zotero/storage/KRNFPYBR/.zotero-ft-cache

Journal of the
OPTICAL SOCIETY

0f
VOLUME 40, NUMBER 10

MnIERICA

OCTOBER, 1950

The Metric of Binocular Visual Space*
RUDOLFK. LUNEBURGt Knapp Memorial Laboratory of Physiological Optics, Collegeof Physicians and Surgeons, Columbia University,
New York, New York (Received March 3, 1950)
The aim of this paper is to show that the so-called visual space has a uniquely determined non-Euclidean metric, or psychometric distance function, the numerical parameters of which depend on the individual observer. Certain well-known phenomena of space perception, such as the horopter, the alley experiment and size constancy, are explained on the basis of the distance function. Methods of measuring the personal parameters of the metric are developed, and applications of the theory to the field of binocular instruments and pictorial representation of space are suggested.

VISUAL1 PURPOSE sensations are established under the influence of many different factors. They are the result of an activation of our mind by physical light stimuli and are distinguished by a remarkable degree of certainty and definiteness which tempts us to believe that the external world itself is revealed to us, and not merely an image of our own making. Through vision, external objects are presented to us in a threedimensional continuum, endowed with certain sensed qualities of color, brightness, form, and localization.
Yet it is well known that the final form of visual sensations actually depends on the observer's psychological condition which in turn is the result of his entire personal development, his former experience, his prejudices, even his instantaneous wishes and in-
tentions. Recognition of this fact does not mean that all
attempts of a quantitative investigation of visual
perception must be abandoned. Visual sensations are
not merely the chance result of coincidental psycho-
* This investigation is being conducted under Project NR143638, Contract N6onr-27119.
t This is the last paper Dr. Luneburg wrote before his sudden, unexpected death in August, 1949. It was not written for publication and has been edited by the Knapp Laboratory staff working on this project and chiefly by Paul Boeder, consultant to the project. The paper is believed to be of great significance because here for the first time, Luneburg removes the restriction on freedom of observation and introduces the probable mathematical function involved in motion.

logical conditions but are governed by certain constant factors which, though not determining the sensations completely, play an integral part in their formation. These factors must be related to the external physical situation as well as to certain constant elements in the personality of the observer. The denial of the existence of such constant elements in the relationship of a living organism to external nature would be tantamount to denying all objectivity in visual sensations and therefore would be absurd.
In the following we are concerned with a special group of constant elements in visual sensations, namely, those which influence the sensed qualities of form and localization in the three-dimensional continuum of visual perception. Our aim is to show that this continuum, the so-called Visual Space, is a metric mathematical space. It possesses a uniquely determined metric, which varies in its numerical parameters from observer to observer. But its general form is invariant; it is the metric of the three-dimensional hyperbolic geometry.
2. CONSIDERATION LIMITED TO BINOCULAR FACTORS OF VISUAL PERCEPTION
This surprising conclusion is arrived at by an inductive analysis of certain experimental facts which are obtained under simplified laboratory conditions. The observations are made in a dark room with the observer's head, and therefore approximately also the rotational centers of his eyes, fixed by means of a

627

628

RUDOLF K. LUNEBURG

z
X
R
FIG. 1. Cartesian coordinate system for physical space; L and R represent centers of rotation of left and right eyes.
headrest. Let us assume that the centers of rotation, R and L, are the points y= :1 1 of the y axis of a Cartesian coordinate system whose x, y-plane coincides with the physical horizontal plane (Fig. 1).
Simple geometrical configurations are presented to the observer; for instance, isolated light points, lighted lines or surfaces. In judging such configurations, the observer has to rely almost entirely on his primitive ability to see; intellectual clues of interpretation are practically excluded. We may assume that visual sensations such as these reveal more clearly the underlying constant factors which we wish to investigate.
There is a great difference between monocular and binocular vision under the artificial conditions just described. With one eye our judgment of spatial form and localization is erratic and inconsistent. Binocular observation, on the other hand, retains a high degree of certainty. True, in ordinary situations monocular observation is not necessarily inferior to binocular vision. But from experiments in the dark room we conclude that binocular vision adds certain elements to visual perception which monocular vision lacks. This is also demonstrated in a stereoscope which provides a situation of clueless vision comparable to that of the dark room. With a stereoscope, the monocular observer is dependent on intellectual clues of perspective and interpretation, whereas the binocular observer obtains a striking sensation of depth.
The following investigation is concerned with these special binocular or stereoscopic factors of visual perception. No claim is made of a general theory of space perception, nor even of a theory of visual perception integrating binocular as well as other psychological factors. In spite of this limitation, the end result is not without significancefor the broader aim inasmuch as it will make possible a precise characterization of the variations which may be caused by the instantaneous psychological condition of the observer.
3. 'A(TS kPSQUIRtNG PSCOGlN fON
A theory of binocular vision must account for the following experimental facts.

(a) A single isolated light point P* is given in the dark room. Let x, y, z, be its physical Cartesian coordinates. The observer sees the point P* as a "sensed" point P in his three-dimensional visual space. It has, at any instant, a definite localization in this space, relative to certain subjective planes of reference of the observer, and relative to an apparent center of observation. The fact that two eyes are involved in creating the sensation of the point P* is not represented in the visual space. We may say that binocular observation in the physical space corresponds to an apparent monocular observation in the visual space. Besides an apparent center of observation, there are three planes of reference of which we are conscious, namely, the subjective horizontal plane, the subjective median plane, and the subjective frontal plane. Mathematically we represent this in a Euclidean space by means of a Cartesian coordinate system, , , . The origin = 7= = 0 represents the apparent center of observation, the , 1-plane the subjective horizontal plane, the I, c-plane the median plane, and the , c-plane the frontal plane of the visual space. In general, these planes are correlated approximately to the corresponding physical planes, i.e. a point P* in the x, y-plane is seen as a point P near the , 7-plane, a point P* in the x, z-plane as a point near to the median plane (Fig. 2).
Since the point P* is seen, at any instant, as a point P in some definite localization relative to the three subjective planes of reference, we might be tempted to search for a constant and necessary one-to-one correspondence between physical points and sensed points in the visual space. However such an attempt would fail, since it would mean that visual sensations are uniquely determined by the physical stimuli, independent of psychological factors. It would also be in disagreement with actual observations. Visual localization of an isolated, repeatedly shown, physical point P* is found to be highly erratic. The point P seems to be floating in space indeterminedly, now it seems near, now far. This uncertainty of correlating visual and physical localizations becomes obvious if a similar experiment is made with a stereoscope. It is even possible to disturb the observer's judgment of localization with respect to the subjective reference planes, for example, by placing him on a slowly rotating chair.
Or
FIG. 2. Subjective coordinate system for visual space.

