A toroidal Maxwell-Cremona-Delaunay correspondence

With Patrick Lin*.

cite>Proceedings of the 36th International Symposium on Computational Geometry, 40:1–40:17, 2020.

🔥 Full version invited and submitted to the special issue of Journal of Computational Geomettry devoted to the conference.


We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: The classical Maxwell-Cremona correspondence and the well-known correspondence between convex hulls and weighted Delaunay triangulations imply that the analogous concepts for plane graphs (with convex outer faces) are equivalent. Indeed, all three conditions are equivalent to G being the projection of the 1-skeleton of the lower convex hull of points in R3. However, this three-way equivalence does not extend directly to geodesic graphs on flat tori. On any flat torus, reciprocal and coherent graphs are equivalent, and every reciprocal graph is equilibrium, but not every equilibrium graph is reciprocal. We establish a weaker correspondence: Every equilibrium graph on any flat torus is affinely equivalent to a reciprocal/coherent graph on some flat torus.

Publications - Jeff Erickson (jeffe@illinois.edu) 18 Sep 2020