Configurations of lines in 3-space and rigidity of planar structures

Orit E. Raz*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

Let L be a sequence (ℓ1, ℓ2, ..., ℓn) of n lines in double-struck C3. We define the intersection graph GL = ([n], E) of L, where [n] := {1, ..., n}, and with {i, j} ∈ E if and only if i ≠ j and the corresponding lines ℓi and ℓj intersect, or are parallel (or coincide). For a graph G = ([n], E), we say that a sequence L is a realization of G if G ⊂ GL. One of the main results of this paper is to provide a combinatorial characterization of graphs G = ([n], E) that have the following property: For every generic (see Definition 4.1) realization L of G, that consists of n pairwise distinct lines, we have GL = Kn, in which case the lines of L are either all concurrent or all coplanar. The general statements that we obtain about lines, apart from their independent interest, turns out to be closely related to the notion of graph rigidity. The connection is established due to the so-called Elekes-Sharir framework, which allows us to transform the problem into an incidence problem involving lines in three dimensions. By exploiting the geometry of contacts between lines in 3D, we can obtain alternative, simpler, and more precise characterizations of the rigidity of graphs.

Original languageAmerican English
Title of host publication32nd International Symposium on Computational Geometry, SoCG 2016
EditorsSandor Fekete, Anna Lubiw
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Pages58.1-58.14
ISBN (Electronic)9783959770095
DOIs
StatePublished - 1 Jun 2016
Externally publishedYes
Event32nd International Symposium on Computational Geometry, SoCG 2016 - Boston, United States
Duration: 14 Jun 201617 Jun 2016

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume51
ISSN (Print)1868-8969

Conference

Conference32nd International Symposium on Computational Geometry, SoCG 2016
Country/TerritoryUnited States
CityBoston
Period14/06/1617/06/16

Bibliographical note

Publisher Copyright:
© Orit Esther Raz.

Keywords

  • Global rigidity
  • Laman graphs
  • Line configurations
  • Rigidity

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