论文标题
分析规范平面中的独特图形
Uniquely realisable graphs in analytic normed planes
论文作者
论文摘要
欧几里得空间中的Bar-Concoint Framework $（G，P）$，如果它是唯一的实现，可以固定，直到刚度的一致性，$ g $ in $ \ mathbb {e} e}^d $，带有$ g $ in $ g $（G，p）$。杰克逊（Jackson）和乔丹（Jordán）以亨德里克森（Hendrickson）和康纳利（Connelly）的关键结果为基础，对$ \ mathbb {e}^2 $中的全局刚度进行了完整的组合表征。当欧几里得规范被$ \ mathbb {r}^2 \ setMinus \ {0 \} $取代时，我们证明了类似的结果。更确切地说，我们证明了图$ g =（v，e）$在非欧亚人分析规范平面上是全球刚性的，并且只有当$ g $是2连接的，并且$ g-e $包含2个边缘 - 偶数跨度的树，用于E $中的所有$ e \。主要的技术工具是在分析规范平面中递归构造2个连接和冗余的刚性图。我们还获得了一些全球刚度的足够条件，这是我们主要结果的推论，并证明了类似的必要条件在$ d $维分析规范的空间中。
A bar-joint framework $(G,p)$ in the Euclidean space $\mathbb{E}^d$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ in $\mathbb{E}^d$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson and Connelly, Jackson and Jordán gave a complete combinatorial characterisation of when a generic framework is global rigidity in $\mathbb{E}^2$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $\mathbb{R}^2 \setminus \{0\}$. More precisely, we show that a graph $G=(V,E)$ is globally rigid in a non-Euclidean analytic normed plane if and only if $G$ is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all $e\in E$. The main technical tool is a recursive construction of 2-connected and redundantly rigid graphs in analytic normed planes. We also obtain some sufficient conditions for global rigidity as corollaries of our main result and prove that the analogous necessary conditions hold in $d$-dimensional analytic normed spaces.
