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

  • Authors:

  • Zhangjin Huang;Jiansong Deng;Guoping Wang

  • Affiliations:

  • School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China and Department of Computer Science and Technology, University of Science and Technology of China, H ...;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China;School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China

  • Venue:
  • Computer Aided Geometric Design
  • Year:
  • 2008

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Abstract

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.