Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Symmetry as a sufficient condition for a finite flex
AU - Schulze, Bernd
PY - 2010
Y1 - 2010
N2 - 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.
AB - 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.
U2 - 10.1137/090776238
DO - 10.1137/090776238
M3 - Journal article
VL - 24
SP - 1291
EP - 1312
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
SN - 0895-4801
IS - 4
ER -