Rigidity of Frameworks Supported on Surfaces

TL;DR

This paper extends rigidity theory to frameworks constrained on surfaces in three-dimensional space, providing combinatorial characterizations for minimal rigidity on specific surfaces like spheres, planes, and cylinders.

Contribution

It develops a general theory for frameworks on surfaces in R^3 and offers necessary and sufficient conditions for minimal rigidity on certain surfaces.

Findings

Characterization of minimally rigid frameworks on surfaces in R^3

Necessary and sufficient conditions for frameworks on spheres, planes, cylinders

Extension of Laman's theorem to surface-constrained frameworks

Abstract

A theorem of Laman gives a combinatorial characterisation of the graphs that admit a realisation as a minimally rigid generic bar-joint framework in $\bR^2$. A more general theory is developed for frameworks in $\bR^3$ whose vertices are constrained to move on a two-dimensional smooth submanifold $\M$. Furthermore, when $\M$ is a union of concentric spheres, or a union of parallel planes or a union of concentric cylinders, necessary and sufficient combinatorial conditions are obtained for the minimal rigidity of generic frameworks.