(C&wyUsp+S;%u zCMFcr)XWM9F=cjpzaBU)4~7lKkzf?K;+tXfbiEQKHz}dHbLUR}I~fIK0GL0&R$=>G zyN1BvvS)_)4n5v(jF zG~~8-)?@sptpsoi@lTyPCE~GLe2h*Q$lDH?Vivf`^XJB(tEi{52r0*9OZ>I$l=El^2&a@12> z9k@3W(rV>ZQ1zpuw79sqyNj6-J@0Ld;JTHIR>(ITc77UY>F6|dbrEXhQfzN;>*vVg zFuUE>(0B&F&EsI@{olfcB^9n4 zY(kI6acpc>wcD=l39?1NV{G)tr;bcY>bOJHya_w3p}#*GDN1b01=_QdC&$8vhll)* zi#HvA43j=R-ecF$(BOCdHRo)NpRpszrccEtBqS6R8rqa7Xf-uI9{`}_^kX(e7$XAX<7oh|AUjPzM@Ho%0E;9Fwz5iE=4M?0e@sQyzwo?{_soN2FTN77 zJKfOGkpJGwSyPnok+n4`9bM?$+a>L%mC5WCXtlJdZYyV>s9LQJ<*Pd+KzsNVb(`O8XKqM>Q%cHx7n)?<&# z=Q^62NF{IV>MB70gfa{`Dp2UX>BsJ!b(e?bv=WT!IqSI0KSZA%OC0ZYKgDEe7(>UW z5p(1ix-9q>aH(FwK~GDLbc~XJp+QyS=g(>jscd6(HqAk#6<$YTeSLk!_dk&(6E`{h z@?Z*AyJOW`m|nx!O2sEGP6yR;aiT`t8~e$Zm!Iyt9PceOENqB&N36J5yN3Ib9OdQa zHoSe%VY@OaC@U)~FtIWe8Gi&FBwIBpV02XP$JqObWv!qnQTJ_b4^L0Jy4X+m;es}? zlYrC18}r|*M-2}TPa)^$A`1&)A;&3^;TIE;h3&5NsonDW`t-5fcb*y6xRrmOmuWt< z5TJb};YtQ4G%TXGQ3n&0&T;wjbrlr?PA)FHsrpkcuC8{;eVPeda8$kz7SLs6W+MFQ zI?i_%%?odMA8sR=*2J zK*V!j;6-$FiS0B7ej1BcT)gVjfCviFK_17%#$LR5@ibJCx|3sZL$~j&g_?bdBdex; zFLE_=Fx*ruTwDO3f_%ii4n>fk@J;KrD0W`aBVZ7tu&LRK?WJV(W|UU}^+F?e?r`72 z`i`F7MJXx9;NalJp&}-?ou6G!O3f%rYHFGEn`uUMUaE|7N?B#gT+FgXO3KQl=g;E; zp#WNyc~q74o|N1la|*E$CN92>4 zpb=#d!;d$;5_Pcs)=v3jxTI+@+d455F2GdJ=vT$Zj|vpj)GXrSNpSC{E^}~rmzQsm z_lwl~Hu@n-&Lc7BA2h%Yrh#Qfv#Te{-a5VL%GpNqPXkd$lP^-mBt&n_?e{f0of7(X3lbD|=h(!qd z!^p@a9Z4YzFbvF!l#wwU`X;iMe0+TLb6z{RCAXuuLwNc5NeT-K;UX5qC>u*=+aj92 zer1q!-}abs|GR0>-1?dw6>r|)(Q+t-_jrSCO7J16JlZ)Mf1rN%WfJbZzdm+0>M z$Nsw9^I{B*KH^V)Uf|;+K0ZF~@=C=c#{?B>X=&Z-j7N(+`AL7gJ4^hWUi8KCOI`U; zqp^Cp)b#Xpc=AwLp*U~;Xk&WXTwyocQ)1d?R;P|{gv8v4irL}W_M;7hgH&*;0hqNyfvZppTx=kgKJ-?MW|Q2a$<Dy`XTZpG(U`?8eZym?a(B^5M)=@I~FB4T3i#ElA5ZLDUXoJZFxd6T?L zhK*uuZ0vG!v{T}-XJ6_ieFx|4=>oYcqv8?;y%zUuY$j=BhMKib?Iwksto* z;KRYkzxwKsQ7sv%sF!OPAGM5?niFTMr-nlrmJ&2HFkk>EI2KdHEHOFQJiD85&8Mt`qeH`%Z@nskH;%~K&V0Fj)b;8 zTG$#wOUuh>_w7eG{+L%{P9Y&L0xiOV!-q#knAzB{l7y@=s;Y!=t{BGP4JO={3#j-= zy(pkIMVjZnd2{XBH55`rm%d!CsH$>Ee$kND;qL@8E54VPmx7Yg=_Y>IQjV`*zdoCz zmfQ$!`KZ(Md_a0jkzZtv7<8ThB8CvOY3r2q$sV!Tlb@YVN?88c_KuH6%Pij%6tvWO z9#p9LFMdRj``Vf<)O?8z-2@dZK-N~gILuw3k~IGM))5sSA9OoUJ7MeMP;8G@c=-Ci z^%S^((cH2!3b+8brGSSK8NFl}1!cf#K~E+a*3N2YdS{`iDR0=V0^{ zGkfp#XR9LRmYch*&fyHQ+*mO&F*ta5m@9?e>CVmc|DkCg!uCFs<$E$m{CTyaOT0nW=M3 zhR;73lkN?_|4yNlv$M0Dtn69nV7GL1+ML8>@qyPEJv|bPxl2T@YOZoBR(R1(y z09?pnFwgEBuUHY+P_oUxT`Ly_n&DklC9a{VS*hl4!F>5LE;*YDpmOEB?kwm78{2S- zXF-f_KR6YwI^dV$efiy2*vLZGJELlATj z$rDtZ#r7oUG*WWOlDBU`Q#lQq3`h&DPGYam17%3{_=T%oWLPjTxWUr`M0Lo{%#TE- zm7hSnP(K(n)ajkcr)|o7HO)>+a;i4|!Il5z4$uVV&>(Yp*VRd55z>B{rx7p#59Ag+ zdf+uTb#;T_(NFnP8%~L_X6C%lrFURLTawVw;OOY+yw1sSNXBz*hRtgWRPk^l_2iKQ zL6!zCeBHf!m%$7gffih(Xfx6wJJ3XLG1xJ36ka2Em^ew4WL5B{nZ{$W{7QTpPNlIs9_-aY<`SEAeLBxa_{ z8)y2hJtkiK2`{jiG0ISA1D#DVTsnYTZB>%RD1q8#CpjvVMs9^# z8#e&@OKfa2-{Oo0KRxg)TZVK1+hm=`ae{rY7O`j{?ziOpm! zq0hCnQVGJNg&hltay``~1=8pz+duB8yb>Wb@>oBYt(M#lZ`XkVpGq*XHzh0+>Kcf^ zeRCcLHnI=$*QvO5y@{oEQ6N*St*`f5dlBb=!R5l(y1!wS=XtP=2&-f7tbH>xO{-=V zFOfkjXzJ~~#qLVG-@Ur-BJfgG;IxJV307mV;$2EibtFVGqjsp`l~Ox`^jd3Ga|ikXWGAHEZ&;HUW5kc_Maz+D*Q-*;yQcI~9yFk2&%qB~habcN zwgvPGkR(1epCSBdv;%BHZoM)b06;!PMSNV=J%ooiXogg;{mVp5QqN%5tT z$1VO{`CHzsj8~Fd4d(mY~t3NvzEZmnrc4{QW_=!h%)z4+|q4 zx}3n1*p9viL=BX_kDZ;ew{DR_WlsjdBwxRh4E(uzQ17?iS`=hw2S^!^n!E-zfZww3 zlCGYf$xxxe=JiBV$I4=#r9VjkGg&am$Z#MmUdG0n0Jj9Vf2MYCo=!wm^m~>8=qG43 zCnqOF5_a2Kx*FrMG|w9upBYd3;@jBEIBaZnV$eVajDi-3)PbSF3hIOd;UHGsUZ2Ho7a zBbEob{T!XO1Z6;*li~+GA%veO4M3c^~TRyHE^Tw^u&+AsZpohLkQWZ;?V3w5~ z+5Z-u=fx;%szBR8@C~3slg-86;c_eL)2B}pWN@5f)Q|bS#|i-yH%v}4A=ebh$jBWj3tVorJH}9uPsp1NEaB+)+aN8V!IyKUP*MK^*UU z4xW%Ptd#D%N9O^E1CN*RHN+#QMfWmLQ&al|2C|5Xo`=^6ew6N8xov0|-r1>g-^{G> z(-n>*|)#9WC z-2@si$*-W98I$L9!bGW_+vi~mLVuK1{#$|_`_(oLIwkCCnbL)&{fIEpmft`Wipu8q z{+yPBdK|{uLVKRE0kDPh1~|C6o8gb`e$6S4*Lu=51rRpDb4J7~P~jy=n{FCKz+Od= z5>`^G6u{smp@N;LPxpt*EXk2t0Z`=o6-kqM0Oa9zv#rK0A(UD#Y0l$A^&be5yKD3P z^TGx=qB-oJ2ex=-x_B5-mo8nphm;Kw+Y4@+i|cczh;w@K%Przo7P#U%Zr!};eR$}B zD0jQF5zGT7!Jnw%;;?U`T4B%wTZz3fIFh}AX#$c*p@sz4e*#zu)5Cd_bnO{Vs zNr92hiLsi@eNK@KU7u~6%`2*H1WE)ut&G$lrirv%k1$yFG<;R+hbJb|ei<0Jo~v~l z^>eb082)OiA?pp{!Z;M*YyGKQ;5;By3zWpBk&()hFykEcGbsDI!?z>!umk7pT}|l- zngrhQ#C3p(#T=tVcT%9mISuwPD41z*Mr<7&{d|4TAhq(x4+3*@bEv5ejg8M@a=Bz! zqZCxkHsw=6D@oZGWG=AV-?a1#2rz*jP9yCl0rkK+Etj4l4@~lvvG;b%gFNFOobbUH z4FE$LzO*=RI5`P|C< z_k&&mbQ`y#q5^p>sz$p0>0J2hB z+I3%G+Vb$e^`Aczf!4ip%rZG190$7@&}M9HO{2C$%hYZ-+qdNOIPDUp%9wDb_f z!U!;?(8^B_*RcWVh5_c+xMgw?xLYg*-nrw$4fR)&?o_~KTL8TKFQm8=@zrdS?QgH( z0~dzVjjg7p)|FBeA$ z>2N>4v!Fps>gE~WM662&a^dvBaAmZf`%HEH8^xP4GaiT6;DYA83Jo>OJC6~YVAY^sf*{2&mFfJy4r^2?x8paReXp^HpRRJsv- z-ng9KmqV)uHb{t}=dYUG`xwa#QXAp`G%7OcmzI^udU!}&xpL*EhDIacRsVOH zHdSB^LqQRoI0=rF|F*C&+a)|H`|u&>vuDo`m54O&FUCs=Od_dE0>O}w`)UDJ7y+=Y zdS}Aj?CF4CY3WrcK^MSG`b~w|eC1R|*?>!c`H85!$O=F%z&4o-ec|cR7Bym<5Yh<& zmJJZh}1tc&RE@I1w*c>+I559Wil|xy?S1D;(JL4{c*%q20e7hpbi@l}!4?|=SIwD8aLN)NTXsa2 zLx~OxQ!QQiZjU3u%6bkt{2=i3&k0svE6$dJnih zFXMdyMOA$Ga8Xh+d0FdutVXV%My^0bl?u6u3R2FdrtanImYfEaoeGhW?|rdSfwP4a zQV^qXp)Ti_oBr`{jW*;~#ulqzSvNS!K>QK?X&S^0V2&mibZc~olB#O6U%y5iuU&8o z!1&MG*mH2WXKLz;@HmK+03A!uQ`UncmTeaE5|yWImyCrb0~KahU60lo`|MOqpT?5- z>!@3Xyw#>{Ll&Hxs^0lJeT?U=s5pC6*Fz!Q!K_7>MRLTN!nUMvj z7xGZXAaZNHjLf-`*x^4^a5pf{&@EVK!Z_;R=?EaGSde~61e>6@5u86#5)u>uE#zE) z&*yVBq{IK})vLEaym-(N`~3pfFM)qz6Tq>idV#fcYinqnhOX$b)_|?`lB9sN;25Na2+WFQe(VT17`zZ{_s;1CcX)n~pl0j{K3fo7AI90C*& z+(BvxV(B}@pLeAd7Z(R-1kv=~2-glZ#lAF{SQ1 zk5To&tqSs92F3o>xn94%#7}_SYoIYrVmpLW0TzN7*ob(5u$>T_s`XxuJ)lHnHGg9U zZ0-I_Jn=%-v=6PUKxB~uE-Lc7V|hO|l>cb@#~*+2gBFQgW*Ki}S#X2eHzeEza`ocZ zs`rnhFY1A^476Xy%#00OY=lPkSl}|NbHYWZrZRyh69QQQ6(~ueG05 zt8XJIEyDmAGmDAQ!T|)5x$H=Q1Z`gD!lJ^~M0^CW+Na>4G!s1LggXN1xw*Bqv9&cI zt5d;hcXzjbY>e*Hr%zBGkp%DDQ#koktE&+srRJKqFr!^lRCSCJE+oKnVGT?!bM;aE zySdPu1!C@+mY+|6;BQHHp%qPxn0D@RL;8*Gs`D4-t8X#%1To1uG zxD430dTkJvWCAkTQ)E=9T|N+d*vXtSHNDI^^!BTH%kppu3j)uPTp%dU-TKn^e@NHm z;6BK>j8&6``#O@=NQ!j0FG~Egs1-K*cTt3gP zgKS^ou%MQ2R9CAiGxvc}PdM0xIHworkd?I0eUJnVkiHM5} zxAK=AZZbshCJFJu-G(LQ^eW)?3Jzx5>s8fWg6;-P5>h4ixL7k-{?2;{4dvR({nlX7 z%5X&9m1xJ8nGqvf9Y`}{sacl^7ej{3;NT!U*lQqY!UKcFBLBbe#`PdWgE}t;2;#{B z)2C$h6U?V#8x4=A7k4Hw_~7UR(#~D57#!^Eu0!0{55o*MSE(ky3W<&QBxKiMW3ybj z@}jVoIp4&IQ+|vz8j`Arq4X#Vqf7QL)iuQMa*Ps0_J&7C!9-(r14%d_E$spVO!t0$ zgC43vzhI?uyuZx%GA_;(Gy~)p0H6Z}f%k&2aNM%iW`LY>kZ7gz3M~XPAR8qkJjdeJ zu&4DIh%(ys*L|S6#W8rsQBb>PW@d_=mbHPC754YybE*VYRzP_K1!4{4u8U$~iBN$O zrX1Io0okrY*S+^^?$e{J>zn^^+?vF|R_<&V$kSPeFbfOtG>7?4C=w=eBFXKdkQxV2 zAy+r2dQCwA2ZUk>uQWJu=4X*GaE;S&4#8jR-6WhQFptAO0mBNZ^kR9l)rU^?3xRyM*qO~GyjCUyhBU{VT-X6+3>nSYmMe@YNIRk{+cnv+ma z_yHgTT8xxOhgbRNO2_YHU=gk`& zNFKXP`%@1;Or(jl?9Wbr_l^qi@~jwTGDP?x(J-~K2ZoPA(h&rtkT-W;od_E;bPofo z&Jwe#DLOJTBA{m?x9cGVn=LXSAs7yIVZWs?Ee~_+P+A&GOqKR|x z@!7m;?cHM2rzpDom-YE*eh4yJ)!fC0F7sX6-h)pyv^Xe01$gxK^Cd`()YjH6j+W!n0Ip&Z(*LF=K6~y+>TB+2r#yt?>rB|=jUN;!BMt1wmIcMUISq^#;Zm~A-TF^ambd!5C@Yi#b2RIW_d(PFhzxj`Br`F$dY3oO z^^zi`9cp5a)5dGuKebI64vLK0+LUF-G7Jk_^UbedVj|D1KZh71Yi~H13=w^Rmh5e*^Az&0Y&!v@Af_#y7Vz*MJBk2oZ|9Le=*NoPl`czXbjY%fT;$ ztbV0h&J9>u83#y&Ao9`jU;)B~pyXwcyWKg2bbu=r%n`s}VK^ut{$!V|Rtj-A^d;$| zO>!ZdiRaIs6Z~0un8lj*wZc`{WNzQ41UfWg5fTrx3x8#SeV4ToZy-yOfA>8RvH?OU zxd5#oF_J}!Wld2erw2gryCLmd0#qxAmP3TW7hY(W{Fxl!a-$%uH2^?FF#x(!cJ2&X zF<<=~VDNgdWx~vYaNqvNg|fu_uE!#pMn(}(B$3LsyZZD5UszbUQ1hZ6Wxf%wzOJr< zKdvR^f0A#sw{>*-wTPzuKq3Pj6qLGfMoFce8fLbzEkU+8J|hI14_gGp7ysLS>I;G( z2Q?=rC;x+X3J6OX0|Qkb|LIGj*|bC(4Y~l-g22b+s(NAn$+=;p3ee2^PmT{=3C>?R z3ueY}vGEyhZf@A1B!eX%TIYZ#*g80X+>2Mg3a42{jNs{VAO*0fl=ZUF5{f^8@Pl`IsPxY%y`^Uvm8+xp_IJa3@djV1h z%E^;rx+pFEZnjtI#tInAfp<#Hz)PG0&Gyg0elUqzv0LRDvLZR^Ovs8LV=F+hKw6%; zUxN#%`w=I^R3XHGXpG2s4xqQv*3p?E$gV@(2MS-(dloJ8cW~kD#8`HtU-36+B8a^N z^#;*#KG4U*~&JY2Aq@b)U1DOKQ2*0<}VHo>d zjT`w1V1KoDbiiMWnOj;01783eD(&@aS(_E5ERE5vm_>pneq>9;@)?DLPdg0)82ITJ zf`HHXe?`;4c$)#2y~oJ~g9!YfWdm6NVfuTPDlMA%gFQ?gAifJAcQ`sOTLU_ku(f;} zm5eWv;M~;H!-Lvo5XgQO#s5#ib+e4XBhSExV~J!nAcV0td2&o$U5o!s*G`bkXhuFoP9V}FgLLH5EM`AJ+cJ{9QyU=agP zv2AZ^Is=-lzrv+w;1}rRgGf)C$Q}khIbtbwOC1m(ZX-xaXHTCRvLrRR;%FGJaET0@ z$R+^ZaMhME|J2--5E&M>*VntVgs|U2$888(5Hd#O z^zR19T_T%?(Tj$hRIddtf^6hu(S;gkQiHXyUyHV6;QtAdt%XB zs3wmnX|m{I@k1r|t6CL|=lNN?Z11T7Dm4`R{qoBbgq(Hg-N0h`4H67l*uH-hdM zXYqfZsHrM!BI^TS?Ih#qC;*QVoUx(-(Xz_KJ!C8bL|9Ov*T8!)S)XYsF7DY|X~tph zPTN3+gH!OxO^W|Y27p9Y5T4W6yon$;;H5ND9wGq*=bwRAY`QkpfV|;IMx$SRGTe0r zOc=sgNw-8S9(g}V4T!@BLE%~O6fnT0!(NayqLZl*^S?m5pO&1wb#|Th1%J3jDEFkYlIj=KMhG1P3=Mv@8jH(q@?OjsU0wGr|&Ey1$i=#IzSL$Pi-- z$%w!aLqsLGZM$SD=CfkFDqt|dw)SikFb0dy{nN_-&YRYQJd8piayr;TQE_pB5Ew<+ z_r6-?9Z9#%Yrs1H1kQRA;c_Y}p^$`tXvx&_a_Fs0IcA8h!YTSD#)J9D0uLKT%J%&% z&w{?@i%8O_HE5Jblpd8B9#keE*7pjzQ5esG_)QCdLfVsfCWym;nSqS&LM#jdE}aw6 z53@*FYcw$XB{-nyqxQT5+C;-X*_bvZDJgbYIvsK%}dgP<@* zxF{tmcL9m8v8%}f@^fAp^+#M|lwQrLky#iF4ueP{GM0z}-Sr$hJ3Go79D`;Py92VJ z638@j-+6l$2po;z!!P8MaSE5TEa=bx^#GQ5>`n)Q%q9%rD+COmgyk|EX7%)b(rEC{ z!En-uW5!+eh;2uq2p%Dy3!t2zpI;1la%46kPrKmZ;oR^bDLLxnr%yH$`jf#(iQ^M5$Ojvq&Kp+C?;lL#z zJ>m_JKDm71i#9JWFHFe>!kD3_r)TtPZ&0w^t-Jx45Agx<8mSa~wY#T5e=dV@0G7U= zxvv0_K*k{*f|^f$j$a2T0fyum06j1P*;f|s6A}^821!ql3;f`f$VGvi?K$rn5uPG& zm}E=?HVFy^)1P#8MlkmP0A^3e1(J6M=^Vc_x!@?J-F9wC;F|4++XKh%I`^pT*#GjNXn%dPpDgVne;J>7`Qk<2?rKj@kJ*rpKu%U~bQ=>qd1P7%(x9&gyqE*h(%*nu z>M?5yeg8`;p)SOEVWI&{CQ$i`9lvX+BwX?TX)PN>YUN<80RSJOU1=a2NER9e?i?R2 z-oA-Kacr#B@2$@=J3BN%Y>kA9N-k9(_&Q>TLsh_oK)-(7#F>K~_sq03YE*qwCc$i5 zYG6`Q{C=qq2McOd+c`9PU~U_Zp$#$c-4ICNKjUyPoh^I*1UT+6qfTIh-65(A0AmTD5%xj zqgMJkayZ5b|LUvo5C~rRv-;8C0vAB}TwCir!!(&V2A&p7#NWtDqI0975^`62usVT4 zxw{{*zuOwz(w;l(ayu$y7J zj~~h#!SBgJ#I1$_sP-wLSr)%V`J|!vO)p)?R|z7dOJ?uC}{`p<$=#v9V+G*}SPgNnsO zLP>cJP9%iKPa#uE&!0C#yI%*Z0<`|b5Hd`sZW2B@e2xOBuNklpei)SDWcgxUx3nhA zDi>6elq`ADdlA`0z~K2UdeY2He~gyn!-;!*=yq~^1ZX?}5+M?jlIzeyPrDSy?EO!> zItbQ<>50eNAFV-nRH>ed2f`M$FIG(Bs2;AB6|01#a*0K+ zFPHuBkM0O5Mk=boN76Qz=S%lpVJc;Fw3`y|91^O8YRql$fdb|^Y;A4pVP@?2cm<%# zT__sJd=f~RQ=tE2KyACz0fS-^F3)3`k`n3g9zXUimv{O(ejY%Em9q>dX!B1dT(ObH zw>I5$3I$^zZA$}r2s1;b;~*&`4|y#xl#xgZjDgJ>o$ONsf{5U``%WMjl1U;S+au=c zuR3Pe6OZl>YjnPPMUQkBOG`^R!Iu;HxAPK@-`A|`HPFo3eNAQZ^nffhjBKcHZp4~M z<)wBfi&H~f-z@_Yx@+x~+sB7;VYL3z$Gd03!^0~8_kq@V&)hr^!4DTm(jXbKI8w?E zZu@xlHV*TArh#~!7|b?va}yvkz}E5*U6D5OX$zot04G`h;=Tam0eDc>!;Dt{@8cRC zR=bWu?TkuF!iV+;X|4M1?$uyj_;`(nhd1^XD+%{tE6kk0AYHNDtlTCnLrXmT1_(D> zx2gNna1|bGWJa{Lwq6tw*$o!b3mJ(tS3ar+X(WH!fiVIbY?{bHN1j1hFv`j@#N7n&Lw6tKhKCd~vSl6}IE#kfXH{X+GPD8GE)9UT*1$idw(3Y-k zSlBC_{Rgg)@hm;6mh9dEfD&c`%Q`Uc;z!cw-5((UB>hc({-a;IAhW}y8YvS~1f--< zC}fTRd>j<AD^szHR5o8 zN`t+IZ8X-)#`}?@Dk~O0Bv}0Q%ZvwAFS1|ifAW7SETnX45Wty+lKD5a7hJ-HmtZ{kFr zMfv$_A3p|3$jS0iDZbc+4+B|VTGAysTUb=I07Jg6M-hw%n>D=_a{&QZQet8zW?rT9 zYis;Pg@tS_xxU2HT56c(7V*@~HmipZ$*y$Ij!aslO2+h0OnjDmbi`!tLqndI)L zL+-zVcZfrl&?nN}b$?8l(?vgfc6$5ZsI$|<-o6paDcam;NK%lks;cT$YHIh3OSufe z2E0S1*?Nk;S42cULgbL10IN5V z^qCv(A3GzhvY*bakcd}(<*nPLBf#2rtN8^=wP|QeBizCt_uW^6K23%xZ(e~6)o|J> z$&K}qjk^X0M~}uI32kg{mbG6lU#V$5yR&iXTI1!KGzJrvkc;T6){h^*_;f#`u#nfa zIyX0%0P9RbmsyNvEj-#_rb+&ej@?}m*kQNK&2vWA?>ko|$0xuCDm*YZ?{9AQli&@x zsQ5xQD=UjLN5#-W1$ANiC2W{=R~MJG+FB`E1_st-HTDQTUS16m`fLybZ`(h(f4^sM zPxQvcDB>XlL&NOKN>M!{BlgVRcyGtltAc_zwDV%X$(}2TjimxAo{^Qsy7i1zRDcZ? zX-iH{UiSXIJVewSZEdg3&dxrR-?xj6i&HEbif)>YY@;W_eqd(y!mua@9LC%BfVo&bENV_;$eMyaGyp^l_RiiVU;O(C$>y^slWj(WPf7plG%nux80)UqrT zR5{G-cNd{hCHx60W##2r&d-^;zB|6?|L!5g7Lp-moT1B2B_<)EB`!A(F8Sj8{QSsp z|7Av(q!$a=FvN&LUAx*758u4a($J^id)GD7)9@{EePMe$wsE>RBj$`xecWUEt=jh~ zYm=*6+SAU_o=yc AfB*mh diff --git a/docs/images/snippets/2020-09-18.html/15225da473048d8c7b5b473b89de0b66.ascii b/docs/images/snippets/2020-09-18.html/15225da473048d8c7b5b473b89de0b66.ascii index c67e57c7..4bd5e5a6 100644 --- a/docs/images/snippets/2020-09-18.html/15225da473048d8c7b5b473b89de0b66.ascii +++ b/docs/images/snippets/2020-09-18.html/15225da473048d8c7b5b473b89de0b66.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n=3 n-k k - B(t) = ❯ \ P \binomnk(1-t) t - cubic ‾‾ k=0 - k - - 3 2 2 3 - = P (1-t) + 3 P (1-t) t + 3P (1-t)t + P t - - 0 1 2 3 + __ n=3 n-k k +B(t) = ❯ \ P \binomnk(1-t) t + cubic ‾‾ k=0 + k + + 3 2 2 3 + = P (1-t) + 3 P (1-t) t + 3P (1-t)t + P t + + 0 1 2 3 diff --git a/docs/images/snippets/abc/131454dcbac04e567f322979f4af80c6.ascii b/docs/images/snippets/abc/131454dcbac04e567f322979f4af80c6.ascii index f8eb0655..41504a40 100644 --- a/docs/images/snippets/abc/131454dcbac04e567f322979f4af80c6.ascii +++ b/docs/images/snippets/abc/131454dcbac04e567f322979f4af80c6.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - distance(B,C) - ratio(t) = ────────────── = Constant - distance(A,B) + distance(B,C) +ratio(t) = ────────────── = Constant + distance(A,B) diff --git a/docs/images/snippets/abc/3166afa345aec1abda432c39b68d39a0.ascii b/docs/images/snippets/abc/3166afa345aec1abda432c39b68d39a0.ascii index 604d47b4..0cdedb32 100644 --- a/docs/images/snippets/abc/3166afa345aec1abda432c39b68d39a0.ascii +++ b/docs/images/snippets/abc/3166afa345aec1abda432c39b68d39a0.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ e - t · A - │ 1 - ╭ e = (1-t) · v + t · A │ v = ─────────── - ╡ 1 1 ==> ╡ 1 1-t - │ e = (1-t) · A + t · v │ e - (1-t) · A - ╰ 2 2 │ 2 - │ v = ─────────────── - ╰ 2 t + ╭ e - t · A + │ 1 +╭ e = (1-t) · v + t · A │ v = ─────────── +╡ 1 1 ==> ╡ 1 1-t +│ e = (1-t) · A + t · v │ e - (1-t) · A +╰ 2 2 │ 2 + │ v = ─────────────── + ╰ 2 t diff --git a/docs/images/snippets/abc/51a9d0588be822a5c80ea38f7d348641.ascii b/docs/images/snippets/abc/51a9d0588be822a5c80ea38f7d348641.ascii index b9fa5fa4..8c32fb0d 100644 --- a/docs/images/snippets/abc/51a9d0588be822a5c80ea38f7d348641.ascii +++ b/docs/images/snippets/abc/51a9d0588be822a5c80ea38f7d348641.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C - B B - C - A = B - ───────── = B + ───────── - ratio(t) ratio(t) + C - B B - C +A = B - ───────── = B + ───────── + ratio(t) ratio(t) diff --git a/docs/images/snippets/abc/5924e162b50272c40c842fad14b8fa48.ascii b/docs/images/snippets/abc/5924e162b50272c40c842fad14b8fa48.ascii index 8c413b3c..415597fe 100644 --- a/docs/images/snippets/abc/5924e162b50272c40c842fad14b8fa48.ascii +++ b/docs/images/snippets/abc/5924e162b50272c40c842fad14b8fa48.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 2 - t + (1-t) - 1 - ratio(t) = |───────────────| - quadratic 2 2 - t + (1-t) + 2 2 + t + (1-t) - 1 +ratio(t) = |───────────────| + quadratic 2 2 + t + (1-t) diff --git a/docs/images/snippets/abc/8bd3e6fed5bf8d871d30221ae400fd93.ascii b/docs/images/snippets/abc/8bd3e6fed5bf8d871d30221ae400fd93.ascii index 84d56a34..d1525456 100644 --- a/docs/images/snippets/abc/8bd3e6fed5bf8d871d30221ae400fd93.ascii +++ b/docs/images/snippets/abc/8bd3e6fed5bf8d871d30221ae400fd93.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ v - (1-t) · start - │ 1 - ╭ v = (1-t) · start + t · C │ C = ──────────────────── - ╡ 1 1 ==> ╡ 1 t - │ v = (1-t) · C + t · end │ v - t · end - ╰ 2 2 │ 2 - │ C = ────────────── - ╰ 2 1-t + ╭ v - (1-t) · start + │ 1 +╭ v = (1-t) · start + t · C │ C = ──────────────────── +╡ 1 1 ==> ╡ 1 t +│ v = (1-t) · C + t · end │ v - t · end +╰ 2 2 │ 2 + │ C = ────────────── + ╰ 2 1-t diff --git a/docs/images/snippets/abc/8c6662f605722fb2ff6cd7f65243a126.ascii b/docs/images/snippets/abc/8c6662f605722fb2ff6cd7f65243a126.ascii index a4d5fb8d..a48497ca 100644 --- a/docs/images/snippets/abc/8c6662f605722fb2ff6cd7f65243a126.ascii +++ b/docs/images/snippets/abc/8c6662f605722fb2ff6cd7f65243a126.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C = u(t) · P + (1-u(t)) · P - start end +C = u(t) · P + (1-u(t)) · P + start end diff --git a/docs/images/snippets/abc/8cd992c1ceaae2e67695285beef23a24.ascii b/docs/images/snippets/abc/8cd992c1ceaae2e67695285beef23a24.ascii index 32c7f760..7c103667 100644 --- a/docs/images/snippets/abc/8cd992c1ceaae2e67695285beef23a24.ascii +++ b/docs/images/snippets/abc/8cd992c1ceaae2e67695285beef23a24.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 3 - t + (1-t) - 1 - ratio(t) = |───────────────| - cubic 3 3 - t + (1-t) + 3 3 + t + (1-t) - 1 +ratio(t) = |───────────────| + cubic 3 3 + t + (1-t) diff --git a/docs/images/snippets/abc/8e7cfee39c98f2ddf9b635a914066cf6.ascii b/docs/images/snippets/abc/8e7cfee39c98f2ddf9b635a914066cf6.ascii index afc09ab9..4fffc33a 100644 --- a/docs/images/snippets/abc/8e7cfee39c98f2ddf9b635a914066cf6.ascii +++ b/docs/images/snippets/abc/8e7cfee39c98f2ddf9b635a914066cf6.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 - (1-t) - u(t) = ─────────── - quadratic 2 2 - t + (1-t) + 2 + (1-t) +u(t) = ─────────── + quadratic 2 2 + t + (1-t) diff --git a/docs/images/snippets/abc/a0b99054cc82ca1fb147f077e175ef10.ascii b/docs/images/snippets/abc/a0b99054cc82ca1fb147f077e175ef10.ascii index 1858a2ac..87bb17d3 100644 --- a/docs/images/snippets/abc/a0b99054cc82ca1fb147f077e175ef10.ascii +++ b/docs/images/snippets/abc/a0b99054cc82ca1fb147f077e175ef10.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 - (1-t) - u(t) = ─────────── - cubic 3 3 - t + (1-t) + 3 + (1-t) +u(t) = ─────────── + cubic 3 3 + t + (1-t) diff --git a/docs/images/snippets/aligning/00480d8ea1d0b86eb66939bced85e14b.ascii b/docs/images/snippets/aligning/00480d8ea1d0b86eb66939bced85e14b.ascii deleted file mode 100644 index 256a97aa..00000000 --- a/docs/images/snippets/aligning/00480d8ea1d0b86eb66939bced85e14b.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 3 2 2 3 - ╡ x = \colorblue120 · (1-t) \colorblue + 35 · 3 · (1-t) · t \colorblue + 220 · 3 · (1-t) · t \colorblue + 220 · t - │ 3 2 2 3 - ╰ y = \colorblue160 · (1-t) \colorblue + 200 · 3 · (1-t) · t \colorblue + 260 · 3 · (1-t) · t \colorblue + 40 · t diff --git a/docs/images/snippets/aligning/016f37843b2af2ae34dc8b2975505f3d.ascii b/docs/images/snippets/aligning/016f37843b2af2ae34dc8b2975505f3d.ascii new file mode 100644 index 00000000..df5cd28c --- /dev/null +++ b/docs/images/snippets/aligning/016f37843b2af2ae34dc8b2975505f3d.ascii @@ -0,0 +1,5 @@ + +╭ 2 2 3 +╡ x = \colorblue - 85 · 3 · (1-t) · t \colorblue - 12 · 3 · (1-t) · t \colorblue + 156 · t +│ 2 2 +╰ y = \colorblue - 40 · 3 · (1-t) · t \colorblue + 140 · 3 · (1-t) · t diff --git a/docs/images/snippets/aligning/1564fdc8d17a064fea88b9d508878e62.ascii b/docs/images/snippets/aligning/1564fdc8d17a064fea88b9d508878e62.ascii new file mode 100644 index 00000000..795ef8fd --- /dev/null +++ b/docs/images/snippets/aligning/1564fdc8d17a064fea88b9d508878e62.ascii @@ -0,0 +1,5 @@ + +╭ 3 2 2 3 +╡ x = \colorred 0 · (1-t) \colorblue - 85 · 3 · (1-t) · t \colorblue + 100 · 3 · (1-t) · t \colorblue + 100 · t +│ 3 2 2 3 +╰ y = \colorred 0 · (1-t) \colorblue + 40 · 3 · (1-t) · t \colorblue + 100 · 3 · (1-t) · t \colorblue - 120 · t diff --git a/docs/images/snippets/aligning/64fa084c1df7300f06af13f53681463c.ascii b/docs/images/snippets/aligning/64fa084c1df7300f06af13f53681463c.ascii new file mode 100644 index 00000000..b7fa1ef1 --- /dev/null +++ b/docs/images/snippets/aligning/64fa084c1df7300f06af13f53681463c.ascii @@ -0,0 +1,5 @@ + +╭ 3 2 2 3 +╡ x = \colorred 0 · (1-t) \colorblue - 85 · 3 · (1-t) · t \colorblue - 12 · 3 · (1-t) · t \colorblue + 156 · t +│ 3 2 2 3 +╰ y = \colorred 0 · (1-t) \colorblue - 40 · 3 · (1-t) · t \colorblue + 140 · 3 · (1-t) · t \colorred + 0 · t diff --git a/docs/images/snippets/aligning/6acf1a1e496f47c11e079a1d13f0a368.ascii b/docs/images/snippets/aligning/6acf1a1e496f47c11e079a1d13f0a368.ascii deleted file mode 100644 index 22433e00..00000000 --- a/docs/images/snippets/aligning/6acf1a1e496f47c11e079a1d13f0a368.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 3 2 2 3 - ╡ x = \colorblue0 · (1-t) \colorblue - 85 · 3 · (1-t) · t \colorblue + 100 · 3 · (1-t) · t \colorblue + 100 · t - │ 3 2 2 3 - ╰ y = \colorblue0 · (1-t) \colorblue + 40 · 3 · (1-t) · t \colorblue + 100 · 3 · (1-t) · t \colorblue - 120 · t diff --git a/docs/images/snippets/aligning/a75137c250be63877a30f4bda8d801f8.ascii b/docs/images/snippets/aligning/a75137c250be63877a30f4bda8d801f8.ascii deleted file mode 100644 index e5d247ae..00000000 --- a/docs/images/snippets/aligning/a75137c250be63877a30f4bda8d801f8.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 2 2 3 - ╡ x = \colorblue - 85 · 3 · (1-t) · t \colorblue - 12 · 3 · (1-t) · t \colorblue + 156 · t - │ 2 2 - ╰ y = \colorblue - 40 · 3 · (1-t) · t \colorblue + 140 · 3 · (1-t) · t diff --git a/docs/images/snippets/aligning/d412c72ed342c2036965ff251c8fb443.ascii b/docs/images/snippets/aligning/d412c72ed342c2036965ff251c8fb443.ascii new file mode 100644 index 00000000..1339c62f --- /dev/null +++ b/docs/images/snippets/aligning/d412c72ed342c2036965ff251c8fb443.ascii @@ -0,0 +1,5 @@ + +╭ 3 2 2 3 +╡ x = \colorblue 0 · (1-t) \colorblue - 85 · 3 · (1-t) · t \colorblue + 100 · 3 · (1-t) · t \colorblue + 100 · t +│ 3 2 2 3 +╰ y = \colorblue 0 · (1-t) \colorblue + 40 · 3 · (1-t) · t \colorblue + 100 · 3 · (1-t) · t \colorblue - 120 · t diff --git a/docs/images/snippets/aligning/e49d7bb45a18d14fbb12df4d91e2c67b.ascii b/docs/images/snippets/aligning/e49d7bb45a18d14fbb12df4d91e2c67b.ascii new file mode 100644 index 00000000..d0530484 --- /dev/null +++ b/docs/images/snippets/aligning/e49d7bb45a18d14fbb12df4d91e2c67b.ascii @@ -0,0 +1,5 @@ + +╭ 3 2 2 3 +╡ x = \colorblue 0 · (1-t) \colorblue - 85 · 3 · (1-t) · t \colorblue - 12 · 3 · (1-t) · t \colorblue + 156 · t +│ 3 2 2 3 +╰ y = \colorblue 0 · (1-t) \colorblue - 40 · 3 · (1-t) · t \colorblue + 140 · 3 · (1-t) · t \colorblue + 0 · t diff --git a/docs/images/snippets/aligning/e5db5e945606d3429e4475ff92283a9c.ascii b/docs/images/snippets/aligning/e5db5e945606d3429e4475ff92283a9c.ascii new file mode 100644 index 00000000..13b9376d --- /dev/null +++ b/docs/images/snippets/aligning/e5db5e945606d3429e4475ff92283a9c.ascii @@ -0,0 +1,5 @@ + +╭ 3 2 2 3 +╡ x = \colorblue 120 · (1-t) \colorblue + 35 · 3 · (1-t) · t \colorblue + 220 · 3 · (1-t) · t \colorblue + 220 · t +│ 3 2 2 3 +╰ y = \colorblue 160 · (1-t) \colorblue + 200 · 3 · (1-t) · t \colorblue + 260 · 3 · (1-t) · t \colorblue + 40 · t diff --git a/docs/images/snippets/aligning/f6767b16ff8e04646f45fb9a1f3e4024.ascii b/docs/images/snippets/aligning/f6767b16ff8e04646f45fb9a1f3e4024.ascii deleted file mode 100644 index a9a2facb..00000000 --- a/docs/images/snippets/aligning/f6767b16ff8e04646f45fb9a1f3e4024.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 3 2 2 3 - ╡ x = \colorblue0 · (1-t) \colorblue - 85 · 3 · (1-t) · t \colorblue - 12 · 3 · (1-t) · t \colorblue + 156 · t - │ 3 2 2 3 - ╰ y = \colorblue0 · (1-t) \colorblue - 40 · 3 · (1-t) · t \colorblue + 140 · 3 · (1-t) · t \colorblue + 0 · t diff --git a/docs/images/snippets/arclength/046bbb52e8c8ed617fdf3a4fd18d62e1.ascii b/docs/images/snippets/arclength/046bbb52e8c8ed617fdf3a4fd18d62e1.ascii index 6f4b03da..6a8fc7d3 100644 --- a/docs/images/snippets/arclength/046bbb52e8c8ed617fdf3a4fd18d62e1.ascii +++ b/docs/images/snippets/arclength/046bbb52e8c8ed617fdf3a4fd18d62e1.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌─────────────────┐ - ╭ z │ 2 2 z ┌ ╭ z -1 z ╮ ╭ z 1 z ╮ ┐ - | ⟍│(dx/dt) +(dy/dt) dt ≃ ─ · │ f│ ─ · ──── + ─ │ + f│ ─ · ──── + ─ │ │ - ╯ 0 2 │ │ 2 ┌─┐ 2 │ │ 2 ┌─┐ 2 │ │ - └ ╰ ⟍│3 ╯ ╰ ⟍│3 ╯ ┘ + ┌─────────────────┐ +╭ z │ 2 2 z ┌ ╭ z -1 z ╮ ╭ z 1 z ╮ ┐ +| ⟍│(dx/dt) +(dy/dt) dt ≃ ─ · │ f│ ─ · ──── + ─ │ + f│ ─ · ──── + ─ │ │ +╯ 0 2 │ │ 2 ┌─┐ 2 │ │ 2 ┌─┐ 2 │ │ + └ ╰ ⟍│3 ╯ ╰ ⟍│3 ╯ ┘ diff --git a/docs/images/snippets/arclength/0748ad25185548150b6c1c4c7039207e.ascii b/docs/images/snippets/arclength/0748ad25185548150b6c1c4c7039207e.ascii index ffc62b78..b5ea8b58 100644 --- a/docs/images/snippets/arclength/0748ad25185548150b6c1c4c7039207e.ascii +++ b/docs/images/snippets/arclength/0748ad25185548150b6c1c4c7039207e.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌─────────────────┐ - ╭ z │ 2 2 - | ⟍│(dx/dt) +(dy/dt) dt - ╯ 0 - z ┌ ╭ z z ╮ ╭ z z ╮ ┐ - ≃ \ ─ · │ C · f│ ─ · t + ─ │ + ... + C · f│ ─ · t + ─ │ │ - 2 └ 1 ╰ 2 1 2 ╯ n ╰ 2 n 2 ╯ ┘ - z __ n ╭ z z ╮ - = \ ─ · ❯ C · f│ ─ · t + ─ │ - 2 ‾‾ i=1 i ╰ 2 i 2 ╯ + ┌─────────────────┐ + ╭ z │ 2 2 + | ⟍│(dx/dt) +(dy/dt) dt + ╯ 0 + z ┌ ╭ z z ╮ ╭ z z ╮ ┐ + ≃ \ ─ · │ C · f│ ─ · t + ─ │ + ... + C · f│ ─ · t + ─ │ │ + 2 └ 1 ╰ 2 1 2 ╯ n ╰ 2 n 2 ╯ ┘ + z __ n ╭ z z ╮ += \ ─ · ❯ C · f│ ─ · t + ─ │ + 2 ‾‾ i=1 i ╰ 2 i 2 ╯ diff --git a/docs/images/snippets/arclength/2f80643c66d8f1448b13537a7b24eb45.ascii b/docs/images/snippets/arclength/2f80643c66d8f1448b13537a7b24eb45.ascii deleted file mode 100644 index 714299cc..00000000 --- a/docs/images/snippets/arclength/2f80643c66d8f1448b13537a7b24eb45.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌─────────────────┐ - ╭ z │ 2 2 - length = | ⟍│(dx/dt) +(dy/dt) dt - ╯ 0 diff --git a/docs/images/snippets/arclength/59ebc3a7c3547a50998d1ea3664fb688.ascii b/docs/images/snippets/arclength/59ebc3a7c3547a50998d1ea3664fb688.ascii index c0556eee..e80f1340 100644 --- a/docs/images/snippets/arclength/59ebc3a7c3547a50998d1ea3664fb688.ascii +++ b/docs/images/snippets/arclength/59ebc3a7c3547a50998d1ea3664fb688.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌───────────────┐ - ╭ z │ 2 2 - | │f '(t) +f '(t) dt - ╯ 0⟍│ x y + ┌───────────────┐ +╭ z │ 2 2 +| │f '(t) +f '(t) dt +╯ 0⟍│ x y diff --git a/docs/images/snippets/arclength/85620f0332fcf16f56c580794fd094c5.ascii b/docs/images/snippets/arclength/85620f0332fcf16f56c580794fd094c5.ascii index 714299cc..e822d55e 100644 --- a/docs/images/snippets/arclength/85620f0332fcf16f56c580794fd094c5.ascii +++ b/docs/images/snippets/arclength/85620f0332fcf16f56c580794fd094c5.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌─────────────────┐ - ╭ z │ 2 2 - length = | ⟍│(dx/dt) +(dy/dt) dt - ╯ 0 + ┌─────────────────┐ + ╭ z │ 2 2 +length = | ⟍│(dx/dt) +(dy/dt) dt + ╯ 0 diff --git a/docs/images/snippets/arclength/a91fbfb7abc38ff712ef660d85679f2e.ascii b/docs/images/snippets/arclength/a91fbfb7abc38ff712ef660d85679f2e.ascii index bfce7dba..35c644b8 100644 --- a/docs/images/snippets/arclength/a91fbfb7abc38ff712ef660d85679f2e.ascii +++ b/docs/images/snippets/arclength/a91fbfb7abc38ff712ef660d85679f2e.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C = 1 - 1 - C = 1 - 2 - 1 - t = - ──── - 1 ┌─┐ - ⟍│3 - 1 - t = + ──── - 2 ┌─┐ - ⟍│3 +C = 1 + 1 +C = 1 + 2 + 1 +t = - ──── + 1 ┌─┐ + ⟍│3 + 1 +t = + ──── + 2 ┌─┐ + ⟍│3 diff --git a/docs/images/snippets/arclength/b76753476ad6ecfe4b8f39bcf9432980.ascii b/docs/images/snippets/arclength/b76753476ad6ecfe4b8f39bcf9432980.ascii index 10c3da8f..28e13d1c 100644 --- a/docs/images/snippets/arclength/b76753476ad6ecfe4b8f39bcf9432980.ascii +++ b/docs/images/snippets/arclength/b76753476ad6ecfe4b8f39bcf9432980.ascii @@ -1,5 +1,4 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - + ┌─────────────────┐ ╭ 1 │ 2 2 ╭ 1 | ⟍│(dx/dt) +(dy/dt) dt = | f(t) dt ≃ ┌ \underset strip 1 \underbrace C · f(t ) + ... + \underset strip n \underbrace C · f(t ) ┐ = diff --git a/docs/images/snippets/arclength/f251e86158649c0e57f7a772ebff83b4.ascii b/docs/images/snippets/arclength/f251e86158649c0e57f7a772ebff83b4.ascii deleted file mode 100644 index c221a164..00000000 --- a/docs/images/snippets/arclength/f251e86158649c0e57f7a772ebff83b4.ascii +++ /dev/null @@ -1,9 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌─────────────────┐ -╭ 1 │ 2 2 ╭ 1 -| ⟍│(dx/dt) +(dy/dt) dt = | f(t) dt ≃ ┌ \undersetstrip 1 \underbrace C · f(t ) + ... + \undersetstrip n \underbrace C · f(t ) ┐ = -╯ -1 ╯ -1 └ 1 1 n n ┘ - __ n - \undersetstrips 1 through n \underbrace ❯ C · f(t ) - ‾‾ i=1 i i diff --git a/docs/images/snippets/bsplines/20e910bbea2e6eff511cb13cef18ef3b.ascii b/docs/images/snippets/bsplines/20e910bbea2e6eff511cb13cef18ef3b.ascii index 926fb0f9..c7ab885e 100644 --- a/docs/images/snippets/bsplines/20e910bbea2e6eff511cb13cef18ef3b.ascii +++ b/docs/images/snippets/bsplines/20e910bbea2e6eff511cb13cef18ef3b.ascii @@ -1,30 +1,29 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ ╭ ╭ 1 0 0 - │ │ │ α × d , with d either 0 or 1 - │ │ 2 1 1 │ 3 3 3 - │ │ α × d , with d = ╡ + - │ │ 3 3 3 │ ╭ 1 ╮ 0 0 - │ │ │ │ 1 - α │ × d , with d either 0 or 1 - │ 3 2 2 │ ╰ ╰ 3 ╯ 2 2 - │ α × d , with d = ╡ + - │ 3 3 3 │ ╭ 1 0 - │ │ │ α × d - │ │ ╭ 2 ╮ 1 1 │ 2 2 - │ │ │ 1 - α │ × d , with d = ╡ + - 3 │ │ ╰ 3 ╯ 2 2 │ ╭ 1 ╮ 0 0 - d = ╡ │ │ │ 1 - α │ × d , with d either 0 or 1 - 3 │ ╰ ╰ ╰ 2 ╯ 1 1 - │ + - │ ╭ 2 1 - │ │ α × d - │ │ 2 2 - │ │ - │ ╭ 3 ╮ 2 2 │ + - │ │ 1 - α │ × d , with d = ╡ ╭ 1 0 - │ ╰ 3 ╯ 2 2 │ │ α × d - │ │ ╭ 2 ╮ 1 1 │ 1 1 - │ │ │ 1 - α │ × d , with d = ╡ + - │ │ ╰ 2 ╯ 1 1 │ ╭ 1 ╮ 0 0 - │ │ │ │ 1 - α │ × d , with d either 0 or 1 - ╰ ╰ ╰ ╰ 1 ╯ 0 0 + ╭ ╭ ╭ 1 0 0 + │ │ │ α × d , with d either 0 or 1 + │ │ 2 1 1 │ 3 3 3 + │ │ α × d , with d = ╡ + + │ │ 3 3 3 │ ╭ 1 ╮ 0 0 + │ │ │ │ 1 - α │ × d , with d either 0 or 1 + │ 3 2 2 │ ╰ ╰ 3 ╯ 2 2 + │ α × d , with d = ╡ + + │ 3 3 3 │ ╭ 1 0 + │ │ │ α × d + │ │ ╭ 2 ╮ 1 1 │ 2 2 + │ │ │ 1 - α │ × d , with d = ╡ + + 3 │ │ ╰ 3 ╯ 2 2 │ ╭ 1 ╮ 0 0 +d = ╡ │ │ │ 1 - α │ × d , with d either 0 or 1 + 3 │ ╰ ╰ ╰ 2 ╯ 1 1 + │ + + │ ╭ 2 1 + │ │ α × d + │ │ 2 2 + │ │ + │ ╭ 3 ╮ 2 2 │ + + │ │ 1 - α │ × d , with d = ╡ ╭ 1 0 + │ ╰ 3 ╯ 2 2 │ │ α × d + │ │ ╭ 2 ╮ 1 1 │ 1 1 + │ │ │ 1 - α │ × d , with d = ╡ + + │ │ ╰ 2 ╯ 1 1 │ ╭ 1 ╮ 0 0 + │ │ │ │ 1 - α │ × d , with d either 0 or 1 + ╰ ╰ ╰ ╰ 1 ╯ 0 0 diff --git a/docs/images/snippets/bsplines/2421f47aa4fe1c0d830d53b2e6563c04.ascii b/docs/images/snippets/bsplines/2421f47aa4fe1c0d830d53b2e6563c04.ascii index a5531f0d..1b1e07e9 100644 --- a/docs/images/snippets/bsplines/2421f47aa4fe1c0d830d53b2e6563c04.ascii +++ b/docs/images/snippets/bsplines/2421f47aa4fe1c0d830d53b2e6563c04.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ t- knot ╮ ╭ knot -t ╮ - │ i │ │ (i+k) │ - N (t) = │ ───────────────────── │ · N (t) + │ ─────────────────────── │ · N (t) - i,k │ knot - knot │ i,k-1 │ knot - knot │ i+1,k-1 - ╰ (i+k-1) i ╯ ╰ (i+k) (i+1) ╯ + ╭ t- knot ╮ ╭ knot -t ╮ + │ i │ │ (i+k) │ +N (t) = │ ───────────────────── │ · N (t) + │ ─────────────────────── │ · N (t) + i,k │ knot - knot │ i,k-1 │ knot - knot │ i+1,k-1 + ╰ (i+k-1) i ╯ ╰ (i+k) (i+1) ╯ diff --git a/docs/images/snippets/bsplines/2514e1aa0565840e33fde0b146e3efe2.ascii b/docs/images/snippets/bsplines/2514e1aa0565840e33fde0b146e3efe2.ascii index 6a4fc7bd..f9ca53c2 100644 --- a/docs/images/snippets/bsplines/2514e1aa0565840e33fde0b146e3efe2.ascii +++ b/docs/images/snippets/bsplines/2514e1aa0565840e33fde0b146e3efe2.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ 1 if t ∈[ knot , knot ) - N (t) = ╡ i i+1 - i,1 ╰ 0 otherwise + ╭ 1 if t ∈[ knot , knot ) +N (t) = ╡ i i+1 + i,1 ╰ 0 otherwise diff --git a/docs/images/snippets/bsplines/49af474c33ce0ee0733626ea3d988570.ascii b/docs/images/snippets/bsplines/49af474c33ce0ee0733626ea3d988570.ascii index ae16d003..e86227fc 100644 --- a/docs/images/snippets/bsplines/49af474c33ce0ee0733626ea3d988570.ascii +++ b/docs/images/snippets/bsplines/49af474c33ce0ee0733626ea3d988570.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - k 0 ╭ 1 if t ∈[ knot , knot ) - d (t) = 0, d (t) = N (t) = ╡ i i+1 - 0 i i,1 ╰ 0 otherwise + k 0 ╭ 1 if t ∈[ knot , knot ) +d (t) = 0, d (t) = N (t) = ╡ i i+1 + 0 i i,1 ╰ 0 otherwise diff --git a/docs/images/snippets/bsplines/89f8e37237d066fa70ccf6d37b3a4922.ascii b/docs/images/snippets/bsplines/89f8e37237d066fa70ccf6d37b3a4922.ascii index 2e97737a..86e4cbb7 100644 --- a/docs/images/snippets/bsplines/89f8e37237d066fa70ccf6d37b3a4922.ascii +++ b/docs/images/snippets/bsplines/89f8e37237d066fa70ccf6d37b3a4922.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n - Point(t) = ❯ P · N (t) - ‾‾ i=0 i i,k + __ n +Point(t) = ❯ P · N (t) + ‾‾ i=0 i i,k diff --git a/docs/images/snippets/bsplines/a88566be442b67fb71f727de6bdb66df.ascii b/docs/images/snippets/bsplines/a88566be442b67fb71f727de6bdb66df.ascii deleted file mode 100644 index 270d3194..00000000 --- a/docs/images/snippets/bsplines/a88566be442b67fb71f727de6bdb66df.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - k 0 ╭ 1 if t ∈[knot ,knot ) - d (t) = 0, d (t) = N (t) = ╡ i i+1 - 0 i i,1 ╰ 0 otherwise diff --git a/docs/images/snippets/bsplines/c7af721e5e201fc3742bce67ff6cd560.ascii b/docs/images/snippets/bsplines/c7af721e5e201fc3742bce67ff6cd560.ascii deleted file mode 100644 index cbbf5e80..00000000 --- a/docs/images/snippets/bsplines/c7af721e5e201fc3742bce67ff6cd560.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - t - knots[i] - α = ───────────────────────── - i,k knots[i+1+n-k] - knots[i] diff --git a/docs/images/snippets/bsplines/cbdf5a61de10eeb6f23be077cf047ab5.ascii b/docs/images/snippets/bsplines/cbdf5a61de10eeb6f23be077cf047ab5.ascii deleted file mode 100644 index 19782167..00000000 --- a/docs/images/snippets/bsplines/cbdf5a61de10eeb6f23be077cf047ab5.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ t-knot ╮ ╭ knot -t ╮ - │ i │ │ (i+k) │ - N (t) = │ ─────────────────── │ · N (t) + │ ───────────────────── │ · N (t) - i,k │ knot - knot │ i,k-1 │ knot - knot │ i+1,k-1 - ╰ (i+k-1) i ╯ ╰ (i+k) (i+1) ╯ diff --git a/docs/images/snippets/bsplines/e62558cdfd8abaf22511e8e68c7afb4a.ascii b/docs/images/snippets/bsplines/e62558cdfd8abaf22511e8e68c7afb4a.ascii index 340f3fb5..f8b4e146 100644 --- a/docs/images/snippets/bsplines/e62558cdfd8abaf22511e8e68c7afb4a.ascii +++ b/docs/images/snippets/bsplines/e62558cdfd8abaf22511e8e68c7afb4a.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - t - knots[i] - α = ─────────────────────────── - i,k knots[i+1+n-k] - knots[i] + t - knots[i] +α = ─────────────────────────── + i,k knots[i+1+n-k] - knots[i] diff --git a/docs/images/snippets/bsplines/ee203de6e554936588eb93adead0a3e5.ascii b/docs/images/snippets/bsplines/ee203de6e554936588eb93adead0a3e5.ascii deleted file mode 100644 index 7a9f5f2c..00000000 --- a/docs/images/snippets/bsplines/ee203de6e554936588eb93adead0a3e5.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 1 if t ∈[knot ,knot ) - N (t) = ╡ i i+1 - i,1 ╰ 0 otherwise diff --git a/docs/images/snippets/bsplines/f0bf7d0f1931060cd801ff707f482c16.ascii b/docs/images/snippets/bsplines/f0bf7d0f1931060cd801ff707f482c16.ascii index e45573bf..ca6d844f 100644 --- a/docs/images/snippets/bsplines/f0bf7d0f1931060cd801ff707f482c16.ascii +++ b/docs/images/snippets/bsplines/f0bf7d0f1931060cd801ff707f482c16.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - k k-1 k-1 - d (t) = α · d (t) + (1-α ) · d (t) - i i,k i i,k i-1 + k k-1 k-1 +d (t) = α · d (t) + (1-α ) · d (t) + i i,k i i,k i-1 diff --git a/docs/images/snippets/canonical/2b6478075f2f9f5e5973e01b3b3a0c8b.ascii b/docs/images/snippets/canonical/2b6478075f2f9f5e5973e01b3b3a0c8b.ascii index 71b8dc0a..9b1736c3 100644 --- a/docs/images/snippets/canonical/2b6478075f2f9f5e5973e01b3b3a0c8b.ascii +++ b/docs/images/snippets/canonical/2b6478075f2f9f5e5973e01b3b3a0c8b.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 S 0 ┐ ┌ x ┐ ┌ x + S · y ┐ - │ 0 1 0 │ · │ y │ = │ y │ - └ 0 0 1 ┘ └ 1 ┘ └ 1 ┘ +┌ 1 S 0 ┐ ┌ x ┐ ┌ x + S · y ┐ +│ 0 1 0 │ · │ y │ = │ y │ +└ 0 0 1 ┘ └ 1 ┘ └ 1 ┘ diff --git a/docs/images/snippets/canonical/464b4ec0b67f248459792752be86d46d.ascii b/docs/images/snippets/canonical/464b4ec0b67f248459792752be86d46d.ascii index ba69aeb7..3093f7c6 100644 --- a/docs/images/snippets/canonical/464b4ec0b67f248459792752be86d46d.ascii +++ b/docs/images/snippets/canonical/464b4ec0b67f248459792752be86d46d.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 - -x + 3x - y = ────────, { x ≤0 } - 3 + 2 + -x + 3x +y = ────────, { x ≤0 } + 3 diff --git a/docs/images/snippets/canonical/5f174bc5019245f467ca63ae84b90a4b.ascii b/docs/images/snippets/canonical/5f174bc5019245f467ca63ae84b90a4b.ascii index de7d596d..5495c160 100644 --- a/docs/images/snippets/canonical/5f174bc5019245f467ca63ae84b90a4b.ascii +++ b/docs/images/snippets/canonical/5f174bc5019245f467ca63ae84b90a4b.ascii @@ -1,5 +1,4 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - + ╭ x · y x · y ╮ │ 2 4 2 3 │ │ x - ──────── / x -──────── │ ╭ x ╮ diff --git a/docs/images/snippets/canonical/674d251590411398d06fb99cba7920f7.ascii b/docs/images/snippets/canonical/674d251590411398d06fb99cba7920f7.ascii index 83453eff..dbc1bba7 100644 --- a/docs/images/snippets/canonical/674d251590411398d06fb99cba7920f7.ascii +++ b/docs/images/snippets/canonical/674d251590411398d06fb99cba7920f7.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 - -x + 2x + 3 - y = ────────────, { x ≤1 } - 4 + 2 + -x + 2x + 3 +y = ────────────, { x ≤1 } + 4 diff --git a/docs/images/snippets/canonical/6959a552f2c90a2bcaa787c23e19f488.ascii b/docs/images/snippets/canonical/6959a552f2c90a2bcaa787c23e19f488.ascii index 04359123..597cc8cb 100644 --- a/docs/images/snippets/canonical/6959a552f2c90a2bcaa787c23e19f488.ascii +++ b/docs/images/snippets/canonical/6959a552f2c90a2bcaa787c23e19f488.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌──────────┐ - │ 2 - ⟍│3(4x - x ) - x - y = ─────────────────, { 0 ≤x ≤1 } - 2 + ┌──────────┐ + │ 2 + ⟍│3(4x - x ) - x +y = ─────────────────, { 0 ≤x ≤1 } + 2 diff --git a/docs/images/snippets/canonical/7ed8b53100737cbf7d87aa6267395d2b.ascii b/docs/images/snippets/canonical/7ed8b53100737cbf7d87aa6267395d2b.ascii index 903d8cc6..e716a32d 100644 --- a/docs/images/snippets/canonical/7ed8b53100737cbf7d87aa6267395d2b.ascii +++ b/docs/images/snippets/canonical/7ed8b53100737cbf7d87aa6267395d2b.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 a ┐ ┌ x ┐ ┌ 1 · x + 0 · y + a · z ┐ ┌ x + a · 1 ┐ ┌ x + a ┐ - │ 0 1 b │ · │ y │ = │ 0 · x + 1 · y + b · z │ = │ y + b · 1 │ = │ y + b │ - └ 0 0 1 ┘ └ z=1 ┘ └ 0 · x + 0 · y + 1 · z ┘ └ 1 · z ┘ └ z=1 ┘ +┌ 1 0 a ┐ ┌ x ┐ ┌ 1 · x + 0 · y + a · z ┐ ┌ x + a · 1 ┐ ┌ x + a ┐ +│ 0 1 b │ · │ y │ = │ 0 · x + 1 · y + b · z │ = │ y + b · 1 │ = │ y + b │ +└ 0 0 1 ┘ └ z=1 ┘ └ 0 · x + 0 · y + 1 · z ┘ └ 1 · z ┘ └ z=1 ┘ diff --git a/docs/images/snippets/canonical/88e3fae7aeef6d7614290587422542c9.ascii b/docs/images/snippets/canonical/88e3fae7aeef6d7614290587422542c9.ascii index 9b093744..0b504b2f 100644 --- a/docs/images/snippets/canonical/88e3fae7aeef6d7614290587422542c9.ascii +++ b/docs/images/snippets/canonical/88e3fae7aeef6d7614290587422542c9.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ x ╮ ╭ x - x · y ╮ y y - │ 43 │ │ 4 2 42 │ 4 3 - ... = │ │, where x = │ ────────────── │, y = ──, and y = ── - │ y + x (1 - y ) │ 43 │ x - x · y │ 42 y 32 y - ╰ 42 43 32 ╯ ╰ 3 2 32 ╯ 2 2 + ╭ x ╮ ╭ x - x · y ╮ y y + │ 43 │ │ 4 2 42 │ 4 3 +... = │ │, where x = │ ────────────── │, y = ──, and y = ── + │ y + x (1 - y ) │ 43 │ x - x · y │ 42 y 32 y + ╰ 42 43 32 ╯ ╰ 3 2 32 ╯ 2 2 diff --git a/docs/images/snippets/canonical/8e39a9e0c7469b4b45a260dd23bd4c6a.ascii b/docs/images/snippets/canonical/8e39a9e0c7469b4b45a260dd23bd4c6a.ascii index 8a0335a8..03a8fce5 100644 --- a/docs/images/snippets/canonical/8e39a9e0c7469b4b45a260dd23bd4c6a.ascii +++ b/docs/images/snippets/canonical/8e39a9e0c7469b4b45a260dd23bd4c6a.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 ┐ - │ ─── 0 0 │ - │ V │ - │ 3x │ - T = │ 1 │ - 3 │ 0 ─── 0 │ - │ V │ - │ 2y │ - └ 0 0 1 ┘ + ┌ 1 ┐ + │ ─── 0 0 │ + │ V │ + │ 3x │ +T = │ 1 │ + 3 │ 0 ─── 0 │ + │ V │ + │ 2y │ + └ 0 0 1 ┘ diff --git a/docs/images/snippets/canonical/9420fd9d7a8de30714e23b8f31b3aa6d.ascii b/docs/images/snippets/canonical/9420fd9d7a8de30714e23b8f31b3aa6d.ascii index 90cad65a..dd189514 100644 --- a/docs/images/snippets/canonical/9420fd9d7a8de30714e23b8f31b3aa6d.ascii +++ b/docs/images/snippets/canonical/9420fd9d7a8de30714e23b8f31b3aa6d.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 0 ┐ - │ 1 - W │ - │ 3y │ - T = │ ─────── 1 0 │ - 4 │ W │ - │ 3x │ - └ 0 0 1 ┘ + ┌ 1 0 0 ┐ + │ 1 - W │ + │ 3y │ +T = │ ─────── 1 0 │ + 4 │ W │ + │ 3x │ + └ 0 0 1 ┘ diff --git a/docs/images/snippets/canonical/ccbfd22cbccf633d182f7f451dee5164.ascii b/docs/images/snippets/canonical/ccbfd22cbccf633d182f7f451dee5164.ascii index 3ff4fb6f..4e1ae4ab 100644 --- a/docs/images/snippets/canonical/ccbfd22cbccf633d182f7f451dee5164.ascii +++ b/docs/images/snippets/canonical/ccbfd22cbccf633d182f7f451dee5164.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ U ┐ - │ 2x │ - │ 1 -─── 0 │ - T = │ U │ - 2 │ 2y │ - │ 0 1 0 │ - └ 0 0 1 ┘ + ┌ U ┐ + │ 2x │ + │ 1 -─── 0 │ +T = │ U │ + 2 │ 2y │ + │ 0 1 0 │ + └ 0 0 1 ┘ diff --git a/docs/images/snippets/canonical/e61fd49e554a0ffc7d64893c75cd376d.ascii b/docs/images/snippets/canonical/e61fd49e554a0ffc7d64893c75cd376d.ascii deleted file mode 100644 index e6d576a7..00000000 --- a/docs/images/snippets/canonical/e61fd49e554a0ffc7d64893c75cd376d.ascii +++ /dev/null @@ -1,24 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ (-x +x )(-y +y ) ╮ - │ 1 2 1 4 │ - │ -x + x - ──────────────── │ - │ 1 4 -y +y │ - │ 1 2 │ - │ ─────────────────────────── │ - │ (-x +x )(-y +y ) │ - │ 1 2 1 3 │ - │ -x +x -──────────────── │ - │ 1 3 -y +y │ - mapped = (x) = │ 1 2 │ - 4 y │ ╭ -y +y ╮ ╭ (-x +x )(-y +y ) ╮ │ - │ │ 1 3 │ │ 1 2 1 4 │ │ - │ │ 1 - ────── │ │ -x + x - ──────────────── │ │ - │ (-y +y ) │ -y +y │ │ 1 4 -y +y │ │ - │ 1 4 ╰ 1 2 ╯ ╰ 1 2 ╯ │ - │ ──────── + ────────────────────────────────────────────── │ - │ -y +y (-x +x )(-y +y ) │ - │ 1 2 1 2 1 3 │ - │ -x +x -──────────────── │ - │ 1 3 -y +y │ - ╰ 1 2 ╯ diff --git a/docs/images/snippets/canonical/f855cbf1d73e4bb7bccbbd4721d95f41.ascii b/docs/images/snippets/canonical/f855cbf1d73e4bb7bccbbd4721d95f41.ascii index 91cf90e7..053265d3 100644 --- a/docs/images/snippets/canonical/f855cbf1d73e4bb7bccbbd4721d95f41.ascii +++ b/docs/images/snippets/canonical/f855cbf1d73e4bb7bccbbd4721d95f41.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 -P ┐ ┌ 1 · x + 0 · y - P · 1 ┐ ┌ x - P ┐ - │ 1x │ ┌ x ┐ │ 1x │ │ 1x │ - T = │ 0 1 -P │ · │ y │ = │ 0 · x + 1 · y - P · 1 │ = │ y - P │ - 1 │ 1y │ └ 1 ┘ │ 1y │ │ 1y │ - └ 0 0 1 ┘ └ 0 · x + 0 · y + 1 · 1 ┘ └ 1 ┘ + ┌ 1 0 -P ┐ ┌ 1 · x + 0 · y - P · 1 ┐ ┌ x - P ┐ + │ 1x │ ┌ x ┐ │ 1x │ │ 1x │ +T = │ 0 1 -P │ · │ y │ = │ 0 · x + 1 · y - P · 1 │ = │ y - P │ + 1 │ 1y │ └ 1 ┘ │ 1y │ │ 1y │ + └ 0 0 1 ┘ └ 0 · x + 0 · y + 1 · 1 ┘ └ 1 ┘ diff --git a/docs/images/snippets/canonical/fff37fa4275e43302f71cf052417a19f.ascii b/docs/images/snippets/canonical/fff37fa4275e43302f71cf052417a19f.ascii index 51769205..486bbc64 100644 --- a/docs/images/snippets/canonical/fff37fa4275e43302f71cf052417a19f.ascii +++ b/docs/images/snippets/canonical/fff37fa4275e43302f71cf052417a19f.ascii @@ -1,24 +1,23 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ (-x +x )(-y +y ) ╮ - │ 1 2 1 4 │ - │ -x + x - ──────────────── │ - │ 1 4 -y +y │ - │ 1 2 │ - │ ─────────────────────────── │ - │ (-x +x )(-y +y ) │ - │ 1 2 1 3 │ - │ -x +x -──────────────── │ - │ 1 3 -y +y │ - mapped = (x) = │ 1 2 │ - 4 y │ ╭ -y +y ╮ ╭ (-x +x )(-y +y ) ╮ │ - │ │ 1 3 │ │ 1 2 1 4 │ │ - │ │ 1 - ────── │ │ -x + x - ──────────────── │ │ - │ (-y +y ) │ -y +y │ │ 1 4 -y +y │ │ - │ 1 4 ╰ 1 2 ╯ ╰ 1 2 ╯ │ - │ ──────── + ────────────────────────────────────────────── │ - │ -y +y (-x +x )(-y +y ) │ - │ 1 2 1 2 1 3 │ - │ -x +x -──────────────── │ - │ 1 3 -y +y │ - ╰ 1 2 ╯ + ╭ (-x +x )(-y +y ) ╮ + │ 1 2 1 4 │ + │ -x + x - ──────────────── │ + │ 1 4 -y +y │ + │ 1 2 │ + │ ─────────────────────────── │ + │ (-x +x )(-y +y ) │ + │ 1 2 1 3 │ + │ -x +x -──────────────── │ + │ 1 3 -y +y │ +mapped = (x) = │ 1 2 │ + 4 y │ ╭ -y +y ╮ ╭ (-x +x )(-y +y ) ╮ │ + │ │ 1 3 │ │ 1 2 1 4 │ │ + │ │ 1 - ────── │ │ -x + x - ──────────────── │ │ + │ (-y +y ) │ -y +y │ │ 1 4 -y +y │ │ + │ 1 4 ╰ 1 2 ╯ ╰ 1 2 ╯ │ + │ ──────── + ────────────────────────────────────────────── │ + │ -y +y (-x +x )(-y +y ) │ + │ 1 2 1 2 1 3 │ + │ -x +x -──────────────── │ + │ 1 3 -y +y │ + ╰ 1 2 ╯ diff --git a/docs/images/snippets/catmullconv/00357d2a2168fe313cd0b38d95a1a681.ascii b/docs/images/snippets/catmullconv/00357d2a2168fe313cd0b38d95a1a681.ascii deleted file mode 100644 index cc82ede4..00000000 --- a/docs/images/snippets/catmullconv/00357d2a2168fe313cd0b38d95a1a681.ascii +++ /dev/null @@ -1,14 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ V = P ┐ - │ 1 2 │ - ┌ P ┐ │ V = P │ - │ 1 │ │ 2 3 │ - │ P │ │ P - P │ - │ 2 │ = │ 3 1 │ - │ P │points │ V' = ─────── │point-tangent - │ 3 │ │ 1 2 │ - │ P │ │ P - P │ - └ 4 ┘ │ 4 2 │ - │ V' = ─────── │ - └ 2 2 ┘ diff --git a/docs/images/snippets/catmullconv/012a8ab7a4de935c1c8d61dcd14fc62c.ascii b/docs/images/snippets/catmullconv/012a8ab7a4de935c1c8d61dcd14fc62c.ascii index 224d8fa8..8524a72e 100644 --- a/docs/images/snippets/catmullconv/012a8ab7a4de935c1c8d61dcd14fc62c.ascii +++ b/docs/images/snippets/catmullconv/012a8ab7a4de935c1c8d61dcd14fc62c.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ ┌ P + 6(P - P ) ┐ - │ 1 │ │ 4 1 2 │ - │ P │ │ P │ - │ 2 │ \Rightarrow │ 1 │ - │ P │ Bézier │ P │ CatmullRom - │ 3 │ │ 4 │ - │ P │ │ P + 6(P - P ) │ - └ 4 ┘ └ 1 4 3 ┘ +┌ P ┐ ┌ P + 6(P - P ) ┐ +│ 1 │ │ 4 1 2 │ +│ P │ │ P │ +│ 2 │ ==> │ 1 │ +│ P │ Bézier │ P │ CatmullRom +│ 3 │ │ 4 │ +│ P │ │ P + 6(P - P ) │ +└ 4 ┘ └ 1 4 3 ┘ diff --git a/docs/images/snippets/catmullconv/032409c03915a6ba75864e1dceae416d.ascii b/docs/images/snippets/catmullconv/032409c03915a6ba75864e1dceae416d.ascii index eaea3f72..60d4255b 100644 --- a/docs/images/snippets/catmullconv/032409c03915a6ba75864e1dceae416d.ascii +++ b/docs/images/snippets/catmullconv/032409c03915a6ba75864e1dceae416d.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ ┌ P ┐ - │ 0 0 1 0 │ │ 1 │ - ┌ 1 0 0 0 ┐ │ -1 1 │ │ P │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ ── 0 ── 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ 2τ 2τ │ │ P │ - └ 2 -2 1 1 ┘ │ -1 1 │ │ 3 │ - │ 0 ── 0 ── │ │ P │ - └ 2τ 2τ ┘ └ 4 ┘ + ┌ 0 1 0 0 ┐ ┌ P ┐ + │ 0 0 1 0 │ │ 1 │ + ┌ 1 0 0 0 ┐ │ -1 1 │ │ P │ +CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ ── 0 ── 0 │ · │ 2 │ + └ 1 t t t ┘ │ -3 3 -2 -1 │ │ 2τ 2τ │ │ P │ + └ 2 -2 1 1 ┘ │ -1 1 │ │ 3 │ + │ 0 ── 0 ── │ │ P │ + └ 2τ 2τ ┘ └ 4 ┘ diff --git a/docs/images/snippets/catmullconv/157b287d6b74109d8c8b634990ea6549.ascii b/docs/images/snippets/catmullconv/157b287d6b74109d8c8b634990ea6549.ascii index 1653a4a2..39e486ec 100644 --- a/docs/images/snippets/catmullconv/157b287d6b74109d8c8b634990ea6549.ascii +++ b/docs/images/snippets/catmullconv/157b287d6b74109d8c8b634990ea6549.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 2 │ - │ P -P │ - │ 3 1 │ - ┌ 1 0 0 0 ┐ │ P + ────── │ - = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 6 · τ │ - └ 1 t t t ┘ │ 3 -6 3 0 │ │ P -P │ - └ -1 3 -3 1 ┘ │ 4 2 │ - │ P - ────── │ - │ 3 6 · τ │ - │ P │ - └ 3 ┘ + ┌ P ┐ + │ 2 │ + │ P -P │ + │ 3 1 │ + ┌ 1 0 0 0 ┐ │ P + ────── │ += ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 6 · τ │ + └ 1 t t t ┘ │ 3 -6 3 0 │ │ P -P │ + └ -1 3 -3 1 ┘ │ 4 2 │ + │ P - ────── │ + │ 3 6 · τ │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/catmullconv/1cef6bbf7b3d10e8c0aaecfac816cc86.ascii b/docs/images/snippets/catmullconv/1cef6bbf7b3d10e8c0aaecfac816cc86.ascii index f7b87005..539f2618 100644 --- a/docs/images/snippets/catmullconv/1cef6bbf7b3d10e8c0aaecfac816cc86.ascii +++ b/docs/images/snippets/catmullconv/1cef6bbf7b3d10e8c0aaecfac816cc86.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ - │ ── 1 ── 0 │ - │ 6τ 6τ │ = V - │ 1 -1 │ - │ 0 ── 1 ── │ - │ 6τ 6τ │ - └ 0 0 1 0 ┘ +┌ 0 1 0 0 ┐ +│ -1 1 │ +│ ── 1 ── 0 │ +│ 6τ 6τ │ = V +│ 1 -1 │ +│ 0 ── 1 ── │ +│ 6τ 6τ │ +└ 0 0 1 0 ┘ diff --git a/docs/images/snippets/catmullconv/1f9fc156aeed9eb092573cd7446593d9.ascii b/docs/images/snippets/catmullconv/1f9fc156aeed9eb092573cd7446593d9.ascii deleted file mode 100644 index 11de3347..00000000 --- a/docs/images/snippets/catmullconv/1f9fc156aeed9eb092573cd7446593d9.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ V ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ V │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ V' │ - └ 2 -2 1 1 ┘ │ 1 │ - │ V' │ - └ 2 ┘ diff --git a/docs/images/snippets/catmullconv/211dadbb9d0f6b2e381f18ea3c4d12fb.ascii b/docs/images/snippets/catmullconv/211dadbb9d0f6b2e381f18ea3c4d12fb.ascii index 413c8fed..8e6e5a51 100644 --- a/docs/images/snippets/catmullconv/211dadbb9d0f6b2e381f18ea3c4d12fb.ascii +++ b/docs/images/snippets/catmullconv/211dadbb9d0f6b2e381f18ea3c4d12fb.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 2 │ - │ P │ - │ 3 │ - ┌ 1 0 0 0 ┐ │ P - P │ - = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 3 1 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ ─────── │ - └ 2 -2 1 1 ┘ │ 2τ │ - │ P - P │ - │ 4 2 │ - │ ─────── │ - └ 2τ ┘ + ┌ P ┐ + │ 2 │ + │ P │ + │ 3 │ + ┌ 1 0 0 0 ┐ │ P - P │ += ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 3 1 │ + └ 1 t t t ┘ │ -3 3 -2 -1 │ │ ─────── │ + └ 2 -2 1 1 ┘ │ 2τ │ + │ P - P │ + │ 4 2 │ + │ ─────── │ + └ 2τ ┘ diff --git a/docs/images/snippets/catmullconv/389a1ea8c9e92df9a2b38718e34bae7b.ascii b/docs/images/snippets/catmullconv/389a1ea8c9e92df9a2b38718e34bae7b.ascii index b1b40ccd..47135003 100644 --- a/docs/images/snippets/catmullconv/389a1ea8c9e92df9a2b38718e34bae7b.ascii +++ b/docs/images/snippets/catmullconv/389a1ea8c9e92df9a2b38718e34bae7b.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ P │ - Bézier(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ - └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ - └ -1 3 -3 1 ┘ │ 3 │ - │ P │ - └ 4 ┘ + ┌ P ┐ + │ 1 │ + ┌ 1 0 0 0 ┐ │ P │ +Bézier(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ + └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ + └ -1 3 -3 1 ┘ │ 3 │ + │ P │ + └ 4 ┘ diff --git a/docs/images/snippets/catmullconv/4c8684109149b0dc79f5583a5912fcd9.ascii b/docs/images/snippets/catmullconv/4c8684109149b0dc79f5583a5912fcd9.ascii index e7697682..09b787ee 100644 --- a/docs/images/snippets/catmullconv/4c8684109149b0dc79f5583a5912fcd9.ascii +++ b/docs/images/snippets/catmullconv/4c8684109149b0dc79f5583a5912fcd9.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ ┌ P ┐ - │ -1 1 │ │ 1 │ - ┌ 1 0 0 0 ┐ │ ── 1 ── 0 │ │ P │ - = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 6τ 6τ │ · │ 2 │ - └ 1 t t t ┘ │ 3 -6 3 0 │ │ 1 -1 │ │ P │ - └ -1 3 -3 1 ┘ │ 0 ── 1 ── │ │ 3 │ - │ 6τ 6τ │ │ P │ - └ 0 0 1 0 ┘ └ 4 ┘ + ┌ 0 1 0 0 ┐ ┌ P ┐ + │ -1 1 │ │ 1 │ + ┌ 1 0 0 0 ┐ │ ── 1 ── 0 │ │ P │ += ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 6τ 6τ │ · │ 2 │ + └ 1 t t t ┘ │ 3 -6 3 0 │ │ 1 -1 │ │ P │ + └ -1 3 -3 1 ┘ │ 0 ── 1 ── │ │ 3 │ + │ 6τ 6τ │ │ P │ + └ 0 0 1 0 ┘ └ 4 ┘ diff --git a/docs/images/snippets/catmullconv/4e0da16710a7339f04dd844c7705423e.ascii b/docs/images/snippets/catmullconv/4e0da16710a7339f04dd844c7705423e.ascii index 94fc8cad..2f3ab124 100644 --- a/docs/images/snippets/catmullconv/4e0da16710a7339f04dd844c7705423e.ascii +++ b/docs/images/snippets/catmullconv/4e0da16710a7339f04dd844c7705423e.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ - │ ── 0 ── 0 │ - ┌ 1 0 0 0 ┐ -1 │ 2τ 2τ │ - │ -3 3 0 0 │ · │ 1 1 1 -1 │ = V - │ 3 -6 3 0 │ │ ─ ── - 3 3 - ─ ── │ - └ -1 3 -3 1 ┘ │ τ 2t t 2t │ - │ -1 1 1 1 │ - │ ── 2 - ── ── - 2 ── │ - └ 2t 2τ 2τ 2t ┘ + ┌ 0 1 0 0 ┐ + │ -1 1 │ + │ ── 0 ── 0 │ +┌ 1 0 0 0 ┐ -1 │ 2τ 2τ │ +│ -3 3 0 0 │ · │ 1 1 1 -1 │ = V +│ 3 -6 3 0 │ │ ─ ── - 3 3 - ─ ── │ +└ -1 3 -3 1 ┘ │ τ 2t t 2t │ + │ -1 1 1 1 │ + │ ── 2 - ── ── - 2 ── │ + └ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/53f216327c0bbcf02b2a331fbf44d389.ascii b/docs/images/snippets/catmullconv/53f216327c0bbcf02b2a331fbf44d389.ascii index 3919d9f2..8c3ed285 100644 --- a/docs/images/snippets/catmullconv/53f216327c0bbcf02b2a331fbf44d389.ascii +++ b/docs/images/snippets/catmullconv/53f216327c0bbcf02b2a331fbf44d389.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ ┌ P ┐ - │ ── 0 ── 0 │ │ 1 │ - │ 2τ 2τ │ │ P │ - = ┌ 2 3 ┐ · │ 1 1 1 -1 │ · │ 2 │ - └ 1 t t t ┘ │ ─ ── - 3 3 - ─ ── │ │ P │ - │ τ 2t t 2t │ │ 3 │ - │ -1 1 1 1 │ │ P │ - │ ── 2 - ── ── - 2 ── │ └ 4 ┘ - └ 2t 2τ 2τ 2t ┘ + ┌ 0 1 0 0 ┐ + │ -1 1 │ ┌ P ┐ + │ ── 0 ── 0 │ │ 1 │ + │ 2τ 2τ │ │ P │ += ┌ 2 3 ┐ · │ 1 1 1 -1 │ · │ 2 │ + └ 1 t t t ┘ │ ─ ── - 3 3 - ─ ── │ │ P │ + │ τ 2t t 2t │ │ 3 │ + │ -1 1 1 1 │ │ P │ + │ ── 2 - ── ── - 2 ── │ └ 4 ┘ + └ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/574bed6665be06b309b8da722c616a41.ascii b/docs/images/snippets/catmullconv/574bed6665be06b309b8da722c616a41.ascii index cb112bdd..2c760b2a 100644 --- a/docs/images/snippets/catmullconv/574bed6665be06b309b8da722c616a41.ascii +++ b/docs/images/snippets/catmullconv/574bed6665be06b309b8da722c616a41.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ ┌ P ┐ - │ ── 0 ── 0 │ │ 1 │ - │ 2τ 2τ │ │ P │ - │ 1 1 1 -1 │ · │ 2 │ - │ ─ ── - 3 3 - ─ ── │ │ P │ - │ τ 2t t 2t │ │ 3 │ - │ -1 1 1 1 │ │ P │ - │ ── 2 - ── ── - 2 ── │ └ 4 ┘ - └ 2t 2τ 2τ 2t ┘ +┌ 0 1 0 0 ┐ +│ -1 1 │ ┌ P ┐ +│ ── 0 ── 0 │ │ 1 │ +│ 2τ 2τ │ │ P │ +│ 1 1 1 -1 │ · │ 2 │ +│ ─ ── - 3 3 - ─ ── │ │ P │ +│ τ 2t t 2t │ │ 3 │ +│ -1 1 1 1 │ │ P │ +│ ── 2 - ── ── - 2 ── │ └ 4 ┘ +└ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/639ca0b74a805c3aebac79b181eac908.ascii b/docs/images/snippets/catmullconv/639ca0b74a805c3aebac79b181eac908.ascii index 8a301512..8fc996ef 100644 --- a/docs/images/snippets/catmullconv/639ca0b74a805c3aebac79b181eac908.ascii +++ b/docs/images/snippets/catmullconv/639ca0b74a805c3aebac79b181eac908.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ V = P ┐ - │ 1 2 │ - ┌ P ┐ │ V = P │ - │ 1 │ │ 2 3 │ - │ P │ │ P - P │ - │ 2 │ = │ 3 1 │ - │ P │points │ V' = ─────── │ point-tangent - │ 3 │ │ 1 2 │ - │ P │ │ P - P │ - └ 4 ┘ │ 4 2 │ - │ V' = ─────── │ - └ 2 2 ┘ + ┌ V = P ┐ + │ 1 2 │ +┌ P ┐ │ V = P │ +│ 1 │ │ 2 3 │ +│ P │ │ P - P │ +│ 2 │ = │ 3 1 │ +│ P │points │ V' = ─────── │ point-tangent +│ 3 │ │ 1 2 │ +│ P │ │ P - P │ +└ 4 ┘ │ 4 2 │ + │ V' = ─────── │ + └ 2 2 ┘ diff --git a/docs/images/snippets/catmullconv/65e589eafae8ff2f39392d8143d2845c.ascii b/docs/images/snippets/catmullconv/65e589eafae8ff2f39392d8143d2845c.ascii index da62f3a7..961db63f 100644 --- a/docs/images/snippets/catmullconv/65e589eafae8ff2f39392d8143d2845c.ascii +++ b/docs/images/snippets/catmullconv/65e589eafae8ff2f39392d8143d2845c.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ - │ ── 0 ── 0 │ - │ 2τ 2τ │ ┌ 1 0 0 0 ┐ - │ 1 1 1 -1 │ = │ -3 3 0 0 │ · V - │ ─ ── - 3 3 - ─ ── │ │ 3 -6 3 0 │ - │ τ 2t t 2t │ └ -1 3 -3 1 ┘ - │ -1 1 1 1 │ - │ ── 2 - ── ── - 2 ── │ - └ 2t 2τ 2τ 2t ┘ +┌ 0 1 0 0 ┐ +│ -1 1 │ +│ ── 0 ── 0 │ +│ 2τ 2τ │ ┌ 1 0 0 0 ┐ +│ 1 1 1 -1 │ = │ -3 3 0 0 │ · V +│ ─ ── - 3 3 - ─ ── │ │ 3 -6 3 0 │ +│ τ 2t t 2t │ └ -1 3 -3 1 ┘ +│ -1 1 1 1 │ +│ ── 2 - ── ── - 2 ── │ +└ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/7bab9dd3da654b05fa065076894e2d82.ascii b/docs/images/snippets/catmullconv/7bab9dd3da654b05fa065076894e2d82.ascii index b35d0216..d6adc2ef 100644 --- a/docs/images/snippets/catmullconv/7bab9dd3da654b05fa065076894e2d82.ascii +++ b/docs/images/snippets/catmullconv/7bab9dd3da654b05fa065076894e2d82.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ 0 0 1 0 │ - │ -1 1 │ - T = │ ── 0 ─ 0 │ - │ 2 2 │ - │ -1 1 │ - │ 0 ── 0 ─ │ - └ 2 2 ┘ + ┌ 0 1 0 0 ┐ + │ 0 0 1 0 │ + │ -1 1 │ +T = │ ── 0 ─ 0 │ + │ 2 2 │ + │ -1 1 │ + │ 0 ── 0 ─ │ + └ 2 2 ┘ diff --git a/docs/images/snippets/catmullconv/8a2a00812363fe1a6cfa7f81b48d31d1.ascii b/docs/images/snippets/catmullconv/8a2a00812363fe1a6cfa7f81b48d31d1.ascii deleted file mode 100644 index e4c8099d..00000000 --- a/docs/images/snippets/catmullconv/8a2a00812363fe1a6cfa7f81b48d31d1.ascii +++ /dev/null @@ -1,12 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ 0 1 0 0 ┐ - │ -1 1 │ ┌ P ┐ - │ ── 0 ── 0 │ │ 1 │ - │ 2τ 2τ │ │ P │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 1 1 1 -1 │ · │ 2 │ - └ 1 t t t ┘ │ ─ ── - 3 3 - ─ ── │ │ P │ - │ τ 2t t 2t │ │ 3 │ - │ -1 1 1 1 │ │ P │ - │ ── 2 - ── ── - 2 ── │ └ 4 ┘ - └ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/8d3a5ca7188f53b914229133b3dbe5fe.ascii b/docs/images/snippets/catmullconv/8d3a5ca7188f53b914229133b3dbe5fe.ascii deleted file mode 100644 index 11de3347..00000000 --- a/docs/images/snippets/catmullconv/8d3a5ca7188f53b914229133b3dbe5fe.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ V ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ V │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ V' │ - └ 2 -2 1 1 ┘ │ 1 │ - │ V' │ - └ 2 ┘ diff --git a/docs/images/snippets/catmullconv/8de53f207d68b25854a5f0b924ac6010.ascii b/docs/images/snippets/catmullconv/8de53f207d68b25854a5f0b924ac6010.ascii index 4a82242c..5502bbdc 100644 --- a/docs/images/snippets/catmullconv/8de53f207d68b25854a5f0b924ac6010.ascii +++ b/docs/images/snippets/catmullconv/8de53f207d68b25854a5f0b924ac6010.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 2 │ - ┌ V ┐ │ P │ ┌ 0 1 0 0 ┐ - │ 1 │ │ 3 │ │ 0 0 1 0 │ - │ V │ │ P - P │ │ -1 1 │ - │ 2 │ = │ 3 1 │ ,\ T = │ ── 0 ── 0 │ - │ V' │ │ ─────── │ │ 2τ 2τ │ - │ 1 │ │ 2τ │ │ -1 1 │ - │ V' │ │ P - P │ │ 0 ── 0 ── │ - └ 2 ┘ │ 4 2 │ └ 2τ 2τ ┘ - │ ─────── │ - └ 2τ ┘ + ┌ P ┐ + │ 2 │ +┌ V ┐ │ P │ ┌ 0 1 0 0 ┐ +│ 1 │ │ 3 │ │ 0 0 1 0 │ +│ V │ │ P - P │ │ -1 1 │ +│ 2 │ = │ 3 1 │ ,\ T = │ ── 0 ── 0 │ +│ V' │ │ ─────── │ │ 2τ 2τ │ +│ 1 │ │ 2τ │ │ -1 1 │ +│ V' │ │ P - P │ │ 0 ── 0 ── │ +└ 2 ┘ │ 4 2 │ └ 2τ 2τ ┘ + │ ─────── │ + └ 2τ ┘ diff --git a/docs/images/snippets/catmullconv/902c290a790b4d44d10236f4a1456cdc.ascii b/docs/images/snippets/catmullconv/902c290a790b4d44d10236f4a1456cdc.ascii index 83393491..bd0157b5 100644 --- a/docs/images/snippets/catmullconv/902c290a790b4d44d10236f4a1456cdc.ascii +++ b/docs/images/snippets/catmullconv/902c290a790b4d44d10236f4a1456cdc.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ V ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ V │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ V' │ - └ 2 -2 1 1 ┘ │ 1 │ - │ V' │ - └ 2 ┘ + ┌ V ┐ + │ 1 │ + ┌ 1 0 0 0 ┐ │ V │ +CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 2 │ + └ 1 t t t ┘ │ -3 3 -2 -1 │ │ V' │ + └ 2 -2 1 1 ┘ │ 1 │ + │ V' │ + └ 2 ┘ diff --git a/docs/images/snippets/catmullconv/917b176a45959b026c56f81999505dc7.ascii b/docs/images/snippets/catmullconv/917b176a45959b026c56f81999505dc7.ascii index d309e7a2..a540ac1e 100644 --- a/docs/images/snippets/catmullconv/917b176a45959b026c56f81999505dc7.ascii +++ b/docs/images/snippets/catmullconv/917b176a45959b026c56f81999505dc7.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ - │ ── 0 ── 0 │ - ┌ 1 0 0 0 ┐ -1 │ 2τ 2τ │ ┌ 1 0 0 0 ┐ -1 ┌ 1 0 0 0 ┐ - │ -3 3 0 0 │ · │ 1 1 1 -1 │ = │ -3 3 0 0 │ · │ -3 3 0 0 │ · V - │ 3 -6 3 0 │ │ ─ ── - 3 3 - ─ ── │ │ 3 -6 3 0 │ │ 3 -6 3 0 │ - └ -1 3 -3 1 ┘ │ τ 2t t 2t │ └ -1 3 -3 1 ┘ └ -1 3 -3 1 ┘ - │ -1 1 1 1 │ - │ ── 2 - ── ── - 2 ── │ - └ 2t 2τ 2τ 2t ┘ + ┌ 0 1 0 0 ┐ + │ -1 1 │ + │ ── 0 ── 0 │ +┌ 1 0 0 0 ┐ -1 │ 2τ 2τ │ ┌ 1 0 0 0 ┐ -1 ┌ 1 0 0 0 ┐ +│ -3 3 0 0 │ · │ 1 1 1 -1 │ = │ -3 3 0 0 │ · │ -3 3 0 0 │ · V +│ 3 -6 3 0 │ │ ─ ── - 3 3 - ─ ── │ │ 3 -6 3 0 │ │ 3 -6 3 0 │ +└ -1 3 -3 1 ┘ │ τ 2t t 2t │ └ -1 3 -3 1 ┘ └ -1 3 -3 1 ┘ + │ -1 1 1 1 │ + │ ── 2 - ── ── - 2 ── │ + └ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/9593c057c84ebf9beb70fd57a11c7e12.ascii b/docs/images/snippets/catmullconv/9593c057c84ebf9beb70fd57a11c7e12.ascii deleted file mode 100644 index c9d316c4..00000000 --- a/docs/images/snippets/catmullconv/9593c057c84ebf9beb70fd57a11c7e12.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ ┌ P + 6(P - P ) ┐ - │ 1 │ │ 4 1 2 │ - │ P │ │ P │ - │ 2 │ \Rightarrow │ 1 │ - │ P │ Bézier │ P │CatmullRom - │ 3 │ │ 4 │ - │ P │ │ P + 6(P - P ) │ - └ 4 ┘ └ 1 4 3 ┘ diff --git a/docs/images/snippets/catmullconv/98ddf6415bd9827a6d899b21d0a5f736.ascii b/docs/images/snippets/catmullconv/98ddf6415bd9827a6d899b21d0a5f736.ascii index 83393491..bd0157b5 100644 --- a/docs/images/snippets/catmullconv/98ddf6415bd9827a6d899b21d0a5f736.ascii +++ b/docs/images/snippets/catmullconv/98ddf6415bd9827a6d899b21d0a5f736.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ V ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ V │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ V' │ - └ 2 -2 1 1 ┘ │ 1 │ - │ V' │ - └ 2 ┘ + ┌ V ┐ + │ 1 │ + ┌ 1 0 0 0 ┐ │ V │ +CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ 2 │ + └ 1 t t t ┘ │ -3 3 -2 -1 │ │ V' │ + └ 2 -2 1 1 ┘ │ 1 │ + │ V' │ + └ 2 ┘ diff --git a/docs/images/snippets/catmullconv/9ae99b090883023a485be7be098858e9.ascii b/docs/images/snippets/catmullconv/9ae99b090883023a485be7be098858e9.ascii index 1d57baff..d4fe2786 100644 --- a/docs/images/snippets/catmullconv/9ae99b090883023a485be7be098858e9.ascii +++ b/docs/images/snippets/catmullconv/9ae99b090883023a485be7be098858e9.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ ┌ P ┐ - │ 1 │ │ 1 │ - │ P │ │ P │ - │ 2 │ \Rightarrow │ 4 │ - │ P │ Bézier │ P + 3(P - P ) │ CatmullRom - │ 3 │ │ 4 1 2 │ - │ P │ │ P + 3(P - P ) │ - └ 4 ┘ └ 1 4 3 ┘ +┌ P ┐ ┌ P ┐ +│ 1 │ │ 1 │ +│ P │ │ P │ +│ 2 │ ==> │ 4 │ +│ P │ Bézier │ P + 3(P - P ) │ CatmullRom +│ 3 │ │ 4 1 2 │ +│ P │ │ P + 3(P - P ) │ +└ 4 ┘ └ 1 4 3 ┘ diff --git a/docs/images/snippets/catmullconv/a323848e706c473833cda0b02bc220ef.ascii b/docs/images/snippets/catmullconv/a323848e706c473833cda0b02bc220ef.ascii index 68fcdb6a..10bcbc46 100644 --- a/docs/images/snippets/catmullconv/a323848e706c473833cda0b02bc220ef.ascii +++ b/docs/images/snippets/catmullconv/a323848e706c473833cda0b02bc220ef.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 2 │ - ┌ P ┐ │ P -P │ - │ 1 │ │ 3 1 │ - │ P │ │ P + ────── │ - │ 2 │ \Rightarrow │ 2 6 · τ │ - │ P │ CatmullRom │ P -P │ Bézier - │ 3 │ │ 4 2 │ - │ P │ │ P - ────── │ - └ 4 ┘ │ 3 6 · τ │ - │ P │ - └ 3 ┘ + ┌ P ┐ + │ 2 │ +┌ P ┐ │ P -P │ +│ 1 │ │ 3 1 │ +│ P │ │ P + ────── │ +│ 2 │ ==> │ 2 6 · τ │ +│ P │ CatmullRom │ P -P │ Bézier +│ 3 │ │ 4 2 │ +│ P │ │ P - ────── │ +└ 4 ┘ │ 3 6 · τ │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/catmullconv/a55773fdcdfd99947acc4f86ad2d4a3d.ascii b/docs/images/snippets/catmullconv/a55773fdcdfd99947acc4f86ad2d4a3d.ascii index 854a3b1d..c152d703 100644 --- a/docs/images/snippets/catmullconv/a55773fdcdfd99947acc4f86ad2d4a3d.ascii +++ b/docs/images/snippets/catmullconv/a55773fdcdfd99947acc4f86ad2d4a3d.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 2 │ - ┌ P ┐ │ P │ ┌ 0 · P1 + 1 · P2 + 0 · P3 + 0 · P4 ┐ ┌ 0 1 0 0 ┐ ┌ P ┐ - │ 1 │ │ 3 │ │ 0 · P1 + 0 · P2 + 1 · P3 + 0 · P4 │ │ 0 0 1 0 │ │ 1 │ - │ P │ │ P - P │ │ -1 1 │ │ -1 1 │ │ P │ - T · │ 2 │ = │ 3 1 │ = │ ── · P1 + 0 · P2 + ─ · P3 + 0 · P4 │ = │ ── 0 ─ 0 │ · │ 2 │ - │ P │ │ ─────── │ │ 2 2 │ │ 2 2 │ │ P │ - │ 3 │ │ 2 │ │ -1 1 │ │ -1 1 │ │ 3 │ - │ P │ │ P - P │ │ 0 · P1 ── · P2 + 0 · P3 + ─ · P4 │ │ 0 ── 0 ─ │ │ P │ - └ 4 ┘ │ 4 2 │ └ 2 2 ┘ └ 2 2 ┘ └ 4 ┘ - │ ─────── │ - └ 2 ┘ + ┌ P ┐ + │ 2 │ + ┌ P ┐ │ P │ ┌ 0 · P1 + 1 · P2 + 0 · P3 + 0 · P4 ┐ ┌ 0 1 0 0 ┐ ┌ P ┐ + │ 1 │ │ 3 │ │ 0 · P1 + 0 · P2 + 1 · P3 + 0 · P4 │ │ 0 0 1 0 │ │ 1 │ + │ P │ │ P - P │ │ -1 1 │ │ -1 1 │ │ P │ +T · │ 2 │ = │ 3 1 │ = │ ── · P1 + 0 · P2 + ─ · P3 + 0 · P4 │ = │ ── 0 ─ 0 │ · │ 2 │ + │ P │ │ ─────── │ │ 2 2 │ │ 2 2 │ │ P │ + │ 3 │ │ 2 │ │ -1 1 │ │ -1 1 │ │ 3 │ + │ P │ │ P - P │ │ 0 · P1 ── · P2 + 0 · P3 + ─ · P4 │ │ 0 ── 0 ─ │ │ P │ + └ 4 ┘ │ 4 2 │ └ 2 2 ┘ └ 2 2 ┘ └ 4 ┘ + │ ─────── │ + └ 2 ┘ diff --git a/docs/images/snippets/catmullconv/a8158b35ec221cccff51a53cdc7f440b.ascii b/docs/images/snippets/catmullconv/a8158b35ec221cccff51a53cdc7f440b.ascii index 68ecf8b9..3578d277 100644 --- a/docs/images/snippets/catmullconv/a8158b35ec221cccff51a53cdc7f440b.ascii +++ b/docs/images/snippets/catmullconv/a8158b35ec221cccff51a53cdc7f440b.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ ┌ P ┐ - │ 0 0 1 0 │ │ 1 │ - ┌ 1 0 0 0 ┐ │ -1 1 │ │ P │ - = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ ── 0 ── 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ 2τ 2τ │ │ P │ - └ 2 -2 1 1 ┘ │ -1 1 │ │ 3 │ - │ 0 ── 0 ── │ │ P │ - └ 2τ 2τ ┘ └ 4 ┘ + ┌ 0 1 0 0 ┐ ┌ P ┐ + │ 0 0 1 0 │ │ 1 │ + ┌ 1 0 0 0 ┐ │ -1 1 │ │ P │ += ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ ── 0 ── 0 │ · │ 2 │ + └ 1 t t t ┘ │ -3 3 -2 -1 │ │ 2τ 2τ │ │ P │ + └ 2 -2 1 1 ┘ │ -1 1 │ │ 3 │ + │ 0 ── 0 ── │ │ P │ + └ 2τ 2τ ┘ └ 4 ┘ diff --git a/docs/images/snippets/catmullconv/b59ff8d654e65df4c874901983208893.ascii b/docs/images/snippets/catmullconv/b59ff8d654e65df4c874901983208893.ascii index a2f1f45e..1271b652 100644 --- a/docs/images/snippets/catmullconv/b59ff8d654e65df4c874901983208893.ascii +++ b/docs/images/snippets/catmullconv/b59ff8d654e65df4c874901983208893.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ ┌ P ┐ ┌ P ┐ - │ ── 0 ── 0 │ │ 1 │ │ 1 │ - │ 2τ 2τ │ │ P │ ┌ 1 0 0 0 ┐ │ P │ - │ 1 1 1 -1 │ · │ 2 │ = │ -3 3 0 0 │ · V · │ 2 │ - │ ─ ── - 3 3 - ─ ── │ │ P │ │ 3 -6 3 0 │ │ P │ - │ τ 2t t 2t │ │ 3 │ └ -1 3 -3 1 ┘ │ 3 │ - │ -1 1 1 1 │ │ P │ │ P │ - │ ── 2 - ── ── - 2 ── │ └ 4 ┘ └ 4 ┘ - └ 2t 2τ 2τ 2t ┘ +┌ 0 1 0 0 ┐ +│ -1 1 │ ┌ P ┐ ┌ P ┐ +│ ── 0 ── 0 │ │ 1 │ │ 1 │ +│ 2τ 2τ │ │ P │ ┌ 1 0 0 0 ┐ │ P │ +│ 1 1 1 -1 │ · │ 2 │ = │ -3 3 0 0 │ · V · │ 2 │ +│ ─ ── - 3 3 - ─ ── │ │ P │ │ 3 -6 3 0 │ │ P │ +│ τ 2t t 2t │ │ 3 │ └ -1 3 -3 1 ┘ │ 3 │ +│ -1 1 1 1 │ │ P │ │ P │ +│ ── 2 - ── ── - 2 ── │ └ 4 ┘ └ 4 ┘ +└ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/b94a4dafc12ba7e4fbf3aff924f55464.ascii b/docs/images/snippets/catmullconv/b94a4dafc12ba7e4fbf3aff924f55464.ascii index f9ff95ba..4ad29332 100644 --- a/docs/images/snippets/catmullconv/b94a4dafc12ba7e4fbf3aff924f55464.ascii +++ b/docs/images/snippets/catmullconv/b94a4dafc12ba7e4fbf3aff924f55464.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 2 │ - ┌ V ┐ ┌ P ┐ │ P │ - │ 1 │ │ 1 │ │ 3 │ - │ V │ │ P │ │ P - P │ - │ 2 │ = T · │ 2 │ = │ 3 1 │ - │ V' │ │ P │ │ ─────── │ - │ 1 │ │ 3 │ │ 2 │ - │ V' │ │ P │ │ P - P │ - └ 2 ┘ └ 4 ┘ │ 4 2 │ - │ ─────── │ - └ 2 ┘ + ┌ P ┐ + │ 2 │ +┌ V ┐ ┌ P ┐ │ P │ +│ 1 │ │ 1 │ │ 3 │ +│ V │ │ P │ │ P - P │ +│ 2 │ = T · │ 2 │ = │ 3 1 │ +│ V' │ │ P │ │ ─────── │ +│ 1 │ │ 3 │ │ 2 │ +│ V' │ │ P │ │ P - P │ +└ 2 ┘ └ 4 ┘ │ 4 2 │ + │ ─────── │ + └ 2 ┘ diff --git a/docs/images/snippets/catmullconv/c0e30b49fbfce6f6b3c81eaa6ca5154f.ascii b/docs/images/snippets/catmullconv/c0e30b49fbfce6f6b3c81eaa6ca5154f.ascii deleted file mode 100644 index f1c2493f..00000000 --- a/docs/images/snippets/catmullconv/c0e30b49fbfce6f6b3c81eaa6ca5154f.ascii +++ /dev/null @@ -1,14 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ - │ 2 │ - ┌ P ┐ │ P -P │ - │ 1 │ │ 3 1 │ - │ P │ │ P + ────── │ - │ 2 │ \Rightarrow │ 2 6 · τ │ - │ P │CatmullRom │ P -P │ Bézier - │ 3 │ │ 4 2 │ - │ P │ │ P - ────── │ - └ 4 ┘ │ 3 6 · τ │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/catmullconv/c1f8861583b4176a9b607aa6a05f9356.ascii b/docs/images/snippets/catmullconv/c1f8861583b4176a9b607aa6a05f9356.ascii deleted file mode 100644 index b1b40ccd..00000000 --- a/docs/images/snippets/catmullconv/c1f8861583b4176a9b607aa6a05f9356.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ P │ - Bézier(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ - └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ - └ -1 3 -3 1 ┘ │ 3 │ - │ P │ - └ 4 ┘ diff --git a/docs/images/snippets/catmullconv/d09e7466c267614c89ead28d6a900ba1.ascii b/docs/images/snippets/catmullconv/d09e7466c267614c89ead28d6a900ba1.ascii deleted file mode 100644 index ed131d68..00000000 --- a/docs/images/snippets/catmullconv/d09e7466c267614c89ead28d6a900ba1.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ ┌ P ┐ - │ 1 │ │ 1 │ - │ P │ │ P │ - │ 2 │ \Rightarrow │ 4 │ - │ P │ Bézier │ P + 3(P - P ) │CatmullRom - │ 3 │ │ 4 1 2 │ - │ P │ │ P + 3(P - P ) │ - └ 4 ┘ └ 1 4 3 ┘ diff --git a/docs/images/snippets/catmullconv/defc6fa4b51fa3c1945d15449f0f392d.ascii b/docs/images/snippets/catmullconv/defc6fa4b51fa3c1945d15449f0f392d.ascii deleted file mode 100644 index 5989deb7..00000000 --- a/docs/images/snippets/catmullconv/defc6fa4b51fa3c1945d15449f0f392d.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ 0 1 0 0 ┐ ┌ P ┐ - │ 0 0 1 0 │ │ 1 │ - ┌ 1 0 0 0 ┐ │ -1 1 │ │ P │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 0 0 1 0 │ · │ ── 0 ── 0 │ · │ 2 │ - └ 1 t t t ┘ │ -3 3 -2 -1 │ │ 2τ 2τ │ │ P │ - └ 2 -2 1 1 ┘ │ -1 1 │ │ 3 │ - │ 0 ── 0 ── │ │ P │ - └ 2τ 2τ ┘ └ 4 ┘ diff --git a/docs/images/snippets/catmullconv/e653724c11600cbf682f1c809c8c6508.ascii b/docs/images/snippets/catmullconv/e653724c11600cbf682f1c809c8c6508.ascii index e4c8099d..c6b1e685 100644 --- a/docs/images/snippets/catmullconv/e653724c11600cbf682f1c809c8c6508.ascii +++ b/docs/images/snippets/catmullconv/e653724c11600cbf682f1c809c8c6508.ascii @@ -1,12 +1,11 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 0 0 ┐ - │ -1 1 │ ┌ P ┐ - │ ── 0 ── 0 │ │ 1 │ - │ 2τ 2τ │ │ P │ - CatmullRom(t) = ┌ 2 3 ┐ · │ 1 1 1 -1 │ · │ 2 │ - └ 1 t t t ┘ │ ─ ── - 3 3 - ─ ── │ │ P │ - │ τ 2t t 2t │ │ 3 │ - │ -1 1 1 1 │ │ P │ - │ ── 2 - ── ── - 2 ── │ └ 4 ┘ - └ 2t 2τ 2τ 2t ┘ + ┌ 0 1 0 0 ┐ + │ -1 1 │ ┌ P ┐ + │ ── 0 ── 0 │ │ 1 │ + │ 2τ 2τ │ │ P │ +CatmullRom(t) = ┌ 2 3 ┐ · │ 1 1 1 -1 │ · │ 2 │ + └ 1 t t t ┘ │ ─ ── - 3 3 - ─ ── │ │ P │ + │ τ 2t t 2t │ │ 3 │ + │ -1 1 1 1 │ │ P │ + │ ── 2 - ── ── - 2 ── │ └ 4 ┘ + └ 2t 2τ 2τ 2t ┘ diff --git a/docs/images/snippets/catmullconv/fe2e6fd487df224b2f55a601898ce333.ascii b/docs/images/snippets/catmullconv/fe2e6fd487df224b2f55a601898ce333.ascii index 0594ba9a..cc9a1dc8 100644 --- a/docs/images/snippets/catmullconv/fe2e6fd487df224b2f55a601898ce333.ascii +++ b/docs/images/snippets/catmullconv/fe2e6fd487df224b2f55a601898ce333.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ P │ - │ -3 3 0 0 │ · │ 2 │ - │ 3 -6 3 0 │ │ P │ - └ -1 3 -3 1 ┘ │ 3 │ - │ P │ - └ 4 ┘ + ┌ P ┐ + │ 1 │ +┌ 1 0 0 0 ┐ │ P │ +│ -3 3 0 0 │ · │ 2 │ +│ 3 -6 3 0 │ │ P │ +└ -1 3 -3 1 ┘ │ 3 │ + │ P │ + └ 4 ┘ diff --git a/docs/images/snippets/circleintersection/2f42c862a0a9d0764727d42b16cf68a0.ascii b/docs/images/snippets/circleintersection/2f42c862a0a9d0764727d42b16cf68a0.ascii index 6cb05741..b821bc58 100644 --- a/docs/images/snippets/circleintersection/2f42c862a0a9d0764727d42b16cf68a0.ascii +++ b/docs/images/snippets/circleintersection/2f42c862a0a9d0764727d42b16cf68a0.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - r= dist(B(t), c) - ┌─────────────────────────┐ - │ 2 2 - = │(B t - c ) + (B t - c ) - ⟍│ x x y y - ┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐ - │╭ 3 2 2 3 ╮2 ╭ 3 2 2 3 ╮2 - = ││ x (1-t) + 3 x (1-t) t + 2 x (1-t) t + x t - c │ + │ y (1-t) + 3 y (1-t) t + 2 y (1-t) t + y t - c │ - ⟍│╰ 1 2 3 4 x ╯ ╰ 1 2 3 4 y ╯ +r= dist(B(t), c) + ┌─────────────────────────┐ + │ 2 2 + = │(B t - c ) + (B t - c ) + ⟍│ x x y y + ┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐ + │╭ 3 2 2 3 ╮2 ╭ 3 2 2 3 ╮2 + = ││ x (1-t) + 3 x (1-t) t + 2 x (1-t) t + x t - c │ + │ y (1-t) + 3 y (1-t) t + 2 y (1-t) t + y t - c │ + ⟍│╰ 1 2 3 4 x ╯ ╰ 1 2 3 4 y ╯ diff --git a/docs/images/snippets/circleintersection/373248ec6a579bacf6c6a317e6db597a.ascii b/docs/images/snippets/circleintersection/373248ec6a579bacf6c6a317e6db597a.ascii index b7afcee9..bc856249 100644 --- a/docs/images/snippets/circleintersection/373248ec6a579bacf6c6a317e6db597a.ascii +++ b/docs/images/snippets/circleintersection/373248ec6a579bacf6c6a317e6db597a.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - dist(B(t), c) = r +dist(B(t), c) = r diff --git a/docs/images/snippets/circleintersection/3e0594855ca99fb87dcc65a693e1ad22.ascii b/docs/images/snippets/circleintersection/3e0594855ca99fb87dcc65a693e1ad22.ascii deleted file mode 100644 index 7555f8c8..00000000 --- a/docs/images/snippets/circleintersection/3e0594855ca99fb87dcc65a693e1ad22.ascii +++ /dev/null @@ -1,11 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - r= dist(B(t), c) - ┌─────────────────────────┐ - │ 2 2 - = │(B t - c ) + (B t - c ) - ⟍│ x x y y - ┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐ - │╭ 3 2 2 3 ╮2 ╭ 3 2 2 3 ╮2 - = ││ x (1-t) + 3 x (1-t) t + 2 x (1-t) t + x t - c │ + │ y (1-t) + 3 y (1-t) t + 2 y (1-t) t + y t - c │ - ⟍│╰ 1 2 3 4 x ╯ ╰ 1 2 3 4 y ╯ diff --git a/docs/images/snippets/circleintersection/674c42035da16a426ef7fe23277eea11.ascii b/docs/images/snippets/circleintersection/674c42035da16a426ef7fe23277eea11.ascii deleted file mode 100644 index f12763c4..00000000 --- a/docs/images/snippets/circleintersection/674c42035da16a426ef7fe23277eea11.ascii +++ /dev/null @@ -1,3 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - dist(B(t), c) = r diff --git a/docs/images/snippets/circleintersection/699e6ea5bffbe8fe377af21a2bd28532.ascii b/docs/images/snippets/circleintersection/699e6ea5bffbe8fe377af21a2bd28532.ascii deleted file mode 100644 index c66f9b8c..00000000 --- a/docs/images/snippets/circleintersection/699e6ea5bffbe8fe377af21a2bd28532.ascii +++ /dev/null @@ -1,11 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - r = dist(B(t), c) - ┌─────────────────────────┐ - │ 2 2 - = │(B t - c ) + (B t - c ) - ⟍│ x x y y - ┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐ - │╭ 3 2 2 3 ╮2 ╭ 3 2 2 3 ╮2 - = ││ x (1-t) + 3 x (1-t) t + 2 x (1-t) t + x t - c │ + │ y (1-t) + 3 y (1-t) t + 2 y (1-t) t + y t - c │ - ⟍│╰ 1 2 3 4 x ╯ ╰ 1 2 3 4 y ╯ diff --git a/docs/images/snippets/circleintersection/9271faa4684904073127482aa91e3bce.ascii b/docs/images/snippets/circleintersection/9271faa4684904073127482aa91e3bce.ascii deleted file mode 100644 index 79e29766..00000000 --- a/docs/images/snippets/circleintersection/9271faa4684904073127482aa91e3bce.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌─────────────────────────┐ - │ 2 2 - dist(B(t), c = r │(B t - c ) + (B t - c ) = r - ⟍│ x x y y diff --git a/docs/images/snippets/circleintersection/b480dae093b572dc707b76eef7fbca04.ascii b/docs/images/snippets/circleintersection/b480dae093b572dc707b76eef7fbca04.ascii deleted file mode 100644 index ab638449..00000000 --- a/docs/images/snippets/circleintersection/b480dae093b572dc707b76eef7fbca04.ascii +++ /dev/null @@ -1,11 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - r = dist(B(t), c = r - ┌─────────────────────────┐ - │ 2 2 - = │(B t - c ) + (B t - c ) - ⟍│ x x y y - ┌─────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────┐ - │╭ 3 2 2 3 ╮2 ╭ 3 2 2 3 ╮2 - = ││ x (1-t) + 3 x (1-t) t + 2 x (1-t) t + x t - c │ + │ y (1-t) + 3 y (1-t) t + 2 y (1-t) t + y t - c │ - ⟍│╰ 1 2 3 4 x ╯ ╰ 1 2 3 4 y ╯ diff --git a/docs/images/snippets/circles/06369b00338310df0a810c592485aa0a.ascii b/docs/images/snippets/circles/06369b00338310df0a810c592485aa0a.ascii index c333eaa0..5d66d731 100644 --- a/docs/images/snippets/circles/06369b00338310df0a810c592485aa0a.ascii +++ b/docs/images/snippets/circles/06369b00338310df0a810c592485aa0a.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - cos(φ)-1 - b = ──────── - sin(φ) + cos(φ)-1 +b = ──────── + sin(φ) diff --git a/docs/images/snippets/circles/1829e42ea956ee4df0e45d9ac5334ef7.ascii b/docs/images/snippets/circles/1829e42ea956ee4df0e45d9ac5334ef7.ascii index 2a326654..2b5c964a 100644 --- a/docs/images/snippets/circles/1829e42ea956ee4df0e45d9ac5334ef7.ascii +++ b/docs/images/snippets/circles/1829e42ea956ee4df0e45d9ac5334ef7.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 1 = cos(φ) + b · -sin(φ) → \ 1 - cos(φ) = -b · sin(φ) → \ -1 + cos(φ) = b · sin(φ) +1 = cos(φ) + b · -sin(φ) → \ 1 - cos(φ) = -b · sin(φ) → \ -1 + cos(φ) = b · sin(φ) diff --git a/docs/images/snippets/circles/222d374252584ac37d967e3ea0f8f28b.ascii b/docs/images/snippets/circles/222d374252584ac37d967e3ea0f8f28b.ascii deleted file mode 100644 index dc81ed32..00000000 --- a/docs/images/snippets/circles/222d374252584ac37d967e3ea0f8f28b.ascii +++ /dev/null @@ -1,34 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - d(θ)= |B - P | - t θ - ─ - 2 - ╭ 3 1 ╮ ╭ ╭ θ ╮ ╮ - │ ─ + ─cos(θ) │ │ cos│ ─ │ │ - = |│ 4 4 │ - │ ╰ 2 ╯ │| - │ 1 ╭ θ ╮ 1 │ │ ╭ θ ╮ │ - │ ─tan│ ─ │ + ─sin(θ) │ │ sin│ ─ │ │ - ╰ 2 ╰ 2 ╯ 4 ╯ ╰ ╰ 2 ╯ ╯ - ╭ 3 1 ╭ θ ╮ ╮ - │ ─ + ─cos(θ) - cos│ ─ │ │ - = |│ 4 4 ╰ 2 ╯ │| - │ 1 ╭ θ ╮ 1 ╭ θ ╮ │ - │ ─tan│ ─ │ + ─sin(θ) - sin│ ─ │ │ - ╰ 2 ╰ 2 ╯ 4 ╰ 2 ╯ ╯ - ╭ 4╭ θ ╮ ╮ - │ 2sin │ ─ │ │ - = |│ ╰ 4 ╯ │| - │ 1 ╭ θ ╮ 1 ╭ θ ╮ │ - │ ─tan│ ─ │ + ─sin(θ) - sin│ ─ │ │ - ╰ 2 ╰ 2 ╯ 4 ╰ 2 ╯ ╯ - ┌─────────────────────────────────────────────────────┐ - │╭ 4╭ θ ╮ ╮2 ╭ 1 ╭ θ ╮ 1 ╭ θ ╮ ╮2 - = ││ 2sin │ ─ │ │ + │ ─tan│ ─ │ + ─sin(θ) - sin│ ─ │ │ - ⟍│╰ ╰ 4 ╯ ╯ ╰ 2 ╰ 2 ╯ 4 ╰ 2 ╯ ╯ - - = we give this to Wolfram Alpha because why would we simplify this by hand? - - 4╭ θ ╮ ╭ θ ╮ - = 2sin │ ─ │ · |sec│ ─ │| - ╰ 4 ╯ ╰ 2 ╯ diff --git a/docs/images/snippets/circles/23d4cd81c759c5374773fae149207c5b.ascii b/docs/images/snippets/circles/23d4cd81c759c5374773fae149207c5b.ascii deleted file mode 100644 index b1c3f3a4..00000000 --- a/docs/images/snippets/circles/23d4cd81c759c5374773fae149207c5b.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 1 2 cos(θ) ╮ ╭ 3 1 ╮ - │ ─ + ─ + ────── │ │ ─ + ─cos(θ) │ - B = │ 4 4 4 │ = │ 4 4 │ - 1 │ 0 2k sin(θ) │ │ 1 ╭ θ ╮ 1 │ - t=─ │ ─ + ── + ────── │ │ ─tan│ ─ │ + ─sin(θ) │ - 2 ╰ 4 4 4 ╯ ╰ 2 ╰ 2 ╯ 4 ╯ diff --git a/docs/images/snippets/circles/474c72b5d1ad5c154d79312c15aee47a.ascii b/docs/images/snippets/circles/474c72b5d1ad5c154d79312c15aee47a.ascii deleted file mode 100644 index 0b51cdd6..00000000 --- a/docs/images/snippets/circles/474c72b5d1ad5c154d79312c15aee47a.ascii +++ /dev/null @@ -1,17 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 4╭ θ ╮ ╭ θ ╮ - = 2sin │ ─ │ · sec│ ─ │ - ╰ 4 ╯ ╰ 2 ╯ - 4╭ θ ╮ 1 - = 2sin │ ─ │ · ──────── - ╰ 4 ╯ ╭ θ ╮ - cos│ ─ │ - ╰ 2 ╯ - 4╭ θ ╮ - sin │ ─ │ - ╰ 4 ╯ - d(θ)= 2───────── - ╭ θ ╮ - cos│ ─ │ - ╰ 2 ╯ diff --git a/docs/images/snippets/circles/485d678b1d63ed7a4c7c8bb282467f02.ascii b/docs/images/snippets/circles/485d678b1d63ed7a4c7c8bb282467f02.ascii deleted file mode 100644 index 679bc7c2..00000000 --- a/docs/images/snippets/circles/485d678b1d63ed7a4c7c8bb282467f02.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - P = S + k · (0, 1) = (1,0) + (0,k) - 2 relative to start - P = E + k · (E , -E ) = (cos(θ), sin(θ)) + k · (sin(θ), -cos(θ)) - 2 relative to end y x diff --git a/docs/images/snippets/circles/5a23f3bc298c85540c6dd18e304d9224.ascii b/docs/images/snippets/circles/5a23f3bc298c85540c6dd18e304d9224.ascii index 26ae6031..a5a1f67f 100644 --- a/docs/images/snippets/circles/5a23f3bc298c85540c6dd18e304d9224.ascii +++ b/docs/images/snippets/circles/5a23f3bc298c85540c6dd18e304d9224.ascii @@ -1,17 +1,16 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - a= sin(φ) + b · cos(φ) - -1 + cos(φ) - ..= sin(φ) + ─────────── · cos(φ) - sin(φ) - 2 - -cos(φ) + cos (φ) - ..= sin(φ) + ───────────────── - sin(φ) - 2 2 - sin (φ) + cos (φ) - cos(φ) - ..= ────────────────────────── - sin(φ) - 1 - cos(φ) - a= ────────── - sin(φ) + a= sin(φ) + b · cos(φ) + -1 + cos(φ) +..= sin(φ) + ─────────── · cos(φ) + sin(φ) + 2 + -cos(φ) + cos (φ) +..= sin(φ) + ───────────────── + sin(φ) + 2 2 + sin (φ) + cos (φ) - cos(φ) +..= ────────────────────────── + sin(φ) + 1 - cos(φ) + a= ────────── + sin(φ) diff --git a/docs/images/snippets/circles/6e455a6733dbca47acc0a36e7e490ba6.ascii b/docs/images/snippets/circles/6e455a6733dbca47acc0a36e7e490ba6.ascii deleted file mode 100644 index a259346a..00000000 --- a/docs/images/snippets/circles/6e455a6733dbca47acc0a36e7e490ba6.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ ┌────────────────┐ ╮ - │ │ ┌─────┐ │ - │ │ │ 2 │ - -1 │ ⟍│2 + ε - ⟍│2ε+ε │ - θ = f(ε) = 4 · cos │ ─────────────────── │ - │ ┌─┐ │ - ╰ ⟍│2 ╯ diff --git a/docs/images/snippets/circles/7ab3da0922477af4cc09f5852100976b.ascii b/docs/images/snippets/circles/7ab3da0922477af4cc09f5852100976b.ascii index 46a53bf1..b759ecee 100644 --- a/docs/images/snippets/circles/7ab3da0922477af4cc09f5852100976b.ascii +++ b/docs/images/snippets/circles/7ab3da0922477af4cc09f5852100976b.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - S = \begin{pmatrix} 1 - 0 \end{pmatrix} , \ E = \begin{pmatrix} cos(φ) - sin(φ) \end{pmatrix} + S = \begin{pmatrix} 1 +0 \end{pmatrix} , \ E = \begin{pmatrix} cos(φ) + sin(φ) \end{pmatrix} diff --git a/docs/images/snippets/circles/8237af1396bb567d70c8b5e4dd7f8115.ascii b/docs/images/snippets/circles/8237af1396bb567d70c8b5e4dd7f8115.ascii index 3a98c249..08e3854e 100644 --- a/docs/images/snippets/circles/8237af1396bb567d70c8b5e4dd7f8115.ascii +++ b/docs/images/snippets/circles/8237af1396bb567d70c8b5e4dd7f8115.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ C = 1 = cos(φ) + b · -sin(φ) - ╡ x - │ C = a = sin(φ) + b · cos(φ) - ╰ y +╭ C = 1 = cos(φ) + b · -sin(φ) +╡ x +│ C = a = sin(φ) + b · cos(φ) +╰ y diff --git a/docs/images/snippets/circles/8b4e1d0a62380ed011f27c645ed13b28.ascii b/docs/images/snippets/circles/8b4e1d0a62380ed011f27c645ed13b28.ascii index 32b2c66c..96a4ec33 100644 --- a/docs/images/snippets/circles/8b4e1d0a62380ed011f27c645ed13b28.ascii +++ b/docs/images/snippets/circles/8b4e1d0a62380ed011f27c645ed13b28.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - φ φ - P = cos(─) , \ P = sin(─) - x 2 y 2 + φ φ +P = cos(─) , \ P = sin(─) + x 2 y 2 diff --git a/docs/images/snippets/circles/942c90bc8311e49d94059b3127fc78d5.ascii b/docs/images/snippets/circles/942c90bc8311e49d94059b3127fc78d5.ascii index d2d9edf9..c661ec57 100644 --- a/docs/images/snippets/circles/942c90bc8311e49d94059b3127fc78d5.ascii +++ b/docs/images/snippets/circles/942c90bc8311e49d94059b3127fc78d5.ascii @@ -1,15 +1,14 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 1 φ 4╭ φ ╮ - d (φ)= T - P = ─(3 + cos(φ)) - cos(─) = 2sin │ ─ │ , - x x x 4 2 ╰ 4 ╯ - 1╭ ╭ φ ╮ ╮ φ - d (φ)= T - P = ─│ 2tan│ ─ │ + sin(φ) │ - sin(─) , - y y y 4╰ ╰ 2 ╯ ╯ 2 - ⇓ - ┌───────┐ ┌───────┐ - │ 2 2 4 φ │ 1 - d(φ)= │d + d = ... = 2sin (─) │─────── - ⟍│ x y 4 │ 2 φ - │cos (─) - ⟍│ 2 + 1 φ 4╭ φ ╮ +d (φ)= T - P = ─(3 + cos(φ)) - cos(─) = 2sin │ ─ │ , + x x x 4 2 ╰ 4 ╯ + 1╭ ╭ φ ╮ ╮ φ +d (φ)= T - P = ─│ 2tan│ ─ │ + sin(φ) │ - sin(─) , + y y y 4╰ ╰ 2 ╯ ╯ 2 + ⇓ + ┌───────┐ ┌───────┐ + │ 2 2 4 φ │ 1 + d(φ)= │d + d = ... = 2sin (─) │─────── + ⟍│ x y 4 │ 2 φ + │cos (─) + ⟍│ 2 diff --git a/docs/images/snippets/circles/986ae9104e0bc52e95689ae7ae4504db.ascii b/docs/images/snippets/circles/986ae9104e0bc52e95689ae7ae4504db.ascii index 6bbf31e1..56602456 100644 --- a/docs/images/snippets/circles/986ae9104e0bc52e95689ae7ae4504db.ascii +++ b/docs/images/snippets/circles/986ae9104e0bc52e95689ae7ae4504db.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ 1 - │ T = ─(3 + cos(φ)) - ╡ x 4 - │ 1╭ 2-2cos(φ) ╮ 1╭ ╭ φ ╮ ╮ - │ T = ─│ ───────── + sin(φ) │ = ─│ 2tan│ ─ │ + sin(φ) │ - ╰ y 4╰ sin(φ) ╯ 4╰ ╰ 2 ╯ ╯ +╭ 1 +│ T = ─(3 + cos(φ)) +╡ x 4 +│ 1╭ 2-2cos(φ) ╮ 1╭ ╭ φ ╮ ╮ +│ T = ─│ ───────── + sin(φ) │ = ─│ 2tan│ ─ │ + sin(φ) │ +╰ y 4╰ sin(φ) ╯ 4╰ ╰ 2 ╯ ╯ diff --git a/docs/images/snippets/circles/9e19c65e23c32c802d2d77edb9812122.ascii b/docs/images/snippets/circles/9e19c65e23c32c802d2d77edb9812122.ascii deleted file mode 100644 index 277f38bc..00000000 --- a/docs/images/snippets/circles/9e19c65e23c32c802d2d77edb9812122.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - P = (1, 0) - 1 - P = (cos(θ), sin(θ)) - 3 diff --git a/docs/images/snippets/circles/a04bd1558a76e60b8ca6e1fe4fa38c00.ascii b/docs/images/snippets/circles/a04bd1558a76e60b8ca6e1fe4fa38c00.ascii index b443cc9d..228a5c10 100644 --- a/docs/images/snippets/circles/a04bd1558a76e60b8ca6e1fe4fa38c00.ascii +++ b/docs/images/snippets/circles/a04bd1558a76e60b8ca6e1fe4fa38c00.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 1 2 1 1 - T = ─S + ─C + ─E = ─(S + 2C + E) - 4 4 4 4 + 1 2 1 1 +T = ─S + ─C + ─E = ─(S + 2C + E) + 4 4 4 4 diff --git a/docs/images/snippets/circles/b5799015ebae3ab9b6712106b3b20ee6.ascii b/docs/images/snippets/circles/b5799015ebae3ab9b6712106b3b20ee6.ascii deleted file mode 100644 index 207829d9..00000000 --- a/docs/images/snippets/circles/b5799015ebae3ab9b6712106b3b20ee6.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - P = (1,0) + (0,k) = 1 - 2 x x - x - P = (cos(θ), sin(θ)) + k · (sin(θ), -cos(θ)) = cos(θ) + k · sin(θ) - 2 x x - x diff --git a/docs/images/snippets/circles/cf8fc61992d2a75416894dee46ee5ffe.ascii b/docs/images/snippets/circles/cf8fc61992d2a75416894dee46ee5ffe.ascii deleted file mode 100644 index 6049e647..00000000 --- a/docs/images/snippets/circles/cf8fc61992d2a75416894dee46ee5ffe.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - cos(θ) + k · sin(θ)= 1 - k · sin(θ)= 1 - cos(θ) - 1 - cos(θ) - k= ────────── - sin(θ) - ╭ θ ╮ - k= tan│ ─ │ - ╰ 2 ╯ diff --git a/docs/images/snippets/circles/d4bfb47b623c968e3231566c9705c6c4.ascii b/docs/images/snippets/circles/d4bfb47b623c968e3231566c9705c6c4.ascii index dd503155..da349114 100644 --- a/docs/images/snippets/circles/d4bfb47b623c968e3231566c9705c6c4.ascii +++ b/docs/images/snippets/circles/d4bfb47b623c968e3231566c9705c6c4.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C = S + a · \begin{pmatrix} 0 - 1 \end{pmatrix} , \ C = E + b · \begin{pmatrix} -sin(φ) - cos(φ) \end{pmatrix} + C = S + a · \begin{pmatrix} 0 +1 \end{pmatrix} , \ C = E + b · \begin{pmatrix} -sin(φ) + cos(φ) \end{pmatrix} diff --git a/docs/images/snippets/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.ascii b/docs/images/snippets/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.ascii deleted file mode 100644 index 1d1ab917..00000000 --- a/docs/images/snippets/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ ╭ θ ╮ ╮ - P = │ 1, tan│ ─ │ │ - 2 ╰ ╰ 2 ╯ ╯ diff --git a/docs/images/snippets/circles/f9d15462df31186feef8c3d53c0f6163.ascii b/docs/images/snippets/circles/f9d15462df31186feef8c3d53c0f6163.ascii index 5fef99cf..56787a82 100644 --- a/docs/images/snippets/circles/f9d15462df31186feef8c3d53c0f6163.ascii +++ b/docs/images/snippets/circles/f9d15462df31186feef8c3d53c0f6163.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ ┌─────────────┐ ╮ - │ │ ┌──────┐ │ - │ ⟍│2+ε-⟍│ε(2+ε) │ - φ = 4 · arccos │ ──────────────── │ - │ ┌─┐ │ - ╰ ⟍│2 ╯ + ╭ ┌─────────────┐ ╮ + │ │ ┌──────┐ │ + │ ⟍│2+ε-⟍│ε(2+ε) │ +φ = 4 · arccos │ ──────────────── │ + │ ┌─┐ │ + ╰ ⟍│2 ╯ diff --git a/docs/images/snippets/circles_cubic/610a69d8be6f86744ffb88d12eda701b.ascii b/docs/images/snippets/circles_cubic/610a69d8be6f86744ffb88d12eda701b.ascii index 1900c5f5..a9c1526d 100644 --- a/docs/images/snippets/circles_cubic/610a69d8be6f86744ffb88d12eda701b.ascii +++ b/docs/images/snippets/circles_cubic/610a69d8be6f86744ffb88d12eda701b.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 2 - B + B = r - x y + 2 2 +B + B = r + x y diff --git a/docs/images/snippets/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.ascii b/docs/images/snippets/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.ascii index 894c25b7..e1fee48c 100644 --- a/docs/images/snippets/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.ascii +++ b/docs/images/snippets/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌───────────────────────┐ - ╭ 1 │ 2 2 - erf (θ, k) = | \| │B (t,θ,k) + B (t,θ,k) - r\|dt - ╯ 0 ⟍│ x y + ┌───────────────────────┐ + ╭ 1 │ 2 2 +erf (θ, k) = | \| │B (t,θ,k) + B (t,θ,k) - r\|dt + ╯ 0 ⟍│ x y diff --git a/docs/images/snippets/circles_cubic/9054528132317434ae2c0be27572d86b.ascii b/docs/images/snippets/circles_cubic/9054528132317434ae2c0be27572d86b.ascii index 563547cf..ac6f3577 100644 --- a/docs/images/snippets/circles_cubic/9054528132317434ae2c0be27572d86b.ascii +++ b/docs/images/snippets/circles_cubic/9054528132317434ae2c0be27572d86b.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - P = (1, 0) - 1 - P = (1, k) - 2 - P = P + k · (sin(θ), -cos(θ)) - 3 4 - P = (cos(θ), sin(θ)) - 4 +P = (1, 0) + 1 +P = (1, k) + 2 +P = P + k · (sin(θ), -cos(θ)) + 3 4 +P = (cos(θ), sin(θ)) + 4 diff --git a/docs/images/snippets/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.ascii b/docs/images/snippets/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.ascii index 0c779da6..7c16bf1c 100644 --- a/docs/images/snippets/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.ascii +++ b/docs/images/snippets/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ ┌───────┐ ╮ - │ ╭ 1 │ 2 2 │ d - │ | \| │B + B - r\|dt │ ── = 0 - ╰ ╯ 0 ⟍│ x y ╯ dt +╭ ┌───────┐ ╮ +│ ╭ 1 │ 2 2 │ d +│ | \| │B + B - r\|dt │ ── = 0 +╰ ╯ 0 ⟍│ x y ╯ dt diff --git a/docs/images/snippets/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.ascii b/docs/images/snippets/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.ascii index e480582f..282e3666 100644 --- a/docs/images/snippets/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.ascii +++ b/docs/images/snippets/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ \pi ╮ 4 ╭ \pi ╮ 4 ┌─┐ - f│ ─── │ = ─ · tan│ ─── │ = ─(⟍│2 -1) ≅ 0.55228474983[...] - ╰ 2 ╯ 3 ╰ 8 ╯ 3 + ╭ \pi ╮ 4 ╭ \pi ╮ 4 ┌─┐ +f│ ─── │ = ─ · tan│ ─── │ = ─(⟍│2 -1) ≅ 0.55228474983[...] + ╰ 2 ╯ 3 ╰ 8 ╯ 3 diff --git a/docs/images/snippets/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.ascii b/docs/images/snippets/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.ascii index 5ace4589..9291dce4 100644 --- a/docs/images/snippets/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.ascii +++ b/docs/images/snippets/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - start= (r, 0) - control = (r, 0.55228 · r) - 1 - control = (0.55228 · r, r) - 2 - end= (0, r) + start= (r, 0) +control = (r, 0.55228 · r) + 1 +control = (0.55228 · r, r) + 2 + end= (0, r) diff --git a/docs/images/snippets/circles_cubic/b985384d01cb32d422f5d1123707ebc8.ascii b/docs/images/snippets/circles_cubic/b985384d01cb32d422f5d1123707ebc8.ascii deleted file mode 100644 index f2fd0f2b..00000000 --- a/docs/images/snippets/circles_cubic/b985384d01cb32d422f5d1123707ebc8.ascii +++ /dev/null @@ -1,24 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 3c + 4sin(θ)) - 3k · cos(θ) θ - ────────────────────────────= sin(─) - 8 2 - ╭ θ ╮ - 3k + 4sin(θ)) - 3k · cos(θ)= 8sin│ ─ │ - ╰ 2 ╯ - ╭ θ ╮ - 3k - 3k · cos(θ)= 8sin│ ─ │ - 4sin(θ) - ╰ 2 ╯ - ╭ ╭ θ ╮ ╮ - 3k (1 - cos(θ))= 4 │ 2sin│ ─ │ - sin(θ) │ - ╰ ╰ 2 ╯ ╯ - θ - 2sin(─) - sin(θ) - 2 - 3k= 4 · ──────────────── - 1 - cos(θ) - ╭ θ ╮ - 2sin│ ─ │ - sin(θ) - 4 ╰ 2 ╯ - k= ─ · ────────────────── - 3 1 - cos(θ) diff --git a/docs/images/snippets/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.ascii b/docs/images/snippets/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.ascii index 14af6068..edef6df5 100644 --- a/docs/images/snippets/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.ascii +++ b/docs/images/snippets/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌───────┐ - │ 2 2 - \| │B + B - r\| = 0, t ∈[0,1] - ⟍│ x y + ┌───────┐ + │ 2 2 +\| │B + B - r\| = 0, t ∈[0,1] + ⟍│ x y diff --git a/docs/images/snippets/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.ascii b/docs/images/snippets/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.ascii index 4fba34b6..1cdbdd3c 100644 --- a/docs/images/snippets/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.ascii +++ b/docs/images/snippets/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.ascii @@ -1,24 +1,23 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3k + 4sin(θ)) - 3k · cos(θ) θ - ────────────────────────────= sin(─) - 8 2 - ╭ θ ╮ - 3k + 4sin(θ)) - 3k · cos(θ)= 8sin│ ─ │ - ╰ 2 ╯ - ╭ θ ╮ - 3k - 3k · cos(θ)= 8sin│ ─ │ - 4sin(θ) - ╰ 2 ╯ - ╭ ╭ θ ╮ ╮ - 3k (1 - cos(θ))= 4 │ 2sin│ ─ │ - sin(θ) │ - ╰ ╰ 2 ╯ ╯ - θ - 2sin(─) - sin(θ) - 2 - 3k= 4 · ──────────────── - 1 - cos(θ) - ╭ θ ╮ - 2sin│ ─ │ - sin(θ) - 4 ╰ 2 ╯ - k= ─ · ────────────────── - 3 1 - cos(θ) +3k + 4sin(θ)) - 3k · cos(θ) θ +────────────────────────────= sin(─) + 8 2 + ╭ θ ╮ +3k + 4sin(θ)) - 3k · cos(θ)= 8sin│ ─ │ + ╰ 2 ╯ + ╭ θ ╮ + 3k - 3k · cos(θ)= 8sin│ ─ │ - 4sin(θ) + ╰ 2 ╯ + ╭ ╭ θ ╮ ╮ + 3k (1 - cos(θ))= 4 │ 2sin│ ─ │ - sin(θ) │ + ╰ ╰ 2 ╯ ╯ + θ + 2sin(─) - sin(θ) + 2 + 3k= 4 · ──────────────── + 1 - cos(θ) + ╭ θ ╮ + 2sin│ ─ │ - sin(θ) + 4 ╰ 2 ╯ + k= ─ · ────────────────── + 3 1 - cos(θ) diff --git a/docs/images/snippets/circles_cubic/d880c4a1b3d7b651b054b008e952b493.ascii b/docs/images/snippets/circles_cubic/d880c4a1b3d7b651b054b008e952b493.ascii index a583d0af..9ab5373a 100644 --- a/docs/images/snippets/circles_cubic/d880c4a1b3d7b651b054b008e952b493.ascii +++ b/docs/images/snippets/circles_cubic/d880c4a1b3d7b651b054b008e952b493.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - start= (r, 0) - control = (r, k) - 1 - control = r · (cos(θ) + k · sin(θ), sin(θ) - k · cos(θ)) - 2 - end= r · (cos(θ), sin(θ)) + start= (r, 0) +control = (r, k) + 1 +control = r · (cos(θ) + k · sin(θ), sin(θ) - k · cos(θ)) + 2 + end= r · (cos(θ), sin(θ)) diff --git a/docs/images/snippets/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.ascii b/docs/images/snippets/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.ascii index 887d55e2..ef8fcc86 100644 --- a/docs/images/snippets/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.ascii +++ b/docs/images/snippets/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.ascii @@ -1,25 +1,24 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - P + P - 2 3 - y y k + sin(θ) - k · cos(θ) - A = ───────── = ──────────────────────── - y 2 2 - 1 - A + ─P k + sin(θ) - k · cos(θ) - y 2 2 ──────────────────────── + ─ - y 2 2 2k + sin(θ) + k · cos(θ) - e = ───────── = ──────────────────────────── = ───────────────────────── - 1 2 2 4 - y - k - A + mid(P , P ) A + sin(θ) - ─ cos(θ) - y 4 3 y 2 k + 3sin(θ) 2k · cos(θ) - e = ───────────────── = ────────────────────── = ──────────────────────── - 2 2 2 4 - y - e + e - 1 2 - y y 3k + 4sin(θ) - 3k · cos(θ) - B = ───────── = ─────────────────────────── - y 2 8 + P + P + 2 3 + y y k + sin(θ) - k · cos(θ) + A = ───────── = ──────────────────────── + y 2 2 + 1 + A + ─P k + sin(θ) - k · cos(θ) + y 2 2 ──────────────────────── + ─ + y 2 2 2k + sin(θ) + k · cos(θ) +e = ───────── = ──────────────────────────── = ───────────────────────── + 1 2 2 4 + y + k + A + mid(P , P ) A + sin(θ) - ─ cos(θ) + y 4 3 y 2 k + 3sin(θ) 2k · cos(θ) +e = ───────────────── = ────────────────────── = ──────────────────────── + 2 2 2 4 + y + e + e + 1 2 + y y 3k + 4sin(θ) - 3k · cos(θ) + B = ───────── = ─────────────────────────── + y 2 8 diff --git a/docs/images/snippets/circles_cubic/f47561c3870425499e31c7527c38dc94.ascii b/docs/images/snippets/circles_cubic/f47561c3870425499e31c7527c38dc94.ascii index 299e5962..53798213 100644 --- a/docs/images/snippets/circles_cubic/f47561c3870425499e31c7527c38dc94.ascii +++ b/docs/images/snippets/circles_cubic/f47561c3870425499e31c7527c38dc94.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 4 ╭ θ ╮ - k = f(θ) = ─ tan│ ─ │ - 3 ╰ 4 ╯ + 4 ╭ θ ╮ +k = f(θ) = ─ tan│ ─ │ + 3 ╰ 4 ╯ diff --git a/docs/images/snippets/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.ascii b/docs/images/snippets/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.ascii index eec98a2c..f62ab393 100644 --- a/docs/images/snippets/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.ascii +++ b/docs/images/snippets/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.ascii @@ -1,18 +1,17 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ θ ╮ - 2sin│ ─ │ - sin(θ) - 4 ╰ 2 ╯ - k= ─ · ────────────────── - 3 1 - cos(θ) - ╭ ╭ θ ╮ ╮ - │ 2sin│ ─ │ │ - 4 │ ╰ 2 ╯ sin(θ) │ - k= ─ · │ ────────── - ────────── │ - 3 ╰ 1 - cos(θ) 1 - cos(θ) ╯ - 4 ╭ ╭ θ ╮ ╭ θ ╮ ╮ - k= ─ · │ csc│ ─ │ - cot│ ─ │ │ - 3 ╰ ╰ 2 ╯ ╰ 2 ╯ ╯ - 4 ╭ θ ╮ - k= ─ · tan│ ─ │ - 3 ╰ 4 ╯ + ╭ θ ╮ + 2sin│ ─ │ - sin(θ) + 4 ╰ 2 ╯ +k= ─ · ────────────────── + 3 1 - cos(θ) + ╭ ╭ θ ╮ ╮ + │ 2sin│ ─ │ │ + 4 │ ╰ 2 ╯ sin(θ) │ +k= ─ · │ ────────── - ────────── │ + 3 ╰ 1 - cos(θ) 1 - cos(θ) ╯ + 4 ╭ ╭ θ ╮ ╭ θ ╮ ╮ +k= ─ · │ csc│ ─ │ - cot│ ─ │ │ + 3 ╰ ╰ 2 ╯ ╰ 2 ╯ ╯ + 4 ╭ θ ╮ +k= ─ · tan│ ─ │ + 3 ╰ 4 ╯ diff --git a/docs/images/snippets/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.ascii b/docs/images/snippets/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.ascii deleted file mode 100644 index c4a7482b..00000000 --- a/docs/images/snippets/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - P = (1, 0) - 1 - P = (1, c) - 2 - P = P + k · (sin(θ), -cos(θ)) - 3 4 - P = (cos(θ), sin(θ)) - 4 diff --git a/docs/images/snippets/control/20b0be6397fbd726298de6ec70a8544b.ascii b/docs/images/snippets/control/20b0be6397fbd726298de6ec70a8544b.ascii deleted file mode 100644 index ed26833c..00000000 --- a/docs/images/snippets/control/20b0be6397fbd726298de6ec70a8544b.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ 3 2 2 3 - ╡ x = \colordarkred110 · (1-t) + \colordarkgreen25 · 3 · (1-t) · t + \colordarkblue210 · 3 · (1-t) · t + \coloramber210 · t - │ 3 2 2 3 - ╰ y = \colordarkred150 · (1-t) + \colordarkgreen190 · 3 · (1-t) · t + \colordarkblue250 · 3 · (1-t) · t + \coloramber30 · t diff --git a/docs/images/snippets/control/3ec466bf6e1aff44b35b8e37cc86cc3e.ascii b/docs/images/snippets/control/3ec466bf6e1aff44b35b8e37cc86cc3e.ascii deleted file mode 100644 index 6b0389a1..00000000 --- a/docs/images/snippets/control/3ec466bf6e1aff44b35b8e37cc86cc3e.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - __ n n-i i - Bézier(n,t) = ❯ \underset 二項係数部分の項\underbrace\binomni · \ \underset 多項式部分の項\underbrace(1-t) · t · \ \underset重み\underbracew - ‾‾ i=0 i diff --git a/docs/images/snippets/control/501494295f07ba5049286489206d98f0.ascii b/docs/images/snippets/control/501494295f07ba5049286489206d98f0.ascii deleted file mode 100644 index b3ba22fe..00000000 --- a/docs/images/snippets/control/501494295f07ba5049286489206d98f0.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i -Bézier(n,t) = ❯ \undersetбиноминальный термин\underbrace\binomni · \ \undersetполиноминальный термин\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetвес\underbracew - i diff --git a/docs/images/snippets/control/6890c4028109e7d30a0b4b89f6fbe292.ascii b/docs/images/snippets/control/6890c4028109e7d30a0b4b89f6fbe292.ascii deleted file mode 100644 index 84c3fd7c..00000000 --- a/docs/images/snippets/control/6890c4028109e7d30a0b4b89f6fbe292.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i -Bézier(n,t) = ❯ \underset binomial term\underbrace\binomni · \ \underset polynomial term\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetweight\underbracew - i diff --git a/docs/images/snippets/control/80cdfeab6ed6038f0e550ef5c1dcb7dd.ascii b/docs/images/snippets/control/80cdfeab6ed6038f0e550ef5c1dcb7dd.ascii index 6a5dfe6b..fbb94210 100644 --- a/docs/images/snippets/control/80cdfeab6ed6038f0e550ef5c1dcb7dd.ascii +++ b/docs/images/snippets/control/80cdfeab6ed6038f0e550ef5c1dcb7dd.ascii @@ -1,5 +1,4 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - __ n n-i i - Bézier(n,t) = ❯ \underset 二項係数部分の項\underbrace\binomni · \ \underset 多項式部分の項\underbrace(1-t) · t · \ \underset 重み\underbracew - ‾‾ i=0 i + __ n n-i i +Bézier(n,t) = ❯ \underset 二項係数部分の項\underbrace\binomni · \ \underset 多項式部分の項\underbrace(1-t) · t · \ \underset 重み\underbracew + ‾‾ i=0 i diff --git a/docs/images/snippets/control/87d587388add62445ec0f3e7d7295094.ascii b/docs/images/snippets/control/87d587388add62445ec0f3e7d7295094.ascii deleted file mode 100644 index ee160ac1..00000000 --- a/docs/images/snippets/control/87d587388add62445ec0f3e7d7295094.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - - __ n n-i i -Bézier(n,t) = ❯ \underset биноминальный термин\underbrace\binomni · \ \underset полиноминальный термин\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetвес\underbracew - i diff --git a/docs/images/snippets/control/9df7dc66b51db8e3046e1f359874c38e.ascii b/docs/images/snippets/control/9df7dc66b51db8e3046e1f359874c38e.ascii deleted file mode 100644 index bc0af7b5..00000000 --- a/docs/images/snippets/control/9df7dc66b51db8e3046e1f359874c38e.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - __ n n-i i - Bézier(n,t) = ❯ \underset二項係数部分の項\underbrace\binomni · \ \underset多項式部分の項\underbrace(1-t) · t · \ \underset重み\underbracew - ‾‾ i=0 i diff --git a/docs/images/snippets/control/a337e3f97387b52d387fc01605314497.ascii b/docs/images/snippets/control/a337e3f97387b52d387fc01605314497.ascii index 6c8d9694..d1d93188 100644 --- a/docs/images/snippets/control/a337e3f97387b52d387fc01605314497.ascii +++ b/docs/images/snippets/control/a337e3f97387b52d387fc01605314497.ascii @@ -1,5 +1,4 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - + __ n n-i i Bézier(n,t) = ❯ \underset биноминальный термин\underbrace\binomni · \ \underset полиноминальный термин\underbrace(1-t) · t · \ \underset ‾‾ i=0 diff --git a/docs/images/snippets/control/b58fb122c5c8159938182c185f287142.ascii b/docs/images/snippets/control/b58fb122c5c8159938182c185f287142.ascii deleted file mode 100644 index 596ece8d..00000000 --- a/docs/images/snippets/control/b58fb122c5c8159938182c185f287142.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i -Bézier(n,t) = ❯ \undersetbinomial term\underbrace\binomni · \ \undersetpolynomial term\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetweight\underbracew - i diff --git a/docs/images/snippets/control/be73034ac382b54863c7a18c2932cbbc.ascii b/docs/images/snippets/control/be73034ac382b54863c7a18c2932cbbc.ascii new file mode 100644 index 00000000..84b079c3 --- /dev/null +++ b/docs/images/snippets/control/be73034ac382b54863c7a18c2932cbbc.ascii @@ -0,0 +1,5 @@ + +╭ 3 2 2 3 +╡ x = \colordarkred 110 · (1-t) + \colordarkgreen 25 · 3 · (1-t) · t + \colordarkblue 210 · 3 · (1-t) · t + \coloramber 210 · t +│ 3 2 2 3 +╰ y = \colordarkred 150 · (1-t) + \colordarkgreen 190 · 3 · (1-t) · t + \colordarkblue 250 · 3 · (1-t) · t + \coloramber 30 · t diff --git a/docs/images/snippets/control/c2f2fe0ef5d0089d9dd8e5e3999405cb.ascii b/docs/images/snippets/control/c2f2fe0ef5d0089d9dd8e5e3999405cb.ascii index 2e1eac5e..752cc1ff 100644 --- a/docs/images/snippets/control/c2f2fe0ef5d0089d9dd8e5e3999405cb.ascii +++ b/docs/images/snippets/control/c2f2fe0ef5d0089d9dd8e5e3999405cb.ascii @@ -1,5 +1,4 @@ - \usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontgbsn00lp.ttf - + __ n n-i i Bézier(n,t) = ❯ \underset binomial term\underbrace\binomni · \ \underset polynomial term\underbrace(1-t) · t · \ \underset ‾‾ i=0 diff --git a/docs/images/snippets/curvature/060acd6ff0a050fe4d98a7802a2b3a3f.ascii b/docs/images/snippets/curvature/060acd6ff0a050fe4d98a7802a2b3a3f.ascii index b2eb70f2..2e931420 100644 --- a/docs/images/snippets/curvature/060acd6ff0a050fe4d98a7802a2b3a3f.ascii +++ b/docs/images/snippets/curvature/060acd6ff0a050fe4d98a7802a2b3a3f.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - x'y'' - x''y' - \kappa = ───────────── - 3 - ─ - 2 2 2 - (x' +y' ) + x'y'' - x''y' +\kappa = ───────────── + 3 + ─ + 2 2 2 + (x' +y' ) diff --git a/docs/images/snippets/curvature/561ab3a938d655550de0abf458ac2494.ascii b/docs/images/snippets/curvature/561ab3a938d655550de0abf458ac2494.ascii index 75180c28..ac56dd24 100644 --- a/docs/images/snippets/curvature/561ab3a938d655550de0abf458ac2494.ascii +++ b/docs/images/snippets/curvature/561ab3a938d655550de0abf458ac2494.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 1 - R(t) = ───────── - \kappa(t) + 1 +R(t) = ───────── + \kappa(t) diff --git a/docs/images/snippets/curvature/afd8cb8b0fe291ff703752c1c9cc33d4.ascii b/docs/images/snippets/curvature/afd8cb8b0fe291ff703752c1c9cc33d4.ascii index af5474f0..ce02bb6e 100644 --- a/docs/images/snippets/curvature/afd8cb8b0fe291ff703752c1c9cc33d4.ascii +++ b/docs/images/snippets/curvature/afd8cb8b0fe291ff703752c1c9cc33d4.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B '(t)B ''(t) - B ''(t)B '(t) - x y x y - \kappa(t) = ───────────────────────────── - 3 - ─ - 2 2 2 - (B '(t) +B '(t) ) - x y + B '(t)B ''(t) - B ''(t)B '(t) + x y x y +\kappa(t) = ───────────────────────────── + 3 + ─ + 2 2 2 + (B '(t) +B '(t) ) + x y diff --git a/docs/images/snippets/curvefitting/06605e008956609e8844ef95697c9096.ascii b/docs/images/snippets/curvefitting/06605e008956609e8844ef95697c9096.ascii deleted file mode 100644 index 5650f150..00000000 --- a/docs/images/snippets/curvefitting/06605e008956609e8844ef95697c9096.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 3 2 2 3 - B =a(1-t) + 3b(1-t) t + 3c(1-t)t + dt - cubic diff --git a/docs/images/snippets/curvefitting/097aa1948b6cdbf9dc7579643a7af246.ascii b/docs/images/snippets/curvefitting/097aa1948b6cdbf9dc7579643a7af246.ascii index 9272dba9..f1be0478 100644 --- a/docs/images/snippets/curvefitting/097aa1948b6cdbf9dc7579643a7af246.ascii +++ b/docs/images/snippets/curvefitting/097aa1948b6cdbf9dc7579643a7af246.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n 2 - E(C) = ❯ (p - Bézier(s )) - ‾‾ i=1 i i + __ n 2 +E(C) = ❯ (p - Bézier(s )) + ‾‾ i=1 i i diff --git a/docs/images/snippets/curvefitting/134baa1043d0849f31a1943d6d5bc607.ascii b/docs/images/snippets/curvefitting/134baa1043d0849f31a1943d6d5bc607.ascii deleted file mode 100644 index 1a6bcb00..00000000 --- a/docs/images/snippets/curvefitting/134baa1043d0849f31a1943d6d5bc607.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ a 0 0 ┐ ┌ 1 0 0 ┐ ┌ a ┐ - B = T · M · C = ┌ 2 ┐ · │ -2a 2b 0 │ = ┌ 2 ┐ · │ -2 2 0 │ · │ b │ - quadratic └ 1 t t ┘ └ a -2b c ┘ └ 1 t t ┘ └ 1 -2 1 ┘ └ c ┘ diff --git a/docs/images/snippets/curvefitting/17d5fbeffcdcceca98cdba537295d258.ascii b/docs/images/snippets/curvefitting/17d5fbeffcdcceca98cdba537295d258.ascii index 4a235f5a..f894fc32 100644 --- a/docs/images/snippets/curvefitting/17d5fbeffcdcceca98cdba537295d258.ascii +++ b/docs/images/snippets/curvefitting/17d5fbeffcdcceca98cdba537295d258.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 2 3 - B =a(1-t) + 3b(1-t) t + 3c(1-t)t + dt - cubic + 3 2 2 3 +B =a(1-t) + 3b(1-t) t + 3c(1-t)t + dt + cubic diff --git a/docs/images/snippets/curvefitting/2f82371abb7835f9b9d440dc5dd151a8.ascii b/docs/images/snippets/curvefitting/2f82371abb7835f9b9d440dc5dd151a8.ascii index 5baf680d..cb60fd60 100644 --- a/docs/images/snippets/curvefitting/2f82371abb7835f9b9d440dc5dd151a8.ascii +++ b/docs/images/snippets/curvefitting/2f82371abb7835f9b9d440dc5dd151a8.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ d = 0 - D = ┌ d d ... d ┐, where ╡ 1 - └ 1 2 n ┘ │ d = d + length(p , p ) - ╰ i i-1 i-1 i + ╭ d = 0 +D = ┌ d d ... d ┐, where ╡ 1 + └ 1 2 n ┘ │ d = d + length(p , p ) + ╰ i i-1 i-1 i diff --git a/docs/images/snippets/curvefitting/31d659cbc72bf304abf4c9a75b6b81de.ascii b/docs/images/snippets/curvefitting/31d659cbc72bf304abf4c9a75b6b81de.ascii deleted file mode 100644 index 63badfeb..00000000 --- a/docs/images/snippets/curvefitting/31d659cbc72bf304abf4c9a75b6b81de.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 2 2 - B = a (1-t) + 2 b (1-t) t + c t - quadratic - 2 2 2 - = a - 2at + at + 2bt - 2bt + ct diff --git a/docs/images/snippets/curvefitting/38bb81bdd3eaa72c2336514187aa374b.ascii b/docs/images/snippets/curvefitting/38bb81bdd3eaa72c2336514187aa374b.ascii index 623a3fec..642acb18 100644 --- a/docs/images/snippets/curvefitting/38bb81bdd3eaa72c2336514187aa374b.ascii +++ b/docs/images/snippets/curvefitting/38bb81bdd3eaa72c2336514187aa374b.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 ... 0 0 ┐ - │ n-2 n-1 │ - │ 1 s s s │ - │ 2 2 2 │ - 𝕋 = │ \vdots ... \vdots │ - │ n-2 n-1 │ - │ 1 s s s │ - │ n-1 n-1 n-1 │ - └ 1 1 ... 1 1 ┘ + ┌ 1 0 ... 0 0 ┐ + │ n-2 n-1 │ + │ 1 s s s │ + │ 2 2 2 │ +𝕋 = │ \vdots ... \vdots │ + │ n-2 n-1 │ + │ 1 s s s │ + │ n-1 n-1 n-1 │ + └ 1 1 ... 1 1 ┘ diff --git a/docs/images/snippets/curvefitting/409d10c3005b0c93489d72a5dba692d7.ascii b/docs/images/snippets/curvefitting/409d10c3005b0c93489d72a5dba692d7.ascii deleted file mode 100644 index f44cabd1..00000000 --- a/docs/images/snippets/curvefitting/409d10c3005b0c93489d72a5dba692d7.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ 1 0 0 0 0 ┐ ┌ a ┐ - │ -4 4 0 0 0 │ │ b │ - B = T · M · C = ┌ 2 3 4 ┐ · │ 6 -12 6 0 0 │ · │ c │ - quartic └ 1 t t t t ┘ │ -4 12 -12 4 0 │ │ d │ - └ 1 -4 6 -4 1 ┘ └ e ┘ diff --git a/docs/images/snippets/curvefitting/464dbfb5adb6233108053dfac6fa4fe5.ascii b/docs/images/snippets/curvefitting/464dbfb5adb6233108053dfac6fa4fe5.ascii deleted file mode 100644 index 0bb85076..00000000 --- a/docs/images/snippets/curvefitting/464dbfb5adb6233108053dfac6fa4fe5.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 2 - E(C) = (p - Bézier(s )) - i i i diff --git a/docs/images/snippets/curvefitting/4dd55c228a26bb50da912a45e8721024.ascii b/docs/images/snippets/curvefitting/4dd55c228a26bb50da912a45e8721024.ascii index 87ecd45d..2376aa27 100644 --- a/docs/images/snippets/curvefitting/4dd55c228a26bb50da912a45e8721024.ascii +++ b/docs/images/snippets/curvefitting/4dd55c228a26bb50da912a45e8721024.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 0 1 n-2 n-1 ┐ - │ s s ... s s │ - │ 1 1 1 1 │ - │ │ - 𝕋 = │ \vdots ... \vdots │ - │ │ - │ 0 1 n-2 n-1 │ - │ s s ... s s │ - └ n n n n ┘ + ┌ 0 1 n-2 n-1 ┐ + │ s s ... s s │ + │ 1 1 1 1 │ + │ │ +𝕋 = │ \vdots ... \vdots │ + │ │ + │ 0 1 n-2 n-1 │ + │ s s ... s s │ + └ n n n n ┘ diff --git a/docs/images/snippets/curvefitting/4e6e20c823c8cc72e0cc00e4ab5b7556.ascii b/docs/images/snippets/curvefitting/4e6e20c823c8cc72e0cc00e4ab5b7556.ascii index 0a85fa31..4a39c9a9 100644 --- a/docs/images/snippets/curvefitting/4e6e20c823c8cc72e0cc00e4ab5b7556.ascii +++ b/docs/images/snippets/curvefitting/4e6e20c823c8cc72e0cc00e4ab5b7556.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 0 0 ┐ ┌ a ┐ - B = T · M · C = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ b │ - cubic └ 1 t t t ┘ │ 3 -6 3 0 │ │ c │ - └ -1 3 -3 1 ┘ └ d ┘ + ┌ 1 0 0 0 ┐ ┌ a ┐ +B = T · M · C = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ b │ + cubic └ 1 t t t ┘ │ 3 -6 3 0 │ │ c │ + └ -1 3 -3 1 ┘ └ d ┘ diff --git a/docs/images/snippets/curvefitting/505ab1ada6a187e9ba392d19739ac2c5.ascii b/docs/images/snippets/curvefitting/505ab1ada6a187e9ba392d19739ac2c5.ascii deleted file mode 100644 index f3555f4c..00000000 --- a/docs/images/snippets/curvefitting/505ab1ada6a187e9ba392d19739ac2c5.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ a 0 0 ┐ ┌ 1 0 0 ┐ ┌ a ┐ - B = T · M · C = ┌ 2 ┐ · │ -2a 2b 0 │ = ┌ 2 ┐ · │ -2 2 0 │ · │ b │ - quadratic └ 1 t t ┘ └ a -2b c ┘ └ 1 t t ┘ └ 1 -2 1 ┘ └ c ┘ diff --git a/docs/images/snippets/curvefitting/6f734d319a1cfe0de76574a65abb07e1.ascii b/docs/images/snippets/curvefitting/6f734d319a1cfe0de76574a65abb07e1.ascii index b71ce883..e662c19a 100644 --- a/docs/images/snippets/curvefitting/6f734d319a1cfe0de76574a65abb07e1.ascii +++ b/docs/images/snippets/curvefitting/6f734d319a1cfe0de76574a65abb07e1.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ s = 0 - │ 1 - S = ┌ s s ... s ┐, where ╡ s = d / d - └ 1 2 n ┘ │ i i n - │ s = 1 - ╰ n + ╭ s = 0 + │ 1 +S = ┌ s s ... s ┐, where ╡ s = d / d + └ 1 2 n ┘ │ i i n + │ s = 1 + ╰ n diff --git a/docs/images/snippets/curvefitting/7b0199bb515d2754c03d8f796b29febf.ascii b/docs/images/snippets/curvefitting/7b0199bb515d2754c03d8f796b29febf.ascii index 86355628..2cb664a6 100644 --- a/docs/images/snippets/curvefitting/7b0199bb515d2754c03d8f796b29febf.ascii +++ b/docs/images/snippets/curvefitting/7b0199bb515d2754c03d8f796b29febf.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 - E(C) = (P - TMC) + 2 +E(C) = (P - TMC) diff --git a/docs/images/snippets/curvefitting/7c6b50cee5dc685515943a199d7a65fc.ascii b/docs/images/snippets/curvefitting/7c6b50cee5dc685515943a199d7a65fc.ascii deleted file mode 100644 index 05afbf6d..00000000 --- a/docs/images/snippets/curvefitting/7c6b50cee5dc685515943a199d7a65fc.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - B =a - cubic - - 3at + 3bt - 2 2 2 - + 3at - 6bt +3ct - 3 3 3 3 - - at + 3bt -3ct + dt diff --git a/docs/images/snippets/curvefitting/8068231b915832938136d5833f74751d.ascii b/docs/images/snippets/curvefitting/8068231b915832938136d5833f74751d.ascii index 4376580d..8cf9e306 100644 --- a/docs/images/snippets/curvefitting/8068231b915832938136d5833f74751d.ascii +++ b/docs/images/snippets/curvefitting/8068231b915832938136d5833f74751d.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - T - E(C) = (P - TMC) (P - TMC) + T +E(C) = (P - TMC) (P - TMC) diff --git a/docs/images/snippets/curvefitting/8928f757abd1376abdc4069e1aa774f2.ascii b/docs/images/snippets/curvefitting/8928f757abd1376abdc4069e1aa774f2.ascii index 39708cf8..0163d52d 100644 --- a/docs/images/snippets/curvefitting/8928f757abd1376abdc4069e1aa774f2.ascii +++ b/docs/images/snippets/curvefitting/8928f757abd1376abdc4069e1aa774f2.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B =a - cubic - - 3at + 3bt - 2 2 2 - + 3at - 6bt +3ct - 3 3 3 3 - - at + 3bt -3ct + dt +B =a + cubic + - 3at + 3bt + 2 2 2 + + 3at - 6bt +3ct + 3 3 3 3 + - at + 3bt -3ct + dt diff --git a/docs/images/snippets/curvefitting/8a66af7570bac674966f6316820ea31b.ascii b/docs/images/snippets/curvefitting/8a66af7570bac674966f6316820ea31b.ascii index e8fcab3b..899a47a2 100644 --- a/docs/images/snippets/curvefitting/8a66af7570bac674966f6316820ea31b.ascii +++ b/docs/images/snippets/curvefitting/8a66af7570bac674966f6316820ea31b.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 0 0 0 ┐ ┌ a ┐ - │ -4 4 0 0 0 │ │ b │ - B = T · M · C = ┌ 2 3 4 ┐ · │ 6 -12 6 0 0 │ · │ c │ - quartic └ 1 t t t t ┘ │ -4 12 -12 4 0 │ │ d │ - └ 1 -4 6 -4 1 ┘ └ e ┘ + ┌ 1 0 0 0 0 ┐ ┌ a ┐ + │ -4 4 0 0 0 │ │ b │ +B = T · M · C = ┌ 2 3 4 ┐ · │ 6 -12 6 0 0 │ · │ c │ + quartic └ 1 t t t t ┘ │ -4 12 -12 4 0 │ │ d │ + └ 1 -4 6 -4 1 ┘ └ e ┘ diff --git a/docs/images/snippets/curvefitting/940455f4016ab1be6d46c6f176fd2f76.ascii b/docs/images/snippets/curvefitting/940455f4016ab1be6d46c6f176fd2f76.ascii deleted file mode 100644 index 31a201a4..00000000 --- a/docs/images/snippets/curvefitting/940455f4016ab1be6d46c6f176fd2f76.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n 2 - E(C) = ❯ (p - Bézier(s )) - ‾‾ i=1 i i diff --git a/docs/images/snippets/curvefitting/9651a687e1522b00bcba063881230902.ascii b/docs/images/snippets/curvefitting/9651a687e1522b00bcba063881230902.ascii deleted file mode 100644 index 94c84155..00000000 --- a/docs/images/snippets/curvefitting/9651a687e1522b00bcba063881230902.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ 1 0 0 0 ┐ ┌ a ┐ - B = T · M · C = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ b │ - cubic └ 1 t t t ┘ │ 3 -6 3 0 │ │ c │ - └ -1 3 -3 1 ┘ └ d ┘ diff --git a/docs/images/snippets/curvefitting/989f2ad06ae308f71cef527a5594129a.ascii b/docs/images/snippets/curvefitting/989f2ad06ae308f71cef527a5594129a.ascii index e738aeae..9522aa93 100644 --- a/docs/images/snippets/curvefitting/989f2ad06ae308f71cef527a5594129a.ascii +++ b/docs/images/snippets/curvefitting/989f2ad06ae308f71cef527a5594129a.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - T - E(C) = (P - 𝕋MC) (P - 𝕋MC) + T +E(C) = (P - 𝕋MC) (P - 𝕋MC) diff --git a/docs/images/snippets/curvefitting/9dec10b81a61b456ca1550cd9b7ba513.ascii b/docs/images/snippets/curvefitting/9dec10b81a61b456ca1550cd9b7ba513.ascii index dae89d92..2bf5b1e3 100644 --- a/docs/images/snippets/curvefitting/9dec10b81a61b456ca1550cd9b7ba513.ascii +++ b/docs/images/snippets/curvefitting/9dec10b81a61b456ca1550cd9b7ba513.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - -1 T -1 T - C = M (𝕋 𝕋) 𝕋 P + -1 T -1 T +C = M (𝕋 𝕋) 𝕋 P diff --git a/docs/images/snippets/curvefitting/a6faaf6083c818431988fef49421cc47.ascii b/docs/images/snippets/curvefitting/a6faaf6083c818431988fef49421cc47.ascii deleted file mode 100644 index 216f8113..00000000 --- a/docs/images/snippets/curvefitting/a6faaf6083c818431988fef49421cc47.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - B =a - quadratic - - 2at+ 2bt - 2 2 2 - + at - 2bt + ct diff --git a/docs/images/snippets/curvefitting/b017da988c9a778a4ce6a6f4ea4790d4.ascii b/docs/images/snippets/curvefitting/b017da988c9a778a4ce6a6f4ea4790d4.ascii index 5700c1a5..d353d47a 100644 --- a/docs/images/snippets/curvefitting/b017da988c9a778a4ce6a6f4ea4790d4.ascii +++ b/docs/images/snippets/curvefitting/b017da988c9a778a4ce6a6f4ea4790d4.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ p ┐ - │ 1 │ - │ p │ - P = │ 2 │ - │ ... │ - │ p │ - └ n ┘ + ┌ p ┐ + │ 1 │ + │ p │ +P = │ 2 │ + │ ... │ + │ p │ + └ n ┘ diff --git a/docs/images/snippets/curvefitting/d39ca235454ced9681b523be056864d2.ascii b/docs/images/snippets/curvefitting/d39ca235454ced9681b523be056864d2.ascii index 4cf5c5e4..f24753c0 100644 --- a/docs/images/snippets/curvefitting/d39ca235454ced9681b523be056864d2.ascii +++ b/docs/images/snippets/curvefitting/d39ca235454ced9681b523be056864d2.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 - E(C) = (p - Bézier(s )) - i i i + 2 +E(C) = (p - Bézier(s )) + i i i diff --git a/docs/images/snippets/curvefitting/db85a7a46b869c892662c26b6aea15a1.ascii b/docs/images/snippets/curvefitting/db85a7a46b869c892662c26b6aea15a1.ascii index e281317a..005d5287 100644 --- a/docs/images/snippets/curvefitting/db85a7a46b869c892662c26b6aea15a1.ascii +++ b/docs/images/snippets/curvefitting/db85a7a46b869c892662c26b6aea15a1.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ a + 0 + 0 ┐ ┌ 1 0 0 ┐ ┌ a ┐ - B = T · M · C = ┌ 2 ┐ · │ -2a + 2b + 0 │ = ┌ 2 ┐ · │ -2 2 0 │ · │ b │ - quadratic └ 1 t t ┘ └ a + -2b + c ┘ └ 1 t t ┘ └ 1 -2 1 ┘ └ c ┘ + ┌ a + 0 + 0 ┐ ┌ 1 0 0 ┐ ┌ a ┐ +B = T · M · C = ┌ 2 ┐ · │ -2a + 2b + 0 │ = ┌ 2 ┐ · │ -2 2 0 │ · │ b │ + quadratic └ 1 t t ┘ └ a + -2b + c ┘ └ 1 t t ┘ └ 1 -2 1 ┘ └ c ┘ diff --git a/docs/images/snippets/curvefitting/dd303afb51d580fb2bf1b914c010f83d.ascii b/docs/images/snippets/curvefitting/dd303afb51d580fb2bf1b914c010f83d.ascii index 2b38a216..2c046c53 100644 --- a/docs/images/snippets/curvefitting/dd303afb51d580fb2bf1b914c010f83d.ascii +++ b/docs/images/snippets/curvefitting/dd303afb51d580fb2bf1b914c010f83d.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 2 - B = a (1-t) + 2 b (1-t) t + c t - quadratic - 2 2 2 - = a - 2at + at + 2bt - 2bt + ct + 2 2 +B = a (1-t) + 2 b (1-t) t + c t + quadratic + 2 2 2 + = a - 2at + at + 2bt - 2bt + ct diff --git a/docs/images/snippets/curvefitting/ea24b0e42f0a89464bda275ac8f9bacf.ascii b/docs/images/snippets/curvefitting/ea24b0e42f0a89464bda275ac8f9bacf.ascii index 9d7d848d..a3f4f0de 100644 --- a/docs/images/snippets/curvefitting/ea24b0e42f0a89464bda275ac8f9bacf.ascii +++ b/docs/images/snippets/curvefitting/ea24b0e42f0a89464bda275ac8f9bacf.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ∂E T - ── = 0 = -2𝕋 (P - 𝕋MC) - ∂C +∂E T +── = 0 = -2𝕋 (P - 𝕋MC) +∂C diff --git a/docs/images/snippets/curvefitting/ff701138fd7a6e35700a2e1ee3e9c020.ascii b/docs/images/snippets/curvefitting/ff701138fd7a6e35700a2e1ee3e9c020.ascii index 17c63901..b774c811 100644 --- a/docs/images/snippets/curvefitting/ff701138fd7a6e35700a2e1ee3e9c020.ascii +++ b/docs/images/snippets/curvefitting/ff701138fd7a6e35700a2e1ee3e9c020.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B =a - quadratic - - 2at+ 2bt - 2 2 2 - + at - 2bt + ct +B =a + quadratic + - 2at+ 2bt + 2 2 2 + + at - 2bt + ct diff --git a/docs/images/snippets/derivatives/02cecadc92b8ff681edc8edb0ace53ce.ascii b/docs/images/snippets/derivatives/02cecadc92b8ff681edc8edb0ace53ce.ascii index 27521999..af6dbfc5 100644 --- a/docs/images/snippets/derivatives/02cecadc92b8ff681edc8edb0ace53ce.ascii +++ b/docs/images/snippets/derivatives/02cecadc92b8ff681edc8edb0ace53ce.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - Bézier (t) = B (t) · w + B (t) · w + B (t) · w + B (t) · w + ... - n,k n,0 0 n,1 1 n,2 2 n,3 3 - d - Bézier (t) ── = n · (B (t) - B (t)) · w + - n,k dt n-1,-1 n-1,0 0 - n · (B (t) - B (t)) · w + - n-1,0 n-1,1 1 - n · (B (t) - B (t)) · w + - n-1,1 n-1,2 2 - n · (B (t) - B (t)) · w + - n-1,2 n-1,3 3 - ... +Bézier (t) = B (t) · w + B (t) · w + B (t) · w + B (t) · w + ... + n,k n,0 0 n,1 1 n,2 2 n,3 3 + d +Bézier (t) ── = n · (B (t) - B (t)) · w + + n,k dt n-1,-1 n-1,0 0 + n · (B (t) - B (t)) · w + + n-1,0 n-1,1 1 + n · (B (t) - B (t)) · w + + n-1,1 n-1,2 2 + n · (B (t) - B (t)) · w + + n-1,2 n-1,3 3 + ... diff --git a/docs/images/snippets/derivatives/12fa7f83f055ef2078cc9f04e1468663.ascii b/docs/images/snippets/derivatives/12fa7f83f055ef2078cc9f04e1468663.ascii index 16fd8bea..200b9352 100644 --- a/docs/images/snippets/derivatives/12fa7f83f055ef2078cc9f04e1468663.ascii +++ b/docs/images/snippets/derivatives/12fa7f83f055ef2078cc9f04e1468663.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - kn! k-1 n-k (n-k)n! k n-1-k - ... = ──────── t (1-t) - ──────── t (1-t) - k!(n-k)! k!(n-k)! + kn! k-1 n-k (n-k)n! k n-1-k +... = ──────── t (1-t) - ──────── t (1-t) + k!(n-k)! k!(n-k)! diff --git a/docs/images/snippets/derivatives/153d99ce571bd664945394a1203a9eba.ascii b/docs/images/snippets/derivatives/153d99ce571bd664945394a1203a9eba.ascii index 6bafb566..d4369ad4 100644 --- a/docs/images/snippets/derivatives/153d99ce571bd664945394a1203a9eba.ascii +++ b/docs/images/snippets/derivatives/153d99ce571bd664945394a1203a9eba.ascii @@ -1,5 +1,4 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - + __ n n-i i Bézier(n,t) = ❯ \underset binomial term\underbrace\binomni · \ \underset polynomial term\underbrace(1-t) · t · \ \underset ‾‾ i=0 diff --git a/docs/images/snippets/derivatives/171357d936dee742b43b9ffb7600c742.ascii b/docs/images/snippets/derivatives/171357d936dee742b43b9ffb7600c742.ascii index c0002991..126c7673 100644 --- a/docs/images/snippets/derivatives/171357d936dee742b43b9ffb7600c742.ascii +++ b/docs/images/snippets/derivatives/171357d936dee742b43b9ffb7600c742.ascii @@ -1,5 +1,4 @@ -\usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - d __ n-1 __ n-1 - Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \underset вес производной \underbracen · (w - w ) - n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k + d __ n-1 __ n-1 +Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \underset вес производной \underbracen · (w - w ) + n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k diff --git a/docs/images/snippets/derivatives/18c6e782012234a2c7425204505c8888.ascii b/docs/images/snippets/derivatives/18c6e782012234a2c7425204505c8888.ascii deleted file mode 100644 index 8f08c5ad..00000000 --- a/docs/images/snippets/derivatives/18c6e782012234a2c7425204505c8888.ascii +++ /dev/null @@ -1,13 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - n · B (t) · w + - n-1,-1 0 - n · B (t) · w - n · B (t) · w + - n-1,\colorblue0 1 n-1,\colorblue0 0 - n · B (t) · w - n · B (t) · w + - n-1,\colorred1 2 n-1,\colorred1 1 - n · B (t) · w - n · B (t) · w + - n-1,\colormagenta2 3 n-1,\colormagenta2 2 - ... - n · B (t) · w + - n-1,3 3 - ... diff --git a/docs/images/snippets/derivatives/2368534c6e964e6d4a54904cc99b8986.ascii b/docs/images/snippets/derivatives/2368534c6e964e6d4a54904cc99b8986.ascii index 4c419367..131b5df7 100644 --- a/docs/images/snippets/derivatives/2368534c6e964e6d4a54904cc99b8986.ascii +++ b/docs/images/snippets/derivatives/2368534c6e964e6d4a54904cc99b8986.ascii @@ -1,5 +1,4 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - + __ k k-i i Bézier'(n,t) = ❯ \underset binomial term\underbrace\binomki · \ \underset polynomial term\underbrace(1-t) · t · \ \underset derivative ‾‾ i=0 diff --git a/docs/images/snippets/derivatives/2d733684f81b65a42c4cdb3f1e589c8b.ascii b/docs/images/snippets/derivatives/2d733684f81b65a42c4cdb3f1e589c8b.ascii index d5f076bb..a4d0f48d 100644 --- a/docs/images/snippets/derivatives/2d733684f81b65a42c4cdb3f1e589c8b.ascii +++ b/docs/images/snippets/derivatives/2d733684f81b65a42c4cdb3f1e589c8b.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B(n,t), w = {A,B,C,D} - B'(n,t), n = 3, w' = {A',B',C'} = {3 · (B-A), 3 · (C-B), 3 · (D-C)} - B''(n,t), n = 2, w'' = {A'',B''} = {2 · (B'-A'), 2 · (C'-B')} - B'''(n,t), n = 1, w''' = {A'''} = {1 · (B''-A'')} +B(n,t), w = {A,B,C,D} +B'(n,t), n = 3, w' = {A',B',C'} = {3 · (B-A), 3 · (C-B), 3 · (D-C)} +B''(n,t), n = 2, w'' = {A'',B''} = {2 · (B'-A'), 2 · (C'-B')} +B'''(n,t), n = 1, w''' = {A'''} = {1 · (B''-A'')} diff --git a/docs/images/snippets/derivatives/2e90e21710bf1bcbbfecbb464d27244a.ascii b/docs/images/snippets/derivatives/2e90e21710bf1bcbbfecbb464d27244a.ascii deleted file mode 100644 index fd6b9ea1..00000000 --- a/docs/images/snippets/derivatives/2e90e21710bf1bcbbfecbb464d27244a.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - - __ k k-i i -Bézier'(n,t) = ❯ \undersetбиноминальный термин\underbrace\binomki · \ \undersetполиноминальный термин\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetвес производной\underbracen · (w - w ) , with k=n-1 - i+1 i diff --git a/docs/images/snippets/derivatives/2fc50617b6886534d1ab4638ed8a24ac.ascii b/docs/images/snippets/derivatives/2fc50617b6886534d1ab4638ed8a24ac.ascii deleted file mode 100644 index 38ca7e22..00000000 --- a/docs/images/snippets/derivatives/2fc50617b6886534d1ab4638ed8a24ac.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ k k-i i -Bézier'(n,t) = ❯ \undersetbinomial term\underbrace\binomki · \ \undersetpolynomial term\underbrace(1-t) · t · \ \undersetderivative - ‾‾ i=0 - weight\underbracen · (w - w ) , with k=n-1 - i+1 i diff --git a/docs/images/snippets/derivatives/3cd7b36839a248eb35f0b678d7bf5508.ascii b/docs/images/snippets/derivatives/3cd7b36839a248eb35f0b678d7bf5508.ascii index 17eccb78..b7569c1a 100644 --- a/docs/images/snippets/derivatives/3cd7b36839a248eb35f0b678d7bf5508.ascii +++ b/docs/images/snippets/derivatives/3cd7b36839a248eb35f0b678d7bf5508.ascii @@ -1,5 +1,4 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - + __ n n-i i Bézier(n,t) = ❯ \underset биноминальный термин\underbrace\binomni · \ \underset полиноминальный термин\underbrace(1-t) · t · \ \underset ‾‾ i=0 diff --git a/docs/images/snippets/derivatives/4eeb75f5de2d13a39f894625d3222443.ascii b/docs/images/snippets/derivatives/4eeb75f5de2d13a39f894625d3222443.ascii index fec66bd6..b4926ea3 100644 --- a/docs/images/snippets/derivatives/4eeb75f5de2d13a39f894625d3222443.ascii +++ b/docs/images/snippets/derivatives/4eeb75f5de2d13a39f894625d3222443.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - d __ n-1 __ n-1 - Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \underset derivative weights \underbracen · (w - w ) - n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k + d __ n-1 __ n-1 +Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \underset derivative weights \underbracen · (w - w ) + n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k diff --git a/docs/images/snippets/derivatives/501494295f07ba5049286489206d98f0.ascii b/docs/images/snippets/derivatives/501494295f07ba5049286489206d98f0.ascii deleted file mode 100644 index b3ba22fe..00000000 --- a/docs/images/snippets/derivatives/501494295f07ba5049286489206d98f0.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i -Bézier(n,t) = ❯ \undersetбиноминальный термин\underbrace\binomni · \ \undersetполиноминальный термин\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetвес\underbracew - i diff --git a/docs/images/snippets/derivatives/50616f9c922967c0c9c179af9b091947.ascii b/docs/images/snippets/derivatives/50616f9c922967c0c9c179af9b091947.ascii deleted file mode 100644 index 43a04070..00000000 --- a/docs/images/snippets/derivatives/50616f9c922967c0c9c179af9b091947.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - d __ n-1 __ n-1 - Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \undersetderivative weights \underbracen · (w - w ) - n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k diff --git a/docs/images/snippets/derivatives/5d7af72e00fb0390af5281d918d77055.ascii b/docs/images/snippets/derivatives/5d7af72e00fb0390af5281d918d77055.ascii deleted file mode 100644 index df3609a0..00000000 --- a/docs/images/snippets/derivatives/5d7af72e00fb0390af5281d918d77055.ascii +++ /dev/null @@ -1,6 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - d -Bézier (t) ── = n · B (t) · (w - w ) + n · B (t) · (w - w ) + n · B (t) · (w - w ) + - n,k dt (n-1),\colorblue0 1 0 (n-1),\colorred1 2 1 (n-1),\colormagenta2 3 2 - ... diff --git a/docs/images/snippets/derivatives/64c06c61727d0912a67c0f287a395e47.ascii b/docs/images/snippets/derivatives/64c06c61727d0912a67c0f287a395e47.ascii index 6acb8f0e..d65c0018 100644 --- a/docs/images/snippets/derivatives/64c06c61727d0912a67c0f287a395e47.ascii +++ b/docs/images/snippets/derivatives/64c06c61727d0912a67c0f287a395e47.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - n! k-1 n-k (n-k)n! k n-1-k - ... = ──────────── t (1-t) - ──────── t (1-t) - (k-1)!(n-k)! k!(n-k)! - ╭ (n-1)! k-1 n-k (n-k)(n-1)! k n-1-k ╮ - ... = n │ ──────────── t (1-t) - ─────────── t (1-t) │ - ╰ (k-1)!(n-k)! k!(n-k)! ╯ - ╭ (n-1)! (k-1) (n-1)-(k-1) (n-1)! k (n-1)-k ╮ - ... = n │ ──────────────────── t (1-t) - ──────────── t (1-t) │ - ╰ (k-1)!((n-1)-(k-1))! k!((n-1)-k)! ╯ + n! k-1 n-k (n-k)n! k n-1-k +... = ──────────── t (1-t) - ──────── t (1-t) + (k-1)!(n-k)! k!(n-k)! + ╭ (n-1)! k-1 n-k (n-k)(n-1)! k n-1-k ╮ +... = n │ ──────────── t (1-t) - ─────────── t (1-t) │ + ╰ (k-1)!(n-k)! k!(n-k)! ╯ + ╭ (n-1)! (k-1) (n-1)-(k-1) (n-1)! k (n-1)-k ╮ +... = n │ ──────────────────── t (1-t) - ──────────── t (1-t) │ + ╰ (k-1)!((n-1)-(k-1))! k!((n-1)-k)! ╯ diff --git a/docs/images/snippets/derivatives/67ca2710769505572e097ffb40de099f.ascii b/docs/images/snippets/derivatives/67ca2710769505572e097ffb40de099f.ascii deleted file mode 100644 index d6bd6578..00000000 --- a/docs/images/snippets/derivatives/67ca2710769505572e097ffb40de099f.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n-1 - Bézier'(n,t) = ❯ Bézier(n-1,t) · n · (w -w ) - ‾‾ i=0 i i+1 i diff --git a/docs/images/snippets/derivatives/6a3672344bb571eadb72669f60a93ff4.ascii b/docs/images/snippets/derivatives/6a3672344bb571eadb72669f60a93ff4.ascii index c4b9ddf2..f348c233 100644 --- a/docs/images/snippets/derivatives/6a3672344bb571eadb72669f60a93ff4.ascii +++ b/docs/images/snippets/derivatives/6a3672344bb571eadb72669f60a93ff4.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ x! y x-y x! k x-k ╮ - ... = n │ ──────── t (1-t) - ──────── t (1-t) │ , with x=n-1, y=k-1 - ╰ y!(x-y)! k!(x-k)! ╯ - ... = n (B (t) - B (t)) - (n-1),(k-1) (n-1),k + ╭ x! y x-y x! k x-k ╮ +... = n │ ──────── t (1-t) - ──────── t (1-t) │ , with x=n-1, y=k-1 + ╰ y!(x-y)! k!(x-k)! ╯ +... = n (B (t) - B (t)) + (n-1),(k-1) (n-1),k diff --git a/docs/images/snippets/derivatives/8324bf1885267fe157bf316e261d1b30.ascii b/docs/images/snippets/derivatives/8324bf1885267fe157bf316e261d1b30.ascii deleted file mode 100644 index 2df56415..00000000 --- a/docs/images/snippets/derivatives/8324bf1885267fe157bf316e261d1b30.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ k k-i i -Bézier'(n,t) = ❯ \undersetбиноминальный термин\underbrace\binomki · \ \undersetполиноминальный термин\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetвес производной\underbracen · (w - w ) , with k=n-1 - i+1 i diff --git a/docs/images/snippets/derivatives/869b60a8e6b992e6f62bc6a50b36deeb.ascii b/docs/images/snippets/derivatives/869b60a8e6b992e6f62bc6a50b36deeb.ascii index 85f341ff..e631aa59 100644 --- a/docs/images/snippets/derivatives/869b60a8e6b992e6f62bc6a50b36deeb.ascii +++ b/docs/images/snippets/derivatives/869b60a8e6b992e6f62bc6a50b36deeb.ascii @@ -1,5 +1,4 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - + __ k k-i i Bézier'(n,t) = ❯ \underset биноминальный термин\underbrace\binomki · \ \underset полиноминальный термин\underbrace(1-t) · t · \ ‾‾ i=0 diff --git a/docs/images/snippets/derivatives/87403e5b0da3dcc4ceca74fae058fe69.ascii b/docs/images/snippets/derivatives/87403e5b0da3dcc4ceca74fae058fe69.ascii new file mode 100644 index 00000000..8188ec01 --- /dev/null +++ b/docs/images/snippets/derivatives/87403e5b0da3dcc4ceca74fae058fe69.ascii @@ -0,0 +1,12 @@ + +n · B (t) · w + + n-1,-1 0 +n · B (t) · w - n · B (t) · w + + n-1,\colorblue 0 1 n-1,\colorblue 0 0 +n · B (t) · w - n · B (t) · w + + n-1,\colorred 1 2 n-1,\colorred 1 1 +n · B (t) · w - n · B (t) · w + + n-1,\colormagenta 2 3 n-1,\colormagenta 2 2 +... - n · B (t) · w + + n-1,3 3 +... diff --git a/docs/images/snippets/derivatives/897cfd8648720dc21463a9358cc65ab4.ascii b/docs/images/snippets/derivatives/897cfd8648720dc21463a9358cc65ab4.ascii deleted file mode 100644 index 27521999..00000000 --- a/docs/images/snippets/derivatives/897cfd8648720dc21463a9358cc65ab4.ascii +++ /dev/null @@ -1,14 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - Bézier (t) = B (t) · w + B (t) · w + B (t) · w + B (t) · w + ... - n,k n,0 0 n,1 1 n,2 2 n,3 3 - d - Bézier (t) ── = n · (B (t) - B (t)) · w + - n,k dt n-1,-1 n-1,0 0 - n · (B (t) - B (t)) · w + - n-1,0 n-1,1 1 - n · (B (t) - B (t)) · w + - n-1,1 n-1,2 2 - n · (B (t) - B (t)) · w + - n-1,2 n-1,3 3 - ... diff --git a/docs/images/snippets/derivatives/8f78fdb9ef54b1bc4dbc00f07263cc97.ascii b/docs/images/snippets/derivatives/8f78fdb9ef54b1bc4dbc00f07263cc97.ascii index 91f9ff0b..cada557c 100644 --- a/docs/images/snippets/derivatives/8f78fdb9ef54b1bc4dbc00f07263cc97.ascii +++ b/docs/images/snippets/derivatives/8f78fdb9ef54b1bc4dbc00f07263cc97.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n-1 - Bézier'(n,t) = ❯ Bézier(n-1,t) · n · (w -w ) - ‾‾ i=0 i i+1 i + __ n-1 +Bézier'(n,t) = ❯ Bézier(n-1,t) · n · (w -w ) + ‾‾ i=0 i i+1 i diff --git a/docs/images/snippets/derivatives/a7b79877822a8f60e45552dcafc0815d.ascii b/docs/images/snippets/derivatives/a7b79877822a8f60e45552dcafc0815d.ascii index 6585b9d4..5dedef5b 100644 --- a/docs/images/snippets/derivatives/a7b79877822a8f60e45552dcafc0815d.ascii +++ b/docs/images/snippets/derivatives/a7b79877822a8f60e45552dcafc0815d.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n-1 - Bézier'(n,t) = n · ❯ (b -b ) · Bézier(n-1,t) - ‾‾ i=0 i+1 i i + __ n-1 +Bézier'(n,t) = n · ❯ (b -b ) · Bézier(n-1,t) + ‾‾ i=0 i+1 i i diff --git a/docs/images/snippets/derivatives/a7c61e0e8b42010df6dab641c92ef13d.ascii b/docs/images/snippets/derivatives/a7c61e0e8b42010df6dab641c92ef13d.ascii deleted file mode 100644 index e95fb088..00000000 --- a/docs/images/snippets/derivatives/a7c61e0e8b42010df6dab641c92ef13d.ascii +++ /dev/null @@ -1,6 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - d -Bézier (t) ── = n · B (t) · (w - w ) + n · B (t) · (w - w ) + n · B (t) · (w - w ) + - n,k dt (n-1),\colorblue0 1 0 (n-1),\colorred1 2 1 (n-1),\colormagenta2 3 2 - ... diff --git a/docs/images/snippets/derivatives/a992185a346518b5ca159484019b6917.ascii b/docs/images/snippets/derivatives/a992185a346518b5ca159484019b6917.ascii index cd821581..52e4295d 100644 --- a/docs/images/snippets/derivatives/a992185a346518b5ca159484019b6917.ascii +++ b/docs/images/snippets/derivatives/a992185a346518b5ca159484019b6917.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - d ╭ n ╮ k n-k d - B (t) ── = │ │ t (1-t) ── - n,k dt ╰ k ╯ dt + d ╭ n ╮ k n-k d +B (t) ── = │ │ t (1-t) ── + n,k dt ╰ k ╯ dt diff --git a/docs/images/snippets/derivatives/b58fb122c5c8159938182c185f287142.ascii b/docs/images/snippets/derivatives/b58fb122c5c8159938182c185f287142.ascii deleted file mode 100644 index 596ece8d..00000000 --- a/docs/images/snippets/derivatives/b58fb122c5c8159938182c185f287142.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i -Bézier(n,t) = ❯ \undersetbinomial term\underbrace\binomni · \ \undersetpolynomial term\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetweight\underbracew - i diff --git a/docs/images/snippets/derivatives/bbe9d45ab271549dea5ef54982fcaaa5.ascii b/docs/images/snippets/derivatives/bbe9d45ab271549dea5ef54982fcaaa5.ascii deleted file mode 100644 index cd0ec93e..00000000 --- a/docs/images/snippets/derivatives/bbe9d45ab271549dea5ef54982fcaaa5.ascii +++ /dev/null @@ -1,7 +0,0 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - - __ n n-i i -Bézier(n,t) = ❯ \undersetбиноминальный термин\underbrace\binomni · \ \undersetполиноминальный термин\underbrace(1-t) · t · \ - ‾‾ i=0 - \undersetвес\underbracew - i diff --git a/docs/images/snippets/derivatives/bec017d11e19b45df562c5a223761a69.ascii b/docs/images/snippets/derivatives/bec017d11e19b45df562c5a223761a69.ascii index fe90410a..991a1856 100644 --- a/docs/images/snippets/derivatives/bec017d11e19b45df562c5a223761a69.ascii +++ b/docs/images/snippets/derivatives/bec017d11e19b45df562c5a223761a69.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ x! y x-y x! k x-k ╮ - ... = n │ ──────── t (1-t) - ──────── t (1-t) │ , где x=n-1, y=k-1 - ╰ y!(x-y)! k!(x-k)! ╯ - ... = n (B (t) - B (t)) - (n-1),(k-1) (n-1),k + ╭ x! y x-y x! k x-k ╮ +... = n │ ──────── t (1-t) - ──────── t (1-t) │ , где x=n-1, y=k-1 + ╰ y!(x-y)! k!(x-k)! ╯ +... = n (B (t) - B (t)) + (n-1),(k-1) (n-1),k diff --git a/docs/images/snippets/derivatives/c3ac18fe4ba0606a15bc111e52b17a9a.ascii b/docs/images/snippets/derivatives/c3ac18fe4ba0606a15bc111e52b17a9a.ascii index 69c7944f..eb23e332 100644 --- a/docs/images/snippets/derivatives/c3ac18fe4ba0606a15bc111e52b17a9a.ascii +++ b/docs/images/snippets/derivatives/c3ac18fe4ba0606a15bc111e52b17a9a.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ n ╮ k-1 n-k k n-k-1 - ... = │ │ (k · t (1-t) + t · (1-t) · (n-k) · -1) - ╰ k ╯ + ╭ n ╮ k-1 n-k k n-k-1 +... = │ │ (k · t (1-t) + t · (1-t) · (n-k) · -1) + ╰ k ╯ diff --git a/docs/images/snippets/derivatives/d575699ab7d13c62f47d3071c0b00da3.ascii b/docs/images/snippets/derivatives/d575699ab7d13c62f47d3071c0b00da3.ascii deleted file mode 100644 index 49ef779a..00000000 --- a/docs/images/snippets/derivatives/d575699ab7d13c62f47d3071c0b00da3.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - d __ n-1 __ n-1 - Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \undersetвес производной \underbracen · (w - w ) - n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k diff --git a/docs/images/snippets/derivatives/de486728b41299269cc990ed09d5d286.ascii b/docs/images/snippets/derivatives/de486728b41299269cc990ed09d5d286.ascii new file mode 100644 index 00000000..1f4ea8ff --- /dev/null +++ b/docs/images/snippets/derivatives/de486728b41299269cc990ed09d5d286.ascii @@ -0,0 +1,5 @@ + + d +Bézier (t) ── = n · B (t) · (w - w ) + n · B (t) · (w - w ) + n · B (t) · (w - w ) + n,k dt (n-1),\colorblue 0 1 0 (n-1),\colorred 1 2 1 (n-1),\colormagenta 2 3 2 + + ... diff --git a/docs/images/snippets/derivatives/f00423aaceac6ade478aba2a761664d8.ascii b/docs/images/snippets/derivatives/f00423aaceac6ade478aba2a761664d8.ascii new file mode 100644 index 00000000..f4c4efe2 --- /dev/null +++ b/docs/images/snippets/derivatives/f00423aaceac6ade478aba2a761664d8.ascii @@ -0,0 +1,8 @@ + +n · B (t) · w - n · B (t) · w + + n-1,\colorblue 0 1 n-1,\colorblue 0 0 +n · B (t) · w - n · B (t) · w + + n-1,\colorred 1 2 n-1,\colorred 1 1 +n · B (t) · w - n · B (t) · w + + n-1,\colormagenta 2 3 n-1,\colormagenta 2 2 +... diff --git a/docs/images/snippets/derivatives/f29a9d52897d2060a0c8a37073ed04fc.ascii b/docs/images/snippets/derivatives/f29a9d52897d2060a0c8a37073ed04fc.ascii deleted file mode 100644 index f600e037..00000000 --- a/docs/images/snippets/derivatives/f29a9d52897d2060a0c8a37073ed04fc.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - n · B (t) · w - n · B (t) · w + - n-1,\colorblue0 1 n-1,\colorblue0 0 - n · B (t) · w - n · B (t) · w + - n-1,\colorred1 2 n-1,\colorred1 1 - n · B (t) · w - n · B (t) · w + - n-1,\colormagenta2 3 n-1,\colormagenta2 2 - ... diff --git a/docs/images/snippets/derivatives/f67d2d379ba6dfaa7f7686a7d1eae367.ascii b/docs/images/snippets/derivatives/f67d2d379ba6dfaa7f7686a7d1eae367.ascii deleted file mode 100644 index c1292623..00000000 --- a/docs/images/snippets/derivatives/f67d2d379ba6dfaa7f7686a7d1eae367.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n-1 - Bézier'(n,t) = n · ❯ (b -b ) · Bézier(n-1,t) - ‾‾ i=0 i+1 i i diff --git a/docs/images/snippets/explanation/05c2d5954eb9dec5ce9f6eb7e89f1e0c.ascii b/docs/images/snippets/explanation/05c2d5954eb9dec5ce9f6eb7e89f1e0c.ascii deleted file mode 100644 index 73a8e836..00000000 --- a/docs/images/snippets/explanation/05c2d5954eb9dec5ce9f6eb7e89f1e0c.ascii +++ /dev/null @@ -1,8 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - линийный= \colorbluea + \colorredb - квадратый= \colorbluea · \colorbluea + \colorbluea · \colorredb + \colorredb · \colorredb -кубический= \colorbluea · \colorbluea · \colorbluea + \colorbluea · \colorbluea · \colorredb + \colorbluea · \colorredb · \colorredb + \colorred - - - b · \colorredb · \colorredb diff --git a/docs/images/snippets/explanation/0cc876c56200446c60114c1b0eeeb2cc.ascii b/docs/images/snippets/explanation/0cc876c56200446c60114c1b0eeeb2cc.ascii index bdae38d7..cb4bfb02 100644 --- a/docs/images/snippets/explanation/0cc876c56200446c60114c1b0eeeb2cc.ascii +++ b/docs/images/snippets/explanation/0cc876c56200446c60114c1b0eeeb2cc.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - f(x) = cos (x) +f(x) = cos (x) diff --git a/docs/images/snippets/explanation/2493468e73b73f43eba8f66f0c189d1a.ascii b/docs/images/snippets/explanation/2493468e73b73f43eba8f66f0c189d1a.ascii deleted file mode 100644 index a4711580..00000000 --- a/docs/images/snippets/explanation/2493468e73b73f43eba8f66f0c189d1a.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - linear= (1-t) + t - 2 2 - square= (1-t) + 2 · (1-t) · t + t - 3 2 2 3 - cubic= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/explanation/263d97ef5db7e823d8d374a2c91b3bf0.ascii b/docs/images/snippets/explanation/263d97ef5db7e823d8d374a2c91b3bf0.ascii new file mode 100644 index 00000000..c6608fed --- /dev/null +++ b/docs/images/snippets/explanation/263d97ef5db7e823d8d374a2c91b3bf0.ascii @@ -0,0 +1,7 @@ + +1次= \colorblue a + \colorred b +2次= \colorblue a · \colorblue a + \colorblue a · \colorred b + \colorred b · \colorred b +3次= \colorblue a · \colorblue a · \colorblue a + \colorblue a · \colorblue a · \colorred b + \colorblue a · \colorred b · \colorred b + \colorr + + + ed b · \colorred b · \colorred b diff --git a/docs/images/snippets/explanation/29695045f04fd06c75bfda7845121213.ascii b/docs/images/snippets/explanation/29695045f04fd06c75bfda7845121213.ascii index cd4a8901..5e5eca2b 100644 --- a/docs/images/snippets/explanation/29695045f04fd06c75bfda7845121213.ascii +++ b/docs/images/snippets/explanation/29695045f04fd06c75bfda7845121213.ascii @@ -1,5 +1,4 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - __ n n-i i - Bézier(n,t) = ❯ \underset 二項係数部分の項\underbrace\binomni · \ \underset 多項式部分の項\underbrace(1-t) · t - ‾‾ i=0 + __ n n-i i +Bézier(n,t) = ❯ \underset 二項係数部分の項\underbrace\binomni · \ \underset 多項式部分の項\underbrace(1-t) · t + ‾‾ i=0 diff --git a/docs/images/snippets/explanation/2c47081c2a9c20d2110f13daa482a3ab.ascii b/docs/images/snippets/explanation/2c47081c2a9c20d2110f13daa482a3ab.ascii deleted file mode 100644 index bfa97e16..00000000 --- a/docs/images/snippets/explanation/2c47081c2a9c20d2110f13daa482a3ab.ascii +++ /dev/null @@ -1,8 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - -linear= \colorbluea + \colorredb -square= \colorbluea · \colorbluea + \colorbluea · \colorredb + \colorredb · \colorredb - cubic= \colorbluea · \colorbluea · \colorbluea + \colorbluea · \colorbluea · \colorredb + \colorbluea · \colorredb · \colorredb + \colorredb · - - - \colorredb · \colorredb diff --git a/docs/images/snippets/explanation/39330ef5591cf0f3205564ad47255d4f.ascii b/docs/images/snippets/explanation/39330ef5591cf0f3205564ad47255d4f.ascii index a1de268f..cc7194d1 100644 --- a/docs/images/snippets/explanation/39330ef5591cf0f3205564ad47255d4f.ascii +++ b/docs/images/snippets/explanation/39330ef5591cf0f3205564ad47255d4f.ascii @@ -1,7 +1,6 @@ -\usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - линийный= (1-t) + t - 2 2 - квадратый= (1-t) + 2 · (1-t) · t + t - 3 2 2 3 - кубический= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t + линийный= (1-t) + t + 2 2 + квадратый= (1-t) + 2 · (1-t) · t + t + 3 2 2 3 +кубический= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/explanation/39d33ea94e7527ed221a809ca6054174.ascii b/docs/images/snippets/explanation/39d33ea94e7527ed221a809ca6054174.ascii index 389a9490..ab3fd7a2 100644 --- a/docs/images/snippets/explanation/39d33ea94e7527ed221a809ca6054174.ascii +++ b/docs/images/snippets/explanation/39d33ea94e7527ed221a809ca6054174.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n n-i i - Bézier(n,t) = ❯ \underset binomial term\underbrace\binomni · \ \underset polynomial term\underbrace(1-t) · t - ‾‾ i=0 + __ n n-i i +Bézier(n,t) = ❯ \underset binomial term\underbrace\binomni · \ \underset polynomial term\underbrace(1-t) · t + ‾‾ i=0 diff --git a/docs/images/snippets/explanation/4319b2e361960c842a4308a610a35048.ascii b/docs/images/snippets/explanation/4319b2e361960c842a4308a610a35048.ascii new file mode 100644 index 00000000..04bb3b36 --- /dev/null +++ b/docs/images/snippets/explanation/4319b2e361960c842a4308a610a35048.ascii @@ -0,0 +1,7 @@ + +linear= \colorblue a + \colorred b +square= \colorblue a · \colorblue a + \colorblue a · \colorred b + \colorred b · \colorred b + cubic= \colorblue a · \colorblue a · \colorblue a + \colorblue a · \colorblue a · \colorred b + \colorblue a · \colorred b · \colorred b + \co + + + lorred b · \colorred b · \colorred b diff --git a/docs/images/snippets/explanation/4bf2d790d2f50bf7767c948e0b9f9822.ascii b/docs/images/snippets/explanation/4bf2d790d2f50bf7767c948e0b9f9822.ascii index de7e8f83..f94ed7fb 100644 --- a/docs/images/snippets/explanation/4bf2d790d2f50bf7767c948e0b9f9822.ascii +++ b/docs/images/snippets/explanation/4bf2d790d2f50bf7767c948e0b9f9822.ascii @@ -1,7 +1,6 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - 1次= (1-t) + t - 2 2 - 2次= (1-t) + 2 · (1-t) · t + t - 3 2 2 3 - 3次= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t +1次= (1-t) + t + 2 2 +2次= (1-t) + 2 · (1-t) · t + t + 3 2 2 3 +3次= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/explanation/4def87a6683264d420f84562776f4b6c.ascii b/docs/images/snippets/explanation/4def87a6683264d420f84562776f4b6c.ascii deleted file mode 100644 index 3cc491c8..00000000 --- a/docs/images/snippets/explanation/4def87a6683264d420f84562776f4b6c.ascii +++ /dev/null @@ -1,8 +0,0 @@ - \usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - -1次= \colorbluea + \colorredb -2次= \colorbluea · \colorbluea + \colorbluea · \colorredb + \colorredb · \colorredb -3次= \colorbluea · \colorbluea · \colorbluea + \colorbluea · \colorbluea · \colorredb + \colorbluea · \colorredb · \colorredb + \colorredb · \c - - - olorredb · \colorredb diff --git a/docs/images/snippets/explanation/516214a32b170186c3f3f4d34c9fe8fe.ascii b/docs/images/snippets/explanation/516214a32b170186c3f3f4d34c9fe8fe.ascii new file mode 100644 index 00000000..eadf2ec1 --- /dev/null +++ b/docs/images/snippets/explanation/516214a32b170186c3f3f4d34c9fe8fe.ascii @@ -0,0 +1,7 @@ + + линийный= \colorblue a + \colorred b + квадратый= \colorblue a · \colorblue a + \colorblue a · \colorred b + \colorred b · \colorred b +кубический= \colorblue a · \colorblue a · \colorblue a + \colorblue a · \colorblue a · \colorred b + \colorblue a · \colorred b · \colorred b + + + + \colorred b · \colorred b · \colorred b diff --git a/docs/images/snippets/explanation/668d140df9db486e5ff2d7c127eaa9d4.ascii b/docs/images/snippets/explanation/668d140df9db486e5ff2d7c127eaa9d4.ascii index ef6d3f1a..38d58a41 100644 --- a/docs/images/snippets/explanation/668d140df9db486e5ff2d7c127eaa9d4.ascii +++ b/docs/images/snippets/explanation/668d140df9db486e5ff2d7c127eaa9d4.ascii @@ -1,5 +1,4 @@ -\usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - __ n n-i i - Bézier(n,t) = ❯ \underset биноминальный термин\underbrace\binomni · \ \underset полиноминальный термин\underbrace(1-t) · t - ‾‾ i=0 + __ n n-i i +Bézier(n,t) = ❯ \underset биноминальный термин\underbrace\binomni · \ \underset полиноминальный термин\underbrace(1-t) · t + ‾‾ i=0 diff --git a/docs/images/snippets/explanation/6914ba615733c387251682db7a3db045.ascii b/docs/images/snippets/explanation/6914ba615733c387251682db7a3db045.ascii index a359fa43..c1d682cb 100644 --- a/docs/images/snippets/explanation/6914ba615733c387251682db7a3db045.ascii +++ b/docs/images/snippets/explanation/6914ba615733c387251682db7a3db045.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - { x = cos (t) - y = sin (t) +{ x = cos (t) + y = sin (t) diff --git a/docs/images/snippets/explanation/6aa5d4e20e83be1c95eaad792517dde9.ascii b/docs/images/snippets/explanation/6aa5d4e20e83be1c95eaad792517dde9.ascii deleted file mode 100644 index 599535f2..00000000 --- a/docs/images/snippets/explanation/6aa5d4e20e83be1c95eaad792517dde9.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - 1次= 1 + 1 - 2次= 1 + 2 + 1 - 3次= 1 + 3 + 3 + 1 - 4次= 1 + 4 + 6 + 4 + 1 diff --git a/docs/images/snippets/explanation/6d58ec36bfb3fcff24248dc46889428a.ascii b/docs/images/snippets/explanation/6d58ec36bfb3fcff24248dc46889428a.ascii deleted file mode 100644 index 93178bfc..00000000 --- a/docs/images/snippets/explanation/6d58ec36bfb3fcff24248dc46889428a.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - линийный= 1 + 1 - квадратый= 1 + 2 + 1 - кубический= 1 + 3 + 3 + 1 - квартический= 1 + 4 + 6 + 4 + 1 diff --git a/docs/images/snippets/explanation/79832780f9209be5569447c4d988e54b.ascii b/docs/images/snippets/explanation/79832780f9209be5569447c4d988e54b.ascii deleted file mode 100644 index 7e4a09a6..00000000 --- a/docs/images/snippets/explanation/79832780f9209be5569447c4d988e54b.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - __ n n-i i - Bézier(n,t) = ❯ \underset 二項係数部分の項\underbrace\binomni · \ \underset 多項式部分の項\underbrace(1-t) · t - ‾‾ i=0 diff --git a/docs/images/snippets/explanation/7a44f3eaa167a5022e2281c62e90fff8.ascii b/docs/images/snippets/explanation/7a44f3eaa167a5022e2281c62e90fff8.ascii deleted file mode 100644 index bf17a8c5..00000000 --- a/docs/images/snippets/explanation/7a44f3eaa167a5022e2281c62e90fff8.ascii +++ /dev/null @@ -1,8 +0,0 @@ - \usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - - линийный= \colorbluea + \colorredb - квадратый= \colorbluea · \colorbluea + \colorbluea · \colorredb + \colorredb · \colorredb -кубический= \colorbluea · \colorbluea · \colorbluea + \colorbluea · \colorbluea · \colorredb + \colorbluea · \colorredb · \colorredb + \colorre - - - db · \colorredb · \colorredb diff --git a/docs/images/snippets/explanation/7acc94ec70f053fd10dab69d424b02a6.ascii b/docs/images/snippets/explanation/7acc94ec70f053fd10dab69d424b02a6.ascii index 204db2a1..c0b35997 100644 --- a/docs/images/snippets/explanation/7acc94ec70f053fd10dab69d424b02a6.ascii +++ b/docs/images/snippets/explanation/7acc94ec70f053fd10dab69d424b02a6.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ f (t) = cos (t) - ╡ a - │ f (t) = sin (t) - ╰ b +╭ f (t) = cos (t) +╡ a +│ f (t) = sin (t) +╰ b diff --git a/docs/images/snippets/explanation/7f74178029422a35267fd033b392fe4c.ascii b/docs/images/snippets/explanation/7f74178029422a35267fd033b392fe4c.ascii index ff567ba6..95ff9f4a 100644 --- a/docs/images/snippets/explanation/7f74178029422a35267fd033b392fe4c.ascii +++ b/docs/images/snippets/explanation/7f74178029422a35267fd033b392fe4c.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - linear= (1-t) + t - 2 2 - square= (1-t) + 2 · (1-t) · t + t - 3 2 2 3 - cubic= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t +linear= (1-t) + t + 2 2 +square= (1-t) + 2 · (1-t) · t + t + 3 2 2 3 + cubic= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/explanation/855a34c7f72733be6529c3fb33fa1a23.ascii b/docs/images/snippets/explanation/855a34c7f72733be6529c3fb33fa1a23.ascii index 9c13e82c..25d963d2 100644 --- a/docs/images/snippets/explanation/855a34c7f72733be6529c3fb33fa1a23.ascii +++ b/docs/images/snippets/explanation/855a34c7f72733be6529c3fb33fa1a23.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 - f(x) = a · x + b · x + c · x + d + 3 2 +f(x) = a · x + b · x + c · x + d diff --git a/docs/images/snippets/explanation/8986c536df8153b30197c3a5407d233a.ascii b/docs/images/snippets/explanation/8986c536df8153b30197c3a5407d233a.ascii deleted file mode 100644 index 271cd244..00000000 --- a/docs/images/snippets/explanation/8986c536df8153b30197c3a5407d233a.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - linear= 1 + 1 - square= 1 + 2 + 1 - cubic= 1 + 3 + 3 + 1 - quartic= 1 + 4 + 6 + 4 + 1 diff --git a/docs/images/snippets/explanation/9229934d71b0b02921bc92594ef11a98.ascii b/docs/images/snippets/explanation/9229934d71b0b02921bc92594ef11a98.ascii deleted file mode 100644 index bfd645d7..00000000 --- a/docs/images/snippets/explanation/9229934d71b0b02921bc92594ef11a98.ascii +++ /dev/null @@ -1,8 +0,0 @@ - \usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - -1次= \colorbluea + \colorredb -2次= \colorbluea · \colorbluea + \colorbluea · \colorredb + \colorredb · \colorredb -3次= \colorbluea · \colorbluea · \colorbluea + \colorbluea · \colorbluea · \colorredb + \colorbluea · \colorredb · \colorredb + \colorredb · \co - - - lorredb · \colorredb diff --git a/docs/images/snippets/explanation/9734aff037ac23a73504ff7cc846eab7.ascii b/docs/images/snippets/explanation/9734aff037ac23a73504ff7cc846eab7.ascii index 5c482a92..24d122f3 100644 --- a/docs/images/snippets/explanation/9734aff037ac23a73504ff7cc846eab7.ascii +++ b/docs/images/snippets/explanation/9734aff037ac23a73504ff7cc846eab7.ascii @@ -1,6 +1,5 @@ -\usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - линийный= 1 + 1 - квадратый= 1 + 2 + 1 - кубический= 1 + 3 + 3 + 1 - квартический= 1 + 4 + 6 + 4 + 1 + линийный= 1 + 1 + квадратый= 1 + 2 + 1 + кубический= 1 + 3 + 3 + 1 +квартический= 1 + 4 + 6 + 4 + 1 diff --git a/docs/images/snippets/explanation/9c921b7b8a8db831f787c1329e29f7cb.ascii b/docs/images/snippets/explanation/9c921b7b8a8db831f787c1329e29f7cb.ascii index 9284a2ad..e4bd3954 100644 --- a/docs/images/snippets/explanation/9c921b7b8a8db831f787c1329e29f7cb.ascii +++ b/docs/images/snippets/explanation/9c921b7b8a8db831f787c1329e29f7cb.ascii @@ -1,6 +1,5 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - 1次= 1 + 1 - 2次= 1 + 2 + 1 - 3次= 1 + 3 + 3 + 1 - 4次= 1 + 4 + 6 + 4 + 1 +1次= 1 + 1 +2次= 1 + 2 + 1 +3次= 1 + 3 + 3 + 1 +4次= 1 + 4 + 6 + 4 + 1 diff --git a/docs/images/snippets/explanation/a2891980850ddbb27d308ac112d69f74.ascii b/docs/images/snippets/explanation/a2891980850ddbb27d308ac112d69f74.ascii index 0fdf3d82..9ebd5996 100644 --- a/docs/images/snippets/explanation/a2891980850ddbb27d308ac112d69f74.ascii +++ b/docs/images/snippets/explanation/a2891980850ddbb27d308ac112d69f74.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - f(a) = cos (a) - f(b) = sin (b) +f(a) = cos (a) +f(b) = sin (b) diff --git a/docs/images/snippets/explanation/a5bb1312adc5e9e23bee6b47555a6e8f.ascii b/docs/images/snippets/explanation/a5bb1312adc5e9e23bee6b47555a6e8f.ascii deleted file mode 100644 index 553bf649..00000000 --- a/docs/images/snippets/explanation/a5bb1312adc5e9e23bee6b47555a6e8f.ascii +++ /dev/null @@ -1,8 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - -linear= \colorbluea + \colorredb -square= \colorbluea · \colorbluea + \colorbluea · \colorredb + \colorredb · \colorredb - cubic= \colorbluea · \colorbluea · \colorbluea + \colorbluea · \colorbluea · \colorredb + \colorbluea · \colorredb · \colorredb + \colorredb - - - · \colorredb · \colorredb diff --git a/docs/images/snippets/explanation/adc7729f7872d71f3fbb1a79741ce10f.ascii b/docs/images/snippets/explanation/adc7729f7872d71f3fbb1a79741ce10f.ascii deleted file mode 100644 index 92184ce2..00000000 --- a/docs/images/snippets/explanation/adc7729f7872d71f3fbb1a79741ce10f.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - 1次= (1-t) + t - 2 2 - 2次= (1-t) + 2 · (1-t) · t + t - 3 2 2 3 - 3次= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/explanation/af40980136c291814e8970dc2a3d8e63.ascii b/docs/images/snippets/explanation/af40980136c291814e8970dc2a3d8e63.ascii index 271cd244..ae942567 100644 --- a/docs/images/snippets/explanation/af40980136c291814e8970dc2a3d8e63.ascii +++ b/docs/images/snippets/explanation/af40980136c291814e8970dc2a3d8e63.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - linear= 1 + 1 - square= 1 + 2 + 1 - cubic= 1 + 3 + 3 + 1 - quartic= 1 + 4 + 6 + 4 + 1 + linear= 1 + 1 + square= 1 + 2 + 1 + cubic= 1 + 3 + 3 + 1 +quartic= 1 + 4 + 6 + 4 + 1 diff --git a/docs/images/snippets/explanation/b5977078d36d847fb299cbe3e7e2c3ba.ascii b/docs/images/snippets/explanation/b5977078d36d847fb299cbe3e7e2c3ba.ascii deleted file mode 100644 index c5a4987d..00000000 --- a/docs/images/snippets/explanation/b5977078d36d847fb299cbe3e7e2c3ba.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - __ n n-i i - Bézier(n,t) = ❯ \underset二項係数部分の項\underbrace\binomni · \ \underset多項式部分の項\underbrace(1-t) · t - ‾‾ i=0 diff --git a/docs/images/snippets/explanation/dc48cdf8f492b44c7602eb64ce2b9986.ascii b/docs/images/snippets/explanation/dc48cdf8f492b44c7602eb64ce2b9986.ascii deleted file mode 100644 index c54bc68a..00000000 --- a/docs/images/snippets/explanation/dc48cdf8f492b44c7602eb64ce2b9986.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - линейный= (1-t) + t - 2 2 - квадратный= (1-t) + 2 · (1-t) · t + t - 3 2 2 3 - кубический= (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/explanation/f79dd2f2d992e22b8d057fdc641290b0.ascii b/docs/images/snippets/explanation/f79dd2f2d992e22b8d057fdc641290b0.ascii deleted file mode 100644 index 6c385df0..00000000 --- a/docs/images/snippets/explanation/f79dd2f2d992e22b8d057fdc641290b0.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i - Bézier(n,t) = ❯ \undersetbinomial term\underbrace\binomni · \ \undersetpolynomial term\underbrace(1-t) · t - ‾‾ i=0 diff --git a/docs/images/snippets/explanation/fa3ed9a4ab61d80ec175d29533b5728e.ascii b/docs/images/snippets/explanation/fa3ed9a4ab61d80ec175d29533b5728e.ascii deleted file mode 100644 index 2427f1f1..00000000 --- a/docs/images/snippets/explanation/fa3ed9a4ab61d80ec175d29533b5728e.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i - Bézier(n,t) = ❯ \undersetбиноминальный термин\underbrace\binomni · \ \undersetполиноминальный термин\underbrace(1-t) · t - ‾‾ i=0 diff --git a/docs/images/snippets/extended/08cd4a8bf4557862c095066728e6ed5e.ascii b/docs/images/snippets/extended/08cd4a8bf4557862c095066728e6ed5e.ascii index cb9ef993..6be5b647 100644 --- a/docs/images/snippets/extended/08cd4a8bf4557862c095066728e6ed5e.ascii +++ b/docs/images/snippets/extended/08cd4a8bf4557862c095066728e6ed5e.ascii @@ -1,4 +1,3 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - 混ぜ合わさった値 = a · 値 + (1 - a) · 値 - 1 2 +混ぜ合わさった値 = a · 値 + (1 - a) · 値 + 1 2 diff --git a/docs/images/snippets/extended/4e0fa763b173e3a683587acf83733353.ascii b/docs/images/snippets/extended/4e0fa763b173e3a683587acf83733353.ascii index a6d2320b..6c3fda37 100644 --- a/docs/images/snippets/extended/4e0fa763b173e3a683587acf83733353.ascii +++ b/docs/images/snippets/extended/4e0fa763b173e3a683587acf83733353.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - m = a · value + (1 - a) · value - 1 2 +m = a · value + (1 - a) · value + 1 2 diff --git a/docs/images/snippets/extended/5a7a12213ca36f2f833e638ea0174d4a.ascii b/docs/images/snippets/extended/5a7a12213ca36f2f833e638ea0174d4a.ascii index cf911ef2..71f36e59 100644 --- a/docs/images/snippets/extended/5a7a12213ca36f2f833e638ea0174d4a.ascii +++ b/docs/images/snippets/extended/5a7a12213ca36f2f833e638ea0174d4a.ascii @@ -1,4 +1,3 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - 混ぜ合わさった値 = a · 値 + b · 値 - 1 2 +混ぜ合わさった値 = a · 値 + b · 値 + 1 2 diff --git a/docs/images/snippets/extended/b0eb0b24e7fa29c545ab1479d2df0554.ascii b/docs/images/snippets/extended/b0eb0b24e7fa29c545ab1479d2df0554.ascii deleted file mode 100644 index 879cd5b7..00000000 --- a/docs/images/snippets/extended/b0eb0b24e7fa29c545ab1479d2df0554.ascii +++ /dev/null @@ -1,4 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - 混ぜ合わさった値 = a · 値 + b · 値 - 1 2 diff --git a/docs/images/snippets/extended/dfd6ded3f0addcf43e0a1581627a2220.ascii b/docs/images/snippets/extended/dfd6ded3f0addcf43e0a1581627a2220.ascii index bf6e7bb5..b1e3b874 100644 --- a/docs/images/snippets/extended/dfd6ded3f0addcf43e0a1581627a2220.ascii +++ b/docs/images/snippets/extended/dfd6ded3f0addcf43e0a1581627a2220.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - mixture = a · value + b · value - 1 2 +mixture = a · value + b · value + 1 2 diff --git a/docs/images/snippets/extended/e2e71b397009b51af8a3ee848bc727b4.ascii b/docs/images/snippets/extended/e2e71b397009b51af8a3ee848bc727b4.ascii deleted file mode 100644 index 026f43ad..00000000 --- a/docs/images/snippets/extended/e2e71b397009b51af8a3ee848bc727b4.ascii +++ /dev/null @@ -1,4 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - m = a · value + (1 - a) · value - 1 2 diff --git a/docs/images/snippets/extended/f56f59f3c0b057c8ec79a477e4e38bec.ascii b/docs/images/snippets/extended/f56f59f3c0b057c8ec79a477e4e38bec.ascii deleted file mode 100644 index 2e6e9bd6..00000000 --- a/docs/images/snippets/extended/f56f59f3c0b057c8ec79a477e4e38bec.ascii +++ /dev/null @@ -1,4 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - mixture = a · value + b · value - 1 2 diff --git a/docs/images/snippets/extended/fd520a6e2c7f39e90496e5cf494cce2e.ascii b/docs/images/snippets/extended/fd520a6e2c7f39e90496e5cf494cce2e.ascii deleted file mode 100644 index fa9b04d0..00000000 --- a/docs/images/snippets/extended/fd520a6e2c7f39e90496e5cf494cce2e.ascii +++ /dev/null @@ -1,4 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - 混ぜ合わさった値 = a · 値 + (1 - a) · 値 - 1 2 diff --git a/docs/images/snippets/extremities/1fab66c84e7df38a2edda147f939bd80.ascii b/docs/images/snippets/extremities/1fab66c84e7df38a2edda147f939bd80.ascii index ba19851b..db969d9c 100644 --- a/docs/images/snippets/extremities/1fab66c84e7df38a2edda147f939bd80.ascii +++ b/docs/images/snippets/extremities/1fab66c84e7df38a2edda147f939bd80.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - (b-a)t + a= 0, - (b-a)t= -a, - -a - t= ─── - b-a +(b-a)t + a= 0, + (b-a)t= -a, + -a + t= ─── + b-a diff --git a/docs/images/snippets/extremities/2c398b492aadc90eb4e4853fc20b23e9.ascii b/docs/images/snippets/extremities/2c398b492aadc90eb4e4853fc20b23e9.ascii deleted file mode 100644 index 14f1d267..00000000 --- a/docs/images/snippets/extremities/2c398b492aadc90eb4e4853fc20b23e9.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌────────┐ - │ 2 - 2 -b ±⟍│b - 4ac - Given f(t) = at + bt + c, f(t)=0 when t = ─────────────── - 2a diff --git a/docs/images/snippets/extremities/3125ab785fb039994582552790a2674b.ascii b/docs/images/snippets/extremities/3125ab785fb039994582552790a2674b.ascii index cae6081b..f6d40698 100644 --- a/docs/images/snippets/extremities/3125ab785fb039994582552790a2674b.ascii +++ b/docs/images/snippets/extremities/3125ab785fb039994582552790a2674b.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌────────┐ - │ 2 - 2 -b ±⟍│b - 4ac - Given f(t) = at + bt + c, f(t)=0 when t = ─────────────── - 2a + ┌────────┐ + │ 2 + 2 -b ±⟍│b - 4ac +Given f(t) = at + bt + c, f(t)=0 when t = ─────────────── + 2a diff --git a/docs/images/snippets/extremities/53e67a29f134bd561aca550a2091a196.ascii b/docs/images/snippets/extremities/53e67a29f134bd561aca550a2091a196.ascii index d9315d4c..1f3dca7e 100644 --- a/docs/images/snippets/extremities/53e67a29f134bd561aca550a2091a196.ascii +++ b/docs/images/snippets/extremities/53e67a29f134bd561aca550a2091a196.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - f (x ) - y n - x = x - ─────── - n+1 n f' (x ) - y n + f (x ) + y n +x = x - ─────── + n+1 n f' (x ) + y n diff --git a/docs/images/snippets/extremities/55e16ef652d30face0f6586b675a6c7b.ascii b/docs/images/snippets/extremities/55e16ef652d30face0f6586b675a6c7b.ascii index f1978f06..2bb7fefd 100644 --- a/docs/images/snippets/extremities/55e16ef652d30face0f6586b675a6c7b.ascii +++ b/docs/images/snippets/extremities/55e16ef652d30face0f6586b675a6c7b.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B'(t) = a(1-t) + b(t)= 0, - a - at + bt= 0, - (b-a)t + a= 0 +B'(t) = a(1-t) + b(t)= 0, + a - at + bt= 0, + (b-a)t + a= 0 diff --git a/docs/images/snippets/extremities/a6acf08f43aa1f48c08a40e76bdd2a31.ascii b/docs/images/snippets/extremities/a6acf08f43aa1f48c08a40e76bdd2a31.ascii index e01ab1b7..7e3d9487 100644 --- a/docs/images/snippets/extremities/a6acf08f43aa1f48c08a40e76bdd2a31.ascii +++ b/docs/images/snippets/extremities/a6acf08f43aa1f48c08a40e76bdd2a31.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - a= v -2v +v = 3(-p + 3p - 3p + p ) - 1 2 3 1 2 3 4 - b= 2(v -v ) = 6(p - 2p + p ) - 2 1 1 2 3 - c= v = 3(p -p ) - 1 2 1 +a= v -2v +v = 3(-p + 3p - 3p + p ) + 1 2 3 1 2 3 4 +b= 2(v -v ) = 6(p - 2p + p ) + 2 1 1 2 3 +c= v = 3(p -p ) + 1 2 1 diff --git a/docs/images/snippets/extremities/bf0ad4611c47f8548396e40595c02b55.ascii b/docs/images/snippets/extremities/bf0ad4611c47f8548396e40595c02b55.ascii index 98718442..12543453 100644 --- a/docs/images/snippets/extremities/bf0ad4611c47f8548396e40595c02b55.ascii +++ b/docs/images/snippets/extremities/bf0ad4611c47f8548396e40595c02b55.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B(t) uses { p ,p ,p ,p } - 1 2 3 4 - B'(t) uses { v ,v ,v }, where v = 3(p -p ), v = 3(p -p ), v = 3(p -p ) - 1 2 3 1 2 1 2 3 2 3 4 3 +B(t) uses { p ,p ,p ,p } + 1 2 3 4 +B'(t) uses { v ,v ,v }, where v = 3(p -p ), v = 3(p -p ), v = 3(p -p ) + 1 2 3 1 2 1 2 3 2 3 4 3 diff --git a/docs/images/snippets/extremities/c4858be225d004441b2aefedacda89a3.ascii b/docs/images/snippets/extremities/c4858be225d004441b2aefedacda89a3.ascii deleted file mode 100644 index 02bf844f..00000000 --- a/docs/images/snippets/extremities/c4858be225d004441b2aefedacda89a3.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - B(t) uses { p ,p ,p ,p } - 1 2 3 4 - B'(t) uses { v ,v ,v }, where v = 3(p -p ), v = 3(p -p ), v = 3(p -p ) - 1 2 3 1 2 1 2 3 2 3 4 3 diff --git a/docs/images/snippets/extremities/d1c65d927825f20c3c358d1ff96ce881.ascii b/docs/images/snippets/extremities/d1c65d927825f20c3c358d1ff96ce881.ascii index 7422ce79..b87bd499 100644 --- a/docs/images/snippets/extremities/d1c65d927825f20c3c358d1ff96ce881.ascii +++ b/docs/images/snippets/extremities/d1c65d927825f20c3c358d1ff96ce881.ascii @@ -1,17 +1,16 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 2 - B'(t)= v (1-t) + 2v (1-t)t + v t - 1 2 3 - 2 2 2 - ...= v (t - 2t + 1) + 2v (t-t ) + v t - 1 2 3 - 2 2 2 - ...= v t - 2v t + v + 2v t - 2v t + v t - 1 1 1 2 2 3 - 2 2 2 - ...= v t - 2v t + v t - 2v t + v + 2v t - 1 2 3 1 1 2 - 2 - ...= (v -2v +v )t + 2(v -v )t + v - 1 2 3 2 1 1 + 2 2 +B'(t)= v (1-t) + 2v (1-t)t + v t + 1 2 3 + 2 2 2 + ...= v (t - 2t + 1) + 2v (t-t ) + v t + 1 2 3 + 2 2 2 + ...= v t - 2v t + v + 2v t - 2v t + v t + 1 1 1 2 2 3 + 2 2 2 + ...= v t - 2v t + v t - 2v t + v + 2v t + 1 2 3 1 1 2 + 2 + ...= (v -2v +v )t + 2(v -v )t + v + 1 2 3 2 1 1 diff --git a/docs/images/snippets/extremities/d31432533bd7940545d4a269eefbabf2.ascii b/docs/images/snippets/extremities/d31432533bd7940545d4a269eefbabf2.ascii index 150a20ac..d4686e3b 100644 --- a/docs/images/snippets/extremities/d31432533bd7940545d4a269eefbabf2.ascii +++ b/docs/images/snippets/extremities/d31432533bd7940545d4a269eefbabf2.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 - very hard: solve at + bt + ct + d = 0 - 3 - easier: solve t + pt + q = 0 + 3 2 +very hard: solve at + bt + ct + d = 0 + 3 + easier: solve t + pt + q = 0 diff --git a/docs/images/snippets/inflections/2dbf3071d74e2ba37ab888aaa3c1a17c.ascii b/docs/images/snippets/inflections/2dbf3071d74e2ba37ab888aaa3c1a17c.ascii index 6e8fa798..d041dc4d 100644 --- a/docs/images/snippets/inflections/2dbf3071d74e2ba37ab888aaa3c1a17c.ascii +++ b/docs/images/snippets/inflections/2dbf3071d74e2ba37ab888aaa3c1a17c.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - a = x · y ╮ - 3 2 │ - b = x · y │ 2 - 4 2 ╞ C(t) = (-3a + 2b + 3c - d)t + (3a - b - 3c)t + (c - a) - c = x · y │ - 2 3 │ - d = x · y │ - 4 3 ╯ +a = x · y ╮ + 3 2 │ +b = x · y │ 2 + 4 2 ╞ C(t) = (-3a + 2b + 3c - d)t + (3a - b - 3c)t + (c - a) +c = x · y │ + 2 3 │ +d = x · y │ + 4 3 ╯ diff --git a/docs/images/snippets/inflections/35299f4eb8e0bed76b68c7beb2038031.ascii b/docs/images/snippets/inflections/35299f4eb8e0bed76b68c7beb2038031.ascii index 2e8de6e9..ad122496 100644 --- a/docs/images/snippets/inflections/35299f4eb8e0bed76b68c7beb2038031.ascii +++ b/docs/images/snippets/inflections/35299f4eb8e0bed76b68c7beb2038031.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 2 3 - Bézier(t) = x (1-t) + 3x (1-t) t + 3x (1-t)t + x t - 1 2 3 4 - \prime 2 2 - Bézier (t) = a(1-t) + 2b(1-t)t + ct { a=3(x -x ),b=3(x -x ),c=3(x -x ) } - 2 1 3 2 4 3 - \prime\prime - Bézier (t) = u(1-t) + vt {u=2(b-a),v=2(c-b)}\ + 3 2 2 3 +Bézier(t) = x (1-t) + 3x (1-t) t + 3x (1-t)t + x t + 1 2 3 4 + \prime 2 2 +Bézier (t) = a(1-t) + 2b(1-t)t + ct { a=3(x -x ),b=3(x -x ),c=3(x -x ) } + 2 1 3 2 4 3 + \prime\prime +Bézier (t) = u(1-t) + vt {u=2(b-a),v=2(c-b)}\ diff --git a/docs/images/snippets/inflections/75fae2d0a94eae4addf074c294855fc7.ascii b/docs/images/snippets/inflections/75fae2d0a94eae4addf074c294855fc7.ascii index 2e50620c..889a84de 100644 --- a/docs/images/snippets/inflections/75fae2d0a94eae4addf074c294855fc7.ascii +++ b/docs/images/snippets/inflections/75fae2d0a94eae4addf074c294855fc7.ascii @@ -1,15 +1,14 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 2 2 2 2 - -18 t x y + 36 t x y - 18 t x y + 18 t x y - 54 t x y - 2 1 3 1 4 1 1 2 3 2 - 2 2 2 2 2 - +36 t x y - 36 t x y + 54 t x y - 18 t x y + 18 t x y - 4 2 1 3 2 3 4 3 1 4 - 2 2 - -36 t x y + 18 t x y + 36 t x y - 54 t x y + 18 t x y - 36 t x y - 2 4 3 4 2 1 3 1 4 1 1 2 - +54 t x y - 18 t x y + 54 t x y - 54 t x y - 18 t x y + 18 t x y - 3 2 4 2 1 3 2 3 1 4 2 4 - -18 x y + 18 x y + 18 x y - 18 x y - 18 x y + 18 x y - 2 1 3 1 1 2 3 2 1 3 2 3 + 2 2 2 2 2 +-18 t x y + 36 t x y - 18 t x y + 18 t x y - 54 t x y + 2 1 3 1 4 1 1 2 3 2 + 2 2 2 2 2 ++36 t x y - 36 t x y + 54 t x y - 18 t x y + 18 t x y + 4 2 1 3 2 3 4 3 1 4 + 2 2 +-36 t x y + 18 t x y + 36 t x y - 54 t x y + 18 t x y - 36 t x y + 2 4 3 4 2 1 3 1 4 1 1 2 ++54 t x y - 18 t x y + 54 t x y - 54 t x y - 18 t x y + 18 t x y + 3 2 4 2 1 3 2 3 1 4 2 4 +-18 x y + 18 x y + 18 x y - 18 x y - 18 x y + 18 x y + 2 1 3 1 1 2 3 2 1 3 2 3 diff --git a/docs/images/snippets/inflections/8278b9bec92ae49927283396692b51d5.ascii b/docs/images/snippets/inflections/8278b9bec92ae49927283396692b51d5.ascii index d6e925a0..aa201115 100644 --- a/docs/images/snippets/inflections/8278b9bec92ae49927283396692b51d5.ascii +++ b/docs/images/snippets/inflections/8278b9bec92ae49927283396692b51d5.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 2 3 - Bézier(t) = y (1-t) + 3y (1-t) t + 3y (1-t)t + y t - 1 2 3 4 - \prime 2 2 - Bézier (t) = d(1-t) + 2e(1-t)t + ft - \prime\prime - Bézier (t) = w(1-t) + zt + 3 2 2 3 +Bézier(t) = y (1-t) + 3y (1-t) t + 3y (1-t)t + y t + 1 2 3 4 + \prime 2 2 +Bézier (t) = d(1-t) + 2e(1-t)t + ft + \prime\prime +Bézier (t) = w(1-t) + zt diff --git a/docs/images/snippets/inflections/852f0346f025c671b8a1ce6b628028aa.ascii b/docs/images/snippets/inflections/852f0346f025c671b8a1ce6b628028aa.ascii index 08c35d78..69163bae 100644 --- a/docs/images/snippets/inflections/852f0346f025c671b8a1ce6b628028aa.ascii +++ b/docs/images/snippets/inflections/852f0346f025c671b8a1ce6b628028aa.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C(t) = Bézier \prime(t) · Bézier \prime\prime(t) - Bézier \prime(t) · Bézier \prime\prime(t) - x y y x +C(t) = Bézier \prime(t) · Bézier \prime\prime(t) - Bézier \prime(t) · Bézier \prime\prime(t) + x y y x diff --git a/docs/images/snippets/inflections/a283e01df17f3d763ec89621f2af6c5c.ascii b/docs/images/snippets/inflections/a283e01df17f3d763ec89621f2af6c5c.ascii deleted file mode 100644 index 1ac847c9..00000000 --- a/docs/images/snippets/inflections/a283e01df17f3d763ec89621f2af6c5c.ascii +++ /dev/null @@ -1,4 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - C(t) = Bézier \prime(t) · Bézier \prime\prime(t) - Bézier \prime(t) · Bézier \prime\prime(t) - x y y x diff --git a/docs/images/snippets/inflections/be9e409d619ecd735b0fbc219bec6d07.ascii b/docs/images/snippets/inflections/be9e409d619ecd735b0fbc219bec6d07.ascii deleted file mode 100644 index d6e925a0..00000000 --- a/docs/images/snippets/inflections/be9e409d619ecd735b0fbc219bec6d07.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 3 2 2 3 - Bézier(t) = y (1-t) + 3y (1-t) t + 3y (1-t)t + y t - 1 2 3 4 - \prime 2 2 - Bézier (t) = d(1-t) + 2e(1-t)t + ft - \prime\prime - Bézier (t) = w(1-t) + zt diff --git a/docs/images/snippets/inflections/d7d564126099bc0740058a7cdd744772.ascii b/docs/images/snippets/inflections/d7d564126099bc0740058a7cdd744772.ascii index b9414ad2..97e86e01 100644 --- a/docs/images/snippets/inflections/d7d564126099bc0740058a7cdd744772.ascii +++ b/docs/images/snippets/inflections/d7d564126099bc0740058a7cdd744772.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌──────────┐ - │ 2 - x = -3a + 2b + 3c - d ╮ -y ±⟍│y - 4 x z - y = 3a - b - 3c ╞ C(t) = 0 \Rightarrow t = ───────────────── - z = c - a ╯ 2x + ┌──────────┐ + │ 2 +x = -3a + 2b + 3c - d ╮ -y ±⟍│y - 4 x z +y = 3a - b - 3c ╞ C(t) = 0 ==> t = ───────────────── +z = c - a ╯ 2x diff --git a/docs/images/snippets/inflections/e50243eaa99b5acc08533dd2e9b71a74.ascii b/docs/images/snippets/inflections/e50243eaa99b5acc08533dd2e9b71a74.ascii index d1153936..0be1cb8d 100644 --- a/docs/images/snippets/inflections/e50243eaa99b5acc08533dd2e9b71a74.ascii +++ b/docs/images/snippets/inflections/e50243eaa99b5acc08533dd2e9b71a74.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 2 - (3 x y +2 x y +3 x y -x y ) t + (3 x y -x y -3 x y ) t + (x y -x y ) - 3 2 4 2 2 3 4 3 3 2 4 2 2 3 2 3 3 2 + 2 +(3 x y +2 x y +3 x y -x y ) t + (3 x y -x y -3 x y ) t + (x y -x y ) + 3 2 4 2 2 3 4 3 3 2 4 2 2 3 2 3 3 2 diff --git a/docs/images/snippets/inflections/ed68dcfb203517ca080fe48914769fb0.ascii b/docs/images/snippets/inflections/ed68dcfb203517ca080fe48914769fb0.ascii index 7f18e4de..bdad299a 100644 --- a/docs/images/snippets/inflections/ed68dcfb203517ca080fe48914769fb0.ascii +++ b/docs/images/snippets/inflections/ed68dcfb203517ca080fe48914769fb0.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C(t) = 0 +C(t) = 0 diff --git a/docs/images/snippets/inflections/f9f2258e59b038659087a5e87ba2e0af.ascii b/docs/images/snippets/inflections/f9f2258e59b038659087a5e87ba2e0af.ascii deleted file mode 100644 index 1c95bead..00000000 --- a/docs/images/snippets/inflections/f9f2258e59b038659087a5e87ba2e0af.ascii +++ /dev/null @@ -1,10 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - 3 2 2 3 - Bézier(t) = x (1-t) + 3x (1-t) t + 3x (1-t)t + x t - 1 2 3 4 - \prime 2 2 - Bézier (t) = a(1-t) + 2b(1-t)t + ct { a=3(x -x ),b=3(x -x ),c=3(x -x ) } - 2 1 3 2 4 3 - \prime\prime - Bézier (t) = u(1-t) + vt {u=2(b-a),v=2(c-b)}\ diff --git a/docs/images/snippets/matrix/1a64ed455c6dd2f8cacca5e5e12bdcc1.ascii b/docs/images/snippets/matrix/1a64ed455c6dd2f8cacca5e5e12bdcc1.ascii index 6520d57c..1551851d 100644 --- a/docs/images/snippets/matrix/1a64ed455c6dd2f8cacca5e5e12bdcc1.ascii +++ b/docs/images/snippets/matrix/1a64ed455c6dd2f8cacca5e5e12bdcc1.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 0 0 ┐ - ┌ 2 3 ┐ · │ -3 3 0 0 │ - └ 1 t t t ┘ │ 3 -6 3 0 │ - └ -1 3 -3 1 ┘ + ┌ 1 0 0 0 ┐ +┌ 2 3 ┐ · │ -3 3 0 0 │ +└ 1 t t t ┘ │ 3 -6 3 0 │ + └ -1 3 -3 1 ┘ diff --git a/docs/images/snippets/matrix/1bae50fefa43210b3a6259d1984f6cbc.ascii b/docs/images/snippets/matrix/1bae50fefa43210b3a6259d1984f6cbc.ascii index 5fcb39cb..a90a68c7 100644 --- a/docs/images/snippets/matrix/1bae50fefa43210b3a6259d1984f6cbc.ascii +++ b/docs/images/snippets/matrix/1bae50fefa43210b3a6259d1984f6cbc.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ 1 0 0 ┐ │ 1 │ - B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ - └ 1 t t ┘ └ 1 -2 1 ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + ┌ 1 0 0 ┐ │ 1 │ +B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ + └ 1 t t ┘ └ 1 -2 1 ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrix/67a5ea33d6c6558f7d954b18226f4956.ascii b/docs/images/snippets/matrix/67a5ea33d6c6558f7d954b18226f4956.ascii index e363cc6b..ba4f3c20 100644 --- a/docs/images/snippets/matrix/67a5ea33d6c6558f7d954b18226f4956.ascii +++ b/docs/images/snippets/matrix/67a5ea33d6c6558f7d954b18226f4956.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 - ... = -1 · t + 3 · t - 3 · t + 1 - 3 2 - + +3 · t - 6 · t + 3 · t + 0 - 3 2 - + -3 · t + 3 · t + 0 · t + 0 - 3 2 - + +1 · t + 0 · t + 0 · t + 0 + 3 2 +... = -1 · t + 3 · t - 3 · t + 1 + 3 2 + + +3 · t - 6 · t + 3 · t + 0 + 3 2 + + -3 · t + 3 · t + 0 · t + 0 + 3 2 + + +1 · t + 0 · t + 0 · t + 0 diff --git a/docs/images/snippets/matrix/87cfac83cb8a4b0bee68ef006effc611.ascii b/docs/images/snippets/matrix/87cfac83cb8a4b0bee68ef006effc611.ascii index 2eac0924..15954b16 100644 --- a/docs/images/snippets/matrix/87cfac83cb8a4b0bee68ef006effc611.ascii +++ b/docs/images/snippets/matrix/87cfac83cb8a4b0bee68ef006effc611.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 2 3 - B(t) = (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t + 3 2 2 3 +B(t) = (1-t) + 3 · (1-t) · t + 3 · (1-t) · t + t diff --git a/docs/images/snippets/matrix/8ecff6b8a37d60385d287ea2b26876db.ascii b/docs/images/snippets/matrix/8ecff6b8a37d60385d287ea2b26876db.ascii index 59035dac..90e4ab73 100644 --- a/docs/images/snippets/matrix/8ecff6b8a37d60385d287ea2b26876db.ascii +++ b/docs/images/snippets/matrix/8ecff6b8a37d60385d287ea2b26876db.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ -1 ┐ ┌ 3 ┐ ┌ -3 ┐ ┌ 1 ┐ - ┌ 3 2 ┐ · │ 3 │ + ┌ 3 2 ┐ · │ -6 │ + ┌ 3 2 ┐ · │ 3 │ + ┌ 3 2 ┐ · │ 0 │ - └ t t t 1 ┘ │ -3 │ └ t t t 1 ┘ │ 3 │ └ t t t 1 ┘ │ 0 │ └ t t t 1 ┘ │ 0 │ - └ 1 ┘ └ 0 ┘ └ 0 ┘ └ 0 ┘ + ┌ -1 ┐ ┌ 3 ┐ ┌ -3 ┐ ┌ 1 ┐ +┌ 3 2 ┐ · │ 3 │ + ┌ 3 2 ┐ · │ -6 │ + ┌ 3 2 ┐ · │ 3 │ + ┌ 3 2 ┐ · │ 0 │ +└ t t t 1 ┘ │ -3 │ └ t t t 1 ┘ │ 3 │ └ t t t 1 ┘ │ 0 │ └ t t t 1 ┘ │ 0 │ + └ 1 ┘ └ 0 ┘ └ 0 ┘ └ 0 ┘ diff --git a/docs/images/snippets/matrix/9a9a55f5b0323d9ea88f82fc6be58ad3.ascii b/docs/images/snippets/matrix/9a9a55f5b0323d9ea88f82fc6be58ad3.ascii index 33bc4b3d..c1e32001 100644 --- a/docs/images/snippets/matrix/9a9a55f5b0323d9ea88f82fc6be58ad3.ascii +++ b/docs/images/snippets/matrix/9a9a55f5b0323d9ea88f82fc6be58ad3.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 2 3 - B(t) = P · (1-t) + P · 3 · (1-t) · t + P · 3 · (1-t) · t + P · t - 1 2 3 4 + 3 2 2 3 +B(t) = P · (1-t) + P · 3 · (1-t) · t + P · 3 · (1-t) · t + P · t + 1 2 3 4 diff --git a/docs/images/snippets/matrix/b32cae2dfc47d5f36df0bc3defb7dfa8.ascii b/docs/images/snippets/matrix/b32cae2dfc47d5f36df0bc3defb7dfa8.ascii index a6db7a5e..d55012c1 100644 --- a/docs/images/snippets/matrix/b32cae2dfc47d5f36df0bc3defb7dfa8.ascii +++ b/docs/images/snippets/matrix/b32cae2dfc47d5f36df0bc3defb7dfa8.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ P │ - B(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ - └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ - └ -1 3 -3 1 ┘ │ 3 │ - │ P │ - └ 4 ┘ + ┌ P ┐ + │ 1 │ + ┌ 1 0 0 0 ┐ │ P │ +B(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ + └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ + └ -1 3 -3 1 ┘ │ 3 │ + │ P │ + └ 4 ┘ diff --git a/docs/images/snippets/matrix/b9527f7d5a0f5d2d737eac118d69243e.ascii b/docs/images/snippets/matrix/b9527f7d5a0f5d2d737eac118d69243e.ascii index 0ffbebab..4f79e409 100644 --- a/docs/images/snippets/matrix/b9527f7d5a0f5d2d737eac118d69243e.ascii +++ b/docs/images/snippets/matrix/b9527f7d5a0f5d2d737eac118d69243e.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ -1 3 -3 1 ┐ - ┌ 3 2 ┐ · │ 3 -6 3 0 │ - └ t t t 1 ┘ │ -3 3 0 0 │ - └ 1 0 0 0 ┘ + ┌ -1 3 -3 1 ┐ +┌ 3 2 ┐ · │ 3 -6 3 0 │ +└ t t t 1 ┘ │ -3 3 0 0 │ + └ 1 0 0 0 ┘ diff --git a/docs/images/snippets/matrix/cdd88611833f3b178df91278359a4193.ascii b/docs/images/snippets/matrix/cdd88611833f3b178df91278359a4193.ascii index 5f431800..db9dca3f 100644 --- a/docs/images/snippets/matrix/cdd88611833f3b178df91278359a4193.ascii +++ b/docs/images/snippets/matrix/cdd88611833f3b178df91278359a4193.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 - ... = (1-t) - 2 - + 3 · (1-t) · t - 2 - + 3 · (1-t) · t - 3 - + t + 3 +... = (1-t) + 2 + + 3 · (1-t) · t + 2 + + 3 · (1-t) · t + 3 + + t diff --git a/docs/images/snippets/matrix/ec118f296511c6e9ac8727be3703a7ce.ascii b/docs/images/snippets/matrix/ec118f296511c6e9ac8727be3703a7ce.ascii index bd260ce0..2302cfff 100644 --- a/docs/images/snippets/matrix/ec118f296511c6e9ac8727be3703a7ce.ascii +++ b/docs/images/snippets/matrix/ec118f296511c6e9ac8727be3703a7ce.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 - ... = (1-t) · (1-t) · (1-t) = -t + 3 · t - 3 · t + 1 - 3 2 - + 3 · (1-t) · (1-t) · t = 3 · t - 6 · t + 3 · t - 3 2 - + 3 · (1-t) · t · t = -3 · t + 3 · t - 3 - + t · t · t = t + 3 2 +... = (1-t) · (1-t) · (1-t) = -t + 3 · t - 3 · t + 1 + 3 2 + + 3 · (1-t) · (1-t) · t = 3 · t - 6 · t + 3 · t + 3 2 + + 3 · (1-t) · t · t = -3 · t + 3 · t + 3 + + t · t · t = t diff --git a/docs/images/snippets/matrixsplit/11505e0215ef026f2e49383ebb4a1abb.ascii b/docs/images/snippets/matrixsplit/11505e0215ef026f2e49383ebb4a1abb.ascii index e6a3008f..329df55e 100644 --- a/docs/images/snippets/matrixsplit/11505e0215ef026f2e49383ebb4a1abb.ascii +++ b/docs/images/snippets/matrixsplit/11505e0215ef026f2e49383ebb4a1abb.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math ┌ P ┐ ┌ P ┐ │ 1 │ diff --git a/docs/images/snippets/matrixsplit/17e308aa6d459b1d06d3160cc8e2e786.ascii b/docs/images/snippets/matrixsplit/17e308aa6d459b1d06d3160cc8e2e786.ascii index c92f6cb1..b5ac450d 100644 --- a/docs/images/snippets/matrixsplit/17e308aa6d459b1d06d3160cc8e2e786.ascii +++ b/docs/images/snippets/matrixsplit/17e308aa6d459b1d06d3160cc8e2e786.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 2 2 ┐ - ┌ P ┐ │ z · P - 2 · z · (z-1) · P + (z-1) · P │ - ┌ 2 2 ┐ │ 1 │ │ 3 2 1 │ - │ (z-1) -2 · z · (z-1) z │ · │ P │ = │ z · P - (z-1) · P │ - │ 0 -(z-1) z │ │ 2 │ │ 3 2 │ - └ 0 0 1 ┘ │ P │ │ P │ - └ 3 ┘ └ 3 ┘ + ┌ 2 2 ┐ + ┌ P ┐ │ z · P - 2 · z · (z-1) · P + (z-1) · P │ +┌ 2 2 ┐ │ 1 │ │ 3 2 1 │ +│ (z-1) -2 · z · (z-1) z │ · │ P │ = │ z · P - (z-1) · P │ +│ 0 -(z-1) z │ │ 2 │ │ 3 2 │ +└ 0 0 1 ┘ │ P │ │ P │ + └ 3 ┘ └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/1bae50fefa43210b3a6259d1984f6cbc.ascii b/docs/images/snippets/matrixsplit/1bae50fefa43210b3a6259d1984f6cbc.ascii index 5fcb39cb..a90a68c7 100644 --- a/docs/images/snippets/matrixsplit/1bae50fefa43210b3a6259d1984f6cbc.ascii +++ b/docs/images/snippets/matrixsplit/1bae50fefa43210b3a6259d1984f6cbc.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ 1 0 0 ┐ │ 1 │ - B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ - └ 1 t t ┘ └ 1 -2 1 ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + ┌ 1 0 0 ┐ │ 1 │ +B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ + └ 1 t t ┘ └ 1 -2 1 ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/206f539367fa1aaefc230709e4f2068e.ascii b/docs/images/snippets/matrixsplit/206f539367fa1aaefc230709e4f2068e.ascii index 0cc1d356..66689a74 100644 --- a/docs/images/snippets/matrixsplit/206f539367fa1aaefc230709e4f2068e.ascii +++ b/docs/images/snippets/matrixsplit/206f539367fa1aaefc230709e4f2068e.ascii @@ -1,8 +1,7 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · M \underset …こうじゃ！\underbrace \kern 1.25em · \kern 1.25em Q \kern 1.25em · \kern 1.25em │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + │ 1 │ += ┌ 2 ┐ · M \underset …こうじゃ！\underbrace \kern 1.25em · \kern 1.25em Q \kern 1.25em · \kern 1.25em │ P │ + └ 1 t t ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/266b71339b55ad3a312a9f41e6bcf988.ascii b/docs/images/snippets/matrixsplit/266b71339b55ad3a312a9f41e6bcf988.ascii index 42df6a71..c85691f1 100644 --- a/docs/images/snippets/matrixsplit/266b71339b55ad3a312a9f41e6bcf988.ascii +++ b/docs/images/snippets/matrixsplit/266b71339b55ad3a312a9f41e6bcf988.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · M \underset ...to get this!\underbrace \kern 1.25em · \kern 1.25em Q \kern 1.25em · \kern 1.25em │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + │ 1 │ += ┌ 2 ┐ · M \underset ...to get this!\underbrace \kern 1.25em · \kern 1.25em Q \kern 1.25em · \kern 1.25em │ P │ + └ 1 t t ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/3b5e41808b6c3bc66f3da2f40651410e.ascii b/docs/images/snippets/matrixsplit/3b5e41808b6c3bc66f3da2f40651410e.ascii index 6b773faa..44c4e376 100644 --- a/docs/images/snippets/matrixsplit/3b5e41808b6c3bc66f3da2f40651410e.ascii +++ b/docs/images/snippets/matrixsplit/3b5e41808b6c3bc66f3da2f40651410e.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ │ 1 │ - = ┌ 2 ┐ · │ 0 z 0 │ · │ -2 2 0 │ · │ P │ - └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ - └ 0 0 z ┘ │ P │ - └ 3 ┘ + ┌ P ┐ + ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ │ 1 │ += ┌ 2 ┐ · │ 0 z 0 │ · │ -2 2 0 │ · │ P │ + └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ + └ 0 0 z ┘ │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/4549b95450db3c73479e8902e4939427.ascii b/docs/images/snippets/matrixsplit/4549b95450db3c73479e8902e4939427.ascii index 4e981e34..94e845cc 100644 --- a/docs/images/snippets/matrixsplit/4549b95450db3c73479e8902e4939427.ascii +++ b/docs/images/snippets/matrixsplit/4549b95450db3c73479e8902e4939427.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - -1 │ 1 │ - = ┌ 2 ┐ · \underset into this...\underbrace M · M · Z · M · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + -1 │ 1 │ += ┌ 2 ┐ · \underset into this...\underbrace M · M · Z · M · │ P │ + └ 1 t t ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/4764868f43815e471bb1ea95a81e1633.ascii b/docs/images/snippets/matrixsplit/4764868f43815e471bb1ea95a81e1633.ascii index 25f48e95..b17ffca2 100644 --- a/docs/images/snippets/matrixsplit/4764868f43815e471bb1ea95a81e1633.ascii +++ b/docs/images/snippets/matrixsplit/4764868f43815e471bb1ea95a81e1633.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 2 ┐ ┌ P ┐ - │ 1 z z │ ┌ 1 0 0 ┐ │ 1 │ - = ┌ 2 ┐ · │ 0 1-z 2 · z · (1-z) │ · │ -2 2 0 │ · │ P │ - └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ - └ 0 0 (1-z) ┘ │ P │ - └ 3 ┘ + ┌ 2 ┐ ┌ P ┐ + │ 1 z z │ ┌ 1 0 0 ┐ │ 1 │ += ┌ 2 ┐ · │ 0 1-z 2 · z · (1-z) │ · │ -2 2 0 │ · │ P │ + └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ + └ 0 0 (1-z) ┘ │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/480ebd0234e2fe1adc94926e8ed4339c.ascii b/docs/images/snippets/matrixsplit/480ebd0234e2fe1adc94926e8ed4339c.ascii index 8f826f26..47bc5a71 100644 --- a/docs/images/snippets/matrixsplit/480ebd0234e2fe1adc94926e8ed4339c.ascii +++ b/docs/images/snippets/matrixsplit/480ebd0234e2fe1adc94926e8ed4339c.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ P ┐ │ 1 │ - ┌ 1 0 0 ┐ │ 1 │ │ z · P - (z-1) · P │ - │ -(z-1) z 0 │ · │ P │ = │ 2 1 │ - │ 2 2 │ │ 2 │ │ 2 2 │ - └ (z - 1) -2 · (z-1) · z z ┘ │ P │ │ z · P - 2 · z · (z-1) · P + (z - 1) · P │ - └ 3 ┘ └ 3 2 1 ┘ + ┌ P ┐ + ┌ P ┐ │ 1 │ +┌ 1 0 0 ┐ │ 1 │ │ z · P - (z-1) · P │ +│ -(z-1) z 0 │ · │ P │ = │ 2 1 │ +│ 2 2 │ │ 2 │ │ 2 2 │ +└ (z - 1) -2 · (z-1) · z z ┘ │ P │ │ z · P - 2 · z · (z-1) · P + (z - 1) · P │ + └ 3 ┘ └ 3 2 1 ┘ diff --git a/docs/images/snippets/matrixsplit/4ce218bc968cbd98da0ca6ab66d415ed.ascii b/docs/images/snippets/matrixsplit/4ce218bc968cbd98da0ca6ab66d415ed.ascii index cec76d25..145875b2 100644 --- a/docs/images/snippets/matrixsplit/4ce218bc968cbd98da0ca6ab66d415ed.ascii +++ b/docs/images/snippets/matrixsplit/4ce218bc968cbd98da0ca6ab66d415ed.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ ┌ P ┐ ╮ - ┌ 1 0 0 ┐ │ ┌ 2 2 ┐ │ 1 │ │ - = ┌ 2 ┐ · │ -2 2 0 │ · │ │ (z-1) -2 · z · (z-1) z │ · │ P │ │ - └ 1 t t ┘ └ 1 -2 1 ┘ │ │ 0 -(z-1) z │ │ 2 │ │ - │ └ 0 0 1 ┘ │ P │ │ - ╰ └ 3 ┘ ╯ + ╭ ┌ P ┐ ╮ + ┌ 1 0 0 ┐ │ ┌ 2 2 ┐ │ 1 │ │ += ┌ 2 ┐ · │ -2 2 0 │ · │ │ (z-1) -2 · z · (z-1) z │ · │ P │ │ + └ 1 t t ┘ └ 1 -2 1 ┘ │ │ 0 -(z-1) z │ │ 2 │ │ + │ └ 0 0 1 ┘ │ P │ │ + ╰ └ 3 ┘ ╯ diff --git a/docs/images/snippets/matrixsplit/55b45214ba90c96978cdc9cdfee24fef.ascii b/docs/images/snippets/matrixsplit/55b45214ba90c96978cdc9cdfee24fef.ascii deleted file mode 100644 index 6e6618e1..00000000 --- a/docs/images/snippets/matrixsplit/55b45214ba90c96978cdc9cdfee24fef.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ - -1 │ 1 │ - = ┌ 2 ┐ · \underset...into this...\underbrace M · M · Z · M · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/598739d23cf8dbcdebe033e4c7f1d28a.ascii b/docs/images/snippets/matrixsplit/598739d23cf8dbcdebe033e4c7f1d28a.ascii deleted file mode 100644 index 25acefd4..00000000 --- a/docs/images/snippets/matrixsplit/598739d23cf8dbcdebe033e4c7f1d28a.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · M \underset...to get this!\underbrace \kern 1.25em · \kern 1.25em Q \kern 1.25em · \kern 1.25em │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/5e008143622c66bb5e9cc4d5d6a8ea62.ascii b/docs/images/snippets/matrixsplit/5e008143622c66bb5e9cc4d5d6a8ea62.ascii index 352387b9..fbfdbc98 100644 --- a/docs/images/snippets/matrixsplit/5e008143622c66bb5e9cc4d5d6a8ea62.ascii +++ b/docs/images/snippets/matrixsplit/5e008143622c66bb5e9cc4d5d6a8ea62.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 0 ┐ - -1 │ 1 │ ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ - Q = M · Z · M = │ 1 ─ 0 │ · │ 0 z 0 │ · │ -2 2 0 │ = │ -(z-1) z 0 │ - │ 2 │ │ 2 │ └ 1 -2 1 ┘ │ 2 2 │ - └ 1 1 1 ┘ └ 0 0 z ┘ └ (z - 1) -2 · (z-1) · z z ┘ + ┌ 1 0 0 ┐ + -1 │ 1 │ ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ +Q = M · Z · M = │ 1 ─ 0 │ · │ 0 z 0 │ · │ -2 2 0 │ = │ -(z-1) z 0 │ + │ 2 │ │ 2 │ └ 1 -2 1 ┘ │ 2 2 │ + └ 1 1 1 ┘ └ 0 0 z ┘ └ (z - 1) -2 · (z-1) · z z ┘ diff --git a/docs/images/snippets/matrixsplit/63d1337d275abf7b296d500b9b5821fd.ascii b/docs/images/snippets/matrixsplit/63d1337d275abf7b296d500b9b5821fd.ascii deleted file mode 100644 index 8eb21c3a..00000000 --- a/docs/images/snippets/matrixsplit/63d1337d275abf7b296d500b9b5821fd.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · \undersetこれを…\underbrace\kern 2.25em Z · M \kern 2.25em · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/6a22184e6ca869d28f4a252b64f23eff.ascii b/docs/images/snippets/matrixsplit/6a22184e6ca869d28f4a252b64f23eff.ascii index 8a598201..043a59ba 100644 --- a/docs/images/snippets/matrixsplit/6a22184e6ca869d28f4a252b64f23eff.ascii +++ b/docs/images/snippets/matrixsplit/6a22184e6ca869d28f4a252b64f23eff.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ 1 0 0 ┐ │ 1 │ - B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ - └ 1 ( 0 + z · t) ( 0 + z · t) ┘ └ 1 -2 1 ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + ┌ 1 0 0 ┐ │ 1 │ +B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ + └ 1 ( 0 + z · t) ( 0 + z · t) ┘ └ 1 -2 1 ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/7a1e792cd3fa6f3482459e154abf2e7d.ascii b/docs/images/snippets/matrixsplit/7a1e792cd3fa6f3482459e154abf2e7d.ascii deleted file mode 100644 index 22ed8149..00000000 --- a/docs/images/snippets/matrixsplit/7a1e792cd3fa6f3482459e154abf2e7d.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · \undersetwe turn this...\underbrace\kern 2.25em Z · M \kern 2.25em · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/8349aa18563bb43427ae2383f1e212ae.ascii b/docs/images/snippets/matrixsplit/8349aa18563bb43427ae2383f1e212ae.ascii index f845e1fd..63ea8b6f 100644 --- a/docs/images/snippets/matrixsplit/8349aa18563bb43427ae2383f1e212ae.ascii +++ b/docs/images/snippets/matrixsplit/8349aa18563bb43427ae2383f1e212ae.ascii @@ -1,8 +1,7 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · \underset これを…\underbrace\kern 2.25em Z · M \kern 2.25em · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + │ 1 │ += ┌ 2 ┐ · \underset これを…\underbrace\kern 2.25em Z · M \kern 2.25em · │ P │ + └ 1 t t ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/9a4899b69e03cd4ad02c5eedffaa6a2f.ascii b/docs/images/snippets/matrixsplit/9a4899b69e03cd4ad02c5eedffaa6a2f.ascii index f326f553..cf5b0c7a 100644 --- a/docs/images/snippets/matrixsplit/9a4899b69e03cd4ad02c5eedffaa6a2f.ascii +++ b/docs/images/snippets/matrixsplit/9a4899b69e03cd4ad02c5eedffaa6a2f.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 2 2 ┐ - │ z · P - 2 · z · (z-1) · P + (z-1) · P │ - ┌ 1 0 0 ┐ │ 3 2 1 │ - = ┌ 2 ┐ · │ -2 2 0 │ · │ z · P - (z-1) · P │ - └ 1 t t ┘ └ 1 -2 1 ┘ │ 3 2 │ - │ P │ - └ 3 ┘ + ┌ 2 2 ┐ + │ z · P - 2 · z · (z-1) · P + (z-1) · P │ + ┌ 1 0 0 ┐ │ 3 2 1 │ += ┌ 2 ┐ · │ -2 2 0 │ · │ z · P - (z-1) · P │ + └ 1 t t ┘ └ 1 -2 1 ┘ │ 3 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/a899891096d82b7fdb23a90e6106b6df.ascii b/docs/images/snippets/matrixsplit/a899891096d82b7fdb23a90e6106b6df.ascii index 28c1101d..f5fc2290 100644 --- a/docs/images/snippets/matrixsplit/a899891096d82b7fdb23a90e6106b6df.ascii +++ b/docs/images/snippets/matrixsplit/a899891096d82b7fdb23a90e6106b6df.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math ┌ 3 2 2 3 ┐ ┌ P ┐ │ z · P - 3 · z · (z-1) · P + 3 · z · (z-1) · P - (z-1) · P │ diff --git a/docs/images/snippets/matrixsplit/aa17f7e82cf50498f90deb6a21a2489a.ascii b/docs/images/snippets/matrixsplit/aa17f7e82cf50498f90deb6a21a2489a.ascii index 34b7315b..0629b51e 100644 --- a/docs/images/snippets/matrixsplit/aa17f7e82cf50498f90deb6a21a2489a.ascii +++ b/docs/images/snippets/matrixsplit/aa17f7e82cf50498f90deb6a21a2489a.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · \underset we turn this...\underbrace\kern 2.25em Z · M \kern 2.25em · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + │ 1 │ += ┌ 2 ┐ · \underset we turn this...\underbrace\kern 2.25em Z · M \kern 2.25em · │ P │ + └ 1 t t ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/baedd4067516584d425b93331b7ce04f.ascii b/docs/images/snippets/matrixsplit/baedd4067516584d425b93331b7ce04f.ascii deleted file mode 100644 index ba9bb74d..00000000 --- a/docs/images/snippets/matrixsplit/baedd4067516584d425b93331b7ce04f.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - ┌ P ┐ - │ 1 │ - = ┌ 2 ┐ · M \underset…こうじゃ！\underbrace \kern 1.25em · \kern 1.25em Q \kern 1.25em · \kern 1.25em │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/c32007be095224e0d157a8f71c62c90e.ascii b/docs/images/snippets/matrixsplit/c32007be095224e0d157a8f71c62c90e.ascii index daf1f127..92e4c051 100644 --- a/docs/images/snippets/matrixsplit/c32007be095224e0d157a8f71c62c90e.ascii +++ b/docs/images/snippets/matrixsplit/c32007be095224e0d157a8f71c62c90e.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ │ 1 │ - B(t) = ┌ 2 ┐ · │ 0 z 0 │ · │ -2 2 0 │ · │ P │ - └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ - └ 0 0 z ┘ │ P │ - └ 3 ┘ + ┌ P ┐ + ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ │ 1 │ +B(t) = ┌ 2 ┐ · │ 0 z 0 │ · │ -2 2 0 │ · │ P │ + └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ + └ 0 0 z ┘ │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/c341532f693c2c1adfd298597bbfb5b5.ascii b/docs/images/snippets/matrixsplit/c341532f693c2c1adfd298597bbfb5b5.ascii index 6012b538..75e95c25 100644 --- a/docs/images/snippets/matrixsplit/c341532f693c2c1adfd298597bbfb5b5.ascii +++ b/docs/images/snippets/matrixsplit/c341532f693c2c1adfd298597bbfb5b5.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 0 ┐ ┌ 2 ┐ - -1 │ 1 │ │ 1 z z │ ┌ 1 0 0 ┐ ┌ 2 2 ┐ - Q' = M · Z' · M = │ 1 ─ 0 │ · │ 0 1-z 2 · z · (1-z) │ · │ -2 2 0 │ = │ (z-1) -2 · z · (z-1) z │ - │ 2 │ │ 2 │ └ 1 -2 1 ┘ │ 0 -(z-1) z │ - └ 1 1 1 ┘ └ 0 0 (1-z) ┘ └ 0 0 1 ┘ + ┌ 1 0 0 ┐ ┌ 2 ┐ + -1 │ 1 │ │ 1 z z │ ┌ 1 0 0 ┐ ┌ 2 2 ┐ +Q' = M · Z' · M = │ 1 ─ 0 │ · │ 0 1-z 2 · z · (1-z) │ · │ -2 2 0 │ = │ (z-1) -2 · z · (z-1) z │ + │ 2 │ │ 2 │ └ 1 -2 1 ┘ │ 0 -(z-1) z │ + └ 1 1 1 ┘ └ 0 0 (1-z) ┘ └ 0 0 1 ┘ diff --git a/docs/images/snippets/matrixsplit/d9d04b9b6c66788d18832a383d6f7ea0.ascii b/docs/images/snippets/matrixsplit/d9d04b9b6c66788d18832a383d6f7ea0.ascii deleted file mode 100644 index baf1ab1c..00000000 --- a/docs/images/snippets/matrixsplit/d9d04b9b6c66788d18832a383d6f7ea0.ascii +++ /dev/null @@ -1,8 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - ┌ P ┐ - -1 │ 1 │ - = ┌ 2 ┐ · \underset…こうして…\underbrace M · M · Z · M · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/da4ebf090f84d9b5d48b0f1e79bb3e7b.ascii b/docs/images/snippets/matrixsplit/da4ebf090f84d9b5d48b0f1e79bb3e7b.ascii index 7bc93369..3e37b5bd 100644 --- a/docs/images/snippets/matrixsplit/da4ebf090f84d9b5d48b0f1e79bb3e7b.ascii +++ b/docs/images/snippets/matrixsplit/da4ebf090f84d9b5d48b0f1e79bb3e7b.ascii @@ -1,8 +1,7 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - ┌ P ┐ - -1 │ 1 │ - = ┌ 2 ┐ · \underset …こうして…\underbrace M · M · Z · M · │ P │ - └ 1 t t ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + -1 │ 1 │ += ┌ 2 ┐ · \underset …こうして…\underbrace M · M · Z · M · │ P │ + └ 1 t t ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/dceed84990aaf6878bcc67ddbaa8d8d9.ascii b/docs/images/snippets/matrixsplit/dceed84990aaf6878bcc67ddbaa8d8d9.ascii index ba5384cc..129f5cd0 100644 --- a/docs/images/snippets/matrixsplit/dceed84990aaf6878bcc67ddbaa8d8d9.ascii +++ b/docs/images/snippets/matrixsplit/dceed84990aaf6878bcc67ddbaa8d8d9.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ ╭ ┌ P ┐ ╮ - │ 1 │ │ │ 1 │ │ - B(t) = ┌ 2 ┐ · M · Q · │ P │ = ┌ 2 ┐ · M · │ Q · │ P │ │ - └ 1 t t ┘ │ 2 │ └ 1 t t ┘ │ │ 2 │ │ - │ P │ │ │ P │ │ - └ 3 ┘ ╰ └ 3 ┘ ╯ + ┌ P ┐ ╭ ┌ P ┐ ╮ + │ 1 │ │ │ 1 │ │ +B(t) = ┌ 2 ┐ · M · Q · │ P │ = ┌ 2 ┐ · M · │ Q · │ P │ │ + └ 1 t t ┘ │ 2 │ └ 1 t t ┘ │ │ 2 │ │ + │ P │ │ │ P │ │ + └ 3 ┘ ╰ └ 3 ┘ ╯ diff --git a/docs/images/snippets/matrixsplit/e079f44b56e07c8d7f83c17c8ebf1ecf.ascii b/docs/images/snippets/matrixsplit/e079f44b56e07c8d7f83c17c8ebf1ecf.ascii index c0f1f715..fade1f8f 100644 --- a/docs/images/snippets/matrixsplit/e079f44b56e07c8d7f83c17c8ebf1ecf.ascii +++ b/docs/images/snippets/matrixsplit/e079f44b56e07c8d7f83c17c8ebf1ecf.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - ┌ 1 0 0 ┐ │ 1 │ - B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ - └ 1 ( z + (1-z) · t) ( z + (1-z) · t) ┘ └ 1 -2 1 ┘ │ 2 │ - │ P │ - └ 3 ┘ + ┌ P ┐ + ┌ 1 0 0 ┐ │ 1 │ +B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ + └ 1 ( z + (1-z) · t) ( z + (1-z) · t) ┘ └ 1 -2 1 ┘ │ 2 │ + │ P │ + └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/e2622175dadafecc015f15c79ddf3002.ascii b/docs/images/snippets/matrixsplit/e2622175dadafecc015f15c79ddf3002.ascii index 75077427..51d3b6ed 100644 --- a/docs/images/snippets/matrixsplit/e2622175dadafecc015f15c79ddf3002.ascii +++ b/docs/images/snippets/matrixsplit/e2622175dadafecc015f15c79ddf3002.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ ╭ ┌ P ┐ ╮ - │ 1 │ │ │ 1 │ │ - B(t) = ┌ 2 ┐ · M · Q · │ P │ = ┌ 2 ┐ · M · │ Q' · │ P │ │ - └ 1 t t ┘ │ 2 │ └ 1 t t ┘ │ │ 2 │ │ - │ P │ │ │ P │ │ - └ 3 ┘ ╰ └ 3 ┘ ╯ + ┌ P ┐ ╭ ┌ P ┐ ╮ + │ 1 │ │ │ 1 │ │ +B(t) = ┌ 2 ┐ · M · Q · │ P │ = ┌ 2 ┐ · M · │ Q' · │ P │ │ + └ 1 t t ┘ │ 2 │ └ 1 t t ┘ │ │ 2 │ │ + │ P │ │ │ P │ │ + └ 3 ┘ ╰ └ 3 ┘ ╯ diff --git a/docs/images/snippets/matrixsplit/e58196b82b78f584779208cce88137f5.ascii b/docs/images/snippets/matrixsplit/e58196b82b78f584779208cce88137f5.ascii index 95e764a2..ec5c9036 100644 --- a/docs/images/snippets/matrixsplit/e58196b82b78f584779208cce88137f5.ascii +++ b/docs/images/snippets/matrixsplit/e58196b82b78f584779208cce88137f5.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - ┌ 1 0 0 ┐ │ z · P - (z-1) · P │ - = ┌ 2 ┐ · │ -2 2 0 │ · │ 2 1 │ - └ 1 t t ┘ └ 1 -2 1 ┘ │ 2 2 │ - │ z · P - 2 · z · (z-1) · P + (z - 1) · P │ - └ 3 2 1 ┘ + ┌ P ┐ + │ 1 │ + ┌ 1 0 0 ┐ │ z · P - (z-1) · P │ += ┌ 2 ┐ · │ -2 2 0 │ · │ 2 1 │ + └ 1 t t ┘ └ 1 -2 1 ┘ │ 2 2 │ + │ z · P - 2 · z · (z-1) · P + (z - 1) · P │ + └ 3 2 1 ┘ diff --git a/docs/images/snippets/matrixsplit/ebf8d72c6056476172deeb89726b75c8.ascii b/docs/images/snippets/matrixsplit/ebf8d72c6056476172deeb89726b75c8.ascii index 4ffa192c..02e6c8bb 100644 --- a/docs/images/snippets/matrixsplit/ebf8d72c6056476172deeb89726b75c8.ascii +++ b/docs/images/snippets/matrixsplit/ebf8d72c6056476172deeb89726b75c8.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ ┌ P ┐ - │ 1 │ ┌ 1 0 0 0 ┐ │ 1 │ - ┌ 1 0 0 0 ┐ │ P │ │ 0 z 0 0 │ ┌ 1 0 0 0 ┐ │ P │ - B(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ = ┌ 2 3 ┐ · │ 2 │ · │ -3 3 0 0 │ · │ 2 │ - └ 1 (z · t) (z · t) (z · t) ┘ │ 3 -6 3 0 │ │ P │ └ 1 t t t ┘ │ 0 0 z 0 │ │ 3 -6 3 0 │ │ P │ - └ -1 3 -3 1 ┘ │ 3 │ │ 3 │ └ -1 3 -3 1 ┘ │ 3 │ - │ P │ └ 0 0 0 z ┘ │ P │ - └ 4 ┘ └ 4 ┘ + ┌ P ┐ ┌ P ┐ + │ 1 │ ┌ 1 0 0 0 ┐ │ 1 │ + ┌ 1 0 0 0 ┐ │ P │ │ 0 z 0 0 │ ┌ 1 0 0 0 ┐ │ P │ +B(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ = ┌ 2 3 ┐ · │ 2 │ · │ -3 3 0 0 │ · │ 2 │ + └ 1 (z · t) (z · t) (z · t) ┘ │ 3 -6 3 0 │ │ P │ └ 1 t t t ┘ │ 0 0 z 0 │ │ 3 -6 3 0 │ │ P │ + └ -1 3 -3 1 ┘ │ 3 │ │ 3 │ └ -1 3 -3 1 ┘ │ 3 │ + │ P │ └ 0 0 0 z ┘ │ P │ + └ 4 ┘ └ 4 ┘ diff --git a/docs/images/snippets/matrixsplit/f565e66677138927335535d009409c3d.ascii b/docs/images/snippets/matrixsplit/f565e66677138927335535d009409c3d.ascii index 92f70d22..aa4e7fcc 100644 --- a/docs/images/snippets/matrixsplit/f565e66677138927335535d009409c3d.ascii +++ b/docs/images/snippets/matrixsplit/f565e66677138927335535d009409c3d.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ ┌ P ┐ - ┌ 1 0 0 ┐ │ 1 │ ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ │ 1 │ - B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ = ┌ 2 ┐ · │ 0 z 0 │ · │ -2 2 0 │ · │ P │ - └ 1 (z · t) (z · t) ┘ └ 1 -2 1 ┘ │ 2 │ └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ - │ P │ └ 0 0 z ┘ │ P │ - └ 3 ┘ └ 3 ┘ + ┌ P ┐ ┌ P ┐ + ┌ 1 0 0 ┐ │ 1 │ ┌ 1 0 0 ┐ ┌ 1 0 0 ┐ │ 1 │ +B(t) = ┌ 2 ┐ · │ -2 2 0 │ · │ P │ = ┌ 2 ┐ · │ 0 z 0 │ · │ -2 2 0 │ · │ P │ + └ 1 (z · t) (z · t) ┘ └ 1 -2 1 ┘ │ 2 │ └ 1 t t ┘ │ 2 │ └ 1 -2 1 ┘ │ 2 │ + │ P │ └ 0 0 z ┘ │ P │ + └ 3 ┘ └ 3 ┘ diff --git a/docs/images/snippets/matrixsplit/f63067c2c3042c374a58dfa7f692309e.ascii b/docs/images/snippets/matrixsplit/f63067c2c3042c374a58dfa7f692309e.ascii index 1c003746..86be891e 100644 --- a/docs/images/snippets/matrixsplit/f63067c2c3042c374a58dfa7f692309e.ascii +++ b/docs/images/snippets/matrixsplit/f63067c2c3042c374a58dfa7f692309e.ascii @@ -1,8 +1,7 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ ┌ P ┐ ╮ - ┌ 1 0 0 ┐ │ ┌ 1 0 0 ┐ │ 1 │ │ - = ┌ 2 ┐ · │ -2 2 0 │ · │ │ -(z-1) z 0 │ · │ P │ │ - └ 1 t t ┘ └ 1 -2 1 ┘ │ │ 2 2 │ │ 2 │ │ - │ └ (z - 1) -2 · (z-1) · z z ┘ │ P │ │ - ╰ └ 3 ┘ ╯ + ╭ ┌ P ┐ ╮ + ┌ 1 0 0 ┐ │ ┌ 1 0 0 ┐ │ 1 │ │ += ┌ 2 ┐ · │ -2 2 0 │ · │ │ -(z-1) z 0 │ · │ P │ │ + └ 1 t t ┘ └ 1 -2 1 ┘ │ │ 2 2 │ │ 2 │ │ + │ └ (z - 1) -2 · (z-1) · z z ┘ │ P │ │ + ╰ └ 3 ┘ ╯ diff --git a/docs/images/snippets/matrixsplit/f690ff0502d9fd7d4697cc43d98afd5d.ascii b/docs/images/snippets/matrixsplit/f690ff0502d9fd7d4697cc43d98afd5d.ascii index a6db7a5e..d55012c1 100644 --- a/docs/images/snippets/matrixsplit/f690ff0502d9fd7d4697cc43d98afd5d.ascii +++ b/docs/images/snippets/matrixsplit/f690ff0502d9fd7d4697cc43d98afd5d.ascii @@ -1,10 +1,9 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ P ┐ - │ 1 │ - ┌ 1 0 0 0 ┐ │ P │ - B(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ - └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ - └ -1 3 -3 1 ┘ │ 3 │ - │ P │ - └ 4 ┘ + ┌ P ┐ + │ 1 │ + ┌ 1 0 0 0 ┐ │ P │ +B(t) = ┌ 2 3 ┐ · │ -3 3 0 0 │ · │ 2 │ + └ 1 t t t ┘ │ 3 -6 3 0 │ │ P │ + └ -1 3 -3 1 ┘ │ 3 │ + │ P │ + └ 4 ┘ diff --git a/docs/images/snippets/molding/2e65bc9c934380c2de6a24bcd5c1c7b7.ascii b/docs/images/snippets/molding/2e65bc9c934380c2de6a24bcd5c1c7b7.ascii index 4da8e4ab..b4b2b170 100644 --- a/docs/images/snippets/molding/2e65bc9c934380c2de6a24bcd5c1c7b7.ascii +++ b/docs/images/snippets/molding/2e65bc9c934380c2de6a24bcd5c1c7b7.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C - B B - C - A = B - ────────── = B + ────────── - ratio(t) ratio(t) - q q + C - B B - C +A = B - ────────── = B + ────────── + ratio(t) ratio(t) + q q diff --git a/docs/images/snippets/molding/48887d68a861a0acdf8313e23fb19880.ascii b/docs/images/snippets/molding/48887d68a861a0acdf8313e23fb19880.ascii index 25ab67d6..4a19ce97 100644 --- a/docs/images/snippets/molding/48887d68a861a0acdf8313e23fb19880.ascii +++ b/docs/images/snippets/molding/48887d68a861a0acdf8313e23fb19880.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - C = u(t) · Start + (1-u(t) ) · End - q q +C = u(t) · Start + (1-u(t) ) · End + q q diff --git a/docs/images/snippets/molding/6f12fcc00f4106bbc920d7451398d3b2.ascii b/docs/images/snippets/molding/6f12fcc00f4106bbc920d7451398d3b2.ascii deleted file mode 100644 index 59f99d3f..00000000 --- a/docs/images/snippets/molding/6f12fcc00f4106bbc920d7451398d3b2.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - C - B B - C - A = B - ───────── = B + ───────── - ratio(t) ratio(t) - q q diff --git a/docs/images/snippets/molding/70262c533569a7da06cc1b950e932d6f.ascii b/docs/images/snippets/molding/70262c533569a7da06cc1b950e932d6f.ascii deleted file mode 100644 index c2870a14..00000000 --- a/docs/images/snippets/molding/70262c533569a7da06cc1b950e932d6f.ascii +++ /dev/null @@ -1,4 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - C = u(t) · Start + (1-u(t) ) · End - q q diff --git a/docs/images/snippets/offsetting/3c80407cfd0bd8c8ebea239272aeabe5.ascii b/docs/images/snippets/offsetting/3c80407cfd0bd8c8ebea239272aeabe5.ascii index 83033f55..d32d2e06 100644 --- a/docs/images/snippets/offsetting/3c80407cfd0bd8c8ebea239272aeabe5.ascii +++ b/docs/images/snippets/offsetting/3c80407cfd0bd8c8ebea239272aeabe5.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - O(t) = B(t) + d +O(t) = B(t) + d diff --git a/docs/images/snippets/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.ascii b/docs/images/snippets/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.ascii index b9a2c822..d72c7ced 100644 --- a/docs/images/snippets/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.ascii +++ b/docs/images/snippets/offsetting/57e62f3f2f7526b2cf7c1b276c17e472.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ B'(t) ╮ - N(t) \bot │ ────────── │ - ╰ || B'(t)|| ╯ + ╭ B'(t) ╮ +N(t) \bot │ ────────── │ + ╰ || B'(t)|| ╯ diff --git a/docs/images/snippets/offsetting/af4b584bb280cc941603255f62c9cc1a.ascii b/docs/images/snippets/offsetting/af4b584bb280cc941603255f62c9cc1a.ascii index 0b2ba0b8..45d92eeb 100644 --- a/docs/images/snippets/offsetting/af4b584bb280cc941603255f62c9cc1a.ascii +++ b/docs/images/snippets/offsetting/af4b584bb280cc941603255f62c9cc1a.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌───────────────────┐ - ╭ 1 │ 2 2 - || B'(t)|| = | │B ''(t) + B ''(t) - ╯ 0 ⟍│ x y + ┌───────────────────┐ + ╭ 1 │ 2 2 +|| B'(t)|| = | │B ''(t) + B ''(t) + ╯ 0 ⟍│ x y diff --git a/docs/images/snippets/offsetting/cf8e602eb0595cf4d9b851c6bda741af.ascii b/docs/images/snippets/offsetting/cf8e602eb0595cf4d9b851c6bda741af.ascii index 7c52ee92..002b3529 100644 --- a/docs/images/snippets/offsetting/cf8e602eb0595cf4d9b851c6bda741af.ascii +++ b/docs/images/snippets/offsetting/cf8e602eb0595cf4d9b851c6bda741af.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌───────────┐ - ╭ b │ 2 2 - || f(x,y)|| = | │f ' + f ' - ╯ a ⟍│ x y + ┌───────────┐ + ╭ b │ 2 2 +|| f(x,y)|| = | │f ' + f ' + ╯ a ⟍│ x y diff --git a/docs/images/snippets/offsetting/de8cdb128273beff2d98534b9f090b85.ascii b/docs/images/snippets/offsetting/de8cdb128273beff2d98534b9f090b85.ascii index 05fe638f..81875f1d 100644 --- a/docs/images/snippets/offsetting/de8cdb128273beff2d98534b9f090b85.ascii +++ b/docs/images/snippets/offsetting/de8cdb128273beff2d98534b9f090b85.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - O(t) = B(t) + d · N(t) +O(t) = B(t) + d · N(t) diff --git a/docs/images/snippets/pointcurves/4fe687c8a65265a2a755ba5841d0e31d.ascii b/docs/images/snippets/pointcurves/4fe687c8a65265a2a755ba5841d0e31d.ascii index ba200658..00f3c813 100644 --- a/docs/images/snippets/pointcurves/4fe687c8a65265a2a755ba5841d0e31d.ascii +++ b/docs/images/snippets/pointcurves/4fe687c8a65265a2a755ba5841d0e31d.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - d = { d if 0 ≤\phi ≤\pi - -d if \phi < 0 \lor \phi > \pi +d = { d if 0 ≤\phi ≤\pi + -d if \phi < 0 \lor \phi > \pi diff --git a/docs/images/snippets/pointcurves/8ffdd4a58cbd0fc24caef781f23a7950.ascii b/docs/images/snippets/pointcurves/8ffdd4a58cbd0fc24caef781f23a7950.ascii index 0801f31d..0042cdb0 100644 --- a/docs/images/snippets/pointcurves/8ffdd4a58cbd0fc24caef781f23a7950.ascii +++ b/docs/images/snippets/pointcurves/8ffdd4a58cbd0fc24caef781f23a7950.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ e = B + t · d - ╡ 1 - │ e = B - (1-t) · d - ╰ 2 +╭ e = B + t · d +╡ 1 +│ e = B - (1-t) · d +╰ 2 diff --git a/docs/images/snippets/pointcurves/a5cd63b54be6b554290c38787cfbbabd.ascii b/docs/images/snippets/pointcurves/a5cd63b54be6b554290c38787cfbbabd.ascii index aa3647b6..3510cb22 100644 --- a/docs/images/snippets/pointcurves/a5cd63b54be6b554290c38787cfbbabd.ascii +++ b/docs/images/snippets/pointcurves/a5cd63b54be6b554290c38787cfbbabd.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - \phi = (atan2(E -S , E -S ) - atan2(B -S , B -S ) + 2 \pi) mod 2 \pi - y y x x y y x x +\phi = (atan2(E -S , E -S ) - atan2(B -S , B -S ) + 2 \pi) mod 2 \pi + y y x x y y x x diff --git a/docs/images/snippets/pointcurves/f8182445c1cd7ae9f368b88fa7090e53.ascii b/docs/images/snippets/pointcurves/f8182445c1cd7ae9f368b88fa7090e53.ascii index 927d4218..0cbe3380 100644 --- a/docs/images/snippets/pointcurves/f8182445c1cd7ae9f368b88fa7090e53.ascii +++ b/docs/images/snippets/pointcurves/f8182445c1cd7ae9f368b88fa7090e53.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ d = || Start - B|| - │ 1 - │ d = || End - B|| - │ 2 - ╡ d - │ 1 - │ t= ───── - │ d +d - ╰ 1 2 +╭ d = || Start - B|| +│ 1 +│ d = || End - B|| +│ 2 +╡ d +│ 1 +│ t= ───── +│ d +d +╰ 1 2 diff --git a/docs/images/snippets/pointvectors/1df6c055ae8e41a46bfdebc55a4f17c0.ascii b/docs/images/snippets/pointvectors/1df6c055ae8e41a46bfdebc55a4f17c0.ascii index cc4c2f12..251f4efc 100644 --- a/docs/images/snippets/pointvectors/1df6c055ae8e41a46bfdebc55a4f17c0.ascii +++ b/docs/images/snippets/pointvectors/1df6c055ae8e41a46bfdebc55a4f17c0.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - x' = x · cos (\phi) - y · sin (\phi) - y' = x · sin (\phi) + y · cos (\phi) +x' = x · cos (\phi) - y · sin (\phi) +y' = x · sin (\phi) + y · cos (\phi) diff --git a/docs/images/snippets/pointvectors/33afd1a141ec444989c393b3e51ec9ca.ascii b/docs/images/snippets/pointvectors/33afd1a141ec444989c393b3e51ec9ca.ascii index a21a609d..c0387e4d 100644 --- a/docs/images/snippets/pointvectors/33afd1a141ec444989c393b3e51ec9ca.ascii +++ b/docs/images/snippets/pointvectors/33afd1a141ec444989c393b3e51ec9ca.ascii @@ -1,16 +1,15 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - ┌─────────────────┐ - │ 2 2 - d = \| tangent(t)\| = │B' (t) + B' (t) - ⟍│ x y - - tangent (t) B' (t) - ^ x x - x(t) = \| tangent (t)\| =─────────────── = ────── - x \| tangent(t)\| d - - tangent (t) B' (t) - ^ y y - y(t) = \| tangent (t)\| = ─────────────── = ────── - y \| tangent(t)\| d + ┌─────────────────┐ + │ 2 2 + d = \| tangent(t)\| = │B' (t) + B' (t) + ⟍│ x y + + tangent (t) B' (t) +^ x x +x(t) = \| tangent (t)\| =─────────────── = ────── + x \| tangent(t)\| d + + tangent (t) B' (t) +^ y y +y(t) = \| tangent (t)\| = ─────────────── = ────── + y \| tangent(t)\| d diff --git a/docs/images/snippets/pointvectors/4d89f0042fc367f1614537c7f05389fb.ascii b/docs/images/snippets/pointvectors/4d89f0042fc367f1614537c7f05389fb.ascii deleted file mode 100644 index 25b4285d..00000000 --- a/docs/images/snippets/pointvectors/4d89f0042fc367f1614537c7f05389fb.ascii +++ /dev/null @@ -1,16 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌─────────────────┐ - │ 2 2 - d = \|tangent(t)\| = │B' (t) + B' (t) - ⟍│ x y - - tangent (t) B' (t) - ^ x x - x(t) = \| tangent (t)\| =────────────── = ────── - x \|tangent(t)\| d - - tangent (t) B' (t) - ^ y y - y(t) = \| tangent (t)\| = ────────────── = ────── - y \|tangent(t)\| d diff --git a/docs/images/snippets/pointvectors/58b19accb8a68c665ff5cbed610eea4e.ascii b/docs/images/snippets/pointvectors/58b19accb8a68c665ff5cbed610eea4e.ascii deleted file mode 100644 index a2c7d26d..00000000 --- a/docs/images/snippets/pointvectors/58b19accb8a68c665ff5cbed610eea4e.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ^ \pi ^ \pi ^ - normal (t) = x(t) · cos ─── - y(t) · sin ─── = - y(t) - x 2 2 - - ^ \pi ^ \pi ^ - normal (t) = \undersetquarter circle rotation \underbrace x(t) · sin ─── + y(t) · cos ─── = x(t) - y 2 2 diff --git a/docs/images/snippets/pointvectors/8b15a314beca97071b0ccb22c969355d.ascii b/docs/images/snippets/pointvectors/8b15a314beca97071b0ccb22c969355d.ascii deleted file mode 100644 index 37b640a4..00000000 --- a/docs/images/snippets/pointvectors/8b15a314beca97071b0ccb22c969355d.ascii +++ /dev/null @@ -1,7 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - tangent (t) = B' (t) - x x - - tangent (t) = B' (t) - y y diff --git a/docs/images/snippets/pointvectors/afdd2dbe7690ed09ea91df63471b480b.ascii b/docs/images/snippets/pointvectors/afdd2dbe7690ed09ea91df63471b480b.ascii deleted file mode 100644 index 5f6e0673..00000000 --- a/docs/images/snippets/pointvectors/afdd2dbe7690ed09ea91df63471b480b.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ^ \pi ^ \pi ^ - normal (t) = x(t) · cos ─── - y(t) · sin ─── = - y(t) - x 2 2 - - ^ \pi ^ \pi ^ - normal (t) = \undersetquarter circle rotation \underbrace x(t) · sin ─── + y(t) · cos ─── = x(t) - y 2 2 diff --git a/docs/images/snippets/pointvectors/b60cdba673c2c9fc84c800f07fd18145.ascii b/docs/images/snippets/pointvectors/b60cdba673c2c9fc84c800f07fd18145.ascii deleted file mode 100644 index af541b1a..00000000 --- a/docs/images/snippets/pointvectors/b60cdba673c2c9fc84c800f07fd18145.ascii +++ /dev/null @@ -1,16 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌─────────────────┐ - │ 2 2 - d = \|tangent(t)\| = │B' (t) + B' (t) - ⟍│ x y - - tangent (t) B' (t) - ^ x x - x(t) = \|tangent (t)\| =────────────── = ────── - x \|tangent(t)\| d - - tangent (t) B' (t) - ^ y y - y(t) = \|tangent (t)\| = ────────────── = ────── - y \|tangent(t)\| d diff --git a/docs/images/snippets/pointvectors/c3d5f3506b763b718e567f90dbb78324.ascii b/docs/images/snippets/pointvectors/c3d5f3506b763b718e567f90dbb78324.ascii index 4585c7ae..66d6644d 100644 --- a/docs/images/snippets/pointvectors/c3d5f3506b763b718e567f90dbb78324.ascii +++ b/docs/images/snippets/pointvectors/c3d5f3506b763b718e567f90dbb78324.ascii @@ -1,9 +1,8 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ^ \pi ^ \pi ^ - normal (t) = x(t) · cos ─── - y(t) · sin ─── = - y(t) - x 2 2 - - ^ \pi ^ \pi ^ - normal (t) = \underset quarter circle rotation \underbrace x(t) · sin ─── + y(t) · cos ─── = x(t) - y 2 2 + ^ \pi ^ \pi ^ +normal (t) = x(t) · cos ─── - y(t) · sin ─── = - y(t) + x 2 2 + + ^ \pi ^ \pi ^ +normal (t) = \underset quarter circle rotation \underbrace x(t) · sin ─── + y(t) · cos ─── = x(t) + y 2 2 diff --git a/docs/images/snippets/pointvectors/da069c7d6cbdb516c5454371dae84e7f.ascii b/docs/images/snippets/pointvectors/da069c7d6cbdb516c5454371dae84e7f.ascii index 37b640a4..6e60341c 100644 --- a/docs/images/snippets/pointvectors/da069c7d6cbdb516c5454371dae84e7f.ascii +++ b/docs/images/snippets/pointvectors/da069c7d6cbdb516c5454371dae84e7f.ascii @@ -1,7 +1,6 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - tangent (t) = B' (t) - x x - - tangent (t) = B' (t) - y y +tangent (t) = B' (t) + x x + +tangent (t) = B' (t) + y y diff --git a/docs/images/snippets/pointvectors/f02e359a5e47667919738fff69d2625b.ascii b/docs/images/snippets/pointvectors/f02e359a5e47667919738fff69d2625b.ascii index 15a3c12b..8cc33d58 100644 --- a/docs/images/snippets/pointvectors/f02e359a5e47667919738fff69d2625b.ascii +++ b/docs/images/snippets/pointvectors/f02e359a5e47667919738fff69d2625b.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ x' ┐ = ┌ cos (\phi) -sin (\phi) ┐ ┌ x ┐ - └ y' ┘ └ sin (\phi) cos (\phi) ┘ └ y ┘ +┌ x' ┐ = ┌ cos (\phi) -sin (\phi) ┐ ┌ x ┐ +└ y' ┘ └ sin (\phi) cos (\phi) ┘ └ y ┘ diff --git a/docs/images/snippets/polybezier/2249056953a47ab1944bb5a41dcbed8c.ascii b/docs/images/snippets/polybezier/2249056953a47ab1944bb5a41dcbed8c.ascii index 36d62e0d..8beecf48 100644 --- a/docs/images/snippets/polybezier/2249056953a47ab1944bb5a41dcbed8c.ascii +++ b/docs/images/snippets/polybezier/2249056953a47ab1944bb5a41dcbed8c.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ B + (B - A ) ┐ ┌ 2B - A ┐ - Mirrored = │ x x x │ = │ x x │ - │ B + (B - A ) │ │ 2B - A │ - └ y y y ┘ └ y y ┘ + ┌ B + (B - A ) ┐ ┌ 2B - A ┐ +Mirrored = │ x x x │ = │ x x │ + │ B + (B - A ) │ │ 2B - A │ + └ y y y ┘ └ y y ┘ diff --git a/docs/images/snippets/polybezier/a37252ff55837b918d9d64078ae92ae7.ascii b/docs/images/snippets/polybezier/a37252ff55837b918d9d64078ae92ae7.ascii index 89c0386b..75af9f82 100644 --- a/docs/images/snippets/polybezier/a37252ff55837b918d9d64078ae92ae7.ascii +++ b/docs/images/snippets/polybezier/a37252ff55837b918d9d64078ae92ae7.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - B'(1) = B'(0) - n n+1 +B'(1) = B'(0) + n n+1 diff --git a/docs/images/snippets/polybezier/ec93d3c42f0ae52a05d0aff9739675e5.ascii b/docs/images/snippets/polybezier/ec93d3c42f0ae52a05d0aff9739675e5.ascii deleted file mode 100644 index 36d62e0d..00000000 --- a/docs/images/snippets/polybezier/ec93d3c42f0ae52a05d0aff9739675e5.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ┌ B + (B - A ) ┐ ┌ 2B - A ┐ - Mirrored = │ x x x │ = │ x x │ - │ B + (B - A ) │ │ 2B - A │ - └ y y y ┘ └ y y ┘ diff --git a/docs/images/snippets/reordering/056e25c397c524d80f378ce3823c7e78.ascii b/docs/images/snippets/reordering/056e25c397c524d80f378ce3823c7e78.ascii index 3e8a8c4e..34e9c9a6 100644 --- a/docs/images/snippets/reordering/056e25c397c524d80f378ce3823c7e78.ascii +++ b/docs/images/snippets/reordering/056e25c397c524d80f378ce3823c7e78.ascii @@ -1,21 +1,20 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ┌ 1 0 . . . . . . ┐ - │ 1 k-1 │ - │ ─ ─── 0 . . . 0 . │ - │ k k │ - │ 2 k-2 │ - │ 0 ─ ─── 0 . . . . │ - │ k k │ - │ 3 k-3 │ - │ . 0 ─ ─── 0 . . . │ - M = │ k k │ - │ . . 0 ... ... 0 . . │ - │ . . . 0 ... ... 0 . │ - │ n-1 k-n+1 │ - │ . . . . 0 ─── ───── 0 │ - │ k k │ - │ n k-n │ - │ . 0 . . . 0 ─ ─── │ - │ k k │ - └ . . . . . . 0 1 ┘ + ┌ 1 0 . . . . . . ┐ + │ 1 k-1 │ + │ ─ ─── 0 . . . 0 . │ + │ k k │ + │ 2 k-2 │ + │ 0 ─ ─── 0 . . . . │ + │ k k │ + │ 3 k-3 │ + │ . 0 ─ ─── 0 . . . │ +M = │ k k │ + │ . . 0 ... ... 0 . . │ + │ . . . 0 ... ... 0 . │ + │ n-1 k-n+1 │ + │ . . . . 0 ─── ───── 0 │ + │ k k │ + │ n k-n │ + │ . 0 . . . 0 ─ ─── │ + │ k k │ + └ . . . . . . 0 1 ┘ diff --git a/docs/images/snippets/reordering/0cf0d5f856ec204dc32e0e42691cc70a.ascii b/docs/images/snippets/reordering/0cf0d5f856ec204dc32e0e42691cc70a.ascii index 087ea198..c2631ff1 100644 --- a/docs/images/snippets/reordering/0cf0d5f856ec204dc32e0e42691cc70a.ascii +++ b/docs/images/snippets/reordering/0cf0d5f856ec204dc32e0e42691cc70a.ascii @@ -1,6 +1,5 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - Bézier(n,t)= (1-t) B(n,t) + t B(n,t) - __ n n __ n n - = ❯ w (1 - t) B (t) + ❯ w t B (t) - ‾‾ i=0 i i ‾‾ i=0 i i +Bézier(n,t)= (1-t) B(n,t) + t B(n,t) + __ n n __ n n + = ❯ w (1 - t) B (t) + ❯ w t B (t) + ‾‾ i=0 i i ‾‾ i=0 i i diff --git a/docs/images/snippets/reordering/0f5698b31598b2390e966fc5e43ab53e.ascii b/docs/images/snippets/reordering/0f5698b31598b2390e966fc5e43ab53e.ascii index 3e1f720d..ad45d3ac 100644 --- a/docs/images/snippets/reordering/0f5698b31598b2390e966fc5e43ab53e.ascii +++ b/docs/images/snippets/reordering/0f5698b31598b2390e966fc5e43ab53e.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - n n! n-i i - (1 - t) B (t)= (1-t) ──────── (1-t) t - i (n-i)!i! - n+1-i (n+1)! n+1-i i - = ───── ────────── (1-t) t - n+1 (n+1-i)!i! - k-i k! k-i i - = ─── ──────── (1-t) t , where k = n + 1 - k (k-i)!i! - k-i k - = ─── B (t) - k i + n n! n-i i +(1 - t) B (t)= (1-t) ──────── (1-t) t + i (n-i)!i! + n+1-i (n+1)! n+1-i i + = ───── ────────── (1-t) t + n+1 (n+1-i)!i! + k-i k! k-i i + = ─── ──────── (1-t) t , where k = n + 1 + k (k-i)!i! + k-i k + = ─── B (t) + k i diff --git a/docs/images/snippets/reordering/1f5b60d190a1c7099b3411e4cc477291.ascii b/docs/images/snippets/reordering/1f5b60d190a1c7099b3411e4cc477291.ascii index ecbb65a0..042e0ebf 100644 --- a/docs/images/snippets/reordering/1f5b60d190a1c7099b3411e4cc477291.ascii +++ b/docs/images/snippets/reordering/1f5b60d190a1c7099b3411e4cc477291.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - M B = B - n k +M B = B + n k diff --git a/docs/images/snippets/reordering/46e64dc07502e14217ec83d755f736ee.ascii b/docs/images/snippets/reordering/46e64dc07502e14217ec83d755f736ee.ascii index 48eaba44..b7a1ea08 100644 --- a/docs/images/snippets/reordering/46e64dc07502e14217ec83d755f736ee.ascii +++ b/docs/images/snippets/reordering/46e64dc07502e14217ec83d755f736ee.ascii @@ -1,16 +1,15 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - M B = B - n k - T T - (M M) B = M B - n k - T -1 T T -1 T - (M M) (M M) B = (M M) M B - n k - T -1 T - I B = (M M) M B - n k - T -1 T - B = (M M) M B - n k + M B = B + n k + T T + (M M) B = M B + n k + T -1 T T -1 T +(M M) (M M) B = (M M) M B + n k + T -1 T + I B = (M M) M B + n k + T -1 T + B = (M M) M B + n k diff --git a/docs/images/snippets/reordering/4ff41e183d60d5fd10a5d3d30dd63358.ascii b/docs/images/snippets/reordering/4ff41e183d60d5fd10a5d3d30dd63358.ascii index 01aba82b..be3f2608 100644 --- a/docs/images/snippets/reordering/4ff41e183d60d5fd10a5d3d30dd63358.ascii +++ b/docs/images/snippets/reordering/4ff41e183d60d5fd10a5d3d30dd63358.ascii @@ -1,17 +1,16 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n+1 n __ n+1 n - Bézier(n,t)= ❯ w (1 - t) B (t) + ❯ w t B (t) - ‾‾ i=0 i i ‾‾ i=0 i i - __ n+1 k-i k __ n+1 i+1 k - = ❯ w ─── B (t) + ❯ w ─── B (t), where k = n + 1 - ‾‾ i=0 i k i ‾‾ i=0 i k i+1 - __ n+1 k-i k __ n+1 i k - = ❯ w ─── B (t) + ❯ p ─ B (t) - ‾‾ i=0 i k i ‾‾ i=0 i-1 k i - __ n+1 ╭ k-i i ╮ k - = ❯ │ w ─── + p ─ │ B (t) - ‾‾ i=0 ╰ i k i-1 k ╯ i - __ n+1 k i - = ❯ (w (1-s) + p s) B (t), where s = ─ - ‾‾ i=0 i i-1 i k + __ n+1 n __ n+1 n +Bézier(n,t)= ❯ w (1 - t) B (t) + ❯ w t B (t) + ‾‾ i=0 i i ‾‾ i=0 i i + __ n+1 k-i k __ n+1 i+1 k + = ❯ w ─── B (t) + ❯ w ─── B (t), where k = n + 1 + ‾‾ i=0 i k i ‾‾ i=0 i k i+1 + __ n+1 k-i k __ n+1 i k + = ❯ w ─── B (t) + ❯ p ─ B (t) + ‾‾ i=0 i k i ‾‾ i=0 i-1 k i + __ n+1 ╭ k-i i ╮ k + = ❯ │ w ─── + p ─ │ B (t) + ‾‾ i=0 ╰ i k i-1 k ╯ i + __ n+1 k i + = ❯ (w (1-s) + p s) B (t), where s = ─ + ‾‾ i=0 i i-1 i k diff --git a/docs/images/snippets/reordering/56130afc4cb313e0e74cf670d34590f6.ascii b/docs/images/snippets/reordering/56130afc4cb313e0e74cf670d34590f6.ascii deleted file mode 100644 index 25164636..00000000 --- a/docs/images/snippets/reordering/56130afc4cb313e0e74cf670d34590f6.ascii +++ /dev/null @@ -1,6 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - Bézier(n,t)= (1-t) B(n,t) + t B(n,t) - __ n n __ n n - = ❯ w (1 - t) B (t) + ❯ w t B (t) - ‾‾ i=0 i i ‾‾ i=0 i i diff --git a/docs/images/snippets/reordering/74e038deabd9e240606fa3f07ba98269.ascii b/docs/images/snippets/reordering/74e038deabd9e240606fa3f07ba98269.ascii index da33d525..bc745bf4 100644 --- a/docs/images/snippets/reordering/74e038deabd9e240606fa3f07ba98269.ascii +++ b/docs/images/snippets/reordering/74e038deabd9e240606fa3f07ba98269.ascii @@ -1,5 +1,4 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - + __ k k-i i Bézier(k,t) = ❯ \underset binomial term\underbrace\binomki · \ \underset polynomial term\underbrace(1-t) · t · \ \underset new ‾‾ i=0 diff --git a/docs/images/snippets/reordering/8090b63b005bf3edb916b97bda317a0e.ascii b/docs/images/snippets/reordering/8090b63b005bf3edb916b97bda317a0e.ascii deleted file mode 100644 index 9c7b7b40..00000000 --- a/docs/images/snippets/reordering/8090b63b005bf3edb916b97bda317a0e.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n n n-i i - Bézier(n,t) = ❯ w B (t) , where B (t) = \binomni · (1-t) · t - ‾‾ i=0 i i i diff --git a/docs/images/snippets/reordering/9fc4ecc087d389dd0111bcba165cd5d0.ascii b/docs/images/snippets/reordering/9fc4ecc087d389dd0111bcba165cd5d0.ascii index 1fa197dc..967f92c2 100644 --- a/docs/images/snippets/reordering/9fc4ecc087d389dd0111bcba165cd5d0.ascii +++ b/docs/images/snippets/reordering/9fc4ecc087d389dd0111bcba165cd5d0.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n n n n-i i - Bézier(n,t) = ❯ w B (t) , where B (t) = \binomni · (1-t) · t - ‾‾ i=0 i i i + __ n n n n-i i +Bézier(n,t) = ❯ w B (t) , where B (t) = \binomni · (1-t) · t + ‾‾ i=0 i i i diff --git a/docs/images/snippets/reordering/ab7c087f7c070d43a42f3f03010a7427.ascii b/docs/images/snippets/reordering/ab7c087f7c070d43a42f3f03010a7427.ascii index 46d6d7ba..2a41342a 100644 --- a/docs/images/snippets/reordering/ab7c087f7c070d43a42f3f03010a7427.ascii +++ b/docs/images/snippets/reordering/ab7c087f7c070d43a42f3f03010a7427.ascii @@ -1,14 +1,13 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - n n! n-i i - t B (t)= t ──────── (1-t) t - i (n-i)!i! - i+1 (n+1)! (n+1)-(i+1) i+1 - = ─── ──────────────────── (1-t) t - n+1 ((n+1)-(i+1))!(i+1)! - i+1 k! k-(i+1) i+1 - = ─── ──────────────── (1-t) t , where k = n + 1 - k (k-(i+1))!(i+1)! - i+1 k - = ─── B (t) - k i+1 + n n! n-i i +t B (t)= t ──────── (1-t) t + i (n-i)!i! + i+1 (n+1)! (n+1)-(i+1) i+1 + = ─── ──────────────────── (1-t) t + n+1 ((n+1)-(i+1))!(i+1)! + i+1 k! k-(i+1) i+1 + = ─── ──────────────── (1-t) t , where k = n + 1 + k (k-(i+1))!(i+1)! + i+1 k + = ─── B (t) + k i+1 diff --git a/docs/images/snippets/reordering/e9fc9c715bb55a702db68b2bb6da0a68.ascii b/docs/images/snippets/reordering/e9fc9c715bb55a702db68b2bb6da0a68.ascii deleted file mode 100644 index 62289a0c..00000000 --- a/docs/images/snippets/reordering/e9fc9c715bb55a702db68b2bb6da0a68.ascii +++ /dev/null @@ -1,9 +0,0 @@ - \setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ k k-i i -Bézier(k,t) = ❯ \undersetbinomial term\underbrace\binomki · \ \undersetpolynomial term\underbrace(1-t) · t · \ \undersetnew - ‾‾ i=0 - ╭ (k-i) · w + i · w ╮ - │ i i-1 │ - weights\underbrace│ ─────────────────────── │ , with k = n+1 and w =0 when i = 0 - ╰ k ╯ i-1 diff --git a/docs/images/snippets/reordering/ff224ded6bbbc94b43130f5f8eeb5d29.ascii b/docs/images/snippets/reordering/ff224ded6bbbc94b43130f5f8eeb5d29.ascii index 055e773a..aa49edd1 100644 --- a/docs/images/snippets/reordering/ff224ded6bbbc94b43130f5f8eeb5d29.ascii +++ b/docs/images/snippets/reordering/ff224ded6bbbc94b43130f5f8eeb5d29.ascii @@ -1,3 +1,2 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - x = 1 x = ((1-t) + t) x = (1-t) x + t x = x (1-t) + x t +x = 1 x = ((1-t) + t) x = (1-t) x + t x = x (1-t) + x t diff --git a/docs/images/snippets/weightcontrol/2ec04091c55fe31bf85ac28c5b6d95cb.ascii b/docs/images/snippets/weightcontrol/2ec04091c55fe31bf85ac28c5b6d95cb.ascii deleted file mode 100644 index d08624b7..00000000 --- a/docs/images/snippets/weightcontrol/2ec04091c55fe31bf85ac28c5b6d95cb.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i - ❯ \binomni · (1-t) · t · w · \colorblueratio - ‾‾ i=0 i i - Rational Bézier(n,t) = ─────────────────────────────────────────────────────────── - __ n n-i i - \colorblue ❯ \binomni · (1-t) · t · ratio - ‾‾ i=0 i diff --git a/docs/images/snippets/weightcontrol/55f079880a77c126c70106e62ff941d9.ascii b/docs/images/snippets/weightcontrol/55f079880a77c126c70106e62ff941d9.ascii deleted file mode 100644 index 28f15b93..00000000 --- a/docs/images/snippets/weightcontrol/55f079880a77c126c70106e62ff941d9.ascii +++ /dev/null @@ -1,9 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i - ❯ \binomni · (1-t) · t · w · \colorblueratio - ‾‾ i=0 i i - Rational Bézier(n,t) = ─────────────────────────────────────────────────────────── - __ n n-i i - \colorblue ❯ \binomni · (1-t) · t · ratio - ‾‾ i=0 i diff --git a/docs/images/snippets/weightcontrol/85d526fb17f9e859dcd7d40d22192e37.ascii b/docs/images/snippets/weightcontrol/85d526fb17f9e859dcd7d40d22192e37.ascii deleted file mode 100644 index f1659f8a..00000000 --- a/docs/images/snippets/weightcontrol/85d526fb17f9e859dcd7d40d22192e37.ascii +++ /dev/null @@ -1,5 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n n-i i - Bézier(n,t) = ❯ \binomni · (1-t) · t · w - ‾‾ i=0 i diff --git a/docs/images/snippets/weightcontrol/942e3b3cacc7f403ad95fcd4acce7d19.ascii b/docs/images/snippets/weightcontrol/942e3b3cacc7f403ad95fcd4acce7d19.ascii new file mode 100644 index 00000000..8dae6879 --- /dev/null +++ b/docs/images/snippets/weightcontrol/942e3b3cacc7f403ad95fcd4acce7d19.ascii @@ -0,0 +1,8 @@ + + __ n n-i i + ❯ \binomni · (1-t) · t · w · \colorblue ratio + ‾‾ i=0 i i +Rational Bézier(n,t) = ──────────────────────────────────────────────────────────── + __ n n-i i + \colorblue ❯ \binomni · (1-t) · t · ratio + ‾‾ i=0 i diff --git a/docs/images/snippets/weightcontrol/ceac4259d2aed0767c7765d2237ca1a3.ascii b/docs/images/snippets/weightcontrol/ceac4259d2aed0767c7765d2237ca1a3.ascii index f1659f8a..482ba041 100644 --- a/docs/images/snippets/weightcontrol/ceac4259d2aed0767c7765d2237ca1a3.ascii +++ b/docs/images/snippets/weightcontrol/ceac4259d2aed0767c7765d2237ca1a3.ascii @@ -1,5 +1,4 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - __ n n-i i - Bézier(n,t) = ❯ \binomni · (1-t) · t · w - ‾‾ i=0 i + __ n n-i i +Bézier(n,t) = ❯ \binomni · (1-t) · t · w + ‾‾ i=0 i diff --git a/docs/images/snippets/whatis/06bbc5c11ad3fd88ff93eb2c06177b66.ascii b/docs/images/snippets/whatis/06bbc5c11ad3fd88ff93eb2c06177b66.ascii deleted file mode 100644 index f50f94c4..00000000 --- a/docs/images/snippets/whatis/06bbc5c11ad3fd88ff93eb2c06177b66.ascii +++ /dev/null @@ -1,11 +0,0 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - - ╭ p = 一方の点 ╮ - │ 1 │ - │ p = もう一方の点 │ - │ 2 │ - │ 距離= (p - p ) │のとき、新しい点 = p + 距離 · 比率 - │ 2 1 │ 1 - │ 百分率 │ - │ 比率= ─── │ - ╰ 100 ╯ diff --git a/docs/images/snippets/whatis/3437a38af1218ca206e921e48678c07e.ascii b/docs/images/snippets/whatis/3437a38af1218ca206e921e48678c07e.ascii index fc0843cf..a9a79b23 100644 --- a/docs/images/snippets/whatis/3437a38af1218ca206e921e48678c07e.ascii +++ b/docs/images/snippets/whatis/3437a38af1218ca206e921e48678c07e.ascii @@ -1,11 +1,10 @@ -\usepackagexeCJK \xeCJKsetupCJKmath=true \setCJKmainfontipaexm.ttf - ╭ p = 一方の点 ╮ - │ 1 │ - │ p = もう一方の点 │ - │ 2 │ - │ 距離= (p - p ) │ のとき、新しい点 = p + 距離 · 比率 - │ 2 1 │ 1 - │ 百分率 │ - │ 比率= ──── │ - ╰ 100 ╯ +╭ p = 一方の点 ╮ +│ 1 │ +│ p = もう一方の点 │ +│ 2 │ +│ 距離= (p - p ) │ のとき、新しい点 = p + 距離 · 比率 +│ 2 1 │ 1 +│ 百分率 │ +│ 比率= ──── │ +╰ 100 ╯ diff --git a/docs/images/snippets/whatis/35964d0485747082c0a8bedc0a16822b.ascii b/docs/images/snippets/whatis/35964d0485747082c0a8bedc0a16822b.ascii index c118242f..0ae0d8c4 100644 --- a/docs/images/snippets/whatis/35964d0485747082c0a8bedc0a16822b.ascii +++ b/docs/images/snippets/whatis/35964d0485747082c0a8bedc0a16822b.ascii @@ -1,11 +1,10 @@ -\usepackageunicode-math \setmainfont[Ligatures=TeX]Linux Libertine O \setmathfontXITS Math - ╭ p = неикая точка ╮ - │ 1 │ - │ p = неикая другая точка │ - │ 2 │ - Дано │ расстояние= (p - p ) │, наша новая точка = p + расстояние · соотношение - │ 2 1 │ 1 - │ процентаж │ - │ соотношение= ────────── │ - ╰ 100 ╯ + ╭ p = неикая точка ╮ + │ 1 │ + │ p = неикая другая точка │ + │ 2 │ +Дано │ расстояние= (p - p ) │, наша новая точка = p + расстояние · соотношение + │ 2 1 │ 1 + │ процентаж │ + │ соотношение= ────────── │ + ╰ 100 ╯ diff --git a/docs/images/snippets/whatis/c3f06301f5ce610df1217bc633257297.ascii b/docs/images/snippets/whatis/c3f06301f5ce610df1217bc633257297.ascii deleted file mode 100644 index 5e5a7f9c..00000000 --- a/docs/images/snippets/whatis/c3f06301f5ce610df1217bc633257297.ascii +++ /dev/null @@ -1,11 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ p = some point ╮ - │ 1 │ - │ p = some other point │ - │ 2 │ - Given │ distance= (p - p ) │, our new point = p + distance · ratio - │ 2 1 │ 1 - │ percentage │ - │ ratio= ────────── │ - ╰ 100 ╯ diff --git a/docs/images/snippets/whatis/c7f8cdd755d744412476b87230d0400d.ascii b/docs/images/snippets/whatis/c7f8cdd755d744412476b87230d0400d.ascii index 3a98fb94..26679da8 100644 --- a/docs/images/snippets/whatis/c7f8cdd755d744412476b87230d0400d.ascii +++ b/docs/images/snippets/whatis/c7f8cdd755d744412476b87230d0400d.ascii @@ -1,11 +1,10 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - ╭ p = some point ╮ - │ 1 │ - │ p = some other point │ - │ 2 │ - Given │ distance= (p - p ) │, our new point = p + distance · ratio - │ 2 1 │ 1 - │ percentage │ - │ ratio= ─────────── │ - ╰ 100 ╯ + ╭ p = some point ╮ + │ 1 │ + │ p = some other point │ + │ 2 │ +Given │ distance= (p - p ) │, our new point = p + distance · ratio + │ 2 1 │ 1 + │ percentage │ + │ ratio= ─────────── │ + ╰ 100 ╯ diff --git a/docs/images/snippets/whatis/e0600b3be5b95f105a1cf2a2c0378b98.ascii b/docs/images/snippets/whatis/e0600b3be5b95f105a1cf2a2c0378b98.ascii deleted file mode 100644 index 38ef5523..00000000 --- a/docs/images/snippets/whatis/e0600b3be5b95f105a1cf2a2c0378b98.ascii +++ /dev/null @@ -1,11 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - ╭ p = неикая точка ╮ - │ 1 │ - │ p = неикая другая точка │ - │ 2 │ - Дано │ расстояние= (p - p ) │, наша новая точка = p + расстояние · соотношение - │ 2 1 │ 1 - │ процентаж │ - │ соотношение= ───────── │ - ╰ 100 ╯ diff --git a/docs/images/snippets/yforx/021718d3b46893b271f90083ccdceaf8.ascii b/docs/images/snippets/yforx/021718d3b46893b271f90083ccdceaf8.ascii index 6bdb24e2..3d4dc5a6 100644 --- a/docs/images/snippets/yforx/021718d3b46893b271f90083ccdceaf8.ascii +++ b/docs/images/snippets/yforx/021718d3b46893b271f90083ccdceaf8.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 - x(t) = (-a + 3b- 3c + d)t + (3a - 6b + 3c)t + (-3a + 3b)t + a + 3 2 +x(t) = (-a + 3b- 3c + d)t + (3a - 6b + 3c)t + (-3a + 3b)t + a diff --git a/docs/images/snippets/yforx/058a76e3e7d67c03f733e075829a6252.ascii b/docs/images/snippets/yforx/058a76e3e7d67c03f733e075829a6252.ascii index 64fa9f99..368b09cb 100644 --- a/docs/images/snippets/yforx/058a76e3e7d67c03f733e075829a6252.ascii +++ b/docs/images/snippets/yforx/058a76e3e7d67c03f733e075829a6252.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 - (-a + 3b - 3c + d)t + (3a - 6b + 3c)t + (-3a + 3b)t + (a-x) = 0 + 3 2 +(-a + 3b - 3c + d)t + (3a - 6b + 3c)t + (-3a + 3b)t + (a-x) = 0 diff --git a/docs/images/snippets/yforx/316e7fae61e10014000d770209779ab6.ascii b/docs/images/snippets/yforx/316e7fae61e10014000d770209779ab6.ascii deleted file mode 100644 index 19e4c64f..00000000 --- a/docs/images/snippets/yforx/316e7fae61e10014000d770209779ab6.ascii +++ /dev/null @@ -1,3 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - x(t) = a(1-t)³ + 3b(1-t)²t + 3c(1-t)t² + dt³ diff --git a/docs/images/snippets/yforx/378d0fd8cefa688d530ac38930d66844.ascii b/docs/images/snippets/yforx/378d0fd8cefa688d530ac38930d66844.ascii index 57a30ec8..0b5dc18d 100644 --- a/docs/images/snippets/yforx/378d0fd8cefa688d530ac38930d66844.ascii +++ b/docs/images/snippets/yforx/378d0fd8cefa688d530ac38930d66844.ascii @@ -1,4 +1,3 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - 3 2 2 3 - x(t) = a(1-t) + 3b(1-t) t + 3c(1-t)t + dt + 3 2 2 3 +x(t) = a(1-t) + 3b(1-t) t + 3c(1-t)t + dt diff --git a/docs/images/snippets/yforx/4d23ee228c5b1cbc40e380496c2184d1.ascii b/docs/images/snippets/yforx/4d23ee228c5b1cbc40e380496c2184d1.ascii deleted file mode 100644 index 2f0edae6..00000000 --- a/docs/images/snippets/yforx/4d23ee228c5b1cbc40e380496c2184d1.ascii +++ /dev/null @@ -1,3 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - x(t) = (-a + 3b- 3c + d)t³ + (3a - 6b + 3c)t² + (-3a + 3b)t + a diff --git a/docs/images/snippets/yforx/699459d89ca6622c90c1e42e4aa03f32.ascii b/docs/images/snippets/yforx/699459d89ca6622c90c1e42e4aa03f32.ascii deleted file mode 100644 index 898c5617..00000000 --- a/docs/images/snippets/yforx/699459d89ca6622c90c1e42e4aa03f32.ascii +++ /dev/null @@ -1,3 +0,0 @@ -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - (-a + 3b - 3c + d)t³ + (3a - 6b + 3c)t² + (-3a + 3b)t + (a-x) = 0 diff --git a/docs/index.html b/docs/index.html index 63a69db5..abf85cff 100644 --- a/docs/index.html +++ b/docs/index.html @@ -1,161 +1,174 @@ + + + + A Primer on Bézier Curves - - - - A Primer on Bézier Curves + + + + + - + - + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + +

