MATHEMATICAL ANALYSIS OF BINOCULAR VISION
R. K. Luneburg
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This is an authorized facsimile of the original book, and was produced in 1970 by microfilm-xerography by University Microfilms, A Xerox Company, Ann Arbor, Michigan, U.S.A.
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MATHEMATICAL ANALYSIS OF BINOCULAR VISION
by RUDOLF K.LUNEBURG
a yyysss
/V -/^ 3 / /
Published for the HANOVER INSTITUTE BY PRINCETON UNIVERSITY PRESS
Princeton, New Jesrey 1947

Copyright 1947 RUDOLF K. LUNEBURG
Lithoprinted in U.S.A. EDWAi D S BROTHERS, INC.
ANN ARBOR, MICHIGAN
1948
149419

FOREWORD

I wish to express ray gratitude to the Dartmouth Eye Institute especially to Professor Adelbert Ames, Jr., and to Mr. John Pearson. The following pages would not have been written without their hospi¬ tality which I had the privilege of enjoying at the Institute and without the interest they accorded this mathematical theory.

In particular I have to thank Professor Ames for many stimu¬ lating discussions and demonstrations. His thesis that our sensations are related to the outside stimulus patterns but cannot be derived from them has been a guiding line in the following considerations.
To Dr. Anna Stein (Bureau of Visual Science, American Optical Company), assistant to Professor Ames, I am greatly indebted on the mathematical side. Her critical help has been invaluable in the dis¬ cussion and solution of the following mathematical problems, and last but not least, in the preparation of the manuscript.

I also wish to express my thanks to Mrs. Alice Weymouth for her work of typing the various drafts of the following text with its many mathematical formulae.

December 26, 19^6

Rudolf Luneburg Dartmouth Eye Institute Hanover, N. H.

CONTENTS MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Page

INTRODUCTION .

1

Section

1

PSYCHOMETRIC COORDINATION

1.1. Contrast of Sensations .

5

1.2. Psychometric Coordination in Case of One Dimension ....

5

1.3- Psychometric Coordination in Case of Two Dimensions .

6

1.4. Psychometric Coordination in Case of Three Dimensions ...

8

2

BIPOLAR COORDINATES

2.1. The Angular Coordinates 0^ 3

.

]_0

2.2. Modified Bipolar Coordinates y, <p> 0. 13

2.3. Simplified Relations between Cartesian and Bipolar

Coordinates.

]_£

2.4. Equivalent Configurations; Iseikonic Transformations ... 17 2.3. Significance of Iseikonic Transformations . 19

2.6. The Distorted Room Equivalent to a Rectangular Room .... 21

2.7. The Distorted Room Equivalent to a Rectangular Room (Analytical Derivation) . 23

2.8. Angular Coordinates for Observation with Head Movements . . 25

3

CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

3-1. The Metric of Plane Euclidean Geometry . 29

3-2. Two-dimensional Metric Manifolds in General . 30

3-3- General Coordinates in the Euclidean Plane . 31

3*4. Non-Euclidean Geometries in Two Dimensions . 32

3.5. The Metric of Three-dimensional Manifolds . 34

3-6. Non-Euclidean Geometries in Three Dimensions . 33

3-7- Geometries of Constant Curvature . 36

3.8. Constancy of Size. Rigid Transformations . 38

4

THE PSYCHOMETRIC OF VISUAL SENSATIONS

4.1. Assignment of Linear Size to Angular Differentials .... 40

4.2. Size Assignment by Observation with Fixed Head. 40

4.3. Size Assignment by Observation with Moving Head.

4.4. Relation of Both Methods of Observation.

42

4.5. Spherical Symmetry of the Metric.* *

4-5

4.6. Parallelism of Line Elements on the x-axis. 44

4.7. Vertical Rods on a Vieth-Muller Circle.

45

4.8. The Hypothesis M(y) = 1.

’

^

4.9. The Hypothesis of Constant Curvature.* * 43

MATHEMATICAL ANALYSIS OP BINOCULAR VISION

Section

5

DERIVATION OF THE HYPERBOLIC METRIC OF VISUAL SENSATIONS

5.1. Observation of Objects in the Horizontal Plane .

5.2. Observations on Vieth-Muller Circles .

5.5. Observations on Vieth-Muller Circles (continued) .

5-4. The Hyperbolic Metric of Visual Sensations and the Relativ¬

istic Metric of Space-Time Manifolds .

5-5- Applications of the Result .

6

GEODESIC LINES: THE HOROPTER PROBLEM .

6.1. Formulation of the Problem .

6.2. The Corresponding Problem in the c, rj, C Space.

6.5. Relation of the x,y-plane and the

r]-plane.

6.4. Geodesic Lines of the Horizontal Plane (Horopters) .

6.5- Frontal Plane Horopters .

6.6. Images of Geodesics in the £, r^-plane.

6.7. General Shape of Frontal Plane Horopters .

6.8. Vertex Curvature of Frontal Plane Horopters .

6.9. Frontal Plane Horopters in the Euclidean and Elliptic

Geometry .

7

THE ALLEY PROBLEM

7.1. Distance Curves .

7.2. Discussion of the Distance Curves .

7-7. Parallel Curves in General .

7.4. Parallel Curves (1st Type) .

7-5* Parallel Curves (2nd Type) .

7.6. Interpretation of the Parallelism of 2nd Type .

7.7. The Projective Map of Geometries of Constant Curvature . . .

7.8. A Method for Determining cr and f.

8

RIGID TRANSFORMATIONS OF THE HYPERBOLIC SPACE

8.1. General Statement of the Problem .

8.2. Hyperbolic Rotations .

8.3. Hyperbolic Reflections .

8.4. Inversions: Hyperbolic Translatory Shifts .

8.5. Special Group of Rigid Transformations for Design of

Distorted Rooms .

8.6. Numerical Calculation of Distorted Rooms .

8.7. Topological Discussion of Distorted Rooms Congruent to a

Rectangular Room .

8.8. Congruent and Equivalent Rooms . .

CONCLUSION .

Page
51 51 57
58 60
61 62 64 65
66
67 69 70
73
74 76 77 78 80 83 84
86
89 89 90 91
94 95
98 102
103

INTRODUCTION

0ur a^m ln the following investigation is to develop a mathematical theory oi visual perception. In particular we are concerned with binocular vision, i.e. with the perception provided by the concerted action of two eyes. We hope to dem¬ onstrate that certain observations analyzed from a general geometrical point of * view lead to a theory of binocular vision, which has some rather interesting con¬ sequences and.which gives a natural explanation to certain well-known phenomena oi visual optics. Before developing this theory in detail, we shall outline the general premises upon which our solution of the -problem is based.
1* i‘,T<3 recognize, by binocular vision, that we are surrounded by a threedimensional manifold of objects. These objects have, besides characteristic qual¬ ities of color and brightness, form and localization. In a visual sensation we thus are not only immediately aware of a distribution of colors and brightnesses but also of the fact that certain of these qualities are combined to unities namely, objects, which have a definite geometrical form and a definite localiza¬ tion in a three-dimensional space. We shall call this space the visual space. Our problem is to investigate its geometrical character, i.e., the qualities of for>rn and localization in visual sensations.

The concept of the visual space becomes clearer from the following consid¬ eration. .We can coordinate the "sensed" points in a particular visual sensation

to the points of a three-dimensional geometrical manifold. This of course can be done in many different ways. We shall call the result of such a coordination a geometrical map of the visual sensation. Consider, for example, the coordination which is the basis of the projection theory of binocular vision. A sensed point

s represented by.the intersection point of two projection lines which are drawn from lwo fixed points of the Euclidean space. The base points are the centers of rotation of the eyes and the projection lines the optical axes. 'We obtain by this

construction a Euclidean map of the visual sensation. However, we cannot be sure that the map.represents truly the sensed qualities of form and localization of the

objects* Thls would be the case

the apparent distance of any two sensed points

were aiways proportional to the geometrical distance of the associated points of the Euclidean map.

Clearly, this is not true. Astronomical objects like the sun or the moon are seen at finite distances; their sensed size is also finite and in no way pro¬ portional to astronomical dimensions. Even the sky itself gives the impression of a dome ol unite radius. It certainly does not introduce any special size sensa¬ tion comparable to Euclidean infinity in its relation to finite Euclidean size.

These considerations indicate our actual problem: To find a coordination

° .-

i'°into 1 a v]’-:jual sensation to the points of a geometrical manifold

suci that the apparent distance of any two sensed points is always proportional to

the geometrical distance of the correlated points. A coordination of this' type is

called a psychometric coordination. Whether or not such a coordination is possible is a psychological problem which requires a special consideration. Certain basic

psychological facts which we shall discuss in § 1 indicate, however that in the
case of visual sensations, a psychometric coordination is possible! Moreover, we

2

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

shall prove from these facts that the geometric manifold in which a psychometric map of visual sensations can be obtained Is uniquely determined. The geometry In this manifold then represents the visual qualities of form and localization in mathematical formulation. It establishes the possibility of measuring in the vis¬ ual space.
Our above discussion of the Euclidean map obtained by projection from two centers does not prove that the visual space is a non-Euclidean manifold. It only shows that this method of coordination yields a map which is not psychometric. There could still exist other coordinations of sensed points to points of the Euclidean space which lead to psychometric map3. However, such a coordination will be impossible, If the visual space should be non-Euclidean.
Let us illustrate this situation by the two-dimensional non-Euclidean man¬ ifold of points on the sphere. We can coordinate the points of the sphere to the points of the Euclidean plane and thus construct a plane map of the sphere. We may study the spherical geometry by the plane map and Its principle of construc¬ tion. However, we must not try to judge the actual 3ize or shape of objects on the sphere by the Euclidean size and shape of their images on the map. A map on which this is allowed is called Isometric and the coordination an Isometric trans¬ formation. In our example such a transformation Is imposoible: A sphere cannot be mapped isometrically to a plane.
Instead of the spherical geometry let us consider the geometry on a cylin¬ der or on a cone. In these cases it is possible to construct isometric maps. We also may say that a cylinder or cone can be applied to a plane without "stretch¬ ing" the material from which It is made.
The problem of Isometric mapping occupies a significant position in mathe¬ matics. In fact, we may consider this originally practical problem as the begin¬ ning of one of the roads which have led to the establishment of geometries differ¬ ent from the Euclidean geometry. In Gauss' theory of curved surfaces conditions were given for isometric transformation of surfaces onto each other. Riemann, after Gauss, generalized thbse results to manifolds of three and more dimensions and formulated their significance for the general space problem. The general re¬ sult is as follows : The geometry In a manifold can be derived from its metric, i.e., from a rule for measuring the size of small line elements. Two such mani¬ folds can be coordinated Isometrically to each other only under certain conditions which the metric of the first manifold must satisfy in relation to the metric of the other. If the second manifold is Euclidean, then these conditions give the answer to the question of whether an isometric Euclidean map of the first manifold can be constructed. Manifolds where the answer Is negative are called non-Euclidean; In this case no Euclidean map can be considered as true in all respects.
Suppose now that we study visual sensations by the Euclidean map obtained by projection from two centers, or, in fact, by any other Euclidean map. This means that we try to interpret visual observations by applying indiscriminately the relations of Euclidean geometry. If the visual space should be non-Euclidean, then any conclusion we draw from our results must be questioned and we must expect eventually to find contradictions with observations. Such contradictions then can be eliminated only by reinterpreting our observations in a non-Euclidean visual
space.
2. We can explore our environment in an entirely different way, namely, by physical measurements. With the aid of certain general principles the results of these measurements are combined mathematically and the environment Is recognized

INTRODUCTION

5

as a manifold of physical objects. Their qualities are principally different from the sensed qualities of visual perception. Instead of colors and brightness we obtain optical qualities referring to reflection or transmission of light waves. Instead of sensed form and localization we have measured physical form and physical localization in a three-dimensional space. We shall call this space the physical space and have to distinguish it carefully from the visual space. We assume the physical space to be Euclidean in what is to follow. This is certainly justified in the environment where sensory depth perception by binocular vision is effective.
Of course there is a certain relationship between the two spaces. This relation is established by the stimuli provided by the light which is emitted or reflected by physical objects. A small part of this radiated energy Is picked up by the dioptric system of our eyes and, by certain electrical and chemical dis¬ turbances, transmitted to the brain. The immediate and definite character of the associated visual sensation may tempt us to the belief that it is determined In all its qualities by these light stimuli. Indeed, if we subscribe to the projec¬ tion theory of binocular vision, we tacitly make this assumption, since we Identify physical and visual space. But even by considering these spaces as metrically different we can still believe in a necessary one-to-one correspondence of physical and visual space. A configuration of physical objects seems to create, by neces¬ sity, one and the same visual sensation for a given observer.
However, this belief does not stand a critical test. Actually, a visual sensation is the response of a living organism to physical stimuli. Thus we can scarcely hope to find the explanation of visual sensations and their sensed quali¬ ties in the complicated chain of physical events by which the organism is stimu¬ lated. We must take account of other factors which are given by the organism it¬ self and not by the stimuli. These are psychological factors determined by the purposes, expectations, and the experiential background of the observer.
By adopting this point of view we have to consider the following possibil¬ ity. Objects can be identical in certain aspects of physical form and localiza¬ tion but are seen as objects which differ in these aspects. Vice versa, two sen¬ sations can be Identical in all their qualities though related to different physi¬ cal objects. That this is true even in the realm of binocular vision Is clearly shown by some experiments carried out at the Dartmouth Eye Institute. A set of rooms with curved walls has been constructed; the walls are provided with curved window patterns. Every one of these distorted rooms gives the appearance of the same rectangular room, i.e., the same sensation is related to an Infinite set of physically different rooms. In a second experiment, perspective patterns are drawn upon a vertical board. The apparent localization of the board changes strikingly if the pattern is varied though physically the board is not moved. An Infinite set of apparently different localizations thus can be related to the same physical localization. ^We stress the point that in both demonstrations the ob¬ servation is binocular.
We conclude from the above experiments that It "would be futile to attempt to express the relation of visual and physical space in the form of a necessary one-to-one correspondence. The qualities of visual sensations are not uniquely determined by the physical stimuli. Since, on the other hand, we cannot consider sensations and stimuli as entirely unrelated, we are forced to the conclusion that only certain special elements of visual sensations are determined by the stimuli.

*The80 demonstrations have been designed by A. Ames, Jr. The mathematical analysis of the exper¬ iments has been given by A. Stein.

4

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The, essence of the following theory will he that, in fact, there exist immutable relationships but that they are confined to the assignments of apparent size to physical line elements. Consider two infinitely close luminous points in the physical 3pace. Our hypothesis is that their apparent relative distance is deter¬ mined by the physical coordinates of the two points. For these differential or primitive size sensations and only for these we shall assume a necessary function¬ al dependence upon the cooresponding differentials of the stimuli.
We stress the point that this relation of apparent size to physical qual¬ ities of localization is purely a mathematical relation. It does not Introduce the concept of physical causality and thus does not express sensory size qualities by physical units. In fact, it Is a relation of two geometrical manifolds to each other: The visual space obtained by psychometric coordination and the geometrical manifold which represents the physical space.
Our postulate Is clearly compatible with the fact that visual sensations are not uniquely determined by the stimuli. Indeed, an actual sensation requires in addition to assigning size to its differential elements the Integration of these elements to a unity. Thus arbitrary parameters of Integration are available. These parameters are chosen by the observer and, in the choice, he depends on his psy¬ chological condition.
The mathematical expression for the apparent size of a line element In terms of Its physical coordinates can be found by analyzing certain observations, i.e., by an inductive empirical investigation. On the other hand, this expression establishes a Riemannian metric for the visual space, namely, a rule for measuring the size of lnifinltely small line elements. Though referring to the infinitely small, it nevertheless already determines the general character of the visual space. It thus must give us the answer to the question whether or not the visual space Is Euclidean. Our answer will be that, in fact, the metric of visual sensa¬ tions is non-Euclidean. In particular we hope to demonstrate that the geometry of the visual space is the hyperbolic geometry of Lobachevski.

Section 1 PSYCHOMETRIC COORDINATION

1.1. It seems to be paradox, at first sight, to Introduce the concept of a metric in a manifold of sensations, i.e., the concept of measuring psychological qualities by coordination of numbers. Indeed, psychological manifolds, like heat sensations, or sensations of brightness, do not seem to be provided with a metric. We may say that sensation' Si is greater or smaller than sensation S2, but not how much greater or smaller. How then can we speak about a manifold of sensations of space which has a non-Euclidean metric?

It is, however, problematic whether greater or smaller is the only prop¬

erty of psychological manifolds which we may recognize. Consider, for example,

the sensation of pitch in sound perception. pare three pitch sensations

If trained, we are well able to com¬

Si, S2, S3

i.e., to judge whether the contrast (S2S3) is greater or smaller than the contrast (SiS2).

In space sensations a similar phenomenon may be observed. Let us consider

for example, the sensation of height. The contrast (SiS2), between two such sen-

,, g3 sations, is Interpreted as the size of the object between Sx and S2.

Obviously we are in the position to judge whether (SiS2) is greater or

.

smaller than (S2S3). We also may say that, by our sensations of height,

we assign vertical size independent of vertical localization.

S|

^ ^

We shall show next that recognition of greater and smaller and

recognition of greater and smaller contrast implies the existence of a

metric and that this metric is in essence uniquely determined.

1 ■ ^‘ Let us first, tor simplicity's sake, consider a one-dimensional man¬ ifold of sensations. Our problem is to coordinate numbers, x, to the sensations, S, of this manifold. This, of course, can be done in a great variety of ways; let

x = x(S)

(1.21)

be such a coordination, and let us also assume that the manifold of coordinated
numbers, x, 1orms a continuous manifold. However, we require that this coordina¬ tion shall be such that

and that if the contrast

x(S2) > x(Si) if S2 > Si
x(S3) - x(S2) > x(S2) - x(Si) (s3s2) > (S2Si)

(1.22)

Only if these conditions are satisfied can we consider the coordination as repre sentlng the characteristics 01 the sensations in question.

6

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

We 3how next that the "function" x(S) is In Its essentials uniquely deter¬ mined. Indeed, let X = X(S) be another coordination of the type (1.22). Then we may consider X as a mathematical function of x. This follows from the fact that to every sensation S there belongs one and only one number x and also one and only one number X. Consequently to a given number x there belongs one and
only one number X, l.e., X = f(x). Furthermore, from x2 > Xi it follows that
S2 > Si and thus X2 > Xi. Finally, from x3 - x2 > x^ - x± that (S3S2) >(S2Si) and thus that X3 - X2 > X2 - Xi. Thi3 means that the function X = f(x) must satisfy the conditions

f(x2) > f(xi) if x2 > xi

(1.25)

f(x3) - f(x2) > f(x2) - f(xO if x3 - x2 > x2 - Xi

whatever number Xi and x2 may be. From the last condition (1.25) and from the continuity of the function
f(x), it follows that

f(x3) - f(x2) = f(x2) - f(xi)

(1.24)

if x3 - x2 = x2 - xi.

