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 - Symmetric Versions of Laman's Theorem
AU - Schulze, Bernd
PY - 2010/12
Y1 - 2010/12
N2 - Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are "fixed" by various symmetry operations of the framework. For the group C-3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46: 762-773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C-3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's theorem and Crapo's theorem for C-3 which provide alternate characterizations of "generically" isostatic graphs with C-3 symmetry. As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups C-2 and C-s, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups C-2 and C-s which are generated by a half-turn and a reflection in the plane, respectively.
AB - Recent work has shown that if an isostatic bar-and-joint framework possesses nontrivial symmetries, then it must satisfy some very simply stated restrictions on the number of joints and bars that are "fixed" by various symmetry operations of the framework. For the group C-3 which describes 3-fold rotational symmetry in the plane, we verify the conjecture proposed by Connelly et al. (Int. J. Solids Struct. 46: 762-773, 2009) that these restrictions on the number of fixed structural components, together with the Laman conditions, are also sufficient for a framework with C-3 symmetry to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In addition, we establish symmetric versions of Henneberg's theorem and Crapo's theorem for C-3 which provide alternate characterizations of "generically" isostatic graphs with C-3 symmetry. As shown in (Schulze, Combinatorial and geometric rigidity with symmetry constraints, Ph.D. thesis, York University, Toronto, Canada, 2009; Schulze, Symmetrized Laman theorems for the groups C-2 and C-s, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups C-2 and C-s which are generated by a half-turn and a reflection in the plane, respectively.
KW - Generic rigidity
KW - Infinitesimal rigidity
KW - Bar-and-joint frameworks
KW - Symmetric frameworks
KW - Laman graphs
KW - Henneberg construction
KW - 3Tree2 partition
U2 - 10.1007/s00454-009-9231-x
DO - 10.1007/s00454-009-9231-x
M3 - Journal article
VL - 44
SP - 946
EP - 972
JO - Discrete and Computational Geometry
JF - Discrete and Computational Geometry
SN - 0179-5376
IS - 4
ER -