plotG

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

plotG

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

4 appVersion 0.98.6179.21440 ≡

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 3.52 - 2.358 - 3 - 2.213 - 2 - 1.256 - 1 - 1.739 - 0 1.482 - 1 1.277 - 62 mat :

X XY 1 col :

Y XY 2 col :

N Y rows 1 - : 5

r 2 N range :

Prepare the Tridiagonal Matrices

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 0 0.678153846153846 1.44 - 0.74 0.052 - 061 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 10000000000000000000000000000000000166 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 100000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00000166 mat

slin 0 2.419 3.285 - 2.0808 0.5982 - 061 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 - 0000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00001 - 166 mat

sp 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 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 1 1.52 - 0.52 000 0.0867 0.5067 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 0000 0.1667 0.6667 0.1667 00012 - 166 mat

sc 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

linear slin 0 2.419 3.285 - 2.0808 0.5982 - 061 mat

parabolic sp 2.0344 2.0344 3.1736 - 2.02 0.4664 - 0.4664 - 61 mat

cubic sc 4.0766 1.6415 3.0412 - 1.8833 0.052 - 1.9873 - 61 mat

◄ Sanity Mathcad 11

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 sc :

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

Part 2. Classroom journey cubic splines

abcd

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 0.7805 - 2.0383 0.57 - 2.358 - 0.7805 - 0.8208 0.9167 2.213 - 0.8208 1.5206 - 0.2169 1.256 - 0.3226 - 0.9417 0.3621 - 1.739 - 0.3226 - 0.026 - 0.5536 1.482 - 54 mat

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

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 :

ptsSpec

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 :

plot 1 x C 2 x C 3 x C 4 x C 5 x C support 61 sys :

Sym . :

size 14 :

clr black :

support X Y ptsSpec augment :

J 4 :

Type cubic

x λ 12 - x ≤ (x 2 ≤ (& cases :

Cubic spline segments through data points

plot

cubic segment(s) observer 'J' "cubic" C(4,x)= C(5,x)

5 x C x λ * J x C support 3 1 sys Superfous Disabled

Demo: collect the 'j' segments u Spline j local assignment n X rows : u X 1 el ≤ n 1 : u X n el ≥ n n 1 - : i 1 n range u X i el ≤ break return 'j' < (i-1) if for if if i 1 - (41 line : j u C 21 line :

x segments x X 1 el ≥ (x X X rows el ≤ (& x Spline Not interpolated if :

Entrance/departure observer [slin, sp, sc]

1 x C 5 x C support 3 1 sys Mathcad 8 Pro. code conversion

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' 1.5 - cubic 1.4251 -

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 100 : 13 mat U L H Npts xd : 21 mat

... I don't have Akima code x Akima X Y x ainterp :

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

Discrete ranged data user

Ω x Akima support 3 1 sys

x blue 1 3.75 - x ≤ (x 2 - ≤ (& cases :

x fushia 1 1.75 - x ≤ (x 1.5 ≤ (& cases :

Cubic spline segments through data points ... [slin .. sp] join 2 points entrance ... departure.

1 x C x blue * 2 x C 3 x C 4 x C x fushia * 5 x C support 6 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 c 1 3.5 - x ≤ (x 1.5 ≤ (& cases : 21 sys

x D x c *

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 7.3287 -

Sources and credits . . .

1. Mathsoft: for the l_p_c types, MCD 8 Pro.

2. Mathsoft: dspline(s,X,Y,t), Advisor

3. Robert Adair: interpspline(s,X,Y,t),

4. Robert Adair: fintegrate(s,X,Y),

5. Jean Giraud : Smath document . . . suite

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Part 5. linear interpolation

Linear interpolation/extrapolation data x Linterp vx data 1 col : vy data 2 col : x vx 1 el ≤ k 1 : x vx vx rows el ≥ k vx rows 1 - : k 1 : x vx k 1 + el ≥ k k 1 + : while 21 line if if vy k el vy k 1 + el vy k el - vx k 1 + el vx k el - / x vx k el - (* + 41 line :

Smath plugin
Smath code X Y 2.5 - linterp XY 2.5 - Linterp 21 mat 2 - 2 - 21 mat

It would not be stupid to 'linterp' that presumed inaccurate data set

3.52 - 2.358 - . 14 black 3 - 2.213 - . 14 black 2 - 1.256 - . 14 black 1 - 1.739 - . 14 black 0 1.482 - . 14 black 1 1.277 - . 14 black 6 5 mat XY x Linterp 1.5 - 0.237 x * + 3 1 sys

support 3.52 - 2.358 - . 14 black 3 - 2.213 - . 14 black 2 - 1.256 - . 14 black 1 - 1.739 - . 14 black 0 1.482 - . 14 black 1 1.277 - . 14 black 65 mat

x f X Y x linterp :

x f x X 1 el X 5 el int 6.031 -

Example 2 . . . Hermite cubic interpolation

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t0 1 time :

XY 3.52 - 2.358 - 3 - 2.213 - 2 - 1.256 - 1 - 1.739 - 0 1.482 - 1 1.277 - 62 mat :

@ this point ... OriginPro 7.5 offers two styles 'connect' 1. B_Spline ... 2. Spline [Bézier] As well, Excel connect Bézier. Bézier code in Mathcad 11 ... NOT Smath. You can mesh Hermite Export in 't'.

user original polygon Xo 3.52 - 3 - 2 - 1 - 0116 mat transpose : Yo 2.358 - 2.213 - 1.256 - 1.739 - 1.482 - 1.277 - 16 mat transpose : 21 sys

populate reasonably for export t 01 0.1 range :

Hermite interpolation Utilities

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 * :

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 : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 3.52 - 2.358 - 3.4907 - 2.361 - 3.4563 - 2.3619 - 3.4168 - 2.3601 - 3.3722 - 2.3548 - 3.3225 - 2.3453 - 3.2678 - 2.3311 - 3.2082 - 2.3114 - 3.1437 - 2.2856 - 3.0743 - 2.253 - 3 - 2.213 - 2.9137 - 2.1167 - 2.823 - 2.0145 - 2.7286 - 1.9087 - 2.6307 - 1.8015 - 2.53 - 1.6952 - 2.4269 - 1.5921 - 2.3218 - 1.4944 - 2.2154 - 1.4043 - 2.1079 - 1.3241 - 2 - 1.256 - 1.9 - 1.2755 - 1.8 - 1.3072 - 1.7 - 1.3489 - 1.6 - 1.3985 - 1.5 - 1.4538 - 1.4 - 1.5125 - 1.3 - 1.5726 - 1.2 - 1.6318 - 1.1 - 1.688 - 1 - 1.739 - 0.9 - 1.7335 - 0.8 - 1.7215 - 0.7 - 1.7036 - 0.6 - 1.6806 - 0.5 - 1.6535 - 0.4 - 1.6229 - 0.3 - 1.5897 - 0.2 - 1.5546 - 0.1 - 1.5184 - 0 1.482 - 0.1165 1.4566 - 0.232 1.4319 - 0.3455 1.4081 - 0.456 1.3852 - 0.5625 1.3634 - 0.664 1.343 - 0.7595 1.3239 - 0.848 1.3065 - 0.9285 1.2908 - 1 1.277 - 512 mat 11 mat

