Coning, symmetry and spherical frameworks

TL;DR

This paper develops unified methods to transfer rigidity results across Euclidean, spherical, Minkowskian, and hyperbolic geometries, simplifying the analysis of symmetric frameworks in these diverse metrics.

Contribution

It reworks and simplifies techniques for transferring infinitesimal and finite rigidity results across multiple geometries, including Minkowskian and hyperbolic spaces, for symmetric frameworks.

Findings

New transfer results for infinitesimal and finite motions in various geometries.

Unified approach applicable to multiple metrics including Minkowskian and hyperbolic.

Extensions to Cayley-Klein geometries.

Abstract

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, (b) the further transfer of these results to spherical space via associated rigidity matrices, and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix. Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric $\M^{d}$ and the hyperbolic metric $\H^{d}$. This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among $\E^{d}$, cones in $E^{d+1}$, $\SS^{d}$, $\M^{d}$, and $\H^{d}$. We also consider the further extensions associated with the other Cayley-Klein geometries overlaid on the shared underlying projective geometry.