SVG(ndp,p) ... Compacted from original Davide for Jean

01a7be3a-aded-41bf-8181-b5563dacb89b 17 4 5 decimal

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

de Casteljau . . .

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Paul de Casteljau (born 1930 in Besançon, France.) is a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, and the curves called De Casteljau curve or Bézier curves. De Casteljau's algorithm is widely used, with some modifications, as it is the most robust and numerically stable method for evaluating polynomials. Other methods, such as Horner's method and forward differencing, are faster for calculating single points but are less robust De Casteljau's algorithm is still very fast for subdividing a De Casteljau curve or Bézier curve into two curve segments at an arbitrary parametric location.

This doument implements in discrete mode the Mathcad scalar equivalent. The Mathcad code is much shorter, not necessarily easier to work with.

SVG(ndp,p) ... Compacted from original Davide for Jean

ndp p SVG t-> interpolation points [0,1] ... p-> control points t p DeCasteljau M 1 t - (p 1 row * t p 2 row * + : p rows 2 > i 2 p rows 1 - range M M 1 t - (p i row * t p i 1 + row * + stack : for t M DeCasteljau 21 line M if 21 line : t-> range of interpolation points [0,1] ... p-> control points t p DeCasteljauRange M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 31 line : ndp-> number of discretised points [>1] ... p-> control points t 011 ndp 1 - / range : M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 91 line :

NDP 50 :

svg 1 :

p svg 3 - 4 - 4 - 2 - 2 - 00242 mat 022042 - 34 - 42 mat 21 mat * :

p1 svg 3 - 4 - 4 - 2 - 2 - 0022042 - 34 - 72 mat * :

Observe this specific construction

Minaret quadruplet is more visual minaret NDP p 1 el SVG : i 2 p length range minaret minaret NDP p i el SVG stack : for 31 line

p1 minaret p1 3 1 sys p 1 el p 2 el 2 1 sys

svg 1 :

NDP 100 :

B svg 600603003006021560350603502035010 - 28550 - 92 mat * :

p svg 606000303023 mat 00030306023 mat 0021560606023 mat 21535035060602023 mat 3503502852010 - 50 - 23 mat 15 mat transpose * :

bezier user companion: NDP, p bezier NDP p 1 el SVG : i 2 p length range bezier bezier NDP p i el SVG stack : for 31 line

B bezier p 3 1 sys

svg 1

The all thing is about the construction of the nest-vectors: join straight or join convex. 0. Each segment is a triplet of paired coordinates. 1. Bezier joins 1rst and last pairs. 2. Pairs that don't line up are joined convex.

p 1 el p 2 el p 3 el p 4 el p 5 el p 6 1 sys

svg 1 :

NDP 100 :

p svg 1010015 - 15 - 023 mat 000005023 mat 0102050605023 mat 202020500023 mat 201010015 - 15 - 23 mat 15 mat transpose * :

bezier user companion: NDP, p bin NDP p 1 el SVG : i 2 p length range bin bin NDP p i el SVG stack : for 31 line

C 0200020000505025 mat transpose :

level 510151022 mat :

top bin 20130012 submatrix :

C bin level 3 1 sys 10 15 - 10 15 - 0 0 3 2 mat 0 0 0 0 0 50 3 2 mat 0 50 10 60 20 50 3 2 mat 20 50 20 0 20 0 3 2 mat 20 0 10 15 - 10 15 - 3 2 mat top 6 1 sys

1015 - 1015 - 0032 mat 000005032 mat 0501060205032 mat 205020020032 mat 2001015 - 1015 - 32 mat 51 mat

1100 range

101200 range

201300 range

301400 range

401500 range

Read more from Wikipedia ... Sergei Natanovich Bernstein [1880 Odessa] Pierre Bézier [1910] Paul De Casteljau [1930]. Paul and Pierre doing same objective didn't know each other until ~ 1983/4 date all that stuff circled in conference(s). From where Autodesk, Adobe ... started, followed by Microsoft fonts. All having in common the ancestor Hermite cubic. In the mathematical subfield of numerical analysis de Boor's algorithm[1] is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de Casteljau's algorithm for Bézier curves. The algorithm was devised by Carl R. de Boor. Simplified, potentially faster variants of the de Boor algorithm have been created but they suffer from comparatively lower stability. ... those goodies are commonly known as NURBS [Non Uniform Rational B-splines]

Bernstein . . .

