Rigidity of Frameworks Supported on Surfaces

Abstract

A theorem of Laman gives a combinatorial characterization of the graphs that admit a realization as a minimally rigid generic bar-joint framework in $\mathbb{R}^2$. A more general theory is developed for frameworks in $\mathbb{R}^3$ whose vertices are constrained to move on a two-dimensional smooth submanifold $\mathcal{M}$. Furthermore, when $\mathcal{M}$ 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.