Symmetric Laman Theorems for the Groups $\mathcal{C}_2$ and $\mathcal{C}_s$
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Bibliographic record
Abstract
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 (Schulze, Discret. Comp. Geom. 44:946-972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46:762-773) 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 in (Schulze, Discret. Comp. Geom. 44:946-972), 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.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.010 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.003 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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