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Bernstein draw arc

n 2,3,4

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

start 1 : end start n + : 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 4

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 - 2.4 s *

De Boor . . .

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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 :

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".

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 eval : for

vx rows 101

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

DeBoor SX SY augment :

1 time t0 - 14.8 s *

SX SX :

SY SY :

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

Hermite . . .

]]> plotG(vx,vy,char,size,color)

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 :

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 :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

You must experiment each set of control points to see what's going on wrt the duplicated

control points X 033814 mat transpose : Y 7102414 mat transpose : 21 sys

X 0333815 mat transpose : Y 71022415 mat transpose : 21 sys

user fillet X 03333816 mat transpose : Y 710842416 mat transpose : 21 sys

svg 1 :

vary center point to shift fillet X svg 0333815 mat * transpose : Y svg 71062415 mat * transpose : 21 sys

XY X Y + 15 blue plotG :

Polyline

r 2 : X X r RepEnd : Y Y r RepEnd : 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Hermite PolyLine t P 16 / t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Hermite basis cubics 0...1

x P 1 el x c * x P 2 el x c * x P 3 el x c * x P 4 el x c * 4 1 sys

X Y augment 070707310363284848492 mat

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

t 01 0.025 range :

<= populate for more points at will

j runs 1..rows(X)-(r+1)

U j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

X rows r 1 + (- 6

collect the project xy U Unwrap V Unwrap augment : xy xy Strip : 21 sys

01 0.025 range

Hermite Parametric Polyline, not subdivided

XY xy X Y augment 3 1 sys

Each "j" segment is collected in matrix of 't' rows * 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export. From this basis Hermite Polyline you can draw from whatever control points XY vectors. The all thing is within the triplet programs 'U,V', Unwrap, strip.

Explain:

Bernstein is NOT a polyline. It collects a single

set of values. Unlike Bernstein, Hermite polyline

collects over the sliding 'j' columns, which are

afterwards stacked [Unwrap]. and stripped of

repeated values.

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Hermite polyline . . .

]]> plotG(vx,vy,char,size,color) ... Hermite Polyline modules

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 :

data 0 :

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 :

data Strip data 0 : nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - (range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 91 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

data sub subspace vx 0 : vy 0 : 12 mat j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 21 line eval : 11 line for v Vsub 1 el : j 2 Vsub rows range v v Vsub j el stack : for v 31 line 31 line :

Round OFF Matrix/Vector M n Round i 1 M rows range m i el M i el n round : for m 21 line :

X 114414 mat transpose : Y 622614 mat transpose : 21 sys

Xo 114414 mat transpose : Yo 622614 mat transpose : 21 sys

Equally sub-divide  the control polygon sub 3 :

<=  populate Hermite at will t 01 0.05 range :

XY X Y augment : origin X Y + 15 blue plotG : 21 line

poly Xo sub subspace Yo sub subspace augment : poly Xo 1 el Yo 1 el 12 mat poly stack : 21 line

isolate poly poly :

X poly 1 col : Y poly 2 col : 12 mat SUB X Y . 12 black plotG : 21 line

Polyline Construct

r 2 : X X r RepEnd : Y Y r RepEnd : 31 sys

x c 10 x ≤ (x 1 ≤ (& cases :

Hermite PolyLine t P 16 / t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Hermite basis cubics 0...1

x P 1 el x c * x P 2 el x c * x P 3 el x c * x P 4 el x c * 4 1 sys

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

U j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

collect the project xy U Unwrap V Unwrap augment : 11 sys

xy xy Xo Xo rows el Yo Yo rows el 12 mat stack :

Hermite Parametric Polyline CAD style ... adjust fillet

origin xy XY SUB 4 1 sys

i 251 range H i el xy i 2 el : d i el 0 i * : 21 line for

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 :

data H d augment :

aggregation @ end points

data + 10 blue plot plotG, RepEnd, Unwrap, Strip

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 :

