A camera's angle of view can be measured horizontally, vertically, or diagonally.

In photography, angle of view describes the angular extent of a given scene that is imaged by a camera. It is used interchangeably with the more general term field of view.

It is important to distinguish the angle of view from the angle of coverage, which describes the angle range that a lens can image. Typically the image circle produced by a lens is large enough to cover the film or sensor completely, possibly including some vignetting toward the edge. If the angle of coverage of the lens does not fill the sensor, the image circle will be visible, typically with strong vignetting toward the edge, and the effective angle of view will be limited to the angle of coverage.

Calculating a camera's angle of view[link]

For lenses projecting rectilinear (non-spatially-distorted) images of distant objects, the effective focal length and the image format dimensions completely define the angle of view. Calculations for lenses producing non-rectilinear images are much more complex and in the end not very useful in most practical applications. (In the case of a lens with distortion, e.g., a fisheye lens, a longer lens with distortion can have a wider angle of view than a shorter lens with low distortion)^[2] Angle of view may be measured horizontally (from the left to right edge of the frame), vertically (from the top to bottom of the frame), or diagonally (from one corner of the frame to its opposite corner).

For a lens projecting a rectilinear image, the angle of view (α) can be calculated from the chosen dimension (d), and effective focal length (f) as follows:^[3]

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {d} {2 f}

Failed to parse (Missing texvc executable; please see math/README to configure.): d

represents the size of the film (or sensor) in the direction measured. For example, for film that is 36 mm wide, Failed to parse (Missing texvc executable; please see math/README to configure.): d = 36
mm would be used to obtain the horizontal angle of view.

Because this is a trigonometric function, the angle of view does not vary quite linearly with the reciprocal of the focal length. However, except for wide-angle lenses, it is reasonable to approximate Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha\approx \frac{d}{f}

radians or Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{180d}{\pi f}
degrees.

The effective focal length is nearly equal to the stated focal length of the lens (F), except in macro photography where the lens-to-object distance is comparable to the focal length. In this case, the magnification factor (m) must be taken into account:

Failed to parse (Missing texvc executable; please see math/README to configure.): f = F \cdot ( 1 + m )

(In photography Failed to parse (Missing texvc executable; please see math/README to configure.): m

is usually defined to be positive, despite the inverted image.)  For example, with a magnification ratio of 1:2, we find Failed to parse (Missing texvc executable; please see math/README to configure.): f = 1.5 \cdot F
and thus the angle of view is reduced by 33% compared to focusing on a distant object with the same lens.

A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:^[4]

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}

Angle of view can also be determined using FOV tables or paper or software lens calculators.^[5]

Example[link]

Consider a 35 mm camera with a lens having a focal length of F = 50 mm. The dimensions of the 35 mm image format are 24 mm (vertically) × 36 mm (horizontal), giving a diagonal of about 43.3 mm.

At infinity focus, f = F, and the angles of view are:

• horizontally, Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_h = 2\arctan\frac{h}{2f} = 2\arctan\frac{36}{2 \times 50}\approx 39.6^\circ

• vertically, Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_v = 2\arctan\frac{v}{2f} = 2\arctan\frac{24}{2 \times 50}\approx 27.0^\circ

• diagonally, Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha_d = 2\arctan\frac{d}{2f} = 2\arctan\frac{43.3}{2 \times 50}\approx 46.8^\circ

Derivation of the angle-of-view formula[link]

Consider a rectilinear lens in a camera used to photograph an object at a distance Failed to parse (Missing texvc executable; please see math/README to configure.): S_1 , and forming an image that just barely fits in the dimension, Failed to parse (Missing texvc executable; please see math/README to configure.): d , of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance Failed to parse (Missing texvc executable; please see math/README to configure.): S_2

from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil):[6]

Now Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha/2

is the angle between the optical axis of the lens and the ray joining its optical center to the edge of the film.  Here Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha
is defined to be the angle-of-view, since it is the angle enclosing the largest object whose image can fit on the film.  We want to find the relationship between:

the angle Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

the "opposite" side of the right triangle, Failed to parse (Missing texvc executable; please see math/README to configure.): d/2

