Global Rigidity and Symmetry of Direction-length Frameworks

Abstract

A two-dimensional direction-length framework (G; p) consists of a multigraph G = (V ;D;L) whose edge set is formed of \direction" edges D and \length" edges L, and a realisation p of this graph in the plane. The edges of the framework represent geometric constraints: length edges x the distance between their endvertices, whereas direction edges specify the gradient of the line through both endvertices. In this thesis, we consider two problems for direction-length frameworks. Firstly, given a framework (G; p), is it possible to nd a di erent realisation of G which satis es the same direction and length constraints but cannot be obtained by translating (G; p) in the plane, and/or rotating (G; p) by 180 ? If no other such realisation exists, we say (G; p) is globally rigid. Our main result on this topic is a characterisation of the direction-length graphs G which are globally rigid for all \generic" realisations p (where p is generic if it is algebraically independent over Q). Secondly, we consider direction-length frameworks (G; p) which are symmetric in the plane, and ask whether we can move the framework whilst preserving both the edge constraints and the symmetry of the framework. If the only possible motions of the framework are translations, we say the framework is symmetry-forced rigid. Our main result here is for frameworks with single mirror symmetry: we characterise symmetry-forced in nitesimal rigidity for such frameworks which are as generic as possible. We also obtain partial results for frameworks with rotational or dihedral symmetry.