### 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))).

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

Pages | 51-61 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 77 |

DOIs | |

Status | Published - 2019 |

### Fingerprint

### 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

*Computational Geometry: Theory and Applications*,

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

}

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

T2 - Computational geometry

JF - Computational geometry

SN - 0925-7721

ER -