Collect some Bernstein polynomials

998f5823-c5cc-40ef-9d59-6b48337df4c5 23 15 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 [omnipresent] 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. .................. "Today all mathematicians and physicists agree the field of applications for mathematics knows no limits except those of knowledge itself... Sergei Natanovich Bernstein"

... approximate min .... 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

j n X B 0.1762

Collect some Bernstein polynomials

x plot n x E 120 x B 220 x B 320 x B 420 x B 520 x B 620 x B 720 x B 820 x B 920 x B 1020 x B 1120 x B 1220 x B 1320 x B 1420 x B 1520 x B 1620 x B 1720 x B 1820 x B 1920 x B 201 sys :

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

n x E j n x B XY 3 1 sys

Collect some Bernstein polynomials

x plot

Sergei Natanovich Bernstein: 1880 . . . 1968

Alma Mater: Université de Paris

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A minimalist application Bézier-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 :

shiftXdomain 3 :

ω 1 : svgX 1 : svgY 1.125 : 13 mat

X svgX 3 - 2 - 1 - 00012319 mat transpose shiftXdomain + (* (: Y svgY 0122 - 2 - 2 - 21019 mat transpose * (: Wy 111 ω ω ω 11119 mat transpose : Wx 11111111119 mat transpose : n X rows 1 - : 51 sys

XY X Y augment : Cntrl XY . 14 black plot : 21 line

Bezier-Bernstein procedure

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 eval :

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

or simply 10^-15 zero 1015 -^eval :

'0' is uncertainty t zero 1 0.0125 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 :

vx BezBern 1 col : vy BezBern 2 col : 12 mat x interp vx vy x cinterp : 21 sys

U 3 :

U interp 1.037 -

Bézier-Bernstein cntrl points. Experiment center point ... single/duplicate/triplicate. Triplicate supports Wy

XY BezBern Cntrl U U interp + 20 black 1 5 mat 4 1 sys

Explore simple lobe project

scale 1.25 :

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

X p 1 col : Y p 2 col : 21 sys

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 :

lobeX i 1 t rows range LobeX i el t i el QX eval : for t LobeX augment 21 line :

lobeY i 1 t rows range LobeY i el t i el QY eval : for t LobeY augment 21 line :

zero 11015 -^*

Lobe t zero 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 :

specifi drop lobeY 2 col Min : 1.562500000000000 -

Exact Bézier constant for this lobe-polygon βez 1.5625 :

junction βez Yo * ≡

Tabulate sanity t lobeY 2 col augment 11 mat :

Center of Gravity

x lobeY 1 col : y lobeY 2 col : 12 mat

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

A P 21 mat 1.25 5.785 21 mat

Xo Yo 21 mat 0.583352869162012 0.229073898952896 - 21 mat

p Lobe 0 drop + 40 black 1 5 mat 0 βez Yo * + 40 black 1 5 mat 4 1 sys lobeX lobeY 2 1 sys

scale drop Yo βez Yo * 41 mat 1.25 1.5625 - 0.2291 - 0.3579 - 41 mat