Constructing the Exact Voronoi Diagram of Arbitrary Lines in Three-Dimensional Space

with Fast Point-Location
  • Michael Hemmer
  • Ophir Setter
  • Dan Halperin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6346)


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(n 3 + 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.


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


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Michael Hemmer
    • 1
  • Ophir Setter
    • 2
  • Dan Halperin
    • 2
  1. 1.INRIASophia AntipolisFrance
  2. 2.Tel-Aviv UniversityIsrael

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