Arrangements of Approaching Pseudo-Lines

by   Stefan Felsner, et al.

We consider arrangements of n pseudo-lines in the Euclidean plane where each pseudo-line ℓ_i is represented by a bi-infinite connected x-monotone curve f_i(x), x ∈ℝ, s.t. for any two pseudo-lines ℓ_i and ℓ_j with i < j, the function 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: 1. There are arrangements of pseudo-lines that are not realizable with approaching pseudo-lines. 2. 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: 1. There are 2^Θ(n^2) isomorphism classes of arrangements of approaching pseudo-lines (while there are only 2^Θ(n log n) isomorphism classes of line arrangements). 2. 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.


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