1 Arrangements of Approaching Pseudo-Lines ∗1 †2 3 2 Stefan Felsner , Alexander Pilz , and Patrick Schnider 1 3 Institut f¨urMathematik, Technische Universit¨atBerlin, 4
[email protected] 2 5 Institute of SoftwareTechnology, Graz University of Technology, Austria, 6
[email protected] 3 7 Department of Computer Science, ETH Z¨urich, Switzerland, 8
[email protected] 9 August 23, 2020 10 Abstract 11 We consider arrangements of n pseudo-lines in the Euclidean plane where each 12 pseudo-line `i is represented by a bi-infinite connected x-monotone curve fi(x), x 2 R, 13 such that for any two pseudo-lines `i and `j with i < j, the function x 7! fj (x)−fi(x) is 14 monotonically decreasing and surjective (i.e., the pseudo-lines approach each other until 15 they cross, and then move away from each other). We show that such arrangements of 16 approaching pseudo-lines, under some aspects, behave similar to arrangements of lines, 17 while for other aspects, they share the freedom of general pseudo-line arrangements. 18 For the former, we prove: 19 • There are arrangements of pseudo-lines that are not realizable with approaching 20 pseudo-lines. 21 • Every arrangement of approaching pseudo-lines has a dual generalized configura- 22 tion of points with an underlying arrangement of approaching pseudo-lines. 23 For the latter, we show: Θ(n2) 24 • There are 2 isomorphism classes of arrangements of approaching pseudo-lines Θ(n log n) 25 (while there are only 2 isomorphism classes of line arrangements).