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Pomax
2022-01-04 11:50:23 -08:00
parent ce1ee672a4
commit b243e9830f
508 changed files with 59429 additions and 32670 deletions
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39
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@@ -1,21 +1,20 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
┌ 1 0 . . . . . . ┐
│ 1 k-1 │
│ ─ ─── 0 . . . 0 . │
│ k k │
│ 2 k-2 │
│ 0 ─ ─── 0 . . . . │
│ k k │
│ 3 k-3 │
│ . 0 ─ ─── 0 . . . │
M = │ k k │
│ . . 0 ... ... 0 . . │
│ . . . 0 ... ... 0 . │
│ n-1 k-n+1 │
│ . . . . 0 ─── ───── 0 │
│ k k │
│ n k-n │
│ . 0 . . . 0 ─ ─── │
│ k k │
└ . . . . . . 0 1 ┘
┌ 1 0 . . . . . . ┐
│ 1 k-1 │
│ ─ ─── 0 . . . 0 . │
│ k k │
│ 2 k-2 │
│ 0 ─ ─── 0 . . . . │
│ k k │
│ 3 k-3 │
│ . 0 ─ ─── 0 . . . │
M = │ k k │
│ . . 0 ... ... 0 . . │
│ . . . 0 ... ... 0 . │
│ n-1 k-n+1 │
│ . . . . 0 ─── ───── 0 │
│ k k │
│ n k-n │
│ . 0 . . . 0 ─ ─── │
│ k k │
└ . . . . . . 0 1 ┘

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@@ -1,6 +1,5 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
Bézier(n,t)= (1-t) B(n,t) + t B(n,t)
__ n n __ n n
= ❯ w (1 - t) B (t) + ❯ w t B (t)
‾‾ i=0 i i ‾‾ i=0 i i
Bézier(n,t)= (1-t) B(n,t) + t B(n,t)
__ n n __ n n
= ❯ w (1 - t) B (t) + ❯ w t B (t)
‾‾ i=0 i i ‾‾ i=0 i i

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@@ -1,14 +1,13 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
n n! n-i i
(1 - t) B (t)= (1-t) ──────── (1-t) t
i (n-i)!i!
n+1-i (n+1)! n+1-i i
= ───── ────────── (1-t) t
n+1 (n+1-i)!i!
k-i k! k-i i
= ─── ──────── (1-t) t , where k = n + 1
k (k-i)!i!
k-i k
= ─── B (t)
k i
n n! n-i i
(1 - t) B (t)= (1-t) ──────── (1-t) t
i (n-i)!i!
n+1-i (n+1)! n+1-i i
= ───── ────────── (1-t) t
n+1 (n+1-i)!i!
k-i k! k-i i
= ─── ──────── (1-t) t , where k = n + 1
k (k-i)!i!
k-i k
= ─── B (t)
k i

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@@ -1,4 +1,3 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
M B = B
n k
M B = B
n k

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@@ -1,16 +1,15 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
M B = B
n k
T T
(M M) B = M B
n k
T -1 T T -1 T
(M M) (M M) B = (M M) M B
n k
T -1 T
I B = (M M) M B
n k
T -1 T
B = (M M) M B
n k
M B = B
n k
T T
(M M) B = M B
n k
T -1 T T -1 T
(M M) (M M) B = (M M) M B
n k
T -1 T
I B = (M M) M B
n k
T -1 T
B = (M M) M B
n k

