Evaluation of piecewise smooth subdivision surfaces

In this paper we consider the constant-time evaluation of subdivision surfaces at arbitrary points. Our work extends the work of J. Stam by considering the subdivision rules for piecewise smooth surfaces with boundaries depending on parameters. The main innovation described in this paper is the idea of using a different set of basis vectors for evaluation, which, unlike eigenvectors, depend continuously on the coefficients of the subdivision rules. The advantage of this approach is that it becomes possible to define evaluation for parametric families of rules without considering an excessive number of special cases and while improving the numerical stability of calculations. We demonstrate how such bases are computed for a particular parametric family of subdivision rules extending Loop subdivision to meshes with boundaries, and we provide a detailed description of the evaluation algorithms.