From 3ba0ed40aec100a20bbdebdac0132033d21c4642 Mon Sep 17 00:00:00 2001
From: Pomax <pomax@nihongoresources.com>
Date: Fri, 1 Jan 2016 17:52:50 -0800
Subject: [PATCH] 1/3rd there

---
 article.js                                    | 1334 ++++++++++++++++-
 components/Graphic.jsx                        |   81 +-
 components/sections/control/index.js          |   34 -
 components/sections/flattening/index.js       |  101 ++
 components/sections/index.js                  |   51 +-
 components/sections/matrixsplit/index.js      |  631 ++++++++
 components/sections/reordering/index.js       |  182 +++
 components/sections/splitting/index.js        |  150 ++
 ...2a1516ca2ce995a0ddebf218b7528bd762b686.svg |  328 ++++
 ...ad48dc956ff2551caae8d7d35f299b395d6a28.svg |  156 ++
 ...3ca089b9e3928414a23931298dadf2886f1d71.svg |  174 +++
 ...cf32cf7e00f9e2a5ef18f352f93a10c55ebc99.svg |  156 ++
 ...efec3a2fe4620906cbf25bc11a28b354097b2c.svg |  146 ++
 ...11ca44d7a85ba755774fc82dfbf9fa14f94f71.svg |  246 +++
 ...44abca018b8ce799a19e39e45bb1021989c00c.svg |  218 +++
 ...d82c66c21d9d4778726bde0ded0070e6a917ab.svg |  202 +++
 ...90d865f1093a974f5aa1c8525c45649137a688.svg |  149 ++
 ...4f49fdad56663b8955f3eb1b9b42a2435d9b13.svg |  315 ++++
 ...d685fe5c634305918f67db4ed693c0095de0d2.svg |  164 ++
 ...d34f4126772cb9ba06e36b9e40df95994cf847.svg |  145 ++
 ...6fb6adcdca2ac5372e04d38becbfe355988ba9.svg |  202 +++
 ...9cf2edc06c8efabdf614f554b21fb929b0765d.svg |  164 ++
 ...d3c6b5b68d86adf40db56f7723a45aa5a0f7a6.svg |  258 ++++
 ...88c77e6c5b3e13ad44665606ad0352040ce89a.svg |   75 +
 ...f197498c75c7bf59bf4e9791e4e948a5699d8e.svg |  128 ++
 ...061bb273e862f3a91b83c6a8b4713a959e0dbf.svg |  225 +++
 ...25370c4e7896c34d1a315db126364538918ea3.svg |  318 ++++
 ...6d8b52374301de6eb2aca5ba798799f0759b5a.svg |  200 +++
 ...7b68b7fbd11a041232c144e31122309fb9d782.svg |  200 +++
 ...9f128759d56fd5b1e6099da9d7e11e83bea104.svg |  122 ++
 ...4671db313e31b678512b59f5aff9de01dedf45.svg |  160 ++
 ...4f3c7fda60e80707370b02140840311bf33d75.svg |  138 ++
 index.html                                    |    2 -
 stylesheets/figure.less                       |    4 +
 tools/mathjax.js                              |   87 +-
 35 files changed, 7090 insertions(+), 156 deletions(-)
 create mode 100644 components/sections/flattening/index.js
 create mode 100644 components/sections/matrixsplit/index.js
 create mode 100644 components/sections/reordering/index.js
 create mode 100644 components/sections/splitting/index.js
 create mode 100644 images/latex/1e2a1516ca2ce995a0ddebf218b7528bd762b686.svg
 create mode 100644 images/latex/2aad48dc956ff2551caae8d7d35f299b395d6a28.svg
 create mode 100644 images/latex/373ca089b9e3928414a23931298dadf2886f1d71.svg
 create mode 100644 images/latex/38cf32cf7e00f9e2a5ef18f352f93a10c55ebc99.svg
 create mode 100644 images/latex/42efec3a2fe4620906cbf25bc11a28b354097b2c.svg
 create mode 100644 images/latex/4511ca44d7a85ba755774fc82dfbf9fa14f94f71.svg
 create mode 100644 images/latex/4744abca018b8ce799a19e39e45bb1021989c00c.svg
 create mode 100644 images/latex/5bd82c66c21d9d4778726bde0ded0070e6a917ab.svg
 create mode 100644 images/latex/5e90d865f1093a974f5aa1c8525c45649137a688.svg
 create mode 100644 images/latex/6c4f49fdad56663b8955f3eb1b9b42a2435d9b13.svg
 create mode 100644 images/latex/6dd685fe5c634305918f67db4ed693c0095de0d2.svg
 create mode 100644 images/latex/6fd34f4126772cb9ba06e36b9e40df95994cf847.svg
 create mode 100644 images/latex/836fb6adcdca2ac5372e04d38becbfe355988ba9.svg
 create mode 100644 images/latex/8b9cf2edc06c8efabdf614f554b21fb929b0765d.svg
 create mode 100644 images/latex/a2d3c6b5b68d86adf40db56f7723a45aa5a0f7a6.svg
 create mode 100644 images/latex/ab88c77e6c5b3e13ad44665606ad0352040ce89a.svg
 create mode 100644 images/latex/b2f197498c75c7bf59bf4e9791e4e948a5699d8e.svg
 create mode 100644 images/latex/c6061bb273e862f3a91b83c6a8b4713a959e0dbf.svg
 create mode 100644 images/latex/c925370c4e7896c34d1a315db126364538918ea3.svg
 create mode 100644 images/latex/d16d8b52374301de6eb2aca5ba798799f0759b5a.svg
 create mode 100644 images/latex/d67b68b7fbd11a041232c144e31122309fb9d782.svg
 create mode 100644 images/latex/dc9f128759d56fd5b1e6099da9d7e11e83bea104.svg
 create mode 100644 images/latex/e04671db313e31b678512b59f5aff9de01dedf45.svg
 create mode 100644 images/latex/e94f3c7fda60e80707370b02140840311bf33d75.svg

diff --git a/article.js b/article.js
index c97a751a..13f321c4 100644
--- a/article.js
+++ b/article.js
@@ -62,7 +62,7 @@
 	var React = __webpack_require__(8);
 	var ReactDOM = __webpack_require__(165);
 	var Article = __webpack_require__(166);
-	var style = __webpack_require__(183);
+	var style = __webpack_require__(187);
 
 	ReactDOM.render(React.createElement(Article, null), document.getElementById("article"));
 
@@ -19754,16 +19754,15 @@
 	  explanation: __webpack_require__(178),
 	  control: __webpack_require__(180),
 	  matrix: __webpack_require__(181),
-	  decasteljau: __webpack_require__(182)
+	  decasteljau: __webpack_require__(182),
+	  flattening: __webpack_require__(183),
+	  splitting: __webpack_require__(184),
+	  matrixsplit: __webpack_require__(185),
+	  reordering: __webpack_require__(186)
 	};
 
 	/*
 
-	  flattening: require("./flattening"),
-	  splitting: require("./splitting"),
-	  matrixsplit: require("./matrixsplit"),
-	  reordering: require("./reordering"),
-
 	  derivatives: require("./derivatives"),
 	  pointvectors: require("./pointvectors"),
 	  components: require("./components"),
@@ -19798,6 +19797,46 @@
 	  arcapproximation: require("./arcapproximation")
 	*/
 
+	/*
+
+	  A lightning introduction
+	  What is a Bézier curve?
+	  The basics of Bézier curves
+	  Controlling Bézier curvatures
+	  Bézier curvatures as matrix operations
+	  de Casteljau's algorithm
+	  Simplified drawing
+	  Splitting curves
+	  Splitting curves using matrices
+	  Lowering and elevating curve order
+	  Derivatives
+	  Tangents and normals
+	  Component functions
+	  Finding extremities
+	  Bounding boxes
+	  Aligning curves
+	  Tight boxes
+	  The canonical form (for cubic curves)
+	  Arc length
+	  Approximated arc length
+	  Tracing a curve at fixed distance intervals
+	  Intersections
+	  Curve/curve intersection
+	  Curve moulding (using the projection ratio)
+	  Creating a curve from three points
+	  Bézier curves and Catmull-Rom curves
+	  Creating a Catmull-Rom curve from three points
+	  Forming poly-Bézier curves
+	  Boolean shape operations
+	  Projecting a point onto a Bézier curve
+	  Curve offsetting
+	  Graduated curve offsetting
+	  Circles and quadratic Bézier curves
+	  Circles and cubic Bézier curves
+	  Approximating Bézier curves with circular arcs
+
+	 */
+
 /***/ },
 /* 168 */
 /***/ function(module, exports, __webpack_require__) {
@@ -20023,6 +20062,31 @@
 	  offset: { x: 0, y: 0 },
 	  lpts: [],
 	  colorSeed: 0,
+	  playing: false,
+	  frame: 0,
+	  playinterval: 33,
+
+	  animate: function animate() {
+	    if (this.playing) {
+	      this.frame++;
+	      // requestAnimationFrame is spectacularly too fast
+	      setTimeout(this.animate, this.playinterval);
+	      this.props.draw(this, this.curve);
+	    }
+	  },
+
+	  getFrame: function getFrame() {
+	    return this.frame;
+	  },
+	  getPlayInterval: function getPlayInterval() {
+	    return this.playinterval;
+	  },
+	  play: function play() {
+	    this.playing = true;this.animate();
+	  },
+	  pause: function pause() {
+	    this.playing = false;
+	  },
 
 	  render: function render() {
 	    var href = "data:text/plain," + this.props.code;
@@ -20030,10 +20094,14 @@
 	      "figure",
 	      { className: this.props.inline ? "inline" : false },
 	      React.createElement("canvas", { ref: "canvas",
+	        tabIndex: "0",
 	        onMouseDown: this.mouseDown,
 	        onMouseMove: this.mouseMove,
 	        onMouseUp: this.mouseUp,
-	        onClick: this.mouseClick
+	        onClick: this.onClick,
+	        onKeyUp: this.onKeyUp,
+	        onKeyDown: this.onKeyDown,
+	        onKeyPress: this.onKeyPress
 	      }),
 	      React.createElement(
 	        "figcaption",
@@ -20058,6 +20126,10 @@
 	    if (this.props.draw) {
 	      this.props.draw(this, this.curve);
 	    }
+
+	    if (this.props.autoplay) {
+	      this.play();
+	    }
 	  },
 
 	  mouseDown: function mouseDown(evt) {
@@ -20074,6 +20146,10 @@
 	        _this.cy = p.y;
 	      }
 	    });
+
+	    if (this.props.mouseDown) {
+	      this.props.mouseDown(evt, this);
+	    }
 	  },
 
 	  mouseMove: function mouseMove(evt) {
@@ -20101,7 +20177,11 @@
 	      this.curve.update();
 	    }
 
-	    if (this.props.draw) {
+	    if (this.props.mouseMove) {
+	      this.props.mouseMove(evt, this);
+	    }
+
+	    if (!this.playing && this.props.draw) {
 	      this.props.draw(this, this.curve);
 	    }
 	  },
@@ -20110,12 +20190,45 @@
 	    if (!this.moving) return;
 	    this.moving = false;
 	    this.mp = false;
+	    if (this.props.onMouseUp) {
+	      this.props.onMouseUp(evt, this);
+	    }
 	  },
 
-	  mouseClick: function mouseClick(evt) {
+	  onClick: function onClick(evt) {
 	    fix(evt);
 	    this.mx = evt.offsetX;
 	    this.my = evt.offsetY;
+	    if (this.props.onClick) {
+	      this.props.onClick(evt, this);
+	    }
+	  },
+
+	  onKeyUp: function onKeyUp(evt) {
+	    if (this.props.onKeyUp) {
+	      this.props.onKeyUp(evt, this);
+	      if (!this.playing && this.props.draw) {
+	        this.props.draw(this, this.curve);
+	      }
+	    }
+	  },
+
+	  onKeyDown: function onKeyDown(evt) {
+	    if (this.props.onKeyDown) {
+	      this.props.onKeyDown(evt, this);
+	      if (!this.playing && this.props.draw) {
+	        this.props.draw(this, this.curve);
+	      }
+	    }
+	  },
+
+	  onKeyPress: function onKeyPress(evt) {
+	    if (this.props.onKeyPress) {
+	      this.props.onKeyPress(evt, this);
+	      if (!this.playing && this.props.draw) {
+	        this.props.draw(this, this.curve);
+	      }
+	    }
 	  },
 