This last result we may formulate as follows:

If x2 is the arithmetic mean

VWj!

x2 = ?(xi + x3)

(1-25)

of two numbers xi, x3, then the value f(x2) of the function f(x) at x2 is also the arithmetic mean of the values f(xi) and f(x3),
i . e .,

f(x2) = \ [f(xi) + f(x3)]

(1.26)

The only continuous functions f(x) satisfying this condition for any two values Xi, x3 are the linear functions

X = f(x) = ax + b

(1.27)

where a and b are arbitrary constants. The arbitrariness of the constant a means that no absolute size is given
but only relative size. (Change of unit of size.) The-arbitrariness of b means that the origin or the scale Is undetermined.
In these limits, however, we see that the psychometric coordination o_f numbers To sensations "is uniquely determined, if the sensations allow recognition of greater and smaller and of greater and smaller contrast.
We mention that this result can be obtained under very much weaker condi¬ tions. We need only to require that contrasts (SiS2) (S2S3) can be compared if S2 and S3 lie in the immediate neighborhood of Si.
1 5. In the case of sensation manifolds of more than one dimension, we have to proceed a little differently. Let us consider a two-dimensional manifold of sensations and coordinate these sensations S to the points P of a two-

PSYCHOMETRIC COORDINATION

7

dimensional manifold m of points. The geometric relations in the point manifold shall be determined by a quadratic differential.

da2 = edx2 + 2fdxdy + gdy2

(1-51)

where x, y are the coordinates
The above differential determines the distance of two

oi a point P and e, f, g functions of x, y.
*
is called the metric of the point manifold; it neighboring points P = (x,y) and P' = (x+dx, y + dy).

We assume again that it is possible to compare contrasts of sensations

l.e., to recognize whether the contrast (SiS{) of two sensations Si and S{ is

greater, and S2.

equal

to,

or smaller than the contrast

(S2S2)

of

two

other

sensations

S2

We now require the coordination of sensations S and points P to be such

R

disi_P

tha _ -fie distances day and da2 of the points Pi, p{ and P2, P2

give a true measure of the contrasts (SiS|) and S2S2), of two

pairs

coordinated "neighboring" sensations Si, Si and S2, S2.

R.

d-52

In other vopd3, ve require that

da?

eidx? + 2fidxidyi + gidy? = e2dx| + 2f£dx2dy2 + g2dyf = dal

(1-32)

if

(Sisi) = (s2s2)

and vice versa.

Only with this condition satisfied can the coordination of sensations and points be considered as representing truly the characteristics of the sensations in question. We shall call it a psychometric coordination.
Let us now assume that, for a given manifold of sensations, such a psycho¬ metric coordination is possible. Then we can show that, in essence, the point manifold m and its metric are uniquely determined.
Indeed, let M be another manifold of points and

dl2 = EdX2 + 2FdXdY + GdY2 its metric expressed In certain coordinates X, Y.

(1-33)

Let us assume that our given manifold of sensations can be coordinated psychometrlcally to M so that always

df! | dzi if (SiS{) | (s2si)

(1.34)

and vice versa. Since the sensations o are coordinated in one-to-one correspond-
ence to the points (x, y) of m as well as to the points (X, Y) of M, it follows
that X, Y must be functions of x, y:

x = x(*,y)

Y = Y(*,y) We conclude furthermore by (1.32) $nd (1.34);
♦In § 3 the concept of a metric is explained in greater detail.

(1-35)

8

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The inequalities
t eidx? + 2f1dx1dy1 + gidy? e2dxf + 2f2dx2dy2 + g2dyl

laply the corresponding inequalities

EidX? + BFidXidYi + GdY? = E2dX§ + 2F2dX2dY2 + G2dY§
whatever dxx. Uyi; dx2, dy2; and xi, yij x2, y2 may be. It is not difficult to see that this is possible only if the quadratic
differentials da and dZ2 are related by the identity df2 = ada2 or

EdX + 2FdXdY + GdY2 = a(edx2 + 2fdxdy + gdy2)

(I.36)

’•'here a is a constant.

pother words: By submitting the differential dZ2 of M to the transfor£a|_lon (JL • 35 j. the differential dq^ is obtained multiplied with a constant a. As ~e_ci"e, we may interpret thi3 appearance of an arbitrary constant a as indicat■Ln& the arbitrariness of the unit of 3ize In psychometric evaluation.
In general It Is not possible to transform a given quadratic differen¬ tial edx + 2fdxdy + gdz2 into another one EdX2 + 2FdXdY + GdY2 with arbitrarily chosen coefficients E, F. G. The result that, in the case of the above differen¬ tials dcr and dX2, such a transformation is possible, points to the fact that, geometrically, the point manlfolds rn and M are identical. The points of M are
, Identical with the points of m but characterized by different numbers X, Y instead 1 I»e., by different coordinates. It is clear that the geometrical characteristics of a point manifold must be independent of the choice of the coordinate system.

It may be mentioned again that the above result already follows If all the three sensations Sj, S2, S2 lie in an infinitesimal neighborhood of Si; recogni¬ tion of greater or smaller contrast thus Is required only if two pairs of sensa¬ tions Si, S{; S2, S^ are sufficiently near to each other.

1.4. A similar consideration for sensational manifolds of three dimensions leads to an analogous result. If a psychometric coordination of sensations to a point manifold is at all possible, then there exists only one such manifold. The geometrical distance of two neighboring points P and P' given by a quadratic dif¬ ferential

• da2 = gudx2 + g22dy2 + g33dz2 + 2gi2dxdy + 2gi3dxdz + 2g23dydz

(1.4l)

measures the contrast of the two associated neighboring sensations S and S'. The unit of the contrast size is the only Indeterminacy In this coordination.
The question whether or not a geometrical manifold actually fits a given manifold of sensations according to the above contrast requirements can only be answered by an empirical investigation. For example, as to the space sensations oL binocular vision, we have no right to assume a priori that the Euclidean space, l.e., a manifold with the metric
da2 = dx2 + dy2 + dz2
truly represents its characteristics.

PSYCHOMETRIC COORDINATION

9

In the following, we shall assume the possibility of psychometric coordin¬ ation o_ 7ioual space sensations to a geometrical manifold. We shall also assume cnat contrasts of space sensations can be compared. This means we may compare the sizes of two arbitrary line elements in space even if these line elements are not atiachen _o the same base point. We then know from the above result that there exists cmy one geometrical manifold which represents the characteristics of binoc¬ ular vision psychometrically. Our aim is to determine this manifold.

Section 2 BIPOLAR COORDINATES

In order to facilitate the mathematical investigation of our problem, we Introduce first a suitable bipolar coordinate system. Thi3 system allows us to characterize a point of the physical, space by three angles a, 3,9 instead of by three Cartesian coordinates x, y, z. We shall discuss in this section the rela¬ tion of these two coordinate systems.

2.1. The Cartesian coordinate system is oriented relative to the observer

as follows: His eyes are at the points y = — 1 of the y-axis, the x,y-plane is

z

his horizontal plane, the x,z-plane his median

plane. We assume first that the observer

views his environment without head movements

so that--to be more precise--the centers of

rotation of his two eyes remain at the points

y = — 1 of the y-axis. We now construct a

plane through the y-axis and through a point

P of coordinates x, y, z. This plane is

called the plane of elevation of the observed

point. Let 9 be the angle of elevation, i.e.,

the angle of the plane of elevation with the

horizontal plane. We draw next, in the plane

of elevation, two lines from the eyes to the

point P. a and 3 are the angles of these lines with the y-axis. As indicated in Fig. 1, we measure the angle a from the positive direction of the y-axis, but 3

from the negative direction.

One verifies easily that the relation between the linear coordinates

x, y, z and the angular coordinates a, 3> 9

given by the formulae

2 cos 9

y+ 1

x =

9 'cot a =

cot a + cot 3

\fix2 + z‘

/! , y =

cot a cot a

- cot 3 + cot 3 ’

.cot

3 =

l - y X2 + Z2

(2.11)

2 sin 0 Z = cot a + cot 3 ’ cot 9 = —z

The transformation of the x, y,
everywhere regular except on the y-axis. treats each plane of elevation alike, we of the horizontal plane. In thi3 case z = 0; 9 = 0, and thus

2

1 +y

X = cot a + cot 3 * cot a =

cot a - cot 3 y = cot a + cot 3

cot p

X

3, 9 space is
(2.12)

BIPOLAR COORDINATES

11

V

t—

0>

C* = 0,(3=TT

Fig. 2

We Illustrate this transformation of the x,y-plane into the angular a, 3-plane by deter¬ mining the region of the a, 3-plane which cor¬ responds to the right half-plane x = 0.

This region is bounded by the y-axis and an ideal curve infinitely far away. The values of the angular coordinates a, 3 on the boundary are given In Fig. 2. They determine immediately the boundary of the domain of the a, 3-plane which corresponds to the half-plane x = 0 (Fig. 3).

We conclude that the half-plane x = 0 is transformed into the Interior of the triangle shown in Fig. 3. To every point x > 0 there belongs one and only one point a, 3 in the Interior of this triangle. The transformation thus is regular in all interior points. As to the transformation of the boundary elements, we notice, however, a striking irregularity. The two eyes are transformed into two lines of the a, 3-plane. The sections -1< y < 1, y < -1, y > 1 of the y-axis, on the other hand, are compressed into 3 separate points (0, 0); (k, 0); (0 n) respectively.

The significance of the bipolar coordinates for the physiological aspects of binocular vision is easily understood. Let us assume that the two eyes are in the 'primary position," l.e., the optical axes are parallel to the x-axis. Then a point P with coordinates a, 3, 0 is projected onto the retina of the right e-e with spherical coordinates (a, 0) and onto the retina of the left eye with soherlcal coordinates (3, 0). Indeed, the planes of elevation 0 = const. Intersect the two retinae in longitudinal sections, l.e., great circles through the retinal points on the y-axis. The cones a = const, intersect the retina of the right e^e in lateral sections, l.e., circles of latitude around the y-axis. The cones 3 = const, give.the lateral sections of the left eye. (Fig. 4)

Fig. 4

12

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

We also notice that a line element (dx, dy, dz) attached to a point P will be projected as a line element (da, d0) onto the right retina and as (-dp, d0) onto the left retina. da, dp, d9 thereby may be found from dx, dy, dz by differ¬ entiating the equations (2.11).
The quantity d0 determines the vertical extension of the line element, the quantity da + dp., the horizontal disparity, and finally -?(dp - da), the horizon¬ tal extension.
As is well known in binocular vision, the sensation of depth is directly related to the horizontal disparity. Clearly, it would vanish if we let the dis¬ tance of the two eyes converge to zero. By the mechanism of our vision the two retinal images are seen as one fused image, provided that the horizontal disparity is not too great. The disparity in the horizontal direction, i.e., the difference of extension of the two images parallel to the y-axis, is perceived as a new space dimension of the line element, namely, depth, in addition to vertical and horizon¬ tal extension.
It is characteristic for our bipolar coordinates that the two Images of a line element in space (dx, dy, dz) must have the same vertical extension d0 on the retina. It i3, however, easily possible by artificial means, for example in a stereoscope or haploscope, to provide the eyes with individual images which have different vertical extensions d9i and d02. Mathematically this would mean offer¬ ing to the observer line elements from a four-dimensional manifold da, dp, d9i and d92. However, even if this vertical disparity d9i - dO2 is small enough that a fused image is seen, there is no sensation of a new space dimension. In other words, our mind refuses to digest the well-meant offer of four-dimensional line elements--either the two images are not fused or, if fused, the vertical disparity is completely ignored.
The bipolar coordinates a, P, 0 preserve their good physiological meaning if the eyes do not remain in the primary position but view objects directly by convergence. The angles a, p, 9 then determine the position of the two optical axes of the eyes. A point P then is projected into the center of the retina (fovea), the region of clearest vision. Also a line element (dx, dy, dz) attached to P is observed in the center of the retina. Its Images, however, are still characterized by the bipolar differentials (da, d0) and (dp, d0). d0 determines the vertical extension; -^(dp - da), the horizontal extension; da + dp, the hori¬ zontal disparity. Again da + dp is sensed as depth.
It is true that eye movements are of a more complicated nature than as¬ sumed above. If the optical axes of the byes are moved to converge at a point P which is not in the horizontal plane, then this movement Is accompanied by welldetermined rotational movements of the retina around the optic axis. This tor¬ sional movement of the eyes has the effect that retinal points on the horizontal section (0 = 0) in Fig. 4, do not lie, after the movement towards P, in the plane of elevation through P. In other word3, our differentials (da, d9); (dp, d0) do not determine directly the location of the images of a line element in a coordi¬ nate system solidly engraved on the retina. In such a system we would obtain two sets of differentials--(da*, d0i*) and (-dp*, d02*) for the right and left eye respectively--which, of course, can be’ found from (da, dp, d9) if the mechanism of torsion is known. It is-a problem which of these different coordinates are sig¬ nificant for the interpretation and localization of the original line element: the coordinates da, dp, d0 in cur bipolar system oriented on the base line between both eyes, or the coordinates (da*, d0i*), (-dp*, dO2*) in systems solidly at¬ tached to the retinae.

BIPOLAR COORDINATES

13

The following consideration seems to favor the assumption that the bipolar

differentials

give

the significant clues for interpretation and localization, if

the line element is viewed with both eyes. Consider

two line elements with d9 = 0, I.e., line elements

which lie in a plane of elevation. It is quite In¬

conceivable that the fact of their lying in a plane

/
/
/
/

of elevation will not be recognized directly in spite of the fact that none of the quantities d9i*. d02 In the solidly attached coordinate systems vanishes. If the same combination of line elements,

however, is viewed only with one eye, we can judge

Its orientation probably only by referring to fixed

coordinate lines on the retina. Binocular vision

Big. 5

thus gives us, in addition to depth perception, a greater certainty In directional localization,

namely, orientation relative to the base line of the two eye3. Exactly this fact.

however, is expressed in our bipolar coordinates.

For this reason we shall base our considerations in the following upon the bipolar coordinates a, 3, 0 and disregard torsional movements of the eyes.

2.2. Modified bipolar coordinates. For many purposes it Is advantageous to use a modification of the bipolar coordinates a, {3, 0 which expresses more di¬ rectly their physiological meaning.
We introduce the bipolar latitude

and the bipolar parallax

9 = i(P - a) V=n - a- 3

(2.21) (2.22)

For a discussion of these modified coordinates y, 9, 9 we may confine ourselves to the horizontal plane 9=0.
The meaning of yis clear; the angle subtended by the lines of sight at the point of convergence, P. Obviously, y may vary from 0 to n. For the inter¬ pretation of 9 construct the circle through P and the eyes R and L. (Vieth-Muller circle.) The x-axls is intersected at P0 and Q0 by this circle. One easily proves that the angle under which the arc P0P appears from either R, L, or Q0 is equal to 9 = -§(3 - a). This shows that it is justified to interpret 9 as determining the lateral position of a point P.
A line element (dx, dy, dz) attached to a point P can be characterized by the differentials dY, d9, d9 . From our former considerations it follows that 19 and d0 determine the lateral and vertical extension of the retinal Images and that dy measures the horizontal dis¬ parity. The latter is sensed as depth extension of the line element.

14

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

One proves readily that the relation between Cartesian coordinates x, y, z and bipolar coordinates y, 9, 0 is given by the formulae:

cos 29 + cos y

X

cos 0,

sin y

tan Y

2 /x2 + z2 x2 + y2 + z2- 1

y -

sin 2 9 sIn y ’

cos 29 + cos Y

z

sin 0,

sin y

tan 29 = 2y y/x2 + z2 x2 + z2 - y2+ 1
tan 0 =

(2.23)

the horizontal plane
cos 29 + cos y
X
sin y

y =

s in 29 sin y

2x tan y = x 2 + y 2 - ,1

tan

_ 29

=

—g2x-y—2

x -y +1

(2.24)

It follows that the curves y = const, are given by the Vieth-Muller cir¬ cles through the eyes:

x2 + y2 - 2x cot Y = 1

(2.25)

and the curves 9 = const, by the hyperbolae of Hlllebrand

-x2 + y2 + 2xy cot 29 = 1

(2.26)

through the eyes. These hyperbolae have the asymptotes

y = x tan 9

(2.27)

i.e., lines through the origin of direction 9. At any practical distance the hyperbolae 9 = const, coincide with these lines, as can be seen in Fig. 7* This demonstrates again the justification of Interpreting 9 as characteristic for the lateral position of a point P and d9 as lateral extension of a line element.

In order to investigate the regularity of the transformation (2.23), let us determine the domain of the Y, 9-plane into which the half-plane x = 0 is transformed. Fig. 8 shows the values of 9 and Y on the boundary of the half-plane x = 0. At infinity we have y = 0.

At the right eye:

y - 29 = rc

At the left eye:

Y + 29 = n;

This gives as domain in the y, 9-plane a triangle bounded by sections of the lines Y - 0, y - 29 = 7t, y + 29 = 71. The eyes are stretched into the lines

and

y - 29 = 71 Y + 29 = n

BIPOLAR COORDINATES

Fig.

15

16

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The remaining sections of the y-axis are compressed into three single points (0, 71/2); (0, -n/2) ; (n, 0). The transformation is regular at interior points but highly irregular on the boundary.

2.3. Simplified relations between Cartesian and bipolar coordinates. In many practical applications we can replace the relations (2.23) by simplified approximate formulae by considering the distance of the eyes and thus the bipolar parallax y as small. We may replace in (2.23) sin y by y and cos y by 1, and ob¬
tain

2 cos2 9 cos 9 x = ---—
Y
2 sin 9 cos 9 y
Y

2 /x2 -t z2 Y
x2 + y2 + z2

tan 9

y yrx~2 +: zi2

(2..31)

- 2 - cos2£ 9 si- n 9 ,

t. an

a
0

z

Y

X

These relations may be used safely lor objects which are far enough away from the eyes (x > 30, for example).
In the horizontal plane we havd

2 cos 9 x = -:- ,
Y
2 sin 9 cos 9 y -
Y

2x 2, 2 x +y

tan 9 =

I
x

(2.32)

The curves y = const, are now circles

x2 + y2 - 2y x

0

(2.35)

through the point x = y = 0 with centers on the x-axis. The curves 9 = const, are the straight lines> y = x tan 9
through the origin (Fig. 10).

(2.34)

BIPOLAR COORDINATES

17

This result allows us easily to determine the domain in the y, 9-plane into which the half plane x = 0 is trans¬ formed. y can assume all values between 00 and 0, and 9 all values between - tc/2 and + n/2. The transformation is regular at all Interior points x > 0.

The irregularity of the boundary coordination is illustrated In Fig. 11 and Fig. 12.

dot! = da2

or

d Yi = d y2

C\J CCL
II

dPi

d 9i = d9s

d0x = d02

dGi = d02

(2.41)

Su^h line elements give to an observer the same binocular clues on the retinae, i.e., the same horizontal disparity dy, the same lateral extension d9 and the same vertical extension d0. For this reason we call such line elements binocularly equivalent.
Since most external objects may be considered as configurations of line elenients--a curve in space is a one-parameter manifold of line elements, a surface in space a two-parameter manifold of line elements--we may extend the concept of equivalence to such configurations.

18

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Consider, for example, a curve in space. We may characterize it by three functions

a = a (t)

3 = 0(t)

(2.42)

0 = 0 (t)

■where t is a parameter. Another curve in space must be considered equivalent to this curve it its line elements da1, d3 1 , d01 are equal to the line elements of the original curve. This obviously is the case then and only then if

a' = a(t) + 6

P' = 0(t) + e

(2.43)

6 • = 0 (t) + X

6, e, X being arbitrary constants.
In a similar manner we obtain equivalent surfaces. In parametric form by three functions.

