## Abstract

We present a new type of Voronoi diagram in R2 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 of R2 into Rm, 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.

by smoothly and injectively embedding of R2 into Rm, 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.

Original language | English |
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Title of host publication | Proceeding 20th European Workshop on Computational Geometry (EuroCG'13) |

Place of Publication | Braunschweig |

Pages | 185-188 |

Publication status | Published - 2013 |