Symmetric Laman theorems for the groups \(\mathcal C_2\) and \(\mathcal C_s\)
Summary: For a bar and joint framework \((G, p)\) with point group \({\mathcal C}_3\) which describes 3-fold rotational symmetry in the plane, it was recently shown in [\textit{B. Schulze}, Discrete Comput. Geom. 44, No.~4, 946--972 (2010; Zbl 1211.52023)] that the standard Laman conditions, together with the condition derived in [\textit{R. Connelly} et al., Int. J. Solids Struct. 46, 762--773 (2009)] that no vertices are fixed by the automorphism corresponding to the 3-fold rotation (geometrically, no vertices are placed on the center of rotation), are both necessary and sufficient for \((G,p)\) to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Laman-type conjectures for the groups \({\mathcal C}_2\) and \({\mathcal C}_s\) which are generated by a half-turn and a reflection in the plane, respectively. In addition, analogously to the results by Schulze [loc. cit.], we also characterize symmetry generic isostatic graphs for the groups \({\mathcal C}_2\) and \({\mathcal C}_s\) in terms of inductive Henneberg-type constructions, as well as Crapo-type 3Tree2 partitions -- the full sweep of methods used for the simpler problem without symmetry.
scientific article; zbMATH DE number 5827403