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Rounding Voronoi Diagram

  • Olivier Devillers
  • Pierre-Marie Gandoin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1568)

Abstract

Computational geometry classically assumes real-number arithmetic which does not exist in actual computers. A solution consists in using integer coordinates for data and exact arithmetic for computations. This approach implies that if the results of an algorithm are the input of another, these results must be rounded to match this hypothesis of integer coordinates. In this paper, we treat the case of two-dimensional Voronoi diagrams and are interested in rounding the Voronoi vertices to grid points while interesting properties of the Voronoi diagram are preserved. These properties are the planarity of the embedding and the convexity of the cells. We give a condition on the grid size to ensure that rounding to the nearest grid point preserves the properties. We also present heuristics to round vertices (not to the nearest grid point) and preserve these properties.

Keywords

geometric computing Voronoi diagram integer coordinates exact computations 

References

  1. 1.
    J.-D. Boissonnat. Shape reconstruction from planar cross-sections. Comput. Vision Graph. Image Process., 44(1):1–29, October 1988.CrossRefGoogle Scholar
  2. 2.
    J.-D. Boissonnat and B. Geiger. Three dimensional reconstruction of complex shapes based on the Delaunay triangulation. In R.S. Acharya and D.B. Goldgof, editors, Biomedical Image Proc. and Biomedical Visualization, volume 1905, pages 964–975. SPIE, 1993.Google Scholar
  3. 3.
    J.-D. Boissonnat and F.P. Preparata. Robust plane sweep for intersecting segments. Research Report 3270, INRIA, 1997.Google Scholar
  4. 4.
    O. Devillers. Computational geometry and discrete computations. In Proc. 6th Discrete Geometry for Computer Imagery conf., 1996.Google Scholar
  5. 5.
    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
  6. 6.
    Steven Fortune. Vertex-rounding a three-dimensional polyhedral subdivision. In Proc. 14th Annu. ACM Sympos. Comput. Geom., pages 116–125, 1998.Google Scholar
  7. 7.
    M. Goodrich, L.J. Guibas, J. Hershberger, and P. Tanenbaum. Snap rounding line segments efficiently in two and three dimensions. In Proc. 13th Annu. ACM Sympos. Comput. Geom., page 284, 1997.Google Scholar
  8. 8.
    L. Guibas and D. Marimont. Rounding arrangements dynamically. In Proc. 11th Annu. ACM Sympos. Comput. Geom., page 190, 1995.Google Scholar
  9. 9.
    G. Liotta, F.P. Preparata, R. Tamassia. Robust proximity queries in implicit Voronoi diagrams. Technical Report CS-96-16, Center for Geometric Computing, Comput. Sci. Dept., Brown Univ., Providence, RI, 1996.Google Scholar
  10. 10.
    R.E. Miles. On the homogenous planar Poisson point-process. Math. Biosci., 6:85–127, 1970.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Okabe, B. Boots, and K. Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Chichester, UK, 1992.zbMATHGoogle Scholar
  12. 12.
    C. Yap. Towards exact geometric computation. Comput. Geom. Theory Appl., 7(1):3–23, 1997.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Olivier Devillers
    • 1
  • Pierre-Marie Gandoin
    • 1
  1. 1.INRIASophia Antipolis

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