September 23, 2008 - Peter Schroder

Title: Discrete Conformal Equivalence of Triangle Meshes

Abstract:
Many tasks in surface processing presuppose the existence of a "nice" parameterization. One such class of parameterizations are conformal, that is, angle preserving (though they may suffer from large area distortion which has a very satisfying resolution, as the talk will show). Since most applications use triangle meshes rather than the smooth surfaces presupposed by classical differential geometry, one must first come up with suitable discrete definitions, which then hopefully lead to practical algorithms. Guided by the tenets of "Discrete Differential Geometry," which aims to discretize entire theories rather than just particular equations, I will discuss a notion of discrete conformal equivalence for triangle meshes. It shares many properties with the smooth definition, is very simple, and, most important, leads to very practical algorithms, requiring only standard convex optimization. What's more, the theory accommodates so called cone singularities (all this will be explained in the talk), which give a powerful tool to reduce the area distortion to moderate levels. I will take the audience through the underlying ideas and present results.

Joint work with Boris Springborn and Ulrich Pinkall (both TU Berlin and Matheon).