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All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion, electric-magnetic fields on the surface. Good parameterization is crucial for simulation and visualization. This paper gives an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on finite dimensional optimization method. The optimal pa- rameterization is intrinsic to the geometry, preserving angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visu- alizing the Riemann surface structure of real surfaces represented as triangle meshes.

Optimal Global Conformal Surface Parameterization for Visualization