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Tilt and Shift Lenses
Table of Content
The Tilt Function
The Shift Function
This page is essentially my efforts to understand the possibilities available with tilt and shift lenses, otherwise known as perspective control lenses, that are available for 35mm SLR/DSLR cameras..
The methods of calculating the focus effects are the same as view cameras.
The basic geometry is shown in Fig 1 above.
SF=Sharp Focus Plane
A= Lens rear node to Image Plane separation normal to the image plane
J=Lens axis to Pivot Point Distance below lens parrallel to the image plane
Alpha=inclination angle of LP plane (lens tilt)
Gamma=inclination angle of the plane of sharp focus
The following equations define the basic geometry  , many other relations can be derived from basic trigonometry :
J=f/sin(Alpha) Eq 1
Gamma=atan(sin(Alpha)/(cos(Alpha)-f/A)) Eq 2
where f is the lens focal length.
Two rules arise from this geometry:
Rule 1: The Scheimpflug Rule
The Image Plane, the Lens Plane and the Plane of Sharp Focus all intersect at a line, this is the Scheimpflug line, which is vertical into the paper in Fig 1 above. 
Rule 2: The Pivot Rule
As the lens to image plane distance A changes, the Plane of Sharp Focus pivots about the pivot point J below the lens in Fig 1 above. 
Applying the two rules above allows the depth of field to be defined.
So referring to Fig 2.
Here we see two blue light rays being brought to focus behind the Image Plane IP at a distance A' from the lens. The lens f-stop number is N and the lens is A away from the IP, so the image point has a circle of confusion (CoC) diameter c.
Similarly in Fig 3 the red rays are brought to a focus in front of the image plane causing an image point to have a circle of confusion (CoC) diameter c.
By noting that the triangles are similar the following expressions can be derived from the ratio of the lengths of their sides :
A'=A/(s-1)+A Eq 3
A''=A-A/(s+1) Eq 4
where s=f/c/N Eq 5
We can then reapply Eq 2 substituting A' and A'' to derive the angles of the planes of acceptable focus resulting in the geometry in Fig 4, where NPAF is the near plane of acceptable focus and FPAF is the far plane of acceptable focus.
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The shift function is a lot easier to understand, all the shift function is doing is moving the image circle from the lens and thus changing the direction of the centre of the angle of view.
We can modify the standard angle of view equation to provide two limits of angle of view relative to the un-shifted optical axis for small magnifications:
AoV_up=atan((Li/2+Shift)/f), AoV_down=atan((Li/2-Shift)/f) Eq 6
This effect can be seen in Fig 5, where the black un-shifted lens has an angle of view that is symmetrical about the optical axis. After shifting down (blue) the angle of view remains the same but swings up effectively shifting the image formed upwards without changing the convergence of parallel lines in the subject when projected on the image plane.
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All the equations and diagrams above make two assumptions:
(i) That the lens is a thin lens:
This is not true, any practical lens will have the entry and exit nodes separated. The distance between them is known as the nodal separation  and can be positive or negative. For a wide-angle SLR lens that must clear the lens mirror the exit node may be some way behind the front element.
All this means is that the diagrams need to be cut along the lens nodal axis and separated by the necessary amount. This is likely to only start becoming significant if doing close-up work at high magnification.
(ii) That the lens is focused by moving the whole lens away from the image plane (Linear Group Extension):
Many modern lenses have internal or rear element focusing, or some form of front group extension and floating elements. The practical result of this is that the focal length may change slightly along with the nodal separation and pupilary magnification. However, the overall effect of the tilt and shift functions is unlikely to be significantly different.
A further issue is that the tilt axis may not be exactly aligned with the entry node and in any event there will almost certainly be some nodal separation. This may result in some inherent shift when the tilt control is operated.
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The following examples are for a 24mm lens on a 1.6 crop factor 35mm DLSR camera. The following plots were created using MATLAB code which can be downloaded here.
First some plots showing how lens tilt and extension affects the planes of sharp focus and acceptable sharpness.
We see from Fig 7 why almost all T&S lenses for 35mm have an available tilt of up to 8 degrees. This is the tilt required to make the plane of sharp focus (SF) be at 90 degrees to the image plane (IP) when focused at infinity.
As the lens is moved away from the image plane the SF plane tilt is reduced however.
Additionally Fig 10 shows that maximum angular separation of the planes of acceptable focus occurs when the both the lens distance A and the lens tilt Alpha are small.
The computer generated plane plots below need some explanation.
The red lines are the Image, Lens and Sharp Focus Planes.
The Dark Blue Diverging lines are the Planes of Acceptable Sharpness.
The green lines are the lens un-shifted angle of view at infinity focus. (Solid: Long side of frame; Dotted: Short side of frame).
The light blue and magenta lines are the +11 mm and -11 mm shifts respectively.
The vertical black line is at the hyperfocal distance, the vertical magenta and cyan lines at the un-titled planes near and far planes respectively, dash for focus set to hyperfocal distance, dash-dot focus as distance Z.
