<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

az el pView α az : ε el : 12 mat ε ε 180 ° * + 360 ° * mod 360 ° * + 360 ° * mod 180 ° * - : ε 90 ° * > ε 180 ° * ε - : α α 180 ° * + : 12 mat ε 90 ° * - < ε 180 ° * - ε - : α α 180 ° * + : 12 mat 0 if if α α 360 ° * mod 360 ° * + 360 ° * mod : α cos ε sin α sin * - α sin ε sin α cos * 0 ε cos 32 mat 51 line :

γ 0.8232 - 0.4194 - 0.5677 0.6187 - 0.009 0.6643 32 mat :

Blessed algebra! The last column is not needed so you don't have to use col (X * λ, n) Simply multiply by lambda.

az γ 1 el acos : ° 145.4064

el γ 32 el acos : ° 48.3714

145 ° * 48 ° * pView 0.8192 - 0.4263 - 0.5736 0.6087 - 0 0.6691 32 mat

More or less the same. Matlab default view is -37.5, 30 deg.

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 :

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 0.8182 - 0.1552 - 4.9328 13 mat

a 0.6026 - 0.5742 - 0.9846 13 mat

dist 3.9762

a f 1.9619 109 -^* -

dot 1.5543 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 0.2587 - 1.9345 3.522 13 mat

a 0.2582 - 1.91 0.0098 13 mat

dist 3.5123

a f 0.0046 -

dot 0.0005 -

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 1.6526 1.0783 - 8.8183 13 mat

a 3.9468 1.3126 7.6319 13 mat

dist 3.5195

a f 8.991 109 -^* -

dot 2.4425 1015 -^*

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 2.5648 - 1.5551 - 5.5385 - 13 mat

a 1.6976 - 1.1877 1.8401 - 13 mat

dist 4.6854

a f 3.5668 109 -^*

dot 2.2204 1016 -^*

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 4.9704 2.8765 - 6.1564 13 mat

a 6.1896 3.1908 - 1.9931 13 mat

dist 4.3495

a f 5.7161 1010 -^*

dot 1.3323 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 0.9231 1.8659 5.0861 13 mat

a 0.7693 3.1094 4.6862 13 mat

dist 1.3152

a f 4.3108 109 -^* -

dot 0

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 0.4217 - 1.3792 - 9.4802 13 mat

a 0.7463 2.3246 - 1.995 13 mat

dist 7.6346

a f 5.8969 1011 -^*

dot 4.4409 1016 -^*

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 0.2069 - 0.5346 - 2.7774 13 mat

a 0.5372 0.2971 - 2.6252 13 mat

dist 0.7958

a f 2.4393 109 -^*

dot 1.5543 1015 -^* -