# Angle of view: Wikis

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# Encyclopedia

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 of projection by the lens onto the focal plane. For most cameras, it may be assumed that the image circle produced by the lens is large enough to cover the film or sensor completely.[1] If the angle of view exceeds the angle of coverage, however, then vignetting will be present in the resulting photograph. For an example of this, see below.

## Calculating a camera's angle of view

In 1916, Northey showed how to calculate the angle of view using ordinary carpenter's tools.[2] The angle that he labels as the angle of view is the half-angle or "the angle that a straight line would take from the extreme outside of the field of view to the center of the lens;" he notes that manufacturers of lenses use twice this angle.

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.

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]

$\alpha = 2 \arctan \frac {d} {2 f}$

d represents the size of the film (or sensor) in the direction measured. For example, for film that is 36 mm wide, 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 $\alpha\approx \frac{d}{f}$ radians or $\frac{180d}{\pi f}$ 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:

$f = F \cdot ( 1 + m )$

(In photography m is usually defined to be positive, despite the inverted image.) For example, with a magnification ratio of 1:2, we find $f = 1.5 \cdot F$ 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]:

$\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}$

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

### Example

Consider a 35 mm camera with a normal 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, $\alpha_h = 2\arctan\frac{h}{2f} = 2\arctan\frac{36}{2 \times 50}\approx 39.6^\circ$
• vertically, $\alpha_v = 2\arctan\frac{v}{2f} = 2\arctan\frac{24}{2 \times 50}\approx 27.0^\circ$
• diagonally, $\alpha_d = 2\arctan\frac{d}{2f} = 2\arctan\frac{43.3}{2 \times 50}\approx 46.8^\circ$

### Derivation of the angle-of-view formula

Consider a rectilinear lens in a camera used to photograph an object at a distance S1, and forming an image that just barely fits in the dimension, d, of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance S2 from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil[6]):

Now α / 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 α 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 α
the "opposite" side of the right triangle, d / 2 (half the film-format dimension)
the "adjacent" side, S2 (distance from the lens to the image plane)

Using basic trigonometry, we find:

$\tan ( \alpha / 2 ) = \frac {d/2} {S_2} .$

which we can solve for α, giving:

$\alpha = 2 \arctan \frac {d} {2 S_2}$

To project a sharp image of distant objects, S2 needs to be equal to the focal length, F, which is attained by setting the lens for infinity focus. Then the angle of view is given by:

$\alpha = 2 \arctan \frac {d} {2 f}$ where f = F

#### Macro photography

For macro photography, we cannot neglect the difference between S2 and F. From the thin lens formula,

$\frac{1}{F} = \frac{1}{S_1} + \frac{1}{S_2}$.

We substitute for the magnification, m = S2 / S1, and with some algebra find:

$S_2 = F\cdot(1+m)$

Defining f = S2 as the "effective focal length", we get the formula presented above:

$\alpha = 2 \arctan \frac {d} {2 f}$ where $f=F\cdot(1+m)$.

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[7]:

$\alpha = 2 \arctan \frac {d} {2 F\cdot ( 1 + m/P )}$

## Measuring a camera's field of view

Schematic of collimator-based optical apparatus used in measuring the FOV of a camera.

In the optical instrumentation industry the term field of view (FOV) is most often used, though the measurements are still expressed as angles.[8] 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.[9]

Monitor display of sensed image from the camera under test

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).

D = dimension of full image
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.[10]

The target's angular extent is:

$\alpha = 2 \arctan \frac {L} {2 f_c}$
where L is the dimension of the target and fc is the focal length of collimator.

The total field of view is then approximately:

$\mathrm{FOV} = \alpha \frac{D}{d}$

or more precisely, if the imaging system is rectilinear:

$\mathrm{FOV} = 2 \arctan \frac {LD} {2 f_c d}$

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

## Lens types and effects

### Focal length

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:

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

Longer lenses magnify the subject more, apparently compressing distance and (when focused on the foreground) blurring the background because of their shallower depth of field. Wider lenses tend to magnify distance between objects while allowing greater depth of field.

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.

### Examples

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

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[11]). Digital compact cameras state their focal lengths in 35mm equivalents, which can be used in this table.

 Focal Length (mm) Diagonal (°) Vertical (°) Horizontal (°) 13 15 18 21 24 28 35 43.3 50 70 85 105 135 180 200 300 400 500 600 800 1200 118 111 100 91.7 84.1 75.4 63.4 53.1 46.8 34.4 28.6 23.3 18.2 13.7 12.4 8.25 6.19 4.96 4.13 3.1 2.07 85.4 77.3 67.4 59.5 53.1 46.4 37.8 31 27 19.5 16.1 13 10.2 7.63 6.87 4.58 3.44 2.75 2.29 1.72 1.15 108 100.4 90 81.2 73.7 65.5 54.4 45.1 39.6 28.8 23.9 19.5 15.2 11.4 10.3 6.87 5.15 4.12 3.44 2.58 1.72
Five images using 24, 28, 35, 50 and 72mm equivalent zoom lengths, portrait format, to illustrate angles of view [12]
Five images using 24, 28, 35, 50 and 72mm equivalent step zoom function, to illustrate angles of view

## Three-dimensional digital art

Displaying 3d graphics requires 3d projection of the models onto a 2d surface, and 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 a fish eye lens effect.

## Cinematography and video gaming

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

As 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.

## References and notes

1. ^ One exception is technical photography involving view camera movements, in which the optical axis of the lens may not be aligned with the center of the frame. The photographer must ensure that the angle of coverage of the lens is large enough to cover the frame in this case.
2. ^ Neil Wayne Northey (September 1916). Frank V. Chambers. ed. "The Angle of View of your Lens". The Camera (Columbia Photographic Society) 20 (9).
3. ^ Ernest McCollough (1893). "Photographic Topography". Industry: A Monthly Magazine Devoted to Science, Engineering and Mechanic Arts (Industrial Publishing Company, San Francisco): 399–406.
4. ^ Paul van Walree (2009). "Center of perspective". 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. Retrieved 2007-01-14.
7. ^ Paul van Walree (2009). "Center of perspective". Retrieved 24 Januari 2010.
8. ^ Holst, G.C. (1998). Testing and Evaluation of Infrared Imaging Systems (2nd ed.). Florida:JCD Publishing, Washington:SPIE.
9. ^ 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
10. ^ Electro Optical Industries, Inc.(2005). EO TestLab Methadology. In Education/Ref. http://www.electro-optical.com/html/toplevel/educationref.asp.
11. ^ 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.
12. ^ The image examples uses a 5.1-15.3mm lens which is called a 24mm 3x zoom by the producer (Ricoh Caplio GX100)