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A linearly constrained framework in $mathbb{R}^d$ is a point configuration together with a system of constraints which 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, and is rigid if it is uniquely defined within some small open neighbourhood. Streinu and Theran characterised generic rigidity of linearly constrained frameworks in $mathbb{R}^2$ in 2010. We obtain an analagous characterisation for generic global rigidity in $mathbb{R}^2$. More precisely 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 generic frameworks which are not balanced, we determine the precise number of solutions to the constraint system whenever the underlying rigidity matroid of the given framework is connected. We also obtain a stress matrix sufficient condition and a Hendrickson type necessary condition for a generic linearly constrained framework to be globally rigid in $mathbb{R}^d$.
We consider the problem of characterising the generic rigidity of bar-joint frameworks in $mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Ther
We consider the problem of characterising the generic rigidity of bar-joint frameworks in $mathbb{R}^d$ in which each vertex is constrained to lie in a given affine subspace. The special case when $d=2$ was previously solved by I. Streinu and L. Ther
We show that a generic framework $(G,p)$ on the cylinder is globally rigid if and only if $G$ is a complete graph on at most four vertices or $G$ is both redundantly rigid and $2$-connected. To prove the theorem we also derive a new recursive constru
Fekete, Jordan and Kaszanitzky [4] characterised the graphs which can be realised as 2-dimensional, infinitesimally rigid, bar-joint frameworks in which two given vertices are coincident. We formulate a conjecture which would extend their characteris
A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in $mathbb{R}^d$ and those in $mathbb{S}^d$ is a classical observation by Pogorelov, and further connections among different rigidity models in various different sp