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 realizations and the other has only collapsed realizations. In terms of infinitesimal rigidity, realizations of the former produce a framework and the latter certifies that this framework is infinitesimally rigid.

Note: The content of this preprint appeared as part of “Frameworks with forced symmetry I”