Discrete conformal mappings and Riemann surfaces

The general idea of discrete differential geometry is to find and investigate discrete models that exibit properties and structures characterisitic of the corresponding smooth geometric objects. We focus on a discrete notion of conformal equivalence of polyhedral metrics. 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 to convex variational principles. We establish a connection between conformal geometry for triangulated surfaces and the geometry of ideal hyperbolic polyhedra. This synthesis enables us to derive a companion theory of discrete uniformization of Riemann surfaces. Applications in geometry processing and computer graphics will be demonstrated.