Crossings of complex line segments

Abstract

The crossing lemma holds in ${ℝ}^2$ because a real line separates the plane into two disjoint regions. In ${ℂ}^2$ removing a complex line keeps the remaining point-set connected. We investigate the crossing structure of affine line segment-like objects in ${ℂ}^2$ by defining two notions of line segments between two points and give computational results on combinatorics of crossings of line segments induced by a set of points. One way we define the line segments motivates a related problem in ${ℝ}^3$, which we introduce and solve.