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## A Discrete parametrization of a surface

Maple

In this post I want to present an easy method to obtain a discrete parametrization of a surface S defined implicitly (f(x,y,z)=0).
This problem was discussed here several times, the most recent post is
http://www.mapleprimes.com/posts/207661-Isolation-Of-Sides-Of-The-Surface-On-The-Graph

S is supposed to be the boundary of a convex body having (x0,y0,z0) an interior point and contained in a ball of radius R centered at (x0,y0,z0).
Actually, the procedure also works if the body is only star-shaped with respect to the interior point, and it is also possible to plot only a part of the surface
inside a solid angle centered at (x0,y0,z0).

Usage:
Par3d(f, x=x0, y=y0, z=z0, R, m, n,  theta1 .. theta2,  phi1 .. phi2)

f           is an expression depending on the variables x, y, z
x0, y0, z0  are the coordinates of the interior point
R           is the radius of the ball which contains the surface,
m, n        are the numbers of the grid lines which will be generated
The last two parameters are optional and are used when only a part of S will be parametrized.

The procedure Par3d returns a MESH structure M, which can be plotted with PLOT3D(M).

```Par3d :=proc(f,x::`=`,y::`=`,z::`=`,R,m,n,th:=0..2*Pi,ph:=0..Pi)
local A,i,j, rij,fij,Cth,Sth,Cph,Sph, theta,phi, r;
A:=Array(1..m+1,1..n+1,1..3,datatype=float);
for i from 0 to m do for j from 0 to n do
theta:=op(1,th)+i/m*(op(2,th)-op(1,th));
phi:=op(1,ph)+j/n*(op(2,ph)-op(1,ph));
Cth:=evalf(cos(theta)); Sth:=evalf(sin(theta));
Cph:=evalf(cos(phi));   Sph:=evalf(sin(phi));
fij:= eval(f, [lhs(x)=rhs(x)+r*Sph*Cth, lhs(y)=rhs(y)+r*Sph*Sth, lhs(z)=rhs(z)+r*Cph]);
rij:=fsolve(fij,r=0..R);  if [rij]::list(numeric) then rij:=min(rij) fi;
if [rij]=[] or not(type(rij,numeric)) then print(['i'=i,'j'=j], fij); rij:=undefined fi;
A[i+1,j+1,1]:=evalf(rhs(x)+rij*Sph*Cth);
A[i+1,j+1,2]:=evalf(rhs(y)+rij*Sph*Sth);
A[i+1,j+1,3]:=evalf(rhs(z)+rij*Cph);
od;od:
MESH(A);
end:
```

The procedure is not optimized, e.g.
- Cth, etc could be Vectors computed outside the loops
- Some small changes to use evalhf.

###### EXAMPLES ######

f1 := x^2+3*y^2+4*z^2 - x*y - 2*y*z - 10:
plots:-implicitplot3d(f1, x=-5..5, y=-5..5, z=-2..2);