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@@ -1,17 +1,16 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ n+1 n __ n+1 n
Bézier(n,t)= ❯ w (1 - t) B (t) + ❯ w t B (t)
‾‾ i=0 i i ‾‾ i=0 i i
__ n+1 k-i k __ n+1 i+1 k
= ❯ w ─── B (t) + ❯ w ─── B (t), where k = n + 1
‾‾ i=0 i k i ‾‾ i=0 i k i+1
__ n+1 k-i k __ n+1 i k
= ❯ w ─── B (t) + ❯ p ─ B (t)
‾‾ i=0 i k i ‾‾ i=0 i-1 k i
__ n+1 ╭ k-i i ╮ k
= ❯ │ w ─── + p ─ │ B (t)
‾‾ i=0 ╰ i k i-1 k ╯ i
__ n+1 k i
= ❯ (w (1-s) + p s) B (t), where s = ─
‾‾ i=0 i i-1 i k
__ n+1 n __ n+1 n
Bézier(n,t)= ❯ w (1 - t) B (t) + ❯ w t B (t)
‾‾ i=0 i i ‾‾ i=0 i i
__ n+1 k-i k __ n+1 i+1 k
= ❯ w ─── B (t) + ❯ w ─── B (t), where k = n + 1
‾‾ i=0 i k i ‾‾ i=0 i k i+1
__ n+1 k-i k __ n+1 i k
= ❯ w ─── B (t) + ❯ p ─ B (t)
‾‾ i=0 i k i ‾‾ i=0 i-1 k i
__ n+1 ╭ k-i i ╮ k
= ❯ │ w ─── + p ─ │ B (t)
‾‾ i=0 ╰ i k i-1 k ╯ i
__ n+1 k i
= ❯ (w (1-s) + p s) B (t), where s = ─
‾‾ i=0 i i-1 i k

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@@ -1,6 +0,0 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
Bézier(n,t)= (1-t) B(n,t) + t B(n,t)
__ n n __ n n
= ❯ w (1 - t) B (t) + ❯ w t B (t)
‾‾ i=0 i i ‾‾ i=0 i i

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@@ -1,5 +1,4 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ k k-i i
Bézier(k,t) = ❯ \underset binomial term\underbrace\binomki · \ \underset polynomial term\underbrace(1-t) · t · \ \underset new
‾‾ i=0

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@@ -1,5 +0,0 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ n n n n-i i
Bézier(n,t) = ❯ w B (t) , where B (t) = \binomni · (1-t) · t
‾‾ i=0 i i i

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@@ -1,5 +1,4 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ n n n n-i i
Bézier(n,t) = ❯ w B (t) , where B (t) = \binomni · (1-t) · t
‾‾ i=0 i i i
__ n n n n-i i
Bézier(n,t) = ❯ w B (t) , where B (t) = \binomni · (1-t) · t
‾‾ i=0 i i i

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@@ -1,14 +1,13 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
n n! n-i i
t B (t)= t ──────── (1-t) t
i (n-i)!i!
i+1 (n+1)! (n+1)-(i+1) i+1
= ─── ──────────────────── (1-t) t
n+1 ((n+1)-(i+1))!(i+1)!
i+1 k! k-(i+1) i+1
= ─── ──────────────── (1-t) t , where k = n + 1
k (k-(i+1))!(i+1)!
i+1 k
= ─── B (t)
k i+1
n n! n-i i
t B (t)= t ──────── (1-t) t
i (n-i)!i!
i+1 (n+1)! (n+1)-(i+1) i+1
= ─── ──────────────────── (1-t) t
n+1 ((n+1)-(i+1))!(i+1)!
i+1 k! k-(i+1) i+1
= ─── ──────────────── (1-t) t , where k = n + 1
k (k-(i+1))!(i+1)!
i+1 k
= ─── B (t)
k i+1

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@@ -1,9 +0,0 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
__ k k-i i
Bézier(k,t) = ❯ \undersetbinomial term\underbrace\binomki · \ \undersetpolynomial term\underbrace(1-t) · t · \ \undersetnew
‾‾ i=0
╭ (k-i) · w + i · w ╮
│ i i-1 │
weights\underbrace│ ─────────────────────── │ , with k = n+1 and w =0 when i = 0
╰ k ╯ i-1

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@@ -1,3 +1,2 @@
\setmainfont[Ligatures=TeX]TeX Gyre Pagella \setmathfontTeX Gyre Pagella Math
x = 1 x = ((1-t) + t) x = (1-t) x + t x = x (1-t) + x t
x = 1 x = ((1-t) + t) x = (1-t) x + t x = x (1-t) + x t