Arrangements of Approaching Pseudo-Lines

Abstract

We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line $$\ell _i$$ ℓ i is represented by a bi-infinite connected x-monotone curve $$f_i(x)$$ f i ( x ) , $$x \in \mathbb {R}$$ x ∈ R , such that for any two pseudo-lines $$\ell _i$$ ℓ i and $$\ell _j$$ ℓ j with $$i \!<\! j$$ i < j , the function $$x \!\mapsto \! f_j(x) \!-\! f_i(x)$$ x ↦ f j ( x ) - f i ( 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 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 $$2^{\Theta (n^2)}$$ 2 Θ ( n 2 ) isomorphism classes of arrangements of approaching pseudo-lines (while there are only $$2^{\Theta (n \log n)}$$ 2 Θ ( n log n ) 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.