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Closed-form representations of the physical optics (PO) field scattered in the far zone by plane penetrable dielectric angular sectors of arbitrary opening angle featuring a conical-section boundary are derived in terms of incomplete cylindrical functions (ICFs). The proposed expressions, possibly in combination with PO formulas for the scattering from polygonal plates, allow one to evaluate the scattering from flat dielectric plates with both convex and concave curved edges in a very efficient manner and constitute a useful tool to improve the accuracy of the geometrical characterization of a class of natural scatterers. In order to check the correctness and computational effectiveness of the proposed scattered-field solutions, comparisons with results obtained by accurate completely numerical PO calculations are provided. Simulation data by the method of moments are also presented to assess the applicability of the PO approximation to the considered sample scattering geometries.