# Field of View

The field of view of a lens is of paramount importance to a photographer. Selection of a wide-angle, telephoto, or normal lens directly determines the boundary of the scene captured in a photograph. The field of view is the solid angle which the scene subtends at the camera lens. The same solid angle is subtended at the lens by the frame of the image. The field of view derives both from the focal length of the lens and the size of the image frame. Although the field of view varies slightly depending on the distance of the subject, for practical purposes we need only consider distant subjects where the image is in the focal plane of the lens.

For a rectangular image centred on the optical axis, three angular measurements of the field of view are of interest: horizontal, vertical, and diagonal.   The field of view can be specified precisely as HFOV × VFOV. For a 50 mm lens on a 35 mm camera with frame size 36 × 24 mm, the field of view is 40° × 27°. This takes into account the aspect ratio of the frame as well as the focal length of the lens. Two significant figures show the angles in degrees with sufficient accuracy. The field of view of a 300 mm lens on a 35 mm camera would be specified as 6.9° × 4.6°.

For digital cameras, it is common to give the "35 mm equivalent" focal length as an indicator of the field of view, although the aspect ratio of most digital cameras is different from 35 mm cameras. Whatever the aspect ratio, the frame diagonal is the diameter of the circular image containing the frame. Optical equivalence is therefore best measured by the DFOV. Cameras have the same DFOV if they have the same ratio of focal length to frame diagonal. The diagonal of a 36 × 24 mm frame is 43.27 mm. For a digital camera with focal length f and frame diagonal D, the 35 mm equivalent focal length is therefore f × 43.27 / D. The ratio 43.27 / D of frame diagonals can be considered a focal length multiplier.

Digital camera specifications do not usually give the diagonal measurement of the image. They indicate the size of the image sensor using an outdated method in which the diameter of a Vidicon tube was expressed as a fraction of an inch. The diagonal measurement of the image cannot be derived from this diameter, but can be obtained from a table such as the one provided by Digital Photography Review. For example, the specification for the Canon PowerShot A20 digital camera gives the size of the image sensor as 1/2.7 inches, so from the table the diagonal is 6.592 mm and the equivalent 35 mm focal length is f × 43.27 / 6.592. The specification gives f = 5.4 - 16.2 mm, equivalent to 35 - 105 mm, consistent with the formula.

For zoom lenses, a zoom factor describes the ratio of two focal lengths. There is a 3× zoom factor between focal length 35 and 105 mm, for example. The zoom factor also indicates the linear magnification of the image: a 3× zoom makes an object 3 times wider and higher (hence 9 times bigger in area).

It is useful to have a single figure to enable comparison of zoom and field of view between different lens and camera combinations. The 35 mm equivalent focal length has been used for this purpose. For a rectilinear lens, however, this is not a satisfactory metric for short focal lengths. Although halving the focal length halves the magnification of the image, it has progressively less effect on the field of view as it approaches 180° × 180°. At long focal lengths, halving the focal length increases the field of view, measured as a solid angle, by about 4 times. The widest available rectilinear lens for 35 mm cameras, the Voigtländer Ultra Wide Heliar 12 mm, has a field of view of 113° × 90°, a solid angle of 2.52 steradians. A hypothetical 6 mm lens would have a field of view of 143° × 127°, a solid angle of 4.05 steradians, only 1.6 times bigger.

I propose a new metric, the zoom index, which indicates successive halving of the solid angle of the field of view. The starting point Z0 is 4π steradians, representing a 360° spherical panorama. Z1 (2π steradians) represents the hemispherical field of view approached by very short focal lengths. Z2 (π steradians) is a 9 mm lens on a 35 mm camera, Z3 an 18 mm lens. (The series Z0, Z1, Z2,... is analogous to the series of standard paper sizes A0, A1, A2,... where a sheet of paper is successively halved in area.)

The solid angle ψ can be calculated from the field of view by ψ = 4arcsin(sin(HFOV/2) × sin(VFOV/2)).

The zoom index Z is given by Z = log2(4π / ψ). Two significant figures are sufficient in practice.

For long focal lengths, HFOV and VFOV are small, so:-

HFOV ≈ 2w / f
VFOV ≈ 2h / f
ψ ≈ 4w × h / f 2
Z ≈ 2log2f + log2(π / (w × h))
An s× zoom factor then increases the zoom index by 2log2s, which is close to the zoom factor increase in the following cases:-
 2× ≡ +Z2 3× ≡ +Z3.2 4× ≡ +Z4 5× ≡ +Z4.6

The calculations for field of view and zoom index can conveniently be performed by this Javascript calculator. It also displays the field of view by superimposing its outline on a wide-angle photograph. The outline can be dragged, as an alternative to keying in the focal length of the lens. The calculator has been tested with Internet Explorer 6 and Firefox. It requires that Javascript is enabled in the browser.

© Chris Jones, 2004