Structure Theorems for Systems of Segments

We study intersection properties of systems of segments in the plane. In particular, we show that there exists a constant c > 0 such that every system S of n straight-line segments in the plane has two at least cn-element subsystems S1, S2 ? S such that either every segment in S1 intersects all elements of S2, or no segment in S1 intersects any element of S2. We also propose a fast approximate solution for reporting most intersections among n segments in the plane.

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