Non-Crossing Frameworks with Non-Crossing Reciprocals (English)

In: Discrete & Computational Geometry   ;  32 ,  4  ;  567-600  ;  2004

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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.

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Guest Editors#8217 Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
Guest Editors’ Preface
Bayer, Margaret M. / Lee, Carl W. / Sturmfels, Bernd | 2004
The Partial-Fractions Method for Counting Solutions to Integral Linear Systems
Beck, Matthias | 2004
Isoradial Bodies
Brandenberg, René / Dattasharma, Abhi / Gritzmann, Peter / Larman, David | 2004
Vertices of Gelfand--Tsetlin Polytopes
Loera, Jesús A. / McAllister, Tyrrell B. | 2004
A Complexity Bound on Faces of the Hull Complex
Develin, Mike | 2004
Enumerative Properties of Ferrers Graphs
Ehrenborg, Richard / Willigenburg, Stephanie | 2004
Tchebyshev Posets
Hetyei, Gábor | 2004
Mixed Fibre Polytopes
McMullen, Peter | 2004
Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices
Nyman, Kathryn / Swartz, Ed | 2004
Convex Combinatorial Optimization
Onn, Shmuel / Rothblum, Uriel G. | 2004
Non-Crossing Frameworks with Non-Crossing Reciprocals
Orden, David / Rote, Günter / Santos, Francisco / Servatius, Brigitte / Servatius, Herman / Whiteley, Walter | 2004
The Et-Construction for Lattices, Spheres and Polytopes
Paffenholz, Andreas / Ziegler, Günter M. | 2004
Graphs, Syzygies, and Multivariate Splines
Rose, Lauren L. | 2004