I’ve been meaning to write this up properly for a long time, but for now, this link will have to do:
The idea is to use some fairly straightforward vector geometry to generate uniform polyhedra and their derivatives, using the kaleidoscopic construction to generate Schwarz triangles that tile the sphere. We use spherical trilinear coordinates within each Schwarz triangle to determine polyhedron vertices (with the trilinear coordinates being converted to barycentric for actually generating the points). Vertices for snub polyhedra are found by iterative approximation.
We also can use Schwarz triangles to apply other symmetry operations to the basic polyhedra to generate compound polyhedra, including all of the uniform compounds enumerated by John Skilling (as well as many others).
There are some other features including construction of dual figures, final stellations, inversions, subdividing polyhedra faces using a Sierpinksi construction, as well as various colouring effects, exploding faces etc..