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  1. Optimal transportation maps play fundamental roles in many engineering and medical fields. The computation of optimal transportation maps can be reduced to solve highly non-linear Monge-Ampere equations. This work summarizes the geometric variational frameworks for spherical optimal transportation maps, which offers solutions to the Minkowski problem in convex differential geometry, reflector design and refractor design problems in optics. The method is rigorous, robust and efficient. The algorithm can directly generalized to higher dimensions. 
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  2. Optimal transportation (OT) maps play fundamental roles in many engineering and medical fields. The computation of optimal transportation maps can be reduced to solve highly non-linear Monge-Ampere equations. In this work, we summarize the geometric variational framework to solve optimal transportation maps in Euclidean spaces. We generalize the method to solve worst transportation maps and discuss about the symmetry between the optimal and the worst transportation maps. Many algorithms from computational geometry are incorporated into the method to improve the efficiency, the accuracy and the robustness of computing optimal transportation. 
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