Companion Bezier

55c71594-606d-4057-8121-92e2d3f63a5c 7 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.

Companion Bezier

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

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Fourier "ifft" from padded created vector [Numeric] v ifft N v length : s 1 N sqrt / : 12 mat n 0 N range I n 1 + el s (v k 1 + el e i 2 π * n * N / (* k * -^* (k 0 N 1 - sum eval * : for I 31 line :

add/remove '0' Y 100000117 mat transpose :

shape/scale directions sX 1 : sY 1 : 12 mat

U Y ifft mat2sys sys2mat : a U Re mat2sys sys2mat : b U Im mat2sys sys2mat : 31 line

Draw odd polygon

sX a * sY b * augment

a transpose #

b transpose #

. . . Close the open regular heptagon . . .

Delete the last RHS coordinates

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xx 0.756 0.614 0.294 0.037 0.037 0.294 0.614 17 mat transpose : yy 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 17 mat transpose : 21 sys

z Close n z rows : w 1 el z n el : i 2 n 1 + (range w i el z i 1 - el : for w w n 2 + el z 1 el : w n 3 + el z 2 el : w 71 line :

XY xx yy augment :

can be automated

CLOSE xx Close yy Close augment :

XY CLOSE 2 1 sys

1. closing a shape is borrowed from Mathcad "Create Amazing Images". 2. The system runs anti-clockwise

CLOSE transpose 1 row CLOSE transpose 2 row 21 mat 0.614 0.756 0.614 0.294 0.037 0.037 0.294 0.614 0.756 0.614 110 mat 0.296 - 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 0 0.296 110 mat 21 mat

bSpline modules

u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :

The Gaussian b-spline

Gaussian basis 'b-spline'

2 - x ≤ x 2 ≤ & x b Undefined if

0 b 0.6667

u v Bsp v m el u m - b * (m 1 v rows sum :

bSpline t 2 X length 1 - (2.1 range : i 1 t rows range vx i el t i el ox eval : vy i el t i el oy eval : 21 line for vx vy augment 31 line :

3 appVersion 0.98.6179 ≡ timing 55 s ≡ 21 mat

once the bSpline is plotted the module is free again for next data set ox(u), oy(u)

t0 1 time :

plot(data,char,size,color) ... Hermite Polyline modules

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 :

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 :

data sub subspace 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 21 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 :

special construct wrt  
closing square-circle Xo 0.614 0.756 0.614 0.294 0.037 0.037 0.294 0.614 0.756 0.614 110 mat transpose : Yo 0.296 - 0 0.296 0.368 0.164 0.164 - 0.368 - 0.296 - 0 0.296 110 mat transpose : 21 sys

sub 1 :

subspaces is 1 for interpolation

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : ctrl Xo Yo augment : 31 line

isolate New New :

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

<=  populate Hermite at will t 01 0.01 range :

Clean the system

Hermite 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 t P 1 el X j el * t P 2 el X j 1 + el * + t P 3 el X j 2 + el * + t P 4 el X j 3 + el * + : t j YP t P 1 el Y j el * t P 2 el Y j 1 + el * + t P 3 el Y j 2 + el * + t P 4 el Y j 3 + el * + : 21 sys

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 : Polyline xy : 21 sys

X Xo 2 RepEnd : Y Yo 2 RepEnd : 21 line

s 2 t rows * :

Polyline Polyline s Polyline rows s - (range 12 range el :

u ox u X Bsp : u oy u Y Bsp : 21 line

u 21 :

bSpline bSpline u bSpline rows u - (range 12 range el :

Bspline bSpline o 5 black plot : XoYo Xo Yo augment + 20 red plot : 21 line

rows bSpline
rows Hermite Bspline rows Polyline rows 21 mat 7070821 mat

ctrl Polyline Bspline XoYo 0.378188 0 + 40 green 1 5 mat Bspline 1 1 el 0 . 15 black 1 5 mat Bspline 36 1 el 0 . 15 black 1 5 mat 7 1 sys

1 time t0 - 46 s *

Bspline 11 el Bspline 361 el 21 mat 0.708667 0.047708 21 mat

0.708667 0.047708 + 2 / 0.378188

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 :

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 :

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 11 - 1 - 111 - 16 mat transpose : Y 111 - 1 - 1116 mat transpose : 21 sys

Xo 11 - 1 - 111 - 16 mat transpose : Yo 111 - 1 - 1116 mat transpose : 21 sys

Equally sub-divide  the control polygon sub 4 :

<=  populate Hermite at will t 01 0.1 range :

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

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

isolate New New :

X New 1 col : Y New 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

s sub 1 - (t rows * :

origin Xo Yo augment :

xy xy s xy rows s - (range 12 range el :

origin origin :

xy rows 177

x y axes x y / - :

x r x - :

origin New xy x r 0 x ≤ ( * axes 5 1 sys

more maths to calculate the radius of the machined piece. Can we add the arrow radius ? More maths for ... 1. calculate the perimeter 2. the inscribed area. 3. ... other things.

J xy 146 range 12 range el :

xy 46 row 1 - 012 mat

xy 21 row xy 22 row xy 23 row 31 mat 0.857333333333333 - 0.957333333333333 12 mat 0.88925 - 0.93925 12 mat 0.916666666666667 - 0.916666666666667 12 mat 31 mat

radius r 0.916666666666667 - (2^1 0.916666666666667 - (2^+ sqrt : 0.920446751432272

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 :

x c 1 x 0 ≤ (x r - ≥ (& cases :

data 0.916666666666667 - 0.916666666666667 augment :

slope 1 0.88925 sqrt - / : 1.0604447908041 -

J data + 20 red plot 0 x - ( x c * x - 4 1 sys