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Home > Research > Publications & Outputs > Coning, symmetry, and spherical frameworks
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Coning, symmetry, and spherical frameworks

Research output: Contribution to journalJournal article

Published

Journal publication date1/10/2012
JournalDiscrete & computational geometry
Journal number3
Volume48
Number of pages36
Pages622-657
Original languageEnglish

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 (Whiteley in Topol. Struct. 8:53–70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561–598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric Md 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 Ed , cones in Ed+1 , Sd , Md , and ℍ d . We also consider the further extensions associated with the other Cayley–Klein geometries overlaid on the shared underlying projective geometry.