# Voronoi diagrams based on general metrics in the plane

## Abstract

Voronoi diagrams based on metrics others than the Euclidean metric or convex distance functions have recently received considerable interest in robotics and in computational geometry. Since the number of relevant metrics is large (and likely to increase, as new applications come up) a general theory should be developed, leading to results on the structure and on the computation of Voronoi diagrams that hold for large *classes* of metrics, rather than investigating each case separately.

In this paper we make the first steps in this direction. First we propose a uniform way of *defining* the Voronoi diagram of *n* points; the result is always a *partition* of the plane into *n Voronoi regions*, even if the bisectors of two points are two-dimensional regions. Then we present a simple axiom for metrics that guarantees that each possible Voronoi region is *connected*. In this case the *normalized* Voronoi diagram is a planar graph with *n* faces and *O*(*n*) edges and vertices, if the bisectors behave well. Next we pose two additional axioms each of which ensures that all Voronoi regions are *simply-connected*. One of them also implies that the bisector of two sets of points divided by a suitable line, contains no loops. Then the normalized Voronoi diagram can be computed within optimal *O*(*n* log *n*) steps, using the *divide-and-conquer* algorithm. The class of metrics thereby specified does not only contain all symmetric convex distance functions but also many *composite* metrics like, for example, the combination of the “grid” metric *L*_{1} in midtown Manhattan and the Euclidean metric in Central Park and on the Hudson River.

## Keywords

Computational geometry convex distance function metric norm robotics shortest path straight curve Voronoi diagram## Preview

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