Arrangements of Approaching Pseudo-Lines

Abstract

AbstractWe consider arrangements ofnpseudo-lines in the Euclidean plane where each pseudo-line$$\ell _i$$ℓiis represented by a bi-infinite connectedx-monotone curve$$f_i(x)$$fi(x),$$x \in \mathbb {R}$$x∈R, such that for any two pseudo-lines$$\ell _i$$ℓiand$$\ell _j$$ℓjwith$$i \!<\! j$$i<j, the function$$x \!\mapsto \! f_j(x) \!-\! f_i(x)$$x↦fj(x)-fi(x)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 sucharrangements 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$$2^{\Theta (n^2)}$$2Θ(n2)isomorphism classes of arrangements of approaching pseudo-lines (while there are only$$2^{\Theta (n \log n)}$$2Θ(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.