### Abstract

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 2013 |

Place of Publication | Braunschweig |

Pages | 185-188 |

Publication status | Published - 2013 |

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### Cite this

*Proceeding 20th European Workshop on Computational Geometry EUROCG 2013*(pp. 185-188). Braunschweig.

**Voronoi Diagrams from Distance Graphs.** / Aurenhammer, Franz; Kapl, Mario; Jüttler, Bert.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research

*Proceeding 20th European Workshop on Computational Geometry EUROCG 2013.*Braunschweig, pp. 185-188.

}

TY - GEN

T1 - Voronoi Diagrams from Distance Graphs

AU - Aurenhammer, Franz

AU - Kapl, Mario

AU - Jüttler, Bert

PY - 2013

Y1 - 2013

N2 - 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 obtainedby 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.

AB - 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 obtainedby 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.

M3 - Conference contribution

SP - 185

EP - 188

BT - Proceeding 20th European Workshop on Computational Geometry EUROCG 2013

CY - Braunschweig

ER -