An eye surgery apparatus has imaging optics for generating an observation image of a patient eye. A device determines the azimuthal orientation of the patient eye with respect to a reference which is fixed with respect to the patient eye. The device includes a display unit for simultaneously displaying a section of an observation image of the eye and a reference image thereof. An input interface permits an observer to move the displayed section of the observation image relative to the displayed section of the reference image. A measuring system determines the azimuthal orientation of the observation image and reference image. A computer program calculates the center of the limbus of the observation image and reference image. The display unit displays the pixels of the observation image of the patient eye with polar coordinates. The pixels of the reference image of the patient eye are displayed with polar coordinates.