c83d4eeb-b075-422e-90b6-f4d6a1f6cdfb 28 6 12 decimal

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

x f x erf :

Part 1 ... Infinitesimal Analysis basics

Infinitesimal Derivative x f deriv h 106 -^: x 2 h * - f x 2 h * + f - 4 h * / - 21 line :

Note . . . Some functions have no symbolic derivative.

The only alternative is infinitesimal from the CAS ancestors.

In this example,'h' is paired with erf. In common Engineering

functions, too low 'h' will generate noise. Infinitesimal is the

sole Smath method to get the derivative of splined functions.

Differently, Mathcad has an equivalent scalar method, about

equivalent to infinitesimal. Mathcad 1rst derivative comes out

scalar from the spline(s) coefficients.

]]> x f x f deriv 2 1 sys

Derivative erf(x) x D 2 π sqrt / e x 2^-^* :

π erf 0.999991

x π :

x D x f deriv 21 mat 0.000058 0.000058 21 mat

Observe vectorize x erf vectorize x 02 int 1.437

a 8 π 3 - (* 3 π * 4 π - (* / : 0.140012

Approximation erf(x)

x erf x sign 1 x 2^4 π / a x 2^* + 1 a x 2^* + / (* - exp - sqrt * :

x D x erf 2 1 sys

[L,H] range Npts L 5 - : H 5 : N 10000 : ε H L - N / : 14 mat 11 sys

pilot the mesh

<= Sanity check function x f 11 x 2^+ / :

x c 11 - x ≤ (x 1 ≤ (& cases :

less noisy ... 10 decimals x ε D1 x 2 ε * + f - 8 x ε + f * + 8 x ε - f * - x 2 ε * - f + 12 ε * / :

expect 8 Decimals not "noisy" x ε D2 x 2 ε * + f - 16 x ε + f * + 30 x f * - 16 x ε - f * + x 2 ε * - f - 12 ε 2^* / :

x f x 1 diff x f x 2 diff 3 1 sys x f x ε D2 2 1 sys x f x 2 diff x c * x ε D2 2 1 sys x f x 2 diff x ε D2 - x f x 1 diff x ε D1 -

https://www.geometrictools.com/Documentation/FiniteDifferences.pdf

https://www.geometrictools.com/Documentation/DocUpdateHistory.pdf

https://www.geometrictools.com/Documentation/ReconstructHeightFromNormals.pdf

1rst derivative  applied on interpolating functions x h D1 x 2 h * + f - 8 x h + f * + 8 x h - f * - x 2 h * - f + 12 h * / :

2nd derivative applied on interpolating functions x h D2 x 2 h * + f - 16 x h + f * + 30 x f * - 16 x h - f * + x 2 h * - f - 12 h 2^* / :

x h D3 x 3 h * - f 8 x 2 h * - f * - 13 x h - f * + 13 x h + f * - 8 x 2 h * + f * + x 3 h * + f - 8 h 3^* / :

x h D4 x 3 h * - f - 12 x 2 h * - f * + 39 x h - f * - 56 x f * + 39 x h + f * - 12 x 2 h * + f * + x 3 h * + f - 6 h 4^* / :

Part 2 ... Rendering a Project

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 : 12 mat x spline X Y x cinterp : 21 mat

x c 1 X 1 el x ≤ (x X 6 el ≤ (& cases :

ptsSpec k 1 : i k X rows range data i el Sy 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 :

Sy . :

size 14 :

clr black :

support X Y ptsSpec augment :

"Proofing" the cubic spline project

x spline x c * support 3 1 sys

[L,H] range Npts L 3.52 - : H 1 : N 10000 : ε H L - N / : 14 mat 11 sys

pilot the mesh

<= Sanity check function x f x spline :

less noisy ... 10 decimals x ε D1 x 2 ε * + f - 8 x ε + f * + 8 x ε - f * - x 2 ε * - f + 12 ε * / :

"Proofing" the cubic spline project

x ε D1 x c * x spline x c * support 3 1 sys

x spline x X 1 el X 6 el int x ε D1 x X 1 el X 6 el int 21 mat 7.381 - 1.081 21 mat