Abstract

A linearly constrained framework in $\mathbb{R}^d$ is a point configuration together with a system of constraints that fixes the distances between some pairs of points and additionally restricts some of the points to lie in given affine subspaces. It is globally rigid if the configuration is uniquely defined by the constraint system. We show that a generic linearly constrained framework in $\mathbb{R}^2$ is globally rigid if and only if it is redundantly rigid and “balanced”. For unbalanced generic frameworks, we determine the precise number of solutions to the constraint system whenever the rigidity matroid of the framework is connected. We obtain a stress matrix sufficient condition and a Hendrickson type necessary condition for a generic linearly constrained framework to be globally rigid in $\mathbb{R}^d$.

中文翻译：

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二维线性约束框架的全局刚性

摘要

$\mathbb{R}^d$ 中的线性约束框架是一个点配置以及一个约束系统，该系统固定一些点对之间的距离，并另外将一些点限制在给定的仿射子空间中。如果配置由约束系统唯一定义，则它是全局刚性的。我们表明，$\mathbb{R}^2$ 中的通用线性约束框架是全局刚性的，当且仅当它是冗余刚性和“平衡”的。对于不平衡的通用框架，只要框架的刚性拟阵连接，我们就会确定约束系统解的精确数量。我们获得了一个应力矩阵充分条件和一个 Hendrickson 类型的必要条件，以使通用线性约束框架在 $\mathbb{R}^d$ 中具有全局刚性。