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 :

Hermite contour :

a b c spec . 10 black 13 mat : data Hermite a b c spec pts : XoYo Xo Yo augment + 20 blue plot : x spline Xo Yo x cinterp : 41 line

plot Hermite x spline data XoYo 41 sys :

Hermite Hermite rows row 1 1.277 - 12 mat

1 time t0 - 4 s *

Hermite generates a discrete export, each segment populated by 't'.

plot

Hermite 3.52 - 2.358 - 3.491 - 2.361 - 3.456 - 2.362 - 3.417 - 2.36 - 3.372 - 2.355 - 3.322 - 2.345 - 3.268 - 2.331 - 3.208 - 2.311 - 3.144 - 2.286 - 3.074 - 2.253 - 3 - 2.213 - 2.914 - 2.117 - 2.823 - 2.014 - 2.729 - 1.909 - 2.631 - 1.802 - 152 mat

B-spline interpolation . . . roams around

]]> Utilities ... Global plot spec

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 :

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 :

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 :

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

Plot 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 :

r 2 :

XY 3.52 - 3 - 2 - 1 - 01 2.358 - 2.213 - 1.256 - 1.739 - 1.482 - 1.277 - 26 mat transpose :

X 3.52 - 3 - 2 - 1 - 0116 mat : Y 2.358 - 2.213 - 1.256 - 1.739 - 1.482 - 1.277 - 16 mat : XY X transpose Y transpose augment : 31 line

X XY 1 col r RepEnd : Y XY 2 col r RepEnd : 21 line

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

polygon X Y . 12 black plotG :

XY spline

XY Plot polygon 3 1 sys

B_spline

Utilities ... Global plot spec

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 :

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 :

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 :

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

Plot 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 :

r 2 :

XY 2 - 1.37 - 0.89 - 0.55 - 0.25 - 0 0.22 0.45 0.75 1.3 2 0.519 0.414 - 1.061 - 0.987 - 0.347 - 2.791 1.081 0.765 1.181 0.588 0.466 - 211 mat transpose :

X XY 1 col r RepEnd : Y XY 2 col r RepEnd : 21 line

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

polygon X Y . 12 black plotG :

XY spline

X Y augment Plot polygon 3 1 sys

XY spline

plotG

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 :

XY 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X XY 1 col : Y XY 2 col : 12 mat data X Y + 15 black plotG : 21 line

x Akima X Y x ainterp : x Cinterp X Y x cinterp : x λ 10 x ≤ (x 1 ≤ (& cases : 31 sys

Akima has about unique application for some Gaussian foot

x Akima x λ * x Cinterp x λ * data 3 1 sys

Example 3 . . . supplementary visit

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L H N xd range discretise U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 51 line :

L 0 : H 10 : Num 100 : 31 line

U L H Num xd :

x φ 11 x 5 - (2^+ / :

Data i 1 Num 1 + range vy i el U i el φ eval : for U vy augment 21 line :

Data

===== generalise from here =====

X Data 1 col : Y Data 2 col : n Data rows : 31 line

N n 1 - : r 2 N range : 21 line

Prepare the cubic splines

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

Initialise the blank diagonal matrix

100000000000000000000000155 mat

M 11 el 1 :

M n n el 1 :

Fill-in the respective lspline tridiagonal matrix

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

s M 1 -^br * eval :

CWD C:\SmathFile CurrentDirectory :

Spool to file the spline coefficcients s CWD spline wfile 1

s CWD spline rfile :

Prepare the pseudo-integrand 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 :

Σ s X Y fintegrate :

Export the cumulative integral Σcumul n X rows 1 - : Sum 1 el 0 : Sum n 1 + el 0 : i 1 n range Sum i 1 + el Sum i el Σ i el + eval : for X Sum augment 51 line :

lspline cumulative integration

Σcumul 0 2 1 sys

Σ 0 0.0039 0.0041 0.0042 0.0044 0.0046 0.0048 0.005 81 mat

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 :

3 D 0.16

x plotD L x ≤ (x H ≤ (& (x D out spline if :

does NOT sub-derive

x D2 x D x diff :

The scalar dspline

x plotD 0 2 1 sys

2nd derivative of [lpc] spline

X s augment 0 2 1 sys

x f s X Y x dspline :

h 0.25 :

x h f D1 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* - :

x h f D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

x plotf L x ≤ (x H ≤ (& (x f out spline if :

x plotD1 L x ≤ (x H ≤ (& (x h f D1 - out spline if :

x plotD2 L x ≤ (x H ≤ (& (x h f D2 - out spline if :

The scalar dspline

x plotf 0 2 1 sys

The scalar 2nd derivative

x plotD1 0 2 1 sys

The scalar 3rd derivative

x plotD2 0 2 1 sys

The long cubic spline [lpc], scalar interpolation. s X Y t InterpSpline 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 + 41 line :

s X Y 3.1 InterpSpline 0.217

Example 4 . . . successive discrete convolution-derivatives

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Discrete derivative by convolution ... [-1 point at each end]. data scale convol vx data 1 col : vy data 2 col : 12 mat f 1 - 0113 mat transpose : Dy vy f al_convr1d eval : Dy Dy 2 Dy rows 1 - 11 submatrix : Deriv vx scale Dy * augment : Deriv Deriv 2 Deriv rows 1 - 12 submatrix : 61 line :

sanity vs Symbolic Derivative_1 Data 1 - convol : Derivative_2 Derivative_1 1 - convol : Derivative_3 Derivative_2 1 - convol : 31 sys

Derivative_1 Derivative_2 Derivative_3

Example 5 . . . Hermite meritas

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Hermite smooth travel

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 :

user original polygon Xo 0241363306015 mat transpose : Yo 5396363416315371685 - 15 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 : 21 line

isolate New New :

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

populate at will for more points 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 :

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

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 :

XoYo Xo Yo + 15 black plotG :

contour rows 120

population of each segment t rows 21

H contour . 9 red plot :

polygon Xo Yo augment :

Otain scalar Hermite ... Akima(x) over Hermite upper/lower branches ...

