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Combining 4- and 3-direction subdivision

Published: 01 October 2004 Publication History

Abstract

4-3 direction subdivision combines quad and triangle meshes. On quad submeshes it applies a 4-direction alternative to Catmull-Clark subdivision and on triangle submeshes a modification of Loop's scheme. Remarkably, 4-3 surfaces can be <i>proven to be C<sup>1</sup> and have bounded curvature everywhere</i>. In regular mesh regions, they are <i>C</i><sup>2</sup> and correspond to two closely-related box-splines of degree four. The box-spline in quad regions has a smaller stencil than Catmull-Clark and defines the unique scheme with a 3 × 3 stencil that can model constant features <i>without ripples</i> both aligned with the quad grid and diagonal to it. From a theoretical point of view, 4-3 subdivision near extraordinary points is remarkable in that the eigenstructure of the local subdivision matrix is easy to determine and a complete analysis is possible. Without tweaking the rules artificially to force a specific spectrum, the leading eigenvalues ordered by modulus of all local subdivision matrices are 1, 1/2, 1/2, 1/4 where the multiplicity of the eigenvalue 1/4 depends on the valence of the extraordinary point and the number of quads surrounding it. This implies equal refinement of the mesh, regardless of the number of neighbors of a mesh node.

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cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 23, Issue 4
October 2004
145 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/1027411
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 01 October 2004
Published in TOG Volume 23, Issue 4

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Author Tags

  1. CAD
  2. curves and surfaces
  3. geometric modeling
  4. subdivision

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