# 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.

## 22 Citations

### Generic Rigidity with Forced Symmetry and Sparse Colored Graphs

- Mathematics
- 2014

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

- Mathematics
- 2013

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…

### CONSTRUCTING ISOSTATIC FRAMEWORKS FOR THE `∞ PLANE

- Mathematics
- 2017

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

- MathematicsDiscret. Comput. Geom.
- 2016

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

- MathematicsSIAM J. Appl. Algebra Geom.
- 2022

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

- MathematicsEur. J. Comb.
- 2023

### Symmetry-forced rigidity of frameworks on surfaces

- Mathematics
- 2013

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

- MathematicsArXiv
- 2014

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

- MathematicsLinear Algebra and its Applications
- 2020

## References

SHOWING 1-10 OF 40 REFERENCES

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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

- Mathematics
- 2013

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…

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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

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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…

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- 2016

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.

### Periodic Rigidity on a Variable Torus Using Inductive Constructions

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This paper proves a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice with variants of the Henneberg operations used frequently in rigidity theory.

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SummaryIn this paper the combinatorial properties of rigid plane skeletal structures are investigated. Those properties are found to be adequately described by a class of graphs.

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