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865633de-b85b-4c6b-96b7-0bc9807674ee 13 6 12 decimal

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Explore the Bernstein polynomials

Bernstein polynomials j n t B n! j! (n j - (! (* / t j^* 1 t - (n j -^* 0 t ≤ (t 1 ≤ (& * :

Approximate 'n' Envelope n t E 12 π * n * t * 1 t - (* sqrt / 0 t ≤ (t 1 ≤ (& * :

Bézier curves are ubiquitous in computer graphics. They were introduced implicitly into theoretical mathematics long before computers, primarily by the French mathematician Charles Hermite and the Russian mathematician Sergei Bernstein. But it was only the work of Pierre Bézier, an employee of the automobile maker Renault, and of Paul de Casteljau, of Citroen, that made these curves familiar to graphics specialists. Recently, the polynomials defined by Bernstein have become again of interest to mathematicians.

... approximate max .... n 20 : j 10 : j n < 13 mat X j n / : Y j n X B : 12 mat XY X Y + 20 black 15 mat : 31 mat

Bernstein polynomials Explorer... n, j, X,Y

n x E j n x B XY 3 1 sys

Collect some Bernstein polynomials

n x E 2 20 x B 3 20 x B 4 20 x B 5 20 x B 6 20 x B 7 20 x B 15 20 x B 17 20 x B 9 1 sys

n j X Y 14 mat 2010 0.5 0.1762 14 mat

Sergei Natanovich Bernstein: 1880 . . . 1968

Alma Mater: Université de Paris

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Today all mathematicians and physicists agree the field of applications for mathematics knows no limits except those of knowledge itself. Sergei Natanovich Bernstein

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

data 1 - 0.242 0.888 - 0.157 0.756 - 0.085 0.609 - 0.003 0.453 - 0.005 0.294 - 0 0.142 - 0.015 0 0.035 0.134 0.068 0.266 0.099 0.41 0.129 0.513 0.183 0.627 0.235 0.729 0.307 0.824 0.376 0.902 0.462 0.948 0.557 0.99 0.649 1 0.735 1 0.809 0.987 0.871 0.962 0.918 0.939 0.952 0.91 0.981 0.879 0.992 0.847 1 0.819 0.997 0.799 0.991 0.78 0.979 0.761 0.961 0.744 0.937 0.721 0.907 0.698 0.872 0.675 0.83 0.659 0.781 0.649 0.725 0.649 0.659 0.66 0.582 0.676 0.487 0.695 0.369 402 mat :

p 1 - 0.5 - 2.95 0.3 0.565 0.73 0.242 0.5 - 0.15 2.18 0.69 0.3 26 mat transpose :

The Lobe project ----------

X p 1 col :

Y p 2 col :

n p rows 1 - :

k 0 :

If there is a way to create and optimise the control polygon, ... NO idea ! Just as is .

r n k - (n ≤ n k - (0 if :

r n 21 mat 5521 mat

Observe: In literature, QX(t) & QY(t) do run over [j=0..r] and Xj,Yj. Smath ORIGIN is '1', implying Xj+1, Yj+1.

Bézier-Bernstein procedure j n t Q n! j! (n j - (! (* / t j^* 1 t - (n j -^* :

t QX j n t Q X j 1 + el * (j 0 r sum eval :

t QY j n t Q Y j 1 + el * (j 0 r sum eval :

Lobe t 1015 -^1 0.01 range : i 1 t rows range LX i el t i el QX eval : LY i el t i el QY eval : 21 line for LX LY augment 31 line :

XY X Y augment : Cntrl XY . 14 black plot : data data o 5 black plot : 31 sys

XY Lobe Cntrl data 4 1 sys

Practice your own project

scale 1.5 :

p scale 1 - 22 - 112 - 2 - 124 mat * transpose :

X p 1 col :

Y p 2 col :

n p rows 1 - : 3

Bézier-Bernstein procedure j n t Q n! j! (n j - (! (* / t j^* 1 t - (n j -^* :

t QX j n t Q X j 1 + el * (j 0 n sum eval :

t QY j n t Q Y j 1 + el * (j 0 n sum eval :

Lobe t 1015 -^1 0.01 range : i 1 t rows range LX i el t i el QX eval : LY i el t i el QY eval : 21 line for LX LY augment 31 line :

p Lobe 2 1 sys

LobeComponents t 1015 -^1 0.01 range : i 1 t rows range LX i el t i el QX eval : LY i el t i el QY eval : 21 line for t LX augment t LY augment 21 mat 31 line :

LobeComponents 1 el LobeComponents 2 el 2 1 sys

Bézier parametric Cubic

scale 2 : (scale at will ≡

p scale 1 - 22 - 112 - 2 - 124 mat * transpose :

For more pairs of control points Bernstein procedure extends Bézier. Similarly, De Boor is not limited by the number of points.

X p 1 col :

Y p 2 col :

t b0 t 3^- 3 t 2^* + 3 t * - 1 + (: t b1 3 t 3^* 6 t 2^* - 3 t * + (: t b2 3 t 3^* - 3 t 2^* + (: t b3 t 3^: 41 line

p 2 - 244 - 4 - 4 - 2242 mat

t bX t b0 X 1 el * t b1 X 2 el * + t b2 X 3 el * + t b3 X 4 el * + : t bY t b0 Y 1 el * t b1 Y 2 el * + t b2 Y 3 el * + t b3 Y 4 el * + : 21 line

t 0.125 - 1.25 0.12 - range : (range 't' at will ≡

Bz i 1 t rows range bX i el t i el bX eval : bY i el t i el bY eval : 21 line for bX bY augment 21 line :

Bézier Cubic 4 cntrl points

p Bz 2 1 sys