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By David W Knight (Cameras Underwater)

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 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'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)

α = 2Arctan(d / 2f) |

The 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.

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. For a flat port, the modified angle of coverage α' can be obtained using

Referring to the diagram above, if the half-angle of coverage in air is α/2, then the new half-angle of coverage in the port material (β/2) is given by the expression:

Sin(β/2) / Sin(α/2) = n

Where n

The expression can be rearranged:

n

Similarly, the half-angle of coverage in the water (α'/2) is given by:

Sin(α'/2) / Sin(β/2) = n

or

n

(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

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

Sin(α'/2) = Sin(α/2) / n

where n

Rearranging gives the working formula:

α' = 2 Arcsin[ Sin(α/2) / n_{w} ] |

Using the formula above; the coverage of a 35mm lens (for example) is reduced from 63.4 to 46.5°, giving it almost exactly the same perspective as a 50mm 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.

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.

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

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.

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 for lens-to-image-plane distance a,

α

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

The relationship between what you put into a system and what you get out is called the

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 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 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.

ExampleIkelite made a 3" radius dome port for the Olympus C-5060 camera and the Olympus WCON-07C wide-angle adapter lens. 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?' |

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.

David Knight

© Cameras Underwater Ltd. 2001, 2006, 2012

David W Knight asserts the right to be recognised as the author of this work.