Rigidity of frameworks on expanding spheres

TL;DR

This paper develops a rigidity theory for frameworks constrained on expanding spheres, providing combinatorial characterizations for generic rigidity in specific dimensions and radii variations, bridging Euclidean and spherical rigidity concepts.

Contribution

It introduces new combinatorial criteria for the rigidity of frameworks on expanding spheres, including symmetry considerations, extending classical rigidity theory to variable-radius spherical frameworks.

Findings

Characterization of rigidity for 1D frameworks with arbitrary radii

Rigidity criteria for 2D frameworks with up to two variable radii

Symmetry-adapted counts for detecting flexibility

Abstract

A rigidity theory is developed for bar-joint frameworks in $\mathbb{R}^{d+1}$ whose vertices are constrained to lie on concentric $d$-spheres with independently variable radii. In particular, combinatorial characterisations are established for the rigidity of generic frameworks for $d=1$ with an arbitrary number of independently variable radii, and for $d=2$ with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in $\mathbb{R}^{3}$. Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the detection of symmetry-induced continuous flexibility in frameworks on spheres with variable radii are also provided.