The geometry processing group at Carnegie Mellon University recently developed an algorithm called Boundary First Flattening that allows you to efficiently and interactively compute conformal parameterizations of triangle meshes. You can download the software here: https://geometrycollective.github.io/boundary-first-flattening/
It is incredibly powerful and easy to use. Unlike almost all previous algorithms for conformal parameterization it allows for significant control over the boundary shape.