polygon XoYo H 3 1 sys

Hermite interpolation

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 051015202530354019 mat transpose : Yo 50 63.2 68.5 70.6 71.4 71.7 71.9 727219 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 : 21 line

isolate New New :

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

populate reasonably t 01 0.1 range :

Polyline construct

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 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 * :

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 : contour contour t rows contour rows range 12 range el : 31 sys

contour is the Hermite data set used 
to interpolate in-situ at user point x Akima Xo Yo x ainterp : x c Xo Yo x cinterp : User U Akima : 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 :

a b c spec o 5 black 13 mat : Hermite contour a b c spec pts : XoYo Xo Yo augment + 20 black plot : 31 line

Labels construct

V(sizechar,n)... V(colorchar,n) var n V n 1 ≡ var ct 1 : j 1 n range b ct 1 el var : ct ct 1 + : 21 line for U b 1 el : k 2 b rows range U U b k el stack : for U 51 line if :

nodes Xo : NodesLabel nodes var2str vectorize : n Xo rows : 31 line

plot symbol converted2Numbers

symbol ▼ size

symbol color

nodes_label Xo Yo NodesLabel 12 n V blue n V augment :

XoYo x Akima nodes_label Hermite 4 1 sys

t0 1 time :

Example 6 ... 4 ≤ support points

Hermite cubic interpolation

4 appVersion 0.98.6179.21440 ≡

EXPLAIN: Hermite interpolation is cubic, the ancestor of more popular/universal "cspline, pspine, lspline" Mathcad style. Either of these three styles will fail miserably this demo. The all things about Hermite is in the collapsed area. As offered, you give the data set and the piecewise population. At this point, observe the population is equally distributed between the support points Xo, Yo ... unique feature for this type of spline fit, stricking of realism for data collection. So to speak, each segment can be populated at will equally. That is most interesting for further finer interpolation of the published data table wrt the region of interest. The published data table can be interpolated "conventional" whereas it is no more "freaked". X_Y plot is nice for publishing but useless as not extractable.

user original polygon Xo 366.59 379.74 387.46 390.186 391.3 391.508 16 mat transpose : Yo 49904882 4441.6 4036.8 3662.7 3316.8 16 mat transpose : 21 sys

recast the data for easy handling plot Xo Xo Xo 1 el - : Yo Yo Yo 6 el - : 12 mat

populate reasonably for export t 01 0.25 range :

Hermite interpolation

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 * :

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 : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 0 1673.2 2.780469 1665.090625 6.0925 1646.725 9.645781 1614.596875 13.15 1565.2 15.572031 1471.8875 17.6615 1363.55 19.425219 1245.9375 20.87 1124.8 21.887578 1020.453906 22.645875 918.25625 23.197734 818.130469 23.596 720 23.998047 623.694531 24.310375 529.26875 24.544016 436.683594 24.71 345.9 24.819672 244.371094 24.883625 149.56875 24.912766 66.457031 24.918 0212 mat 11 mat

a b c spec . 12 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 6 el ≤ (& cases : x spline Xo Yo x cinterp : 41 mat

data set from Mollier under populated

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

1 time t0 - 3 s *

XoYo Xo Yo augment :

17 f 1400.7

XoYo XoYo 2 1 sys plotG Hatch

vx vy chr sz clr plotG n vx length : plot vx 1 el vy 1 el chr sz clr augment : i 2 n range plot plot vx i el vy i el chr sz clr augment stack : for plot 41 line :

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*N vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f1 g1 x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f1 x.1 x.1 g1 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g1 x dx 2 / + x dx 2 / + f1 x dx + x dx + f1 x dx + x dx + g1 42 mat stack : for h 41 line :

Quadratic spline 4 points

This Smath document implements the "Quadratic spline", the basis construction. Once the basis 'D' vector in hand, some supplementary fabrication is applied. QuadSpline supports the numerical integrator(s) and 'hatch', piecewise.

M9 366.59 4990 375.88 4882 380.525 4441.6 381.419 4036.8 42 mat :

X M9 1 col : Y M9 2 col : 12 mat

recast the data for easy handling plot X X X 1 el - : Y Y Y 4 el - : 12 mat

data X Y . 14 black plotG :

D 1 el 0 : i 2 X rows range D i el 2 Y i el Y i 1 - el - X i el X i 1 - el - / (* D i 1 - el - : for 21 line

D 0 23.2508 - 166.3724 - 739.2204 - 41 mat

data 0 953.2 . 14 black 9.29 845.2 . 14 black 13.935 404.8 . 14 black 14.829 0.14 black 45 mat

j x Q i j : D i el 2 / x X i 1 - el - (2^X i el X i 1 - el - (/ * D i 1 - el 2 / x X i el - (2^X i el X i 1 - el - (/ * - Y i el + D i el 2 / X i el X i 1 - el - (* - 21 line :

x m9 2 x Q X 1 el x ≤ (x X 2 el ≤ (& 3 x Q X 2 el x ≤ (x X 3 el ≤ (& 4 x Q X 3 el x ≤ (x X 4 el ≤ (& cases :

recast x f x m9 :

data x f 2 1 sys

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline f a b n Int w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : α 0 : β 1 : h β α - n / : 14 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

x f x X 1 el X 4 el int 11900

f X 1 el X 4 el 20 Int 11900

Low High step # f inc U Low High step range : i 1 U rows range f i el U i el f : for U f augment 31 line :

Plot X 1 el X 4 el X 1 el X 4 el 25 / + x f inc :

Plot rows 26

U Plot 1 col : V Plot 2 col : 12 mat

x λ 1 X 1 el x ≤ (x X 4 el ≤ (& cases :

x Quadratic U V x cinterp :

Plot range

data x Quadratic x λ * 2 1 sys

Example 7 ... join ifinitely many quadratic segments add/remove segments wrt project.

plotG ...Hatch

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 :

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f g u v N hatch dx v u - N / (: h u u f u u g 22 mat : x u v dx - u dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

XY 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X XY 1 col : Y XY 2 col : data X Y . 14 black plotG : 31 line

D 1 el 0 : i 26 range D i el 2 Y i el Y i 1 - el - X i el X i 1 - el - / (* D i 1 - el - : for 21 line

Model function x φ 0.002 0.002 x 0.5 - (2^+ / :

Select/display segments algo dtyle. Transit via local definition j x Q i j : D i el 2 / x X i 1 - el - (2^X i el X i 1 - el - (/ * D i 1 - el 2 / x X i el - (2^X i el X i 1 - el - (/ * - Y i el + D i el 2 / X i el X i 1 - el - (* - 21 line :

