On polyhedral approximations to a sphere

The authors investigate methods by which successive approximations to a sphere can be generated from polyhedra. Each approximation can be obtained by bevel-cutting each edge of the previous approximation with a plane tangent to the sphere. They show that each member of the sequence of polyhedra can be associated with a Voronoi tessellation of the sphere. Under this formulation, the bevel-cutting operation can be defined by the insertion of points into the Voronoi tessellation. The algorithm is defined such that affine combinations of the polyhedra will converge to affine operations of the sphere. The method is useful as a modeling operation and as a level-of-detail representation for a sphere