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A formulation based on the physical theory of diffraction (PTD) is presented for finite conical surfaces with circular and elliptic cross sections. The base-rim discontinuity is represented by equivalent currents, including second-order terms extended for elliptic boundaries. Tip-rim interactions are examined as a function of the tip-rim distance, cone angle, and illumination angle for circular cones; and their implication for elliptic cones is noted. The diffraction contribution from tip-rim interactions is shown to be dependent on the cone angle and the illumination angle but to be relatively insensitive to the tip-rim distance. The Fock Ansatz is used to enlarge the validity of the PTD formulation to cases where nonspecular effects arising from surface curvature and shadow boundaries are significant. The formulation is applied to cones with varying ellipticity for axial and oblique illumination. Correlation is made with published results for circular cones and with experimental data for an elliptic cone.