as shown in the section Proof of the Scheimpflug principle. Changing the plane of focus When the lens and image planes are not parallel, adjusting focus rotates the PoF rather than merely displacing it along the lens axis. The axis of rotation is the intersection of the lens's front focal plane and a plane through the center of the lens parallel to the image plane, as shown in Figure 3. As the image plane is moved from IP1 to IP2, the PoF rotates about the axis G from position PoF1 to position PoF2; the "Scheimpflug line" moves from position S1 to position S2. The axis of rotation has been given many different names: "counter axis" (Scheimpflug 1904), "hinge line" (Merklinger 1996), and "pivot point" (Wheeler). Refer to Figure 4; if a lens with focal length f is tilted by an angle θ relative to the image plane, the distance J from the center of the lens to the axis G is given by If v′ is the distance along the line of sight from the image plane to the center of the lens, the angle ψ between the image plane and the PoF is given by Equivalently, on the object side of the lens, if u′ is the distance along the line of sight from the center of the lens to the PoF, the angle ψ is given by The angle ψ increases with focus distance; when the focus is at infinity, the PoF is perpendicular to the image plane for any nonzero value of tilt. The distances u′ and v′ along the line of sight are not the object and image distances u and v used in the thin-lens formula where the distances are perpendicular to the lens plane. Distances u and v are related to the line-of-sight distances by and . For an essentially planar subject, such as a roadway extending for miles from the camera on flat terrain, the tilt can be set to place the axis G in the subject plane, and the focus then adjusted to rotate the PoF so that it coincides with the subject plane. The entire subject can be in focus, even if it is not parallel to the image plane. The plane of focus also can be rotated so that it does not coincide with the subject plane, and so that only a small part of the subject is in focus.