Reducing simple polygons to triangles - A proof for an improved conjecture -

  • Thorsten Graf
  • Kamakoti Veezhinathan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1443)


An edge of a simple closed polygon is called eliminating if it can be translated in parallel towards the interior of the polygon to eliminate itself or one of its neighbor edges without violating simplicity.

In this paper we prove that in each simple closed polygon there exist at least two eliminating edges; this lower bound is tight since for all n ≥ 5 there exists a polygon with only two eliminating edges. Furthermore we present an algorithm that computes in total O(n log n) time using O(n) space an eliminating edge for each elimination step. We thus obtain the first non-trivial algorithm that computes for P a sequence of n − 3 edge translations reducing P to a triangle.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F. Aurenhammer. Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Comput. Surv., 23:345–405, 1991.CrossRefGoogle Scholar
  2. 2.
    Francis Chin, Jack Snoeyink, and Cao-An Wang. Finding the medial axis of a simple polygon in linear time. In Proc. 6th Annu. Internat. Sympos. Algorithms Comput. (ISAAC 95), volume 1004 of Lecture Notes in Computer Science, pages 382–391. Springer-Verlag, 1995.Google Scholar
  3. 3.
    L. Guibas and J. Hershberger. Morphing simple polygons. In Proc. 10th Annu. ACM Sympos. Comput. Geom., pages 267–276, 1994.Google Scholar
  4. 4.
    Atsuyuki Okabe, Barry Boots, and Kokichki Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Chichester, England, 1992.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Thorsten Graf
    • 1
  • Kamakoti Veezhinathan
    • 2
  1. 1.Research Center JülichJülichGermany
  2. 2.Institute of Mathematical SciencesChennaiIndia

Personalised recommendations