Rigid cylindrical frameworks with two coincident points

TL;DR

This paper develops a rigidity theory for 3D frameworks with two coincident points, focusing on infinitesimal motions tangent to cylinders, and shows vertex splitting preserves global rigidity on concentric cylinders.

Contribution

It introduces a new rigidity framework for frameworks with coincident points and demonstrates that vertex splitting maintains global rigidity under certain conditions.

Findings

Rigidity theory for frameworks with coincident points in 3D.

Vertex splitting preserves global rigidity on concentric cylinders.

Frameworks with two coincident points are characterized by tangent infinitesimal motions.

Abstract

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.