论文标题
气缸上对称框架的刚度
Rigidity of symmetric frameworks on the cylinder
论文作者
论文摘要
一个bar-contracework $（g，p）$是有限的简单图$ g =（v，e）$和位置$ p：v \ rightarrow \ mathbb {r}^d $的组合。如果唯一的边缘长度保存顶点的连续变形是由空间的同位物产生的，则该框架是刚性的。本文结合了刚性和灵活图的通用理论的两个最新扩展，通过考虑$ \ mathbb {r}^3 $限制在表面上移动的对称框架。在任何有限点组对称性下，在圆柱体上的对称框架中给出了对称框架的特定组合条件。在每种情况下，当对称组都是循环的时，我们证明该组限制为反转，半转或反射对称性时，这些条件在适当的通用假设下被证明是足够的，在这些上下文中提供了对称性等静态图的精确组合描述。
A bar-joint framework $(G,p)$ is the combination of a finite simple graph $G=(V,E)$ and a placement $p:V\rightarrow \mathbb{R}^d$. The framework is rigid if the only edge-length preserving continuous deformations of the vertices arise from isometries of the space. This article combines two recent extensions of the generic theory of rigid and flexible graphs by considering symmetric frameworks in $\mathbb{R}^3$ restricted to move on a surface. In particular necessary combinatorial conditions are given for a symmetric framework on the cylinder to be isostatic (i.e. minimally infinitesimally rigid) under any finite point group symmetry. In every case when the symmetry group is cyclic, which we prove restricts the group to being inversion, half-turn or reflection symmetry, these conditions are then shown to be sufficient under suitable genericity assumptions, giving precise combinatorial descriptions of symmetric isostatic graphs in these contexts.
