Apparatus for a laser-assisted ophthalmic surgical treatment system, comprising: - a first image capturing unit, configured to capture a first image of an eye to be treated, - a computer arrangement, configured to identify at least a first feature (40, 40') of the eye by image processing of the first image and to ascertain a position and orientation of the first feature in a coordinate system (S') of the treatment system, wherein the computer arrangement is further configured to ascertain the position and orientation of an incision figure, to be produced in the eye, in the coordinate system (S') of the treatment system as a function of the ascertained position and orientation of the first feature in the coordinate system and as a function of a relative position and orientation, ascertained in advance, of at least one second feature (64, 64') of the eye in relation to the first feature (40, 40'), characterized in that the incision figure defines a corneal flap with a flap hinge, wherein the computer arrangement is configured to ascertain the position and orientation of a figure element of the incision figure, defining the flap hinge, as a function of the position and orientation of the first feature (40, 40') in the coordinate system (S') of the treatment system and as a function of the relative position and orientation of the second feature (64, 64') in relation to the first feature (40, 40'), in that the at least one second feature (64, 64') comprises a corneal region with astigmatic curvature and the computer arrangement is configured to ascertain the position and orientation of the incision figure taking account of a predetermined desired relative orientation stipulation between the flap and the corneal region with astigmatic curvature, and/or in that a hinge axis (Q 1 '-Q 2 ') is assigned to the flap hinge and an astigmatism axis (P 1 '-P 2 ') is assigned to the corneal region with astigmatic curvature, wherein the computer arrangement is configured to ascertain the position and orientation of the incision figure taking account of a predetermined desired relative orientation stipulation between the hinge axis and the astigmatism axis, wherein, in particular, the desired relative orientation stipulation between the hinge axis (Q 1 '-Q 2 ') and the astigmatism axis (P 1 '-P 2 ') predetermines a relative orientation of the two axes that are substantially perpendicular to one another. - a first image capturing unit, configured to capture a first image of an eye to be treated, - a computer arrangement, configured to identify at least a first feature (40, 40') of the eye by image processing of the first image and to ascertain a position and orientation of the first feature in a coordinate system (S') of the treatment system, wherein the computer arrangement is further configured to ascertain the position and orientation of an incision figure, to be produced in the eye, in the coordinate system (S') of the treatment system as a function of the ascertained position and orientation of the first feature in the coordinate system and as a function of a relative position and orientation, ascertained in advance, of at least one second feature (64, 64') of the eye in relation to the first feature (40, 40'),