2 x Q 3 x Q 4 x Q 5 x Q 6 x Q data 6 1 sys Scalar interpolation

The companion scalar interpolation

Quadratic scalar interpolation x Quad 2 x Q X 1 el x ≤ (x X 2 el ≤ (& 3 x Q X 2 el x ≤ (x X 3 el ≤ (& 4 x Q X 3 el x ≤ (x X 4 el ≤ (& 5 x Q X 4 el x ≤ (x X 5 el ≤ (& 6 x Q X 5 el x ≤ (x X 6 el ≤ (& add segments cases :

segment X or Y ... wrt 'x' x Xdata J search index n 6 : x Xdata 2 el ≤ n 1 : x Xdata n el ≥ n n : resume next if if i 2 n range x Xdata i el ≤ break close if for J i 1 - (: 51 line :

x freak X Y x cinterp :

An excellent Quadratic spline scalar interpolation.

x Quad x freak x φ data 4 1 sys

0.375 Quad 0.1225

Locate [X, Y] segment
wrt to 'x' 0.375 X J 0.375 Y J 21 mat 4521 mat

recast interpolation x f x Quad :

Lo Hi step # f inc U Lo Hi step range : i 1 U rows range f i el U i el f : for U f augment 31 line :

Export X 1 el X 6 el X 1 el 0.01 + x f inc :

Export rows 51

Export 0 0.0079 0.01 0.008 0.02 0.0081 0.03 0.0083 0.04 0.0086 0.05 0.009 0.06 0.0094 0.07 0.0099 0.08 0.0105 0.09 0.0112 0.1 0.012 0.11 0.0128 0.12 0.0137 132 mat

Example 8 ... Hermite-Akima to rescue Mollier

Mollier

M 358.93 2 363.82 4.8 368.75 10.35 373.68 20.44 378.58 37.47 383.39 64.49 384.35 71.4 385.29 78.9 386.23 87 387.17 95.75 387.72 101.32 388.09 105.18 389.01 115.33 389.93 126.23 390.84 137.93 391.74 150.45 392.63 163.84 393.52 178.15 394.39 193.4 395.26 209.65 396.12 226.93 396.67 245.29 397.81 264.77 398.64 285.42 399.46 307.28 400.27 330.4 401.07 354.82 401.85 380.59 402.63 407.75 403.39 436.36 404.14 466.46 404.87 498.11 405.59 531.34 406.3 566.22 406.99 602.79 407.67 641.09 408.33 681.19 408.97 723.14 409.6 766.98 410.21 812.77 410.8 860.56 411.38 910.41 411.93 962.36 412.46 1016.5 412.98 1072.8 413.46 1131.4 413.93 1192.4 414.37 1255.7 414.79 1321.5 415.18 1389.8 415.54 1460.6 415.87 1534.1 416.17 1610.3 416.44 1689.2 416.68 1770.9 416.87 1855.5 417.03 1943.1 417.24 2175.3 417.14 2427.4 416.65 2700.8 415.69 2996.7 414.09 3316.8 411.6 3662.7 407.72 4036.8 401.33 4441.6 387.46 4882 366.59 4990672 mat :

X M 1 col M 11 el - :

M X M 2 col augment :

jct 58.31 2175.3 + 40 red 15 mat :

M jct 2 1 sys

M rows 67

N M 58 M rows range 12 range el :

58.31 2175.3 58.21 2427.4 57.72 2700.8 56.76 2996.7 55.16 3316.8 52.67 3662.7 48.79 4036.8 42.4 4441.6 28.53 4882 7.66 4990102 mat

N 58.31 2175.3 58.21 2427.4 57.72 2700.8 56.76 2996.7 55.16 3316.8 52.67 3662.7 48.79 4036.8 42.4 4441.6 28.53 4900 7.66 4990102 mat :

So much naive 'N' looks ... so much advanced techniques it needs to rescue/interpolate.

user original polygon Xo N 1 col : Yo N 2 col : 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 : 21 line

isolate New New :

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

populate from observing the discrete derivative t 01 0.1 range :

Hermite interpolation

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 * :

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 : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 58.31 2175.3 58.309285 2192.9737 58.30728 2212.9376 58.303695 2234.9609 58.29824 2258.8128 58.290625 2284.2625 58.28056 2311.0792 58.267755 2339.0321 58.25192 2367.8904 58.232765 2397.4233 58.21 2427.4 58.17987 2453.7617 58.14576 2480.3376 58.10759 2507.1289 58.06528 2534.1368 58.01875 2561.3625 57.96792 2588.8072 57.91271 2616.4721 57.85304 2644.3584 57.78883 2672.4673 57.72 2700.8 57.647955 2729.34945 57.57104 2758.1256 57.489085 2787.13015 57.40192 2816.3648 57.309375 2845.83125 57.21128 2875.5312 57.107465 2905.46635 56.99776 2935.6384 56.881995 2966.04905 56.76 2996.7 56.632925 3027.5946 56.4992 3058.7328 56.358575 3090.1162 56.2108 3121.7464 56.055625 3153.625 55.8928 3185.7536 55.722075 3218.1338 55.5432 3250.7672 55.355925 3283.6554 55.16 3316.8 54.9593 3350.1894 54.7492 3383.8392 54.5292 3417.7518 54.2988 3451.9296 54.0575 3486.375 53.8048 3521.0904 53.5402 3556.0782 53.2632 3591.3408 52.9733 3626.8806 52.67 3662.7 52.36303 3698.79975 52.04104 3735.184 51.70291 3771.85525 51.34752 3808.816 50.97375 3846.06875 50.58048 3883.616 50.16659 3921.46025 49.73096 3959.604 49.27247 3998.04975 48.79 4036.8 48.345955 4075.52065 47.87184 4114.5712 47.362685 4153.97455 46.81352 4193.7536 46.219375 4233.93125 45.57528 4274.5304 44.876265 4315.57395 44.11736 4357.0848 43.293595 4399.08585 42.4 4441.6 41.34168 4491.991 40.20904 4542.496 39.00256 4592.693 37.72272 4642.16 36.37 4690.475 34.94488 4737.216 33.44784 4781.961 31.87936 4824.288 30.23992 4863.775 28.53 4900 26.298145 4920.9844 24.02416 4938.5632 21.735915 4953.0148 19.46128 4964.6176 17.228125 4973.65 15.06432 4980.3904 12.997735 4985.1172 11.05624 4988.1088 9.267705 4989.6436 7.66 4990912 mat 11 mat

vx contour 1 col : vy contour 2 col : 12 mat x Akima vx vy x ainterp : XoYo Xo Yo augment : 31 mat

m 35 :

mollier m m Akima augment :

N x Akima vx vx rows el x ≤ ( x vx 1 el ≤ ( & * XoYo mollier 4 1 sys

Not sanity checked ... x Akima x 7.66 58.31 int 2.3 105^*

N x Akima vx vx rows el x ≤ (* x vx 1 el ≤ (& XoYo mollier 41 sys

m Akima 4735.5

Lo Hi step # f inc U Lo Hi step range : i 1 U rows range f i el U i el f eval : for U f augment 31 line :

x f x Akima :

