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

95ac401f-35af-4d24-8d05-e76a5f1dbe48 37 6 12 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 :

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 :

Xo 0.5 - 123456718 mat transpose : Yo 6 7.5 7.5 1.25 1.25 64718 mat transpose : 21 sys

t0 1 time :

X 0.5 - 123456718 mat transpose : Y 6 7.5 7.5 1.25 1.25 64718 mat transpose : 21 sys

Equally sub-divide  the control polygon sub 3 :

<=  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 :

X rows r 1 + (- 23

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

Hermite Parametric Polyline CAD style ... adjust fillet

origin xy XY SUB 4 1 sys

X Y augment transpose 0.5 - 0.5 - 0.5 - 0 0.5 1 1.3333 1.6667 2 2.3333 2.6667 3 3.3333 666 6.5 7 7.5 7.5 7.5 7.5 5.4167 3.3333 1.25 1.25 213 mat

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 :

Xo 0.5 - 123456718 mat transpose : Yo 6 7.5 7.5 1.25 1.25 64718 mat transpose : 21 sys

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 Vsub 21 line :

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

sub 3 :

Equally sub-divide ctrl polygon

New Xo sub subspace Unest Yo sub subspace Unest augment :

New Xo 1 el Yo 1 el 12 mat New stack :

New 0.5 - 60 6.5 0.5 71 7.5 1.3333 7.5 1.6667 7.5 2 7.5 2.3333 5.4167 2.6667 3.3333 3 1.25 3.3333 1.25 3.6667 1.25 4 1.25 4.3333 2.8333 4.6667 4.4167 56 5.3333 5.3333 5.6667 4.6667 64 6.3333 5 6.6667 677222 mat

XoYo Xo Yo + 15 black plotG : Sub New 1 col New 2 col . 14 red plotG : 21 line

Xo Yo augment XoYo Sub 3 1 sys

1 time t0 - 10 s *

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 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 :

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

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

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 :

1 time t0 - 19 s *

data H d augment :

aggregation @ end points

data + 10 blue plot 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 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 :

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 3 :

<=  populate Hermite at will t 01 0.05 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 * :

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

1. Hermite Parametric Polyline CAD style ... adjust fillet. 2. Delicious square fillet ... brute force cleanup applied ! 3. bSpline interpolation would draw an inscribed circle. Hermite interpolation as well ... both from a closed data set

origin xy XY SUB 4 1 sys 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 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 :

XY 1 0.25 203 0.25 - 4 0.5 - 5 0.25 - 607 0.25 8 0.5 9 0.25 10011 0.25 - 12 0.5 - 13 0.25 - 14015 0.25 16 0.5 17 0.25 18019 0.25 - 20 0.5 - 21 0.25 - 22023 0.25 232 mat :

X XY 1 col : Y XY 2 col : 12 mat Xo XY 1 col : Yo XY 2 col : 12 mat 21 sys

Equally sub-divide  the control polygon sub 2 :

<=  populate Hermite at will t 01 0.05 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

1. Hermite Parametric Polyline CAD style ... adjust fillet.

origin xy XY SUB 4 1 sys

1 time t0 - 70 s *

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 37 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 ε 1 - 1 - 111 - 1 - 117 mat * transpose : Yo 11 - 1 - 111 - 1 - 17 mat transpose : 21 sys

ε 1.375 :

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 20 :

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 4240521 mat

ctrl Polyline Bspline XoYo 4 1 sys

1 time t0 - 38 s *

circle/oblate/prolate ellipse from 'ε' above

. . . Close the square . . .

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xx 1 - 111 - 14 mat transpose : yy 1 - 1 - 1114 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 :

CLOSE xx Close yy Close augment :

XY CLOSE 2 1 sys

CLOSE 1 - 11 - 1 - 11 - 111 - 11 - 1 - 11 - 72 mat

XY 1 - 1 - 11 - 111 - 142 mat

. . . a freak of no interest . . .

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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 :

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

bSpline t 05 0.05 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 0 + vy augment 31 line :

vy 11 - 24 - 0116 mat transpose :

n 16 range :

X n :

Y vy :

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

n vy augment bSpline 2 1 sys

. . . Close the square . . .

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xx 1 - 111 - 14 mat transpose : yy 1 - 1 - 0114 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 :

CLOSE xx Close yy Close augment :

XY CLOSE 2 1 sys

XY 1 - 1 - 11 - 101 - 142 mat

CLOSE transpose 1 - 1 - 111 - 1 - 111 - 1 - 011 - 1 - 27 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 37 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 ε 1 - 1 - 111 - 1 - 117 mat * transpose : Yo 11 - 1 - 011 - 1 - 17 mat transpose : 21 sys

CLOSE transpose 1 row CLOSE transpose 2 row 21 mat 1 - 1 - 111 - 1 - 117 mat 11 - 1 - 011 - 1 - 17 mat 21 mat

ε 1 :

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 20 :

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 4240521 mat

ctrl Polyline Bspline XoYo 4 1 sys

1 time t0 - 38 s *