Bisecting three classes of lines

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.