Constructing the exact Voronoi diagram of arbitrary lines in three-dimensional space: With fast point-location

Michael Hemmer*, Ophir Setter, Dan Halperin

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We introduce a new, efficient, and complete algorithm, and its exact implementation, to compute the Voronoi diagram of lines in space. This is a major milestone towards the robust construction of the Voronoi diagram of polyhedra. As we follow the exact geometric-computation paradigm, it is guaranteed that we always compute the mathematically correct result. The algorithm is complete in the sense that it can handle all configurations, in particular all degenerate ones. The algorithm requires O(n3+ε ) time and space, where n is the number of lines. The Voronoi diagram is represented by a data structure that permits answering point-location queries in O(log2 n) expected time. The implementation employs the Cgal packages for constructing arrangements and lower envelopes together with advanced algebraic tools.

Original languageEnglish
Title of host publicationAlgorithms, ESA 2010 - 18th Annual European Symposium, Proceedings
Number of pages12
EditionPART 1
StatePublished - 2010
Event18th Annual European Symposium on Algorithms, ESA 2010 - Liverpool, United Kingdom
Duration: 6 Sep 20108 Sep 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume6346 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference18th Annual European Symposium on Algorithms, ESA 2010
Country/TerritoryUnited Kingdom


  • CGAL
  • Computational Geometry
  • Lower Envelopes
  • Point Location
  • Robust Geometric Computing
  • Voronoi Diagrams


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