Discrete & Computational Geometry

, Volume 2, Issue 2, pp 99–111

Triangulating point sets in space

  • David Avis
  • Hossam ElGindy

DOI: 10.1007/BF02187874

Cite this article as:
Avis, D. & ElGindy, H. Discrete Comput Geom (1987) 2: 99. doi:10.1007/BF02187874


A setP ofn points inRd is called simplicial if it has dimensiond and contains exactlyd + 1 extreme points. We show that whenP containsn′ interior points, there is always one point, called a splitter, that partitionsP intod + 1 simplices, none of which contain more thandn′/(d + 1) points. A splitter can be found inO(d4 +nd2) time. Using this result, we give anO(nd4 log1+1/dn) algorithm for triangulating simplicial point sets that are in general position. InR3 we give anO(n logn +k) algorithm for triangulating arbitrary point sets, wherek is the number of simplices produced. We exhibit sets of 2n + 1 points inR3 for which the number of simplices produced may vary between (n − 1)2 + 1 and 2n − 2. We also exhibit point sets for which every triangulation contains a quadratic number of simplices.

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • David Avis
    • 1
  • Hossam ElGindy
    • 2
  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Department of Computer and Information ScienceUniversity of PennsylvaniaPhiladelphiaUSA