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The problem of the scattering of electromagnetic waves by a rotating perfectly conducting cylinder of arbitrary geometrical cross section is solved relativistically by the point-matching method. Since the formulation contains the relativistic Doppler effect due to the rotational motion of the cylinder, it can be applied to the case where the angular velocity of the cylinder is of arbitrary values. The general relativistic field-transformation formulas under the arbitrary coordinate transformations are employed to solve the problem. Numerical calculations are performed for the case of the rotating elliptic cylinder. The numerical illustrations of the angular distribution of the power density of various frequency components of the scattered waves are shown. The relativistic effect which is the Doppler effect due to the rotational motion of the cylindrical surface are discussed and compared with the results obtained by the quasistationary approximation method.