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Symmetric Versions of Laman's Theorem

Symmetric Versions of Laman's Theorem,10.1007/s00454-009-9231-x,Discrete & Computational Geometry,Bernd Schulze

Symmetric Versions of Laman's Theorem   (Citations: 5)
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Recent work has shown that if an isostatic bar and joint framework possesses non-trivial 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 in [4] 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 [19], 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.
Journal: Discrete & Computational Geometry - DCG , vol. 44, no. 4, pp. 946-972, 2010
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    • ...Within the context of symmetric frameworks, there are generalizations for key plane groups (C3, Cs , and C2) presented in [32, 36]...
    • ...This has been extended to fully symmetric inductive techniques, still with the standard rigidity matrix, in [32, 36]...
    • ...For 3-fold rotational symmetry in the plane, the symmetric inductive techniques in [36] do provide a full characterization for the corresponding orbit matrix to be independent and of full rank...

    Bernd Schulzeet al. The Orbit Rigidity Matrix of a Symmetric Framework

    • ...The present paper therefore aims to provide the mathematical foundation that is necessary to give rigorous proofs of these results, as well as additional results and conjectures relating to the rigidity of symmetric frameworks (such as the ones stated in [5, 20, 21, 22, 23], for example)...
    • ...The definitions and results presented in this paper have already been used to establish symmetrized versions of a variety of famous theorems in each of the above-mentioned theories (such as Maxwell’s rule from 1864 and Laman’s Theorem, for example) and are fundamental to the results in [20, 21, 22, 23]...
    • ...The results in [22] and many of the results in [20] are based on this definition of generic...

    Bernd Schulze. Injective and non-injective realizations with symmetry

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