 	  /**
@@ -20207,7 +20320,7 @@
 	    this.ctx.fillStyle = "transparent";
 	  },
 
-	  drawSkeleton: function drawSkeleton(curve, offset) {
+	  drawSkeleton: function drawSkeleton(curve, offset, nocoords) {
 	    offset = offset || { x: 0, y: 0 };
 	    var pts = curve.points;
 	    this.ctx.strokeStyle = "lightgrey";
@@ -20219,7 +20332,9 @@
 	    }
 	    this.ctx.strokeStyle = "black";
 	    this.drawPoints(pts, offset);
-	    this.drawCoordinates(curve, offset);
+	    if (!nocoords) {
+	      this.drawCoordinates(curve, offset);
+	    }
 	  },
 
 	  drawHull: function drawHull(curve, t, offset) {
@@ -20246,7 +20361,7 @@
 	    var pts = curve.points;
 	    this.setFill("black");
 	    pts.forEach(function (p) {
-	      _this2.text("(" + p.x + "," + p.y + ")", { x: p.x + 5, y: p.y + 10 });
+	      _this2.text("(" + p.x + "," + p.y + ")", { x: p.x + offset.x + 5, y: p.y + offset.y + 10 });
 	    });
 	  },
 
@@ -23378,7 +23493,9 @@
 	   */
 	  var Bezier = function(coords) {
 	    var args = (coords && coords.forEach) ? coords : [].slice.call(arguments);
+	    var coordlen = false;
 	    if(typeof args[0] === "object") {
+	      coordlen = args.length;
 	      var newargs = [];
 	      args.forEach(function(point) {
 	        ['x','y','z'].forEach(function(d) {
@@ -23389,12 +23506,23 @@
 	      });
 	      args = newargs;
 	    }
+	    var higher = false;
 	    var len = args.length;
-	    if(len!==6 && len!==8 && len!==9 && len!==12) {
-	      console.log(coords);
-	      throw new Error("This Bezier curve library only supports quadratic and cubic curves (in 2d and 3d)");
+	    if (coordlen) {
+	      if(coordlen>4) {
+	        if (arguments.length !== 1) {
+	          throw new Error("Only new Bezier(point[]) is accepted for 4th and higher order curves");
+	        }
+	        higher = true;
+	      }
+	    } else {
+	      if(len!==6 && len!==8 && len!==9 && len!==12) {
+	        if (arguments.length !== 1) {
+	          throw new Error("Only new Bezier(point[]) is accepted for 4th and higher order curves");
+	        }
+	      }
 	    }
-	    var _3d = (len === 9 || len === 12);
+	    var _3d = (!higher && (len === 9 || len === 12)) || (coords && coords[0] && typeof coords[0].z !== "undefined");
 	    this._3d = _3d;
 	    var points = [];
 	    for(var idx=0, step=(_3d ? 3 : 2); idx<len; idx+=step) {
@@ -25366,35 +25494,6 @@
 
 	module.exports = Control;
 
-	/**
-
-	        <textarea class="sketch-code" data-sketch-preset="ratios" data-sketch-title="Quadratic interpolations">
-	        int order = 3;
-	        </textarea>
-
-	        <textarea class="sketch-code" data-sketch-preset="ratios" data-sketch-title="Cubic interpolations">
-	        int order = 4;
-	        </textarea>
-
-	        <textarea class="sketch-code" data-sketch-preset="ratios" data-sketch-title="15th order interpolations">
-	        int order = 15;
-	        </textarea>
-
-	 **/
-
-	/**
-
-	        void setupCurve() {
-	          setupDefaultCubic();
-	        }
-
-	        void drawCurve(BezierCurve curve) {
-	          curve.draw();
-	        }</Graphic>
-
-
-	**/
-
 /***/ },
 /* 181 */
 /***/ function(module, exports, __webpack_require__) {
@@ -25892,15 +25991,1142 @@
 
