Discrete & Computational Geometry

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

The Voronoi Diagram of Curved Objects

  • Helmut Alt
  • Otfried Cheong
  • Antoine Vigneron


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).


Computational Mathematic Systematic Study Arbitrary Point Topological Property Voronoi Diagram 
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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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