In origami theory, the problem of rigid maps consists in finding a paper folding from the two-dimensional space onto the three-dimensional space. This problem is an example of a first-order fully nonlinear equation. In this article, we present a general variational framework to solve the problem of rigid maps with Dirichlet boundary conditions. The numerical framework relies on the introduction of a regularized objective function and the penalization of the constraints. A splitting algorithm is advocated for the corresponding flow problem. The iterations sequence consists of local nonlinear problems and a global linear variational problem at each step. Numerical experiments validate the efficiency of the method for piecewise smooth exact solutions.