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arXiv · 1806.08688

Generic Unlabeled Global Rigidity

Steven J. Gortler·Louis Theran·Dylan P. Thurston
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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.

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BibTeXRIS

Steven J. Gortler, Louis Theran, Dylan P. Thurston. 2019-06-14. Generic Unlabeled Global Rigidity. https://doi.org/10.1017/fms.2019.16

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