METRIC OF BINOCULAR VISUAL SPACE

629

A point P seems to remain in the median plane, though

Distance Alley

the physical x, y-plane has been rotated into a different

position.

The net result of experiments with an isolated point is purely negative: Binocular observation of a single

Parallel Alley

point does not differ from monocular observation. Both

are equally uncertain as to correlating a sensed point

P to the physical coordinates of the stimulating

point P*.

(b) The situation is quite different when the object

consists of several isolated points P1 *, P 2*, P3*,---Aside from sensing these points as points P, P2, P3, ***, in a three-dimensional continuum, we have a

FIG. 4. Distance and parallel alleys.

rather definite sensation of distance of the points from
one another. We also find that the sizes of these sensed
distances can be compared. Thus, denoting the' sensed
distance between two points P1 and P2 by (P1 P2 ), our visual sensation of three points Pi, P2, P3 contains an immediate judgment of inequalities (PlP 2)> (P2P3) or (PdP2) <(P2 P3 ). The same is true for two pairs of points P1 , P2 and P3 , P4. There is an immediate perception of inequalities (PlP 2)> (P3P4) or (PIP2) < (P3P4).
Moreover it is found, for binocular observation, that the inequality signs are determined by the physical
coordinates of the stimulating points P1*, P2 *, P3 *, P4 *. This means that the scores of repeated tests with the
same two pairs of stimulating points are highly corre-
lated, regardless of other physical qualities of the points
such as brightness and color. A corresponding correlation is not found in monocular vision where such addi-
tional qualities may easily change our judgment. We
note here that in general the sensed inequalities do not
correspond to those between the physical distances
(P 1*P2*) and (P 3*P4*). Moreover, there is no sensation of absolute localization of the whole configuration, that is, there is no certainty in judging the physical
coordinates of the points P* from the sensed coordinates of the points P.
(c) The following extremely interesting observation is due to Helmholtz.1 An observer, placed into the
position described previously, is given the task of arranging a number of light points so that they appear to lie on a horizontal straight line, symmetric to the median plane. It is found that the points are set

consistently on certain physical curves which are, in general, not straight in the physical sense. The form of these so-called horopter curves depends on the distance x0, of the center point. At a certain distance xo=a the horopter is practically straight. At nearer distances, xo<a, the horopters are concave to the observer, at greater distances, xo>a, they are convex (Fig. 3). The distance, a, of the straight horopter depends on the individual observer. The Helmholtz horopter experiment reveals a new element in visual sensations, namely, that there is an immediatefeelingfor straightness which allows us to judge whether three points PRQ lie on a subjective straight line.
(d) A related experiment was made by Hillebrand; it is the so-called Alley experiment2. It is well known that physically parallel lines like railroad tracks, or the curbs of a straight road, seem to converge in the distance. The same is true for two physically parallel rows of light points observed in the dark room. In the Hillebrand experiment the rows of light points are arranged so that they no longer converge but seem to form a parallel "alley". The resulting divergent physical curves have a characteristic form comparable to that of hyperbolas.
(e) Hillebrand's experiment was repeated by Blumenfeld3 under improved technical conditions. The general result was the same but an additional observation was made. When the observer was instructed to pay predominant attention to the distance between opposite points, and, by making them equal, to set up alleys of apparently equidistant walls, the resulting curves

had the general form found by Hillebrand, but did not

appear parallel and straight. The "distance alleys"

differed from the "parallel alleys"and were found to lie

consistently outside the parallel alleys (Fig. 4).

P - .r
R

a' , an

(

l

_ -

IX _-i

II

/2
FIG. 3. Horopter curves for different fixations.
IH. V. Helmholtz, Treatise on Psychological Optics, J. P. C. Southall, Editor (Optical Society of America, 1925), Vol. 3, pp. 482 f.

4. THE PSYCHOMETRIC DISTANCE FUNCTION
The preceding experiments indicate strongly that the visual space, as revealed in dark room experiments, can be characterized mathematically as a metric space. This means that we may assume the existence of a dis-
2 F. Hillebrand, Denkschr. Akad. Wiss. Wien, math.-nat. Kl. 72, 255 (1902).
3 W. Blumenfeld, Zeits. f. Physiol. d. Sinnesorgane 65, 241 (1913).

630

RUDOLF K. LUNEBURG

5. THE INDETERMINACY OF THE DISTANCE FUNCTION

First we ask whether such a function is uniquely

determined by the preceding conditions. For if a multi-

plicity of possible distance functions existed, any one

of them would give us merely an empirical description

of certain experimental findings. But if the function is

uniquely determined, the establishment of its explicit

form must represent an interesting psychological result

L

concerning the nature of visual sensations. Obviously, D is not uniquely determined since along

with D(P1 , P2) any other function D(P1 , P2 )

R

=C- D(P1 , P2), C being a positive constant, satisfies the conditions. It can be shown, however, that this

indeterminacy is the only one possible.

In order to show this mathematically, we formulate

certain basic assumptions which are concerned with the

compactnessand the convexity of the visual space.

(a) The visual space is finitely compact: To any in-

FIG. 5. Equivalent configurations.

finite sequence P, such that D(P1 , P,) is bounded, there exists a subsequence Pn, and a point P of the visual
space such that D(Pnv, P)-4O.

tance function, D(P1 , P2 ), which accounts quantitatively for the observations just described. In other
words, we assume that to any two points P1 , P2 of the visual space, a non-negative number can be assigned
which measures the sensed distance, (PlP 2 ), between these two points.
Any such distance function must satisfy the follow-
ing conditions:

(b) The visual space is internally convex: To any pair of points P1, P2 (P1 #P2 ) there exists a third point Q such that D(P1 , Q)+D(Q, P2 )=D(P1 , P2 ).
General postulates of this nature, of course, cannot
be tested by experiments since an infinity of points is
involved. Yet, they are not arbitrary assumptions but
formulations of certain observations which we make
with regard to our own visual sensations. The uniqueness of the psychometric distance func-

(1) D(P1, P2) =D(P2, PI)>0, if P1, P2 are sensed as different.

