The angle of
coverage (or field of view) of a lens
By David W Knight.
Finding
angle of coverage from focal length
The focal length quoted for a camera lens is often an approximation.
This is the source of the discrepancy in angle of coverage figures
quoted by different manufacturers for a given focal length and image
format, particularly in the case of wideangle lenses, where small
changes in focal length cause large variations in angle of coverage.
The standard or 'nominal' figure quoted is the angle of coverage on the
diagonal of the image format, with the lens focused at infinity. This
means that the lens pupil (of an equivalent symmetric lens) is at a
distance f from the film, where f is the focal length. The diagonal of
the film format, d, can be calculated using Pythagoras's theorem (the
square on the hypotenuse of a right angled triangle is equal to the sum
of the squares on the other two sides).
For the 35mm film stills format (36 × 24mm):
d = √(36² + 24² ) = 43.3mm.
From the diagram below, note that an angle equal to half the angle of
coverage has its tangent equal to half the format diagonal divided by
the focal length; i.e.:
Tan(α/2) = (d/2) / f
Hence:
Angle of coverage (field of view, FOV)
where "Arctan" means "the angle for which the tangent is". Arctan (the
inverse tangent) is also sometimes written as: Tan
^{1}.
The formula derived above gives 46.8° for a 50 mm lens,
63.4° for a 35 mm lens, and 94.5° for a 20 mm lens
(assuming the 35 mm film format). Note that these figures apply only to
bare lenses. Any supplementary lens, flat underwater port, or dome port
operated with the lens entrance pupil not exactly at the centre of
curvature of the dome, will alter the coverage.
Effect of a
flat airwater boundary
If the lens is situated behind a port, the change in coverage due to
refraction at the airwater boundary must be taken into account. For a
flat port, the modified angle of coverage α' can be obtained
using
Snell's law as
follows:
Referring to the diagram above, if the halfangle of coverage in air
is α/2, then the new halfangle of coverage in the
port material (β/2) is given by the expression:
Sin(β/2) / Sin(α/2) = n
_{a} / n
_{p}
Where n
_{a} and n
_{p} are
respectively the refractive indeces of the air and the port material.
The expression can be rearranged:
n
_{p} Sin(β/2) = n
_{a}
Sin(α/2) . . . . . . . . (
1)
Similarly, the halfangle of coverage in the water
(α'/2) is given by:
Sin(α'/2) / Sin(β/2) = n
_{p} / n
_{w}
or
n
_{w} Sin(α'/2) = n
_{p}
Sin(β/2)
(α' should be pronounced "alpha prime", where a prime
indicates a modified version of an original quantity without a prime).
Substitution using (
1)
gives:
n
_{w} Sin(α'/2) = n
_{a}
Sin(α/2)
i.e., the effect of the intervening port port material is cancelled,
and the port itself has no effect on the angle of coverage (although it
does alter the effective position of the lens entrance pupil as seen
from outside the housing).
If we assume that the refractive index of air (n
_{a})
is 1 we get:
Sin(α'/2) = Sin(α/2) / n
_{w}
where n
_{w} is about 1.334 for fresh water
and 1.339 for sea water
Rearranging gives the working formula:
α' = 2 Arcsin[ Sin(α/2) / n_{w}
] 
(divide the 'in air' angle of coverage by 2, take the sine, divide the
result by the refractive index of water, take the inversesine,
multiply the result by 2).
Using the formula above; the coverage of a 35 mm lens (for example) is
reduced from 63.4 to 46.5°, giving it almost exactly the same
perspective as a 50 mm lens in air.
Note that for a dome port, when the entrance pupil of the lens is
located exactly at the centre of curvature of the dome, the angle of
coverage is the same as in air.
Diagonal
coverage of lenses on the 35mm format
This table applies to lenses designed for use in air. The 35 mm format
diagonal is 43.267 mm. The refractive index of (sea) water was taken to
be 1.339 for the underwater coverage figures.
Focal length / mm

Coverage in Air

Coverage UW (flat port)