A surface Is obtained

a = a(s, t)

3 = 3(3, t) 0 = 0 (s, t) which determine a two-parameter manifold of line elements

(2.44)

da = a8 ds + at dt

d0 = 3S ds + 3t dt d0 = 0a ds + 01 dt

(2.45)

Another surface a1, 3', 9' ds equivalent to this surface if its line ele¬ ments da', dp', d01 can be coordinated to the original line elements such that

da' = da d 3' = dp d01 = d0

This leads immediately to the result that an equivalent surface must have the form

a1 = a(s, t) + 6 3' = 3(3, t) + e 0' = 0(s, t) + X where §, e, X are arbitrary constants.

(2.46)

BIPOLAR COORDINATES

19

We may integrate the above results from a more general point of view by interpreting the relations

a' = a + 5

0' = 3 + e-

(2.47)

O' =0 + X

as a group of transformations of the angular a, (3, 0 3pace. In fact, they repre¬
sent the simplest type of transformations of this space, namely, translatory shifts. If these transformations, however, are formulated in the Cartesian
y> z coordinates, an interesting and in no way trivial group of transformations of the physical x, y, z space is obtained. A general investigation of these transformations should give us many general results interesting for binocular vision. Any configuration submitted to such transformation will be seen by the same sequence of retinal images before and after the transformation. For this reason we shall call the transformations (2.47) iseikonic transformations. Math¬ ematically we recognize Immediately a characteristic feature of these transformations. A cone a = const, through the right eye is transformed into another cone through this eye. Similarly a cone 0 = const, through the left eye into another such cone. These two basic sets of cones thus are transformed without distorting but only interchanging the individual cones.

ii" we prel er the use of the modified coordinates y, iseikonic transformations by the relations

0 we can express

Y' = Y + x
9' = <p + a

(2.48)

0’ = 0 + X
where x, o, X are arbitrary constants.
Two configurations of objects such that the one can be transformed into the other by an iseikonic transformation are called equivalent configurations.

2•5• Significance of iseikon!c transformations. We ask the question: Are two equivalent configurations of objects indistinguishable in binocular vision? Inoeed, both can be observed by identical sequences of images on the retinae.
With regard to the above question, we can have two extremely opposite points of view. If we subscribe to the "projection theory" that our eyes are a kind of measuring device for the angular coordinates Y, 9, 0 and that the results of the measurements are directly transformed by our mind into space sensations, then equivalent configurations are sensed as different. However, if we believe the actual values of y , <p , 0 (i.e., the convergence of the eyes, the relation of the optical axes to the median and horizontal plane) are of no consequence and the sequence of retinal Images provides the only stimulus for sensations, then equivalent configurations are absolutely indistinguishable, even in binocular vision. The coserver depends upon intellectual clues such as perspective to choose the one or the other physical realization of equivalent configurations.
The second hypothesis can be supported by actual experiments. Ames (Dartmouth Eye Institute) has shown experimentally that to a given surface a whole

20

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

family of surfaces belongs which may he interpreted as the original surface if suitable perspective patterns are drawn on the "wrong" surface. For example, to a rectangular room there belongs a family of non-rectangular rooms which are binocularly indistinguishable from the original room. If the walls are provided with certain distorted windows which correspond to rectangular windows in the or¬ iginal room, then the visual sensation of the distorted room is that of a rectan¬ gular room. Indeed, the observer knows that windows are rectangular and thus in his sensation he chooses the physical realization of the impinging pattern which fits this notion.
It is possible to construct mathematically such a family of distorted rooms with the aid of iseikonic transformations. The result is a set of distorted rooms equivalent to the original rectangular room. Rooms which have been con¬ structed on this basis. Indeed, give, at least approximately, the above-described effect. For this reason we shall derive In the two following sections the mathe¬ matical equations for distorted rooms equivalent to a given rectangular room.
The experiments with distorted rooms seem thus to be fully explained by the above hypothesis that the actual values of 9, 0 and especially the value of the convergence y are Insignificant for the visual sensations. However, this the¬ ory leads us Into difficulties when we try to understand the psychological fact of judging size Independent of localization. Indeed, two line elements (dxi, dyi, dzi)and (dX2, dy2, dZ2) can have the same bipolar characteristics dy, dq>, d0, and still be of entirely different physical size. If these "local signs" dy, dq>, d9 are the only basis for visual sensations, then It Is hard to understand how we can judge the difference in actual size with such remarkable accuracy. Is this judg¬ ment obtained purely by former experience, or is it at least partly an element of direct visual sensation? In other words, is judgment of size only the result of training, or can we assume that it has developed from a seed which Is an Immutable part of primitive sensation of space?
The-fact that two line elements can have the same impinging characteris¬ tics dy, dcp, d0, but different apparent linear size, forces us to reconsider the significance of the absolute values y, 9, 0, especially of the convergence y. We shall. In §4, relate the apparent size ds of a line element dy, d9, d0 to the bipolar parallax, y, i.e., to the convergence of the eyes. We shall not attempt to explain this relation of size estimation to convergence physiologically, but shall consider it as a hypothesis necessary for the solution of our problem: To establish a metric for the manifold of visual sensations.
By the introduction of the convergence y as a significant element of bi¬ nocular vision, we have to conclude that equivalent configurations can not be truly indistinguishable. However, we shall see that our postulated relation of apparent size and convergence does not necessarily mean absolute localization In space. On the contrary, the special functional relation of both which we shall establish In §6 allows an even greater group of configurations metrically equiva¬ lent to a given configuration. This means that a group of transformations of the space exists which transforms a given configuration Into other ones with identical binocular characteristics. These transformations we shall call rigid transformations, and two configurations of this type, congruent configurations. Instead of a three-parameter group as the iseikonic transiormatlons, we shall -ind a sixparameter group of rigid transformations. Ames's postulate of the existence of a group of surfaces Indistinguishable from a given surface thus Is evqn more guaran¬ teed, if we Introduce into binocular vision the convergence y as a significant
factor.

BIPOLAR COORDINATES

21

In. §8 we shall derive a set of distorted rooms with Vails congruent to the plane walls of a rectangular room. The result will be, that these congruent rooms are nearly identical with the equivalent rooms to he derived in §2.6 and 3 2.7, though obtained from entirely different mathematical principles. We thus may consider equivalent rooms as a good approximation of congruent rooms. However, we shall see that the differences between both types are great enough to be easily observable. To compare the Impressions of both types of rooms as to the convic¬ tion of seeing an ordinary rectangular room can be considered a direct test of the two theories.

2.6. The distorted room equivalent to a rectangular room. We assume that the walls of an originally rectangular rooms are given by the planes
x = x0

y - -y°

(2.61)

Z = ±Zo

We consider a special iseikonlc transformations of the space represented by the relations

Y' = Y + T

<P' = <J>

(2.62)

0' = 0

where t is an arbitrary constant. The plane walls of the original room are trans¬ formed into equivalent curved surfaces. Any pattern drawn on these plane surfaces is transformed into an equivalent pattern on the, curved surfaces and will be seen by the same binocular characteristics dy' = dy; dtp' = d<p; d0 ' = d0 as the original pattern. The only difference is the absolute value of the convergence of the eyes. 12 2- special part of the plane walls is observed with an angle of convergence y then the corresponding part on the equivalent curved wall is seen with an angle' Y1 = Y + t. The assumption that the convergence of the lines of sight is immater2ai for binocular space sensation then leads to the consequence that the observer sees the equivalent curved walls as plane if a suitable pattern on the walls in¬ duces him to this interpretation.

Fig. 13

Before we derive analytic expre3-

B

sions for the curved walls, let us con-

sider a simple topological method which

allows us easily to determine the general

shape of these walls. We determine for

this purpose the curves in the y, 9-plane

which correspond to the rectangular cross

x0

section of the original room with the hori¬

zontal plane. The front wall x = x0 be¬

comes a curve symmetrical to the y-axls

which reaches infinity (y = 0) at the

points cp = ± n/2. The side wall y = yQ

:

may be considered as a line connecting a

point y0 > 1 of the y-axis with a point

at 00 for which 9 = 0. Consequently, its

22

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

image in the y, 9-plane is a curve from the point (0, x/2) to the point (0, 0). This consideration leads to Fig. 14, where the shaded region corresponds to the interior of the rectangle of Fig. 15*
We now submit the y, 9-plane to the transformation y' = y + x. Let us first assume that x < 0. The curve ABCD is shifted to the left by this tranalormatlon and thus is located in the basic triangle as shown in Fig. 15* The front wall extends from B to 00 (y = 0) and reaches this line at a value of 9c* < 71 /2 • Consequently it must be a curve of hyperbolic shape in the x,y-plane, a curve which is symmetric to the x-axis and approaches °° asymptotically with the angles ± 9c* where 9c* < 7t/2. The side walls go from B (or C) directly to y = 0, and ^ reach it at 9-values smaller than n/2 but greater than the values 9c* of he front wall. This means that the side walls also give hyperbola-shaped curves which

Fig. 16

Fig. IT

BIPOLAR COORDINATES

25

approach °= with an asymptote steeper than the asymptotes of the front wall. From these considerations it follows that the equivalent room must have a cross sec¬ tion with the horizontal plane as illustrated in Fig. 16. The analytic investi¬ gation shows that the curves are in fact true hyperbolae.

We consider next the case t> 0. Now the curve ABCD in Fig. 14 is shifted

towards the rh

and located in the basic triangle as shown In Fig. 17* The

extension of the front wall beyond B goes di¬

rectly to the upper or lower boundary line and

\

thus in the x,y-plane to one of the eyes. It

goes through the other eye if extended beyond

C. Thus it must be an elliptically shaped

curve symmetric to the x-axis and passing

through the eyes. The upper side wall goes to

the upper boundary line and thus to the left

eye in the x,y-plane. Its extension beyond B

reaches y = 0 at a negative value of 9<= > -x/2.

This leads to the equivalent room whose cross

section with the horizontal plane Is shown In

Fig. 18

The analytical treatment will show that the curved walls really intersect the horizontal plane in an ellipse and two hyper¬ bolae .

2.7. The distorted room equivalent to a rectangular room. (Analytical derivation) Since the dimensions of the room are quite large compared with the distances of the eyes, we shall use the simplified relations (2.31) for our pur¬ pose.
From these formulae it follows that the transformation
y' = y + x

9' = 9

0' = 0

y, In the

<p, 0 space may be written in Cartesian coordinates as follows:

y/ X ' 2 + Z ' 2 x'2 + y'2 + z'2

/2 ,

2

\f X + Z

2 ,

2 ,

2

x +y +z

(2.71)

y
/x*2 + X'2
z'
x1

y
—
/2 , 2 /x + z
z X

(2.72)

24

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The last tvo equations may be replaced by

y

We now Introduce y'

z'

z

X' X

(2.73)

y

z

x' and z' = — x' In the first equation, and obtain

x

2 ,

2 ,

2

/ , = 1 + ixr x + y + z

2

2

2

X +z

Combined with (2.73) this leads to the formulae

(2.74)

x' 1 + I. 2

x 2 + y2 + z 2 /x2 + z2

y

y

2 ,

2

X2 +

+ Z

+ I , 2

2

1

/X + Z

(2.75)

z 1

2 ,

2

2

X +y +2

i +1

X 2 +, z 2

between Cartesian coordinates.
The inversion of these formulae Is simply obtained by replacing x by - x, 1. e., we have

x
. 2 -f z 2
+ z 72

y

y

_T

+

+ z

2 — X' 2 + z' 2

(2.76)

z 1

Z = -3-^

r x' + y'2 + z'

' y + 2

x-2 z's

The last equations give us Immediately the equations for the curved walls of our distorted room. Indeed, since the plane walls of the original room are given by the planes x = Xo, y = yo, z = z0, where x0, yo, z0 are constants, it follows by (2.76) that (we use x, y, z instead of x', y', z') the equivalent walls are determined by the surfaces

BIPOLAR COORDINATES

x =

1

_

2
1 x

+

2
y

+

2
z

2 r2

2

s/X + Z

A y =
/o Vv

X x2 + y2 + z2 ’ 2 yA x2~7 +— z2-

25
(2.77)

Z = Zn

T X2 + V2 + Z2 1 --

2

/“ 2

2

/X + Z

with t as arbitrary parameter.
The cross sections with the horizontal plane (z = 0) are given by the curves:

(2.78)

One recognizes immediately that these curves are conic sections, namely, hyper¬ bolae if x < 0, and ellipses (front wall) and hyperbolae (side walls) if x > 0 in agreement with our former results.
We remark in general that each plane of elevation z = x tan 9 is inter¬ sected in conic sections by our surfaces. Furthermore: The front wall Inter¬ sects the median plane y = 0 in a conic section, namely

Finally we remark that the side walls are surfaces of revolution

y = Jo

2 \ + y

with the y-axis as axis of revolution.

p = /x2 + z2

(2.79) (2.791)

2-8. Angular coordinates lor observa11on with head movements, By movements of the head we are in a position to view the neighborhood of any point, not only in d: rect vision but also in symmetrical convergence of the eyes. We
assume that the head rotates about a center of rotation so that the eyes are moved on a sphere around this center. The base line of the eyes loses its significance for directional orientation. It is replaced by a reference line given by the position of our shoulders, and we can assume that this refer¬ ence line remains in our consciousness if we move our head. Similarly we are conscious of the position of the horizontal plane normal to the direction of the gravitational force.
For this manner of observation we introduce suitable systems of Cartesian and angular coordinates.
The x,y-plane shall be the horizontal plane and the y-axis shall coincide with the direction of the'shoulders.

26

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

We place the origin of the coordinate system at the center of rotation of our head.
We replace this Cartesian system by an angu¬ lar system as follows. We determine the plane of elevation of a point by the plane through P and the y-axis. Let 0* be its angle with the horizontal plane. We draw then In this plane the line which connects the origin 0 with P. We characterize this line by its angle (90 - 9 *) with the y-axis. The angles of elevation 0* and the lateral angle 9* then correspond to longitude and latitude on a sphere with two poles on the y-axis. The distance OP of a point from the origin finally is characterized by the parallax Y* at P with the eyes as basi3. Accord¬ ing to our assumption, we consider the lines of sight RP and LP as always symmetrical to the radius vector OP.
The relations of the Cartesian coordinates ‘x, y, z to the angular coordinates are simple. Let d be the distance of the base line RL from the center of rotation 0. Then
OP = d + cot Y* /2

x = (d + cot y* /2) cos tp* cos 0* y = (d + cot y * /2) sin 9 *

(2.81)

z = (d + cot Y * /2) cos 9 * sin0*

A line element (dx, dy. dz) attached to a point P, can be expressed in terms of our angular di f f erentials dy*, dcp*, d0 *. These differentials determine at the same tine the si gnificant characteristics of the retinal images of the line element, namely, t disparity dy* Interpreted as depth and the lateral and vertical extensions d<? + and d6 *.
We may assume that the base line RL of our eyes remains approximately horizontal if we move the head according to a habit established in the past. Thus we may call the disparity dy* also in this case the horizontal disparity.
The assumption of a base line which remains horizontal Is, however, not essential for our theory. 9^0 existence of a well-established habit we can cer¬ tainly expect, since otherwise repeated fixation of the same line element would result in erratic judgment of depth In accordance with different disparities dy*.
'If a given configuration of objects is ob¬ served with the head In fixed position and then with moving head, then it is not self-evident that the two Interpretations are identical. Indeed, one can easi¬ ly demonstrate that this Is not the case. For this purpose we construct a number of marks (for example, pins) arranged at equal distances on a Vieth-Muller

BIPOLAR COORDINATES

27

circle through the eyes. If these pins are observed with the head in fixed posi¬ tion, they give the impression of being arranged on a circle with the observer at its center. This impression does not remain if the pins are viewed with moving head.
On the other hand, let us consider two configurations of points which are physically entirely different, namely

= (d + cot Yi/2) cos

cos 91

y j* = (d + coty1/2) sincp^

(2.82)

and

zp = (d + coty^/2) cos 9^ sin 9 ^

cos 2cp^ + cos y i

*1

sin y ^

cos 0

sin 2<p1

siny

cos 2 + cos yi

zi

sinYi

sin 9 i

(2.8J)

where the quantities y i, cp^, 0 ^ are in both configurations the same.
We observe the configuration (2.82) with moving head and symmetrical con¬ vergence of the eyes, but the configuration (2.85) with the head in fixed position and asymmetrical convergence of the eyes.
We shall adhere in the following jto the hypothesis that the observer is led to an identical sensation in both configurations and that he interprets both configurations as identical.
This hypothesis may be tested easily enough by experiment. For example, we can construct a network of wires obtained by rotating the Vieth-Muller circle in Fig. 21 around the base line of the eyes. The result is a network on a torus surface as illustrated in Fig. 22. According to our hypothesis this network would appear, if observed with fixed head, as a spherical network with the observer as center. The vires indicate the circles of longitude and latitude on this sphere. The same sensation would be obtained if an actual spherical network (Fig. 25) is
observed with moving head so that the individual parts of the network are seen in symmetrical con¬ vergence .
The above-formulated principle seems to con¬ tradict, at first sight, our common experience. It certainly implies that from the same physical con¬ figuration of objects different sensations may be ob¬ tained, depending upon the manner of observation: with head in fixed position or with moving head. Moreover, since head and eye movements may be com¬ bined in an Infinity of variations, we must expect that the same physical configuration can lead the observer to an infinite variety of sensations. Still,

28

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

we are convinced that all these differ¬ ent views are associated with the same and unchanged environment.
Is this conviction the result only of experience, i.e., correlated to a judgment of probability based on repitition of similar observations in the past? This may be the explanation, but we are by no means forced to consider this the only possible explanation. It is possible that all these different interpretations have an invariant ele¬ ment in common, and that this invariant element is the criterion which gives us the conviction of seeing the same exter¬ nal configuration. We may compare this with constructing two different maps of the same configuration by using two different principles of mapping. In fact, we may describe an interpretation of a configuration of objects as a coordination or a mapping of sensations to the Euclidean screen of our intuitive mathematical thinking. To formulate mathematically the invariant element which both Euclidean maps have In common--namely, the same non-Euclidean metric relations--will be one of the subjects of the next sections.

Section 3 CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

3-1 ^ie 11161 ri c of a. man if old of _ point3 is defined as a rule for determln-

ing tne size of objects in this manifold. It Is a significant mathematical fact

that it is sufficient to formulate this rule for small objects, indeed, "infini¬

tesimally" small objects. We illustrate this by the example of the Euclidean

plane. We introduce Cartesian coordinates x, y and consider two points Pi = (x, y)

and P2 = (x + dx, y + dy) where dx and dy are the differential increments of

the coordinates x, y. The line element ds connecting

y

the two points then is characterized by the differential

(dx, dy), and its size ds Is given by Pythagoras' theo¬ rem :

ds = \/dx‘ + dy

(3-11)

We also may say that the size is determined by the quadratic differential

ds = dx2 + dy‘

(3-12)

which again is an analytical formulation of Pythagoras' theorem. This quadratic differential (3-12) is the mathematical expression for the metric of the Euclidean plane. By purely analytical methods one is in the position to derive the theorems of Euclidean geometry from the fundamental differential (3.12). It is clear that this fact is by no means self evident, since it means that a metric (3.12) refer¬ ring to points which are infinitely close also determines metric relations of points which are far apart. We may consider the differential (3-12) as an Incon¬ spicuous seed out of which we may develop the whole organism of Euclidean geometry with all its variety of relations.
To illustrate th is, we 3how how Intimately the measurement of angles is related to the measureme nt of size through the quadratic differential (3.12). Let
us consider two line elements PXP2 and PiP3 attached to the point Pi. We denote their differential coordinates by (dx, dy) and (6 x, 6y). Then it follows that the. angle included by the two line elements is given by the formula:

_dx 6x + dy fry 003 u = yfa* + dy2 . /6x2 + by2

(3-!3)

Fig. 25

The expression dx • 5x + dy • 6y is called the mixed quadratic differential associated with (3.12). Hence the angular metric follows directly from the original metric of size.

