Generic unlabeled global rigidity

Abstract

Let p be a configuration of n points in ℝd for some n and some d ≥ 2. Each pair of points has a Euclidean length in the configuration. Given some graph G on n vertices, we measure the point-pair lengths corresponding to the edges of G. In this paper, we study the question of when a generic p in d dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair lengths together with knowledge of d and n. In this setting the lengths are given simply as a set of real numbers; they are not labeled with the combinatorial data that describes which point-pair gave rise to which distance, nor is data about G given. We show, perhaps surprisingly, that in terms of generic uniqueness, labels have no effect. A generic configuration is determined by an unlabeled set of point-pair distances (together with d and n) if and only if it is determined by the labeled distances.