好的点集可能有糟糕的德洛内三角剖分
Nice point sets can have nasty Delaunay triangulations
摘要：
本文研究了在一定的几何约束下R^3中点集的德洛内三角剖分的复杂性。一组点的散布是最长和最短成对距离之间的比率。我们证明了在最坏的情况下，R^3中n个点的德洛内三角剖分的复杂性为Omega（min{D^3，nD，n^2}）和O（min{D^4，n^2}）。对于D = Theta（sqrt{n}）的情况，我们的下界构造由具有有界曲率的光滑凸曲面的均匀样本组成。我们还构造了一族光滑的连通曲面，使得任何好的点样本的德洛内三角剖分具有近二次的复杂性。
英文摘要：
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 D has complexity Omega(min{D^3, nD, n^2}) and O(min{D^4, n^2}). For the case D = Theta(sqrt{n}), our lower bound construction consists of a uniform sample of a smooth convex surface with bounded curvature. We also construct a family of smooth connected surfaces such that the Delaunay triangulation of any good point sample has near-quadratic complexity.
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