### Abstract

Language | English |
---|---|

Pages | 51-61 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 77 |

DOIs | |

Status | Published - 2019 |

### Cite this

*Computational Geometry: Theory and Applications*,

*77*, 51-61. DOI: https://doi.org/10.1016/j.comgeo.2018.02.005

**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.

Research output: Contribution to journal › Article › Research › peer-review

*Computational Geometry: Theory and Applications*, vol. 77, pp. 51-61. DOI: https://doi.org/10.1016/j.comgeo.2018.02.005

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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 $L = 1, n$ 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 $On3+nd(d-2)$.

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

T2 - Computational geometry

JF - Computational geometry

SN - 0925-7721

ER -