 /***/ },
 /* 183 */
+/***/ function(module, exports, __webpack_require__) {
+
+	"use strict";
+
+	var React = __webpack_require__(8);
+	var Graphic = __webpack_require__(170);
+	var SectionHeader = __webpack_require__(175);
+
+	var Flattening = React.createClass({
+	  displayName: "Flattening",
+
+	  statics: {
+	    title: "Simplified drawing"
+	  },
+
+	  setupQuadratic: function setupQuadratic(api) {
+	    var curve = api.getDefaultQuadratic();
+	    api.setCurve(curve);
+	    api.steps = 3;
+	  },
+
+	  setupCubic: function setupCubic(api) {
+	    var curve = api.getDefaultCubic();
+	    api.setCurve(curve);
+	    api.steps = 5;
+	  },
+
+	  drawFlattened: function drawFlattened(api, curve) {
+	    api.reset();
+	    api.setColor("#DDD");
+	    api.drawSkeleton(curve);
+	    api.setColor("#DDD");
+	    api.drawCurve(curve);
+	    var step = 1 / api.steps;
+	    var p0 = curve.points[0],
+	        pc;
+	    for (var t = step; t < 1.0 + step; t += step) {
+	      pc = curve.get(Math.min(t, 1));
+	      api.setColor("red");
+	      api.drawLine(p0, pc);
+	      p0 = pc;
+	    }
+	    api.setFill("black");
+	    api.text("Curve approximation using " + api.steps + " segments", { x: 10, y: 15 });
+	  },
+
+	  values: {
+	    "+": 1,
+	    "-": -1
+	  },
+
+	  onKeyDown: function onKeyDown(e, api) {
+	    e.preventDefault();
+	    var v = this.values[e.key];
+	    if (v) {
+	      api.steps += v;
+	      if (api.steps < 1) {
+	        api.steps = 1;
+	      }
+	    }
+	  },
+
+	  render: function render() {
+	    return React.createElement(
+	      "section",
+	      null,
+	      React.createElement(
+	        SectionHeader,
+	        this.props,
+	        Flattening.title
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "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."
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "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 for doing true intersection detection, or curvature alignment."
+	      ),
+	      React.createElement(Graphic, { preset: "twopanel", title: "Flattening a quadratic curve", setup: this.setupQuadratic, draw: this.drawFlattened, onKeyDown: this.onKeyDown }),
+	      React.createElement(Graphic, { preset: "twopanel", title: "Flattening a cubic curve", setup: this.setupCubic, draw: this.drawFlattened, onKeyDown: this.onKeyDown }),
+	      React.createElement(
+	        "p",
+	        null,
+	        "Try clicking on the sketch and using your '+' and '-' 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."
+	      ),
+	      React.createElement(
+	        "div",
+	        { className: "howtocode" },
+	        React.createElement(
+	          "h3",
+	          null,
+	          "How to implement curve flattening"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "Let's just use the algorithm we just specified, and implement that:"
+	        ),
+	        React.createElement(
+	          "pre",
+	          null,
+	          "function flattenCurve(curve, segmentCount):",
+	          '\n',
+	          "  step = 1/segmentCount;",
+	          '\n',
+	          "  coordinates = [curve.getXValue(0), curve.getYValue(0)]",
+	          '\n',
+	          "  for(i=1; i <= segmentCount; i++):",
+	          '\n',
+	          "    t = i*step;",
+	          '\n',
+	          "    coordinates.push[curve.getXValue(t), curve.getYValue(t)]",
+	          '\n',
+	          "  return coordinates;"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "And done, that's the algorithm implemented. That just leaves drawing the resulting \"curve\" as a sequence of lines:"
+	        ),
+	        React.createElement(
+	          "pre",
+	          null,
+	          "function drawFlattenedCurve(curve, segmentCount):",
+	          '\n',
+	          "  coordinates = flattenCurve(curve, segmentCount)",
+	          '\n',
+	          "  coord = coordinates[0], _coords;",
+	          '\n',
+	          "  for(i=1; i < coordinates.length; i++):",
+	          '\n',
+	          "    _coords = coordinates[i]",
+	          '\n',
+	          "    line(coords, _coords)",
+	          '\n',
+	          "    coords = _coords"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "We start with the first coordinate as reference point, and then just draw lines between each point and its next point."
+	        )
+	      )
+	    );
+	  }
+	});
+
+	module.exports = Flattening;
+
+/***/ },
+/* 184 */
+/***/ function(module, exports, __webpack_require__) {
+
+	"use strict";
+
+	var React = __webpack_require__(8);
+	var Graphic = __webpack_require__(170);
+	var SectionHeader = __webpack_require__(175);
+
+	var Splitting = React.createClass({
+	  displayName: "Splitting",
+
+	  statics: {
+	    title: "Splitting curves"
+	  },
+
+	  setupCubic: function setupCubic(api) {
+	    var curve = api.getDefaultCubic();
+	    api.setCurve(curve);
+	    api.forward = true;
+	  },
+
+	  drawSplit: function drawSplit(api, curve) {
+	    api.setPanelCount(2);
+	    api.reset();
+	    api.drawSkeleton(curve);
+	    api.drawCurve(curve);
+
+	    var offset = { x: 0, y: 0 };
+	    var t = 0.5;
+	    var pt = curve.get(0.5);
+	    var split = curve.split(t);
+	    api.drawCurve(split.left);
+	    api.drawCurve(split.right);
+	    api.setColor("red");
+	    api.drawCircle(pt, 3);
+
+	    api.setColor("black");
+	    offset.x = api.getPanelWidth();
+	    api.drawLine({ x: 0, y: 0 }, { x: 0, y: api.getPanelHeight() }, offset);
+
+	    api.setColor("lightgrey");
+	    api.drawCurve(curve, offset);
+	    api.drawCircle(pt, 4);
+
+	    offset.x -= 20;
+	    offset.y -= 20;
+	    api.drawSkeleton(split.left, offset, true);
+	    api.drawCurve(split.left, offset);
+
+	    offset.x += 40;
+	    offset.y += 40;
+	    api.drawSkeleton(split.right, offset, true);
+	    api.drawCurve(split.right, offset);
+	  },
+
+	  drawAnimated: function drawAnimated(api, curve) {
+	    api.setPanelCount(3);
+	    api.reset();
+
+	    var frame = api.getFrame();
+	    var interval = 5 * api.getPlayInterval();
+	    var t = frame % interval / interval;
+	    var forward = frame % (2 * interval) < interval;
+	    if (forward) {
+	      t = t % 1;
+	    } else {
+	      t = 1 - t % 1;
+	    }
+	    var offset = { x: 0, y: 0 };
+
+	    api.setColor("lightblue");
+	    api.drawHull(curve, t);
+	    api.drawSkeleton(curve);
+	    api.drawCurve(curve);
+	    var pt = curve.get(t);
+	    api.drawCircle(pt, 4);
+
+	    api.setColor("black");
+	    offset.x += api.getPanelWidth();
+	    api.drawLine({ x: 0, y: 0 }, { x: 0, y: api.getPanelHeight() }, offset);
+
+	    var split = curve.split(t);
+
+	    api.setColor("lightgrey");
+	    api.drawCurve(curve, offset);
+	    api.drawHull(curve, t, offset);
+	    api.setColor("black");
+	    api.drawCurve(split.left, offset);
+	    api.drawPoints(split.left.points, offset);
+	    api.setFill("black");
+	    api.text("Left side of curve split at t = " + (100 * t | 0) / 100, { x: 10 + offset.x, y: 15 + offset.y });
+
+	    offset.x += api.getPanelWidth();
+	    api.drawLine({ x: 0, y: 0 }, { x: 0, y: api.getPanelHeight() }, offset);
+
+	    api.setColor("lightgrey");
+	    api.drawCurve(curve, offset);
+	    api.drawHull(curve, t, offset);
+	    api.setColor("black");
+	    api.drawCurve(split.right, offset);
+	    api.drawPoints(split.right.points, offset);
+	    api.setFill("black");
+	    api.text("Right side of curve split at t = " + (100 * t | 0) / 100, { x: 10 + offset.x, y: 15 + offset.y });
+	  },
+
+	  togglePlay: function togglePlay(evt, api) {
+	    if (api.playing) {
+	      api.pause();
+	    } else {
+	      api.play();
+	    }
+	  },
+
+	  render: function render() {
+	    return React.createElement(
+	      "section",
+	      null,
+	      React.createElement(
+	        SectionHeader,
+	        this.props,
+	        Splitting.title
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "With de Casteljau's algorithm we 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",
+	        React.createElement(
+	          "i",
+	          null,
+	          "t"
+	        ),
+	        ", the procedure gives us all the points we need to split a curve at that ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "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."
+	      ),
+	      React.createElement(Graphic, { title: "Splitting a curve", setup: this.setupCubic, draw: this.drawSplit }),
+	      React.createElement(
+	        "div",
+	        { className: "howtocode" },
+	        React.createElement(
+	          "h3",
+	          null,
+	          "implementing curve splitting"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "We can implement curve splitting by bolting some extra logging onto the de Casteljau function:"
+	        ),
+	        React.createElement(
+	          "pre",
+	          null,
+	          "left=[]",
+	          '\n',
+	          "right=[]",
+	          '\n',
+	          "function drawCurve(points[], t):",
+	          '\n',
+	          "  if(points.length==1):",
+	          '\n',
+	          "    left.add(points[0])",
+	          '\n',
+	          "    right.add(points[0])",
+	          '\n',
+	          "    draw(points[0])",
+	          '\n',
+	          "  else:",
+	          '\n',
+	          "    newpoints=array(points.size-1)",
+	          '\n',
+	          "    for(i=0; i<newpoints.length; i++):",
+	          '\n',
+	          "      if(i==0):",
+	          '\n',
+	          "        left.add(points[i])",
+	          '\n',
+	          "      if(i==newpoints.length-1):",
+	          '\n',
+	          "        right.add(points[i+1])",
+	          '\n',
+	          "      newpoints[i] = (1-t) * points[i] + t * points[i+1]",
+	          '\n',
+	          "    drawCurve(newpoints, t)"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "After running this function for some value ",
+	          React.createElement(
+	            "i",
+	            null,
+	            "t"
+	          ),
+	          ", the ",
+	          React.createElement(
+	            "i",
+	            null,
+	            "left"
+	          ),
+	          " and ",
+	          React.createElement(
+	            "i",
+	            null,
+	            "right"
+	          ),
+	          " arrays will contain all the coordinates for two new curves - one to the \"left\" of our ",
+	          React.createElement(
+	            "i",
+	            null,
+	            "t"
+	          ),
+	          " value, the other on the \"right\", of the same order as the original curve, and overlayed exactly on the original curve."
+	        )
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "This is best illustrated with an animated graphic (click to play/pause):"
+	      ),
+	      React.createElement(Graphic, { preset: "threepanel", title: "Bézier curve splitting", setup: this.setupCubic, draw: this.drawAnimated, onClick: this.togglePlay })
+	    );
+	  }
+	});
+
+	module.exports = Splitting;
+
+/***/ },
+/* 185 */
+/***/ function(module, exports, __webpack_require__) {
+
+	"use strict";
+
+	var React = __webpack_require__(8);
+	var Graphic = __webpack_require__(170);
+	var SectionHeader = __webpack_require__(175);
+
+	var MatrixSplit = React.createClass({
+	  displayName: "MatrixSplit",
+
+	  statics: {
+	    title: "Splitting curves using matrices"
+	  },
+
+	  render: function render() {
+	    return React.createElement(
+	      "section",
+	      null,
+	      React.createElement(
+	        SectionHeader,
+	        this.props,
+	        MatrixSplit.title
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "Another way to split curves is to exploit the matrix representation of a Bézier curve. In ",
+	        React.createElement(
+	          "a",
+	          { href: "#matrix" },
+	          "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 (using the reversed Bézier coefficients vector for legibility):"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/8fa7f5a1b32d06d66b0a4d0f000b2496e0bacccd.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "and"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/6fd34f4126772cb9ba06e36b9e40df95994cf847.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "Let's say we want to split the curve at some point ",
+	        React.createElement(
+	          "em",
+	          null,
+	          "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 split out the the actual \"point on the curve\" information as a new matrix multiplication:"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/a2d3c6b5b68d86adf40db56f7723a45aa5a0f7a6.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "and"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/1e2a1516ca2ce995a0ddebf218b7528bd762b686.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "If we could compact these matrices back to a form ",
+	        React.createElement(
+	          "strong",
+	          null,
+	          "[t values] · [bezier 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 ",
+	        React.createElement(
+	          "em",
+	          null,
+	          "t = 0"
+	        ),
+	        " to ",
+	        React.createElement(
+	          "em",
+	          null,
+	          "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!)."
+	      ),
+	      React.createElement(
+	        "div",
+	        { className: "note" },
+	        React.createElement(
+	          "h2",
+	          null,
+	          "Deriving new hull coordinates"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "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:"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/2aad48dc956ff2551caae8d7d35f299b395d6a28.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/dc9f128759d56fd5b1e6099da9d7e11e83bea104.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/b2f197498c75c7bf59bf4e9791e4e948a5699d8e.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/ab88c77e6c5b3e13ad44665606ad0352040ce89a.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "We do this, because [",
+	          React.createElement(
+	            "em",
+	            null,
+	            "M · M",
+	            React.createElement(
+	              "sup",
+	              null,
+	              "-1"
+	            )
+	          ),
+	          "] is the identity matrix (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). Adding that as matrix multiplication has no effect on the total formula, but it does allow us to change the matrix sequence [",
+	          React.createElement(
+	            "em",
+	            null,
+	            "something · M"
+	          ),
+	          "] to a sequence [",
+	          React.createElement(
+	            "em",
+	            null,
+	            "M · something"
+	          ),
+	          "], and that makes a world of difference: if we know what [",
+	          React.createElement(
+	            "em",
+	            null,
+	            "M",
+	            React.createElement(
+	              "sup",
+	              null,
+	              "-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 [",
+	          React.createElement(
+	            "em",
+	            null,
+	            "T · M · P"
+	          ),
+	          "]), with a new set of coordinates that represent the curve from",
+	          React.createElement(
+	            "em",
+	            null,
+	            "t = 0"
+	          ),
+	          " to ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "t = z"
+	          ),
+	          ". So let's get computing:"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/4744abca018b8ce799a19e39e45bb1021989c00c.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "Excellent! Now we can form our new quadratic curve:"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/38cf32cf7e00f9e2a5ef18f352f93a10c55ebc99.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/d67b68b7fbd11a041232c144e31122309fb9d782.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/8b9cf2edc06c8efabdf614f554b21fb929b0765d.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement(
+	            "strong",
+	            null,
+	            React.createElement(
+	              "em",
+	              null,
+	              "Brilliant"
+	            )
+	          ),
+	          ": if we want a subcurve from ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "t = 0"
+	          ),
+	          "to ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "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, except it uses (z-1) rather than (1-z)... These new coordinates are actually really easy to compute directly!"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "Of course, that's only one of the two curves. Getting the section from ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "t = z"
+	          ),
+	          "to ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "t = 1"
+	          ),
+	          " requires doing this again. We first observe what what we just did is actually evaluate the general interval [0,",
+	          React.createElement(
+	            "em",
+	            null,
+	            "z"
+	          ),
+	          "], which we wrote down simplified becuase of that zero, but we actually evaluated this:"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/e94f3c7fda60e80707370b02140840311bf33d75.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/5e90d865f1093a974f5aa1c8525c45649137a688.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "If we want the interval [",
+	          React.createElement(
+	            "em",
+	            null,
+	            "z"
+	          ),
+	          ",1], we will be evaluating this instead:"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/42efec3a2fe4620906cbf25bc11a28b354097b2c.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/373ca089b9e3928414a23931298dadf2886f1d71.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "We're going to do the same trick, to turn ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "[something · M]"
+	          ),
+	          " into ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "[M · something]"
+	          ),
+	          ":"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/4511ca44d7a85ba755774fc82dfbf9fa14f94f71.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          "So, our final second curve looks like:"
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/e04671db313e31b678512b59f5aff9de01dedf45.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/d16d8b52374301de6eb2aca5ba798799f0759b5a.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement("img", { className: "LaTeX SVG", src: "images/latex/6dd685fe5c634305918f67db4ed693c0095de0d2.svg" })
+	        ),
+	        React.createElement(
+	          "p",
+	          null,
+	          React.createElement(
+	            "strong",
+	            null,
+	            React.createElement(
+	              "em",
+	              null,
+	              "Nice"
+	            )
+	          ),
+	          ": we see the same as before; 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 it uses (z-1) rather than (1-z). These new coordinates are ",
+	          React.createElement(
+	            "em",
+	            null,
+	            "also"
+	          ),
+	          "really easy to compute directly!"
+	        )
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "So, using linear algebra rather than de Casteljau's algorithm, we have determined that for any quadratic curve split at some value ",
+	        React.createElement(
+	          "em",
+	          null,
+	          "t = z"
+	        ),
+	        ", we get two subcurves that are described as Bézier curves with simple-to-derive coordinates."
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/5bd82c66c21d9d4778726bde0ded0070e6a917ab.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "and"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/836fb6adcdca2ac5372e04d38becbfe355988ba9.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "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:"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/6c4f49fdad56663b8955f3eb1b9b42a2435d9b13.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "and"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/c925370c4e7896c34d1a315db126364538918ea3.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "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 ",
+	        React.createElement(
+	          "strong",
+	          null,
+	          React.createElement(
+	            "em",
+	            null,
+	            "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\" ",
+	        React.createElement(
+	          "strong",
+	          null,
+	          React.createElement(
+	            "em",
+	            null,
+	            "Q'"
+	          )
+	        ),
+	        "."
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "Implementing curve splitting this way requires less recursion, and is just straight arithmetic with cached values, so can be cheaper on systems were 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."
+	      )
+	    );
+	  }
+	});
+
+	module.exports = MatrixSplit;
+
+/***/ },
+/* 186 */
+/***/ function(module, exports, __webpack_require__) {
+
+	"use strict";
+
+	var React = __webpack_require__(8);
+	var Graphic = __webpack_require__(170);
+	var SectionHeader = __webpack_require__(175);
+
+	var Reordering = React.createClass({
+	  displayName: "Reordering",
+
+	  statics: {
+	    title: "Lowering and elevating curve order"
+	  },
+
+	  getInitialState: function getInitialState() {
+	    return {
+	      order: 0
+	    };
+	  },
+
+	  setup: function setup(api) {
+	    var points = [];
+	    var w = api.getPanelWidth(),
+	        h = api.getPanelHeight();
+	    for (var i = 0; i < 10; i++) {
+	      points.push({
+	        x: w / 2 + Math.random() * 20 + Math.cos(Math.PI * 2 * i / 10) * (w / 2 - 40),
+	        y: h / 2 + Math.random() * 20 + Math.sin(Math.PI * 2 * i / 10) * (h / 2 - 40)
+	      });
+	    }
+	    var curve = new api.Bezier(points);
+	    api.setCurve(curve);
+	  },
+
+	  draw: function draw(api, curve) {
+	    api.reset();
+	    var pts = curve.points;
+
+	    this.setState({
+	      order: pts.length
+	    });
+
+	    var p0 = pts[0];
+
+	    // we can't "just draw" this curve, since it'll be an arbitrary order,
+	    // And the canvas only does 2nd and 3rd - we use de Casteljau's algorithm:
+	    for (var t = 0; t <= 1; t += 0.01) {
+	      var q = JSON.parse(JSON.stringify(pts));
+	      while (q.length > 1) {
+	        for (var i = 0; i < q.length - 1; i++) {
+	          q[i] = {
+	            x: q[i].x + (q[i + 1].x - q[i].x) * t,
+	            y: q[i].y + (q[i + 1].y - q[i].y) * t
+	          };
+	        }
+	        q.splice(q.length - 1, 1);
+	      }
+	      api.drawLine(p0, q[0]);
+	      p0 = q[0];
+	    }
+
+	    p0 = pts[0];
+	    api.setColor("black");
+	    api.drawCircle(p0, 3);
+	    pts.forEach(function (p) {
+	      if (p === p0) return;
+	      api.setColor("#DDD");
+	      api.drawLine(p0, p);
+	      api.setColor("black");
+	      api.drawCircle(p, 3);
+	      p0 = p;
+	    });
+	  },
+
+	  // Improve this based on http://www.sirver.net/blog/2011/08/23/degree-reduction-of-bezier-curves/
+	  lower: function lower(curve) {
+	    var pts = curve.points,
+	        q = [],
+	        n = pts.length;
+	    pts.forEach(function (p, k) {
+	      if (!k) {
+	        return q[k] = p;
+	      }
+	      var f1 = k / n,
+	          f2 = 1 - f1;
+	      q[k] = {
+	        x: f1 * p.x + f2 * pts[k - 1].x,
+	        y: f1 * p.y + f2 * pts[k - 1].y
+	      };
+	    });
+	    q.splice(n - 1, 1);
+	    q[n - 2] = pts[n - 1];
+	    curve.points = q;
+	    return curve;
+	  },
+
+	  onKeyDown: function onKeyDown(evt, api) {
+	    evt.preventDefault();
+	    if (evt.key === "ArrowUp") {
+	      api.setCurve(api.curve.raise());
+	    } else if (evt.key === "ArrowDown") {
+	      api.setCurve(this.lower(api.curve));
+	    }
+	  },
+
+	  getOrder: function getOrder() {
+	    var order = this.state.order;
+	    if (order % 10 === 1 && order !== 11) {
+	      order += "st";
+	    } else if (order % 10 === 2 && order !== 12) {
+	      order += "nd";
+	    } else if (order % 10 === 3 && order !== 13) {
+	      order += "rd";
+	    } else {
+	      order += "th";
+	    }
+	    return order;
+	  },
+
+	  render: function render() {
+	    return React.createElement(
+	      "section",
+	      null,
+	      React.createElement(
+	        SectionHeader,
+	        this.props,
+	        Reordering.title
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "One interesting property of Bézier curves is that an ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "n",
+	          React.createElement(
+	            "sup",
+	            null,
+	            "th"
+	          )
+	        ),
+	        " order curve can always be perfectly represented by an ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "(n+1)",
+	          React.createElement(
+	            "sup",
+	            null,
+	            "th"
+	          )
+	        ),
+	        " order curve, by giving the higher order curve specific control points."
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "If we have a curve with three points, then we can create a four point curve that exactly reproduce the original curve as long as we give it the same start and end points, and for its two control points we pick \"1/3",
+	        React.createElement(
+	          "sup",
+	          null,
+	          "rd"
+	        ),
+	        " start + 2/3",
+	        React.createElement(
+	          "sup",
+	          null,
+	          "rd"
+	        ),
+	        " control\" and \"2/3",
+	        React.createElement(
+	          "sup",
+	          null,
+	          "rd"
+	        ),
+	        " control + 1/3",
+	        React.createElement(
+	          "sup",
+	          null,
+	          "rd"
+	        ),
+	        " end\", and now we have exactly the same curve as before, except represented as a cubic curve, rather than a quadratic curve."
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "The general rule for raising an ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "n",
+	          React.createElement(
+	            "sup",
+	            null,
+	            "th"
+	          )
+	        ),
+	        " order curve to an ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "(n+1)",
+	          React.createElement(
+	            "sup",
+	            null,
+	            "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):"
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        React.createElement("img", { className: "LaTeX SVG", src: "images/latex/c6061bb273e862f3a91b83c6a8b4713a959e0dbf.svg" })
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "However, this rule also has as direct consequence that you ",
+	        React.createElement(
+	          "strong",
+	          null,
+	          "cannot"
+	        ),
+	        " generally safely lower a curve from ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "n",
+	          React.createElement(
+	            "sup",
+	            null,
+	            "th"
+	          )
+	        ),
+	        " order to ",
+	        React.createElement(
+	          "i",
+	          null,
+	          "(n-1)",
+	          React.createElement(
+	            "sup",
+	            null,
+	            "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."
+	      ),
+	      React.createElement(
+	        "p",
+	        null,
+	        "We can apply this to a (semi) random curve, as is done in the following graphic. Select the sketch and press your up and down cursor keys to elevate or lower the curve order."
+	      ),
+	      React.createElement(Graphic, { preset: "simple", title: "A " + this.getOrder() + " order Bézier curve", setup: this.setup, draw: this.draw, onKeyDown: this.onKeyDown }),
+	      React.createElement(
+	        "p",
+	        null,
+	        "There is a good, if mathematical, explanation on the matrices necessary for optimal reduction over on ",
+	        React.createElement(
+	          "a",
+	          { href: "http://www.sirver.net/blog/2011/08/23/degree-reduction-of-bezier-curves/" },
+	          "Sirver's Castle"
+	        ),
+	        ", which given time will find its way in a more direct description into this article."
+	      )
+	    );
+	  }
+	});
+
+	module.exports = Reordering;
+
+	/*
+	void setupCurve() {
+	          int d = dim - 2*pad;
+	          int order = 10;
+	          ArrayList<Point> pts = new ArrayList<Point>();
+
+	          float dst = d/2.5, nx, ny, a=0, step = 2*PI/order, r;
+	          for(a=0; a<2*PI; a+=step) {
+	            r = random(-dst/4,dst/4);
+	            pts.add(new Point(d/2 + cos(a) * (r+dst), d/2 + sin(a) * (r+dst)));
+	            dst -= 1.2;
+	          }
+
+	          Point[] points = new Point[pts.size()];
+	          for(int p=0,last=points.length; p<last; p++) { points[p] = pts.get(p); }
+	          curves.add(new BezierCurve(points));
+	          reorder();
+	        }
+
+	        void drawCurve(BezierCurve curve) {
+	          curve.draw();
+	        }</textarea>
+	 */
+
+/***/ },
+/* 187 */
 /***/ function(module, exports, __webpack_require__) {
 
