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


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.


Computational Mathematic Special Interest Geometric Property Planar Graph Sign Pattern 
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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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