To execute curved cuts in a transparent material, and especially a cornea, optical perforations are formed in the material by a focused laser beam. The focal point is given a three-dimension offset in one spatial direction by a lens, and the focal point movement is moved in the other two spatial directions (x,y) to follow the contours (17) of the cut in a concentric pattern in the contour map (16) with transits (18). The contours of the cut lie on a plane at right angles to the first spatial direction.