# Arrangements of Approaching Pseudo-Lines

Discrete Comput Geom. 2022;67(2):380-402. doi: 10.1007/s00454-021-00361-w. Epub 2022 Jan 22.

## Abstract

We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line ${\ell }_{i}$ is represented by a bi-infinite connected x-monotone curve ${f}_{i}\left(x\right)$ , $x\in R$ , such that for any two pseudo-lines ${\ell }_{i}$ and ${\ell }_{j}$ with $i\phantom{\rule{0ex}{0ex}}<\phantom{\rule{0ex}{0ex}}j$ , the function $x\phantom{\rule{0ex}{0ex}}↦\phantom{\rule{0ex}{0ex}}{f}_{j}\left(x\right)\phantom{\rule{0ex}{0ex}}-\phantom{\rule{0ex}{0ex}}{f}_{i}\left(x\right)$ is monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until they cross, and then move away from each other). We show that such arrangements of approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, while for other aspects, they share the freedom of general pseudo-line arrangements. For the former, we prove:There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines.Every arrangement of approaching pseudo-lines has a dual generalized configuration of points with an underlying arrangement of approaching pseudo-lines. For the latter, we show:There are ${2}^{\Theta \left({n}^{2}\right)}$ isomorphism classes of arrangements of approaching pseudo-lines (while there are only ${2}^{\Theta \left(nlogn\right)}$ isomorphism classes of line arrangements).It can be decided in polynomial time whether an allowable sequence is realizable by an arrangement of approaching pseudo-lines. Furthermore, arrangements of approaching pseudo-lines can be transformed into each other by flipping triangular cells, i.e., they have a connected flip graph, and every bichromatic arrangement of this type contains a bichromatic triangular cell.

Keywords: Discrete geometry; Order types; Pseudo-line arrangements.