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 :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

r 2 :

svg 0.75 :

Hermite Polyline Project

t0 1 time :

XW svg 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 92 mat * :

X XW 1 col r RepEnd : Y XW 2 col r RepEnd : XY X Y augment : 31 line

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

α 16 / :

at α = 1/6 first & last points join, other α we have a zoom ± effect

Hermite PolyLine t P α t 3^- 3 t 2^* + 3 t * - 1 + (3 t 3^* 6 t 2^* - 4 + (3 t 3^* - 3 t 2^* + 3 t * + 1 + (t 3^41 mat * :

Each "j" segment is collected in matrix of 't' rows x 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export.

t 01 0.05 range :

<= populate at will

Hermite Procedure

t j XP μ j 11 line : t P 1 el X μ el * t P 2 el X μ 1 + el * + t P 3 el X μ 2 + el * + t P 4 el X μ 3 + el * + 21 line :

t j YP μ j 11 line : t P 1 el Y μ el * t P 2 el Y μ 1 + el * + t P 3 el Y μ 2 + el * + t P 4 el Y μ 3 + el * + 21 line :

j runs 1..rows(X)-(r+1)

U j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

xy U Unwrap V Unwrap augment :

xy xy Strip :

polygon XW 1 col XW 2 col . 10 black plotG :

Hermite Polyline to Esplina

xy XW polygon 3 1 sys

Hermite PolyLine 0...1

x P 1 el x P 2 el x P 3 el x P 4 el 4 1 sys

1 time t0 - 14 s *

Pierre Étienne Bézier (1 September 1910 – 25 November 1999) was a French engineer and one of the founders of the fields of solid, geometric and physical modelling as well as in the field of representing curves, especially in computer-aided design and manufacturing systems. As an engineer at Renault, he became a leader in the transformation of design and manufacturing, through mathematics and computing tools, into computer-aided design and three-dimensional modeling.[1] Bézier patented and popularized the Bézier curves and Bézier surfaces that are now used in most computer-aided design and computer graphics systems. Sergei Natanovich Bernstein (Russian: Серге́й Ната́нович Бернште́йн, sometimes Romanized as Bernshtein; 5 March 1880 – 26 October 1968) was a Russian and Soviet mathematician of Jewish origin known for contributions to partial differential equations, differential geometry, probability theory, and approximation theory.

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

P 1.5 0 3.5 - 4.1 - 1.5 - 0 3.2 4.7 2.6 4.8 3.4 0 1.5 - 4.1 - 3.1 - 028 mat transpose :

t0 1 time :

X P 1 col : Y P 2 col : 12 mat XY P o 7 red plotG : 21 line

N 200 :

Linear vector 'u' for cspline S n P rows : X X X X stack : Y Y Y Y stack : 21 line u 1 el 0 : i 2 n 2 n * + (range u i el u i 1 - el X i el X i 1 - el - Y i el Y i 1 - el - 22 mat det + : for |,| determinant uN u n 1 + el u 1 el - N / : (increments ≡ i 1 N range ux i el i uN * : for i 1 N range Sx i el u X ux i el cinterp : Sy i el u Y ux i el cinterp : Sx 1 el P 11 el : Sy 1 el P 12 el : 12 mat 31 line for Sx Sy augment 31 line eval 71 line :

S S :

S XY 2 1 sys

"Esplina": courtesy Mathcad 11. Permission granted from Mathsoft Collab AI2000. Smath code is demonstrative ... NOT 1/1 MCD NOT 1/1 MCD ... is immaterial whereas Esplina is just a smoothing circulation shaper.

The cspline proceeds in sorted order of 'vx', thus excluding splining closed data sets. The vector 'u' is created from the descending vector of the determinants |X,Y|... thus: directly parent to X,Y. This condition is necessary and sufficient to spline a closed data set in Cartesian coordinates system. 'uN' [incremental] determines the spacing of the interpolated points over the N points.

x S 1 col : y S 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

Area
Perimeter A P 21 mat 52.1 28.9 21 mat

Center of gravity Xo Yo 21 mat 0.046 0.354 21 mat

1 time t0 - 1.9 s *