As noted above, a camera's angle level of view depends not only on the lens, but also on the sensor used. Digital sensors are usually smaller than 35 mm film, causing the lens to usually behave as a longer focal length lens would behave, and have a narrower angle of view than with 35 mm film, by a constant factor for each sensor (called the crop factor). In everyday digital cameras, the crop factor can range from around 1 (professional digital SLRs), to 1.6 (mid-market SLRs), to around 3 to 6 for compact cameras. So a standard 50 mm lens for 35 mm photography acts like a 50 mm standard "film" lens even on a professional digital SLR, but would act closer to a 75 mm (1.5×50 mm Nikon) or 80 mm lens (1.6×50mm Canon) on many mid-market DSLRs, and the 40-degree angle of view of a standard 50 mm lens on a film camera is equivalent to a 28–35 mm lens on many digital SLRs.

To project a sharp image of distant objects, S 2 {\displaystyle S_{2}} needs to be equal to the focal length, F {\displaystyle F} , which is attained by setting the lens for infinity focus. Then the angle of view is given by:

Consider a rectilinear lens in a camera used to photograph an object at a distance S 1 {\displaystyle S_{1}} , and forming an image that just barely fits in the dimension, d {\displaystyle d} , of the frame (the film or image sensor). Treat the lens as if it were a pinhole at distance S 2 {\displaystyle S_{2}} from the image plane (technically, the center of perspective of a rectilinear lens is at the center of its entrance pupil):[6]

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.

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.

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.

Fov definitioncamera

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.

Let’s say you take a picture of an automobile with two cameras, a 35mm film camera and a smartphone camera. You stand in the same spot and take two photos, one with each camera. In both cases you want to take a photo of the automobile that fills the frame. If the 35mm film camera lens has a 50mm focal length, the digital camera’s focal length might be 4mm. So even though they are very different numbers they produce the same result because of the size of the imaging surface. So the “equivalent 35mm focal length” for this smartphone camera at 4mm is 50mm.

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 α ≈ d f {\displaystyle \alpha \approx {\frac {d}{f}}} radians or 180 d π f {\displaystyle {\frac {180d}{\pi f}}} degrees.

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.[15]

For macro photography, we cannot neglect the difference between S 2 {\displaystyle S_{2}} and F {\displaystyle F} . From the thin lens formula, 1 F = 1 S 1 + 1 S 2 . {\displaystyle {\frac {1}{F}}={\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}.}

(In photography m {\displaystyle 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 ⋅ F {\displaystyle 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.

d {\displaystyle d} represents the size of the film (or sensor) in the direction measured (see below: sensor effects). For example, for 35 mm film which is 36 mm wide and 24 mm high, d = 36 m m {\displaystyle d=36\,\mathrm {mm} } would be used to obtain the horizontal angle of view and d = 24 m m {\displaystyle d=24\,\mathrm {mm} } for the vertical angle.

FOVcalculator

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.

FOVurban dictionary

From the definition of magnification, m = S 2 / S 1 {\displaystyle m=S_{2}/S_{1}} , we can substitute S 1 {\displaystyle S_{1}} and with some algebra find: S 2 = F ⋅ ( 1 + m ) {\displaystyle S_{2}=F\cdot (1+m)}

In many cases though, the advantages of using focus (i.e. crisp targets and distinct features) with subtle effects on principal distance/focal length, outweigh the advantages of keeping focus and principal distance constant (i.e. potentially causing blur in some photos taken at a different distance).

Defining f = S 2 {\displaystyle f=S_{2}} as the "effective focal length", we get the formula presented above: α = 2 arctan ⁡ d 2 f {\displaystyle \alpha =2\arctan {\frac {d}{2f}}} where f = F ⋅ ( 1 + m ) {\displaystyle f=F\cdot (1+m)} .

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)[3] 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).

PhotoModeler is one of the leading tools for photogrammetry (the science of generating measurements and accurate 3d data from photography).

