### Global Conformal Surface Parameterization

Xianfeng Gu and Shing-Tung Yau

ACM Symposium on Geometry Processing 2003, pages 127-137

We solve the problem of computing global conformal parametrizations for surfaces with arbitrary topologies, with or without boundaries. The parameterization preserves the conformality everywhere except for a few points, and it has no boundary of discontinuity. We analyze the structure of the space of all global conformal parameterizations of a given surface and find all possible solutions instead of finding just one. This space has a natural structure solely determined by the surface geometry. So our computing result is independent of connectivity, insensitive to resolution, and independent of the algorithms to discover it. Our algorithm is based on the properties of gradient fields of conformal maps, which are closedness, harmonity, conjugacy, duality and symmetry. These properties can be formulated by sparse linear systems, so the method is easy to implement and the whole process is automatic.