Generic unlabeled global rigidity

Authors: Steven J. Gortler, Louis Theran, and Dylan P. Thurston
Preprint: 1806.08688, 2018
Full text: arXiv

Let be a configuration of $n$ points in for some and some . Each pair of points has a Euclidean length in the configuration. Given some
graph on vertices, we measure the point-pair lengths corresponding to the edges of .

In this paper, we study the question of when a generic in dimensions will be uniquely determined (up to an unknowable Euclidean transformation) from a given set of point-pair lengths together with knowledge of and . 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 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 and ) iff it is determined by the labeled edge lengths.