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Topologically consistent algorithms related to convex polyhedra

  • Kokichi Sugihara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 650)

Abstract

The paper presents a general method for the design of numerically robust and topologically consistent geometric algorithms concerning convex polyhedra in the three-dimensional space. A graph is the vertex-edge graph of a convex polyhedron if and only if it is planar and triply connected (Steinitz' theorem). On the basis of this theorem, conventional geometric algorithms are revised in such a way that, no matter how poor the precision in numerical computation may be, the output graph is at least planar and triply connected. The resultant algorithms are robust in the sense that they do not fail in finiteprecision arithmetic, and are consistent in the sense that the output is the true solution to a perturbed input.

Keywords

Convex Hull Half Space Voronoi Diagram Convex Polyhedron Geometric Problem 
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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References

  1. [I]
    F. Aurenhammer: Power diagrams — Properties, algorithms and applications. SIAM Journal on Computing, Vol. 16 (1987), pp. 78–96.CrossRefGoogle Scholar
  2. [2]
    K. Q. Brown: Voronoi diagrams from convex hulls. Information Processing Letters, Vol. 9 (1979), pp. 223–228.CrossRefGoogle Scholar
  3. [3]
    M. B. Dillencourt: Toughness and Delaunay triangulations. Discrete and Computational Geometry, Vol. 5 (1990), pp. 575–601.Google Scholar
  4. [4]
    L. Guibas, D. Salesin and J. Stolfi: Epsilon geometry — Building robust algorithms from imprecise computations. Proceedings of the 5th ACM Annual Symposium on Computational Geometry, Saarbrücken, 1989, pp. 208–217.Google Scholar
  5. [5]
    C. M. Hoffmann: The problems of accuracy and robustness in geometric computation. IEEE Computer, Vol. 22, No. 3 (March 1989), pp. 31–41.Google Scholar
  6. [6]
    J. E. Hopcroft and P. J. Kahn: A paradigm for robust geometric algorithms. Technical Report, TR89-1044, Department of Computer Science, Cornell University, Ithaca, 1989.Google Scholar
  7. [7]
    V. Milenkovic: Verifiable implementations of geometric algorithms using finite precision arithmetic. Artificial Intelligence, Vol. 37 (December 1988), pp. 377–401.CrossRefGoogle Scholar
  8. [8]
    T. Ottmann, G. Thiemt and C. Ullrich: Numerical stability of geometric algorithms. Proceedings of the 3rd ACM Annual Conference on Computational Geometry, Waterloo, pp. 119–125, 1987.Google Scholar
  9. [9]
    E. Steinitz: Polyeder und Raumeinteilungen. EncyklopÄdie der mathematischen Wissenschaften, Band III, Teil 1, 2. HÄlfte, IIIAB12, pp. 1–139, 1916.Google Scholar
  10. [10]
    K. Sugihara and M. Iri: Two design principles of geometric algorithms in finiteprecision arithmetic. Applied Mathematics Letters, Vol. 2 (1989), pp. 203–206.CrossRefGoogle Scholar
  11. [11]
    K. Sugihara and M. Iri: A solid modelling system free from topological inconsistency. Journal of Information Processing, Vol. 12 (1989), pp. 380–393.Google Scholar
  12. [12]
    K. Sugihara and M. Iri: Construction of the Voronoi diagram for “one million” generators in single-precision arithmetic. Proceedings of the IEEE (to appear).Google Scholar
  13. [13]
    H. Whitney: Congruent graphs and the connectivity of graphs. American Journal of Mathematics, Vol. 54 (1932), pp. 150–168.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Kokichi Sugihara
    • 1
  1. 1.Department of Mathematical Engineering and Information Physics Faculty of EngineeringUniversity of TokyoTokyoJapan

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