Affine Rigidity and Conics at Infinity
Affine Rigidity and Conics at Infinity
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仿射刚度和无穷远圆锥曲线
DOI：
10.1093/imrn/rnx014
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Louis Theran
中科院分区：
文献类型：
--
作者：
R. Connelly;S. Gortler;Louis Theran
我们证明了，如果一个图的框架是$d维邻域仿射刚性的(或具有更强的具有秩为$n-d-1$的平衡应力矩阵的性质)，则它有仿射挠性(保持所有边长度的仿射但非欧几里德空间变换)当且仅当该框架在单个二次曲面上被规则。这加强并简化了Alfakih的一个相关结果。它还使我们证明了超稳定的性质对于射影变换是不变的，对于圆锥和切片运算也是不变的。最后，我们可以统一以前关于矩阵的强Arnold性质的一些结果。
We prove that if a framework of a graph is neighborhood affine rigid in $d$-dimensions (or has the stronger property of having an equilibrium stress matrix of rank $n-d-1$) then it has an affine flex (an affine, but non Euclidean, transform of space that preserves all of the edge lengths) if and only if the framework is ruled on a single quadric. This strengthens and also simplifies a related result by Alfakih. It also allows us to prove that the property of super stability is invariant with respect to projective transforms and also to the coning and slicing operations. Finally this allows us to unify some previous results on the Strong Arnold Property of matrices.