A new algorithm for conformal mesh parameterization is presented here. It is based on a precise notion of discrete conformal equivalence for triangle meshes which mimics the notion of conformal equivalence for smooth surfaces. The problem of finding a flat mesh that is discretely conformally equivalent to a given mesh can be solved efficiently by minimizing a convex energy function, whose Hessian turns out to be the well known $cot$-Laplace operator. This method can also be used to map a surface mesh to a parameter domain which is flat except for isolated cone singularities.
Examples are shown of how these can be placed automatically in order to reduce the distortion of the parameterization.
B. Springborn, P. Schröder, and U. Pinkall.
Conformal equivalence of triangle meshes.
ACM Transactions on Graphics, 2008.
URL: http://www.multires.caltech.edu/pubs/ConfEquiv.pdf, doi:10.1145/1360612.1360676.