Configurations of Lines in 3-Space and Rigidity of Planar Structures

Author Orit E. Raz



PDF
Thumbnail PDF

File

LIPIcs.SoCG.2016.58.pdf
  • Filesize: 479 kB
  • 14 pages

Document Identifiers

Author Details

Orit E. Raz

Cite As Get BibTex

Orit E. Raz. Configurations of Lines in 3-Space and Rigidity of Planar Structures. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 58:1-58:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.58

Abstract

Let L be a sequence (l_1,l_2,...,l_n) of n lines in C^3. We define the intersection graph G_L=([n],E) of L, where [n]:={1,..., n}, and with {i,j} in E if and only if i\neq j and the corresponding lines l_i and l_j 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 subset G_L. 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 realization L of G, that consists of n pairwise distinct lines, we have G_L=K_n, 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.

Subject Classification

Keywords
  • Line configurations
  • Rigidity
  • Global Rigidity
  • Laman graphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. L. Asimow and B. Roth. The rigidity of graphs. Trans. Amer. Math. Soc., 245:279-289, 1978. Google Scholar
  2. R. Connelly. Generic global rigidity. Discrete Comput. Geom., 33:549-563, 2005. Google Scholar
  3. H. Crapo. Structural rigidity. Structural Topology, 1:26-45, 1979. Google Scholar
  4. Gy. Elekes and M. Sharir. Incidences in three dimensions and distinct distances in the plane. Combinat. Probab. Comput., 20:571-608, 2011. Google Scholar
  5. L. Guth and N. H. Katz. On the Erdős distinct distances problem in the plane. Annals Math., 18:155-190, 2015. Google Scholar
  6. R. Hartshorne. Algebraic Geometry. Springer-Verlag, New York, 1977. Google Scholar
  7. B. Hendrickson. Conditions for unique graph embeddings. Technical Report 88-950, Department of Computer Science, Cornell University, 1988. Google Scholar
  8. B. Jackson and T. Jordán. Connected rigidity matroids and unique realizations of graphs. J. Combinat. Theory, Ser. B, 94:1-29, 2005. Google Scholar
  9. B. Jackson, T. Jordán, and Z. Szabadka. Globally linked pairs of vertices in equivalent realizations of graphs. Discrete Comput. Geom., 35:493-512, 2006. Google Scholar
  10. G. Laman. On graphs and rigidity of plane skeletal structures. J. Engrg. Math., 4:333-338, 1970. Google Scholar
  11. J.L. Martin. Geometry of graph varieties. Trans. Amer. Math. Soc., 355:4151-4169, 2003. Google Scholar
  12. T. Tao. Lines in the Euclidean group SE(2). Blog post, available at URL: https://terrytao.wordpress.com/2011/03/05/lines-in-the-euclidean-group-se2/.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail