Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$

Oswin Aichholzer, Ruy Fabila-Monroy, Ferran Hurtado, Pablo Perez-Lantero, Andres J. Ruiz-Vargas, Jorge Urrutia Galicia, Birgit Vogtenhuber

Research output: Contribution to journalArticleResearchpeer-review

Abstract

We consider sets L={ℓ 1,…,ℓ n} of n labeled lines in general position in R 3, and study the order types of point sets {p 1,…,p n} that stem from the intersections of the lines in L with (directed) planes Π not parallel to any line of L, that is, the proper cross-sections of L. As two main results, we show that the number of different order types that can be obtained as cross-sections of L is O(n 9) when considering all possible planes Π and O(n 3) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R 2 moving with constant (but possibly different) speeds along straight lines forms at most O(n 3) different order types over time. We further generalize the setting from R 3 to R d with d>3, showing that the number of order types that can be obtained as cross-sections of a set of n labeled (d−2)-flats in R d with planes is O(((n3)+nd(d−2))).

Original languageEnglish
Pages (from-to)51-61
Number of pages11
JournalComputational Geometry: Theory and Applications
Volume77
DOIs
Publication statusPublished - 2019

Fingerprint

Order Type
Cross section
Configuration
Line
Straight Line
Set of points
Pairwise
Intersection
Imply
Generalise

Keywords

  • Cross-section
  • Lines in 3-space
  • Moving points in the plane
  • Order type

ASJC Scopus subject areas

  • Computational Mathematics
  • Control and Optimization
  • Geometry and Topology
  • Computer Science Applications
  • Computational Theory and Mathematics

Cite this

Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$. / Aichholzer, Oswin; Fabila-Monroy, Ruy; Hurtado, Ferran; Perez-Lantero, Pablo; Ruiz-Vargas, Andres J.; Urrutia Galicia, Jorge; Vogtenhuber, Birgit.

In: Computational Geometry: Theory and Applications, Vol. 77, 2019, p. 51-61.

Research output: Contribution to journalArticleResearchpeer-review

Aichholzer, Oswin ; Fabila-Monroy, Ruy ; Hurtado, Ferran ; Perez-Lantero, Pablo ; Ruiz-Vargas, Andres J. ; Urrutia Galicia, Jorge ; Vogtenhuber, Birgit. / Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$. In: Computational Geometry: Theory and Applications. 2019 ; Vol. 77. pp. 51-61.
@article{c5cd140850234d869d2f729aaf18dd2a,
title = "Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$",
abstract = "We consider sets L={ℓ 1,…,ℓ n} of n labeled lines in general position in R 3, and study the order types of point sets {p 1,…,p n} that stem from the intersections of the lines in L with (directed) planes Π not parallel to any line of L, that is, the proper cross-sections of L. As two main results, we show that the number of different order types that can be obtained as cross-sections of L is O(n 9) when considering all possible planes Π and O(n 3) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R 2 moving with constant (but possibly different) speeds along straight lines forms at most O(n 3) different order types over time. We further generalize the setting from R 3 to R d with d>3, showing that the number of order types that can be obtained as cross-sections of a set of n labeled (d−2)-flats in R d with planes is O(((n3)+nd(d−2))).",
keywords = "Cross-section, Lines in 3-space, Moving points in the plane, Order type",
author = "Oswin Aichholzer and Ruy Fabila-Monroy and Ferran Hurtado and Pablo Perez-Lantero and Ruiz-Vargas, {Andres J.} and {Urrutia Galicia}, Jorge and Birgit Vogtenhuber",
note = "Special Issue of CCCG 2014",
year = "2019",
doi = "https://doi.org/10.1016/j.comgeo.2018.02.005",
language = "English",
volume = "77",
pages = "51--61",
journal = "Computational geometry",
issn = "0925-7721",
publisher = "Elsevier B.V.",

}

TY - JOUR

T1 - Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$

AU - Aichholzer, Oswin

AU - Fabila-Monroy, Ruy

AU - Hurtado, Ferran

AU - Perez-Lantero, Pablo

AU - Ruiz-Vargas, Andres J.

AU - Urrutia Galicia, Jorge

AU - Vogtenhuber, Birgit

N1 - Special Issue of CCCG 2014

PY - 2019

Y1 - 2019

N2 - We consider sets L={ℓ 1,…,ℓ n} of n labeled lines in general position in R 3, and study the order types of point sets {p 1,…,p n} that stem from the intersections of the lines in L with (directed) planes Π not parallel to any line of L, that is, the proper cross-sections of L. As two main results, we show that the number of different order types that can be obtained as cross-sections of L is O(n 9) when considering all possible planes Π and O(n 3) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R 2 moving with constant (but possibly different) speeds along straight lines forms at most O(n 3) different order types over time. We further generalize the setting from R 3 to R d with d>3, showing that the number of order types that can be obtained as cross-sections of a set of n labeled (d−2)-flats in R d with planes is O(((n3)+nd(d−2))).

AB - We consider sets L={ℓ 1,…,ℓ n} of n labeled lines in general position in R 3, and study the order types of point sets {p 1,…,p n} that stem from the intersections of the lines in L with (directed) planes Π not parallel to any line of L, that is, the proper cross-sections of L. As two main results, we show that the number of different order types that can be obtained as cross-sections of L is O(n 9) when considering all possible planes Π and O(n 3) when restricting considerations to sets of pairwise parallel planes, where both bounds are tight. The result for parallel planes implies that any set of n points in R 2 moving with constant (but possibly different) speeds along straight lines forms at most O(n 3) different order types over time. We further generalize the setting from R 3 to R d with d>3, showing that the number of order types that can be obtained as cross-sections of a set of n labeled (d−2)-flats in R d with planes is O(((n3)+nd(d−2))).

KW - Cross-section

KW - Lines in 3-space

KW - Moving points in the plane

KW - Order type

UR - http://www.scopus.com/inward/record.url?scp=85042560536&partnerID=8YFLogxK

U2 - https://doi.org/10.1016/j.comgeo.2018.02.005

DO - https://doi.org/10.1016/j.comgeo.2018.02.005

M3 - Article

VL - 77

SP - 51

EP - 61

JO - Computational geometry

JF - Computational geometry

SN - 0925-7721

ER -