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On the Density of Iterated Line Segment Intersections

Ansgar Gruene·Sanaz Kamali Sarvestani
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Abstract

Given S_1, a finite set of points in the plane, we define a sequence of point sets S_i as follows: With S_i already determined, let L_i be the set of all the line segments connecting pairs of points of the union of S_1,...,S_i, and let S_i+1 be the set of intersection points of those line segments in L_i, which cross but do not overlap. We show that with the exception of some starting configurations the set of all crossing points is dense in a particular subset of the plane with nonempty interior. This region is the intersection of all closed half planes which contain all but at most one point from S_1.

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Ansgar Gruene, Sanaz Kamali Sarvestani. 2007-07-02. On the Density of Iterated Line Segment Intersections. https://arxiv.org/abs/math/0512158

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