Angle of view (photography)

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, the angles of view are:

• horizontally, ${\displaystyle \alpha _{h}=2\arctan {\frac {h}{2f}}=2\arctan {\frac {36}{2\times 50}}\approx 39.6^{\circ }}$
• vertically, ${\displaystyle \alpha _{v}=2\arctan {\frac {v}{2f}}=2\arctan {\frac {24}{2\times 50}}\approx 27.0^{\circ }}$
• diagonally, ${\displaystyle \alpha _{d}=2\arctan {\frac {d}{2f}}=2\arctan {\frac {43.3}{2\times 50}}\approx 46.8^{\circ }}$

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

Now ${\displaystyle \alpha /2}$ is the angle between the optical axis of the lens and the ray joining its optical center to the edge of the film. Here ${\displaystyle \alpha }$ 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:

• the angle ${\displaystyle \alpha }$
• the "opposite" side of the right triangle, ${\displaystyle d/2}$ (half the film-format dimension)
• the "adjacent" side, ${\displaystyle S_{2}}$ (distance from the lens to the image plane)

Using basic trigonometry, we find:

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

which we can solve for α, giving:

$${\displaystyle \alpha =2\arctan {\frac {d}{2S_{2}}}}$$

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

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

where ${\displaystyle f=F}$.

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

For macro photography, we cannot neglect the difference between ${\displaystyle S_{2}}$ and ${\displaystyle F}$. From the thin lens formula,

$${\displaystyle {\frac {1}{F}}={\frac {1}{S_{1}}}+{\frac {1}{S_{2}}}.}$$

From the definition of magnification, ${\displaystyle m=S_{2}/S_{1}}$, we can substitute ${\displaystyle S_{1}}$ and with some algebra find:

$${\displaystyle S_{2}=F\cdot (1+m)}$$

Defining ${\displaystyle f=S_{2}}$ as the "effective focal length", we get the formula presented above:

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

where ${\displaystyle f=F\cdot (1+m)}$.

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]

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

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 24 mm of focal length in 35 mm film format, here 14 mm gives 114° and 24 mm gives 84° .
• Wide-angle lenses (24–35 mm in 35 mm film format) cover between 84° and 64°
• Normal, or Standard lenses (36–60 mm in 35 mm 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)[11] generally have an angle of view of 35° or less.[12] Since photographers usually only encounter the telephoto lens sub-type,[13] they are referred to in common photographic parlance as:
• "Medium telephoto", a focal length of 85 mm to 250 mm in 35 mm film format covering between 30° and 10°[14]
• "Super telephoto" (over 300 mm in 35 mm film format) generally cover between 8° through less than 1°[14]

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.

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.[15]

An example of how lens choice affects angle of view.

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°