Export 7.66 58.31 7.66 0.1 + x f inc :

Export rows 507

Discrete derivative by convolution ... [-1 point at each end]. data scale convol vx data 1 col : vy data 2 col : 12 mat f 1 - 0113 mat transpose : Dy vy f al_convr1d eval : Dy Dy 2 Dy rows 1 - 11 submatrix : Deriv vx scale Dy * augment : Deriv Deriv 2 Deriv rows 1 - 12 submatrix : 61 line :

Export 1 convol 0.00003 x 4 ^ ( 0.85 ^ * 0.0000000000004 x 4 ^ ( 2 ^ * 3 1 sys

The discrete derivative is not smooth, not detrimental to the project. You may apply the tecchnique exposed above. Once smoothed ... linterp/cinterp. You may attempt fitting a model function. Discretize 'red' up to jct, stack last part of the discrete convolution ... Spline that new construction ... not offered.

D Export 1 convol :

jct 0.00003 x 4^(0.85^* 0.0000000000004 x 4^(2^* - 0 ≡ x 4856 solve :

jct 51.52

D 439 row 51.56 20.888 12 mat

At this point of the project, spline 'new' ... evaluate in-situ.

sub D 439 D rows range 12 range el :

x f 0.00003 x 4^(0.85^* :

Lo Hi step # f inc U Lo Hi step range : i 1 U rows range f i el U i el f eval : for U f augment 31 line :

fit 0 jct 1 x f inc : new fit sub stack eval : vx new 1 col : vy new 2 col : 12 mat x DerivativeMollier vx vy x cinterp : 41 sys

user 45 :

user DerivativeMollier 12.53

fit sub user user DerivativeMollier + 20 black 1 5 mat 3 1 sys

new rows 119

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 :

data ListCollect collects index list select 92541497510516 mat : ct 1 : i select aa ct el data i 1 el : bb ct el data i 2 el : ct ct 1 + : 31 line for aa bb augment 51 line :

Th new ListCollect :

Th Th eval :

Thiele Th . 14 black plot :

fit sub user user DerivativeMollier + 20 black 1 5 mat Thiele 4 1 sys

Inadequate for 
Thiele suite...
 ► discontinuous Th 8 0.0353 24 1.4786 40 8.397 48 15.6078 53.76 26.7807 56.76 49.4518 62 mat

Example 9 ... Esplina

plotG, RepEnd, Unwrap, Strip

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 :

XW 1.5 2.6 0 4.8 3.5 - 3.4 4.1 - 0 1.5 - 1.5 - 0 4.1 - 3.2 3.1 - 4.7 0 1.5 2.6 92 mat :

r 2 :

Hermite Polyline Project

t0 1 time :

X XW 1 col r RepEnd : Y XW 2 col r RepEnd : XY X Y augment : 31 line

X XY 1 col : Y XY 2 col : Cntrl X Y . 14 black plotG : 31 line

α 16 / :

at α = 1/6 first & last points join, other α we have a zoom ± effect

Hermite Interpolation t P α 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 * :

Each "j" segment is collected in matrix of 't' rows x 'j' cols. Each U,V is Unwrapped in column, augmented in 'xy', then stripped of repeated pairs for export.

t 01 0.05 range :

Hermite Procedure

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 :

j runs 1..rows(X)-(r+1)

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 :

xy U Unwrap V Unwrap augment :

xy xy Strip :

polygon XW 1 col XW 2 col . 10 black plotG :

strip both ends
for clean junction span xy rows t rows - 2 + : xy xy t rows span range 12 range el : 21 sys

Hermite Polyline to Esplina

xy XW polygon 3 1 sys

Hermite PolyLine 0...1

x P 1 el x P 2 el x P 3 el x P 4 el 4 1 sys

1 time t0 - 12 s *

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. data char size clr plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line for data r3 r4 r5 augment 21 line :

P 1.5 0 3.5 - 4.1 - 1.5 - 0 3.2 4.7 2.6 4.8 3.4 0 1.5 - 4.1 - 3.1 - 028 mat transpose :

t0 1 time :

X P 1 col : Y P 2 col : 12 mat XY P o 7 red plotG : 21 line

N 200 :

Experiment ... 'cinterp' ... 'ainterp' S n P rows : X X X X stack : Y Y Y Y stack : 21 line u 1 el 0 : i 2 n 2 n * + (range u i el u i 1 - el X i el X i 1 - el - Y i el Y i 1 - el - 22 mat det + : for |,| determinant uN u n 1 + el u 1 el - N / : (increments ≡ i 1 N range ux i el i uN * : for i 1 N range Sx i el u X ux i el cinterp : Sy i el u Y ux i el cinterp : Sx 1 el P 11 el : Sy 1 el P 12 el : 12 mat 31 line for Sx Sy augment 31 line eval 71 line :

S S eval :

S XY 2 1 sys

"Esplina": courtesy Mathcad 11. Permission granted from Mathsoft Collab AI2000.

Helper 'cinterp' regions S 1 col Min S 1 col Max 21 mat 4.8465 - 4.8974 21 mat

The cspline proceeds in sorted order of 'vx', thus excluding splining closed data sets. The vector 'u' is created from the descending vector of the determinants |X,Y|... thus: directly parent to X,Y. This condition is necessary and sufficient to spline a closed data set in Cartesian coordinates system. 'uN' [incremental] determines the spacing of the interpolated points over the N points.

x S 1 col : y S 2 col : 12 mat

Area Polygon wrt vertices coordinates A 12 / x i el y i 1 + el * x i 1 + el y i el * - (i 1 x rows 1 - sum * abs :

Perimeter  Polygon wrt vertices coordinates P x i 1 + el x i el - (2^y i 1 + el y i el - (2^+ sqrt i 1 x rows 1 - sum :

Center of gravity Xo 16 A * / x i el x i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : Yo 16 A * / y i el y i 1 + el + (x i el y i 1 + el * x i 1 + el y i el * - (* i 1 x rows 1 - sum * : 21 mat