(half the film-format dimension)

the "adjacent" side, Failed to parse (Missing texvc executable; please see math/README to configure.): S_2

(distance from the lens to the image plane)

Using basic trigonometry, we find:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan ( \alpha / 2 ) = \frac {d/2} {S_2} .

which we can solve for α, giving:

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {d} {2 S_2}

To project a sharp image of distant objects, Failed to parse (Missing texvc executable; please see math/README to configure.): S_2

needs to be equal to the focal length, Failed to parse (Missing texvc executable; please see math/README to configure.): F

, which is attained by setting the lens for infinity focus. Then the angle of view is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {d} {2 f}

where Failed to parse (Missing texvc executable; please see math/README to configure.): f=F

Macro photography[link]

For macro photography, we cannot neglect the difference between Failed to parse (Missing texvc executable; please see math/README to configure.): S_2

and Failed to parse (Missing texvc executable; please see math/README to configure.): F

. From the thin lens formula,

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{1}{F} = \frac{1}{S_1} + \frac{1}{S_2}

.

We substitute for the magnification, Failed to parse (Missing texvc executable; please see math/README to configure.): m = S_2/S_1 , and with some algebra find:

Failed to parse (Missing texvc executable; please see math/README to configure.): S_2 = F\cdot(1+m)

Defining Failed to parse (Missing texvc executable; please see math/README to configure.): f=S_2

as the "effective focal length", we get the formula presented above:

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {d} {2 f}

where Failed to parse (Missing texvc executable; please see math/README to configure.): f=F\cdot(1+m)

.

A second effect which comes into play in macro photography is lens asymmetry (an asymmetric lens is a lens where the aperture appears to have different dimensions when viewed from the front and from the back). The lens asymmetry causes an offset between the nodal plane and pupil positions. The effect can be quantified using the ratio (P) between apparent exit pupil diameter and entrance pupil diameter. The full formula for angle of view now becomes:^[4]

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}

Measuring a camera's field of view[link]

In the optical instrumentation industry the term field of view (FOV) is most often used, though the measurements are still expressed as angles.^[7] Optical tests are commonly used for measuring the FOV of UV, visible, and infrared (wavelengths about 0.1–20 µm in the electromagnetic spectrum) sensors and cameras.

The purpose of this test is to measure the horizontal and vertical FOV of a lens and sensor used in an imaging system, when the lens focal length or sensor size is not known (that is, when the calculation above is not immediately applicable). Although this is one typical method that the optics industry uses to measure the FOV, there exist many other possible methods.

UV/visible light from an integrating sphere (and/or other source such as a black body) is focused onto a square test target at the focal plane of a collimator (the mirrors in the diagram), such that a virtual image of the test target will be seen infinitely far away by the camera under test. The camera under test senses a real image of the virtual image of the target, and the sensed image is displayed on a monitor.^[8]

The sensed image, which includes the target, is displayed on a monitor, where it can be measured. Dimensions of the full image display and of the portion of the image that is the target are determined by inspection (measurements are typically in pixels, but can just as well be inches or cm).

Failed to parse (Missing texvc executable; please see math/README to configure.): D

= dimension of full image

Failed to parse (Missing texvc executable; please see math/README to configure.): d

= dimension of image of target

The collimator's distant virtual image of the target subtends a certain angle, referred to as the angular extent of the target, that depends on the collimator focal length and the target size. Assuming the sensed image includes the whole target, the angle seen by the camera, its FOV, is this angular extent of the target times the ratio of full image size to target image size.^[9]

The target's angular extent is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha = 2 \arctan \frac {L} {2 f_c}

where Failed to parse (Missing texvc executable; please see math/README to configure.): L

is the dimension of the target and Failed to parse (Missing texvc executable; please see math/README to configure.): f_c
is the focal length of collimator.

The total field of view is then approximately:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{FOV} = \alpha \frac{D}{d}

or more precisely, if the imaging system is rectilinear:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{FOV} = 2 \arctan \frac {LD} {2 f_c d}

This calculation could be a horizontal or a vertical FOV, depending on how the target and image are measured.

