Triangulating point sets in space |
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Authors: | David Avis Hossam ElGindy |
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Affiliation: | (1) School of Computer Science, McGill University, 805 Sherbrooke St. W., H3A 2K6 Montreal, Canada;(2) Department of Computer and Information Science, University of Pennsylvania, 19104 Philadelphia, PA, USA |
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Abstract: | 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.Research supported by the Natural Science and Engineering Research Council grant A3013 and the F.C.A.R. grant EQ1678. |
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