Voronoi Diagrams on the Sphere
نویسندگان
چکیده
Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorithm for the spherical Voronoi diagram of a set of circular arcs on the sphere, and an O(n logn) algorithm for the furthest-site Voronoi diagram for a set of circular arcs in the plane.
منابع مشابه
Exact Implementation of Arrangements of Geodesic Arcs on the Sphere with Applications
Recently, the Arrangement 2 package of Cgal, the Computational Geometry Algorithms Library, has been greatly extended to support arrangements of curves embedded on two-dimensional parametric surfaces. The general framework for sweeping a set of curves embedded on a two-dimensional parametric surface was introduced in [3]. In this paper we concentrate on the specific algorithms and implementatio...
متن کاملAlgorithms for Complex Shapes with Certified Numerics and Topology Arrangements of Geodesic Arcs on the Sphere
In this report we concentrate on exact construction and maintenance of arrangements induced by arcs of great circles embedded on the sphere, also known as geodesic arcs, and exact computation of Voronoi diagrams on the sphere, the bisectors of which are geodesic arcs. This class of Voronoi diagrams includes the subclass of Voronoi diagrams of points and its generalization, power diagrams, also ...
متن کاملVoronoi diagrams on the spher
Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorit...
متن کاملA Plane Sweep Algorithm for the Voronoi Tessellation of the Sphere
We have extended Fortune’s sweep-line algorithm for the construction Voronoi diagrams in the plane to the surface of a sphere. Although the extension is straightforward, it requires interesting modifications. The main difference between the sweep line algorithms on plane and on the sphere is that that the beach line on the sphere is a closed curve. We have implemented this algorithm and tessell...
متن کاملThe geometric stability of Voronoi diagrams in normed spaces which are not uniformly convex
The Voronoi diagram is a geometric object which is widely used in many areas. Recently it has been shown that under mild conditions Voronoi diagrams have a certain continuity property: small perturbations of the sites yield small perturbations in the shapes of the corresponding Voronoi cells. However, this result is based on the assumption that the ambient normed space is uniformly convex. Unfo...
متن کامل