(2) D(P 1 , P2) = 0 if PI= P2.

(4.1)

(3) D(P1, P2)+D(P2, P3)-D(P1 , P3 )

for any 3 points P1 , P2, P3-

The physical distance D*(P1 *, P2*) though it ob-

tion can now be readily deduced. Suppose there are two
such functions D(P1 , P2) and D(P1 , P2 ), both satisfying all requirements. Then, whenever (P1P2) > (P3P4 ) then D(P1, P2)>D(P3 , P4 ) and fD(P1, P2 )>D(P3 , P4). ItfollowsthatD(PI, P2 )>D(P3 , P4 ) wheneverD(Pi,P2 ) >D(P 3 , P4 ). This is possible only if D=f(D), for all points where D and D are defined. Consider next any

viously satisfies the condition (4.1) cannot be consid- two points P1, P2 of the visual space and assume that ered as a distance function of the corresponding sensa- D(P1 , P2 )= a> 0. As a consequence of compactness and tions because it would not measure the sensed distance internal convexity, it can be shown that to any number

correctly. In fact, we have already seen that it would t of the interval 0_ta there exists a point Q such that

not satisfy the following further condition which must D(P1 , Q)= t and D(Q, P2)= a-t. On account of the

be fulfilled by the distance function:

straight line condition (4.2) it follows that D(P, Q)

+D(Q, P2 )=D(P1 , P2), or f(t)+f(a-t)=f(a) for any t

(1) If (PI, P2) and (P3 , P 4 ) are any two pairs of

in 0_t~a and for any a determined by a pair of points

sensed points and (P1, P 2)> (P3 , P4 ) then

P1 , P2. This leads to the conclusion f(t) = Ct and hence

D(P1, P2 )>D(P 3, P4).

(4.2) D(P1 , P2 )=C-D(P 1 , P2 ), where C is a constant.

(2) If P, P2 , P3 are sensed on an apparently

Thus, the only possible indeterminacy of the psycho-

straight line, then D(P1 , P2 )+D(P2 , P3 )

metric distance function is a multiplicative constant.

= D(P1 , P3) and vice versa.

This means that the ratio D(P1 , P2 )/D(P3 , P4 ) of two

distances is a uniquely determined function of the four

A distance function D(P1 , P2) which satisfies both points in question. This result is in agreement with the

conditions (4.1) and (4.2) is called a psychometricdis- observation made previously that the inequality sign in

tancefunction. Our problem is to determine this func- the sen5ation (lP2)> (P3PI) i determined by the

tion for the visual space, and, if possible, to express physical coordinates of the points P1 *, P2*, P3*, P4 *.

it in terms of the physical coordinates of the stimulating We conclude that the ratio D(P 1, P2 )/D(P 3 , P4 ) will be

points Pi*, 2*.

a function of the physical coordinates of the four stimu-

METRIC OF BINOCULAR VISUAL SPACE

631

lating points which is invariant for an individual observer. It establishes a constant relationship between the objective physical space and the subjective visual space of the observer which is unaffected by other psychological factors. Any parameters in this relation, which are not the physical coordinates of the four points, must be constant factors of the personality of the observer, characterizing his visual reaction to external nature.
6. THE FORM OF THE DISTANCE FUNCTION
In order to find an explicit expression for the psychometric function D(P1 , P2 ), we make use of certain general psychological observations and of certain instructive experiments. These observations are concerned with the thesis that there is no absolute localization even in binocular vision. A given physical configuration provides no immediate visual clues which could determine localization and absolute phenomenological size of the sensed configuration. Localization and size are determined by other non-visual psychological factors, in particular, by the intellectual experience of the observer.
Depending upon such -factors, the same physical configuration may be seen in different localizations and different phenomenological sizes. Vice versa, entirely different physical configurations may give rise to the same visual sensation.
This principle is readily seen to apply to monocular vision, in fact, this is the tacit assumption in reproducing a visual sensation by painting. Its validity in binocular vision is indicated by observations of an isolated point. But there are more convincing demonstrations.
Ames of the Dartmouth Eye Institute has shown that to a given ordinary rectangular room, a set of distorted rooms can be constructed, which, if viewed from a fixed position, are indistinguishable from the original room (Fig. 5). In fact, the illusion of seeing the original room in all cases is inescapable if suitable suggestive details, for example, window frames, are included. Obviously, it is our experience of living in rectangular
rooms which causes us to see the rectangular form even though the physical reality is quite different. The distorted rooms, of course, cannot be constructed arbitrarily. Certain metric clues must be identical in all of them.
In order to combine this principle of the lack of absolute localization with our previous result concerning the existence of a psychometric distance function we are forced to the conclusion that D(P1 , P2 ) must be a distance function of a homogeneous metric space, that is, a Riemannian space of constant curvature. For only in these spaces form and localization are completely uncorrelated, which means that the form of a configuration provides no clues for its localization.
The preceding considerations warrant the introduction of the hypothesis of constant curvature, but there

is another observation which adds strong support to the hypothesis. Observations in the dark room reveal that we not only have an immediate sensation of straightness but also of plane surfaces. If light points or pliable wires are arranged in apparently plane surfaces, the resulting physical surfaces are curved. If any two points of a "visual plane" are connected by a visually straight line, it would not leave the visual plane. It is well known that two arbitrarily located dimensional submanifolds of this type exist only in the homogeneous spaces. Thus, the very existence of an immediate sensation of "planeness" can be considered as additional evidence in favor of the hypothesis of constant curvature.
The mathematical consequence of this hypothesis is that there exists in the visual space a subjective oordinate system, such that the psychometric distance function D(P1 , P2) is given by the formula:

2 sin--1(-K) -D-

(-K)l

2

C

[(1-4 2)2+ (X41-772)2+(12)

(6.)12]

Lf [1+SP?2 (1±ŽP22)]

4 )(

4 )

where pn2=n2+

2
_+n +¢n

2, and K and C are real valued

constants which are as yet undetermined. We interpret

(, 7, r as Cartesian coordinates in a Euclidean space.

