We build a universal r.v. generator from the intrinsic geometry induced by what we define to be the Kendall quantile ordering of probability distribution functions on R 2 , having its own statistical interest. Using this generator we define a closed form transport map τ F G between any smooth distributions F and G. This τ F G is optimal when reduced to the Kendall quantile curves, for a large class of coordinate-wise convex costs. It coincides with the optimal transport if F and G share the same copula. The empirical counterpart τ n,m of τ F G is a non parametric transport plan that is easy to compute even for large samples. We illustrate the probabilistic geometry of τ F G by simulations of τ n,m that exhibit good performance with respect to the L 2 Wasserstein distance, and point out some statistical applications.