Modeling Coverage Areas of Anisotropic Transmitters by Voronoi-like Structures

Introduction: Consider a set S of points in the plane, called sites, and a signal that is sent out from each site. Now assume that each signal starts at the same time, say time t := 0, and propagates with unit speed uniformly in all directions. The locations at time t ≥ 0 that are reached by a signal sent out from a site s ∈ S is given by a circle ( o set circle ) of radius t centered at s, and the area that has been covered by that signal by time t is the corresponding circular disc. For t su ciently small, no pair of these discs will intersect. However, as t increases, intersections will occur. Apparently, intersections of two such circles correspond to points of the plane that are reached by two di erent signals at the same time. Assigning each locus of the plane to the site whose signal reached it rst yields a partition of the plane that is well-known as the Voronoi diagram of S; cf. Fig. 1(a). Adjacent regions of this partition are separated by straight-line segments. (We refer to the textbook [11] for more background information on Voronoi diagrams.) The boundary of the area covered by at least one signal by time t is called the wavefront of S at time t. It is easy to see that every wavefront of S consists of circular arcs whose endpoints lie on the Voronoi diagram of S. Voronoi diagrams can be generalized to settings where the signals no longer all travel at the same speed. To each site s a weight σ(s) is assigned that speci es how fast the signal travels: In this modi ed setting, the signal has reached points at distance σ(s) · t (from s) at time t. The corresponding Voronoi diagram is known asmultiplicatively weighted Voronoi diagram [5]. The common boundary of two adjacent regions is no longer a line segment but is a circular arc. Also, the region associated with a speci c site s can now be disconnected or multiply-connected; cf. Fig. 1(b). In a similar way, one can generalize the Voronoi diagram by allowing the sites to start emitting their signals at di erent points in time. This leads to the concept of additively weighted Voronoi diagrams. Voronoi diagrams have become an important geometric tool for modeling and analyzing coverage areas of sensors and transmitters. We refer to [4, 10, 12] for sample publications on this application. Common to these publications is the fact that the signal propagation is assumed to be uniform both over all sites and over all directions for each site.