Area
Perimeter A P 21 mat 52.1 28.9 21 mat

Center of gravity Xo Yo 21 mat 0.046 0.354 21 mat

1 time t0 - 1.8 s *

Example 10 ... Hermite bird travel

Hermite Smooth Travel

data 05396 2.0033 - 5410.6466 3.9765 - 5425.073 5.8894 - 5439.0589 7.712 - 5452.384 9.4141 - 5464.8281 10.9655 - 5476.171 12.3362 - 5486.1924 13.496 - 5494.672 14.4148 - 5501.3896 15.0625 - 5506.125 15.4089 - 5508.6579 15.424 - 5508.768 15.0776 - 5506.2351 14.3395 - 5500.839 13.1797 - 5492.3594 11.568 - 5480.576 9.4743 - 5465.2686 6.8685 - 5446.217 3.7204 - 5423.2009 05396 9.3188 5337.1042 19.195 5273.9935 29.5838 5206.8581 40.44 5135.888 51.7188 5061.2734 63.375 4983.2045 75.3638 4901.8713 87.64 4817.464 100.1588 4730.1727 112.875 4640.1875 125.7438 4547.6986 138.72 4452.896 151.7588 4355.9699 164.815 4257.1105 177.8438 4156.5078 190.8 4054.352 203.6387 3950.8332 216.315 3846.1415 228.7838 3740.4671 2413634 250.425 3530.0144 259.545 3425.57 268.3525 3320.8106 276.84 3215.88 285 3110.9219 292.825 3006.08 300.3075 2901.4981 307.44 2797.32 314.215 2693.6894 320.625 2590.75 326.6625 2488.6456 332.32 2387.52 337.59 2287.5169 342.465 2188.78 346.9375 2091.4531 351 1995.68 354.645 1901.6044 357.865 1809.37 360.6525 1719.1206 3631631 364.9831 1571.6438 366.51 1514.3055 367.5719 1458.7304 368.16 1404.664 368.2656 1351.8516 367.88 1300.0385 366.9944 1248.9702 365.6 1198.392 363.6881 1148.0493 361.25 1097.6875 358.2769 1047.0519 354.76 995.888 350.6906 943.9411 346.06 890.9565 340.8594 836.6797 335.08 780.856 328.7131 723.2308 321.75 663.5495 314.1819 601.5574 306537 292.0003 424.8431 277.4475 310.285 262.4109 193.7444 246.96 75.64 231.1641 43.6094 - 215.0925 163.585 - 198.8147 283.8681 - 182.4 404.04 - 165.9178 523.6819 - 149.4375 642.375 - 133.0284 759.7006 - 116.76 875.24 - 100.7016 988.5744 - 84.9225 1099.285 - 69.4922 1206.9531 - 54.48 1311.16 - 39.9553 1411.4869 - 25.9875 1507.515 - 12.6459 1598.8256 - 01685 - 4.7239 - 1719.3021 - 8.721 - 1748.327 - 12.0296 - 1772.3524 - 14.688 - 1791.656 - 16.7344 - 1806.5156 - 18.207 - 1817.209 - 19.1441 - 1824.0139 - 19.584 - 1827.208 - 19.5649 - 1827.0691 - 19.125 - 1823.875 - 18.3026 - 1817.9034 - 17.136 - 1809.432 - 15.6634 - 1798.7386 - 13.923 - 1786.101 - 11.9531 - 1771.7969 - 9.792 - 1756.104 - 7.4779 - 1739.3001 - 5.049 - 1721.663 - 2.5436 - 1703.4704 - 1202 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 0241363306015 mat transpose : Yo 5396363416315371685 - 15 mat transpose : 21 sys

Smooth data from Hermite cubic

XoYo Xo Yo + 15 black plotG :

Linear interpolation/extrapolation data x Linterp vx data 1 col : vy data 2 col : x vx 1 el ≤ k 1 : x vx vx rows el ≥ k vx rows 1 - : k 1 : x vx k 1 + el ≥ k k 1 + : while 21 line if if vy k el vy k 1 + el vy k el - vx k 1 + el vx k el - / x vx k el - (* + 41 line :

limit data 1 col Max : 368.2656

data 66 row 368.2656 1351.8516 12 mat

sub1 data 166 range 12 range el : sub2 data 67 data rows range 12 range el : 21 sys

sub1 128 Linterp data 128 Linterp 21 mat 4531.2151 755.5469 - 21 mat

vx sub2 1 col : vy sub2 2 col : 12 mat x f2 vx vy x ainterp : vx sub1 1 col : vy sub1 2 col : 12 mat x f1 vx vy x ainterp : x λ 10 x ≤ (x limit ≤ (& cases : 51 mat

x plot x f1 x λ * x f2 x λ * 12861441282048 - 22 mat XoYo 41 sys :

x plot

128 f1 128 f2 21 mat 4531.3323 795.4913 - 21 mat

X data 1 col : Y data 2 col : 12 mat

x Akima X Y x ainterp : x lin X Y x linterp : x cs X Y x cinterp : 31 sys

128 Akima 3157.9943

◄ The 3 are freak f1(x)freak cinterp ... ▲

128 lin 2903.8809

128 cs #

Example 11 ...Hermite splines the closed data set

Spiral data set

Spiral 0.5 0 0.4169 0.1354 - 0.3156 0.2293 - 0.2063 0.2839 - 0.0983 0.3026 - 1.5822 107 -^* 0.2902 - 0.0822 - 0.253 - 0.1439 - 0.198 - 0.1828 - 0.1328 - 0.1988 - 0.0646 - 0.1938 - 2.4509 108 -^* - 0.1712 - 0.0556 0.1355 - 0.0984 0.0917 - 0.1262 0.0449 - 0.1383 6.8716 108 -^* 0.1357 0.0392 0.1206 0.0697 0.0959 0.0897 0.0651 0.0986 0.032 0.0971 1.1793 107 -^* - 212 mat :

user original polygon Xo Spiral 1 col : Yo Spiral 2 col : 21 sys

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 :

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 : 21 line

isolate New New :

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

populate segments t 01 0.1 range :

Hermite interpolation

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 * :