 	// style-loader: Adds some css to the DOM by adding a <style> tag
 
 	// load the styles
-	var content = __webpack_require__(184);
+	var content = __webpack_require__(188);
 	if(typeof content === 'string') content = [[module.id, content, '']];
 	// add the styles to the DOM
-	var update = __webpack_require__(187)(content, {});
+	var update = __webpack_require__(191)(content, {});
 	if(content.locals) module.exports = content.locals;
 	// Hot Module Replacement
 	if(false) {
@@ -25917,21 +27143,21 @@
 	}
 
 /***/ },
-/* 184 */
+/* 188 */
 /***/ function(module, exports, __webpack_require__) {
 
-	exports = module.exports = __webpack_require__(185)();
+	exports = module.exports = __webpack_require__(189)();
 	// imports
 
 
 	// module
-	exports.push([module.id, "html,\nbody {\n  font-family: Verdana;\n  margin: 0;\n  padding: 0;\n}\nbody {\n  background: url(" + __webpack_require__(186) + ");\n}\nheader,\nsection,\nfooter {\n  width: 960px;\n  margin: 0 auto;\n}\nheader {\n  font-family: Times;\n  text-align: center;\n  margin-bottom: 2rem;\n}\nheader h1 {\n  font-size: 360%;\n  margin: 0;\n  margin-bottom: 1rem;\n}\nheader h2 {\n  font-size: 125%;\n  margin: 0;\n}\narticle {\n  font-family: Verdana;\n  width: 960px;\n  margin: auto;\n  background: rgba(255, 255, 255, 0.74);\n  border: solid rgba(255, 0, 0, 0.35);\n  border-width: 0;\n  border-left-width: 1px;\n  padding: 1em;\n  box-shadow: 25px 0px 25px 25px rgba(255, 255, 255, 0.74);\n}\na,\na:visited {\n  color: #0000c8;\n  text-decoration: none;\n}\n#ribbonimg {\n  position: fixed;\n  top: 0;\n  right: 0;\n  z-index: 999;\n}\nfooter {\n  font-style: italic;\n  margin: 2em 0 1em 0;\n  background: inherit;\n}\nnavigation {\n  font-family: Georgia;\n  display: block;\n  width: 70%;\n  margin: 0 auto;\n  padding: 0;\n  border: 1px solid grey;\n}\nnavigation ul {\n  background: #F2F2F9;\n  list-style: none;\n  margin: 0;\n  padding: 0.5em 1em;\n}\nnavigation ul li:nth-child(n+2):before {\n  content: \"\\A7\" attr(data-number) \". \";\n}\nsection p {\n  text-align: justify;\n}\nsection h2[data-num]:before {\n  content: \"\\A7\" attr(data-num) \" \\2014   \";\n}\nsection h2 a,\nsection h2 a:active,\nsection h2 a:hover,\nsection h2 a:visited {\n  text-decoration: none;\n  color: inherit;\n}\ndiv.note {\n  font-size: 90%;\n  margin: 1em 2em;\n  padding: 1em;\n  border: 1px solid grey;\n  background: rgba(150, 150, 50, 0.05);\n}\ndiv.note * {\n  margin: 0;\n  padding: 0;\n}\ndiv.note p {\n  margin: 1em 0;\n}\ndiv.note div.MathJax_Display {\n  margin: 1em 0;\n}\n.howtocode {\n  border: 1px solid #8d94bd;\n  padding: 0 1em;\n  margin: 0 2em;\n  overflow-x: hidden;\n}\n.howtocode h3 {\n  margin: 0 -1em;\n  padding: 0;\n  background: #91bef7;\n  padding-left: 0.5em;\n  color: white;\n  text-shadow: 1px 1px 0 #000000;\n  cursor: pointer;\n}\n.howtocode pre {\n  border: 1px solid #8d94bd;\n  background: rgba(223, 226, 243, 0.32);\n  margin: 0.5em;\n  padding: 0.5em;\n}\nfigure {\n  display: inline-block;\n  border: 1px solid grey;\n  background: #F0F0F0;\n  padding: 0.5em 0.5em 0 0.5em;\n  text-align: center;\n}\nfigure.inline {\n  border: none;\n  margin: 0;\n}\nfigure canvas {\n  display: inline-block;\n  background: white;\n  border: 1px solid lightgrey;\n}\nfigure figcaption {\n  text-align: center;\n  padding: 0.5em 0;\n  font-style: italic;\n  font-size: 90%;\n}\ndiv.figure {\n  display: inline-block;\n  border: 1px solid grey;\n  text-align: center;\n}\ngithub-issues {\n  position: relative;\n  display: block;\n  width: 100%;\n  border: 1px solid #EEE;\n  border-left: 0.3em solid #e5ecf3;\n  background: white;\n  padding: 0 0.3em;\n  width: 95%;\n  margin: auto;\n  min-height: 33px;\n  font: 13px Helvetica, arial, freesans, clean, sans-serif;\n}\ngithub-issues github-issue + github-issue {\n  margin-top: 1em;\n}\ngithub-issues github-issue h3 {\n  font-size: 100%;\n  background: #e5ecf3;\n  margin: 0;\n  position: relative;\n  left: -0.5%;\n  width: 101%;\n  font-weight: bold;\n  border-bottom: 1px solid #999;\n}\ngithub-issues github-issue a {\n  position: absolute;\n  top: 2px;\n  right: 10px;\n  padding: 0 4px;\n  color: #4183C4!important;\n  background: white;\n  line-height: 10px;\n  font-size: 10px;\n}\nimg.LaTeX {\n  display: block;\n  margin-left: 2em;\n}\n", ""]);
+	exports.push([module.id, "html,\nbody {\n  font-family: Verdana;\n  margin: 0;\n  padding: 0;\n}\nbody {\n  background: url(" + __webpack_require__(190) + ");\n}\nheader,\nsection,\nfooter {\n  width: 960px;\n  margin: 0 auto;\n}\nheader {\n  font-family: Times;\n  text-align: center;\n  margin-bottom: 2rem;\n}\nheader h1 {\n  font-size: 360%;\n  margin: 0;\n  margin-bottom: 1rem;\n}\nheader h2 {\n  font-size: 125%;\n  margin: 0;\n}\narticle {\n  font-family: Verdana;\n  width: 960px;\n  margin: auto;\n  background: rgba(255, 255, 255, 0.74);\n  border: solid rgba(255, 0, 0, 0.35);\n  border-width: 0;\n  border-left-width: 1px;\n  padding: 1em;\n  box-shadow: 25px 0px 25px 25px rgba(255, 255, 255, 0.74);\n}\na,\na:visited {\n  color: #0000c8;\n  text-decoration: none;\n}\n#ribbonimg {\n  position: fixed;\n  top: 0;\n  right: 0;\n  z-index: 999;\n}\nfooter {\n  font-style: italic;\n  margin: 2em 0 1em 0;\n  background: inherit;\n}\nnavigation {\n  font-family: Georgia;\n  display: block;\n  width: 70%;\n  margin: 0 auto;\n  padding: 0;\n  border: 1px solid grey;\n}\nnavigation ul {\n  background: #F2F2F9;\n  list-style: none;\n  margin: 0;\n  padding: 0.5em 1em;\n}\nnavigation ul li:nth-child(n+2):before {\n  content: \"\\A7\" attr(data-number) \". \";\n}\nsection p {\n  text-align: justify;\n}\nsection h2[data-num]:before {\n  content: \"\\A7\" attr(data-num) \" \\2014   \";\n}\nsection h2 a,\nsection h2 a:active,\nsection h2 a:hover,\nsection h2 a:visited {\n  text-decoration: none;\n  color: inherit;\n}\ndiv.note {\n  font-size: 90%;\n  margin: 1em 2em;\n  padding: 1em;\n  border: 1px solid grey;\n  background: rgba(150, 150, 50, 0.05);\n}\ndiv.note * {\n  margin: 0;\n  padding: 0;\n}\ndiv.note p {\n  margin: 1em 0;\n}\ndiv.note div.MathJax_Display {\n  margin: 1em 0;\n}\n.howtocode {\n  border: 1px solid #8d94bd;\n  padding: 0 1em;\n  margin: 0 2em;\n  overflow-x: hidden;\n}\n.howtocode h3 {\n  margin: 0 -1em;\n  padding: 0;\n  background: #91bef7;\n  padding-left: 0.5em;\n  color: white;\n  text-shadow: 1px 1px 0 #000000;\n  cursor: pointer;\n}\n.howtocode pre {\n  border: 1px solid #8d94bd;\n  background: rgba(223, 226, 243, 0.32);\n  margin: 0.5em;\n  padding: 0.5em;\n}\nfigure {\n  display: inline-block;\n  border: 1px solid grey;\n  background: #F0F0F0;\n  padding: 0.5em 0.5em 0 0.5em;\n  text-align: center;\n}\nfigure.inline {\n  border: none;\n  margin: 0;\n}\nfigure canvas {\n  display: inline-block;\n  background: white;\n  border: 1px solid lightgrey;\n}\nfigure canvas:focus {\n  border: 1px solid grey;\n}\nfigure figcaption {\n  text-align: center;\n  padding: 0.5em 0;\n  font-style: italic;\n  font-size: 90%;\n}\ndiv.figure {\n  display: inline-block;\n  border: 1px solid grey;\n  text-align: center;\n}\ngithub-issues {\n  position: relative;\n  display: block;\n  width: 100%;\n  border: 1px solid #EEE;\n  border-left: 0.3em solid #e5ecf3;\n  background: white;\n  padding: 0 0.3em;\n  width: 95%;\n  margin: auto;\n  min-height: 33px;\n  font: 13px Helvetica, arial, freesans, clean, sans-serif;\n}\ngithub-issues github-issue + github-issue {\n  margin-top: 1em;\n}\ngithub-issues github-issue h3 {\n  font-size: 100%;\n  background: #e5ecf3;\n  margin: 0;\n  position: relative;\n  left: -0.5%;\n  width: 101%;\n  font-weight: bold;\n  border-bottom: 1px solid #999;\n}\ngithub-issues github-issue a {\n  position: absolute;\n  top: 2px;\n  right: 10px;\n  padding: 0 4px;\n  color: #4183C4!important;\n  background: white;\n  line-height: 10px;\n  font-size: 10px;\n}\nimg.LaTeX {\n  display: block;\n  margin-left: 2em;\n}\n", ""]);
 
 	// exports
 
 
 /***/ },
-/* 185 */
+/* 189 */
 /***/ function(module, exports) {
 
 	/*
@@ -25987,13 +27213,13 @@
 
 
 /***/ },
-/* 186 */
+/* 190 */
 /***/ function(module, exports, __webpack_require__) {
 
 	module.exports = __webpack_require__.p + "images/packed/7d3b28205544712db60d1bb7973f10f3.png";
 
 /***/ },
-/* 187 */
+/* 191 */
 /***/ function(module, exports, __webpack_require__) {
 
 	/*
diff --git a/components/Graphic.jsx b/components/Graphic.jsx
index 0043cab8..7fe74955 100644
--- a/components/Graphic.jsx
+++ b/components/Graphic.jsx
@@ -24,16 +24,37 @@ var Graphic = React.createClass({
   offset: { x: 0, y: 0},
   lpts: [],
   colorSeed: 0,
+  playing: false,
+  frame: 0,
+  playinterval: 33,
+
+  animate: function animate() {
+    if (this.playing) {
+      this.frame++;
+      // requestAnimationFrame is spectacularly too fast
+      setTimeout(this.animate, this.playinterval);
+      this.props.draw(this,this.curve);
+    }
+  },
+
+  getFrame: function() { return this.frame; },
+  getPlayInterval: function() { return this.playinterval; },
+  play: function() { this.playing = true; this.animate(); },
+  pause: function() { this.playing = false; },
 
   render: function() {
     var href = "data:text/plain," + this.props.code;
     return (
       <figure className={this.props.inline ? "inline": false}>
         <canvas ref="canvas"
+                tabIndex="0"
                 onMouseDown={this.mouseDown}
                 onMouseMove={this.mouseMove}
                 onMouseUp={this.mouseUp}
-                onClick={this.mouseClick}
+                onClick={this.onClick}
+                onKeyUp={this.onKeyUp}
+                onKeyDown={this.onKeyDown}
+                onKeyPress={this.onKeyPress}
                 />
         <figcaption>{this.props.title}</figcaption>
       </figure>
@@ -55,6 +76,10 @@ var Graphic = React.createClass({
     if (this.props.draw) {
       this.props.draw(this,this.curve);
     }
+
+    if (this.props.autoplay) {
+      this.play();
+    }
   },
 
   mouseDown: function(evt) {
@@ -69,6 +94,10 @@ var Graphic = React.createClass({
         this.cy = p.y;
       }
     });
+
+    if (this.props.mouseDown) {
+      this.props.mouseDown(evt, this);
+    }
   },
 
   mouseMove: function(evt) {
@@ -96,7 +125,11 @@ var Graphic = React.createClass({
       this.curve.update();
     }
 
-    if (this.props.draw) {
+    if (this.props.mouseMove) {
+      this.props.mouseMove(evt, this);
+    }
+
+    if (!this.playing && this.props.draw) {
       this.props.draw(this, this.curve);
     }
   },
@@ -105,12 +138,46 @@ var Graphic = React.createClass({
     if(!this.moving) return;
     this.moving = false;
     this.mp = false;
+    if (this.props.onMouseUp) {
+      this.props.onMouseUp(evt, this);
+    }
   },
 
-  mouseClick: function(evt) {
+  onClick: function(evt) {
     fix(evt);
     this.mx = evt.offsetX;
     this.my = evt.offsetY;
+    if (this.props.onClick) {
+      this.props.onClick(evt, this);
+    }
+  },
+
+
+  onKeyUp: function(evt) {
+    if (this.props.onKeyUp) {
+      this.props.onKeyUp(evt, this);
+      if (!this.playing && this.props.draw) {
+        this.props.draw(this, this.curve);
+      }
+    }
+  },
+
+  onKeyDown: function(evt) {
+    if (this.props.onKeyDown) {
+      this.props.onKeyDown(evt, this);
+      if (!this.playing && this.props.draw) {
+        this.props.draw(this, this.curve);
+      }
+    }
+  },
+
+  onKeyPress: function(evt) {
+    if (this.props.onKeyPress) {
+      this.props.onKeyPress(evt, this);
+      if (!this.playing && this.props.draw) {
+        this.props.draw(this, this.curve);
+      }
+    }
   },
 
 
@@ -204,7 +271,7 @@ var Graphic = React.createClass({
     this.ctx.fillStyle = "transparent";
   },
 
-  drawSkeleton: function(curve, offset) {
+  drawSkeleton: function(curve, offset, nocoords) {
     offset = offset || { x:0, y:0 };
     var pts = curve.points;
     this.ctx.strokeStyle = "lightgrey";
@@ -213,7 +280,9 @@ var Graphic = React.createClass({
     else {this.drawLine(pts[2], pts[3], offset); }
     this.ctx.strokeStyle = "black";
     this.drawPoints(pts, offset);
-    this.drawCoordinates(curve, offset);
+    if (!nocoords) {
+      this.drawCoordinates(curve, offset);
+    }
   },
 
   drawHull: function(curve, t, offset) {
@@ -238,7 +307,7 @@ var Graphic = React.createClass({
     var pts = curve.points;
     this.setFill("black");
     pts.forEach(p => {
-      this.text(`(${p.x},${p.y})`, {x:p.x + 5, y:p.y + 10});
+      this.text(`(${p.x},${p.y})`, {x:p.x + offset.x + 5, y:p.y + offset.y + 10});
     });
   },
 
diff --git a/components/sections/control/index.js b/components/sections/control/index.js
index 9e152ff8..d210e429 100644
--- a/components/sections/control/index.js
+++ b/components/sections/control/index.js
@@ -276,37 +276,3 @@ function Bezier(3,t,w[]):
 });
 
