A Meta Scheme for Iterative Refinement of Meshes

  • Markus Kohler


Subdivision is a powerful method for smooth visualization of coarse meshes. An initial mesh of arbitrary topology is subsequently subdivided until the surface appears smooth and visually pleasing. Because of the simplicity of the algorithms and their data structure they have achieved much attention. Most schemes exhibit undesirable artifacts in the case of irregular topology. A more general description of subdivision algorithms is required to be able to avoid artifacts by changing parameters and also the topology of subdivision. We develop a partition of the algorithms to sub-algorithms and combinatoric mappings. An algebraic notation enables the description of calculations on the mesh and steering the algorithm by a configuration file. Opposite to index based notation of tensor-product surfaces the algebraic notation is even applicable for irregular and non-tensor-product surfaces. Variations of these sub-algorithms and mappings lead to known and new subdivision schemes. Further, the unified refinement scheme makes the efficient depth-first subdivision available to all these subdivision schemes.


Type Relation Subdivision Scheme Configuration File Dual Graph Initial Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    S. ABRAMOWSKI AND H. MÜLLER, Geometrisches Modellieren, Reihe Informatik, B. I. Wissenschaftsverlag, Mannheim, 1991.Google Scholar
  2. 2.
    G. R. BADURA, Generierung und Implementierung neuer Zuordnungsverfahren mittels eines kombinatorischen Ansatzes durch die objektorientierte Analyse der Unterteilungsverfahren von Catmull-Clark und Doo-Sabin, Master’s thesis, Lehrstuhl Informatik VII, Universität Dortmund, Germany, May 1996.Google Scholar
  3. 3.
    W. BOEHM, H. PRAUTZSCH, AND ARNER, On triangular splines, Constructive Approximation 3 (1987), 157–167.MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    E. CATMULL AND J. CLARK, Recursively generated B-spline surfaces on arbitrary topological meshes,Computer Aided Design 10:6 (1978), 350 – 355.Google Scholar
  5. 5.
    A. S. CAVARETTA, W. DAHMEN, AND C. A. MICCHELLI, Stationary subdivision, Memoirs of the American Mathematical Society 93: 453 (1991), 1–186.Google Scholar
  6. 6.
    G. M. CHAIKIN, An algorithm for high speed curve generation, Computer Graphics and Image Processing 3 (1974), 346–349.CrossRefGoogle Scholar
  7. 7.
    D. Doo AND M. SABIN, Behaviour of recursive division surfaces near extraordinary points, Computer Aided Design 10 (1978), 356–360.CrossRefGoogle Scholar
  8. 8.
    N. DYN AND D. LEVIN, Subdivision Schemes for Surface Interpolation, Workshop in Computational Geometry (1993), 97–118.Google Scholar
  9. 9.
    N. DYN, D. LEVIN, AND C. A. MICCHELLI, Using parameters to increase smoothness of curves and surfaces generated by subdivision, Computer Aided Geometric Design 7 (1990), 129–140.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    G. FARIN, Curves and Surfaces for CAGD, 3rd edition ed., Academic Press, San Diego, 1993.Google Scholar
  11. 11.
    M. HALSTEAD, M. KASS, AND T. DERosE, Efficient, Fair Interpolation using Catmull-Clark Surfaces, ACM Computer Graphics Proceedings, Annual Conference Series, August 1993, pp. 35–44.Google Scholar
  12. 12.
    L. KOBBELT, Iterative Erzeugung glatter Interpolanten, Dissertation, Fakultät für Informatik, Universität Karlsruhe, December 1994.Google Scholar
  13. 13.
    M. KOHLER AND H. MÜLLER, Efficient calculation of subdivision surfaces for visualization, Visualization and Mathematics (H.-C. HEGE AND K. POLTHIER, eds.), Springer, Heidelberg, 1997, pp. 165–179.Google Scholar
  14. 14.
    C. Loop, Smooth Subdivision Surfaces Based on Triangles, Master’s thesis, Department of Mathematics, University of Utah, 1987.Google Scholar
  15. 15.
    A. H. NASRI, Polyhedral subdivision methods for free-form surfaces, ACM Transactions on Graphics 6: 1 (1987), 29–73.MATHCrossRefGoogle Scholar
  16. 16.
    H. PRAUTZSCH AND L. KOBBELT, Convergence of subdivision and degree elevation, Advances in Computational Mathematics 2: 2 (1994), 143–154.MathSciNetGoogle Scholar
  17. 17.
    J. A. ROULIER, A convexity preserving grid refinement algorithm for interpolation of bivariate functions, IEEE CG & A 7: 1 (1987), 57–62.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Markus Kohler
    • 1
  1. 1.Lehrstuhl für Graphische SystemeUniversität DortmundGermany

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