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Surface segmentation is a fundamental problem in computer graphics. It has various applications such as metamorphosis, surface matching, surface compression, 3D shape retrieval, texture mapping, etc. All orientable surfaces are Riemann surfaces, and admit conformal structures. This paper introduces a novel surface segmentation algorithm based on its conformal structure. Each segment can be conformally mapped to a planar rectangle, and the transition maps are planar translations. The segmentation is intrinsic to the surface, independent of the embedding, and consistent for surfaces with similar geometries. By using segmentation based on conformal structure, the mapping between surfaces with arbitrary topologies can be constructed explicitly. The method is rigorous, efficient and automatic. The segmentation can be applied to surface morphing, construct conformal geometry image, convert mesh to Spline

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