A toroidal maxwell-cremona-delaunay correspondence

Jeff Erickson, Patrick Lin

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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.

Original languageEnglish (US)
Title of host publication36th International Symposium on Computational Geometry, SoCG 2020
EditorsSergio Cabello, Danny Z. Chen
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771436
DOIs
StatePublished - Jun 1 2020
Event36th International Symposium on Computational Geometry, SoCG 2020 - Zurich, Switzerland
Duration: Jun 23 2020Jun 26 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume164
ISSN (Print)1868-8969

Conference

Conference36th International Symposium on Computational Geometry, SoCG 2020
CountrySwitzerland
CityZurich
Period6/23/206/26/20

Keywords

  • Combinatorial topology
  • Flat torus
  • Geometric graphs
  • Homology
  • Intrinsic Delaunay
  • Spring embedding

ASJC Scopus subject areas

  • Software

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