Modifying the angle of view over time (known as zooming), is a frequently used cinematic technique, often combined with camera movement to produce a "dolly zoom" effect, made famous by the film Vertigo. Using a wide angle of view can exaggerate the camera's perceived speed, and is a common technique in tracking shots, phantom rides, and racing video games. See also Field of view in video games.

Note: A technical photogrammetry term that you may come across is the “Principal Distance”.  Strictly, the Principal Distance is the distance mentioned above (i.e. distance from imaging plane to the lens optical sensor), and the focal length is the principal distance when the lens is focused at infinity. See below for more information on focus vs focal length. When PhotoModeler lists focal length for a camera, it is actually the Principal Distance that is shown.

Note that the angle of view varies slightly when the focus is not at infinity (See breathing (lens)), given by S 2 = S 1 f S 1 − f {\displaystyle S_{2}={\frac {S_{1}f}{S_{1}-f}}} rearranging the lens equation.

Using basic trigonometry, we find: tan ⁡ ( α / 2 ) = d / 2 S 2 . {\displaystyle \tan(\alpha /2)={\frac {d/2}{S_{2}}}.} which we can solve for α, giving: α = 2 arctan ⁡ d 2 S 2 {\displaystyle \alpha =2\arctan {\frac {d}{2S_{2}}}}

All lenses have a stated or specified focal length value (or range of values for a zoom lens). This printed number is actually its nominal length or the principal distance when the lens is focused at infinity. As you focus on objects that are closer to the camera, the principal distance changes.  So for example, a 50mm lens focused on an object a few feet away might have a principal distance of 55mm lens at that time. The most extreme example of this is with a macro setting (a lens setting that allows you to focus on very close, very small objects, under 5″ in size for example). A lens that has a 50mm nominal focal length (so a 50mm principal distance when focused at infinity) might in fact have a 100mm principal distance when focused at a few inches! This is why it is good with photogrammetry (where precise geometry is needed) to calibrate a camera at the distance you will be working with.

Learn how to use PhotoModeler with your camera to create detailed digital models: www.photomodeler.com/products/why.html

The table below shows the horizontal, vertical and diagonal angles of view, in degrees, when used with 22.2 mm × 14.8 mm format (that is Canon's DSLR APS-C frame size) and a diagonal of 26.7 mm.

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).

What isFOVin games

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 ⋅ ( 1 + m ) {\displaystyle f=F\cdot (1+m)}

FOVto focal length calculator

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] α = 2 arctan ⁡ d 2 F ⋅ ( 1 + m / P ) {\displaystyle \alpha =2\arctan {\frac {d}{2F\cdot (1+m/P)}}}

Above we mention that focal length is related to focus distance.  Focal length is the principal distance of a camera when it is focused at infinity. In photogrammetry we are interested in the camera’s internal geometry at the time photos were taken – so it is the principal distance that we want to know precisely in photogrammetry.

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 35 mm digital using width 36 mm, height 24 mm, and diagonal 43.3 mm for d in the formula above).[16] Digital compact cameras sometimes state the focal lengths of their lenses in 35 mm equivalents, which can be used in this table.

There is some ability to calibrate a camera (which solves the principal distance) at one focus and execute your photogrammetric project at another focus.  The actual discrepancy that is acceptable depends on your accuracy requirements and how much the focus changes.  Generally a calibration done at 2m/6ft focus distance is acceptable for projects up to infinite focus (again depending on accuracy requirements), but may not be acceptable for a project where the focus distance was 50cm/20in.

A camera's angle of view depends not only on the lens, but also on the sensor. Digital sensors are usually smaller than 35 mm film, and this causes the lens to have a narrower angle of view than with 35 mm film, by a constant factor for each sensor (called the crop factor). In everyday digital cameras, the crop factor can range from around 1 (professional digital SLRs), to 1.6 (consumer SLR), to 2 (Micro Four Thirds ILC) to 6 (most compact cameras). So a standard 50 mm lens for 35 mm photography acts like a 50 mm standard "film" lens on a professional digital SLR, but would act closer to an 80 mm lens (1.6×50mm) on many mid-market DSLRs, and the 40-degree angle of view of a standard 50 mm lens on a film camera is equivalent to an 80 mm lens on many digital SLRs.

