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

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Abstract

We consider sets $L = 1, n$ 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 $, 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 $, 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 $$ 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 $On3+nd(d-2)$.
LanguageEnglish
Pages51-61
Number of pages11
JournalComputational Geometry: Theory and Applications
Volume77
DOIs
StatusPublished - 2019

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 O, Fabila-Monroy R, Hurtado F, Perez-Lantero P, Ruiz-Vargas AJ, Urrutia Galicia J et al. Cross-sections of line configurations in $R^3$ and $(d-2)$-flat configurations in $R^d$. Computational Geometry: Theory and Applications. 2019;77:51-61. Available from, DOI: https://doi.org/10.1016/j.comgeo.2018.02.005
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
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abstract = "We consider sets $L = 1, n$ 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 $, 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 $, 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 $$ 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 $On3+nd(d-2)$.",
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",
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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 $L = 1, n$ 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 $, 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 $, 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 $$ 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 $On3+nd(d-2)$.

AB - We consider sets $L = 1, n$ 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 $, 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 $, 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 $$ 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 $On3+nd(d-2)$.

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