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Pomax
2020-10-18 14:27:57 -07:00
parent 133105d44e
commit 0f6683ca6d
314 changed files with 11182 additions and 266 deletions
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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
kn! k-1 n-k (n-k)n! k n-1-k
... = ──────── t (1-t) - ──────── t (1-t)
k!(n-k)! k!(n-k)!

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
n · B (t) · w +
n-1,-1 0
n · B (t) · w - n · B (t) · w +
n-1,\colorblue0 1 n-1,\colorblue0 0
n · B (t) · w - n · B (t) · w +
n-1,\colorred1 2 n-1,\colorred1 1
n · B (t) · w - n · B (t) · w +
n-1,\colormagenta2 3 n-1,\colormagenta2 2
... - n · B (t) · w +
n-1,3 3
...

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
B(n,t), w = {A,B,C,D}
B'(n,t), n = 3, w' = {A',B',C'} = {3 · (B-A), 3 · (C-B), 3 · (D-C)}
B''(n,t), n = 2, w'' = {A'',B''} = {2 · (B'-A'), 2 · (C'-B')}
B'''(n,t), n = 1, w''' = {A'''} = {1 · (B''-A'')}

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ k k-i i
Bézier'(n,t) = ❯ \undersetbinomial term\underbrace\binomki · \ \undersetpolynomial term\underbrace(1-t) · t · \ \undersetderivative
‾‾ i=0
weight\underbracen · (w - w ) , with k=n-1
i+1 i

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
d __ n-1 __ n-1
Bézier (t) ── = ❯ n · B (t) · (w - w ) = ❯ B (t) · \undersetderivative weights \underbracen · (w - w )
n,k dt ‾‾ k=0 n-1,k k+1 k ‾‾ k=0 n-1,k k+1 k

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
n! k-1 n-k (n-k)n! k n-1-k
... = ──────────── t (1-t) - ──────── t (1-t)
(k-1)!(n-k)! k!(n-k)!
╭ (n-1)! k-1 n-k (n-k)(n-1)! k n-1-k ╮
... = n │ ──────────── t (1-t) - ─────────── t (1-t) │
╰ (k-1)!(n-k)! k!(n-k)! ╯
╭ (n-1)! (k-1) (n-1)-(k-1) (n-1)! k (n-1)-k ╮
... = n │ ──────────────────── t (1-t) - ──────────── t (1-t) │
╰ (k-1)!((n-1)-(k-1))! k!((n-1)-k)! ╯

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ n-1
Bézier'(n,t) = ❯ Bézier(n-1,t) · n · (w -w )
‾‾ i=0 i i+1 i

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
╭ x! y x-y x! k x-k ╮
... = n │ ──────── t (1-t) - ──────── t (1-t) │ , with x=n-1, y=k-1
╰ y!(x-y)! k!(x-k)! ╯
... = n (B (t) - B (t))
(n-1),(k-1) (n-1),k

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
Bézier (t) = B (t) · w + B (t) · w + B (t) · w + B (t) · w + ...
n,k n,0 0 n,1 1 n,2 2 n,3 3
d
Bézier (t) ── = n · (B (t) - B (t)) · w +
n,k dt n-1,-1 n-1,0 0
n · (B (t) - B (t)) · w +
n-1,0 n-1,1 1
n · (B (t) - B (t)) · w +
n-1,1 n-1,2 2
n · (B (t) - B (t)) · w +
n-1,2 n-1,3 3
...

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
d
Bézier (t) ── = n · B (t) · (w - w ) + n · B (t) · (w - w ) + n · B (t) · (w - w ) +
n,k dt (n-1),\colorblue0 1 0 (n-1),\colorred1 2 1 (n-1),\colormagenta2 3 2
...

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
d ╭ n ╮ k n-k d
B (t) ── = │ │ t (1-t) ──
n,k dt ╰ k ╯ dt

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ n n-i i
Bézier(n,t) = ❯ \undersetbinomial term\underbrace\binomni · \ \undersetpolynomial term\underbrace(1-t) · t · \
‾‾ i=0
\undersetweight\underbracew
i

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
╭ n ╮ k-1 n-k k n-k-1
... = │ │ (k · t (1-t) + t · (1-t) · (n-k) · -1)
╰ k ╯

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
n · B (t) · w - n · B (t) · w +
n-1,\colorblue0 1 n-1,\colorblue0 0
n · B (t) · w - n · B (t) · w +
n-1,\colorred1 2 n-1,\colorred1 1
n · B (t) · w - n · B (t) · w +
n-1,\colormagenta2 3 n-1,\colormagenta2 2
...

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\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ n-1
Bézier'(n,t) = n · ❯ (b -b ) · Bézier(n-1,t)
‾‾ i=0 i+1 i i