Non-manifold Decomposition in Arbitrary Dimensions

  • Leila De Floriani
  • Mostefa Mohammed Mesmoudi
  • Franco Morando
  • Enrico Puppo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2301)


In this paper we consider the problem of decomposing a nonmanifold n-dimensional object described by an abstract simplicial complex into an assembly of ‘more-regular’ components. Manifolds, which would be natural candidates for components, cannot be used to this aim in high dimensions because they are not decidable sets. Therefore, we define d-quasi-manifolds, a decidable superset of the class of combinatorial d-manifolds that coincides with d-manifolds in dimension less or equal than two. We first introduce the notion of d-quasi-manifold complexes, then we sketch an algorithm to decompose an arbitrary complex into an assembly of quasi-manifold components abutting at non-manifold joints. This result provides a rigorous starting point for our future work, which includes designing efficient data structures for non-manifold modeling, as well as defining a notion of measure of shape complexity of such models.


Arbitrary Dimension Abstract Simplicial Complex Singular Vertex Combinatorial Topology Combinatorial Manifold 
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]
    M.K. Agoston. Algebraic Topology, A First Course,. Pure and Applied Mathematics, Marcel Dekker, 1976.Google Scholar
  2. [2]
    Bruce G. Baumgart. Winged edge polyhedron representation. Technical Report CS-TR-72-320, Stanford University, Department of Computer Science, October 1972.Google Scholar
  3. [3]
    L. De Floriani, P. Magillo, F. Morando, and E. Puppo. Non-manifold multi-Tessellation: from meshes to iconic representations of 3d objects. In C. Arcelli, L.P. Cordella, and G. Sanniti di Baja, editors, Proceedings of the 4th International Workshop on Visual Form (IWVF4), volume 2059 of Springer-Verlag (LNCS), page 654 ff, Berlin, 2001. Springer-Verlag.Google Scholar
  4. [4]
    L. De Floriani, M. M. Mesmoudi, F. Morando, and E. Puppo. Decomposition of n-dimensional complex into quasi-manifold components. Technical Report DISITR-01-11, Department of Computer and Information Sciences of Genova (DISI), Genova-Italy, 2001.Google Scholar
  5. [5]
    H. Desaulnier and N. Stewart. An extension of manifold boundary representation to r-sets. ACM Trans. on Graphics, 11(1):40–60, 1992.CrossRefGoogle Scholar
  6. [6]
    D. Dobkin and M. Laszlo. Primitives for the manipulation of three-dimensional subdivisions. Algorithmica, 5(4):3–32, 1989.CrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Edelsbrunner. Algorithms in combinatorial geometry. In Brauer, W., Rozenberg, G., and Salomaa, A., editors, EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1987.Google Scholar
  8. [8]
    H. Elter and P. Lienhardt. Different combinatorial models based on the map concept for the representation of sunsets of cellular complexes. In Proc. IFIP TC 5/WG 5.10 Working Conference on Geometric Modeling in Computer Graphics, pages 193–212, 1993.Google Scholar
  9. [9]
    R. Engelking and K. Svekhicki. Topology: A Geometric Approach. Heldermann Verlag-Berlin, 1992.zbMATHGoogle Scholar
  10. [10]
    B. Falcidieno and O. Ratto. Two-manifold cell-decomposition of r-sets. In A. Kilgour and L. Kjelldahl, editors, Computer Graphics Forum (EUROGRAPHICS’ 92 Proceedings), volume 11 n 3, pages 391–404, sep 1992.Google Scholar
  11. [11]
    A. Gueziec, G. Taubin, F. Lazarus, and W. Horn. Converting sets of polygons to manifold surfaces by cutting and stitching. In Scott Grisson, Janet McAndless, Omar Ahmad, Christopher Stapleton, Adele Newton, Celia Pearce, Ryan Ulyate, and Rick Parent, editors, Conference abstracts and applications: SIGGRAPH 98, July 14–21, 1998, Orlando, FL, Computer Graphics, pages 245-245, New York, NY 10036, USA, 1998. ACM Press.Google Scholar
  12. [12]
