Subdivision by WAVES – Weighted AVEraging Schemes

The Catmull-Clark subdivision algorithm consists of an operator that can be decomposed into a refinement and a smoothing operator, where the refinement operator splits each face with m vertices into m quadrilateral subfaces and the smoothing operator replaces each internal vertex with an affine combination of its neighboring vertices and itself. In this paper, we generalize the Catmull-Clark scheme. We consider an arbitrarily fixed number r of weighted averaging steps and allow that these r smoothing operators are different. These w(eighted) ave(raging) s(chemes) form an infinite class of stationary subdivision schemes, which we call wave shemes. This class includes the Catmull-Clark scheme and the midpoint schemes. For regular meshes, wave schemes generalize the tensor product Lane-Riesenfeld subdivision algorithm. We analyze the smoothness of stationary wave surfaces at extraordinary points using established methods for analyzing midpoint subdivision. For regular meshes, we analyze the smoothness of non-stationary wave schemes which need not be asymptotically equivalent to stationary schemes. Wave surfaces are smooth at their regular and, in most cases, extraordinary points.