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

Research output: Contribution to journalArticleResearchpeer-review

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 languageEnglish
Pages (from-to)51-61
Number of pages11
JournalComputational Geometry: Theory and Applications
Volume77
DOIs
Publication statusPublished - 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

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