A camera typically has focal length in a range of 10mm to 500mm.  Different types of camera can have different ranges and speciality lenses can extend outside this range as well.  A 10mm focal length would be a very wide lens (capturing a lot of the scene), and 500mm would be a very narrow lens (capturing only a small part of the scene – giving a large magnification like binoculars or a telescope).

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]

Cameras can have fixed lenses (sometimes called ‘prime’ lenses) which have just one focal length, or zoom lenses which allow the focal length to be varied (for example between 18mm-55mm, or 55mm-200mm). For high accuracy photogrammetric work in PhotoModeler, a fixed (or prime) wide lens (such as a 20mm lens on an APS-C frame camera) is recommended as the primary option, but different applications may require different focal lengths, and cameras with adjustable zoom lenses can still be used to achieve very good results with some extra procedural care over the focal length.

Modern digital cameras can have imaging chips that are as small as 6mm by 4mm; some Smartphone cameras are even smaller, and then up to full 24mm by 35mm size.  A very common size is the APS-C format at 16mm by 24mm. This smaller size affects what is considered to be a ‘normal’ focal length.

Camera manufacturers sometimes list these equivalents because some photographers are more familiar with 35mm cameras and they want to make it easier to understand. It also gives us a standard of reference for all the different format sizes. They may also list the multiplier factor. For example, the APS-C multiplier is around 1.6x.  So a 32mm lens on an APS-C camera (like the Nikon D3200) would act like a 50mm lens on a 35mm film camera. Does focusing affect the focal length?

Now α / 2 {\displaystyle \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 α {\displaystyle \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:

When you buy a digital camera you will often see the specification “equivalent 35mm focal length”. What does this mean? Most digital cameras have imaging chips that cover much less area than a standard 35mm film frame. Since 35mm film cameras were the standard for so long in photography, much of the techniques and methods were developed around them. A 35mm film camera has a negative that is about 36mm wide by 24mm high (the “35” comes from the physical width of the film stock that is exactly 35mm wide).  A ‘normal lens’ (has a field of view that appears ‘natural’ to humans) on a 35mm film camera has a focal length of 50mm.

Focal length is a number that is vital to photography and photogrammetry but often misunderstood. What is focal length?

Fov definitionphotography

The total field of view is then approximately: F O V = α D d {\displaystyle \mathrm {FOV} =\alpha {\frac {D}{d}}} or more precisely, if the imaging system is rectilinear: F O V = 2 arctan ⁡ L D 2 f c d {\displaystyle \mathrm {FOV} =2\arctan {\frac {LD}{2f_{c}d}}}

In photography, angle of view (AOV)[1] 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.

Field of viewdefinitionmicroscope

Field of view human eye

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.

The focal length number tells us how much of the scene is captured in the picture. The lower the number the wider the view, and the more we can see.  The higher the number, the narrower the view, and the less we can see.   This is illustrated below – where the camera is stationary and the focal length (in white numerals) changes:

For a lens projecting a rectilinear image (focused at infinity, see derivation), the angle of view (α) can be calculated from the chosen dimension (d), and effective focal length (f) as follows:[4] α = 2 arctan ⁡ d 2 f {\displaystyle \alpha =2\arctan {\frac {d}{2f}}}

The target's angular extent is: α = 2 arctan ⁡ L 2 f c {\displaystyle \alpha =2\arctan {\frac {L}{2f_{c}}}} where L {\displaystyle L} is the dimension of the target and f c {\displaystyle f_{c}} is the focal length of collimator.

A strict technical definition of focal length is difficult without providing a lot of background in lens theory, so we will use a simplification. You can think of focal length as the distance between the imaging plane (e.g. the image chip in a digital camera) and a point where all light rays intersect inside the lens (the ‘optical center’). So a focal length of 20mm means that the distance from the optical center to the imaging plane is 20mm long (about ¾ of an inch). What does the focal length number mean?

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 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.

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.