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A Toroidal Maxwell-Cremona-Delaunay Correspondence

Authors Jeff Erickson , Patrick Lin

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Jeff Erickson
  • University of Illinois, Urbana-Champaign, IL, USA
Patrick Lin
  • University of Illinois, Urbana-Champaign, IL, USA

Funding

Portions of this work were supported by NSF grant CCF-1408763.

Acknowledgements

We thank the anonymous reviewers for their helpful comments and suggestions.

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Jeff Erickson and Patrick Lin. A Toroidal Maxwell-Cremona-Delaunay Correspondence. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.40

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graphs and surfaces
Keywords
  • combinatorial topology
  • geometric graphs
  • homology
  • flat torus
  • spring embedding
  • intrinsic Delaunay

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