An explicit expression of the high-frequency surface current excited on a perfectly conducting spherical cap by the edge-diffraction of an obliquely incident wave is derived. The result is given in the GTD terminology and shows the influence of the incidence angles on the transfer functions. To this end the real (physical) space is embedded into a twofold extended abstract space in which an equivalent canonical problem is established, This latter yields a system of dual integral equations whose kernels were not previously treated. By inverting these new kernels, the system of dual equations is reduced to two independent Wiener-Hopf problems. From the asymptotic solutions of these problems, one derives the expression of the edge-excited current along with the known geometrical optics term. The resutlt can be used directly in practical applications. If the incident ray becomes normal to the edge, then the resulting expressions are reduced to the already known expressions. The basic idea and general formulas used here can also be used to solve other diffraction problems with similar geometry.