Partitioning axis-parallel lines in 3D
| Autor: | Aronov, Boris, Basit, Abdul, de Berg, Mark, Gudmundsson, Joachim |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Computing in Geometry and Topology, 2.1(2023), 9:1-9:20 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.57717/cgt.v2i1.14 |
| Popis: | Let $L$ be a set of $n$ axis-parallel lines in $\mathbb{R}^3$. We are are interested in partitions of $\mathbb{R}^3$ by a set $H$ of three planes such that each open cell in the arrangement $\mathcal{A}(H)$ is intersected by as few lines from $L$ as possible. We study such partitions in three settings, depending on the type of splitting planes that we allow. We obtain the following results. $\bullet$ There are sets $L$ of $n$ axis-parallel lines such that, for any set $H$ of three splitting planes, there is an open cell in $\mathcal{A}(H)$ that intersects at least~$\lfloor n/3 \rfloor-1 \approx \frac{1}{3}n$ lines. $\bullet$ If we require the splitting planes to be axis-parallel, then there are sets $L$ of $n$ axis-parallel lines such that, for any set $H$ of three splitting planes, there is an open cell in $\mathcal{A}(H)$ that intersects at least $\frac{3}{2}\lfloor n/4 \rfloor -1 \approx \left( \frac{1}{3}+\frac{1}{24}\right) n$ lines. Furthermore, for any set $L$ of $n$ axis-parallel lines, there exists a set $H$ of three axis-parallel splitting planes such that each open cell in $\mathcal{A}(H)$ intersects at most $\frac{7}{18} n = \left( \frac{1}{3}+\frac{1}{18}\right) n$ lines. $\bullet$ For any set $L$ of $n$ axis-parallel lines, there exists a set $H$ of three axis-parallel and mutually orthogonal splitting planes, such that each open cell in $\mathcal{A}(H)$ intersects at most $\lceil \frac{5}{12} n \rceil \approx \left( \frac{1}{3}+\frac{1}{12}\right) n$ lines. Comment: 21 pages, minor changes, accepted to Computing in Geometry and Topology |
| Databáze: | arXiv |
| Externí odkaz: |
|