Symmetry-forced rigidity of frameworks on surfaces
Symmetry-forced rigidity of frameworks on surfaces
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表面框架的对称强制刚度
DOI：
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发表时间：
2013
期刊：
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通讯作者：
B. Schulze
中科院分区：
文献类型：
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作者：
A. Nixon;B. Schulze
拉曼基本定理描述了当在欧几里得平面上实现的杆节点框架允许其顶点的非平凡连续变形时。最近，这一观点以两种方式得到了扩展。首先是对某些点群对称但其他方面是一般的框架，其次是欧几里得三维空间中约束于二维代数变体上的框架。我们将这两种设置结合起来，并考虑在这些表面上实现的对称框架的刚性。首先，我们通过为此类框架建立一个适应对称的刚度矩阵，并将Jordán等人（2012）中的方法扩展到这个新背景，为框架对任何组和任何表面都具有对称强制刚性建立必要条件。由此在群标记商图上得到了几个新的对称自适应刚性拟阵。在表面为球体、圆柱体或锥体的情况下，我们还提供了一些对称群的一般对称强制刚性框架的组合特征，包括旋转、反射、反转和二面体对称。这些结果的证明是基于群标记商图上的一些新的henneberg型归纳构造，这些构造对应于所讨论的拟阵的基。对于三维空间中剩余的对称群，以及其他类型的曲面，我们提供了一些观察和猜想。
A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been extended in two ways. Firstly to frameworks that are symmetric with respect to some point group but are otherwise generic, and secondly to frameworks in Euclidean 3-space that are constrained to lie on 2-dimensional algebraic varieties. We combine these two settings and consider the rigidity of symmetric frameworks realised on such surfaces. First we establish necessary conditions for a framework to be symmetry-forced rigid for any group and any surface by setting up a symmetry-adapted rigidity matrix for such frameworks and by extending the methods in Jordán et al. (2012) to this new context. This gives rise to several new symmetry-adapted rigidity matroids on group-labelled quotient graphs. In the cases when the surface is a sphere, a cylinder or a cone we then also provide combinatorial characterisations of generic symmetry-forced rigid frameworks for a number of symmetry groups, including rotation, reflection, inversion and dihedral symmetry. The proofs of these results are based on some new Henneberg-type inductive constructions on the group-labelled quotient graphs that correspond to the bases of the matroids in question. For the remaining symmetry groups in 3-space—as well as for other types of surfaces—we provide some observations and conjectures.