19

97.42°

68.27°

20

94.49°

66.51°

24

84.06°

60.00°

27

77.41°

55.68°

28

75.38°

54.34°

30

71.59°

51.80°

32

68.12°

49.45°

35

63.44°

46.24°

38

59.31°

43.37°

40

56.81°

41.62°

50

46.79°

34.50°

55

42.94°

31.73°

60

39.65°

29.35°

70

34.35°

25.48°

85

28.56°

21.23°

105

23.28°

17.34°

Angle of
Coverage when not focused at Infinity
If you need to know the exact coverage of a camera system in a
particular application; you should be mindful of the fact the angle of
coverage is reduced when the lens is focused closer than infinity, and
is reduced substantially when making macro photographs. The reason for
the change is that the iris of the lens has moved away from the film
plane to a new distance f+x, where x is the lens extension required to
bring the image into focus. For a 1:1 macro photograph, the distance
f+x is equal to 2f, and so it should be obvious that the capture angle
will be much reduced from its infinity value. The modified coverage
formula becomes:
Angle of coverage for lens extension x,
α
_{(f+x)} = 2 Arctan[ d / 2(f+x) ]
Calculating actual coverage thus requires information which may be
difficult to determine (measuring the lens extension is tricky), but
there is a pragmatic solution (see below) which stems from the fact
that the formula given earlier can perfectly well be used backwards.
Estimating
angle of coverage
Assuming that you are interested in determining exact coverage for some
scientific purpose, you will probably be more interested in the
horizontal or vertical angle than in the more marketingorientated
diagonal figure. All you have to do is lay a ruler or a tapemeasure
across the subject area, take a photograph of it, and record the
distance from the subject plane to the lens entrance pupil (the point
where the iris appears to be when you look into the front of the lens).
If d is the width, height, or diagonal of the subject plane (whichever
you want to use), and a is the distance from the subject plane to the
entrance pupil, the new formula is:
Angle of coverage for lenstoimageplane distance a,
α
_{a} = 2 Arctan(d / 2a)
Effect of
Radial Transfer Function on Coverage Determination
Barrel (fisheye) distortion 
Pincushion distortion 
In all lens systems, the relationship between radial distance from the
lensaxis in the object and radial distance from the lens axis in the
image is
nonlinear (in principle if not in
practice). This nonlinear relationship gives rise to barrel distortion
in wideangle lenses (unless extra lens elements are included to
correct for it), and pincushion distortion in underwater optical
systems that have a flat airwater boundary (port) in front of the
lens. Both phenomena can be understood by considering what happens when
a photograph is taken of a graduated ruler that passes through the
centre of the picture: If a lens suffers from barrel distortion, the
ruler graduations in the image will get closer together the further
they are away from the lens axis. If the lens suffers from pincushion
distortion, the ruler graduations will get further apart. Now; consider
what happens if you take a photograph of a rectangular object that
nearly fills the frame. The top and bottom, and left and right, edges
of the rectangle will be closer to the middle of the picture (the lens
axis) than the corners. Therefore, if the lens exhibits barrel
distortion, the corners of the rectangle will appear to have been
dragged in, while if the lens exhibits pincushion distortion, the
corners will appear to have been pushed out.
The relationship
between what you put into a system and what you get out is called the
transfer
function. Hence the relationship between distance from the
lens axis in the object and distance from the lens axis in the image is
called the
radial transfer function. This
transfer function for simple lenses follows an approximately cubic law;
which means that for light rays that have only a small angular
displacement from the lens axis, the associated scale distortion will
be barely perceptible, but as the angular displacement increases, the
gradient of the transfer function will become progressively steeper and
the scale distortion will soon become severe. It is for this reason
that normal and telephoto lenses show little distortion (and are in any
case relatively easy for the designer to correct), whereas wide angle
lenses, and especially zoom lenses set in the wideangle position, will
often show pronounced distortion and are inherently difficult to
correct.
Scale distortion, of
course, will affect any measurements that have to be made from a
photograph. If, for example, you take a photograph of a ruler that is
horizontal in the picture, and try to use this measurement and the
picture aspect ratio to find the diagonal distance, you will get the
wrong answer! You can only calculate the diagonal distance if you know
the radial transfer function, and you're unlikely to have that
information unless you designed the entire lens system yourself. The
pragmatic solution therefore is to apply a radial correction. The
radial correction process is simply a matter of multiplying the direct
distances of each of the pixels from the lens axis by a factor
calculated from an inverse transfer function (which is determined by
trial and error); and the only guidance needed in obtaining this
function is that it should cause the image to reproduce straight lines
and rightangles correctly (rectilinear correction). Once the
correction has been performed, scale distortion is removed, rectangles
are rectangles once more, and the image becomes a nice friendly linear
twodimensional space.

