<p> Hermite interpolation </p>

2014f3bb-4fae-4e7d-a6c2-f46deabd0394 15 4 5 decimal

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

Hermite interpolation

4 appVersion 0.98.6179.21440 ≡

t0 1 time :

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 pop[ulated 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". The 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 t 01 0.125 range :

t rows 9

<p> Hermite interpolation </p>

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 1.3056 1670.2082 2.7805 1665.0906 4.3882 1657.409 6.0925 1646.725 7.8571 1632.6004 9.6458 1614.5969 11.4222 1592.2762 13.15 1565.2 14.403 1520.7812 15.572 1471.8875 16.6579 1419.2375 17.6615 1363.55 18.5837 1305.5438 19.4252 1245.9375 20.1871 1185.45 20.87 1124.8 21.4145 1072.3536 21.8876 1020.4539 22.2958 969.0913 22.6459 918.2562 22.9443 867.9392 23.1977 818.1305 23.4128 768.8206 23.596 720 23.8089 671.6099 23.998 623.6945 24.1647 576.2491 24.3104 529.2688 24.4363 482.7485 24.544 436.6836 24.6348 391.069 24.71 345.9 24.7712 294.605 24.8197 244.3711 24.8567 195.8188 24.8836 149.5688 24.9019 106.2413 24.9128 66.457 24.9177 30.8364 24.918 0412 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

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

1 time t0 - 6 s *

XYdata Xo Yo augment :

XYdata XYdata 2 1 sys

blue ► HermiteCubic red ► freak cinterp

uncertified ...
wrt 't' population x f x Xo 1 el Xo 6 el int 35479

h Xo 6 el t 2 el * : x h deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* x λ * : 21 sys

x h deriv Xo 6 el 50 Xo 6 el 350 - 2 2 mat 2 1 sys

vx 11 mat 0 1.3056 2.7805 4.3882 6.0925 7.8571 9.6458 11.4222 13.15 14.403 15.572 16.6579 17.6615 18.5837 19.4252 20.1871 20.87 21.4145 21.8876 22.2958 22.6459 22.9443 23.1977 23.4128 23.596 23.8089 23.998 24.1647 24.3104 24.4363 24.544 24.6348 24.71 24.7712 24.8197 24.8567 24.8836 24.9019 24.9128 24.9177 24.918 411 mat 11 mat

contour 11 mat 0 1673.2 1.3056 1670.2082 2.7805 1665.0906 4.3882 1657.409 6.0925 1646.725 7.8571 1632.6004 9.6458 1614.5969 11.4222 1592.2762 13.15 1565.2 14.403 1520.7812 15.572 1471.8875 16.6579 1419.2375 17.6615 1363.55 18.5837 1305.5438 19.4252 1245.9375 20.1871 1185.45 20.87 1124.8 21.4145 1072.3536 21.8876 1020.4539 22.2958 969.0913 22.6459 918.2562 22.9443 867.9392 23.1977 818.1305 23.4128 768.8206 23.596 720 23.8089 671.6099 23.998 623.6945 24.1647 576.2491 24.3104 529.2688 24.4363 482.7485 24.544 436.6836 24.6348 391.069 24.71 345.9 24.7712 294.605 24.8197 244.3711 24.8567 195.8188 24.8836 149.5688 24.9019 106.2413 24.9128 66.457 24.9177 30.8364 24.918 0412 mat 11 mat

Xo 6 el 24.918