Arrangements of Approaching Pseudo-Lines

We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line [Formula: see text] is represented by a bi-infinite connected x-monotone curve [Formula: see text] , [Formula: see text] , such that for any two pseudo-lines [Formula: see text] and [Formula: see text] with [Formula: see text] , the function [Formula: see text] 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. There are [Formula: see text] isomorphism classes of arrangements of approaching pseudo-lines (while there are only [Formula: see text] 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.