Empty-ellipse graphs

Olivier Devillers (1, 2) , Jeff Erickson (3) , Xavier Goaoc (4)

1 GEOMETRICA - Geometric computing
2 CRISAM - Inria Sophia Antipolis - Méditerranée
3 Department. of Computer Science [Illinois]
4 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility

We define and study a geometric graph over points in the plane that captures the local behavior of Delaunay triangulations of points on smooth surfaces in 3-space. Two points in a planar point set P are neighbors in the empty-ellipse graph if they lie on an axis-aligned ellipse with no point of P in its interior. The empty-ellipse graph can be a clique in the worst case, but it is usually much less dense. Specifically, the emptyellipse graph of n points has complexity Θ(Δn) in the worst case, where Δ is the ratio between the largest and smallest pairwise distances. For points generated uniformly at random in a rectangle, the empty-ellipse graph has expected complexity Θ(n log n). As an application of our proof techniques, we show that the Delaunay triangulation of n random points on a circular cylinder has expected complexity Θ(n log n).

Domains: Computational Geometry [cs.CG]

Olivier Devillers, Jeff Erickson, Xavier Goaoc. Empty-ellipse graphs. 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'08), 2008, San Francisco, United States. pp.1249--1256. ⟨inria-00176204⟩