Symmetric Versions of Laman’s Theorem
Symmetric Versions of Laman’s Theorem
复制标题
拉曼定理的对称版本
DOI：
--
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发表时间：
2009
影响因子：
0.8
通讯作者：
B. Schulze
中科院分区：
文献类型：
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作者：
B. Schulze
摘要最近的研究表明，如果一个等静力杆节点框架具有非平凡的对称性，那么它必须满足一些非常简单的关于节点和杆的数量的限制，这些限制是由框架的各种对称操作“固定”的。对于小组来说
AbstractRecent 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
$mathcal{C}_{3}$
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
$mathcal{C}_{3}$
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
$mathcal{C}_{3}$
which provide alternate characterizations of “generically” isostatic graphs with
$mathcal{C}_{3}$
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
$mathcal{C}_{2}$
and
$mathcal{C}_{s}$
, in preparation, 2009), our techniques can be extended to establish analogous results for the symmetry groups
$mathcal{C}_{2}$
and
$mathcal{C}_{s}$
which are generated by a half-turn and a reflection in the plane, respectively.