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

0c95df99-7017-45fa-91fc-1f112f776886 26 4 5 decimal

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Hermite . . . Polyline/interpolation, [Xo,Yo] . . .

]]> 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 :

Xo 2 - 1.37 - 0.89 - 0.5 - 0.25 - 0 0.22 0.45 0.75 1.3 2111 mat :

Yo 0.519 0.414 - 1.061 - 1 - 0.347 - 2.791 1.081 0.765 1.181 0.588 0.466 - 111 mat :

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

ctrl Xo Yo augment : XoYo ctrl + 15 black plot : 21 sys

Polyline Notes ... 1. 'sub ≡ 1" interpolates exactly 'b-Spline' thus, 'b_spline' not given. 2. 'sub 2,3 ...' roams closer to support points. 3. 'sub 3' produces duplicates top/end.

lovely, smooth contour ! sub 2 :

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 0.1 range :

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 Strip : Polyline Polyline Xo 11 el Yo 11 el 12 mat stack : 31 sys

Polyline Polyline 2 Polyline rows 1 - (range 12 range el sub 3 ≡ Polyline cases :

Polyline rows 221

ctrl Polyline XoYo 3 1 sys

b_spline . . . Polyline/interpolation, [Xo,Yo] . . .

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t0 1 time :

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 ≡ WStiming 2 min ≡ 21 mat

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

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 :

user original polygon Xo 0.5 - 0.4 - 0.3 - 0.2 - 0.1 - 0.04 - 0 0.04 0.1 0.2 0.3 0.4 0.5 113 mat transpose : Yo 0.007937 0.012 0.022 0.048 0.167 0.556 1 0.556 0.167 0.048 0.022 0.012 0.007937 113 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 0.05 range :

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 Strip : 21 sys

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

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

bx bSpline 1 col : by bSpline 2 col : 21 line

Bspline bSpline o 5 black plot : XoYo Xo Yo augment + 15 black plot : 21 line

rows bSpline
rows Hermite Bspline rows Polyline rows 21 mat 14128021 mat

adapt the export Polyline Polyline 3 Polyline rows 1 - (range 12 range el :

ctrl Polyline Bspline XoYo 4 1 sys

1 time t0 - 52 s *

Xo 2 - 1.37 - 0.89 - 0.5 - 0.25 - 0 0.22 0.45 0.75 1.3 2111 mat :

Yo 0.519 0.414 - 1.061 - 1 - 0.347 - 2.791 1.081 0.765 1.181 0.588 0.466 - 111 mat :

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

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

bx bSpline 1 col : by bSpline 2 col : 21 line

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 0.05 range :

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 Strip : 21 sys

Bspline bSpline o 5 black plot : XoYo ctrl + 15 black plot : 21 line

search 70 : user bSpline search row : User search user augment : 31 line

rows bSpline
rows Hermite Bspline rows Polyline rows 21 mat 12124021 mat

Adapt the export Polyline Polyline 2 Polyline rows range 12 range el :

Polyline 1 row Polyline Polyline rows row 21 sys 2 - 0.519 12 mat 2 0.466 - 12 mat 21 sys

ctrl Polyline Bspline XoYo user 1 1 el user 1 2 el + 50 green 1 5 mat 5 1 sys

User 70 0.1992 1.4206 13 mat

1 time t0 - 101 s *

Pretty good  ... x f Xo Yo x ainterp :

ctrl x f XoYo 3 1 sys

Hermite . . . interpolation . . . [Xo Yo]

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New 2 - 0.519 1.37 - 0.414 - 0.89 - 1.061 - 0.5 - 1 - 0.25 - 0.347 - 0 2.791 0.22 1.081 0.45 0.765 0.75 1.181 1.3 0.588 2 0.466 - 112 mat :

Xo New 1 col : Yo New 2 col : 12 mat support New . 14 black plot : 21 mat

Global definition 
fetched from below 
collapsed area ... New . 14 black plot

0.01 over populate demo t 01 0.1 range :

t rows 11

truncate t rows 2 - : 9

Hermite Interpolation 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 :

subspaces is '1' for interpolation sub 1 :

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

isolate Hermite Hermite :

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

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

x c 10 x ≤ (x 1 ≤ (& cases :

interpolate 16 / t 3^- 3 t 2^* + 2 t * - 3 t 3^* 6 t 2^* - 3 t * - 6 + 3 t 3^* - 3 t 2^* + 6 t * + t 3^t - 41 mat *

Hermite PolyLine t P 16 / t 3^- 3 t 2^* + 2 t * - 3 t 3^* 6 t 2^* - 3 t * - 6 + 3 t 3^* - 3 t 2^* + 6 t * + t 3^t - 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

polyline 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 *

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 : contour xy Strip : 21 sys

Ω contour :

Ω rows 120

Ω Ω t rows Ω rows truncate - (range 12 range el :

Ω Ω eval :

Ω rows 101

Hermite produces discrete export interpolated values. At this point for convenience, you can interpolate cinterp/ainterp.

ctrl Ω support 3 1 sys

vx Ω 1 col : vy Ω 2 col : 12 mat x f vx vy x ainterp : 21 mat

x f support 3 1 sys

Hermite Interpolation 16 / t 3^- 3 t 2^* + 2 t * - 3 t 3^* 6 t 2^* - 3 t * - 6 + 3 t 3^* - 3 t 2^* + 6 t * + t 3^t - 41 mat *

Hermite Polyline 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 *

jmG: 20230223