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Discrete conformal mappings and Riemann surfaces

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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 thevertices. This simple definition leads to a surprisingly rich theory. We establish a connection between conformal geometry for triangulated surfaces, the geometry of ideal hyperbolic polyhedra and discrete uniformization of Riemann surfaces. Applications in geometry processing and computer graphics will be demonstrated. Fragments from a new movie "Conform!" will be shown.