Thus we obtain a Euclidean map of the visual space,

which, in general, is not isometric, since the distance

of two points is given by a distance function (6.1) which

is non Euclidean if KOO.

The question whether the geometry of the visual

space is elliptic, hyperbolic or Euclidean, i.e. whether

in (6.1) K>O, K<O or K=O can only be answered by

2/

/Visual Straight Line
FIG. 6. A visually straight line represented in hyperbolic space.

632
Z

RUDOLF K. LUNEBURG P *

X R

FIG. 7. Bipolar coordinates of a physical point.
experiments. All experiments carried out thus far indicate that K is negative, which means that the visual space must be a part of a hyperbolic or Lobachevski space.
The total hyperbolic space is represented in our map by the interior of the sphere 2+n2+ 2 =- 4/K. The visual straight lines are given by circles which intersect the basic sphere at right angles; all of these geodesics are thus represented by plane curves which lie in the planes through the origin (Fig. 6).
The visual planes are represented by spheres which intersect the basic sphere at right angles.
Because of the symmetry of the metric (6.1) with regard to the origin, we identify the origin = n= = 0 with the apparent center of observation in the visual space. Furthermore we coordinate the , -plane to the subjective horizontal plane, the , i-plane to the median plane, and the -q, -plane to the frontal plane.

FIG. 8. Bipolar parallax and bipolar latitude of a physical point.

plane are seen in the subjective horizontal , -plane, and also, that points in the x, z-plane are placed into the median , c-plane. For the further description of the required relationship we introduce, in the physical space, a special bipolar system of angular coordinates which is better adapted to the physiological mechanism of binocular vision. Consider the plane through a point P* and through the eyes R and L. Its angle with the x, y-plane is called the angleof elevation.In this plane of elevation, two lines are drawn from R and L toward P*. These lines determine the positions of the optical axes of the eyes if the point P* is fixated. We characterize these lines by their angles a and ,3 with the y axis as shown in Fig. 7. The localization of the point P* is then determined by the three angles a, ,3,0. Instead of at and j3 we shall use the linear combinations:

y= - a-

(7.1)

7. RELATION OF VISUAL TO PHYSICAL SPACE
Next we establish a relation of the subjective coordinates , , of a sensed point P to the physical coordinates of its stimulating point P*. As stated before, no such relation can be constant or necessary since there is no absolute localization in the visual space. Hence, if we found one possible relation (x, y, z) =f(% , ) between the coordinates, any other relation (x, y, z) = f(', ', ') would be visually equivalent provided that (', 7/ 7', )=g(, , t) represents a motion, i.e. a congruence transformation of the visual space.
The problem of finding a coordination of the two spaces can actually be solved only individually since the physiological condition of the eyes is a determining factor. In case of defective vision, the results are different for observations with or without glasses. The visual space and its metric, of course, are independent of these conditions; only the correlation of visual sensations to the physical qualities of external objects is affected. Since such differences, therefore, can be removed to a certain degree by corrective glasses, we are justified in
choosing among the possible coordinations one which
applies to a normal observer under usual conditions. For an observer whose eyes are functioning equally
well, we may assume that objects in the physical x, y-

which are called the bipolar parallax and the bipolar latitude of a point P* (Fig. 8). In any plane of elevation, the curves y=const. are circles through the eyes, the so-called Vieth-Mueller circles, whereas the curves 4=const. are hyperbolas, the Hillebrand hyperbolas, whose asymptotes go through the point x=y=z=0 (Fig. 9).
If a Vieth-Mueller circle is rotated about the y axis, a torus surface y=const. is obtained which shall be called a Vieth-Mueller torus.
, (
l
Ai
FIG. 9. Curves of constant bipolar parallax and latitude.

METRIC OF BINOCULAR VISUAL SPACE
0t= const.

633
a = const.

FIG. 10. Testing correlation between physical and visual spaces on Vieth-Mueller circle by changing convergence.

For infinitely distant points of the physical space, FIG. 11. Torus of constant binocular parallax, showing curves of

we have -y=0. Since these points give the visual im-

constant elevation and constant binocular latitude.

pression of forming an apparently spherical dome of finite radius around the observer,: we represent them, in the (, q, i-space, as points on a sphere concentric to the origin. This sphere will include the total visual part of the visual space. We choose arbitrarily a sphere of radius 2 to representphysical infinity.
This normation makes the constant K in the expression (6.1) for D(P1 , P2 ) a dimensionless parameter, with individual differences in K indicating differences in visual perception.
Since we have no visual sensations of infinite size, we conclude that K can never be smaller than -1, so that the basic sphere, 2/(-K)I, in case K 0Ol,ies always outside the sphere of radius 2 which represents physical infinity. The two spheres coincide if K= -1; in which case astronomical objects would give an immediate sensation of immense size and distance. Though this limiting case is never reached, we conclude

original observation for a set of such circles. If the point configuration is rotated about the y axis, and arrested at a number of equidistant 0 positions, a set of points is obtained which lie on certain curves 4= const. and 0= const. of the Vieth-Mueller torus y= const. Wires placed along these curves would form a network like that shown in Fig. 11. If this is observed from the points R and L, the resulting sensation is approximately that of meridians and circles of latitude on a sphere around the observer. This means that the torus surfaces y= const. appear as concentric spheres around the center of observation, and the bipolar angles 4 and 0 as polar coordinates in the I, 7, c-space. The observation may be verified for a number of such torus configurations by simply overconverging or diverging the eyes as before. Mathematically, these observations are expressed by the equations:

that visual sensation of size and distance relations agrees with physical reality better the nearer the value of K is to -1. Thus, individual differences in the parameter K may correlate with individual differences

t=f(7) cos4 cos0

77=f(y) sino

(7.2)

A=f(-y) coso sinG

in certain mechanical skills.

