<p>f(u,v) Plot γ</p>

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Distance and projection of a point onto a parametric surface

Calculation procedure 1.system of equations. Given the equation paramatric surface f (x) = 0 and the coordinates of point A. It is required to find the orthogonal projection (a) of point A onto this surface. We will use the condition that the vector product of the unit normal vector eN to the surface by the vector (Ax) is equal to zero: As a result, we obtain a system of three equations (the equation of the surface and two equations of the projections of the vector product) 2.Initial guess and solution of the system of equations. Denote by "Mnodes" the mesh node matrix CreateMesh.Using the sorting method, we find the node(a0) that is at the minimum distance from point A. We take the coordinates of this node as the initial guess and solution using al_nlegsolve. 3.The calculation accuracy was checked through the dot product of two unit vectors eN and Aa. The black star is the Initial guess ,the red line is normal to the surface.

and solution using programs

Razonar code x rndBox x length 1 ≡ x 1 el a b 12 mat x min x max 12 mat eval : a b a - (109 -^* 109^random * + eval 21 line if 11 line :

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

x y z rndBox x rndBox y rndBox z rndBox 13 mat eval 11 line :

N 15 :

u v f8 u v 2 u * v * u 2^v 2^+ 0.01 + / 31 mat :

u v f1 u v 8 u 2^v 2^+ 2 / - exp * 31 mat :

u v f3 u v 2 u 2^7 / v 2^14 / + cos * 31 mat :

u v f5 u v v (2^2 / u (2^4 / - 31 mat :

u v f9 u v 2 u * sin 2 v * sin + 31 mat :

u v f7 u v 2 u sin * v sin * 31 mat :

u v f6 u v 2 u * v sin * 31 mat :

u v f4 u v 4 u * u 2^- v 2^- exp * 31 mat :

Plot3 Σγ Mnodes γ * eval : aAγ A a stack γ * eval : 12 mat A γ * (1 el A γ * (2 el A 14 black 15 mat a0 γ * (1 el a0 γ * (2 el * 17 black 15 mat A γ * (1 el A γ * (2 el . 15 green 15 mat Σγ 1 col Σγ 2 col augment a γ * (1 el a γ * (2 el . 15 red 15 mat aAγ 1 col aAγ 2 col augment No γ * 1 col No γ * 2 col augment 81 sys 21 line :

Pr c 13 range : g c el x f x c el diff : eN g g norme / : x G g : k 1 Mnodes rows range : 15 mat d k el A Mnodes k row - norme : a0 Mnodes d augment 4 csort 1 row c el transpose eval : 12 mat x F x f 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 : a a0 transpose x F al_nleqsolve transpose eval : 12 mat eN a G transpose a G norme / : dist A a - norme : No a a A f sign 2 * dist * eN * + stack : 13 mat a dist No a0 eN transpose A a - (transpose dist / * abs 1 - 15 mat 51 line :

x f 2 x 1 el (* x 2 el * x 1 el (2^x 2 el (2^+ 0.01 + / x 3 el - : Mnodes f8 3 - 33 - 3 N N CreateMesh : A 3 - 312 mat 3 - 312 mat 5 - 712 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 1.7846 - 1.0135 - 4.9086 - 13 mat

a 0.6764 - 0.5449 0.9643 - 13 mat

dist 4.3834

a f 1.3108 1010 -^*

dot 1.7764 1015 -^* -

x f 8 x 1 el (2^2 x 2 el * (2^+ 2 / - exp * x 3 el - : Mnodes f1 3 - 33 - 3 N N CreateMesh : A 2 - 412 mat 2 - 412 mat 3712 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 0.6368 2.7805 3.5778 13 mat

a 0.6368 2.7804 1.2596 106 -^* 13 mat

dist 3.5778

a f 1.1775 1010 -^*

dot 0

x f 2 x 1 el (* x 2 el sin * x 3 el - : Mnodes f6 4 - 4 π - π N N CreateMesh : A 2 - 212 mat 3 - 312 mat 21012 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 0.1638 - 2.7399 6.1162 13 mat

a 2.4316 1.7296 4.802 13 mat

dist 3.0796

a f 2.1344 1010 -^* -

dot 4.4409 1016 -^* -

x f 2 x 1 el sin * x 2 el sin * x 3 el - : Mnodes f7 3 - 33 - 3 N N CreateMesh : A 3 - 312 mat 3 - 312 mat 10 - 1012 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 2.1824 - 2.0523 5.4815 - 13 mat

a 1.6476 - 1.6313 1.9905 - 13 mat

dist 3.5568

a f 3.3858 1010 -^* -

dot 2.1094 1015 -^* -

x f 2 x 1 el (2^7 / x 2 el (2^14 / + cos * x 3 el - : Mnodes f3 8 - 88 - 8 N N CreateMesh : A 7 - 712 mat 7 - 712 mat 51012 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 4.877 5.7918 - 5.6127 13 mat

a 5.1207 5.9329 - 1.9995 13 mat

dist 3.6242

a f 2.9403 109 -^* -

dot 1.1102 1015 -^* -

x f x 2 el (2^2 / x 1 el (2^4 / - x 3 el - : Mnodes f5 4 - 45 - 5 N N CreateMesh : A 1 - 112 mat 2 - 212 mat 31212 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 0.2071 - 0.3401 10.6597 13 mat

a 0.1417 - 4.414 9.7368 13 mat

dist 4.1777

a f 1.2215 108 -^* -

dot 3.5527 1015 -^*

x f 2 x 1 el * sin 2 x 2 el * sin + x 3 el - : Mnodes f9 π - π π - π N N CreateMesh : A 3 - 312 mat 3 - 312 mat 01212 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 1.0043 - 1.8338 - 8.512 13 mat

a 2.3062 - 2.3369 - 1.9943 13 mat

dist 6.6655

a f 1.68 109 -^* -

dot 0

x f 4 x 1 el * x - 1 el 2^x 2 el 2^- exp * x 3 el - : Mnodes f4 3 - 33 - 3 N N CreateMesh : A 1 - 112 mat 2 - 212 mat 0512 mat rndBox : a dist No a0 dot 15 mat Pr : 41 sys

Plot3

A 0.974 - 1.0317 - 1.9223 13 mat

a 0.4128 0.481 - 1.5538 13 mat

dist 1.537

a f 4.9147 109 -^*

dot 2.4425 1015 -^*