Optimal Global Conformal Surface Parameterization for Visualization

Abstract

All orientable metric surfaces are Riemann surfaces and admitglobal conformal parameterizations. Riemann surface structureis a fundamental structure and governs many natural physicalphenomena, such as heat diffusion, electric-magnetic fields onthe surface. Good parameterization is crucial for simulationand visualization. This paper gives an explicit method forfinding optimal global conformal parameterizations of arbitrarysurfaces. It relies on certain holomorphic differential formsand conformal mappings from differential geometry and Riemannsurface theories. Algorithms are developed to modify topology,locate zero points, and determine cohomology types of differentialforms. The implementation is based on finite dimensional optimizationmethod. The optimal parameterization is intrinsic to the geometry,preserving angular structure, and can play an important role invarious applications including texture mapping, remeshing, morphingand simulation. The method is demonstrated by visualizing the Riemannsurface structure of real surfaces represented as triangle meshes.