which relate physical and subjective coordinates. The

Further indication of the correlation of the I, n, function f( 7y) is a non-negative function of y which

c-space to the physical space is obtained by the follow- increases monotonically if y decreases from positive

ing observation. We arrange a number of equidistant values to y=0. From the preceding convention it fol-

light points on a Vieth-Mueller circle, as shown in lowsthat f(O)=V

Fig. 10. These points are seen in the visual space as Even for the neighborhood of the horizontal plane,

equidistant points on a circle about the center of obser- this formulation should be considered no more than a

vation. If the distance between the points Po*, Pi* is first approximation. There is experimental evidence of

made equal to the interpupillary distance of the eyes, small Vieth-Mueller circles giving an elliptical rather

the configuration can be "projected" toward infinity than a circular impression. Therefore, it is possible

(-y=O) by simply diverging the eyes to a position of that a more precise formulation may require the re-

parallel optical axes. By overconverging, the configura- placement of the relations (7.2) by the following:

tion may be projected on smaller Vieth-Mueller circles (Fig. 10). In this way we can conveniently verify the
1In a more careful investigation, the fact that the sky appears to be a flattened dome could be considered. Thus, the following arguments are to be taken as approximations referring to visual sensation in the neighborhood of the horizontal plane.

§ A more careful experimentation may lead to the result that, at greater distances, the network configuration is not seen as a sphere but as an oval, flattened at the top. The previously mentioned impression of the sky, and also the so-called moon phenomenon, seem to indicate this possibility. In this case, the relation =f(-y) cost sino would have to be replaced by a relation
r=f'(y) cosp sinGwith fi&) <ffy).

634

RUDOLF K. LUNEBURG

t=fi~y) cos4 cosO, =f2(y) sin+, =f3 (y) cos4 sinG,
with three different functions f('y), f2Qy), f 3 (y).

Ii

s, Horopter

8. BOUNDARIES OF VISUAL AND PHYSICAL SPACE

IN THE HORIZONTAL PLANE

N

A

-It is interesting to note that the experiments of

N /

Helmholtz, Hillebrand, and Blumenfeld can be fully

explained on the basis of these results, that is, without

any further information about the function f(y).

Consider the transformations (7.2) in the horizontal

plane:

7t==f7(f-lyy) )csois4n+.

(8.1)

This transformation is continuous and one-to-one in the interior of the half-plane x> 0; it is, however, quite irregular on the boundary (Fig. 12). In fact, the boundary of this half-plane is characterized by the followingconditions for the bipolar coordinates y and k:

A: ,y=O, 4=7r/2, or x= 0, y>

L: -y= r-20,
B: y= +=, 0

or x= 0, y= + (left eye)

orx=0,- I<y<

(8.2)

R: y= 7r+ 10,

or x= 0, y= -1 (right eye)

C: or= O.0 = --7r12, or x=0, y<-l.

The Sections A, B, C of the y axis are contracted into

three isolated points A: t=O, q= 2 ; B: t=f(ir),

7=0,

C: = 0, = -2 whereas the two eyes are stretched out

into two curves pl=f(r-2k), P2=f(7r+20) which

connect the point B with A and C respectively. These

two curves join at B at an acute angle, as follows from

the fact that f(-y) increases with decreasing y, i.e.

f'(y)<O, and hence dpil/d>O, dp2 /d0<O. The half-plane x>0 is thus imaged into a scythe-

shaped domain of the I, 7-plane as shown in Fig. 13.

The image of the half-space x> 0 is obtained by simply

rotating this domain around the 'q-axis.

l
A/
C

FIG. 13. Boundaries of the horizontal half-plane of visual space and a frontal plane horopter, S, mapped on a ,7-diagram.

9. HOROPTERS AND ALLEYS
We interpret Helmholtz' horopter curves as those geodesics of the I, -plane which are symmetric to the t axis (Fig. 13). Their equation is:

K_(02+ 272)-I= Ct.

(9.1)

4

If K<0, they are circles normal to the basic circle 2/(-K)+ and normal to the axis. Two different groups of circles can be distinguished. Starting from a point to of the axis, they either reach directly the circle C'-+n 2 = 4 representing infinity, or they are intercepted by the curves representing the eyes.
Their counterparts in the physical x, y-plane, therefore, must be curves which, starting from x0, either go

FIG. 12. Boundaries of the horizontal half-plane in physical space.

FIG. 14. Frontal plane horopters.

METRIC OF BINOCULAR VISUAL SPACE

635

If K<0, they are circles through the points = J2/

A q

(-K) 1 of the t axis. Since these curves lie to the left of

the radius vector OP to any of their points, it follows

N.
N
,\ Parallel Alley

that their physical counterparts must lie above the corresponding sections of the hyperbolas = const. This means that they must have the tendency of bending away from the curves 4= const. which is charac-

teristic for the alley curves in the Blumenfeld experi-

ment.

The parallel alleys of Hillebrand, given by the geo-

\Distance Alley desics:

K _(2+

q2)-1=

-

C-1

(9.3)

4

/

FIG. 15. Distance and parallel alleys in map of subjective horizontal plane.

asymptotically toward infinity or approach the eyes R and L. Thus they are curves of precisely the type observed in the horopter experiment. In particular, there must be one curve, which, in the neighborhood of the
x axis, is practically straight (Fig. 14). This situation not only exists for the case K<O, but
also for K=O or K>O. If K=O, the horopter curves (9.1) are straight lines, if K> 0, they are circles through the points q= L2/(K) 1 of the q axis. In either case, as before, the two different groups can be distinguished.
The distance alleys of Blumenfeld are given by the
equation:

K

_(V2+ )+1= Cn-

(9.2)

4

are symmetric to the q axis. Hence, for K<0 they are circles normal to the basic circle 2/(-K) I and normal to the axis. Also these curves lie above any radius vector OP and thus have the same characteristic form
as the distance alleys. If a parallel alley and a distance alley are started
from the same point P, the parallel alley lies inside the
distance alley, in agreement with actual observations
(Figs. 15 and 16). If K= O, parallel alleys and distance alleys are
identical, namely, straight lines = const. in the I, 1-plane. For K> 0, distance alleys are represented by circles normal to the sphere p 2/(K) 1, parallel alleys by circles through the points 27(K)1 of the t axis. In this case, therefore, the distance alley should lie inside the parallel alley. The Blumenfeld experiment contradicts this, thus giving a direct demonstration that the space of binocular vision is a Lobachevski space.

