A Note on Toroidal Maxwell-Cremona Correspondences

TL;DR

This paper extends the toroidal Maxwell-Cremona correspondence to include non-positive stresses, linking orthogonal dual graph drawings and parallel representations, broadening the theoretical framework for toroidal graph embeddings.

Contribution

It generalizes existing toroidal Maxwell-Cremona correspondence results to non-positive stresses and establishes new links between equilibrium stresses and dual graph drawings.

Findings

Extended the correspondence to non-positive stresses.

Linked equilibrium stresses with orthogonal dual graph drawings.

Connected equilibrium stresses to parallel dual drawings.

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.