The step from the infinitely small to the finite is represented by the problem of geodesic lines, i.e., by the problem of finding among all curves connecting two given points P0 and ?1 the special curve which

30

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

has

the

shortest

length. The length of such a summation of the size of tegral

connecting curve is obtained by its line element, i.e., by the in-

J S = /dx2 + dy2

(3*1^0

and our problem means to determine among all curves connect¬ ing P0 and Pi one for which this integral assumes a minimum value. This problem

J /dx2 + dy2 = Minimum

(3.15)

Fig. 26

can be solved by purely analytical methods, and leads, of

course, in our case. to the Euclidean straight line between Po and Pi.

^.2. Let us next consider a metric manifold of two dimensions in general. We characterize the points P of the manifold by coordinates x, y in a suitably chosen coordinate system. The size d3 of a line element connecting two neighboring points (x, y) and (x + dx, y + dy) is given by the general quadratic differential

ds2 = Edx2 + 2Fdxdy + Gdy2

(3.21)

where E(x, y), F(x, y), G(x, y) are given functions of x, y.
In the above example of the Euclidean plane we have E=G=1, F = 0.
2
If the manifold is a sphere of radius one, we have E = 1, F = 0, G = sin x so that

ds2 = dx2 + sin2 x • dy2

(3.22)

The coordinates x, y in this case are related to the latitude cp and the longitude 0 on the sphere, namely:

x = 7l/2 - 9
Y = 9

(3.23)

The geometry in our manifold can be developed by purely analytical methods

from the differential (3.21). For example, the angle co included by two line ele¬ ments (dx, dy) and (6x, 6 y) attached to the same point (Fig. 25) is now given by

the expression

F,dx6x + F(dx5y + 6xdy) + GdySy_

fb.24)

cos a) = yEdy,2 + 2Fdxdy + Gdy2 • /e&x2 + 2F6x6y + G6y2

and thus directly related to the basic quadratic differential (3.21). From the infinitely small we go to the finite by the problem of geodesics: To find, among all curves connecting two points Po,Pi, the one t op which the length

S = f v^dx2 + 2Fdxdy + Gdy2

(3-25)

assumes a minimum value. These curves are called Geodesics; they have a similar significance in our
general metric manifold as the straight lines in the Euclidean plane.

CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

31

We mention another property of Geodesics which Is significant for a prob¬ lem in space perception treated later on. It is possible to relate the concept of parallelism of line elements which are not attached to the same point to the quadratic differential (3-21). Consider a line element attached to a point Po-
We wish to transfer this line element from P0 to another point Pi along a given path C, but in such a way that it does not change its direction. The analytic conditions for this parallelism of two line elements (Parallelism of Levi-Civita) can be shown to be directly related to the basic quadratic differential (3-21), i.e., to the metric of size. Without formulating this condition here, we only *-X mention the fact that the line elements of geodesic lines are always parallel. This means the result of constructing a curve by attaching line elements to each other without change of direction Is again a geodesic line, i.e., the shortest connection between two of its points (Fig. 28). This is,y of course, quite clear In the Euclidean plane, but it is true also in non-Euclidean geometries charac¬ terized by other metric differentials.

3*3* The above considerations illustrate the basic ^ significance of the metric differential (3.21) for the
geometry of a manifold of points. We may call it the "cede script" of this geometry. It allows us to replace geometrical relations by relations of numbers and geometrical methods by analytical methods. And in the end this is based upon the Cartesian idea of coordinating geometrical elements, namely, points, to numbers by introducing a coordinate sys¬ tem. Obviously, a certain arbitrariness is revealed here. The coordination of numbers to points can be dene in a great variety of ways. Let us illustrate this situation again by the Euclidean plane. If, instead of Cartesian coordinates, we choose, for example, polar coordinates, then the same point of the plane receives an entirely different pair of numbers. There seems to be an almost unlimited range of different coordinate systems which we may introduce in cur Euclidean plane. Mathematically this is expressed by the fact that we may introduce new coordinates ^, rj by a transformation

X = f U, R )

(3-31)

y = sU, t))
of the original Cartesian coordinates x, y into new coordinates Jj, tj . (The func¬ tions f and g are completely arbitrary.) A point P of the Euclidean plane then is located In a system of curved coordinated lines instead of a system of rectilinear
lines (Fig. 29). By submitting the quadratic differential (3-12) to the transformation (3-31) we obtain again a quad¬ ratic differential, but of a more general type

ds = Ed£2 + 2Fd£dr, + Gdr]2

(3.32)

where the functions E(£, r] ) , F(£, rj), G(5 , rj) are given by

Fig* 29

52

MATHEMATICAL ANALYSIS C? BINOCULAR VISION

E = f£2 + g^2

(3-33)

F = f^fT; +

G = fn2 + Sr)
This shows that the differential (5-12) may appear in an infinite number of mathematical variations (5*52), depending upon the coordinate system chosen for assigning numbers to the points of the Euclidean plane. In polar coordinates
r = £, 9 = 7), for example, we have

ds' = d£2 + £2 dr,2

By cnoosing the bipo]ar coordinates explained In 2.5, we have, letting

ds2 = 4

Y = £ > <p = ’I •

2 2 s in n cps
Q- +

dE,. drj

dr]'

Obviously the relations of the Euclidean geometry itself cannot depend upon the special choice of the coordinate system and thus upon the special analytic
form of the differential (5*12). Hence we conclude that all 'these variations (5-32) must have an element in common which expresses the fact that the geometry developed from any of them is Euclidean. This means, for example, that the angle <o between two line elements characterized by its differentials in a £, r, system: d£, drj; 6 £ , 6 r], and computed by the formula:

003 “

Ed£&£ + F(d£6; + dr]5£) + Gdr]6r]
-F 3-3*0
>/Ed£2 + 2Fd^dr] + Gdr)2 /e6£2 + 2F5E,bri + Gbr)

based upon the quadratic differential (5-32) must be the same as the angle found by (5-13).
Furthermore, the geodesic lines of the differential (3-32), i.e., uie curves for which the Integral

Ed£2 + 2Fdcdr] + Gdr]'

assumes a minimum, must be given by functions vhlch In the curved coordinate system lead to the straight lines of the Euclidean plane.
The property which all the quadratic diit erentials (5.52) have in common obviously is that they are obtained from (5.12) by a transformation of the coordinates. Vice versa, we may say that, by a suitable transformation of the coordinates, the line element (5-32) can be transformed Into the normal form (5.12).

5.4. The above consideration leads us to the ques¬ tion: Suppose a geometrical manifold is given, and, after choosing a coordinate system in it, we obtain a metric differential

CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

33

ds2 = Ed%2 + 2Fd!-dr) + Gdrj2

(3-41)

where E, F, G, are known functions of % and rj . Is it then always possible to find a transformation

x = ffe , tj )

(3.42)

y = g(E , rj )
of the coordinates such that in the new coordinate system the line element (3*41) assumes the Euclidean form ds2 = dx2 + dy2? Since the transformation (3*42) is the analytical expression for drawing in the Euclidean x,y-plane a map of the given metric manifold (3*41), it then would be possible to obtain a plane isomet¬ ric map of the given two-dimensional manifold. •In other words : Euclidean meas¬ urements of size on the plane map would give the size of objects in the original manifold (3.4l).
The answer to our question, however, is negative: Unless the functions E, F, G -satisfy a certain mathematical condition, the desired transformation is impossible. Indeed, if the functions E, F, G are given, then the equations (3*33) represent a system of three partial differential equations of first order for two unknown functions f(l-, rj ) and g(£ , j] ) . The system, obviously. Is overdetermined and in general will have no solution. A solution can be expected only If the functions E, F, G satisfy a certain mathematical condition. If we interpret the geometry associated with a quadratic differential

ds2 = Edx2+ 2Fdxdy + Gdy2

(3.43)

as the geometry on a curved surface in the three-dimensional Euclidean space—as we always can-then the formulation of the above condition is given by Gauss' theorems egregium: The Gaussian Curvature K of a surface can be derived from the coefficients E, F, G of the metric differential (3.43). The curvature K of the Euclidean plane is zero, and, as a geometrical quantity, must be zero for any choice of the coordinate system. This means, for a differential (3*43) which has been obtained from the Euclidean differential ‘ds2 = dx2 + dy2 by transforma¬ tion of the coordinates, that the result of introducing the functions E, F, G into the Gaussian expression for E mus l be identically zero. Also the reverse is true: If the i unctions E, F, G satisfy the condition K = 0, then a transformation (3*42) can be found which transforms the differential (3.43) into the normal Euclidean form ds2 = dx2 + dy2. A line element (3*43) of this special type is called a Euclidean line element.
It is not difficult to give examples of non-Euclidean line elements. The geometry on a sphere of radius one is characterized by the metric differential

ds2 = dx2 + sin2 x dy2

(3.44)

where <p = n/2 - x is the latitude and 9 = y the longitude on the sphere. The Gaussian curvature K has the constant value K = + 1, and it is impossible to trans form this line element into the Euclidean form (3*12) for which K = 0. The nonEuclidean geometry developed from (3*44) is called the elliptic geometry.

Another example is given by the quadratic differential

ds2 = dx2 + slnh2 x dy2

(3.45)

34

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

and I3 distinguished by the fact that its Gaussian curvature has the constant

value K = -1. geometry.

The corresponding special non-EuclIdean geometry is the hyperbolic

The three geometries based on the line elements (3.12) (3 lit) and (3 4n)

™it6d

££ £2il£tant curvature, since the Gaussi^ curvature^K Is ^

t‘“ ' Wt“ay aYect that thase geometries are of special Interest as compared
great variety of other non-Euclldean geometries In vhlch K Is variable Irom point to point.

By introducing other coordinates in the manifolds of constant curvature we obtain a great variety of diflerent forms of the associated quadratic differentials. We mention Riemann' s normal form

ds 2

di/ + dr/

+ | k(52 + p2) 2

(3-46)

by which these geometries can be represented simultaneously K being the constant Gaussian curvature. For K = 1, 0, -1 we obtain the elliptic Euclidean, and hyperbolic geometry respectively.

3-5- Geometrical manifolds of three dimensions may be treated in a similar
manner. The solid Euclidean geometry, for example, can be derived from the quad¬ ratic differential

ds2 = dx2 + dy2 + dz2
The Euclidean angle gj between two line elements (dx, dy, dz) and (5 x, 8y, 8z) is given by the expression

(3-51)

_dxSx + dySy + dz5z_
cos (i) /dx2 + dy2 + dz2 • /6x® + 6 y2 + Sz2
z

(3-52)

The problem of finding the shortest connection of two

points P0 and Pi introduces finite elements into the

geometry, namely, the geodesic lines. On these geode¬

sics the integral

/ /dx2 + dy2 + dz2 assumes a minimum
value; the geodesics of the Euclidean space are, of course, the straight lines in space.

tesimally distant points quadratic differential

We may characterize in general the points of a three-dimensional manifold by three suitable coordinates x, y, z and determine the distance ds of two infini(xj Y, z) and (x + dx, y + dy, z + dz) by the general

ds - gu dx2 + g22dy2 + g33dz2 + 2g12dxdy + 2g13dxdz + 2g23dydz

(3-53)

where the six coefficients gilc are given functions of x, y, z. The angle of two line elements (dx, dy, dz) and (5x, 6 y, 8z) attached to the same point P can* then be found with the aid of (3*53), namely, by the expression

cos «

(ds, 6 s) /[da, diy*/(6 s, 6sT]

(3.54)

CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

35

where (d3, da) and (5 s, 6 s) are the quadratic differentials

(ds, ds) = gudx + g22dy2 + g33dz2 + 2g12dxdy + 2g13dxdz -t 2g33dydz

(6s, 6s) = gu6x2 + g226y2 + g336z2 + 2g126x6y + 2g135x6z + 2g23&y6z and (ds, 6s) the mixed quadratic differential

(ds, 6s) = giidx&x + g22dy6y + g33dz6z + gl2(dx6y + 6xdy) + g13 (dx6z + 6xdz) + g23 (dy6z + 5ydz)

The geodesics finally are determined by the solution of the problem of variation

S

I^glldx2 + ^22dy + g33dz + 2g12dxdy + 2g13dxdz + 2g23dydz = Minimum (3.55)

3-6. The special analytic form of the differential (3.51) of the Euclidean ^ °n the 3PeCial Ch0iGe °f a octangular Cartesian coordinate
system. By introducing other coordinates by a transformation

X =

TJ, 0

y = z(z> t} , 0

z = M? , 7] , c)

(3.61)

°f analytlc representations of (3-51) may be obtained, namely

quadratic differentials of the general form

’

ds = gud? + g22d7j + g33d t,2 + 2g12d£dT) + 2g13d^dC + 2g23d71d£

(3-62)

tion. of tv,*

p’i^v , cceerrttaaiinn 1fnunnoc1t1ionnnqs of £c , q, £y , name-«ly, quadratic combina¬

tions of the first derivatives of the functions f, g, h:

hi = fp2 + g>c?2 + h 2
g22 = fn2 + gA^2 + h^2
g33 = f ^ + g £ + h £

gl2 = f?fq +
613 = fB,fC + g^Sc + h?hc
gsa = f Tjf ? + gqg^ +

(3.63)

All these different forms have in common the fact that a transformation can be

found which transforms (3.62) into the Euclidean normal form [”

h 1 e

tries derived t rom these differentials thus have the character of the soliS

Euclidean geometry.

iiU

ferentlalh(3

“wh^h^ U’))0wlns probleffl; Consider a general quadratic dif-

^ M 52 possible to

8111

general such a transformation is impossible. InTa-t

—hZ-t

' 1

obtain for the three functions f, g, h, of a transformation (3*61), a sjlTem ™

(3.63) of six quadratic differential equations. This system is overdetenSined

and has thus, in general, no solution. In order to insure the existence of a

solution, three conditions must be satisfied by the coefficients gik. These

36

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

conditions can be derived from Riernann13 Tensor of Curvature. a generalization of Gau33 curvature to the case of manifolds of three dimensions. The three essen¬ tial components of Riernann's tensor can be expressed in terms of the functions glk and their derivatives. If the differential (3-62) represents a Euclidean line element, then all components of Riernann'3 tensor are Identically zero. Vice versa if the components of this tensor are Identically zero, then a transformation (3.61) may be found which transforms (3-62) Into the Euclidean normal form (3-51), so that the manifold in question is isometric to the Euclidean space. This re¬ sult implies the remarkable fact that the question whether or not a given metric manifold of three dimensions is Euclidean can be answered by measurements in the manifold Itself. It means the curvature of such a manifold can be recognized even if observation from a viewpoint In an additional fourth dimension is impos¬ sible.

3.7. Among the general metric differentials

ds = gudx2 + g2sdy2 + g3sdz2 + 2g12dxdy + 2g13dxdz + 2g23dydz

(3-71)

there exists a special group distinguished by the property that Riernann's tensor 0l curvature I3 constant, l.e., its three components are Independent of the lo¬ calization. Riernann ha3 shown that all these line elements of constant curvature can be transformed into a normal form similar to (3*^6) In case of two dimensions:

ds2 =

d£2 + dn2 + dC

[L + £ K (^2 + T!2 + C2)]

(3.72)

where K is a constant, namely, the cobstant Riemannian curvature of the threedimensional manifold.

For K = 0 we obtain the Euclidean geometry:

ds2 = dc2 + dr]2 + dC2 For K < 0, for example K = + 1, the elliptic geometry:

(3.73)

ds2 =
E

d^2 + dii2 + df
n2 + f)]

For K > 0, for example K = -1, the hyperbolic geometry

(3.74)

ds2 = d e + d t] + d
E-i< f n + 2 + t2)]

(3.75)

Instead of Riernann's normal fora, we shall use later on another form which Is found by (3-72) by introducing polar coordinates

^ = R cos 9 cos 0

N = R sin 9

(3-76)

£ = R cos 9 sin 0

CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

This gives

^ 2 _ dR + R (d cp2 + cos2^ d 02)
(i +f R2)2

37 (3-77)

The geodesic distance D of a point P from the origin ob¬ viously Is obtained by summing up line elements ds which have no lateral or vertical extension d <p and d 0. Hence

4 D °

dR

2

=

K

2

1+4 R

arc tan (\ /K R)

(3-78)

By introducing this geodesic distance D instead of R in (3.77), we find the line element

where M Is the function

ds = dD2 + M2 (d<p2 + cos2<p d 0 2)

(3.781)

M

sin /K D

/K

(3-782)

This function may be interpreted as determining the linear size of a line element with no depth extension (dD = 0) but only lateral extension given by the angular coordinates dq>, d0 . -The dependence of this size factor M on the distance D Is given by (3.782), I.e., it Is

M = sin D M= D M = sinh D

In the elliptic case K = 1 in the Euclidean case K = 0 in the hyperbolic case K = -1

(3.783)

element is advantageous. We submit (3.78I) to the f

K = 1:

- tan D = a (y + n)

K = 0: K = -1:

- log -jy D = a (y + p) - t a nh-^ D = 0 (Y + H)

(3.79)

where a and g are constants and y a variable replacing the distance D. It follows quite easily that

K = 1: ds2 =

1

cosh a (y +

K = 0: ds2 = 2 e_2cI ^7 + P)

K = -1: ds2 = -—— 1 ---sinh2 a (y + p )

2d y2 + d<p2 + cos2qp d02j|a2d y2 + dqj2 + cos2cp d02}
2d y2 + d0 2 + cos2q; d02J

(3-791)

28

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

All these line elements have the same general form ds2 = M2 (a2 dy2 + dtp2 + cos2<p dO^)
where the size factor M(y) is given by the functions

(3.792)

cosh or ( y + |i ) M = 2e"7(Y+|i)

in case of the elliptic geometry in case of the the Euclidean geometry

s i nil a ( y + jj. )

in case of the hyperbolic geometry.

2-8. Cons tancy ol s 1 ze . R1 g i d t ransf orrna t ions . The Euclidean geometry is distinguished by the fact that objects can be moved without changing their size oi their other metric characteristics. A triangle in the Euclidean plane can be moved freely in thi3 plane to any other position without distortion of it3 char¬ acteristics. The result is a triangle congruent to the original one.

