Discrete conformal maps and ideal hyperbolic polyhedra

TL;DR

This paper links discrete conformal geometry of triangulated surfaces with hyperbolic polyhedra, introducing a unified theory that encompasses circle packings and variational principles for constructing hyperbolic structures.

Contribution

It establishes a novel connection between discrete conformal maps and ideal hyperbolic polyhedra, providing new variational principles and reinterpretations of conformality.

Findings

Unified theory of discrete conformal maps and hyperbolic polyhedra

Variational principles for constructing hyperbolic structures

Relation to circle packing definitions of conformality

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 M\"obius 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 addresses 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.