<content> plotG </content>

37cf7936-56b2-4cb5-9acd-2c778763a47f 35 4 5 false false 0 0 decimal

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

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

Example 1 . . . l_p_c cubic spline coefficients

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Part 1. l_p_c cubic scalar splines

Essentially, the coefficients of the cubic spline consists in solving
the basis tridiagonal matrix wrt the 'br' vector of derivatives .
Independently of the type of cubic spline [type: l, p, c]
two steps of preparation are needed [collapsed area]:
0. Install the constant vector 'br'
1. Install the blank tridiagonal matrix.

XY 142537 4.5 956 6.1 7711824 8.9 34104011371247122 mat :

X XY 1 col :

Y XY 2 col :

N XY rows 1 - : 11

r 2 N range :

<content> Prepare the Tridiagonal Matrices </content>

Prep.0 Install the constant vector 'br'

i r br i el 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 - / (- eval : br N 1 + el 0 : 21 line for

br 01 0.666666666666667 - 7.33333333333333 - 6.90909090909091 3.53535353535354 8.55555555555556 1.88888888888889 - 5.65656565656566 - 8.45454545454546 - 130121 mat

Prep.1 Install the tridiagonal matrix

M 11 el 1 :

M N 1 + N 1 + el 1 :

The basis tridiagonal matrix [l, p, c] M 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011212 mat

0. Construct the tridiagonal linear spline matrix [lspline]

i r M i i 1 - el X i el X i 1 - el - 6 / : for

i r M i i el X i 1 + el X i 1 - el - 3 / : for

i r M i i 1 + el X i 1 + el X i el - 6 / : for

slin M 1 -^br * eval :

M 10000000 0.1667 0.6667 0.1667 000000 0.1667 0.8333 0.25 000000 0.25 0.6667 0.0833 000000 0.0833 0.5333 0.1833 000000 0.1833 0.6667 0.15 000000 0.15 0.6333 0.1667 000000 0.1667 0.6333 0000000 0.15 000000000000000000000000128 mat

slin 0 0.6629 3.3484 14.2699 - 16.1137 2.7041 - 15.8926 6.6248 - 2.2796 - 17.1442 - 23.7861 0121 mat

1. Construct the tridiagonal parabolic spline matrix [pspline]

M 12 el 1 - :

M N 1 + N el 1 - :

sp M 1 -^br * eval :

M 11 - 000000 0.1667 0.6667 0.1667 000000 0.1667 0.8333 0.25 000000 0.25 0.6667 0.0833 000000 0.0833 0.5333 0.1833 000000 0.1833 0.6667 0.15 000000 0.15 0.6333 0.1667 000000 0.1667 0.6333 0000000 0.15 000000000000000000000000128 mat

sp 0.5241 0.5241 3.3797 14.2816 - 16.1135 2.698 - 15.8658 6.5284 - 2.6569 - 15.8512 - 18.7702 18.7702 121 mat

2. Construct the tridiagonal cubic spline matrix [cspline]

M 12 el X 1 el X 3 el - (X 3 el X 2 el - / (:

M N 1 + N el X N el X N 2 - el - (X N 2 - el X N 1 - el - / :

M 13 el X 2 el X 1 el - (X 3 el X 2 el - (/ :

M N 1 + N 1 - el X N 1 - el X N el - (X N 1 - el X N el - (/ :

sc M 1 -^br * eval :

M 12 - 100000 0.1667 0.6667 0.1667 000000 0.1667 0.8333 0.25 000000 0.25 0.6667 0.0833 000000 0.0833 0.5333 0.1833 000000 0.1833 0.6667 0.15 000000 0.15 0.6333 0.1667 000000 0.1667 0.6333 0000000 0.15 000000000000000000000000128 mat

sc 1.272 - 1 3.272 14.24 - 16.1037 2.6884 - 15.8353 6.4211 - 3.0759 - 14.4152 - 13.2 39.6152 121 mat

<content> ptsSpec </content>

ptsSpec k 1 : i k X rows range data i el Sym size clr 13 mat : k k 1 + : 21 line for v data 1 el : j 2 data rows range v v data j el stack : for v 51 line :

Sym . :

size 4 :

clr black :

linear slin 0 0.6629 3.3484 14.2699 - 16.1137 2.7041 - 15.8926 6.6248 - 2.2796 - 17.1442 - 23.7861 0121 mat

parabolic sp 0.5241 0.5241 3.3797 14.2816 - 16.1135 2.698 - 15.8658 6.5284 - 2.6569 - 15.8512 - 18.7702 18.7702 121 mat

cubic sc 1.272 - 1 3.272 14.24 - 16.1037 2.6884 - 15.8353 6.4211 - 3.0759 - 14.4152 - 13.2 39.6152 121 mat

support X Y ptsSpec augment :

support 14.4 black 25.4 black 37.4 black 4.5 9.4 black 56.4 black 6.1 7.4 black 711.4 black 824.4 black 8.9 34.4 black 1040.4 black 1137.4 black 1247.4 black 125 mat

