The trajectory of the polarization state of a monochromatic beam passing through a fixed linear polarizer and a rotating elliptical retarder on the Poincaré sphere is found to be a three-dimensional 8-shaped contour, which is determined as the line of intersection of a right-circular cylinder with the Poincaré sphere. The cylinder is parallel to the S3 axis, and the projection of the contour on the S1S2 plane is a circle whose center and radius are determined. A method of projecting the three-dimensional geometric relationships to the two-dimensional S1S2 plane to locate the position of the polarization state of the emerging beam on the Poincaré sphere for a given azimuth of the elliptical retarder is presented, and applied to solve a problem of polarization optics. The proposed graphic method substantially simplifies the polarization state analysis involving elliptical retarders.
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