The Scheimpflug principle is a description of the geometric relationship between the orientation of the plane of focus, the lens plane, and the image plane of an optical system (such as a camera) when the lens plane is not parallel to the image plane. It is applicable to the use of some camera movements on a view camera. It is also the principle used in corneal tomography, often performed prior to refractive eye surgery such as LASIK, and used for early detection of keratoconus. The principle is named after Austrian army Captain Theodor Scheimpflug, who used it in devising a systematic method and apparatus for correcting perspective distortion in aerial photographs, although Captain Scheimpflug himself credits Jules Carpentier with the rule, thus making it an example of Stigler's law of eponymy.
Description
Normally, the lens and image (film or sensor) planes of a camera are parallel, and the plane of focus (PoF) is parallel to the lens and image planes. If a planar subject (such as the side of a building) is also parallel to the image plane, it can coincide with the PoF, and the entire subject can be rendered sharply. If the subject plane is not parallel to the image plane, it will be in focus only along a line where it intersects the PoF, as illustrated in Figure 1. But when a lens is tilted with respect to the image plane, an oblique tangent extended from the image plane and another extended from the lens plane meet at a line through which the PoF also passes, as illustrated in Figure 2. With this condition, a planar subject that is not parallel to the image plane can be completely in focus. While many photographers were/are unaware of the exact geometric relationship between the PoF, lens plane, and film plane, swinging and tilting the lens to swing and tilt the PoF was practiced since the middle of the 19th century. But, when Carpentier and Scheimpflug wanted to produce equipment to automate the process, they needed to find a geometric relationship. Scheimpflug referenced this concept in his 1904 British patent; Jules Carpentier also described the concept in an earlier 1901 British patent for a perspective-correcting photographic enlarger. The concept can be inferred from a theorem in projective geometry of Gérard Desargues; the principle also readily derives from simple geometric considerations and application of the Gaussian thin-lens formula, 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", "hinge line", and "pivot point". 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
J = f sin θ . {\displaystyle J={\frac {f}{\sin \theta }}.}
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
tan ψ = u ′ f sin θ . {\displaystyle \tan {\psi }={u' \over f}\sin {\theta }.}
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
1 u + 1 v = 1 f , {\displaystyle {1 \over u}+{1 \over v}={1 \over f},}
where the distances are perpendicular to the lens plane. Distances u and v are related to the line-of-sight distances by
u = u ′ cos θ v = v ′ cos θ . {\displaystyle {\begin{aligned}u&=u'\cos {\theta }\\v&=v'\cos {\theta }.\end{aligned}}}
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