Lens types and effects[link]

Focal length[link]

How focal length affects perspective: Varying focal lengths at identical field size achieved by different camera-subject distances. Notice that the shorter the focal length and the larger the angle of view, perspective distortion and size differences increase.

Lenses are often referred to by terms that express their angle of view:

• fisheye lenses, typical focal lengths are between 8 mm and 10 mm for circular images, and 15–16 mm for full-frame images. Up to 180° and beyond.
  • A circular fisheye lens (as opposed to a full-frame fisheye) is an example of a lens where the angle of coverage is less than the angle of view. The image projected onto the film is circular because the diameter of the image projected is narrower than that needed to cover the widest portion of the film.
• Ultra wide angle lens is a rectilinear which is less than 24mm of focal length in 35mm film format, here 14mm is 114° and 24mm is 84° .
• Wide-angle lenses (24–35mm in 35mm film format) cover between 84° and 64°
• Normal, or Standard lenses (36–60mm in 35mm film format) cover between 62° and 40°
• Long focus lenses (any lens with a focal length greater than the diagonal of the film or sensor used)^[10] generally have an angle of view of 35° or less.^[11] Since photographers usually only encounter the telephoto lens sub-type,^[12] they are referred to in common photographic parlance as:
• "Medium telephoto", a focal length of 85mm to 135mm in 35mm film format covering between 30° and 10°^[13]
• "Super telephoto" (over 300mm in 35mm film format) generally cover between 8° through less than 1°^[13]

Zoom lenses are a special case wherein the focal length, and hence angle of view, of the lens can be altered mechanically without removing the lens from the camera.

Characteristics[link]

For a given camera–subject distance, longer lenses magnify the subject more. For a given subject magnification (and thus different camera–subject distances), longer lenses appear to compress distance; wider lenses appear to expand the distance between objects.

Another result of using a wide angle lens is a greater apparent perspective distortion when the camera is not aligned perpendicularly to the subject: parallel lines converge at the same rate as with a normal lens, but converge more due to the wider total field. For example, buildings appear to be falling backwards much more severely when the camera is pointed upward from ground level than they would if photographed with a normal lens at the same distance from the subject, because more of the subject building is visible in the wide-angle shot.

Because different lenses generally require a different camera–subject distance to preserve the size of a subject, changing the angle of view can indirectly distort perspective, changing the apparent relative size of the subject and foreground.

If the subject image size remains the same, then at any given aperture all lenses, wide angle and long lenses, will give the same depth of field.^[14]

Examples[link]

An example of how lens choice affects angle of view. The photos below were taken by a 35 mm still camera at a constant distance from the subject:

28 mm lens, 65.5° × 46.4°	50 mm lens, 39.6° × 27.0°
70 mm lens, 28.9° × 19.5°	210 mm lens, 9.8° × 6.5°

Common lens angles of view[link]

This table shows the diagonal, horizontal, and vertical angles of view, in degrees, for lenses producing rectilinear images, when used with 36 mm × 24 mm format (that is, 135 film or full-frame 35mm digital using width 36 mm, height 24 mm, and diagonal 43.3 mm for d in the formula above).^[15] Digital compact cameras sometimes state the focal lengths of their lenses in 35mm equivalents, which can be used in this table.

Five images using 24, 28, 35, 50 and 72mm equivalent zoom lengths, portrait format, to illustrate angles of view ^[16]

Five images using 24, 28, 35, 50 and 72mm equivalent step zoom function, to illustrate angles of view

Three-dimensional digital art[link]

This unreferenced section requires citations to ensure verifiability.

Displaying 3d graphics as 3d projection of the models onto a 2d surface uses a series of mathematical calculations to render the scene. The angle of view of the scene is thus readily set and changed; some renderers even measure the angle of view as the focal length of an imaginary lens. The angle of view can also be projected onto the surface at an angle greater than 90°, effectively creating an ultra-wide lens effect.

Cinematography and video gaming[link]

This unreferenced section requires citations to ensure verifiability.

Modifying the angle of view over time, or zooming, is a frequently used cinematic technique.

