It has been recently shown that reconstructing an isometric surface from a single 2D input image matched to a 3D template was a well-posed problem. This however does not tell us how reconstruction algorithms will behave in practical conditions, where the amount of perspective is generally small and the projection thus behaves like weak-perspective or orthography. We here bring answers to what is theoretically recoverable in such imaging conditions, and explain why existing convex numerical solutions and analytical solutions to 3D reconstruction may be unstable. We then propose a new algorithm which works under all imaging conditions, from strong to loose perspective. We empirically show that the gain in stability is tremendous, bringing our results close to the iterative minimization of a statisticallyoptimal cost. Our algorithm has a low complexity, is simple and uses only one round of linear least-squares.