## Abstract

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, such that for any two pseudo-lines ℓ_{i} and ℓ_{j} with i<j, the function 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 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 2Θ(n2) isomorphism classes of arrangements of approaching pseudo-lines (while there are only 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.

Original language | English |
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Pages (from-to) | 380-402 |

Number of pages | 23 |

Journal | Discrete and Computational Geometry |

Volume | 67 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2022 |

## Keywords

- Discrete geometry
- Order types
- Pseudo-line arrangements

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics