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GENERIC UNLABELED GLOBAL RIGIDITY

Part of: Distance geometry Discrete geometry

Published online by Cambridge University Press:  30 July 2019

STEVEN J. GORTLER ,
LOUIS THERAN Open the ORCID record for LOUIS THERAN [Opens in a new window]  and
DYLAN P. THURSTON
STEVEN J. GORTLER
Affiliation:
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA; sjg@cs.harvard.edu
LOUIS THERAN
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews, Scotland; lst6@st-andrews.ac.uk
DYLAN P. THURSTON
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN, USA; dpthurst@indiana.edu

Abstract

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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.

MSC classification

Primary: 52C25: Rigidity and flexibility of structures 51K05: General theory
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e21
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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