Volume 19, issue 4 (2015)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 20
Issue 4, 1807–2438
Issue 3, 1257–1806
Issue 2, 629–1255
Issue 1, 1–627

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Subscriptions
Author Index
To Appear
Contacts
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Discrete conformal maps and ideal hyperbolic polyhedra

Alexander I Bobenko, Ulrich Pinkall and Boris A Springborn

Geometry & Topology 19 (2015) 2155–2215
Abstract

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.

Keywords
discrete conformal geometry, polyhedron, hyperbolic geometry
Mathematical Subject Classification 2010
Primary: 52C26
Secondary: 52B10, 57M50
References
Publication
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
Authors
Alexander I Bobenko
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
10623 Berlin
Germany
http://page.math.tu-berlin.de/~bobenko
Ulrich Pinkall
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
10623 Berlin
Germany
http://page.math.tu-berlin.de/~pinkall
Boris A Springborn
Technische Universität Berlin
Institut für Mathematik
Strasse des 17. Juni 136
10623 Berlin
Germany
http://page.math.tu-berlin.de/~springb