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...

Consider a set of n points in d-dimensional Euclidean space, d 2, each of which is continuously moving along a given individual trajectory. At each instant in time, the points deene a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we prese...

We revisit the L∞ Hausdorff Voronoi diagram of clusters of points, equivalently, the L∞ Hausdorff Voronoi diagram of rectangles, and present a plane sweep algorithm for its construction that generalizes and improves upon previous results. We show that the structural complexity of the L∞ Hausdorff Voronoi diagram is Θ(n+m), where n is the number of given clusters and m is the number of essential...

We present a graphics hardware implementation of the tangent-plane algorithm for computing the kth-order Voronoi diagram of a set of point sites in image space. Correct and efficient implementation of this algorithm using graphics hardware is possible only with the use of an appropriate shader program on the GPU. This is achieved by rendering in k passes the parallel projection of the top k lev...

In this paper we consider fast nearest-neighbor search techniques based on the projections of Voronoi regions. The Voronoi diagram of a given set of points provides an implicit geometric interpretation of nearest-neighbor search and serves as an important basis for several proximity search algorithms in computational geometry and in developing structure-based fast vector quantization techniques...

Restricted Voronoi diagrams are a fundamental geometric structure used in many applications such as surface reconstruction from point sets or optimal transport. Given a set of sites V = {vk}k=1 ⊂ R d and a mesh X with vertices in Rd connected by triangles, the restricted Voronoi diagram partitions X by computing for each site the portion of X for which the site is the nearest. The restricted Vo...

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...

In this paper, we introduce the fuzzy Voronoi diagram as an extension of the Voronoi diagram. We assume Voronoi sites to be fuzzy points and then define the Voronoi diagram for this kind of sites, then we provide an algorithm for computing this diagram based on Fortune’s algorithm which costs O(n log n) time. Also we introduce the fuzzy Voronoi diagram for a set of fuzzy circles, rather than fu...