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 : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 0.5 0 0.494359 0.010366 - 0.487951 0.021909 - 0.480843 0.034452 - 0.473098 0.047818 - 0.464781 0.061831 - 0.455958 0.076314 - 0.446694 0.091089 - 0.437053 0.105979 - 0.4271 0.120809 - 0.4169 0.1354 - 0.407421 0.146621 - 0.39777 0.15743 - 0.387957 0.167827 - 0.377993 0.177817 - 0.367888 0.1874 - 0.357651 0.196579 - 0.347294 0.205357 - 0.336826 0.213734 - 0.326258 0.221715 - 0.3156 0.2293 - 0.304877 0.236472 - 0.294082 0.243255 - 0.283227 0.249652 - 0.272319 0.255666 - 0.261369 0.2613 - 0.250385 0.266558 - 0.239377 0.271444 - 0.228354 0.275961 - 0.217325 0.280112 - 0.2063 0.2839 - 0.195303 0.287306 - 0.184327 0.290358 - 0.173381 0.293061 - 0.162474 0.295419 - 0.151612 0.297438 - 0.140806 0.299121 - 0.130064 0.300474 - 0.119393 0.301502 - 0.108802 0.302209 - 0.0983 0.3026 - 0.087928 0.302656 - 0.077659 0.302406 - 0.0675 0.301859 - 0.057458 0.301019 - 0.047538 0.299894 - 0.037747 0.298489 - 0.028091 0.296811 - 0.018577 0.294866 - 0.009211 0.29266 - 1.5822 107 -^* 0.2902 - 0.009017 - 0.28748 - 0.017869 - 0.28452 - 0.026551 - 0.281326 - 0.035058 - 0.277904 - 0.043387 - 0.274262 - 0.051534 - 0.270408 - 0.059492 - 0.266348 - 0.067259 - 0.262088 - 0.07483 - 0.257636 - 0.0822 - 0.253 - 0.08933 - 0.248176 - 0.096254 - 0.243181 - 0.102967 - 0.238023 - 0.109469 - 0.23271 - 0.115756 - 0.22725 - 0.121827 - 0.22165 - 0.127679 - 0.215917 - 0.13331 - 0.210059 - 0.138718 - 0.204084 - 0.1439 - 0.198 - 0.148818 - 0.19182 - 0.153507 - 0.185546 - 0.157969 - 0.179183 - 0.162202 - 0.172741 - 0.166206 - 0.166225 - 0.169982 - 0.159643 - 0.17353 - 0.153003 - 0.176849 - 0.14631 - 0.179939 - 0.139574 - 0.1828 - 0.1328 - 0.185399 - 0.126006 - 0.187771 - 0.119189 - 0.189918 - 0.112355 - 0.191842 - 0.10551 - 0.193544 - 0.098662 - 0.195026 - 0.091818 - 0.196291 - 0.084982 - 0.197341 - 0.078163 - 0.198176 - 0.071367 - 0.1988 - 0.0646 - 0.199189 - 0.057889 - 0.199371 - 0.051219 - 0.19935 - 0.044596 - 0.19913 - 0.038026 - 0.198712 - 0.031513 - 0.198102 - 0.025062 - 0.197303 - 0.018681 - 0.196317 - 0.012373 - 0.195148 - 0.006144 - 0.1938 - 2.4509 108 -^* - 0.192258 - 0.006028 0.190544 - 0.011962 0.188663 - 0.017798 0.18662 - 0.023533 0.184419 - 0.029162 0.182064 - 0.034683 0.17956 - 0.040091 0.176912 - 0.045382 0.174124 - 0.050553 0.1712 - 0.0556 0.168137 - 0.060492 0.164948 - 0.065254 0.161638 - 0.069884 0.158212 - 0.074379 0.154675 - 0.078738 0.151032 - 0.082957 0.147288 - 0.087035 0.143448 - 0.09097 0.139517 - 0.094759 0.1355 - 0.0984 0.1314 - 0.101867 0.127225 - 0.105182 0.122978 - 0.108347 0.118666 - 0.111359 0.114294 - 0.114219 0.109866 - 0.116925 0.105387 - 0.119477 0.100863 - 0.121874 0.096299 - 0.124115 0.0917 - 0.1262 0.087074 - 0.1281 0.082423 - 0.129844 0.077752 - 0.131433 0.073066 - 0.132868 0.068369 - 0.13415 0.063666 - 0.13528 0.058963 - 0.136259 0.054265 - 0.137088 0.049575 - 0.137768 0.0449 - 0.1383 0.040262 - 0.138665 0.035646 - 0.138886 0.031058 - 0.138963 0.026499 - 0.138901 0.021975 - 0.1387 0.017489 - 0.138363 0.013044 - 0.137893 0.008646 - 0.13729 0.004296 - 0.136559 6.8716 108 -^* 0.1357 0.004226 0.134705 0.008392 0.133587 0.012495 0.132351 0.016532 0.130998 0.0205 0.129531 0.024396 0.127954 0.028217 0.12627 0.03196 0.124481 0.035622 0.12259 0.0392 0.1206 0.042671 0.118504 0.046054 0.116316 0.049345 0.114039 0.052545 0.111676 0.05565 0.109231 0.058659 0.106708 0.061571 0.10411 0.064382 0.10144 0.067093 0.098702 0.0697 0.0959 0.072182 0.093032 0.074559 0.090106 0.07683 0.087128 0.078994 0.084099 0.08105 0.081025 0.082998 0.077909 0.084838 0.074754 0.086569 0.071566 0.08819 0.068346 0.0897 0.0651 0.091078 0.061837 0.092346 0.058555 0.093504 0.055257 0.094553 0.051946 0.095494 0.048625 0.096327 0.045298 0.097054 0.041969 0.097674 0.038641 0.09819 0.035317 0.0986 0.032 0.098722 0.028241 0.098751 0.024523 0.098701 0.020879 0.098582 0.017338 0.098406 0.013931 0.098186 0.01069 0.097934 0.007646 0.097661 0.004829 0.097379 0.00227 0.0971 1.1793 107 -^* - 2012 mat 11 mat

XoYo Xo Yo augment : XoYo XoYo o 5 black plot : 21 mat

x f 0.8147 - 1.6294 x * + :

