Optimal Global Conformal Surface Parameterization

Miao Jin, Yalin Wang, Shing-Tung Yau and Xianfeng Gu
IEEE Visualization, Austin, TX, Oct. 2004, pp. 267-274
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 and electro-magnetic .elds on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for .nding 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 a .nite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.