Ck Continuity of Subdivision Surfaces

Abstract

Stationary subdivision is an important tool for generating smooth free-form surfaces for CAGD and computer graphics. One of the challenges in construction of subdivision schemes for arbitrary meshes is to guarantee that the limit surface will have smooth regular parameterization in a neighborhood of any point. First results in this direction were obtained only recently. In this paper we derive necessary and sufficient criteria for Ck -continuity that generalize and extend most known conditions. We create a general mathematical framework that can be used for analysis of more general types of schemes. Finally, we prove a degree estimate for Ck -continuous polynomial schemes generalizing an estimate of Reif [20] and give a practical sufficient condition for smoothness.