Editing Scheimpflug principle (section)

==Changing the plane of focus==
When the lens and image planes are not parallel, adjusting focus{{Efn|Strictly, the PoF rotation axis remains fixed only when focus is adjusted by moving the camera back, as on a view camera. When focusing by moving the lens, there is a slight motion of the rotation axis, but except for very small camera-to-subject distances, the motion is usually insignificant.}} 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#Focal points and planes|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 IP<sub>1</sub> to IP<sub>2</sub>, the PoF rotates about the axis G from position PoF<sub>1</sub> to position PoF<sub>2</sub>; the "Scheimpflug line" moves from position S<sub>1</sub> to position S<sub>2</sub>. 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 <var>f</var> is tilted by an angle <var>θ</var> relative to the image plane, the distance <var>J</var>{{Efn|The symbol <var>J</var> for the distance from the center of the lens to the PoF rotation axis was introduced by Merklinger (1996), and apparently has no particular significance.}} from the center of the lens to the axis G is given by

: <math>J = \frac f {\sin \theta}.</math>

If <var>v′</var> is the distance along the line of sight from the image plane to the center of the lens, the angle <var>ψ</var> between the image plane and the PoF is given by{{Efn|Merklinger (1996, 24) gives the formula for the angle of the plane of focus as

: <math>\frac {v'} {f} = \sin \theta \left [ \frac {1} {\tan \left ( \psi - \theta \right ) } + \frac {1} {\tan \theta } \right ] \,;</math>
by application of the angle-difference formula for the tangent and rearrangement, it can be transformed into the form given in this article.}}

: <span id="PoF_ImageSide"><math>\tan \psi = \frac {v'} {v' \cos \theta - f} \sin \theta.</math></span>

Equivalently, on the object side of the lens, if <var>u′</var> is the distance along the line of sight from the center of the lens to the PoF, the angle <var>ψ</var> is given by

: <math>\tan \psi = \frac {u'} f \sin \theta.</math>

The angle <var>ψ</var> 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 <var>u′</var> and <var>v′</var> along the line of sight are ''not'' the object and image distances <var>u</var> and <var>v</var> used in the thin-lens formula

: <math>\frac 1 u + \frac 1 v = \frac 1 f,</math>

<!-- There ought to be a cleaner way to do this and still use the running-text font. -->
where the distances are perpendicular to the lens plane. Distances <var>u</var> and <var>v</var> are related to the line-of-sight distances by
{{nowrap|1=<var>u</var> = <var>u′</var>&thinsp;cos&thinsp;<var>θ</var>}} and
{{nowrap|1=<var>v</var> = <var>v′</var>&thinsp;cos&thinsp;<var>θ</var>}}.

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. This technique sometimes is referred to as "anti-Scheimpflug", though it actually relies on the Scheimpflug principle.

Rotation of the plane of focus can be accomplished by rotating either the lens plane or the image plane. Rotating the lens (as by adjusting the front standard on a [[view camera]]) does not alter [[linear perspective]]{{Efn|Strictly, keeping the image plane parallel to a planar subject maintains perspective in that subject only when the lens is of symmetrical design, i.e., the [[entrance pupil|entrance]] and [[exit pupil]]s coincide with the [[Cardinal point (optics)#Nodal points|nodal planes]]. Most view-camera lenses are nearly symmetrical, but this is not always the case with tilt/shift lenses used on small- and medium-format cameras, especially with [[wide-angle lens]]es of [[retrofocus]] design. If a retrofocus or [[telephoto]] lens is tilted, the angle of the camera back may need to be adjusted to maintain perspective.}} in a planar subject such as the face of a building, but requires a lens with a large [[image circle]] to avoid [[vignetting]]. Rotating the image plane (as by adjusting the back or rear standard on a view camera) alters perspective (e.g., the sides of a building converge), but works with a lens that has a smaller image circle. Rotation of the lens or back about a horizontal axis is commonly called ''tilt'', and rotation about a vertical axis is commonly called ''swing''.

[[File:Rolleiflex SL 66 Retro Scheimpflug.jpg|thumb|Medium format camera with built in tilt]]