- - +
+ This page on GitHub + + This page on GitHub + +
-
- This page on GitHub - - This page on GitHub - -
+
+ + + submit to reddit + submit to hacker news + tweet your read + +
-
- - - submit to reddit - submit to hacker news - tweet your read - -
+ - +
+

+ A Primer on Bézier Curves +

+

A free, online book for when you really need to know how to do Bézier things.

+
+ Read this in your own language: + +

+ (Don't see your language listed, or want to see it reach 100%? + Help translate this content!) +

+
+

+ Welcome to the Primer on Bezier Curves. This is a free website/ebook dealing with both the maths and programming aspects of Bezier Curves, + covering a wide range of topics relating to drawing and working with that curve that seems to pop up everywhere, from Photoshop paths to CSS + easing functions to Font outline descriptions. +

+

+ If this is your first time here: welcome! Let me know if you were looking for anything in particular that the primer doesn't cover over on the + issue tracker! +

-
+

Donations and sponsorship

-

A Primer on Bézier Curves

-

A free, online book for when you really need to know how to do Bézier things.

-
- Read this in your own language: - -

(Don't see your language listed, or want to see it reach 100%? Help translate this content!)

-
+

+ If this is a resource that you're using for research, as work reference, or even writing your own software, please consider + donating (any amount helps) or + signing up as a patron on Patreon. I don't get paid to work on this, so if you find this site + valuable, and you'd like it to stick around for a long time to come, a lot of coffee went into writing this over the years, and a lot more + coffee will need to go into it yet: if you can spare a coffee, you'd be helping keep a resource alive and well. +

+

+ Also, if you are a company and your staff uses this book as a resource, or you use it as an onboarding resource, then please: consider + sponsoring the site! I am more than happy to work with your finance department on sponsorship invoicing and recognition. +

-

- Welcome to the Primer on Bezier Curves. This is a free website/ebook dealing with both - the maths and programming aspects of Bezier Curves, covering a wide range of topics - relating to drawing and working with that curve that seems to pop up everywhere, from - Photoshop paths to CSS easing functions to Font outline descriptions. -

-

- If this is your first time here: welcome! Let me know if you were looking for anything - in particular that the primer doesn't cover over on the issue tracker! -

- -

Donations and sponsorship

- -

- If this is a resource that you're using for research, as work reference, or even writing your own software, - please consider donating - (any amount helps) or signing up as a patron on Patreon. - I don't get paid to work on this, so if you find this site valuable, and you'd like it - to stick around for a long time to come, a lot of coffee went into writing this over the - years, and a lot more coffee will need to go into it yet: if you can spare a coffee, you'd - be helping keep a resource alive and well. -

-

- Also, if you are a company and your staff uses this book as a resource, or you use it as an - onboarding resource, then please: consider sponsoring the site! I am more than happy to work - with your finance department on sponsorship invoicing and recognition. -

- - - -
+
-

- — Pomax -

- + +
+
+
+

Preface

+

+ In order to draw things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The + first of these are as easy to draw as they are easy to make a computer draw. Give a computer the first and last point in the line, and BAM! + straight line. No questions asked. +

+

+ Curves, however, are a much bigger problem. While we can draw curves with ridiculous ease freehand, computers are a bit handicapped in that + they can't draw curves unless there is a mathematical function that describes how it should be drawn. In fact, they even need this for + straight lines, but the function is ridiculously easy, so we tend to ignore that as far as computers are concerned; all lines are + "functions", regardless of whether they're straight or curves. However, that does mean that we need to come up with fast-to-compute + functions that lead to nice looking curves on a computer. There are a number of these, and in this article we'll focus on a particular + function that has received quite a bit of attention and is used in pretty much anything that can draw curves: Bézier curves. +

+

+ They're named after Pierre Bézier, who is principally responsible for making + them known to the world as a curve well-suited for design work (publishing his investigations in 1962 while working for Renault), although + he was not the first, or only one, to "invent" these type of curves. One might be tempted to say that the mathematician + Paul de Casteljau was first, as he began investigating the nature of these + curves in 1959 while working at Citroën, and came up with a really elegant way of figuring out how to draw them. However, de Casteljau did + not publish his work, making the question "who was first" hard to answer in any absolute sense. Or is it? Bézier curves are, at their core, + "Bernstein polynomials", a family of mathematical functions investigated by + Sergei Natanovich Bernstein, whose publications on them date back at + least as far as 1912. +

+

+ Anyway, that's mostly trivia, what you are more likely to care about is that these curves are handy: you can link up multiple Bézier curves + so that the combination looks like a single curve. If you've ever drawn Photoshop "paths" or worked with vector drawing programs like Flash, + Illustrator or Inkscape, those curves you've been drawing are Bézier curves. +

+

+ But what if you need to program them yourself? What are the pitfalls? How do you draw them? What are the bounding boxes, how do you + determine intersections, how can you extrude a curve, in short: how do you do everything that you might want to do with these curves? That's + what this page is for. Prepare to be mathed! +

+
+

Virtually all Bézier graphics are interactive.

+

+ This page uses interactive examples, relying heavily on Bezier.js, as well as maths + formulae which are typeset into SVG using the XeLaTeX typesetting system and + pdf2svg by David Barton. +

+

This book is open source.

+

+ This book is an open source software project, and lives on two github repositories. The first is + https://github.com/pomax/bezierinfo and is the purely-for-presentation version you are + viewing right now. The other repository is https://github.com/pomax/BezierInfo-2, + which is the development version, housing all the code that gets turned into the web version, and is also where you should file + issues if you find bugs or have ideas on what to change or add to the primer. +

+

How complicated is the maths going to be?

+

+ Most of the mathematics in this Primer are early high school maths. If you understand basic arithmetic, and you know how to read English, + you should be able to get by just fine. There will at times be far more complicated maths, but if you don't feel like digesting + them, you can safely skip over them by either skipping over the "detail boxes" in section or by just jumping to the end of a section with + maths that looks too involving. The end of sections typically simply list the conclusions so you can just work with those values directly. +

+

What language is all this example code in?

+

+ There are way too many programming languages to favour one of all others, soo all the example code in this Primer uses a form of + pseudo-code that uses a syntax that's close enough to, but not actually, modern scripting languages like JS, Python, etc. That means you + won't be able to copy-paste any of it without giving it any thought, but that's intentional: if you're reading this primer, presumably you + want to learn, and you don't learn by copy-pasting. You learn by doing things yourself, making mistakes, and then fixing + those mistakes. Now, of course, I didn't intentionally add errors in the example code just to trick you into making mistakes (that would + be horrible!) but I did intentionally keep the code from favouring one programming language over another. Don't worry though, if + you know even a single procedural programming language, you should be able to read the examples without any difficulties. +

+

Questions, comments:

+

+ If you have suggestions for new sections, hit up the Github issue tracker (also + reachable from the repo linked to in the upper right). If you have questions about the material, there's currently no comment section + while I'm doing the rewrite, but you can use the issue tracker for that as well. Once the rewrite is done, I'll add a general comment + section back in, and maybe a more topical "select this section of text and hit the 'question' button to ask a question about it" system. + We'll see. +

+

Help support the book!

+

+ If you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me + know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 100-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing! +

+
+
+
+

What's new?

+

+ This primer is a living document, and so depending on when you last look at it, there may be new content. Click the following link to expand + this section to have a look at what got added, when, or click through to the News posts for more detailed updates. (RSS feed + available) +

+ + + +
+

November 2020

+
    +

  • Added a section on finding curve/circle intersections

    • +
+

October 2020

+
    +

  • Added the Ukranian locale! Help out in getting its localization to 100%!

    • +
+

August-September 2020

+
    +
  • +

    + Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS + turned on. +

    +
    • +
+

June 2020

+
    +

  • Added automatic CI/CD using Github Actions

    • +
+

January 2020

+
    +

  • Added reset buttons to all graphics

    • +

  • Updated to preface to correctly describe the on-page maths

    • +

  • Fixed the Catmull-Rom section because it had glaring maths errors

    • +
+

August 2019

+
    +

  • Added a section on (plain) rational Bezier curves

    • +

  • Improved the Graphic component to allow for sliders

    • +
+

December 2018

+
    +

  • Added a section on curvature and calculating kappa.

    • +
  • +

    + Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this + site! +

    +
    • +
+

August 2018

+
    +

  • Added a section on finding a curve's y, if all you have is the x coordinate.

    • +
+

July 2018

+
    +

  • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

    • +

  • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

    • +
  • +

    (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

    +
    • +
+

June 2018

+
    +

  • Added a section on direct curve fitting.

    • +

  • Added source links for all graphics.

    • +

  • Added this "What's new?" section.

    • +
+

April 2017

+
    +

  • Added a section on 3d normals.

    • +

  • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

    • +
+

February 2017

+
    +

  • Finished rewriting the entire codebase for localization.

    • +
+

January 2016

+
    +

  • Added a section to explain the Bezier interval.

    • +

  • Rewrote the Primer as a React application.

    • +
+

December 2015

+
    +

  • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

    • +
  • +

    + Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics. +

    +
    • +
+

May 2015

+
    +

  • Switched over to pure JS rather than Processing-through-Processing.js

    • +

  • Added Cardano's algorithm for finding the roots of a cubic polynomial.

    • +
+

April 2015

+
    +

  • Added a section on arc length approximations.

    • +
+

February 2015

+
    +

  • Added a section on the canonical cubic Bezier form.

    • +
+

November 2014

+
    +

  • Switched to HTTPS.

    • +
+

July 2014

+
    +

  • Added the section on arc approximation.

    • +
+

April 2014

+
    +

  • Added the section on Catmull-Rom fitting.

    • +
+

November 2013

+
    +

  • Added the section on Catmull-Rom / Bezier conversion.

    • +

  • Added the section on Bezier cuves as matrices.

    • +
+

April 2013

+
    +

  • Added a section on poly-Beziers.

    • +

  • Added a section on boolean shape operations.

    • +
+

March 2013

+
    +

  • First drastic rewrite.

    • +

  • Added sections on circle approximations.

    • +

  • Added a section on projecting a point onto a curve.

    • +

  • Added a section on tangents and normals.

    • +

  • Added Legendre-Gauss numerical data tables.

    • +
+

October 2011

+
    +
  • +

    + First commit for the bezierinfo site, based on the pre-Primer webpage that covered + the basics of Bezier curves in HTML with Processing.js examples. +

    +
    • +
+
+
+
+
+

+
table of contentsnext
+ A lightning introduction +

+

+ Let's start with the good stuff: when we're talking about Bézier curves, we're talking about the things that you can see in the following + graphics. They run from some start point to some end point, with their curvature influenced by one or more "intermediate" control points. + Now, because all the graphics on this page are interactive, go manipulate those curves a bit: click-drag the points, and see how their + shape changes based on what you do. +

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
-
- -
- -

Preface

-

In order to draw things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The first of these are as easy to draw as they are easy to make a computer draw. Give a computer the first and last point in the line, and BAM! straight line. No questions asked.

-

Curves, however, are a much bigger problem. While we can draw curves with ridiculous ease freehand, computers are a bit handicapped in that they can't draw curves unless there is a mathematical function that describes how it should be drawn. In fact, they even need this for straight lines, but the function is ridiculously easy, so we tend to ignore that as far as computers are concerned; all lines are "functions", regardless of whether they're straight or curves. However, that does mean that we need to come up with fast-to-compute functions that lead to nice looking curves on a computer. There are a number of these, and in this article we'll focus on a particular function that has received quite a bit of attention and is used in pretty much anything that can draw curves: Bézier curves.

-

They're named after Pierre Bézier, who is principally responsible for making them known to the world as a curve well-suited for design work (publishing his investigations in 1962 while working for Renault), although he was not the first, or only one, to "invent" these type of curves. One might be tempted to say that the mathematician Paul de Casteljau was first, as he began investigating the nature of these curves in 1959 while working at Citroën, and came up with a really elegant way of figuring out how to draw them. However, de Casteljau did not publish his work, making the question "who was first" hard to answer in any absolute sense. Or is it? Bézier curves are, at their core, "Bernstein polynomials", a family of mathematical functions investigated by Sergei Natanovich Bernstein, whose publications on them date back at least as far as 1912.

-

Anyway, that's mostly trivia, what you are more likely to care about is that these curves are handy: you can link up multiple Bézier curves so that the combination looks like a single curve. If you've ever drawn Photoshop "paths" or worked with vector drawing programs like Flash, Illustrator or Inkscape, those curves you've been drawing are Bézier curves.

-

But what if you need to program them yourself? What are the pitfalls? How do you draw them? What are the bounding boxes, how do you determine intersections, how can you extrude a curve, in short: how do you do everything that you might want to do with these curves? That's what this page is for. Prepare to be mathed!

-
- -

Virtually all Bézier graphics are interactive.

-

This page uses interactive examples, relying heavily on Bezier.js, as well as maths formulae which are typeset into SVG using the XeLaTeX typesetting system and pdf2svg by David Barton.

-

This book is open source.

-

This book is an open source software project, and lives on two github repositories. The first is https://github.com/pomax/bezierinfo and is the purely-for-presentation version you are viewing right now. The other repository is https://github.com/pomax/BezierInfo-2, which is the development version, housing all the code that gets turned into the web version, and is also where you should file issues if you find bugs or have ideas on what to change or add to the primer.

-

How complicated is the maths going to be?

-

Most of the mathematics in this Primer are early high school maths. If you understand basic arithmetic, and you know how to read English, you should be able to get by just fine. There will at times be far more complicated maths, but if you don't feel like digesting them, you can safely skip over them by either skipping over the "detail boxes" in section or by just jumping to the end of a section with maths that looks too involving. The end of sections typically simply list the conclusions so you can just work with those values directly.

-

What language is all this example code in?

-

There are way too many programming languages to favour one of all others, soo all the example code in this Primer uses a form of pseudo-code that uses a syntax that's close enough to, but not actually, modern scripting languages like JS, Python, etc. That means you won't be able to copy-paste any of it without giving it any thought, but that's intentional: if you're reading this primer, presumably you want to learn, and you don't learn by copy-pasting. You learn by doing things yourself, making mistakes, and then fixing those mistakes. Now, of course, I didn't intentionally add errors in the example code just to trick you into making mistakes (that would be horrible!) but I did intentionally keep the code from favouring one programming language over another. Don't worry though, if you know even a single procedural programming language, you should be able to read the examples without any difficulties.

-

Questions, comments:

-

If you have suggestions for new sections, hit up the Github issue tracker (also reachable from the repo linked to in the upper right). If you have questions about the material, there's currently no comment section while I'm doing the rewrite, but you can use the issue tracker for that as well. Once the rewrite is done, I'll add a general comment section back in, and maybe a more topical "select this section of text and hit the 'question' button to ask a question about it" system. We'll see.

-

Help support the book!

-

If you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 100-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing!

-
-
-
-

What's new?

-

This primer is a living document, and so depending on when you last look at it, there may be new content. Click the following link to expand this section to have a look at what got added, when, or click through to the News posts for more detailed updates. (RSS feed available)

- - - -

November 2020

  • Added a section on finding curve/circle intersections

    -
-

October 2020

  • Added the Ukranian locale! Help out in getting its localization to 100%!

    -
-

August-September 2020

  • Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS turned on.

    -
-

June 2020

  • Added automatic CI/CD using Github Actions

    -
-

January 2020

  • Added reset buttons to all graphics

    -
    • -

  • Updated to preface to correctly describe the on-page maths

    -
    • -

  • Fixed the Catmull-Rom section because it had glaring maths errors

    -
-

August 2019

  • Added a section on (plain) rational Bezier curves

    -
    • -

  • Improved the Graphic component to allow for sliders

    -
-

December 2018

  • Added a section on curvature and calculating kappa.

    -
    • -

  • Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this site!

    -
-

August 2018

  • Added a section on finding a curve's y, if all you have is the x coordinate.

    -
-

July 2018

  • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

    -
    • -

  • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

    -
    • -

  • (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

    -
-

June 2018

  • Added a section on direct curve fitting.

    -
    • -

  • Added source links for all graphics.

    -
    • -

  • Added this "What's new?" section.

    -
-

April 2017

  • Added a section on 3d normals.

    -
    • -

  • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

    -
-

February 2017

  • Finished rewriting the entire codebase for localization.

    -
-

January 2016

  • Added a section to explain the Bezier interval.

    -
    • -

  • Rewrote the Primer as a React application.

    -
-

December 2015

  • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

    -
    • -

  • Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics.

    -
-

May 2015

  • Switched over to pure JS rather than Processing-through-Processing.js

    -
    • -

  • Added Cardano's algorithm for finding the roots of a cubic polynomial.

    -
-

April 2015

  • Added a section on arc length approximations.

    -
-

February 2015

  • Added a section on the canonical cubic Bezier form.

    -
-

November 2014

  • Switched to HTTPS.

    -
-

July 2014

  • Added the section on arc approximation.

    -
-

April 2014

  • Added the section on Catmull-Rom fitting.

    -
-

November 2013

  • Added the section on Catmull-Rom / Bezier conversion.

    -
    • -

  • Added the section on Bezier cuves as matrices.

    -
-

April 2013

  • Added a section on poly-Beziers.

    -
    • -

  • Added a section on boolean shape operations.

    -
-

March 2013

  • First drastic rewrite.

    -
    • -

  • Added sections on circle approximations.

    -
    • -

  • Added a section on projecting a point onto a curve.

    -
    • -

  • Added a section on tangents and normals.

    -
    • -

  • Added Legendre-Gauss numerical data tables.

    -
-

October 2011

  • First commit for the bezierinfo site, based on the pre-Primer webpage that covered the basics of Bezier curves in HTML with Processing.js examples.

    -
-
-
-

table of contentsnext
A lightning introduction

-

Let's start with the good stuff: when we're talking about Bézier curves, we're talking about the things that you can see in the following graphics. They run from some start point to some end point, with their curvature influenced by one or more "intermediate" control points. Now, because all the graphics on this page are interactive, go manipulate those curves a bit: click-drag the points, and see how their shape changes based on what you do.

-
- - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
- -

These curves are used a lot in computer aided design and computer aided manufacturing (CAD/CAM) applications, as well as in graphic design programs like Adobe Illustrator and Photoshop, Inkscape, GIMP, etc. and in graphic technologies like scalable vector graphics (SVG) and OpenType fonts (TTF/OTF). A lot of things use Bézier curves, so if you want to learn more about them... prepare to get your learn on!

- -
-
-

previoustable of contentsnext
So what makes a Bézier Curve?

-

Playing with the points for curves may have given you a feel for how Bézier curves behave, but what are Bézier curves, really? There are two ways to explain what a Bézier curve is, and they turn out to be the entirely equivalent, but one of them uses complicated maths, and the other uses really simple maths. So... let's start with the simple explanation:

-

Bézier curves are the result of linear interpolations. That sounds complicated but you've been doing linear interpolation since you were very young: any time you had to point at something between two other things, you've been applying linear interpolation. It's simply "picking a point between two points".

-

If we know the distance between those two points, and we want a new point that is, say, 20% the distance away from the first point (and thus 80% the distance away from the second point) then we can compute that really easily:

- - -

So let's look at that in action: the following graphic is interactive in that you can use your up and down arrow keys to increase or decrease the interpolation ratio, to see what happens. We start with three points, which gives us two lines. Linear interpolation over those lines gives us two points, between which we can again perform linear interpolation, yielding a single point. And that point —and all points we can form in this way for all ratios taken together— form our Bézier curve:

- - - Scripts are disabled. Showing fallback image. - - - - - -

And that brings us to the complicated maths: calculus.

-

While it doesn't look like that's what we've just done, we actually just drew a quadratic curve, in steps, rather than in a single go. One of the fascinating parts about Bézier curves is that they can both be described in terms of polynomial functions, as well as in terms of very simple interpolations of interpolations of [...]. That, in turn, means we can look at what these curves can do based on both "real maths" (by examining the functions, their derivatives, and all that stuff), as well as by looking at the "mechanical" composition (which tells us, for instance, that a curve will never extend beyond the points we used to construct it).

-

So let's start looking at Bézier curves a bit more in depth: their mathematical expressions, the properties we can derive from them, and the various things we can do to, and with, Bézier curves.

- -
-
-

previoustable of contentsnext
The mathematics of Bézier curves

-

Bézier curves are a form of "parametric" function. Mathematically speaking, parametric functions are cheats: a "function" is actually a well defined term representing a mapping from any number of inputs to a single output. Numbers go in, a single number comes out. Change the numbers that go in, and the number that comes out is still a single number.

-

Parametric functions cheat. They basically say "alright, well, we want multiple values coming out, so we'll just use more than one function". An illustration: Let's say we have a function that maps some value, let's call it x, to some other value, using some kind of number manipulation:

- - -

The notation f(x) is the standard way to show that it's a function (by convention called f if we're only listing one) and its output changes based on one variable (in this case, x). Change x, and the output for f(x) changes.

-

So far, so good. Now, let's look at parametric functions, and how they cheat. Let's take the following two functions:

- - -

There's nothing really remarkable about them, they're just a sine and cosine function, but you'll notice the inputs have different names. If we change the value for a, we're not going to change the output value for f(b), since a isn't used in that function. Parametric functions cheat by changing that. In a parametric function all the different functions share a variable, like this:

- - -

Multiple functions, but only one variable. If we change the value for t, we change the outcome of both fa(t) and fb(t). You might wonder how that's useful, and the answer is actually pretty simple: if we change the labels fa(t) and fb(t) with what we usually mean with them for parametric curves, things might be a lot more obvious:

- - -

There we go. x/y coordinates, linked through some mystery value t.

-

So, parametric curves don't define a y coordinate in terms of an x coordinate, like normal functions do, but they instead link the values to a "control" variable. If we vary the value of t, then with every change we get two values, which we can use as (x,y) coordinates in a graph. The above set of functions, for instance, generates points on a circle: We can range t from negative to positive infinity, and the resulting (x,y) coordinates will always lie on a circle with radius 1 around the origin (0,0). If we plot it for t from 0 to 5, we get this:

- - - Scripts are disabled. Showing fallback image. - - - - - -

Bézier curves are just one out of the many classes of parametric functions, and are characterised by using the same base function for all of the output values. In the example we saw above, the x and y values were generated by different functions (one uses a sine, the other a cosine); but Bézier curves use the "binomial polynomial" for both the x and y outputs. So what are binomial polynomials?

-

You may remember polynomials from high school. They're those sums that look like this:

- - -

If the highest order term they have is x³, they're called "cubic" polynomials; if it's x², it's a "square" polynomial; if it's just x, it's a line (and if there aren't even any terms with x it's not a polynomial!)

-

Bézier curves are polynomials of t, rather than x, with the value for t being fixed between 0 and 1, with coefficients a, b etc. taking the "binomial" form, which sounds fancy but is actually a pretty simple description for mixing values:

- - -

I know what you're thinking: that doesn't look too simple! But if we remove t and add in "times one", things suddenly look pretty easy. Check out these binomial terms:

- - -

Notice that 2 is the same as 1+1, and 3 is 2+1 and 1+2, and 6 is 3+3... As you can see, each time we go up a dimension, we simply start and end with 1, and everything in between is just "the two numbers above it, added together", giving us a simple number sequence known as Pascal's triangle. Now that's easy to remember.

-

There's an equally simple way to figure out how the polynomial terms work: if we rename (1-t) to a and t to b, and remove the weights for a moment, we get this:

- - -

It's basically just a sum of "every combination of a and b", progressively replacing a's with b's after every + sign. So that's actually pretty simple too. So now you know binomial polynomials, and just for completeness I'm going to show you the generic function for this:

- - -

And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ).

-
+ +

+ And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable + listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ). +

+
+

How to implement the basis function

+

We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:

-

How to implement the basis function

-

We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:

- - - - - -
1 - -
2
3
4
5
+ return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers in between being the sum of the previous row's elements on either side "above" the one we're computing.

-

We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup table:

+

+ I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above + explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then + [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers + in between being the sum of the previous row's elements on either side "above" the one we're computing. +

+

+ We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup + table: +

- - - - - - - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
+ return lut[n][k] + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + -

So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, we expand it first. Our basis function now looks like this:

+

+ So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, + we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, + we expand it first. Our basis function now looks like this: +

- - - - -
1 - -
2
3
4
5
+ return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves (although we will also provide code for arbitrary curves in this primer); we need quadratic and cubic curves, and that means we can drastically simplify the code:

+

+ Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves (although we will also + provide code for arbitrary curves in this primer); we need quadratic and cubic curves, and that means we can drastically simplify the + code: +

- - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
+ return mt3 + 3*mt2*t + 3*mt*t2 + t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

And now we know how to program the basis function. Excellent.

-
+

And now we know how to program the basis function. Excellent.

+
-

So, now we know what the basis function looks like, time to add in the magic that makes Bézier curves so special: control points.

+

So, now we know what the basis function looks like, time to add in the magic that makes Bézier curves so special: control points.

+
+
+

+
previoustable of contentsnext
+ Controlling Bézier curvatures +

+

+ Bézier curves are, like all "splines", interpolation functions. This means that they take a set of points, and generate values somewhere + "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline + for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point + contributes to the value generated by the function, so we can see which points are important, where, in the curve. +

+

+ The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's + contribution to the total sum of the Bézier function. Click-and-drag to see the interpolation percentages for each curve-defining point at + a specific t value. +

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
-
-
-

previoustable of contentsnext
Controlling Bézier curvatures

-

Bézier curves are, like all "splines", interpolation functions. This means that they take a set of points, and generate values somewhere "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point contributes to the value generated by the function, so we can see which points are important, where, in the curve.

-

The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's contribution to the total sum of the Bézier function. Click-and-drag to see the interpolation percentages for each curve-defining point at a specific t value.

-
- - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
- -

Also shown is the interpolation function for a 15th order Bézier function. As you can see, the start and end point contribute considerably more to the curve's shape than any other point in the control point set.

-

If we want to change the curve, we need to change the weights of each point, effectively changing the interpolations. The way to do this is about as straightforward as possible: just multiply each point with a value that changes its strength. These values are conventionally called "weights", and we can add them to our original Bézier function:

- - -

That looks complicated, but as it so happens, the "weights" are actually just the coordinate values we want our curve to have: for an nth order curve, w0 is our start coordinate, wn is our last coordinate, and everything in between is a controlling coordinate. Say we want a cubic curve that starts at (110,150), is controlled by (25,190) and (210,250) and ends at (210,30), we use this Bézier curve:

- - -

Which gives us the curve we saw at the top of the article:

- - - Scripts are disabled. Showing fallback image. - - - -

What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and algorithms that we can apply, and the tasks they achieve.

-
+ +

Which gives us the curve we saw at the top of the article:

+ + + Scripts are disabled. Showing fallback image. + + + +

+ What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and + algorithms that we can apply, and the tasks they achieve. +

+
+

How to implement the weighted basis function

+

Given that we already know how to implement basis function, adding in the control points is remarkably easy:

-

How to implement the weighted basis function

-

Given that we already know how to implement basis function, adding in the control points is remarkably easy:

- - - - - -
1 - -
2
3
4
5
+ return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

And now for the optimized versions:

+

And now for the optimized versions:

- - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
+ return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

And now we know how to program the weighted basis function.

-
- -
-
-

previoustable of contentsnext
Controlling Bézier curvatures, part 2: Rational Béziers

-

We can further control Bézier curves by "rationalising" them: that is, adding a "ratio" value in addition to the weight value discussed in the previous section, thereby gaining control over "how strongly" each coordinate influences the curve.

-

Adding these ratio values to the regular Bézier curve function is fairly easy. Where the regular function is the following:

- - -

The function for rational Bézier curves has two more terms:

- - -

In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just using different weight. So far, nothing too special.

-

However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights.

-

This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly describing circles (which we'll see in a later section is literally impossible using standard Bézier curves).

-

But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects what gets drawn:

- - - Scripts are disabled. Showing fallback image. - - - - - - - - -

You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence relative to all other points.

-
+ +

+ In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] + then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just + using different weight. So far, nothing too special. +

+

+ However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a + fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the + "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights. +

+

+ This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve + that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly + describing circles (which we'll see in a later section is literally impossible using standard Bézier curves). +

+

+ But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an + interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with + ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate + exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects + what gets drawn: +

+ + + Scripts are disabled. Showing fallback image. + + + + + + + + +

+ You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will + want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like + with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence + relative to all other points. +

+
+

How to implement rational curves

+

Extending the code of the previous section to include ratios is almost trivial:

-

How to implement rational curves

-

Extending the code of the previous section to include ratios is almost trivial:

- - - - - - - - - - - - - - - - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+ return (f[0] * w[0] + f[1] * w[1] + f[2] * w[2] + f[3] * w[3])/basis + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + -

And that's all we have to do.

-
- -
-
-

previoustable of contentsnext
The Bézier interval [0,1]

-

Now that we know the mathematics behind Bézier curves, there's one curious thing that you may have noticed: they always run from t=0 to t=1. Why that particular interval?

-

It all has to do with how we run from "the start" of our curve to "the end" of our curve. If we have a value that is a mixture of two other values, then the general formula for this is:

- - -

The obvious start and end values here need to be a=1, b=0, so that the mixed value is 100% value 1, and 0% value 2, and a=0, b=1, so that the mixed value is 0% value 1 and 100% value 2. Additionally, we don't want "a" and "b" to be independent: if they are, then we could just pick whatever values we like, and end up with a mixed value that is, for example, 100% value 1 and 100% value 2. In principle that's fine, but for Bézier curves we always want mixed values between the start and end point, so we need to make sure we can never set "a" and "b" to some values that lead to a mix value that sums to more than 100%. And that's easy:

- - -

With this we can guarantee that we never sum above 100%. By restricting a to values in the interval [0,1], we will always be somewhere between our two values (inclusively), and we will always sum to a 100% mix.

-

But... what if we use this form, which is based on the assumption that we will only ever use values between 0 and 1, and instead use values outside of that interval? Do things go horribly wrong? Well... not really, but we get to "see more".

-

In the case of Bézier curves, extending the interval simply makes our curve "keep going". Bézier curves are simply segments of some polynomial curve, so if we pick a wider interval we simply get to see more of the curve. So what do they look like?

-

The following two graphics show you Bézier curves rendered "the usual way", as well as the curves they "lie on" if we were to extend the t values much further. As you can see, there's a lot more "shape" hidden in the rest of the curve, and we can model those parts by moving the curve points around.

-
- - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
+ +

+ With this we can guarantee that we never sum above 100%. By restricting a to values in the interval [0,1], we will always be + somewhere between our two values (inclusively), and we will always sum to a 100% mix. +

+

+ But... what if we use this form, which is based on the assumption that we will only ever use values between 0 and 1, and instead use + values outside of that interval? Do things go horribly wrong? Well... not really, but we get to "see more". +

+

+ In the case of Bézier curves, extending the interval simply makes our curve "keep going". Bézier curves are simply segments of some + polynomial curve, so if we pick a wider interval we simply get to see more of the curve. So what do they look like? +

+

+ The following two graphics show you Bézier curves rendered "the usual way", as well as the curves they "lie on" if we were to extend the + t values much further. As you can see, there's a lot more "shape" hidden in the rest of the curve, and we can model those + parts by moving the curve points around. +

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
-

In fact, there are curves used in graphics design and computer modelling that do the opposite of Bézier curves; rather than fixing the interval, and giving you freedom to choose the coordinates, they fix the coordinates, but give you freedom over the interval. A great example of this is the "Spiro" curve, which is a curve based on part of a Cornu Spiral, also known as Euler's Spiral. It's a very aesthetically pleasing curve and you'll find it in quite a few graphics packages like FontForge and Inkscape. It has even been used in font design, for example for the Inconsolata typeface.

- -
-
-

previoustable of contentsnext
Bézier curvatures as matrix operations

-

We can also represent Bézier curves as matrix operations, by expressing the Bézier formula as a polynomial basis function and a coefficients matrix, and the actual coordinates as a matrix. Let's look at what this means for the cubic curve, using P... to refer to coordinate values "in one or more dimensions":

- - -

Disregarding our actual coordinates for a moment, we have:

- - -

We can write this as a sum of four expressions:

- - -

And we can expand these expressions:

- - -

Furthermore, we can make all the 1 and 0 factors explicit:

- - -

And that, we can view as a series of four matrix operations:

- - -

If we compact this into a single matrix operation, we get:

- - -

This kind of polynomial basis representation is generally written with the bases in increasing order, which means we need to flip our t matrix horizontally, and our big "mixing" matrix upside down:

- - -

And then finally, we can add in our original coordinates as a single third matrix:

- - -

We can perform the same trick for the quadratic curve, in which case we end up with:

- - -

If we plug in a t value, and then multiply the matrices, we will get exactly the same values as when we evaluate the original polynomial function, or as when we evaluate the curve using progressive linear interpolation.

-

So: why would we bother with matrices? Matrix representations allow us to discover things about functions that would otherwise be hard to tell. It turns out that the curves form triangular matrices, and they have a determinant equal to the product of the actual coordinates we use for our curve. It's also invertible, which means there's a ton of properties that are all satisfied. Of course, the main question is "why is this useful to us now?", and the answer to that is that it's not immediately useful, but you'll be seeing some instances where certain curve properties can be either computed via function manipulation, or via clever use of matrices, and sometimes the matrix approach can be (drastically) faster.

-

So for now, just remember that we can represent curves this way, and let's move on.

+ +

+ If we plug in a t value, and then multiply the matrices, we will get exactly the same values as when we evaluate the original + polynomial function, or as when we evaluate the curve using progressive linear interpolation. +

+

+ So: why would we bother with matrices? Matrix representations allow us to discover things about functions that would + otherwise be hard to tell. It turns out that the curves form + triangular matrices, and they have a determinant equal to the product of the + actual coordinates we use for our curve. It's also invertible, which means there's + a ton of properties that are all satisfied. Of + course, the main question is "why is this useful to us now?", and the answer to that is that it's not immediately useful, but + you'll be seeing some instances where certain curve properties can be either computed via function manipulation, or via clever use of + matrices, and sometimes the matrix approach can be (drastically) faster. +

+

So for now, just remember that we can represent curves this way, and let's move on.

+
+
+

+
previoustable of contentsnext
+ de Casteljau's algorithm +

+

+ If we want to draw Bézier curves, we can run through all values of t from 0 to 1 and then compute the weighted basis function + at each value, getting the x/y values we need to plot. Unfortunately, the more complex the curve gets, the more expensive + this computation becomes. Instead, we can use de Casteljau's algorithm to draw curves. This is a geometric approach to curve + drawing, and it's really easy to implement. So easy, in fact, you can do it by hand with a pencil and ruler. +

+

Rather than using our calculus function to find x/y values for t, let's do this instead:

+
    +
  • treat t as a ratio (which it is). t=0 is 0% along a line, t=1 is 100% along a line.
    • +
  • Take all lines between the curve's defining points. For an order n curve, that's n lines.
    • +
  • + Place markers along each of these line, at distance t. So if t is 0.2, place the mark at 20% from the start, + 80% from the end. +
    • +
  • Now form lines between those points. This gives n-1 lines.
    • +
  • Place markers along each of these line at distance t.
    • +
  • Form lines between those points. This'll be n-2 lines.
    • +
  • Place markers, form lines, place markers, etc.
    • +
  • + Repeat this until you have only one line left. The point t on that line coincides with the original curve point at + t. +
    • +
+

+ To see this in action, move the slider for the following sketch to changes which curve point is explicitly evaluated using de Casteljau's + algorithm. +

+ + + Scripts are disabled. Showing fallback image. + + + + + +
+

How to implement de Casteljau's algorithm

+

+ Let's just use the algorithm we just specified, and implement that as a function that can take a list of curve-defining points, and a + t value, and draws the associated point on the curve for that t value: +

-
-
-

previoustable of contentsnext
de Casteljau's algorithm

-

If we want to draw Bézier curves, we can run through all values of t from 0 to 1 and then compute the weighted basis function at each value, getting the x/y values we need to plot. Unfortunately, the more complex the curve gets, the more expensive this computation becomes. Instead, we can use de Casteljau's algorithm to draw curves. This is a geometric approach to curve drawing, and it's really easy to implement. So easy, in fact, you can do it by hand with a pencil and ruler.

-

Rather than using our calculus function to find x/y values for t, let's do this instead:

-
    -
  • treat t as a ratio (which it is). t=0 is 0% along a line, t=1 is 100% along a line.
    • -
  • Take all lines between the curve's defining points. For an order n curve, that's n lines.
    • -
  • Place markers along each of these line, at distance t. So if t is 0.2, place the mark at 20% from the start, 80% from the end.
    • -
  • Now form lines between those points. This gives n-1 lines.
    • -
  • Place markers along each of these line at distance t.
    • -
  • Form lines between those points. This'll be n-2 lines.
    • -
  • Place markers, form lines, place markers, etc.
    • -
  • Repeat this until you have only one line left. The point t on that line coincides with the original curve point at t.
    • -
-

To see this in action, move the slider for the following sketch to changes which curve point is explicitly evaluated using de Casteljau's algorithm.

- - - Scripts are disabled. Showing fallback image. - - - - - -
- -

How to implement de Casteljau's algorithm

-

Let's just use the algorithm we just specified, and implement that as a function that can take a list of curve-defining points, and a t value, and draws the associated point on the curve for that t value:

- - - - - - - - -
1 - -
2
3
4
5
6
7
8
+ drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

And done, that's the algorithm implemented. Although: usually you don't get the luxury of overloading the "+" operator, so let's also give the code for when you need to work with x and y values separately:

+

+ And done, that's the algorithm implemented. Although: usually you don't get the luxury of overloading the "+" operator, so let's also + give the code for when you need to work with x and y values separately: +

- - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
+ drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

So what does this do? This draws a point, if the passed list of points is only 1 point long. Otherwise it will create a new list of points that sit at the t ratios (i.e. the "markers" outlined in the above algorithm), and then call the draw function for this new list.

-
+

+ So what does this do? This draws a point, if the passed list of points is only 1 point long. Otherwise it will create a new list of + points that sit at the t ratios (i.e. the "markers" outlined in the above algorithm), and then call the draw function for this + new list. +

+ +
+
+

+
previoustable of contentsnext
+ Simplified drawing +

+

+ We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, + a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines. +

+

+ We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the + number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we + can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we + lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or + curvature alignment. +

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
+

+ Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. + You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the + cubic curve), a higher number is required to capture the curvature changes properly. +

+
+

How to implement curve flattening

+

Let's just use the algorithm we just specified, and implement that:

-
-
-

previoustable of contentsnext
Simplified drawing

-

We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines.

-

We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or curvature alignment.

-
- - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
- -

Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the cubic curve), a higher number is required to capture the curvature changes properly.

-
- -

How to implement curve flattening

-

Let's just use the algorithm we just specified, and implement that:

- - - - - - - -
1 - -
2
3
4
5
6
7
+ return coordinates; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

+

And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

- - - - - - -
1 - -
2
3
4
5
6
7
+ coord = _coord + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

-
+

We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

+ +
+
+

+
previoustable of contentsnext
+ Splitting curves +

+

+ Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken + together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the + points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our + on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point. +

+ + + Scripts are disabled. Showing fallback image. + + + + + +
+

implementing curve splitting

+

We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

-
-
-

previoustable of contentsnext
Splitting curves

-

Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point.

- - - Scripts are disabled. Showing fallback image. - - - - - -
- -

implementing curve splitting

-

We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

- - - - - - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
+ drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + -

After running this function for some value t, the left and right arrays will contain all the coordinates for two new curves - one to the "left" of our t value, the other on the "right". These new curves will have the same order as the original curve, and can be overlaid exactly on the original curve.

-
- -
-
-

previoustable of contentsnext
Splitting curves using matrices

-

Another way to split curves is to exploit the matrix representation of a Bézier curve. In the section on matrices, we saw that we can represent curves as matrix multiplications. Specifically, we saw these two forms for the quadratic and cubic curves respectively: (we'll reverse the Bézier coefficients vector for legibility)

- - -

and

- + +

+ Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the + coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual + "point on the curve" information into a new matrix multiplication: +

+ + +

and

+ + +

+ If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two + staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, + from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear + algebra (and if you don't care about the derivations, just skip to the end of the box for the results!). +

+
+

Deriving new hull coordinates

+

+ Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look + at the quadratic curve first: +

+ + + - -

Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual "point on the curve" information into a new matrix multiplication:

- - -

and

- - -

If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear algebra (and if you don't care about the derivations, just skip to the end of the box for the results!).

-
- -

Deriving new hull coordinates

-

Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look at the quadratic curve first:

- - - - - - - - -

We can do this because [M · M-1] is the identity matrix. It's a bit like multiplying something by x/x in calculus: it doesn't do anything to the function, but it does allow you to rewrite it to something that may be easier to work with, or can be broken up differently. In the same way, multiplying our matrix by [M · M-1] has no effect on the total formula, but it does allow us to change the matrix sequence [something · M] to a sequence [M · something], and that makes a world of difference: if we know what [M-1 · Z · M] is, we can apply that to our coordinates, and be left with a proper matrix representation of a quadratic Bézier curve (which is [T · M · P]), with a new set of coordinates that represent the curve from t = 0 to t = z. So let's get computing:

- - -

Excellent! Now we can form our new quadratic curve:

- - - - - - -

Brilliant: if we want a subcurve from t = 0 to t = z, we can keep the first coordinate the same (which makes sense), our control point becomes a z-ratio mixture of the original control point and the start point, and the new end point is a mixture that looks oddly similar to a Bernstein polynomial of degree two. These new coordinates are actually really easy to compute directly!

-

Of course, that's only one of the two curves. Getting the section from t = z to t = 1 requires doing this again. We first observe that in the previous calculation, we actually evaluated the general interval [0,z]. We were able to write it down in a more simple form because of the zero, but what we actually evaluated, making the zero explicit, was:

- - - - -

If we want the interval [z,1], we will be evaluating this instead:

- - - - -

We're going to do the same trick of multiplying by the identity matrix, to turn [something · M] into [M · something]:

- - -

So, our final second curve looks like:

- - - - - - -

Nice. We see the same thing as before: we can keep the last coordinate the same (which makes sense); our control point becomes a z-ratio mixture of the original control point and the end point, and the new start point is a mixture that looks oddly similar to a bernstein polynomial of degree two, except this time it uses (z-1) rather than (1-z). These new coordinates are also really easy to compute directly!

-
- -

So, using linear algebra rather than de Casteljau's algorithm, we have determined that, for any quadratic curve split at some value t = z, we get two subcurves that are described as Bézier curves with simple-to-derive coordinates:

- - -

and

- - -

We can do the same for cubic curves. However, I'll spare you the actual derivation (don't let that stop you from writing that out yourself, though) and simply show you the resulting new coordinate sets:

- - -

and

- - -

So, looking at our matrices, did we really need to compute the second segment matrix? No, we didn't. Actually having one segment's matrix means we implicitly have the other: push the values of each row in the matrix Q to the right, with zeroes getting pushed off the right edge and appearing back on the left, and then flip the matrix vertically. Presto, you just "calculated" Q'.

-

Implementing curve splitting this way requires less recursion, and is just straight arithmetic with cached values, so can be cheaper on systems where recursion is expensive. If you're doing computation with devices that are good at matrix multiplication, chopping up a Bézier curve with this method will be a lot faster than applying de Casteljau.

- -
-
-

previoustable of contentsnext
Lowering and elevating curve order

-

One interesting property of Bézier curves is that an nth order curve can always be perfectly represented by an (n+1)th order curve, by giving the higher-order curve specific control points.

-

If we have a curve with three points, then we can create a curve with four points that exactly reproduces the original curve. First, we give it the same start and end points, and for its two control points we pick "1/3rd start + 2/3rd control" and "2/3rd control + 1/3rd end". Now we have exactly the same curve as before, except represented as a cubic curve rather than a quadratic curve.

-

The general rule for raising an nth order curve to an (n+1)th order curve is as follows (observing that the start and end weights are the same as the start and end weights for the old curve):

- - -

However, this rule also has as direct consequence that you cannot generally safely lower a curve from nth order to (n-1)th order, because the control points cannot be "pulled apart" cleanly. We can try to, but the resulting curve will not be identical to the original, and may in fact look completely different.

-

However, there is a surprisingly good way to ensure that a lower order curve looks "as close as reasonably possible" to the original curve: we can optimise the "least-squares distance" between the original curve and the lower order curve, in a single operation (also explained over on Sirver's Castle). However, to use it, we'll need to do some calculus work and then switch over to linear algebra. As mentioned in the section on matrix representations, some things can be done much more easily with matrices than with calculus functions, and this is one of those things. So... let's go!

-

We start by taking the standard Bézier function, and condensing it a little:

- - -

Then, we apply one of those silly (actually, super useful) calculus tricks: since our t value is always between zero and one (inclusive), we know that (1-t) plus t always sums to 1. As such, we can express any value as a sum of t and 1-t:

- - -

So, with that seemingly trivial observation, we rewrite that Bézier function by splitting it up into a sum of a (1-t) and t component:

- - -

So far so good. Now, to see why we did this, let's write out the (1-t) and t parts, and see what that gives us. I promise, it's about to make sense. We start with (1-t):

- - -

So by using this seemingly silly trick, we can suddenly express part of our nth order Bézier function in terms of an (n+1)th order Bézier function. And that sounds a lot like raising the curve order! Of course we need to be able to repeat that trick for the t part, but that's not a problem:

- - -

So, with both of those changed from an order n expression to an order (n+1) expression, we can put them back together again. Now, where the order n function had a summation from 0 to n, the order n+1 function uses a summation from 0 to n+1, but this shouldn't be a problem as long as we can add some new terms that "contribute nothing". In the next section on derivatives, there is a discussion about why "higher terms than there is a binomial for" and "lower than zero terms" both "contribute nothing". So as long as we can add terms that have the same form as the terms we need, we can just include them in the summation, they'll sit there and do nothing, and the resulting function stays identical to the lower order curve.

-

Let's do this:

- - -

And this is where we switch over from calculus to linear algebra, and matrices: we can now express this relation between Bézier(n,t) and Bézier(n+1,t) as a very simple matrix multiplication:

- - -

where the matrix M is an n+1 by n matrix, and looks like:

- - -

That might look unwieldy, but it's really just a mostly-zeroes matrix, with a very simply fraction on the diagonal, and an even simpler fraction to the left of it. Multiplying a list of coordinates with this matrix means we can plug the resulting transformed coordinates into the one-order-higher function and get an identical looking curve.

-

Not too bad!

-

Equally interesting, though, is that with this matrix operation established, we can now use an incredibly powerful and ridiculously simple way to find out a "best fit" way to reverse the operation, called the normal equation. What it does is minimize the sum of the square differences between one set of values and another set of values. Specifically, if we can express that as some function A x = b, we can use it. And as it so happens, that's exactly what we're dealing with, so:

- - -

The steps taken here are:

-
    -
  1. We have a function in a form that the normal equation can be used with, so
    2. -
  2. apply the normal equation!
    3. -
  3. Then, we want to end up with just Bn on the left, so we start by left-multiply both sides such that we'll end up with lots of stuff on the left that simplified to "a factor 1", which in matrix maths is the identity matrix.
    4. -
  4. In fact, by left-multiplying with the inverse of what was already there, we've effectively "nullified" (but really, one-inified) that big, unwieldy block into the identity matrix I. So we substitute the mess with I, and then
    5. -
  5. because multiplication with the identity matrix does nothing (like multiplying by 1 does nothing in regular algebra), we just drop it.
    6. -
-

And we're done: we now have an expression that lets us approximate an n+1th order curve with a lower nth order curve. It won't be an exact fit, but it's definitely a best approximation. So, let's implement these rules for raising and lowering curve order to a (semi) random curve, using the following graphic. Select the sketch, which has movable control points, and press your up and down arrow keys to raise or lower the curve order.

- - - Scripts are disabled. Showing fallback image. - - - - - - - -
-
-

previoustable of contentsnext
Derivatives

-

There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straightforward, although we do need a bit of math.

-

First, let's look at the derivative rule for Bézier curves, which is:

- - -

which we can also write (observing that b in this formula is the same as our w weights, and that n times a summation is the same as a summation where each term is multiplied by n) as:

- - -

Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi). So for a 3rd degree curve, with four weights, the derivative has three new weights: w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2).

-
- -

"Slow down, why is that true?"

-

Sometimes just being told "this is the derivative" is nice, but you might want to see why this is indeed the case. As such, let's have a look at the proof for this derivative. First off, the weights are independent of the full Bézier function, so the derivative involves only the derivative of the polynomial basis function. So, let's find that:

- - -

Applying the product and chain rules gives us:

- - -

Which is hard to work with, so let's expand that properly:

- - -

Now, the trick is to turn this expression into something that has binomial coefficients again, so we want to end up with things that look like "x! over y!(x-y)!". If we can do that in a way that involves terms of n-1 and k-1, we'll be on the right track.

- - -

And that's the first part done: the two components inside the parentheses are actually regular, lower-order Bézier expressions:

- - -

Now to apply this to our weighted Bézier curves. We'll write out the plain curve formula that we saw earlier, and then work our way through to its derivative:

- - -

If we expand this (with some color to show how terms line up), and reorder the terms by increasing values for k we see the following:

- - -

Two of these terms fall way: the first term falls away because there is no -1st term in a summation. As such, it always contributes "nothing", so we can safely completely ignore it for the purpose of finding the derivative function. The other term is the very last term in this expansion: one involving Bn-1,n. This term would have a binomial coefficient of [i choose i+1], which is a non-existent binomial coefficient. Again, this term would contribute "nothing", so we can ignore it, too. This means we're left with:

- - -

And that's just a summation of lower order curves:

- - -

We can rewrite this as a normal summation, and we're done:

- - -
- -

Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for Bézier curves, and then the derivative:

- + +

And that's just a summation of lower order curves:

+ + +

We can rewrite this as a normal summation, and we're done:

+ + + +

+ Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for + Bézier curves, and then the derivative: +

+ - - - -

What are the differences? In terms of the actual Bézier curve, virtually nothing! We lowered the order (rather than n, it's now n-1), but it's still the same Bézier function. The only real difference is in how the weights change when we derive the curve's function. If we have four points A, B, C, and D, then the derivative will have three points, the second derivative two, and the third derivative one:

- - -

We can keep performing this trick for as long as we have more than one weight. Once we have one weight left, the next step will see k = 0, and the result of our "Bézier function" summation is zero, because we're not adding anything at all. As such, a quadratic curve has no second derivative, a cubic curve has no third derivative, and generalized: an nth order curve has n-1 (meaningful) derivatives, with any further derivative being zero.

- -
-
-

previoustable of contentsnext
Tangents and normals

-

If you want to move objects along a curve, or "away from" a curve, the two vectors you're most interested in are the tangent vector and normal vector for curve points. These are actually really easy to find. For moving and orienting along a curve, we use the tangent, which indicates the direction of travel at specific points, and is literally just the first derivative of our curve:

- - -

This gives us the directional vector we want. We can normalize it to give us uniform directional vectors (having a length of 1.0) at each point, and then do whatever it is we want to do based on those directions:

- - -

The tangent is very useful for moving along a line, but what if we want to move away from the curve instead, perpendicular to the curve at some point t? In that case we want the normal vector. This vector runs at a right angle to the direction of the curve, and is typically of length 1.0, so all we have to do is rotate the normalized directional vector and we're done:

- - -
- -

Rotating coordinates is actually very easy, if you know the rule for it. You might find it explained as "applying a rotation matrix, which is what we'll look at here, too. Essentially, the idea is to take the circles over which we can rotate, and simply "sliding the coordinates" over these circles by the desired -angle. If we want a quarter circle turn, we take the coordinate, slide it along the circle by a quarter turn, and done.

-

To turn any point (x,y) into a rotated point (x',y') (over 0,0) by some angle φ, we apply this nice and easy computation:

- - -

Which is the "long" version of the following matrix transformation:

- - -

And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1.

-

But why does this work? Why this matrix multiplication? Wikipedia (technically, Thomas Herter and Klaus Lott) tells us that a rotation matrix can be -treated as a sequence of three (elementary) shear operations. When we combine this into a single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. DataGenetics have an excellent article about this very thing: it's really quite cool, and I strongly recommend taking a quick break from this primer to read that article.

-
+ +

+ And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even + easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1. +

+

+ But why does this work? Why this matrix multiplication? + Wikipedia (technically, Thomas Herter and Klaus + Lott) tells us that a rotation matrix can be treated as a sequence of three (elementary) shear operations. When we combine this into a + single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. + DataGenetics have an excellent article about this very thing: it's + really quite cool, and I strongly recommend taking a quick break from this primer to read that article. +

+ -

The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant).

-
- - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
+

+ The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the + normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant). +

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
+
+
+

+
previoustable of contentsnext
+ Working with 3D normals +

+

+ Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things + this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but + for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, + it's not "super hard", but there are more steps involved and we should have a look at those. +

+

+ Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. + However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, + not a single vector, so we basically need to define what "the" normal is in the 3D case. +

+

+ The "naïve" approach is to construct what is known as the + Frenet normal, where we follow a simple recipe that works in + many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are + perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each + point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the + local tangent, as well as the tangents "right next to it". +

+

+ Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the + vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know + lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same + logic we used for the 2D case: "just rotate it 90 degrees". +

+

So let's do that! And in a twist surprise, we can do this in four lines:

+
    +
  • a = normalize(B'(t))
    • +
  • b = normalize(a + B''(t))
    • +
  • r = normalize(b × a)
    • +
  • normal = normalize(r × a)
    • +
+

Let's unpack that a little:

+
    +
  • + We start by taking the normalized vector for the derivative at some point on the + curve. We normalize it so the maths is less work. Less work is good. +
    • +
  • + Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and + just had the same derivative and second derivative from that point on. +
    • +
  • + This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which + means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the + cross product. (Note that while that operation uses the × operator, it's most + definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent + a quarter circle to get our normal, just like we did in the 2D case. +
    • +
  • + Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal + vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second + time, and immediately get our normal vector. +
    • +
+

+ And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can + move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor + position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc: +

+ + + Scripts are disabled. Showing fallback image. + + + + + +

+ However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the + curve" between t=0.65 and t=0.75... Why is it doing that? +

+

+ As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are + "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute + normals that just... look good. +

+

Thankfully, Frenet normals are not our only option.

+

+ Another option is to take a slightly more algorithmic approach and compute a form of + Rotation Minimising Frame + (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational + axis, and the normal vector, centered on an on-curve point. +

+

+ These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we + could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at + the same time that you're building lookup tables for your curve. +

+

+ The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by + applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are: +

+
    +
  • Take a point on the curve for which we know the RM frame already,
    • +
  • take a next point on the curve for which we don't know the RM frame yet, and
    • +
  • + reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous + points as a "mirror". +
    • +
  • + This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal + that's slightly off-kilter, so: +
    • +
  • + reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror". +
    • +
  • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
    • +
+

So, let's write some code for that!

+
+

Implementing Rotation Minimising Frames

+

+ We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it + yields a frame with properties: +

-
-
-

previoustable of contentsnext
Working with 3D normals

-

Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, it's not "super hard", but there are more steps involved and we should have a look at those.

-

Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, not a single vector, so we basically need to define what "the" normal is in the 3D case.

-

The "naïve" approach is to construct what is known as the Frenet normal, where we follow a simple recipe that works in many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the local tangent, as well as the tangents "right next to it".

-

Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same logic we used for the 2D case: "just rotate it 90 degrees".

-

So let's do that! And in a twist surprise, we can do this in four lines:

-
    -
  • a = normalize(B'(t))
    • -
  • b = normalize(a + B''(t))
    • -
  • r = normalize(b × a)
    • -
  • normal = normalize(r × a)
    • -
-

Let's unpack that a little:

-
    -
  • We start by taking the normalized vector for the derivative at some point on the curve. We normalize it so the maths is less work. Less work is good.
    • -
  • Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on.
    • -
  • This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the cross product. (Note that while that operation uses the × operator, it's most definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent a quarter circle to get our normal, just like we did in the 2D case.
    • -
  • Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second time, and immediately get our normal vector.
    • -
-

And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc:

- - - Scripts are disabled. Showing fallback image. - - - - - -

However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the curve" between t=0.65 and t=0.75... Why is it doing that?

-

As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute normals that just... look good.

-

Thankfully, Frenet normals are not our only option.

-

Another option is to take a slightly more algorithmic approach and compute a form of Rotation Minimising Frame (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational axis, and the normal vector, centered on an on-curve point.

-

These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at the same time that you're building lookup tables for your curve.

-

The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are:

-
    -
  • Take a point on the curve for which we know the RM frame already,
    • -
  • take a next point on the curve for which we don't know the RM frame yet, and
    • -
  • reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror".
    • -
  • This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter, so:
    • -
  • reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
    • -
  • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
    • -
-

So, let's write some code for that!

-
- -

Implementing Rotation Minimising Frames

-

We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it yields a frame with properties:

- - - - - - -
1 - -
2
3
4
5
6
+} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + -

Then, we can write a function that generates a sequence of RM frames in the following manner:

+

Then, we can write a function that generates a sequence of RM frames in the following manner:

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
+ frames.add(x1) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + -

Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more code to get much better looking normals.

-
+

+ Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more + code to get much better looking normals. +

+ -

Speaking of better looking, what does this actually look like? Let's revisit that earlier curve, but this time use rotation minimising frames rather than Frenet frames:

- - - Scripts are disabled. Showing fallback image. - - - - - -

That looks so much better!

-

For those reading along with the code: we don't even strictly speaking need a Frenet frame to start with: we could, for instance, treat the z-axis as our initial axis of rotation, so that our initial normal is (0,0,1) × tangent, and then take things from there, but having that initial "mathematically correct" frame so that the initial normal seems to line up based on the curve's orientation in 3D space is just nice.

- -
-
-

previoustable of contentsnext
Component functions

-

One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that function, but this poses a problem, since our function is parametric: every axis has its own function.

-

The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.

-

Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis value, respectively.

-

If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a change in the right graph.

- - - Scripts are disabled. Showing fallback image. - - - -

 

- - - Scripts are disabled. Showing fallback image. - - - - -
-
-

previoustable of contentsnext
Finding extremities: root finding

-

Now that we understand (well, superficially anyway) the component functions, we can find the extremities of our Bézier curve by finding maxima and minima on the component functions, by solving the equation B'(t) = 0. We've already seen that the derivative of a Bézier curve is a simpler Bézier curve, but how do we solve the equality? Fairly easily, actually, until our derivatives are 4th order or higher... then things get really hard. But let's start simple:

-

Quadratic curves: linear derivatives.

-

The derivative of a quadratic Bézier curve is a linear Bézier curve, interpolating between just two terms, which means finding the solution for "where is this line 0" is effectively trivial by rewriting it to a function of t and solving. First we turn our quadratic Bézier function into a linear one, by following the rule mentioned at the end of the derivatives section:

- - -

And then we turn this into our solution for t using basic arithmetics:

- - -

Done.

-

Although with the caveat that if b-a is zero, there is no solution and we probably shouldn't try to perform that division.

-

Cubic curves: the quadratic formula.

-

The derivative of a cubic Bézier curve is a quadratic Bézier curve, and finding the roots for a quadratic polynomial means we can apply the Quadratic formula. If you've seen it before, you'll remember it, and if you haven't, it looks like this:

- - -

So, if we can rewrite the Bézier component function as a plain polynomial, we're done: we just plug in the values into the quadratic formula, check if that square root is negative or not (if it is, there are no roots) and then just compute the two values that come out (because of that plus/minus sign we get two). Any value between 0 and 1 is a root that matters for Bézier curves, anything below or above that is irrelevant (because Bézier curves are only defined over the interval [0,1]). So, how do we convert?

-

First we turn our cubic Bézier function into a quadratic one, by following the rule mentioned at the end of the derivatives section:

- - -

And then, using these v values, we can find out what our a, b, and c should be:

- - -

This gives us three coefficients {a, b, c} that are expressed in terms of v values, where the v values are expressions of our original coordinate values, so we can do some substitution to get:

- - -

Easy-peasy. We can now almost trivially find the roots by plugging those values into the quadratic formula.

-

And as a cubic curve, there is also a meaningful second derivative, which we can compute by simple taking the derivative of the derivative.

-

Quartic curves: Cardano's algorithm.

-

We haven't really looked at them before now, but the next step up would be a Quartic curve, a fourth degree Bézier curve. As expected, these have a derivative that is a cubic function, and now things get much harder. Cubic functions don't have a "simple" rule to find their roots, like the quadratic formula, and instead require quite a bit of rewriting to a form that we can even start to try to solve.

-

Back in the 16th century, before Bézier curves were a thing, and even before calculus itself was a thing, Gerolamo Cardano figured out that even if the general cubic function is really hard to solve, it can be rewritten to a form for which finding the roots is "easier" (even if not "easy"):

- - -

We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather than three: this makes things considerably easier to solve because it lets us use regular calculus to find the values that satisfy the equation.

-

Now, there is one small hitch: as a cubic function, the solutions may be complex numbers rather than plain numbers... And Cardano realised this, centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number.

-

So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at Ken J. Ward's page for solving the cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a read-through.

-
+ +

+ We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather + than three: this makes things considerably easier to solve because it lets us use + regular calculus to find the values that satisfy the equation. +

+

+ Now, there is one small hitch: as a cubic function, the solutions may be + complex numbers rather than plain numbers... And Cardano realised this, + centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these + numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we + have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're + going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number. +

+

+ So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at + Ken J. Ward's page for solving the + cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the + complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a + read-through. +

+
+

Implementing Cardano's algorithm for finding all real roots

+

+ The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" + number space (denoted with ℝ), this is perfectly fine in the + "complex" number space (denoted with ℂ). And, as it so happens, Cardano is + also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm! +

-

Implementing Cardano's algorithm for finding all real roots

-

The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" number space (denoted with ℝ), this is perfectly fine in the "complex" number space (denoted with ℂ). And, as it so happens, Cardano is also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm!

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 - -
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
+} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + + 37 + + + 38 + + + 39 + + + 40 + + + 41 + + + 42 + + + 43 + + + 44 + + + 45 + + + 46 + + + 47 + + + 48 + + + 49 + + + 50 + + + 51 + + + 52 + + + 53 + + + 54 + + + 55 + + + 56 + + + 57 + + + 58 + + + 59 + + + 60 + + + 61 + + + 62 + + + 63 + + + 64 + + + 65 + + + 66 + + + 67 + + + 68 + + + 69 + + + 70 + + + 71 + + + 72 + + + 73 + + + 74 + + + 75 + + + 76 + + + 77 + + + 78 + + + 79 + + + 80 + + + 81 + + +
-
- -

And that's it. The maths is complicated, but the code is pretty much just "follow the maths, while caching as many values as we can to prevent recomputing things as much as possible" and now we have a way to find all roots for a cubic function and can just move on with using that to find extremities of our curves.

-

And of course, as a quartic curve also has meaningful second and third derivatives, we can quite easily compute those by using the derivative of the derivative (of the derivative), just as for cubic curves.

-

Quintic and higher order curves: finding numerical solutions

-

And this is where thing stop, because we cannot find the roots for polynomials of degree 5 or higher using algebra (a fact known as the Abel–Ruffini theorem). Instead, for occasions like these, where algebra simply cannot yield an answer, we turn to numerical analysis.

-

That's a fancy term for saying "rather than trying to find exact answers by manipulating symbols, find approximate answers by describing the underlying process as a combination of steps, each of which can be assigned a number via symbolic manipulation". For example, trying to mathematically compute how much water fits in a completely crazy three dimensional shape is very hard, even if it got you the perfect, precise answer. A much easier approach, which would be less perfect but still entirely useful, would be to just grab a buck and start filling the shape until it was full: just count the number of buckets of water you used. And if we want a more precise answer, we can use smaller buckets.

-

So that's what we're going to do here, too: we're going to treat the problem as a sequence of steps, and the smaller we can make each step, the closer we'll get to that "perfect, precise" answer. And as it turns out, there is a really nice numerical root-finding algorithm, called the Newton-Raphson root finding method (yes, after that Newton), which we can make use of. The Newton-Raphson approach consists of taking our impossible-to-solve function f(x), picking some initial value x (literally any value will do), and calculating f(x). We can think of that value as the "height" of the function at x. If that height is zero, we're done, we have found a root. If it isn't, we calculate the tangent line at f(x) and calculate at which x value its height is zero (which we've already seen is very easy). That will give us a new x and we repeat the process until we find a root.

-

Mathematically, this means that for some x, at step n=1, we perform the following calculation until fy(x) is zero, so that the next t is the same as the one we already have:

- - -

(The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

-

Now, this works well only if we can pick good starting points, and our curve is continuously differentiable and doesn't have oscillations. Glossing over the exact meaning of those terms, the curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: which starting points do we pick?

-

As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as high as you'll ever need to go.

-

In conclusion:

-

So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red:

- - - Scripts are disabled. Showing fallback image. - - - -

And for cubic curves, that means first and second derivatives, in red and purple respectively:

- - - Scripts are disabled. Showing fallback image. - - - + +

(The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

+

+ Now, this works well only if we can pick good starting points, and our curve is + continuously differentiable and doesn't have + oscillations. Glossing over the exact meaning of those terms, the + curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: + which starting points do we pick? +

+

+ As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from + t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may + pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: + there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order + curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as + high as you'll ever need to go. +

+

In conclusion:

+

+ So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those + roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red: +

+ + + Scripts are disabled. Showing fallback image. + + +

And for cubic curves, that means first and second derivatives, in red and purple respectively:

+ + + Scripts are disabled. Showing fallback image. + + +
+
+

+
previoustable of contentsnext
+ Bounding boxes +

+

+ If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four + values we need to box in our curve: +

+

Computing the bounding box for a Bézier curve:

+
    +
  1. Find all t value(s) for the curve derivative's x- and y-roots.
    2. +
  2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
    3. +
  3. + Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original + functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct. +
    4. +
+

+ Applying this approach to our previous root finding, we get the following + axis-aligned bounding boxes (with all curve extremity points + shown on the curve): +

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
-
-
-

previoustable of contentsnext
Bounding boxes

-

If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four values we need to box in our curve:

-

Computing the bounding box for a Bézier curve:

-
    -
  1. Find all t value(s) for the curve derivative's x- and y-roots.
    2. -
  2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
    3. -
  3. Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct.
    4. -
-

Applying this approach to our previous root finding, we get the following axis-aligned bounding boxes (with all curve extremity points shown on the curve):

-
- - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
- -

We can construct even nicer boxes by aligning them along our curve, rather than along the x- and y-axis, but in order to do so we first need to look at how aligning works.

- -
-
-

previoustable of contentsnext
Aligning curves

-

While there are an incredible number of curves we can define by varying the x- and y-coordinates for the control points, not all curves are actually distinct. For instance, if we define a curve, and then rotate it 90 degrees, it's still the same curve, and we'll find its extremities in the same spots, just at different draw coordinates. As such, one way to make sure we're working with a "unique" curve is to "axis-align" it.

-

Aligning also simplifies a curve's functions. We can translate (move) the curve so that the first point lies on (0,0), which turns our n term polynomial functions into n-1 term functions. The order stays the same, but we have less terms. Then, we can rotate the curves so that the last point always lies on the x-axis, too, making its coordinate (...,0). This further simplifies the function for the y-component to an n-2 term function. For instance, if we have a cubic curve such as this:

- - -

Then translating it so that the first coordinate lies on (0,0), moving all x coordinates by -120, and all y coordinates by -160, gives us:

- - -

If we then rotate the curve so that its end point lies on the x-axis, the coordinates (integer-rounded for illustrative purposes here) become:

- - -

If we drop all the zero-terms, this gives us:

- - -

We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae:

- - - Scripts are disabled. Showing fallback image. - - - -

 

- - - Scripts are disabled. Showing fallback image. - - - + +

+ We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning + our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae: +

+ + + Scripts are disabled. Showing fallback image. + + +

 

+ + + Scripts are disabled. Showing fallback image. + + +
+
+

+
previoustable of contentsnext
+ Tight bounding boxes +

+

+ With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, + recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding + box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T. +

+

We now have nice tight bounding boxes for our curves:

+
+ + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
-
-
-

previoustable of contentsnext
Tight bounding boxes

-

With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T.

-

We now have nice tight bounding boxes for our curves:

-
- - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
- -

These are, strictly speaking, not necessarily the tightest possible bounding boxes. It is possible to compute the optimal bounding box by determining which spanning lines we need to effect a minimal box area, but because of the parametric nature of Bézier curves this is actually a rather costly operation, and the gain in bounding precision is often not worth it.

- -
-
-

previoustable of contentsnext
Curve inflections

-

Now that we know how to align a curve, there's one more thing we can calculate: inflection points. Imagine we have a variable size circle that we can slide up against our curve. We place it against the curve and adjust its radius so that where it touches the curve, the curvatures of the curve and the circle are the same, and then we start to slide the circle along the curve - for quadratic curves, we can always do this without the circle behaving oddly: we might have to change the radius of the circle as we slide it along, but it'll always sit against the same side of the curve.

-

But what happens with cubic curves? Imagine we have an S curve and we place our circle at the start of the curve, and start sliding it along. For a while we can simply adjust the radius and things will be fine, but once we get to the midpoint of that S, something odd happens: the circle "flips" from one side of the curve to the other side, in order for the curvatures to keep matching. This is called an inflection, and we can find out where those happen relatively easily.

-

What we need to do is solve a simple equation:

- - -

What we're saying here is that given the curvature function C(t), we want to know for which values of t this function is zero, meaning there is no "curvature", which will be exactly at the point between our circle being on one side of the curve, and our circle being on the other side of the curve. So what does C(t) look like? Actually something that seems not too hard:

- - -

The function C(t) is the cross product between the first and second derivative functions for the parametric dimensions of our curve. And, as already shown, derivatives of Bézier curves are just simpler Bézier curves, with very easy to compute new coefficients, so this should be pretty easy.

-

However as we've seen in the section on aligning, aligning lets us simplify things a lot, by completely removing the contributions of the first coordinate from most mathematical evaluations, and removing the last y coordinate as well by virtue of the last point lying on the x-axis. So, while we can evaluate C(t) = 0 for our curve, it'll be much easier to first axis-align the curve and then evaluating the curvature function.

-
- -

Let's derive the full formula anyway

-

Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:

- - -

And of course the same functions for y:

- - -

Asking a computer to now compose the C(t) function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:

- - -

That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all disappear if we axis-align our curve, which is why aligning is a great idea.

-
+ +

+ That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all + disappear if we axis-align our curve, which is why aligning is a great idea. +

+ -

Aligning our curve so that three of the eight coefficients become zero, and observing that scale does not affect finding t values, we end up with the following simple term function for C(t):

- - -

That's a lot easier to work with: we see a fair number of terms that we can compute and then cache, giving us the following simplification:

- - -

This is a plain quadratic curve, and we know how to solve C(t) = 0; we use the quadratic formula:

- - -

We can easily compute this value if the discriminator isn't a negative number (because we only want real roots, not complex roots), and if x is not zero, because divisions by zero are rather useless.

-

Taking that into account, we compute t, we disregard any t value that isn't in the Bézier interval [0,1], and we now know at which t value(s) our curve will inflect.

- - - Scripts are disabled. Showing fallback image. - - - - -
-
-

previoustable of contentsnext
The canonical form (for cubic curves)

-

While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?

-

As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper "A Geometric Characterization of Parametric Cubic curves". It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?

-

The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown:

- - - Scripts are disabled. Showing fallback image. - - - -

This is a fairly funky image, so let's see what the various parts of it mean...

-

We see the three fixed points at (0,0), (0,1) and (1,1). The various regions and boundaries indicate what property the original curve will have, if the fourth point is in/on that region or boundary. Specifically, if the fourth point is...

-
    -

  1. ...anywhere inside the red zone, but not on its boundaries, the curve will be self-intersecting (yielding a loop). We won't know where it self-intersects (in terms of t values), but we are guaranteed that it does.

    -
    2. -

  2. ...on the left (red) edge of the red zone, the curve will have a cusp. We again don't know where, but we know there is one. This edge is described by the function:

    - - -
    3. -

  3. ...on the almost circular, lower right (pink) edge, the curve's end point touches the curve, forming a loop. This edge is described by the function:

    - - -
    4. -

  4. ...on the top (blue) edge, the curve's start point touches the curve, forming a loop. This edge is described by the function:

    - - -
    5. -

  5. ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

    -
    6. -

  6. ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two inflections (switching from concave to convex and then back again, or from convex to concave and then back again).

    -
    7. -

  7. ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

    -
    8. -
-

Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

-
+ + +
  • +

    ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

    +
    • +
  • +

    + ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two + inflections (switching from concave to convex and then back again, or from convex to concave and then back again). +

    +
    • +

  • ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

    • + +

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    +
    +

    Wait, where do those lines come from?

    +

    + Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, + and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield + formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic + expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the + properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval. +

    +

    + The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general + cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to + it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general + curve does over the interval t=0 to t=1. +

    +

    + For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you + can probably follow along with this paper, although you might have to read it a few times before all the bits "click". +

    +
    -

    Wait, where do those lines come from?

    -

    Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval.

    -

    The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general curve does over the interval t=0 to t=1.

    -

    For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you can probably follow along with this paper, although you might have to read it a few times before all the bits "click".

    -
    - -

    So now the question becomes: how do we manipulate our curve so that it fits this canonical form, with three fixed points, and one "free" point? Enter linear algebra. Don't worry, I'll be doing all the math for you, as well as show you what the effect is on our curves, but basically we're going to be using linear algebra, rather than calculus, because "it's way easier". Sometimes a calculus approach is very hard to work with, when the equivalent geometrical solution is super obvious.

    -

    The approach is going to start with a curve that doesn't have all-colinear points (so we need to make sure the points don't all fall on a straight line), and then applying three graphics operations that you will probably have heard of: translation (moving all points by some fixed x- and y-distance), scaling (multiplying all points by some x and y scale factor), and shearing (an operation that turns rectangles into parallelograms).

    -

    Step 1: we translate any curve by -p1.x and -p1.y, so that the curve starts at (0,0). We're going to make use of an interesting trick here, by pretending our 2D coordinates are 3D, with the z coordinate simply always being 1. This is an old trick in graphics to overcome the limitations of 2D transformations: without it, we can only turn (x,y) coordinates into new coordinates of the form (ax + by, cx + dy), which means we can't do translation, since that requires we end up with some kind of (x + a, y + b). If we add a bogus z coordinate that is always 1, then we can suddenly add arbitrary values. For example:

    - - -

    Sweet! z stays 1, so we can effectively ignore it entirely, but we added some plain values to our x and y coordinates. So, if we want to subtract p1.x and p1.y, we use:

    - - -

    Running all our coordinates through this transformation gives a new set of coordinates, let's call those U, where the first coordinate lies on (0,0), and the rest is still somewhat free. Our next job is to make sure point 2 ends up lying on the x=0 line, so what we want is a transformation matrix that, when we run it, subtracts x from whatever x we currently have. This is called shearing, and the typical x-shear matrix and its transformation looks like this:

    - - -

    So we want some shearing value that, when multiplied by y, yields -x, so our x coordinate becomes zero. That value is simply -x/y, because *-x/y * y = -x*. Done:

    - - -

    Now, running this on all our points generates a new set of coordinates, let's call those V, which now have point 1 on (0,0) and point 2 on (0, some-value), and we wanted it at (0,1), so we need to do some scaling to make sure it ends up at (0,1). Additionally, we want point 3 to end up on (1,1), so we can also scale x to make sure its x-coordinate will be 1 after we run the transform. That means we'll be x-scaling by 1/point3x, and y-scaling by point2y. This is really easy:

    - - -

    Then, finally, this generates a new set of coordinates, let's call those W, of which point 1 lies on (0,0), point 2 lies on (0,1), and point three lies on (1, ...) so all that's left is to make sure point 3 ends up at (1,1) - but we can't scale! Point 2 is already in the right place, and y-scaling would move it out of (0,1) again, so our only option is to y-shear point three, just like how we x-sheared point 2 earlier. In this case, we do the same trick, but with y/x rather than x/y because we're not x-shearing but y-shearing. Additionally, we don't actually want to end up at zero (which is what we did before) so we need to shear towards an offset, in this case 1:

    - - -

    And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will be:

    - - -

    Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just pull this apart a little and end up with an easy-to-calculate bit of code!

    -

    First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does that leave?

    - - -

    Suddenly things look a lot simpler: the mapped x is fairly straight forward to compute, and we see that the mapped y actually contains the mapped x in its entirety, so we'll have that part already available when we need to evaluate it. In fact, let's pull out all those common factors to see just how simple this is:

    - - -

    That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it look!

    -
    + +

    + That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it + look! +

    +
    +

    How do you track all that?

    +

    + Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply + use Mathematica. Tracking all this math by hand is insane, and we invented computers, + literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the + program do the math for me. And real math, too, with symbols, not with numbers. In fact, + here's the Mathematica notebook if you want to see how + this works for yourself. +

    +

    + Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's + $344 for home use, up from $295 when I original wrote this, but it's + also free when you buy a $35 raspberry pi. Obviously, I bought a + raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do + it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for. +

    +
    -

    How do you track all that?

    -

    Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply use Mathematica. Tracking all this math by hand is insane, and we invented computers, literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the program do the math for me. And real math, too, with symbols, not with numbers. In fact, here's the Mathematica notebook if you want to see how this works for yourself.

    -

    Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's $344 for home use, up from $295 when I original wrote this, but it's also free when you buy a $35 raspberry pi. Obviously, I bought a raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for.

    -
    - -

    So, let's write up a sketch that'll show us the canonical form for any curve drawn in blue, overlaid on our canonical map, so that we can immediately tell which features our curve must have, based on where the fourth coordinate is located on the map:

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    previoustable of contentsnext
    Finding Y, given X

    -

    One common task that pops up in things like CSS work, or parametric equalizers, or image leveling, or any other number of applications where Bézier curves are used as control curves in a way that there is really only ever one "y" value associated with one "x" value, you might want to cut out the middle man, as it were, and compute "y" directly based on "x". After all, the function looks simple enough, finding the "y" value should be simple too, right? Unfortunately, not really. However, it is possible and as long as you have some code in place to help, it's not a lot of a work either.

    -

    We'll be tackling this problem in two stages: the first, which is the hard part, is figuring out which "t" value belongs to any given "x" value. For instance, have a look at the following graphic. On the left we have a Bézier curve that looks for all intents and purposes like it fits our criteria: every "x" has one and only one associated "y" value. On the right we see the function for just the "x" values: that's a cubic curve, but not a really crazy cubic curve. If you move the graphic's slider, you will see a red line drawn that corresponds to the x coordinate: this is a vertical line in the left graphic, and a horizontal line on the right.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Now, if you look more closely at that right graphic, you'll notice something interesting: if we treat the red line as "the x axis", then the point where the function crosses our line is really just a root for the cubic function x(t) through a shifted "x-axis"... and we've already seen how to calculate roots, so let's just run cubic root finding - and not even the complicated cubic case either: because of the kind of curve we're starting with, we know there is only root, simplifying the code we need!

    -

    First, let's look at the function for x(t):

    - - -

    We can rewrite this to a plain polynomial form, by just fully writing out the expansion and then collecting the polynomial factors, as:

    - - -

    Nothing special here: that's a standard cubic polynomial in "power" form (i.e. all the terms are ordered by their power of t). So, given that a, b, c, d, and x(t) are all known constants, we can trivially rewrite this (by moving the x(t) across the equal sign) as:

    - - -

    You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using Cardano's algorithm, and we're left with some rather short code:

    + +

    + You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they + all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using + Cardano's algorithm, and we're left with some rather short code: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +y = curve.get(t).y + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So the procedure is fairly straight forward: pick an x, find the associated t value, evaluate our curve for that t value, which gives us the curve's {x,y} coordinate, which means we know y for this x. Move the slider for the following graphic to see this in action:

    - - - Scripts are disabled. Showing fallback image. - - - - - - -
    -
    -

    previoustable of contentsnext
    Arc length

    -

    How long is a Bézier curve? As it turns out, that's not actually an easy question, because the answer requires maths that —much like root finding— cannot generally be solved the traditional way. If we have a parametric curve with fx(t) and fy(t), then the length of the curve, measured from start point to some point t = z, is computed using the following seemingly straight forward (if a bit overwhelming) formula:

    - - -

    or, more commonly written using Leibnitz notation as:

    - - -

    This formula says that the length of a parametric curve is in fact equal to the area underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an unwieldy computation, and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates".

    -

    Seriously: It cannot be done.

    -

    So we turn to numerical approaches again. The method we'll look at here is the Gauss quadrature. This approximation is a really neat trick, because for any nth degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:

    - - -

    In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the better the approximation:

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    + +

    + In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips + sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The + more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the + better the approximation: +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    Now, infinitely many terms to sum and infinitely thin rectangles are not something that computers can work with, so instead we're going to approximate the infinite summation by using a sum of a finite number of "just thin" rectangular strips. As long as we use a high enough number of thin enough rectangular strips, this will give us an approximation that is pretty close to what the real value is.

    -

    So, the trick is to come up with useful rectangular strips. A naive way is to simply create n strips, all with the same width, but there is a far better way using special values for C and f(t) depending on the value of n, which indicates how many strips we'll use, and it's called the Legendre-Gauss quadrature.

    -

    This approach uses strips that are not spaced evenly, but instead spaces them in a special way based on describing the function as a polynomial (the more strips, the more accurate the polynomial), and then computing the exact integral for that polynomial. We're essentially performing arc length computation on a flattened curve, but flattening it based on the intervals dictated by the Legendre-Gauss solution.

    -
    - -

    Note that one requirement for the approach we'll use is that the integral must run from -1 to 1. That's no good, because we're dealing with Bézier curves, and the length of a section of curve applies to values which run from 0 to "some value smaller than or equal to 1" (let's call that value z). Thankfully, we can quite easily transform any integral interval to any other integral interval, by shifting and scaling the inputs. Doing so, we get the following:

    - - -

    That may look a bit more complicated, but the fraction involving z is a fixed number, so the summation, and the evaluation of the f(t) values are still pretty simple.

    -

    So, what do we need to perform this calculation? For one, we'll need an explicit formula for f(t), because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these Ci and ti values should be. Luckily, that's less work because there are actually many tables available that give these values, for any n, so if we want to approximate our integral with only two terms (which is a bit low, really) then these tables would tell us that for n=2 we must use the following values:

    - - -

    Which means that in order for us to approximate the integral, we must plug these values into the approximate function, which gives us:

    - - -

    We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the Bézier curve functions Bx(t) and By(t).

    -
    + +

    + We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the + Bézier curve functions Bx(t) and By(t). +

    + -

    If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the start/end points on the inside?

    - - - Scripts are disabled. Showing fallback image. - - - +

    + If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), + we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, + with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make + a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the + start/end points on the inside? +

    + + + Scripts are disabled. Showing fallback image. + + + +
    +
    +

    +
    previoustable of contentsnext
    + Approximated arc length +

    +

    + Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length + instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This + will come with an error, but this can be made arbitrarily small by increasing the segment count. +

    +

    + If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal + effort: +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    previoustable of contentsnext
    Approximated arc length

    -

    Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This will come with an error, but this can be made arbitrarily small by increasing the segment count.

    -

    If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal effort:

    -
    - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    You may notice that even though the error in length is actually pretty significant in absolute terms, even at a low number of segments we get a length that agrees with the true length when it comes to just the integer part of the arc length. Quite often, approximations can drastically speed things up!

    - -
    -
    -

    previoustable of contentsnext
    Curvature of a curve

    -

    If we have two curves, and we want to line them in up in a way that "looks right", what would we use as metric to let a computer decide what "looks right" means?

    -

    For instance, we can start by ensuring that the two curves share an end coordinate, so that there is no "gap" between the end of one and the start of the next curve, but that won't guarantee that things look right: both curves can be going in wildly different directions, and the resulting joined geometry will have a corner in it, rather than a smooth transition from one curve to the next.

    -

    What we want is to ensure that the curvature at the transition from one curve to the next "looks good". So, we start with a shared coordinate, and then also require that derivatives for both curves match at that coordinate. That way, we're assured that their tangents line up, which must mean the curve transition is perfectly smooth. We can even make the second, third, etc. derivatives match up for better and better transitions.

    -

    Problem solved!

    -

    However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on reordering curves showed us that the same looking curve can have an infinite number of curve expressions of arbitrarily high Bézier degree, and each of those will have wildly different derivative values.

    -

    So what we really want is some kind of expression that's not based on any particular expression of t, but is based on something that is invariant to the kind of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being the same function for "coordinate at some distance D along the curve".

    -

    We've seen this before... that's the arc length function.

    -

    So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the form of the arc length function, which we saw above and is fairly simple, rather than needing to solve the arc length function. If we start with the arc length expression and the run through the steps necessary to determine its derivative (with an alternative, shorter demonstration of how to do this found over on Stackexchange), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the fundamental theorem of calculus), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.

    -

    Let me highlight what just happened, because it's pretty special:

    -
      -
    1. we wanted to make curves line up, and initially thought to match the curves' derivatives, but
      2. -
    2. that turned out to be a really bad choice, so instead
      3. -
    3. we picked a function that is basically impossible to work with, and then worked with that, which
      4. -
    4. gives us a simple formula that is and expression using the curves' derivatives.
      5. -
    -

    That's crazy!

    -

    But that's also one of the things that makes maths so powerful: even if your initial ideas are off the mark, you might be much closer than you thought you were, and the journey from "thinking we're completely wrong" to "actually being remarkably close to being right" is where we can find a lot of insight.

    -

    So, what does the function look like? This:

    - - -

    Which is really just a "short form" that glosses over the fact that we're dealing with functions of t, so let's expand that a tiny bit:

    - - -

    And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so evaluating this function for any t value is just a matter of basic arithematics.

    -

    In fact, let's just implement it right now:

    + +

    + And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any + (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that + looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier + curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so + evaluating this function for any t value is just a matter of basic arithematics. +

    +

    In fact, let's just implement it right now:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return numerator / denominator + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    That was easy! (Well okay, that "not a number" value will need to be taken into account by downstream code, but that's a reality of programming anyway)

    -

    With all of that covered, let's line up some curves! The following graphic gives you two curves that look identical, but use quadratic and cubic functions, respectively. As you can see, despite their derivatives being necessarily different, their curvature (thanks to being derived based on maths that "ignores" specific function derivative, and instead gives a formula that smooths out any differences) is exactly the same. And because of that, we can put them together such that the point where they overlap has the same curvature for both curves, giving us the smoothest transition.

    - - - Scripts are disabled. Showing fallback image. - - - -

    One thing you may have noticed in this sketch is that sometimes the curvature looks fine, but seems to be pointing in the wrong direction, making it hard to line up the curves properly. A way around that, of course, is to show the curvature on both sides of the curve, so let's just do that. But let's take it one step further: we can also compute the associated "radius of curvature", which gives us the implicit circle that "fits" the curve's curvature at any point, using what is possibly the simplest bit of maths found in this entire primer:

    - - -

    So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" our curve at some point that we can control by using a slider:

    - - - Scripts are disabled. Showing fallback image. - - - - - + +

    + So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" + our curve at some point that we can control by using a slider: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +
    +

    +
    previoustable of contentsnext
    + Tracing a curve at fixed distance intervals +

    +

    + Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance + intervals over time, like a train along a track, and you want to use Bézier curves. +

    +

    Now you have a problem.

    +

    + The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a + track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick + t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In + fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another. +

    +

    + The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be + straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To + make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for + this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length + function, which can have more inflection points. +

    + + + Scripts are disabled. Showing fallback image. + + +

    + So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through + the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and + then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of + points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two + points, but if we have a high enough number of samples, we don't even need to bother. +

    +

    + So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t + curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks + like, by coloring each section of curve between two distance markers differently: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    +

    + However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to + constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT). +

    +
    +
    +

    +
    previoustable of contentsnext
    + Intersections +

    +

    + Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to + work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to + perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box + overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the + actual curve. +

    +

    + We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by + implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both + lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution. +

    +

    Line-line intersections

    +

    + If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are + an intervals on by linear algebra, using the procedure outlined in this + top coder + article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line + segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments. +

    +

    + The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on + (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection + point). +

    + + + Scripts are disabled. Showing fallback image. + + + +
    +

    Implementing line-line intersections

    +

    + Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, + but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe + you're using point structs for the line. Let's get coding: +

    -
    -
    -

    previoustable of contentsnext
    Tracing a curve at fixed distance intervals

    -

    Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance intervals over time, like a train along a track, and you want to use Bézier curves.

    -

    Now you have a problem.

    -

    The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another.

    -

    The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.

    - - - Scripts are disabled. Showing fallback image. - - - -

    So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two points, but if we have a high enough number of samples, we don't even need to bother.

    -

    So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks like, by coloring each section of curve between two distance markers differently:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    -

    However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT).

    - -
    -
    -

    previoustable of contentsnext
    Intersections

    -

    Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the actual curve.

    -

    We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution.

    -

    Line-line intersections

    -

    If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are an intervals on by linear algebra, using the procedure outlined in this top coder article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments.

    -

    The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection point).

    - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    Implementing line-line intersections

    -

    Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe you're using point structs for the line. Let's get coding:

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    + return lli4(line1.p1, line1.p2, line2.p1, line2.p2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + +
    - +

    What about curve-line intersections?

    +

    + Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we + translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in + a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a + curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the + section on finding extremities. +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    What about curve-line intersections?

    -

    Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the section on finding extremities.

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    + Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the + curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and + curve splitting. +

    +
    +
    +

    +
    previoustable of contentsnext
    + Curve/curve intersection +

    +

    + Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" + technique: +

    +
      +
    1. + Take two curves C1 and C2, and treat them as a pair. +
      2. +
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. +
    3. + With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2). +
      4. +
    4. + For each pair, check whether their bounding boxes overlap. +
        +
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. +
      2. If there is overlap, rerun all steps for this pair.
        3. +
      +
      5. +
    5. + Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we + might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value + (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative). +
      6. +
    +

    + This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then + taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections. +

    +

    + The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click + the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value + that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the + algorithm, so you can try this with lots of different curves. +

    +

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    + + + Scripts are disabled. Showing fallback image. + + + + + + +

    + Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into + two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at + t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each + iteration, and each successive step homing in on the curve's self-intersection points. +

    +
    +
    +

    +
    previoustable of contentsnext
    + The projection identity +

    +

    + De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to + draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. + Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point + around to change the curve's shape. +

    +

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    +

    + In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want + to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know + has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for + reasons that will become obvious. +

    +

    + So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following + graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which + taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what + happens to that value? +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -

    Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and curve splitting.

    - -
    -
    -

    previoustable of contentsnext
    Curve/curve intersection

    -

    Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" technique:

    -
      -
    1. Take two curves C1 and C2, and treat them as a pair.
      2. -
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. -
    3. With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2).
      4. -
    4. For each pair, check whether their bounding boxes overlap.
        -
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. -
      2. If there is overlap, rerun all steps for this pair.
        3. -
      -
      5. -
    5. Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative).
      6. -
    -

    This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections.

    -

    The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the algorithm, so you can try this with lots of different curves.

    -

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    - - - Scripts are disabled. Showing fallback image. - - - - - - -

    Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each iteration, and each successive step homing in on the curve's self-intersection points.

    - -
    -
    -

    previoustable of contentsnext
    The projection identity

    -

    De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point around to change the curve's shape.

    -

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    -

    In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for reasons that will become obvious.

    -

    So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what happens to that value?

    -
    - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    So these graphics show us several things:

    -
      -
    1. a point at the tip of the curve construction's "hat": let's call that A, as well as
      2. -
    2. our on-curve point give our chosen t value: let's call that B, and finally,
      3. -
    3. a point that we get by projecting A, through B, onto the line between the curve's start and end points: let's call that C.
      4. -
    4. for both quadratic and cubic curves, two points e1 and e2, which represent the single-to-last step in de Casteljau's algorithm: in the last step, we find B at (1-t) * e1 + t * e2.
      5. -
    5. for cubic curves, also the points v1 and v2, which together with A represent the first step in de Casteljau's algorithm: in the next step, we find e1 and e2.
      6. -
    -

    These three values A, B, and C allow us to derive an important identity formula for quadratic and cubic Bézier curves: for any point on the curve with some t value, the ratio of distances from A to B and B to C is fixed: if some t value sets up a C that is 20% away from the start and 80% away from the end, then it doesn't matter where the start, end, or control points are; for that t value, C will always lie at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic and then move all the other points around: if you only move the control points, start and end won't move, and so neither will C, and if you move either start or end point, C will move but its relative position will not change.

    -

    So, how can we compute C? We start with our observation that C always lies somewhere between the start and end points, so logically C will have a function that interpolates between those two coordinates:

    - - -

    If we can figure out what the function u(t) looks like, we'll be done. Although we do need to remember that this u(t) will have a different form depending on whether we're working with quadratic or cubic curves. Running through the maths (with thanks to Boris Zbarsky) shows us the following two formulae:

    - - -

    And

    - - -

    So, if we know the start and end coordinates and the t value, we know C without having to calculate the A or even B coordinates. In fact, we can do the same for the ratio function. As another function of t, we technically don't need to know what A or B or C are. It, too, can be expressed as a pure function of t.

    -

    We start by observing that, given A, B, and C, the following always holds:

    - - -

    Working out the maths for this, we see the following two formulae for quadratic and cubic curves:

    - - -

    And

    - - -

    Which now leaves us with some powerful tools: given three points (start, end, and "some point on the curve"), as well as a t value, we can construct curves. We can compute C using the start and end points and our u(t) function, and once we have C, we can use our on-curve point (B) and the ratio(t) function to find A:

    - - -

    With A found, finding e1 and e2 for quadratic curves is a matter of running the linear interpolation with t between start and A to yield e1, and between A and end to yield e2. For cubic curves, there is no single pair of points that can act as e1 and e2 (there are infinitely many, because the tangent at B is a free parameter for cubic curves) so as long as the distance ratio between e1 to B and B to e2 is the Bézier ratio (1-t):t, we are free to pick any pair, after which we can reverse engineer v1 and v2:

    - - -

    And then reverse engineer the curve's control points:

    - - -

    So: if we have a curve's start and end points, as well as some third point B that we want the curve to pass through, then for any t value we implicitly know all the ABC values, which (combined with an educated guess on appropriate e1 and e2 coordinates for cubic curves) gives us the necessary information to reconstruct a curve's "de Casteljau skeleton". Which means that we can now do several things: we can "fit" curves using only three points, which means we can also "mold" curves by moving an on-curve point but leaving its start and end points, and then reconstruct the curve based on where we moved the on-curve point to. These are very useful things, and we'll look at both in the next few sections.

    - -
    -
    -

    previoustable of contentsnext
    Creating a curve from three points

    -

    Given the preceding section, you might be wondering if we can use that knowledge to just "create" curves by placing some points and having the computer do the rest, to which the answer is: that's exactly what we can now do!

    -

    For quadratic curves, things are pretty easy. Technically, we'll need a t value in order to compute the ratio function used in computing the ABC coordinates, but we can just as easily approximate one by treating the distance between the start and B point, and B and end point as a ratio, using

    - - -

    With this code in place, creating a quadratic curve from three points is literally just computing the ABC values, and using A as our curve's control point:

    - - - Scripts are disabled. Showing fallback image. - - - -

    For cubic curves we need to do a little more work, but really only just a little. We're first going to assume that a decent curve through the three points should approximate a circular arc, which first requires knowing how to fit a circle to three points. You may remember (if you ever learned it!) that a line between two points on a circle is called a chord, and that one property of chords is that the line from the center of any chord, perpendicular to that chord, passes through the center of the circle.

    -

    That means that if we have three points on a circle, we have three (different) chords, and consequently, three (different) lines that go from those chords through the center of the circle: if we find two of those lines, then their intersection will be our circle's center, and the circle's radius will—by definition!—be the distance from the center to any of our three points:

    - - - Scripts are disabled. Showing fallback image. - - - -

    With that covered, we now also know the tangent line to our point B, because the tangent to any point on the circle is a line through that point, perpendicular to the line from that point to the center. That just leaves marking appropriate points e1 and e2 on that tangent, so that we can construct a new cubic curve hull. We use the approach as we did for quadratic curves to automatically determine a reasonable t value, and then our e1 and e2 coordinates must obey the standard de Casteljau rule for linear interpolation:

    - - -

    Where d is the total length of the line segment from e1 to e2. So how long do we make that? There are again all kinds of approaches we can take, and a simple-but-effective one is to set the length of that segment to "one third the length of the baseline". This forces e1 and e2 to always be the "linear curve" distance apart, which means if we place our three points on a line, it will actually look like a line. Nice! The last thing we'll need to do is make sure to flip the sign of d depending on which side of the baseline our B is located, so we don't end up creating a funky curve with a loop in it. To do this, we can use the atan2 function:

    - - -

    This angle φ will be between 0 and π if B is "above" the baseline (rotating all three points so that the start is on the left and the end is the right), so we can use a relatively straight forward check to make sure we're using the correct sign for our value d:

    - - -

    The result of this approach looks as follows:

    - - - Scripts are disabled. Showing fallback image. - - - -

    It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; more on that later), we're only trying to come up with some reasonable e1 and e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives us:

    - - - Scripts are disabled. Showing fallback image. - - - -

    That looks perfectly serviceable!

    -

    Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at implementing curve molding in the section after that, so read on!

    + +

    The result of this approach looks as follows:

    + + + Scripts are disabled. Showing fallback image. + + + +

    + It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, + we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; + more on that later), we're only trying to come up with some reasonable e1 and + e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives + us: +

    + + + Scripts are disabled. Showing fallback image. + + + +

    That looks perfectly serviceable!

    +

    + Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" + curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding + the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at + implementing curve molding in the section after that, so read on! +

    +
    +
    +

    +
    previoustable of contentsnext
    + Projecting a point onto a Bézier curve +

    +

    + Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're + trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the + curve our cursor will be closest to. So, how do we project points onto a curve? +

    +

    + If the Bézier curve is of low enough order, we might be able to + work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a + numerical approach again. We'll be finding our ideal t value using a + binary search. First, we do a coarse distance-check based on + t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast: +

    -
    -
    -

    previoustable of contentsnext
    Projecting a point onto a Bézier curve

    -

    Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the curve our cursor will be closest to. So, how do we project points onto a curve?

    -

    If the Bézier curve is of low enough order, we might be able to work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a numerical approach again. We'll be finding our ideal t value using a binary search. First, we do a coarse distance-check based on t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    After this runs, we know that LUT[i] is the coordinate on the curve in our LUT that is closest to the point we want to project, so that's a pretty good initial guess as to what the best projection onto our curve is. To refine it, we note that LUT[i] is a better guess than both LUT[i-1] and LUT[i+1], but there might be an even better projection somewhere else between those two values, so that's what we're going to be testing for, using a variation of the binary search.

    -
      -
    1. we start with our point p, and the t values t1=LUT[i-1].t and t2=LUT[i+1].t, which span an interval v = t2-t1.
      2. -
    2. we test this interval in five spots: the start, middle, and end (which we already have), and the two points in between the middle and start/end points
      3. -
    3. we then check which of these five points is the closest to our original point p, and then repeat step 1 with the points before and after the closest point we just found.
      4. -
    -

    This makes the interval we check smaller and smaller at each iteration, and we can keep running the three steps until the interval becomes so small as to lead to distances that are, for all intents and purposes, the same for all points.

    -

    So, let's see that in action: in this case, I'm going to arbitrarily say that if we're going to run the loop until the interval is smaller than 0.001, and show you what that means for projecting your mouse cursor or finger tip onto a rather complex Bézier curve (which, of course, you can reshape as you like). Also shown are the original three points that our coarse check finds.

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    previoustable of contentsnext
    Intersections with a circle

    -

    It might seem odd to cover this subject so much later than the line/line, line/curve, and curve/curve intersection topics from several sections earlier, but the reason we can't cover circle/curve intersections is that we can't really discuss circle/curve intersection until we've covered the kind of lookup table (LUT) walking that the section on projecting a point onto a curve uses. To see why, let's look at what we would have to do if we wanted to find the intersections between a curve and a circle using calculus.

    -

    First, we observe that "finding intersections" in this case means that, given a circle defined by a center point c = (x,y) and a radius r, we want to find all points on the Bezier curve for which the distance to the circle's center point is equal to the circle radius, which by definition means those points lie on the circle, and so count as intersections. In maths, that means we're trying to solve:

    - - -

    Which seems simple enough. Unfortunately, when we expand that dist function, things get a lot more problematic:

    - - -

    And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the Abel-Ruffini theorem that we saw before, we can't solve this by just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left with a monstrous function to solve.

    -

    So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a bit more code to make sure we find all of them, but let's look at the steps involved:

    + +

    + And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the + Abel-Ruffini theorem that we saw before, we can't solve this by + just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to + actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and + all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left + with a monstrous function to solve. +

    +

    + So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the + on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance + r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a + bit more code to make sure we find all of them, but let's look at the steps involved: +

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + -

    This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is exactly the radius, so the coordinate lies on both the curve and the circle.

    -

    So far so good.

    -

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    +

    + This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor + change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between + that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is + exactly the radius, so the coordinate lies on both the curve and the circle. +

    +

    So far so good.

    +

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + start = i + 2; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points ("current", "previous" and "before previous") and then check whether they form a local minimum:

    +

    + After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we + can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now + not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of + course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in + finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points + ("current", "previous" and "before previous") and then check whether they form a local minimum: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return start + i + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    In words: given a start index, the circle center and radius, and our LUT, we check where (closest to our start index) we can find a local minimum for the difference between "the distance from the curve to the circle center", and the circle's radius. We track this by looking at three values (associated with the indices index-2, index-1, and index), and we know we've found a local minimum if the three values show that the middle value (pd1) is less than either value beside it. When we do, we can set our "best guess, relative to start" as index-1. Of course, since we're now checking values relative to some start value, we might not find another candidate value at all, in which case we return start - 1, so that a simple "is the result less than start?" lets us determine that there are no more intersections to find.

    -

    Finally, while not necessary for point projection, there is one more step we need to perform when we run the binary refinement function on our candidate LUT indices, because we've so far only been testing using distances "closest to the radius of the circle", and that's actually not good enough... we need distances that are the radius of the circle. So, after running the refinement for each of these indices, we need to discard any final value that isn't the circle radius. And because we're working with floating point numbers, what this really means is that we need to discard any value that's a pixel or more "off". Or, if we want to get really fancy, "some small epsilon value".

    -

    Based on all of that, the following graphic shows this off for the standard cubic curve (which you can move the coordinates around for, of course) and a circle with a controllable radius centered on the graphic's center, using the code approach described above.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    And of course, for the full details, click that "view source" link.

    - -
    -
    -

    previoustable of contentsnext
    Molding a curve

    -

    Armed with knowledge of the "ABC" relation, point-on-curve projection, and guestimating reasonable looking helper values for cubic curve construction, we can finally cover curve molding: updating a curve's shape interactively, by dragging points on the curve around.

    -

    For quadratic curve, this is a really simple trick: we project our cursor onto the curve, which gives us a t value and initial B coordinate. We don't even need the latter: with our t value and "wherever the cursor is" as target B, we can compute the associated C:

    - - -

    And then the associated A:

    - - -

    And we're done, because that's our new quadratic control point!

    - - - Scripts are disabled. Showing fallback image. - - - -

    As before, cubic curves are a bit more work, because while it's easy to find our initial t value and ABC values, getting those all-important e1 and e2 coordinates is going to pose a bit of a problem... in the section on curve creation, we were free to pick an appropriate t value ourselves, which allowed us to find appropriate e1 and e2 coordinates. That's great, but when we're curve molding we don't have that luxury: whatever point we decide to start moving around already has its own t value, and its own e1 and e2 values, and those may not make sense for the rest of the curve.

    -

    For example, let's see what happens if we just "go with what we get" when we pick a point and start moving it around, preserving its t value and e1/e2 coordinates:

    - - - Scripts are disabled. Showing fallback image. - - - -

    That looks reasonable, close to the original point, but the further we drag our point, the less "useful" things become. Especially if we drag our point across the baseline, rather than turning into a nice curve.

    -

    One way to combat this might be to combine the above approach with the approach from the creating curves section: generate both the "unchanged t/e1/e2" curve, as well as the "idealized" curve through the start/cursor/end points, with idealized t value, and then interpolating between those two curves:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    The slide controls the "falloff distance" relative to where the original point on the curve is, so that as we drag our point around, it interpolates with a bias towards "preserving t/e1/e2" closer to the original point, and bias towards "idealized" form the further away we move our point, with anything that's further than our falloff distance simply being the idealized curve. We don't even try to interpolate at that point.

    -

    A more advanced way to try to smooth things out is to implement continuous molding, where we constantly update the curve as we move around, and constantly change what our B point is, based on constantly projecting the cursor on the curve as we're updating it - this is, you won't be surprised to learn, tricky, and beyond the scope of this section: interpolation (with a reasonable distance) will do for now!

    - -
    -
    -

    previoustable of contentsnext
    Curve fitting

    -

    Given the previous section, one question you might have is "what if I don't want to guess t values?". After all, plenty of graphics packages do automated curve fitting, so how can we implement that in a way that just finds us reasonable t values all on its own?

    -

    And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a least squares polynomial regression. In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an nth order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".

    -

    Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the squared distance between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep-dive the specific merits of least squared error metrics versus least absolute error metrics, but those are well beyond the scope of this material.

    -

    So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his "Least Squares Bézier Fit" article, and end with some nice interactive graphics for doing some curve fitting.

    -

    Before we begin, we're going to use the curve in matrix form. In the section on matrices, I mentioned that some things are easier if we use the matrix representation of a Bézier curve rather than its calculus form, and this is one of those things.

    -

    As such, the first step in the process is expressing our Bézier curve as powers/coefficients/coordinate matrix T x M x C, by expanding the Bézier functions.

    -
    - -

    Revisiting the matrix representation

    -

    Rewriting Bézier functions to matrix form is fairly easy, if you first expand the function, and then arrange them into a multiple line form, where each line corresponds to a power of t, and each column is for a specific coefficient. First, we expand the function:

    - - -

    And then we (trivially) rearrange the terms across multiple lines:

    - - -

    This rearrangement has "factors of t" at each row (the first row is t⁰, i.e. "1", the second row is t¹, i.e. "t", the third row is t²) and "coefficient" at each column (the first column is all terms involving "a", the second all terms involving "b", the third all terms involving "c").

    -

    With that arrangement, we can easily decompose this as a matrix multiplication:

    - - -

    We can do the same for the cubic curve, of course. We know the base function for cubics:

    - - -

    So we write out the expansion and rearrange:

    - - -

    Which we can then decompose:

    - - -

    And, of course, we can do this for quartic curves too (skipping the expansion step):

    - - -

    And so and on so on. Now, let's see how to use these T, M, and C, to do some curve fitting.

    -
    + +

    + And so and on so on. Now, let's see how to use these T, M, and C, to do some curve + fitting. +

    + -

    Let's get started: we're going to assume we picked the right order curve: for n points we're fitting an n-1th order curve, so we "start" with a vector P that represents the coordinates we already know, and for which we want to do curve fitting:

    - - -

    Next, we need to figure out appropriate t values for each point in the curve, because we need something that lets us tie "the actual coordinate" to "some point on the curve". There's a fair number of different ways to do this (and a large part of optimizing "the perfect fit" is about picking appropriate t values), but in this case let's look at two "obvious" choices:

    -
      -
    1. equally spaced t values, and
      2. -
    2. t values that align with distance along the polygon.
      3. -
    -

    The first one is really simple: if we have n points, then we'll just assign each point i a t value of (i-1)/(n-1). So if we have four points, the first point will have t=(1-1)/(4-1)=0/3, the second point will have t=(2-1)/(4-1)=1/3, the third point will have t=2/3, and the last point will be t=1. We're just straight up spacing the t values to match the number of points we have.

    -

    The second one is a little more interesting: since we're doing polynomial regression, we might as well exploit the fact that our base coordinates just constitute a collection of line segments. At the first point, we're fixing t=0, and the last point, we want t=1, and anywhere in between we're simply going to say that t is equal to the distance along the polygon, scaled to the [0,1] domain.

    -

    To get these values, we first compute the general "distance along the polygon" matrix:

    - - -

    Where length() is literally just that: the length of the line segment between the point we're looking at, and the previous point. This isn't quite enough, of course: we still need to make sure that all the values between i=1 and i=n fall in the [0,1] interval, so we need to scale all values down by whatever the total length of the polygon is:

    - - -

    And now we can move on to the actual "curve fitting" part: what we want is a function that lets us compute "ideal" control point values such that if we build a Bézier curve with them, that curve passes through all our original points. Or, failing that, have an overall error distance that is as close to zero as we can get it. So, let's write out what the error distance looks like.

    -

    As mentioned before, this function is really just "the distance between the actual coordinate, and the coordinate that the curve evaluates to for the associated t value", which we'll square to get rid of any pesky negative signs:

    - - -

    Since this function only deals with individual coordinates, we'll need to sum over all coordinates in order to get the full error function. So, we literally just do that; the total error function is simply the sum of all these individual errors:

    - - -

    And here's the trick that justifies using matrices: while we can work with individual values using calculus, with matrices we can compute as many values as we make our matrices big, all at the "same time", We can replace the individual terms pi with the full P coordinate matrix, and we can replace Bézier(si) with the matrix representation T x M x C we talked about before, which gives us:

    - - -

    In which we can replace the rather cumbersome "squaring" operation with a more conventional matrix equivalent:

    - - -

    Here, the letter T is used instead of the number 2, to represent the matrix transpose; each row in the original matrix becomes a column in the transposed matrix instead (row one becomes column one, row two becomes column two, and so on).

    -

    This leaves one problem: T isn't actually the matrix we want: we don't want symbolic t values, we want the actual numerical values that we computed for S, so we need to form a new matrix, which we'll call 𝕋, that makes use of those, and then use that 𝕋 instead of T in our error function:

    - - -

    Which, because of the first and last values in S, means:

    - - -

    Now we can properly write out the error function as matrix operations:

    - - -

    So, we have our error function: we now need to figure out the expression for where that function has minimal value, e.g. where the error between the true coordinates and the coordinates generated by the curve fitting is smallest. Like in standard calculus, this requires taking the derivative, and determining where that derivative is zero:

    - - -
    + +
    +

    Where did this derivative come from?

    +

    + That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I + straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know. +

    +

    + So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific + case, I + posted a question + to Math.stackexchange, and received a answer that goes into way more detail than I had + hoped to receive. +

    +

    + Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean + maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that + true?". There are answers. They might just require some time to come to understand. +

    +
    -

    Where did this derivative come from?

    -

    That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know.

    -

    So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific case, I posted a question to Math.stackexchange, and received a answer that goes into way more detail than I had hoped to receive.

    -

    Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that true?". There are answers. They might just require some time to come to understand.

    -
    - -

    Now, given the above derivative, we can rearrange the terms (following the rules of matrix algebra) so that we end up with an expression for C:

    - - -

    Here, the "to the power negative one" is the notation for the matrix inverse. But that's all we have to do: we're done. Starting with P and inventing some t values based on the polygon the coordinates in P define, we can compute the corresponding Bézier coordinates C that specify a curve that goes through our points. Or, if it can't go through them exactly, as near as possible.

    -

    So before we try that out, how much code is involved in implementing this? Honestly, that answer depends on how much you're going to be writing yourself. If you already have a matrix maths library available, then really not that much code at all. On the other hand, if you are writing this from scratch, you're going to have to write some utility functions for doing your matrix work for you, so it's really anywhere from 50 lines of code to maybe 200 lines of code. Not a bad price to pay for being able to fit curves to pre-specified coordinates.

    -

    So let's try it out! The following graphic lets you place points, and will start computing exact-fit curves once you've placed at least three. You can click for more points, and the code will simply try to compute an exact fit using a Bézier curve of the appropriate order. Four points? Cubic Bézier. Five points? Quartic. And so on. Of course, this does break down at some point: depending on where you place your points, it might become mighty hard for the fitter to find an exact fit, and things might actually start looking horribly off once there's enough points for compound floating point rounding errors to start making a difference (which is around 10~11 points).

    - - - Scripts are disabled. Showing fallback image. - - - - - - -

    You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be able to freely manipulate them and see what the effect on your curve is.

    - -
    -
    -

    previoustable of contentsnext
    Bézier curves and Catmull-Rom curves

    -

    Taking an excursion to different splines, the other common design curve is the Catmull-Rom spline, which unlike Bézier curves pass through each control point, so they offer a kind of "built-in" curve fitting.

    -

    In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what looks like a single Catmull-Rom curve is actually a spline: a single curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is defined by the point's coordinates, and the tangent for those points, the latter of which can trivially be derived from knowing the previous and next point:

    - + +

    + You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio + along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get + direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be + able to freely manipulate them and see what the effect on your curve is. +

    +
    +
    +

    +
    previoustable of contentsnext
    + Bézier curves and Catmull-Rom curves +

    +

    + Taking an excursion to different splines, the other common design curve is the + Catmull-Rom spline, which unlike Bézier curves + pass through each control point, so they offer a kind of "built-in" curve fitting. +

    +

    + In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets + you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the + tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what + looks like a single Catmull-Rom curve is actually a spline: a single + curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, + and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is + defined by the point's coordinates, and the tangent for those points, the latter of which + can trivially be derived from knowing the + previous and next point: +

    + - -

    One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on that in the next section.

    -

    In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of variations that make things considerable more complex):

    + +

    + One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up + with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent + should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on + that in the next section. +

    +

    + In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of + variations that make things + considerable more + complex): +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + point(c0 * v1 + c1 * dv1 + c2 * v2 + c3 * dv2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Now, since a Catmull-Rom curve is a form of cubic Hermite spline, and as cubic Bézier curves are also a form of cubic Hermite spline, we run into an interesting bit of maths programming: we can convert one to the other and back, and the maths for doing so is surprisingly simple!

    -

    The main difference between Catmull-Rom curves and Bézier curves is "what the points mean":

    -
      -
    • A cubic Bézier curve is defined by a start point, a control point that implies the tangent at the start, a control point that implies the tangent at the end, and an end point, plus a characterizing matrix that we can multiply by that point vector to get on-curve coordinates.
      • -
    • A Catmull-Rom curve is defined by a start point, a tangent that for that starting point, an end point, and a tangent for that end point, plus a characteristic matrix that we can multiple by the point vector to get on-curve coordinates.
      • -
    -

    Those are very similar, so let's see exactly how similar they are. We've already see the matrix form for Bézier curves, so how different is the matrix form for Catmull-Rom curves?:

    - - -

    That's pretty dang similar. So the question is: how can we convert that expression with Catmull-Rom matrix and vector into an expression of the Bézier matrix and vector? The short answer is of course "by using linear algebra", but the longer answer is the rest of this section, and involves some maths that you may not even care for: if you just want to know the (incredibly simple) conversions between the two curve forms, feel free to skip to the end of the following explanation, but if you want to how we can get one from the other... let's get mathing!

    -
    - -

    Deriving the conversion formulae

    -

    In order to convert between Catmull-Rom curves and Bézier curves, we need to know two things. Firstly, how to express the Catmull-Rom curve using a "set of four coordinates", rather than a mix of coordinates and tangents, and secondly, how to convert those Catmull-Rom coordinates to and from Bézier form.

    -

    We start with the first part, to figure out how we can go from Catmull-Rom V coordinates to Bézier P coordinates, by applying "some matrix T". We don't know what that T is yet, but we'll get to that:

    - - -

    So, this mapping says that in order to map a Catmull-Rom "point + tangent" vector to something based on an "all coordinates" vector, we need to determine the mapping matrix such that applying T yields P2 as start point, P3 as end point, and two tangents based on the lines between P1 and P3, and P2 nd P4, respectively.

    -

    Computing T is really more "arranging the numbers":

    - - -

    Thus:

    - - -

    However, we're not quite done, because Catmull-Rom curves have that "tension" parameter, written as τ (a lowercase"tau"), which is a scaling factor for the tangent vectors: the bigger the tension, the smaller the tangents, and the smaller the tension, the bigger the tangents. As such, the tension factor goes in the denominator for the tangents, and before we continue, let's add that tension factor into both our coordinate vector representation, and mapping matrix T:

    - - -

    With the mapping matrix properly done, let's rewrite the "point + tangent" Catmull-Rom matrix form to a matrix form in terms of four coordinates, and see what we end up with:

    - - -

    Replace point/tangent vector with the expression for all-coordinates:

    - - -

    and merge the matrices:

    - - -

    This looks a lot like the Bézier matrix form, which as we saw in the chapter on Bézier curves, should look like this:

    - - -

    So, if we want to express a Catmull-Rom curve using a Bézier curve, we'll need to turn this Catmull-Rom bit:

    - - -

    Into something that looks like this:

    - - -

    And the way we do that is with a fairly straight forward bit of matrix rewriting. We start with the equality we need to ensure:

    - - -

    Then we remove the coordinate vector from both sides without affecting the equality:

    - - -

    Then we can "get rid of" the Bézier matrix on the right by left-multiply both with the inverse of the Bézier matrix:

    - - -

    A matrix times its inverse is the matrix equivalent of 1, and because "something times 1" is the same as "something", so we can just outright remove any matrix/inverse pair:

    - - -

    And now we're basically done. We just multiply those two matrices and we know what V is:

    - - -

    We now have the final piece of our function puzzle. Let's run through each step.

    -
      -
    1. Start with the Catmull-Rom function:
      2. -
    - - -
      -
    1. rewrite to pure coordinate form:
      2. -
    - - -
      -
    1. rewrite for "normal" coordinate vector:
      2. -
    - - -
      -
    1. merge the inner matrices:
      2. -
    - - -
      -
    1. rewrite for Bézier matrix form:
      2. -
    - - -
      -
    1. and transform the coordinates so we have a "pure" Bézier expression:
      2. -
    - - -

    And we're done: we finally know how to convert these two curves!

    -
    + +

    And we're done: we finally know how to convert these two curves!

    + -

    If we have a Catmull-Rom curve defined by four coordinates P1 through P4, then we can draw that curve using a Bézier curve that has the vector:

    - - -

    Similarly, if we have a Bézier curve defined by four coordinates P1 through P4, we can draw that using a standard tension Catmull-Rom curve with the following coordinate values:

    - - -

    Or, if your API allows you to specify Catmull-Rom curves using plain coordinates:

    - - - -
    -
    -

    previoustable of contentsnext
    Creating a Catmull-Rom curve from three points

    -

    Much shorter than the previous section: we saw that Catmull-Rom curves need at least 4 points to draw anything sensible, so how do we create a Catmull-Rom curve from three points?

    -

    Short and sweet: we don't.

    -

    We run through the maths that lets us create a cubic Bézier curve, and then convert its coordinates to Catmull-Rom form using the conversion formulae we saw above.

    - -
    -
    -

    previoustable of contentsnext
    Forming poly-Bézier curves

    -

    Much like lines can be chained together to form polygons, Bézier curves can be chained together to form poly-Béziers, and the only trick required is to make sure that:

    -
      -
    1. the end point of each section is the starting point of the following section, and
      2. -
    2. the derivatives across that dual point line up.
      3. -
    -

    Unless you want sharp corners, of course. Then you don't even need 2.

    -

    We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind that just follows point 1. where the end point of a segment is the same point as the start point of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're the easiest to implement:

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Dragging the control points around only affects the curve segments that the control point belongs to, and moving an on-curve point leaves the control points where they are, which is not the most useful for practical modelling purposes. So, let's add in the logic we need to make things a little better. We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve point is the same as it's "outgoing" derivative:

    - - -

    We can effect this quite easily, because we know that the vector from a curve's last control point to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control point of the next curve matches that, all we have to do is mirror that last control point through the last on-curve point. And mirroring any point A through any point B is really simple:

    - - -

    So let's implement that and see what it gets us. The following two graphics show a quadratic and a cubic poly-Bézier curve again, but this time moving the control points around moves others, too. However, you might see something unexpected going on for quadratic curves...

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all the control points are effectively linked. Move one of them, and you move all of them. Not only that, but if we move the on-curve points, it's possible to get a situation where a control point cannot satisfy the constraint that it's the reflection of its two neighbouring control points... This means that we cannot use quadratic poly-Béziers for anything other than really, really simple shapes. And even then, they're probably the wrong choice. Cubic curves are pretty decent, but the fact that the derivatives are linked means we can't manipulate curves as well as we might if we relaxed the constraints a little.

    -

    So: let's relax the requirement a little.

    -

    We can change the constraint so that we still preserve the angle of the derivatives across sections (so transitions from one section to the next will still look natural), but give up the requirement that they should also have the same vector length. Doing so will give us a much more useful kind of poly-Bézier curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Cubic curves are now better behaved when it comes to dragging control points around, but the quadratic poly-Bézier still has the problem that moving one control points will move the control points and may ending up defining "the next" control point in a way that doesn't work. Quadratic curves really aren't very useful to work with...

    -

    Finally, we also want to make sure that moving the on-curve coordinates preserves the relative positions of the associated control points. With that, we get to the kind of curve control that you might be familiar with from applications like Photoshop, Inkscape, Blender, etc.

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a new, very serious problem: we can move an on-curve point in such a way that we can't compute what needs to "happen next". Move the top point down, below the left and right points, for instance. There is no way to preserve correct control points without a kink at the bottom point. Quadratic curves: just not that good...

    -

    A final improvement is to offer fine-level control over which points behave which, so that you can have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves for the rest of the path. Implementing that, is left as an exercise for the reader.

    - -
    -
    -

    previoustable of contentsnext
    Curve offsetting

    -

    Perhaps you're like me, and you've been writing various small programs that use Bézier curves in some way or another, and at some point you make the step to implementing path extrusion. But you don't want to do it pixel based; you want to stay in the vector world. You find that extruding lines is relatively easy, and tracing outlines is coming along nicely (although junction caps and fillets are a bit of a hassle), and then you decide to do things properly and add Bézier curves to the mix. Now you have a problem.

    -

    Unlike lines, you can't simply extrude a Bézier curve by taking a copy and moving it around, because of the curvatures; rather than a uniform thickness, you get an extrusion that looks too thin in places, if you're lucky, but more likely will self-intersect. The trick, then, is to scale the curve, rather than simply copying it. But how do you scale a Bézier curve?

    -

    Bottom line: you can't. So you cheat. We're not going to do true curve scaling, or rather curve offsetting, because that's impossible. Instead we're going to try to generate 'looks good enough' offset curves.

    -
    - -

    "What do you mean, you can't? Prove it."

    -

    First off, when I say "you can't," what I really mean is "you can't offset a Bézier curve with another Bézier curve", not even by using a really high order curve. You can find the function that describes the offset curve, but it won't be a polynomial, and as such it cannot be represented as a Bézier curve, which has to be a polynomial. Let's look at why this is:

    -

    From a mathematical point of view, an offset curve O(t) is a curve such that, given our original curve B(t), any point on O(t) is a fixed distance d away from coordinate B(t). So let's math that:

    - - -

    However, we're working in 2D, and d is a single value, so we want to turn it into a vector. If we want a point distance d "away" from the curve B(t) then what we really mean is that we want a point at d times the "normal vector" from point B(t), where the "normal" is a vector that runs perpendicular ("at a right angle") to the tangent at B(t). Easy enough:

    - - -

    Now this still isn't very useful unless we know what the formula for N(t) is, so let's find out. N(t) runs perpendicular to the original curve tangent, and we know that the tangent is simply B'(t), so we could just rotate that 90 degrees and be done with it. However, we need to ensure that N(t) has the same magnitude for every t, or the offset curve won't be at a uniform distance, thus not being an offset curve at all. The easiest way to guarantee this is to make sure N(t) always has length 1, which we can achieve by dividing B'(t) by its magnitude:

    - - -

    Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc length from some start a to end b, we must use the formula we saw earlier. Noting that "length" is usually denoted with double vertical bars:

    - - -

    So if we want the length of the tangent, we plug in B'(t), with t = 0 as start and -t = 1 as end:

    - - -

    And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.

    -

    There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine because they do, but they don't; remember that only the base function has binomial coefficients. That's before we factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

    -

    So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve.

    -

    And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the real offset curve would look like, if we could compute it.

    -
    + +

    + And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is + screwing everything up, because it turns our nice polynomials into things that are no longer polynomials. +

    +

    + There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with + unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine + because they do, but they don't; remember that only the base function has binomial coefficients. That's before we + factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make + all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call + them circles, and they have much simpler functions than Bézier curves. +

    +

    + So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means + N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means + that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for + O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve. +

    +

    + And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier + curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, + much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the + real offset curve would look like, if we could compute it. +

    + -

    So, you cannot offset a Bézier curve perfectly with another Bézier curve, no matter how high-order you make that other Bézier curve. However, we can chop up a curve into "safe" sub-curves (where "safe" means that all the control points are always on a single side of the baseline, and the midpoint of the curve at t=0.5 is roughly in the center of the polygon defined by the curve coordinates) and then point-scale each sub-curve with respect to its scaling origin (which is the intersection of the point normals at the start and end points).

    -

    A good way to do this reduction is to first find the curve's extreme points, as explained in the earlier section on curve extremities, and use these as initial splitting points. After this initial split, we can check each individual segment to see if it's "safe enough" based on where the center of the curve is. If the on-curve point for t=0.5 is too far off from the center, we simply split the segment down the middle. Generally this is more than enough to end up with safe segments.

    -

    The following graphics show off curve offsetting, and you can use the slider to control the distance at which the curve gets offset. The curve first gets reduced to safe segments, each of which is then offset at the desired distance. Especially for simple curves, particularly easily set up for quadratic curves, no reduction is necessary, but the more twisty the curve gets, the more the curve needs to be reduced in order to get segments that can safely be scaled.

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -

    You may notice that this may still lead to small 'jumps' in the sub-curves when moving the curve around. This is caused by the fact that we're still performing a naive form of offsetting, moving the control points the same distance as the start and end points. If the curve is large enough, this may still lead to incorrect offsets.

    - -
    -
    -

    previoustable of contentsnext
    Graduated curve offsetting

    -

    What if we want to do graduated offsetting, starting at some distance s but ending at some other distance e? Well, if we can compute the length of a curve (which we can if we use the Legendre-Gauss quadrature approach) then we can also determine how far "along the line" any point on the curve is. With that knowledge, we can offset a curve so that its offset curve is not uniformly wide, but graduated between with two different offset widths at the start and end.

    -

    Like normal offsetting we cut up our curve in sub-curves, and then check at which distance along the original curve each sub-curve starts and ends, as well as to which point on the curve each of the control points map. This gives us the distance-along-the-curve for each interesting point in the sub-curve. If we call the total length of all sub-curves seen prior to seeing "the current" sub-curve S (and if the current sub-curve is the first one, S is zero), and we call the full length of our original curve L, then we get the following graduation values:

    -
      -
    • start: map S from interval (0,L) to interval (s,e)
      • -
    • c1: map(<strong>S+d1</strong>, 0,L, s,e), d1 = distance along curve to projection of c1
      • -
    • c2: map(<strong>S+d2</strong>, 0,L, s,e), d2 = distance along curve to projection of c2
      • -
    • ...
      • -
    • end: map(<strong>S+length(subcurve)</strong>, 0,L, s,e)
      • -
    -

    At each of the relevant points (start, end, and the projections of the control points onto the curve) we know the curve's normal, so offsetting is simply a matter of taking our original point, and moving it along the normal vector by the offset distance for each point. Doing so will give us the following result (these have with a starting width of 0, and an end width of 40 pixels, but can be controlled with your up and down arrow keys):

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - - -
    -
    -

    previoustable of contentsnext
    Circles and quadratic Bézier curves

    -

    Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. For instance, if you're doing font design, fonts have no concept of geometric shapes, they only know straight lines, and Bézier curves. OpenType fonts with TrueType outlines only know quadratic Bézier curves, and OpenType fonts with Type 2 outlines only know cubic Bézier curves. So how do you draw a circle, or an ellipse, or an arc?

    -

    You approximate.

    -

    We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.

    -

    Since arcs are mid-point-symmetrical, we need the control points to set up a symmetrical curve. For quadratic curves this means that the control point will be somewhere on a line that intersects the baseline at a right angle. And we don't get any choice on where that will be, since the derivatives at the start and end point have to line up, so our control point will lie at the intersection of the tangents at the start and end point.

    -

    First, let's try to fit the quadratic curve onto a circular arc. In the following sketch you can move the mouse around over a unit circle, to see how well, or poorly, a quadratic curve can approximate the arc from (1,0) to where your mouse cursor is:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    As you can see, things go horribly wrong quite quickly; even trying to approximate a quarter circle using a quadratic curve is a bad idea. An eighth of a turns might look okay, but how okay is okay? Let's apply some maths and find out. What we're interested in is how far off our on-curve coordinates are with respect to a circular arc, given a specific start and end angle. We'll be looking at how much space there is between the circular arc, and the quadratic curve's midpoint.

    -

    We start out with our start and end point, and for convenience we will place them on a unit circle (a circle around 0,0 with radius 1), at some angle φ:

    - - -

    What we want to find is the intersection of the tangents, so we want a point C such that:

    - - -

    i.e. we want a point that lies on the vertical line through S (at some distance a from S) and also lies on the tangent line through E (at some distance b from E). Solving this gives us:

    - - -

    First we solve for b:

    - - -

    which yields:

    - - -

    which we can then substitute in the expression for a:

    - - -

    A quick check shows that plugging these values for a and b into the expressions for Cx and Cy give the same x/y coordinates for both "a away from A" and "b away from B", so let's continue: now that we know the coordinate values for C, we know where our on-curve point T for t=0.5 (or angle φ/2) is, because we can just evaluate the Bézier polynomial, and we know where the circle arc's actual point P is for angle φ/2:

    - - -

    We compute T, observing that if t=0.5, the polynomial values (1-t)², 2(1-t)t, and t² are 0.25, 0.5, and 0.25 respectively:

    - - -

    Which, worked out for the x and y components, gives:

    - - -

    And the distance between these two is the standard Euclidean distance:

    - - -

    So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter circle and eighth circle?

    -
    - - plotted for 0 ≤ φ ≤ π: - - - plotted for 0 ≤ φ ≤ ½π: - - - - - plotted for 0 ≤ φ ≤ ¼π: -
    + +

    + So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter + circle and eighth circle? +

    + + + + + + + + +
    + + plotted for 0 ≤ φ ≤ π: + + + plotted for 0 ≤ φ ≤ ½π: + + + + + plotted for 0 ≤ φ ≤ ¼π: +
    -

    We now see why the eighth circle arc looks decent, but the quarter circle arc doesn't: an error of roughly 0.06 at t=0.5 means we're 6% off the mark... we will already be off by one pixel on a circle with pixel radius 17. Any decent sized quarter circle arc, say with radius 100px, will be way off if approximated by a quadratic curve! For the eighth circle arc, however, the error is only roughly 0.003, or 0.3%, which explains why it looks so close to the actual eighth circle arc. In fact, if we want a truly tiny error, like 0.001, we'll have to contend with an angle of (rounded) 0.593667, which equates to roughly 34 degrees. We'd need 11 quadratic curves to form a full circle with that precision! (technically, 10 and ten seventeenth, but we can't do partial curves, so we have to round up). That's a whole lot of curves just to get a shape that can be drawn using a simple function!

    -

    In fact, let's flip the function around, so that if we plug in the precision error, labelled ε, we get back the maximum angle for that precision:

    - - -

    And frankly, things are starting to look a bit ridiculous at this point, we're doing way more maths than we've ever done, but thankfully this is as far as we need the maths to take us: If we plug in the precisions 0.1, 0.01, 0.001 and 0.0001 we get the radians values 1.748, 1.038, 0.594 and 0.3356; in degrees, that means we can cover roughly 100 degrees (requiring four curves), 59.5 degrees (requiring six curves), 34 degrees (requiring 11 curves), and 19.2 degrees (requiring a whopping nineteen curves).

    -

    The bottom line? Quadratic curves are kind of lousy if you want circular (or elliptical, which are circles that have been squashed in one dimension) curves. We can do better, even if it's just by raising the order of our curve once. So let's try the same thing for cubic curves.

    - -
    -
    -

    previoustable of contentsnext
    Circular arcs and cubic Béziers

    -

    Let's look at approximating circles and circular arcs using cubic Béziers. How much better is that?

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    At cursory glance, a fair bit better, but let's find out how much better by looking at how to construct the Bézier curve.

    -

    A construction diagram for a cubic approximation of a circular arc

    -

    The start and end points are trivial, but the mid point requires a bit of work, but it's mostly basic trigonometry once we know the angle θ for our circular arc: if we scale our circular arc to a unit circle, we can always start our arc, with radius 1, at (1,0) and then given our arc angle θ, we also know that the circular arc has length θ (because unit circles are nice that way). We also know our end point, because that's just (cos(θ), sin(θ)), and so the challenge is to figure out what control points we need in order for the curve at t=0.5 to exactly touch the circular arc at the angle θ/2:

    -

    So let's again formally describe this:

    - - -

    Only P3 isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P4, P3) is a right angled triangle, with the distance between the origin and P4 being 1 (because we're working with a unit circle), and the distance between P4 and P3 being k, so that we can represent P3 as "The point P4 plus the vector from the origin to P4 but then rotated a quarter circle, counter-clockwise, and scaled by k".

    -

    With that, we can determine the y-coordinates for A, B, e1, and e2, after which we have all the information we need to determine what the value of k is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P2 and P3, e1 is between A and "midway between P1 and P2" (which is "half height" P2), and so forth:

    - - -

    Which now gives us two identities for B, because in addition to determining B through linear interpolation, we also know that B's y coordinate is just sin(θ/2): we started this exercise by saying we were going to approximate the circular arc using a Bézier curve that had its midpoint, which is point B, touching the unit circle at the arc's half-angle, by definition making B the point at (cos(θ/2), sin(θ/2)).

    -

    This means we can equate the two identities we now have for By and solve for k.

    -
    - -

    Deriving k

    -

    Solving for k is fairly straight forward, but it's a fair few steps, and if you just the immediate result: using a tool like Wolfram Alpha is definitely the way to go. That said, let's get going:

    - - -

    And finally, we can take further advantage of several trigonometric identities to drastically simplify our formula for k:

    - - -

    And we're done.

    -
    + +

    And we're done.

    + -

    So, the distance of our control points to the start/end points can be expressed as a number that we get from an almost trivial expression involving the circular arc's angle:

    - - -

    Which means that for any circular arc with angle θ and radius r, our Bézier approximation based on three points of incidence is:

    - - -

    Which also gives us the commonly found value of 0.55228 for quarter circles, based on them having an angle of half π:

    - - -

    And thus giving us the following Bézier coordinates for a quarter circle of radius r:

    - - -
    + +
    +

    So, how accurate is this?

    +

    + Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points + that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum + deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but + rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we + get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff. +

    +

    + So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, + finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around + that t value, repeating as many times as necessary to get us an arbitrarily precise value of t: +

    -

    So, how accurate is this?

    -

    Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff.

    -

    So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around that t value, repeating as many times as necessary to get us an arbitrarily precise value of t:

    - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return getMostWrongT(radius, bezier, worst_t - stepsize, worst_t + stepsize) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    Plus, how often do you get to write a function with that name?

    -

    Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity first, and then coming back down as we approach 2π)

    - - -
    - - - - - -
    - error plotted for 0 ≤ φ ≤ 2π - - error plotted for 0 ≤ φ ≤ π - - error plotted for 0 ≤ φ ≤ ½π -
    +

    Plus, how often do you get to write a function with that name?

    +

    + Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see + what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity + first, and then coming back down as we approach 2π) +

    + + + + + + + + + + + + + +
    + + + + + +
    + error plotted for 0 ≤ φ ≤ 2π + + error plotted for 0 ≤ φ ≤ π + + error plotted for 0 ≤ φ ≤ ½π +
    -

    That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again:

    -

    +

    + That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we + can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again: +

    +

    -

    Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves.

    -
    +

    + Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At + approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over + quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves. +

    +
    -

    So that's it, kappa is 4/3 · tan(θ/4) , we're done! ...or are we?

    -

    Can we do better?

    -

    Technically: yes, we can. But I'm going to prefix this section with "we can, and we should investigate that possibility, but let me warn you up front that the result is only better if we're going to hard-code the values". We're about to get into the weeds and the standard three-points-of-incidence value is so good already that for most applications, trying to do better won't make any sense at all.

    -

    So with that said: what we calculated above is an upper bound for a best fit Bézier curve for a circular arc: anywhere we don't touch the circular arc in our approximation, we've "overshot" the arc. What if we dropped our value for k just a little, so that the curve starts out as an over-estimation, but then crosses the circular arc, yielding an region of underestimation, and then crosses the circular arc again, with another region of overestimation. This might give us a lower overall error, so let's see what we can do.

    -

    First, let's express the total error (given circular arc angle θ, and some k) using standard calculus notation:

    - - -

    This says that the error function for a given angle and value of k is equal to the "infinite" sum of differences between our curve and the circular arc, as we run t from 0 to 1, using an infinitely small step size. between subsequent t values.

    -

    Now, since we want to find the minimal error, that means we want to know where along this function things go from "error is getting progressively less" to "error is increasing again", which means we want to know where its derivative is zero, which as mathematical expression looks like:

    - - -

    And here we have the most direct application of the Fundamental Theorem of Calculus: the derivative and integral are each other's inverse operations, so they cancel out, leaving us with our original function:

    - - -

    And now we just solve for that... oh wait. We've seen this before. In order to solve this, we'd end up needing to solve this:

    - - -

    And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section on arc lengths—there is no symbolic solution for this equasion. Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer!

    -
    + +

    + And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section + on arc lengths—there is no symbolic solution for this equasion. + Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer! +

    +
    +

    Iterating on a solution

    +

    + By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the + value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's + center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just + binary search our way to the most accurate value for c that gets us that middle case. +

    +

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    -

    Iterating on a solution

    -

    By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just binary search our way to the most accurate value for c that gets us that middle case.

    -

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    - - - - -
    1 - -
    2
    3
    4
    + return binarySearch(radius, angle, points, lowerBound, upperBound) + + + + 2 + + + 3 + + + 4 + + -

    And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and blog posts than you can ever read in a life time:

    +

    + And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and + blog posts than you can ever read in a life time: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return binarySearch(radius, angle, points, value, upperBound) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Using the following radialError function, which samples the curve's approximation of the circular arc over several points (although the first and last point will never contribute anything, so we skip them):

    +

    + Using the following radialError function, which samples the curve's approximation of the circular arc over several points + (although the first and last point will never contribute anything, so we skip them): +

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return err + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a vector, we can get its length to the origin using a magnitude call.

    -

    Examining the result

    -

    Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, for each angle, plot what the difference between the circular arc and the Bézier approximation looks like:

    -

    image-20210419085430711

    -

    Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from 0 to θ:

    -

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    - - -
    - - - - - -
    - max deflection using unit scale - - max deflection at 10x scale - - max deflection at 100x scale -
    +

    + In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a + vector, we can get its length to the origin using a magnitude call. +

    +

    Examining the result

    +

    + Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, + for each angle, plot what the difference between the circular arc and the Bézier approximation looks like: +

    +

    image-20210419085430711

    +

    + Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the + plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from + 0 to θ: +

    +

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    + + + + + + + + + + + + + +
    + + + + + +
    + max deflection using unit scale + + max deflection at 10x scale + + max deflection at 100x scale +
    -

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    -

    As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference.

    -
    +

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    +

    + As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by + 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an + almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference. +

    +
    -

    At this point it should be clear that while, yes, there are improvement to be had, they're essentially insignificant while also being much more computationally expensive.

    -

    TL;DR: just tell me which value I should be using

    -

    It depends on what we need to do. If we just want the best value for quarter circles, and we're going to hard code the value for k, then there is no reason to hard-code the constant k=4/3*tan(pi/8) when you can just as easily hard-code the constant as k=0.551784777779014 instead.

    -

    If you need "the" value for quarter circles, use 0.551785 instead of 0.55228

    -

    However, for dynamic arc approximation, in code that tries to fit circular paths using Bézier paths instead, it should be fairly obvious that the simple function involving a tangent computation, two divisions, and one multiplication, is vastly more performant than running all the code we ended writing just to get a 25% lower error value, and most certainly worth preferring over getting the "more accurate" value.

    -

    If you need to fit Béziers to circular arcs on the fly, use 4/3 * tan(θ/4)

    -

    However, always remember that if you're writing for humans, you can typically use the best of both worlds: as the user interacts with their curves, you should draw their curves instead of drawing approximations of them. If they need to draw circles or circular arcs, draw those, and only approximate them with a Bézier curve when the data needs to be exported to a format that doesn't support those. Ideally with a preview mechanism that highlights where the errors will be, and how large they will be.

    -

    If you're writing code for graphics design by humans, use circular arcs for circular arcs

    -

    And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best k values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).

    - -
    -
    -

    previoustable of contentsnext
    Approximating Bézier curves with circular arcs

    -

    Let's look at doing the exact opposite of the previous section: rather than approximating circular arc using Bézier curves, let's approximate Bézier curves using circular arcs.

    -

    We already saw in the section on circle approximation that this will never yield a perfect equivalent, but sometimes you need circular arcs, such as when you're working with fabrication machinery, or simple vector languages that understand lines and circles, but not much else.

    -

    The approach is fairly simple: pick a starting point on the curve, and pick two points that are further along the curve. Determine the circle that goes through those three points, and see if it fits the part of the curve we're trying to approximate. Decent fit? Try spacing the points further apart. Bad fit? Try spacing the points closer together. Keep doing this until you've found the "good approximation/bad approximation" boundary, record the "good" arc, and then move the starting point up to overlap the end point we previously found. Rinse and repeat until we've covered the entire curve.

    -

    We already saw how to fit a circle through three points in the section on creating a curve from three points, and finding the arc through those points is straight-forward: pick one of the three points as start point, pick another as an end point, and the arc has to necessarily go from the start point, to the end point, over the remaining point.

    -

    So, how can we convert a Bézier curve into a (sequence of) circular arc(s)?

    -
      -
    • Start at t=0
      • -
    • Pick two points further down the curve at some value m = t + n and e = t + 2n
      • -
    • Find the arc that these points define
      • -
    • Determine how close the found arc is to the curve:
        -
      • Pick two additional points e1 = t + n/2 and e2 = t + n + n/2.
        • -
      • These points, if the arc is a good approximation of the curve interval chosen, should - lie on the circle, so their distance to the center of the circle should be the - same as the distance from any of the three other points to the center.
        • -
      • For point points, determine the (absolute) error between the radius of the circle, and the -actual distance from the center of the circle to the point on the curve.
        • -
      • If this error is too high, we consider the arc bad, and try a smaller interval.
        • -
      -
      • -
    -

    The result of this is shown in the next graphic: we start at a guaranteed failure: s=0, e=1. That's the entire curve. The midpoint is simply at t=0.5, and then we start performing a binary search.

    -
      -
    1. We start with low=0, mid=0.5 and high=1
      2. -
    2. That'll fail, so we retry with the interval halved: {0, 0.25, 0.5}
        -
      • If that arc's good, we move back up by half distance: {0, 0.375, 0.75}.
        • -
      • However, if the arc was still bad, we move down by half the distance: {0, 0.125, 0.25}.
        • -
      -
      3. -
    3. We keep doing this over and over until we have two arcs, in sequence, of which the first arc is good, and the second arc is bad. When we find that pair, we've found the boundary between a good approximation and a bad approximation, and we pick the good arc.
      4. -
    -

    The following graphic shows the result of this approach, with a default error threshold of 0.5, meaning that if an arc is off by a combined half pixel over both verification points, then we treat the arc as bad. This is an extremely simple error policy, but already works really well. Note that the graphic is still interactive, and you can use your up and down arrow keys keys to increase or decrease the error threshold, to see what the effect of a smaller or larger error threshold is.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    With that in place, all that's left now is to "restart" the procedure by treating the found arc's end point as the new to-be-determined arc's starting point, and using points further down the curve. We keep trying this until the found end point is for t=1, at which point we are done. Again, the following graphic allows for up and down arrow key input to increase or decrease the error threshold, so you can see how picking a different threshold changes the number of arcs that are necessary to reasonably approximate a curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    So... what is this good for? Obviously, if you're working with technologies that can't do curves, but can do lines and circles, then the answer is pretty straightforward, but what else? There are some reasons why you might need this technique: using circular arcs means you can determine whether a coordinate lies "on" your curve really easily (simply compute the distance to each circular arc center, and if any of those are close to the arc radii, at an angle between the arc start and end, bingo, this point can be treated as lying "on the curve"). Another benefit is that this approximation is "linear": you can almost trivially travel along the arcs at fixed speed. You can also trivially compute the arc length of the approximated curve (it's a bit like curve flattening). The only thing to bear in mind is that this is a lossy equivalence: things that you compute based on the approximation are guaranteed "off" by some small value, and depending on how much precision you need, arc approximation is either going to be super useful, or completely useless. It's up to you to decide which, based on your application!

    - -
    -
    -

    previoustable of contentsnext
    B-Splines

    -

    No discussion on Bézier curves is complete without also giving mention of that other beast in the curve design space: B-Splines. Easily confused to mean Bézier splines, that's not actually what they are; they are "basis function" splines, which makes a lot of difference, and we'll be looking at those differences in this section. We're not going to dive as deep into B-Splines as we have for Bézier curves (that would be an entire primer on its own) but we'll be looking at how B-Splines work, what kind of maths is involved in computing them, and how to draw them based on a number of parameters that you can pick for individual B-Splines.

    -

    First off: B-Splines are piecewise, polynomial interpolation curves, where the "single curve" is built by performing polynomial interpolation over a set of points, using a sliding window of a fixed number of points. For instance, a "cubic" B-Spline defined by twelve points will have its curve built by evaluating the polynomial interpolation of four points, and the curve can be treated as a lot of different sections, each controlled by four points at a time, such that the full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, ..., {8,9,10,11}, and finally {9,10,11,12}, for eight sections.

    -

    What do they look like? They look like this! Tap on the graphic to add more points, and move points around to see how they map to the spline curve drawn.

    - - - Scripts are disabled. Showing fallback image. - - - -

    The important part to notice here is that we are not doing the same thing with B-Splines that we do for poly-Béziers or Catmull-Rom curves: both of the latter simply define new sections as literally "new sections based on new points", so a 12 point cubic poly-Bézier curve is actually impossible, because we start with a four point curve, and then add three more points for each section that follows, so we can only have 4, 7, 10, 13, 16, etc. point Poly-Béziers. Similarly, while Catmull-Rom curves can grow by adding single points, this addition of a single point introduces three implicit Bézier points. Cubic B-Splines, on the other hand, are smooth interpolations of each possible curve involving four consecutive points, such that at any point along the curve except for our start and end points, our on-curve coordinate is defined by four control points.

    -

    Consider the difference to be this:

    -
      -
    • for Bézier curves, the curve is defined as an interpolation of points, but:
      • -
    • for B-Splines, the curve is defined as an interpolation of curves.
      • -
    -

    In fact, let's look at that again, but this time with the base curves shown, too. Each consecutive four points define one curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have a look at how things work.

    -

    How to compute a B-Spline curve: some maths

    -

    Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the end, just like for Bézier curves), by evaluating the following function:

    - + +

    + In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have + a look at how things work. +

    +

    How to compute a B-Spline curve: some maths

    +

    + Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic + B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the + end, just like for Bézier curves), by evaluating the following function: +

    + - -

    Which, honestly, doesn't tell us all that much. All we can see is that a point on a B-Spline curve is defined as "a mix of all the control points, weighted somehow", where the weighting is achieved through the N(...) function, subscripted with an obvious parameter i, which comes from our summation, and some magical parameter k. So we need to know two things: 1. what does N(t) do, and 2. what is that k? Let's cover both, in reverse order.

    -

    The parameter k represents the "knot interval" over which a section of curve is defined. As we learned earlier, a B-Spline curve is itself an interpolation of curves, and we can treat each transition where a control point starts or stops influencing the total curvature as a "knot on the curve". -Doing so for a degree d B-Spline with n control point gives us d + n + 1 knots, defining d + n intervals along the curve, and it is these intervals that the above k subscript to the N() function applies to.

    -

    Then the N() function itself. What does it look like?

    - - -

    So this is where we see the interpolation: N(t) for an (i,k) pair (that is, for a step in the above summation, on a specific knot interval) is a mix between N(t) for (i,k-1) and N(t) for (i+1,k-1), so we see that this is a recursive iteration where i goes up, and k goes down, so it seem reasonable to expect that this recursion has to stop at some point; obviously, it does, and specifically it does so for the following i/k values:

    - - -

    And this function finally has a straight up evaluation: if a t value lies within a knot-specific interval once we reach a k=1 value, it "counts", otherwise it doesn't. We did cheat a little, though, because for all these values we need to scale our t value first, so that it lies in the interval bounded by knots[d] and knots[n], which are the start point and end point where curvature is controlled by exactly order control points. For instance, for degree 3 (=order 4) and 7 control points, with knot vector [1,2,3,4,5,6,7,8,9,10,11], we map t from [the interval 0,1] to the interval [4,8], and then use that value in the functions above, instead.

    -

    Can we simplify that?

    -

    We can, yes.

    -

    People far smarter than us have looked at this work, and two in particular — Maurice Cox and Carl de Boor — came to a mathematically pleasing solution: to compute a point P(t), we can compute this point by evaluating d(t) on a curve section between knots i and i+1:

    - - -

    This is another recursive function, with k values decreasing from the curve order to 1, and the value α (alpha) defined by:

    - - -

    That looks complicated, but it's not. Computing alpha is just a fraction involving known, plain numbers. And, once we have our alpha value, we also have (1-alpha) because it's a trivial subtraction. Computing the d() function is thus mostly a matter of computing pretty simple arithmetical statements, with some caching of results so we can refer to them as we recurve. While the recursion might see computationally expensive, the total algorithm is cheap, as each step only involves very simple maths.

    -

    Of course, the recursion does need a stop condition:

    - - -

    So, we actually see two stopping conditions: either i becomes 0, in which case d() is zero, or k becomes zero, in which case we get the same "either 1 or 0" that we saw in the N() function above.

    -

    Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily using the same kind of linear interpolation we saw in de Casteljau's algorithm. For instance, if we write out d() for i=3 and k=3, we get the following recursion diagram:

    - - -

    That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. It's really quite elegant.

    -

    One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then summing back up "to the left". We can just start on the right and work our way left immediately.

    -

    Running the computation

    -

    Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on this Michigan Tech page, but an easier to read version is implemented by b-spline.js, so we'll look at its code.

    -

    Given an input value t, we first map the input to a value from the domain [0,1] to the domain [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped t value lies on:

    + +

    + That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a + mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are + themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. + It's really quite elegant. +

    +

    + One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the + d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so + in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point + vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then + summing back up "to the left". We can just start on the right and work our way left immediately. +

    +

    Running the computation

    +

    + Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, + and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on + this Michigan Tech page, but an easier to read version + is implemented by b-spline.js, so we'll look at its code. +

    +

    + Given an input value t, we first map the input to a value from the domain [0,1] to the domain + [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped + t value lies on: +

    - - -
    1 - -
    2
    3
    +} + + + + 2 + + + 3 + + -

    after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU page (updated to use this description's variable names):

    +

    + after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU + page (updated to use this description's variable names): +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those v[i-1] don't try to use an array index that doesn't exist)

    -

    Open vs. closed paths

    -

    Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need more than just a single point.

    -

    Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than separate coordinate objects.

    -

    Manipulating the curve through the knot vector

    -

    The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at:

    -
      -
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. -
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. -
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. -
    4. we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a uniform vector in between.
      5. -
    -

    Uniform B-Splines

    -

    The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to the curvature.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve.

    -

    The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get rid of these, by being clever about how we apply the following uniformity-breaking approach instead...

    -

    Reducing local curve complexity by collapsing intervals

    -

    Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost and the curve "kinks".

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Open-Uniform B-Splines

    -

    By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the curve not starting and ending where we'd kind of like it to:

    -

    For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences that determine the curve shape.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Non-uniform B-Splines

    -

    This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than that any value knots[k+1] should be greater than or equal to knots[k].

    -

    One last thing: Rational B-Splines

    -

    While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like for rational Bézier curves.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in arbitrary-precision 2D design due to the level of control a NURBS curve offers designers.

    -

    While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the last.

    -

    Extending our implementation to cover rational splines

    -

    The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the extended dimension.

    -

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    -

    We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how many dimensions it needs to interpolate.

    -

    In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight information.

    -

    Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing (x'/w', y'/w'). And that's it, we're done!

    +

    + (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the + interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those + v[i-1] don't try to use an array index that doesn't exist) +

    +

    Open vs. closed paths

    +

    + Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last + point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first + and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As + such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly + more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need + more than just a single point. +

    +

    + Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for + points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same + coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing + references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than + separate coordinate objects. +

    +

    Manipulating the curve through the knot vector

    +

    + The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As + mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the + values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on + intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting + things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at: +

    +
      +
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. +
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. +
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. +
    4. + we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a + uniform vector in between. +
      5. +
    +

    Uniform B-Splines

    +

    + The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire + curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or + [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines + the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to + the curvature. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every + interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve. +

    +

    + The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can + perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get + rid of these, by being clever about how we apply the following uniformity-breaking approach instead... +

    +

    Reducing local curve complexity by collapsing intervals

    +

    + Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the + sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, + and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost + and the curve "kinks". +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Open-Uniform B-Splines

    +

    + By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the + curve not starting and ending where we'd kind of like it to: +

    +

    + For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in + which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values + N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector + [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences + that determine the curve shape. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Non-uniform B-Splines

    +

    + This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to + pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than + that any value knots[k+1] should be greater than or equal to knots[k]. +

    +

    One last thing: Rational B-Splines

    +

    + While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's + "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a + ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a + control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like + for rational Bézier curves. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the + NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, + the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an + important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in + arbitrary-precision 2D design due to the level of control a NURBS curve offers designers. +

    +

    + While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform + Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, + they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the + last. +

    +

    Extending our implementation to cover rational splines

    +

    + The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the + control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to + one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the + extended dimension. +

    +

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    +

    + We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate + interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how + many dimensions it needs to interpolate. +

    +

    + In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the + final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight + information. +

    +

    + Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing + (x'/w', y'/w'). And that's it, we're done! +

    +
    +
    + -
    -
    - +

    +
    previoustable of contents
    + Comments and questions +

    +

    + First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how + to let me know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing. +

    +

    With that said, on to the comments!

    +
    +
    +
    +
    -

    previoustable of contents
    Comments and questions

    -

    First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing.

    -

    With that said, on to the comments!

    -
    +
    +
    + This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are + almost no rights reserved. Go do something cool with it! +
    - - - - -
    - -
    - - This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are - almost no rights reserved. - Go do something cool with it! - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/ja-JP/index.html b/docs/ja-JP/index.html index b439df7d..7d1d41bb 100644 --- a/docs/ja-JP/index.html +++ b/docs/ja-JP/index.html @@ -1,161 +1,177 @@ + + + + ベジェ曲線入門 - - - - ベジェ曲線入門 + - + - + - + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    + - +
    +

    + ベジェ曲線入門 +

    +

    A free, online book for when you really need to know how to do Bézier things.

    +
    + Read this in your own language: + +

    + (Don't see your language listed, or want to see it reach 100%? + Help translate this content!) +

    +
    +

    + Welcome to the Primer on Bezier Curves. This is a free website/ebook dealing with both the maths and programming aspects of Bezier Curves, + covering a wide range of topics relating to drawing and working with that curve that seems to pop up everywhere, from Photoshop paths to CSS + easing functions to Font outline descriptions. +

    +

    + If this is your first time here: welcome! Let me know if you were looking for anything in particular that the primer doesn't cover over on the + issue tracker! +

    -
    +

    Donations and sponsorship

    -

    ベジェ曲線入門

    -

    A free, online book for when you really need to know how to do Bézier things.

    -
    - Read this in your own language: - -

    (Don't see your language listed, or want to see it reach 100%? Help translate this content!)

    -
    +

    + If this is a resource that you're using for research, as work reference, or even writing your own software, please consider + donating (any amount helps) or + signing up as a patron on Patreon. I don't get paid to work on this, so if you find this site + valuable, and you'd like it to stick around for a long time to come, a lot of coffee went into writing this over the years, and a lot more + coffee will need to go into it yet: if you can spare a coffee, you'd be helping keep a resource alive and well. +

    +

    + Also, if you are a company and your staff uses this book as a resource, or you use it as an onboarding resource, then please: consider + sponsoring the site! I am more than happy to work with your finance department on sponsorship invoicing and recognition. +

    -

    - Welcome to the Primer on Bezier Curves. This is a free website/ebook dealing with both - the maths and programming aspects of Bezier Curves, covering a wide range of topics - relating to drawing and working with that curve that seems to pop up everywhere, from - Photoshop paths to CSS easing functions to Font outline descriptions. -

    -

    - If this is your first time here: welcome! Let me know if you were looking for anything - in particular that the primer doesn't cover over on the issue tracker! -

    - -

    Donations and sponsorship

    - -

    - If this is a resource that you're using for research, as work reference, or even writing your own software, - please consider donating - (any amount helps) or signing up as a patron on Patreon. - I don't get paid to work on this, so if you find this site valuable, and you'd like it - to stick around for a long time to come, a lot of coffee went into writing this over the - years, and a lot more coffee will need to go into it yet: if you can spare a coffee, you'd - be helping keep a resource alive and well. -

    -

    - Also, if you are a company and your staff uses this book as a resource, or you use it as an - onboarding resource, then please: consider sponsoring the site! I am more than happy to work - with your finance department on sponsorship invoicing and recognition. -

    - - - -
    +
    -

    - — Pomax -

    - + +
    +
    +
    +

    まえがき

    +

    + 2次元上になにかを描くとき、普通は線を使いますが、これは直線と曲線の2つに分類することができます。直線を描くのはとても簡単で、これをコンピュータに描かせるのも容易です。直線の始点と終点をコンピュータに与えてやれば、ポン！直線が描けました。疑問の余地もありません。 +

    +

    + しかしながら、曲線の方はもっと大きな問題です。私たちはフリーハンドでいとも簡単に曲線を描くことができますが、コンピュータの方は少し不利です。曲線の描き方を表した数学的な関数が与えられないと、コンピュータは曲線を描くことができないのです。実際には、直線でさえも関数が必要になります。直線の関数はとても簡単なので、わたしたちはよく無視してしまいますが、コンピュータにとっては直線であれ曲線であれ、線はすべて「関数」なのです。しかしこれは、コンピュータで速く計算できて、きれいな曲線が得られるような関数を見つける必要がある、ということになります。そのような関数はたくさんありますが、多くの関心を集め続け、そしてどんな場面でも使われている、ある特定の関数に対してこの記事では焦点を絞ります。この関数は「ベジエ」曲線を描きます。 +

    +

    + ベジエ曲線はPierre Bézierから名付けられました。この曲線がデザイン作業に適していることを世界に知らしめたのが、彼なのです（ルノーに勤務し、1962年にその研究を発表しました）。ただし、この曲線を「発明」したのは彼が最初で唯一というわけではありません。数学者Paul de Casteljauはシトロエンで働いていた1959年、この曲線の性質について研究し、ベジエ曲線の非常にエレガントな描き方を発見しました。これが最初だと言う人もいます。しかしながら、de + Casteljauは自分の成果を発表しなかったため、「誰が最初か？」という問いに答えるのがとても難しくなっています。またベジエ曲線は、核心的にはSergei Natanovich Bernsteinが研究した「ベルンシュタイン多項式」という数学関数の一種ですが、こちらの公刊に関しては少なくとも1912年まで遡ることができます。いずれにせよ、これらはほとんど瑣末なことです。より注目すべきなのは、ベジエ曲線は取り扱いに便利だいうことです。たとえば複数のベジエ曲線を繋いで、1つの曲線に見えるようにすることができます。もしあなたがPhotoshopで「パス」を描いたり、FlashやIllustrator、Inkscapeのようなベクタードローイングソフトを使ったことがあるのであれば、そこで描いてきた曲線はベジエ曲線です。 +

    +

    + では、これを自分でプログラムしなければならないとなったらどうでしょう？ハマりどころは何でしょうか？どうやってベジエ曲線を描くのでしょう？バウンディングボックスとは何で、どうやって交点を求め、どうやったら曲線を押し出せるのでしょうか？つまるところ、ベジエ曲線に対して行いたいあらゆる操作は、どのようにすればいいのでしょう？このページはそれに答えるためにあります。数学にとりかかりましょう！ +

    +
    +

    追伸：コーヒーをおごってくれませんか？

    +

    + この本のことを印刷するほど気に入ったのであれば、あなたは「何かお礼をする方法はないか」と考えているかもしれません。この質問に対する答えはこうです：いつでもわたしにコーヒーをおごってください。あなたが住んでいるところの、コーヒー1杯の値段でかまいません。この本にかなりの金額を支払いたいのであれば、非常に高価なコーヒーをおごってくれてもかまいません + =) +

    +

    + この本は小さな入門からはじまり、印刷85ページ以上に相当するようなベジエ曲線の電子書籍へと、年々成長してきています。そして、多くのコーヒーがその執筆に費やされてきました。わたしは執筆に使った時間が惜しいとは思いませんが、もう少しコーヒーがあれば、ずっと書き続けることができるのです！https://pomax.github.io/bezierinfo/にアクセスしてオンライン版のまえがきにあるリンクをクリックし、コーヒー代を寄付してください。 +

    +
    -
    - -
    - -

    まえがき

    -

    2次元上になにかを描くとき、普通は線を使いますが、これは直線と曲線の2つに分類することができます。直線を描くのはとても簡単で、これをコンピュータに描かせるのも容易です。直線の始点と終点をコンピュータに与えてやれば、ポン！直線が描けました。疑問の余地もありません。

    -

    しかしながら、曲線の方はもっと大きな問題です。私たちはフリーハンドでいとも簡単に曲線を描くことができますが、コンピュータの方は少し不利です。曲線の描き方を表した数学的な関数が与えられないと、コンピュータは曲線を描くことができないのです。実際には、直線でさえも関数が必要になります。直線の関数はとても簡単なので、わたしたちはよく無視してしまいますが、コンピュータにとっては直線であれ曲線であれ、線はすべて「関数」なのです。しかしこれは、コンピュータで速く計算できて、きれいな曲線が得られるような関数を見つける必要がある、ということになります。そのような関数はたくさんありますが、多くの関心を集め続け、そしてどんな場面でも使われている、ある特定の関数に対してこの記事では焦点を絞ります。この関数は「ベジエ」曲線を描きます。

    -

    ベジエ曲線はPierre Bézierから名付けられました。この曲線がデザイン作業に適していることを世界に知らしめたのが、彼なのです（ルノーに勤務し、1962年にその研究を発表しました）。ただし、この曲線を「発明」したのは彼が最初で唯一というわけではありません。数学者Paul de Casteljauはシトロエンで働いていた1959年、この曲線の性質について研究し、ベジエ曲線の非常にエレガントな描き方を発見しました。これが最初だと言う人もいます。しかしながら、de Casteljauは自分の成果を発表しなかったため、「誰が最初か？」という問いに答えるのがとても難しくなっています。またベジエ曲線は、核心的にはSergei Natanovich Bernsteinが研究した「ベルンシュタイン多項式」という数学関数の一種ですが、こちらの公刊に関しては少なくとも1912年まで遡ることができます。いずれにせよ、これらはほとんど瑣末なことです。より注目すべきなのは、ベジエ曲線は取り扱いに便利だいうことです。たとえば複数のベジエ曲線を繋いで、1つの曲線に見えるようにすることができます。もしあなたがPhotoshopで「パス」を描いたり、FlashやIllustrator、Inkscapeのようなベクタードローイングソフトを使ったことがあるのであれば、そこで描いてきた曲線はベジエ曲線です。

    -

    では、これを自分でプログラムしなければならないとなったらどうでしょう？ハマりどころは何でしょうか？どうやってベジエ曲線を描くのでしょう？バウンディングボックスとは何で、どうやって交点を求め、どうやったら曲線を押し出せるのでしょうか？つまるところ、ベジエ曲線に対して行いたいあらゆる操作は、どのようにすればいいのでしょう？このページはそれに答えるためにあります。数学にとりかかりましょう！

    -
    - -

    追伸：コーヒーをおごってくれませんか？

    -

    この本のことを印刷するほど気に入ったのであれば、あなたは「何かお礼をする方法はないか」と考えているかもしれません。この質問に対する答えはこうです：いつでもわたしにコーヒーをおごってください。あなたが住んでいるところの、コーヒー1杯の値段でかまいません。この本にかなりの金額を支払いたいのであれば、非常に高価なコーヒーをおごってくれてもかまいません =)

    -

    この本は小さな入門からはじまり、印刷85ページ以上に相当するようなベジエ曲線の電子書籍へと、年々成長してきています。そして、多くのコーヒーがその執筆に費やされてきました。わたしは執筆に使った時間が惜しいとは思いませんが、もう少しコーヒーがあれば、ずっと書き続けることができるのです！https://pomax.github.io/bezierinfo/にアクセスしてオンライン版のまえがきにあるリンクをクリックし、コーヒー代を寄付してください。

    -
    - -
    - -

    注：ベジエの図はすべてインタラクティブになっています。

    -

    このページではBezier.jsを活用し、インタラクティブな例示を行っています。

    - -

    本書はオープンソースです。

    -

    この本はオープンソースソフトウェアのプロジェクトで、2つのGitHubリポジトリ上に存在しています。1つ目のhttps://github.com/pomax/bezierinfoは表示のためだけに用意されたバージョンで、あなたが今読んでいるものです。もう一方のhttps://github.com/pomax/BezierInfo-2は開発版で、すべてのHTML・JavaScript・CSSが含まれています。どちらのリポジトリもフォークすることができますし、あなたの好きなように使ってかまいません。ただし、これを自分の成果だと騙って売ることはもちろん除きます。=)

    -

    数学はどこまで難しくなりますか？

    -

    この入門に出てくる数学は、大半が高校初年度程度です。基本的な計算を理解していて、英語の読み方が分かっていれば、こなすことができるはずです。時にはずっと難しい数学も出てきますが、もし理解できないように感じたら、そこは読み飛ばしても大丈夫です。節の中の「詳細欄」を飛ばしてもいいですし、厄介そうな数学があれば節の最後まで読み飛ばしてもかまいません。各節の最後にはたいてい結論を並べてありますので、これを直に利用することもできます。

    -

    質問・コメント：

    -

    新しい節の提案があれば、GitHubのissueトラッカーをクリックしてください（右上にあるリポジトリのリンクからでもたどり着けます）。改訂中のため、現在はこのページにはコメント欄がありませんが、内容に関する質問がある場合にもissueトラッカーを使ってかまいません。改訂が完了したら、総合的なコメント欄を復活させる予定です。あるいは、「質問対象の節を選択して『質問』ボタンをクリック」するような項目別のシステムになるかもしれません。いずれわかります。

    -

    コーヒーをおごってくれませんか？

    -

    この本が気に入った場合や、取り組んでいたことに役に立つと思った場合、あるいは、この本への感謝をわたしに伝えるにはどうすればいいのかわからない場合。そのような場合には、わたしにコーヒーをおごってください。あなたが住んでいるところのコーヒー1杯の値段でかまいません。この本は小さな入門からはじまり、印刷で70ページほどに相当するようなベジエ曲線の読み物へと、年々成長してきています。そして、多くのコーヒーが執筆に費やされてきました。わたしは執筆に使った時間が惜しいとは思いませんが、もう少しコーヒーがあれば、ずっと書き続けることができるのです！

    -
    -
    -
    -

    What's new?

    -

    This primer is a living document, and so depending on when you last look at it, there may be new content. Click the following link to expand this section to have a look at what got added, when, or click through to the News posts for more detailed updates. (RSS feed available)

    - - - -

    November 2020

    • Added a section on finding curve/circle intersections

      -
    -

    October 2020

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      -
    -

    August-September 2020

    • Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS turned on.

      -
    -

    June 2020

    • Added automatic CI/CD using Github Actions

      -
    -

    January 2020

    • Added reset buttons to all graphics

      -
      • -

    • Updated to preface to correctly describe the on-page maths

      -
      • -

    • Fixed the Catmull-Rom section because it had glaring maths errors

      -
    -

    August 2019

    • Added a section on (plain) rational Bezier curves

      -
      • -

    • Improved the Graphic component to allow for sliders

      -
    -

    December 2018

    • Added a section on curvature and calculating kappa.

      -
      • -

    • Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this site!

      -
    -

    August 2018

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      -
    -

    July 2018

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      -
      • -

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      -
      • -

    • (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      -
    -

    June 2018

    • Added a section on direct curve fitting.

      -
      • -

    • Added source links for all graphics.

      -
      • -

    • Added this "What's new?" section.

      -
    -

    April 2017

    • Added a section on 3d normals.

      -
      • -

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      -
    -

    February 2017

    • Finished rewriting the entire codebase for localization.

      -
    -

    January 2016

    • Added a section to explain the Bezier interval.

      -
      • -

    • Rewrote the Primer as a React application.

      -
    -

    December 2015

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      -
      • -

    • Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics.

      -
    -

    May 2015

    • Switched over to pure JS rather than Processing-through-Processing.js

      -
      • -

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      -
    -

    April 2015

    • Added a section on arc length approximations.

      -
    -

    February 2015

    • Added a section on the canonical cubic Bezier form.

      -
    -

    November 2014

    • Switched to HTTPS.

      -
    -

    July 2014

    • Added the section on arc approximation.

      -
    -

    April 2014

    • Added the section on Catmull-Rom fitting.

      -
    -

    November 2013

    • Added the section on Catmull-Rom / Bezier conversion.

      -
      • -

    • Added the section on Bezier cuves as matrices.

      -
    -

    April 2013

    • Added a section on poly-Beziers.

      -
      • -

    • Added a section on boolean shape operations.

      -
    -

    March 2013

    • First drastic rewrite.

      -
      • -

    • Added sections on circle approximations.

      -
      • -

    • Added a section on projecting a point onto a curve.

      -
      • -

    • Added a section on tangents and normals.

      -
      • -

    • Added Legendre-Gauss numerical data tables.

      -
    -

    October 2011

    • First commit for the bezierinfo site, based on the pre-Primer webpage that covered the basics of Bezier curves in HTML with Processing.js examples.

      -
    -
    -
    -

    目次次
    バッとした導入

    -

    まずは良い例から始めましょう。ベジエ曲線というのは、下の図に表示されているもののことです。ベジエ曲線はある始点からある終点へと延びており、その曲率は1個以上の「中間」制御点に左右されています。さて、このページの図はどれもインタラクティブになっていますので、ここで曲線をちょっと操作してみましょう。点をドラッグしたとき、曲線の形がそれに応じてどう変化するのか、確かめてみてください。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    本書はオープンソースです。

    +

    + この本はオープンソースソフトウェアのプロジェクトで、2つのGitHubリポジトリ上に存在しています。1つ目のhttps://github.com/pomax/bezierinfoは表示のためだけに用意されたバージョンで、あなたが今読んでいるものです。もう一方のhttps://github.com/pomax/BezierInfo-2は開発版で、すべてのHTML・JavaScript・CSSが含まれています。どちらのリポジトリもフォークすることができますし、あなたの好きなように使ってかまいません。ただし、これを自分の成果だと騙って売ることはもちろん除きます。=) +

    +

    数学はどこまで難しくなりますか？

    +

    + この入門に出てくる数学は、大半が高校初年度程度です。基本的な計算を理解していて、英語の読み方が分かっていれば、こなすことができるはずです。時にはずっと難しい数学も出てきますが、もし理解できないように感じたら、そこは読み飛ばしても大丈夫です。節の中の「詳細欄」を飛ばしてもいいですし、厄介そうな数学があれば節の最後まで読み飛ばしてもかまいません。各節の最後にはたいてい結論を並べてありますので、これを直に利用することもできます。 +

    +

    質問・コメント：

    +

    + 新しい節の提案があれば、GitHubのissueトラッカーをクリックしてください（右上にあるリポジトリのリンクからでもたどり着けます）。改訂中のため、現在はこのページにはコメント欄がありませんが、内容に関する質問がある場合にもissueトラッカーを使ってかまいません。改訂が完了したら、総合的なコメント欄を復活させる予定です。あるいは、「質問対象の節を選択して『質問』ボタンをクリック」するような項目別のシステムになるかもしれません。いずれわかります。 +

    +

    コーヒーをおごってくれませんか？

    +

    + この本が気に入った場合や、取り組んでいたことに役に立つと思った場合、あるいは、この本への感謝をわたしに伝えるにはどうすればいいのかわからない場合。そのような場合には、わたしにコーヒーをおごってください。あなたが住んでいるところのコーヒー1杯の値段でかまいません。この本は小さな入門からはじまり、印刷で70ページほどに相当するようなベジエ曲線の読み物へと、年々成長してきています。そして、多くのコーヒーが執筆に費やされてきました。わたしは執筆に使った時間が惜しいとは思いませんが、もう少しコーヒーがあれば、ずっと書き続けることができるのです！ +

    +
    + +
    +

    What's new?

    +

    + This primer is a living document, and so depending on when you last look at it, there may be new content. Click the following link to expand + this section to have a look at what got added, when, or click through to the News posts for more detailed updates. (RSS feed + available) +

    + + + +
    +

    November 2020

    +
      +

    • Added a section on finding curve/circle intersections

      • +
    +

    October 2020

    +
      +

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      • +
    +

    August-September 2020

    +
      +
    • +

      + Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS + turned on. +

      +
      • +
    +

    June 2020

    +
      +

    • Added automatic CI/CD using Github Actions

      • +
    +

    January 2020

    +
      +

    • Added reset buttons to all graphics

      • +

    • Updated to preface to correctly describe the on-page maths

      • +

    • Fixed the Catmull-Rom section because it had glaring maths errors

      • +
    +

    August 2019

    +
      +

    • Added a section on (plain) rational Bezier curves

      • +

    • Improved the Graphic component to allow for sliders

      • +
    +

    December 2018

    +
      +

    • Added a section on curvature and calculating kappa.

      • +
    • +

      + Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this + site! +

      +
      • +
    +

    August 2018

    +
      +

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      • +
    +

    July 2018

    +
      +

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      • +

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      • +
    • +

      (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      +
      • +
    +

    June 2018

    +
      +

    • Added a section on direct curve fitting.

      • +

    • Added source links for all graphics.

      • +

    • Added this "What's new?" section.

      • +
    +

    April 2017

    +
      +

    • Added a section on 3d normals.

      • +

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      • +
    +

    February 2017

    +
      +

    • Finished rewriting the entire codebase for localization.

      • +
    +

    January 2016

    +
      +

    • Added a section to explain the Bezier interval.

      • +

    • Rewrote the Primer as a React application.

      • +
    +

    December 2015

    +
      +

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      • +
    • +

      + Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics. +

      +
      • +
    +

    May 2015

    +
      +

    • Switched over to pure JS rather than Processing-through-Processing.js

      • +

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      • +
    +

    April 2015

    +
      +

    • Added a section on arc length approximations.

      • +
    +

    February 2015

    +
      +

    • Added a section on the canonical cubic Bezier form.

      • +
    +

    November 2014

    +
      +

    • Switched to HTTPS.

      • +
    +

    July 2014

    +
      +

    • Added the section on arc approximation.

      • +
    +

    April 2014

    +
      +

    • Added the section on Catmull-Rom fitting.

      • +
    +

    November 2013

    +
      +

    • Added the section on Catmull-Rom / Bezier conversion.

      • +

    • Added the section on Bezier cuves as matrices.

      • +
    +

    April 2013

    +
      +

    • Added a section on poly-Beziers.

      • +

    • Added a section on boolean shape operations.

      • +
    +

    March 2013

    +
      +

    • First drastic rewrite.

      • +

    • Added sections on circle approximations.

      • +

    • Added a section on projecting a point onto a curve.

