Bézier interpolation Utilities

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Bézier . . . interpolation for native Smath

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data 0 0.00794 0.1 0.012 0.2 0.02 0.3 0.048 0.4 0.17 0.46 0.556 62 mat :

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

reserve the end point Xn X X length el :

mesh the discrete data set export t 01 0.025 range :

all Bézier stuff

Bézier interpolation Utilities

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 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 :

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 :

Polyline 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

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

Bézier interpolation 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 * :

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

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 2 + (- (range : ct 1 : j j i 1 t rows range Bx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Bx 41 line :

V j 1 X rows r 2 + (- (range : ct 1 : j j i 1 t rows range By i ct el t i el j YP eval : for ct ct 1 + : 21 line for By 41 line :

collect the project xy U Unwrap V Unwrap augment : contour xy Strip : contour contour t rows contour rows range 12 range el : contour contour data data rows row stack : 41 sys

contour contour eval :

XY data o 7 black plot : x λ 1 X 1 el x ≤ (x Xn ≤ (& cases : 21 line

vx contour 1 col : vy contour 2 col : 12 mat x f vx vy x cinterp : 21 line

contour rows 201

∫Adaptive(f,a,b,N,ε) Adapted from Carlos

f a b N ε ∫Adaptive sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 41 line eval :

Bézier spline borrowed from OriginLab 6.0f(x) reconnected scalar wrt 'x' for Smath. OriginLab B_Spline roams around points, indirect way to 'smooth/rectify' data. "Interpolate bSpline" document. That the data set does not look clean, not much can be done directly. We simply best manage interpolation, not knowing more about this data.

iter 100 ... plenty
'ε' canvas accuracy
'a' governable limit
'b' governable limit iter 100 : ε 0.001 : a X 1 el : b Xn : 41 sys

x Σ f a x iter ε ∫Adaptive 11 line :

in-situ command line u 0.375 :

f a b iter 1012 -^∫Adaptive 0.035037796

0 0.5 0.25 0.5 u u f 3 2 mat x f x λ * XY 3 1 sys

acc 1000 x f x X 1 el Xn int 0.035037796

@ accuracy of canvas u u f 12 mat 0.375 0.1177 12 mat

f a u iter 1012 -^∫Adaptive 0.011057808

Cumulative integration ... @ accuracy of canvas plot.

x Σ x λ * 0.125 - 0.025 0.25 0.025 u u Σ 3 2 mat u u Σ + 20 black 1 5 mat 3 1 sys

@ accuracy of canvas u u Σ 12 mat 0.375 0.011 12 mat

Ad Absurdum . . . glorious freaks !

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data 0 0.00794 0.1 0.012 0.2 0.02 0.3 0.048 0.4 0.17 0.46 0.556 62 mat :

alternate cinterp/ainterp
Experiment X5 ... 0.25
Seemingly this point is typo.
likely X3 as well. X data 1 col : Y data 2 col : 12 mat XY data o 7 black plot : x spline X Y x cinterp : 31 sys

XY x spline x λ * 2 1 sys

jmG 20230106