Nixon, Anthony and Owen, J. C. and Power, Stephen (2012) Rigidity of frameworks supported on surfaces. SIAM Journal on Discrete Mathematics, 26 (4). pp. 1733-1757. ISSN 0895-4801
NixOwenPowSIDMA.pdf - Submitted Version
Available under License Creative Commons Attribution Non-commercial No Derivatives.
Download (306kB)
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.