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