Computational geometry and discrete computations

  • Olivier Devillers
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1176)


In this talk we describe some problems arising in practical implementation of algorithms from computational geometry. Going to robust algorithms needs to solve issues such as rounding errors and degeneracies. Most of the problems are closely related to the incompatibility between on one side algorithms designed for continuous data and on the other side the discrete nature of the data and the computations in an actual computer.


Voronoi Diagram Computational Geometry Delaunay Triangulation Exact Computation Float Point Number 
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.
    F. Avnaim, J.-D. Boissonnat, O. Devillers, F. Preparata, and M. Yvinec. Evaluation of a new method to compute signs of determinants. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C16–C17, 1995.Google Scholar
  2. 2.
    J.-D. Boissonnat and M. Yvinec. Géométrie algorithmique. Ediscience international, Paris, 1995.Google Scholar
  3. 3.
    C. Burnikel. Exact Computation of Voronoi Diagrams and Line Segment Intersections. Ph.D thesis, Universität des Saarlandes, March 1996.Google Scholar
  4. 4.
    Christoph Burnikel, Jochen Könnemann, Kurt Mehlhorn, Stefan Näher, Stefan Schirra, and Christian Uhrig. Exact geometric computation in LEDA. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages C18–C19, 1995.Google Scholar
  5. 5.
    K. L. Clarkson. Safe and effective determinant evaluation. In Proc. 33rd Annu. IEEE Sympos. Found. Comput. Sci., pages 387–395, 1992.Google Scholar
  6. 6.
    O. Devillers and F. Preparata. A probabilistic analysis of the power of arithmetic filters. Research Report to appear, INRIA, BP93, 06902 Sophia-Antipolis, France, 1996.Google Scholar
  7. 7.
    M. B. Dillencourt. Realizability of Delaunay triangulations. Inform. Process. Lett., 33:283–287, 1990.Google Scholar
  8. 8.
    H. Edelsbrunner. Algorithms in Combinatorial Geometry, volume 10 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Heidelberg, West Germany, 1987.Google Scholar
  9. 9.
    Jeff Erickson. New lower bounds for convex hull problems in odd dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 1–9, 1996.Google Scholar
  10. 10.
    Jeff Erickson and Raimund Seidel. Better lower bounds on detecting affine and spherical degeneracies. In Proc. 34th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS 93), pages 528–536, 1993.Google Scholar
  11. 11.
    Stefan Felsner. On the number of arrangements of pseudolines. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 30–37, 1996.Google Scholar
  12. 12.
    S. Fortune and C. J. Van Wyk. Efficient exact arithmetic for computational geometry. In Proc. 9th Annu. ACM Sympos. Comput. Geom., pages 163–172, 1993.Google Scholar
  13. 13.
    Donald E. Knuth. Axioms and Hulls, volume 606 of Lecture Notes in Computer Science. Springer-Verlag, Heidelberg, Germany, 1992.Google Scholar
  14. 14.
    G. Liotta, F. P. Preparata, and R. Tamassia. Robust proximity queries in implicit Voronoi diagrams. Research Report CS-96-16, Brown University, Providence, RI, 1996.Google Scholar
  15. 15.
    K. Mehlhorn. Data Structures and Algorithms 3: Multi-dimensional Searching and Computational Geometry, volume 3 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Heidelberg, West Germany, 1984.Google Scholar
  16. 16.
    K. Mehlhorn and S. Näher. The implementation of geometric algorithms. In Proc. 13th World Computer Congress IFIP94, volume 1, pages 223–231, 1994.Google Scholar
  17. 17.
    D. Michelucci. Arithmetic isuues in geometric computations. In Proc. 2nd Real Numbers and Computer Conf., pages 43–69, Marseille, France, 1996.Google Scholar
  18. 18.
    F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, New York, NY, 1985.Google Scholar
  19. 19.
    D. Priest. Algorithms for arbitrary precision floating point arithmetic. In Proc. 10th Symp. on coputer arithmetic, pages 132–143, 1991.Google Scholar
  20. 20.
    Jonathan R. Shewchuk. Robust adaptive floating-point geometric predicates. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 141–150, 1996.Google Scholar
  21. 21.
    K. Sugihara and M. Iri. A robust topology-oriented incremental algorithm for Voronoi diagrams. Internat. J. Comput. Geom. Appl., 4:179–228, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Olivier Devillers
    • 1
  1. 1.INRIASophia-Antipolis cedex

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