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Aspects in topology-based geometric modeling Possible tools for discrete geometry?

  • Pascal Lienhardt
Invited Speakers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1347)

Abstract

Topology-based Geometric Modeling is concerned with modeling subdivisions of geometric spaces. Methods are close to that of combinatorial topology, but for different purposes. We discuss some of these methods, their interests and drawbacks for Geometric Modeling, mainly aspects we think that could be of possible interest for Discrete Geometry.

Keywords

Geometric Modeling Boundary Representation Algebraic Topology Topological Information Combinatorial Structure 
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.

References

  1. [Ago]
    Agoston, M.: Algebraic Topology: A First Course. Pure and Applied Mathematics, Marcel Dekker Ed., New-York, U.S.A. (1976)Google Scholar
  2. [AFF]
    Ansaldi, S., de Floriani, L., Falcidieno, B.: Geometric Modeling of Solid Objects by Using a Face Adjacency Graph Representation. Computer Graphics 19,3 (1985) 131–139Google Scholar
  3. [Ale]
    Alexandroff: Combinatorial Topology. Graylock Press, Rochester, New-York, USA (1957)Google Scholar
  4. [ArKo]
    Arquèus, D., Koch, P.: Modélisation de solides par les pavages. Proc. of Pixim'89, Paris, France (1989) 47–61Google Scholar
  5. [Bau]
    Baumgart, B.: A Polyhedron Representation for Computer Vision. AMPS Nat. Conf. Proc. 44 (1975) 589–596Google Scholar
  6. [BD]
    Bertrand, Y., Dufourd, J.-F.: Algebraic specification of a 3D-modeler based on hypermaps. CGVIP, 1 (1994) 29–60Google Scholar
  7. [BDFL]
    Bertrand, Y., Dufourd, J.-F., Françon, J., Lienhardt, P.: Algebraic Specification and Development in Geometric Modeling. Proc. of TAPSOFT'93 Orsay, France (1993)Google Scholar
  8. [BeLi]
    Bertrand, Y., Lienhardt, P.: Spécification algébrique et sortes ordonnées pour un multi-modeleur d'extensions des cartes. Journées du GDR/PRC de Programmation, Orléans, France (1996).Google Scholar
  9. [BF]
    Bertrand, Y., Françon, J.: 2D manifolds for Boundary Representation: A statistical study of the cells. Research Report R97-8, Université Louis Pasteur, Strasbourg, France (1997)Google Scholar
  10. [Bor]
    Borianne, P.: Conception of a Modeller of Subdivisions of Surfaces based on Generalized Maps. PhD Thesis 037, Université Louis Pasteur, Strasbourg, France (1991)Google Scholar
  11. [Bra]
    Braid, I.: The Synthesis of Solids Bounded by Many Faces. Communications of the A.C.M. 18,4 (1975) 209–216Google Scholar
  12. [BrGu]
    Braquelaire, J.-P., Guitton, P.: A Model for Image Structuration. Proc. of Computer Graphics International'88 Genève, Switzerland (1988)Google Scholar
  13. [Bri]
    Brisson, E.: Representing Geometric Structures in D Dimensions: Topology and Order. Discrete and Computational Geometry 9 (1993) 387–426Google Scholar
  14. [Bru]
    Brun, L.: Segmentation d'images à base topologique. PhD Thesis 1651, Université de Bordeaux 1, Bordeaux, France (1996)Google Scholar
  15. [BrSi]
    Bryant, R., Singerman, D.: Foundations of the Theory of Maps on Surfaces with Boundaries. Quart. Journal of Math. Oxford 2,36 (1985) 17–41Google Scholar
  16. [CCM]
    Cavalcanti, P.R.. Carvalho, P.C.P., Martha, L.F.: Non-manifold modeling: an approach based on spatial subdivision. Computer-Aided Design 29, 3 (1997) 299–220Google Scholar
  17. [Cor]
    Cori, R. Un code pour les graphes planaires et ses applications. Astérisque 27 (1975)Google Scholar
  18. [CrRe]
    Crocker, G., Reinke, W.: An Editable Non-Manifold Boundary Representation. Computer Graphics and Applications 11,2 (1991)Google Scholar
  19. [Dola]
    Dobkin, D., Laszlo, M.: Primitives for the Manipulation of Three-Dimensional Subdivisions. Proc. of 3rd Symposium on Computational Geometry Waterloo, Canada (1987) 86–99Google Scholar
