Volume 19, issue 4 (2015)

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Discrete conformal maps and ideal hyperbolic polyhedra

Alexander I Bobenko, Ulrich Pinkall and Boris A Springborn

Geometry & Topology 19 (2015) 2155–2215

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle-preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.

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discrete conformal geometry, polyhedron, hyperbolic geometry
Mathematical Subject Classification 2010
Primary: 52C26
Secondary: 52B10, 57M50
Received: 16 September 2013
Revised: 4 August 2014
Accepted: 12 October 2014
Published: 29 July 2015
Proposed: David Gabai
Seconded: Danny Calegari, Jean-Pierre Otal
Alexander I Bobenko
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
10623 Berlin
Ulrich Pinkall
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
10623 Berlin
Boris A Springborn
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
10623 Berlin