In 's' , plug the the selected type of spline [slin, sp,sc]'slin & sp' join cubic 2 points entrance and along and departutre.'sc' joins cubic 3 points entrance and along and departure. s sp :

Type linear s slin ≡ parabolic s sp ≡ cubic cases :

Type parabolic

Part 2. Classroom journey cubic splines

Recover segments coeff's [a,b,c,d]

Collect the cubic 's' components for C(j,u) abcd i 1 X rows 1 - range hs i el s i 1 + el s i el - : hy i el Y i 1 + el Y i el - : hx i el X i 1 + el X i el - : 31 line for i 1 X rows 1 - range a i el hs i el 6 hx i el * / : b i el s i el 2 / : c i el hy i el hx i el / hx i el 6 / s i 1 + el 2 s i el * + (* - (: d i el Y i el : 41 line for a b c d augment 31 line :

abcd 0 0.262 0.738 4 0.4759 0.262 1.262 5 1.9624 - 1.6898 3.2139 7 10.1317 7.1408 - 4.9625 - 9 2.8502 - 8.0567 4.5046 - 6 3.4377 1.349 - 2.874 7 3.7324 - 7.9329 8.7995 11 0.7169 3.2642 - 13.4682 24 1.9991 - 1.3284 - 9.3348 34 5.7702 7.9256 - 0.8446 - 400 9.3851 0.6149 37114 mat

a abcd 1 col :

c abcd 3 col :

b abcd 2 col :

d abcd 4 col :

j u C i j : a i el u X i el - (3^* b i el u X i el - (2^* + c i el u X i el - (* + d i el + 21 line :

j support j x C 2 1 sys <content> Mathcad 8 Pro. code conversion </content>

s X Y t InterpSpline Mathcad 8 Pro. code conversion t X 1 el ≤ idx 1221 mat : t X N el ≥ idx N N 1 + 21 mat : i 1 : X i el t < i i 1 + : while idx i 1 - i 21 mat : 31 line if if hx X idx 2 el el X idx 1 el el - (: hy Y idx 2 el el Y idx 1 el el - (: hs s idx 2 el el s idx 1 el el - (: 31 line a hs 6 hx * / : b s idx 1 el el 2 / : c hy hx / hx 6 / s idx 2 el el 2 s idx 1 el el * + (* - (: d Y idx 1 el el : 41 line a t X idx 1 el el - (3^* b t X idx 1 el el - (2^* + c t X idx 1 el el - (* + d + 51 line :

◄ 's' from your collection above

recast the 'interpolation' code x cubic s X Y x InterpSpline :

interpolate within 'X range' 5 cubic 6

Discretise for export L H N xd U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx - (dx i * + : for U 41 line :

L X 1 el : H X X rows el : Npts 499 : 13 mat U L H Npts xd : 21 mat

Ω i 1 U rows range vy i el U i el cubic eval : for U vy augment 21 line :

Ω 14 1.022 4.0164 1.0441 4.033 1.0661 4.0499 1.0882 4.0671 1.1102 4.0845 1.1323 4.1022 1.1543 4.1201 1.1764 4.1383 1.1984 4.1567 1.2204 4.1754 1.2425 4.1944 1.2645 4.2135 1.2866 4.233 1.3086 4.2527 1.3307 4.2727 1.3527 4.2929 1.3747 4.3134 182 mat

j x C support 2 1 sys

Ω Ω rows row 124712 mat

Ω rows 500

<content> Mathcad 8 Pro. code conversion </content>

s X Y t InterpSpline Mathcad 8 Pro. code conversion t X 1 el ≤ idx 1221 mat : t X N el ≥ idx N N 1 + 21 mat : i 1 : X i el t < i i 1 + : while idx i 1 - i 21 mat : 31 line if if hx X idx 2 el el X idx 1 el el - (: hy Y idx 2 el el Y idx 1 el el - (: hs s idx 2 el el s idx 1 el el - (: 31 line a hs 6 hx * / : b s idx 1 el el 2 / : c hy hx / hx 6 / s idx 2 el el 2 s idx 1 el el * + (* - (: d Y idx 1 el el : 41 line a t X idx 1 el el - (3^* b t X idx 1 el el - (2^* + c t X idx 1 el el - (* + d + 51 line :

x λ 1 X 1 el x ≤ (x X X rows el ≤ (& cases :