 module.exports = Control;
-
-
-
-
-
-/**
-
-        <textarea class="sketch-code" data-sketch-preset="ratios" data-sketch-title="Quadratic interpolations">
-        int order = 3;
-        </textarea>
-
-        <textarea class="sketch-code" data-sketch-preset="ratios" data-sketch-title="Cubic interpolations">
-        int order = 4;
-        </textarea>
-
-        <textarea class="sketch-code" data-sketch-preset="ratios" data-sketch-title="15th order interpolations">
-        int order = 15;
-        </textarea>
-
- **/
-
-/**
-
-        void setupCurve() {
-          setupDefaultCubic();
-        }
-
-        void drawCurve(BezierCurve curve) {
-          curve.draw();
-        }</Graphic>
-
-
-**/
-
diff --git a/components/sections/flattening/index.js b/components/sections/flattening/index.js
new file mode 100644
index 00000000..40353dd4
--- /dev/null
+++ b/components/sections/flattening/index.js
@@ -0,0 +1,101 @@
+var React = require("react");
+var Graphic = require("../../Graphic.jsx");
+var SectionHeader = require("../../SectionHeader.jsx");
+
+var Flattening = React.createClass({
+  statics: {
+    title: "Simplified drawing"
+  },
+
+  setupQuadratic: function(api) {
+    var curve = api.getDefaultQuadratic();
+    api.setCurve(curve);
+    api.steps = 3;
+  },
+
+  setupCubic: function(api) {
+    var curve = api.getDefaultCubic();
+    api.setCurve(curve);
+    api.steps = 5;
+  },
+
+  drawFlattened: function(api, curve) {
+    api.reset();
+    api.setColor("#DDD");
+    api.drawSkeleton(curve);
+    api.setColor("#DDD");
+    api.drawCurve(curve);
+    var step = 1 / api.steps;
+    var p0 = curve.points[0], pc;
+    for(var t=step; t<1.0+step; t+=step) {
+      pc = curve.get(Math.min(t,1));
+      api.setColor("red");
+      api.drawLine(p0,pc);
+      p0 = pc;
+    }
+    api.setFill("black");
+    api.text("Curve approximation using "+api.steps+" segments", {x:10, y:15});
+  },
+
+  values: {
+    "+": 1,
+    "-": -1
+  },
+
+  onKeyDown: function(e, api) {
+    e.preventDefault();
+    var v = this.values[e.key];
+    if(v) {
+      api.steps += v;
+      if (api.steps < 1) {
+        api.steps = 1;
+      }
+    }
+  },
+
+  render: function() {
+    return (
+      <section>
+        <SectionHeader {...this.props}>{ Flattening.title }</SectionHeader>
+
+        <p>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.</p>
+
+        <p>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 for doing true intersection detection, or curvature alignment.</p>
+
+        <Graphic preset="twopanel" title="Flattening a quadratic curve" setup={this.setupQuadratic} draw={this.drawFlattened} onKeyDown={this.onKeyDown}/>
+        <Graphic preset="twopanel" title="Flattening a cubic curve" setup={this.setupCubic} draw={this.drawFlattened} onKeyDown={this.onKeyDown} />
+
+        <p>Try clicking on the sketch and using your '+' and '-' 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.</p>
+
+        <div className="howtocode">
+          <h3>How to implement curve flattening</h3>
+
+          <p>Let's just use the algorithm we just specified, and implement that:</p>
+
+<pre>function flattenCurve(curve, segmentCount):
+  step = 1/segmentCount;
+  coordinates = [curve.getXValue(0), curve.getYValue(0)]
+  for(i=1; i &lt;= segmentCount; i++):
+    t = i*step;
+    coordinates.push[curve.getXValue(t), curve.getYValue(t)]
+  return coordinates;</pre>
+
+          <p>And done, that's the algorithm implemented. That just leaves drawing the resulting "curve" as a sequence of lines:</p>
+
+<pre>function drawFlattenedCurve(curve, segmentCount):
+  coordinates = flattenCurve(curve, segmentCount)
+  coord = coordinates[0], _coords;
+  for(i=1; i &lt; coordinates.length; i++):
+    _coords = coordinates[i]
+    line(coords, _coords)
+    coords = _coords</pre>
+
+          <p>We start with the first coordinate as reference point, and then just draw lines between each point and its next point.</p>
+        </div>
+
+      </section>
+    );
+  }
+});
+
+module.exports = Flattening;
diff --git a/components/sections/index.js b/components/sections/index.js
index ff31ba17..2af35fb9 100644
--- a/components/sections/index.js
+++ b/components/sections/index.js
@@ -9,17 +9,16 @@ module.exports = {
   explanation: require("./explanation"),
   control: require("./control"),
   matrix: require("./matrix"),
-  decasteljau: require("./decasteljau")
+  decasteljau: require("./decasteljau"),
+  flattening: require("./flattening"),
+  splitting: require("./splitting"),
+  matrixsplit: require("./matrixsplit"),
+  reordering: require("./reordering")
 };
 
