, Volume 2, Issue 1, pp 99-111

Triangulating point sets in space

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Abstract

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.

Communicated by David Dobkin
Research supported by the Natural Science and Engineering Research Council grant A3013 and the F.C.A.R. grant EQ1678.