The physical theory of diffraction with transition currents (PTD-TC) is used to predict the electromagnetic fields diffracted by obstacles with or without edges, and its usefulness is verified by applying it to perfectly conducting rectangular and ogive cylinders. Edge and transition currents are borrowed from the solution of canonical problems of wedge and circular cylinders. Transition currents of curved surfaces are represented in the form of the Fock-type integral. For an obstacle with edges, the currents near the edges are expressed by eigenfunction expansion. The currents of other parts are represented by the Fresnel integral, which has a slightly different form than that derived by Kouyoumjian and Pathak. Numerical results of far-scattered patterns are obtained and compared with those obtained by the method of moments. It was found that the present method gives very precise results.