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doi:10.1016/0167-8396(91)90038-D 
Copyright © 1991 Published by Elsevier Science B.V. All rights reserved.
Construction of three-dimensional Delaunay triangulations using local transformations
Available online 21 March 2002.
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Abstract
In [Joe '89], we presented an algorithm which uses local transformations to construct a triangulation of a set of n three-dimensional points that is pseudo-locally-optimal with respect to the sphere criterion. We conjectured that this algorithm always constructs a Delaunay triangulation, and supported our conjecture with experimental results. The empirical time complexity of this algorithm is O(n4/3) or O(n(log n)2) for sets of random points, and O(n2) in the worst case (even for Delaunay triangulations containing O(n2) tetrahedrons). These time complexities are the same or better than those of other algorithms for constructing a three-dimensional Delaunay triangulation.
In this paper, we prove that the conjecture is true, i.e., local transformations can be used to construct a Delaunay triangulation. From our proof, it follows that the algorithm can be improved by removing unnecessary tests. The empirical time complexities of the improved algorithm are the same as before. We also compare the improved algorithm with a related algorithm in which the local transformations are not explicitly performed. We show that both of these algorithms have a worst case time complexity of O(n2), which is worst case optimal.
Author Keywords: 3-D triangulation; Delaunay triangulation; Voronoi tessellation; computational geometry; mesh generation.
