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Topology-Oriented Approach to Robust Geometric Computation

  • Kokichi Sugihara
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1741)

Abstract

The topology-oriented approach is a principle for translating geometric algorithms into practically valid computer software. In this principle, the highest priority is placed on the topological consistency of the geometric objects; numerical values are used as lower-priority information. The resulting software is completely robust in the sense that no matter how large numerical errors arise, the algorithm never fail. The basic idea of this approach and various examples are surveyed.

Keywords

Convex Hull Voronoi Diagram Computational Geometry Convex Polyhedron Voronoi Region 
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. 1.
    M. Benouamer, D. Michelucci and B. Peroche: Error-free boundary evaluation using lazy rational arithmetic-A detailed implementation. Proceedings of the 2nd Symposium on Solid Modeling and Applications, Montreal, 1993, pp. 115–126.Google Scholar
  2. 2.
    S. Fortune: Stable maintenance of point-set triangulations in two dimensions. Proceedings of the 30th IEEE Annual Symposium on Foundations of Computer Science, Research Triangle Park, California, 1989, pp. 494–499.CrossRefGoogle Scholar
  3. 3.
    S. Fortune and C. von Wyk: Efficient exact arithmetic for computational geometry. Proceedings of the 9th ACM Annual Symposium on Computational Geometry, San Diego, 1993, pp. 163–172.Google Scholar
  4. 4.
    L. Guibas, D. Salesin and J. Stolfi: Epsilon geometry-Building robust algorithms from imprecise computations. Proc. 5th ACM Annual Symposium on Computational Geometry (Saarbrücken, May 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.
    C. M. Hoffmann: Geometric and Solid Modeling. Morgan Kaufmann Publisher, San Mateo, 1989.Google Scholar
  7. 7.
    T. Imai: A topology-oriented algorithm for the Voronoi diagram of polygon. Proceedings of the 8th Canadian Conference on Computational Geometry, 1996, pp. 107–112.Google Scholar
  8. 8.
    H. Inagaki, K. Sugihara and N. Sugie, N.: Numerically robust incremental algorithm for constructing three-dimensional Voronoi diagrams. Proceedings of the 4th Canadian Conference Computational Geometry, Newfoundland, August 1992, pp. 334–339.Google Scholar
  9. 9.
    M. Karasick, D. Lieber and L. R. Nackman: Efficient Delaunay triangulation using rational arithmetic. ACM Transactions on Graphics, vol. 10 (1991), pp. 71–91.Google Scholar
  10. 10.
    V. Milenkovic: Verifiable implementations of geometric algorithms using finite precision arithmetic. Artificial Intelligence, vol. 37 (1988), pp. 377–401.Google Scholar
  11. 11.
    T. Minakawa and K. Sugihara: Topology oriented vs. exact arithmetic—experience in implementing the three-dimensional convex hull algorithm. H. W. Leong, H. Imai and S. Jain (eds.): Algorithms and Computation, 8th International Symposium, ISAAC’97 (Lecture Notes in Computer Science 1350), (December, 1997, Singapore), pp. 273–282.Google Scholar
  12. 12.
    T. Minakawa and K. Sugihara: Topology-oriented construction of three-dimensional convex hulls. Optimization Methods and Software, vol. 10 (1998), pp. 357–371.Google Scholar
  13. 13.
    Y. Oishi and K. Sugihara: Topology-oriented divide-and-conquer algorithm for Voronoi diagrams. Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing, vol. 57 (1995), pp. 303–314.Google Scholar
  14. 14.
    T. Ottmann, G. Thiemt and C. Ullrich: Numerical stability of geometric algorithms. Proceedings of the 3rd ACM Annual Symposium on Computational Geometry, Waterloo, 1987, pp. 119–125.Google Scholar
  15. 15.
    P. Schorn: Robust algorithms in a program library for geometric computation. Dissertation submitted to the Swiss Federal Institute of Technology (ETH) Zürich for the degree of Doctor of Technical Sciences, 1991.Google Scholar
  16. 16.
    M. Segal and C. H. Sequin: Consistent calculations for solid modeling. Proceedings of the ACM Annual Symposium on Computational Geometry, Baltimore, 1985, pp. 29–38.Google Scholar
  17. 17.
    K. Sugihara: A simple method for avoiding numerical errors and degeneracy in Voronoi diagram construction. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E75-A (1992), pp. 468–477.Google Scholar
  18. 18.
    K. Sugihara: Approximation of generalized Voronoi diagrams by ordinary Voronoi diagrams. CVGIP: Graphical Models and Image Processing, vol. 55 (1993), pp. 522–531.Google Scholar
  19. 19.
    K. Sugihara: A robust and consistent algorithm for intersecting convex polyhedra. Computer Graphics Forum, EUROGRAPHICS’94, Oslo, 1994, pp. C–45–C–54.Google Scholar
  20. 20.
    K. Sugihara: Robust gift wrapping for the three-dimensional convex hull. J. Computer and System Sciences, vol. 49 (1994), pp. 391–407.Google Scholar
  21. 21.
    K. Sugihara: Experimental study on acceleration of an exact-arithmetic geometric algorithm. Proceedings of the 1997 International Conference on Shape Modeling and Applications, Aizu-Wakamatsu, 1997, pp. 160–168.Google Scholar
  22. 22.
    K. Sugihara and H. Inagaki: Why is the 3d Delaunay triangulation difficult to construct? Information Processing Letters, vol. 54 (1995), pp. 275–280.Google Scholar
  23. 23.
    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
  24. 24.
    K. Sugihara and M. Iri: Construction of the Voronoi diagram for “one million” generators in single-precision arithmetic. Proceedings of the IEEE, vol. 80 (1992), pp. 1471–1484.Google Scholar
  25. 25.
    K. Sugihara and M. Iri: A robust topology-oriented incremental algorithm for Voronoi diagrams. International Journal of Computational Geometry and Applications, vol. 4 (1994), pp. 179–228.Google Scholar
  26. 26.
    C. K. Yap: The exact computation paradigm. D.-Z. Du and F. Hwang (eds.): Computing in Euclidean Geometry, 2nd edition. World Scientific, Singapore, 1995, pp. 452–492.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Kokichi Sugihara
    • 1
  1. 1.Department of Mathematical Engineering and Information PhysicsUniversity of TokyoBunkyo-kuJapan

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