Frameworks with forced symmetry I: Reflections and rotations

Abstract

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.

27 pages, 5 figures. All the results in this paper have appeared previously in our other papers available at arXiv:1108.2518 and arXiv:1203.2276 (v2, added some examples, discussion of non-free actions and expanded references)

References in corpus (4)

• Generic combinatorial rigidity of periodic frameworks
• Generic rigidity of frameworks with orientation-preserving crystallographic symmetry
• Generic rigidity of reflection frameworks
• Matroids of Gain Graphs in Applied Discrete Geometry

Cited by in corpus (2)

• Ultrarigid periodic frameworks
• Symmetry-forced rigidity of frameworks on surfaces