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To enlarge the scope of the geometrical theory of diffraction, the diffraction matrix for a surface singularity where the curvature but not the slope is discontinuous is rigorously derived. The model that is employed consists of two parabolic cylinders of different latus recta joined together at the front, thereby creating a line discontinuity of the required form. For each of the two principal polarizations, asymptotic developments of the surface fields in the vicinity of the join are calculated, from which the diffraction coefficients are then obtained by integration. The results differ significantly from the physical optics estimates and are analogous to those for a wedge-like singularity. This analogy permits a trivial deduction of the complete diffraction matrix.