对称形式的拉曼定理
Symmetric versions of Laman's Theorem
摘要：
最近的工作表明，如果一个均衡酒吧和联合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规定限制的数量关节和酒吧是“固定”的各种对称操作的框架。对于描述平面内三重旋转对称的群$C_3$，我们验证了在[4]中提出的猜想，即对固定结构构件数目的这些限制，连同Laman条件，也足以使具有$C_3$对称性的框架是均衡的，只要它的结点在给定的对称性约束下尽可能一般地定位。此外，我们建立了对称版本的Henneberg定理和Crapo定理的$C_3$提供交替的“一般”均衡图与$C_3$对称性的表征。如[19]所示，我们的技术可以扩展到建立对称群$C_2$和$C_s$的类似结果，它们分别由半圈和平面反射产生。
英文摘要：
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.
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