Triangulating point sets in space
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A setP ofn points inR d 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(d 4 +nd 2) time. Using this result, we give anO(nd 4 log1+1/d n) algorithm for triangulating simplicial point sets that are in general position. InR 3 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 inR 3 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.
- D. Avis and H. ElGindy, Triangulating simplicial point sets in space,Proceedings of the Second ACM Conference on Computational Geometry, IBM, Yorktown Heights, New York, 1986.
- H. Edelsbrunner, F. P. Preparata, and D. B. West, Tetrahedrizing point sets in three dimensions, Report No. UIUC DCS-R-86-1310, Department of Computer Science, University of Illinois at Urbana-Champaign, 1986.
- B. Grünbaum,Convex Polytopes, Wiley, New York, 1967.
- D. Kirkpatrick, Optimal search in planar subdivisions,SIAM J. Comput. 12 (1983), 28–35.
- F. P. Preparata and S. J. Hong, Convex hulls of finite sets of points in two and three dimensions,Comm. ACM 22 (1977), 87–93.
- F. P. Preparata and M. I. Shamos,Computational Geometry, Springer-Verlag, New York, 1985.
- Triangulating point sets in space
Discrete & Computational Geometry
Volume 2, Issue 1 , pp 99-111
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