      • +

    • Added a section on tangents and normals.

      • +

    • Added Legendre-Gauss numerical data tables.

      • +
    +

    October 2011

    +
      +
    • +

      + First commit for the bezierinfo site, based on the pre-Primer webpage that covered + the basics of Bezier curves in HTML with Processing.js examples. +

      +
      • +
    +
    +
    +
    +
    +

    +
    目次次
    + バッとした導入 +

    +

    + まずは良い例から始めましょう。ベジエ曲線というのは、下の図に表示されているもののことです。ベジエ曲線はある始点からある終点へと延びており、その曲率は1個以上の「中間」制御点に左右されています。さて、このページの図はどれもインタラクティブになっていますので、ここで曲線をちょっと操作してみましょう。点をドラッグしたとき、曲線の形がそれに応じてどう変化するのか、確かめてみてください。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    ベジエ曲線は、CAD（computer aided designやCAM（computer aided manufacturing）のアプリケーションで多用されています。もちろん、Adobe Illustrator・Photoshop・Inkscape・Gimp などのグラフィックデザインアプリケーションや、SVG（scalable vector graphics）・OpenTypeフォント（otf/ttf）のようなグラフィック技術でも利用されています。ベジエ曲線はたくさんのものに使われていますので、これについてもっと詳しく学びたいのであれば……さあ、準備しましょう！

    - -
    -
    -

    前目次次
    ではベジエ曲線はどうやってできるのでしょう？

    -

    ベジエ曲線がどのように動くのか、点を触ってみて感覚が摑めたかもしれません。では、実際のところベジエ曲線とはいったい何でしょうか？これを説明する方法は2通りありますが、どちらの説明でも行き着く先はまったく同じです。一方は複雑な数学を使うのに対し、もう一方はとても簡単です。というわけで……簡単な説明の方から始めましょう。

    -

    ベジエ曲線は、線形補間の結果です。というと難しそうに聞こえますが、誰でも幼い頃から線形補間をやってきています。例えば、何か2つのものの間を指し示すときには、いつも線形補間を行っています。線形補間とは、単純に「2点の間から点を得る」ことなのです。

    -

    例えば、2点間の距離がわかっているとして、一方の点から距離の20%だけ離れた（すなわち、もう一方の点から80%離れた）新しい点を求めたい場合、次のようにとても簡単に計算できます。

    - - -

    では、実際に見てみましょう。下の図はインタラクティブになっています。上下キーで補間の比率が増減しますので、どうなるか確かめてみましょう。最初に3点があり、それを結んで2本の直線が引かれています。この直線の上でそれぞれ線形補間を行うと、2つの点が得られます。この2点の間でさらに線形補間を行うと、1つの点を得ることができます。そして、あらゆる比率に対して同様に点を求め、それをすべて集めると、このようにベジエ曲線ができるのです。

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    また、これが複雑な方の数学につながっていきます。微積分です。

    -

    いま上で行ったものとは似つかないように思えますが、実はあれは2次曲線を描いていたのです。ただし一発で描きあげるのではなく、手順を追って描いていきました。ベジエ曲線は多項式関数で表現できますが、その一方で、とても単純な補間の補間の補間の……というふうにも説明できます。これがベジエ曲線のおもしろいところです。これはまた、ベジェ曲線は「本当の数学」で見る（関数を調べたり微分を調べたり、あらゆる方法で）ことも可能ですし、「機械的」な組み立て操作として見る（例えば、ベジエ曲線は組み立てに使う点の間からは決してはみ出ないということがわかります）ことも可能だということを意味しています。

    -

    それでは、もう少し詳しくベジエ曲線を見ていきましょう。数学的な表現やそこから導かれる性質、さらには、ベジエ曲線に対して／ベジエ曲線を使ってできるさまざまな内容についてです。

    - -
    -
    -

    前目次次
    ベジエ曲線の数学

    -

    ベジエ曲線は「パラメトリック」関数の一種です。数学的に言えば、パラメトリック関数というのはインチキです。というのも、「関数」はきっちり定義された用語であり、いくつかの入力を1つの出力に対応させる写像を表すものだからです。いくつかの数値を入れると、1つの数値が出てきます。入れる数値が変わっても、出てくる数値はやはり1つだけです。パラメトリック関数はインチキです。基本的には「じゃあわかった、値を複数個出したいから、関数を複数個使うことにするよ」ということです。例として、ある値xに何らかの操作を行い、別の値へと写す関数があるとします。

    - - -

    f(x)という記法は、これが関数（1つしかない場合は慣習的にfと呼びます）であり、その出力が1つの変数（この場合はxです）に応じて変化する、ということを示す標準的な方法です。xを変化させると、f(x)の出力が変化します。

    -

    ここまでは順調です。では、パラメトリック関数について、これがどうインチキなのかを見てみましょう。以下の2つの関数を考えます。

    - - -

    注目すべき箇所は特に何もありません。ただの正弦関数と余弦関数です。ただし、入力が別々の名前になっていることに気づくでしょう。仮にaの値を変えたとしても、f(b)の出力の値は変わらないはずです。なぜなら、こちらの関数にはaは使われていないからです。パラメトリック関数は、これを変えてしまうのでインチキなのです。パラメトリック関数においては、どの関数も変数を共有しています。例えば、

    - - -

    複数の関数がありますが、変数は1つだけです。tの値を変えた場合、fa(t)とfb(t)の両方の出力が変わります。これがどのように役に立つのか、疑問に思うかもしれません。しかし、実際には答えは至ってシンプルです。fa(t)とfb(t)のラベルを、パラメトリック曲線の表示によく使われているもので置き換えてやれば、ぐっとはっきりするかと思います。

    - - -

    きました。x/y座標です。謎の値tを通して繫がっています。

    -

    というわけで、普通の関数ではy座標をx座標によって定義しますが、パラメトリック曲線ではそうではなく、座標の値を「制御」変数と結びつけます。tの値を変化させるたびに2つの値が変化するので、これをグラフ上の座標 (x,y)として使うことができます。例えば、先ほどの関数の組は円周上の点を生成します。負の無限大から正の無限大へとtを動かすと、得られる座標(x,y)は常に中心(0,0)・半径1の円の上に乗ります。tを0から5まで変化させてプロットした場合は、このようになります。

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    ベジエ曲線はパラメトリック関数の一種であり、どの次元に対しても同じ基底関数を使うという点で特徴づけられます。先ほどの例では、xの値とyの値とで異なる関数（正弦関数と余弦関数）を使っていましたが、ベジエ曲線ではxとyの両方で「二項係数多項式」を使います。では、二項係数多項式とは何でしょう？

    -

    高校で習った、こんな形の多項式を思い出すかもしれません。

    - - -

    最高次の項がx³であれば3次多項式、x²であれば2次多項式と呼び、xだけの場合は1次多項式――ただの直線です。（そしてxの入った項が何もなければ、多項式ではありません！）

    -

    ベジエ曲線はxの多項式ではなく、tの多項式です。tの値は0から1までの間に制限され、その係数a、bなどは「二項係数」の形をとります。というと複雑そうに聞こえますが、実際には値を組み合わせて、とてもシンプルに記述できます。

    - - -

    「そこまでシンプルには見えないよ」と思っていることでしょう。しかし仮に、tを取り去って係数に1を掛けることにしてしまえば、急激に簡単になります。これが二項係数部分の項です。

    - - -

    2は1+1に等しく、3は2+1や1+2に等しく、6は3+3に等しく、……ということに注目してください。見てわかるように、先頭と末尾は単に1になっていますが、中間はどれも次数が増えるたびに「上の2つの数を足し合わせた」ものになっています。これなら覚えやいですね。

    -

    多項式部分の項がどうなっているのか、同じぐらい簡単な方法で考えることができます。仮に、(1-t)をaに、tをbに書き換え、さらに重みを一旦削除してしまえば、このようになります。

    - - -

    これは要するに、「aとbのすべての組み合わせ」の単なる和です。プラスが出てくるたびに、aをbへと1つずつ置き換えていけばよいのです。こちらも本当に単純です。さて、これで「二項係数多項式」がわかりました。完璧を期するため、この関数の一般の形を示しておきます。

    - - -

    そして、これがベジエ曲線の完全な表現です。この関数中のΣは、加算の繰り返し（Σの下にある変数を使って、...=<値>から始めてΣの下にある値まで）を表します。

    -
    + +

    + そして、これがベジエ曲線の完全な表現です。この関数中のΣは、加算の繰り返し（Σの下にある変数を使って、...=<値>から始めてΣの下にある値まで）を表します。 +

    +
    +

    基底関数の実装方法

    +

    上で説明した関数を使えば、数学的な組み立て方で、基底関数をナイーブに実装することもできます。

    -

    基底関数の実装方法

    -

    上で説明した関数を使えば、数学的な組み立て方で、基底関数をナイーブに実装することもできます。

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    「こともできる」と書いたのは、この方法では実装しない方が良いからです。階乗はとてつもなく重い計算なのです。また、先ほどの説明からわかるように、実際は階乗を使わなくても、かなり簡単にパスカルの三角形を作ることができます。[1]から始めて[1,1]、[1,2,1]、[1,3,3,1]、……としていくだけです。下の段は上の段よりも1つ要素が増え、各段の先頭と末尾は1になります。中間の数はどれも、左右斜め上にある両要素の和になります。

    -

    このパスカルの三角形は、「リストのリスト」として瞬時に生成できます。そして、これをルックアップテーブルとして利用すれば、二項係数を計算する必要はまったくなくなります。

    +

    + 「こともできる」と書いたのは、この方法では実装しない方が良いからです。階乗はとてつもなく重い計算なのです。また、先ほどの説明からわかるように、実際は階乗を使わなくても、かなり簡単にパスカルの三角形を作ることができます。[1]から始めて[1,1]、[1,2,1]、[1,3,3,1]、……としていくだけです。下の段は上の段よりも1つ要素が増え、各段の先頭と末尾は1になります。中間の数はどれも、左右斜め上にある両要素の和になります。 +

    +

    + このパスカルの三角形は、「リストのリスト」として瞬時に生成できます。そして、これをルックアップテーブルとして利用すれば、二項係数を計算する必要はまったくなくなります。 +

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    + return lut[n][k] + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + -

    これはどのように動くのでしょう？最初に、十分に大きなサイズのルックアップテーブルを宣言します。次に、求めたい値を得るための関数を定義します。この関数は、求めたい値のn/kのペアがテーブル中にまだ存在しない場合、先にテーブルを拡張するようになっています。さて、これで基底関数は次のようになりました。

    +

    + これはどのように動くのでしょう？最初に、十分に大きなサイズのルックアップテーブルを宣言します。次に、求めたい値を得るための関数を定義します。この関数は、求めたい値のn/kのペアがテーブル中にまだ存在しない場合、先にテーブルを拡張するようになっています。さて、これで基底関数は次のようになりました。 +

    - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    完璧です。もちろん、さらなる最適化を施すこともできます。コンピュータグラフィクス用途ではたいてい、任意の次数の曲線が必要になるわけではありません。2次と3次の曲線だけが必要であれば、以下のようにコードを劇的に単純化することができます（実際、この入門では任意の次数までは扱いませんので、これに似たようなコードが出てきます）。

    +

    + 完璧です。もちろん、さらなる最適化を施すこともできます。コンピュータグラフィクス用途ではたいてい、任意の次数の曲線が必要になるわけではありません。2次と3次の曲線だけが必要であれば、以下のようにコードを劇的に単純化することができます（実際、この入門では任意の次数までは扱いませんので、これに似たようなコードが出てきます）。 +

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return mt3 + 3*mt2*t + 3*mt*t2 + t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    これで基底関数をプログラムする方法がわかりました。すばらしい。

    -
    +

    これで基底関数をプログラムする方法がわかりました。すばらしい。

    +
    -

    というわけで、基底関数がどのようなものか理解できました。今度はベジエ曲線を特別にする魔法――制御点を導入する時間です。

    +

    というわけで、基底関数がどのようなものか理解できました。今度はベジエ曲線を特別にする魔法――制御点を導入する時間です。

    +
    +
    +

    +
    前目次次
    + ベジエ曲線の曲率の制御 +

    +

    + ベジエ曲線は（すべての「スプライン」と同様に）補間関数です。これは点の集合を受け取って、それらの点のどこか「内側」の値を生成するということです。（このことから、制御点同士を結んで輪郭をつくったとき、その外側に位置する点は決して生成されないことがわかります。なお、この輪郭を曲線の「包」と呼びます。お役立ち情報でした！）実際に、補間関数によって生成された値に対する、各点の寄与の大きさを可視化することができますが、これを見れば、ベジエ曲線のどの場所でどの点が重要になるのかがわかります。 +

    +

    + 下のグラフは、2次ベジエ曲線や3次ベジエ曲線の補間関数を表しています。ここでSは、ベジエ関数全体に対しての、その点の寄与の大きさを示します。あるtにおいて、ベジエ曲線を定義する各点の補間率がどのようになっているのか、クリックドラッグをして確かめてみてください。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    前目次次
    ベジエ曲線の曲率の制御

    -

    ベジエ曲線は（すべての「スプライン」と同様に）補間関数です。これは点の集合を受け取って、それらの点のどこか「内側」の値を生成するということです。（このことから、制御点同士を結んで輪郭をつくったとき、その外側に位置する点は決して生成されないことがわかります。なお、この輪郭を曲線の「包」と呼びます。お役立ち情報でした！）実際に、補間関数によって生成された値に対する、各点の寄与の大きさを可視化することができますが、これを見れば、ベジエ曲線のどの場所でどの点が重要になるのかがわかります。

    -

    下のグラフは、2次ベジエ曲線や3次ベジエ曲線の補間関数を表しています。ここでSは、ベジエ関数全体に対しての、その点の寄与の大きさを示します。あるtにおいて、ベジエ曲線を定義する各点の補間率がどのようになっているのか、クリックドラッグをして確かめてみてください。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    あわせて、15次ベジエ関数における補間関数も示しています。始点と終点は他の制御点と比較して、曲線の形に対してかなり大きな影響を与えていることがわかります。

    -

    曲線を変更したい場合は、各点の重みを変える（実質的には補間率を変える）必要があります。これはとても単純で、寄与の大きさを変えるための値を、各点にただ掛ければいいのです。この値は「重み」と呼ばれていますが、これを元のベジエ関数に組み込めば、次のようになります。

    - - -

    複雑そうに見えますが、運がいいことに「重み」というのは実はただの座標値です。というのはn次の曲線の場合、w0が始点の座標、wnが終点の座標となり、その間はどれも制御点の座標になります。例えば、始点が(120,160)、制御点が(35,200)と(220,260)、終点が(220,40)となる3次ベジエ曲線は、次のようになります。

    - - -

    この式からは、記事の冒頭に出てきた曲線が得られます。

    - + +

    この式からは、記事の冒頭に出てきた曲線が得られます。

    + -

    ベジエ曲線で、他にはどんなことができるでしょうか？実は、非常にたくさんのことが可能です。この記事の残りの部分では、実現可能な各種操作や適用可能なアルゴリズム、そしてこれによって達成できるタスクについて扱います。

    -
    +

    + ベジエ曲線で、他にはどんなことができるでしょうか？実は、非常にたくさんのことが可能です。この記事の残りの部分では、実現可能な各種操作や適用可能なアルゴリズム、そしてこれによって達成できるタスクについて扱います。 +

    +
    +

    重みつき基底関数の実装方法

    +

    基底関数の実装方法はすでに知っていますし、これに制御点を組み込むのは非常に簡単です。

    -

    重みつき基底関数の実装方法

    -

    基底関数の実装方法はすでに知っていますし、これに制御点を組み込むのは非常に簡単です。

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    そして、最適化を行ったバージョンは以下のようになります。

    +

    そして、最適化を行ったバージョンは以下のようになります。

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    これで、重みつき基底関数をプログラムする方法がわかりました。

    -
    - -
    -
    -

    前目次次
    Controlling Bézier curvatures, part 2: Rational Béziers

    -

    We can further control Bézier curves by "rationalising" them: that is, adding a "ratio" value in addition to the weight value discussed in the previous section, thereby gaining control over "how strongly" each coordinate influences the curve.

    -

    Adding these ratio values to the regular Bézier curve function is fairly easy. Where the regular function is the following:

    - - -

    The function for rational Bézier curves has two more terms:

    - - -

    In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just using different weight. So far, nothing too special.

    -

    However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights.

    -

    This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly describing circles (which we'll see in a later section is literally impossible using standard Bézier curves).

    -

    But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects what gets drawn:

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - -

    You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence relative to all other points.

    -
    + +

    + In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] + then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just + using different weight. So far, nothing too special. +

    +

    + However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a + fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the + "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights. +

    +

    + This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve + that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly + describing circles (which we'll see in a later section is literally impossible using standard Bézier curves). +

    +

    + But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an + interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with + ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate + exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects + what gets drawn: +

    + + + Scripts are disabled. Showing fallback image. + + + + + + + + +

    + You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will + want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like + with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence + relative to all other points. +

    +
    +

    How to implement rational curves

    +

    Extending the code of the previous section to include ratios is almost trivial:

    -

    How to implement rational curves

    -

    Extending the code of the previous section to include ratios is almost trivial:

    - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    + return (f[0] * w[0] + f[1] * w[1] + f[2] * w[2] + f[3] * w[3])/basis + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + -

    And that's all we have to do.

    -
    - -
    -
    -

    前目次次
    ベジエ曲線の区間 [0,1]

    -

    ここまでの説明で、ベジエ曲線の裏側にある数学がわかりました。しかし、気づいているかもしれませんが、ひとつ引っかかる点があります。ベジエ曲線はいつもt=0からt=1の間を動いていますが、どうしてこの特定の区間なのでしょう？

    -

    このことは、曲線の「始点」から曲線の「終点」までどうやって動かすか、ということにすべて関係しています。2つの値を混ぜ合わせて1つの値をつくる場合、一般の式は次のようになります。

    - - -

    明らかに、始点ではa=1, b=0とする必要があります。こうすれば、値1が100%、値2が0%で混ぜ合わさるからです。また、終点ではa=0, b=1とする必要があります。こうすれば、値1が0%、値2が100%で混ぜ合わさります。これに加えて、aとbを独立にしておきたくはありません。独立になっている場合、何でも好きな値にすることできますが、こうすると例えば「値1が100%かつ値2が100%」のようなことが可能になってしまいます。これはこれで原則としてはかまいませんが、ベジエ曲線の場合は混ぜ合わさった値が常に始点と終点の間になってほしいのです。というわけで、混ぜ合わせの和が100%を決して超えないように、aとbの値を設定する必要があります。これは次のようにすれば簡単です。

    - - -

    こうすれば、和が100%を超えることはないと保証できます。aの値を区間[0,1]に制限してしまえば、混ぜ合わさった値は常に2つの値の間のどこか（両端を含む）になり、また和は常に100%になります。

    -

    しかし……この式を0と1の間の値だけで使うのではなく、もし仮にこの区間の外の値で使うとしたら、どうなるのでしょう？めちゃくちゃになってしまうのでしょうか？……実はそうではありません。「その先」が見えるのです。

    -

    ベジエ曲線の場合、区間を広げると曲線は単純に「そのまま延びて」いきます。ベジエ曲線はある多項式曲線の一部分にすぎませんので、単純に区間を広くとればとるほど、曲線のより多くの部分が現れるようになります。では、どのように見えるのでしょうか？

    -

    下の2つの図は「いつもの方法」で描いたベジエ曲線ですが、これと一緒に、tの値をずっと先まで広げた場合の「延びた」ベジエ曲線も表示しています。見てわかるように、曲線の残りの部分には多くの「かたち」が隠れています。そして曲線の点を動かせば、その部分の形状も変わります。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    + +

    + こうすれば、和が100%を超えることはないと保証できます。aの値を区間[0,1]に制限してしまえば、混ぜ合わさった値は常に2つの値の間のどこか（両端を含む）になり、また和は常に100%になります。 +

    +

    + しかし……この式を0と1の間の値だけで使うのではなく、もし仮にこの区間の外の値で使うとしたら、どうなるのでしょう？めちゃくちゃになってしまうのでしょうか？……実はそうではありません。「その先」が見えるのです。 +

    +

    + ベジエ曲線の場合、区間を広げると曲線は単純に「そのまま延びて」いきます。ベジエ曲線はある多項式曲線の一部分にすぎませんので、単純に区間を広くとればとるほど、曲線のより多くの部分が現れるようになります。では、どのように見えるのでしょうか？ +

    +

    + 下の2つの図は「いつもの方法」で描いたベジエ曲線ですが、これと一緒に、tの値をずっと先まで広げた場合の「延びた」ベジエ曲線も表示しています。見てわかるように、曲線の残りの部分には多くの「かたち」が隠れています。そして曲線の点を動かせば、その部分の形状も変わります。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    実際に、グラフィックデザインやコンピュータモデリングで使われている曲線の中には、座標が固定されていて、区間は自由に動かせるような曲線があります。これは、区間が固定されていて、座標を自由に動かすことのできるベジエ曲線とは反対になっています。すばらしい例が「Spiro」曲線で、これはオイラー螺旋とも呼ばれるクロソイド曲線の一部分に基づいた曲線です。非常に美しく心地よい曲線で、FontForgeやInkscapeなど多くのグラフィックアプリに実装されており、フォントデザインにも利用されています（Inconsolataフォントなど）。

    - -
    -
    -

    前目次次
    行列演算としてのベジエ曲線の曲率

    -

    ベジエ曲線は、行列演算の形でも表現することができます。ベジエ曲線の式を多項式基底と係数行列で表し、実際の座標も行列として表現するのです。これがどういうことを意味しているのか、3次ベジエ曲線について見てみましょう。

    - - -

    実際の座標を一旦無視すると、次のようになります。

    - - -

    これは、4つの項の和になっています。

    - - -

    それぞれの項を展開します。

    - - -

    その上で、係数の0や1もすべて明示的に書けば、このようになります。

    - - -

    さらに、これは4つの行列演算の和として見ることができます。

    - - -

    これを1つの行列演算にまとめると、以下のようになります。

    - - -

    多項式基底をこのような形で表現する場合、通常はその基底を昇冪の順に並べます。したがって、tの行列を左右反転させ、大きな「混合」行列は上下に反転させる必要があります。

    - - -

    そして最後に、もともとあった座標を3番目の行列として付け加えます。

    - - -

    2次ベジエ曲線の場合も同様に変形することができ、最終的には以下のようになります。

    - - -

    tの値を代入して行列の乗算を行えば、もともとの多項式関数から計算したときの値や、線形補間によって順次求めたときの値と、まったく同じものが得られます。

    -

    では、なぜわざわざ行列を使うのでしょう？ 行列表現を使うことによって、他の表現ではわからなかった関数の性質を発見できるようになります。ベジエ曲線は三角行列の形になり、その行列式は実際の座標の積に等しくなることがわかります。また、この行列には逆行列が存在しますが、これはさまざまな性質が満たされることを意味します。もっとも、疑問の中心は「それでなぜこれが役に立つのか？」という点でしょうが、「ただちには役に立たない」というのが回答です。しかしながら、この先に出てくる曲線のプロパティの中には、関数を操作して求めることも、また行列をうまいこと利用して求めることも、どちらでも可能な例があります。そしてときには、行列による手法の方が（劇的に）速くなる場合があるのです。

    -

    というわけで、現時点では「ベジエ曲線は行列で表現可能」ということだけを覚えて、次に進みましょう。

    + +

    + tの値を代入して行列の乗算を行えば、もともとの多項式関数から計算したときの値や、線形補間によって順次求めたときの値と、まったく同じものが得られます。 +

    +

    + では、なぜわざわざ行列を使うのでしょう？ + 行列表現を使うことによって、他の表現ではわからなかった関数の性質を発見できるようになります。ベジエ曲線は三角行列の形になり、その行列式は実際の座標の積に等しくなることがわかります。また、この行列には逆行列が存在しますが、これはさまざまな性質が満たされることを意味します。もっとも、疑問の中心は「それでなぜこれが役に立つのか？」という点でしょうが、「ただちには役に立たない」というのが回答です。しかしながら、この先に出てくる曲線のプロパティの中には、関数を操作して求めることも、また行列をうまいこと利用して求めることも、どちらでも可能な例があります。そしてときには、行列による手法の方が（劇的に）速くなる場合があるのです。 +

    +

    というわけで、現時点では「ベジエ曲線は行列で表現可能」ということだけを覚えて、次に進みましょう。

    +
    +
    +

    +
    前目次次
    + ド・カステリョのアルゴリズム +

    +

    + ベジエ曲線を描く場合は、tの値を0から1まで動かしながら重みつき基底関数を計算し、プロットに必要なx/yの値を求めます。しかし、曲線が複雑になればなるほど、計算コストがかかるようになってしまいます。そこでその代わりに、「ド・カステリョのアルゴリズム」を使って曲線を描くこともできます。こちらは幾何学的に曲線を描く方法で、実装も非常に簡単です。実際、鉛筆と定規を使って手描きすることもできるほど、とても簡単な方法なのです。 +

    +

    x/yの値を求めるために関数を使うのではなく、次のようにします。

    +
      +
    • tを（そのまま）比率として考えます。t=0は直線上の0%の位置、t=1は100%の位置です。
      • +
    • 曲線を定める点同士を結ぶように、それぞれ直線を引きます。n次の曲線であれば、n本の直線を引きます。
      • +
    • + 各直線上において、距離がtとなる点にそれぞれ印をつけます。例えばtが0.2なら、始点から20%、終点から80%の位置になります。 +
      • +
    • 今度は、それらの点同士を直線で結びます。n-1本の直線が得られます。
      • +
    • 各直線上において、距離がtとなる点にそれぞれ印をつけます。
      • +
    • それらの点同士を直線で結びます。n-2本の直線が得られます。
      • +
    • 印をつけ、直線で結び、印をつけ、……
      • +
    • 1本の直線になるまで繰り返します。その直線上のtの点は、元の曲線上でtとなる点に一致しています。
      • +
    +

    + 下の図にマウスを乗せると、この様子を実際に見ることができます。ド・カステリョのアルゴリズムによって曲線上の点を明示的に計算していますが、マウスを動かすと求める点が変わります。マウスカーソルを左から右へ（もちろん、右から左へでも）動かせば、このアルゴリズムによって曲線が生成される様子がわかります。 +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +

    ド・カステリョのアルゴリズムの実装方法

    +

    いま説明したアルゴリズムを実装すると、以下のようになります。

    -
    -
    -

    前目次次
    ド・カステリョのアルゴリズム

    -

    ベジエ曲線を描く場合は、tの値を0から1まで動かしながら重みつき基底関数を計算し、プロットに必要なx/yの値を求めます。しかし、曲線が複雑になればなるほど、計算コストがかかるようになってしまいます。そこでその代わりに、「ド・カステリョのアルゴリズム」を使って曲線を描くこともできます。こちらは幾何学的に曲線を描く方法で、実装も非常に簡単です。実際、鉛筆と定規を使って手描きすることもできるほど、とても簡単な方法なのです。

    -

    x/yの値を求めるために関数を使うのではなく、次のようにします。

    -
      -
    • tを（そのまま）比率として考えます。t=0は直線上の0%の位置、t=1は100%の位置です。
      • -
    • 曲線を定める点同士を結ぶように、それぞれ直線を引きます。n次の曲線であれば、n本の直線を引きます。
      • -
    • 各直線上において、距離がtとなる点にそれぞれ印をつけます。例えばtが0.2なら、始点から20%、終点から80%の位置になります。
      • -
    • 今度は、それらの点同士を直線で結びます。n-1本の直線が得られます。
      • -
    • 各直線上において、距離がtとなる点にそれぞれ印をつけます。
      • -
    • それらの点同士を直線で結びます。n-2本の直線が得られます。
      • -
    • 印をつけ、直線で結び、印をつけ、……
      • -
    • 1本の直線になるまで繰り返します。その直線上のtの点は、元の曲線上でtとなる点に一致しています。
      • -
    -

    下の図にマウスを乗せると、この様子を実際に見ることができます。ド・カステリョのアルゴリズムによって曲線上の点を明示的に計算していますが、マウスを動かすと求める点が変わります。マウスカーソルを左から右へ（もちろん、右から左へでも）動かせば、このアルゴリズムによって曲線が生成される様子がわかります。

    - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    ド・カステリョのアルゴリズムの実装方法

    -

    いま説明したアルゴリズムを実装すると、以下のようになります。

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    これで実装完了です。ただし、+演算子のオーバーロードなどという贅沢品はたいてい無いでしょうから、xやyの値を直接扱う場合のコードも示しておきます。

    +

    + これで実装完了です。ただし、+演算子のオーバーロードなどという贅沢品はたいてい無いでしょうから、xやyの値を直接扱う場合のコードも示しておきます。 +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    さて、これは何をしているのでしょう？関数に渡す点のリストが長さ1であれば、点を1つ描きます。それ以外であれば、比率tの位置の点（すなわち、さきほどの説明に出てきた「印」）のリストを作り、そしてこの新しいリストを引数にして関数を呼び出します。

    -
    +

    + さて、これは何をしているのでしょう？関数に渡す点のリストが長さ1であれば、点を1つ描きます。それ以外であれば、比率tの位置の点（すなわち、さきほどの説明に出てきた「印」）のリストを作り、そしてこの新しいリストを引数にして関数を呼び出します。 +

    + +
    +
    +

    +
    前目次次
    + 簡略化した描画 +

    +

    + 曲線を複数点で「サンプリング」し、さらにそれを直線で繫げてしまえば、描画の手順を簡略化することができます。単なる一連の直線、つまり「平坦」な線へと曲線を単純化するので、この処理は「平坦化」という名前で知られています。 +

    +

    + 例えば「X個の線分がほしい」場合には、分割数がそうなるようにサンプリング間隔を選び、曲線をサンプリングします。この方法の利点は速さです。曲線の座標を100個だの1000個だの計算するのではなく、ずっと少ない回数のサンプリングでも、十分きれいに見えるような曲線を作ることができるのです。欠点はもちろん、「本物の曲線」に比べて精度が損なわれてしまうことです。したがって、交点の検出や曲線の位置揃えを正しく行いたい場合には、平坦化した曲線は普通利用できません。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    前目次次
    簡略化した描画

    -

    曲線を複数点で「サンプリング」し、さらにそれを直線で繫げてしまえば、描画の手順を簡略化することができます。単なる一連の直線、つまり「平坦」な線へと曲線を単純化するので、この処理は「平坦化」という名前で知られています。

    -

    例えば「X個の線分がほしい」場合には、分割数がそうなるようにサンプリング間隔を選び、曲線をサンプリングします。この方法の利点は速さです。曲線の座標を100個だの1000個だの計算するのではなく、ずっと少ない回数のサンプリングでも、十分きれいに見えるような曲線を作ることができるのです。欠点はもちろん、「本物の曲線」に比べて精度が損なわれてしまうことです。したがって、交点の検出や曲線の位置揃えを正しく行いたい場合には、平坦化した曲線は普通利用できません。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    +

    + 2次ベジエ曲線も3次ベジエ曲線も、図をクリックして上下キーを押すと曲線の分割数が増減しますので、試してみてください。ある曲線では分割数が少なくてもうまくいきますが、曲線が複雑になればなるほど、曲率の変化を正確に捉えるためにはより多くの分割数が必要になることがわかります（3次ベジエ曲線で試してみてください）。 +

    +
    +

    曲線平坦化の実装方法

    +

    上でいま解説したアルゴリズムを使って実装するだけです。

    -

    2次ベジエ曲線も3次ベジエ曲線も、図をクリックして上下キーを押すと曲線の分割数が増減しますので、試してみてください。ある曲線では分割数が少なくてもうまくいきますが、曲線が複雑になればなるほど、曲率の変化を正確に捉えるためにはより多くの分割数が必要になることがわかります（3次ベジエ曲線で試してみてください）。

    -
    - -

    曲線平坦化の実装方法

    -

    上でいま解説したアルゴリズムを使って実装するだけです。

    - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return coordinates; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    これで完了です。実装できました。あとは、一連の直線で結果の「曲線」を描画するだけです。

    +

    これで完了です。実装できました。あとは、一連の直線で結果の「曲線」を描画するだけです。

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + coord = _coord + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    先頭の座標を参照点にしてスタートし、あとはそれぞれの点からその次の点へと、直線を引いていくだけです。

    -
    +

    先頭の座標を参照点にしてスタートし、あとはそれぞれの点からその次の点へと、直線を引いていくだけです。

    +
    +
    +
    +

    +
    前目次次
    + 曲線の分割 +

    +

    + ベジエ曲線を分割して、繫ぎ合わせたときに元に戻るような小さい2曲線にしたい場合にも、ド・カステリョのアルゴリズムを使えば、これに必要な点をすべて求めることができます。ある値tに対してド・カステリョの骨格を組み立てると、そのtで曲線を分割する際に必要になる点がすべて得られます。骨格内部の点のうち、曲線上の点から見て手前側にある点によって一方の曲線が定義され、向こう側にある点によってもう一方の曲線が定義されます。 +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +

    曲線分割の実装方法

    +

    ド・カステリョの関数の中に記録する処理を追加すれば、曲線の分割が実装できます。

    -
    -
    -

    前目次次
    曲線の分割

    -

    ベジエ曲線を分割して、繫ぎ合わせたときに元に戻るような小さい2曲線にしたい場合にも、ド・カステリョのアルゴリズムを使えば、これに必要な点をすべて求めることができます。ある値tに対してド・カステリョの骨格を組み立てると、そのtで曲線を分割する際に必要になる点がすべて得られます。骨格内部の点のうち、曲線上の点から見て手前側にある点によって一方の曲線が定義され、向こう側にある点によってもう一方の曲線が定義されます。

    - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    曲線分割の実装方法

    -

    ド・カステリョの関数の中に記録する処理を追加すれば、曲線の分割が実装できます。

    - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + -

    ある値tに対してこの関数を実行すると、leftとrightに新しい2曲線の座標が入ります。一方はtの「左」側、もう一方は「右」側の曲線です。この2曲線は元の曲線と同じ次数になり、また元の曲線とぴったり重なります。

    -
    - -
    -
    -

    前目次次
    行列による曲線の分割

    -

    曲線分割には、ベジエ曲線の行列表現を利用する方法もあります。行列についての節では、行列の乗算で曲線が表現できることを確認しました。特に2次・3次のベジエ曲線に関しては、それぞれ以下のような形になりました（読みやすさのため、ベジエの係数ベクトルを反転させています）。

    - - -

    ならびに

    - - -

    曲線をある点t = zで分割し、新しく2つの（自明ですが、より短い）ベジエ曲線を作ることを考えましょう。曲線の行列表現と線形代数を利用すると、この2つのベジエ曲線の座標を求めることができます。まず、実際の「曲線上の点」の情報を分解し、新しい行列の積のかたちにします。

    - - -

    ならびに

    - - -

    これらの行列をまとめて、仮に**[tの値たち] · [ベジエ行列] · [列ベクトル]**の形式にできたとしましょう。ただし、先頭2つの行列は変わらずそのままだとします。このとき、右端の列ベクトルは、前半部分すなわちt = 0からt = zを表す、新しいベジエ曲線の座標となります。結論からいうと、線形代数の簡単な規則を使えば、この変形は非常に容易です（そして、導出過程を気にしないのであれば、囲みの末尾まで飛ばして結果に行ってもかまいません！）。

    -
    - -

    新しい凸包の座標の導出

    -

    曲線を分割して2つの部分を得るためには、いくつかの段階を経る必要があります。曲線の次数が高くなるほど手間がかかるようになりますので、まずは2次の曲線で見てみましょう。

    - - - - - - - - -

    [M · M-1]は単位行列なので、このような操作ができるのです（これは微積分でいえば、なにかにx/xを掛けるようなものです。関数自体はなにも変わりませんが、解きやすいかたちに変形したり、別のかたちに分解したりといったことが可能になります）。この行列を掛けると、式全体としてはなにも変わりませんが、[なにか · M]という行列の並びを[M · なにか]という並びに変えることができます。そして、これが大きな違いを生み出します。[M-1 · Z · M]の値が分かれば、それを座標に掛け合わせることによって、2次ベジエ曲線の正しい行列表現（すなわち[T · M · P]）と、t = 0からt = zまでの曲線を表す座標の組とが得られます。では、計算してみましょう。

    - - -

    いいですね！これで、新しい2次ベジエ曲線が得られます。

    - - - - - - -

    すばらしい。t = 0からt = zの部分曲線を求める場合、始点の座標はそのままになります（もっともです）。制御点は、元々の制御点と始点を、比率zで混ぜ合わせたものになります。そして不思議なことに、新たな終点は2次のベルンシュタイン多項式に似た混ぜ合わせになります。これらの新しい座標は、とても簡単に直接計算ができるのです！

    -

    もちろん、これは2曲線のうちの片方にすぎません。t = zからt = 1の部分を得るためには、同様の計算をする必要があります。まず、今さっき行ったのは、一般の区間[0,z]についての計算でした。これは0があるので簡単な形になっていましたが、実際には、次の式を計算していたということがわかります。

    - - - - -

    区間[z,1]を求めたい場合は、かわりに次のような計算になります。

    - - - - -

    先ほどと同じ手法を使い、[なにか · M]を[M · なにか]に変えます。

    - - -

    よって、後半部分の曲線は結局のところ以下のようになります。

    - - - - - - -

    おみごと。先ほどと同じようになっていることがわかります。終点の座標はそのままで（もっともです）、制御点は、元々の制御点と終点を比率zで混ぜ合わせたものになります。そして不思議なことに、新たな始点は2次のベルンシュタイン多項式に似た混ぜ合わせになります。ただし、(1-z)の代わりに(z-1)になっています。これらの新しい座標も、とても簡単に直接計算ができるのです！

    -
    + +

    + おみごと。先ほどと同じようになっていることがわかります。終点の座標はそのままで（もっともです）、制御点は、元々の制御点と終点を比率zで混ぜ合わせたものになります。そして不思議なことに、新たな始点は2次のベルンシュタイン多項式に似た混ぜ合わせになります。ただし、(1-z)の代わりに(z-1)になっています。これらの新しい座標も、とても簡単に直接計算ができるのです！ +

    + -

    というわけで、ド・カステリョのアルゴリズムではなく線形代数の方を使うと、どのような2次ベジエ曲線でもある値t = zで分割すれば2つのベジエ曲線となり、しかもその座標は簡単に求められる、ということがわかりました。

    - - -

    および

    - - -

    3次の曲線についても同様です。ただし、実際の導出はあなたにとっておきますので（自力で書き下してみてください）、新しい座標の組の結果を示すだけにします。

    - - -

    および

    - - -

    さて、これらの行列を見るに、後半部分の曲線の行列は本当に計算する必要があったのでしょうか？いえ、ありませんでした。片方の行列が得られれば、実はもう一方の行列も暗に得られたことになります。まず、行列*Q**の各行の値を右側に寄せ、右側にあった0を左側に押しのけます。次に行列を上下に反転させます。これでたちまちQ'***が「計算」できるのです。

    -

    この方法で曲線の分割を実装すれば、再帰が少なくて済みます。また、数値のキャッシュを利用した単純な演算になるので、再帰の計算コストが大きいシステムにおいては、コストが抑えられるかもしれません。行列の乗算に適したデバイスで計算を行えば、ド・カステリョのアルゴリズムに比べてかなり速くなるでしょう。

    - -
    -
    -

    前目次次
    Lowering and elevating curve order

    -

    One interesting property of Bézier curves is that an nth order curve can always be perfectly represented by an (n+1)th order curve, by giving the higher-order curve specific control points.

    -

    If we have a curve with three points, then we can create a curve with four points that exactly reproduces the original curve. First, we give it the same start and end points, and for its two control points we pick "1/3rd start + 2/3rd control" and "2/3rd control + 1/3rd end". Now we have exactly the same curve as before, except represented as a cubic curve rather than a quadratic curve.

    -

    The general rule for raising an nth order curve to an (n+1)th order curve is as follows (observing that the start and end weights are the same as the start and end weights for the old curve):

    - - -

    However, this rule also has as direct consequence that you cannot generally safely lower a curve from nth order to (n-1)th order, because the control points cannot be "pulled apart" cleanly. We can try to, but the resulting curve will not be identical to the original, and may in fact look completely different.

    -

    However, there is a surprisingly good way to ensure that a lower order curve looks "as close as reasonably possible" to the original curve: we can optimise the "least-squares distance" between the original curve and the lower order curve, in a single operation (also explained over on Sirver's Castle). However, to use it, we'll need to do some calculus work and then switch over to linear algebra. As mentioned in the section on matrix representations, some things can be done much more easily with matrices than with calculus functions, and this is one of those things. So... let's go!

    -

    We start by taking the standard Bézier function, and condensing it a little:

    - - -

    Then, we apply one of those silly (actually, super useful) calculus tricks: since our t value is always between zero and one (inclusive), we know that (1-t) plus t always sums to 1. As such, we can express any value as a sum of t and 1-t:

    - - -

    So, with that seemingly trivial observation, we rewrite that Bézier function by splitting it up into a sum of a (1-t) and t component:

    - - -

    So far so good. Now, to see why we did this, let's write out the (1-t) and t parts, and see what that gives us. I promise, it's about to make sense. We start with (1-t):

    - - -

    So by using this seemingly silly trick, we can suddenly express part of our nth order Bézier function in terms of an (n+1)th order Bézier function. And that sounds a lot like raising the curve order! Of course we need to be able to repeat that trick for the t part, but that's not a problem:

    - - -

    So, with both of those changed from an order n expression to an order (n+1) expression, we can put them back together again. Now, where the order n function had a summation from 0 to n, the order n+1 function uses a summation from 0 to n+1, but this shouldn't be a problem as long as we can add some new terms that "contribute nothing". In the next section on derivatives, there is a discussion about why "higher terms than there is a binomial for" and "lower than zero terms" both "contribute nothing". So as long as we can add terms that have the same form as the terms we need, we can just include them in the summation, they'll sit there and do nothing, and the resulting function stays identical to the lower order curve.

    -

    Let's do this:

    - - -

    And this is where we switch over from calculus to linear algebra, and matrices: we can now express this relation between Bézier(n,t) and Bézier(n+1,t) as a very simple matrix multiplication:

    - - -

    where the matrix M is an n+1 by n matrix, and looks like:

    - - -

    That might look unwieldy, but it's really just a mostly-zeroes matrix, with a very simply fraction on the diagonal, and an even simpler fraction to the left of it. Multiplying a list of coordinates with this matrix means we can plug the resulting transformed coordinates into the one-order-higher function and get an identical looking curve.

    -

    Not too bad!

    -

    Equally interesting, though, is that with this matrix operation established, we can now use an incredibly powerful and ridiculously simple way to find out a "best fit" way to reverse the operation, called the normal equation. What it does is minimize the sum of the square differences between one set of values and another set of values. Specifically, if we can express that as some function A x = b, we can use it. And as it so happens, that's exactly what we're dealing with, so:

    - - -

    The steps taken here are:

    -
      -
    1. We have a function in a form that the normal equation can be used with, so
      2. -
    2. apply the normal equation!
      3. -
    3. Then, we want to end up with just Bn on the left, so we start by left-multiply both sides such that we'll end up with lots of stuff on the left that simplified to "a factor 1", which in matrix maths is the identity matrix.
      4. -
    4. In fact, by left-multiplying with the inverse of what was already there, we've effectively "nullified" (but really, one-inified) that big, unwieldy block into the identity matrix I. So we substitute the mess with I, and then
      5. -
    5. because multiplication with the identity matrix does nothing (like multiplying by 1 does nothing in regular algebra), we just drop it.
      6. -
    -

    And we're done: we now have an expression that lets us approximate an n+1th order curve with a lower nth order curve. It won't be an exact fit, but it's definitely a best approximation. So, let's implement these rules for raising and lowering curve order to a (semi) random curve, using the following graphic. Select the sketch, which has movable control points, and press your up and down arrow keys to raise or lower the curve order.

    - - - Scripts are disabled. Showing fallback image. - - - - - - - -
    -
    -

    前目次次
    Derivatives

    -

    There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straightforward, although we do need a bit of math.

    -

    First, let's look at the derivative rule for Bézier curves, which is:

    - - -

    which we can also write (observing that b in this formula is the same as our w weights, and that n times a summation is the same as a summation where each term is multiplied by n) as:

    - - -

    Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi). So for a 3rd degree curve, with four weights, the derivative has three new weights: w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2).

    -
    - -

    "Slow down, why is that true?"

    -

    Sometimes just being told "this is the derivative" is nice, but you might want to see why this is indeed the case. As such, let's have a look at the proof for this derivative. First off, the weights are independent of the full Bézier function, so the derivative involves only the derivative of the polynomial basis function. So, let's find that:

    - - -

    Applying the product and chain rules gives us:

    - - -

    Which is hard to work with, so let's expand that properly:

    - - -

    Now, the trick is to turn this expression into something that has binomial coefficients again, so we want to end up with things that look like "x! over y!(x-y)!". If we can do that in a way that involves terms of n-1 and k-1, we'll be on the right track.

    - - -

    And that's the first part done: the two components inside the parentheses are actually regular, lower-order Bézier expressions:

    - - -

    Now to apply this to our weighted Bézier curves. We'll write out the plain curve formula that we saw earlier, and then work our way through to its derivative:

    - - -

    If we expand this (with some color to show how terms line up), and reorder the terms by increasing values for k we see the following:

    - - -

    Two of these terms fall way: the first term falls away because there is no -1st term in a summation. As such, it always contributes "nothing", so we can safely completely ignore it for the purpose of finding the derivative function. The other term is the very last term in this expansion: one involving Bn-1,n. This term would have a binomial coefficient of [i choose i+1], which is a non-existent binomial coefficient. Again, this term would contribute "nothing", so we can ignore it, too. This means we're left with:

    - - -

    And that's just a summation of lower order curves:

    - - -

    We can rewrite this as a normal summation, and we're done:

    - - -
    - -

    Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for Bézier curves, and then the derivative:

    - + +

    And that's just a summation of lower order curves:

    + + +

    We can rewrite this as a normal summation, and we're done:

    + + + +

    + Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for + Bézier curves, and then the derivative: +

    + - - - -

    What are the differences? In terms of the actual Bézier curve, virtually nothing! We lowered the order (rather than n, it's now n-1), but it's still the same Bézier function. The only real difference is in how the weights change when we derive the curve's function. If we have four points A, B, C, and D, then the derivative will have three points, the second derivative two, and the third derivative one:

    - - -

    We can keep performing this trick for as long as we have more than one weight. Once we have one weight left, the next step will see k = 0, and the result of our "Bézier function" summation is zero, because we're not adding anything at all. As such, a quadratic curve has no second derivative, a cubic curve has no third derivative, and generalized: an nth order curve has n-1 (meaningful) derivatives, with any further derivative being zero.

    - -
    -
    -

    前目次次
    Tangents and normals

    -

    If you want to move objects along a curve, or "away from" a curve, the two vectors you're most interested in are the tangent vector and normal vector for curve points. These are actually really easy to find. For moving and orienting along a curve, we use the tangent, which indicates the direction of travel at specific points, and is literally just the first derivative of our curve:

    - - -

    This gives us the directional vector we want. We can normalize it to give us uniform directional vectors (having a length of 1.0) at each point, and then do whatever it is we want to do based on those directions:

    - - -

    The tangent is very useful for moving along a line, but what if we want to move away from the curve instead, perpendicular to the curve at some point t? In that case we want the normal vector. This vector runs at a right angle to the direction of the curve, and is typically of length 1.0, so all we have to do is rotate the normalized directional vector and we're done:

    - - -
    - -

    Rotating coordinates is actually very easy, if you know the rule for it. You might find it explained as "applying a rotation matrix, which is what we'll look at here, too. Essentially, the idea is to take the circles over which we can rotate, and simply "sliding the coordinates" over these circles by the desired -angle. If we want a quarter circle turn, we take the coordinate, slide it along the circle by a quarter turn, and done.

    -

    To turn any point (x,y) into a rotated point (x',y') (over 0,0) by some angle φ, we apply this nice and easy computation:

    - - -

    Which is the "long" version of the following matrix transformation:

    - - -

    And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1.

    -

    But why does this work? Why this matrix multiplication? Wikipedia (technically, Thomas Herter and Klaus Lott) tells us that a rotation matrix can be -treated as a sequence of three (elementary) shear operations. When we combine this into a single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. DataGenetics have an excellent article about this very thing: it's really quite cool, and I strongly recommend taking a quick break from this primer to read that article.

    -
    + +

    + And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even + easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1. +

    +

    + But why does this work? Why this matrix multiplication? + Wikipedia (technically, Thomas Herter and Klaus + Lott) tells us that a rotation matrix can be treated as a sequence of three (elementary) shear operations. When we combine this into a + single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. + DataGenetics have an excellent article about this very thing: it's + really quite cool, and I strongly recommend taking a quick break from this primer to read that article. +

    + -

    The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant).

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    + The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the + normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant). +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    +
    +
    +

    +
    前目次次
    + Working with 3D normals +

    +

    + Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things + this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but + for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, + it's not "super hard", but there are more steps involved and we should have a look at those. +

    +

    + Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. + However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, + not a single vector, so we basically need to define what "the" normal is in the 3D case. +

    +

    + The "naïve" approach is to construct what is known as the + Frenet normal, where we follow a simple recipe that works in + many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are + perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each + point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the + local tangent, as well as the tangents "right next to it". +

    +

    + Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the + vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know + lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same + logic we used for the 2D case: "just rotate it 90 degrees". +

    +

    So let's do that! And in a twist surprise, we can do this in four lines:

    +
      +
    • a = normalize(B'(t))
      • +
    • b = normalize(a + B''(t))
      • +
    • r = normalize(b × a)
      • +
    • normal = normalize(r × a)
      • +
    +

    Let's unpack that a little:

    +
      +
    • + We start by taking the normalized vector for the derivative at some point on the + curve. We normalize it so the maths is less work. Less work is good. +
      • +
    • + Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and + just had the same derivative and second derivative from that point on. +
      • +
    • + This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which + means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the + cross product. (Note that while that operation uses the × operator, it's most + definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent + a quarter circle to get our normal, just like we did in the 2D case. +
      • +
    • + Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal + vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second + time, and immediately get our normal vector. +
      • +
    +

    + And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can + move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor + position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the + curve" between t=0.65 and t=0.75... Why is it doing that? +

    +

    + As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are + "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute + normals that just... look good. +

    +

    Thankfully, Frenet normals are not our only option.

    +

    + Another option is to take a slightly more algorithmic approach and compute a form of + Rotation Minimising Frame + (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational + axis, and the normal vector, centered on an on-curve point. +

    +

    + These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we + could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at + the same time that you're building lookup tables for your curve. +

    +

    + The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by + applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are: +

    +
      +
    • Take a point on the curve for which we know the RM frame already,
      • +
    • take a next point on the curve for which we don't know the RM frame yet, and
      • +
    • + reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous + points as a "mirror". +
      • +
    • + This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal + that's slightly off-kilter, so: +
      • +
    • + reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror". +
      • +
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • +
    +

    So, let's write some code for that!

    +
    +

    Implementing Rotation Minimising Frames

    +

    + We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it + yields a frame with properties: +

    -
    -
    -

    前目次次
    Working with 3D normals

    -

    Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, it's not "super hard", but there are more steps involved and we should have a look at those.

    -

    Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, not a single vector, so we basically need to define what "the" normal is in the 3D case.

    -

    The "naïve" approach is to construct what is known as the Frenet normal, where we follow a simple recipe that works in many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the local tangent, as well as the tangents "right next to it".

    -

    Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same logic we used for the 2D case: "just rotate it 90 degrees".

    -

    So let's do that! And in a twist surprise, we can do this in four lines:

    -
      -
    • a = normalize(B'(t))
      • -
    • b = normalize(a + B''(t))
      • -
    • r = normalize(b × a)
      • -
    • normal = normalize(r × a)
      • -
    -

    Let's unpack that a little:

    -
      -
    • We start by taking the normalized vector for the derivative at some point on the curve. We normalize it so the maths is less work. Less work is good.
      • -
    • Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on.
      • -
    • This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the cross product. (Note that while that operation uses the × operator, it's most definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent a quarter circle to get our normal, just like we did in the 2D case.
      • -
    • Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second time, and immediately get our normal vector.
      • -
    -

    And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the curve" between t=0.65 and t=0.75... Why is it doing that?

    -

    As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute normals that just... look good.

    -

    Thankfully, Frenet normals are not our only option.

    -

    Another option is to take a slightly more algorithmic approach and compute a form of Rotation Minimising Frame (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational axis, and the normal vector, centered on an on-curve point.

    -

    These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at the same time that you're building lookup tables for your curve.

    -

    The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are:

    -
      -
    • Take a point on the curve for which we know the RM frame already,
      • -
    • take a next point on the curve for which we don't know the RM frame yet, and
      • -
    • reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror".
      • -
    • This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter, so:
      • -
    • reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
      • -
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • -
    -

    So, let's write some code for that!

    -
    - -

    Implementing Rotation Minimising Frames

    -

    We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it yields a frame with properties:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + -

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    +

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    + frames.add(x1) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + -

    Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more code to get much better looking normals.

    -
    +

    + Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more + code to get much better looking normals. +

    + -

    Speaking of better looking, what does this actually look like? Let's revisit that earlier curve, but this time use rotation minimising frames rather than Frenet frames:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    That looks so much better!

    -

    For those reading along with the code: we don't even strictly speaking need a Frenet frame to start with: we could, for instance, treat the z-axis as our initial axis of rotation, so that our initial normal is (0,0,1) × tangent, and then take things from there, but having that initial "mathematically correct" frame so that the initial normal seems to line up based on the curve's orientation in 3D space is just nice.

    - -
    -
    -

    前目次次
    Component functions

    -

    One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that function, but this poses a problem, since our function is parametric: every axis has its own function.

    -

    The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.

    -

    Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis value, respectively.

    -

    If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a change in the right graph.

    - - - Scripts are disabled. Showing fallback image. - - - -

     

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目次次
    Finding extremities: root finding

    -

    Now that we understand (well, superficially anyway) the component functions, we can find the extremities of our Bézier curve by finding maxima and minima on the component functions, by solving the equation B'(t) = 0. We've already seen that the derivative of a Bézier curve is a simpler Bézier curve, but how do we solve the equality? Fairly easily, actually, until our derivatives are 4th order or higher... then things get really hard. But let's start simple:

    -

    Quadratic curves: linear derivatives.

    -

    The derivative of a quadratic Bézier curve is a linear Bézier curve, interpolating between just two terms, which means finding the solution for "where is this line 0" is effectively trivial by rewriting it to a function of t and solving. First we turn our quadratic Bézier function into a linear one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then we turn this into our solution for t using basic arithmetics:

    - - -

    Done.

    -

    Although with the caveat that if b-a is zero, there is no solution and we probably shouldn't try to perform that division.

    -

    Cubic curves: the quadratic formula.

    -

    The derivative of a cubic Bézier curve is a quadratic Bézier curve, and finding the roots for a quadratic polynomial means we can apply the Quadratic formula. If you've seen it before, you'll remember it, and if you haven't, it looks like this:

    - - -

    So, if we can rewrite the Bézier component function as a plain polynomial, we're done: we just plug in the values into the quadratic formula, check if that square root is negative or not (if it is, there are no roots) and then just compute the two values that come out (because of that plus/minus sign we get two). Any value between 0 and 1 is a root that matters for Bézier curves, anything below or above that is irrelevant (because Bézier curves are only defined over the interval [0,1]). So, how do we convert?

    -

    First we turn our cubic Bézier function into a quadratic one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then, using these v values, we can find out what our a, b, and c should be:

    - - -

    This gives us three coefficients {a, b, c} that are expressed in terms of v values, where the v values are expressions of our original coordinate values, so we can do some substitution to get:

    - - -

    Easy-peasy. We can now almost trivially find the roots by plugging those values into the quadratic formula.

    -

    And as a cubic curve, there is also a meaningful second derivative, which we can compute by simple taking the derivative of the derivative.

    -

    Quartic curves: Cardano's algorithm.

    -

    We haven't really looked at them before now, but the next step up would be a Quartic curve, a fourth degree Bézier curve. As expected, these have a derivative that is a cubic function, and now things get much harder. Cubic functions don't have a "simple" rule to find their roots, like the quadratic formula, and instead require quite a bit of rewriting to a form that we can even start to try to solve.

    -

    Back in the 16th century, before Bézier curves were a thing, and even before calculus itself was a thing, Gerolamo Cardano figured out that even if the general cubic function is really hard to solve, it can be rewritten to a form for which finding the roots is "easier" (even if not "easy"):

    - - -

    We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather than three: this makes things considerably easier to solve because it lets us use regular calculus to find the values that satisfy the equation.

    -

    Now, there is one small hitch: as a cubic function, the solutions may be complex numbers rather than plain numbers... And Cardano realised this, centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number.

    -

    So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at Ken J. Ward's page for solving the cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a read-through.

    -
    + +

    + We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather + than three: this makes things considerably easier to solve because it lets us use + regular calculus to find the values that satisfy the equation. +

    +

    + Now, there is one small hitch: as a cubic function, the solutions may be + complex numbers rather than plain numbers... And Cardano realised this, + centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these + numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we + have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're + going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number. +

    +

    + So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at + Ken J. Ward's page for solving the + cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the + complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a + read-through. +

    +
    +

    Implementing Cardano's algorithm for finding all real roots

    +

    + The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" + number space (denoted with ℝ), this is perfectly fine in the + "complex" number space (denoted with ℂ). And, as it so happens, Cardano is + also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm! +

    -

    Implementing Cardano's algorithm for finding all real roots

    -

    The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" number space (denoted with ℝ), this is perfectly fine in the "complex" number space (denoted with ℂ). And, as it so happens, Cardano is also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm!

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    60
    61
    62
    63
    64
    65
    66
    67
    68
    69
    70
    71
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + + 37 + + + 38 + + + 39 + + + 40 + + + 41 + + + 42 + + + 43 + + + 44 + + + 45 + + + 46 + + + 47 + + + 48 + + + 49 + + + 50 + + + 51 + + + 52 + + + 53 + + + 54 + + + 55 + + + 56 + + + 57 + + + 58 + + + 59 + + + 60 + + + 61 + + + 62 + + + 63 + + + 64 + + + 65 + + + 66 + + + 67 + + + 68 + + + 69 + + + 70 + + + 71 + + + 72 + + + 73 + + + 74 + + + 75 + + + 76 + + + 77 + + + 78 + + + 79 + + + 80 + + + 81 + + +
    -
    - -

    And that's it. The maths is complicated, but the code is pretty much just "follow the maths, while caching as many values as we can to prevent recomputing things as much as possible" and now we have a way to find all roots for a cubic function and can just move on with using that to find extremities of our curves.

    -

    And of course, as a quartic curve also has meaningful second and third derivatives, we can quite easily compute those by using the derivative of the derivative (of the derivative), just as for cubic curves.

    -

    Quintic and higher order curves: finding numerical solutions

    -

    And this is where thing stop, because we cannot find the roots for polynomials of degree 5 or higher using algebra (a fact known as the Abel–Ruffini theorem). Instead, for occasions like these, where algebra simply cannot yield an answer, we turn to numerical analysis.

    -

    That's a fancy term for saying "rather than trying to find exact answers by manipulating symbols, find approximate answers by describing the underlying process as a combination of steps, each of which can be assigned a number via symbolic manipulation". For example, trying to mathematically compute how much water fits in a completely crazy three dimensional shape is very hard, even if it got you the perfect, precise answer. A much easier approach, which would be less perfect but still entirely useful, would be to just grab a buck and start filling the shape until it was full: just count the number of buckets of water you used. And if we want a more precise answer, we can use smaller buckets.

    -

    So that's what we're going to do here, too: we're going to treat the problem as a sequence of steps, and the smaller we can make each step, the closer we'll get to that "perfect, precise" answer. And as it turns out, there is a really nice numerical root-finding algorithm, called the Newton-Raphson root finding method (yes, after that Newton), which we can make use of. The Newton-Raphson approach consists of taking our impossible-to-solve function f(x), picking some initial value x (literally any value will do), and calculating f(x). We can think of that value as the "height" of the function at x. If that height is zero, we're done, we have found a root. If it isn't, we calculate the tangent line at f(x) and calculate at which x value its height is zero (which we've already seen is very easy). That will give us a new x and we repeat the process until we find a root.

    -

    Mathematically, this means that for some x, at step n=1, we perform the following calculation until fy(x) is zero, so that the next t is the same as the one we already have:

    - - -

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    -

    Now, this works well only if we can pick good starting points, and our curve is continuously differentiable and doesn't have oscillations. Glossing over the exact meaning of those terms, the curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: which starting points do we pick?

    -

    As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as high as you'll ever need to go.

    -

    In conclusion:

    -

    So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red:

    - - - Scripts are disabled. Showing fallback image. - - - -

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    - - - Scripts are disabled. Showing fallback image. - - - + +

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    +

    + Now, this works well only if we can pick good starting points, and our curve is + continuously differentiable and doesn't have + oscillations. Glossing over the exact meaning of those terms, the + curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: + which starting points do we pick? +

    +

    + As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from + t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may + pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: + there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order + curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as + high as you'll ever need to go. +

    +

    In conclusion:

    +

    + So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those + roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red: +

    + + + Scripts are disabled. Showing fallback image. + + +

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    + + + Scripts are disabled. Showing fallback image. + + +
    +
    +

    +
    前目次次
    + Bounding boxes +

    +

    + If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four + values we need to box in our curve: +

    +

    Computing the bounding box for a Bézier curve:

    +
      +
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. +
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. +
    3. + Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original + functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct. +
      4. +
    +

    + Applying this approach to our previous root finding, we get the following + axis-aligned bounding boxes (with all curve extremity points + shown on the curve): +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -
    -
    -

    前目次次
    Bounding boxes

    -

    If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four values we need to box in our curve:

    -

    Computing the bounding box for a Bézier curve:

    -
      -
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. -
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. -
    3. Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct.
      4. -
    -

    Applying this approach to our previous root finding, we get the following axis-aligned bounding boxes (with all curve extremity points shown on the curve):

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    We can construct even nicer boxes by aligning them along our curve, rather than along the x- and y-axis, but in order to do so we first need to look at how aligning works.

    - -
    -
    -

    前目次次
    Aligning curves

    -

    While there are an incredible number of curves we can define by varying the x- and y-coordinates for the control points, not all curves are actually distinct. For instance, if we define a curve, and then rotate it 90 degrees, it's still the same curve, and we'll find its extremities in the same spots, just at different draw coordinates. As such, one way to make sure we're working with a "unique" curve is to "axis-align" it.

    -

    Aligning also simplifies a curve's functions. We can translate (move) the curve so that the first point lies on (0,0), which turns our n term polynomial functions into n-1 term functions. The order stays the same, but we have less terms. Then, we can rotate the curves so that the last point always lies on the x-axis, too, making its coordinate (...,0). This further simplifies the function for the y-component to an n-2 term function. For instance, if we have a cubic curve such as this:

    - - -

    Then translating it so that the first coordinate lies on (0,0), moving all x coordinates by -120, and all y coordinates by -160, gives us:

    - - -

    If we then rotate the curve so that its end point lies on the x-axis, the coordinates (integer-rounded for illustrative purposes here) become:

    - - -

    If we drop all the zero-terms, this gives us:

    - - -

    We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae:

    - - - Scripts are disabled. Showing fallback image. - - - -

     

    - - - Scripts are disabled. Showing fallback image. - - - + +

    + We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning + our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae: +

    + + + Scripts are disabled. Showing fallback image. + + +

     

    + + + Scripts are disabled. Showing fallback image. + + +
    +
    +

    +
    前目次次
    + Tight bounding boxes +

    +

    + With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, + recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding + box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T. +

    +

    We now have nice tight bounding boxes for our curves:

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -
    -
    -

    前目次次
    Tight bounding boxes

    -

    With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T.

    -

    We now have nice tight bounding boxes for our curves:

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    These are, strictly speaking, not necessarily the tightest possible bounding boxes. It is possible to compute the optimal bounding box by determining which spanning lines we need to effect a minimal box area, but because of the parametric nature of Bézier curves this is actually a rather costly operation, and the gain in bounding precision is often not worth it.

    - -
    -
    -

    前目次次
    Curve inflections

    -

    Now that we know how to align a curve, there's one more thing we can calculate: inflection points. Imagine we have a variable size circle that we can slide up against our curve. We place it against the curve and adjust its radius so that where it touches the curve, the curvatures of the curve and the circle are the same, and then we start to slide the circle along the curve - for quadratic curves, we can always do this without the circle behaving oddly: we might have to change the radius of the circle as we slide it along, but it'll always sit against the same side of the curve.

    -

    But what happens with cubic curves? Imagine we have an S curve and we place our circle at the start of the curve, and start sliding it along. For a while we can simply adjust the radius and things will be fine, but once we get to the midpoint of that S, something odd happens: the circle "flips" from one side of the curve to the other side, in order for the curvatures to keep matching. This is called an inflection, and we can find out where those happen relatively easily.

    -

    What we need to do is solve a simple equation:

    - - -

    What we're saying here is that given the curvature function C(t), we want to know for which values of t this function is zero, meaning there is no "curvature", which will be exactly at the point between our circle being on one side of the curve, and our circle being on the other side of the curve. So what does C(t) look like? Actually something that seems not too hard:

    - - -

    The function C(t) is the cross product between the first and second derivative functions for the parametric dimensions of our curve. And, as already shown, derivatives of Bézier curves are just simpler Bézier curves, with very easy to compute new coefficients, so this should be pretty easy.

    -

    However as we've seen in the section on aligning, aligning lets us simplify things a lot, by completely removing the contributions of the first coordinate from most mathematical evaluations, and removing the last y coordinate as well by virtue of the last point lying on the x-axis. So, while we can evaluate C(t) = 0 for our curve, it'll be much easier to first axis-align the curve and then evaluating the curvature function.

    -
    - -

    Let's derive the full formula anyway

    -

    Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:

    - - -

    And of course the same functions for y:

    - - -

    Asking a computer to now compose the C(t) function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:

    - - -

    That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all disappear if we axis-align our curve, which is why aligning is a great idea.

    -
    + +

    + That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all + disappear if we axis-align our curve, which is why aligning is a great idea. +

    + -

    Aligning our curve so that three of the eight coefficients become zero, and observing that scale does not affect finding t values, we end up with the following simple term function for C(t):

    - - -

    That's a lot easier to work with: we see a fair number of terms that we can compute and then cache, giving us the following simplification:

    - - -

    This is a plain quadratic curve, and we know how to solve C(t) = 0; we use the quadratic formula:

    - - -

    We can easily compute this value if the discriminator isn't a negative number (because we only want real roots, not complex roots), and if x is not zero, because divisions by zero are rather useless.

    -

    Taking that into account, we compute t, we disregard any t value that isn't in the Bézier interval [0,1], and we now know at which t value(s) our curve will inflect.

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目次次
    The canonical form (for cubic curves)

    -

    While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?

    -

    As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper "A Geometric Characterization of Parametric Cubic curves". It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?

    -

    The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown:

    - - - Scripts are disabled. Showing fallback image. - - - -

    This is a fairly funky image, so let's see what the various parts of it mean...

    -

    We see the three fixed points at (0,0), (0,1) and (1,1). The various regions and boundaries indicate what property the original curve will have, if the fourth point is in/on that region or boundary. Specifically, if the fourth point is...

    -
      -

    1. ...anywhere inside the red zone, but not on its boundaries, the curve will be self-intersecting (yielding a loop). We won't know where it self-intersects (in terms of t values), but we are guaranteed that it does.

      -
      2. -

    2. ...on the left (red) edge of the red zone, the curve will have a cusp. We again don't know where, but we know there is one. This edge is described by the function:

      - - -
      3. -

    3. ...on the almost circular, lower right (pink) edge, the curve's end point touches the curve, forming a loop. This edge is described by the function:

      - - -
      4. -

    4. ...on the top (blue) edge, the curve's start point touches the curve, forming a loop. This edge is described by the function:

      - - -
      5. -

    5. ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

      -
      6. -

    6. ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two inflections (switching from concave to convex and then back again, or from convex to concave and then back again).

      -
      7. -

    7. ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

      -
      8. -
    -

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    -
    + + +
  • +

    ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

    +
    • +
  • +

    + ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two + inflections (switching from concave to convex and then back again, or from convex to concave and then back again). +

    +
    • +

  • ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

    • + +

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    +
    +

    Wait, where do those lines come from?

    +

    + Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, + and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield + formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic + expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the + properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval. +

    +

    + The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general + cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to + it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general + curve does over the interval t=0 to t=1. +

    +

    + For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you + can probably follow along with this paper, although you might have to read it a few times before all the bits "click". +

    +
    -

    Wait, where do those lines come from?

    -

    Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval.

    -

    The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general curve does over the interval t=0 to t=1.

    -

    For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you can probably follow along with this paper, although you might have to read it a few times before all the bits "click".

    -
    - -

    So now the question becomes: how do we manipulate our curve so that it fits this canonical form, with three fixed points, and one "free" point? Enter linear algebra. Don't worry, I'll be doing all the math for you, as well as show you what the effect is on our curves, but basically we're going to be using linear algebra, rather than calculus, because "it's way easier". Sometimes a calculus approach is very hard to work with, when the equivalent geometrical solution is super obvious.

    -

    The approach is going to start with a curve that doesn't have all-colinear points (so we need to make sure the points don't all fall on a straight line), and then applying three graphics operations that you will probably have heard of: translation (moving all points by some fixed x- and y-distance), scaling (multiplying all points by some x and y scale factor), and shearing (an operation that turns rectangles into parallelograms).

    -

    Step 1: we translate any curve by -p1.x and -p1.y, so that the curve starts at (0,0). We're going to make use of an interesting trick here, by pretending our 2D coordinates are 3D, with the z coordinate simply always being 1. This is an old trick in graphics to overcome the limitations of 2D transformations: without it, we can only turn (x,y) coordinates into new coordinates of the form (ax + by, cx + dy), which means we can't do translation, since that requires we end up with some kind of (x + a, y + b). If we add a bogus z coordinate that is always 1, then we can suddenly add arbitrary values. For example:

    - - -

    Sweet! z stays 1, so we can effectively ignore it entirely, but we added some plain values to our x and y coordinates. So, if we want to subtract p1.x and p1.y, we use:

    - - -

    Running all our coordinates through this transformation gives a new set of coordinates, let's call those U, where the first coordinate lies on (0,0), and the rest is still somewhat free. Our next job is to make sure point 2 ends up lying on the x=0 line, so what we want is a transformation matrix that, when we run it, subtracts x from whatever x we currently have. This is called shearing, and the typical x-shear matrix and its transformation looks like this:

    - - -

    So we want some shearing value that, when multiplied by y, yields -x, so our x coordinate becomes zero. That value is simply -x/y, because *-x/y * y = -x*. Done:

    - - -

    Now, running this on all our points generates a new set of coordinates, let's call those V, which now have point 1 on (0,0) and point 2 on (0, some-value), and we wanted it at (0,1), so we need to do some scaling to make sure it ends up at (0,1). Additionally, we want point 3 to end up on (1,1), so we can also scale x to make sure its x-coordinate will be 1 after we run the transform. That means we'll be x-scaling by 1/point3x, and y-scaling by point2y. This is really easy:

    - - -

    Then, finally, this generates a new set of coordinates, let's call those W, of which point 1 lies on (0,0), point 2 lies on (0,1), and point three lies on (1, ...) so all that's left is to make sure point 3 ends up at (1,1) - but we can't scale! Point 2 is already in the right place, and y-scaling would move it out of (0,1) again, so our only option is to y-shear point three, just like how we x-sheared point 2 earlier. In this case, we do the same trick, but with y/x rather than x/y because we're not x-shearing but y-shearing. Additionally, we don't actually want to end up at zero (which is what we did before) so we need to shear towards an offset, in this case 1:

    - - -

    And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will be:

    - - -

    Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just pull this apart a little and end up with an easy-to-calculate bit of code!

    -

    First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does that leave?

    - - -

    Suddenly things look a lot simpler: the mapped x is fairly straight forward to compute, and we see that the mapped y actually contains the mapped x in its entirety, so we'll have that part already available when we need to evaluate it. In fact, let's pull out all those common factors to see just how simple this is:

    - - -

    That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it look!

    -
    + +

    + That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it + look! +

    +
    +

    How do you track all that?

    +

    + Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply + use Mathematica. Tracking all this math by hand is insane, and we invented computers, + literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the + program do the math for me. And real math, too, with symbols, not with numbers. In fact, + here's the Mathematica notebook if you want to see how + this works for yourself. +

    +

    + Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's + $344 for home use, up from $295 when I original wrote this, but it's + also free when you buy a $35 raspberry pi. Obviously, I bought a + raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do + it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for. +

    +
    -

    How do you track all that?

    -

    Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply use Mathematica. Tracking all this math by hand is insane, and we invented computers, literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the program do the math for me. And real math, too, with symbols, not with numbers. In fact, here's the Mathematica notebook if you want to see how this works for yourself.

    -

    Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's $344 for home use, up from $295 when I original wrote this, but it's also free when you buy a $35 raspberry pi. Obviously, I bought a raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for.

    -
    - -

    So, let's write up a sketch that'll show us the canonical form for any curve drawn in blue, overlaid on our canonical map, so that we can immediately tell which features our curve must have, based on where the fourth coordinate is located on the map:

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目次次
    Finding Y, given X

    -

    One common task that pops up in things like CSS work, or parametric equalizers, or image leveling, or any other number of applications where Bézier curves are used as control curves in a way that there is really only ever one "y" value associated with one "x" value, you might want to cut out the middle man, as it were, and compute "y" directly based on "x". After all, the function looks simple enough, finding the "y" value should be simple too, right? Unfortunately, not really. However, it is possible and as long as you have some code in place to help, it's not a lot of a work either.

    -

    We'll be tackling this problem in two stages: the first, which is the hard part, is figuring out which "t" value belongs to any given "x" value. For instance, have a look at the following graphic. On the left we have a Bézier curve that looks for all intents and purposes like it fits our criteria: every "x" has one and only one associated "y" value. On the right we see the function for just the "x" values: that's a cubic curve, but not a really crazy cubic curve. If you move the graphic's slider, you will see a red line drawn that corresponds to the x coordinate: this is a vertical line in the left graphic, and a horizontal line on the right.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Now, if you look more closely at that right graphic, you'll notice something interesting: if we treat the red line as "the x axis", then the point where the function crosses our line is really just a root for the cubic function x(t) through a shifted "x-axis"... and we've already seen how to calculate roots, so let's just run cubic root finding - and not even the complicated cubic case either: because of the kind of curve we're starting with, we know there is only root, simplifying the code we need!

    -

    First, let's look at the function for x(t):

    - - -

    We can rewrite this to a plain polynomial form, by just fully writing out the expansion and then collecting the polynomial factors, as:

    - - -

    Nothing special here: that's a standard cubic polynomial in "power" form (i.e. all the terms are ordered by their power of t). So, given that a, b, c, d, and x(t) are all known constants, we can trivially rewrite this (by moving the x(t) across the equal sign) as:

    - - -

    You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using Cardano's algorithm, and we're left with some rather short code:

    + +

    + You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they + all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using + Cardano's algorithm, and we're left with some rather short code: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +y = curve.get(t).y + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So the procedure is fairly straight forward: pick an x, find the associated t value, evaluate our curve for that t value, which gives us the curve's {x,y} coordinate, which means we know y for this x. Move the slider for the following graphic to see this in action:

    - - - Scripts are disabled. Showing fallback image. - - - - - - -
    -
    -

    前目次次
    Arc length

    -

    How long is a Bézier curve? As it turns out, that's not actually an easy question, because the answer requires maths that —much like root finding— cannot generally be solved the traditional way. If we have a parametric curve with fx(t) and fy(t), then the length of the curve, measured from start point to some point t = z, is computed using the following seemingly straight forward (if a bit overwhelming) formula:

    - - -

    or, more commonly written using Leibnitz notation as:

    - - -

    This formula says that the length of a parametric curve is in fact equal to the area underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an unwieldy computation, and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates".

    -

    Seriously: It cannot be done.

    -

    So we turn to numerical approaches again. The method we'll look at here is the Gauss quadrature. This approximation is a really neat trick, because for any nth degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:

    - - -

    In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the better the approximation:

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    + +

    + In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips + sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The + more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the + better the approximation: +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    Now, infinitely many terms to sum and infinitely thin rectangles are not something that computers can work with, so instead we're going to approximate the infinite summation by using a sum of a finite number of "just thin" rectangular strips. As long as we use a high enough number of thin enough rectangular strips, this will give us an approximation that is pretty close to what the real value is.

    -

    So, the trick is to come up with useful rectangular strips. A naive way is to simply create n strips, all with the same width, but there is a far better way using special values for C and f(t) depending on the value of n, which indicates how many strips we'll use, and it's called the Legendre-Gauss quadrature.

    -

    This approach uses strips that are not spaced evenly, but instead spaces them in a special way based on describing the function as a polynomial (the more strips, the more accurate the polynomial), and then computing the exact integral for that polynomial. We're essentially performing arc length computation on a flattened curve, but flattening it based on the intervals dictated by the Legendre-Gauss solution.

    -
    - -

    Note that one requirement for the approach we'll use is that the integral must run from -1 to 1. That's no good, because we're dealing with Bézier curves, and the length of a section of curve applies to values which run from 0 to "some value smaller than or equal to 1" (let's call that value z). Thankfully, we can quite easily transform any integral interval to any other integral interval, by shifting and scaling the inputs. Doing so, we get the following:

    - - -

    That may look a bit more complicated, but the fraction involving z is a fixed number, so the summation, and the evaluation of the f(t) values are still pretty simple.

    -

    So, what do we need to perform this calculation? For one, we'll need an explicit formula for f(t), because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these Ci and ti values should be. Luckily, that's less work because there are actually many tables available that give these values, for any n, so if we want to approximate our integral with only two terms (which is a bit low, really) then these tables would tell us that for n=2 we must use the following values:

    - - -

    Which means that in order for us to approximate the integral, we must plug these values into the approximate function, which gives us:

    - - -

    We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the Bézier curve functions Bx(t) and By(t).

    -
    + +

    + We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the + Bézier curve functions Bx(t) and By(t). +

    + -

    If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the start/end points on the inside?

    - - - Scripts are disabled. Showing fallback image. - - - +

    + If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), + we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, + with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make + a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the + start/end points on the inside? +

    + + + Scripts are disabled. Showing fallback image. + + + +
    +
    +

    +
    前目次次
    + Approximated arc length +

    +

    + Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length + instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This + will come with an error, but this can be made arbitrarily small by increasing the segment count. +

    +

    + If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal + effort: +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    前目次次
    Approximated arc length

    -

    Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This will come with an error, but this can be made arbitrarily small by increasing the segment count.

    -

    If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal effort:

    -
    - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    You may notice that even though the error in length is actually pretty significant in absolute terms, even at a low number of segments we get a length that agrees with the true length when it comes to just the integer part of the arc length. Quite often, approximations can drastically speed things up!

    - -
    -
    -

    前目次次
    Curvature of a curve

    -

    If we have two curves, and we want to line them in up in a way that "looks right", what would we use as metric to let a computer decide what "looks right" means?

    -

    For instance, we can start by ensuring that the two curves share an end coordinate, so that there is no "gap" between the end of one and the start of the next curve, but that won't guarantee that things look right: both curves can be going in wildly different directions, and the resulting joined geometry will have a corner in it, rather than a smooth transition from one curve to the next.

    -

    What we want is to ensure that the curvature at the transition from one curve to the next "looks good". So, we start with a shared coordinate, and then also require that derivatives for both curves match at that coordinate. That way, we're assured that their tangents line up, which must mean the curve transition is perfectly smooth. We can even make the second, third, etc. derivatives match up for better and better transitions.

    -

    Problem solved!

    -

    However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on reordering curves showed us that the same looking curve can have an infinite number of curve expressions of arbitrarily high Bézier degree, and each of those will have wildly different derivative values.

    -

    So what we really want is some kind of expression that's not based on any particular expression of t, but is based on something that is invariant to the kind of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being the same function for "coordinate at some distance D along the curve".

    -

    We've seen this before... that's the arc length function.

    -

    So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the form of the arc length function, which we saw above and is fairly simple, rather than needing to solve the arc length function. If we start with the arc length expression and the run through the steps necessary to determine its derivative (with an alternative, shorter demonstration of how to do this found over on Stackexchange), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the fundamental theorem of calculus), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.

    -

    Let me highlight what just happened, because it's pretty special:

    -
      -
    1. we wanted to make curves line up, and initially thought to match the curves' derivatives, but
      2. -
    2. that turned out to be a really bad choice, so instead
      3. -
    3. we picked a function that is basically impossible to work with, and then worked with that, which
      4. -
    4. gives us a simple formula that is and expression using the curves' derivatives.
      5. -
    -

    That's crazy!

    -

    But that's also one of the things that makes maths so powerful: even if your initial ideas are off the mark, you might be much closer than you thought you were, and the journey from "thinking we're completely wrong" to "actually being remarkably close to being right" is where we can find a lot of insight.

    -

    So, what does the function look like? This:

    - - -

    Which is really just a "short form" that glosses over the fact that we're dealing with functions of t, so let's expand that a tiny bit:

    - - -

    And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so evaluating this function for any t value is just a matter of basic arithematics.

    -

    In fact, let's just implement it right now:

    + +

    + And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any + (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that + looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier + curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so + evaluating this function for any t value is just a matter of basic arithematics. +

    +

    In fact, let's just implement it right now:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return numerator / denominator + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    That was easy! (Well okay, that "not a number" value will need to be taken into account by downstream code, but that's a reality of programming anyway)

    -

    With all of that covered, let's line up some curves! The following graphic gives you two curves that look identical, but use quadratic and cubic functions, respectively. As you can see, despite their derivatives being necessarily different, their curvature (thanks to being derived based on maths that "ignores" specific function derivative, and instead gives a formula that smooths out any differences) is exactly the same. And because of that, we can put them together such that the point where they overlap has the same curvature for both curves, giving us the smoothest transition.

    - - - Scripts are disabled. Showing fallback image. - - - -

    One thing you may have noticed in this sketch is that sometimes the curvature looks fine, but seems to be pointing in the wrong direction, making it hard to line up the curves properly. A way around that, of course, is to show the curvature on both sides of the curve, so let's just do that. But let's take it one step further: we can also compute the associated "radius of curvature", which gives us the implicit circle that "fits" the curve's curvature at any point, using what is possibly the simplest bit of maths found in this entire primer:

    - - -

    So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" our curve at some point that we can control by using a slider:

    - - - Scripts are disabled. Showing fallback image. - - - - - + +

    + So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" + our curve at some point that we can control by using a slider: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +
    +

    +
    前目次次
    + Tracing a curve at fixed distance intervals +

    +

    + Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance + intervals over time, like a train along a track, and you want to use Bézier curves. +

    +

    Now you have a problem.

    +

    + The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a + track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick + t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In + fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another. +

    +

    + The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be + straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To + make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for + this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length + function, which can have more inflection points. +

    + + + Scripts are disabled. Showing fallback image. + + +

    + So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through + the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and + then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of + points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two + points, but if we have a high enough number of samples, we don't even need to bother. +

    +

    + So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t + curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks + like, by coloring each section of curve between two distance markers differently: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    +

    + However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to + constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT). +

    +
    +
    +

    +
    前目次次
    + Intersections +

    +

    + Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to + work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to + perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box + overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the + actual curve. +

    +

    + We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by + implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both + lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution. +

    +

    Line-line intersections

    +

    + If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are + an intervals on by linear algebra, using the procedure outlined in this + top coder + article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line + segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments. +

    +

    + The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on + (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection + point). +

    + + + Scripts are disabled. Showing fallback image. + + + +
    +

    Implementing line-line intersections

    +

    + Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, + but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe + you're using point structs for the line. Let's get coding: +

    -
    -
    -

    前目次次
    Tracing a curve at fixed distance intervals

    -

    Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance intervals over time, like a train along a track, and you want to use Bézier curves.

    -

    Now you have a problem.

    -

    The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another.

    -

    The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.

    - - - Scripts are disabled. Showing fallback image. - - - -

    So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two points, but if we have a high enough number of samples, we don't even need to bother.

    -

    So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks like, by coloring each section of curve between two distance markers differently:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    -

    However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT).

    - -
    -
    -

    前目次次
    Intersections

    -

    Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the actual curve.

    -

    We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution.

    -

    Line-line intersections

    -

    If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are an intervals on by linear algebra, using the procedure outlined in this top coder article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments.

    -

    The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection point).

    - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    Implementing line-line intersections

    -

    Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe you're using point structs for the line. Let's get coding:

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    + return lli4(line1.p1, line1.p2, line2.p1, line2.p2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + +
    - +

    What about curve-line intersections?

    +

    + Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we + translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in + a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a + curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the + section on finding extremities. +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    What about curve-line intersections?

    -

    Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the section on finding extremities.

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    + Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the + curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and + curve splitting. +

    +
    +
    +

    +
    前目次次
    + Curve/curve intersection +

    +

    + Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" + technique: +

    +
      +
    1. + Take two curves C1 and C2, and treat them as a pair. +
      2. +
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. +
    3. + With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2). +
      4. +
    4. + For each pair, check whether their bounding boxes overlap. +
        +
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. +
      2. If there is overlap, rerun all steps for this pair.
        3. +
      +
      5. +
    5. + Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we + might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value + (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative). +
      6. +
    +

    + This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then + taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections. +

    +

    + The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click + the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value + that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the + algorithm, so you can try this with lots of different curves. +

    +

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    + + + Scripts are disabled. Showing fallback image. + + + + + + +

    + Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into + two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at + t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each + iteration, and each successive step homing in on the curve's self-intersection points. +

    +
    +
    +

    +
    + 前目次次 +
    + The projection identity +

    +

    + De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to + draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. + Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point + around to change the curve's shape. +

    +

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    +

    + In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want + to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know + has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for + reasons that will become obvious. +

    +

    + So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following + graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which + taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what + happens to that value? +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -

    Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and curve splitting.

    - -
    -
    -

    前目次次
    Curve/curve intersection

    -

    Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" technique:

    -
      -
    1. Take two curves C1 and C2, and treat them as a pair.
      2. -
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. -
    3. With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2).
      4. -
    4. For each pair, check whether their bounding boxes overlap.
        -
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. -
      2. If there is overlap, rerun all steps for this pair.
        3. -
      -
      5. -
    5. Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative).
      6. -
    -

    This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections.

    -

    The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the algorithm, so you can try this with lots of different curves.

    -

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    - - - Scripts are disabled. Showing fallback image. - - - - - - -

    Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each iteration, and each successive step homing in on the curve's self-intersection points.

    - -
    -
    -

    前目次次
    The projection identity

    -

    De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point around to change the curve's shape.

    -

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    -

    In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for reasons that will become obvious.

    -

    So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what happens to that value?

    -
    - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    So these graphics show us several things:

    -
      -
    1. a point at the tip of the curve construction's "hat": let's call that A, as well as
      2. -
    2. our on-curve point give our chosen t value: let's call that B, and finally,
      3. -
    3. a point that we get by projecting A, through B, onto the line between the curve's start and end points: let's call that C.
      4. -
    4. for both quadratic and cubic curves, two points e1 and e2, which represent the single-to-last step in de Casteljau's algorithm: in the last step, we find B at (1-t) * e1 + t * e2.
      5. -
    5. for cubic curves, also the points v1 and v2, which together with A represent the first step in de Casteljau's algorithm: in the next step, we find e1 and e2.
      6. -
    -

    These three values A, B, and C allow us to derive an important identity formula for quadratic and cubic Bézier curves: for any point on the curve with some t value, the ratio of distances from A to B and B to C is fixed: if some t value sets up a C that is 20% away from the start and 80% away from the end, then it doesn't matter where the start, end, or control points are; for that t value, C will always lie at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic and then move all the other points around: if you only move the control points, start and end won't move, and so neither will C, and if you move either start or end point, C will move but its relative position will not change.

    -

    So, how can we compute C? We start with our observation that C always lies somewhere between the start and end points, so logically C will have a function that interpolates between those two coordinates:

    - - -

    If we can figure out what the function u(t) looks like, we'll be done. Although we do need to remember that this u(t) will have a different form depending on whether we're working with quadratic or cubic curves. Running through the maths (with thanks to Boris Zbarsky) shows us the following two formulae:

    - - -

    And

    - - -

    So, if we know the start and end coordinates and the t value, we know C without having to calculate the A or even B coordinates. In fact, we can do the same for the ratio function. As another function of t, we technically don't need to know what A or B or C are. It, too, can be expressed as a pure function of t.

    -

    We start by observing that, given A, B, and C, the following always holds:

    - - -

    Working out the maths for this, we see the following two formulae for quadratic and cubic curves:

    - - -

    And

    - - -

    Which now leaves us with some powerful tools: given three points (start, end, and "some point on the curve"), as well as a t value, we can construct curves. We can compute C using the start and end points and our u(t) function, and once we have C, we can use our on-curve point (B) and the ratio(t) function to find A:

    - - -

    With A found, finding e1 and e2 for quadratic curves is a matter of running the linear interpolation with t between start and A to yield e1, and between A and end to yield e2. For cubic curves, there is no single pair of points that can act as e1 and e2 (there are infinitely many, because the tangent at B is a free parameter for cubic curves) so as long as the distance ratio between e1 to B and B to e2 is the Bézier ratio (1-t):t, we are free to pick any pair, after which we can reverse engineer v1 and v2:

    - - -

    And then reverse engineer the curve's control points:

    - - -

    So: if we have a curve's start and end points, as well as some third point B that we want the curve to pass through, then for any t value we implicitly know all the ABC values, which (combined with an educated guess on appropriate e1 and e2 coordinates for cubic curves) gives us the necessary information to reconstruct a curve's "de Casteljau skeleton". Which means that we can now do several things: we can "fit" curves using only three points, which means we can also "mold" curves by moving an on-curve point but leaving its start and end points, and then reconstruct the curve based on where we moved the on-curve point to. These are very useful things, and we'll look at both in the next few sections.

    - -
    -
    -

    前目次次
    Creating a curve from three points

    -

    Given the preceding section, you might be wondering if we can use that knowledge to just "create" curves by placing some points and having the computer do the rest, to which the answer is: that's exactly what we can now do!

    -

    For quadratic curves, things are pretty easy. Technically, we'll need a t value in order to compute the ratio function used in computing the ABC coordinates, but we can just as easily approximate one by treating the distance between the start and B point, and B and end point as a ratio, using

    - - -

    With this code in place, creating a quadratic curve from three points is literally just computing the ABC values, and using A as our curve's control point:

    - - - Scripts are disabled. Showing fallback image. - - - -

    For cubic curves we need to do a little more work, but really only just a little. We're first going to assume that a decent curve through the three points should approximate a circular arc, which first requires knowing how to fit a circle to three points. You may remember (if you ever learned it!) that a line between two points on a circle is called a chord, and that one property of chords is that the line from the center of any chord, perpendicular to that chord, passes through the center of the circle.

    -

    That means that if we have three points on a circle, we have three (different) chords, and consequently, three (different) lines that go from those chords through the center of the circle: if we find two of those lines, then their intersection will be our circle's center, and the circle's radius will—by definition!—be the distance from the center to any of our three points:

    - - - Scripts are disabled. Showing fallback image. - - - -

    With that covered, we now also know the tangent line to our point B, because the tangent to any point on the circle is a line through that point, perpendicular to the line from that point to the center. That just leaves marking appropriate points e1 and e2 on that tangent, so that we can construct a new cubic curve hull. We use the approach as we did for quadratic curves to automatically determine a reasonable t value, and then our e1 and e2 coordinates must obey the standard de Casteljau rule for linear interpolation:

    - - -

    Where d is the total length of the line segment from e1 to e2. So how long do we make that? There are again all kinds of approaches we can take, and a simple-but-effective one is to set the length of that segment to "one third the length of the baseline". This forces e1 and e2 to always be the "linear curve" distance apart, which means if we place our three points on a line, it will actually look like a line. Nice! The last thing we'll need to do is make sure to flip the sign of d depending on which side of the baseline our B is located, so we don't end up creating a funky curve with a loop in it. To do this, we can use the atan2 function:

    - - -

    This angle φ will be between 0 and π if B is "above" the baseline (rotating all three points so that the start is on the left and the end is the right), so we can use a relatively straight forward check to make sure we're using the correct sign for our value d:

    - - -

    The result of this approach looks as follows:

    - - - Scripts are disabled. Showing fallback image. - - - -

    It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; more on that later), we're only trying to come up with some reasonable e1 and e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives us:

    - - - Scripts are disabled. Showing fallback image. - - - -

    That looks perfectly serviceable!

    -

    Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at implementing curve molding in the section after that, so read on!

    + +

    The result of this approach looks as follows:

    + + + Scripts are disabled. Showing fallback image. + + + +

    + It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, + we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; + more on that later), we're only trying to come up with some reasonable e1 and + e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives + us: +

    + + + Scripts are disabled. Showing fallback image. + + + +

    That looks perfectly serviceable!

    +

    + Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" + curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding + the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at + implementing curve molding in the section after that, so read on! +

    +
    +
    +

    +
    + 前目次次 +
    + Projecting a point onto a Bézier curve +

    +

    + Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're + trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the + curve our cursor will be closest to. So, how do we project points onto a curve? +

    +

    + If the Bézier curve is of low enough order, we might be able to + work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a + numerical approach again. We'll be finding our ideal t value using a + binary search. First, we do a coarse distance-check based on + t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast: +

    -
    -
    -

    前目次次
    Projecting a point onto a Bézier curve

    -

    Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the curve our cursor will be closest to. So, how do we project points onto a curve?

    -

    If the Bézier curve is of low enough order, we might be able to work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a numerical approach again. We'll be finding our ideal t value using a binary search. First, we do a coarse distance-check based on t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    After this runs, we know that LUT[i] is the coordinate on the curve in our LUT that is closest to the point we want to project, so that's a pretty good initial guess as to what the best projection onto our curve is. To refine it, we note that LUT[i] is a better guess than both LUT[i-1] and LUT[i+1], but there might be an even better projection somewhere else between those two values, so that's what we're going to be testing for, using a variation of the binary search.

    -
      -
    1. we start with our point p, and the t values t1=LUT[i-1].t and t2=LUT[i+1].t, which span an interval v = t2-t1.
      2. -
    2. we test this interval in five spots: the start, middle, and end (which we already have), and the two points in between the middle and start/end points
      3. -
    3. we then check which of these five points is the closest to our original point p, and then repeat step 1 with the points before and after the closest point we just found.
      4. -
    -

    This makes the interval we check smaller and smaller at each iteration, and we can keep running the three steps until the interval becomes so small as to lead to distances that are, for all intents and purposes, the same for all points.

    -

    So, let's see that in action: in this case, I'm going to arbitrarily say that if we're going to run the loop until the interval is smaller than 0.001, and show you what that means for projecting your mouse cursor or finger tip onto a rather complex Bézier curve (which, of course, you can reshape as you like). Also shown are the original three points that our coarse check finds.

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目次次
    Intersections with a circle

    -

    It might seem odd to cover this subject so much later than the line/line, line/curve, and curve/curve intersection topics from several sections earlier, but the reason we can't cover circle/curve intersections is that we can't really discuss circle/curve intersection until we've covered the kind of lookup table (LUT) walking that the section on projecting a point onto a curve uses. To see why, let's look at what we would have to do if we wanted to find the intersections between a curve and a circle using calculus.

    -

    First, we observe that "finding intersections" in this case means that, given a circle defined by a center point c = (x,y) and a radius r, we want to find all points on the Bezier curve for which the distance to the circle's center point is equal to the circle radius, which by definition means those points lie on the circle, and so count as intersections. In maths, that means we're trying to solve:

    - - -

    Which seems simple enough. Unfortunately, when we expand that dist function, things get a lot more problematic:

    - - -

    And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the Abel-Ruffini theorem that we saw before, we can't solve this by just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left with a monstrous function to solve.

    -

    So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a bit more code to make sure we find all of them, but let's look at the steps involved:

    + +

    + And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the + Abel-Ruffini theorem that we saw before, we can't solve this by + just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to + actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and + all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left + with a monstrous function to solve. +

    +

    + So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the + on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance + r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a + bit more code to make sure we find all of them, but let's look at the steps involved: +

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + -

    This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is exactly the radius, so the coordinate lies on both the curve and the circle.

    -

    So far so good.

    -

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    +

    + This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor + change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between + that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is + exactly the radius, so the coordinate lies on both the curve and the circle. +

    +

    So far so good.

    +

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + start = i + 2; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points ("current", "previous" and "before previous") and then check whether they form a local minimum:

    +

    + After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we + can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now + not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of + course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in + finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points + ("current", "previous" and "before previous") and then check whether they form a local minimum: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return start + i + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    In words: given a start index, the circle center and radius, and our LUT, we check where (closest to our start index) we can find a local minimum for the difference between "the distance from the curve to the circle center", and the circle's radius. We track this by looking at three values (associated with the indices index-2, index-1, and index), and we know we've found a local minimum if the three values show that the middle value (pd1) is less than either value beside it. When we do, we can set our "best guess, relative to start" as index-1. Of course, since we're now checking values relative to some start value, we might not find another candidate value at all, in which case we return start - 1, so that a simple "is the result less than start?" lets us determine that there are no more intersections to find.

    -

    Finally, while not necessary for point projection, there is one more step we need to perform when we run the binary refinement function on our candidate LUT indices, because we've so far only been testing using distances "closest to the radius of the circle", and that's actually not good enough... we need distances that are the radius of the circle. So, after running the refinement for each of these indices, we need to discard any final value that isn't the circle radius. And because we're working with floating point numbers, what this really means is that we need to discard any value that's a pixel or more "off". Or, if we want to get really fancy, "some small epsilon value".

    -

    Based on all of that, the following graphic shows this off for the standard cubic curve (which you can move the coordinates around for, of course) and a circle with a controllable radius centered on the graphic's center, using the code approach described above.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    And of course, for the full details, click that "view source" link.

    - -
    -
    -

    前目次次
    Molding a curve

    -

    Armed with knowledge of the "ABC" relation, point-on-curve projection, and guestimating reasonable looking helper values for cubic curve construction, we can finally cover curve molding: updating a curve's shape interactively, by dragging points on the curve around.

    -

    For quadratic curve, this is a really simple trick: we project our cursor onto the curve, which gives us a t value and initial B coordinate. We don't even need the latter: with our t value and "wherever the cursor is" as target B, we can compute the associated C:

    - - -

    And then the associated A:

    - - -

    And we're done, because that's our new quadratic control point!

    - - - Scripts are disabled. Showing fallback image. - - - -

    As before, cubic curves are a bit more work, because while it's easy to find our initial t value and ABC values, getting those all-important e1 and e2 coordinates is going to pose a bit of a problem... in the section on curve creation, we were free to pick an appropriate t value ourselves, which allowed us to find appropriate e1 and e2 coordinates. That's great, but when we're curve molding we don't have that luxury: whatever point we decide to start moving around already has its own t value, and its own e1 and e2 values, and those may not make sense for the rest of the curve.

    -

    For example, let's see what happens if we just "go with what we get" when we pick a point and start moving it around, preserving its t value and e1/e2 coordinates:

    - - - Scripts are disabled. Showing fallback image. - - - -

    That looks reasonable, close to the original point, but the further we drag our point, the less "useful" things become. Especially if we drag our point across the baseline, rather than turning into a nice curve.

    -

    One way to combat this might be to combine the above approach with the approach from the creating curves section: generate both the "unchanged t/e1/e2" curve, as well as the "idealized" curve through the start/cursor/end points, with idealized t value, and then interpolating between those two curves:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    The slide controls the "falloff distance" relative to where the original point on the curve is, so that as we drag our point around, it interpolates with a bias towards "preserving t/e1/e2" closer to the original point, and bias towards "idealized" form the further away we move our point, with anything that's further than our falloff distance simply being the idealized curve. We don't even try to interpolate at that point.

    -

    A more advanced way to try to smooth things out is to implement continuous molding, where we constantly update the curve as we move around, and constantly change what our B point is, based on constantly projecting the cursor on the curve as we're updating it - this is, you won't be surprised to learn, tricky, and beyond the scope of this section: interpolation (with a reasonable distance) will do for now!

    - -
    -
    -

    前目次次
    Curve fitting

    -

    Given the previous section, one question you might have is "what if I don't want to guess t values?". After all, plenty of graphics packages do automated curve fitting, so how can we implement that in a way that just finds us reasonable t values all on its own?

    -

    And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a least squares polynomial regression. In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an nth order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".

    -

    Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the squared distance between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep-dive the specific merits of least squared error metrics versus least absolute error metrics, but those are well beyond the scope of this material.

    -

    So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his "Least Squares Bézier Fit" article, and end with some nice interactive graphics for doing some curve fitting.

    -

    Before we begin, we're going to use the curve in matrix form. In the section on matrices, I mentioned that some things are easier if we use the matrix representation of a Bézier curve rather than its calculus form, and this is one of those things.

    -

    As such, the first step in the process is expressing our Bézier curve as powers/coefficients/coordinate matrix T x M x C, by expanding the Bézier functions.

    -
    - -

    Revisiting the matrix representation

    -

    Rewriting Bézier functions to matrix form is fairly easy, if you first expand the function, and then arrange them into a multiple line form, where each line corresponds to a power of t, and each column is for a specific coefficient. First, we expand the function:

    - - -

    And then we (trivially) rearrange the terms across multiple lines:

    - - -

    This rearrangement has "factors of t" at each row (the first row is t⁰, i.e. "1", the second row is t¹, i.e. "t", the third row is t²) and "coefficient" at each column (the first column is all terms involving "a", the second all terms involving "b", the third all terms involving "c").

    -

    With that arrangement, we can easily decompose this as a matrix multiplication:

    - - -

    We can do the same for the cubic curve, of course. We know the base function for cubics:

    - - -

    So we write out the expansion and rearrange:

    - - -

    Which we can then decompose:

    - - -

    And, of course, we can do this for quartic curves too (skipping the expansion step):

    - - -

    And so and on so on. Now, let's see how to use these T, M, and C, to do some curve fitting.

    -
    + +

    + And so and on so on. Now, let's see how to use these T, M, and C, to do some curve + fitting. +

    + -

    Let's get started: we're going to assume we picked the right order curve: for n points we're fitting an n-1th order curve, so we "start" with a vector P that represents the coordinates we already know, and for which we want to do curve fitting:

    - - -

    Next, we need to figure out appropriate t values for each point in the curve, because we need something that lets us tie "the actual coordinate" to "some point on the curve". There's a fair number of different ways to do this (and a large part of optimizing "the perfect fit" is about picking appropriate t values), but in this case let's look at two "obvious" choices:

    -
      -
    1. equally spaced t values, and
      2. -
    2. t values that align with distance along the polygon.
      3. -
    -

    The first one is really simple: if we have n points, then we'll just assign each point i a t value of (i-1)/(n-1). So if we have four points, the first point will have t=(1-1)/(4-1)=0/3, the second point will have t=(2-1)/(4-1)=1/3, the third point will have t=2/3, and the last point will be t=1. We're just straight up spacing the t values to match the number of points we have.

    -

    The second one is a little more interesting: since we're doing polynomial regression, we might as well exploit the fact that our base coordinates just constitute a collection of line segments. At the first point, we're fixing t=0, and the last point, we want t=1, and anywhere in between we're simply going to say that t is equal to the distance along the polygon, scaled to the [0,1] domain.

    -

    To get these values, we first compute the general "distance along the polygon" matrix:

    - - -

    Where length() is literally just that: the length of the line segment between the point we're looking at, and the previous point. This isn't quite enough, of course: we still need to make sure that all the values between i=1 and i=n fall in the [0,1] interval, so we need to scale all values down by whatever the total length of the polygon is:

    - - -

    And now we can move on to the actual "curve fitting" part: what we want is a function that lets us compute "ideal" control point values such that if we build a Bézier curve with them, that curve passes through all our original points. Or, failing that, have an overall error distance that is as close to zero as we can get it. So, let's write out what the error distance looks like.

    -

    As mentioned before, this function is really just "the distance between the actual coordinate, and the coordinate that the curve evaluates to for the associated t value", which we'll square to get rid of any pesky negative signs:

    - - -

    Since this function only deals with individual coordinates, we'll need to sum over all coordinates in order to get the full error function. So, we literally just do that; the total error function is simply the sum of all these individual errors:

    - - -

    And here's the trick that justifies using matrices: while we can work with individual values using calculus, with matrices we can compute as many values as we make our matrices big, all at the "same time", We can replace the individual terms pi with the full P coordinate matrix, and we can replace Bézier(si) with the matrix representation T x M x C we talked about before, which gives us:

    - - -

    In which we can replace the rather cumbersome "squaring" operation with a more conventional matrix equivalent:

    - - -

    Here, the letter T is used instead of the number 2, to represent the matrix transpose; each row in the original matrix becomes a column in the transposed matrix instead (row one becomes column one, row two becomes column two, and so on).

    -

    This leaves one problem: T isn't actually the matrix we want: we don't want symbolic t values, we want the actual numerical values that we computed for S, so we need to form a new matrix, which we'll call 𝕋, that makes use of those, and then use that 𝕋 instead of T in our error function:

    - - -

    Which, because of the first and last values in S, means:

    - - -

    Now we can properly write out the error function as matrix operations:

    - - -

    So, we have our error function: we now need to figure out the expression for where that function has minimal value, e.g. where the error between the true coordinates and the coordinates generated by the curve fitting is smallest. Like in standard calculus, this requires taking the derivative, and determining where that derivative is zero:

    - - -
    + +
    +

    Where did this derivative come from?

    +

    + That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I + straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know. +

    +

    + So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific + case, I + posted a question + to Math.stackexchange, and received a answer that goes into way more detail than I had + hoped to receive. +

    +

    + Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean + maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that + true?". There are answers. They might just require some time to come to understand. +

    +
    -

    Where did this derivative come from?

    -

    That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know.

    -

    So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific case, I posted a question to Math.stackexchange, and received a answer that goes into way more detail than I had hoped to receive.

    -

    Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that true?". There are answers. They might just require some time to come to understand.

    -
    - -

    Now, given the above derivative, we can rearrange the terms (following the rules of matrix algebra) so that we end up with an expression for C:

    - - -

    Here, the "to the power negative one" is the notation for the matrix inverse. But that's all we have to do: we're done. Starting with P and inventing some t values based on the polygon the coordinates in P define, we can compute the corresponding Bézier coordinates C that specify a curve that goes through our points. Or, if it can't go through them exactly, as near as possible.

    -

    So before we try that out, how much code is involved in implementing this? Honestly, that answer depends on how much you're going to be writing yourself. If you already have a matrix maths library available, then really not that much code at all. On the other hand, if you are writing this from scratch, you're going to have to write some utility functions for doing your matrix work for you, so it's really anywhere from 50 lines of code to maybe 200 lines of code. Not a bad price to pay for being able to fit curves to pre-specified coordinates.

    -

    So let's try it out! The following graphic lets you place points, and will start computing exact-fit curves once you've placed at least three. You can click for more points, and the code will simply try to compute an exact fit using a Bézier curve of the appropriate order. Four points? Cubic Bézier. Five points? Quartic. And so on. Of course, this does break down at some point: depending on where you place your points, it might become mighty hard for the fitter to find an exact fit, and things might actually start looking horribly off once there's enough points for compound floating point rounding errors to start making a difference (which is around 10~11 points).

    - - - Scripts are disabled. Showing fallback image. - - - - - - -

    You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be able to freely manipulate them and see what the effect on your curve is.

    - -
    -
    -

    前目次次
    Bézier curves and Catmull-Rom curves

    -

    Taking an excursion to different splines, the other common design curve is the Catmull-Rom spline, which unlike Bézier curves pass through each control point, so they offer a kind of "built-in" curve fitting.

    -

    In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what looks like a single Catmull-Rom curve is actually a spline: a single curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is defined by the point's coordinates, and the tangent for those points, the latter of which can trivially be derived from knowing the previous and next point:

    - + +

    + You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio + along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get + direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be + able to freely manipulate them and see what the effect on your curve is. +

    +
    +
    +

    +
    + 前目次次 +
    + Bézier curves and Catmull-Rom curves +

    +

    + Taking an excursion to different splines, the other common design curve is the + Catmull-Rom spline, which unlike Bézier curves + pass through each control point, so they offer a kind of "built-in" curve fitting. +

    +

    + In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets + you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the + tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what + looks like a single Catmull-Rom curve is actually a spline: a single + curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, + and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is + defined by the point's coordinates, and the tangent for those points, the latter of which + can trivially be derived from knowing the + previous and next point: +

    + - -

    One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on that in the next section.

    -

    In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of variations that make things considerable more complex):

    + +

    + One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up + with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent + should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on + that in the next section. +

    +

    + In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of + variations that make things + considerable more + complex): +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + point(c0 * v1 + c1 * dv1 + c2 * v2 + c3 * dv2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Now, since a Catmull-Rom curve is a form of cubic Hermite spline, and as cubic Bézier curves are also a form of cubic Hermite spline, we run into an interesting bit of maths programming: we can convert one to the other and back, and the maths for doing so is surprisingly simple!

    -

    The main difference between Catmull-Rom curves and Bézier curves is "what the points mean":

    -
      -
    • A cubic Bézier curve is defined by a start point, a control point that implies the tangent at the start, a control point that implies the tangent at the end, and an end point, plus a characterizing matrix that we can multiply by that point vector to get on-curve coordinates.
      • -
    • A Catmull-Rom curve is defined by a start point, a tangent that for that starting point, an end point, and a tangent for that end point, plus a characteristic matrix that we can multiple by the point vector to get on-curve coordinates.
      • -
    -

    Those are very similar, so let's see exactly how similar they are. We've already see the matrix form for Bézier curves, so how different is the matrix form for Catmull-Rom curves?:

    - - -

    That's pretty dang similar. So the question is: how can we convert that expression with Catmull-Rom matrix and vector into an expression of the Bézier matrix and vector? The short answer is of course "by using linear algebra", but the longer answer is the rest of this section, and involves some maths that you may not even care for: if you just want to know the (incredibly simple) conversions between the two curve forms, feel free to skip to the end of the following explanation, but if you want to how we can get one from the other... let's get mathing!

    -
    - -

    Deriving the conversion formulae

    -

    In order to convert between Catmull-Rom curves and Bézier curves, we need to know two things. Firstly, how to express the Catmull-Rom curve using a "set of four coordinates", rather than a mix of coordinates and tangents, and secondly, how to convert those Catmull-Rom coordinates to and from Bézier form.

    -

    We start with the first part, to figure out how we can go from Catmull-Rom V coordinates to Bézier P coordinates, by applying "some matrix T". We don't know what that T is yet, but we'll get to that:

    - - -

    So, this mapping says that in order to map a Catmull-Rom "point + tangent" vector to something based on an "all coordinates" vector, we need to determine the mapping matrix such that applying T yields P2 as start point, P3 as end point, and two tangents based on the lines between P1 and P3, and P2 nd P4, respectively.

    -

    Computing T is really more "arranging the numbers":

    - - -

    Thus:

    - - -

    However, we're not quite done, because Catmull-Rom curves have that "tension" parameter, written as τ (a lowercase"tau"), which is a scaling factor for the tangent vectors: the bigger the tension, the smaller the tangents, and the smaller the tension, the bigger the tangents. As such, the tension factor goes in the denominator for the tangents, and before we continue, let's add that tension factor into both our coordinate vector representation, and mapping matrix T:

    - - -

    With the mapping matrix properly done, let's rewrite the "point + tangent" Catmull-Rom matrix form to a matrix form in terms of four coordinates, and see what we end up with:

    - - -

    Replace point/tangent vector with the expression for all-coordinates:

    - - -

    and merge the matrices:

    - - -

    This looks a lot like the Bézier matrix form, which as we saw in the chapter on Bézier curves, should look like this:

    - - -

    So, if we want to express a Catmull-Rom curve using a Bézier curve, we'll need to turn this Catmull-Rom bit:

    - - -

    Into something that looks like this:

    - - -

    And the way we do that is with a fairly straight forward bit of matrix rewriting. We start with the equality we need to ensure:

    - - -

    Then we remove the coordinate vector from both sides without affecting the equality:

    - - -

    Then we can "get rid of" the Bézier matrix on the right by left-multiply both with the inverse of the Bézier matrix:

    - - -

    A matrix times its inverse is the matrix equivalent of 1, and because "something times 1" is the same as "something", so we can just outright remove any matrix/inverse pair:

    - - -

    And now we're basically done. We just multiply those two matrices and we know what V is:

    - - -

    We now have the final piece of our function puzzle. Let's run through each step.

    -
      -
    1. Start with the Catmull-Rom function:
      2. -
    - - -
      -
    1. rewrite to pure coordinate form:
      2. -
    - - -
      -
    1. rewrite for "normal" coordinate vector:
      2. -
    - - -
      -
    1. merge the inner matrices:
      2. -
    - - -
      -
    1. rewrite for Bézier matrix form:
      2. -
    - - -
      -
    1. and transform the coordinates so we have a "pure" Bézier expression:
      2. -
    - - -

    And we're done: we finally know how to convert these two curves!

    -
    + +

    And we're done: we finally know how to convert these two curves!

    + -

    If we have a Catmull-Rom curve defined by four coordinates P1 through P4, then we can draw that curve using a Bézier curve that has the vector:

    - - -

    Similarly, if we have a Bézier curve defined by four coordinates P1 through P4, we can draw that using a standard tension Catmull-Rom curve with the following coordinate values:

    - - -

    Or, if your API allows you to specify Catmull-Rom curves using plain coordinates:

    - - - -
    -
    -

    前目次次
    Creating a Catmull-Rom curve from three points

    -

    Much shorter than the previous section: we saw that Catmull-Rom curves need at least 4 points to draw anything sensible, so how do we create a Catmull-Rom curve from three points?

    -

    Short and sweet: we don't.

    -

    We run through the maths that lets us create a cubic Bézier curve, and then convert its coordinates to Catmull-Rom form using the conversion formulae we saw above.

    - -
    -
    -

    前目次次
    Forming poly-Bézier curves

    -

    Much like lines can be chained together to form polygons, Bézier curves can be chained together to form poly-Béziers, and the only trick required is to make sure that:

    -
      -
    1. the end point of each section is the starting point of the following section, and
      2. -
    2. the derivatives across that dual point line up.
      3. -
    -

    Unless you want sharp corners, of course. Then you don't even need 2.

    -

    We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind that just follows point 1. where the end point of a segment is the same point as the start point of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're the easiest to implement:

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Dragging the control points around only affects the curve segments that the control point belongs to, and moving an on-curve point leaves the control points where they are, which is not the most useful for practical modelling purposes. So, let's add in the logic we need to make things a little better. We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve point is the same as it's "outgoing" derivative:

    - - -

    We can effect this quite easily, because we know that the vector from a curve's last control point to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control point of the next curve matches that, all we have to do is mirror that last control point through the last on-curve point. And mirroring any point A through any point B is really simple:

    - - -

    So let's implement that and see what it gets us. The following two graphics show a quadratic and a cubic poly-Bézier curve again, but this time moving the control points around moves others, too. However, you might see something unexpected going on for quadratic curves...

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all the control points are effectively linked. Move one of them, and you move all of them. Not only that, but if we move the on-curve points, it's possible to get a situation where a control point cannot satisfy the constraint that it's the reflection of its two neighbouring control points... This means that we cannot use quadratic poly-Béziers for anything other than really, really simple shapes. And even then, they're probably the wrong choice. Cubic curves are pretty decent, but the fact that the derivatives are linked means we can't manipulate curves as well as we might if we relaxed the constraints a little.

    -

    So: let's relax the requirement a little.

    -

    We can change the constraint so that we still preserve the angle of the derivatives across sections (so transitions from one section to the next will still look natural), but give up the requirement that they should also have the same vector length. Doing so will give us a much more useful kind of poly-Bézier curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Cubic curves are now better behaved when it comes to dragging control points around, but the quadratic poly-Bézier still has the problem that moving one control points will move the control points and may ending up defining "the next" control point in a way that doesn't work. Quadratic curves really aren't very useful to work with...

    -

    Finally, we also want to make sure that moving the on-curve coordinates preserves the relative positions of the associated control points. With that, we get to the kind of curve control that you might be familiar with from applications like Photoshop, Inkscape, Blender, etc.

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a new, very serious problem: we can move an on-curve point in such a way that we can't compute what needs to "happen next". Move the top point down, below the left and right points, for instance. There is no way to preserve correct control points without a kink at the bottom point. Quadratic curves: just not that good...

    -

    A final improvement is to offer fine-level control over which points behave which, so that you can have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves for the rest of the path. Implementing that, is left as an exercise for the reader.

    - -
    -
    -

    前目次次
    Curve offsetting

    -

    Perhaps you're like me, and you've been writing various small programs that use Bézier curves in some way or another, and at some point you make the step to implementing path extrusion. But you don't want to do it pixel based; you want to stay in the vector world. You find that extruding lines is relatively easy, and tracing outlines is coming along nicely (although junction caps and fillets are a bit of a hassle), and then you decide to do things properly and add Bézier curves to the mix. Now you have a problem.

    -

    Unlike lines, you can't simply extrude a Bézier curve by taking a copy and moving it around, because of the curvatures; rather than a uniform thickness, you get an extrusion that looks too thin in places, if you're lucky, but more likely will self-intersect. The trick, then, is to scale the curve, rather than simply copying it. But how do you scale a Bézier curve?

    -

    Bottom line: you can't. So you cheat. We're not going to do true curve scaling, or rather curve offsetting, because that's impossible. Instead we're going to try to generate 'looks good enough' offset curves.

    -
    - -

    "What do you mean, you can't? Prove it."

    -

    First off, when I say "you can't," what I really mean is "you can't offset a Bézier curve with another Bézier curve", not even by using a really high order curve. You can find the function that describes the offset curve, but it won't be a polynomial, and as such it cannot be represented as a Bézier curve, which has to be a polynomial. Let's look at why this is:

    -

    From a mathematical point of view, an offset curve O(t) is a curve such that, given our original curve B(t), any point on O(t) is a fixed distance d away from coordinate B(t). So let's math that:

    - - -

    However, we're working in 2D, and d is a single value, so we want to turn it into a vector. If we want a point distance d "away" from the curve B(t) then what we really mean is that we want a point at d times the "normal vector" from point B(t), where the "normal" is a vector that runs perpendicular ("at a right angle") to the tangent at B(t). Easy enough:

    - - -

    Now this still isn't very useful unless we know what the formula for N(t) is, so let's find out. N(t) runs perpendicular to the original curve tangent, and we know that the tangent is simply B'(t), so we could just rotate that 90 degrees and be done with it. However, we need to ensure that N(t) has the same magnitude for every t, or the offset curve won't be at a uniform distance, thus not being an offset curve at all. The easiest way to guarantee this is to make sure N(t) always has length 1, which we can achieve by dividing B'(t) by its magnitude:

    - - -

    Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc length from some start a to end b, we must use the formula we saw earlier. Noting that "length" is usually denoted with double vertical bars:

    - - -

    So if we want the length of the tangent, we plug in B'(t), with t = 0 as start and -t = 1 as end:

    - - -

    And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.

    -

    There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine because they do, but they don't; remember that only the base function has binomial coefficients. That's before we factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

    -

    So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve.

    -

    And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the real offset curve would look like, if we could compute it.

    -
    + +

    + And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is + screwing everything up, because it turns our nice polynomials into things that are no longer polynomials. +

    +

    + There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with + unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine + because they do, but they don't; remember that only the base function has binomial coefficients. That's before we + factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make + all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call + them circles, and they have much simpler functions than Bézier curves. +

    +

    + So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means + N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means + that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for + O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve. +

    +

    + And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier + curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, + much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the + real offset curve would look like, if we could compute it. +

    + -

    So, you cannot offset a Bézier curve perfectly with another Bézier curve, no matter how high-order you make that other Bézier curve. However, we can chop up a curve into "safe" sub-curves (where "safe" means that all the control points are always on a single side of the baseline, and the midpoint of the curve at t=0.5 is roughly in the center of the polygon defined by the curve coordinates) and then point-scale each sub-curve with respect to its scaling origin (which is the intersection of the point normals at the start and end points).

    -

    A good way to do this reduction is to first find the curve's extreme points, as explained in the earlier section on curve extremities, and use these as initial splitting points. After this initial split, we can check each individual segment to see if it's "safe enough" based on where the center of the curve is. If the on-curve point for t=0.5 is too far off from the center, we simply split the segment down the middle. Generally this is more than enough to end up with safe segments.

    -

    The following graphics show off curve offsetting, and you can use the slider to control the distance at which the curve gets offset. The curve first gets reduced to safe segments, each of which is then offset at the desired distance. Especially for simple curves, particularly easily set up for quadratic curves, no reduction is necessary, but the more twisty the curve gets, the more the curve needs to be reduced in order to get segments that can safely be scaled.

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -

    You may notice that this may still lead to small 'jumps' in the sub-curves when moving the curve around. This is caused by the fact that we're still performing a naive form of offsetting, moving the control points the same distance as the start and end points. If the curve is large enough, this may still lead to incorrect offsets.

    - -
    -
    -

    前目次次
    Graduated curve offsetting

    -

    What if we want to do graduated offsetting, starting at some distance s but ending at some other distance e? Well, if we can compute the length of a curve (which we can if we use the Legendre-Gauss quadrature approach) then we can also determine how far "along the line" any point on the curve is. With that knowledge, we can offset a curve so that its offset curve is not uniformly wide, but graduated between with two different offset widths at the start and end.

    -

    Like normal offsetting we cut up our curve in sub-curves, and then check at which distance along the original curve each sub-curve starts and ends, as well as to which point on the curve each of the control points map. This gives us the distance-along-the-curve for each interesting point in the sub-curve. If we call the total length of all sub-curves seen prior to seeing "the current" sub-curve S (and if the current sub-curve is the first one, S is zero), and we call the full length of our original curve L, then we get the following graduation values:

    -
      -
    • start: map S from interval (0,L) to interval (s,e)
      • -
    • c1: map(<strong>S+d1</strong>, 0,L, s,e), d1 = distance along curve to projection of c1
      • -
    • c2: map(<strong>S+d2</strong>, 0,L, s,e), d2 = distance along curve to projection of c2
      • -
    • ...
      • -
    • end: map(<strong>S+length(subcurve)</strong>, 0,L, s,e)
      • -
    -

    At each of the relevant points (start, end, and the projections of the control points onto the curve) we know the curve's normal, so offsetting is simply a matter of taking our original point, and moving it along the normal vector by the offset distance for each point. Doing so will give us the following result (these have with a starting width of 0, and an end width of 40 pixels, but can be controlled with your up and down arrow keys):

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - - -
    -
    -

    前目次次
    Circles and quadratic Bézier curves

    -

    Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. For instance, if you're doing font design, fonts have no concept of geometric shapes, they only know straight lines, and Bézier curves. OpenType fonts with TrueType outlines only know quadratic Bézier curves, and OpenType fonts with Type 2 outlines only know cubic Bézier curves. So how do you draw a circle, or an ellipse, or an arc?

    -

    You approximate.

    -

    We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.

    -

    Since arcs are mid-point-symmetrical, we need the control points to set up a symmetrical curve. For quadratic curves this means that the control point will be somewhere on a line that intersects the baseline at a right angle. And we don't get any choice on where that will be, since the derivatives at the start and end point have to line up, so our control point will lie at the intersection of the tangents at the start and end point.

    -

    First, let's try to fit the quadratic curve onto a circular arc. In the following sketch you can move the mouse around over a unit circle, to see how well, or poorly, a quadratic curve can approximate the arc from (1,0) to where your mouse cursor is:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    As you can see, things go horribly wrong quite quickly; even trying to approximate a quarter circle using a quadratic curve is a bad idea. An eighth of a turns might look okay, but how okay is okay? Let's apply some maths and find out. What we're interested in is how far off our on-curve coordinates are with respect to a circular arc, given a specific start and end angle. We'll be looking at how much space there is between the circular arc, and the quadratic curve's midpoint.

    -

    We start out with our start and end point, and for convenience we will place them on a unit circle (a circle around 0,0 with radius 1), at some angle φ:

    - - -

    What we want to find is the intersection of the tangents, so we want a point C such that:

    - - -

    i.e. we want a point that lies on the vertical line through S (at some distance a from S) and also lies on the tangent line through E (at some distance b from E). Solving this gives us:

    - - -

    First we solve for b:

    - - -

    which yields:

    - - -

    which we can then substitute in the expression for a:

    - - -

    A quick check shows that plugging these values for a and b into the expressions for Cx and Cy give the same x/y coordinates for both "a away from A" and "b away from B", so let's continue: now that we know the coordinate values for C, we know where our on-curve point T for t=0.5 (or angle φ/2) is, because we can just evaluate the Bézier polynomial, and we know where the circle arc's actual point P is for angle φ/2:

    - - -

    We compute T, observing that if t=0.5, the polynomial values (1-t)², 2(1-t)t, and t² are 0.25, 0.5, and 0.25 respectively:

    - - -

    Which, worked out for the x and y components, gives:

    - - -

    And the distance between these two is the standard Euclidean distance:

    - - -

    So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter circle and eighth circle?

    -
    - - plotted for 0 ≤ φ ≤ π: - - - plotted for 0 ≤ φ ≤ ½π: - - - - - plotted for 0 ≤ φ ≤ ¼π: -
    + +

    + So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter + circle and eighth circle? +

    + + + + + + + + +
    + + plotted for 0 ≤ φ ≤ π: + + + plotted for 0 ≤ φ ≤ ½π: + + + + + plotted for 0 ≤ φ ≤ ¼π: +
    -

    We now see why the eighth circle arc looks decent, but the quarter circle arc doesn't: an error of roughly 0.06 at t=0.5 means we're 6% off the mark... we will already be off by one pixel on a circle with pixel radius 17. Any decent sized quarter circle arc, say with radius 100px, will be way off if approximated by a quadratic curve! For the eighth circle arc, however, the error is only roughly 0.003, or 0.3%, which explains why it looks so close to the actual eighth circle arc. In fact, if we want a truly tiny error, like 0.001, we'll have to contend with an angle of (rounded) 0.593667, which equates to roughly 34 degrees. We'd need 11 quadratic curves to form a full circle with that precision! (technically, 10 and ten seventeenth, but we can't do partial curves, so we have to round up). That's a whole lot of curves just to get a shape that can be drawn using a simple function!

    -

    In fact, let's flip the function around, so that if we plug in the precision error, labelled ε, we get back the maximum angle for that precision:

    - - -

    And frankly, things are starting to look a bit ridiculous at this point, we're doing way more maths than we've ever done, but thankfully this is as far as we need the maths to take us: If we plug in the precisions 0.1, 0.01, 0.001 and 0.0001 we get the radians values 1.748, 1.038, 0.594 and 0.3356; in degrees, that means we can cover roughly 100 degrees (requiring four curves), 59.5 degrees (requiring six curves), 34 degrees (requiring 11 curves), and 19.2 degrees (requiring a whopping nineteen curves).

    -

    The bottom line? Quadratic curves are kind of lousy if you want circular (or elliptical, which are circles that have been squashed in one dimension) curves. We can do better, even if it's just by raising the order of our curve once. So let's try the same thing for cubic curves.

    - -
    -
    -

    前目次次
    Circular arcs and cubic Béziers

    -

    Let's look at approximating circles and circular arcs using cubic Béziers. How much better is that?

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    At cursory glance, a fair bit better, but let's find out how much better by looking at how to construct the Bézier curve.

    -

    A construction diagram for a cubic approximation of a circular arc

    -

    The start and end points are trivial, but the mid point requires a bit of work, but it's mostly basic trigonometry once we know the angle θ for our circular arc: if we scale our circular arc to a unit circle, we can always start our arc, with radius 1, at (1,0) and then given our arc angle θ, we also know that the circular arc has length θ (because unit circles are nice that way). We also know our end point, because that's just (cos(θ), sin(θ)), and so the challenge is to figure out what control points we need in order for the curve at t=0.5 to exactly touch the circular arc at the angle θ/2:

    -

    So let's again formally describe this:

    - - -

    Only P3 isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P4, P3) is a right angled triangle, with the distance between the origin and P4 being 1 (because we're working with a unit circle), and the distance between P4 and P3 being k, so that we can represent P3 as "The point P4 plus the vector from the origin to P4 but then rotated a quarter circle, counter-clockwise, and scaled by k".

    -

    With that, we can determine the y-coordinates for A, B, e1, and e2, after which we have all the information we need to determine what the value of k is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P2 and P3, e1 is between A and "midway between P1 and P2" (which is "half height" P2), and so forth:

    - - -

    Which now gives us two identities for B, because in addition to determining B through linear interpolation, we also know that B's y coordinate is just sin(θ/2): we started this exercise by saying we were going to approximate the circular arc using a Bézier curve that had its midpoint, which is point B, touching the unit circle at the arc's half-angle, by definition making B the point at (cos(θ/2), sin(θ/2)).

    -

    This means we can equate the two identities we now have for By and solve for k.

    -
    - -

    Deriving k

    -

    Solving for k is fairly straight forward, but it's a fair few steps, and if you just the immediate result: using a tool like Wolfram Alpha is definitely the way to go. That said, let's get going:

    - - -

    And finally, we can take further advantage of several trigonometric identities to drastically simplify our formula for k:

    - - -

    And we're done.

    -
    + +

    And we're done.

    + -

    So, the distance of our control points to the start/end points can be expressed as a number that we get from an almost trivial expression involving the circular arc's angle:

    - - -

    Which means that for any circular arc with angle θ and radius r, our Bézier approximation based on three points of incidence is:

    - - -

    Which also gives us the commonly found value of 0.55228 for quarter circles, based on them having an angle of half π:

    - - -

    And thus giving us the following Bézier coordinates for a quarter circle of radius r:

    - - -
    + +
    +

    So, how accurate is this?

    +

    + Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points + that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum + deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but + rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we + get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff. +

    +

    + So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, + finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around + that t value, repeating as many times as necessary to get us an arbitrarily precise value of t: +

    -

    So, how accurate is this?

    -

    Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff.

    -

    So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around that t value, repeating as many times as necessary to get us an arbitrarily precise value of t:

    - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return getMostWrongT(radius, bezier, worst_t - stepsize, worst_t + stepsize) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    Plus, how often do you get to write a function with that name?

    -

    Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity first, and then coming back down as we approach 2π)

    - - -
    - - - - - -
    - error plotted for 0 ≤ φ ≤ 2π - - error plotted for 0 ≤ φ ≤ π - - error plotted for 0 ≤ φ ≤ ½π -
    +

    Plus, how often do you get to write a function with that name?

    +

    + Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see + what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity + first, and then coming back down as we approach 2π) +

    + + + + + + + + + + + + + +
    + + + + + +
    + error plotted for 0 ≤ φ ≤ 2π + + error plotted for 0 ≤ φ ≤ π + + error plotted for 0 ≤ φ ≤ ½π +
    -

    That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again:

    -

    +

    + That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we + can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again: +

    +

    -

    Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves.

    -
    +

    + Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At + approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over + quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves. +

    +
    -

    So that's it, kappa is 4/3 · tan(θ/4) , we're done! ...or are we?

    -

    Can we do better?

    -

    Technically: yes, we can. But I'm going to prefix this section with "we can, and we should investigate that possibility, but let me warn you up front that the result is only better if we're going to hard-code the values". We're about to get into the weeds and the standard three-points-of-incidence value is so good already that for most applications, trying to do better won't make any sense at all.

    -

    So with that said: what we calculated above is an upper bound for a best fit Bézier curve for a circular arc: anywhere we don't touch the circular arc in our approximation, we've "overshot" the arc. What if we dropped our value for k just a little, so that the curve starts out as an over-estimation, but then crosses the circular arc, yielding an region of underestimation, and then crosses the circular arc again, with another region of overestimation. This might give us a lower overall error, so let's see what we can do.

    -

    First, let's express the total error (given circular arc angle θ, and some k) using standard calculus notation:

    - - -

    This says that the error function for a given angle and value of k is equal to the "infinite" sum of differences between our curve and the circular arc, as we run t from 0 to 1, using an infinitely small step size. between subsequent t values.

    -

    Now, since we want to find the minimal error, that means we want to know where along this function things go from "error is getting progressively less" to "error is increasing again", which means we want to know where its derivative is zero, which as mathematical expression looks like:

    - - -

    And here we have the most direct application of the Fundamental Theorem of Calculus: the derivative and integral are each other's inverse operations, so they cancel out, leaving us with our original function:

    - - -

    And now we just solve for that... oh wait. We've seen this before. In order to solve this, we'd end up needing to solve this:

    - - -

    And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section on arc lengths—there is no symbolic solution for this equasion. Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer!

    -
    + +

    + And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section + on arc lengths—there is no symbolic solution for this equasion. + Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer! +

    +
    +

    Iterating on a solution

    +

    + By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the + value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's + center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just + binary search our way to the most accurate value for c that gets us that middle case. +

    +

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    -

    Iterating on a solution

    -

    By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just binary search our way to the most accurate value for c that gets us that middle case.

    -

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    - - - - -
    1 - -
    2
    3
    4
    + return binarySearch(radius, angle, points, lowerBound, upperBound) + + + + 2 + + + 3 + + + 4 + + -

    And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and blog posts than you can ever read in a life time:

    +

    + And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and + blog posts than you can ever read in a life time: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return binarySearch(radius, angle, points, value, upperBound) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Using the following radialError function, which samples the curve's approximation of the circular arc over several points (although the first and last point will never contribute anything, so we skip them):

    +

    + Using the following radialError function, which samples the curve's approximation of the circular arc over several points + (although the first and last point will never contribute anything, so we skip them): +

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return err + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a vector, we can get its length to the origin using a magnitude call.

    -

    Examining the result

    -

    Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, for each angle, plot what the difference between the circular arc and the Bézier approximation looks like:

    -

    image-20210419085430711

    -

    Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from 0 to θ:

    -

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    - - -
    - - - - - -
    - max deflection using unit scale - - max deflection at 10x scale - - max deflection at 100x scale -
    +

    + In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a + vector, we can get its length to the origin using a magnitude call. +

    +

    Examining the result

    +

    + Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, + for each angle, plot what the difference between the circular arc and the Bézier approximation looks like: +

    +

    image-20210419085430711

    +

    + Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the + plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from + 0 to θ: +

    +

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    + + + + + + + + + + + + + +
    + + + + + +
    + max deflection using unit scale + + max deflection at 10x scale + + max deflection at 100x scale +
    -

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    -

    As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference.

    -
    +

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    +

    + As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by + 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an + almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference. +

    +
    -

    At this point it should be clear that while, yes, there are improvement to be had, they're essentially insignificant while also being much more computationally expensive.

    -

    TL;DR: just tell me which value I should be using

    -

    It depends on what we need to do. If we just want the best value for quarter circles, and we're going to hard code the value for k, then there is no reason to hard-code the constant k=4/3*tan(pi/8) when you can just as easily hard-code the constant as k=0.551784777779014 instead.

    -

    If you need "the" value for quarter circles, use 0.551785 instead of 0.55228

    -

    However, for dynamic arc approximation, in code that tries to fit circular paths using Bézier paths instead, it should be fairly obvious that the simple function involving a tangent computation, two divisions, and one multiplication, is vastly more performant than running all the code we ended writing just to get a 25% lower error value, and most certainly worth preferring over getting the "more accurate" value.

    -

    If you need to fit Béziers to circular arcs on the fly, use 4/3 * tan(θ/4)

    -

    However, always remember that if you're writing for humans, you can typically use the best of both worlds: as the user interacts with their curves, you should draw their curves instead of drawing approximations of them. If they need to draw circles or circular arcs, draw those, and only approximate them with a Bézier curve when the data needs to be exported to a format that doesn't support those. Ideally with a preview mechanism that highlights where the errors will be, and how large they will be.

    -

    If you're writing code for graphics design by humans, use circular arcs for circular arcs

    -

    And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best k values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).

    - -
    -
    -

    前目次次
    Approximating Bézier curves with circular arcs

    -

    Let's look at doing the exact opposite of the previous section: rather than approximating circular arc using Bézier curves, let's approximate Bézier curves using circular arcs.

    -

    We already saw in the section on circle approximation that this will never yield a perfect equivalent, but sometimes you need circular arcs, such as when you're working with fabrication machinery, or simple vector languages that understand lines and circles, but not much else.

    -

    The approach is fairly simple: pick a starting point on the curve, and pick two points that are further along the curve. Determine the circle that goes through those three points, and see if it fits the part of the curve we're trying to approximate. Decent fit? Try spacing the points further apart. Bad fit? Try spacing the points closer together. Keep doing this until you've found the "good approximation/bad approximation" boundary, record the "good" arc, and then move the starting point up to overlap the end point we previously found. Rinse and repeat until we've covered the entire curve.

    -

    We already saw how to fit a circle through three points in the section on creating a curve from three points, and finding the arc through those points is straight-forward: pick one of the three points as start point, pick another as an end point, and the arc has to necessarily go from the start point, to the end point, over the remaining point.

    -

    So, how can we convert a Bézier curve into a (sequence of) circular arc(s)?

    -
      -
    • Start at t=0
      • -
    • Pick two points further down the curve at some value m = t + n and e = t + 2n
      • -
    • Find the arc that these points define
      • -
    • Determine how close the found arc is to the curve:
        -
      • Pick two additional points e1 = t + n/2 and e2 = t + n + n/2.
        • -
      • These points, if the arc is a good approximation of the curve interval chosen, should - lie on the circle, so their distance to the center of the circle should be the - same as the distance from any of the three other points to the center.
        • -
      • For point points, determine the (absolute) error between the radius of the circle, and the -actual distance from the center of the circle to the point on the curve.
        • -
      • If this error is too high, we consider the arc bad, and try a smaller interval.
        • -
      -
      • -
    -

    The result of this is shown in the next graphic: we start at a guaranteed failure: s=0, e=1. That's the entire curve. The midpoint is simply at t=0.5, and then we start performing a binary search.

    -
      -
    1. We start with low=0, mid=0.5 and high=1
      2. -
    2. That'll fail, so we retry with the interval halved: {0, 0.25, 0.5}
        -
      • If that arc's good, we move back up by half distance: {0, 0.375, 0.75}.
        • -
      • However, if the arc was still bad, we move down by half the distance: {0, 0.125, 0.25}.
        • -
      -
      3. -
    3. We keep doing this over and over until we have two arcs, in sequence, of which the first arc is good, and the second arc is bad. When we find that pair, we've found the boundary between a good approximation and a bad approximation, and we pick the good arc.
      4. -
    -

    The following graphic shows the result of this approach, with a default error threshold of 0.5, meaning that if an arc is off by a combined half pixel over both verification points, then we treat the arc as bad. This is an extremely simple error policy, but already works really well. Note that the graphic is still interactive, and you can use your up and down arrow keys keys to increase or decrease the error threshold, to see what the effect of a smaller or larger error threshold is.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    With that in place, all that's left now is to "restart" the procedure by treating the found arc's end point as the new to-be-determined arc's starting point, and using points further down the curve. We keep trying this until the found end point is for t=1, at which point we are done. Again, the following graphic allows for up and down arrow key input to increase or decrease the error threshold, so you can see how picking a different threshold changes the number of arcs that are necessary to reasonably approximate a curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    So... what is this good for? Obviously, if you're working with technologies that can't do curves, but can do lines and circles, then the answer is pretty straightforward, but what else? There are some reasons why you might need this technique: using circular arcs means you can determine whether a coordinate lies "on" your curve really easily (simply compute the distance to each circular arc center, and if any of those are close to the arc radii, at an angle between the arc start and end, bingo, this point can be treated as lying "on the curve"). Another benefit is that this approximation is "linear": you can almost trivially travel along the arcs at fixed speed. You can also trivially compute the arc length of the approximated curve (it's a bit like curve flattening). The only thing to bear in mind is that this is a lossy equivalence: things that you compute based on the approximation are guaranteed "off" by some small value, and depending on how much precision you need, arc approximation is either going to be super useful, or completely useless. It's up to you to decide which, based on your application!

    - -
    -
    -

    前目次次
    B-Splines

    -

    No discussion on Bézier curves is complete without also giving mention of that other beast in the curve design space: B-Splines. Easily confused to mean Bézier splines, that's not actually what they are; they are "basis function" splines, which makes a lot of difference, and we'll be looking at those differences in this section. We're not going to dive as deep into B-Splines as we have for Bézier curves (that would be an entire primer on its own) but we'll be looking at how B-Splines work, what kind of maths is involved in computing them, and how to draw them based on a number of parameters that you can pick for individual B-Splines.

    -

    First off: B-Splines are piecewise, polynomial interpolation curves, where the "single curve" is built by performing polynomial interpolation over a set of points, using a sliding window of a fixed number of points. For instance, a "cubic" B-Spline defined by twelve points will have its curve built by evaluating the polynomial interpolation of four points, and the curve can be treated as a lot of different sections, each controlled by four points at a time, such that the full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, ..., {8,9,10,11}, and finally {9,10,11,12}, for eight sections.

    -

    What do they look like? They look like this! Tap on the graphic to add more points, and move points around to see how they map to the spline curve drawn.

    - - - Scripts are disabled. Showing fallback image. - - - -

    The important part to notice here is that we are not doing the same thing with B-Splines that we do for poly-Béziers or Catmull-Rom curves: both of the latter simply define new sections as literally "new sections based on new points", so a 12 point cubic poly-Bézier curve is actually impossible, because we start with a four point curve, and then add three more points for each section that follows, so we can only have 4, 7, 10, 13, 16, etc. point Poly-Béziers. Similarly, while Catmull-Rom curves can grow by adding single points, this addition of a single point introduces three implicit Bézier points. Cubic B-Splines, on the other hand, are smooth interpolations of each possible curve involving four consecutive points, such that at any point along the curve except for our start and end points, our on-curve coordinate is defined by four control points.

    -

    Consider the difference to be this:

    -
      -
    • for Bézier curves, the curve is defined as an interpolation of points, but:
      • -
    • for B-Splines, the curve is defined as an interpolation of curves.
      • -
    -

    In fact, let's look at that again, but this time with the base curves shown, too. Each consecutive four points define one curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have a look at how things work.

    -

    How to compute a B-Spline curve: some maths

    -

    Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the end, just like for Bézier curves), by evaluating the following function:

    - + +

    + In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have + a look at how things work. +

    +

    How to compute a B-Spline curve: some maths

    +

    + Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic + B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the + end, just like for Bézier curves), by evaluating the following function: +

    + - -

    Which, honestly, doesn't tell us all that much. All we can see is that a point on a B-Spline curve is defined as "a mix of all the control points, weighted somehow", where the weighting is achieved through the N(...) function, subscripted with an obvious parameter i, which comes from our summation, and some magical parameter k. So we need to know two things: 1. what does N(t) do, and 2. what is that k? Let's cover both, in reverse order.

    -

    The parameter k represents the "knot interval" over which a section of curve is defined. As we learned earlier, a B-Spline curve is itself an interpolation of curves, and we can treat each transition where a control point starts or stops influencing the total curvature as a "knot on the curve". -Doing so for a degree d B-Spline with n control point gives us d + n + 1 knots, defining d + n intervals along the curve, and it is these intervals that the above k subscript to the N() function applies to.

    -

    Then the N() function itself. What does it look like?

    - - -

    So this is where we see the interpolation: N(t) for an (i,k) pair (that is, for a step in the above summation, on a specific knot interval) is a mix between N(t) for (i,k-1) and N(t) for (i+1,k-1), so we see that this is a recursive iteration where i goes up, and k goes down, so it seem reasonable to expect that this recursion has to stop at some point; obviously, it does, and specifically it does so for the following i/k values:

    - - -

    And this function finally has a straight up evaluation: if a t value lies within a knot-specific interval once we reach a k=1 value, it "counts", otherwise it doesn't. We did cheat a little, though, because for all these values we need to scale our t value first, so that it lies in the interval bounded by knots[d] and knots[n], which are the start point and end point where curvature is controlled by exactly order control points. For instance, for degree 3 (=order 4) and 7 control points, with knot vector [1,2,3,4,5,6,7,8,9,10,11], we map t from [the interval 0,1] to the interval [4,8], and then use that value in the functions above, instead.

    -

    Can we simplify that?

    -

    We can, yes.

    -

    People far smarter than us have looked at this work, and two in particular — Maurice Cox and Carl de Boor — came to a mathematically pleasing solution: to compute a point P(t), we can compute this point by evaluating d(t) on a curve section between knots i and i+1:

    - - -

    This is another recursive function, with k values decreasing from the curve order to 1, and the value α (alpha) defined by:

    - - -

    That looks complicated, but it's not. Computing alpha is just a fraction involving known, plain numbers. And, once we have our alpha value, we also have (1-alpha) because it's a trivial subtraction. Computing the d() function is thus mostly a matter of computing pretty simple arithmetical statements, with some caching of results so we can refer to them as we recurve. While the recursion might see computationally expensive, the total algorithm is cheap, as each step only involves very simple maths.

    -

    Of course, the recursion does need a stop condition:

    - - -

    So, we actually see two stopping conditions: either i becomes 0, in which case d() is zero, or k becomes zero, in which case we get the same "either 1 or 0" that we saw in the N() function above.

    -

    Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily using the same kind of linear interpolation we saw in de Casteljau's algorithm. For instance, if we write out d() for i=3 and k=3, we get the following recursion diagram:

    - - -

    That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. It's really quite elegant.

    -

    One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then summing back up "to the left". We can just start on the right and work our way left immediately.

    -

    Running the computation

    -

    Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on this Michigan Tech page, but an easier to read version is implemented by b-spline.js, so we'll look at its code.

    -

    Given an input value t, we first map the input to a value from the domain [0,1] to the domain [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped t value lies on:

    + +

    + That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a + mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are + themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. + It's really quite elegant. +

    +

    + One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the + d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so + in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point + vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then + summing back up "to the left". We can just start on the right and work our way left immediately. +

    +

    Running the computation

    +

    + Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, + and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on + this Michigan Tech page, but an easier to read version + is implemented by b-spline.js, so we'll look at its code. +

    +

    + Given an input value t, we first map the input to a value from the domain [0,1] to the domain + [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped + t value lies on: +

    - - -
    1 - -
    2
    3
    +} + + + + 2 + + + 3 + + -

    after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU page (updated to use this description's variable names):

    +

    + after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU + page (updated to use this description's variable names): +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those v[i-1] don't try to use an array index that doesn't exist)

    -

    Open vs. closed paths

    -

    Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need more than just a single point.

    -

    Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than separate coordinate objects.

    -

    Manipulating the curve through the knot vector

    -

    The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at:

    -
      -
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. -
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. -
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. -
    4. we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a uniform vector in between.
      5. -
    -

    Uniform B-Splines

    -

    The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to the curvature.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve.

    -

    The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get rid of these, by being clever about how we apply the following uniformity-breaking approach instead...

    -

    Reducing local curve complexity by collapsing intervals

    -

    Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost and the curve "kinks".

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Open-Uniform B-Splines

    -

    By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the curve not starting and ending where we'd kind of like it to:

    -

    For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences that determine the curve shape.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Non-uniform B-Splines

    -

    This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than that any value knots[k+1] should be greater than or equal to knots[k].

    -

    One last thing: Rational B-Splines

    -

    While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like for rational Bézier curves.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in arbitrary-precision 2D design due to the level of control a NURBS curve offers designers.

    -

    While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the last.

    -

    Extending our implementation to cover rational splines

    -

    The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the extended dimension.

    -

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    -

    We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how many dimensions it needs to interpolate.

    -

    In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight information.

    -

    Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing (x'/w', y'/w'). And that's it, we're done!

    +

    + (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the + interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those + v[i-1] don't try to use an array index that doesn't exist) +

    +

    Open vs. closed paths

    +

    + Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last + point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first + and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As + such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly + more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need + more than just a single point. +

    +

    + Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for + points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same + coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing + references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than + separate coordinate objects. +

    +

    Manipulating the curve through the knot vector

    +

    + The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As + mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the + values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on + intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting + things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at: +

    +
      +
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. +
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. +
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. +
    4. + we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a + uniform vector in between. +
      5. +
    +

    Uniform B-Splines

    +

    + The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire + curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or + [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines + the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to + the curvature. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every + interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve. +

    +

    + The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can + perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get + rid of these, by being clever about how we apply the following uniformity-breaking approach instead... +

    +

    Reducing local curve complexity by collapsing intervals

    +

    + Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the + sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, + and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost + and the curve "kinks". +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Open-Uniform B-Splines

    +

    + By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the + curve not starting and ending where we'd kind of like it to: +

    +

    + For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in + which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values + N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector + [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences + that determine the curve shape. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Non-uniform B-Splines

    +

    + This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to + pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than + that any value knots[k+1] should be greater than or equal to knots[k]. +

    +

    One last thing: Rational B-Splines

    +

    + While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's + "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a + ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a + control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like + for rational Bézier curves. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the + NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, + the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an + important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in + arbitrary-precision 2D design due to the level of control a NURBS curve offers designers. +

    +

    + While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform + Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, + they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the + last. +

    +

    Extending our implementation to cover rational splines

    +

    + The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the + control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to + one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the + extended dimension. +

    +

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    +

    + We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate + interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how + many dimensions it needs to interpolate. +

    +

    + In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the + final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight + information. +

    +

    + Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing + (x'/w', y'/w'). And that's it, we're done! +

    +
    +
    + -
    -
    - +

    +
    前目次
    + Comments and questions +

    +

    + First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how + to let me know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing. +

    +

    With that said, on to the comments!

    +
    +
    +
    + -

    前目次
    Comments and questions

    -

    First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing.

    -

    With that said, on to the comments!

    -
    +
    +
    + This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are + almost no rights reserved. Go do something cool with it! +
    - - - - -
    - -
    - - This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are - almost no rights reserved. - Go do something cool with it! - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/ko-KR/index.html b/docs/ko-KR/index.html index f4b41522..f041f238 100644 --- a/docs/ko-KR/index.html +++ b/docs/ko-KR/index.html @@ -1,165 +1,177 @@ + + + + 베지에 곡선 입문 - - - - 베지에 곡선 입문 + - + - + - + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    + - +
    +

    + 베지에 곡선 입문 +

    +

    베지에 곡선이 꼭 필요할 때 읽기 좋은 무료 온라인 서적.

    +
    + 다른 언어로 읽기: + +

    + (원하는 언어가 없거나, 100%까지 올리고 싶다면 + 내용 번역을 도와주세요!) +

    +
    +

    + 베지에 곡선 입문서에 오신 것을 환영합니다. 이곳은 베지에 곡선의 수학과 프로그래밍을 모두 다루는 무료 웹사이트/eBook으로, 포토샵 패스부터 CSS + easing 함수와 글꼴 테두리까지 어느 곳에서든 튀어나오는 바로 그 곡선을 다루고 그리는 데 대한 다양한 주제를 다룹니다. +

    +

    + 이 책은 처음인가요? 어서 오세요! 이 입문서에 없는 것 중 꼭 다루었으면 좋겠는 것이 있다면 + 이슈 트래커로 의견을 나누어 주세요! +

    -
    +

    기부와 후원

    -

    베지에 곡선 입문

    -

    베지에 곡선이 꼭 필요할 때 읽기 좋은 무료 온라인 서적.

    -
    - 다른 언어로 읽기: - -

    (원하는 언어가 없거나, 100%까지 올리고 싶다면 내용 번역을 도와주세요!)

    -
    +

    + If this is a resource that you're using for research, as work reference, or even writing your own software, please consider + donating (any amount helps) or + signing up as a patron on Patreon. I don't get paid to work on this, so if you find this site + valuable, and you'd like it to stick around for a long time to come, a lot of coffee went into writing this over the years, and a lot more + coffee will need to go into it yet: if you can spare a coffee, you'd be helping keep a resource alive and well. +

    +

    + Also, if you are a company and your staff uses this book as a resource, or you use it as an onboarding resource, then please: consider + sponsoring the site! I am more than happy to work with your finance department on sponsorship invoicing and recognition. +

    -

    - 베지에 곡선 입문서에 오신 것을 환영합니다. 이곳은 베지에 곡선의 수학과 프로그래밍을 - 모두 다루는 무료 웹사이트/eBook으로, 포토샵 패스부터 CSS easing 함수와 글꼴 테두리까지 - 어느 곳에서든 튀어나오는 바로 그 곡선을 다루고 그리는 데 대한 다양한 주제를 다룹니다. -

    -

    - 이 책은 처음인가요? 어서 오세요! 이 입문서에 없는 것 중 꼭 다루었으면 좋겠는 것이 있다면 - 이슈 트래커로 의견을 나누어 주세요! -

    - -

    기부와 후원

    - -

    - If this is a resource that you're using for research, as work reference, or even writing your own software, - please consider donating - (any amount helps) or signing up as a patron on Patreon. - I don't get paid to work on this, so if you find this site valuable, and you'd like it - to stick around for a long time to come, a lot of coffee went into writing this over the - years, and a lot more coffee will need to go into it yet: if you can spare a coffee, you'd - be helping keep a resource alive and well. -

    -

    - Also, if you are a company and your staff uses this book as a resource, or you use it as an - onboarding resource, then please: consider sponsoring the site! I am more than happy to work - with your finance department on sponsorship invoicing and recognition. -

    - - - -
    +
    -

    - — Pomax -

    - + +
    +
    +
    +

    머리말

    +

    + 2D에 그림을 그리려면 보통 선을 사용하고, 선이라고 하면 직선과 곡선의 두 종류로 나뉩니다. 전자는 컴퓨터에게 그리기를 시킬 줄 안다면 쉽게 그릴 + 수 있습니다. 시작점과 끝점만 입력하면 짜잔! 직선이 나왔습니다. 어렵지 않죠. +

    +

    + 하지만 곡선은 훨씬 까다롭습니다. 손그림으로는 곡선 그리기가 그렇게 쉬울 수 없었겠지만, 컴퓨터는 곡선을 어떻게 그릴지 기술하는 수학적 함수 + 없이 곡선을 그릴 수 없다는 제한이 있습니다. 실은 직선을 그릴 때조차도 함수가 필요하지만, 어차피 직선의 함수는 간단하니 컴퓨터를 사용할 때는 + 보통 이 사실을 무시합니다. 직선이든 곡선이든 모든 선은 "함수"입니다. 그러나 선 그리기에 함수가 필요하기에, 빠르게 계산할 수 있고 컴퓨터로 + 그렸을 때 보기도 좋은 함수를 만들어야 할 필요성이 생깁니다. 이런 함수는 여러 종류가 있는데, 이 글에서 다룰 것은 이 중 많은 주목을 받고 + 곡선을 그릴 수 있으면 어디서나 쓰이는 것, 바로 베지에 곡선입니다. +

    +

    + 베지에 곡선이라는 이름은 이 곡선이 설계 작업을 하기에 좋다고 세상에 알린 + 피에르 베지에의 이름을 땄지만(1962년에 르노 소속으로서 연구 결과를 발표함), + 베지에가 이 곡선을 "발명"한 최초의, 혹은 유일한 사람은 아닙니다. 베지에 곡선을 처음 발견한 사람은 1959년에 시트로엥에서 일하면서 이 곡선의 + 성질을 연구하고 이 곡선을 그리는 정말 우아한 방법을 찾아낸 폴 드 카스텔죠라고 + 하는 경우도 있습니다. 그러나 드 카스텔죠는 본인의 연구 결과를 발표하지 않았기 때문에 누가 먼저인지를 단언하기는 어렵습니다... 정말일까요? + 사실 베지에 곡선은 근본적으로 "베른시테인 다항식"으로, + 세르게이 나타노비치 베른시테인이 연구하여 자그마치 1912년에 발표한 수학적 함수의 집합입니다. +

    +

    + 아무튼 위의 내용은 잡설이었고, 독자가 관심을 가질 만한 내용은 이 곡선이 편리하다는 것입니다. 여러 개의 베지에 곡선을 연결해서 하나의 + 곡선처럼 보이게 만들 수도 있습니다. 포토샵에서 패스를 한 번이라도 그려봤거나, 플래시, 일러스트레이터, 잉크스케이프 등 벡터 드로잉 프로그램을 + 써본 적이 있다면, 이 프로그램에서 그동안 그리던 곡선이 전부 베지에 곡선입니다. +

    +

    + 그런데 이를 직접 프로그래밍하려면 어떻게 해야 할까요? 함정이 어디에 도사리고 있을까요? 곡선의 바운딩 박스는 어떻게 구하고, 교차 판정은 + 어떻게 하고, 밖으로 튀어나오게는 어떻게 하고... 간단히 말해서, 이 곡선으로 할 만한 온갖 것들을 어떻게 하는 것이 좋을까요? 그것이 바로 이 + 책의 존재 이유입니다. 수학 할 준비 되셨나요? +

    +
    +

    거의 모든 베지에 그래픽이 인터랙티브입니다.

    +

    + 이 페이지에서는 Bezier.js를 사용하여 인터랙티브 예제를 제공하며, 수학 공식은 + XeLaTeX 조판 체계를 이용해 조판하고 David Barton님의 + pdf2svg를 이용해 SVG로 변환하여 표시하고 있습니다. +

    +

    이 책은 오픈 소스입니다.

    +

    + 이 책은 오픈 소스 소프트웨어 프로젝트의 일환으로, 소스 코드가 GitHub 레포지토리 두 곳에 보관되어 있습니다. 하나는 + https://github.com/pomax/bezierinfo로, 지금 읽고 계시는 완전히 공개용인 버전입니다. 다른 + 하나는 개발 버전인 https://github.com/pomax/BezierInfo-2로 + 웹 버전으로 변환되는 코드가 보관되어 있습니다. 버그를 찾아서 제보하려고 하거나 입문서의 내용을 수정하거나 추가할 아이디어가 + 있다면 이곳으로 보내 주세요. +

    +

    이 책의 수식은 얼마나 어렵나요?

    +

    + 입문서에 등장하는 대부분의 수식은 고등학교 수준입니다. 기본적인 산수 계산을 할 수 있고 영어를 읽을 줄 안다면 대부분 이해할 수 있을 + 정도입니다. 훨씬 어려운 수식도 가끔 등장하지만, 너무 어려워 보이는 상자 안의 세부 내용을 건너뛰거나, 읽고 있는 장의 끝으로 + 건너뛰어도 좋습니다. 각 장의 끝에는 도출된 값만 활용할 수 있도록 결론을 단순 나열합니다. +

    +

    예제 코드는 무슨 언어로 되어 있나요?

    +

    + 특정한 프로그래밍 언어 하나를 선호하기에는 프로그래밍 언어가 너무 많기 때문에, 입문서에 수록된 모든 예제는 JS나 Python 등 현대적인 + 스크립트 언어와 문법이 어느 정도 비슷하지만 같지는 않은 의사코드로 작성했습니다. 그렇기 때문에 예제 코드를 생각 없이 복사해서 쓸 수 + 없지만, 이는 의도적인 결정입니다. 이 입문서를 보고 계시는 것은 웬만하면 배우기 위해서이고, 배움은 복붙으로 이루어지지 않죠. + 학습은 무언가를 직접 해 보고, 실수를 하며, 그 실수를 고치면서 이루어집니다. 독자들이 실수를 하라고 일부러 예제 코드에 오류를 + 넣는다거나 하는 것은 당연히 아니지만(그건 끔찍하겠죠!), 어떤 한 프로그래밍 언어에 편중된 코드는 의도적으로 지양하고 있습니다. + 절차적 프로그래밍 언어를 하나라도 알고 있다면 예제 코드를 읽는 데는 전혀 어려움이 없으니 걱정은 하지 말아주세요. +

    +

    질문이나 의견이 있다면

    +

    + 새로운 장을 제안하고 싶다면, GitHub 이슈 트래커를 찾아 주세요(오른쪽 위에 + 링크한 레포지토리에서도 확인할 수 있습니다). 코드 재작성이 끝날 때까지는 댓글창이 없을 예정이지만, 내용에 대한 질문이 있다면 역시 이슈 + 트래커를 이용할 수 있습니다. 재작성을 마치면 내용 전반에 대한 댓글창을 추가하고, 확실치는 않지만 아마 "이 부분을 마우스로 선택하고 질문 + 버튼을 눌러서 질문하기" 시스템을 만들 수도 있겠습니다. +

    +

    이 책을 지원해 주세요!

    +

    + 이 책을 재미있게 읽었거나 하려고 했던 작업에 도움이 되었고, 제게 고마움을 표하고 싶다면 두 가지 방법이 있습니다. 이 책의 + Patreon 페이지에서 정기 후원을 하거나, 일회성 후원을 하고 싶다면 + buy Pomax a coffee 페이지를 찾아 + 주세요. 이 책은 지난 몇 년 동안 베지에 곡선에 관한 조그마한 입문서에서 종이책 100페이지짜리 교재로 커졌고, 책을 쓰면서 수백 잔의 커피가 + 들었습니다. 저는 이 책을 쓰게 된 것이 전혀 후회되지 않지만, 약간의 커피만 더 있으면 집필을 이어나갈 수 있겠죠! +

    +
    +
    +
    +

    업데이트 내역

    +

    + 이 입문서는 "살아 숨쉬는" 문서로, 오랜만에 다시 찾아왔다면 새로운 내용이 추가되었을 수도 있습니다. 아래 링크를 눌러서 무엇이 언제 + 추가되었는지 확인할 수도 있고, 뉴스 포스트(영문, RSS 피드도 있습니다)에서 자세한 내역을 + 확인할 수도 있습니다. +

    + + + +
    +

    November 2020

    +
      +

    • Added a section on finding curve/circle intersections

      • +
    +

    October 2020

    +
      +

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      • +
    +

    August-September 2020

    +
      +
    • +

      + Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS + turned on. +

      +
      • +
    +

    June 2020

    +
      +

    • Added automatic CI/CD using Github Actions

      • +
    +

    January 2020

    +
      +

    • Added reset buttons to all graphics

      • +

    • Updated to preface to correctly describe the on-page maths

      • +

    • Fixed the Catmull-Rom section because it had glaring maths errors

      • +
    +

    August 2019

    +
      +

    • Added a section on (plain) rational Bezier curves

      • +

    • Improved the Graphic component to allow for sliders

      • +
    +

    December 2018

    +
      +

    • Added a section on curvature and calculating kappa.

      • +
    • +

      + Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this + site! +

      +
      • +
    +

    August 2018

    +
      +

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      • +
    +

    July 2018

    +
      +

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      • +

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      • +
    • +

      (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      +
      • +
    +

    June 2018

    +
      +

    • Added a section on direct curve fitting.

      • +

    • Added source links for all graphics.

      • +

    • Added this "What's new?" section.

      • +
    +

    April 2017

    +
      +

    • Added a section on 3d normals.

      • +

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      • +
    +

    February 2017

    +
      +

    • Finished rewriting the entire codebase for localization.

      • +
    +

    January 2016

    +
      +

    • Added a section to explain the Bezier interval.

      • +

    • Rewrote the Primer as a React application.

      • +
    +

    December 2015

    +
      +

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      • +
    • +

      + Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics. +

      +
      • +
    +

    May 2015

    +
      +

    • Switched over to pure JS rather than Processing-through-Processing.js

      • +

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      • +
    +

    April 2015

    +
      +

    • Added a section on arc length approximations.

      • +
    +

    February 2015

    +
      +

    • Added a section on the canonical cubic Bezier form.

      • +
    +

    November 2014

    +
      +

    • Switched to HTTPS.

      • +
    +

    July 2014

    +
      +

    • Added the section on arc approximation.

      • +
    +

    April 2014

    +
      +

    • Added the section on Catmull-Rom fitting.

      • +
    +

    November 2013

    +
      +

    • Added the section on Catmull-Rom / Bezier conversion.

      • +

    • Added the section on Bezier cuves as matrices.

      • +
    +

    April 2013

    +
      +

    • Added a section on poly-Beziers.

      • +

    • Added a section on boolean shape operations.

      • +
    +

    March 2013

    +
      +

    • First drastic rewrite.

      • +

    • Added sections on circle approximations.

      • +

    • Added a section on projecting a point onto a curve.

      • +

    • Added a section on tangents and normals.

      • +

    • Added Legendre-Gauss numerical data tables.

      • +
    +

    October 2011

    +
      +
    • +

      + First commit for the bezierinfo site, based on the pre-Primer webpage that covered + the basics of Bezier curves in HTML with Processing.js examples. +

      +
      • +
    +
    +
    +
    +
    +

    +
    목차다음
    + A lightning introduction +

    +

    + Let's start with the good stuff: when we're talking about Bézier curves, we're talking about the things that you can see in the following + graphics. They run from some start point to some end point, with their curvature influenced by one or more "intermediate" control points. + Now, because all the graphics on this page are interactive, go manipulate those curves a bit: click-drag the points, and see how their + shape changes based on what you do. +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    -
    - -
    - -

    머리말

    -

    2D에 그림을 그리려면 보통 선을 사용하고, 선이라고 하면 직선과 곡선의 두 종류로 나뉩니다. 전자는 컴퓨터에게 그리기를 시킬 줄 안다면 쉽게 그릴 수 있습니다. 시작점과 끝점만 입력하면 짜잔! 직선이 나왔습니다. 어렵지 않죠.

    -

    하지만 곡선은 훨씬 까다롭습니다. 손그림으로는 곡선 그리기가 그렇게 쉬울 수 없었겠지만, 컴퓨터는 곡선을 어떻게 그릴지 기술하는 수학적 함수 없이 곡선을 그릴 수 없다는 제한이 있습니다. 실은 직선을 그릴 때조차도 함수가 필요하지만, 어차피 직선의 함수는 간단하니 컴퓨터를 사용할 때는 보통 이 사실을 무시합니다. 직선이든 곡선이든 모든 선은 "함수"입니다. 그러나 선 그리기에 함수가 필요하기에, 빠르게 계산할 수 있고 컴퓨터로 그렸을 때 보기도 좋은 함수를 만들어야 할 필요성이 생깁니다. 이런 함수는 여러 종류가 있는데, 이 글에서 다룰 것은 이 중 많은 주목을 받고 곡선을 그릴 수 있으면 어디서나 쓰이는 것, 바로 베지에 곡선입니다.

    -

    베지에 곡선이라는 이름은 이 곡선이 설계 작업을 하기에 좋다고 세상에 알린 피에르 베지에의 이름을 땄지만(1962년에 르노 소속으로서 연구 결과를 발표함), 베지에가 이 곡선을 "발명"한 최초의, 혹은 유일한 사람은 아닙니다. 베지에 곡선을 처음 발견한 사람은 1959년에 시트로엥에서 일하면서 이 곡선의 성질을 연구하고 이 곡선을 그리는 정말 우아한 방법을 찾아낸 폴 드 카스텔죠라고 하는 경우도 있습니다. 그러나 드 카스텔죠는 본인의 연구 결과를 발표하지 않았기 때문에 누가 먼저인지를 단언하기는 어렵습니다... 정말일까요? 사실 베지에 곡선은 근본적으로 "베른시테인 다항식"으로, 세르게이 나타노비치 베른시테인이 연구하여 자그마치 1912년에 발표한 수학적 함수의 집합입니다.

    -

    아무튼 위의 내용은 잡설이었고, 독자가 관심을 가질 만한 내용은 이 곡선이 편리하다는 것입니다. 여러 개의 베지에 곡선을 연결해서 하나의 곡선처럼 보이게 만들 수도 있습니다. 포토샵에서 패스를 한 번이라도 그려봤거나, 플래시, 일러스트레이터, 잉크스케이프 등 벡터 드로잉 프로그램을 써본 적이 있다면, 이 프로그램에서 그동안 그리던 곡선이 전부 베지에 곡선입니다.

    -

    그런데 이를 직접 프로그래밍하려면 어떻게 해야 할까요? 함정이 어디에 도사리고 있을까요? 곡선의 바운딩 박스는 어떻게 구하고, 교차 판정은 어떻게 하고, 밖으로 튀어나오게는 어떻게 하고... 간단히 말해서, 이 곡선으로 할 만한 온갖 것들을 어떻게 하는 것이 좋을까요? 그것이 바로 이 책의 존재 이유입니다. 수학 할 준비 되셨나요?

    -
    - -

    거의 모든 베지에 그래픽이 인터랙티브입니다.

    -

    이 페이지에서는 Bezier.js를 사용하여 인터랙티브 예제를 제공하며, 수학 공식은 XeLaTeX 조판 체계를 이용해 조판하고 David Barton님의 pdf2svg를 이용해 SVG로 변환하여 표시하고 있습니다.

    -

    이 책은 오픈 소스입니다.

    -

    이 책은 오픈 소스 소프트웨어 프로젝트의 일환으로, 소스 코드가 GitHub 레포지토리 두 곳에 보관되어 있습니다. 하나는 https://github.com/pomax/bezierinfo로, 지금 읽고 계시는 완전히 공개용인 버전입니다. 다른 하나는 개발 버전인 https://github.com/pomax/BezierInfo-2로 웹 버전으로 변환되는 코드가 보관되어 있습니다. 버그를 찾아서 제보하려고 하거나 입문서의 내용을 수정하거나 추가할 아이디어가 있다면 이곳으로 보내 주세요.

    -

    이 책의 수식은 얼마나 어렵나요?

    -

    입문서에 등장하는 대부분의 수식은 고등학교 수준입니다. 기본적인 산수 계산을 할 수 있고 영어를 읽을 줄 안다면 대부분 이해할 수 있을 정도입니다. 훨씬 어려운 수식도 가끔 등장하지만, 너무 어려워 보이는 상자 안의 세부 내용을 건너뛰거나, 읽고 있는 장의 끝으로 건너뛰어도 좋습니다. 각 장의 끝에는 도출된 값만 활용할 수 있도록 결론을 단순 나열합니다.

    -

    예제 코드는 무슨 언어로 되어 있나요?

    -

    특정한 프로그래밍 언어 하나를 선호하기에는 프로그래밍 언어가 너무 많기 때문에, 입문서에 수록된 모든 예제는 JS나 Python 등 현대적인 스크립트 언어와 문법이 어느 정도 비슷하지만 같지는 않은 의사코드로 작성했습니다. 그렇기 때문에 예제 코드를 생각 없이 복사해서 쓸 수 없지만, 이는 의도적인 결정입니다. 이 입문서를 보고 계시는 것은 웬만하면 배우기 위해서이고, 배움은 복붙으로 이루어지지 않죠. 학습은 무언가를 직접 해 보고, 실수를 하며, 그 실수를 고치면서 이루어집니다. 독자들이 실수를 하라고 일부러 예제 코드에 오류를 넣는다거나 하는 것은 당연히 아니지만(그건 끔찍하겠죠!), 어떤 한 프로그래밍 언어에 편중된 코드는 의도적으로 지양하고 있습니다. 절차적 프로그래밍 언어를 하나라도 알고 있다면 예제 코드를 읽는 데는 전혀 어려움이 없으니 걱정은 하지 말아주세요.

    -

    질문이나 의견이 있다면

    -

    새로운 장을 제안하고 싶다면, GitHub 이슈 트래커를 찾아 주세요(오른쪽 위에 링크한 레포지토리에서도 확인할 수 있습니다). 코드 재작성이 끝날 때까지는 댓글창이 없을 예정이지만, 내용에 대한 질문이 있다면 역시 이슈 트래커를 이용할 수 있습니다. 재작성을 마치면 내용 전반에 대한 댓글창을 추가하고, 확실치는 않지만 아마 "이 부분을 마우스로 선택하고 질문 버튼을 눌러서 질문하기" 시스템을 만들 수도 있겠습니다.

    -

    이 책을 지원해 주세요!

    -

    이 책을 재미있게 읽었거나 하려고 했던 작업에 도움이 되었고, 제게 고마움을 표하고 싶다면 두 가지 방법이 있습니다. 이 책의 Patreon 페이지에서 정기 후원을 하거나, 일회성 후원을 하고 싶다면 buy Pomax a coffee 페이지를 찾아 주세요. 이 책은 지난 몇 년 동안 베지에 곡선에 관한 조그마한 입문서에서 종이책 100페이지짜리 교재로 커졌고, 책을 쓰면서 수백 잔의 커피가 들었습니다. 저는 이 책을 쓰게 된 것이 전혀 후회되지 않지만, 약간의 커피만 더 있으면 집필을 이어나갈 수 있겠죠!

    -
    -
    -
    -

    업데이트 내역

    -

    이 입문서는 "살아 숨쉬는" 문서로, 오랜만에 다시 찾아왔다면 새로운 내용이 추가되었을 수도 있습니다. 아래 링크를 눌러서 무엇이 언제 추가되었는지 확인할 수도 있고, 뉴스 포스트(영문, RSS 피드도 있습니다)에서 자세한 내역을 확인할 수도 있습니다.

    - - - -

    November 2020

    • Added a section on finding curve/circle intersections

      -
    -

    October 2020

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      -
    -

    August-September 2020

    • Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS turned on.

      -
    -

    June 2020

    • Added automatic CI/CD using Github Actions

      -
    -

    January 2020

    • Added reset buttons to all graphics

      -
      • -

    • Updated to preface to correctly describe the on-page maths

      -
      • -

    • Fixed the Catmull-Rom section because it had glaring maths errors

      -
    -

    August 2019

    • Added a section on (plain) rational Bezier curves

      -
      • -

    • Improved the Graphic component to allow for sliders

      -
    -

    December 2018

    • Added a section on curvature and calculating kappa.

      -
      • -

    • Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this site!

      -
    -

    August 2018

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      -
    -

    July 2018

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      -
      • -

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      -
      • -

    • (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      -
    -

    June 2018

    • Added a section on direct curve fitting.

      -
      • -

    • Added source links for all graphics.

      -
      • -

    • Added this "What's new?" section.

      -
    -

    April 2017

    • Added a section on 3d normals.

      -
      • -

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      -
    -

    February 2017

    • Finished rewriting the entire codebase for localization.

      -
    -

    January 2016

    • Added a section to explain the Bezier interval.

      -
      • -

    • Rewrote the Primer as a React application.

      -
    -

    December 2015

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      -
      • -

    • Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics.

      -
    -

    May 2015

    • Switched over to pure JS rather than Processing-through-Processing.js

      -
      • -

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      -
    -

    April 2015

    • Added a section on arc length approximations.

      -
    -

    February 2015

    • Added a section on the canonical cubic Bezier form.

      -
    -

    November 2014

    • Switched to HTTPS.

      -
    -

    July 2014

    • Added the section on arc approximation.

      -
    -

    April 2014

    • Added the section on Catmull-Rom fitting.

      -
    -

    November 2013

    • Added the section on Catmull-Rom / Bezier conversion.

      -
      • -

    • Added the section on Bezier cuves as matrices.

      -
    -

    April 2013

    • Added a section on poly-Beziers.

      -
      • -

    • Added a section on boolean shape operations.

      -
    -

    March 2013

    • First drastic rewrite.

      -
      • -

    • Added sections on circle approximations.

      -
      • -

    • Added a section on projecting a point onto a curve.

      -
      • -

    • Added a section on tangents and normals.

      -
      • -

    • Added Legendre-Gauss numerical data tables.

      -
    -

    October 2011

    • First commit for the bezierinfo site, based on the pre-Primer webpage that covered the basics of Bezier curves in HTML with Processing.js examples.

      -
    -
    -
    -

    목차다음
    A lightning introduction

    -

    Let's start with the good stuff: when we're talking about Bézier curves, we're talking about the things that you can see in the following graphics. They run from some start point to some end point, with their curvature influenced by one or more "intermediate" control points. Now, because all the graphics on this page are interactive, go manipulate those curves a bit: click-drag the points, and see how their shape changes based on what you do.

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    - -

    These curves are used a lot in computer aided design and computer aided manufacturing (CAD/CAM) applications, as well as in graphic design programs like Adobe Illustrator and Photoshop, Inkscape, GIMP, etc. and in graphic technologies like scalable vector graphics (SVG) and OpenType fonts (TTF/OTF). A lot of things use Bézier curves, so if you want to learn more about them... prepare to get your learn on!

    - -
    -
    -

    이전목차다음
    So what makes a Bézier Curve?

    -

    Playing with the points for curves may have given you a feel for how Bézier curves behave, but what are Bézier curves, really? There are two ways to explain what a Bézier curve is, and they turn out to be the entirely equivalent, but one of them uses complicated maths, and the other uses really simple maths. So... let's start with the simple explanation:

    -

    Bézier curves are the result of linear interpolations. That sounds complicated but you've been doing linear interpolation since you were very young: any time you had to point at something between two other things, you've been applying linear interpolation. It's simply "picking a point between two points".

    -

    If we know the distance between those two points, and we want a new point that is, say, 20% the distance away from the first point (and thus 80% the distance away from the second point) then we can compute that really easily:

    - - -

    So let's look at that in action: the following graphic is interactive in that you can use your up and down arrow keys to increase or decrease the interpolation ratio, to see what happens. We start with three points, which gives us two lines. Linear interpolation over those lines gives us two points, between which we can again perform linear interpolation, yielding a single point. And that point —and all points we can form in this way for all ratios taken together— form our Bézier curve:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    And that brings us to the complicated maths: calculus.

    -

    While it doesn't look like that's what we've just done, we actually just drew a quadratic curve, in steps, rather than in a single go. One of the fascinating parts about Bézier curves is that they can both be described in terms of polynomial functions, as well as in terms of very simple interpolations of interpolations of [...]. That, in turn, means we can look at what these curves can do based on both "real maths" (by examining the functions, their derivatives, and all that stuff), as well as by looking at the "mechanical" composition (which tells us, for instance, that a curve will never extend beyond the points we used to construct it).

    -

    So let's start looking at Bézier curves a bit more in depth: their mathematical expressions, the properties we can derive from them, and the various things we can do to, and with, Bézier curves.

    - -
    -
    -

    이전목차다음
    The mathematics of Bézier curves

    -

    Bézier curves are a form of "parametric" function. Mathematically speaking, parametric functions are cheats: a "function" is actually a well defined term representing a mapping from any number of inputs to a single output. Numbers go in, a single number comes out. Change the numbers that go in, and the number that comes out is still a single number.

    -

    Parametric functions cheat. They basically say "alright, well, we want multiple values coming out, so we'll just use more than one function". An illustration: Let's say we have a function that maps some value, let's call it x, to some other value, using some kind of number manipulation:

    - - -

    The notation f(x) is the standard way to show that it's a function (by convention called f if we're only listing one) and its output changes based on one variable (in this case, x). Change x, and the output for f(x) changes.

    -

    So far, so good. Now, let's look at parametric functions, and how they cheat. Let's take the following two functions:

    - - -

    There's nothing really remarkable about them, they're just a sine and cosine function, but you'll notice the inputs have different names. If we change the value for a, we're not going to change the output value for f(b), since a isn't used in that function. Parametric functions cheat by changing that. In a parametric function all the different functions share a variable, like this:

    - - -

    Multiple functions, but only one variable. If we change the value for t, we change the outcome of both fa(t) and fb(t). You might wonder how that's useful, and the answer is actually pretty simple: if we change the labels fa(t) and fb(t) with what we usually mean with them for parametric curves, things might be a lot more obvious:

    - - -

    There we go. x/y coordinates, linked through some mystery value t.

    -

    So, parametric curves don't define a y coordinate in terms of an x coordinate, like normal functions do, but they instead link the values to a "control" variable. If we vary the value of t, then with every change we get two values, which we can use as (x,y) coordinates in a graph. The above set of functions, for instance, generates points on a circle: We can range t from negative to positive infinity, and the resulting (x,y) coordinates will always lie on a circle with radius 1 around the origin (0,0). If we plot it for t from 0 to 5, we get this:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Bézier curves are just one out of the many classes of parametric functions, and are characterised by using the same base function for all of the output values. In the example we saw above, the x and y values were generated by different functions (one uses a sine, the other a cosine); but Bézier curves use the "binomial polynomial" for both the x and y outputs. So what are binomial polynomials?

    -

    You may remember polynomials from high school. They're those sums that look like this:

    - - -

    If the highest order term they have is x³, they're called "cubic" polynomials; if it's x², it's a "square" polynomial; if it's just x, it's a line (and if there aren't even any terms with x it's not a polynomial!)

    -

    Bézier curves are polynomials of t, rather than x, with the value for t being fixed between 0 and 1, with coefficients a, b etc. taking the "binomial" form, which sounds fancy but is actually a pretty simple description for mixing values:

    - - -

    I know what you're thinking: that doesn't look too simple! But if we remove t and add in "times one", things suddenly look pretty easy. Check out these binomial terms:

    - - -

    Notice that 2 is the same as 1+1, and 3 is 2+1 and 1+2, and 6 is 3+3... As you can see, each time we go up a dimension, we simply start and end with 1, and everything in between is just "the two numbers above it, added together", giving us a simple number sequence known as Pascal's triangle. Now that's easy to remember.

    -

    There's an equally simple way to figure out how the polynomial terms work: if we rename (1-t) to a and t to b, and remove the weights for a moment, we get this:

    - - -

    It's basically just a sum of "every combination of a and b", progressively replacing a's with b's after every + sign. So that's actually pretty simple too. So now you know binomial polynomials, and just for completeness I'm going to show you the generic function for this:

    - - -

    And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ).

    -
    + +

    + And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable + listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ). +

    +
    +

    How to implement the basis function

    +

    We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:

    -

    How to implement the basis function

    -

    We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers in between being the sum of the previous row's elements on either side "above" the one we're computing.

    -

    We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup table:

    +

    + I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above + explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then + [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers + in between being the sum of the previous row's elements on either side "above" the one we're computing. +

    +

    + We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup + table: +

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    + return lut[n][k] + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + -

    So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, we expand it first. Our basis function now looks like this:

    +

    + So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, + we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, + we expand it first. Our basis function now looks like this: +

    - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves (although we will also provide code for arbitrary curves in this primer); we need quadratic and cubic curves, and that means we can drastically simplify the code:

    +

    + Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves (although we will also + provide code for arbitrary curves in this primer); we need quadratic and cubic curves, and that means we can drastically simplify the + code: +

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return mt3 + 3*mt2*t + 3*mt*t2 + t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    And now we know how to program the basis function. Excellent.

    -
    +

    And now we know how to program the basis function. Excellent.

    +
    -

    So, now we know what the basis function looks like, time to add in the magic that makes Bézier curves so special: control points.

    +

    So, now we know what the basis function looks like, time to add in the magic that makes Bézier curves so special: control points.

    +
    +
    +

    +
    + 이전목차다음 +
    + Controlling Bézier curvatures +

    +

    + Bézier curves are, like all "splines", interpolation functions. This means that they take a set of points, and generate values somewhere + "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline + for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point + contributes to the value generated by the function, so we can see which points are important, where, in the curve. +

    +

    + The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's + contribution to the total sum of the Bézier function. Click-and-drag to see the interpolation percentages for each curve-defining point at + a specific t value. +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    -
    -
    -

    이전목차다음
    Controlling Bézier curvatures

    -

    Bézier curves are, like all "splines", interpolation functions. This means that they take a set of points, and generate values somewhere "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point contributes to the value generated by the function, so we can see which points are important, where, in the curve.

    -

    The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's contribution to the total sum of the Bézier function. Click-and-drag to see the interpolation percentages for each curve-defining point at a specific t value.

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -
    - -

    Also shown is the interpolation function for a 15th order Bézier function. As you can see, the start and end point contribute considerably more to the curve's shape than any other point in the control point set.

    -

    If we want to change the curve, we need to change the weights of each point, effectively changing the interpolations. The way to do this is about as straightforward as possible: just multiply each point with a value that changes its strength. These values are conventionally called "weights", and we can add them to our original Bézier function:

    - - -

    That looks complicated, but as it so happens, the "weights" are actually just the coordinate values we want our curve to have: for an nth order curve, w0 is our start coordinate, wn is our last coordinate, and everything in between is a controlling coordinate. Say we want a cubic curve that starts at (110,150), is controlled by (25,190) and (210,250) and ends at (210,30), we use this Bézier curve:

    - - -

    Which gives us the curve we saw at the top of the article:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and algorithms that we can apply, and the tasks they achieve.

    -
    + +

    Which gives us the curve we saw at the top of the article:

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +

    + What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and + algorithms that we can apply, and the tasks they achieve. +

    +
    +

    How to implement the weighted basis function

    +

    Given that we already know how to implement basis function, adding in the control points is remarkably easy:

    -

    How to implement the weighted basis function

    -

    Given that we already know how to implement basis function, adding in the control points is remarkably easy:

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    And now for the optimized versions:

    +

    And now for the optimized versions:

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    And now we know how to program the weighted basis function.

    -
    - -
    -
    -

    이전목차다음
    Controlling Bézier curvatures, part 2: Rational Béziers

    -

    We can further control Bézier curves by "rationalising" them: that is, adding a "ratio" value in addition to the weight value discussed in the previous section, thereby gaining control over "how strongly" each coordinate influences the curve.

    -

    Adding these ratio values to the regular Bézier curve function is fairly easy. Where the regular function is the following:

    - - -

    The function for rational Bézier curves has two more terms:

    - - -

    In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just using different weight. So far, nothing too special.

    -

    However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights.

    -

    This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly describing circles (which we'll see in a later section is literally impossible using standard Bézier curves).

    -

    But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects what gets drawn:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - -

    You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence relative to all other points.

    -
    + +

    + In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] + then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just + using different weight. So far, nothing too special. +

    +

    + However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a + fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the + "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights. +

    +

    + This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve + that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly + describing circles (which we'll see in a later section is literally impossible using standard Bézier curves). +

    +

    + But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an + interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with + ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate + exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects + what gets drawn: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + + + +

    + You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will + want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like + with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence + relative to all other points. +

    +
    +

    How to implement rational curves

    +

    Extending the code of the previous section to include ratios is almost trivial:

    -

    How to implement rational curves

    -

    Extending the code of the previous section to include ratios is almost trivial:

    - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    + return (f[0] * w[0] + f[1] * w[1] + f[2] * w[2] + f[3] * w[3])/basis + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + -

    And that's all we have to do.

    -
    - -
    -
    -

    이전목차다음
    The Bézier interval [0,1]

    -

    Now that we know the mathematics behind Bézier curves, there's one curious thing that you may have noticed: they always run from t=0 to t=1. Why that particular interval?

    -

    It all has to do with how we run from "the start" of our curve to "the end" of our curve. If we have a value that is a mixture of two other values, then the general formula for this is:

    - - -

    The obvious start and end values here need to be a=1, b=0, so that the mixed value is 100% value 1, and 0% value 2, and a=0, b=1, so that the mixed value is 0% value 1 and 100% value 2. Additionally, we don't want "a" and "b" to be independent: if they are, then we could just pick whatever values we like, and end up with a mixed value that is, for example, 100% value 1 and 100% value 2. In principle that's fine, but for Bézier curves we always want mixed values between the start and end point, so we need to make sure we can never set "a" and "b" to some values that lead to a mix value that sums to more than 100%. And that's easy:

    - - -

    With this we can guarantee that we never sum above 100%. By restricting a to values in the interval [0,1], we will always be somewhere between our two values (inclusively), and we will always sum to a 100% mix.

    -

    But... what if we use this form, which is based on the assumption that we will only ever use values between 0 and 1, and instead use values outside of that interval? Do things go horribly wrong? Well... not really, but we get to "see more".

    -

    In the case of Bézier curves, extending the interval simply makes our curve "keep going". Bézier curves are simply segments of some polynomial curve, so if we pick a wider interval we simply get to see more of the curve. So what do they look like?

    -

    The following two graphics show you Bézier curves rendered "the usual way", as well as the curves they "lie on" if we were to extend the t values much further. As you can see, there's a lot more "shape" hidden in the rest of the curve, and we can model those parts by moving the curve points around.

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    + +

    + With this we can guarantee that we never sum above 100%. By restricting a to values in the interval [0,1], we will always be + somewhere between our two values (inclusively), and we will always sum to a 100% mix. +

    +

    + But... what if we use this form, which is based on the assumption that we will only ever use values between 0 and 1, and instead use + values outside of that interval? Do things go horribly wrong? Well... not really, but we get to "see more". +

    +

    + In the case of Bézier curves, extending the interval simply makes our curve "keep going". Bézier curves are simply segments of some + polynomial curve, so if we pick a wider interval we simply get to see more of the curve. So what do they look like? +

    +

    + The following two graphics show you Bézier curves rendered "the usual way", as well as the curves they "lie on" if we were to extend the + t values much further. As you can see, there's a lot more "shape" hidden in the rest of the curve, and we can model those + parts by moving the curve points around. +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    -

    In fact, there are curves used in graphics design and computer modelling that do the opposite of Bézier curves; rather than fixing the interval, and giving you freedom to choose the coordinates, they fix the coordinates, but give you freedom over the interval. A great example of this is the "Spiro" curve, which is a curve based on part of a Cornu Spiral, also known as Euler's Spiral. It's a very aesthetically pleasing curve and you'll find it in quite a few graphics packages like FontForge and Inkscape. It has even been used in font design, for example for the Inconsolata typeface.

    - -
    -
    -

    이전목차다음
    Bézier curvatures as matrix operations

    -

    We can also represent Bézier curves as matrix operations, by expressing the Bézier formula as a polynomial basis function and a coefficients matrix, and the actual coordinates as a matrix. Let's look at what this means for the cubic curve, using P... to refer to coordinate values "in one or more dimensions":

    - - -

    Disregarding our actual coordinates for a moment, we have:

    - - -

    We can write this as a sum of four expressions:

    - - -

    And we can expand these expressions:

    - - -

    Furthermore, we can make all the 1 and 0 factors explicit:

    - - -

    And that, we can view as a series of four matrix operations:

    - - -

    If we compact this into a single matrix operation, we get:

    - - -

    This kind of polynomial basis representation is generally written with the bases in increasing order, which means we need to flip our t matrix horizontally, and our big "mixing" matrix upside down:

    - - -

    And then finally, we can add in our original coordinates as a single third matrix:

    - - -

    We can perform the same trick for the quadratic curve, in which case we end up with:

    - - -

    If we plug in a t value, and then multiply the matrices, we will get exactly the same values as when we evaluate the original polynomial function, or as when we evaluate the curve using progressive linear interpolation.

    -

    So: why would we bother with matrices? Matrix representations allow us to discover things about functions that would otherwise be hard to tell. It turns out that the curves form triangular matrices, and they have a determinant equal to the product of the actual coordinates we use for our curve. It's also invertible, which means there's a ton of properties that are all satisfied. Of course, the main question is "why is this useful to us now?", and the answer to that is that it's not immediately useful, but you'll be seeing some instances where certain curve properties can be either computed via function manipulation, or via clever use of matrices, and sometimes the matrix approach can be (drastically) faster.

    -

    So for now, just remember that we can represent curves this way, and let's move on.

    + +

    + If we plug in a t value, and then multiply the matrices, we will get exactly the same values as when we evaluate the original + polynomial function, or as when we evaluate the curve using progressive linear interpolation. +

    +

    + So: why would we bother with matrices? Matrix representations allow us to discover things about functions that would + otherwise be hard to tell. It turns out that the curves form + triangular matrices, and they have a determinant equal to the product of the + actual coordinates we use for our curve. It's also invertible, which means there's + a ton of properties that are all satisfied. Of + course, the main question is "why is this useful to us now?", and the answer to that is that it's not immediately useful, but + you'll be seeing some instances where certain curve properties can be either computed via function manipulation, or via clever use of + matrices, and sometimes the matrix approach can be (drastically) faster. +

    +

    So for now, just remember that we can represent curves this way, and let's move on.

    +
    +
    +

    +
    이전목차다음
    + de Casteljau's algorithm +

    +

    + If we want to draw Bézier curves, we can run through all values of t from 0 to 1 and then compute the weighted basis function + at each value, getting the x/y values we need to plot. Unfortunately, the more complex the curve gets, the more expensive + this computation becomes. Instead, we can use de Casteljau's algorithm to draw curves. This is a geometric approach to curve + drawing, and it's really easy to implement. So easy, in fact, you can do it by hand with a pencil and ruler. +

    +

    Rather than using our calculus function to find x/y values for t, let's do this instead:

    +
      +
    • treat t as a ratio (which it is). t=0 is 0% along a line, t=1 is 100% along a line.
      • +
    • Take all lines between the curve's defining points. For an order n curve, that's n lines.
      • +
    • + Place markers along each of these line, at distance t. So if t is 0.2, place the mark at 20% from the start, + 80% from the end. +
      • +
    • Now form lines between those points. This gives n-1 lines.
      • +
    • Place markers along each of these line at distance t.
      • +
    • Form lines between those points. This'll be n-2 lines.
      • +
    • Place markers, form lines, place markers, etc.
      • +
    • + Repeat this until you have only one line left. The point t on that line coincides with the original curve point at + t. +
      • +
    +

    + To see this in action, move the slider for the following sketch to changes which curve point is explicitly evaluated using de Casteljau's + algorithm. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    +

    How to implement de Casteljau's algorithm

    +

    + Let's just use the algorithm we just specified, and implement that as a function that can take a list of curve-defining points, and a + t value, and draws the associated point on the curve for that t value: +

    -
    -
    -

    이전목차다음
    de Casteljau's algorithm

    -

    If we want to draw Bézier curves, we can run through all values of t from 0 to 1 and then compute the weighted basis function at each value, getting the x/y values we need to plot. Unfortunately, the more complex the curve gets, the more expensive this computation becomes. Instead, we can use de Casteljau's algorithm to draw curves. This is a geometric approach to curve drawing, and it's really easy to implement. So easy, in fact, you can do it by hand with a pencil and ruler.

    -

    Rather than using our calculus function to find x/y values for t, let's do this instead:

    -
      -
    • treat t as a ratio (which it is). t=0 is 0% along a line, t=1 is 100% along a line.
      • -
    • Take all lines between the curve's defining points. For an order n curve, that's n lines.
      • -
    • Place markers along each of these line, at distance t. So if t is 0.2, place the mark at 20% from the start, 80% from the end.
      • -
    • Now form lines between those points. This gives n-1 lines.
      • -
    • Place markers along each of these line at distance t.
      • -
    • Form lines between those points. This'll be n-2 lines.
      • -
    • Place markers, form lines, place markers, etc.
      • -
    • Repeat this until you have only one line left. The point t on that line coincides with the original curve point at t.
      • -
    -

    To see this in action, move the slider for the following sketch to changes which curve point is explicitly evaluated using de Casteljau's algorithm.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -
    - -

    How to implement de Casteljau's algorithm

    -

    Let's just use the algorithm we just specified, and implement that as a function that can take a list of curve-defining points, and a t value, and draws the associated point on the curve for that t value:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    And done, that's the algorithm implemented. Although: usually you don't get the luxury of overloading the "+" operator, so let's also give the code for when you need to work with x and y values separately:

    +

    + And done, that's the algorithm implemented. Although: usually you don't get the luxury of overloading the "+" operator, so let's also + give the code for when you need to work with x and y values separately: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So what does this do? This draws a point, if the passed list of points is only 1 point long. Otherwise it will create a new list of points that sit at the t ratios (i.e. the "markers" outlined in the above algorithm), and then call the draw function for this new list.

    -
    +

    + So what does this do? This draws a point, if the passed list of points is only 1 point long. Otherwise it will create a new list of + points that sit at the t ratios (i.e. the "markers" outlined in the above algorithm), and then call the draw function for this + new list. +

    +
    + +
    +

    +
    이전목차다음
    + Simplified drawing +

    +

    + We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, + a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines. +

    +

    + We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the + number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we + can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we + lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or + curvature alignment. +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    +

    + Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. + You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the + cubic curve), a higher number is required to capture the curvature changes properly. +

    +
    +

    How to implement curve flattening

    +

    Let's just use the algorithm we just specified, and implement that:

    -
    -
    -

    이전목차다음
    Simplified drawing

    -

    We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines.

    -

    We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or curvature alignment.

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -
    - -

    Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the cubic curve), a higher number is required to capture the curvature changes properly.

    -
    - -

    How to implement curve flattening

    -

    Let's just use the algorithm we just specified, and implement that:

    - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return coordinates; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

    +

    And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + coord = _coord + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

    -
    +

    We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

    + +
    +
    +

    +
    + 이전목차다음 +
    + Splitting curves +

    +

    + Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken + together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the + points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our + on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    +

    implementing curve splitting

    +

    We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

    -
    -
    -

    이전목차다음
    Splitting curves

    -

    Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -
    - -

    implementing curve splitting

    -

    We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

    - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + -

    After running this function for some value t, the left and right arrays will contain all the coordinates for two new curves - one to the "left" of our t value, the other on the "right". These new curves will have the same order as the original curve, and can be overlaid exactly on the original curve.

    -
    - -
    -
    -

    이전목차다음
    Splitting curves using matrices

    -

    Another way to split curves is to exploit the matrix representation of a Bézier curve. In the section on matrices, we saw that we can represent curves as matrix multiplications. Specifically, we saw these two forms for the quadratic and cubic curves respectively: (we'll reverse the Bézier coefficients vector for legibility)

    - - -

    and

    - + +

    + Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the + coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual + "point on the curve" information into a new matrix multiplication: +

    + + +

    and

    + + +

    + If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two + staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, + from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear + algebra (and if you don't care about the derivations, just skip to the end of the box for the results!). +

    +
    +

    Deriving new hull coordinates

    +

    + Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look + at the quadratic curve first: +

    + + + - -

    Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual "point on the curve" information into a new matrix multiplication:

    - - -

    and

    - - -

    If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear algebra (and if you don't care about the derivations, just skip to the end of the box for the results!).

    -
    - -

    Deriving new hull coordinates

    -

    Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look at the quadratic curve first:

    - - - - - - - - -

    We can do this because [M · M-1] is the identity matrix. It's a bit like multiplying something by x/x in calculus: it doesn't do anything to the function, but it does allow you to rewrite it to something that may be easier to work with, or can be broken up differently. In the same way, multiplying our matrix by [M · M-1] has no effect on the total formula, but it does allow us to change the matrix sequence [something · M] to a sequence [M · something], and that makes a world of difference: if we know what [M-1 · Z · M] is, we can apply that to our coordinates, and be left with a proper matrix representation of a quadratic Bézier curve (which is [T · M · P]), with a new set of coordinates that represent the curve from t = 0 to t = z. So let's get computing:

    - - -

    Excellent! Now we can form our new quadratic curve:

    - - - - - - -

    Brilliant: if we want a subcurve from t = 0 to t = z, we can keep the first coordinate the same (which makes sense), our control point becomes a z-ratio mixture of the original control point and the start point, and the new end point is a mixture that looks oddly similar to a Bernstein polynomial of degree two. These new coordinates are actually really easy to compute directly!

    -

    Of course, that's only one of the two curves. Getting the section from t = z to t = 1 requires doing this again. We first observe that in the previous calculation, we actually evaluated the general interval [0,z]. We were able to write it down in a more simple form because of the zero, but what we actually evaluated, making the zero explicit, was:

    - - - - -

    If we want the interval [z,1], we will be evaluating this instead:

    - - - - -

    We're going to do the same trick of multiplying by the identity matrix, to turn [something · M] into [M · something]:

    - - -

    So, our final second curve looks like:

    - - - - - - -

    Nice. We see the same thing as before: we can keep the last coordinate the same (which makes sense); our control point becomes a z-ratio mixture of the original control point and the end point, and the new start point is a mixture that looks oddly similar to a bernstein polynomial of degree two, except this time it uses (z-1) rather than (1-z). These new coordinates are also really easy to compute directly!

    -
    - -

    So, using linear algebra rather than de Casteljau's algorithm, we have determined that, for any quadratic curve split at some value t = z, we get two subcurves that are described as Bézier curves with simple-to-derive coordinates:

    - - -

    and

    - - -

    We can do the same for cubic curves. However, I'll spare you the actual derivation (don't let that stop you from writing that out yourself, though) and simply show you the resulting new coordinate sets:

    - - -

    and

    - - -

    So, looking at our matrices, did we really need to compute the second segment matrix? No, we didn't. Actually having one segment's matrix means we implicitly have the other: push the values of each row in the matrix Q to the right, with zeroes getting pushed off the right edge and appearing back on the left, and then flip the matrix vertically. Presto, you just "calculated" Q'.

    -

    Implementing curve splitting this way requires less recursion, and is just straight arithmetic with cached values, so can be cheaper on systems where recursion is expensive. If you're doing computation with devices that are good at matrix multiplication, chopping up a Bézier curve with this method will be a lot faster than applying de Casteljau.

    - -
    -
    -

    이전목차다음
    Lowering and elevating curve order

    -

    One interesting property of Bézier curves is that an nth order curve can always be perfectly represented by an (n+1)th order curve, by giving the higher-order curve specific control points.

    -

    If we have a curve with three points, then we can create a curve with four points that exactly reproduces the original curve. First, we give it the same start and end points, and for its two control points we pick "1/3rd start + 2/3rd control" and "2/3rd control + 1/3rd end". Now we have exactly the same curve as before, except represented as a cubic curve rather than a quadratic curve.

    -

    The general rule for raising an nth order curve to an (n+1)th order curve is as follows (observing that the start and end weights are the same as the start and end weights for the old curve):

    - - -

    However, this rule also has as direct consequence that you cannot generally safely lower a curve from nth order to (n-1)th order, because the control points cannot be "pulled apart" cleanly. We can try to, but the resulting curve will not be identical to the original, and may in fact look completely different.

    -

    However, there is a surprisingly good way to ensure that a lower order curve looks "as close as reasonably possible" to the original curve: we can optimise the "least-squares distance" between the original curve and the lower order curve, in a single operation (also explained over on Sirver's Castle). However, to use it, we'll need to do some calculus work and then switch over to linear algebra. As mentioned in the section on matrix representations, some things can be done much more easily with matrices than with calculus functions, and this is one of those things. So... let's go!

    -

    We start by taking the standard Bézier function, and condensing it a little:

    - - -

    Then, we apply one of those silly (actually, super useful) calculus tricks: since our t value is always between zero and one (inclusive), we know that (1-t) plus t always sums to 1. As such, we can express any value as a sum of t and 1-t:

    - - -

    So, with that seemingly trivial observation, we rewrite that Bézier function by splitting it up into a sum of a (1-t) and t component:

    - - -

    So far so good. Now, to see why we did this, let's write out the (1-t) and t parts, and see what that gives us. I promise, it's about to make sense. We start with (1-t):

    - - -

    So by using this seemingly silly trick, we can suddenly express part of our nth order Bézier function in terms of an (n+1)th order Bézier function. And that sounds a lot like raising the curve order! Of course we need to be able to repeat that trick for the t part, but that's not a problem:

    - - -

    So, with both of those changed from an order n expression to an order (n+1) expression, we can put them back together again. Now, where the order n function had a summation from 0 to n, the order n+1 function uses a summation from 0 to n+1, but this shouldn't be a problem as long as we can add some new terms that "contribute nothing". In the next section on derivatives, there is a discussion about why "higher terms than there is a binomial for" and "lower than zero terms" both "contribute nothing". So as long as we can add terms that have the same form as the terms we need, we can just include them in the summation, they'll sit there and do nothing, and the resulting function stays identical to the lower order curve.

    -

    Let's do this:

    - - -

    And this is where we switch over from calculus to linear algebra, and matrices: we can now express this relation between Bézier(n,t) and Bézier(n+1,t) as a very simple matrix multiplication:

    - - -

    where the matrix M is an n+1 by n matrix, and looks like:

    - - -

    That might look unwieldy, but it's really just a mostly-zeroes matrix, with a very simply fraction on the diagonal, and an even simpler fraction to the left of it. Multiplying a list of coordinates with this matrix means we can plug the resulting transformed coordinates into the one-order-higher function and get an identical looking curve.

    -

    Not too bad!

    -

    Equally interesting, though, is that with this matrix operation established, we can now use an incredibly powerful and ridiculously simple way to find out a "best fit" way to reverse the operation, called the normal equation. What it does is minimize the sum of the square differences between one set of values and another set of values. Specifically, if we can express that as some function A x = b, we can use it. And as it so happens, that's exactly what we're dealing with, so:

    - - -

    The steps taken here are:

    -
      -
    1. We have a function in a form that the normal equation can be used with, so
      2. -
    2. apply the normal equation!
      3. -
    3. Then, we want to end up with just Bn on the left, so we start by left-multiply both sides such that we'll end up with lots of stuff on the left that simplified to "a factor 1", which in matrix maths is the identity matrix.
      4. -
    4. In fact, by left-multiplying with the inverse of what was already there, we've effectively "nullified" (but really, one-inified) that big, unwieldy block into the identity matrix I. So we substitute the mess with I, and then
      5. -
    5. because multiplication with the identity matrix does nothing (like multiplying by 1 does nothing in regular algebra), we just drop it.
      6. -
    -

    And we're done: we now have an expression that lets us approximate an n+1th order curve with a lower nth order curve. It won't be an exact fit, but it's definitely a best approximation. So, let's implement these rules for raising and lowering curve order to a (semi) random curve, using the following graphic. Select the sketch, which has movable control points, and press your up and down arrow keys to raise or lower the curve order.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - -
    -
    -

    이전목차다음
    Derivatives

    -

    There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straightforward, although we do need a bit of math.

    -

    First, let's look at the derivative rule for Bézier curves, which is:

    - - -

    which we can also write (observing that b in this formula is the same as our w weights, and that n times a summation is the same as a summation where each term is multiplied by n) as:

    - - -

    Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi). So for a 3rd degree curve, with four weights, the derivative has three new weights: w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2).

    -
    - -

    "Slow down, why is that true?"

    -

    Sometimes just being told "this is the derivative" is nice, but you might want to see why this is indeed the case. As such, let's have a look at the proof for this derivative. First off, the weights are independent of the full Bézier function, so the derivative involves only the derivative of the polynomial basis function. So, let's find that:

    - - -

    Applying the product and chain rules gives us:

    - - -

    Which is hard to work with, so let's expand that properly:

    - - -

    Now, the trick is to turn this expression into something that has binomial coefficients again, so we want to end up with things that look like "x! over y!(x-y)!". If we can do that in a way that involves terms of n-1 and k-1, we'll be on the right track.

    - - -

    And that's the first part done: the two components inside the parentheses are actually regular, lower-order Bézier expressions:

    - - -

    Now to apply this to our weighted Bézier curves. We'll write out the plain curve formula that we saw earlier, and then work our way through to its derivative:

    - - -

    If we expand this (with some color to show how terms line up), and reorder the terms by increasing values for k we see the following:

    - - -

    Two of these terms fall way: the first term falls away because there is no -1st term in a summation. As such, it always contributes "nothing", so we can safely completely ignore it for the purpose of finding the derivative function. The other term is the very last term in this expansion: one involving Bn-1,n. This term would have a binomial coefficient of [i choose i+1], which is a non-existent binomial coefficient. Again, this term would contribute "nothing", so we can ignore it, too. This means we're left with:

    - - -

    And that's just a summation of lower order curves:

    - - -

    We can rewrite this as a normal summation, and we're done:

    - - -
    - -

    Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for Bézier curves, and then the derivative:

    - + +

    And that's just a summation of lower order curves:

    + + +

    We can rewrite this as a normal summation, and we're done:

    + + + +

    + Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for + Bézier curves, and then the derivative: +

    + - - - -

    What are the differences? In terms of the actual Bézier curve, virtually nothing! We lowered the order (rather than n, it's now n-1), but it's still the same Bézier function. The only real difference is in how the weights change when we derive the curve's function. If we have four points A, B, C, and D, then the derivative will have three points, the second derivative two, and the third derivative one:

    - - -

    We can keep performing this trick for as long as we have more than one weight. Once we have one weight left, the next step will see k = 0, and the result of our "Bézier function" summation is zero, because we're not adding anything at all. As such, a quadratic curve has no second derivative, a cubic curve has no third derivative, and generalized: an nth order curve has n-1 (meaningful) derivatives, with any further derivative being zero.

    - -
    -
    -

    이전목차다음
    Tangents and normals

    -

    If you want to move objects along a curve, or "away from" a curve, the two vectors you're most interested in are the tangent vector and normal vector for curve points. These are actually really easy to find. For moving and orienting along a curve, we use the tangent, which indicates the direction of travel at specific points, and is literally just the first derivative of our curve:

    - - -

    This gives us the directional vector we want. We can normalize it to give us uniform directional vectors (having a length of 1.0) at each point, and then do whatever it is we want to do based on those directions:

    - - -

    The tangent is very useful for moving along a line, but what if we want to move away from the curve instead, perpendicular to the curve at some point t? In that case we want the normal vector. This vector runs at a right angle to the direction of the curve, and is typically of length 1.0, so all we have to do is rotate the normalized directional vector and we're done:

    - - -
    - -

    Rotating coordinates is actually very easy, if you know the rule for it. You might find it explained as "applying a rotation matrix, which is what we'll look at here, too. Essentially, the idea is to take the circles over which we can rotate, and simply "sliding the coordinates" over these circles by the desired -angle. If we want a quarter circle turn, we take the coordinate, slide it along the circle by a quarter turn, and done.

    -

    To turn any point (x,y) into a rotated point (x',y') (over 0,0) by some angle φ, we apply this nice and easy computation:

    - - -

    Which is the "long" version of the following matrix transformation:

    - - -

    And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1.

    -

    But why does this work? Why this matrix multiplication? Wikipedia (technically, Thomas Herter and Klaus Lott) tells us that a rotation matrix can be -treated as a sequence of three (elementary) shear operations. When we combine this into a single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. DataGenetics have an excellent article about this very thing: it's really quite cool, and I strongly recommend taking a quick break from this primer to read that article.

    -
    + +

    + And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even + easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1. +

    +

    + But why does this work? Why this matrix multiplication? + Wikipedia (technically, Thomas Herter and Klaus + Lott) tells us that a rotation matrix can be treated as a sequence of three (elementary) shear operations. When we combine this into a + single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. + DataGenetics have an excellent article about this very thing: it's + really quite cool, and I strongly recommend taking a quick break from this primer to read that article. +

    + -

    The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant).

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    +

    + The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the + normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant). +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    +
    +
    +

    +
    + 이전목차다음 +
    + Working with 3D normals +

    +

    + Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things + this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but + for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, + it's not "super hard", but there are more steps involved and we should have a look at those. +

    +

    + Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. + However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, + not a single vector, so we basically need to define what "the" normal is in the 3D case. +

    +

    + The "naïve" approach is to construct what is known as the + Frenet normal, where we follow a simple recipe that works in + many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are + perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each + point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the + local tangent, as well as the tangents "right next to it". +

    +

    + Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the + vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know + lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same + logic we used for the 2D case: "just rotate it 90 degrees". +

    +

    So let's do that! And in a twist surprise, we can do this in four lines:

    +
      +
    • a = normalize(B'(t))
      • +
    • b = normalize(a + B''(t))
      • +
    • r = normalize(b × a)
      • +
    • normal = normalize(r × a)
      • +
    +

    Let's unpack that a little:

    +
      +
    • + We start by taking the normalized vector for the derivative at some point on the + curve. We normalize it so the maths is less work. Less work is good. +
      • +
    • + Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and + just had the same derivative and second derivative from that point on. +
      • +
    • + This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which + means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the + cross product. (Note that while that operation uses the × operator, it's most + definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent + a quarter circle to get our normal, just like we did in the 2D case. +
      • +
    • + Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal + vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second + time, and immediately get our normal vector. +
      • +
    +

    + And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can + move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor + position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    + However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the + curve" between t=0.65 and t=0.75... Why is it doing that? +

    +

    + As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are + "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute + normals that just... look good. +

    +

    Thankfully, Frenet normals are not our only option.

    +

    + Another option is to take a slightly more algorithmic approach and compute a form of + Rotation Minimising Frame + (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational + axis, and the normal vector, centered on an on-curve point. +

    +

    + These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we + could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at + the same time that you're building lookup tables for your curve. +

    +

    + The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by + applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are: +

    +
      +
    • Take a point on the curve for which we know the RM frame already,
      • +
    • take a next point on the curve for which we don't know the RM frame yet, and
      • +
    • + reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous + points as a "mirror". +
      • +
    • + This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal + that's slightly off-kilter, so: +
      • +
    • + reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror". +
      • +
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • +
    +

    So, let's write some code for that!

    +
    +

    Implementing Rotation Minimising Frames

    +

    + We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it + yields a frame with properties: +

    -
    -
    -

    이전목차다음
    Working with 3D normals

    -

    Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, it's not "super hard", but there are more steps involved and we should have a look at those.

    -

    Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, not a single vector, so we basically need to define what "the" normal is in the 3D case.

    -

    The "naïve" approach is to construct what is known as the Frenet normal, where we follow a simple recipe that works in many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the local tangent, as well as the tangents "right next to it".

    -

    Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same logic we used for the 2D case: "just rotate it 90 degrees".

    -

    So let's do that! And in a twist surprise, we can do this in four lines:

    -
      -
    • a = normalize(B'(t))
      • -
    • b = normalize(a + B''(t))
      • -
    • r = normalize(b × a)
      • -
    • normal = normalize(r × a)
      • -
    -

    Let's unpack that a little:

    -
      -
    • We start by taking the normalized vector for the derivative at some point on the curve. We normalize it so the maths is less work. Less work is good.
      • -
    • Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on.
      • -
    • This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the cross product. (Note that while that operation uses the × operator, it's most definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent a quarter circle to get our normal, just like we did in the 2D case.
      • -
    • Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second time, and immediately get our normal vector.
      • -
    -

    And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the curve" between t=0.65 and t=0.75... Why is it doing that?

    -

    As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute normals that just... look good.

    -

    Thankfully, Frenet normals are not our only option.

    -

    Another option is to take a slightly more algorithmic approach and compute a form of Rotation Minimising Frame (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational axis, and the normal vector, centered on an on-curve point.

    -

    These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at the same time that you're building lookup tables for your curve.

    -

    The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are:

    -
      -
    • Take a point on the curve for which we know the RM frame already,
      • -
    • take a next point on the curve for which we don't know the RM frame yet, and
      • -
    • reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror".
      • -
    • This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter, so:
      • -
    • reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
      • -
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • -
    -

    So, let's write some code for that!

    -
    - -

    Implementing Rotation Minimising Frames

    -

    We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it yields a frame with properties:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + -

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    +

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    + frames.add(x1) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + -

    Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more code to get much better looking normals.

    -
    +

    + Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more + code to get much better looking normals. +

    + -

    Speaking of better looking, what does this actually look like? Let's revisit that earlier curve, but this time use rotation minimising frames rather than Frenet frames:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    That looks so much better!

    -

    For those reading along with the code: we don't even strictly speaking need a Frenet frame to start with: we could, for instance, treat the z-axis as our initial axis of rotation, so that our initial normal is (0,0,1) × tangent, and then take things from there, but having that initial "mathematically correct" frame so that the initial normal seems to line up based on the curve's orientation in 3D space is just nice.

    - -
    -
    -

    이전목차다음
    Component functions

    -

    One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that function, but this poses a problem, since our function is parametric: every axis has its own function.

    -

    The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.

    -

    Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis value, respectively.

    -

    If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a change in the right graph.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

     

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - -
    -
    -

    이전목차다음
    Finding extremities: root finding

    -

    Now that we understand (well, superficially anyway) the component functions, we can find the extremities of our Bézier curve by finding maxima and minima on the component functions, by solving the equation B'(t) = 0. We've already seen that the derivative of a Bézier curve is a simpler Bézier curve, but how do we solve the equality? Fairly easily, actually, until our derivatives are 4th order or higher... then things get really hard. But let's start simple:

    -

    Quadratic curves: linear derivatives.

    -

    The derivative of a quadratic Bézier curve is a linear Bézier curve, interpolating between just two terms, which means finding the solution for "where is this line 0" is effectively trivial by rewriting it to a function of t and solving. First we turn our quadratic Bézier function into a linear one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then we turn this into our solution for t using basic arithmetics:

    - - -

    Done.

    -

    Although with the caveat that if b-a is zero, there is no solution and we probably shouldn't try to perform that division.

    -

    Cubic curves: the quadratic formula.

    -

    The derivative of a cubic Bézier curve is a quadratic Bézier curve, and finding the roots for a quadratic polynomial means we can apply the Quadratic formula. If you've seen it before, you'll remember it, and if you haven't, it looks like this:

    - - -

    So, if we can rewrite the Bézier component function as a plain polynomial, we're done: we just plug in the values into the quadratic formula, check if that square root is negative or not (if it is, there are no roots) and then just compute the two values that come out (because of that plus/minus sign we get two). Any value between 0 and 1 is a root that matters for Bézier curves, anything below or above that is irrelevant (because Bézier curves are only defined over the interval [0,1]). So, how do we convert?

    -

    First we turn our cubic Bézier function into a quadratic one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then, using these v values, we can find out what our a, b, and c should be:

    - - -

    This gives us three coefficients {a, b, c} that are expressed in terms of v values, where the v values are expressions of our original coordinate values, so we can do some substitution to get:

    - - -

    Easy-peasy. We can now almost trivially find the roots by plugging those values into the quadratic formula.

    -

    And as a cubic curve, there is also a meaningful second derivative, which we can compute by simple taking the derivative of the derivative.

    -

    Quartic curves: Cardano's algorithm.

    -

    We haven't really looked at them before now, but the next step up would be a Quartic curve, a fourth degree Bézier curve. As expected, these have a derivative that is a cubic function, and now things get much harder. Cubic functions don't have a "simple" rule to find their roots, like the quadratic formula, and instead require quite a bit of rewriting to a form that we can even start to try to solve.

    -

    Back in the 16th century, before Bézier curves were a thing, and even before calculus itself was a thing, Gerolamo Cardano figured out that even if the general cubic function is really hard to solve, it can be rewritten to a form for which finding the roots is "easier" (even if not "easy"):

    - - -

    We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather than three: this makes things considerably easier to solve because it lets us use regular calculus to find the values that satisfy the equation.

    -

    Now, there is one small hitch: as a cubic function, the solutions may be complex numbers rather than plain numbers... And Cardano realised this, centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number.

    -

    So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at Ken J. Ward's page for solving the cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a read-through.

    -
    + +

    + We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather + than three: this makes things considerably easier to solve because it lets us use + regular calculus to find the values that satisfy the equation. +

    +

    + Now, there is one small hitch: as a cubic function, the solutions may be + complex numbers rather than plain numbers... And Cardano realised this, + centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these + numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we + have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're + going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number. +

    +

    + So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at + Ken J. Ward's page for solving the + cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the + complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a + read-through. +

    +
    +

    Implementing Cardano's algorithm for finding all real roots

    +

    + The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" + number space (denoted with ℝ), this is perfectly fine in the + "complex" number space (denoted with ℂ). And, as it so happens, Cardano is + also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm! +

    -

    Implementing Cardano's algorithm for finding all real roots

    -

    The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" number space (denoted with ℝ), this is perfectly fine in the "complex" number space (denoted with ℂ). And, as it so happens, Cardano is also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm!

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    60
    61
    62
    63
    64
    65
    66
    67
    68
    69
    70
    71
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + + 37 + + + 38 + + + 39 + + + 40 + + + 41 + + + 42 + + + 43 + + + 44 + + + 45 + + + 46 + + + 47 + + + 48 + + + 49 + + + 50 + + + 51 + + + 52 + + + 53 + + + 54 + + + 55 + + + 56 + + + 57 + + + 58 + + + 59 + + + 60 + + + 61 + + + 62 + + + 63 + + + 64 + + + 65 + + + 66 + + + 67 + + + 68 + + + 69 + + + 70 + + + 71 + + + 72 + + + 73 + + + 74 + + + 75 + + + 76 + + + 77 + + + 78 + + + 79 + + + 80 + + + 81 + + +
    -
    - -

    And that's it. The maths is complicated, but the code is pretty much just "follow the maths, while caching as many values as we can to prevent recomputing things as much as possible" and now we have a way to find all roots for a cubic function and can just move on with using that to find extremities of our curves.

    -

    And of course, as a quartic curve also has meaningful second and third derivatives, we can quite easily compute those by using the derivative of the derivative (of the derivative), just as for cubic curves.

    -

    Quintic and higher order curves: finding numerical solutions

    -

    And this is where thing stop, because we cannot find the roots for polynomials of degree 5 or higher using algebra (a fact known as the Abel–Ruffini theorem). Instead, for occasions like these, where algebra simply cannot yield an answer, we turn to numerical analysis.

    -

    That's a fancy term for saying "rather than trying to find exact answers by manipulating symbols, find approximate answers by describing the underlying process as a combination of steps, each of which can be assigned a number via symbolic manipulation". For example, trying to mathematically compute how much water fits in a completely crazy three dimensional shape is very hard, even if it got you the perfect, precise answer. A much easier approach, which would be less perfect but still entirely useful, would be to just grab a buck and start filling the shape until it was full: just count the number of buckets of water you used. And if we want a more precise answer, we can use smaller buckets.

    -

    So that's what we're going to do here, too: we're going to treat the problem as a sequence of steps, and the smaller we can make each step, the closer we'll get to that "perfect, precise" answer. And as it turns out, there is a really nice numerical root-finding algorithm, called the Newton-Raphson root finding method (yes, after that Newton), which we can make use of. The Newton-Raphson approach consists of taking our impossible-to-solve function f(x), picking some initial value x (literally any value will do), and calculating f(x). We can think of that value as the "height" of the function at x. If that height is zero, we're done, we have found a root. If it isn't, we calculate the tangent line at f(x) and calculate at which x value its height is zero (which we've already seen is very easy). That will give us a new x and we repeat the process until we find a root.

    -

    Mathematically, this means that for some x, at step n=1, we perform the following calculation until fy(x) is zero, so that the next t is the same as the one we already have:

    - - -

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    -

    Now, this works well only if we can pick good starting points, and our curve is continuously differentiable and doesn't have oscillations. Glossing over the exact meaning of those terms, the curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: which starting points do we pick?

    -

    As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as high as you'll ever need to go.

    -

    In conclusion:

    -

    So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - + +

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    +

    + Now, this works well only if we can pick good starting points, and our curve is + continuously differentiable and doesn't have + oscillations. Glossing over the exact meaning of those terms, the + curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: + which starting points do we pick? +

    +

    + As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from + t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may + pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: + there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order + curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as + high as you'll ever need to go. +

    +

    In conclusion:

    +

    + So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those + roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + +

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + +
    +
    +

    +
    이전목차다음
    + Bounding boxes +

    +

    + If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four + values we need to box in our curve: +

    +

    Computing the bounding box for a Bézier curve:

    +
      +
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. +
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. +
    3. + Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original + functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct. +
      4. +
    +

    + Applying this approach to our previous root finding, we get the following + axis-aligned bounding boxes (with all curve extremity points + shown on the curve): +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    -
    -
    -

    이전목차다음
    Bounding boxes

    -

    If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four values we need to box in our curve:

    -

    Computing the bounding box for a Bézier curve:

    -
      -
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. -
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. -
    3. Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct.
      4. -
    -

    Applying this approach to our previous root finding, we get the following axis-aligned bounding boxes (with all curve extremity points shown on the curve):

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    - -

    We can construct even nicer boxes by aligning them along our curve, rather than along the x- and y-axis, but in order to do so we first need to look at how aligning works.

    - -
    -
    -

    이전목차다음
    Aligning curves

    -

    While there are an incredible number of curves we can define by varying the x- and y-coordinates for the control points, not all curves are actually distinct. For instance, if we define a curve, and then rotate it 90 degrees, it's still the same curve, and we'll find its extremities in the same spots, just at different draw coordinates. As such, one way to make sure we're working with a "unique" curve is to "axis-align" it.

    -

    Aligning also simplifies a curve's functions. We can translate (move) the curve so that the first point lies on (0,0), which turns our n term polynomial functions into n-1 term functions. The order stays the same, but we have less terms. Then, we can rotate the curves so that the last point always lies on the x-axis, too, making its coordinate (...,0). This further simplifies the function for the y-component to an n-2 term function. For instance, if we have a cubic curve such as this:

    - - -

    Then translating it so that the first coordinate lies on (0,0), moving all x coordinates by -120, and all y coordinates by -160, gives us:

    - - -

    If we then rotate the curve so that its end point lies on the x-axis, the coordinates (integer-rounded for illustrative purposes here) become:

    - - -

    If we drop all the zero-terms, this gives us:

    - - -

    We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

     

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - + +

    + We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning + our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + +

     

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + +
    +
    +

    +
    이전목차다음
    + Tight bounding boxes +

    +

    + With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, + recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding + box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T. +

    +

    We now have nice tight bounding boxes for our curves:

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    -
    -
    -

    이전목차다음
    Tight bounding boxes

    -

    With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T.

    -

    We now have nice tight bounding boxes for our curves:

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    - -

    These are, strictly speaking, not necessarily the tightest possible bounding boxes. It is possible to compute the optimal bounding box by determining which spanning lines we need to effect a minimal box area, but because of the parametric nature of Bézier curves this is actually a rather costly operation, and the gain in bounding precision is often not worth it.

    - -
    -
    -

    이전목차다음
    Curve inflections

    -

    Now that we know how to align a curve, there's one more thing we can calculate: inflection points. Imagine we have a variable size circle that we can slide up against our curve. We place it against the curve and adjust its radius so that where it touches the curve, the curvatures of the curve and the circle are the same, and then we start to slide the circle along the curve - for quadratic curves, we can always do this without the circle behaving oddly: we might have to change the radius of the circle as we slide it along, but it'll always sit against the same side of the curve.

    -

    But what happens with cubic curves? Imagine we have an S curve and we place our circle at the start of the curve, and start sliding it along. For a while we can simply adjust the radius and things will be fine, but once we get to the midpoint of that S, something odd happens: the circle "flips" from one side of the curve to the other side, in order for the curvatures to keep matching. This is called an inflection, and we can find out where those happen relatively easily.

    -

    What we need to do is solve a simple equation:

    - - -

    What we're saying here is that given the curvature function C(t), we want to know for which values of t this function is zero, meaning there is no "curvature", which will be exactly at the point between our circle being on one side of the curve, and our circle being on the other side of the curve. So what does C(t) look like? Actually something that seems not too hard:

    - - -

    The function C(t) is the cross product between the first and second derivative functions for the parametric dimensions of our curve. And, as already shown, derivatives of Bézier curves are just simpler Bézier curves, with very easy to compute new coefficients, so this should be pretty easy.

    -

    However as we've seen in the section on aligning, aligning lets us simplify things a lot, by completely removing the contributions of the first coordinate from most mathematical evaluations, and removing the last y coordinate as well by virtue of the last point lying on the x-axis. So, while we can evaluate C(t) = 0 for our curve, it'll be much easier to first axis-align the curve and then evaluating the curvature function.

    -
    - -

    Let's derive the full formula anyway

    -

    Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:

    - - -

    And of course the same functions for y:

    - - -

    Asking a computer to now compose the C(t) function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:

    - - -

    That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all disappear if we axis-align our curve, which is why aligning is a great idea.

    -
    + +

    + That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all + disappear if we axis-align our curve, which is why aligning is a great idea. +

    + -

    Aligning our curve so that three of the eight coefficients become zero, and observing that scale does not affect finding t values, we end up with the following simple term function for C(t):

    - - -

    That's a lot easier to work with: we see a fair number of terms that we can compute and then cache, giving us the following simplification:

    - - -

    This is a plain quadratic curve, and we know how to solve C(t) = 0; we use the quadratic formula:

    - - -

    We can easily compute this value if the discriminator isn't a negative number (because we only want real roots, not complex roots), and if x is not zero, because divisions by zero are rather useless.

    -

    Taking that into account, we compute t, we disregard any t value that isn't in the Bézier interval [0,1], and we now know at which t value(s) our curve will inflect.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - -
    -
    -

    이전목차다음
    The canonical form (for cubic curves)

    -

    While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?

    -

    As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper "A Geometric Characterization of Parametric Cubic curves". It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?

    -

    The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    This is a fairly funky image, so let's see what the various parts of it mean...

    -

    We see the three fixed points at (0,0), (0,1) and (1,1). The various regions and boundaries indicate what property the original curve will have, if the fourth point is in/on that region or boundary. Specifically, if the fourth point is...

    -
      -

    1. ...anywhere inside the red zone, but not on its boundaries, the curve will be self-intersecting (yielding a loop). We won't know where it self-intersects (in terms of t values), but we are guaranteed that it does.

      -
      2. -

    2. ...on the left (red) edge of the red zone, the curve will have a cusp. We again don't know where, but we know there is one. This edge is described by the function:

      - - -
      3. -

    3. ...on the almost circular, lower right (pink) edge, the curve's end point touches the curve, forming a loop. This edge is described by the function:

      - - -
      4. -

    4. ...on the top (blue) edge, the curve's start point touches the curve, forming a loop. This edge is described by the function:

      - - -
      5. -

    5. ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

      -
      6. -

    6. ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two inflections (switching from concave to convex and then back again, or from convex to concave and then back again).

      -
      7. -

    7. ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

      -
      8. -
    -

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    -
    + + +
  • +

    ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

    +
    • +
  • +

    + ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two + inflections (switching from concave to convex and then back again, or from convex to concave and then back again). +

    +
    • +

  • ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

    • + +

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    +
    +

    Wait, where do those lines come from?

    +

    + Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, + and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield + formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic + expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the + properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval. +

    +

    + The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general + cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to + it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general + curve does over the interval t=0 to t=1. +

    +

    + For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you + can probably follow along with this paper, although you might have to read it a few times before all the bits "click". +

    +
    -

    Wait, where do those lines come from?

    -

    Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval.

    -

    The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general curve does over the interval t=0 to t=1.

    -

    For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you can probably follow along with this paper, although you might have to read it a few times before all the bits "click".

    -
    - -

    So now the question becomes: how do we manipulate our curve so that it fits this canonical form, with three fixed points, and one "free" point? Enter linear algebra. Don't worry, I'll be doing all the math for you, as well as show you what the effect is on our curves, but basically we're going to be using linear algebra, rather than calculus, because "it's way easier". Sometimes a calculus approach is very hard to work with, when the equivalent geometrical solution is super obvious.

    -

    The approach is going to start with a curve that doesn't have all-colinear points (so we need to make sure the points don't all fall on a straight line), and then applying three graphics operations that you will probably have heard of: translation (moving all points by some fixed x- and y-distance), scaling (multiplying all points by some x and y scale factor), and shearing (an operation that turns rectangles into parallelograms).

    -

    Step 1: we translate any curve by -p1.x and -p1.y, so that the curve starts at (0,0). We're going to make use of an interesting trick here, by pretending our 2D coordinates are 3D, with the z coordinate simply always being 1. This is an old trick in graphics to overcome the limitations of 2D transformations: without it, we can only turn (x,y) coordinates into new coordinates of the form (ax + by, cx + dy), which means we can't do translation, since that requires we end up with some kind of (x + a, y + b). If we add a bogus z coordinate that is always 1, then we can suddenly add arbitrary values. For example:

    - - -

    Sweet! z stays 1, so we can effectively ignore it entirely, but we added some plain values to our x and y coordinates. So, if we want to subtract p1.x and p1.y, we use:

    - - -

    Running all our coordinates through this transformation gives a new set of coordinates, let's call those U, where the first coordinate lies on (0,0), and the rest is still somewhat free. Our next job is to make sure point 2 ends up lying on the x=0 line, so what we want is a transformation matrix that, when we run it, subtracts x from whatever x we currently have. This is called shearing, and the typical x-shear matrix and its transformation looks like this:

    - - -

    So we want some shearing value that, when multiplied by y, yields -x, so our x coordinate becomes zero. That value is simply -x/y, because *-x/y * y = -x*. Done:

    - - -

    Now, running this on all our points generates a new set of coordinates, let's call those V, which now have point 1 on (0,0) and point 2 on (0, some-value), and we wanted it at (0,1), so we need to do some scaling to make sure it ends up at (0,1). Additionally, we want point 3 to end up on (1,1), so we can also scale x to make sure its x-coordinate will be 1 after we run the transform. That means we'll be x-scaling by 1/point3x, and y-scaling by point2y. This is really easy:

    - - -

    Then, finally, this generates a new set of coordinates, let's call those W, of which point 1 lies on (0,0), point 2 lies on (0,1), and point three lies on (1, ...) so all that's left is to make sure point 3 ends up at (1,1) - but we can't scale! Point 2 is already in the right place, and y-scaling would move it out of (0,1) again, so our only option is to y-shear point three, just like how we x-sheared point 2 earlier. In this case, we do the same trick, but with y/x rather than x/y because we're not x-shearing but y-shearing. Additionally, we don't actually want to end up at zero (which is what we did before) so we need to shear towards an offset, in this case 1:

    - - -

    And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will be:

    - - -

    Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just pull this apart a little and end up with an easy-to-calculate bit of code!

    -

    First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does that leave?

    - - -

    Suddenly things look a lot simpler: the mapped x is fairly straight forward to compute, and we see that the mapped y actually contains the mapped x in its entirety, so we'll have that part already available when we need to evaluate it. In fact, let's pull out all those common factors to see just how simple this is:

    - - -

    That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it look!

    -
    + +

    + That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it + look! +

    +
    +

    How do you track all that?

    +

    + Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply + use Mathematica. Tracking all this math by hand is insane, and we invented computers, + literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the + program do the math for me. And real math, too, with symbols, not with numbers. In fact, + here's the Mathematica notebook if you want to see how + this works for yourself. +

    +

    + Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's + $344 for home use, up from $295 when I original wrote this, but it's + also free when you buy a $35 raspberry pi. Obviously, I bought a + raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do + it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for. +

    +
    -

    How do you track all that?

    -

    Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply use Mathematica. Tracking all this math by hand is insane, and we invented computers, literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the program do the math for me. And real math, too, with symbols, not with numbers. In fact, here's the Mathematica notebook if you want to see how this works for yourself.

    -

    Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's $344 for home use, up from $295 when I original wrote this, but it's also free when you buy a $35 raspberry pi. Obviously, I bought a raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for.

    -
    - -

    So, let's write up a sketch that'll show us the canonical form for any curve drawn in blue, overlaid on our canonical map, so that we can immediately tell which features our curve must have, based on where the fourth coordinate is located on the map:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - -
    -
    -

    이전목차다음
    Finding Y, given X

    -

    One common task that pops up in things like CSS work, or parametric equalizers, or image leveling, or any other number of applications where Bézier curves are used as control curves in a way that there is really only ever one "y" value associated with one "x" value, you might want to cut out the middle man, as it were, and compute "y" directly based on "x". After all, the function looks simple enough, finding the "y" value should be simple too, right? Unfortunately, not really. However, it is possible and as long as you have some code in place to help, it's not a lot of a work either.

    -

    We'll be tackling this problem in two stages: the first, which is the hard part, is figuring out which "t" value belongs to any given "x" value. For instance, have a look at the following graphic. On the left we have a Bézier curve that looks for all intents and purposes like it fits our criteria: every "x" has one and only one associated "y" value. On the right we see the function for just the "x" values: that's a cubic curve, but not a really crazy cubic curve. If you move the graphic's slider, you will see a red line drawn that corresponds to the x coordinate: this is a vertical line in the left graphic, and a horizontal line on the right.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Now, if you look more closely at that right graphic, you'll notice something interesting: if we treat the red line as "the x axis", then the point where the function crosses our line is really just a root for the cubic function x(t) through a shifted "x-axis"... and we've already seen how to calculate roots, so let's just run cubic root finding - and not even the complicated cubic case either: because of the kind of curve we're starting with, we know there is only root, simplifying the code we need!

    -

    First, let's look at the function for x(t):

    - - -

    We can rewrite this to a plain polynomial form, by just fully writing out the expansion and then collecting the polynomial factors, as:

    - - -

    Nothing special here: that's a standard cubic polynomial in "power" form (i.e. all the terms are ordered by their power of t). So, given that a, b, c, d, and x(t) are all known constants, we can trivially rewrite this (by moving the x(t) across the equal sign) as:

    - - -

    You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using Cardano's algorithm, and we're left with some rather short code:

    + +

    + You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they + all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using + Cardano's algorithm, and we're left with some rather short code: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +y = curve.get(t).y + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So the procedure is fairly straight forward: pick an x, find the associated t value, evaluate our curve for that t value, which gives us the curve's {x,y} coordinate, which means we know y for this x. Move the slider for the following graphic to see this in action:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - -
    -
    -

    이전목차다음
    Arc length

    -

    How long is a Bézier curve? As it turns out, that's not actually an easy question, because the answer requires maths that —much like root finding— cannot generally be solved the traditional way. If we have a parametric curve with fx(t) and fy(t), then the length of the curve, measured from start point to some point t = z, is computed using the following seemingly straight forward (if a bit overwhelming) formula:

    - - -

    or, more commonly written using Leibnitz notation as:

    - - -

    This formula says that the length of a parametric curve is in fact equal to the area underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an unwieldy computation, and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates".

    -

    Seriously: It cannot be done.

    -

    So we turn to numerical approaches again. The method we'll look at here is the Gauss quadrature. This approximation is a really neat trick, because for any nth degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:

    - - -

    In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the better the approximation:

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    + +

    + In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips + sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The + more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the + better the approximation: +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    -

    Now, infinitely many terms to sum and infinitely thin rectangles are not something that computers can work with, so instead we're going to approximate the infinite summation by using a sum of a finite number of "just thin" rectangular strips. As long as we use a high enough number of thin enough rectangular strips, this will give us an approximation that is pretty close to what the real value is.

    -

    So, the trick is to come up with useful rectangular strips. A naive way is to simply create n strips, all with the same width, but there is a far better way using special values for C and f(t) depending on the value of n, which indicates how many strips we'll use, and it's called the Legendre-Gauss quadrature.

    -

    This approach uses strips that are not spaced evenly, but instead spaces them in a special way based on describing the function as a polynomial (the more strips, the more accurate the polynomial), and then computing the exact integral for that polynomial. We're essentially performing arc length computation on a flattened curve, but flattening it based on the intervals dictated by the Legendre-Gauss solution.

    -
    - -

    Note that one requirement for the approach we'll use is that the integral must run from -1 to 1. That's no good, because we're dealing with Bézier curves, and the length of a section of curve applies to values which run from 0 to "some value smaller than or equal to 1" (let's call that value z). Thankfully, we can quite easily transform any integral interval to any other integral interval, by shifting and scaling the inputs. Doing so, we get the following:

    - - -

    That may look a bit more complicated, but the fraction involving z is a fixed number, so the summation, and the evaluation of the f(t) values are still pretty simple.

    -

    So, what do we need to perform this calculation? For one, we'll need an explicit formula for f(t), because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these Ci and ti values should be. Luckily, that's less work because there are actually many tables available that give these values, for any n, so if we want to approximate our integral with only two terms (which is a bit low, really) then these tables would tell us that for n=2 we must use the following values:

    - - -

    Which means that in order for us to approximate the integral, we must plug these values into the approximate function, which gives us:

    - - -

    We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the Bézier curve functions Bx(t) and By(t).

    -
    + +

    + We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the + Bézier curve functions Bx(t) and By(t). +

    + -

    If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the start/end points on the inside?

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - +

    + If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), + we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, + with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make + a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the + start/end points on the inside? +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    +
    +

    +
    이전목차다음
    + Approximated arc length +

    +

    + Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length + instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This + will come with an error, but this can be made arbitrarily small by increasing the segment count. +

    +

    + If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal + effort: +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    -
    -
    -

    이전목차다음
    Approximated arc length

    -

    Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This will come with an error, but this can be made arbitrarily small by increasing the segment count.

    -

    If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal effort:

    -
    - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -
    - -

    You may notice that even though the error in length is actually pretty significant in absolute terms, even at a low number of segments we get a length that agrees with the true length when it comes to just the integer part of the arc length. Quite often, approximations can drastically speed things up!

    - -
    -
    -

    이전목차다음
    Curvature of a curve

    -

    If we have two curves, and we want to line them in up in a way that "looks right", what would we use as metric to let a computer decide what "looks right" means?

    -

    For instance, we can start by ensuring that the two curves share an end coordinate, so that there is no "gap" between the end of one and the start of the next curve, but that won't guarantee that things look right: both curves can be going in wildly different directions, and the resulting joined geometry will have a corner in it, rather than a smooth transition from one curve to the next.

    -

    What we want is to ensure that the curvature at the transition from one curve to the next "looks good". So, we start with a shared coordinate, and then also require that derivatives for both curves match at that coordinate. That way, we're assured that their tangents line up, which must mean the curve transition is perfectly smooth. We can even make the second, third, etc. derivatives match up for better and better transitions.

    -

    Problem solved!

    -

    However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on reordering curves showed us that the same looking curve can have an infinite number of curve expressions of arbitrarily high Bézier degree, and each of those will have wildly different derivative values.

    -

    So what we really want is some kind of expression that's not based on any particular expression of t, but is based on something that is invariant to the kind of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being the same function for "coordinate at some distance D along the curve".

    -

    We've seen this before... that's the arc length function.

    -

    So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the form of the arc length function, which we saw above and is fairly simple, rather than needing to solve the arc length function. If we start with the arc length expression and the run through the steps necessary to determine its derivative (with an alternative, shorter demonstration of how to do this found over on Stackexchange), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the fundamental theorem of calculus), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.

    -

    Let me highlight what just happened, because it's pretty special:

    -
      -
    1. we wanted to make curves line up, and initially thought to match the curves' derivatives, but
      2. -
    2. that turned out to be a really bad choice, so instead
      3. -
    3. we picked a function that is basically impossible to work with, and then worked with that, which
      4. -
    4. gives us a simple formula that is and expression using the curves' derivatives.
      5. -
    -

    That's crazy!

    -

    But that's also one of the things that makes maths so powerful: even if your initial ideas are off the mark, you might be much closer than you thought you were, and the journey from "thinking we're completely wrong" to "actually being remarkably close to being right" is where we can find a lot of insight.

    -

    So, what does the function look like? This:

    - - -

    Which is really just a "short form" that glosses over the fact that we're dealing with functions of t, so let's expand that a tiny bit:

    - - -

    And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so evaluating this function for any t value is just a matter of basic arithematics.

    -

    In fact, let's just implement it right now:

    + +

    + And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any + (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that + looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier + curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so + evaluating this function for any t value is just a matter of basic arithematics. +

    +

    In fact, let's just implement it right now:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return numerator / denominator + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    That was easy! (Well okay, that "not a number" value will need to be taken into account by downstream code, but that's a reality of programming anyway)

    -

    With all of that covered, let's line up some curves! The following graphic gives you two curves that look identical, but use quadratic and cubic functions, respectively. As you can see, despite their derivatives being necessarily different, their curvature (thanks to being derived based on maths that "ignores" specific function derivative, and instead gives a formula that smooths out any differences) is exactly the same. And because of that, we can put them together such that the point where they overlap has the same curvature for both curves, giving us the smoothest transition.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    One thing you may have noticed in this sketch is that sometimes the curvature looks fine, but seems to be pointing in the wrong direction, making it hard to line up the curves properly. A way around that, of course, is to show the curvature on both sides of the curve, so let's just do that. But let's take it one step further: we can also compute the associated "radius of curvature", which gives us the implicit circle that "fits" the curve's curvature at any point, using what is possibly the simplest bit of maths found in this entire primer:

    - - -

    So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" our curve at some point that we can control by using a slider:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - + +

    + So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" + our curve at some point that we can control by using a slider: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    +
    +

    +
    + 이전목차다음 +
    + Tracing a curve at fixed distance intervals +

    +

    + Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance + intervals over time, like a train along a track, and you want to use Bézier curves. +

    +

    Now you have a problem.

    +

    + The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a + track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick + t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In + fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another. +

    +

    + The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be + straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To + make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for + this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length + function, which can have more inflection points. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + +

    + So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through + the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and + then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of + points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two + points, but if we have a high enough number of samples, we don't even need to bother. +

    +

    + So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t + curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks + like, by coloring each section of curve between two distance markers differently: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    +

    + However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to + constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT). +

    +
    +
    +

    +
    + 이전목차다음 +
    + Intersections +

    +

    + Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to + work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to + perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box + overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the + actual curve. +

    +

    + We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by + implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both + lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution. +

    +

    Line-line intersections

    +

    + If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are + an intervals on by linear algebra, using the procedure outlined in this + top coder + article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line + segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments. +

    +

    + The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on + (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection + point). +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    +

    Implementing line-line intersections

    +

    + Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, + but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe + you're using point structs for the line. Let's get coding: +

    -
    -
    -

    이전목차다음
    Tracing a curve at fixed distance intervals

    -

    Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance intervals over time, like a train along a track, and you want to use Bézier curves.

    -

    Now you have a problem.

    -

    The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another.

    -

    The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two points, but if we have a high enough number of samples, we don't even need to bother.

    -

    So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks like, by coloring each section of curve between two distance markers differently:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    -

    However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT).

    - -
    -
    -

    이전목차다음
    Intersections

    -

    Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the actual curve.

    -

    We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution.

    -

    Line-line intersections

    -

    If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are an intervals on by linear algebra, using the procedure outlined in this top coder article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments.

    -

    The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection point).

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    - -

    Implementing line-line intersections

    -

    Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe you're using point structs for the line. Let's get coding:

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    + return lli4(line1.p1, line1.p2, line2.p1, line2.p2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + +
    - +

    What about curve-line intersections?

    +

    + Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we + translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in + a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a + curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the + section on finding extremities. +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +
    -

    What about curve-line intersections?

    -

    Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the section on finding extremities.

    -
    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -
    +

    + Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the + curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and + curve splitting. +

    +
    +
    +

    +
    이전목차다음
    + Curve/curve intersection +

    +

    + Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" + technique: +

    +
      +
    1. + Take two curves C1 and C2, and treat them as a pair. +
      2. +
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. +
    3. + With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2). +
      4. +
    4. + For each pair, check whether their bounding boxes overlap. +
        +
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. +
      2. If there is overlap, rerun all steps for this pair.
        3. +
      +
      5. +
    5. + Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we + might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value + (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative). +
      6. +
    +

    + This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then + taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections. +

    +

    + The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click + the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value + that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the + algorithm, so you can try this with lots of different curves. +

    +

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + +

    + Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into + two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at + t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each + iteration, and each successive step homing in on the curve's self-intersection points. +

    +
    +
    +

    +
    + 이전목차다음 +
    + The projection identity +

    +

    + De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to + draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. + Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point + around to change the curve's shape. +

    +

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    +

    + In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want + to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know + has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for + reasons that will become obvious. +

    +

    + So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following + graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which + taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what + happens to that value? +

    +
    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +
    -

    Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and curve splitting.

    - -
    -
    -

    이전목차다음
    Curve/curve intersection

    -

    Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" technique:

    -
      -
    1. Take two curves C1 and C2, and treat them as a pair.
      2. -
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. -
    3. With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2).
      4. -
    4. For each pair, check whether their bounding boxes overlap.
        -
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. -
      2. If there is overlap, rerun all steps for this pair.
        3. -
      -
      5. -
    5. Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative).
      6. -
    -

    This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections.

    -

    The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the algorithm, so you can try this with lots of different curves.

    -

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - -

    Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each iteration, and each successive step homing in on the curve's self-intersection points.

    - -
    -
    -

    이전목차다음
    The projection identity

    -

    De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point around to change the curve's shape.

    -

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    -

    In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for reasons that will become obvious.

    -

    So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what happens to that value?

    -
    - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -
    - -

    So these graphics show us several things:

    -
      -
    1. a point at the tip of the curve construction's "hat": let's call that A, as well as
      2. -
    2. our on-curve point give our chosen t value: let's call that B, and finally,
      3. -
    3. a point that we get by projecting A, through B, onto the line between the curve's start and end points: let's call that C.
      4. -
    4. for both quadratic and cubic curves, two points e1 and e2, which represent the single-to-last step in de Casteljau's algorithm: in the last step, we find B at (1-t) * e1 + t * e2.
      5. -
    5. for cubic curves, also the points v1 and v2, which together with A represent the first step in de Casteljau's algorithm: in the next step, we find e1 and e2.
      6. -
    -

    These three values A, B, and C allow us to derive an important identity formula for quadratic and cubic Bézier curves: for any point on the curve with some t value, the ratio of distances from A to B and B to C is fixed: if some t value sets up a C that is 20% away from the start and 80% away from the end, then it doesn't matter where the start, end, or control points are; for that t value, C will always lie at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic and then move all the other points around: if you only move the control points, start and end won't move, and so neither will C, and if you move either start or end point, C will move but its relative position will not change.

    -

    So, how can we compute C? We start with our observation that C always lies somewhere between the start and end points, so logically C will have a function that interpolates between those two coordinates:

    - - -

    If we can figure out what the function u(t) looks like, we'll be done. Although we do need to remember that this u(t) will have a different form depending on whether we're working with quadratic or cubic curves. Running through the maths (with thanks to Boris Zbarsky) shows us the following two formulae:

    - - -

    And

    - - -

    So, if we know the start and end coordinates and the t value, we know C without having to calculate the A or even B coordinates. In fact, we can do the same for the ratio function. As another function of t, we technically don't need to know what A or B or C are. It, too, can be expressed as a pure function of t.

    -

    We start by observing that, given A, B, and C, the following always holds:

    - - -

    Working out the maths for this, we see the following two formulae for quadratic and cubic curves:

    - - -

    And

    - - -

    Which now leaves us with some powerful tools: given three points (start, end, and "some point on the curve"), as well as a t value, we can construct curves. We can compute C using the start and end points and our u(t) function, and once we have C, we can use our on-curve point (B) and the ratio(t) function to find A:

    - - -

    With A found, finding e1 and e2 for quadratic curves is a matter of running the linear interpolation with t between start and A to yield e1, and between A and end to yield e2. For cubic curves, there is no single pair of points that can act as e1 and e2 (there are infinitely many, because the tangent at B is a free parameter for cubic curves) so as long as the distance ratio between e1 to B and B to e2 is the Bézier ratio (1-t):t, we are free to pick any pair, after which we can reverse engineer v1 and v2:

    - - -

    And then reverse engineer the curve's control points:

    - - -

    So: if we have a curve's start and end points, as well as some third point B that we want the curve to pass through, then for any t value we implicitly know all the ABC values, which (combined with an educated guess on appropriate e1 and e2 coordinates for cubic curves) gives us the necessary information to reconstruct a curve's "de Casteljau skeleton". Which means that we can now do several things: we can "fit" curves using only three points, which means we can also "mold" curves by moving an on-curve point but leaving its start and end points, and then reconstruct the curve based on where we moved the on-curve point to. These are very useful things, and we'll look at both in the next few sections.

    - -
    -
    -

    이전목차다음
    Creating a curve from three points

    -

    Given the preceding section, you might be wondering if we can use that knowledge to just "create" curves by placing some points and having the computer do the rest, to which the answer is: that's exactly what we can now do!

    -

    For quadratic curves, things are pretty easy. Technically, we'll need a t value in order to compute the ratio function used in computing the ABC coordinates, but we can just as easily approximate one by treating the distance between the start and B point, and B and end point as a ratio, using

    - - -

    With this code in place, creating a quadratic curve from three points is literally just computing the ABC values, and using A as our curve's control point:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    For cubic curves we need to do a little more work, but really only just a little. We're first going to assume that a decent curve through the three points should approximate a circular arc, which first requires knowing how to fit a circle to three points. You may remember (if you ever learned it!) that a line between two points on a circle is called a chord, and that one property of chords is that the line from the center of any chord, perpendicular to that chord, passes through the center of the circle.

    -

    That means that if we have three points on a circle, we have three (different) chords, and consequently, three (different) lines that go from those chords through the center of the circle: if we find two of those lines, then their intersection will be our circle's center, and the circle's radius will—by definition!—be the distance from the center to any of our three points:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    With that covered, we now also know the tangent line to our point B, because the tangent to any point on the circle is a line through that point, perpendicular to the line from that point to the center. That just leaves marking appropriate points e1 and e2 on that tangent, so that we can construct a new cubic curve hull. We use the approach as we did for quadratic curves to automatically determine a reasonable t value, and then our e1 and e2 coordinates must obey the standard de Casteljau rule for linear interpolation:

    - - -

    Where d is the total length of the line segment from e1 to e2. So how long do we make that? There are again all kinds of approaches we can take, and a simple-but-effective one is to set the length of that segment to "one third the length of the baseline". This forces e1 and e2 to always be the "linear curve" distance apart, which means if we place our three points on a line, it will actually look like a line. Nice! The last thing we'll need to do is make sure to flip the sign of d depending on which side of the baseline our B is located, so we don't end up creating a funky curve with a loop in it. To do this, we can use the atan2 function:

    - - -

    This angle φ will be between 0 and π if B is "above" the baseline (rotating all three points so that the start is on the left and the end is the right), so we can use a relatively straight forward check to make sure we're using the correct sign for our value d:

    - - -

    The result of this approach looks as follows:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; more on that later), we're only trying to come up with some reasonable e1 and e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives us:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    That looks perfectly serviceable!

    -

    Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at implementing curve molding in the section after that, so read on!

    + +

    The result of this approach looks as follows:

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +

    + It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, + we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; + more on that later), we're only trying to come up with some reasonable e1 and + e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives + us: +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + +

    That looks perfectly serviceable!

    +

    + Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" + curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding + the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at + implementing curve molding in the section after that, so read on! +

    +
    +
    +

    +
    + 이전목차다음 +
    + Projecting a point onto a Bézier curve +

    +

    + Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're + trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the + curve our cursor will be closest to. So, how do we project points onto a curve? +

    +

    + If the Bézier curve is of low enough order, we might be able to + work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a + numerical approach again. We'll be finding our ideal t value using a + binary search. First, we do a coarse distance-check based on + t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast: +

    -
    -
    -

    이전목차다음
    Projecting a point onto a Bézier curve

    -

    Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the curve our cursor will be closest to. So, how do we project points onto a curve?

    -

    If the Bézier curve is of low enough order, we might be able to work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a numerical approach again. We'll be finding our ideal t value using a binary search. First, we do a coarse distance-check based on t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    After this runs, we know that LUT[i] is the coordinate on the curve in our LUT that is closest to the point we want to project, so that's a pretty good initial guess as to what the best projection onto our curve is. To refine it, we note that LUT[i] is a better guess than both LUT[i-1] and LUT[i+1], but there might be an even better projection somewhere else between those two values, so that's what we're going to be testing for, using a variation of the binary search.

    -
      -
    1. we start with our point p, and the t values t1=LUT[i-1].t and t2=LUT[i+1].t, which span an interval v = t2-t1.
      2. -
    2. we test this interval in five spots: the start, middle, and end (which we already have), and the two points in between the middle and start/end points
      3. -
    3. we then check which of these five points is the closest to our original point p, and then repeat step 1 with the points before and after the closest point we just found.
      4. -
    -

    This makes the interval we check smaller and smaller at each iteration, and we can keep running the three steps until the interval becomes so small as to lead to distances that are, for all intents and purposes, the same for all points.

