Orthogonal maps are the solutions of the mathematical model of paper-folding, also called the origami problem. They consist of a system of first-order fully nonlinear equations involving the gradient of the solution. The Dirichlet problem for orthogonal maps is considered here. A variational approach is advocated for the numerical approximation of the maps. The introduction of a suitable objective function allows us to enforce the uniqueness of the solution. A strategy based on a splitting algorithm for the corresponding flow problem is presented and leads to decoupling the time-dependent problem into a sequence of local nonlinear problems and a global linear variational problem at each time step. Numerical experiments validate the accuracy and the efficiency of the method for various domains and meshes.