◄ 's' from your collection above

recast the 'interpolation' code x cubic s X Y x InterpSpline :

interpolate within 'X range' 5.5 cubic 5.4056

x cubic x λ * support 2 1 sys

Part 3. The complimentary scalar dspline

s X Y t dspline t X 2 el ≤ k 1 : t X X rows 1 - el ≥ k X rows 1 - : k 1 : t X k 1 + el > k k 1 + : while 21 line if if c Y k 1 + el Y k el - X k 1 + el X k el - / s k 1 + el 2 s k el * + (X k 1 + el X k el - 6 / (* - : s k 1 + el s k el - X k 1 + el X k el - / (t X k el - (2^2 / * s k el t X k el - (* + c + 31 line eval :

x D s X Y x dspline :

x D x λ * 0 2 1 sys s x el 0 2 1 sys

N 11 :

Part 4. The complimentary fintegrate

Prepare the pseudo-integrand ... 'N' from initial  spline setup s X Y fintegrate i 1 N range h i el X i 1 + el X i el - : f i 1 + el h i el 2 / Y i el Y i 1 + el + (* h i el (3^24 / s i el s i 1 + el + (* - : 21 line for f 21 line :

s X Y fintegrate sum 205.7038

206.1809 205.7038 205.2389 31 sys linear parabolic cubic 31 sys ≡

∫ X s X Y fintegrate augment :

∫ 102 4.4563 3 5.8373 4.5 13.5331 5 3.7405 6.1 6.406 7 7.7 8 17.1109 8.9 26.379 10 41.7264 11 38.3784 12 40.4358 122 mat

∫ 2 col sum 205.7038

∫explore

∫ 2 col i el i 1 ∫explore sum 126.8896

data Σ Finite differences Cumul Integration Y 0 : 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 71 line :

x cubic x λ * support 2 1 sys

Σ Ω Σ :

vx Σ 1 col : vy Σ 2 col : 12 mat x ∫ vx vy x cinterp : 21 mat

ω 10 :

track 0 ω ∫ ω ω ∫ ω 032 mat :

ω ω ∫ 12 mat 10 126.8895 12 mat

Σ x ∫ x λ * track 2 1 sys

jmG: 20231211

XY 142537 4.5 956 6.1 7711824 8.9 34104011371247122 mat :

X XY 1 col : Y XY 2 col : 12 mat k X Y lspline : 21 mat

k 0300 0.6629 3.3484 14.2699 - 16.1137 2.7041 - 15.8926 6.6248 - 2.2796 - 17.1442 - 23.7861 0151 mat

0 0.6629 3.3484 14.2699 - 16.1137 2.7041 - 15.8926 6.6248 - 2.2796 - 17.1442 - 23.7861 0121 mat

MCD style

Hermite cubic to rescue the freak Smath cinterp

XY 366.59 4990 375.88 4882 380.525 4441.6 381.419 4036.8 42 mat :

Xo XY 1 col transpose : 366.59 375.88 380.525 381.419 14 mat

Yo XY 2 col transpose : 49904882 4441.6 4036.8 14 mat

user original polygon Xo 366.59 375.88 380.525 381.419 14 mat transpose : Yo 49904882 4441.6 4036.8 14 mat transpose : 21 sys

recast the data for easy handling plot Xo Xo Xo 1 el - : Yo Yo Yo 4 el - : 12 mat

recast artefact

populate reasonably for export t 01 0.125 range :

<content> Hermite interpolation </content>

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 :

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

x c 10 x ≤ (x 1 ≤ (& cases :

Hermite interpolate 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 * :

x P 1 el x c * x P 2 el x c * x P 3 el x c * x P 4 el x c * 0.5 1 0.5 0.25 - 2 2 mat 5 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 : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 0 953.2 0.939 950.2082 1.9959 945.0906 3.1435 937.409 4.3547 926.725 5.6021 912.6004 6.8586 894.5969 8.097 872.2762 9.29 845.2 10.1063 800.7812 10.8518 751.8875 11.5282 699.2375 12.1372 643.55 12.6807 585.5438 13.1603 525.9375 13.5778 465.45 13.935 404.8 14.1933 344.6816 14.3986 285.8406 14.5564 228.998 14.6723 174.875 14.752 124.1926 14.8009 77.6719 14.8247 36.034 14.829 0252 mat 11 mat

a b c spec . 3 black 13 mat : data contour a b c spec pts : XoYo Xo Yo augment + 20 black plot : 31 line

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

vx contour 1 col : vy contour 2 col : 12 mat x f vx vy x cinterp : x λ 1 Xo 1 el x ≤ (x Xo 4 el ≤ (& cases : x spline Xo Yo x cinterp : 41 mat

XYdata Xo Yo augment :

XYdata 0 953.2 9.29 845.2 13.935 404.8 14.829 042 mat

XYdata

contour contour rows row 14.829 012 mat

x f x λ * x spline x λ * XoYo data 4 1 sys

x f x λ * x spline x λ * XoYo data 41 sys

blue ► HermiteCubic
red ► freak cinterp