    -

    So, let's see that in action: in this case, I'm going to arbitrarily say that if we're going to run the loop until the interval is smaller than 0.001, and show you what that means for projecting your mouse cursor or finger tip onto a rather complex Bézier curve (which, of course, you can reshape as you like). Also shown are the original three points that our coarse check finds.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - -
    -
    -

    이전목차다음
    Intersections with a circle

    -

    It might seem odd to cover this subject so much later than the line/line, line/curve, and curve/curve intersection topics from several sections earlier, but the reason we can't cover circle/curve intersections is that we can't really discuss circle/curve intersection until we've covered the kind of lookup table (LUT) walking that the section on projecting a point onto a curve uses. To see why, let's look at what we would have to do if we wanted to find the intersections between a curve and a circle using calculus.

    -

    First, we observe that "finding intersections" in this case means that, given a circle defined by a center point c = (x,y) and a radius r, we want to find all points on the Bezier curve for which the distance to the circle's center point is equal to the circle radius, which by definition means those points lie on the circle, and so count as intersections. In maths, that means we're trying to solve:

    - - -

    Which seems simple enough. Unfortunately, when we expand that dist function, things get a lot more problematic:

    - - -

    And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the Abel-Ruffini theorem that we saw before, we can't solve this by just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left with a monstrous function to solve.

    -

    So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a bit more code to make sure we find all of them, but let's look at the steps involved:

    + +

    + And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the + Abel-Ruffini theorem that we saw before, we can't solve this by + just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to + actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and + all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left + with a monstrous function to solve. +

    +

    + So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the + on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance + r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a + bit more code to make sure we find all of them, but let's look at the steps involved: +

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + -

    This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is exactly the radius, so the coordinate lies on both the curve and the circle.

    -

    So far so good.

    -

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    +

    + This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor + change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between + that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is + exactly the radius, so the coordinate lies on both the curve and the circle. +

    +

    So far so good.

    +

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + start = i + 2; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points ("current", "previous" and "before previous") and then check whether they form a local minimum:

    +

    + After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we + can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now + not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of + course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in + finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points + ("current", "previous" and "before previous") and then check whether they form a local minimum: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return start + i + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    In words: given a start index, the circle center and radius, and our LUT, we check where (closest to our start index) we can find a local minimum for the difference between "the distance from the curve to the circle center", and the circle's radius. We track this by looking at three values (associated with the indices index-2, index-1, and index), and we know we've found a local minimum if the three values show that the middle value (pd1) is less than either value beside it. When we do, we can set our "best guess, relative to start" as index-1. Of course, since we're now checking values relative to some start value, we might not find another candidate value at all, in which case we return start - 1, so that a simple "is the result less than start?" lets us determine that there are no more intersections to find.

    -

    Finally, while not necessary for point projection, there is one more step we need to perform when we run the binary refinement function on our candidate LUT indices, because we've so far only been testing using distances "closest to the radius of the circle", and that's actually not good enough... we need distances that are the radius of the circle. So, after running the refinement for each of these indices, we need to discard any final value that isn't the circle radius. And because we're working with floating point numbers, what this really means is that we need to discard any value that's a pixel or more "off". Or, if we want to get really fancy, "some small epsilon value".

    -

    Based on all of that, the following graphic shows this off for the standard cubic curve (which you can move the coordinates around for, of course) and a circle with a controllable radius centered on the graphic's center, using the code approach described above.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    And of course, for the full details, click that "view source" link.

    - -
    -
    -

    이전목차다음
    Molding a curve

    -

    Armed with knowledge of the "ABC" relation, point-on-curve projection, and guestimating reasonable looking helper values for cubic curve construction, we can finally cover curve molding: updating a curve's shape interactively, by dragging points on the curve around.

    -

    For quadratic curve, this is a really simple trick: we project our cursor onto the curve, which gives us a t value and initial B coordinate. We don't even need the latter: with our t value and "wherever the cursor is" as target B, we can compute the associated C:

    - - -

    And then the associated A:

    - - -

    And we're done, because that's our new quadratic control point!

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    As before, cubic curves are a bit more work, because while it's easy to find our initial t value and ABC values, getting those all-important e1 and e2 coordinates is going to pose a bit of a problem... in the section on curve creation, we were free to pick an appropriate t value ourselves, which allowed us to find appropriate e1 and e2 coordinates. That's great, but when we're curve molding we don't have that luxury: whatever point we decide to start moving around already has its own t value, and its own e1 and e2 values, and those may not make sense for the rest of the curve.

    -

    For example, let's see what happens if we just "go with what we get" when we pick a point and start moving it around, preserving its t value and e1/e2 coordinates:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    That looks reasonable, close to the original point, but the further we drag our point, the less "useful" things become. Especially if we drag our point across the baseline, rather than turning into a nice curve.

    -

    One way to combat this might be to combine the above approach with the approach from the creating curves section: generate both the "unchanged t/e1/e2" curve, as well as the "idealized" curve through the start/cursor/end points, with idealized t value, and then interpolating between those two curves:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    The slide controls the "falloff distance" relative to where the original point on the curve is, so that as we drag our point around, it interpolates with a bias towards "preserving t/e1/e2" closer to the original point, and bias towards "idealized" form the further away we move our point, with anything that's further than our falloff distance simply being the idealized curve. We don't even try to interpolate at that point.

    -

    A more advanced way to try to smooth things out is to implement continuous molding, where we constantly update the curve as we move around, and constantly change what our B point is, based on constantly projecting the cursor on the curve as we're updating it - this is, you won't be surprised to learn, tricky, and beyond the scope of this section: interpolation (with a reasonable distance) will do for now!

    - -
    -
    -

    이전목차다음
    Curve fitting

    -

    Given the previous section, one question you might have is "what if I don't want to guess t values?". After all, plenty of graphics packages do automated curve fitting, so how can we implement that in a way that just finds us reasonable t values all on its own?

    -

    And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a least squares polynomial regression. In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an nth order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".

    -

    Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the squared distance between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep-dive the specific merits of least squared error metrics versus least absolute error metrics, but those are well beyond the scope of this material.

    -

    So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his "Least Squares Bézier Fit" article, and end with some nice interactive graphics for doing some curve fitting.

    -

    Before we begin, we're going to use the curve in matrix form. In the section on matrices, I mentioned that some things are easier if we use the matrix representation of a Bézier curve rather than its calculus form, and this is one of those things.

    -

    As such, the first step in the process is expressing our Bézier curve as powers/coefficients/coordinate matrix T x M x C, by expanding the Bézier functions.

    -
    - -

    Revisiting the matrix representation

    -

    Rewriting Bézier functions to matrix form is fairly easy, if you first expand the function, and then arrange them into a multiple line form, where each line corresponds to a power of t, and each column is for a specific coefficient. First, we expand the function:

    - - -

    And then we (trivially) rearrange the terms across multiple lines:

    - - -

    This rearrangement has "factors of t" at each row (the first row is t⁰, i.e. "1", the second row is t¹, i.e. "t", the third row is t²) and "coefficient" at each column (the first column is all terms involving "a", the second all terms involving "b", the third all terms involving "c").

    -

    With that arrangement, we can easily decompose this as a matrix multiplication:

    - - -

    We can do the same for the cubic curve, of course. We know the base function for cubics:

    - - -

    So we write out the expansion and rearrange:

    - - -

    Which we can then decompose:

    - - -

    And, of course, we can do this for quartic curves too (skipping the expansion step):

    - - -

    And so and on so on. Now, let's see how to use these T, M, and C, to do some curve fitting.

    -
    + +

    + And so and on so on. Now, let's see how to use these T, M, and C, to do some curve + fitting. +

    + -

    Let's get started: we're going to assume we picked the right order curve: for n points we're fitting an n-1th order curve, so we "start" with a vector P that represents the coordinates we already know, and for which we want to do curve fitting:

    - - -

    Next, we need to figure out appropriate t values for each point in the curve, because we need something that lets us tie "the actual coordinate" to "some point on the curve". There's a fair number of different ways to do this (and a large part of optimizing "the perfect fit" is about picking appropriate t values), but in this case let's look at two "obvious" choices:

    -
      -
    1. equally spaced t values, and
      2. -
    2. t values that align with distance along the polygon.
      3. -
    -

    The first one is really simple: if we have n points, then we'll just assign each point i a t value of (i-1)/(n-1). So if we have four points, the first point will have t=(1-1)/(4-1)=0/3, the second point will have t=(2-1)/(4-1)=1/3, the third point will have t=2/3, and the last point will be t=1. We're just straight up spacing the t values to match the number of points we have.

    -

    The second one is a little more interesting: since we're doing polynomial regression, we might as well exploit the fact that our base coordinates just constitute a collection of line segments. At the first point, we're fixing t=0, and the last point, we want t=1, and anywhere in between we're simply going to say that t is equal to the distance along the polygon, scaled to the [0,1] domain.

    -

    To get these values, we first compute the general "distance along the polygon" matrix:

    - - -

    Where length() is literally just that: the length of the line segment between the point we're looking at, and the previous point. This isn't quite enough, of course: we still need to make sure that all the values between i=1 and i=n fall in the [0,1] interval, so we need to scale all values down by whatever the total length of the polygon is:

    - - -

    And now we can move on to the actual "curve fitting" part: what we want is a function that lets us compute "ideal" control point values such that if we build a Bézier curve with them, that curve passes through all our original points. Or, failing that, have an overall error distance that is as close to zero as we can get it. So, let's write out what the error distance looks like.

    -

    As mentioned before, this function is really just "the distance between the actual coordinate, and the coordinate that the curve evaluates to for the associated t value", which we'll square to get rid of any pesky negative signs:

    - - -

    Since this function only deals with individual coordinates, we'll need to sum over all coordinates in order to get the full error function. So, we literally just do that; the total error function is simply the sum of all these individual errors:

    - - -

    And here's the trick that justifies using matrices: while we can work with individual values using calculus, with matrices we can compute as many values as we make our matrices big, all at the "same time", We can replace the individual terms pi with the full P coordinate matrix, and we can replace Bézier(si) with the matrix representation T x M x C we talked about before, which gives us:

    - - -

    In which we can replace the rather cumbersome "squaring" operation with a more conventional matrix equivalent:

    - - -

    Here, the letter T is used instead of the number 2, to represent the matrix transpose; each row in the original matrix becomes a column in the transposed matrix instead (row one becomes column one, row two becomes column two, and so on).

    -

    This leaves one problem: T isn't actually the matrix we want: we don't want symbolic t values, we want the actual numerical values that we computed for S, so we need to form a new matrix, which we'll call 𝕋, that makes use of those, and then use that 𝕋 instead of T in our error function:

    - - -

    Which, because of the first and last values in S, means:

    - - -

    Now we can properly write out the error function as matrix operations:

    - - -

    So, we have our error function: we now need to figure out the expression for where that function has minimal value, e.g. where the error between the true coordinates and the coordinates generated by the curve fitting is smallest. Like in standard calculus, this requires taking the derivative, and determining where that derivative is zero:

    - - -
    + +
    +

    Where did this derivative come from?

    +

    + That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I + straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know. +

    +

    + So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific + case, I + posted a question + to Math.stackexchange, and received a answer that goes into way more detail than I had + hoped to receive. +

    +

    + Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean + maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that + true?". There are answers. They might just require some time to come to understand. +

    +
    -

    Where did this derivative come from?

    -

    That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know.

    -

    So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific case, I posted a question to Math.stackexchange, and received a answer that goes into way more detail than I had hoped to receive.

    -

    Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that true?". There are answers. They might just require some time to come to understand.

    -
    - -

    Now, given the above derivative, we can rearrange the terms (following the rules of matrix algebra) so that we end up with an expression for C:

    - - -

    Here, the "to the power negative one" is the notation for the matrix inverse. But that's all we have to do: we're done. Starting with P and inventing some t values based on the polygon the coordinates in P define, we can compute the corresponding Bézier coordinates C that specify a curve that goes through our points. Or, if it can't go through them exactly, as near as possible.

    -

    So before we try that out, how much code is involved in implementing this? Honestly, that answer depends on how much you're going to be writing yourself. If you already have a matrix maths library available, then really not that much code at all. On the other hand, if you are writing this from scratch, you're going to have to write some utility functions for doing your matrix work for you, so it's really anywhere from 50 lines of code to maybe 200 lines of code. Not a bad price to pay for being able to fit curves to pre-specified coordinates.

    -

    So let's try it out! The following graphic lets you place points, and will start computing exact-fit curves once you've placed at least three. You can click for more points, and the code will simply try to compute an exact fit using a Bézier curve of the appropriate order. Four points? Cubic Bézier. Five points? Quartic. And so on. Of course, this does break down at some point: depending on where you place your points, it might become mighty hard for the fitter to find an exact fit, and things might actually start looking horribly off once there's enough points for compound floating point rounding errors to start making a difference (which is around 10~11 points).

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - -

    You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be able to freely manipulate them and see what the effect on your curve is.

    - -
    -
    -

    이전목차다음
    Bézier curves and Catmull-Rom curves

    -

    Taking an excursion to different splines, the other common design curve is the Catmull-Rom spline, which unlike Bézier curves pass through each control point, so they offer a kind of "built-in" curve fitting.

    -

    In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what looks like a single Catmull-Rom curve is actually a spline: a single curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is defined by the point's coordinates, and the tangent for those points, the latter of which can trivially be derived from knowing the previous and next point:

    - + +

    + You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio + along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get + direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be + able to freely manipulate them and see what the effect on your curve is. +

    +
    +
    +

    +
    + 이전목차다음 +
    + Bézier curves and Catmull-Rom curves +

    +

    + Taking an excursion to different splines, the other common design curve is the + Catmull-Rom spline, which unlike Bézier curves + pass through each control point, so they offer a kind of "built-in" curve fitting. +

    +

    + In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets + you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the + tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    + Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what + looks like a single Catmull-Rom curve is actually a spline: a single + curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, + and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is + defined by the point's coordinates, and the tangent for those points, the latter of which + can trivially be derived from knowing the + previous and next point: +

    + - -

    One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on that in the next section.

    -

    In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of variations that make things considerable more complex):

    + +

    + One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up + with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent + should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on + that in the next section. +

    +

    + In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of + variations that make things + considerable more + complex): +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + point(c0 * v1 + c1 * dv1 + c2 * v2 + c3 * dv2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Now, since a Catmull-Rom curve is a form of cubic Hermite spline, and as cubic Bézier curves are also a form of cubic Hermite spline, we run into an interesting bit of maths programming: we can convert one to the other and back, and the maths for doing so is surprisingly simple!

    -

    The main difference between Catmull-Rom curves and Bézier curves is "what the points mean":

    -
      -
    • A cubic Bézier curve is defined by a start point, a control point that implies the tangent at the start, a control point that implies the tangent at the end, and an end point, plus a characterizing matrix that we can multiply by that point vector to get on-curve coordinates.
      • -
    • A Catmull-Rom curve is defined by a start point, a tangent that for that starting point, an end point, and a tangent for that end point, plus a characteristic matrix that we can multiple by the point vector to get on-curve coordinates.
      • -
    -

    Those are very similar, so let's see exactly how similar they are. We've already see the matrix form for Bézier curves, so how different is the matrix form for Catmull-Rom curves?:

    - - -

    That's pretty dang similar. So the question is: how can we convert that expression with Catmull-Rom matrix and vector into an expression of the Bézier matrix and vector? The short answer is of course "by using linear algebra", but the longer answer is the rest of this section, and involves some maths that you may not even care for: if you just want to know the (incredibly simple) conversions between the two curve forms, feel free to skip to the end of the following explanation, but if you want to how we can get one from the other... let's get mathing!

    -
    - -

    Deriving the conversion formulae

    -

    In order to convert between Catmull-Rom curves and Bézier curves, we need to know two things. Firstly, how to express the Catmull-Rom curve using a "set of four coordinates", rather than a mix of coordinates and tangents, and secondly, how to convert those Catmull-Rom coordinates to and from Bézier form.

    -

    We start with the first part, to figure out how we can go from Catmull-Rom V coordinates to Bézier P coordinates, by applying "some matrix T". We don't know what that T is yet, but we'll get to that:

    - - -

    So, this mapping says that in order to map a Catmull-Rom "point + tangent" vector to something based on an "all coordinates" vector, we need to determine the mapping matrix such that applying T yields P2 as start point, P3 as end point, and two tangents based on the lines between P1 and P3, and P2 nd P4, respectively.

    -

    Computing T is really more "arranging the numbers":

    - - -

    Thus:

    - - -

    However, we're not quite done, because Catmull-Rom curves have that "tension" parameter, written as τ (a lowercase"tau"), which is a scaling factor for the tangent vectors: the bigger the tension, the smaller the tangents, and the smaller the tension, the bigger the tangents. As such, the tension factor goes in the denominator for the tangents, and before we continue, let's add that tension factor into both our coordinate vector representation, and mapping matrix T:

    - - -

    With the mapping matrix properly done, let's rewrite the "point + tangent" Catmull-Rom matrix form to a matrix form in terms of four coordinates, and see what we end up with:

    - - -

    Replace point/tangent vector with the expression for all-coordinates:

    - - -

    and merge the matrices:

    - - -

    This looks a lot like the Bézier matrix form, which as we saw in the chapter on Bézier curves, should look like this:

    - - -

    So, if we want to express a Catmull-Rom curve using a Bézier curve, we'll need to turn this Catmull-Rom bit:

    - - -

    Into something that looks like this:

    - - -

    And the way we do that is with a fairly straight forward bit of matrix rewriting. We start with the equality we need to ensure:

    - - -

    Then we remove the coordinate vector from both sides without affecting the equality:

    - - -

    Then we can "get rid of" the Bézier matrix on the right by left-multiply both with the inverse of the Bézier matrix:

    - - -

    A matrix times its inverse is the matrix equivalent of 1, and because "something times 1" is the same as "something", so we can just outright remove any matrix/inverse pair:

    - - -

    And now we're basically done. We just multiply those two matrices and we know what V is:

    - - -

    We now have the final piece of our function puzzle. Let's run through each step.

    -
      -
    1. Start with the Catmull-Rom function:
      2. -
    - - -
      -
    1. rewrite to pure coordinate form:
      2. -
    - - -
      -
    1. rewrite for "normal" coordinate vector:
      2. -
    - - -
      -
    1. merge the inner matrices:
      2. -
    - - -
      -
    1. rewrite for Bézier matrix form:
      2. -
    - - -
      -
    1. and transform the coordinates so we have a "pure" Bézier expression:
      2. -
    - - -

    And we're done: we finally know how to convert these two curves!

    -
    + +

    And we're done: we finally know how to convert these two curves!

    + -

    If we have a Catmull-Rom curve defined by four coordinates P1 through P4, then we can draw that curve using a Bézier curve that has the vector:

    - - -

    Similarly, if we have a Bézier curve defined by four coordinates P1 through P4, we can draw that using a standard tension Catmull-Rom curve with the following coordinate values:

    - - -

    Or, if your API allows you to specify Catmull-Rom curves using plain coordinates:

    - - - -
    -
    -

    이전목차다음
    Creating a Catmull-Rom curve from three points

    -

    Much shorter than the previous section: we saw that Catmull-Rom curves need at least 4 points to draw anything sensible, so how do we create a Catmull-Rom curve from three points?

    -

    Short and sweet: we don't.

    -

    We run through the maths that lets us create a cubic Bézier curve, and then convert its coordinates to Catmull-Rom form using the conversion formulae we saw above.

    - -
    -
    -

    이전목차다음
    Forming poly-Bézier curves

    -

    Much like lines can be chained together to form polygons, Bézier curves can be chained together to form poly-Béziers, and the only trick required is to make sure that:

    -
      -
    1. the end point of each section is the starting point of the following section, and
      2. -
    2. the derivatives across that dual point line up.
      3. -
    -

    Unless you want sharp corners, of course. Then you don't even need 2.

    -

    We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind that just follows point 1. where the end point of a segment is the same point as the start point of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're the easiest to implement:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    Dragging the control points around only affects the curve segments that the control point belongs to, and moving an on-curve point leaves the control points where they are, which is not the most useful for practical modelling purposes. So, let's add in the logic we need to make things a little better. We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve point is the same as it's "outgoing" derivative:

    - - -

    We can effect this quite easily, because we know that the vector from a curve's last control point to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control point of the next curve matches that, all we have to do is mirror that last control point through the last on-curve point. And mirroring any point A through any point B is really simple:

    - - -

    So let's implement that and see what it gets us. The following two graphics show a quadratic and a cubic poly-Bézier curve again, but this time moving the control points around moves others, too. However, you might see something unexpected going on for quadratic curves...

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all the control points are effectively linked. Move one of them, and you move all of them. Not only that, but if we move the on-curve points, it's possible to get a situation where a control point cannot satisfy the constraint that it's the reflection of its two neighbouring control points... This means that we cannot use quadratic poly-Béziers for anything other than really, really simple shapes. And even then, they're probably the wrong choice. Cubic curves are pretty decent, but the fact that the derivatives are linked means we can't manipulate curves as well as we might if we relaxed the constraints a little.

    -

    So: let's relax the requirement a little.

    -

    We can change the constraint so that we still preserve the angle of the derivatives across sections (so transitions from one section to the next will still look natural), but give up the requirement that they should also have the same vector length. Doing so will give us a much more useful kind of poly-Bézier curve:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    Cubic curves are now better behaved when it comes to dragging control points around, but the quadratic poly-Bézier still has the problem that moving one control points will move the control points and may ending up defining "the next" control point in a way that doesn't work. Quadratic curves really aren't very useful to work with...

    -

    Finally, we also want to make sure that moving the on-curve coordinates preserves the relative positions of the associated control points. With that, we get to the kind of curve control that you might be familiar with from applications like Photoshop, Inkscape, Blender, etc.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a new, very serious problem: we can move an on-curve point in such a way that we can't compute what needs to "happen next". Move the top point down, below the left and right points, for instance. There is no way to preserve correct control points without a kink at the bottom point. Quadratic curves: just not that good...

    -

    A final improvement is to offer fine-level control over which points behave which, so that you can have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves for the rest of the path. Implementing that, is left as an exercise for the reader.

    - -
    -
    -

    이전목차다음
    Curve offsetting

    -

    Perhaps you're like me, and you've been writing various small programs that use Bézier curves in some way or another, and at some point you make the step to implementing path extrusion. But you don't want to do it pixel based; you want to stay in the vector world. You find that extruding lines is relatively easy, and tracing outlines is coming along nicely (although junction caps and fillets are a bit of a hassle), and then you decide to do things properly and add Bézier curves to the mix. Now you have a problem.

    -

    Unlike lines, you can't simply extrude a Bézier curve by taking a copy and moving it around, because of the curvatures; rather than a uniform thickness, you get an extrusion that looks too thin in places, if you're lucky, but more likely will self-intersect. The trick, then, is to scale the curve, rather than simply copying it. But how do you scale a Bézier curve?

    -

    Bottom line: you can't. So you cheat. We're not going to do true curve scaling, or rather curve offsetting, because that's impossible. Instead we're going to try to generate 'looks good enough' offset curves.

    -
    - -

    "What do you mean, you can't? Prove it."

    -

    First off, when I say "you can't," what I really mean is "you can't offset a Bézier curve with another Bézier curve", not even by using a really high order curve. You can find the function that describes the offset curve, but it won't be a polynomial, and as such it cannot be represented as a Bézier curve, which has to be a polynomial. Let's look at why this is:

    -

    From a mathematical point of view, an offset curve O(t) is a curve such that, given our original curve B(t), any point on O(t) is a fixed distance d away from coordinate B(t). So let's math that:

    - - -

    However, we're working in 2D, and d is a single value, so we want to turn it into a vector. If we want a point distance d "away" from the curve B(t) then what we really mean is that we want a point at d times the "normal vector" from point B(t), where the "normal" is a vector that runs perpendicular ("at a right angle") to the tangent at B(t). Easy enough:

    - - -

    Now this still isn't very useful unless we know what the formula for N(t) is, so let's find out. N(t) runs perpendicular to the original curve tangent, and we know that the tangent is simply B'(t), so we could just rotate that 90 degrees and be done with it. However, we need to ensure that N(t) has the same magnitude for every t, or the offset curve won't be at a uniform distance, thus not being an offset curve at all. The easiest way to guarantee this is to make sure N(t) always has length 1, which we can achieve by dividing B'(t) by its magnitude:

    - - -

    Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc length from some start a to end b, we must use the formula we saw earlier. Noting that "length" is usually denoted with double vertical bars:

    - - -

    So if we want the length of the tangent, we plug in B'(t), with t = 0 as start and -t = 1 as end:

    - - -

    And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.

    -

    There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine because they do, but they don't; remember that only the base function has binomial coefficients. That's before we factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

    -

    So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve.

    -

    And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the real offset curve would look like, if we could compute it.

    -
    + +

    + And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is + screwing everything up, because it turns our nice polynomials into things that are no longer polynomials. +

    +

    + There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with + unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine + because they do, but they don't; remember that only the base function has binomial coefficients. That's before we + factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make + all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call + them circles, and they have much simpler functions than Bézier curves. +

    +

    + So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means + N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means + that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for + O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve. +

    +

    + And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier + curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, + much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the + real offset curve would look like, if we could compute it. +

    + -

    So, you cannot offset a Bézier curve perfectly with another Bézier curve, no matter how high-order you make that other Bézier curve. However, we can chop up a curve into "safe" sub-curves (where "safe" means that all the control points are always on a single side of the baseline, and the midpoint of the curve at t=0.5 is roughly in the center of the polygon defined by the curve coordinates) and then point-scale each sub-curve with respect to its scaling origin (which is the intersection of the point normals at the start and end points).

    -

    A good way to do this reduction is to first find the curve's extreme points, as explained in the earlier section on curve extremities, and use these as initial splitting points. After this initial split, we can check each individual segment to see if it's "safe enough" based on where the center of the curve is. If the on-curve point for t=0.5 is too far off from the center, we simply split the segment down the middle. Generally this is more than enough to end up with safe segments.

    -

    The following graphics show off curve offsetting, and you can use the slider to control the distance at which the curve gets offset. The curve first gets reduced to safe segments, each of which is then offset at the desired distance. Especially for simple curves, particularly easily set up for quadratic curves, no reduction is necessary, but the more twisty the curve gets, the more the curve needs to be reduced in order to get segments that can safely be scaled.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    You may notice that this may still lead to small 'jumps' in the sub-curves when moving the curve around. This is caused by the fact that we're still performing a naive form of offsetting, moving the control points the same distance as the start and end points. If the curve is large enough, this may still lead to incorrect offsets.

    - -
    -
    -

    이전목차다음
    Graduated curve offsetting

    -

    What if we want to do graduated offsetting, starting at some distance s but ending at some other distance e? Well, if we can compute the length of a curve (which we can if we use the Legendre-Gauss quadrature approach) then we can also determine how far "along the line" any point on the curve is. With that knowledge, we can offset a curve so that its offset curve is not uniformly wide, but graduated between with two different offset widths at the start and end.

    -

    Like normal offsetting we cut up our curve in sub-curves, and then check at which distance along the original curve each sub-curve starts and ends, as well as to which point on the curve each of the control points map. This gives us the distance-along-the-curve for each interesting point in the sub-curve. If we call the total length of all sub-curves seen prior to seeing "the current" sub-curve S (and if the current sub-curve is the first one, S is zero), and we call the full length of our original curve L, then we get the following graduation values:

    -
      -
    • start: map S from interval (0,L) to interval (s,e)
      • -
    • c1: map(<strong>S+d1</strong>, 0,L, s,e), d1 = distance along curve to projection of c1
      • -
    • c2: map(<strong>S+d2</strong>, 0,L, s,e), d2 = distance along curve to projection of c2
      • -
    • ...
      • -
    • end: map(<strong>S+length(subcurve)</strong>, 0,L, s,e)
      • -
    -

    At each of the relevant points (start, end, and the projections of the control points onto the curve) we know the curve's normal, so offsetting is simply a matter of taking our original point, and moving it along the normal vector by the offset distance for each point. Doing so will give us the following result (these have with a starting width of 0, and an end width of 40 pixels, but can be controlled with your up and down arrow keys):

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - - -
    -
    -

    이전목차다음
    Circles and quadratic Bézier curves

    -

    Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. For instance, if you're doing font design, fonts have no concept of geometric shapes, they only know straight lines, and Bézier curves. OpenType fonts with TrueType outlines only know quadratic Bézier curves, and OpenType fonts with Type 2 outlines only know cubic Bézier curves. So how do you draw a circle, or an ellipse, or an arc?

    -

    You approximate.

    -

    We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.

    -

    Since arcs are mid-point-symmetrical, we need the control points to set up a symmetrical curve. For quadratic curves this means that the control point will be somewhere on a line that intersects the baseline at a right angle. And we don't get any choice on where that will be, since the derivatives at the start and end point have to line up, so our control point will lie at the intersection of the tangents at the start and end point.

    -

    First, let's try to fit the quadratic curve onto a circular arc. In the following sketch you can move the mouse around over a unit circle, to see how well, or poorly, a quadratic curve can approximate the arc from (1,0) to where your mouse cursor is:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    As you can see, things go horribly wrong quite quickly; even trying to approximate a quarter circle using a quadratic curve is a bad idea. An eighth of a turns might look okay, but how okay is okay? Let's apply some maths and find out. What we're interested in is how far off our on-curve coordinates are with respect to a circular arc, given a specific start and end angle. We'll be looking at how much space there is between the circular arc, and the quadratic curve's midpoint.

    -

    We start out with our start and end point, and for convenience we will place them on a unit circle (a circle around 0,0 with radius 1), at some angle φ:

    - - -

    What we want to find is the intersection of the tangents, so we want a point C such that:

    - - -

    i.e. we want a point that lies on the vertical line through S (at some distance a from S) and also lies on the tangent line through E (at some distance b from E). Solving this gives us:

    - - -

    First we solve for b:

    - - -

    which yields:

    - - -

    which we can then substitute in the expression for a:

    - - -

    A quick check shows that plugging these values for a and b into the expressions for Cx and Cy give the same x/y coordinates for both "a away from A" and "b away from B", so let's continue: now that we know the coordinate values for C, we know where our on-curve point T for t=0.5 (or angle φ/2) is, because we can just evaluate the Bézier polynomial, and we know where the circle arc's actual point P is for angle φ/2:

    - - -

    We compute T, observing that if t=0.5, the polynomial values (1-t)², 2(1-t)t, and t² are 0.25, 0.5, and 0.25 respectively:

    - - -

    Which, worked out for the x and y components, gives:

    - - -

    And the distance between these two is the standard Euclidean distance:

    - - -

    So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter circle and eighth circle?

    -
    - - plotted for 0 ≤ φ ≤ π: - - - plotted for 0 ≤ φ ≤ ½π: - - - - - plotted for 0 ≤ φ ≤ ¼π: -
    + +

    + So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter + circle and eighth circle? +

    + + + + + + + + +
    + + plotted for 0 ≤ φ ≤ π: + + + plotted for 0 ≤ φ ≤ ½π: + + + + + plotted for 0 ≤ φ ≤ ¼π: +
    -

    We now see why the eighth circle arc looks decent, but the quarter circle arc doesn't: an error of roughly 0.06 at t=0.5 means we're 6% off the mark... we will already be off by one pixel on a circle with pixel radius 17. Any decent sized quarter circle arc, say with radius 100px, will be way off if approximated by a quadratic curve! For the eighth circle arc, however, the error is only roughly 0.003, or 0.3%, which explains why it looks so close to the actual eighth circle arc. In fact, if we want a truly tiny error, like 0.001, we'll have to contend with an angle of (rounded) 0.593667, which equates to roughly 34 degrees. We'd need 11 quadratic curves to form a full circle with that precision! (technically, 10 and ten seventeenth, but we can't do partial curves, so we have to round up). That's a whole lot of curves just to get a shape that can be drawn using a simple function!

    -

    In fact, let's flip the function around, so that if we plug in the precision error, labelled ε, we get back the maximum angle for that precision:

    - - -

    And frankly, things are starting to look a bit ridiculous at this point, we're doing way more maths than we've ever done, but thankfully this is as far as we need the maths to take us: If we plug in the precisions 0.1, 0.01, 0.001 and 0.0001 we get the radians values 1.748, 1.038, 0.594 and 0.3356; in degrees, that means we can cover roughly 100 degrees (requiring four curves), 59.5 degrees (requiring six curves), 34 degrees (requiring 11 curves), and 19.2 degrees (requiring a whopping nineteen curves).

    -

    The bottom line? Quadratic curves are kind of lousy if you want circular (or elliptical, which are circles that have been squashed in one dimension) curves. We can do better, even if it's just by raising the order of our curve once. So let's try the same thing for cubic curves.

    - -
    -
    -

    이전목차다음
    Circular arcs and cubic Béziers

    -

    Let's look at approximating circles and circular arcs using cubic Béziers. How much better is that?

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    At cursory glance, a fair bit better, but let's find out how much better by looking at how to construct the Bézier curve.

    -

    A construction diagram for a cubic approximation of a circular arc

    -

    The start and end points are trivial, but the mid point requires a bit of work, but it's mostly basic trigonometry once we know the angle θ for our circular arc: if we scale our circular arc to a unit circle, we can always start our arc, with radius 1, at (1,0) and then given our arc angle θ, we also know that the circular arc has length θ (because unit circles are nice that way). We also know our end point, because that's just (cos(θ), sin(θ)), and so the challenge is to figure out what control points we need in order for the curve at t=0.5 to exactly touch the circular arc at the angle θ/2:

    -

    So let's again formally describe this:

    - - -

    Only P3 isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P4, P3) is a right angled triangle, with the distance between the origin and P4 being 1 (because we're working with a unit circle), and the distance between P4 and P3 being k, so that we can represent P3 as "The point P4 plus the vector from the origin to P4 but then rotated a quarter circle, counter-clockwise, and scaled by k".

    -

    With that, we can determine the y-coordinates for A, B, e1, and e2, after which we have all the information we need to determine what the value of k is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P2 and P3, e1 is between A and "midway between P1 and P2" (which is "half height" P2), and so forth:

    - - -

    Which now gives us two identities for B, because in addition to determining B through linear interpolation, we also know that B's y coordinate is just sin(θ/2): we started this exercise by saying we were going to approximate the circular arc using a Bézier curve that had its midpoint, which is point B, touching the unit circle at the arc's half-angle, by definition making B the point at (cos(θ/2), sin(θ/2)).

    -

    This means we can equate the two identities we now have for By and solve for k.

    -
    - -

    Deriving k

    -

    Solving for k is fairly straight forward, but it's a fair few steps, and if you just the immediate result: using a tool like Wolfram Alpha is definitely the way to go. That said, let's get going:

    - - -

    And finally, we can take further advantage of several trigonometric identities to drastically simplify our formula for k:

    - - -

    And we're done.

    -
    + +

    And we're done.

    + -

    So, the distance of our control points to the start/end points can be expressed as a number that we get from an almost trivial expression involving the circular arc's angle:

    - - -

    Which means that for any circular arc with angle θ and radius r, our Bézier approximation based on three points of incidence is:

    - - -

    Which also gives us the commonly found value of 0.55228 for quarter circles, based on them having an angle of half π:

    - - -

    And thus giving us the following Bézier coordinates for a quarter circle of radius r:

    - - -
    + +
    +

    So, how accurate is this?

    +

    + Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points + that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum + deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but + rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we + get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff. +

    +

    + So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, + finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around + that t value, repeating as many times as necessary to get us an arbitrarily precise value of t: +

    -

    So, how accurate is this?

    -

    Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff.

    -

    So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around that t value, repeating as many times as necessary to get us an arbitrarily precise value of t:

    - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return getMostWrongT(radius, bezier, worst_t - stepsize, worst_t + stepsize) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    Plus, how often do you get to write a function with that name?

    -

    Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity first, and then coming back down as we approach 2π)

    - - -
    - - - - - -
    - error plotted for 0 ≤ φ ≤ 2π - - error plotted for 0 ≤ φ ≤ π - - error plotted for 0 ≤ φ ≤ ½π -
    +

    Plus, how often do you get to write a function with that name?

    +

    + Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see + what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity + first, and then coming back down as we approach 2π) +

    + + + + + + + + + + + + + +
    + + + + + +
    + error plotted for 0 ≤ φ ≤ 2π + + error plotted for 0 ≤ φ ≤ π + + error plotted for 0 ≤ φ ≤ ½π +
    -

    That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again:

    -

    +

    + That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we + can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again: +

    +

    -

    Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves.

    -
    +

    + Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At + approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over + quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves. +

    +
    -

    So that's it, kappa is 4/3 · tan(θ/4) , we're done! ...or are we?

    -

    Can we do better?

    -

    Technically: yes, we can. But I'm going to prefix this section with "we can, and we should investigate that possibility, but let me warn you up front that the result is only better if we're going to hard-code the values". We're about to get into the weeds and the standard three-points-of-incidence value is so good already that for most applications, trying to do better won't make any sense at all.

    -

    So with that said: what we calculated above is an upper bound for a best fit Bézier curve for a circular arc: anywhere we don't touch the circular arc in our approximation, we've "overshot" the arc. What if we dropped our value for k just a little, so that the curve starts out as an over-estimation, but then crosses the circular arc, yielding an region of underestimation, and then crosses the circular arc again, with another region of overestimation. This might give us a lower overall error, so let's see what we can do.

    -

    First, let's express the total error (given circular arc angle θ, and some k) using standard calculus notation:

    - - -

    This says that the error function for a given angle and value of k is equal to the "infinite" sum of differences between our curve and the circular arc, as we run t from 0 to 1, using an infinitely small step size. between subsequent t values.

    -

    Now, since we want to find the minimal error, that means we want to know where along this function things go from "error is getting progressively less" to "error is increasing again", which means we want to know where its derivative is zero, which as mathematical expression looks like:

    - - -

    And here we have the most direct application of the Fundamental Theorem of Calculus: the derivative and integral are each other's inverse operations, so they cancel out, leaving us with our original function:

    - - -

    And now we just solve for that... oh wait. We've seen this before. In order to solve this, we'd end up needing to solve this:

    - - -

    And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section on arc lengths—there is no symbolic solution for this equasion. Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer!

    -
    + +

    + And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section + on arc lengths—there is no symbolic solution for this equasion. + Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer! +

    +
    +

    Iterating on a solution

    +

    + By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the + value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's + center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just + binary search our way to the most accurate value for c that gets us that middle case. +

    +

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    -

    Iterating on a solution

    -

    By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just binary search our way to the most accurate value for c that gets us that middle case.

    -

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    - - - - -
    1 - -
    2
    3
    4
    + return binarySearch(radius, angle, points, lowerBound, upperBound) + + + + 2 + + + 3 + + + 4 + + -

    And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and blog posts than you can ever read in a life time:

    +

    + And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and + blog posts than you can ever read in a life time: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return binarySearch(radius, angle, points, value, upperBound) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Using the following radialError function, which samples the curve's approximation of the circular arc over several points (although the first and last point will never contribute anything, so we skip them):

    +

    + Using the following radialError function, which samples the curve's approximation of the circular arc over several points + (although the first and last point will never contribute anything, so we skip them): +

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return err + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a vector, we can get its length to the origin using a magnitude call.

    -

    Examining the result

    -

    Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, for each angle, plot what the difference between the circular arc and the Bézier approximation looks like:

    -

    image-20210419085430711

    -

    Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from 0 to θ:

    -

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    - - -
    - - - - - -
    - max deflection using unit scale - - max deflection at 10x scale - - max deflection at 100x scale -
    +

    + In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a + vector, we can get its length to the origin using a magnitude call. +

    +

    Examining the result

    +

    + Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, + for each angle, plot what the difference between the circular arc and the Bézier approximation looks like: +

    +

    image-20210419085430711

    +

    + Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the + plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from + 0 to θ: +

    +

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    + + + + + + + + + + + + + +
    + + + + + +
    + max deflection using unit scale + + max deflection at 10x scale + + max deflection at 100x scale +
    -

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    -

    As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference.

    -
    +

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    +

    + As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by + 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an + almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference. +

    +
    -

    At this point it should be clear that while, yes, there are improvement to be had, they're essentially insignificant while also being much more computationally expensive.

    -

    TL;DR: just tell me which value I should be using

    -

    It depends on what we need to do. If we just want the best value for quarter circles, and we're going to hard code the value for k, then there is no reason to hard-code the constant k=4/3*tan(pi/8) when you can just as easily hard-code the constant as k=0.551784777779014 instead.

    -

    If you need "the" value for quarter circles, use 0.551785 instead of 0.55228

    -

    However, for dynamic arc approximation, in code that tries to fit circular paths using Bézier paths instead, it should be fairly obvious that the simple function involving a tangent computation, two divisions, and one multiplication, is vastly more performant than running all the code we ended writing just to get a 25% lower error value, and most certainly worth preferring over getting the "more accurate" value.

    -

    If you need to fit Béziers to circular arcs on the fly, use 4/3 * tan(θ/4)

    -

    However, always remember that if you're writing for humans, you can typically use the best of both worlds: as the user interacts with their curves, you should draw their curves instead of drawing approximations of them. If they need to draw circles or circular arcs, draw those, and only approximate them with a Bézier curve when the data needs to be exported to a format that doesn't support those. Ideally with a preview mechanism that highlights where the errors will be, and how large they will be.

    -

    If you're writing code for graphics design by humans, use circular arcs for circular arcs

    -

    And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best k values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).

    - -
    -
    -

    이전목차다음
    Approximating Bézier curves with circular arcs

    -

    Let's look at doing the exact opposite of the previous section: rather than approximating circular arc using Bézier curves, let's approximate Bézier curves using circular arcs.

    -

    We already saw in the section on circle approximation that this will never yield a perfect equivalent, but sometimes you need circular arcs, such as when you're working with fabrication machinery, or simple vector languages that understand lines and circles, but not much else.

    -

    The approach is fairly simple: pick a starting point on the curve, and pick two points that are further along the curve. Determine the circle that goes through those three points, and see if it fits the part of the curve we're trying to approximate. Decent fit? Try spacing the points further apart. Bad fit? Try spacing the points closer together. Keep doing this until you've found the "good approximation/bad approximation" boundary, record the "good" arc, and then move the starting point up to overlap the end point we previously found. Rinse and repeat until we've covered the entire curve.

    -

    We already saw how to fit a circle through three points in the section on creating a curve from three points, and finding the arc through those points is straight-forward: pick one of the three points as start point, pick another as an end point, and the arc has to necessarily go from the start point, to the end point, over the remaining point.

    -

    So, how can we convert a Bézier curve into a (sequence of) circular arc(s)?

    -
      -
    • Start at t=0
      • -
    • Pick two points further down the curve at some value m = t + n and e = t + 2n
      • -
    • Find the arc that these points define
      • -
    • Determine how close the found arc is to the curve:
        -
      • Pick two additional points e1 = t + n/2 and e2 = t + n + n/2.
        • -
      • These points, if the arc is a good approximation of the curve interval chosen, should - lie on the circle, so their distance to the center of the circle should be the - same as the distance from any of the three other points to the center.
        • -
      • For point points, determine the (absolute) error between the radius of the circle, and the -actual distance from the center of the circle to the point on the curve.
        • -
      • If this error is too high, we consider the arc bad, and try a smaller interval.
        • -
      -
      • -
    -

    The result of this is shown in the next graphic: we start at a guaranteed failure: s=0, e=1. That's the entire curve. The midpoint is simply at t=0.5, and then we start performing a binary search.

    -
      -
    1. We start with low=0, mid=0.5 and high=1
      2. -
    2. That'll fail, so we retry with the interval halved: {0, 0.25, 0.5}
        -
      • If that arc's good, we move back up by half distance: {0, 0.375, 0.75}.
        • -
      • However, if the arc was still bad, we move down by half the distance: {0, 0.125, 0.25}.
        • -
      -
      3. -
    3. We keep doing this over and over until we have two arcs, in sequence, of which the first arc is good, and the second arc is bad. When we find that pair, we've found the boundary between a good approximation and a bad approximation, and we pick the good arc.
      4. -
    -

    The following graphic shows the result of this approach, with a default error threshold of 0.5, meaning that if an arc is off by a combined half pixel over both verification points, then we treat the arc as bad. This is an extremely simple error policy, but already works really well. Note that the graphic is still interactive, and you can use your up and down arrow keys keys to increase or decrease the error threshold, to see what the effect of a smaller or larger error threshold is.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    With that in place, all that's left now is to "restart" the procedure by treating the found arc's end point as the new to-be-determined arc's starting point, and using points further down the curve. We keep trying this until the found end point is for t=1, at which point we are done. Again, the following graphic allows for up and down arrow key input to increase or decrease the error threshold, so you can see how picking a different threshold changes the number of arcs that are necessary to reasonably approximate a curve:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    So... what is this good for? Obviously, if you're working with technologies that can't do curves, but can do lines and circles, then the answer is pretty straightforward, but what else? There are some reasons why you might need this technique: using circular arcs means you can determine whether a coordinate lies "on" your curve really easily (simply compute the distance to each circular arc center, and if any of those are close to the arc radii, at an angle between the arc start and end, bingo, this point can be treated as lying "on the curve"). Another benefit is that this approximation is "linear": you can almost trivially travel along the arcs at fixed speed. You can also trivially compute the arc length of the approximated curve (it's a bit like curve flattening). The only thing to bear in mind is that this is a lossy equivalence: things that you compute based on the approximation are guaranteed "off" by some small value, and depending on how much precision you need, arc approximation is either going to be super useful, or completely useless. It's up to you to decide which, based on your application!

    - -
    -
    -

    이전목차다음
    B-Splines

    -

    No discussion on Bézier curves is complete without also giving mention of that other beast in the curve design space: B-Splines. Easily confused to mean Bézier splines, that's not actually what they are; they are "basis function" splines, which makes a lot of difference, and we'll be looking at those differences in this section. We're not going to dive as deep into B-Splines as we have for Bézier curves (that would be an entire primer on its own) but we'll be looking at how B-Splines work, what kind of maths is involved in computing them, and how to draw them based on a number of parameters that you can pick for individual B-Splines.

    -

    First off: B-Splines are piecewise, polynomial interpolation curves, where the "single curve" is built by performing polynomial interpolation over a set of points, using a sliding window of a fixed number of points. For instance, a "cubic" B-Spline defined by twelve points will have its curve built by evaluating the polynomial interpolation of four points, and the curve can be treated as a lot of different sections, each controlled by four points at a time, such that the full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, ..., {8,9,10,11}, and finally {9,10,11,12}, for eight sections.

    -

    What do they look like? They look like this! Tap on the graphic to add more points, and move points around to see how they map to the spline curve drawn.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - -

    The important part to notice here is that we are not doing the same thing with B-Splines that we do for poly-Béziers or Catmull-Rom curves: both of the latter simply define new sections as literally "new sections based on new points", so a 12 point cubic poly-Bézier curve is actually impossible, because we start with a four point curve, and then add three more points for each section that follows, so we can only have 4, 7, 10, 13, 16, etc. point Poly-Béziers. Similarly, while Catmull-Rom curves can grow by adding single points, this addition of a single point introduces three implicit Bézier points. Cubic B-Splines, on the other hand, are smooth interpolations of each possible curve involving four consecutive points, such that at any point along the curve except for our start and end points, our on-curve coordinate is defined by four control points.

    -

    Consider the difference to be this:

    -
      -
    • for Bézier curves, the curve is defined as an interpolation of points, but:
      • -
    • for B-Splines, the curve is defined as an interpolation of curves.
      • -
    -

    In fact, let's look at that again, but this time with the base curves shown, too. Each consecutive four points define one curve:

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have a look at how things work.

    -

    How to compute a B-Spline curve: some maths

    -

    Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the end, just like for Bézier curves), by evaluating the following function:

    - + +

    + In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have + a look at how things work. +

    +

    How to compute a B-Spline curve: some maths

    +

    + Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic + B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the + end, just like for Bézier curves), by evaluating the following function: +

    + - -

    Which, honestly, doesn't tell us all that much. All we can see is that a point on a B-Spline curve is defined as "a mix of all the control points, weighted somehow", where the weighting is achieved through the N(...) function, subscripted with an obvious parameter i, which comes from our summation, and some magical parameter k. So we need to know two things: 1. what does N(t) do, and 2. what is that k? Let's cover both, in reverse order.

    -

    The parameter k represents the "knot interval" over which a section of curve is defined. As we learned earlier, a B-Spline curve is itself an interpolation of curves, and we can treat each transition where a control point starts or stops influencing the total curvature as a "knot on the curve". -Doing so for a degree d B-Spline with n control point gives us d + n + 1 knots, defining d + n intervals along the curve, and it is these intervals that the above k subscript to the N() function applies to.

    -

    Then the N() function itself. What does it look like?

    - - -

    So this is where we see the interpolation: N(t) for an (i,k) pair (that is, for a step in the above summation, on a specific knot interval) is a mix between N(t) for (i,k-1) and N(t) for (i+1,k-1), so we see that this is a recursive iteration where i goes up, and k goes down, so it seem reasonable to expect that this recursion has to stop at some point; obviously, it does, and specifically it does so for the following i/k values:

    - - -

    And this function finally has a straight up evaluation: if a t value lies within a knot-specific interval once we reach a k=1 value, it "counts", otherwise it doesn't. We did cheat a little, though, because for all these values we need to scale our t value first, so that it lies in the interval bounded by knots[d] and knots[n], which are the start point and end point where curvature is controlled by exactly order control points. For instance, for degree 3 (=order 4) and 7 control points, with knot vector [1,2,3,4,5,6,7,8,9,10,11], we map t from [the interval 0,1] to the interval [4,8], and then use that value in the functions above, instead.

    -

    Can we simplify that?

    -

    We can, yes.

    -

    People far smarter than us have looked at this work, and two in particular — Maurice Cox and Carl de Boor — came to a mathematically pleasing solution: to compute a point P(t), we can compute this point by evaluating d(t) on a curve section between knots i and i+1:

    - - -

    This is another recursive function, with k values decreasing from the curve order to 1, and the value α (alpha) defined by:

    - - -

    That looks complicated, but it's not. Computing alpha is just a fraction involving known, plain numbers. And, once we have our alpha value, we also have (1-alpha) because it's a trivial subtraction. Computing the d() function is thus mostly a matter of computing pretty simple arithmetical statements, with some caching of results so we can refer to them as we recurve. While the recursion might see computationally expensive, the total algorithm is cheap, as each step only involves very simple maths.

    -

    Of course, the recursion does need a stop condition:

    - - -

    So, we actually see two stopping conditions: either i becomes 0, in which case d() is zero, or k becomes zero, in which case we get the same "either 1 or 0" that we saw in the N() function above.

    -

    Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily using the same kind of linear interpolation we saw in de Casteljau's algorithm. For instance, if we write out d() for i=3 and k=3, we get the following recursion diagram:

    - - -

    That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. It's really quite elegant.

    -

    One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then summing back up "to the left". We can just start on the right and work our way left immediately.

    -

    Running the computation

    -

    Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on this Michigan Tech page, but an easier to read version is implemented by b-spline.js, so we'll look at its code.

    -

    Given an input value t, we first map the input to a value from the domain [0,1] to the domain [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped t value lies on:

    + +

    + That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a + mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are + themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. + It's really quite elegant. +

    +

    + One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the + d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so + in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point + vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then + summing back up "to the left". We can just start on the right and work our way left immediately. +

    +

    Running the computation

    +

    + Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, + and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on + this Michigan Tech page, but an easier to read version + is implemented by b-spline.js, so we'll look at its code. +

    +

    + Given an input value t, we first map the input to a value from the domain [0,1] to the domain + [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped + t value lies on: +

    - - -
    1 - -
    2
    3
    +} + + + + 2 + + + 3 + + -

    after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU page (updated to use this description's variable names):

    +

    + after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU + page (updated to use this description's variable names): +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those v[i-1] don't try to use an array index that doesn't exist)

    -

    Open vs. closed paths

    -

    Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need more than just a single point.

    -

    Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than separate coordinate objects.

    -

    Manipulating the curve through the knot vector

    -

    The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at:

    -
      -
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. -
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. -
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. -
    4. we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a uniform vector in between.
      5. -
    -

    Uniform B-Splines

    -

    The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to the curvature.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve.

    -

    The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get rid of these, by being clever about how we apply the following uniformity-breaking approach instead...

    -

    Reducing local curve complexity by collapsing intervals

    -

    Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost and the curve "kinks".

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Open-Uniform B-Splines

    -

    By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the curve not starting and ending where we'd kind of like it to:

    -

    For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences that determine the curve shape.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Non-uniform B-Splines

    -

    This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than that any value knots[k+1] should be greater than or equal to knots[k].

    -

    One last thing: Rational B-Splines

    -

    While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like for rational Bézier curves.

    - - - 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. - - - - - -

    Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in arbitrary-precision 2D design due to the level of control a NURBS curve offers designers.

    -

    While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the last.

    -

    Extending our implementation to cover rational splines

    -

    The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the extended dimension.

    -

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    -

    We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how many dimensions it needs to interpolate.

    -

    In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight information.

    -

    Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing (x'/w', y'/w'). And that's it, we're done!

    +

    + (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the + interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those + v[i-1] don't try to use an array index that doesn't exist) +

    +

    Open vs. closed paths

    +

    + Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last + point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first + and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As + such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly + more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need + more than just a single point. +

    +

    + Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for + points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same + coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing + references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than + separate coordinate objects. +

    +

    Manipulating the curve through the knot vector

    +

    + The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As + mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the + values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on + intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting + things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at: +

    +
      +
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. +
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. +
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. +
    4. + we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a + uniform vector in between. +
      5. +
    +

    Uniform B-Splines

    +

    + The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire + curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or + [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines + the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to + the curvature. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    + This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every + interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve. +

    +

    + The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can + perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get + rid of these, by being clever about how we apply the following uniformity-breaking approach instead... +

    +

    Reducing local curve complexity by collapsing intervals

    +

    + Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the + sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, + and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost + and the curve "kinks". +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    Open-Uniform B-Splines

    +

    + By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the + curve not starting and ending where we'd kind of like it to: +

    +

    + For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in + which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values + N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector + [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences + that determine the curve shape. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    Non-uniform B-Splines

    +

    + This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to + pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than + that any value knots[k+1] should be greater than or equal to knots[k]. +

    +

    One last thing: Rational B-Splines

    +

    + While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's + "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a + ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a + control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like + for rational Bézier curves. +

    + + + 스크립트가 꺼져 있어 대체 이미지를 대신 표시합니다. + + + + + +

    + Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the + NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, + the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an + important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in + arbitrary-precision 2D design due to the level of control a NURBS curve offers designers. +

    +

    + While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform + Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, + they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the + last. +

    +

    Extending our implementation to cover rational splines

    +

    + The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the + control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to + one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the + extended dimension. +

    +

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    +

    + We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate + interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how + many dimensions it needs to interpolate. +

    +

    + In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the + final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight + information. +

    +

    + Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing + (x'/w', y'/w'). And that's it, we're done! +

    +
    +
    + -
    -
    - +

    +
    이전목차
    + Comments and questions +

    +

    + First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how + to let me know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing. +

    +

    With that said, on to the comments!

    +
    +
    + + -

    이전목차
    Comments and questions

    -

    First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing.

    -

    With that said, on to the comments!

    -
    +
    +
    + This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are + almost no rights reserved. Go do something cool with it! +
    - - - - -
    - -
    - - This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are - almost no rights reserved. - Go do something cool with it! - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/news/2020-09-18.html b/docs/news/2020-09-18.html index 5d5af600..f80113fa 100644 --- a/docs/news/2020-09-18.html +++ b/docs/news/2020-09-18.html @@ -1,177 +1,208 @@ + + + + A Primer on Bézier Curves - Rewriting the tech stack - - - - A Primer on Bézier Curves - Rewriting the tech stack + - + - + - + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    +

    Rewriting the tech stack

    +
    Fri, 18 Sep 2020
    +
    +
    + +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    +

    + Once upon a time, I needed to draw some Bezier curves because I was trying to create a Japanese kanji composition system that turned strokes + into outlines, and that required knowing how to offset Bezier curves and... at the time (2011, time flies) there was no good single source of + information for Bezier curves on the web. So I made one. Sure it started small, but it turns out that if you just keep adding bits to + something, several years later you have quite the monster, and a single HTML file becomes intractible. +

    +

    + So, in 2016, when React.js exploded onto the scene, I rewrote the primer as a React app, and it became a + lot easier to maintain. Like, a lot a lot. However, there was a downside: no JS meant no content. Sure, server-side rendering sort of + existed, but not really, and because the Primer is hosted through github, there was no "server" to run. Plus, trying to rehydrate an app the + size of the Primer from a giant HTML file had truly dire performance. +

    +

    + So I left it a regular React app, and every time I thought "wouldn't it be nice if it was just... a web page again?" the browser landscape + just hadn't caught up. Finally, in 2020, things are different: with a global pandemic, and some vacation time, and something random causing me + to look up the state of HTML custom elements, everything was pointing at it being time to finally, finally, turn the Primer back into + a normal web page. +

    +

    + The new tech stack is, frankly, pretty amazing. It does some things that weren't possible even half a year before I started the rewrite, let + alone being possible in 2016, and so because so much has changed, this post will be the first in a series of posts on how the new tech stack + works. +

    +

    To give a bit of a teaser, some of the things I'll be writing about:

    +
      +
    • Essentially reinventing (a limited form of) Processing.js
      • +
    • Writing a custom build system, because I'm exhausted with Webpack and hope to never use it again.
      • +
    • Using modern ES module code that runs in both the browser and Node.js.
      • +
    • + Chapter content written as easy to read and write markdown format: view this blog post's source file. +
      • +
    • A custom <graphics-element> element that turns a src="blah.js" into an interactive canvas graphic...
      • +
    • + ...with that same source code being read in and run by Node.js on a canvas to generate fallback images so that even without JS, + graphics work. +
      • +
    +
    + + + Scripts are disabled. Showing fallback image. + + + +
    -
    - -

    Rewriting the tech stack

    -
    Fri, 18 Sep 2020
    - -
    - -
    - - -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    - - -

    Once upon a time, I needed to draw some Bezier curves because I was trying to create a Japanese kanji composition system that turned strokes into outlines, and that required knowing how to offset Bezier curves and... at the time (2011, time flies) there was no good single source of information for Bezier curves on the web. So I made one. Sure it started small, but it turns out that if you just keep adding bits to something, several years later you have quite the monster, and a single HTML file becomes intractible.

    -

    So, in 2016, when React.js exploded onto the scene, I rewrote the primer as a React app, and it became a lot easier to maintain. Like, a lot a lot. However, there was a downside: no JS meant no content. Sure, server-side rendering sort of existed, but not really, and because the Primer is hosted through github, there was no "server" to run. Plus, trying to rehydrate an app the size of the Primer from a giant HTML file had truly dire performance.

    -

    So I left it a regular React app, and every time I thought "wouldn't it be nice if it was just... a web page again?" the browser landscape just hadn't caught up. Finally, in 2020, things are different: with a global pandemic, and some vacation time, and something random causing me to look up the state of HTML custom elements, everything was pointing at it being time to finally, finally, turn the Primer back into a normal web page.

    -

    The new tech stack is, frankly, pretty amazing. It does some things that weren't possible even half a year before I started the rewrite, let alone being possible in 2016, and so because so much has changed, this post will be the first in a series of posts on how the new tech stack works.

    -

    To give a bit of a teaser, some of the things I'll be writing about:

    -
      -
    • Essentially reinventing (a limited form of) Processing.js
      • -
    • Writing a custom build system, because I'm exhausted with Webpack and hope to never use it again.
      • -
    • Using modern ES module code that runs in both the browser and Node.js.
      • -
    • Chapter content written as easy to read and write markdown format: view this blog post's source file.
      • -
    • A custom <graphics-element> element that turns a src="blah.js" into an interactive canvas graphic...
      • -
    • ...with that same source code being read in and run by Node.js on a canvas to generate fallback images so that even without JS, graphics work.
      • -
    -
    - - - Scripts are disabled. Showing fallback image. - - - -
    - -
      -
    • Real LaTeX code, that gets saved as .tex file, so it can be compiled into optimized SVG using xelatex, pdfcrop, pdf2svg, and svgo:
      • -
    -
    - - -
    + + -
      -
    • "Lazy loaded everything", so that you get what you need, only when or even just before you need it.
      • -
    • Localized content based on a simple filenaming scheme.
      • -
    • Nicely formatted HTML, CSS, and JS thanks to prettier.
      • -
    • with some code formatting so that there are line numbers without needing JS:
      • -
    +
      +
    • "Lazy loaded everything", so that you get what you need, only when or even just before you need it.
      • +
    • Localized content based on a simple filenaming scheme.
      • +
    • Nicely formatted HTML, CSS, and JS thanks to prettier.
      • +
    • with some code formatting so that there are line numbers without needing JS:
      • +
    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -
      -
    • Responsive CSS, so the content intelligently reflows where possible.
      • -
    • A "Live build" setup for working on the content and code.
      • -
    • Automatic link-checking to make sure none of the links in the Primer lead you to a 404.
      • -
    • This "news" section, so that I can write posts to go along with new sections getting added, or notable changes being made.
      • -
    -

    It's going to take me a while to detail the entire tech stack, but ultimately what matters is that you get a Primer that is a normal "vanilla" HTML, CSS, and JS page again, that "just works" even with JS disabled.

    -

    Enjoy The new Primer on Bézier Curves, and if you find any problems, you know where to go.

    -

    See you in the next post!

    -

    — Pomax

    +
      +
    • Responsive CSS, so the content intelligently reflows where possible.
      • +
    • A "Live build" setup for working on the content and code.
      • +
    • Automatic link-checking to make sure none of the links in the Primer lead you to a 404.
      • +
    • This "news" section, so that I can write posts to go along with new sections getting added, or notable changes being made.
      • +
    +

    + It's going to take me a while to detail the entire tech stack, but ultimately what matters is that you get a Primer that is a normal "vanilla" + HTML, CSS, and JS page again, that "just works" even with JS disabled. +

    +

    + Enjoy The new Primer on Bézier Curves, and if you find any problems, + you know where to go. +

    +

    See you in the next post!

    +

    — Pomax

    +
    +
    -
    - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/news/2020-11-22.html b/docs/news/2020-11-22.html index 4e217d0b..0784c77a 100644 --- a/docs/news/2020-11-22.html +++ b/docs/news/2020-11-22.html @@ -1,142 +1,143 @@ + + + + A Primer on Bézier Curves - Curve-circle intersections - - - - A Primer on Bézier Curves - Curve-circle intersections + - + - + - + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    +

    Curve-circle intersections

    +
    Sun, 22 Nov 2020
    +
    +
    + +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    +

    + While the primer covered line/line, line/curve, and curve/curve intersections, there was one other obvious intersection conspicuously missing: + circle/curve intersections. You'd think those were just an extension on the maths used for the other three, but unfortunately, this is not the + case. Rather than using calculus, the only real way to determine where a polynomial curve intersects it is to sample the curve at a resolution + high enough to find you intervals on the curve where there likely is an intersection, then refining that interval until you find actual + intersections. +

    +

    + It is, in fact, rather similar to projecting a point onto a bezier curve where + the point is the circle's center, and where the projection distance actually needs to match the circle radius, so: + let's see how to do that! +

    +

    — Pomax

    +
    -
    +
    -

    Curve-circle intersections

    -
    Sun, 22 Nov 2020
    - -
    - -
    - - -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    - - -

    While the primer covered line/line, line/curve, and curve/curve intersections, there was one other obvious intersection conspicuously missing: circle/curve intersections. You'd think those were just an extension on the maths used for the other three, but unfortunately, this is not the case. Rather than using calculus, the only real way to determine where a polynomial curve intersects it is to sample the curve at a resolution high enough to find you intervals on the curve where there likely is an intersection, then refining that interval until you find actual intersections.

    -

    It is, in fact, rather similar to projecting a point onto a bezier curve where the point is the circle's center, and where the projection distance actually needs to match the circle radius, so: let's see how to do that!

    -

    — Pomax

    - - -
    - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/news/index.html b/docs/news/index.html index ffc2af04..675d2722 100644 --- a/docs/news/index.html +++ b/docs/news/index.html @@ -1,138 +1,113 @@ + + + + A Primer on Bézier Curves - News - - - - A Primer on Bézier Curves - News + - + - + - + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    +

    News posts

    +
    +
    +

    + Every now and then the Primer gets updated - these posts chronicle the evolution of the site, and hopefully offer interesting information not + just about the process of maintaining a resource like this, but also neat tech tricks, implementation approaches, maths that didn't make it + into the primer itself, etc. +

    +

    + This section is still very new, so for the moment there aren't all that many posts up yet, but there's a series of posts planned already, and + if you're the kind of person who likes to keep tabs on updates by using RSS: good news, have an RSS link!. +

    -
    + +
    -

    News posts

    +
    - - -
    - -

    - Every now and then the Primer gets updated - these posts chronicle the evolution of the site, - and hopefully offer interesting information not just about the process of maintaining a resource - like this, but also neat tech tricks, implementation approaches, maths that didn't make it into - the primer itself, etc. -

    - -

    - This section is still very new, so for the moment there aren't all that many posts up yet, but - there's a series of posts planned already, and if you're the kind of person who likes to keep - tabs on updates by using RSS: good news, have an RSS link!. -

    - - - -
    - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/news/rss.xml b/docs/news/rss.xml index e1d011e4..8eee0b2a 100644 --- a/docs/news/rss.xml +++ b/docs/news/rss.xml @@ -6,7 +6,7 @@ News updates for the primer on Bézier Curves by Pomax en-GB - Tue Jan 04 2022 03:29:08 +00:00 + Tue Jan 04 2022 11:49:37 +00:00 https://pomax.github.io/bezierinfo/images/og-image.png A Primer on Bézier Curves @@ -17,19 +17,19 @@ Curve-circle intersections https://pomax.github.io/bezierinfo/news/2020-11-22.html - + <p>While the primer covered line/line, line/curve, and curve/curve intersections, there was one other obvious intersection conspicuously missing: circle/curve intersections. You'd think those were just an extension on the maths used for the other three, but unfortunately, this is not the case. Rather than using calculus, the only real way to determine where a polynomial curve intersects it is to sample the curve at a resolution high enough to find you intervals on the curve where there likely is an intersection, then refining that interval until you find actual intersections.</p> <p>It is, in fact, rather similar to <a href="https://pomax.github.io/bezierinfo/#projections">projecting a point onto a bezier curve</a> where the point is the circle's center, and where the projection distance actually needs to match the circle radius, so: <a href="https://pomax.github.io/bezierinfo/#circleintersection">let's see how to do that</a>!</p> <p>— <a href="https://twitter.com/TheRealPomax">Pomax</a></p> - Sun Nov 22 2020 09:00:00 +00:00 + Sat Nov 21 2020 16:00:00 +00:00 2020-11-22.html Rewriting the tech stack https://pomax.github.io/bezierinfo/news/2020-09-18.html - + <p>Once upon a time, I needed to draw some Bezier curves because I was trying to create a Japanese kanji composition system that turned strokes into outlines, and that required knowing how to offset Bezier curves and... at the time (2011, time flies) there was no good single source of information for Bezier curves on the web. So I made one. Sure it started small, but it turns out that if you just keep adding bits to something, several years later you have quite the monster, and a single HTML file becomes intractible.</p> <p>So, in 2016, when <a href="https://reactjs.org/">React.js</a> exploded onto the scene, I rewrote the primer as a React app, and it became a lot easier to maintain. Like, <em>a lot</em> a lot. However, there was a downside: no JS meant no content. Sure, server-side rendering sort of existed, but not really, and because the Primer is hosted through github, there was no "server" to run. Plus, trying to rehydrate an app the size of the Primer from a giant HTML file had truly <em>dire</em> performance.</p> <p>So I left it a regular React app, and every time I thought "wouldn't it be nice if it was just... a web page again?" the browser landscape just hadn't caught up. Finally, in 2020, things are different: with a global pandemic, and some vacation time, and something random causing me to look up the state of HTML custom elements, everything was pointing at it being time to finally, <em>finally</em>, turn the Primer back into a normal web page.</p> @@ -57,17 +57,16 @@ </ul> <blockquote> <!-- -\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math - - __ n=3 n-k k - B(t) = ❯ \ P \binomnk(1-t) t - cubic ‾‾ k=0 - k - - 3 2 2 3 - = P (1-t) + 3 P (1-t) t + 3P (1-t)t + P t - - 0 1 2 3 + + __ n=3 n-k k +B(t) = ❯ \ P \binomnk(1-t) t + cubic ‾‾ k=0 + k + + 3 2 2 3 + = P (1-t) + 3 P (1-t) t + 3P (1-t)t + P t + + 0 1 2 3 --> <img class="LaTeX SVG" src="./images/news/2020-09-18.html/15225da473048d8c7b5b473b89de0b66.svg" width="401px" height="97px" loading="lazy"> </blockquote> @@ -106,7 +105,7 @@ draw() { <tr><td>11</td></tr> <tr><td>12</td></tr> <tr><td>13</td></tr></table> - + <ul> <li>Responsive CSS, so the content intelligently reflows where possible.</li> <li>A "Live build" setup for working on the content and code.</li> @@ -119,7 +118,7 @@ draw() { <p>— <a href="https://twitter.com/TheRealPomax">Pomax</a></p> - Fri Sep 18 2020 09:00:00 +00:00 + Thu Sep 17 2020 17:00:00 +00:00 2020-09-18.html diff --git a/docs/ru-RU/index.html b/docs/ru-RU/index.html index 5fc9ff8b..1f48d7a6 100644 --- a/docs/ru-RU/index.html +++ b/docs/ru-RU/index.html @@ -1,161 +1,169 @@ + + + + Основы кривых Безье - - - - Основы кривых Безье + - + - + - + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    + - +
    +

    + Основы кривых Безье +

    +

    Бесплатная онлайн-книга для тех, кому действительно важно знать, как работают кривые Безье

    +
    + Читайте на своём языке: + +

    + (Не нашли свой язык или хотите, чтобы он достиг до 100%? + Помогите нам с переводом!) +

    +
    +

    + Приветствуем на Основах кривых Безье. Это бесплатная веб-страница/электронная книга, посвященная как математическим, так и программным + аспектам кривых Безье, охватывает широкий спектр тем касающихся рисования и работы с кривой, которая, кажется, появляется повсюду: от кривых в + Photoshop до функций плавности (easing functions) CSS и начертания шрифтов. +

    +

    + Если вы здесь впервые: Добро пожаловать! Обязательно напишите мне, если вы ищете + что-то конкретное, но не нашли ответ в тексте. +

    -
    +

    Пожертвования и спонсорство

    -

    Основы кривых Безье

    -

    Бесплатная онлайн-книга для тех, кому действительно важно знать, как работают кривые Безье

    -
    - Читайте на своём языке: - -

    (Не нашли свой язык или хотите, чтобы он достиг до 100%? Помогите нам с переводом!)

    -
    +

    + Если вы используете этот источник для исследований, в качестве справочника по работе, или даже для написания собственного программного + обеспечения рассмотрите возможность + задонатить + (любая сумма годится) или станьте патроном на Patreon. Мне не платят за эту работу, так что, + если вы нашли что-то полезное для себя и хотите, чтобы он оставался с нами надолго, то вот вам мысль: на создание этих страниц ушло много + кофе, и уйдет ещё больше на их совершенствование; если вы можете помочь с кофе, то поможете долгой и счастливой жизни этого ресурса. +

    +

    + Кроме того, если вы являетесь компанией и ваши сотрудники используют эту книгу в качестве источника, или вы используете её в качестве + вспомогательного ресурса, пожалуйста: подумайте о финансировании сайта! Я более чем счастлив, работая с вашим финансовым отделом над + выставлением счетов и признанием спонсорства. +

    -

    - Приветствуем на Основах кривых Безье. Это бесплатная веб-страница/электронная книга, посвященная как математическим, - так и программным аспектам кривых Безье, охватывает широкий спектр тем касающихся рисования и работы с кривой, - которая, кажется, появляется повсюду: от кривых в Photoshop до функций плавности (easing functions) CSS и начертания шрифтов. -

    -

    - Если вы здесь впервые: Добро пожаловать! Обязательно напишите мне, - если вы ищете что-то конкретное, но не нашли ответ в тексте. -

    - -

    Пожертвования и спонсорство

    - - -

    - Если вы используете этот источник для исследований, в качестве справочника по работе, или даже для написания - собственного программного обеспечения рассмотрите возможность - задонатить - (любая сумма годится) или станьте патроном на Patreon. Мне не - платят за эту работу, так что, если вы нашли что-то полезное для себя и хотите, чтобы он оставался с нами надолго, - то вот вам мысль: на создание этих страниц ушло много кофе, и уйдет ещё больше на их совершенствование; - если вы можете помочь с кофе, то поможете долгой и счастливой жизни этого ресурса. -

    -

    - Кроме того, если вы являетесь компанией и ваши сотрудники используют эту книгу в качестве источника, или вы используете - её в качестве вспомогательного ресурса, пожалуйста: подумайте о финансировании сайта! Я более чем счастлив, работая - с вашим финансовым отделом над выставлением счетов и признанием спонсорства. -

    - - - -
    +
    -

    - — Pomax -

    - + +
    +
    +
    +

    Предисловие

    +

    + Чтобы рисовать объекты в 2D, мы обычно полагаемся на линии, которые обычно подразделяются на две категории: прямые линии и кривые. Первые из + них так же легко нарисовать, как их легко нарисовать на компьютере. Дайте компьютеру первую и последнюю точку в строке, и БАЦ! Прямая линия. + Никаких сложностей не должно возникнуть. +

    +

    + Кривые, однако, представляют собой гораздо большую проблему. В то время как мы можем рисовать кривые с невероятной легкостью от руки, + компьютеры немного ограничены в том, что они не могут рисовать кривые, если нет математической функции, описывающей, как они должны быть + нарисованы. На самом деле, они также нуждаются в этом и для прямых линий, но функция до смешного проста, поэтому мы склонны игнорировать это + в том, что касается компьютеров; все линии являются "функциями", независимо от того, прямые они или кривые. Однако это означает, что нам + нужно придумать быстрые в вычислении функции, которые приводят к красивым кривым на компьютере. Их существует несколько, и в этой статье мы + сосредоточимся на конкретной функции, которая получила довольно много внимания и используется практически во всем, что может рисовать + кривые: кривые Безье. +

    +

    + Они названы в честь Пьера Безье, + который сделал так, чтобы они стали известны миру как кривая, хорошо подходящая для проектных работ (опубликовал свои исследования в 1962 + году, работая в Renault), хотя он не был первым или единственным, кто "изобрел" эти типы кривых. Может возникнуть соблазн сказать, что + математик + Поль де Кастельжо + был первым, поскольку он начал исследовать природу этих кривых в 1959 году, работая в Citroën, и открыл действительно элегантный способ, как + их нарисовать. Однако, Поль де Кастельжо не опубликовал свою работу, поэтому на вопрос, «кто был первым», трудно ответить в каком-либо + абсолютном смысле. Либо это? Кривые Безье – это, по сути, «полиномы Бернштейна», семейство математических функций, исследованных + Сергеем Натановичем Бернштейном, чьи публикации о них датируются, по крайней мере, до 1912 года. +

    +

    + В любом случае это в основном мелочи, вас, скорее всего, будет волновать то, что эти кривые удобны: вы можете связать несколько кривых + Безье, чтобы комбинация выглядела как одна кривая. Если вы когда-либо рисовали "Путь" (Path – прим. пер.) в Photoshop или работали с + программами векторного рисования, такими как Flash, Illustrator или Inkscape, эти кривые, которые вы рисовали, являются кривыми Безье. +

    +

    + Но что, если вам нужно будет запрограммировать их самостоятельно? В чем заключаются подводные камни? Как вы их рисуете? Что такое + ограничительные рамки, как определяются пересечения, как можно вытянуть кривую, короче говоря: какие возможности они предоставляют и что + можно сделать с этими кривыми? Вот для чего предназначена эта страница. Приготовьтесь к матчу! +

    +
    +

    Практически вся графика Безье интерактивная.

    +

    + На этой странице используются интерактивные примеры, в значительной степени опирающиеся на + Bezier.js, а также математические формулы, которые набираются в SVG с помощью + XeLaTeX система набора текста и pdf2svg + Дэвид Бартон. +

    +

    Эта книга с открытым исходным кодом.

    +

    + Эта книга представляет собой проект с открытым исходным кодом и хранится в двух репозиториях github. Во-первых, это + https://github.com/pomax/bezierinfo и является версией исключительно для презентаций, + которую вы просматриваете прямо сейчас. Другим хранилищем является + https://github.com/pomax/BezierInfo-2 – это версия для разработки, в которой + содержится весь код, и он превращается в веб-версию. Также в нём вы можете сообщать о проблемах, если обнаружите ошибки или у вас есть + идеи, что улучшить или добавить в учебник. +

    +

    Насколько сложной будет математика?

    +

    + Большая часть математики в этом учебнике – это математика для начальной школы. Если вы разбираетесь в элементарной арифметике и умеете + читать, у вас все должно получиться просто отлично. Иногда будут гораздо более сложные математические задачи, но если вам не + хочется их переваривать, вы можете спокойно пропустить их, либо пропустив "подробные поля" в разделе, либо просто перейдя к концу раздела + с математикой, которая выглядит слишком увлекательной. В конце разделов обычно просто перечисляются выводы, чтобы вы могли просто работать + непосредственно с этими знаниями. +

    +

    На каком языке написаны все примеры кода?

    +

    + Существует слишком много языков программирования, чтобы отдать предпочтение одному из них, поэтому во всех примерах в этом учебнике + используется форма псевдокода, которая использует синтаксис, достаточно близкий к современным скриптовым языкам, таким как JS, Python и т. + д., но на самом деле это не они. Поэтому вы не сможете копировать-вставить что-либо, не задумываясь, это сделано намеренно: если читаете + этот учебник, предположительно, вы хотите учиться, но вы не учитесь путем копирования-вставки. Вы учитесь, делая что-то + самостоятельно, совершая ошибки, а затем исправляя эти ошибки. Конечно, я намеренно не добавлял ошибки в пример кода только для того, + чтобы заставить вас ошибаться (это было бы ужасно!), Но я намеренно не позволял примерам кода отдавать предпочтение одному языку + программирования перед другим. Не волнуйтесь, если вы знаете хотя бы один процедурный язык программирования, вы сможете прочитать примеры + без каких-либо трудностей. +

    +

    Вопросы, комментарии:

    +

    + Если у вас есть предложения по новым разделам, перейдите по ссылке + на Github (также доступно из репозитория, ссылка в правом верхнем углу). Если у + вас есть вопросы по материалу, в настоящее время нет раздела комментариев, пока я его переписываю, но вы также можете использовать github + для этого. Как только переписывание будет закончено, я снова добавлю раздел общих комментариев и, возможно, более актуальную систему + "выберите этот раздел текста и нажмите кнопку "вопрос", чтобы задать вопрос об этом". Посмотрим. +

    +

    Помогите поддержать книгу!

    +

    + Если вам понравилась эта книга или просто сочли её полезной, сообщите об этом, чтобы оценить эту книгу, есть два варианта: можете либо + перейти на страницу Patreon или, если вы предпочитаете сделать единовременное + пожертвование, зайдите на страницу + купить кофе Pomax. За эти годы + эта работа из небольшого букваря превратилась в пособие, эквивалентное 100 с лишним печатным страницам, посвященное кривым Безье, и в его + создание было вложено много кофе. Я не жалею ни минуты, потраченной на его написание, но мне всегда не помешает еще немного кофе, чтобы + продолжать писать! +

    +
    +
    +
    +

    Что нового?

    +

    + Этот документ живое пособие, в зависимости от даты вашего последнего посещения, может появиться новый материал. Кликайте + здесь для просмотра лога изменений. (также доступен RSS-канал) +

    + + + +
    +

    November 2020

    +
      +

    • Added a section on finding curve/circle intersections

      • +
    +

    October 2020

    +
      +

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      • +
    +

    August-September 2020

    +
      +
    • +

      + Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS + turned on. +

      +
      • +
    +

    June 2020

    +
      +

    • Added automatic CI/CD using Github Actions

      • +
    +

    January 2020

    +
      +

    • Added reset buttons to all graphics

      • +

    • Updated to preface to correctly describe the on-page maths

      • +

    • Fixed the Catmull-Rom section because it had glaring maths errors

      • +
    +

    August 2019

    +
      +

    • Added a section on (plain) rational Bezier curves

      • +

    • Improved the Graphic component to allow for sliders

      • +
    +

    December 2018

    +
      +

    • Added a section on curvature and calculating kappa.

      • +
    • +

      + Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this + site! +

      +
      • +
    +

    August 2018

    +
      +

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      • +
    +

    July 2018

    +
      +

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      • +

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      • +
    • +

      (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      +
      • +
    +

    June 2018

    +
      +

    • Added a section on direct curve fitting.

      • +

    • Added source links for all graphics.

      • +

    • Added this "What's new?" section.

      • +
    +

    April 2017

    +
      +

    • Added a section on 3d normals.

      • +

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      • +
    +

    February 2017

    +
      +

    • Finished rewriting the entire codebase for localization.

      • +
    +

    January 2016

    +
      +

    • Added a section to explain the Bezier interval.

      • +

    • Rewrote the Primer as a React application.

      • +
    +

    December 2015

    +
      +

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      • +
    • +

      + Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics. +

      +
      • +
    +

    May 2015

    +
      +

    • Switched over to pure JS rather than Processing-through-Processing.js

      • +

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      • +
    +

    April 2015

    +
      +

    • Added a section on arc length approximations.

      • +
    +

    February 2015

    +
      +

    • Added a section on the canonical cubic Bezier form.

      • +
    +

    November 2014

    +
      +

    • Switched to HTTPS.

      • +
    +

    July 2014

    +
      +

    • Added the section on arc approximation.

      • +
    +

    April 2014

    +
      +

    • Added the section on Catmull-Rom fitting.

      • +
    +

    November 2013

    +
      +

    • Added the section on Catmull-Rom / Bezier conversion.

      • +

    • Added the section on Bezier cuves as matrices.

      • +
    +

    April 2013

    +
      +

    • Added a section on poly-Beziers.

      • +

    • Added a section on boolean shape operations.

      • +
    +

    March 2013

    +
      +

    • First drastic rewrite.

      • +

    • Added sections on circle approximations.

      • +

    • Added a section on projecting a point onto a curve.

      • +

    • Added a section on tangents and normals.

      • +

    • Added Legendre-Gauss numerical data tables.

      • +
    +

    October 2011

    +
      +
    • +

      + First commit for the bezierinfo site, based on the pre-Primer webpage that covered + the basics of Bezier curves in HTML with Processing.js examples. +

      +
      • +
    +
    +
    +
    +
    +

    +
    оглавлениеследующий
    + Быстрое введение +

    +

    + Начнем с приятного: говоря о кривых Безье, мы говорим о штуках доступных нашему лицезрению на картинках ниже. Отрезки протекают от + начальной до конечной точек, искривление обусловленно одной или более вспомогательными контрольными точками. Теперь, поскольку вся графика + на данной странице интереактивна, вы можете манипулировать кривыми посредством перемещения вышеупомянутых точек и наблюдать производимый + эффект соотносительно вашим действиям. +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    -
    - -
    - -

    Предисловие

    -

    Чтобы рисовать объекты в 2D, мы обычно полагаемся на линии, которые обычно подразделяются на две категории: прямые линии и кривые. Первые из них так же легко нарисовать, как их легко нарисовать на компьютере. Дайте компьютеру первую и последнюю точку в строке, и БАЦ! Прямая линия. Никаких сложностей не должно возникнуть.

    -

    Кривые, однако, представляют собой гораздо большую проблему. В то время как мы можем рисовать кривые с невероятной легкостью от руки, компьютеры немного ограничены в том, что они не могут рисовать кривые, если нет математической функции, описывающей, как они должны быть нарисованы. На самом деле, они также нуждаются в этом и для прямых линий, но функция до смешного проста, поэтому мы склонны игнорировать это в том, что касается компьютеров; все линии являются "функциями", независимо от того, прямые они или кривые. Однако это означает, что нам нужно придумать быстрые в вычислении функции, которые приводят к красивым кривым на компьютере. Их существует несколько, и в этой статье мы сосредоточимся на конкретной функции, которая получила довольно много внимания и используется практически во всем, что может рисовать кривые: кривые Безье.

    -

    Они названы в честь Пьера Безье, который сделал так, чтобы они стали известны миру как кривая, хорошо подходящая для проектных работ (опубликовал свои исследования в 1962 году, работая в Renault), хотя он не был первым или единственным, кто "изобрел" эти типы кривых. Может возникнуть соблазн сказать, что математик Поль де Кастельжо был первым, поскольку он начал исследовать природу этих кривых в 1959 году, работая в Citroën, и открыл действительно элегантный способ, как их нарисовать. Однако, Поль де Кастельжо не опубликовал свою работу, поэтому на вопрос, «кто был первым», трудно ответить в каком-либо абсолютном смысле. Либо это? Кривые Безье – это, по сути, «полиномы Бернштейна», семейство математических функций, исследованных Сергеем Натановичем Бернштейном, чьи публикации о них датируются, по крайней мере, до 1912 года.

    -

    В любом случае это в основном мелочи, вас, скорее всего, будет волновать то, что эти кривые удобны: вы можете связать несколько кривых Безье, чтобы комбинация выглядела как одна кривая. Если вы когда-либо рисовали "Путь" (Path – прим. пер.) в Photoshop или работали с программами векторного рисования, такими как Flash, Illustrator или Inkscape, эти кривые, которые вы рисовали, являются кривыми Безье.

    -

    Но что, если вам нужно будет запрограммировать их самостоятельно? В чем заключаются подводные камни? Как вы их рисуете? Что такое ограничительные рамки, как определяются пересечения, как можно вытянуть кривую, короче говоря: какие возможности они предоставляют и что можно сделать с этими кривыми? Вот для чего предназначена эта страница. Приготовьтесь к матчу!

    -
    - -

    Практически вся графика Безье интерактивная.

    -

    На этой странице используются интерактивные примеры, в значительной степени опирающиеся на Bezier.js, а также математические формулы, которые набираются в SVG с помощью XeLaTeX система набора текста и pdf2svg Дэвид Бартон.

    -

    Эта книга с открытым исходным кодом.

    -

    Эта книга представляет собой проект с открытым исходным кодом и хранится в двух репозиториях github. Во-первых, это https://github.com/pomax/bezierinfo и является версией исключительно для презентаций, которую вы просматриваете прямо сейчас. Другим хранилищем является https://github.com/pomax/BezierInfo-2 – это версия для разработки, в которой содержится весь код, и он превращается в веб-версию. Также в нём вы можете сообщать о проблемах, если обнаружите ошибки или у вас есть идеи, что улучшить или добавить в учебник.

    -

    Насколько сложной будет математика?

    -

    Большая часть математики в этом учебнике – это математика для начальной школы. Если вы разбираетесь в элементарной арифметике и умеете читать, у вас все должно получиться просто отлично. Иногда будут гораздо более сложные математические задачи, но если вам не хочется их переваривать, вы можете спокойно пропустить их, либо пропустив "подробные поля" в разделе, либо просто перейдя к концу раздела с математикой, которая выглядит слишком увлекательной. В конце разделов обычно просто перечисляются выводы, чтобы вы могли просто работать непосредственно с этими знаниями.

    -

    На каком языке написаны все примеры кода?

    -

    Существует слишком много языков программирования, чтобы отдать предпочтение одному из них, поэтому во всех примерах в этом учебнике используется форма псевдокода, которая использует синтаксис, достаточно близкий к современным скриптовым языкам, таким как JS, Python и т. д., но на самом деле это не они. Поэтому вы не сможете копировать-вставить что-либо, не задумываясь, это сделано намеренно: если читаете этот учебник, предположительно, вы хотите учиться, но вы не учитесь путем копирования-вставки. Вы учитесь, делая что-то самостоятельно, совершая ошибки, а затем исправляя эти ошибки. Конечно, я намеренно не добавлял ошибки в пример кода только для того, чтобы заставить вас ошибаться (это было бы ужасно!), Но я намеренно не позволял примерам кода отдавать предпочтение одному языку программирования перед другим. Не волнуйтесь, если вы знаете хотя бы один процедурный язык программирования, вы сможете прочитать примеры без каких-либо трудностей.

    -

    Вопросы, комментарии:

    -

    Если у вас есть предложения по новым разделам, перейдите по ссылке на Github (также доступно из репозитория, ссылка в правом верхнем углу). Если у вас есть вопросы по материалу, в настоящее время нет раздела комментариев, пока я его переписываю, но вы также можете использовать github для этого. Как только переписывание будет закончено, я снова добавлю раздел общих комментариев и, возможно, более актуальную систему "выберите этот раздел текста и нажмите кнопку "вопрос", чтобы задать вопрос об этом". Посмотрим.

    -

    Помогите поддержать книгу!

    -

    Если вам понравилась эта книга или просто сочли её полезной, сообщите об этом, чтобы оценить эту книгу, есть два варианта: можете либо перейти на страницу Patreon или, если вы предпочитаете сделать единовременное пожертвование, зайдите на страницу купить кофе Pomax. За эти годы эта работа из небольшого букваря превратилась в пособие, эквивалентное 100 с лишним печатным страницам, посвященное кривым Безье, и в его создание было вложено много кофе. Я не жалею ни минуты, потраченной на его написание, но мне всегда не помешает еще немного кофе, чтобы продолжать писать!

    -
    -
    -
    -

    Что нового?

    -

    Этот документ живое пособие, в зависимости от даты вашего последнего посещения, может появиться новый материал. Кликайте здесь для просмотра лога изменений. (также доступен RSS-канал)

    - - - -

    November 2020

    • Added a section on finding curve/circle intersections

      -
    -

    October 2020

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      -
    -

    August-September 2020

    • Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS turned on.

      -
    -

    June 2020

    • Added automatic CI/CD using Github Actions

      -
    -

    January 2020

    • Added reset buttons to all graphics

      -
      • -

    • Updated to preface to correctly describe the on-page maths

      -
      • -

    • Fixed the Catmull-Rom section because it had glaring maths errors

      -
    -

    August 2019

    • Added a section on (plain) rational Bezier curves

      -
      • -

    • Improved the Graphic component to allow for sliders

      -
    -

    December 2018

    • Added a section on curvature and calculating kappa.

      -
      • -

    • Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this site!

      -
    -

    August 2018

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      -
    -

    July 2018

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      -
      • -

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      -
      • -

    • (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      -
    -

    June 2018

    • Added a section on direct curve fitting.

      -
      • -

    • Added source links for all graphics.

      -
      • -

    • Added this "What's new?" section.

      -
    -

    April 2017

    • Added a section on 3d normals.

      -
      • -

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      -
    -

    February 2017

    • Finished rewriting the entire codebase for localization.

      -
    -

    January 2016

    • Added a section to explain the Bezier interval.

      -
      • -

    • Rewrote the Primer as a React application.

      -
    -

    December 2015

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      -
      • -

    • Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics.

      -
    -

    May 2015

    • Switched over to pure JS rather than Processing-through-Processing.js

      -
      • -

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      -
    -

    April 2015

    • Added a section on arc length approximations.

      -
    -

    February 2015

    • Added a section on the canonical cubic Bezier form.

      -
    -

    November 2014

    • Switched to HTTPS.

      -
    -

    July 2014

    • Added the section on arc approximation.

      -
    -

    April 2014

    • Added the section on Catmull-Rom fitting.

      -
    -

    November 2013

    • Added the section on Catmull-Rom / Bezier conversion.

      -
      • -

    • Added the section on Bezier cuves as matrices.

      -
    -

    April 2013

    • Added a section on poly-Beziers.

      -
      • -

    • Added a section on boolean shape operations.

      -
    -

    March 2013

    • First drastic rewrite.

      -
      • -

    • Added sections on circle approximations.

      -
      • -

    • Added a section on projecting a point onto a curve.

      -
      • -

    • Added a section on tangents and normals.

      -
      • -

    • Added Legendre-Gauss numerical data tables.

      -
    -

    October 2011

    • First commit for the bezierinfo site, based on the pre-Primer webpage that covered the basics of Bezier curves in HTML with Processing.js examples.

      -
    -
    -
    -

    оглавлениеследующий
    Быстрое введение

    -

    Начнем с приятного: говоря о кривых Безье, мы говорим о штуках доступных нашему лицезрению на картинках ниже. Отрезки протекают от начальной до конечной точек, искривление обусловленно одной или более вспомогательными контрольными точками. Теперь, поскольку вся графика на данной странице интереактивна, вы можете манипулировать кривыми посредством перемещения вышеупомянутых точек и наблюдать производимый эффект соотносительно вашим действиям.

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    - -

    Данный тип кривых активно используется в компьютерном дизайне и производственных (CAD/CAM) применениях, в графических программах, таких как Adobe Illustrator and Photoshop, Inkscape, GIMP ... в графических технологиях, таких как маштабируемая векторная графика (SVG) и шрифты OpenType (TTF/OTF). Области использования кривых Безье довольно широки, и если вы желаете узнать больше о составляющих их аспектах: пристегните ремни!

    - -
    -
    -

    предыдущийоглавлениеследующий
    Итак, из чего сделаны кривые Безье?

    -

    Играя с позициями точек, вы, возможно, получили общее представление о том, как кривые Безье себя ведут. Но чем же является кривая Безье? Ее можно объяснять двумя способами. Оба эквиваленты, однако один из них затрагивает объемные математические описания, тогда как второй пользуется довольно простой математикой. Итак... давайте начнем с простейшего:

    -

    Кривая Безье есть результатом линейной интерполяции (*в оригинале текста другая ссылка, тут и далее пр. переводчика). Звучит заморочено, но в действительности вы занимались линейной интерполяций с юных лет: каждый раз когда вам приходилось указывать на что либо лежащее между двумя другими "чем либо" — вы применяли линейную интерполяцию. Т.е. попросту "выбор точки лежащей на линии между двумя другими точками".

    -

    Вот, скажем, зная расстояние между двух точек и желая поставить третью на удалении 20ти % этого расстояния до первой и, соответственно, 80ти % до второй, вычислить результат можно следующим образом:

    - - -

    Что же, посмотрим на это в действии: ниже представлена интерактивная проекция, кликнув на ползунок, можно пользоваться клавишами вниз-вверх для увеличения и уменьшения соотношения интерполяции и наблюдать получаемый результат. Сначала, основываясь на трех точках, задаем два отрезка, затем производим линейную интерполяцию по длине каждого из них, получая еще две точки. Далее мы опять производим линейную интерполяцию уже между полученными точками и в итоге получаем искомую точку. (* на изображении видим 3 проекции соответственно этим трем действиям).

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    И это приводит нас к более замороченному математическому анализу.

    -

    Хотя и не очевидно, но мы только что прорисовали квадратную кривую Безье, просто пошагово вместо цельной линии. Один увлекательный аспект кривых Безье является тем, что кривая может одновременно быть описана как полиномиальная функция и так же как набор линейных интерполяций. Это предоставляет возможность рассматривать эти типы кривых с двух планов: математического (анализа функции, ее производных, и все такое) а так же с механической композиции (которая, к примеру, позволяет легко заметить, что точка никогда не выйдет за рамки заданных точек, используемых для ее конструкции).

    -

    С этим, давайте рассмотрим кривые Безье более подробно: в частности их математические выражения, свойства, доступные к вычислению на их основе, и вариации действий применимых к и производимых из кривых Безье.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Математика под кривыми Безье

    -

    Кривые Безье по своей форме представляют собой параметрические функции. С точки зрения математики, эти функции хитрят, поскольку разнятся с каноническим определением функции. Последнее есть преобразованием любого количества вводных в единый результат. Как бы мы не меняли вводные значения переменных, на выходе всегда получим одно значение.

    -

    Параметрические функции хитрят, потому как говоря: "что ж, ладно, нам нужно произвести более одного вывода", заключают, попросту: "давайте использовать более одной функции". Проиллюстрируем: представьте что есть функция, которая преобразует вводную, назовем ее x, в другое значение следуя определенной логике

    - - -

    Нотация f(x) есть стандартным способом записи функции (по конвенции, f, когда представлена всего одна) и ее вывод меняется в зависимости от значения одной переменной (в данном случае, x). Так, изменив x и получим другой результат f(x).

    -

    Пока все логично. Теперь, давайте рассмотрим параметрические функции в действии и разберемся в чем их фокус. И, для начала, примите ко внимаю следующий пример:

    - - -

    Итак, нам представлены две функции. Ничего особо выдающегося: просто функции синуса и косинуса. Отметим, что, как вы можете видеть, вводные переменные разные. Затем, меняя значение a, мы не влияем на вывод f(b), поскольку a никак не задействована в этой функции. Параметрические функции хитрят именно с этим: весь набор функций вывода делит между собой одну или более переменную вводную. Как здесь:

    - - -

    Что же, несколько функций, и всего одна вводная. Меняя значение t — меняем вывод обеих ф-ций, fa(t) и fb(t). Возможно вы спросите: "в чем же польза?". Ответ прост и очевиден, если уточнить, что мы имеем ввиду под записью наших функций:

    - - -

    Так-то. Координаты x/y. Связанные мистическим значением t.

    -

    Таким образом, мы видим, что параметрические функции не определяют значение y через значение x, как обычные ф-ции, вместо этого они выводят и x и y из мистического значения t. Значит, для каждого значения t, существуют два соответствующих значения x и y, которые мы можем использовать для зарисовки кривой на графике. К примеру, приведенная выше параметрическая функция выводит координаты точки на круге: можно подставить любое значение t (от негативной до позитивной бесконечности), и на выводе мы всегда получим координаты точки на круге с центром в (0,0) и радиусом в 1. Зарисовав точки для t от 0 до 5, получаем следующее:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Кривые Безье — всего один из многих классов параметрических функций. Их главной характеристикой есть использование одной и той же базовой функции для генерации всех выводов. До сих пор, в использованном нами примере, мы производили значения x и y с помощью разных функций (ф-цией синуса и ф-цией косинуса); Безье же использует единый "биноминальный полином" для вывода обоих значений. Но что же такое "биноминальный полином"?

    -

    Возможно, вы помните полиномы из школьной программы. Они выглядят следующим образом:

    - - -

    Так, если наивысшая степень в уравнении составляет x³, мы называем такой полином кубическим, если x² — квадратным. Если это только x¹ — значит это просто линия (когда же x не упоминается вовсе — это не полином!).

    -

    Кривые Безье являются полиномом t (вместо x), где значение t — варъируется между 0 и 1, и коэффициентами a, b и т.д. принимающими "биноминальную" форму. Звучит сложно, но на практике выглядит понятнее:

    - - -

    Я знаю что вы думаете: это не выглядит таким уж простым. Но если мы уберем t и вместо этого поставим *1, внезапно, все становится ясно. Примите ко внимаю такие биномиальные термины:

    - - -

    Обратите внимание, что 2 это то же, что 1+1, и 3 это 2+1 и 1+2 и 6 это 3+3... Как вы можете видеть, каждый раз поднимаясь на одно разрешение выше, мы начинаем и заканчиваем с единицы, а все промежуточные значения равны суме двух значений над ними в предыдущей строке. Так получаем последовательность цифр, называемой треугольником Паскаля (*в оригинале другая ссылка ).

    -

    Так-же существует простой способ для выяснения подоплотной работы полиноминальных терминов: если мы переименуем (1-t) в a и t в b, убрав "веса", получим следующее:

    - - -

    В целом это просто сума "каждого сочетания a и b", получаемая прогрессивной заменой a на b по ходу уравнения. Потому, это так-же довольно просто. Итак теперь вы знаете что такое биноминальные полиномы. Для полноты картины, ниже привожу их общую функцию:

    - - -

    И теперь, это полное объяснение. Σ в этой функции означает, что это серия сум (с использованием переменной приведенной под Σ, со стартовым значением в ...=<value> и максимальным значением представленным над Σ)

    -
    + +

    + И теперь, это полное объяснение. Σ в этой функции означает, что это серия сум (с использованием переменной приведенной под Σ, со стартовым + значением в ...=<value> и максимальным значением представленным над Σ) +

    +
    +

    Имплементация базовых функций

    +

    Наивно, мы могли бы воплотить базис этой функции как математическую конструкцию, используя функцию как путеводитель:

    -

    Имплементация базовых функций

    -

    Наивно, мы могли бы воплотить базис этой функции как математическую конструкцию, используя функцию как путеводитель:

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    Я говорю "мы могли бы", поскольку это не входит в наши планы: факторные функции невероятно дорого стоят. И, как мы можем понять из выше приведенного текста, треугольник Паскаля можно получить и без таких ухищрений: попросту начнем с [1], далее [1,1], затем [1,2,1], потом [1,3,3,1] и т.д., с каждым новым рядком, записывая на 1 цифру больше, чем в предыдущем, начиная и заканчивая с единицы, а промежуточные цифры определяя суммируя две над ними в предыдущем рядке.

    -

    Такой ряд можно сгенерировать предельно быстро, и с этим нам не придется компилировать биноминальные термины, поскольку можно пользоваться значениями из таблицы:

    +

    + Я говорю "мы могли бы", поскольку это не входит в наши планы: факторные функции невероятно дорого стоят. И, как мы можем понять + из выше приведенного текста, треугольник Паскаля можно получить и без таких ухищрений: попросту начнем с [1], далее [1,1], затем + [1,2,1], потом [1,3,3,1] и т.д., с каждым новым рядком, записывая на 1 цифру больше, чем в предыдущем, начиная и заканчивая с единицы, а + промежуточные цифры определяя суммируя две над ними в предыдущем рядке. +

    +

    + Такой ряд можно сгенерировать предельно быстро, и с этим нам не придется компилировать биноминальные термины, поскольку можно + пользоваться значениями из таблицы: +

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    + return lut[n][k] + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + -

    Итак, что же здесь происходит? Сначала мы декларируем таблицу достаточного размера для удовлетворения большинства запросов. Далее мы заявляем функцию вывода необходимого значения, вытаскивая его из таблицы, предварительно убедившись, что значения для запрашиваемых n/k присутствуют в наборе и расширяя набор по необходимости (если не присутствуют). Наша базовая функция теперь выглядит типа этого:

    +

    + Итак, что же здесь происходит? Сначала мы декларируем таблицу достаточного размера для удовлетворения большинства запросов. Далее мы + заявляем функцию вывода необходимого значения, вытаскивая его из таблицы, предварительно убедившись, что значения для запрашиваемых + n/k присутствуют в наборе и расширяя набор по необходимости (если не присутствуют). Наша базовая функция теперь выглядит типа + этого: +

    - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    Отлично. Конечно, мы можем оптимизировать ее и далее. Для большинства задач компьютерной графики нам не потребуются кривые произвольного порядка (хотя мы приводим код для произвольных кривых в этом пособии); зачастую нам нужны квадратные и кубические кривые, а это значит, мы можем значительно упростить весь наш код:

    +

    + Отлично. Конечно, мы можем оптимизировать ее и далее. Для большинства задач компьютерной графики нам не потребуются кривые произвольного + порядка (хотя мы приводим код для произвольных кривых в этом пособии); зачастую нам нужны квадратные и кубические кривые, а это значит, + мы можем значительно упростить весь наш код: +

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return mt3 + 3*mt2*t + 3*mt*t2 + t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    И вот теперь мы знаем как программировать базовую функцию. Превосходно.

    -
    +

    И вот теперь мы знаем как программировать базовую функцию. Превосходно.

    +
    -

    Итак, зная как выглядят базовые функции, время добавить магию делающую кривые Безье такими особенными: контрольные точки.

    +

    Итак, зная как выглядят базовые функции, время добавить магию делающую кривые Безье такими особенными: контрольные точки.

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Контроль кривых Безье +

    +

    + Кривые Безье, как и все отрезки кривых, являются интерполярными функциями. Это значит, что они имеют заданный набор контрольных точек и + генерируют значения где-то "между" ними. (Одним следствием этого есть то, что невозможно сгенерировать точки вне этого "каркаса". Полезная + информация!). На практике мы можем проиллюстрировать влияние каждой контрольной точки на течение кривой по отдельности и проследить какие + из точек имеют наибольший эффект в тех и иных частях кривой. +

    +

    + Следующие зарисовки демонстрируют интерполяции функций для квадратной и кубической кривых, где "S" представляет степень влияния + контрольной точки на положение участка кривой по ходу развития t. Потяните ползунки, чтобы отследить проценты интерполяций каждой + контрольной точки на конкретном t. +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + + + Скрипты отключены. Показываем резервное изображение. + + + + + + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    -
    -
    -

    предыдущийоглавлениеследующий
    Контроль кривых Безье

    -

    Кривые Безье, как и все отрезки кривых, являются интерполярными функциями. Это значит, что они имеют заданный набор контрольных точек и генерируют значения где-то "между" ними. (Одним следствием этого есть то, что невозможно сгенерировать точки вне этого "каркаса". Полезная информация!). На практике мы можем проиллюстрировать влияние каждой контрольной точки на течение кривой по отдельности и проследить какие из точек имеют наибольший эффект в тех и иных частях кривой.

    -

    Следующие зарисовки демонстрируют интерполяции функций для квадратной и кубической кривых, где "S" представляет степень влияния контрольной точки на положение участка кривой по ходу развития t. Потяните ползунки, чтобы отследить проценты интерполяций каждой контрольной точки на конкретном t.

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - -
    - -

    Также представлена интерполяция кривой Безье 15го порядка. Как вы можете видеть, начальная и конечная точки влияют на положение кривой значительно больше, чем любая другая контрольная точка.

    -

    Желая изменить кривую, нам нужно изменить "вес"-а (* коофициенты) для каждой точки, по сути меняя интерполяцию. Имплементация оного предельно логична: нужно просто перемножить каждую точку на значение, меняющее ее 'силу'. Эти значения, по конвенции называют "весом" и мы можем учесть их запись в нашей оригинальной функции Безье:

    - - -

    Хоть и выглядит заморочено, но, так уж получается, в реальности "веса" просто значения координат на графике, к которым мы бы хотели, чтобы наша функция стремилась. Так, для кривой n-го порядка, w0 есть начальной координатой, wn конечной координатой, а все между ними — контрольными координатами. Например, чтобы кубическая кривая начиналась в (110,150), стремилась к точкам (25,190) и (210,250) заканчиваясь на (210,30), мы запишем это следующим образом:

    - - -

    Это производит кривую, график которой мы видели в первой главе этой статьи.

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    Что еще можно сделать с кривой Безье? На самом деле — вполне не мало. Остальная часть статьи описывает множества доступных операций и задач, которые они решают.

    -
    + +

    Это производит кривую, график которой мы видели в первой главе этой статьи.

    + + + Скрипты отключены. Показываем резервное изображение. + + + +

    + Что еще можно сделать с кривой Безье? На самом деле — вполне не мало. Остальная часть статьи описывает множества доступных операций и + задач, которые они решают. +

    +
    +

    Имплементация весовых функций.

    +

    + Отталкиваясь от того, что мы уже знаем как воплотить базис функции, добавление контрольных точек представляет собой удивительно легкую + задачу: +

    -

    Имплементация весовых функций.

    -

    Отталкиваясь от того, что мы уже знаем как воплотить базис функции, добавление контрольных точек представляет собой удивительно легкую задачу:

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    И оптимизированная версия:

    +

    И оптимизированная версия:

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    И вот, мы знаем как программировать базис функции с весами.

    -
    - -
    -
    -

    предыдущийоглавлениеследующий
    Контроль над кривыми Безье, часть 2: Соотносительные Безье

    -

    Мы можем расширить степень влияния на кривые Безье, "соотнеся" их составляющие: подразумевается добавление в формулу коэффициента соотношений (* в оригинале "ratio" — соотношение) в довесок к уже используемым весам.

    -

    Как и прочее, воплощение этого коэффициента не должно составить нам особого труда. Тогда как обычная функция:

    - - -

    Функция для соотносительных кривых Безье имеет два дополнительных термина:

    - - -

    Первый из добавочных терминов, представляет собой дополнительный "вес" для каждой координаты, Например если наши значения соотношений [1, 0.5, 0.5, 1], тогда частица0 = 1, частица1 = 0.5 и т.д., и на практике ничем не отличается от использования дополнительного "вес"-а. Пока ничего особо выдающегося.

    -

    Тем не менее, второй из добавленных терминов, составляет всю важность разницы. В процессе компиляции точек на кривой, мы компилируем "нормальные" значения Безье, а далее делим их на значение Безье с учетом только соотношений (частиц), без учета весов.

    -

    Это создает неожиданный эффект: превращает наш полиноминал во что-то, что полиноминалом более не является. Теперь этот тип кривой есть супер классом полиноминала, что позволяет производить крутые вещи, на которые кривые Безье не способны сами по себе. Например обозначить круг посредством такой кривой (что, как мы убедимся чуть погодя, категорически невозможно используя стандартную кривую Безье).

    -

    Но лучшим объяснением, как и всегда, является наглядная демонстрация: давайте посмотрим на эффект "соотносительности" нашей кривой Безье на примере интерактивного графика соотнесительной кривой. Следующий график показывает кривую Безье из предыдущей секции, "обогащенную" факторами соотношений для каждой координаты. Чем ближе к нулю мы выставляем значения этих факторов — тем меньше влияния оказывают соответственные контрольные точки на начертания кривой, и, соответственно, чем выше эти показатели — тем больше влияния контрольная точка оказывает на изгиб кривой. Попробуйте сами:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - -

    Вы можете думать о значениях соотношений, как о показателе силы притяжения соответствующей точки. Чем выше сила притяжения, тем больше наша кривая будет стремится к этой точке. Вы также можете наблюдать, что одинаковое увеличение или уменьшение всех показателей не оказывает никакого эффекта на результат... схоже с гравитацией: если значения остаются одинаковыми относительно друг-друга, на выводе ничего не меняется. Значения соотношений определяют влияние каждой координаты относительно всех остальных координат.

    -
    + +

    + Первый из добавочных терминов, представляет собой дополнительный "вес" для каждой координаты, Например если наши значения соотношений [1, + 0.5, 0.5, 1], тогда частица0 = 1, частица1 = 0.5 и т.д., и на практике ничем не отличается + от использования дополнительного "вес"-а. Пока ничего особо выдающегося. +

    +

    + Тем не менее, второй из добавленных терминов, составляет всю важность разницы. В процессе компиляции точек на кривой, мы компилируем + "нормальные" значения Безье, а далее делим их на значение Безье с учетом только соотношений (частиц), без учета весов. +

    +

    + Это создает неожиданный эффект: превращает наш полиноминал во что-то, что полиноминалом более не является. Теперь этот тип кривой есть + супер классом полиноминала, что позволяет производить крутые вещи, на которые кривые Безье не способны сами по себе. Например обозначить + круг посредством такой кривой (что, как мы убедимся чуть погодя, категорически невозможно используя стандартную кривую Безье). +

    +

    + Но лучшим объяснением, как и всегда, является наглядная демонстрация: давайте посмотрим на эффект "соотносительности" нашей кривой Безье + на примере интерактивного графика соотнесительной кривой. Следующий график показывает кривую Безье из предыдущей секции, "обогащенную" + факторами соотношений для каждой координаты. Чем ближе к нулю мы выставляем значения этих факторов — тем меньше влияния оказывают + соответственные контрольные точки на начертания кривой, и, соответственно, чем выше эти показатели — тем больше влияния контрольная точка + оказывает на изгиб кривой. Попробуйте сами: +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + + + +

    + Вы можете думать о значениях соотношений, как о показателе силы притяжения соответствующей точки. Чем выше сила притяжения, тем больше + наша кривая будет стремится к этой точке. Вы также можете наблюдать, что одинаковое увеличение или уменьшение всех показателей не + оказывает никакого эффекта на результат... схоже с гравитацией: если значения остаются одинаковыми относительно друг-друга, на выводе + ничего не меняется. Значения соотношений определяют влияние каждой координаты относительно всех остальных координат. +

    +
    +

    Имплементация соотносительных кривых

    +

    Дополнение кода из предыдущей секции для учета этой функциональности фактически тривиально:

    -

    Имплементация соотносительных кривых

    -

    Дополнение кода из предыдущей секции для учета этой функциональности фактически тривиально:

    - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    + return (f[0] * w[0] + f[1] * w[1] + f[2] * w[2] + f[3] * w[3])/basis + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + -

    Вот и все что требуется.

    -
    - -
    -
    -

    предыдущийоглавлениеследующий
    Интервал Безье [0,1]

    -

    В математике кривых Безье, вы могли заметить одну любопытную деталь — кривые Безье всегда считают вдоль одного и того же интервала t, t=0 to t=1. Почему же именно этот интервал?

    -

    Последнее обусловленно тем, как мы определяем "начало" и "конец" нашей кривой. Если у нас есть значение, которое представляет собой сочетание двух других значений, тогда общая формула для этого будет:

    - - -

    Очевидно, что начальное и конечное значения a и b должны быть a=1, b=0, чтобы в начале получать вывод 100% первого показателя и 0% второго; и a=0, b=1, чтобы в конце получать 0% value 1 и 100% value 2. В дополнение, мы не хотим чтобы "a" и "b" были независимыми, в коем случае можно было бы присвоить им любые значения и на выводе получить, например, 100% первого показателя и 100% второго. В принципе, с последним все ок, но в случае кривых Безье, мы всегда должны получать значение между двух крайностей, потому нельзя присвоить a и b значения, которые бы вместе составляли суму более 100% на выводе, что можно записать как:

    - - -

    С этим у нас есть гарантия, что мы не получим суму значений пропорций более 100%. Мы ограничиваем значение a интервалом [0,1], потому всегда получаем вывод из пропорционального смешения двух показателей, с сумой смесителей не превышающей 100%.

    -

    Но... что если мы воспользуемся этой формой, предполагающей что мы всегда будем использовать значения а между 0 и 1, для подсчета точек лежащих вне стандартного интервала? Наши вычисления пойдут по откосной? Что ж... не то чтобы, мы попросту увидим больше.

    -

    В случае кривой Безье, продление интервала просто продляет развитие кривой. Кривые Безье всего лишь сегменты полиноминальных кривых. Выбирая более широкий отрезок — мы можем проследить дальнейшее развитие кривой. И как же это выглядит?

    -

    Ниже представлены графики кривых Безье, зарисованных "обычным путем", по верх зарисовки кривых на которых они лежат, полученных путем расширения интервала значений t. Как мы можем наблюдать, кривая Безье есть частью большей формы, скрытой за границами нашего интервала, которую мы так-же можем регулировать путем перемещения контрольных точек.

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    + +

    + С этим у нас есть гарантия, что мы не получим суму значений пропорций более 100%. Мы ограничиваем значение a интервалом + [0,1], потому всегда получаем вывод из пропорционального смешения двух показателей, с сумой смесителей не превышающей 100%. +

    +

    + Но... что если мы воспользуемся этой формой, предполагающей что мы всегда будем использовать значения а между 0 и 1, для подсчета точек + лежащих вне стандартного интервала? Наши вычисления пойдут по откосной? Что ж... не то чтобы, мы попросту увидим больше. +

    +

    + В случае кривой Безье, продление интервала просто продляет развитие кривой. Кривые Безье всего лишь сегменты полиноминальных кривых. + Выбирая более широкий отрезок — мы можем проследить дальнейшее развитие кривой. И как же это выглядит? +

    +

    + Ниже представлены графики кривых Безье, зарисованных "обычным путем", по верх зарисовки кривых на которых они лежат, полученных путем + расширения интервала значений t. Как мы можем наблюдать, кривая Безье есть частью большей формы, скрытой за границами нашего + интервала, которую мы так-же можем регулировать путем перемещения контрольных точек. +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    -

    В области компьютерной графики, существуют множество кривых, которые действуют по противоположному кривым Безье принципу: вместо фиксированного интервала и свободного выбора контрольных точек формирующих развитие искривлений, они фиксируют форму кривой, предоставляя возможность выбора интервала. Отличным примером последней есть кривая "Spiro", которая частично базируется на спирали Корню, также известной как спираль Эйлера (* в оригинале другая ссылка на Корню и Эйлера ). Эту эстетически приятную кривую можно встретить в нескольких графических пакетах: FontForge и Inkscape. Ее даже используют в дизайне шрифтов, например в начертания шрифта Inconsolata.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Кривые Безье как матричные уравнения

    -

    Мы также можем представить кривые Безье как матричную операцию, выразив формулу Безье как функцию с полиноминальноминальной основой, матрицу коэффициентов и матрицу конкретных координат. Давайте рассмотрим что это значит для уравнений кубических кривых Безье, используя P... для обозначения координат в "одном или более пространстве".

    - - -

    Обобщив, игнорируя конкретные значения, мы получим:

    - - -

    Это, в свою очередь, может быть записано как:

    - - -

    Последнее мы можем раскрыть, записав как:

    - - -

    Более того можно записать с коэффициентами 1 и 0, включив нивелированные термины:

    - - -

    И уже это можно рассматривать как серию четырех матричных операций:

    - - -

    Скомбинировав в единую матричную операцию, получим:

    - - -

    Такой тип функций полиноминальной основы зачастую записывается с основой в возрастающем порядке, что значит мы должны обернуть нашу t матрицу горизонтально, а нашу "большую" матрицу — вертикально:

    - - -

    Наконец, мы можем добавить оригинальные координаты единой третьей матрицей:

    - - -

    Такой же фокус может быть проделан с квадратной кривой, в коем случае мы получаем:

    - - -

    Подставив t и перемножив матрицы, мы получим такие-же значения, как при подсчете с использованием исходной полиноминальной функции или графического метода интерполяции.

    -

    Итак: зачем нам возится с матрицами? Матричное произведение раскрывает свойства функции кривых, которые в противном случае, было бы сложно обнаружить. Например, мы видим, что наша функция принадлежит к типу треугольных матриц (* в оригинале другая ссылка), определенные количеством контрольных координат и обладают всеми соответствующими свойствами. Также, что они могут быть обернуты, что в свою очередь определяет тонну других свойств (* в оригинале другая ссылка), применимых к нашим кривым. Конечно же, основным вопросом остается: "В чем состоит польза?". Тогда как ответ не становится очевидным немедленно, чуть далее мы увидим определенные случаи, где некоторые свойства кривых могут быть исчислены посредством манипуляции функцией, либо остроумным использованием матриц, и иногда последнее намного быстрее.

    -

    Потому пока давайте запомним, что функции могут быть описаны таким образом, и будем двигаться дальше.

    + +

    + Подставив t и перемножив матрицы, мы получим такие-же значения, как при подсчете с использованием исходной полиноминальной + функции или графического метода интерполяции. +

    +

    + Итак: зачем нам возится с матрицами? Матричное произведение раскрывает свойства функции кривых, которые в противном + случае, было бы сложно обнаружить. Например, мы видим, что наша функция принадлежит к типу + треугольных матриц + (* в оригинале другая ссылка), определенные количеством контрольных + координат и обладают всеми соответствующими свойствами. Также, что они могут быть обернуты, что в свою очередь определяет + тонну других свойств + (* в оригинале другая ссылка), применимых к + нашим кривым. Конечно же, основным вопросом остается: "В чем состоит польза?". Тогда как ответ не становится + очевидным немедленно, чуть далее мы увидим определенные случаи, где некоторые свойства кривых могут быть исчислены посредством + манипуляции функцией, либо остроумным использованием матриц, и иногда последнее намного быстрее. +

    +

    Потому пока давайте запомним, что функции могут быть описаны таким образом, и будем двигаться дальше.

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Алгоритм де Кастельжо +

    +

    + Для зарисовки кривой Безье, мы можем пробежаться по всем значениям t от 0 до 1 и скомпилировать вывод базовой функции с + подставленными весами для каждого значения. К сожалению, чем замысловатее кривая, тем дороже обходиться это обчисление. Вместо этого, мы + можем воспользоваться Алгоритмом де Кастельжо для прорисовки кривых. Этот алгоритм позволяет прорисовывать кривые базируясь на + геометрических вычислениях и довольно прост в применении. На деле, настолько прост, что его можно воплотить при помощи карандаша и + линейки. +

    +

    + Вместо использования функции математического анализа для нахождения значений x/y для t, давайте сделаем + следующее: +

    +
      +
    • Примем t за пропорцию(чем оно и является). t=0 будет 0% вдоль линии, t=1, соответсвенно, 100% вдоль линии.
      • +
    • Возьмем все линии между заданными контрольными точками. Для кривой n-го порядка это n линий
      • +
    • + Разместим маркеры вдоль линий, соответсвенно значению t. Так, если t 0.2, разместим маркер на 20% от начала, + и, соответственно, 80% от конца. +
      • +
    • Теперь соединим полученные точки линиями. Это дает нам n-1 линий.
      • +
    • Далее разместим маркеры на новых линиях, соответсвенно тому-же значению t.
      • +
    • + Продолжим повторять два предыдущих шага, пока на выводе у нас останется всего одна линия. Маркер t на этой линии будет + соответствовать точке для t на нашей кривой. +
      • +
    +

    + Чтобы проверить это в действии, ведите ползунок под ниже представленным графиком слева направо и наоборот, задавая разные значения + t для иллюстрации геометрического построения. +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    +

    Имплементация Алгоритма де Кастельжо

    +

    Запишем согласно предложенному алгоритму:

    -
    -
    -

    предыдущийоглавлениеследующий
    Алгоритм де Кастельжо

    -

    Для зарисовки кривой Безье, мы можем пробежаться по всем значениям t от 0 до 1 и скомпилировать вывод базовой функции с подставленными весами для каждого значения. К сожалению, чем замысловатее кривая, тем дороже обходиться это обчисление. Вместо этого, мы можем воспользоваться Алгоритмом де Кастельжо для прорисовки кривых. Этот алгоритм позволяет прорисовывать кривые базируясь на геометрических вычислениях и довольно прост в применении. На деле, настолько прост, что его можно воплотить при помощи карандаша и линейки.

    -

    Вместо использования функции математического анализа для нахождения значений x/y для t, давайте сделаем следующее:

    -
      -
    • Примем t за пропорцию(чем оно и является). t=0 будет 0% вдоль линии, t=1, соответсвенно, 100% вдоль линии.
      • -
    • Возьмем все линии между заданными контрольными точками. Для кривой n-го порядка это n линий
      • -
    • Разместим маркеры вдоль линий, соответсвенно значению t. Так, если t 0.2, разместим маркер на 20% от начала, и, соответственно, 80% от конца.
      • -
    • Теперь соединим полученные точки линиями. Это дает нам n-1 линий.
      • -
    • Далее разместим маркеры на новых линиях, соответсвенно тому-же значению t.
      • -
    • Продолжим повторять два предыдущих шага, пока на выводе у нас останется всего одна линия. Маркер t на этой линии будет соответствовать точке для t на нашей кривой.
      • -
    -

    Чтобы проверить это в действии, ведите ползунок под ниже представленным графиком слева направо и наоборот, задавая разные значения t для иллюстрации геометрического построения.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -
    - -

    Имплементация Алгоритма де Кастельжо

    -

    Запишем согласно предложенному алгоритму:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    Готово, алгоритм воплощен. Помимо того факта, что зачастую мы не располагаем роскошью перенагрузки оператора "+", так давайте совместим вычисление для значений x и y:

    +

    + Готово, алгоритм воплощен. Помимо того факта, что зачастую мы не располагаем роскошью перенагрузки оператора "+", так давайте совместим + вычисление для значений x и y: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    И что это делает? Это зарисует точку на график, если длинна вводной points составляет всего одну точку. В противном случае — это создает новый список для вводной, составленный из "маркеров", обчисленых пропорционально значению t между поточного списка точек вводной и вызовет саму себя с новой вводной.

    -
    +

    + И что это делает? Это зарисует точку на график, если длинна вводной points составляет всего одну точку. В противном случае — это создает + новый список для вводной, составленный из "маркеров", обчисленых пропорционально значению t между поточного списка точек вводной + и вызовет саму себя с новой вводной. +

    +
    + +
    +

    +
    + предыдущийоглавлениеследующий +
    + Simplified drawing +

    +

    + We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, + a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines. +

    +

    + We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the + number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we + can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we + lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or + curvature alignment. +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    -
    -
    -

    предыдущийоглавлениеследующий
    Simplified drawing

    -

    We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines.

    -

    We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or curvature alignment.

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - -
    +

    + Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. + You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the + cubic curve), a higher number is required to capture the curvature changes properly. +

    +
    +

    How to implement curve flattening

    +

    Let's just use the algorithm we just specified, and implement that:

    -

    Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the cubic curve), a higher number is required to capture the curvature changes properly.

    -
    - -

    How to implement curve flattening

    -

    Let's just use the algorithm we just specified, and implement that:

    - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return coordinates; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

    +

    And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + coord = _coord + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

    -
    +

    We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

    +
    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Splitting curves +

    +

    + Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken + together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the + points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our + on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point. +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    +

    implementing curve splitting

    +

    We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

    -
    -
    -

    предыдущийоглавлениеследующий
    Splitting curves

    -

    Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -
    - -

    implementing curve splitting

    -

    We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

    - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + -

    After running this function for some value t, the left and right arrays will contain all the coordinates for two new curves - one to the "left" of our t value, the other on the "right". These new curves will have the same order as the original curve, and can be overlaid exactly on the original curve.

    -
    - -
    -
    -

    предыдущийоглавлениеследующий
    Splitting curves using matrices

    -

    Another way to split curves is to exploit the matrix representation of a Bézier curve. In the section on matrices, we saw that we can represent curves as matrix multiplications. Specifically, we saw these two forms for the quadratic and cubic curves respectively: (we'll reverse the Bézier coefficients vector for legibility)

    - - -

    and

    - + +

    + Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the + coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual + "point on the curve" information into a new matrix multiplication: +

    + + +

    and

    + + +

    + If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two + staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, + from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear + algebra (and if you don't care about the derivations, just skip to the end of the box for the results!). +

    +
    +

    Deriving new hull coordinates

    +

    + Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look + at the quadratic curve first: +

    + + + - -

    Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual "point on the curve" information into a new matrix multiplication:

    - - -

    and

    - - -

    If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear algebra (and if you don't care about the derivations, just skip to the end of the box for the results!).

    -
    - -

    Deriving new hull coordinates

    -

    Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look at the quadratic curve first:

    - - - - - - - - -

    We can do this because [M · M-1] is the identity matrix. It's a bit like multiplying something by x/x in calculus: it doesn't do anything to the function, but it does allow you to rewrite it to something that may be easier to work with, or can be broken up differently. In the same way, multiplying our matrix by [M · M-1] has no effect on the total formula, but it does allow us to change the matrix sequence [something · M] to a sequence [M · something], and that makes a world of difference: if we know what [M-1 · Z · M] is, we can apply that to our coordinates, and be left with a proper matrix representation of a quadratic Bézier curve (which is [T · M · P]), with a new set of coordinates that represent the curve from t = 0 to t = z. So let's get computing:

    - - -

    Excellent! Now we can form our new quadratic curve:

    - - - - - - -

    Brilliant: if we want a subcurve from t = 0 to t = z, we can keep the first coordinate the same (which makes sense), our control point becomes a z-ratio mixture of the original control point and the start point, and the new end point is a mixture that looks oddly similar to a Bernstein polynomial of degree two. These new coordinates are actually really easy to compute directly!

    -

    Of course, that's only one of the two curves. Getting the section from t = z to t = 1 requires doing this again. We first observe that in the previous calculation, we actually evaluated the general interval [0,z]. We were able to write it down in a more simple form because of the zero, but what we actually evaluated, making the zero explicit, was:

    - - - - -

    If we want the interval [z,1], we will be evaluating this instead:

    - - - - -

    We're going to do the same trick of multiplying by the identity matrix, to turn [something · M] into [M · something]:

    - - -

    So, our final second curve looks like:

    - - - - - - -

    Nice. We see the same thing as before: we can keep the last coordinate the same (which makes sense); our control point becomes a z-ratio mixture of the original control point and the end point, and the new start point is a mixture that looks oddly similar to a bernstein polynomial of degree two, except this time it uses (z-1) rather than (1-z). These new coordinates are also really easy to compute directly!

    -
    - -

    So, using linear algebra rather than de Casteljau's algorithm, we have determined that, for any quadratic curve split at some value t = z, we get two subcurves that are described as Bézier curves with simple-to-derive coordinates:

    - - -

    and

    - - -

    We can do the same for cubic curves. However, I'll spare you the actual derivation (don't let that stop you from writing that out yourself, though) and simply show you the resulting new coordinate sets:

    - - -

    and

    - - -

    So, looking at our matrices, did we really need to compute the second segment matrix? No, we didn't. Actually having one segment's matrix means we implicitly have the other: push the values of each row in the matrix Q to the right, with zeroes getting pushed off the right edge and appearing back on the left, and then flip the matrix vertically. Presto, you just "calculated" Q'.

    -

    Implementing curve splitting this way requires less recursion, and is just straight arithmetic with cached values, so can be cheaper on systems where recursion is expensive. If you're doing computation with devices that are good at matrix multiplication, chopping up a Bézier curve with this method will be a lot faster than applying de Casteljau.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Lowering and elevating curve order

    -

    One interesting property of Bézier curves is that an nth order curve can always be perfectly represented by an (n+1)th order curve, by giving the higher-order curve specific control points.

    -

    If we have a curve with three points, then we can create a curve with four points that exactly reproduces the original curve. First, we give it the same start and end points, and for its two control points we pick "1/3rd start + 2/3rd control" and "2/3rd control + 1/3rd end". Now we have exactly the same curve as before, except represented as a cubic curve rather than a quadratic curve.

    -

    The general rule for raising an nth order curve to an (n+1)th order curve is as follows (observing that the start and end weights are the same as the start and end weights for the old curve):

    - - -

    However, this rule also has as direct consequence that you cannot generally safely lower a curve from nth order to (n-1)th order, because the control points cannot be "pulled apart" cleanly. We can try to, but the resulting curve will not be identical to the original, and may in fact look completely different.

    -

    However, there is a surprisingly good way to ensure that a lower order curve looks "as close as reasonably possible" to the original curve: we can optimise the "least-squares distance" between the original curve and the lower order curve, in a single operation (also explained over on Sirver's Castle). However, to use it, we'll need to do some calculus work and then switch over to linear algebra. As mentioned in the section on matrix representations, some things can be done much more easily with matrices than with calculus functions, and this is one of those things. So... let's go!

    -

    We start by taking the standard Bézier function, and condensing it a little:

    - - -

    Then, we apply one of those silly (actually, super useful) calculus tricks: since our t value is always between zero and one (inclusive), we know that (1-t) plus t always sums to 1. As such, we can express any value as a sum of t and 1-t:

    - - -

    So, with that seemingly trivial observation, we rewrite that Bézier function by splitting it up into a sum of a (1-t) and t component:

    - - -

    So far so good. Now, to see why we did this, let's write out the (1-t) and t parts, and see what that gives us. I promise, it's about to make sense. We start with (1-t):

    - - -

    So by using this seemingly silly trick, we can suddenly express part of our nth order Bézier function in terms of an (n+1)th order Bézier function. And that sounds a lot like raising the curve order! Of course we need to be able to repeat that trick for the t part, but that's not a problem:

    - - -

    So, with both of those changed from an order n expression to an order (n+1) expression, we can put them back together again. Now, where the order n function had a summation from 0 to n, the order n+1 function uses a summation from 0 to n+1, but this shouldn't be a problem as long as we can add some new terms that "contribute nothing". In the next section on derivatives, there is a discussion about why "higher terms than there is a binomial for" and "lower than zero terms" both "contribute nothing". So as long as we can add terms that have the same form as the terms we need, we can just include them in the summation, they'll sit there and do nothing, and the resulting function stays identical to the lower order curve.

    -

    Let's do this:

    - - -

    And this is where we switch over from calculus to linear algebra, and matrices: we can now express this relation between Bézier(n,t) and Bézier(n+1,t) as a very simple matrix multiplication:

    - - -

    where the matrix M is an n+1 by n matrix, and looks like:

    - - -

    That might look unwieldy, but it's really just a mostly-zeroes matrix, with a very simply fraction on the diagonal, and an even simpler fraction to the left of it. Multiplying a list of coordinates with this matrix means we can plug the resulting transformed coordinates into the one-order-higher function and get an identical looking curve.

    -

    Not too bad!

    -

    Equally interesting, though, is that with this matrix operation established, we can now use an incredibly powerful and ridiculously simple way to find out a "best fit" way to reverse the operation, called the normal equation. What it does is minimize the sum of the square differences between one set of values and another set of values. Specifically, if we can express that as some function A x = b, we can use it. And as it so happens, that's exactly what we're dealing with, so:

    - - -

    The steps taken here are:

    -
      -
    1. We have a function in a form that the normal equation can be used with, so
      2. -
    2. apply the normal equation!
      3. -
    3. Then, we want to end up with just Bn on the left, so we start by left-multiply both sides such that we'll end up with lots of stuff on the left that simplified to "a factor 1", which in matrix maths is the identity matrix.
      4. -
    4. In fact, by left-multiplying with the inverse of what was already there, we've effectively "nullified" (but really, one-inified) that big, unwieldy block into the identity matrix I. So we substitute the mess with I, and then
      5. -
    5. because multiplication with the identity matrix does nothing (like multiplying by 1 does nothing in regular algebra), we just drop it.
      6. -
    -

    And we're done: we now have an expression that lets us approximate an n+1th order curve with a lower nth order curve. It won't be an exact fit, but it's definitely a best approximation. So, let's implement these rules for raising and lowering curve order to a (semi) random curve, using the following graphic. Select the sketch, which has movable control points, and press your up and down arrow keys to raise or lower the curve order.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Производные кривых Безье

    -

    Есть целый ряд вещей, которые мы можем сотворить с кривыми Безье базируясь на их производных, и одним восхитительным наблюдением на счет первых, является то, что их производные по сути тоже являются кривыми Безье. На практике, дифференциация кривых Безье относительно логична, хотя нам и потребуется немного математики.

    -

    Для начала рассмотрим правило производных для кривых Безье:

    - - -

    которое мы так же можем записать (отметив что b в этой формуле, то-же что наш w "вес", а n умноженная на функцию сумы — то-же что функция сумы, с каждой составляющей умноженной на n) как:

    - - -

    Другими словами, производная кривой Безье nго порядка равна кривой Безье на порядок ниже (n-1), соответственно имея на один термин составляющих меньше, где новые веса w'0...w'n-1 произведены из оригинальных по принципу n(wi+1 - wi). Так, для кривой 3го порядка, с 4мя весами, производная имеет 3 веса: w'0 = 3(w1-w0), w'1 = 3(w2-w1) и w'2 = 3(w3-w2).

    -
    - -

    "Погодите, а почему это работает?"

    -

    Порой, когда говорят "вот она производная", наше счастье приходит немедленно и остается неизменным вне зависимости от следующих аргументов. Но, возможно, вам захочется покопаться в доказательствах производной. Если так, то для начала оговоримся: веса независимы от общей формулы функции, потому производная кривой затрагивает только производные полиноминалов базовой функции. Найдем сперва это:

    - - -

    Приминение правил продукта и цепного правила (* в оригинале другие ссылки) дает нам следующее:

    - - -

    В таком виде с этой формулой работать сложно, потому раскроем:

    - - -

    Теперь фокус заключается в трансформации этой функции во что-то, что бы имело биноминальные коэффициенты, потому как мы хотим получить выражение в форме вроде "x! деленное на y!(x-y)!". Если нам удастся сделать это так, чтобы учесть в запись термины n-1 и k-1 — мы на верном пути.

    - - -

    Первая часть готова: компоненты в скобках на самом то деле обычные выражения кривых Безье на порядок ниже:

    - - -

    Теперь применим это к записям наших формул с "весами". Начнем с формулы кривой Безье приведенной выше и пройдемся по ее производным.

    - - -

    Раскрыв это (немного раскрасив, чтобы подчеркнуть соотношение терминов), и выстроив термины по возрастанию значений, мы увидим следующее:

    - - -

    Два из приведенных терминов отпадают. Первый — потому что степень -1 не присутствует в суме, оттого каждый раз приводя "ничего" в результат. Можем смело игнорировать это значение в нахождении производной функции. Второй — самый последний термин этой записи, затрагивающий Bn-1,n. Этот термин имел-бы биноминальный коэффициент [i указывая на i+1], который не существуюет. И опять — термин ничего не привносит в результат, затем также может быть проигнорирован. Это значит, нам остается:

    - - -

    И это по сути формула функции сумы на 1 порядок ниже:

    - - -

    Можно переписать по стандартной форме сумы, и готово:

    - - -
    - -

    Давайте перепишем это по форме схожей с нашей исходной формулой, чтобы легче было разглядеть разницу. Сначала оригинальная формула, за ней производная:

    - + +

    И это по сути формула функции сумы на 1 порядок ниже:

    + + +

    Можно переписать по стандартной форме сумы, и готово:

    + + + +

    + Давайте перепишем это по форме схожей с нашей исходной формулой, чтобы легче было разглядеть разницу. Сначала оригинальная формула, за ней + производная: +

    + - - - -

    И в чем же разница? В терминах формулы кривой Безье, по сути, никакой! Мы уменьшили порядок (вместо порядка n, он теперь n-1), но это все та же функция Безье. Единственное отличие в подсчете изменений в "весах" при нахождении производной. К примеру, исходя из 4-х контрольных точек A, B, C и D, первая производная получит 3 точки, вторая — 2, третья — 1:

    - - -

    Можно продолжать производить этот фокус, до тех пор пока у нас имеется более одного веса. Когда же остается один вес, следующим шагом будет k = 0, и результат сложения "функции сумы Безье" будет равен 0, поскольку мы ничего ни с чем не слагаем. По этому у квадратной функций нету второй производной, у кубической — третей, и, обобщая, кривая Безье nго порядка, имеет n-1 (внятных) производных, с каждой следующей производной равной нулю.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Tangents and normals

    -

    If you want to move objects along a curve, or "away from" a curve, the two vectors you're most interested in are the tangent vector and normal vector for curve points. These are actually really easy to find. For moving and orienting along a curve, we use the tangent, which indicates the direction of travel at specific points, and is literally just the first derivative of our curve:

    - - -

    This gives us the directional vector we want. We can normalize it to give us uniform directional vectors (having a length of 1.0) at each point, and then do whatever it is we want to do based on those directions:

    - - -

    The tangent is very useful for moving along a line, but what if we want to move away from the curve instead, perpendicular to the curve at some point t? In that case we want the normal vector. This vector runs at a right angle to the direction of the curve, and is typically of length 1.0, so all we have to do is rotate the normalized directional vector and we're done:

    - - -
    - -

    Rotating coordinates is actually very easy, if you know the rule for it. You might find it explained as "applying a rotation matrix, which is what we'll look at here, too. Essentially, the idea is to take the circles over which we can rotate, and simply "sliding the coordinates" over these circles by the desired -angle. If we want a quarter circle turn, we take the coordinate, slide it along the circle by a quarter turn, and done.

    -

    To turn any point (x,y) into a rotated point (x',y') (over 0,0) by some angle φ, we apply this nice and easy computation:

    - - -

    Which is the "long" version of the following matrix transformation:

    - - -

    And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1.

    -

    But why does this work? Why this matrix multiplication? Wikipedia (technically, Thomas Herter and Klaus Lott) tells us that a rotation matrix can be -treated as a sequence of three (elementary) shear operations. When we combine this into a single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. DataGenetics have an excellent article about this very thing: it's really quite cool, and I strongly recommend taking a quick break from this primer to read that article.

    -
    + +

    + And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even + easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1. +

    +

    + But why does this work? Why this matrix multiplication? + Wikipedia (technically, Thomas Herter and Klaus + Lott) tells us that a rotation matrix can be treated as a sequence of three (elementary) shear operations. When we combine this into a + single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. + DataGenetics have an excellent article about this very thing: it's + really quite cool, and I strongly recommend taking a quick break from this primer to read that article. +

    + -

    The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant).

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    +

    + The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the + normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant). +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Working with 3D normals +

    +

    + Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things + this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but + for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, + it's not "super hard", but there are more steps involved and we should have a look at those. +

    +

    + Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. + However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, + not a single vector, so we basically need to define what "the" normal is in the 3D case. +

    +

    + The "naïve" approach is to construct what is known as the + Frenet normal, where we follow a simple recipe that works in + many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are + perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each + point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the + local tangent, as well as the tangents "right next to it". +

    +

    + Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the + vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know + lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same + logic we used for the 2D case: "just rotate it 90 degrees". +

    +

    So let's do that! And in a twist surprise, we can do this in four lines:

    +
      +
    • a = normalize(B'(t))
      • +
    • b = normalize(a + B''(t))
      • +
    • r = normalize(b × a)
      • +
    • normal = normalize(r × a)
      • +
    +

    Let's unpack that a little:

    +
      +
    • + We start by taking the normalized vector for the derivative at some point on the + curve. We normalize it so the maths is less work. Less work is good. +
      • +
    • + Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and + just had the same derivative and second derivative from that point on. +
      • +
    • + This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which + means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the + cross product. (Note that while that operation uses the × operator, it's most + definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent + a quarter circle to get our normal, just like we did in the 2D case. +
      • +
    • + Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal + vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second + time, and immediately get our normal vector. +
      • +
    +

    + And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can + move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor + position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc: +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    + However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the + curve" between t=0.65 and t=0.75... Why is it doing that? +

    +

    + As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are + "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute + normals that just... look good. +

    +

    Thankfully, Frenet normals are not our only option.

    +

    + Another option is to take a slightly more algorithmic approach and compute a form of + Rotation Minimising Frame + (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational + axis, and the normal vector, centered on an on-curve point. +

    +

    + These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we + could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at + the same time that you're building lookup tables for your curve. +

    +

    + The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by + applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are: +

    +
      +
    • Take a point on the curve for which we know the RM frame already,
      • +
    • take a next point on the curve for which we don't know the RM frame yet, and
      • +
    • + reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous + points as a "mirror". +
      • +
    • + This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal + that's slightly off-kilter, so: +
      • +
    • + reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror". +
      • +
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • +
    +

    So, let's write some code for that!

    +
    +

    Implementing Rotation Minimising Frames

    +

    + We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it + yields a frame with properties: +

    -
    -
    -

    предыдущийоглавлениеследующий
    Working with 3D normals

    -

    Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, it's not "super hard", but there are more steps involved and we should have a look at those.

    -

    Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, not a single vector, so we basically need to define what "the" normal is in the 3D case.

    -

    The "naïve" approach is to construct what is known as the Frenet normal, where we follow a simple recipe that works in many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the local tangent, as well as the tangents "right next to it".

    -

    Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same logic we used for the 2D case: "just rotate it 90 degrees".

    -

    So let's do that! And in a twist surprise, we can do this in four lines:

    -
      -
    • a = normalize(B'(t))
      • -
    • b = normalize(a + B''(t))
      • -
    • r = normalize(b × a)
      • -
    • normal = normalize(r × a)
      • -
    -

    Let's unpack that a little:

    -
      -
    • We start by taking the normalized vector for the derivative at some point on the curve. We normalize it so the maths is less work. Less work is good.
      • -
    • Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on.
      • -
    • This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the cross product. (Note that while that operation uses the × operator, it's most definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent a quarter circle to get our normal, just like we did in the 2D case.
      • -
    • Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second time, and immediately get our normal vector.
      • -
    -

    And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the curve" between t=0.65 and t=0.75... Why is it doing that?

    -

    As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute normals that just... look good.

    -

    Thankfully, Frenet normals are not our only option.

    -

    Another option is to take a slightly more algorithmic approach and compute a form of Rotation Minimising Frame (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational axis, and the normal vector, centered on an on-curve point.

    -

    These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at the same time that you're building lookup tables for your curve.

    -

    The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are:

    -
      -
    • Take a point on the curve for which we know the RM frame already,
      • -
    • take a next point on the curve for which we don't know the RM frame yet, and
      • -
    • reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror".
      • -
    • This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter, so:
      • -
    • reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
      • -
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • -
    -

    So, let's write some code for that!

    -
    - -

    Implementing Rotation Minimising Frames

    -

    We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it yields a frame with properties:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + -

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    +

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    + frames.add(x1) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + -

    Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more code to get much better looking normals.

    -
    +

    + Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more + code to get much better looking normals. +

    + -

    Speaking of better looking, what does this actually look like? Let's revisit that earlier curve, but this time use rotation minimising frames rather than Frenet frames:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    That looks so much better!

    -

    For those reading along with the code: we don't even strictly speaking need a Frenet frame to start with: we could, for instance, treat the z-axis as our initial axis of rotation, so that our initial normal is (0,0,1) × tangent, and then take things from there, but having that initial "mathematically correct" frame so that the initial normal seems to line up based on the curve's orientation in 3D space is just nice.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Component functions

    -

    One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that function, but this poses a problem, since our function is parametric: every axis has its own function.

    -

    The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.

    -

    Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis value, respectively.

    -

    If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a change in the right graph.

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

     

    - - - Скрипты отключены. Показываем резервное изображение. - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Finding extremities: root finding

    -

    Now that we understand (well, superficially anyway) the component functions, we can find the extremities of our Bézier curve by finding maxima and minima on the component functions, by solving the equation B'(t) = 0. We've already seen that the derivative of a Bézier curve is a simpler Bézier curve, but how do we solve the equality? Fairly easily, actually, until our derivatives are 4th order or higher... then things get really hard. But let's start simple:

    -

    Quadratic curves: linear derivatives.

    -

    The derivative of a quadratic Bézier curve is a linear Bézier curve, interpolating between just two terms, which means finding the solution for "where is this line 0" is effectively trivial by rewriting it to a function of t and solving. First we turn our quadratic Bézier function into a linear one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then we turn this into our solution for t using basic arithmetics:

    - - -

    Done.

    -

    Although with the caveat that if b-a is zero, there is no solution and we probably shouldn't try to perform that division.

    -

    Cubic curves: the quadratic formula.

    -

    The derivative of a cubic Bézier curve is a quadratic Bézier curve, and finding the roots for a quadratic polynomial means we can apply the Quadratic formula. If you've seen it before, you'll remember it, and if you haven't, it looks like this:

    - - -

    So, if we can rewrite the Bézier component function as a plain polynomial, we're done: we just plug in the values into the quadratic formula, check if that square root is negative or not (if it is, there are no roots) and then just compute the two values that come out (because of that plus/minus sign we get two). Any value between 0 and 1 is a root that matters for Bézier curves, anything below or above that is irrelevant (because Bézier curves are only defined over the interval [0,1]). So, how do we convert?

    -

    First we turn our cubic Bézier function into a quadratic one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then, using these v values, we can find out what our a, b, and c should be:

    - - -

    This gives us three coefficients {a, b, c} that are expressed in terms of v values, where the v values are expressions of our original coordinate values, so we can do some substitution to get:

    - - -

    Easy-peasy. We can now almost trivially find the roots by plugging those values into the quadratic formula.

    -

    And as a cubic curve, there is also a meaningful second derivative, which we can compute by simple taking the derivative of the derivative.

    -

    Quartic curves: Cardano's algorithm.

    -

    We haven't really looked at them before now, but the next step up would be a Quartic curve, a fourth degree Bézier curve. As expected, these have a derivative that is a cubic function, and now things get much harder. Cubic functions don't have a "simple" rule to find their roots, like the quadratic formula, and instead require quite a bit of rewriting to a form that we can even start to try to solve.

    -

    Back in the 16th century, before Bézier curves were a thing, and even before calculus itself was a thing, Gerolamo Cardano figured out that even if the general cubic function is really hard to solve, it can be rewritten to a form for which finding the roots is "easier" (even if not "easy"):

    - - -

    We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather than three: this makes things considerably easier to solve because it lets us use regular calculus to find the values that satisfy the equation.

    -

    Now, there is one small hitch: as a cubic function, the solutions may be complex numbers rather than plain numbers... And Cardano realised this, centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number.

    -

    So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at Ken J. Ward's page for solving the cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a read-through.

    -
    + +

    + We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather + than three: this makes things considerably easier to solve because it lets us use + regular calculus to find the values that satisfy the equation. +

    +

    + Now, there is one small hitch: as a cubic function, the solutions may be + complex numbers rather than plain numbers... And Cardano realised this, + centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these + numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we + have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're + going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number. +

    +

    + So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at + Ken J. Ward's page for solving the + cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the + complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a + read-through. +

    +
    +

    Implementing Cardano's algorithm for finding all real roots

    +

    + The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" + number space (denoted with ℝ), this is perfectly fine in the + "complex" number space (denoted with ℂ). And, as it so happens, Cardano is + also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm! +

    -

    Implementing Cardano's algorithm for finding all real roots

    -

    The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" number space (denoted with ℝ), this is perfectly fine in the "complex" number space (denoted with ℂ). And, as it so happens, Cardano is also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm!

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    60
    61
    62
    63
    64
    65
    66
    67
    68
    69
    70
    71
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + + 37 + + + 38 + + + 39 + + + 40 + + + 41 + + + 42 + + + 43 + + + 44 + + + 45 + + + 46 + + + 47 + + + 48 + + + 49 + + + 50 + + + 51 + + + 52 + + + 53 + + + 54 + + + 55 + + + 56 + + + 57 + + + 58 + + + 59 + + + 60 + + + 61 + + + 62 + + + 63 + + + 64 + + + 65 + + + 66 + + + 67 + + + 68 + + + 69 + + + 70 + + + 71 + + + 72 + + + 73 + + + 74 + + + 75 + + + 76 + + + 77 + + + 78 + + + 79 + + + 80 + + + 81 + + +
    -
    - -

    And that's it. The maths is complicated, but the code is pretty much just "follow the maths, while caching as many values as we can to prevent recomputing things as much as possible" and now we have a way to find all roots for a cubic function and can just move on with using that to find extremities of our curves.

    -

    And of course, as a quartic curve also has meaningful second and third derivatives, we can quite easily compute those by using the derivative of the derivative (of the derivative), just as for cubic curves.

    -

    Quintic and higher order curves: finding numerical solutions

    -

    And this is where thing stop, because we cannot find the roots for polynomials of degree 5 or higher using algebra (a fact known as the Abel–Ruffini theorem). Instead, for occasions like these, where algebra simply cannot yield an answer, we turn to numerical analysis.

    -

    That's a fancy term for saying "rather than trying to find exact answers by manipulating symbols, find approximate answers by describing the underlying process as a combination of steps, each of which can be assigned a number via symbolic manipulation". For example, trying to mathematically compute how much water fits in a completely crazy three dimensional shape is very hard, even if it got you the perfect, precise answer. A much easier approach, which would be less perfect but still entirely useful, would be to just grab a buck and start filling the shape until it was full: just count the number of buckets of water you used. And if we want a more precise answer, we can use smaller buckets.

    -

    So that's what we're going to do here, too: we're going to treat the problem as a sequence of steps, and the smaller we can make each step, the closer we'll get to that "perfect, precise" answer. And as it turns out, there is a really nice numerical root-finding algorithm, called the Newton-Raphson root finding method (yes, after that Newton), which we can make use of. The Newton-Raphson approach consists of taking our impossible-to-solve function f(x), picking some initial value x (literally any value will do), and calculating f(x). We can think of that value as the "height" of the function at x. If that height is zero, we're done, we have found a root. If it isn't, we calculate the tangent line at f(x) and calculate at which x value its height is zero (which we've already seen is very easy). That will give us a new x and we repeat the process until we find a root.

    -

    Mathematically, this means that for some x, at step n=1, we perform the following calculation until fy(x) is zero, so that the next t is the same as the one we already have:

    - - -

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    -

    Now, this works well only if we can pick good starting points, and our curve is continuously differentiable and doesn't have oscillations. Glossing over the exact meaning of those terms, the curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: which starting points do we pick?

    -

    As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as high as you'll ever need to go.

    -

    In conclusion:

    -

    So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    - - - Скрипты отключены. Показываем резервное изображение. - - - + +

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    +

    + Now, this works well only if we can pick good starting points, and our curve is + continuously differentiable and doesn't have + oscillations. Glossing over the exact meaning of those terms, the + curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: + which starting points do we pick? +

    +

    + As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from + t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may + pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: + there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order + curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as + high as you'll ever need to go. +

    +

    In conclusion:

    +

    + So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those + roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red: +

    + + + Скрипты отключены. Показываем резервное изображение. + + +

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    + + + Скрипты отключены. Показываем резервное изображение. + + +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Bounding boxes +

    +

    + If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four + values we need to box in our curve: +

    +

    Computing the bounding box for a Bézier curve:

    +
      +
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. +
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. +
    3. + Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original + functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct. +
      4. +
    +

    + Applying this approach to our previous root finding, we get the following + axis-aligned bounding boxes (with all curve extremity points + shown on the curve): +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    -
    -
    -

    предыдущийоглавлениеследующий
    Bounding boxes

    -

    If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four values we need to box in our curve:

    -

    Computing the bounding box for a Bézier curve:

    -
      -
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. -
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. -
    3. Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct.
      4. -
    -

    Applying this approach to our previous root finding, we get the following axis-aligned bounding boxes (with all curve extremity points shown on the curve):

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    - -

    We can construct even nicer boxes by aligning them along our curve, rather than along the x- and y-axis, but in order to do so we first need to look at how aligning works.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Выравнивание (пересчет) кривых

    -

    В то время как есть бесконечное множество кривых мы можем задать перемещая x и y кооординаты контрольных точек, не все кривые являются различными между собой. К примеру, если мы зададим кривую, затем повернем ее на 90 градусов, это будет все та же кривая, и все свойста, как, к примеру, точки экстримов, будут на тех же ее участках, только зарисованы на полотне координат в другой точке. Таким образом, одним из способов убедится, что мы работаем с "уникальными" кривыми, является выравнивание их с осями координат.

    -

    Выравнивание так же упрощает уравнение кривой. Мы можем транслитерировать (переместить) кривую таким образом, что первая ее точка будет лежать на точке начала координат (0,0), что, в свою очередь, переведет наш полиноминальную функцию n порядка, на порядок ниже, (n-1). Последовательность останется такой же, но у нас будет меньше условий. Далее мы развернем кривую так, чтобы последняя точка тоже лежала на оси координат, переводя ее значение в (..., 0). Это далее упрощает ф-цию для ее формулы по Y переводя ее еще на порядок ниже. Так, если у нас была кубическая ф-ция:

    - - -

    Мы переместим первую точку в начало координат, смещая все значания x на -120 и y на -160:

    - -

    Затем, повернем кривую до оси X (новые значания x и y округлены в иллюстративных целях):

    - -

    Итак, опуская все нулевые значения, получаем

    - - -

    Как мы видим, наше изначальное определение кривой было существенно упрощено. Следущие зарисовки демострируют результат выравнивание кривой с осью X. первый рисунок соотвествует приведеному выше примеру кривой.

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

     

    - - - Скрипты отключены. Показываем резервное изображение. - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Tight bounding boxes

    -

    With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T.

    -

    We now have nice tight bounding boxes for our curves:

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    - -

    These are, strictly speaking, not necessarily the tightest possible bounding boxes. It is possible to compute the optimal bounding box by determining which spanning lines we need to effect a minimal box area, but because of the parametric nature of Bézier curves this is actually a rather costly operation, and the gain in bounding precision is often not worth it.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Curve inflections

    -

    Now that we know how to align a curve, there's one more thing we can calculate: inflection points. Imagine we have a variable size circle that we can slide up against our curve. We place it against the curve and adjust its radius so that where it touches the curve, the curvatures of the curve and the circle are the same, and then we start to slide the circle along the curve - for quadratic curves, we can always do this without the circle behaving oddly: we might have to change the radius of the circle as we slide it along, but it'll always sit against the same side of the curve.

    -

    But what happens with cubic curves? Imagine we have an S curve and we place our circle at the start of the curve, and start sliding it along. For a while we can simply adjust the radius and things will be fine, but once we get to the midpoint of that S, something odd happens: the circle "flips" from one side of the curve to the other side, in order for the curvatures to keep matching. This is called an inflection, and we can find out where those happen relatively easily.

    -

    What we need to do is solve a simple equation:

    - - -

    What we're saying here is that given the curvature function C(t), we want to know for which values of t this function is zero, meaning there is no "curvature", which will be exactly at the point between our circle being on one side of the curve, and our circle being on the other side of the curve. So what does C(t) look like? Actually something that seems not too hard:

    - - -

    The function C(t) is the cross product between the first and second derivative functions for the parametric dimensions of our curve. And, as already shown, derivatives of Bézier curves are just simpler Bézier curves, with very easy to compute new coefficients, so this should be pretty easy.

    -

    However as we've seen in the section on aligning, aligning lets us simplify things a lot, by completely removing the contributions of the first coordinate from most mathematical evaluations, and removing the last y coordinate as well by virtue of the last point lying on the x-axis. So, while we can evaluate C(t) = 0 for our curve, it'll be much easier to first axis-align the curve and then evaluating the curvature function.

    -
    + +

    + Как мы видим, наше изначальное определение кривой было существенно упрощено. Следущие зарисовки демострируют результат выравнивание кривой + с осью X. первый рисунок соотвествует приведеному выше примеру кривой. +

    + + + Скрипты отключены. Показываем резервное изображение. + + +

     

    + + + Скрипты отключены. Показываем резервное изображение. + + +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Tight bounding boxes +

    +

    + With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, + recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding + box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T. +

    +

    We now have nice tight bounding boxes for our curves:

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    -

    Let's derive the full formula anyway

    -

    Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:

    - - -

    And of course the same functions for y:

    - - -

    Asking a computer to now compose the C(t) function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:

    - - -

    That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all disappear if we axis-align our curve, which is why aligning is a great idea.

    - - -

    Aligning our curve so that three of the eight coefficients become zero, and observing that scale does not affect finding t values, we end up with the following simple term function for C(t):

    - - -

    That's a lot easier to work with: we see a fair number of terms that we can compute and then cache, giving us the following simplification:

    - - -

    This is a plain quadratic curve, and we know how to solve C(t) = 0; we use the quadratic formula:

    - - -

    We can easily compute this value if the discriminator isn't a negative number (because we only want real roots, not complex roots), and if x is not zero, because divisions by zero are rather useless.

    -

    Taking that into account, we compute t, we disregard any t value that isn't in the Bézier interval [0,1], and we now know at which t value(s) our curve will inflect.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - -
    -
    -

    предыдущийоглавлениеследующий
    The canonical form (for cubic curves)

    -

    While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?

    -

    As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper "A Geometric Characterization of Parametric Cubic curves". It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?

    -

    The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    This is a fairly funky image, so let's see what the various parts of it mean...

    -

    We see the three fixed points at (0,0), (0,1) and (1,1). The various regions and boundaries indicate what property the original curve will have, if the fourth point is in/on that region or boundary. Specifically, if the fourth point is...

    -
      -

    1. ...anywhere inside the red zone, but not on its boundaries, the curve will be self-intersecting (yielding a loop). We won't know where it self-intersects (in terms of t values), but we are guaranteed that it does.

      -
      2. -

    2. ...on the left (red) edge of the red zone, the curve will have a cusp. We again don't know where, but we know there is one. This edge is described by the function:

      - - -
      3. -

    3. ...on the almost circular, lower right (pink) edge, the curve's end point touches the curve, forming a loop. This edge is described by the function:

      - - -
      4. -

    4. ...on the top (blue) edge, the curve's start point touches the curve, forming a loop. This edge is described by the function:

      - - -
      5. -

    5. ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

      -
      6. -

    6. ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two inflections (switching from concave to convex and then back again, or from convex to concave and then back again).

      -
      7. -

    7. ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

      -
      8. -
    -

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    -
    - -

    Wait, where do those lines come from?

    -

    Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval.

    -

    The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general curve does over the interval t=0 to t=1.

    -

    For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you can probably follow along with this paper, although you might have to read it a few times before all the bits "click".

    -
    - -

    So now the question becomes: how do we manipulate our curve so that it fits this canonical form, with three fixed points, and one "free" point? Enter linear algebra. Don't worry, I'll be doing all the math for you, as well as show you what the effect is on our curves, but basically we're going to be using linear algebra, rather than calculus, because "it's way easier". Sometimes a calculus approach is very hard to work with, when the equivalent geometrical solution is super obvious.

    -

    The approach is going to start with a curve that doesn't have all-colinear points (so we need to make sure the points don't all fall on a straight line), and then applying three graphics operations that you will probably have heard of: translation (moving all points by some fixed x- and y-distance), scaling (multiplying all points by some x and y scale factor), and shearing (an operation that turns rectangles into parallelograms).

    -

    Step 1: we translate any curve by -p1.x and -p1.y, so that the curve starts at (0,0). We're going to make use of an interesting trick here, by pretending our 2D coordinates are 3D, with the z coordinate simply always being 1. This is an old trick in graphics to overcome the limitations of 2D transformations: without it, we can only turn (x,y) coordinates into new coordinates of the form (ax + by, cx + dy), which means we can't do translation, since that requires we end up with some kind of (x + a, y + b). If we add a bogus z coordinate that is always 1, then we can suddenly add arbitrary values. For example:

    - - -

    Sweet! z stays 1, so we can effectively ignore it entirely, but we added some plain values to our x and y coordinates. So, if we want to subtract p1.x and p1.y, we use:

    - - -

    Running all our coordinates through this transformation gives a new set of coordinates, let's call those U, where the first coordinate lies on (0,0), and the rest is still somewhat free. Our next job is to make sure point 2 ends up lying on the x=0 line, so what we want is a transformation matrix that, when we run it, subtracts x from whatever x we currently have. This is called shearing, and the typical x-shear matrix and its transformation looks like this:

    - - -

    So we want some shearing value that, when multiplied by y, yields -x, so our x coordinate becomes zero. That value is simply -x/y, because *-x/y * y = -x*. Done:

    - - -

    Now, running this on all our points generates a new set of coordinates, let's call those V, which now have point 1 on (0,0) and point 2 on (0, some-value), and we wanted it at (0,1), so we need to do some scaling to make sure it ends up at (0,1). Additionally, we want point 3 to end up on (1,1), so we can also scale x to make sure its x-coordinate will be 1 after we run the transform. That means we'll be x-scaling by 1/point3x, and y-scaling by point2y. This is really easy:

    - - -

    Then, finally, this generates a new set of coordinates, let's call those W, of which point 1 lies on (0,0), point 2 lies on (0,1), and point three lies on (1, ...) so all that's left is to make sure point 3 ends up at (1,1) - but we can't scale! Point 2 is already in the right place, and y-scaling would move it out of (0,1) again, so our only option is to y-shear point three, just like how we x-sheared point 2 earlier. In this case, we do the same trick, but with y/x rather than x/y because we're not x-shearing but y-shearing. Additionally, we don't actually want to end up at zero (which is what we did before) so we need to shear towards an offset, in this case 1:

    - - -

    And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will be:

    - - -

    Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just pull this apart a little and end up with an easy-to-calculate bit of code!

    -

    First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does that leave?

    - + +

    + And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and + only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will + be: +

    + + +

    + Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also + notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just + pull this apart a little and end up with an easy-to-calculate bit of code! +

    +

    + First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does + that leave? +

    + - -

    Suddenly things look a lot simpler: the mapped x is fairly straight forward to compute, and we see that the mapped y actually contains the mapped x in its entirety, so we'll have that part already available when we need to evaluate it. In fact, let's pull out all those common factors to see just how simple this is:

    - - -

    That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it look!

    -
    + +

    + That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it + look! +

    +
    +

    How do you track all that?

    +

    + Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply + use Mathematica. Tracking all this math by hand is insane, and we invented computers, + literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the + program do the math for me. And real math, too, with symbols, not with numbers. In fact, + here's the Mathematica notebook if you want to see how + this works for yourself. +

    +

    + Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's + $344 for home use, up from $295 when I original wrote this, but it's + also free when you buy a $35 raspberry pi. Obviously, I bought a + raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do + it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for. +

    +
    -

    How do you track all that?

    -

    Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply use Mathematica. Tracking all this math by hand is insane, and we invented computers, literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the program do the math for me. And real math, too, with symbols, not with numbers. In fact, here's the Mathematica notebook if you want to see how this works for yourself.

    -

    Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's $344 for home use, up from $295 when I original wrote this, but it's also free when you buy a $35 raspberry pi. Obviously, I bought a raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for.

    -
    - -

    So, let's write up a sketch that'll show us the canonical form for any curve drawn in blue, overlaid on our canonical map, so that we can immediately tell which features our curve must have, based on where the fourth coordinate is located on the map:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Finding Y, given X

    -

    One common task that pops up in things like CSS work, or parametric equalizers, or image leveling, or any other number of applications where Bézier curves are used as control curves in a way that there is really only ever one "y" value associated with one "x" value, you might want to cut out the middle man, as it were, and compute "y" directly based on "x". After all, the function looks simple enough, finding the "y" value should be simple too, right? Unfortunately, not really. However, it is possible and as long as you have some code in place to help, it's not a lot of a work either.

    -

    We'll be tackling this problem in two stages: the first, which is the hard part, is figuring out which "t" value belongs to any given "x" value. For instance, have a look at the following graphic. On the left we have a Bézier curve that looks for all intents and purposes like it fits our criteria: every "x" has one and only one associated "y" value. On the right we see the function for just the "x" values: that's a cubic curve, but not a really crazy cubic curve. If you move the graphic's slider, you will see a red line drawn that corresponds to the x coordinate: this is a vertical line in the left graphic, and a horizontal line on the right.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Now, if you look more closely at that right graphic, you'll notice something interesting: if we treat the red line as "the x axis", then the point where the function crosses our line is really just a root for the cubic function x(t) through a shifted "x-axis"... and we've already seen how to calculate roots, so let's just run cubic root finding - and not even the complicated cubic case either: because of the kind of curve we're starting with, we know there is only root, simplifying the code we need!

    -

    First, let's look at the function for x(t):

    - - -

    We can rewrite this to a plain polynomial form, by just fully writing out the expansion and then collecting the polynomial factors, as:

    - - -

    Nothing special here: that's a standard cubic polynomial in "power" form (i.e. all the terms are ordered by their power of t). So, given that a, b, c, d, and x(t) are all known constants, we can trivially rewrite this (by moving the x(t) across the equal sign) as:

    - - -

    You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using Cardano's algorithm, and we're left with some rather short code:

    + +

    + You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they + all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using + Cardano's algorithm, and we're left with some rather short code: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +y = curve.get(t).y + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So the procedure is fairly straight forward: pick an x, find the associated t value, evaluate our curve for that t value, which gives us the curve's {x,y} coordinate, which means we know y for this x. Move the slider for the following graphic to see this in action:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Arc length

    -

    How long is a Bézier curve? As it turns out, that's not actually an easy question, because the answer requires maths that —much like root finding— cannot generally be solved the traditional way. If we have a parametric curve with fx(t) and fy(t), then the length of the curve, measured from start point to some point t = z, is computed using the following seemingly straight forward (if a bit overwhelming) formula:

    - - -

    or, more commonly written using Leibnitz notation as:

    - - -

    This formula says that the length of a parametric curve is in fact equal to the area underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an unwieldy computation, and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates".

    -

    Seriously: It cannot be done.

    -

    So we turn to numerical approaches again. The method we'll look at here is the Gauss quadrature. This approximation is a really neat trick, because for any nth degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:

    - - -

    In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the better the approximation:

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    + +

    + In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips + sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The + more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the + better the approximation: +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    -

    Now, infinitely many terms to sum and infinitely thin rectangles are not something that computers can work with, so instead we're going to approximate the infinite summation by using a sum of a finite number of "just thin" rectangular strips. As long as we use a high enough number of thin enough rectangular strips, this will give us an approximation that is pretty close to what the real value is.

    -

    So, the trick is to come up with useful rectangular strips. A naive way is to simply create n strips, all with the same width, but there is a far better way using special values for C and f(t) depending on the value of n, which indicates how many strips we'll use, and it's called the Legendre-Gauss quadrature.

    -

    This approach uses strips that are not spaced evenly, but instead spaces them in a special way based on describing the function as a polynomial (the more strips, the more accurate the polynomial), and then computing the exact integral for that polynomial. We're essentially performing arc length computation on a flattened curve, but flattening it based on the intervals dictated by the Legendre-Gauss solution.

    -
    - -

    Note that one requirement for the approach we'll use is that the integral must run from -1 to 1. That's no good, because we're dealing with Bézier curves, and the length of a section of curve applies to values which run from 0 to "some value smaller than or equal to 1" (let's call that value z). Thankfully, we can quite easily transform any integral interval to any other integral interval, by shifting and scaling the inputs. Doing so, we get the following:

    - - -

    That may look a bit more complicated, but the fraction involving z is a fixed number, so the summation, and the evaluation of the f(t) values are still pretty simple.

    -

    So, what do we need to perform this calculation? For one, we'll need an explicit formula for f(t), because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these Ci and ti values should be. Luckily, that's less work because there are actually many tables available that give these values, for any n, so if we want to approximate our integral with only two terms (which is a bit low, really) then these tables would tell us that for n=2 we must use the following values:

    - - -

    Which means that in order for us to approximate the integral, we must plug these values into the approximate function, which gives us:

    - - -

    We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the Bézier curve functions Bx(t) and By(t).

    -
    + +

    + We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the + Bézier curve functions Bx(t) and By(t). +

    + -

    If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the start/end points on the inside?

    - - - Скрипты отключены. Показываем резервное изображение. - - - +

    + If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), + we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, + with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make + a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the + start/end points on the inside? +

    + + + Скрипты отключены. Показываем резервное изображение. + + + +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Approximated arc length +

    +

    + Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length + instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This + will come with an error, but this can be made arbitrarily small by increasing the segment count. +

    +

    + If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal + effort: +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    -
    -
    -

    предыдущийоглавлениеследующий
    Approximated arc length

    -

    Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This will come with an error, but this can be made arbitrarily small by increasing the segment count.

    -

    If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal effort:

    -
    - - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - -
    - -

    You may notice that even though the error in length is actually pretty significant in absolute terms, even at a low number of segments we get a length that agrees with the true length when it comes to just the integer part of the arc length. Quite often, approximations can drastically speed things up!

    - -
    -
    -

    предыдущийоглавлениеследующий
    Curvature of a curve

    -

    If we have two curves, and we want to line them in up in a way that "looks right", what would we use as metric to let a computer decide what "looks right" means?

    -

    For instance, we can start by ensuring that the two curves share an end coordinate, so that there is no "gap" between the end of one and the start of the next curve, but that won't guarantee that things look right: both curves can be going in wildly different directions, and the resulting joined geometry will have a corner in it, rather than a smooth transition from one curve to the next.

    -

    What we want is to ensure that the curvature at the transition from one curve to the next "looks good". So, we start with a shared coordinate, and then also require that derivatives for both curves match at that coordinate. That way, we're assured that their tangents line up, which must mean the curve transition is perfectly smooth. We can even make the second, third, etc. derivatives match up for better and better transitions.

    -

    Problem solved!

    -

    However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on reordering curves showed us that the same looking curve can have an infinite number of curve expressions of arbitrarily high Bézier degree, and each of those will have wildly different derivative values.

    -

    So what we really want is some kind of expression that's not based on any particular expression of t, but is based on something that is invariant to the kind of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being the same function for "coordinate at some distance D along the curve".

    -

    We've seen this before... that's the arc length function.

    -

    So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the form of the arc length function, which we saw above and is fairly simple, rather than needing to solve the arc length function. If we start with the arc length expression and the run through the steps necessary to determine its derivative (with an alternative, shorter demonstration of how to do this found over on Stackexchange), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the fundamental theorem of calculus), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.

    -

    Let me highlight what just happened, because it's pretty special:

    -
      -
    1. we wanted to make curves line up, and initially thought to match the curves' derivatives, but
      2. -
    2. that turned out to be a really bad choice, so instead
      3. -
    3. we picked a function that is basically impossible to work with, and then worked with that, which
      4. -
    4. gives us a simple formula that is and expression using the curves' derivatives.
      5. -
    -

    That's crazy!

    -

    But that's also one of the things that makes maths so powerful: even if your initial ideas are off the mark, you might be much closer than you thought you were, and the journey from "thinking we're completely wrong" to "actually being remarkably close to being right" is where we can find a lot of insight.

    -

    So, what does the function look like? This:

    - - -

    Which is really just a "short form" that glosses over the fact that we're dealing with functions of t, so let's expand that a tiny bit:

    - - -

    And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so evaluating this function for any t value is just a matter of basic arithematics.

    -

    In fact, let's just implement it right now:

    + +

    + And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any + (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that + looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier + curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so + evaluating this function for any t value is just a matter of basic arithematics. +

    +

    In fact, let's just implement it right now:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return numerator / denominator + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    That was easy! (Well okay, that "not a number" value will need to be taken into account by downstream code, but that's a reality of programming anyway)

    -

    With all of that covered, let's line up some curves! The following graphic gives you two curves that look identical, but use quadratic and cubic functions, respectively. As you can see, despite their derivatives being necessarily different, their curvature (thanks to being derived based on maths that "ignores" specific function derivative, and instead gives a formula that smooths out any differences) is exactly the same. And because of that, we can put them together such that the point where they overlap has the same curvature for both curves, giving us the smoothest transition.

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    One thing you may have noticed in this sketch is that sometimes the curvature looks fine, but seems to be pointing in the wrong direction, making it hard to line up the curves properly. A way around that, of course, is to show the curvature on both sides of the curve, so let's just do that. But let's take it one step further: we can also compute the associated "radius of curvature", which gives us the implicit circle that "fits" the curve's curvature at any point, using what is possibly the simplest bit of maths found in this entire primer:

    - - -

    So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" our curve at some point that we can control by using a slider:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - + +

    + So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" + our curve at some point that we can control by using a slider: +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Tracing a curve at fixed distance intervals +

    +

    + Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance + intervals over time, like a train along a track, and you want to use Bézier curves. +

    +

    Now you have a problem.

    +

    + The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a + track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick + t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In + fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another. +

    +

    + The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be + straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To + make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for + this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length + function, which can have more inflection points. +

    + + + Скрипты отключены. Показываем резервное изображение. + + +

    + So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through + the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and + then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of + points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two + points, but if we have a high enough number of samples, we don't even need to bother. +

    +

    + So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t + curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks + like, by coloring each section of curve between two distance markers differently: +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    +

    + However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to + constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT). +

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Intersections +

    +

    + Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to + work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to + perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box + overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the + actual curve. +

    +

    + We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by + implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both + lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution. +

    +

    Line-line intersections

    +

    + If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are + an intervals on by linear algebra, using the procedure outlined in this + top coder + article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line + segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments. +

    +

    + The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on + (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection + point). +

    + + + Скрипты отключены. Показываем резервное изображение. + + + +
    +

    Implementing line-line intersections

    +

    + Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, + but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe + you're using point structs for the line. Let's get coding: +

    -
    -
    -

    предыдущийоглавлениеследующий
    Tracing a curve at fixed distance intervals

    -

    Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance intervals over time, like a train along a track, and you want to use Bézier curves.

    -

    Now you have a problem.

    -

    The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another.

    -

    The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two points, but if we have a high enough number of samples, we don't even need to bother.

    -

    So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks like, by coloring each section of curve between two distance markers differently:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    -

    However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT).

    - -
    -
    -

    предыдущийоглавлениеследующий
    Intersections

    -

    Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the actual curve.

    -

    We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution.

    -

    Line-line intersections

    -

    If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are an intervals on by linear algebra, using the procedure outlined in this top coder article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments.

    -

    The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection point).

    - - - Скрипты отключены. Показываем резервное изображение. - - - -
    - -

    Implementing line-line intersections

    -

    Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe you're using point structs for the line. Let's get coding:

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    + return lli4(line1.p1, line1.p2, line2.p1, line2.p2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + +
    - +

    What about curve-line intersections?

    +

    + Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we + translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in + a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a + curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the + section on finding extremities. +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + Скрипты отключены. Показываем резервное изображение. + + + +
    -

    What about curve-line intersections?

    -

    Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the section on finding extremities.

    -
    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -
    +

    + Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the + curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and + curve splitting. +

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Curve/curve intersection +

    +

    + Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" + technique: +

    +
      +
    1. + Take two curves C1 and C2, and treat them as a pair. +
      2. +
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. +
    3. + With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2). +
      4. +
    4. + For each pair, check whether their bounding boxes overlap. +
        +
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. +
      2. If there is overlap, rerun all steps for this pair.
        3. +
      +
      5. +
    5. + Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we + might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value + (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative). +
      6. +
    +

    + This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then + taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections. +

    +

    + The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click + the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value + that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the + algorithm, so you can try this with lots of different curves. +

    +

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + +

    + Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into + two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at + t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each + iteration, and each successive step homing in on the curve's self-intersection points. +

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + The projection identity +

    +

    + De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to + draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. + Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point + around to change the curve's shape. +

    +

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    +

    + In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want + to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know + has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for + reasons that will become obvious. +

    +

    + So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following + graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which + taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what + happens to that value? +

    +
    + + + Скрипты отключены. Показываем резервное изображение. + + + + + + + + Скрипты отключены. Показываем резервное изображение. + + + + + +
    -

    Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and curve splitting.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Curve/curve intersection

    -

    Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" technique:

    -
      -
    1. Take two curves C1 and C2, and treat them as a pair.
      2. -
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. -
    3. With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2).
      4. -
    4. For each pair, check whether their bounding boxes overlap.
        -
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. -
      2. If there is overlap, rerun all steps for this pair.
        3. -
      -
      5. -
    5. Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative).
      6. -
    -

    This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections.

    -

    The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the algorithm, so you can try this with lots of different curves.

    -

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - -

    Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each iteration, and each successive step homing in on the curve's self-intersection points.

    - -
    -
    -

    предыдущийоглавлениеследующий
    The projection identity

    -

    De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point around to change the curve's shape.

    -

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    -

    In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for reasons that will become obvious.

    -

    So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what happens to that value?

    -
    - - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - -
    - -

    So these graphics show us several things:

    -
      -
    1. a point at the tip of the curve construction's "hat": let's call that A, as well as
      2. -
    2. our on-curve point give our chosen t value: let's call that B, and finally,
      3. -
    3. a point that we get by projecting A, through B, onto the line between the curve's start and end points: let's call that C.
      4. -
    4. for both quadratic and cubic curves, two points e1 and e2, which represent the single-to-last step in de Casteljau's algorithm: in the last step, we find B at (1-t) * e1 + t * e2.
      5. -
    5. for cubic curves, also the points v1 and v2, which together with A represent the first step in de Casteljau's algorithm: in the next step, we find e1 and e2.
      6. -
    -

    These three values A, B, and C allow us to derive an important identity formula for quadratic and cubic Bézier curves: for any point on the curve with some t value, the ratio of distances from A to B and B to C is fixed: if some t value sets up a C that is 20% away from the start and 80% away from the end, then it doesn't matter where the start, end, or control points are; for that t value, C will always lie at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic and then move all the other points around: if you only move the control points, start and end won't move, and so neither will C, and if you move either start or end point, C will move but its relative position will not change.

    -

    So, how can we compute C? We start with our observation that C always lies somewhere between the start and end points, so logically C will have a function that interpolates between those two coordinates:

    - - -

    If we can figure out what the function u(t) looks like, we'll be done. Although we do need to remember that this u(t) will have a different form depending on whether we're working with quadratic or cubic curves. Running through the maths (with thanks to Boris Zbarsky) shows us the following two formulae:

    - - -

    And

    - - -

    So, if we know the start and end coordinates and the t value, we know C without having to calculate the A or even B coordinates. In fact, we can do the same for the ratio function. As another function of t, we technically don't need to know what A or B or C are. It, too, can be expressed as a pure function of t.

    -

    We start by observing that, given A, B, and C, the following always holds:

    - - -

    Working out the maths for this, we see the following two formulae for quadratic and cubic curves:

    - - -

    And

    - - -

    Which now leaves us with some powerful tools: given three points (start, end, and "some point on the curve"), as well as a t value, we can construct curves. We can compute C using the start and end points and our u(t) function, and once we have C, we can use our on-curve point (B) and the ratio(t) function to find A:

    - - -

    With A found, finding e1 and e2 for quadratic curves is a matter of running the linear interpolation with t between start and A to yield e1, and between A and end to yield e2. For cubic curves, there is no single pair of points that can act as e1 and e2 (there are infinitely many, because the tangent at B is a free parameter for cubic curves) so as long as the distance ratio between e1 to B and B to e2 is the Bézier ratio (1-t):t, we are free to pick any pair, after which we can reverse engineer v1 and v2:

    - - -

    And then reverse engineer the curve's control points:

    - - -

    So: if we have a curve's start and end points, as well as some third point B that we want the curve to pass through, then for any t value we implicitly know all the ABC values, which (combined with an educated guess on appropriate e1 and e2 coordinates for cubic curves) gives us the necessary information to reconstruct a curve's "de Casteljau skeleton". Which means that we can now do several things: we can "fit" curves using only three points, which means we can also "mold" curves by moving an on-curve point but leaving its start and end points, and then reconstruct the curve based on where we moved the on-curve point to. These are very useful things, and we'll look at both in the next few sections.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Creating a curve from three points

    -

    Given the preceding section, you might be wondering if we can use that knowledge to just "create" curves by placing some points and having the computer do the rest, to which the answer is: that's exactly what we can now do!

    -

    For quadratic curves, things are pretty easy. Technically, we'll need a t value in order to compute the ratio function used in computing the ABC coordinates, but we can just as easily approximate one by treating the distance between the start and B point, and B and end point as a ratio, using

    - - -

    With this code in place, creating a quadratic curve from three points is literally just computing the ABC values, and using A as our curve's control point:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    For cubic curves we need to do a little more work, but really only just a little. We're first going to assume that a decent curve through the three points should approximate a circular arc, which first requires knowing how to fit a circle to three points. You may remember (if you ever learned it!) that a line between two points on a circle is called a chord, and that one property of chords is that the line from the center of any chord, perpendicular to that chord, passes through the center of the circle.

    -

    That means that if we have three points on a circle, we have three (different) chords, and consequently, three (different) lines that go from those chords through the center of the circle: if we find two of those lines, then their intersection will be our circle's center, and the circle's radius will—by definition!—be the distance from the center to any of our three points:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    With that covered, we now also know the tangent line to our point B, because the tangent to any point on the circle is a line through that point, perpendicular to the line from that point to the center. That just leaves marking appropriate points e1 and e2 on that tangent, so that we can construct a new cubic curve hull. We use the approach as we did for quadratic curves to automatically determine a reasonable t value, and then our e1 and e2 coordinates must obey the standard de Casteljau rule for linear interpolation:

    - - -

    Where d is the total length of the line segment from e1 to e2. So how long do we make that? There are again all kinds of approaches we can take, and a simple-but-effective one is to set the length of that segment to "one third the length of the baseline". This forces e1 and e2 to always be the "linear curve" distance apart, which means if we place our three points on a line, it will actually look like a line. Nice! The last thing we'll need to do is make sure to flip the sign of d depending on which side of the baseline our B is located, so we don't end up creating a funky curve with a loop in it. To do this, we can use the atan2 function:

    - - -

    This angle φ will be between 0 and π if B is "above" the baseline (rotating all three points so that the start is on the left and the end is the right), so we can use a relatively straight forward check to make sure we're using the correct sign for our value d:

    - - -

    The result of this approach looks as follows:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; more on that later), we're only trying to come up with some reasonable e1 and e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives us:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    That looks perfectly serviceable!

    -

    Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at implementing curve molding in the section after that, so read on!

    + +

    The result of this approach looks as follows:

    + + + Скрипты отключены. Показываем резервное изображение. + + + +

    + It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, + we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; + more on that later), we're only trying to come up with some reasonable e1 and + e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives + us: +

    + + + Скрипты отключены. Показываем резервное изображение. + + + +

    That looks perfectly serviceable!

    +

    + Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" + curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding + the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at + implementing curve molding in the section after that, so read on! +

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Projecting a point onto a Bézier curve +

    +

    + Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're + trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the + curve our cursor will be closest to. So, how do we project points onto a curve? +

    +

    + If the Bézier curve is of low enough order, we might be able to + work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a + numerical approach again. We'll be finding our ideal t value using a + binary search. First, we do a coarse distance-check based on + t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast: +

    -
    -
    -

    предыдущийоглавлениеследующий
    Projecting a point onto a Bézier curve

    -

    Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the curve our cursor will be closest to. So, how do we project points onto a curve?

    -

    If the Bézier curve is of low enough order, we might be able to work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a numerical approach again. We'll be finding our ideal t value using a binary search. First, we do a coarse distance-check based on t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    After this runs, we know that LUT[i] is the coordinate on the curve in our LUT that is closest to the point we want to project, so that's a pretty good initial guess as to what the best projection onto our curve is. To refine it, we note that LUT[i] is a better guess than both LUT[i-1] and LUT[i+1], but there might be an even better projection somewhere else between those two values, so that's what we're going to be testing for, using a variation of the binary search.

    -
      -
    1. we start with our point p, and the t values t1=LUT[i-1].t and t2=LUT[i+1].t, which span an interval v = t2-t1.
      2. -
    2. we test this interval in five spots: the start, middle, and end (which we already have), and the two points in between the middle and start/end points
      3. -
    3. we then check which of these five points is the closest to our original point p, and then repeat step 1 with the points before and after the closest point we just found.
      4. -
    -

    This makes the interval we check smaller and smaller at each iteration, and we can keep running the three steps until the interval becomes so small as to lead to distances that are, for all intents and purposes, the same for all points.

    -

    So, let's see that in action: in this case, I'm going to arbitrarily say that if we're going to run the loop until the interval is smaller than 0.001, and show you what that means for projecting your mouse cursor or finger tip onto a rather complex Bézier curve (which, of course, you can reshape as you like). Also shown are the original three points that our coarse check finds.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Intersections with a circle

    -

    It might seem odd to cover this subject so much later than the line/line, line/curve, and curve/curve intersection topics from several sections earlier, but the reason we can't cover circle/curve intersections is that we can't really discuss circle/curve intersection until we've covered the kind of lookup table (LUT) walking that the section on projecting a point onto a curve uses. To see why, let's look at what we would have to do if we wanted to find the intersections between a curve and a circle using calculus.

    -

    First, we observe that "finding intersections" in this case means that, given a circle defined by a center point c = (x,y) and a radius r, we want to find all points on the Bezier curve for which the distance to the circle's center point is equal to the circle radius, which by definition means those points lie on the circle, and so count as intersections. In maths, that means we're trying to solve:

    - - -

    Which seems simple enough. Unfortunately, when we expand that dist function, things get a lot more problematic:

    - - -

    And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the Abel-Ruffini theorem that we saw before, we can't solve this by just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left with a monstrous function to solve.

    -

    So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a bit more code to make sure we find all of them, but let's look at the steps involved:

    + +

    + And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the + Abel-Ruffini theorem that we saw before, we can't solve this by + just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to + actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and + all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left + with a monstrous function to solve. +

    +

    + So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the + on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance + r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a + bit more code to make sure we find all of them, but let's look at the steps involved: +

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + -

    This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is exactly the radius, so the coordinate lies on both the curve and the circle.

    -

    So far so good.

    -

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    +

    + This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor + change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between + that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is + exactly the radius, so the coordinate lies on both the curve and the circle. +

    +

    So far so good.

    +

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + start = i + 2; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points ("current", "previous" and "before previous") and then check whether they form a local minimum:

    +

    + After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we + can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now + not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of + course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in + finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points + ("current", "previous" and "before previous") and then check whether they form a local minimum: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return start + i + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    In words: given a start index, the circle center and radius, and our LUT, we check where (closest to our start index) we can find a local minimum for the difference between "the distance from the curve to the circle center", and the circle's radius. We track this by looking at three values (associated with the indices index-2, index-1, and index), and we know we've found a local minimum if the three values show that the middle value (pd1) is less than either value beside it. When we do, we can set our "best guess, relative to start" as index-1. Of course, since we're now checking values relative to some start value, we might not find another candidate value at all, in which case we return start - 1, so that a simple "is the result less than start?" lets us determine that there are no more intersections to find.

    -

    Finally, while not necessary for point projection, there is one more step we need to perform when we run the binary refinement function on our candidate LUT indices, because we've so far only been testing using distances "closest to the radius of the circle", and that's actually not good enough... we need distances that are the radius of the circle. So, after running the refinement for each of these indices, we need to discard any final value that isn't the circle radius. And because we're working with floating point numbers, what this really means is that we need to discard any value that's a pixel or more "off". Or, if we want to get really fancy, "some small epsilon value".

    -

    Based on all of that, the following graphic shows this off for the standard cubic curve (which you can move the coordinates around for, of course) and a circle with a controllable radius centered on the graphic's center, using the code approach described above.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    And of course, for the full details, click that "view source" link.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Molding a curve

    -

    Armed with knowledge of the "ABC" relation, point-on-curve projection, and guestimating reasonable looking helper values for cubic curve construction, we can finally cover curve molding: updating a curve's shape interactively, by dragging points on the curve around.

    -

    For quadratic curve, this is a really simple trick: we project our cursor onto the curve, which gives us a t value and initial B coordinate. We don't even need the latter: with our t value and "wherever the cursor is" as target B, we can compute the associated C:

    - - -

    And then the associated A:

    - - -

    And we're done, because that's our new quadratic control point!

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    As before, cubic curves are a bit more work, because while it's easy to find our initial t value and ABC values, getting those all-important e1 and e2 coordinates is going to pose a bit of a problem... in the section on curve creation, we were free to pick an appropriate t value ourselves, which allowed us to find appropriate e1 and e2 coordinates. That's great, but when we're curve molding we don't have that luxury: whatever point we decide to start moving around already has its own t value, and its own e1 and e2 values, and those may not make sense for the rest of the curve.

    -

    For example, let's see what happens if we just "go with what we get" when we pick a point and start moving it around, preserving its t value and e1/e2 coordinates:

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    That looks reasonable, close to the original point, but the further we drag our point, the less "useful" things become. Especially if we drag our point across the baseline, rather than turning into a nice curve.

    -

    One way to combat this might be to combine the above approach with the approach from the creating curves section: generate both the "unchanged t/e1/e2" curve, as well as the "idealized" curve through the start/cursor/end points, with idealized t value, and then interpolating between those two curves:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    The slide controls the "falloff distance" relative to where the original point on the curve is, so that as we drag our point around, it interpolates with a bias towards "preserving t/e1/e2" closer to the original point, and bias towards "idealized" form the further away we move our point, with anything that's further than our falloff distance simply being the idealized curve. We don't even try to interpolate at that point.

    -

    A more advanced way to try to smooth things out is to implement continuous molding, where we constantly update the curve as we move around, and constantly change what our B point is, based on constantly projecting the cursor on the curve as we're updating it - this is, you won't be surprised to learn, tricky, and beyond the scope of this section: interpolation (with a reasonable distance) will do for now!

    - -
    -
    -

    предыдущийоглавлениеследующий
    Curve fitting

    -

    Given the previous section, one question you might have is "what if I don't want to guess t values?". After all, plenty of graphics packages do automated curve fitting, so how can we implement that in a way that just finds us reasonable t values all on its own?

    -

    And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a least squares polynomial regression. In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an nth order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".

    -

    Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the squared distance between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep-dive the specific merits of least squared error metrics versus least absolute error metrics, but those are well beyond the scope of this material.

    -

    So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his "Least Squares Bézier Fit" article, and end with some nice interactive graphics for doing some curve fitting.

    -

    Before we begin, we're going to use the curve in matrix form. In the section on matrices, I mentioned that some things are easier if we use the matrix representation of a Bézier curve rather than its calculus form, and this is one of those things.

    -

    As such, the first step in the process is expressing our Bézier curve as powers/coefficients/coordinate matrix T x M x C, by expanding the Bézier functions.

    -
    - -

    Revisiting the matrix representation

    -

    Rewriting Bézier functions to matrix form is fairly easy, if you first expand the function, and then arrange them into a multiple line form, where each line corresponds to a power of t, and each column is for a specific coefficient. First, we expand the function:

    - - -

    And then we (trivially) rearrange the terms across multiple lines:

    - - -

    This rearrangement has "factors of t" at each row (the first row is t⁰, i.e. "1", the second row is t¹, i.e. "t", the third row is t²) and "coefficient" at each column (the first column is all terms involving "a", the second all terms involving "b", the third all terms involving "c").

    -

    With that arrangement, we can easily decompose this as a matrix multiplication:

    - - -

    We can do the same for the cubic curve, of course. We know the base function for cubics:

    - - -

    So we write out the expansion and rearrange:

    - - -

    Which we can then decompose:

    - - -

    And, of course, we can do this for quartic curves too (skipping the expansion step):

    - - -

    And so and on so on. Now, let's see how to use these T, M, and C, to do some curve fitting.

    -
    + +

    + And so and on so on. Now, let's see how to use these T, M, and C, to do some curve + fitting. +

    + -

    Let's get started: we're going to assume we picked the right order curve: for n points we're fitting an n-1th order curve, so we "start" with a vector P that represents the coordinates we already know, and for which we want to do curve fitting:

    - - -

    Next, we need to figure out appropriate t values for each point in the curve, because we need something that lets us tie "the actual coordinate" to "some point on the curve". There's a fair number of different ways to do this (and a large part of optimizing "the perfect fit" is about picking appropriate t values), but in this case let's look at two "obvious" choices:

    -
      -
    1. equally spaced t values, and
      2. -
    2. t values that align with distance along the polygon.
      3. -
    -

    The first one is really simple: if we have n points, then we'll just assign each point i a t value of (i-1)/(n-1). So if we have four points, the first point will have t=(1-1)/(4-1)=0/3, the second point will have t=(2-1)/(4-1)=1/3, the third point will have t=2/3, and the last point will be t=1. We're just straight up spacing the t values to match the number of points we have.

    -

    The second one is a little more interesting: since we're doing polynomial regression, we might as well exploit the fact that our base coordinates just constitute a collection of line segments. At the first point, we're fixing t=0, and the last point, we want t=1, and anywhere in between we're simply going to say that t is equal to the distance along the polygon, scaled to the [0,1] domain.

    -

    To get these values, we first compute the general "distance along the polygon" matrix:

    - - -

    Where length() is literally just that: the length of the line segment between the point we're looking at, and the previous point. This isn't quite enough, of course: we still need to make sure that all the values between i=1 and i=n fall in the [0,1] interval, so we need to scale all values down by whatever the total length of the polygon is:

    - - -

    And now we can move on to the actual "curve fitting" part: what we want is a function that lets us compute "ideal" control point values such that if we build a Bézier curve with them, that curve passes through all our original points. Or, failing that, have an overall error distance that is as close to zero as we can get it. So, let's write out what the error distance looks like.

    -

    As mentioned before, this function is really just "the distance between the actual coordinate, and the coordinate that the curve evaluates to for the associated t value", which we'll square to get rid of any pesky negative signs:

    - - -

    Since this function only deals with individual coordinates, we'll need to sum over all coordinates in order to get the full error function. So, we literally just do that; the total error function is simply the sum of all these individual errors:

    - - -

    And here's the trick that justifies using matrices: while we can work with individual values using calculus, with matrices we can compute as many values as we make our matrices big, all at the "same time", We can replace the individual terms pi with the full P coordinate matrix, and we can replace Bézier(si) with the matrix representation T x M x C we talked about before, which gives us:

    - - -

    In which we can replace the rather cumbersome "squaring" operation with a more conventional matrix equivalent:

    - - -

    Here, the letter T is used instead of the number 2, to represent the matrix transpose; each row in the original matrix becomes a column in the transposed matrix instead (row one becomes column one, row two becomes column two, and so on).

    -

    This leaves one problem: T isn't actually the matrix we want: we don't want symbolic t values, we want the actual numerical values that we computed for S, so we need to form a new matrix, which we'll call 𝕋, that makes use of those, and then use that 𝕋 instead of T in our error function:

    - - -

    Which, because of the first and last values in S, means:

    - - -

    Now we can properly write out the error function as matrix operations:

    - - -

    So, we have our error function: we now need to figure out the expression for where that function has minimal value, e.g. where the error between the true coordinates and the coordinates generated by the curve fitting is smallest. Like in standard calculus, this requires taking the derivative, and determining where that derivative is zero:

    - - -
    + +
    +

    Where did this derivative come from?

    +

    + That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I + straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know. +

    +

    + So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific + case, I + posted a question + to Math.stackexchange, and received a answer that goes into way more detail than I had + hoped to receive. +

    +

    + Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean + maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that + true?". There are answers. They might just require some time to come to understand. +

    +
    -

    Where did this derivative come from?

    -

    That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know.

    -

    So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific case, I posted a question to Math.stackexchange, and received a answer that goes into way more detail than I had hoped to receive.

    -

    Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that true?". There are answers. They might just require some time to come to understand.

    -
    - -

    Now, given the above derivative, we can rearrange the terms (following the rules of matrix algebra) so that we end up with an expression for C:

    - - -

    Here, the "to the power negative one" is the notation for the matrix inverse. But that's all we have to do: we're done. Starting with P and inventing some t values based on the polygon the coordinates in P define, we can compute the corresponding Bézier coordinates C that specify a curve that goes through our points. Or, if it can't go through them exactly, as near as possible.

    -

    So before we try that out, how much code is involved in implementing this? Honestly, that answer depends on how much you're going to be writing yourself. If you already have a matrix maths library available, then really not that much code at all. On the other hand, if you are writing this from scratch, you're going to have to write some utility functions for doing your matrix work for you, so it's really anywhere from 50 lines of code to maybe 200 lines of code. Not a bad price to pay for being able to fit curves to pre-specified coordinates.

    -

    So let's try it out! The following graphic lets you place points, and will start computing exact-fit curves once you've placed at least three. You can click for more points, and the code will simply try to compute an exact fit using a Bézier curve of the appropriate order. Four points? Cubic Bézier. Five points? Quartic. And so on. Of course, this does break down at some point: depending on where you place your points, it might become mighty hard for the fitter to find an exact fit, and things might actually start looking horribly off once there's enough points for compound floating point rounding errors to start making a difference (which is around 10~11 points).

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - -

    You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be able to freely manipulate them and see what the effect on your curve is.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Bézier curves and Catmull-Rom curves

    -

    Taking an excursion to different splines, the other common design curve is the Catmull-Rom spline, which unlike Bézier curves pass through each control point, so they offer a kind of "built-in" curve fitting.

    -

    In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what looks like a single Catmull-Rom curve is actually a spline: a single curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is defined by the point's coordinates, and the tangent for those points, the latter of which can trivially be derived from knowing the previous and next point:

    - + +

    + You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio + along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get + direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be + able to freely manipulate them and see what the effect on your curve is. +

    +
    +
    +

    +
    + предыдущийоглавлениеследующий +
    + Bézier curves and Catmull-Rom curves +

    +

    + Taking an excursion to different splines, the other common design curve is the + Catmull-Rom spline, which unlike Bézier curves + pass through each control point, so they offer a kind of "built-in" curve fitting. +

    +

    + In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets + you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the + tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next. +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    + Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what + looks like a single Catmull-Rom curve is actually a spline: a single + curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, + and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is + defined by the point's coordinates, and the tangent for those points, the latter of which + can trivially be derived from knowing the + previous and next point: +

    + - -

    One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on that in the next section.

    -

    In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of variations that make things considerable more complex):

    + +

    + One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up + with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent + should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on + that in the next section. +

    +

    + In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of + variations that make things + considerable more + complex): +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + point(c0 * v1 + c1 * dv1 + c2 * v2 + c3 * dv2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Now, since a Catmull-Rom curve is a form of cubic Hermite spline, and as cubic Bézier curves are also a form of cubic Hermite spline, we run into an interesting bit of maths programming: we can convert one to the other and back, and the maths for doing so is surprisingly simple!

    -

    The main difference between Catmull-Rom curves and Bézier curves is "what the points mean":

    -
      -
    • A cubic Bézier curve is defined by a start point, a control point that implies the tangent at the start, a control point that implies the tangent at the end, and an end point, plus a characterizing matrix that we can multiply by that point vector to get on-curve coordinates.
      • -
    • A Catmull-Rom curve is defined by a start point, a tangent that for that starting point, an end point, and a tangent for that end point, plus a characteristic matrix that we can multiple by the point vector to get on-curve coordinates.
      • -
    -

    Those are very similar, so let's see exactly how similar they are. We've already see the matrix form for Bézier curves, so how different is the matrix form for Catmull-Rom curves?:

    - - -

    That's pretty dang similar. So the question is: how can we convert that expression with Catmull-Rom matrix and vector into an expression of the Bézier matrix and vector? The short answer is of course "by using linear algebra", but the longer answer is the rest of this section, and involves some maths that you may not even care for: if you just want to know the (incredibly simple) conversions between the two curve forms, feel free to skip to the end of the following explanation, but if you want to how we can get one from the other... let's get mathing!

    -
    - -

    Deriving the conversion formulae

    -

    In order to convert between Catmull-Rom curves and Bézier curves, we need to know two things. Firstly, how to express the Catmull-Rom curve using a "set of four coordinates", rather than a mix of coordinates and tangents, and secondly, how to convert those Catmull-Rom coordinates to and from Bézier form.

    -

    We start with the first part, to figure out how we can go from Catmull-Rom V coordinates to Bézier P coordinates, by applying "some matrix T". We don't know what that T is yet, but we'll get to that:

    - - -

    So, this mapping says that in order to map a Catmull-Rom "point + tangent" vector to something based on an "all coordinates" vector, we need to determine the mapping matrix such that applying T yields P2 as start point, P3 as end point, and two tangents based on the lines between P1 and P3, and P2 nd P4, respectively.

    -

    Computing T is really more "arranging the numbers":

    - - -

    Thus:

    - - -

    However, we're not quite done, because Catmull-Rom curves have that "tension" parameter, written as τ (a lowercase"tau"), which is a scaling factor for the tangent vectors: the bigger the tension, the smaller the tangents, and the smaller the tension, the bigger the tangents. As such, the tension factor goes in the denominator for the tangents, and before we continue, let's add that tension factor into both our coordinate vector representation, and mapping matrix T:

    - - -

    With the mapping matrix properly done, let's rewrite the "point + tangent" Catmull-Rom matrix form to a matrix form in terms of four coordinates, and see what we end up with:

    - - -

    Replace point/tangent vector with the expression for all-coordinates:

    - - -

    and merge the matrices:

    - - -

    This looks a lot like the Bézier matrix form, which as we saw in the chapter on Bézier curves, should look like this:

    - - -

    So, if we want to express a Catmull-Rom curve using a Bézier curve, we'll need to turn this Catmull-Rom bit:

    - - -

    Into something that looks like this:

    - - -

    And the way we do that is with a fairly straight forward bit of matrix rewriting. We start with the equality we need to ensure:

    - - -

    Then we remove the coordinate vector from both sides without affecting the equality:

    - - -

    Then we can "get rid of" the Bézier matrix on the right by left-multiply both with the inverse of the Bézier matrix:

    - - -

    A matrix times its inverse is the matrix equivalent of 1, and because "something times 1" is the same as "something", so we can just outright remove any matrix/inverse pair:

    - - -

    And now we're basically done. We just multiply those two matrices and we know what V is:

    - - -

    We now have the final piece of our function puzzle. Let's run through each step.

    -
      -
    1. Start with the Catmull-Rom function:
      2. -
    - - -
      -
    1. rewrite to pure coordinate form:
      2. -
    - - -
      -
    1. rewrite for "normal" coordinate vector:
      2. -
    - - -
      -
    1. merge the inner matrices:
      2. -
    - - -
      -
    1. rewrite for Bézier matrix form:
      2. -
    - - -
      -
    1. and transform the coordinates so we have a "pure" Bézier expression:
      2. -
    - - -

    And we're done: we finally know how to convert these two curves!

    -
    + +

    And we're done: we finally know how to convert these two curves!

    + -

    If we have a Catmull-Rom curve defined by four coordinates P1 through P4, then we can draw that curve using a Bézier curve that has the vector:

    - - -

    Similarly, if we have a Bézier curve defined by four coordinates P1 through P4, we can draw that using a standard tension Catmull-Rom curve with the following coordinate values:

    - - -

    Or, if your API allows you to specify Catmull-Rom curves using plain coordinates:

    - - - -
    -
    -

    предыдущийоглавлениеследующий
    Creating a Catmull-Rom curve from three points

    -

    Much shorter than the previous section: we saw that Catmull-Rom curves need at least 4 points to draw anything sensible, so how do we create a Catmull-Rom curve from three points?

    -

    Short and sweet: we don't.

    -

    We run through the maths that lets us create a cubic Bézier curve, and then convert its coordinates to Catmull-Rom form using the conversion formulae we saw above.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Forming poly-Bézier curves

    -

    Much like lines can be chained together to form polygons, Bézier curves can be chained together to form poly-Béziers, and the only trick required is to make sure that:

    -
      -
    1. the end point of each section is the starting point of the following section, and
      2. -
    2. the derivatives across that dual point line up.
      3. -
    -

    Unless you want sharp corners, of course. Then you don't even need 2.

    -

    We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind that just follows point 1. where the end point of a segment is the same point as the start point of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're the easiest to implement:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -

    Dragging the control points around only affects the curve segments that the control point belongs to, and moving an on-curve point leaves the control points where they are, which is not the most useful for practical modelling purposes. So, let's add in the logic we need to make things a little better. We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve point is the same as it's "outgoing" derivative:

    - - -

    We can effect this quite easily, because we know that the vector from a curve's last control point to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control point of the next curve matches that, all we have to do is mirror that last control point through the last on-curve point. And mirroring any point A through any point B is really simple:

    - - -

    So let's implement that and see what it gets us. The following two graphics show a quadratic and a cubic poly-Bézier curve again, but this time moving the control points around moves others, too. However, you might see something unexpected going on for quadratic curves...

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -

    As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all the control points are effectively linked. Move one of them, and you move all of them. Not only that, but if we move the on-curve points, it's possible to get a situation where a control point cannot satisfy the constraint that it's the reflection of its two neighbouring control points... This means that we cannot use quadratic poly-Béziers for anything other than really, really simple shapes. And even then, they're probably the wrong choice. Cubic curves are pretty decent, but the fact that the derivatives are linked means we can't manipulate curves as well as we might if we relaxed the constraints a little.

    -

    So: let's relax the requirement a little.

    -

    We can change the constraint so that we still preserve the angle of the derivatives across sections (so transitions from one section to the next will still look natural), but give up the requirement that they should also have the same vector length. Doing so will give us a much more useful kind of poly-Bézier curve:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -

    Cubic curves are now better behaved when it comes to dragging control points around, but the quadratic poly-Bézier still has the problem that moving one control points will move the control points and may ending up defining "the next" control point in a way that doesn't work. Quadratic curves really aren't very useful to work with...

    -

    Finally, we also want to make sure that moving the on-curve coordinates preserves the relative positions of the associated control points. With that, we get to the kind of curve control that you might be familiar with from applications like Photoshop, Inkscape, Blender, etc.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - Скрипты отключены. Показываем резервное изображение. - - - -

    Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a new, very serious problem: we can move an on-curve point in such a way that we can't compute what needs to "happen next". Move the top point down, below the left and right points, for instance. There is no way to preserve correct control points without a kink at the bottom point. Quadratic curves: just not that good...

    -

    A final improvement is to offer fine-level control over which points behave which, so that you can have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves for the rest of the path. Implementing that, is left as an exercise for the reader.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Curve offsetting

    -

    Perhaps you're like me, and you've been writing various small programs that use Bézier curves in some way or another, and at some point you make the step to implementing path extrusion. But you don't want to do it pixel based; you want to stay in the vector world. You find that extruding lines is relatively easy, and tracing outlines is coming along nicely (although junction caps and fillets are a bit of a hassle), and then you decide to do things properly and add Bézier curves to the mix. Now you have a problem.

    -

    Unlike lines, you can't simply extrude a Bézier curve by taking a copy and moving it around, because of the curvatures; rather than a uniform thickness, you get an extrusion that looks too thin in places, if you're lucky, but more likely will self-intersect. The trick, then, is to scale the curve, rather than simply copying it. But how do you scale a Bézier curve?

    -

    Bottom line: you can't. So you cheat. We're not going to do true curve scaling, or rather curve offsetting, because that's impossible. Instead we're going to try to generate 'looks good enough' offset curves.

    -
    - -

    "What do you mean, you can't? Prove it."

    -

    First off, when I say "you can't," what I really mean is "you can't offset a Bézier curve with another Bézier curve", not even by using a really high order curve. You can find the function that describes the offset curve, but it won't be a polynomial, and as such it cannot be represented as a Bézier curve, which has to be a polynomial. Let's look at why this is:

    -

    From a mathematical point of view, an offset curve O(t) is a curve such that, given our original curve B(t), any point on O(t) is a fixed distance d away from coordinate B(t). So let's math that:

    - - -

    However, we're working in 2D, and d is a single value, so we want to turn it into a vector. If we want a point distance d "away" from the curve B(t) then what we really mean is that we want a point at d times the "normal vector" from point B(t), where the "normal" is a vector that runs perpendicular ("at a right angle") to the tangent at B(t). Easy enough:

    - - -

    Now this still isn't very useful unless we know what the formula for N(t) is, so let's find out. N(t) runs perpendicular to the original curve tangent, and we know that the tangent is simply B'(t), so we could just rotate that 90 degrees and be done with it. However, we need to ensure that N(t) has the same magnitude for every t, or the offset curve won't be at a uniform distance, thus not being an offset curve at all. The easiest way to guarantee this is to make sure N(t) always has length 1, which we can achieve by dividing B'(t) by its magnitude:

    - - -

    Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc length from some start a to end b, we must use the formula we saw earlier. Noting that "length" is usually denoted with double vertical bars:

    - - -

    So if we want the length of the tangent, we plug in B'(t), with t = 0 as start and -t = 1 as end:

    - - -

    And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.

    -

    There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine because they do, but they don't; remember that only the base function has binomial coefficients. That's before we factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

    -

    So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve.

    -

    And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the real offset curve would look like, if we could compute it.

    -
    + +

    + And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is + screwing everything up, because it turns our nice polynomials into things that are no longer polynomials. +

    +

    + There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with + unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine + because they do, but they don't; remember that only the base function has binomial coefficients. That's before we + factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make + all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call + them circles, and they have much simpler functions than Bézier curves. +

    +

    + So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means + N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means + that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for + O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve. +

    +

    + And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier + curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, + much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the + real offset curve would look like, if we could compute it. +

    + -

    So, you cannot offset a Bézier curve perfectly with another Bézier curve, no matter how high-order you make that other Bézier curve. However, we can chop up a curve into "safe" sub-curves (where "safe" means that all the control points are always on a single side of the baseline, and the midpoint of the curve at t=0.5 is roughly in the center of the polygon defined by the curve coordinates) and then point-scale each sub-curve with respect to its scaling origin (which is the intersection of the point normals at the start and end points).

    -

    A good way to do this reduction is to first find the curve's extreme points, as explained in the earlier section on curve extremities, and use these as initial splitting points. After this initial split, we can check each individual segment to see if it's "safe enough" based on where the center of the curve is. If the on-curve point for t=0.5 is too far off from the center, we simply split the segment down the middle. Generally this is more than enough to end up with safe segments.

    -

    The following graphics show off curve offsetting, and you can use the slider to control the distance at which the curve gets offset. The curve first gets reduced to safe segments, each of which is then offset at the desired distance. Especially for simple curves, particularly easily set up for quadratic curves, no reduction is necessary, but the more twisty the curve gets, the more the curve needs to be reduced in order to get segments that can safely be scaled.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    You may notice that this may still lead to small 'jumps' in the sub-curves when moving the curve around. This is caused by the fact that we're still performing a naive form of offsetting, moving the control points the same distance as the start and end points. If the curve is large enough, this may still lead to incorrect offsets.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Graduated curve offsetting

    -

    What if we want to do graduated offsetting, starting at some distance s but ending at some other distance e? Well, if we can compute the length of a curve (which we can if we use the Legendre-Gauss quadrature approach) then we can also determine how far "along the line" any point on the curve is. With that knowledge, we can offset a curve so that its offset curve is not uniformly wide, but graduated between with two different offset widths at the start and end.

    -

    Like normal offsetting we cut up our curve in sub-curves, and then check at which distance along the original curve each sub-curve starts and ends, as well as to which point on the curve each of the control points map. This gives us the distance-along-the-curve for each interesting point in the sub-curve. If we call the total length of all sub-curves seen prior to seeing "the current" sub-curve S (and if the current sub-curve is the first one, S is zero), and we call the full length of our original curve L, then we get the following graduation values:

    -
      -
    • start: map S from interval (0,L) to interval (s,e)
      • -
    • c1: map(<strong>S+d1</strong>, 0,L, s,e), d1 = distance along curve to projection of c1
      • -
    • c2: map(<strong>S+d2</strong>, 0,L, s,e), d2 = distance along curve to projection of c2
      • -
    • ...
      • -
    • end: map(<strong>S+length(subcurve)</strong>, 0,L, s,e)
      • -
    -

    At each of the relevant points (start, end, and the projections of the control points onto the curve) we know the curve's normal, so offsetting is simply a matter of taking our original point, and moving it along the normal vector by the offset distance for each point. Doing so will give us the following result (these have with a starting width of 0, and an end width of 40 pixels, but can be controlled with your up and down arrow keys):

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - - - - Скрипты отключены. Показываем резервное изображение. - - - - - - -
    -
    -

    предыдущийоглавлениеследующий
    Circles and quadratic Bézier curves

    -

    Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. For instance, if you're doing font design, fonts have no concept of geometric shapes, they only know straight lines, and Bézier curves. OpenType fonts with TrueType outlines only know quadratic Bézier curves, and OpenType fonts with Type 2 outlines only know cubic Bézier curves. So how do you draw a circle, or an ellipse, or an arc?

    -

    You approximate.

    -

    We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.

    -

    Since arcs are mid-point-symmetrical, we need the control points to set up a symmetrical curve. For quadratic curves this means that the control point will be somewhere on a line that intersects the baseline at a right angle. And we don't get any choice on where that will be, since the derivatives at the start and end point have to line up, so our control point will lie at the intersection of the tangents at the start and end point.

    -

    First, let's try to fit the quadratic curve onto a circular arc. In the following sketch you can move the mouse around over a unit circle, to see how well, or poorly, a quadratic curve can approximate the arc from (1,0) to where your mouse cursor is:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    As you can see, things go horribly wrong quite quickly; even trying to approximate a quarter circle using a quadratic curve is a bad idea. An eighth of a turns might look okay, but how okay is okay? Let's apply some maths and find out. What we're interested in is how far off our on-curve coordinates are with respect to a circular arc, given a specific start and end angle. We'll be looking at how much space there is between the circular arc, and the quadratic curve's midpoint.

    -

    We start out with our start and end point, and for convenience we will place them on a unit circle (a circle around 0,0 with radius 1), at some angle φ:

    - - -

    What we want to find is the intersection of the tangents, so we want a point C such that:

    - - -

    i.e. we want a point that lies on the vertical line through S (at some distance a from S) and also lies on the tangent line through E (at some distance b from E). Solving this gives us:

    - - -

    First we solve for b:

    - - -

    which yields:

    - - -

    which we can then substitute in the expression for a:

    - - -

    A quick check shows that plugging these values for a and b into the expressions for Cx and Cy give the same x/y coordinates for both "a away from A" and "b away from B", so let's continue: now that we know the coordinate values for C, we know where our on-curve point T for t=0.5 (or angle φ/2) is, because we can just evaluate the Bézier polynomial, and we know where the circle arc's actual point P is for angle φ/2:

    - - -

    We compute T, observing that if t=0.5, the polynomial values (1-t)², 2(1-t)t, and t² are 0.25, 0.5, and 0.25 respectively:

    - - -

    Which, worked out for the x and y components, gives:

    - - -

    And the distance between these two is the standard Euclidean distance:

    - - -

    So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter circle and eighth circle?

    -
    - - plotted for 0 ≤ φ ≤ π: - - - plotted for 0 ≤ φ ≤ ½π: - - - - - plotted for 0 ≤ φ ≤ ¼π: -
    + +

    + So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter + circle and eighth circle? +

    + + + + + + + + +
    + + plotted for 0 ≤ φ ≤ π: + + + plotted for 0 ≤ φ ≤ ½π: + + + + + plotted for 0 ≤ φ ≤ ¼π: +
    -

    We now see why the eighth circle arc looks decent, but the quarter circle arc doesn't: an error of roughly 0.06 at t=0.5 means we're 6% off the mark... we will already be off by one pixel on a circle with pixel radius 17. Any decent sized quarter circle arc, say with radius 100px, will be way off if approximated by a quadratic curve! For the eighth circle arc, however, the error is only roughly 0.003, or 0.3%, which explains why it looks so close to the actual eighth circle arc. In fact, if we want a truly tiny error, like 0.001, we'll have to contend with an angle of (rounded) 0.593667, which equates to roughly 34 degrees. We'd need 11 quadratic curves to form a full circle with that precision! (technically, 10 and ten seventeenth, but we can't do partial curves, so we have to round up). That's a whole lot of curves just to get a shape that can be drawn using a simple function!

    -

    In fact, let's flip the function around, so that if we plug in the precision error, labelled ε, we get back the maximum angle for that precision:

    - - -

    And frankly, things are starting to look a bit ridiculous at this point, we're doing way more maths than we've ever done, but thankfully this is as far as we need the maths to take us: If we plug in the precisions 0.1, 0.01, 0.001 and 0.0001 we get the radians values 1.748, 1.038, 0.594 and 0.3356; in degrees, that means we can cover roughly 100 degrees (requiring four curves), 59.5 degrees (requiring six curves), 34 degrees (requiring 11 curves), and 19.2 degrees (requiring a whopping nineteen curves).

    -

    The bottom line? Quadratic curves are kind of lousy if you want circular (or elliptical, which are circles that have been squashed in one dimension) curves. We can do better, even if it's just by raising the order of our curve once. So let's try the same thing for cubic curves.

    - -
    -
    -

    предыдущийоглавлениеследующий
    Circular arcs and cubic Béziers

    -

    Let's look at approximating circles and circular arcs using cubic Béziers. How much better is that?

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    At cursory glance, a fair bit better, but let's find out how much better by looking at how to construct the Bézier curve.

    -

    A construction diagram for a cubic approximation of a circular arc

    -

    The start and end points are trivial, but the mid point requires a bit of work, but it's mostly basic trigonometry once we know the angle θ for our circular arc: if we scale our circular arc to a unit circle, we can always start our arc, with radius 1, at (1,0) and then given our arc angle θ, we also know that the circular arc has length θ (because unit circles are nice that way). We also know our end point, because that's just (cos(θ), sin(θ)), and so the challenge is to figure out what control points we need in order for the curve at t=0.5 to exactly touch the circular arc at the angle θ/2:

    -

    So let's again formally describe this:

    - - -

    Only P3 isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P4, P3) is a right angled triangle, with the distance between the origin and P4 being 1 (because we're working with a unit circle), and the distance between P4 and P3 being k, so that we can represent P3 as "The point P4 plus the vector from the origin to P4 but then rotated a quarter circle, counter-clockwise, and scaled by k".

    -

    With that, we can determine the y-coordinates for A, B, e1, and e2, after which we have all the information we need to determine what the value of k is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P2 and P3, e1 is between A and "midway between P1 and P2" (which is "half height" P2), and so forth:

    - - -

    Which now gives us two identities for B, because in addition to determining B through linear interpolation, we also know that B's y coordinate is just sin(θ/2): we started this exercise by saying we were going to approximate the circular arc using a Bézier curve that had its midpoint, which is point B, touching the unit circle at the arc's half-angle, by definition making B the point at (cos(θ/2), sin(θ/2)).

    -

    This means we can equate the two identities we now have for By and solve for k.

    -
    - -

    Deriving k

    -

    Solving for k is fairly straight forward, but it's a fair few steps, and if you just the immediate result: using a tool like Wolfram Alpha is definitely the way to go. That said, let's get going:

    - - -

    And finally, we can take further advantage of several trigonometric identities to drastically simplify our formula for k:

    - - -

    And we're done.

    -
    + +

    And we're done.

    + -

    So, the distance of our control points to the start/end points can be expressed as a number that we get from an almost trivial expression involving the circular arc's angle:

    - - -

    Which means that for any circular arc with angle θ and radius r, our Bézier approximation based on three points of incidence is:

    - - -

    Which also gives us the commonly found value of 0.55228 for quarter circles, based on them having an angle of half π:

    - - -

    And thus giving us the following Bézier coordinates for a quarter circle of radius r:

    - - -
    + +
    +

    So, how accurate is this?

    +

    + Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points + that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum + deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but + rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we + get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff. +

    +

    + So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, + finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around + that t value, repeating as many times as necessary to get us an arbitrarily precise value of t: +

    -

    So, how accurate is this?

    -

    Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff.

    -

    So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around that t value, repeating as many times as necessary to get us an arbitrarily precise value of t:

    - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return getMostWrongT(radius, bezier, worst_t - stepsize, worst_t + stepsize) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    Plus, how often do you get to write a function with that name?

    -

    Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity first, and then coming back down as we approach 2π)

    - - -
    - - - - - -
    - error plotted for 0 ≤ φ ≤ 2π - - error plotted for 0 ≤ φ ≤ π - - error plotted for 0 ≤ φ ≤ ½π -
    +

    Plus, how often do you get to write a function with that name?

    +

    + Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see + what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity + first, and then coming back down as we approach 2π) +

    + + + + + + + + + + + + + +
    + + + + + +
    + error plotted for 0 ≤ φ ≤ 2π + + error plotted for 0 ≤ φ ≤ π + + error plotted for 0 ≤ φ ≤ ½π +
    -

    That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again:

    -

    +

    + That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we + can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again: +

    +

    -

    Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves.

    -
    +

    + Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At + approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over + quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves. +

    +
    -

    So that's it, kappa is 4/3 · tan(θ/4) , we're done! ...or are we?

    -

    Can we do better?

    -

    Technically: yes, we can. But I'm going to prefix this section with "we can, and we should investigate that possibility, but let me warn you up front that the result is only better if we're going to hard-code the values". We're about to get into the weeds and the standard three-points-of-incidence value is so good already that for most applications, trying to do better won't make any sense at all.

    -

    So with that said: what we calculated above is an upper bound for a best fit Bézier curve for a circular arc: anywhere we don't touch the circular arc in our approximation, we've "overshot" the arc. What if we dropped our value for k just a little, so that the curve starts out as an over-estimation, but then crosses the circular arc, yielding an region of underestimation, and then crosses the circular arc again, with another region of overestimation. This might give us a lower overall error, so let's see what we can do.

    -

    First, let's express the total error (given circular arc angle θ, and some k) using standard calculus notation:

    - - -

    This says that the error function for a given angle and value of k is equal to the "infinite" sum of differences between our curve and the circular arc, as we run t from 0 to 1, using an infinitely small step size. between subsequent t values.

    -

    Now, since we want to find the minimal error, that means we want to know where along this function things go from "error is getting progressively less" to "error is increasing again", which means we want to know where its derivative is zero, which as mathematical expression looks like:

    - - -

    And here we have the most direct application of the Fundamental Theorem of Calculus: the derivative and integral are each other's inverse operations, so they cancel out, leaving us with our original function:

    - - -

    And now we just solve for that... oh wait. We've seen this before. In order to solve this, we'd end up needing to solve this:

    - - -

    And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section on arc lengths—there is no symbolic solution for this equasion. Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer!

    -
    + +

    + And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section + on arc lengths—there is no symbolic solution for this equasion. + Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer! +

    +
    +

    Iterating on a solution

    +

    + By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the + value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's + center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just + binary search our way to the most accurate value for c that gets us that middle case. +

    +

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    -

    Iterating on a solution

    -

    By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just binary search our way to the most accurate value for c that gets us that middle case.

    -

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    - - - - -
    1 - -
    2
    3
    4
    + return binarySearch(radius, angle, points, lowerBound, upperBound) + + + + 2 + + + 3 + + + 4 + + -

    And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and blog posts than you can ever read in a life time:

    +

    + And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and + blog posts than you can ever read in a life time: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return binarySearch(radius, angle, points, value, upperBound) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Using the following radialError function, which samples the curve's approximation of the circular arc over several points (although the first and last point will never contribute anything, so we skip them):

    +

    + Using the following radialError function, which samples the curve's approximation of the circular arc over several points + (although the first and last point will never contribute anything, so we skip them): +

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return err + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a vector, we can get its length to the origin using a magnitude call.

    -

    Examining the result

    -

    Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, for each angle, plot what the difference between the circular arc and the Bézier approximation looks like:

    -

    image-20210419085430711

    -

    Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from 0 to θ:

    -

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    - - -
    - - - - - -
    - max deflection using unit scale - - max deflection at 10x scale - - max deflection at 100x scale -
    +

    + In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a + vector, we can get its length to the origin using a magnitude call. +

    +

    Examining the result

    +

    + Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, + for each angle, plot what the difference between the circular arc and the Bézier approximation looks like: +

    +

    image-20210419085430711

    +

    + Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the + plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from + 0 to θ: +

    +

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    + + + + + + + + + + + + + +
    + + + + + +
    + max deflection using unit scale + + max deflection at 10x scale + + max deflection at 100x scale +
    -

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    -

    As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference.

    -
    +

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    +

    + As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by + 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an + almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference. +

    +
    -

    At this point it should be clear that while, yes, there are improvement to be had, they're essentially insignificant while also being much more computationally expensive.

    -

    TL;DR: just tell me which value I should be using

    -

    It depends on what we need to do. If we just want the best value for quarter circles, and we're going to hard code the value for k, then there is no reason to hard-code the constant k=4/3*tan(pi/8) when you can just as easily hard-code the constant as k=0.551784777779014 instead.

    -

    If you need "the" value for quarter circles, use 0.551785 instead of 0.55228

    -

    However, for dynamic arc approximation, in code that tries to fit circular paths using Bézier paths instead, it should be fairly obvious that the simple function involving a tangent computation, two divisions, and one multiplication, is vastly more performant than running all the code we ended writing just to get a 25% lower error value, and most certainly worth preferring over getting the "more accurate" value.

    -

    If you need to fit Béziers to circular arcs on the fly, use 4/3 * tan(θ/4)

    -

    However, always remember that if you're writing for humans, you can typically use the best of both worlds: as the user interacts with their curves, you should draw their curves instead of drawing approximations of them. If they need to draw circles or circular arcs, draw those, and only approximate them with a Bézier curve when the data needs to be exported to a format that doesn't support those. Ideally with a preview mechanism that highlights where the errors will be, and how large they will be.

    -

    If you're writing code for graphics design by humans, use circular arcs for circular arcs

    -

    And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best k values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).

    - -
    -
    -

    предыдущийоглавлениеследующий
    Approximating Bézier curves with circular arcs

    -

    Let's look at doing the exact opposite of the previous section: rather than approximating circular arc using Bézier curves, let's approximate Bézier curves using circular arcs.

    -

    We already saw in the section on circle approximation that this will never yield a perfect equivalent, but sometimes you need circular arcs, such as when you're working with fabrication machinery, or simple vector languages that understand lines and circles, but not much else.

    -

    The approach is fairly simple: pick a starting point on the curve, and pick two points that are further along the curve. Determine the circle that goes through those three points, and see if it fits the part of the curve we're trying to approximate. Decent fit? Try spacing the points further apart. Bad fit? Try spacing the points closer together. Keep doing this until you've found the "good approximation/bad approximation" boundary, record the "good" arc, and then move the starting point up to overlap the end point we previously found. Rinse and repeat until we've covered the entire curve.

    -

    We already saw how to fit a circle through three points in the section on creating a curve from three points, and finding the arc through those points is straight-forward: pick one of the three points as start point, pick another as an end point, and the arc has to necessarily go from the start point, to the end point, over the remaining point.

    -

    So, how can we convert a Bézier curve into a (sequence of) circular arc(s)?

    -
      -
    • Start at t=0
      • -
    • Pick two points further down the curve at some value m = t + n and e = t + 2n
      • -
    • Find the arc that these points define
      • -
    • Determine how close the found arc is to the curve:
        -
      • Pick two additional points e1 = t + n/2 and e2 = t + n + n/2.
        • -
      • These points, if the arc is a good approximation of the curve interval chosen, should - lie on the circle, so their distance to the center of the circle should be the - same as the distance from any of the three other points to the center.
        • -
      • For point points, determine the (absolute) error between the radius of the circle, and the -actual distance from the center of the circle to the point on the curve.
        • -
      • If this error is too high, we consider the arc bad, and try a smaller interval.
        • -
      -
      • -
    -

    The result of this is shown in the next graphic: we start at a guaranteed failure: s=0, e=1. That's the entire curve. The midpoint is simply at t=0.5, and then we start performing a binary search.

    -
      -
    1. We start with low=0, mid=0.5 and high=1
      2. -
    2. That'll fail, so we retry with the interval halved: {0, 0.25, 0.5}
        -
      • If that arc's good, we move back up by half distance: {0, 0.375, 0.75}.
        • -
      • However, if the arc was still bad, we move down by half the distance: {0, 0.125, 0.25}.
        • -
      -
      3. -
    3. We keep doing this over and over until we have two arcs, in sequence, of which the first arc is good, and the second arc is bad. When we find that pair, we've found the boundary between a good approximation and a bad approximation, and we pick the good arc.
      4. -
    -

    The following graphic shows the result of this approach, with a default error threshold of 0.5, meaning that if an arc is off by a combined half pixel over both verification points, then we treat the arc as bad. This is an extremely simple error policy, but already works really well. Note that the graphic is still interactive, and you can use your up and down arrow keys keys to increase or decrease the error threshold, to see what the effect of a smaller or larger error threshold is.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    With that in place, all that's left now is to "restart" the procedure by treating the found arc's end point as the new to-be-determined arc's starting point, and using points further down the curve. We keep trying this until the found end point is for t=1, at which point we are done. Again, the following graphic allows for up and down arrow key input to increase or decrease the error threshold, so you can see how picking a different threshold changes the number of arcs that are necessary to reasonably approximate a curve:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    So... what is this good for? Obviously, if you're working with technologies that can't do curves, but can do lines and circles, then the answer is pretty straightforward, but what else? There are some reasons why you might need this technique: using circular arcs means you can determine whether a coordinate lies "on" your curve really easily (simply compute the distance to each circular arc center, and if any of those are close to the arc radii, at an angle between the arc start and end, bingo, this point can be treated as lying "on the curve"). Another benefit is that this approximation is "linear": you can almost trivially travel along the arcs at fixed speed. You can also trivially compute the arc length of the approximated curve (it's a bit like curve flattening). The only thing to bear in mind is that this is a lossy equivalence: things that you compute based on the approximation are guaranteed "off" by some small value, and depending on how much precision you need, arc approximation is either going to be super useful, or completely useless. It's up to you to decide which, based on your application!

    - -
    -
    -

    предыдущийоглавлениеследующий
    B-Splines

    -

    No discussion on Bézier curves is complete without also giving mention of that other beast in the curve design space: B-Splines. Easily confused to mean Bézier splines, that's not actually what they are; they are "basis function" splines, which makes a lot of difference, and we'll be looking at those differences in this section. We're not going to dive as deep into B-Splines as we have for Bézier curves (that would be an entire primer on its own) but we'll be looking at how B-Splines work, what kind of maths is involved in computing them, and how to draw them based on a number of parameters that you can pick for individual B-Splines.

    -

    First off: B-Splines are piecewise, polynomial interpolation curves, where the "single curve" is built by performing polynomial interpolation over a set of points, using a sliding window of a fixed number of points. For instance, a "cubic" B-Spline defined by twelve points will have its curve built by evaluating the polynomial interpolation of four points, and the curve can be treated as a lot of different sections, each controlled by four points at a time, such that the full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, ..., {8,9,10,11}, and finally {9,10,11,12}, for eight sections.

    -

    What do they look like? They look like this! Tap on the graphic to add more points, and move points around to see how they map to the spline curve drawn.

    - - - Скрипты отключены. Показываем резервное изображение. - - - -

    The important part to notice here is that we are not doing the same thing with B-Splines that we do for poly-Béziers or Catmull-Rom curves: both of the latter simply define new sections as literally "new sections based on new points", so a 12 point cubic poly-Bézier curve is actually impossible, because we start with a four point curve, and then add three more points for each section that follows, so we can only have 4, 7, 10, 13, 16, etc. point Poly-Béziers. Similarly, while Catmull-Rom curves can grow by adding single points, this addition of a single point introduces three implicit Bézier points. Cubic B-Splines, on the other hand, are smooth interpolations of each possible curve involving four consecutive points, such that at any point along the curve except for our start and end points, our on-curve coordinate is defined by four control points.

    -

    Consider the difference to be this:

    -
      -
    • for Bézier curves, the curve is defined as an interpolation of points, but:
      • -
    • for B-Splines, the curve is defined as an interpolation of curves.
      • -
    -

    In fact, let's look at that again, but this time with the base curves shown, too. Each consecutive four points define one curve:

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have a look at how things work.

    -

    How to compute a B-Spline curve: some maths

    -

    Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the end, just like for Bézier curves), by evaluating the following function:

    - + +

    + In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have + a look at how things work. +

    +

    How to compute a B-Spline curve: some maths

    +

    + Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic + B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the + end, just like for Bézier curves), by evaluating the following function: +

    + - -

    Which, honestly, doesn't tell us all that much. All we can see is that a point on a B-Spline curve is defined as "a mix of all the control points, weighted somehow", where the weighting is achieved through the N(...) function, subscripted with an obvious parameter i, which comes from our summation, and some magical parameter k. So we need to know two things: 1. what does N(t) do, and 2. what is that k? Let's cover both, in reverse order.

    -

    The parameter k represents the "knot interval" over which a section of curve is defined. As we learned earlier, a B-Spline curve is itself an interpolation of curves, and we can treat each transition where a control point starts or stops influencing the total curvature as a "knot on the curve". -Doing so for a degree d B-Spline with n control point gives us d + n + 1 knots, defining d + n intervals along the curve, and it is these intervals that the above k subscript to the N() function applies to.

    -

    Then the N() function itself. What does it look like?

    - - -

    So this is where we see the interpolation: N(t) for an (i,k) pair (that is, for a step in the above summation, on a specific knot interval) is a mix between N(t) for (i,k-1) and N(t) for (i+1,k-1), so we see that this is a recursive iteration where i goes up, and k goes down, so it seem reasonable to expect that this recursion has to stop at some point; obviously, it does, and specifically it does so for the following i/k values:

    - - -

    And this function finally has a straight up evaluation: if a t value lies within a knot-specific interval once we reach a k=1 value, it "counts", otherwise it doesn't. We did cheat a little, though, because for all these values we need to scale our t value first, so that it lies in the interval bounded by knots[d] and knots[n], which are the start point and end point where curvature is controlled by exactly order control points. For instance, for degree 3 (=order 4) and 7 control points, with knot vector [1,2,3,4,5,6,7,8,9,10,11], we map t from [the interval 0,1] to the interval [4,8], and then use that value in the functions above, instead.

    -

    Can we simplify that?

    -

    We can, yes.

    -

    People far smarter than us have looked at this work, and two in particular — Maurice Cox and Carl de Boor — came to a mathematically pleasing solution: to compute a point P(t), we can compute this point by evaluating d(t) on a curve section between knots i and i+1:

    - - -

    This is another recursive function, with k values decreasing from the curve order to 1, and the value α (alpha) defined by:

    - - -

    That looks complicated, but it's not. Computing alpha is just a fraction involving known, plain numbers. And, once we have our alpha value, we also have (1-alpha) because it's a trivial subtraction. Computing the d() function is thus mostly a matter of computing pretty simple arithmetical statements, with some caching of results so we can refer to them as we recurve. While the recursion might see computationally expensive, the total algorithm is cheap, as each step only involves very simple maths.

    -

    Of course, the recursion does need a stop condition:

    - - -

    So, we actually see two stopping conditions: either i becomes 0, in which case d() is zero, or k becomes zero, in which case we get the same "either 1 or 0" that we saw in the N() function above.

    -

    Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily using the same kind of linear interpolation we saw in de Casteljau's algorithm. For instance, if we write out d() for i=3 and k=3, we get the following recursion diagram:

    - - -

    That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. It's really quite elegant.

    -

    One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then summing back up "to the left". We can just start on the right and work our way left immediately.

    -

    Running the computation

    -

    Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on this Michigan Tech page, but an easier to read version is implemented by b-spline.js, so we'll look at its code.

    -

    Given an input value t, we first map the input to a value from the domain [0,1] to the domain [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped t value lies on:

    + +

    + That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a + mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are + themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. + It's really quite elegant. +

    +

    + One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the + d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so + in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point + vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then + summing back up "to the left". We can just start on the right and work our way left immediately. +

    +

    Running the computation

    +

    + Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, + and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on + this Michigan Tech page, but an easier to read version + is implemented by b-spline.js, so we'll look at its code. +

    +

    + Given an input value t, we first map the input to a value from the domain [0,1] to the domain + [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped + t value lies on: +

    - - -
    1 - -
    2
    3
    +} + + + + 2 + + + 3 + + -

    after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU page (updated to use this description's variable names):

    +

    + after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU + page (updated to use this description's variable names): +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those v[i-1] don't try to use an array index that doesn't exist)

    -

    Open vs. closed paths

    -

    Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need more than just a single point.

    -

    Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than separate coordinate objects.

    -

    Manipulating the curve through the knot vector

    -

    The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at:

    -
      -
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. -
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. -
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. -
    4. we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a uniform vector in between.
      5. -
    -

    Uniform B-Splines

    -

    The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to the curvature.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve.

    -

    The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get rid of these, by being clever about how we apply the following uniformity-breaking approach instead...

    -

    Reducing local curve complexity by collapsing intervals

    -

    Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost and the curve "kinks".

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Open-Uniform B-Splines

    -

    By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the curve not starting and ending where we'd kind of like it to:

    -

    For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences that determine the curve shape.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Non-uniform B-Splines

    -

    This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than that any value knots[k+1] should be greater than or equal to knots[k].

    -

    One last thing: Rational B-Splines

    -

    While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like for rational Bézier curves.

    - - - Скрипты отключены. Показываем резервное изображение. - - - - - -

    Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in arbitrary-precision 2D design due to the level of control a NURBS curve offers designers.

    -

    While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the last.

    -

    Extending our implementation to cover rational splines

    -

    The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the extended dimension.

    -

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    -

    We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how many dimensions it needs to interpolate.

    -

    In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight information.

    -

    Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing (x'/w', y'/w'). And that's it, we're done!

    +

    + (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the + interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those + v[i-1] don't try to use an array index that doesn't exist) +

    +

    Open vs. closed paths

    +

    + Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last + point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first + and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As + such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly + more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need + more than just a single point. +

    +

    + Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for + points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same + coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing + references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than + separate coordinate objects. +

    +

    Manipulating the curve through the knot vector

    +

    + The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As + mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the + values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on + intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting + things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at: +

    +
      +
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. +
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. +
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. +
    4. + we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a + uniform vector in between. +
      5. +
    +

    Uniform B-Splines

    +

    + The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire + curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or + [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines + the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to + the curvature. +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    + This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every + interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve. +

    +

    + The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can + perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get + rid of these, by being clever about how we apply the following uniformity-breaking approach instead... +

    +

    Reducing local curve complexity by collapsing intervals

    +

    + Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the + sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, + and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost + and the curve "kinks". +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    Open-Uniform B-Splines

    +

    + By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the + curve not starting and ending where we'd kind of like it to: +

    +

    + For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in + which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values + N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector + [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences + that determine the curve shape. +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    Non-uniform B-Splines

    +

    + This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to + pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than + that any value knots[k+1] should be greater than or equal to knots[k]. +

    +

    One last thing: Rational B-Splines

    +

    + While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's + "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a + ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a + control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like + for rational Bézier curves. +

    + + + Скрипты отключены. Показываем резервное изображение. + + + + + +

    + Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the + NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, + the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an + important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in + arbitrary-precision 2D design due to the level of control a NURBS curve offers designers. +

    +

    + While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform + Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, + they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the + last. +

    +

    Extending our implementation to cover rational splines

    +

    + The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the + control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to + one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the + extended dimension. +

    +

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    +

    + We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate + interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how + many dimensions it needs to interpolate. +

    +

    + In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the + final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight + information. +

    +

    + Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing + (x'/w', y'/w'). And that's it, we're done! +

    +
    +
    + -
    -
    - +

    +
    предыдущийоглавление
    + Comments and questions +

    +

    + First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how + to let me know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing. +

    +

    With that said, on to the comments!

    +
    +
    + + -

    предыдущийоглавление
    Comments and questions

    -

    First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing.

    -

    With that said, on to the comments!

    -
    +
    +
    + This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are + almost no rights reserved. Go do something cool with it! +
    - - - - -
    - -
    - - This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are - almost no rights reserved. - Go do something cool with it! - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/uk-UA/index.html b/docs/uk-UA/index.html index a74aa96e..e0f5718b 100644 --- a/docs/uk-UA/index.html +++ b/docs/uk-UA/index.html @@ -1,161 +1,168 @@ + + + + Підручник з кривих Безьє - - - - Підручник з кривих Безьє + - + - + - + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    + - +
    +

    + Підручник з кривих Безьє +

    +

    Безкоштовна онлайн-книга, яка навчить вас всьому необхідному, щоб працювати з кривими Безьє.

    +
    + Читати рідною мовою: + +

    + (Не бачите своєї мови у списку або хочете, щоб вона досягла 100%? + Допоможіть перекласти цей контент!) +

    +
    +

    + Ласкаво прошу до Підручника з кривих Безьє. Це безкоштовний вебсайт/електронна книга, що пояснює і математичні, і програмувальні аспекти + кривих Безьє, покриваючи широкий спектр областей, які стосуються малювання та роботи з цими кривими. Криві Безьє застосовуються всюди, + починаючи з кривих у Photoshop, і закінчуючи функціями пом'якшення (easing function) CSS та описом контурів популярних шрифтів. +

    +

    + Якщо ви тут вперше: ласкаво прошу! Напишіть мені сюди, якщо вас цікавить щось, + пов'язане з кривими Безьє, чого немає у підручнику. +

    -
    +

    Donations and sponsorship

    -

    Підручник з кривих Безьє

    -

    Безкоштовна онлайн-книга, яка навчить вас всьому необхідному, щоб працювати з кривими Безьє.

    -
    - Читати рідною мовою: - -

    (Не бачите своєї мови у списку або хочете, щоб вона досягла 100%? Допоможіть перекласти цей контент!)

    -
    +

    + Якщо ви використовуєте цей ресурс для наукових досліджень, або пишете власне програмне забезпечення, будь ласка, розгляньте можливість + фінансової підтримки (будь-яка сума + вітається), або станьте патроном на Patreon. Мені не платять за цю роботу, тому якщо ви + вважаєте цей сайт корисним, і хотіли б, щоб він підтримувався протягом тривалого часу, знайте: багато кави потрібно було за ці роки, і ще + багато буде потрібно. Тому, якщо можете допомогти з кавою, то можете бути впевнені, що цей ресурс буде підтримуватися ще довго! +

    -

    - Ласкаво прошу до Підручника з кривих Безьє. Це безкоштовний вебсайт/електронна книга, що пояснює і математичні, - і програмувальні аспекти кривих Безьє, покриваючи широкий спектр областей, які стосуються малювання та роботи з - цими кривими. Криві Безьє застосовуються всюди, починаючи з кривих у Photoshop, і закінчуючи функціями пом'якшення - (easing function) CSS та описом контурів популярних шрифтів. -

    -

    - Якщо ви тут вперше: ласкаво прошу! Напишіть мені сюди, - якщо вас цікавить щось, пов'язане з кривими Безьє, чого немає у підручнику. -

    - -

    Donations and sponsorship

    - - -

    - Якщо ви використовуєте цей ресурс для наукових досліджень, або пишете власне програмне забезпечення, будь ласка, - розгляньте можливість фінансової - підтримки (будь-яка сума вітається), або станьте патроном на Patreon. - Мені не платять за цю роботу, тому якщо ви вважаєте цей сайт корисним, і хотіли б, щоб він підтримувався протягом - тривалого часу, знайте: багато кави потрібно було за ці роки, і ще багато буде потрібно. Тому, якщо можете допомогти - з кавою, то можете бути впевнені, що цей ресурс буде підтримуватися ще довго! -

    - - - -
    +
    -

    - — Pomax -

    - + +
    +
    +
    +

    Preface

    +

    + In order to draw things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The + first of these are as easy to draw as they are easy to make a computer draw. Give a computer the first and last point in the line, and BAM! + straight line. No questions asked. +

    +

    + Curves, however, are a much bigger problem. While we can draw curves with ridiculous ease freehand, computers are a bit handicapped in that + they can't draw curves unless there is a mathematical function that describes how it should be drawn. In fact, they even need this for + straight lines, but the function is ridiculously easy, so we tend to ignore that as far as computers are concerned; all lines are + "functions", regardless of whether they're straight or curves. However, that does mean that we need to come up with fast-to-compute + functions that lead to nice looking curves on a computer. There are a number of these, and in this article we'll focus on a particular + function that has received quite a bit of attention and is used in pretty much anything that can draw curves: Bézier curves. +

    +

    + They're named after Pierre Bézier, who is principally responsible for making + them known to the world as a curve well-suited for design work (publishing his investigations in 1962 while working for Renault), although + he was not the first, or only one, to "invent" these type of curves. One might be tempted to say that the mathematician + Paul de Casteljau was first, as he began investigating the nature of these + curves in 1959 while working at Citroën, and came up with a really elegant way of figuring out how to draw them. However, de Casteljau did + not publish his work, making the question "who was first" hard to answer in any absolute sense. Or is it? Bézier curves are, at their core, + "Bernstein polynomials", a family of mathematical functions investigated by + Sergei Natanovich Bernstein, whose publications on them date back at + least as far as 1912. +

    +

    + Anyway, that's mostly trivia, what you are more likely to care about is that these curves are handy: you can link up multiple Bézier curves + so that the combination looks like a single curve. If you've ever drawn Photoshop "paths" or worked with vector drawing programs like Flash, + Illustrator or Inkscape, those curves you've been drawing are Bézier curves. +

    +

    + But what if you need to program them yourself? What are the pitfalls? How do you draw them? What are the bounding boxes, how do you + determine intersections, how can you extrude a curve, in short: how do you do everything that you might want to do with these curves? That's + what this page is for. Prepare to be mathed! +

    +
    +

    Virtually all Bézier graphics are interactive.

    +

    + This page uses interactive examples, relying heavily on Bezier.js, as well as maths + formulae which are typeset into SVG using the XeLaTeX typesetting system and + pdf2svg by David Barton. +

    +

    This book is open source.

    +

    + This book is an open source software project, and lives on two github repositories. The first is + https://github.com/pomax/bezierinfo and is the purely-for-presentation version you are + viewing right now. The other repository is https://github.com/pomax/BezierInfo-2, + which is the development version, housing all the code that gets turned into the web version, and is also where you should file + issues if you find bugs or have ideas on what to change or add to the primer. +

    +

    How complicated is the maths going to be?

    +

    + Most of the mathematics in this Primer are early high school maths. If you understand basic arithmetic, and you know how to read English, + you should be able to get by just fine. There will at times be far more complicated maths, but if you don't feel like digesting + them, you can safely skip over them by either skipping over the "detail boxes" in section or by just jumping to the end of a section with + maths that looks too involving. The end of sections typically simply list the conclusions so you can just work with those values directly. +

    +

    What language is all this example code in?

    +

    + There are way too many programming languages to favour one of all others, soo all the example code in this Primer uses a form of + pseudo-code that uses a syntax that's close enough to, but not actually, modern scripting languages like JS, Python, etc. That means you + won't be able to copy-paste any of it without giving it any thought, but that's intentional: if you're reading this primer, presumably you + want to learn, and you don't learn by copy-pasting. You learn by doing things yourself, making mistakes, and then fixing + those mistakes. Now, of course, I didn't intentionally add errors in the example code just to trick you into making mistakes (that would + be horrible!) but I did intentionally keep the code from favouring one programming language over another. Don't worry though, if + you know even a single procedural programming language, you should be able to read the examples without any difficulties. +

    +

    Questions, comments:

    +

    + If you have suggestions for new sections, hit up the Github issue tracker (also + reachable from the repo linked to in the upper right). If you have questions about the material, there's currently no comment section + while I'm doing the rewrite, but you can use the issue tracker for that as well. Once the rewrite is done, I'll add a general comment + section back in, and maybe a more topical "select this section of text and hit the 'question' button to ask a question about it" system. + We'll see. +

    +

    Help support the book!

    +

    + If you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me + know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 100-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing! +

    +
    +
    +
    +

    Що нового?

    +

    + Цей підручник постійно розвививається, тож залежно від того, коли ви востаннє його переглядали, тут можуть бути оновлення. Перейдіть за + цим посиланням, щоб побачити, що було додано. (Також доступний RSS-канал) +

    + + + +
    +

    November 2020

    +
      +

    • Added a section on finding curve/circle intersections

      • +
    +

    October 2020

    +
      +

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      • +
    +

    August-September 2020

    +
      +
    • +

      + Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS + turned on. +

      +
      • +
    +

    June 2020

    +
      +

    • Added automatic CI/CD using Github Actions

      • +
    +

    January 2020

    +
      +

    • Added reset buttons to all graphics

      • +

    • Updated to preface to correctly describe the on-page maths

      • +

    • Fixed the Catmull-Rom section because it had glaring maths errors

      • +
    +

    August 2019

    +
      +

    • Added a section on (plain) rational Bezier curves

      • +

    • Improved the Graphic component to allow for sliders

      • +
    +

    December 2018

    +
      +

    • Added a section on curvature and calculating kappa.

      • +
    • +

      + Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this + site! +

      +
      • +
    +

    August 2018

    +
      +

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      • +
    +

    July 2018

    +
      +

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      • +

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      • +
    • +

      (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      +
      • +
    +

    June 2018

    +
      +

    • Added a section on direct curve fitting.

      • +

    • Added source links for all graphics.

      • +

    • Added this "What's new?" section.

      • +
    +

    April 2017

    +
      +

    • Added a section on 3d normals.

      • +

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      • +
    +

    February 2017

    +
      +

    • Finished rewriting the entire codebase for localization.

      • +
    +

    January 2016

    +
      +

    • Added a section to explain the Bezier interval.

      • +

    • Rewrote the Primer as a React application.

      • +
    +

    December 2015

    +
      +

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      • +
    • +

      + Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics. +

      +
      • +
    +

    May 2015

    +
      +

    • Switched over to pure JS rather than Processing-through-Processing.js

      • +

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      • +
    +

    April 2015

    +
      +

    • Added a section on arc length approximations.

      • +
    +

    February 2015

    +
      +

    • Added a section on the canonical cubic Bezier form.

      • +
    +

    November 2014

    +
      +

    • Switched to HTTPS.

      • +
    +

    July 2014

    +
      +

    • Added the section on arc approximation.

      • +
    +

    April 2014

    +
      +

    • Added the section on Catmull-Rom fitting.

      • +
    +

    November 2013

    +
      +

    • Added the section on Catmull-Rom / Bezier conversion.

      • +

    • Added the section on Bezier cuves as matrices.

      • +
    +

    April 2013

    +
      +

    • Added a section on poly-Beziers.

      • +

    • Added a section on boolean shape operations.

      • +
    +

    March 2013

    +
      +

    • First drastic rewrite.

      • +

    • Added sections on circle approximations.

      • +

    • Added a section on projecting a point onto a curve.

      • +

    • Added a section on tangents and normals.

      • +

    • Added Legendre-Gauss numerical data tables.

      • +
    +

    October 2011

    +
      +
    • +

      + First commit for the bezierinfo site, based on the pre-Primer webpage that covered + the basics of Bezier curves in HTML with Processing.js examples. +

      +
      • +
    +
    +
    +
    +
    +

    +
    змістнаступний
    + Короткий вступ +

    +

    + Давайте розпочнемо з добрих новин: криві Безьє, про які ми говоритимемо, ви зможете побачити далі на графіках. Ці криві розпочинаються у + якійсь певній точці, і закінчуються у якійсь певній точці. Їх кривизна залежить від однієї або кількох "проміжних" контрольних точок. + Зараз, оскільки всі графіки на цій сторінці інтерактивні, поекспериментуйте трохи з цими кривими. Клікніть на точку мишкою й потягніть - + так ви зможете відчути, як форма кривої змінюється в залежності від ваших дій. +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    -
    - -
    - -

    Preface

    -

    In order to draw things in 2D, we usually rely on lines, which typically get classified into two categories: straight lines, and curves. The first of these are as easy to draw as they are easy to make a computer draw. Give a computer the first and last point in the line, and BAM! straight line. No questions asked.

    -

    Curves, however, are a much bigger problem. While we can draw curves with ridiculous ease freehand, computers are a bit handicapped in that they can't draw curves unless there is a mathematical function that describes how it should be drawn. In fact, they even need this for straight lines, but the function is ridiculously easy, so we tend to ignore that as far as computers are concerned; all lines are "functions", regardless of whether they're straight or curves. However, that does mean that we need to come up with fast-to-compute functions that lead to nice looking curves on a computer. There are a number of these, and in this article we'll focus on a particular function that has received quite a bit of attention and is used in pretty much anything that can draw curves: Bézier curves.

    -

    They're named after Pierre Bézier, who is principally responsible for making them known to the world as a curve well-suited for design work (publishing his investigations in 1962 while working for Renault), although he was not the first, or only one, to "invent" these type of curves. One might be tempted to say that the mathematician Paul de Casteljau was first, as he began investigating the nature of these curves in 1959 while working at Citroën, and came up with a really elegant way of figuring out how to draw them. However, de Casteljau did not publish his work, making the question "who was first" hard to answer in any absolute sense. Or is it? Bézier curves are, at their core, "Bernstein polynomials", a family of mathematical functions investigated by Sergei Natanovich Bernstein, whose publications on them date back at least as far as 1912.

    -

    Anyway, that's mostly trivia, what you are more likely to care about is that these curves are handy: you can link up multiple Bézier curves so that the combination looks like a single curve. If you've ever drawn Photoshop "paths" or worked with vector drawing programs like Flash, Illustrator or Inkscape, those curves you've been drawing are Bézier curves.

    -

    But what if you need to program them yourself? What are the pitfalls? How do you draw them? What are the bounding boxes, how do you determine intersections, how can you extrude a curve, in short: how do you do everything that you might want to do with these curves? That's what this page is for. Prepare to be mathed!

    -
    - -

    Virtually all Bézier graphics are interactive.

    -

    This page uses interactive examples, relying heavily on Bezier.js, as well as maths formulae which are typeset into SVG using the XeLaTeX typesetting system and pdf2svg by David Barton.

    -

    This book is open source.

    -

    This book is an open source software project, and lives on two github repositories. The first is https://github.com/pomax/bezierinfo and is the purely-for-presentation version you are viewing right now. The other repository is https://github.com/pomax/BezierInfo-2, which is the development version, housing all the code that gets turned into the web version, and is also where you should file issues if you find bugs or have ideas on what to change or add to the primer.

    -

    How complicated is the maths going to be?

    -

    Most of the mathematics in this Primer are early high school maths. If you understand basic arithmetic, and you know how to read English, you should be able to get by just fine. There will at times be far more complicated maths, but if you don't feel like digesting them, you can safely skip over them by either skipping over the "detail boxes" in section or by just jumping to the end of a section with maths that looks too involving. The end of sections typically simply list the conclusions so you can just work with those values directly.

