Geometry & Topology

Discrete conformal maps and ideal hyperbolic polyhedra

Alexander I Bobenko, Ulrich Pinkall, and Boris A Springborn

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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.

Article information

Geom. Topol., Volume 19, Number 4 (2015), 2155-2215.

Received: 16 September 2013
Revised: 4 August 2014
Accepted: 12 October 2014
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C26: Circle packings and discrete conformal geometry
Secondary: 52B10: Three-dimensional polytopes 57M50: Geometric structures on low-dimensional manifolds

discrete conformal geometry polyhedron hyperbolic geometry


Bobenko, Alexander I; Pinkall, Ulrich; Springborn, Boris A. Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19 (2015), no. 4, 2155--2215. doi:10.2140/gt.2015.19.2155.

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