CIMMS Lunchtime Series: Parameterizing Surface Meshes Using Circle Patterns

Boris Springborn, Institut für Mathematik, Technische Universität Berlin

In computer graphics and numerics, surfaces in space are often represented by triangulations. A fundamental problem that appears in many applications is to parameterize such surface meshes, i.e. to map them to planar domains. For example, such parameterizations are used for texture mapping in computer graphics. I will describe a new method to construct parameterizations of surface meshes that are nearly conformal (angle preserving). Our approach is based on circle patterns, i.e., arrangements of circles in the plane, one for each face in the mesh, with prescribed intersection angles. A variational principle involving the logarithmic radii as variables is used to construct these circle patterns efficiently (joint work with A. Bobenko, TU Berlin). I will show examples of our parameterizations, explain how the area distortion of global conformal parameterizations can be mitigated with cone singularities, propose a strict definition for discrete conformal maps of  planar domains, and mention other applications of circle patterns and circle packings in mathematics and visualization.  

Joint work with Peter Schroder and Liliya Kharevych (Caltech)