Bisecting three classes of lines
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We consider the following problem: Let L be an arrangement of n lines in 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 L∩ 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^2log^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.
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