In this paper, we propose an approximate "fuzzy Voronoi" diagram (FVD)for fuzzy numbers of dimension two (FNDT) by designing an extension of crisp Voronoi diagram for fuzzy numbers. The fuzzy Voronoi sites are defined as fuzzy numbers of dimension two. In this approach, the fuzzy numbers have a convex continuous differentiable shape. The proposed algorithm has two stages: in the first stage we ...

The farthest line segment Voronoi diagram shows properties different from both the closest-segment Voronoi diagram and the farthest-point Voronoi diagram. Surprisingly, this structure did not receive attention in the computational geometry literature. We analyze its combinatorial and topological properties and outline an O(n log n) time construction algorithm that is easy to implement. No restr...

We present techniques for fast motion planning by using discrete approximations of generalized Voronoi diagrams, computed with graphics hardware. Approaches based on this diagram computation are applicable to both static and dynamic environments of fairly high complexity. We compute a discrete Voronoi diagram by rendering a three-dimensional distance mesh for each Voronoi site. The sites can be...

Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicativelyweighted Voronoi diagram in O((n+m)2 log(nm)) time and for ...

This paper applies the randomized incremental construction (RIC) framework to computing the Hausdorff Voronoi diagram of a family of k clusters of points in the plane. The total number of points is n. The diagram is a generalization of Voronoi diagrams based on the Hausdorff distance function. The combinatorial complexity of the Hausdorff Voronoi diagram is O(n + m), where m is the total number...

In this paper, we construct an algorithm for determining whether a given tessellation on a sphere is a spherical Laguerre Voronoi diagram or not. For spherical Laguerre tessellations, not only the locations of the Voronoi generators, but also their weights are required to recover. However, unlike the ordinary spherical Voronoi diagram, the generator set is not unique, which makes the problem di...

Voronoi diagrams have several important applications in science and engineering. While the properties and algorithms for the ordinary Voronoi diagram of point set have been well-known, their counterparts for a set of spheres have not been sufficiently studied. In this paper, we present the definition, properties and algorithms for the Voronoi diagram of 3D spheres based on the Euclidean distanc...

G. Leibon and D. Letscher showed that for general and sufficiently dense point set its Delaunay triangulation and Voronoi diagram in Riemannian manifold exist. They also proposed an algorithm to construct them for a given set. In this paper we estimate the necessary number of points for computing the Voronoi diagram in the manifold by using sectional curvature of the manifold. Moreover, we show...

In the Hausdorff Voronoi diagram of a set of point-clusters in the plane, the distance between a point t and a cluster P is measured as the maximum distance between t and any point in P while the diagram is defined in a nearest sense. This diagram finds direct applications in VLSI computer-aided design. In this paper, we consider “non-crossing” clusters, for which the combinatorial complexity o...

One of most famous theorems in computational geometry is the duality between Voronoi diagram and Delaunay triangulation in Euclidean space. This paper proposes an extension of that theorem to the Voronoi diagram and Delaunay-type triangulation in dually at space. In that space, the Voronoi diagram and the triangulation can be computed e ciently by using potential functions. We also propose high...