Nice point sets can have nasty delaunay triangulations

We consider the complexity of Delaunay triangulations of sets of points in R-3 under certain practical geometric constraints. The spread of a set of points is the ratio between the longest and shortest pairwise distances. We show that in the worst case, the Delaunay triangulation of n points in R-3 with spread Delta has complexity Omega(min{Delta(3), nDelta, n(2)}) and O(min{Delta(4), n(2)}). For the case Delta = Theta(rootn), our lower bound construction consists of a grid-like sample of a right circular cylinder with constant height and radius. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.