A Toroidal MaxwellCremonaDelaunay Correspondence
Abstract
We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: (1) 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. (2) 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 $G^*$. (3) 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 MaxwellCremona correspondence and the wellknown 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 1skeleton of the lower convex hull of points in $\mathbb{R}^3$. However, this threeway 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2020
 arXiv:
 arXiv:2003.10057
 Bibcode:
 2020arXiv200310057E
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Computational Geometry;
 52C25;
 05C10
 EPrint:
 19 pages, 3 figures, extended abstract to appear at SOCG 2020