## Abstract

We consider the following problem: Let L be an arrangement of n lines in R^{3} 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(n^{2}log^{2}(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 language | English |
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Article number | 101775 |

Journal | Computational Geometry: Theory and Applications |

Volume | 98 |

DOIs | |

Publication status | Published - Oct 2021 |

## Keywords

- 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