the human
e acts as
such in-
you'' or
- which sug-
t is quite
e frame
radial
r angle
' equation
es himself
* line of
'al world
ivVironment
.y of lights
on a circle
vhich the
rcle passing
pter is
t along the
e displaced
3 the circum-
ular axes is
ctive direction
o retinal
ory rays from
bipolar axi-
te the sensory
r from the
on a circle
numerical
not depend
purely on convergence, but on convergence disparity and we write
(12) ray)
where it is convenient to choose y, as the maximum value of y in the stimulus
so that y - y, is non-negative for all visible points. The function f is the same
for all binocularly equivalent stimuli, that is, for all stimuli which can be trans-
formed into each other by the rule of (1l. We cannot conclude, a priori, that f is
a universal function for a given observer and takes on the same values for stimuli
which are not binocularly equivalent. Fortunately, our present experimental informa-
tion on this point does indicate that for any given observer the function f is universal
and does not particularly depend on the stimulus, | S].
It is not sufficient to know that the circles r = constant, (p - constant of the
polar coordinate grid in visual space correspond to the loci y = constant, ()- constant
of physical space: we must describe the correspondence by a precise numerical trans-
formation. Even accepting the relations,
refly Vo)
929
we cannot determine the function f without knowing the intrinsic geometry of the per-
(13)
ceptual space. There are a number of experiments which show that the perceptual
geometry is not euclidean [5], [1], and which determine the type of geometry involved.
Most observers give experimental results which can be explained consistently only on
the hypothesis that the visual geometry is the so-called hyperbolic non-Euclidean
geometry of Lobachevski. In this geometry the distance D between two points with
sensory polar coordinates (ry, ®, and (r,, Pa) is given by
(14) cosh d » cosh r, cosh r, - sinh r, sinh r, cos (9, -9)
Equation (14) is the hyperbolic analog of the Euclidean law of cosines and is used in
much the same way. With (13) and (14) it is possible to design experiments to deter-
mine the function f(y - y9) and a number of such experimental determinations have
been made. Once the numerical values of the function f have been determined there
is no longer any difficulty in calculating the perceptual effect of stereoscopic presen-
tations.
For the purpose of seeing how this knowledge may be applied, let us take the
example of the preceding section. We take as our object of regard in real space, an
open rectangular chamber symmetric with respect to the median, with its farther
wall in the background plane, 4 meters wide and 5 meters deep (Fig. 9). We shall
-Al-