Triangulating point sets in space

Abstract

A set P of n points in Rd is called simplicial if it has dimension d and contains exactly d + 1 extreme points. We show that when P contains n′ interior points, there is always one point, called a splitter, that partitions P into d + 1 simplices, none of which contain more than dn′/(d + 1) points. A splitter can be found in O(d4 +nd2) time. Using this result, we give an O(nd4 log1+1/dn) algorithm for triangulating simplicial point sets that are in general position. In R3 we give an O(n log n +k) algorithm for triangulating arbitrary point sets, where k is the number of simplices produced. We exhibit sets of 2 n + 1 points in R3 for which the number of simplices produced may vary between (n - 1)2 + 1 and 2 n - 2. We also exhibit point sets for which every triangulation contains a quadratic number of simplices. © 1987 Springer-Verlag New York Inc.