Arrangements of Approaching Pseudo-Lines

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 $i<j$ , the function $x\mapsto f_j\left(x\right)-f_i\left(x\right)$ 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 $2^{\Theta (nlogn)}$ 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.