Matrix for skewed rays matrix methods may be generalized to describe skewed paraxial rays in circularly symmetric systems, and to astigmatic (non-circularly symmetric) systems. a ray crossing the plane z 0 is generally characterized by four variables the coordinates (x, y) of its position in the plane, and the angles (x, y) that its projections in the v r and y z. planes make with the r axis. the emerging ray is also characterized by four variables linearly related to the initial four variables. the optical system may then be characterized completely, within the paraxial approximation, by a 4 times 4 matrix. determine the 4 times 4 ray-transfer matrix of a distance d in free space, determine the 4 times 4 ray-transfer matrix of a thin cylindrical lens with focal length f oriented in the y direction. the cylindrical lens has focal length/for rays in the y-z plane, and no focusing power for rays in the x-z plane.