Voronoi Diagrams from Distance Graphs ∗

We present a new type of Voronoi diagram in R that respects the anisotropy exerted on the plane by a given distance graph. It is based on a metric obtained by smoothly and injectively embedding R into R, and a scalar-valued function for re-scaling the distances. A spline representation of the embedding surface is constructed with the Gauß-Newton algorithm, which approximates the given distance graph in the sense of least squares. The graph is required to satisfy the generalized polygon inequality. We explain a simple method to compute the Voronoi diagrams for such metrics, and give conditions under which Voronoi cells stay connected. Several examples of diagrams resulting from different metrics are presented.