# Scheimpflug principle

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Scheimpflug principle

The Scheimpflug principle is a geometric rule that describes the orientation of the plane of focus of an optical system (such as a camera) when the lens plane is not parallel to the image plane. It is commonly applied to the use of camera movements on a view camera. 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.

Description of the Scheimpflug principle

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.

When an oblique tangent is extended from the image plane, and another is extended from the lens plane, they 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.

Scheimpflug (1904) referenced this concept in his British patent; Carpentier (1901) also described the concept in an earlier 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 [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 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: &ldquo;counter axis&rdquo; (Scheimpflug 1904), &ldquo;hinge line&rdquo; (Merklinger 1996), and &ldquo;pivot point&rdquo; (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 [The symbol J 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

:$J = frac f \left\{sin heta\right\}$&thinsp;.

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

:$an psi = frac \left\{v\text{'}\right\} \left\{v\text{'} cos heta - f\right\} sin heta$&thinsp;.

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

:$an psi = frac \left\{u\text{'}\right\} f sin heta$&thinsp;.

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. Note that 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

:$frac 1 u + frac 1 v = frac 1 f$,

where the distances are perpendicular to the lens plane. Distances u and v are related to the line-of-sight distances bynowrap|1=u = u'&thinsp;cos&thinsp;θ andnowrap|1=v = v'&thinsp;cos&thinsp;θ.

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 &ldquo;anti-Scheimpflug&rdquo;, 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 [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 and exit pupils coincide with the 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 lenses 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".

Camera movements

Tilt and swing are available on most view cameras, often on both the front and rear standards, and on some small- and medium-format cameras with special lenses that partially emulate view-camera movements. Such lenses are often called “tilt/shift” lenses (the Canon 24 mm, 45 mm, and 90 mm TS-E) or “perspective control” [The earliest Nikon perspective-control lenses included only shift, hence, the designation “PC”; the 85 mm PC (introduced in 1999) and the three PC-E lenses (introduced in 2008) also include tilt.] lenses (the Nikon 24 mm, 45 mm, and 85 mm PC-E). Some cameras use adapters (like the HTS 1.5 tilt-and-shift adapter, announced in July 2008, for Hasselblad H-Series cameras [The Hasselblad HTS 1.5 is compatible with the 28 mm, 35 mm, 50 mm, 80 mm, and 100 mm H-System lenses. Source: Hasselblad [http://www.hasselblad.se/media/1332322/uk_hts_datasheet_v3.pdf HTS 1.5 product data sheet] 09.08 (PDF). Retrieved 8 October 2008.] ) that provide movements with some of the manufacturer’s regular lenses.

The Arri professional motion-picture camera company offers tilt-focus lenses (which provide ±8° of tilt but no shift) in focal lengths of 24 mm, 45 mm, and 90 mm. It also manufactures a shift/tilt bellows system that utilizes standard PL-mount motion-picture camera lenses.

Depth of field

When the lens and image planes are parallel, the depth of field (DoF) extends between parallel planes on either side of the plane of focus. When the Scheimpflug principle is employed, the DoF becomes wedge shaped, [When the lens plane is not parallel to the image plane, the blur spots are ellipses rather than circles, and the limits of DoF are not exactly planar. There is little data on human perception of elliptical rather than circular blurs, but taking the major axis of the ellipse as the governing dimension is arguably the worst-case condition. Using this assumption, Robert Wheeler examines the effect of elliptical blur spots on DoF limits for a tilted lens in his Notes on View Camera Geometry; he concludes that in typical applications, the effect is negligible, and that the assumption of planar DoF limits is reasonable. His analysis considers only points on a vertical plane through the center of the lens, however.] with the apex of the wedge at the PoF rotation axis, as shown in Figure 5. The DoF is zero at the apex, remains shallow at the edge of the lens’s field of view, and increases with distance from the camera. The shallow DoF near the camera requires the PoF to be positioned carefully if near objects are to be rendered sharply.

On a plane parallel to the image plane, the DoF is equally distributed above and below the PoF; in Figure 5, the distances yn and yf on the plane VP are equal. This distribution can be helpful in determining the best position for the PoF; if a scene includes a distant tall feature, the best fit of the DoF to the scene often results from having the PoF pass through the vertical midpoint of that feature. Note that the angular DoF is not equally distributed about the PoF.

With some subjects, such as landscapes, the wedge-shaped DoF is a good fit to the scene, and satisfactory sharpness can often be achieved with a smaller lens f-number (larger aperture) than would be required if the PoF were parallel to the image plane.

Proof of the Scheimpflug principle

In a two-dimensional representation, an object plane inclined to the lensplane is a line described by

:$y=au+b$ .

From the thin-lens equation,

:$u=frac\left\{vf\right\}\left\{v-f\right\}$

and

:$m=-frac\left\{v\right\}\left\{u\right\}=-frac\left\{v-f\right\}\left\{f\right\}$ .

On the image side of the lens, from Figure 6,

:

giving

:$y_\left\{v\right\}=-left\left( a-frac\left\{b\right\}\left\{f\right\} ight\right)v+b$ .

The locus of focus for the inclined object plane is a plane; intwo-dimensional representation, the y-intercept is the same as that for theline describing the object plane, so the object plane, lens plane, and imageplane have a common intersection.

A similar proof is given by Larmore (1965, 171&ndash;173).

Notes

References

* Carpentier, Jules. 1901. Improvements in Enlarging or like Cameras. GB Patent No. 1139. Filed 17 January 1901, and issued 2 November 1901. [http://www.trenholm.org/hmmerk/CARPNTR.pdf Available for download] (PDF).
* Larmore, Lewis. 1965. "Introduction to Photographic Principles". New York: Dover Publications, Inc.
* Scheimpflug, Theodor. 1904. Improved Method and Apparatus for the Systematic Alteration or Distortion of Plane Pictures and Images by Means of Lenses and Mirrors for Photography and for other purposes. GB Patent No. 1196. Filed 16 January 1904, and issued 12 May 1904. [http://www.trenholm.org/hmmerk/TSBP.pdf Available for download] (PDF).

* [http://www.largeformatphotography.info/how-to-focus.html How to Focus the View Camera] by Quang-Tuan Luong. Includes discussion of how to set the plane of focus.
* [http://www.trenholm.org/hmmerk/#SR The Scheimpflug Principle] by Harold Merklinger.
* [http://www.trenholm.org/hmmerk/FVCADNDM.pdf Addendum to "Focusing the View Camera"] (PDF) by Harold Merklinger.
* [http://cgi.cae.wisc.edu/~ram/Wiki/data/MyWork_attach/msthesis.pdf Unilateral Real-time Scheimpflug Videography to Study Accommodation Dynamics in Human Eyes] (PDF) by Ram Subramanian.
* [http://www.bobwheeler.com/photo/ViewCam.pdf Notes on View Camera Geometry] (PDF) by Robert Wheeler.
* [http://www.zen20934.zen.co.uk/photography/tiltshift.htm Tilt and Shift Lenses] : Tailored towards 35 mm and dSLR perspective-control lenses, although the principles apply equally to press and view cameras with lens movements.

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