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

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Summary

Introduction

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

A combinatorial proof
Dual interpretation in cross-sections
Bi-chromatic sign sequences
Transfer
The algorithm
A general framework
Applying the framework
Conclusion
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