BD046177

CHAPTER V.
ON THE THEORY OF STEREOSCOPIC VISION.
HAVING, in the preceding chapter, described the ocular, the reflecting, and the lenticular stereoscopes, and explained the manner in which the two binocular pictures are combined or laid upon one another in the last of these instruments, we shall now proceed to consider the theory of stereoscopic vision.
In order to understand how the two pictures, vhen placed the one above the other, rise into relief; we must first explain the manner in which a solid object itself is, in ordinary vision, seen in relief; and we shall then shew how this process takes place in the two forms of the ocular stereoscope, and in the lenticular stereoscope. For this purpose, let A B CD, Fig. 1 9, be a section of the frustum of a cone, that is, a cone with its top cut off by a plane c eng, and having A E B G for its base. In order that the figure may not be complicated, it will be sufficient to consider how we see, with two eyes, L and R, the cone as projected upon a plane passing through its summit ce D g. The points L, R being the points of sight, draw the lines R A, RB, which will cut the plane on which the projection is to be made in the points a, b, 80 that a b will represent the line A B, and a circle, whose diameter is a b, will represent the base of the cone, as seen by the right surface, as seen by the left eye L, consists of two circles, whose thameters are A B, CD; and, as seen by the right eye, of other two circles, whose diameters are a b, CD, which, in Fig. 20, are represented by A B, CD, and a b, cci. These
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plane figures being also the representation of the solid on the retina of the two eyes, how comes it that we see the solid and not the plane pictures ? When we look at the point B, Fig. 19, with both eyes, we converge upon it the optic axes L B, RB, and we therefore see the point single, and at the distances L B, RB from each eye. When we look at the point D, we withdraw the optic axes from B, and converge them upon D. We therefore see the point D single, and at the distances L D, RD from each eye; and in like manner the eyes run over the whole solid, seeing every point single and distinct upon which they converge their axes, and at the distance of the point of convergence from the observer. During this rapid survey of the object, tbe whole of it is seen distinctly as a solid, although every point of it is seen double and indistinct, excepting the point upon which the axes are for the instant converged.
From these observations it is obvious, that when we look
with both eyes at any solid or body in relief, we see more of
the right side of it by the right eye, and more of the left side of it by the left eye. The right side of the frustum A B CD,
Fig. 19, is represented by the line nb, as seen by the right
eye, and by the shorter line PB’, as seen by the left eye.
In like manner, the left side AC is represented by CA’, as
seen by the left eye, and by the shorter line Ca’, as seen by
the right eye.
When the body is hollow, like a wine glass, we see more
of the right siide with the left eye, and more of the left side
with the right eye.
If we now separate, as in Fig. 20, the two projections
shewn together on Fig. 19, we shall see that the two
summits, CD, cd, of the frustum are farther from one
another than the more distant bases, AB, ab, and it is true
generally that in the two pictures of any solid ‘in relief, the
similar parts that are near the observer are more distant in
the two pictures than the remoter parts, when the plane of
perspective is beyond the object. In the binocular picture
of the human face the distance between the two noses is
greater than the distance between the two right or left
eyes, and the distance between the two right or left eyes
greater than the distance between the two remoter ears.
We are now in a condition to explain the process by
which, with the eyes alone, we can see a solid in relief by
uniting the right and left eye pictures of it,—or the theory
ocular stereoscope. In order to obtain the proper relief
we must place the right eye picture on the left side, and
the left eye picture on the right side, as shewn in Fig. 21, by the pictures A B cr, abc ci, of the frustum of a cone, as
obtained from Fig. 19.
In order to unite these two dissimilar projections, we
must converge the optical axes to a point nearer the obrelief at MN, the place where the optic axes meet or cross each other. At first the solid figure will appear in the middie, between the two pictures from which it is formed and of the same size, but after some practice it will appear smaller and nearer the eye. Its smallness is an optical illusion, as it has the same angle of apparent magnitude as the plane figures, namely, rn fl L A B L but its position
at MN is a reality, for if we look at the point of our finger held beyond M the solid figure will be seen nearer the eye. The difficulty which we experience in seeing it of the size and in the position shewn in Fig. 21, arises from its being
seen along with its two plane representations, as we shall prove experimentally when we treat in a future chapter of
the union of similar figures by the eye.
The two images being thus superimposed, or united, we shall now see that the combined images are seen in relief in the very same way that in ordinary vision we saw the real solid, ABCD, Fig. 19, in relief; by the union of the two pictures of it on the retina. From the points A, B, C, D, a, b, c, d, draw lines to L and R, the centres of visible direction of each eye, and it will be seen that the circles A B, a
representing the base of the cone, can be fluted by converging the optical axes to points in the line m n, and that the circles c D, cd, which are more distant, can be united only by converging the optic axes to points iii the line op. The points A, a, for example, united by converging the axes to m, are seen at that point single ; the points B, 6 at n single, the points c, c at o single, the points D, d at p single, the centres s, s of the base at M single,
and the centres s’, s’ of the summit plane at N single.
Hence the eyes L and R see the combined pictures at
MN in relief, exactly in the same manner as they saw in
relief the original solid MN in Fig. 19.
