Bisecting three classes of lines

TL;DR

This paper proves the existence of a vertical plane bisecting three classes of lines in 3D space using discrete geometry, and provides an efficient algorithm to find such a plane, with potential applications to similar geometric problems.

Contribution

It offers a new combinatorial proof of a topological result and introduces an algorithm with quadratic-logarithmic complexity for finding the bisecting plane.

Findings

Existence of a bisecting vertical plane for three line classes in 3D.

An $O(n^2 ext{log}^2 n)$ algorithm to find the bisecting plane.

A general framework applicable to similar geometric problems.

Abstract

We consider the following problem: Let $\mathcal{L}$ be an arrangement of $n$ lines in $\mathbb{R}^3$ colored red, green, and blue. Does there exist a vertical plane $P$ such that a line on $P$ simultaneously bisects all three classes of points in the cross-section $\mathcal{L} \cap P$? Recently, Schnider [SoCG 2019] 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 the 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.