Symmetry as a sufficient condition for a finite flex - Mathematics > Metric GeometryReport as inadecuate




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Abstract: We show that if the joints of a bar and joint framework $G,p$ arepositioned as `generically- as possible subject to given symmetry constraintsand $G,p$ possesses a `fully-symmetric- infinitesimal flex i.e., thevelocity vectors of the infinitesimal flex remain unaltered under all symmetryoperations of $G,p$, then $G,p$ also possesses a finite flex whichpreserves the symmetry of $G,p$ throughout the path. This and other relatedresults are obtained by symmetrizing techniques described by L. Asimov and B.Roth in their paper `The Rigidity Of Graphs- from 1978 and by using the factthat the rigidity matrix of a symmetric framework can be transformed into ablock-diagonalized form by means of group representation theory. The finiteflexes that can be detected with these symmetry-based methods can in generalnot be found with the analogous non-symmetric methods.



Author: Bernd Schulze

Source: https://arxiv.org/







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