Topological Quadrangulations of Closed Triangulated Surfaces Using the Reeb Graph

  • Franck Hétroy
  • Dominique Attali
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2301)


Although surfaces are more and more often represented by dense triangulations, it can be useful to convert them to B-spline surface patches, lying on quadrangles. This paper presents a method to construct coarse topological quadrangulations of closed triangulated surfaces, based on theoretical results about topological classification of surfaces and Morse theory. In order to compute a canonical set of generators, a Reeb graph is constructed on the surface using Dijkstra’s algorithm. Some results are shown on different surfaces.


Subdivision Scheme Morse Theory Morse Function Triangulate Surface Canonical Generator 
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.


  1. 1.
    S. Biasotti, B. Falcidieno, M. Spagnuolo. Extended Reeb Graphs for Surface Understanding and Description. Proceedings of DGCI’00, Lecture Notes in Computer Science, Vol. 1953, pp.185–197, 2000.Google Scholar
  2. 2.
    H. Carr, J. Snoeyink, U. Axen. Computing Contour Trees in All Dimensions. Proceedings of ACM 11th Symposium on Discrete Algorithms, pp. 918–926. San Francisco, California, USA, Jan. 2000.Google Scholar
  3. 3.
    T. Dey, H. Schipper. A new Technique to Compute Polygonal Schema for 2-Manifolds with Application to Null-Homotopy Detection. Discrete and Computational Geometry,14:93–110, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    T. Dey, S. Guha. Computing Homology Groups of Simplicial Complexes in IR3. Journal of ACM, Vol. 45, No. 2, pp. 266–287, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Eck, H. Hoppe. Automatic Reconstruction of B-Spline Surfaces Of Arbitrary Topological Type. Proceedings of SIGGRAPH’96, pp. 325–334, August 1996.Google Scholar
  6. 6.
    A.T. Fomenko, T.L. Kunii. Topological Modeling for Visualization. Springer, 1997.Google Scholar
  7. 7.
    M. Gondran, M. Minoux. Graphs and Algorithms. Wiley, 1995.Google Scholar
  8. 8.
    N. Hartsfield, G. Ringel. Minimal Quadrangulations of Orientable Surfaces. Journal of Combinatorial Theory, Series B, Vol. 46, No. 1, pp. 84–95, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    P.S. Heckbert, M. Garland. Survey of Polygonal Surface Simplification Algorithms. Multiresolution Surface Modeling Course, SIGGRAPH’97, 1997.Google Scholar
  10. 10.
    M. Hilaga, Y. Shinigawa, T. Kohmura, T.L. Kunii. Topology Matching for Fully Automatic Similarity Estimation of 3D Shapes. Proceedings of SIGGRAPH’01, August 2001.Google Scholar
  11. 11.
    M. van Kreveld, R. van Ostrum, C. Bajaj, V. Pascucci, D. Schikore. Contour Trees and Small Seed Sets for Isosurface Traversal. Proceedings of ACM 13th Symposium on Computational Geometry, pp. 212–220. Nice, France, June 1997.Google Scholar
  12. 12.
    V. Krishnamurthy, M. Levoy. Fitting Smooth Surfaces to Dense Polygon Meshes. Proceedings of SIGGRAPH’96, pp. 313–324, August 1996.Google Scholar
  13. 13.
    F. Lazarus, A. Verroust. Level Set Diagrams of Polyhedral Objects. Proceedings of ACM 5th Symposium on Solid Modeling and Applications, pp. 130–140. Ann Arbor, Michigan, USA, June 1999.Google Scholar
  14. 14.
    F. Lazarus, M. Pocchiola, G. Vegter, A. Verroust. Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface. Proceedings of ACM 17th Symposium on Computational Geometry, pp. 80–89. Tufts University, Medford, USA, June 2001.Google Scholar
  15. 15.
    S. Owen. A Survey of Unstructured Mesh Generation Technology. Proceedings of the 7th International Meshing Roundtable, Sandia National labs, pp. 239–267. Dearborn, Michigan, U.S.A., October 1998.Google Scholar
  16. 16.
    G. Reeb. Sur les Points Singuliers d’une Forme de Pfa. Complètement Intégrable ou d’une Fonction Numérique. Comptes Rendus Acad. Sciences, Paris, France, 222:847–849, 1946.zbMATHMathSciNetGoogle Scholar
  17. 17.
    Y. Shinagawa, T.L. Kunii. Surface Coding based on Morse Theory. IEEE Computer Graphics and Applications, pp. 66–78, September 1991.Google Scholar
  18. 18.
    S. Takahashi, Y. Shinagawa, T.L. Kunii. A Feature-based Approach for Smooth Surfaces. Proceedings of ACM 4th Symposium on Solid Modeling and Applications, pp. 97–110. Atlanta, Georgia, USA, May 1997.Google Scholar
  19. 19.
    G. Vegter, C.K. Yap. Computational Complexity of Combinatorial Surfaces. Proceedings of ACM 6th Symposium on Computational Geometry, pp. 102–111. Berkeley, California, USA, June 1990.Google Scholar
  20. 20.
    Z. Wood. Semi-Regular Mesh Extraction from Volumes. Master’s thesis, Caltech, Pasadena, California, USA, 2000.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Franck Hétroy
    • 1
  • Dominique Attali
    • 1
  1. 1.LIS LaboratoryDomaine UniversitaireSaint Martin d’Hères cedexFrance

Personalised recommendations