Smoothing polyhedra made easy

A mesh of points outlining a surface is polyhedral if all cells are either quadrilateral or planar. A mesh is vertex-degree bounded if at most four cells meet at every vertex. This paper shows that if a mesh has both properties then simple averaging of its points yields the Bemstein-B6zier coefficients of a smooth, at most cubic, surface that consists of twice as many three-sided polynomial pieces as there are interior edges in the mesh. Meshes with checkerboard structure, that is, rectilinear meshes, are a special case and result in a quadratic surface. Since any bivariate mesh and, in particular, any wireframe of a polyhedron can be refined, by averaging, to a vertex-degree-bounded polyhedral mesh the above allows reinterpretation of a number of algorithms that construct smooth surfaces and advertises the corresponding averaging formulas as a model for a wider class of algorithms.