Example
Ikelite made a 3" radius dome port for the Olympus C5060 camera and
the
Olympus WCON07C 0.7× wideangle converter lens. Peter Schulz
posed the
question; 'what coverage do you get if you leave out the wideconverter
and just use the port with the bare camera lens?' 
If the entrance pupil of the lens is at
the exact centre of curvature of the dome, the coverage should be the
same as in air, i.e., 77° for this camera at the maximum wide
zoom setting. In this configuration however, the entrance pupil is
slightly behind the dome centre, and so it was known that the coverage
would not be exactly conserved. Given all the vagaries of optical
calculation and of locating the entrance pupil exactly; the easiest
(and most convincing) way to find the answer was therefore to
photograph a measuring scale, and measure the distance from the lens
pupil to the object plane. For practical reasons also, the scale was
placed horizontally in the picture. The procedure was therefore to take
several pictures of a rectangular test card underwater and use them to
determine a radial correction function. A scale was then photographed,
and the predetermined radial correction was applied to the picture.
The resulting (corrected) image is shown below (the original was of
higher resolution and the scales were readable):
The ruler seen protruding from the camera was strapped to the lens port
casing with cableties, and the scale zero was adjusted to be in the
same plane as the entrance pupil with the equipment submerged, as far a
could be determined by eye (about ±1 cm). In this
photograph, the object plane is 88.2 cm wide after radial correction,
and the distance from the lens pupil is 74±1 cm. Since the
aspect ratio of the C5060 CCD sensor is 4:3. this makes the height of
the object 88.2 x 3 / 4 = 66.15 cm, and the diagonal, by Pythagoras'
theorem, is √(88.2² + 66.15²) = 110.25 cm.
The angle of coverage, using the formula given earlier is thus
73.4°, but before accepting this figure we should consider the
errors inherent in the determination. By far the greatest source of
error in this case lies in the determination of the entrance pupil
position. There will be some error in the diagonal measurement also,
due to possible imperfections in the radial correction function, but
since the diagonal measurement will also vary as the lens is moved
backwards and forwards, this can be lumped with the error in the
entrance pupil distance. Hence if we increase the uncertainty in the
entrance pupil position to about ±1.5 cm, this should
provide a reasonable confidence interval (standard deviation) for the
measurement. The edges of this confidence interval can be found by
calculating the angles of coverage for distances of 74+1.5 cm and
741.5 cm, and the two values so obtained are 72.3° and
74.5°. So the determined angle of coverage is 73.4
±1.1°.
Note that the lens
system is focused at 74 cm for this determination (the camera lens is
actually focused on a nearby virtual image created by the dome port).
Due to a change in the lens focusing extension, the angle of coverage
will be very slightly wider when the system is focused at infinity.
The Olympus C5060 + WCON07c
combination is used as an example in the
radial
correction
article. In air, this system has a
coverage of 97°, but when used underwater with a flat port, the
coverage is only 68°. Using the camera underwater
with a dome port and no wideconverter turns out to give a greater FOV
(73°), and the flat port also produces severe chromatic aberration.
If the wide converter is used with the dome port, the FOV is, of
course, approximately the same as in air.
DWK
© D W Knight. 2001, 2006, 2012. Updated Feb. 2018.
David Knight asserts the right to be recognised as the author of this
work.