A Theoretical Framework for Computing Generalized Weighted Voronoi Diagrams Based on Lower Envelopes

This paper presents a theoretical framework for constructing generalized weighted Voronoi diagrams (GWVDs) of weighted points and straight-line segments (“sites”) in the Euclidean plane, based on lower envelopes constructed in three-dimensional space. Central to our approach is an algebraic distance function that defines the minimum weighted distance from a point to a site. We also introduce a parameterization for the bisectors, ensuring a precise representation of Voronoi edges. The connection to lower envelopes allows us to derive (almost tight) bounds on the combinatorial complexity of a GWVD. We conclude with a short discussion of implementation strategies, ranging from leveraging computational geometry libraries to employing graphics hardware for approximate solutions.