For a visual effect, some first person video games (especially racing games), widen the angle of view beyond 90° to exaggerate the distance the player is travelling, thus exaggerating the player's perceived speed and giving a tunnel effect (like pincushion distortion). Narrowing the view angle gives a zoom in effect. Also see Field of view in video games.

References and notes[link]

1. ^ Neil Wayne Northey (September 1916). Frank V. Chambers. ed. "The Angle of View of your Lens". The Camera (Columbia Photographic Society) 20 (9). http://books.google.com/?id=kiUEAAAAYAAJ&pg=PA481&dq=%22field-of+view%22+%22focal+length%22+camera. 
2. ^ http://www.the-digital-picture.com/reviews/canon-ef-15mm-f-2.8-fisheye-lens-review.aspx
3. ^ Ernest McCollough (1893). "Photographic Topography". Industry: A Monthly Magazine Devoted to Science, Engineering and Mechanic Arts (Industrial Publishing Company, San Francisco): 399–406. http://books.google.com/?id=eCkAAAAAMAAJ&pg=PA402&dq=%22field-of+view%22+%22focal+length%22+camera+tangent+%22length+of+the+plate%22. 
4. ^ ^{a b} Paul van Walree (2009). "Center of perspective". http://toothwalker.org/optics/cop.html#fov. Retrieved 24 Januari 2010. 
5. ^ CCTV Field of View Camera Lens Calculations by JVSG, December, 2007
6. ^ Kerr, Douglas A. (2005). "The Proper Pivot Point for Panoramic Photography" (PDF). The Pumpkin. http://doug.kerr.home.att.net/pumpkin/Pivot_Point.pdf. Retrieved 2007-01-14. 
7. ^ Holst, G.C. (1998). Testing and Evaluation of Infrared Imaging Systems (2nd ed.). Florida:JCD Publishing, Washington:SPIE.
8. ^ Mazzetta, J.A.; Scopatz, S.D. (2007). Automated Testing of Ultraviolet, Visible, and Infrared Sensors Using Shared Optics. Infrared Imaging Systems: Design Analysis, Modeling, and Testing XVIII,Vol. 6543, pp. 654313-1 654313-14
9. ^ Electro Optical Industries, Inc.(2005). EO TestLab Methadology. In Education/Ref. http://www.electro-optical.com/html/toplevel/educationref.asp.
10. ^ Sidney F. Ray, Applied photographic optics: lenses and optical systems for photography, page 294
11. ^ Lynne Warren, Encyclopedia of 20th century photography, page 211
12. ^ Michael Langford, Basic photography, page 83
13. ^ ^{a b} photographywebsite.co.uk - Lens Types Explained
14. ^ Reichmann, Michael. "Do Wide Angle Lenses Really Have Greater Depth of Field Than Telephotos?". http://www.luminous-landscape.com/tutorials/dof2.shtml. 
15. ^ However, most interchangeable-lens digital cameras do not use 24x36 mm image sensors and therefore produce narrower angles of view than set out in the table. See crop factor and the subtopic digital camera issues in the article on wide-angle lenses for further discussion.
16. ^ The image examples uses a 5.1-15.3mm lens which is called a 24mm 3x zoom by the producer (Ricoh Caplio GX100)

See also[link]

• 35 mm equivalent focal length
• Camera angle
• Camera coverage
• Camera operator
• Cinematic techniques
• Field of view
• Filmmaking
• Multiple-camera setup
• Single-camera setup
• Video production

External links[link]

• Angle of View on digital SLR cameras with reduced sensor size
• Focal Length and Angle of View

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2010)

The field of view (also field of vision, abbreviated FOV) is the extent of the observable world that is seen at any given moment.

Different animals have different fields of view, depending on the placement of the eyes. Humans have an almost 180-degree forward-facing horizontal field of view, while some birds have a complete or nearly complete 360-degree field of view. In addition, the vertical range of the field of view in humans is typically around 100 degrees.

