Non-Crossing Frameworks with Non-Crossing Reciprocals

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 straightline 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 pseudotriangulation with one non-pointed vertex. We show that for such pseudo-triangulation embeddings of planar Laman circuits which are sufficiently generic, the reciprocal is noncrossing 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 re∗ The first and third authors were supported by Grant BFM2001-1153, Spanish Ministry of Science and Technology, the second author was partly supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant RO 2338/2-1, and the last author was supported by grants from NSERC (Canada) and NIH (US). 568 D. Orden, G. Rote, F. Santos, B. Servatius, H. Servatius, and W. Whiteley ciprocal 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.