It is easily seen that the image is deformed relatively to the object. In particular the points on the line X, the intersection of the object plane with the anterior focal plane, will be imaged at infinity, the secondary axes NX, N'X' being parallel. In the same way infinitely distant points of the object plane in the direction NY will be imaged on the straight line Y', the intersection of the image plane with the posterior focal plane. On this straight line all the vanish ing points of parallel straight lines in the object plane will be imaged, while all lines meeting in X will be parallels in the image plane.
This manner of distorting is made use of for correcting the perspective of photographs acci dentally taken on an inclined plane (by making all the vanishing points of vertical lines in such a perspective meet in X, they will be corrected in the image), or for making lantern slides which are to be projected obliquely. We shall return to this subject in more detail in Chapter XLV.
65. Experimental Determination of the Focal Length of a Lens. The method of measuring the focal length 1 usually described consists in focussing a distant object (distant at least L000 times the focal length to be measured) in the camera, then focussing an equal size image of an easily measurable geometrical figure (e.g. a circle or equilateral triangle). The amount the camera has to be extended between the two positions is exactly the focal length. 2 Unfortunately, it is not always possible to have a distant view at hand, and, on the other hand, to get an image exactly full size often requires a large number of trials when the use of a special camera such as a process camera is not avail able. 3 Instead of measuring the focal length directly, as above (distance of a focus from the corre sponding symmetrical point), it is easy to cal culate it by focussing a test object on two different known scales, and measuring the dis placement of the focussing screen between the two positions.
As far as possible, the precautions recom mended in process work (Chapter XLV) should be taken to ensure parallelism between the object plane and the focussing screen (this being supposed perpendicular to the axis), and accu rate focussing should be done in the manner suggested in § 308. Let n and n' be the two scales, which may be chosen arbitrarily but very previously considered). Replacing p and p' by
their values in terms of n and F, we obtain, after simplification 1 / F 2 ± n iin If n is very small (which will always he the case in small cameras) Tin will be very large, and consequently the error arising from neglect ing the internodal distance will be divided by a number generally greater than Do and will therefore nearly always be negligible, except for telephotos and single lenses of convertible sets. If, for example, a lens of 6 in. focal length has an internodal distance of 0.12 in. if the scale of reduction is i/io, and the total separation between object and image is 71-5 in., the for mula gives a focal length of 5-9 in., which is sufficient approximation for all practical pur poses.
67. A variation of this method will avoid the error arising from neglecting the internodal distance, whatever its value, and at the same time determine it. Having measured the total distances 1 and 1' for two scales n and n', the focal length F and internodal distance i are given by 1-1' F — + n respectively.
If, for example, it has been found that for n =1/5 1 = 43-6 in.
n' 1/3 1' = 32.4 it follows that F= 866 6-o in.
and i = 43-6 - 6(2 + 5 + 1/5)= - = in.
'a 68. Direct Determination of the Position of the Nodal Points and the Focal Length. If the lens is rotated about an axis perpendicular to the optical axis, and containing the nodal point of emergence, the images of very distant points le remain fixed during the rotation, at least if it 3, accurately measured, and e the change in camera extension between the two positions.
We know that the extra-focal distances at scales n and n' are Fn, Fn' respectively ; their difference is the length e, which is known. Thus, e - ')F, whence F = In other words, the focal length is the differ ence in extension measured, divided by the difference between the two scales of reproduc tion.' To get sufficient precision, the two scales chosen should be as widely different as possible.
If, for example, the test object is an equilateral of which the sides are 4-8 in. long, reduced in the two positions to 3.6 and i-6 in. (scales of reduction 0.75 and 0-33 respectively) and that the increase in camera extension is 2-8 in., the focal length will be given by 2-8/0.42 = 6.67 in.