x f contour XoYo 3 1 sys

Xo 1 el Xo 2 el 21 mat Yo 1 el Yo 2 el 21 mat Intercept 0.8147 -

Xo 1 el Xo 2 el 21 mat Yo 1 el Yo 2 el 21 mat Slope 1.6294

Example 12 ... Interpolate spiral from Hermite

Spiral data set to interpolate

Ψ 0.5 0 0.494 0.01 - 0.488 0.022 - 0.481 0.034 - 0.473 0.048 - 0.465 0.062 - 0.456 0.076 - 0.447 0.091 - 0.437 0.106 - 0.427 0.121 - 0.417 0.135 - 0.407 0.147 - 0.398 0.157 - 0.388 0.168 - 0.378 0.178 - 0.368 0.187 - 0.358 0.197 - 0.347 0.205 - 0.337 0.214 - 0.326 0.222 - 0.316 0.229 - 0.305 0.236 - 0.294 0.243 - 0.283 0.25 - 0.272 0.256 - 0.261 0.261 - 0.25 0.267 - 0.239 0.271 - 0.228 0.276 - 0.217 0.28 - 0.206 0.284 - 0.195 0.287 - 0.184 0.29 - 0.173 0.293 - 0.162 0.295 - 0.152 0.297 - 0.141 0.299 - 0.13 0.3 - 0.119 0.302 - 0.109 0.302 - 0.098 0.303 - 0.088 0.303 - 0.078 0.302 - 0.068 0.302 - 0.057 0.301 - 0.048 0.3 - 0.038 0.298 - 0.028 0.297 - 0.019 0.295 - 0.009 0.293 - 1.582 107 -^* 0.29 - 0.009 - 0.287 - 0.018 - 0.285 - 0.027 - 0.281 - 0.035 - 0.278 - 0.043 - 0.274 - 0.052 - 0.27 - 0.059 - 0.266 - 0.067 - 0.262 - 0.075 - 0.258 - 0.082 - 0.253 - 0.089 - 0.248 - 0.096 - 0.243 - 0.103 - 0.238 - 0.109 - 0.233 - 0.116 - 0.227 - 0.122 - 0.222 - 0.128 - 0.216 - 0.133 - 0.21 - 0.139 - 0.204 - 0.144 - 0.198 - 0.149 - 0.192 - 0.154 - 0.186 - 0.158 - 0.179 - 0.162 - 0.173 - 0.166 - 0.166 - 0.17 - 0.16 - 0.174 - 0.153 - 0.177 - 0.146 - 0.18 - 0.14 - 0.183 - 0.133 - 0.185 - 0.126 - 0.188 - 0.119 - 0.19 - 0.112 - 0.192 - 0.106 - 0.194 - 0.099 - 0.195 - 0.092 - 0.196 - 0.085 - 0.197 - 0.078 - 0.198 - 0.071 - 0.199 - 0.065 - 0.199 - 0.058 - 0.199 - 0.051 - 0.199 - 0.045 - 0.199 - 0.038 - 0.199 - 0.032 - 0.198 - 0.025 - 0.197 - 0.019 - 0.196 - 0.012 - 0.195 - 0.006 - 0.194 - 2.451 108 -^* - 0.192 - 0.006 0.191 - 0.012 0.189 - 0.018 0.187 - 0.024 0.184 - 0.029 0.182 - 0.035 0.18 - 0.04 0.177 - 0.045 0.174 - 0.051 0.171 - 0.056 0.168 - 0.06 0.165 - 0.065 0.162 - 0.07 0.158 - 0.074 0.155 - 0.079 0.151 - 0.083 0.147 - 0.087 0.143 - 0.091 0.14 - 0.095 0.136 - 0.098 0.131 - 0.102 0.127 - 0.105 0.123 - 0.108 0.119 - 0.111 0.114 - 0.114 0.11 - 0.117 0.105 - 0.119 0.101 - 0.122 0.096 - 0.124 0.092 - 0.126 0.087 - 0.128 0.082 - 0.13 0.078 - 0.131 0.073 - 0.133 0.068 - 0.134 0.064 - 0.135 0.059 - 0.136 0.054 - 0.137 0.05 - 0.138 0.045 - 0.138 0.04 - 0.139 0.036 - 0.139 0.031 - 0.139 0.026 - 0.139 0.022 - 0.139 0.017 - 0.138 0.013 - 0.138 0.009 - 0.137 0.004 - 0.137 6.872 108 -^* 0.136 0.004 0.135 0.008 0.134 0.012 0.132 0.017 0.131 0.021 0.13 0.024 0.128 0.028 0.126 0.032 0.124 0.036 0.123 0.039 0.121 0.043 0.119 0.046 0.116 0.049 0.114 0.053 0.112 0.056 0.109 0.059 0.107 0.062 0.104 0.064 0.101 0.067 0.099 0.07 0.096 0.072 0.093 0.075 0.09 0.077 0.087 0.079 0.084 0.081 0.081 0.083 0.078 0.085 0.075 0.087 0.072 0.088 0.068 0.09 0.065 0.091 0.062 0.092 0.059 0.094 0.055 0.095 0.052 0.095 0.049 0.096 0.045 0.097 0.042 0.098 0.039 0.098 0.035 0.099 0.032 0.099 0.028 0.099 0.025 0.099 0.021 0.099 0.017 0.098 0.014 0.098 0.011 0.098 0.008 0.098 0.005 0.097 0.002 0.097 1.179 107 -^* - 2012 mat :

region of interpolation j 125 :

Ψ rows 201

X Ψ j 1 el Ψ j 1 + 1 el Ψ j 2 + 1 el Ψ j 3 + 1 el 41 mat : 0.119 - 0.114 - 0.11 - 0.105 - 41 mat

Y Ψ j 2 el Ψ j 1 + 2 el Ψ j 2 + 2 el Ψ j 3 + 2 el 41 mat : 0.111 0.114 0.117 0.119 41 mat

x sol X Y x ainterp :

or select 'U' somewhere in 'X' U X Mean : 0.112 -

U U sol + 40 blue 1 5 mat Ψ XoYo 3 1 sys

U U sol 12 mat 0.112 - 0.1155 12 mat

t Clear 1

X Clear 1

Example 13 ... Spline/Interpolate Van der Pol

Parametric Rkadapt RKF45 ... wrt 'β'  ►  D(t,X,β)) β 0.5 0.5 21 mat : t X β D X 2 el β 1 el X 2 el * X 1 el (2^1 - (* - β 2 el (2^X 1 el * - 21 mat : t.max 20 : N 2000 : init 0 β 1 el - 21 mat : t.0 init 1 el : 14 mat res init t.0 t.max N t X β D Rkadapt : X res 1 col : Y res 2 col : Z res 3 col : 13 mat IntegralPath Y Z augment : 61 sys

IntegralPath init 1 el init 2 el + 300 red 1 5 mat 2 1 sys

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

Hermite cubic is happy of sparse data... van IntegralPath 30 Decimate eval :

van rows 69

user original polygon Xo van 1 col : Yo van 2 col : 21 sys

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 :

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 : 21 line

isolate New New :

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

populate segments t 01 0.1 range :

Hermite interpolation

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 * :

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 : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

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XoYo Xo Yo augment : XoYo XoYo o 5 black plot : 21 mat

Ω XoYo :

region of interpolation j 47 :

X Ω j 1 el Ω j 1 + 1 el Ω j 2 + 1 el Ω j 3 + 1 el 41 mat eval : 0.4398 - 0.8204 - 1.2006 - 1.533 - 41 mat

Y Ω j 2 el Ω j 1 + 2 el Ω j 2 + 2 el Ω j 3 + 2 el 41 mat eval : 1.2697 - 1.3366 - 1.2569 - 1.0115 - 41 mat

x sol X Y x ainterp :

or select 'U' somewhere in 'X' U X Mean : 0.9985 -

XY X Y augment : support XY . 14 black plot : 21 sys

U U sol + 40 blue 1 5 mat contour XoYo support 4 1 sys

U U sol 12 mat 0.9985 - 1.3215 - 12 mat

specific per example region ... x λ 1 x X 1 el ≤ (X 4 el x ≤ (& cases :

x sol x λ * contour support 3 1 sys