Article

Distance between a Catmull-Clark subdivision surface and its limit mesh

Authors:

Zhangjin Huang,

Guoping WangAuthors Info & Claims

SPM '07: Proceedings of the 2007 ACM symposium on Solid and physical modeling

Pages 233 - 240

https://doi.org/10.1145/1236246.1236280

Published: 04 June 2007 Publication History

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Abstract

In geometry processing a refined control mesh is often used to approximate a Catmull-Clark subdivision surface (CCSS). By pushing the control points to their limit positions, a limit mesh of the subdivision surface is obtained. We present 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. The bound shows that the limit mesh may approximate the limit surface better than the corresponding control mesh in general. Consequently, for a given error tolerance, fewer subdivision steps are needed if the refined control mesh is replaced with the corresponding limit mesh.

References

[1]

Catmull, E., and Clark, J. 1978. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 6, 350--355.

Crossref

Google Scholar

[2]

Chen, G., and Cheng, F. 2006. Matrix based subdivision depth computation for extra-ordinary Catmull-Clark subdivision surface patches. In Proceedings of GMP 2006 (LNCS 4077), 545--552.

Digital Library

Google Scholar

[3]

Cheng, F., and Yong, J. 2006. Subdivision depth computation for Catmull-Clark subdivision surfaces. Computer Aided Design & Applications 3, 1--4, 485--494.

Crossref

Google Scholar

[4]

Cheng, F., Chen, G., and Yong, J. 2006. Subdivision depth computation for extra-ordinary Catmull-Clark subdivision surface patches. In Proceedings of CGI 2006 (LNCS 4035), 404--416.

Digital Library

Google Scholar

[5]

Doo, D., and Sabin, M. A. 1978. Behaviour of recursive subdivision surfaces near extraordinary points. Computer-Aided Design 10, 6, 356--360.

Crossref

Google Scholar

[6]

Farin, G. 2002. Curves and Surfaces for Computer-Aided Geometric Design --- A Practical Guide, fifth ed. Morgan Kaufmann Publishers.

Digital Library

Google Scholar

[7]

Halstead, M., Kass, M., and DeRose, T. 1993. Efficient, fair interpolation using Catmull-Clark surfaces. In Proceedings of SIGGRAPH 93, ACM, 35--44.

Digital Library

Google Scholar

[8]

Hoppe, H., DeRose, T., Duchamp, T., Halstead, M., Jin, H., McDonald, J., Schweitzer, J., and Stuetzle, W. 1994. Piecewise smooth surface reconsruction. In Proceedings of SIGGRAPH 94, ACM, 295--302.

Digital Library

Google Scholar

[9]

Huang, Z., and Wang, G. 2006. Improved error estimate for extraordinary Catmull-Clark subdivision surface patches. Submitted, available at http://graphics.pku.edu.cn/hzj/pub.htm.

Google Scholar

[10]

Loop, C. T. 1987. Smooth subdivision surfaces based on triangles. Master's thesis, Department of Mathematics, University of Utah.

Google Scholar

[11]

Stam, J. 1998. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In Proceedings of SIG-GRAPH 98, Annual Conference Series, ACM, 395--404.

Digital Library

Google Scholar

Cited By

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Abdul Karim SKhan FMustafa GShahzad AAsghar M(2023)An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision SchemesMathematics10.3390/math1111244911:11(2449)Online publication date: 25-May-2023
https://doi.org/10.3390/math11112449
Cashman T(2012)Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface MethodsComputer Graphics Forum10.1111/j.1467-8659.2011.02083.x31:1(42-61)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1111/j.1467-8659.2011.02083.x
Peters JWu X(2009)The distance of a subdivision surface to its control polyhedronJournal of Approximation Theory10.1016/j.jat.2008.10.012161:2(491-507)Online publication date: 1-Dec-2009
https://dl.acm.org/doi/10.1016/j.jat.2008.10.012
Show More Cited By

Index Terms

Distance between a Catmull-Clark subdivision surface and its limit mesh
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models

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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 ...

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Information

Published In

SPM '07: Proceedings of the 2007 ACM symposium on Solid and physical modeling

June 2007

455 pages

ISBN:9781595936660

DOI:10.1145/1236246

Program Chairs:
Bruno Levy
INRIA
,
Dinesh Manocha
University of North Carolina

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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SIGGRAPH: ACM Special Interest Group on Computer Graphics and Interactive Techniques

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 04 June 2007

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SPM07

Sponsor:

Tsinghua University

SPM07: Symposium on Solid and Physical Modeling

June 4 - 6, 2007

Beijing, China

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Cited By

View all

Abdul Karim SKhan FMustafa GShahzad AAsghar M(2023)An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision SchemesMathematics10.3390/math1111244911:11(2449)Online publication date: 25-May-2023
https://doi.org/10.3390/math11112449
Cashman T(2012)Beyond Catmull–Clark? A Survey of Advances in Subdivision Surface MethodsComputer Graphics Forum10.1111/j.1467-8659.2011.02083.x31:1(42-61)Online publication date: 1-Feb-2012
https://dl.acm.org/doi/10.1111/j.1467-8659.2011.02083.x
Peters JWu X(2009)The distance of a subdivision surface to its control polyhedronJournal of Approximation Theory10.1016/j.jat.2008.10.012161:2(491-507)Online publication date: 1-Dec-2009
https://dl.acm.org/doi/10.1016/j.jat.2008.10.012
Jiang DStewart N(2009)Robustness of Boolean Operations on Subdivision-Surface ModelsNumerical Validation in Current Hardware Architectures10.1007/978-3-642-01591-5_10(161-174)Online publication date: 29-Apr-2009
https://dl.acm.org/doi/10.1007/978-3-642-01591-5_10
Huang ZWang G(2008)Bounding the distance between a loop subdivision surface and its limit meshProceedings of the 5th international conference on Advances in geometric modeling and processing10.5555/1792279.1792283(33-46)Online publication date: 23-Apr-2008
https://dl.acm.org/doi/10.5555/1792279.1792283
Huang ZWang G(2008)Bounding the Distance between a Loop Subdivision Surface and Its Limit MeshAdvances in Geometric Modeling and Processing10.1007/978-3-540-79246-8_3(33-46)Online publication date: 2008
https://doi.org/10.1007/978-3-540-79246-8_3
Huang ZWang G(2007)Improved error estimate for extraordinary Catmull–Clark subdivision surface patchesThe Visual Computer10.1007/s00371-007-0173-023:12(1005-1014)Online publication date: 8-Aug-2007
https://doi.org/10.1007/s00371-007-0173-0

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Approximating Catmull-Clark subdivision surfaces with bicubic patches

Feature-adaptive GPU rendering of Catmull-Clark subdivision surfaces

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