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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.

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Notes

  1. Colored graphs in this sense are also called “gain graphs” in the literature, e.g. [36].

  2. This terminology is from [1]. Elissa Ross introduced this class in [23], and we introduced this terminology in light of her contribution. In [23], they are called “constructive periodic orbit graphs”.

  3. The matroid of Ross graphs has more circuits, but these are the ones we are interested in here. See Sect. 2.5.

  4. Recall that here we are using Ross circuit to refer to only one kind of circuit in the Ross matroid. The other type of circuit cannot appear since reflection-Laman graphs do not have (2, 2) blocks with trivial \(\rho \)-image.

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Acknowledgments

LT was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 247029-SDModels. JM was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 226135. LT and JM had been supported by NSF CDI-I Grant DMR 0835586 to Igor Rivin and M. M. J. Treacy.

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Correspondence to Louis Theran.

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Malestein, J., Theran, L. Frameworks with Forced Symmetry I: Reflections and Rotations. Discrete Comput Geom 54, 339–367 (2015). https://doi.org/10.1007/s00454-015-9692-z

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Keywords

  • Combinatorial rigidity
  • Matroids
  • Forced symmetry
  • Colored graph