On dimensional rigidity of bar-and-joint frameworks
On dimensional rigidity of bar-and-joint frameworks
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关于杆节点框架的尺寸刚度
DOI：
10.1016/j.dam.2006.11.011
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发表时间：
2007
期刊：
影响因子：
--
通讯作者：
A. Alfakih
中科院分区：
文献类型：
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作者：
A. Alfakih
设V={1，2，...，n}。一个映射p：V→Rr，其中p1，.，pn不包含在一个真超平面中，称为r-构形.设G=（V，E）是n阶简单连通图.则一个r-构型p与图G（其中G的相邻顶点被约束为保持相同的距离）一起被称为Rr中的一个杆-关节框架（或框架），记为G（p）。本文引入了框架维数刚性的概念，并研究了判定给定G（p）是否维数刚性的问题。称Rris中的给定框架G（p）是维数刚性的当且仅当对于s <$r+1不存在Rris中的框架G（q），使得对所有（i，j）∈E，有<$qi-qj <$2 =<$pi-pj <$2.我们提出了G（p）是维数刚性的必要和充分条件，我们制定了检查这些条件的有效性的问题作为一个半定规划（SDP）问题。本文还研究了给定r-构型的点p1，…，pn处于一般位置的情况。
Let V={1,2,…,n}. A mapping p:V→Rr, where p1,…,pnare not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in Rr, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in Rris said to be dimensionally rigid iff there does not exist a framework G(q) in Rsfor s⩾r+1, such that ∥qi-qj∥2=∥pi-pj∥2for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p1,…,pnof the given r-configuration are in general position, is also investigated.