An Algorithm for Decomposing Multi-dimensional Non-manifold Objects into Nearly Manifold Components

  • M. Mostefa Mesmoudi
  • Leila De Floriani
  • Franco Morando
  • Enrico Puppo
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


In this paper we address the problem of building valid representations for non-manifold d-dimensional objects. To this aim, we have developed a combinatorial approach based on decomposing a non-manifold d-dimensional object into an assembly of more regular components, that we call initial quasi-manifolds. We present a decomposition algorithm, whose complexity is slightly super-linear in the total number of simplexes. Our approach provides a rigorous basis for designing efficient dimension-independent data structures for describing non-manifold objects.


Simplicial Complex Hasse Diagram External Vertex Standard Decomposition Abstract Simplicial Complex 


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  1. 1.
    De Floriani, L., Morando, F., Puppo, E.: A Representation for Abstract Simplicial Complexes: An Analysis and a Comparison. In: Proc. 11th Int. Conf. on Discrete Geometry for Computer Imagery (2003).Google Scholar
  2. 2.
    De Floriani, L., Magillo, P., Morando, F., Puppo, E.: Non-manifold Multi-Tessellation: from meshes to iconic representation of 3D objects. In: Proceed. of 4th Intern. Workshop on Visual Form (IWVF4), C. Arcelli, L.P. Cordella, and G. Sannitidi Baja, editors, LNCS 2059 page 654, Berlin (2001), Springer-Verlag.Google Scholar
  3. 3.
    De Floriani, L., Mesmoudi, M.M., Morando, F., Puppo, E.: Decomposing Nonmanifold Objects in arbitrary Dimensions. Graphical Models, 65, 2–22 (2003)CrossRefGoogle Scholar
  4. 4.
    De Floriani, L., Magillo, P., Puppo, P., Sobrero, D.: A multi-resolution topological representation for non-manifold meshes, Computer-Aided Design, 36(2):141–159.Google Scholar
  5. 5.
    Desaulnier, H., Stewart, N.: An extension of manifold boundary representation to r-sets. ACM Trans. on Graphics, 11(1), 40–60, (1992)CrossRefGoogle Scholar
  6. 6.
    Elter, H., Lienhardt, P.: 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, 193–212 (1993)Google Scholar
  7. 7.
    Falcidieno, B., Ratto, O.: Two-manifold cell-decomposition of r-sets. In: A. Kilgour and L. Kjelldahl, Eds., Proceedings EUROGRAPHICS '92, 11, 391–404, September (1992)Google Scholar
  8. 8.
    Gueziec, A., Bossen, F., Lazarus, F., Horn, W.: Converting sets of polygons to manifold surfaces by cutting and stitching In: Conference abstracts and applications: SIGGRAPH '98, July 14–21, (1998)Google Scholar
  9. 9.
    Gursoz, E. L., Choi, Y., Prinz, F. B.: Vertex-based representation of non-manifold boundaries, In: M. J. Wozny, J. U. Turner, and K. Preiss, Eds., Geometric Modeling for Product Engineering, North Holland, 107–130, (1990)Google Scholar
  10. 10.
    Hudson, J.F.P,: Piecewise Linear Topology. W.A. Benjamin, Inc., New York (1969)Google Scholar
  11. 11.
    Lee S.H., Lee K., Partial Entity structure: a fast and compact non-manifold boundary representation based on partial topological entities, in Proceedings of the Sixth ACM Symposium on Solid Modeling and Applications, Ann Arbor, Michigan, 2001, pp.159–170Google Scholar
  12. 12.
    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Int. Journal of Comp. Geom. and Appl., 4(3), 275–324, (1994)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Melhorn, K.: Data Structures and Algorithms. Springer Publishing Company (1984)Google Scholar
  14. 14.
    Morando, F.: Decomposition and Modeling in the Non-Manifold domain, PhD Thesis, Department of Computer and Information Science, University of Genova, Genova (Italy), February 2003Google Scholar
  15. 15.
    Nabutovsky, A.: Geometry of the space of triangulations of a compact manifold. Comm. Math. Phys., 181, 303–330 (1996)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Paoluzzi, A., Bernardini, F., Cattani, C., Ferrucci, V.: Dimension-independent modeling with simplicial complexes, ACM Transactions on Graphics, 12(1), 56–102, (1993)CrossRefGoogle Scholar
  17. 17.
    Rossignac, J., Cardoze, D.: Matchmaker: Manifold BReps for non-manifold rsets. In: Willem F. Bronsvoort and David C. Anderson, editors, Proceedings of the Fifth ACM Symposium on Solid Modeling and Applications, 31–41, ACM, June (1999)Google Scholar
  18. 18.
    Rossignac, J.R., O'Connor, M.A.: SGC: A dimension-independent model for point sets with internal structures and incomplete boundaries. In: J.U. Turner, M. J. Wozny and K. Preiss, Eds., Geometric Modeling for Product Engineering, North-Holland, 145–180 (1990)Google Scholar
  19. 19.
    Weiler, K.: The Radial Edge structure: A topological representation for non-manifold geometric boundary modeling. In: M.J. Wozny, H.W. McLauglin, J.L. Encarna\({\tilde c}\)ao (eds), Geometric Modeling for CAD Applications, North-Holland, 1988, 3–36.Google Scholar
  20. 20.
    Weiler, K.: Topological Structures for Geometric Modeling. PhD Thesis, Troy, NY, August (1986)Google Scholar
  21. 21.
    Yamaguchi, Y., Kimura, F.: Non-manifold topology based on coupling entities. IEEE Computer Graphics and Applications, 15(1):42–50, (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • M. Mostefa Mesmoudi
    • 1
  • Leila De Floriani
    • 1
  • Franco Morando
    • 1
  • Enrico Puppo
    • 1
  1. 1.Department of Computer Science (DISI)University of GenovaItaly

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