Equivalence Between Regular n-G-Maps and n-Surfaces

  • Sylvie Alayrangues
  • Xavier Daragon
  • Jacques-Olivier Lachaud
  • Pascal Lienhardt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3322)


Many combinatorial structures have been designed to represent the topology of space subdivisions and images. We focus here on two particular models, namely the n-G-maps used in geometric modeling and computational geometry and the n-surfaces used in discrete imagery. We show that a subclass of n-G-maps is equivalent to n-surfaces. We exhibit a local property characterising this subclass, which is easy to check algorithmatically. Finally, the proofs being constructive, we show how to switch from one representation to another effectively.


Computational Geometry Incidence Relation Switch Property Incidence Graph Barycentric Subdivision 
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.


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  1. 1.
    Alayrangues, S., Lachaud, J.-O.: Equivalence Between Order and Cell Complex Representations. In: Proc. Computer Vision Winter Workshop, CVWW 2002 (2002)Google Scholar
  2. 2.
    Alayrangues, S., Daragon, X., Lachaud, J.-O., Lienhardt, P.: Equivalence between Regular n-G-maps and n-surfaces, Research Report,
  3. 3.
    Bertrand, G.: New Notions for Discrete Geometry. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, p. 218. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  4. 4.
    Bertrand, G.: A Model for Digital Topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, p. 229. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  5. 5.
    Björner, A.: Topological methods. In: Handbook of combinatorics, vol. 2. MIT Press, Cambridge (1995)Google Scholar
  6. 6.
    Brisson, E.: Representing Geometric Structures in d Dimensions: Topology and Order. In: Proceedings of the Fifth Annual Symposium on Computational Geometry (1989)Google Scholar
  7. 7.
    Brun, L., Kropatsch, W.: Contraction Kernels and Combinatorial Maps. In: 3rd IAPR-TC15 Workshop on Graph-based Representations in Pattern Recognition (2001)Google Scholar
  8. 8.
    Daragon, X., Couprie, M., Bertrand, G.: New “marching-cubes-like” algorithm for Alexandroff-Khalimsky spaces. In: Proc. of SPIE: Vision Geometry, vol. XI (2002)Google Scholar
  9. 9.
    Daragon, X., Couprie, M., Bertrand, G.: Discrete Frontiers. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 236–245. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Edelsbrunner, H.: Algorithms in combinatorial geometry. Springer-Verlag New York, Inc. (1987)Google Scholar
  11. 11.
    Elter, H.: Etude de structures combinatoires pour la representation de complexes cellulaires, Universit Louis Pasteur, Strasbourg, France (1994)Google Scholar
  12. 12.
    Evako, A.V., Kopperman, R., Mukhin, Y.V.: Dimensional properties of graphs and digital spaces. Journal of Mathematical Imaging and Vision (1996)Google Scholar
  13. 13.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  14. 14.
    Lienhardt, P.: Subdivisions of n-dimensional spaces and n-dimensional generalized maps. In: Proc. 5th Annual ACM Symp. on Computational Geometry (1989)Google Scholar
  15. 15.
    Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Computer-Aided Design (1991)Google Scholar
  16. 16.
    Lienhardt, P.: N-dimensional generalized combinatorial maps and cellular quasi-manifolds. International Journal of Computational Geometry and Applications (1994)Google Scholar
  17. 17.
    May, P.: Simplicial objects in algebraic topology, von Nostrand (1967)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Sylvie Alayrangues
    • 1
  • Xavier Daragon
    • 2
  • Jacques-Olivier Lachaud
    • 1
  • Pascal Lienhardt
    • 3
  1. 1.LaBRITalence Cedex
  2. 2.ESIEE – Laboratoire A2SINoisy le Grand Cedex
  3. 3.SIC – bât SP2MIFuturoscope Chasseneuil Cedex

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