A movement of objects in a metric manifold can be described mathematical¬ ly by a point transformation of the manifold in itself. This means, for example. In a Euclidean plane, that to any point (x, y) another point (x1, y') is coor¬ dinated

x' = f(x, y) Y' = g(x, y)

(3.81)

namely, the point (x1, y') to which an object located at x, y Is moved. Such a point transformation is principally different from a transformation (3-31) of a coordinate system, although both are mathematically represented in a similar way. In (3-21) the same point of a plane Is associated with different numbers, i.e., the points are considered fixed but the coordinate system is changed. In (3.81), however, we keep the same coordinate system but Interchange the points of the plane. Suppose now that a configuration of objects x, y i3 transformed into a configuration x', y', by (3.81). Since the coordinate system Is unchanged, we have the same metric differential ds2 = dx2 + dy2 before and after the transforma¬ tion (3.8I), and thus we must expect in general that the new configuration x', y' is metrically different from the original configuration. This distortion is a consequence of the fact that

dx'2 + dy'2= ds'2= (f* + gx) dx2 + 2(fxfy + SxSy) dxdy + (fy2 + g2) dy2 is in general not Identical with dx 2 + dy 2 = ds 2 .

There exist, however, in the Euclidean plane, special transformations (3*8l) which preserve the metric of a configuration, i.e., transformations for which

dx' + dy' = dx2 + dy2

(2-82)

These special transformation are called rigid transformations of the plane, and are given by the three-parameter group of transformations

x' = x cos to - y sin to + a y' = x sin to + y cos to + b

(2-82)

CHARACTERIZATION OF A METRIC BY QUADRATIC DIFFERENTIALS

39

where <o, a, b are arbitrary constants. The transformations (3.83) can be de¬ scribed as rotations around the origin by an angle a) plus an additional translatory shift. If any configuration of objects is submitted to such a transforma¬ tion, then the resulting configuration is congruent to the original one.
In a similar manner we may determine the rigid transformations of the three-dimensional Euclidean space. It is a six-parameter group of transforma¬ tions characterized by rotations around the origin (3 parameters) plus an addi¬ tional translatory shift (3 parameters).
By these groups of transformations we thus may move any line element dx, dy, dz to any other position and direction without changing its size. We can say that the existence of such transformations is a necessary and sufficient condition for the existence of objects independent of localization, i.e., objects may be moved freely without distorting their metric characteristics.
The existence of a group of rigid transformations of sufficiently great number of parameters (6 in three dimensions; 3 in two dimensions) so that complete movability of objects Is insured, is by no means self-evident. Indeed, in the case of a general quadratic differential ds2 we must expect that no transforma¬ tion of this type exists; objects in such a manifold are frozen to their positiors any movement will result In a distortion of their metric characteristics.
The remarkable feature of the geometries of constant curvature is now that In all three types, elliptical, Euclidean, hyperbolic, the same complete movabili¬ ty ot objects is found. Indeed, this is Rlemann's result: These geometries are the only ones which have this character. Other metric differentials may allow partial movability, i.e., possess a group of rigid transformations of lower number of parameters with the result that objects may be moved Into certain restricted positions, but only the three quadratic differentials of constant curvature allow objects to be moved into any position without distortion of their metric characterlstlcs. These three geometries thus are the only ones with true size constancy, We shall, later on, discuss the rigid transformations of the hyperbolic geometry in mathematical detail, and recognize their significance for binocular vision.

Section 4 THE PSYCHOMETRIC OF VISUAL SENSATIONS

4.1. We have recognized in $2 the physiological significance of the angu¬ lar differentials dy, dcp, d0 for the binocular observation of a physical line ele¬ ment (dx, dy dz). These angular differentials determine directly the essential characteristics of the images of the line element on the retinae of the observer, provided his head remains in a fixed position. If, on the other hand, the line •element is observed with moving head, then the eyes are directed in symmetrical convergence towards the line element, and its retinal images are given by the dif¬ ferentials dy*, d<p*, d0* as explained in §2.8.

Our problem in this section is the sensation of size which we have when

observing a given physical line element. Our basic assumption is that assignment

of size to physical line elements is a primitive sensation, i.e., a sensation com¬

pletely determined by the physical characteristics of the line element. Though

the retinal images of a line element are determined by the above angular differ¬

entials, we do not sense these differentials as angles but as linear quantities.

Even objects as far away as the moon or clouds appear to have a certain linear

size which, of course, is in no way identical with their actual physical size.

This transformation of angular into linear distances is quite characteristic for

our visual sensations, and, in the case of uniocular vision, can be demonstrated

by a simple experiment. An angular wedge made from cardboard is placed in front

of one eye of the observer so that the vertex of the angle

coincides with the center of rotation of the eye. The wedge

may be vertical or horizontal or in intermediate position.

With the other eye closed, the legs of the wedge do not ap¬

pear converging but parallel to each other, i.e., as two

lines which have a constant linear distance from each other.

Physically they include, however, a constant angle. We may

Fig. 35

express this peculiar situation by saying that the angular

coordinates of a polar system are interpreted as linear co¬

ordinates of a Cartesian system.

The assignment of linear size to angular impingements should not be under¬ stood as assigning physical length measurable in cm. to the objects. We may do so by intellectual association of sensation and- physical experience, but the primi¬ tive act of transforming angles into size is based on physiological units quite different from units of physical length. However, the character of these physio¬ logical units is not the problem we are here concerned with. From a geometrical point of view we need not be interested in the unit employed in size assignments. The metric characteristics of the visual apparent space are entirely independent of the physiological dimensions of this unit.

4.2. Let us now consider a physical line element (dx, dy, dz) attached to a point P - (x, y, z) of the physical space. By using bipolar coordinates we may characterize the base point P by the three angles y, <p, 0 and the line element by the differentials dy, dcp, d0 . These differentials can be found from dx, dy, dz by differentiating the formulae

THE PSYCHOMETRIC OF VISUAL SENSATIONS

41

cos 2 cp + cos y

x

cos 0

sin y

sin 2 9 sin y

(4.21)

CDS 2 Cp + COS Y
z = —-L-L sin 0 sin y
which relate Cartesian and bipolar coordinates.
We observe this line element with fixed head so that the optical axes of the eyes converge at P. Our assumption is that the observer is led to a definite sensation of the size and that this apparent size ds is a function of the differ¬ entials dy, dcp, d0, namely, given by a quadratic, differential
ds = C2 (gn dy2 + g22 dcp2 + g33 d02 + 2g12 dydcp + 2g13 dyd0 + 2g23 dcpd0)
(4.22)
The coefficients gik are functions of the coordinates y, cp, 0 of the base point P. The constant C determines the unit of the apparent size, which, as we point out again, is not a unit of physical length. Since this constant is immaterial for the geometry based upon the differential (4.22), we shall omit it in the follow¬ ing and thus assume C = 1. The size ds of the line element dy, dcp, d0 thus will be a pure number of our number system.

4.3. Let us next observe the above line element with moving head so that the eyes are directed In symmetrical convergence towards the point P. We now characterize the point P by the angular coordinates y*, cp*, 0* of §2.8 and the line element by the differentials dy*, dcp*, d0*, which are found from dx, dy, dz by differentiating the equations

X = (a + cot i Y*) cos 9 * cos 0 *

y = (a + cot ir*) sin 9 *

(4.31)

z = (a + cot i Y*) cos 9 * sin 0 *
relating the coordinates x. y> z and Y*> 9*, 0 * ^
The apparent size ds* which we assign to our line element by this manner 01 observation is based upon the differentials dy*, dcp*, d0* and upon the coor¬ dinates 9*, y*, 0* of the point P. Indeed, the differentials dy*, dcp*, d0 * de¬ termine the psychologically significant characteristics of the retinal images of the line element and y*, 9*, 0*, the position of the eyes and their optical axes. The principle formulated in §2.8 about the relationship of observations with fixed and moving head enforces the implication that ds* can be found from y*, 9* Q* by t*16.35"6 formula as ds from y, 9,0. This means that ds* is given by the quadratic differential

ds*2 = gn dy*2 + g22 d9*2 + g33 d0*2 + 2g12 dy*d9* + 2g13 dy* d0 * + 2g23

where the coefficients g Sik (Y, ? ■ 6 ) in (4.22) .

d9* d0*

(4.32)

(y*, y*> 9*) are the same functions as the coefficients

42

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Indeed, consider the following pair of two different physical line ele¬ ments :

(dx, dy, dz) at a point P = (x, y, z) and
(dx*, dy*, dz*) at a point P* = (x*, y*, z*)
Let the first one be chosen arbitrarily and be given by the bipolar dif¬ ferentials dy, dcp, d0 and the base point P = (y, cp , 0). We determine the second base point P* by its coordinates (y *, <p*, 0 *) for head movements requiring that

Y* =Y cp * = <p 0* =0

We next construct the line element (dx*, dy*, dz*) at P* by the condition that, with regard to the coordinates (y*, cp*, 0*) it shall have the differentials

d y* = d y

dcp* = d<p do* = d0

The second line element thus has the same coordinates in the y*, <p*, 0* system for head movements as the first one in the bipolar y, cp, 0 system for observation with fixed head. Obviously, the second line element is uniquely determined by the first one. From the general principle of §2.8, it now follows that the same size sensation must be obtained if the first line element is observed with fixed head and the second with moving head. If this be true for any such pair of line ele¬ ments, then the quadratic differentials (4.22) and (4.32) must be formally identi¬ cal .

4.4. The above result has the consequence that to the same physical line element (dx, dy, dz) a different size will be assigned if it is first viewed with fixed and second with moving head. Let us first consider an observation with fixed head. With the aid of the relations (4.21) we express the quadratic dif¬ ferential (4.22) by the Cartesian coordinates x, y, z and their differentials dx, dy, dz. The result of this transformation is a quadratic differential of the form

ds2 ^ Andx2 + A22dy2 + A33dz2 + 2Ai2dxdy + 2Ai3dxdz + 2A23dydz

(4.4l)

where the coefficients Alk are certain functions of x, y, z. It determines di¬
rectly the apparent size of the physical line element dx, dy, dz observed at the point P = (x, y, z).
On the other hand, let us observe the same line element with moving head and thus direct the eyes in symmetrical convergence towards the point P. We nov transform the differential (4.32) which Is formally identical with (4.22) with the aid of the relations (4.31), and obtain another quadratic differential

THE PSYCHOMETRIC OF VISUAL SENSATIONS

43

ds*2 = Bi idx + B22dy + B33dz + 2Bi2dxdy + 2Bisdxdz + 2B23dydz

(4.42)

The coefficients B^ (x, y, z) are of course not the same as the coefficients Aik(x, y, z) in (4.4l), and consequently the apparent size ds* of our line ele¬ ment differs in general from its apparent size ds.
However, we know that the two quadratic differentials (4.4l) and (4.42) have been obtained by transformation from two differentials (4.22) and (4.32) which are formally identical. In other words, they are the result of two differ¬ ent transformations of the same basic quadratic differential

ds2 = gudy2 + g22dcp2 + g33d0 2 + 2gi2dyd0 + 2g13dyd0 + 2g23d<pd0

(4.43)

We may also say that both differentials (4.4l) and (4.42) can be transformed into each other. We have seen In §3 that this property of two metric differentials es¬ tablishes an intimate relationship between them, namely, that both differentials characterize the same invariant geometry but in different coordinate systems. They are different analytic expressions for the same metric, and the geometries developed from them are identical.
By observing the same physical configuration, whether with fixed or moving head, we find in ourselves the conviction that the resulting sensations, though actually different, belong to the same unchanged environment of objects. In our theory, this conviction of observing the same objects but by different methods of observation finds its mathematical expression in the fact that the two quadratic differentials associated with these sensations represent the same Invariant geometrical relations.

4.5. Our problem Is to determine the coefficients gllc(y, 9, 0) of the basic quadratic differential

ds2 = gudy2 + g22dq>2 + g33d02 + 2g12dyd9 + 2gi3dyd0 + 2g23dq>d0

(4.51)

In general we shall interpret 9, 9, 0 as bipolar coordinates referring to observa¬ tion with fixed head. However, in this section, we shall base our discussion upon observation with moving head. We know the corresponding differential

ds2 = gudy*2 + g22d9*2 + g33d0*2 + 2g12dy*d0* + 2gi3dy*d0* + 2g23d9*d0* (4.52)
must be formally identical with (4.51). Hence we are sure that any information obtained about the functions glk by observation with moving head can be used di¬ rectly for the metric (4.51) of the sensations associated with observation with fixed head.
Observation with moving head as described in §2.8 must be considered as spherically symmetrical to the origin. Indeed, we can safely assume that two line elements (dx, dy, dz) and (dx', dy1, dz1) of equal length, located at the same distance R from the observer and including the same angle with the corresponding lines of view are judged to be of equal size (provided that we turn our head towards them). Mathematically we may express this as follows: Rotations of the space around the origin must not change the metric differential (4.52). In other words, these rotations must be rigid transformations of the differential (4.52).

44

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

It is not difficult to prove that this condi¬ tion i3 satisfied then and only then if (4.52) has the form

ds*2 = A2(y*) dy*2+M2(y*) (dtp*2 + cos2<p* d0*2)

(4.53)
where A(y*) and M(y*) are arbitrary functions of y* alone.*)

differential form

(4.51)

Since the differentials formally identical, we conclude for observation with fixed head must also have

ds* and ds are that the metric the general

ds‘

A2(y) d y2 + M2(y) (dcp2 + cos2cp d02)

(4.54)

where y, <p, 0 are the ordinary bipolar coordinates. Instead of six unknown func¬ tions gijj- of three variables y, cp, 0 there are only two functions A(y) and M(y) unknown in (4.54). With this simplification we abandon the discussion of ob¬ servations with moving head and confine ourselves in the following to the in¬ vestigation of visual sensations associated with observation with fixed head.

4.6. The function M(y) in (4.54) determines the apparent size of a line element dcp, d0 which has no disparity dy. The function A(y) gives the depth ex¬ tension of a line element with disparity dy but with no lateral or vertical ex¬ tension d<p, d0. By summing up line elements of the latter type, we obtain the apparent distance D of two points P0 and P from each other

D = Y A(y) dy

' (4.6l)

J Yo

The arbitrariness of the functions M(y) and A(y) illustrates the fact that there

Is no general relation between the concepts of size and distance, i.e., of size

and localization.

We show next that in the case of binocular vision there exists such a relation. Let us consider two line elements (dx, dy, dz) and (dx1, dy', dz1) located at two different points of the same radius vector. Without loss of gen¬ erality, we may assume that the line elements are lying in the horizontal plane on the x-axis. Let us assume that both line elements have the same bipolar dif¬
ferentials, i.e..

d,c' - dlC d<p1 = dcp

(4.62)

Fig. 37

so that we observe the same Impinging angular characteristics in the two cases. We determine the apparent angle co which these line elements Include with the x-axis, i.e., with two line

elements

6y' = by 6<p' = 69 = 0

(4.63)

*A proof can. be found in: Levi-Civita, Der absolute Differential Calcul, Berlin: 1928, pp. 278-285.

THE PSYCHOMETRIC OF VISUAL SENSATIONS

45

The apparent angle w Ip given by the formula (refer to 3-5^) •
A2(Y)dy5y+ M2(y)d<p5<p cos co =
V^A2 (y )dy2 + M2(y)dcp2 • sJ~P?(y~) b^~+ M2 ( y) Sep5"

i.e., on account of (4.63) by
A (Y)dy cos co —
\fhZ (Y) by2 + M2 (y) a<p‘

1
( ?)‘ 3 i +£ A k (yl) Vdy.

Similarly we find co' by

(4.64

COS (0
^G?)‘

(4.65)

We notice that the two angles co and co' are different unless the ratio A(y)/M(y) is Independent of y, i.e., equal to a constant q.
Actually, two such line elements on the same line of view which give the same angular impingement dy, d<p are seen as parallel. Hence we conclude that A(y) =uM(y), and thus that the binocular metric must have the form

ds2 = M2(y) (q2dy2 + dtp2 + cos2cp d02)

(4.66)

which reduces our problem to the problem of finding only one function M(y ), the size factor of a line element with no disparity dy.

The constant a Is a constant depending upon the individual observer; It determines the sensitivity of depth perception through disparity dy as compared with size perception through lateral and vertical angles d9 and d@. We know that the angular threshold of disparity which gives the sensation of depth is con¬ siderably smaller than the angular threshold for recognition of size difference. This means that cr must be expected to be considerably greater thdn 1.

The apparent distance D of two points P0 and Pi on the x-axis (or any other line to the origin) is given by the integral

yf
D = a / M( y)dy

(4.67)

Ao which establishes the relation of apparent distance to apparent size in binocular vision.

We mention finally a simple demonstration which shows that line elements attached to points of the x-axis with the same angular coordinates dy, d^ are seen as parallel. We construct a .simple chessboard pattern of the type shown in Fig. 38 and place it normal to the horizontal plane so that an angle w ^ 9O0 is Included with the x-axis. By crossing our eyes at a point P0 of the x axis so that the squares of the pattern are fused in pairs (1, 2). (2, 3), (3, 4), etc., a smaller pattern is seen at some other position (not necessarily P0). However, the resulting pattern appears to be parallel to the original one. The angular impingement dy, d<p obviously is unchanged by this observation, the angles y of convergence, however, are different.

46

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

4.7- The result (4.66) leads us to an

Interesting demonstration. We construct on a

Vieth-Muller circle a number of small vertical

rods. Since, for these rods, dy = dq? = 0, we

have by (4.66) for their apparent size the ex¬

pression

ds = M(y) cos 9 d 0

(4.71)

Since M(y) Is constant on the Vieth-Muller cir¬ cle, it follows from (4.71) that the apparent size of the rods is proportional to the expression cos 9 d0. If new the physical height, h, of the rods is chosen such that cos 9 dO is a constant, then the rods will have the same apparent height and thus appear to form a circular fence of equal height with the observer as center.

Since the physical height of a rod d0 at the latitude 9 on the ViethMuller circle is approximately given by the relation

h = 2R cos2cp dO

(4.72)

we conclude that our rods must decrease according to the law

h = h0 cos 9

(4.73)

if they are to have the same apparent height. rod on the x-axis.

