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Discrete & Computational Geometry

, Volume 32, Issue 4, pp 567–600 | Cite as

Non-Crossing Frameworks with Non-Crossing Reciprocals

  • David Orden
  • Günter Rote
  • Francisco Santos
  • Brigitte Servatius
  • Herman Servatius
  • Walter  Whiteley
Article

Abstract

We study non-crossing frameworks in the plane for which the classical reciprocal on the dual graph is also non-crossing. We give a complete description of the self-stresses on non-crossing frameworks G whose reciprocals are non-crossing, in terms of: the types of faces (only pseudo-triangles and pseudo-quadrangles are allowed); the sign patterns in the stress on G; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of non-crossingness and rigidity of straight-line plane graphs is studied, pseudo-triangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudo-triangulation with one non-pointed vertex. We show that for such pseudo-triangulation embeddings of planar Laman circuits which are sufficiently generic, the reciprocal is non-crossing and again a pseudo-triangulation embedding of a planar Laman circuit. For a singular (non-generic) pseudo-triangulation embedding of a planar Laman circuit, the reciprocal is still non-crossing and a pseudo-triangulation, but its underlying graph may not be a Laman circuit. Moreover, all the pseudo-triangulations which admit a non-crossing reciprocal arise as the reciprocals of such, possibly singular, stresses on pseudo-triangulation Laman circuits. All self-stresses on a planar graph correspond to liftings to piecewise linear surfaces in 3-space. We prove characteristic geometric properties of the lifts of such non-crossing reciprocal pairs.

Keywords

Computational Mathematic Special Interest Geometric Property Planar Graph Sign Pattern 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer 2004

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria, E-39005 Santander Spain
  2. 2.Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195 BerlinGermany
  3. 3.Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609USA
  4. 4.Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3Canada

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