end DeBoor

d64f37d1-2e32-459c-8d7d-32d7e98d91fe 92 4 5 decimal

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

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

http://www.chemicalaid.com/tools/reactionstoichiometry.php?equation=S+%2B+O2+%3D+SO3

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

De Boor convex closed path

This example suggests manipulating the vectors X, Y of control points [add pts, position ... ]. The idea is to track a secondary set of data points from experiment or fabricate some sort of lobe as you wish, for instance: this "data" to make things simple and demonstrative.

data 00 0.0312 0.0625 0.125 0.25 0.2812 0.5625 0.5 1 0.6562 1.4688 0.625 1.875 0.4062 2.2188 0 2.5 0.4062 - 2.6562 0.625 - 2.625 0.6562 - 2.4062 0.5 - 2 0.1875 - 1.5312 0.25 1.125 0.8125 0.7812 1.5 0.5 2.1562 0.2812 2.625 0.125 2.9062 0.0312 30212 mat :

t0 1 time :

X 011 - 0351 mat :

Y 0231051 mat :

λ X length : 5

Cntrl X Y . 14 black plotG :

Polygon X Y augment :

Data data 1 col data 2 col + 8 black plotG :

r 1 : repeat end points ≡

n 2 : DeBoor spline order ≡

X X r RepEnd :

Y Y r RepEnd :

DeBoor 'u' vector

u 1 X rows n 2 + (+ range :

Isolate the repeated polygon.
Those X, Y used in  Sx(v), Sy(v) X 0011 - 03317 mat transpose : Y 002310017 mat transpose : 21 line

Notes: 1. spline order 'n' = 1 is stricly linear interpolation around the control polygon. 2. spline order 'n' = 2 is "convex hull" within the control polygon. 3. spline order 'n' = 3 is "convex hull shrinked internal.

Observe: Smath does not support an index as argument, similar to Mathcad 8 and earlier versions. The "index argument" is rendered via the RHS in local definition. All N_(v,j)are in local mode, including Sx(v), Sy(v). The discrete SX, SY are not in eval(,) mode. The computation time depends directly upon the size of the original polygon [open or closed]. Besides geometric construction [CAD style], DeBoor & Hermite are used to smooth data usually after a ksmooth pass, by far superior to Matlab "spline smooth".

# μ j 11 line : # 21 line :

v j N0 μ j 11 line : u μ el v ≤ (v u μ 1 + el < (& 10 if 21 line :

v j N1 μ j 11 line : v u μ el - u μ 1 + el u μ el - / v μ N0 * u μ 2 + el v - u μ 2 + el u μ 1 + el - / v μ 1 + N0 * + 21 line :

v j N2 μ j 11 line : v u μ el - u μ 2 + el u μ el - / v μ N1 * u μ 3 + el v - u μ 3 + el u μ 1 + el - / v μ 1 + N1 * + 21 line :

v j N3 μ j 11 line : v u μ el - u μ 3 + el u μ el - / v μ N2 * u μ 4 + el v - u μ 4 + el u μ 1 + el - / v μ 1 + N2 * + 21 line :

v j N μ j 11 line : v μ N1 21 line 11 mat :

activate spline order 'n' = 1

v j N μ j 11 line : v μ N1 21 line μ j 11 line : v μ N2 21 line 21 mat :

activate spline order 'n' = 2

v j N μ j 11 line : v μ N1 21 line μ j 11 line : v μ N2 21 line μ j 11 line : v μ N3 21 line 31 mat :

activate spline order 'n' = 3

λ 5

Defined as local definition v Sx v k N n el X k el * (k 1 X rows sum 11 line :

Defined as local definition v Sy v k N n el Y k el * (k 1 X rows sum 11 line :

vx 33 λ + (3.05 range :

As it looks, 'vx' always starts @ 3 ... ends @ (3+λ) 'λ' length of the original polygon

i 1 vx rows range SX i el vx i el Sx : for

vx rows 101

i 1 vx rows range SY i el vx i el Sy : for

DeBoor SX SY augment :

1 time t0 - 13.8 s *

De Boor parametric order 2 wrt Smath ... kind of Convex Hull

Polygon DeBoor Data Cntrl 4 1 sys

De Boor components

vx SX augment vx SY augment 2 1 sys x Sx x Sy 2 1 sys end DeBoor

Bézier-Bernstein CAD tool

X 011 - 0315 mat transpose : Y 0231015 mat transpose : 21 line

XY X Y augment :

Cntrl X Y . 14 black plotG :

t0 1 time :

n X rows 1 - :

n 4

'n 'determines the end point of the segment. 'n' can only be n≤rows(X)-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 n sum :

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

t 1015 -^1 0.01 range :

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

Bézier-Bernstein cntrl points

XY BezBern Cntrl 3 1 sys

The components

t BezBern 1 col augment t BezBern 2 col augment 2 1 sys x QX x QY 2 1 sys

1 time t0 - 2.5 s *

end Bézier-Bernstein

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

Bézier-Bernstein draw arc

X 011 - 0315 mat transpose : Y 0231015 mat transpose : 21 line

start 1 : end start 2 + : 21 line

XY X Y augment :

XY XY start end 12 submatrix :

X XY 1 col : Y XY 2 col : 21 line

t0 1 time :

Cntrl X Y . 14 black plotG :

n X rows 1 - :

n 2

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 :

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

t 1015 -^1 0.01 range :

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

Bézier-Bernstein cntrl points

XY BezBern Cntrl 3 1 sys

The components

t BezBern 1 col augment t BezBern 2 col augment 2 1 sys x QX x QY 2 1 sys

1 time t0 - 2 s *

jmGiraud: 2016/09/12