 
 /*
 
-  flattening: require("./flattening"),
-  splitting: require("./splitting"),
-  matrixsplit: require("./matrixsplit"),
-  reordering: require("./reordering"),
-
   derivatives: require("./derivatives"),
   pointvectors: require("./pointvectors"),
   components: require("./components"),
@@ -53,3 +52,43 @@ module.exports = {
   circles_cubic: require("./circles_cubic"),
   arcapproximation: require("./arcapproximation")
 */
+
+/*
+
+  A lightning introduction
+  What is a Bézier curve?
+  The basics of Bézier curves
+  Controlling Bézier curvatures
+  Bézier curvatures as matrix operations
+  de Casteljau's algorithm
+  Simplified drawing
+  Splitting curves
+  Splitting curves using matrices
+  Lowering and elevating curve order
+  Derivatives
+  Tangents and normals
+  Component functions
+  Finding extremities
+  Bounding boxes
+  Aligning curves
+  Tight boxes
+  The canonical form (for cubic curves)
+  Arc length
+  Approximated arc length
+  Tracing a curve at fixed distance intervals
+  Intersections
+  Curve/curve intersection
+  Curve moulding (using the projection ratio)
+  Creating a curve from three points
+  Bézier curves and Catmull-Rom curves
+  Creating a Catmull-Rom curve from three points
+  Forming poly-Bézier curves
+  Boolean shape operations
+  Projecting a point onto a Bézier curve
+  Curve offsetting
+  Graduated curve offsetting
+  Circles and quadratic Bézier curves
+  Circles and cubic Bézier curves
+  Approximating Bézier curves with circular arcs
+
+ */
\ No newline at end of file
diff --git a/components/sections/matrixsplit/index.js b/components/sections/matrixsplit/index.js
new file mode 100644
index 00000000..9102e3e5
--- /dev/null
+++ b/components/sections/matrixsplit/index.js
@@ -0,0 +1,631 @@
+var React = require("react");
+var Graphic = require("../../Graphic.jsx");
+var SectionHeader = require("../../SectionHeader.jsx");
+
+var MatrixSplit = React.createClass({
+  statics: {
+    title: "Splitting curves using matrices"
+  },
+
+  render: function() {
+    return (
+      <section>
+        <SectionHeader {...this.props}>{ MatrixSplit.title }</SectionHeader>
+
+        <p>Another way to split curves is to exploit the matrix representation of
+        a Bézier curve. In <a href="#matrix">the section on matrices</a> we saw that
+        we can represent curves as matrix multiplications. Specifically, we saw these
+        two forms for the quadratic, and cubic curves, respectively (using the reversed
+        Bézier coefficients vector for legibility):</p>
+
+        <p>\[
+          B(t) = \begin{bmatrix}
+          1 & t & t^2
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           1 &  0 & 0 \\
+          -2 &  2 & 0 \\
+           1 & -2 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3
+          \end{bmatrix}
+        \]</p>
+
+        <p>and</p>
+
+        <p>\[
+          B(t) = \begin{bmatrix}
+          1 & t & t^2 & t^3
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           1 &  0 &  0 & 0\\
+          -3 &  3 &  0 & 0\\
+           3 & -6 &  3 & 0\\
+          -1 &  3 & -3 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3 \\ P_4
+          \end{bmatrix}
+        \]</p>
+
+        <p>Let's say we want to split the curve at some point <em>t = z</em>, 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 split out the the actual "point on the curve" information as a new matrix
+        multiplication:</p>
+
+        <p>\[
+          B(t) =
+          \begin{bmatrix}
+          1 & (z \cdot t) & (z \cdot t)^2
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           1 &  0 & 0 \\
+          -2 &  2 & 0 \\
+           1 & -2 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3
+          \end{bmatrix}
+          =
+          \begin{bmatrix}
+          1 & t & t^2
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          1 & 0 & 0 \\
+          0 & z & 0 \\
+          0 & 0 & z^2
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           1 &  0 & 0 \\
+          -2 &  2 & 0 \\
+           1 & -2 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3
+          \end{bmatrix}
+        \]</p>
+
+        <p>and</p>
+
+        <p>\[
+          B(t) =
+          \begin{bmatrix}
+          1 & (z \cdot t) & (z \cdot t)^2 & (z \cdot t)^3
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           1 &  0 &  0 & 0 \\
+          -3 &  3 &  0 & 0 \\
+           3 & -6 &  3 & 0 \\
+          -1 &  3 & -3 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3 \\ P_4
+          \end{bmatrix}
+          =
+          \begin{bmatrix}
+          1 & t & t^2 & t^3
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          1 & 0 & 0   & 0\\
+          0 & z & 0   & 0\\
+          0 & 0 & z^2 & 0\\
+          0 & 0 & 0   & z^3
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           1 &  0 &  0 & 0 \\
+          -3 &  3 &  0 & 0 \\
+           3 & -6 &  3 & 0 \\
+          -1 &  3 & -3 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3 \\ P_4
+          \end{bmatrix}
+        \]</p>
+
+        <p>If we could compact these matrices back to a form <strong>[t values] · [bezier matrix] · [column matrix]</strong>,
+        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 <em>t = 0</em> to <em>t = z</em>.
+        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!).</p>
+
+        <div className="note">
+          <h2>Deriving new hull coordinates</h2>
+
+          <p>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:</p>
+
+          <p>\[
+            B(t) =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            0 & z & 0 \\
+            0 & 0 & z^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \underset{we\ turn\ this...}{\underbrace{Z \cdot M}}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \underset{...into\ this!}{\underbrace{ M \cdot M^{-1} \cdot Z \cdot M }}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            M
+            \cdot
+            Q
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>We do this, because [<em>M · M<sup>-1</sup></em>] is the identity matrix (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). Adding that as matrix multiplication has no effect on the total
+          formula, but it does allow us to change the matrix sequence [<em>something · M</em>] to
+          a sequence [<em>M · something</em>], and that makes a world of difference: if we know
+          what [<em>M<sup>-1</sup> · Z · M</em>] is, we can apply that to our coordinates, and be
+          left with a proper matrix representation of a quadratic Bézier curve (which is
+          [<em>T · M · P</em>]), with a new set of coordinates that represent the curve from
+          <em>t = 0</em> to <em>t = z</em>. So let's get computing:</p>
+
+          <p>\[
+            Q = M^{-1} \cdot Z \cdot M =
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            1 & \frac{1}{2} & 0 \\
+            1 & 1 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            0 & z & 0 \\
+            0 & 0 & z^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            =
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            -(z-1) & z & 0 \\
+            (z - 1)^2 & -2 \cdot (z-1) \cdot z & z^2
+            \end{bmatrix}
+          \]</p>
+
+          <p>Excellent! Now we can form our new quadratic curve:</p>
+
+          <p>\[
+            B(t) =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot M \cdot Q \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            M
+            \cdot
+            \left (
+            Q
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+            \right )
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \left (
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            -(z-1) & z & 0 \\
+            (z - 1)^2 & -2 \cdot (z-1) \cdot z & z^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+            \right )
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+              P_1 \\
+              z \cdot P_2 - (z-1) \cdot P_1 \\
+              z^2 \cdot P_3 - 2 \cdot z \cdot (z-1) \cdot P_2 + (z - 1)^2 \cdot P_1
+            \end{bmatrix}
+          \]</p>
+
+          <p><strong><em>Brilliant</em></strong>: if we want a subcurve from <em>t = 0</em>
+          to <em>t = z</em>, 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, except it uses (z-1) rather than (1-z)... These new
+          coordinates are actually really easy to compute directly!</p>
+
+          <p>Of course, that's only one of the two curves. Getting the section from <em>t = z</em>
+          to <em>t = 1</em> requires doing this again. We first observe what what we just did is
+          actually evaluate the general interval [0,<em>z</em>], which we wrote down simplified
+          becuase of that zero, but we actually evaluated this:</p>
+
+          <p>\[
+            B(t) =
+            \begin{bmatrix}
+            1 & ( 0 + z \cdot t) & ( 0 + z \cdot t)^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            0 & z & 0 \\
+            0 & 0 & z^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>If we want the interval [<em>z</em>,1], we will be evaluating this instead:</p>
+
+          <p>\[
+            B(t) =
+            \begin{bmatrix}
+            1 & ( z + (1-z) \cdot t) & ( z + (1-z) \cdot t)^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            1 & z & z^2 \\
+            0 & 1-z & 2 \cdot z \cdot (1-z) \\
+            0 & 0 & (1-z)^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p>We're going to do the same trick, to turn <em>[something · M]</em> into <em>[M · something]</em>:</p>
+
+          <p>\[
+            Q' = M^{-1} \cdot Z' \cdot M =
+            \begin{bmatrix}
+            1 & 0 & 0 \\
+            1 & \frac{1}{2} & 0 \\
+            1 & 1 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            1 & z & z^2 \\
+            0 & 1-z & 2 \cdot z \cdot (1-z) \\
+            0 & 0 & (1-z)^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            =
+            \begin{bmatrix}
+            (z-1)^2 & -2 \cdot z \cdot (z-1) & z^2 \\
+            0 & -(z-1) & z \\
+            0 & 0 & 1
+            \end{bmatrix}
+          \]</p>
+
+          <p>So, our final second curve looks like:</p>
+
+          <p>\[
+            B(t) =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot M \cdot Q \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            M
+            \cdot
+            \left (
+            Q'
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+            \right )
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \left (
+            \begin{bmatrix}
+            (z-1)^2 & -2 \cdot z \cdot (z-1) & z^2 \\
+            0 & -(z-1) & z \\
+            0 & 0 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+            P_1 \\ P_2 \\ P_3
+            \end{bmatrix}
+            \right )
+          \]</p>
+
+          <p>\[
+            =
+            \begin{bmatrix}
+            1 & t & t^2
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+             1 &  0 & 0 \\
+            -2 &  2 & 0 \\
+             1 & -2 & 1
+            \end{bmatrix}
+            \cdot
+            \begin{bmatrix}
+              z^2 \cdot P_3 - 2 \cdot z \cdot (z-1) \cdot P_2 + (z-1)^2 \cdot P_1 \\
+              z \cdot P_3  - (z-1) \cdot P_2 \\
+              P_3
+            \end{bmatrix}
+          \]</p>
+
+          <p><strong><em>Nice</em></strong>: we see the same as before; 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 it uses (z-1) rather than (1-z). These new coordinates are <em>also</em>
+          really easy to compute directly!</p>
+        </div>
+
+        <p>So, using linear algebra rather than de Casteljau's algorithm, we have determined
+        that for any quadratic curve split at some value <em>t = z</em>, we get two subcurves
+        that are described as Bézier curves with simple-to-derive coordinates.</p>
+
+        <p>\[
+          \begin{bmatrix}
+          1 & 0 & 0 \\
+          -(z-1) & z & 0 \\
+          (z - 1)^2 & -2 \cdot (z-1) \cdot z & z^2
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3
+          \end{bmatrix}
+          =
+          \begin{bmatrix}
+            P_1 \\
+            z \cdot P_2 - (z-1) \cdot P_1 \\
+            z^2 \cdot P_3 - 2 \cdot z \cdot (z-1) \cdot P_2 + (z - 1)^2 \cdot P_1
+          \end{bmatrix}
+        \]</p>
+
+        <p>and</p>
+
+        <p>\[
+          \begin{bmatrix}
+          (z-1)^2 & -2 \cdot z \cdot (z-1) & z^2 \\
+          0 & -(z-1) & z \\
+          0 & 0 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+          P_1 \\ P_2 \\ P_3
+          \end{bmatrix}
+          =
+          \begin{bmatrix}
+            z^2 \cdot P_3 - 2 \cdot z \cdot (z-1) \cdot P_2 + (z-1)^2 \cdot P_1 \\
+            z \cdot P_3  - (z-1) \cdot P_2 \\
+            P_3
+          \end{bmatrix}
+        \]</p>
+
+        <p>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:</p>
+
+        <p>\[
+          \begin{bmatrix}
+            1 & 0 & 0 & 0 \\
+            -(z-1) & z & 0 & 0 \\
+            (z-1)^2 & -2 \cdot (z-1) \cdot z & z^2 & 0 \\
+            -(z-1)^3 & 3 \cdot (z-1)^2 \cdot z & -3 \cdot (z-1) \cdot z^2 & z^3
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           P_1 \\ P_2 \\ P_3 \\ P_4
+          \end{bmatrix}
+          =
+          \begin{bmatrix}
+            P_1 \\
