Abstract
We consider the following problem: Let L be an arrangement of n lines in R3 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(n2log2(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.
Highlights
The Ham-Sandwich theorem is a fundamental result in discrete and computational geometry
Given an input to the framework described above, a solution point set P can be found in time O (n2 log2(n)+ T (n) log2(n))
The underlying structures that we discovered allowed us to give an O (n2 log2(n)) time algorithm
Summary
The Ham-Sandwich theorem is a fundamental result in discrete and computational geometry. Given an arrangement L of n lines in R3 in general position colored red, green, and blue, does there exist a cross-section determined by a vertical plane in which there exists a line that simultaneously bisects all three classes of points?. For any color class which contains an even number of lines, remove an arbitrary line, and consider a solution for the remaining arrangement This solution consists of a plane P and a line in the cross-section defined by P. Problems that involve points moving at individual constant speed can be modeled as problems on line arrangements in R3 and cross-sections with vertically translated planes (see, e.g., [5] for a historical account) For such problems, the number of changes in the combinatorial structure of the point set is crucial. Of a line arrangement in which no point triple changes its orientation; in our setting, after rotating the plane around the z-axis by 180◦, the orientation of all point triples in the cross-section has changed
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