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Discrete & Computational Geometry

, Volume 34, Issue 3, pp 439–453 | Cite as

The Voronoi Diagram of Curved Objects

  • Helmut Alt
  • Otfried Cheong
  • Antoine Vigneron
Article

Abstract

Voronoi diagrams of curved objects can show certain phenomena that are often considered artifacts: The Voronoi diagram is not connected; there are pairs of objects whose bisector is a closed curve or even a two-dimensional object; there are Voronoi edges between different parts of the same site (so-called self-Voronoi-edges); these self-Voronoi-edges may end at seemingly arbitrary points not on a site, and, in the case of a circular site, even degenerate to a single isolated point. We give a systematic study of these phenomena, characterizing their differential-geometric and topological properties. We show how a given set of curves can be refined such that the resulting curves define a “well-behaved” Voronoi diagram. We also give a randomized incremental algorithm to compute this diagram. The expected running time of this algorithm is O(n log n).

Keywords

Computational Mathematic Systematic Study Arbitrary Point Topological Property Voronoi Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Institut fur Informatik, Freie Universitat Berlin, Takustrasse 9, 14195 Berlin Germany
  2. 2.Division of Computer Science, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701 Korea
  3. 3.Department of Computer Science, National University of Singapore, 3 Science Drive 2, 117543Singapore

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