The construction of anisotropic triangulations is desirable for various applications, such as the numerical solving of partial differential equations and the representation of surfaces in graphics. To solve this notoriously difficult problem in a practical way, we introduce the discrete Riemannian Voronoi diagram, a discrete structure that approximates the Riemannian Voronoi diagram. This struc...

The MIR Voronoi diagram appears with the notion of good illumination introduced in [3, 4]. This illumination concept generalizes well-covering [7] and triangle guarding [9]. The MIR Voronoi diagram merges the notions of proximity and convex dependency. Given a set S of planar light sources, for each point q in the plane we search for the subset Sq ⊂ S nearest to q and such that q is an interior...

a new model is proposed for two-dimensional simulation of the concrete fracture in compression. the model generated by using the voronoi diagram method and with considering random shape and distribution of full graded aggregates at the mesoscopic level. the aggregates modeled by combining irregular polygons, which then is placed into the concrete with no intersection between them. by this new m...

We propose an incremental approach to compute the order-k maximum Voronoi diagram of disks in the plane. In our approach, we start with an order-k Voronoi diagram of disk centers and iteratively expand disks and update the changes of the diagram until all disks reach their targeted size. When disks expand continuously, the structure of the diagram changes discretely. The algorithm takes O (⌈ rm...

We show that the combinatorial complexity of the Euclidean Voronoi diagram of n lines in R3 that have at most c distinct orientations is O(c3n2+ε) for any ε > 0. This result is a step toward proving the long-standing conjecture that the Euclidean Voronoi diagram of lines in three dimensions has near-quadratic complexity. It provides the first natural instance in which this conjecture is shown t...

F. Labelle and J. Shewchuk have proposed a discrete definition of anisotropic Voronoi diagrams. These diagrams are parametrized by a metric field. Under mild hypotheses on the metric field, such Voronoi diagrams can be refined so that their dual is a triangulation, with elements shaped according to the specified anisotropic metric field. We propose an alternative view of the construction of the...

The power pow (x, s) of a point x with respect to a sphere s in Euclidean d-space E d is given by d2(x, z) 2, where d denotes the Euclidean distance function, and z and are the center and the radius of s. The power diagram of a finite set S of spheres in E d is a cell complex that associates each s S with the convex domain {x E d Ipow (x, s) < pow (x, t), for all S-{s}}. The close relationship ...

Abstract Voronoi diagrams were introduced by R . Klein [Kle89b, Kle88a, Kle88b] as an axiomatic basis ofVoronoi diagrams. We show how to construct abstract Voronoi diagrams in time O(nlogn) by a randomized algorithm, which is based on Clarkson and Shor's randomized incremental construction technique [CS89]. The new algorithm has the following advantages over previous algorithms: • It can handle...