Although flattening a cortical surface necessarily introduces metric distortion due to the non-constant Gaussian curvature of the surface, the Riemann Mapping Theorem states that continuously differentiable surfaces can be mapped without angular distortion. Several techniques have been proposed for flattening polygonal representations of surfaces while substantially minimizing metric distortion, and methods for conformal flattening of polygonal surfaces have also been proposed. We describe an efficient method for generating conformal flat maps of triangulated surfaces while minimizing metric distortion within the class of conformal maps. Our method, which controls both angular and metric distortion, involves the solution of a linear system and a small scale nonlinear minimization. It can be applied to user-defined "patches" or to an entire cortical surface.