Symmetry as a sufficient condition for a finite flex

Schulze, Bernd (2010) Symmetry as a sufficient condition for a finite flex. SIAM Journal on Discrete Mathematics, 24 (4). pp. 1291-1312. ISSN 0895-4801

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We show that if the joints of a bar and joint framework $(G,p)$ are positioned as “generically” as possible subject to given symmetry constraints and $(G,p)$ possesses a “fully symmetric” infinitesimal flex (i.e., the velocity vectors of the infinitesimal flex remain unaltered under all symmetry operations of $(G,p)$), then $(G,p)$ also possesses a finite flex which preserves the symmetry of $(G,p)$ throughout the path. This and other related results are obtained by symmetrizing techniques described by L. Asimov and B. Roth in their 1978 paper “The Rigidity of Graphs” [Trans. Amer. Math. Soc., 245 (1978), pp. 279–289] and by using the fact that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form by means of group representation theory. The finite flexes that can be detected with these symmetry-based methods can in general not be found with the analogous nonsymmetric methods.

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Journal Article
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SIAM Journal on Discrete Mathematics
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26 Apr 2013 09:58
Last Modified:
18 Sep 2023 00:39