Discrete Conformal Equivalence of Triangle Meshes

Peter Schroder, CalTech

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).

Speaker Biography

Peter Schröder is a professor of computer science and applied & computational mathematics at Caltech where he has been on the faculty since 1995. He is well known for his work on hierarchical methods in computer graphics and is considered one of the founders of the field of Digital Geometry Processing. More recently he has devoted his research to the development of Discrete Differential Geometry. His work has been recognized by a number of awards among them a Packard Foundation Fellowship, the ACM SIGGRAPH Computer Graphics Achievement award, and, most recently, a Humboldt Foundation Forschungspreis.