The voronoi diagram of curved objects

Helmut Alt*, Otfried Cheong, Antoine Vigneron

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Scopus citations


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).

Original languageEnglish (US)
Pages (from-to)439-453
Number of pages15
JournalDiscrete and Computational Geometry
Issue number3
StatePublished - Sep 2005
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics


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