On Constructing the Voronoi Diagram for Lines in the Plane under a Linear-Function Distance

Abstract

We consider the problem of constructing the Voronoi diagram for a set of weighted lines in the plane. First, we examine the case when each of the given lines is assigned an additive real weight, and next---a more general case when each line is endowed with a linear function, and the distance between any point in the plane and a weighted line is given by the value of the associated linear function of the latter, to which the Euclidean distance between the point and the line is passed as the argument.Our proposed method is based on the wavefront propagation, and allows for an efficient computation of the respective Voronoi diagrams, as well as for the analysis of their structure and properties. Its advantage over the general approach to studying and constructing Voronoi diagrams, which, in our case, would require the computation of the lower envelope of a set of half-planes in three-dimensional space, lies in its relative simplicity.