When it is stated that a lens of short focus gives " faulty perspective," which should be translated as " geometrically correct but un pleasant perspective," it is understood, then, that the photographer has clumsily tried to compensate for the smallness of the scale of his image by approaching too close to his subject.
The position of the camera should be chosen without any consideration of scale. If, when the viewpoint is chosen, it is found that the lens is not of sufficient focal length to give directly as large an image as desired, the small image should be subsequently enlarged.
76. Depth of Field. Knowing that the image of a point outside the plane focussed on is a circular patch on the image plane, the limits within which the objects should lie in order that these patches (circles of confusion) should be practically indistinguishable from points, can be determined. 1 First of all, a standard of latitude must be agreed upon ; usually a maximum value is assigned to the diameter of the circle of con fusion, e.g. 1/250th in. (it is then said that a sharpness of 1/250th in. is required), a blur which, viewed at 12 in., is indistinguishable from a point by a person having very good sight (§ 34). This convention is purely arbitrary, and is too severe for pictures which arc to be viewed at a greater distance, as when placed on a wall, and is not sufficiently severe for a small image which has to be subsequently enlarged by projection or examination under a magnifier (the case of stereoscopic pictures). The image being normally examined from its viewpoint (§ 25), it is logical, at any rate in pictorial photo graphy, to fix the diameter of the circle of confusion which can be tolerated as I/2,000th of the ultra-nodal distance of the image (J. Thovert, 1904).
77. Relative Depth of Field. Let q and q' (Fig.58) be the ultra-nodal distances of the object plane Q and its conjugate Q' on which the photo graphic image is recorded, respectively, Flirt the relative aperture of the lens, r and s the ultra nodal distances of point-objects respectively in front of and behind the plane Q, r' and s' the ultra-nodal distances of their focussed images.
The dimensions of the circles of confusion in the plane of the image (r) and (s) are ex pressed by (r) r' - q'_(s) q' - s' F FJn In order that the blurs (r) and (s) should have the maximum permissible diameter aq' (the coefficient a being, for example, 1/2,000), the distances r' and s' must be such that nag' - q' q' - s' F s' which may be written - 1/7' = xis' - na iF and as (§ 6o) I/F - i/q, - 1/7 'is' = I/F - its it follows that iir - i/q = i/q 'is = nalF The difference between the extreme conver gences (§60) of R and S and the convergence of the plane Q focussed on is then represented by ni2,000ths of the power of the lens, all measure ments being expressed in diopters ; and the total depth of field (distances between R and S measured parallel to the optical axis) corre sponds with a difference in convergence equal to n/i,000ths of the power.
For a lens of 4-4 in. focal length, i.e. o-II m., or a power of I/0-ii = 9-09 diopters, with an aperture of F/6, the total tolerance of conver gence will be (9.09 x 6)/L000, or 0-05454 diop
ters, which has to be divided between the near and far points. If the object focussed on was at 197 in. (5 m.) with a convergence of 1/5 ----- 0•2 diopters, the convergences of the two limits of depth of field will be 0*20 ± o-o2727, correspond ing with object distances of 1/0-22727 and 1/0-17273, or 4-40 and 5-80 m. (173 and 229 in.) respectively.
A table of reciprocals of numbers from x to 1,000 will be useful for making calculations rapidly and of sufficient accuracy in problems relating to depth of field.
It may be said that the depth of field is that portion of space from which the lens aperture is viewed under an angle approximately con stant within the limiting angle of confusion (A. Jonon, 1925).
It should be noted that the sharp field extends less in front of the plane focussed on than behind it (24 and 36 in. in the example above).
Calculations of a similar degree of simplicity enable us to work out the aperture at which the lens must be used in order to give a sharp image of objects at different distances from the lens, and on what plane the lens ought to be focussed. The convergences of the extreme points at distances of 8o and 320 in. (2 and 8 m.) respec tively, are = o-5 and = 0-125 diopters. The difference is thus 0-375 diopters. In order that the tolerance in convergence may be equal to this, which represents 375/9-09 thousandths of the power of the lens (say 41/1,000) the lens must be stopped down to 1741. In practice, the nearest marked aperture of the iris, 1745, is used, and this will give ample guarantee of the sharpness and depth required. The distance to focus on will be given by the mean of the extreme convergences (0-5o0 0-125)/2 = 0-312 corresponding with a distance of 1/0-312 m. = 3-2 m. (126 in.).
78. Absolute Depth of Field. To conform to tradition we shall deduce the formulae for depth of field in terms of an absolute diameter' of the circle of confusion e (e.g. e = 1/250th in.), and not, as above, a constant fraction of the ultra nodal distance of the plane of the photographic plate.
Keeping the same notation as in the preceding section, and calling (R) and (S) the image patches projected on the plane Q by beams having their apices at R and S, and bounded by the diaphragm, these are given (§ 58) by the relations Rq-r S s - q r s but if the image Q` of the plane Q is reduced on a scale On, which implies that q (m the diameters (r) and (s) of the images are equal to (R)/m and (S)/ni respectively. If these are to be equal to the maximum diameter e, then (1?) and (S) are equal to me, and r and s (the ultra-nodal distances of the near and far planes which will be rendered sharply) will be calcu lated from em _q-r s-q F/n r q/r = I -I- me/F, q/s = i - nme/F whence, after simplification (m (m sFf = tone — F rtme Using the same numerical values as in the previous example, all distances being reduced to metres and taking 1/250th in. (0-01 cm.) as the maximum diameter of the. circle of eon fusion, we shall find, for the case of an object at 197 in. (5 m.), 500 + 1= = 45'45 m = 4445