    -

    What language is all this example code in?

    -

    There are way too many programming languages to favour one of all others, soo all the example code in this Primer uses a form of pseudo-code that uses a syntax that's close enough to, but not actually, modern scripting languages like JS, Python, etc. That means you won't be able to copy-paste any of it without giving it any thought, but that's intentional: if you're reading this primer, presumably you want to learn, and you don't learn by copy-pasting. You learn by doing things yourself, making mistakes, and then fixing those mistakes. Now, of course, I didn't intentionally add errors in the example code just to trick you into making mistakes (that would be horrible!) but I did intentionally keep the code from favouring one programming language over another. Don't worry though, if you know even a single procedural programming language, you should be able to read the examples without any difficulties.

    -

    Questions, comments:

    -

    If you have suggestions for new sections, hit up the Github issue tracker (also reachable from the repo linked to in the upper right). If you have questions about the material, there's currently no comment section while I'm doing the rewrite, but you can use the issue tracker for that as well. Once the rewrite is done, I'll add a general comment section back in, and maybe a more topical "select this section of text and hit the 'question' button to ask a question about it" system. We'll see.

    -

    Help support the book!

    -

    If you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 100-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing!

    -
    -
    -
    -

    Що нового?

    -

    Цей підручник постійно розвививається, тож залежно від того, коли ви востаннє його переглядали, тут можуть бути оновлення. Перейдіть за цим посиланням, щоб побачити, що було додано. (Також доступний RSS-канал)

    - - - -

    November 2020

    • Added a section on finding curve/circle intersections

      -
    -

    October 2020

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      -
    -

    August-September 2020

    • Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS turned on.

      -
    -

    June 2020

    • Added automatic CI/CD using Github Actions

      -
    -

    January 2020

    • Added reset buttons to all graphics

      -
      • -

    • Updated to preface to correctly describe the on-page maths

      -
      • -

    • Fixed the Catmull-Rom section because it had glaring maths errors

      -
    -

    August 2019

    • Added a section on (plain) rational Bezier curves

      -
      • -

    • Improved the Graphic component to allow for sliders

      -
    -

    December 2018

    • Added a section on curvature and calculating kappa.

      -
      • -

    • Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this site!

      -
    -

    August 2018

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      -
    -

    July 2018

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      -
      • -

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      -
      • -

    • (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      -
    -

    June 2018

    • Added a section on direct curve fitting.

      -
      • -

    • Added source links for all graphics.

      -
      • -

    • Added this "What's new?" section.

      -
    -

    April 2017

    • Added a section on 3d normals.

      -
      • -

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      -
    -

    February 2017

    • Finished rewriting the entire codebase for localization.

      -
    -

    January 2016

    • Added a section to explain the Bezier interval.

      -
      • -

    • Rewrote the Primer as a React application.

      -
    -

    December 2015

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      -
      • -

    • Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics.

      -
    -

    May 2015

    • Switched over to pure JS rather than Processing-through-Processing.js

      -
      • -

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      -
    -

    April 2015

    • Added a section on arc length approximations.

      -
    -

    February 2015

    • Added a section on the canonical cubic Bezier form.

      -
    -

    November 2014

    • Switched to HTTPS.

      -
    -

    July 2014

    • Added the section on arc approximation.

      -
    -

    April 2014

    • Added the section on Catmull-Rom fitting.

      -
    -

    November 2013

    • Added the section on Catmull-Rom / Bezier conversion.

      -
      • -

    • Added the section on Bezier cuves as matrices.

      -
    -

    April 2013

    • Added a section on poly-Beziers.

      -
      • -

    • Added a section on boolean shape operations.

      -
    -

    March 2013

    • First drastic rewrite.

      -
      • -

    • Added sections on circle approximations.

      -
      • -

    • Added a section on projecting a point onto a curve.

      -
      • -

    • Added a section on tangents and normals.

      -
      • -

    • Added Legendre-Gauss numerical data tables.

      -
    -

    October 2011

    • First commit for the bezierinfo site, based on the pre-Primer webpage that covered the basics of Bezier curves in HTML with Processing.js examples.

      -
    -
    -
    -

    змістнаступний
    Короткий вступ

    -

    Давайте розпочнемо з добрих новин: криві Безьє, про які ми говоритимемо, ви зможете побачити далі на графіках. Ці криві розпочинаються у якійсь певній точці, і закінчуються у якійсь певній точці. Їх кривизна залежить від однієї або кількох "проміжних" контрольних точок. Зараз, оскільки всі графіки на цій сторінці інтерактивні, поекспериментуйте трохи з цими кривими. Клікніть на точку мишкою й потягніть - так ви зможете відчути, як форма кривої змінюється в залежності від ваших дій.

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    - -

    Ці криві інтенсивно використовуються у системах автоматизованого проектування та виробництва (CAD/CAM), а також у програмах для графічного дизайну, таких як Adobe Illustrator, Photoshop, Inkscape, GIMP, тощо. Також криві Безьє використовуються у графічних технологіях, таких як масштабована векторна графіка (SVG) та шрифти OpenType (TTF/OTF). Криві Безьє використовуються багато де, тому якщо хочете дізнатись про них більше, приготуйтесь трохи повчитися!

    - -
    -
    -

    попереднійзмістнаступний
    So what makes a Bézier Curve?

    -

    Playing with the points for curves may have given you a feel for how Bézier curves behave, but what are Bézier curves, really? There are two ways to explain what a Bézier curve is, and they turn out to be the entirely equivalent, but one of them uses complicated maths, and the other uses really simple maths. So... let's start with the simple explanation:

    -

    Bézier curves are the result of linear interpolations. That sounds complicated but you've been doing linear interpolation since you were very young: any time you had to point at something between two other things, you've been applying linear interpolation. It's simply "picking a point between two points".

    -

    If we know the distance between those two points, and we want a new point that is, say, 20% the distance away from the first point (and thus 80% the distance away from the second point) then we can compute that really easily:

    - - -

    So let's look at that in action: the following graphic is interactive in that you can use your up and down arrow keys to increase or decrease the interpolation ratio, to see what happens. We start with three points, which gives us two lines. Linear interpolation over those lines gives us two points, between which we can again perform linear interpolation, yielding a single point. And that point —and all points we can form in this way for all ratios taken together— form our Bézier curve:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    And that brings us to the complicated maths: calculus.

    -

    While it doesn't look like that's what we've just done, we actually just drew a quadratic curve, in steps, rather than in a single go. One of the fascinating parts about Bézier curves is that they can both be described in terms of polynomial functions, as well as in terms of very simple interpolations of interpolations of [...]. That, in turn, means we can look at what these curves can do based on both "real maths" (by examining the functions, their derivatives, and all that stuff), as well as by looking at the "mechanical" composition (which tells us, for instance, that a curve will never extend beyond the points we used to construct it).

    -

    So let's start looking at Bézier curves a bit more in depth: their mathematical expressions, the properties we can derive from them, and the various things we can do to, and with, Bézier curves.

    - -
    -
    -

    попереднійзмістнаступний
    The mathematics of Bézier curves

    -

    Bézier curves are a form of "parametric" function. Mathematically speaking, parametric functions are cheats: a "function" is actually a well defined term representing a mapping from any number of inputs to a single output. Numbers go in, a single number comes out. Change the numbers that go in, and the number that comes out is still a single number.

    -

    Parametric functions cheat. They basically say "alright, well, we want multiple values coming out, so we'll just use more than one function". An illustration: Let's say we have a function that maps some value, let's call it x, to some other value, using some kind of number manipulation:

    - - -

    The notation f(x) is the standard way to show that it's a function (by convention called f if we're only listing one) and its output changes based on one variable (in this case, x). Change x, and the output for f(x) changes.

    -

    So far, so good. Now, let's look at parametric functions, and how they cheat. Let's take the following two functions:

    - - -

    There's nothing really remarkable about them, they're just a sine and cosine function, but you'll notice the inputs have different names. If we change the value for a, we're not going to change the output value for f(b), since a isn't used in that function. Parametric functions cheat by changing that. In a parametric function all the different functions share a variable, like this:

    - - -

    Multiple functions, but only one variable. If we change the value for t, we change the outcome of both fa(t) and fb(t). You might wonder how that's useful, and the answer is actually pretty simple: if we change the labels fa(t) and fb(t) with what we usually mean with them for parametric curves, things might be a lot more obvious:

    - - -

    There we go. x/y coordinates, linked through some mystery value t.

    -

    So, parametric curves don't define a y coordinate in terms of an x coordinate, like normal functions do, but they instead link the values to a "control" variable. If we vary the value of t, then with every change we get two values, which we can use as (x,y) coordinates in a graph. The above set of functions, for instance, generates points on a circle: We can range t from negative to positive infinity, and the resulting (x,y) coordinates will always lie on a circle with radius 1 around the origin (0,0). If we plot it for t from 0 to 5, we get this:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Bézier curves are just one out of the many classes of parametric functions, and are characterised by using the same base function for all of the output values. In the example we saw above, the x and y values were generated by different functions (one uses a sine, the other a cosine); but Bézier curves use the "binomial polynomial" for both the x and y outputs. So what are binomial polynomials?

    -

    You may remember polynomials from high school. They're those sums that look like this:

    - - -

    If the highest order term they have is x³, they're called "cubic" polynomials; if it's x², it's a "square" polynomial; if it's just x, it's a line (and if there aren't even any terms with x it's not a polynomial!)

    -

    Bézier curves are polynomials of t, rather than x, with the value for t being fixed between 0 and 1, with coefficients a, b etc. taking the "binomial" form, which sounds fancy but is actually a pretty simple description for mixing values:

    - - -

    I know what you're thinking: that doesn't look too simple! But if we remove t and add in "times one", things suddenly look pretty easy. Check out these binomial terms:

    - - -

    Notice that 2 is the same as 1+1, and 3 is 2+1 and 1+2, and 6 is 3+3... As you can see, each time we go up a dimension, we simply start and end with 1, and everything in between is just "the two numbers above it, added together", giving us a simple number sequence known as Pascal's triangle. Now that's easy to remember.

    -

    There's an equally simple way to figure out how the polynomial terms work: if we rename (1-t) to a and t to b, and remove the weights for a moment, we get this:

    - - -

    It's basically just a sum of "every combination of a and b", progressively replacing a's with b's after every + sign. So that's actually pretty simple too. So now you know binomial polynomials, and just for completeness I'm going to show you the generic function for this:

    - - -

    And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ).

    -
    + +

    + And that's the full description for Bézier curves. Σ in this function indicates that this is a series of additions (using the variable + listed below the Σ, starting at ...=<value> and ending at the value listed on top of the Σ). +

    +
    +

    How to implement the basis function

    +

    We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:

    -

    How to implement the basis function

    -

    We could naively implement the basis function as a mathematical construct, using the function as our guide, like this:

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers in between being the sum of the previous row's elements on either side "above" the one we're computing.

    -

    We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup table:

    +

    + I say we could, because we're not going to: the factorial function is incredibly expensive. And, as we can see from the above + explanation, we can actually create Pascal's triangle quite easily without it: just start at [1], then [1,1], then [1,2,1], then + [1,3,3,1], and so on, with each next row fitting 1 more number than the previous row, starting and ending with "1", with all the numbers + in between being the sum of the previous row's elements on either side "above" the one we're computing. +

    +

    + We can generate this as a list of lists lightning fast, and then never have to compute the binomial terms because we have a lookup + table: +

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    + return lut[n][k] + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + -

    So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, we expand it first. Our basis function now looks like this:

    +

    + So what's going on here? First, we declare a lookup table with a size that's reasonably large enough to accommodate most lookups. Then, + we declare a function to get us the values we need, and we make sure that if an n/k pair is requested that isn't in the LUT yet, + we expand it first. Our basis function now looks like this: +

    - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves (although we will also provide code for arbitrary curves in this primer); we need quadratic and cubic curves, and that means we can drastically simplify the code:

    +

    + Perfect. Of course, we can optimize further. For most computer graphics purposes, we don't need arbitrary curves (although we will also + provide code for arbitrary curves in this primer); we need quadratic and cubic curves, and that means we can drastically simplify the + code: +

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return mt3 + 3*mt2*t + 3*mt*t2 + t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    And now we know how to program the basis function. Excellent.

    -
    +

    And now we know how to program the basis function. Excellent.

    +
    -

    So, now we know what the basis function looks like, time to add in the magic that makes Bézier curves so special: control points.

    +

    So, now we know what the basis function looks like, time to add in the magic that makes Bézier curves so special: control points.

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + Controlling Bézier curvatures +

    +

    + Bézier curves are, like all "splines", interpolation functions. This means that they take a set of points, and generate values somewhere + "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline + for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point + contributes to the value generated by the function, so we can see which points are important, where, in the curve. +

    +

    + The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's + contribution to the total sum of the Bézier function. Click-and-drag to see the interpolation percentages for each curve-defining point at + a specific t value. +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + + + Скрипти вимкнено. показує резервний. + + + + + + + + Скрипти вимкнено. показує резервний. + + + + + +
    -
    -
    -

    попереднійзмістнаступний
    Controlling Bézier curvatures

    -

    Bézier curves are, like all "splines", interpolation functions. This means that they take a set of points, and generate values somewhere "between" those points. (One of the consequences of this is that you'll never be able to generate a point that lies outside the outline for the control points, commonly called the "hull" for the curve. Useful information!). In fact, we can visualize how each point contributes to the value generated by the function, so we can see which points are important, where, in the curve.

    -

    The following graphs show the interpolation functions for quadratic and cubic curves, with "S" being the strength of a point's contribution to the total sum of the Bézier function. Click-and-drag to see the interpolation percentages for each curve-defining point at a specific t value.

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - -
    - -

    Also shown is the interpolation function for a 15th order Bézier function. As you can see, the start and end point contribute considerably more to the curve's shape than any other point in the control point set.

    -

    If we want to change the curve, we need to change the weights of each point, effectively changing the interpolations. The way to do this is about as straightforward as possible: just multiply each point with a value that changes its strength. These values are conventionally called "weights", and we can add them to our original Bézier function:

    - - -

    That looks complicated, but as it so happens, the "weights" are actually just the coordinate values we want our curve to have: for an nth order curve, w0 is our start coordinate, wn is our last coordinate, and everything in between is a controlling coordinate. Say we want a cubic curve that starts at (110,150), is controlled by (25,190) and (210,250) and ends at (210,30), we use this Bézier curve:

    - - -

    Which gives us the curve we saw at the top of the article:

    - - - Скрипти вимкнено. показує резервний. - - - -

    What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and algorithms that we can apply, and the tasks they achieve.

    -
    + +

    Which gives us the curve we saw at the top of the article:

    + + + Скрипти вимкнено. показує резервний. + + + +

    + What else can we do with Bézier curves? Quite a lot, actually. The rest of this article covers a multitude of possible operations and + algorithms that we can apply, and the tasks they achieve. +

    +
    +

    How to implement the weighted basis function

    +

    Given that we already know how to implement basis function, adding in the control points is remarkably easy:

    -

    How to implement the weighted basis function

    -

    Given that we already know how to implement basis function, adding in the control points is remarkably easy:

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    And now for the optimized versions:

    +

    And now for the optimized versions:

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    And now we know how to program the weighted basis function.

    -
    - -
    -
    -

    попереднійзмістнаступний
    Controlling Bézier curvatures, part 2: Rational Béziers

    -

    We can further control Bézier curves by "rationalising" them: that is, adding a "ratio" value in addition to the weight value discussed in the previous section, thereby gaining control over "how strongly" each coordinate influences the curve.

    -

    Adding these ratio values to the regular Bézier curve function is fairly easy. Where the regular function is the following:

    - - -

    The function for rational Bézier curves has two more terms:

    - - -

    In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just using different weight. So far, nothing too special.

    -

    However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights.

    -

    This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly describing circles (which we'll see in a later section is literally impossible using standard Bézier curves).

    -

    But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects what gets drawn:

    - - - Скрипти вимкнено. показує резервний. - - - - - - - - -

    You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence relative to all other points.

    -
    + +

    + In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] + then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just + using different weight. So far, nothing too special. +

    +

    + However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a + fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the + "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights. +

    +

    + This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve + that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly + describing circles (which we'll see in a later section is literally impossible using standard Bézier curves). +

    +

    + But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an + interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with + ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate + exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects + what gets drawn: +

    + + + Скрипти вимкнено. показує резервний. + + + + + + + + +

    + You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will + want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like + with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence + relative to all other points. +

    +
    +

    How to implement rational curves

    +

    Extending the code of the previous section to include ratios is almost trivial:

    -

    How to implement rational curves

    -

    Extending the code of the previous section to include ratios is almost trivial:

    - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    + return (f[0] * w[0] + f[1] * w[1] + f[2] * w[2] + f[3] * w[3])/basis + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + -

    And that's all we have to do.

    -
    - -
    -
    -

    попереднійзмістнаступний
    The Bézier interval [0,1]

    -

    Now that we know the mathematics behind Bézier curves, there's one curious thing that you may have noticed: they always run from t=0 to t=1. Why that particular interval?

    -

    It all has to do with how we run from "the start" of our curve to "the end" of our curve. If we have a value that is a mixture of two other values, then the general formula for this is:

    - - -

    The obvious start and end values here need to be a=1, b=0, so that the mixed value is 100% value 1, and 0% value 2, and a=0, b=1, so that the mixed value is 0% value 1 and 100% value 2. Additionally, we don't want "a" and "b" to be independent: if they are, then we could just pick whatever values we like, and end up with a mixed value that is, for example, 100% value 1 and 100% value 2. In principle that's fine, but for Bézier curves we always want mixed values between the start and end point, so we need to make sure we can never set "a" and "b" to some values that lead to a mix value that sums to more than 100%. And that's easy:

    - - -

    With this we can guarantee that we never sum above 100%. By restricting a to values in the interval [0,1], we will always be somewhere between our two values (inclusively), and we will always sum to a 100% mix.

    -

    But... what if we use this form, which is based on the assumption that we will only ever use values between 0 and 1, and instead use values outside of that interval? Do things go horribly wrong? Well... not really, but we get to "see more".

    -

    In the case of Bézier curves, extending the interval simply makes our curve "keep going". Bézier curves are simply segments of some polynomial curve, so if we pick a wider interval we simply get to see more of the curve. So what do they look like?

    -

    The following two graphics show you Bézier curves rendered "the usual way", as well as the curves they "lie on" if we were to extend the t values much further. As you can see, there's a lot more "shape" hidden in the rest of the curve, and we can model those parts by moving the curve points around.

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    + +

    + With this we can guarantee that we never sum above 100%. By restricting a to values in the interval [0,1], we will always be + somewhere between our two values (inclusively), and we will always sum to a 100% mix. +

    +

    + But... what if we use this form, which is based on the assumption that we will only ever use values between 0 and 1, and instead use + values outside of that interval? Do things go horribly wrong? Well... not really, but we get to "see more". +

    +

    + In the case of Bézier curves, extending the interval simply makes our curve "keep going". Bézier curves are simply segments of some + polynomial curve, so if we pick a wider interval we simply get to see more of the curve. So what do they look like? +

    +

    + The following two graphics show you Bézier curves rendered "the usual way", as well as the curves they "lie on" if we were to extend the + t values much further. As you can see, there's a lot more "shape" hidden in the rest of the curve, and we can model those + parts by moving the curve points around. +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    -

    In fact, there are curves used in graphics design and computer modelling that do the opposite of Bézier curves; rather than fixing the interval, and giving you freedom to choose the coordinates, they fix the coordinates, but give you freedom over the interval. A great example of this is the "Spiro" curve, which is a curve based on part of a Cornu Spiral, also known as Euler's Spiral. It's a very aesthetically pleasing curve and you'll find it in quite a few graphics packages like FontForge and Inkscape. It has even been used in font design, for example for the Inconsolata typeface.

    - -
    -
    -

    попереднійзмістнаступний
    Bézier curvatures as matrix operations

    -

    We can also represent Bézier curves as matrix operations, by expressing the Bézier formula as a polynomial basis function and a coefficients matrix, and the actual coordinates as a matrix. Let's look at what this means for the cubic curve, using P... to refer to coordinate values "in one or more dimensions":

    - - -

    Disregarding our actual coordinates for a moment, we have:

    - - -

    We can write this as a sum of four expressions:

    - - -

    And we can expand these expressions:

    - - -

    Furthermore, we can make all the 1 and 0 factors explicit:

    - - -

    And that, we can view as a series of four matrix operations:

    - - -

    If we compact this into a single matrix operation, we get:

    - - -

    This kind of polynomial basis representation is generally written with the bases in increasing order, which means we need to flip our t matrix horizontally, and our big "mixing" matrix upside down:

    - - -

    And then finally, we can add in our original coordinates as a single third matrix:

    - - -

    We can perform the same trick for the quadratic curve, in which case we end up with:

    - - -

    If we plug in a t value, and then multiply the matrices, we will get exactly the same values as when we evaluate the original polynomial function, or as when we evaluate the curve using progressive linear interpolation.

    -

    So: why would we bother with matrices? Matrix representations allow us to discover things about functions that would otherwise be hard to tell. It turns out that the curves form triangular matrices, and they have a determinant equal to the product of the actual coordinates we use for our curve. It's also invertible, which means there's a ton of properties that are all satisfied. Of course, the main question is "why is this useful to us now?", and the answer to that is that it's not immediately useful, but you'll be seeing some instances where certain curve properties can be either computed via function manipulation, or via clever use of matrices, and sometimes the matrix approach can be (drastically) faster.

    -

    So for now, just remember that we can represent curves this way, and let's move on.

    + +

    + If we plug in a t value, and then multiply the matrices, we will get exactly the same values as when we evaluate the original + polynomial function, or as when we evaluate the curve using progressive linear interpolation. +

    +

    + So: why would we bother with matrices? Matrix representations allow us to discover things about functions that would + otherwise be hard to tell. It turns out that the curves form + triangular matrices, and they have a determinant equal to the product of the + actual coordinates we use for our curve. It's also invertible, which means there's + a ton of properties that are all satisfied. Of + course, the main question is "why is this useful to us now?", and the answer to that is that it's not immediately useful, but + you'll be seeing some instances where certain curve properties can be either computed via function manipulation, or via clever use of + matrices, and sometimes the matrix approach can be (drastically) faster. +

    +

    So for now, just remember that we can represent curves this way, and let's move on.

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + de Casteljau's algorithm +

    +

    + If we want to draw Bézier curves, we can run through all values of t from 0 to 1 and then compute the weighted basis function + at each value, getting the x/y values we need to plot. Unfortunately, the more complex the curve gets, the more expensive + this computation becomes. Instead, we can use de Casteljau's algorithm to draw curves. This is a geometric approach to curve + drawing, and it's really easy to implement. So easy, in fact, you can do it by hand with a pencil and ruler. +

    +

    Rather than using our calculus function to find x/y values for t, let's do this instead:

    +
      +
    • treat t as a ratio (which it is). t=0 is 0% along a line, t=1 is 100% along a line.
      • +
    • Take all lines between the curve's defining points. For an order n curve, that's n lines.
      • +
    • + Place markers along each of these line, at distance t. So if t is 0.2, place the mark at 20% from the start, + 80% from the end. +
      • +
    • Now form lines between those points. This gives n-1 lines.
      • +
    • Place markers along each of these line at distance t.
      • +
    • Form lines between those points. This'll be n-2 lines.
      • +
    • Place markers, form lines, place markers, etc.
      • +
    • + Repeat this until you have only one line left. The point t on that line coincides with the original curve point at + t. +
      • +
    +

    + To see this in action, move the slider for the following sketch to changes which curve point is explicitly evaluated using de Casteljau's + algorithm. +

    + + + Скрипти вимкнено. показує резервний. + + + + + +
    +

    How to implement de Casteljau's algorithm

    +

    + Let's just use the algorithm we just specified, and implement that as a function that can take a list of curve-defining points, and a + t value, and draws the associated point on the curve for that t value: +

    -
    -
    -

    попереднійзмістнаступний
    de Casteljau's algorithm

    -

    If we want to draw Bézier curves, we can run through all values of t from 0 to 1 and then compute the weighted basis function at each value, getting the x/y values we need to plot. Unfortunately, the more complex the curve gets, the more expensive this computation becomes. Instead, we can use de Casteljau's algorithm to draw curves. This is a geometric approach to curve drawing, and it's really easy to implement. So easy, in fact, you can do it by hand with a pencil and ruler.

    -

    Rather than using our calculus function to find x/y values for t, let's do this instead:

    -
      -
    • treat t as a ratio (which it is). t=0 is 0% along a line, t=1 is 100% along a line.
      • -
    • Take all lines between the curve's defining points. For an order n curve, that's n lines.
      • -
    • Place markers along each of these line, at distance t. So if t is 0.2, place the mark at 20% from the start, 80% from the end.
      • -
    • Now form lines between those points. This gives n-1 lines.
      • -
    • Place markers along each of these line at distance t.
      • -
    • Form lines between those points. This'll be n-2 lines.
      • -
    • Place markers, form lines, place markers, etc.
      • -
    • Repeat this until you have only one line left. The point t on that line coincides with the original curve point at t.
      • -
    -

    To see this in action, move the slider for the following sketch to changes which curve point is explicitly evaluated using de Casteljau's algorithm.

    - - - Скрипти вимкнено. показує резервний. - - - - - -
    - -

    How to implement de Casteljau's algorithm

    -

    Let's just use the algorithm we just specified, and implement that as a function that can take a list of curve-defining points, and a t value, and draws the associated point on the curve for that t value:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    And done, that's the algorithm implemented. Although: usually you don't get the luxury of overloading the "+" operator, so let's also give the code for when you need to work with x and y values separately:

    +

    + And done, that's the algorithm implemented. Although: usually you don't get the luxury of overloading the "+" operator, so let's also + give the code for when you need to work with x and y values separately: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So what does this do? This draws a point, if the passed list of points is only 1 point long. Otherwise it will create a new list of points that sit at the t ratios (i.e. the "markers" outlined in the above algorithm), and then call the draw function for this new list.

    -
    +

    + So what does this do? This draws a point, if the passed list of points is only 1 point long. Otherwise it will create a new list of + points that sit at the t ratios (i.e. the "markers" outlined in the above algorithm), and then call the draw function for this + new list. +

    +
    + +
    +

    +
    + попереднійзмістнаступний +
    + Simplified drawing +

    +

    + We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, + a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines. +

    +

    + We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the + number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we + can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we + lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or + curvature alignment. +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + + + Скрипти вимкнено. показує резервний. + + + + + +
    +

    + Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. + You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the + cubic curve), a higher number is required to capture the curvature changes properly. +

    +
    +

    How to implement curve flattening

    +

    Let's just use the algorithm we just specified, and implement that:

    -
    -
    -

    попереднійзмістнаступний
    Simplified drawing

    -

    We can also simplify the drawing process by "sampling" the curve at certain points, and then joining those points up with straight lines, a process known as "flattening", as we are reducing a curve to a simple sequence of straight, "flat" lines.

    -

    We can do this is by saying "we want X segments", and then sampling the curve at intervals that are spaced such that we end up with the number of segments we wanted. The advantage of this method is that it's fast: instead of evaluating 100 or even 1000 curve coordinates, we can sample a much lower number and still end up with a curve that sort-of-kind-of looks good enough. The disadvantage of course is that we lose the precision of working with "the real curve", so we usually can't use the flattened for doing true intersection detection, or curvature alignment.

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - -
    - -

    Try clicking on the sketch and using your up and down arrow keys to lower the number of segments for both the quadratic and cubic curve. You'll notice that for certain curvatures, a low number of segments works quite well, but for more complex curvatures (try this for the cubic curve), a higher number is required to capture the curvature changes properly.

    -
    - -

    How to implement curve flattening

    -

    Let's just use the algorithm we just specified, and implement that:

    - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return coordinates; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

    +

    And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + coord = _coord + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

    -
    +

    We start with the first coordinate as reference point, and then just draw lines between each point and its next point.

    + +
    +
    +

    +
    + попереднійзмістнаступний +
    + Splitting curves +

    +

    + Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken + together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the + points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our + on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point. +

    + + + Скрипти вимкнено. показує резервний. + + + + + +
    +

    implementing curve splitting

    +

    We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

    -
    -
    -

    попереднійзмістнаступний
    Splitting curves

    -

    Using de Casteljau's algorithm, we can also find all the points we need to split up a Bézier curve into two, smaller curves, which taken together form the original curve. When we construct de Casteljau's skeleton for some value t, the procedure gives us all the points we need to split a curve at that t value: one curve is defined by all the inside skeleton points found prior to our on-curve point, with the other curve being defined by all the inside skeleton points after our on-curve point.

    - - - Скрипти вимкнено. показує резервний. - - - - - -
    - -

    implementing curve splitting

    -

    We can implement curve splitting by bolting some extra logging onto the de Casteljau function:

    - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + -

    After running this function for some value t, the left and right arrays will contain all the coordinates for two new curves - one to the "left" of our t value, the other on the "right". These new curves will have the same order as the original curve, and can be overlaid exactly on the original curve.

    -
    - -
    -
    -

    попереднійзмістнаступний
    Splitting curves using matrices

    -

    Another way to split curves is to exploit the matrix representation of a Bézier curve. In the section on matrices, we saw that we can represent curves as matrix multiplications. Specifically, we saw these two forms for the quadratic and cubic curves respectively: (we'll reverse the Bézier coefficients vector for legibility)

    - - -

    and

    - + +

    + Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the + coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual + "point on the curve" information into a new matrix multiplication: +

    + + +

    and

    + + +

    + If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two + staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, + from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear + algebra (and if you don't care about the derivations, just skip to the end of the box for the results!). +

    +
    +

    Deriving new hull coordinates

    +

    + Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look + at the quadratic curve first: +

    + + + - -

    Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual "point on the curve" information into a new matrix multiplication:

    - - -

    and

    - - -

    If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear algebra (and if you don't care about the derivations, just skip to the end of the box for the results!).

    -
    - -

    Deriving new hull coordinates

    -

    Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look at the quadratic curve first:

    - - - - - - - - -

    We can do this because [M · M-1] is the identity matrix. It's a bit like multiplying something by x/x in calculus: it doesn't do anything to the function, but it does allow you to rewrite it to something that may be easier to work with, or can be broken up differently. In the same way, multiplying our matrix by [M · M-1] has no effect on the total formula, but it does allow us to change the matrix sequence [something · M] to a sequence [M · something], and that makes a world of difference: if we know what [M-1 · Z · M] is, we can apply that to our coordinates, and be left with a proper matrix representation of a quadratic Bézier curve (which is [T · M · P]), with a new set of coordinates that represent the curve from t = 0 to t = z. So let's get computing:

    - - -

    Excellent! Now we can form our new quadratic curve:

    - - - - - - -

    Brilliant: if we want a subcurve from t = 0 to t = z, we can keep the first coordinate the same (which makes sense), our control point becomes a z-ratio mixture of the original control point and the start point, and the new end point is a mixture that looks oddly similar to a Bernstein polynomial of degree two. These new coordinates are actually really easy to compute directly!

    -

    Of course, that's only one of the two curves. Getting the section from t = z to t = 1 requires doing this again. We first observe that in the previous calculation, we actually evaluated the general interval [0,z]. We were able to write it down in a more simple form because of the zero, but what we actually evaluated, making the zero explicit, was:

    - - - - -

    If we want the interval [z,1], we will be evaluating this instead:

    - - - - -

    We're going to do the same trick of multiplying by the identity matrix, to turn [something · M] into [M · something]:

    - - -

    So, our final second curve looks like:

    - - - - - - -

    Nice. We see the same thing as before: we can keep the last coordinate the same (which makes sense); our control point becomes a z-ratio mixture of the original control point and the end point, and the new start point is a mixture that looks oddly similar to a bernstein polynomial of degree two, except this time it uses (z-1) rather than (1-z). These new coordinates are also really easy to compute directly!

    -
    - -

    So, using linear algebra rather than de Casteljau's algorithm, we have determined that, for any quadratic curve split at some value t = z, we get two subcurves that are described as Bézier curves with simple-to-derive coordinates:

    - - -

    and

    - - -

    We can do the same for cubic curves. However, I'll spare you the actual derivation (don't let that stop you from writing that out yourself, though) and simply show you the resulting new coordinate sets:

    - - -

    and

    - - -

    So, looking at our matrices, did we really need to compute the second segment matrix? No, we didn't. Actually having one segment's matrix means we implicitly have the other: push the values of each row in the matrix Q to the right, with zeroes getting pushed off the right edge and appearing back on the left, and then flip the matrix vertically. Presto, you just "calculated" Q'.

    -

    Implementing curve splitting this way requires less recursion, and is just straight arithmetic with cached values, so can be cheaper on systems where recursion is expensive. If you're doing computation with devices that are good at matrix multiplication, chopping up a Bézier curve with this method will be a lot faster than applying de Casteljau.

    - -
    -
    -

    попереднійзмістнаступний
    Lowering and elevating curve order

    -

    One interesting property of Bézier curves is that an nth order curve can always be perfectly represented by an (n+1)th order curve, by giving the higher-order curve specific control points.

    -

    If we have a curve with three points, then we can create a curve with four points that exactly reproduces the original curve. First, we give it the same start and end points, and for its two control points we pick "1/3rd start + 2/3rd control" and "2/3rd control + 1/3rd end". Now we have exactly the same curve as before, except represented as a cubic curve rather than a quadratic curve.

    -

    The general rule for raising an nth order curve to an (n+1)th order curve is as follows (observing that the start and end weights are the same as the start and end weights for the old curve):

    - - -

    However, this rule also has as direct consequence that you cannot generally safely lower a curve from nth order to (n-1)th order, because the control points cannot be "pulled apart" cleanly. We can try to, but the resulting curve will not be identical to the original, and may in fact look completely different.

    -

    However, there is a surprisingly good way to ensure that a lower order curve looks "as close as reasonably possible" to the original curve: we can optimise the "least-squares distance" between the original curve and the lower order curve, in a single operation (also explained over on Sirver's Castle). However, to use it, we'll need to do some calculus work and then switch over to linear algebra. As mentioned in the section on matrix representations, some things can be done much more easily with matrices than with calculus functions, and this is one of those things. So... let's go!

    -

    We start by taking the standard Bézier function, and condensing it a little:

    - - -

    Then, we apply one of those silly (actually, super useful) calculus tricks: since our t value is always between zero and one (inclusive), we know that (1-t) plus t always sums to 1. As such, we can express any value as a sum of t and 1-t:

    - - -

    So, with that seemingly trivial observation, we rewrite that Bézier function by splitting it up into a sum of a (1-t) and t component:

    - - -

    So far so good. Now, to see why we did this, let's write out the (1-t) and t parts, and see what that gives us. I promise, it's about to make sense. We start with (1-t):

    - - -

    So by using this seemingly silly trick, we can suddenly express part of our nth order Bézier function in terms of an (n+1)th order Bézier function. And that sounds a lot like raising the curve order! Of course we need to be able to repeat that trick for the t part, but that's not a problem:

    - - -

    So, with both of those changed from an order n expression to an order (n+1) expression, we can put them back together again. Now, where the order n function had a summation from 0 to n, the order n+1 function uses a summation from 0 to n+1, but this shouldn't be a problem as long as we can add some new terms that "contribute nothing". In the next section on derivatives, there is a discussion about why "higher terms than there is a binomial for" and "lower than zero terms" both "contribute nothing". So as long as we can add terms that have the same form as the terms we need, we can just include them in the summation, they'll sit there and do nothing, and the resulting function stays identical to the lower order curve.

    -

    Let's do this:

    - - -

    And this is where we switch over from calculus to linear algebra, and matrices: we can now express this relation between Bézier(n,t) and Bézier(n+1,t) as a very simple matrix multiplication:

    - - -

    where the matrix M is an n+1 by n matrix, and looks like:

    - - -

    That might look unwieldy, but it's really just a mostly-zeroes matrix, with a very simply fraction on the diagonal, and an even simpler fraction to the left of it. Multiplying a list of coordinates with this matrix means we can plug the resulting transformed coordinates into the one-order-higher function and get an identical looking curve.

    -

    Not too bad!

    -

    Equally interesting, though, is that with this matrix operation established, we can now use an incredibly powerful and ridiculously simple way to find out a "best fit" way to reverse the operation, called the normal equation. What it does is minimize the sum of the square differences between one set of values and another set of values. Specifically, if we can express that as some function A x = b, we can use it. And as it so happens, that's exactly what we're dealing with, so:

    - - -

    The steps taken here are:

    -
      -
    1. We have a function in a form that the normal equation can be used with, so
      2. -
    2. apply the normal equation!
      3. -
    3. Then, we want to end up with just Bn on the left, so we start by left-multiply both sides such that we'll end up with lots of stuff on the left that simplified to "a factor 1", which in matrix maths is the identity matrix.
      4. -
    4. In fact, by left-multiplying with the inverse of what was already there, we've effectively "nullified" (but really, one-inified) that big, unwieldy block into the identity matrix I. So we substitute the mess with I, and then
      5. -
    5. because multiplication with the identity matrix does nothing (like multiplying by 1 does nothing in regular algebra), we just drop it.
      6. -
    -

    And we're done: we now have an expression that lets us approximate an n+1th order curve with a lower nth order curve. It won't be an exact fit, but it's definitely a best approximation. So, let's implement these rules for raising and lowering curve order to a (semi) random curve, using the following graphic. Select the sketch, which has movable control points, and press your up and down arrow keys to raise or lower the curve order.

    - - - Скрипти вимкнено. показує резервний. - - - - - - - -
    -
    -

    попереднійзмістнаступний
    Derivatives

    -

    There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straightforward, although we do need a bit of math.

    -

    First, let's look at the derivative rule for Bézier curves, which is:

    - - -

    which we can also write (observing that b in this formula is the same as our w weights, and that n times a summation is the same as a summation where each term is multiplied by n) as:

    - - -

    Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi). So for a 3rd degree curve, with four weights, the derivative has three new weights: w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2).

    -
    - -

    "Slow down, why is that true?"

    -

    Sometimes just being told "this is the derivative" is nice, but you might want to see why this is indeed the case. As such, let's have a look at the proof for this derivative. First off, the weights are independent of the full Bézier function, so the derivative involves only the derivative of the polynomial basis function. So, let's find that:

    - - -

    Applying the product and chain rules gives us:

    - - -

    Which is hard to work with, so let's expand that properly:

    - - -

    Now, the trick is to turn this expression into something that has binomial coefficients again, so we want to end up with things that look like "x! over y!(x-y)!". If we can do that in a way that involves terms of n-1 and k-1, we'll be on the right track.

    - - -

    And that's the first part done: the two components inside the parentheses are actually regular, lower-order Bézier expressions:

    - - -

    Now to apply this to our weighted Bézier curves. We'll write out the plain curve formula that we saw earlier, and then work our way through to its derivative:

    - - -

    If we expand this (with some color to show how terms line up), and reorder the terms by increasing values for k we see the following:

    - - -

    Two of these terms fall way: the first term falls away because there is no -1st term in a summation. As such, it always contributes "nothing", so we can safely completely ignore it for the purpose of finding the derivative function. The other term is the very last term in this expansion: one involving Bn-1,n. This term would have a binomial coefficient of [i choose i+1], which is a non-existent binomial coefficient. Again, this term would contribute "nothing", so we can ignore it, too. This means we're left with:

    - - -

    And that's just a summation of lower order curves:

    - - -

    We can rewrite this as a normal summation, and we're done:

    - - -
    - -

    Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for Bézier curves, and then the derivative:

    - + +

    And that's just a summation of lower order curves:

    + + +

    We can rewrite this as a normal summation, and we're done:

    + + + +

    + Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for + Bézier curves, and then the derivative: +

    + - - - -

    What are the differences? In terms of the actual Bézier curve, virtually nothing! We lowered the order (rather than n, it's now n-1), but it's still the same Bézier function. The only real difference is in how the weights change when we derive the curve's function. If we have four points A, B, C, and D, then the derivative will have three points, the second derivative two, and the third derivative one:

    - - -

    We can keep performing this trick for as long as we have more than one weight. Once we have one weight left, the next step will see k = 0, and the result of our "Bézier function" summation is zero, because we're not adding anything at all. As such, a quadratic curve has no second derivative, a cubic curve has no third derivative, and generalized: an nth order curve has n-1 (meaningful) derivatives, with any further derivative being zero.

    - -
    -
    -

    попереднійзмістнаступний
    Tangents and normals

    -

    If you want to move objects along a curve, or "away from" a curve, the two vectors you're most interested in are the tangent vector and normal vector for curve points. These are actually really easy to find. For moving and orienting along a curve, we use the tangent, which indicates the direction of travel at specific points, and is literally just the first derivative of our curve:

    - - -

    This gives us the directional vector we want. We can normalize it to give us uniform directional vectors (having a length of 1.0) at each point, and then do whatever it is we want to do based on those directions:

    - - -

    The tangent is very useful for moving along a line, but what if we want to move away from the curve instead, perpendicular to the curve at some point t? In that case we want the normal vector. This vector runs at a right angle to the direction of the curve, and is typically of length 1.0, so all we have to do is rotate the normalized directional vector and we're done:

    - - -
    - -

    Rotating coordinates is actually very easy, if you know the rule for it. You might find it explained as "applying a rotation matrix, which is what we'll look at here, too. Essentially, the idea is to take the circles over which we can rotate, and simply "sliding the coordinates" over these circles by the desired -angle. If we want a quarter circle turn, we take the coordinate, slide it along the circle by a quarter turn, and done.

    -

    To turn any point (x,y) into a rotated point (x',y') (over 0,0) by some angle φ, we apply this nice and easy computation:

    - - -

    Which is the "long" version of the following matrix transformation:

    - - -

    And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1.

    -

    But why does this work? Why this matrix multiplication? Wikipedia (technically, Thomas Herter and Klaus Lott) tells us that a rotation matrix can be -treated as a sequence of three (elementary) shear operations. When we combine this into a single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. DataGenetics have an excellent article about this very thing: it's really quite cool, and I strongly recommend taking a quick break from this primer to read that article.

    -
    + +

    + And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even + easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1. +

    +

    + But why does this work? Why this matrix multiplication? + Wikipedia (technically, Thomas Herter and Klaus + Lott) tells us that a rotation matrix can be treated as a sequence of three (elementary) shear operations. When we combine this into a + single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. + DataGenetics have an excellent article about this very thing: it's + really quite cool, and I strongly recommend taking a quick break from this primer to read that article. +

    + -

    The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant).

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    +

    + The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the + normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant). +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    +
    +
    +

    +
    + попереднійзмістнаступний +
    + Working with 3D normals +

    +

    + Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things + this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but + for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, + it's not "super hard", but there are more steps involved and we should have a look at those. +

    +

    + Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. + However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, + not a single vector, so we basically need to define what "the" normal is in the 3D case. +

    +

    + The "naïve" approach is to construct what is known as the + Frenet normal, where we follow a simple recipe that works in + many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are + perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each + point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the + local tangent, as well as the tangents "right next to it". +

    +

    + Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the + vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know + lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same + logic we used for the 2D case: "just rotate it 90 degrees". +

    +

    So let's do that! And in a twist surprise, we can do this in four lines:

    +
      +
    • a = normalize(B'(t))
      • +
    • b = normalize(a + B''(t))
      • +
    • r = normalize(b × a)
      • +
    • normal = normalize(r × a)
      • +
    +

    Let's unpack that a little:

    +
      +
    • + We start by taking the normalized vector for the derivative at some point on the + curve. We normalize it so the maths is less work. Less work is good. +
      • +
    • + Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and + just had the same derivative and second derivative from that point on. +
      • +
    • + This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which + means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the + cross product. (Note that while that operation uses the × operator, it's most + definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent + a quarter circle to get our normal, just like we did in the 2D case. +
      • +
    • + Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal + vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second + time, and immediately get our normal vector. +
      • +
    +

    + And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can + move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor + position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc: +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    + However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the + curve" between t=0.65 and t=0.75... Why is it doing that? +

    +

    + As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are + "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute + normals that just... look good. +

    +

    Thankfully, Frenet normals are not our only option.

    +

    + Another option is to take a slightly more algorithmic approach and compute a form of + Rotation Minimising Frame + (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational + axis, and the normal vector, centered on an on-curve point. +

    +

    + These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we + could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at + the same time that you're building lookup tables for your curve. +

    +

    + The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by + applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are: +

    +
      +
    • Take a point on the curve for which we know the RM frame already,
      • +
    • take a next point on the curve for which we don't know the RM frame yet, and
      • +
    • + reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous + points as a "mirror". +
      • +
    • + This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal + that's slightly off-kilter, so: +
      • +
    • + reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror". +
      • +
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • +
    +

    So, let's write some code for that!

    +
    +

    Implementing Rotation Minimising Frames

    +

    + We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it + yields a frame with properties: +

    -
    -
    -

    попереднійзмістнаступний
    Working with 3D normals

    -

    Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, it's not "super hard", but there are more steps involved and we should have a look at those.

    -

    Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, not a single vector, so we basically need to define what "the" normal is in the 3D case.

    -

    The "naïve" approach is to construct what is known as the Frenet normal, where we follow a simple recipe that works in many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the local tangent, as well as the tangents "right next to it".

    -

    Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same logic we used for the 2D case: "just rotate it 90 degrees".

    -

    So let's do that! And in a twist surprise, we can do this in four lines:

    -
      -
    • a = normalize(B'(t))
      • -
    • b = normalize(a + B''(t))
      • -
    • r = normalize(b × a)
      • -
    • normal = normalize(r × a)
      • -
    -

    Let's unpack that a little:

    -
      -
    • We start by taking the normalized vector for the derivative at some point on the curve. We normalize it so the maths is less work. Less work is good.
      • -
    • Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on.
      • -
    • This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the cross product. (Note that while that operation uses the × operator, it's most definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent a quarter circle to get our normal, just like we did in the 2D case.
      • -
    • Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second time, and immediately get our normal vector.
      • -
    -

    And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the curve" between t=0.65 and t=0.75... Why is it doing that?

    -

    As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute normals that just... look good.

    -

    Thankfully, Frenet normals are not our only option.

    -

    Another option is to take a slightly more algorithmic approach and compute a form of Rotation Minimising Frame (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational axis, and the normal vector, centered on an on-curve point.

    -

    These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at the same time that you're building lookup tables for your curve.

    -

    The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are:

    -
      -
    • Take a point on the curve for which we know the RM frame already,
      • -
    • take a next point on the curve for which we don't know the RM frame yet, and
      • -
    • reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror".
      • -
    • This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter, so:
      • -
    • reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
      • -
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • -
    -

    So, let's write some code for that!

    -
    - -

    Implementing Rotation Minimising Frames

    -

    We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it yields a frame with properties:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + -

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    +

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    + frames.add(x1) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + -

    Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more code to get much better looking normals.

    -
    +

    + Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more + code to get much better looking normals. +

    + -

    Speaking of better looking, what does this actually look like? Let's revisit that earlier curve, but this time use rotation minimising frames rather than Frenet frames:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    That looks so much better!

    -

    For those reading along with the code: we don't even strictly speaking need a Frenet frame to start with: we could, for instance, treat the z-axis as our initial axis of rotation, so that our initial normal is (0,0,1) × tangent, and then take things from there, but having that initial "mathematically correct" frame so that the initial normal seems to line up based on the curve's orientation in 3D space is just nice.

    - -
    -
    -

    попереднійзмістнаступний
    Component functions

    -

    One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that function, but this poses a problem, since our function is parametric: every axis has its own function.

    -

    The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.

    -

    Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis value, respectively.

    -

    If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a change in the right graph.

    - - - Скрипти вимкнено. показує резервний. - - - -

     

    - - - Скрипти вимкнено. показує резервний. - - - - -
    -
    -

    попереднійзмістнаступний
    Finding extremities: root finding

    -

    Now that we understand (well, superficially anyway) the component functions, we can find the extremities of our Bézier curve by finding maxima and minima on the component functions, by solving the equation B'(t) = 0. We've already seen that the derivative of a Bézier curve is a simpler Bézier curve, but how do we solve the equality? Fairly easily, actually, until our derivatives are 4th order or higher... then things get really hard. But let's start simple:

    -

    Quadratic curves: linear derivatives.

    -

    The derivative of a quadratic Bézier curve is a linear Bézier curve, interpolating between just two terms, which means finding the solution for "where is this line 0" is effectively trivial by rewriting it to a function of t and solving. First we turn our quadratic Bézier function into a linear one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then we turn this into our solution for t using basic arithmetics:

    - - -

    Done.

    -

    Although with the caveat that if b-a is zero, there is no solution and we probably shouldn't try to perform that division.

    -

    Cubic curves: the quadratic formula.

    -

    The derivative of a cubic Bézier curve is a quadratic Bézier curve, and finding the roots for a quadratic polynomial means we can apply the Quadratic formula. If you've seen it before, you'll remember it, and if you haven't, it looks like this:

    - - -

    So, if we can rewrite the Bézier component function as a plain polynomial, we're done: we just plug in the values into the quadratic formula, check if that square root is negative or not (if it is, there are no roots) and then just compute the two values that come out (because of that plus/minus sign we get two). Any value between 0 and 1 is a root that matters for Bézier curves, anything below or above that is irrelevant (because Bézier curves are only defined over the interval [0,1]). So, how do we convert?

    -

    First we turn our cubic Bézier function into a quadratic one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then, using these v values, we can find out what our a, b, and c should be:

    - - -

    This gives us three coefficients {a, b, c} that are expressed in terms of v values, where the v values are expressions of our original coordinate values, so we can do some substitution to get:

    - - -

    Easy-peasy. We can now almost trivially find the roots by plugging those values into the quadratic formula.

    -

    And as a cubic curve, there is also a meaningful second derivative, which we can compute by simple taking the derivative of the derivative.

    -

    Quartic curves: Cardano's algorithm.

    -

    We haven't really looked at them before now, but the next step up would be a Quartic curve, a fourth degree Bézier curve. As expected, these have a derivative that is a cubic function, and now things get much harder. Cubic functions don't have a "simple" rule to find their roots, like the quadratic formula, and instead require quite a bit of rewriting to a form that we can even start to try to solve.

    -

    Back in the 16th century, before Bézier curves were a thing, and even before calculus itself was a thing, Gerolamo Cardano figured out that even if the general cubic function is really hard to solve, it can be rewritten to a form for which finding the roots is "easier" (even if not "easy"):

    - - -

    We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather than three: this makes things considerably easier to solve because it lets us use regular calculus to find the values that satisfy the equation.

    -

    Now, there is one small hitch: as a cubic function, the solutions may be complex numbers rather than plain numbers... And Cardano realised this, centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number.

    -

    So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at Ken J. Ward's page for solving the cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a read-through.

    -
    + +

    + We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather + than three: this makes things considerably easier to solve because it lets us use + regular calculus to find the values that satisfy the equation. +

    +

    + Now, there is one small hitch: as a cubic function, the solutions may be + complex numbers rather than plain numbers... And Cardano realised this, + centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these + numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we + have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're + going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number. +

    +

    + So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at + Ken J. Ward's page for solving the + cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the + complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a + read-through. +

    +
    +

    Implementing Cardano's algorithm for finding all real roots

    +

    + The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" + number space (denoted with ℝ), this is perfectly fine in the + "complex" number space (denoted with ℂ). And, as it so happens, Cardano is + also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm! +

    -

    Implementing Cardano's algorithm for finding all real roots

    -

    The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" number space (denoted with ℝ), this is perfectly fine in the "complex" number space (denoted with ℂ). And, as it so happens, Cardano is also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm!

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    60
    61
    62
    63
    64
    65
    66
    67
    68
    69
    70
    71
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + + 37 + + + 38 + + + 39 + + + 40 + + + 41 + + + 42 + + + 43 + + + 44 + + + 45 + + + 46 + + + 47 + + + 48 + + + 49 + + + 50 + + + 51 + + + 52 + + + 53 + + + 54 + + + 55 + + + 56 + + + 57 + + + 58 + + + 59 + + + 60 + + + 61 + + + 62 + + + 63 + + + 64 + + + 65 + + + 66 + + + 67 + + + 68 + + + 69 + + + 70 + + + 71 + + + 72 + + + 73 + + + 74 + + + 75 + + + 76 + + + 77 + + + 78 + + + 79 + + + 80 + + + 81 + + +
    -
    - -

    And that's it. The maths is complicated, but the code is pretty much just "follow the maths, while caching as many values as we can to prevent recomputing things as much as possible" and now we have a way to find all roots for a cubic function and can just move on with using that to find extremities of our curves.

    -

    And of course, as a quartic curve also has meaningful second and third derivatives, we can quite easily compute those by using the derivative of the derivative (of the derivative), just as for cubic curves.

    -

    Quintic and higher order curves: finding numerical solutions

    -

    And this is where thing stop, because we cannot find the roots for polynomials of degree 5 or higher using algebra (a fact known as the Abel–Ruffini theorem). Instead, for occasions like these, where algebra simply cannot yield an answer, we turn to numerical analysis.

    -

    That's a fancy term for saying "rather than trying to find exact answers by manipulating symbols, find approximate answers by describing the underlying process as a combination of steps, each of which can be assigned a number via symbolic manipulation". For example, trying to mathematically compute how much water fits in a completely crazy three dimensional shape is very hard, even if it got you the perfect, precise answer. A much easier approach, which would be less perfect but still entirely useful, would be to just grab a buck and start filling the shape until it was full: just count the number of buckets of water you used. And if we want a more precise answer, we can use smaller buckets.

    -

    So that's what we're going to do here, too: we're going to treat the problem as a sequence of steps, and the smaller we can make each step, the closer we'll get to that "perfect, precise" answer. And as it turns out, there is a really nice numerical root-finding algorithm, called the Newton-Raphson root finding method (yes, after that Newton), which we can make use of. The Newton-Raphson approach consists of taking our impossible-to-solve function f(x), picking some initial value x (literally any value will do), and calculating f(x). We can think of that value as the "height" of the function at x. If that height is zero, we're done, we have found a root. If it isn't, we calculate the tangent line at f(x) and calculate at which x value its height is zero (which we've already seen is very easy). That will give us a new x and we repeat the process until we find a root.

    -

    Mathematically, this means that for some x, at step n=1, we perform the following calculation until fy(x) is zero, so that the next t is the same as the one we already have:

    - - -

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    -

    Now, this works well only if we can pick good starting points, and our curve is continuously differentiable and doesn't have oscillations. Glossing over the exact meaning of those terms, the curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: which starting points do we pick?

    -

    As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as high as you'll ever need to go.

    -

    In conclusion:

    -

    So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red:

    - - - Скрипти вимкнено. показує резервний. - - - -

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    - - - Скрипти вимкнено. показує резервний. - - - + +

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    +

    + Now, this works well only if we can pick good starting points, and our curve is + continuously differentiable and doesn't have + oscillations. Glossing over the exact meaning of those terms, the + curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: + which starting points do we pick? +

    +

    + As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from + t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may + pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: + there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order + curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as + high as you'll ever need to go. +

    +

    In conclusion:

    +

    + So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those + roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red: +

    + + + Скрипти вимкнено. показує резервний. + + +

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    + + + Скрипти вимкнено. показує резервний. + + +
    +
    +

    +
    + попереднійзмістнаступний +
    + Bounding boxes +

    +

    + If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four + values we need to box in our curve: +

    +

    Computing the bounding box for a Bézier curve:

    +
      +
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. +
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. +
    3. + Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original + functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct. +
      4. +
    +

    + Applying this approach to our previous root finding, we get the following + axis-aligned bounding boxes (with all curve extremity points + shown on the curve): +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    -
    -
    -

    попереднійзмістнаступний
    Bounding boxes

    -

    If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four values we need to box in our curve:

    -

    Computing the bounding box for a Bézier curve:

    -
      -
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. -
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. -
    3. Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct.
      4. -
    -

    Applying this approach to our previous root finding, we get the following axis-aligned bounding boxes (with all curve extremity points shown on the curve):

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    - -

    We can construct even nicer boxes by aligning them along our curve, rather than along the x- and y-axis, but in order to do so we first need to look at how aligning works.

    - -
    -
    -

    попереднійзмістнаступний
    Aligning curves

    -

    While there are an incredible number of curves we can define by varying the x- and y-coordinates for the control points, not all curves are actually distinct. For instance, if we define a curve, and then rotate it 90 degrees, it's still the same curve, and we'll find its extremities in the same spots, just at different draw coordinates. As such, one way to make sure we're working with a "unique" curve is to "axis-align" it.

    -

    Aligning also simplifies a curve's functions. We can translate (move) the curve so that the first point lies on (0,0), which turns our n term polynomial functions into n-1 term functions. The order stays the same, but we have less terms. Then, we can rotate the curves so that the last point always lies on the x-axis, too, making its coordinate (...,0). This further simplifies the function for the y-component to an n-2 term function. For instance, if we have a cubic curve such as this:

    - - -

    Then translating it so that the first coordinate lies on (0,0), moving all x coordinates by -120, and all y coordinates by -160, gives us:

    - - -

    If we then rotate the curve so that its end point lies on the x-axis, the coordinates (integer-rounded for illustrative purposes here) become:

    - - -

    If we drop all the zero-terms, this gives us:

    - - -

    We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae:

    - - - Скрипти вимкнено. показує резервний. - - - -

     

    - - - Скрипти вимкнено. показує резервний. - - - + +

    + We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning + our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae: +

    + + + Скрипти вимкнено. показує резервний. + + +

     

    + + + Скрипти вимкнено. показує резервний. + + +
    +
    +

    +
    + попереднійзмістнаступний +
    + Tight bounding boxes +

    +

    + With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, + recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding + box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T. +

    +

    We now have nice tight bounding boxes for our curves:

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    -
    -
    -

    попереднійзмістнаступний
    Tight bounding boxes

    -

    With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T.

    -

    We now have nice tight bounding boxes for our curves:

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    - -

    These are, strictly speaking, not necessarily the tightest possible bounding boxes. It is possible to compute the optimal bounding box by determining which spanning lines we need to effect a minimal box area, but because of the parametric nature of Bézier curves this is actually a rather costly operation, and the gain in bounding precision is often not worth it.

    - -
    -
    -

    попереднійзмістнаступний
    Curve inflections

    -

    Now that we know how to align a curve, there's one more thing we can calculate: inflection points. Imagine we have a variable size circle that we can slide up against our curve. We place it against the curve and adjust its radius so that where it touches the curve, the curvatures of the curve and the circle are the same, and then we start to slide the circle along the curve - for quadratic curves, we can always do this without the circle behaving oddly: we might have to change the radius of the circle as we slide it along, but it'll always sit against the same side of the curve.

    -

    But what happens with cubic curves? Imagine we have an S curve and we place our circle at the start of the curve, and start sliding it along. For a while we can simply adjust the radius and things will be fine, but once we get to the midpoint of that S, something odd happens: the circle "flips" from one side of the curve to the other side, in order for the curvatures to keep matching. This is called an inflection, and we can find out where those happen relatively easily.

    -

    What we need to do is solve a simple equation:

    - - -

    What we're saying here is that given the curvature function C(t), we want to know for which values of t this function is zero, meaning there is no "curvature", which will be exactly at the point between our circle being on one side of the curve, and our circle being on the other side of the curve. So what does C(t) look like? Actually something that seems not too hard:

    - - -

    The function C(t) is the cross product between the first and second derivative functions for the parametric dimensions of our curve. And, as already shown, derivatives of Bézier curves are just simpler Bézier curves, with very easy to compute new coefficients, so this should be pretty easy.

    -

    However as we've seen in the section on aligning, aligning lets us simplify things a lot, by completely removing the contributions of the first coordinate from most mathematical evaluations, and removing the last y coordinate as well by virtue of the last point lying on the x-axis. So, while we can evaluate C(t) = 0 for our curve, it'll be much easier to first axis-align the curve and then evaluating the curvature function.

    -
    - -

    Let's derive the full formula anyway

    -

    Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:

    - - -

    And of course the same functions for y:

    - - -

    Asking a computer to now compose the C(t) function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:

    - - -

    That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all disappear if we axis-align our curve, which is why aligning is a great idea.

    -
    + +

    + That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all + disappear if we axis-align our curve, which is why aligning is a great idea. +

    + -

    Aligning our curve so that three of the eight coefficients become zero, and observing that scale does not affect finding t values, we end up with the following simple term function for C(t):

    - - -

    That's a lot easier to work with: we see a fair number of terms that we can compute and then cache, giving us the following simplification:

    - - -

    This is a plain quadratic curve, and we know how to solve C(t) = 0; we use the quadratic formula:

    - - -

    We can easily compute this value if the discriminator isn't a negative number (because we only want real roots, not complex roots), and if x is not zero, because divisions by zero are rather useless.

    -

    Taking that into account, we compute t, we disregard any t value that isn't in the Bézier interval [0,1], and we now know at which t value(s) our curve will inflect.

    - - - Скрипти вимкнено. показує резервний. - - - - -
    -
    -

    попереднійзмістнаступний
    The canonical form (for cubic curves)

    -

    While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?

    -

    As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper "A Geometric Characterization of Parametric Cubic curves". It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?

    -

    The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown:

    - - - Скрипти вимкнено. показує резервний. - - - -

    This is a fairly funky image, so let's see what the various parts of it mean...

    -

    We see the three fixed points at (0,0), (0,1) and (1,1). The various regions and boundaries indicate what property the original curve will have, if the fourth point is in/on that region or boundary. Specifically, if the fourth point is...

    -
      -

    1. ...anywhere inside the red zone, but not on its boundaries, the curve will be self-intersecting (yielding a loop). We won't know where it self-intersects (in terms of t values), but we are guaranteed that it does.

      -
      2. -

    2. ...on the left (red) edge of the red zone, the curve will have a cusp. We again don't know where, but we know there is one. This edge is described by the function:

      - - -
      3. -

    3. ...on the almost circular, lower right (pink) edge, the curve's end point touches the curve, forming a loop. This edge is described by the function:

      - - -
      4. -

    4. ...on the top (blue) edge, the curve's start point touches the curve, forming a loop. This edge is described by the function:

      - - -
      5. -

    5. ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

      -
      6. -

    6. ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two inflections (switching from concave to convex and then back again, or from convex to concave and then back again).

      -
      7. -

    7. ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

      -
      8. -
    -

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    -
    + + +
  • +

    ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

    +
    • +
  • +

    + ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two + inflections (switching from concave to convex and then back again, or from convex to concave and then back again). +

    +
    • +

  • ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

    • + +

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    +
    +

    Wait, where do those lines come from?

    +

    + Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, + and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield + formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic + expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the + properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval. +

    +

    + The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general + cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to + it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general + curve does over the interval t=0 to t=1. +

    +

    + For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you + can probably follow along with this paper, although you might have to read it a few times before all the bits "click". +

    +
    -

    Wait, where do those lines come from?

    -

    Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval.

    -

    The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general curve does over the interval t=0 to t=1.

    -

    For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you can probably follow along with this paper, although you might have to read it a few times before all the bits "click".

    -
    - -

    So now the question becomes: how do we manipulate our curve so that it fits this canonical form, with three fixed points, and one "free" point? Enter linear algebra. Don't worry, I'll be doing all the math for you, as well as show you what the effect is on our curves, but basically we're going to be using linear algebra, rather than calculus, because "it's way easier". Sometimes a calculus approach is very hard to work with, when the equivalent geometrical solution is super obvious.

    -

    The approach is going to start with a curve that doesn't have all-colinear points (so we need to make sure the points don't all fall on a straight line), and then applying three graphics operations that you will probably have heard of: translation (moving all points by some fixed x- and y-distance), scaling (multiplying all points by some x and y scale factor), and shearing (an operation that turns rectangles into parallelograms).

    -

    Step 1: we translate any curve by -p1.x and -p1.y, so that the curve starts at (0,0). We're going to make use of an interesting trick here, by pretending our 2D coordinates are 3D, with the z coordinate simply always being 1. This is an old trick in graphics to overcome the limitations of 2D transformations: without it, we can only turn (x,y) coordinates into new coordinates of the form (ax + by, cx + dy), which means we can't do translation, since that requires we end up with some kind of (x + a, y + b). If we add a bogus z coordinate that is always 1, then we can suddenly add arbitrary values. For example:

    - - -

    Sweet! z stays 1, so we can effectively ignore it entirely, but we added some plain values to our x and y coordinates. So, if we want to subtract p1.x and p1.y, we use:

    - - -

    Running all our coordinates through this transformation gives a new set of coordinates, let's call those U, where the first coordinate lies on (0,0), and the rest is still somewhat free. Our next job is to make sure point 2 ends up lying on the x=0 line, so what we want is a transformation matrix that, when we run it, subtracts x from whatever x we currently have. This is called shearing, and the typical x-shear matrix and its transformation looks like this:

    - - -

    So we want some shearing value that, when multiplied by y, yields -x, so our x coordinate becomes zero. That value is simply -x/y, because *-x/y * y = -x*. Done:

    - - -

    Now, running this on all our points generates a new set of coordinates, let's call those V, which now have point 1 on (0,0) and point 2 on (0, some-value), and we wanted it at (0,1), so we need to do some scaling to make sure it ends up at (0,1). Additionally, we want point 3 to end up on (1,1), so we can also scale x to make sure its x-coordinate will be 1 after we run the transform. That means we'll be x-scaling by 1/point3x, and y-scaling by point2y. This is really easy:

    - - -

    Then, finally, this generates a new set of coordinates, let's call those W, of which point 1 lies on (0,0), point 2 lies on (0,1), and point three lies on (1, ...) so all that's left is to make sure point 3 ends up at (1,1) - but we can't scale! Point 2 is already in the right place, and y-scaling would move it out of (0,1) again, so our only option is to y-shear point three, just like how we x-sheared point 2 earlier. In this case, we do the same trick, but with y/x rather than x/y because we're not x-shearing but y-shearing. Additionally, we don't actually want to end up at zero (which is what we did before) so we need to shear towards an offset, in this case 1:

    - - -

    And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will be:

    - - -

    Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just pull this apart a little and end up with an easy-to-calculate bit of code!

    -

    First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does that leave?

    - - -

    Suddenly things look a lot simpler: the mapped x is fairly straight forward to compute, and we see that the mapped y actually contains the mapped x in its entirety, so we'll have that part already available when we need to evaluate it. In fact, let's pull out all those common factors to see just how simple this is:

    - - -

    That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it look!

    -
    + +

    + That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it + look! +

    +
    +

    How do you track all that?

    +

    + Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply + use Mathematica. Tracking all this math by hand is insane, and we invented computers, + literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the + program do the math for me. And real math, too, with symbols, not with numbers. In fact, + here's the Mathematica notebook if you want to see how + this works for yourself. +

    +

    + Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's + $344 for home use, up from $295 when I original wrote this, but it's + also free when you buy a $35 raspberry pi. Obviously, I bought a + raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do + it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for. +

    +
    -

    How do you track all that?

    -

    Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply use Mathematica. Tracking all this math by hand is insane, and we invented computers, literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the program do the math for me. And real math, too, with symbols, not with numbers. In fact, here's the Mathematica notebook if you want to see how this works for yourself.

    -

    Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's $344 for home use, up from $295 when I original wrote this, but it's also free when you buy a $35 raspberry pi. Obviously, I bought a raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for.

    -
    - -

    So, let's write up a sketch that'll show us the canonical form for any curve drawn in blue, overlaid on our canonical map, so that we can immediately tell which features our curve must have, based on where the fourth coordinate is located on the map:

    - - - Скрипти вимкнено. показує резервний. - - - - -
    -
    -

    попереднійзмістнаступний
    Finding Y, given X

    -

    One common task that pops up in things like CSS work, or parametric equalizers, or image leveling, or any other number of applications where Bézier curves are used as control curves in a way that there is really only ever one "y" value associated with one "x" value, you might want to cut out the middle man, as it were, and compute "y" directly based on "x". After all, the function looks simple enough, finding the "y" value should be simple too, right? Unfortunately, not really. However, it is possible and as long as you have some code in place to help, it's not a lot of a work either.

    -

    We'll be tackling this problem in two stages: the first, which is the hard part, is figuring out which "t" value belongs to any given "x" value. For instance, have a look at the following graphic. On the left we have a Bézier curve that looks for all intents and purposes like it fits our criteria: every "x" has one and only one associated "y" value. On the right we see the function for just the "x" values: that's a cubic curve, but not a really crazy cubic curve. If you move the graphic's slider, you will see a red line drawn that corresponds to the x coordinate: this is a vertical line in the left graphic, and a horizontal line on the right.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Now, if you look more closely at that right graphic, you'll notice something interesting: if we treat the red line as "the x axis", then the point where the function crosses our line is really just a root for the cubic function x(t) through a shifted "x-axis"... and we've already seen how to calculate roots, so let's just run cubic root finding - and not even the complicated cubic case either: because of the kind of curve we're starting with, we know there is only root, simplifying the code we need!

    -

    First, let's look at the function for x(t):

    - - -

    We can rewrite this to a plain polynomial form, by just fully writing out the expansion and then collecting the polynomial factors, as:

    - - -

    Nothing special here: that's a standard cubic polynomial in "power" form (i.e. all the terms are ordered by their power of t). So, given that a, b, c, d, and x(t) are all known constants, we can trivially rewrite this (by moving the x(t) across the equal sign) as:

    - - -

    You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using Cardano's algorithm, and we're left with some rather short code:

    + +

    + You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they + all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using + Cardano's algorithm, and we're left with some rather short code: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +y = curve.get(t).y + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So the procedure is fairly straight forward: pick an x, find the associated t value, evaluate our curve for that t value, which gives us the curve's {x,y} coordinate, which means we know y for this x. Move the slider for the following graphic to see this in action:

    - - - Скрипти вимкнено. показує резервний. - - - - - - -
    -
    -

    попереднійзмістнаступний
    Arc length

    -

    How long is a Bézier curve? As it turns out, that's not actually an easy question, because the answer requires maths that —much like root finding— cannot generally be solved the traditional way. If we have a parametric curve with fx(t) and fy(t), then the length of the curve, measured from start point to some point t = z, is computed using the following seemingly straight forward (if a bit overwhelming) formula:

    - - -

    or, more commonly written using Leibnitz notation as:

    - - -

    This formula says that the length of a parametric curve is in fact equal to the area underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an unwieldy computation, and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates".

    -

    Seriously: It cannot be done.

    -

    So we turn to numerical approaches again. The method we'll look at here is the Gauss quadrature. This approximation is a really neat trick, because for any nth degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:

    - - -

    In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the better the approximation:

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    + +

    + In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips + sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The + more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the + better the approximation: +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    -

    Now, infinitely many terms to sum and infinitely thin rectangles are not something that computers can work with, so instead we're going to approximate the infinite summation by using a sum of a finite number of "just thin" rectangular strips. As long as we use a high enough number of thin enough rectangular strips, this will give us an approximation that is pretty close to what the real value is.

    -

    So, the trick is to come up with useful rectangular strips. A naive way is to simply create n strips, all with the same width, but there is a far better way using special values for C and f(t) depending on the value of n, which indicates how many strips we'll use, and it's called the Legendre-Gauss quadrature.

    -

    This approach uses strips that are not spaced evenly, but instead spaces them in a special way based on describing the function as a polynomial (the more strips, the more accurate the polynomial), and then computing the exact integral for that polynomial. We're essentially performing arc length computation on a flattened curve, but flattening it based on the intervals dictated by the Legendre-Gauss solution.

    -
    - -

    Note that one requirement for the approach we'll use is that the integral must run from -1 to 1. That's no good, because we're dealing with Bézier curves, and the length of a section of curve applies to values which run from 0 to "some value smaller than or equal to 1" (let's call that value z). Thankfully, we can quite easily transform any integral interval to any other integral interval, by shifting and scaling the inputs. Doing so, we get the following:

    - - -

    That may look a bit more complicated, but the fraction involving z is a fixed number, so the summation, and the evaluation of the f(t) values are still pretty simple.

    -

    So, what do we need to perform this calculation? For one, we'll need an explicit formula for f(t), because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these Ci and ti values should be. Luckily, that's less work because there are actually many tables available that give these values, for any n, so if we want to approximate our integral with only two terms (which is a bit low, really) then these tables would tell us that for n=2 we must use the following values:

    - - -

    Which means that in order for us to approximate the integral, we must plug these values into the approximate function, which gives us:

    - - -

    We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the Bézier curve functions Bx(t) and By(t).

    -
    + +

    + We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the + Bézier curve functions Bx(t) and By(t). +

    + -

    If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the start/end points on the inside?

    - - - Скрипти вимкнено. показує резервний. - - - +

    + If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), + we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, + with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make + a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the + start/end points on the inside? +

    + + + Скрипти вимкнено. показує резервний. + + + +
    +
    +

    +
    + попереднійзмістнаступний +
    + Approximated arc length +

    +

    + Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length + instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This + will come with an error, but this can be made arbitrarily small by increasing the segment count. +

    +

    + If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal + effort: +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + + + Скрипти вимкнено. показує резервний. + + + + + +
    -
    -
    -

    попереднійзмістнаступний
    Approximated arc length

    -

    Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This will come with an error, but this can be made arbitrarily small by increasing the segment count.

    -

    If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal effort:

    -
    - - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - -
    - -

    You may notice that even though the error in length is actually pretty significant in absolute terms, even at a low number of segments we get a length that agrees with the true length when it comes to just the integer part of the arc length. Quite often, approximations can drastically speed things up!

    - -
    -
    -

    попереднійзмістнаступний
    Curvature of a curve

    -

    If we have two curves, and we want to line them in up in a way that "looks right", what would we use as metric to let a computer decide what "looks right" means?

    -

    For instance, we can start by ensuring that the two curves share an end coordinate, so that there is no "gap" between the end of one and the start of the next curve, but that won't guarantee that things look right: both curves can be going in wildly different directions, and the resulting joined geometry will have a corner in it, rather than a smooth transition from one curve to the next.

    -

    What we want is to ensure that the curvature at the transition from one curve to the next "looks good". So, we start with a shared coordinate, and then also require that derivatives for both curves match at that coordinate. That way, we're assured that their tangents line up, which must mean the curve transition is perfectly smooth. We can even make the second, third, etc. derivatives match up for better and better transitions.

    -

    Problem solved!

    -

    However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on reordering curves showed us that the same looking curve can have an infinite number of curve expressions of arbitrarily high Bézier degree, and each of those will have wildly different derivative values.

    -

    So what we really want is some kind of expression that's not based on any particular expression of t, but is based on something that is invariant to the kind of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being the same function for "coordinate at some distance D along the curve".

    -

    We've seen this before... that's the arc length function.

    -

    So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the form of the arc length function, which we saw above and is fairly simple, rather than needing to solve the arc length function. If we start with the arc length expression and the run through the steps necessary to determine its derivative (with an alternative, shorter demonstration of how to do this found over on Stackexchange), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the fundamental theorem of calculus), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.

    -

    Let me highlight what just happened, because it's pretty special:

    -
      -
    1. we wanted to make curves line up, and initially thought to match the curves' derivatives, but
      2. -
    2. that turned out to be a really bad choice, so instead
      3. -
    3. we picked a function that is basically impossible to work with, and then worked with that, which
      4. -
    4. gives us a simple formula that is and expression using the curves' derivatives.
      5. -
    -

    That's crazy!

    -

    But that's also one of the things that makes maths so powerful: even if your initial ideas are off the mark, you might be much closer than you thought you were, and the journey from "thinking we're completely wrong" to "actually being remarkably close to being right" is where we can find a lot of insight.

    -

    So, what does the function look like? This:

    - - -

    Which is really just a "short form" that glosses over the fact that we're dealing with functions of t, so let's expand that a tiny bit:

    - - -

    And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so evaluating this function for any t value is just a matter of basic arithematics.

    -

    In fact, let's just implement it right now:

    + +

    + And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any + (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that + looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier + curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so + evaluating this function for any t value is just a matter of basic arithematics. +

    +

    In fact, let's just implement it right now:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return numerator / denominator + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    That was easy! (Well okay, that "not a number" value will need to be taken into account by downstream code, but that's a reality of programming anyway)

    -

    With all of that covered, let's line up some curves! The following graphic gives you two curves that look identical, but use quadratic and cubic functions, respectively. As you can see, despite their derivatives being necessarily different, their curvature (thanks to being derived based on maths that "ignores" specific function derivative, and instead gives a formula that smooths out any differences) is exactly the same. And because of that, we can put them together such that the point where they overlap has the same curvature for both curves, giving us the smoothest transition.

    - - - Скрипти вимкнено. показує резервний. - - - -

    One thing you may have noticed in this sketch is that sometimes the curvature looks fine, but seems to be pointing in the wrong direction, making it hard to line up the curves properly. A way around that, of course, is to show the curvature on both sides of the curve, so let's just do that. But let's take it one step further: we can also compute the associated "radius of curvature", which gives us the implicit circle that "fits" the curve's curvature at any point, using what is possibly the simplest bit of maths found in this entire primer:

    - - -

    So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" our curve at some point that we can control by using a slider:

    - - - Скрипти вимкнено. показує резервний. - - - - - + +

    + So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" + our curve at some point that we can control by using a slider: +

    + + + Скрипти вимкнено. показує резервний. + + + + + +
    +
    +

    +
    + попереднійзмістнаступний +
    + Tracing a curve at fixed distance intervals +

    +

    + Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance + intervals over time, like a train along a track, and you want to use Bézier curves. +

    +

    Now you have a problem.

    +

    + The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a + track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick + t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In + fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another. +

    +

    + The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be + straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To + make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for + this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length + function, which can have more inflection points. +

    + + + Скрипти вимкнено. показує резервний. + + +

    + So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through + the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and + then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of + points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two + points, but if we have a high enough number of samples, we don't even need to bother. +

    +

    + So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t + curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks + like, by coloring each section of curve between two distance markers differently: +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    +

    + However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to + constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT). +

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + Intersections +

    +

    + Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to + work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to + perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box + overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the + actual curve. +

    +

    + We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by + implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both + lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution. +

    +

    Line-line intersections

    +

    + If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are + an intervals on by linear algebra, using the procedure outlined in this + top coder + article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line + segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments. +

    +

    + The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on + (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection + point). +

    + + + Скрипти вимкнено. показує резервний. + + + +
    +

    Implementing line-line intersections

    +

    + Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, + but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe + you're using point structs for the line. Let's get coding: +

    -
    -
    -

    попереднійзмістнаступний
    Tracing a curve at fixed distance intervals

    -

    Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance intervals over time, like a train along a track, and you want to use Bézier curves.

    -

    Now you have a problem.

    -

    The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another.

    -

    The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.

    - - - Скрипти вимкнено. показує резервний. - - - -

    So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two points, but if we have a high enough number of samples, we don't even need to bother.

    -

    So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks like, by coloring each section of curve between two distance markers differently:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    -

    However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT).

    - -
    -
    -

    попереднійзмістнаступний
    Intersections

    -

    Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the actual curve.

    -

    We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution.

    -

    Line-line intersections

    -

    If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are an intervals on by linear algebra, using the procedure outlined in this top coder article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments.

    -

    The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection point).

    - - - Скрипти вимкнено. показує резервний. - - - -
    - -

    Implementing line-line intersections

    -

    Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe you're using point structs for the line. Let's get coding:

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    + return lli4(line1.p1, line1.p2, line2.p1, line2.p2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + +
    - +

    What about curve-line intersections?

    +

    + Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we + translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in + a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a + curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the + section on finding extremities. +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + Скрипти вимкнено. показує резервний. + + + +
    -

    What about curve-line intersections?

    -

    Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the section on finding extremities.

    -
    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -
    +

    + Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the + curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and + curve splitting. +

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + Curve/curve intersection +

    +

    + Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" + technique: +

    +
      +
    1. + Take two curves C1 and C2, and treat them as a pair. +
      2. +
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. +
    3. + With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2). +
      4. +
    4. + For each pair, check whether their bounding boxes overlap. +
        +
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. +
      2. If there is overlap, rerun all steps for this pair.
        3. +
      +
      5. +
    5. + Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we + might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value + (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative). +
      6. +
    +

    + This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then + taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections. +

    +

    + The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click + the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value + that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the + algorithm, so you can try this with lots of different curves. +

    +

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    + + + Скрипти вимкнено. показує резервний. + + + + + + +

    + Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into + two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at + t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each + iteration, and each successive step homing in on the curve's self-intersection points. +

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + The projection identity +

    +

    + De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to + draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. + Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point + around to change the curve's shape. +

    +

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    +

    + In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want + to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know + has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for + reasons that will become obvious. +

    +

    + So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following + graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which + taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what + happens to that value? +

    +
    + + + Скрипти вимкнено. показує резервний. + + + + + + + + Скрипти вимкнено. показує резервний. + + + + + +
    -

    Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and curve splitting.

    - -
    -
    -

    попереднійзмістнаступний
    Curve/curve intersection

    -

    Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" technique:

    -
      -
    1. Take two curves C1 and C2, and treat them as a pair.
      2. -
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. -
    3. With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2).
      4. -
    4. For each pair, check whether their bounding boxes overlap.
        -
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. -
      2. If there is overlap, rerun all steps for this pair.
        3. -
      -
      5. -
    5. Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative).
      6. -
    -

    This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections.

    -

    The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the algorithm, so you can try this with lots of different curves.

    -

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    - - - Скрипти вимкнено. показує резервний. - - - - - - -

    Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each iteration, and each successive step homing in on the curve's self-intersection points.

    - -
    -
    -

    попереднійзмістнаступний
    The projection identity

    -

    De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point around to change the curve's shape.

    -

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    -

    In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for reasons that will become obvious.

    -

    So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what happens to that value?

    -
    - - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - -
    - -

    So these graphics show us several things:

    -
      -
    1. a point at the tip of the curve construction's "hat": let's call that A, as well as
      2. -
    2. our on-curve point give our chosen t value: let's call that B, and finally,
      3. -
    3. a point that we get by projecting A, through B, onto the line between the curve's start and end points: let's call that C.
      4. -
    4. for both quadratic and cubic curves, two points e1 and e2, which represent the single-to-last step in de Casteljau's algorithm: in the last step, we find B at (1-t) * e1 + t * e2.
      5. -
    5. for cubic curves, also the points v1 and v2, which together with A represent the first step in de Casteljau's algorithm: in the next step, we find e1 and e2.
      6. -
    -

    These three values A, B, and C allow us to derive an important identity formula for quadratic and cubic Bézier curves: for any point on the curve with some t value, the ratio of distances from A to B and B to C is fixed: if some t value sets up a C that is 20% away from the start and 80% away from the end, then it doesn't matter where the start, end, or control points are; for that t value, C will always lie at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic and then move all the other points around: if you only move the control points, start and end won't move, and so neither will C, and if you move either start or end point, C will move but its relative position will not change.

    -

    So, how can we compute C? We start with our observation that C always lies somewhere between the start and end points, so logically C will have a function that interpolates between those two coordinates:

    - - -

    If we can figure out what the function u(t) looks like, we'll be done. Although we do need to remember that this u(t) will have a different form depending on whether we're working with quadratic or cubic curves. Running through the maths (with thanks to Boris Zbarsky) shows us the following two formulae:

    - - -

    And

    - - -

    So, if we know the start and end coordinates and the t value, we know C without having to calculate the A or even B coordinates. In fact, we can do the same for the ratio function. As another function of t, we technically don't need to know what A or B or C are. It, too, can be expressed as a pure function of t.

    -

    We start by observing that, given A, B, and C, the following always holds:

    - - -

    Working out the maths for this, we see the following two formulae for quadratic and cubic curves:

    - - -

    And

    - - -

    Which now leaves us with some powerful tools: given three points (start, end, and "some point on the curve"), as well as a t value, we can construct curves. We can compute C using the start and end points and our u(t) function, and once we have C, we can use our on-curve point (B) and the ratio(t) function to find A:

    - - -

    With A found, finding e1 and e2 for quadratic curves is a matter of running the linear interpolation with t between start and A to yield e1, and between A and end to yield e2. For cubic curves, there is no single pair of points that can act as e1 and e2 (there are infinitely many, because the tangent at B is a free parameter for cubic curves) so as long as the distance ratio between e1 to B and B to e2 is the Bézier ratio (1-t):t, we are free to pick any pair, after which we can reverse engineer v1 and v2:

    - - -

    And then reverse engineer the curve's control points:

    - - -

    So: if we have a curve's start and end points, as well as some third point B that we want the curve to pass through, then for any t value we implicitly know all the ABC values, which (combined with an educated guess on appropriate e1 and e2 coordinates for cubic curves) gives us the necessary information to reconstruct a curve's "de Casteljau skeleton". Which means that we can now do several things: we can "fit" curves using only three points, which means we can also "mold" curves by moving an on-curve point but leaving its start and end points, and then reconstruct the curve based on where we moved the on-curve point to. These are very useful things, and we'll look at both in the next few sections.

    - -
    -
    -

    попереднійзмістнаступний
    Creating a curve from three points

    -

    Given the preceding section, you might be wondering if we can use that knowledge to just "create" curves by placing some points and having the computer do the rest, to which the answer is: that's exactly what we can now do!

    -

    For quadratic curves, things are pretty easy. Technically, we'll need a t value in order to compute the ratio function used in computing the ABC coordinates, but we can just as easily approximate one by treating the distance between the start and B point, and B and end point as a ratio, using

    - - -

    With this code in place, creating a quadratic curve from three points is literally just computing the ABC values, and using A as our curve's control point:

    - - - Скрипти вимкнено. показує резервний. - - - -

    For cubic curves we need to do a little more work, but really only just a little. We're first going to assume that a decent curve through the three points should approximate a circular arc, which first requires knowing how to fit a circle to three points. You may remember (if you ever learned it!) that a line between two points on a circle is called a chord, and that one property of chords is that the line from the center of any chord, perpendicular to that chord, passes through the center of the circle.

    -

    That means that if we have three points on a circle, we have three (different) chords, and consequently, three (different) lines that go from those chords through the center of the circle: if we find two of those lines, then their intersection will be our circle's center, and the circle's radius will—by definition!—be the distance from the center to any of our three points:

    - - - Скрипти вимкнено. показує резервний. - - - -

    With that covered, we now also know the tangent line to our point B, because the tangent to any point on the circle is a line through that point, perpendicular to the line from that point to the center. That just leaves marking appropriate points e1 and e2 on that tangent, so that we can construct a new cubic curve hull. We use the approach as we did for quadratic curves to automatically determine a reasonable t value, and then our e1 and e2 coordinates must obey the standard de Casteljau rule for linear interpolation:

    - - -

    Where d is the total length of the line segment from e1 to e2. So how long do we make that? There are again all kinds of approaches we can take, and a simple-but-effective one is to set the length of that segment to "one third the length of the baseline". This forces e1 and e2 to always be the "linear curve" distance apart, which means if we place our three points on a line, it will actually look like a line. Nice! The last thing we'll need to do is make sure to flip the sign of d depending on which side of the baseline our B is located, so we don't end up creating a funky curve with a loop in it. To do this, we can use the atan2 function:

    - - -

    This angle φ will be between 0 and π if B is "above" the baseline (rotating all three points so that the start is on the left and the end is the right), so we can use a relatively straight forward check to make sure we're using the correct sign for our value d:

    - - -

    The result of this approach looks as follows:

    - - - Скрипти вимкнено. показує резервний. - - - -

    It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; more on that later), we're only trying to come up with some reasonable e1 and e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives us:

    - - - Скрипти вимкнено. показує резервний. - - - -

    That looks perfectly serviceable!

    -

    Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at implementing curve molding in the section after that, so read on!

    + +

    The result of this approach looks as follows:

    + + + Скрипти вимкнено. показує резервний. + + + +

    + It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, + we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; + more on that later), we're only trying to come up with some reasonable e1 and + e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives + us: +

    + + + Скрипти вимкнено. показує резервний. + + + +

    That looks perfectly serviceable!

    +

    + Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" + curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding + the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at + implementing curve molding in the section after that, so read on! +

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + Projecting a point onto a Bézier curve +

    +

    + Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're + trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the + curve our cursor will be closest to. So, how do we project points onto a curve? +

    +

    + If the Bézier curve is of low enough order, we might be able to + work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a + numerical approach again. We'll be finding our ideal t value using a + binary search. First, we do a coarse distance-check based on + t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast: +

    -
    -
    -

    попереднійзмістнаступний
    Projecting a point onto a Bézier curve

    -

    Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the curve our cursor will be closest to. So, how do we project points onto a curve?

    -

    If the Bézier curve is of low enough order, we might be able to work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a numerical approach again. We'll be finding our ideal t value using a binary search. First, we do a coarse distance-check based on t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    After this runs, we know that LUT[i] is the coordinate on the curve in our LUT that is closest to the point we want to project, so that's a pretty good initial guess as to what the best projection onto our curve is. To refine it, we note that LUT[i] is a better guess than both LUT[i-1] and LUT[i+1], but there might be an even better projection somewhere else between those two values, so that's what we're going to be testing for, using a variation of the binary search.

    -
      -
    1. we start with our point p, and the t values t1=LUT[i-1].t and t2=LUT[i+1].t, which span an interval v = t2-t1.
      2. -
    2. we test this interval in five spots: the start, middle, and end (which we already have), and the two points in between the middle and start/end points
      3. -
    3. we then check which of these five points is the closest to our original point p, and then repeat step 1 with the points before and after the closest point we just found.
      4. -
    -

    This makes the interval we check smaller and smaller at each iteration, and we can keep running the three steps until the interval becomes so small as to lead to distances that are, for all intents and purposes, the same for all points.

    -

    So, let's see that in action: in this case, I'm going to arbitrarily say that if we're going to run the loop until the interval is smaller than 0.001, and show you what that means for projecting your mouse cursor or finger tip onto a rather complex Bézier curve (which, of course, you can reshape as you like). Also shown are the original three points that our coarse check finds.

    - - - Скрипти вимкнено. показує резервний. - - - - -
    -
    -

    попереднійзмістнаступний
    Intersections with a circle

    -

    It might seem odd to cover this subject so much later than the line/line, line/curve, and curve/curve intersection topics from several sections earlier, but the reason we can't cover circle/curve intersections is that we can't really discuss circle/curve intersection until we've covered the kind of lookup table (LUT) walking that the section on projecting a point onto a curve uses. To see why, let's look at what we would have to do if we wanted to find the intersections between a curve and a circle using calculus.

    -

    First, we observe that "finding intersections" in this case means that, given a circle defined by a center point c = (x,y) and a radius r, we want to find all points on the Bezier curve for which the distance to the circle's center point is equal to the circle radius, which by definition means those points lie on the circle, and so count as intersections. In maths, that means we're trying to solve:

    - - -

    Which seems simple enough. Unfortunately, when we expand that dist function, things get a lot more problematic:

    - - -

    And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the Abel-Ruffini theorem that we saw before, we can't solve this by just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left with a monstrous function to solve.

    -

    So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a bit more code to make sure we find all of them, but let's look at the steps involved:

    + +

    + And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the + Abel-Ruffini theorem that we saw before, we can't solve this by + just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to + actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and + all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left + with a monstrous function to solve. +

    +

    + So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the + on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance + r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a + bit more code to make sure we find all of them, but let's look at the steps involved: +

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + -

    This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is exactly the radius, so the coordinate lies on both the curve and the circle.

    -

    So far so good.

    -

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    +

    + This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor + change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between + that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is + exactly the radius, so the coordinate lies on both the curve and the circle. +

    +

    So far so good.

    +

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + start = i + 2; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points ("current", "previous" and "before previous") and then check whether they form a local minimum:

    +

    + After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we + can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now + not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of + course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in + finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points + ("current", "previous" and "before previous") and then check whether they form a local minimum: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return start + i + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    In words: given a start index, the circle center and radius, and our LUT, we check where (closest to our start index) we can find a local minimum for the difference between "the distance from the curve to the circle center", and the circle's radius. We track this by looking at three values (associated with the indices index-2, index-1, and index), and we know we've found a local minimum if the three values show that the middle value (pd1) is less than either value beside it. When we do, we can set our "best guess, relative to start" as index-1. Of course, since we're now checking values relative to some start value, we might not find another candidate value at all, in which case we return start - 1, so that a simple "is the result less than start?" lets us determine that there are no more intersections to find.

    -

    Finally, while not necessary for point projection, there is one more step we need to perform when we run the binary refinement function on our candidate LUT indices, because we've so far only been testing using distances "closest to the radius of the circle", and that's actually not good enough... we need distances that are the radius of the circle. So, after running the refinement for each of these indices, we need to discard any final value that isn't the circle radius. And because we're working with floating point numbers, what this really means is that we need to discard any value that's a pixel or more "off". Or, if we want to get really fancy, "some small epsilon value".

    -

    Based on all of that, the following graphic shows this off for the standard cubic curve (which you can move the coordinates around for, of course) and a circle with a controllable radius centered on the graphic's center, using the code approach described above.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    And of course, for the full details, click that "view source" link.

    - -
    -
    -

    попереднійзмістнаступний
    Molding a curve

    -

    Armed with knowledge of the "ABC" relation, point-on-curve projection, and guestimating reasonable looking helper values for cubic curve construction, we can finally cover curve molding: updating a curve's shape interactively, by dragging points on the curve around.

    -

    For quadratic curve, this is a really simple trick: we project our cursor onto the curve, which gives us a t value and initial B coordinate. We don't even need the latter: with our t value and "wherever the cursor is" as target B, we can compute the associated C:

    - - -

    And then the associated A:

    - - -

    And we're done, because that's our new quadratic control point!

    - - - Скрипти вимкнено. показує резервний. - - - -

    As before, cubic curves are a bit more work, because while it's easy to find our initial t value and ABC values, getting those all-important e1 and e2 coordinates is going to pose a bit of a problem... in the section on curve creation, we were free to pick an appropriate t value ourselves, which allowed us to find appropriate e1 and e2 coordinates. That's great, but when we're curve molding we don't have that luxury: whatever point we decide to start moving around already has its own t value, and its own e1 and e2 values, and those may not make sense for the rest of the curve.

    -

    For example, let's see what happens if we just "go with what we get" when we pick a point and start moving it around, preserving its t value and e1/e2 coordinates:

    - - - Скрипти вимкнено. показує резервний. - - - -

    That looks reasonable, close to the original point, but the further we drag our point, the less "useful" things become. Especially if we drag our point across the baseline, rather than turning into a nice curve.

    -

    One way to combat this might be to combine the above approach with the approach from the creating curves section: generate both the "unchanged t/e1/e2" curve, as well as the "idealized" curve through the start/cursor/end points, with idealized t value, and then interpolating between those two curves:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    The slide controls the "falloff distance" relative to where the original point on the curve is, so that as we drag our point around, it interpolates with a bias towards "preserving t/e1/e2" closer to the original point, and bias towards "idealized" form the further away we move our point, with anything that's further than our falloff distance simply being the idealized curve. We don't even try to interpolate at that point.

    -

    A more advanced way to try to smooth things out is to implement continuous molding, where we constantly update the curve as we move around, and constantly change what our B point is, based on constantly projecting the cursor on the curve as we're updating it - this is, you won't be surprised to learn, tricky, and beyond the scope of this section: interpolation (with a reasonable distance) will do for now!

    - -
    -
    -

    попереднійзмістнаступний
    Curve fitting

    -

    Given the previous section, one question you might have is "what if I don't want to guess t values?". After all, plenty of graphics packages do automated curve fitting, so how can we implement that in a way that just finds us reasonable t values all on its own?

    -

    And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a least squares polynomial regression. In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an nth order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".

    -

    Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the squared distance between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep-dive the specific merits of least squared error metrics versus least absolute error metrics, but those are well beyond the scope of this material.

    -

    So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his "Least Squares Bézier Fit" article, and end with some nice interactive graphics for doing some curve fitting.

    -

    Before we begin, we're going to use the curve in matrix form. In the section on matrices, I mentioned that some things are easier if we use the matrix representation of a Bézier curve rather than its calculus form, and this is one of those things.

    -

    As such, the first step in the process is expressing our Bézier curve as powers/coefficients/coordinate matrix T x M x C, by expanding the Bézier functions.

    -
    - -

    Revisiting the matrix representation

    -

    Rewriting Bézier functions to matrix form is fairly easy, if you first expand the function, and then arrange them into a multiple line form, where each line corresponds to a power of t, and each column is for a specific coefficient. First, we expand the function:

    - - -

    And then we (trivially) rearrange the terms across multiple lines:

    - - -

    This rearrangement has "factors of t" at each row (the first row is t⁰, i.e. "1", the second row is t¹, i.e. "t", the third row is t²) and "coefficient" at each column (the first column is all terms involving "a", the second all terms involving "b", the third all terms involving "c").

    -

    With that arrangement, we can easily decompose this as a matrix multiplication:

    - - -

    We can do the same for the cubic curve, of course. We know the base function for cubics:

    - - -

    So we write out the expansion and rearrange:

    - - -

    Which we can then decompose:

    - - -

    And, of course, we can do this for quartic curves too (skipping the expansion step):

    - - -

    And so and on so on. Now, let's see how to use these T, M, and C, to do some curve fitting.

    -
    + +

    + And so and on so on. Now, let's see how to use these T, M, and C, to do some curve + fitting. +

    + -

    Let's get started: we're going to assume we picked the right order curve: for n points we're fitting an n-1th order curve, so we "start" with a vector P that represents the coordinates we already know, and for which we want to do curve fitting:

    - - -

    Next, we need to figure out appropriate t values for each point in the curve, because we need something that lets us tie "the actual coordinate" to "some point on the curve". There's a fair number of different ways to do this (and a large part of optimizing "the perfect fit" is about picking appropriate t values), but in this case let's look at two "obvious" choices:

    -
      -
    1. equally spaced t values, and
      2. -
    2. t values that align with distance along the polygon.
      3. -
    -

    The first one is really simple: if we have n points, then we'll just assign each point i a t value of (i-1)/(n-1). So if we have four points, the first point will have t=(1-1)/(4-1)=0/3, the second point will have t=(2-1)/(4-1)=1/3, the third point will have t=2/3, and the last point will be t=1. We're just straight up spacing the t values to match the number of points we have.

    -

    The second one is a little more interesting: since we're doing polynomial regression, we might as well exploit the fact that our base coordinates just constitute a collection of line segments. At the first point, we're fixing t=0, and the last point, we want t=1, and anywhere in between we're simply going to say that t is equal to the distance along the polygon, scaled to the [0,1] domain.

    -

    To get these values, we first compute the general "distance along the polygon" matrix:

    - - -

    Where length() is literally just that: the length of the line segment between the point we're looking at, and the previous point. This isn't quite enough, of course: we still need to make sure that all the values between i=1 and i=n fall in the [0,1] interval, so we need to scale all values down by whatever the total length of the polygon is:

    - - -

    And now we can move on to the actual "curve fitting" part: what we want is a function that lets us compute "ideal" control point values such that if we build a Bézier curve with them, that curve passes through all our original points. Or, failing that, have an overall error distance that is as close to zero as we can get it. So, let's write out what the error distance looks like.

    -

    As mentioned before, this function is really just "the distance between the actual coordinate, and the coordinate that the curve evaluates to for the associated t value", which we'll square to get rid of any pesky negative signs:

    - - -

    Since this function only deals with individual coordinates, we'll need to sum over all coordinates in order to get the full error function. So, we literally just do that; the total error function is simply the sum of all these individual errors:

    - - -

    And here's the trick that justifies using matrices: while we can work with individual values using calculus, with matrices we can compute as many values as we make our matrices big, all at the "same time", We can replace the individual terms pi with the full P coordinate matrix, and we can replace Bézier(si) with the matrix representation T x M x C we talked about before, which gives us:

    - - -

    In which we can replace the rather cumbersome "squaring" operation with a more conventional matrix equivalent:

    - - -

    Here, the letter T is used instead of the number 2, to represent the matrix transpose; each row in the original matrix becomes a column in the transposed matrix instead (row one becomes column one, row two becomes column two, and so on).

    -

    This leaves one problem: T isn't actually the matrix we want: we don't want symbolic t values, we want the actual numerical values that we computed for S, so we need to form a new matrix, which we'll call 𝕋, that makes use of those, and then use that 𝕋 instead of T in our error function:

    - - -

    Which, because of the first and last values in S, means:

    - - -

    Now we can properly write out the error function as matrix operations:

    - - -

    So, we have our error function: we now need to figure out the expression for where that function has minimal value, e.g. where the error between the true coordinates and the coordinates generated by the curve fitting is smallest. Like in standard calculus, this requires taking the derivative, and determining where that derivative is zero:

    - - -
    + +
    +

    Where did this derivative come from?

    +

    + That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I + straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know. +

    +

    + So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific + case, I + posted a question + to Math.stackexchange, and received a answer that goes into way more detail than I had + hoped to receive. +

    +

    + Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean + maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that + true?". There are answers. They might just require some time to come to understand. +

    +
    -

    Where did this derivative come from?

    -

    That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know.

    -

    So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific case, I posted a question to Math.stackexchange, and received a answer that goes into way more detail than I had hoped to receive.

    -

    Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that true?". There are answers. They might just require some time to come to understand.

    -
    - -

    Now, given the above derivative, we can rearrange the terms (following the rules of matrix algebra) so that we end up with an expression for C:

    - - -

    Here, the "to the power negative one" is the notation for the matrix inverse. But that's all we have to do: we're done. Starting with P and inventing some t values based on the polygon the coordinates in P define, we can compute the corresponding Bézier coordinates C that specify a curve that goes through our points. Or, if it can't go through them exactly, as near as possible.

    -

    So before we try that out, how much code is involved in implementing this? Honestly, that answer depends on how much you're going to be writing yourself. If you already have a matrix maths library available, then really not that much code at all. On the other hand, if you are writing this from scratch, you're going to have to write some utility functions for doing your matrix work for you, so it's really anywhere from 50 lines of code to maybe 200 lines of code. Not a bad price to pay for being able to fit curves to pre-specified coordinates.

    -

    So let's try it out! The following graphic lets you place points, and will start computing exact-fit curves once you've placed at least three. You can click for more points, and the code will simply try to compute an exact fit using a Bézier curve of the appropriate order. Four points? Cubic Bézier. Five points? Quartic. And so on. Of course, this does break down at some point: depending on where you place your points, it might become mighty hard for the fitter to find an exact fit, and things might actually start looking horribly off once there's enough points for compound floating point rounding errors to start making a difference (which is around 10~11 points).

    - - - Скрипти вимкнено. показує резервний. - - - - - - -

    You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be able to freely manipulate them and see what the effect on your curve is.

    - -
    -
    -

    попереднійзмістнаступний
    Bézier curves and Catmull-Rom curves

    -

    Taking an excursion to different splines, the other common design curve is the Catmull-Rom spline, which unlike Bézier curves pass through each control point, so they offer a kind of "built-in" curve fitting.

    -

    In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what looks like a single Catmull-Rom curve is actually a spline: a single curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is defined by the point's coordinates, and the tangent for those points, the latter of which can trivially be derived from knowing the previous and next point:

    - + +

    + You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio + along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get + direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be + able to freely manipulate them and see what the effect on your curve is. +

    +
    +
    +

    +
    + попереднійзмістнаступний +
    + Bézier curves and Catmull-Rom curves +

    +

    + Taking an excursion to different splines, the other common design curve is the + Catmull-Rom spline, which unlike Bézier curves + pass through each control point, so they offer a kind of "built-in" curve fitting. +

    +

    + In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets + you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the + tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next. +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    + Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what + looks like a single Catmull-Rom curve is actually a spline: a single + curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, + and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is + defined by the point's coordinates, and the tangent for those points, the latter of which + can trivially be derived from knowing the + previous and next point: +

    + - -

    One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on that in the next section.

    -

    In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of variations that make things considerable more complex):

    + +

    + One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up + with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent + should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on + that in the next section. +

    +

    + In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of + variations that make things + considerable more + complex): +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + point(c0 * v1 + c1 * dv1 + c2 * v2 + c3 * dv2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Now, since a Catmull-Rom curve is a form of cubic Hermite spline, and as cubic Bézier curves are also a form of cubic Hermite spline, we run into an interesting bit of maths programming: we can convert one to the other and back, and the maths for doing so is surprisingly simple!

    -

    The main difference between Catmull-Rom curves and Bézier curves is "what the points mean":

    -
      -
    • A cubic Bézier curve is defined by a start point, a control point that implies the tangent at the start, a control point that implies the tangent at the end, and an end point, plus a characterizing matrix that we can multiply by that point vector to get on-curve coordinates.
      • -
    • A Catmull-Rom curve is defined by a start point, a tangent that for that starting point, an end point, and a tangent for that end point, plus a characteristic matrix that we can multiple by the point vector to get on-curve coordinates.
      • -
    -

    Those are very similar, so let's see exactly how similar they are. We've already see the matrix form for Bézier curves, so how different is the matrix form for Catmull-Rom curves?:

    - - -

    That's pretty dang similar. So the question is: how can we convert that expression with Catmull-Rom matrix and vector into an expression of the Bézier matrix and vector? The short answer is of course "by using linear algebra", but the longer answer is the rest of this section, and involves some maths that you may not even care for: if you just want to know the (incredibly simple) conversions between the two curve forms, feel free to skip to the end of the following explanation, but if you want to how we can get one from the other... let's get mathing!

    -
    - -

    Deriving the conversion formulae

    -

    In order to convert between Catmull-Rom curves and Bézier curves, we need to know two things. Firstly, how to express the Catmull-Rom curve using a "set of four coordinates", rather than a mix of coordinates and tangents, and secondly, how to convert those Catmull-Rom coordinates to and from Bézier form.

    -

    We start with the first part, to figure out how we can go from Catmull-Rom V coordinates to Bézier P coordinates, by applying "some matrix T". We don't know what that T is yet, but we'll get to that:

    - - -

    So, this mapping says that in order to map a Catmull-Rom "point + tangent" vector to something based on an "all coordinates" vector, we need to determine the mapping matrix such that applying T yields P2 as start point, P3 as end point, and two tangents based on the lines between P1 and P3, and P2 nd P4, respectively.

    -

    Computing T is really more "arranging the numbers":

    - - -

    Thus:

    - - -

    However, we're not quite done, because Catmull-Rom curves have that "tension" parameter, written as τ (a lowercase"tau"), which is a scaling factor for the tangent vectors: the bigger the tension, the smaller the tangents, and the smaller the tension, the bigger the tangents. As such, the tension factor goes in the denominator for the tangents, and before we continue, let's add that tension factor into both our coordinate vector representation, and mapping matrix T:

    - - -

    With the mapping matrix properly done, let's rewrite the "point + tangent" Catmull-Rom matrix form to a matrix form in terms of four coordinates, and see what we end up with:

    - - -

    Replace point/tangent vector with the expression for all-coordinates:

    - - -

    and merge the matrices:

    - - -

    This looks a lot like the Bézier matrix form, which as we saw in the chapter on Bézier curves, should look like this:

    - - -

    So, if we want to express a Catmull-Rom curve using a Bézier curve, we'll need to turn this Catmull-Rom bit:

    - - -

    Into something that looks like this:

    - - -

    And the way we do that is with a fairly straight forward bit of matrix rewriting. We start with the equality we need to ensure:

    - - -

    Then we remove the coordinate vector from both sides without affecting the equality:

    - - -

    Then we can "get rid of" the Bézier matrix on the right by left-multiply both with the inverse of the Bézier matrix:

    - - -

    A matrix times its inverse is the matrix equivalent of 1, and because "something times 1" is the same as "something", so we can just outright remove any matrix/inverse pair:

    - - -

    And now we're basically done. We just multiply those two matrices and we know what V is:

    - - -

    We now have the final piece of our function puzzle. Let's run through each step.

    -
      -
    1. Start with the Catmull-Rom function:
      2. -
    - - -
      -
    1. rewrite to pure coordinate form:
      2. -
    - - -
      -
    1. rewrite for "normal" coordinate vector:
      2. -
    - - -
      -
    1. merge the inner matrices:
      2. -
    - - -
      -
    1. rewrite for Bézier matrix form:
      2. -
    - - -
      -
    1. and transform the coordinates so we have a "pure" Bézier expression:
      2. -
    - - -

    And we're done: we finally know how to convert these two curves!

    -
    + +

    And we're done: we finally know how to convert these two curves!

    + -

    If we have a Catmull-Rom curve defined by four coordinates P1 through P4, then we can draw that curve using a Bézier curve that has the vector:

    - - -

    Similarly, if we have a Bézier curve defined by four coordinates P1 through P4, we can draw that using a standard tension Catmull-Rom curve with the following coordinate values:

    - - -

    Or, if your API allows you to specify Catmull-Rom curves using plain coordinates:

    - - - -
    -
    -

    попереднійзмістнаступний
    Creating a Catmull-Rom curve from three points

    -

    Much shorter than the previous section: we saw that Catmull-Rom curves need at least 4 points to draw anything sensible, so how do we create a Catmull-Rom curve from three points?

    -

    Short and sweet: we don't.

    -

    We run through the maths that lets us create a cubic Bézier curve, and then convert its coordinates to Catmull-Rom form using the conversion formulae we saw above.

    - -
    -
    -

    попереднійзмістнаступний
    Forming poly-Bézier curves

    -

    Much like lines can be chained together to form polygons, Bézier curves can be chained together to form poly-Béziers, and the only trick required is to make sure that:

    -
      -
    1. the end point of each section is the starting point of the following section, and
      2. -
    2. the derivatives across that dual point line up.
      3. -
    -

    Unless you want sharp corners, of course. Then you don't even need 2.

    -

    We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind that just follows point 1. where the end point of a segment is the same point as the start point of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're the easiest to implement:

    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -

    Dragging the control points around only affects the curve segments that the control point belongs to, and moving an on-curve point leaves the control points where they are, which is not the most useful for practical modelling purposes. So, let's add in the logic we need to make things a little better. We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve point is the same as it's "outgoing" derivative:

    - - -

    We can effect this quite easily, because we know that the vector from a curve's last control point to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control point of the next curve matches that, all we have to do is mirror that last control point through the last on-curve point. And mirroring any point A through any point B is really simple:

    - - -

    So let's implement that and see what it gets us. The following two graphics show a quadratic and a cubic poly-Bézier curve again, but this time moving the control points around moves others, too. However, you might see something unexpected going on for quadratic curves...

    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -

    As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all the control points are effectively linked. Move one of them, and you move all of them. Not only that, but if we move the on-curve points, it's possible to get a situation where a control point cannot satisfy the constraint that it's the reflection of its two neighbouring control points... This means that we cannot use quadratic poly-Béziers for anything other than really, really simple shapes. And even then, they're probably the wrong choice. Cubic curves are pretty decent, but the fact that the derivatives are linked means we can't manipulate curves as well as we might if we relaxed the constraints a little.

    -

    So: let's relax the requirement a little.

    -

    We can change the constraint so that we still preserve the angle of the derivatives across sections (so transitions from one section to the next will still look natural), but give up the requirement that they should also have the same vector length. Doing so will give us a much more useful kind of poly-Bézier curve:

    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -

    Cubic curves are now better behaved when it comes to dragging control points around, but the quadratic poly-Bézier still has the problem that moving one control points will move the control points and may ending up defining "the next" control point in a way that doesn't work. Quadratic curves really aren't very useful to work with...

    -

    Finally, we also want to make sure that moving the on-curve coordinates preserves the relative positions of the associated control points. With that, we get to the kind of curve control that you might be familiar with from applications like Photoshop, Inkscape, Blender, etc.

    - - - Скрипти вимкнено. показує резервний. - - - - - - Скрипти вимкнено. показує резервний. - - - -

    Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a new, very serious problem: we can move an on-curve point in such a way that we can't compute what needs to "happen next". Move the top point down, below the left and right points, for instance. There is no way to preserve correct control points without a kink at the bottom point. Quadratic curves: just not that good...

    -

    A final improvement is to offer fine-level control over which points behave which, so that you can have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves for the rest of the path. Implementing that, is left as an exercise for the reader.

    - -
    -
    -

    попереднійзмістнаступний
    Curve offsetting

    -

    Perhaps you're like me, and you've been writing various small programs that use Bézier curves in some way or another, and at some point you make the step to implementing path extrusion. But you don't want to do it pixel based; you want to stay in the vector world. You find that extruding lines is relatively easy, and tracing outlines is coming along nicely (although junction caps and fillets are a bit of a hassle), and then you decide to do things properly and add Bézier curves to the mix. Now you have a problem.

    -

    Unlike lines, you can't simply extrude a Bézier curve by taking a copy and moving it around, because of the curvatures; rather than a uniform thickness, you get an extrusion that looks too thin in places, if you're lucky, but more likely will self-intersect. The trick, then, is to scale the curve, rather than simply copying it. But how do you scale a Bézier curve?

    -

    Bottom line: you can't. So you cheat. We're not going to do true curve scaling, or rather curve offsetting, because that's impossible. Instead we're going to try to generate 'looks good enough' offset curves.

    -
    - -

    "What do you mean, you can't? Prove it."

    -

    First off, when I say "you can't," what I really mean is "you can't offset a Bézier curve with another Bézier curve", not even by using a really high order curve. You can find the function that describes the offset curve, but it won't be a polynomial, and as such it cannot be represented as a Bézier curve, which has to be a polynomial. Let's look at why this is:

    -

    From a mathematical point of view, an offset curve O(t) is a curve such that, given our original curve B(t), any point on O(t) is a fixed distance d away from coordinate B(t). So let's math that:

    - - -

    However, we're working in 2D, and d is a single value, so we want to turn it into a vector. If we want a point distance d "away" from the curve B(t) then what we really mean is that we want a point at d times the "normal vector" from point B(t), where the "normal" is a vector that runs perpendicular ("at a right angle") to the tangent at B(t). Easy enough:

    - - -

    Now this still isn't very useful unless we know what the formula for N(t) is, so let's find out. N(t) runs perpendicular to the original curve tangent, and we know that the tangent is simply B'(t), so we could just rotate that 90 degrees and be done with it. However, we need to ensure that N(t) has the same magnitude for every t, or the offset curve won't be at a uniform distance, thus not being an offset curve at all. The easiest way to guarantee this is to make sure N(t) always has length 1, which we can achieve by dividing B'(t) by its magnitude:

    - - -

    Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc length from some start a to end b, we must use the formula we saw earlier. Noting that "length" is usually denoted with double vertical bars:

    - - -

    So if we want the length of the tangent, we plug in B'(t), with t = 0 as start and -t = 1 as end:

    - - -

    And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.

    -

    There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine because they do, but they don't; remember that only the base function has binomial coefficients. That's before we factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

    -

    So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve.

    -

    And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the real offset curve would look like, if we could compute it.

    -
    + +

    + And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is + screwing everything up, because it turns our nice polynomials into things that are no longer polynomials. +

    +

    + There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with + unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine + because they do, but they don't; remember that only the base function has binomial coefficients. That's before we + factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make + all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call + them circles, and they have much simpler functions than Bézier curves. +

    +

    + So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means + N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means + that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for + O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve. +

    +

    + And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier + curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, + much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the + real offset curve would look like, if we could compute it. +

    + -

    So, you cannot offset a Bézier curve perfectly with another Bézier curve, no matter how high-order you make that other Bézier curve. However, we can chop up a curve into "safe" sub-curves (where "safe" means that all the control points are always on a single side of the baseline, and the midpoint of the curve at t=0.5 is roughly in the center of the polygon defined by the curve coordinates) and then point-scale each sub-curve with respect to its scaling origin (which is the intersection of the point normals at the start and end points).

    -

    A good way to do this reduction is to first find the curve's extreme points, as explained in the earlier section on curve extremities, and use these as initial splitting points. After this initial split, we can check each individual segment to see if it's "safe enough" based on where the center of the curve is. If the on-curve point for t=0.5 is too far off from the center, we simply split the segment down the middle. Generally this is more than enough to end up with safe segments.

    -

    The following graphics show off curve offsetting, and you can use the slider to control the distance at which the curve gets offset. The curve first gets reduced to safe segments, each of which is then offset at the desired distance. Especially for simple curves, particularly easily set up for quadratic curves, no reduction is necessary, but the more twisty the curve gets, the more the curve needs to be reduced in order to get segments that can safely be scaled.

    - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - -

    You may notice that this may still lead to small 'jumps' in the sub-curves when moving the curve around. This is caused by the fact that we're still performing a naive form of offsetting, moving the control points the same distance as the start and end points. If the curve is large enough, this may still lead to incorrect offsets.

    - -
    -
    -

    попереднійзмістнаступний
    Graduated curve offsetting

    -

    What if we want to do graduated offsetting, starting at some distance s but ending at some other distance e? Well, if we can compute the length of a curve (which we can if we use the Legendre-Gauss quadrature approach) then we can also determine how far "along the line" any point on the curve is. With that knowledge, we can offset a curve so that its offset curve is not uniformly wide, but graduated between with two different offset widths at the start and end.

    -

    Like normal offsetting we cut up our curve in sub-curves, and then check at which distance along the original curve each sub-curve starts and ends, as well as to which point on the curve each of the control points map. This gives us the distance-along-the-curve for each interesting point in the sub-curve. If we call the total length of all sub-curves seen prior to seeing "the current" sub-curve S (and if the current sub-curve is the first one, S is zero), and we call the full length of our original curve L, then we get the following graduation values:

    -
      -
    • start: map S from interval (0,L) to interval (s,e)
      • -
    • c1: map(<strong>S+d1</strong>, 0,L, s,e), d1 = distance along curve to projection of c1
      • -
    • c2: map(<strong>S+d2</strong>, 0,L, s,e), d2 = distance along curve to projection of c2
      • -
    • ...
      • -
    • end: map(<strong>S+length(subcurve)</strong>, 0,L, s,e)
      • -
    -

    At each of the relevant points (start, end, and the projections of the control points onto the curve) we know the curve's normal, so offsetting is simply a matter of taking our original point, and moving it along the normal vector by the offset distance for each point. Doing so will give us the following result (these have with a starting width of 0, and an end width of 40 pixels, but can be controlled with your up and down arrow keys):

    - - - Скрипти вимкнено. показує резервний. - - - - - - - - Скрипти вимкнено. показує резервний. - - - - - - -
    -
    -

    попереднійзмістнаступний
    Circles and quadratic Bézier curves

    -

    Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. For instance, if you're doing font design, fonts have no concept of geometric shapes, they only know straight lines, and Bézier curves. OpenType fonts with TrueType outlines only know quadratic Bézier curves, and OpenType fonts with Type 2 outlines only know cubic Bézier curves. So how do you draw a circle, or an ellipse, or an arc?

    -

    You approximate.

    -

    We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.

    -

    Since arcs are mid-point-symmetrical, we need the control points to set up a symmetrical curve. For quadratic curves this means that the control point will be somewhere on a line that intersects the baseline at a right angle. And we don't get any choice on where that will be, since the derivatives at the start and end point have to line up, so our control point will lie at the intersection of the tangents at the start and end point.

    -

    First, let's try to fit the quadratic curve onto a circular arc. In the following sketch you can move the mouse around over a unit circle, to see how well, or poorly, a quadratic curve can approximate the arc from (1,0) to where your mouse cursor is:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    As you can see, things go horribly wrong quite quickly; even trying to approximate a quarter circle using a quadratic curve is a bad idea. An eighth of a turns might look okay, but how okay is okay? Let's apply some maths and find out. What we're interested in is how far off our on-curve coordinates are with respect to a circular arc, given a specific start and end angle. We'll be looking at how much space there is between the circular arc, and the quadratic curve's midpoint.

    -

    We start out with our start and end point, and for convenience we will place them on a unit circle (a circle around 0,0 with radius 1), at some angle φ:

    - - -

    What we want to find is the intersection of the tangents, so we want a point C such that:

    - - -

    i.e. we want a point that lies on the vertical line through S (at some distance a from S) and also lies on the tangent line through E (at some distance b from E). Solving this gives us:

    - - -

    First we solve for b:

    - - -

    which yields:

    - - -

    which we can then substitute in the expression for a:

    - - -

    A quick check shows that plugging these values for a and b into the expressions for Cx and Cy give the same x/y coordinates for both "a away from A" and "b away from B", so let's continue: now that we know the coordinate values for C, we know where our on-curve point T for t=0.5 (or angle φ/2) is, because we can just evaluate the Bézier polynomial, and we know where the circle arc's actual point P is for angle φ/2:

    - - -

    We compute T, observing that if t=0.5, the polynomial values (1-t)², 2(1-t)t, and t² are 0.25, 0.5, and 0.25 respectively:

    - - -

    Which, worked out for the x and y components, gives:

    - - -

    And the distance between these two is the standard Euclidean distance:

    - - -

    So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter circle and eighth circle?

    -
    - - plotted for 0 ≤ φ ≤ π: - - - plotted for 0 ≤ φ ≤ ½π: - - - - - plotted for 0 ≤ φ ≤ ¼π: -
    + +

    + So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter + circle and eighth circle? +

    + + + + + + + + +
    + + plotted for 0 ≤ φ ≤ π: + + + plotted for 0 ≤ φ ≤ ½π: + + + + + plotted for 0 ≤ φ ≤ ¼π: +
    -

    We now see why the eighth circle arc looks decent, but the quarter circle arc doesn't: an error of roughly 0.06 at t=0.5 means we're 6% off the mark... we will already be off by one pixel on a circle with pixel radius 17. Any decent sized quarter circle arc, say with radius 100px, will be way off if approximated by a quadratic curve! For the eighth circle arc, however, the error is only roughly 0.003, or 0.3%, which explains why it looks so close to the actual eighth circle arc. In fact, if we want a truly tiny error, like 0.001, we'll have to contend with an angle of (rounded) 0.593667, which equates to roughly 34 degrees. We'd need 11 quadratic curves to form a full circle with that precision! (technically, 10 and ten seventeenth, but we can't do partial curves, so we have to round up). That's a whole lot of curves just to get a shape that can be drawn using a simple function!

    -

    In fact, let's flip the function around, so that if we plug in the precision error, labelled ε, we get back the maximum angle for that precision:

    - - -

    And frankly, things are starting to look a bit ridiculous at this point, we're doing way more maths than we've ever done, but thankfully this is as far as we need the maths to take us: If we plug in the precisions 0.1, 0.01, 0.001 and 0.0001 we get the radians values 1.748, 1.038, 0.594 and 0.3356; in degrees, that means we can cover roughly 100 degrees (requiring four curves), 59.5 degrees (requiring six curves), 34 degrees (requiring 11 curves), and 19.2 degrees (requiring a whopping nineteen curves).

    -

    The bottom line? Quadratic curves are kind of lousy if you want circular (or elliptical, which are circles that have been squashed in one dimension) curves. We can do better, even if it's just by raising the order of our curve once. So let's try the same thing for cubic curves.

    - -
    -
    -

    попереднійзмістнаступний
    Circular arcs and cubic Béziers

    -

    Let's look at approximating circles and circular arcs using cubic Béziers. How much better is that?

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    At cursory glance, a fair bit better, but let's find out how much better by looking at how to construct the Bézier curve.

    -

    A construction diagram for a cubic approximation of a circular arc

    -

    The start and end points are trivial, but the mid point requires a bit of work, but it's mostly basic trigonometry once we know the angle θ for our circular arc: if we scale our circular arc to a unit circle, we can always start our arc, with radius 1, at (1,0) and then given our arc angle θ, we also know that the circular arc has length θ (because unit circles are nice that way). We also know our end point, because that's just (cos(θ), sin(θ)), and so the challenge is to figure out what control points we need in order for the curve at t=0.5 to exactly touch the circular arc at the angle θ/2:

    -

    So let's again formally describe this:

    - - -

    Only P3 isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P4, P3) is a right angled triangle, with the distance between the origin and P4 being 1 (because we're working with a unit circle), and the distance between P4 and P3 being k, so that we can represent P3 as "The point P4 plus the vector from the origin to P4 but then rotated a quarter circle, counter-clockwise, and scaled by k".

    -

    With that, we can determine the y-coordinates for A, B, e1, and e2, after which we have all the information we need to determine what the value of k is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P2 and P3, e1 is between A and "midway between P1 and P2" (which is "half height" P2), and so forth:

    - - -

    Which now gives us two identities for B, because in addition to determining B through linear interpolation, we also know that B's y coordinate is just sin(θ/2): we started this exercise by saying we were going to approximate the circular arc using a Bézier curve that had its midpoint, which is point B, touching the unit circle at the arc's half-angle, by definition making B the point at (cos(θ/2), sin(θ/2)).

    -

    This means we can equate the two identities we now have for By and solve for k.

    -
    - -

    Deriving k

    -

    Solving for k is fairly straight forward, but it's a fair few steps, and if you just the immediate result: using a tool like Wolfram Alpha is definitely the way to go. That said, let's get going:

    - - -

    And finally, we can take further advantage of several trigonometric identities to drastically simplify our formula for k:

    - - -

    And we're done.

    -
    + +

    And we're done.

    + -

    So, the distance of our control points to the start/end points can be expressed as a number that we get from an almost trivial expression involving the circular arc's angle:

    - - -

    Which means that for any circular arc with angle θ and radius r, our Bézier approximation based on three points of incidence is:

    - - -

    Which also gives us the commonly found value of 0.55228 for quarter circles, based on them having an angle of half π:

    - - -

    And thus giving us the following Bézier coordinates for a quarter circle of radius r:

    - - -
    + +
    +

    So, how accurate is this?

    +

    + Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points + that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum + deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but + rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we + get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff. +

    +

    + So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, + finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around + that t value, repeating as many times as necessary to get us an arbitrarily precise value of t: +

    -

    So, how accurate is this?

    -

    Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff.

    -

    So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around that t value, repeating as many times as necessary to get us an arbitrarily precise value of t:

    - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return getMostWrongT(radius, bezier, worst_t - stepsize, worst_t + stepsize) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    Plus, how often do you get to write a function with that name?

    -

    Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity first, and then coming back down as we approach 2π)

    - - -
    - - - - - -
    - error plotted for 0 ≤ φ ≤ 2π - - error plotted for 0 ≤ φ ≤ π - - error plotted for 0 ≤ φ ≤ ½π -
    +

    Plus, how often do you get to write a function with that name?

    +

    + Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see + what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity + first, and then coming back down as we approach 2π) +

    + + + + + + + + + + + + + +
    + + + + + +
    + error plotted for 0 ≤ φ ≤ 2π + + error plotted for 0 ≤ φ ≤ π + + error plotted for 0 ≤ φ ≤ ½π +
    -

    That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again:

    -

    +

    + That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we + can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again: +

    +

    -

    Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves.

    -
    +

    + Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At + approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over + quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves. +

    +
    -

    So that's it, kappa is 4/3 · tan(θ/4) , we're done! ...or are we?

    -

    Can we do better?

    -

    Technically: yes, we can. But I'm going to prefix this section with "we can, and we should investigate that possibility, but let me warn you up front that the result is only better if we're going to hard-code the values". We're about to get into the weeds and the standard three-points-of-incidence value is so good already that for most applications, trying to do better won't make any sense at all.

    -

    So with that said: what we calculated above is an upper bound for a best fit Bézier curve for a circular arc: anywhere we don't touch the circular arc in our approximation, we've "overshot" the arc. What if we dropped our value for k just a little, so that the curve starts out as an over-estimation, but then crosses the circular arc, yielding an region of underestimation, and then crosses the circular arc again, with another region of overestimation. This might give us a lower overall error, so let's see what we can do.

    -

    First, let's express the total error (given circular arc angle θ, and some k) using standard calculus notation:

    - - -

    This says that the error function for a given angle and value of k is equal to the "infinite" sum of differences between our curve and the circular arc, as we run t from 0 to 1, using an infinitely small step size. between subsequent t values.

    -

    Now, since we want to find the minimal error, that means we want to know where along this function things go from "error is getting progressively less" to "error is increasing again", which means we want to know where its derivative is zero, which as mathematical expression looks like:

    - - -

    And here we have the most direct application of the Fundamental Theorem of Calculus: the derivative and integral are each other's inverse operations, so they cancel out, leaving us with our original function:

    - - -

    And now we just solve for that... oh wait. We've seen this before. In order to solve this, we'd end up needing to solve this:

    - - -

    And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section on arc lengths—there is no symbolic solution for this equasion. Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer!

    -
    + +

    + And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section + on arc lengths—there is no symbolic solution for this equasion. + Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer! +

    +
    +

    Iterating on a solution

    +

    + By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the + value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's + center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just + binary search our way to the most accurate value for c that gets us that middle case. +

    +

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    -

    Iterating on a solution

    -

    By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just binary search our way to the most accurate value for c that gets us that middle case.

    -

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    - - - - -
    1 - -
    2
    3
    4
    + return binarySearch(radius, angle, points, lowerBound, upperBound) + + + + 2 + + + 3 + + + 4 + + -

    And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and blog posts than you can ever read in a life time:

    +

    + And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and + blog posts than you can ever read in a life time: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return binarySearch(radius, angle, points, value, upperBound) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Using the following radialError function, which samples the curve's approximation of the circular arc over several points (although the first and last point will never contribute anything, so we skip them):

    +

    + Using the following radialError function, which samples the curve's approximation of the circular arc over several points + (although the first and last point will never contribute anything, so we skip them): +

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return err + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a vector, we can get its length to the origin using a magnitude call.

    -

    Examining the result

    -

    Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, for each angle, plot what the difference between the circular arc and the Bézier approximation looks like:

    -

    image-20210419085430711

    -

    Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from 0 to θ:

    -

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    - - -
    - - - - - -
    - max deflection using unit scale - - max deflection at 10x scale - - max deflection at 100x scale -
    +

    + In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a + vector, we can get its length to the origin using a magnitude call. +

    +

    Examining the result

    +

    + Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, + for each angle, plot what the difference between the circular arc and the Bézier approximation looks like: +

    +

    image-20210419085430711

    +

    + Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the + plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from + 0 to θ: +

    +

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    + + + + + + + + + + + + + +
    + + + + + +
    + max deflection using unit scale + + max deflection at 10x scale + + max deflection at 100x scale +
    -

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    -

    As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference.

    -
    +

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    +

    + As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by + 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an + almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference. +

    +
    -

    At this point it should be clear that while, yes, there are improvement to be had, they're essentially insignificant while also being much more computationally expensive.

    -

    TL;DR: just tell me which value I should be using

    -

    It depends on what we need to do. If we just want the best value for quarter circles, and we're going to hard code the value for k, then there is no reason to hard-code the constant k=4/3*tan(pi/8) when you can just as easily hard-code the constant as k=0.551784777779014 instead.

    -

    If you need "the" value for quarter circles, use 0.551785 instead of 0.55228

    -

    However, for dynamic arc approximation, in code that tries to fit circular paths using Bézier paths instead, it should be fairly obvious that the simple function involving a tangent computation, two divisions, and one multiplication, is vastly more performant than running all the code we ended writing just to get a 25% lower error value, and most certainly worth preferring over getting the "more accurate" value.

    -

    If you need to fit Béziers to circular arcs on the fly, use 4/3 * tan(θ/4)

    -

    However, always remember that if you're writing for humans, you can typically use the best of both worlds: as the user interacts with their curves, you should draw their curves instead of drawing approximations of them. If they need to draw circles or circular arcs, draw those, and only approximate them with a Bézier curve when the data needs to be exported to a format that doesn't support those. Ideally with a preview mechanism that highlights where the errors will be, and how large they will be.

    -

    If you're writing code for graphics design by humans, use circular arcs for circular arcs

    -

    And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best k values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).

    - -
    -
    -

    попереднійзмістнаступний
    Approximating Bézier curves with circular arcs

    -

    Let's look at doing the exact opposite of the previous section: rather than approximating circular arc using Bézier curves, let's approximate Bézier curves using circular arcs.

    -

    We already saw in the section on circle approximation that this will never yield a perfect equivalent, but sometimes you need circular arcs, such as when you're working with fabrication machinery, or simple vector languages that understand lines and circles, but not much else.

    -

    The approach is fairly simple: pick a starting point on the curve, and pick two points that are further along the curve. Determine the circle that goes through those three points, and see if it fits the part of the curve we're trying to approximate. Decent fit? Try spacing the points further apart. Bad fit? Try spacing the points closer together. Keep doing this until you've found the "good approximation/bad approximation" boundary, record the "good" arc, and then move the starting point up to overlap the end point we previously found. Rinse and repeat until we've covered the entire curve.

    -

    We already saw how to fit a circle through three points in the section on creating a curve from three points, and finding the arc through those points is straight-forward: pick one of the three points as start point, pick another as an end point, and the arc has to necessarily go from the start point, to the end point, over the remaining point.

    -

    So, how can we convert a Bézier curve into a (sequence of) circular arc(s)?

    -
      -
    • Start at t=0
      • -
    • Pick two points further down the curve at some value m = t + n and e = t + 2n
      • -
    • Find the arc that these points define
      • -
    • Determine how close the found arc is to the curve:
        -
      • Pick two additional points e1 = t + n/2 and e2 = t + n + n/2.
        • -
      • These points, if the arc is a good approximation of the curve interval chosen, should - lie on the circle, so their distance to the center of the circle should be the - same as the distance from any of the three other points to the center.
        • -
      • For point points, determine the (absolute) error between the radius of the circle, and the -actual distance from the center of the circle to the point on the curve.
        • -
      • If this error is too high, we consider the arc bad, and try a smaller interval.
        • -
      -
      • -
    -

    The result of this is shown in the next graphic: we start at a guaranteed failure: s=0, e=1. That's the entire curve. The midpoint is simply at t=0.5, and then we start performing a binary search.

    -
      -
    1. We start with low=0, mid=0.5 and high=1
      2. -
    2. That'll fail, so we retry with the interval halved: {0, 0.25, 0.5}
        -
      • If that arc's good, we move back up by half distance: {0, 0.375, 0.75}.
        • -
      • However, if the arc was still bad, we move down by half the distance: {0, 0.125, 0.25}.
        • -
      -
      3. -
    3. We keep doing this over and over until we have two arcs, in sequence, of which the first arc is good, and the second arc is bad. When we find that pair, we've found the boundary between a good approximation and a bad approximation, and we pick the good arc.
      4. -
    -

    The following graphic shows the result of this approach, with a default error threshold of 0.5, meaning that if an arc is off by a combined half pixel over both verification points, then we treat the arc as bad. This is an extremely simple error policy, but already works really well. Note that the graphic is still interactive, and you can use your up and down arrow keys keys to increase or decrease the error threshold, to see what the effect of a smaller or larger error threshold is.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    With that in place, all that's left now is to "restart" the procedure by treating the found arc's end point as the new to-be-determined arc's starting point, and using points further down the curve. We keep trying this until the found end point is for t=1, at which point we are done. Again, the following graphic allows for up and down arrow key input to increase or decrease the error threshold, so you can see how picking a different threshold changes the number of arcs that are necessary to reasonably approximate a curve:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    So... what is this good for? Obviously, if you're working with technologies that can't do curves, but can do lines and circles, then the answer is pretty straightforward, but what else? There are some reasons why you might need this technique: using circular arcs means you can determine whether a coordinate lies "on" your curve really easily (simply compute the distance to each circular arc center, and if any of those are close to the arc radii, at an angle between the arc start and end, bingo, this point can be treated as lying "on the curve"). Another benefit is that this approximation is "linear": you can almost trivially travel along the arcs at fixed speed. You can also trivially compute the arc length of the approximated curve (it's a bit like curve flattening). The only thing to bear in mind is that this is a lossy equivalence: things that you compute based on the approximation are guaranteed "off" by some small value, and depending on how much precision you need, arc approximation is either going to be super useful, or completely useless. It's up to you to decide which, based on your application!

    - -
    -
    -

    попереднійзмістнаступний
    B-Splines

    -

    No discussion on Bézier curves is complete without also giving mention of that other beast in the curve design space: B-Splines. Easily confused to mean Bézier splines, that's not actually what they are; they are "basis function" splines, which makes a lot of difference, and we'll be looking at those differences in this section. We're not going to dive as deep into B-Splines as we have for Bézier curves (that would be an entire primer on its own) but we'll be looking at how B-Splines work, what kind of maths is involved in computing them, and how to draw them based on a number of parameters that you can pick for individual B-Splines.

    -

    First off: B-Splines are piecewise, polynomial interpolation curves, where the "single curve" is built by performing polynomial interpolation over a set of points, using a sliding window of a fixed number of points. For instance, a "cubic" B-Spline defined by twelve points will have its curve built by evaluating the polynomial interpolation of four points, and the curve can be treated as a lot of different sections, each controlled by four points at a time, such that the full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, ..., {8,9,10,11}, and finally {9,10,11,12}, for eight sections.

    -

    What do they look like? They look like this! Tap on the graphic to add more points, and move points around to see how they map to the spline curve drawn.

    - - - Скрипти вимкнено. показує резервний. - - - -

    The important part to notice here is that we are not doing the same thing with B-Splines that we do for poly-Béziers or Catmull-Rom curves: both of the latter simply define new sections as literally "new sections based on new points", so a 12 point cubic poly-Bézier curve is actually impossible, because we start with a four point curve, and then add three more points for each section that follows, so we can only have 4, 7, 10, 13, 16, etc. point Poly-Béziers. Similarly, while Catmull-Rom curves can grow by adding single points, this addition of a single point introduces three implicit Bézier points. Cubic B-Splines, on the other hand, are smooth interpolations of each possible curve involving four consecutive points, such that at any point along the curve except for our start and end points, our on-curve coordinate is defined by four control points.

    -

    Consider the difference to be this:

    -
      -
    • for Bézier curves, the curve is defined as an interpolation of points, but:
      • -
    • for B-Splines, the curve is defined as an interpolation of curves.
      • -
    -

    In fact, let's look at that again, but this time with the base curves shown, too. Each consecutive four points define one curve:

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have a look at how things work.

    -

    How to compute a B-Spline curve: some maths

    -

    Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the end, just like for Bézier curves), by evaluating the following function:

    - + +

    + In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have + a look at how things work. +

    +

    How to compute a B-Spline curve: some maths

    +

    + Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic + B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the + end, just like for Bézier curves), by evaluating the following function: +

    + - -

    Which, honestly, doesn't tell us all that much. All we can see is that a point on a B-Spline curve is defined as "a mix of all the control points, weighted somehow", where the weighting is achieved through the N(...) function, subscripted with an obvious parameter i, which comes from our summation, and some magical parameter k. So we need to know two things: 1. what does N(t) do, and 2. what is that k? Let's cover both, in reverse order.

    -

    The parameter k represents the "knot interval" over which a section of curve is defined. As we learned earlier, a B-Spline curve is itself an interpolation of curves, and we can treat each transition where a control point starts or stops influencing the total curvature as a "knot on the curve". -Doing so for a degree d B-Spline with n control point gives us d + n + 1 knots, defining d + n intervals along the curve, and it is these intervals that the above k subscript to the N() function applies to.

    -

    Then the N() function itself. What does it look like?

    - - -

    So this is where we see the interpolation: N(t) for an (i,k) pair (that is, for a step in the above summation, on a specific knot interval) is a mix between N(t) for (i,k-1) and N(t) for (i+1,k-1), so we see that this is a recursive iteration where i goes up, and k goes down, so it seem reasonable to expect that this recursion has to stop at some point; obviously, it does, and specifically it does so for the following i/k values:

    - - -

    And this function finally has a straight up evaluation: if a t value lies within a knot-specific interval once we reach a k=1 value, it "counts", otherwise it doesn't. We did cheat a little, though, because for all these values we need to scale our t value first, so that it lies in the interval bounded by knots[d] and knots[n], which are the start point and end point where curvature is controlled by exactly order control points. For instance, for degree 3 (=order 4) and 7 control points, with knot vector [1,2,3,4,5,6,7,8,9,10,11], we map t from [the interval 0,1] to the interval [4,8], and then use that value in the functions above, instead.

    -

    Can we simplify that?

    -

    We can, yes.

    -

    People far smarter than us have looked at this work, and two in particular — Maurice Cox and Carl de Boor — came to a mathematically pleasing solution: to compute a point P(t), we can compute this point by evaluating d(t) on a curve section between knots i and i+1:

    - - -

    This is another recursive function, with k values decreasing from the curve order to 1, and the value α (alpha) defined by:

    - - -

    That looks complicated, but it's not. Computing alpha is just a fraction involving known, plain numbers. And, once we have our alpha value, we also have (1-alpha) because it's a trivial subtraction. Computing the d() function is thus mostly a matter of computing pretty simple arithmetical statements, with some caching of results so we can refer to them as we recurve. While the recursion might see computationally expensive, the total algorithm is cheap, as each step only involves very simple maths.

    -

    Of course, the recursion does need a stop condition:

    - - -

    So, we actually see two stopping conditions: either i becomes 0, in which case d() is zero, or k becomes zero, in which case we get the same "either 1 or 0" that we saw in the N() function above.

    -

    Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily using the same kind of linear interpolation we saw in de Casteljau's algorithm. For instance, if we write out d() for i=3 and k=3, we get the following recursion diagram:

    - - -

    That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. It's really quite elegant.

    -

    One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then summing back up "to the left". We can just start on the right and work our way left immediately.

    -

    Running the computation

    -

    Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on this Michigan Tech page, but an easier to read version is implemented by b-spline.js, so we'll look at its code.

    -

    Given an input value t, we first map the input to a value from the domain [0,1] to the domain [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped t value lies on:

    + +

    + That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a + mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are + themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. + It's really quite elegant. +

    +

    + One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the + d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so + in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point + vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then + summing back up "to the left". We can just start on the right and work our way left immediately. +

    +

    Running the computation

    +

    + Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, + and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on + this Michigan Tech page, but an easier to read version + is implemented by b-spline.js, so we'll look at its code. +

    +

    + Given an input value t, we first map the input to a value from the domain [0,1] to the domain + [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped + t value lies on: +

    - - -
    1 - -
    2
    3
    +} + + + + 2 + + + 3 + + -

    after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU page (updated to use this description's variable names):

    +

    + after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU + page (updated to use this description's variable names): +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those v[i-1] don't try to use an array index that doesn't exist)

    -

    Open vs. closed paths

    -

    Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need more than just a single point.

    -

    Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than separate coordinate objects.

    -

    Manipulating the curve through the knot vector

    -

    The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at:

    -
      -
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. -
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. -
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. -
    4. we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a uniform vector in between.
      5. -
    -

    Uniform B-Splines

    -

    The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to the curvature.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve.

    -

    The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get rid of these, by being clever about how we apply the following uniformity-breaking approach instead...

    -

    Reducing local curve complexity by collapsing intervals

    -

    Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost and the curve "kinks".

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Open-Uniform B-Splines

    -

    By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the curve not starting and ending where we'd kind of like it to:

    -

    For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences that determine the curve shape.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Non-uniform B-Splines

    -

    This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than that any value knots[k+1] should be greater than or equal to knots[k].

    -

    One last thing: Rational B-Splines

    -

    While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like for rational Bézier curves.

    - - - Скрипти вимкнено. показує резервний. - - - - - -

    Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in arbitrary-precision 2D design due to the level of control a NURBS curve offers designers.

    -

    While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the last.

    -

    Extending our implementation to cover rational splines

    -

    The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the extended dimension.

    -

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    -

    We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how many dimensions it needs to interpolate.

    -

    In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight information.

    -

    Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing (x'/w', y'/w'). And that's it, we're done!

    +

    + (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the + interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those + v[i-1] don't try to use an array index that doesn't exist) +

    +

    Open vs. closed paths

    +

    + Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last + point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first + and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As + such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly + more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need + more than just a single point. +

    +

    + Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for + points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same + coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing + references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than + separate coordinate objects. +

    +

    Manipulating the curve through the knot vector

    +

    + The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As + mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the + values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on + intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting + things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at: +

    +
      +
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. +
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. +
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. +
    4. + we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a + uniform vector in between. +
      5. +
    +

    Uniform B-Splines

    +

    + The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire + curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or + [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines + the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to + the curvature. +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    + This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every + interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve. +

    +

    + The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can + perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get + rid of these, by being clever about how we apply the following uniformity-breaking approach instead... +

    +

    Reducing local curve complexity by collapsing intervals

    +

    + Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the + sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, + and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost + and the curve "kinks". +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    Open-Uniform B-Splines

    +

    + By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the + curve not starting and ending where we'd kind of like it to: +

    +

    + For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in + which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values + N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector + [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences + that determine the curve shape. +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    Non-uniform B-Splines

    +

    + This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to + pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than + that any value knots[k+1] should be greater than or equal to knots[k]. +

    +

    One last thing: Rational B-Splines

    +

    + While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's + "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a + ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a + control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like + for rational Bézier curves. +

    + + + Скрипти вимкнено. показує резервний. + + + + + +

    + Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the + NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, + the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an + important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in + arbitrary-precision 2D design due to the level of control a NURBS curve offers designers. +

    +

    + While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform + Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, + they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the + last. +

    +

    Extending our implementation to cover rational splines

    +

    + The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the + control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to + one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the + extended dimension. +

    +

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    +

    + We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate + interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how + many dimensions it needs to interpolate. +

    +

    + In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the + final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight + information. +

    +

    + Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing + (x'/w', y'/w'). And that's it, we're done! +

    +
    +
    + -
    -
    - +

    +
    попереднійзміст
    + Comments and questions +

    +

    + First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how + to let me know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing. +

    +

    With that said, on to the comments!

    +
    +
    + + -

    попереднійзміст
    Comments and questions

    -

    First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing.

    -

    With that said, on to the comments!

    -
    +
    +
    + This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are + almost no rights reserved. Go do something cool with it! +
    - - - - -
    - -
    - - This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are - almost no rights reserved. - Go do something cool with it! - -
    - - - - - - - \ No newline at end of file + + + diff --git a/docs/zh-CN/index.html b/docs/zh-CN/index.html index c9e58a16..5c9834a2 100644 --- a/docs/zh-CN/index.html +++ b/docs/zh-CN/index.html @@ -1,161 +1,177 @@ + + + + 贝塞尔曲线底漆 - - - - 贝塞尔曲线底漆 + - + - + - + + + + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + - - + + + - + + - - +
    + This page on GitHub + + This page on GitHub + +
    -
    - This page on GitHub - - This page on GitHub - -
    +
    + + + submit to reddit + submit to hacker news + tweet your read + +
    -
    - - - submit to reddit - submit to hacker news - tweet your read - -
    + - +
    +

    + 贝塞尔曲线底漆 +

    +

    A free, online book for when you really need to know how to do Bézier things.

    +
    + Read this in your own language: + +

    + (Don't see your language listed, or want to see it reach 100%? + Help translate this content!) +

    +
    +

    + Welcome to the Primer on Bezier Curves. This is a free website/ebook dealing with both the maths and programming aspects of Bezier Curves, + covering a wide range of topics relating to drawing and working with that curve that seems to pop up everywhere, from Photoshop paths to CSS + easing functions to Font outline descriptions. +

    +

    + If this is your first time here: welcome! Let me know if you were looking for anything in particular that the primer doesn't cover over on the + issue tracker! +

    -
    +

    Donations and sponsorship

    -

    贝塞尔曲线底漆

    -

    A free, online book for when you really need to know how to do Bézier things.

    -
    - Read this in your own language: - -

    (Don't see your language listed, or want to see it reach 100%? Help translate this content!)

    -
    +

    + If this is a resource that you're using for research, as work reference, or even writing your own software, please consider + donating (any amount helps) or + signing up as a patron on Patreon. I don't get paid to work on this, so if you find this site + valuable, and you'd like it to stick around for a long time to come, a lot of coffee went into writing this over the years, and a lot more + coffee will need to go into it yet: if you can spare a coffee, you'd be helping keep a resource alive and well. +

    +

    + Also, if you are a company and your staff uses this book as a resource, or you use it as an onboarding resource, then please: consider + sponsoring the site! I am more than happy to work with your finance department on sponsorship invoicing and recognition. +

    -

    - Welcome to the Primer on Bezier Curves. This is a free website/ebook dealing with both - the maths and programming aspects of Bezier Curves, covering a wide range of topics - relating to drawing and working with that curve that seems to pop up everywhere, from - Photoshop paths to CSS easing functions to Font outline descriptions. -

    -

    - If this is your first time here: welcome! Let me know if you were looking for anything - in particular that the primer doesn't cover over on the issue tracker! -

    - -

    Donations and sponsorship

    - -

    - If this is a resource that you're using for research, as work reference, or even writing your own software, - please consider donating - (any amount helps) or signing up as a patron on Patreon. - I don't get paid to work on this, so if you find this site valuable, and you'd like it - to stick around for a long time to come, a lot of coffee went into writing this over the - years, and a lot more coffee will need to go into it yet: if you can spare a coffee, you'd - be helping keep a resource alive and well. -

    -

    - Also, if you are a company and your staff uses this book as a resource, or you use it as an - onboarding resource, then please: consider sponsoring the site! I am more than happy to work - with your finance department on sponsorship invoicing and recognition. -

    - - - -
    +
    -

    - — Pomax -

    - + +
    - -
    - -
    - -

    序言

    -

    我们通常用线条来绘制2D图形，大致分为两种线条：直线和曲线。不论我们动手还是用电脑，都能很容易地画出第一种线条。只要给电脑起点和终点，砰！直线就画出来了。没什么好疑问的。

    -

    然而，绘制曲线却是个大问题。虽然我们可以很容易地徒手画出曲线，但除非给出描述曲线的数学函数，不然计算机无法画出曲线。实际上，画直线时也需要数学函数，但画直线所需的方程式很简单，我们在这里不去考虑。在计算机看来，所有线条都是“函数”，不管它们是直线还是曲线。然而，这就表示我们需要找到能在计算机上表现良好的曲线方程。这样的曲线有很多种，在本文我们主要关注一类特殊的、备受关注的函数，基本上任何画曲线的地方都会用到它：贝塞尔曲线。

    -

    它们是以Pierre Bézier命名的，尽管他并不是第一个，或者说唯一“发明”了这种曲线的人，但他让世界知道了这种曲线十分适合设计工作（在1962年为Renault工作并发表了他的研究）。有人也许会说数学家Paul de Casteljau是第一个发现这类曲线特性的人，在Citroën工作时，他提出了一种很优雅的方法来画这些曲线。然而，de Casteljau没有发表他的工作，这使得“谁先发现”这一问题很难有一个确切的答案。 -贝塞尔曲线本质上是伯恩斯坦多项式，这是Sergei Natanovich Bernstein研究的一种数学函数，关于它们的出版物至少可以追溯到1912年。无论如何，这些都只是一些冷知识，你可能更在意的是这些曲线很方便：你可以连接多条贝塞尔曲线，并且连接起来的曲线看起来就像是一条曲线。甚至，在你在Photoshop中画“路径”或使用一些像Flash、Illustrator和Inkscape这样的矢量绘图程序时，所画的曲线都是贝塞尔曲线。

    -

    那么，要是你自己想编程实现它们呢？有哪些陷阱？你怎么画它们？包围盒是怎么样的，怎么确定交点，怎么拉伸曲线，简单来说：你怎么对曲线做一切你想做的事？这就是这篇文章想说的。准备好学习一些数学吧!

    -
    - -

    注意：几乎所有的贝塞尔图形都是可交互的。

    -

    这个页面使用了基于Bezier.js 的可交互例子。

    - -

    这本书是开源的。

    -

    这本书是开源的软件项目，现有两个github仓库。第一个https://github.com/pomax/bezierinfo，它是你现在在看的这个，纯粹用来展示的版本。另外一个https://github.com/pomax/BezierInfo-2，是带有所有html, javascript和css的开发版本。你可以fork任意一个，随便做些什么，当然除了把它当作自己的作品来商用。 =)

    -

    用到的数学将有多复杂？

    -

    这份入门读物用到的大部分数学知识都是高中所学的。如果你理解基本的计算并能看懂英文的话，就能上手这份材料。有时候会用到复杂一点的数学，但如果你不想深究它们，可以选择跳过段落里的“详解”部分，或者直接跳到章节末尾，避开那些看起来很深入的数学。章节的末尾往往会列出一些结论，因此你可以直接利用这些结论。

    -

    问题，评论：

    -

    如果你有对于新章节的一些建议，点击 Github issue tracker （也可以点右上角的repo链接）。如果你有关于材料的一些问题，由于我现在在做改写工作，目前没有评论功能，但你可以用issue跟踪来发表评论。一旦完成重写工作，我会把评论功能加上，或者会有“选择文字段落，点击‘问题’按钮来提问”的系统。到时候我们看看。

    -

    给我买杯咖啡？

    -

    如果你很喜欢这本书，或发现它对你要做的事很有帮助，或者你想知道怎么表达自己对这本书的感激，你可以 给我买杯咖啡 ，所少钱取决于你。这份工作持续了很多年，从一份小小的简要到70多页关于贝塞尔曲线的读物，在完成它的过程中倾注了很多咖啡。我从未后悔花在这上面的每一分钟，但如果有更多咖啡的话，我可以坚持写下去!

    -
    -
    -
    -

    What's new?

    -

    This primer is a living document, and so depending on when you last look at it, there may be new content. Click the following link to expand this section to have a look at what got added, when, or click through to the News posts for more detailed updates. (RSS feed available)

    - - - -

    November 2020

    • Added a section on finding curve/circle intersections

      -
    -

    October 2020

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      -
    -

    August-September 2020

    • Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS turned on.

      -
    -

    June 2020

    • Added automatic CI/CD using Github Actions

      -
    -

    January 2020

    • Added reset buttons to all graphics

      -
      • -

    • Updated to preface to correctly describe the on-page maths

      -
      • -

    • Fixed the Catmull-Rom section because it had glaring maths errors

      -
    -

    August 2019

    • Added a section on (plain) rational Bezier curves

      -
      • -

    • Improved the Graphic component to allow for sliders

      -
    -

    December 2018

    • Added a section on curvature and calculating kappa.

      -
      • -

    • Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this site!

      -
    -

    August 2018

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      -
    -

    July 2018

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      -
      • -

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      -
      • -

    • (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      -
    -

    June 2018

    • Added a section on direct curve fitting.

      -
      • -

    • Added source links for all graphics.

      -
      • -

    • Added this "What's new?" section.

      -
    -

    April 2017

    • Added a section on 3d normals.

      -
      • -

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      -
    -

    February 2017

    • Finished rewriting the entire codebase for localization.

      -
    -

    January 2016

    • Added a section to explain the Bezier interval.

      -
      • -

    • Rewrote the Primer as a React application.

      -
    -

    December 2015

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      -
      • -

    • Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics.

      -
    -

    May 2015

    • Switched over to pure JS rather than Processing-through-Processing.js

      -
      • -

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      -
    -

    April 2015

    • Added a section on arc length approximations.

      -
    -

    February 2015

    • Added a section on the canonical cubic Bezier form.

      -
    -

    November 2014

    • Switched to HTTPS.

      -
    -

    July 2014

    • Added the section on arc approximation.

      -
    -

    April 2014

    • Added the section on Catmull-Rom fitting.

      -
    -

    November 2013

    • Added the section on Catmull-Rom / Bezier conversion.

      -
      • -

    • Added the section on Bezier cuves as matrices.

      -
    -

    April 2013

    • Added a section on poly-Beziers.

      -
      • -

    • Added a section on boolean shape operations.

      -
    -

    March 2013

    • First drastic rewrite.

      -
      • -

    • Added sections on circle approximations.

      -
      • -

    • Added a section on projecting a point onto a curve.

      -
      • -

    • Added a section on tangents and normals.

      -
      • -

    • Added Legendre-Gauss numerical data tables.

      -
    -

    October 2011

    • First commit for the bezierinfo site, based on the pre-Primer webpage that covered the basics of Bezier curves in HTML with Processing.js examples.

      -
    -
    -
    -

    目录下
    简单介绍

    -

    让我们有个好的开始：当我们在谈论贝塞尔曲线的时候，所指的就是你在如下图像看到的东西。它们从某些起点开始，到终点结束，并且受到一个或多个的“中间”控制点的影响。本页面上的图形都是可交互的，你可以拖动这些点，看看这些形状在你的操作下会怎么变化。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    这本书是开源的。

    +

    + 这本书是开源的软件项目，现有两个github仓库。第一个https://github.com/pomax/bezierinfo，它是你现在在看的这个，纯粹用来展示的版本。另外一个https://github.com/pomax/BezierInfo-2，是带有所有html, javascript和css的开发版本。你可以fork任意一个，随便做些什么，当然除了把它当作自己的作品来商用。 =) +

    +

    用到的数学将有多复杂？

    +

    + 这份入门读物用到的大部分数学知识都是高中所学的。如果你理解基本的计算并能看懂英文的话，就能上手这份材料。有时候会用到复杂一点的数学，但如果你不想深究它们，可以选择跳过段落里的“详解”部分，或者直接跳到章节末尾，避开那些看起来很深入的数学。章节的末尾往往会列出一些结论，因此你可以直接利用这些结论。 +

    +

    问题，评论：

    +

    + 如果你有对于新章节的一些建议，点击 + Github issue tracker + （也可以点右上角的repo链接）。如果你有关于材料的一些问题，由于我现在在做改写工作，目前没有评论功能，但你可以用issue跟踪来发表评论。一旦完成重写工作，我会把评论功能加上，或者会有“选择文字段落，点击‘问题’按钮来提问”的系统。到时候我们看看。 +

    +

    给我买杯咖啡？

    +

    + 如果你很喜欢这本书，或发现它对你要做的事很有帮助，或者你想知道怎么表达自己对这本书的感激，你可以 + 给我买杯咖啡 + ，所少钱取决于你。这份工作持续了很多年，从一份小小的简要到70多页关于贝塞尔曲线的读物，在完成它的过程中倾注了很多咖啡。我从未后悔花在这上面的每一分钟，但如果有更多咖啡的话，我可以坚持写下去! +

    +
    + +
    +

    What's new?

    +

    + This primer is a living document, and so depending on when you last look at it, there may be new content. Click the following link to expand + this section to have a look at what got added, when, or click through to the News posts for more detailed updates. (RSS feed + available) +

    + + + +
    +

    November 2020

    +
      +

    • Added a section on finding curve/circle intersections

      • +
    +

    October 2020

    +
      +

    • Added the Ukranian locale! Help out in getting its localization to 100%!

      • +
    +

    August-September 2020

    +
      +
    • +

      + Completely overhauled the site: the Primer is now a normal web page that works fine with JS disabled, but obviously better with JS + turned on. +

      +
      • +
    +

    June 2020

    +
      +

    • Added automatic CI/CD using Github Actions

      • +
    +

    January 2020

    +
      +

    • Added reset buttons to all graphics

      • +

    • Updated to preface to correctly describe the on-page maths

      • +

    • Fixed the Catmull-Rom section because it had glaring maths errors

      • +
    +

    August 2019

    +
      +

    • Added a section on (plain) rational Bezier curves

      • +

    • Improved the Graphic component to allow for sliders

      • +
    +

    December 2018

    +
      +

    • Added a section on curvature and calculating kappa.

      • +
    • +

      + Added a Patreon page! Head on over to patreon.com/bezierinfo to help support this + site! +

      +
      • +
    +

    August 2018

    +
      +

    • Added a section on finding a curve's y, if all you have is the x coordinate.

      • +
    +

    July 2018

    +
      +

    • Rewrote the 3D normals section, implementing and explaining Rotation Minimising Frames.

      • +

    • Updated the section on curve order raising/lowering, showing how to get a least-squares optimized lower order curve.

      • +
    • +

      (Finally) updated 'npm test' so that it automatically rebuilds when files are changed while the dev server is running.

      +
      • +
    +

    June 2018

    +
      +

    • Added a section on direct curve fitting.

      • +

    • Added source links for all graphics.

      • +

    • Added this "What's new?" section.

      • +
    +

    April 2017

    +
      +

    • Added a section on 3d normals.

      • +

    • Added live-updating for the social link buttons, so they always link to the specific section you're reading.

      • +
    +

    February 2017

    +
      +

    • Finished rewriting the entire codebase for localization.

      • +
    +

    January 2016

    +
      +

    • Added a section to explain the Bezier interval.

      • +

    • Rewrote the Primer as a React application.

      • +
    +

    December 2015

    +
      +

    • Set up the split repository between BezierInfo-2 as development repository, and bezierinfo as live page.

      • +
    • +

      + Removed the need for client-side LaTeX parsing entirely, so the site doesn't take a full minute or more to load all the graphics. +

      +
      • +
    +

    May 2015

    +
      +

    • Switched over to pure JS rather than Processing-through-Processing.js

      • +

    • Added Cardano's algorithm for finding the roots of a cubic polynomial.

      • +
    +

    April 2015

    +
      +

    • Added a section on arc length approximations.

      • +
    +

    February 2015

    +
      +

    • Added a section on the canonical cubic Bezier form.

      • +
    +

    November 2014

    +
      +

    • Switched to HTTPS.

      • +
    +

    July 2014

    +
      +

    • Added the section on arc approximation.

      • +
    +

    April 2014

    +
      +

    • Added the section on Catmull-Rom fitting.

      • +
    +

    November 2013

    +
      +

    • Added the section on Catmull-Rom / Bezier conversion.

      • +

    • Added the section on Bezier cuves as matrices.

      • +
    +

    April 2013

    +
      +

    • Added a section on poly-Beziers.

      • +

    • Added a section on boolean shape operations.

      • +
    +

    March 2013

    +
      +

    • First drastic rewrite.

      • +

    • Added sections on circle approximations.

      • +

    • Added a section on projecting a point onto a curve.

      • +

    • Added a section on tangents and normals.

      • +

    • Added Legendre-Gauss numerical data tables.

      • +
    +

    October 2011

    +
      +
    • +

      + First commit for the bezierinfo site, based on the pre-Primer webpage that covered + the basics of Bezier curves in HTML with Processing.js examples. +

      +
      • +
    +
    +
    +
    +
    +

    +
    目录下
    + 简单介绍 +

    +

    + 让我们有个好的开始：当我们在谈论贝塞尔曲线的时候，所指的就是你在如下图像看到的东西。它们从某些起点开始，到终点结束，并且受到一个或多个的“中间”控制点的影响。本页面上的图形都是可交互的，你可以拖动这些点，看看这些形状在你的操作下会怎么变化。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    这些曲线在计算机辅助设计和计算机辅助制造应用（CAD/CAM）中用的很多。在图形设计软件中也常用到，像Adobe Illustrator, Photoshop, Inkscape, Gimp等等。还可以应用在一些图形技术中，像矢量图形（SVG）和OpenType字体（ttf/otf）。许多东西都用到贝塞尔曲线，如果你想更了解它们...准备好继续往下学吧！

    - -
    -
    -

    前目录下
    什么构成了贝塞尔曲线？

    -

    操作点的移动，看看曲线的变化，可能让你感受到了贝塞尔曲线是如何表现的。但贝塞尔曲线究竟是什么呢？有两种方式来解释贝塞尔曲线，并且可以证明它们完全相等，但是其中一种用到了复杂的数学，另外一种比较简单。所以...我们先从简单的开始吧：

    -

    贝塞尔曲线是线性插值的结果。这听起来很复杂，但你在很小的时候就做过线性插值：当你指向两个物体中的另外一个物体时，你就用到了线性插值。它就是很简单的“选出两点之间的一个点”。

    -

    如果我们知道两点之间的距离，并想找出离第一个点20%间距的一个新的点(也就是离第二个点80%的间距)，我们可以通过简单的计算来得到：

    - - -

    让我们来通过实际操作看一下：下面的图形都是可交互的，因此你可以通过上下键来增加或减少插值距离，来观察图形的变化。我们从三个点构成的两条线段开始。通过对各条线段进行线性插值得到两个点，对点之间的线段再进行线性插值，产生一个新的点。最终这些点——所有的点都可以通过选取不同的距离插值产生——构成了贝塞尔曲线 -： - - - Scripts are disabled. Showing fallback image. - - - - - -

    这为我们引出了复杂的数学：微积分。

    -

    虽然我们刚才好像没有用到这个，我们实际上只是逐步地画了一条二次曲线，而不是一次画好。贝塞尔曲线的一个很棒的特性就是它们可以通过多项式方程表示，也可以用很简单的插值形式表示。因此，反过来说，我们可以基于“真正的数学”（检查方程式，导数之类的东西），也可以通过观察曲线的“机械”构成（比如说，可以得知曲线永远不会延伸超过我们用来构造它的点），来看看这些曲线能够做什么。

    -

    让我们从更深的层次来观察贝塞尔曲线。看看它们的数学表达式，从这些表达式衍生得到的属性，以及我们可以对贝塞尔曲线做的事。

    + +

    + 让我们来通过实际操作看一下：下面的图形都是可交互的，因此你可以通过上下键来增加或减少插值距离，来观察图形的变化。我们从三个点构成的两条线段开始。通过对各条线段进行线性插值得到两个点，对点之间的线段再进行线性插值，产生一个新的点。最终这些点——所有的点都可以通过选取不同的距离插值产生——构成了贝塞尔曲线 + ： + + + Scripts are disabled. Showing fallback image. + + + + + +

    -
    -
    -

    前目录下
    贝塞尔曲线的数学原理

    -

    贝塞尔曲线是“参数”方程的一种形式。从数学上讲，参数方程作弊了：“方程”实际上是一个从输入到唯一输出的、良好定义的映射关系。几个输入进来，一个输出返回。改变输入变量，还是只有一个输出值。参数方程在这里作弊了。它们基本上干了这么件事，“好吧，我们想要更多的输出值，所以我们用了多个方程”。举个例子：假如我们有一个方程，通过一些计算，将假设为x的一些值映射到另外的值:

    - - -

    记号f(x)是表示函数的标准方式（为了方便起见，如果只有一个的话，我们称函数为f），函数的输出根据一个变量（本例中是x）变化。改变x，f(x)的输出值也会变。

    -

    到目前没什么问题。现在，让我们来看一下参数方程，以及它们是怎么作弊的。我们取以下两个方程：

    - - -

    这俩方程没什么让人印象深刻的，只不过是正弦函数和余弦函数，但正如你所见，输入变量有两个不同的名字。如果我们改变了a的值，f(b)的输出不会有变化，因为这个方程没有用到a。参数方程通过改变这点来作弊。在参数方程中，所有不同的方程共用一个变量，如下所示：

    - - -

    多个方程，但只有一个变量。如果我们改变了t的值，fa(t)和fb(t)的输出都会发生变化。你可能会好奇这有什么用，答案其实很简单：对于参数曲线，如果我们用常用的标记来替代fa(t)和fb(t)，看起来就有些明朗了：

    - - -

    好了，通过一些神秘的t值将x/y坐标系联系起来。

    -

    所以，参数曲线不像一般函数那样，通过x坐标来定义y坐标，而是用一个“控制”变量将它们连接起来。如果改变t的值，每次变化时我们都能得到两个值，这可以作为图形中的(x,y)坐标。比如上面的方程组，生成位于一个圆上的点：我们可以使t在正负极值间变化，得到的输出(x,y)都会位于一个以原点(0,0)为中心且半径为1的圆上。如果我们画出t从0到5时的值，将得到如下图像：

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    贝塞尔曲线是（一种）参数方程，并在它的多个维度上使用相同的基本方程。在上述的例子中x值和y值使用了不同的方程，与此不同的是，贝塞尔曲线的x和y都用了“二项多项式”。那什么是二项多项式呢？

    -

    你可能记得高中所学的多项式，看起来像这样：

    - - -

    如果它的最高次项是x³就称为“三次”多项式，如果最高次项是x²，称为“二次”多项式，如果只含有x的项，它就是一条线（不过不含任何x的项它就不是一个多项式！）

    -

    贝塞尔曲线不是x的多项式，它是t的多项式，t的值被限制在0和1之间，并且含有a，b等参数。它采用了二次项的形式，听起来很神奇但实际上就是混合不同值的简单描述：

    - - -

    我明白你在想什么：这看起来并不简单，但如果我们拿掉t并让系数乘以1，事情就会立马简单很多，看看这些二次项：

    - - -

    需要注意的是，2与1+1相同，3相当于2+1或1+2，6相当于3+3...如你所见，每次我们增加一个维度，只要简单地将头尾置为1，中间的操作都是“将上面的两个数字相加”。现在就能很容易地记住了。

    -

    还有一个简单的办法可以弄清参数项怎么工作的：如果我们将(1-t)重命名为a，将t重命名为b，暂时把权重删掉，可以得到这个：

    - - -

    基本上它就是“每个a和b结合项”的和，在每个加号后面逐步的将a换成b。因此这也很简单。现在你已经知道了二次多项式，为了叙述的完整性，我将给出一般方程：

    - - -

    这就是贝塞尔曲线完整的描述。在这个函数中的Σ表示了这是一系列的加法（用Σ下面的变量，从...=<值>开始，直到Σ上面的数字结束）。

    -
    + +

    这就是贝塞尔曲线完整的描述。在这个函数中的Σ表示了这是一系列的加法（用Σ下面的变量，从...=<值>开始，直到Σ上面的数字结束）。

    +
    +

    如何实现基本方程

    +

    我们可以用之前说过的方程，来简单地实现基本方程作为数学构造，如下：

    -

    如何实现基本方程

    -

    我们可以用之前说过的方程，来简单地实现基本方程作为数学构造，如下：

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    我说我们“可以用”是因为我们不会这么去做：因为阶乘函数开销非常大。并且，正如我们在上面所看到的，我们不用阶乘也能够很容易地构造出帕斯卡三角形：一开始是[1]，接着是[1,2,1],然后是[1,3,3,1]等等。下一行都比上一行多一个数，首尾都为1，中间的数字是上一行两边元素的和。

    -

    我们可以很快的生成这个列表，并在之后使用这个查找表而不用再计算二次多项式的系数：

    +

    + 我说我们“可以用”是因为我们不会这么去做：因为阶乘函数开销非常大。并且，正如我们在上面所看到的，我们不用阶乘也能够很容易地构造出帕斯卡三角形：一开始是[1]，接着是[1,2,1],然后是[1,3,3,1]等等。下一行都比上一行多一个数，首尾都为1，中间的数字是上一行两边元素的和。 +

    +

    我们可以很快的生成这个列表，并在之后使用这个查找表而不用再计算二次多项式的系数：

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    + return lut[n][k] + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + -

    这里做了些什么？首先，我们声明了一个足够大的查找表。然后，我们声明了一个函数来获取我们想要的值，并且确保当一个请求的n/k对不在LUT查找表中时，先将表扩大。我们的基本函数如下所示：

    +

    + 这里做了些什么？首先，我们声明了一个足够大的查找表。然后，我们声明了一个函数来获取我们想要的值，并且确保当一个请求的n/k对不在LUT查找表中时，先将表扩大。我们的基本函数如下所示： +

    - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    完美。当然我们可以进一步优化。为了大部分的计算机图形学目的，我们不需要任意的曲线。我们需要二次曲线和三次曲线（实际上这篇文章没有涉及任意次的曲线，因此你会在其他地方看到与这些类似的代码），这说明我们可以彻底简化代码:

    +

    + 完美。当然我们可以进一步优化。为了大部分的计算机图形学目的，我们不需要任意的曲线。我们需要二次曲线和三次曲线（实际上这篇文章没有涉及任意次的曲线，因此你会在其他地方看到与这些类似的代码），这说明我们可以彻底简化代码: +

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return mt3 + 3*mt2*t + 3*mt*t2 + t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    现在我们知道如何代用码实现基本方程了。很好。

    -
    +

    现在我们知道如何代用码实现基本方程了。很好。

    +
    -

    既然我们已经知道基本函数的样子，是时候添加一些魔法来使贝塞尔曲线变得特殊了：控制点。

    +

    既然我们已经知道基本函数的样子，是时候添加一些魔法来使贝塞尔曲线变得特殊了：控制点。

    +
    +
    +

    +
    前目录下
    + 控制贝塞尔的曲率 +

    +

    + 贝塞尔曲线是插值方程（就像所有曲线一样），这表示它们取一系列的点，生成一些处于这些点之间的值。（一个推论就是你永远无法生成一个位于这些控制点轮廓线外面的点，更普遍是称为曲线的外壳。这信息很有用！）实际上，我们可以将每个点对方程产生的曲线做出的贡献进行可视化，因此可以看出曲线上哪些点是重要的，它们处于什么位置。 +

    +

    + 下面的图形显示了二次曲线和三次曲线的差值方程，“S”代表了点对贝塞尔方程总和的贡献。点击拖动点来看看在特定的t值时，每个曲线定义的点的插值百分比。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    前目录下
    控制贝塞尔的曲率

    -

    贝塞尔曲线是插值方程（就像所有曲线一样），这表示它们取一系列的点，生成一些处于这些点之间的值。（一个推论就是你永远无法生成一个位于这些控制点轮廓线外面的点，更普遍是称为曲线的外壳。这信息很有用！）实际上，我们可以将每个点对方程产生的曲线做出的贡献进行可视化，因此可以看出曲线上哪些点是重要的，它们处于什么位置。

    -

    下面的图形显示了二次曲线和三次曲线的差值方程，“S”代表了点对贝塞尔方程总和的贡献。点击拖动点来看看在特定的t值时，每个曲线定义的点的插值百分比。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    上面有一张是15th阶的插值方程。如你所见，在所有控制点中，起点和终点对曲线形状的贡献比其他点更大些。

    -

    如果我们要改变曲线，就需要改变每个点的权重，有效地改变插值。可以很直接地做到这个：只要用一个值乘以每个点，来改变它的强度。这个值照惯例称为“权重”，我们可以将它加入我们原始的贝塞尔函数：

    - - -

    看起来很复杂，但实际上“权重”只是我们想让曲线所拥有的坐标值：对于一条nth阶曲线，w0是起始坐标，wn是终点坐标，中间的所有点都是控制点坐标。假设说一条曲线的起点为（110，150），终点为（210，30），并受点（25，190）和点（210，250）的控制，贝塞尔曲线方程就为：

    - - -

    这就是我们在文章开头看到的曲线：

    - + +

    这就是我们在文章开头看到的曲线：

    + -

    我们还能对贝塞尔曲线做些什么？实际上还有很多。文章接下来涉及到我们可能运用到的一系列操作和算法，以及它们可以完成的任务。

    -
    +

    我们还能对贝塞尔曲线做些什么？实际上还有很多。文章接下来涉及到我们可能运用到的一系列操作和算法，以及它们可以完成的任务。

    +
    +

    如何实现权重基本函数

    +

    鉴于我们已经知道怎样实现基本函数，在其加入控制点是非常简单的：

    -

    如何实现权重基本函数

    -

    鉴于我们已经知道怎样实现基本函数，在其加入控制点是非常简单的：

    - - - - - -
    1 - -
    2
    3
    4
    5
    + return sum + + + + 2 + + + 3 + + + 4 + + + 5 + + -

    下面是优化过的版本：

    +

    下面是优化过的版本：

    - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return w[0]*mt3 + 3*w[1]*mt2*t + 3*w[2]*mt*t2 + w[3]*t3 + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    现在我们知道如何编程实现基本权重函数了。

    -
    - -
    -
    -

    前目录下
    Controlling Bézier curvatures, part 2: Rational Béziers

    -

    We can further control Bézier curves by "rationalising" them: that is, adding a "ratio" value in addition to the weight value discussed in the previous section, thereby gaining control over "how strongly" each coordinate influences the curve.

    -

    Adding these ratio values to the regular Bézier curve function is fairly easy. Where the regular function is the following:

    - - -

    The function for rational Bézier curves has two more terms:

    - - -

    In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just using different weight. So far, nothing too special.

    -

    However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights.

    -

    This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly describing circles (which we'll see in a later section is literally impossible using standard Bézier curves).

    -

    But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects what gets drawn:

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - -

    You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence relative to all other points.

    -
    + +

    + In this, the first new term represents an additional weight for each coordinate. For example, if our ratio values are [1, 0.5, 0.5, 1] + then ratio0 = 1, ratio1 = 0.5, and so on, and is effectively identical as if we were just + using different weight. So far, nothing too special. +

    +

    + However, the second new term is what makes the difference: every point on the curve isn't just a "double weighted" point, it is a + fraction of the "doubly weighted" value we compute by introducing that ratio. When computing points on the curve, we compute the + "normal" Bézier value and then divide that by the Bézier value for the curve that only uses ratios, not weights. +

    +

    + This does something unexpected: it turns our polynomial into something that isn't a polynomial anymore. It is now a kind of curve + that is a super class of the polynomials, and can do some really cool things that Bézier curves can't do "on their own", such as perfectly + describing circles (which we'll see in a later section is literally impossible using standard Bézier curves). +

    +

    + But the best way to show what this does is to do literally that: let's look at the effect of "rationalising" our Bézier curves using an + interactive graphic for a rationalised curves. The following graphic shows the Bézier curve from the previous section, "enriched" with + ratio factors for each coordinate. The closer to zero we set one or more terms, the less relative influence the associated coordinate + exerts on the curve (and of course the higher we set them, the more influence they have). Try to change the values and see how it affects + what gets drawn: +

    + + + Scripts are disabled. Showing fallback image. + + + + + + + + +

    + You can think of the ratio values as each coordinate's "gravity": the higher the gravity, the closer to that coordinate the curve will + want to be. You'll also notice that if you simply increase or decrease all the ratios by the same amount, nothing changes... much like + with gravity, if the relative strengths stay the same, nothing really changes. The values define each coordinate's influence + relative to all other points. +

    +
    +

    How to implement rational curves

    +

    Extending the code of the previous section to include ratios is almost trivial:

    -

    How to implement rational curves

    -

    Extending the code of the previous section to include ratios is almost trivial:

    - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    + return (f[0] * w[0] + f[1] * w[1] + f[2] * w[2] + f[3] * w[3])/basis + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + -

    And that's all we have to do.

    -
    - -
    -
    -

    前目录下
    贝塞尔区间[0,1]

    -

    既然我们知道了贝塞尔曲线背后的数学原理，你可能会注意到一件奇怪的事：它们都是从t=0到t=1。为什么是这个特殊区间？

    -

    这一切都与我们如何从曲线的“起点”变化到曲线“终点”有关。如果有一个值是另外两个值的混合，一般方程如下：

    - - -

    很显然，起始值需要a=1, b=0，混合值就为100%的value 1和0%的value 2。终点值需要a=0, b=1，则混合值是0%的value 1和100%的value 2。另外，我们不想让“a”和“b”是互相独立的:如果它们是互相独立的话，我们可以任意选出自己喜欢的值，并得到混合值，比如说100%的value1和100%的value2。原则上这是可以的，但是对于贝塞尔曲线来说，我们通常想要的是起始值和终点值之间的混合值，所以要确保我们不会设置一些“a”和"b"而导致混合值超过100%。这很简单：

    - - -

    用这个式子我们可以保证相加的值永远不会超过100%。通过将a限制在区间[0，1],我们将会一直处于这两个值之间（包括这两个端点），并且相加为100%。

    -

    但是...如果我们没有假定只使用0到1之间的数，而是用一些区间外的值呢，事情会变得很糟糕吗？好吧...不全是，我们接下来看看。

    -

    对于贝塞尔曲线的例子，扩展区间只会使我们的曲线“保持延伸”。贝塞尔曲线是多项式曲线上简单的片段，如果我们选一个更大的区间，会看到曲线更多部分。它们看起来是什么样的呢？

    -

    下面两个图形给你展示了以“普通方式”来渲染的贝塞尔曲线，以及如果我们扩大t值时它们所“位于”的曲线。如你所见，曲线的剩余部分隐藏了很多“形状”，我们可以通过移动曲线的点来建模这部分。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    + +

    + 用这个式子我们可以保证相加的值永远不会超过100%。通过将a限制在区间[0，1],我们将会一直处于这两个值之间（包括这两个端点），并且相加为100%。 +

    +

    但是...如果我们没有假定只使用0到1之间的数，而是用一些区间外的值呢，事情会变得很糟糕吗？好吧...不全是，我们接下来看看。

    +

    + 对于贝塞尔曲线的例子，扩展区间只会使我们的曲线“保持延伸”。贝塞尔曲线是多项式曲线上简单的片段，如果我们选一个更大的区间，会看到曲线更多部分。它们看起来是什么样的呢？ +

    +

    + 下面两个图形给你展示了以“普通方式”来渲染的贝塞尔曲线，以及如果我们扩大t值时它们所“位于”的曲线。如你所见，曲线的剩余部分隐藏了很多“形状”，我们可以通过移动曲线的点来建模这部分。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    实际上，图形设计和计算机建模中还用了一些和贝塞尔曲线相反的曲线，这些曲线没有固定区间和自由的坐标，相反，它们固定座标但给你自由的区间。"Spiro"曲线就是一个很好的例子，它的构造是基于羊角螺线,也就是欧拉螺线的一部分。这是在美学上很令人满意的曲线，你可以在一些图形包中看到它，比如FontForge和Inkscape，它也被用在一些字体设计中（比如Inconsolata字体）。

    - -
    -
    -

    前目录下
    用矩阵运算来表示贝塞尔曲率

    -

    通过将贝塞尔公式表示成一个多项式基本方程、系数矩阵以及实际的坐标，我们也可以用矩阵运算来表示贝塞尔。让我们看一下这对三次曲线来说有什么含义：

    - - -

    暂时不用管我们具体的坐标，现在有：

    - - -

    可以将它写成四个表达式之和：

    - - -

    我们可以扩展这些表达式：

    - - -

    更进一步，我们可以加上所有的1和0系数，以便看得更清楚：

    - - -

    现在，我们可以将它看作四个矩阵运算：

    - - -

    如果我们将它压缩到一个矩阵操作里，就能得到：

    - - -

    这种多项式表达式一般是以递增的顺序来写的，所以我们应该将t矩阵水平翻转，并将大的那个“混合”矩阵上下颠倒：

    - - -

    最终，我们可以加入原始的坐标，作为第三个单独矩阵：

    - - -

    我们可以对二次曲线运用相同的技巧，可以得到：

    - - -

    如果我们代入t值并乘以矩阵来计算，得到的值与解原始多项式方程或用逐步线性插值计算的结果一样。

    -

    因此：为什么我们要用矩阵来计算？ 用矩阵形式来表达曲线可以让我们去探索函数的一些很难被发现的性质。可以证明的是曲线构成了三角矩阵，并且它与我们用在曲线中的实际坐标的求积相同。它还是可颠倒的，这说明可以满足大量特性。当然，主要问题是：“现在，为什么这些对我们很有用？”，答案就是这些并不是立刻就很有用，但是以后你会看到在一些例子中，曲线的一些属性可以用函数式来计算，也可以巧妙地用矩阵运算来得到，有时候矩阵方法要快得多。

    -

    所以，现在只要记着我们可以用这种形式来表示曲线，让我们接着往下看看。

    + +

    如果我们代入t值并乘以矩阵来计算，得到的值与解原始多项式方程或用逐步线性插值计算的结果一样。

    +

    + 因此：为什么我们要用矩阵来计算？ 用矩阵形式来表达曲线可以让我们去探索函数的一些很难被发现的性质。可以证明的是曲线构成了三角矩阵，并且它与我们用在曲线中的实际坐标的求积相同。它还是可颠倒的，这说明可以满足大量特性。当然，主要问题是：“现在，为什么这些对我们很有用？”，答案就是这些并不是立刻就很有用，但是以后你会看到在一些例子中，曲线的一些属性可以用函数式来计算，也可以巧妙地用矩阵运算来得到，有时候矩阵方法要快得多。 +

    +

    所以，现在只要记着我们可以用这种形式来表示曲线，让我们接着往下看看。

    +
    +
    +

    +
    前目录下
    + de Casteljau's 算法 +

    +

    + 要绘制贝塞尔曲线，我们可以从0到1遍历t的所有值，计算权重函数，得到需要画的x/y值。但曲线越复杂，计算量也变得越大。我们可以利用“de + Casteljau算法"，这是一种几何画法，并且易于实现。实际上，你可以轻易地用笔和尺画出曲线。 +

    +

    我们用以下步骤来替代用t计算x/y的微积分算法：

    +
      +
    • 把t看做一个比例（实际上它就是），t=0代表线段的0%，t=1代表线段的100%。
      • +
    • 画出所有点的连线，对n阶曲线来说可以画出n条线。
      • +
    • 在每条线的t处做一个记号。比如t是0.2，就在离起点20%（离终点80%）的地方做个记号。
      • +
    • 连接这些点，得到n-1条线。
      • +
    • 在这些新得到的线上同样用t为比例标记。
      • +
    • 把相邻的那些点连线，得到n-2条线。
      • +
    • 取记号，连线，取记号，等等。
      • +
    • 重复这些步骤，直到剩下一条线。这条线段上的t点就是原始曲线在t处的点。
      • +
    +

    + 我们通过实际操作来观察这个过程。在以下的图表中，移动鼠标来改变用de + Casteljau算法计算得到的曲线点，左右移动鼠标，可以实时看到曲线是如何生成的。 +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +

    如何实现de Casteljau算法

    +

    让我们使用刚才描述过的算法，并实现它：

    -
    -
    -

    前目录下
    de Casteljau's 算法

    -

    要绘制贝塞尔曲线，我们可以从0到1遍历t的所有值，计算权重函数，得到需要画的x/y值。但曲线越复杂，计算量也变得越大。我们可以利用“de Casteljau算法"，这是一种几何画法，并且易于实现。实际上，你可以轻易地用笔和尺画出曲线。

    -

    我们用以下步骤来替代用t计算x/y的微积分算法：

    -
      -
    • 把t看做一个比例（实际上它就是），t=0代表线段的0%，t=1代表线段的100%。
      • -
    • 画出所有点的连线，对n阶曲线来说可以画出n条线。
      • -
    • 在每条线的t处做一个记号。比如t是0.2，就在离起点20%（离终点80%）的地方做个记号。
      • -
    • 连接这些点，得到n-1条线。
      • -
    • 在这些新得到的线上同样用t为比例标记。
      • -
    • 把相邻的那些点连线，得到n-2条线。
      • -
    • 取记号，连线，取记号，等等。
      • -
    • 重复这些步骤，直到剩下一条线。这条线段上的t点就是原始曲线在t处的点。
      • -
    -

    我们通过实际操作来观察这个过程。在以下的图表中，移动鼠标来改变用de Casteljau算法计算得到的曲线点，左右移动鼠标，可以实时看到曲线是如何生成的。

    - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    如何实现de Casteljau算法

    -

    让我们使用刚才描述过的算法，并实现它：

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    好了，这就是算法的实现。一般来说你不能随意重载“+”操作符，因此我们给出计算x和y坐标的实现：

    +

    好了，这就是算法的实现。一般来说你不能随意重载“+”操作符，因此我们给出计算x和y坐标的实现：

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    以上算法做了什么？如果参数points列表只有一个点， 就画出一个点。如果有多个点，就生成以t为比例的一系列点（例如，以上算法中的"标记点"），然后为新的点列表调用绘制函数。

    -
    +

    + 以上算法做了什么？如果参数points列表只有一个点， + 就画出一个点。如果有多个点，就生成以t为比例的一系列点（例如，以上算法中的"标记点"），然后为新的点列表调用绘制函数。 +

    + +
    +
    +

    +
    前目录下
    + 简化绘图 +

    +

    + 我们可以简化绘制的过程，先在具体的位置“采样”曲线，然后用线段把这些点连接起来。由于我们是将曲线转换成一系列“平整的”直线，故将这个过程称之为“拉平(flattening)”。 +

    +

    + 我们可以先确定“想要X个分段”，然后在间隔的地方采样曲线，得到一定数量的分段。这种方法的优点是速度很快：比起遍历100甚至1000个曲线坐标，我们可以采样比较少的点，仍然得到看起来足够好的曲线。这么做的缺点是，我们失去了“真正的曲线”的精度，因此不能用此方法来做真实的相交检测或曲率对齐。 +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    前目录下
    简化绘图

    -

    我们可以简化绘制的过程，先在具体的位置“采样”曲线，然后用线段把这些点连接起来。由于我们是将曲线转换成一系列“平整的”直线，故将这个过程称之为“拉平(flattening)”。

    -

    我们可以先确定“想要X个分段”，然后在间隔的地方采样曲线，得到一定数量的分段。这种方法的优点是速度很快：比起遍历100甚至1000个曲线坐标，我们可以采样比较少的点，仍然得到看起来足够好的曲线。这么做的缺点是，我们失去了“真正的曲线”的精度，因此不能用此方法来做真实的相交检测或曲率对齐。

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    +

    + 试着点击图形，并用上下键来降低二次曲线和三次曲线的分段数量。你会发现对某些曲率来说，数量少的分段也能做的很好，但对于复杂的曲率（在三次曲线上试试），足够多的分段才能很好地满足曲率的变化。 +

    +
    +

    如何实现曲线的拉平

    +

    让我们来实现刚才简述过的算法：

    -

    试着点击图形，并用上下键来降低二次曲线和三次曲线的分段数量。你会发现对某些曲率来说，数量少的分段也能做的很好，但对于复杂的曲率（在三次曲线上试试），足够多的分段才能很好地满足曲率的变化。

    -
    - -

    如何实现曲线的拉平

    -

    让我们来实现刚才简述过的算法：

    - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return coordinates; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    好了，这就是算法的实现。它基本上是画出一系列的线段来模拟“曲线”。

    +

    好了，这就是算法的实现。它基本上是画出一系列的线段来模拟“曲线”。

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + coord = _coord + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    我们将第一个坐标作为参考点，然后在相邻两个点之间画线。

    -
    +

    我们将第一个坐标作为参考点，然后在相邻两个点之间画线。

    +
    +
    +
    +

    +
    前目录下
    + 分割曲线 +

    +

    + 使用 de Casteljau 算法我们也可以将一条贝塞尔曲线分割成两条更小的曲线，二者拼接起来即可形成原来的曲线。当采用某个 t 值构造 de + Casteljau 算法时，该过程会给到我们在 t 点分割曲线的所有点: + 一条曲线包含该曲线上点之前的所有点，另一条曲线包含该曲线上点之后的所有点。 +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +

    分割曲线的代码实习

    +

    通过在 de Casteljau 函数里插入一些额外的输出代码，我们就可以实现曲线的分割：

    -
    -
    -

    前目录下
    分割曲线

    -

    使用 de Casteljau 算法我们也可以将一条贝塞尔曲线分割成两条更小的曲线，二者拼接起来即可形成原来的曲线。当采用某个 t 值构造 de Casteljau 算法时，该过程会给到我们在 t 点分割曲线的所有点: 一条曲线包含该曲线上点之前的所有点，另一条曲线包含该曲线上点之后的所有点。

    - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    分割曲线的代码实习

    -

    通过在 de Casteljau 函数里插入一些额外的输出代码，我们就可以实现曲线的分割：

    - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    + drawCurvePoint(newpoints, t) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + -

    对某个给定 t 值，该函数执行后，数组 left 和 right 将包含两条曲线的所有点的坐标 -- 一条是t值左侧的曲线，一条是t值右侧的曲线， 与原始曲线同序且完全重合。

    -
    - -
    -
    -

    前目录下
    Splitting curves using matrices

    -

    Another way to split curves is to exploit the matrix representation of a Bézier curve. In the section on matrices, we saw that we can represent curves as matrix multiplications. Specifically, we saw these two forms for the quadratic and cubic curves respectively: (we'll reverse the Bézier coefficients vector for legibility)

    - - -

    and

    - + +

    + Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the + coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual + "point on the curve" information into a new matrix multiplication: +

    + + +

    and

    + + +

    + If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two + staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, + from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear + algebra (and if you don't care about the derivations, just skip to the end of the box for the results!). +

    +
    +

    Deriving new hull coordinates

    +

    + Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look + at the quadratic curve first: +

    + + + - -

    Let's say we want to split the curve at some point t = z, forming two new (obviously smaller) Bézier curves. To find the coordinates for these two Bézier curves, we can use the matrix representation and some linear algebra. First, we separate out the actual "point on the curve" information into a new matrix multiplication:

    - - -

    and

    - - -

    If we could compact these matrices back to the form [t values] · [Bézier matrix] · [column matrix], with the first two staying the same, then that column matrix on the right would be the coordinates of a new Bézier curve that describes the first segment, from t = 0 to t = z. As it turns out, we can do this quite easily, by exploiting some simple rules of linear algebra (and if you don't care about the derivations, just skip to the end of the box for the results!).

    -
    - -

    Deriving new hull coordinates

    -

    Deriving the two segments upon splitting a curve takes a few steps, and the higher the curve order, the more work it is, so let's look at the quadratic curve first:

    - - - - - - - - -

    We can do this because [M · M-1] is the identity matrix. It's a bit like multiplying something by x/x in calculus: it doesn't do anything to the function, but it does allow you to rewrite it to something that may be easier to work with, or can be broken up differently. In the same way, multiplying our matrix by [M · M-1] has no effect on the total formula, but it does allow us to change the matrix sequence [something · M] to a sequence [M · something], and that makes a world of difference: if we know what [M-1 · Z · M] is, we can apply that to our coordinates, and be left with a proper matrix representation of a quadratic Bézier curve (which is [T · M · P]), with a new set of coordinates that represent the curve from t = 0 to t = z. So let's get computing:

    - - -

    Excellent! Now we can form our new quadratic curve:

    - - - - - - -

    Brilliant: if we want a subcurve from t = 0 to t = z, we can keep the first coordinate the same (which makes sense), our control point becomes a z-ratio mixture of the original control point and the start point, and the new end point is a mixture that looks oddly similar to a Bernstein polynomial of degree two. These new coordinates are actually really easy to compute directly!

    -

    Of course, that's only one of the two curves. Getting the section from t = z to t = 1 requires doing this again. We first observe that in the previous calculation, we actually evaluated the general interval [0,z]. We were able to write it down in a more simple form because of the zero, but what we actually evaluated, making the zero explicit, was:

    - - - - -

    If we want the interval [z,1], we will be evaluating this instead:

    - - - - -

    We're going to do the same trick of multiplying by the identity matrix, to turn [something · M] into [M · something]:

    - - -

    So, our final second curve looks like:

    - - - - - - -

    Nice. We see the same thing as before: we can keep the last coordinate the same (which makes sense); our control point becomes a z-ratio mixture of the original control point and the end point, and the new start point is a mixture that looks oddly similar to a bernstein polynomial of degree two, except this time it uses (z-1) rather than (1-z). These new coordinates are also really easy to compute directly!

    -
    - -

    So, using linear algebra rather than de Casteljau's algorithm, we have determined that, for any quadratic curve split at some value t = z, we get two subcurves that are described as Bézier curves with simple-to-derive coordinates:

    - - -

    and

    - - -

    We can do the same for cubic curves. However, I'll spare you the actual derivation (don't let that stop you from writing that out yourself, though) and simply show you the resulting new coordinate sets:

    - - -

    and

    - - -

    So, looking at our matrices, did we really need to compute the second segment matrix? No, we didn't. Actually having one segment's matrix means we implicitly have the other: push the values of each row in the matrix Q to the right, with zeroes getting pushed off the right edge and appearing back on the left, and then flip the matrix vertically. Presto, you just "calculated" Q'.

    -

    Implementing curve splitting this way requires less recursion, and is just straight arithmetic with cached values, so can be cheaper on systems where recursion is expensive. If you're doing computation with devices that are good at matrix multiplication, chopping up a Bézier curve with this method will be a lot faster than applying de Casteljau.

    - -
    -
    -

    前目录下
    Lowering and elevating curve order

    -

    One interesting property of Bézier curves is that an nth order curve can always be perfectly represented by an (n+1)th order curve, by giving the higher-order curve specific control points.

    -

    If we have a curve with three points, then we can create a curve with four points that exactly reproduces the original curve. First, we give it the same start and end points, and for its two control points we pick "1/3rd start + 2/3rd control" and "2/3rd control + 1/3rd end". Now we have exactly the same curve as before, except represented as a cubic curve rather than a quadratic curve.

    -

    The general rule for raising an nth order curve to an (n+1)th order curve is as follows (observing that the start and end weights are the same as the start and end weights for the old curve):

    - - -

    However, this rule also has as direct consequence that you cannot generally safely lower a curve from nth order to (n-1)th order, because the control points cannot be "pulled apart" cleanly. We can try to, but the resulting curve will not be identical to the original, and may in fact look completely different.

    -

    However, there is a surprisingly good way to ensure that a lower order curve looks "as close as reasonably possible" to the original curve: we can optimise the "least-squares distance" between the original curve and the lower order curve, in a single operation (also explained over on Sirver's Castle). However, to use it, we'll need to do some calculus work and then switch over to linear algebra. As mentioned in the section on matrix representations, some things can be done much more easily with matrices than with calculus functions, and this is one of those things. So... let's go!

    -

    We start by taking the standard Bézier function, and condensing it a little:

    - - -

    Then, we apply one of those silly (actually, super useful) calculus tricks: since our t value is always between zero and one (inclusive), we know that (1-t) plus t always sums to 1. As such, we can express any value as a sum of t and 1-t:

    - - -

    So, with that seemingly trivial observation, we rewrite that Bézier function by splitting it up into a sum of a (1-t) and t component:

    - - -

    So far so good. Now, to see why we did this, let's write out the (1-t) and t parts, and see what that gives us. I promise, it's about to make sense. We start with (1-t):

    - - -

    So by using this seemingly silly trick, we can suddenly express part of our nth order Bézier function in terms of an (n+1)th order Bézier function. And that sounds a lot like raising the curve order! Of course we need to be able to repeat that trick for the t part, but that's not a problem:

    - - -

    So, with both of those changed from an order n expression to an order (n+1) expression, we can put them back together again. Now, where the order n function had a summation from 0 to n, the order n+1 function uses a summation from 0 to n+1, but this shouldn't be a problem as long as we can add some new terms that "contribute nothing". In the next section on derivatives, there is a discussion about why "higher terms than there is a binomial for" and "lower than zero terms" both "contribute nothing". So as long as we can add terms that have the same form as the terms we need, we can just include them in the summation, they'll sit there and do nothing, and the resulting function stays identical to the lower order curve.

    -

    Let's do this:

    - - -

    And this is where we switch over from calculus to linear algebra, and matrices: we can now express this relation between Bézier(n,t) and Bézier(n+1,t) as a very simple matrix multiplication:

    - - -

    where the matrix M is an n+1 by n matrix, and looks like:

    - - -

    That might look unwieldy, but it's really just a mostly-zeroes matrix, with a very simply fraction on the diagonal, and an even simpler fraction to the left of it. Multiplying a list of coordinates with this matrix means we can plug the resulting transformed coordinates into the one-order-higher function and get an identical looking curve.

    -

    Not too bad!

    -

    Equally interesting, though, is that with this matrix operation established, we can now use an incredibly powerful and ridiculously simple way to find out a "best fit" way to reverse the operation, called the normal equation. What it does is minimize the sum of the square differences between one set of values and another set of values. Specifically, if we can express that as some function A x = b, we can use it. And as it so happens, that's exactly what we're dealing with, so:

    - - -

    The steps taken here are:

    -
      -
    1. We have a function in a form that the normal equation can be used with, so
      2. -
    2. apply the normal equation!
      3. -
    3. Then, we want to end up with just Bn on the left, so we start by left-multiply both sides such that we'll end up with lots of stuff on the left that simplified to "a factor 1", which in matrix maths is the identity matrix.
      4. -
    4. In fact, by left-multiplying with the inverse of what was already there, we've effectively "nullified" (but really, one-inified) that big, unwieldy block into the identity matrix I. So we substitute the mess with I, and then
      5. -
    5. because multiplication with the identity matrix does nothing (like multiplying by 1 does nothing in regular algebra), we just drop it.
      6. -
    -

    And we're done: we now have an expression that lets us approximate an n+1th order curve with a lower nth order curve. It won't be an exact fit, but it's definitely a best approximation. So, let's implement these rules for raising and lowering curve order to a (semi) random curve, using the following graphic. Select the sketch, which has movable control points, and press your up and down arrow keys to raise or lower the curve order.

    - - - Scripts are disabled. Showing fallback image. - - - - - - - -
    -
    -

    前目录下
    Derivatives

    -

    There's a number of useful things that you can do with Bézier curves based on their derivative, and one of the more amusing observations about Bézier curves is that their derivatives are, in fact, also Bézier curves. In fact, the differentiation of a Bézier curve is relatively straightforward, although we do need a bit of math.

    -

    First, let's look at the derivative rule for Bézier curves, which is:

    - - -

    which we can also write (observing that b in this formula is the same as our w weights, and that n times a summation is the same as a summation where each term is multiplied by n) as:

    - - -

    Or, in plain text: the derivative of an nth degree Bézier curve is an (n-1)th degree Bézier curve, with one fewer term, and new weights w'0...w'n-1 derived from the original weights as n(wi+1 - wi). So for a 3rd degree curve, with four weights, the derivative has three new weights: w'0 = 3(w1-w0), w'1 = 3(w2-w1) and w'2 = 3(w3-w2).

    -
    - -

    "Slow down, why is that true?"

    -

    Sometimes just being told "this is the derivative" is nice, but you might want to see why this is indeed the case. As such, let's have a look at the proof for this derivative. First off, the weights are independent of the full Bézier function, so the derivative involves only the derivative of the polynomial basis function. So, let's find that:

    - - -

    Applying the product and chain rules gives us:

    - - -

    Which is hard to work with, so let's expand that properly:

    - - -

    Now, the trick is to turn this expression into something that has binomial coefficients again, so we want to end up with things that look like "x! over y!(x-y)!". If we can do that in a way that involves terms of n-1 and k-1, we'll be on the right track.

    - - -

    And that's the first part done: the two components inside the parentheses are actually regular, lower-order Bézier expressions:

    - - -

    Now to apply this to our weighted Bézier curves. We'll write out the plain curve formula that we saw earlier, and then work our way through to its derivative:

    - - -

    If we expand this (with some color to show how terms line up), and reorder the terms by increasing values for k we see the following:

    - - -

    Two of these terms fall way: the first term falls away because there is no -1st term in a summation. As such, it always contributes "nothing", so we can safely completely ignore it for the purpose of finding the derivative function. The other term is the very last term in this expansion: one involving Bn-1,n. This term would have a binomial coefficient of [i choose i+1], which is a non-existent binomial coefficient. Again, this term would contribute "nothing", so we can ignore it, too. This means we're left with:

    - - -

    And that's just a summation of lower order curves:

    - - -

    We can rewrite this as a normal summation, and we're done:

    - - -
    - -

    Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for Bézier curves, and then the derivative:

    - + +

    And that's just a summation of lower order curves:

    + + +

    We can rewrite this as a normal summation, and we're done:

    + + + +

    + Let's rewrite that in a form similar to our original formula, so we can see the difference. We will first list our original formula for + Bézier curves, and then the derivative: +

    + - - - -

    What are the differences? In terms of the actual Bézier curve, virtually nothing! We lowered the order (rather than n, it's now n-1), but it's still the same Bézier function. The only real difference is in how the weights change when we derive the curve's function. If we have four points A, B, C, and D, then the derivative will have three points, the second derivative two, and the third derivative one:

    - - -

    We can keep performing this trick for as long as we have more than one weight. Once we have one weight left, the next step will see k = 0, and the result of our "Bézier function" summation is zero, because we're not adding anything at all. As such, a quadratic curve has no second derivative, a cubic curve has no third derivative, and generalized: an nth order curve has n-1 (meaningful) derivatives, with any further derivative being zero.

    - -
    -
    -

    前目录下
    Tangents and normals

    -

    If you want to move objects along a curve, or "away from" a curve, the two vectors you're most interested in are the tangent vector and normal vector for curve points. These are actually really easy to find. For moving and orienting along a curve, we use the tangent, which indicates the direction of travel at specific points, and is literally just the first derivative of our curve:

    - - -

    This gives us the directional vector we want. We can normalize it to give us uniform directional vectors (having a length of 1.0) at each point, and then do whatever it is we want to do based on those directions:

    - - -

    The tangent is very useful for moving along a line, but what if we want to move away from the curve instead, perpendicular to the curve at some point t? In that case we want the normal vector. This vector runs at a right angle to the direction of the curve, and is typically of length 1.0, so all we have to do is rotate the normalized directional vector and we're done:

    - - -
    - -

    Rotating coordinates is actually very easy, if you know the rule for it. You might find it explained as "applying a rotation matrix, which is what we'll look at here, too. Essentially, the idea is to take the circles over which we can rotate, and simply "sliding the coordinates" over these circles by the desired -angle. If we want a quarter circle turn, we take the coordinate, slide it along the circle by a quarter turn, and done.

    -

    To turn any point (x,y) into a rotated point (x',y') (over 0,0) by some angle φ, we apply this nice and easy computation:

    - - -

    Which is the "long" version of the following matrix transformation:

    - - -

    And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1.

    -

    But why does this work? Why this matrix multiplication? Wikipedia (technically, Thomas Herter and Klaus Lott) tells us that a rotation matrix can be -treated as a sequence of three (elementary) shear operations. When we combine this into a single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. DataGenetics have an excellent article about this very thing: it's really quite cool, and I strongly recommend taking a quick break from this primer to read that article.

    -
    + +

    + And that's all we need to rotate any coordinate. Note that for quarter, half, and three-quarter turns these functions become even + easier, since sin and cos for these angles are, respectively: 0 and 1, -1 and 0, and 0 and -1. +

    +

    + But why does this work? Why this matrix multiplication? + Wikipedia (technically, Thomas Herter and Klaus + Lott) tells us that a rotation matrix can be treated as a sequence of three (elementary) shear operations. When we combine this into a + single matrix operation (because all matrix multiplications can be collapsed), we get the matrix that you see above. + DataGenetics have an excellent article about this very thing: it's + really quite cool, and I strongly recommend taking a quick break from this primer to read that article. +

    + -

    The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant).

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    + The following two graphics show the tangent and normal along a quadratic and cubic curve, with the direction vector coloured blue, and the + normal vector coloured red (the markers are spaced out evenly as t-intervals, not spaced equidistant). +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    +
    +
    +

    +
    前目录下
    + Working with 3D normals +

    +

    + Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things + this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but + for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, + it's not "super hard", but there are more steps involved and we should have a look at those. +

    +

    + Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. + However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, + not a single vector, so we basically need to define what "the" normal is in the 3D case. +

    +

    + The "naïve" approach is to construct what is known as the + Frenet normal, where we follow a simple recipe that works in + many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are + perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each + point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the + local tangent, as well as the tangents "right next to it". +

    +

    + Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the + vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know + lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same + logic we used for the 2D case: "just rotate it 90 degrees". +

    +

    So let's do that! And in a twist surprise, we can do this in four lines:

    +
      +
    • a = normalize(B'(t))
      • +
    • b = normalize(a + B''(t))
      • +
    • r = normalize(b × a)
      • +
    • normal = normalize(r × a)
      • +
    +

    Let's unpack that a little:

    +
      +
    • + We start by taking the normalized vector for the derivative at some point on the + curve. We normalize it so the maths is less work. Less work is good. +
      • +
    • + Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and + just had the same derivative and second derivative from that point on. +
      • +
    • + This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which + means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the + cross product. (Note that while that operation uses the × operator, it's most + definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent + a quarter circle to get our normal, just like we did in the 2D case. +
      • +
    • + Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal + vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second + time, and immediately get our normal vector. +
      • +
    +

    + And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can + move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor + position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the + curve" between t=0.65 and t=0.75... Why is it doing that? +

    +

    + As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are + "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute + normals that just... look good. +

    +

    Thankfully, Frenet normals are not our only option.

    +

    + Another option is to take a slightly more algorithmic approach and compute a form of + Rotation Minimising Frame + (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational + axis, and the normal vector, centered on an on-curve point. +

    +

    + These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we + could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at + the same time that you're building lookup tables for your curve. +

    +

    + The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by + applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are: +

    +
      +
    • Take a point on the curve for which we know the RM frame already,
      • +
    • take a next point on the curve for which we don't know the RM frame yet, and
      • +
    • + reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous + points as a "mirror". +
      • +
    • + This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal + that's slightly off-kilter, so: +
      • +
    • + reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror". +
      • +
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • +
    +

    So, let's write some code for that!

    +
    +

    Implementing Rotation Minimising Frames

    +

    + We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it + yields a frame with properties: +

    -
    -
    -

    前目录下
    Working with 3D normals

    -

    Before we move on to the next section we need to spend a little bit of time on the difference between 2D and 3D. While for many things this difference is irrelevant and the procedures are identical (for instance, getting the 3D tangent is just doing what we do for 2D, but for x, y, and z, instead of just for x and y), when it comes to normals things are a little more complex, and thus more work. Mind you, it's not "super hard", but there are more steps involved and we should have a look at those.

    -

    Getting normals in 3D is in principle the same as in 2D: we take the normalised tangent vector, and then rotate it by a quarter turn. However, this is where things get that little more complex: we can turn in quite a few directions, since "the normal" in 3D is a plane, not a single vector, so we basically need to define what "the" normal is in the 3D case.

    -

    The "naïve" approach is to construct what is known as the Frenet normal, where we follow a simple recipe that works in many cases (but does super bizarre things in some others). The idea is that even though there are infinitely many vectors that are perpendicular to the tangent (i.e. make a 90 degree angle with it), the tangent itself sort of lies on its own plane already: since each point on the curve (no matter how closely spaced) has its own tangent vector, we can say that each point lies in the same plane as the local tangent, as well as the tangents "right next to it".

    -

    Even if that difference in tangent vectors is minute, "any difference" is all we need to find out what that plane is - or rather, what the vector perpendicular to that plane is. Which is what we need: if we can calculate that vector, and we have the tangent vector that we know lies on a plane, then we can rotate the tangent vector over the perpendicular, and presto. We have computed the normal using the same logic we used for the 2D case: "just rotate it 90 degrees".

    -

    So let's do that! And in a twist surprise, we can do this in four lines:

    -
      -
    • a = normalize(B'(t))
      • -
    • b = normalize(a + B''(t))
      • -
    • r = normalize(b × a)
      • -
    • normal = normalize(r × a)
      • -
    -

    Let's unpack that a little:

    -
      -
    • We start by taking the normalized vector for the derivative at some point on the curve. We normalize it so the maths is less work. Less work is good.
      • -
    • Then, we compute b which represents what a next point's tangent would be if the curve stopped changing at our point and just had the same derivative and second derivative from that point on.
      • -
    • This lets us find two vectors (the derivative, and the second derivative added to the derivative) that lie on the same plane, which means we can use them to compute a vector perpendicular to that plane, using an elementary vector operation called the cross product. (Note that while that operation uses the × operator, it's most definitely not a multiplication!) The result of that gives us a vector that we can use as the "axis of rotation" for turning the tangent a quarter circle to get our normal, just like we did in the 2D case.
      • -
    • Since the cross product lets us find a vector that is perpendicular to some plane defined by two other vectors, and since the normal vector should be perpendicular to the plane that the tangent and the axis of rotation lie in, we can use the cross product a second time, and immediately get our normal vector.
      • -
    -

    And then we're done, we found "the" normal vector for a 3D curve. Let's see what that looks like for a sample curve, shall we? You can move your cursor across the graphic from left to right, to show the normal at a point with a t value that is based on your cursor position: all the way on the left is 0, all the way on the right = 1, midway is t=0.5, etc:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    However, if you've played with that graphic a bit, you might have noticed something odd. The normal seems to "suddenly twist around the curve" between t=0.65 and t=0.75... Why is it doing that?

    -

    As it turns out, it's doing that because that's how the maths works, and that's the problem with Frenet normals: while they are "mathematically correct", they are "practically problematic", and so for any kind of graphics work what we really want is a way to compute normals that just... look good.

    -

    Thankfully, Frenet normals are not our only option.

    -

    Another option is to take a slightly more algorithmic approach and compute a form of Rotation Minimising Frame (also known as "parallel transport frame" or "Bishop frame") instead, where a "frame" is a set made up of the tangent, the rotational axis, and the normal vector, centered on an on-curve point.

    -

    These type of frames are computed based on "the previous frame", so we cannot simply compute these "on demand" for single points, as we could for Frenet frames; we have to compute them for the entire curve. Thankfully, the procedure is pretty simple, and can be performed at the same time that you're building lookup tables for your curve.

    -

    The idea is to take a starting "tangent/rotation axis/normal" frame at t=0, and then compute what the next frame "should" look like by applying some rules that yield a good looking next frame. In the case of the RMF paper linked above, those rules are:

    -
      -
    • Take a point on the curve for which we know the RM frame already,
      • -
    • take a next point on the curve for which we don't know the RM frame yet, and
      • -
    • reflect the known frame onto the next point, by treating the plane through the curve at the point exactly between the next and previous points as a "mirror".
      • -
    • This gives the next point a tangent vector that's essentially pointing in the opposite direction of what it should be, and a normal that's slightly off-kilter, so:
      • -
    • reflect the vectors of our "mirrored frame" a second time, but this time using the plane through the "next point" itself as "mirror".
      • -
    • Done: the tangent and normal have been fixed, and we have a good looking frame to work with.
      • -
    -

    So, let's write some code for that!

    -
    - -

    Implementing Rotation Minimising Frames

    -

    We first assume we have a function for calculating the Frenet frame at a point, which we already discussed above, inn a way that it yields a frame with properties:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + -

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    +

    Then, we can write a function that generates a sequence of RM frames in the following manner:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    + frames.add(x1) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + -

    Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more code to get much better looking normals.

    -
    +

    + Ignoring comments, this is certainly more code than when we were just computing a single Frenet frame, but it's not a crazy amount more + code to get much better looking normals. +

    + -

    Speaking of better looking, what does this actually look like? Let's revisit that earlier curve, but this time use rotation minimising frames rather than Frenet frames:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    That looks so much better!

    -

    For those reading along with the code: we don't even strictly speaking need a Frenet frame to start with: we could, for instance, treat the z-axis as our initial axis of rotation, so that our initial normal is (0,0,1) × tangent, and then take things from there, but having that initial "mathematically correct" frame so that the initial normal seems to line up based on the curve's orientation in 3D space is just nice.

    - -
    -
    -

    前目录下
    Component functions

    -

    One of the first things people run into when they start using Bézier curves in their own programs is "I know how to draw the curve, but how do I determine the bounding box?". It's actually reasonably straightforward to do so, but it requires having some knowledge on exploiting math to get the values we need. For bounding boxes, we aren't actually interested in the curve itself, but only in its "extremities": the minimum and maximum values the curve has for its x- and y-axis values. If you remember your calculus (provided you ever took calculus, otherwise it's going to be hard to remember) we can determine function extremities using the first derivative of that function, but this poses a problem, since our function is parametric: every axis has its own function.

    -

    The solution: compute the derivative for each axis separately, and then fit them back together in the same way we do for the original.

    -

    Let's look at how a parametric Bézier curve "splits up" into two normal functions, one for the x-axis and one for the y-axis. Note the leftmost figure is again an interactive curve, without labeled axes (you get coordinates in the graph instead). The center and rightmost figures are the component functions for computing the x-axis value, given a value for t (between 0 and 1 inclusive), and the y-axis value, respectively.

    -

    If you move points in a curve sideways, you should only see the middle graph change; likewise, moving points vertically should only show a change in the right graph.

    - - - Scripts are disabled. Showing fallback image. - - - -

     

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目录下
    Finding extremities: root finding

    -

    Now that we understand (well, superficially anyway) the component functions, we can find the extremities of our Bézier curve by finding maxima and minima on the component functions, by solving the equation B'(t) = 0. We've already seen that the derivative of a Bézier curve is a simpler Bézier curve, but how do we solve the equality? Fairly easily, actually, until our derivatives are 4th order or higher... then things get really hard. But let's start simple:

    -

    Quadratic curves: linear derivatives.

    -

    The derivative of a quadratic Bézier curve is a linear Bézier curve, interpolating between just two terms, which means finding the solution for "where is this line 0" is effectively trivial by rewriting it to a function of t and solving. First we turn our quadratic Bézier function into a linear one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then we turn this into our solution for t using basic arithmetics:

    - - -

    Done.

    -

    Although with the caveat that if b-a is zero, there is no solution and we probably shouldn't try to perform that division.

    -

    Cubic curves: the quadratic formula.

    -

    The derivative of a cubic Bézier curve is a quadratic Bézier curve, and finding the roots for a quadratic polynomial means we can apply the Quadratic formula. If you've seen it before, you'll remember it, and if you haven't, it looks like this:

    - - -

    So, if we can rewrite the Bézier component function as a plain polynomial, we're done: we just plug in the values into the quadratic formula, check if that square root is negative or not (if it is, there are no roots) and then just compute the two values that come out (because of that plus/minus sign we get two). Any value between 0 and 1 is a root that matters for Bézier curves, anything below or above that is irrelevant (because Bézier curves are only defined over the interval [0,1]). So, how do we convert?

    -

    First we turn our cubic Bézier function into a quadratic one, by following the rule mentioned at the end of the derivatives section:

    - - -

    And then, using these v values, we can find out what our a, b, and c should be:

    - - -

    This gives us three coefficients {a, b, c} that are expressed in terms of v values, where the v values are expressions of our original coordinate values, so we can do some substitution to get:

    - - -

    Easy-peasy. We can now almost trivially find the roots by plugging those values into the quadratic formula.

    -

    And as a cubic curve, there is also a meaningful second derivative, which we can compute by simple taking the derivative of the derivative.

    -

    Quartic curves: Cardano's algorithm.

    -

    We haven't really looked at them before now, but the next step up would be a Quartic curve, a fourth degree Bézier curve. As expected, these have a derivative that is a cubic function, and now things get much harder. Cubic functions don't have a "simple" rule to find their roots, like the quadratic formula, and instead require quite a bit of rewriting to a form that we can even start to try to solve.

    -

    Back in the 16th century, before Bézier curves were a thing, and even before calculus itself was a thing, Gerolamo Cardano figured out that even if the general cubic function is really hard to solve, it can be rewritten to a form for which finding the roots is "easier" (even if not "easy"):

    - - -

    We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather than three: this makes things considerably easier to solve because it lets us use regular calculus to find the values that satisfy the equation.

    -

    Now, there is one small hitch: as a cubic function, the solutions may be complex numbers rather than plain numbers... And Cardano realised this, centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number.

    -

    So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at Ken J. Ward's page for solving the cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a read-through.

    -
    + +

    + We can see that the easier formula only has two constants, rather than four, and only two expressions involving t, rather + than three: this makes things considerably easier to solve because it lets us use + regular calculus to find the values that satisfy the equation. +

    +

    + Now, there is one small hitch: as a cubic function, the solutions may be + complex numbers rather than plain numbers... And Cardano realised this, + centuries before complex numbers were a well-understood and established part of number theory. His interpretation of them was "these + numbers are impossible but that's okay because they disappear again in later steps", allowing him to not think about them too much, but we + have it even easier: as we're trying to find the roots for display purposes, we don't even care about complex numbers: we're + going to simplify Cardano's approach just that tiny bit further by throwing away any solution that's not a plain number. +

    +

    + So, how do we rewrite the hard formula into the easier formula? This is explained in detail over at + Ken J. Ward's page for solving the + cubic equation, so instead of showing the maths, I'm simply going to show the programming code for solving the cubic equation, with the + complex roots getting totally ignored, but if you're interested you should definitely head over to Ken's page and give the procedure a + read-through. +

    +
    +

    Implementing Cardano's algorithm for finding all real roots

    +

    + The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" + number space (denoted with ℝ), this is perfectly fine in the + "complex" number space (denoted with ℂ). And, as it so happens, Cardano is + also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm! +

    -

    Implementing Cardano's algorithm for finding all real roots

    -

    The "real roots" part is fairly important, because while you cannot take a square, cube, etc. root of a negative number in the "real" number space (denoted with ℝ), this is perfectly fine in the "complex" number space (denoted with ℂ). And, as it so happens, Cardano is also attributed as the first mathematician in history to have made use of complex numbers in his calculations. For this very algorithm!

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    20
    21
    22
    23
    24
    25
    26
    27
    28
    29
    30
    31
    32
    33
    34
    35
    36
    37
    38
    39
    40
    41
    42
    43
    44
    45
    46
    47
    48
    49
    50
    51
    52
    53
    54
    55
    56
    57
    58
    59
    60
    61
    62
    63
    64
    65
    66
    67
    68
    69
    70
    71
    72
    73
    74
    75
    76
    77
    78
    79
    80
    81
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + + 20 + + + 21 + + + 22 + + + 23 + + + 24 + + + 25 + + + 26 + + + 27 + + + 28 + + + 29 + + + 30 + + + 31 + + + 32 + + + 33 + + + 34 + + + 35 + + + 36 + + + 37 + + + 38 + + + 39 + + + 40 + + + 41 + + + 42 + + + 43 + + + 44 + + + 45 + + + 46 + + + 47 + + + 48 + + + 49 + + + 50 + + + 51 + + + 52 + + + 53 + + + 54 + + + 55 + + + 56 + + + 57 + + + 58 + + + 59 + + + 60 + + + 61 + + + 62 + + + 63 + + + 64 + + + 65 + + + 66 + + + 67 + + + 68 + + + 69 + + + 70 + + + 71 + + + 72 + + + 73 + + + 74 + + + 75 + + + 76 + + + 77 + + + 78 + + + 79 + + + 80 + + + 81 + + +
    -
    - -

    And that's it. The maths is complicated, but the code is pretty much just "follow the maths, while caching as many values as we can to prevent recomputing things as much as possible" and now we have a way to find all roots for a cubic function and can just move on with using that to find extremities of our curves.

    -

    And of course, as a quartic curve also has meaningful second and third derivatives, we can quite easily compute those by using the derivative of the derivative (of the derivative), just as for cubic curves.

    -

    Quintic and higher order curves: finding numerical solutions

    -

    And this is where thing stop, because we cannot find the roots for polynomials of degree 5 or higher using algebra (a fact known as the Abel–Ruffini theorem). Instead, for occasions like these, where algebra simply cannot yield an answer, we turn to numerical analysis.

    -

    That's a fancy term for saying "rather than trying to find exact answers by manipulating symbols, find approximate answers by describing the underlying process as a combination of steps, each of which can be assigned a number via symbolic manipulation". For example, trying to mathematically compute how much water fits in a completely crazy three dimensional shape is very hard, even if it got you the perfect, precise answer. A much easier approach, which would be less perfect but still entirely useful, would be to just grab a buck and start filling the shape until it was full: just count the number of buckets of water you used. And if we want a more precise answer, we can use smaller buckets.

    -

    So that's what we're going to do here, too: we're going to treat the problem as a sequence of steps, and the smaller we can make each step, the closer we'll get to that "perfect, precise" answer. And as it turns out, there is a really nice numerical root-finding algorithm, called the Newton-Raphson root finding method (yes, after that Newton), which we can make use of. The Newton-Raphson approach consists of taking our impossible-to-solve function f(x), picking some initial value x (literally any value will do), and calculating f(x). We can think of that value as the "height" of the function at x. If that height is zero, we're done, we have found a root. If it isn't, we calculate the tangent line at f(x) and calculate at which x value its height is zero (which we've already seen is very easy). That will give us a new x and we repeat the process until we find a root.

    -

    Mathematically, this means that for some x, at step n=1, we perform the following calculation until fy(x) is zero, so that the next t is the same as the one we already have:

    - - -

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    -

    Now, this works well only if we can pick good starting points, and our curve is continuously differentiable and doesn't have oscillations. Glossing over the exact meaning of those terms, the curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: which starting points do we pick?

    -

    As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as high as you'll ever need to go.

    -

    In conclusion:

    -

    So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red:

    - - - Scripts are disabled. Showing fallback image. - - - -

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    - - - Scripts are disabled. Showing fallback image. - - - + +

    (The Wikipedia article has a decent animation for this process, so I will not add a graphic for that here)

    +

    + Now, this works well only if we can pick good starting points, and our curve is + continuously differentiable and doesn't have + oscillations. Glossing over the exact meaning of those terms, the + curves we're dealing with conform to those constraints, so as long as we pick good starting points, this will work. So the question is: + which starting points do we pick? +

    +

    + As it turns out, Newton-Raphson is so blindingly fast that we could get away with just not picking: we simply run the algorithm from + t=0 to t=1 at small steps (say, 1/200th) and the result will be all the roots we want. Of course, this may + pose problems for high order Bézier curves: 200 steps for a 200th order Bézier curve is going to go wrong, but that's okay: + there is no reason (at least, none that I know of) to ever use Bézier curves of crazy high orders. You might use a fifth order + curve to get the "nicest still remotely workable" approximation of a full circle with a single Bézier curve, but that's pretty much as + high as you'll ever need to go. +

    +

    In conclusion:

    +

    + So now that we know how to do root finding, we can determine the first and second derivative roots for our Bézier curves, and show those + roots overlaid on the previous graphics. For the quadratic curve, that means just the first derivative, in red: +

    + + + Scripts are disabled. Showing fallback image. + + +

    And for cubic curves, that means first and second derivatives, in red and purple respectively:

    + + + Scripts are disabled. Showing fallback image. + + +
    +
    +

    +
    前目录下
    + Bounding boxes +

    +

    + If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four + values we need to box in our curve: +

    +

    Computing the bounding box for a Bézier curve:

    +
      +
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. +
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. +
    3. + Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original + functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct. +
      4. +
    +

    + Applying this approach to our previous root finding, we get the following + axis-aligned bounding boxes (with all curve extremity points + shown on the curve): +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -
    -
    -

    前目录下
    Bounding boxes

    -

    If we have the extremities, and the start/end points, a simple for-loop that tests for min/max values for x and y means we have the four values we need to box in our curve:

    -

    Computing the bounding box for a Bézier curve:

    -
      -
    1. Find all t value(s) for the curve derivative's x- and y-roots.
      2. -
    2. Discard any t value that's lower than 0 or higher than 1, because Bézier curves only use the interval [0,1].
      3. -
    3. Determine the lowest and highest value when plugging the values t=0, t=1 and each of the found roots into the original functions: the lowest value is the lower bound, and the highest value is the upper bound for the bounding box we want to construct.
      4. -
    -

    Applying this approach to our previous root finding, we get the following axis-aligned bounding boxes (with all curve extremity points shown on the curve):

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    We can construct even nicer boxes by aligning them along our curve, rather than along the x- and y-axis, but in order to do so we first need to look at how aligning works.

    - -
    -
    -

    前目录下
    Aligning curves

    -

    While there are an incredible number of curves we can define by varying the x- and y-coordinates for the control points, not all curves are actually distinct. For instance, if we define a curve, and then rotate it 90 degrees, it's still the same curve, and we'll find its extremities in the same spots, just at different draw coordinates. As such, one way to make sure we're working with a "unique" curve is to "axis-align" it.

    -

    Aligning also simplifies a curve's functions. We can translate (move) the curve so that the first point lies on (0,0), which turns our n term polynomial functions into n-1 term functions. The order stays the same, but we have less terms. Then, we can rotate the curves so that the last point always lies on the x-axis, too, making its coordinate (...,0). This further simplifies the function for the y-component to an n-2 term function. For instance, if we have a cubic curve such as this:

    - - -

    Then translating it so that the first coordinate lies on (0,0), moving all x coordinates by -120, and all y coordinates by -160, gives us:

    - - -

    If we then rotate the curve so that its end point lies on the x-axis, the coordinates (integer-rounded for illustrative purposes here) become:

    - - -

    If we drop all the zero-terms, this gives us:

    - - -

    We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae:

    - - - Scripts are disabled. Showing fallback image. - - - -

     

    - - - Scripts are disabled. Showing fallback image. - - - + +

    + We can see that our original curve definition has been simplified considerably. The following graphics illustrate the result of aligning + our example curves to the x-axis, with the cubic case using the coordinates that were just used in the example formulae: +

    + + + Scripts are disabled. Showing fallback image. + + +

     

    + + + Scripts are disabled. Showing fallback image. + + +
    +
    +

    +
    前目录下
    + Tight bounding boxes +

    +

    + With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, + recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding + box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T. +

    +

    We now have nice tight bounding boxes for our curves:

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -
    -
    -

    前目录下
    Tight bounding boxes

    -

    With our knowledge of bounding boxes, and curve alignment, We can now form the "tight" bounding box for curves. We first align our curve, recording the translation we performed, "T", and the rotation angle we used, "R". We then determine the aligned curve's normal bounding box. Once we have that, we can map that bounding box back to our original curve by rotating it by -R, and then translating it by -T.

    -

    We now have nice tight bounding boxes for our curves:

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    These are, strictly speaking, not necessarily the tightest possible bounding boxes. It is possible to compute the optimal bounding box by determining which spanning lines we need to effect a minimal box area, but because of the parametric nature of Bézier curves this is actually a rather costly operation, and the gain in bounding precision is often not worth it.

    - -
    -
    -

    前目录下
    Curve inflections

    -

    Now that we know how to align a curve, there's one more thing we can calculate: inflection points. Imagine we have a variable size circle that we can slide up against our curve. We place it against the curve and adjust its radius so that where it touches the curve, the curvatures of the curve and the circle are the same, and then we start to slide the circle along the curve - for quadratic curves, we can always do this without the circle behaving oddly: we might have to change the radius of the circle as we slide it along, but it'll always sit against the same side of the curve.

    -

    But what happens with cubic curves? Imagine we have an S curve and we place our circle at the start of the curve, and start sliding it along. For a while we can simply adjust the radius and things will be fine, but once we get to the midpoint of that S, something odd happens: the circle "flips" from one side of the curve to the other side, in order for the curvatures to keep matching. This is called an inflection, and we can find out where those happen relatively easily.

    -

    What we need to do is solve a simple equation:

    - - -

    What we're saying here is that given the curvature function C(t), we want to know for which values of t this function is zero, meaning there is no "curvature", which will be exactly at the point between our circle being on one side of the curve, and our circle being on the other side of the curve. So what does C(t) look like? Actually something that seems not too hard:

    - - -

    The function C(t) is the cross product between the first and second derivative functions for the parametric dimensions of our curve. And, as already shown, derivatives of Bézier curves are just simpler Bézier curves, with very easy to compute new coefficients, so this should be pretty easy.

    -

    However as we've seen in the section on aligning, aligning lets us simplify things a lot, by completely removing the contributions of the first coordinate from most mathematical evaluations, and removing the last y coordinate as well by virtue of the last point lying on the x-axis. So, while we can evaluate C(t) = 0 for our curve, it'll be much easier to first axis-align the curve and then evaluating the curvature function.

    -
    - -

    Let's derive the full formula anyway

    -

    Of course, before we do our aligned check, let's see what happens if we compute the curvature function without axis-aligning. We start with the first and second derivatives, given our basis functions:

    - - -

    And of course the same functions for y:

    - - -

    Asking a computer to now compose the C(t) function for us (and to expand it to a readable form of simple terms) gives us this rather overly complicated set of arithmetic expressions:

    - - -

    That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all disappear if we axis-align our curve, which is why aligning is a great idea.

    -
    + +

    + That is... unwieldy. So, we note that there are a lot of terms that involve multiplications involving x1, y1, and y4, which would all + disappear if we axis-align our curve, which is why aligning is a great idea. +

    + -

    Aligning our curve so that three of the eight coefficients become zero, and observing that scale does not affect finding t values, we end up with the following simple term function for C(t):

    - - -

    That's a lot easier to work with: we see a fair number of terms that we can compute and then cache, giving us the following simplification:

    - - -

    This is a plain quadratic curve, and we know how to solve C(t) = 0; we use the quadratic formula:

    - - -

    We can easily compute this value if the discriminator isn't a negative number (because we only want real roots, not complex roots), and if x is not zero, because divisions by zero are rather useless.

    -

    Taking that into account, we compute t, we disregard any t value that isn't in the Bézier interval [0,1], and we now know at which t value(s) our curve will inflect.

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目录下
    The canonical form (for cubic curves)

    -

    While quadratic curves are relatively simple curves to analyze, the same cannot be said of the cubic curve. As a curvature is controlled by more than one control point, it exhibits all kinds of features like loops, cusps, odd colinear features, and as many as two inflection points because the curvature can change direction up to three times. Now, knowing what kind of curve we're dealing with means that some algorithms can be run more efficiently than if we have to implement them as generic solvers, so is there a way to determine the curve type without lots of work?

    -

    As it so happens, the answer is yes, and the solution we're going to look at was presented by Maureen C. Stone from Xerox PARC and Tony D. deRose from the University of Washington in their joint paper "A Geometric Characterization of Parametric Cubic curves". It was published in 1989, and defines curves as having a "canonical" form (i.e. a form that all curves can be reduced to) from which we can immediately tell what features a curve will have. So how does it work?

    -

    The first observation that makes things work is that if we have a cubic curve with four points, we can apply a linear transformation to these points such that three of the points end up on (0,0), (0,1) and (1,1), with the last point then being "somewhere". After applying that transformation, the location of that last point can then tell us what kind of curve we're dealing with. Specifically, we see the following breakdown:

    - - - Scripts are disabled. Showing fallback image. - - - -

    This is a fairly funky image, so let's see what the various parts of it mean...

    -

    We see the three fixed points at (0,0), (0,1) and (1,1). The various regions and boundaries indicate what property the original curve will have, if the fourth point is in/on that region or boundary. Specifically, if the fourth point is...

    -
      -

    1. ...anywhere inside the red zone, but not on its boundaries, the curve will be self-intersecting (yielding a loop). We won't know where it self-intersects (in terms of t values), but we are guaranteed that it does.

      -
      2. -

    2. ...on the left (red) edge of the red zone, the curve will have a cusp. We again don't know where, but we know there is one. This edge is described by the function:

      - - -
      3. -

    3. ...on the almost circular, lower right (pink) edge, the curve's end point touches the curve, forming a loop. This edge is described by the function:

      - - -
      4. -

    4. ...on the top (blue) edge, the curve's start point touches the curve, forming a loop. This edge is described by the function:

      - - -
      5. -

    5. ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

      -
      6. -

    6. ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two inflections (switching from concave to convex and then back again, or from convex to concave and then back again).

      -
      7. -

    7. ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

      -
      8. -
    -

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    -
    + + +
  • +

    ...inside the lower (green) zone, past y=1, the curve will have a single inflection (switching concave/convex once).

    +
    • +
  • +

    + ...between the left and lower boundaries (below the cusp line but above the single-inflection line), the curve will have two + inflections (switching from concave to convex and then back again, or from convex to concave and then back again). +

    +
    • +

  • ...anywhere on the right of self-intersection zone, the curve will have no inflections. It'll just be a simple arch.

    • + +

    Of course, this map is fairly small, but the regions extend to infinity, with well defined boundaries.

    +
    +

    Wait, where do those lines come from?

    +

    + Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, + and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield + formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic + expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the + properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval. +

    +

    + The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general + cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to + it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general + curve does over the interval t=0 to t=1. +

    +

    + For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you + can probably follow along with this paper, although you might have to read it a few times before all the bits "click". +

    +
    -

    Wait, where do those lines come from?

    -

    Without repeating the paper mentioned at the top of this section, the loop-boundaries come from rewriting the curve into canonical form, and then solving the formulae for which constraints must hold for which possible curve properties. In the paper these functions yield formulae for where you will find cusp points, or loops where we know t=0 or t=1, but those functions are derived for the full cubic expression, meaning they apply to t=-∞ to t=∞... For Bézier curves we only care about the "clipped interval" t=0 to t=1, so some of the properties that apply when you look at the curve over an infinite interval simply don't apply to the Bézier curve interval.

    -

    The right bound for the loop region, indicating where the curve switches from "having inflections" to "having a loop", for the general cubic curve, is actually mirrored over x=1, but for Bézier curves this right half doesn't apply, so we don't need to pay attention to it. Similarly, the boundaries for t=0 and t=1 loops are also nice clean curves but get "cut off" when we only look at what the general curve does over the interval t=0 to t=1.

    -

    For the full details, head over to the paper and read through sections 3 and 4. If you still remember your high school pre-calculus, you can probably follow along with this paper, although you might have to read it a few times before all the bits "click".

    -
    - -

    So now the question becomes: how do we manipulate our curve so that it fits this canonical form, with three fixed points, and one "free" point? Enter linear algebra. Don't worry, I'll be doing all the math for you, as well as show you what the effect is on our curves, but basically we're going to be using linear algebra, rather than calculus, because "it's way easier". Sometimes a calculus approach is very hard to work with, when the equivalent geometrical solution is super obvious.

    -

    The approach is going to start with a curve that doesn't have all-colinear points (so we need to make sure the points don't all fall on a straight line), and then applying three graphics operations that you will probably have heard of: translation (moving all points by some fixed x- and y-distance), scaling (multiplying all points by some x and y scale factor), and shearing (an operation that turns rectangles into parallelograms).

    -

    Step 1: we translate any curve by -p1.x and -p1.y, so that the curve starts at (0,0). We're going to make use of an interesting trick here, by pretending our 2D coordinates are 3D, with the z coordinate simply always being 1. This is an old trick in graphics to overcome the limitations of 2D transformations: without it, we can only turn (x,y) coordinates into new coordinates of the form (ax + by, cx + dy), which means we can't do translation, since that requires we end up with some kind of (x + a, y + b). If we add a bogus z coordinate that is always 1, then we can suddenly add arbitrary values. For example:

    - - -

    Sweet! z stays 1, so we can effectively ignore it entirely, but we added some plain values to our x and y coordinates. So, if we want to subtract p1.x and p1.y, we use:

    - - -

    Running all our coordinates through this transformation gives a new set of coordinates, let's call those U, where the first coordinate lies on (0,0), and the rest is still somewhat free. Our next job is to make sure point 2 ends up lying on the x=0 line, so what we want is a transformation matrix that, when we run it, subtracts x from whatever x we currently have. This is called shearing, and the typical x-shear matrix and its transformation looks like this:

    - - -

    So we want some shearing value that, when multiplied by y, yields -x, so our x coordinate becomes zero. That value is simply -x/y, because *-x/y * y = -x*. Done:

    - - -

    Now, running this on all our points generates a new set of coordinates, let's call those V, which now have point 1 on (0,0) and point 2 on (0, some-value), and we wanted it at (0,1), so we need to do some scaling to make sure it ends up at (0,1). Additionally, we want point 3 to end up on (1,1), so we can also scale x to make sure its x-coordinate will be 1 after we run the transform. That means we'll be x-scaling by 1/point3x, and y-scaling by point2y. This is really easy:

    - - -

    Then, finally, this generates a new set of coordinates, let's call those W, of which point 1 lies on (0,0), point 2 lies on (0,1), and point three lies on (1, ...) so all that's left is to make sure point 3 ends up at (1,1) - but we can't scale! Point 2 is already in the right place, and y-scaling would move it out of (0,1) again, so our only option is to y-shear point three, just like how we x-sheared point 2 earlier. In this case, we do the same trick, but with y/x rather than x/y because we're not x-shearing but y-shearing. Additionally, we don't actually want to end up at zero (which is what we did before) so we need to shear towards an offset, in this case 1:

    - - -

    And this generates our final set of four coordinates. Of these, we already know that points 1 through 3 are (0,0), (0,1) and (1,1), and only the last coordinate is "free". In fact, given any four starting coordinates, the resulting "transformation mapped" coordinate will be:

    - - -

    Okay, well, that looks plain ridiculous, but: notice that every coordinate value is being offset by the initial translation, and also notice that a lot of terms in that expression are repeated. Even though the maths looks crazy as a single expression, we can just pull this apart a little and end up with an easy-to-calculate bit of code!

    -

    First, let's just do that translation step as a "preprocessing" operation so we don't have to subtract the values all the time. What does that leave?

    - - -

    Suddenly things look a lot simpler: the mapped x is fairly straight forward to compute, and we see that the mapped y actually contains the mapped x in its entirety, so we'll have that part already available when we need to evaluate it. In fact, let's pull out all those common factors to see just how simple this is:

    - - -

    That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it look!

    -
    + +

    + That's kind of super-simple to write out in code, I think you'll agree. Coding math tends to be easier than the formulae initially make it + look! +

    +
    +

    How do you track all that?

    +

    + Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply + use Mathematica. Tracking all this math by hand is insane, and we invented computers, + literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the + program do the math for me. And real math, too, with symbols, not with numbers. In fact, + here's the Mathematica notebook if you want to see how + this works for yourself. +

    +

    + Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's + $344 for home use, up from $295 when I original wrote this, but it's + also free when you buy a $35 raspberry pi. Obviously, I bought a + raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do + it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for. +

    +
    -

    How do you track all that?

    -

    Doing maths can be a pain, so whenever possible, I like to make computers do the work for me. Especially for things like this, I simply use Mathematica. Tracking all this math by hand is insane, and we invented computers, literally, to do this for us. I have no reason to use pen and paper when I can write out what I want to do in a program, and have the program do the math for me. And real math, too, with symbols, not with numbers. In fact, here's the Mathematica notebook if you want to see how this works for yourself.

    -

    Now, I know, you're thinking "but Mathematica is super expensive!" and that's true, it's $344 for home use, up from $295 when I original wrote this, but it's also free when you buy a $35 raspberry pi. Obviously, I bought a raspberry pi, and I encourage you to do the same. With that, as long as you know what you want to do, Mathematica can just do it for you. And we don't have to be geniuses to work out what the maths looks like. That's what we have computers for.

    -
    - -

    So, let's write up a sketch that'll show us the canonical form for any curve drawn in blue, overlaid on our canonical map, so that we can immediately tell which features our curve must have, based on where the fourth coordinate is located on the map:

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目录下
    Finding Y, given X

    -

    One common task that pops up in things like CSS work, or parametric equalizers, or image leveling, or any other number of applications where Bézier curves are used as control curves in a way that there is really only ever one "y" value associated with one "x" value, you might want to cut out the middle man, as it were, and compute "y" directly based on "x". After all, the function looks simple enough, finding the "y" value should be simple too, right? Unfortunately, not really. However, it is possible and as long as you have some code in place to help, it's not a lot of a work either.

    -

    We'll be tackling this problem in two stages: the first, which is the hard part, is figuring out which "t" value belongs to any given "x" value. For instance, have a look at the following graphic. On the left we have a Bézier curve that looks for all intents and purposes like it fits our criteria: every "x" has one and only one associated "y" value. On the right we see the function for just the "x" values: that's a cubic curve, but not a really crazy cubic curve. If you move the graphic's slider, you will see a red line drawn that corresponds to the x coordinate: this is a vertical line in the left graphic, and a horizontal line on the right.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Now, if you look more closely at that right graphic, you'll notice something interesting: if we treat the red line as "the x axis", then the point where the function crosses our line is really just a root for the cubic function x(t) through a shifted "x-axis"... and we've already seen how to calculate roots, so let's just run cubic root finding - and not even the complicated cubic case either: because of the kind of curve we're starting with, we know there is only root, simplifying the code we need!

    -

    First, let's look at the function for x(t):

    - - -

    We can rewrite this to a plain polynomial form, by just fully writing out the expansion and then collecting the polynomial factors, as:

    - - -

    Nothing special here: that's a standard cubic polynomial in "power" form (i.e. all the terms are ordered by their power of t). So, given that a, b, c, d, and x(t) are all known constants, we can trivially rewrite this (by moving the x(t) across the equal sign) as:

    - - -

    You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using Cardano's algorithm, and we're left with some rather short code:

    + +

    + You might be wondering "where did all the other 'minus x' for all the other values a, b, c, and d go?" and the answer there is that they + all cancel out, so the only one we actually need to subtract is the one at the end. Handy! So now we just solve this equation using + Cardano's algorithm, and we're left with some rather short code: +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +y = curve.get(t).y + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    So the procedure is fairly straight forward: pick an x, find the associated t value, evaluate our curve for that t value, which gives us the curve's {x,y} coordinate, which means we know y for this x. Move the slider for the following graphic to see this in action:

    - - - Scripts are disabled. Showing fallback image. - - - - - - -
    -
    -

    前目录下
    Arc length

    -

    How long is a Bézier curve? As it turns out, that's not actually an easy question, because the answer requires maths that —much like root finding— cannot generally be solved the traditional way. If we have a parametric curve with fx(t) and fy(t), then the length of the curve, measured from start point to some point t = z, is computed using the following seemingly straight forward (if a bit overwhelming) formula:

    - - -

    or, more commonly written using Leibnitz notation as:

    - - -

    This formula says that the length of a parametric curve is in fact equal to the area underneath a function that looks a remarkable amount like Pythagoras' rule for computing the diagonal of a straight angled triangle. This sounds pretty simple, right? Sadly, it's far from simple... cutting straight to after the chase is over: for quadratic curves, this formula generates an unwieldy computation, and we're simply not going to implement things that way. For cubic Bézier curves, things get even more fun, because there is no "closed form" solution, meaning that due to the way calculus works, there is no generic formula that allows you to calculate the arc length. Let me just repeat this, because it's fairly crucial: for cubic and higher Bézier curves, there is no way to solve this function if you want to use it "for all possible coordinates".

    -

    Seriously: It cannot be done.

    -

    So we turn to numerical approaches again. The method we'll look at here is the Gauss quadrature. This approximation is a really neat trick, because for any nth degree polynomial it finds approximated values for an integral really efficiently. Explaining this procedure in length is way beyond the scope of this page, so if you're interested in finding out why it works, I can recommend the University of South Florida video lecture on the procedure, linked in this very paragraph. The general solution we're looking for is the following:

    - - -

    In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the better the approximation:

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    + +

    + In plain text: an integral function can always be treated as the sum of an (infinite) number of (infinitely thin) rectangular strips + sitting "under" the function's plotted graph. To illustrate this idea, the following graph shows the integral for a sinusoid function. The + more strips we use (and of course the more we use, the thinner they get) the closer we get to the true area under the curve, and thus the + better the approximation: +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    Now, infinitely many terms to sum and infinitely thin rectangles are not something that computers can work with, so instead we're going to approximate the infinite summation by using a sum of a finite number of "just thin" rectangular strips. As long as we use a high enough number of thin enough rectangular strips, this will give us an approximation that is pretty close to what the real value is.

    -

    So, the trick is to come up with useful rectangular strips. A naive way is to simply create n strips, all with the same width, but there is a far better way using special values for C and f(t) depending on the value of n, which indicates how many strips we'll use, and it's called the Legendre-Gauss quadrature.

    -

    This approach uses strips that are not spaced evenly, but instead spaces them in a special way based on describing the function as a polynomial (the more strips, the more accurate the polynomial), and then computing the exact integral for that polynomial. We're essentially performing arc length computation on a flattened curve, but flattening it based on the intervals dictated by the Legendre-Gauss solution.

    -
    - -

    Note that one requirement for the approach we'll use is that the integral must run from -1 to 1. That's no good, because we're dealing with Bézier curves, and the length of a section of curve applies to values which run from 0 to "some value smaller than or equal to 1" (let's call that value z). Thankfully, we can quite easily transform any integral interval to any other integral interval, by shifting and scaling the inputs. Doing so, we get the following:

    - - -

    That may look a bit more complicated, but the fraction involving z is a fixed number, so the summation, and the evaluation of the f(t) values are still pretty simple.

    -

    So, what do we need to perform this calculation? For one, we'll need an explicit formula for f(t), because that derivative notation is handy on paper, but not when we have to implement it. We'll also need to know what these Ci and ti values should be. Luckily, that's less work because there are actually many tables available that give these values, for any n, so if we want to approximate our integral with only two terms (which is a bit low, really) then these tables would tell us that for n=2 we must use the following values:

    - - -

    Which means that in order for us to approximate the integral, we must plug these values into the approximate function, which gives us:

    - - -

    We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the Bézier curve functions Bx(t) and By(t).

    -
    + +

    + We can program that pretty easily, provided we have that f(t) available, which we do, as we know the full description for the + Bézier curve functions Bx(t) and By(t). +

    + -

    If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the start/end points on the inside?

    - - - Scripts are disabled. Showing fallback image. - - - +

    + If we use the Legendre-Gauss values for our C values (thickness for each strip) and t values (location of each strip), + we can determine the approximate length of a Bézier curve by computing the Legendre-Gauss sum. The following graphic shows a cubic curve, + with its computed lengths; Go ahead and change the curve, to see how its length changes. One thing worth trying is to see if you can make + a straight line, and see if the length matches what you'd expect. What if you form a line with the control points on the outside, and the + start/end points on the inside? +

    + + + Scripts are disabled. Showing fallback image. + + + +
    +
    +

    +
    前目录下
    + Approximated arc length +

    +

    + Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length + instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This + will come with an error, but this can be made arbitrarily small by increasing the segment count. +

    +

    + If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal + effort: +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -
    -
    -

    前目录下
    Approximated arc length

    -

    Sometimes, we don't actually need the precision of a true arc length, and we can get away with simply computing the approximate arc length instead. The by far fastest way to do this is to flatten the curve and then simply calculate the linear distance from point to point. This will come with an error, but this can be made arbitrarily small by increasing the segment count.

    -

    If we combine the work done in the previous sections on curve flattening and arc length computation, we can implement these with minimal effort:

    -
    - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    You may notice that even though the error in length is actually pretty significant in absolute terms, even at a low number of segments we get a length that agrees with the true length when it comes to just the integer part of the arc length. Quite often, approximations can drastically speed things up!

    - -
    -
    -

    前目录下
    Curvature of a curve

    -

    If we have two curves, and we want to line them in up in a way that "looks right", what would we use as metric to let a computer decide what "looks right" means?

    -

    For instance, we can start by ensuring that the two curves share an end coordinate, so that there is no "gap" between the end of one and the start of the next curve, but that won't guarantee that things look right: both curves can be going in wildly different directions, and the resulting joined geometry will have a corner in it, rather than a smooth transition from one curve to the next.

    -

    What we want is to ensure that the curvature at the transition from one curve to the next "looks good". So, we start with a shared coordinate, and then also require that derivatives for both curves match at that coordinate. That way, we're assured that their tangents line up, which must mean the curve transition is perfectly smooth. We can even make the second, third, etc. derivatives match up for better and better transitions.

    -

    Problem solved!

    -

    However, there's a problem with this approach: if we think about this a little more, we realise that "what a curve looks like" and its derivative values are pretty much entirely unrelated. After all, the section on reordering curves showed us that the same looking curve can have an infinite number of curve expressions of arbitrarily high Bézier degree, and each of those will have wildly different derivative values.

    -

    So what we really want is some kind of expression that's not based on any particular expression of t, but is based on something that is invariant to the kind of function(s) we use to draw our curve. And the prime candidate for this is our curve expression, reparameterised for distance: no matter what order of Bézier curve we use, if we were able to rewrite it as a function of distance-along-the-curve, all those different degree Bézier functions would end up being the same function for "coordinate at some distance D along the curve".

    -

    We've seen this before... that's the arc length function.

    -

    So you might think that in order to find the curvature of a curve, we now need to solve the arc length function itself, and that this would be quite a problem because we just saw that there is no way to actually do that. Thankfully, we don't. We only need to know the form of the arc length function, which we saw above and is fairly simple, rather than needing to solve the arc length function. If we start with the arc length expression and the run through the steps necessary to determine its derivative (with an alternative, shorter demonstration of how to do this found over on Stackexchange), then the integral that was giving us so much problems in solving the arc length function disappears entirely (because of the fundamental theorem of calculus), and what we're left with us some surprisingly simple maths that relates curvature (denoted as κ, "kappa") to—and this is the truly surprising bit—a specific combination of derivatives of our original function.

    -

    Let me highlight what just happened, because it's pretty special:

    -
      -
    1. we wanted to make curves line up, and initially thought to match the curves' derivatives, but
      2. -
    2. that turned out to be a really bad choice, so instead
      3. -
    3. we picked a function that is basically impossible to work with, and then worked with that, which
      4. -
    4. gives us a simple formula that is and expression using the curves' derivatives.
      5. -
    -

    That's crazy!

    -

    But that's also one of the things that makes maths so powerful: even if your initial ideas are off the mark, you might be much closer than you thought you were, and the journey from "thinking we're completely wrong" to "actually being remarkably close to being right" is where we can find a lot of insight.

    -

    So, what does the function look like? This:

    - - -

    Which is really just a "short form" that glosses over the fact that we're dealing with functions of t, so let's expand that a tiny bit:

    - - -

    And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so evaluating this function for any t value is just a matter of basic arithematics.

    -

    In fact, let's just implement it right now:

    + +

    + And while that's a little more verbose, it's still just as simple to work with as the first function: the curvature at some point on any + (and this cannot be overstated: any) curve is a ratio between the first and second derivative cross product, and something that + looks oddly similar to the standard Euclidean distance function. And nothing in these functions is hard to calculate either: for Bézier + curves, simply knowing our curve coordinates means we know what the first and second derivatives are, and so + evaluating this function for any t value is just a matter of basic arithematics. +

    +

    In fact, let's just implement it right now:

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return numerator / denominator + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    That was easy! (Well okay, that "not a number" value will need to be taken into account by downstream code, but that's a reality of programming anyway)

    -

    With all of that covered, let's line up some curves! The following graphic gives you two curves that look identical, but use quadratic and cubic functions, respectively. As you can see, despite their derivatives being necessarily different, their curvature (thanks to being derived based on maths that "ignores" specific function derivative, and instead gives a formula that smooths out any differences) is exactly the same. And because of that, we can put them together such that the point where they overlap has the same curvature for both curves, giving us the smoothest transition.

    - - - Scripts are disabled. Showing fallback image. - - - -

    One thing you may have noticed in this sketch is that sometimes the curvature looks fine, but seems to be pointing in the wrong direction, making it hard to line up the curves properly. A way around that, of course, is to show the curvature on both sides of the curve, so let's just do that. But let's take it one step further: we can also compute the associated "radius of curvature", which gives us the implicit circle that "fits" the curve's curvature at any point, using what is possibly the simplest bit of maths found in this entire primer:

    - - -

    So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" our curve at some point that we can control by using a slider:

    - - - Scripts are disabled. Showing fallback image. - - - - - + +

    + So let's revisit the previous graphic with the curvature visualised on both sides of our curves, as well as showing the circle that "fits" + our curve at some point that we can control by using a slider: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +
    +
    +

    +
    前目录下
    + Tracing a curve at fixed distance intervals +

    +

    + Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance + intervals over time, like a train along a track, and you want to use Bézier curves. +

    +

    Now you have a problem.

    +

    + The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a + track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick + t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In + fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another. +

    +

    + The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be + straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To + make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for + this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length + function, which can have more inflection points. +

    + + + Scripts are disabled. Showing fallback image. + + +

    + So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through + the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and + then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of + points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two + points, but if we have a high enough number of samples, we don't even need to bother. +

    +

    + So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t + curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks + like, by coloring each section of curve between two distance markers differently: +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    +

    + However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to + constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT). +

    +
    +
    +

    +
    前目录下
    + Intersections +

    +

    + Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to + work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to + perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box + overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the + actual curve. +

    +

    + We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by + implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both + lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution. +

    +

    Line-line intersections

    +

    + If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are + an intervals on by linear algebra, using the procedure outlined in this + top coder + article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line + segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments. +

    +

    + The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on + (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection + point). +

    + + + Scripts are disabled. Showing fallback image. + + + +
    +

    Implementing line-line intersections

    +

    + Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, + but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe + you're using point structs for the line. Let's get coding: +

    -
    -
    -

    前目录下
    Tracing a curve at fixed distance intervals

    -

    Say you want to draw a curve with a dashed line, rather than a solid line, or you want to move something along the curve at fixed distance intervals over time, like a train along a track, and you want to use Bézier curves.

    -

    Now you have a problem.

