Another Metascheme of Subdivision Surfaces

  • Heinrich Müller
  • Markus Rips
Part of the Mathematics and Visualization book series (MATHVISUAL)


A subdivision surface is defined by a polygonal mesh which is iteratively refined into an infinite sequence of meshes converging to the desired smooth surface. A framework of systematic classification and construction of subdivision schemes is presented which is based on a single operation, the calculation of the average of the vertices incident to a vertex, edge, or face. More complex subdivision schemes are constructed by concatenation of a collection of elementary subdivision schemes. Properties of these schemes are discussed. In particular it is shown how known subdivision schemes fit into this framework.


subdivision metascheme sudivision surfaces geometric modelling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. de Boor, K. Höllig, S.D. Riemenschneider, Box Splines, Springer-Verlag, New-York, 1993zbMATHCrossRefGoogle Scholar
  2. 2.
    E. Catmull, J. Clark, Recursively generated B-spline surfaces of arbitrary topological meshes, Computer Aided Design 10 (6) (1978) 350–355CrossRefGoogle Scholar
  3. 3.
    G.M. Chaikin, Art algorithm for high speed curve generation, Computer Graphics and Image Processing 3 (1974) 346 349Google Scholar
  4. 4.
    D. Doo, M. Sabin, Behaviour of recursive division surfaces near extraordinary points, Computer Aided Design 10 (6) (1978) 356–360CrossRefGoogle Scholar
  5. 5.
    N. Dyn, D. Levin, J.A. Gregory, A butterfly subdivision scheme for surface interpolation with tension control ACM Trans. on Graphics 9(2) (1990) 160169Google Scholar
  6. 6.
    G. Farin, Curves and Surfaces for CAGD, 3rd edition, Academic Press, 1993Google Scholar
  7. 7.
    A. Habib, J. Warren, Edge and vertex insertion for a class of C l subdivision surfaces, Comp. Aided Geom. Des. 16 (4) (1999) 223–247MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    L. Kobbelt, subdivision, Proc. SIGGRAPH 2000, 103–112, 2000zbMATHGoogle Scholar
  9. 9.
    M. Kohler, A Meta Scheme for Interactive Refinement of Meshes, Visualization and Mathematics (Ch. Hege, K. Polthier, eds. ), Springer-Verlag, 1998Google Scholar
  10. 10.
    U. Labsik, G. G.einer, Interpolatory 0-subdivision, Proc. Eurographics 2000, Computer Graphics Forum 19 (3) (2000) 131–138CrossRefGoogle Scholar
  11. 11.
    C. Loop, Smooth Subdivision Surfaces Based on Triangles, Master’s Thesis, Department of Mathematics, University of Utah, 1987Google Scholar
  12. 12.
    H. Müller, R. Jaeschke, Adaptive Subdivision Curves and Surfaces, Proc. Computer Graphics International 1998 (CGI’98), IEEE Computer Society Press, 1998, 48–58Google Scholar
  13. 13.
    H. Müller, M. Rips, Another Metascheme of Subdivision Surfaces, Research Report 713, Department of Computer Science, Univ. of Dortmund, Germany, 1999Google Scholar
  14. 14.
    P. Oswald, P. Schröder, Composite primal/dual 0-subdivision schemes submitted, Scholar
  15. 15.
    J. Peters, U. Reif, The simplest subdivision scheme for smoothing polyhedra, ACM Trans. on Graphics 16 (4), 1997, 420–431CrossRefGoogle Scholar
  16. 16.
    H. Prautzsch, Smoothness of subdivision surfaces at extraordinary points, Advances in Computational Mathematics 9 (1998) 377–389MathSciNetCrossRefGoogle Scholar
  17. 17.
    J. Stain, On subdivision schemes generalizing uniform B-spline surfaces of arbitrary degree, Comp. Aided Geom. Des. 18 (5) (2001) 383–396zbMATHGoogle Scholar
  18. 18.
    G. Taubin, Dual mesh resampling, Proc. Pacific Graphics 2001, 2001Google Scholar
  19. 19.
    G. Umlauf, Smooth free-form surfaces and optimized subdivision algorithms (in German), Shaker Verlag, 1999Google Scholar
  20. 20.
    L. Velho, Quasi 4–8 subdivision, Comp. Aided Geom. Des. 18 (4) (2001) 345–357zbMATHGoogle Scholar
  21. 21.
    L. Velho, D. Zorin, 4–8 subdivision, Comp. Aided Geom. Des. 18 (5) (2001) 397–422MathSciNetzbMATHGoogle Scholar
  22. 22.
    D. Zorin, P. Schröder, A unified framework for primal/dual quadrilateral subdivision surfaces, Comp. Aided Geom. Des. 18 (5) (2001) 429–454MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Heinrich Müller
    • 1
  • Markus Rips
    • 1
  1. 1.Informatik VIIUniversity of DortmundDortmundGermany

Personalised recommendations