Frameworks with Forced Symmetry I: Reflections and Rotations

@article{Malestein2013FrameworksWF,
  title={Frameworks with Forced Symmetry I: Reflections and Rotations},
  author={Justin Malestein and Louis Theran},
  journal={Discrete \& Computational Geometry},
  year={2013},
  volume={54},
  pages={339-367}
}
We give a combinatorial characterization of generic frameworks that are minimally rigid under the additional constraint of maintaining symmetry with respect to a finite order rotation or a reflection. To establish these results, we develop a new technique for deriving linear representations of sparsity matroids on colored graphs and extend the direction network method of proving rigidity characterizations to handle reflections. 
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22 Citations
Background Citations
11
Methods Citations
3
Results Citations
3

22 Citations

Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of…

Frameworks with forced symmetry II: orientation-preserving crystallographic groups

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are…

Motions of grid-like reflection frameworks

CONSTRUCTING ISOSTATIC FRAMEWORKS FOR THE `∞ PLANE

We use a new coloured multi-graph constructive method to prove that every 2-tree decomposition can be realised in the plane as a bar-joint framework which is minimally rigid (isostatic) with respect…

Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

It is shown that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroid defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group.

Generic Symmetry-Forced Infinitesimal Rigidity: Translations and Rotations

We characterize the combinatorial types of symmetric frameworks in the plane that are minimally generically symmetry-forced infinitesimally rigid when the underlying symmetry group consists of…

Sufficient connectivity conditions for rigidity of symmetric frameworks

Symmetry-forced rigidity of frameworks on surfaces

A fundamental theorem of Laman characterises when a bar-joint framework realised generically in the Euclidean plane admits a non-trivial continuous deformation of its vertices. This has recently been…

Ultrarigid periodic frameworks

We give an algebraic characterization of when a $d$-dimensional periodic framework has no non-trivial, symmetry preserving, motion for any choice of periodicity lattice. Our condition is decidable,…

Rigidity of symmetric frameworks in normed spaces

References

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Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

We review some recent results in the generic rigidity theory of planar frameworks with forced symmetry, giving a uniform treatment to the topic. We also give new combinatorial characterizations of…

Frameworks with forced symmetry II: orientation-preserving crystallographic groups

We give a combinatorial characterization of minimally rigid planar frameworks with orientation-preserving crystallographic symmetry, under the constraint of forced symmetry. The main theorems are…

Generic rigidity of reflection frameworks

We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful…

Generic rigidity of frameworks with orientation-preserving crystallographic symmetry

We extend our generic rigidity theory for periodic frameworks in the plane to frameworks with a broader class of crystallographic symmetry. Along the way we introduce a new class of combinatorial…

Generic combinatorial rigidity of periodic frameworks

Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

It is shown that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroid defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group.

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