Local differentiability and monotonicity properties of Voronoi diagrams for disjoint convex sites in three dimensions
نویسنده
چکیده
This paper studies local properties of Voronoi diagrams of sets of disjoint compact convex sites in R. It is established that bisectors are C surfaces and trisectors are C curves, and that as a point moves along a trisector its clearance sphere develops monotonically (Lemma 2.4). This monotonicity property is useful in establishing the existence of Voronoi vertices bounding edges in certain situations.1 The paper then considers the diagram for a set of disjoint spheres. Considerations about general position are covered in detail. By letting the spheres grow from point sites till they reach their true radius, it is shown that the Voronoi cell for the smallest site has complexity O(n), assuming that the sites are of at most k distinct radii. It follows that the Voronoi diagram is O(n). Although this is weaker than Aurenhammer’s result [1] establishing O(n) complexity with no restriction on radius, the techniques may be of value for studying more general Voronoi diagrams. Finally, the paper shows that without the bound on the number of different radii, the cell owned by a point site can have complexity Ω(n). 1 Voronoi diagrams: differentiability properties This paper considers the Voronoi diagrams of spherical sites in R. For a general survey of Voronoi diagrams see [2]. The current state of knowledge about the complexity of Voronoi diagrams, in 3 dimensions, is scanty. e-mail: [email protected]. Mathematics department website: http://www.maths.tcd.ie. Some of this work was presented at the eighteenth European Conference on Computational Geometry, Bonn, Germany, April 2003. 1 Chee Yap [8] seems to have been first to exploit such monotonicity properties.
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