A particularly interesting case is that in which the object S has an ultra-nodal distance twice the focal length. In this case FS == MF, and of all points on a plane perpendicular to the optical axis lie on another plane also perpen dicular to the optical axis. Knowing also that the exterior parts of a secondary axis are parallel straight lines (lines joining object and image points to the corresponding nodal points), we can determine the scale of the image (relation between corresponding dimensions of object and the straight line MS is inclined at 45°, and S' is such that NS' NS. The two points S and S' at equal distances from their respective nodal points are called the symmetrical points of the lens ; their separation is the shortest distance that can exist between a point object and its real image. Another particular case, but of no practical interest, is that where the points R and R' coincide with A T this leads to the fact image) of an object of which the position is known relatively to a lens of known focal length.
Let us suppose that the length of the image of an arrow RQ (Fig. 46) of length 1 perpen dicular to the axis, at an ultra-nodal distance p, is to be determined. Join by a straight line the point Q to the nodal point of incidence N, and draw through the nodal point of emergence N' a line parallel to QN to meet in Q' the straight line R'Q' through R' (the image of R), parallel to RQ. The point Q' is the image of Q, and the element of line R'Q' is the image of RQ. The length l' of this image can be ascertained from 1, since the triangles RQN and R'Q'N' are similar.
The ratio of similitude or scale (n = i/N) of reproduction is then equal to the ratio of the ultra-nodal distances of image and object n = i/N = = From this relation can be deduced p = (1 N)F (1 ± iin)F = (I n)F =(I + I1N)F If we replace the ultra-nodal distances by the ultra-focal distances, we obtain the simpler and more easily expressed relations d NF FIn 61' n? To obtain an image of a plane object placed perpendicular to the axis at a reduction of IlN, it should be placed at an ultra-focal distance equal to N times the focal length ; the focussed farther from the lens than its image," and magnification whenever the object is closer to the lens than to its image.
When the object photographed has a certain depth, it is no longer possible to speak of the scale of the image, since this will vary from point to point. It should be mentioned here that in the most general case, where such an image is photographed on a plane perpendicular to the axis, the relative dimensions of the different parts of the image are inversely proportional to the ultra-nodal distances of the corresponding point objects, and not to their ultra-focal dis tances, the point images being all on the same plane and no longer the conjugates of the point objects.
image is formed at an ultra-focal distance IiNth the focal length.
For enlargement n times, the object should be placed at an ultra-focal distance equal to I/nth the focal length, and the image will be at an ultra-focal distance equal to n times the focal length.
For example, if it is desired to reduce a square of 12 in. sides to an image of 4 in. sides, with a lens of 8 in. focal length, N = 3. The original must be placed at 3X 8 = 24 in. from the anterior focus, and the image will be formed on a plane at 8/3 = 21 in. from the posterior focus. Reversing the two positions, the image would be enlarged three times (n = 3).
For a reproduction same size (N n 1) the planes of copy and image cut the axis at the symmetrical points' S and S'.
There is reduction whenever the object is 63. Graphical Construction of the Image Formed by an Optical System. Knowing the position of a point object Q relatively to the foci F and F', and the nodal points N and N' of an optical system, the position of the image Q' can be determined as follows : Draw the optical axis FF' (Fig. 47), and at N and N' draw per pendiculars to it to indicate the nodal planes. From Q draw a straight line parallel to the axis, meeting the nodal plane of emergence in the point a' ; a' is the image of a, the intersection of the ray with the nodal plane of incidence. The emergent ray will then pass through the focus F', since all incident rays parallel to the axis, after refraction, meet the optical axis at the posterior focus. Draw another line QF and produce it to meet the nodal plane of incidence in b; the image of b is b' on a line through b parallel to the axis, which is also the emergent ray. Q', the intersection of a'F' and bb', is the image required. The accuracy of the construc tion can be tested by seeing whether QN, of the secondary axis are parallel to one another.
64. Image of a Plane Inclined to the Axis. Consider a lens of which the foci and nodal points are and NN' respectively (Fig. 48), and let R and R' be two conjugate points. If a plane perpendicular to the plane of the paper meets the optical axis obliquely at R, all the points in this plane (at least all those not far from the axis) form their images on another plane, also meeting the optical axis obliquely at and perpendicular to the plane of the paper. This image plane is defined by the condition that its intersection M' with the nodal plane of emergence should be contained in the plane MM' parallel to the axis through ill, the intersection of the object plane with the nodal plane of emergence.