<content> <p style="font-family: Courier New; font-size: 8px;">Proj</p> </content>

b7746fc8-6dc8-4722-8521-9a9d1b018047 71 4 15 false false 0 decimal

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

Projection of a curve onto a surface

t0 0 time :

Computing the projection of a point onto a surface is to find a closest point on the surface, and projection of a curve onto a surface is the locus of all points on the curve project onto the surface.

γ 0.8232 - 0.4194 - 0.3827 0.5677 0.6187 - 0.543 0.009 0.6643 0.7474 33 mat :

x Proj.γ x γ * 1 col x γ * 3 col augment eval 11 line :

PMesh returns a maitrx suitable for 3D plots. F# pMesh F F# : nu F rows 1 - : nv F cols 1 - : 13 mat np 5 : r.np 1 np range : r.nv 1 nv 1 - (range : 13 mat M 03 matrix : c 13 range : 12 mat u\032E\ 1 nu range v 1 nv range r M rows : k r r.np + : 12 mat M k c el F 0011015 mat k r - el u\032E\ + 0110015 mat k r - el v + el c el : 21 line for r M rows : j r r.nv + : 12 mat M j c el M u\032E\ 1 - (np nv * nv + 1 - (* np nv 1 - (* + j r - 1 - (np * - c el : 31 line for M 51 line :

CreateMesh replicate several times the same points.
Given that those points are use all for evaluating d[k≡] for
a good guess for the solver, it's faster take only all
the surface points and not the mesh.

1 λ b 2 φ B ProjC2S u v x S L C a d Clear c 13 range : b rng 1 b 3 el range b 1 el b 2 el b 1 el - b 3 el 1 - / 1 b 3 el range * + 12 mat : 13 mat nx X 12 mat B 1 row rng : ny Y 12 mat B 2 row rng : 12 mat S nx ny el X nx el Y ny el X nx el Y ny el f augment eval : k 1 S length range : 12 mat x φ x 1 el x 2 el f x 3 el - : φ' c el x φ x c el diff : eN φ' φ' norme / : 13 mat nt T 12 mat b rng : C nt c el T nt el λ c el : 12 mat n nt A C n c el : ao S d k el A S k el - norme : k augment 1 csort 2 el el : 12 mat x F x φ eN 2 el A 3 el x 3 el - (* eN 3 el A 2 el x 2 el - (* - eN 3 el A 1 el x 1 el - (* eN 1 el A 3 el x 3 el - (* - 31 mat : α ao transpose x F al_nleqsolve : 12 mat a n c el α c el : L n el α transpose A stack : 12 mat 31 line for C S pMesh a L 14 mat 71 line :

Surface

Normal

Curve

for each point in C

Points and Lines

α A stack α α A f sign A α - norme * eN x α ≡ at transpose * + stack ≡

Returns the coordinates of the Curve (C), Surface (S), Projection (a) and Line segments between the point in the curve and the point in the surface (L).

Notice that the coordinates are for 3D Plots, and use Proj γ for
show them in a 2D plot

A\032E\ pCycleC B\0324\ A\032E\ sys2mat eval : M\0324\ B\0324\ 1 el 11 mat : 12 mat k 2 : k B\0324\ length ≤ k k 1 + : n 15 range M\0324\ M\0324\ rows 1 + el 105^105^12 mat : for M\0324\ M\0324\ rows 1 + el B\0324\ k el : 21 line for M\0324\ mat2sys.1 eval 31 line :

Shorthand for plots

X Plot C S A L 14 mat X : C Proj.γ A Proj.γ \002E\ 12 green augment L Proj.γ vectorize pCycleC S Proj.γ 51 sys 21 line :

Number of points:

nt 50 :

nx 30 :

ny 30 :

Curve given by the
parametric equations

t λ t x t y t z 13 mat ≡

and

box t1 t2 nt 13 mat :

Surface given by the
explicit equation

z x y f ≡

and

Box x1 x2 nx y1 y2 ny 23 mat ≡

Curve t λ 0.2 t * 2 t sin * 4 t cos * 13 mat : Bt 2030 nt 13 mat : Surface x y f x y * 4 x * - : Bxy 1 - 6 nx 1 - 6 ny 23 mat : 61 line