Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values

Abstract

In this paper we disprove the belief widespread within the computer graphics community that Catmull-Clark subdivision surfaces cannot be evaluated directly without explicitly subdividing. We show that the surface and all its derivatives can be evaluated in terms of a set of eigenbasis functions which depend only on the subdivision scheme and we derive analytical expressions for these basis functions. In particular, on the regular part of the control mesh where Catmull-Clark surfaces are bi-cubic B-splines, the eigenbasis is equal to the power basis. Also, our technique is both easy to implement and efficient. We have used our implementation to compute high quality curvature plots of subdivision surfaces. The cost of our evaluation scheme is comparable to that of a bi-cubic spline. Therefore, our method allows many algorithms developed for parametric surfaces to be applied to Catmull-Clark subdivision surfaces. This makes subdivision surfaces an even more attractive tool for free-form surface modeling. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, Surface, Solid, and Object Representations J.6 [Computer Applications]: Computer-Aided Engineering—Computer Aided Design (CAD)