We present in this article numerical techniques — the Julia control-toolbox package — for computing geometric optimal control concepts: Hamiltonian flows associated to the optimal control problem, Jacobi flows, Poisson brackets of Hamiltonians to define the singular control associated to a singular arc, etc. With these tools, it becomes easy to solve an optimal control problem by indirect methods, and to compute conjugate times together with the cut locus. We present the numerical tools on two test bed examples: the surface of revolution of minimum area and the Goddard problem. The first problem comes from calculus of variations and thus is regular. We also compute the conjugate locus for this example. On the other hand, the optimal solution of the Goddard problem contains bang and singular arcs.