 
Summary: TRIANGULATING POINT SETS IN SPACE
David Avis*
School of Computer Science
McGill University
805 Sherbrooke St. W.
Montreal, Canada, H3A 2K6
Hossam ElGindy
Department of Computer and Information Science
University of Pennsylvania
Philadelphia, PA 19104,
U.S.A.
ABSTRACT
A set P of n points in R d is called simplicial if it has dimension d and con
tains exactly d + 1 extreme points. We show that when P contains ną interior
points, there is always one point, called a splitter, that partitions P into d + 1 sim
plices, none of which contain more than dną/(d + 1) points. A splitter can be
found in O(d 4 + nd 2 ) time. Using this result, we give a O(nd 4 log 1+1/d n) algo
rithm for triangulating simplicial point sets that are in general position. In R 3 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 2n + 1 points in R 3 for
