Straightened hinged frameworks

The properties of systems of pairwise distances between points thrown into a Euclidean space have been poorly studied to date. These properties are described in terms of a map called the “edge function” (the “rigidity mapping”) and defining the behaviour of actual frameworks of levers and hinges. The simplest non-trivial case is that of rigidity mappings corresponding to plane frameworks (hingers) all fastened hinges of which lie on one straight line. In this paper rigidity mappings are investigated and examples of such straightened hingers with peculiar properties are presented.