    Leonidas Guibas and Jorge Stolfi. Primitives for the manipulation of general subdivisions and the computation of voronoi diagrams. ACM Transaction on Graphics, 4(2):74–123, April 1985.zbMATHCrossRefGoogle Scholar
  13. [13]
    E. L. Gursoz, Y. Choi, and F. B. Prinz. Vertex-based representation of nonmanifold boundaries. In M. J. Wozny, J. U. Turner, and K. Preiss, editors, Geometric Modeling for Product Engineering, pages 107–130. Elsevier Science Publishers B.V., North Holland, 1990.Google Scholar
  14. [14]
    V. E. Kuznetsov I. A. Volodin and A. T. Fomenko. The problem of discriminating algorithmically the standard three-dimensional sphere. Russisan Math. Surveys, 29(5):71–172, 1974. Original Russian article in Uspekhi Mat. Nauk, 29(1), (1974), pp. 72-168.CrossRefGoogle Scholar
  15. [15]
    W. R. B. Lickorish. Simplicial moves on the complexes and manifolds. Geometry and Topology Monographs: Proceedings of the Kirbyfest, 2:299–320, 1999.Google Scholar
  16. [16]
    P. Lienhardt. Topological models for boundary representation: a comparison with n-dimensional generalized maps. CAD, 23(1):59–82, 1991.zbMATHGoogle Scholar
  17. [17]
    P. Lienhardt. Aspects in Topology-Based Geometric Modeling: Possible Tools for Discrete Geometry? In Proceedings of Discrete Geometry in Computer Science LNCS 1347, pages 33–48, 1997.Google Scholar
  18. [18]
    M. Mantyla. An introduction to solid modeling. Computer Science Press, 1983.Google Scholar
  19. [19]
    A. A. Markov. Unsolvability of the problem of homeomorphy. In International Congress of Mathematics, pages 300–306, 1958. In Russian.Google Scholar
  20. [20]
    J. Popovic and H. Hoppe. Progressive simplicial complexes. In ACM Computer Graphics Proc., Annual Conference Series, (Siggraph’ 97), 1997. (to appear).Google Scholar
  21. [21]
    J. Rossignac and D. Cardoze. Matchmaker: Manifold BReps for non-manifold R-Sets. In Willem F. Bronsvoort and David C. Anderson, editors, Proceedings of the Fifth Symposium on Solid Modeling and Applications (SSMA-99), pages 31–41, New York, June 9–11 1999. ACM Press.Google Scholar
  22. [22]
    J.R. Rossignac and M.A. O’Connor. SGC: A dimension-indipendent model for pointsets with internal structures and incomplete boundaries. In J.U. Turner M. J. Woznyand K. Preiss, editors, Geometric Modeling for Product Engineering, pages 145–180. Elsevier Science Publishers B.V. (North-Holland), Amsterdam, 1990.Google Scholar
  23. [23]
    John Stillwell. Classical Topology and Combinatorial Group Theory. Number 72 in Graduate Texts in Mathematics. Springer-Verlag, New York, 1993.Google Scholar
  24. [24]
    A. Thompson. Thin position and the recognition problem for s 3. Mat. Res. Lett., 1:613–630, 1994.zbMATHGoogle Scholar
  25. [25]
    K. Weiler. Boundary graph operators for non-manifold geometric modeling topology representations. In J.L. Encarnacao M.J. Wozny, H.W. McLaughlin, editor, Geometric Modeling for CAD Applications, pages 37–66, North-Holland, 1988. Elsevier Science.Google Scholar
  26. [26]
    K. Weiler. The radial edge data structure: A topological representation for non-manifold geometric boundary modeling. In J.L. Encarnacao M.J. Wozny, H.W. McLaughlin, editor, Geometric Modeling for CAD Applications, pages 3–36, North-Holland, 1988. Elsevier Science.Google Scholar
  27. [27]
    Kevin Weiler. Topological Structures for Geometric Modeling. Ph.D. thesis, Computer and Systems Engineering, Rennselaer Polytechnic Institute, Troy, NY, August 1986.Google Scholar
  28. [28]
    Tony C. Woo. A combinatorial analysis of boundary data structure schemata. IEEE Computer Graphics and Applications, 5(3):19–27, March 1985.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Leila De Floriani
    • 1
  • Mostefa Mohammed Mesmoudi
    • 1
  • Franco Morando
    • 1
  • Enrico Puppo
    • 1
  1. 1.Department of Computer and Information SciencesUniversità di GenovaGenovaItaly

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