Bisecting three classes of lines

Alexander Pilz, Patrick Schnider*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider the following problem: Let L be an arrangement of n lines in R3 in general position colored red, green, and blue. Does there exist a vertical plane P such that a line in P simultaneously bisects all three classes of points induced by the intersection of lines in L with P? Recently, Schnider used topological methods to prove that such a cross-section always exists. In this work, we give an alternative proof of this fact, using only methods from discrete geometry. With this combinatorial proof at hand, we devise an O(n2log2⁡(n)) time algorithm to find such a plane and a bisector of the induced cross-section. We do this by providing a general framework, from which we expect that it can be applied to solve similar problems on cross-sections and kinetic points.

Original languageEnglish
Article number101775
JournalComputational Geometry: Theory and Applications
Publication statusPublished - Oct 2021


  • Algorithmic framework
  • Line arrangements
  • Mass partitions
  • Parametric search

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics


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