+            z \cdot P_2 - (z-1) \cdot P_1 \\
+            z^2 \cdot P_3 - 2 \cdot z \cdot (z-1) \cdot P_2 + (z-1)^2 \cdot P_1 \\
+            z^3 \cdot P_4 - 3 \cdot z^2 \cdot (z-1) \cdot P_3 + 3 \cdot z \cdot (z-1)^2 \cdot P_2 - (z-1)^3 \cdot P_1
+          \end{bmatrix}
+        \]</p>
+
+        <p>and</p>
+
+        <p>\[
+          \begin{bmatrix}
+            -(z-1)^3 & 3 \cdot (z-1)^2 \cdot z & -3 \cdot (z-1)^3 \cdot z^2 & z^3 \\
+            0 & (z-1)^2 & -2 \cdot (z-1) \cdot z & z^2 \\
+            0 & 0 & -(z-1) & z \\
+            0 & 0 & 0 & 1
+          \end{bmatrix}
+          \cdot
+          \begin{bmatrix}
+           P_1 \\ P_2 \\ P_3 \\ P_4
+          \end{bmatrix}
+          =
+          \begin{bmatrix}
+            z^3 \cdot P_4 - 3 \cdot z^2 \cdot (z-1) \cdot P_3 + 3 \cdot z \cdot (z-1)^2 \cdot P_2 - (z-1)^3 \cdot P_1 \\
+            z^2 \cdot P_4 - 2 \cdot z \cdot (z-1) \cdot P_3 + (z-1)^2 \cdot P_2 \\
+            z \cdot P_4 - (z-1) \cdot P_3 \\
+            P_4
+          \end{bmatrix}
+        \]</p>
+
+        <p>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 <strong><em>Q</em></strong> 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" <strong><em>Q'</em></strong>.</p>
+
+        <p>Implementing curve splitting this way requires less recursion, and is just straight
+        arithmetic with cached values, so can be cheaper on systems were 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.</p>
+      </section>
+    );
+  }
+});
+
+module.exports = MatrixSplit;
diff --git a/components/sections/reordering/index.js b/components/sections/reordering/index.js
new file mode 100644
index 00000000..e8c7c11e
--- /dev/null
+++ b/components/sections/reordering/index.js
@@ -0,0 +1,182 @@
+var React = require("react");
+var Graphic = require("../../Graphic.jsx");
+var SectionHeader = require("../../SectionHeader.jsx");
+
+var Reordering = React.createClass({
+  statics: {
+    title: "Lowering and elevating curve order"
+  },
+
+  getInitialState: function() {
+    return {
+      order: 0
+    };
+  },
+
+  setup: function(api) {
+    var points = [];
+    var w = api.getPanelWidth(),
+        h = api.getPanelHeight();
+    for (var i=0; i<10; i++) {
+      points.push({
+        x: w/2 + (Math.random() * 20) + Math.cos(Math.PI*2 * i/10) * (w/2 - 40),
+        y: h/2 + (Math.random() * 20) + Math.sin(Math.PI*2 * i/10) * (h/2 - 40)
+      });
+    }
+    var curve = new api.Bezier(points);
+    api.setCurve(curve);
+  },
+
+  draw: function(api, curve) {
+    api.reset();
+    var pts = curve.points;
+
+    this.setState({
+      order: pts.length
+    });
+
+    var p0 = pts[0];
+
+    // we can't "just draw" this curve, since it'll be an arbitrary order,
+    // And the canvas only does 2nd and 3rd - we use de Casteljau's algorithm:
+    for(var t=0; t<=1; t+=0.01) {
+      var q = JSON.parse(JSON.stringify(pts));
+      while(q.length > 1) {
+        for (var i=0; i<q.length-1; i++) {
+          q[i] = {
+            x: q[i].x + (q[i+1].x - q[i].x) * t,
+            y: q[i].y + (q[i+1].y - q[i].y) * t
+          }
+        }
+        q.splice(q.length-1, 1);
+      }
+      api.drawLine(p0, q[0]);
+      p0 = q[0];
+    }
+
+    p0 = pts[0];
+    api.setColor("black");
+    api.drawCircle(p0,3);
+    pts.forEach(p => {
+      if(p===p0) return;
+      api.setColor("#DDD");
+      api.drawLine(p0,p);
+      api.setColor("black");
+      api.drawCircle(p,3);
+      p0 = p;
+    });
+  },
+
+  // Improve this based on http://www.sirver.net/blog/2011/08/23/degree-reduction-of-bezier-curves/
+  lower: function(curve) {
+    var pts = curve.points, q=[], n = pts.length;
+    pts.forEach((p,k) => {
+      if (!k) { return (q[k] = p); }
+      var f1 = k/n, f2 = 1 - f1;
+      q[k] = {
+        x: f1 * p.x + f2 * pts[k-1].x,
+        y: f1 * p.y + f2 * pts[k-1].y
+      };
+    });
+    q.splice(n-1,1);
+    q[n-2] = pts[n-1];
+    curve.points = q;
+    return curve;
+  },
+
+  onKeyDown: function(evt, api) {
+    evt.preventDefault();
+    if(evt.key === "ArrowUp") {
+      api.setCurve(api.curve.raise());
+    } else if(evt.key === "ArrowDown") {
+      api.setCurve(this.lower(api.curve));
+    }
+  },
+
+  getOrder: function() {
+    var order = this.state.order;
+    if (order%10 === 1 && order !== 11) {
+      order += "st";
+    } else if (order%10 === 2 && order !== 12) {
+      order += "nd";
+    } else if (order%10 === 3 && order !== 13) {
+      order += "rd";
+    } else {
+      order += "th";
+    }
+    return order;
+  },
+
+  render: function() {
+    return (
+      <section>
+        <SectionHeader {...this.props}>{ Reordering.title }</SectionHeader>
+
+        <p>One interesting property of Bézier curves is that an <i>n<sup>th</sup></i> order curve can
+        always be perfectly represented by an <i>(n+1)<sup>th</sup></i> order curve, by giving the
+        higher order curve specific control points.</p>
+
+        <p>If we have a curve with three points, then we can create a four point curve that exactly
+        reproduce the original curve as long as we give it the same start and end points, and for
+        its two control points we pick "1/3<sup>rd</sup> start + 2/3<sup>rd</sup> control" and
+        "2/3<sup>rd</sup> control + 1/3<sup>rd</sup> end", and now we have exactly the same curve as
+        before, except represented as a cubic curve, rather than a quadratic curve.</p>
+
+        <p>The general rule for raising an <i>n<sup>th</sup></i> order curve to an <i>(n+1)<sup>th</sup></i>
+        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):</p>
+
+        <p>\[
+          Bézier(k,t) = \sum_{i=0}^{k}
+                        \underset{binomial\ term}{\underbrace{\binom{k}{i}}}
+                        \cdot\
+                        \underset{polynomial\ term}{\underbrace{(1-t)^{k-i} \cdot t^{i}}}
+                        \ \cdot \
+                        \underset{new\ weights}{\underbrace{\left ( \frac{(k-i) \cdot w_i + i \cdot w_{i-1}}{k} \right )}}
+          \ ,\ with\ k = n+1
+        \]</p>
+
+        <p>However, this rule also has as direct consequence that you <strong>cannot</strong> generally
+        safely lower a curve from <i>n<sup>th</sup></i> order to <i>(n-1)<sup>th</sup></i> 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.</p>
+
+        <p>We can apply this to a (semi) random curve, as is done in the following graphic. Select the sketch
+        and press your up and down cursor keys to elevate or lower the curve order.</p>
+
+        <Graphic preset="simple" title={"A " + this.getOrder() + " order Bézier curve"} setup={this.setup} draw={this.draw} onKeyDown={this.onKeyDown} />
+
+        <p>There is a good, if mathematical, explanation on the matrices necessary for optimal reduction
+        over on <a href="http://www.sirver.net/blog/2011/08/23/degree-reduction-of-bezier-curves/">Sirver's Castle</a>,
+        which given time will find its way in a more direct description into this article.</p>
+
+      </section>
+    );
+  }
+});
+
+module.exports = Reordering;
+
+/*
+void setupCurve() {
+          int d = dim - 2*pad;
+          int order = 10;
+          ArrayList<Point> pts = new ArrayList<Point>();
+
+          float dst = d/2.5, nx, ny, a=0, step = 2*PI/order, r;
+          for(a=0; a<2*PI; a+=step) {
+            r = random(-dst/4,dst/4);
+            pts.add(new Point(d/2 + cos(a) * (r+dst), d/2 + sin(a) * (r+dst)));
+            dst -= 1.2;
+          }
+
+          Point[] points = new Point[pts.size()];
+          for(int p=0,last=points.length; p<last; p++) { points[p] = pts.get(p); }
+          curves.add(new BezierCurve(points));
+          reorder();
+        }
+
+        void drawCurve(BezierCurve curve) {
+          curve.draw();
+        }</textarea>
+ */
\ No newline at end of file
diff --git a/components/sections/splitting/index.js b/components/sections/splitting/index.js
new file mode 100644
index 00000000..39e222f9
--- /dev/null
+++ b/components/sections/splitting/index.js
@@ -0,0 +1,150 @@
+var React = require("react");
+var Graphic = require("../../Graphic.jsx");
+var SectionHeader = require("../../SectionHeader.jsx");
+
+var Splitting = React.createClass({
+  statics: {
+    title: "Splitting curves"
+  },
+
+  setupCubic: function(api) {
+    var curve = api.getDefaultCubic();
+    api.setCurve(curve);
+    api.forward = true;
+  },
+
+  drawSplit: function(api, curve) {
+    api.setPanelCount(2);
+    api.reset();
+    api.drawSkeleton(curve);
+    api.drawCurve(curve);
+
+    var offset = {x:0, y:0};
+    var t = 0.5;
+    var pt = curve.get(0.5);
+    var split = curve.split(t);
+    api.drawCurve(split.left);
+    api.drawCurve(split.right);
+    api.setColor("red");
+    api.drawCircle(pt,3);
+
+    api.setColor("black");
+    offset.x = api.getPanelWidth();
+    api.drawLine({x:0,y:0},{x:0,y:api.getPanelHeight()}, offset);
+
+    api.setColor("lightgrey");
+    api.drawCurve(curve, offset);
+    api.drawCircle(pt,4);
+
+    offset.x -= 20;
+    offset.y -= 20;
+    api.drawSkeleton(split.left, offset, true);
+    api.drawCurve(split.left, offset);
+
+    offset.x += 40;
+    offset.y += 40;
+    api.drawSkeleton(split.right, offset, true);
+    api.drawCurve(split.right, offset);
+  },
+
+  drawAnimated: function(api, curve) {
+    api.setPanelCount(3);
+    api.reset();
+
+    var frame = api.getFrame();
+    var interval = 5 * api.getPlayInterval();
+    var t = (frame%interval)/interval;
+    var forward = (frame%(2*interval)) < interval;
+    if (forward) { t = t%1; } else { t = 1 - t%1; }
+    var offset = {x:0, y:0};
+
+    api.setColor("lightblue");
+    api.drawHull(curve, t);
+    api.drawSkeleton(curve);
+    api.drawCurve(curve);
+    var pt = curve.get(t);
+    api.drawCircle(pt, 4);
+
+    api.setColor("black");
+    offset.x += api.getPanelWidth();
+    api.drawLine({x:0,y:0},{x:0,y:api.getPanelHeight()}, offset);
+
+    var split = curve.split(t);
+
+    api.setColor("lightgrey");
+    api.drawCurve(curve, offset);
+    api.drawHull(curve, t, offset);
+    api.setColor("black");
+    api.drawCurve(split.left, offset);
+    api.drawPoints(split.left.points, offset);
+    api.setFill("black");
+    api.text("Left side of curve split at t = " + (((100*t)|0)/100), {x: 10 + offset.x, y: 15 + offset.y});
+
+    offset.x += api.getPanelWidth();
+    api.drawLine({x:0,y:0},{x:0,y:api.getPanelHeight()}, offset);
+
+    api.setColor("lightgrey");
+    api.drawCurve(curve, offset);
+    api.drawHull(curve, t, offset);
+    api.setColor("black");
+    api.drawCurve(split.right, offset);
+    api.drawPoints(split.right.points, offset);
+    api.setFill("black");
+    api.text("Right side of curve split at t = " + (((100*t)|0)/100), {x: 10 + offset.x, y: 15 + offset.y});
+  },
+
+  togglePlay: function(evt, api) {
+    if (api.playing) { api.pause(); } else { api.play(); }
+  },
+
+  render: function() {
+    return (
+      <section>
+        <SectionHeader {...this.props}>{ Splitting.title }</SectionHeader>
+
+        <p>With de Casteljau's algorithm we 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
+        <i>t</i>, the procedure gives us all the points we need to split a curve at that <i>t</i> 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.</p>
+
+        <Graphic title="Splitting a curve" setup={this.setupCubic} draw={this.drawSplit} />
+
+        <div className="howtocode">
+          <h3>implementing curve splitting</h3>
+
+          <p>We can implement curve splitting by bolting some extra logging onto the de Casteljau function:</p>
+
+<pre>left=[]
+right=[]
+function drawCurve(points[], t):
+  if(points.length==1):
+    left.add(points[0])
+    right.add(points[0])
+    draw(points[0])
+  else:
+    newpoints=array(points.size-1)
+    for(i=0; i&lt;newpoints.length; i++):
+      if(i==0):
+        left.add(points[i])
+      if(i==newpoints.length-1):
+        right.add(points[i+1])
+      newpoints[i] = (1-t) * points[i] + t * points[i+1]
+    drawCurve(newpoints, t)</pre>
+
+          <p>After running this function for some value <i>t</i>, the <i>left</i> and <i>right</i> arrays
+          will contain all the coordinates for two new curves - one to the "left" of our <i>t</i> value,
+          the other on the "right", of the same order as the original curve, and overlayed exactly on the
+          original curve.</p>
+        </div>
+
+        <p>This is best illustrated with an animated graphic (click to play/pause):</p>
+
+        <Graphic preset="threepanel" title="Bézier curve splitting" setup={this.setupCubic} draw={this.drawAnimated} onClick={this.togglePlay} />
+
+      </section>
+    );
+  }
+});
+
+module.exports = Splitting;
diff --git a/images/latex/1e2a1516ca2ce995a0ddebf218b7528bd762b686.svg b/images/latex/1e2a1516ca2ce995a0ddebf218b7528bd762b686.svg
new file mode 100644
index 00000000..ba6d5886
--- /dev/null
+++ b/images/latex/1e2a1516ca2ce995a0ddebf218b7528bd762b686.svg
@@ -0,0 +1,328 @@
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\ No newline at end of file
diff --git a/index.html b/index.html
index 87fb2bca..b28b330f 100644
--- a/index.html
+++ b/index.html
@@ -17,7 +17,6 @@
     <meta property="og:tag" content="Bézier Curves">
   </head>
   <body>
-
     <!-- This page lives on github -->
     <img id="ribbonimg" src="images/ribbon.png" alt="This page on GitHub" border=0 usemap="#githubmap">
     <map name="githubmap">
@@ -25,7 +24,6 @@
     </map>
 