    -

    The reason you have a problem is that Bézier curves are parametric functions with non-linear behaviour, whereas moving a train along a track is about as close to a practical example of linear behaviour as you can get. The problem we're faced with is that we can't just pick t values at some fixed interval and expect the Bézier functions to generate points that are spaced a fixed distance apart. In fact, let's look at the relation between "distance along a curve" and "t value", by plotting them against one another.

    -

    The following graphic shows a particularly illustrative curve, and its distance-for-t plot. For linear traversal, this line needs to be straight, running from (0,0) to (length,1). That is, it's safe to say, not what we'll see: we'll see something very wobbly, instead. To make matters even worse, the distance-for-t function is also of a much higher order than our curve is: while the curve we're using for this exercise is a cubic curve, which can switch concave/convex form twice at best, the distance function is our old friend the arc length function, which can have more inflection points.

    - - - Scripts are disabled. Showing fallback image. - - - -

    So, how do we "cut up" the arc length function at regular intervals, when we can't really work with it? We basically cheat: we run through the curve using t values, determine the distance-for-this-t-value at each point we generate during the run, and then we find "the closest t value that matches some required distance" using those values instead. If we have a low number of points sampled, we can then even refine which t value "should" work for our desired distance by interpolating between two points, but if we have a high enough number of samples, we don't even need to bother.

    -

    So let's do exactly that: the following graph is similar to the previous one, showing how we would have to "chop up" our distance-for-t curve in order to get regularly spaced points on the curve. It also shows what using those t values on the real curve looks like, by coloring each section of curve between two distance markers differently:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Use the slider to increase or decrease the number of equidistant segments used to colour the curve.

    -

    However, are there better ways? One such way is discussed in "Moving Along a Curve with Specified Speed" by David Eberly of Geometric Tools, LLC, but basically because we have no explicit length function (or rather, one we don't have to constantly compute for different intervals), you may simply be better off with a traditional lookup table (LUT).

    - -
    -
    -

    前目录下
    Intersections

    -

    Let's look at some more things we will want to do with Bézier curves. Almost immediately after figuring out how to get bounding boxes to work, people tend to run into the problem that even though the minimal bounding box (based on rotation) is tight, it's not sufficient to perform true collision detection. It's a good first step to make sure there might be a collision (if there is no bounding box overlap, there can't be one), but in order to do real collision detection we need to know whether or not there's an intersection on the actual curve.

    -

    We'll do this in steps, because it's a bit of a journey to get to curve/curve intersection checking. First, let's start simple, by implementing a line-line intersection checker. While we can solve this the traditional calculus way (determine the functions for both lines, then compute the intersection by equating them and solving for two unknowns), linear algebra actually offers a nicer solution.

    -

    Line-line intersections

    -

    If we have two line segments with two coordinates each, segments A-B and C-D, we can find the intersection of the lines these segments are an intervals on by linear algebra, using the procedure outlined in this top coder article. Of course, we need to make sure that the intersection isn't just on the lines our line segments lie on, but actually on our line segments themselves. So after we find the intersection, we need to verify that it lies without the bounds of our original line segments.

    -

    The following graphic implements this intersection detection, showing a red point for an intersection on the lines our segments lie on (thus being a virtual intersection point), and a green point for an intersection that lies on both segments (being a real intersection point).

    - - - Scripts are disabled. Showing fallback image. - - - -
    - -

    Implementing line-line intersections

    -

    Let's have a look at how to implement a line-line intersection checking function. The basics are covered in the article mentioned above, but sometimes you need more function signatures, because you might not want to call your function with eight distinct parameters. Maybe you're using point structs for the line. Let's get coding:

    - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    + return lli4(line1.p1, line1.p2, line2.p1, line2.p2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + +
    - +

    What about curve-line intersections?

    +

    + Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we + translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in + a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a + curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the + section on finding extremities. +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + Scripts are disabled. Showing fallback image. + + + +
    -

    What about curve-line intersections?

    -

    Curve/line intersection is more work, but we've already seen the techniques we need to use in order to perform it: first we translate/rotate both the line and curve together, in such a way that the line coincides with the x-axis. This will position the curve in a way that makes it cross the line at points where its y-function is zero. By doing this, the problem of finding intersections between a curve and a line has now become the problem of performing root finding on our translated/rotated curve, as we already covered in the section on finding extremities.

    -
    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -
    +

    + Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the + curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and + curve splitting. +

    +
    +
    +

    +
    前目录下
    + Curve/curve intersection +

    +

    + Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" + technique: +

    +
      +
    1. + Take two curves C1 and C2, and treat them as a pair. +
      2. +
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. +
    3. + With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2). +
      4. +
    4. + For each pair, check whether their bounding boxes overlap. +
        +
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. +
      2. If there is overlap, rerun all steps for this pair.
        3. +
      +
      5. +
    5. + Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we + might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value + (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative). +
      6. +
    +

    + This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then + taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections. +

    +

    + The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click + the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value + that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the + algorithm, so you can try this with lots of different curves. +

    +

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    + + + Scripts are disabled. Showing fallback image. + + + + + + +

    + Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into + two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at + t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each + iteration, and each successive step homing in on the curve's self-intersection points. +

    +
    +
    +

    +
    + 前目录下 +
    + The projection identity +

    +

    + De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to + draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. + Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point + around to change the curve's shape. +

    +

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    +

    + In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want + to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know + has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for + reasons that will become obvious. +

    +

    + So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following + graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which + taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what + happens to that value? +

    +
    + + + Scripts are disabled. Showing fallback image. + + + + + + + + Scripts are disabled. Showing fallback image. + + + + + +
    -

    Curve/curve intersection, however, is more complicated. Since we have no straight line to align to, we can't simply align one of the curves and be left with a simple procedure. Instead, we'll need to apply two techniques we've met before: de Casteljau's algorithm, and curve splitting.

    - -
    -
    -

    前目录下
    Curve/curve intersection

    -

    Using de Casteljau's algorithm to split the curve we can now implement curve/curve intersection finding using a "divide and conquer" technique:

    -
      -
    1. Take two curves C1 and C2, and treat them as a pair.
      2. -
    2. If their bounding boxes overlap, split up each curve into two sub-curves
      3. -
    3. With C1.1, C1.2, C2.1 and C2.2, form four new pairs (C1.1,C2.1), (C1.1, C2.2), (C1.2,C2.1), and (C1.2,C2.2).
      4. -
    4. For each pair, check whether their bounding boxes overlap.
        -
      1. If their bounding boxes do not overlap, discard the pair, as there is no intersection between this pair of curves.
        2. -
      2. If there is overlap, rerun all steps for this pair.
        3. -
      -
      5. -
    5. Once the sub-curves we form are so small that they effectively occupy sub-pixel areas, we consider an intersection found, noting that we might have a cluster of multiple intersections at the sub-pixel level, out of which we pick one to act as "found" t value (we can either throw all but one away, we can average the cluster's t values, or you can do something even more creative).
      6. -
    -

    This algorithm will start with a single pair, "balloon" until it runs in parallel for a large number of potential sub-pairs, and then taper back down as it homes in on intersection coordinates, ending up with as many pairs as there are intersections.

    -

    The following graphic applies this algorithm to a pair of cubic curves, one step at a time, so you can see the algorithm in action. Click the button to run a single step in the algorithm, after setting up your curves in some creative arrangement. You can also change the value that is used in step 5 to determine whether the curves are small enough. Manipulating the curves or changing the threshold will reset the algorithm, so you can try this with lots of different curves.

    -

    (can you find the configuration that yields the maximum number of intersections between two cubic curves? Nine intersections!)

    - - - Scripts are disabled. Showing fallback image. - - - - - - -

    Finding self-intersections is effectively the same procedure, except that we're starting with a single curve, so we need to turn that into two separate curves first. This is trivially achieved by splitting at an inflection point, or if there are none, just splitting at t=0.5 first, and then running the exact same algorithm as above, with all non-overlapping curve pairs getting removed at each iteration, and each successive step homing in on the curve's self-intersection points.

    - -
    -
    -

    前目录下
    The projection identity

    -

    De Casteljau's algorithm is the pivotal algorithm when it comes to Bézier curves. You can use it not just to split curves, but also to draw them efficiently (especially for high-order Bézier curves), as well as to come up with curves based on three points and a tangent. Particularly this last thing is really useful because it lets us "mold" a curve, by picking it up at some point, and dragging that point around to change the curve's shape.

    -

    How does that work? Succinctly: we run de Casteljau's algorithm in reverse!

    -

    In order to run de Casteljau's algorithm in reverse, we need a few basic things: a start and end point, a point on the curve that we want to be moving around, which has an associated t value, and a point we've not explicitly talked about before, and as far as I know has no explicit name, but lives one iteration higher in the de Casteljau process then our on-curve point does. I like to call it "A" for reasons that will become obvious.

    -

    So let's use graphics instead of text to see where this "A" is, because text only gets us so far: move the sliders for the following graphics to see what, given a specific t value, our A coordinate is. As well as some other coordinates, which taken together let us derive a value that the graphics call "ratio": if you move the curve's points around, A, B, and C will move, what happens to that value?

    -
    - - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -
    - -

    So these graphics show us several things:

    -
      -
    1. a point at the tip of the curve construction's "hat": let's call that A, as well as
      2. -
    2. our on-curve point give our chosen t value: let's call that B, and finally,
      3. -
    3. a point that we get by projecting A, through B, onto the line between the curve's start and end points: let's call that C.
      4. -
    4. for both quadratic and cubic curves, two points e1 and e2, which represent the single-to-last step in de Casteljau's algorithm: in the last step, we find B at (1-t) * e1 + t * e2.
      5. -
    5. for cubic curves, also the points v1 and v2, which together with A represent the first step in de Casteljau's algorithm: in the next step, we find e1 and e2.
      6. -
    -

    These three values A, B, and C allow us to derive an important identity formula for quadratic and cubic Bézier curves: for any point on the curve with some t value, the ratio of distances from A to B and B to C is fixed: if some t value sets up a C that is 20% away from the start and 80% away from the end, then it doesn't matter where the start, end, or control points are; for that t value, C will always lie at 20% from the start and 80% from the end point. Go ahead, pick an on-curve point in either graphic and then move all the other points around: if you only move the control points, start and end won't move, and so neither will C, and if you move either start or end point, C will move but its relative position will not change.

    -

    So, how can we compute C? We start with our observation that C always lies somewhere between the start and end points, so logically C will have a function that interpolates between those two coordinates:

    - - -

    If we can figure out what the function u(t) looks like, we'll be done. Although we do need to remember that this u(t) will have a different form depending on whether we're working with quadratic or cubic curves. Running through the maths (with thanks to Boris Zbarsky) shows us the following two formulae:

    - - -

    And

    - - -

    So, if we know the start and end coordinates and the t value, we know C without having to calculate the A or even B coordinates. In fact, we can do the same for the ratio function. As another function of t, we technically don't need to know what A or B or C are. It, too, can be expressed as a pure function of t.

    -

    We start by observing that, given A, B, and C, the following always holds:

    - - -

    Working out the maths for this, we see the following two formulae for quadratic and cubic curves:

    - - -

    And

    - - -

    Which now leaves us with some powerful tools: given three points (start, end, and "some point on the curve"), as well as a t value, we can construct curves. We can compute C using the start and end points and our u(t) function, and once we have C, we can use our on-curve point (B) and the ratio(t) function to find A:

    - - -

    With A found, finding e1 and e2 for quadratic curves is a matter of running the linear interpolation with t between start and A to yield e1, and between A and end to yield e2. For cubic curves, there is no single pair of points that can act as e1 and e2 (there are infinitely many, because the tangent at B is a free parameter for cubic curves) so as long as the distance ratio between e1 to B and B to e2 is the Bézier ratio (1-t):t, we are free to pick any pair, after which we can reverse engineer v1 and v2:

    - - -

    And then reverse engineer the curve's control points:

    - - -

    So: if we have a curve's start and end points, as well as some third point B that we want the curve to pass through, then for any t value we implicitly know all the ABC values, which (combined with an educated guess on appropriate e1 and e2 coordinates for cubic curves) gives us the necessary information to reconstruct a curve's "de Casteljau skeleton". Which means that we can now do several things: we can "fit" curves using only three points, which means we can also "mold" curves by moving an on-curve point but leaving its start and end points, and then reconstruct the curve based on where we moved the on-curve point to. These are very useful things, and we'll look at both in the next few sections.

    - -
    -
    -

    前目录下
    Creating a curve from three points

    -

    Given the preceding section, you might be wondering if we can use that knowledge to just "create" curves by placing some points and having the computer do the rest, to which the answer is: that's exactly what we can now do!

    -

    For quadratic curves, things are pretty easy. Technically, we'll need a t value in order to compute the ratio function used in computing the ABC coordinates, but we can just as easily approximate one by treating the distance between the start and B point, and B and end point as a ratio, using

    - - -

    With this code in place, creating a quadratic curve from three points is literally just computing the ABC values, and using A as our curve's control point:

    - - - Scripts are disabled. Showing fallback image. - - - -

    For cubic curves we need to do a little more work, but really only just a little. We're first going to assume that a decent curve through the three points should approximate a circular arc, which first requires knowing how to fit a circle to three points. You may remember (if you ever learned it!) that a line between two points on a circle is called a chord, and that one property of chords is that the line from the center of any chord, perpendicular to that chord, passes through the center of the circle.

    -

    That means that if we have three points on a circle, we have three (different) chords, and consequently, three (different) lines that go from those chords through the center of the circle: if we find two of those lines, then their intersection will be our circle's center, and the circle's radius will—by definition!—be the distance from the center to any of our three points:

    - - - Scripts are disabled. Showing fallback image. - - - -

    With that covered, we now also know the tangent line to our point B, because the tangent to any point on the circle is a line through that point, perpendicular to the line from that point to the center. That just leaves marking appropriate points e1 and e2 on that tangent, so that we can construct a new cubic curve hull. We use the approach as we did for quadratic curves to automatically determine a reasonable t value, and then our e1 and e2 coordinates must obey the standard de Casteljau rule for linear interpolation:

    - - -

    Where d is the total length of the line segment from e1 to e2. So how long do we make that? There are again all kinds of approaches we can take, and a simple-but-effective one is to set the length of that segment to "one third the length of the baseline". This forces e1 and e2 to always be the "linear curve" distance apart, which means if we place our three points on a line, it will actually look like a line. Nice! The last thing we'll need to do is make sure to flip the sign of d depending on which side of the baseline our B is located, so we don't end up creating a funky curve with a loop in it. To do this, we can use the atan2 function:

    - - -

    This angle φ will be between 0 and π if B is "above" the baseline (rotating all three points so that the start is on the left and the end is the right), so we can use a relatively straight forward check to make sure we're using the correct sign for our value d:

    - - -

    The result of this approach looks as follows:

    - - - Scripts are disabled. Showing fallback image. - - - -

    It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; more on that later), we're only trying to come up with some reasonable e1 and e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives us:

    - - - Scripts are disabled. Showing fallback image. - - - -

    That looks perfectly serviceable!

    -

    Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at implementing curve molding in the section after that, so read on!

    + +

    The result of this approach looks as follows:

    + + + Scripts are disabled. Showing fallback image. + + + +

    + It is important to remember that even though we're using a circular arc to come up with decent e1 and e2 terms, + we're not trying to perfectly create a circular arc with a cubic curve (which is good, because we can't; + more on that later), we're only trying to come up with some reasonable e1 and + e2 points so we can construct a new cubic curve... so now that we have those: let's see what kind of cubic curve that gives + us: +

    + + + Scripts are disabled. Showing fallback image. + + + +

    That looks perfectly serviceable!

    +

    + Of course, we can take this one step further: we can't just "create" curves, we also have (almost!) all the tools available to "mold" + curves, where we can reshape a curve by dragging a point on the curve around while leaving the start and end fixed, effectively molding + the shape as if it were clay or the like. We'll see the last tool we need to do that in the next section, and then we'll look at + implementing curve molding in the section after that, so read on! +

    +
    +
    +

    +
    + 前目录下 +
    + Projecting a point onto a Bézier curve +

    +

    + Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're + trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the + curve our cursor will be closest to. So, how do we project points onto a curve? +

    +

    + If the Bézier curve is of low enough order, we might be able to + work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a + numerical approach again. We'll be finding our ideal t value using a + binary search. First, we do a coarse distance-check based on + t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast: +

    -
    -
    -

    前目录下
    Projecting a point onto a Bézier curve

    -

    Before we can move on to actual curve molding, it'll be good if know how to actually be able to find "some point on the curve" that we're trying to click on. After all, if all we have is our Bézier coordinates, that is not in itself enough to figure out which point on the curve our cursor will be closest to. So, how do we project points onto a curve?

    -

    If the Bézier curve is of low enough order, we might be able to work out the maths for how to do this, and get a perfect t value back, but in general this is an incredibly hard problem and the easiest solution is, really, a numerical approach again. We'll be finding our ideal t value using a binary search. First, we do a coarse distance-check based on t values associated with the curve's "to draw" coordinates (using a lookup table, or LUT). This is pretty fast:

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + -

    After this runs, we know that LUT[i] is the coordinate on the curve in our LUT that is closest to the point we want to project, so that's a pretty good initial guess as to what the best projection onto our curve is. To refine it, we note that LUT[i] is a better guess than both LUT[i-1] and LUT[i+1], but there might be an even better projection somewhere else between those two values, so that's what we're going to be testing for, using a variation of the binary search.

    -
      -
    1. we start with our point p, and the t values t1=LUT[i-1].t and t2=LUT[i+1].t, which span an interval v = t2-t1.
      2. -
    2. we test this interval in five spots: the start, middle, and end (which we already have), and the two points in between the middle and start/end points
      3. -
    3. we then check which of these five points is the closest to our original point p, and then repeat step 1 with the points before and after the closest point we just found.
      4. -
    -

    This makes the interval we check smaller and smaller at each iteration, and we can keep running the three steps until the interval becomes so small as to lead to distances that are, for all intents and purposes, the same for all points.

    -

    So, let's see that in action: in this case, I'm going to arbitrarily say that if we're going to run the loop until the interval is smaller than 0.001, and show you what that means for projecting your mouse cursor or finger tip onto a rather complex Bézier curve (which, of course, you can reshape as you like). Also shown are the original three points that our coarse check finds.

    - - - Scripts are disabled. Showing fallback image. - - - - -
    -
    -

    前目录下
    Intersections with a circle

    -

    It might seem odd to cover this subject so much later than the line/line, line/curve, and curve/curve intersection topics from several sections earlier, but the reason we can't cover circle/curve intersections is that we can't really discuss circle/curve intersection until we've covered the kind of lookup table (LUT) walking that the section on projecting a point onto a curve uses. To see why, let's look at what we would have to do if we wanted to find the intersections between a curve and a circle using calculus.

    -

    First, we observe that "finding intersections" in this case means that, given a circle defined by a center point c = (x,y) and a radius r, we want to find all points on the Bezier curve for which the distance to the circle's center point is equal to the circle radius, which by definition means those points lie on the circle, and so count as intersections. In maths, that means we're trying to solve:

    - - -

    Which seems simple enough. Unfortunately, when we expand that dist function, things get a lot more problematic:

    - - -

    And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the Abel-Ruffini theorem that we saw before, we can't solve this by just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left with a monstrous function to solve.

    -

    So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a bit more code to make sure we find all of them, but let's look at the steps involved:

    + +

    + And now we have a problem because that's a sixth degree polynomial inside the square root. So, thanks to the + Abel-Ruffini theorem that we saw before, we can't solve this by + just going "square both sides because we don't care about signs"... we can't solve a sixth degree polynomial. So, we're going to have to + actually evaluate that expression. We can "simplify" this by translating all our coordinates so that the center of the circle is (0,0) and + all our coordinates are shifted accordingly, which makes the cx and cy terms fall away, but then we're still left + with a monstrous function to solve. +

    +

    + So instead, we turn to the same kind of "LUT walking" that we saw for projecting points onto a curve, with a twist: instead of finding the + on-curve point with the smallest distance to our projection point, we want to find the on-curve point that has the exact distance + r to our projection point (namely, our circle center). Of course, there can be more than one such point, so there's also a + bit more code to make sure we find all of them, but let's look at the steps involved: +

    - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    + i = index + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + -

    This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is exactly the radius, so the coordinate lies on both the curve and the circle.

    -

    So far so good.

    -

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    +

    + This is very similar to the code in the previous section, with an extra input r for the circle radius, and a minor + change in the "distance for this coordinate": rather than just distance(coordinate, p) we want to know the difference between + that distance and the circle radius. After all, if that difference is zero, then the distance from the coordinate to the circle center is + exactly the radius, so the coordinate lies on both the curve and the circle. +

    +

    So far so good.

    +

    However, we also want to make sure we find all the points, not just a single one, so we need a little more code for that:

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    + start = i + 2; + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points ("current", "previous" and "before previous") and then check whether they form a local minimum:

    +

    + After running this code, values will be the list of all LUT coordinates that are closest to the distance r: we + can use those values to run the same kind of refinement lookup we used for point projection (with the caveat that we're now + not checking for smallest distance, but for "distance closest to r"), and we'll have all our intersection points. Of + course, that does require explaining what findClosest does: rather than looking for a global minimum, we're now interested in + finding a local minimum, so instead of checking a single point and looking at its distance value, we check three points + ("current", "previous" and "before previous") and then check whether they form a local minimum: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return start + i + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    In words: given a start index, the circle center and radius, and our LUT, we check where (closest to our start index) we can find a local minimum for the difference between "the distance from the curve to the circle center", and the circle's radius. We track this by looking at three values (associated with the indices index-2, index-1, and index), and we know we've found a local minimum if the three values show that the middle value (pd1) is less than either value beside it. When we do, we can set our "best guess, relative to start" as index-1. Of course, since we're now checking values relative to some start value, we might not find another candidate value at all, in which case we return start - 1, so that a simple "is the result less than start?" lets us determine that there are no more intersections to find.

    -

    Finally, while not necessary for point projection, there is one more step we need to perform when we run the binary refinement function on our candidate LUT indices, because we've so far only been testing using distances "closest to the radius of the circle", and that's actually not good enough... we need distances that are the radius of the circle. So, after running the refinement for each of these indices, we need to discard any final value that isn't the circle radius. And because we're working with floating point numbers, what this really means is that we need to discard any value that's a pixel or more "off". Or, if we want to get really fancy, "some small epsilon value".

    -

    Based on all of that, the following graphic shows this off for the standard cubic curve (which you can move the coordinates around for, of course) and a circle with a controllable radius centered on the graphic's center, using the code approach described above.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    And of course, for the full details, click that "view source" link.

    - -
    -
    -

    前目录下
    Molding a curve

    -

    Armed with knowledge of the "ABC" relation, point-on-curve projection, and guestimating reasonable looking helper values for cubic curve construction, we can finally cover curve molding: updating a curve's shape interactively, by dragging points on the curve around.

    -

    For quadratic curve, this is a really simple trick: we project our cursor onto the curve, which gives us a t value and initial B coordinate. We don't even need the latter: with our t value and "wherever the cursor is" as target B, we can compute the associated C:

    - - -

    And then the associated A:

    - - -

    And we're done, because that's our new quadratic control point!

    - - - Scripts are disabled. Showing fallback image. - - - -

    As before, cubic curves are a bit more work, because while it's easy to find our initial t value and ABC values, getting those all-important e1 and e2 coordinates is going to pose a bit of a problem... in the section on curve creation, we were free to pick an appropriate t value ourselves, which allowed us to find appropriate e1 and e2 coordinates. That's great, but when we're curve molding we don't have that luxury: whatever point we decide to start moving around already has its own t value, and its own e1 and e2 values, and those may not make sense for the rest of the curve.

    -

    For example, let's see what happens if we just "go with what we get" when we pick a point and start moving it around, preserving its t value and e1/e2 coordinates:

    - - - Scripts are disabled. Showing fallback image. - - - -

    That looks reasonable, close to the original point, but the further we drag our point, the less "useful" things become. Especially if we drag our point across the baseline, rather than turning into a nice curve.

    -

    One way to combat this might be to combine the above approach with the approach from the creating curves section: generate both the "unchanged t/e1/e2" curve, as well as the "idealized" curve through the start/cursor/end points, with idealized t value, and then interpolating between those two curves:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    The slide controls the "falloff distance" relative to where the original point on the curve is, so that as we drag our point around, it interpolates with a bias towards "preserving t/e1/e2" closer to the original point, and bias towards "idealized" form the further away we move our point, with anything that's further than our falloff distance simply being the idealized curve. We don't even try to interpolate at that point.

    -

    A more advanced way to try to smooth things out is to implement continuous molding, where we constantly update the curve as we move around, and constantly change what our B point is, based on constantly projecting the cursor on the curve as we're updating it - this is, you won't be surprised to learn, tricky, and beyond the scope of this section: interpolation (with a reasonable distance) will do for now!

    - -
    -
    -

    前目录下
    Curve fitting

    -

    Given the previous section, one question you might have is "what if I don't want to guess t values?". After all, plenty of graphics packages do automated curve fitting, so how can we implement that in a way that just finds us reasonable t values all on its own?

    -

    And really this is just a variation on the question "how do I get the curve through these X points?", so let's look at that. Specifically, let's look at the answer: "curve fitting". This is in fact a rather rich field in geometry, applying to anything from data modelling to path abstraction to "drawing", so there's a fair number of ways to do curve fitting, but we'll look at one of the most common approaches: something called a least squares polynomial regression. In this approach, we look at the number of points we have in our data set, roughly determine what would be an appropriate order for a curve that would fit these points, and then tackle the question "given that we want an nth order curve, what are the coordinates we can find such that our curve is "off" by the least amount?".

    -

    Now, there are many ways to determine how "off" points are from the curve, which is where that "least squares" term comes in. The most common tool in the toolbox is to minimise the squared distance between each point we have, and the corresponding point on the curve we end up "inventing". A curve with a snug fit will have zero distance between those two, and a bad fit will have non-zero distances between every such pair. It's a workable metric. You might wonder why we'd need to square, rather than just ensure that distance is a positive value (so that the total error is easy to compute by just summing distances) and the answer really is "because it tends to be a little better". There's lots of literature on the web if you want to deep-dive the specific merits of least squared error metrics versus least absolute error metrics, but those are well beyond the scope of this material.

    -

    So let's look at what we end up with in terms of curve fitting if we start with the idea of performing least squares Bézier fitting. We're going to follow a procedure similar to the one described by Jim Herold over on his "Least Squares Bézier Fit" article, and end with some nice interactive graphics for doing some curve fitting.

    -

    Before we begin, we're going to use the curve in matrix form. In the section on matrices, I mentioned that some things are easier if we use the matrix representation of a Bézier curve rather than its calculus form, and this is one of those things.

    -

    As such, the first step in the process is expressing our Bézier curve as powers/coefficients/coordinate matrix T x M x C, by expanding the Bézier functions.

    -
    - -

    Revisiting the matrix representation

    -

    Rewriting Bézier functions to matrix form is fairly easy, if you first expand the function, and then arrange them into a multiple line form, where each line corresponds to a power of t, and each column is for a specific coefficient. First, we expand the function:

    - - -

    And then we (trivially) rearrange the terms across multiple lines:

    - - -

    This rearrangement has "factors of t" at each row (the first row is t⁰, i.e. "1", the second row is t¹, i.e. "t", the third row is t²) and "coefficient" at each column (the first column is all terms involving "a", the second all terms involving "b", the third all terms involving "c").

    -

    With that arrangement, we can easily decompose this as a matrix multiplication:

    - - -

    We can do the same for the cubic curve, of course. We know the base function for cubics:

    - - -

    So we write out the expansion and rearrange:

    - - -

    Which we can then decompose:

    - - -

    And, of course, we can do this for quartic curves too (skipping the expansion step):

    - - -

    And so and on so on. Now, let's see how to use these T, M, and C, to do some curve fitting.

    -
    + +

    + And so and on so on. Now, let's see how to use these T, M, and C, to do some curve + fitting. +

    + -

    Let's get started: we're going to assume we picked the right order curve: for n points we're fitting an n-1th order curve, so we "start" with a vector P that represents the coordinates we already know, and for which we want to do curve fitting:

    - - -

    Next, we need to figure out appropriate t values for each point in the curve, because we need something that lets us tie "the actual coordinate" to "some point on the curve". There's a fair number of different ways to do this (and a large part of optimizing "the perfect fit" is about picking appropriate t values), but in this case let's look at two "obvious" choices:

    -
      -
    1. equally spaced t values, and
      2. -
    2. t values that align with distance along the polygon.
      3. -
    -

    The first one is really simple: if we have n points, then we'll just assign each point i a t value of (i-1)/(n-1). So if we have four points, the first point will have t=(1-1)/(4-1)=0/3, the second point will have t=(2-1)/(4-1)=1/3, the third point will have t=2/3, and the last point will be t=1. We're just straight up spacing the t values to match the number of points we have.

    -

    The second one is a little more interesting: since we're doing polynomial regression, we might as well exploit the fact that our base coordinates just constitute a collection of line segments. At the first point, we're fixing t=0, and the last point, we want t=1, and anywhere in between we're simply going to say that t is equal to the distance along the polygon, scaled to the [0,1] domain.

    -

    To get these values, we first compute the general "distance along the polygon" matrix:

    - - -

    Where length() is literally just that: the length of the line segment between the point we're looking at, and the previous point. This isn't quite enough, of course: we still need to make sure that all the values between i=1 and i=n fall in the [0,1] interval, so we need to scale all values down by whatever the total length of the polygon is:

    - - -

    And now we can move on to the actual "curve fitting" part: what we want is a function that lets us compute "ideal" control point values such that if we build a Bézier curve with them, that curve passes through all our original points. Or, failing that, have an overall error distance that is as close to zero as we can get it. So, let's write out what the error distance looks like.

    -

    As mentioned before, this function is really just "the distance between the actual coordinate, and the coordinate that the curve evaluates to for the associated t value", which we'll square to get rid of any pesky negative signs:

    - - -

    Since this function only deals with individual coordinates, we'll need to sum over all coordinates in order to get the full error function. So, we literally just do that; the total error function is simply the sum of all these individual errors:

    - - -

    And here's the trick that justifies using matrices: while we can work with individual values using calculus, with matrices we can compute as many values as we make our matrices big, all at the "same time", We can replace the individual terms pi with the full P coordinate matrix, and we can replace Bézier(si) with the matrix representation T x M x C we talked about before, which gives us:

    - - -

    In which we can replace the rather cumbersome "squaring" operation with a more conventional matrix equivalent:

    - - -

    Here, the letter T is used instead of the number 2, to represent the matrix transpose; each row in the original matrix becomes a column in the transposed matrix instead (row one becomes column one, row two becomes column two, and so on).

    -

    This leaves one problem: T isn't actually the matrix we want: we don't want symbolic t values, we want the actual numerical values that we computed for S, so we need to form a new matrix, which we'll call 𝕋, that makes use of those, and then use that 𝕋 instead of T in our error function:

    - - -

    Which, because of the first and last values in S, means:

    - - -

    Now we can properly write out the error function as matrix operations:

    - - -

    So, we have our error function: we now need to figure out the expression for where that function has minimal value, e.g. where the error between the true coordinates and the coordinates generated by the curve fitting is smallest. Like in standard calculus, this requires taking the derivative, and determining where that derivative is zero:

    - - -
    + +
    +

    Where did this derivative come from?

    +

    + That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I + straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know. +

    +

    + So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific + case, I + posted a question + to Math.stackexchange, and received a answer that goes into way more detail than I had + hoped to receive. +

    +

    + Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean + maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that + true?". There are answers. They might just require some time to come to understand. +

    +
    -

    Where did this derivative come from?

    -

    That... is a good question. In fact, when trying to run through this approach, I ran into the same question! And you know what? I straight up had no idea. I'm decent enough at calculus, I'm decent enough at linear algebra, and I just don't know.

    -

    So I did what I always do when I don't understand something: I asked someone to help me understand how things work. In this specific case, I posted a question to Math.stackexchange, and received a answer that goes into way more detail than I had hoped to receive.

    -

    Is that answer useful to you? Probably: no. At least, not unless you like understanding maths on a recreational level. And I do mean maths in general, not just basic algebra. But it does help in giving us a reference in case you ever wonder "Hang on. Why was that true?". There are answers. They might just require some time to come to understand.

    -
    - -

    Now, given the above derivative, we can rearrange the terms (following the rules of matrix algebra) so that we end up with an expression for C:

    - - -

    Here, the "to the power negative one" is the notation for the matrix inverse. But that's all we have to do: we're done. Starting with P and inventing some t values based on the polygon the coordinates in P define, we can compute the corresponding Bézier coordinates C that specify a curve that goes through our points. Or, if it can't go through them exactly, as near as possible.

    -

    So before we try that out, how much code is involved in implementing this? Honestly, that answer depends on how much you're going to be writing yourself. If you already have a matrix maths library available, then really not that much code at all. On the other hand, if you are writing this from scratch, you're going to have to write some utility functions for doing your matrix work for you, so it's really anywhere from 50 lines of code to maybe 200 lines of code. Not a bad price to pay for being able to fit curves to pre-specified coordinates.

    -

    So let's try it out! The following graphic lets you place points, and will start computing exact-fit curves once you've placed at least three. You can click for more points, and the code will simply try to compute an exact fit using a Bézier curve of the appropriate order. Four points? Cubic Bézier. Five points? Quartic. And so on. Of course, this does break down at some point: depending on where you place your points, it might become mighty hard for the fitter to find an exact fit, and things might actually start looking horribly off once there's enough points for compound floating point rounding errors to start making a difference (which is around 10~11 points).

    - - - Scripts are disabled. Showing fallback image. - - - - - - -

    You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be able to freely manipulate them and see what the effect on your curve is.

    - -
    -
    -

    前目录下
    Bézier curves and Catmull-Rom curves

    -

    Taking an excursion to different splines, the other common design curve is the Catmull-Rom spline, which unlike Bézier curves pass through each control point, so they offer a kind of "built-in" curve fitting.

    -

    In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what looks like a single Catmull-Rom curve is actually a spline: a single curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is defined by the point's coordinates, and the tangent for those points, the latter of which can trivially be derived from knowing the previous and next point:

    - + +

    + You'll note there is a convenient "toggle" buttons that lets you toggle between equidistant t values, and distance ratio + along the polygon formed by the points. Arguably more interesting is that once you have points to abstract a curve, you also get + direct control over the time values through sliders for each, because if the time values are our degree of freedom, you should be + able to freely manipulate them and see what the effect on your curve is. +

    +
    +
    +

    +
    + 前目录下 +
    + Bézier curves and Catmull-Rom curves +

    +

    + Taking an excursion to different splines, the other common design curve is the + Catmull-Rom spline, which unlike Bézier curves + pass through each control point, so they offer a kind of "built-in" curve fitting. +

    +

    + In fact, let's start with just playing with one: the following graphic has a predefined curve that you manipulate the points for, and lets + you add points by clicking/tapping the background, as well as let you control "how fast" the curve passes through its point using the + tension slider. The tenser the curve, the more the curve tends towards straight lines from one point to the next. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + Now, it may look like Catmull-Rom curves are very different from Bézier curves, because these curves can get very long indeed, but what + looks like a single Catmull-Rom curve is actually a spline: a single + curve built up of lots of identically-computed pieces, similar to if you just took a whole bunch of Bézier curves, placed them end to end, + and lined up their control points so that things look like a single curve. For a Catmull-Rom curve, each "piece" between two points is + defined by the point's coordinates, and the tangent for those points, the latter of which + can trivially be derived from knowing the + previous and next point: +

    + - -

    One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on that in the next section.

    -

    In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of variations that make things considerable more complex):

    + +

    + One downside of this is that—as you may have noticed from the graphic—the first and last point of the overall curve don't actually join up + with the rest of the curve: they don't have a previous/next point respectively, and so there is no way to calculate what their tangent + should be. Which also makes it rather tricky to fit a Catmull-Rom curve to three points like we were able to do for Bézier curves. More on + that in the next section. +

    +

    + In fact, before we move on, let's look at how to actually draw the basic form of these curves (I say basic, because there are a number of + variations that make things + considerable more + complex): +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + point(c0 * v1 + c1 * dv1 + c2 * v2 + c3 * dv2) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Now, since a Catmull-Rom curve is a form of cubic Hermite spline, and as cubic Bézier curves are also a form of cubic Hermite spline, we run into an interesting bit of maths programming: we can convert one to the other and back, and the maths for doing so is surprisingly simple!

    -

    The main difference between Catmull-Rom curves and Bézier curves is "what the points mean":

    -
      -
    • A cubic Bézier curve is defined by a start point, a control point that implies the tangent at the start, a control point that implies the tangent at the end, and an end point, plus a characterizing matrix that we can multiply by that point vector to get on-curve coordinates.
      • -
    • A Catmull-Rom curve is defined by a start point, a tangent that for that starting point, an end point, and a tangent for that end point, plus a characteristic matrix that we can multiple by the point vector to get on-curve coordinates.
      • -
    -

    Those are very similar, so let's see exactly how similar they are. We've already see the matrix form for Bézier curves, so how different is the matrix form for Catmull-Rom curves?:

    - - -

    That's pretty dang similar. So the question is: how can we convert that expression with Catmull-Rom matrix and vector into an expression of the Bézier matrix and vector? The short answer is of course "by using linear algebra", but the longer answer is the rest of this section, and involves some maths that you may not even care for: if you just want to know the (incredibly simple) conversions between the two curve forms, feel free to skip to the end of the following explanation, but if you want to how we can get one from the other... let's get mathing!

    -
    - -

    Deriving the conversion formulae

    -

    In order to convert between Catmull-Rom curves and Bézier curves, we need to know two things. Firstly, how to express the Catmull-Rom curve using a "set of four coordinates", rather than a mix of coordinates and tangents, and secondly, how to convert those Catmull-Rom coordinates to and from Bézier form.

    -

    We start with the first part, to figure out how we can go from Catmull-Rom V coordinates to Bézier P coordinates, by applying "some matrix T". We don't know what that T is yet, but we'll get to that:

    - - -

    So, this mapping says that in order to map a Catmull-Rom "point + tangent" vector to something based on an "all coordinates" vector, we need to determine the mapping matrix such that applying T yields P2 as start point, P3 as end point, and two tangents based on the lines between P1 and P3, and P2 nd P4, respectively.

    -

    Computing T is really more "arranging the numbers":

    - - -

    Thus:

    - - -

    However, we're not quite done, because Catmull-Rom curves have that "tension" parameter, written as τ (a lowercase"tau"), which is a scaling factor for the tangent vectors: the bigger the tension, the smaller the tangents, and the smaller the tension, the bigger the tangents. As such, the tension factor goes in the denominator for the tangents, and before we continue, let's add that tension factor into both our coordinate vector representation, and mapping matrix T:

    - - -

    With the mapping matrix properly done, let's rewrite the "point + tangent" Catmull-Rom matrix form to a matrix form in terms of four coordinates, and see what we end up with:

    - - -

    Replace point/tangent vector with the expression for all-coordinates:

    - - -

    and merge the matrices:

    - - -

    This looks a lot like the Bézier matrix form, which as we saw in the chapter on Bézier curves, should look like this:

    - - -

    So, if we want to express a Catmull-Rom curve using a Bézier curve, we'll need to turn this Catmull-Rom bit:

    - - -

    Into something that looks like this:

    - - -

    And the way we do that is with a fairly straight forward bit of matrix rewriting. We start with the equality we need to ensure:

    - - -

    Then we remove the coordinate vector from both sides without affecting the equality:

    - - -

    Then we can "get rid of" the Bézier matrix on the right by left-multiply both with the inverse of the Bézier matrix:

    - - -

    A matrix times its inverse is the matrix equivalent of 1, and because "something times 1" is the same as "something", so we can just outright remove any matrix/inverse pair:

    - - -

    And now we're basically done. We just multiply those two matrices and we know what V is:

    - - -

    We now have the final piece of our function puzzle. Let's run through each step.

    -
      -
    1. Start with the Catmull-Rom function:
      2. -
    - - -
      -
    1. rewrite to pure coordinate form:
      2. -
    - - -
      -
    1. rewrite for "normal" coordinate vector:
      2. -
    - - -
      -
    1. merge the inner matrices:
      2. -
    - - -
      -
    1. rewrite for Bézier matrix form:
      2. -
    - - -
      -
    1. and transform the coordinates so we have a "pure" Bézier expression:
      2. -
    - - -

    And we're done: we finally know how to convert these two curves!

    -
    + +

    And we're done: we finally know how to convert these two curves!

    + -

    If we have a Catmull-Rom curve defined by four coordinates P1 through P4, then we can draw that curve using a Bézier curve that has the vector:

    - - -

    Similarly, if we have a Bézier curve defined by four coordinates P1 through P4, we can draw that using a standard tension Catmull-Rom curve with the following coordinate values:

    - - -

    Or, if your API allows you to specify Catmull-Rom curves using plain coordinates:

    - - - -
    -
    -

    前目录下
    Creating a Catmull-Rom curve from three points

    -

    Much shorter than the previous section: we saw that Catmull-Rom curves need at least 4 points to draw anything sensible, so how do we create a Catmull-Rom curve from three points?

    -

    Short and sweet: we don't.

    -

    We run through the maths that lets us create a cubic Bézier curve, and then convert its coordinates to Catmull-Rom form using the conversion formulae we saw above.

    - -
    -
    -

    前目录下
    Forming poly-Bézier curves

    -

    Much like lines can be chained together to form polygons, Bézier curves can be chained together to form poly-Béziers, and the only trick required is to make sure that:

    -
      -
    1. the end point of each section is the starting point of the following section, and
      2. -
    2. the derivatives across that dual point line up.
      3. -
    -

    Unless you want sharp corners, of course. Then you don't even need 2.

    -

    We'll cover three forms of poly-Bézier curves in this section. First, we'll look at the kind that just follows point 1. where the end point of a segment is the same point as the start point of the next segment. This leads to poly-Béziers that are pretty hard to work with, but they're the easiest to implement:

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Dragging the control points around only affects the curve segments that the control point belongs to, and moving an on-curve point leaves the control points where they are, which is not the most useful for practical modelling purposes. So, let's add in the logic we need to make things a little better. We'll start by linking up control points by ensuring that the "incoming" derivative at an on-curve point is the same as it's "outgoing" derivative:

    - - -

    We can effect this quite easily, because we know that the vector from a curve's last control point to its last on-curve point is equal to the derivative vector. If we want to ensure that the first control point of the next curve matches that, all we have to do is mirror that last control point through the last on-curve point. And mirroring any point A through any point B is really simple:

    - - -

    So let's implement that and see what it gets us. The following two graphics show a quadratic and a cubic poly-Bézier curve again, but this time moving the control points around moves others, too. However, you might see something unexpected going on for quadratic curves...

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    As you can see, quadratic curves are particularly ill-suited for poly-Bézier curves, as all the control points are effectively linked. Move one of them, and you move all of them. Not only that, but if we move the on-curve points, it's possible to get a situation where a control point cannot satisfy the constraint that it's the reflection of its two neighbouring control points... This means that we cannot use quadratic poly-Béziers for anything other than really, really simple shapes. And even then, they're probably the wrong choice. Cubic curves are pretty decent, but the fact that the derivatives are linked means we can't manipulate curves as well as we might if we relaxed the constraints a little.

    -

    So: let's relax the requirement a little.

    -

    We can change the constraint so that we still preserve the angle of the derivatives across sections (so transitions from one section to the next will still look natural), but give up the requirement that they should also have the same vector length. Doing so will give us a much more useful kind of poly-Bézier curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Cubic curves are now better behaved when it comes to dragging control points around, but the quadratic poly-Bézier still has the problem that moving one control points will move the control points and may ending up defining "the next" control point in a way that doesn't work. Quadratic curves really aren't very useful to work with...

    -

    Finally, we also want to make sure that moving the on-curve coordinates preserves the relative positions of the associated control points. With that, we get to the kind of curve control that you might be familiar with from applications like Photoshop, Inkscape, Blender, etc.

    - - - Scripts are disabled. Showing fallback image. - - - - - - Scripts are disabled. Showing fallback image. - - - -

    Again, we see that cubic curves are now rather nice to work with, but quadratic curves have a new, very serious problem: we can move an on-curve point in such a way that we can't compute what needs to "happen next". Move the top point down, below the left and right points, for instance. There is no way to preserve correct control points without a kink at the bottom point. Quadratic curves: just not that good...

    -

    A final improvement is to offer fine-level control over which points behave which, so that you can have "kinks" or individually controlled segments when you need them, with nicely well-behaved curves for the rest of the path. Implementing that, is left as an exercise for the reader.

    - -
    -
    -

    前目录下
    Curve offsetting

    -

    Perhaps you're like me, and you've been writing various small programs that use Bézier curves in some way or another, and at some point you make the step to implementing path extrusion. But you don't want to do it pixel based; you want to stay in the vector world. You find that extruding lines is relatively easy, and tracing outlines is coming along nicely (although junction caps and fillets are a bit of a hassle), and then you decide to do things properly and add Bézier curves to the mix. Now you have a problem.

    -

    Unlike lines, you can't simply extrude a Bézier curve by taking a copy and moving it around, because of the curvatures; rather than a uniform thickness, you get an extrusion that looks too thin in places, if you're lucky, but more likely will self-intersect. The trick, then, is to scale the curve, rather than simply copying it. But how do you scale a Bézier curve?

    -

    Bottom line: you can't. So you cheat. We're not going to do true curve scaling, or rather curve offsetting, because that's impossible. Instead we're going to try to generate 'looks good enough' offset curves.

    -
    - -

    "What do you mean, you can't? Prove it."

    -

    First off, when I say "you can't," what I really mean is "you can't offset a Bézier curve with another Bézier curve", not even by using a really high order curve. You can find the function that describes the offset curve, but it won't be a polynomial, and as such it cannot be represented as a Bézier curve, which has to be a polynomial. Let's look at why this is:

    -

    From a mathematical point of view, an offset curve O(t) is a curve such that, given our original curve B(t), any point on O(t) is a fixed distance d away from coordinate B(t). So let's math that:

    - - -

    However, we're working in 2D, and d is a single value, so we want to turn it into a vector. If we want a point distance d "away" from the curve B(t) then what we really mean is that we want a point at d times the "normal vector" from point B(t), where the "normal" is a vector that runs perpendicular ("at a right angle") to the tangent at B(t). Easy enough:

    - - -

    Now this still isn't very useful unless we know what the formula for N(t) is, so let's find out. N(t) runs perpendicular to the original curve tangent, and we know that the tangent is simply B'(t), so we could just rotate that 90 degrees and be done with it. However, we need to ensure that N(t) has the same magnitude for every t, or the offset curve won't be at a uniform distance, thus not being an offset curve at all. The easiest way to guarantee this is to make sure N(t) always has length 1, which we can achieve by dividing B'(t) by its magnitude:

    - - -

    Determining the length requires computing an arc length, and this is where things get Tricky with a capital T. First off, to compute arc length from some start a to end b, we must use the formula we saw earlier. Noting that "length" is usually denoted with double vertical bars:

    - - -

    So if we want the length of the tangent, we plug in B'(t), with t = 0 as start and -t = 1 as end:

    - - -

    And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is screwing everything up, because it turns our nice polynomials into things that are no longer polynomials.

    -

    There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine because they do, but they don't; remember that only the base function has binomial coefficients. That's before we factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call them circles, and they have much simpler functions than Bézier curves.

    -

    So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve.

    -

    And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the real offset curve would look like, if we could compute it.

    -
    + +

    + And that's where things go wrong. It doesn't even really matter what the second derivative for B(t) is, that square root is + screwing everything up, because it turns our nice polynomials into things that are no longer polynomials. +

    +

    + There is a small class of polynomials where the square root is also a polynomial, but they're utterly useless to us: any polynomial with + unweighted binomial coefficients has a square root that is also a polynomial. Now, you might think that Bézier curves are just fine + because they do, but they don't; remember that only the base function has binomial coefficients. That's before we + factor in our coordinates, which turn it into a non-binomial polygon. The only way to make sure the functions stay binomial is to make + all our coordinates have the same value. And that's not a curve, that's a point. We can already create offset curves for points, we call + them circles, and they have much simpler functions than Bézier curves. +

    +

    + So, since the tangent length isn't a polynomial, the normalised tangent won't be a polynomial either, which means + N(t) won't be a polynomial, which means that d times N(t) won't be a polynomial, which means + that, ultimately, O(t) won't be a polynomial, which means that even if we can determine the function for + O(t) just fine (and that's far from trivial!), it simply cannot be represented as a Bézier curve. +

    +

    + And that's one reason why Bézier curves are tricky: there are actually a lot of curves that cannot be represented as a Bézier + curve at all. They can't even model their own offset curves. They're weird that way. So how do all those other programs do it? Well, + much like we're about to do, they cheat. We're going to approximate an offset curve in a way that will look relatively close to what the + real offset curve would look like, if we could compute it. +

    + -

    So, you cannot offset a Bézier curve perfectly with another Bézier curve, no matter how high-order you make that other Bézier curve. However, we can chop up a curve into "safe" sub-curves (where "safe" means that all the control points are always on a single side of the baseline, and the midpoint of the curve at t=0.5 is roughly in the center of the polygon defined by the curve coordinates) and then point-scale each sub-curve with respect to its scaling origin (which is the intersection of the point normals at the start and end points).

    -

    A good way to do this reduction is to first find the curve's extreme points, as explained in the earlier section on curve extremities, and use these as initial splitting points. After this initial split, we can check each individual segment to see if it's "safe enough" based on where the center of the curve is. If the on-curve point for t=0.5 is too far off from the center, we simply split the segment down the middle. Generally this is more than enough to end up with safe segments.

    -

    The following graphics show off curve offsetting, and you can use the slider to control the distance at which the curve gets offset. The curve first gets reduced to safe segments, each of which is then offset at the desired distance. Especially for simple curves, particularly easily set up for quadratic curves, no reduction is necessary, but the more twisty the curve gets, the more the curve needs to be reduced in order to get segments that can safely be scaled.

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - -

    You may notice that this may still lead to small 'jumps' in the sub-curves when moving the curve around. This is caused by the fact that we're still performing a naive form of offsetting, moving the control points the same distance as the start and end points. If the curve is large enough, this may still lead to incorrect offsets.

    - -
    -
    -

    前目录下
    Graduated curve offsetting

    -

    What if we want to do graduated offsetting, starting at some distance s but ending at some other distance e? Well, if we can compute the length of a curve (which we can if we use the Legendre-Gauss quadrature approach) then we can also determine how far "along the line" any point on the curve is. With that knowledge, we can offset a curve so that its offset curve is not uniformly wide, but graduated between with two different offset widths at the start and end.

    -

    Like normal offsetting we cut up our curve in sub-curves, and then check at which distance along the original curve each sub-curve starts and ends, as well as to which point on the curve each of the control points map. This gives us the distance-along-the-curve for each interesting point in the sub-curve. If we call the total length of all sub-curves seen prior to seeing "the current" sub-curve S (and if the current sub-curve is the first one, S is zero), and we call the full length of our original curve L, then we get the following graduation values:

    -
      -
    • start: map S from interval (0,L) to interval (s,e)
      • -
    • c1: map(<strong>S+d1</strong>, 0,L, s,e), d1 = distance along curve to projection of c1
      • -
    • c2: map(<strong>S+d2</strong>, 0,L, s,e), d2 = distance along curve to projection of c2
      • -
    • ...
      • -
    • end: map(<strong>S+length(subcurve)</strong>, 0,L, s,e)
      • -
    -

    At each of the relevant points (start, end, and the projections of the control points onto the curve) we know the curve's normal, so offsetting is simply a matter of taking our original point, and moving it along the normal vector by the offset distance for each point. Doing so will give us the following result (these have with a starting width of 0, and an end width of 40 pixels, but can be controlled with your up and down arrow keys):

    - - - Scripts are disabled. Showing fallback image. - - - - - - - - Scripts are disabled. Showing fallback image. - - - - - - -
    -
    -

    前目录下
    Circles and quadratic Bézier curves

    -

    Circles and Bézier curves are very different beasts, and circles are infinitely easier to work with than Bézier curves. Their formula is much simpler, and they can be drawn more efficiently. But, sometimes you don't have the luxury of using circles, or ellipses, or arcs. Sometimes, all you have are Bézier curves. For instance, if you're doing font design, fonts have no concept of geometric shapes, they only know straight lines, and Bézier curves. OpenType fonts with TrueType outlines only know quadratic Bézier curves, and OpenType fonts with Type 2 outlines only know cubic Bézier curves. So how do you draw a circle, or an ellipse, or an arc?

    -

    You approximate.

    -

    We already know that Bézier curves cannot model all curves that we can think of, and this includes perfect circles, as well as ellipses, and their arc counterparts. However, we can certainly approximate them to a degree that is visually acceptable. Quadratic and cubic curves offer us different curvature control, so in order to approximate a circle we will first need to figure out what the error is if we try to approximate arcs of increasing degree with quadratic and cubic curves, and where the coordinates even lie.

    -

    Since arcs are mid-point-symmetrical, we need the control points to set up a symmetrical curve. For quadratic curves this means that the control point will be somewhere on a line that intersects the baseline at a right angle. And we don't get any choice on where that will be, since the derivatives at the start and end point have to line up, so our control point will lie at the intersection of the tangents at the start and end point.

    -

    First, let's try to fit the quadratic curve onto a circular arc. In the following sketch you can move the mouse around over a unit circle, to see how well, or poorly, a quadratic curve can approximate the arc from (1,0) to where your mouse cursor is:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    As you can see, things go horribly wrong quite quickly; even trying to approximate a quarter circle using a quadratic curve is a bad idea. An eighth of a turns might look okay, but how okay is okay? Let's apply some maths and find out. What we're interested in is how far off our on-curve coordinates are with respect to a circular arc, given a specific start and end angle. We'll be looking at how much space there is between the circular arc, and the quadratic curve's midpoint.

    -

    We start out with our start and end point, and for convenience we will place them on a unit circle (a circle around 0,0 with radius 1), at some angle φ:

    - - -

    What we want to find is the intersection of the tangents, so we want a point C such that:

    - - -

    i.e. we want a point that lies on the vertical line through S (at some distance a from S) and also lies on the tangent line through E (at some distance b from E). Solving this gives us:

    - - -

    First we solve for b:

    - - -

    which yields:

    - - -

    which we can then substitute in the expression for a:

    - - -

    A quick check shows that plugging these values for a and b into the expressions for Cx and Cy give the same x/y coordinates for both "a away from A" and "b away from B", so let's continue: now that we know the coordinate values for C, we know where our on-curve point T for t=0.5 (or angle φ/2) is, because we can just evaluate the Bézier polynomial, and we know where the circle arc's actual point P is for angle φ/2:

    - - -

    We compute T, observing that if t=0.5, the polynomial values (1-t)², 2(1-t)t, and t² are 0.25, 0.5, and 0.25 respectively:

    - - -

    Which, worked out for the x and y components, gives:

    - - -

    And the distance between these two is the standard Euclidean distance:

    - - -

    So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter circle and eighth circle?

    -
    - - plotted for 0 ≤ φ ≤ π: - - - plotted for 0 ≤ φ ≤ ½π: - - - - - plotted for 0 ≤ φ ≤ ¼π: -
    + +

    + So, what does this distance function look like when we plot it for a number of ranges for the angle φ, such as a half circle, quarter + circle and eighth circle? +

    + + + + + + + + +
    + + plotted for 0 ≤ φ ≤ π: + + + plotted for 0 ≤ φ ≤ ½π: + + + + + plotted for 0 ≤ φ ≤ ¼π: +
    -

    We now see why the eighth circle arc looks decent, but the quarter circle arc doesn't: an error of roughly 0.06 at t=0.5 means we're 6% off the mark... we will already be off by one pixel on a circle with pixel radius 17. Any decent sized quarter circle arc, say with radius 100px, will be way off if approximated by a quadratic curve! For the eighth circle arc, however, the error is only roughly 0.003, or 0.3%, which explains why it looks so close to the actual eighth circle arc. In fact, if we want a truly tiny error, like 0.001, we'll have to contend with an angle of (rounded) 0.593667, which equates to roughly 34 degrees. We'd need 11 quadratic curves to form a full circle with that precision! (technically, 10 and ten seventeenth, but we can't do partial curves, so we have to round up). That's a whole lot of curves just to get a shape that can be drawn using a simple function!

    -

    In fact, let's flip the function around, so that if we plug in the precision error, labelled ε, we get back the maximum angle for that precision:

    - - -

    And frankly, things are starting to look a bit ridiculous at this point, we're doing way more maths than we've ever done, but thankfully this is as far as we need the maths to take us: If we plug in the precisions 0.1, 0.01, 0.001 and 0.0001 we get the radians values 1.748, 1.038, 0.594 and 0.3356; in degrees, that means we can cover roughly 100 degrees (requiring four curves), 59.5 degrees (requiring six curves), 34 degrees (requiring 11 curves), and 19.2 degrees (requiring a whopping nineteen curves).

    -

    The bottom line? Quadratic curves are kind of lousy if you want circular (or elliptical, which are circles that have been squashed in one dimension) curves. We can do better, even if it's just by raising the order of our curve once. So let's try the same thing for cubic curves.

    - -
    -
    -

    前目录下
    Circular arcs and cubic Béziers

    -

    Let's look at approximating circles and circular arcs using cubic Béziers. How much better is that?

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    At cursory glance, a fair bit better, but let's find out how much better by looking at how to construct the Bézier curve.

    -

    A construction diagram for a cubic approximation of a circular arc

    -

    The start and end points are trivial, but the mid point requires a bit of work, but it's mostly basic trigonometry once we know the angle θ for our circular arc: if we scale our circular arc to a unit circle, we can always start our arc, with radius 1, at (1,0) and then given our arc angle θ, we also know that the circular arc has length θ (because unit circles are nice that way). We also know our end point, because that's just (cos(θ), sin(θ)), and so the challenge is to figure out what control points we need in order for the curve at t=0.5 to exactly touch the circular arc at the angle θ/2:

    -

    So let's again formally describe this:

    - - -

    Only P3 isn't quite straight-forward here, and its description is based on the fact that the triangle (origin, P4, P3) is a right angled triangle, with the distance between the origin and P4 being 1 (because we're working with a unit circle), and the distance between P4 and P3 being k, so that we can represent P3 as "The point P4 plus the vector from the origin to P4 but then rotated a quarter circle, counter-clockwise, and scaled by k".

    -

    With that, we can determine the y-coordinates for A, B, e1, and e2, after which we have all the information we need to determine what the value of k is. We can find these values by using (no surprise here) linear interpolation between known points, as A is midway between P2 and P3, e1 is between A and "midway between P1 and P2" (which is "half height" P2), and so forth:

    - - -

    Which now gives us two identities for B, because in addition to determining B through linear interpolation, we also know that B's y coordinate is just sin(θ/2): we started this exercise by saying we were going to approximate the circular arc using a Bézier curve that had its midpoint, which is point B, touching the unit circle at the arc's half-angle, by definition making B the point at (cos(θ/2), sin(θ/2)).

    -

    This means we can equate the two identities we now have for By and solve for k.

    -
    - -

    Deriving k

    -

    Solving for k is fairly straight forward, but it's a fair few steps, and if you just the immediate result: using a tool like Wolfram Alpha is definitely the way to go. That said, let's get going:

    - - -

    And finally, we can take further advantage of several trigonometric identities to drastically simplify our formula for k:

    - - -

    And we're done.

    -
    + +

    And we're done.

    + -

    So, the distance of our control points to the start/end points can be expressed as a number that we get from an almost trivial expression involving the circular arc's angle:

    - - -

    Which means that for any circular arc with angle θ and radius r, our Bézier approximation based on three points of incidence is:

    - - -

    Which also gives us the commonly found value of 0.55228 for quarter circles, based on them having an angle of half π:

    - - -

    And thus giving us the following Bézier coordinates for a quarter circle of radius r:

    - - -
    + +
    +

    So, how accurate is this?

    +

    + Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points + that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum + deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but + rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we + get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff. +

    +

    + So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, + finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around + that t value, repeating as many times as necessary to get us an arbitrarily precise value of t: +

    -

    So, how accurate is this?

    -

    Unlike for the quadratic curve, we can't use t=0.5 as our reference point because by its very nature it's one of the three points that are actually guaranteed to be on the circular arc itself. Instead, we need a different t value that will give us the maximum deflection - there are two possible choices (as our curve is still strictly "overshoots" the circular arc, and it's symmetrical) but rather than trying to use calculus to find the perfect t value—which we could! the maths is perfectly reasonable as long as we get to use computers—we can also just perform a binary search for the biggest deflection and not bother with all this maths stuff.

    -

    So let's do that instead: we can run a maximum deflection check that just runs through t from 0 to 1 at some coarse interval, finds a t value that has "the highest deflection of the bunch", then reruns the same check with a much smaller interval around that t value, repeating as many times as necessary to get us an arbitrarily precise value of t:

    - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    + return getMostWrongT(radius, bezier, worst_t - stepsize, worst_t + stepsize) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + -

    Plus, how often do you get to write a function with that name?

    -

    Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity first, and then coming back down as we approach 2π)

    - - -
    - - - - - -
    - error plotted for 0 ≤ φ ≤ 2π - - error plotted for 0 ≤ φ ≤ π - - error plotted for 0 ≤ φ ≤ ½π -
    +

    Plus, how often do you get to write a function with that name?

    +

    + Using this code, we find that our t values are approximately 0.211325 and 0.788675, so let's pick the lower of the two and see + what the maximum deflection is across our domain of angles, with the original quadratic error show in green (rocketing off to infinity + first, and then coming back down as we approach 2π) +

    + + + + + + + + + + + + + +
    + + + + + +
    + error plotted for 0 ≤ φ ≤ 2π + + error plotted for 0 ≤ φ ≤ π + + error plotted for 0 ≤ φ ≤ ½π +
    -

    That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again:

    -

    +

    + That last image is probably not quite clear enough: the cubic approximation of a quarter circle is so incredibly much better that we + can't even really see it at the same scale of our quadratic curve. Let's scale the y-axis a little, and try that again: +

    +

    -

    Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves.

    -
    +

    + Yeah... the error of a cubic approximation for a quarter circle turns out to be two orders of magnitude better. At + approximately 0.00027 (or: just shy of being 2.7 pixels off for a circle with a radius of 10,000 pixels) the increase in precision over + quadratic curves is quite spectacular - certainly good enough that no one in their right mind should ever use quadratic curves. +

    +
    -

    So that's it, kappa is 4/3 · tan(θ/4) , we're done! ...or are we?

    -

    Can we do better?

    -

    Technically: yes, we can. But I'm going to prefix this section with "we can, and we should investigate that possibility, but let me warn you up front that the result is only better if we're going to hard-code the values". We're about to get into the weeds and the standard three-points-of-incidence value is so good already that for most applications, trying to do better won't make any sense at all.

    -

    So with that said: what we calculated above is an upper bound for a best fit Bézier curve for a circular arc: anywhere we don't touch the circular arc in our approximation, we've "overshot" the arc. What if we dropped our value for k just a little, so that the curve starts out as an over-estimation, but then crosses the circular arc, yielding an region of underestimation, and then crosses the circular arc again, with another region of overestimation. This might give us a lower overall error, so let's see what we can do.

    -

    First, let's express the total error (given circular arc angle θ, and some k) using standard calculus notation:

    - - -

    This says that the error function for a given angle and value of k is equal to the "infinite" sum of differences between our curve and the circular arc, as we run t from 0 to 1, using an infinitely small step size. between subsequent t values.

    -

    Now, since we want to find the minimal error, that means we want to know where along this function things go from "error is getting progressively less" to "error is increasing again", which means we want to know where its derivative is zero, which as mathematical expression looks like:

    - - -

    And here we have the most direct application of the Fundamental Theorem of Calculus: the derivative and integral are each other's inverse operations, so they cancel out, leaving us with our original function:

    - - -

    And now we just solve for that... oh wait. We've seen this before. In order to solve this, we'd end up needing to solve this:

    - - -

    And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section on arc lengths—there is no symbolic solution for this equasion. Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer!

    -
    + +

    + And both of those terms on the left of the equal sign are 6th degree polynomials, which means—as we've covered in the section + on arc lengths—there is no symbolic solution for this equasion. + Instead, we'll have to use a numerical approach to find the solutions here, so... to the computer! +

    +
    +

    Iterating on a solution

    +

    + By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the + value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's + center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just + binary search our way to the most accurate value for c that gets us that middle case. +

    +

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    -

    Iterating on a solution

    -

    By which I really mean "to the binary search algorithm", because we're dealing with a reasonably well behaved function: depending on the value for k , we're either going to end up with a Bézier curve that's on average "not at distance r from the arc's center", "exactly distance r from the arc's center", or "more than distance r from the arc's center", so we can just binary search our way to the most accurate value for c that gets us that middle case.

    -

    First our setup, where we determine our upper and lower bounds, before entering our binary search:

    - - - - -
    1 - -
    2
    3
    4
    + return binarySearch(radius, angle, points, lowerBound, upperBound) + + + + 2 + + + 3 + + + 4 + + -

    And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and blog posts than you can ever read in a life time:

    +

    + And then the binary search algorithm, which can be found in pretty much any CS textbook, as well as more online articles, tutorials, and + blog posts than you can ever read in a life time: +

    - - - - - - - - - - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    11
    12
    13
    14
    15
    16
    17
    18
    19
    + return binarySearch(radius, angle, points, value, upperBound) + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + + 11 + + + 12 + + + 13 + + + 14 + + + 15 + + + 16 + + + 17 + + + 18 + + + 19 + + -

    Using the following radialError function, which samples the curve's approximation of the circular arc over several points (although the first and last point will never contribute anything, so we skip them):

    +

    + Using the following radialError function, which samples the curve's approximation of the circular arc over several points + (although the first and last point will never contribute anything, so we skip them): +

    - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    + return err + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + -

    In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a vector, we can get its length to the origin using a magnitude call.

    -

    Examining the result

    -

    Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, for each angle, plot what the difference between the circular arc and the Bézier approximation looks like:

    -

    image-20210419085430711

    -

    Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from 0 to θ:

    -

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    - - -
    - - - - - -
    - max deflection using unit scale - - max deflection at 10x scale - - max deflection at 100x scale -
    +

    + In this, getOnCurvePoint is just the standard Bézier evaluation function, yielding a point. Treating that point as a + vector, we can get its length to the origin using a magnitude call. +

    +

    Examining the result

    +

    + Running the above code we can get a list of k values associated with a list of angles θ from 0 to π, and we can use that to, + for each angle, plot what the difference between the circular arc and the Bézier approximation looks like: +

    +

    image-20210419085430711

    +

    + Here we see the difference between an arc and its Bézier approximation plotted as we run t from 0 to 1. Just by looking at the + plot we can tell that there is maximum deflection at t = 0.5, so let's plot the maximum deflection "function", for angles from + 0 to θ: +

    +

    In fact, let's plot the maximum deflections for both approaches as a functions over θ:

    + + + + + + + + + + + + + +
    + + + + + +
    + max deflection using unit scale + + max deflection at 10x scale + + max deflection at 100x scale +
    -

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    -

    As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference.

    -
    +

    That doesn't actually appear to be all that much better, so let's look at some numbers, to see what the improvement actually is:

    + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    angle"improved" deflection"upper bound" deflectiondifference
    1/8 π6.202833502388927E-86.657161222278773E-84.5432771988984655E-9
    1/4 π3.978021202111215E-64.246252911066506E-62.68231708955291E-7
    3/8 π4.547652269037972E-54.8397483513262785E-52.9209608228830675E-6
    1/2 π2.569196199214696E-42.7251652752280364E-41.559690760133403E-5
    5/8 π9.877526288810667E-40.00104441758597118025.666495709011343E-5
    3/4 π0.002981649786796270.00314556284145806051.6391305466179062E-4
    7/8 π0.00763231828070198850.0080477779099483734.1545962924638413E-4
    π0.0173621859640437080.0183490165195459029.86830555502194E-4
    +

    + As we can see, the increase in precision is not particularly big: for a quarter circle (π/2) the traditional k will be off by + 2.75 pixels on a circle with radius 10,000 pixels, whereas this "better" fit will be off by 2.56 pixels. And while that's certainly an + almost 10% improvement, it's also nowhere near enough of an improvement to make a discernible difference. +

    +
    -

    At this point it should be clear that while, yes, there are improvement to be had, they're essentially insignificant while also being much more computationally expensive.

    -

    TL;DR: just tell me which value I should be using

    -

    It depends on what we need to do. If we just want the best value for quarter circles, and we're going to hard code the value for k, then there is no reason to hard-code the constant k=4/3*tan(pi/8) when you can just as easily hard-code the constant as k=0.551784777779014 instead.

    -

    If you need "the" value for quarter circles, use 0.551785 instead of 0.55228

    -

    However, for dynamic arc approximation, in code that tries to fit circular paths using Bézier paths instead, it should be fairly obvious that the simple function involving a tangent computation, two divisions, and one multiplication, is vastly more performant than running all the code we ended writing just to get a 25% lower error value, and most certainly worth preferring over getting the "more accurate" value.

    -

    If you need to fit Béziers to circular arcs on the fly, use 4/3 * tan(θ/4)

    -

    However, always remember that if you're writing for humans, you can typically use the best of both worlds: as the user interacts with their curves, you should draw their curves instead of drawing approximations of them. If they need to draw circles or circular arcs, draw those, and only approximate them with a Bézier curve when the data needs to be exported to a format that doesn't support those. Ideally with a preview mechanism that highlights where the errors will be, and how large they will be.

    -

    If you're writing code for graphics design by humans, use circular arcs for circular arcs

    -

    And that's it. We have pretty well exhausted this subject. There are different metrics we could use to find "different best k values", like trying to match arc length (e.g. when we're optimizing for material cost), or minimizing the area between the circular arc and the Bézier curve (e.g. when we're optimizing for inking), or minimizing the rate of change of the Bézier's curvature (e.g. when we're optimizing for curve traversal) and they all yield values that are so similar that it's almost certainly not worth it. (For instance, for quarter circle approximations those values are 0.551777, 0.5533344, and 0.552184 respectively. Much like the 0.551785 we get from minimizing the maximum deflection, none of these values are significantly better enough to prefer them over the upper bound value).

    - -
    -
    -

    前目录下
    Approximating Bézier curves with circular arcs

    -

    Let's look at doing the exact opposite of the previous section: rather than approximating circular arc using Bézier curves, let's approximate Bézier curves using circular arcs.

    -

    We already saw in the section on circle approximation that this will never yield a perfect equivalent, but sometimes you need circular arcs, such as when you're working with fabrication machinery, or simple vector languages that understand lines and circles, but not much else.

    -

    The approach is fairly simple: pick a starting point on the curve, and pick two points that are further along the curve. Determine the circle that goes through those three points, and see if it fits the part of the curve we're trying to approximate. Decent fit? Try spacing the points further apart. Bad fit? Try spacing the points closer together. Keep doing this until you've found the "good approximation/bad approximation" boundary, record the "good" arc, and then move the starting point up to overlap the end point we previously found. Rinse and repeat until we've covered the entire curve.

    -

    We already saw how to fit a circle through three points in the section on creating a curve from three points, and finding the arc through those points is straight-forward: pick one of the three points as start point, pick another as an end point, and the arc has to necessarily go from the start point, to the end point, over the remaining point.

    -

    So, how can we convert a Bézier curve into a (sequence of) circular arc(s)?

    -
      -
    • Start at t=0
      • -
    • Pick two points further down the curve at some value m = t + n and e = t + 2n
      • -
    • Find the arc that these points define
      • -
    • Determine how close the found arc is to the curve:
        -
      • Pick two additional points e1 = t + n/2 and e2 = t + n + n/2.
        • -
      • These points, if the arc is a good approximation of the curve interval chosen, should - lie on the circle, so their distance to the center of the circle should be the - same as the distance from any of the three other points to the center.
        • -
      • For point points, determine the (absolute) error between the radius of the circle, and the -actual distance from the center of the circle to the point on the curve.
        • -
      • If this error is too high, we consider the arc bad, and try a smaller interval.
        • -
      -
      • -
    -

    The result of this is shown in the next graphic: we start at a guaranteed failure: s=0, e=1. That's the entire curve. The midpoint is simply at t=0.5, and then we start performing a binary search.

    -
      -
    1. We start with low=0, mid=0.5 and high=1
      2. -
    2. That'll fail, so we retry with the interval halved: {0, 0.25, 0.5}
        -
      • If that arc's good, we move back up by half distance: {0, 0.375, 0.75}.
        • -
      • However, if the arc was still bad, we move down by half the distance: {0, 0.125, 0.25}.
        • -
      -
      3. -
    3. We keep doing this over and over until we have two arcs, in sequence, of which the first arc is good, and the second arc is bad. When we find that pair, we've found the boundary between a good approximation and a bad approximation, and we pick the good arc.
      4. -
    -

    The following graphic shows the result of this approach, with a default error threshold of 0.5, meaning that if an arc is off by a combined half pixel over both verification points, then we treat the arc as bad. This is an extremely simple error policy, but already works really well. Note that the graphic is still interactive, and you can use your up and down arrow keys keys to increase or decrease the error threshold, to see what the effect of a smaller or larger error threshold is.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    With that in place, all that's left now is to "restart" the procedure by treating the found arc's end point as the new to-be-determined arc's starting point, and using points further down the curve. We keep trying this until the found end point is for t=1, at which point we are done. Again, the following graphic allows for up and down arrow key input to increase or decrease the error threshold, so you can see how picking a different threshold changes the number of arcs that are necessary to reasonably approximate a curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    So... what is this good for? Obviously, if you're working with technologies that can't do curves, but can do lines and circles, then the answer is pretty straightforward, but what else? There are some reasons why you might need this technique: using circular arcs means you can determine whether a coordinate lies "on" your curve really easily (simply compute the distance to each circular arc center, and if any of those are close to the arc radii, at an angle between the arc start and end, bingo, this point can be treated as lying "on the curve"). Another benefit is that this approximation is "linear": you can almost trivially travel along the arcs at fixed speed. You can also trivially compute the arc length of the approximated curve (it's a bit like curve flattening). The only thing to bear in mind is that this is a lossy equivalence: things that you compute based on the approximation are guaranteed "off" by some small value, and depending on how much precision you need, arc approximation is either going to be super useful, or completely useless. It's up to you to decide which, based on your application!

    - -
    -
    -

    前目录下
    B-Splines

    -

    No discussion on Bézier curves is complete without also giving mention of that other beast in the curve design space: B-Splines. Easily confused to mean Bézier splines, that's not actually what they are; they are "basis function" splines, which makes a lot of difference, and we'll be looking at those differences in this section. We're not going to dive as deep into B-Splines as we have for Bézier curves (that would be an entire primer on its own) but we'll be looking at how B-Splines work, what kind of maths is involved in computing them, and how to draw them based on a number of parameters that you can pick for individual B-Splines.

    -

    First off: B-Splines are piecewise, polynomial interpolation curves, where the "single curve" is built by performing polynomial interpolation over a set of points, using a sliding window of a fixed number of points. For instance, a "cubic" B-Spline defined by twelve points will have its curve built by evaluating the polynomial interpolation of four points, and the curve can be treated as a lot of different sections, each controlled by four points at a time, such that the full curve consists of smoothly connected sections defined by points {1,2,3,4}, {2,3,4,5}, ..., {8,9,10,11}, and finally {9,10,11,12}, for eight sections.

    -

    What do they look like? They look like this! Tap on the graphic to add more points, and move points around to see how they map to the spline curve drawn.

    - - - Scripts are disabled. Showing fallback image. - - - -

    The important part to notice here is that we are not doing the same thing with B-Splines that we do for poly-Béziers or Catmull-Rom curves: both of the latter simply define new sections as literally "new sections based on new points", so a 12 point cubic poly-Bézier curve is actually impossible, because we start with a four point curve, and then add three more points for each section that follows, so we can only have 4, 7, 10, 13, 16, etc. point Poly-Béziers. Similarly, while Catmull-Rom curves can grow by adding single points, this addition of a single point introduces three implicit Bézier points. Cubic B-Splines, on the other hand, are smooth interpolations of each possible curve involving four consecutive points, such that at any point along the curve except for our start and end points, our on-curve coordinate is defined by four control points.

    -

    Consider the difference to be this:

    -
      -
    • for Bézier curves, the curve is defined as an interpolation of points, but:
      • -
    • for B-Splines, the curve is defined as an interpolation of curves.
      • -
    -

    In fact, let's look at that again, but this time with the base curves shown, too. Each consecutive four points define one curve:

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have a look at how things work.

    -

    How to compute a B-Spline curve: some maths

    -

    Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the end, just like for Bézier curves), by evaluating the following function:

    - + +

    + In order to make this interpolation of curves work, the maths is necessarily more complex than the maths for Bézier curves, so let's have + a look at how things work. +

    +

    How to compute a B-Spline curve: some maths

    +

    + Given a B-Spline of degree d and thus order k=d+1 (so a quadratic B-Spline is degree 2 and order 3, a cubic + B-Spline is degree 3 and order 4, etc) and n control points P0 through Pn-1, we can compute a point on the curve for some value t in the interval [0,1] (where 0 is the start of the curve, and 1 the + end, just like for Bézier curves), by evaluating the following function: +

    + - -

    Which, honestly, doesn't tell us all that much. All we can see is that a point on a B-Spline curve is defined as "a mix of all the control points, weighted somehow", where the weighting is achieved through the N(...) function, subscripted with an obvious parameter i, which comes from our summation, and some magical parameter k. So we need to know two things: 1. what does N(t) do, and 2. what is that k? Let's cover both, in reverse order.

    -

    The parameter k represents the "knot interval" over which a section of curve is defined. As we learned earlier, a B-Spline curve is itself an interpolation of curves, and we can treat each transition where a control point starts or stops influencing the total curvature as a "knot on the curve". -Doing so for a degree d B-Spline with n control point gives us d + n + 1 knots, defining d + n intervals along the curve, and it is these intervals that the above k subscript to the N() function applies to.

    -

    Then the N() function itself. What does it look like?

    - - -

    So this is where we see the interpolation: N(t) for an (i,k) pair (that is, for a step in the above summation, on a specific knot interval) is a mix between N(t) for (i,k-1) and N(t) for (i+1,k-1), so we see that this is a recursive iteration where i goes up, and k goes down, so it seem reasonable to expect that this recursion has to stop at some point; obviously, it does, and specifically it does so for the following i/k values:

    - - -

    And this function finally has a straight up evaluation: if a t value lies within a knot-specific interval once we reach a k=1 value, it "counts", otherwise it doesn't. We did cheat a little, though, because for all these values we need to scale our t value first, so that it lies in the interval bounded by knots[d] and knots[n], which are the start point and end point where curvature is controlled by exactly order control points. For instance, for degree 3 (=order 4) and 7 control points, with knot vector [1,2,3,4,5,6,7,8,9,10,11], we map t from [the interval 0,1] to the interval [4,8], and then use that value in the functions above, instead.

    -

    Can we simplify that?

    -

    We can, yes.

    -

    People far smarter than us have looked at this work, and two in particular — Maurice Cox and Carl de Boor — came to a mathematically pleasing solution: to compute a point P(t), we can compute this point by evaluating d(t) on a curve section between knots i and i+1:

    - - -

    This is another recursive function, with k values decreasing from the curve order to 1, and the value α (alpha) defined by:

    - - -

    That looks complicated, but it's not. Computing alpha is just a fraction involving known, plain numbers. And, once we have our alpha value, we also have (1-alpha) because it's a trivial subtraction. Computing the d() function is thus mostly a matter of computing pretty simple arithmetical statements, with some caching of results so we can refer to them as we recurve. While the recursion might see computationally expensive, the total algorithm is cheap, as each step only involves very simple maths.

    -

    Of course, the recursion does need a stop condition:

    - - -

    So, we actually see two stopping conditions: either i becomes 0, in which case d() is zero, or k becomes zero, in which case we get the same "either 1 or 0" that we saw in the N() function above.

    -

    Thanks to Cox and de Boor, we can compute points on a B-Spline pretty easily using the same kind of linear interpolation we saw in de Casteljau's algorithm. For instance, if we write out d() for i=3 and k=3, we get the following recursion diagram:

    - - -

    That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. It's really quite elegant.

    -

    One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then summing back up "to the left". We can just start on the right and work our way left immediately.

    -

    Running the computation

    -

    Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on this Michigan Tech page, but an easier to read version is implemented by b-spline.js, so we'll look at its code.

    -

    Given an input value t, we first map the input to a value from the domain [0,1] to the domain [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped t value lies on:

    + +

    + That is, we compute d(3,3) as a mixture of d(2,3) and d(2,2), where those two are themselves a + mixture of d(1,3) and d(1,2), and d(1,2) and d(1,1), respectively, which are + themselves a mixture of etc. etc. We simply keep expanding our terms until we reach the stop conditions, and then sum everything back up. + It's really quite elegant. +

    +

    + One thing we need to keep in mind is that we're working with a spline that is constrained by its control points, so even though the + d(..., k) values are zero or one at the lowest level, they are really "zero or one, times their respective control point", so + in the next section you'll see the algorithm for running through the computation in a way that starts with a copy of the control point + vector and then works its way up to that single point, rather than first starting "on the left", working our way "to the right" and then + summing back up "to the left". We can just start on the right and work our way left immediately. +

    +

    Running the computation

    +

    + Unlike the de Casteljau algorithm, where the t value stays the same at every iteration, for B-Splines that is not the case, + and so we end having to (for each point we evaluate) run a fairly involving bit of recursive computation. The algorithm is discussed on + this Michigan Tech page, but an easier to read version + is implemented by b-spline.js, so we'll look at its code. +

    +

    + Given an input value t, we first map the input to a value from the domain [0,1] to the domain + [knots[degree], knots[knots.length - 1 - degree]. Then, we find the section number s that this mapped + t value lies on: +

    - - -
    1 - -
    2
    3
    +} + + + + 2 + + + 3 + + -

    after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU page (updated to use this description's variable names):

    +

    + after running this code, s is the index for the section the point will lie on. We then run the algorithm mentioned on the MU + page (updated to use this description's variable names): +

    - - - - - - - - - -
    1 - -
    2
    3
    4
    5
    6
    7
    8
    9
    10
    +} + + + + 2 + + + 3 + + + 4 + + + 5 + + + 6 + + + 7 + + + 8 + + + 9 + + + 10 + + -

    (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those v[i-1] don't try to use an array index that doesn't exist)

    -

    Open vs. closed paths

    -

    Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need more than just a single point.

    -

    Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than separate coordinate objects.

    -

    Manipulating the curve through the knot vector

    -

    The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at:

    -
      -
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. -
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. -
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. -
    4. we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a uniform vector in between.
      5. -
    -

    Uniform B-Splines

    -

    The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to the curvature.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve.

    -

    The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get rid of these, by being clever about how we apply the following uniformity-breaking approach instead...

    -

    Reducing local curve complexity by collapsing intervals

    -

    Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost and the curve "kinks".

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Open-Uniform B-Splines

    -

    By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the curve not starting and ending where we'd kind of like it to:

    -

    For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences that determine the curve shape.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Non-uniform B-Splines

    -

    This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than that any value knots[k+1] should be greater than or equal to knots[k].

    -

    One last thing: Rational B-Splines

    -

    While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like for rational Bézier curves.

    - - - Scripts are disabled. Showing fallback image. - - - - - -

    Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in arbitrary-precision 2D design due to the level of control a NURBS curve offers designers.

    -

    While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the last.

    -

    Extending our implementation to cover rational splines

    -

    The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the extended dimension.

    -

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    -

    We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how many dimensions it needs to interpolate.

    -

    In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight information.

    -

    Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing (x'/w', y'/w'). And that's it, we're done!

    +

    + (A nice bit of behaviour in this code is that we work the interpolation "backwards", starting at i=s at each level of the + interpolation, and we stop when i = s - order + level, so we always end up with a value for i such that those + v[i-1] don't try to use an array index that doesn't exist) +

    +

    Open vs. closed paths

    +

    + Much like poly-Béziers, B-Splines can be either open, running from the first point to the last point, or closed, where the first and last + point are the same coordinate. However, because B-Splines are an interpolation of curves, not just points, we can't simply make the first + and last point the same, we need to link as many points as are necessary to form "a curve" that the spline performs interpolation with. As + such, for an order d B-Spline, we need to make the first and last d points the same. This is of course hardly + more work than before (simply append points.splice(0,d) to points) but it's important to remember that you need + more than just a single point. +

    +

    + Of course if we want to manipulate these kind of curves we need to make sure to mark them as "closed" so that we know the coordinate for + points[0] and points[n-k] etc. don't just happen to have the same x/y values, but really are the same + coordinate, so that manipulating one will equally manipulate the other, but programming generally makes this really easy by storing + references to points, rather than copies (or other linked values such as coordinate weights, discussed in the NURBS section) rather than + separate coordinate objects. +

    +

    Manipulating the curve through the knot vector

    +

    + The most important thing to understand when it comes to B-Splines is that they work because of the concept of a knot vector. As + mentioned above, knots represent "where individual control points start/stop influencing the curve", but we never looked at the + values that go in the knot vector. If you look back at the N() and a() functions, you see that interpolations are based on + intervals in the knot vector, rather than the actual values in the knot vector, and we can exploit this to do some pretty interesting + things with clever manipulation of the knot vector. Specifically there are four things we can do that are worth looking at: +

    +
      +
    1. we can use a uniform knot vector, with equally spaced intervals,
      2. +
    2. we can use a non-uniform knot vector, without enforcing equally spaced intervals,
      3. +
    3. we can collapse sequential knots to the same value, locally lowering curve complexity using "null" intervals, and
      4. +
    4. + we can form a special case non-uniform vector, by combining (1) and (3) to for a vector with collapsed start and end knots, with a + uniform vector in between. +
      5. +
    +

    Uniform B-Splines

    +

    + The most straightforward type of B-Spline is the uniform spline. In a uniform spline, the knots are distributed uniformly over the entire + curve interval. For instance, if we have a knot vector of length twelve, then a uniform knot vector would be [0,1,2,3,...,9,10,11]. Or + [4,5,6,...,13,14,15], which defines the same intervals, or even [0,2,3,...,18,20,22], which also defines + the same intervals, just scaled by a constant factor, which becomes normalised during interpolation and so does not contribute to + the curvature. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + This is an important point: the intervals that the knot vector defines are relative intervals, so it doesn't matter if every + interval is size 1, or size 100 - the relative differences between the intervals is what shapes any particular curve. +

    +

    + The problem with uniform knot vectors is that, as we need order control points before we have any curve with which we can + perform interpolation, the curve does not "start" at the first point, nor "ends" at the last point. Instead there are "gaps". We can get + rid of these, by being clever about how we apply the following uniformity-breaking approach instead... +

    +

    Reducing local curve complexity by collapsing intervals

    +

    + Collapsing knot intervals, by making two or more consecutive knots have the same value, allows us to reduce the curve complexity in the + sections that are affected by the knots involved. This can have drastic effects: for every interval collapse, the curve order goes down, + and curve continuity goes down, to the point where collapsing order knots creates a situation where all continuity is lost + and the curve "kinks". +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Open-Uniform B-Splines

    +

    + By combining knot interval collapsing at the start and end of the curve, with uniform knots in between, we can overcome the problem of the + curve not starting and ending where we'd kind of like it to: +

    +

    + For any curve of degree D with control points N, we can define a knot vector of length N+D+1 in + which the values 0 ... D+1 are the same, the values D+1 ... N+1 follow the "uniform" pattern, and the values + N+1 ... N+D+1 are the same again. For example, a cubic B-Spline with 7 control points can have a knot vector + [0,0,0,0,1,2,3,4,4,4,4], or it might have the "identical" knot vector [0,0,0,0,2,4,6,8,8,8,8], etc. Again, it is the relative differences + that determine the curve shape. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    Non-uniform B-Splines

    +

    + This is essentially the "free form" version of a B-Spline, and also the least interesting to look at, as without any specific reason to + pick specific knot intervals, there is nothing particularly interesting going on. There is one constraint to the knot vector, other than + that any value knots[k+1] should be greater than or equal to knots[k]. +

    +

    One last thing: Rational B-Splines

    +

    + While it is true that this section on B-Splines is running quite long already, there is one more thing we need to talk about, and that's + "Rational" splines, where the rationality applies to the "ratio", or relative weights, of the control points themselves. By introducing a + ratio vector with weights to apply to each control point, we greatly increase our influence over the final curve shape: the more weight a + control point carries, the closer to that point the spline curve will lie, a bit like turning up the gravity of a control point, just like + for rational Bézier curves. +

    + + + Scripts are disabled. Showing fallback image. + + + + + +

    + Of course this brings us to the final topic that any text on B-Splines must touch on before calling it a day: the + NURBS, or Non-Uniform Rational B-Spline (NURBS is not a plural, + the capital S actually just stands for "spline", but a lot of people mistakenly treat it as if it is, so now you know better). NURBS is an + important type of curve in computer-facilitated design, used a lot in 3D modelling (typically as NURBS surfaces) as well as in + arbitrary-precision 2D design due to the level of control a NURBS curve offers designers. +

    +

    + While a true non-uniform rational B-Spline would be hard to work with, when we talk about NURBS we typically mean the Open-Uniform + Rational B-Spline, or OURBS, but that doesn't roll off the tongue nearly as nicely, and so remember that when people talk about NURBS, + they typically mean open-uniform, which has the useful property of starting the curve at the first control point, and ending it at the + last. +

    +

    Extending our implementation to cover rational splines

    +

    + The algorithm for working with Rational B-Splines is virtually identical to the regular algorithm, and the extension to work in the + control point weights is fairly simple: we extend each control point from a point in its original number of dimensions (2D, 3D, etc.) to + one dimension higher, scaling the original dimensions by the control point's weight, and then assigning that weight as its value for the + extended dimension. +

    +

    For example, a 2D point (x,y) with weight w becomes a 3D point (w * x, w * y, w).

    +

    + We then run the same algorithm as before, which will automatically perform weight interpolation in addition to regular coordinate + interpolation, because all we've done is pretended we have coordinates in a higher dimension. The algorithm doesn't really care about how + many dimensions it needs to interpolate. +

    +

    + In order to recover our "real" curve point, we take the final result of the point generation algorithm, and "unweigh" it: we take the + final point's derived weight w' and divide all the regular coordinate dimensions by it, then throw away the weight + information. +

    +

    + Based on our previous example, we take the final 3D point (x', y', w'), which we then turn back into a 2D point by computing + (x'/w', y'/w'). And that's it, we're done! +

    +
    +
    + -
    -
    - +

    +
    前目录
    + Comments and questions +

    +

    + First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how + to let me know you appreciated this book, you have two options: you can either head on over to the + Patreon page for this book, or if you prefer to make a one-time donation, head on over to + the buy Pomax a coffee page. This + work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of + coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on + writing. +

    +

    With that said, on to the comments!

    +
    +
    +
    + -

    前目录
    Comments and questions

    -

    First off, if you enjoyed this book, or you simply found it useful for something you were trying to get done, and you were wondering how to let me know you appreciated this book, you have two options: you can either head on over to the Patreon page for this book, or if you prefer to make a one-time donation, head on over to the buy Pomax a coffee page. This work has grown from a small primer to a 70-plus print-page-equivalent reader on the subject of Bézier curves over the years, and a lot of coffee went into the making of it. I don't regret a minute I spent on writing it, but I can always do with some more coffee to keep on writing.

    -

    With that said, on to the comments!

    -
    +
    +
    + This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are + almost no rights reserved. Go do something cool with it! +
    - - - - -
    - -
    - - This article is © 2011-2020 to me, Mike "Pomax" Kamermans, but the text, code, and images are - almost no rights reserved. - Go do something cool with it! - -
    - - - - - - - \ No newline at end of file + + + diff --git a/package-lock.json b/package-lock.json index e7952777..d933ec15 100644 --- a/package-lock.json +++ b/package-lock.json @@ -12,7 +12,7 @@ "write-good": "^1.0.3" }, "devDependencies": { - "canvas": "git://github.com/Pomax/node-canvas#master", + "canvas": "^2.8.0", "chokidar-cli": "^2.1.0", "fs-extra": "^9.0.1", "glob": "^7.1.6", @@ -52,6 +52,41 @@ "js-tokens": "^4.0.0" } }, + "node_modules/@mapbox/node-pre-gyp": { + "version": "1.0.8", + "resolved": "https://registry.npmjs.org/@mapbox/node-pre-gyp/-/node-pre-gyp-1.0.8.tgz", + "integrity": "sha512-CMGKi28CF+qlbXh26hDe6NxCd7amqeAzEqnS6IHeO6LoaKyM/n+Xw3HT1COdq8cuioOdlKdqn/hCmqPUOMOywg==", + "dev": true, + "dependencies": { + "detect-libc": "^1.0.3", + "https-proxy-agent": "^5.0.0", + "make-dir": "^3.1.0", + "node-fetch": "^2.6.5", + "nopt": "^5.0.0", + "npmlog": "^5.0.1", + "rimraf": "^3.0.2", + "semver": "^7.3.5", + "tar": "^6.1.11" + }, + "bin": { + "node-pre-gyp": "bin/node-pre-gyp" + } + }, + "node_modules/@mapbox/node-pre-gyp/node_modules/semver": { + "version": "7.3.5", + "resolved": "https://registry.npmjs.org/semver/-/semver-7.3.5.tgz", + "integrity": "sha512-PoeGJYh8HK4BTO/a9Tf6ZG3veo/A7ZVsYrSA6J8ny9nb3B1VrpkuN+z9OE5wfE5p6H4LchYZsegiQgbJD94ZFQ==", + "dev": true, + "dependencies": { + "lru-cache": "^6.0.0" + }, + "bin": { + "semver": "bin/semver.js" + }, + "engines": { + "node": ">=10" + } + }, "node_modules/@tokenizer/token": { "version": "0.1.1", "resolved": "https://registry.npmjs.org/@tokenizer/token/-/token-0.1.1.tgz", @@ -109,13 +144,42 @@ "npm": ">=5" } }, + "node_modules/agent-base": { + "version": "6.0.2", + "resolved": "https://registry.npmjs.org/agent-base/-/agent-base-6.0.2.tgz", + "integrity": "sha512-RZNwNclF7+MS/8bDg70amg32dyeZGZxiDuQmZxKLAlQjr3jGyLx+4Kkk58UO7D2QdgFIQCovuSuZESne6RG6XQ==", + "dev": true, + "dependencies": { + "debug": "4" + }, + "engines": { + "node": ">= 6.0.0" + } + }, + "node_modules/agent-base/node_modules/debug": { + "version": "4.3.3", + "resolved": "https://registry.npmjs.org/debug/-/debug-4.3.3.tgz", + "integrity": "sha512-/zxw5+vh1Tfv+4Qn7a5nsbcJKPaSvCDhojn6FEl9vupwK2VCSDtEiEtqr8DFtzYFOdz63LBkxec7DYuc2jon6Q==", + "dev": true, + "dependencies": { + "ms": "2.1.2" + }, + "engines": { + "node": ">=6.0" + }, + "peerDependenciesMeta": { + "supports-color": { + "optional": true + } + } + }, "node_modules/ansi-regex": { - "version": "2.1.1", - "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-2.1.1.tgz", - "integrity": "sha1-w7M6te42DYbg5ijwRorn7yfWVN8=", + "version": "5.0.1", + "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-5.0.1.tgz", + "integrity": "sha512-quJQXlTSUGL2LH9SUXo8VwsY4soanhgo6LNSm84E1LBcE8s3O0wpdiRzyR9z/ZZJMlMWv37qOOb9pdJlMUEKFQ==", "dev": true, "engines": { - "node": ">=0.10.0" + "node": ">=8" } }, "node_modules/ansi-styles": { @@ -144,19 +208,36 @@ } }, "node_modules/aproba": { - "version": "1.2.0", - "resolved": "https://registry.npmjs.org/aproba/-/aproba-1.2.0.tgz", - "integrity": "sha512-Y9J6ZjXtoYh8RnXVCMOU/ttDmk1aBjunq9vO0ta5x85WDQiQfUF9sIPBITdbiiIVcBo03Hi3jMxigBtsddlXRw==", + "version": "2.0.0", + "resolved": "https://registry.npmjs.org/aproba/-/aproba-2.0.0.tgz", + "integrity": "sha512-lYe4Gx7QT+MKGbDsA+Z+he/Wtef0BiwDOlK/XkBrdfsh9J/jPPXbX0tE9x9cl27Tmu5gg3QUbUrQYa/y+KOHPQ==", "dev": true }, "node_modules/are-we-there-yet": { - "version": "1.1.5", - "resolved": "https://registry.npmjs.org/are-we-there-yet/-/are-we-there-yet-1.1.5.tgz", - "integrity": "sha512-5hYdAkZlcG8tOLujVDTgCT+uPX0VnpAH28gWsLfzpXYm7wP6mp5Q/gYyR7YQ0cKVJcXJnl3j2kpBan13PtQf6w==", + "version": "2.0.0", + "resolved": "https://registry.npmjs.org/are-we-there-yet/-/are-we-there-yet-2.0.0.tgz", + "integrity": "sha512-Ci/qENmwHnsYo9xKIcUJN5LeDKdJ6R1Z1j9V/J5wyq8nh/mYPEpIKJbBZXtZjG04HiK7zV/p6Vs9952MrMeUIw==", "dev": true, "dependencies": { "delegates": "^1.0.0", - "readable-stream": "^2.0.6" + "readable-stream": "^3.6.0" + }, + "engines": { + "node": ">=10" + } + }, + "node_modules/are-we-there-yet/node_modules/readable-stream": { + "version": "3.6.0", + "resolved": "https://registry.npmjs.org/readable-stream/-/readable-stream-3.6.0.tgz", + "integrity": "sha512-BViHy7LKeTz4oNnkcLJ+lVSL6vpiFeX6/d3oSH8zCW7UxP2onchk+vTGB143xuFjHS3deTgkKoXXymXqymiIdA==", + "dev": true, + "dependencies": { + "inherits": "^2.0.3", + "string_decoder": "^1.1.1", + "util-deprecate": "^1.0.1" + }, + "engines": { + "node": ">= 6" } }, "node_modules/argparse": { @@ -286,15 +367,14 @@ } }, "node_modules/canvas": { - "version": "2.7.0", - "resolved": "git+ssh://git@github.com/Pomax/node-canvas.git#20defd4f41e23b9bbcfdf219f304fd49ebcc18f8", - "integrity": "sha512-wIIEeJpcdESW3UvKYjOruWyqLIrwuO22PKxBr9H3kubXgIch/mBWhOrYIPSD3hNqT0pbP7NK8HehLp0WBywxJw==", + "version": "2.8.0", + "resolved": "https://registry.npmjs.org/canvas/-/canvas-2.8.0.tgz", + "integrity": "sha512-gLTi17X8WY9Cf5GZ2Yns8T5lfBOcGgFehDFb+JQwDqdOoBOcECS9ZWMEAqMSVcMYwXD659J8NyzjRY/2aE+C2Q==", "dev": true, "hasInstallScript": true, - "license": "MIT", "dependencies": { + "@mapbox/node-pre-gyp": "^1.0.0", "nan": "^2.14.0", - "node-pre-gyp": "^0.15.0", "simple-get": "^3.0.3" }, "engines": { @@ -419,10 +499,13 @@ } }, "node_modules/chownr": { - "version": "1.1.4", - "resolved": "https://registry.npmjs.org/chownr/-/chownr-1.1.4.tgz", - "integrity": "sha512-jJ0bqzaylmJtVnNgzTeSOs8DPavpbYgEr/b0YL8/2GO3xJEhInFmhKMUnEJQjZumK7KXGFhUy89PrsJWlakBVg==", - "dev": true + "version": "2.0.0", + "resolved": "https://registry.npmjs.org/chownr/-/chownr-2.0.0.tgz", + "integrity": "sha512-bIomtDF5KGpdogkLd9VspvFzk9KfpyyGlS8YFVZl7TGPBHL5snIOnxeshwVgPteQ9b4Eydl+pVbIyE1DcvCWgQ==", + "dev": true, + "engines": { + "node": ">=10" + } }, "node_modules/cliui": { "version": "5.0.0", @@ -493,15 +576,6 @@ "node": ">= 4.0" } }, - "node_modules/code-point-at": { - "version": "1.1.0", - "resolved": "https://registry.npmjs.org/code-point-at/-/code-point-at-1.1.0.tgz", - "integrity": "sha1-DQcLTQQ6W+ozovGkDi7bPZpMz3c=", - "dev": true, - "engines": { - "node": ">=0.10.0" - } - }, "node_modules/color-convert": { "version": "1.9.3", "resolved": "https://registry.npmjs.org/color-convert/-/color-convert-1.9.3.tgz", @@ -517,6 +591,15 @@ "integrity": "sha1-p9BVi9icQveV3UIyj3QIMcpTvCU=", "dev": true }, + "node_modules/color-support": { + "version": "1.1.3", + "resolved": "https://registry.npmjs.org/color-support/-/color-support-1.1.3.tgz", + "integrity": "sha512-qiBjkpbMLO/HL68y+lh4q0/O1MZFj2RX6X/KmMa3+gJD3z+WwI1ZzDHysvqHGS3mP6mznPckpXmw1nI9cJjyRg==", + "dev": true, + "bin": { + "color-support": "bin.js" + } + }, "node_modules/colors": { "version": "1.4.0", "resolved": "https://registry.npmjs.org/colors/-/colors-1.4.0.tgz", @@ -1037,12 +1120,15 @@ } }, "node_modules/fs-minipass": { - "version": "1.2.7", - "resolved": "https://registry.npmjs.org/fs-minipass/-/fs-minipass-1.2.7.tgz", - "integrity": "sha512-GWSSJGFy4e9GUeCcbIkED+bgAoFyj7XF1mV8rma3QW4NIqX9Kyx79N/PF61H5udOV3aY1IaMLs6pGbH71nlCTA==", + "version": "2.1.0", + "resolved": "https://registry.npmjs.org/fs-minipass/-/fs-minipass-2.1.0.tgz", + "integrity": "sha512-V/JgOLFCS+R6Vcq0slCuaeWEdNC3ouDlJMNIsacH2VtALiu9mV4LPrHc5cDl8k5aw6J8jwgWWpiTo5RYhmIzvg==", "dev": true, "dependencies": { - "minipass": "^2.6.0" + "minipass": "^3.0.0" + }, + "engines": { + "node": ">= 8" } }, "node_modules/fs.realpath": { @@ -1073,19 +1159,23 @@ "dev": true }, "node_modules/gauge": { - "version": "2.7.4", - "resolved": "https://registry.npmjs.org/gauge/-/gauge-2.7.4.tgz", - "integrity": "sha1-LANAXHU4w51+s3sxcCLjJfsBi/c=", + "version": "3.0.2", + "resolved": "https://registry.npmjs.org/gauge/-/gauge-3.0.2.tgz", + "integrity": "sha512-+5J6MS/5XksCuXq++uFRsnUd7Ovu1XenbeuIuNRJxYWjgQbPuFhT14lAvsWfqfAmnwluf1OwMjz39HjfLPci0Q==", "dev": true, "dependencies": { - "aproba": "^1.0.3", + "aproba": "^1.0.3 || ^2.0.0", + "color-support": "^1.1.2", "console-control-strings": "^1.0.0", - "has-unicode": "^2.0.0", - "object-assign": "^4.1.0", + "has-unicode": "^2.0.1", + "object-assign": "^4.1.1", "signal-exit": "^3.0.0", - "string-width": "^1.0.1", - "strip-ansi": "^3.0.1", - "wide-align": "^1.1.0" + "string-width": "^4.2.3", + "strip-ansi": "^6.0.1", + "wide-align": "^1.1.2" + }, + "engines": { + "node": ">=10" } }, "node_modules/get-caller-file": { @@ -1281,6 +1371,36 @@ "node": ">=6" } }, + "node_modules/https-proxy-agent": { + "version": "5.0.0", + "resolved": "https://registry.npmjs.org/https-proxy-agent/-/https-proxy-agent-5.0.0.tgz", + "integrity": "sha512-EkYm5BcKUGiduxzSt3Eppko+PiNWNEpa4ySk9vTC6wDsQJW9rHSa+UhGNJoRYp7bz6Ht1eaRIa6QaJqO5rCFbA==", + "dev": true, + "dependencies": { + "agent-base": "6", + "debug": "4" + }, + "engines": { + "node": ">= 6" + } + }, + "node_modules/https-proxy-agent/node_modules/debug": { + "version": "4.3.3", + "resolved": "https://registry.npmjs.org/debug/-/debug-4.3.3.tgz", + "integrity": "sha512-/zxw5+vh1Tfv+4Qn7a5nsbcJKPaSvCDhojn6FEl9vupwK2VCSDtEiEtqr8DFtzYFOdz63LBkxec7DYuc2jon6Q==", + "dev": true, + "dependencies": { + "ms": "2.1.2" + }, + "engines": { + "node": ">=6.0" + }, + "peerDependenciesMeta": { + "supports-color": { + "optional": true + } + } + }, "node_modules/iconv-lite": { "version": "0.4.24", "resolved": "https://registry.npmjs.org/iconv-lite/-/iconv-lite-0.4.24.tgz", @@ -1299,15 +1419,6 @@ "integrity": "sha512-4vf7I2LYV/HaWerSo3XmlMkp5eZ83i+/CDluXi/IGTs/O1sejBNhTtnxzmRZfvOUqj7lZjqHkeTvpgSFDlWZTg==", "dev": true }, - "node_modules/ignore-walk": { - "version": "3.0.3", - "resolved": "https://registry.npmjs.org/ignore-walk/-/ignore-walk-3.0.3.tgz", - "integrity": "sha512-m7o6xuOaT1aqheYHKf8W6J5pYH85ZI9w077erOzLje3JsB1gkafkAhHHY19dqjulgIZHFm32Cp5uNZgcQqdJKw==", - "dev": true, - "dependencies": { - "minimatch": "^3.0.4" - } - }, "node_modules/indent-string": { "version": "4.0.0", "resolved": "https://registry.npmjs.org/indent-string/-/indent-string-4.0.0.tgz", @@ -1410,15 +1521,12 @@ } }, "node_modules/is-fullwidth-code-point": { - "version": "1.0.0", - "resolved": "https://registry.npmjs.org/is-fullwidth-code-point/-/is-fullwidth-code-point-1.0.0.tgz", - "integrity": "sha1-754xOG8DGn8NZDr4L95QxFfvAMs=", + "version": "3.0.0", + "resolved": "https://registry.npmjs.org/is-fullwidth-code-point/-/is-fullwidth-code-point-3.0.0.tgz", + "integrity": "sha512-zymm5+u+sCsSWyD9qNaejV3DFvhCKclKdizYaJUuHA83RLjb7nSuGnddCHGv0hk+KY7BMAlsWeK4Ueg6EV6XQg==", "dev": true, - "dependencies": { - "number-is-nan": "^1.0.0" - }, "engines": { - "node": ">=0.10.0" + "node": ">=8" } }, "node_modules/is-glob": { @@ -1595,15 +1703,6 @@ "link-checker": "cli" } }, - "node_modules/link-checker/node_modules/ansi-regex": { - "version": "5.0.0", - "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-5.0.0.tgz", - "integrity": "sha512-bY6fj56OUQ0hU1KjFNDQuJFezqKdrAyFdIevADiqrWHwSlbmBNMHp5ak2f40Pm8JTFyM2mqxkG6ngkHO11f/lg==", - "dev": true, - "engines": { - "node": ">=8" - } - }, "node_modules/link-checker/node_modules/ansi-styles": { "version": "4.3.0", "resolved": "https://registry.npmjs.org/ansi-styles/-/ansi-styles-4.3.0.tgz", @@ -1654,21 +1753,6 @@ "integrity": "sha512-GpVkmM8vF2vQUkj2LvZmD35JxeJOLCwJ9cUkugyk2nuhbv3+mJvpLYYt+0+USMxE+oj+ey/lJEnhZw75x/OMcQ==", "dev": true }, - "node_modules/link-checker/node_modules/emoji-regex": { - "version": "8.0.0", - "resolved": "https://registry.npmjs.org/emoji-regex/-/emoji-regex-8.0.0.tgz", - "integrity": "sha512-MSjYzcWNOA0ewAHpz0MxpYFvwg6yjy1NG3xteoqz644VCo/RPgnr1/GGt+ic3iJTzQ8Eu3TdM14SawnVUmGE6A==", - "dev": true - }, - "node_modules/link-checker/node_modules/is-fullwidth-code-point": { - "version": "3.0.0", - "resolved": "https://registry.npmjs.org/is-fullwidth-code-point/-/is-fullwidth-code-point-3.0.0.tgz", - "integrity": "sha512-zymm5+u+sCsSWyD9qNaejV3DFvhCKclKdizYaJUuHA83RLjb7nSuGnddCHGv0hk+KY7BMAlsWeK4Ueg6EV6XQg==", - "dev": true, - "engines": { - "node": ">=8" - } - }, "node_modules/link-checker/node_modules/mkdirp": { "version": "1.0.4", "resolved": "https://registry.npmjs.org/mkdirp/-/mkdirp-1.0.4.tgz", @@ -1681,32 +1765,6 @@ "node": ">=10" } }, - "node_modules/link-checker/node_modules/string-width": { - "version": "4.2.2", - "resolved": "https://registry.npmjs.org/string-width/-/string-width-4.2.2.tgz", - "integrity": "sha512-XBJbT3N4JhVumXE0eoLU9DCjcaF92KLNqTmFCnG1pf8duUxFGwtP6AD6nkjw9a3IdiRtL3E2w3JDiE/xi3vOeA==", - "dev": true, - "dependencies": { - "emoji-regex": "^8.0.0", - "is-fullwidth-code-point": "^3.0.0", - "strip-ansi": "^6.0.0" - }, - "engines": { - "node": ">=8" - } - }, - "node_modules/link-checker/node_modules/strip-ansi": { - "version": "6.0.0", - "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-6.0.0.tgz", - "integrity": "sha512-AuvKTrTfQNYNIctbR1K/YGTR1756GycPsg7b9bdV9Duqur4gv6aKqHXah67Z8ImS7WEz5QVcOtlfW2rZEugt6w==", - "dev": true, - "dependencies": { - "ansi-regex": "^5.0.0" - }, - "engines": { - "node": ">=8" - } - }, "node_modules/link-checker/node_modules/wrap-ansi": { "version": "7.0.0", "resolved": "https://registry.npmjs.org/wrap-ansi/-/wrap-ansi-7.0.0.tgz", @@ -1814,6 +1872,18 @@ "integrity": "sha1-wj6RtxAkKscMN/HhzaknTMOb8vQ=", "dev": true }, + "node_modules/lru-cache": { + "version": "6.0.0", + "resolved": "https://registry.npmjs.org/lru-cache/-/lru-cache-6.0.0.tgz", + "integrity": "sha512-Jo6dJ04CmSjuznwJSS3pUeWmd/H0ffTlkXXgwZi+eq1UCmqQwCh+eLsYOYCwY991i2Fah4h1BEMCx4qThGbsiA==", + "dev": true, + "dependencies": { + "yallist": "^4.0.0" + }, + "engines": { + "node": ">=10" + } + }, "node_modules/make-dir": { "version": "3.1.0", "resolved": "https://registry.npmjs.org/make-dir/-/make-dir-3.1.0.tgz", @@ -2030,22 +2100,28 @@ } }, "node_modules/minipass": { - "version": "2.9.0", - "resolved": "https://registry.npmjs.org/minipass/-/minipass-2.9.0.tgz", - "integrity": "sha512-wxfUjg9WebH+CUDX/CdbRlh5SmfZiy/hpkxaRI16Y9W56Pa75sWgd/rvFilSgrauD9NyFymP/+JFV3KwzIsJeg==", + "version": "3.1.6", + "resolved": "https://registry.npmjs.org/minipass/-/minipass-3.1.6.tgz", + "integrity": "sha512-rty5kpw9/z8SX9dmxblFA6edItUmwJgMeYDZRrwlIVN27i8gysGbznJwUggw2V/FVqFSDdWy040ZPS811DYAqQ==", "dev": true, "dependencies": { - "safe-buffer": "^5.1.2", - "yallist": "^3.0.0" + "yallist": "^4.0.0" + }, + "engines": { + "node": ">=8" } }, "node_modules/minizlib": { - "version": "1.3.3", - "resolved": "https://registry.npmjs.org/minizlib/-/minizlib-1.3.3.tgz", - "integrity": "sha512-6ZYMOEnmVsdCeTJVE0W9ZD+pVnE8h9Hma/iOwwRDsdQoePpoX56/8B6z3P9VNwppJuBKNRuFDRNRqRWexT9G9Q==", + "version": "2.1.2", + "resolved": "https://registry.npmjs.org/minizlib/-/minizlib-2.1.2.tgz", + "integrity": "sha512-bAxsR8BVfj60DWXHE3u30oHzfl4G7khkSuPW+qvpd7jFRHm7dLxOjUk1EHACJ/hxLY8phGJ0YhYHZo7jil7Qdg==", "dev": true, "dependencies": { - "minipass": "^2.9.0" + "minipass": "^3.0.0", + "yallist": "^4.0.0" + }, + "engines": { + "node": ">= 8" } }, "node_modules/mkdirp": { @@ -2072,23 +2148,6 @@ "integrity": "sha512-isWHgVjnFjh2x2yuJ/tj3JbwoHu3UC2dX5G/88Cm24yB6YopVgxvBObDY7n5xW6ExmFhJpSEQqFPvq9zaXc8Jw==", "dev": true }, - "node_modules/needle": { - "version": "2.5.2", - "resolved": "https://registry.npmjs.org/needle/-/needle-2.5.2.tgz", - "integrity": "sha512-LbRIwS9BfkPvNwNHlsA41Q29kL2L/6VaOJ0qisM5lLWsTV3nP15abO5ITL6L81zqFhzjRKDAYjpcBcwM0AVvLQ==", - "dev": true, - "dependencies": { - "debug": "^3.2.6", - "iconv-lite": "^0.4.4", - "sax": "^1.2.4" - }, - "bin": { - "needle": "bin/needle" - }, - "engines": { - "node": ">= 4.4.x" - } - }, "node_modules/nice-try": { "version": "1.0.5", "resolved": "https://registry.npmjs.org/nice-try/-/nice-try-1.0.5.tgz", @@ -2104,39 +2163,31 @@ "npm": ">=5" } }, - "node_modules/node-pre-gyp": { - "version": "0.15.0", - "resolved": "https://registry.npmjs.org/node-pre-gyp/-/node-pre-gyp-0.15.0.tgz", - "integrity": "sha512-7QcZa8/fpaU/BKenjcaeFF9hLz2+7S9AqyXFhlH/rilsQ/hPZKK32RtR5EQHJElgu+q5RfbJ34KriI79UWaorA==", - "deprecated": "Please upgrade to @mapbox/node-pre-gyp: the non-scoped node-pre-gyp package is deprecated and only the @mapbox scoped package will recieve updates in the future", + "node_modules/node-fetch": { + "version": "2.6.6", + "resolved": "https://registry.npmjs.org/node-fetch/-/node-fetch-2.6.6.tgz", + "integrity": "sha512-Z8/6vRlTUChSdIgMa51jxQ4lrw/Jy5SOW10ObaA47/RElsAN2c5Pn8bTgFGWn/ibwzXTE8qwr1Yzx28vsecXEA==", "dev": true, "dependencies": { - "detect-libc": "^1.0.2", - "mkdirp": "^0.5.3", - "needle": "^2.5.0", - "nopt": "^4.0.1", - "npm-packlist": "^1.1.6", - "npmlog": "^4.0.2", - "rc": "^1.2.7", - "rimraf": "^2.6.1", - "semver": "^5.3.0", - "tar": "^4.4.2" + "whatwg-url": "^5.0.0" }, - "bin": { - "node-pre-gyp": "bin/node-pre-gyp" + "engines": { + "node": "4.x || >=6.0.0" } }, "node_modules/nopt": { - "version": "4.0.3", - "resolved": "https://registry.npmjs.org/nopt/-/nopt-4.0.3.tgz", - "integrity": "sha512-CvaGwVMztSMJLOeXPrez7fyfObdZqNUK1cPAEzLHrTybIua9pMdmmPR5YwtfNftIOMv3DPUhFaxsZMNTQO20Kg==", + "version": "5.0.0", + "resolved": "https://registry.npmjs.org/nopt/-/nopt-5.0.0.tgz", + "integrity": "sha512-Tbj67rffqceeLpcRXrT7vKAN8CwfPeIBgM7E6iBkmKLV7bEMwpGgYLGv0jACUsECaa/vuxP0IjEont6umdMgtQ==", "dev": true, "dependencies": { - "abbrev": "1", - "osenv": "^0.1.4" + "abbrev": "1" }, "bin": { "nopt": "bin/nopt.js" + }, + "engines": { + "node": ">=6" } }, "node_modules/normalize-package-data": { @@ -2160,32 +2211,6 @@ "node": ">=0.10.0" } }, - "node_modules/npm-bundled": { - "version": "1.1.1", - "resolved": "https://registry.npmjs.org/npm-bundled/-/npm-bundled-1.1.1.tgz", - "integrity": "sha512-gqkfgGePhTpAEgUsGEgcq1rqPXA+tv/aVBlgEzfXwA1yiUJF7xtEt3CtVwOjNYQOVknDk0F20w58Fnm3EtG0fA==", - "dev": true, - "dependencies": { - "npm-normalize-package-bin": "^1.0.1" - } - }, - "node_modules/npm-normalize-package-bin": { - "version": "1.0.1", - "resolved": "https://registry.npmjs.org/npm-normalize-package-bin/-/npm-normalize-package-bin-1.0.1.tgz", - "integrity": "sha512-EPfafl6JL5/rU+ot6P3gRSCpPDW5VmIzX959Ob1+ySFUuuYHWHekXpwdUZcKP5C+DS4GEtdJluwBjnsNDl+fSA==", - "dev": true - }, - "node_modules/npm-packlist": { - "version": "1.4.8", - "resolved": "https://registry.npmjs.org/npm-packlist/-/npm-packlist-1.4.8.tgz", - "integrity": "sha512-5+AZgwru5IevF5ZdnFglB5wNlHG1AOOuw28WhUq8/8emhBmLv6jX5by4WJCh7lW0uSYZYS6DXqIsyZVIXRZU9A==", - "dev": true, - "dependencies": { - "ignore-walk": "^3.0.1", - "npm-bundled": "^1.0.1", - "npm-normalize-package-bin": "^1.0.1" - } - }, "node_modules/npm-run-all": { "version": "4.1.5", "resolved": "https://registry.npmjs.org/npm-run-all/-/npm-run-all-4.1.5.tgz", @@ -2212,15 +2237,15 @@ } }, "node_modules/npmlog": { - "version": "4.1.2", - "resolved": "https://registry.npmjs.org/npmlog/-/npmlog-4.1.2.tgz", - "integrity": "sha512-2uUqazuKlTaSI/dC8AzicUck7+IrEaOnN/e0jd3Xtt1KcGpwx30v50mL7oPyr/h9bL3E4aZccVwpwP+5W9Vjkg==", + "version": "5.0.1", + "resolved": "https://registry.npmjs.org/npmlog/-/npmlog-5.0.1.tgz", + "integrity": "sha512-AqZtDUWOMKs1G/8lwylVjrdYgqA4d9nu8hc+0gzRxlDb1I10+FHBGMXs6aiQHFdCUUlqH99MUMuLfzWDNDtfxw==", "dev": true, "dependencies": { - "are-we-there-yet": "~1.1.2", - "console-control-strings": "~1.1.0", - "gauge": "~2.7.3", - "set-blocking": "~2.0.0" + "are-we-there-yet": "^2.0.0", + "console-control-strings": "^1.1.0", + "gauge": "^3.0.0", + "set-blocking": "^2.0.0" } }, "node_modules/nth-check": { @@ -2232,15 +2257,6 @@ "boolbase": "~1.0.0" } }, - "node_modules/number-is-nan": { - "version": "1.0.1", - "resolved": "https://registry.npmjs.org/number-is-nan/-/number-is-nan-1.0.1.tgz", - "integrity": "sha1-CXtgK1NCKlIsGvuHkDGDNpQaAR0=", - "dev": true, - "engines": { - "node": ">=0.10.0" - } - }, "node_modules/nunjucks": { "version": "3.2.2", "resolved": "https://registry.npmjs.org/nunjucks/-/nunjucks-3.2.2.tgz", @@ -2390,34 +2406,6 @@ "opener": "bin/opener-bin.js" } }, - "node_modules/os-homedir": { - "version": "1.0.2", - "resolved": "https://registry.npmjs.org/os-homedir/-/os-homedir-1.0.2.tgz", - "integrity": "sha1-/7xJiDNuDoM94MFox+8VISGqf7M=", - "dev": true, - "engines": { - "node": ">=0.10.0" - } - }, - "node_modules/os-tmpdir": { - "version": "1.0.2", - "resolved": "https://registry.npmjs.org/os-tmpdir/-/os-tmpdir-1.0.2.tgz", - "integrity": "sha1-u+Z0BseaqFxc/sdm/lc0VV36EnQ=", - "dev": true, - "engines": { - "node": ">=0.10.0" - } - }, - "node_modules/osenv": { - "version": "0.1.5", - "resolved": "https://registry.npmjs.org/osenv/-/osenv-0.1.5.tgz", - "integrity": "sha512-0CWcCECdMVc2Rw3U5w9ZjqX6ga6ubk1xDVKxtBQPK7wis/0F2r9T6k4ydGYhecl7YUBxBVxhL5oisPsNxAPe2g==", - "dev": true, - "dependencies": { - "os-homedir": "^1.0.0", - "os-tmpdir": "^1.0.0" - } - }, "node_modules/p-defer": { "version": "1.0.0", "resolved": "https://registry.npmjs.org/p-defer/-/p-defer-1.0.0.tgz", @@ -2826,15 +2814,18 @@ } }, "node_modules/rimraf": { - "version": "2.7.1", - "resolved": "https://registry.npmjs.org/rimraf/-/rimraf-2.7.1.tgz", - "integrity": "sha512-uWjbaKIK3T1OSVptzX7Nl6PvQ3qAGtKEtVRjRuazjfL3Bx5eI409VZSqgND+4UNnmzLVdPj9FqFJNPqBZFve4w==", + "version": "3.0.2", + "resolved": "https://registry.npmjs.org/rimraf/-/rimraf-3.0.2.tgz", + "integrity": "sha512-JZkJMZkAGFFPP2YqXZXPbMlMBgsxzE8ILs4lMIX/2o0L9UBw9O/Y3o6wFw/i9YLapcUJWwqbi3kdxIPdC62TIA==", "dev": true, "dependencies": { "glob": "^7.1.3" }, "bin": { "rimraf": "bin.js" + }, + "funding": { + "url": "https://github.com/sponsors/isaacs" } }, "node_modules/safe-buffer": { @@ -2904,9 +2895,9 @@ "dev": true }, "node_modules/signal-exit": { - "version": "3.0.3", - "resolved": "https://registry.npmjs.org/signal-exit/-/signal-exit-3.0.3.tgz", - "integrity": "sha512-VUJ49FC8U1OxwZLxIbTTrDvLnf/6TDgxZcK8wxR8zs13xpx7xbG60ndBlhNrFi2EMuFRoeDoJO7wthSLq42EjA==", + "version": "3.0.6", + "resolved": "https://registry.npmjs.org/signal-exit/-/signal-exit-3.0.6.tgz", + "integrity": "sha512-sDl4qMFpijcGw22U5w63KmD3cZJfBuFlVNbVMKje2keoKML7X2UzWbc4XrmEbDwg0NXJc3yv4/ox7b+JWb57kQ==", "dev": true }, "node_modules/simple-concat": { @@ -3003,19 +2994,25 @@ } }, "node_modules/string-width": { - "version": "1.0.2", - "resolved": "https://registry.npmjs.org/string-width/-/string-width-1.0.2.tgz", - "integrity": "sha1-EYvfW4zcUaKn5w0hHgfisLmxB9M=", + "version": "4.2.3", + "resolved": "https://registry.npmjs.org/string-width/-/string-width-4.2.3.tgz", + "integrity": "sha512-wKyQRQpjJ0sIp62ErSZdGsjMJWsap5oRNihHhu6G7JVO/9jIB6UyevL+tXuOqrng8j/cxKTWyWUwvSTriiZz/g==", "dev": true, "dependencies": { - "code-point-at": "^1.0.0", - "is-fullwidth-code-point": "^1.0.0", - "strip-ansi": "^3.0.0" + "emoji-regex": "^8.0.0", + "is-fullwidth-code-point": "^3.0.0", + "strip-ansi": "^6.0.1" }, "engines": { - "node": ">=0.10.0" + "node": ">=8" } }, + "node_modules/string-width/node_modules/emoji-regex": { + "version": "8.0.0", + "resolved": "https://registry.npmjs.org/emoji-regex/-/emoji-regex-8.0.0.tgz", + "integrity": "sha512-MSjYzcWNOA0ewAHpz0MxpYFvwg6yjy1NG3xteoqz644VCo/RPgnr1/GGt+ic3iJTzQ8Eu3TdM14SawnVUmGE6A==", + "dev": true + }, "node_modules/string.prototype.padend": { "version": "3.1.0", "resolved": "https://registry.npmjs.org/string.prototype.padend/-/string.prototype.padend-3.1.0.tgz", @@ -3059,15 +3056,15 @@ } }, "node_modules/strip-ansi": { - "version": "3.0.1", - "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-3.0.1.tgz", - "integrity": "sha1-ajhfuIU9lS1f8F0Oiq+UJ43GPc8=", + "version": "6.0.1", + "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-6.0.1.tgz", + "integrity": "sha512-Y38VPSHcqkFrCpFnQ9vuSXmquuv5oXOKpGeT6aGrr3o3Gc9AlVa6JBfUSOCnbxGGZF+/0ooI7KrPuUSztUdU5A==", "dev": true, "dependencies": { - "ansi-regex": "^2.0.0" + "ansi-regex": "^5.0.1" }, "engines": { - "node": ">=0.10.0" + "node": ">=8" } }, "node_modules/strip-bom": { @@ -3176,21 +3173,32 @@ } }, "node_modules/tar": { - "version": "4.4.13", - "resolved": "https://registry.npmjs.org/tar/-/tar-4.4.13.tgz", - "integrity": "sha512-w2VwSrBoHa5BsSyH+KxEqeQBAllHhccyMFVHtGtdMpF4W7IRWfZjFiQceJPChOeTsSDVUpER2T8FA93pr0L+QA==", + "version": "6.1.11", + "resolved": "https://registry.npmjs.org/tar/-/tar-6.1.11.tgz", + "integrity": "sha512-an/KZQzQUkZCkuoAA64hM92X0Urb6VpRhAFllDzz44U2mcD5scmT3zBc4VgVpkugF580+DQn8eAFSyoQt0tznA==", "dev": true, "dependencies": { - "chownr": "^1.1.1", - "fs-minipass": "^1.2.5", - "minipass": "^2.8.6", - "minizlib": "^1.2.1", - "mkdirp": "^0.5.0", - "safe-buffer": "^5.1.2", - "yallist": "^3.0.3" + "chownr": "^2.0.0", + "fs-minipass": "^2.0.0", + "minipass": "^3.0.0", + "minizlib": "^2.1.1", + "mkdirp": "^1.0.3", + "yallist": "^4.0.0" }, "engines": { - "node": ">=4.5" + "node": ">= 10" + } + }, + "node_modules/tar/node_modules/mkdirp": { + "version": "1.0.4", + "resolved": "https://registry.npmjs.org/mkdirp/-/mkdirp-1.0.4.tgz", + "integrity": "sha512-vVqVZQyf3WLx2Shd0qJ9xuvqgAyKPLAiqITEtqW0oIUjzo3PePDd6fW9iFz30ef7Ysp/oiWqbhszeGWW2T6Gzw==", + "dev": true, + "bin": { + "mkdirp": "bin/cmd.js" + }, + "engines": { + "node": ">=10" } }, "node_modules/temp-dir": { @@ -3252,6 +3260,12 @@ "npm": ">=5" } }, + "node_modules/tr46": { + "version": "0.0.3", + "resolved": "https://registry.npmjs.org/tr46/-/tr46-0.0.3.tgz", + "integrity": "sha1-gYT9NH2snNwYWZLzpmIuFLnZq2o=", + "dev": true + }, "node_modules/trim-newlines": { "version": "3.0.0", "resolved": "https://registry.npmjs.org/trim-newlines/-/trim-newlines-3.0.0.tgz", @@ -3379,6 +3393,22 @@ "resolved": "https://registry.npmjs.org/weasel-words/-/weasel-words-0.1.1.tgz", "integrity": "sha1-cTeUZYXHP+RIggE4U70ADF1oek4=" }, + "node_modules/webidl-conversions": { + "version": "3.0.1", + "resolved": "https://registry.npmjs.org/webidl-conversions/-/webidl-conversions-3.0.1.tgz", + "integrity": "sha1-JFNCdeKnvGvnvIZhHMFq4KVlSHE=", + "dev": true + }, + "node_modules/whatwg-url": { + "version": "5.0.0", + "resolved": "https://registry.npmjs.org/whatwg-url/-/whatwg-url-5.0.0.tgz", + "integrity": "sha1-lmRU6HZUYuN2RNNib2dCzotwll0=", + "dev": true, + "dependencies": { + "tr46": "~0.0.3", + "webidl-conversions": "^3.0.0" + } + }, "node_modules/which": { "version": "1.3.1", "resolved": "https://registry.npmjs.org/which/-/which-1.3.1.tgz", @@ -3398,12 +3428,12 @@ "dev": true }, "node_modules/wide-align": { - "version": "1.1.3", - "resolved": "https://registry.npmjs.org/wide-align/-/wide-align-1.1.3.tgz", - "integrity": "sha512-QGkOQc8XL6Bt5PwnsExKBPuMKBxnGxWWW3fU55Xt4feHozMUhdUMaBCk290qpm/wG5u/RSKzwdAC4i51YigihA==", + "version": "1.1.5", + "resolved": "https://registry.npmjs.org/wide-align/-/wide-align-1.1.5.tgz", + "integrity": "sha512-eDMORYaPNZ4sQIuuYPDHdQvf4gyCF9rEEV/yPxGfwPkRodwEgiMUUXTx/dex+Me0wxx53S+NgUHaP7y3MGlDmg==", "dev": true, "dependencies": { - "string-width": "^1.0.2 || 2" + "string-width": "^1.0.2 || 2 || 3 || 4" } }, "node_modules/wrap-ansi": { @@ -3504,9 +3534,9 @@ "dev": true }, "node_modules/yallist": { - "version": "3.1.1", - "resolved": "https://registry.npmjs.org/yallist/-/yallist-3.1.1.tgz", - "integrity": "sha512-a4UGQaWPH59mOXUYnAG2ewncQS4i4F43Tv3JoAM+s2VDAmS9NsK8GpDMLrCHPksFT7h3K6TOoUNn2pb7RoXx4g==", + "version": "4.0.0", + "resolved": "https://registry.npmjs.org/yallist/-/yallist-4.0.0.tgz", + "integrity": "sha512-3wdGidZyq5PB084XLES5TpOSRA3wjXAlIWMhum2kRcv/41Sn2emQ0dycQW4uZXLejwKvg6EsvbdlVL+FYEct7A==", "dev": true }, "node_modules/yargs": { @@ -3668,6 +3698,34 @@ "js-tokens": "^4.0.0" } }, + "@mapbox/node-pre-gyp": { + "version": "1.0.8", + "resolved": "https://registry.npmjs.org/@mapbox/node-pre-gyp/-/node-pre-gyp-1.0.8.tgz", + "integrity": "sha512-CMGKi28CF+qlbXh26hDe6NxCd7amqeAzEqnS6IHeO6LoaKyM/n+Xw3HT1COdq8cuioOdlKdqn/hCmqPUOMOywg==", + "dev": true, + "requires": { + "detect-libc": "^1.0.3", + "https-proxy-agent": "^5.0.0", + "make-dir": "^3.1.0", + "node-fetch": "^2.6.5", + "nopt": "^5.0.0", + "npmlog": "^5.0.1", + "rimraf": "^3.0.2", + "semver": "^7.3.5", + "tar": "^6.1.11" + }, + "dependencies": { + "semver": { + "version": "7.3.5", + "resolved": "https://registry.npmjs.org/semver/-/semver-7.3.5.tgz", + "integrity": "sha512-PoeGJYh8HK4BTO/a9Tf6ZG3veo/A7ZVsYrSA6J8ny9nb3B1VrpkuN+z9OE5wfE5p6H4LchYZsegiQgbJD94ZFQ==", + "dev": true, + "requires": { + "lru-cache": "^6.0.0" + } + } + } + }, "@tokenizer/token": { "version": "0.1.1", "resolved": "https://registry.npmjs.org/@tokenizer/token/-/token-0.1.1.tgz", @@ -3721,10 +3779,30 @@ "resolved": "https://registry.npmjs.org/adverb-where/-/adverb-where-0.2.1.tgz", "integrity": "sha512-IAveFBziMRMNPKFdWRcdIKaJvJG1cAfU9/tf9MzqQ84Dh4QjD9eqwnt4hNSt9cbrcEJD74BMIOaRVgVDEU7MwQ==" }, + "agent-base": { + "version": "6.0.2", + "resolved": "https://registry.npmjs.org/agent-base/-/agent-base-6.0.2.tgz", + "integrity": "sha512-RZNwNclF7+MS/8bDg70amg32dyeZGZxiDuQmZxKLAlQjr3jGyLx+4Kkk58UO7D2QdgFIQCovuSuZESne6RG6XQ==", + "dev": true, + "requires": { + "debug": "4" + }, + "dependencies": { + "debug": { + "version": "4.3.3", + "resolved": "https://registry.npmjs.org/debug/-/debug-4.3.3.tgz", + "integrity": "sha512-/zxw5+vh1Tfv+4Qn7a5nsbcJKPaSvCDhojn6FEl9vupwK2VCSDtEiEtqr8DFtzYFOdz63LBkxec7DYuc2jon6Q==", + "dev": true, + "requires": { + "ms": "2.1.2" + } + } + } + }, "ansi-regex": { - "version": "2.1.1", - "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-2.1.1.tgz", - "integrity": "sha1-w7M6te42DYbg5ijwRorn7yfWVN8=", + "version": "5.0.1", + "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-5.0.1.tgz", + "integrity": "sha512-quJQXlTSUGL2LH9SUXo8VwsY4soanhgo6LNSm84E1LBcE8s3O0wpdiRzyR9z/ZZJMlMWv37qOOb9pdJlMUEKFQ==", "dev": true }, "ansi-styles": { @@ -3747,19 +3825,32 @@ } }, "aproba": { - "version": "1.2.0", - "resolved": "https://registry.npmjs.org/aproba/-/aproba-1.2.0.tgz", - "integrity": "sha512-Y9J6ZjXtoYh8RnXVCMOU/ttDmk1aBjunq9vO0ta5x85WDQiQfUF9sIPBITdbiiIVcBo03Hi3jMxigBtsddlXRw==", + "version": "2.0.0", + "resolved": "https://registry.npmjs.org/aproba/-/aproba-2.0.0.tgz", + "integrity": "sha512-lYe4Gx7QT+MKGbDsA+Z+he/Wtef0BiwDOlK/XkBrdfsh9J/jPPXbX0tE9x9cl27Tmu5gg3QUbUrQYa/y+KOHPQ==", "dev": true }, "are-we-there-yet": { - "version": "1.1.5", - "resolved": "https://registry.npmjs.org/are-we-there-yet/-/are-we-there-yet-1.1.5.tgz", - "integrity": "sha512-5hYdAkZlcG8tOLujVDTgCT+uPX0VnpAH28gWsLfzpXYm7wP6mp5Q/gYyR7YQ0cKVJcXJnl3j2kpBan13PtQf6w==", + "version": "2.0.0", + "resolved": "https://registry.npmjs.org/are-we-there-yet/-/are-we-there-yet-2.0.0.tgz", + "integrity": "sha512-Ci/qENmwHnsYo9xKIcUJN5LeDKdJ6R1Z1j9V/J5wyq8nh/mYPEpIKJbBZXtZjG04HiK7zV/p6Vs9952MrMeUIw==", "dev": true, "requires": { "delegates": "^1.0.0", - "readable-stream": "^2.0.6" + "readable-stream": "^3.6.0" + }, + "dependencies": { + "readable-stream": { + "version": "3.6.0", + "resolved": "https://registry.npmjs.org/readable-stream/-/readable-stream-3.6.0.tgz", + "integrity": "sha512-BViHy7LKeTz4oNnkcLJ+lVSL6vpiFeX6/d3oSH8zCW7UxP2onchk+vTGB143xuFjHS3deTgkKoXXymXqymiIdA==", + "dev": true, + "requires": { + "inherits": "^2.0.3", + "string_decoder": "^1.1.1", + "util-deprecate": "^1.0.1" + } + } } }, "argparse": { @@ -3865,13 +3956,13 @@ } }, "canvas": { - "version": "git+ssh://git@github.com/Pomax/node-canvas.git#20defd4f41e23b9bbcfdf219f304fd49ebcc18f8", - "integrity": "sha512-wIIEeJpcdESW3UvKYjOruWyqLIrwuO22PKxBr9H3kubXgIch/mBWhOrYIPSD3hNqT0pbP7NK8HehLp0WBywxJw==", + "version": "2.8.0", + "resolved": "https://registry.npmjs.org/canvas/-/canvas-2.8.0.tgz", + "integrity": "sha512-gLTi17X8WY9Cf5GZ2Yns8T5lfBOcGgFehDFb+JQwDqdOoBOcECS9ZWMEAqMSVcMYwXD659J8NyzjRY/2aE+C2Q==", "dev": true, - "from": "canvas@git://github.com/Pomax/node-canvas#master", "requires": { + "@mapbox/node-pre-gyp": "^1.0.0", "nan": "^2.14.0", - "node-pre-gyp": "^0.15.0", "simple-get": "^3.0.3" } }, @@ -3975,9 +4066,9 @@ } }, "chownr": { - "version": "1.1.4", - "resolved": "https://registry.npmjs.org/chownr/-/chownr-1.1.4.tgz", - "integrity": "sha512-jJ0bqzaylmJtVnNgzTeSOs8DPavpbYgEr/b0YL8/2GO3xJEhInFmhKMUnEJQjZumK7KXGFhUy89PrsJWlakBVg==", + "version": "2.0.0", + "resolved": "https://registry.npmjs.org/chownr/-/chownr-2.0.0.tgz", + "integrity": "sha512-bIomtDF5KGpdogkLd9VspvFzk9KfpyyGlS8YFVZl7TGPBHL5snIOnxeshwVgPteQ9b4Eydl+pVbIyE1DcvCWgQ==", "dev": true }, "cliui": { @@ -4036,12 +4127,6 @@ "q": "^1.1.2" } }, - "code-point-at": { - "version": "1.1.0", - "resolved": "https://registry.npmjs.org/code-point-at/-/code-point-at-1.1.0.tgz", - "integrity": "sha1-DQcLTQQ6W+ozovGkDi7bPZpMz3c=", - "dev": true - }, "color-convert": { "version": "1.9.3", "resolved": "https://registry.npmjs.org/color-convert/-/color-convert-1.9.3.tgz", @@ -4057,6 +4142,12 @@ "integrity": "sha1-p9BVi9icQveV3UIyj3QIMcpTvCU=", "dev": true }, + "color-support": { + "version": "1.1.3", + "resolved": "https://registry.npmjs.org/color-support/-/color-support-1.1.3.tgz", + "integrity": "sha512-qiBjkpbMLO/HL68y+lh4q0/O1MZFj2RX6X/KmMa3+gJD3z+WwI1ZzDHysvqHGS3mP6mznPckpXmw1nI9cJjyRg==", + "dev": true + }, "colors": { "version": "1.4.0", "resolved": "https://registry.npmjs.org/colors/-/colors-1.4.0.tgz", @@ -4471,12 +4562,12 @@ } }, "fs-minipass": { - "version": "1.2.7", - "resolved": "https://registry.npmjs.org/fs-minipass/-/fs-minipass-1.2.7.tgz", - "integrity": "sha512-GWSSJGFy4e9GUeCcbIkED+bgAoFyj7XF1mV8rma3QW4NIqX9Kyx79N/PF61H5udOV3aY1IaMLs6pGbH71nlCTA==", + "version": "2.1.0", + "resolved": "https://registry.npmjs.org/fs-minipass/-/fs-minipass-2.1.0.tgz", + "integrity": "sha512-V/JgOLFCS+R6Vcq0slCuaeWEdNC3ouDlJMNIsacH2VtALiu9mV4LPrHc5cDl8k5aw6J8jwgWWpiTo5RYhmIzvg==", "dev": true, "requires": { - "minipass": "^2.6.0" + "minipass": "^3.0.0" } }, "fs.realpath": { @@ -4499,19 +4590,20 @@ "dev": true }, "gauge": { - "version": "2.7.4", - "resolved": "https://registry.npmjs.org/gauge/-/gauge-2.7.4.tgz", - "integrity": "sha1-LANAXHU4w51+s3sxcCLjJfsBi/c=", + "version": "3.0.2", + "resolved": "https://registry.npmjs.org/gauge/-/gauge-3.0.2.tgz", + "integrity": "sha512-+5J6MS/5XksCuXq++uFRsnUd7Ovu1XenbeuIuNRJxYWjgQbPuFhT14lAvsWfqfAmnwluf1OwMjz39HjfLPci0Q==", "dev": true, "requires": { - "aproba": "^1.0.3", + "aproba": "^1.0.3 || ^2.0.0", + "color-support": "^1.1.2", "console-control-strings": "^1.0.0", - "has-unicode": "^2.0.0", - "object-assign": "^4.1.0", + "has-unicode": "^2.0.1", + "object-assign": "^4.1.1", "signal-exit": "^3.0.0", - "string-width": "^1.0.1", - "strip-ansi": "^3.0.1", - "wide-align": "^1.1.0" + "string-width": "^4.2.3", + "strip-ansi": "^6.0.1", + "wide-align": "^1.1.2" } }, "get-caller-file": { @@ -4662,6 +4754,27 @@ "union": "~0.5.0" } }, + "https-proxy-agent": { + "version": "5.0.0", + "resolved": "https://registry.npmjs.org/https-proxy-agent/-/https-proxy-agent-5.0.0.tgz", + "integrity": "sha512-EkYm5BcKUGiduxzSt3Eppko+PiNWNEpa4ySk9vTC6wDsQJW9rHSa+UhGNJoRYp7bz6Ht1eaRIa6QaJqO5rCFbA==", + "dev": true, + "requires": { + "agent-base": "6", + "debug": "4" + }, + "dependencies": { + "debug": { + "version": "4.3.3", + "resolved": "https://registry.npmjs.org/debug/-/debug-4.3.3.tgz", + "integrity": "sha512-/zxw5+vh1Tfv+4Qn7a5nsbcJKPaSvCDhojn6FEl9vupwK2VCSDtEiEtqr8DFtzYFOdz63LBkxec7DYuc2jon6Q==", + "dev": true, + "requires": { + "ms": "2.1.2" + } + } + } + }, "iconv-lite": { "version": "0.4.24", "resolved": "https://registry.npmjs.org/iconv-lite/-/iconv-lite-0.4.24.tgz", @@ -4677,15 +4790,6 @@ "integrity": "sha512-4vf7I2LYV/HaWerSo3XmlMkp5eZ83i+/CDluXi/IGTs/O1sejBNhTtnxzmRZfvOUqj7lZjqHkeTvpgSFDlWZTg==", "dev": true }, - "ignore-walk": { - "version": "3.0.3", - "resolved": "https://registry.npmjs.org/ignore-walk/-/ignore-walk-3.0.3.tgz", - "integrity": "sha512-m7o6xuOaT1aqheYHKf8W6J5pYH85ZI9w077erOzLje3JsB1gkafkAhHHY19dqjulgIZHFm32Cp5uNZgcQqdJKw==", - "dev": true, - "requires": { - "minimatch": "^3.0.4" - } - }, "indent-string": { "version": "4.0.0", "resolved": "https://registry.npmjs.org/indent-string/-/indent-string-4.0.0.tgz", @@ -4754,13 +4858,10 @@ "dev": true }, "is-fullwidth-code-point": { - "version": "1.0.0", - "resolved": "https://registry.npmjs.org/is-fullwidth-code-point/-/is-fullwidth-code-point-1.0.0.tgz", - "integrity": "sha1-754xOG8DGn8NZDr4L95QxFfvAMs=", - "dev": true, - "requires": { - "number-is-nan": "^1.0.0" - } + "version": "3.0.0", + "resolved": "https://registry.npmjs.org/is-fullwidth-code-point/-/is-fullwidth-code-point-3.0.0.tgz", + "integrity": "sha512-zymm5+u+sCsSWyD9qNaejV3DFvhCKclKdizYaJUuHA83RLjb7nSuGnddCHGv0hk+KY7BMAlsWeK4Ueg6EV6XQg==", + "dev": true }, "is-glob": { "version": "4.0.1", @@ -4897,12 +4998,6 @@ "yargs": "^16.0.3" }, "dependencies": { - "ansi-regex": { - "version": "5.0.0", - "resolved": "https://registry.npmjs.org/ansi-regex/-/ansi-regex-5.0.0.tgz", - "integrity": "sha512-bY6fj56OUQ0hU1KjFNDQuJFezqKdrAyFdIevADiqrWHwSlbmBNMHp5ak2f40Pm8JTFyM2mqxkG6ngkHO11f/lg==", - "dev": true - }, "ansi-styles": { "version": "4.3.0", "resolved": "https://registry.npmjs.org/ansi-styles/-/ansi-styles-4.3.0.tgz", @@ -4944,44 +5039,12 @@ "integrity": "sha512-GpVkmM8vF2vQUkj2LvZmD35JxeJOLCwJ9cUkugyk2nuhbv3+mJvpLYYt+0+USMxE+oj+ey/lJEnhZw75x/OMcQ==", "dev": true }, - "emoji-regex": { - "version": "8.0.0", - "resolved": "https://registry.npmjs.org/emoji-regex/-/emoji-regex-8.0.0.tgz", - "integrity": "sha512-MSjYzcWNOA0ewAHpz0MxpYFvwg6yjy1NG3xteoqz644VCo/RPgnr1/GGt+ic3iJTzQ8Eu3TdM14SawnVUmGE6A==", - "dev": true - }, - "is-fullwidth-code-point": { - "version": "3.0.0", - "resolved": "https://registry.npmjs.org/is-fullwidth-code-point/-/is-fullwidth-code-point-3.0.0.tgz", - "integrity": "sha512-zymm5+u+sCsSWyD9qNaejV3DFvhCKclKdizYaJUuHA83RLjb7nSuGnddCHGv0hk+KY7BMAlsWeK4Ueg6EV6XQg==", - "dev": true - }, "mkdirp": { "version": "1.0.4", "resolved": "https://registry.npmjs.org/mkdirp/-/mkdirp-1.0.4.tgz", "integrity": "sha512-vVqVZQyf3WLx2Shd0qJ9xuvqgAyKPLAiqITEtqW0oIUjzo3PePDd6fW9iFz30ef7Ysp/oiWqbhszeGWW2T6Gzw==", "dev": true }, - "string-width": { - "version": "4.2.2", - "resolved": "https://registry.npmjs.org/string-width/-/string-width-4.2.2.tgz", - "integrity": "sha512-XBJbT3N4JhVumXE0eoLU9DCjcaF92KLNqTmFCnG1pf8duUxFGwtP6AD6nkjw9a3IdiRtL3E2w3JDiE/xi3vOeA==", - "dev": true, - "requires": { - "emoji-regex": "^8.0.0", - "is-fullwidth-code-point": "^3.0.0", - "strip-ansi": "^6.0.0" - } - }, - "strip-ansi": { - "version": "6.0.0", - "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-6.0.0.tgz", - "integrity": "sha512-AuvKTrTfQNYNIctbR1K/YGTR1756GycPsg7b9bdV9Duqur4gv6aKqHXah67Z8ImS7WEz5QVcOtlfW2rZEugt6w==", - "dev": true, - "requires": { - "ansi-regex": "^5.0.0" - } - }, "wrap-ansi": { "version": "7.0.0", "resolved": "https://registry.npmjs.org/wrap-ansi/-/wrap-ansi-7.0.0.tgz", @@ -5069,6 +5132,15 @@ "integrity": "sha1-wj6RtxAkKscMN/HhzaknTMOb8vQ=", "dev": true }, + "lru-cache": { + "version": "6.0.0", + "resolved": "https://registry.npmjs.org/lru-cache/-/lru-cache-6.0.0.tgz", + "integrity": "sha512-Jo6dJ04CmSjuznwJSS3pUeWmd/H0ffTlkXXgwZi+eq1UCmqQwCh+eLsYOYCwY991i2Fah4h1BEMCx4qThGbsiA==", + "dev": true, + "requires": { + "yallist": "^4.0.0" + } + }, "make-dir": { "version": "3.1.0", "resolved": "https://registry.npmjs.org/make-dir/-/make-dir-3.1.0.tgz", @@ -5221,22 +5293,22 @@ } }, "minipass": { - "version": "2.9.0", - "resolved": "https://registry.npmjs.org/minipass/-/minipass-2.9.0.tgz", - "integrity": "sha512-wxfUjg9WebH+CUDX/CdbRlh5SmfZiy/hpkxaRI16Y9W56Pa75sWgd/rvFilSgrauD9NyFymP/+JFV3KwzIsJeg==", + "version": "3.1.6", + "resolved": "https://registry.npmjs.org/minipass/-/minipass-3.1.6.tgz", + "integrity": "sha512-rty5kpw9/z8SX9dmxblFA6edItUmwJgMeYDZRrwlIVN27i8gysGbznJwUggw2V/FVqFSDdWy040ZPS811DYAqQ==", "dev": true, "requires": { - "safe-buffer": "^5.1.2", - "yallist": "^3.0.0" + "yallist": "^4.0.0" } }, "minizlib": { - "version": "1.3.3", - "resolved": "https://registry.npmjs.org/minizlib/-/minizlib-1.3.3.tgz", - "integrity": "sha512-6ZYMOEnmVsdCeTJVE0W9ZD+pVnE8h9Hma/iOwwRDsdQoePpoX56/8B6z3P9VNwppJuBKNRuFDRNRqRWexT9G9Q==", + "version": "2.1.2", + "resolved": "https://registry.npmjs.org/minizlib/-/minizlib-2.1.2.tgz", + "integrity": "sha512-bAxsR8BVfj60DWXHE3u30oHzfl4G7khkSuPW+qvpd7jFRHm7dLxOjUk1EHACJ/hxLY8phGJ0YhYHZo7jil7Qdg==", "dev": true, "requires": { - "minipass": "^2.9.0" + "minipass": "^3.0.0", + "yallist": "^4.0.0" } }, "mkdirp": { @@ -5260,17 +5332,6 @@ "integrity": "sha512-isWHgVjnFjh2x2yuJ/tj3JbwoHu3UC2dX5G/88Cm24yB6YopVgxvBObDY7n5xW6ExmFhJpSEQqFPvq9zaXc8Jw==", "dev": true }, - "needle": { - "version": "2.5.2", - "resolved": "https://registry.npmjs.org/needle/-/needle-2.5.2.tgz", - "integrity": "sha512-LbRIwS9BfkPvNwNHlsA41Q29kL2L/6VaOJ0qisM5lLWsTV3nP15abO5ITL6L81zqFhzjRKDAYjpcBcwM0AVvLQ==", - "dev": true, - "requires": { - "debug": "^3.2.6", - "iconv-lite": "^0.4.4", - "sax": "^1.2.4" - } - }, "nice-try": { "version": "1.0.5", "resolved": "https://registry.npmjs.org/nice-try/-/nice-try-1.0.5.tgz", @@ -5282,32 +5343,22 @@ "resolved": "https://registry.npmjs.org/no-cliches/-/no-cliches-0.2.2.tgz", "integrity": "sha512-iEOqDAOFl6uN5jZGRj39Jdo8qALzf2HPXtpFso8+BMaDylDrUMYMwhFbfYGgxdnMlsRnxYTwv68kaXEpsHIapg==" }, - "node-pre-gyp": { - "version": "0.15.0", - "resolved": "https://registry.npmjs.org/node-pre-gyp/-/node-pre-gyp-0.15.0.tgz", - "integrity": "sha512-7QcZa8/fpaU/BKenjcaeFF9hLz2+7S9AqyXFhlH/rilsQ/hPZKK32RtR5EQHJElgu+q5RfbJ34KriI79UWaorA==", + "node-fetch": { + "version": "2.6.6", + "resolved": "https://registry.npmjs.org/node-fetch/-/node-fetch-2.6.6.tgz", + "integrity": "sha512-Z8/6vRlTUChSdIgMa51jxQ4lrw/Jy5SOW10ObaA47/RElsAN2c5Pn8bTgFGWn/ibwzXTE8qwr1Yzx28vsecXEA==", "dev": true, "requires": { - "detect-libc": "^1.0.2", - "mkdirp": "^0.5.3", - "needle": "^2.5.0", - "nopt": "^4.0.1", - "npm-packlist": "^1.1.6", - "npmlog": "^4.0.2", - "rc": "^1.2.7", - "rimraf": "^2.6.1", - "semver": "^5.3.0", - "tar": "^4.4.2" + "whatwg-url": "^5.0.0" } }, "nopt": { - "version": "4.0.3", - "resolved": "https://registry.npmjs.org/nopt/-/nopt-4.0.3.tgz", - "integrity": "sha512-CvaGwVMztSMJLOeXPrez7fyfObdZqNUK1cPAEzLHrTybIua9pMdmmPR5YwtfNftIOMv3DPUhFaxsZMNTQO20Kg==", + "version": "5.0.0", + "resolved": "https://registry.npmjs.org/nopt/-/nopt-5.0.0.tgz", + "integrity": "sha512-Tbj67rffqceeLpcRXrT7vKAN8CwfPeIBgM7E6iBkmKLV7bEMwpGgYLGv0jACUsECaa/vuxP0IjEont6umdMgtQ==", "dev": true, "requires": { - "abbrev": "1", - "osenv": "^0.1.4" + "abbrev": "1" } }, "normalize-package-data": { @@ -5328,32 +5379,6 @@ "integrity": "sha512-6eZs5Ls3WtCisHWp9S2GUy8dqkpGi4BVSz3GaqiE6ezub0512ESztXUwUB6C6IKbQkY2Pnb/mD4WYojCRwcwLA==", "dev": true }, - "npm-bundled": { - "version": "1.1.1", - "resolved": "https://registry.npmjs.org/npm-bundled/-/npm-bundled-1.1.1.tgz", - "integrity": "sha512-gqkfgGePhTpAEgUsGEgcq1rqPXA+tv/aVBlgEzfXwA1yiUJF7xtEt3CtVwOjNYQOVknDk0F20w58Fnm3EtG0fA==", - "dev": true, - "requires": { - "npm-normalize-package-bin": "^1.0.1" - } - }, - "npm-normalize-package-bin": { - "version": "1.0.1", - "resolved": "https://registry.npmjs.org/npm-normalize-package-bin/-/npm-normalize-package-bin-1.0.1.tgz", - "integrity": "sha512-EPfafl6JL5/rU+ot6P3gRSCpPDW5VmIzX959Ob1+ySFUuuYHWHekXpwdUZcKP5C+DS4GEtdJluwBjnsNDl+fSA==", - "dev": true - }, - "npm-packlist": { - "version": "1.4.8", - "resolved": "https://registry.npmjs.org/npm-packlist/-/npm-packlist-1.4.8.tgz", - "integrity": "sha512-5+AZgwru5IevF5ZdnFglB5wNlHG1AOOuw28WhUq8/8emhBmLv6jX5by4WJCh7lW0uSYZYS6DXqIsyZVIXRZU9A==", - "dev": true, - "requires": { - "ignore-walk": "^3.0.1", - "npm-bundled": "^1.0.1", - "npm-normalize-package-bin": "^1.0.1" - } - }, "npm-run-all": { "version": "4.1.5", "resolved": "https://registry.npmjs.org/npm-run-all/-/npm-run-all-4.1.5.tgz", @@ -5372,15 +5397,15 @@ } }, "npmlog": { - "version": "4.1.2", - "resolved": "https://registry.npmjs.org/npmlog/-/npmlog-4.1.2.tgz", - "integrity": "sha512-2uUqazuKlTaSI/dC8AzicUck7+IrEaOnN/e0jd3Xtt1KcGpwx30v50mL7oPyr/h9bL3E4aZccVwpwP+5W9Vjkg==", + "version": "5.0.1", + "resolved": "https://registry.npmjs.org/npmlog/-/npmlog-5.0.1.tgz", + "integrity": "sha512-AqZtDUWOMKs1G/8lwylVjrdYgqA4d9nu8hc+0gzRxlDb1I10+FHBGMXs6aiQHFdCUUlqH99MUMuLfzWDNDtfxw==", "dev": true, "requires": { - "are-we-there-yet": "~1.1.2", - "console-control-strings": "~1.1.0", - "gauge": "~2.7.3", - "set-blocking": "~2.0.0" + "are-we-there-yet": "^2.0.0", + "console-control-strings": "^1.1.0", + "gauge": "^3.0.0", + "set-blocking": "^2.0.0" } }, "nth-check": { @@ -5392,12 +5417,6 @@ "boolbase": "~1.0.0" } }, - "number-is-nan": { - "version": "1.0.1", - "resolved": "https://registry.npmjs.org/number-is-nan/-/number-is-nan-1.0.1.tgz", - "integrity": "sha1-CXtgK1NCKlIsGvuHkDGDNpQaAR0=", - "dev": true - }, "nunjucks": { "version": "3.2.2", "resolved": "https://registry.npmjs.org/nunjucks/-/nunjucks-3.2.2.tgz", @@ -5500,28 +5519,6 @@ "integrity": "sha512-goYSy5c2UXE4Ra1xixabeVh1guIX/ZV/YokJksb6q2lubWu6UbvPQ20p542/sFIll1nl8JnCyK9oBaOcCWXwvA==", "dev": true }, - "os-homedir": { - "version": "1.0.2", - "resolved": "https://registry.npmjs.org/os-homedir/-/os-homedir-1.0.2.tgz", - "integrity": "sha1-/7xJiDNuDoM94MFox+8VISGqf7M=", - "dev": true - }, - "os-tmpdir": { - "version": "1.0.2", - "resolved": "https://registry.npmjs.org/os-tmpdir/-/os-tmpdir-1.0.2.tgz", - "integrity": "sha1-u+Z0BseaqFxc/sdm/lc0VV36EnQ=", - "dev": true - }, - "osenv": { - "version": "0.1.5", - "resolved": "https://registry.npmjs.org/osenv/-/osenv-0.1.5.tgz", - "integrity": "sha512-0CWcCECdMVc2Rw3U5w9ZjqX6ga6ubk1xDVKxtBQPK7wis/0F2r9T6k4ydGYhecl7YUBxBVxhL5oisPsNxAPe2g==", - "dev": true, - "requires": { - "os-homedir": "^1.0.0", - "os-tmpdir": "^1.0.0" - } - }, "p-defer": { "version": "1.0.0", "resolved": "https://registry.npmjs.org/p-defer/-/p-defer-1.0.0.tgz", @@ -5822,9 +5819,9 @@ } }, "rimraf": { - "version": "2.7.1", - "resolved": "https://registry.npmjs.org/rimraf/-/rimraf-2.7.1.tgz", - "integrity": "sha512-uWjbaKIK3T1OSVptzX7Nl6PvQ3qAGtKEtVRjRuazjfL3Bx5eI409VZSqgND+4UNnmzLVdPj9FqFJNPqBZFve4w==", + "version": "3.0.2", + "resolved": "https://registry.npmjs.org/rimraf/-/rimraf-3.0.2.tgz", + "integrity": "sha512-JZkJMZkAGFFPP2YqXZXPbMlMBgsxzE8ILs4lMIX/2o0L9UBw9O/Y3o6wFw/i9YLapcUJWwqbi3kdxIPdC62TIA==", "dev": true, "requires": { "glob": "^7.1.3" @@ -5888,9 +5885,9 @@ "dev": true }, "signal-exit": { - "version": "3.0.3", - "resolved": "https://registry.npmjs.org/signal-exit/-/signal-exit-3.0.3.tgz", - "integrity": "sha512-VUJ49FC8U1OxwZLxIbTTrDvLnf/6TDgxZcK8wxR8zs13xpx7xbG60ndBlhNrFi2EMuFRoeDoJO7wthSLq42EjA==", + "version": "3.0.6", + "resolved": "https://registry.npmjs.org/signal-exit/-/signal-exit-3.0.6.tgz", + "integrity": "sha512-sDl4qMFpijcGw22U5w63KmD3cZJfBuFlVNbVMKje2keoKML7X2UzWbc4XrmEbDwg0NXJc3yv4/ox7b+JWb57kQ==", "dev": true }, "simple-concat": { @@ -5970,14 +5967,22 @@ } }, "string-width": { - "version": "1.0.2", - "resolved": "https://registry.npmjs.org/string-width/-/string-width-1.0.2.tgz", - "integrity": "sha1-EYvfW4zcUaKn5w0hHgfisLmxB9M=", + "version": "4.2.3", + "resolved": "https://registry.npmjs.org/string-width/-/string-width-4.2.3.tgz", + "integrity": "sha512-wKyQRQpjJ0sIp62ErSZdGsjMJWsap5oRNihHhu6G7JVO/9jIB6UyevL+tXuOqrng8j/cxKTWyWUwvSTriiZz/g==", "dev": true, "requires": { - "code-point-at": "^1.0.0", - "is-fullwidth-code-point": "^1.0.0", - "strip-ansi": "^3.0.0" + "emoji-regex": "^8.0.0", + "is-fullwidth-code-point": "^3.0.0", + "strip-ansi": "^6.0.1" + }, + "dependencies": { + "emoji-regex": { + "version": "8.0.0", + "resolved": "https://registry.npmjs.org/emoji-regex/-/emoji-regex-8.0.0.tgz", + "integrity": "sha512-MSjYzcWNOA0ewAHpz0MxpYFvwg6yjy1NG3xteoqz644VCo/RPgnr1/GGt+ic3iJTzQ8Eu3TdM14SawnVUmGE6A==", + "dev": true + } } }, "string.prototype.padend": { @@ -6011,12 +6016,12 @@ } }, "strip-ansi": { - "version": "3.0.1", - "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-3.0.1.tgz", - "integrity": "sha1-ajhfuIU9lS1f8F0Oiq+UJ43GPc8=", + "version": "6.0.1", + "resolved": "https://registry.npmjs.org/strip-ansi/-/strip-ansi-6.0.1.tgz", + "integrity": "sha512-Y38VPSHcqkFrCpFnQ9vuSXmquuv5oXOKpGeT6aGrr3o3Gc9AlVa6JBfUSOCnbxGGZF+/0ooI7KrPuUSztUdU5A==", "dev": true, "requires": { - "ansi-regex": "^2.0.0" + "ansi-regex": "^5.0.1" } }, "strip-bom": { @@ -6100,18 +6105,25 @@ } }, "tar": { - "version": "4.4.13", - "resolved": "https://registry.npmjs.org/tar/-/tar-4.4.13.tgz", - "integrity": "sha512-w2VwSrBoHa5BsSyH+KxEqeQBAllHhccyMFVHtGtdMpF4W7IRWfZjFiQceJPChOeTsSDVUpER2T8FA93pr0L+QA==", + "version": "6.1.11", + "resolved": "https://registry.npmjs.org/tar/-/tar-6.1.11.tgz", + "integrity": "sha512-an/KZQzQUkZCkuoAA64hM92X0Urb6VpRhAFllDzz44U2mcD5scmT3zBc4VgVpkugF580+DQn8eAFSyoQt0tznA==", "dev": true, "requires": { - "chownr": "^1.1.1", - "fs-minipass": "^1.2.5", - "minipass": "^2.8.6", - "minizlib": "^1.2.1", - "mkdirp": "^0.5.0", - "safe-buffer": "^5.1.2", - "yallist": "^3.0.3" + "chownr": "^2.0.0", + "fs-minipass": "^2.0.0", + "minipass": "^3.0.0", + "minizlib": "^2.1.1", + "mkdirp": "^1.0.3", + "yallist": "^4.0.0" + }, + "dependencies": { + "mkdirp": { + "version": "1.0.4", + "resolved": "https://registry.npmjs.org/mkdirp/-/mkdirp-1.0.4.tgz", + "integrity": "sha512-vVqVZQyf3WLx2Shd0qJ9xuvqgAyKPLAiqITEtqW0oIUjzo3PePDd6fW9iFz30ef7Ysp/oiWqbhszeGWW2T6Gzw==", + "dev": true + } } }, "temp-dir": { @@ -6157,6 +6169,12 @@ "resolved": "https://registry.npmjs.org/too-wordy/-/too-wordy-0.2.2.tgz", "integrity": "sha512-ePZfjs1ajL4b8jT4MeVId+9Ci5hJCzAtNIEXIHyFYmKmQuX+eHC/RNv6tuLMUhrGrhJ+sYWW/lBF/LKILHGZEA==" }, + "tr46": { + "version": "0.0.3", + "resolved": "https://registry.npmjs.org/tr46/-/tr46-0.0.3.tgz", + "integrity": "sha1-gYT9NH2snNwYWZLzpmIuFLnZq2o=", + "dev": true + }, "trim-newlines": { "version": "3.0.0", "resolved": "https://registry.npmjs.org/trim-newlines/-/trim-newlines-3.0.0.tgz", @@ -6262,6 +6280,22 @@ "resolved": "https://registry.npmjs.org/weasel-words/-/weasel-words-0.1.1.tgz", "integrity": "sha1-cTeUZYXHP+RIggE4U70ADF1oek4=" }, + "webidl-conversions": { + "version": "3.0.1", + "resolved": "https://registry.npmjs.org/webidl-conversions/-/webidl-conversions-3.0.1.tgz", + "integrity": "sha1-JFNCdeKnvGvnvIZhHMFq4KVlSHE=", + "dev": true + }, + "whatwg-url": { + "version": "5.0.0", + "resolved": "https://registry.npmjs.org/whatwg-url/-/whatwg-url-5.0.0.tgz", + "integrity": "sha1-lmRU6HZUYuN2RNNib2dCzotwll0=", + "dev": true, + "requires": { + "tr46": "~0.0.3", + "webidl-conversions": "^3.0.0" + } + }, "which": { "version": "1.3.1", "resolved": "https://registry.npmjs.org/which/-/which-1.3.1.tgz", @@ -6278,12 +6312,12 @@ "dev": true }, "wide-align": { - "version": "1.1.3", - "resolved": "https://registry.npmjs.org/wide-align/-/wide-align-1.1.3.tgz", - "integrity": "sha512-QGkOQc8XL6Bt5PwnsExKBPuMKBxnGxWWW3fU55Xt4feHozMUhdUMaBCk290qpm/wG5u/RSKzwdAC4i51YigihA==", + "version": "1.1.5", + "resolved": "https://registry.npmjs.org/wide-align/-/wide-align-1.1.5.tgz", + "integrity": "sha512-eDMORYaPNZ4sQIuuYPDHdQvf4gyCF9rEEV/yPxGfwPkRodwEgiMUUXTx/dex+Me0wxx53S+NgUHaP7y3MGlDmg==", "dev": true, "requires": { - "string-width": "^1.0.2 || 2" + "string-width": "^1.0.2 || 2 || 3 || 4" } }, "wrap-ansi": { @@ -6365,9 +6399,9 @@ "dev": true }, "yallist": { - "version": "3.1.1", - "resolved": "https://registry.npmjs.org/yallist/-/yallist-3.1.1.tgz", - "integrity": "sha512-a4UGQaWPH59mOXUYnAG2ewncQS4i4F43Tv3JoAM+s2VDAmS9NsK8GpDMLrCHPksFT7h3K6TOoUNn2pb7RoXx4g==", + "version": "4.0.0", + "resolved": "https://registry.npmjs.org/yallist/-/yallist-4.0.0.tgz", + "integrity": "sha512-3wdGidZyq5PB084XLES5TpOSRA3wjXAlIWMhum2kRcv/41Sn2emQ0dycQW4uZXLejwKvg6EsvbdlVL+FYEct7A==", "dev": true }, "yargs": { diff --git a/package.json b/package.json index da70119f..6db85355 100644 --- a/package.json +++ b/package.json @@ -63,7 +63,7 @@ ], "------": "There are a number of personal forks here, as they include patches not landed upstream yet", "devDependencies": { - "canvas": "git://github.com/Pomax/node-canvas#master", + "canvas": "^2.8.0", "chokidar-cli": "^2.1.0", "fs-extra": "^9.0.1", "glob": "^7.1.6", diff --git a/src/build/markdown/processors/latex/latex-to-svg.js b/src/build/markdown/processors/latex/latex-to-svg.js index 0659bb47..8936410f 100644 --- a/src/build/markdown/processors/latex/latex-to-svg.js +++ b/src/build/markdown/processors/latex/latex-to-svg.js @@ -58,7 +58,7 @@ export default async function latexToSVG(latex, pathdata, localeStrings, block) fonts = ` \\usepackage{unicode-math} \\setmainfont[Ligatures=TeX]{Linux Libertine O} - \\\setmathfont{XITS Math} + \\setmathfont{XITS Math} `; // For some reason https://tex.stackexchange.com/a/201244/8406 yields xetex errors... } @@ -201,16 +201,15 @@ function runCmd(cmd, thenRunThis) { // Remove latex preamble and superfluous indents from "ASCII" graphics. function cleanASCII(filename) { - let data = fs.readFileSync(filename).toString(`utf8`); - let lines = data.split(`\n`).slice(3); - let indent = lines.reduce((t, e) => { + const lines = fs.readFileSync(filename).toString(`utf8`).split(`\n`); + const indent = lines.reduce((t, e) => { if (!e.trim()) return t; - let m = e.match(/^\s+/); - let len = m ? m[0].length : 0; + let m = e.match(/^ +/); + let len = m ? m[0].length : t; return len < t ? len : t; }, 1000); - let re = new RegExp(`^${" ".repeat(indent)}`); - data = lines.map((l) => l.replace(re, ``)).join(`\n`); + let re = new RegExp(`^ {${indent}}`); + const data = lines.map((l) => l.replace(re, ``)).join(`\n`); fs.writeFileSync(filename, data); return data; } @@ -222,7 +221,7 @@ function colorPreProcess(input) { if (content.indexOf(` `) !== -1) { content = ` ${content}`; } - return `{\\color{${color.toLowerCase()}}${content.replace(/ /g, "~")}}`; + return `{\\color{${color.toLowerCase()}} ${content.replace(/ /g, "~")}}`; }); return output; } diff --git a/src/tex2utf/tex2utf.pl b/src/tex2utf/tex2utf.pl index 2894dd11..83d32c07 100644 --- a/src/tex2utf/tex2utf.pl +++ b/src/tex2utf/tex2utf.pl @@ -1,12 +1,13 @@ #!/usr/local/bin/perl -# $Id: tex2utf.pl, v 1.0 2020/10/16 16:09:00 Pomax $ +# $Id: tex2utf.pl, v 1.1 2022/01/04 11:37:00 Pomax $ # # UTF8-massaged version of https://ctan.org/pkg/tex2mail # -# Updated October 2020 by pomax@nihongoressources.com, +# Rewritten October 2020 by pomax@nihongoressources.com, # original header immediately follows this comment block, # with spacing updated to something that looks uniform. +# See the end of this file for the change log. # $Id: tex2mail.in,v 1.1 2000/10/27 19:13:53 karim Exp $ # @@ -88,7 +89,7 @@ sub debug_print_record { local($lead) = ($i++ == $b) ? 'b [' : ' ['; print STDERR "$lead$_]\n"; } - while ($i < $h) { # Empty lines may skipped + while ($i < $h) { # Empty lines may skipped local($lead) = ($i++ == $b) ? 'b' : ''; print STDERR "$lead\n"; } @@ -150,8 +151,7 @@ sub join { @str[$b-$b1 .. $b-$b1+$h1-1]=split(/\n/,$str1,$h1); @str2[0..$h2-1]=split(/\n/,$str2,$h2); unless (length($str2[$b2])) { - $str2[$b2] = ' ' x $l2; # Needed for length=0 "color" strings - # in the baseline. + $str2[$b2] = ' ' x $l2; # Needed for length=0 "color" strings in the baseline. } if ($debug & $debug_record && (grep(/\n/,@str) || grep(/\n/,@str2))) { warn "\\n found in \@str or \@str2"; @@ -360,11 +360,11 @@ sub prepare_cut { $out[$#out-$_]=$p[0]; @p=($p[1],@out[$#out-$_+1..$#out]); @out=@out[0..$#out-$_]; -warn "\@p is !", join('!', @p), "!\n\@out is !", join('!', @out), "!\n" - if $debug & $debug_flow; + warn "\@p is !", join('!', @p), "!\n\@out is !", join('!', @out), "!\n" + if $debug & $debug_flow; &print(); - warn "did reach that\n" - if $debug & $debug_length; + warn "did reach that\n" + if $debug & $debug_length; @out=@p; $good=1; $curlength=0; @@ -372,7 +372,7 @@ warn "\@p is !", join('!', @p), "!\n\@out is !", join('!', @out), "!\n" last; } warn "did reach wow-this\n" - if $debug & $debug_length; + if $debug & $debug_length; } warn "did reach this\n" if $debug & $debug_length; @@ -789,7 +789,7 @@ sub center { # +--+ #\|12 #EOF -<; + $par=<>; # load stdio input into $par return 0 unless defined $par; - return 1 unless $par =~ /\S/; # whitespace only - print "\n" if $secondtime++ && !$opt_by_par; + return 1 unless $par =~ /\S/; # whitespace only + + # TEX2UTF Edit: remove everything before "\begin{document}" + $par =~ s/^[\w\W]*(\\begin{document})/\1/g; + + print "\n" if $secondtime++ && !$opt_by_par; #$par =~ s/(^|[^\\])%.*\n[ \t]*/\1/g; $par =~ s/((^|[^\\])(\\\\)*)(%.*\n[ \t]*)+/\1/g; $par =~ s/\n\s*\n/\\par /g; @@ -909,7 +913,9 @@ sub paragraph { $par =~ s/\s+$//; $par =~ s/(\$\$)\s+/\1/g; $par =~ s/\\par\s*$//; + local($defcount,$piece,$pure,$type,$sub,@t,$arg)=(0); + &commit("1,5,0,0, ") unless $opt_noindent || ($par =~ s/^\s*\\noindent\s*([^a-zA-Z\s]|$)/\1/); while ($tokenByToken[$#level] ? @@ -924,22 +930,20 @@ sub paragraph { ($pure = $piece) =~ s/\s+$//; if (defined ($type=$type{$pure})) { if ($type eq "def") { - warn "To many def expansions in a paragraph" if $defcount++==$maxdef; - last if $defcount>$maxdef; - @t=(0); - for (1..$args{$pure}) { - push(@t,&get_balanced()); - } - warn "Defined token `$pure' found with $args{$pure} arguments @t[1..$#t]\n" - if $debug & $debug_parsing; - $sub=$def{$pure}; - $sub =~ s/(^|[^\\#])#(\d)/$1 . $t[$2]/ge if $args{$pure}; - $par=$sub . $par; + warn "To many def expansions in a paragraph" if $defcount++==$maxdef; + last if $defcount>$maxdef; + @t=(0); + for (1..$args{$pure}) { + push(@t,&get_balanced()); + } + warn "Defined token `$pure' found with $args{$pure} arguments @t[1..$#t]\n" + if $debug & $debug_parsing; + $sub=$def{$pure}; + $sub =~ s/(^|[^\\#])#(\d)/$1 . $t[$2]/ge if $args{$pure}; + $par=$sub . $par; } elsif ($type eq "sub") { - $sub=$contents{$pure}; - index($sub,";")>=0? - (($sub,$arg)=split(";",$sub,2), &$sub($pure,$arg)): - &$sub($pure); + $sub=$contents{$pure}; + index($sub,";")>=0 ? (($sub,$arg)=split(";",$sub,2), &$sub($pure,$arg)) : &$sub($pure); } elsif ($type =~ /^sub(\d+)$/) { &start($1,"f_$contents{$pure}"); $tokenByToken[$#level]=1; @@ -955,14 +959,14 @@ sub paragraph { } elsif ($type eq "self") { &puts(substr($pure,1) . ($pure =~ /^\\[a-zA-Z]/ ? " ": "")); } elsif ($type eq "par_self") { - &finishBuffer; - &commit("1,5,0,0, "); + &finishBuffer; + &commit("1,5,0,0, "); &puts($pure . ($pure =~ /^\\[a-zA-Z]/ ? " ": "")); } elsif ($type eq "self_par") { &puts($pure . ($pure =~ /^\\[a-zA-Z]/ ? " ": "")); - &finishBuffer; - &commit("1,5,0,0, ") - unless $par =~ s/^\s*\\noindent(\s+|([^a-zA-Z\s])|$)/\2/; + &finishBuffer; + &commit("1,5,0,0, ") + unless $par =~ s/^\s*\\noindent(\s+|([^a-zA-Z\s])|$)/\2/; } elsif ($type eq "string") { &puts($contents{$pure},1); } elsif ($type eq "nothing") { @@ -1294,7 +1298,7 @@ sub bbackslash { &ddollar(); &ddollar(); } elsif ($wait[$#level] eq 'endCell') { - return if $par =~ /^\s*\\end/; # Ignore the last one + return if $par =~ /^\s*\\end/; # Ignore the last one &finish('endCell', 1); &trim(1); &collapse(1); @@ -1343,11 +1347,12 @@ sub endmatrixArg { # and truncates the rest # I'm trying to add parameters: -# length to insert between columns -# Array of centering options one for a column (last one repeated if needed) -# Currently supported: c for center -# r for right -# l for left +# length to insert between columns +# Array of centering options one for a column (last one repeated if needed) +# Currently supported: +# c for center +# r for right +# l for left sub halign { local($explength)=(shift); @@ -1734,7 +1739,7 @@ sub makecompound { sub arg2stack {push(@argStack,&get_balanced());} sub par {&finishBuffer;&commit("1,5,0,0, ") - unless $par =~ s/^\s*\\noindent\s*(\s+|([^a-zA-Z\s])|$)/\2/;} + unless $par =~ s/^\s*\\noindent\s*(\s+|([^a-zA-Z\s])|$)/\2/;} $type{"\\sum"}="record"; $contents{"\\sum"}="3,3,1,0," . <<'EOF'; @@ -2239,32 +2244,40 @@ $debug_matrix=16; # is nothing left to parse. # ============================= - $/ = $opt_by_par ? "\n\n" : ''; # whole paragraph mode while (¶graph()) { } &finishBuffer; __END__ -# History: Jul 98: \choose added, fixed RE for \noindent, \eqalign and \cr. -# \proclaim and better \noindent added. +# History: +# Jul 98: \choose added, fixed RE for \noindent, \eqalign and \cr. +# \proclaim and better \noindent added. # Sep 98: last was used inside an if block, was leaking out. # Jan 00: \sb \sp # Feb 00: remove extraneous second EOF needed at end. - remove an empty line at end of output - New option -by_par to support per-paragraph processing - New option -TeX which support a different \pmatrix - New option -ragged to not insert whitespace to align right margin. - New option -noindent to not insert whitespace at beginning. - Ignore \\ and \cr if followed by \end{whatever}. - Ignore \noindent if not important. - Ignore whitespace paragraphs. + remove an empty line at end of output + New option -by_par to support per-paragraph processing + New option -TeX which support a different \pmatrix + New option -ragged to not insert whitespace to align right margin. + New option -noindent to not insert whitespace at beginning. + Ignore \\ and \cr if followed by \end{whatever}. + Ignore \noindent if not important. + Ignore whitespace paragraphs. # Apr 00: Finishing a level 1 would not merge things into one chunk. # May 00: Additional argument to finish() to distinguish finishing - things which cannot be broken between lines. + things which cannot be broken between lines. # Sep 00: Add support for new macro for strings with screen escapes sequences: - \LITERALnoLENGTH{escapeseq}. + \LITERALnoLENGTH{escapeseq}. # Oct 00: \LITERALnoLENGTH can have a chance to work in the baseline only; - in fact the previous version did not work even there... - If the added record is longer than line length, do not try to - break the line before it... + in fact the previous version did not work even there... + If the added record is longer than line length, do not try to + break the line before it... + +# --------------------- switchover from tex2mail to tex2utf -------------------- + +# Oct 20: rewrote parts of the script to yield a better text form by using + modern unicode characters instead of ASCII art. +# Jan 22: added a ¶graph preprocess step to remove everything before + the \begin{document} command, as it doesn't get consulted anyway, + but will happily pollute the output. diff --git a/tmp-pdfcrop-1411-img.pdf b/tmp-pdfcrop-1411-img.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6ef48e40e64a615d5c5d5c532c242bdd23b6ab58 GIT binary patch literal 3826 zcma)9c{CLK*H>9bvL*)EvxHfgu}ijyj4h0@4F)mUX0or@_dJ9N6EU(zNRqK;iO9ZJ zcFJ1WHM~9Nyyra6`#!((yzhVa-19y6p8L6Sr3P9yxYH9!+-Uoxe5AdIkPJ3%e4Y{Zt z)Vy%2{xI(~X^2C&n(b2pKF&eg*LBk2dMLk3Hj6idM;_nc;=`2?6j-Oi2A$kp$6;5Z7|T#HblfM z*tiA1SWVx!OHva`xOPz-ZhUn67Vm1#NrEiadHFOPVbwsN^i6^;cTN_m_jMtln|(h_ zuqv%gENdKyevu~aQdMpps8~D{HNd}TK#o;0%K0VbHx;-bjosb=n!ls%KV;$&pDs#0 z16AmBxlcB2bQDI@V&=n{$PtpVHM@c{Sx=b_+9p6HY*aN43ZUH_{ zhlpP)_25bHcvmuJI;J;Hl#5~BzY=ZmC z`c^c%_03Mc&BM8C_jpl^hqr+_Nrgr)@$^vEws4rt)&nHccOs-(>fuRfE8U3$(woNm zrndSsv*c~_y1j_h$&&t#V_i4L>6hv6U2!ss_+porX9^*?vbX`&mhA(|z@qZcPaSb0 zY^)t%*~Fs<Vre95E|zn}&-r*V$Ks(V1YLB3zWBkYFTh14DbC13D(E zA}b+PSM)4c#U0;Uq32jT_;hMYh!uAuxt^U1K+!w~P_))bHw@+qhbOe&vuh=}K3SuC zmZgpS5bP{!x^Gu(T`*|Aa|Ez;G~<>QZTg%)?A8#C^pvvAC?7-aN@7T9s5<-kXCDlv zW#-}Td)5%UK2dlp1(l+|=3SonvjM=EmEK7)X)e9(4mT?eZNH51c27E1vL%-em~_5} zB~Y6b{o&rVIjf>%L1QOj;kiVIkPADW3Eq?l3 zsG2hX(p0S`Jyv<`w)wv%^X0fgvpkJwM)_z=#Tj4WJIOQ010gG0HqzioX8a1+ zFZ&}Q=!53U_1%@C=jp`wdt9!1mr2*o3$Z>RKo`6=OfS$K z>nwRriY(HF&=KpK4MnuF?^2U*+Qk*6)$v04bd`D$#hdq`e5O780P1difh92qDTy{y zDq*}Tp?-CT!$2_O(XR3PFY|85_D96X=jn2}E4U!l#m@D8G$buo*fc!fJHSk%m<13W zjQ44b3h&Z2CUaf4Dk7i~GGZAK)^R7&;&GpQAp0iQ5FsW`TnOilC=7&~@qUDuG zhc3@Di&IWcp6hcqnI9%rB-!26u4tU7b_w#^1P@P5`J>CByPp;NyjRf;jH!s+mI&HZ zU*1?#@v({6+7|%#{)=jd>QW^k(6#FawU|(pvb9E-K{NgLVq5-W1@7M9yEwdpFoTzE z-<7t)9-wuBaBLS`JK0~p)rZr32G|Q6QdTSJsyac^cx0qWw%$GD&N-$>w9*lVUVjfk zI-*Q@2B&Aohhw^kBCjTMHr3tJ$wMhoVp<1h+@d&$_~igkEchU^9lIy~01HjquHgo?3xKV!-aHhOAfRQXZ&u9#N^daTGy4{#^ol{WAdTarD~#O6&V+i`cS_<@2D zF&hN9g>E>TcrNoL;Y(BGQ<{VN{pd^2p6f-t@bYsN`w?T4T+xP!gsRU7q=;_300qVx zYVdDVrE(jg@=s{>5kuZc7MpHu@8Ui7m(}JanMe)Jmdiip`!TzctIxKnb6~xh9(R>| zYkR3l_m9(;Vo zF8}dH18$jGG@3WV>HC;MxsgPv6JHKlCs25I=vZcwE_NZTsr&QyQG%Idxk9Lf&2J*Mg zZGF6@N`JH};zuFW1%=h&J zm6n!BtHN=sGBm6~jcFmipAr!<*h=gPJ?q|K-97eIG-f7>xep)9c~nl3_LoPZioT=T zFDNWaaWe5Gb~d7d?5Th0e{zD}zTa6j-0&%{Q^)Q@2dmYUN1kZxRbY@P!8bbvtZk>; zncNUBvgoOv9YNj8=4`Lj&OvH#50nhvk+1RYQ28UdMfASH4-L47&>e;N;0unjL`f|0Wq3_rMsT&snovF8WHLOK!(_&boD34AQmJo_CKh+++QmH zFHRNycQtp{okKeaFjkkyK`!zPg5~n0&rzMqo`?Liy24+@|MUzXgoB$Y{yxZ1K=HSE z@WJ5)V6y+>BH&QS-`hSmoiUtIX9?N1_`nk{S!-eU(5T4z37ep}P=OKm4Xz+?CaCa24Zrw;NK`C^juixg^m`23Z~Y-Wc%gXd3swmDHFFz$L5CBJ&r zjkHfJ;c86dnZ4fKjJ+CGJamSt$IXtY8o#K^8#NzhEcdv4qXP&7{iqLmsplik4@r>nqCrnw2lhmBE8Pnr7w|DDbn>{iP)z> zWPGEn7aW1wsPxNOf1Akv^lP&nGgITka!K;-%|yRMZ3|NJ&lz5;#J3M9*uYDPHLv>7 za((^mDO0bnbr)A84k?KZ+z#S0?WHug=Lu(w#5HK?V{40N*_k|wtu-?b1cV2+KDF)@ z?%tYe`HmfTmN5~(uwwYBMNG3FZZ_hVeKv*3n!2{4=^5f0)x+(VSc9b+Rapm zr~>BRR0Un`V+IsE9|m5Y2#Bb?`CHNQr}7EL%+7>QNsIp?)`b}nmy4(x{C}B^H)XTEV80> zSbkX!Aogx3OZ=0`5ZM2svZV>e8KkF$Qjvv1oSa}VXJ8nE z+aI1ijgN?opwR+H((ACaofDem;y&c&QkSCD@qo}L#-|NkXm@aNYNSTLg!*h!NzR9~ iwo*||jL?Mrb9fvc?SuFC!8ikCVK68_R8$+G1NbjL?gje* literal 0 HcmV?d00001 diff --git a/tmp-pdfcrop-1411.tex b/tmp-pdfcrop-1411.tex new file mode 100644 index 00000000..e69de29b

