Global Rigidity and Symmetry of Direction-length Frameworks
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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.
Authors
Clinch, KatharineCollections
- Theses [3592]