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We develop a rigidity theory for frameworks in $\mathbb{R}^3$ which have twocoincident points but are otherwise generic and only infinitesimal motionswhich are tangential to a family of cylinders induced by the realisation areconsidered. We then apply our results to show that vertex splitting, under theadditional assumption that the new edge is redundant, preserves the property ofbeing generically globally rigid on families of concentric cylinders.

Rigid cylindrical frameworks with two coincident points