The constant h0 is the height of a

4.8. The remaining problem is to determine the size factor M(y) of the quadratic differential (4.66). We shall do this in the following sections by evaluating certain observable facts. In this section, however, we shall first in¬ vestigate the simplest hypothesis about M(y), namely M(y) = const. This assump¬ tion would mean that the angular impingements dy, d9, d0 and their retinal Images are the only significant clues for visual sensations; the absolute value of y, i.e., the convergence of the eyes, is without significance. Without loss of gen¬ erality, we may assume M = 1 and thus have

ds2 = a2dy2 + d92 + cos29 d02

(4.8l)

We notice that, in every plane of elevation,© = const., a Euclidean differential

ds2 = u2dy2 + d02

(4.82)

is obtained so that the plane geometries In these planes must be Euclidean. Yet these plane Euclidean geometries do not supplement each other to a Euclidean geometry of the three-dimensional space. In fact, one can show without difficulty that the quadratic differential (4.8l) is non-Euclldean.
We notice readily that the differential (4.8l) preserves Its mathematical form if sub¬ mitted to the transformation

y' = y + T

9• = 9

(4.83)

Fig- 39

0i = 0 + X

THE PSYCHOMETRIC CF VISUAL SENSATIONS

47

depending on the arbitrary parameters t and X . These transformations form a sub¬ group of the general iseikonic transformations

Y' = Y + X 9 r = 9 + a* 0> =0 + X

(4.84)

which we have discussed in §2.5, and we conclude that the transformations of this subgroup (4.8^) represent rigid movements in the non-Euclidean geometry based upon the differential (4.8l). However, we see also that the complete group (4.84) of iseikonic transformations does not give the rigid movements of our geometry. In¬ deed the differential (4.8l) changes its form in case a* ^ 0.
It is not difficult to determine the complete group of rigid transforma¬ tions which belong to the quadratic differential (4.8l). It consists of the spe¬ cial iseikonic transformations

Y- = Y + T

(4.85)

and those transformations of 9, 9 into 9' , 0 ' which do not change the quadratic differential d<p2 + cos2qpd92. Since d<p2 + cos2cpd02 is the line element on a sphere of radius one, and since this line element preserves its form by any rotation of the sphere about its origin, we recognize that the desired transformations of 9, 9 Into 9', 0' must be isomorphic to the three-parameter group of spherical rota¬ tions. We thus obtain a four-parameter group of rigid transformations for our geometry (4.8l). Complete movability of objects requires, however, the existence of a six-parameter group of rigid movements, as we have seen in §3.8. Hence we conclude that the geometry based upon (4.8l) does not provide complete movability. This already seems to indicate that our above hypothesis M(y) = 1, which denies the significance of convergence for space perception, has to be revised: It is contradictory to the conviction which accompanies our visual sensations that ob¬ jects can be moved to any position without changing their metric characteristics.
We can easily find another Indication that the assumption M(y ) = 1 cannot be defended. The geodesic lines in the horizontal plane, i.e., the solutions of the minimum problem

Minimum

are given by the curves

T = ay + b

(4.86)

where a and b are arbitrary constants.
Since geodesic lines have the property that all their line elements have the same unchanged direction, we conclude that a number of marks arranged on such a curve will give the impression of arrangement on an apparent straight line. Curves of this type are called Horopters; in particular those curves which are syra metrical to the x-axls are known as Frontal plane horopters. From (4.86) it fol¬ lows readily that these frontal plane horopters are given by the curves y = const, by the Vleth-Muller circles. But we know already that Vieth-Muller circles appear

48

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

flatter than, they are but not yet plane, namely, as circles with the observer as center. Furthermore, It Is well known that the actually observed horopters are curves of the type shown in Fig. 40. We conclude that the horopter phenomenon
cannot be explained by the hypothesis M(y) = 1.
4.9* We base our next hypothesis about M(y) upon the psychological conviction that the visual shape and 3ize of objects can be repeated in any position and orientation. Thl3 conviction makes U3 consider shape and form of an object as qualities which are independent of localization. We also may say that we are convinced that any ob¬ ject can be moved as a visually rigid body to any desired position and orientation and the result of this movement is an object metrically congruent to the original object. We must, however, not assume that such a movement of an ob¬ ject is necessarily the same as a physical movement in a Euclidean space. In fact, we know the opposite is true: apparent shape and size of an object do not remain unchanged, when it is moved physically. Consequently we prefer to formu¬ late the conviction about the complete movability of visual objects as follows: To any conflguration of objects there exists a six-parameter set of configurations which, in all their metrical characteristics, are visually congruent to the origi¬ nal configuration. This psychological fact, that metrical form and localization of objects are considered as Independent, would be hard to understand If the geometry of our visual sensations did not Itself provide what we have called above complete movability of objects. This, however, is possible only If the quadratic differential (4.66) represents a geometry of constant curvature. It follows from §3*79 that we should expect the factor M(y) to be one of the three functions

M(y)
M(y)

1_ cosh a (y + p)
2e-o(Y+n)

(4.91)

sinh a (y + p)
In the first case the geometry is elliptic, in the second. Euclidean, in the last, hyperbolic. In addition to a another individual constant, P, enters these expressions. It determines the limit which the size factor M(y) approaches if y _► 0, i.e., if the object is physically moved far away. It also is closely related to the apparent distance at which we place objects 01 infinite physical distance.
We have yet no means of deciding which one of the three functions (4.91) has to be chosen for our problem. We shall, however, in the following, accumulate evidence obtained by theoretical considerations and by experimental results that M(y ) is given by the last function

^ sinh a (y + p )

(4.92)

so that the metric of our space sensations is hyperbolic, namely, given by the quadratic differential

THE PSYCHOMETRIC OF VISUAL SENSATIONS

49

d32 =
sinh

a ( y + n) rP

dy + dtp' + co329 d02^

(4.93)

We may take as a first indication the fact that the hyperbolic metric la best 3uited for approximate coincidence of physical and apparent size In a suf¬ ficiently large Interval.

Let us assume that an observer, by systematic training or experience, is in the position to change his constants cr and n. We shall demonstrate that the observer can Improve his size judgment by changing the constant p from positive values gradually to zero if the metric is hyperbolic but not if it I3 elliptic or Euclidean.

It Is clear that the limiting value M(0) for objects at infinity is the greater In all three cases the smaller H is. Its greatest value is assumed for M- = 0, and obviously this represents the best approximation of apparent size to physical size. Let us therefore consider this best case in the three metrics (4.91)• We have

M(y) m(y )

1_ cosh a y
2

(4.94)

M(y)

1 sinh a y

The limits at y = 0 are M(0) - 1 in case of the elliptic, M(0) = 2 in case of the Euclidean, but M(0) = 00 in case of the hyperbolic geometry. Only the hy¬ perbolic metric thus may approach the physical situation that the size of objects subtending equal angles increases beyond all bounds.

To illustrate this, let us consider the apparent size ds of an object of constant physical size dh If moved in the physical space. We have d9 = dh tan y/2 and hence, assuming 9=0, d9 = dy =0:

ds

dh tan y/2

cosh ct y

tan y/2 ds 2dh
" ~~e° V

(4.95)

ds

tan y/2

sinh a y

We conclude from (4.95) that the ratio ds/dh of apparent to physical size con¬ verges to zero if the object is moved towards infinity and if the metric is ellip¬ tic or Euclidean. In the hyperbolic case, however. It reaches asymptotically a
constant value ds/dh = 7^, so that ds/dh remains constant in a great Interval.
This situation is shown in Fig. 4l where the three curves ds/dh as function of x = cot y/2 are drawn. In case ji / 0 all three functions approach zero, but only the hyperbolic metric allows by variation of |x to retard this approach effectively and thus to improve subjective 3ize estimation in regard to agreement with objec¬ tive physical size.

50

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

X
o I
I
E u o o o

Relation between Apparent and Physical Si

Section 5 DERIVATION OF THE HYPERBOLIC METRIC OF VISUAL SENSATIONS

5.1. We shall discuss in this section facts of ohservation which support the hypothesis that the psychometric of visual sensations Is the hyperbolic metric of constant curvature. Our problem is to determine the factor M(y) in the quad¬ ratic differential

ds2 = M2(y ) (<j 2dy2 + dqp2 + cos 2<pd0 2)

(5.11)

Without loss of generality we may confine ourselves to observation of objects distributed in the horizontal plane 9=0. The apparent geometry In this plane is represented by the differential

d32 = M2 (Y)(cr2dY2 + d<p2)

(5-12)

It determines the apparent size ds of a horizontal line element given by the bi¬ polar differentials dy, dq> and observed at a point (y, 9) of the horizontal plane.

5.2. Observations on Vieth-Muller circle's. Our first demonstration is based upon certain observations referring to properties of Vieth-Muller circles. We construct two horizontal bundles of straight lines from the two eye points y = 11 of the y-axis. The angles Act and Af3 between neighboring lines shall be constant, i.e.. Act = A3 = constant. (Fig. 42) For the actual demonstration it is advisable to use two bundles of illuminated threads of different color (for exam¬ ple red and green) stretched out in a dark room. However, the effect described in what is to follow can already be observed by simply drawing two bundles of red and green lines on a plane sheet of paper. Let us assume in the following that the red bundle has Its vertex at the left eye L and the green bundle at the right eye R. We have indicated in Fig. 42 the green bundle by solid lines and the red bundle by dotted lines. The lines of the two bundles intersect each other in pairs on a set of Vieth-Muller circles which determine on the x-axis points PQ, Pi, P2, P3, P4, etc., accumulating near the origin.
We now bring our eyes into a position exactly above the points R and L, and observe the bundles from this position by converging at the points of the Vieth-Muller circle through P0. Instead of two bundles we see one bundle of fused lines which intersect the Vieth-Muller circle through P0 at regular distances. However, these lines do not lie in the horizontal piano; they are space curves in¬ tersecting the horizontal plane on points of the Vieth-Muller circle. This ef¬ fect can be made very striking by observing in a dark room the Illuminated threads with a green filter in front of the right eye and with a red filter in front of the left eye. This, of course, has the result of weakening the stimulus of the red lines on the right eye and the stimulus of the green lines on the left eye. If, on the other hand, we observe the colored lines against a white background-as is the case when the lines are drawn on a white sheet of paper--a red filter has to be used in front of the right eye and a green filter In front of the left eye.

52

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The appearance of the fused lines is that of straight lines arranged on a circle around the observer. In fact, we know that the Vieth-Muller circle it¬ self appears to the observer as a circle around the point 0 of the horizontal plane vertically below the apparent center C of observation. We also observe that these straight lines are perpendicular to the lines of view connecting the "egocenter" C with the points of the Vieth-Muller circle. Now, the lines of view form a cone with C as vertex and the Vieth-Muller circle as base; conse¬ quently our fused lines are normal to this cone. The surface formed by them must also be a cone with the Vieth-Muller circle as base and a vertex 0 at some position above the origin. This result is illustrated in Fig. 45. We have to interpret this figure as a Euclidean map of an apparent surface formed in a visu¬ al sensation. It represents the interpretation which we give to our sensation, namely, that of a circular cone with Its vertex on the Z-axis of a Euclidean,
Y, Z space. We denote the Cartesian coordinates in this space intentionally X, Y, Z in order to indicate that it is by no means identical with the physical
space. Let us now converge at the points of the Vieth-Muller circle through Pi.
We observe the same phenomenon with the difference that the fused lines intersect

DERIVATION OF THE HYPERBOLIC METRIC OF VISUAL SENSATIONS

53

the horizontal plane at the VIeth-Muller circle through Pi. Again they form a surface inter¬ pretable as a Euclidean circular cone with its vertex Qi on the Z-axis. This cone is normal to the cone through the observation center C and the Vieth-Muller circle. However, the ap¬ parent radius of the base circle has decreased with the effect that the fused lines appear to have a somewhat smaller distance from each other. By changing the convergence of the eye3 in suc¬ cession to the points of the Vieth-Muller cir¬ cles through Po, Pi, P2 . • • -we observe a sequence of circular cones with decreasing ap¬ parent radii Ro, Ri, R2 • • • • and distances of the fused lines decreasing In the same ratio. We point out that this sequence of apparent radii is not proportional to the sequence of physical radii of the Vieth-Muller circles.
The significance of a cone artificially created in the above manner Is that it provides the observer with a background towards which he may project the horizontal plane. It estab¬ lishes the possibility of observing objects in the horizontal plane without chang¬ ing the convergence of the eyes simply by taking care that the background does not change its position. With a little practice it is possible to scrutinize even objects which are far away from the Vieth-Muller circle of convergence without varying the position of the background, i.e., without changing the convergence of the eyes. The impression associated with this manner of observation is that of a uniocular projection of the horizontal plane towards the conical background, the point C being the center of projection. We shall refer in the following to this peculiar projection as Cyclopean projection. Binocular vision has a twofold office in Cyclopean projection: To create a conical background for uniocular pro¬ jection, and to determine on the background a linear scale of size which Is pro¬ portional to the apparent radius R of the base circle of the cone.
In general, any actual point of the horizontal plane is seen as double when observed by Cyclopean projection, i.e., we see two images on the background. Only the points of the basic Vieth-Muller circle itself are seen as single. There exist, however, certain extended curves which do not appear doubled. The ViethMuller circles themselves are such curves, since their two projected Images co¬ incide on the background. Yet every individual point of these curves has two different points of the image curve a3 projection. The Vieth-Muller circles ap¬ pear on the conical background as a system of circles of latitude, which--at least in the neighborhood of the base circle--are nearly equidistant.
In addition to the Vieth-Muller circles there are two other sets of curves which appear single: The two bundles of red and green lines In Fig. ■4-2. Though each line of these bundles has two separated Images, we fuse one of them with a generating line of the background and thus see only the other image as an object of the horizontal plane projected upon the background. It becomes a curve on the background which intersects the fused lines at equal apparent angles. The result of viewing the total system of curves drawn In Fig. 42 In Cyclopean projection-namely, the Vieth-Muller and the two bundles through the eyes--is shown In Fig. 44.

54

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Ne remark that the oblique red and green lines in Fig. 44 are the result of a unlocular pro¬ jection. The green lines are obtained by project¬ ing the green lines of Fig. 42 with the left eye and similarly the red lines by projecting the red lines of Fig. 42 with the right eye. The impres¬ sion of the observer, however, is in both cases that ol uniocular projection from the same apparent projection center C.
In general, it is possible to interpret the above curves also as curves of the horizontal x>Y~Plane and not as curves on the cone. However, this alternative disappears more and more when other clues are eliminated in the observation, i.e., when the experiment is carried out in a dark room. We also notice that the compulsion to see the curves on the background becomes greater and greater the lower our eyes are placed with respect to the hori¬ zontal plane. Moreover, we base our judgment of size of subdivisions of these lines more and more on their apparent size upon the background, the greater this compulsion becomes to localize our curves on the cone.

The above observations indicate that we may consider the act of assigning size to the horizontal line elements by binocular observation in the horizontal plane as a limiting case of Cyclopean observations from above the plane.

This intimate relationship of binocular vision and Cyclopean projection is more readily understood by the following consideration. Consider the two points P0 and P in Fig. 42. The point P is chosen intentionally on one of the lines through the left eye. By Cyclopean projection we obtain, on the background through P0, one image of PG but two images Q and Q' of P (Fig. 45). The image Q' is produced by the left eye; it lies on one of the fused lines of ’the cone and thus helps to establish the background. Let us now assume that the right eye
dominates in localizing objects relative to the ob¬ server. Then the image Q will be interpreted as the apparent projection of the point P onto the background, and thus the line element~P0Q on”the cone as the projection of the line element P0P. Since this projection is uniocular, we determine the size of PQP by the size of its projection P0Q on the background (Emmert's law).

Fig. 45

The above analysis of the mechanism of as¬ signing 3lze to horizontal line elements by binocu¬ lar vision can be applied to any such line element. Let us consider two arbitrary neighboring points Po and P of the physical x,y-plane. We connect the two points with the left eye and the point P0 with the right eye. Next we construct the Vieth-Muller circle through P0 and draw a line from it3 inter¬ section point A with LP towards the right eye (Fig. 46). This configuration is viewed from above with the effect that the point P is seen as

DERIVATION OF THE HYPERBOLIC METRIC OF VISUAL SENSATIONS

55

y

a point Q on a Cyclopean background. This background is indicated by two fused lines which intersect the horizontal plane at the points P0 and A of the ViethMuller circle. Indeed, as above, the image Q' of P produced by the left eye lies on the fused line through A and thus supports the establishment of the background. The image Q from the dominating right eye determines the Cyclopean projection of P onto the background, and the distance of P0 from. P is estimated by the distance of Q from PQ. A similar construction can obviously be made if the left eye should be the dominating eye and determines the localization of objects relative to the observer. By gradually lowering our point of observation, the result of Cyclopean size estimation becomes gradually the binocular size ds of P0P based upon the as¬ sociated differentials dy and dtp.
We shall now demonstrate that the above interpretation of binocular vision as uniocular projection against a variable conical background induces in the hori¬ zontal plane a hyperbolic metric. First we introduce, on the cone, certain angu¬ lar coordinates co and y. The declination w is the angle which the line through C and a point Q on the cone includes with the horizontal plane. The azimuth y Is the angle which a plane through the Z-axis and the point Q includes with the X,Z-plane (Fig. 48). Since u and y are actually spherical coordinates determining the direction of the projection lines through C, we can also characterize the points P of the X,Y-plane by these coordinates. The angle u determines the ap¬ parent declination and v the apparent azimuth of a point P of this plane If ob¬ served from C.
It is easy to express the distance ds of two neithboring points P0 and Q on the cone by the above coordinates. We assume that P0 lies on the base circle of the cone. We find the formula

Fig. U8

ds2 = h2 ( d(0g • + cot2ud\y2N\

yain w

)

(5.21)

where h is the apparent height of C above the X,Y-plane. Since the distance of P0Q.measures also the apparent distance of the points P0P in the horizontal plane, we find that the metric introduced in this plane by Cyclopean projection has the form

+ cot^o dy2 sin2w

(5.22)

56

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The constant factor h2 in (5.21) has been dropped, 3ince it is insignificant for metric relations. Thi3 means, in other words, that the Cyclopean metric has, in¬ dependent of the position of the center C of observation, the same mathematical form (5.22) if expressed in the coordinates w, y of the plane.
We transform the differential (5-22) by introducing the apparent distance D:

D - AlWV - 106

|

(5-23)

The result is

ds2 = dD2 + sinh2Ddv2

(5.24)

and thus the line element of the hyperbolic geometry. Since the binocular metric of the horizontal plane can be ■ considered as a limit Of metrics of Cyclopean pro¬ jections with decreasing projection heights, and since these Cyclopean metrics all
have the same form (5*24), It follows that the binocular metric itself must have the form of the hyperbolic geometry.

It remains to express the differential (5*24) by the bipolar coordinates Y, 9 of the x,y-plane. The apparent azimuth W, obviously, is identical with the bipolar latitude 9, so that we.may write

ds2 = dD2 +'sinh2Dd92

{5.25)

We know, on the other hand, the general form of ds2 in terms of y and <p, namely,

ds2 = M2(y)(a 2dy2 + d<p2)

(5-12)

By identifying the two differentials (5-12) and (5.25) we obtain the relations

M(y) = sinh D

The last equation yields

— dD = -asl,nh, D^ dy

tanh AD = e-a(Y^) where n is a certain constant. Since

2 tanh D M(y) = sinh D = -rr~
1 - tanh D
we find by (5.27) that

and hence the differential ds 2

^ sinh a (y + |x)

1_ 3 Inh2tr (y +|i)

( ct 2 dy 2 + d9 2)

(5-26)
(5.27)
(5-28)
(5.29)

DERIVATION OF THE HYPERBOLIC METRIC OF VISUAL SENSATIONS

57

This Implies that the three-dimensional differential (5-11) must have the form

ds = -—- (a2 dy2 + d<j>2 + cos^dG2) sinh a (y + fi)

(5.291)

and this means that binocular vision establishes a hyperbolic manifold of sensa¬ tions.
5-3- The ■ interpretation of the Cyclopean background as a cone is not es¬ sential for the result of the preceding section. Only a small part around the base circle of the background Is used for Cyclopean projection in any instance. Thus any surface which coincides with this part of the cone would lead to the same result, i.e., to the same metric differential

d(o‘ ds2 =

+ cot2o d92

sin2o

(5.31)

expressed in apparent declination « and apparent azimuth vj/ • For example, a sphere around the center of observation, C, which intersects the X,Y-plane along the base circle serves our purpose equally well. The fused lines would be interpreted as meridians on this sphere and the Vieth-Muller circles as circles of latitude. This is readily understood by the fact that in a uniocular viewr such a system of curves allows a multitude of interpretations, for example, that of a sphere or a cone. As we have seen, Cyclopean projection is Interpreted by the observer as such a uni’oculap view.
In order to transform the quadratic differential (5.31) into the final form (5.29)-, ve have made use of a former result, namely, that In the bipolar system Y, 9, It must have the general form (5-12). Independent of this result we can derive the desired transformation as follows. We remark as before that the apparent azimuth _is identical with the bipolar latitude, i.e. , we have 9 = cp . We also know by observation that the Vieth-Muller circles y = const, are imaged as circles of latitude « = const, upon the background. Hence it follows that u must be a function of y alone: o = o> (y) . The problem is to determine this relation of the apparent declination co to the bipolar parallax y.
We consider for this purpose the two bundles of red and green lines in • ^2. Since a = const, and 3 = const, in these bundles, we conclude that their equations in the y, 9-plane are

y - 2 9 = cons t. Y + 29 = const.
In differential form we may write
Of. _ + 2
d9

(5-32) (5-33)

In they, 9-plane these curves thus are represented by two systems of equidistant parallel lines (Fig. 49). If the two red and green bundles of Fig. 42 are ob¬ served by Cyclopean projection from above the plane, then they appear, as we have seen, as curves on the background which Intersect the meridians 9 ^ const. at constant apparent angles. By Interpreting the background as a sphere around C,

58

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

the apparent angles are equal to the corresponding an¬ gles on the sphere itself and our curves - may be inter¬ preted as loxodromes on the sphere. These are the curves on which a ship travels without changing its course, i.e., its angle with the -meridians. They are characterized by the condition

dco = const.
:os (j d<p.

