Triangulating point sets in space
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- Avis, D. & ElGindy, H. Discrete Comput Geom (1987) 2: 99. doi:10.1007/BF02187874
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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.