below the optical axis negative. The relationship between the object distance u, the image distance v, and the lens focal length f is given by the thin-lens equation solving for u gives so that . The magnification m is the ratio of image height yv to object height yu and yv are of opposite sense, so the magnification is negative, indicating an inverted image. From similar triangles in Figure 6, the magnification also relates the image and object distances, so that . On the image side of the lens, giving . The locus of focus for the inclined object plane is a plane; in two-dimensional representation, the y-intercept is the same as that for the line describing the object plane, so the object plane, lens plane, and image plane have a common intersection. A similar proof is given by Larmore (1965, 171–173). Angle of the PoF with the image plane From Figure 7, where u′ and v′ are the object and image distances along the line of sight and S is the distance from the line of sight to the Scheimpflug intersection at S. Again from Figure 7, combining the previous two equations gives From the thin-lens equation, Solving for u′ gives substituting this result into the equation for gives or Similarly, the thin-lens equation can be solved for v′, and the result substituted into the equation for to give the object-side relationship Noting that the relationship between ψ and θ can be expressed in terms of the magnification m of the object in the line of sight: Proof of the "hinge rule" From Figure 7, combining with the previous result for the object side and eliminating ψ gives Again from Figure 7, so the distance d is the lens focal length f, and the point G is at the intersection the lens's front focal plane with a line parallel to the image plane. The distance J depends only on the lens tilt and the lens focal length; in particular, it is not affected by changes in focus. From Figure 7, so the distance to the Scheimpflug intersection at S varies as the focus is changed. Thus the PoF rotates about the axis at G as focus is adjusted. Notes References Carpentier, Jules. 1901. Improvements in Enlarging or like Cameras. GB Patent No. 1139. Filed 17 January 1901, and issued 2 November 1901. Available for download (PDF). Larmore, Lewis. 1965.