Assur decompositions of direction-length frameworks

A bar-joint framework is a realisation of a graph consisting of stiff bars linked by universal joints. The framework is rigid if the only bar-length preserving continuous motions of the joints arise from isometries. A rigid framework is isostatic if deleting any single edge results in a flexible framework. Generically, rigidity depends only on the graph and we say an Assur graph is a pinned isostatic graph with no proper pinned isostatic subgraphs. Any pinned isostatic graph can be decomposed into Assur components which may be of use for mechanical engineers in decomposing mechanisms for simpler analysis and synthesis. A direction-length framework is a generalisation of bar-joint framework where some distance constraints are replaced by direction constraints. We initiate a theory of Assur graphs and Assur decompositions for direction-length frameworks using graph orientations and spanning trees and then analyse choices of pinning set

Global Rigidity and Symmetry of Direction-length Frameworks

PhDA 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.EpSRC Studentshi