    ZXtnyhG%%(|FQzMSU{o&X6wRx>Uba#zA+`$>6_Ix0aD>ztGf|!XNqL z$B!2=g@uJlo?_zS>H7k@b%Ij+2NaB?YGC4`AMvn{AeWu_#=cBn2?j?;$A;_ktv6{x zw<%S01UOl0rt&W`3u6JQK>G;e}CjA&Wj&R?3k*Pbs zEbevq#9i;?PFO(&fYH&bY16#7H9QO~A6*^vHybb3SoIq)&-8td8ho71?Xhb}?{{hh zYPz}hhTBZ3c^x@iipWb#V_#jInE(ESNB-WSZ3*T+_s4TBU9Q>hDw*H<{l+rK`yZgi zgR@ofCa$McQCD)*iFk%gqN zn3%Q7PI{BmT_lB{a`>Rc#b+48jmPKI0SpEM56v;Nb$Nw_RmOiJoEUZ{N@+@Uel0EB zT)Tg(ob{zHm#5(GvX_dieOP!YCWu?mK{)5Z{)fHrGELYSH3YYv8If$hI-V)Kyh*@q z8}v6&_3iW@*h>+x?ZM0;R6K5NX%69@$vxlXd!g0jb4HNKpb6a4yN5)o9&4uPX@^D7 zIQHeWvamjdWy3lRV2r*X+n&W2J_}H7)x&X_%BZK+xp9IzZ80FA-jHp`$ks@I7X7(4wLo;`&BRENrQ= zKazV4EM(_SO7^5BGBD6d4-M+_?_EFq6Aq|X!p^0+GrS8cmX;X#DW0y6J%*6J$G;)c{CYREcl{Vk~D?;7Q~JWD)tfdVu7rY+DtrXhCC*)bebrF*A}>8cuQn z%NYY%%&gz=!VL&%VU)HDcKU&OeLyBO2?a7_dn7>ap!$I<_#T*s>Zf3Y7>AZg<=k>v z?GIrep>@Eot|{Mr87W5+7eGOoDtlOn|KLjArh2%2P=r|&}Ic`_$ws@opl)Dc5h|C6 zTI<>yBZDn#cr`SLjRrd`NXWl-)c0kgAEbGwHC`gq`?bO>3TOk``z3B@2gDBYkVw|~ zyAJ~N8x2<#ZjV2;RLe-;TC7zRYow|WASzp}R~7KWIM}72ENKyu6pUr#Kd4w(Q&N4P zP-%2Yc^?Ueh-SKs#khft5d=^MqC0=fq@lAGNa;s>T|nKmS_-1#2Dv;B3kzz?kwx)$ z$TUzeu4NKu`hlmWfO?rku)F6)4FLej|Eq3QFS!Hsh;j^5Wl-p>>4w_+FwU2F;5Jp~ z!4HP`zJ>*5;1`y+L}EE1gDX5LZ*NO!-g~eC?(z8q8@Pt9K|x&%6k2SB)(H#ekxRZE zZeI)JC=aT0w$+3{>~=>KxqWKUPj9Ps&^?+v&zo%kKM8m9lk7`1pHJ|L6~xKY6?EQi zNiqN+1!&IHTBeK|Xay- z{Nx-XAFeF;A#NL3&%Wc_Kk|Rs8l3V_vqr{;h5ML=Wth_k{l)*q! z{#^RD*CEV$iL*W`r|Tt~=cMmNz*Womeb<4^9{+C3=^i%#!90bsq=;QS`%Ghw0B?~q z;@oMOZ>(p7Lb0pNh)m7A_P0v_46nVTbzC#mog4xweBmwdnaX6Kd0*&?2=tlWcH*Uz zlMQfg%3sro${URJd7)U?EmY=GH`h9ZBysVn!lQ3s|CGc)lbMjrjYaQI3nsihXl3=c z7H5C>Kp1(NmV0_}6Ud#gzd z&|^WxXY z3IDavx#ht$%3^4-@)LY4D)Rjq@|@ZM&@+aH*#Nev<+nks-dd zvoymImVN&lZ+T@nEPNS28l2I;+HL+C4qQ-@?C3Pn!flt=zpMTOG01z8l67ZtW3j3L zQ^Woqw%|Ania%hm2fAzkjappS(EJUzosN6Fq`}~K*ZVlxRh2ynI}ftA=WsH|{UPGk z@j~z!_{y;=D}nW%;dOO?routE{noX={Se5;xF#}`LCTte?Y2Yy7^d5lwvuX(+AYkK zR{Mf>V)+CB`Kdm_!g+3J1x(*J@`Az{uFm0xIWqB~J-;h!p*7lMK6bw4;D8Ro8;{L! z%eSU6^n4G1hBNAY4gnABl#7cRcM8r zkTNN34Fr5@#Cufft)9*8!udIngQ}~fW~sZAq_fIcYE;XeGEvmrwkg0duq4oXw*Rm% zp7hMO0GIp(&^Noe#Sz8q4~hWtD6e5VZ8}eGg=`pc!Bf*#N^NZqM_I8{&<#PB)cu;Z z`I>M1PFJ7XYLdAEcw;SEpldL6EfZ*f!qor#(3ULqm(RBZb3EeV*%l{@&3h zz@R{%TJPvPuU2|go_+hOFuI^qQSZXDwxZ{L3)$uUy!7KG^6lrp?L`>NtT$S{rY_%S0ji0M{!tWp#N2GTKsq+vCFmv0zx93_`I3V{#xR!{qmmT z5b)6&oI1&=F|R%c%<2If>grxuY3YjFUC(vspaSqhFL3PJM}qgKYbLd|Amb~1qR(Q_h5bI(9qL08u1eJSUCM&B5{mR|?6sBcaRZZYM)`o4pAv z>nhyV2{1$ovm#607@7Dzz@*f-7BX{9I-OiDJ!V$rV}a50#}Y^6d(WHS954NeO)y2! zP6F7tLsX3;^C6lr^5M%GyMdB_xY3!;i%kyR^aiKL)t2v$_R_AebRMa^(?X6kiHjYP z-=hOB_yXh%iAwlXBTZRtEA3aUKfAFJmNfQ4L^Nit%by5$Id$@s&4Co@FFAgPSRfLW z1OuoU%~1;Z+BrYbG=B+pX1NsRMOhY~{R5-*x8OVX`9dEnXaTGUa<6l5vN#{c2tlr# zq>A=!jWh4W_*cWC*q%s1m{>z*%`^E+d5_v362x@JBWLW9nK$bGF?WZl<%3S+zbqzd z_C>t`ZmNw^mW2=1_mR>dNTY;Nm@1kcU~O-gD0)qHx+~+{De?qJcgQJoqaO0pQxl87 zLvH*TIiT?BhnxVjnC0(38fo~Kdxjz$T3+X@q&xkgw~vM|{0Uwdk?nL#Yp??{_-$#* zMkyC0@!)vIVa9fM-g~Gx5p5;*nnMYmm;al_KAB8f@)B4TGW=l}@S1}TnoV4*IPCTU zdhx}&HGo1=@>U`d{-sgU=Qc-az=zKJLyZ?^o1|y#W)AaWf2M!yaOU0zN(WUh{3Y(u z0**cSEm%-H`RH(3;Uy4B zyK}bIZ&z}%Ac&tB!TdxjEzjAU3=(+frOh)v*F;-E-(`%l;1U+W?Tc*I`w7=iS zhdEhj0}^Yx2tesMS&+;XZBFoOLe9>(4MDUbrU*z}*WYG_0a4(; zi`OY4iKl||i(L(#(;M*izEpqXB>XQ9zM&K85GCNBtAEr&OxdK$ZQVV>#n_$`3bSLLMhgU$=_G+ zpDms=3(93(9@i8g0Lst;E)gN5;V1DX#dN^O`$*&9j(pBnpgM*%`&Ag>Y)|2&i~ODj zJf4)@;8lpQC6g%4MCm(^+5mt)w;JPaB%${JaWy?EP{9WvPz}f{$4cQZrbCR*iI;u6 zZyW2;#Voi0g&YVU2xrg(jK1Ke&qUHx&q zq~ETzns4-nLVHM({)@ri$7rKfh2T&^KuYk0>Gy2oH3oq~wfS8c0<^G3xTej#4Yfpn z8bR;Z-wEu~vq&}Y13?ae`8UQr#Mn;A-#TJq?f7smN_e6$X5&MZOHrAKo-)>w8xM~I z`j?p!P$C|*U9Bbo;)|NIB|>MDEjfnY#TqE#!oUYG6|5x++v((uAnP)o&Nc&Y&h0sC z#M5xMn+t}|bcBKUsj1km<4AV!gXipZ)Mq>X*|LOHCe};P(Lh)psn66Ds&2H1tlG{t zdW=f+y(S%fE@&?&fxPy@ju%~FPF!tmt}MgJ1&4hTf;sZ^rMZvjXKn->Ga7X1h zg2JJf(=4Jyj@aYE7C4@F4Oaqb&i4Vnde)8RhqjCnK!O7*6F@CSLS}+7U8o&ycsuKd z5H4A7BevCKg+J3ac6&ePh(GJeW;1P?!OY_|^*}~^IA5=+L5$RFkQHE8z=IO0mos*` z>uJt5NffI3=y3WPD~j;ZLbBAjm)Gc;cuBL*(Yi@aXP4J4%5%<�F8C-FrW)kU(s} ziaOeA3nFmX$S?Gvr2e(A<9Ljo!I!!^KazQ&{{iaLuw?Aqer^);`y-B7vDM)^n6fkX z`W9!SQkl$gaw;lvIOsP^MKQ-Da|3FtSw|m0II)>me4p;c&DSWJwlrZ@DYzZKZ}+b% z1?87mkdKNgf`rBPO_l?qR+INJ^}Uz+IlTo+1x(rqH&*U>uind}Cdu>*N(0`yjJ^AiOVThxRp3FUx<2e3xU3H zB*6;{ny5_InOg3AcYxUGQ=hCbdO$2Th1I~G- zt<4hbkQ9CQi4;rP^e^7g$Hd~MsA&T)f&s~e%Ni^k!$qlsHLLOM393@yimpzKh9O$D z50;kRXbq+lri;#$DT_&ALE;>br%=noaz=9^2Lm4GE|2(oXwzRfWSs$AmRtS++?Au@ zAe!JaK(%1zkbpoh?3QB_*?R;4v|k6!&rJ*Szj3wz?hxDLz7Wio5pd~KrTE=kt=_xxmXJXLFTIP;C;=P~uuqw;?I7Waj3m~cRc z1`xK#Xouq|7FV=Ynfk~`mZ$ScdZ)VHy#*f}>&ey>e)WrW=7WoeoNp+*vt^L>{=A|K zP+Dm$$iT?=IPFV1iP_7uAto|bK*EMG(rTVR%(hsV=WE{_`t4I~OCzqUcCyVwIKMU7 z7DB}Or0Kk8HfcB;$7K_GU0OrxjtoXg(2#_x0fNi%V!XgRaY-4<#@&${zXQ$aKHSPX zXX{Yk6Wc=j$6p_yP#K|L76k3?xo)MzO#f=r-T8=vV&)}tL%+~`3lv`lihc&UA(j`@ z*@4>I^!zEl6y+9r#^?|DAX<`l)Q48$ zp6lk@MmJk$6==GMWcrzU$ye>%*9xBivdh`2Z<(DvHkHuR6V=i1g}24YNue$0m01A+ zd~@8Nx<-|m%oM0A{!reKTj3DIjhAyIaWIuUKaMC0^<6SWz4D)WeF;+6?1PvUe9R%tO)U*;u)Xqwx}Z zVVF}i-C729izDXx$Fty2kLX_bPXM0E`**$at;n~f1W@^6Q~X0FcVoW9JKP40!aZ!d z@|7qJMuSNrfNmM}NTdVm!T(HP{om-W|Knlcb8gkn5H>)s0Z5)T02MMIKo%8NG7A4c z>AC;ob^mXo@PA+A2ufK5sGTFz(*BV=YhoP%&>8j43SaQ!|6W!8?+g5+L0`SWnF&Ev VQQ=KI0|fS<*RsmcGHJt4{{z4ytUv$& diff --git a/docs/images/chapters/circles/12f78070db3292742f66cf3cd9c7ed92.png b/docs/images/chapters/circles/12f78070db3292742f66cf3cd9c7ed92.png deleted file mode 100644 index 6e77186ece8df0661fd422ebee69cebbfc291c17..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 16785 zcmcJ%bx_q^7%oamNtZ~YbSg-92~rBuB_iG3AzjklBHi8H9n#$(xq(gRU3@3z%-p~3 z%su0Py4kGXiZ`C;dEd2zKg)kaM}3V70|SFDEhVl90|OiR{Du4iyyD%)ISl@K@m=Pl zILy=YpUjrL7#J7|7-?~lug3(RDYhyj*M?I zZrGNrUSZ~9j=>QccwHd=CEj?9f>QL0ddk52@2~VPlc0B-?_=Jvhv^%(8o3AwjQZZB z+oX*pZ8wiOXs<#X=1DLDeZNb|$h=`=E9cJ8U@}!WYH)Q!K|z5Lmz0c`aw8NyDiClo z9@ANRF*!NOT1ZiocR7{5pyZZ5ty2LnE-t<$i;9JfZ4&|q{+*fygN==yM<*;U-cA4i zdU;M!Nr{x&13M&$R_fCyxJ<1ETLM+8jbwLtc=%9W{YX@rS- zAD4#yG2;7b8~Jd>q+qGi-|!G!9{IMoL02(I*hFm3=~DSo*4O>M79(S0)4=%njy#op zhBfEJ#S;;kQ&Jp09hRMx)OgKJVtrrFlO7G%7v3@P$B$IN6poJRFvHPo%A40A({lN~ zE)t8em6wWI9+v8u_8oSurT@|5$f{;t=kYQPiLGw7oo@YNU0^<4{?~ye`71Jc0{v(< z9!gm6wf=r+d~EFEN`=15Z&9sOl=4+i-!UPsrR(ra@33TjaQUO6W89^*ibR>DambuytNmUnJ zE;&(;6Pqv-2HT99wrf^5g)-e>Xc*0;$#F-*ZjP<~-uC2z`luTP?$ZgsQ-Xe7ijIu$ zi}51DJo(g2cWXBD6>}f^?N83PCMOsEDOLf*RvfU|H+U71aN>jUU3xoADy1C1D-1$W zLM@lSbt6OJ7y>fLt1T@oBLq#uA0C@-7Ua@UeP49ot0uDTElj`k3XY{j42l&Dg!jCL zC1CRQM;n=!*KX>Zs~G%s(*>hb1oIX*0ly@(_w)nwv$BdyN7)fxbY9G+ESi-iRkaE}z9RElO1+sb zNx4aOIxuv|+4#NWOrxmvH(h|ul0;8cOV9NcI&m_}>6{w-%ReV{0w2th%WTY<+@A!d z$}shNh5tw*v|eZN_^V|j&e4+wB8;O*_ZQj$!gkZTc$H% z)u)>ySBB#hiTp&S(>`DRe4Aw{D#GQ;g^5C|)ALQLKAM|IRcSOtcfW|^$wjf5#eani z|Dv61HF3Zx1--PPmt?ViVt>CI9qfuTaKB}z5tBbxOYqB0U5^iJp3U+t4j<8pJN-}$ zb+XM=U%l^kw_5d>buW_$5rL)JjZb|ULzDFGqCjbVfOcg1uhXhUsS?)J4#cW8LFma` z-Zk-k4-+oC0ct9*frG=w1|EfC{nS@GQ{#u*lO94gBaC8gw7S*fm7yT1^+QKH^Z`tv z_Vc}R_lBDlh1BaY+bb%giS&Qb5l36na5M_NPiap5`{O^W-|6?C{zb8|TE$&9+Ledv zjp%c_F#SvJnHYvd){c$$!$EyR)EpzQ>r1M_d8~l@w9a}o26^t%;e76_%~8$H`C_Qr z>#pexE1Lxw?v_-yAMfJruKad2vr9DB{R!8aAj7uo5&WIOVJ~M36pl<3>?_7Mdi-=8 z;)rV1%G;N!OQW|;SC;N&o1X-kv{6S=Hm`1WRaB{2gk_SPtH?dsR9O~}vWHWpjq~JH zRNs~s+Zoy2oqmK1K=*yR6LQ`}H=gL+AMB>u|LOdvD{v%yph3% zutYPa*))zPt;tLkT4nEAuh1L{*PXJpmB`quS?xXkxoWC^*~x5LrNP><>1KYuaat>L zE(>uuQBOsOpzJp#lhx^>mG!EJA$oQkOZyyiJT+mBX{laY*u8=&~O%(3mowD z#&i2Bs)L%@WM;kD5+!G;fu;@A>(h^R5^i^wJ4W&eNY4C-L9}<5ns%JiKj95`h0s-{ zO)BSy$L8XhcZ^Dv8P(cW_6xJ|=!N;eSe9*;Hm&s#j@(Kr7piH;IfoW1kBlt=rz|22 zBhI_1;g5yup1eN`osZFpfa_ zcj^s!UWfNWO)?3c%yr-RJeT%Ex>3OMs`TKsZ2Z-mOmA%2rRg`qsi@YVW9xrqP+E?8 ztC&LS@ezagZr0%I_h_8g5&!mv!IF~~G2%3Ex3;o*^6$No6i#5-ITW!W)<7H5zQ1ZS z{PjDIk&^edd}?_kRTNH)&2Ab(jU~E(Yob|VZ&*d+TlyD>K}9k$?5eTk{KzR>DVA+P zw5HR8sR5n`CFS~oE?rv?D+gs;j0N0f|M_q->L2Yewpm$HN&NFnDa;1Fi&x@xAJj}M; z4yH2M{+#7%ScW=GUUCE2V^E&$d78^Nxt0AJ={3{}2p5>~vX&BOJ)uyf;g(qr9 zQECV^oP;{x7`);h-!~O6{@}_XE}7gir&{d2Gg4A2HJS-yL#^w9_;8n!Q{LoI)%Y5F zmhDH-{LM)K@o*v9(E=DJHJ6e}wZfKq$t8~mQ)itrdUb>=NEi=s>ueq@VALsT>5zzJ2t@ae1PeiWImFmwv6%}FISR5^O zZS?H;bhQirTUp)@X-;;fP|>IzPBp19jhmPFEIwLhp$Y9hIBpv0Zd8+m{T^r4_94bM zlrZH1TRi)Eq556RrHJp^9i*hR*Nk4cxLgnZ{J@9?v8;6gEOSX|v6juNwPDV2Q?jV& z^b=}iWvvVQk}cYBn%SrYA@;7r-uIc|#tRiY*wR>fxvYeodT#a`LR|e(6T+xar-PEn zL}Qw%*;4=g>5|fZGdkhHprdP>e+&4M;T)PSAyNZG9LV7$KS6H!U{-yvKs14+qAsOC z>)Z}ge=6y`g$-7CD*NVRvflclrEl0vFa=c9$iF+*?Q0#}on0g2aY3=^!MMNOU3?_N z${TUYGGXW~#srY=E~9uoc;_y*3Fwst1wrr-1kgu@RbBg6-z15YiAFh$;i#X(fweI^ zT~sCZ>raFCH-5{`Z#XPW1FEyX{$sW=nXg3$nOMt0=TE@&bdYmgYCc_6?NZyLL2X~0 z;%^vwbgw6i(u+h+^yClEM-9OvwcyOzUm*=At7&{4ku5MGqv6O&S+^?hSY?>qa^f>R z-7hho{x6-<_}uug?P;_Cq<@m&*jGovI)9IT5gUz)J)Oy%JQ#%x80;&#^;jynWw}YY zB_&xeJM7@C8f7@N+%DA%fDO*x9vaEM>HT%nDJIUewTlGZNjkz=s9`50{uDe;A|)*? z<&8wfhW6IR3Rdy1?g&~=q3^;8-^YQBXFMj=yzcCpUj}5{l7hjZS2;WH)(O9 zx!`BGw8=Iz6Msrd#l=Sr4%N7A;BsXv8gU{rf8K`H7@#%ZSI%#WsV0ZkbL#4{$4=*> z+y8?ZNzv4f(wt;m>5-E2=Y#mE>io>717{Ysk*_)1yhqH|evc74iAO=Gzr=0)w>zvL zBHSaQ-9+q$iIl}UEyI!VT5K1>sZq(OVgZRc zpE|orN+rcpv9V!eXgZdi#59tpr|YQ^gV-PMbfKd;Zcasv*ioSqMTA3V~B(14T6#SyqLMf3fiT0z6Ic9eh7ScRtxftOKH|JL|WB{>6+ zDePDa2F6G1_Xj5nz_%?j)dtVw1uOI_jA&+N-4KtWCoa+0m24qdr5k#Eow$0mo6fFU zC{yt-uDm|o{jZj@{Q$L!Diu`@MRkWg=JsB4Ymyllu2w@eejE8>Cr1cHmMLpLR_>gF$O41Mo|ltghuDw4s*W^6u>ZfO7f zhC@Qp8?VD_hRJvuRin1L5f8fpq`MUMid==dMx2D4j+V~sY^;-BV(Rzy)Kr#-H@oXI zniUtv=+xZeTl7=#^ut>NKWa?iN0QaGgM-ux3}isaVyB`+q!S)>+{f$gjubSFosJ6% zVszf8zh8A%T3jU%g|)49TKS1V+CQV5PX#w#Nj=Z3**Tn~qPk&=M`1KL>^G~!u9_7R zlp6ndy?L@<6d7mWK+hrpYn$tj_FGS^DnB`>-(-Gb;&T;^`&saGPCXuCf0B+HIsnA` z4C3AR0*~AvWlk%D*QH7!t?V&B7IwA7QSA;&tk2#StLr;Zvxvb)Gt#^u^T0Fb1Sf{A z`lG7jO+;c8Dl8sFZ@NIaCFx5NV+U%gjlttaG?lV|hr5!}ZfZJVm=#ar7wGTr(n5pq zs5z#ZCsH{__CIO)z`bS1-5Gwz7a1IsItGu~UaY=T(=|jdEck$z%<)t5S21w7C6O4K z&t+uAYI+XT@k-vZ8Ko};_<@)sa-(-uy zWKYDbxrZ0HyOs-Ep_X#x~OS40xA+61?CZF7BN^lxtBIqt;R z?;{X%CNf*K1UVu0C+La-AX&|a9Tkm(181=SScjUnbkbQV%)|*uIPs<0EtU~j5Y$N4 zR&Qdf>PA%TQ*o{pblx$)6aeTS4IT8&ILJBqS z>&wHRxU|Kk*|N?by^A%GN;G~(MEtiwP^fP_^N3kE;Yo5xnoHWnbiRSF-O(JATQ*An1+lYgG`r=~0BkGN{EAmu~i>J5w z;)?`r!%|T~Y!dD8!;{k4weaM{cOKWDHC{s%+{Du;(n zZ|1O{$W=i$Tm>&KfQ4Q0DUEmkXZ;5t{;1_xYF`L1E$DiQ{CcgTN=dbnFNqF)z16|b zCk#+Z5b}RR;$iFQ`0^(jA@xuE0U}$4q`038`q{U1VhSRn$YfON`hk!jsS4#{V-~~F z!OF&H#mxp-+rz}07p6o`*cCp@mMiTvG#X&$r*i@m9*)Ovolm~WKc4tb=LDtvP5)3n{1iOB2g|4p2Uf9Yi&Ar&KI=%Qs)Nf?@~e4FQkH8%-jxSOSScCP36Il{yon2ifE6qH*UJnkfV_thX3FT?;^31B{&n)p7 zzS5n)i)YsC#aGq(k~tz4h+(t&i%T)*6Mmz^LGN#|z%x>-xpKYMO}P}#ZRnT~>v-$# z3n|GL&k+jvCA*gwd#X;*)v8wpv1L6aWRu5kr~F$7TsR3&C;J$GSNl(R=<#Bu0h(3I zh00G(5)33DkZ$%x5Z=C0%9UMDb6U~6I$q|#{rAVkBAk>jUWT?8lpMO>U%fP&$h~sja`I*kT{zqdVkV|IaRFA4%zfaBZQ;D zJ9Om3&MVWCQBW{CS?d^1VyD$z^XBPz^Y*RZOo=Ac>*>ntYnhI~>cdbx^U$4BUpOh$ zm4Q*U^vA>P#n%Ewf9yAsQyucvt_I&B>Z(q(dnf?k~K^FgJgSu zZNs2inhHrAG3>IeW4HhJIae;_MSF~b-~g%9Qpd%tZU)58&W$XE%PQA&y70!8$%xPX z9~OAV<_Om+!WyLQDfd$x1I*VFjW|C)KRdFBu&}okk&{sZT*4fF)Kmn#wp++nM@s{m zx*k1IqVXIBpMQ(~+yvWqcxVnD)w|Z=3mX(1tT&wa&IL+tH0nEd>qt%Se0&eoD%pA^400#*6>cafC%h;eHWw z8E)-QhiO&%_5LE>ZAlq(m{wNBCL~O_I(-wtAI)WQWX*jh|7L7-bTs9lzXc*otB}4m zt*Sf1MVVnXQ>=b|I3Ke&nYVlJ8$8lKG4WM&OiXZa@OtAx>5$VsIK!LpzMh^5umHHD zyYz?bkm&Sud1)PMwS^=9w=|-LG*3c0y!u&ov+6jqI?X}|;*9f<%SJ2A)%99J`23^{k-Ib%Gt3?G2 z9v5;#M|6n@GJ$$X>)n|k8QV7)_!lp^jhY%dtS{Su;EzK0qyKSuj7LdI3YpyF{L9bH zsHiCS^NEj$3wPTIy3U6ceW>oOBVXp&8T}Tud3f}E8oZ4<{XhbzDiF9bV+BXH3g_AN zMYbb5xYXf!^kB<`R zg0zZSB0SVrRBu=Qh;E|}-&v5A{7JEu3ne&QYN%=_QcZ8S3oIYCd90DLLFDm~qU_m! z-;M6C{PNG27CI{PZ8QPj0o;5fDqD>GpZz`7KlsY;QoS%nv_U9$UVDF`SnETAie;lC z?}#Bt&7A}><^Ru&soVS3#b6pKlvDozedFg(iBcbWTS`&nibez*rSLn*ZgSRVGz;J7`q1 z>vOcd6`2$#X6=uP=sCl`eg)6(f;==!C@S_lD6LUEWGFn1g5QZm^V2|q(huJDRo}X} zlO-6~prCQ_#@L{5|NBSZ8SSIQ6xQcq6$_%_#6S7b{N45ea))cU80^ez9NMz>E9)`(l=yDQ8FTsRDSnD-CUbXP67&Ezz7_a;ot@+Es$;^j;|Zk5xgC z$S|?Qu^cg<9HOG*9*y8c$oSn6)5$0t4j6>9#plZRWSJT$lJ1^p=6uUUkrXho(DkHI z{<6hu1y#3&JM18m8mf~^t`HZPCj+*YpoC^NVN@;zrF#;2Bwe+ev8jJ68EQ`pTEEQb27h&F`0n#e4O8B z=!QqY@@7qMs`HJ_ymk&p^25T}n788P7ZOhX;A^o6KqtQ+3tj96@*I>fko0q@7Eh53 zy)wWWBPm@oCC^qM3QN_4eSV;Wk&{g)yrffQcrwHqnP%ZYy#dRCNwpgn1Fw1c?dqau zb#-;$^i$j}l9bx3uLW>Tj)1Q?!ypEw&aQW*Rx4*qeg!lomUj#ji>*s787T@Ulah(2 ztnHDi!2N^iR6fA$Iiq}VTW3RhP{Zd{t$E!>NPW4#?6%3*LinOH4quUzo68&0MBCVO z_Hd~QM!OV2QqW}B&;1^%kk#TxD zRj51!HX_Xl0a>`ovE1+%K&nXNNotCCi>1rl`&#lUDg>ausFI+x-%KmKP^;`M75v#- z>We7nle|{ii>VUL*?q-9KREF%xSKUGnDmy(RPK)T%ucS9d3*v%kjP3(b7h?so^2Br z_T&x%7y_Aa3l{y_n#f=tV);Qo_UjL(KO8<7bh^PIWV$)2)eb+K0iE!cd9B4F{c0t< z*ZparD+O7BCf9fkg(93iO?)czj}ZcM?Z!&8S(eva05R|+oV#f!$GhhS+G3CQ0=^TcH zlc7>=EysBS6;&m}W=3J*E;U7}!#R|!tx|h}D}s|rR=?V&C{jMh@oZ__q9J;Az(2T_ zU>H^Z^E$a~nbMOu;8cbGzo75WObmdl@FXyi_y(5lAl=ds|% z>sZ~de}hv>0kz3B>(3^qriUK46%Gd~$sf!N9V83}`QWz(85JO!%{{;ke<5P~VG61w zP^v7nW%PGp7*F_s^!>L`nX!nGK?%?^m=?D#A4K3fmcs#ZM&Dg{GA@V<=8g^Dzkfg1 zU`IQW!c_;JyxjsD%@^k2+Q)`%WK7l_%0nXVelKt-_;9_Kmtj0tPDx3*@}YRI1r~?Y zf@Z2_AXW=OJLW<;$ZK|WjIKMh%PLe}Pd2mvRL0Hh z4U-nU?Qbq*nW^zh1&e3^z*${mXJ>c_1{R=IzD_F@ji@&Ijbz<-U$3vt;23L9*=$MzW0!5H+WFk#4~GC0V|L8w=$vRAOepFW|2qqIL-nBM3t&Vk{X1loA;Ii$4;49PBnQ8IWiaOia} z@=Lp5r`qe=#s2iOZ|b;RDzgBE$DH$(Ajyhz7qIdXKW>hw;{gWGTlDo7$k?6p=|&F# znUkF%TE1{wU{g989j-w^2k>U%FPPoFN2>})W1azg5QH^=kPlZ{QZA1cHMJ5ifZl|L z5m2Fzau4x(4kSUCf(W*c64E^nej_)D<9q>Kdp&qUH(R)ttv81E4-d?x!>*{2ol9Oq zV1V^Nav)GF1(ix&34>XeuzJygLK95?=&0d*rNNTxK?woj&TBjhe-va2k1Z+Z9inG% z&q6~LP?%fl`TF{v`}zjf@T}QzBpG1fzJ*RLKe%jUB6y_EQf=NhG9hxX8+x91zckD? zhMWJHKzGOT6fy(}2(~rwC?rmyz7u~yMeun7egWnDGXClrpH0^O75Dveo0+ zevZs&E(}isr5FVZhTy{#*f?|q+%WW%X#**g@`viA+X|%h^ChJF$-9%@Uw=Zr%gD(& z09UoQ+PZtVLJ+uTqn#}a{BHihnEnPNu{9qSg1dQ$k)j246JcE}v!>*W4-itBl8(&* zUXQ}#i9a73u2$S!u9lrp(9jqiH8VUeiD2rMY-I|jOg}(Ya9g;KmTl3Rp%~jn!`>D8 z+GT<@s-{QyMhfRui$$^q;nrJO2X;*YodWp+cZ9lM#>%i^xt3#{%+Iz4o(V*LemHp=}uE+~_M zl0Yab_0&F2xBA`X7cY6HRz|NuzS!8`g$|sc{fWKhkNLSS*EH@PR@pQ7#yNR|Ap9d*_}|- z=-dTmqFC+l@q66b16D3sV>->mFnj^t{Pk#o%`j~57dHvey;-Di_bj{DDU27iOqU+n zrV8VnoL5^tiZyCN0FT#DOFdBX=H9I6qYxl99C`L^5dhvW7x&eY&1HLcJJxT)6_PoE zJA9Fpev2Z8GT33FVj0aHoy=&0CIga&>(JirDuBDh6>+{irO3>oStYLSO?uv+esqMa z$N@EiiD^`TPB_(}2j4I)6^ZOQ%)Ns}_$SMxx@(Pf<^@q~+CCc3l2}*^Nej z>zLO7lTr;Fud_S!_8GRYFEn#lusMz6GrXcuv5I7 z=1e?Xh*Hw^S333QiY0N>E_f8B1CXu=6Wco3Nc8zd#In09klp3fb`*_~?_r|on{Wjm|9vDH&*LzUbHVQ~)VK5Xen@mbNIE({I z>9lO%_ztgZ$LThrKIqTKg41qIzcx-Z2IWDgSzzdUGbO|$9|B!E@F=MF7R*MJ_v@D6 z=;^_!6DYBEOJwKao^6QluW*no)C@Vz0$M-U<|PF9Y*jw@6{rT8MDTwfWo&!$1W_rJ zURcZb>DHX}FWqZee|5$R$ znBESGNu9@ikVUyfxV%25ABYC1&;!amQRWfs@uDY=$xz5h<6wI`@w--hxkPL){z#-PQr7SpsP`GR+ z+0>vvqKMbCZ6JgXCI(atMt1#tTCA!!cfy)i5b~PQ#z^;ibV=qg=>*b|-j|>0A7K>< z4jmOi@FeA1yV)Iky4iSV&tNrws(w$Dz`Uri#%BjTg21v_EO20u@^&Y_(`PCiWjDdd z4&#P{U;h&^>@Rz9kkUigvsr+o$Z2jiJ8`}}hOB*#4jR}bC09IT5?Bmw&bEis1vF>f zV%ZR-)C<*dTW@-`Dlky7S+Qi_6Yy>NFb(QQlTT6K>B zx{PAQcZpEcb1hIpR%sJ-zoE82Hov3--A(%;*?996xZ+s!#wIC>6?*W$(i0`7iH3%W ztJx3dDtz}z3b4f1Qw;s1{0jbzx`X&?D4k!p@q%s&;tEeTiyD8+`$F$IH-Vhpm3>|DFZlgJs);BX}MQDwgVa#T)laqL*YZ$-T@SJ}7oRCk^tVA#1l< zIiZQQ3F#^KZTR|~WcMrTw{K6@d@=u487Uk$?2?5toC_w4JK?M9n9$5LIR(bh>WybY zpU7>{IL#5T#orI8By7~0o6Y<^8Q}Svb|0*sfBw?O#^xn4XV~Mz9lLN|Zvs7H5MyvS zC#CJA+}b8QJd%cXRBYroAAAb=$`W;%u8OLB-n~C?>lH?0WC57uqaIhweHKQ{sdPCC zup6dXe?}armYc*x;lf_&IY=nNabRwctZo+>-*!?ljNVqaS01+ApP_*h`PLnT&AT{A z1r`g+rR63uNouRL=%9R7jo_;ZVteMWEL`sBMx`A^~s;~0|CrCfPIRi`fk)SNT^gM`lJ_bYLs#TQTLKVr10EL<2gMd9LfT|*_K z(gs}JIp)k=t_%zfad2_FO0}Duff}B#)llglM-cHp6nO&NLnmVoVt0H4h;F;EIb{s?}t6*^!i<~bo zNTlrTIo`c{w~LNR&j+@Jw#7Bc!OQi6%DD%%>gA^(o!BLRxNNMwy=w)p1jA6Typ4OS zwPMNzAo{Zbe30&bCSA&Nupv^B?n-f-t0x9JsYpCM4ebiGn~F=NLZr>Ur){^xc1j9< zJP#K>gbI!jl1WQT^WSX6Fljf&f{MNK4`*%?jN`@wmZBlLcz_%TT(v;g5gfd<#{qLa zRNi=%to6DRc=@X-_?tkBMMqOi&wwG~bDFghr?oJbxueB8yWKP=MWEmq4kwC?R9)Vj z-~p-qkL0h{F63n(f@oSc_=EpU084+?DIrjxAH z&7OaV$$vP1X&xaM@1!a(kMRrvw)&&OUcCd&iIVHPyGD8yIcCR84fY2!%#cm++9|~A zDP?;oVe(n~M-zPnafbiyu<9)h%?_!Itn3d^r+8))%$l`$x345>c-*h8L2BxZrj{`V zy7$jb0NTf#nl&%4Fm8fpT?`MW<0Of;_A?v-UXtq7=S${i*=dH){ z^Ybr>-ur={R0$KO@j7HF<;izP&NRwS3sneDw(-}ayTU%S@Ak3tDT&lmoBpc3>2t5;!{9>_ilnaTBjs_Bz?<*ls z_ts4yYTtyb{*RIf+!*L~lfFHFXJmr@3Z_W4tJ(yybOYbaWtqRfK*>)wT`or13+MfX zlJPhNq!d(*&Nf0>afpZpfB~7f&uFL~v%GoZ3sesH@jo9i@5eO4={PuI!A`kTR=QnT z0+8Md=JBk#&(+(!Q4o6IfQ5yXP2ogmvm*zj{m=KF5SXupUxVqDa>qg8LG*yGSk7C! z$#J5|$?EpvAb~Z}@o1sPqHej@8y5a7oR)?Lhk!sO(*j-3B^ji&WnQ~ll~96QKwY|0 z;JPn2UzlDJOWWIuiK@6~5Rw6#{{i}0?{|fs^gx+PHkpHdH8h`AN?tw}7-RF}?ZNXG zhvib@*4BN|yE^mv=?&7xn zr!}O7&81R3a-wTxz2Qw+B0P_M(Araq*k;L|pPW`S;q&bV;33c%O{K0Lq0>$Vmw`l|4p7M|7!tGYisNE-QDv5 zuv-@#w%eYmd>nCUX>xH4%87hVmMf?Fec@?UBbrt%_`15fAOiH3>vh3Ym3=xsje7I( zEaT~(7IN*h>hTc}8#I0e(0kc*uBcY^B3A({a_x8Ew?nm(Lo-TpcjUK`9Djj$zGYyb zr`hVWuOyT+di3VRG@Wsnl?VAxo8110K_FOI+7#Q#i2q18Mp<&6^E+l~M<>ggbS$P&u z#sift$y(rdJvaVY>+9sxxdGuc!u<#l2?@I5 z2HED@+1*vm?6b*bhwwpHX#oYeyGo)|`kEpkAbD)lw483mTnvb5*?7$u9>q$QjsEqe z<|Ogn47z0lV!9Vu;n#}B_hq=UY#-^bF~*tGxJBB1s5zW>i5Udg2z1(z%sD}hO=35~ z0D)~YobPxI=!q;dStX?u^z^|GcUQaq;+A8bV`C8-HKyB$62MaPL9LZif{C`;WHPsz z@=qYSXZ?8SL&$9#;ieYsR&mz;S8wN-rIr5npKK=4Jo;w+Sg*K7_XnFpXor30{!Ugf z>x$Iy6y-0UTHd#M;QIJVfBN)&rP+l^t3DENu5{oRY^&3grEdiU(pucEb`HLD_w~g@ zM~i|W`uylGU%qJBjeLk@)lCN#TkX$xK-%2;#kE40v#;$7*G-Iw)CC%U>0>MOL1poT zoCgQA6#9k57Je2jVd-$eKW3u4D<1E7u#GF3JRQ`CND3e~IiK1~rRO7`jyM5RDop?26)#>QF ziR??@K|Ek98K7i+^SBFcmxt>o>s@|&L}VUB|<<7TSy- zleu!|`_rM|K+C_;e+7fg1a!1{Naz=)iJ2#Q(INmWjp)y=^s&s$ z=udl-D4_AZQls|mLq%VI>pnx#5Q7qR3lNeY&u&7wGwN}y>fX*dL2@_xbUzyn7>eHA zhi*RHK`$0+p3zZQ*h_Fk&z6eHG-C4(=;+s=C4$F#4dyurKRrH>a9eu=c>`FF1W+^! zbXs_8%w}`J?<~ZPOU#jAe5`o!UBCcSx5nw`5xB%stPa}ku!vAVN371E3(ey4AkvGvm1lYs8(Z&KQS?J zezceXdK>hs=2IU%@tj2Y7#IrsQI0lF{4#``j1|PxY%PEP-Z?A=h6L`26n}89PhxaA ze3$>ZV;^-ndZ;)!F52fIv@D-fpJ&>aIMNP6r=pYgEPhFAo$?5{o&#PYNT(8%zBi0= zv=kW>5wQz$y1%^f-OWwI;a)2s*Fku?pvkIM55D>GwU%vgL+)0(TP7%^6xlKUMEKq( zkHIs8XMIBPxLTUCkg6>H^_QX2SYJ|w8R3DBd(?AV#xwN+T>;J8EukCM+@MC!z3b=r z^(&sLwgW!cOGaB!&Afp|TErl&dd4yiK1wp~Zvl0djVPdXNaSx_*znUl{9an~{#P0o8wcc`nUA*@OU|3W!id;&`*||h zX{){Nj<_($1&2Xcwc#cKwX;xwqjuzs0T@w+;^{n>po?^twR#@AaZjdIPe%C+QO zPWKiGE$c;6G|~Wc+I<=6Ff@aGtw*> zknq^wS~#NqIfrHS*61@-jfzWWKA8to5EQ)Q&f;H7%gE@1^af&6mEEp92pVGFzf%Fw zH&v>g0xatb0EW*hAh7crV13UbEVP7!aRpq2Asn*|JSgL;^FwgH()N)I_lu$Ztt(W& ziNU<076cvfZGJkt3rjJJPm<`zWOd^zm{po(K(XYLs$QdZ*lKQAC3t2r{8pb$DGY(JpfT4O&Z-_pT2y> z_UxSLE;~eqW9a}MTOea#7>=g$gpmuSgTfyc35l#fDt^9N1r;%;c_w%R3BPkBNK^n{ zlLFQWBW!PPzv5W1Aut6W_1A8I`egMfc`;XR2d;-jd)Q;SQRM1)XH!uvYKk|}7e;r-u)wmOovE3!ZH9!njyZQ+W)MY%sLYKirB_|6?V zf(;r3n($n;2`(`D&1ADuN|4XNsXuQu5GDZT7*69$uv~6j-`wmn%JAqNP2+RfFUY$F zTMzSZp~h@w)5y>eh%-ISh%d~h3wr>r?Ot|T)sEJsvavzMq9xWIu$0Dd_CgRq|U;s?WaCmIIQHVJqCQiWBO;~ov~bkR^uhIz-> zYDW3l^bPv~OhHo8%e~1!&`nceJVEs=BCfQ!-8|kwh~EE@`Z;OScDv6==6S96_;_bk z+Inkxd%o8NaNMhR-w~<|N4^7r@+~cGOk`vv>(irKI}_#OL79Z^ELFNR zES15KA-h{moiXem$s{!Vw#Nl?`wu2K*?D43j^ni@4WRMV`5&(&x!3(WyM&RP-kcBr zZa$B>-am$b6_)<6m##h3$2Mkm7jshML+xUR6uvo9~PBi11ty-xtFf zLK%KQi9f1aMaIN58i=MwKtMR@##M++O&tbW7vLuRfRejJtXp#fN{R#`A_brppi%4u zP>#Q~7MixUwy!vhvB2?UXJ>CctO7fe3?~@?G~wnRItHa#!30jyF$?!IoCu8(o(rH-vA%;xVxkV za|OiS8nmo|ly&9kwQIfM^nM*6bW<$#8(a>H(90E`--qkORc{^p;iq}y|^^8nO;wifGk;qwlJUo=DX(v1O)6{Fo7VHaHR$uy3ew1GDn6m zS3YeJP>e(l6GeK@iEexoW~Q25&m(Z@*CqD(e(hpyN2&Hs^?IYjNQr!!*CQ#uL41DM zR}}fOlCP#_dZc`YmLDU)t=c`2qeY{~(Fdmn+v-o!dP-+Pv)%w~M-TxqxdE)F%p{I*zY;r?(j`&{#;{;MPX+09@QC^s9P3@U!% zq$5YN0|Pg@#aG844xC82zv+xekbR)UC6ia{3BkgKU0*j0lw><_oj1pNg)AcCmz%0% zDZu?o5!_fFVd8cE?@nVV4>IU7nz*N-6K%uV}nPp6uPfoB)k^bEsS zy`Iu84rY}=4<^g3fBZ%O7+2ffo#K6Tsy7U`0-?b6IhKm?yMS^MT6sJ^e%tB%FTUWw zAABAof*5TeO6uS`{QLj&I=sMyM9Al4aK+mhZWXk@H*X`4_9%^Ns({Ph^+(&S6%3=B zuANY2H&ck{;{Uq!ZNc&4Bly?=mgmm}_`h=R{r|XA5AKdqJ>M3h+RYW1m`-0{^!e{5 zI%Mg&&-4GnmHYqW(f{K*KWGM*!Uo^^8vqgpm+wO||C&nD*Dam{{hxXJ5K-r diff --git a/docs/images/chapters/circles/222d374252584ac37d967e3ea0f8f28b.svg b/docs/images/chapters/circles/222d374252584ac37d967e3ea0f8f28b.svg deleted file mode 100644 index 44228cc7..00000000 --- a/docs/images/chapters/circles/222d374252584ac37d967e3ea0f8f28b.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/22bda43c696b48e49a2aa83f0c4a4a0e.png b/docs/images/chapters/circles/22bda43c696b48e49a2aa83f0c4a4a0e.png deleted file mode 100644 index 07525b85f9d9512afff3ab0d6e228fbf8a9a4391..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 12798 zcmd73bx@RV{5A?GAdA#eisVX)fRuo=F0yp9D4~dgbi>jp5(|>j-Jo>G(v8v$OD;+; z-FY6q@67qr*Fb?>{ouB&?*L6Kebyay15FH2y2Zuxf2GzvD!Hu~6caH$L z;@kYZ5BNp!K}jBpb94KX-kA3T2ZtF)0V=DFNXE{&B@g_&?%td98gv+xBsP7<^^H8I ztl$}>D991w5CzFr{j(lTDV;Ln{Wj%Aj#g~8YEqUC+mk0IL~KeLEMYt!&1m{&o283i z$3C<jflU(_NcIr3l>$_@do^Lua%#x}|23Bg&D%>BHX6c(Kp+mB_(4HI-=MdD zvuOAR1;Joc%n%3|{68*BxVgD~v0L~C!Q4f;VDZvl`h#M*DUhmrK-aQ!SE0sA=i_Keqb6>FodfvgELwVn*s3q&qit4VY}q zOUYs3)GqPoD#pnF{oMY~yZayOb_Av6@^{Y+p-*zpw3G;igN~q+2)dmqA%C|2dxbI9 zw9J~tAY$W9-3H5vmg}ohXYRkNNzLe{xy!^j!m~5hooSQnZeG&6EAO5j z-|zP&h=^9MFqq#b2NY#6T*yF~8CjtbiS&)Ii|AlZ7kx7XrpXP^;rJ3+>r^?zTQj>_$)vXgn+17Iwc$_@g6Z-;q;7nD9 zYD)Wprl!^9X3)13Jm`nPV{9pc|EZycw_(Er-D067^-EU6%Z`*35@8FOf4^6-H3fll zvQNR=(+v-O|MvAQ$5s{~8m!Tz0U*oCM4XEy!qZcegNyNkjFE8f*N!(!o|%y57Eo(?^TzI9j9u45yZzqk0!h&vxkh)p z0Wr4b!2ps{&$=^%rZWN~ccP|?fx8B}G(zStJ%6yZBpEekBAo}ba3b^ z*_@Cq&>bpgyHEf5<#Od{%cJ%SX;rb#2IsBQ3lllHQVZ&*Z|^R6`t2l`y>G$Ee?vJ_ z`K8)A!`brl)R1xo5{io-G*x9rrkrMzi;%h;l8CIfwoDbWttU!+?thBdcirxw?ef~z z=-w2D^Bir+hBI!>)%^(W?Jc#-AR76pwUp|?EXBx+y3bqJaW`T@s{?``6kQ_IQuszW z#p2~M+Ezu+Y2$9qf=#MSCk*CWG5Flp=Pbl+tDC96dG27ay=wC1!3w)FZ{wJ<4>I~O zeh}1%8OL$8tpoYYJZc*2w4U?KQ?$~OP$~@zGU;B=ORCYg;YYQZ4hYH4qH5|AQ5ewL z_@8fKx%I4VRLtOqgrqHhdv_OZ+M>CJ(s2*B%yUgF#!3xS1X|bx!nD;aEup)T znXxe{YVlXW`>y*Hr`vliZmS91G*5wvcQE4z-8H-MMx1@Bus*EcOm{TXs~})>gX-Iw zXk4A|u4S1T{vA=u@bNjQ@;du;P(?%Z6%QI_J;;byc_^57eR;5Z(wv0KDXHt4&0{X( z;UD(oq2&=sVodcIpr<^Tbu4u)!EUhr=90<&=;mf>`FXB>x0mCYF;4k zP%=6DO2C!(I|DJ3*T<$~HOZq3`xA(~x|);Nce4{4&#c#xqNeu=XR03iOg#%WsAt|j zPF=JA6SwT~mjYupki8hIKP7^AhTFX0@&+s?@kNH2yifc^PGCtF_jIvJQvU9U(#o%_ zSA-Q;)%WF1pcku;`VdJz;scU`LCdo#yJxW$dmj+HvZokg$JxH3nsUd5rnR4H%Rf2l z^$xdD*{L4I3DxnU!wu^OY+MtyWoM~~E9cY2JD+FczlBRz8^%C!89rf59aj^@;~I?| z-g!oj9SqhOMnWsX82Zr-Zl^Z&MAuigh;#GLBONZEJ|K%_Wjf}}P}>rXWw?Q_{y;uBs_YF0TxqBvz{v-5jq`RB8VvU3x&UKiKB&a3{7JaF$n z8NhU|5OlJuyq^_y+%K;U$Ss}41TwuJVs<1L8T*;&60w{Amil%0^CCAwvVfaQ!mjVb z{W-iD+~j%OoXeZ5v3)+Kt$?oTp#u7j&&OGfUbLn`2x=fXF#6E*&e9EZPZ1 z&(ubBzzU4vXN(ue`UjGe`l7Z+&fb#d;MgbbGzm(I=@vd)eRQ;Cj)K@1NMH4Ol6l`qR@#vKi9;8|+LUoI zRQ~>UJB@bNXDcxa*$NSn#;7OusNXS5DX#C=)GJ6&07Dc>7yXLYL8M#!%>}ER7Kl&m zG|n=dXQt-3yz#d;*YRJyb2WF=Gj-pN`t{NNL%_yc^U@Mk9U;T{F4m)nLUXN(mee06 zCM!4r5?X#cn(g8ZTN2hpMo6Aji}PMHtAk8EY)cy|~5{*pZ+u(z+~26H8yA`pKmKsL<`ybVKJjM4EK6{R0K zuTGK#19DaUP5a-ks-mfp-};-Rymr&Po?kAJtuW>qw_C-r%BuT9QJ0>M@@HM4fTOM3 zqP5SA<>`NZXfyYITHxR-7yqz}j>G*8Bx9$9eod6axDM5LzU+3s#>$ixmx#>{(k6sM zg!aZ47jeAJb`k@>_pMB);9Hz~ru-(N0G@OdmMUr75tKzLTWOR=+@+=F6tYZBlX$(E z=4JXkBzf^lh5zKNYYp)h8vc60hYX8u_r+a7fK$@BZ76BYX8s*xExl>HPg`B&H1UeQ zK$HB5`(wII!du7T=;GHM#Tl5qq60WB%3F)^OJ5VJh9q_!rdYhx`|d)Qpfo)RurFlXf`cGwVdG#a|In#OK`! z{h^XHv7eP*9_aBPXdWp;A8}zpNkGj)wP9#F*V7BBQyiNHig_QVw^ikap)jXE0|BlJ z!XotSY_h8Z)!YukhF_)*mjX@m#w*QpL(+rj-z7a?@t+i)jQ4jgdyV_y&uUsyXDjf^ z6$O+Pv6^gBJX6w0({wrFB3ztF-yvxV>;g z{5S)=nt2Np9fcX4K!-B%J8!Wqv>A#H4%p0FW}K^Yik_ZlT5t$hOidD>nj6!6<${i> z{ffsrvW+6sEn&Ca(%aog(`U`ZU>2osSZn~-cxsFu7fqdzftX{Uh&{P@!9d?cS!8th zn6|?^kI+gb-YT*N0#4Su4t zm|(V;Xvr+S;3CYgfC~=3*byl)c;Q-YhzyFB3um-jN!T5F1+=V?*!Du>osTuIm#{M% z=Ki3hnGw0mjt%Mi==62aU+C6?tqs(DS2ka_|8}&b9BEbqJ#-QeK*ZK?N-Zj8jt_^Q zVXDu&S%QKbXYKsXoM%N*&K(c%VWri!B5+0v6>f}#z5R;WMbkxJ$_AK9l9FI^$45kj zF~uzxs|aNU2iF{K%txe~Q%F7l%*bP>^<)8E42440ffqtLWc|`L9;3&hl7?UIE?1)$ zQ$1u-)}hHfl5>5}XI+x8iXXn>{YA`s4E7-{mJB)=Y-(|zTMLk2{>G0{Ug;h0a1^OL z+Gq6merGk!7DdNpwt?I>+8fP{bQ+Z0>h0BV9X3%{J2VMBSxI_k4pvU0N!`o5Iv7U; zYN$EIg}VZQDgMXxRaQk4=L4Xg%h?5L7W^uo%XSwo$cOfJvZP1%g8`+1WvtkyJIIcc^4NXZ-^SE z@H!OJBDN}{8w)H)M=lLdx(>JCV-|T0UwZQ$qX%M9U&VJSflY(@p?28!-rZGZ^>Xy5;^U5g8s zh8%ED;YYf+mt%fnP>(GrN!Tw@7znmjabO)P=0Nv-1>kA4P7q9}9(e209<2kV#)p;h zH9ZUYxZ%hPVOyUdaoGuYQ(*-M3sW(7z7G4=e2Fi?h>ihX*+(o+==Ur8_T?pmA%yKV{fw=}Cx%@gq)kiOreZY}pHe)V;7*%boLxPQJpgXSJ%Qp2G7)VXm$C>e|pwor3f?6>9}0^j(fSR1+P@ zp^7~06QE}x6-rey_??2Qsr|8OtG1^^8BEJVr3gk-AlPsqGGirK}2Lk?; ztHQ!XbepjwskiS(ASwH~2`M7W>oFilYRt=R;s}`*A^-K^c3_n;cO~6h(6;E}HqZ%v zPZZlqQ@G&Zt<{Li{$MvXbedC}Ucz)+47YONe2RJOi<8|WFM)%G~2iH^+71SzoTj_;%F-rB?VDj zG?nNoeLO!fgnz#caWmHecqCeh&)G1X+!eovBdvukZmRetGHUllS|^(O~?(Od@r?JG@srmLL;#;Fw4$ zYHS1!&)fJ_dRjXUqokO2nC;W!>Y zS-;(smhO;s0Lte5WyjkIOqKJeYxU00+=O|gj!S(}W4>ROp0! z|NXIsQ`irXMgaJSG?RefZBVdr*zYx^)wL{` z*f2FrS-+X_7ZBfp8EYJEtcdlrCNnVWSIfjaxiu$|Uz(i~gktd(cRp_F0!Z4FMozFB zsl&5Xf}SR?h&V!Ht8xGht2}ga6g>EH2M9Y4fIHRJE5!LGvvM5-C6f)lb4_zDY!u5= z|CM<^W=oh*s{>Vr!5y4O(719%%kkjGk|xFB^*l5$8zdP`*_e*=OPA!#q+@oqV6ROo z6#$x{FnB6X?KBc zk>btuRXE*39d5ZT)2ZiwBT`Zp0!DPY|A%jV4t3Hh-F$O>@tsjJady_A-Ua*nU{Km; zcZbpIP(!y+&wlK|7a3m}=9Vx9amC9AKiE|;^By~&mfHL)ZRXAwfdqCcK}zMRcRND%^V}vtB22Le?|Ut$u_|^p2Co zh7G5fkV@NyRDP4LQpeSScE-Vh0jaIMtt~6X8LNM@I9#0S0K{t8U5ke+da+eOc2Uv2 z%pxU~>N@WhXxtuvJ5+0LvN2gQI8|ms=CL5dEF&w6o91==u(!9@X?-MTEL_m*-0A#e z$4BoTbs!voVM#p}q8M%G>Yb6xFGOT+F-DsHRLP_j+eW#AKqk}Ut!cKV!3MVj(hEm> zPABY4j69>;UEg=!XZ}pyr%Q|D?HQZQNF>r}YbugV>eTm6&TJ`SHO1AT6`yt};}uXa zKXlyPOHQW8F&#*Do4SZQJWUpJRHR2w<7RS`Qr_-2G_756)tUa*;gsD4G02A~M?&MDdvP(Q@_=)epc16Z*V+%EI!6BNg#= zzR@G@T&i=WKdF1B$`Y_nk&BZZ4&bK9Ji-syT?YpTMb8GE0^PBacWIMG9@x?{jNdga zg?$FwPo*HG7w0i^k)*8j%u-UG(yDZP)!l!NRv1sb*p=gD+5I(va{YRzJGGutL99 zj7VFs@8*6SO6##~J>QUU*x`g{-rWov{PTx+Q^okU$BWJUW(O^F5~+vPSiT08^a72~ z&A)&D{BxRKHK&CqW1egxWPUooa3x{oFB`);LR$A}JG)Zaz=i`Bjx@_OVOg>%%5TRC zd)xep%|~-XvE^+8by|AeQ$9T4r#>xeyxnPGhUPV z2n@f#-XMsN#oa~nEADiB=^j%$+b0CD$QOgxcfy*O`s;8rxtS~ru9rt6$iLmF(4A3cJs)+N;M;`#Bh1oM>5L&0^oA;ai#lcD&5PMnJ28XWzyf6?)KSDgE zw=_F=KV6ZUCtFF6^@kPeE>hUzBr0CJ)rFek-xkhKgEzPk;|zZ%F=LMt%mJ6o2Y4u5 zsY?;|Ne35`nB5r@%68~nPLbgnTc5?u_aRuaPx$_OOI0# z=ufbxdNpBxM+eFca`g5Uvbi5Zdpp;c zxRBOepPLG|ZF|vGLYOX%=LrK9%}gurDL=5`|r>0O#& z0uZ?v0D;0n)I?A+3#I~sAx6P?z7DrZa4c=J_W|gG2Hviip!Y;AW zkt3{yZ$p}8wa}lUiMx-X(%P4+Y0Dmm8po;$A5c_CmQKlGZf((qy!Mm5g^E9@Kd%+Q zcYe(s-96&14ZSlxU9E%^RbwWf6=UL`Yj-`d5&cIsYVd@=79P%jb!O2}`n~3?hVs6E zpdgutCmZ*h!Si;;kYL7Hy1{G2#fjsF|D=Y;ZvhyuSN$lFG}B@%S7D_Pz@9Xnb=|J} zTf#`qqBpNBIybuSlX2L~`{Gnu6IYl{-&%xtS+-CrHTO^L<`}HN?@$lCY?VgUbt^EBhuCIOJ0bucPN+iG~&<>76*JMxoq&4`@n4z?reaZcZP z>}iNNnm5Y;$eno`$eYeV$6IBF$c|cVP3t)3VOGBI-|wFKlQLMrHH&AdnYp+Cg2r>t zAFIgPb+)(MYJ$3tr1=x4b(Ss2@?C_l3(pn*-e)?cwzd#j>23G&h>^3Vv#_!1xGCPO z**ZQM-lhk+s#UjUM>CS6*EM#wQrG!umX`U=8+CP@1s_`!7-|0&MvF61^w`7%H`zV_ zNQ_f1@H&>iaF=`&nFmUW4pICd9K`Cto{Zo3%tAOa6ZI=3R(yDeUvJe{GR&R zTmr(w$Q%Xco_)w%^j21b!tSmH z*}HaNSHgQO>9WsEzhj@U2CRFNpZq*Lc8WT`UuqcO42n=qc|XcNM+srk@~!Q=xt6c8 z^|Ajq4VTyS?dWW+nry2D@ZZP>IYkZIQKFK8PiT(`N>Zhc_;3IK^9^Bde-*GJ%!XfT zVY0X!jZzGQRaZcfO-;p&0~EXr;ubv6mzg;BWIMEiKfM*X^L73J<-9Derd1C)VuS0p zx@QBMfMLv1RaWLkIt@pFKVv0eBdjFFAc$$yy?k;l z$yA>r@&2m2&f=;`7^6T)SrB<3@m-kDzdM=)3pS{z{7f9Nu}^Uc{)c=`w>I;@M<#Fo zMZ!s5T7DB`(CQ-255Nt8%73NWP_G9Eha&G*3y`$5o(Gg8{OoM${!Sc9K0X8w;Rd#- z@A+9cVk5vxB#U);ouu5_(Hf|ub)qn1;;-CLUN}}UGYs!9thX0n_Y_VRZcUiQz9SME zg%x1M8)0uy*+{bqc*Z!eMg@w9GVqa{&rveoHbBR~?hiAplOB$$C7-W{Z*8I{0Te_5 zR}-8T{*~p``;L46Jwf6>@ZN46#Jhn)ByK= zKSu2>mKg?o`?8wkKh{5e@9l*sJNyJn_WCmJ9j5sojf>9_HLH1CXwuzL$}3n1m?m{mmIekZ+2H|#up{rguciuLZffp>B(?V9i?X1GS=W!LjzdT5L=D=z-^&VJ0hqr`C zd;J>Ct9Q0b9xyWu{wrMY_j>1kci6*XnGOajfV#8CpvJ7`h&RT{3>B5tDgEA)Fp-ix zY7<$IIasMKh~$h>{DlRxY8Sw@TwjvzX*HJ_D)a%MRq4oRf9!mc&DTr_dK?z52tkTg zVg*;Is3ZCJ*?!I2iF~u?XLa!nwlQ(w>VP^Z6>M(1=; zsTil^4`*fQaGx$SWf(~UjZrDG_)7wwGvKb}@@rME{pQndA5&CRd=_^ErA`6O{YR$7 zufL*kd3dw@46qE71?poReOBVYkuzM$lGL71)#*|joa zY=8*Pbqj@pN@!EPXtwK#tD~=S`R!@&Vc(~!Lg;P~{*{Dj&!tBl+73%-ey#NPi;4LW z0uW8+_ghyWhm!G)_GwlwqM6H3phS;%JO#j7iObSD@vCVwz}z)npZGyV2F*szcRTqI zJ36bw0-)5G2_9a_eIrRv??za#C{{DFXAd2|jxm8?g?` zE(C~VpUXSN^`LzxP{9LW_cQ?m*&AN{={0T#017~-68JFD`}Frh>DCLMiotNfDd2g} zqyP^~OxU>+A?)y2_LZG7+9P^+ak0`EB zxq@+|reysCXL0N<+6T;5ZIg3;BdOH&WBE3>92X71V@-bH^vw0AD;88&S6x8y$B7!u z1_Hs{*0?sD>K@KiR~d$8amut|0no2$f3pNQ2zt3VNK&N9J9h3G-uYIM8QxC&F-l21 ziMqOMr>&kQ+&fbupChmitk8@@sPvfWj;U^E3CK*r;VanGQ?-Iay7OX&Fe3URioW}Q zVQxV_dU=PtJaKzZbiXP5*sZY*Z}#%|JpQD9V-z79`oH8h`XdOV2CI)Iup>FG@ZER@ z+vV3Al9%l3%egHI95Ln+9G=qhA0$+aDwgB0lOY zA=9ZeshA@H#fF|TB;|xt*%&zvr zpE>`+TCn0d;e%g?&}f(~z8ZTdMR3b#gl=-lWXfI~KXVfRC==UeYZM((1#48$;{lh4 z{DTKaB4ZbS1jjrtNkn%f(EsiM6eeTQ+XVlkffVBtn_6ATq?9(mIe8zKb^{LLkD%rA zqZ_F&7FqNBgfWT$G7bG<+OHhsw2o(QZ+A3)OE)X9zrh7V1`DUhn50rI1~1`D6rZS= zhjZTuI;O!^2TI)pXT4F_VAlJUq~OHU@Lc;YKif#DYcGaBjQ+rLYkh)CnGLV?KBtanR}lwtUMzC=FmKYPR1 z2=KW?UvkS7$^b}| z3}z~>*Zo9(Gw`!Hed>eEz)JFo%t?cOoL*qXf9*@Q7HWdZY|yZH`)S!5mFW?PgLLBg@Nu#LUPy zz-XKqUbKwllp5a=vId3FOc%q#TE|EAEGMJ959Wl@Qq*$H52zSP({b#-f~Xh&#VyTy zER84usSry#CBu}wxxPj#dsZ#u_$T<4i1 z`*+476e8>UCf>&#BhA;1(022i$N$bwZi6GW!h>~ChdF@$iaGY#6tSx)S3E(|*?`Cg z8_vcW9PafWk8ynZ9S0&K9H9=PWMfZ6O_ns#5s-e<$m;v@{YC%S3wYs|%YEYBn^YOU zPW)`iqtvhMtxouukD+v}g>Ot13k=08UY8Gciu-2D(A%>v4e=BZk}kUW2#lAYl1znB zk;lgzAhN(w>Y!1#M!Tz{Tu4BbKwHp1AW%{@-9+i!bgA?>XS?*+eJb ze-)Tm9Qf#YOM+R{>4=iL_D7Te9L6^8LX<&#_x*~p1!;XqMRQNqe6uvmQke0IhoXVJ zC9{ogRzMxD{kap^who_NMW8#wX$`Ojc_sFTWQ=$nQ~jwxQMT7vBWiuE+u-9t_3l+xhAA(H^?Z+b3v8%~3!!rHY8QuZ|e-y`g#zc3Knf2^U zY4{`3`50~mp9@Nh=9-AHR1tN684lBr1H0jUSyY{KR3v*B>iTsVJAG%lJLwLcprCON zIgovYUpx^qm0oydH4ms70kp03yKDZtvzVQ;rm<$e6ULPYTHM1XHFi9Q96aAnQ?lZ)SyG#}Hn#%Q4>DutlpRj9P zk@@6~n+Sg15A?_v@ZwLEUdT?-aWHrbja&R=mQ-_g0PvPeeS-aN`jdu)B#&l_d2?AK zztQr^>CL=ow113D8(jHIG%9-bcZGFc5ch3$rB(=#h6N?p>J}}Aiu6gcE_c6*;|27S z>@fEu0XUC?GNT0Rk3PoZ=mo<5!EQkxT%GyzuRm5xvM`^#cE6|DG`T=nh5#&RJ6db8 z7_|s!?>g=|4geg^v-ddmj;p_Zj@w875z=MGk-mcF43}u$TyC06x)Yf>HU3Z(Q`);+ zu=5X$2DDw-TMb6FYmkF#H^GKWfoqmmeH2^(;plVzgVgq^EcDJ_pPR?V$DY3$j_`F{ z&vvVJ0(|yi`I>AFOj-u=xR9kIv}%#0AAyGbmAl~hcmZukr|ix=yRxeEoJ))b%a+-P zt9@c*1+BU>=Kin3KY49(lDiVS1VUCl11aKrf%b7>_W+nHYCwW=yLq9(>q@xc`h@`^ z;|!&m0%-dVR--wmBee&9d*W-}f&<|@n>pIoYup0QX(GJmnRf3pkn|_|=fNUP)O}%R zpNHETB(FuE8r|2*OY~3z*xQLBm3XIZtG9dg-^iHkJ2YU%yEgz@jG*Dmz|byeo{jo{ z{P?_MVf#bvj&9HIV1qJ%M+F+%M<|yZ0~x@8cz(NE-#VOL34W#~@sc6ufQ2!6})0r2B`wMskX zgdt{yCo5`)^+~EduYD1!)iFFF#TKFg;A_Jlc^C-af!Wh_ z!*+H`8U687!lQ>KpOiLkMYv*5-@NqaQR+pbg2!y(5iTui8@)$+leT^Hv3Fn(z4<>c z>1W2{K{NGp6o+z%&!(qOgVU62Y(+LXmEscLEg;(ItI9oU>R$ZRBD9#izB&=Mh=07n z4G%ax?3qAWC%K{PsV9ns;p%`M>s^S?yNs{q7l|>Q3ZY5%BX7Knp(Y1)*DEapDIhYW zJBS=9xxPNs)aWKYmMR%2{pRLU@bBMv_@#bLV$FY==v%T)4f?uk$4pA+*T;#^QBf^B z9Kt6x-VD9H03&yHX8P;r>HX5#SwCY9K$lHvWp3UauHRTsR8By$IHKf3CRC&!^omoa zLrha^sj#9?-~2!Q_Hpr>S2NdF>^S@TeGAqLHr`h}PlLf=Z>E!IEpO?OhSMhig(e#r z*}DDh?vt@%>upyGdm{XxS|tPekEWV`4;Wm7IZtYqXSMu92P6T;w(XTc_e1?RP-yln zmkboj$_gTZ5f{9dDyEcr2MGIB zoq+uQ|36aq{|}Y*|Kp1UfUnM`+CcI;SZyGSfsYQjfD6FK1||Sp_y6vbhW`c^bwe0D Yo=IvY&leAT(1D`>@D0T;-+%qT0Ktc@=l}o! diff --git a/docs/images/chapters/circles/23d4cd81c759c5374773fae149207c5b.svg b/docs/images/chapters/circles/23d4cd81c759c5374773fae149207c5b.svg deleted file mode 100644 index 43d9a205..00000000 --- a/docs/images/chapters/circles/23d4cd81c759c5374773fae149207c5b.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/474c72b5d1ad5c154d79312c15aee47a.svg b/docs/images/chapters/circles/474c72b5d1ad5c154d79312c15aee47a.svg deleted file mode 100644 index 22939061..00000000 --- a/docs/images/chapters/circles/474c72b5d1ad5c154d79312c15aee47a.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/485d678b1d63ed7a4c7c8bb282467f02.svg b/docs/images/chapters/circles/485d678b1d63ed7a4c7c8bb282467f02.svg deleted file mode 100644 index 8706225d..00000000 --- a/docs/images/chapters/circles/485d678b1d63ed7a4c7c8bb282467f02.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/571246ffef745298873a78473257d871.png b/docs/images/chapters/circles/571246ffef745298873a78473257d871.png deleted file mode 100644 index 6e77186ece8df0661fd422ebee69cebbfc291c17..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 16785 zcmcJ%bx_q^7%oamNtZ~YbSg-92~rBuB_iG3AzjklBHi8H9n#$(xq(gRU3@3z%-p~3 z%su0Py4kGXiZ`C;dEd2zKg)kaM}3V70|SFDEhVl90|OiR{Du4iyyD%)ISl@K@m=Pl zILy=YpUjrL7#J7|7-?~lug3(RDYhyj*M?I zZrGNrUSZ~9j=>QccwHd=CEj?9f>QL0ddk52@2~VPlc0B-?_=Jvhv^%(8o3AwjQZZB z+oX*pZ8wiOXs<#X=1DLDeZNb|$h=`=E9cJ8U@}!WYH)Q!K|z5Lmz0c`aw8NyDiClo z9@ANRF*!NOT1ZiocR7{5pyZZ5ty2LnE-t<$i;9JfZ4&|q{+*fygN==yM<*;U-cA4i zdU;M!Nr{x&13M&$R_fCyxJ<1ETLM+8jbwLtc=%9W{YX@rS- zAD4#yG2;7b8~Jd>q+qGi-|!G!9{IMoL02(I*hFm3=~DSo*4O>M79(S0)4=%njy#op zhBfEJ#S;;kQ&Jp09hRMx)OgKJVtrrFlO7G%7v3@P$B$IN6poJRFvHPo%A40A({lN~ zE)t8em6wWI9+v8u_8oSurT@|5$f{;t=kYQPiLGw7oo@YNU0^<4{?~ye`71Jc0{v(< z9!gm6wf=r+d~EFEN`=15Z&9sOl=4+i-!UPsrR(ra@33TjaQUO6W89^*ibR>DambuytNmUnJ zE;&(;6Pqv-2HT99wrf^5g)-e>Xc*0;$#F-*ZjP<~-uC2z`luTP?$ZgsQ-Xe7ijIu$ zi}51DJo(g2cWXBD6>}f^?N83PCMOsEDOLf*RvfU|H+U71aN>jUU3xoADy1C1D-1$W zLM@lSbt6OJ7y>fLt1T@oBLq#uA0C@-7Ua@UeP49ot0uDTElj`k3XY{j42l&Dg!jCL zC1CRQM;n=!*KX>Zs~G%s(*>hb1oIX*0ly@(_w)nwv$BdyN7)fxbY9G+ESi-iRkaE}z9RElO1+sb zNx4aOIxuv|+4#NWOrxmvH(h|ul0;8cOV9NcI&m_}>6{w-%ReV{0w2th%WTY<+@A!d z$}shNh5tw*v|eZN_^V|j&e4+wB8;O*_ZQj$!gkZTc$H% z)u)>ySBB#hiTp&S(>`DRe4Aw{D#GQ;g^5C|)ALQLKAM|IRcSOtcfW|^$wjf5#eani z|Dv61HF3Zx1--PPmt?ViVt>CI9qfuTaKB}z5tBbxOYqB0U5^iJp3U+t4j<8pJN-}$ zb+XM=U%l^kw_5d>buW_$5rL)JjZb|ULzDFGqCjbVfOcg1uhXhUsS?)J4#cW8LFma` z-Zk-k4-+oC0ct9*frG=w1|EfC{nS@GQ{#u*lO94gBaC8gw7S*fm7yT1^+QKH^Z`tv z_Vc}R_lBDlh1BaY+bb%giS&Qb5l36na5M_NPiap5`{O^W-|6?C{zb8|TE$&9+Ledv zjp%c_F#SvJnHYvd){c$$!$EyR)EpzQ>r1M_d8~l@w9a}o26^t%;e76_%~8$H`C_Qr z>#pexE1Lxw?v_-yAMfJruKad2vr9DB{R!8aAj7uo5&WIOVJ~M36pl<3>?_7Mdi-=8 z;)rV1%G;N!OQW|;SC;N&o1X-kv{6S=Hm`1WRaB{2gk_SPtH?dsR9O~}vWHWpjq~JH zRNs~s+Zoy2oqmK1K=*yR6LQ`}H=gL+AMB>u|LOdvD{v%yph3% zutYPa*))zPt;tLkT4nEAuh1L{*PXJpmB`quS?xXkxoWC^*~x5LrNP><>1KYuaat>L zE(>uuQBOsOpzJp#lhx^>mG!EJA$oQkOZyyiJT+mBX{laY*u8=&~O%(3mowD z#&i2Bs)L%@WM;kD5+!G;fu;@A>(h^R5^i^wJ4W&eNY4C-L9}<5ns%JiKj95`h0s-{ zO)BSy$L8XhcZ^Dv8P(cW_6xJ|=!N;eSe9*;Hm&s#j@(Kr7piH;IfoW1kBlt=rz|22 zBhI_1;g5yup1eN`osZFpfa_ zcj^s!UWfNWO)?3c%yr-RJeT%Ex>3OMs`TKsZ2Z-mOmA%2rRg`qsi@YVW9xrqP+E?8 ztC&LS@ezagZr0%I_h_8g5&!mv!IF~~G2%3Ex3;o*^6$No6i#5-ITW!W)<7H5zQ1ZS z{PjDIk&^edd}?_kRTNH)&2Ab(jU~E(Yob|VZ&*d+TlyD>K}9k$?5eTk{KzR>DVA+P zw5HR8sR5n`CFS~oE?rv?D+gs;j0N0f|M_q->L2Yewpm$HN&NFnDa;1Fi&x@xAJj}M; z4yH2M{+#7%ScW=GUUCE2V^E&$d78^Nxt0AJ={3{}2p5>~vX&BOJ)uyf;g(qr9 zQECV^oP;{x7`);h-!~O6{@}_XE}7gir&{d2Gg4A2HJS-yL#^w9_;8n!Q{LoI)%Y5F zmhDH-{LM)K@o*v9(E=DJHJ6e}wZfKq$t8~mQ)itrdUb>=NEi=s>ueq@VALsT>5zzJ2t@ae1PeiWImFmwv6%}FISR5^O zZS?H;bhQirTUp)@X-;;fP|>IzPBp19jhmPFEIwLhp$Y9hIBpv0Zd8+m{T^r4_94bM zlrZH1TRi)Eq556RrHJp^9i*hR*Nk4cxLgnZ{J@9?v8;6gEOSX|v6juNwPDV2Q?jV& z^b=}iWvvVQk}cYBn%SrYA@;7r-uIc|#tRiY*wR>fxvYeodT#a`LR|e(6T+xar-PEn zL}Qw%*;4=g>5|fZGdkhHprdP>e+&4M;T)PSAyNZG9LV7$KS6H!U{-yvKs14+qAsOC z>)Z}ge=6y`g$-7CD*NVRvflclrEl0vFa=c9$iF+*?Q0#}on0g2aY3=^!MMNOU3?_N z${TUYGGXW~#srY=E~9uoc;_y*3Fwst1wrr-1kgu@RbBg6-z15YiAFh$;i#X(fweI^ zT~sCZ>raFCH-5{`Z#XPW1FEyX{$sW=nXg3$nOMt0=TE@&bdYmgYCc_6?NZyLL2X~0 z;%^vwbgw6i(u+h+^yClEM-9OvwcyOzUm*=At7&{4ku5MGqv6O&S+^?hSY?>qa^f>R z-7hho{x6-<_}uug?P;_Cq<@m&*jGovI)9IT5gUz)J)Oy%JQ#%x80;&#^;jynWw}YY zB_&xeJM7@C8f7@N+%DA%fDO*x9vaEM>HT%nDJIUewTlGZNjkz=s9`50{uDe;A|)*? z<&8wfhW6IR3Rdy1?g&~=q3^;8-^YQBXFMj=yzcCpUj}5{l7hjZS2;WH)(O9 zx!`BGw8=Iz6Msrd#l=Sr4%N7A;BsXv8gU{rf8K`H7@#%ZSI%#WsV0ZkbL#4{$4=*> z+y8?ZNzv4f(wt;m>5-E2=Y#mE>io>717{Ysk*_)1yhqH|evc74iAO=Gzr=0)w>zvL zBHSaQ-9+q$iIl}UEyI!VT5K1>sZq(OVgZRc zpE|orN+rcpv9V!eXgZdi#59tpr|YQ^gV-PMbfKd;Zcasv*ioSqMTA3V~B(14T6#SyqLMf3fiT0z6Ic9eh7ScRtxftOKH|JL|WB{>6+ zDePDa2F6G1_Xj5nz_%?j)dtVw1uOI_jA&+N-4KtWCoa+0m24qdr5k#Eow$0mo6fFU zC{yt-uDm|o{jZj@{Q$L!Diu`@MRkWg=JsB4Ymyllu2w@eejE8>Cr1cHmMLpLR_>gF$O41Mo|ltghuDw4s*W^6u>ZfO7f zhC@Qp8?VD_hRJvuRin1L5f8fpq`MUMid==dMx2D4j+V~sY^;-BV(Rzy)Kr#-H@oXI zniUtv=+xZeTl7=#^ut>NKWa?iN0QaGgM-ux3}isaVyB`+q!S)>+{f$gjubSFosJ6% zVszf8zh8A%T3jU%g|)49TKS1V+CQV5PX#w#Nj=Z3**Tn~qPk&=M`1KL>^G~!u9_7R zlp6ndy?L@<6d7mWK+hrpYn$tj_FGS^DnB`>-(-Gb;&T;^`&saGPCXuCf0B+HIsnA` z4C3AR0*~AvWlk%D*QH7!t?V&B7IwA7QSA;&tk2#StLr;Zvxvb)Gt#^u^T0Fb1Sf{A z`lG7jO+;c8Dl8sFZ@NIaCFx5NV+U%gjlttaG?lV|hr5!}ZfZJVm=#ar7wGTr(n5pq zs5z#ZCsH{__CIO)z`bS1-5Gwz7a1IsItGu~UaY=T(=|jdEck$z%<)t5S21w7C6O4K z&t+uAYI+XT@k-vZ8Ko};_<@)sa-(-uy zWKYDbxrZ0HyOs-Ep_X#x~OS40xA+61?CZF7BN^lxtBIqt;R z?;{X%CNf*K1UVu0C+La-AX&|a9Tkm(181=SScjUnbkbQV%)|*uIPs<0EtU~j5Y$N4 zR&Qdf>PA%TQ*o{pblx$)6aeTS4IT8&ILJBqS z>&wHRxU|Kk*|N?by^A%GN;G~(MEtiwP^fP_^N3kE;Yo5xnoHWnbiRSF-O(JATQ*An1+lYgG`r=~0BkGN{EAmu~i>J5w z;)?`r!%|T~Y!dD8!;{k4weaM{cOKWDHC{s%+{Du;(n zZ|1O{$W=i$Tm>&KfQ4Q0DUEmkXZ;5t{;1_xYF`L1E$DiQ{CcgTN=dbnFNqF)z16|b zCk#+Z5b}RR;$iFQ`0^(jA@xuE0U}$4q`038`q{U1VhSRn$YfON`hk!jsS4#{V-~~F z!OF&H#mxp-+rz}07p6o`*cCp@mMiTvG#X&$r*i@m9*)Ovolm~WKc4tb=LDtvP5)3n{1iOB2g|4p2Uf9Yi&Ar&KI=%Qs)Nf?@~e4FQkH8%-jxSOSScCP36Il{yon2ifE6qH*UJnkfV_thX3FT?;^31B{&n)p7 zzS5n)i)YsC#aGq(k~tz4h+(t&i%T)*6Mmz^LGN#|z%x>-xpKYMO}P}#ZRnT~>v-$# z3n|GL&k+jvCA*gwd#X;*)v8wpv1L6aWRu5kr~F$7TsR3&C;J$GSNl(R=<#Bu0h(3I zh00G(5)33DkZ$%x5Z=C0%9UMDb6U~6I$q|#{rAVkBAk>jUWT?8lpMO>U%fP&$h~sja`I*kT{zqdVkV|IaRFA4%zfaBZQ;D zJ9Om3&MVWCQBW{CS?d^1VyD$z^XBPz^Y*RZOo=Ac>*>ntYnhI~>cdbx^U$4BUpOh$ zm4Q*U^vA>P#n%Ewf9yAsQyucvt_I&B>Z(q(dnf?k~K^FgJgSu zZNs2inhHrAG3>IeW4HhJIae;_MSF~b-~g%9Qpd%tZU)58&W$XE%PQA&y70!8$%xPX z9~OAV<_Om+!WyLQDfd$x1I*VFjW|C)KRdFBu&}okk&{sZT*4fF)Kmn#wp++nM@s{m zx*k1IqVXIBpMQ(~+yvWqcxVnD)w|Z=3mX(1tT&wa&IL+tH0nEd>qt%Se0&eoD%pA^400#*6>cafC%h;eHWw z8E)-QhiO&%_5LE>ZAlq(m{wNBCL~O_I(-wtAI)WQWX*jh|7L7-bTs9lzXc*otB}4m zt*Sf1MVVnXQ>=b|I3Ke&nYVlJ8$8lKG4WM&OiXZa@OtAx>5$VsIK!LpzMh^5umHHD zyYz?bkm&Sud1)PMwS^=9w=|-LG*3c0y!u&ov+6jqI?X}|;*9f<%SJ2A)%99J`23^{k-Ib%Gt3?G2 z9v5;#M|6n@GJ$$X>)n|k8QV7)_!lp^jhY%dtS{Su;EzK0qyKSuj7LdI3YpyF{L9bH zsHiCS^NEj$3wPTIy3U6ceW>oOBVXp&8T}Tud3f}E8oZ4<{XhbzDiF9bV+BXH3g_AN zMYbb5xYXf!^kB<`R zg0zZSB0SVrRBu=Qh;E|}-&v5A{7JEu3ne&QYN%=_QcZ8S3oIYCd90DLLFDm~qU_m! z-;M6C{PNG27CI{PZ8QPj0o;5fDqD>GpZz`7KlsY;QoS%nv_U9$UVDF`SnETAie;lC z?}#Bt&7A}><^Ru&soVS3#b6pKlvDozedFg(iBcbWTS`&nibez*rSLn*ZgSRVGz;J7`q1 z>vOcd6`2$#X6=uP=sCl`eg)6(f;==!C@S_lD6LUEWGFn1g5QZm^V2|q(huJDRo}X} zlO-6~prCQ_#@L{5|NBSZ8SSIQ6xQcq6$_%_#6S7b{N45ea))cU80^ez9NMz>E9)`(l=yDQ8FTsRDSnD-CUbXP67&Ezz7_a;ot@+Es$;^j;|Zk5xgC z$S|?Qu^cg<9HOG*9*y8c$oSn6)5$0t4j6>9#plZRWSJT$lJ1^p=6uUUkrXho(DkHI z{<6hu1y#3&JM18m8mf~^t`HZPCj+*YpoC^NVN@;zrF#;2Bwe+ev8jJ68EQ`pTEEQb27h&F`0n#e4O8B z=!QqY@@7qMs`HJ_ymk&p^25T}n788P7ZOhX;A^o6KqtQ+3tj96@*I>fko0q@7Eh53 zy)wWWBPm@oCC^qM3QN_4eSV;Wk&{g)yrffQcrwHqnP%ZYy#dRCNwpgn1Fw1c?dqau zb#-;$^i$j}l9bx3uLW>Tj)1Q?!ypEw&aQW*Rx4*qeg!lomUj#ji>*s787T@Ulah(2 ztnHDi!2N^iR6fA$Iiq}VTW3RhP{Zd{t$E!>NPW4#?6%3*LinOH4quUzo68&0MBCVO z_Hd~QM!OV2QqW}B&;1^%kk#TxD zRj51!HX_Xl0a>`ovE1+%K&nXNNotCCi>1rl`&#lUDg>ausFI+x-%KmKP^;`M75v#- z>We7nle|{ii>VUL*?q-9KREF%xSKUGnDmy(RPK)T%ucS9d3*v%kjP3(b7h?so^2Br z_T&x%7y_Aa3l{y_n#f=tV);Qo_UjL(KO8<7bh^PIWV$)2)eb+K0iE!cd9B4F{c0t< z*ZparD+O7BCf9fkg(93iO?)czj}ZcM?Z!&8S(eva05R|+oV#f!$GhhS+G3CQ0=^TcH zlc7>=EysBS6;&m}W=3J*E;U7}!#R|!tx|h}D}s|rR=?V&C{jMh@oZ__q9J;Az(2T_ zU>H^Z^E$a~nbMOu;8cbGzo75WObmdl@FXyi_y(5lAl=ds|% z>sZ~de}hv>0kz3B>(3^qriUK46%Gd~$sf!N9V83}`QWz(85JO!%{{;ke<5P~VG61w zP^v7nW%PGp7*F_s^!>L`nX!nGK?%?^m=?D#A4K3fmcs#ZM&Dg{GA@V<=8g^Dzkfg1 zU`IQW!c_;JyxjsD%@^k2+Q)`%WK7l_%0nXVelKt-_;9_Kmtj0tPDx3*@}YRI1r~?Y zf@Z2_AXW=OJLW<;$ZK|WjIKMh%PLe}Pd2mvRL0Hh z4U-nU?Qbq*nW^zh1&e3^z*${mXJ>c_1{R=IzD_F@ji@&Ijbz<-U$3vt;23L9*=$MzW0!5H+WFk#4~GC0V|L8w=$vRAOepFW|2qqIL-nBM3t&Vk{X1loA;Ii$4;49PBnQ8IWiaOia} z@=Lp5r`qe=#s2iOZ|b;RDzgBE$DH$(Ajyhz7qIdXKW>hw;{gWGTlDo7$k?6p=|&F# znUkF%TE1{wU{g989j-w^2k>U%FPPoFN2>})W1azg5QH^=kPlZ{QZA1cHMJ5ifZl|L z5m2Fzau4x(4kSUCf(W*c64E^nej_)D<9q>Kdp&qUH(R)ttv81E4-d?x!>*{2ol9Oq zV1V^Nav)GF1(ix&34>XeuzJygLK95?=&0d*rNNTxK?woj&TBjhe-va2k1Z+Z9inG% z&q6~LP?%fl`TF{v`}zjf@T}QzBpG1fzJ*RLKe%jUB6y_EQf=NhG9hxX8+x91zckD? zhMWJHKzGOT6fy(}2(~rwC?rmyz7u~yMeun7egWnDGXClrpH0^O75Dveo0+ zevZs&E(}isr5FVZhTy{#*f?|q+%WW%X#**g@`viA+X|%h^ChJF$-9%@Uw=Zr%gD(& z09UoQ+PZtVLJ+uTqn#}a{BHihnEnPNu{9qSg1dQ$k)j246JcE}v!>*W4-itBl8(&* zUXQ}#i9a73u2$S!u9lrp(9jqiH8VUeiD2rMY-I|jOg}(Ya9g;KmTl3Rp%~jn!`>D8 z+GT<@s-{QyMhfRui$$^q;nrJO2X;*YodWp+cZ9lM#>%i^xt3#{%+Iz4o(V*LemHp=}uE+~_M zl0Yab_0&F2xBA`X7cY6HRz|NuzS!8`g$|sc{fWKhkNLSS*EH@PR@pQ7#yNR|Ap9d*_}|- z=-dTmqFC+l@q66b16D3sV>->mFnj^t{Pk#o%`j~57dHvey;-Di_bj{DDU27iOqU+n zrV8VnoL5^tiZyCN0FT#DOFdBX=H9I6qYxl99C`L^5dhvW7x&eY&1HLcJJxT)6_PoE zJA9Fpev2Z8GT33FVj0aHoy=&0CIga&>(JirDuBDh6>+{irO3>oStYLSO?uv+esqMa z$N@EiiD^`TPB_(}2j4I)6^ZOQ%)Ns}_$SMxx@(Pf<^@q~+CCc3l2}*^Nej z>zLO7lTr;Fud_S!_8GRYFEn#lusMz6GrXcuv5I7 z=1e?Xh*Hw^S333QiY0N>E_f8B1CXu=6Wco3Nc8zd#In09klp3fb`*_~?_r|on{Wjm|9vDH&*LzUbHVQ~)VK5Xen@mbNIE({I z>9lO%_ztgZ$LThrKIqTKg41qIzcx-Z2IWDgSzzdUGbO|$9|B!E@F=MF7R*MJ_v@D6 z=;^_!6DYBEOJwKao^6QluW*no)C@Vz0$M-U<|PF9Y*jw@6{rT8MDTwfWo&!$1W_rJ zURcZb>DHX}FWqZee|5$R$ znBESGNu9@ikVUyfxV%25ABYC1&;!amQRWfs@uDY=$xz5h<6wI`@w--hxkPL){z#-PQr7SpsP`GR+ z+0>vvqKMbCZ6JgXCI(atMt1#tTCA!!cfy)i5b~PQ#z^;ibV=qg=>*b|-j|>0A7K>< z4jmOi@FeA1yV)Iky4iSV&tNrws(w$Dz`Uri#%BjTg21v_EO20u@^&Y_(`PCiWjDdd z4&#P{U;h&^>@Rz9kkUigvsr+o$Z2jiJ8`}}hOB*#4jR}bC09IT5?Bmw&bEis1vF>f zV%ZR-)C<*dTW@-`Dlky7S+Qi_6Yy>NFb(QQlTT6K>B zx{PAQcZpEcb1hIpR%sJ-zoE82Hov3--A(%;*?996xZ+s!#wIC>6?*W$(i0`7iH3%W ztJx3dDtz}z3b4f1Qw;s1{0jbzx`X&?D4k!p@q%s&;tEeTiyD8+`$F$IH-Vhpm3>|DFZlgJs);BX}MQDwgVa#T)laqL*YZ$-T@SJ}7oRCk^tVA#1l< zIiZQQ3F#^KZTR|~WcMrTw{K6@d@=u487Uk$?2?5toC_w4JK?M9n9$5LIR(bh>WybY zpU7>{IL#5T#orI8By7~0o6Y<^8Q}Svb|0*sfBw?O#^xn4XV~Mz9lLN|Zvs7H5MyvS zC#CJA+}b8QJd%cXRBYroAAAb=$`W;%u8OLB-n~C?>lH?0WC57uqaIhweHKQ{sdPCC zup6dXe?}armYc*x;lf_&IY=nNabRwctZo+>-*!?ljNVqaS01+ApP_*h`PLnT&AT{A z1r`g+rR63uNouRL=%9R7jo_;ZVteMWEL`sBMx`A^~s;~0|CrCfPIRi`fk)SNT^gM`lJ_bYLs#TQTLKVr10EL<2gMd9LfT|*_K z(gs}JIp)k=t_%zfad2_FO0}Duff}B#)llglM-cHp6nO&NLnmVoVt0H4h;F;EIb{s?}t6*^!i<~bo zNTlrTIo`c{w~LNR&j+@Jw#7Bc!OQi6%DD%%>gA^(o!BLRxNNMwy=w)p1jA6Typ4OS zwPMNzAo{Zbe30&bCSA&Nupv^B?n-f-t0x9JsYpCM4ebiGn~F=NLZr>Ur){^xc1j9< zJP#K>gbI!jl1WQT^WSX6Fljf&f{MNK4`*%?jN`@wmZBlLcz_%TT(v;g5gfd<#{qLa zRNi=%to6DRc=@X-_?tkBMMqOi&wwG~bDFghr?oJbxueB8yWKP=MWEmq4kwC?R9)Vj z-~p-qkL0h{F63n(f@oSc_=EpU084+?DIrjxAH z&7OaV$$vP1X&xaM@1!a(kMRrvw)&&OUcCd&iIVHPyGD8yIcCR84fY2!%#cm++9|~A zDP?;oVe(n~M-zPnafbiyu<9)h%?_!Itn3d^r+8))%$l`$x345>c-*h8L2BxZrj{`V zy7$jb0NTf#nl&%4Fm8fpT?`MW<0Of;_A?v-UXtq7=S${i*=dH){ z^Ybr>-ur={R0$KO@j7HF<;izP&NRwS3sneDw(-}ayTU%S@Ak3tDT&lmoBpc3>2t5;!{9>_ilnaTBjs_Bz?<*ls z_ts4yYTtyb{*RIf+!*L~lfFHFXJmr@3Z_W4tJ(yybOYbaWtqRfK*>)wT`or13+MfX zlJPhNq!d(*&Nf0>afpZpfB~7f&uFL~v%GoZ3sesH@jo9i@5eO4={PuI!A`kTR=QnT z0+8Md=JBk#&(+(!Q4o6IfQ5yXP2ogmvm*zj{m=KF5SXupUxVqDa>qg8LG*yGSk7C! z$#J5|$?EpvAb~Z}@o1sPqHej@8y5a7oR)?Lhk!sO(*j-3B^ji&WnQ~ll~96QKwY|0 z;JPn2UzlDJOWWIuiK@6~5Rw6#{{i}0?{|fs^gx+PHkpHdH8h`AN?tw}7-RF}?ZNXG zhvib@*4BN|yE^mv=?&7xn zr!}O7&81R3a-wTxz2Qw+B0P_M(Araq*k;L|pPW`S;q&bV;33c%O{K0Lq0>$Vmw`l|4p7M|7!tGYisNE-QDv5 zuv-@#w%eYmd>nCUX>xH4%87hVmMf?Fec@?UBbrt%_`15fAOiH3>vh3Ym3=xsje7I( zEaT~(7IN*h>hTc}8#I0e(0kc*uBcY^B3A({a_x8Ew?nm(Lo-TpcjUK`9Djj$zGYyb zr`hVWuOyT+di3VRG@Wsnl?VAxo8110K_FOI+7#Q#i2q18Mp<&6^E+l~M<>ggbS$P&u z#sift$y(rdJvaVY>+9sxxdGuc!u<#l2?@I5 z2HED@+1*vm?6b*bhwwpHX#oYeyGo)|`kEpkAbD)lw483mTnvb5*?7$u9>q$QjsEqe z<|Ogn47z0lV!9Vu;n#}B_hq=UY#-^bF~*tGxJBB1s5zW>i5Udg2z1(z%sD}hO=35~ z0D)~YobPxI=!q;dStX?u^z^|GcUQaq;+A8bV`C8-HKyB$62MaPL9LZif{C`;WHPsz z@=qYSXZ?8SL&$9#;ieYsR&mz;S8wN-rIr5npKK=4Jo;w+Sg*K7_XnFpXor30{!Ugf z>x$Iy6y-0UTHd#M;QIJVfBN)&rP+l^t3DENu5{oRY^&3grEdiU(pucEb`HLD_w~g@ zM~i|W`uylGU%qJBjeLk@)lCN#TkX$xK-%2;#kE40v#;$7*G-Iw)CC%U>0>MOL1poT zoCgQA6#9k57Je2jVd-$eKW3u4D<1E7u#GF3JRQ`CND3e~IiK1~rRO7`jyM5RDop?26)#>QF ziR??@K|Ek98K7i+^SBFcmxt>o>s@|&L}VUB|<<7TSy- zleu!|`_rM|K+C_;e+7fg1a!1{Naz=)iJ2#Q(INmWjp)y=^s&s$ z=udl-D4_AZQls|mLq%VI>pnx#5Q7qR3lNeY&u&7wGwN}y>fX*dL2@_xbUzyn7>eHA zhi*RHK`$0+p3zZQ*h_Fk&z6eHG-C4(=;+s=C4$F#4dyurKRrH>a9eu=c>`FF1W+^! zbXs_8%w}`J?<~ZPOU#jAe5`o!UBCcSx5nw`5xB%stPa}ku!vAVN371E3(ey4AkvGvm1lYs8(Z&KQS?J zezceXdK>hs=2IU%@tj2Y7#IrsQI0lF{4#``j1|PxY%PEP-Z?A=h6L`26n}89PhxaA ze3$>ZV;^-ndZ;)!F52fIv@D-fpJ&>aIMNP6r=pYgEPhFAo$?5{o&#PYNT(8%zBi0= zv=kW>5wQz$y1%^f-OWwI;a)2s*Fku?pvkIM55D>GwU%vgL+)0(TP7%^6xlKUMEKq( zkHIs8XMIBPxLTUCkg6>H^_QX2SYJ|w8R3DBd(?AV#xwN+T>;J8EukCM+@MC!z3b=r z^(&sLwgW!cOGaB!&Afp|TErl&dd4yiK1wp~Zvl0djVPdXNaSx_*znUl{9an~{#P0o8wcc`nUA*@OU|3W!id;&`*||h zX{){Nj<_($1&2Xcwc#cKwX;xwqjuzs0T@w+;^{n>po?^twR#@AaZjdIPe%C+QO zPWKiGE$c;6G|~Wc+I<=6Ff@aGtw*> zknq^wS~#NqIfrHS*61@-jfzWWKA8to5EQ)Q&f;H7%gE@1^af&6mEEp92pVGFzf%Fw zH&v>g0xatb0EW*hAh7crV13UbEVP7!aRpq2Asn*|JSgL;^FwgH()N)I_lu$Ztt(W& ziNU<076cvfZGJkt3rjJJPm<`zWOd^zm{po(K(XYLs$QdZ*lKQAC3t2r{8pb$DGY(JpfT4O&Z-_pT2y> z_UxSLE;~eqW9a}MTOea#7>=g$gpmuSgTfyc35l#fDt^9N1r;%;c_w%R3BPkBNK^n{ zlLFQWBW!PPzv5W1Aut6W_1A8I`egMfc`;XR2d;-jd)Q;SQRM1)XH!uvYKk|}7e;r-u)wmOovE3!ZH9!njyZQ+W)MY%sLYKirB_|6?V zf(;r3n($n;2`(`D&1ADuN|4XNsXuQu5GDZT7*69$uv~6j-`wmn%JAqNP2+RfFUY$F zTMzSZp~h@w)5y>eh%-ISh%d~h3wr>r?Ot|T)sEJsvavzMq9xWIu$0Dd_CgRq|U;s?WaCmIIQHVJqCQiWBO;~ov~bkR^uhIz-> zYDW3l^bPv~OhHo8%e~1!&`nceJVEs=BCfQ!-8|kwh~EE@`Z;OScDv6==6S96_;_bk z+Inkxd%o8NaNMhR-w~<|N4^7r@+~cGOk`vv>(irKI}_#OL79Z^ELFNR zES15KA-h{moiXem$s{!Vw#Nl?`wu2K*?D43j^ni@4WRMV`5&(&x!3(WyM&RP-kcBr zZa$B>-am$b6_)<6m##h3$2Mkm7jshML+xUR6uvo9~PBi11ty-xtFf zLK%KQi9f1aMaIN58i=MwKtMR@##M++O&tbW7vLuRfRejJtXp#fN{R#`A_brppi%4u zP>#Q~7MixUwy!vhvB2?UXJ>CctO7fe3?~@?G~wnRItHa#!30jyF$?!IoCu8(o(rH-vA%;xVxkV za|OiS8nmo|ly&9kwQIfM^nM*6bW<$#8(a>H(90E`--qkORc{^p;iq}y|^^8nO;wifGk;qwlJUo=DX(v1O)6{Fo7VHaHR$uy3ew1GDn6m zS3YeJP>e(l6GeK@iEexoW~Q25&m(Z@*CqD(e(hpyN2&Hs^?IYjNQr!!*CQ#uL41DM zR}}fOlCP#_dZc`YmLDU)t=c`2qeY{~(Fdmn+v-o!dP-+Pv)%w~M-TxqxdE)F%p{I*zY;r?(j`&{#;{;MPX+09@QC^s9P3@U!% zq$5YN0|Pg@#aG844xC82zv+xekbR)UC6ia{3BkgKU0*j0lw><_oj1pNg)AcCmz%0% zDZu?o5!_fFVd8cE?@nVV4>IU7nz*N-6K%uV}nPp6uPfoB)k^bEsS zy`Iu84rY}=4<^g3fBZ%O7+2ffo#K6Tsy7U`0-?b6IhKm?yMS^MT6sJ^e%tB%FTUWw zAABAof*5TeO6uS`{QLj&I=sMyM9Al4aK+mhZWXk@H*X`4_9%^Ns({Ph^+(&S6%3=B zuANY2H&ck{;{Uq!ZNc&4Bly?=mgmm}_`h=R{r|XA5AKdqJ>M3h+RYW1m`-0{^!e{5 zI%Mg&&-4GnmHYqW(f{K*KWGM*!Uo^^8vqgpm+wO||C&nD*Dam{{hxXJ5K-r diff --git a/docs/images/chapters/circles/5ef0824098ef73efbd89accb8db408e8.png b/docs/images/chapters/circles/5ef0824098ef73efbd89accb8db408e8.png deleted file mode 100644 index fe6c01c00b7d4d060d85ee358e74e2baf7e11b43..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 17047 zcmdVCbySpZ7&bVh2qN8$A|gt6D&_qO`K3^Ho_0D?5Pageg>BDwJ^zf89FDWJq|nY>|$xX=aLNvTV;u zFLTHJpf>faVWVY|cQ#{k^Y+ebfjxEJwoV<(!omU_1A}Bk5=VN~Ld1=Zos*Nm@5hhl zi4ktRJQ?ugK!g{zu%U;CM=bUq7^MohCeo}FflWit$jI2hA|WN!#e$7bK;RIDDkUYA z_8me%Kv3}fe}36mK}o5m>FOCV0jq%L*-J>CLLy%SA;I|?88PwuA377mX@Zi;T&B8b zp}ZMzr*nj(z81;G@yF#$OOZ&onpF#Ih5z-v{})64zj^rwbE7QEUn0G=KJabTq8Syu z03U;C=Ci!~e_G-Ho8kVi+tj6GGglw`kmy;G-j~kuw!6Ny-ncwCX)j~X(ouHqb7 zfWS6{s+$5%4yG6}a5^oXlL~j`ahvY~qezih1rk?BbIVZdpk5k9y#Rey{uV24n zi#+~mG4>2Pl-3*0-uhcb{5JkzQ@g>D39L#q;f2O`-}^g}i=_bMu=x0pcEoDWbc-kd zU5j z>bV{r9o=1MnN3j95J^InYF79_`jlz8%2)go^o<%}ZXQK+x{&EjatgMEMPQn^T{9gi z=C5Bd+~ZUPMmH4Hk78r-DMAMq7cT@17T+XRnhn$PSx-q=TfcnuvHJJ1mX;QIf#bj4 z^_*xi-!VB!)-*rlhk^#%tC^%$$$OV0LoEUu zLHIzS&pN;0nskSec-`NgcXf5?B4EIR!DP`5_vad3YF7rz#XTdr$P)(91BZ+)S0Rxt z%7o~dqG7uq%EbnDcVHlT3XjFgZd;621gezCaH0B7cW&!>%bNyw?%f5OK*=HY?KFcq zn;8kkNAa($n)O6H1vEpXJC4bGQG(Yz@(g0C1Xt5Vwy{VE2_qoat9V+Ce=qL6ud|#j zWS%m|rKRm&T-&wX@~y%npYZSyBLn33oE$;tE&K#lP3S_id-LT+YT%bIe|{QCwrd@1 zO_b`i_K07u5%mbkn4AAF@%KV5L#H-^MLnOxZC@7?ml9p#<3~zSkCUZ}9$K`et zqh974L=T;N#^$j2{lGE!R%d{Rc#MtjV8^W7p2CaneJ^Ug_*kcw97V01md6?kmr~+j zLABuV+DC4V!ETmUmVf;e^7E)4(`|T#lg6-0u<6#fE7{Bvk`TU02*I-aPz>Lr5#lf# zqRw#Npzt`^`qdl9eEewj;6Oz}LZY?DX$M$1|DqQ+go1)1YN_#fNEjJf3>A~t?>u8; zV{a}uGvi-q^f~MTug!8l-jI@!aoiY4F23mqAz_i|@}An0kuA|i6TdURaX+Nzv3oq2 z@~ab%Dw}}pff%h9o^_tQ2%j~1j!Yz@pkVUSl3|(MLRw+Y+p4o!yXkrxD%QzzW=`%n zoc0y=R{BOoihtvUoi2Mb?_RM*0pBGcC&yY67<(=&Apv0^E7Q^a8pBA?WJ9XW z?-+(IIK1*xFX6wxg`FnRqB5$g_`G(l$v(QxkHwz`c9F1Z-H&UQEY?O2_9q^XAFZQf zVy=);D@sYByX=gEg-`-p#iycDu(sYMu{&x~GryrRyU)-{M3s{NHg8*^?R8s2K$*%% zVKmynEpl7UdAiE9-+FBezV<(hw=ZV?P)^_aQ%EGJFsm)>`qOEMny*Z^cV^Mo?6TGS zhMZh>`e^+j7S59=PXdnC`-zB%Bz1MEIXOAOsB6h#gbpMaA;jWMG!tb(7qBG$;rf~F z+F8RjS@T&^Hs$W$i^rvUsWa@&5;jW<)jXA&rN)~#m)5#X-y)}{t+8+d{=V<%=$LBr z6&H5fU;7>VBu&KqD>lgsd@{1P*48CYw7!4;{&U(ID*g(gijB`gOUBmzU7Evf(KLiotD~BfPG2?VUtjsy{xFH&~+N*Wt%6JEKxvVEUPiYQ_NH@ zj28d$Cofx*ot(E13!g>5qT=Eajr@GJWMPxHaKOr{pr9L@Rolg94V^l!S4=^alllrx zu6wYp?BE}|4P3pwy-%M${Rx8TV_K2lDZEz5rM9-Vy1!w1_UucMMlq@7q6mT(EUhu( zk=9R0zH-o_Pe7};Txc|NNmjpB-oU1by1L|0hp@h=AZVFhpER9{Wde2?^byM9c!PL;Be|J2MbW zk;%cNCyIcA!hO2KTTwA$Li43E3Ztgoyb$bEL#qaRzm4ug`B_nMfyf6{{F7}o@2jOY zC+1fLN;&JnbVtLkds<`PF%%}tH^CZLu6D``JkR&5%|{uBp^;z{-~Ukc2ezV`BZJ!` z)Ek6DCg8Y+^#G4{ZRuA)$;X;703L)$_nq?F9Xq6P&L^@Vu^qhwfhyublLq{E_nG%R zLn1sb%gL4#fr$oRDohZW+`&`^?m<+yw{kwaoULr?U)&bG6G8fR+DvzI?4fa6o32Nw z`S|!m6_(7Ysi_Hgocu6}TnDL)k%7TKBH|GU$HyfVfx*F6%L`~M5^H@RpHXlxMe@_X zz|Qb|+~{_`EBtGn9R&rh{b{lF@NzRyIqmm6vx?*SNcPQW5y`zyW4!OpDjko-pFsT< z4~N^+NwTY5hNn*h>+1zpR#wa>%hFe2Lml2jl>JCss1*IUj z1^SSAT zhtZ221@-}ETBxb&^) zC==dngA;n|Egy<{fn&Cf&L5cw`s=T(&dov7qN9TwO(qZg*;@h^@x){9*5W=rqU<`m zy-w!0qfOul4vikui4EM{UJHWJmY0{`7rhUC zp;4q-VJr=PNI5$@XSh(geM?E{18F=-)U$R~%hi?7Y&e7I>C?4Jzk%PsmkyUZ-_3AC-0?7n8vG&v^&$?%m6 zCH`M+_cpuEJI_v&IrWvrBUuOU4F+MFou%;EW(s&@Kmu#z9uyOAV>l4K_2x+78=Iay zz>f_Ft=IJ%DOU0&tqNrOPWvXk+V#;bNBKqzI&IkB-MAw&A7lZxiVKgT zQ2=;qb2xLeW>P=Lpv@<$jT?4oxZtt#%F)Mk)|w9-I^+rLY=7Q!dJ|x2kBoC}E-qPF z+2@BBU7g-h#UY!3NHp- zr*(T)4WtVvChB|iBJQ#!g9YrD02X7o0S<`_1yPVo0K~PLVF*(bB)A2^hKrl~_?Eh1 z7SH#1kl%Xh=WidVk9Rp+a}7?}83CwWsik^2l3)Cy@mT1*B($#2kTc%hNR6yB z@L@Pa^eoT2vE0I`M6R}SPPWH_ksv#m)*}R98gBcgcE-40M_b!ATUX_{F8?;*=U9Qt@x^L1x9{DJ$UOb(%1Zs+ z)eczecW^=j8KS8mw2@p~K2uBrRO5yux~Hp4pd*)qn0{evigT4~&a{6iFEPtNypnC` zJp86~r+o8cAL)iMz7kd9c13EuIey%^5)U*8!Ly{8_kmU$V_f zE(q^C)jCcCw*}khPP+dD0AszTT3{{1nc^y4-H($60e;@ndVq8hLq)yQ?XjK;(mIem zfE`eq5ll}{BT3E!T9_X)kzF?c%j*M{<2l{do!0~ICIDbvc4okYM6*TnuC+;1>}byN zev{FzR?c8?)89?#ss(N?{wj^Q?XF9$d2EI9(TJpBl>Xz_o($=v>hi_DglyT2>cq z>zsf&3eP<7y;u&u0f5haA{aSxtM@fE%AXcbw=Z~f%J1J3_b0NurID;X#z`IAfcXM) z1xXWji{dna+pk9Q3q@_dEJJ&#-<dU``GW2~vPhJ$iOvhGCh_*y zAJ-d*+aJ>zA|xjD9<<#U?<$Yvq!nF=eF8t%y}fe1o~eoI1eSTm-gfiHXV#`}8h{Ziai@(4!PA6x7;w6};K0Bs z6k$U=T-+Mp!(R{F;r&BHwRnIZuAg>ycMFORWxWpA1-A(DY0YrxiPTL$aVF>N5SYCf z_Fd}4%YK7VRZM6{Kx5-I97+99)PvmWX>#(EHD9Cao;(l*faGwpI~~WQ@GVKtrebTn zm<%8}peGR!6H6&7J_6Le@0gP>H4@nN@OtL;pVQc41n^S5apD}*F3Vb*04iHQy{3o- zYEjP=y;xYW=K*i)1K)d^kF|j`!jR`0MZ}%Kc$*2D`bL0fh+MA4QHy%~T{G5Wefcut zl}_#9$(YJ9_^LAPCw(I+Imp((PE7wHHUWdZ8L~G^RerNEh{j*Fb^-`$Kxeo3Ati_VjVjl#Brfd_=(&NW`q1q+3G#4Yh0R zm4U4y-W)8)!6fUU+kF_Z{CLsfyxT}UCyy1T_}v!lIU8G)ir8f!a=1D>gAXNIM*8V$ z=GgcurgfdH0(Q>_>Xn(yorJU(1^1jW-TMRJM zp$yS}ki$Vjj(MR$@~@ql8<~^2A7J8rk&BaUx^+eM_8u&g%{9Nd0YO`~ZG-Z& zuro)a;eu22>N3cq?+TP@fVU6T+RVaEM&F9&8^!aFv;`Kx5Yi?O=F8jot~A57WZ~ZD z6lQ-VzmV^^H}IH`_#w{<$U;ZpwBx>hHC9V-WWfk&Qc&P#^S{}knyM#oJVnE=h=&eF z^p9)~2S1?Y;Ja&n&c*fBZNWVlh)tM}$lEJSdK6#2cLb^)RTUEx!At?(G$&_n^DSSz zL>$#+c-^@ae>mU~ODJeRnV92_M&C}cPk-?G^%(;Mg<$aJjcLYt8sN`Zen8QfUy~1$ zGvc4-@^s=Vr?NZHZ)&mCYw|uZZxR)5+FHHEjdT;V$v0ntk+Dtf1w5GDJN-AkcYSVf zhG5k!t9IL+8g2DBwRHN&ntO2p^Yrp+?uWC6XalXYpXT;XxZ3Ig8YcN{ocXm2g0Hpp z5KaaElC?KOack|^fPhoE0|AAc8~O-j+Zs9Pu(A$Eek%SqlVu!6pGo_rzu*P`c%DOc z{guhSLPbCVcp>b3-pzd_lv!3zZW$;sTotMZx!m|}N5CEni4-%-4)J>tO933FVTh^< z50{K!kn@tbOtHD41BPe2)6Yn@T8QcQf#of?UPtIT3%l_(Q4*a0DNS{tdml{;Ckbr; zJetd9hDYrF#tGyzQ#JnuIh2*h8^in`{n$Q!8aNO8y>8ps3-ceZWB0mq&j<$G5;-UB zpN1%OUWfjIr|t|GA=9llgm(w`5g88$Q&aUKLp-gA&8fujpQI#8Ub(uLhNGVgDgDH>ca625~O1)ey+xjOOJvE!wD6B@FEO9fTLS#;6N zC|yDUi3;fT=sIt_{zhr1m^$;HKnCH?*Vn{ST#y+F2U zTML+-vtrHOGw%Q~$$ajo6(vDVH+>Gn(wYtGx~XMic?Vwb)~QKME) z^&U>i@sQaT*-%3HZe2gNU+s=z`* zLT_%DSEB(UU!Ju)7WoTQ@S^vH9U%PJs5xXPp{%?J4V8)hEa+5i&bDG()&PeYoga24 zmIfo4CdkS^-H)MhVaK}I_?kawBiSsXMf0ORGwv5jKFGOOUTlRA?@9wWBq5=kAbPMw zSdK@Kd3#@xi+7R%?>iom#N(9noFGX-&3!Ht`FW-$Qc!~cL4A8IdHRpT!#Sf9gs~OC zEngb6&ZRYqM4)zQ7i&^^>M5mLx-v1YL~M^m_Ivu?O*{M^9*#EZ{DRm>wX=NPf=^68 zB&!tkfR3m3MQ8*M7}-Il!SQfH$Nv5{=G%sV7a;oTEgmjn*Sp+A;m41#@y(hq7PEc>tn|hpYD{y3lG+Mp_+`0Wrst;hYOxR z9dcGx$BQ(!{}vJf@Wo%jab5svpafERr0gpEwR83Kct{BA`x5irSYC7dPo{#8#cvKf zk@7Ym-$|OOsp@{Zk>`uSG599J=mZr}Z|QjN23+8P#q`d*Ee=*&j48h8yJeBU-ucV2 zIv`nUAzjilIe9Tn%MGtrzx@utbS0o$QM96##t!=mZ4YsF8a5-SqW?|sHW3oAJjSJ> z4FA`#kISOh$b;NXz?JyG6!Rb~08AM!cl6r|5fX$`aAS&HKdQBHqW%Z)ocP^EPL6oQ zjWmiHxnyw1@o<39Sg4kcR3PUphx3nBRTxYF6h)Rh@x*z+t#Tok!Z>7d2x`mWIcL{x~et_pwGX;Y>@D6(|{Q z79);+yBldF4O<0)-yOH7i9p`xGVnsd6Z;MPb!)LrTr&8v=~BD@#nlz5Wm0>)#M03L zL!D==BUTBsL0HAJXsl6|SqUPsE z!JVbeZ9en&zYAaKylL@(-!hN^z8Ku#w3(t6@A=MejhQK^oqvJDmYSI0fYq?>d}OrMw{L^4y$RID!w z9GAdtq?$!rDYfGP9R$gHEGNqbL48hVgth_u|1D6wvgrn!F z25f_(su-WZ?PP%au#Xj4^c!h(u>vBua9lW7BI?tgz63a9@>N>6tPe> z89Kh(c<1*C+EJp-VV^af5E8;n&VvOV@EcJ4SZ)4mC|$Va%K;&8Zv-5&hD zyZ!0#WX>u2zah5AaezQC8Eg(rzU}Uc707K+O^uaiVF~Eq=Y^JK<4F#J*f$R!f1wp! z%83pM388<+OjOVGfAj)8`vTPRR6UK*<9``jc_};$rV}qKt6f6ZBt=Kp<-I`7WUZn{ z;Jlw*FJ%^FaY$LR}EgKcn9pl&4 zDUa=35}@kI|KoztXnuXckPzZ=(erk*%+XvqB0k$Wb5~xbzH8JjTmF=elPx2pQ(%aC zv~Omrm+Omy9hQbd^T8a*4{%~$=|o(M{g1d-@mVGktR?35y6|Jo3m|{f+pY*oc91Pdhei#|AY3;)eVx^4k`(MuAbim3*t z0&r`rgNHGCP3_Qt6yCuyDXEel`l7N*Ol%^$X7t~xJb0^ZZIFvFE&cdzbywTvJMu1(9d?WEfpVuK*QUVV!TO%QVGQc%(S%?Y4smA4)Byx9hD>?4<9~+ zYO3!xe`od>@x4QH6}A`_HFITIc@0`5(g=8&8*sb%lVYU3OdM;Xu#5{Y!)8jQl;nxg}?wn3z=?81XY6+ zJ-Lc1AHOs)T(cpGgcr!kEYPz>g@uMvi#h+^l@CilMf=oG`omzc9!F0qxBl`%wb9k) z_YnNwPX7cRzLG(xten+U%+``&PZ0SB(_6@NB?cuA2RLZNGh9migmjxroB2{YdYw8- zbuC6lraT|-zD)6Q2m~DW4Vsi0J)`kguJNMSZ?Q(M6y5SDjEriJ`u-M4&jjIo+gCK- zg9tifz0gEjDdJ5lsD4dbe}Y0&^yDilT>YrUAFucQH{Be^bu4s+gJ9D=h@8BdUr2~x zU&50STTMS$Zp0&5B@8k?Z2|(8>JTi{UDrRayqLbZpslURg@lao?K2ubrlY&D;p0+Q z8z{@2y2#7RBe^2p0eqM3cCA}5 z_rJ|KK0jx!^5cyGikkhTo?C*w^w9QvlUwU~l@`B*ntC?A*#}0(?(UwR;9VTDBkz}D zVnBJJ5?(Nvu>Dhw&6=%eI+6yvN&7;Ll^_1+t?2m z@i=)7v`SWz7w@wGegy&&&oBa1a5e<>fe-;yT#w7+rm*rLv)S- zeTt0r0LsO!-fCAt3FzpCK|P8BydP9HTtsR?AIdY846kW&*Cra!*07E0dg~2}RY^iF zFPhvAph%UF^KBHh96(xA&FcPOTj>0jk)aGK2+6!wA2Qleo&zYC zot>@gdwbe^z25}TIm_^BI2pU^jV~xCQJcL03IVV9{S7x*{?c|qWRHuy!(0GEZy`@#?{sT=6NM#tb5D@tRAxsshhY;|8ppBz>1p!(n zlz~<~T|a?~uVfS&5*kG(KJ+aRo$vONxhI;gZ5MHPc-S8`yXRV2SqTGeAlm@4L6Fi~ zR|rAWY4qF`3#%HpMin*Xs;HKVNxR|Ysd#yQT*r5}bKPDz-3Fdpa^eal* z%Cqd1Dj1*IibgPc-F1IX@R@R2lG$sqo(Kx=El`sI@^-(|mp|E(51uKdbX^{ACi7Sj z*3CI8WUPX4;U2~O^y$cV>F|E|dDDp;$x~6$`8>}?;&WIwFns8<-XJd@Uw-{}79zt` z?1F+6OO%Q6NB; zF`ubs0ozqzJKxxRI-zsvbFfa`pKRHujpn|dpov`Jn;`6{st{1I!J;2NnamLNgq=;B zxr5FU7R@qhAeuURm8GpZI5@aRv4a{hxR2h4Xcw|kX6k)x|FbJEpMWgG-DFt z896`YZ|kiipkz zhQeYrbRIHF{H@-vd_Ga4U z>kk)~5LOkV%gfhVaBGMovRllAl=K2z_5L&~6?RobWDK6)siB#vt;9ky+)a zHZcjNu1v-#A0NcFC8R4&*);)`y^58^bpkI&Eida0hN=mR0}4jSO{q!G^K~*KT?|VH zfNnUHnQ9(0vSqQnywnMPOhBsFx%)ys&EEC~WR+A2ux>q@4;{$;hpse1EHy3x6fqFp?iRokuL3 zIDUq@2@GztK02*aF43cK1@+{pBW0jEewmoO@N zmW!2E%M5adoteH7chy?)FY^NRS0)sZ0kIedn1~JR-{(Nv8=^L{_hz5A~j1dHJ++jF3Uh7SGN!L15FgptT1g zX64K9UC*tmpCjlT152V@|8Uw2*dML3y4L4;4c;XVYFO#r{6q1=QW)poUWW=ngJmKI%DRXbDCi;Qb7%+z@(u%Elr)Xq1s5|XiAEd2@{-`5 zXiv>3i|uc#rH=Ak@Y$ht>ifqU`oi8A3)J*VDdaf+EV-DZvxNV&Of9s;t5i$2ctACc zDa*G)^<&x-4d9zooNCNUTusCdk{G8_!+X`k7Ooe2&kMhHMFx6Zunb0hU^3ZKhb9_R z9w6V6^ET3|?))HF!AGuM+M9CtF9Q2tXoil^_RYZX#P&AmN9Wg<$F7*ai?(#~`NHbu zOWcLDy+L&?$ml4j95?1_rjg2Km31AxsWq5*SK-KA(9tOm7&rv;abTf3@O+V^{RarK zMZ9nP?&n-}29$TNH-Li9KvVDz$icj`{ zf3c(nAGFMOd0v#<`1d8GNP;+(xI#)yFZcyFVA3EMhYWc#5kP#Kz8zqb4Ac2Q`c9pU zkmE$Viq(M%JBE5Ng`)fgD6cdTA|x1GO9QkPu1?<%MI6mU5*_D2diRpe3|FLpt>fiF zDg>I+L!w)sI**J&HCH|m(>J*6KwsvT>skQ2GWrbQ#blqC>wfN0+56Q>Uk9K|L92cd zskBqsC4mJdsWy@RDSf#TP9n8%95QWMPlNG0j}``*DEZ|!%ZK!e@7(w}vu&9#!hHg+ z_qm%0>vA9s?`5$fsO3h&*;BwR@D02kGXCD>$|a9aITbWI7eis2OB;PbOn(XbtG#M% zgKDe*?-dEzm#xc67jA$~XMQ@_F$0?=hw?SEUE)tLZop{He^=N(ug*}TdzEEkm1i(lD#j6y0Xx|R(gbOwI=&Rj|#MI?38ggQ97s_1t#l#7O+|rD;mBc zXe;LjXWKJEA!{TpMV@#fb|>)33{>3e3+X=}N+)`Ir0rTWG7sk7hOgnTb~&BE)OkL2lUUbe4ql>*{a>_fr}4&y)vKDBz*tnPkuhWUW0J#>Km5)OO_ zK^bG$QX~qsi$XH^jfc=c5>SFxzEm@R+x&TZ_@rhsv*-LigQJN!Iv3*bB^pmL<57;x zFU0pno@W!^Aqaf4VZ!6Z=;yrqxT|I#%Tih`hZ^^VOfu762LPX%Q>6EABTB?C7>QPKjQ8S1bluwjv2UhJ;glU%7XRctAdHXoQvkHp&aS#;-6t1!+2ts zT~o|EaH?!0Nzfp}PoT(QCSc-9@ij9L-T2uM!)`hgeE6q;k{P|UMvKtRdD}<;RckkdHvTZ&xWReKWJFUg7p^mH~t%%dSsCI1WVH+ktnRE=4Zhs2}S< zIalGGU;WI|FSyHkw1_c)CR`XHo>)m|47h4VtK3`n18m)ne~DVir_|K2S}vhnn2+d9 zmOut3U=|!B60MWUlsdcEb1;>2Ot{hXQi;<{T;{%3z{2KgqV7lFU}H1>z&Fq;J~@09 zi|C8@cR$9mos3#?-OF-!IxkXLC9!NyBl{3^*?bwf8G!dzVCB8{{@*8aCOyKAft0J_ zPmHe!f2d~5u4*q|HweJTh(9{m%;+6JyWix#^nE~p4ybf-;9elCr2TkF>6-wL>6z8YOy;JQ3}$!L9tjrI6T#G@*OBW&!L zuf$Y1tRKkTN&?<=;RND;#-(!I1=&%rSX15UgjovC9w47EPzEo4E+PN%^xscwiaD>U z(eNxy_A1t|#%lq1ztH18*VeL-?o2rBwL$Sky(yc5 zXgi4&@=uXPG0y+?YB>Ne%gs)!X=2whHwpN4x9h<(*D0#^w1{t`${!VTAe?}|!N;}d zn>r^AK*%9ROZTrIv1@y`w|uW^v-W=L53tbY4Ad?f&;kMaZ0_$suP&$ovyn>Sw_c>^ zed(#XZS<#*62h7?n9}1VLg?l zccCf6gN#>u_Y_nKVLGaQ^*~jKz~7!B@$$~LGXQ=}_K-meW##GYzsw(kj+mo%gBq}U z3ea9+@u#a^C0Bt=Rxz+3ldp-<0i`rRpy#02?gzK67y)ihE;5(o@x4j7DLo5-CEs7DUsqRqTQ?}HSdS>B_dxP- zLD~7u*w195o$!h?_Gp#a$>#5%E(|I#WerVys)^KLITn2HaYjeu|4j_md$!0Cofi#% zUnQwwzRK%+7gd$^>b*DZsMFl|Dvhwr-w-)n&#@;LB3D|m@T!jUOy-CJmX|7=pRg2D zKJ8}mI^~1lL&OuQ!Ev8Cp;~o7Zh7ur@q3wp>23H{M($fYlf_J0Q7rAst|?5W_igba zL8wpZwztt0uVXNA?w%T-VUG}(z8$3qDJaw?^0~xlYHJ%!vjoC8>(@7bz_StpUP)@5 z@R3iR_+oUv*@uS`paTk&`u?r;+An&8#yzj=(rFSohOI|y?6t3S3jg@>L|V;BF0B=+ zhPY@sh^KTMz2~F4InIQWK0jZybLOWOCcU{GEK|f@iWWyfkMSp~(`m8y_q)LO>$9WN zVUTRw3?Yop05mGQvkdrJsMBsC}fn+ZAn6_kr;IB)f+1)639Qd$W7M!VzsxZ z(q+Y6Sb_6-QMY92kWdH>K?;J$`C_Le9>r{@_z@PhFw|8;zgKNc%WMc=;8gP8sh2+{ z?iGHsd2&Z*_;ZP7xuNOb`}3+TjSk%tAp|mbt}L7%u<&&}glaD{)e9Hx%Y=0TDi`hi zQP8Sy8x!C8!E$2o$;IZjJYW|be@hkG2a;pbZCp+D86r5U6zntRLG@f>xBZ=N8&?A5 zflv57_mhYN7KG48Gz65B6OJ3wpGb7L7ROG&7et+~u20zb{;75wH__s1&}lA{uwrX%azR4dw)x$@m%U#i4L{2%lMrC^J#E&v;*^Xt<9nD{R0(I(_M{fu3LfOuuh%;-`O~Sn zs<#f?9PW&ufMZ6!d&gl-?tE}pSN=xy>~iC4QWBQL`}gq*@cun?=8N=29Ixgs&!pKs zP^mN(yAA7Wqs`kD4fuKL`?I~?Qh2!wND*4t_zfO| zses8#S!*l0zEPgL(ijlG?Mhcm z(I4q6Ou%f8t&pZ!NEkthA?((%H3}Eg(I`SbEm)>kod3Z$M=HN!*LA8P?(keZ8+!fL z{PK?U-`w5HWpUf?v+`JMJQl)qf7#K?3yGd6xQ;QS*9F6`&My=?xhMO$=L^MyRLAd1 zP#l*acjBcvb2ka`;#9JRO~P3EJP-R4SX+l#^xuM~m|PI@rN?Lw)HX&KOp1y%*$+>H zd~Z6j;;hPfo8;@3jzlY>XqlD@PUE8uUN;0-#P3g`%}xobHt`{!0htQY+z#jjLBq3H{Mm!r8Sy zk=(J<2!F9TChaSklbSZA3XLdIvY%DvfO7Ddt9o^ZtpV+YsUOPiQNwSE4!RNMe$;z4 zR6y=0bl}DRCQ7q4SXWCOOb)B|>nxsUr>$Gx;XzGbzd@~?_r$ujAv67OE3@sy*T-Y6 z&_0N^Y{2iiH=SDlVCU{rm>GB)O?aPU0rHsUGo25|M}%_bW6$$9WBKU3R?xIU9!pR% z)P{;bd+Rr~XMZC=EfTW0fZlgs8F^?>OGK;X&*fsjar~gal1gp$CGaZih{PEPU02R@aVqK2pjg3%F`{%TOQGN z@a=v78)Vq=V&Z8%RPt*oF1_<7=8=xNdQOax`np5=t(k;sJ{zooOH+@3@fA^(F2#sV zV5A>dH-{+9n8%A=j~Dsg`YJ|LNPACL;uE}UKFtuZXl7!1{pPa+W{q7Y!ffb1k+tg2I@=%!oQu!@b}r>??NQx187jfuQmrKNo;~ zr^R!*hb~g{Motb6p8mn7kSh$qTFnUg{k!eV3mcDr!u<1RdV!>$@!_{ZZ{E=Jl7Oz3 z!u&MRb~QC>qjy3qy;!wsd3Tg`j;2q3TaJI8wY_E?skTK*lmt8A46aTi>pp)neXT~= zdws6{f_n_h@9+E73iW~rarQ6LwQctXs~?I^CckY6nsNU6C8G*TgF}m%e>93I&znME z4>3YuWCjKyVT{5Yhjs-aYHYTfv>!tD=Jw2no{MztXsxP|f?-?3r(lkUq6a-j->;KLQKERNF>g|E%IKuzCPZ#+A{_Oq#^yvcNp&m8JsT5fQ u7qVG`vK;a+v<$(sknsPr&mp+STOL{IC$K5~06u8~A}6gZRrbd4^Zy19R!E@$ diff --git a/docs/images/chapters/circles/6e455a6733dbca47acc0a36e7e490ba6.svg b/docs/images/chapters/circles/6e455a6733dbca47acc0a36e7e490ba6.svg deleted file mode 100644 index 005b8a7d..00000000 --- a/docs/images/chapters/circles/6e455a6733dbca47acc0a36e7e490ba6.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/9e19c65e23c32c802d2d77edb9812122.svg b/docs/images/chapters/circles/9e19c65e23c32c802d2d77edb9812122.svg deleted file mode 100644 index c738b623..00000000 --- a/docs/images/chapters/circles/9e19c65e23c32c802d2d77edb9812122.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/b5799015ebae3ab9b6712106b3b20ee6.svg b/docs/images/chapters/circles/b5799015ebae3ab9b6712106b3b20ee6.svg deleted file mode 100644 index a2eefe40..00000000 --- a/docs/images/chapters/circles/b5799015ebae3ab9b6712106b3b20ee6.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/cf8fc61992d2a75416894dee46ee5ffe.svg b/docs/images/chapters/circles/cf8fc61992d2a75416894dee46ee5ffe.svg deleted file mode 100644 index c65a49ad..00000000 --- a/docs/images/chapters/circles/cf8fc61992d2a75416894dee46ee5ffe.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.svg b/docs/images/chapters/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.svg deleted file mode 100644 index 60750b92..00000000 --- a/docs/images/chapters/circles/e1fbb58c3730b2f4a378e39c8dc5e61f.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/610a69d8be6f86744ffb88d12eda701b.svg b/docs/images/chapters/circles_cubic/610a69d8be6f86744ffb88d12eda701b.svg index d44e2774..7ef84d3c 100644 --- a/docs/images/chapters/circles_cubic/610a69d8be6f86744ffb88d12eda701b.svg +++ b/docs/images/chapters/circles_cubic/610a69d8be6f86744ffb88d12eda701b.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.svg b/docs/images/chapters/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.svg index 8e181ceb..8a148414 100644 --- a/docs/images/chapters/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.svg +++ b/docs/images/chapters/circles_cubic/85c3dc0187f786b37f5fa5a7cc74d642.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/891d50a946936c9701adc855de12623d.png b/docs/images/chapters/circles_cubic/891d50a946936c9701adc855de12623d.png index 4d4632289ea344be86b363b8640a2a334deb9784..8b2566b5cd1e4ce2ea5903e17d06791860bb897e 100644 GIT binary patch literal 15592 zcmb_@1yt2v*Crr{gmkwcB3(*%m!N=vbR+Gh;}R0mBGQdgBHaxFS3pTYO1NC4F5R7T z?*E;6*P6BFTl0QvW~r#ZILj-^ZoFMMFcoud1S;jfRH)0rltJU2rA1 zQ)m$Ub=Oi|Spn@9^((KvI1vqv2~AZ&R@WzM8}1JsKECbS?;Fl^e*NCYJ*^+2sJyDK zLa_hxVp3nN@UfU`7(r)80nrP!2g|go%*2W%bRDX?FrsAo!Kw#B5!v3?H)mC`ahn%g zf25}V`rgi0%yv{%j5t<0Hb@3&?xw~M+16S_l$CLXgobYWmYC>j9UHfs8XD4|O_b_; zt4S(<$=PU@mo)Rov$eBJ{rX@+M-uh>u*}a3J2En|znP4Qnb|TLLqS17<3%vI!j4Ns zOw6x{&dkhQ^jJng0Y-)euBI&}rl;GxpFDfaoWsPzawj;NoYQbFD$?f4r$GRiG{KUdo|M$;*Mk6LB*7*|m zIbQDJ+(TnykRyU%;}Y$2H8+1qvJUGxk6*TC;lstf7Z(}qFp;N5z622>`*sgEC&I+cBC0+D-1mcVCrZs~l2JBy zeo6c3V-`LRj?(Mvy*tpOg#BAW`VghKJYV-E$nivPU&d4Q%VeVJFbst`c};?=v;2KL z-=4cmR)JD57*%JOYX^0XA8v-wwZGQqw(xf}@nW8N&oc3D{cRs@&Ed>8(poe+Bohas zqmQ;ch4s=f8Z~g;&MRs>4~}&FIS?90`?s{x{MwJhWd`FuElsEO?J=*<;j^XZ3F0_7 zZ#OFh8IXS(2|I4*eLCHd-MK*|G^BUv#bl7fqhFIDJ{qv^k7Zz%dwf}yd+OL7N=Orb z1vTiLKl=ms$%lOU2KVkU@oq*ukc9ZRBs~AAtVaPodr;?Smv!gt+M7z8LR)(Q?j46D zXGmQ|VsFp()Y%Cw}N!+K+> zN2~MY@}FE?#mWZ?+R<1XEGH_dC@Xd?bX8eJOwdP1=mbX|uB;GO`BX!WpEtVn($5+- zd`mU!ruSa6ZK)w5GN;2eZYBx5q2|zkS78!25F&ju-SqVt><1asq69T{Z*%O$G0xS( z9iJQDrMoRJ@?$yU3ZTDB?&K@enq4NaSX_8?8`T91&-10&6GKR_B9lzrY9OA^J)T@0 z;6YEQi&J=oD~iS*aBQBPcS+ykI8M3w?HEtD&4{c^L7N~{FA9>tYRu0p?Qk1X16YUz_4QPk5RqXH zWpu^x(jco9whGeM-0aWKF^4mi_nIKfUh8rvk;G&^8#T0fDCP7cqtEX3nK|8EG3h2xqvY&wg z_x{l(l(Y_0&Q}7LD?SGXoUPI>ebBo+={4CA6-7NNgTA=!wG&3`3iIwGSc3Kiqc1F3 zdFn4ZK%udhiVZH|+(!2{r+M1uEN zrfq1-jP}&c*!{lKG}?U{19(DHso!zzWf&MX70JnVBWlt#&=kI|F+^`&sqA z=7~r$l?mo=yzU|Vdvh7*|IR~Yz!7|`D~hz&^Ex-c*7)O0MfL=JRIxo`UZkMjgZ*gB zzAW<*gBJSY`pMSYOqURjm$CACG+PgC3iY^`NcJ8*TX;w7F%n7nJlLm(xARFw5;cE! znk)m=Yft6H^{eZ+mx0Rf!wBdXN-N8DRDOQIfa!pd5lZ9d20GAXQm|`XUvwdJX&Dg# zQyVS?v9I|RwXABXy>Ul!UFkGOU+P&V)4GQ3c1Fy)2oe-_?7vHiJmtHXJ$g7ZGg?eo zX6l^9rSB6t@i{-v-F3F13+^p%TLijK$}S_JsC3t312Zsi5ANLy6D8IsBDToF!?_pi z0*@`l_|EcJMxtwfX|!4Ji8HP{l7>Z1(OC645%h={xb)O321y6de@t8=@6;6AGP{We zZjH<$tHH$wn}x%8l||J7W(+5XM(>5j7BqE zNkoFAic`4rWnz3irhK zB;@p$s$^?JxRsyg-pYjI_MYywcosy2H|qs`Elyu!*xa*_^FIzzX5e2+@=f_#+tPXayo;S-PUA}Un_W*YbZ0bI{%$%a_BadAdF74A^TGUP5SNTe^F%-Rd zwPjqTv&7=y30c{<)j2P>|LmR2qMm?CN$m}bKhM7_ah7Kr-GqY`DNeK_lE^bYXFa%mSy{80xx@2Wf@+b^I?aNQj3!6UJQk-6PJQwSDq{|UCK{r z5F1YY?%Qj|tKFa$F~v6JWR700Od;o3qV;+B#`^+Hv9Oh|UfTliv3N~z^-b+Pg_&14 z#b_vag>3(9%~*K~Yhz$Va*B!fK(BnPib5iV9pE&kLGZNdK-*W;5N=!|ePCxgb7CHR zQqexO9RZ(A#C;afP109F$e1c@=R{Og->N2-?I^&WYGmU^5YqayENXj@z>|T^4aU#6 z%{0*{D#RYwP;l`PAWglqtLfUBFt^_}MgP{d628ulml-gMAtNWZnPVAitgIGh=8;8T z#B~x{Lm+(WoE<@DE7ntt7UumurC{NYWsJLim%Lf~02zshOxjA;kA*btw%d8Kg$%zn zaGm0z61)qJf00Q}W>&m=!RLo@nfSByp-OROJN>2hgfKPKUWkWuD)+oYtpr0w8;NP) zj7U~;BA5kJV2j*j5-xNd;$Cn3ZlI+`36=m=I^u1^Guu+Qe@vP`9XOkII?0YZNTTlG z(QyT?%nfE16mD2=Tq!-l!VQqp#zd)jVGfJ#2mv+$m7v08+1p3Yz)jvS#qtJvj~@ps zGlo}NJqwSBG&kP48B_1|dxeFi=e?9n#-!)7$-QK4etVz355Cv|>L*Gu<+AkC+&eL(Ho@aBR-lf$d^7R-8nQrSi&eh1@EeVLKo)iPLB(qt|V>vfb8`li``ZMRM}8 zwLue+*KR`#De>;nocm}UJ3g%~SKDLFV#yCR2jDQRWYjGOy<(m(QM%u>E9Gp1a)_lUY>njR0ycA#JaEKkBSV25Lko5%1$RsP4HJXm-#Pd4CL_2?}HV+M&?3*8gf|bjD+1T zPTBMwe~BjBeim;vE!Eq*I_!NzYxkS;?M8Bd@9@$!q*p{fIB~a zhG5~v`b#UojwP=6pl6@qlESvkkIp(O?EJu_6Vb2T}Z+5=5=+IN+xVV0+{W0Bhq+<|!RXRSPr(b;vauMAy<9OS7KCo6`@2D#d?}=l zWz?1yWUYL|gq(pdQlR@)K@aPrmF}_3z$kd4R}p69Ix)t|ywf{E2$1ur77+k>3M!OX1iO`Wn1TMIeUTaHKv7X&pV}(Nl5>;Yyw|H zSqXfN?bY5LSg#>GJW@p7i3@t$K4H?tiWNysD+EXe2}P%}eL-#lo#3=;2uq!VjVC4Z zB7Svs?+YwQ8V2kIa9zR-6BW*2kc;@9n{t{21x3IXW?ux0I9z%)T{x7s5)s=uOyVyt zmhE(JKbE1Z4-kjJ3{b7aY@uV7TO-NDEGmBFft7PZuObi&Yswy}h!@>-}7822U{aCjIAIthJw-UW=hfGaG5w)DW(<`tL~KthP07ZnisZI)+EH;-@V6=Z=^ z;}pmBcchNvl=k!5qKWb*tRaQQMpLMSZmI2sdlUTfC2C*y`b~T#&dGKjRP$D9C-G5} zr0=eoN?_^b*CyUKepyN*z4`s@Y$<*n(HSHvtXm@+dbiqPftg1bLcV&2z(@zz9K?n zWXxrbKO}D)DtearMZc2acw=IzI5&`Ezd4GOO~7Z*{3Capn+CdN<)8-l=f{idA)zsn zQBs~8x(v`gSvh%m^y!WO@$~d`E|Zod^gDOas+?zqG;^hfuo7P><58tZ4A!vAR? zhPR{tY=4Pb+>491{YX1%Bs@GEa(jI$Xwi*{47^X5hgPlcC4_kwWvNMpc}z=cWrt<2 zYTxa^TPb~c*w`Wp3Lg3$t-V-2Ut4?K-P1#VQHOLn+^p;zD>o8$Y26kIJYPvA$-T4o zD<8X5ztU^XT{IB`W-2P$E1VkXQ}In^i7^JLs;c@xOsu8fT~<~WpHAdMj-)?@lwIwN z&);xLDoH<&zwS~}Qok!qGS3f@l$Y!yh)0r=*`uS+!AL~|uY96L(0`X3rCw9;$X0^x zYQ8@KHaz_+U)k60)MfsBNBn(`x4Z=;A{Q4I*Pz9N{SGz(7j#H)M50IXbgt(Pp~Uku z5eTsNJ*R+onQ3r)W`-g{l zBovOP%z}P3I2(CwO~2R@;x?=^KR;MGJ3AZwtQZ3(N<~G5iG$VmauF6)P*}K+xV@naCT3AW`;d{L z1~93Lni?|)M>G!UqYo)5y}}5nd_04s+MMm$=qHJTeipRBEYZZex;o3;I|fZ}9ytH0 zd%(id^RY4)%sG#x@-zW7lsBM2n3st#CC>k&v(B3knv#+dN{Q3!Q-s;xTne|**;MDv%S9n>qlRT`P}q2BebqV)G` z&xynInD!`|XM8bd_~+&D!97KV*K>ubxU9c7CbWkT3TA(8*JXCe(31BVt#eH zxjp6cl^f(Y#%DHX3Dj@a56#jwrId^OR?K(>JJC8vM=d*r=dLfCxi&!L*nZ6M5<^-V z(2w$8mTs9nTTe^ac|V}|63nxG> zHtKsu`uj@Fx3#~XT$fAvt$>uW_+ zwIvZsVdgI_Kke_s9iYFs3}!!;K`K`b(+9Ass75XtLM4t*0FPfizQbW5epNq}7e zfW-1(fE_+8hpUZ88+OC+0UoKY#wFq{MBA76VRY(toc!mKPafU!-HA*a08fVxuLIN4 z2$WH{LdHK<~yDRz0qmE z2oLH52g|qcAY{(owK2spfEi`O*%lKV;CRMclZZr4w=SUx&rQNNg#N*VdgV;)I0OuJ zEu`#rEtD|9EH#b4d+X2)xNk8)3MeEQ7&!Xr4)nTyB3ngtoFUR~y4{0*>gCtS+w>m2 zMG*jsJ+^T<^}(5$Bq)7#Pp;Er4xV%D|JolO2UcW~jH&QfJ!hHe{qu7Wo2_X0KiJ`x zW=n~>-aJM)Vvu0>Y0RKP;~!P^^fTB&QYg9Bi8KND8-TX~IK7{mj`H6kJ|jo{I< zC`iBkr`Mva-mx4!@#5ZtvjrG}@at_KCAldji?CJu-KtLyOU0?PLrlXosL0=4P;JFN+6`yzT zq-Br?_q^wEq;H+d;#o~)BdX@qFLp3V#s${-Jibhn`v4oQ#l)7+mD;z7rinYZ(XTZ(5(#ovHgS;UJW$GmlPML!jH%wJp`~kshTOogEa+DyuqEf zb#c7;aD`v9BsDEymy|1E8yR{XCh{h*@vvG#H6J1;6*&M zMyDA8W)_yH=xEFD(d1|nw^xT6nL=^fvyG`2*Xf05?40!u)WQ@UCmCR$QjZt5RoKIu z&OKscKdUYE7FIlU{`2H6K2>mMr!*P_0@;GMrqYXh@wv=3F^@RU)Pw_^*8XO;-T#zA z3N_B#^N}E*t?E7%5hT$WfXuI|EwKhtxR>e$J9VzEuc-l~N#t(Z&_UNu$?+#x9ekTa z2KKH%fh5tCyVyCAw6o~9P^&)C!ufaiX?dnYr>m{7-I2pKYrg9FSyBNRf9f46Z$zb} zZbIaBDUbYzg%AO!^x0yb9B%KeMsg&Lz2Tn5LD$Dyf9gy3&W&8#lRY;l(*w;4!7$Z< zBqdL1&kO5emoXl>?F|@o_S}5Yu1Z?`8{i>M5?nM%{}b!60#)5Fd#4Pq_4FuEV@VUR z!A1M^>zBvI@2D?d9{0ww-#q_wJ_|4*WY)P6OmYvcYa9BvABk)@xVX%ZA4ev#X;HY) zl9PY`Bp;D=z%T?1s2;FY`SE-3MnGj0#9|T!cMc&ogqCBiI$8HN-#@u%7wY=7y|}oT zCKVtyUW|C~hyV?ZE&-JbpvA_+Q_&xUckkZ9)#b{`4t!v;+z4;P#M;IN0!F^yjmJA$ zZPnL3l_mMxbLUf0fL$OXS>rPR#%CIwXDv?kmwQRI|tiNCxB1 z9eTgq;ey?Z8ohcBd^(j;SX8g7=e`QL6M4Vlo$tn7Q~5}U-8E-a+JVZ>8?~c!ntI2n)p+Tfgw;p5VER-v2`tJ705WTr>Tye5?a8Bk zOiWA{ayI;1Snq$jV>4Z4@$k_j?N%@6)sbAnZ@qD}&6^di0NDe>kT_pXw6?W9=)z`1 z+q>ATJUmNhkKS!N)U-gTr*KstRrrE#-XmnYJV)YCOHg&(GC-j~0?7iR42F|X2UTm@ z+ih_Y zrEtA+>vG4z!8rn~EX*1F(Dy0XX&wmxA){iB`m<*?ATqiF0d}g7w~aDV=BMg#`zath zxJ;cqIkoS~IT=pE9cB?ma+-cT-m&zXdktXgvx$hLRaNnE=#{C!TRibjNxq3-^Ly{i zb-R-k@;gmeF(dg0X+X3aL&5E~LGtMR7=w!M{&6_F&O(m=Lc?rc$PWG9Lvp8!%R23y zALGz{#ICRJ0_oX>_=zL8pkN{hh!R1FUbPqfRml0z0#e^2Dr0)u0N7CWq`Q4~yTgDQ z?uUclEl=lsSTr&)Z%=~&kFB{J!6=KF5U1DB(49ePM@(uMwRw5${<<|J1Y_QBlRM+7 zqtaA`$-o!!kwSTc73Z;sfaIL6?c30V{qT^Ed`fWJmKaA%LUr*DVC!5#I|ATxkHA>* z#SWpMH41Vp-geA*2@Jc6Goq0^H1?o|5DY1Q3kE{?@92%8O!2zyMnvGidwOx7i}hk& z5o6Yq7r_R9tNZfu(B}QY)`dj@u6egOicwBU>a=ZrdcLp0c@`Dkfl&A16Wt(J+jY7~ za%h0XB-iWNNs&(sC{`ONN*@lAZI-I>dzXgjT)kx!e+YxX%22f8n_=c&O2-+HyH+*8js|puOX*Om7Wsl zTs*McdE(yIwt0QL>C$oGBre$ZEKhAH<9kO@QzwF2(Dr$=o6W_^c1(O|sqDYwmC~m! z`1wj*GvX89%k7(VXE^$YExH72eb0&vHgaFJPcq~NoKWxY?@R2%NC_e$lE5~>W$?$Z zXu?p62izz-YqkonQE9}pLWe-60JH_cqr8oehrFB& zy2W#x1Ugev$cfC);oTU$!-4o8vA(yRUb~79A3o%BU)3lsmu6K2u3zzZW3LOl_fwg> z(yM5HoClcF9y%Gdg||8V-BX>`gZZJ6U(65{&cF_O!G!xB{ZcM*MG7ra0d1*yg@Q_g zYNw6V!zRn2?*qu=@2B>qeLQN?4f zK`M!fifl;wuBhOL@5dkp68+e5mBTB2lYaWBP4jFeb+&$}#dCAI3Btmn6wAcKlou;b z24t+SN^#~p#%mLJfw%At7Z`d=QsdG4?W;szUS5l%JJ*pei1`XOFVxD?4!EEnR=d5OA&)mPjU(sF%2}?3I0k7M?J)d?K&blCYQ(iXdcM zjT4?X-&^QXP*hBYM7V+AXhQOeZ_YEdn(bqVeo%;HGnH z!;is1>#MVU;QnG(35GEf#$E(Bg;raA(&Jv68cAbe<^fW5Q^*xm%GaFk=|Lz}i}|O3 zSu!#{o1bKsmX>Q<-@djj(g9r;X!Uc2fomVCuUa2bw@3hl{b8v00G+g7{fG6IUGA(T z!fH56bQy_s2S;l%BvhLYE3)+-!8>wJ^F9J90oaDiGD#iwV%jlFQDV*JJ=eg7SQ&Ax z;7a4>kJ;khsNEb_P#=i_%MN=-^WdQKmRKUNZ35n}_*iDE(F-Pi)nG5Z8(8;&Pn-?Y zWws%XRXv5z5_XS6cXU4ICD3q-v{?=6qk{|8|JF~H?I@`cy;%)%gCOEFhIkRzKQ?s` ztjuD6Dx8Pk7tjkir74XWY6z3GIa>?Di!}|8AkFG^4 zh-$Tllk@efItSre+uQ}=_RPo2+8G0BC{3$SG2c3TUDtl}+s-o-_N}R{r51UU zSA^Sgk1=zo!%OS=Fc8d- z%;zPnhA=M1r83$&n{+zreriV2}mDljH(z?7GKIpOvn>b zdq+8;LT{&@6!kxixZcWhrr{Rx&Q#%8cbA#fYwA|Ccf&WNVt=6uSRi zVC{L9DjkFM!%Bi1ez@U+Gb z6ZP*IPw6WlJ1T~?!meuPK;3ZtyL6}LU03~-9HQgRZabeXK8j^e@55tF8qaEIi4(9G z__2~{hp;##RE_nuW)dc#GXFXZFy51ollOu5o2_#t?7VR>Ui9CtC!)0TRAv~_840DE z?TA=6F^c2iDgEw|dSoliXsr0IosBJrM--%n}lGnM1vA?_i5~tUXVd$`*lyBqb%` zHE5};=Qr__VPRnv6coGxK0U@oXAU`AV}l2PMfWh)Pb{p+tj(XCaq7_BRzjDK83U4) zIy+NI-TAqFKE3%U9jK%=t`4?Fv3h9A;3eWNjn|IxMhiPTyGD*gIx4`l>cxi!1OU-U z`u40(`t?`=j`zGT8)%Qqur89bqA3KB;A8cU`Ui)H=x6K2xlQM*nKK|3!m8hGKQ;XW z7^?D5CPm;dz4)VhF}E{39O*0m>2m?okLM&{j|By0m88QeOc zLkwT!ytrnxS`>ENVi5Ib2ZPkecoGG`_J^!25 z8>j^ZrtN-cTt*GCwnG`oXTE@+cvWMKv;C)@)A)O1fitQf<%adl?Cep%i;69A`%7R& z8qeJqrrNZ>Ph-tB2=o#|g7296h&!S;MiRM4)ID1khwH!E$a9^}5@@rU(29p_tuBt% zQPNQBPD{-1-?|P{6#-qF?+6)(2Y5R*Kx}Fc789VIwzI{UCoXf-JNykmP_kI=Pr$^) zGES}rh*pT}Rd=*(=U)Q`|{)$vO_B-xIP8#8Wm;j@)J!KB0 zJ^4@b;SSSHCpAb*>3BFHCS;NW`H@i83 z{Ejt#uS72{u~^fjX1;LV)xm)SM4l_7pX3#kl@kF<-uv4FjO7~$MZr{1l6$_~@A>2p0cg9NdSV&iU`QE99nIr-2s%8GzR)?Xd<@;)yZu-WBlPF|Mhp zc|B9ZYSiHL_ov|Lm#oeqz+=TgJGv`>l;|ukFB|XO{7WS`$HvAcrGu`Q*aO8}|5)us zr8cY2_ERdUtZ$8oM|+I7e0mr3PjQFwNVI%9?e4&GMjW-;9dudfF9Z7r{N9~M1z?{e z9Py^>MAKdLjk}5ur5}B^kZ;{I*|@f~vH4YPrL3)^^J8d;m5B-D6(M!X4WFXXJbC>M z>~LI12@rzsAp<1#E5F&Yb<3S=dC?5c{ z$LE)&dQR|`WDo~^NKXE~I+XciU?3kv7{CbjL!6b~)BOP40~l>y9uv?CzncYJE9NW3 zMMK*q{-Tr-knAo}p+?T~#A_=UmmK6^`&`(IxOZzuBXN_kYkIyKAxg|N@e%#;~ahb;BQCuU~i$+)=i;8F5$L7#z54s2TWZ!Re*>8s8_Nk9*M z1}h5~G)7eTBp*SrKc>#h4?+T-)-8c=_cDQV5CrJ$=4^?Ogp|}C-trU>;V6y_h*XEe z;=6pPz*vzQ2&yRJX#~5<@;feYFq^;kPOPPb#CiQrl3=>+!0`5%+Kwp(Gi_F-WA+h?^M7Is(sJCPM5?fFVN^{S-f z$7AE;ICywVU^%uBow)&*Cka_ugCO8L16uJtTwKeWtMhPwNMK;#(Jmr)eZ2Ti?=?-i zQR6C*ON*an3Kd>lUIV8DK9TP{+n{*iUS|puT}k9Q?Mgv?2J4PJ-D;R(AP83>)$wY{I} z@JR=xCtZanzy>F~2rV}^0T93ewlM+7qQEjDV`Cozbsls!2lyFca&ko(88jfaE&>7l z8v?;7EiIknzti092iy~vXob5KPuU3Y+MF!coy|SoY27^ltd=nmgSNo-+b=nX zn=302zj+Cc!3-=_)zQjae(lpio3xZuJ3C1*kINqil!Q`@msG@bA~bB)jy_F8or_Lo z#=P08e0$x5u0O~WJo&qozEQS>H2pMUD?=B24DHQ&tj+`0Qf9WNν31MYT<@^IGhp?q#+lYCg_q5HMeQ9vb$93OwJQiJ z`@z-(o3{`c2FQ)z3%Y?exWLt9Wzm7c{0Z=+r~q;~oYK_%VEKm!(4ti;fzwRIC(Qfp zPr3L9l{cRu6fQg;yuW1egCtD~-c(#*r`2@nvNlbe^(LaW^&V@|5?$xQIXA-j@A7`) z-1M;$nxUub_xstd3kDnfy>sI499=rz2}#b5*2oWZBP(3Sv&Lt=1W(dWBN4g1(wR^H z{u19QSgNwX%og>X19@+#$uu^myK@)c{BU&$(5ApA%D#ES=h}HKw0|0o z&-mEcWrUy|ZyVZgq;QN&Wy`ze4Tztz^bZdfnU4D{+aPPacMSpiRx~G?a`D#!-(mc1 z^K8Q}*TK@?kC;0TYo~8cV)Hm z7^p;P0#c~BXUoSO6=|ZN;L=&sfUZN@&S{mEfo_q5W2HcllwKn)dIyon7Ilwc=|^AA zf@ZlaqF%RNHd9wi(>`b3OLqrkT3H2^JgExN~Aw^q~Ptm5-XbB*V9A;SWNL%|S;(q9FU6A1AgU{$v z);CdzHwP7z8p@y+7HDZMpjW+eTcf0bk4BD02Sup5c=2?R$I|-MzNbT;^4|0njR@BU z_V9fEyis+22-l7LA{Wjf3$x2w?s$9W2M$5e3th6tIX66MfH{G)Ed~z4qNCVH0<$tH6Lva^p zXSR_SQ20fL^$t&*ra$~l=G3XSR0PMjL9Ok5aA;Ysj}-!#Lgq4FWg)-uyF}-#r4}&A zC9i62X96KGpzy+;o}bH>>h<2~7x7C{+u!ev{-&Nv`B^GLN=VmCb5#F1}mXqgD@8AS) z&n$2CJ_K|*Kgc{HV3g7*7EDA#0QTIl!RZG}yhJzfrasFF$_7>D=wK&35EV^#nE1NC zoTya;z$rkRP1h&0_H#}A0HrSisH3ElmXoD5zT~nFkXK>qXyCx8cu- zojWfiPs>wN?t-eO_$HC1r5*V5`mc}?Rd`EpU+t|l2kzIGp;c8i&4jQiN{|k_Y;E;@ z@yut*N)BT-@HV@lq{KSwaB1?1SVkcp#q-;XO_T%X=jTVIN1?PUfCboE*oLWsY;zVqqVAqJ8gdSq=R_8E#r%3-%(u3)uHBXHF0Xd~$`6lMQ;wID z34T$lSKb*~`cL|JhLYlo<4L0*YN`)Fnit5nGNloI&*|EP9t`Mp+rhLh9WBZ4fR|_j z0RZnAI)U^o!j~qe@%>9+OCSYtDXc9wmk}r=ih6pIx5swb;BWxhmXnZ!W!1X`bOvLH zU>{MeNFR!z+s+{0)Q94kI#E$!$YrXxZbx5;@7dkH4BQ|CiI{Iru4D|-i23XY8Q6ZWmY;P literal 16228 zcmbVz1yt2vv?eJf4bm+o(%lFM2uOD;N=RP1L0W0)mJ;cbE@_bN?!HKO&$<8iX4bq} zujZ*NyuY}o_SyUUzP-;E6(w0rbW(IUI5^BVa?)yWaPZNvAJiw{ieD?=0QmLfy@ISX z+#~Go_vZXqI5=v!H`1>(+|u?JJ-vp`AA62_s=4RMx2=g|2sU+So(ekK-@N!5DecX2 zqxJK!mbR5?>`QKRHvKWCs+@O0sK$ndnwe?G76Bl3CYR3_Lpy&Xj@h``eNbYShR=Ml%z$}RdjTc)N70L^YhhPyxeQOO(xYv zmCIZ&h*-?UO4*x?z(?)9U7Upf;`?92)6&xZVEXTM=0A9t2j4IkHRTJN10>~3sYjtU z(x9}q=wzb+MLXVqIIjQTtN#5-_)i};%MK%W#Xb6(O^JI{Q~CVs_v9Tg(?t1FI{)5$ z{-%P%dsX0XG~x zypd zd=F;eTkoGt7CoLzH5!TTjCNdADN(GWc)1CXD5X6YiBCb7m@U*~pOyFU5ZVmz4_6Vi zhr8J4mfUMRT{Cgl(Q@iwBisEqHS*l3Y86Dwxy2WTTY7BQ8NL_C_`n&V+$ zIK*r)oT@!ts4XX|Jzo2vno)JGy(6Nw!8yzO7GpUx8LiQ!?P6bMrbO>!ps_v2FB^>g zITGI*3)P$Euq4ps;kZN;GX8ex!93_gnb*yzvYPx{GWi7NjAynbbg`LtnbCXvFXFKq`pLFKSdNT>A z-mtL!7BXVNeR(_>jce;byQ3vHbiQ zhYZ86`6KyBDhIsJ{iMJ#B^d?;;=4FZtJV&htaSOy5xG8HYGW`Y^eGOP)UFlNuz`u^ zD;>v(zy`ln0xm<8$ooJiC!TO}G+|f0#jGm<VBcOX(Ec-^cWfvPbe438ysf3@I4Lz@PcXT_~G);bn%dhGucT6E*VnCtMDt32<|5CSo zce6jDsK6=okI?7O_IoAewiM=Dg8_r75wQn3vScB+ufM*5r`<{yJy>e0KfWx{kBF^n zh@D&C;J3rVct#ALnP~nphsz-s@})fV4LS~FaWtnpE2gL{*bmOsJ(5}7iKKx4r~b6S zK%HkT#Qcs>`}ff*IvywR0z(SQ{6khew$=tO8cb1|9&X(E@4h64e)T6Z4m9@pJfczu zeQ`DH8-A)ot2CzQRR8PFOLA+N(R%E;gXbBw=K%5FmWJY9!8=O^4+#gZEcO<;qZHl` z_qWR%3-0+9uFMX&uy|s>^`(WYL;KvIxYm|Jmli6cHCR>9b>JkqMYU3Uzc9c(e$@#2tp}HSh+8MwkfmB(A zmdewolL~FQ4L`D2m;sm7c`V?JeJshxzDDpKI1KkyG_V}Qq2u`0S`is|vce0yl5PtJ zVP)d@lNW{SzcIQ9SEDg!N4WjcuHizuRCLpn z{(&Qa#)nuh0CFvuwCDx;j%| zA12{0K9G?QeWrfKixmRNw>L8M={zBlGi&G%+7RCg7~pV%9n#K>kw@GrF?96QIBX|+qB93D{|kI zFGh2c#-0(;bplH4`oV@_ZXPAy%gZ_1(}+noSh&5s^NhO7TJhZbS&vo%ki;GYxmeOzex46&t}2q8rpZ{`Z3W)cwmy{K zb`B@co%Wtk2t|vUg-C36a@9@kayPNBgx6La*;+0XTS>da)mYQ-5s+Y@2omCn5i4lW1M%Xe`fu z_9JYV4bbV3uCRz?zSes)XMQy#yB*-Ei&UjMb=a-i5Z3K&i?q5Oj@EF0p{y~hsf90|VT&WN+Oayc>{zT*#?VNG zH{%|oqT?v9*#92M>edgz0=->JZg2?_KAgD5!On&|MjFrWSncQ*rJ_~)^>{08N`4@G zsI&OxyCe0~?IuwY2mDkSf8_aBxl#WB8n&UjzH(Dpm}82(OSzCJj!VaMU3j-}IQlaN zES6hHKU=_dCO?+>=7!End6GqV66zShX>1kiWsu~EH1I@BMYmPo6_F0YkTVrZf zg%7UUA6Ung4IG^(c>q_INcV8ZJyE#neqTUYP*4D;QKC1=dTx;Ap(Nlol0;MK!pdX0 z=6-+uN|d6#(ti=FBc!AA%a3*`X%045-wR#S=9i6&P}{q22%YG_F{eEx7n%U`*fbx2 zEvbUjbYC->enoE>q&bI&zZyie(s;;2E}vV6fDo?WA?y;UZ~J+!y5yj@%mO?2*ZrRH z(TwRxdDdil=v6RFsb2`CKV#;v2K?3>IOexj)9q6wGJs&*hLgZhS;utl1e1ikf=^4u z_VGzS&NE;Iv@LmxB&PIxF?7z7%>>L6@^5-?52d=kj+A9imV{pY_>4$Hi?HOo3{|=d zR?1Ecj%7qk6(P|oE^olcUB0;4vmbvuW7~wg;AR|1pyWxN_YMTYM(2P8){+D1&RSkO zti?urA@2q-BxGc{H^LqSX2T31N)`#@9%HZf;B#3eYz4<5fLo~oxCeEJjRyim%oNVh z==s{RZplR04I!sqfvo(lC24uhsaWOqG{Np%WoAzF=x=AfkLJbWInCHHYMVce7IhBp z^Ad|~yYhEm6n}GAfqQ9y+UTmk_$AcW1Ik=?BJZ|_X1e+sBE0`OE^gW=!1FNZCljn2 z%9Dj{D{UM$DWH!pL3_Que(t9$F|x&A0Id!s3a=~;V zoy|4}4w6{>Qm6cP-KCVY$ngDT2MQJOa?#o|bR1XRU#N4xJ~ueSUO~slN1Qf5R7eCP z7*_GC>;F5;;q-ITFtoSpDVZ7hr)Ra`V#mY%(O#MT>7bS?EiKV_Bfs_hlfFJJ*EM_s z>VqQ?fs0R9Ou($hgrT8vN`4rnEfPLP0}bia;n+3~LfND{NdOwY8)NMH0moeGeH$*N zLm>nXS;}DL|`qa)%^&jvtP>q z)_hra??Tb-fQT#yAs71aGHRry8Ek@37T~6?+}1sH6v3lCf8s|y(9yO-ZF2(wnLMoe_yK3Oj*}4|?^i-P+N*c(-T{@t@C1eq zr&-VWYy)K?n6!%$JfX?oJ?M=O|FUZhc^MeHHT&q6dAS3KeS(HIQW(E|zP)_I-{c0D}}l$I-S~#?m&naVDZ&Q_?m>7Ix8D44PK; ztNV28Vlg!UTh;UwUhbCKE_(3n{tW!vRK3NZ&^q<;A9IMDui)1T_vZ;g2V^{{f&O3+ zlZxSc2mdw&w(ND|09_m`cE6Zk<3SE(!Nbqe<7HCshD~&Odq%<5-DSYKyIQ<) z@WEWts^PtVAASgdfF&bD1Ga^0Oo(sl6%{s4KwT;)$6+gkhr6s^mVwau8gvH89bSo8 z@9Gud*U*7Tacn3QxXA??`t~iHxWJ!G0xn=& zO?LL^VFpi6!!@Q8&ktg}Qd6*5wlFibVDgdeV($!q##*^?U_l?hST7`v z?@hfcGg^ddHTi>w^x4HI3pF_JIhzj73p2pxH*J2Q89Fe;XO`EPV}$FKmx1+8aBoL- zowy+Ms`Mw*x@R*^zEb+yoK+dUxOnE>otIv%?OR2~r_Ruc)#Dzjo>hDMmEZ6UF8lKP zbJdn3-{HsemDy)n8nNowy+ga4MjX8*_AREwo1L2UyFGCreH)5g~W?~9$_PnfgJv8F8UC`O- zVY1e5TpUVwUFvIcKb>ju7Vb}AHSGw-+!k$l*AeXB)6vmUe=%pJTVt);=M_1W1qabG z`|(Vb;Mn=yyVn`gQDI^0rHuy&zs;&t?}tyHUteF(?7LX=k&TXyI_-?W`xR;yjnzR+ zu_#4$CPLUg&1d>PB!gKP{bi}{9}rM~IwY)H`Qc4W6yn+bd?L2k)pwmtbaN(&wD@#P z8X7HE8d`d0o{2iVJ-f=vN;xGZ1|gxFrXDihPtqD11aDLM^7LzMx0?^a&}hWOGWJRv zu{ivHfQ}ouZ^jR#3twj2q&4oh1)vmY7HPHJX6NL*QBq<%f%sXK)(0Q9+zwSw+X4`+ zD=64aln7Qf8fQJRt5wkBjEN9KFtFKtm=cRZsB@{=bGFIdv2ESb(()-OPbdku84v^E ze{T(@Uf*9X>sGu+9{=&y^#0~Ns77U`rW73$lb)SDDowI|B%OeI=uh7$WoRh2u*+X! zyMv{{)W=(@o(C7HcNP{*uNMzgvLulkTn~p%Hv8dw2L^JLGsVKm1;0^>dJu?|EoJWQ zbai%)6lt^j+#UB|d?Ku|8BF1QBV?}B_i%p;8&ynHW_)6zti64CQRb6GFmriuxW~G( zQIvU*^=^@RA5mpCeVO&{2tkJpxwhM!>}=4;%c}@6ditQwPU+t>rdMa<%9dlfXzuRr zPss(MLBsaT-j{^dzIRs;HFb52n){P}cFUV>i@(R~J+VniIvX|929B6LQB>x)7yBDq zTX;l7vYBGuU%kkw1qG8m&Y|zBX64ULpE+H;Hk5a9afv~*4z8WVP*w<7Fi;Y{zHZQH zUQ=uSR^|$f94_se@;GREZ}@y`ICFS+Jikvwc@7c-amG0c6Qe3U?G`aApwVs zOu@@b#^APKDhIIuS`m}FsBHemi+`I$a1oaVt$IlOP2z;2Y*(t1|A=t34j%Dc8Rnu8o= zZf52HzHR)cDjj%Dx50@yUC7x1YE>(c*-EWG`TDMNC^Q`@(9lCxlwBa|JN2cvaDmb8 zlme>K@5uY6o`;9|+fZD(BmoF*MMVW{%RBE)8LH%a-(HZ5dDV?XK@Q&RSBa;^#>QGs z783GWP34*mro6VD2TIjcsiFSHIv9CihGV45?pR*a*_guXCcR6vLcQujb>*{oVaB)9 zMLM`+W2mktr_OyIUAZ9cu=Dc9qtg)DOAE2-vU79Ca@jTw=%35i8r2Fn;9g3SUki3q z%6<6o+;!QTU)6U>BrPLDX}-p${(395K%+pdqT}H2wDDx2rf!oPq~cS$J~lNST>xz6 zvR`A=PBapyBSN&~Bc;;hGJzmy3z&|;|<&X?tyK`1eL z5#G|V$(e;GD1w}f6-M4KznB7Y>Q5uFCgRo4m^WYGe0@$|C zof)yoe^{GHd?_>8Qvjp?sfmF_UfzIF;Wrd;)pB+fApHbcZ!uUKgm+qoConuF`BLSX z;oRCmUdNvL+{J?s&dCWva=!Hps#H&p_v~o6!HX%N z$&wkp28{?gQ_~|4S%?LSf2D4RKFbZ<$Pjt$Q5tYkZ3U3@KV1|cCgt@v_5p~z!cQsp zwv*WQ0!5x2f%s6d(#b z1Nu&B+%lGfyOU_q{1FW9SYmhD1>wo#q-br2VI2y~WkTeh!C#Ze&=oBiyy4TNPmaGW_d`ZMKiP_|A4 z@UAD`w9V!p2fg;rr^uls`*Xm6;t)D7$!8i8i`d7PLP*eYxB*eF1z*Gn8OBX+zEF3sOuEFdrfMqg&oI^auY)tfdTXeFCdUIMH zU#A4GMlR~v{wE%+Uf#=Suw}6+>@Mf|TSv1dtHF{?5>R)I0z9sZczq3ETb`mP%@V6P zW8xypn5FNqs8;P7>&;~>A`uC7dr z8l%;I;O{Qk^qdSj)fXmD+PH}V(t$~i`gqU2v8lWGC60Db_R=VFLB!?3BXH;vyBZ84 zAJ7{i=kN}ly(!oe*<5jL!Gln~&cJ4cX*FJs=$*Y%AT@x;AR+U%0YMlg2a@utaPV<$ z-mXdtyVvf!muF24L6bIDK&26WN(`rvC~13;m&qa~#W>ciyp;UX<*%U3rSHQyR0}56 zQ2-!YSY&eTP$V3qmJ51ozI``rK|m@w`Qt4Z4I`7p!p9wzB;I}~TP|SvO~X?k#YgiM z_<_L#dvo1grt-NP)uB?j^T&GUo_Mx+DHpTd-yInu+*3{TIl8L( z)s1UvlQ2$*-wr^Pa^rSHLS1JBA00m-KN&6ebOq+i47%cZrs}py2%GfmQ|hI6>;9Ne zp#^QK2WI^P15N_M_=aiGnaUsf6Vj5oL+)TQgUr{}f6}Fmy_~1`@QgDQzZ!q(;o+;b zEdKHLWv4x;ibpbff6;Ry&fe9!nV>J9!&{c0yhE>gI4b)kNpzlAU^CJqNCjX9?Fq#* z#~@r8+Am_^ zuAlvQ_srMFUmr>#L|h$Z3UBX)av$VgAW*|ZH{>$ligLC9LCdlVxB10EHXGQb<) zHv2Ch)%gtX7AsCa4%k9Vn#tq7V!tm$M`{4{x9_yKC7@(>A zs6gp+wd%hQaP8pHY6pp!mq3neRLl2=u9}FbsQlMq&tM32szkra=iUSE8LM8WrnX_% ze(my5G>ug6@Nkb=)o81)FBdm=*nJ8+{m)Ku|5lGP)w(TGV&P|n8Wjy>%&*XKhMiul zQj6bvh30Sf)aNaQlW-G@xN*RJ{`~pxVk1%NpiWG^cA1eEfKmX&f3|D64%Vyw6cZaO z1#m8a!rV4!!(WEdl@{W`4uQ%9_1EZ(kpcuyCqsj?8iK z(|Q&-I4Vq7EkFmhI1VM{(DzpXI@O;D`4^q=0b|(!S2rGR4_*7M*uP##$jg`Lbw>oT zH-7-zzTJtr%>2efjlB7jtorp=mIXDtho(OE!k; zaKYwse_j@lwl5J8d4S0LvYL+U2*Hm1_D!$R|gI{AkyOuER1C!G< znuQizGcP2)(hxdNy@Yq^Yzv8`CK=u?sIjxMmW&jAYq?s(0CNFYj#<}NLgc8Po*r)N z8IHBJHB>CDC*h9d`ZdNL8$^nXKN_E|6+e0bY~Ng#peKE zr=_L&?>jg+pqJ8OJmq2g%e1S{LqJV$GeCTGB!%#`4l3DX%+ZD<=p;0%o_pUAmFfD! z;ZPygaLyJW^no z1NGky3Jio(0W`C;^}*xfVA1ncNqbF+el443p#}{D!xKM1jKTUz0303~6~#Vt5d|Qq z?P3F*06Esl=_v+6UM|p0^)v{=iC^u+F~e5uXR*}wE0Hdu_3iQ2wvQTx#>=}lV2HAf zF8ex-F05A5rFn*pF5CC^r8*U~CTSf4w|6i?_3c|bw)oxeg?dNJ=~4<35|T6MWLOB5 zPZHQ|(FqAMN=jILeSKbl9*_$<1fK6rSAFQio2#|M{$2jAW2VYnIc8o)R@QO$V`2rk zp_C@jc6->W47d`EWwo`nd7bnzKBuQYUh+KTv|o|n{&w4kO$J_#v^$(;r=LAy zbE%7wSwckWq=9O-ia)})iDNWxU#nQ>74SM(A&4Oz;6De*AxTfL$aU=VJycY14x2A2 zVLNFzKXWVDJR{3|IKz6mCH>bg9tPzM#UiZ|3&1m2v`e4!^CtqfoP4m@$o;Mz33d}O zeux0M*ZHKjMxjPQuMND|%?{l&b|b|oO3}KLKP)7I4$t8f5?<1a`Ub)IlpA;@S+4h_BqwPH3HQ_%(f4&U`C7Q`II|y7^uf3P) zbfJ0q;!oaN7LA`062FaGl~It9RkI|6PBTt-a-tlDMK3wPi~yQ`4L-8{94|F&l*E1u z|5CrEc;#`}sM`~WDlobR$UoTXuWj|4J)GCp*qX=%85n{eA1+&o76e<|j^Bfoa*EOZ zOF4(gOfTrVgeE9h?p85bB8SGbK2mw0HUBWgxRJX(=9+EgNd*TO8W8kA(a)=Ths~eb zWt50do(dca@_w2y`8!jw(Fie+0!HTsbp8b`?3ICJ?xv%53?c(gExcbSYI+C=E9cYi zMyTs{8JLQsC|jgM%8TPYp-(fLYcsZGMHD5Xt;DB*J9gi%p1xWMz-GGLzWfM`0i*PT zkhV2|e>bde%$3IypCHRwewu#w&?^$q6yn2RSsSty==>JXRsn4DV-@Zg6Z4jD>Dg$a1@mGy5;EFTD;L4?GI_ zY@lN`#@^?I;KdIUKUIS$OTQ_o`{}G^h5WV+aa{sv9zqA+xqbHs#hxqkRat$_CCn2) zV>9TTsW3&uo^DOm6+(F*2j2?@B@T6sVY>ZFFXTV!&r<8VZ z-~s~Kc0~`RupKKiBpoD#9IiLJGkV1zEu9(H7(OxuU@oq!zaYmFyPg(aaJeSr`Mb2Z zJIKYwl_cWMB^CTM>r3iX--$WE)_G~_v)WRLG{;UX&pj$L_n2mVVL%~eZjYn!$a@dTDR;Uo%bz9^1{aF zS#>J|uddtx1bCO&A@vpHzLBs+k$A1zua1F3>jI#W-f9qk{L3Hg)yV1eRwxvBmhCY1 z7L&7`QTmpq`c{wxrjxb+WBU{6HN~%47d2 z75I}MiplK&nOLrD@U4-8^yKawp-g1p!~HAmZg+YzzCYjN-4~g~xx?y0%s9$*9rdt5 zGKR0dzFskzD+nkqBOol|9gO$__cpjQs#05u`+P0liMYR6>~`BqqbdY)I6qXfuwpNe z1hkrJtLd#NeLL3GgS%7&SW&{!L1XtyfuJKuv@3?f_@-BPl=}LWsVw> z6xg&p+*kogv^R;<6og8)Rjk(_-WNhlllZZ3&I3d}L5l|872LzvZKkOPd>`<;m%Z7~ z)5+IkQ+*WT4UDFM;2qyUL?Cjtjs0^w3xi#D#hQK3+i^decENaAMv}Es0YvR@& zpyCR;;DvG>MOZ4?#3m>Afn}F#^|`lzt|{U$Q-Wl5k^t_fNq^5n-mTEk9*`^;%E%n< zaaGRN^I?i&*V$R%a}h3UeGmfrVNN4P(aBd&CW)rcgJuhPK3C-xca01r9wO;-t}z{L zC+}kW6ox*meN#@)IOK%+jnh*S5jT4@ex@K`U}5DPIu!^m2Z245RSjhINc*4bwo*B{ z-#lzGm9gr|gvK)d)-E;Z1HRksZ$%H!w<%hXYR99r4eYI5#=BSx1BqiIpd&9eaN3>h zUA`H(vZujSQbJt-#SfWYCupYK;aS$Y`}?{;U9kC+7@}E(YSchr}iM2U?YesAT2FC?1cnq@Fe z>to$dP5$Z>&BsXYaA=KAiHgQjY3-X`6boutO$`*-0DtPX8Lw>tU;_Vwy%d@&)6j^b z6T7*vL*S9meQ)c55Er8|U+D~9EeALLy+HwdK_^OAV-@}3w*ArIPVLo4M=o{qfn0}< zU?sbogT}*e>FI+&HssM9k(wdL+Ac*qk0)sn(>?4&Tem&`@piSOHoQYGV>{4Jq)AQWm=o98Bfs22eEr z2l`F>bSD4AIL%9Al`Mo}Bb(x6=N^g%(?zKrqv+O+^uG72zqj8!?ju!+8?K(AIdaI1 zmFiZ>&{>LtiVAIMnf5|WP#IZKp(osrh=eCc{7$R(U41cKPx8DS`Nyca+S zWTTJt&EBdLRcKkmp_nOrH$0-r*z@&6HRv6$gpUV(Bq7y(?^WYLUURHMW#%UEE1afJ zKz0K55|D3-Jzdw0VtnpR7KK3O*nT};z`^pt1+_mTy<_{)D`>eUpUY}ZVd8LZa+8$G z=ZXfT0PeQMADKU6b)|!B-ZZZ@DjC0|3*Rq0Y&0}2*A4AIYR||XAj6U}ckoLs*Wzij zo?ovsOF0NBjeYNVQ~7MxKze9ePLKUN+%XC+?$M|$heb5R;l7_8)C<@l;Yg5p`Igxo zG8!M>eAnL5*hXKv`b4o=sO?(cT~B8@xYWd)rw~$p7c}@pSp=R{zdf49emkAx9YSEB z&SLT~24fm`T17(y-fC~(WFo1Hl+}YheEKZhq zRb!0?%M`{8pH7}qTh9BMnhpCpI$6D-V$-n@5&^I!na_sw{`L~q;~cnSSyk0?r-F*F zcH2WoH+v}#I{seA^2Pgh4~hK2L%G`@2i+VO`88Vpa;ZBHMdsQ7yt~-o?C6-NHU{P@ z6Kt?wmWZxPZ>|b8Ijo*d063s6){}_vobCZGGd!iQAnv7E%MmT6T~me5Ip8B|HBiYCUN$@U>lZMxE9POx`jk2Xee_edixT3V5v`F?l zIJ-IS=67L5#j|Lij4O+CdR-m)$NXYuX3ovcH8@%TP$L#F2_>Z=_bBrZA8@FshP$Q$ zMct050j{5Uyl^(0s0li^-x;AvG)iq%@k0#^1QkY4y)G$U8a?G-fNQ;b4?5lvJXloL zDRgscHvm275%;MKsAW*x$iiMjW!oa21s;jtW=`6~gc_v8c&w(v0aY0-(dP%W|4%Y^ zcMyu>)6XYzFCCBVLK2`6h#X#5_Rz^9*&LY&Hq?YFkBh%=fNXHxBmU6FJ8dAP zrUpp1QE^CxY0n9W`$T)Zx3tKicbk>{;R;?T*uhHa;q;|Ll{z=>Q@ThG2XC~)o~EI0YAR7n)8jU3w1Hl&+*ZR(*7{pF}nLT^>6lS%@rJ#dSMbT)w$*G3}G3Hb_0$ zuQ_Z;4LnOAp$dw}2@3A9tq(D}q8V#NvwI}`XEZ;&>Zf*NsuI5?CiYAes9P?#hyt&U zn96VWLzku+Fms(IH_qB+A7QvEv%&Z6foNyD;~{{_LblUg8_pn^J~x)OK;l)(5a|O> zJZ*04P!tIT1=L6KK3uJ1CyLdSsr&VIRa=@C=bx_xbhn3JF63;gC?1`OM&q}3pB_`- zC!I=@*YY%HMDbf^N01S%I0=x$c(lXjA5N3saBz_n!ZJbVghQT7fK2a9mr<$bE9HSq zFOaTe!in4Fs;yw!4B*8A*rI3A3RIqRzbyaGRfn+|HgEv$sUY@vD4M2O@&0wB#n6zu zKgg)syD)focr@sX*FoJ?n|&D(UOJZ@=Al;?G!39ECr~e_xtC#4r~> z$jeB?|0X4NJ;aNG9HP9CSMHIIk5&2Zk7UhXxjbb>#D}UC);}5r?FHlc?7D|)W&9w` z2qYNCtD|?&=`v*?xB)ebEs2*3Xh2y~AwOZndm%#1VubqU@=*EmVDXKzG9$=jfz;#+ z%|dBV+78Q%$1=PXIy4m~#~K?SXHU|)x_khV!6&nUB*>_ZqoX7H&gy~((5`H6h0ZPx zvHkw3Qfy}v>|otmm( z4@!<85TJeKtq6h$%lI$ZPoN;8$wZRB0d&HBAn~iKtBM&SuoUWOt(`GoQgYVTFF_L13_RonodDb%Y}|gZ z0oZr40Gv(}fcWR;KG*wYv6}Fj(tGt55gc?F_-QbU8j&sxFqFj{flP!yOa>juac_5z z{$;$T%jd+!ewnEl^g$S7+xvHHe?lVxWg-lBd?jQNfUF?_pze2^dfy&|G3>bnD+{Rn z*ble9aRcfUbecVQOZ2Lk*{)z7}t2#cPHU`d@Dj17G9*Aeeq8?5$^BE%Ul_QjJJ~z9JKtw4c z^na9b%?@)16go2>i#+faJMYqG^F#6E2ucgp9R^Q>Otw+b#@`J=Tyh#4Nf zVybUb2#U;rd7)raK4)hB0z#G}&t8VwVZ6&2g>_iu^p?ChcW7Q5w^rn{5= z59a3Tr3QjPY|`K9v0QB6?N4NHT{$+-a1RDQTTMOWSSUbdFz~%UM?yyCw3;fGw0c&< zs`3H*bbHtw#)(^oGnplFuRcw1{GfceUFI;y0_D4_yjG-EHj54zNFJAlGY4vOp41m0 z?=W;59)ZE{@RBw46|3IynE5dN@pfKusholW9q2G33Kob%%J1LPf-n`wZkz#jvwEY8 z4XmyNp#0FtNLVHtOy23Nc_uCa0W7j8i=I!o^CBW5CJQ<;fcd2D#Y{vEfUx&~QjP^5 zvG2yMJ_Z2Q+Lr8M5Kym);o@>QJij~5+!)AI9*JWvIcpQy+h&n@IIyrU@o2p>2#SD3r%eD(KySr1zt40i{ zYxHE-6FtL;FVuJkY}D&!H-BSubF@I6F~jpfP*PG7#?XNPMa#$-qLeNKyMaY6@EZtf zy^-SD)-7Nm0J-1*kz{M$u5~kle~GECESrPM`wSO|l{CN^0L_lbqhlGwK*YL7r1D1R zzcZuIXCmJaq!-wkLKl4Q!&hdiG0cYf4(KeG$b+$hmYVM!u-+VHK5l-~-?_!!7w(*@ z)+c#bvj>1iFcuOVGcJQijyG`ic^zpua_l-7rOooS=dvg2_2_5~#lcP6x(z~3z*M9 zn&pD}SR+5MA*WpgTOqGi%yzVGBt~@aWg;rCS5b|g1b6ojXqQBHzC8~V*WP=FPNX9R zD-c;*N^K*fu+8=5b>G;;Q30Mn-}4)kR?3u)-fIZxUvg8E2&|4JL(~%p(gMfF$3T9J ziHUgy!fNaN1zEGl`FpT;CrS-d0j1!CtVx4)dJU2jLwjXlvp;8KM1)nK2)Q13^}*pT zZ;3D0zb_e4G0i>r{wg}ndSO+}r-_rI!p<0!`|XUhZFg+Ehp%+r*P`?}U&WxTo!|GW zT@yZVzo0y+bewIprVtjt{Y=qH`L5?cbiq>!h0=2s8R63eHP{7zNw||lJzd%gD?zl+ z0e-L(6u)Fip#Ue-1Gp=wCgFn#ky~j}VJ?m7ge+PF2BSzE9%mMSz3=&N2SBi8L5Twx z;6$16@?a+Y@w1lu#TOX`TCD4p%c?mhk3Jw-2ofO|2fP<@9%md&pQe5_sMj{mxJSp` zGi#KpUiKk>CWCg44J?5HG{7X2(Zvn?k!VBPcP!g01L7gjzSm*s3FH8_k$X>v#gzdB-W^>p6V2+O^bBu2XaF6ZF4PPQ z3I+UXimg=oyVD*fCAhbP{K*v|D(*XF=4Vu zTloAA{P7p0Tmz@7)&vVR0~!zcmQGIJoto)1lQJkXoPRjU+TK2A3NFwoNstSA~QcMTpb(u zTyKKRhqrgDRT#bza2;v2XjeMZQ=YyFUNCsp^Cnjot7;MkLwg)v=2nYy6`qnCU zQ3%paQYtvVY;Bq%$Sp+@QqbpX$?VAZe(T${9W3Gyeq3-=E6X_Eo_&mp41S>*eyb@j zDf0vrNTOg7vpfUC_C1)Foy!nKO0P=F>RA2mApIONo468;Xg*6PJLZ>4Fx;Bq`{4e&Zd{0yGX^BgK1~&W88icjqX=}X z%C`|;0=@Efp-!*NjS)H2t*lTp%SwBFzqqNX)yMDgx+9gnmFGcGp3CR=_7Tj$*1D#4+kcmpGXfV31YC+&O71F5amcMFiZ%DjR-iMu z2Q8^AIXPdTQxU}*A3#0|6&rhd_DyWv-{Dm`*je2LP%8#2ef$&8G7iv+q_K-?)O4|K zY)X)U8_)|6@gBB~bWKcFkQtk8`1E6VF+9b710v&gP`{17D03HrAnm1~GGZ*=6!tBB za5oT%*3ltcLeOOZH4vOAy3bdZQo_2Z!q^5fBD~|ph|Y&x+-It)$nbHL1k%@FAKPNg z)LG)I|4@7fbX6ev27;`X@lg{B5Z3Jv(`Ny(bG=Pp0(mg!PlR^IHo!i`v$AH2v|Bec z4b9cm;X~`>I0)?5P@I6vOXm9cRh{*#8zSpA+rK>s{QiHZ);lxP(iaT@YzYH(@rwU? z#MgCL+BFDTknxYQ1I{V`*GFaluS3oMgNH#CtsLxd?Z( \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.svg b/docs/images/chapters/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.svg index 0593eef0..f796681f 100644 --- a/docs/images/chapters/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.svg +++ b/docs/images/chapters/circles_cubic/a7dc2e51b90e89ec62e4a328d2b24635.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.svg b/docs/images/chapters/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.svg index 8e8890d9..79c1687e 100644 --- a/docs/images/chapters/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.svg +++ b/docs/images/chapters/circles_cubic/acbde4be3cde3838b99b0ffc933f1f89.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.svg b/docs/images/chapters/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.svg index 2c1af2d2..7b312e84 100644 --- a/docs/images/chapters/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.svg +++ b/docs/images/chapters/circles_cubic/ad4d512cb92280ac88af531309cdbb8c.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/b985384d01cb32d422f5d1123707ebc8.svg b/docs/images/chapters/circles_cubic/b985384d01cb32d422f5d1123707ebc8.svg deleted file mode 100644 index 8cd29e58..00000000 --- a/docs/images/chapters/circles_cubic/b985384d01cb32d422f5d1123707ebc8.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.svg b/docs/images/chapters/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.svg index 32d3ada9..31000ca5 100644 --- a/docs/images/chapters/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.svg +++ b/docs/images/chapters/circles_cubic/b996b7c1af4c9187004af7d04b3740a5.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.svg b/docs/images/chapters/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.svg index 00f95bd3..3e393329 100644 --- a/docs/images/chapters/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.svg +++ b/docs/images/chapters/circles_cubic/cb6686f1aff26d9f47ed4c695109fd5f.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/d880c4a1b3d7b651b054b008e952b493.svg b/docs/images/chapters/circles_cubic/d880c4a1b3d7b651b054b008e952b493.svg index b6168dc9..4032c038 100644 --- a/docs/images/chapters/circles_cubic/d880c4a1b3d7b651b054b008e952b493.svg +++ b/docs/images/chapters/circles_cubic/d880c4a1b3d7b651b054b008e952b493.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.svg b/docs/images/chapters/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.svg index 30722c09..cfd5327b 100644 --- a/docs/images/chapters/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.svg +++ b/docs/images/chapters/circles_cubic/e0d46f5fd4bf01e72f23495757f64448.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/f47561c3870425499e31c7527c38dc94.svg b/docs/images/chapters/circles_cubic/f47561c3870425499e31c7527c38dc94.svg index c4c12810..304fbd64 100644 --- a/docs/images/chapters/circles_cubic/f47561c3870425499e31c7527c38dc94.svg +++ b/docs/images/chapters/circles_cubic/f47561c3870425499e31c7527c38dc94.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.svg b/docs/images/chapters/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.svg index c586da5f..4ba10f93 100644 --- a/docs/images/chapters/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.svg +++ b/docs/images/chapters/circles_cubic/f65f4e30a9f7a08c5c0092a1a3853922.svg @@ -1 +1 @@ - \ No newline at end of file + \ No newline at end of file diff --git a/docs/images/chapters/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.svg b/docs/images/chapters/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.svg deleted file mode 100644 index 4fa187ca..00000000 --- a/docs/images/chapters/circles_cubic/fe6cc524978eaa4f35d8de32c3b9ad94.svg +++ /dev/null @@ -1 +0,0 @@ - \ No newline at end of file diff --git a/docs/images/chapters/components/1e6e38f6403dbe4c8b80295a94fc6748.png b/docs/images/chapters/components/1e6e38f6403dbe4c8b80295a94fc6748.png index e8741a4be13a7f8badbaeba08bb5ffd08463e131..864f6ad2367f741da52327cccac98415e8cd8691 100644 GIT binary patch literal 18898 zcmcJ%cRbhs{y+Q{sjN^$_B$a;$c~7&ks>6iY#K%xDI=jIZ#zk3RA#hLDT=Hz8%T?+ zP(~T0NaDIbea<=Gb6wx-cf0<&oX_oc&hdGV=j%BhkNbLxFf}n?XIafcp-|Yj8Pd!t z6h<`)g<-`~X8eZdQN2C>i`j9z0gW!!vYdE0n>PXF>TbVmJE*_X6;v>fw%Qzn+MMGBX1Nrm$+ z8z~YV;WsH)KR!&Mn46ooDKm5NUwwOfKO@EDipDy{C0s*V+su|ffBDjKr1Dl>0RExj zC&jd1BAiZs3`GhX#Hi#awEADaVq-}aT=XkC5&+KUuL>S>r3)-}iII+y1GT|D!eg%_F$x_Wp;>kB+;Q`Tx-O5f6Cv=H?!+ z{(7z!c|+q9qaPy}7@0+tl`VZ%7Jn+GO-)!kS}C5Abx6>q1$^tCpHtC~<~yC2C)IXy z&+^rqzo4Zm@1PpY)xi4)2Sn>J|$q#5N@U*2@+{&~954d1b@!!4O%gM6-a$0CAy zN}qZjp0TFxmRhOzr0A-(?4FSxGEay2ep*N;?sHp}6cMrQ^!(08S$a{tgPo6k=L*I; zABPKVKC#m8YeRyhgNsYVw~mK3&!6wu&lTY(+rp@kWNNnUfC9g>!Uu!p7n=qOA3U(^ zSeW}msy}H#e0jFz0dp#MmWynm{TzN1b|9m=ynAVd{}1Mkwq;V|R$)2}0s;a@o>CkC z8Q$4QBWZOCmzm4t*AFX>wBO&lZ{O;p-A`gJZP*_fOB*Y?cdw>3XS>&6vzX7fb{>l9 z!=pB5&z=qRbhot?c64(3{U@&`R#5GX%$_?vzkhtelP^wDJ|eAtVc+=wOrWPR2iH*X z%+C?0?kA@Xzb_OE`ulT_$+qUL>6?9PQjSz{EXD- zv=RBQ&q>aURK0B5o%^a#c%dKDdl{8vT)sQPK zJHh+dv7#d#%E`&e_Vwq*>6hXdK#+Uhy-c7xAQ z>tY@m%as?;T=8r{|Nxxaa|>n{_I>8R7>pcg*}$4*Zi z-N})6aeI8oh}!6G_yW-_TZHkBl9TUv9UMBnMl6l{`win0);zntUx|zN*W%HxVm*C* z=1+ZndUb2but=p=6kWS^F@;m;FTeLOV=cceDtpch4(^=yS&wPq%3p$_&hGcHLEHPT zaYmiEWXB>33Q5&lq#WGv`^{@SWc?C@&!qre#2Wy|zvw+>l*ZGvF*LGMj)h zo2{*_-^}DXN@draOGbhJR9a$UqNM^iKA)d;^p3v8qE}b7>f+ZJjE@vlQ03*hE?>Sp z-Jh_t`PSY|(W{kMczJnC0)Hz6{3@ew_TBQ^-X_)}C0tljb`j<4w{NV1PG4T8s|L-g z22J-aiK)0cWt=R>pn38a4Zkkl^3UQ<7;%_!mHAC9TC!aDX*B<)<{~EppRq2X{SVx? zhOzL-9eOD!sNu)i($cb&h2>0K+;VGcYkL=$WlO@ZkN@mke(1}e^9^0v5+@n)zsN7} z`@;M8&qhTxE$KsXU(Ue5U{&h9mU7MIb*|K&JA$rnZ$%vGc)&AbKda zG01cZU2U+de6zE{-Z`hm$?Wc6rwMZvGDYcV15xn!&iwq<^&1yqjqyf1R1 zM@7}(y}%U$%1;X&YMP7P-f49{{n4qn)~Syk`;rNeFhk@>rp2g~XI{yxx> zvwhXYb-O-?exa^eBOf^JX=P@8XQ530{&KaYA~=|hUm?4)ZqcGeoZQ@ta?B%8 zGzGWJe)0P9>S~;R%xVXe^xttjldHFH-_9naz7pS}ho#*Xbe61eHR?=tNABle-#SV> zdl$XDq8WAnzM}KNgVk@}^4{K8tn==UP0O{djE3TR71Mi*9Q7uLUN2Gc=%SgMbK5=% zTt>+=%~&KTDA@D$tF?V7BSq3At#Y*Uv6kP*sw#$orlgINMd&=&+}_alVjZ#^p8oS^ zjg(Ygb6vl;+WPhDubJn6PracPFgeII_v>MW7*0yZgCn}Hu4uBJGEP@NKHgIm>Nr5^ zVC!Oc?|P}S$L1?UMB+%-#8aF%%JKW&zf7EaS^IsLP;`*07cXu%-;`;etk{q@KQ#hq z;n;j_>-49njV!00H#V~D-MhEGsYw?*a!Vez=3YT{pQUK;+NsJ%&R@FptK}UhcF2%` zvYPYxx%X|>eEjbmH@UXDO1Y5KhyJZ=tDkrOW2(RC; z!Qse}4eBQ{$fN)LGxbG94NtKM+wV7n-$|b9*XquMhRS>PXj3Zh*gU3Cu8zy_Lic3x_}{saS6vY-EfZ8fk|ZZr8rzjOw4I~`I0w1G|vhP#l2#M z7s}|z5--Nm($c&=1;WGc3gT|vT2D5@$I@eYHjkC}6}v9>@bJLu>-l`8KK30G4*2yo zV$-3Q{bPle<~U!q=f!j*kMEWMOut(Ce;}aVw_Fvx@ z>C{S6f09Y2^z=6*=+u5GJzMXI{_)}J=w&m$QtLZ!J)Z21`x)8SResWMdc61Qoi;`H z_nfk_vSg{pzkb*xs2Q*lOGyT%e&Q>A==Jq1*EcuzHC!JJIC0{{^sf#N`QW+0TXq#s zEQ)n>bRyF|fMw{>(X^U#?5u%ve*>^?IgN)6y}iB1zYlCL3;fMpU{_I##*$$wH9qsx zjiPO7$qS&wP}Z4T{qdtXcUGWJL{5C z_Q>ne>Y?m2Z`%}I+m>RygRMEN7+6@xLyx}k-kPPXthxh@i7lv}?i~-LU@*}1u&0D8|i4M^?EWs}}>ht18+ z4ZG{}ykw%0R*7^{LP97Sp0l$v&eM|L#w~adv81N>D_3IiYpebHYfx6++}N#K?E1$3 z(4lZFX2ywc?Z1uDLgMe7{5j0?Ypgr&)Sq7&Ch6*)Cr-S5pXBfEZ78|&e(vYa#1vVZ zN0m5T=*#3datNuP#b)Y#k*YHIv2;^rW~S+N6QLH5(`?N@zPuv!@@R{8&yDXK?5@x%D=RC}Ki&2jtEL(}rvc*N)mO`|s;XM~<_*uOnaQp8_M(lKHpr-d zGDwu!!-3+U_d3`3he%UnV>RaAYoL+mY=SfXV}oF^UN|Z#Rd$?wB&h01LHj;$C_dPdwFLE! zDc|D8>-}32f=s0oOS}hI2scU7JjsjwTl3{=a4d#g^X%DT^L&Ri?xsr{SbcqcF&~T+ z92&20?MhE;YaP4F{Og}TKYRVUB&iUZ`D4S)DsHP&1*PLNkFFfwgO(eCh5Oq1 zconb|=%~sfM#ivP{-QOcTh)Xw??ZpaxzWc4mOu7+M|tv@T$e=Y-8yl?n$1}T%cWMb zfKc_=&Cjet)$aZFEe16^6HA@x#LX4YfgK|Vs(0q=S9byxs;a8gl{SSKbMS2Qde=7l zCNXJx+VAU!drJuy3sF{k>a-UBp) z*LxK%vwdqX)&<`>fBAC63P}@DX=xU8UR@NHmfQPPEbdtE_1S^?#bA`fT^A=(2WpGs z6pEuW?_!)sw1LIa4f+hhYb7O@)Wr(bV9evgZ)5KriWxh@9>9V?+3!so z35I%feB>ORzVv;2dt_{mFS=>?(auL1`yU=9b%T@`d8gXSmKH8lnrDZuo~jjMkDy04 zr>c0M0y4OFKM}$%wkq)y1G1sj&1`q~UaH-o!ZMvaIq5Sy-4{3gMQc8S&}HmBnA(qZmA}7tXt{=l1`xq%ROpD<-Qw=v(H1VsEK-q#hJ3ra8I5JEV*ZNCSMit| zmbX_62(sgJjE|3pgoiH?)(VP7C35PiJbn1x9U*4+)!7rNrKqOA4jeq_h*^IB*mq<5 zgCkKQLDO5ua%Jx|$a4?Bm%1h;Yh4lop@u3Z20@0c^2w<`s%vBFwFAF5NG7Wvm#&TG z55-Q~0SF!=pi+yT0qDC?U6_k2(fr^+2^8%xQl@uZH+f&?zX`2KRNXS7utDB*_@h*h z(?z4TCTXgtQ5kgckRbO<+1OS-Qc>q}F#%APx38p-#lu>MS5#;bjNOuDP}9(`%|`?E zD_2Wd#pdyeXE+Q$=eD~lt1;_z3M^i!cl4+%Dfy^989HGsWOB>O%N-mYtN*lf@-sxx z&cq6B&O}2{JvF0<-Rd?bSyBrd-t}7Os zX`5lD?SWd}-#x$zl+VS zfq{!#R!F9w@4!J6VYDSK%Uf2bu^-(IxKmrT#&U~_k96>o9#$W}1~a>S_U~A5I1m*X zuD(Hzc`xc_EkIuH@Ngus8Q<(^vb(p>S!E3N22K&pz`*J6%un6P;(A*x`546^mhHWkmiA48G*&Es;y;4t2Dq= zMQY}5`5RPdJ2VVLU;I|0#{7=8+Kv>(W#uQQYJIy;g^L8way@zSRPOZNy$u0;B z3mZI4qcN|6e06hAfxeLui^R?=(Yzb>(PPD#uHST_%et>L6$K)hZ>*T*PLMFxL8YYCp&q&sD2PS({NBGUT3B=S)~#D{ z@H3s3)kqi^`?zF?QP2zL5F-1HG*xuC zu-mt9lfee8RZ&?*nwr+!#PZ3}P5}ZHz|{w>swZB%FB|%#&+u&G^9uo#H%7v&QK*Z& z`t(UZ_gN2)0t8nR$s^-AJMBdZ1i+#G)h(y|)?TLPEj78iC8oaK5Pff;J1xD-F<}?& z*%4Qt58z0Jfxo@mAPEX6yNBTLyi{_aIa{|s^oaZXBGn0GmvDddGp+X_VRU+zk&#jL z_kkv{ctb*LPo6y467R#2nx<1f;Qg`8|FtM^3{fVUn@zK`vn$`dTZN9AxrU2t;NIQ4 z)n^v7dVYS+zvGGqzwSAae38MN+o_nzC{je;8s zD%8L6yO4tO(o5_2o>4f^NRQQ{S%_}jxKYuil?l}O8PG35^cJ`G?Z#A49Z>0hjdn(% zsb@h|OjbK4@y;~(Y|EU_aY6S`Q)yC9- ze6&0`vcIK>PQMSZ&f(zTfF@pxZc;j$OOMWNXqk#x6ge{09cKAO~-%He;$K&2X zlXh&lu$C~O6JuR31io>_+ukcG+Uo2q{@_T*Ij%MO_Py0%r~}+WQFOyKM6Y$K;~=#x zOFu>j2f9SLh>0z?LH;Qj9*DBlBcFAIKmsh)ym_-5!=n~s%kruFNTIf&tcE})&GP7e zlIa9VYPYquAbLGB5Sx4FBc4l_E@iG6+E9MPo6SEfM^ZRUX2{d`T>%8Bvurwa=U zXW5zkex9SVrRU_VCCf|NCLDw5|nH7BzlX`FB2NcsRk*tz8*vDdDNni%g&FjoOR zS0kMILPEliRU`XBWT0;VB*<;rBx7dd)FE|E9H`_R6n5ehKa;sVWM!AC)(vbbe6 zkyRXxIr(HRok&^~hoCAYI5>EEdYT?fqlwe7hbC&4YuN+mmiYgW$Td!7-nDBNc_5Y? zQ>j7--9UYnuU;)bdh}@K8vbnVh38QXoR%Yc!=%KrmV!*LsTVdB?M{W`Ej2fm2-_%>BwQGfu`Ufne0=Xts|pN}~o`n&P8A&VD&M@KMKcAN8PR zO2nhQD8Nfl4$rI*b9EHvGiT1o9c~td?{Uv?Pd%L;eVRtCLjCZ9=1A62+ zrDod8)+HL<+Ix?gkhKeIxAzPWyIypIe!C?X^|SEaJzZ^Wio}lO5F7~jEj0a6%!}2- zqe7_5=U_m1J27ln=aukdRQH6pi8qAR3jaT8*kEt(*zE5a87e3Tqo13bLr;4D;D|s$ z7bn*#8hY!*->L2z)U1Yv20k@!aV-nZ__vxtvwYy&8=MvR0e!;3A?UHr7TNInK79C~ zr(RML6R~V=D$ZzY!EWCyr^7$VCITF}@A8^?jX~^L8uj-s_LkpPaX-7v7KDF`eFXVi zSgZsXn)vz7ZjGpD$g%DSLmBkif7L9f@BIyBnd|8m5(!%+5>8EOms-zVA36Ynjr|vT zivQ%tm!&p6ZeL$~{SkL6*tPx7Pot*h;lGowccRrlgCf&%$7XYS6a2ENx`5eBX+H5V z=&){uL4N~C?L~dW%RLM@#sfBR2omYV<8?qcB-kH zbq`WQpVOER9Xm8_IT03mnGup#@25}St44^?7ZO6bWnIcmX}<5W>mryhn#WUY1q!98 zsHpJa!|?KQO(>6|%1Z|c$5eU!{(iu{{r)nj zQ7cdc>P3Bq+m;>s{JaVa==pc%c#9XkJ_N=k1*oD9m7>u9$3ee<0Bky`XSIw64;~~M z4zcpoE8bPL8STDYDsw|FjC-y&;SZNA;XBT zUE7Q1KnVf1ytrXMFNnj2!YYOkP(gb@3lWXco}Li2dPWKnCQwIpa?BkdH!gmQjVy9& ziQjxt^>EM%K|)veuQbK6=K+UNMK$ z4WqDMi@XPn0E$+i==LZN(k9KuF7XnL1W3z1G^Vr`O55BJm*5VM^5)CNN;P+?AEo|avjpn98J>QrYLoSoH+;ChCUHLPd zg#%5OsyaIM8!}U=PS~60&Y$Om>PO%LWE#M*_*ixS@8USZG2m>X;^MUHPZY18NepSJ#UFd1U=sU^N~my?rM?ps1ZM-0Ii<$cp!^`^HboJ&|Z7W^N=oy(~}Dm7p` zHNLXDTLj|hQ=CLDLBZ!Sc62E_5Lq2v-3c@enx0zyGv-ku2@+wKLKcC!~O^M z(0Yc0e*}xEtFHzru__Nx5oe1m^(_f<5mY-lxYSBoqTbi3o#X zU80F}`cpn7*Oeg1@&_7MVrA&Dy0kO>0|O3tfz<e&hC|fj7ey+xt;xsKA{p zVVRX*Amq@0j=XbI(8cXH4yaQ^QUc?vMQ`^vIhTs$@vL4TWpPj;lM zD+T8=aPh~HD*rrL9y_)g?QsEA#pj*H=cC?NH8d*x@;zVoVb`{wO+PoIjS z9D$FSH7RZe(k%l_+={X>+?Ee-BLtcdz+$!-VNd|z@h~CqOvk<`GG?^Ms8Pm09J(qd zCRQa?!49&unsh91lRH&n)YeTT^N?g06al0!Fi_z@t0iXa8vW=H_)G{Gyzk-Tj8Reh z{Q0xzv12u*_(;u*CfkgR&J+}E1nFaftcgla&toFQMojmdJx1jBb2yPam;?kaLZLAt z0{@_a#HF(G@!{ht=W}oIM-j9>GdYwj@5H?C{$Vk74yY|U08#Q-DKxT}R4z{b5cqRD zE-Ng~&(BBPfKgCL2>XY<*(jm!)2B1|05K({-2yesFacp?n}O^?9KVNer)4w4=+eQW z$sHfIIJ(g|yACoh(Hka*->^br2;Cup#-vD-F^XV6)Uo0de53kCnjqJZy{C; zj#d;l63w-(*lo{3)(3?M4$oQ)l}8te6UzeI7~!M z9#HCR2+YPWuly~hbWB=)zAdneTeZ=)=VrkZ>G9&sABZ!#+_++tWA1i@YV+nF$?|!M z5^MLVDWQkSYg-w2UfrSxNVZz>a0sAf2Brev!VA-8Z@lcAh5>Jw3`u^XsP7P8vvYDX zApGUrxjq`0b9<_3V+HYf76dEHTY1L_R#My>7p2sY3UO7gy=)&!{Fy_tQST=AEpMYMz= zATC|DtoNcZRp`Q{OXnb*QekPqN!0^_z?w7QlOS+bLoS(^m>{1-wja2Z&&XS2pOw^5 z;unJ0_s{RkFzrDm+KoEwpv(R9F(#MDJdzLO}E)C%@g!{VAavp4n3bf zeUdxz)#_i;>gO4OHMwv8DMM@r8{1)n#iylV$q&+Fb)4nkhZ66tJ9u7*Qs~*MGYedJ zW)~OFb4X^VPoFm2wyg?4RY~xaBh6V~;+PhNoa=W=5Fh&SaXaGfE+FT{WsXK+ZErerOhw>bzzd7!81~ z7g#^sRzn}bnNYm$6OI}&4G08=TCGiOnK2M)Ed&d}ge>tL{a6Lr3_wRxoqLFKd*1lr z-MeQJ5>^0*WPZzdBLTq~q$gR!Utylg3{WY6a0Z-$7{g3Tzlv4YO}ECGP5}O#s3jBsc@}2aBq~OD&}x;V>oIwv zV#cwhrD{Znz#)p@d-4>PCeg`0+&h$Io@0au;1`EKxom}m==$|+gy$e=b5t%h@w^vU zRt?mPo{^C#&|p+gHX|T736r#?`4*9A<6T$Jv#fRhX#>JHIU3r8(`cHd&kjq?mTdFm zbF=>U-9K={V+h3v$TWfjM8@h_tcPh{{WM~Q{K;SL5Pw^|qHR&;91a{ZOixhFiGc-8HdJaRpuv8%5|NX zrs&sbr0H0eFJE5%A?)B#YYtJf-QC@32Jmq;TORv*yzl5(qo`eI3U~+Oxw0?qdLvzK&1wyw+Ea^bJeENo=Lt|Keh+iKOvDMc0%Ak}8 z(KcbL$$NdG;hCaaAt2HWM9ou;5~an&#hGv}5k<-$RMijr!u=Qdj&V?GCmXev=biS zjoWu}gMtue%N&^qU2Qf&pfA>jba~`>&|>4|@OZiafPtq@`R`OTm&xBDfu{Xu@H8B3 zaCTRL>A0Ur_nB1$oe~!puWV|%+4j>&2EsHFR5fsTOrHe$7q_+bzl!^5wbSQc**^td zLP8-zll#_gR0Eo zU${H7Or_+|IkT@{M{x0s%CvY)%Vmi=RA0zks=lLX&4R>|!Lz?nFAk?iUiThoEb$rQ zMq#1Dk^??yiKi>p0+-^ELSaM|DCY5FyldfRf`c2NXZM0*buVv~N_X+5RD$jdUQ~j& z0c7KVz)-aUXUl%#LSyW|<3#zxpgLv1OGSVzXHaTkgr1<6q?ZFeR^lw;=&!l9%QzM( z0F;w+C@^AuB8Ub-^DWy5u;&0S6{3RwvsMB17_KUs>SO<{P_ZTDqLT|!r>2#M$WU=m>N$`6pC4Ri(i?(h9uT$`}Jno+n6#nq?Qpo*Pt*lUcozD^vXWg#c%Fa$9 z7#Qae@WI=|@OIGU@^U5R0~-*_ff=+qA7>(a8Nfg-{6sHM*cP*TIwDMt2$=E7*;Ag( ze%}tXvBl_ysZ?a(hy9(eYa`8Hy?n_*I0l}@{m7B+He=~vqoTm8Jkl0YlgFuD8ab1n@>RetM1fR1+)P2X2z&bUcJRTgS49rKOl#N}zRlx*N0OiAzq@;2yY=G} z>?kI?w`(q>IVZqMd1h9}k0qfzcMp3tE{Z>aBw;VOQT02!V8ZV+|KU5QAAz;<%39N* zkl<^zv9qYJe-F7S~@nI_fm&v zP+e2Q#n0ccEBAyHh>3YNv* zA&Ds`=zT=rz`>AK=P-*HG*0O2>kCaz=7Tno_~D! z0#8Yh5h7_d;&wjbndYx=O4Pl6y~4}O%T#J*&W(iR21Z4$G67VHrd5fS)br^GpMNN`Cmyxiyb)L+n;ZHYp44xa$sm_&Qg> zMAUspqsPZRTg(iX1E3IffcX8nJ1$ojH#(b6-e_Pj(mJpKBmw%|Clwwblk?2Wg`@lX zozv8g>Ep>AD(2@lrb+EmdAjs8Oz64U8JelOz*bZqFgos^965rXn0gkvHY|EnH z`5-*3fkwdOQk?GEe_NwobVZ|$se-^y8Yh3T0jxg@I^yaots*3f2m*$?h7US{%c14Ea|-p$ZIEu?oB*-ioL6Bk&Q=uJLNND-YQoeF3J zdxnColf3B=2RWjMumL}1D+P4)tE#K(P~%0{t;?@3@JD<^3`KxwHkV{=BC(y&xw4N3bpA_t$gP+BiW6Sd-oCG9VemVFe|kqpV^P9^ zI#!=Rm~Z{Q`%B?M!{HzgzcgPbLmQ5V3qn_ z^5|C(0<%8FwqEaQx#Ibp>SCYenPv5S`}V?PioFeRqR6>{`<7MFnTO9svm_E5ob~o) zQIne91n|&oexi1Sm6WK$p_$IY?JFKY*(k+3cb?e2b<(uCb;!{%^Fcc;TkSyO#nV7* zp!7wsnHFH2VGM+eLVR@KRHtv54Ku%H`IS4&Wj0AGR;}8TebWq86lwl6UUM>Cx!A}x zIJywWIfcU729Yo)>H>Bl1j#B&ilB^)43y(u#KuH4Ei+6#4=EwT7y$TG4KnM6fHF2H zJBi983W?B2+SDTPnT_QhfQU^^tblAHnz9fMpo;H&@;m|4^h44zDBIa}-RPSSzI+DZ-oKi{d5#9-~Si zeMNblXo?^fpM(>z@%}XRA?9lD>Utx`0$=>-$mC!P2}418kWpvh*|lQ_8+kAEg9UF< zH~SbH>URiaqrl19Yto64kG1i^f0Lm8p*x3qJo(KGAr9ls#CRRiI$*fXovBh(BNrI( ztMe%-(m)hk@m-`QA#Pr@K#=p6zPzdx2gVAJcY&Rw#ZE3+qOPrNC0a7r-MUG>e2d1# zmoUIFeyt!4auwiXMX;UE${AKXLsPe2j2pft1z$tKzKH=h=LwiPuB?VPh(If=fp9~l zoRu>kQW3|IJvti}=78`B#uiz5Ek*Eu43+7?%L)g^QF}LBCl%%tSV10g+BjH}>MZ47 zU^o#I6I{>q_C6&Xu*ox9mGGTILwg6AnGjlJ3Dao;Af`{&fM*jg5gs@Q`Ah+91bf6L zfb-XSOm4GaV}AEvlAegh9zZwL>SczwXOaB}ejL$T;Yve1p^%UOg2lug@8J>5bW|vj zs6!-LSy`DL7dPJnJk=<+W{s{ksl$JJ1_l;G1wt8&jCE`~+Eo?-<3SAS2^<`PJjeK~ z6Bx@hx2{q=@S?Sq1jl*F#UsOD;yp=AN7}KmTtHA67edzWy~~6K!HqX<l-3Lrm ziR2U$>3XdqxDlv>_!FK{LOrqtx9qme&dx$dV#1n``AylXtfpPH>n(0@(oul6=3B5B zkvS4UNhabCYW0Nhv3(?HPKqxiMC_}0B0RjZPfY_IU4u65U8*rJk8}er!gw~qSux{+ zR6>q3qV21Mo?X*KP+`--=OKV!P$R&B6qf!ck2YI(ZmW=@%`3!(D`vm5A{LoCZRu)Me(2V6KVD$c$t;jr9FFp!s&i?tijm5rz+O--jSz ztGmw20H^fo@E?l5ch+z%^Z1KXowMcLeKx$m^u?Vz-gp* zL5&p-Gj}iel!xXtk74Sbk^9JbO!y!|z#xrA#b)}VEf8{w0!A)ykXh2mI_wRw2eTp+ z{xZTcT%cnE+p9=xo7{=Ypl8s$9d!*Ca%v!5Lt;~4(Qo-L>6{V*i34}%^xUPejTKmk zLj!Drv{L1UsfhberBW6MUgOof2R%HlR_K$9(}uf(W+x0tHXNel8JeS_$Yc81O(_*e z{!y+&(J#SSy`Z~7W58X&5Fu!@@ZG^-_(+r=Pb;ne`o9j-8B7@h9mESnc!AV40%!T-)w=a2wTN(oX+fKgHwvEpjm( z1mL4l3PxZ=Lv#yNY#Mqu5!5i;!R66+8@zmc5REyFoCa=MfO8N8kTLe;4+jJ@K6%Hp zQ1d2*|ATKX5uMK3SrayjHEv}A{nEgvjBE;|;1#hI3y-Pk(P|Ik@{Zr!tQruAo`G2A zqW?|rGDpVRj_;>XmU)oB3jh+rMRM&psSuwRj$RF(+Y9|0 zokfh;1;p`#0R@Cgd*(a|8bNGJF!g8Hi&zG{PKXpG2>tk8;tWVLD63I`O2`#`oM-Yz z8tse_qOfQ_BoO3rVn_D%_x%*eJkOwiHIjN$U!Mbk0G=znx6>##*xg2Sa}Tx&SG-i28Rm3F>?*M5Opqb!+uL2CaO83vYOvX z$v&jKt|9qO3HkWZnxg%`uO0IzU|{)`L;Rp>MM*q368YGOr7FZyCn|h6=KH3Wd2&+O>J{8~hGk{?_t0?~VuBNtw?F!1Z^8x~gih+^6e>+3INTD09&!-n(L#o~*H*83bCcBwn(u>NLT zu}^v9t}dg`{IxpWqU5u>Wvbnkos?5o9U1)byYPd%%-e~54fj5{v$3&_zv&en9jnlN z@O#*ksbg+9`Ca2r>7D*J;}50Gtbu7lbKdZNvgwWBuEjgP*agcA%lqCdouDsO$7LwX zj`rQUMmPr{v`dAP8^{3f@*bI?9wOewZObR>>54XG^-2BC`Y#URpC{sy zuYD${cIhU!EM-u^ulxuqco)_2?KALDWxrPYCOe9dkPsr!J)WwAU%#d$5cq-M2Jqw; z1OQk+s|s=maxB3G+a=c?Jxa;Uq&^A!eOLM_06OM1;NH;Hgcbb!+fWiVY}gQ3+~Kmt z_ozuKu6_VeAhOtzc1%f0DGjP+)U!piCq1fB$kcGdhH~^_>i6&7&kt@ws3Q|zhNZr* zuL|YJks~G6vbg&pFXNWd@h$hpjY#}G1yo)mM3^jZCIZWP<6;?n$nxhqNj;-E|LvI6 zg1D~H>1e)rJA0P(T?ju!;+ec+nN?O$%kf({vD95#g&v2 zA-&rVsklx}P4)Sx9}*fF8p?S1FbQ;Z@jc94q1AdecJ{F6@@%24t>)6w(r4I zU>7qx8yL7ngg5HIOIeQ-kJBMb+D5Ygd1G>sEARWMogdB4Vb|(b0v+ zM=bI8AA}XMN=lM}QSd}(>TvPX@5g6$OsAxl77P@=bK&9qA|iGz>37Mr(3H7()S6Zz zG5c<0E##f#kAn90_8|!0*r!52Pf1NJvRaRP1H<^Y4i%($2S-P9uU!klKCnN%>4mcb z*HWf4+u!5w5*&7Qt#(#e{D*PePXmU_tzS8jxAJgF_w~DX;V%xZQBg^1Z*O1Rk#zCm zY;Jr+eL!X@jizH3e{ToWU_~XR#Zj@C0{GzyH;rMO$|gl2+tl3LOsT|P1f*iTcIOVq zwC^V+woqs$8p6vV$0Ecinw*fD%32B7c;@#;yC>qCHXZvZYwpbB=J5u9==ay;Te%XN|cQhV%a_k5a33}Vy%n5C7`fm^gSEP2p;fM0b zS}Le`aY$zI(v*xy~}Y}mMi}9 zed{BO`#*nXPKy6~6xfA*dj(a7>@nnvR|^O*5OxGSC?P8wd*K2%^O7YlZf;o_8K;3t z3+XrGB@Q_NIxi4+sb#N(=6f+XJNGiExFu25g5R@5YTq5=pf#PV8Kl<34k9 zEz{GBrJsEU&rE9ldI)Edl<2s)grp>Nv&|t518cLgvdW;CynGo-rM^~7q8myrd_D^c ziyT{MRn=As@<8hl1H0m@D(I&n%)-j*Rcg(85bfP#cbzoVE$M&0W0L#A@Abc&oF)Fs zaeXLT_^fst@~1|yaaZ!Fk)lSun~_&f&IMOI`!hui4R@wRr#&GPzAd_VaV2F-@SNc1 zcod*Z|17~tt2#02h~3!_?zQg98{3<|{BXGU!Sa^Vz`C9O0+O$s6>eKSAO*|YrIesO qI^CTNp8PoLn#IHa!;e6tRtjaVq{}Wt6n~_NvQ5u~c1Oql%>M_{}4DF*mpMl(O^l51ut{6#Uy!b?ywMHjG6i@WV#IFUJh~mPdq#n<`6B z6!YPKCMKpBWz&jg0YUQ9PV3?sN`7>*ivIPhq@cmcG1F<+bDMZT5NvR1|K;vvu;GG&ZI=cH9XK53et>Ybf$t zywDbt&2#R@=eo+XGt$ReZh0QMb>!$#L1}5mGgE_XSE9t9zkIoFIW0XS?`}zN$u@?{ zYh%IvW!K2~^G|Qxd1GQ@1r?d%v-FdcrDbHq=BL!OF72TBh-OOQt^C7W@2t{9vy^m)`-4=q7W#KWk9UcNp`{QUf% z2Oe>pIC0|m+v@XuBioO@@cvNt;>Eto`N>C$r$1Y`YHxa1N?ki`sS)q6DXw{AT(gaj z?THgo5)u-o@=J3+uCE%imHzYR&+Zf5l9bx~gAbn#hHZX1+Zx>4-u~@o;g*@1P#p}W zO{((RtLL0y3_Gc&EPQ-?I(m9f=Dv;;V`d&dc@h>yWjp?+Oy&TH!=1&FExQ0subr#8bMY6K61c;g38N&N<$~j#fQ>12OdP@ql8&ryf zgvz!@@$O(`X7*3r9g=30uH`*D#`NOl%PlGwnqxPY-iXqv`TJMcO|Qv$b<79jkN<4J zj4bY%4I4J3dWgm6~zn6K9GgnqtIw@^ta#Xh28#>EEra{N=5Q}Dx zX|-2_(blb7FZ1r$KTsU1^Bjwqk>YKD9ldGCjvbpsMA9=dR`~t>&S7)@yIEUIsLtB2 zaJ0n8(LoH7^-t{ z>9f6~j4_QqzuyIhW($)SRwpk^ewLzcDKy3YKSJ}Li?lq;c{DXOg>cBJ_d#j zioSEw!SoyI4ZO(On}+@zWz5-i_^{xbHEY}_uC1$Ypc{Ipx#Q|p&a{ljj2v701yp<~ zCkC7X|NQZ_s`NJUvikR%=&U>Owt5|-8k(~nXA8Qqq zlVdYTR@pVu82R=0L}9u?iW=qO%*R&ySGV8BT{}D!{8v_%!ii^(drAKRp)*Rii)l(tXWXTG8#TSVXdeKFb5MQ3GY zkyj`xF0Q*Gu=`4K^1>U=H9^aUhlh_3RLEr*JN9*!*QC6D==S#B$=4;00T@!=?EGFt zPKMK;O(A3aS9LeX#>IVY;`bYR9L7?1;ji+VwQEyN#oMaSpPT#r``t^0x0qswKBe;V z^488Px|w6IA8uvfSEHv3p`ZJ$_LlQTVj>v<+hojr`zD4Jw6xdG^KRG>hQe0c*2Y8% z=hGOWm}}Se506~Bbm@3c0q+BsK7W+#6?Am=4Z-VLFBczvvgYog8osxFi>m#l&gR$N zPkofKEXn@-X0Yme_CwIaC$goH-QEzeB8*=1$LQ$6=?t;K-@z)s6l&;9WIV0G}|>w71;<(vXh z*2UF1$Oi3yEF9uB_*Ok<&%S++Be@l9j~rQ%X&i=Cb8jd(P z1ld&i*nNHXMA36}mD;aSmW(@h)J~r8@$u;^N9AkX*Rd4rR2NU3kALpx?9tWrN5p>W z$_vqs<|8rueWIr+(XBd+%)DNrJXg87-p7`tn1lq&?oq0?w*Bwf@!sn5^V`?yYexV4 z+{c1^a6`e3Z-1dpB%i8}ih@>(+WBY6kEnu|jB|V^-(Zv6J85EK(w1W)QmT54BqTH2 zxZRe8m3Qv!3ZNxXks$9H-W1J0^11JR(Zh!rhU)bD>xa@X0InMKm-ACK(dpXJX?w3r zQ7*QnYcHqxeE(#2`UQR`?yDjz%gTRYPT+2SK0@pgHV=F4{g*FdH}92p>J+W~`{Olv z05hOqs%hLL!9cSD}pE3mo z1r5iW+w3Z|sggeNMzOcQKUX!S;?scx2Zn1eEf>%GDSJqdD>m!w!Y^MrkBSsg;DPo0 zYE&B=KCBig6r!wap1${F<=`MFCMYPV^_B@T;fEWYu%^Jv|(_w9NO{whIftj=e}x=ik47|JcAIg|zha zHcV8T!6mO7m}fwp$v5Qzot>r`Sy>|sf9K~W3Tv+1yt$f%g~h_sG7Z5lt)Sp{$0J5J zQ(9g=xV&5mDe>FP49)G&e}GR$KYkQaSLa7mK7IDgUnBKmPL3E>-_F@N!rbQU%G0M$ z7u~-f5D>6CXkLQMX#!FakD@yTiSJTX70G8=4_2liYq_%xn2d~$ns#?`a&p$z*DrH) zbnGp4=0Gx8AtWR;J2}9OJ%O9C{FT+A%fw)x#c$Ek)zvUIWiwfHUuEoJ zA8L9Gkar$-Z+R`-FmbmmWOCEeceRTSZt%F16 zH6G-sE+Eo%pL>g!fBpJ(`s>#^1Z3C39oa;l+*9$6P7BrhJb)4y_*6peo1%t)e0e?A zk+q&((sBhd>!ntIuSkv5+7!QqFci*_Kfh)OoCOqad*FJo#JPuoM2Ka{aT*ho4$Ib~ zPEKp+=;+V}sr)>)a^*?~z25PTEG9O#AjDupU*Cp;f`YMEsmpnIcoZ|QgoZlB))SmcS~Kje|kmWhK#*d{>z;RV<+>jJ0HfCNI+a#PUrr6*0IFxEuWBfRA5L5-Q~-3 zJ=aV}P&AN?CD7)1jJI!AR8~&g9m2pV@51?^C28&U?c3X!4&?$vxO;e9iip_TXOBHd z#=kH(vL1I4+Prz?m;TZKq(pRxe_jldTA6zlSsUg1Fq$p*)2ACV_QVBN%Dgm>$nZqm ziiwH6mtr+EGMO>e*JmJ;hD$ZX;}1^{)lH*BPzBrOHmB^F5o29LYQbo8+}uxKAtWiU z$u}D?xF?uGI~NxT)hV_8M|J0c^kdx|DVOhnI;SGavNm`PF#Paj|YI zoyX=kj5l*gy~xr0;>Ba6@}e_SHnONB7}4Xq2kw^~4rg?%M!FrI9&S)P`)w0qCIpv} z5I^;)lZjVPk(6;BMMivp>fs^t?%g}mmrGccF-}fSa>^cO>@RT?QU+3VESj71M$sUt zh2M9Yrn0I^?o;G8d8Jw7uJWu8nkkikfOO`DL-^f)d_I7h#`64{UUQD#yY!mUrTa$8M|js3A*ZGq#w|IEn%N`zx+ ztLHYw6MXP>q{(A;%xHG%Ek6duti0ah;LRI1Zi8qbnQiiIO*E#!7GaBA&{VVVeXF;( z_wmk0UYPtHy1Kg4Gp*|OuOGM`@5*KS@ZzS)WOzyfn2@gHsTZ#|KYLa`Iq;|$b6?`v zL3_ye>&iC+1Le0QbFQYQ3fP?cA_qJ+qgfW-+uLjV;)cxJY=^;Dth7C1&7#6fY~g#M zpB*-|Z-9TO&U#ihHoL`zzrE!rMFzb`)=5}bm^#Z_AB#&5ajG!f>F}oVEX(|lFAX=h z9~&AO2{<*;oeGE~?n6s$J@-D|ymu%YP>@FX)TclMA^`ye`Vn9aEHNCgUTw%!UQm$N z)X?xEQV9*PiYAJ5(TM@&{DXHx^rH9-yEJIxnGhC8YeTK-i;uB$m>3x^qEhD!9bk+( z{@}rb8!BhFHD2BjaDM)WU47tcl48_;j|o+W?V@HZPnK7pW8m4@nykvi%bS1dPv9BaHcUKMnCTR zqwkozmlrE9YF2J`Q4!;gGm~VOzk29)$Yr2{4KL5eo2S(~Iz~Aq!D_f+^XAQpUSo_X zQ$~e0s+ajxxdtAc3Bh#yEEQ;9PD@KmmKrY-)RAQ%v}w}{y#(1hBtY4PU{m0{QBS4Q zU;2Xp_i9m>6BXQ0u*%0Q#Os8$*XEo|z4*-g+vw`UhYvII@qBSLHW!%nnWH+ zL+-UJO@pnZnn>)Pn%G}*Vt@%#wI_z^ zEWdj7s_oQZRd26`jUy0Kipi*V>=l zO=XMK+2lQu|8NK(3TVdeO_>L&Q`-FJR}ggfH8gZK3YjfNNV~{mM2ny#kSh&a3M&Pb zuUQs1w^2%~uo2_&bmZ*Ns36MTTMAuwgaI)?JHY4d)-_Ib#25_V0e;=>BbKkYoA|GgapEUA(}moWOhl@l?$S zcJwN>80735?U45lM&@eRPAH*vBqQ{fh<#{HiAIJrKfBNM*J7QITw#`}xS3_5R|HT+ zV-#-$5UO3@JqN4q~vHchWimQOKasIa0<|9{{ zk8D*1f-%qRNT1C~HqLqcB!c6)wJ6yh`uh52)md8Bt-#IqEvkK$P&Al$c<9fZ`N`E~ za!Rec0BSGQ$v_%(&}H6vJOAw_m>UaL-()udSRit6Tnl1!$vKUJ~L@0HEPTtuk;=qg3g{j zVrNn*t}af=73Y3R;kH+9-MXA(l2zZ)VdAw(Fl`$R zW7wisFVM4saxF{NKen^{~2D>v*o{h6e+V(+hqbux|H zibF%6y&oSB!+?ZTRCsC@es2Ni@kET1n$e@XcW<$Cj}*a(APqulYSgDvFRZTmGkJUl zH#{|5ztZ=6m-!O+>-xHJoS&M(1_&sF7y1KUg;V{43d-LqH#ax(6sLh-Gark+#*Gzs z?65~_wK`oBJDb+r7KCLH2fRkLtH*l|O-)6k1b|egnmV{u{}Lv3DKIc_qr5!(g$ozr z`(HbzF4Ad>?a}=A-g8)sHaZBil!nZV#b?TSf|6YX?t(Dtq__J0eWHm?ByX2U0*|$| z=(sjA4sq!d$fHfD@l49{ZZF_WPY2n4cU^Z8+Ur#!hBF7HlrR2XU$v9nDL?<~t5gA_EKi~SbPO&hJs{(}ruF^Vp->pNkW8~s`c7__JBZMAY2x|?T@#LE1 z!LrmWF0=U^i8pS9`ui_?gg$oh}{|EMT{O(n)JFd}AsU3AetzJ$CKN zPD4O{K=OYx2!90hsk7?A!NE6V9hn#YPWh2CjalCXWT)ALR5Sgy&*1s>t=iby8oPH7 zkBx~c$F}61jys7>LaANQo=XIB@@HYDozzMtg4jF9`kL1t^Oa z_m8eywrm;6XVGfEb;#X!kB^Vfeab7eb#b|jM9GER%F4T^kh2-=02`axr=c1JVcpDo z?EyNQVm%YN%oLO>_D>-RlLF}Tb28FE?HAZa+SX3SBrxw&Ik{HrL!AOZE(r<=6}t~> zytuiYZr83|U6x@+Nojh>r`&`=&M?n;s z7l{h%-}=t1Q(OFV5^?yg!(Grg4=~Uckw^M~>R%tY2h}AFOV4oMd-#x!LcfiFMY`wF zstt9k0$$M3P!i;v_n(M_T3I>MvZE{VHalohmfr;>#yMmsBY}iuXJ>nI^FULmgz#hs zEKBmMv5ARhlYk&ESyk89^81oiHlVe5jA-v2TDAX+a1+uBa$Hb)vCuLu+SDNnAFh= zWQ4xpw%glk9RmZug8h)N5cA3Bw117;FZx!dCAjvN)FFiO=3R4Q-n~1#V&%$VG_}2z zXSbi378W#)AxtiqJOaw44pmDiGcb)`0j?*oPDfwA5t9p`C#lSwN~;%TDy|94M#e^G zPd8`z>u}~5-Mb~$VOu3?E(R1Y$D#pDm<+3oVW2A84P>@$W9F83sRh;cT-lDY3Sa_E z#}x!1W==>^ajx8waHU2yMP|IEcHYtpx}2xMRMpp?Kbw%|83CW5=Cl=$`8ky=M@P&Cy~H1-VWW~#xyzGM z5rgE~$B$R+4yM0UTsZQ;cXHSyG)8x*+I+SD??`hj5 z_0QqRk*fpeSb+G^Q6W7p8jd`(73=x87r=EtSipbS<AP0t%oIY5VNjHd`T3FnTTWb$>JuWlehHHD5i0EcD$pv$Zk0|&$_H@!gLfB6%`xh zk0|;rt#`f1Ow|R@*#o$?`mX7`f6M?B&qkmS@ONW)fyfy%)9sLnP`z_K{$BqDa#$Md zzQm#GTZ=lOp(`MQSX6q;B&(eDN2%VOYDh{F*C)s)oI6fSKz|4;tp2T=`Fh(GpX~hR zuRnfhfvX3cLc#>6)r2QJuSOx068-9xArUpYySqtK!M@doeyFcX zDY>BF$L!bZf2-3HKvWYp-gGX>dk_+I5zrKd#Dk1xHi5!tDzkj~@_i|4yr4x_ty!b7 zXHOVlO2#fwnM@F^(J|VB*Lu@y5`jS#KX?#~C1z%37FRini7G?B?CUWw*O;H*3W^M@ z?mNIlQ3;91=&tD@D|~cPwYL5;RU@9e`gVQ7>eZ_`w;c%;uVHKLedxx&Ki@K_)VZg= zxp|N42qTlxd%atVOlZv?Uq5sq4B4Q~1C8LV9cLKO7cp5rJQ2yZW&dr+8_@XKFc=z& zfk7*W{(d<3CH5$30`cVu5GkTOlGZEsNwhT zS3;TrLmLsbo9a(GO&~l1HjW<`9rRl~U*MY;)Y(7k^`)PUL(1x+m6Z8MY0F`}7}*-o za%_(s6G0|oiXTCvq6s2r=;OyGtq`5{1LY^_KmuriSXDUM0|SFwiXO|*GJoRJ6(S9UgakeCORI~oE|r>)hm{8PcPS->e}92h7`Oleh@Ff< zhoF)QY~8xPt*x!--aTgCFw>p+DWjB&AS*!;=Ukr(-YhIE2+{yM`FQu;n-~YxbS3N< zb%1=Aw^q+-Z1oFf8V>+^ut$GFsLXHgpZ5)oF_tPQp|^BDd2+Xl%iW5Qc2Nbl!KLL} zxt87i=5awm$?qQ$-IhA#t70b8V{JlYQ8iiDtRyfD!?GK$58_nz66N4{CFA7gNdbnK z$m9ZAIKiks$&45Ix7b+%)1iP3js@f>K905F*8A; zKra^ZbMRbr#M-ekOGnR5%&RNB#{EHGb~d`#!%K1)Nty7P*bStJ3bK8ATcb|9o{wH# zT^&qoIChkop`~T`)2BLAO-(zrz}{d)G_?igw!@nvDlVSeSqJK+i!U3vl59;t5^MR7 z#lqj82~`ZjdT|9YjZnXSu3*gfZj;Za+^nd`vthDrP)Y1;Z7JBeq@iCTIKx71fLWFU zBG)??Rs};5Yj~>jC1PzHt@QtN_#9~dy7-==sKY`*G&=NA&Q=Cpznm;lm(PbrL^Oa} zcy8@v>ZJxB)~W81mkAjLzkg0e-@3&Eq$H@Ka%=02P}yFvaz(zsJm-G>sv~_RPp)R^ zHtR29>({SOvbs2giL)s8+zhh+aTP0A7Sb;czkBBoN^nYGmefkSm;btE}v@#$Wo)G@P&(BX^q|3gvAHf32 zyk*~k*Twd95KUxU`(@jVLcKJQ?5A6HObtP(--+gD&}Txj;IJl_o?y&I#8XaA4j8Pw zF~wQ+J9q97${GHKNTjy8xw&b}G`iEDdoO`^Ax$(AEh}%xioEYX9)PW%n|w4SUX_-3 z2lQ+`nu)>|wDQv0lAn;oPSiy~VZTs;lmZK+}0)J`c~`1Iz*3wkg=gw^-;^>sSu z9e?Hd`x{Cph41XUxeSCQ;>PRQv6S<{@1I{EdHD81M@~+D#%n0h;2Wep-cg~8pIpzd zPWr+h@3QjpVU)Gr8o!z`#SGzxCkE=#0Z9XR(DN!|E3|JcRkL0c#*;XG>Lq~{SYuP1g(FK<| zOQ9O~BdHA!4b=uGWhaD3M3C<5abx3Q04x{x?Djx8CCbqI_k_^ionsn)`KvM1kKGV* zkZvWgQ3eU%$;yq$o}TNh5hI@*j-jKY!^rzyDN)|F z9C6dWVcsmF$vZa{S>uG0GaS8;|~ZEqG9N0qELrhXqlS*BImRZ3^C}{*n2gINK zVC7jt)=tS83fb%}*$ws$nE!ur0aGlo^BEb%ykd;zSJ%Q~fSs^|drbDZ31J(0iv);2 z1hXNV91*dWvT^ff&CTD5Yz4lM$gc3|2Ic2}SsRVsLX|>fF7XX1+42%Lm7KtJ4KU~I!&Ys zxdL+&LUREq{Fr*X9*RPIr|HL^XzLBURLyEwT#pWkC-VbNT?y%jXn|g=f~CYtMg&ljs)lMW zVPW`QWeYb!@bByEgBg}(+Xb*H{SV!up;MVXox?3^h+aoPAm9uFFQ=2#0-zPZ<9$Mt zRlJRQ|n8?$BP(XFYVMw|}j78E8 zOixe8`e@7MHq$KlJxegox!dKj2+bd{SJ0+UrPjm9M$Bd?ELx{eKS0Xw+7Gc7_;(0J zvH`FNRxlxD=J5@Y&Yu5iWKHu%(8e)>9Lw&Kf#-0cxut~x&Il!cfB)NtqSPjV+o4JxH3VnnID(ukE&_(nX;ZYiXRuk7(mLx?3q0SY9jUAce;+w z`{%$JOo(b$j9ye}sT@jj%#|yKTSlHdetesB^4r+42^_!#6R-mtO4*sIWX!-;udnYk z33>-A-wg>Cuhag5j9pdL+2NTb+ztgcRpIErbI(h_YnDr59GQ*j-&rX1*(r$SFOqg} z{EKeHJt?wf&Ex&0TcoVYDCJfK-5#(WFaUc#J+iHoKt)qt7z!?8Vr_T(XtZQKz!pP7%SO$kP~dh0Smjam zVI}yaw6qj$aqFvIVLHRbtEJ#*fi-J^gI(=mEbh9B?FAYh5|Ib+e4B?(E0GZ4Ai|9B z?Z0br>7Nob%M*FR6Z$!^I)F&Y&b)vY9~_HOmOgK3r-&frD6mB2#lb*2r99I!WrA54 z3J38U5!jYtoP%M!Zy)=gm^9n_x2#MNl74n3_H#^b>G8H@YHDg0rB3l|FfgqX~9(i3?ou<#kbzFOMbYvDECt|?iHeb{LSmAcGtQSE1Is%3mk%q1vY zc8^1u-l!y%`2WX$*7EYD+4xX=&aq?1LJ)WF(Ci21=etHusKzHGtf3SY6=hGWFtuJs zTI8NsI)NaZ7Oq;Q{_%dR8nIE@xw&mcM3e#O+QI05L-m|IQEUj*d!mCgAx^(Waj?O{ z!Xn#cD7SM0s}9EDRmCai_|D}Q!ByTybz(~v`u6J{--P2qRs91SbFGgaJ$gg!yfO&! zq6>e0203;)orKmzrgANZ6m^LM7s!>3OevuKvv^g`toooY{u%o3E{E^%{yCYz`}7JX z(xt-4+HOQx6X2Mlqc6GS6*Eqiq&0hZc?E)EG&YWX<|K@@@9Hx9=cBK|S%pJ~c?ZHg zE&WQA5@dREt^uCBbO@r@d1g2F%L2;`52xN+xoX???M4N96 zQRXhdL6gdL@X={P(Lj7+AeuLzt)6oNkF&Eg_!&U~fz-xQpy>MYOr*~3=nC3O98Fx_ zR$F_qwi^TCkkud*(?97yKTD3JZXciABhUJgUtZ;!ZDCvThSS3oI0uev8U+?v$5JcR zkPz26q*8}Tv;~)<%xo0tXcpIjX(jG5&+8XLPq|}*}m31Q&%fiZU;cMbRSA_N4bJ#mO z(X$>BGbW!M=8KEa9Busmp7Moh2a}(3b-$d_&X-wtH;afgw6sJ`nUcqIfpY*;Nf23^ zkxl@t*Q=lgX}}aJ?K!H0IWyItuo-{!=+Ow8Fl@8UwQpr8O$yMN*jY!dfym_jhAJKIQTAGj`P zyC3eg&t3-CK|Ir7Y&GF|AQgx^-+74($?G@iCXkKPx+lb&F1z>VNpMAWOm36X>2G04lgauiz{^ueZpS@nJ%n~3g zkrGIchnz$nyCdJ_e;~W*uQ&5Qr=ap{t;->gj)2LLU^l=fR6iuSD_5_w?n_c;tE;Oc zq7XJ}sRkZCw3v0*>{NjVgeER|m+U*Hs4@xf(QSsr#j!x&7gk=g8Xpb3c##5;wwB0` zJZYBcrFv26=|aFa*dyZSTR@~?6RpAymJ@~-xLpEmK?AAmFio@>WwVnG4!q3cG|h-t zFc9L+o#MOYm%_{_bLA?eJs{W2Mm(FE&;)KBmKK6TL8xjrrCMIOa^)g;-NR^(kGG`- zk)iA7CGubjaXQKDw|`P%;#&O3Wnr<~RX~<%BIHQN91*4AbQ&%&0xuyyzk9GDF?JUoibj`LTqoeqAglCGFxG>? zKIC)PN~*9kYd>bV!Udj;IndO^JBff=Nz8UW-<^m$0BRax=(r-E(0d&nEMQ{Jk9$0M z@&trvTbU5Y*J3MRt{@^AQCiq@={+($xZDGPqpmT5kiCm=GT?&$?%DOqZU57sZ&fLPzKH25y&8^(-ar4X zSp>epBk}Ux!Y|!Fb&8Q*3QMX1XEZ@9iug2TCwy#&DrB;ab4JAt@uG|N$B&0%J4z>XzDP(0cne9B5a0}$-#((%n(g-p+Mq<}PW+y%hM&2OtVG!JPgw;X% zC`i+qU8dYIssle1U_MR?=1g;f!5*~T1BG=xi}1dkz`HnWp{bq90R;Ypm|!4V0Cs}D z&S>+x10o!*4Tw$^^#eVTk&!TM#98X26U*pBFdcnygHTN7bK9>zR)$NGsGKNjFj5k{ zADLzHuc&hjQZQV~PU0!(vWV9`fNnkV`XaHBqNh0we^C+Uk2geT%a|#M37qL;liGn+79mAWRl#z#eY|d&qsB*(|0-?q7}WmY;*%2o)Dm78=!`NQ!Ek&P%9F+WIGLZnex%G-=)+fk%tAkx5|Uf0s1j~U6& zh||;WDX{hcJb3#4+c7?*?>+QJ)5ETeWIcEv(%}#>Su{T2M>);9`++8&uIZ5;CSqa{V7% zc~$i&8c{&bZ`BZg({$l83*C^t*f?l-B_$>Inn;_H zoGc~1G;`yiKCuBjAW63wd1-;8CC6WU5&L8PrC?^M%Gz0pJm>+j_lB?-4$)OO5k%bn zEAEi=qGs^3Wq3cmoH?u@Fop~zof*9J@Zm%9!9DmO0nznfjh;_#08i~e4i}ng0>3qp z^B@TCHlq?tP*#LP#o=2b)FT&;OC-7dw-;PN0jy|1%ei8y*r*YIGg`r4FJ#?wRX~}e zm8>Oah3Obb@MVqr{u6u-=RU|fwyz+kiEoi4I*&HY7TJ$*cw?fLV&Ie^!PZ0YLOaxc z)~9XM``kGpDJce=_5eAlMp3!%M~+FF=_7e*YQNG28rO7m6eH6^s3(6LStX=yyp^Vg z#xhDRa2c{J^DX4g(AheK5b1Ov9476ZWa(nAI=+Kc@}JzBc}{ zMiI&)1OWjxH9kqp648UMk>qu|h?q82v&ctC6r6a(g7|2=B`kptIk!O`2vL|WTix?b$q)|_BHSZN{QEVbxxU84d zDva4+D&b#fjkpSR_U-#;)FoEfsbGm$fX}b|+Q^Mv!!%pe3_I>vd*)h#&oKrTQh}C+ zVNA{@g0~@NXc%>gc(Ry&Mvs9|N;Np|hLG_1`}-8$22-~dCklwm1=@uM+}KE$D6!9l zo51oK=dj`s^%8gJI50Lvp{$-D-wQzIu>@YQ2wR+Afp->lfx2kB>H2dVqBG42K+kHX zT=vul#)`)P1|HzrA#ik-**FyOO^iFKW*XBqy{sm^xS_Z z2HUpp#G!)@ZwSzZaNK~|vf?D=$al?utl`IodT@UJG+OVCFf`H9`Y4G7>x_(lso4j# zCF|PHie3tCNT#KQFR6vs(PM7fPQtQe6@ft`j7BSGA1vv{2Az54>tWDbD$g;&dvVAO zZki$(eLM3Hnj@?D&{Uo9tT_E894ySzd36nJ4zAD0JFryMVMDjw-nAUlIktJWVkzOn zSq9dx=`Z;rf@~{tb_vUYZj_w9!gblqxQ^x`If4W-?fTY3P$6)oEIA)W*cPIyJ$a&u zC+BXeawEAM6sSM`kxVt749DyWv_+gM-qJ87U^}IWzncFy383ilE)bjrD!u?%K!W^W zj5!Ru4|!#_@$*O+i@$qn+!q(lA31Ua-+w^vXbWu;Wo7#NcQyZ>?FM6`AQ1e}Co$K29s5p{)l*VaCi|32)myKCq#+){y%X@m8%v0=!cV81ZhJ&I0*hvP>{K{vy&C~(nvf@4?p za2A0;q8X}Lx1K`2eBu9b#F^SiA_vtG^O*JDBhFl#Mbho}(PJ9eDDH)NiHput3mR#< zi;7WxFKT4_6~mQ+3KpfPf^5aizt`UMvlOH^3fEYb7P0Pk0OO_@)7?yR9EaJxmEVEPhbmu1MR0sKIrGrJsgYGh z{!U88qtcOgnv#BBH5wWkK21(0Pfaev$L8p04vkj&{xXg&`7o%uGCET4tg5%mKIh4S zxJ^ntQT%;!xQKF}aayzG)vGI;#7+b+6P`RMAX~hZ{C<(`+jxgaWK`6Jo))R=&$H-$ zhNwJzt@^QP`pLOwq1Kd{g)cu>;movD5Vn$UkH0xCI{8!P(DZec^tNo|-s2lLYi?Pb zv+qAo8ymNI51a@0C-2wX+In{MrKXLc;g!~VG!)80-o=!HYk%@D{yP6r|NQEkKKIMF z3JCnhU-qDaw3lqKx3?D;6T5j|twrnXS*1m5Itpdy&NXv_BHOo{&<5-(G%@+^?2x%& zRb}ZH6f#jb?f&=X&74D9w!Gh`XLjUB1`d0wx;Db9e>ExTOCQ6UH3_po(D9pa05`9w z$Pn=_E+w_!&u{UnMB>Xkkh{TaisIIil9C-q^{21hgw*nAVDf(JYD|fR=(6gk*mWiz z9>oCBO+5`tUOGL_PF_`aw0<1A5VL)=CI}mGaq({zS}ma4%?u5P`c6L@NY*}fEDMak z=pJQ+DLgNk1qD*D$4>yJo80^K=@Wb=a;km{R$$V);dje=nKUGM|Mu;j*zVt(kFCI? zB{tF2aAt49=&~WDgFTOH7=DFSdvoI9Wvz5j?pI@CQegxtoI?L|Nd6 zy0Ki`++A&LsW6}*Iq!qt?Z&;Dmq%w*-d3Ky54KYEHJZ)1I@?#!j~t3Ej>vLV9TgT9 z&P`9Z-=BinrW9Q<8ueviXTS6LvrB8qeZQIAUw}Boq@+&hLXLbyPa6QHY1Fe^R#7n< z57`sj=W?xMd^Vx)?c2g{-#oiIJ9W5f!e_oum@6geV^F+`?mP4I^KG^x1s0&8yRb-} zLXlvmQMZ1{ff30`OG{@~Ra97NNAY}hX=-fz1T+=f7;$5KjnEz^XJ-=|n`HQTbAjIt z=E2Y%kxx7YQePFabK}0O4zalA(L@$9Y3z&zqjS*YlYP#ei@$PZ$Pp!?^Mv1vRMpy<;kDX=) zx233@P0P*A-4MOMLwEP0){~bOk`q_<)$N2EDR;ITJIe0j73)Ju!hveHPh)yc>0lbe0)4Kr4+a;^NaANU(N(lVZcDE#NLRbQ6jTGCd&_-SY0gcNg$ib{yE#UOheJy=_Vm05qBU)L z+P(1R8!6sZDwj7?j%vg#9}0Q)?3wZY{oPl8z}BJHzc9}lH`Tm?(PHVY-0Mj|SeRmD z_>mB2#g{i>H^<+GI$EKvt^M&4KZU&r-`TULJBwnO^tXv{n+5%SlV2P#84Sl?^BqYFP?} zie(iIe!}+rwI%*Tb4*K3nX*X!O(;&gL80(bG?e!lxy255c$$5fUfeTQZJePskS?Oq z&&n2T{LYE-NaXgn)wc`{<^`A}WpCPq^QH+oa|ML0ceZ_Jv+i8nw!lQC)=fzUF0KU+ zyLe-KKFFC(K4*-W|MB!wbJM&>3Gcy=pC(_J=S+VW(idT+y{Y8mqnKh~XU8bU)@-fN z{4v&pk&%&dF)wdN>b)yLc?YV)xv40DYk1e)y%N+IuWPsFX3cR;1I_z_d2s?|R~;`G;HBtK!YW8EdIoHC^wCm)=ubz5NPPGc8Vfo#Wx z87IL_n?l1AU0&Q;Yn=V&#jQs*`~vysCx?tDMmjd!matI80Q3(X5|Lvr7L=7;9~>N< zWPR=W>5k`YyuJ&Py0J3+av=skO_S!_#Ok+3d5e)J+n;H9c$8z;X^t;nzSKRk+RwH7 zyta#=M@=xxrft?$^d++OYv<@T)3i;5{^7|~`J&3&NG&CRv4vALhel8bqd``XAdYg5+T z{IrjyW%q><{GZ98bso95H}q6gD9L+F4CYsV9T=!=Z+|X&j47%+Us++yR4FxelN^Of zUu9ybiSC}9tCEe4@I$l0%e&6B2Zbw3WE^=W{`GxKp!fV#c}ojBMX%Jy`(5<*fhgC4gJyy4)Ofew_471$S;6l!!3{JMRNb^ zMgsabou!JO>%?0rG+P&EW({vqZd!`N6G8r#V%VLVu6crV% zxUUn#7pXp0vnhV>!Gi}~1O*KCV^7&_BKz+8{R8pg$!ZbHC=V@4!w=`$tkzXl9tvt< z7WnZ}I55bl;pv%Aj$_^XUiUlC%*~aIht*X0(`AXQXVST|ZJ&#uL2tgFx~uWmd_TH1 zYa=zEpBrEt9v&w92ag>pgo-X1w{$84L@qps2vLmN}!NLE$;>5&+rlPmdf9{p!EFvOO z_9~P;^6uS`;o;MW_B$~#l{Gc1#l^+zu*TexK|w+4rykevizXw&1tcYz3JVJ*;R%I8dxa#=@1O%8b##~~KD)*;j|8vIiY=T;Z|D)U@dWGU;gWuT7K8#2$ zEUZg1$=>k#?b}LRqg*Y7&opSBb`E;OCnm-~8SC^cO3XSk@n+i9C8RF1X@otTT#=Od zN;oi`{z+~w6{Wd3!$JPONw#mv?He}?g`zRmpg|Rn+1c4^HOEusX4*E$pTIscZ1JFd z^{Hgj>~vQM?Y;d`=A%VJn>(kd$sjY2c^G7{8e~k|TWNFhWF_9-Zj%i@Qte@5WApmm zJ7v{?jYExvzuK*(q`0)g#T;ym-HTJRH$|%7G>O6E(vxJ4hNaVUaWC3g>fRgj|MKM^ z`=+hp&6OF33KlX`CyqQjt72(s+3)WadhOa;L1AH9S!rZNePiqdBbtW~A8y;VOI0<1 zg_&9S%hVZm)^(ctheTfe_$02bekbTE{U(kL8!F$w{}DBimYT{hCB>|DYbzB6@y8Vq z5b!8Bw-TFQGrY)sTzIRPm{y6R)0gF3+}u5bgF%LAy2S?1u;bIsi{6)XV-^IAE4(r9do3(@vGD5J+IVg1sZrkfHT#@JPer+Y;|A&jBX>pFCF-9DKFUnmth@Q~brzHmSE^z`9ZzI>#b zhz98}>aRb1=um=1sZW8+H)aC^gKLqIRFp?gp9b8x!D41+X8x(0QdWLtz*}2b!=f)-{k$(5u+76yN#m>swjTARFKJM8sYhC?k(dJ!k z?Ha_})_J;3czXjDvg`KY?MFsN94}l@b9CHx_3G8NTwDtu22#_~5IW!dt5E{(>)s1V zPfwRtl-Rv{gQ}{ko(SvfH*Y+CNYc>IY}>xQa;PcG!u`YemPA<@89oJt;vEtK&(==_%X1W*m3eHKZI5X^Y9T ztzWi!_3CTtg0}TZD=6JPm4Sn(X*J#o6uuiZns*Zh;>U-*DA(@ZrS15R7un--ZQkF#PVYH2ybm?`?Qf4@%!Cnl z%-@Mf?WEr>FVA)KxywFVTal|_VHVLPQ`#mb&2)zEP$x0IxeCcmy0K+rJ(W}xb_sK@ z)T;4X_wxsI&CPjsoo>CjKaep>R05F{DfdNz%*?Gwu9YiSVsI~<{PUg!g@h;-drZEW zh#ebq+-qEN*%bGX>odQoGwQF1>W|~1j7?8VM5~%K0WJ8&-X_s3iUhti6>l1s z5RHESF~sS<6)RRad|n`H8Dt(6A8(N{+19t#SWjP{37H!?N$QTY1CRL8=SFr*zgVm0 zOfJp%}U+fVP{}BU`g)e@9)2qfv@2A>S7BM9}moQUsS0{$yoXn)vH&pu+_|thT&7`tRtPpbllwBSl2Xz+T`2u@oS>BO){^<|G4{b z>w{{KA#;f>Tj;POHM6GWT{F}4L~fdrRJG${oPuXvvFEhro`^UjQGX**?}1CeE7-V( z?ZJm5CU+#7W14D4BSgPuWqL%(rrjc+x?(0?UAt%TDniY8u8?HAWzOf$X-)D|k=cB@ z+k0-u=g*(@Fx#4Y*vay`EzC$BG}(xG5%os_Ogvc|XY4GFYRon^^V9n83l>-QR!2nM zxQbyD}MdTY#4un`KSCdytarve}c|-Jm065X8(zijQv_v)T_v)#g=X@_LN>& zSjb#>LDGhX`0d{MWHsdRb=>}BRr}3~&NF=d`Ze7+%Oom31}`9c;>3w^;JoM0pC=_R zy+iD1=_1eXxk(o1v$%P`%i!$jGs>7V?FyDb1Zlf=6>43h8bSQM-NQ+U&UO$yz#al z)#&Bjou`!5dR%rU{c^(_;nN8@ITCWTWCLodYieE&3|xtPt*l0hD$loX-_i}!!Us9X z@{3x}PyHC|DlHx4prA>KdW$*Ldv^9{y(DfQWK@B_^4)&h)7K{`M@u1Z?>s+srnRl@ zpQ&Vidb;iMCF+$S;rf@!LZW2G{M}cSU%!4ij(nEQJM+oJT`j2$;K zJG)O!&DE5yNkb_`xX)R3hjBiM+<-P6-U#lph!uWL{Ry9WnFrSI|5f2XRKHmN8VU9BO^mJS6+bpLIj;t zKydJCRH99CZ?=$E*4NaeUD^Z<9x=PlRCJeLQK@zl!@Hh%`FB7+q@<)A2R?ASxw!>} z|1u@7vRC}}#+^IaCcgtLE^X>N?_;DZySgk~u1b@PNMmPb_vp!!FH6NKC7;a2aI2)t zw+0oLn}07jm}9QeTNS2KjRGRT{B?e`RBg1gI58~^!`iTfCr`CJOth-}C#qNL8yWc_ zHznb7Ny#tRuzRJt)kzC!my3Rn2^-WuGBR>d#DT!So2bR4g%Q@d8?-cGM*a8+)+(ui zj#2(UDG|+6l!r6fXPFo9g6gRTgF{0lW1ml!?nxRCe(>+qXjB`yQLQe|&TuJ+e(C;q zO!&;Bbn(w_Bs+8WxJ#F)j!Q+4c?o>i5M)km62?n5<=K4{){fbw<8sx2>{Wy3z%F-p zoe^DHGX1FCo>N;NJb19Q`}8%@AM0=aJv$KZ=;Xx2P0B&~n{10x9x5uTB=%r5Y4%^Qi0BFTZ#ppYO?wmMNVTxUXkmV3Qn8#vNqoUL?x0+3|s& zEf&5#&DlqQzXI>w<0khwdzKM<`p%s@WsQxjlxs0Dq3BWZ*q#sfb|Yvju-aqOUB1a4 zlS8(xkN7!v+Al}{R`C3}^8~sMKhs7r&Yt%CQ)qwozQ4N@GrP~g(2&G5m!cO3 zB0>$s#LsVO@uI(L{W3B(n?5}qd`s-eE&21vo3S!Z%euU$xozIvp<1(MjmwXAw$9E@ zRQ!WOvzW`cXC9OG%V}tkPQ;gyPC%MDvE4Q>7NkL0(^V+@M3DJfb1=g9WzLk zJ$>d(ecAzD)UcJnAu{g21R1$xF1n9@P1K3m1x$M-Cgw<#4L0EW3*Vc?J{MhBV{VQZ zx=DvUbV0^-h!qnoBu8T$`Q^*8;f^Bq4ZBY7O;!!tcJ{MORNHbBpk4%*%BQDi_;1uW z&Wu?-Fe_w^j*cE|$>H_#_I6*hhKXsjzg)x@3K?-`K>{498m$ zD5`Z^+&?=zd+?CMaZAgKYxXRx<@K52Z}|A}v2``$%=~=sl{JbNOH23Iw>%V+L%Pby z0EYcs?8%Nf!hknmz|RWYehQEt`|Me9Ia=c++|sJ2!XLS&!gq1rVkt8l2?M5Z>^{F$ zHJmHh&u{rv9tBQxQXFzFB3xWt%GH1#J6U*nBM{S=Xnr#6FN5{Twjb|@4RRQ1AX%;5 zupyLeCV6?YA5#>!!;?czl8)VbCq}z?S2A*y-P~+gm#~leDwixXP!Ah-L{Ly;S=a{Y zvF>u}{9^Wj=wCUmnkg%=^Qwf^_H*y?;fnW}+kE=;>9NTci!AReK-25tsN`FBq=HR`t|Fz+`w>v@n&4AVVpv8s{R0+gA7vJw>x3?JTyzS`ZYfO_;?L7 z!4PIWyFbXLv+xxP* zx)iWMu)lby(Vq=rk+p>ehmCNJBWJP#^w4lV&cgnS6-o*49G{ z$bpB7J$7#0x>Z??`NNwxwDtA%Gsr7zxw#n-LtayDCu1vAl~Z0Fv$kehzkV}_x555_ zf&IcaqK3LbZ=}Y>P98d7{QdWz`O$QXQbj@Ln%%YY!_#W_EF4t6XAI?~Z8{KVInFv$ch@1a21ToaulllzzHhV!1AI~T5` zjS4$~zUvCl4H(?}SXM@niq_61Pg^PmCby7ZNqg5(bfFsEu}Zc6VGj?FRjXF*MaEvs z!gA&1OJ!_{$c^U8%BD$(D3Z35R6>>2^hJNf+|-?5bS~s7{XMlo&Sj8KjyB2=X&Z|| z8ogaIsmD*93dOv+d3svE3}TwW^r@(kaX&(ky;%G+j~sq$P(%}Ta!?2WHy|vG0Vza9 zEzwzBLu22eLmY~W3+Hxao)C;wugg7o;N`1VR>){USFgSVCSRHY&0l>rET_9lxc3Lp zGjXeY1Bxi_!;1_yXU8Cd7*V5NrD)vPcB*-6So`PP%%)6Zy0^E)t`rsRM#c^S^$;8m z6cc4x8Ne|3F>MX{I~6s3Ic%uf%^>W6ZGM)o(}b%VQ0gsYNzZErqz{eBG5T~}Yf&(f0bY*&ex9H9T^@`D!M{`$$PiKhAX zNy$cJuh+H}U6And@=~r&kh-p!I@npfn{$UPpInG45?Cejhor}Z?bQv^jK~mcxsl98 zy8*Had>2bpyNcdmvMtcO%bXwPcz*teC>HGsfh(&xM~kOBD>IF=9-V3t_nPg$jiSEv z={F>;aY&zy9ps1#(P6AF^_ly&>BV%XCn_i~m->mObbV?vsZmjE%c!ZH0AK1s+ojVZ z`BOYSJ$Ib_e8}bI(kfd3%&Sa~bO37^xYv%emRRI_yQua`PEx^a;|LpV}wR#o8Mp}M!!+^v~d+OPtqVr-NQ^TvZw6yB= zMbW(rQ1&d0QIj7jC@9qH2Q`M|=kGLmXcinB%jM_q?=-F_Ai(B|!Ez~hup(oya;Ivd zL&-jP=ui;YLDCGO1#`77~b;Ui5z(^tQV+OpH9X7Dy(MfUmrH*8^D^r7jQhfPh7VFti8 z?+1wma5t!q=sbnJJ=m7J34lgLEyLinvwGH<&KK)J(OwK^^koaWse$^5ghdbxLiAd6 z^k(v$=0c}l8fYFFJ3|@dBSweSPo1!_5!k$WH7YyWM0INhZaLOVmoBYhWUPwXa!|RN zU!Z{#L2uzRD+ExRvh*>$O*Q~Q85mnYGfbOZ9yt5%P>GT<%zb=(G6u9xo;7H$=e|EfgiHgoO*w$uPv6C9Uo0FeZvW?p zl4)8(0tJ+Mby~a+Vkfd$a>tIBAR9?j5EQhkZ(tx48#zUP>c^**8#ivubLcX6VNGns z4rwbqw~2Gt=^)&l?2srI7nf~z4T1$lMZLMTij|=L$;N6}wT4&m<=o_VNvn$G_4s%?ZEpXPJ0x34tck20M|}{RP|2k)b|uT*CE8H$7+kO z>{(pc9;fInD#NDVAHDq~BQA@QQ#pE+?58IP^Bwk$LCG?r+G=Vm!HQHMt5Q)&P5W8R zyCN;=O+(?iliJZzE3B-n0CtiFWm2rdc@#pxAy8A8nVAVLZ2>}4=Z zeU8N*FohF?_4}d4q#BB_;!{-xPQ3-wUFWc5CWId%g%{(l*H34~iq&NwkxofXwfjA^ zt^H*~1M?P>2W5nSY05en8or?nI((Wr_K z9~c2A5nY=*0P!AO7&X7jz1sxST`3&6!yJ_tz4{fDir{di)C4_|wam#8@d3I)cy*u&hRY&q{nU*bErk87@LtdSb zr-P|3s&#$NEVzM z2p^PDR#xf&bjMolH8Nsl|L8^+-j^N>k%%x&U?g6D_yAOSv9M4!Ha52V%a;o2pE-}6 zdR1lQRZDdeRAZ3$npsQo z1CoCvAVX4;04qED#_ii#(sb`tAS?VFtlwBKY5WHD&sXes zl^XJaIqcK)4TY4etIJ+Z$4O-A~(r^i1k-`tgFf{ z{5;mEsj&~u2p$rMq18JXUq;h2+?FeOx^>$UO}3xSac;t{o=N@Hj?=Ac0n%TQ1P2g_ zy=#BDg}Grlpk*cWk1+oI0YTwO>stHL3+ULz`OtpnpZ&Zy!k^CS!mQK;Ep5USB(?s?|OZ{JaH2@SYef#W_Gv#3YtMEkyeJ%+lNl&rTxNJuC+e6Q&1=H_*8 zZ%b4o$2~q-%Y&4~0OfaR1WhFoNP(r_&pT=c>x73O*-=woJoA)x2@NryEz#9yGX2r{ zyW(;C){Psfot&KR>^j5g;xbbmqi)#r;4lk`#S-s%AbuL`gCJ2I(090D0FI)c!3Vif?+ z=)wL~l?~X~*{$E)+{_{CEI8BxW-r|IubzIkp|;a;3*Gmq&QE`aOcr*Z&d?JPbzwC2 zDDKagu3WQ+4=d(9H?g^?Fe+-aZ8YNf;)19_goC&SE6UNHvbLZx zi8LTTCPV?CII_p?|3dW?H%RW>`3l;6WY1ub+`)%0rtZ+IsZlOQM;|WSp7rLtm_zZ+ zo2*2S1w1;Fam&caM)7a?%l|WLoX0b;@`gIPEHaVn#7zCr?r=d2`SN5AK8U9dUE$~s zwD0cFaAC~{0Vg_yJxySi&+o~Dl(G!-V!a2`Lr+y-d3eY)YVzmi$*n z6wOARR}?-FcNM*LMQ5kEovQ|Bohv0Jg+d{m2b%rzVD%;>64Hr)DNupX0%BPJaewib znHFdui?+u>o!?p?TcfM&4v1$f6IOe5{Nzb%bVCHS12%<1H6sRv^}4$1cVw8J9W~Pv ziAvHFeRYPNT~IqVh^bj<+vf_uU;Sy-D zimtBsk3H#)0rcxHgF&)Jfzgal(ilp6&~hF5AXeU;dEdT$4huyQ=}%D4EDpVJlU%p; z&_!hB(D3A^=G>fo{C}oP7p>smXw5yjhVWpJZ1}crrH97pHZjNqRz0$(`{Hlw>XKh( zU|wftW)jdJ6q*PQ#*6y2V7m_ey{j7{QlMVA&9pd z`S>W~U*A_gIq|mh=MDEF;J+_~G6P=OViahk1%}B9+2eHUBPzhbI?wka!u2tPhXCSR zwR&~=msg=yOUZh*LG!8LxHulz`|1yZgsXlSAXcQMKkVFQS zsN``tXjHU$Z)3mDdV$~|9DJcUFOyvfuoejb7Gdo&RH!<@Y>f9kpte=plM^z&f8nTS-~!qO;2A%28irc zkRrfT92FC@fw*^ol95w_9eC2sAeNXk3=Mt9& zn)RcajAbBzXMRt2IbrqKzs{?wsr5p!0XMQg^?gxZZti|lQ%-Wj`vdv^Ra#~N9o#h| zAFRAATARk&x>vfH>d)c4gcn;9-=n9ms;ZhPU7RmI8)V{T&;fH9HG=IHAB%5OD`=$WJ*Cp;h;UNa|*33xJ=p&o=oZ>t7Dxa{ivbuze!G3Z@@F-LSF>NI4FwoO;X=cWa zjWj3)5v`t3BEH*db9 z=q-otgZSl&j1`}EXCZ@-F*>Cfb_#_ChPR|^)(MCis0$MO8)om9nFY8 zhYYpVG;bB$Ej`YUl#R|p?Irn}n6@A;>|a9boMj(BvYcvuP>wa*=KIHs)HN_sY}}zn zjkz?{5E;->f%DlK-X~I?t8f;6y}GAoju38X=c#6|9u9w{OTgn5SU`kxdPB#l-@kur z-M^Y~xuBpxiaS0w_Nqkb0yEvZt=*dT%0^aj3X&#?JW5$rRX|j9DD3SROO*e!XVNc< zi=8G0^$E0uDMeLJZ!OddK{?KhAXFmlctvjTt(voxyPa2cbXUlK>J^58vE#XGCCTQ; zj=@X>A7D6W*JEfMjN7}@0rVuSzq|~-BXxq-VHjN#+C@mg%T033!(h|_WV!g!Qp2v( zchQ@4+=Lp0yJE$L4I7rPSm89vB1RBw&5cdT(9=*gn7Gw?AJ=RS9GEWn)}FtG1Rbhi z>CYKtOiMKN*w)sm;!+8A;ua-W(IdfFbe(~b$(V4t2qfzA@^Ui!_KoQvP3%^Ik^}lp zM$#a9i_M4FV8CZfgz)L}d3u1xXcI1?{_qm%s@HbKF4@$HUp&&p?6b#bVRxDgOd~K9 zt^^r)B`i#bE!Ef;*-O;Uu^qwlo`Shwve zQ6SLhU92n%`m|N_^v83Pg-G!RX<-SX2MP!ax7w*kHSU4HJc2?(RtPE~&(C?#7aK!q z@B^ZAp=3qH8l8(p8-HWdfhw@o_1GIgwCZ#&gLPDp>%qlwB_${KeE9;1WXFM*2EU(# z5eU5x&(0PD2j^ZJPNM}16$?`p&K-&p6dayx+}ig7xPyxH2MV5330N0A{ALphyyUT$ zt46*VXsU&C?Su4WQyUkCY7q<^Y?=c3<`9q$;$r9?3cR37~QZH)NVC&7IcVhRIoOrmCbw2O$bp!gUjhvV*H=o>*zK@bUqa;$olqkLX( zXBkL@+@&HM9BnEA4rm6-qhQ51G%_nEhW+ufcCF*k`k?b{|Sc2Bhi-<44WtMFh4NOrv*d$3c$LvfSof^3Z=)o=DN@f47ZPbetQtc z4fGW}dseyGNsPRTSBdQ!R4Wk$d)MYUJ~ur)U62t3Wx)Adp|so6vb`W+X&AXg&>YvF zLcf5NLJBO$PJ31aBjw1EBlVBfwLogE0=iTMA+qc2=O6&`BoIP_pPoq&cE5V#!uA;J zZw;vhKELHa2e00;WlMeJpoM9D-^~jVF(Fdhw(SKknHsmhgO!cVYW7#ZjN2IBI&rgA z=+FtT-I<{l!IOTpXcvhrlBJuPj9mm91Q7BhDe@?K1v!-dUbi?~>syy$xCvt7`2F*j zhz9`HSeKE{>(RN!8l)fF@w_HukVl&B1Vj=L2IFB1rG)3k%$9>#0%K$_G_hGA=}mK{ z@fOl~Kv^dkdWo}Vwu?p@=MxlMj#RZ@Nog4w3p!UzwA!}K526R%Oq1b>MP7M@LU9>D z?VQRXw$wDCPby2t;tc z$B^#{4Nv0DvmyPBMydfl$2JN~LHi91m_fo{LlMWFIFNU}zbia6k{qawua1>H*Zt#1 z7}|PHIJ}ZJ)^D&ix-ot_A_iBL*VJSvd43kz4>{-Y*$z2KO!|jt>H)d^(XH=G)U!ml zVa9Ves4>ppeN2Cw-a&q(f2To~Iw@#E8N!?v3u zn>MY498n2QPV!9q4j2xV;jCX7c1Qlsy?bFqh5bJ$z;Blf#S$(8DEh;n3k_`A|(1H3B zOmsecnkvwrIy>?P(JVrQ)jOcC?`uaVIZ0>SPf4+a{y|7h)Vlhj3l8!=3x`~`LRswX z>+{396N-k|8L=Awuql+X&!4%7i4~B{5l1&bi><_iFQX|@oEzK-64La$^!li`{bX{Q zb1c??d_rsGXnOaq;jFM)Baif(l&q{!gfL;SQQ!zsH!)%KM{QA;h>Ykwt^t-2iIy-Z zq8pV{^0jqz9-nTL@|tWqI5V7EyCn7jsU}%76(<18*nTdMMC}b^*`&({Upwi&Jx7ZJ z!5EOLlC`dZFv)0fes)~wO2qZxER>Opm8%7W^x7`8CruaVKLuFY2s>ZA?;lw($^dy5 z2`Jo2)=|;0X@IAH42tpdb62`6E7z#Hxa>$V$QZO$R<8IqIA{s@Gy`u4;J+VaILDkF z1{$b~upTgg2EKxVM%D)2zKtadzqE*tn_JFhFVYgM9s+Xqsbu!gk9LMZGe}0BnHkE; z(F4bX1RQMiutq;h;jHKtBhgoAl7FHv?EQFP&nr~YI@m+t`o)+JJp~RvWqW)3nqn!q z=h4_)4h?;KYH?deJCAgfia@00U|phZ%R}z3AMS+`10--J{(A3CN7gt5=Y_fNIjyC> zd&JFO1QCJP$4`u`F=QcrKM;V-R7WH@M<-POFle$cqdnG#Hx$nINjGLWmj^!7R- zIg!F_;XTPVH$55=p*Tm}R29x0rM{(o#~i1I55rtXggk((f#187l}{YEwGC>^J=t3{ zT2cvel@PRRLe#T~4-!%o5yDJ=@lTP2V_PqPx4GqYu>%G*Qih*CeOmEk9X##mcGY!t ztKg5j1k*K~Ve0{Au~}~ zNsV(n-^@#WxRJ9ez+n=B`uOqVWsqZFyhuNo!!!KjhklwacjLpON9;y7gS{XYnUu7& zjY2{zpf3=>GX~lpLO2Kb+-msfyN8C}7&GSIF^Gx%hhLYbJ=Ps&R%`q$;T(&u7lU?*v~JoI6V!I90pos#%@uDznqv1pq*O+&Ja5rk{Jx#%QkJ= z)Cix3Sy(N|E634J1(E^K_7cPc90L9$0B>B5Mc8ih?joFrnrjqTm@>9NMSF1M86#TU zB^nD5?yR`Fy$-}dQfiQT>X2$Fi#rs7S@+r7ixRkkCwxH&%s}Z01X+a6IRPmSM(ryA z=m4oWQ9{|jzmk1q{Qdj){zyv1O%1q!YmY-`hW(p4csZaGe6WlV>PQ+ufjyTJV4lz@ zyosEyd9?e&G#$f-?J6Mx!#*!ISnaGN_D2Aaz0qkct8lad0P<2q#2TRav!4qtK_Fs* zv$TppZ78CBeSIYJqst}YdvI_KUHmFhR$3Gc4&WbRG{hOR6obw1ome?ItO556PJOgj+AoFT?IW7XrC7`SbMc_aPAMCAC(eyYg^tz_Ty}uCQx=l*zGJp}<-g2S1;2jEl2NQ znt32XbNjuTckk$th#Vh$4cfBeptlVHV+KwR{!yZb1Y1eQ}!@BpxIoBLn{Db-Dswl z1kasFIe$+y1>7iM5jNHJ3%x{_!n_%O%;?Rw)jJEsPTaFReflbCI7LKOA!oe=Lxr3h z3Qlk%a5wVMOGtQiFWiY2nn<%KjmySrH&xXxj>{qF!SQXKmE-59avf^C1U?aVvK17W zf1i3f^s5!DG7~y#VE--S)!=a_bYZ9Dv^t*}C;5;em;Se|N`f=%ir5Uu}x_TGrh$ zzIZP-)_leT3MsMa!L5Y0YYaLdA?6YOI1^Znse0_Y=<^?8V86EZ)$2Z@#HBL;lm!Bg z^zGybD|F(ztb-B{MO2iHXf#WYnf|5h$0jEHQd5P{4JM?etpuKeZ8gbv@sGo2_HNW> z@=4O278J-3TO`UmA&}vr3PLAN<#qvRbQ-Q)_^?+3n&SvAH3f-grG{&)SMwp)nbcLIbiC}|NW6c|e<`NNQpqU7{ z0XTXYM|5her5!q75*Fs%S8bw6zy)pF`9c;j4A$n3gF_JTSt5A%;0o#P-d^hNc<K~=aG!D!`XC>6GmO+~kVQ4a zA)I;liWg%$BmwB-Na`{4T!iWd9tgpwoX(vQNNUl`h>ZP9aVH+%o(H%9ENl-nw<`?}@xJlqx6SAp#(Pa7ls!fUrUequ>~hlD+*9 zohtwf(VMyCT+*G*`EaV?Ivnc!$QS@0RTy6Afjz%jf3)XU`u-W`ZEbH430G4Vys5cU z7JCU0ksK*&esI`uXh20pg@5<%dH-t60y51?vWO^9GT0Itl^;>?NY#&vi<9sgV#D~s zlpyDfM0sHTGhx!^H5_t>Z!nXzy>nAPLKVH|y0Ip$PtR;1h!_Tp z%9fVfTb{tvrC%tV_C)dBF9!-`?E&(=0PEMUPecS0sEbD0v8Q6i0&HL8qcBU?!SDf} zN9p;fUDyZ4IGozu-@gJ10l>g9pndo&;6`ME7ZUbNxrC*2Kg>+VqS>dM6qIS&G7W?u&tw@0U-Pt@xhFi z%rin2QA5uP7)OhgOD=%T$GR8i`onZSoB}<8mWIZ*?THX^nF5P_LH?*lVzPwJ*g!A4 zdv|EKCJg;H0LTP&Vvgx31+L%MVPE>8*jPgOC4a^F8`uw5oH})izz66C$UBsa#GG;U z0P4kQ=gxgg=PcI4l1gM>*#Dg%P=HxWgjbhEQ zIIMaJt}1{eK9DBZgKl#Z8D%L*@s+}9^uQtz>X9J*sHkbWQ3_>z7Gwxz1^@~BX9NTM z_IAsGli_tQ|L{Ql=ntQ&UsefD(3YaI%DwD1;x4SZ+z{bK$~;hmES3T4f{u z|3#5Xziv$N0J0&pu-y6KAf$CZND=U&D1dG4`T8{geQhG}P2ERBbBhd(J$qaqPQy9< zzt9S9$@X8)Zs|Nc3AGCOAQ>zmIg~}d;DPK5Sn2{{wE*bA9yHF3_dL;OtydK8#Qq1?PYKOE&E!UmSN8~g#84dOgN@Kk*Iv;o111I*8twvDp# zl8`DTDLL!