Discrete conformal maps and ideal hyperbolic polyhedra

Bobenko, Alexander I¹ ; Pinkall, Ulrich¹ ; Springborn, Boris A¹

¹ Technische Universität Berlin, Institut für Mathematik, Strasse des 17. Juni 136, 10623 Berlin, Germany

Geometry & topology, Tome 19 (2015) no. 4, pp. 2155-2215.

Résumé

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.

Détail

DOI : 10.2140/gt.2015.19.2155

Classification : 52C26, 52B10, 57M50
Keywords: discrete conformal geometry, polyhedron, hyperbolic geometry

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