Configurations of Lines in Space and Combinatorial Rigidity

Orit E. Raz*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let L be a sequence (ℓ1, ℓ2, … , ℓn) of n lines in C3. We define the intersection graphGL= ([ 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
Pages (from-to)986-1009
Number of pages24
JournalDiscrete and Computational Geometry
Volume58
Issue number4
DOIs
StatePublished - 1 Dec 2017
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

Keywords

  • Elekes–Sharir framework
  • Graph rigidity
  • Line configurations

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