I can create a polyhedral approximation to a sphere by beginning with some polyhedron such as an octahedron or icosahedron, subdividing its faces, and projecting the vertices onto a sphere. This gives a Geodesic Polyhedron. The resulting polyhedron has every vertex lying on the sphere, so it is inscribed in the sphere.
I would like to construct a sphere approximation that is instead circumscribed, where each face of the approximating polyhedron is tangent to the sphere. It seems like it should work to construct the projected dual of some initial polyhedron: project the vertices onto the sphere, compute the tangent plane, and find the intersections of those tangent planes to define the faces. This would produce something like the Goldberg Polyhedra. But this procedure is not very efficient, since it requires solving a lot of systems of 3 equations. Is there a simpler, more efficient, more direct method for producing a circumscribing sphere approximation?