<p> Companion Bezier </p>

b0c7b132-fb27-4ba8-bcb8-5d8971244bc5 44 4 5 decimal

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DeCasteljau SVG algorithm chained style

Chained style adapted from Davide Carpi original.

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 :

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

In these three examples, svg = 1 is the original creation to make it visible. Vary svg coefficient different than '1' and see the all system scaled zoomed [±] in both Cartesian directions X, Y.

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

NDP 100 :

svg 1 :

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

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

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

θ rot θ cos θ sin - θ sin θ cos 22 mat : θ π - 2 / : RotData p1 θ rot * : RotBezier minaret θ rot * : 41 line

RotData RotBezier 2 1 sys

Extrapolate from within the "Bézier companion"

You can extrapolate the RHS by introducing the end range 'ε' of 't' in the 'companion' ... then collect both extrapolated branches.

<p> Companion Bezier </p>

ndp =  number of discretization points (> 1)
p = control points ndp p ε SVG 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 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 : ---------------------- t 0 ε 1 ndp 1 - / range : M t 1 el p DeCasteljau : i 2 t length range M M t i el p DeCasteljau stack : for M 81 line :

NDP 100 :

svg 1 :

ε 1.6 :

Extrapolate the RHS
... in the companion t 0 ε 1 ndp 1 - / range ≡

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

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

minaret NDP p 1 el ε SVG : i 2 p length range minaret minaret NDP p i el ε SVG stack : for 21 line

sub minaret NDP ε * minaret rows 12 submatrix : R sub 1 col 1 - (* sub 2 col augment : 21 line

mid k 2 : sub k 1 + 1 el sub 1 col max ≠ idx k el k : k k 1 + : 21 line while sub idx rows 2 + row 31 line :

minaret rows 318

mid 3.265965 2.909091 - 12 mat

sub R p1 p1 0 10 0 10 - 2 2 mat 0 mid 7 1 sys

ABOUT arc length: For the arc length of "Minaret", we will consider the extrapolated blue. This ½ is split in two segments at the point of discontinuity of the derivative. The total arc length is the Σ of each segm's. The length of each segment is the last value of the cumulative integration... i.e: the algebraic distance between these two extreme datum. --------------------------------- Observe: the "Procedure" proceeds in the descending order of the "Companion"

Procedure to arc length ½ Minaret

data Deriv X data 1 col : Y data 2 col : n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : X deriv augment 91 line :

D sub Deriv :

idx switch index k 2 : D k 1 + 2 el D k 2 el ≤ idx k el k : k k 1 + : 21 line while idx rows 1 + 41 line :

idx 82

sub idx row 3.265965 2.909091 - 12 mat

D1 D 1 idx 12 submatrix : D2 D idx 1 + (D rows 12 submatrix : 21 line

data Σ vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 51 line :

D1 D2 2 1 sys

ArcIntegrand_1 δx D1 2 col : xx D1 1 col : i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for xx yy 1 (* augment 41 line :

ArcIntegrand_2 δx D2 2 col : xx D2 1 col : i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for xx yy augment 41 line :

Σ1 ArcIntegrand_1 Σ :

Σ2 ArcIntegrand_2 Σ :

limit Σ1 idx row 11 el : end D D rows 1 el : 21 line

A1 Σ1 1 col Σ1 2 col augment : A2 Σ2 1 col Σ2 2 col augment : 21 line

limit end 21 mat 3.266 2.619 - 21 mat

Arc length is the algebraic
distance between 'datum' Σ1 idx 2 el Σ2 Σ2 rows 2 el 21 mat 5.923714 8.033065 - 21 mat

Cumulative arc length of each segment

A1 A2 end Σ2 Σ2 rows 2 el . 14 black limit 0 . 14 black limit Σ1 idx 2 el . 14 black 3 5 mat limit 10 limit 10 - 2 2 mat 4 1 sys

Σ1 idx 2 el abs 5.924

UG 1 : SVG UnitGraphic ≡

Absolute value to make sense Σ1 idx 2 el Σ2 Σ2 rows 2 el abs 21 mat sum UG 13.957

Σ2 Σ2 rows 2 el abs 8.033