Items tagged with parametrization

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

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).

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,St