Generic unlabeled global rigidity
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Let p be a configuration of n points in ℝd for some n and some d ≥ 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 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 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.
Gortler , S , Theran , L & Thurston , D 2019 , ' Generic unlabeled global rigidity ' , Forum of Mathematics, Sigma , vol. 7 , e21 , pp. 1-34 . https://doi.org/10.1017/fms.2019.16
Forum of Mathematics, Sigma
Copyright © The Author(s) 2019. 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.
DescriptionThe first author was partially supported by NSF grant DMS-1564473.
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