10. USE OF ALLEY DATA TO DETERMINE
f(y) AND K

The Hillebrand-Blumenfeld alleys can be used to determine the function f(-y) and the constant K. Consider a distance and a parallel curve which starts from

the same point:

o=f(yo) cosko lo= f(,yo) sino.

(10.1)

P

FIG. 16. Distance and parallel alleys in physical horizontal plane.

636

RUDOLF K. LUNEBURG

Distance Alley

tudes, Al and 02 (Fig. 17). We determine the ratios

sing4l

and

sink2 S2=

sinoo

sinfo

by measurements. From (10.2) it follows that

f
-Si
f(-Yo)

fA('o)
-SI
fAy)

fry) S-S2
f(Yo)

f(yu) - S2
f(Y)

or

FIG. 17. Distance and parallel alleys with a common point in a system of Vieth-Mueller circles.

From (9.2) and (9.3) it follows that

K (f) +I
4
f(-y) sinki

K
TAf 2 (YO+)1
4 (distance curve)
f(-yo) sino

1 f(Y) f(Yo) 1+S1 S2
2 fyo) f(T)/ S1+S2

and hence

A70) 1+S1S2+E(Si'-1)(S?-1)J'

(10.3)

fAfY)

S1+S2

and

K _f2(-yt-1I
4
f(-y) sinO2

K-f2(7)- 1
4 f(,yo) sinko

(parallel curve),

or

Since parallel and distance curves are in general not far apart, we may replace Si and S2 by an intermediate value sin4/sin4 0 =S. Then it follows from (10.3) that

f(Yo) S=msino

AY)

sin0

(10.4)

-J(,Yo_) =- + ' 2K

/ fey) sinki

f(lyo) sinok

4

f(,yo) sinou/ f(y) sino (10.2)

K /f -f2 (Yo) I

) sin 2

-

I =

sin42 fYo) -- +

4

\f(gyo) sino/

sinko f(-y)

A Vieth-Mueller circle of parallax y is intersected by the two curves at Pl* and P2 * of different bipolar lati-

i.e. the function logf(y) is obtained directly if a distance or a parallel alley is plotted in the coordinates y and log sino.
The experiments seem to indicate that the function logf(-y) in first approximation can be considered as a linear function of y. Then logf(ry)=- y+C or f(y) = 2e-7f, a being a new parameter of the observer.
The constant (K/4)f 2 (-yo)can be found from (10.2)by introducing the expression (10.3) in one of the Eqs.

Distance Alley s

Parallel Alley

FIG. 18. Tangents to distance and parallel alleys at the common point.

METRIC OF BINOCULAR VISUAL SPACE

637

(10.2). The result may be written as follows:

K -- f2 QYO)+1
4
K -f2(70) + 4

(S 2-1)' (sin 2 i-sin 2oo)i (S2 2-W1) (sin 20 2 - sin2 ko)l

FIG. 20. The subjective

correlation of Fig. 19.

-

This yields at the end point of the alleys

l--Kfl(yo)- 2 do,

I4I(

dy bi

1+K-f2(yo)

dO
-

b2

4

. dy

It is readily shown that 2(dqfi/dry)= b1 and 2(d42/dy)
=b 2 determine the intersection of the tangents of the two curves at P* with the y axis (Fig. 18). Thus,

t(hKe/4r)aft2io(ybo1)/.bI2f

can the

be used directly to find the two alleys start at the same

constant infinitely

distant point, we have f(,yo)=2, and hence the simple

relation

1-K)2 bi

(+K b2

11. METHODS OF MEASURING K AND f(y)
The alley experiments provide a convincing demonstration of the non-Euclidean hyperbolic character of the binocular visual space. But they are not too well suited for the problem of measuring the parameters of the metric. The mechanical construction has to be quite elaborate, if undesirable clues are to be eliminated that would invalidate the measurements. The setting up of a pair of alleys requires time, especially with new observers. Consistency is generally lacking at the beginning so that each new observer must go through a
series of preparatory experiments. In order to remove these difficulties, other methods

\dF

L

of measurement had to be investigated. The following method so far seems to promise the best results.
Three isolated light points P0*, P1*, P2*are given in the horizontal plane. P0 * and Pi* are movable on a fixed Vieth-Mueller circle of parallax y0, P2* on a second fixed Vieth-Mueller circle of parallax ty (Fig. 19).
In the (, q-plane, these points are represented by the two points P0 and Pi on a circle of radius po= f(,yo), and by a third point P2 on a circle pi=f('yi) (Fig. 20). Both circles are concentric to the origin. The task of the observer is to set the point P2* in such a position on its circle that the visual distance D(PoP2 ) is equal to the visual distance D(PoP1 ). This is carried out for a number of settings P0*, P* on the Vieth-Mueller circle yo.From (6.1) it follows that the distances are equal if
{2po 2[1-cos(0I-k0)] i
1+K-po2 4
[po2 + pi 2- 2poPi cos(k2- 0i)] ( 4 )_ (11.1)

i. e. if

-Po 4 Sin2 (02-

1+K-Pi2
Po 4
o)=Pi 1+-KPo2
4

X4 in"20 -

-Pi) -Po-)

(11.2)

R FIG. 19. Points on two Vieth-Mueller circles.

The quantities
1+K-p 2
Po 4
A =Pi 1+-KPo2
4

638

RUDOLF K. LUNEBURG

and

Consider four points Po*, Pi*, P2*, P3*. The first pair

B=6 1/P-o\ i - -P 1 \ 2=-P+O--P21

Piy \ PO)

P1 Po

(11.3)

is movable on a pair on a smaller

Vieth-Mueller circle circle of parallax yi>

yo, the ,yo. The

second task of

the observer is to place the point P3 *, for a given setting

are constants of the apparatus and of the observer.

of P0 *, P*, P2* so that

The quantities

D(Po, P1 ) =D(P2 , P3 ).

