On generalized reciprocal diagrams for self-stressed frameworks

We present the duality between edge lengths and axial forces in self-stressed frameworks, upon which are based reciprocal diagrams, introduced by Maxwell and Cremona in the nineteenth century. The main concepts and principles are simplified by using a graph theoretic approach. We describe some unusual orthogonality relations, involving lengths, axial forces and their rates of change. Reciprocal diagrams, which exist for frameworks with underlying planar graph, are extended also to the non-planar case by introducing a new criterion. When this criterion can be applied, different reciprocals can be obtained as symmetric frameworks. The same criterion can also be applied to planar cases giving new reciprocals as a result. Although reciprocal diagrams cannot be obtained for all self-stressed frameworks, the presented duality always holds and it provides useful insights for design and form-finding purposes.