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Computing smooth and optimal one-to-one maps between surfaces of same topology is a fundamental problem in graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing state-of-the-art intersurface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is solely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt toward rigorously and numerically computing globally optimal maps. Consequently, we demonstrate our mapping framework, offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching.