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This paper deals with the accurate and efficient modal analysis of arbitrarily shaped waveguides whose cross section is defined by a combination of straight, circular, and/or elliptical arcs. A novel technique for considering the presence of circular and/or elliptical segments within the frame of the well-known boundary integral-resonant mode expansion (BI-RME) method is proposed. This new extended BI-RME method will allow a more accurate solution of a wider number of hollow conducting waveguides with arbitrary profiles, which are usually present in most modern passive waveguide components. To show the advantages of this new extended technique, the modal chart of canonical (circular and elliptical) waveguides, as well as of irises with great practical interest (i.e., cross-shaped irises with rounded corners) has been first successfully solved. Next, a computer-aided-design software package based on such a novel modal analysis tool has first been validated with the accurate analysis of a referenced complex dual-mode filter, and then applied to the complete design of a novel twist component for K-band application based on circular and elliptical waveguides. A prototype of this novel device has been manufactured and measured for verification purposes.