We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n2) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newtonﾕs method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining conformal mapping and optimal mass transport mapping, serves as a powerful tool for a broad range of applications in visualization and graphics, especially for medical imaging. We provide a variety of experimental results to demonstrate the efficiency, robustness and efficacy of our novel framework.