We consider the following problem: Let $\mathcal{L}$ be an arrangement of n lines in ${\mathbb{R}}^3$ 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 $\mathcal{L}$ 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.

我们考虑以下问题： $\mathcal{大号}$是n行的排列${\mathbb{[R}}^3$一般情况下，颜色分别为红色，绿色和蓝色。不存在的垂直平面P，使得在线路P同时平分所有三类点的通过线在交叉感应$\mathcal{大号}$与P？最近，施耐德（Schnider）使用拓扑方法证明了这种横截面始终存在。在这项工作中，我们仅使用离散几何中的方法给出了这一事实的替代证明。有了这个组合证明，我们设计了一个$Ø（{ñ}^{\mathrm{2个}}{\mathrm{日志}}^{\mathrm{2个}}（ñ））$时间算法来找到这样一个平面和等分线的横截面。我们通过提供一个通用框架来做到这一点，我们希望可以将其应用于解决横截面和动力学点上的类似问题。