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

TL;DR

This paper extends the concept of rigid body motions to non-Euclidean planes with analytic norms, enabling new insights into the global rigidity of bar-joint frameworks beyond Euclidean spaces.

Contribution

It introduces generalized rigid body motions in non-Euclidean normed planes, advancing the understanding of global rigidity in these spaces.

Findings

New geometric and combinatorial results on global rigidity

Extension of rigid body motion concepts to analytic normed planes

Frameworks in non-Euclidean planes exhibit unique rigidity properties

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 $X$.