Abstract
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.
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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.
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.
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.
T. Dey, S. Guha. Computing Homology Groups of Simplicial Complexes in IR3. Journal of ACM, Vol. 45, No. 2, pp. 266–287, 1998.
M. Eck, H. Hoppe. Automatic Reconstruction of B-Spline Surfaces Of Arbitrary Topological Type. Proceedings of SIGGRAPH’96, pp. 325–334, August 1996.
A.T. Fomenko, T.L. Kunii. Topological Modeling for Visualization. Springer, 1997.
M. Gondran, M. Minoux. Graphs and Algorithms. Wiley, 1995.
N. Hartsfield, G. Ringel. Minimal Quadrangulations of Orientable Surfaces. Journal of Combinatorial Theory, Series B, Vol. 46, No. 1, pp. 84–95, 1989.
P.S. Heckbert, M. Garland. Survey of Polygonal Surface Simplification Algorithms. Multiresolution Surface Modeling Course, SIGGRAPH’97, 1997.
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.
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.
V. Krishnamurthy, M. Levoy. Fitting Smooth Surfaces to Dense Polygon Meshes. Proceedings of SIGGRAPH’96, pp. 313–324, August 1996.
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.
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.
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.
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.
Y. Shinagawa, T.L. Kunii. Surface Coding based on Morse Theory. IEEE Computer Graphics and Applications, pp. 66–78, September 1991.
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.
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.
Z. Wood. Semi-Regular Mesh Extraction from Volumes. Master’s thesis, Caltech, Pasadena, California, USA, 2000.
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Hétroy, F., Attali, D. (2002). Topological Quadrangulations of Closed Triangulated Surfaces Using the Reeb Graph. In: Braquelaire, A., Lachaud, JO., Vialard, A. (eds) Discrete Geometry for Computer Imagery. DGCI 2002. Lecture Notes in Computer Science, vol 2301. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45986-3_5
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DOI: https://doi.org/10.1007/3-540-45986-3_5
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