Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Kinetic convex hulls and delaunay triangulations in the black-box model
Proceedings of the twenty-seventh annual symposium on Computational geometry
Farthest-point optimized point sets with maximized minimum distance
Proceedings of the ACM SIGGRAPH Symposium on High Performance Graphics
Computing convex quadrangulations
Discrete Applied Mathematics
New Bounds on the Size of Optimal Meshes
Computer Graphics Forum
Kinetic 2-centers in the black-box model
Proceedings of the twenty-ninth annual symposium on Computational geometry
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The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity $O(\Delta^3)$. This bound is tight in the worst case for all $\Delta = O(\sqrt{n})$. In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of $k$-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any $n$ and $\Delta = O(n)$, we construct a regular triangulation of complexity $\Omega(n\Delta)$ whose $n$ vertices have spread $\Delta$.