X=4 sin2'(4 1-40); Y=4 sin2'(42-0o) (11.4) This leads to the condition

are measured after each setting. It follows that

Y= AX-B,

(11.5)

so that a graphical plot of Y against X should yield a straight line. The experimental points are actually found scattered about a certain average straight line. The scattering may be considerable for a new observer, but it decreases rapidly with some training. It is interesting to note that young children display a remarkable degree of certainty and consistency in such observations.
From a suitable number of observations we obtain by the method of least squares the best straight line representation of the observation, and from the constant B the ratio

P1 fyi)

po f(Yo)'
and from A the quantity K f2 (,yo). Repetition of these experiments with other pairs of
Vieth-Mueller circles yields further ratios f(y+ 1)/f(yj), and thus, with f(O) = 2, a numerical tabulation of f(y) is obtained. Measurements such as these, as well as other observations, seem to indicate that a first approximation of the ratio f(-yi)/f(yo) depends only on the difference yl-70. This implies that logf(-y) is a linear function of y, and hence

f(y) = 2e7T,

(11.6)

where o-is an individual constant of the observer. The numerical value of o-characterizes the degree of
depth perception of a person. There is no doubt that considerable differences exist in the quality of depth perception; o-values ranging from 4 to 15 have been found.
The parameter can be determined from observations on a single pair of Vieth-Mueller circles. We have 2 sinh(o/2)(71-7o) (B)1, and hence, in sufficient approximation

2 sin2(0)3-4)
or where

K
1+-p

2

PO 4

2 )=-

2 sin-(0 1 -40)

P 1+-Kpo2

4

Y=AX

(11.8)

X=2 sin (Ol-4o) and Y=2 sin (q53-02)2).
A graphical plot of the observed quantities Y against the settings X should yield a straight line through the origin. If the two Vieth-Mueller circles are chosen far enough apart, the direction A of this straight line varies considerably with K. Hence, if is known, we find K from the relation*

A = ea

1+Ke 2 z'1
_._- l-o)
1+Kela'Vo

(11.9)

12. THE HOMOGENEITY OF VISUAL SPACE
The essential feature of the preceding experiments is that distance observations are made on sets of points that are confined to two fixed Vieth-Mueller circles, that is, to curves which are sensed as two subjective circles around the observer. A consequence of this restriction is that certain simple trigonometric functions of the observed angular differences become linear functions of each other, thus providing an elegant method of statistical evaluation. This applies even if the points do not lie in the horizontal plane or the same plane of elevation, provided they are confined to two different Vieth-Mueller torus surfaces, that is, to two surfaces sensed as two subjective spheres around the observer. In this more general case, spherical arcs ik, for instance the arc 12 between two points Pi*, P2* given by
cosQ 12 = cos01 cos42 cos(0 2 - 6i)+sini sinO2,

(B),
71-7O
If o. is known, the constant A can be utilized to calculate K. However, if the two Vieth-Mueller circles are not far apart, the variation of A with respect to K is not sufficient to provide a good measurement of this constant. A more effective arrangement is the following:

replace the differences 4)k- s in the horizontal plane. Hence, with X=4 sin2 2Qo1 ; Y= 4 sin21 0 02, in the
* This formula is based on the approximation fy) =2e. If this approximation is not accepted, the quantity K may be found from
A =L(Y-) I++Kf2(-l) f&' ) 1+ f2(yo)
if fy) is known from observations of the first type.

METRIC OF BINOCULAR VISUAL SPACE

639

A

B

C

FIG. 21. Horizontal cross-section of Ames' experimental room.

experiment of the first type, we should still get points X, Y scattered about the same straight line Y= AX-B.
And with X= 2 sin Qoi; Y= 2 sin'0 23, we should find points about Y=AX in the second experiment. This follows from the special form of the distance
function D(Po, P1):

2 l

D

-~ sink-(-K)--

(-K)l

2

C

(po2+p 2-2popi cosgoi) K" K I
1+-Po 2 1+-pi2
44

,
(t1Z.1)

More generally, this is true for any metric space whose distance function D(Po, P1 ) has the form

ID
FIG. 22. Rectangle of Fig. 21 mapped in the subjective plane.

be a solution of the partial differential equations

DDeeo2++DD0r20

+De 2 +Dr0

2

=n2 = n2

(p) (po).

(12.4)

The problem is therefore to determine those functions n 2(p) for which the differential equations admit solutions of the special form (12.2).
The answer is that this is possible then, and only then,
if n(p) has the form

C

n(p=) K1+K-p2

(12.5)

4

D(Po, Pi)=F[M(po, pi)+N(po, Pi) cosQoi] (12.2)

where M(po, P1) and N(po, P1) are symmetric functions

of p0 and pi.
Since the linearity of the graphical representation is one of the most consistent observations in these experiments, the question may be considered whether this

fact should be interpreted as an experimental verifica-

tion of the hypothesis of constant curvature. In other words: Are the homogeneous spaces the only Rieman-

't

nian spaces where the distance function D(P, Q) can be

written in the form (12.2)? Since the metric, defined by (12.2), is symmetric to
the origin, the associated line element ds2 can be as-

sumed to have the form

ds2=n2(p)(deY+dq2+dr2 )

(12.3)

so that the distance function D(to, qo, to; (, 7, ¢) must FIG.23. Horizontal cross sections of visually congruent rooms.

640

RUDOLF K. LUNEBURG

3 =2.

I FIG. 25. The domain of Fig. 24 after a congruency-transformation.

struct, to any given configuration, a metrically congruent one in any localization or orientation.
The original rectangular room in the Ames experiment may serve as an example. The cross section made with the horizontal plane is shown in Fig. 21, its counterpart in the , -plane in Fig. 22.
Consider the following hyperbolic motion:

-a-(a2+K

±)
-)2

a 2+2

7 (a2+ ±)X
K (a- a)'+ 77+ 2

(13.1)

FIG. 24. A domain in the physical plane and the correlated domain in the subjective plane.

i'= a+_-
K (-a)2+712+t2'

where C and K are constants. In other words, the space must be Riemannian space of constant curvature.
Therefore, the linearity of the relation between X and Y is indeed a direct indication of the homogeneous
character of the visual space.

which transforms median and horizontal planes into

13. CONGRUENT CONFIGURATIONS4

L i

A~~~~~~~_~A

There are a number of practical applications of the

theory of binocular vision which depend on the knowl-

edge of the numerical values of the parameters K and

the function f(y). We consider first the possibility of psychometric reproduction of visual sensations. Since in

I /

ZI --- A.

homogeneous spaces a given configuration can be

moved freely without metric distortion, we can con-

4 R. K. Luneburg, MathemzaticalAnalysis of Binocular Vision (Princeton University Press, Princeton, 1947); Metric Methods in Binocular Visual Perception,Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).

