GENERIC UNLABELED GLOBAL RIGIDITY

Abstract

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^{d}$ for some $n$ and some $d\geqslant 2$ . Each pair of points has a Euclidean distance in the configuration. Given some graph $G$ on $n$ vertices, we measure the point-pair distances corresponding to the edges of $G$ . In this paper, we study the question of when a generic $\mathbf{p}$ in $d$ dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair distances together with knowledge of $d$ and $n$ . In this setting the distances 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.