2014/06/23 by Bill Jackson, Anthony Nixon, Jackson, Bill +1
Biochemistry, Genetics and Molecular Biology · Engineering · #52C25 #Advanced Materials and Mechanics #Cellular Mechanics and Interactions #FOS: Mathematics #Metric Geometry (math.MG) #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.1406.5996
openalex publication_date 2014/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2005, Bob Connelly showed that a generic framework in \bRd 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 \bR3. 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.