Generalised rigid body motions in non-Euclidean planes with applications to global rigidity

Abstract

A bar-joint framework $(G,p)$ in a (non-Euclidean) real normed plane $X$ is the combination of a finite, simple graph $G$ and a placement $p$ of the vertices in $X$. A framework $(G,p)$ is globally rigid in $X$ if every other framework $(G,q)$ in $X$ with the same edge lengths as $(G,p)$ arises from an isometry of $X$. The weaker property of local rigidity in normed planes (where only $(G,q)$ within a neighbourhood of $(G,p)$ are considered) has been studied by several researchers over the last 5 years after being introduced by Kitson and Power for $\ell_p$-norms. However global rigidity is an unexplored area for general normed spaces, despite being intensely studied in the Euclidean context by many groups over the last 40 years. In order to understand global rigidity in $X$, we introduce new generalised rigid body motions in normed planes where the norm is determined by an analytic function. This theory allows us to deduce several geometric and combinatorial results concerning the global rigidity of bar-joint frameworks in $X$.