Application of the delaunay triangulation to geometric intersection problems

  • Kokichi Sugihara
II Mathematical Programming II.1 Computational Geometry
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)


The paper presents a new robust method for finding intersections of line segments in the plane. This method first constructs the Delaunay triangulation spanning the end points of line segments, and next recursively inserts the midpoints in the line segments that are not realized by Delaunay edges, until the descendants of the line segments become realized by Delaunay edges or the areas containing points of intersection are sufficiently localized. The method is robust in the sense that in any imprecise arithmetic it gives a topologically consistent arrangement as the output, and is stable in the sense that it does not miss intersections that can be easily detected by naive pairwise check with the precision at hand.


Line Segment Voronoi Diagram Numerical Error Computational Geometry Delaunay Triangulation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. L. Bentley and T. A. Ottmann: Algorithms for reporting and counting geometric intersections, IEEE Transactions on Computers, vol. 28 (1979), pp. 643–647.zbMATHCrossRefGoogle Scholar
  2. [2]
    T. Ottmann, G. Thiemt and C. Ullrich: Numerical stability of geometric algorithms. Proceedings of the 3rd ACM Annual Conference on Computational Geometry, Waterloo, 1987, pp. 119–125.Google Scholar
  3. [3]
    V. Milenkovic: Verifiable implementations of geometric algorithms using finite precision arithmetic. Artificial Intelligence, vol. 37 (1988), pp. 377–401.zbMATHCrossRefGoogle Scholar
  4. [4]
    F. P. Preparata and M. I. Shamos: Computational Geometry — An Introduction. Springer-Verlag, New York, 1985.Google Scholar
  5. [5]
    A. Ohsawa: Failure free geometrical algorithm against calculation error — Realization by means of space model (in Japanese). Transactions of Information Processing Society of Japan, vol. 31, (1990), pp. 42–55.Google Scholar
  6. [6]
    H. Samet: Design and Analysis of Spatial Data Structures, Addison-Wesley, Reading, 1990.Google Scholar
  7. [7]
    K. Sugihara and M. Iri: Construction of the Voronoi diagram for one million generators in single-precision arithmetic. Paper presented at the First Canadian Conference on Computational Geometry, August 21–25, 1989, Montreal, Canada, and submitted for publication.Google Scholar
  8. [8]
    K. Sugihara and M. Iri: VORONOI2 reference manual. Research Memorandum RMI 89-04, Department of Mathematical Engineering and Information Physics, Faculty of Engineering, University of Tokyo, 1989.Google Scholar
  9. [9]
    J. M. Keil and C. A. Gutwin: The Delaunay triangulation closely approximates the complete Euclidean graph. Proceedings of the First Workshop on Algorithms and Data Structures, 1989, pp. 47–56.Google Scholar
  10. [10]
    D. P. Dobkin, S. J. Friedman and K. J. Supowit: Delaunay graphs are almost as good as complete graphs, Discrete and Computational Geometry, vol. 5 (1990), pp. 399–407.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    T. Ohya, M. Iri and K. Murota: Improvements of the incremental method for the Voronoi diagram with computational comparison of various algorithms. Journal of the Operations Research Society of Japan, vol. 27 (1984), pp. 306–336.zbMATHMathSciNetGoogle Scholar

Copyright information

© International Federation for Information Processing 1992

Authors and Affiliations

  • Kokichi Sugihara
    • 1
  1. 1.Department of Mathematical Engineering and Information PhysicsUniversity of TokyoTokyoJapan

Personalised recommendations