(5-54)

which states nothing but that the tdngent of the angle

with the meridians-' Is a constant along the curve. If

Fi8- 49

the background la Interpreted as a cone, the apparent

angle with the meridians.Is not equal to actual angles

on the cone but still its tangent is given by the expression (5*3*0

We may formulate our observation as follows: A physical curve on which

d v

i

d (i)

r1- = —2 is seen in Cyclopean projection as a curve on which - = const. As

d<P

cosco dcp

w Is a function of y alone, it follows from

d co _ cosiodcp

co' (y) dy cos co(y) d<p

(5*35)

that on any particular straight line dcp

2 of Fig. 49; thb expression co

"

coscoCy)

remains constant. Since this expression Is a function of y alone. It follows

that it must be Independent of y, i.e., we have

1(.Y .
cos co(Y)

. =

<7

(5*36)

where a is a constant.

By integration it follows that

log tan | - f = (y +M-)

(5-37)

where (i. is another constant We may also write

tan ( 4 ' 2

= e - (y+f)

and hence

tan —

= cot <*> = sinh a(y+y.)

(5*38)

By (5.31) and (5*38) it follows that the Cyclopean line element (5*31) ex¬ pressed in y and 9 must have the form

ds2 =

- (a2 dy2 +■dy2)

sinh2 a (y + p0

(5*39)

In agreement with our former result.

5.4. The hyperbolic metric of visual sensations and the relativistic met¬ ric of time-space manlfolds. In the preceding discussion we have tacitly assumed

DERIVATION OF THE HYPERBOLIC METRIC OF VISUAL SENSATIONS

59

that the physical space is a Euclidean manifold. We are certainly justified in making this assumption for the domain in which binocular vision is of practical importance to us. Still, our conclusion that visual sensations form a nonEuclldean manifold seems to imply a result which is unsatisfactory: Visual and physical perception of our external environment are contradictory to each other. From a principal point of view It is therefore significant that It is possible to remove this: contradiction. We shall show in this section that. In fact, the two perceptions are not contradictory: Physical observation leads to a hyperbolic metric If discussed from a relativistic point of view. On 'the other hand, this -result will give us new evidence for the hyperbolic metric of binocular vision.

assume

We confine ourselves to observations of the horizontal plane. Let us that a uniocular observer Is placed at the height h above the horizontal
plane. Objects in this plane are revealed to him by light signals emitted from the point P of the plane. The light signals which reach C at the time t = 0 give to the observer the clues for his visual sensation and are combined to an apparent distribution of objects in this plane at one par¬ ticular time moment of his individual time scale.

Fig. 50

X

It is clear that In our assumed x, y, z,

coordinate system this combination does not refer

to simultaneous events. The light signal which

reaches the observer C at the time t = 0 was

emitted from P at the time

t -

x2 + y2 + h2

(5-41)

where c Is the velocity of light.

Consequently: The physical space-time manifold which is knitted together by the observer at t = 0 to one sensation of space _is_ not the manifold z = 0, f = Q In our physical coordinate system. However, it Is the two-dimensional man¬ ifold.

2,2

2

Ct -X

y =h z = 0

(5-42)

of- the physical x, y, z, t space.

The metric relations of the four-dimensional space-time world are now de¬ termined by the general line element

ds2 = dx2 + dy2 + dz2 - c2dt2

(5.43)

according to the theory of relativity. This is a "pseudo-Euclidean" line ele¬ ment; it has, like Euclidean line elements, the form of an algebraic sum of the squares of the differentials. It would be truly Euclidean if -c2dt2 could be replaced by +c2dt2.
The general line element (5*4?) now determines the metric of any manifold of less dimensions which is embodied Into the four-dimensional space-time world. It does this In a similar way a3 the Euclidean line element of the three-dimen¬ sional space
ds2 = dx2 + dy2 + dz2
In the case of surfaces or curves embodied in the Euclidean space.

b0

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Which raetric ia nov Introduced by (5-43) in the two-dimensional manifold (5- 2) . Ne f ir3t write this manli old In the following parametric form

ct = -h cosh.D

x = h sinh D cos 9 y = h 3inh D sin 9

(5-44)

z = 0 where D and 9 are the variable parameters.
We have

cdt = -h sinh D dD

dx = h( cosh D cos 9 dD - sinh D 3in9d?) dy = h(cosh D sin9 dD + sinh D cos 9 d9)

(5.45)

and hence

dz = 0

ds2 = dx2 + dy2 + dz2 - c2dt2 = h2(dD2 + sinh2 Dd92) i.e., the metric of the hyperbolic geometry.

(5-46)

We thus cannot be surprised that the geometry in which our visual sensa¬ tions form themselves is the non-Euclidean hyperbolic geometry. Indeed, our re¬ sult means that already the physical events which are the direct basis for our Instantaneous sensation are forming in the physical world a manifold with a hyper¬ bolic metric.

5.5. The preceding theoretical considerations give us strong evidence that the psychometric of our spatial sensations is determined by the hyberbolic differ¬ ential

ds2 = -—- (a2dy2 + d92 + cos29dG2) sinh a (Y + M-)
The evidence is certainly strong enough to justify deriving the mathematical Im¬ plications of the above metric for binocular vision. If these consequences then represent observable and measurable phenomena we are given the means of obtaining additional evidence for the support of our theory.
We shall apply our theory In the following to mree significant phenomena of binocular vision; (1) The problem of the Frontal Plane Horopter where curves which appear as straight are physically curved. (2) Hillebrand's Alley Problem where walls which are physically not parallel are seen as a parallel alley. (3) The problem of the distorted room which appears to be rectangular.
We shall see that all these phenomena are a direct consequence of our theory and that they provide us with the possibility of testing the theory by the results of measurements. We shall also find, by comparing the results of measure¬ ments with the implications of the elliptic and Euclidean metrics that only the hyperbolic geometry can explain certain facts of observation.

Section 6 GEODESIC LINES: THE HOROPTER PROBLEM

6.1. With the aid of the quadratic differential

ds2 = M2(y)(CT2dy2 + d?2 + co329d92)

(6.11)

we are In a position to assign a linear size to the impinging characteristics

dYs ^9, d9 of a small section of a line. If we converge at a point P0 of the

physical space, a rectilinear Cartesian coordinate system In the neighborhood of

Po 13 established In which <7Mdy, Md9, Mcos9d0

are the coordinates of a neighboring point Pi.

To this neighborhood of PQ we then apply a

Euclidean metric which leads to the expression

(6.11). By moving our eyes to another point P

a similar Euclidean system is formed for the

neighborhood of this point, but different yard¬

sticks are used for the evaluation of the physi¬

cal quantities dy, d9, d@. Observation with

eye movements thus requires connecting the re¬

Fig. 51

sults of these different measurements to a unity, l.e., to a sensation of a finite section of space.

Mathematically this leads to the problem of Integration of the above dif¬ ferential (6.11). Since mathematical integration always gives a set of solutions, l.e., contains parameters which may be assigned arbitrarily, we recognize the possibility of different interpretations of the same Impinging characteristics. The former experience of the observer, his present purposes and other psychological factors will influence him In the choice of such arbitrary elements in inte¬ grating the primitive sensations (6.11) to a unity. It thu3 can be seen that the hypothesis of immutable elements in our sensations Is not contradictory to an un¬ limited variety of results In integrating a manifold of Immutable primitive sensa¬ tions to a total unity.

There are, however, certain Integration processes in which arbitrary in¬ terpretation has no place. The curves which determine the shortest connection between two points or the curves which are the result of attaching line elements to each other without change of apparent direction are uniquely determined by the differential (6.11). These geodesic lines of the metric (6.11), l.e., the ap¬ parent straight lines in our sensations, are invariant elements’and do not allow other interpretations. For the absolute localization In the apparent space we have stil_L. a tree choice. However, the impression that they are straight, l.e. that they are geodesic lines. Is enforced by the metric differential (6.11) itself and thus beyond our control.

Curves which are apparently straight are of great Interest in visual science. Helmholtz noticed the fact that vertical threads arranged by an observer in fixed head position to form an apparent frontal parallel plane are not actually

62

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

ar.ranged on a physical plane. The shape of the physical surface which appears to be plane varies with the distance from the observer. The shape of their cross section with the horizontal plane is schematically shown in Fig. 40. We shall apply our theory to this problem, and by identifying the horopter curves with geodesic lines of our metric (6.11), we shall find that the geodesic lines of the horizontal plane which are symmetrical to the x-axis form a set of curves exactly of the observed type.

Another classical problem is also closely related to the geodesic lines of our metric: Hillebrand's Alley Problem. If we determine on the frontal plane

horopters points which have the same geodesic dis¬

tance from the x-axls, two curves are obtained

'J

(Fig. 52) which have the same apparent (geodesic)

distance from each other. They thus should give

the impression of an alley formed by equidistant

walls. Again we shall find that the curves which

follow from our theory are of the type observed by

Hillebrand, who firBt made the experiment.

Instead of constructing two equidistant

curves, we may also ask for curves which are

Pig- 52

straight and parallel to each other and symmetric to the x-axis. We shall see that these two sets of

curves are only identical if our metric (6.11) is

Euclidean. They are different curves in the elliptic and hyperbolic geometry.

Our mathematical result will be that in the hyperbolic geometry the curves of

equal distance lie outside the curves of parallel direction, but that the situa¬

tion is reversed In the elliptic geometry. Blumenfeld, who repeated Hillebrand's

experiments with greater precision, found that the two Instructions lead the ob¬

server to different curves, with the curves of equal distance being outside the

curves of parallel direction. This experimental result we then have to consider

as additional strong evidence that the metric of binocular space sensations Is

given by the hyperbolic differential (6.11) with

M(y)

_1
sinh ct(y + n)

(6.12)

6.2. We determine the geodesic lines of the metric (6.11) as solutions of the problem of variation

vA M(y)

dy2 + dcp2 + cos2cpd02 = Minimum

(6.21)

We may solve this problem simultaneously for all the three geometries by writing M(y) in the form

m(y)

_2_
^(y+f) + ee-<x(y+n)

(6.22)

where e = -1, 0, +1 for the hyperbolic. Euclidean, and elliptic geometry respec¬ tively.
For the discussion of the solutions it will be advantageous to use, in¬ stead of y, 9, 0 other variables

GEODESIC LINES: THE HOROPTER PROBLEM

63

£ = p COS 9 cos 0 7] = p sin 9

(6.23)

where

£ = p cos9 sin 0 P = y52 + n* + C1 = e~CT (y U)

(6.24)

The problem (6.21), In these variables, assumes the spherically symmetri¬ cal form

2

—+ liri +-.diL = Minlmura

1 + ep‘

related to Riemann's normal form

(6.25)

ds2 =

(d£2 + dr]2 + dC2)

(1 + £P2)2

of the metric of manifolds of constant curvature.

We see from (6.25) that, in the £, 7), C space, our problem is formally identical to finding the light rays In a medium of index of refraction

1 n =
1 + ep'

(6.26)

It is a medium in which the optical substances are arranged in concentric spheri¬ cal layers around the origin.
In case e = 0 we have a homogeneous medium n = 1, and the light rays are the straight lines of the £, r), £ space.
In case e = 1 (elliptic geometry) we obtain

1 (6.27)

a medium well known as "Maxwell's Fisheye." Finally the hyperbolic geometry, z = -1, may be represented optically by
a medium
(6.28)

in which the Index of refraction increases from 1 to ® if p varies from 0 to 1. (Poincare's model of hyperbolic geometry.)

The Interpretation of the three geometries as an optical medium of spheri¬

cal symmetry makes it quite obvious that, in the £ , *) , C space, the geodesic

lines (light rays) are plane curves. Indeed, a light ray in such a medium will

remain in the plane which is determined by the origin and any one of the line ele¬

ments of the ray. Thus if we have found all the geodesic lines in the

q-plane,

we can find all other geodesic lines simply by rotating this plane around the or¬

igin into any other position. In other words, we may consider our problem as

solved If we know the geodesic lines in the

-plane.

64

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

ine u. e-nark, however, that the actual horopters In the physical x, y, z space

are In general not plane. We obtain these .horopters by expressing the equation of

the geodesic lines in the E, , r,, £ space first by the angular coordinates y, 9, 0

and then with the aid of the relations (2.23) or (2.31) by the physical coordinates

x> Vf z* Tne resulting curves are plane only if. In the

q, £ space, the geo¬

desic line lies in a plane through the q-axls; Indeed, in this case, they lie in

the same plane of elevation 0 - const. However, if a geodesic line is sought which

connects two points in different planes of elevation, then the resulting curve Is

plane only In exceptional cases (for example, if the two points lie in the median plane y = 0).

We shall confine ourselves in the following to the plane geodesic lines, in particular to those which lie in the horizontal plane; the frontal plane horop¬ ter and the alley curves are curves of this type.

6.3- Before solving the problem of variation (6.21) or (6.25), let U3
study the transformation of the horizontal x, y-plane into the £, r]-plane which is expressed by (6.23). As in §2, we do this by constructing the domain in the
0-plane into which the half-plane x = 0 of the horizontal plane is transformed.

from

Since { can vary from ti to zero, it follows that p = y/i-2 + q 2 may vary

t0

Px = e~a^

(6.31)

Consequently the half-plane x > 0 must be imaged Into some domain inside the circular ring enclosed by two circles of radius PD and Pi. The latter circle represents the infinity of the horizontal plane (y = 0). Both p0 and pi are smaller than one.

We have seen that the two eyes. In the y, 9-plane, are given by the straight lines

2<p + y = n -2cp + y = n

(6.32)

By p = e

+

it follows that,

by the logarithmic spirals

in the t, q-plane, these eye3 are represented

P = Poe20? P = P0e"2 °?

(6.33)

This leads to the boundary coordination illustrated in Fig. 53. The half-plane x > 0 is Imaged into a sickle-shaped figure which lies inside the unit circle of the £ , r] -plane.

GEODESIC LINES: THE HOROPTER PROBLEM

65

6.4. The geodesic lines of the horizontal plane (Horopters). The geodesic lines of the horizontal plane are the solutions of the problem of variation

fHY) /^2 dy2 + d<?2 = Minimum

or, in terms of the coordinates £, q :

f v/chp + dr)"

J

1 1 +^e2 p

= Minimum

(6.41) (6.411)

It is mathematically a little simpler to treat the problem in the first form (6.4l). For this purpose we introduce the variable

T = U(Y+ fi)
and consider 9 as function of t. Then it follows

(6.42)

where

/m(x) v^T 91

dx = Minimum

M(t )

1

T

-T

e +£e

(6.43)

The solutions of the problem, l.e., the geodesic lines, must satisfy Euler's differential equation which in our case has the simple form

It follows or

_d_ dx

M9 VI + 9 I 2

= 0

M9
y/l + 9'2

= const. = C

9' = C

1 /

4- £ 6 l \)

N/1 -

+ ee T)

(6.44)

66

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

With the aid of the relations

i(eT

+ ee“T)•

“

1/ T 2 ( e

-

ee~T)

e <D
H|CVI
II

i(eT - ee"1)'

-T
+ ee )

(6.45)

i(eT

+

ee‘T)2

-

1( T T (e

-

e e~T ) 2 =

e

we can write (6.44) in the form

where
is again a constant. to the equation

•1g/(eT - ee -T .) 1 T '
V^2 - "T(e 1 - £e"X):

(6.46)

1

C

k
/r^c2

The condition (6.46) can.be integrated directly and leads

1

T

_ T

9 = T0 + arc cos — (e - ee )

c~ K

or

2 k cos (T - To) = eT - ee~1

(6.47)

in which k and' to are arbitrary constants of integration. We thus obtain, rein¬ troducing y by t = cr(y + (i) the two-parameter set of geodesic lines

2 k cos(t - cpo)

(y + h)

-

-a(y + fi) e e

(6.48)

depending upon the parameters k and cpQ.
By now giving to e the values -1, 0, +1, we have the result that the hori¬ zontal geodesic lines of the three geometries of constant curvature are given by the following curves:

Hyperbolic geometry: Euclidean geometry:

cosh cr (y + u) = k cos(qp - qpo) eCT(Y + M-) = 2k cos(t - To)

(6.49)

Elliptic geometry:

sinh o(y +n) = k cos(t - To)

Since k = ® is a permissible constant, we conclude that the hyperbolae T = const, are geodesic lines in all three geometries.

6.5* We are especially interested in those geodesic lines which are sym¬ metrical to the x-axis, since these curves represent the frontal plane horopters of binocular vision. Since symmetry to the x-axis means that the two values ±9 must belong to any value of y, it follows that the parameter To In (6.49) must have the value To = 0.
The frontal plane horopters thus are given by the curves

Hyperbolic geometry:

cosh ff(Y + n) = k cos T

Euclidean geometry: Elliptic geometry:

ea (y + M-) = 2 k cos t sinh a(y+n) = k cos q>

(6.51)

GEODESIC LINES: THE HOROPTER PROBLEM

67

The constant k characterizes the point x0 vhere the horopter intersects the x-axis. Indeed, for cp = 0 ve have in the three cases

k = cosh cr (y0 + (i) k = iea(Yo +
k = sinh o’ (Yo + 4) vhere yQ then determines the point xQ by
Xo = cot Yo

(6.52) (6.53)

form

V/ith the aid of (6.32) we may write the frontal plane horopters in the

.

cosh a(y + jjl)

Hyperbolic geometry: ---y-= cos q>

cosh cr (Yo + p.)

Euclidean geometry: e°^ Y°) = cos 9

(6.54)

PV] , ,.

,

Elliptic geometry:

sinhcr(y + li) sinha(yo + p.)

