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Arrangement of pseudolines — Wikipedia (en)

Source: Wikipedia (en)
Original: https://en.wikipedia.org/wiki/Arrangement_of_pseudolines
License (as archived): CC-BY-SA-4.0
Archived: 2026-11-09T23:44:06.662Z
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Arrangement of pseudolines — Wikipedia (en) — Wikipedia (en), https://en.wikipedia.org/wiki/Arrangement_of_pseudolines (CC-BY-SA-4.0); archived copy: https://metraipolis.com/documento/dc_236b80247101 (Metraipolis archive dc_236b80247101, sha256 30b7704c2b0609ded2c3f2013d213a3798b461290956496dc69921c50e128277)

BibTeX

@misc{dc236b80247101,
  title = {Arrangement of pseudolines — Wikipedia (en)},
  howpublished = {Wikipedia (en)},
  url = {https://en.wikipedia.org/wiki/Arrangement_of_pseudolines},
  note = {License CC-BY-SA-4.0; archived copy https://metraipolis.com/documento/dc_236b80247101; sha256 30b7704c2b0609ded2c3f2013d213a3798b461290956496dc69921c50e128277},
  year = {2026}
}

Archived text

Reproduced under CC-BY-SA-4.0. Attribution: Wikipedia (en), https://en.wikipedia.org/wiki/Arrangement_of_pseudolines. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

An arrangement of pseudolines is a family of curves that share similar topological properties with a line arrangement. Most commonly, in the study of arrangements of lines, these have the simple property that each crosses every other line exactly once. These can be defined in the projective plane as simple closed curves any two of which meet in a single crossing point. Furthermore, in a simple (or uniform) arrangement, along with all lines being required to cross all others, no 3 pseudolines may cross at the same point. Every arrangement of finitely many pseudolines can be extended so that they become lines in a "spread", a type of non-Euclidean incidence geometry in which every two points of a topological plane are connected by a unique line (as in the Euclidean plane) but in which other axioms of Euclidean geometry may not apply. A common diagram used to represent an arrangement is the wiring diagram, a series of parallel lines with crossings between them drawn as an "X" in a simple crossing. When drawn this way, they can be described with notation for either the order in which each line crosses the other, the state of the orders between each crossing (or allowed groups of crossing whose orders do not matter), or as a list of pairs, each pair being the labels of 2 lines which have crossed, ordered in a given direction (usually left to right). They draw similarities to braids, although without any need to keep track of which crosses atop the other, the crossings may be seen as elements of the Coxeter group. Two arrangements are said to be "related by a triangle-flip" if one of them can be transformed into the other by changing the orientation of a single triangular face, or in other words, by moving one of the three pseudolines that form the triangle across the intersection of the other two. For any two simple wiring diagrams numbered 1 through n in order, one can be transformed into the other with a sequence of these triangle-flips (and vice versa). This fact has counterparts in the terminology of mutations on oriented matroids and Coxeter relations for reduced decomposition. Felsner and Valtr proved in 2009 that for an arrangement of n pseudolines, there are at most 20.657n2 simple arrangements. This improves upon the previous bounds of 20.792n2 in 1992 and 20.697n2 in 1997. They also proved a lower bound of 20.1887n2, which was improved in 2024 by Kühnast et al. to 20.2721n2 for sufficiently large n. The number of simple arrangements of n pseudolines in the projective plane with a marked cell is known up to n=13: 1, 1, 1, 2, 3, 16, 135, 3315, 158830, 14320182, 2343203071, 691470685682, 366477801792538 (sequence A006247 in the OEIS) The growth rate for the number of line arrangements is smaller compared to that of pseudoline arrangements; while for pseudolines An=2Θ(n2), for lines, An=2Θ(nlogn).

Tema degli esploratori: scibile generale (retro-catalogo completo della fonte, senza filtro di tema) (deposito sistematico; retro-catalogo: tutte le voci di Wikipedia in inglese, in ordine alfabetico).
Fonte: Wikipedia in inglese, voce «Arrangement of pseudolines» — https://en.wikipedia.org/wiki/Arrangement_of_pseudolines
Wikidata Q134531054: «pseudolines arranged largely to study arrangements of lines» (dati Wikidata in CC0 1.0) — https://www.wikidata.org/wiki/Q134531054
Autori: i contributori di Wikipedia (cronologia: https://en.wikipedia.org/wiki/Arrangement_of_pseudolines?action=history).
Licenza: Creative Commons Attribuzione-Condividi allo stesso modo 4.0 (CC BY-SA 4.0) — https://creativecommons.org/licenses/by-sa/4.0/. Il riuso richiede attribuzione e stessa licenza.
Estratto dell'introduzione non modificato nel contenuto; revisione 1375425460, consultata il 2026-11-09T23:41:07.377Z tramite l'API ufficiale di Wikipedia. Impronta SHA-256 dell'estratto: ea8ab7641c3061a4f5b4ee5530a522255c5d2d8501387fefab0634666950d968