On the Density of Iterated Line Segment Intersections

Details On the Density of Iterated Line Segment Intersections On the Density of Iterated Line Segment Intersections

2005-12-07

arxiv.org/abs/math/0512158v6

Technical Report 004, Department of Computer Science I, University of Bonn, Germany, 2005

Mathematics

Metric Geometry

20 pages, 17 figures, revised version, still based on Technical Report 004

Scientific paper

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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