We analyze structural properties of the order-k Voronoi diagram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single orderk Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of...

We describe a kinetic data structure for maintaining a compact Voronoi-like diagram of convex polygons moving around in the plane. We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96]. A key feature of this diagram is that its size is only a function of the number of polygons and not of their complexity. We demonstrate a local certifying property of that di...

We present novel exploration algorithms and a control law that enables the construction of Voronoi diagrams over unknown areas using a single vehicle equipped with range sensors. The control law uses range measurements to make the vehicle track Voronoi edges between obstacles. The exploration algorithms make decisions at vertices in the Voronoi diagram to expand the explored area until a comple...

Multiresolution representation of high-dimensional scattered data is a fundamental problem in scientific visualization. This paper introduces a data hierarchy of Voronoi diagrams as a versatile solution. Given an arbitrary set of points in the plane, our goal is the construction of an approximation hierarchy using the Voronoi diagram as the essential building block. We have implemented two Voro...

We present novel exploration algorithms and control law that enable the construction of Voronoi diagram in an initially unknown area using one vehicle equipped with range scanners. Our control law and exploration algorithms are provably complete, i.e., convergence is guaranteed. The control laws use range measurements to make the vehicle track Voronoi edges between two obstacles. The exploratio...

Computational geometry classically assumes real-number arithmetic which does not exist in actual computers. A solution consists in using integer coordinates for data and exact arithmetic for computations. This approach implies that if the results of an algorithm are the input of another, these results must be rounded to match this hypothesis of integer coordinates. In this paper, we treat the c...

The main problem of educational centers in a mega city like Tehran, capital of Iran, is that no enforced service areas exist to guide school selection or allow students to make the most convenient commutes to the nearest schools. Without the defined school service areas, parents seeking better and more reputable schools often have no choice but to send children to schools outside the local area...

Point clusters occur in both spatial and non-spatial data. In the former context they may represent segmented particle data, in the latter context they may represent clusters in scatterplots. In order to visualize such point clusters, enclosing surfaces lead to much better comprehension than pure point renderings. We propose a flexible system for the generation of enclosing surfaces for 3D poin...

Ron Wein et al. [4] introduced the Visibility-Voronoi diagram for clearance c, denoted by V V , which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning natural-looking paths for a robot translating amidst polygonal obsta...