Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

TL;DR

This paper extends Hendrickson's necessary conditions for global rigidity from planar frameworks to frameworks constrained on surfaces like cylinders, cones, and ellipsoids, advancing understanding of rigidity in higher dimensions.

Contribution

It provides the first analogues of Hendrickson's necessary conditions for global rigidity of frameworks on specific surfaces in three-dimensional space.

Findings

Necessary conditions for global rigidity on cylinders, cones, and ellipsoids derived.

Frameworks on these surfaces satisfy similar combinatorial properties as in the plane.

Results contribute to the broader understanding of rigidity on curved surfaces.

Abstract

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jord\'an, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in $\mathbb{R}^3$ when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson's necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.