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 film geometry and the pinning properties of the superconductor. In particular, we investigate the anisotropic vortex-pinning, due to columnar defects introduced by heavy ion irradiation, as revealed in the local current density dependence on the vortex curvature during magnetic flux diffusion inside the superconducting film.