The range of visual abilities is not uniform across a field of view, and varies from animal to animal. For example, binocular vision, which is important for depth perception, only covers 120 degrees (horizontally) of the field of vision in humans^{[citation needed]}; the remaining peripheral 60 degrees have no binocular vision (because of the lack of overlap in the images from either eye for those parts of the field of view). Some birds have a scant 10 or 20 degrees of binocular vision.

Similarly, color vision and the ability to perceive shape and motion vary across the field of view; in humans the former is concentrated in the center of the visual field, while the latter tends to be much stronger in the periphery. This is due to the much higher concentration of color-sensitive cone cells in the fovea, the central region of the retina, in comparison to the higher concentration of motion-sensitive rod cells in the periphery. Since cone cells require considerably brighter light sources to be activated, the result of this distribution is that peripheral vision is much stronger at night relative to binocular vision.

Conversions[link]

Many optical instruments, particularly binoculars or spotting scopes, are advertised with their field of view specified in one of two ways: angular field of view, and linear field of view. Angular field of view is typically specified in degrees, while linear field of view is a ratio of lengths. For example, binoculars with a 5.8 degree (angular) field of view might be advertised as having a (linear) field of view of 305 ft per 1000 yd or 102 mm per meter. As long as the FOV is less than about 10 degrees or so, the following approximation formulas allow one to convert between linear and angular field of view. Let Failed to parse (Missing texvc executable; please see math/README to configure.): A

be the angular field of view in degrees.  Let Failed to parse (Missing texvc executable; please see math/README to configure.): L
be the linear field of view in feet per 1000 yd.  Let Failed to parse (Missing texvc executable; please see math/README to configure.): M
be the linear field of view in millimeters per meter.  Then, using the small-angle approximation:

Failed to parse (Missing texvc executable; please see math/README to configure.): A \approx {360^{\circ}\over 2 \pi} \cdot {L\over 3000 } \approx 0.0191 \times L

Failed to parse (Missing texvc executable; please see math/README to configure.): A \approx {360^{\circ}\over 2 \pi} \cdot {M\over 1000 } \approx 0.0573 \times M

Failed to parse (Missing texvc executable; please see math/README to configure.): L \approx {2 \pi \cdot 3000 \over 360^{\circ}} \cdot A \approx 52.36 \times A

Failed to parse (Missing texvc executable; please see math/README to configure.): M \approx {2 \pi \cdot 1000 \over 360^{\circ}} \cdot A \approx 17.45 \times A

Machine vision[link]

In machine vision the lens focal length sets up the fixed relationship between the field of view and the working distance. Field of view is the area of the inspection captured on the camera’s imager. The size of the field of view and the size of the camera’s imager directly affect the image resolution (one determining factor in accuracy). Working distance is the distance between the back of the lens and the target object.

Astronomy[link]

In astronomy the field of view is usually expressed as an angular area viewed by the instrument, in square degrees, or for higher magnification instruments, in square arc-minutes. For reference the Wide Field Channel on the Advanced Camera for Surveys on the Hubble Space Telescope has a field of view of 10 sq. arc-minutes, and the High Resolution Channel of the same instrument has a field of view of 0.15 sq. arc-minutes. Ground based survey telescopes have much wider fields of view. The photographic plates used by the UK Schmidt Telescope had a field of view of 30 sq. degrees. The 1.8 m (71 in) Pan-STARRS telescope, with the most advanced digital camera to date has a field of view of 7 sq. degrees. In the near infra-red WFCAM on UKIRT has a field of view of 0.2 sq. degrees and the forthcoming VISTA telescope will have a field of view of 0.6 sq. degrees. Until recently digital cameras could only cover a small field of view compared to photographic plates, although they beat photographic plates in quantum efficiency, linearity and dynamic range, as well as being much easier to process.

Photography[link]

In photography, the field of view is that part of the world that is visible through the camera at a particular position and orientation in space; objects outside the FOV when the picture is taken are not recorded in the photograph. It is most often expressed as the angular size of the view cone, as an angle of view.

Video games[link]

The field of view in video games refers to the part you see of a game world, which is dependent on the scaling method used.

See also[link]

• Perimetry
• Peripheral vision
• Visual perception
• Useful field of view

References[link]