We present the experimental observation of magnetic field line curvature at the surface of a superconducting film by local quantitative magneto-optics. In addition to the knowledge of the full induction field at the superconductor surface yielding the quantitative observation of the flux line curvature, our analysis method allows also local value measurements of the electrical current density inside the sample. Thus, we study the interplay between the electrodynamic constraints dictated by the f...