Graph theoretic approaches plays important role in the path planning problems. The usefulness of Graph Theoretic approaches mainly the voronoi diagram is provided for the path planning in static and dynamic environment in this paper. Voronoi Diagram plays important role in computing the optimal path between the source & destination in static as well as dynamic environment. So, through the liter...

In Euclidean geometry, it is well-known that the k-order Voronoi diagram in R can be computed from the vertical projection of the k-level of an arrangement of hyperplanes tangent to a convex potential function in R: the paraboloid. Similarly, we report for the Klein ball model of hyperbolic geometry such a concave potential function: the northern hemisphere. Furthermore, we also show how to bui...

We consider a variation of Voronoi diagram, or time-based Voronoi diagram, for a set S of points in the presence of transportation lines or highways in the plane. A shortest time-distance path from a query point to any given point in S is a path that takes the least travelling time. The travelling speeds and hence travelling times of the subpaths along the highways and in the plane are differen...

This paper concerns the behavior of an SINR diagram of wireless systems, composed of a set S of n stations embedded in R, when restricted to the corresponding Voronoi diagram imposed on S. The diagram obtained by restricting the SINR zones to their corresponding Voronoi cells is referred to hereafter as an SINR+Voronoi diagram. The study of SINR+Voronoi diagrams is motivated by the following tw...

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...

In this paper, we propose a novel image steganographic technique based on the Voronoi diagram. The basic idea is to generate a Voronoi diagram of a graph transformed from cover image; that is, the Voronoi points correspond to every two consecutive pixels in the cover image. According to the secret bits to be embedded, every Voronoi point will either be replaced by one of its neighbors or keep i...

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...