  20. [Edm]
    Edmonds, J.: A Combinatorial Representation for Polyhedral Surfaces. Notices Amer. Math. Soc. 7 (1960)Google Scholar
  21. [ElLil]
    Elter, H., Lienhardt, P.: Different Combinatorial Models based on the Map Concept for the Representation of Different Types of Cellular Complexes. IFIP TC5/WG5.10 Working Conference on Geometric Modeling in Computer Graphics, Genova, Italy (1993)Google Scholar
  22. [ElLi2]
    Etter, H., Lienhardt, P.: Cellular complexes as structured semi-simplicial sets. Int. Journal of Shape Modeling, 1,2 (1994) 191–217Google Scholar
  23. [FB]
    Françon, J., Bertrand, Y.: 3D manifolds for Boundary Representation: A statistical study of the cells. Research Report R96-9, Université Louis Pasteur, Strasbourg, France (1997)Google Scholar
  24. [FePa]
    Ferruci, V., Paoluzzi, A.: Extrusion and Boundary Evaluation for Multidimensional Polyhedra. Computer-Aided Design 23,1 (1991) 40–50Google Scholar
  25. [Fey]
    Fey, C.: Etude de plongements hiérarchisés de subdivisions simpliciales ou cellulaires. PhD Thesis, Université Louis Pasteur, Strasbourg, France (1996)Google Scholar
  26. [Fio]
    Fiorio, C.: A Topologically Consistent Representation for Image Analysis: The Frontiers Topological Graph. Proc. of DGCI'96, Lyon, France (1996) 151–162Google Scholar
  27. [FBDFR]
    Fousse, A., Bertrand, Y., Dufourd, J.-F., Françon, J., Rodriguès, D.: Localisation des points d'un maillage généré en vue de calculs en différences finies. Journées “Modélisation du sous-soi”, Orléans, France (1997).Google Scholar
  28. [Fra]
    Françon, J.: On Recent Trends in Discrete Geometry in Computer Science. Proc. DGCI'96, Lyon, France (1996) 3–16Google Scholar
  29. [FrPi]
    Fritsch, R., Piccinini, R. A.: Cellular Structures in Topology. Cambridge University Press (1990)Google Scholar
  30. [FuLi]
    Fuchs, L., Lienhardt, P.: Topological Structures for d-Dimensional Free-Form Objects. CAGD'97, Lillehammer, Norway (1997)Google Scholar
  31. [GHPT]
    Gangnet, M., Hervé, J.-C., Pudet, T., Van Thong, J.-M.: Incremental Computation of Planar Maps. Computer Graphics 23,3 (1989) 345–354Google Scholar
  32. [Gib]
    Giblin, P.J.: Graphs, surfaces and homology. Chapman and Hall, London, UK (1977)Google Scholar
  33. [Gri]
    Griffiths, H.-B. Surfaces. Cambridge University Press Cambridge, U.K. (1981)Google Scholar
  34. [Gust]
    Guibas, L., Stolfi, J.: Primitives for the Manipulation of General Subdivisions and the Computation of Voronoï Diagrams. A.C.M. Transactions on Graphics 4,2 (1985) 74–123Google Scholar
  35. [GCP]
    Gursoz, E. L., Choi, Y., Prinz, F. B.: Vertex-based Representation of Non-Manifolds Boundaries. in Geometric Modeling for Product Engineering M. Wozny, J. Turner and K. Preiss eds., North-Holland (1989) 107–130Google Scholar
  36. [HC]
    Hansen, O.H., Christensen, N.J.: A model for n-dimensional boundary topology. Proc. 2nd ACM Symp. Solid Modeling Foundations and CAD/CAM Applications (J. Rossignac and J. Turner eds) Montréal, Canada (1993) 65–73Google Scholar
  37. [HCRR]
    Halbwachs, Y., Courrioux, G., Renaud, X., Repusseau, P.: Topological and Geometric Characterization of Fault Networks Using 3-Dimensional Generalized Maps. Mathematical Geology, 28, 5 (1996) 625–656Google Scholar
  38. [Jac]
    Jacque, A.: Constellations et Graphes Topologiques. Colloque Math. Soc. Janos Bolyai (1970) 657–672Google Scholar
  39. [KIE]
    Kenmochi, Y., Imiya, A., Ezquerra, N.: Polyhedra generation from lattice points. Proc. DGCI'96, Lyon, France (1996) 127–138Google Scholar
  40. [Lac]
    Lachaud, J.-O.: Topologically Defined Isosurfaces. Proc. of DGCI'96, Lyon, France (1996) 245–256Google Scholar
  41. [LaLil]
    Lang, V., Lienhardt, P.: Geometric Modeling with Simplicial Sets. Proc. of CGI'96, Pohang, Korea (1996)Google Scholar
