Uniquely Realisable Graphs in Analytic Normed Planes

Abstract A framework (G, p) in Euclidean space E^d is globally rigid if it is the unique realisation, up to rigid congruences, of G with the edge lengths of (G, p). Building on key results of Hendrickson 28 and Connelly 14, Jackson and Jordán 29 gave a complete combinatorial characterisation of when a generic framework is global rigidity in E^2. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on R^2 \0\. Specifically, we show that a graph G= (V, E) has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if G is 2-connected and G-e contains 2 edge-disjoint spanning trees for all e E. We also prove that the analogous necessary conditions hold in d-dimensional normed spaces.