The focal length of the lens used in a camera affects two aspects of the image: the field of view and the magnification.
The
field of view is the size of the object that can be contained on the film and
the magnification is the ratio of the size of the image to the size of the
object that produced it. These two quantities are related to each other –
increasing the magnification while leaving the size of the image unchanged
inevitably decreases the field of view (since the increase in the magnification
implies that the same sized image will be produced by a smaller object), and
vice-versa.
The
field of view is usually expressed as an angle. There are a number of different
ways of estimating this quantity, but all of them yield about the same value.
Since we are going to use the approximation that the camera lens is a simple
single thin lens, the calculation will not be exactly correct anyway. As with
previous calculations, the agreement between this simple model and a real-lens
is surprisingly good.
If
the object is not too close to the lens, then the image is always formed
approximately at the focal distance back from the lens. The size of the image
on the image plane is usually limited by the size of the film, and the angular
size of the field of view is given by those rays that enter the lens at its
center (and so are not deviated) and which then strike the edges of the film as
shown in the following figure:
The
angles on the object and image side of the lens are the same, so that the
tangent of one-half of the angle of the field of view is approximately the
ratio of the half-width of the film to the focal length of the lens. For
example, the half-width of 35 mm film is 18 mm. The horizontal field of view of
a 50 mm lens is then twice the arc-tangent of 18/50 or about 40 degrees. The
film is not square, so that the field of view in the vertical direction is not
the same. The half-height of the film is only 12 mm, so that the vertical field
of view is twice the arc tangent of 12/50 or about 27 degrees.
(Another
method for calculating the field of view would be to consider those rays that
pass through the first focal point on the object side of the lens. These rays
are refracted so that they emerge parallel to the axis on the image side of the
lens. The angular spread of these rays provides an alternate method for
estimating the field of view. This estimate is essentially the same as the
estimate using the method described above unless the object is quite close to
the lens. The two estimates differ in this situation because the lens must be
moved away from the film plane in this situation so that the distance from the
lens to the film in the first figure is somewhat greater than the focal length.)
See the following figure, which compares the field of view calculated in this
way with the calculation above when the object is far away.
The
field of view angle decreases with increasing focal length, but the relationship
is through the arc tangent function and is not exactly linear (except at very
small angles). The horizontal field of view of a telephoto lens whose focal
length was 135 mm would be twice the arc tangent of 18/135 or about 15 degrees.
The
magnification of a lens can be estimated using the same sort of analysis. The
following figure shows the image produced by an object.
The
two shaded triangles are similar to each other, so that the corresponding sides
are proportional. The magnification is the ratio of the size of the image to
the size of the object that produced it. Since the triangles are similar, that
ratio is also equal to the ratio of the distance between the lens and the image
(“I” in the figure) and the distance between the lens and the object (“O” in
the figure). Since the distance “I” is
approximately the focal distance, the magnification of a lens for a fixed
object depends directly on the focal length and is equal to the ratio of the
focal length to the distance of the object from the lens.
The
magnification increases directly with the focal length when the object, the
position of the camera, etc. are unchanged. Thus the same object appears larger
when imaged by a camera with a telephoto lens because the focal length of that
lens is larger than the focal length of a normal lens.
Although
a film image is purely two-dimensional, the variation in the magnification of a
lens with object distance can distort perspective – that is, how we estimate
the size of something in an image by comparing it to other things in the same
image. For example, if an object is sufficiently large so that the distances
between different parts of the object and the lens are significantly different,
then the parts that are closer to the lens will have a greater magnification
than the parts that are further away. This effect can be seen when photographing
a tall building from a point near its base. The bottom of the building is
significantly closer to the camera than the top, is magnified by a larger amount
and appears disproportionately too large as a result.
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