FIG. 26. The domain of Fig. 25 in the physical plane.

METRIC OF BINOCULAR VISUAL SPACE

641

FIG. 27. Illustrating difference

between psychometric representa-

tion and central projection.

'

themselves. Submitting the "room" A, B, C, D to this transformation, we find that all the points of the boundary move either to the right or to the left, depending on the parameter a. The physical counterparts of these visually congruent rooms are no longer rectangular but have the form shown in Fig. 23. They conform with Ames' experimental result, and can be used as a quantitative basis for the design of such rooms.
Next, we consider the problem of psychometric reproduction of landscapes, for example, a scenery which lies between a certain Vieth-Mueller circle y and physical infinity y= 0. The corresponding domain in the
I, -q, i-space lies between two concentric spheres, p=p(-y) and p2 (Fig. 24). Submitting this domain to a transformation (13.1) [a>2/(-K)1], we obtain a domain bounded by two non-concentric spheres as shown in Fig. 25.
The corresponding domain in the physical space is obtained by rotating the shaded area in Fig. 26 about the y axis. The entire original part of the space is condensed into this narrow shell. With the aid of (13.1) a relief reproduction of the original scenery could be constructed which is visually indistinguishable from the
original. II
If the inner boundary p= p(y) is sufficientlyfar away, the relief becomes so flat that it may be replaced by a painting or projected image on the outer screen surface, which is metrically congruent to infinity. In order to find the coordination of a point on this screen, we can use the Blumenfeld alleys since points on these alleys have a constant visual distance from the x axis. If the screen is chosen so small that it lies in the domain where the distance alleys bend away from their asymptotes, the image points depart considerably from the position of central projection (Fig. 27). This explains why the main feature of a landscape appears often disappointingly small compared with the original impression. Therefore in order to obtain a true psychometric projection of a scenery, we should project upon a curved screen designed in accordance with explicit mathematical principles. Furthermore, we should use a projection lens which possesses a strong amount of specified pincushion distortion.

14. PHENOMENOLOGICAL SIZE AND DISTANCE
Our last result is related to a well-known fact. The phenomenological size of physical objects which are moving away from us, does not decrease in proportion to the angle of view. The size seems to remain constant for a while until, at greater distances, it decreases rapidly.
Consider two points P0* and Pi* in the median plane and symmetric to the x axis (Fig. 28). We judge their apparent distance D according to the formula:

2

1

D

sin -(-K)I-

(-K)i

2

C

[2p 2(1-cos20)]1 2 p sino

Since

K 1+-p

2

K

4

4

'z

-y

sinO=

= tan-;

(a 2+nZf)f

2

(14.1)

and p= 2e-o-Yi,t follows that

2

1

D

sin -(-K)'--±=

(-K) i 2

C

4 tan-
2
eaY+ Ke-Yn

(14.2)

If z is kept unchanged, but the position x of the two points is altered, DIC becomes a well-defined function

L-

-

fl

-

P0

^

|

-

^

11This illusion can be accomplished to a remarkable degree, as for instance the so-called Habitual groups of animal life in the Los Angeles County Museum demonstrate. As a matter of fact, it is nearly impossible to judge the actual dimensions of the scenery.

"PI FIG. 28. Showing a line segment in the median plane.

642

RUDOLF K. LUNEBURG

the visual space but merely its coordination to the

physical space. This fact may be expressed by stating

that any optical instrument represents a certain point

transformation of the visual space. Except for trivial cases, these transformations are not isometric, and

therefore a metrically distorted view of external objects

is introduced. A surface which originally gave the im-

pression of a physically plane surface, may become apparently convex or. concave. In a theoretical investiga-

tion of this situation the theory of curved surface em-

FIG. 29. Showing relative apparent size of line segment of Fig. 28 as a function of its position along the x axis.

bedded into a hyperbolic space thus finds a significant application.
16. INCORPORATION OF MOTION IN THEORY

of yythat is, of the distance x of the two points from The artificial restriction of the freedom of observation

the observer. For small values of z we may write ap- which has been introduced at the beginning is not main-

proximately

tained in usual visual sensations. We move our head,

even our body, quite freely. This leads us to the problem

4 tan-

D

2

-= Z *

C ea+Ke-7

of incorporating such motions into the theory of visual
perception. Obviously, this problem is intimately ret~~~~14_la5t)ed to the general problem of visual sensations of
moving objects or of a moving observer. In order to

Thus D approaches zero if x->oo, unless K=-1, in which case a finite limit is reached asymptotically (Fig. 29). We conclude that constancy of size is maintained longer, the nearer to -1 the value of the observer's constant K is found to be.*

approach this subject, a psychometric function should be established which measures the sensation of velocity ofvisual objects. Certain simple observations lead to the conclusion that this function is most probably given by a function of the type

15. APPLICATIONS TO BINOCULAR OPTICAL INSTRUMENTS

c+v
w(v) = C log-,
c-v

(16.1)

The interpretation of the visual space as a metric space provides a theoretical basis for an investigation of binocular optical instruments, such as range finders, binocular field glasses, binocular microscopes, stereoscopic projectors, or even ordinary spectacles. Such instruments do not change the geometric character of
* The same results are obtained qualitatively without the assumption f(-y) = 2e-"y. Instead of (14.3) we have
D 2f(y) K
1+ 4f2(z)

where c is a certain constant representing the extreme velocity of eye motions, whereas v is the image of the physical velocity in the I, -q,c-space. This psychometric function is real valued only if v<c, which means that physical velocities v>c are no longer recognized as velocity sensations. This corresponds to the fact that a moving light point produces the sensation of a streak
of light if a certain velocity is exceeded. A space-time metric introduced to describe this situation mathematically will then be closely related to the space-time metric of special relativity.