Computation of Abelian integrals without quadratures

Formulas that go back to Riemann, represent abelian integrals in terms of theta functions. However, these formulas explicitly contain the Abel-Jacobi mapping, that is Abelian integrals again. One can circumvent the stumbling block by explicit localization of the image of the curve inside its Jacobian under the A-J map. We now have a parametric representation of abelian integrals: an explicit expression for the integral on the one hand, and an explicit expression for some projection of the curve onto the sphere - on the other. This representation of the Abelian integral can be used either for its computation (with the machine precision), or - for its inversion.