Generic Unlabeled Global Rigidity

TL;DR

This paper demonstrates that for generic point configurations in Euclidean space, unlabeled point-pair lengths suffice to determine the configuration uniquely, matching the information provided by labeled lengths, regardless of the graph structure.

Contribution

It proves that in generic cases, unlabeled point-pair lengths are as informative as labeled lengths for configuration determination in Euclidean space.

Findings

Unlabeled lengths determine configurations in generic cases.

Labels have no effect on the uniqueness of configuration recovery.

Generic configurations are uniquely determined by unlabeled lengths.

Abstract

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. 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 $\mathbf{p}$ 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 length, 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 lengths (together with $d$ and $n$) iff it is determined by the labeled edge lengths.