Infinitesimal amths ==>> plot Tangent at user [xo, yo]

a0e112fd-4790-4513-9f49-0a830a098cd8 33 15 12 decimal

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

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 :

Remark . . .

cinterp takes clean data pair from pure functions.

From real project it will take the collected data

that you will have to clean someway.

cinterp for much less noisy Kurvature.

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V 20 : a 50 : b 100 : 13 mat

r a 2^b 2^+ 2 / sqrt :

ω V r / : 0.253

3 appVersion 0.98.6179 ≡

t 08 0.01 range : k 1 t rows range X k el 501 ω t k el * π 2 / + sin - (* eval : Y k el 100 ω t k el * sin * eval : 21 line for 21 line

x f X Y x cinterp :

x c 10 x ≤ (x 71.886 ≤ (& cases :

M X Y augment :

M 1 row 0012 mat

M 801 row 71.8860049221856 89.9111292009186 12 mat

XY M 400 row (: 23.3815 84.6512 12 mat

assign coordinates point of normalcy

M rows 801

xo 23.3815 :

yo 84.6512 :

xy XY + 20 black plot :

Infinitesimal amths ==>> plot Tangent at user [xo, yo]

The function x f x f :

Algo style Memo formula of incremental analysis of this demo example h 0.1 : (increment ≡ x h f deriv 0.5 x h - f x f - x h - (x - / x f x h - f - x x h - (- / + (* : x h f D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / : (1rst Derivative ≡ 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^* / : (2nd Derivative ≡ 41 line

The curvature x K x h f D2 1 x h f D1 (2^+ (32 /^/ eval :

xo h xo K D1

The normal x N xo h xo K D1 (1 -^x xo - (* - xo K + yo + eval :

The tangent x T xo h xo K D1 (x xo - (* xo K + yo + eval :

x λ 10 x ≤ (x 50 ≤ (& cases :

start 0 : end 71.8860049221856 : N 801 : 13 mat

x f x c * x T x λ * xy x N 41 sys

x f x c * x T x λ * xy x N 4 1 sys

Recast curvature @ low 'h' to minimize
the noise from  ...  D2(x,h,f) x h f K x h f D2 1 x h f D1 (2^+ (32 /^/ eval :

x 0.1 f K x c *

Export/populate on image of data set curv u start end end start - N / range : i 1 u rows range Kurv i el u i el 0.1 f K eval : for u Kurv augment 31 line :

c(t) Unknown

t Clear 1

t x 501 ω t * π 2 / + sin - (* :

t y 100 ω t * sin * :

t c t x t diff t y t 2 diff * (t y t diff t x t 2 diff * (- abs t x t diff (2^t y t diff (2^+ (32 /^/ :

x 0 > ( x 8 < ( & x c not defined if

data u start end end start - N / range : i 1 u rows range spline i el u i el f eval : for u spline augment 31 line :

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 :

data data 4 Decimate :

data rows 267

Finite Differences [FD] methods

Finite differences derivative X data 1 col : Y data 2 col : 12 mat n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 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 - / + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : D X deriv augment : Finite differences Cumul Integration vx data 1 col : vy data 2 col : 12 mat 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 cumul vx Y augment (: Arc integrand .................... xx D 1 col : δx D 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 231 line

Function & its 1rst Derivative

data D 50.078 100 + 40 red 1 5 mat 3 1 sys

Cumulaive integration

cumul

Cumulative arc length

For this example . . .

Decimation is adequate/accurate.

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N data 175200 range 12 range el :

N 11 mat 46.847 99.801 47.116 99.834 47.386 99.863 47.655 99.89 47.924 99.914 48.193 99.935 48.462 99.953 48.732 99.968 49.001 99.98 49.27 99.989 49.539 99.996 49.809 99.999 50.078 100 50.347 99.998 50.616 99.992 50.886 99.984 51.155 99.973 51.424 99.959 51.693 99.943 51.963 99.923 52.232 99.9 52.501 99.875 52.77 99.846 53.039 99.815 53.309 99.781 53.578 99.744 262 mat 11 mat

M key FindIndex r M rows : c M cols : 12 mat i 1 : j 1 : 12 mat j c ≤ (M i j el key ≠ (* i r < i i 1 + : i 1 : j j 1 + : 21 line if while j c ≤ i j 21 mat 0 if 41 line :

N N Max FindIndex 13221 mat

N 13 row 50.078 10012 mat