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The results of our recent asymptotic treatment of the double diffraction by a pair of parallel perfectly conducting wedges are applied to polygonal cylinders and strips. The combination of the appropriate nonuniform expression for the doubly diffracted ray field with the nonuniform geometrical theory of diffraction (GTD) expressions for the singly diffracted fields yields a uniform, closed-form solution for the far scattered field valid throughout the overlapping transition regions. The advantage over the existing uniform GTD solution for a polygonal cylinder consists in the extension from strictly grazing to nearly grazing incidence. For a strip, the present solution improves upon the accuracy of the previous uniform high-frequency solutions.