@inproceedings{0ac5381556dc415ba2fae6e2d3c52c8d,

title = "A toroidal maxwell-cremona-delaunay correspondence",

abstract = "We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: A torus graph G is equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors incident to each vertex of G sum to zero. A torus graph G is reciprocal if there is a geodesic embedding of the dual graph G∗ on the same flat torus, where each edge of G is orthogonal to the corresponding dual edge in G∗. A torus graph G is coherent if it is possible to assign weights to the vertices, so that G is the (intrinsic) weighted Delaunay graph of its vertices. 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.",

keywords = "Combinatorial topology, Flat torus, Geometric graphs, Homology, Intrinsic Delaunay, Spring embedding",

author = "Jeff Erickson and Patrick Lin",

note = "Funding Information: Portions of this work were supported by NSF grant CCF-1408763.; 36th International Symposium on Computational Geometry, SoCG 2020 ; Conference date: 23-06-2020 Through 26-06-2020",

year = "2020",

month = jun,

day = "1",

doi = "10.4230/LIPIcs.SoCG.2020.40",

language = "English (US)",

series = "Leibniz International Proceedings in Informatics, LIPIcs",

publisher = "Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing",

editor = "Sergio Cabello and Chen, {Danny Z.}",

booktitle = "36th International Symposium on Computational Geometry, SoCG 2020",

}