Stress matrices and global rigidity of frameworks on surfaces

TL;DR

This paper extends the concept of stress matrices and global rigidity from frameworks in Euclidean space to those on certain surfaces in three-dimensional space, establishing conditions for rigidity on cylinders, cones, and ellipsoids.

Contribution

It introduces a natural surface stress matrix for frameworks on specific surfaces and proves that maximum rank of this matrix guarantees global rigidity on these surfaces, extending prior Euclidean results.

Findings

Maximum rank stress matrices imply global rigidity on cylinders and ellipsoids.

The surface stress matrix concept is natural and applicable to frameworks on these surfaces.

The results are preserved under 1-extension, aiding characterization of rigidity on cylinders.

Abstract

In 2005, Bob Connelly showed that a generic framework in $\bR^d$ is globally rigid if it has a stress matrix of maximum possible rank, and that this sufficient condition for generic global rigidity is preserved by the 1-extension operation. His results gave a key step in the characterisation of generic global rigidity in the plane. We extend these results to frameworks on surfaces in $\bR^3$. For a framework on a family of concentric cylinders, cones or ellipsoids, we show that there is a natural surface stress matrix arising from assigning edge and vertex weights to the framework, in equilibrium at each vertex. In the case of cylinders and ellipsoids, we show that having a maximum rank stress matrix is sufficient to guarantee generic global rigidity on the surface. We then show that this sufficient condition for generic global rigidity is preserved under 1-extension and use this to make progress on the problem of characterising generic global rigidity on the cylinder.