On the universal rigidity of generic bar frameworks

Let V=1, …, n be a finite set. An r-configuration is a mapping p: V→Rr, where p1, …, pn are not contained in a proper hyper-plane. A framework G (p) in Rr is an r-configuration together with a graph G= (V, E) such that every two points corresponding to adjacent vertices of G are constrained to stay the same distance apart. A framework G (p) is said to be generic if all the coordinates of p1, …, pn are algebraically independent over the integers. A framework G (p) in Rr is said to be unique if there does not exist a framework G (q) in Rs, for some s, 1≤s≤n−1, such that ||qi−qj||=||pi−pj|| for all (i, j) ∈E. In this paper we present a sufficient condition for a generic framework G (p) to be unique, and we conjecture that this condition is also necessary. Connections with the closely related problems of global rigidity and dimensional rigidity are also discussed.