  We compute the length of geodesics on a Riemannian manifold by regular
polynomial interpolation of the global solution of the eikonal equation related
to the line element $ds^2=g_{ij}dx^idx^j$ of the manifold. Our algorithm
approximates the length functional in arbitrarily strong Sobolev norms. Error
estimates are obtained where the geometric information is used. It is pointed
out how the algorithm can be used to get accurate approximation of solutions of
parabolic partial differential equations leading obvious applications to
finance and physics.
