#### Vol. 8, No. 2, 2015

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Crossings of complex line segments

### Samuli Leppänen

Vol. 8 (2015), No. 2, 285–294
##### 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.

##### Keywords
discrete geometry, crossing inequality
##### Mathematical Subject Classification 2010
Primary: 51M05, 51M30, 52C35
Secondary: 51M04