The two plots Fig 11 and Fig 12 show the situation with the plane of sharp focus crossing the un-shifted optical axis at a range of Z=1000M for 1 and 8 degree of lens tilt respectively.
Notice how much wider the depth of field is for the 1 degree case of Fig 11. Also the near plane of acceptable focus crossed the un-shifted optical axis at about 1M compared to more than 2.5M. Also notice the plane of acceptable focus crosses the optical axis at the hyperfocal distance.
Looking at the plots not a lot is being gained in terms of pan focus, in the case of Fig 11 it seems about 2 meters of ground depth of field is gained by tilting over the un-tilted case assuming the camera is set so the pivot point is at ground level. However, without shift this area is outside the lens AoV anyway.
Looking at Fig 11b suggests there is more scope for the pan-focus effect with a large aperture, although with modern low noise DLSRs this is less of an issue as the ISO can be increased to maintain fast shutter speeds for freezing motion in high winds.
This effect is probably more pronounced with a longer lens, see Fig 17.
It may be the best use for the tilt function is to bring some things out of focus, tops of trees, or one side of the frame for creative effect.
The two plots above, Fig 13 and Fig 14 the situation with the plane of sharp focus crossing the un-shifted optical axis at a range of Z=0.1M from the lens for 1 and 8 degree of lens tilt respectively.
There is considerable magnification here (31.6/100=0.316 and 32/100=0.320 at the optical axis focus point) and the depth of field is correspondingly small. Some limited focus plane tilt can be obtained however.
NB: The angle of view in this plot has not been corrected for magnification by overall group extension. For M=1, the angle of view of the lens is halved.
The two above figures Fig 15 and Fig 16 show the detail in the region of the depth of field near the optical axis. We can see that there a slightly larger amount of DoF in the 1 degree lens tilt case but the difference is minimal. In both cases the tilted planes of sharp focus cross the optical axis at the same point as the un-tilted planes of sharp focus.
However, if the subject is suitably orientated the apparent depth of field can be increased.
Fig 17 shows the situation with a 90mm lens. The tilt has been set to place the pivot point at about 2m height. The planes of acceptable focus, although not high extend from beneath the camera to infinity. Fig 18 indicates that this crosses the lower limit of un-shifted AoV at about 11m range, where it is about 0.7m high. Un-tilted the near plane of the depth of field would not be until about 27m.
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The problem of pan focus from very close to far away can be considered more specifically.
Consider again Fig 19, if the camera optical centre height h and a distance Zf are defined as a point at which the far plane of acceptable focus can pass then the lens tilt can be selected to arrive at various tilts for the FPAF. At the distance Zf the two planes of acceptable focus will have a vertical separation of hf. This may require non-infinity focus.
We are trading two parallel planes of acceptable focus for two determining planes by tilting the lens.
We can imagine photographing a scene where the land may slope up or down requiring a different tilt to the FPAF and containing a bush or some such object with vertical extent that we want to keep in focus too.
Then the various plots following can be arrived at, specifying Hf, lens tilt and set focus range for the untilted lens (marked focus distance), for three cases Fig 20 a, b & c, Fig 21 a, b & c and Fig 22 a, b & c.
Fig 20 a, b & c
Fig 21 a, b & c
Fig 22 a, b & c
We can see there is an upper limit of the tilt angle of the FPAF. To get around this we can reverse the lens tilt, tiliting the lens up (negative angles) and set the near plane of acceptable focus NPAF passing through the point h and Zf as shown in Fig 23.
Then the various plots following can be arrived at, specifying Hf, lens tilt and set focus range for the untilted lens (marked focus distance), for three cases Fig 24 a, b & c, Fig 25 a, b & c and Fig 26 a, b & c.
Fig 24 a, b & c
Fig 25 a, b & c
Fig 26 a, b & c
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The amount of sharp focus plane tilt has been seen to reduce for close focus. Plotting the tilt of the planes of sharp focus and near and far acceptable focus against magnification at centre of image on the plane of sharp focus for 24mm, 45mm and 90mm lenses. The lens tilt is assumed to be 8 degree and the set f-stop f16 with a circle of confusion on 0.019mm.
This analysis neglects the approximates made for non-closeup work and for asymmetrical lenses and so may not be accurate at higher magnifications. However, the principle of the Gaussian lens equation are not violated so the trend should be representative.
So we can conclude that the plane of sharp focus is the same for a given lens regardless of the focal length. However, the longer the focal length the less the separation between the planes of acceptable focus.
Looking around the macro region of 1X.
We can see the trend continues.
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 "Notes on View Camera Geometry", Robert E. Wheeler May 8, 2003
 "ADDENDUM to FOCUSING the VIEW CAMERA", Harold M. Merklinger, World Wide Web Edition © Harold M. Merklinger, Ottawa, Canada, September 1993 & November 1995.
Last Updated 05/06/2008
All Content © 2005-16 Lester Wareham All Rights
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