c64bdd4e-a417-49c9-8149-1f42d2dfe8d1 20 6 9 decimal

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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 :

This Smath document reproduces Mathcad 11 De Boor spline. Takes much longer to compute. De Boor reproduces De Casteljau more compact algorithm. Bézier-Bernstein were "Alma Mater" companions. Bézier and De Casteljau did not know each other until the founder of ADOBE met Bézier. The idea of De Boor was to code De Casteljau algorithm to be executable on the available computing machinery of his time. A nice lagacy.

t0 1 time :

X 1.5 0 3.5 - 4.1 - 1.5 - 0 3.2 4.7 1.5 91 mat :

Y 2.6 4.8 3.4 0 1.5 - 4.1 - 3.1 - 0 2.6 91 mat :

r 1 :

n 2 :

X X r RepEnd :

Y Y r RepEnd :

m 1 X rows n 2 + (+ range :

# # range

XY 1.5 2.6 1.5 2.6 0 4.8 3.5 - 3.4 4.1 - 0 1.5 - 1.5 - 0 4.1 - 3.2 3.1 - 4.7 0 1.5 2.6 1.5 2.6 112 mat :

X XY 1 col : Y XY 2 col : Cntrl X Y . 14 black plotG : 31 line

Esplina data for De Boor

XY Cntrl 2 1 sys

u i 1 m rows range u i el m i el : for u 21 line :

m u :

v m N0 two errors code u m el v ≤ u m 1 + el < 10 if 21 line :

Smath does not support an index in the calling function [LHS], neither it supports the calling function being indexed.

1.75 1 N0 #

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

The index must be locally assigned in the EXECUTABLE procedure

μ 2 :

u μ el 2

u μ 1 + el 3

5.5 3 N0 1

Not yet instructed: Smath is rigid wrt the bolean. The above Bolean is underdetermined, Ah !

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

Now, the instruction is complete and executable for De Boor spline.

2.5 1 N0 0

11 N0 1

1.5 1 N0 1

21 N0 0

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 :

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

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 :

Not needed for n = 2 then not collected

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

Collect the respective De boor

n 2

<= Spline order, set from above

v Sx v k N n el X k el * (k 1 X rows sum :

v Sy v k N n el Y k el * (k 1 X rows sum :

vx 312 3.1 range :

<= n=2 populate for more points at will

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

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

DeBoor SX SY augment :

DeBoor DeBoor :

U DeBoor 1 col : V DeBoor 2 col : 12 mat

Slightly slower  [10 sec  90 points] DeBoor i 1 vx rows range SX i el vx i el Sx : SY i el vx i el Sy : 21 line for SX SY augment 21 line eval 11 line :

1 time t0 - 167 sec *

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

XY DeBoor Cntrl 3 1 sys

De Boor components

vx U augment vx V augment 2 1 sys

jmGiraud: 2015/09/27