fa9bf39e-3800-448e-bac5-2f020f0be63c 52 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

u v char size clr plotG n u length : plot u 1 el v 1 el char size clr augment : i 2 n range plot plot u i el v i el char size clr augment stack : for plot 41 line :

X 2 - 0231 mat :

Y 02031 mat :

<= User control points

XY X Y augment :

Cntrl X Y . 14 black plotG :

Bézier parametric Quadratic

t b0 1 t - (2^(:

t b1 2 t * 1 t - (* :

t b2 t 2^:

t bX t b0 X 1 el * t b1 X 2 el * + t b2 X 3 el * + :

t bY t b0 Y 1 el * t b1 Y 2 el * + t b2 Y 3 el * + :

t 3 - 3 2.99 - range :

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

Bézier Quadratic 3 cntrl points

XY B Cntrl 3 1 sys

XY 2 - 0022032 mat

Bézier parametric Cubic

X 033841 mat :

Y 7102441 mat :

<= User control points

XY X Y augment :

Cntrl X Y . 14 black plotG :

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^:

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 * + :

t 1 - 10 0.99 - range :

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

Bézier Cubic 4 cntrl points

XY Bz Cntrl 3 1 sys

shiftXdomain 3 :

X 3 - 2 - 1 - 0012318 mat transpose shiftXdomain + :

Bézier-Bernstein

t0 1 time :

Y 0122 - 2 - 21018 mat transpose :

Smath is capricious with Bézier-Bernstein. Because of the indexing starting at '1', it does NOT run with X in the negative domain. In this demo, we shift the X domain to enable Smath compute. Once the plot is done, it's possible to shift the system back. NOT done ! Experiment with the weight vectors Wx, Wy and adapt the control polygon for the plot. -------- Bézier-Bernstein hyperfast ---------

n X rows 1 - :

n 7

Wy 1111111118 mat transpose :

Wx 1111111118 mat transpose :

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 * Wx j 1 + el * (j 0 n sum :

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

zero 1015 -^eval :

t zero 1 0.005 range :

BezBern i 1 t rows range qX i el t i el QX eval : qY i el t i el QY eval : 21 line for qX qY augment 21 line :

XY X Y augment :

Cntrl X Y . 14 black plotG :

Bézier-Bernstein cntrl points

XY BezBern Cntrl 3 1 sys

1 time t0 - 4.6 s *

Explore the Bernstein polynomials

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

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

For a good view, set 'n' high enough, then run over 'j'. Bernstein polynomials run 0 < t < 1. They may be useful to fit symmetric/assymetric statistical data that may not have an immediate model function. for j = n/2 poly is symmetric & centered. A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau.

jmGiraud: 2015/10/01

n 20 :

j 3 :

j n <

X j n / :

Y j n X B :

XY X Y + 20 black 15 mat :

List 1122334455110 mat :

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

x 0 ≥ ( x 1 ≤ ( & j n x B undefined if x 0 ≥ ( x 1 ≤ ( & n x E undefined if 2 1 sys X Y + 20 black 1 5 mat 2 1 sys

j 4 :

"j" steps

1 10 range List transpose augment j List j el + 20 red 1 5 mat 2 1 sys

n j X Y 14 mat 204 0.15 0.2428 14 mat

Collect some Bernstein polynomials

x 0 ≥ ( x 1 ≤ ( & n x E undefined if 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

x 0 ≥ (x 1 ≤ (& n x E undefined if 220 x B 320 x B 420 x B 520 x B 720 x B 1520 x B 1720 x B 91 sys

This plot can't be constrained, however suitable to plot a Bernstein collection.

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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. Sergei Natanovich Bernstein: 1880 .. 1968 Alma Mater: Université de Paris