Symmetry as a Sufficient Condition for a Finite Flex

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 unde...

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Bibliographic Details
Published in:SIAM journal on discrete mathematics 2010-01, Vol.24 (4), p.1291-1312
Main Author: Schulze, Bernd
Format: Article
Language:English
Subjects:
Blocking
Finite element analysis
Graphs
Mathematical analysis
Matrix
Preserves
Representations
Rigidity
Studies
Symmetry
Vectors (mathematics)
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Summary: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. [PUBLICATION ABSTRACT]
ISSN:0895-4801
1095-7146
DOI:10.1137/090776238