The invention relates to an eye surgery apparatus (1) having imaging optics (14, 26, 27) for generating an observation image (40) of a patient's eye (16). The eye surgery apparatus (1) has a device (36) for determining the azimuthal orientation of a patient's eye (16) with respect to a reference (3) which is fixed with respect to the patient's eye (16). The device (36) contains a visual display device (34) for simultaneously displaying a section (92) of an observation image (40) of the patient's eye (16), which is captured by the imaging optics (14, 26, 27) and contains at least one segment of the limbus or of the iris, said section being composed of pixels, and a reference image of the patient's eye (16), which is composed of pixels (106) and contains this segment of the limbus or of the iris. The device (36) has an input interface (28) which makes it possible for an observer to move the displayed section (92) of the captured observation image (40) of the patient's eye (16) relative to the displayed section (94) of the reference image (80). The device (36) comprises a measuring system which can be used to determine the azimuthal orientation of the observation image and of the reference image, which orientation is changed during the relative movement. The device (36) has a computer unit (5) having a computer program for calculating the center (46) of the limbus (44) and/or of the iris (43) of the observation image (40) of the patient's eye (16), as captured by the imaging optics (14, 26, 27), and of the reference image (80). The visual display device (34) displays the pixels (102) of the observation image (40) of the patient's eye (16), as captured by the imaging optics (14, 26, 27), with polar coordinates in a polar coordinate system (96), the center (46) of which coincides with the calculated center of the circular structure (44) of the limbus and/or of the iris (43) of the patient's eye (16). It displays the pixels (106) of the reference image (80) of the patient
