1ffa8888-8b11-42a5-b20c-2c029177c79d 22 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 :

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

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 :

vx data 1 col : vy data 2 col : 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 :

X p 1 col :

Y p 2 col :

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

n p rows 1 - :

k 0 :

r n k - (n ≤ n k - (0 if :

r 5

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 11 line :

t QY j n t Q Y j 1 + el * (j 0 r sum 11 line :

Lobe t 10307 -^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 X Y . 14 red plotG :

data vx vy o 5 green plotG :

XY Lobe Cntrl data 4 1 sys

Lobe #

The two bugs: 1. The local definition of the Σ is viable though not obvious for constructing formulas from classical maths litterature. 2. Bézier-Berntein is parametric wrt 't' in the range 0..1____In Lobe, Smath 5346 did support 't:=0,0.01..1' Smath 6179 does not. It needs to read some default value for '0'. It raises the suggestion that 6179 should have an implicit '0 guard'or there is a more sutile bug. I Lobe, set t:=0,0.01..1, tabulate Lobe => "uncertainty" caused by t[0]^0

Practice your own project

scale 2 :

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 11 line :

t QY j n t Q Y j 1 + el * (j 0 n sum 11 line :

Lobe t 0.000001 1 0.005 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 0 0.571653 - + 20 green 1 5 mat 3 1 sys

Specific wrt scale 2

U 0.016236 - 0.013812 21 mat :

V 0.5398 - 0.59875 - 21 mat :

u zero U V u linterp :

0 zero 0.571653 -

Lobe 2 - 2 1.911 - 1.9104 1.8242 - 1.8218 1.7394 - 1.7341 1.6566 - 1.6473 1.5759 - 1.5613 1.4971 - 1.4763 1.4203 - 1.3922 1.3455 - 1.3089 1.2726 - 1.2266 1.2016 - 1.1451 1.1325 - 1.0646 1.0653 - 0.985 0.9999 - 0.9062 0.9364 - 0.8284 0.8746 - 0.7515 0.8146 - 0.6754 0.7564 - 0.6003 0.7 - 0.5261 0.6452 - 0.4527 0.5922 - 0.3803 0.5408 - 0.3087 0.4911 - 0.2381 232 mat

LobeComponents t 0.000001 1 0.005 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

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

Bézier parametric Cubic

scale 2 :

p 2 - 244 - 4 - 4 - 2242 mat

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

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

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

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 0.124 - range :

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