Determining Angle
of Coverage
Finding angle
of Coverage from focal length:
The focal length quoted for a camera lens is often an approximation,
to the nearest 5mm. 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 wide-angle 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' 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 x 24mm):
D = √(36² + 24² )
= 43.3mm.
A little trigonometry then gives you:
|
Angle of coverage, L = 2Arctan(D/2f) |
where "Arctan" means "the angle for which
the tangent is".
Arctan (the inverse tangent) may also be written as: Tan^-1. |
|
This formula derived above gives 46.8° for a 50mm lens, 63.4°
for a 35mm lens, and 94.5° for a 20mm lens (assuming the
35mm 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 air-water boundary
If the lens is situated behind a port, the change in coverage
due to refraction at the air-water boundary must be taken into
account. If you have a flat port, the modified angle of coverage
L' can be obtained by using Snell's
law, i.e.:
Sin(L'/2) = Sin(L/2)/n,
where n is the refractive index of water (~1.334 fresh, ~1.339
sea).
(L' should be pronounced "L prime", where a prime indicates a
modified version of an original quantity without a prime).
Hence:
|
L' = 2Arcsin[{Sin(L/2)}/n] |
(divide the 'in air' angle of coverage by 2, take the
sine, divide the result by the refractive index of water, take
the inverse-sine, multiply the result by 2).
E.g., the coverage of a 35mm lens is reduced from 63.4 to 46.5°,
giving it almost exactly the same perspective as a 50mm lens
in air. If you have a dome port, and 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 35mm format diagonal is 43.267mm. 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° |
Underwater Lens Comparison Chart (35mm format equivalent
focal length):
(SLR lens must be used with a dome port to achieve
the same coverage as in air)
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. Calculating
actual coverage thus requires information which may be difficult
to obtain, but there is a pragmatic solution (see below) which
stems from the fact that the formula given earlier can perfectly
well be used backwards. |
Measuring
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 marketing-orientated
diagonal figure. All you have to do is lay a ruler or a tape-measure
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, L = 2Arctan(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 lens-axis in the object and radial distance from the
lens axis in the image is non-linear (in principle if
not in practice). This non-linear relationship gives rise to
barrel distortion in wide-angle lenses (unless extra lens
elements are included to correct for it), and pincushion distortion
in underwater optical systems which have a flat air-water 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 which 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 which
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 which 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 wide-angle
position, will often show pronounced distortion and are inherently
difficult to correct.
Scale distortion, of course, will affect any measurements which
have to be made from a photograph. If, for example, you take
a photograph of a ruler which 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 this information unless
you designed the entire lens system yourself. The pragmatic solution
therefore is to digitise the image (if it's not from a digital
camera already), and apply a radial correction using software
(see the radial correction article).
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 right-angles 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 two-dimensional space.
Example:
Ikelite made a 3" radius dome port for the Olympus C-5060
camera and the Olympus WCON-07C wide-angle adapter lens (see C-5060,
C-7070 article). Peter Schulz posed the question; 'what coverage
do you get if you leave out the wide-converter and just use the
port with the bare camera lens?' Now, 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 pre-determined radial correction was applied to the picture.
The resulting (corrected) image is shown below (the original
is of higher resolution and the scales are readable): |
The ruler seen protruding from the camera was strapped to the
lens port casing with cable-ties, 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 ±1cm).
In this photograph, the object plane is 88.2cm wide after radial
correction, and the distance from the lens pupil is 74±1cm.
Since the aspect ratio of the C-5060 CCD sensor is 4:3. this
makes the height of the object 88.2 x 3 / 4 = 66.15cm, and the
diagonal, by Pythagoras' theorem, is √(88.2² + 66.15²)
= 110.25cm. 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 ±1.5cm, 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+1.5cm and 74-1.5cm, 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 74cm 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. |
D. W. Knight. Text & Illustrations ©
Cameras Underwater 2001 - 2006. |