=qvMpe(R4}G&y%cM@Sj^-+i4eRlxZAiLmv@_Ltutr<~!tyCa+3 zt6uuk%l5G(Q|k%;>~Fdb>$UFQsj_Ukb6)k#QoIN!yTfyYFp-7=VN;f46bW` zylHoLXk^4hODlB2rxaX{MB2yAA9om=nB3oNc>mDS)oaIB$w~_I^Y06=dcBQ)a$-XH zq+4cI)=t+UqxAuaaH5GyO4bjVP*Ers?c4Xt zN2m0GqnX|9&D*z4_4SEWRjN%*O*I4K83hH0?%QRir>i?TEpXy#mc!kH=VQ^Y-)#9F z)$+Xw)T+y`K8>c9mbbY2(IY+BvQ6`BBZtoyHDxcu2WtJUZD%}j{J1i8Eoy&bV`B3}! zf2GRj&O>2;Jdl1{J~@ISDyWqpX1CE{Vmvu6jf ziFfF{kKV4fE9U+6KXC_(L0)d=Vyr&xyHS7C&gVW^9=f!wo15iMK%7u?!t40W+b}GK4KMd+TuGfhjUSfl+O0KSRP!|eHgQx zG`H-4yL%y8%Es2#^ykmzX1x?)ID5N8hmDPGvv|0L4p}_za6zkRIJ4Y~&GF}Gp|c8H zl}6s)#a@c>;2w55bjrVv-?RAm;d*LZKDX9+EekBANtXryAW7it-+u+iT{Cb#7I&@b z{b>u|>C(Qq$f5Rn-#d+^E%cw{;Q~?bM_aBM`cFhuYHF&chQ{Bue#xiAc+1rka<7-Xogvt*JBhEy9-Xe!Ks$p^&RX$6PWI%_V!F6ty>}r3f?KZ$Q`Ho z`z$|ykLlSgA%RXNkaNQZY)-KlH+wEIBk@r+?a@4mLh;QZOm(WFhln(mJM~w*-$CI9dsQLfI-? z+}KBn;Fpy>%oCvoZ_#zZ6<4CKgfq~jXNxvx}S6pCOSDM>Oyo-QVvVhHc<`eMR zTVIN$T-tQ)+O-7ipFyx%cXwv(U!18<46H)4eF{Qt2&8P=Hh9?|f{hT_yjkPA3k-l7 z4d~E}4x}BR-!+ z>aD+gjauFxnqOhZS^Uw6@U>b*?{xFVjgO=G>Vip7Ee}B};F<7F)WexGjSyvlq2XaJ z$9H>OPw$E`6Tb}wIURlvdef8W)8KAuxb_>Pze?lBWw~6zEvp~*K(k{9nb;J+O&Epi z0u=MJ#!sIv>FfH+#O&~2Up0g9Q&|t@Bn#_$FSgc%F zH5SaCx^Kf5pImbpl}p9x*?kZUZVPr4o>NndKk)2q#}~`xW!GA7Q#e-fujJSVu-$Sh zr}#c{*qa_ld6zlUc=RkjEi;g|Ey^t)b=`pshMwQ}I*%fQ?sd~6N0Ke3JC2@7ki&^- zboaM$@MO+q!{kj6e!|~6pW2)_LHxkiQ-E-W;0kO(v%P_vnsUX;VwokM{3bjSbKQV< z;&l(qtImfG_C8j2Jo-BnPjiSLQL(xM+gm({hExxFUR;AVu(BN?f^Tf$)0o3_PC|A2 zhKhM>g2r;f(qdV+&$~X#Mi@0t)EibED~EWq^~f}MlYF>=xn&&pUMG29y3=hxD7(yu z51YYtxWah#dBql$4sPf*DK1myGRzLXFh^0t!Sa&jWqVi2Qnvntx@M0KlT3slK_}2U4&uw0QasacgI`0_WWB%B zo3U)>)xSRR*uF^=fpN6d=jipDad8i^NmXtmZnkjHlN$lEc^wz4+!l8#hjNU4o|!_; z-$!}$Qt!fUPWYvKxb9GTmwwtaVvpTQsbW=P#D6YI`d=fM!czM&JYrjz83VINsLMol zLAljvz_)mbjVv#-eF_ywqXDwNy&W$_we-K`R<+{ChP5`i3`fr3o+n>@`|r<~-81Z? Vwe^HoD+w5ihKjE8LnX`0{{#0pw9Ehi literal 20780 zcmZ|12Rzqp-#-2=6_IR7Mp23oB`cD>Wrh+;MU<6FGLlV55)IiQI|oacKS@8dYmZ-~C0Iz25XErmj%*VIrkq)@12D3lc} zG;8oLY)_wC<3DSTYpbhJ{*wPCR-{K$D4QvoDvHML_r86&WD$P)FZIO1KS^(uHrWfU zQPte^Nbkb^rrp2K20x0t9`en+;ZA&WW%%bBGwRz-yE#s77tlTBuVa`LE%E*HOyf20 zm%+D;I%VQp{!CBZ(fcqnGktwY+5c#KIAvQ@F74l`AncuX9d@ormKzBWpV8!gLd$wf-XJ_tF7Vdxy!!qwh*-x+I z!)zKOLLv-B-G#S2FB!@;8(zj&w_cvSTD z89%+e{j*PMduC=P&AN5ZqPHJhm74O=)y1XebApektXkXO-4sYS(j!K?AtPYIrrKpI&|h<*GgN-eB&O7MUDX?1gEMpxH-_?O zT!S^Xa4UFoMGNTG-ri#?cBV_L)g@R%bJ9#H^^W^N>}M)s>G+S{($SvcT{^nDlVj~W zN56jEP`Y?yo3@Y`ca;6*KLSpwjW%neJ_@zitVLWKYB;ffde_#b^6JjssJOV``RT7! z7kl#a#d<3)Z^$`Z^l;#&O6n-NzwYw=E9AF_>InQ9y_$k=Ud_vMy41+Wx2+CWwErlN z_`IH-oz2M08(Cg1C+9WKT4dkO%pDSZlQD9e%EJ&_@&@!=(v0K{u%E3Dbnh})Dj!ra zX64ZiHfC9Vo)1sYQ&HqRe$n`?p=Z7q7<`j$`FBaUW=lEgAtA3eGV`_92?S-Zy1KeD za&w0xiuvW`xsmQ`)~v~Y^yq3z3jgDxB31RR$V@|XJh;|qeb@#X1_pICwT6BD2N8z$ z_Cg;&etc7U&X!}-CORz*By;Nfu3^TQ1A2Pvv5&qlU+To&X{hU-leTYvN==y^V%XPP z(n{adBuTC_-dAx+jT8e7iryAoWN8FGkFOPcEdHVzSB@P@84g^m%@av z*8b&3^6-$QR6k1{UDNPvcyM12GrnU`|Mq2bv(kkN7YZbAX{EVx-|-28KMMeoZO))&|;xi56BC@qzJ{;IuZs4+tN!b{$! zrYCRSz4P<)tGn#*t*4klPfsr(GLlzey{#?To4VFkCfnEdR!nquM@jkg-MV?xUoT#^ z=H<(EN1wVW8XCT>YLLG0je}Fl(ZcNK!rV{Z-p<#r*AEU3x?Tz=wOYj{TG@Ez`}glh z-=@+1`t@tclEGH)$=M@*XL~v}%XxC$h(4Q`mluhGwV_n)bzjBh)&iTx+Qv<7Rr^o9x-*@3_QJO=U#~yEw`|+C z`rN0NzIl1$)HC0^3iy?jl$saj8L?r4D@E6687nn3G>m>IlRAF2yc-MrLMaT3)A~C-m|jzp6jIx=T0SICmY-!Y?h&v2o)@n)U0|$}W6gDtcvV z6`ZH+E3&QI^#lU1=e{WeY8<@0OqxVX3hM1xtmyUU=YleCOX&8t`RCJ&8k zP)ixv*?-q~@$>PCIS*7ns$e7!PJ49*8Et?r(AFOI_mX0j_u`vkr~cZ;@wN6-l6*JN+-3Ui&710r{a)K= zxK5hCER>rfArw-JJZ#QLMJ>cv%q{?elQ&~y8<$auhA*IqPzv^$L3BxzJkhwvo}Uhv zfHjQOE?v5WDu^#F>!usqfs zv#V}vdu-9zigG3N``a3)L%OjWI&9=m=EdH+wT7FUyR`lC-QDdwUnr&GcT#0*>*`w5 zv}0X))&IA;Wx8=pGwp3{KduO|3y6&`I|?bofU_83cqkAr#kHR(hIm+2g-l6|$6r)+YgavB?T7Ki1sTwtv@l zaQbU!$;EM=-rimzHw&`Y)q9pvr|Q1*W32Csova4wf(r2zRDFPxz@m?VGEKVPOFNN@ zr?qLSkA41n|N7oz z`n=2YqMEUmpr9a11QWL(R#;=O+A=Xywlkjpe(x$UWpIQd*=>FAxseTR4^8L-0|RS! zlO;`i_Dsdh%Kw?~URAiTBdG%0Z;o`-aoyYUj?K|1MeW;-8tM+}!m?jvQ$T9PRnXbk*evho)Go3NJNEDAk(887wpqSp z(ZzAg0J$(KazpWderN2)jT_gmU#Fn?w;J??A??-)+*=`%yOj zrqK2v>Qiup;a1#y8{>uzK?fBUjsMB!EeI$kBs4l#XvCBvxz9fto%zTpT`%e6i(9ll zU-sk5$8R*92o4VqsHmv;=keHb6B8+ye}AjFzC~F@eQw9A>}=r^wzeA_Y%-07I%_am z-puSn6W34EV#Dpr3&-#F^R~{ZbVN6uaLv)h3x~leW|-#PAKg(;S}O-h;;yc)#Q%yFlZ@{@#WuswqmpfI20s|3 zs5B3tdpo=8&46&d<+Jm1k8&@FV4bwl*C)tnZyWoESYp={>A2H@id(BIXwPpIX?EkaQkD4J6v%f`S=Aj(Ud-3roj@P^w=h-5>(EhmO zJy_}3g|B2~?Ua+3Z_PH}W9Q)T;=Ct1U2%ZFzja%N0bY!jyjaPF@0{p5DvgP>hfo+W zm>xr^n*G_L&{}c%;_J6Ed$V(LTG5CQ-Tb?EGb=33^4}6RUx{~dzjUdl zrY1>g4V-zQ^XGIR36ZIRcQ0LiFkuJdNU8U!?pE-M0{L<73 zz-xUsVagSR_4_Ah!@$w-?c2AB4B$`Cf4zSHe#CyS`9wU{y%pDkgNKiQ?a1K}Kkh%u zk%n*_rE)!Y&gIYVv%-fTYq+rr*vTj-$i>N?7s7<<{PEdZ&*^s4{KCSzSS#KA~okSmM zI@6WE0yTOAcZv>Kl4^9E*78f|TE#Q&Au(()2$S~zKrfzFU0uC3s#yhLi%`E38@mA@ zN!)ARt?T{!+Q!D9SAZpK+7=dEyLaz?ld8$mYS8SzvxUj)z=;!E4?nisBD*HW4;)TS zPtT@~j*hNMud=sN(oROErs{b20@UEgXM5O}mX>H}X_d6JLPzb;zy_3*m1CBK*AvX4 zs->w}gJ_gKXR<9iK0f^X*N(wXE1&wY_N#H!@NNaWaVY0Gg@k?A(CStOoSY zmM9ICM!@9N=f?Zwhd+I)?d!92ySaPlHditqc&^pCxw$6~_eQa5dwR;1S5zE3e*7wG zzt)F=azeOx&Ht(??)Rnxw8NnDadvX(deezy%*9bF&C4!`05tQ<23kyf*kr(jgu8Mm z=Ws}bqN^*Gd)2B{*6$ym=8$#E(2~p}pgqGNg^`*0>htFs*Eqyi78DdD+mP#&pn_hz zcFo$>cDrn-#f0nLEgp1gp*2^pUL{L@je8&a)2C0DpN<0P)m7qRlXiHw@}8YL>AJhS>))FR#WbIH z2JThO*??+;I8IJX97q|eoLe;b^!Z4KN2o`)(Wz4cr(WM%%fiBveYA}I&YcB|&u3r1 zeoZpt-?~g@QG{sDs2KoX4TjbrkLj_Rw{H#ISW8=PYHY24y23;#rWPZrBUK^KS3%bb zYJyq?nt@s3B)TJ(Hz_@6E@vJK;^(E}YtfEh8nH=A2sSZ_UeAP`8uDR4cvtTo(lcNSh1U^?c$ET((acp z7uE=;6jN1#k!Ffu>`LyqXnfj zGIFdSRaLomhFBH~olqxPYb z{hb(!R7gfJCfn!>E=;}H6+mYt^hfIz-|gGCk4=xYqY~BRAFE?zW7|d^C&mveM6R9z z#%pwkD<-N@5g8eo*)Qn{UBf$)Zr)^l87EGqeM>l#H7ri>D$tsB`$H3&wQChMG-xma ztE!`|%4q88HIO9EGTns~T#Zs%Y*FFChEjm=3mn~HywYcBj?B*aCz_bsvP=)U-P5XX zZl=8}EycBdpT6>mh-Sg)(DQe+LhK4tme{m%u>1sfLYFgF2ukQNRenO-nNW&r&QzzQm z+Xsh+sCapK@m}krqN0-d(Hz82oH${P0uC;fg*&acplQoWg2T2SyidXd&{HrR(7^`N zISn11vY8nN5>f@Loy^enT`cBNuGy+6! zl|@Zn6yf7ri8%4g%E}tu!4^pFFtsc%um z896ya^D5`H+-}m=3j`Uz2h0HTqOB)=V5_=-fB*uo1^|+ggCiKv-WnAgY~0b2a|AG$ z0gPE#gke(G@O7=8vWvUse}AVte||}bu_^k=6G^Nj^}c=kuJOoowj?NQ05hvG@~okO z50H*(%{oTzwA#sd?#%4ZEeWKZC-^L3mQac5KwgQ1JG!=U&zhMZA6U49f;CREva(u# zd7Bm|dG?N$cU+c7OWddv!Mso!a+%N7vxKxgK5c{+Orxl%7#tePOnzw_m$Wm503x(D zRdpt%|5+Rjrt$L5vlGY8ynmee@_hfo(vGR0BI6$ArT6wf0JcN+s~$K&;o0vQytMcS z9K4G0>9o|;fwJ++I`BfcsM`O=8f<^mM~J;-Rl2rJqx6?WZf=ZxuNhdWm)p}uea-mFFWh;RXVDxHk3K|i--s<9yfiD$p-PL*Y&6ags z()vh;82%`Ye77%#Cr+I@j*zUcya9$O^mA8%jhK{F&@ysRLS)AjV!oEJdqV+hi^$I0 z3XF(Wl?aT+vFFq}%X0TEh%|#VLd$*wc=z`9PEMggF*4S|fP3LbZ^*8y1yp*ENCH+7 zjscP5AIgIM%#73NsEnMRofsVL%;QxH=L`U$5E7vrqhV;soNG~PI4dnFd8?86;ja_B zH=b&_Pw(U7qxTT=Wa{wqFU` zFI-SVfw&}`q$PrWMS=`*tg3Fnfh;^pM!_FHevombH{D2JZljFrRv>|ZjEo(cz4P8^zBLl<2=tv6%{SzFaZ0v~D`P7oYprD`t)g&w; zMGRw8=BiBNP>kbsjg9JVtn)M?A|fb*bzp6QVA;8Yg!F;QNxB9H)13M6)WnU;(3Lr4 z6}UxbXXh(9IXR=!DkG$3KPUslAC@8hzyQj8Iyx3JWfPqg;QnCS9 zQnfLO_CZ!w7|JUdd(EDl+4xV*+x8$YujSyAOvB3%#Jpz*sgcKr-B<%4g$Pyjd;g(F zkNCXefTqrUxs3qF_2}aS#ULmlg0bmfxI!)htJQ%Ty&fG+8gW!@Q&Uq-`K6zvpF;Hu zj!1Ft8rDzSz{PcKur3JY9T0jx7;}Ci8u=_5X)A!uukn7~4Wbr|++nNqO-+NUsuX|( zgVwSL41lsq=8s~2Xp+U=sgk1QFgG))$zYEsCMRof6>G>eh=wSnen`XtV^Oec0oSgr^PH^TkesqZ z_YBsD!<$j~FlnhF+OhYTtgTO;Tn(^Y8R8R(hz4A}A9h?Y6V%|Ay?fc9crkMaupYq9 z8DI#TotxXTV+RdArvxK+$jw!e+l+6YIC*l*=FODCQ?IU~6^u&LhTFQ{tPb0-3pDYH z-Vzro6r1NwO^P5_(Ue?<8X$%_Q$S@gavLRW-MZDfztSsC+9erKK?@5E9`)CHWG5-3 zfZu9vZjVQ$Q@6ScCpDh{m4xu|UnU)BBW+z-E&iKs)zDZsH|x^&^dbXgA}F5Q(d}#C ziA|gQgMv~(;K`*>C1T>j1sec?tbsYvtC|c@lM9a3ts%i<Gg0I8Kc zB;*eirg-!yH*#nLcYv1Qu3h65RxB%gL^yx^3g7=9FThj#b|I(%RKO2t6|5XsO>;8Z zlWb2`4NJ^@T|8c2W2#w0#xcskg<%?rzH?VPaNnLNe#({E+1b!*53YkBBydLOu4Fj% ziftFL=yI2)RsqPFRe4un>8^lJVM4BA>H-MjP9C~`^&_g5U+gZck9VCaYakyPpa{0Q z++4HE)W{jrI)xI=AR#8;Bcqlyk|mKzh&`8^q5n5;IpebXc6M*$^~TCj z{c#A>%Of`m)$wqVJWnTYpPP%*JGCorEXgm%p=TWHest3P*C+^J8_&VEX1HP-xQPo>u%6d}JY%7IA8 zo9^$fUiAQ7eM{s>AUG=uCHKO3#KO!%N_<{sNY;lkD$4%3;pmx;<`*xNF^#|2mCc1( zllRPa^`l3R#Lm8#AP^WVyKwpm<5rZ}2&@mLK`kX+U3#LI;laec7u>;NA4c(K+$MiZ z?(*+!u6K-VEiDRvuYvn~8Nbi;2k+L0w30Hk0?2mEP4(5jR4(7T#3jzYXT@;me##-t zW@*X&+rSkIZtmf+(Sd>#{)o9h{M{Yw9v&X3_wG)wAOhAjHDyi|+_`gjFrk=jdNDP1 ztJ_%H+Eb@a5d{+T=F(m@7yHV^!esmR-Q8Q{yzX_nbZ-1+>)efEa zMq%ZYJQEC74ZF>Y+9ve7(&_KF?+F`B3Xw&?qH_~r)3?T}{_>!@)T6nCRxmJ`I08;T z!vcz5_+|z`g$Apof))f=Akz@ZOUmoD((z3Gbjy4!Tfa&-J1lCMWPHG%H7^EZM2XzBn7!^t; zn0PiBHDhB|VG$AS_({ON63@BaAOgieFFiif{Yd4HY1Scn?5bUB^6HLkbH6)#9oWs^ zy?x7!Vjz0xLrIA=6wx)ujvc#a7!}+HZey*yyu41F*a{3v0WmQ#qZ+9yT-@B`frwWE zA}HqIAPB$vOjQ}1jv$+0H}Su6g|a;8aiuj1s8hgXV5}OL7(`=zegI?fh|sF8Js{W~LydrMcFpC|rbm0$ac4Yt?qk&3qnr%gXlsi~L?DDDWb8fc|gl zV0O#D@cpogx;p>Xt*h+q?K$?^x9#IqVQx}JU%(5ja&mG4-U*8^63lPPJTNymSK>a& zSNV78GT6fuz))ehI+&3CpT0$;$ zsZce*>*5)zCaxJ8?2V3&R*>Cy7^*2Lm_*or_pZ>mB_!C`ucAU8u^oyPMANSB?lwo_ zDn&#@5G+S%-tbLQN(iXR%1XPojHppNHB9!otrM0Ce@zz0#C7NE5W*n;w7vhq$a|F$5rHR^ohY@ODPBy};J3 ziKpK^;LR>NUQatZHb!E2o6l~QR98@;1(?~h%?eh60**$vc}AeE%aGc-O_J*zkZ;dI zSeRH?_+{Az3!r}!Lc!6|5oUnmQRx`kmf@Qzoq1M6_#x<92cT|&YVZCyWKJ>=Zp}q6T#Av_k(yXz_P=> zU&`w1moG_}sk@$@-$`LL--Fo{B#iJ>>6!jYZkQRm9-n4uYwu2BF_?@4OM;0zPVUlf zPzbg*sk>bO!ZCd39xV$5Ii_S~#Y@)xO``9pU8)M;sTgXznYsBf%pl1_#yEoeCT+qoY*BYjSmlhv-|Yv z)9hp;uYz=g!c2@-nrg&!(jBPj;QaRVm9;=oB-0o0HXF#jIcpxiWS`v<63@V@iJc!a zPKAk~r{@Sk`Cu5G&!4ZyGv}Y_CO!H%c<%Dk_x6wJ`jHZX2MI=OPo_A_Xu%bPSFBu` zcL1|{4A7Nhzo+{k{cqrhIEmBDP%T&a`23~a$AVn<9k2YO_v5_wQ#)H1&zTPw-@JJ< z+c}eGQF;}rWdmDZhGoUm-o|Zw?fL)|*@ug0Nx3d}7hi_*$y&sRcI3$!#f1G{`RLg? z@v>}?SvGKou^J<@VHu>u0ED0oi7+}{&=ln-<yIS+F3W4%v~M|cn(+zk!K6ZfBa#|sK$T@`$5>Osp<~^VVnWMg(oV0l)BnF~jp|$^vwS?6n>xLw5o_?a?2qtP^aL^ZM_;_FWrHrFxl4=oLYY7r+ z)kp-5K_)-)EeIGc3S4qM~t?W#xVWy?eoliJM_;0I?Nw4YC(xb|IG2YA;A zLZ&tUm@=_Bu5QmXqDJ@h2g!L~!RNKMc1CXoIALH~yLS)hlarGfxwzIrC?9OkRJA#Rl^ec3xWR%tART2?-Vf+(RhZ@{giJ48YbIO?&K+SI+a@5uCnv{6r2GC#d3Iql zN{9+ytS6}6tA&NiW2 zCXU`M`C%orc$yR0GrjIMF+r3J?ZC}(^vBxG&n{Mb$sd|?0T{nxS(4SOA@1kG|? zK6n!7h)7t{7k_YpKWqGw#Acj!#+7tqmhE~sFNNQI1ks&|AS-Oaq1mobqRamM)BiXB zY>y-vuPiMs31uG>nv)}fXeOLG8Lv^>SnvS)AW&PYe0<;^+VJ^xd>zVoR7^|&QSJ&` z_782{II;t5OBlPz)u141;KAwFi@m94LMG8aPlewFrmm^0OVM%YeO_mq;1vj-lNrJk z(=ElvFutr>y&B~B7GYr>jY3V8tq0+uBn&vAtuFsQL*iV@FPd>|Y>Y4>>FL`)4h^|r z_Qw_y35aYj%+RXp*%Fo&9s#H@WN080G9gbielf*-{8-c0W=xnwE1$)^2!(vlIakzt zi`iFJDEokyOyEiRB!XxF1t3Gzfj42(<-bqM`>CKKX^ATf?vL)_z+9pDFMexcVe z)`K^#$7+Jc3WZ5(=xbh81H5he`udw?@4sz^Zszm%kJUO3F$!2xLST0f>{y$&Kl&%8 zIHJh_mDzRXaw7y%oWn%JLEEzk=v)Dx+}!KuikeHnW`g&;AcBLQj*gB90mFJI%jnb@ zM~X!W6AWKgZB31$o3Nn^upe>XzkxhOs1?x0jb!YE1sY>#5=ggXexcPEQ80gR#xwwJ z1Ne~|b}>;|_LwV%k}`iSh8h|hsZqgbI|nES!0Nq$A3*rXlXWONIcw4HzeKf1$67^!BjhT)sUW_6$@qxY2o6*Ylfz`U zZGs{SxtmIW+vG=8ctw*y>YC+SuYf~9RXy2I|Bk?C8;9NBP0>{0#q6LmkAuf|D=UIXF1Tos?{cx6K*N3aGcCIAXAg zz*-*%7Kl^u*$*UjNN~LXc;7l0h@8RnkxYb;FeuEglKKd2I4ONN#bFKB%k0sKWv5Rg zpNBen=bH=zenP_;7#I-!6)lvE@$McTwcXvNhUMylxnSRjV1*r2$0iUN2r6($gsN*| zzHY?mUdQ1ah+_&t#DkfChb((*RR+w(b#+u&gy*kbslrhZ!6P5~59&w+X0YpiwC&-c93ffz zQsP!1lOfRUNs--UwNFXG%bi+6qI{3;<7C@&2@SFt2X$7|QW>^KOfUKM^UZkeTfm#FQ&;aI(2%jMB zYcU%DKjc4oa%l8HI*M@B()`ZX_vOP-v!Lu#0J_YI9l1Lh4Oh52xsv3Wn3(YS;C}zk zolw*&nVFjWVC5%c#^1$pACg?~Km5amsv4icmGg^= z>OqeZS`=p;25<#3^gyqpf$fF3=22Ck)$pJM3-e#|@(T(S zF*g)oY{sCbtRc#*oU4ly0slFY@$7TM&}`OFNvszG#X8+4$Z`IIb@VYm@ONN^j-zq} zM?|OrJh8fl-wk~O;7@QSgni-}K-FGTQd*jX(>5S|R7+c$mY@fbQV216b|ltn;#>C~ z!u63o#^y>_Y-KXoNPGuAf0yifOT>{)A-gJPSHhTLP*OOf!kYr;yR=AAUfwRy3`|g7 z@fKv#0M@F5;6U6AV1F230KjId0fi9rR%t~#0H7MoYTyFNSs5bTARLvk@?#x2VQ8E- zZxR(r*HK6`Ip%uSH8~|E>h|pr>QaV$zVoZ6uA9yl?(4qjyz39o1LK61J?oBe$+(;=hPk|_TIREsqHh_*)0(3P}R|tiN$C(|DK4|UeX!&s-wo#M2Jns?!`6^SM*>9V+^1?H4+$q5f;B^ts@oS^ z@cHxS4VC?blqQZGv~DsfLTV1e8G?UZcwSy!7_Zs7CtjS`4V7TCygWCo{5W3*Fa8P& zI0U#^m~kRP5g>w4wy#fVj9+(%{y8zIOsS4CRqs3P-1l6WC!$>Ts%HE4B=EeW?~AhJ zg7R3&D+0^9jlNk?ZPi)Z2fYPDH^DnR3f}8*whX=+FaR*HZn$&j&bYO-Wa&u z&{~GokkdH0k^lIy3P8@%^iLFNBp@F@e@edZrJsxC<$ry~XMbPbgybj3THd^2#JD6! zr&Z6_De(D~Rom>nS-#>+aeo*oC1xkYD@xgapM_f`I5HBY#7IG_~^R(GQ^!dMYZaFa%O#`w1IE_^mjxV=K2Zhrlc}l8cgtqfszm8ejSEvcuv) z%gNp51=NR(jQ-eF+%Qb@fUYU}`w7H~dd?$J33+^-mw*0A$(R#Lw$${V2^rl5!=e(f z`tKhf=s*r)@Q*JvMPUemms+qRUA?~aUK5HGM6D~~Pw0;q+Vq!&7v0)>rRw!CnZ;15 z2preqDwmkU3Cre``~P@X*(9x0+lJDpRMJw=NfrIpu!+VG3=jLm)9};MgfDsbjj!CF zoY=pA%kAgBGhWK9$8u*Be_cL)v>E;%B6torMHkq%@DN7>9NMUAH*K?bV~pfg$Q0m9 zc7=(_@6}x?W$eFl;JO;ndc?M4XpaoTD3X+wMR!Ex$ZY3CkY5CqTbo;A$^$C(5dH;+ z*}6kLgy1JkF&UqrFhSNKsyBoU71W$9(M^mvL&2d7jG z0}7d+&-9is0X!%kK75BbzqB-SbtAwP|3m4^ciDAYmruH9i(C2a{p29_+K%hYWnoe zHUj7{P!N%ra^Sz|0kFtS{Ndhr0WOwcIV&N!a%Sd#y*pTN>E|hs64me61Zok$glmhU`Rd%XV~8jh9w1$AIwIEPkI?#Z5~MNlu-8XGKS&`!1{ZGBhetlmDx{soygf;q7hUJg_LMqQaZ#tg8pP2aZhrS$DKF zQe1$YE!-6%u&z%pd=JH&+Wh|3O@J51g!Kezf&~Ko{cc~F4f_Z#N<%|a4MB%O!I-L! zaT)fWe8&&6j~zPKQ*hxcfJC?I3b*m`3Gsk6Bxsue3uuh~B=j(@;#4IocQ}rp?68R5 z^AGRMx*s4y4IGRtAiK2DVU*FqPCzxHmXcGyglz?DLl_xMK+~^Z+T%1h0+B=BLqb3& zjzTee>L!SD9aQKCBGiup@Eqv|EkqmM6pk_##Rtn4P-i__$-#;AR6DyY9AbA zH81UrHe&xT!rI5~)8pcb3h84oC6VEhiC=9qWQL1BdVRyg>4^>rh|1@1=kDDhIQte% zEB-z0b<*1rK{%BRn+KsTfG=i$_c)2WPgn!ylmCmIWaN$v*4Po?;3^>@G4bJIe~iap zDf(-m{ori%BP1UtS>n_N^ECltlwDGeAl`z4Np*)?+ZbT2yYv>vDuGR4=3G|1k=}%L zU9+Bx24^^k2iaf0L?B%bG6sGb?Y-pkj41lmWYynA2F$%MNh$)Z&oynp=|vF`AO!nh z-l)ZmC-FeXV9tT0sXeIFsiN{wY1^H9#a~88tLy7mu`i$tsrMo1_^(MI|#25IT8y03fS z@u1suzqi){RsJJ#FK*vOQIJ7{pFXVt%?WeW0XJjpAoCtEv1**Z5W|>35H>`p08mvZ z8YEc>$XmaDJv>aIqZu$;roAOse*Ab~V1NRTxaZtREYY=qyU3IVEwKhKh38kt#?#rJ zhNIb#{a>&f%*1~kM^e!V1~4rXa0EXwxz6RwmkIU*B)|zwcJAB38Esv|bLYsnGS&eovIX~@S`Rag*manef zTJyt)Yk|~&g$2*JqX_un9Z2sZ%LKJO2|Vhp;NY9Ffq`nrek!Y^o|)AvU)Sg8j&V2; z*BXKdN(wJJFFB@wI-tZC6&#_3v`|UiEm8RB5gd9)v%}L2GcBm|bf&j5uZ5M}7u4kd zs!qxIuk6GN1oF2Ar+pD`c##l`P4}CB%?qBZkg8G$y_3f`R$=ZR3!YAc$xdtQjpG%Q zI4CV9CU(82I;QkKczbXO#H%akv&03@+cV5ZlSriGV|9TDeP7Iu-~t7<@_ub9{{MlL zz4PQ5HQY-N%VgnbD^B7uNJ&XyPR46*g4eMYb?=ysjZw*L2KZtUVEBqzdG{V|Dh?VW zbBPbbeQ|y-QPa@fJpX(z4xn%OTXZo`+~C6>&PHDUV+|4fV*a((=~pFJKig;>_(Df} z-#(G)+WopIOuAY04I8`)?@D}^6z`DOa#nYjrg(w%UE|A_l0?ovme?l|mwY<>yY-52 zCAw`cg-7H&(*W+G&Ixw)2g9BEv=mqa8At9j?h26s!%3=}{cHMo&{_W5T3gyb_fa@$Puc@W< zIu_&48fxnHuC5(b{dw5_7xqPK0w3cqySj2#;-Xt3ud^ISU7_-|wX@TJEcGxi&s0-0 zM3`OMdZVd3e&R|2ldDo(jrH}qLHyBoaVa=_d+?;KQ^u)sc}AP9B~^dU_E!4^;$(3%^cXb`-y~*^Wl$ zoxYh_9FP0Jiqa2`IUqT|fB*h4C#N0z(AL?Rj#1!Jh2_aprSk1C*8e;O+8+xV%u`dw>l5(Ws9Lx&z_XY1&t z8;Qnh14iTEVT~gju2{lPBeTcM)3cnGj?To)ECZ8tdwYB4g9mCDmD@lnj7+-Uyn8pR zyU>=p%mlw+{g%=72Qy|E-|Bbo-lagUSp8#2^WecwIHHDs&PkxnSQnE8>19`XeRPMU zNea5xy11QYc zSEuMMWZ@yIOK4;`qFe25_I$t9^SxtWAdz#^CbEe@gXTFM)z_auuU`GnBMA9(_3D%9 zZvrNxOzhkA^t6zmU{w9_wZiz25c=f{QnpYRH&}~3A9Ti_jFJ*FxGAUpEJEYDyKapi zqOROB57q4I2O-@3hauC3U6yUt)z!K=y*)kqGWVtT{mQc~?0$?2o?}*^gksqS0b&~G z{_Ne|F=v`LKY#wlk70iXMv}x|Qf$+rK0ZoCRhb!504bS82y#bt|E3c--S>1b6jX6& zXy~2C+(u5H+RNn5oCWJ`@92oBDVmU*cznT~JGBv{SWzFr3IC6imZu~*_rga0Xi-POgvLR{^{G3HNC$xypYj0<#f9XUEfhLt#2^8I(O#xe1FvQodn6bbA=!*h?W?Wnbj(a0> zGgDHO@nH?9QO4%xx9{ntTZqQuP$fPsDYxf0%!3beb34#t^X~M2_>lAChhx!`C-#^D z%NwatL^Wt+O1?st%F55z$3h^!438d-#~MC9_gPQGvfPX(QXP8s9r?>o&=aO17k>o- zW$#dCYGNX~S1PivKl;}1s`)?tkwP-1Mks%<6|5j*`q6;JfYTKewgyxn-eXxHis+ zZ|u=KUvXs70zdd8l_t7L;uA}1ZhKEpRz^mKW{>!Eh7|W#bdoPvV=cWOhT7U6qw}G`9F>ljRG#JE_jssx zL&9fiLEHh(8chsqB{uSU4`6+Cj#)#Y=*Sr1dl!)1zr0`9Dw*?JJZ46i5 z%^5W8w)7x;lX^n7B#x}ga7Bna;KOP@H)AN(!S*zvvDp_je{mZFXM95~FT!H!TpCzR z2ORSVjg0Js{C^&Y8@~Q_Ir|E-+ELS=%}h~Unw+c%JIm5$x=ECfNpJ#e64tvgjNk9DPgr>Cz?6~s-ehh9f zK~zlC(j6{#j8AR-W6|(5bP6f8Q(pe)cD;N0IGt#DX(Z-rN9mJK=UU`9m|ZCS5mVHd zRSJ_v#owhrKHuw*KQzpJ(Nw*BSx_Xi4t)VyTAXgb~7wtqOLIMJxc5!c~P!8lfZ=;SojsylQdJ2Ej zR!v^7Us1rTsi4w&#RfDC6%~E4aMNR1c@G~yHpV9!%%1;hapdRwi^CAKPJZe{*?_-;lMDn>z=h1@+ig0fDjf1n;--w@5>cyu&H~^W~msj|CiJFb1-G-_t|9duX6z z_b>l2(9yYu&*H&HXRN1ZVE7t+aK8Z#$5?x-topm1YM@FwmJ&zP?`9pEV_v8R%gB12 z$mvx@eSLi{{RgA5Z%&nSr)s?{%Fa4D<>ABE&b`gr1HWN&tc3H5AS9rN zaO;1?p(?sw?Y92@JOD`tFh@7kf?_S6Sl6s9Dk>7h+0h+~D`|A^X;ZERU8l0vGBGuE zzy##+yW4geN)B-HX{g$G&?^?ArUOy!zz5jzoA85d#Ym~C_>J>G#vnGG&!3B8)O@P& zOSn`Yjn3Uyr!-T>82|>?zAQEq4h;!0z~REa%j5DJ!y8;2`a&8lsA$gOraJ!QrYBFF z*ul>~^mHzA|NN-QJJa>zrMK@;euysr05t<@5+%-n2A@bsp`2n>T95x+*|_Z+B9{XL z4$TE#$^&u0D9w2^D!<;fSH66kqLZ&${N0oIdxiCUnAmL{;1DT`g5qLRsIk#cY{m>C tJ@%ycjm#L~QvNHEgZS_NxWmX_3WI8zfz{bh=J+;6Q&mqTSIPS7{{a;t*G>Qc diff --git a/docs/images/chapters/control/1b8c5e574dc67bfb0afc3fb0a8727378.png b/docs/images/chapters/control/1b8c5e574dc67bfb0afc3fb0a8727378.png index 7bc49082978adbc3c4afa7f095ffd9bf7e2fea6f..8f4769cebc9fcedae1a694413c5bf452a517d115 100644 GIT binary patch literal 16602 zcmZvEWl&sAur?tOT!K5nEx3CKmc`vAxVw9Bhu}-lAi>>P+@0X=3xq&$*L!&D`|AF= z#llt{_RN{->1lcT>5WuUkVZo$LWY5XK?8k}P=SGYgZ=u8_zt*&Te95@yuCA)m6m{c zdHu<4FG_-e`2Yiw5L5Ha{I}-esiyuscy@C+R&cY94GTh47l>M{@u;cPuPJXmex8rA zvnx9;wSixubgv04rkDJ_p7aUvp#6p?-8mpYLQ7I+WNcM1>B5_g97*o@iIy#A)#ZeL zBR`QE^cfS*C}P$b{{v!>1*Q=tVvu+Q+k3cf3JEY`&?Y9O9GsXVqIR$8#ol<^<+Lmg zKK`h7*auW%VqSaGg-U$|{zhA)T89mh3BePg>*KZ2a_zcmt&h$rlqB7i`jFdW$O)81 z-0^$7$bEz?YkRVM1`q%4ZEfhVrlDVd!-B!6O;x*;wKe16LRDHB&K4q%-E!aK-GyYX zCt|yS;@7XaGjXiCU^R91_oSp~pKNVE;^N}I4j#H>`>U+%5L}f+9qD#B-;zsdRL#=7 z2}`S(eN?|=ljRQeS1M5pe`?4jNupES3qoPxyLHmmCM5U2-7+$Scwbrj`ub|Md9r)A zudCA_BU9*tB_t(D+lWfa%c&U|Whzxd^7&jFLi>$>wzS+;SN%IUp!oQ44A%@vhI*Kq zl~pk+CMMaws|I0!pRLAeQ2$$tbFfw(_)CYqquG6BSwwk1)eTvyq>NW*ok6+5v1~smg}up zie(7OD=S$Pgw7;XR8`qC#?eJPWD@&Ijl?j+?7K8+HOf*vHvH0V)_pKC#-%hh@lo(u zOt*SM5l-%Txw&h;cEg6_Ga&{!qustc>ctQpW9@u|{Uj##W~IqtaA5)ePln+6IPb>k z5LNv5A3y#s%D{^$)xW`T0iLB@V+`*O$>Mj}Q!+5{KP{01HbA%0c57Oe)oi8F?vHD$ z3E4>~acZun?%A}Ka>>0+jg4KdfU|a@OiyLajbHBb)z!COMG)rgpW-<%i}Hj?^wl01LIvW~@t$&isOQ;;u{ z8Jxjy-z;Y}+BDk!_fK9{76m3SFffKxFm-WJiyD+yT8d0gPF~shd>RW0f97BAb93eotq6?cCe*zlE0>?|+WfLHO=TCtmOF}KDn5ffu9ZpttGhWRKNJF$uy z$~Rw#s59^&>e19CVk%Ti5AVQZ~+6Z7o zMMYr*1O&8NT@L_p?5(iY3H?f)RP;h^kdbd%lCqMg>z~U?o_AGNn*({-;S{PsjTsZa*jB7e=n)Z zTPQMT?FqDwUAbP*PFYoUudYOC?1T#ra{|6GXO&Sz4;}p!N5C89iSv(dpI&N%E!Q*I3w3&X^w$Cu#`ZW?evC zyKdzC1(!ZDGDhX*ez5O+a{q{jf=w&8xf3sHzTFpzV2F*4?b2sbLXCJZRR?JZsd)n; zKqHbGcwpyPKk=wKJZe~*cx0~2D5`}nmUc!OYOIC+0MJ@T? z_O4-n2wcxBs|P8Q$HZ&m(2CQq8*E9y#SB7mqX1bblS40CqXcd4s%^RN{-tDMbEmgB z(P26u_v-pU$!j4fx`c!$;RBH`JP0JTX?o5`3wkbSF!+YPXWbhCfF()iw!zDniZxzs zb{fmz#Q-G|i26S&$K$V9naca|n7mDlyCrhSPrZZ_oyLCTJGr&w+2c8z#JK*cTn6@$ zA(`O_Mz}Y-%+UN!R`o$Rkjn4n)FM$nue~ig_0e4iMUeibBsKLB-Bm;fcE%TxVC&&9 zK*SQ&7@TM7J}l#-2_O6Kj?3*RzeG4E?%3KlCgIkHZqq^L1CuF{ScjG6?D<&ah&}UG zZ!I?$Sht1V{M`%hxiMGivwl>?y7J`Z)Pzq?*M}bHlr*_8UvkU&2ouJ$zlR5JywEu! zj3KbR?w^yZEiUoiZ4vgYJPkM|X-CBmXR*p&Yf%3Eohv4%Y-Ro?~5AXaM|VzOcgcjU=a8ncm_5!t68u@FUc%C8|+&E5&= zz*bpL7L{6ItIKB`9NT^95{&$mqt^2{$i{I`2`tGuEi$ApY(Pllqb^$4YO}T+8X0JV zwClM>7HcV#dDT~@EvfW~>ad^z-k}}!(`XF60t&gJ#;5JvI8~*o?xM00hVmn5;u8B< zHfEegemSp=g&XhH%8!zjc{vvYK|th48JJ&glQ;9%ecs>5-4}`8$c?3HSnIkgt+mDe z*t5I!lzdOf7Z}ln1a}wbs9*E+HI15wXSb#pN(Xz@p@%D)q3mB1EHGaggr{L!vOB@= zr}^UjVj^lriYgg048Bk_Q8gj+D9Gk4*ioPLxHyY9JNjt3`?!-2=H$j=Z9fmgZ!{2$ zU8+Fn?O(T{Gqz^?i4BWHi2V()+#hg^rbWY1utBvmtID6`FL)S<7G`EvpaMwiQj92___7<5ly}01qt(ZGJlo z9BwmcMukwLS%I;zt)Z+gGnTf8yLIZgtLKw;tAf0-W7Gi%8iKAw?LVrK0z>2xse?Pp zQkTc&aac;eK5%-$S=?iLNWmFl!%C*Dze%JP$*NA z;Xw4#NxwQLJ7ZFQ`*QfbP=h7w&J1kJ@t8zEnjiXU99N>NJr`(S%A{$rWMy4$n0M~T z-%6CdD8vM8_l-&v2m9Ocy1BEAXnw&l0<13K z`Z2^ChyCZE!5b#J1_k*(=vPWw1QYw?LSHlHJB$=1Qa$M&ZDomZRI$R}O0L|XTGf@> z;3suihD!Uda?=tDWDi2L!4H>eB?ne}CG9zw`=lcQ18Y%YJBz2fb#>X;1N;_=%Rh0j zX|aP;JnJm^v8KNrtC2@5lRoYS^H(K?2fJX@Jty{8M?rurpTAiD3jvP39YkHczQS!2QC z!upq+eW1bais1KiPR{h=;w_Rw%awm?dN~I1@^#emv!fE zVerm&*k*6Nh3b@>;RWmV+qd@}F%QwK?G<<<{>4U5eoNpUH0n%TP~YtLjkWqK3BW{| z^}a(cO`Eo=q~YI|Qy{$|M{VtRFv4|Pvi+3(kS};qsBu z0mYt)l{v}xFaj{>8jf>;=r$We7kxRc{d9Q&Q2d0 z@CT=xAt{Mw>PKE>;CiN=$aK;@rwdK$VFjx-8E2Udx*^`!m|G^qi^)Vtx!9;3(@xHV z$j;HwupR;DTFJgnT|Tf0t>zJLkx`O%U{$rXE=g6+Ny5?H(B1wPAWG38oI(R`I4I}S zbSD*^wY>#Z98*QRwVz$*uBsiB3Gc{L+0RSi@)QyKBR53SO03og;_`UU&Z4$Mre3-h z2$(%U!RARv9~t$BfBo_k%67u7q22Eora@}l<96v?swCY0)izAevz;+9`mryC9siSWk8E$4ZlG=gz5o&YGB?4GWX| zx}^P5xE$z#iIZ~=;`Tk+W6oveTGOh@{o;I4HS=%) zb|y^3up3uLAbObPj&-daEk?d|sv?{dq1i4+%zFy10oA>;8mme37s;7T-ZrEKK|NbFWLlf z1!LDm7pHMNiB76m>j@)Zn;El8I>==wSN$G`z?Sv)=a)F?)uT6n3+kQUdlE`>drMn! z@v#1=VPGYQl(5c13hdu;I*H?ENnDxN0r%t1-c-H2mZ!NBnMTH=K7@qA_Vb52gPc!& z$zSJRm@}TpQz({pAC{?yC$ekfZ><)DL+;HuEVe&yf40~{5;IOaYWw~I-=D=Tl;l=1 z8Vu*3RlL_z#D_r7AxDc4Z|b7j@o|5?SpC&rM|@!2V;$>WwMi;OmsTxN@5{pLJ_^M? zu&X=2$P%<<6}%?viJPC&ACHF||7xs-zn@E1HP4^pmwD;3@@*RKS zUn%r+^S24`5_H<4MP(t|{`R>S4odv@asD-kSQ@TbeVSHaJY2H~GYbG*5VVhXU`Bhc zcUknLAiu}U(9l8=P4xH?Wo~?h&#%Wu0zuJjxdp1UGBc(R!E;o @gRHkj9e5^aAA z>NS=5{e_47I0=Gk9*49$5v|&Ca%Fq1={lqzCvGLLGw@^vXQeqoI@cwC3OhO{f4Jj;NeYaMw8R> zt!5;W)A6hm|B)?&vH$Ud5LhH_CkT2ntD!U{=31=W81^*^4O?kIY5yCLG-Riz_1hTY z)i~>Bt?u$=P86Lm(%khB%)hwt(#RqNbe5L;bBi9gWfVlrq^Gl!2TzP>fK*uS1r6n{)pJKh(LCyaI_nQ_}03@;$0NQr$MMRXM`|ArrXU? zftTR}yJ%63A2~4j@Q~9aaQYejyDXoIf#a$wM&9a5inRSYdqNupdx#d46-n?rM3s_Aa{y}=>+pds3xKN4dqC%khoG^TCaY6rbFcfH@@xn=bp1pS) zBMm|n91g{8oj?Ck@u=1mJ;+(*?kQdWQ>Q&YTbt!?u{(7XMv_%`{%xUA{Hobwpjg$` zu&z{{vo7iA{AfVZqPDNG+}PuSmQ1Qf-OXBJ2kBM-Rn@l3Iitufl$~|+B-imHm!C`SU@`SSw(}b90MR8zj zxys~0C=E54J-=kmzjKr%Jgg&zvJ95=?0tH;b*Sm^5{N8~J`3shlCEuG$nlW%haOBV zY!|7nhv_4fPuiQ7V&AMBvo;}OCk=f1zMeMCvil6ZxtTgUYWui;*BBDp5(^v`93YU? z%6fvCVQCGq!XO{~b=NX$dpTC+W^>`cPXCx{QCI2W#hMsMUv)@H{ppzky%_}lw4s`n zbLawP*(rwgy%G{+jc!WAtE1ZRTcIUW5W7-w$7%0ADz`woCXNhZkF^-KcDJ z7ZpfUmXt4Pcdu;ty~aAvu8~D=Ee0^(avBhbKV6%<3+2Q^-#i)C>dJajyr5){wS{2@ zpIaQ;4_SR(w!91H+mtfe&gaImKBeo_zY~S;g>Cmd-{K1Zsn^Hx8rYlEGrk~upED|#eCqc|z)&h4)7T4WEJ zZV|CDo*1$Q*O$cQXMU$$3ufrop1U0T_B_@R;QAbV6BWG+vh1Nls5|H;Il|jTkdas9 zjQ@A{Eqr>vRsK7SFXZPB&R1!d=VIt8i<}(MTNBtsoVg#lqO~fp&09wjWWJo?_VDYf->C%;FS2_~ z5zSm5=fp+Uhsv-|V{bHUbHR3aWFy*rb7ytwm|#i|{lUg@>OOmUhyJSEdo zj>Y<9FAKvK8f^{#i{vNc&m%(N@#fHnnUX1wxHfT$=~0dM2512l7>1MH{Ebp^$GKtWyK;7 ztfoil6dB(HTnj$mzP;4hYWg#$Q>oxM*}-I3Ub3kNFO2@o+=P~N(4D`yr1vFl?xP1o zu=7gjaDDyF`Bc1F?Ob)8<(sv3c{v5hLJP4`$=4%&s_OKArk#|A-~6>2!%B~(j%9>3 z8;Qd5R%}hA{ZVFcs*6t?S9#=S8M*6v2deYz!&eJYqjU2{)8B_6HJv;o)m?u8n{GH#m*mvUntA&~0^IE%&!aHZ z><6}-X`+2*++tfjMXx7;xJggch8WSGb2{8fWTUy(`oi3YSqm72d62-O8@~}MLQVRP zl&%`zGnbUS`7M&Tm2gn00~gvJkG0P#B;bo$L_+urSCsVnL}BI9+05}u>&TM9?K{Bd zNad%#6lF+NnV(Y;Lff>z=tk}=Fmr=fJozl%-^8~iYb9~q`=M&+*QNhzG>{r#62x2+ zc{Oc6dQ^CIs$8f+QzEPWM#vZ)>Eb?Vb4G&=i>%^>XXC8unVjNxGf%>latf%U#`@wc?e2q5!O5c` z^`BS-`QyUzpuZ2lg*qpi?xzVWvQhr^4SJFi$o)Ks{N;+`Dp89^v~`f8&a}dvI3CKf zmj@=a=aqkUEawqCbmwKW$vnrW@Rk7onbrA)P{h<2{Pbx&8*^ZG>?@g{1;#Z+N4Gm} zXJ{3B7<7n}L@jHlKN~B~Fep;k0UzUl&mFWK*-p;RPamsFH6U0l;8n4z0U&i0f80g_ z${0I)`~~Q&ISe5OF^jz|mjzmR;%%M>Gu?{kj8%rRCJ!P5bQ?!qBbihe^QJl|0>xVV zx4*Wka9OB6o%GVwaFS}Jx4kN-_DqUs-y00%B(^FTSXU8fgW!u(e%Y$GKk}M6@u&ud zFPe&rGzCOv-8?LMKM_@9gK9_l$fq+o^)b-nm#$7j`oOyTu^j&((%i5K5XOu73s45N zL0Jk7g=pn+9k|al?VrK7O=^s{hqzQj41b^x^4w3)Z3eQh*3eYyUYx-7S)u%Q&1HuG zkX(LU;q(IQegr;{;s#k<(;3^4Hi*#ytp$bsaS3fO6T*u}PC^P~z;T=ZJ-tHV!GNAW zSA2Jo@~U_=K)Mol=MAc@ai8?%*r}xK5*&ud6-aexi&PGfF}--yJ9@I5sm?F_9zPPW zH%T?&cDPg#ZvC%Xsi3+sW^h{>h?uYYE2E4tflHDgi`6q15y0pDa4j7vcOk0YT%o9jb`Wc@rm0i}1b)y2SS7`i~bw*2&ZALvmlHt?R#s-#uli+FA)*XC~h(8rslx^Jo zZcvM6=d-B#%nqC?_i@IrbU+43a`Q<*(oka>{Pe&rYnRIMoc+Y&QIPphqh0sGrv38@ zFvW+LGQpUR#<*-3s_Q27>qPzU;))K2tV_r1v^;ylqcO@D`7?}FZx_B=$rI6zD_RV? zJp8f$bEs?5cyndn&gjpxBm7lmdCRx(_p1;SjIbP##S^KUY!4U7U!B#6(|e)o39^qo zD2LZV1I^Hkvzxdy2v_(;BzuCJG_o$)T5!X{i0HNd*w}AW^A@9%`DZ0(-2s!cA;6cE zB`Goic|+JtrXXk3ts{1P+XY%eAa*2rHlqhf0h@8v|K>I1N89-b-Hf|+{W^fR98uT%ecbX>TV)O8 zq>eBn9X_N#lIv*;E>?1~8PR?$+9Ym~bwVzy2xo?W**havFm(VC=Tu#1rgw(lw_0OE zR-*X#e;yq9x4@LW)l_2|268gznU-PuE56$%As#sxj>qlOIijW2WL7{?n;JIFrt>E0 ze$)MsuA1E3*GDT~<+*iZmT73DNF%pyy71hayOs1w^7ZFhs=^!5Pwll^o(UP2mDQ+y z7}F9-mIjQp{IErJ?jJ#0GOm+?Rx2j`JpmQUNw}`ug;S_}x^HV~ro2y_uFb`f~VMxW0xJf+4H{G^tNX+ilxwr6PiO?FT35g`WKt*p*AAJtOJh# zp5yiR&<qlet?h6%vx(?{dGmbi=__D&RG7uVORGqPBvUX{=ti#D(}>t!zc<9jH8y>ABNRGY zHz*}uV--3as}O43e@jN>wxsOSG?0lCCX@fmIK^j!PUljeVSgf~XXTGUoj!(zmWiPW zbcm5|EA6vXXltpwVx`Prl0=lHMI+G6i~8|_5qp8xA37DFO+3inz=RJF7o+kqHl zm&6li&EQ z*N=T=5n&H|*ccjkJQN_K;aFi7`hG#SFClMaZn^hOgxA&>)8Oa3znZNJ(~+y&Jj(*i zauZ4u_HV?w%X{E98=-Zc`Iap!whYTts`5_%kO$}{iD&n7ob~wX=t9mG1%0ada~~l2 zOEiLt$4}sr?tDHfXQ1%Hz88n{x1;Q5UF}Nu9Dg`I>SnS-fnrM=}Yekn}yTa zDWrJ5hjSN7+yDNLBlBO8T|=L=XA;h?fAn3P`63YLyCD_THrwKwLu&UKpz6L-xgH!8 z{`q0P<%g3peX>35bn zv7s#eTVZBkxW9gU47MM0QmT?5$yC<=7z>-Pr3a{iq;aQ2&2fRIgsUE`ecJFCSPL9c zqOpqA6mJ$r5|_6e07^O%P@Z40c(xKNFk(?Q!f$0v5Qx0~&9@pD7DkUmFe&G+`}CsV zsU2>ne4?%G*aKaX^lGq_o#Ud@oHZ_n0cX@b9%q#w#m0SVQ&d21DI@_wNI7N(bH>Wz{>N{X|bAcx{(tL-;*2uR;_ z$|w)7aopSMH%j&iBVK$BhUE9DTkNV5p9$uTV80*Zhj&i&Nx{3;RaxG`w@;*1fBTr0 zsaTKU1foPZ?7-kjhD&TnT-ppfIFh;&@s&HSuOzrV1imRL_GEHRVtvTM>~VX^K7g!bF-uD{10sYR$}@B1`% z#D*-w>~YtE;pql?!&&<|0`s-DrL9W7?r50 z4ORgn2;@-=Hg~=h)X9^u3SXI3oTj@vKv%}t{8E}w8-Sq))JGk_T<^%n+Y$qI13>Ld z+!Yvyx-Nej8X3$th=C!S>2X=xf9#78XyuX+EH6?Md&Kh@vODBOSaoX&+HxIPI}P6v zt$2Gq_xcHW`-Mq!3IAn%Pa7U_iM%}z<@Dn6U~hNdzkZWIEeOPNM#dvq3c{10fRpKAxE8*ovr74P?U#<=Mmx~`~HHoKai`+@(%VH{4aA+tm+zlVx^RsY0eDF3Rn0SCMN z`ixty1ej!?Aou+k;_&93pKPXg_LKSsCK(@|sA{?DBqsqQ?y8p)Bh`*y%l@u@z#{!q z)AC3CN4uL93F6=usw>;@iUud@6HBU}J8$W=|E?lMvHG{>5Sn~WPbP!TUeG5XCQ&ti zJcgTP#LZFXvta*PRhacy$~qha8DSXoNJg>fcRY(^U%fbIktuy?{#Kb+76|cR-Yg#i zJu+;WO1>L@Lscb{`=?CCTQ{pt=W#|I7tHL=A_(#lCD9WL8CC`gGrk&g|J+At4Fsb% zD5>m^zuzY}kDQr^x4isS7CLulA$ERV5cjZ|@h2O*V-1jpd{62b9MLj*v6LrLVU0FB z5u`yEy}tGvf2;>;1{0)PrzQ}T_q5VKHEq&!oZzc)e8mgj5-T^k;>|2;%NgJPe(-FT zk%_^W+z<(q+(fQ z8eqQu1+4i(q{kG7EHx=>ej^KyZuNQ1uJP=&NAI}BwyzotcITp$yQkb zCw@$-t8#kiuk{jJ?k`xqV zA-XuYG($z0v~B+LFB~x3L1ytV;3>4!TAhCnRxgi1Q~f__Y&W2nk8zhmGHx&8cwg(v z9>2238!Tv@?~}hgKc0{#FECh?1Oyard>`t#k*&Ex^%tJcJCQk^F5&lSwK($ORHKx8 zw>dHEvN)wT^yB_W%+KF`i4utY`iecO2E$q^3Lk|ZNbP0$qi3jSNgkh)XhWPP&8R$2 z$4T6eJ;p=Wq}k#AfcQ8Q{h~+TUEKkg?P6P|O4HNxw zIqC0xcUMi-x%%|kKQ5VSSuU>_IsSR)X5?5IA@S*8N)=(`vyk7YrGd6emq&)RX|dyJ z;zaZ>ds%E!+_x)~`Q?q#yqzq@&kYBE>bIF)zuhXJD&?++KS&w$Jg(j>txdmh3#}l( z^e*;gWUgNhDeEw22{qhc`xh?_MDN&({O#yWppbfY+t<>ttJ*1%^s)M<x7&_m*ShSmuEp=cCgwDDuLMA)^M$YL9m(@kiyVfHKkIyGYOl^J>{M$hCxt47 z#Gy!3s4GJX4~hZSJmfV%8O5U5Lr>3}b%Bzq%gJ%5^no*L zO&Qc~fM(-p`u<&hYf)t9KmQDJ>&Al zZ+!U34+4#dkJ1(R`1o|S22E?a_$$2gCFZhxyp9AgjF?Yze`;zjjC}!Di@qz<8ypJK)I4+8DyMeqe zo;NF4Q-e z<)pI7^TQK|fKDUU5ihbs6=KWi!`r&G9v>Rf+CnQ)z}UyTcJdWv4(2X6&hUdW`B;+o zMMvnk=hM9Q=}frj59!)~ig>Q8XsLhYppWqfMdoGMZ}D-O$Aom1QuiRebiUu+Mv7xx z$IEx`Mx)c{&r3}$WZz#jc19Ug>@=J;@iTw%X97mkeCEgwLBzhu-k3;jm+stR2(K^O zUb?eK^603}Ql*oOFdB!6WWuHc6D9_BK)!2^tgvaEY?t0Hv7C(g(?h6h$_gW@N_nOM zsCep-#`9`UmDir9P1VGt zZ?0HY~&a>?)OXW`=JvY_wY{;*`%i zGC3H6h7ZInXQeULqgw6rKF1WtJ66~=$wKj(YP_X|iJgfekl8u7IlcfdQx09GSN5VaV6_tsyUa9y)cV)VvUqqy2H8ye3T+}k)+&L zXQdybja#mG8T!UE*|e`An3yC2T^t6L@DzpvD;jO^JMw%yhkU%GG`tEaw?(#tgN7O_ zMQj@n?(h1GpUo3RIqN=yGvE7;6@gkD@V2l`sQu%d_Yu&N5wUrV)5T z-p{9THjkLOSw6#|lq2afKQRlK<1Ogd17MY);6gnT@v{Z>#j)+bP%Q~CU)OVK)fYrT zr1p8~V-+}r87~`vl_Yh>doCy7N=GRdHe}ALs`W%_#g0FuAGJw6jNh-++D4S&bvj-Y z{x&=u1W?U&o9xX-Q<>E?eV(;k40rLV!*YTYcGBa`;DO>`Up9%tR2uLwOOks_&$&`{ zJ*R%$zJY=5{e6V4vZ@XlYwJ=f#e4z{T?|R?4X69t1t^Q8V=eR|U)yce9cwb~vkewg zB*x%RqjPAIBEbHva(o7A9_Q6jDVk^e!|yHf=13sYrJL(q21MKY5=xhnG*p$2{-f_a zpude2G&Jz4I_?>P+2uDh%q4_tkd#$qOt+>KNVP1?75xIk6b8v7p|7%$gV*6#MMagBqq{o~QEb>B%Nrfo4f2PwP6~!?R4!}wIltzqKXZLqWMae9)+(&A zDG(A#CrhyHQf(;EEAV@I`m?XE5R*oEm+5)A^shwI{Ngzfki}c_DH$yRQ!YMKL7Comky21qMXY-9^C0!xV)K&sXSXIcaITU8x%h$IKm zEhT9aj9!g@l!$;&wYnW<0-xGd_odZ%hZ6AJ*9bLKwvUISw zIJAFnF^Sw8zaC&FElPkddjTZ|#qqsOktxT+^;ay^c<-STV)-uu2+b3zi6t90l1QAw zRtpi4Qyk5y%{CUkHld#|OG<)C#I4q>^MYujLDoI*ujGwU@~q_n>}zP`r_uEMyp*!? zdzgotlRKwg-0FqYZHZKDg&|VsDelZ+l z?)AXRXnx}Z=e3)YfXB=;hxLvY*llD{^6XckkVsWE1TUwO8Yli>`>M<-Uc9j+=KzsB zR?$-k88*UE^AmT5rWdPiuTY`J$W%~1$9$1VL+b?DZm&5bLQD&buMyZm@&xrE{0Yf^ zhJr`5HIZ!m654>cf`8@CcxymF-EQ@?Xj@6a0Lu>mK3QmbpqO{+h8&bSt}57JuQQ1X zTe3EOPur=yP>bRq4!3TQS5?HuHK6qN7a}W>dlO(!MkCnnV%u?Z67i_W5K!V0oQF?; z^wIV}HA#i%w92zWZetT6INjoKzwV5!>ZY~%g|E1`h#)87Jxr|zLDR}_dDKUu928H-y)fvkgiOJ*7K%tMq~eu*8c zWGb}zeCzq)&`-_XZM8iaQ{dXNKN&%K4Q#)9oK(f&3e#smQ!&htFFseO&loHNXrKc3 zQEx%6_p3PoDgc{XSYh3B!`0~6kTn_g8QQ$yqG7;JZc$pmsn*pXLHb#MJ2=aG3-X*u zr^HmJiroN9;K|pZA>Wt$p>%Z4Zwa30N912!AixiB7}~o$KC$0sM{PS9Az)?S!soVfGPIC*8a9eeI1jNDEPXrup4 zLk3FHGKv{i65>M1aMnd!)5{{)>XX5ytAlt6N`c@QfzK7syH{sr74x-^RYtkmer zIkBgI=c|~3Q<1dNwSL8_8Q&}$3U*}ey?S5xcq#@fjuP*D7}kYFo>C3WESvR>;EAg zKK%T*sy|k<+m1Su5OmOV>_atg(%T0(-i6wX;#|<{Jx^F7{1IVm#}L+NyOPh8Fc+&KtXywN9PMf4)yLLY^yn<2J?hDMXd`t1|u z#v07kXopNBm!!~jo*UiA{RATwji#{0-|2w%AqHdleMVFlOmO9Mm<&cxkPWrK%5jPo5me8UJ4ktUA}o;r#Qfs z{&_Jqtvm%BwWmF0BNe@ee6TL1XAF30ORd~B=VzRBC`fXLwGU1)4T37>7*R|}p;m=$ zwCy}j`f$cA6)Efl7ypyn3Lwf(>ui!SMGDi($yjS*v4-&lg#$}C=8;blqL!mB{lOEB z+Y>Yx3C5E6wEnUvXx_ZaU;!wkc1RYVTpPhyerp{K}D0gAhc z!`Kk@+v&}$15nFK&Ub!bdoNC6#OE0PGfp==a(rFZMfwd;1OP%G%uGi5wfOlUbX#-C zY9UF2Za7G?*W38>FIdLcr|heU-Ohfn=AmyAEeHN(wWesC_k+#0FAGZ{H0YtX)efvx zuJ3xqgSnaY8t!dkb)2mO?MOAQ^e&3V0CT1Gp1Ut>?E9nABUMRWep2U{6RZ+|Ims)DE_b9v$ z_g(U`T!~SWscJIPCk*stU6T^ZX4^l8u4)R5kCHTzty53**5Llr>msxXU@t8%ty@cd zpe(x{068hjCIE-mAk>J}+#S4a-`A=SR^uwC;gPU?VnB76q6<-JP|%K#SYwpa5tL;o z?D=iu_bMsVLT;-wX^Q7fgoDS?mGlWObISlELJflVV=^=Hgt=}NM))zZV|Jl zJN3h(2J{lM6xW$*-ma&h){08GJ873i9218I=z(d-_yJ{k>zKrLa8x2*btYDI1=vS1 zC7i5yyb5J;&Dclm#Z^itE#tkw-;(tu$EQaUYg7IMEN0KQq9 z^~DdnKO9t5NLn@?GmJ?K>|O2`fVr2-LfmeLr+o-8Gu624KL~h60ntHFo}>cWFZ291 zUgNb7a`TbbF+q-~37nscpC#9S#^t@5j+rU_&`yt_S2fPNqwxl)$+6?MNfqxT!$EE{ zJd|(a0WN`GZrJAdAi#oDu|`N7TZT44jl6-#=otR$n1G|RJ*;)T%u{oC?=<|ygzRi z_;*Rd+d24?WeR)jo5-(ssyd&4V2Iqpq{v8q`UDe=&$?I9wB9p0iSZ3+22L!LO(Q#7 zBsyU<{yafEvhV@1hyB%WnwwB{oQ1!1A0rN;CKVSK2ls-Di|MIB_iKY|piwV0-6EU+_C=PZ zCB&w(<=wYv)((PA4Dz>a&-aI8Ke(fn^QGABSNVXjKb#sALmmJXsbsX{P1L9JPD2bz zI`FM8Mk8uH9IGVV(czy4~R;>0sGlr>bJC~sby&(%61->c3Vo;K(M9{@OtVUm0fiTr?wPSGQQ2^W4FK)0LnyqGnK4@?0D NB&i@#`T1+m{{Z>y4ZZ*X literal 16616 zcmZX6WmFtZ6D|o5+=IIZx8N3ndvJFP?kvt5Tml4l2<{ME776YU+!l9O-2D#Ux%byS z>>i+Jx~ICjx~lrAs)_ifB#nklgbV`%gC;8@p$Y@@4(IKQ_yK6aE!}Jeetj^NmzIEe zef#;{UYq~}LjfZzA*Sw`dAtJhRM&VJIKNFCrAb)zaij_Uf=Ho(3I0TNbD^eX0IjkB zS**Bg%+*F&)NEEo`N2zyeT#&L6Z8H~SrEVKQ_bZX5FnwALTh1pk~QQdj1fzj26>D| zB|qW6g`BF$#?w$D2C^&Iw4h+Zb#a8VW5IP%Nu(kM8e>r@z={1q#G(tN&x~jI85u-!-?Mkt*47H?pOOBPCilDQoj3A6 zpoKsn+6^{I`n1SaPQDinwYBkV9e3S~m-MwE5t`)|(?yCE2H<4Dljfk3l20viSmWzm z2;{wEW5)%qbd{BrsS0#+)kd+!#l@(zNU%xS*-GBtt+O%J*ve=qfyPMUtr>L=D*~&Y zThTamw~~^Q%sRE(y}EG#QzIhy#0X42_*v8fH;{SnQ>v<|ail3!7`CTJ;LvX#9m%Vy z;pjCv&@wVco_0LO6co_d&axl|cB6Wao8MpT2;FZd=(~<_!@wcA5F}=PT{q2qdZw2y+0#0Q#QSqu4D0`k`N|g7&ZN6KBIxIfV^|Qg5yW)t64&S z@V0q(0XJpoSkSWye3zxV4VO(cR#Fj0L;uwl{TRK7W+{z2(N=5@Z|=*vaLm$OXRLBp zK_E|MC|-0-r_r;r?K5H!M#5IqSiLSwS67$1re=J6d^bxIq&t}+Pkwp5Fem2np5!1M2{4~ zZyntqfstX$R%j;YtqZE#UsGy&uC`7U2?V3vW^ey#EQ?!7@*21cUo{Xq-^^}op8k~o zk4zl+tjCfg2y78;zSlGaSxM#9mj7VK7d^@z98a3}z}nmkTspLErCn-eJlL-)sw<9( zIUy48wmE>77S@4ds#ZU^>4H$b1N5_CIre)v_Me@mS~m-v`($nZ%5Fs}9fbm)gt`CZ z3JFzd(S$2~RT{zqZ{*|zY~O{m!(mz=8i(3asD#2$>BIgj&tlJ@Jy~r@33fVKaVB>A z-)06>S8oJPX;~T7=g%KOAkf=3U* z3R(sjWAMl7$KQYFmqk^jlR@%PJ^R2Lg!8W>)JwPxnrp? z|4SY>)xZAH+yb{%hVgfn{}<-G?DIx0Mz7AE*>%#&PS$nBr)4=!TZWV{h27*qSLNXT zQWD1RbPV03(M#S9I_B^3C`3hbh7#!2bajU-bn2882D49ww<|5GRgb5Fj)Jwxl$uSy z?4{5wq|N@laFl!3{PKiYRCI^FEB&YBB$O=Ob=i8@HlM#bx+s%J-R4ZrX@IGV^b`)% zhP?u;G1|f1=CHYOsE@_e?D7{gJytY(ug@=c^4&kHA3ym*aL(M7{|>NV;kWyqHjtt% zGPA9`|9g#9m%1s4JPX4XZKNMJFzV^3y4Cj2OV?^pXrC#8-*p$}o2qJeH1DeC=4Etr zbldaIa&v3z#SneYAImYK-LVW%{}#N*!NR0!uq`nJ8Q`o zs}H7CnYhDVIPX>TIJ}hm7qyiCsSAs#1^6b&{xh7Elot~mgSY=!Rr$N{H)#_)><=72 z#MJE|t;~$sFzN0zna>UQ-xiz0hWmx<-8I*b@xq$)NI1msf02RMPFN`ipAjsQM}75N ze`(+fhJS{03V`TiL&Oo>Q4@5iF11l^4GAo=r~+t?@*0|w+mXK6S7gNG`wk{*_&9~J*K}9#dPeQ7QH%jz_^rXCd#>=>ZFzuGqM(QWKBW1!rZ)42- z_tn1|gnDSWQB+5aJsvf`ZoXI}*czmLqEb+;ZvIagtp!3Z3ep}ZFF7*pE8^sgZD0?u z{Dhy48*FHj`~DfHD_UBwbanBIy%dh$u~Z=4BlhQ6{7!Y{8PQUm;Gw^n%nGBsA~t&H zY5+Bnp$7Z)4sc$@IJ<)IqOseQ%D%d4v8A!m+=LZX3u0wfpXWn!gtsyIPm+@E708Z9 zn*Y7D$n>GQR#2`m#qh>fO2-$NC>n-5fX@cqkv=!x)LwvHPVOE>UYx|KN^6F_EhP$7 zHU#@|);O|%eh^@6Em>S1tWh4}|HFr$!$BsYPayFZ&atJs_Be({R-N}a|IY){p)S|C zOEhKg9%W>)vy}`SOB*Re&knYHfm!wm$*9+qYiu56oPZ%i0?p zip|CL(KRP+sa^4-pW#q$%4J0JC(LJlZvqIrSMR}zJze=WSELQ^<^hqc{#l^{;q|A( zX~@R6NRhfYUKC!Zg2Gg*e#qD0E|;S(=(zR2NOpc`=I70uN(=K#1|$7dc;&FG6f})v z=V#>{{CgDpbFZgbuK6hB`RRU_#`LQ`i-)i6H^EY*^d-^l16wZDj)u^-&ctsAdh#=c zlyJ(o;+425OfZ&1GG5Of7fhW6WnH}m!I#aSo)z76d0XU(|8&RLWDmWh68h(R`8^4L zXA!zu^8OQ(wpCFlbDZ)Khwfn2nOV5eVL?30Xv#(%;w}5pY)i;%Q&c(?E`cBD4>3lJD zYL3RM)CmplExRD~HLOVp!>J~?7z>-`$e=~v=5{@O;)^1GGzT89n-ofO>l9O;%-+v6 zyg8sOf3`a&T5vg`eBtGGV1l>$8JRpbxAOT{kXE3eb!_wW%0Ky`ent-=Z-OQ#Sm60M zZpmzn(WC`i>@x8Yk)bOe)%|2`tq5zn@SVHv=A9<*Vg4=7(z=Gg&Mz$YueoY&9&8V1 z9Ncb{H7KRDQcG6SNVnuSO!|m(n7khd6G+oU!hzAkX9;MbjbK4I^gP=Mf(VKx*GY{u zAzFXR^mC_KC6Mh#!xPPgRJ}-IU4BLZrn#n-4X3C+gtMf>mg3^O*>WLEMl*+I~tGt@M)^zbAXg&>m}rI z5jy0cdbJHLEm3EDPv?ol-j$KPD}pS-umz6glzHlu`D*nb_}73UVyd;$JlXe4_=HZT zr~{HWJzGl@QSDYj1^jW6P>wh8kMUKjmm-s2G!a4fiV(k$oiOF%ypm2afZu$#m@$pX zy3hJzBI9jWfBBw-+#!#f{>V&AmHtY$lRo?$-eRZ;If}VF`4M9%H!3xSN+-3U74;4% zN4L+(Wksl#g+lVWqwm4!@!dC!Co8WX`~@%!i5~UJ5sC zspp56FHBwO&l;`UcfBY5>*G#W9w^r3v5;ewz_v502ldBzfpGj6TW19RQ(RYQor}~f z&xbX2i^@9wq^0B@5@uf%{4Db6_?N(9+e8R0+H&Iqb26@fUl7tIfmN4al9kYDuD4~u z(oK6sGduqXeVm1)k*gK757s~2y1Bp4=X^NG`ssd6x}@Z2yY-UF&c)F7Jlw@%3Mb!M zXi08PFQkNXcj~} zLAu@eMYE8wGDw=T==pWg8YGK4}fzm>dO# zJELvJ1>ct-{8R0jiotNEqkUX>f<&U?Wa-M+Q~n2OMAjLvR_&@xk#JrOhVg%WFiBsk zcvcTOE@@zlcZfEHh@V4!cru`%D0|?zC2NU}-#HQkwMt7o!z@*4%F|?UNE9|#Y(IW_ zISb%pb0V3S9!;!_flO(dirb6!z6%LOdUAZ1k;$4(GOCC#aduW7_Nn?fu>nv1Qt85Y zZpv-6-LfNK9mc+J?(tq%Q2zemFKp^)yMAKFM^TJ{rT2wxb=hGU?R|<^drr>(;B;)` zY{OxZ(Gz4lBA^a~aTX<)YH9;_=L5vpUQi?HK}MS-VCd=n*`J1(;==|tP%qA?ww*$} zcDXnjE@}CcI+i2Z{()6VV97rp=5oVol~v5Ck_ldr#nKt}sf&7@ewD?B4q0z96Nd8r zUT2>y&kA=#EBIZhdZH>5d9mIC?mwG|yOWJ5@cua0ay5Y z)`4g!i+frxM(ti9ckl!F&t10q%hRa>k*;^Md1UR<6I#mLZ_491IGY=u+Dqj$Kmrt= z+&6+cgjr>=7cH0o8!*v(w}u6!KDFpvrW5FNC&RjrDkN9|OzYtM z@d9t>Lv*>#x&=p$K^k>RI5O{Jqy5}2l+ZJW$2S>PXQY_V0FKTI7%l+kTK!#Ho}K=i zJQ_07WC2V@iG<^whjgfpI)l%k^@;sRDVEFwxu`cEBi2HzztOq7tf}eCv9i`q0r@v&m%H_;){-SGr9QX2e4oxGT_#ZB= zp!3+IeMqksTlJLhrAjXEsna3?8KqoEt5c<~HQ+BQ5q+)_Z83~i4mxp<| z5f-%DW35yj#Hv*cfBlU`nKf^Va?LboLn*qWJs0m$X&K?84XR5HfMgAqRpQ!>mn4Bn z$}dtEm5ye(OOg$K7=e99`t40zk0~{^pzGwNO(y>sK{YEe)M&LQO@5gM!4YEl^n;5T zo9H_)Bd3RXJ1v3(e@xW~1v!s)SO*w~51ZIJ6QekPl&8r9~E2~65C@6=-kdEqsbQ-*;eJ9id2nl_7mFL%=CCsLv>sH7^KllE1G zk^mE>W&@Ebzw(6ein49Xm_$K#P%{?o%(cc%m%xntyw~toO0Tx>N-y0^mrzRG)ZJ=<^9LWn)-b}ZX66WdLQ|5jCepkp!*g9jtw@K&Py7&qK*VtVDV`yIESil=Jq_^ zO0^b$)ph4453em}SjhVg&Ya33@n|1WJ+ z*ly^FOXb~OJb`!Lf1PX|{+~?hU>B~n>59l6krDw1*l^UKFJXvxsSg+Fi>)+yAvfvI z2#xV^SPxt}Msp&#zEkK3#kfAazj?nhW;g<6=S()%4+M);`{T2d;dU%sZEssq-H z@ak6D5Zg{m>(lItqEOPc(M!A=sY=xkmlQ7JBao)^xw%CpUt>A@yXH^qS&p{o#>t7V zQo^hvkSu9OqQ0Ef(SjO{78Z)JU6Bsa3%;(^1R2x}(e15XLqe%_8b=RNe>m41m)A@p zpJh+xO9L`e_XesZebmGu=JVqYT~;9`nG;q2{*!KrYG>0nEoQ-wcWvVbG}2y90}ui{ zR$&F7l#6tA`S^(A#ekl9>WBis!uho|m-n0l_U&}7HcsKfV?;2`!weqphLbKFt+q- zXfb8rOo<+g2c$qR-zmz;`Gl@LpE=YO@8y#s3DbN}op7RO{ptKtZ^89?+x61FlFp2m-P6uru;UWDV*AX{h+I&fj|_Ujw!msQdBnr@LI@?6F3?Jx|p)%JWvL3 zX4~30jdtZ|0G%7FxupOY*zzn|3o%q(D>MYdLGj&}^|tuVWMh1Cho5}Xt6xLEAK%DH z8YcZm8a8B}qe5cCkT=bD))*+~a*ONicAa{EI!{|W`-kn7dxnt(XDa|<4l=I;Zx(O-8yX_KqSYo%4HUcg2o2@H=;}RMaZ72sLJc%V)5sTmka;qJ2EPMBzUnblBZ1xR)g3LksSyFrye*!+W@bJFI5f zLMxe`yW=m49mq@-O=+Gw&peDo9xfikYV>~JCp2f2Y{qpHzs|;Zm*0r<&l&c`w@r9i zjaLizPxWluHw^ML^*@N=yI3n4yFmGpfU6EQQbZ_ zlVjS0akTQ)?kj8LHsD!bU7>gKhA-M}KT<2R^E~hfp$_ZuoRC>l-1AOCy{Ja0#*(}h z{Fq05!!oF?5_U;#e<1J7E?ez}^<0lCX-V-mOgo3OPnL0yp2=^3m&9FCOtI9|d|RAU zDWX=#`ah3a>_5h`(EjrwG4Y>fR?pgjSHA6W%%qo2#zQ~iqZ=B{%O`g>F&0pRI;A}g z*{vFl{B(Z#&OQ8EUiO5H@ZDih$KxJghQ0tx@V>E@;fwjcEhlo>+~n-C@#LO;1?|Dy z_f}%>Plmi#+`Mdi)s+1#sq*4ssB>>Z~_ znQ^A+fUVDLza8q;4OQ!pY^wzH(nU>$%ENEH@&S-~(q{{d4z86wTrhju7zEF+12-~Z zU~*G4(D&-jdmoU{FC=sdzhA9QeL5AN)EIGv+zj5g^pV;vaGwR#2>Ki@CS76_(Th50 z2_9Rc5z&ccOT4X88Tp+V_EGtb>OaFS@rX;Phx^fx?~5P&BSo4d` z0weOxBJTk_tD75_{@Gi{igaSnuRm{+cd_qUF-v_2gimtov6}avY&zYbJuQ;UujKK zG<~gz`9_H?qO;2G)?LrJxTWwx#Ol8L4Y!?ZGoZDp+>mp$1jZ$x%F&T-R$V*K?dF31 zP}*hSFr3{+tDx-v6-$1HMKWKD>%4l1d&B#Q zjm)3op|Yt_=(D4$%VZP&$5rKbG%vdDOlx5DS6Vh2(4mbZUm2HD6w)BBv@7-Df#YhYm=*jtTme z9s_<~nj1b={P$QLS!aDKC=ZPDPUa^3ul!#B%>y&Qm*$^l)o}zo2x*d?@OQ9b?ObC6YXTb+=u-Vc@%!v^ zlPonGdQtV5%=P$(wDP)4k^D3ZBfOgR3xXDTLcB1H1PUw1q6a$uQJlV&_AwGx|FAl2 zvX{j@v~k^RFjLYk@5ef}V%efYW-B8#kyCuf&6b%p<9{zt9=FY5ID_8hzp^cNPL_oT zavaOemk5jkeKU7hEwo8Q6kr@8GsjYre7+wG-Vdwz)Y6~Qs^&Y2pI#kV@w7oQJ-xdK z40Y0!4byh?D3|06_28@VeTQ62JNXX|HQQ5ghMFW@&kCp6nwk(QN&PC8RfeMRrG^cL z|3rtgb9}!V#mMMKpv`!DR-`R&?&Ln=ZHE8wJlvze5{W5s5Sl^@CyfDlCiN%Ls znP%Lamyvx_Fdhjzb858*T{EIOE!56au$aV*| z|6eY^+)=25VwP}&d6I8$?;XO#-4dXL$sRjZo^}9e%f2>;6vGyr;q^55D8zHP^zPb$ z^w#)&%PUK6`MAG-e-2~yF%1y9aD_sDbf^(>nB-}ewg~QGzi2dzov4QX4sbm`cLwE8 zqgRM~sU{-a8sSbDtqOeAE8a=x)|1yWq7N8)_zz)|jedyQ_siPWc9F&CiZj$Glj)JH zU;KPG{p(3OevqzxPE%ljb#E}zmvj)bpC$EU#`f~5$0s?PruAJCq5a19I8t7&$58f{ zD@!^9;bFhX){&W!?om(+&y(F$`__l+_IEgK9B}+Z_X}CT5uzX7{)>Y(@bFl2uN;*3 z(|o+f&|fie-j2)-MjF#9lNK9?72@$UtGarF!deRfw@*1l&L-H_h!m65nbo#8BL>5S zvhisvEo|&`7n=3yxobRafF>DIA$!{DA(0Fzp^<%}wxZ1!rfLy#+mD6&p2;l%A?xeH zcZmyK99zTdSW*)Q7>M=bW?I#iLP|ds213H0forigZKipn|5fR}_^vg!dqCKKVc=mb z^Lu8!%}cwb=d!H!R3|d;Z#6+@+bgdgkz}7dLBog>lE=}9M|=!v<(#xw-X!Ns$(sPC zyA&z&)VqBMKbtX0PO^OpI8n{!!x=XKPZ_E|{d{L}+@gI9DBd z$XB1@0fcdq7{~f}e=Ay}vAvv@^4z;CrT)hK;`^pX2@3(ufL9+F%PkqBg*;jNX`$zO zyp_tK<ZsFiv|jQ){TBd8xP(o!=g;wzb%FK>fCOwH+(y%ghJ}n0tzrE(cna0 zl=%^Zek7kuBNNr9o|4(Ql~%DrRRH39)Famw{6<4#III_qj2l4Wo;3UDUeaLGFHAB4 z1c)lVvw_&d-azy1a93q+4KFp|sjhrIoARw#^l1B)6gqpL_f|dr|$^&L*yN6Ii`=yJ{)l zRG25a2j5kuMVcc+K`Axfw7%1~15`$~R$(AQRb+nC#OT!Q0JC=p73FzNvf{iILnTkghfxx?yQgR z$gY#LK)2F#L-E=5Q?;L_aom@;zTpmt^}wd)dFthzkK(2(fyjO$2p8DBFvT_>v;qokP|#d))9x`COXM7Hg8QOw6V7r4*<5u?X>V+8n56drK(0nT4=@ zdhni!$Hjl@9kSca?n`rvP(pj4Aro&dkGvE*E(^8riD7xR)gN`nW{^Kg6{~LcGey{H z2bZWl#O!3N1sdx--|D0cA?bq~ayC1O+Sw0?tmxou1Xh*n)XMc$;~dUY@I`1yIf9)( z0!ie2^oqO(z-TJGv)dxMgYT63e`~EaX1{sF_U|XVK9KoeuUGTPy_WW`@{sZ0;Y~V- z38k_tJBz(HMs%V(V>XJM(*^z5jAC+0OT+sctP(R_Z2ns!HM4tmgX>>ugRgpPALGF3 zq5Zb@wYQYCzGKps-f%R&gv&^uiqO@tn2VhMN`XLe!_P4EmkhnU@vZ)FSMw65b!><5 z-{#`n!eet@x=)=(DS56_PS^Y~_t3GlaCZLawLYWq)qWul9&^2?g|MhqONH&R?=ufv zo!XW-3BH~eHncyMo?;1oe>AtY9jF2zR(qn0zCjSxXy6!_usk;M;`eLl@C%dZJ zE{utcp1$>gkD~79YtVdk8d9{TPw2d1#&VLV+>%8}lU6?O)ayc1`!q+3UhbVm#Q9}_E-i3ouHqy@sh1{>GN2ksnO7JuT zf|`6tva4>Ie!cnAAv%t(vpZz^*@?Z~^L5{C>N*XsA=C!vk*@hy85;)n>0{V$At&1j zG|=6i!-ucILoiw7sZ#Z}n2s3vud+EcLC{*<0;v_7mTo(m&dR_w(kJ}+;IZ|xGi&_- zf4KQk^sOJX2t#a+Dd09S`Bv`qC)~m-chu*b*IKrya3trQHl!P_(Kb0de|9*A6$_$0kgpNrUyEi#m%v}Q-?u{XURam!sO~lC&^~n@fexggWS`{3QQFvT} zj1C#dL!Z96h%%E--8v4J>}XpcfX3=h$NE`hgNG~t2coo_A(hP$HN&(3dkLp;wrYJAT$7)Ck?my8rzB*u}7&Sge^ zXMEa8tUsTer+o{g>*NTk5&@KNSGjK}%g+A+5GC}_7*AfYe(=@oA~&pHuS@P*_cMi? z0TLV5?T{J4vss_ai-wMDZ^JgWByspLp+|q7vE$*Bhk#7ELaLoAZS;1|vzsJ@i*)U) z6BRNrKN2X&omhL37>iCEa}TduwEroz5dO%sVLgiLMy%8h5~OtWBWK_{u?C+Tpy+h& zF(6H^Xhrw5m$z@OY_=E6&n}&jwNoXFLjutgA8XMZdj_o;GU4J)t|izkaB_Nkae%|X z_ju3i>G#Q1BK@1-0Zd{>I{g5EXP6;`Po93hug5Eo7;Ul|w?k_`<7u2rz7tw=bT(Lc ziK{|XdSDUO3YzOe3lDBNMgm5E$8s=@9qLGkxG2&T(y0i z`cPx5wgjP)1%2s-7SRAD8!b&(z{31BxHL`6KBQL*xe-eXX!WpcUHfiG;)6h*>_0CG z<+#Q0p&63UT$C=5Sfz0dDQ@g6{SW@pP^{QvbL{Fdl$1u4n`bXL3ZKchnznM{z!gUKt=^)iDR!H!XKBnN?hT43?0vzl=Sw}-zyYLBnC)b5H;Sso zIKDR2ABk7u?D!c$yG6b)>(VSOJ+CbQE>R&k0$1T$@#{+V7;%z)vzED&{7g&jXZ05b z*?7JM)j_Kr(SH_)J3yQS>(F$$lbJCoQocCkSMvH;6w}3mM@ja5T{i*#9>XJxYn_hi z{W!92rlL^%N2xNzcrlzF5+?SH=$U!{uBr3R{GOSyQw$dB;!2VsY~^G$RSPZH(~ki? ziav(M7K!RL5V@svcJ@( zhN-Jb^XdkA1NWv&m%jxq^0it0k)m?8Foh49B}Tf=3M^j@d)#C-(-KD?LGH}U1OFHj z$xq$jO;@k}9_6s~td>r z6|J5vN7Ak_$q~tfpwu+H|HK@#J}lGK_2xcVHj-Ca0rHpo@}WwQH?vOa7QybHxse?2 z9HzVVAC2+T67~3>Oi`=4H;gcy?a%%#a)0>hT4JvWQTPHNBl}tY9X9m?5NRp3cY3y5 zL*H2`x>yo*Ud&Q=Q+}4;Y$y0y(lCOF5MBA7SK#u^O?{qKI;vKc0RU$2LyVcnUu`PX zn}#NPYeo7DE}O;r>d><1X%2A~t>x60iwb9Vd%iO@4`R%qrfYUlI$B=-bh#$SrTmu9neE;0vqJweNI~G2MzsE}QP6XgIR*-+A4qt)WFDGx8~=Dd=jv zVHr_(jW-mNp8f#uH zk-C7D&=Q752yziyvfZzTHFB_z9|hF1IlwUw?(Ow*kaYRfKkLKXu?5Mn8e;mS57Bbb z<#U8VbA(#aWLlM6Rk~Jrs6R&038I?DIZ`}%8?I}ujaX>8v+6;F2T-1=NsGLEFp!cg zn>RtTahTV)A{o`C17uE2{#zQ^HYd0Ic}>kdjuP#?p3nwl$2G2k{&Rgvijg`c(4wD1 z$KkLt6@yPY3O~!?E}{7GqBp$NzD00KI`Dj+qOznChQw7xB+RNST^|HqY5<}ds2@dvoa#h47ejokf{p;c~jV&;ev+1tUJB~rRBql{L5Uvl;p6)$Z ztt+p(Fxw*#gNz$j^{%)11J*wDxsEc{$=iWd_q|_c1d=l?=l6Xv7k$XA$;V0n3fgk@ zq3Z7adcic0O~_76Us(<*M8wMzC)1dmK40KJj zE5X*Hho$K5eSg6?he_%tq8m*R&qq`l6?*0h%b;(-0Ah%w?>0|;P{{f?XGgsTHwON=WChAN*pW0NWub-Za zKA7$OYwCA$13D>Y@WjQ(e^pXKmyRVXY?^=Ju_CH~Txj%$>4$3OKRpMAY)tNRON=2E zt*be^yD$P>wzti1po=~Eu_D78Jz;5rpsGUjN}2Y49=Ssv9CDWyfImFgeqM;UYqL}H zbX7`9{iY`I$;th^Yra7=vZGmoDQLtz+K2xl6V+ZA&M93dOQdgJYN;cnis=q(F_A@x zA=$FjA${EiU~q=>R)3A>xharOY|`;?$Rtmy>+BtUoH%Ds%ev6PGlLz zi?2oBXmh&n3#8HOfIWbW_LyZ9s%lfTbIO!_m8h2ZIH`Os9AVzzJZ{sK zAvPUm1&8L14|)0dUOP!zsXX>Xw?K6VFrQCAMsfU-V($k7DiA}ULf$T%N~5fs*&}U~ zFn5J&Y4cvT&)b*TT}G0zYDFRZ)Z7`t3T}Fz8!mfC{(e`0iWM;tOwxzRWQ-t5)wIcl zExtEm5Kd9Iz$DIpqr=Zxf33kB6(;N9!L!_GUw37Ko7;LoK^BUL`VhCQC$Yb`Sv(*N z`Lfbw*&?rk8WlFwXuo`S)`P*~r!6_iU?i)$!0nPk`AGu8ZSv7qmkvZ2Ng2uyJ2R#8 zHK!DZ{MHkYrH|b}#<3YVqA^|x7y~XUG-C3*(7`|qMUyXXes@(o7*))~v>JJ9!KNl0 zjhTcl$Nzx=C=I!~xw&{as@A*sS9^G1n_IPmXwoqa$j-Oa^@EPj=8c)(=Y-;Q>T~Od z%r^^D48O+Hw6(P{dwtj0(W;_lHIISnX(JEktCohR<0wqxC(*2aWB~Ek*6t_06{g7R z>yrp%#Q+r?Gx1@`(L)e<=P^3dpx!C3jqN;Eq#|Fn<8Y&1%Sk4jO3jg}LU{%j7TQlT zI)V@k4cTMMug^u1k~ZB8)|uRJN;ieRYCwjxK>wrvCuVp)uFT%ZqQ5Dz5UUJQ3N+?CRi*A%vkaI?k^T_B9e8x z<#}$Fv2T65o6fV998HJX6)OWVAz`rkDe&&sNG}z)$-fvjQznuhSql=WPl0a}uhcG} zLp9j$>q_)f3n~SfvzX$TT%h!EXCUGLT0H%|<8&4=Jm2Tdp$z)Z?wcO$zeGzJE~XjYxWzVXgR&tiY9IERf<|_~Bcl^w@C- zf#Qm3T)HvGYQzZvSfM`24M5Bl!`bgu{AclLLJcTEgn>a3?Nn~P5IQ`Q_rTpV4dz%I zqG~6qa0ob@(>bByEIy(X$N9C1W8&gMVyEC498H^7yVY34@&a8LHH1c6y` zqu8WL2EhHe)I18v<-C~q%nk@BjkxX+TdQf7 zq4V5&0#wXEst!`9JsOUj5Lo8EnJXg3LA3`5WY)?7;W5kJU|EnO7XW(a4V3>T0A#^! z&TV=c!URyxVcK0#vdd%|$odP=4|i$dD6kCAtx`ZQ)8OC;36(Lry!Yb?U`*f=kdDRh z+Biqab3}w?qPiR((WHl}q^kw+Dmcg4Wp)$mSgt1y4~RoF0^s{V*&G%=vt}Fe@JIJj zGlgCO-_$Fju1X&wz{>$Dv9}lhGHe72&$>Da?`?A(-SHy`HbHE zR77=yfd>sO3kb-P6I?;>0eY3}>Y2Xp>n%4nqGCnOd&X&a8xZtv%Df0hG8&a787sbDPu0iZyY-DeJ9Y+73THbB`^P;ep$w2%C+eE|wH z+_V6+Z*&*rgNbl$^xZlDVMi3-gf6ck{i!R`+m-w9cCo+#MP;*0;*Tqkfb+jyMi{i| zWTL8hD~x5@@xSKxB~OOI5q7!)asVrGr&|Fjr$CKL0OMta;n>rX0^Bf0$EBln&xkmb<3KF}=(SAP(KDG!r|7MT-RQ>vcz4O2Yp9&EVi*qHm?D)ntYQdztAeRLLL!3r8 zdI3k;36;nS?F8!s*em|+N`XfL!6qU){o4}pE)JyQgdyU{a;nWX$71HlzRdy;*E1P| z+lTnrlvpJUF#$+AzVo8V*R*|OJ+oY;##Yw0SdYmLs2UW`&QP@?QSde#BSa}wz=NC) zu)hz<%}KfC7*nfqf{b z@x~6iz@j_V_HwZkPNku$o`Pl%oXH=iO(G`dh@x!2IN>qefqWD1$EE$l(t#dKAZ{0E zjB^tAq1DD`-~K28oS0C%%+650Ce{492pDEW9$fFnPE+!iI-s@mS;n$b4le@Yx%UR0Ggy7hE3r zbhHhoOekWA8KhtCEiKU8p{xqShn@5gWe7H&RO7$2;Uf%NEt4sTN_e>|rjl;Z+ZYw~ zA6Qslymg}z>u^I2LS&tz7lyxL#-L=7mXy#(95VuH4}`PHrSnDy77#`M)~?5o@v^9x z3Y7k=Gc?v>2IIsyhD^Ft7@VlW2qGucJ{$+%D<%NCA8vLk4@P4Me=88d&RY z~*4O*^oU zYl6+#>;byan{=qX@%5u3(pV8LIooAoAqfS-dK4sesw52hFdWAf~*L z^7$4KHfwez)q8tcOM22ob+#T5?})cG_MkKljIGcMXB1HAWib&WXkNFzKzIL0Oc!sS zSHpsDbW8IlaNzYUsFIozalCB&&Ql~BunQEPga>}BmI6K%CNxm4up7uR4y9(P%`~bp zp#o;f?&$)ILPUo>tN4g?LBs~43^dky8t_p`nPeVm!6l?qJRUf6i~d?UHw^5I1>A-N zqFWg+@61n}3d2VM6#9VCGm+UQIt2cg@R1TDX#G1RGBfyZOTyV?Y7N|L;Px9on9>DM z(AzeTWb~0J2^SBfo2)SusA^}QMaw*vJWBGL*-O9Q{a`l&%9Pnjx~A>dwj@;EIu~Hz zw^zS0*1+{<*6gsxJ{CtZL{_qg^{eD5w%DKWb%qX}6)kDt-T-;+D^@HxdBp4!>De1B z0y-%*jN_GAPXem9X_Ex{6#N)JWX-tdQ&C7&k6NV-Gy#ZoJ0Bw~ISPRa}qe^DhaoOZ9*Q_)JN`hfo^Qv~j_YJ5JHJ$PXWRtIbGBXiz z+6|Tc{Y8xYu4g0I@bK}uKqqfS*J;8&JW}~_*GKc0|HLtBye=&rT6XYYs#I#h|351O`?BnBQe%$34T3MQ1h;v(t^j_OytvRc$zSJ#RDtRAl+&4($a=}PTQq{THuisHY-T8GYu`8@1W{RCZ%@@VqntM zEnm$G4b*y1VR(2rJSOG?g@8c%)Rd~6ynIAVj8y(4CN}n0NlADH1_p@B2+L1Gu25R} zWOmRABnQn$N$T^_D$_&xpOx2Cn$V-n?`EN`dB!`X76bkuaVOk*@h zG>56GIgmlE=-=X^td|#m0=;rkLqie*x$m#paxK@JMXRlo6Z9{@`T!N?#(j~vZI|PM z!05&KE{9(@4rJC=YeZR?_4jh+v;Qx*kpEMu7>+ePsiwtADheud*13 zAq7MR6F6J|)$D;ZvWbaq7{O5BO#vZq?*&M7+KT4Q=KE%5M(x=7%u`iW3o!iW~xYbF7q*&$|>L5-t<@%-V8r-3n9q{>4q@T-++6spnm=pZzDp`)BL zEuQR7ttTkR$jDUY@{qadHeWuu7}6@3FFG?6mtOUQ)!;ixu_h)MQ>5kY-mD}%6mB_u8pDhO5IcR8V&eYi@7cq>{0_9=W zS>Cb)5s}B|=QIoqf2{u19)QEc!^5AwXJd;Q8`Dr%FMyBkU2KDzncIiWsw&O@a!-#BM0}3(p|O#Xk$Uy^Gcv|4?t40Z{{DK^ z7O%6ivMwVS1YVh1;*gQ~n{sRD=*WUVwSX*^_pWuJT_!p0?T>jCBA$n&D=RCP-)RM? zBHuSPH6;}lF~CdpYFJ!cTnO5X*qI_gukEJF>k0}AKD9sI?lHIj{lcy9aX4=irZqOH z-{MhLXxtHyGFWX)9VAL3h5wzXyQXxMWQf?R~M<<&fFB3^n<#l4F zOouC&8H0>YP-)T?jPCicx4WAn25z71R)nlK3i`dc-1(VpwKbfHw7#+7a=a{Ye}5l% z<|fyEyUV;eoT+4M`^y~)R2>z(cTfd}BH^{~wCCk=U6)r>RAduQ43CU7UuySP&ft%K zfr`1jvf|Qvm43Ew)|x;koQNWvC@ZWj3w*9a%@mpm2oiSdpPpjnNUX44#LizCryEvv zTduCHNl8h)K$4J&wj7y7tlgG;|U-ti20(K{p90I zydkLWtHGdB)Ha-_wWnS*_t86p9W7MV)Va8ejIR5EDB?~!j}cd|sU8N`Sd8r<0_p2VW}`*$e%$<-CA$>c9>VC6USj&P?p zCB9TBJ_d~QL;R`07iWTP;eDID#KXj2>^VlSyF2#MF}Sltp4LzYhH=Q(Wkyt=fz&W_ zd=-)O*>l}aJ{#rdA_pR2QFJMMPVXwpZ$5da9o=7KD=Z0bVhZ-tJU)FTu|^b_Eccc; zHW2;E%ll>C$(v58i@lE4Nc~@bL?1CuM6r3{b1P!*Vwv^?WMnGN&L7-12fIu@jQ&)k zNMbiu1y+l_m5%rFlr=^DsDEs=?7nTU!1y6UCttGUgpTR_G?tFpR$m}#=}(M#U2B6N zMGbN@E$XPqO!nZg#8+fn!CM7~ZlC$~`Q+DYcUqPl5`Q5P$x34; zXZNKiuvTq&i#h5izGHS!qcWXxN@uOjI8x(QmM`1=wa`&(cO0eE_G}$T0j0USyA!rwqzZd;c6D_T z`d8ebZ#|3Suwq!6T0gd!ZH#O`)aT)DaAE0)r}mB}9kN^A7Z?v!hc6qMdf935HkFq^ zmNq|=EKpSI!4*%`B{Rocyx(p1XUGqX3j);rs(jV#4Oy9TcH7D1gGJ8}1%0=( z_p{~VNecZF*&@Bgr#uN$2efP;Bz1`?Y=TiTtczX>T$^Jx)9i@!#sHPjM?G^q2%f5Qxl1$+dOwH`K_4b^!8LlTLayQ3oxQvd= z$KOdy8FWaPJY3bfA)bvIDv(`JI7*l)(!Oj1Un#&m=z12ut}-3ffxb#5yzY#vJY->B zl$Dy(doFUoCwcslnFvDE7dQApu>|B-bN7bp(MimIEV$7whqO$dY`ln>>S@p?qNE;& zXg*AovF7!;3IU`Lmb2z8CEB6X)Xg^Kmcpsdln@o&6r=|CI5io2MM-4VT{qauoQf(- z_h?(odEWf0Re0lz%xzs{wB%vq;ft17$p9M}m}gWhQJ{ut_Ki5ECLK~YwJ(?5lQ3QH z!q_DgvJ4c}ARus_+dB^Baav(1mx^YB4G)x(#{08{7WHUb(L zEQBh}QQ*l>j3t0Rk(mO_@mD=W_XM8+y+AhBOsfOyBO{p|+xV8H5E1?YMcr6HzD*bt zZlF5i!F#(?xUUzp(_n+&H{*+N!u($`$lR?Q#CY+yt`Pj@i$AvJH~J5G;FpR`DSry} zSgcN&r=Z%kUyGLawx@1hKlksMM{mMugLQ8HqHbOAjWMj)bGBS9)^f9xW^XT^wmu{B z)nyi#4eLKtD6Cf>$*Ys`YcBknnycEKU}z$Go$OqJi;GaK=6&*zr3P;n4vnWt^{?p<2f z?Lf#pwtMC_zitBYR{vh?#cF4`PnL0p)RZ*XDoT-RR2CXSGiS@i3)OCos-8XCiRX{L zuI&1JkoL~kz;hGng%sCU;ZEo2masK*)Kl*nHf4gX8KWFE{kKQ=RaiDI?ZT(Clt~T@ zeI}r+`@>;-4v231w+mG3eDymQ-|VaJSR}zUc;Pcc!x4Eh6WmtNR8Hu&$sAd_ z4OWY->-IjOR8gPtCzLSqRHS(lzORL}bJG2dR;k57t!Mp<=So;BSWiN~SL;HzF-`*_ z+ENF7V({6lx>uHPce988w4Y_R%OfQv+gnR!G^)Smmq?ZNM#bI0AMCmnMyQvkYUD0o z?op>cCAYs*A=qs7i3^S(Kvd1kN+~_t!y4|g=isb9^3$Mlk~973&Rh@55sj6v$SGO; zJgA?s$}GUMVwHsFgiqFO+bXO*_%5*r3LU%NLY?ueufwBnWnI@g9r1(5lv+0d+scP- zNW9E7(}^n?&&kyF{w;zdBVQS(cIAsUaSR_pc#N60A)M0%S)JvtrhHi?r zI^N9=%Bu1t$L|tS3&u=h91$wSss`QLxnTNM?|FIIbJ`}3gW9C3J}Hwj-+n)B7MY)4 zL%wfya8r%?@^eJTq&8)Hb?aSrw&sIlCd8Qw{&^WTTqup=dva^uc0B&e`~0FiLAxO$ zW*wwItX;iwyjDw;PPAy31t)mj+23oBvWj%27_hEoImY}Ub++D0(VZtRe>zIB*yQ$) zG&yNvkLD4nd3U=aB0fXb<${-&;^D=@>d(=JacYJ{Rj%X1%G1C4wihbrZ$KFiZ!>-= zBhmF^SlPgI5FGa9p89W{gtK5tA=<)s0R+|t(RtJYEFlX0ScS3`pa(bz@!F#df{^ZF2hpwu z)UD+${GlNcqn7d7!RlLY{M$kZ=;$LgZhI1~DY_D*V7Eu|WdyC|hz`x|#cuwYK_a-f*#t3ZHo5>Ew zQlb8|h$06&0pN(Y#>P_sEf>Eytark`X?f|d7nh8B@Kx_KJ zn2r%fK}oyo;kbXBpfqIR5Q|^9#KGnfD0L{XpJ4g?0}B;&p{|LW1O#-k004{DYiG~L zn)9y)2Z?2_OzyNa4vS6v|JOzPgdMI$=>V6AQrY+8o`LS zh}^|{YgM%*CnnR$OHPD~{UNBAkCa(|kX7U{q=VN8L`q3O=wRkIFFt|RKTkMg1Al2E zEluH?rpRDhj6){#-2|*=KjU%llnc-P$n@EhcQ29jmsqrZah~T#dW1R~d)!l=Au?rNb=BIQMv3`9wssTtIYgBamGcl~K#*=+c%jeFzAn@f>_G%Of z%X9nKKPK!(;99~GPo2u`bV-B@8zXQkTorXb)J~PKwLQtpY3-2A=z=4J^6tW8#vIwS z=@dgDs5oI9V&Kee2Fz}?l1;IWJqee>81L+?-}W8_O$-E>nE}bazIDGjxIiPu+7h}S z&{-6C;NgON6WV`S7wJ;W`D~j4(x+$6+)s=yBj@st+?Mt!B`fBqK8y>!X0#zyTJE<6 zZzc7efbu3UiluO*Vlg%u%+!Jpf81geA;zx)O2Bk!;(V5}V1swGQPCSr{lLnSB>414PYHpZ9YO!hV zq$6U+Oxg74kg#Ylw$@fzw^ddjQ9QI7&R=gm&f#2-8_3!GojKYw|4?6fF|Aq9uR=$8 zPtamGrMEqVl0@KSslWOWxCSHONeVe(SkzQLHF*;v9u|8=K_Zf=!Yo-G?Y^X$iSNvkp!yF< zp4aX{en!oTiXK#Nq<&2;{#^jN0(Q5(2bs0bl1CCyxk+eK78pNGDLGuf*v$6#5DUvE zz}j{p?!c9lm>X#I(6pvG(}Y;KX@#P2h90 zG7?Id`_H_(80b)WN6_dbF(3c80K*Wk!w`FAos2NEs+5_;J9f#3mUwbLnC;P??dfom z=B$2qPURb5_;7gg>kP2gz3L-ho;iQ5uVXkhKIW)NmrD#r33WHuzHvmTlcq^N>&PA@YU>&_$o6dg zh`GhS{Mmr)Vf-;r$jNqs+&JOD?Bf@d6{GShI+(-2OB2_7lfRh);`zBUAiS_2A3LE? zLCx{$zW%+)*@wLtLVsy!`FE1$dp*_oj$o^h7I6k(a9aoSOsI-qb@6&cg_!!SfOfPTQQRXp8omAw)4*LQNykf9-+M zKUvVxrztVoyWC@hNNbkh2@6q&NYU@Ey&9<)T zN<_q2jDiu`O@^rqA3q3fU@p3^t#JtcmMU9sA|kRE{MgI4qTcHFwB<{b1afi}}U_p>F@%0Zua@l#<@uL3m3hfF zKjw(Z*oooanm30o6tT*RE2*J+8SX*(@(wE26KA_H_G``L69z-C$LPHPs{=d$6pOfo zcriOg1G{<1jH8C@&ReiQijMnz2wQl7awF~EUFK**1c`S^>RX}j?~dLAVh$Vt_gGBP z-WQIY86AIRU{|>xp9{UbrWs}2jX@XVZ8fH0M2NQDS+;jt#5h>1@D8DnQRkQAlJ#y{ zq~0%w51Z?0HO31zG0#=3XY0uU+s{a$<6&~-!2`r;sV(?-zxBL;*K^Y3i>XAHPnI$X zydiWUndWYMrfpGc9n6Vp@+$IYyAdpt?RCSl zI-Wg+CQSVynL#qv-S1wKf-hGd0(d8y1#0Y7J8oFYvLK;nSGLU{Nu*L{KjM=XkO_{> z_8i1FrXbqyrJ(zHGWeTCfBycpT5pq}PEx>D0;-K6x*lCej<%gw-q|Okfd{;r-dhqD zp%;<8gTonHJ9s=1-#ahaW%sghIv1qOZnmiQ>>swH1_{vFw*vaii}O!3Tc(6b_2>u@ zBJbn5pFZg%G;XB|g`sZ(p0JATc_!IoUBg8z4ew`oFb^`X{N^>__0UhUrww-EA=LL( zlVG&>yf!ZlN@87Z_bYS!u^WJ-x_&os|B&Ieznb%b+o<9-Zz7fq+*{Dzv0F&I@{3bTwHAdLyGh$4T#7 z%dWyGnR+6Z)Z{l(LuYtJ0<>z^qGc?2@VNvWD}l9vaW*|KlpSZ!p4^TCK#YCo z@2e(u2p&h&pPWS<@D+LHb0P$LFc_TT7HxgP|HJJEuaQ{Yzy{$-U6?QU5;zuQgDEK_ zUT3v?tSG`7$IomN`7+IA=k|%s9U~~Gjk{N!&>TE4)EEC445e(7e80fNCzIjVqdNdu zgTTA9X){LtpFwyfjOB~6+H2~!g`$pUb|ReZxO&Jc_OnNVB;H5AU_WG+ddLvv9p5WN zEK)(8+2eg@5-TXXbHfoau0y+NYA&VD9DD%!&?v6IxzJKX9=WH>EF@9-!)RZ9b75bG z82x=8d~x3d7f}VvR@VLD+zAv{GqwT%K_WCeGR9S9xNfqB`O5gsKuTmEb*JOh#kkM$ z;ESEs#`TvMLk=D-<*cn9uK43PjmfO&9(_%g^}07A>C#k2BlQ zJkKY!UtO**H>Xps=iVl2dNKV63E}YKB5$ikczi-#V1!H8=|M3WAOal8z?{v zKbh*oCGp9xe8cB~MJxI{9shuzaTJjupgVH9dN8PPaINt|mma>(bY0Zu#p2FAvgIdP z0`va1S2xszr|m?`<^5xuL71B^)FI6750i7&9evpk-YmG$fO+PE6SlGMAI>Jl)F8&F zh~56BSnzkm{xxp(n?3@Mt@fV4a{-_nE1uSKzPmileJXga+3Ol+zgtti>_{tEgeV0EEcGgncf;A` zuVSr1gBW>h{49EOQ(t^mKl)7=u`o?D2oBJ0XqR@Gam14fNP-fKo65j6S>%fCiyMv1 zQDO|!fs(l<|3NO&G!|TDz>s8SI2JEQRvYH!-z%CcVbd#i7vSYe*J+&uN1CcWW=m=_Qh74d|Feq~{=~up=E|gYq7@5wEWNE8i{Ha!KIoD%8 zq_>TS>5-c@`pLlcd0Ur@N5h<9IvgOhgrTCyiRl$BHdr_K)i}`uAz+V|*xpbQm@Nwq z8EW7QZzsq*7v|hYLm9cIykVDsH+v+*bH?l8G0(*v-5Af#rcV4C?oM>PBVyL@EK)zI z2Zmui4K~Ims|Co0m+OB?g6tlB5ztjgo<5Ou1NB^|uN66dF+U4p$`u$eu)FD)#}`iL zNPItHw=~xaSn{fB;JVA&J!HK7aIbBBm&3URBOa z!AK;2!+KA>Xa1PNoAs&e^Xqx+YGwQPL+nX_Lv(hvMltKJk=|=&W_{VQoqqeuC`-bl zt)Ay4;I?1)Xk{5EL?bMg`VV}dK^+uNae&(ZSvlg59nB17!6n(DfY4cJ%prcud_}#a z;Vg<>D7SGC5kiK{2z^3(g3}@m{YT@IrcwTGyTPx;1=f43Wx?q}!NGlqp0RXR9x(o8msv4Zm(!{Mn6jKaAm1q9-UBw6+~2GT zEAG1LyB?+axJ$yYX`N0TVUYx3cVoiGn*Eo}Zd2`sK^X~FB-P6MEC9agC|Ky8FQwJm zBdOSFyUg!g(~;6{3&(uT39`Q!Tb;5C??M3{=pl%71i*5 ztu#XwVo4>xw5(;cEt)x{<`1i0bs)Q~W8L}%+)%I!o0yiez%pS`%0&EhB|*P{leE*e zR-aS!tdYj-_O4=H#jQn))K_?WWCN{(VSj9xHkSn8h;)pDTX$czc1b`tCm-*o`E%D! zPIR~cwz#!+UAl#wkYrHy5r~g?OjvQSpHbrY{dS-jQ-N_}E0>HN+cy-`+(@6^NgZjp zm#gr&?}(jj_I*#?!H9%Kfpj8V1X@XIq2i zqo)9X3^)b(TKprsHrBN8o}kYZW1+xGkL=OwR{bzLst5>Jf<29K*`Q?EdPii$Hl(HI zL%rQ^DzXp#47}qA-E=wm{L%PA-ErfFMPQsRAF5_QKBfyq0z6Nv>**Ofb4=tf-h2(4 z@91OUsb7{Qr?H}rIQ_7wPVr_3v!*khZ?Mn-IRY#HasjZ;d^TBd{Hb2jgV=#d z#px8TWG@_2&oY0dj~U!jRwC+;5Y1B?IFgogIvJ6U?{g+j+G;Pek^&la~j9; z%p#S)h~g+;If7y^A9bmR-t3K0R#2X~04bOP>l@aI^(AUL5t;u3)|K44lO6WD?m`#s zXy;2YWTjtddInX-Akwg!s?)b5!tInp^v^846Snyshi>@VolQ~9(Zf!c%q?P*iE<}S zKM!+Or$>4N_l|9-B1kw_nScu&{V#M}>vu_;m6__eXif`AHfO? z*1YZ_$Me>j06c@atNcKzkKs-uhuLv#4j6J^>9_e_r(hyrFh9v`h1BZ}=E_<2um>%!eVX#(n1N0;x79l#md6OI1>$%?=u8csUfrx?U^ z`@vk+C-OgqDn8MOHM@y(7iyQ04z=71(!-id1vzJ`Ezvc9C{wbrVS=B|p4OIp_Ey7Z+w|o`Kh$w=&I7RJ4M8o*Sx>p z>F3eEIPCEpQS2w?U`-*fTZh`T4<#8`e!~-eQ}}n+rS|mx!GbQvzstVbJui*JA;w_s zym4SC%PlWURa5e;EZk+Frn>t;>G4+oqRvUG$yJH5F4~8j7-|-pFkEr6e{*O1teu?K z8(b2QKAta!wFt}*$lb6bCaG{xlJwM^OHfkLmWSn`bCWk zh4q|n!1X!;MACU3%+9wFtv8$L$$o95sTtP1BzoX&&-v*M#^-)G4z3u`-oYOO-7L&A zvh+(|`g-QY>9ZNQf5kkpy0cQ_w!y@Wl#ojX+Ubw%1)ysv|Dt^9WIXB`g=`FM*nI{{9E0v~A67_y7(zrRf3@&R;TqG3-(x6+qGuAHU4qPrE%#(vY^!Vd$&Z zZG2JmdqAw0+0Fn#C+&m!u>Szqf`4*MXvhSWL-LXonJzF=ST-pF9Qa)pP2ap3%7Hr7 z#&OxwVMK2srZ71w>Wlhv>PoZ#ds44OW?5Apq3@5|B_>SvYn%@|?}Xln zL^Y2$!|0hpR7`u;mm`xfeW3ykCNEbFYa=6%VDH!~*)m8=`AqOo`;ZNm=M(Ri7XoK+ zlh=nvrB^H9ZShZzlNLXufoSn`ooK1M zF0T|y6@r4DRo<_piT`B63A}SsGr3!_MPt?nC{BfH<*${MSL=4b6!@76g}441O)$~M zs>p9dC;psuSN=A^QLX%vp>#ef+mk!@aVIaoG4os+oF~~fbKl)w1-Y0mP`*4$778|u zKujp=oZ$G-iZKXA%%uTY0GxQTPV@%kdZS0@p`lUpLamM+I|vAL#0QLu&;}w=@Ce2J{T{Q5 z1E91ZSn1Z66;um4F(B@7j$YI6Znl#4{M^Acz2DAqlGDc4mP$;FB2e?EswpSzNdif) z7cP%R@l6Mp>0kA($E-IkY{6>0i9mPfz&Sw)2~y9OJ$bx4;<6c|c{x1c`3`_Q3}rJC z`uhuc`SQumBdk^X{Q6Qd2LcH7j5KXH4GqoLhr25--%I<;{h1Jb)~gcHxGOOq+9!g< zG2T7O0=u6&jHx_=GFeS-(WL<+(j}9pqf%Hfs%HSaMS3sT*s)AAmUw=E^R?H^RIOgv zZOax$QppQMO9Z@b$$UJ0K7sIxGs%gT!f#5>INlBA<@3|((eXkIZW*g`UmRaR!fb>h zkKePST)x_zo^ zfM83h(q;b1M^1kWoTGqLfFmA{w=}byjP6G?@Zb;%egxs|Fk$QxDb@^kVtZ4WgaJ!N z1-Pw2t$aJ_yIFG|n;`;gn%;Y7<0*r#L1Q zJtK$=FI_DJsWlpd@DwU{sBv$+{IO-Bq*?d!NeZYLOvX|mXE1QJBGhTRB%gDit6{C| z8R7#A2AshI3+_7CU-wF5Yjoi;R07|D0IYrS_M06B>jrM&)7fi5kFt@!x5DQb2TH^a{EJyT=FzkyrI?LMfmqEw?Vc*^nW z(IvakfSLlB|< zHO|r%n;mYkQu&Cg;v6gvzr}L;rVZ>&~UU9DiMxA zmAevv>J|bRN)uiZ7^9wq>^(pm($Kp}3>AY`Xz zy>t}TI{mJ;GwbF3>m3lP*+`$a#?E1ZQq=C+F;>jUDNY;!o1V>VB3gZasQBkEBVZ5z z<9)m0e_!onIa4Tx>I2i3VMs*#ff>8i?H&p+vw$O?qoAt{P{@syLH}BB{xC94(As4H zh17Uhq#^;B|G~7u+xxiD!k_C<*{X|p%J}8>rKQGr0*1J>$&Y0kNXc$IPGGP7%JdEp zl={bQuaE9{?j)WYAUJ3U`RtU@j>?8TcWON5o0S!9+q>0Bqyp9Q03Aj~#*X3;9u5tg z|Ch8dmjt`%^`k&gyEFrZ=_#U|;Q)(Z#@n)a+~WW*D6-`Ou8C3G-++GRrM=?585IZF zi*BbyA}v%avk-92(83jB(}BBe-3mYi)-!!L$)uT$mZ=#Oi59D6ud&HwT)BW7;rx*F z4*?-iVwn0zJ)CCNf{^Ur-vDDqNmZFlnkfJK<(=;s5QQuY`z0e+#nixr8`-Pu2vckx z0IE;7b+78R7xxs_dxA^(8W0Sm;UnBXQfOF5s&5+I?r|9CG2?)0Rzu2N zfKW(04bmYQ`3OCUp*3ig*HYtiTlB{#kN{IWsqFU-dLo5Kd(U`8xIqH!MjeJWwzg3u z?aHDlOOpYvUVEO8+rR(?V!~f14aiDK-aSbJ*NTJQ+Pb`2swVFBXCtrf|85r zf!-=K7Zp?Ee5%i#-jSncGp6gEoO^Y0L?uTH=|(%bDzM8FTOxBgM1)5shllR3s>vs- zIn$te9#wiJ`R&c;<0iBHH`29dS3k2PU`d)wkKbeLga_<9uOh2ZSaEsFr5B>#DONw;0vWwaP5OuBDY4W{%GoOjvpCSGyc{%X1o!okapG3l zSRA=p0$2h1e#rt*TZWET93dk|`@U;+=DuO6azKjizd1r%oTP=7GVDRC8kRstMWlE( zjbB?Hp#B28f4RA?wOC6Oo>_=+r9=gt6HZQ?LwqtRNa<=j_SRM=KOC6K!1{d=je3kC zO)@t2zKli!){?bAKa?z)5xNg4wCGX8{#SqZUx3)lKVCCR^>d!`fUW0Bg-Yj?0nlm+ z2WVwhfYp*|0$RnfglM-WIG$+A^vBVpr+BouMX6@A1NA%ljY-%6Y2gvCE~zLr;92(l zVW;FjB05>PPe?$n#yHKbu-u~!>Lsh%)zKjUh zOa`W*!HKhuO9~TFCX4TcVD#phe};@FDQI`A1J?e&)%@SyYx(~AjE-Bp8!;XYkkSc2 zu?LL40L~Rc%g1HuB@E{i+}!SAgcm62M$Op2pRT9@PM?J?px$i^tN%df4gOCSXz8jY zC@~)grlQiy69dpf(*4)?K$1>o`?tQN=Xt?*-(#p0BtK8_qy(dpXPsLRPL7_D}rM*eHLFfWwRCg7gozypxDL7Is$ z*r_%sf;dzo7l1gBuNsW_QLU5_6&?uyF&{LZ5Dv4?$#H$bsNV3eHKr+G@%&mUAiN=f zUGEGZD9AZASnYO{EIa+znl!jivsXn#YNoLR*m0dU13kZsPuUfs$yo)z(+DI~@%0|} zUUG)<*TNYG3JR>~0od&2i}GK`w4)=?C>n6PXP{I|MxL^(lFs8X|0hU+ahjJBDV#SV zyJz#*Qe(mEjgv~{=TUlv zf-c*fHe-n<+&ix?jsT!aBT!rGB?j73#>>QRKpM@3)3jI+8Lj~b3NpKP|6JTfbDykU z8E|m9Glu8|uYtRY2F3_n;dC9Xp}Qy?=tw}RD_oQoxE|JnMHOs2D7G@cFUMJ$eb{BF zSdfyu22iCuM6;<-jo-K0)&{#0X%q-fY2Daro`$HxqY!id$%41hc4DmfW>bnzx zJjsF_+ZUO6hi?JxWsGIM$eXULz{!WT17npdqYc}?=d}p@H(gyAT(dWNK@un)D4%k# z_Y1=Pq8~dBtg)WFrR>`dzaXZsbOG}%O zlx2+dmXJ_RRTa~q!Eqx|*T}~6JDuTc6iswsYw27k&?wxhC%+T}($FG?w_lxKRPNN_ z5x{$$u9)!9fY%bDqN3c~+(>Wn@BpvjW~RT*=Yk60Xn>S@Z*PJ2_V&RPj*#11AEd+o zeMrGB3SJRL@_93XODOn#JMK~E=_T!+9e0LtD)I6PHGlX7c}tD$jCbuf)FWk-_noqZ zW91~3V!(i+Mr}SkFHqi?jOIuTrtzQ$hN2TzyR6CbI?n(3`a0S8qU7TakgKNvlJ6*a zAf-`gj(Xgo%wQ2vGIUEP+?#{H|*P+%er%p3N9yryEoRe+RA mCWX@|=upC?Av7h|;JA6MeS@mIe&8t>l7ft?bh)HS@c#o5@M`Y> literal 13998 zcmZ{LbyQW~^EMVLB1kt#cT1;IUL>Tu8>G9H&P!Z6r9rwIq`9OQknToGLOR~f=llEj zT`raj_L+0`o|!#+=9y=QewLGZjY5cmgoN}O@=;t73F#5m!_Nz3@E4q-)duh%vf(Ew zaisf)f8SdRU`R-Bks#vlmEF?z7hK)qR62WCmSku+cVLU2-=d#>G?$K{cr1;jEULg{ z)uF=FP*%0uROhTSWsC6&SDC3$`MoH%IKQ9Mi|D7AW1iHl!^UkQpIua>F-5K0A+Bp(!yrOJ7u zNyEp-*K*WLL@em`*%%d4_V^da0 zH)c|z+nREFeI5=a*U1tKMw>BB-kYfouBc#J^gI+Y9ZKq$t+kSnh+#4K_vd%Z#cH_V z`Me_@Q{2FS^xf4|yP~=}-P`mD4e$oDv0MmEEBFk2bd;8X0p;uVw)ye#u{eP_bWm7rkS-rg6as_N>|C`GEl(>WD9U0vPwD_>e# zTF2fPrsxU|0uJLB&z?OKyjb=x(P>U-^*j#?qtKDGvt#>|z(h_>9r)fK`O)9garw;q zo14oMdq+nju)=H#>BOVK;i=VC!^v_z+t_Q`&H?c}#Y&Sw+{sd{KJW7dDP!Z@2fYd2 zU92@UHtPJ%favIuc(i8Nw%_8GsxtncHVVKZXgs|GwYiuevFuK}79&BgiQKl*T}PH$ zN8xaT4!>txTVX)?nu{SYDuMBh7&v$!Nx+rG`|9+4`Bb&(Fd;G;aY$4YIu{q$-gKpZ zfB(li>p97C-By`+tyrp8;~SndpAuO%9xQxyx^r=Tt)!5Xc!7!uegziiCX)b1`Jk$Qv;m>Dv8BBy$p-XS9{p{N9{6&a_3mo@rIHIq&M?)^-RZt7l))I=>*4VkVSA ze+Sh}BX0f0>yd3gk?nACt5mP_FOc}t{Wmr|U)aXJLvlcHFboFk`uXh%Jv}`+71dK$ zSJ%yv^ub&R%`J-4JH_PZfYPGR-RaR z90X>y^7VSxgE#v!uLw-LYue4?h$@hhlasZ*|NUOy+TwIr??2%&X653zT+XRS;zh_0 z&yy;JIK6K0(;M(jFSVs)F3-T4sU;Rs`eZF8&`PneLsCjjuldP1`iAJAbbm~!{RH=> zqJpPiY>Jrj?}C-=qdxgLB42Av;e(gFMM9Ad>1n+jxW+!i;g!BfqPv^saZ|UYEKxEv zGD%3Hy1JzGOLt2Mac5_)a6H!D-@oH=7*#7=_BHcmlE;dsTA00_4x+elu=;4Kt!YOW zG^db5gi#^)HzT9Ckz!{%V9$x~vt?$S?ii|wp>f~8r*;~fWu=5&A9@D_qp_~f1kAl~ zIOe`|CQ5Mzs~&8tu{_HtG`5Rng#{yHkgS>>Tg^S@xe|gqM zC}NFi8NYMi5?`Zy5>;cuw0Hid_VxbPXOS{8CLYsv33z*}M-4=>KFmC3IQ}p%{IvD1 zv{a+%bj=%EysGBTb0umuM<@ba<7z}QM0&FJTct*uAFkN zW#fJ)mmjQSlwhSHlN|9^b`@Axxps0T=ZJ02Sp`j6GXH{|qbI5}%(QJmtJC|QU;IO^ z08bPuFk2Z<41am{881O#anTQX%1+;Rl(D+K3p4Mc3|R|))Iw{n9+Pr*>HX4wZ<76) zu7Te8MKOvir+{foG-{U5h#(tZiik{zv8yFcrdR4Gv1(|jm><5-TTx+$1eT8lwlwu$ zUTIJ-Ef+dag6WER3K^X#^xKv)%?}Tgiw)J;M)$ssamYqYO0{}c?L)){q=5Ta*Z3QY`x zKLgcB_lC2SxNfW<{6dt14Xz)kG7b{{32XjIOQ<7+Zm{J4IKaL&G{xFBX(~mFv^Ijj zm%RGd3-7q`J!5h(er)I1|E7&)Hhkne9!cK#)%oe_g2;J+SrQTYut+EE(&pIYA~}D; z1mp_Jt>`FXFdh?pWWNi4eY;fqg972aec5zDYlA{Sv3`ZCMRNdU(cV~qm74U4OYLAj z&+`iv^Vc|4_MLSinZF)*)V8D-Hg7Q%OVH19oALVj!885NuTdedGeSBeWo4&)ld-q({V7cFHW_Jd1n-if>>RU;P zeF!&6hM2`Lx#amRx;Mq4x{!5^X zIG!sJrZ^9|-@a(1K31q|OtT(Zxu$e9pW?C?yO-#GCjTL;;==aZAphrdNN%{u9bsj- zBJ-8?2&)Y`8T@A43V(Y+%n3Wk&R)^}e%-tm^~-u@W*%p5J*(@&8*(#5N=S$PvfE+- zaA1RvH+qOB$uOv~vfifcmb_4Y7}VgB66-4)o=6R}jJSLlhoq;F!e85cH`Hf>XXoOi zQ~8IbmV{Il-*BnQ7uj1*>n4xh{1y{OiCE24|HK?d0~Vj`9CayRyRe1?*5mq6sDTrq z6m`@*=3wO1Q_k~Yqdg(=f(@ci3cq!Ic)1fv$$9Qt@JAgcJqQL9-RBx9S2YprNQ~~O zPu{HIKK;3KMd8t))yrXP)RbRNCe*dyep!u*W;*eAInKP1qA}I!N);{LJ+&#{3)o&n z`(3`k;`>~1ywhQaoh%V$Yy8ff6rhwmF}ZU=u*>u=4TesU?SwanhdsYPtkB*Xe{Duo zJ>#@ap7vW|tEN$=i;q)Of0edX9W=Rj$vC^i;SJNH7-i+YN%mtm-{jPTIm1oGKAX=W-wp~)+h z8oCuqPN=X?A*cE)SoCQ_MQev-8Q;qgRcfEZCuVi~MzBT|HC$uu>0ep5eMCyY*)z^x zcBgJPaW>)jDFF4`l$O@A6qjP>Q!~TyZKwTa;O&*ndHLrK-=%O=LNN?80-(7XP z9rqQn7ZUGz%1H;m?_1$KRP^DFUxBb@IakgPhecSscd+CKr)*t>l~ptW&;PvibZi%; z(6Tl&&=KPs3zJti8B$b>Pe680ql(*M*->=>|@5^TABrR;;-Q6suwcVJ_(ASZ$;F0uwIUcBYSUx4v zPQ2mAxJx+sIHCP#HkJ1?4F2MNF*`>)`gcO8^vch;=nk*gYyWM!4^Q0$*zaSECl44* zXV5RR*OIg~?>OY}%GYMw?t3NH82O=Z$hK@uE&Jb8ljuip&^YRVC5e$nGsKZ%IC@XlW`G_`A^gmzdt1&^LR7KDMXmzA}K zI!lPKGdk34)1@17nyjvVH?Z7(lQ6xEEWLn1us-0sd1`VLq}_H|sqU0EIKXfhG%+7V zuy8KIT~DOx>YY&^pZXzqbyc4vm48Mi8A%3(up3c5gUzi5{1C&dhVbs>qh1{`!xqfe zQKGN0ka8xxzkI%9d;OBv%1gYj@DzR; zUBj32+HoMM)4dgZlD_D$neBw{v^VS)pFim!Nz>|= zW+AsNh8DZO>E5qc2?-=@{)!gy zucl>JKg;6oBa_hUvS;Twg$~ER{3%w6b;;(LBRle9pC$0|0Vx@LnTqW0)P(^g;+<9R zYE`Q)w~YcW(DLQi7p6~$W0(H0IsJWqCP^BWOvf*sIAfd~(0qXyq)dx62^N>C0MvMM;oqXt?_)M6Bu z@e;gqo{Bs|PPVTia-l2m`X~h*v$k@1M=dTbs1W2nDL4eRVdTqHEv{^nY69ND`a$8t z3c`mILYoJYcm0ton0$kQE-ge`@XlpI3>Q87iflP;ReV-EPoEC|LbPw{bs|$zgX}i1Q#g{3Xvh~T zbiP&j79n%AHG?kqtBi9I7a8PEE*WkKFYaT+pX=JzOx1GHTAuLEE&hC?w=108wK@27 zxnui?N$}(~@_No{c%y~*3vWXq_oKCUrzC23&K&rsdvcx_zN2l&MZ?tiY=3>u>ILzw zFN^(H`&b63dYMD)@O)k3$nZ#dvWE%r-iv>K#p<0V*@uH2*72+3peFHGlXxt)eJ}>r zY{vdZU#Q(tebR|vd4LW`_-_@~E(CpvtVveQd^M}CX(X_g_q(n1GO|4#?@(`?{%~oh zKmG`98=XG-BCQn+1!uaw-UK*NB06mbx|Q~O>Dq<8RLu4IjGX%Xl}1R8|DCBN>)bUy z=UkF3Qbw?DzQ>}fW_~DB2HMdxEP4JX_*7qWn(GUTfQ)nMKWjo`Ih1EZ=?MSXAQxO; z*!WJSE7KOGrgJT7nCD|YiMz2MbAK+g_FjxL{WWeos4C`AU{smMurFO8jY8l^=lQu; zd)VhU#IUxt-zX5$p0s~2D~cvR@~z#LjbiTUYfMeXVJED8&1_dw%<%|+V=4G1AEjA{ zG>1_c4F!apZ);{FRffAgWTgdoCEsuM?1RockOOg;lkupVvjY6`r6n)VF+diyci^h~ z&2I+5GyU%fU*}<~cD9D@2`M;tkjzzuwP%4sbBe0%Z9T?zZlD?WUl19xjquYK;$Mog zMzOU_*nsFu&L5`otB}O)@*7p#bx|C1@SP{Ih0P4^NSse!NN%En27J;ld%sk=PitSU zMJj2VPSPw9#7Wx;b>WePh_bG>sJ?9`JX)M`;fY|CITSv-s-VsvWN$e=8knKu5BqKBsCqQ!9qQF0oXY&Xj$LqvdV*oj zt;7N?ysCYT|K9jbIoZ{5VX2P$&tKRDy3CF!I@b!8F^|k?5-= zl(HPL%dd>n`Vb7Z3x|si4N9P2d$;^nX^Hd1ZVGt9t#rS3PwCAp1Akc0Z005Zw+#A# z%UjppO)l@-kI<9#{qps>?r`vcw3d{>+GA^9S3V6Ik<>XOl*~=*ENQvg`}A=My_~7~ zF@fWeCGUeWA_@z0OSAnlibrrJEeR{;%=NpUYeM-UP($1*8J~P-r;WSq-AlDzl*U8& zJzXiExXmTHHp#siOUn}w($UQ)8^ppYMt5Iv2flvu&D8Aas>DL@&-O)HYS7#p zg~+qsaAlORF0%56NfcT!tbd*q(1u++MjHs1mmcS3a9?Q?$1NMluK2LnkF2iN5``2= z`=r$qe4iV9e=TMjQo?u_+DUUQT;3q7HhNHlM?frsZuW0u^#0$`XU;>Yo36H3Y>?^o zB4>U@QS85->m)1zh!b{otjRHv9GWF*RBERPXF?9=^w!?AUY)i~SObaed4t@2O9Pv$ zDKv69u%~zC3PHGo>T#D`nbv5pWascX9BQOmCByO-LiAp5)I4E!Cs`nPv?V;sv)pE% zMg#yF7cX2FuV;lb|9U02ryGx&gIWosXz5I_@C&hZd>6Hhnr5AS&hRP3ysE@fp=Z-A zp>NU~2@`9gUDk?$=>(GroWDjUguWp)!?iq#l!NOO#PJMwm?G)-X}V%o!QMm1Oah(( zdt--Stx=3K^s-rB3rx?9yD~7L#FpDX!s@?Sy?cXOa)3_3S9mX~8bmeZ`P31H6ykaGZd^)Skudw-E z=M7b4-D)?BCBRMSuu(F?+5-bMc8hFnF-iKiN7C(2C|W!7S9W8}Pp9M~#yG5%@mYNw zOw5kde_Wl>EW)pX3RZ0^cO<;h$p>?>{L)&INx9Q_swTk(bCRmUTiL;tRsOnTLkK0o z7IlhdBr6%@p5YRAr*vfI>axj{wuX8n~1}2Y|GvV@ z|NV?atS9oE3};}%^ai!-63B8EzMFYHjNRY*Cfak3Kv z6ct|YNK2TnAWm^M?D+iCuseCh?8CkVzKOjl^nZ?30}@lF`6&s{qj7VB zKdEoQ9*Qv;dmYs)_kc5Dzf%c;j+qoSW-hNS!ro+(cXFWa?0<@4;}O2(R#Oi;U@-ro zZ;g-_#1zl~;f8WBmq=TgkTkqGMvYT-EZY`~q+ndy3QfV~>;)Wg9t!RBhGp|P{V7q| z|29@)yY@RGUT|_vQd{{r?fX)!$(Sri!sdzjBib`A;?B?pGoycxsdjxEdM1+jjLD~| ze&Brvr)%%Y<`0?T>A0p=u|7^(;f;h}2Dxs%AyA_j<7Dk`3U}TO1I@Je1L_tlhxe}^ zQpPrg^VLmaD*hq&_iZEdLc(Ec3AsjaJz@WjpY0r~4z;d=l-7G*`bM!P4uPftufcIK z@*LMINlkKvA&qM>;5rhUWWyD}j4S(Do5lNqz?c)+X_0Ql3(2IWOXC}?QVF3Lg4mqj zRyQpHLZaXlw^UT}X-j5D61iKLfqy_kh@Tl6c`XfRs%RN*QNuaN&uvTzHlzXZ=g2r2 z|J#SGe3it6L10u0b_sH1LkS*Uk`u910pdzOP8CdpQI+60=QqZ;eoKX+tV<^;!k z3#lG^7q$GODP7Jzp9NX7ehzlbTE!j_OtT}2MQ%kaNAbQ#`b#@gA+Itb3asKlfq;Xv zKG2=j(Nf8w0J`w8mE49_4xwT6N$48eYhIO8Fwxt_+rzST-2KmEeTNYocWMtmXaz#t6 z(VWy{Wg_%9pCJB`SntVQKUe#Afn^z=s!?u@dceYv(NO#R{(>s9&gCS-VE((+rY)`T z8lnL%-iVy^Z?;Y}Av6>?0UITVSU#(c)pC{sDXk%F?FJVXo~ipOy_2+(gB8MUUKNX8 z*O@i0+ZWfNqsbWD+H%snG#WHC@A%&)`{=TprcS7;Us9S8IYLk}O^&USXG?Y|`{`}g z`;^DYf=t=)-KdS8-L@JM>@Q7$tXTz2eh(UAh2U?h>hFfl5^4WWJI-IynXov)DH4;NFRhAmzzyFeLM7RQU(h~T z-CWK-;(~LD|NS??ww1tpcc~EsKa0{)X-LgZuq1`1o5xPo(^kn0iGq4brn7Xfgl#yt z4$D1rZ68;v;#@`2*BVhK{g;M&B675Sl;_IIww!llBn(S%+zMGb@J6*k@tDz^hA@wk zm3r6^vZ$IPXUSN~_d($|sbaQldsH!9wQYpy=#g{|>9%tNO*nO<)`iqqx_KN_(o@Hi zX0ST~09h?@yx~)F-X?Jn>(YH^;vp~pte2zNH7m9+q~c@E4&|03aSDp8T8{Tos)(fB zgifsP5Z7{o#5xhtpC4<`nBuMP4UY>Iq^C|yI#BzL6Hccdp@d>^+Pk8$YYKnDS=|~M zVqa;gqFM{9W2CbxS^@qtw96OymezLdS)6flNOW%G@=8+Z4gEhcX!`!5c38>k>q6o* zS&zlYJOTl_H?)yYh|XMWm0eH2aLecWbc198G9ii2mrgY(9{p(1e(2CsB)p2J>YcAt zqD!-_CG(atg0Sq*5iJ%?jg0)_tUs3kN)hke<--1JeD#!efAlIIj`qV_)W2bz1|P1e z2gTotL?%qZ$llf$MI??TjQuzqy8Y_doc}t+^Qlz)ElKtApAy3nX8W7nlaba1i{JGp zinEHxDFg08mya43djpJ z2|b~j?hB2)X1R4K$jwWUifWS*8CBoxk%wu$OiE(Hdd1G}_U2qNa-+y7oKG9(+(9-z zq4^%ftSgCo#Ly1YC|N;og~M*o(gAT9d=)&@DDuaH1{s~hk^h=%u-gyy^q`~I`(R~( za|vjo?@+k?{?tp+&~&XZHi}axO?jw|J!L8}^KM})Ve@c#0k12S6$uGf?STuxxpZuq z>50G0d>EA!4rYS2x7U?Qj}B0rpMyJt7W=%eWYnNJVi_MSJL0fkR*dNl66MN`&wNNA zt7u7UO9PN%OY-sCjuBx$tXVt_8r=Sw_rstOh_z4HZzQ!?@Lf0x>+%!+8v!Y&cj25{ zo9YQ)hA6A6{Q96R@-;<(+2>3da`-EqD9M22B2C&kFv$a%Vb|RF*!H@(Rn{{dFQasX0=| zb*wtzU(>^QIj$uIfu3|P^tepXYjA+?=m(E5QIUHggT5&X8lkRClfOPKSqAUfquPC|i_jE;~}Qvh`_< z;-J4^AQiY0CXz<`zWeKyjD>C!nh^NqFL0q`Y%d(311>R}Jq-17-y3t*5%_DT3cqg1=%to$@;RNEHV zd2$jUR#uFCQDg;KSsh8NR%#$&K>Sja`C_+)HsE6a%TDX!1@3D%Y}a^BbTaqE`l?AK zlb&E-Mx|d3y}-BJEz0M10GvjrO%q`x>~CJQ&Po(j}c${LL6Q$hvtO4fw$xM z$>Hol!zMKs2Q){qr0_5{Fn_K-igI+^{+!mt`v_#sA`6rLuBv7Trlo_VNW_5gU;;B0 zHFZo(%zI5uLThX5UJr}rn3QXF_V;V{&M13lR=f-zi2$RO-)uf?tg!fqH(cd7w^ZrOTh0-|t(a)ri~L&h9UM{z`fY!R z+HSucp#B=#;qUfii*G<_ zN`5r?1)qfi{ZKpV^5kx(+vz__)ZJZ>csHt#DFcV*G#kmSt4rWn@`3?A)A8RfbRsY@ zRnTs@8$W4m+ESu>>IHjL7>5R6Wh`*uzRA?x5-H={rne}gDc_mjHq=u=W<+sO{H^Kg zGod)Q7!uXHwuKQX@?X_&YactJx6|B2tbe!IU@p;w_{wQPH`-kZenNvkavx!9yMaI zBpMlxooq@rCC^QJ#Q4$Lh~-&)Gf1|m0cz@AyZ~B?p-_ZLby<}DRHd}TCqfCSN{A#Y z3{bgT_GeA@5^LEXdI9jC4Sk;nNXCaHH@>eUEbLnv|8x?$d{XP(HmS0jViCT-aWrZ# zk%hA?L`+iu*F#d+lQWEa9`lXGtG!*u2|JZ?X){Tqxh&??&*U_;Ny`bl*?Ba8F5xy> z-l$p*j2liou*mRkYE~W zKg|VnX&iFAJ;Q*FUd6EhKlTN z30I~4h7=mx%;IDt6Bphw9C(rA~9AX006NB zCM>L-ZSyVjk>|PHf&f~ZfcU0ib_Fl3|0;tuS2+iRaR{sS`+)n}26Hb7sTh9vlu}ne zwy&xUBQ{A71Du2XTWk9P%ndtrq-S84W3~9Fe4BC+~5#now>m$3A z7u%`CKrenwSd@h&pK=;YQ|#J7_w%D9H^ofUk6G0*$6fGxw6Q zJ^UE7jhJkG;{$(E<^yRMqf`E=)=f<-qG@`oL+-@sb-KrZ>45g*w&&mUpLd#d*bnbjdB0X`rM_Bl-FNZhnXa6OInW5Qb~y*^;?{b1=3>!0GFh~pCdB1h6R%;DhOJy(qH06nBD zn+>k+#gnf$M<`G+VqmQX)vI;^+8070Alk&?5wh6xqDC?pv2!_iXMCfrA~u9ykC6LBog9UPcoPIh6@+G9$UtOj@d7$(yX@5*Q zFVR~G1%|S2V`4kBRvJME9 z)^A$L@T{(%N1OoWONvWR41I9j7)&88r#J9k01kF#JI`=?PeOGPA@4VA1hqSl>|O#? zw90EpIPMVaZ#f|_Rl!=Q#m#kh9qf81y?#)eH)IH+besJ z!dzd_gGy5w)Yw2z4|jU%y6U+2D~mb!co;P3rKn@=>UHr(!kGW}M>zA6bhMgp<53J% z+T?JI4>vb|Whw`s2*b<7985UJ{MpHNx9Vp4v~2Qvye;J~MhWoZ@GNIXs>a~`2)Vs% z8Pp)OF_Z+jCv|L3<9-vH>KEpk&>VI!FcK+!wUT=V0!6G(&!CR%k~*^%y4HP(dMYcH ze}|~YiUVe+4Hg(>*-oStm=xx(yySovPlvU0Ydm~oV?Qfjz9hj~6%6cfu;ZiV;*~_p z+fjwP&N;6z)gL25WfLJjJ={07t@f&(z=I%(>-$+W^qPY%jJgV6s!a^p@INxD%lT;& zFlmG)8+B=%`Tv8;eW{Z7NEEX)!^1`hU5_YB^91jMn9%h^Xzm=vc=5+k>^i!#m zlOfC`R44@;*1}q4<3mBfH@-_e)(tiOX9|6}HF$VZV(Db4WTN=F@rj7_V-`XY zu(Nlf|fvLLY3SN$bw8W#+Nx*#jG$87u}lwb^nF{gUqcTZioW} z*`+`=6zTv`)d3#(4b=M|!;+{U1Sd-DF}xc2$Uc##sXHfQtM2#Cl?Fe2jrHeT=(fI^ zkDyKCxm-z;%-C__fX)O}v;AC>U4qRB9?4pENu7-J5A@Yd?1*N^hvUH*7ce=WUh&$q z`R+toW#^HNmiaRd%V1XOUWBbm8>QcLIfr>RIU1at(|5eP;$tJc@G`@!Ft><=#hY`8qMqB9?%WISO!ng5bsV0~y#lX25vyI&ve~RyU`t-_^@{x)7KoEn z;cNzWZ?w?We$4__A&S0)Ni2w=FYc`r_{{$s{;nXv2AWBcBJp9+$LbK^1s)vXQy63p z)ep*1upz|gY+8_TathIVWRCOTqTxuRZG|KY56%$2y7K2^rn-FiwA(Dh09B6Nvm+3k z`=Y7B-`&o#soLO(^enCp=GJV!D3g2N!1oL=XCS!r{e7WGwzN>=44jm%)j=v9Qc)I2 zqBCF*B;3Ds@q8T40!s;E;z#vp{4O8MLP^sd>C*bGq!ZbIfC(7%m1^u88Xl3@+>r)n zCw#{GP%R|$N|Ui^MM)h;;rELU?=z{Kdw=RKt|xLS=#Kx7B>A73Dez7qifZ$M904kR zxJa7D$u)IT3La-~Fz)`7=_&NNBmP*PUxv7Og~VJX^PmGFs{ds8{^QJcUy+3m`&frx z+x&K|iPgIE*ZxHdS$GGOn~ENn>iy_@Pdr3!$F%cSa^RVF?1U`ft5Yi8OnZGFDw(P+ zd8r7P%l z%gKnq#N_1TXh5>}d-iJNKIwO{TD8ew0?RXck<_qaSo_1=B{A360x}bWCOB(N8y7D* z5a#cUO--u+0CJK!G(2o_ybjMTC~(}boys2Hxj37%O$K)|wCZg(%9;-z>w}9HU*;QP z2sllDjg57Rp$m!I+1Vxf<>D|=0?r5JZ50*7&rxY_xEAU2zd>JcizU&1t=E40SJp|U zcD3p2`UMxFldVy+i@z&3plo5%X^Pv*@PumFw&H+`3x9TZ`Fw78-QC^AB_t}Hb`)b_ zupO|50x>|%e7}UC0d|Rv`G@LSkk{%zA1m21|C^?DlKLK diff --git a/docs/images/chapters/control/8d158a13e9a86969b99c64057644cbc6.png b/docs/images/chapters/control/8d158a13e9a86969b99c64057644cbc6.png index 06300409b5c9b7af31cb8a74c6cfb29dcf551354..ba7e5b85afb8876a6bc762b0c998a0395a9ea9fa 100644 GIT binary patch literal 9924 zcmaia1yoku*7bvQcXtR1N=hmqA>~5~D4l||fP{d6AT1)I(j_9@QqqkIFDfbBNQ2V( zuk*h5-v55%8{_|nI1ZfWJm>7Q*V=Q=HP?>N(oiNQU?4!DP{gV#iaID11|{+z4+qXr z=gro@0mt&TvLfmn`Ipg<6Nf^vp;Q&+biGs7Cw%YedYs{I?Xz8IWy_fN(6Z5@B&Tn$ z&rVGKC}(Pan`NBe;a2t-&Fa)R1{rlPGs~A@m=l!e~Npb|PUJZ$fiRqO|*u59C%T&N3;Ql`@55*_{ ziS_@#D{|Shf)Qe}pPUV*F>7m%53Vu1e*0FQ4C~|5=ke!{43N&rFcGbDR&TGPVxY%U%b=v#ZU5JB?>_E*Kt{opt`<`J*Wrqf~oabBs zc7cq7T9<={^j%U?Qg3hXgvzXjBGu16g|@3cGBPsX>?<$v>%R*)7vZkH%&*T-?c~y@ zBWZejWm`C``-0ki6D@1&o|{fie*_d9986+kg&MwBsb9Q!k)5BvcrMAYW4zWqHYsV8 zy^M&dgg z6O7lrHXN(^w&%JqPzCx$@_v3d2nY$Cj@&U0atVj5q@&)ve$6H-N`3$SeeS9Ky*(ve zT`IpPPm&wiKmerdPa&&!q zaS^A|khsz?TGD;7`Ri9g_r)GN_ojF6IC6FKl}hk~0%=XXG4S#6uWsBmG$arZ5WovU zrE1u^F?xA<9UL4?pYBh&RB5)km|nod)YjFt{>#Gf**-eF&{~jgde(BXM5n)wAc)QU z?Bp;()Pd$~uezzS#%zCMLfgO~A~KR7E?#r@$=c6~E;@VS{j!CNmbbla z|L7=mq9(xp{pXL6Wjh|KIV(mca(dc)E?f0>Yg=1vN(xaTI&^F6AYHC zp6x)LNx0{cVkM$+F)0KWOca>)2S~^N7U(JHd_90Mjo9gj14h(`E{X}-*;shLLSd; z)%pCXQolD|fRB$9%aa{#)J;!M&k6?O{k$ey9c}HWSBC;Sq>hFlq;;jwCDIliTt1(Mk81DiWDt;*8Ve<6 zlqwYKgDB|UxnrfBAi;~g|Bla-sNjJVd|3HvqN1Wilz^Fqog#5kfS7!Gckf;wZ2f-K zB7w{x!j{_ud70dzq7XK`6ER$4%(uJu`go;MNj5e%#0rLAHVn=&lmay959OVo###oX z8pOm)D(YVn5D;HwwX$snV3Rj6pl|wI`_w|-I;6Z@I4yDfUuP4yaSAJ};4zu+-(#hv zrCC<9<0B*Z^++o8GcoepR?j>sa(w;zb%s$j!0A61{QERq_|M(!d4gL4U%jH}UT`Zj zip2`W*S@~|NDBV?i>?u#JD?FSU>4fc)HLFxg46aaHpprgpttyJr^aZ{Khq?T4U#v z(^$iRRFfEo=5m(R%HS6B=D~uyOba^|R72II(G(R3()5_C@p-Uo{ zE-Wkvq0vB@2I29N9}tQ#7o^Mh{(y{%ip!JlRU1U^roX?8^QUEQ|2V4)2us-ZrFHa* zPhIh3eANygH`lr^33_jt!`QjtY1rO2KZN0!h$YvYC}K;-#B>Gj(=KXm5VPC!Xr>h# zXd>fwJb(1w0z;`s4bBxH_QW4QevmORj5^8!h!kTGnl6$huJ0`O(}_7Iz)RD>Q_-@g zJCySB@(_Y@cnh0vtGg0j8%!>ZRY3gKq5V0UoPvTcIn9iKM1|WTD0wpU3*e1^AlG(( z`C^&y5y=1?NRT)lO^z#9&};f!(Qqczed#ZtUs7q`J*qltRmj6ic1}*rEae!QI^xg& z`BN#H;$MHhtE!?gFfgE#bbCKiq{0G0p#J&ur`9E^t=(PfI_kf_GCStd)N!yq*Wuj8 z51D1SJ>fMagzkaavVRK0Drq{4{Ps<&%&3fzi1HoH`|NCsonPO9k)uY6MzA6NgfEyR zgYYT2!zV%~AT)^?!>hnjErMmJ;lTUSLn7=~uM*42$-Vq+?c~ot99Y!$_w1qpcbb{K z#w~_y-H1n|>U-GjTg?ThlU*8H-mAv7J9iYrSdtEA!dSmmyO`~)4r{LIuOSH)(I+V*8SGx_b#@Ki;OE~WK^{R2*`&5X8IFrOf6kJjb54sBCDKIGL z0?^U)kdX^aU0sJD`-M{9eP?vK40CaDv1PSoH68c-PJTgw>(;a!oHc>LFtj~OnT$8{ zkB|$-VP|8z`L9!^Iyw|Kf2CaZWb}-&kk?r6NffE^I~2CX8-w+gGP<`uCgi?|uBq9& z+hTZv|Em9oeWirY&ZUjoC6+QHS3Dw0H^b|mYd0S~y7tLyli!n%L{8tp;BrD8&Xw4# z+`__kv-6|7^?&}lt}hVu8i)p!Am)^G)v&?{S^g>IlYgaN7xF*x#=^xlGKq3%9-6sN zBFPMzP*e7lid1^%6V(q#blVpOhI&a-mhIBKM2YVzGW1Qg4cI{ zqvIvJg17z;40!o##b!+}NM#OCK*Qv9)M1c9Ib~K|B)Nn3M9fmUYx+zSv@UEpTKf8- z>toe(4f~Una8-v0r5i!Am{aTg=QwgYx=ZjnQMTOkTYxg90q6dHC)-_?)wAD#SxGiw zCAYVc%05q%_Klo~RIo~Tsc?LDycwIE90jZgbK?!8x;HpDc(2y&BF&Y1C~e#(Wmb`s+T zAL_ikJRk*^olp7&A^HWkG&D#cfu@(1cyz*kxC1yiPBk(E#6vdZv*ms(Js5Kw1{>P z$NBs!Ztg-JyY`pd+@|w~+w@ApICyaT^-GjC$Bp6E(*t>TEUWqOefQSn9zT`@a=+-w zrx9Yo;d*#@hzc}De9Le0<<`6F^Z}z3h!#Tq z%KrYbMtzfU)1=O`=#I}TUJws2Uk1W75;0s;S@TSJvAVlDY_Y*mx*JgD6tdgH^KmZ? zt-s$CWIkxCvg|!}#|4BC%ug4@(P7ATEWhb!iq-f0(7*@?MXlZxK6Tw)(WByf^DAKS z`DdrdH)^QBm>A7E-?v}3tEx>+O<81)esO}5{8{Ky>G+cs!G^C)5&4k7i@nL)yTfLf zC}EqPn?(+P1g7jwOk!Q1DX%V&0WR;t*5us8KOErdy|!0YtL zgL-pvAvD*Aj~}TZJRLB;sNQ68&MQ}l+-jLRc&T5}UX`~{!ONGvp|w8qwDlz!mw@3j z%Qpn@X=Y|-@5;i`lGW!|7rj@Ds-KtGU1K;nJ{BJ>&dA94?y$VPti}5VegM7%e>(I_ z2`Ba(q?eSqM0w~J8HOAjcvI2PRGs4vHZ!`^Kh0DKlRC}fZtryErq~9UMarR(rT|%l z%P-&gU_M#@cndVSswzHGX))1-zqU=Yt179k7KcRpR_QQovsq-?z))e^7aAWQzb*Pk zZP2lN-!$OVgN_L2icuBO<;$0Q_k{G6$yPQt2Fx^*^Eu z^w+NkctL?M32~L`gLe7ZQlK&w&}ekQan%CI&F!A2S2R+)eS2nM0b7nN zqkp+)Qj_StEM-;uQco~7*7ZCj=MFKs_MI;v!Q#p%gg&_u�ntz!-8eg# zThH%*o0Po1&0 z1f>9|@2f)?E6$Tu`^oXteEK)}^F{7tg67w-p$NQVg*4FPkx+(x4PoQG-K0n=;G&Z%z5&^C1 zF79#U({z4T_i(xyVkV4%h2tqm?+Zu(9!> z64f7u2}y)`aPbHLN!JPt0wuD1h5g|0ZBwjAsj zWnqcAYo+*tMLH6sjd@2TbuWzo4Hkw#!w$s}@Uxwqj)0h$zK%nN^GIpSyY-6l@~u&a zl9_1n$!T$MicY~Y`BS=BsOiQZFQp=RU*FK6?tHwQlRgXi)Zx3?Fq|i6x#{NSR{G#O z$BpxoKfO3Zu^$D|qnR^wu^>v23Tfh#hlA@etPQS<7cEwYij47lFGxyC9xTb8M}TD@ zsIrD?Ye&n+32|s^YbVGC_(ySSGH&c2t(Lgs%Mqr^JQ=0qs-Eyn?#=0oTNgr)Y8e?t zfmXEKoNVY6;-KWwxOVLD8BPG$AZ0i6c;1|Z|A|DJ?}j7-DwN!4vN~pA%C`5|J`Ix+ z+r^-z;d^gKKY8%6_^t`Ph!E_LOT9k)Hesgb=gZGuy=;#h7_bjg$BVEZJ`kpBnIIYZVLTE9E3Di zn!zI|%747`OB-mJ+~m`vGVS;ms$^J&W=$+e+ShG1Mqyy$gilaZ2KSd>_#Xbg&m`kV z2MA3kVng)yih-r^Uf<{$f=l7ycvj4pnMwv$^z`*DA()pgUv5bfb(o%;>!0=4kv1*H zFzT`!4s?=Ing<`o{+d!d%_%i4tyT9+_D(Z0<)g!&?EWO}d6@>YRa^6f+K zS{z5_oVuQ}N6*7WdHMKiA7~P;xP0vF=rCwdp->IJ^GwyZEBxNqi3#T660^;3eWl_c zSMN3XOCvA})mE}`_1m>Lg|LW9E5*XW5;Iw#kKeS5p`bOdn0%r-J3AYxa!OX$tW)(P zq30$ZCNHyj9$pE?(Kg_n)M=n7)Cig~hggVy0t0G0zUk!F+S^=A3EhN&81kQWUIMzj zZ}hAb9cN(9yKq_k0&hjpL<|Lw{FN>n1HzOZ+zkNflhNU)l8Omn2Le*in?2dptME{4 z9{}Y|#45uCVxuMf{KU3wY+2W*fdzaWe$S2Zh%OdiA~0i^u-B&7q|gk-oWedk58_tC z!<{z9YY)0)&z=IRUhkC-5@&rBZN!uHv0s3jN| zyajqhvrYORJB6JMA3{9^yv3}hrY7hzc>|qx!vNlj@~2+HM=>SEFLwsgfg(;$Hv_t$ zxP1YLGdV+}{N?M{HoyZHYg=1Retr@V1EgQRd}()%l1iL!Q!EtXs2a$=O$$E6bO^0! z^Y4YP+pUBG&4cX<-Rr$SD{RZ``nj#|-=As0Wo`c|&_o2)UD+5%tjeCt#}SyoBETT; zrBtu2fb;!;1QA;j)5iIm=MVB5H*S>r?Ao8e;%!kNm43#P?nkjd>04rgHNWsR-W4EpZM2dQcouP*fLUhA zZ0}h!GXEBrD;egzew~pay3<>rt*fi+;HA?Y%1`gbl-_4`TMPH3Ni)JqZjYqq+xlsn zZt+)&WP*h2yZMdm6A;D+fCaIyUIjrxKknMa3^SZPcw6M9Rr%wDe+k@P8ce7z!29g} z{8>&{_w}~f+)$e6Tz61V&=#=RYaA z#PVDp9bBofNojGao(LUtt|LxNOgvbt=qvR<^?A-DNfF1b-U`HheT}|-7mSD?D2ZoB z%B-MHZiOKTd9L028)F;&YPSli*2&k#4y9bwM%|-(Da-b0K>vDB@iI^lxO)&20 z=mbU43e27y{^^E9gPaZmF`oe(%&GJo!K&@qC;`h~-!eU2NyFZEeXw|ixYr|h8%wlb z4iu2vmzFrPF7Bp;0nS-0_9RxFc!y0dEQEle3Z6PN5at&bXLHj1<)O+T0XG(b$P7~ib^DK8L$A10+3zILRDahUZn&$hZSgh6Qw7vTFb;8RhFq|qRVUHhDy(USXdDn6 zHU6hkAc%$>ry#^_;HN@5{`#JU+w>`$WO&)3`O8*UQ^sCZXZGur-R1tSH7B6ko}=@7 zU%L1fe2g)xUZ@WT?rv{`Z4gpkKw-OMi+0X>bmv0u@hGBxJ7C@c5lBVGs*7cZtQubA zc8Qv=j_hb8>ln~diPYP|TBLr>XDnnQ!;C@j-#T>U@U`t1r2ofo*LCK-^4shg**=+f_l;@xKR3gD@n{YOg1)F@*BZY- zvP<*EHWrA`a9HtK_4Q0J&s_kVuE*=ubD#Cyf%a&JC!a=b~OEbfyt7ZQ<7|qu!AY)~X0Tp9(b}u|72u+67l0w`o zttdUVsWo`>HV)Xpv6%r1dEJ6J;INOi174E z`s_11Fjni%1Hyi5I+%3yJqz3VoKs#}uX6Lz;o(f<@kZNne@<+CJZ`E6E(%gilq6?! zw6frHvK)B&@=#rYq_XKtBaT)I?&(4MZFkTjg9G+JW$<$uGWtyJoc{jp213{JU~2{z z;sve!z{3BC|GBDvWIquoOsC&}*xYECg~hvidx`BU8SE=Z-qDDc`99UZTrra>Ht?IG@oqLPy55IdzLk!r|iNN=J0U#Qx< zx)k{1lTU`Yd3bVuA{8xk=(tZ#Pou^4zJB@g@Z`Xi>AEKoGV#EX2)@o;I})<$yqRTp zV-0#M7m#XZrY-FHA}WF@;e9KYpdcCKU3g4P*9Z0Fo=E|A_ROEv3X_wQ3D@i>xYd)M zL9STa*f0d&2k8_Z9^MQgW-MYx+6@2n0px!W>Qbu4zHh}sSz1{|$Fo*(#D9%$L3#>^ z|0yCu3EeP-n>SIN(TrgbK&!S;VzE(i2KC-T0Q5qjTF?oC7*CG(sO$7NKD)07 z+T}V77k8RRL`Aj5$ex8GjMG7D_RN@H+>}7F?dYT8dW)b?OGgJbz^FVB2&|Q8DAGCL z!N3dUu8mr7@~$0^eW1k;9BA`^4}dU3^dh0tNGV2TWhF8TFt&%lowql%sTETrEG(ZM z9m?~*AplZQ1PA>E^YzeZM)7yRFV*92gux*xG;Qq!kU4mGc!D@zVHOZ6H8a~0rByJ9 zygw}2CVGw>wt~V!svZ-}w>|C)U#Jm16UOQf9ocxKB+|}OvVa))rF_6t)6l>Pv~llF ziSDIU7LKnNsB>E&Xgojj1whKm&BbdABkck*6td}|E6{%)zp^c5CC&drH|!kkSqcNK zp7h}}Ha2GHrDYIvr0aaa90kap1ro`!TFsgo3S#>{KXwk^7D4ie+CL#R4=H~VhDKl| zTsP~c>7(UXxkN-6VdvDS)t4p>$oCXVtmgjyJ8Sw83>FD0_xcazWzSDUz-QkAPHhEv zL>{+97z>?`wP#ULx3erxyJA?oKF9G$nuT)(TUdr!3n#jA%yP!uuw)u4y~gnfHU~H% zdja|m<+->+BQPIG6{i0QQ6VCt#BGC9G;peb^ew%JO$PK)WKXs*_tt&}gWtuNE&vL= z^HAQJ^2BJK%V&2b1aM7JS($N(kQw~D)!~vZ=y3rSJO@Ub$Ul^yAE)Ei$n|g3di3by z_MyCT3`+;ZPQomFG}>Zzer1Vpa7uZ`kKJA%6rbL;kqinQd2ZwwX(8_`_N_~+~b=!NjwyrKKKVPl<)I(p|bWtnp*Ki3d zXmLri?g&A2Zkuc4OLy}Gt(;ju_O6ceE|qIey|m>We?oHq2%=Re(7m+w5{0s0o`Dd9 zOYp&Tolo#-)M|?ePZ8i!C!&1y5fTEh9b9lYVL?P$&Im^*INpY%G#q^w|Dv7!6Cwq@ z5hk_%B|>nPY3%Imdea2jMQFz0hHbCFmoT?G^2=OVu_sxr76SV@I<)O*m4~CstMMgl zGw}2jyT(BLNQxOXUC}2K3>2)Qf;T=sE_L*fD-1mJyk$kj+lz!I{}0=C*dUA(nsshI zZ+mEetnR9)s>jgP+VV?E5~N%oCjGO-!+GhKIOxtVHH}ds56SB=IP&t-1wKAn z)+scozyW2c=;|tjGM{QRV!&2yNMfK}8xto$fdw*^xE}ceX$rv{LVps4ivDmnYRvvx z0|x5v9SxKJ>}U}5A|CniLnql^0y5y>m+zB_#E+46J@e7G#3@J zc|(o)l^;BKP;vU_&z~~u&lL5$Lq@2;(o#WhA0PQlEptHb&UFF;g6YjomlJpB9KTk( z2tUHY!byXoddl2!^ydX;?^<+I^$PUaWn@@FLPGRU9j$}dKKgAB2TG@Tma(jQC^kT& z^!TN7})U8TG9=B{ZA0As21gj9t;j^6-wEb+nrl1I=eSX#1W z(wM4rxEpuOBJ~IlBxtYc-8@}{lFtls`b7z{pa5f&l41yAW0R3-?3hd%`fu;EYk~&p zlGfDL$`$D=6zO9Mo}+UBVR_%=e|G>=5{l8A&?}5Oe)INiD`@T@HoBz?qWAM(pt%vw z1K&Opn)&q|QlJDc@$%YNoDV@iOG^8{w@I<30iBQ~7HS$eYlaP`sySWKl3)6#O!5C1N zNV)M?xV*gFu#hwEvzDITeZQCWc~U5(+e$`eKGA}2^JDqZqVpGIZ_`!pmAss1?d^(l z>DuB)y$9)e!{E>Y4040{MgZfM_wo6}h8Og=QCK-_>QMF<6inDqVOxO1(h4CoD&)iq z0w?LxjnkceeP)5SXS?@GC|ALJXTNkwP{VUC8Z$*8Td*AeUQ7Sltc6_NevYYXDsp{pipLPPrJ+=BX($%U Hn+N?L7(fSw literal 9857 zcmaia1yohvy8Qu_?gnWQ#Gs@kL_$C$M3Jt;!JtKHkWOh3L0UkhJES`Vkq$+qJ0%1p zeBbtW$Gi6*Z#*3i9GkV*T6=vp=loXCJyiu_0tNyU3PpTJQSLqpg++<{!^43m)Y&s1 z;RlY%T?IMRALMUp-G^utiWPN7PFlk$aecy7Lu39FXDcAsQ-H%;pF^&RK%E|6?w*8t z&?Oh;^=ES$6-p}ICBd<8GNr1Y=HDk&i|4sb*zCHnxOKR)^Lxfb8RU2n9E zoJ$iQ8+%>R#}|uXshe2WF5L{OS-ODb^M)3ch4za_FqGdLry^FKT>)D5d;kwMg>vsESJN zH+EmarQ~ua(i**+RZt+Wsv0)7?!fJjA4$VwR8NG$BT@B^iAn3}&CjEP|6`-(78e~K zS)^g&$jqik^7T+CTG~D7>6zzM!xlY$_EuK;?IJ=2d3i&tUv$FvLh_USa}!s}NMBlB zk(aa(7Z)G<`qixLU2*=MPcC^MHeDoLf63|S(3=X_I62MbxP>Z@u%M=vmd2)&iRVwu zAB2CnA*Z`E*&v0QY7M&c^y$;1M^9yagj5sb&Z3q^3sbrn2>h@gWGXM0F^jWva>}o! z7u4`82bLk{X=4*UW5UA0#VaT6CAG3@rEX*s?2XMG6-}F#(TYnEboTW0^s>d;_bN&( zEUc~$ZfaU0ArH$D+b|;#HEkqYP*CbduwbrnRr7;}gOH0u<+kE&G;Br&ER4uidzD zgH23~Vew0nth;;twvi+JA)@0$xx2eBjpQeMTfZ(K&{t_=G(Y?LHP!rNOH0c%ocG45 z$M0n>MsWusYisK`DUX}-p$x+dLX$7|nLIOZm6Vlfgtb4cw3@7cX}dPYPb`Hs41vUC zjKTPLdZNyrJ!||W@y61RI`M_iZ`1oLsHv%++1i@zxT&j?o0ymsJ?@EjsN}tN4LL;5 zOw<)_ZuP_Jg~4>gIu|r-blrM&W#!fO_Vdu+S^;2jG4rQSJ!4{G+6x`+?0l6uE%v_m zrz?;%FuZc!Szu#lPn%eWcvsigCrah^)Hp}kXQwgH*XMa~;8c6O-MwzDcmKXOOl@|| ztbvXrTUH$B!zU)bjYhxQwv~~==2S}#=}8c|QW#st#m2@I zulViVyJTnQwFT_QX=rHJIXL8I_frWV<`lr&Mn>X|_?=d=KSaEFlj$abX(#G56$qif zE`PY*vp!y__cD5Ld-IA`ft;!;iN3!6v(sZ|x7~h)u2DfkJiJ@ap7B*XZiZ1`(PX`O zvwp&AaA06-cTjD0_r_K{eqkx}o3zptd- zyVN%})@k#iQOFn~+wQ))I+mWEUfr<8r?C}VV(k%(B-5(kpHn&$4+9+}ZpE^4at4-I zFfSVU*aoB@K8)EG?D!7BY1eomy5T>6-cwiiVwQ9XobZ>1bh+$_-WGxML($hwSQgZ0t_w953?Ut0G3*!*VryS>rjy^lCoSBnjC1%-vv z^74qz;1Jyc)32fqP1O!vd%iRAA$avr0F0bFeUH3HNXw(|Sy)&IC&Pifu)V##HRVJ43U+S- z3wWW^m|>KxH!G`%2(1qeF+~lvPwQV75wzP|d1$x!gGp0UGrwYEC_|N8CQb7Ji+!2m@Jq=8v(RdF~U1~5=KR6(pN{V=K1gENr0Oe#%^kl!4f>`zXw zulEQke*>!wG%gQ*ASWYZg@Xx&7rd@N`C`1(_s*me-3O!h!CeDOpY#eSZ+M|)>`1Gs zs=&e`iAhOq)@`ZafE&|upQ7$WL);hXKRH615sXKD7xz=>B)Py)+mx29X@_)UD_bk2d_hz|@7lGD&|I-Hk>H8cE` zIc05a`HxRdV985ho*_~9V^M$N^Eh|!p2k1*PN9fo2q0lQw~QU4>A5*m6yMPW#saKvm&ZCb9brs(goG`g zSZC}C+Ti@bXFrGO^K<|@MDjP7>JB)@;Vdv!~oox^GiH?p|WMW8eZk9n{*LZ~$Wes&{ z6CeMx|G=_4JLGXu-bcMDxG4Rd9oYq6{7XN7$(x&BPD@XB_Qg&uE+)5WI1x_vICU-% zo%D0+X>EK2h&nS9$2yh=4KCDjg$q}c;9MRRfz$k)G4fz<;G%|>)&(vug*6A_W-ytt znOV@E<3%YrMNK+RoQ>`IZj|R#28T8`m6f~o`+KaS-1w-7l$6PseEJMX{l>6)?OG>( z+Mo@S<1*s!`FfQo6wmruziuP1!DHPQ!;onnb-$U;{*c`4LwfU)gZSf)>WT`Q_yR0F z9i8maq|zUC_}+cXM|~X?{`YwK`N35qN! z{N{JE0Vos~r;3b^{W$L=LtehE$p8oHR)hPA#t77Z`4Ep*Hn?LFRskTow`TQKWX_!v= zbIkR)H<{RidQ=|5m5humRl=Zu!CUwC?Q;M;#lG8^nPorc$gZgoS5i_E{Gmy97J(=k z8E0ACwy==%JcCP;mB0KKdT+aq)Y-4n{BYY*xoTKTck-tir7~j@XZOzEABXX#8y#W- zScLCU7ggfTmkdEcK~9@>`(%ua-?@+M|AO?@t;?7%U%p)8;qm|Qf$P<)S7jueckkZ) zSW|;J+@9~cz+o?RS4nB(&`McW_RJ5L)oUS4;xyq>r%nMsD3@6Z-`aWo{_O(ro!4nL zR3suY((CQ@CqQ~=#2q(UBQGcv*{_bM5>lY>{vvsFKxKGf;4J{{Ik~yyw6p<85j058 z$svGm^V;rIGU$OMCo!%d(zms4oZVgbT$ zE$zKYNJ~pAvE6xLtlBCv15>&x5x1;%fU=AFY1<6Pgz-+gUxsqL@aQB z%wVPpBae+S9cNN1gskiSMjLFN|M{HaRlN!#2L}g460n}E57jbyNsVOIyDuhZ)N{Z~3@5hXDaB*RJlO>VW`|4cx z9gf}=bnx}3Ln?-Nnt|#U7!uM7r8citQCCL?A8J3)Ds*QA8}L5&*u6)O9{EkwIIh+e zRQPlYzANaR?Fh$ARNDuJhX<6Fmj2kkTaAfZY8uYfDt6p7M82rm7hk*du}Xr+jdRkC z;+wT+09V)kY(LL#Z=P6EQnJ;rz{1AELtN!@3M$C1LCDoiAQqn_F0z=@FT1)w8MdgN*iSf~4FBo%X(_BTzY6f*vrbFYv=} zR;R~1YDg&o`0=R5r#q`0@Se$!TIayf&_@w;SpGr&8_XBB=c3iv`S^-@^k3KDdwT-+ zGXA~0yfpYhjn>}cSQqEhg>Z?q{pFnu529#S0DTC}`Uy3)d2o;tAZ06fh|rJIzIU=& z+1b(>8X8;RR2vQvYye?k5975DG$J2w63uCj|+u!#V~ZTvD1ECcx}-ypG}I&WlNocJXF}D1|UaX(y+TT|ut9BGP0HZQ=3cOQN)~acW5zTdk37u9lu& z@WWiqcfej!+<$X#)NI}?AE#eALLmER2jVd}b%a>ww+AkzH$+8IfES<-7*)hpPe<3Xx&IU?|qmmWIacfGt_Rr~&)3hJ0 zZqIr4*+(?!fqs7W>XOn@St~264;w9Gm#7Hx^eRap=mG(j-h0pXMEK>E|`&Nn! z=g*~I;sXxTg(b2IQAt!$TP@2!QlZ@lyMfBQOu}S6N`?=>gW@cxR zRZM=ZeFg4r{=q1n-Im@yu1sISba$!$@+dj^_+8#QUnH(u0!T#yJy!{ENts)@Qrx_8 zS=`8sUhYwIiN@on6Hz`~4ICO7VbG62HOi7o`re7=L=WdY)UI*Jzth`qRv)f%=MKK1 z>v}oKH3^mW?x)Rjh8Q`b!z*L4hA#Lo7Ctuxu($_8-HZD^eG23fag4+pFK`fF2FwA# z?3#n#P#e)r-iF2OQB%Hjz5-7Xz$?v^`hCfGApW}rHAtO@lC1i$~TUyGwxQHQA)#~V9 zK?{k-z=&zTpm!nU%SejTE;pyHu>9)%B;_RyZ|xQ~K)cWxu`W<$BBP?bTU+Jeyw3DW z>=3nwF?G}TXAE)w;p+TP*6ZBTCl!2_ltBeenOo8H{yb+9n?+-2s3tOt<9>IGIoSs4{* zuASZ82!DA^t2$mY8p6Hv^Cwc2k)Q>cDbJvW22@8#22{Z7Z?EP8U{eTZZUf?F3*3$9 zyfBg4%|nivnHl97F;aWxI{zTx3iU6K?mc+mbNt&aL*xCF+k`X9GbNveQ&id0v0MbA zlr2W7*lN7OVN(c0(%jrkMnmII$)(E5&5aM|r~t|l^UGazUq3(l-^3rfy1EbtM|k1> zq=y0|P8k^)GG^wm_jhBPo0{G|*jLMxEks8pB_X*C5!kTq4keSF9n8ADc9%#mp!-)T3vOL9=)`xDV)8yOinxw@tsHMoNg zHCpZd=+Q+)c<}bd20eHdP!pUNgzz;)u*79{Lfw6HwJBBFCuD|!gdRTs(|mU<7V6H! zJNe!jZ?OqD9$rKz??^x%i9b0j`ViJ=525$r$Fu{gHmLUf{8<^{iynWDSU~!?2Q><+ ztf-h+^N%0QBO@bA{pkTE{`x!$u-?k*s_o8#3P^C;Iyy})EoWXH{kjcUmyDi1=o}%< zC7(N!!HW@x{T6Y8MwUIXi@7g*=6`pi8l_bjwtxQg3J+JyT{Jg02dqyjO@>$pK*;y1 z7Yc_vKz80iaW1N;@CQv80M$oDN_(-o0<7Gg(7bC`uZCEM-j&T+Twc!pX+cfkCx0Y7 zw{Ou2Vvr>z3>pWL^g1S{X}tY>m3`G^qrnC}50r`mr_`V0>+)fYK}VYnMx`6}YvFs^ zh1Wqo8{>4T8yJ!JXc^VQp`9!0Q{pegc|#c+wYxe>Iy^j#1Sr49shj0wJrn2{_bySz z2|Rjrb9_KYSvf>g!9Z?7u@KGvSM(-LvA(H2=E`<#b%M4)6Y3G_DXwQA58N!>SR)k? zuY$US0Y(!i=_(9qOS8~0^yh4cEZJEoy3kO`$jD&j<0Dg5RRydo`fLj2nK_v8#zRAW zf5*tzTcM8>N~A7AR;a!1c|bFHA2i_%AIzrJ|KAZN7g zt&XD6=>Ckrbl1wVjte}koSc$oU)#fx1Hv9Ylh zCx(HagPMo|wcBZ&J{J*p*}atPwtWXdpNQOl^LR@Z1jH%uPz6iNtH>Ili<#Hn8UT}f z?JV}DL$YIIW9uU9_y_Le^x28jFWduBU%Jq+Zg-zENI9a(51KWX;ILxCDJdxUoVQIa zf7I$YtFC~XfUhBp!0Bkc(spHtTU1;e10R4`%N-tR`g|Pv)Ya9Tb~|_r04z3&ot>Qw z06t_Tw_jhMbUng_LnWu67{K3r(`iSiA1ARl^Z;zk8v`E%k8O5l5)pFVk2 zR*J}n(0RRjMNlp@F)>l>@#j={58y8X%9AF~M{-~P$X8ra(!4QIBVPOR#XV6`QC0~F z#_8$lk1vme5hX7qgvjk=Z?wx8+7*7$(Swo}D+8pFReT@$mWh7-`qgEeotxY8we(43 za&j1GAT%IQOW(Uk2CbU5qZ+d1#;eU47eSIgd+r?DrAv6AslZ1ms;PAVh&~66!w@=t zY;as(Id<}zhwSMl4(NfX(9lo;ZAhY#2?@bq=9vk{Nk1S@dyZ-1IaH;t*%-;Bb1_~$Mz2*Kv2b$7cLnU zl`jMeGl{P9{mPNVblXK4800PIO)9FYRLr*4C*`((*n=vwgREz5&9g`t*<>6?$A+jfa&oYM zEa+`miM(?Do_`$Q(!qmbK4oR?ofE(Ia2C?H%6}opwQIx)A~x-SqRF7?Hr*1C4wcZP zQi;A&ajH8U=nn-!l0%>nmIixQyHdXeAQrrF{i7 z{ozAu$X~VxTWB|ka9EqEosa)T&yF89Hny>qRnMW!5Sobc{)`*jz_}J*4df^EJvQzP z9?H}P+b{^RXpAz%^Y#R+$_^c{52{HrASxh6BVrPxB9+{Y zxI?Br@eaT>9pFgOMM8Q$e|&nzhx@c*ZU=mT zq;42>dV+493ZG4Yhv%7;1oQnJLk#m-g666O3pv45yMjr)tw)6RJRn-XuG!?Zn{Gar z2<^U}6F;ZRKtm}7bfEFrWn6W1Dg*Zf?vt!8p7`zhpk*GX&ljC81O+lB@p8m=BOwR~ z3@9lmM8NHvySoYRXHy9a3ma^O5fQfde!kxw6!`56Ep!lNK|uwrL{HKPs0>H}0I<+d zCLW8B_`|en35PQO)?l}H3?ic_0(TY@3K|X$4O)Eh-){Zlk%k!nsQ0YkA#JK=I2va_ zgrK{|@>ESntDWNe&1_mg``J*`@k5wEGfUsjCgrqWbYBEb23zrd`)1i4?3>Rca@pMMR-Dm8g~-GbN1 z`uA5*UaK=iWYGqCdW0&Yp85atMT1byj{@yN+|a7rH7EqOdg;m)pUO(2sw!p#jWrPc z|F0Ps4`fglRaigWd10}Nqa1ICjp{2jl8OFQOD!&2zr$wE@K z7@nMH$ON6aYrRmvihUP4dBJYGZ@vc-80+1)#JM>mKmyrk=+0Zb-S6p1Ij-zQ{kc8++4n{pf`x_Iv zZUyh&NdtfL=X2c9pDylTwy1mzcT?cP!eC}W&VKhGV;axG(h}ZV<&l-DbcJw4X5Lxo zk%fF+GY;2O(m#HryERx&LQ6Z0G1MJFI`}cYdb+yJ6BA6`vAi8Ylw9BPb^dkFV7DIH zg$7q)3h3Q-bwGP2PQqD0*ycxG|26pw|BYW-+uG1TY^@Am6vjhcJAf|gYg+H&;j34# z$|))aKDJR_{~3uBAXIHEIdXl}1vE-pINMA&qST%gKz!M5hGg-o27rgo!Uov{1nwi> zSEA}Fl$(?D+9OtxLq>JvD2Q=iD1fy)P_3Hv@BrPri_W9qI+f|YP21W!X zDe_yzBl9B5-ABa~__^(_6+bFNa|&Zw7#BoArO}Hqc$IIhPg~t_cGlkBPE+v*Vz3EB zXDtkYK}!P-9aj2)2qP(R-6d~EF@Mb8BOoLcGC{Vg`@VPnc6jgg63_d@b<97Vkd@rFK^YX`jrG!sqq0M8|oO?P9gz7;DzQZ^Y#BY;nWuysGnd z2)D4PsLuI#3XLr>;U<938+bE-CZ9=`_(?uv!5S12@?rG)H47^n8wwT{l%S9hI06E~ ze^W5SU%2sLMw$-z2|+j8Z+{3ddW+;Va@ZxO2i*LZd2Z#_ga=@ z4Q;eJLw!_NRO~0n^WUyLmE`ydmCV-}4Z^bGOnxpP%kQZNqKQ$|)zy7pk}x(V{Tlr5 zL^0FN4%}HT?(W`~)(xv+QBj~A>?KJM4zV1vl$4Z+j7(JP{S=R=kWgUl`sM$A{`qnZ zR#{n@(Pq0VDbTe|$7nbP|MPnRclamzf454qy~h)T&5m zX(Q>in=@G>{ex3~nIqbFzfgSp_Kg&%N|Ti7_j>yOUx67Jtl3}|mz*53Y4^GMC@qvZ zfy4iwi9ZQGn@#te+BBMNevYRw4qv(%*J)_9yHHD~FyP<3y1TmGPVroWF8=cU_q0Ca zhb`2_K0sh92A>D4w6t_*r}yv4qgO7Yo5^H48=Xe&pT@?<9QVTiag>Uoo~{W=X=&s6 zV%d@U2~jLv5ntc;f7LQyHhBYH?x$N<@BRz;Ub=XO^4RsUC)_l^Pwm%02c^)3!3Eg2#iojniGGfVe>LE>FN`!*R{!v@TlEb7ZAWSULCqg^Q#A&OM zZupATj}-~VV@NQt#Y=|^{plL|cLi`LLq$fPJ?`duZd)7onvF3ky?!D_^!kZ8{Jy!} zki&sIj2_wjNhXLo_wPH;%?<(i&%s%L(f~4iQW8;tYiJnBWQ8#nR3n3AUz6|kKZKk} zrNKFfq)~;O2yC$|Ot9*~Rr-R#1Eg9NVL6~7B*06w!jWmdHmlf0=sYp8@Ok+i3a}QT z0B}knW3j$qpv*MJ ziP$u#r?ydKOqIC)!5P+CwAlXpKjx--tXgqz1%~`ZDERpJM(SlWG_YM=U41~e!xHf# z5)#m~YL#F_R8-Jl4XYDtP1ap!TF;~RiS8kBMFLsFc2j+}&dMF#!DBTzlHI(?j_QM? zK~c97#J20rGfnnbc_o=rcL*&r<&G%%f8?BhUJfapwY!l9$v6$HnuzF!AF;0QXPr&TN&o872#AI2-(3x4pAx-rc(KXnqf0|#A4Kp>(DxR0wN7xGDj_% zCX&ht(83R%H67k*R9IY>ODhZKmd3hdCxuMGFVPk&Qa~*?#vd?Sj4Qv>Oc^K{3$c3KU$;lN>$sH8fK>B31QqiJ>ZNe`ainw6eKd~9m z(Typp!|BLFODV$oi6vD8scOLVk%wYYFq|(hN6Eww;pzZgPQ+(sigj76v?g2B_%o83 zi+hF&eAYr}w)C+ST}!9caa69-?iYnOmYF${P}v{@#QZ6+Uy$g-FTC)?C=3CbypJ&q zx`(KbjXO|93kNszAPJkQj9pRDyidNs#3m{^$bx>i7iTEQLLv%*jAa2yyeoLBYt!Y> z)wsGD8F^n`p*`l`5BfC^`fqg~&!KOI*H-zC{cc{?Pe!)ZWK_1i?FX#j!^OzhIbr%9 zKe`yci_)J{+iyn*DaE3*XPxB2lQdaUMY~-X?$T9k^=@6^m<1I%$eD0+{dS}x3s2f+ zuk}(!b3Pt$+?}uhy(gjz=t|iCgUp}Kk^vQNgE%sG(LQfkpmsp_(sp9 zzDZNlNRC3xqNU*{MfTjK6132j+MOHq>AFslJ5JQVaZaFv78 zvW?JmL))2?Ghn6`ejQCPjo9T5wud%n_qVQlLCa5&OGFI!NPnNiti(u4wWfi_%Me;g zJPndh`z@#vHO%G&M;X2Ova*S(C0nF|dw1HrV5s_I{s2(~BUVI6fnwjKjtoi+r3PQP zmK!;8{C8emoR&1@q#Eg9v03{?SK)aK}st%QWy*)5JWWCKo&r<$*)n=iH%x##(#v^2 z@yqio75(P%xTsFe$atKjuD3h!YZ_1Em6e^n-gce6rd0P?fgmte+o?g!io&b{QEmplhbHBo{* z#YYh+GpE@P+#CSn-tzrlZ2z!V;$hfka93cRRYH=$C~9q7{(dwvspUI9`w*)cJFh0}F= zA4_xs^$xEcH*n)Lth~AIOo$Wo)5dl&BC)+^n}UW-5q=&|2l!{m=rAn*Q#u;LwHGP7 z==dX@OlP>&=@}FpJiEBaR{PL))qV~Q2X|k7-emd-z*p4uaoQDf!*Z?fDa`P$#^Z44 z%uzr!6OUu!853^qT??>!`hBMECk8Bln;jZNnq`E;3OyGns66D6BXmc)HJs-iDI}yQ zNn*r(y3-vutd>fh!l+XY-EaYmA)vg3B=0STGc*GD4S|J=?V@5h5vRkPK*0s_7lK~j{2I!qNp7Z--^ zlqecGt*$I84mQU)(RixD*S zU3bV>A@&{1u#?uBi<$T#>ZyQAF(GC~#-cXPi*8iG~z!pRlCS9leiYF z_{2PS9GN)&$3|ak*LC{M`n-<9RGV?>K3pLu34_wD+?z>5) zuETm>e@6?OtqBEGJH^xJo*EoRo%uenL|pLALCgRP%yJPVzk7H`+1PYcyMxB(UuRRX58GW(N-4dx_*`#+XZMl zE>6JYfQpt95MdOQSERr6-eC3LmIDpWl7ZN_Xv^@}RqjRm-O?+wCR_(ldrH8pbuo+? z+tW{+8O8E&{5ZBv>8UVKj|a6RO5ageZ795e!6K8~NJ&#~ zuHRzuS+Y8KN?+=Owx;Z&=I_IE5`?_^UQR$E(~pj$c1ukMx3=$QLsomOxvRk|O-X-7 zH1?)H@ri$;g^Xy}*mZAh{A;(R&*UHtR5m{u4Uf7h-#%iATW-znO43JBO6^@>R*&=3 z#H7iJAW+_Zci>IKrj?@de=>cbJ|)w44}}VU=wCDF6&M0svPoOA6BxP;Z9^|2`VxaD zVry&Y)MtmXFc4`IMmKeh{*`HmMn2;XtkR-?A(2bftk!3f!Fj2-tYb zA)UESp%%{7sWs>g)EBv@a=4k%qY3nCGuvpRLias?N_Ft>`wQsuP4a^U^}kcS6~;Vk z4%^Wcj0fsfkrR=`&rr{E-ffTwQOb+@F5L29f1DA`n&bOnZ&ScRJ( zSVrdhAxitcSIKsyxaT=S1xrr5&o35i#CmUR8ts}cCiL>hI=!5}WY7WQMnY9QZaD&P zVle7@=<-_eQ^HFsb^}W$yY_sN5zpEbjmp?Hn7g#)xH`O(BW~S7S8Nm&# z^>|Dl&5m@_c?++%f<*_WBglXk-5e4#xVX?^;rKUcobZTaDQfe&OZ@@kI8-r~dohVznW)@T!g55$jqE6R z$bhPX=(VJAX0=47t7LEJ$+V4S@qfj>^ggwD$v?Svb=Z zm`|`LCHptOhgzR|TjC}qP)W|qks5`Ic8bp8lp75=b64D4AH5tkXX4`>Ja0PNXvsK3 z+9GFSPODqq2qFtMMpU?&U2Z~8e7JdQ3!lhDVKzCnXk@SJdF6v&@+v0Z7He#uu2(t# z>*&H&`c@b+rHfCtl_>LYks7dhFTBatsa6*^k;`c(#~Uf}X#l?j8Tj2OE(Y%Gir?u^ zIpbDuUgVMTz6`38QR>sWiBRBcrOwaa2~DBTVAG+iw(iPxjp(t!yP2V5#pPblkb^r2S!h9fEm(Vm zz?b{!67%i4L%Ef%i+}pm4K7Z-N!q9T3DdBV9l$wsc}c!E#8(s9AHuDuRIgU~ek-3p zsj$;jGfp3gw6qYW{aL4;xSY-#qlt6&CdUtroBkB(t=`SRzSS+7*r~>_!9Kk&1WgRr z!lZG=w>Hs~w70YW6p^2T6~33R2)lFwB94rZPdHP-VG)iJ|yqHSq~2EhA}0nuDWBkAa^|uKd`cD4p=I1{`5yyPQ zAQt@QjDiGkt*qJRpNxV#-emjc=hbS`(Yz{ccsrz(vbK0VoHsAPX~K8;v3F>90v*#l zfO#B)OK&x3lj|HFcjKV=sgUh2x>YYqb;#id zLkF|H?a?p2Tf8rMu=$qE>U^&0^XnaQknOJ9gOT%>O|jgsJ{2ymD(`rA+~B08x8Ds| zxdtYYE--WaboCOo19O|$3m>w^*B>#{fu+rE6D@z;kif$G;u2hO z{vr#$e3?BwkmJox%ie?}2s!$EIV}G;`@vJLMLXiMBpXLruL`IXJA8)70I_fTe2fh1g3Wrf^E^Nl>4Rvgdu3_0ia z^Tu)0poxR#Pf>HaAL5>mhAIZO7}Y%SLn+hnlAKoufaUH3^?V$WImy1H?AzV)jmJn? z^9C%Z!&g%@j`!J5109FjBB$cl5q$1jbYG4L#JhGhnFD}GH}`;{!0_X(H3l`C97>>C z7ilJg{N246#rw*`zTfSktM%X$<25qFvBKPq7=<{E_dwO>3T<*tH6}v~-36a+{!J65 z$VVg+TxgyB)Nq1ifBTnVj=J(o7EWC8{?6Jlg_#O~Bs-sW$@^9`xQzV&Hc8TLdqiO%a*Iq|1|g&x@D^T{YhK?zdE z!qE&E^{k}ZSw$PJeRt?s?hbd(w_INi-rTt6k5P;$w_VR7yb#NJXYpS5`6pD53=h@ptagHt#ym{) zFgbLSgOrH#U|`ALt{_AvQz!EVpA0K$ex5M$m_{Tg&NLS!Sk}lUnD z|2a@*G*n&x?B50IZq1q48@u+Bf+*ho_Jk=AE8KmZiNR6^5M2qL`VCxa8G9nRjzzrr zwz{iq4LrZ^T(ITA;(5w_V)!wq<3Mcby+%Cums5%w%3lg*)GZTV`R)FZPg{$g+`Bo) z;HO6;4sy_>GuqC%M(%QD6OMV*rg*nF;U?W%JozWN3+_xeQLM~>A z^my2zG_qVoDoD?vgRAp#ZF(Me4i(Y%jMiAL72jxo5>+&{yQSW*Zx#7S5GAqyo#y~- z57D5&eQFHL$XO%38|SMEKO*eL@VzJ4WJw5mpSKVrP%k26i)U)oc-O+|_OOW|+>|(P8Lc zbjjjXz9@{;Yk!5afzHsUa8rF1xaC~af!k!sfIK;n-}L$3`jIffYx{l2Kz4?EL~``- z*4TTpWX&QuCa6Mz_eCwPc14H3rlBoQ>caOzZMM}ky421d>+QoyLWQmdli#k4xb)Se zY~^u6Xm)YcefxmTpEj8g)KJ?IqP56gqUqE4iKw{ldeIFuP5TOb1D9{fk$NQnBSUfe zgk^zJWZaf#y<);v?-~a@R~Tqw5qZx%Dvevv=auOcpFSVCn;YJ(h5tu(=D57b&FL&l2BiJ#(< zhYgG&2WL>zIzu+^U6MzE(R}Ig%jOg&^HZ{ z)aYePdtkSi?Yr&Y(SccGWKRGem6X1>Vyn$?G6j2zaHmGQByZ|dFcar%_35pN(nh1$ zf6M2r9o;==$>uI*O{YG^SMo3afY)^jl`xt<^nI_n%Ra)&Ib*qvI|?!GY3G~b>olp5 z+gSd(5+ly~#2%0m7+%B95#`|ji@&gT7U)fpGQd5QktN|J~j`ZIE!K7G9WKR1DC0hq=&4bt2Z z@}A5t#kD9WR(kDYktd*av)RmWN`}dksk+lebuJ$jm-UO8?I|?K#FMFflS*%HL=)hf zBmHz^0`(1l!DFY~t$-OxWyrN)u!KQTg&k&l?^ZiJG3{R%Roe-yNAN)o_sYH7k`cTq zjVt4juWgqs@nnu}I+*9$cT{X_zt|Su5QqQ0Altau{6?K@yq+-9Qu~qjn^ZjsZVHyf zp)xs;Qd%KO_3nZr`o}pdubVRFH~wbJ#aE*IsdU-}$HakXxXfBe9aW!3m!LIRuK6lDMh5b#Kvm*6> zSkrG}NjmD)f6uuZU--NZ1y;PAnf7U<3s8-T^6BTpw`@yrsiFbPgg(VGbl&?K)cPR^ z6ih0I5r*uFHNlFi&=RK_iaz@jHGw~K@{$XlvkwRE%02Zl%^Dn$DQphg2&WV2?CsnG z{;3)N#|3Ci_M&{$58cnv36oNl*0?Go#bmG1y2WfxrWo3OLc$YQE`4hWDR{IT>6=VW ze;NtrMv_#jEqlm`nk{1D`jA56V<1y$LunE=$}QuqySkinY=HcTF@Y z?Jf?0|J{+o${)i#dh}rZh!>|yR|fiLB*%!*jnAo+i3x!WUqYEOs>XvG^S*K*HX2KP z4-wHCp=)vSJKWgl2<(14?wnZXDh3T3%KY@u+M39_`L%JvJhg9Eco`s?@{D!ck%A^T zBx=O@v~i)nz=}F<&5{%GqUdbXpZOzi7$(CfLVgup_Xk?wzGwS^Hn^&v$O?IPjc3%! zj?I)z2U1brlTHE}_~*14boTE`=s^NF>rcny;r358EzKbjwH9y;v`QJ)7rA# zS%Y#K9mCcYOs-oP*e!{PeFNBlrZNH=PKfZKn$h@QTymCMf#(<+IOo{@*FI!Ls2v5@ ze%P5cXa({m>uVyW$9C|^S`$Dwv*OMETg~0Cgrle}s|E+@)?nyGub}2_cGYzMoXvE@ zb6q57zw)gzITg-WF%-|+@~6~Ytjo6_L*(y9CQg0$wSIQwVkIT}G|XDkz8M($C`|D} zBzF-G<=s{|YAuB-@qSP5iCsWcq2$m@I8<|U-jTQ=)kbXCQQ1z2<2uS#zKZSXu0RG)k8`+J=))w* z<@$C<#?MNREJ@=U4+R(tkK^d*Yzx=A9g$l8L^ZJUfFUl{Y=X$2;95qG95(Jw5e(;i9yrWrJ zF27`pNiXc5Bkt!24GOJ^;CiJ^x1If8JqaYew{9MX#Evr2zM9R#Um#Q$#N~=0z^r67 z><-KAU$z6w>sXP-!;I+o~MhYD#HsRvOpI*bG>3xQKjR^7~3(warosYK8sC=mc z`OV2Jb^?z}CU?E$iL&`VuEUq`M$n4fWQi+j3|#I-0yp=0g2daKaAMCr+0Y$nK|ET0 z)c(9K8M=w$E$*+$>N~~JZ`hOKoc)hzN;4k(*C&JIVk9^Gc+Bi~)32&@AJ(w+rXQW} zj*F!S{JQ$XfVVZzkAQHI95k6R-F+&L6XfQjyeDM=cu}z!P(#$H{4r`OCl#z#yhO*Y zo(|@(CTn$%k*}aBc8_#E>Sr(;b_x%MY0gnU9;U4{Yxwi!No&>N4-D~b;c19|_Ufgb z8{wJJ5c>A-w5~<7M#%u-=W$E+{Euxp3O~E)s$D-BDWpS%v}-6WCCP!wh}^4=C6wtn zmwx(2yRAP}lkWO07bfPnMpL7DBM%E5D?SI3Cd|Bw-+oZB@g{iFk6X7WnOv;T>27!C z@Csu9O!O`nO`;>=cgnn-H17Oh4x5UJhk3j&Cj2}8^yp#NL84OHMF^4Jt%a$ zJR@~sfhyv&rGCv5Obg{CiB1X2E58n*w%sc}|FKL5dvt7kji=?i`PK8Xkc3Id9{Y`ZY|5h( zFVS;oj~#(D+gZif_)kMkdH-gw_+i+J`yW@1Nb#`_Oz>6D#hUw`qA zMKcrQa}+zdbmsi4Xl?B@EOOQ2iDZ*-ls86{iP#qGDL5KG2xcTXH}&;6#< zG7^vQ-f9-;|NWs&vu6nk5nE+_ONQoWq?%MJz5+Gu6xlEUKKVs)`?+rO3U#j4(}R&c zh!druIIs0ABFU3?(l)0)xQnXy)O^hCZCnQH)qiMzfaxiY?$I6BjV&89Ki>$3seK@SV(DaG9%i?Y5yBV4{UE4&`vn+UXy6wviK ztmrmmw(jV}_m`j*Nt6HDqjOvGy=eICq4D2CqBh?x!QR?K0blxH1!!BUtYgo@RREicVW>&Z3RBYQ@&geX%EJ~T8r zoK*|Q;$%fmt*`03Z3)??Py#W)O!%Yp2n82;41cu2qxb;#BiRpQP4R~b`5+|AC(^#= zvlz-GK6|Ab2cT2+pR6+~7yWt%H7zx{)3LdsZKKj{HDw+;eO2F_jq#bWrf~c|sE;)D z`5A7Z>NTi2OTY`yS-T3)4`e7rAwzNMC9fZ+hSrya_9>}2?sbQ&j_^}nai2y?d&&8B zz#r8?r+&@VSl(1i1N93&fvs4-zXTa;qvR}H9t4MTAg`MW!z~`A|Qh3sp z%ySYN3(;(5t#A2ve}^?g!BRZ?&Hpc&90lqQP*K`dCo?8uXqOaUxhAT_$#lPTD#>ej zQ>EvVqK|Lo0rP^^Kh4teDNhCOEe}0DUQ-iL-vu(Yk=yQ${tx)YLeOqy^mKq;RA6txxby6{OO|QPHfVZ!{r4U;x43VzI)($brLQV0 z9BHaSawOd7PrdRBPhx|dAhEv9zsY>y;i4I!)#>lg}gOocuY zn-rpCc6`#yai+nCj-C?Z(ct=(p(Ub+kpc$Zs*r1(MUD8Ck5c$V+caQ22``WlykDPt2QkhEwvmBrGI z4I@d-Z1~|Zk<^9f(}^rx7EbLyErPR4tbG398+3P>(rEJ%LmHVw1X?&9(DOL0b)qRE zOgZNLB}gS2t*^XU7Y*l>{gavQV{Qc|6F+VU)VFA3GkQE8(nXi%@ZIb2h95duXz$wH zR7uNnkB|{dX?12C#4`+#AbStv!y;tw_d~y zPbZZ)lGG?%JQ3BxkVf(_qIo!*!o{!)3HLMeDxFU>WGNOH|18t^?VV;?ha}=X_{xQsDff-go*6C`hAE%B+F;+4a?Cf{za3L*HVD0zq(6@U&81=Z4FZTW<=i5Uyfie%}U^P74 zad{suz!gWpjzM%fftYx}w-nQZO{}6Gfjq4Dm@|xVD*M-vgVK9QtN7=>pO#OGE6*5W zZ!@@aM{YM(gE#~esgZl7Kg(jt2nYpIk{-ROm=Wcj#I2%yx{Bh4?Z=Py8jMFJ;f{h2 zi};OgAPe?u1LiQAGH*Xrv|5_yJ1q~g=)s6hXtw>bRy0#))-^JJ5RuLZnebE>M29d1V$v3{U;DM(4k zG2LPf$3JQyPFmwK6a3_F9{wV2yr)mU1@*7b`>dG2@-*{cjS9}yYC({>aGHn8Z_I3SAnGTALXB86KaV&tGc6on<%zVfa) zPXO5m*KGw+0+DO(vIg3ps5&pH^ANin#E`;RiDy}K*9KHZmWd=|y_8YGq^ja8 z{SOpgL2(5+d)TnzfPlT_Y(tR#M*z*t{o$SfMS80`NI))vW0KXY^Rd zwK-Gq;Iyg#zE=4J{RDObw~_y!>*gkht+f6hW%$8m3t{~IJcg|kVhaJE>LUZ8rr=DD z8zX*be+nawOLdod+ZoR!QOhHTQR-ENs_umjHUxgib3=V z{|d6bljH^u%$=zHOJIfLt~newk7;Y0*b{&yZ(uV)Aqj6!gDRb$u*GAjIZg``fnV`Z)Sd~Uz-&T&&O$4L}} zuk+p%V|pVa^2%aNWC7cspp7R|lo8AD?C}cSR%bgfZ)D`T$IRo0o#lQL*;b~|?Oq>o z-jBY^;^$9#l9x44i+$9tv*;|(pUpsucGxF`Mtmz!goG~6EpdX+lraMBeGn)jZd3%V zQY;5uVt6<1t$ow&tPyG!f^yc?VLR;ojhXAn%nE_$gjL-jC96}dEjg5>`XS8*+F>Xa z-YF2(qh|g-;O)7CdyfP>^>lEzTuF)!H1DDIJ~Pt0KjF2DdEuEK=BPf5mVEirdvSF;pKspBUAR z>Umv3z|BS6{nE5enMan|yaI_;t;?OxC%9=7%krJ>`OL`1*^pkg zzS>xV`!O=+MH+fqT>f#oG@8bP6Ux=TkMtKKu}pt)WFUAyuJai{VlIY)Tv2qDfCB)w z+w+rCk)|-uPhpe%0+S?pPlhUe1<9I(X1O^-EM|42u>-xfS%wYu)e)pFS+x+>9Y^}L z)xloG;mL+9p`>IGp@j7~deOR$$-k1+QomhnwCAMQQHNwnkuXACvEkCpc9E#Q=+_F#9ON-5W`TYDR7-M*5hj5K_ zn_=8DBAR@9mW?Voe{rcFl+XC3Tqo2f;ll-3a(OfTDRl%Wt`%RpgU+t;sXh(>(^tkE z{&)xR?ay{5is(@*(Uz7Yoz&8tMbX2e6a`<}U#rxE=>>tJ*|vi3`-H{Ito`;DE-TYIA8J97A|vFX8NRa zl5R(?e<@ak^nl_F{dl<lT)k zK3*daCj6bk1pWy#a78zIJ}I{zC~h1PcwOS#&}B|fO!T1TKW6_#Kt@F#hDbwdXtwxf z<_#s5k|Yi%u>a8rCpka-oH$?3%A4d2QKJSDRJDgH)}xf~Ql8*jjyB#{+0EK((aRFi zZAhBrS#tn#QhRm<-Xaz46LNvB;)SY+to;y888MA%DhlluhYPn{mP?J9_NIFx+WD-p zOR|MDa-YcKI4UjqCl+QzjJ^wTl3zK+U~?ed_8^Fr^!}Fah`xA;ll-S*C6aU6k<+jo zB=rP`5l0Y&rWXiiVBzX>gBKeq_wHS_88AY)!F%}j3u0mjp}lMi@k6BzOvGm zp%d2`UxbwH#XlS%ad93(O5S|DH__-Pyb6bYrwi@sZ616)hC3lvmtTfGSCf zQvLYxyE5tgERB^v=;2^LnNpA6b+)|>Lbn`IxBRI{W_fqhq5w@H`m_Q0b2*s^3|h%D zG?5rPYW7tY40S59@QY*-1Ta+;z1JENo5Xq6#=w@?7*tnyS`{yt(>EyXB{pH0)cgmm zPjN(4e!BN?S*!@RkTmwcsaCX)2A7coQr69m0)AR0w#S7R5)zl3I<&hHJCn8##1US6 zYa5-nm{LG%=>1)3XYy6q=-imlh=5ZEbZ{A?i_8H`Y{}9|jZZn7Ldwv@J|Pz&s9v&! zCZ%>oeobV&ct8^ym--;}%&2727km;W_1o@-)~BR-ekhN~Q4Ym_FNl9DwL?-LkOe+J zTny|;b1pP4S^KwW-^^dRP4O7JFyaY!+xx@dZQ^_gG-MPkdC3CL#Z5{Hv-k?`GpODZ zMdT2QEWL+gjWNA(znZbRVACzXCxTPvd@Tf)j&|Sa>ydpF-uT>0^kftS=~f+kOw9+a zNn>Dgm(TX}_0{VR=C)%?@%!(zu3_ATiE?i)rR-YSlY&I}?EUzC*e{v6eh>0|lHAQ% zkQ4)oZdT(_&`90}O#JBlLU!h-DBaE=(0lG4sH}P)bTcR;dnkSxv7-r18E(b}_ov_f z4H<CE|zKBP!;;F%cg_6|I-k^TJqn~RnagzE4#(p z2V(J2$R0=F;CJzNa^3ECap1BoVsdV!Ox!jn`=4+3i%!;i<5$OIR|gv#d12BpT@ zq1RO_0n9a(;%Jo=hHmKoo!r66h&=kX1YGG2zEG(F|o;$$> zq64)V@^v=VqgS*p3pdmHE*kQomuNL#E^k!dzi|WY`W18p&#phi z^jisGmMQ06+}^{SzH0c|Hm{NB)3?{(f+lnUlA11r8!VL@>q**aDNf1D`Qix$j3hNL z>%^B8J=O>Hs|C^wtVus-&Ee(Ir#__}DFVvDCi*!IzEKoG05swa1odseV4H#*XMm;1 zc4s`zaZE=xH7t1v#XI#{r_^^J)Nq<3m3C|WNsig;p~ohm==a5IW5;JiOdge`=+V7{ z&NAN*jHE(?-#}HmnpG*)n(JvBmlSMEP+nFP?wb!jtmIxDPyMr_J+3y;`x6gZ@C}%A z!k;Ed7Nx?Oeg+vZ+g14G|qU2hUPS+ACJdJx<79NZth0zjU6&JTrTh5-!$tkQIUj{Zq&eKm)BQAvW!-)CS1qn(*00H5gDNYx`I zF0N-CYGuN4A+8hU0zVuG`A@D;q~)b}eeMUZRNC>f7Z0vge}6nn1yv?V;e9)E$H&EsXGJ&5uj+r1{QZMUT&$=y~c`Eg!!6)4!nD znsKlyBV?rNqb{yf={sO6?hs|OEFPV1qB{!7+D*|22d8m4%Rcw4UyDK@>u=xn3MZRY zxcK$xPO$qJQ(CB48@H@jgr1hjQVyXg{=sP`q4Tl^%{JAr?IR`$(Tc_$op}-HTh-8g zp6aO(^jNyocucf#p$V}^g%z!PbjGRZv7%>iNaBoTqAQ-Xxki)YgBlFpV~WS<@NH%v zV-7QX4=PJQ9Br^z$m{IvJ~p#_3E}K~|0$;roqkEVDc^ofdTBQ@GOoM>@JBj_2lr5B zePj)en}g_uOO2Cpe@OvS7)6iYNk4EZ-HL&qFBzHIYolV z?(^Grxh}U$w&RO&-4QM7ZtlHFuw3!(N%E$YO|K^(`o031SnNXYQJE?%aZ-Rnz&H1C z0qe=h7;(1uI%0=ui6mvDCps<9m?1~pWL6Qk2A#ow7JAeNWnoBQ@6J3Y(+7wC*&u+tu$=p>fAc;?_soioCR@6kbjEl1NC14 z_y=S77yX=y9HJUU3RFaSLWN8GrR;biql*hFt7VRTmq#OK9B3R0wS2Jr6MV8P6ae<}d0P!}d z$rB!eV!u)Fvy_h!z$31vLQg=bxPDkWfX)!7D6Tfq!r0T`ewnk&j=RFP(%<2i@zu0=%E2e#__+WzEJERS5UImke_hWzxik=F zfRK6f+LijaaJFt!W*E5HW_>oHy}DH!W(>pzUdrbEig#G4TcF!|hgjIHzgKa96>Z0N9X2y&_pCU@anQb6$J->?7U0$figc0Eym$<<`oTdOGR(ENAk&T=FILH z`Och`vz?P8c_O=)Y@hShVb+h#;yhQRDsj>NK4*DLWISZK|muq(OS z$uE8?BhPQdGgjHkr`t{!ZEf80UAj%#BBjbwH7GAmAL9xAa3mEO9?sJAltk9w0o|RA z)sE!+U)~@avXR@=A7GN8M*E4I#1H zr~byxnTMC=K)F7=a(Q$A0fkiipvamu%_I>!LT zMg*03;+{}h0q<5%&bFNiv?Btj((05p`E~NZ!(rY97@j5g+Fq?c`__uvMU8Wd_ge-t zO%zO7LGReuac?l8{15Np)`Wv5hBG|_6#^0T2#%6sRf%r<8`4(?GZkeU4Is>gb=+Bx zFh)vD*o4_Y!#_ZW`iw*uOtz=r?FOIpgf5dx)tl{wR|Jme>Nwdc%gM_8K7a?6_ZJFC z@zY2{hw)bL{J+hM01*JV96v=DUy)346cp$Ke(!F5O(Mve2_DB>)g|~=>9$?_maSlc zK^kdj!G2Z!F4EL}!?>YUy}3HO>B|Mx3gY;}e)SxC-@2W=d0w78{5TW0qw8kDrB7-5 z(YRK9=hq)9?dr{^3V_SK8{t>S`=I0>`la=ukA&x2WtTxawbmsnS4#M(@_y%|ODod@ zV(U!C1LD#Ke?o$U6(6n7r(4)BemzWcDRx{+(xJcKaw-wIAR9Y&z*lfMuNz6%CyAUb z{;qu#YrfzGED3vD*S{qyUJhnW>6#BIXJg)Yl+)x$&rKR% zuHM0}Cj0sSEd@evqh!oB&biJH_gKqbFA90*7|pT;+y}1Ce`j%-VzLLyhNy)ved4{{ ze#ri_0lb8!vl-NuDiZY?V)gq zg`uL{4a}^nEnu8P{t$jw2w-{NRA!2NuE}VAw-^OdJ5-gFV7u#D((u0|L1n;w>^yt6 zZ1le{@K6ITEd#wx4Nc;SkNsC^&+;qM7LJ03@AC7a+J=K`AW8-D$Dj(2imwJ@XQ6$x zZ=4rlC7l8M!~M0Wn$TVYXEx%p0##Nl`^0w`3|<{_`qrDMs`CVdp-yQWTX%wHvr;64 zuhzK?Pq|fA?^!eU9*sD{JXxyug5O4FZ^zvp7IhpuXu}2GI}ZN-68}!>@53=rmviPmd2Q~l{HJ> zO%q$cGDtcJ4ju@QiW7h8tJ2=T-0DE0xBC#(q$j|jYjA}{WAviE>fpE>EMq-YHQ~4y z-l5?-534}ANU%p_CX~dhNEH+No0?+~CC*|e zcGDN%F)kQQpG!A9uAqKG$vR+jI$+0*kV`6LjkKv<|&Iew^a{7-i-t$O7NV zpL$4nwq44EewVSfeth-oD4|NBMF{z z>`Pwe{7x(4hQn@jLHK^J!D&rqYn6t!We2n2 z(vT1)29CPeh66lmMp6J-8+~pugfe z`~Cq(4|W&bX)APYL*3Q^p~ZHlbHu!g)^aAh)ca_hsYMX0zMA1clgdt+5?dKu8$J5q z{_3DZG|tKBWcIvTolT`$7@$mmD^cR?vxoU&H`77#r*oORrtn~VAK&dB`yj~=ZV}ia zQ=$~Rn{pqJMttj?J+^Ygf;QTbTomCCL&J^x@{eUr$i8VQgqF#`RrYg%&uRQl8I;jTQfwevl5cmC6)=7~`^rmd7y3@nLY%|g zWuVtrOmm;@PK?Wwf;n7TTXYB8;iuVDp+6PfIxDxXu<>l1=rJ`$!jpvlbH46J;K#(3 z@x4J(s=y<*e&u674|>S5G|J{!i=LXs)%)e=Xyxu+>o^N1_SWddZmQa^8juA$#}6WZ zqJp8jZUKqqYbw7PHR?ksZc4%?|rd!;>LarA?O6>C!6H8+lT;+VA@!1*sp8OF`I;oaFW!aQ*e^^?dC0#Lj~>?K6k z1m&vO-bY-hyN%!XQMntnU60$m@B`FHW~>ygu200|*6GDvpU=yHu!TF){zswd0IZ3AnR|}(||;~=>4v9qaRjN{#Q;N z@E#5Cd@!vF0|7j-;FvhMa)*N_u}kl@F&2M_slFJ(Kin1lWf)55i=O83PowkjUF*LP0P&l(-v7>@q;{?AGT(+-;q;iU6nQCVACeJJeS=4*pr7cIA!XL zMMp`lMFhBJpwJC*fOUuAejXDA?7`SHr()-i9=HJV+r75GUJ+wJF**{4?$MV>?B9B? z#%D;ruyN1^m3nn<7tkZ`dT1$SZi}c+9~{(2u%TGLmH4^vR#y6YBkwp4OA-?Hm#~Cp z+66g?GN@k}2y^m%IRC(Gr#Em_u+Qf?OT6lNsQU5>v_H$M2gL=J_3xeud2Qab`80y1 zjTqm)!y`{vyWTh}@JYpb`(TlTho|sV_H- z-tmwj_9AenwwJ@+a?-+X@kycF{8~4tqubhT*3#P9_RjD{=T4AX@q2d;F_U6+&^4t3 zt*+~o4Xyik?bQN`*;Q9dw7Bu5$Z^x%3dMV@(n&M>5#7KVSwT8{yVg(gs^=T8x6_tH zgd)J~PAw<1L87qM28^`1_xI8gEqLl(+!DogA((_=aR1S?&C$(!D2)%b{ag;-FJPro z%XI*}_#zB8RFzoFIJP%)QAJxu?CGaBKN^p!pm-c1ml|YpFDpP}-ZLqDka~NhdJ@g^ zYGbDV)e`Tx!>oU^->z8VX7OZX>sCkhH)%3}rO`xfOS%8u%(w(;K1HAy2FQmB^b37M z^4C(!j&d^x#}4hA)0%+4X6Lds>#ZD#R$AAewn%io@=fvrFA{P*9*W4;d&Q#ixt_Ut zt?QYQwMP-|22JPJOdi$Yl(7tF>OPcqniH@spHI-F6?mM_KAAa|;DM{Cj z=lbPYa@F4jEvTEdeeflMJOB@yBqP|M*lfIWVW!Dr|5qtYqsUzS_R$<~CJ~Xa7sZr7 z8n3wres<0?Mg_evcwxHvLay^3Th@qWPQ?FD8~j^bw40UdxebrWb(>G8y`qYb3IIof zBd{S?An!f?q@|ol_~fdKC0=v_)d#1$p;J2C)M(syIfq+G`7%LJ4nS#hL^sQHnqy?L zaEol0?&B#{GdWg!07`CpQf{+YTK{gEPJUs^{d=dj@b8TwSDi?xCM#$`Vj2hBF&v@_ zNOighvhS!Kmu7`dEFGBAwd&Jb^Anq8@X^)jPEEr0Glq9}uxdXfISMlRGBaWf{qPv) z%MO`5DTZ%;v08S8sgoR3O}|G$-Gh2mpaK2P+Mue%`$RSDEY9b!EuEwE{;sUNW_Ln06;H ztz9+e>jL+s3y>)A-_I{Otq*maM+pL3ZBPe~-8;Njypde}l~YGj__FJMHLDUh12Bx9DP1wa(~#S&vu?5SmrS5JHP5jochZQ zAk3e(C@bx;yFl!NgmrP!6bj9;IG2H}Y9;)AHA8+U_P4b)l!6cD!SpTX%c@7DkEJ}_ z^)dH#TnF~+n<5Qq0FuY(=Lf;>Muygf9c@YFNxsU_Xi>Q`y2k|F4mJks819IFdki4! zbF;C(h(|Nc=9BmOfj)T=kzw+U6|c(lx*i0%DjkiziPc- zV8zRFN6KY_F5VNZA)U4jZOb|N{DzmnRT=NP@!yFVo>=6UQE15Xd{d`(j|KQE8#%>E&S(6mTGFA`rvF7>|qx@0)66H zuBF4;HrnzA6+MUImg$iq)v4?U9q0_tvgf+*yV@_Ybc5g6M!(UmL`)a;E`fh*ISp#r zl@_*qbW4DZyJasJ;4-Po2nQRA`EALTQN)QJQci}6Rh#)vpe8p~g0F9z6D*0p2b#j# zisQ5{tjbAD43X5NK`)?W4f$!Vco$swoe}vUjdZr%84dTH@1Co#HiXPbtquCX_O(2I zZ!9W3VufLr7oX=^s_ec+UmonKD3x+D9|Tc={0=4bL2NK>uy~8H)Lia`qBiSPZQ=Moy#vG@Mec*rob&L|fIEn6i5P zacQ+K1#bqSdx79u(Vfwzz|#kNSlufK0zj{XJo`_iBIj*h?kI(4xqoWz*GDs~HzX7= zJQNLZc`B@(EJwPf@?%vn<w8VWN%xI{<4S zbVud1l3&xn5vm59E@#!je)ntXljd$su7kw77$%%QV$}-y*dou3je07BvPGkpzqI$g z)1!Ym0`05atRo8nhp*-o4;ql9cArF8k}O6Fo6HHzF$;YK9hn+`$WOxtIF3&7dyn>F zg!L44%mnS#bt;mQjwX7$-{S>YiLZ>IFYJ=dio31%O<{(P^V8Y zxK@+kmptQQhoaarHb|s0YEDCC!A;_E<4jm;rOITZCD$HU_xjLx}INK0X~a^%;8VfABB zJL3cy*A$j0SuNwpo+FRz78cAPvto@uiM0I~TvL#Sabz`mVAv;G@f7VBz*h3se50Nn z1HXuC*ys-}uCT35RPN8tj8nK$BvDsdT*l!I5|SmvXBO~g>xlCFkU=H6-)RMAhIaQY zuU6$PJ02$SGy+aQOZL@UttT6;d;9iS%eNT)h{yqI{4z%Nx0(l7GtRu79gi+_%IiY7 zPi0X#pSVE3=D8qZ-bsWrivz)v<{=srF`*u)zm^srIGKJ@z~W4(bYwz$x|NOw?c9By zcl%S`=PPKm(3V<0Tzcd%s5H^1q_3A~5Xx83M-$&dAY{Y-`9BwhS>^U_y^ z&b`4ib;|5;xjR0|`$K7L09Ikiz8~N#g-pcxQ{S0z?xx+oEpPlV_#jbjin6GNJmWHNPxn%TS10kuu?nJbCVno!G$?;9Mf(WB8ma$_%M0rNqop;CM0$@OhgG8IOT=*XR z)q%V^Q84bztXNWklq-@}inMTpy7xU2ClH6V~6s>WD+daCYa zUT;XVPYLO$tD>pthblvQ;)X}uWt_<9-k{ag%X5u|ae!eQqqY79vv^4|#|g?iVj@4o z=+a&9%z@er-*YCq3V~v}LnEhdlet$$U8~+-dP@8hSMItnd(NLde0G?z3YjoylO5T8 z94v!pM?Z%3#OfPSEPiPJ`Jpb!N`5Oy;W6yqa@t|gk*+@<(0@v46@N%`yp&N&bFn3G z+wrQ4AH)~5SUtF6z-`2=E0MD{R=>kSC085_Qn~iv8t>-HXGHLC5OLg|N}lwCx48}kyQ?|oNdn*UzMMwTtSbX_6=Ua?c7I3{~{xtpa zlF@Kuh?=9V?J?$*Fd~i!d-Er=YjHHlE5_tf{FwJ>BBAVQ2ycG;^zv%CgIMrtVXB=( zT~dne3>2jP+~-x<`YJU&*%{xzW3BS!fnB9PPIP&IM|9hnRw&T)S&i$QgOOSiY~aT+ zwc{1^m%@Em;vg~M!mMcyqg?*j<27HYf;|D3E4QKSvSI0G#B4Fyiuw2HqFLAdp(NcI zy`IYP>me?Hv(d7IRHpA_%et>3$^H2H5Y2|B%ib`Ev23GvFr}iEHUP;hhPPOH60rXy z$i3c~da{HaU-3)=4@re6q=*NeCj$ly-yb;SB%Oy5rzr>3mbyS&Y)Q~x4K$Qr*q)n` z=9boh^b`+mU*PF0fgl4wAE885ijX2P_UIam)9NRMqva2iX(x2-D?{^#b3}KJXo5<= zZ=WaAG~7-?ys^xEs~;fgothF~a0_LL>RY#$j>}`QH_jp0(1U-k>iANpN`{0d^@^SG z&x*}T3tFq3<)x|mD*|T6mmb`Ab@ylfl*44tw(u*z zGqc8)hs>w5-zHk3{C3sT4K1%@UP@0^P_^}KKXaO8%wldMh{o5KBK;*uOXR8TgQfzz zi|br9XwjrdGS*6RHymow)`1pPP6O}>Mim3~_m2Ay6?6rBm7recboa@uqQhY0=}Z?L zij5b&(K&0Y&AUZahTuH)hqgN}CFR`SmMk|N-(%m|1bTMm+s@eckswoI);#Kl);mw5zf>RaGKVx>r>iGQY#lKX-|yD2-`>*nQmY*IHr@%OkC*=e8&e-k z3wIf}ZHg3;BZJYf;wjG@)dYet52O2J8p`@F){rGMoyB`uop0teR2M6$xCDA$7!D|b zbYo0D4^^%4HZHJ`CQ5|4Qja8K>;tRrL6flE3)qC^RyWQYNsdbOQM+EA4)`Khmu#Zv zuo%5@QiYqK_CYgt`p3b(zBl~i|zq;6zJhmzL^?_!lKoY6TbnI z#f(tnY7Hzy){MQ|@A7$`@}ME>mGf?FFZU0wBAuo|ZJym|N=M%vP4N{7hw_nu#xii`I$U#mHP4z(~w zIj}hFce0GMK22y+(R^(^AMj81gZ&0JS1R;8ycgXN;G)QjJc=pGKHSlffeH>_hVONs zuyC;3{IKE|6{BK4&7-R8g~KMQkkP;^aNz94zItHJNFR)uX2JeasS%i2pEFjhNUOD^ zL#!IdFBV!n5>_}Tdy*~#SQAJbBC~Bp5Z4x~3~R3;9F9doCY`7N7dJUOW=)G?*htpBI-YZecbz2xud6zR0TY^acrv5RDsSzh8Y#AF_C#q7E_^hzL ze~c!v2%7|>h=cmfI|l9CTBX2DyFY5_5j~39Vd@aLMml!k#qV9}85jQU!zlRtwso*- z!d$m`;+zzu2@?kH5m;sl3E#q(I%}`1*w{thGZg`6yLg=9Fx?689bOe!T1=64A)KGDHs7q;lJSTbg* zU~Vn(2@fCptWMbSINwD?>Fh6kBY7VIw9Pz=E}V<9%8Wm%&7XyvCn3?m7lWr#6b}x& zRiSH=Z~-cFOdk}5H(X3(eKl>}HFM&zGOsF`N{(1K9i;LA-lu0zx*-y#f9RFz&1c%X z(7DNg`zd^xN2gfMVInhLQ0OY3cdq#|?a9tEPC;!(HMh4J73YyeW%HLuEXOR-d=yHTdB9;Y}X#3 z#PCnB14jp15K-6Qn$q0~8N?B=q1idGk@%e_^^vrP|KgD!YOP}+RZeEk(9*xyH6!Qr zQy?=4265ZwrG{7C@uMuSg*Tf~%x3d(g*JCFo!%}>q7w2l`ogf2hC64j5DVVSCk;o?rOWjl{prl559ka5*H8cLKYe?Vw8ge* zhBw}JXC2i)p<@h|<%+}{fr7#*_;*G?A|R5#{T(ab?>Lj?u(capS*T9Scy2P>Hb4@` zQfAyYb=euEowDHJI+%^R>K};Kr~XH+#@PfV1J2bEzh|I*VM=#h!1~Ke2zI&tn|RzH zK^o`Zy75&zV_91XVXkLZr|qxR;@CUgYQ0|&M@`sQ)5p-{4gc0Soh?dLmRqwXj9Lgj zHBb4sl|{Y>YQufew8qv8jnwg&jG<-V=HfVxQC#JAgTcOn^MW(r^_EnQFa@s_E(+Rx zjFI;RaH>fU6sp)9RGwy5Ry|b$N8(24yuaGl5YN(;sLr~>$Ow1A%S$2-MEB&KBafOS znk$qnV)@ryaN8kH4P+a|N#khZv z4XDR+@y>0Biigv&09_^(ijHQGB8?WqEaRRgWu^1P##uJhZ??IY=bPBimOy2Ne#+SZ zf_j@;xN3~WaW&OwHV5zSh9KjRwGu97d}zVnsmb9Y z4OYK9<)o0}Ay3-B^XXKw?A>E8RKF3*vv#VMCIk*^Bg~khxubi0U;#DidSO^1tQJ3a zd2*Ek6e)UtnB82^V#WzNrnoz^fe8fW7 zH!(wSTrZky3#Jv{;8V!VA#P}81mnHZc;JSpuv&pPIh(h?-gXaTk*&vnn(Y7;cOaPi zoE(E2%%u}$pWjXg@7)ATGqFvse}7R6{Rk*boeBDbM`-4@7Dl_`DtMq#oQOqZ~sm2vyKitcB27E`f5AXcb^*ipjOlLNIClMnIv%0GV9 z-}9;^3VuUrjWgSUXvS#VDLcovpR7Rz<4O4l6jZRc&l13lJ!&aOMb#nPWguWTGDTC2 zE`|My^lXwynlRDUv|h}@c*n26zi}&(%Hv>VZ-JXY-}i+_8RIEs#Ds#~iAd2d?b%;g zDow7U;vUI*rRi1zg_Q5___6vTeA3R&;FbDon^9br0K@A$n)?il>6fBB zf9RBJk*h}oAbFIP-+8{$YYj3e&-DbVxzfp$RH0y2iegU6^Q5t{t50j!jqvrUiMZd6 zLU!s8nie)}zOl_t($oOe-zEp~n*D9mJ{wn!JMS?)!$^L1I1h`j40Is@tkAq%Hdxok zZwHs@b4KzZUHjmQkbZqj5y&f1{@@{K-gNCnc0VkHOxx%K3j3Q$UlxxM!u70|T;`l=Na zeu5&UU_$0$K@xRR%_kUBTGb;Ii)nDLs(|Rva(1wI6a-miD6Y=$n9jQVT1DSf>@7Lq~_=2dN`rCJMNNE(? zNrL7o1p%JU(4QUqJSZday>slT)Uhvtxv)pQPnSmo`^vD!k3g*lhRl$3G9>}Bpa`3n zBal(LKg@F>j|>GaR7!gN)B6v|qlC;Kc^guEY_0DwVT-J#`lX$zZquc1Iwuh-Jdzl2 znkVUfLpi?Wr?Hz1Mrf(z_zG0R!3Na`9vm>_R5Qu-NVOsEXeT^nuQ` z9A$+E?v1d4TVw>TJw<(5r}quDWJL4@9_0YNd zw+jB&+SnoDl91tS$lI&=Sk2NY5d@N`KqjsxjCTal%U~5AaSSnfxh%0PA?L@{9!@Y2 zv~YZxaj%JiMvc^p)x6+^`>cYm*3-XbD_#+c@**b63v-}3<=(aoY0@xEgO$xI@)1k) zV4llqyZuIz7Ny*T0$>bl+WghfR;TI!QZ%cJGNC|NGrQSpEZ`^tg651C7|tumh<;LV zf*Mv{(`gGnk;_mFx`;yqt!z+q#?rQ=ILtxv>9RtRD`Y-NRYy!zs6C?*K7*cF}7!RaiR zAtV$Hi0P+WY)i0r&$c_eBAxa+zQ5;o|yqf zL_nFGt+s;q_DC-L^oegFgP&w4yFX4d1n(BjA?F>1yG;r;i)`S2QvU^Izq2e+f`$sT4KU6^hd_#u>zEn8uMSu{)PrJdhtcP;{hj3LY)S}*#haty<^WfC7KA1QN z@nOYGssUnDswnq{d9cN5mV8skE!JDho5pNMBn(vXgQGg-$mYU+YoZPb)WdMT6 zV2bV1DSZLLOF2vjvu5`dWH5b2Ak03ZAbT`S2(6Bx{?Q-S!P06|CHJx{as+W8-rh|Q z1E1>9+&=-LAm%e;6F~T4gaqL9YSfw%Dr7q%+ZMCUj$?$7OBQl&JL`+PoElU{K(zzbPKsr8vDpWM z4DE|QhXh!uggbt=UxZ7l77r}2hLHA_#s1mG=qZBFO8@pLBme*~32<&3FVi&3gLo)% z&Pd1_ac2KCq*HA}Vrx*McxDzdT9|CF(>?y2D;g7oh241Py$8F0cp`eTEKF&Gr`IfIcg3xHs% z3`pAf!*OtrI@7J=0+7ZC%dABPIK~kdixK{3B_)Z+&OSuJzLSt@vnvSdXA8q5)tY`* zIH~4_sB{Y|)XazX1Sq=@#`UlGB?MB0@IS~v=>0xbK>-Of5)d5aRiQ|SLQB*A_w!2| zM)o9PA0!Nzz;1o zqQrgEDJOOj#NP2w?Cl`U{C8cP=qHbZJB5x!tS2qA&HCi+fB zgj6JcX&+1F6F-{#&$5X$!0fO_HqLUPVT!DE?aPqd=%2apmK`Jqry+CO13|oLlnAd* ztZCtW^yyzURR4F;%~6j&30l1gWVi?TH_v@U{Fk+vR3T0uk?4VP_yOx6BSeJxc{ZAW zlQVJq5A(ZCKeRIcKVQI!9!IHQnijcqJQDg}o)@5~GVA}mK}^b&>}&FXC|@47 zaE}Dh6?O=8zCdyqpHf2L7YsOsg~x31^8apum8@;=KlOxsEFvy~k=?>n=%0qPmM(5< z)55@(Nd6B)f~U#nI51&HAq~@lM?~pNNO1pwO}_F!yM&RE`4|`%509NO7A#hc2(i4a zA0^3l&VD%`eg9R1nd=fM5k4xEa&F;m$}avR?4>)}Md*c^p;!EvtHTfOfoYY`#={MP zo^M3HBSQzUhpI#3mFYJd%>i+fbV@If8y}GMH>VsW+N3DSKeH!8>8R#}fq(wb>hPSG z#Z9^3*~TzKpi4CgT)fflazVXxzbX;`Yhlq4-j6jQYK}-0?`wF!EgI54!tW)T1a+i0 zeMA|A|IcAX6|4U4TrI-ikdUJT3@!2^SSlo z)h-my=g&Kr-o7s#R^tQ;CDn+tHC<1U5Ehy6B;MavS69FBqzzL^)IcmCTmdl1-~)H^ z8UU9B23t^-SIFNC)8y4UZ;5pnv6SX`5|G4|4o;?|q-bbrLdWOkPRv*7>o$DFoD{rP z*lZs)lp*nWf8_>}Jrt`FIJXIi5_6E9pL3)OUiM=PK7x~=W@l&PGc%FCXli~+q`H0Z z=?y_N2ztGhN$l?jVYi+(t)z3=;W~A{azlMK_Gk0$T;CtJbfrnu1_yo$>G`3FSBJj& zNR-zDSqGYdQnPS;z3+3qci96^N(LESi;UcUz2Df6;aMl6p!o2OrTZlja(?2#Jt2d^ z<|jx(h=)Q@Z2HA569xr9M@QEl#s>M8Nu1r;Pfy2l?6{ONVE&v0P~0wz6E?d754Rg7 zrKaxPcUrQ4PLzg(PE*#e$sgNY#;sWoNw)vpi-Gr6sDaC$Pf!NVJ+M%nfpFftDcZb{ zv~YNG@|^4DQhp2OpVCG1HSyqXz_pVm)62^vhF)uuc6JPLf>#LV-S5IsX-2+u{{+#R zHEy-@9i9Zr7cIbQBKz=ukMdXS@6k8vRxUaZ!_mFxHaw~%R(v4A+HJvK+Phzb3Gn{_J9kM0 literal 30204 zcmV*yKs~>SP)zLir*Rsm@n6AdGC7UY zIF0`bPLs)LoW^PVS8z(nq^hcl_uhMt0|ySA%5I#-Y5cFp!Gi~xH*X%RR;@Z|_c&!_ z;&3>)=%R}#Dk|cttFEHDy82Xg%dM!P2EmnLmF%FTC)AM?z_DZ|B)(pQXOO{uK1?G)9XlQ>Jj$ zRac$Vdz>;dsjaOgF)F(}6 z1wA~C-wG6`^5&8xoyyyv#wajr)-0ASTgH?rQ#n z|J}vY_>JI{kx6`fJe8G|09001_G>Ap@!Q0SQ5V6%!ES&1`uhI=*`3q)4dRrM$ysNe zMO|GT_uqd%D^{#v&z?P}sw<~40%BrfSiE>KlO|1K`t<1lJoVI5{nr;RT!==aVd>JP zr>cvmF#`IGuZ@k3tXj3|q}WeR8JQ>)3N~-v%*vH3`O}~NL_$Kssp`sUjDSZTc?69{ zRUO_;29Luc-hKDoeqWKLOP8KD#GS@z{2#;rSTk9(W(|)%`Y7YakH_hBvTWHhrcIkh zXlN*1U0rP7zMX`G1h#J7N=8P;$?ErMoW^PV#_&J3nSB5K_sp9&kDG3~iHL{@3riR4CM85z2dnQhtIBFvKaN+I4T>Wou9Sg55{?FHW(vV-p;4PUlRWv<@qk~m&2kotuC8x#GF3E6JmCSMTpKflJm~4h*_0 zIOGE(b19B1xI)4%DL{Zgh8+b$l5omoSe$X<^ z$&fkZU{?1I&nZJ@^*KHa35C@M>?&{x6vB?eX+!RGp>#NV>kAihVLx7f;CGpDA`?zf zIY8m+eQrJuWHLLFytfVt7sz`@EaU>GQeacau_`27vfi;fR05}5!lsbnl*w>OAcPFN z-3OObj?<~Y?o#2DdY_NWslX+Af0tWSsO$<Cwn>N)mRb5}h#=r(K1z$CuCM zJ;=$z|5!6Q@4WMP{q@&5^UO2J%F1Ht(xu#b>#c-^h0)d3g~ejwzWeUOX0tJ8&YWQl zt4}O`hlIw$|7ZXsb-sVfk5F+$+LAmmgy{bv1~(uLqKCm8)9+U!5bgqQ6$f7TzP}}o zzv%$TToQde(8mqZpiW4DOGpSIA&`*wH7J3M<3u9^1@_(sWCFPhdgr`A?v#*R5)u+J zrwkCdWFRCvE*Ur_0g?l!1WqK7>^K3Z3}h-RGSN#!3Smc%K<09xz|l{NE|dtNKnNuc zr3{1$mr&CC3aM~NJ~(__NWy_kW2?v>}&VggRY>S!9G*jM+2qIo zr_6~>>B6C~A(J!mut;mHPn@WMK#|L3m z(dZjWy*7kabpU>LJ6fw1pAH4O%6N>;5%_C$sO1iBY$+%6_6a0kdcF5~{vY6fteKq1 zJwgb6`spW@FJDezU?6Y2@dlrN{yE{{;e7x7_w3uZ&ud#+g@e8RXOJ0iHqYLkfz#@w zW^FU&OX^9yGMbcI;t?(v-T8U6@7+b`!Tkh{pGd;3chTzOAW-I_!%#+>p@_PUTm*uI zuye?axdypHLz`7kO;0-&JuQ?QTPQX(67LsAiYksP6LJWZ2h!SUp{d{PkV1wQ8U7AXS)C|MF6g#nZZKi0H(~3vA%wso$#6OZ@*ov{@&5S7`{NhkgKwA; zrP_)lvm(hXAoN%}8ZcIsV``|uqHD!z?m^%{lB-Z@g768BKphc>CLtM3ToTIQ5PU*H zFo-r1YyD@Q=R~|G2UEgoa}gCXP0)puBx25)vt)XAr4PC(=I|x!#U7ID{{&cXLEn$Fz_H zZVpRkd|(_hN#gsJwX8YT#I2XdG5h=|e0<~-y^@b9wiVyVV6X4ve~SOHW^(JTw{pc5 zSM)D)YHDiGYPA4db=6f|e);7*_Sj?n%N~=-ccM@z`>z`;%~ZAhN^#Q~VuPnJe#}j1 zeWM2Cg`ia5%;ts?4w_2wlPS44GKWheb5JXM2GxVwW)r!`I@o)(lcqK^XHN=aW_Bp4 zV>GA&2kz-KJJFXJ=`PmOd9Vv(wTYmt0K(4=#+g<_Q_o)7y7O_h_7E~QlK^7z?eaxx zGSX94fw8I-O+qqZ)6XSj;u-jcN4h;Dr?6Akc^IqHK#QS-8eIWh<_0YCKpI_sNIqj2 z6Oc)&HkM>f46(i;Xq9TLPCLi+4b=2>&|vPS#L!HKt%oH4a3%+hA;cO%bCn>!N>4?D zk+47&<6<-<#rhNEucD*FhU9|ES|cSD28<>rX-Qh9<^(e(Czt>yv>xc9^@Q+u!X{8Rc ziI(jfXxX_Lm(@&g)}=(AI|=`BDQ=$NdP_40>i$h#*CDc_uOKD-JOT#IVTZ%c(qkK0 zX=~!9F;lrBItRHlu#fWn%32N_?c&uZv++6J*QT|N=#F&ZOjt|!w3$SvO!xXu{wMga z7+?X)%F38AV+OUgwVZ$c`TYjj;WU%Czj=rJU_BpQ`W%nEcZ9ijWDpW85!8Rhf(?$(Mwy5(yk5Lmd+fGWQKM;dElCD5m4kUOEo%1DBJCvo0j! zoQv@dkHjUM>~GCwkG_=you!1T0+<$(OrlDL!q!Ng?jQzB3yGmKNC}%sY)}q;v$)&d zOXPt{H8#78J%#P8FR3KoSdSE?M^f9EtWDsGgbXHyjB%Tk^(|&%!u|TYYqL7py|$6v zyW1#eFk%D7$N6*iSrMFfc?^o<&qBDs+-OD}=jS%YuDqi(tviVBa0|kj2#!z!v$aHB zorME@`1P*C0K3z|uBt`Ub{%5sm|IAXybzh>-k!l>;l=!wq}$~De)3hQ z6)LyuKdh?b;L%QAeliCiRd1h7wLKKQQozJ_r*QP!7m1(!J3?bm&c?|9lb^|&HEXD; zso~aJZ|!$f+P7~X&CShRbImnWRaLQK#flMaCLetD9=jvD_-^(*atk{-P~6Fb*CqE~ z?9!grbpffj+8FdMnQL}K#G?djD*s!CMqZK+Pr3Q1}#Biph z2H~TUV{lm5(^1LJ_Hs%LEzAs0VRlpw3I3t|WMZu8p>9(PO*`8O7^flX%n%}{hhmmN z?vW1m9PXsD#>hF-!nowzNXEwa_mfFaO%=@>*U`3fJKEHA!p=O0&}lPJXtX%3PIL!5 zY1`gP*O4xQCIk~XD;RBxKUS5M(&qJ)HE$%qFP5Z;3mF@FCO%5-kg+>@40IgbN5k&b z7#iy-85c`_`WU7s&*0+7Y@&RFygqBA$-#>44Xi#`O=fsFb1sY}J2?=6gUZf>9Bp1l zqoIV%hzpn=cRha2AgaEvrExeA#>)P8Scn zb%@WNn(9`^Zg8;u&xbkZ?FsmX4lr@Oxt$olP>jVzl)V2oX@7l^fYfx4_tBdh`L*Ot z!nA2jOSqS&4yzlP^ug$`u&8P)rG_RRO`T1OCd%!)t-IIKuL@LRFgf^feF^J#H;|LD zk83YVAvJb3KFR=(edsQzrgf*2rp>LyULH;S4RI*7LnlpvzLuZrcF<($_t5V=~0OePq6c{FhshNDn>CP9`G4HU3ikeZqzvuifQjP19jk*Ka0K9>ry? zAm#4DHzJlXm)%G}QsxjDLyw)GH#M+sM-vyF8Nu1-XvmI?^w2t(?RpM3tR}Z=DM5v4 zObxn$w5xMaY87-FdgsH5WMZf)qwKSVXp_=OyycG`$8N4S;T!G4j_S{7HI#C3+7kr% z#dz#Pk-mwyN`B$-v`fjym|9ZO3S0bp+mWp2lQ}r@`cD`68V+gM1}Yg8aS}jko(B+3-h7$ zyBgvyi0CiZJ4|SmYJ9^Y2%a#B!q?^#Jb4<*fT1t5k5Wr=={`^G@OHRu?m&aA5F3rNj_9dbO;w~n-248 z(Q*QnYHk}dl{rZl5t}lJnDehf>93_`#_09+1{w1D^nxDU>Va?B^ z-1As3`KweO+ckIT*--ZiDuo}HXU@e>={Xdk-qgkX!qvQ(eKpCN2oM5?$$(NbFwYl! zSj5d&jv+b0owN*PdMwQr!e{orH}yMK5OvNhl=71klzB=ulg~bXpY?IAEV=9@0CwcH zQC4TUbt6J=4d&bRTa&WSnoWO<)}9|kz4gC-KILGCEi1{s#mjAY6G!$ zSgHG=hR(h1jJ++9$V;O84VwEp%lW8$BN+j)+?_C!2;V@u%=HBM#UP#F)ZJBhkjkGI zVfT;Z-^Q6l#zk|_4dVz6?0v~y2pY{g;{4o+)@gN8zNiXazMib7($FOM17Nauv$^79 zI!tw(8~Yc+)Z+-$DzSETQ1p*i&?F=??$&#eDZDRb^_FJVZhM`q>^B!>5Y3t$1 zyq|e8?K08>j%&#-mz!p?bx$ih_qX%>UxsR6Ri9U&P1O>1Ztpv@=hZ(k;mNmAdm*m> z7~&M-(?$Of*ROwGI9*O5gb)_1L)`!70grVpT{f}tj$C2wwh1AGs57+*r*OH|={T@Y z9DMWvVejejdQYcQh$EGI#E-dm3ca~;#6Iq7FA=xx`%vWRYD7`-DzSR&{i3<+kjJ_X zr%QajvRd3ecb_P!*1KIZ)z|gj{5V{~B~~})iog8$q1bTec2V??GU2c|hP3GtPN6f^ zxs~wOk3mrE2hZ4lq&-Yxdk zd@CGIdvBdGgV^@aA(8uHfiTn!Jh!@LQ-60W46Wj+yd~m^yq`r~Pv@Y1)QKZ=pAbi1 zcuE+nDthnZ6r!ijG|*0CkC^vvfq3ZoL!zw8AcPRY>9mW(HQ$P#_TMc^n>PrTtM~tC z)9ZvyuX7uNrOhG=<{cG>|9U{wl{AazkNqefJ@SpH>1lJT*J-K~F5wg>!eVuZe|%da zUVY+YvEz}OMdS80LbwLzjZ?TpovG7Jrn^KhcHX{ERD4q{Tn^WXO$s4|sO~x>KH7Vy z*j2GybQR`_eSf`8v>({zvF>2#OpW5}+&e`@!G0l}p4%2eh#tFDJaF(|BDbT;{esge zOwGOXL8rHgJ08y!UB};v6LH|aef@LF<#dQW3+@#?Hl4@1{|%gSGWp$_7lh7c>@R!n z(_^Bp#WNXHt!@;@J}DPMcqS7eglJf|N))^{-|Ib{P9f@AOrlA5RQ#BGr|32{jM&dA zeWSQ>$4lbl-3x@N(>P-Nf{JeOhiCSQwL1pLYA~5}+j_(kd5grn{BK34)gYX9rzrcP zQtZ5ApU@R_yS3$sOoR}^?r@2JzyG~he%noA?~f}z_G2)a^vgLdVrRu?;_JP4iEUr3 z5}WSM6>W#RhR6;klRoTfFBiA%`$BwHwoMd&^QqW(@9m;}uloyhBAN8zP=1%V`^j9f z;FI0ryPfxneHEVxyVITgJdsHswro5kmfo^jYq%iV^gibk$)tWwquB9>q5lU@XxHh**IOPGi@v==bT?FtSij3u zFTT#bL)7bz3R7Lruw?RL!78z;F>mOFL1gmr;&QQOi{}qwPi>Dl_WXdnP2VNASG&P#oQ z4f)SvvUYf_XLs~aR{k1~L`9J&4`*rHp<&wv$Q&2IhtE!8`++vz{<;K*W55x;%+SDJ zj(ozUk&}5L>sEqQ8e|F?$#*4^^K1r%Z+2=6pA0MNx+)Kg2 zdDyyjBlb-p^C820BO%|-p(b_>vZwYDG~Tnl;6$7in#98388mHN%ls%KfeT(H^o+Ac zC7YAs$DM!LM~~|WYks;Nf5%NIWW(E10Cct6SQ5>Ut`Q~|+cJv5KVGcV@Ig1=y>DeJHAvb!jFtAv(c+c@LHNd#tk zHlzFvUNsXTgm9S6V*j7-5r)e0A=mmelRh+b9TLk9-Y;y9q1uqe)-Bc@ zxKGryZtLw^XOEbB=m+u7icO;$pUdSE->#_feileK%A#9goCiu_9VFK6SXRbubMzZ1H`fv?etnu*Kd5=CEEi7of+ z7rF|AaJgJ!f6e#e+x_>57DK69dnalpE~iseefy<2@Ynl#7mI+xxy)-RM3a zy7P4+_s-p-Y2AQU*{7NGm>pvNCk5igPYOhjdFZmS4~CjDkvsQJ(YRxcsA_E$xqm$@ zD!;7}UZ&I#RDAo1*t1~1=xFcNR0~_TiO&z*Crmc|kh)e!kNEE3{i1fz^4ObZQf6oo z_w4^hm>r&%E`6HGhhLY8_1l`f_V3Wc`-Rfp z_EDZ_?l60laXDOK^WFPJWqp%ZG7&-ueR-)k_{jajX|=kQpGYP`2vOR+L9EStN;q8u zlh10?i**O??83a8yEI!$$=uEi+s{`-EhbaVY^9Py9R{P>L1VjxwD_S;N|H=MsZmhBv6V4rMIdW z(=eJ`X#JHKE6d1#@mXT8yq2*y-RibWp^(WrGjtpRc%$SOvI3$BQE6}(9LO}Vv-Dls zjit=Wc@^ymJ7;pa#PkUvXf;Y+{-B8Taauw&U~@TmqjCc}yNUVPHxX1;x(5INAOJ~3 zK~(B9bPfQ>lrq9+M$nzFqyC3lTH>6%_GKN9{2`SKrbT#^IT5BN2enJXP^Xt8vzHJu zVaA9_&MOyxj7`@?#>3BIbM%(=Z4HO}Cm@sLY_I-;?8u9eB#BL>3klOCF){X9w{{$Q zJ4&sR!CPFPSM4M{Ae!^SQ@nRT2)_NHkP9vtN77i&%a+zHE%?Q$2}mEI_K~xGZ!_ml@;oWD32KTyuOjxGaFkvgsQ?1fGN@ks z1=`f~{_SEGLyD`YJd#Je~DTc{E#eWCyyt z`}aWz!J8#ZP{8u&8lDS6E!Blz!64)VUdiP4Mmws*tHjkGzmJV>!F3z93RP9ovDuRDS*u z8Gn18kV#X$$|h;T$q9_*mEyIi6+T2*B{o(+OL#yk=cYc2T;_eOW1`h$r3A3xqhfsh zU3}5Gn#{07{+x0dN||@x7U(_QAY^JNMHSsNe_qe;pPWEOs`oJthN0ZZvH3-0&&|N+ z!l^imkJ5f{2cZ+s^7@QxSA37Tv5w5YK8GY%4DmAoh|;E@voulCzLRcK9X+-VW+o2} z`#+IP%8jioZ#cpeX_t8PzV1Uur-kKf%DDHg@!tFSLp9Nt$D#}xc<#F5!$=xG{9_YN z8kzhC46t$xcC%aA*eHK$Tg^ivl=~pbBqFZ~qiRLdux(3nIVlf6M(Ky|AY86t>jIpc z`Zxs*%jqyykdrW6o)u6?GUjGp%VAwLYnl#wtn*&^N>p+skEC3KBuOO4Ynb;yCNF(j zL~+&d21J1RHZ#vIDJA5YP=b#bX;?e_cn4$bb$@`!nKXa(B9os8At=T37v3EbXN^Y<8YcJOP+Qbr_iR_1BXn&;$bG`S#(mr$-8#wlI5#wLX>~|VWx$O}o zg^Kcj{llY7)8^Ia3iHVL%UmS+@WJ-yCEP))v5Z5FE4d`?i4ptb5`uS1H}Pod#Y19; zPQ;3hja+tKBr=)j=PHE2TxUW(*59q%ZZx>L^Pdcyaxzi5B)T0#CqP1!nyM!66C?D3 zK$>^7_iGE@piN34Ff)gSl}kqKi_Tbqzg$JPU5ClqIbwZMqEKAl^6HW=qMeP9Dv?fz`3sNxU?Dk z(5Ay7Mka88LVW{o*DvFx#}a5LQSs%ERgA`!^1HRQY~I(x8;?yO@|-}%J($9=XO3WQ zG<%ilF4B|#S`iap&PH>>3zvG^UvXM3R4rNLR^GN}8_nC+ll9oV5s3tPcW#{aZiGxh zv$1@{`b+8$F+MPk6wUCHTXGU7cF{%{vf8`ooL^x-PBz(pdxheUUZ*SHvj)}w z3~V;?U->(7wT20|Jja}zH`!kPA?0m5MtlxcZNIX&{3EW+d5srhuSG3a@k0Je>@Mf9 za)-LAc%yhFFX!CMgz#8i`AZVj4JJPLx@=fN`FvRw`Q_a#`0IGozP-zgkm(_$J(WrR zb9q=htV3k|bR6?G@xcBQFnJ z!ok|Y`JA8p5OXq~XKl&bI9=Z7QL{zQ`sRG@OgMAIwm0r*=KL99D29o#(jD$3c+$|Y z2}h3}q$DgQJ^pV#($7;SI zKKy)w{KqjSTnJ zOKJ0Zj?{k76%!Vs^^HQ3BpymWpEPYG&*m@3>a=^5+tpse#}%7+WBl(KtL_cxQz~W5 zdn6N+*~wd{6Jsf+vhzawu z@E<4qG~>t!pUKXe z#a`=wRI!bF#+{3ge0Zb05P}sO8<}%y?6CE^4s{Vcd1yF^y~lvue`Md8{{*L;OjIrz zMu%lcd32Z`EuEHOi9)4S5|pB)b-#BFrVpXh&%|Z7qstrWbk^NdLsRE|vf^*)FPoBZ zCr#aXG{S3%ZK!*pkW!M&A)KPgtzfixf?%kPdJkq zA&ESFY#An}yQAUuwjvhS?BLA_w-Vv&{-R2f#FO{+2GV`JxWY|CS+t~*+9ngv- z0{5Y|>ktjyN12guM}OIjF}G0Ov4{4ap=W6y>Z-x%axo)x+=y-G=5>*ps70&sPCCLR zu(nw6jUK9XIZQ?rej|_lKfx&{69AW&wRLiwhVuHM%K*1JH^fo#bN#69lYGxZRDJbN zgmbX!Eyyi zAjZ#qgWBca)4FFH08WdAWAmON^1@4qyZSnhay|-w=Hx7(ynP3G4J-P~3mTSF-M))i z;}@V(c>da7A2*qcBQtpB*fK1~HI%Kb1^iIEn>WVaN|4gCoJ=P1>_h2v>Fs>+V=rlb zwxp7#b_>r9LnM6&J1d;1%VQ{b;V6ga9wYOKG_)zBQ}cz>N#)lc5qIg0)NTAXF2~5r zU9-KLt!4jUR{C6IgXWQ3reId;qpT}?hryYBox;VZRXceo`TSA!Y00{JZkRoM9ui}@ z0gX39yKtTKv%P;4Wa15zQ-e8JI{Z>6V4Ocf2=vw7U77pf7ZXd!^t0&RIPit7Y~Fy@ zFNUyyq1mSb{GypO_Evr^d2__J7gcRwVo(x)8g~(?+Gswm*f^q0AI7H!5gqMIMTL>` zCWMblMv_b9JePr~&dhzecUv3@UajQ_pY zu>U^DWD-w3kWN{ZfoB)yQ{Q4{?mZdEM;K!t!e>UHFYiGc>qGGP(MQTweDhInE|MGn zL{QpH`kU0vUU5Rf=# zc+H}BiF5LDypDw51ewSsna3~rl(Zm@l#Lv3l5~9>l}m?@XX!)Sb+=&Hy#;%>4u{i9 zVcic*9`}IP`YDl@pi*ckYFzEL&QVwe~N{SsznFF-Mhn#GH_3cGp2`&BwTwphN@z^OOJY$Ioh}yUqt}vgB%=B z#6?N>v#0KR>`p5>n~B{W72GstbVqeAS=Yb~myf*E(jDpaXfF{?2hs>WJO2&*Cdec} zq4v0=6B_74r{0FsHN1f}a(W0I1>HDI!|xnGuQ4|G%A2VA_Df0|mNG7C7AgamY23!^(SQD|kBn#(p(Yk|h3~bv-}V?;}q)bgeuX1yy?1>}lrl-=%To z%xK1^2J!kAC8OFJZ9CLO$%KnN;7|0l7RUXQ7%n$}%gMkfbU zx9=c8sX?yLP~JZBE>9mex8_r8YUaAwseD?wW7KCAZOv_E$Dwwb(* z1mwS7Oz^}YVy}oM?TJj9)-<3m>>gH3S8#;-bxX;7 zrIEab~mhnaPF0P86Frv{Fz+`q(Sk}$73Ep$`00^hR-f2S> zI+Wb)J?_brP8NO>WTI87(K|f*2BeMAP+B{@gO?-8M`!hD(dayB(T*VCRmvVTnJ$M2yE z4kqyrcairXSaKlncT1O_L!(G*USG${6vTx?@ z#Ip%fs(G($Qo)zhY?JfG}}WP-NIMy;kt24oSb$>CV*5B+8PI!b7_>A50$ znn#&&{t-NobP4kdm*O~~V?w~Ii;KB!)>y_w3`|~0g2(SjqomrvnqNJ8yjr_#f8)`kAI02Y6X*C$)@bnN_6?&zrZ#B{uW`kGw#lN0Z;{nkn{9wRDAs*#%k{lR`p#6 zI8yr^mu0_#N})!jP;*W8tK`=Hi<+(jUggS+&3st#3$N$=j$l>qn*ZLU^RT%b{HwSzFgwsjkyYor9{XzW4cz(&FXI_b|tBua11ARW+u!NYvETRL3W;yc&rQDZh#&Usn z=@ZD#fbL+vmpLU*`}poyOCCH5XYO3lf2ScBgPnP`=23?6(<04Ameb9e+J zEPzt0Ks#1V$1pEa9~@3I_3AtV!;|Pd;C(`$h{@4+Qlr~NyP*`T)6OTAYk4$f&ag6P zhNdtjIFWz0?R6_(yuO;)Fh4GuGSur)A(wdZ!7Ns8Yb5UkUmKT0kUPJS)ZdLEDAj#} zsDe~Xd?|~9w@NTn4Aq8Owr-+3|0pRBJ?a)<<`WuD&eN|_{P7#~)DK;jwHb=pUG@R9 zau&F4|Eg6%%*k26wz3asF%%DxtLbUu&Ei$OnsW;IWj8InrIvrk0_exfOLr z-ukM9mmbb0D8N0gCv}{b-(59^`R^6taJsWXXP%Dw)s0MeF$>9q{V4LHD12l5s9x-@ z_0=q0Oz?zhgif9@L}oBjZhw@DAODHdW~IeYLRH6JrpMnptju8S>!`v%=w^YFf84E^-#h z_dHDHH=iR$vGoTam#H{!>@#dFUcfii>zExib;$C@6aEl?4$U?LTiSY$LM^D$v$wF5 z`>yfsU!e9=F#n+}-d$WyUGsoye(a-i!o~*@f0^f`IDXN7jDJ3pqw@=~whVZIb{^VK zji5u`uxH;UeS32Vnda;gqv1zU<CedM@Jx(v1 z8AWzlARjLt(43YF!^G}M_$G;{rVAQR{YYyoXbWwyb6cF(QQW+ z;yIwwZq%bZ$*%P9TOkvr+AEm=q>l}txMt+QxRA^MOwE>IGqDNbq_+K6Qev*cCoG)M zsncm#yK>Zh4f2a8JopTnTUK#h+}Wd&ktB(yqGs}K?JkNsn^^eoQvQBdrq@&geTWS8 z<%I_`cwxy2&Xzjq@Ff%v*M;=MJ zoFt9+q}@JpCG)bb;-iXflp9;=(%X3TvqGNvV>(G=hW81%>&66{S}kna*@C6jLh(P! z82|T7d?SW;d6%hVWImrw>Dxuy3<08kC>m_)!-p#=b}rwC8mbHs168eQnz=%oZ7`ac2Oa zO-`fq!*__h;1aLL67-=jsPOZSLeg9{L}(J-jvpNkGNXyWQ@x;@pV@IC+zG0F0Ma5D1 z*=rmN+|0Dp`-lmiG@_i=CxQUqC^nS7!SY%ocjsKpgrKn_%BkflG6JG`rEo3#SK7Gw zq6DUlANiR|lEfJkLwVz$B`DUlk@EWxqQ=L1&2KXp3at{C(?ZF+TgiFxkJthk zq?~Ykg{Pw~S~fPJO%A}{Lx3MZ=h3~WqsE{JJ07y|zcmE`^ll9grFbQ`~KtN_TO=(IZ zSDA^`Oc=FXRM2FKozbLf_c1BxCy`e1XI*i!Z`7hLiZGD${vVeY8pzOC8IwXu`jj-DMKIGe*w zt31|sItAPIwsGNEqf7GCeMCplFu7{%|Ce@iv`RmZ!G3)R4N##ojLvQ%_mLAmC77mN z?L)2^%=Ng01I-C4d(7;s(H%L6sbP4>CLshr)Nkj$q{|6Xk0C4OTK1HGFk+veHzCtE z>uZ@mdnV)jBlxO%`>5m)kpF1~x89h@xTF9Uelxn10K(;>?znO9Ks{;Q+P1>+KeSBsW2K5{nMf#E{jX&>i9HX zs@#NYlt(!gzFkU6Y5>3geIg%z-ZCoR9DwSdzCtrLiR>57rsnG^ocht(%(j)k&xDw3 z3099`+)ek>vTpH+W2`rIaadQ!RWa_Q9I70HL)hsw51n&6541CNeDIJUMo$n{6^bP#lHgb-U(jRK^ z2~-h3D~fxrPo&#uXT{c`&T}WC@~hAA36CJ=@~cR=c?_yRCFO$y1HE8x#mD&jM35Fe zn=x1a4!_74Nid!#&Uf` zHjyd~pH+?gJnL38;Ltlr{!=2eW=0VZpx}p9btB5P?fVsdSrOxIy$@B0im`Vj()Q8F zIbzD&c3`%5GaJ3R`x*L3<|rO*z94;xDTy6+G*a>M$X)fe#7Y>e;>w<+zyTw zdv^^mR1{OY^jor@n2#hs?z?h+B*FphJ3Sw-0${MTvZwkhF3Wg^!Okm@7vDtd<{!NV z0_N%Ju{j+~3Xb=9Jw0qDWgWY5x(0UZ%Bl?b`O1m(vXgZpbjJ^U9}W(a5&6kJ=7Qe{ znMjg^a|Ca*iK)S(wwnMXUKK<2$^oaZy0#s}hMtLRm>eUi4?aTqryqFSDqho+%bB5R zLt;vNm4VDix{sYFM1}UjU~=&J>MHKI+}#;Ut?=RL^s9NZWHovZTA9sY=kUT(CO?&i zY)~>BAC-)k9?IsMRn&Rsal0(jjH!&`1*03^uF|hjBLr=68exy*_2oDC~J^h~tA z(HyHEni}$pHC0@DUJOAV@=%QPkKm56=Q6)=iF=Ae0f*iwA??<9wDChj^aC_XUV12- zcNUjYQ8%D{nVXv^{a_*4e}4sqhc&WHA!EY)ESk5q(ljVTT|?I)^6QsyapsF2$ySod z$oT8?bRNm2dF#O8X6u@cklR(ka~aoqRO96`1y^Rh$f5e5s2h|Q=9fJ!Y(Ln>iw|ab z%o?Uu`0`5j92S;tq{%#>X&5Su6uwi+#Fw&rlQExw_!u?6zb2kH{#oKyE}RYu-+hsU z8}GzFeyDGcJVZ&%oH(k!s2FnXQ2oyY`o@wN>e-XPV{&k;q?;LDoX<{#a0+a?;q4`Wn}g!X#BYX76p|dr2r2%O^*ZZ3(Kh_Bqc<$p%f z{gk(C=KpK&z5|=C^T&@rvscr!>E5N#LZMJ7D9gP72d=NU_0GFfPdyL!JkL8l*Eu-1 zo}T09oeROqaI#TmDU{Vg_ug&N?EL;H1xk}9VA`bce*C3r^7*_=lULs3^?tn@56J}G zRn|dibr*9_^8QV8N=!BrBhq<(&o>A9HKkuvqtweuIoo@VMtroIhi@Fq-{06tbwf8U zo0VPve4J4?-G(OKd-}CZB_r>N@suyDqH$dl4IR7rx#U&O$$8vsO%ed9T+X;V|3=-a z?`T}Vf)x!r`L1C*e;aoh@&R_FD&<<{<~+{Il2@qj+|9cDCO-SFihulRe80op50#)X zaOv9CCeu~iNyqMkfA_kM-Rx=jkyA$A?DcgrwT`$MXH)aT;scL$*t_|vrhscwr+R%| zuV`%=UA7jw?VT)L(X$W7-(Ey#Q73vIg~DA92NLkv-`4;DAOJ~3K~%YaiFLnL$OIF; zKl3J;5p3S$$F9-+M$Ap5d`TTu4Qok`_U{-=xh{;9OXgAb|NaAj)n#XC-6k$f8n{NM zOseF}v43I3?x(TXJ8-$&ytuH4KVO#V^UZNh+DW)^^M2VXT6c9&vAmw#`B^^OI*nar zLt9PLZVScTonpb=y|}v+cXxLwlmfwuCwPmyySo&3hvIIg6{#!#Pr- zGX2S|7eVCJPJ5JOlPx6k@A?Lf1T1mzanP#%~;d>eA1ofT$*i8Enu{3`J)Jl;H${U z2t;!w*`E-rdR<|(2$gFVKdF&_{iQq>XEE!v*=FWH(!i1HhMN3RpD_oXeGM{v;Xjzs zRhP7u!`z&6_%qfFw1GQJxj|Z_=AiH;v%l-?E)vyB0O86>Ck6)(x7AKibv4A?i$H+w z&(c)a=DHXrYjiM+)Hv7jXOg?K@MMFeze{a@owwdc>WW6kX}uk^JFsb89i0L*TTIE< zA2)a(NY4+Lo$j38L;>RlVTwnPO_xR?;uARRRp%p0v&>*dEcx%r{?255qL7BP!Yo0; zz@4n14i+Y=m`FM4QatU;z72eECdZG+_Z-3gBbr2JMcHy^rOr#Mx+<<;sNA>RUs`Fq z!C(zt;};n0A?cs<4>>Y(ttB6KfH1V8Xj*rSVXAD?i~+3ZGKVQT4lHBZo1O}O1;#kv zAB!Y=8(=`~g2|GUremb{H3tROkF9RP__U9UW$y{H*x=hF#KiW740vhRFBV4%I9Oik zf4r1%pH4l0IYj(vwY=ERi|=+qUU)go_`Tk5S;?QSt|D!3mE6MWoH_Gy4)LGn1{K`T z#~77mcwIX^GLU7w=>vpcNGx;O6=nyyMU0!=G^la{jR&Z@GRAey_pcxC;Mtek$?-Nv z7wpVejgea1U-&g`1s+-#{6lj4j!v4`ms=9ZbE~dTe4>nO7k`*+F1D$9O0uy8#cft8 z2%MOe20bNyYy-J(f^}uMwg@Yf@_yyTT~FRQSHgCidi|;8QzL*o=jK`v3nXZU@N=`oSrTcmpoIKAegCUD=W`dPysVov6l z<~Yu@?1j!L{R-@l1kzFA>3X|)!u4}1kkyD#pZjo zc)RSJ7o+OZi;P|Jx^YcmY`o}Tsx}XoEnNPto>C|j5nWVaK5?@6^;fX&=!QhCQ8R7R z1Ybpkwm4RAp_ut;2y%YgLht@F5<%doY;MtA0Gp>9ZZI(~sK|>sZ*Et3s4!54 zV&I~^|KT}D(l&B8Zqu^HMGiNGyK_^&VxM6F)X_)yW#enzx|n_^on4u*`I7H+lU)c_YvEH^s9enN&w2CS%!$A>I3)Q1 zQS|O}u35_xnzJwm#Rj~w+c(^TknCQWqi{)S$QDo3<_)Xm#+JwRajx0^)b`x; z{a=Z#%^HDr>+@C;r^YvT^p&r zO`b0Ln#k%Ii+4=SJT{#6S>KzKr|3d}_TAa)DjP~Qc@e0g05fH}i*oOuA5F4J2?3N{ z{E9RH3>1HvTH0=T+4yCb9!y~y9Q1i1h5q!h8$vc%q*ay%3gAd+Nm3o6y&Dz2vbg%q zpghKtNU(nWD;kx2G#Qh(+G16B$uPT%(EF_|`GHW4VCe4kPIiI7-;Yjeba}~-O=-=t zu6Cy;LauFF4`m17)4u-)&DV03r+BIL{@4(Z5osZdp*7)l5LILJHQSX&=h>3&I zG-fMJP2aLSbTf*Lr-wHd<%j(_A|kJ{X;EnHDn*f!RCw~}Mf~`zU#$Oz zx-WlmO)yT%(`oc&l)LU}zcyc5iAwM)KzpHctt!Z2{8(DCVco)Ti+^3fh_q+6<;&l$ z(h8=s)$rA=pO-)xV74zqO;P676?X?TA{KJ}!lR(AeSNq}&Q5>ZlRL22IFNlyCg#}Un71>Lw+3+lSdyXFrtUHO>qoGV zKw>YYaZkUJ!lF^`O&+>XqNgW=Km1b0jJ?{e01LmxN0~x(_WkpSrRTpIWQ@r$?QQun zj1%(0ssvA}=2#Q7+MMa+-l8F+>vdR`l?~O;uWo+kdEm(Df(D!&KT3)^rH9*yu92Kx_bdQ$tKJXIQ{AT>Tp2o z&wp#yx37YJRS8||;T6YILO03g$k^%s9Tp3EvaX8(FKFDrr6+_oS#DJhS)ZQ}}Q zy*7qS_bh=4E8Lr$`wQ@@(ms%{a?~ytNA_Uw?CP zs8`D7oeED1r#nT&9^3~FNto~7j5Ns!oM4r98J}KpM?78N;~IEw^|0o^zSrAx0J)zL%##0FK-m!9U=Zj7ClY;bDR60lOMg9e)f z$p)?1>mEzi6`LJ#kUB0bN7-7Uh!>lFs@nU|>x-9>i#4^|lG;bV4IJ&9?@ibjy3ax^ z|HV;EPo2Ma)z4Ne9>Q=$7`*H21G6?9Bz`>Ab`(3-{9?zpPCCD&hf+s|cYK*oRD;(H*1D)g3j<@=^_HN51gJj0v)l z3x%_XTXfS#PN0l)BF@&nXn=2U&lN}+{bvHhk=N5iQoQUZVcouLq>GNE)&=H%84|DI z1XzwL`2wq&d!aaIJ6<=ZUPLlRFPgDd1b=II?oqT#gXZGOgS7m$zx9@}(u0pKn8{3xR8`x1@6%AFFoPKZl>2H0hKkCwgAX z)?SR6O_mKf1lqu76&NOKdRP5QVV$=Iaq&(PFBhQ{uyB8Cr!|r))U3VT(nlvN z*a~mOMJ zjQ?H*oXN96KY4#rX0yG70NAr>@$Zp|VN*7@d-8-4F{VslJ8oo(GG1RT>bulZdT)rv zctGO^Zc3Dut{h`S@Ajvz3Sf5gl*V5ac{-bZF?qN zC{kM=ZcLRA4N@2~0ixO-?*Os$bygYoG`;^V$NsIqkf{~Qz8hy@`di(;a#J+<@pT&R z#a=!)Fsyu)3_4fNwg!tD?mKObnhggiXd- zBiO;eO~)4imBAjui=EFOAsaQ}ZFLhnyhlZTS>!+heC6|NY+KTjt%%3R^7O8hx?3Wn zc1}QO2J@6>xe2EG; zzpa>Uy34*ZP4n}x6*%s?80php*Xh5%ZENN{VXDeGdPu7)Hiig0p+Aq8p0t*fb91`8 zL4ebk3BJh^w6Z{3|jIa{63RTUZj`KCA{)IH8atKe!J0(kSeK zpE#@|@>Y}du}S6Tb@cuC_p@Y{%I)%CLxr|$3+;2?^cb+lh*AsIh$*=lsIO-OKS!z( zoz`}PCa|>>?t3aUa8ogr!M#YW$RC{^r{)gxtlZlj%aCI~-!$1j?JC&VuCJRlR6M9Z zqgySv;8wOca1%t~PXuIGmfaF2_0#=48Ka7AkFjD;0im zpKMyo^7QH&k8o;=j>RQcpQ}%F9}*dEM?6}4JJiP7nI{mQ48CXI_NE%i=}Kz^3io9k zxU)hF#=Tu7lr%_MspQ&@>X{YZRQ#&@BPwO!@0va|Et3xb@+gq9G{KbQLvHnX8`+)= zHV+Qp>F5)MJLnRKGl6fI(4QAM)ffu;iRP2%(`8i-1&y4}mn7;tA!557y|p(UMCrwb zi;}%D-!{>V=U;krI6Hpips%eG3@km9nXXHqyY~!SPe0e#jxTJPtRhP(`dW~<#m3M{ zyKUuzmYXjeMK;|SeQg&L@q|g^O`M(0heY4xEzYTDDm&xioF;X1c6yT@+}-d-qc#fr zQ^bIRp!sL7`1!dp1Vg5~jd?0y7>5QGjiG1#lA>ZX(ZIYdEqjo--6Zcxf$E}#sfsSG z|EM5+C70>3*H5j{>C%FH+GB76v6OUe<)pf^rV%wuTXdy5b{36c{h7^>t=ATtVEoB0 zUGJWx640VwtHfse9v<}`T9#}>`8EB5{&h2hjw%;HVe~U7G00?zMvIP3>ol*<1D&d^ z%~q%*^Y67oCeM?17_l(Kkfz%)t?1fld2&swI=(A5Lm^BYrkUPWZ<2NIB8enn6gSC; zzuNVpN59VZbRcWZ-C5v)0Co5lNz+=g`eyw0#q;fvAVpxM0;Y(TI$H#ki*9s^t+?Q1 z*ZUJ%lL{yB@Z`q4_-su={h?XpN5-A!FJ)CKVw0xZU*#1_bX0M(CRk-0s#bT`?FN66 z*KD-+c<+z_o{ky9Xs&1}?$@#l>t8;fT2OJMB)e(AU;+E9j4(f!@nrAiRa$MJSIs0+ z41jqDPTaG2@K^CpX|7AJ!ivZp$`ng2R)$4qrfQ;Jq9C;`FNV^iDg`^6n;cKPO(vNf zxV*1zX8?O_soT+I{d?xEbV-)+mK<3_fp-n-6*yj+YEDkQr>AK)acbe5oD9*78rJbY zq1E(E>9ENw03v$hB^rJ~kkTh7sLsg-qJ$M7(QoD5hW$8ShHQC>c`Na*Sg__O17RiZ zU{RK_T?Dgqi0s(4zL->2b}!7-iOLpEn-Oqk&mmWRCSAQ|UUOhD_dGf&rwd9ueQ+lr z=8b04+B6M?eQRqZ zZK!jB`88B-rxY$%C9Y@FoUwu#^)T^8Fq{iRoB<5AhYk*;K4e>}obGuoC&sH!j>Top z>+)8;YwwNAfZ7Q>f+6b{Aog=~LZ*_{;g5)&Cn`67TS=5b?(1?E;d?bZS28Zkj*=>8 zT@+JiZgpGJ;${Guc(?F_`Rt`jhY7NK0)J4)Xk7Y_8pB_|wEvQCNAvuQFLQ0pq}$We zQ?jfF`=!#$3}*MVV86+MTbu>xPfk8;vUL`5$JXOJUQ8Ldd_O^ zyBm4jr#D1DYEAJUr+2xYgpx-yH3DP9`5X>t$PJ`|;6LF_47JT% zM+_{oqVp%a4{5XmYU-O4JFI8h&fYP%5A6*~1MT{a*};nK1mgkt{`jYW`{j_XazC1E zFOh~a5XaxY8NSe*#C~S#XU+U5r=)w!r<=$ThlHnF=M2wLwaFB zI%Y@Z=A7uVqACXXy>A%pGCf&uN^m>c>(6$Bd+xN$b@Zm4%^vR2+YH{wGoNHB?a~V?DNk18 z-H`4N`<8LxQA)tEGcgz&9d}NILIhEuBVpBaBb5ukq9z=iT``mj17MC&ASVm7M72rh zj4w3GlWjht5}L9bBys^yFFUcpScq%xvo>Te2WWUKCEM+~t}k=d89x|#U3Gt3oD&8r z=a7vS>(?%MEm}#(W`xZRy*I!rL3v%meta1CF66SPMK-GGmuNR!P#Y*nIGAnrCAN$x zl7Dab!?%h&T&d=ZulRHnxQf(Rh?P`)nMfv2&2(e|=FcHqcTbHK{ZbDiA(dZz!s86B zH95~-nbuU=0{5c=I!nXr3j583PCq*WCB?T-uYqZ<9F`FOob9F!I4L6CEhq3H8f+A8vZe?@fgULr?qa8+3->2W+k7Ed=sz zUGtA)=DfY97B6Avk6sRbLFur70E9*pd~XP zXCu2PuLloyJRG*e9RrRS@nBO@`Rcbda^J|w32#uRD|&BiWPF#se@GyjL~1! z!=nOOb5^l*Kdd}f_bKh_gwzV+8Xj)S6uD{YqF<9FN-t@3Q-J4S?TX72S!ygR$sZQd zH?j=l%%{i2R9zK`taivQEt&nrH&BT_Z}p@`-n64_;R?T0yigFL2Nl&^6`~TTM8lvD zRQLzJc+y~wj1a8OgHKAQ2A>!E6uEb6JGaf>wc58O99huy;{F*5v8fg{RpRFx`s#=6 z)>9GHgZF^-kFCSjr~7Inp)zaRJG2A8p;A zs&?AeFQ2xtg1p}lssv5fm3aj>=zf{!q}f7-r3w?Vgc}Yr3ffwcl*B1TKr&%spj__6 zTy8hz7afDk(2?^-ENJgF9Jb8R7=3(}Q(ae&zOebCX*h;V3fsfzcv z9)gC?dM`O)AE5`6QgghyG&Z%x!zQk`4zl^yS%RE4#^uz^6X*-Z0(d~_T1q(WnrqIt zj?1hK&{~}%BbV*m8wSp%dye|5QVeO2W3U#b@y|uLFZt>tKKmhvl0-Obcf)wc!Z_(4 z5Qxb`7?hRPXF9J2FxuBOYTIg`qbFYZ+eQTI>jpaR>?=2eoD(qKAfP>{Q%Zbo{uOyf}qGDb7 zEwqD=SvdHiQJFaBO=rM);w)G*eD$nwdsSdz!|?MyZi$6q+VNs1B1JtkU9z{PSqaYg zy$+%Kv#RAu#+wRwFjwfQP!ynJ;nY~IUdL%c90$xn(%stWgN3xxZ!)u}|?U+R0ga2mwO zi&&H`bCzxyv?(Yjni{ooMxE(zAwhCD!cO=*kRKz|9Z*w>&{&U0esz+I!q8hkuDKwE z&HwxUI-`OVOoBH_ScNs*%8`PJPyKh+-d|34IV_+97>nsy+?(Q|**Bx{1372eUPKPDQzC zFWsj7$>=e!yYfpnqGW{^Jh|X_`(>ByaBHSVa+(V`GiEgQGq(S3u!*UUg37|xl%ufl zX1EpOs#hcQ<#a-7Ih8tn-sL&-a4QL0w^7AI+JS197e9Lo{$soGcrQ_X9f}S~*6RfS z-N1@~M}x8d_cb=CUgQnzcf70P9VhiC5aoy)hzqv+fzpFkgt9TSKY^NwuI2#$oK5aO3u#NX)-%nn zA5m17kh^D0dR820Xo8p;vi#u!Drp<<@RGx+1l@KPky=bBQ)B}d@-OpC+-aw`mJQsM z8CZjPF4=6IWw`S^5Mfe@9vq}_<=BaoNd)VIF-#vglw-w{iVN%#?q_~-q`(>emK9(` zf^XNpQt+2;6Ud@|)hQM9B6ZbYV_A%+6%RLHd#@_Dr(@VzHc4LEVveu&R5M0Ejw-98 z|EI%c=6JvNeFUEmkn#yjbaWVxLvxb)zwx3}mVSF~9D;G3uGLhe}E ze5_on{&<3HrEyF+?4t1~MgE(PgLo8xOzReUw}Yc(Q!xA)DaiWoMV0)~@wqX{@M*QW zlbwl~BqUe<-S_7e-;=UHAy*Bw()W$|$f30ADao0^yR{eNW-G|TT@h5Zw5FBJS2~$% z)3!x76=oa<_sJAWY~oPtfY^T+0SauzJ2p)Z!ITszuW(&P6pyz2GyM&f=XxH;4=@n& zk}qyYwj=GgT9DFhQEoh!UekeGe=TmX?|zrR&*}Znl>Mv10e)B?eJ^f|KNCoJ^0fbS z&n9KKhSPTKMx$#;VUvO45u&apO-2$bW3z2r=OfyB>(V2u^9=(Pa8-(M{Ns~rg?Hz! z>cOP!&*tC#t!II48JoNgmR25{xtvqjvTsjb6v3TLNnwY-7<(B>GPr<59M0rQTgJ#j zaLI>jvn9rmz?`A<1s*eoBkHa8PG1_}Q;iJQTutL#UYtN}%<^Z15fcsuri5*lO?b_F zyCbGvpW#Qosk)kz645y8tth(S5~40 zEj2}*Qt}b)_jrjt78W`m3Jg7iM7zc{L)ptAQ#6WH2O}9nbyW-fCpn-?-y?w45Ghkt zb5U;6$DGr>Ypwa0>qY|(jW`EpV~l`l>*KV`NjXi{%6yw1CxLsiv0w6hws`ZR0TLr8 zQs6sJ_O}t>aNpBGasLUYE!(3AMfI02mh`R3zt8~`pnABUNo7ZF9y^0)S1;*(x@Tv_ zIN7{^N^sQ4I!u|I1M7ZNi6+;HSyIw8f{&#+*)bWEb4W%VK~|Kay&dSGe^|AWqkxkt zg^&~H6=m_b%KIy z`zK&Oa$9YWNpcf35+``wY#o@lAM(>{qr~x8L1tn;n9xq=f37IJP z$KDSkL>JYM-lKSB*pkC#T_CNEp73xAvY>ZpPb_(SL!cI)qN7S77$k~6!R?>Slv-ce z6EVilBNofE&6zy#)JnlhJ>@?$t^V%7HiDv%Km0>(iDy=XG5|KoPl3_V3p?#8|1-A= z#s~47?C;QsRN=b2g_^w;lj{Ln@pm+W!gb~o+mbjBI0i-GXMa_;96hEf415xeX9gc@ zkd3!0QfL>JvIwYkUI9Kpxt!&jv<9TTq|Z|;6(bcZbL=c0st3oi$1lU&w!A5(Q-0_G zXy$W1R*;Ws;<->n459Epqi2hI{{Faa@NmYKE9J30wSb?GAAr%@BN+-?p(q8)gS_8? zZ*k>g9=6=Lr+ArZ&sd8@(Ns^EW(WhMsY~@ABG%))oZj*yr4d1k&VJTUBrR0lB_bt5 zXD9nWL9E#A-3c5~R{{I|NjqV+58euugs2>2hndLoIkIpwb`o9kZyoEbFmL>rJS$~D zUBQ4C7Xru3(4c@3SFISR#18`ZTPJ3smQ86S@G}cI#4f++$itzzFMh?YscA(a8n-kG z-bi|62nEUznqDP?+wT=ulnP*2kXg%C$4-4Ntk)34>42iPJf1Lfk!QNoS0ae^rA%TV zCTNGFR=zt_48@h!c2?w712k=Jr-QL2D#4FR7@;&?E4o}x2Eb9L>j(Fop8Qr6L-nUm z4HHfAXaI(i4@Ilr`_3NA6OoPqqYDOw!>H4rzm**;;^ zt73ehJ&K7gemqk6G1{*VBtgV7AnPNYXZYOsX#1R((Tbv{f^G!MF0 zSr}fkxS>)_jY=6}nffH0+z`|PrZO)ct6A~@Ky7`R@c=#5F5HxyuR*)?<${FV(B6;!X; zf1Rlfehm~OoPhK;mkqy*fi^B4^;hL(HHm?0sO?EQ7~(0v38Do6A&u*=U6SS=M_l@4 zPeKb?x}m>KBr)+sG?i}XcE2HZFJJiv72 z_GFehU8@%XS#+LA%nO};nNOaZn7lig6b@O|eU#v_$lq2H_G&p=q;(AmY@Xsd2Z363 zAt;7IX{ItkJY}E4umbe}t@$2y+Cl3FR|rZ%q<7ztv7Cy&E%e4pDxtiY^;IJPIDpCY z33(3%(DTEbyiS>c+HUC;q)erHqvX(YR&u9j@oi zwx3y34rP6nTnW~4FGh_v`M^dSganNy-%mN(ixos3USBWDgIofibZz#w50$r;J}uhF z<@l4*+dc{rRg*PjdUSre2*<~6ht>FaYg#_@OnZN(h6#QFf>cLoV=NWl}q z-%ynVr%HER+SUUMTA~)^Q(e9+3^V8AI25VZ#-ixoMxQg%8>)F>Z&_4bpP>vbdgWc)Y>iD>&^GU08 zxP9MCuhF^RusywzNkaA`7J*e?rV&6v7(YL=*Mij=4!O7HH=h$nUZJsk#ymvgIQ(}t z3V@}Fv?KyGDAYwLUe1Eg@XmxcFUJ&Lk*QI8PTssag91UUKttNB_R;tECI+EbPkvn+ zQHOM9FcrxrCAVC?Ux_0YQy&{x6KFz7a{sTzj#kM2quRcvbnaN$`5m$0RiO25;!X41o&Gf^a$n3k!mD>VJguLky@e}#yiS+If@1$q4H3u)r?SPMx(%o{fgizbcS~U zpM~CFil<04MXGM9gaHjNj&%geDZwPTZFR`s4e(E`5X5CwNBgSzTA}>Fge$SRwyl5q z&4DCNE?gkLE|SBG18u8W(ueZ+Z5>4nfnHAvS~>l;hsPll2xi3$dqaT|WJ2w6f@6yb z1upCT(8eYHk0|1off+K&q-v;$ftg|t=P1Dly}$glJ>q(@f%A5j&v$1Yo*#q4mmP5X zNW}Ou#c*Yc3a0O|BR+SB%0cmkbaizNyp}C@cXz8DmTNy761_`mFoI6;7TeabxeK(W z3c6@~Qc~afdfRe~J1tKJvnmMgUu>Eeg?7mO(X@G%(fqKU%v?CB#(b#Ov=8pQ?Lq_z z2?;Fd?evoc_3P-KBm<@=l$`1>%|&XH9*dcLxq$n1@8)%<{fVqmC_od8r)?$+BS#5` z$s%@`IZ&~*e+%hU_KywhY1xKZ4V&w>Boy^0a@xk?@$cSP8Yk_iMLb6S&_9OF0^w-( zFoT~KpAdD}9#88ghbJcw#yBT;M_Idedy!dD<_e_ZfS*u48O$d&dyT;+l~uMz-Q9T^ z1zbs4O=ON#G;I(SD_l$l;r2*k2qC8QH+=nw4V3IMjh$sA`hQ_F4ixts?npo?L(OQ;t%j{AGyFx2F>wMyd?)7;;Qz)@}DbhYq@^nUekoK z{k+7WOT?KPXHhOnI>-qoYg$#3lr-_f2H1BDvtz=N`?)J^-=>M#OMLbRfQ*Cq^Q2?N zOEld!1MffXrduiG4F}=ElJ{!$utJGs0}WUX8WxT6a&to;9=b_VEgQKp$rJG7?Gbxx z7z}VFeD#A=S1~SyUuTO{tfBZ*@7uBm7Gk6=QI>9Dw{&m{V{%6qNalFLUS?dH8a~FV z!(xjOL@L|j^z`GGNl=#~hO!Q=&0Lyb&^{uDI%5GXAl5WK;sW62HMD$nwNf>V7mB1| z*OqTzdfD^80S6YM^eFZfV0*{~EU5wd`cvoRGd=mM3rqmxk3A|q*IoaRlq>pTe2{jk20tJ{!hiN_wPzR6~gXF~tYpOF<7Gnn9 ziIr09Mhe~Zkf$=###9u-g6ha+4)`nS*I7&Kp-`mIoLzMqG!VDVJ^Gdj)=PS7=%)u( zi*La3>F@Ht(Ep>OHhQZ=Vkp+)Jq(xRRQ69*sc^>(lBQ4rX1Y2UaXMfIpB4Z#)atK= znyQxSPb}yW4s4jCy`e6YNcCIp$1qH2$m_)T8wZ$KS9ys$j{^1UtSKI}ghwW~|#A}rWyB2T}F z@!6S*{SHMPH!*X@7RDCilZVb+o#9y8JCg=r-|vrGHXlzguBT7{cV3!NktVzn7~wP( zl;W^SM%w;@cGH}gVR^Y)1c!J%`4Lc-#aYykpk0dh{Fv^3v91#&_VmXh110lkkb9@& zWvQE8&>jP!+XaMBwwEviHdns3gi)>@r|?fG_DCggZL=Kt_b4qUGj)kn(DmM0L>T<* zGW-!cx`VXg-w=$Y%5eU31{Ci?YlKL0d#Hu}8@&=0zA$$A9nU{zj7zL1@*QO_2t#}F zCI0y~Rxf_ZIdoYGB#<$QM@lc1O-qU~PMTJ-aKRQ)Fv5?>an^P%P&>Rq}CkN39XkpPrKk{BPXi$M_*Y~aX1{!k1ABB_U~HO~K7 zQ2l2ZR$Za8f{7oBK_?NoD-{X#Pya^pZkhqbLojaDgz|0=LHsAlVi~`gtq#TqnezQe z6H0{l>~k5f(6?C4YdrbtzWW93n-V4Ze^+F4-4UlufcJwke~$rGa&>*!th}9 zx4$m3u-IjW|D?r)zcLhP2%VbVpZw?H{AR{+pHQI1S#z+VN4cOb($wYVMlmkQG)^g* z93xC@wode*`6%t*a(t5#mgH6vrv`gO2|*1Y_0S@l3==9cluTu6xFaQuHU`|dEBBad z%h`6f8}xUM5|W}(k;8jLXb^cM6F(Krk4?-DpC15i@KZnjaRnu`{BQB# zmR+F+2CCNt0d_59W!E|vX|iXO&sxwQRB>dH5=u4*j#kxu_n|ZlS{-*fgbocHRO?Gy0k!hciJobJ)_%89G~l z0>y9ZwuE#3u)^T(Kjg(s2p>@Htj|8PEmxbu{DU@(#$!Id4vQt`{aY3M8eLmk3-crJ zdYrfY>A0-EsR^dqc~bbcdTV#aLKUF=i<)jDzhLot>TuLas zSp?7kzu6S~Lz8es{e;ZoXV=`425hhY8QA zWe*Pv8#{2rfD*FzgKSGDrG+90Lyzba+81?y;CG-=@XJ+I+j!2+A3>