A bound on the approximation of a Catmull–Clark subdivision surface by its limit mesh

A Catmull–Clark subdivision surface (CCSS) is a smooth surface generated by recursively refining its control meshes, which are often used as linear approximations to the limit surface in geometry processing. For a given control mesh of a CCSS, by pushing the control points to their limit positions, another linear approximation—a limit mesh of the CCSS is obtained. In general a limit mesh might approximate a CCSS better than the corresponding control mesh. We derive a bound on the distance between a CCSS patch and its limit face in terms of the maximum norm of the second order differences of the control points and a constant that depends only on the valence of the patch. A subdivision depth estimation formula for the limit mesh approximation is also proposed. For a given error tolerance, fewer subdivision steps are needed if the refined control mesh is replaced with the corresponding limit mesh.