     <article>
-
       <header>
         <h1>A Primer on Bézier Curves</h1>
         <h2>A free, online book for when you really need to know how to do Bézier things.</h2>
diff --git a/stylesheets/figure.less b/stylesheets/figure.less
index df50740a..a60376ea 100644
--- a/stylesheets/figure.less
+++ b/stylesheets/figure.less
@@ -15,6 +15,10 @@ figure {
     display: inline-block;
     background: white;
     border: 1px solid lightgrey;
+
+    &:focus {
+      border: 1px solid grey;
+    }
   }
 
   figcaption {
diff --git a/tools/mathjax.js b/tools/mathjax.js
index 25c4b472..a7a5d8eb 100644
--- a/tools/mathjax.js
+++ b/tools/mathjax.js
@@ -7,20 +7,41 @@
  *  do async loaders (unfortunately).
  *
  */
+var className="LaTeX SVG";
+var fs = require("fs");
+var API = require("mathjax-node/lib/mj-single");
 
-// fine the --latex option
-var hash, base64, latex;
-
-hash = process.argv.indexOf("--hash");
-if (latex === -1) {
-  hash = Date.now();
-} else {
-  hash = process.argv[hash+1];
+function cleanUp(latex) {
+  // strip any \[ and \], which is an block-level LaTeX markup indicator for MathJax:
+  latex = latex.replace(/^'/,'').replace(/'$/,'').replace('\\[','').replace('\\]','');
+  // Accented letters need shimming. For now, at least, until I figure out
+  // how to make mathjax-node use a full STIX or the like for typesetting.
+  latex = latex.replace(/é/g,'\\acute{e}');
+  // done.
+  return latex;
 }
 
-latex = process.argv.indexOf("--latex");
+// Set up the MathJax processor
+API.config({
+  MathJax: {
+    TeX: {
+      extensions: [
+        "AMSmath.js",
+        "AMSsymbols.js",
+        "autoload-all.js",
+        "color.js"
+      ]
+    }
+  }
+});
+API.start();
+
+// Get the LaTeX we need, and the filename it'll need to write to
+var hash = process.argv.indexOf("--hash");
+var latex = process.argv.indexOf("--latex");
+
 if (latex === -1) {
-  base64 = process.argv.indexOf("--base64");
+  var base64 = process.argv.indexOf("--base64");
   if (base64 === -1) {
     console.error("missing [--latex] or [--base64] runtime option");
     process.exit(1);
@@ -37,55 +58,13 @@ if (latex === -1) {
   }
 }
 
-
-// strip any \[ and \], which is an block-level LaTeX markup indicator for MathJax:
-latex = latex.replace(/^'/,'').replace(/'$/,'').replace('\\[','').replace('\\]','');
-
-// Accented letters need shimming. For now, at least, until I figure out
-// how to make mathjax-node use a full STIX or the like for typesetting.
-latex = latex.replace(/é/g,'\\acute{e}');
-
-// set up the MathJax processor
-var API = require("mathjax-node/lib/mj-single");
-API.config({
-  MathJax: {
-    TeX: {
-      extensions: [
-        "AMSmath.js",
-        "AMSsymbols.js",
-        "autoload-all.js",
-        "color.js"
-      ]
-    }
-  }
-});
-API.start();
-
-
-// find style="..." declarations and rewrite them to style={{...}} format
-var toReactStyle = function(input) {
-  return input.replace(/style="([^"]+)"/g, function(a, b) {
-    var c = {};
-    b = b.split(';');
-    b.forEach(function(v) {
-      v = v.split(':').map(function(s) { return s.trim(); });
-      v[0] = v[0].replace(/-(\w)/g,function(a, b) {
-        return b.toUpperCase();
-      });
-      c[v[0]] = v[1];
-    });
-    return "style={" + JSON.stringify(c) + "}";
-  });
-};
-
-var fs = require("fs");
+var hash = (latex === -1) ? Date.now() : process.argv[hash+1];
 var filename = "images/latex/" + hash + ".svg";
 var destination = __dirname + "/../" + filename;
-var className="LaTeX SVG";
 
 // convert the passed LaTeX to SVG form
 API.typeset({
-  math: latex,
+  math: cleanUp(latex),
   format: "TeX",
   svg: true
 }, function (data) {