## 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 language | English |
---|---|

Pages (from-to) | 51-61 |

Number of pages | 11 |

Journal | Computational Geometry |

Volume | 77 |

DOIs | |

Publication status | Published - 2019 |

## 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