Smoothness of subdivision surfaces at extraordinary points (Q1272515)

The aim of the paper is to investigate the behaviour of \textit{E. Catmull} and \textit{J. Clark}'s subdivision algorithm [Recursively generated \(B\)-spline surfaces on arbitrary topological meshes, Comput. Aided Design 10, No. 6, 350-355 (1978)] as well as other stationary subdivision schemes described by matrix iterations around extraordinary points. The author obtains important results that show how higher-order smoothness of a limiting surface obtained by a stationary subdivision algorithm for tri- and quadrilateral nets depends on the spectral properties of the matrix, and necessary and sufficient conditions are provided. These results have been successfully applied to improve several important subdivision algorithms, and to extend \textit{U. Reif}'s degree estimate [Proc. Am. Math. Soc. 124, No. 7, 2167-2174 (1996; Zbl 0857.65018)] to subdivision surfaces of arbitrary high smoothness.