Generic global rigidity

Suppose a finite configuration of labeled points p = (p(1).....p(n)) in E-d stop is given along with certain pairs of those points determined by a graph G such that the coordinates of the points of p are generic, i.e., algebraically independent over the integers. If another corresponding configuration q = (q(1).....q(n)) in E-d stop is given such that the corresponding edges of G for p and q have the same length, we provide a sufficient condition to ensure that p and q are congruent in E-d stop. This condition, together with recent results of Jackson and Jordan [JJ], give necessary and sufficient conditions for a graph being generically globally rigid in the plane.