  42. [LaLi2]
    Lang, V., Lienhardt, P.: Cartesian Product of Simplicial Sets. Proc. of WSCG'97, Plzen, Czech Republic (1997)Google Scholar
  43. [Liel]
    Lienhardt, P.: Topological Models for Boundary Representation: a Comparison with N-Dimensional Generalized Maps. Computer-Aided Design 23,1 (1991) 59–82Google Scholar
  44. [Lie2]
    Lienhardt, P.: N-Dimensional Generalized Combinatorial Maps and Cellular Quasi-Manifolds. Int. Journal of Computational Geometry and Applications 4,3 (1994) 275–324Google Scholar
  45. [LuLu]
    Luo, Y., Lukacs, G. A.: A Boundary Representation for Form-Features and Non-Manifold Solid Objects. Proc. of 1st ACM/Siggraph Symposium on Solid Modeling Foundations and CAD/CAM Applications Austin, Texas, U.S.A. (1990)Google Scholar
  46. [Män]
    Mäntylä, M.: An Introduction to Solid Modeling. Computer Science Press Rockville, U.S.A. (1988)Google Scholar
  47. [May]
    May, J. P.: Simplicial Objects in Algebraic Topology. Van Nostrand Princeton (1967)Google Scholar
  48. [Mar]
    Marcheix, D.: Modélisation géométrique d'objets non-variétés: construction, représentation and manipulation. PhD Thesis, Université de Bordeaux I, Bordeaux, France (1996)Google Scholar
  49. [Mil]
    Milnor: The geometric realization of semi-Simplicial complexes. Ann. of Math. 65 (1957) 357–362Google Scholar
  50. [MLCF]
    Mallet, J.-L., Levy, B., Conreaux, S., Fousse, A.: G-maps, a new topological model for gOcad. 15th Gocad Meeting, Nancy, France (1997)Google Scholar
  51. [MuHi]
    Murabata, S., Higashi, M.: Non-manifold geometric modeling for set operations and surface operations. IFIP/RPI Geometric Modeling Conference Rensselaerville, N.Y. (1990)Google Scholar
  52. [PBCF]
    Paoluzzi, A., Bernardini, F., Cattani, C., Ferrucci, V.: Dimension Independent Modeling with Simplicial Complexes. ACM Transactions on Graphics, 12,1 (1993)Google Scholar
  53. [RoOC]
    Rossignac, J., O'Connor, M.: SGC: A Dimension-Independent Model for Pointsets with Internal Structures and Incomplete Boundaries. in Geometric Modeling for Product Engineering M. Wozny, J. Turner and K. Preiss eds., North-Holland (1989) 145–180Google Scholar
  54. [Ser]
    Serre,J.-P.: Homologie singulière des espaces fibrés. Ann. of Math. 54 (1951) 425–505Google Scholar
  55. [SeTh]
    Seifert, H., Threlfall, W.: A textbook of topology. Academic Press New York, (1980)Google Scholar
  56. [Som]
    Sommellier, L.: Mise en correspondance d'images stéréoscopiques utilisant un modèle topologique. PhD Thesis, Université Claude Bernard, Lyon, France (1997)Google Scholar
  57. [Spe]
    Spehner, J.-C.: Merging in Maps and Pavings. Theoretical Computer Science 86, (1991) 205–232Google Scholar
  58. [Tak]
    Takala, T.: A taxonomy of geometric and topological models. Computer Graphics and Mathematics (B. Falcidieno, I. Herman and C. Pienovi eds.), Springer (1992)Google Scholar
  59. [Tut]
    Tutte, W. Graph Theory. Encyclopaedia of Mathematics and its Applications, Addison Wesley, Menlo Park, U.S.A. (1984)Google Scholar
  60. [Vin]
    Vince, A.: Combinatorial Maps. Journal of Combinatorial Theory Series B 34 (1983) 1–21Google Scholar
  61. [Weil]
    Weiler, K.: Edge-based Data Structures for Solid Modeling in Curved-Surface Environments. Computer Graphics and Applications 5,1 (1985) 21–40Google Scholar
  62. [Wei2]
    Weiler, K.: The Radial-Edge Data Structure: A Topological Representation for Non-Manifold Geometry Boundary Modeling. Proc. IFIP WG 5.2 Working Conference Rensselaerville, U.S.A. (1986), in Geometric Modeling for CAD Applications Elsevier (1988) 3–36Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Pascal Lienhardt
    • 1
  1. 1.Université de Poitiers, IRCOM-SIC (UNIR CNRS 6615)Futuroscope CedexFrance

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