In order to find the height MN of the conical frustum thus seen, let D distance o d S s, the distance of the two points united at ii ; d s’ s’, the distance of the two points united at and C LB = 2 inches, the distance of the eyes. Then we have—
As the summit plane op rises above the base m n by
the successive convergency of the optic axes to different points in the line o Np, it may be asked how it happens that the conical frustum still appears a solid, and the plane op where it is, when the optic axes are converged to points in the line m Mn, so as to see the base distinctly The reason of this is that the rays emanate from op exactly in the same manner, and form exactly the same image of it, on the two retinas as if it were the summit CD, Fig. 19, of the real solid when seen with both eyes. The only effect of the advance of the point of convergence from N to M is to throw the image of N a little to the right side of the optic axis of the left eye, and a little to the lçft of the optic axis of the right eye. The summit plane op will there
irn .
fore retain its place, and will be seen slightly doubled and
iiidistiiict till the point of convergence again returns to it.
It has been already stated that the two dissimilar piee tures may be united by converging the optical axes to a
Ioint beyond them. In order to do this, the distance s s’ of the pictures, Fig. 21, must be greatly less than the distance of the eyes L R, in order that the optic axes, in passing through similar points of the two plane pictures, may meet at a moderate distance beyond them. In order to explain how the relief is produced in this case, let A B, CD, a b, cd, Fig. 22, be the dissimilar pictures of the frustum of a cone whose summit is aD, as seen by the right eye, and cd as seen by the left eye. From L and R, as before, draw lines through all the leading points of the pictures, and we shall have the points A, a united at m, the points B, b at n, the points a, c at o, and the points r, d at p, the points s, s at M, and the points s’, s’ at N, forming the
cone mno p, with its base rnn towards the observer, and its summit op more remote. If the cone had been formed of
lines drawn from the outline of the summit to the outline of
. . .
the base, it would now appear hollow, the inside of it being
seen in place of the outside as before. If the pictures
A B, a b are made to change places the combined picture
would be iii relief while in the case shewn in Fig. 21
it would have been hollow. Hence the right-eye view of
any solid must be placed on the left hand, and the left-eye
view of it on the rig/it hand, when we wish to obtain it in
relief by converging the optic axes to a point between the
pictures and the eye, and vice versa when we wish to obtain In order to find the height MN, or rather the depth of
the cone in Fig. 22, let D, d, C, c, represent the same quantities as before, and we shall have
M P =
o:dd
When D, c, d, d’ have the same values as before, we shall have
MN = 187 feet
When c = d, MP will be infinite.
We have already explained how the two binocular pictures are combined or laid upon one another in the lenticular stereoscope. Let us now see how the relief is obtained. The two plane pictures abed, A B on, in Fig. 18, are, as we have already explained, combined or simply laid upon one another by the lenses L L, L’ L’, and in this state are shewn by the middle circles at AaBb, ccDd. The images of the bases A B, ab of the cone are accurately united in the double base A B, a b, but the summits of the conical frustum remain separate, as seen at C’ D’, c’ d’. It is now the business of the eyes to unite these, or rather to i nake them appear as united. We have already seen how they are brought into relief when the summits are refracted so as to pass one another, as in Fig. 18. Let us therefore take the case shewn in Fig. 20, where the summits en, cd are more distant than the bases A B, a b. The union of these figures is instantly effected, as shewn in Fig. 23, by converging the optic axes to points rn and n successively, and thus uniting c and c and D and d, and making these points of the summit plane appear at m and n, the pictures 4. B CD, abc d, were taken. We shall, therefore, see in relief the frustum of a cone whose section is m n B.
The theory of the stereoscope may be expressed and
illustrated in the following manner, without any reference
to binocular vision :—
1. When a drawing of any object or series of objects is executed on a plane surface from one oit of sight, according to the principles of geometrical perspective, every point of its surface that is visible from the point of sight will be represented on the plane.
2. If another drawing of the same object or series of objects is similarly executed on the same plane from a second point of sight, sufficiently distant from the first to make the two drawings separate without overlapping, every point of its surface visible from this second point of sight will also be represented on the plane, so that we shall have two different drawings of the object placed, at a short distance from each other, on the same plane.
two points of sight, that the distances 1, 1, on the plane
MN, of any pair of points in the two pictures representing
the point 1 of the object, will be to the distance of any
other pair 2, 2, representing the point 2, as the distances
P, 2’ i of the points of the object from the plane MN, mul
tiplied inversely by the distances of these points from the points of sight L, R, or the middle point o between them.
4. If the sculptor, therefore, or the architect, or the mechanist, or the surveyor, possesses two such pictures, either as drawn by a skilful artist or taken photographically, he can, by measuring the distances of every pair of points, obtain the relief or prominence of the original point, or its distance from the plane MN or A B and without the use of the stereoscope, the sculptor may model the object from its plane picture, and the distances of every point from a given plane. In like manner, the other artists may determine distances in buildings, in machinery, and in the field.
5. If the distance of the points of sight is equal to the
of the eyes L, R, the two plane pictures may be
united and raised into relief by the stereoscope, and thus give the sculptor aild other artists an accurate model, from which they will derive additional aid in the execution of their work.
6. In stereoscopic vision, therefore, when we join the points 1, 1 by converging the optic axes to i’m the line P0, and the points 2, 2 by converging them to 2’ in the same line, we place these points at the distances o 1, 02, and
see the relief, or the various differences of distance which the sculptor and others obtained by the method which we
have described.