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

Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

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))).

Originalspracheenglisch
Seiten (von - bis)51-61
Seitenumfang11
FachzeitschriftComputational Geometry: Theory and Applications
Jahrgang77
DOIs
PublikationsstatusVeröffentlicht - 2019

Fingerprint

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

Schlagwörter

    ASJC Scopus subject areas

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

    Dies zitieren

    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, Jahrgang 77, 2019, S. 51-61.

    Publikation: Beitrag in einer FachzeitschriftArtikelForschungBegutachtung

    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 ; Jahrgang 77. S. 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 -