Rigidity of symmetric frameworks on the cylinder

TL;DR

This paper investigates the rigidity of symmetric frameworks constrained to a cylindrical surface in three-dimensional space, providing combinatorial conditions for minimal rigidity under certain symmetries.

Contribution

It establishes necessary and sufficient combinatorial conditions for symmetric frameworks on a cylinder to be minimally rigid, focusing on cyclic symmetry groups.

Findings

Necessary conditions for isostaticity under symmetry

Sufficient conditions for cyclic symmetry groups

Complete combinatorial characterization for certain symmetries

Abstract

A bar-joint framework $(G,p)$ is the combination of a finite simple graph $G=(V,E)$ and a placement $p:V\rightarrow \mathbb{R}^d$. The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in $\mathbb{R}^3$ restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.