An Approximating-Interpolatory Subdivision scheme

In the last decade, study and construction of quad/triangle subdivision schemes have attracted attention. The quad/triangle subdivision starts with a control mesh consisting of both quads and triangles and produces finer and finer meshes with quads and triangles (Fig. 1). Designers often want to model certain regions with quad meshes and others with triangle meshes to get better visual quality of subdivision surfaces. Smoothness analysis tools exist for regular quad/triangle vertices. Moreover C1 and C2 quad/triangle schemes (for regular vertices) have been constructed. But to our knowledge, there are no quad/triangle schemes that unifies approximating and interpolatory subdivision schemes. In this paper we introduce a new subdivision operator that unifies triangular and quadrilateral subdivision schemes. Our new scheme is a generalization of the well known CatmullClark and Butterfly subdivision algorithms. We show that in the regular case along the quad/triangle boundary where vertices are shared by two adjacent quads and three adjacent triangles our scheme is C2 everywhere except for ordinary Butterfly where our scheme is C1. 1 ha l-0 06 39 05 1, v er si on 1 8 N ov 2 01 1 Author manuscript, published in "International Journal of Pure and Applied Mathematics 71, 1 (2011) 129-147"