The Voronoi diagram is a fundamental geometry structure widely used in various fields, especially in computer graphics and geometry computing. For a set of points in a compact 3D domain (i.e. a finite 3D volume), some Voronoi cells of their Voronoi diagram are infinite, but in practice only the parts of the cells inside the domain are needed, as when computing the centroidal Voronoi tessellatio...

Given a set of spheres in 3D, constructing its Voronoi diagram in Euclidean distance metric is not easy at all even though many mathematical properties of its structure are known. This Voronoi diagram has been known for many important applications from science and engineering. In this paper, we characterize the Voronoi diagram of spheres in three-dimensional Euclidean space, which is also known...

To fill a theory gap of Voronoi diagrams that there have been no reports of extended diagrams in spatial density so far. A new concept of spatial density Voronoi diagram was proposed. An important property was presented and proven. And a construction algorithm was presented. Spatial density can be used to indicate factors related to density such as conveyance and the traffic conditions. Some pr...

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...

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...

To support the need for interactive spatial analysis, it is often necessary to rethink the data structures and algorithms underpinning applications. This paper describes the development of an interactive environment in which a number of different Voronoi models of space can be manipulated together in real time, to (1) study their behaviour, (2) select appropriate models for specific analysis ta...

We study the Hausdorff Voronoi diagram of a set S of polygonal objects in the plane, a generalization of Voronoi diagrams based on the maximum distance of a point from a polygon, and show that it is equivalent to the Voronoi diagram of S under the Hausdorff distance function. We investigate the structural and combinatorial properties of the Hausdorff Voronoi diagram and give a divide and conque...

Irregular arrangements of vesicles filled with excitable and precipitating chemical systems are imitated by Voronoi automata — finite-state machines defined on a planar Voronoi diagram. Every Voronoi cell takes four states: resting, excited, refractory and precipitate. A resting cell excites if it has at least one excited neighbour; the cell precipitates if a ratio of excited cells in its neigh...