A Note on Toroidal Maxwell-Cremona Correspondences

Abstract

We explore toroidal analogues of the Maxwell-Cremona correspondence. Erickson and Lin [arXiv:2003.10057] showed the following correspondence for geodesic torus graphs $G$: a positive equilibrium stress for $G$, an orthogonal embedding of its dual graph $G^*$, and vertex weights such that $G$ is the intrinsic weighted Delaunay graph of its vertices. We extend their results to equilibrium stresses that are not necessarily positive, which correspond to orthogonal drawings of $G^*$ that are not necessarily embeddings. We also give a correspondence between equilibrium stresses and parallel drawings of the dual.