2022/06/15 by Sean Dewar, Dewar, Sean, John F. Hewetson +3
Engineering · #Structural Analysis and Optimization #Advanced Materials and Mechanics
paper · doi:10.48550/arxiv.2206.07426
A bar-joint framework (G,p) in the Euclidean space 𝔼d is globally rigid if it is the unique realisation, up to rigid congruences, of G in 𝔼d with the edge lengths of (G,p). Building on key results of Hendrickson and Connelly, Jackson and Jordán gave a complete combinatorial characterisation of when a generic framework is global rigidity in 𝔼2. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on ℝ2 ∖ \0\. More precisely, we show that a graph G=(V,E) is globally rigid 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. The main technical tool is a recursive construction of 2-connected and redundantly rigid graphs in analytic normed planes. We also obtain some sufficient conditions for global rigidity as corollaries of our main result and prove that the analogous necessary conditions hold in d-dimensional analytic normed spaces.