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.
[Illustration: FIG. 19.]
In order to understand how the two pictures, when 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 ABCD, Fig. 19, be a section of the frustum of a cone, that
is, a cone with its top cut off by a plane C_e_D_g_, and having AEBG
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 C_e_D_g_. The
points L, R being the points of sight, draw the lines RA, RB, which
will cut the plane on which the projection is to be made in the points
_a_, _b_, so that _ab_ will represent the line AB, and a circle, whose
diameter is _ab_, will represent the base of the cone, as seen by the
right eye R. In like manner, by drawing LA, LB, we shall find that
A′B′ will represent the line AB, and a circle, whose diameter is A′B′,
the base AEBG, as seen by the left eye. The summit, C_e_D_g_, of the
frustum being in the plane of projection, will be represented by the
circle C_e_D_g_. The representation of the frustum ABCD, therefore,
upon a plane surface, as seen by the left eye L, consists of two
circles, whose diameters are AB, CD; and, as seen by the right eye, of
other two circles, whose diameters are _ab_, CD, which, in Fig. 20,
are represented by AB, CD, and _ab_, _cd_. These 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 LB, RB, and we therefore see the point single, and at
the distances LB, 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 LD, 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, the 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.
[Illustration: FIG. 20.]
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 ABCD, Fig. 19, is represented by the line
D_b_, as seen by the right eye, and by the shorter line DB′, 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 C_a′_, as seen by the
right eye.
When the body is hollow, like a wine glass, _we see more of the right
side 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 ABCD, _abcd_, of the frustum of a cone, as obtained
from Fig. 19.
[Illustration: FIG. 21.]
In order to unite these two dissimilar projections, we must converge
the optical axes to a point nearer the observer, or look at some point
about M. Both pictures will immediately be doubled. An image of the
figure _ab_ will advance towards P, and an image of AB will likewise
advance towards P; and the instant these images are united, the frustum
of a cone, which they represent, will appear in relief at MN, the
place where the optic axes meet or cross each other. At first the
solid figure will appear in the middle, 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, _mn_L = ABL; 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 AB, _ab_,
representing the base of the cone, can be united by converging the
optical axes to points in the line _mn_, and that the circles CD, _cd_,
which are more distant, can be united only by converging the optic axes
to points in 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, _b_ 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 OP; _d_ = S_s_, the distance of the two points united at M;
_d′_ = S′_s′_, the distance of the two points united at N; and C = LR =
2½ inches, the distance of the eyes. Then we have—
D_d_
MP = ———————
C + _d_
D_d′_
NP = ——————— , and
C + _d′_
D_d_ D_d′_
MN = —————— - ———————
C + _d_ C + _d′_
If D = 9·24 inches,
C = 2·50, then
_d_ = 2·14,
_d′_ = 2·42, and
MN = 0·283, the height of the cone.
DC
When C = _d_, MP = ————
2C.
As the summit plane _op_ rises above the base _mn_ by the successive
convergency of the optic axes to different points in the line _o_N_p_,
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_M_n_, 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 _left_ of the optic axis of the right eye. The summit plane _op_
will therefore retain its place, and will be seen slightly doubled and
indistinct till the point of convergence again returns to it.
It has been already stated that the two dissimilar pictures may be
united by converging the optical axes to a point beyond them. In order
to do this, the distance SS′ 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 AB, CD, _ab_, _cd_, Fig. 22, be the
dissimilar pictures of the frustum of a cone whose summit is CD, 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 C, _c_ at _o_, and the points D, _d_ at _p_, the points
S, _s_ at M, and the points S′, _s′_ at N, forming the cone _mnop_,
with its base _mn_ 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 AB, _ab_ are made to change places the combined picture would
be in 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 _right_ 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 it in relief by converging the optic axes to a point beyond the
pictures. In every case when we wish the combined pictures to represent
a _hollow_, or the converse of relief, their places must be exchanged.
[Illustration: FIG. 22.]
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
D_d_
MP = ———————
C - _d_
D_d′_
NP = ——————— , and
C - _d′_
D_d′_ D_d_
OP = ——————— - ——————
C - _d′_ C - _d_
When D, C, _d_, _d′_ have the same values as before, we shall have MN =
18·7 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 _abcd_, ABCD, in
Fig. 18, are, as we have already explained, combined or simply laid
upon one another by the lenses LL, L′L′, and in this state are shewn by
the middle circles at A_a_B_b_, C_c_D_d_. The images of the bases AB,
_ab_ of the cone are accurately united in the double base AB, _ab_, 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
make 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 CD, _cd_ are more distant than the bases AB, _ab_. The
union of these figures is instantly effected, as shewn in Fig. 23, by
converging the optic axes to points _m_ 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 points of convergence of the axes
L_m_, R_m_, and L_n_, R_n_. In like manner, every pair of points in the
summit plane, and in the sides A_m_, B_n_ of the frustum, are converged
to points corresponding to their distance from the base AB of the
original solid frustum, from which the plane pictures ABCD, _abcd_,
were taken. We shall, therefore, see in relief the frustum of a cone
whose section is A_mn_B.
[Illustration: FIG. 23.]
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 point 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.
[Illustration: FIG. 24.]
3. Calling these different points of the object 1, 2, 3, 4, &c., it
will be seen from Fig. 24, in which L, R are the 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 1′P, 2′P of the points of the object from the plane MN,
multiplied 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 AB; 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 distance of
the eyes L, R, the two plane pictures may be united and raised into
relief by the stereoscope, and thus give the sculptor and 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 1′ in the line PQ, and the points 2, 2 by
converging them to 2′ in the same line, we place these points at the
distances O1, OO2, and _see the relief_, or the various differences of
distance which the sculptor and others obtained by the method which we
have described.
7. Hence we infer, that if the stereoscopic vision of relief had never
been thought of, the principles of the instrument are involved in the
geometrical relief which is embodied in the two pictures of an object
taken from two points of sight, and in the prominence of every part of
it obtained geometrically.