Subdivision as a Sequence of Sampled Cp Surfaces

  • Cédric Gérot
  • Loïc Barthe
  • Neil A. Dodgson
  • Malcolm Sabin
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


This article deals with practical conditions for tuning a subdivision scheme in order to control its artifacts in the vicinity of a mark point. To do so, we look for good behaviour of the limit vertices rather than good mathematical properties of the limit surface. The good behaviour of the limit vertices is characterised with the definition of C2-convergence of a scheme. We propose necessary explicit conditions for C2-convergence of a scheme in the vicinity of any mark point being a vertex of valency greater or equal to three.


Limit Surface Mark Point Subdivision Scheme Polygonal Mesh Subdivision Surface 
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  1. 1.
    A. A. Ball and D. J. T. Storry. Conditions for tangent plane continuity over recursively generated B-spline surfaces. ACM Transactions on Graphics, 7(2):83–102, 1988.CrossRefGoogle Scholar
  2. 2.
    L. Barthe, C. Gérot, M. A. Sabin, and L. Kobbelt. Simple computation of the eigencomponents of a subdivision matrix in the fourier domain. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 245–257 (this book). Springer-Verlag, 2004.Google Scholar
  3. 3.
    L. Barthe and L. Kobbelt. Subdivision scheme tuning around extraordinary vertices. Submitted.Google Scholar
  4. 4.
    E. Catmull and J. Clark. Recursively generated B-spline surfaces on arbitrary topological meshes. Computer Aided Design, 10(6):350–355, 1978.CrossRefGoogle Scholar
  5. 5.
    D. Doo and M. A. Sabin. Behaviour of recursive division surface near extraordinary points. Computer Aided Design, 10(6):356–360, 1978.CrossRefGoogle Scholar
  6. 6.
    C. Gérot, L. Barthe, N. A. Dodgson, and M. A. Sabin. Subdivision as sequence of sampled Cp-surfaces and conditions for tuning schemes. Technical Report 583, University of Cambridge Computer Laboratory, Mar 2004. Scholar
  7. 7.
    C. T. Loop. Smooth subdivision surfaces based on triangles. Master's thesis, University of Utah, 1987.Google Scholar
  8. 8.
    J. Peters and U. Reif. Analysis of algorithms generalizing B-spline subdivision. SIAM J. Numer. Anal., 35(2):728–748, 1998.MathSciNetCrossRefGoogle Scholar
  9. 9.
    H. Prautzsch. Smoothness of subdivision surfaces at extraordinary points. Adv. Comput. Math, 9:377–389, 1998.zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    H. Prautzsch and G. Umlauf. Improved triangular subdivision schemes. In Proc. Computer Graphics International, pages 626–632, 1998.Google Scholar
  11. 11.
    U. Reif. A unified approach to subdivision algorithm near extraordinary vertices. Computer Geometric Aided Design, 12:153–174, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    M. A. Sabin. Eigenanalysis and artifacts of subdivision curves and surfaces. In A. Iske, E. Quak, and M. S. Floater, editors, Tutorials on Multiresolution in Geometric Modelling, pages 69–97. Springer-Verlag, 2002.Google Scholar
  13. 13.
    M. A. Sabin. Recent progress in subdivision — a survey. In N. A. Dodgson, M. S. Floater, and M. A. Sabin, editors, Advances in Multiresolution for Geometric Modelling, pages 203–230 (this book). Springer-Verlag, 2004.Google Scholar
  14. 14.
    M. A. Sabin and L. Barthe. Artifacts in recursive subdivision surfaces. In A. Cohen, J.-L. Merrien, and L. L. Schumaker, editors, Curve and Surface Fitting: Saint-Malo 2002, pages 353–362. Nashboro Press, 2003.Google Scholar
  15. 15.
    J. Stam. Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. Proc. ACM SIGGRAPH '98, pages 395–404, 1998.Google Scholar
  16. 16.
    D. Zorin. Stationary subdivision and multiresolution surface representations. PhD thesis, California Institute of Technology, 1997Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Cédric Gérot
    • 1
  • Loïc Barthe
    • 2
  • Neil A. Dodgson
    • 3
  • Malcolm Sabin
    • 4
  1. 1.Laboratoire des Images et des SignauxDomaine UniversitaireGrenobleFrance
  2. 2.Computer Graphics Group, IRIT/UPSToulouseFrance
  3. 3.Computer LaboratoryUniversity of CambridgeUK
  4. 4.Numerical Geometry Ltd.CambridgeUK

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