Coning, Symmetry and Spherical Frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning [28], (b) the further transfer of these results to spherical space via associated rigidity matrices [19], and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetryderived orbit rigidity matrix [23]. Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metricMd and the hyperbolic metricHd. This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among Ed, cones in Ed+1, Sd, Md, and Hd. We also consider the further extensions associated with the other Cayley-Klein geometries overlaid on the shared underlying projective geometry.