= cos © r

Since the right side cannot be larger than one, and since on the left side we have functions of y which are monotically increasing, we conclude that on all of these curves

Y * Yo

(6.55)

In other words: The frontal plane horopters lie outside the Vieth-Muller circle through the median point xQ of the horopters (Fig. 54).

Fig. 54

6.6. The geodesic lines of the horizontal plane assume a remarkably simple form if the £ , r\ map of our geometries is chosen. Ve know that these geodesic lines then are given by the llghc rays In a medium of index of refraction

n = 1 + ep

(6.61)

We obtain these "light rays" by Introducing p = e~a^Y+4) Jn thQ general equation (6.48). This gives

66
or where

MATHEMATICAL ANALYSIS OF BINOCULAR VISION
2k p cos(9 - To) = 1-1 - sp 2 S2 + n2) + 2(S.So + n.rio) = 1 So = k C03 To, >1o = .k sin cp0.

(6.62) (6.63)

Obviously these curves are circles in case of the hyperbolic or elliptic geometries (e = — l), and straight lines in the Euclidean case (e = 0).

Without 103s of generality we may assume rj0 = 0 and thus discuss only the frontal plane horopters symmetrical to the ^-axis. Indeed, all other geodesic lines may be obtained from these special geodesic lines simply by rotation around the origin £ = 7] = 0.

form

In the hyperbolic case, e = -1, we write the resulting equation in the

(5 - S0)2 + V2 =

- l

(6.64)

and recognize a circle around a point f;0 > 1 of the %-axis which Intersects the unit circle at right angles (Fig. 55)• The geodesic- lines of the hyberbollc

Fig. 55

geometry thus are resresented in the i-, rj map of this geometry by the circles which Intersect the unit circle at right' angles.
In the elliptic case we have

(S + So)2 + 42 = 1 + S o

(6.65)

and recognize a circle around the point -£0 which goes through the points r) = -tl of the unit circle (Fig. 56). The geodeslc lines of the elliptic geometry thus are given by the circles which Intersect the unit circle at two points at oppo¬ site ends of a diameter.

The geodesic lines of the Euclidean geometry finally are simply the

straight lines of the

r)-plane (Fig. 57)*

GEODESIC LINES: THE HOROPTER PROBLEM

69

6.J. The above results can be used in a simple

way to determine the general form of the horopter

curves in the physical x,y-plane. We have seen

(Fig. 53) that the half-plane x * 0 is represented in

the

’l-plane by a moon-shaped domain. If we draw In

this domain the horopter curves of the hyperbolic ge¬

ometry, for example, we observe Immediately that these

curves are divided into two groups (Fig. 58)• The

curves of the first group go from the median point,

directly towards the boundary circle p = px (infin¬

ity of the x,y-plane) without leaving the domain. The

curves of the second group, however, cross first the

red lines (eyes), leave the domain, cross the same

red lines again, and then reach p = p 1 in the original domain. These two groups are separated by one horopter which just touches the red lines.
This means, in the x,y-plane, that the horopter curves of the first group go from a point x m of the x-axis without leaving the half-plane x = 0, towards infinity which is reached, asymptotically at a certain angle
?co < ti/2.
The curves of the second group go from their median point x m to the eyes, cross Into the left half-plane x < 0, loop back through the eyes into the right half¬ plane, and approach infinity asymptotically. This general behavior Implies that, for great values of xm, the horopters must be convex towards the left and for small values of xm concave to the left. At a certain median point xm = xQ, the horopter will be nearly flat. This is Illustrated In Fig. 59, where a schematic drawing oi these curves is given. For a numerically correct picture refer to Fig. 71.
The geodesic lines of our hyperbolic geometry thus have, in the x,y-plane a general form which agrees perfectly with the horopter curves determined'by'ex¬ periment in the neighborhood of the x-axis. However, we cannot consider this as

70

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

evidence for the hyperbolic geometry, since the two other geometries furnish horopter curves with the same phenomenon: changing. In the neighborhood of a point

X? J ® X"aXi!' fr0m.COncave to convex curves. This can be seen by replacing in

~g‘ 5

e geodesic lines of the hyperbolic geometry by the proper geodesic lines

oi the Euclidean and elliptic geometry (Fig. 60).

Euclidean Geometry

Fig. 60

Ell.pt.ca/ Geometry

6.8. The phenomenon that frontal plane horopters are concave In the neigh¬ borhood of the observer but convex at greater distances can obviously be used for a numerical test of the theory. We determine for this purpose in this section the curvature of the horopter curves on the x-axis. We carry out the calculation only for the hyperbolic metric.

The frontal plane horopters of the hyperbolic metric are given by the first equation (6-54) :

In the neighborhood of 9

cosh g(y +F ) cosh a ( Yq + p)
0, we have

cos 9

(6.81)

and

cosh a(Y + h) = cosh cr(Y0 + N) + a sinh cr(Yo + F) Ay +
cos 9 = 1 - -|92 +. Hence we may replace (6.8l) by

AY = - ^ coth a ( Yq +N)?2

(6.82)

If we are Interested only in the shape which the curves have in the immediate neighborhood of the x-axis.

GEODESIC LINES: THE HOROPTER PROBLEM

71

We next have to translate the relation (6.82) Into Cartesian x,y coor¬ dinates. From the transformation formulae

x = cos 29 + cos Y slnl'
3 in 2q> y = sin y we conclude for small values of 9 that

(6.83)

■ _ 1 - 2y2 + cos Yo - sin YpAy

sin Yo + cos YqAy

+

1 + cos Yo •4 1
sin Yo

29:
1 + cos Yo

ay
} +
Sin Yo .

(6.84)

The expression on the right side contains all terms of second order in 9 since
Ay Is by (6.82) of second order in 9 so that (AY)2, etc., may be omitted for our
purpose.

By subtracting xG =

from (6.84), we find

Ax Now, if using (6.81),

1 + cos Y( sin Yo

29'

Ay

+

1 + cos Yo sin Yo

we obtain

Ay = C'9

A. x = _ - 1 - + -cL os ° Yo sin yo

(2 tan A1 yo + c) 2

The second equation (6.83) gives to a sufficient approximation

9 = sin yo y and thus, by introducing this in (6.87), we have

Ax = -I 0032

(2 tan iYo + 0)y=

(6.85)
(6.86)
(6.87)
(6.88)

(6 86) Thi3 13 3 Parab°la whlch ha3 the 3ame curvature K as the original curve

We thus may formulate the theorem: If a curve which is symmetrical to

the x-axls has, for small values of 9, the development

-

ay = C92 + . then the curvature of this curve on the x-axls is given by the expression

72

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

With the aid of this result we obtain the curvature of the horopter curves (6.8l) by Introducing from (6.82) the value

C = - — coth c(y6 + p)

It follows

K = + cos2 -|yo { -2 tan j?yo + 7^ coth o(yo + p)J

(6.891)

in which Y0 characterizes the point x0 = cot -|yo where the horopter Intersects the x-axis.

The sign of K is given by the bracket in (6.891); this bracket approaches -°° if Yo -*• n , i.e., if x0 -*• 0. It approaches asymptotically the positive value

— coth au if x0 -> <*,, i.e., y0 -> 0. Consequently it must be zero at some point

2tx

~

x0 in between. For the computation of thl3 special value x0, we have the equation

K = 0, i.e., by (6.89I):

Y

1

tan -f tanh a (yo + f) = ^

(6.892)

Since x0 = cot -§Yo, we may write this equation as follows

x0 = 4o tanh ct(yo + F )

(6.893)

and conclude immediately that

x0 < 4

(6.894)

We can use the equation (6.892) and, more generally, the observable curva¬ ture relation (6.891) In order to determine the individual constants a and p of our hyperbolic metric.

In the next section we shall find another method based on the alley exper¬ iments. For one of Blumenfeld's observers (Lo) the constants a and p have been calculated on the basis of Blumenfeld's data:

a = 14.58 p = 0.0809

(6.895)

We use these values for the calculation of yo from the transcendental equation (6.892).

The result is

Yo = 2.°36

and hence

x0 = cot Yo/2 = 48.7

The Interpupillary distance of Blumenfeld's subject is 3O.5 mm. Hence we find that for this observer a frontal plane horopter at Xo = 148.2 cm. should have the curvature K = 0, i.e., coincide practically with a curve whloh Is phys1ca11y straight.

GEODESIC LINES: THE HOROPTER PROBLEM

75

6.9. It is easy to carry out similar calculation, for the Euclidean and elliptic geometry. From (6.54) it follows

a

1 2

AY = -^ 9

Ay

tanh a(yo + hOt* 2ct

(6.91)

for the Euclidean and elliptic horopters respectively. We Introduce the constants C which follow from (6.91) in our general theorem (6.89) and obtain formulae for the curvature K of the horopters:

Euclidean Geometry:

K = cos2 ^Yo (72^a - 2 tan -gYoj

Elliptic Geometry:

tanh cr( Yq + M-)

K = cos2 -§Yo

2a

- tan -§Yo,

(6.92)

The curvature K Is zero for a value Yo which satisfies the equations

.

iv

1

tan ¥Yo = 7^:

tan -§Y0 coth a(y0 + p) = -^

(6.95)

respectively. By introducing Xo = cot -“Yo, we obtain

Euclidean Geometry: x0 = 4a

Elliptic Geometry:

xQ - 4a coth cr(Yo + P-)

(6.94)

The difference between the three geometries thus reflects upon the rela¬ tion of the constant a to the position x0 where the frontal plane horopter Is physically plane. Indeed, we have the Inequalities

Xo < 4a

(Hyperbolic geometry)

x0 = 4a

(Euclidean geometry)

(6.95)

■=tot
A
O X

(Elliptic geometry)

Section 7 THE ALLEY PROBLEM

Fop the application of our theory to the alley problem, we distinguish with Blumenfeld*) two types of alleys: alleys which are formed by walls of equal apparent distance, and alleys formed by apparently parallel walls. The curves where the walls Intersect the horizontal plane are, for brevity, called Pis tance Curves and Parallel Curves.
Our first problem is to interpret these curves mathematically, i.e., to find mathematical conditions which describe adequately the psychological impres¬ sions of equality of distance and of parallelism. Thi3 problem is easily solved in the case of the distance curves, but we shall find that the concept of paral¬ lelism Involves certain difficulties. The reason for this difficulty can be seen in the fact that parallelism of lines has an immediate and intuitive meaning only in the Euclidean geometry.

7.1. Distance Curves. The basis for the construction of aidlstance alley

are the frontal plane horopters, i.e., the geodesic lines symmetrical to the

y

x-axis. Consider, for example, the geodesic

through P0 and determine a point Q0 on this line

which has a given geodesic distance 5 from P0-

Similarly, on another geodesic through Pi we can

find a point Qi which ha3 the same geodesic dis¬

tance from Pi. By doing this for the whole set

of geodesic lines, a curve is obtained by the

points Q which has a constant geodestic distance

S from the x-axis. If we determine a similar

curve symmetrically located on the other side of

the x-axis, an alley is found in which the two

walls have the constant geodesic distance 2S from

each other. We expect that these alleys are the mathematical expression for the

observed Distance Alleys.

We determine the equations of the distance curves simultaneously for all three geometries in question. On account of (6.48) we have for the frontal plane horopters the equation

2 k cos 9 = e Y*H)

e e -a(y+n)

(7.11)

where e = -1, 0, + 1 for the hyperbolic, Euclidean, and elliptic geometry respec¬ tively. By introducing temporarily the variable

ve may write

= ff(Y M-)

2 k cos 9 = e T - ee T

(7-12)

♦Blumenfeld, Walter: Untersuchungen liber die scheinbare Gr'dase lm Sehraume. Z.f. Psychol. u. Physiol, d. Sinnescrg., 65: 24l-Ul6, 1915*

THE ALLEY PROBLEM

75

The geodesic distance S of a point Q on such a curve from the median point P ( 9 = 0) is given by the Integral

9 S = 2

\/dT2 + dtp2 - e 1 + e ed 9

(7-13)

We consider x as function of 9 (given by 7*12), and thus have to evaluate the integral

p y 1 + 7T

S = 2/

e_x + e e x d 9

(7.131)

Wow, from (7.12) It follows that

and hence

-2 k sin 9 = (ex + ee-T) x'

1 +x

k2 sin2 9 + ir(eT + se T)2 T(ex + e'e-x)«

With the aid of the Identity

(7.14)

i(eT + e~T)^ = i(eT - e e_x)2 + e and by taking (7-12) into account, we may write (7-l4) as follow:

(7-141)

1 +X
so that our integral (7.131) becomes

k +e ~(eT + e e x)

S = v/k2 + From (7«l4l) it follows that

9 d 9
i(e'l+ e e~ T)2

(7.15)
(7.16)

and hence

1r(eT + e e x)2 = k2 cos29 +

S - \/ k

+ e

9

d 9

/ ,2

2

/ k cos 9 + e

(7.161)

= — arc tan v42 + - tan *

We write this last result in the form

1 tan
A

S =

e tan 9

(7.17)

It determines the geodesic distance S of a point of latitude 9 which lies on the horopter determined by the constant k.

76

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Finally we eliminate the constant k with the aid of (7.11). We first 'rite (7-17) a little differently by using the identity

>in /F S

tan y/e~ S 1 + tan2 s/T S

It foilow3

3in yTT S =

3 in 9

v/T

e + k cos m

and hence by (7.11) and (7.l4l):

— sin y/T S ve
or by reintroducing y by t = o-(y + fi) :

2 s in 9 eT + ee T

(7.18)

sin vT" S =

2 s in 9

VT

' '

Pff(Y + n) + e

cr(Y+fJ- )

(7-181)

The equation (7-181) determines the geodesic distance of any point of the hori¬ zontal plane from the x-axis in terms of Its bipolar coordinates y, 9. Finally,
we assign to e Its values -1, 0, + 1 and have the result that the geodesic distance of a point Q from the x-axis is given by the functions:

Hyperbolic Geometry: sinh S =

sin 9

sinh cr( y + M-)

Euclidean Geometry:

S = 2 sin 9 e~ a(7+v) (7-19)

Elliptic Geometry:

sin 9 sin S =
cosh a ( y + (j.)

The distance curves are characterized by the condition that the geodesic distance from the x-axis is constant. Thus these curves are given by the equatIons:

Hyperbolic Geometry: 3inh <r(y + n) - sin 9

Euclidean Geometry: ea^+M^ = 2 C 3ln 9 Elliptic Geometry: cosh a(y + n) = C sin 9

(7.191)

where C is a constant.

7.2. For the discussion of the distance curves (7.191), we introduce

again the

r) map of the three geometries obtained by the transformation

£= e

r ^ cos 9

n = e-°(Y+n) am 9 o = yfTV =, e-ff(Y+n)

(T.21)

Ve write the three distance curves (7.191) In the unified form

THE ALLEY PROBLEM
etf(y+fO + e e v 1 ' = 2 C sin <p
and find, in the £, r]-plane:

77 (7-22)

1 + ep2 = 2 C p sin 9 or

eU2 + 712) — 2 C rj +1 = 0

(7.23)

For e = -1 (Hyperbolic geometry) this is the equation of circles

£2 + (tj + C)2 = 1 + C2

(7.231)

which all go through the points 7^ = 0, £ = ± 1 of the unit circle and thus have their center -C on the r)-axis.

In case e = 0 (Euclidean geometry), we find the straight lines

parallel to the £-axls.

1 4 = 2C

(7.232)

In case of e = +1 (Elliptic geometry) we obtain

^2 - (n - c)2 = c2 - 1

(7.233)

i■e* i circles around points C of the 7]-axis which intersect the unit circle at right angles.

The three types of curves are shown in Fig. 63. We conclude immediately tnat the actual alleys in the x,y-plane must go to the eyes and approach infinity asymptotically with a certain direction

n/2

It follows that these curves must have the general form shown In Fig. 64, which is in agreement with experimental results.

7-3- Parallel Curves. The mathematical formulation of the psycho¬ logical impression that an alley has S parallel walls Is not immediately as clear as in case of distance alleys. In fact, we can easily give different mathematical concepts of parallelism and defend them with equally good argu¬ ments .
The statement that two lines elements not attached to the same point are parallel has no absolute meaning in non-Euclidean geometries.

78

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

It is true that Levl-Civita'3 concept of parallel transfer allows us to move a given line element d/0 along a given curve C to another position Pi "parallel with itself," i.e., without apparent change of direction. However, the result
H

of this transfer depends on the chosen curve C. A different result will he obtained if another curve is chosen on which the line element is transferred. The Euclidean geometry is the only one in which par¬ allel transfer is independent of the path C so that a statement regarding the parellelism of two line elements not attached to the same point has an abso¬ lute meaning only in this special geometry.

From this consideration it follows that

only in the Euclidean geometry will the instruction

to construct alleys with parallel walls without pay¬

Fig. 65

ing attention to the distance of the walls lead the observer to a uniquely determined reaction. If,

however, as we have good reason to assume, the metric of binocular vision is non-

Euclidean, a uniquely determined reaction cannot be expected, and thus a more

specified instruction must be given. We shall discuss in the following two dif¬

ferent types of "Parallel Alleys" which may be considered as two different inter¬

pretations of Blumenfeld's experimental parallel alleys. We are forced to this

ambiguity since the experiments have apparently been carried out without such

specified instructions. We shall consider the second of the following interpre¬

tations (7.5) as the one which represents Blumenfeld's experiments with greater

probability.

7.4 Parallel Curves (1st type). We assume that P0 and Q0 are two fixed points symmetrical to the x-axis. Let <9 = 90 be the "hyperbola of sight" through P0 so that the straight line 0Po includes an angle with the x-axis prac¬ tically equal to 90- The observer is asked to move a point Pi in such a position that the line P0Pi includes with 0Po the same apparent angle 9o as 0Po with the x-axis (Fig. 66). The process is continued with Pi, Qi as fixed, and two other points P2, Q2 moved into position. As a result two alleys are obtained with walls which are parallel in the sense that two individual opposite wall sections seem

THE ALLEY PROBLEM

79

to be parallel to the median plane. Note that nothing is required vlth regard to the impression which the curves give in their total extension.

It is easy to find the equations for these curves. Let (dy, dtp) be the line element of the curve at P0 and (6 y, &T = 0) the line element of the curve 9 = 9o at P0. The two line elements in¬ clude an apparent angle w with each other. This angle is determine by the quadratic differential

namely, according to (3-24) given by

ds2 = M2 (y) (a2 dy2 + dtp2) formula

(7-41)

_M2 (y) (q2 d y6y + d tpSqj )_

- \/m~2 (y) (a2 dy2 + d92)

/m2 (y) (a2 Sy2 + 6<p2)

(7-42)

We notice immediately that M(y) cancels in this formula so that the ap¬ parent angle o is independent of the choice of M. It is the same not only for the three geometries of constant curvature, but also for any other non-Euclidean geometry which has a metric of the type (7.41).

Since 69 = 0, we find

as expression const. If we cpndition

for the require

COS (0

gdy Vct2 dy2 + d92

apparent angle of the curve with the that u = 9o, or in general w = 9, we

(7.4-5)
lines of sight 9 = obtain from (7-43) the

cos 9

o' d Y_

\/o2 dy2 + d92

(7-44)

or

d 9

dj y = crtan 9T

(7.45)

The solution of this differential equation is

eCTY = C sin 9 where C is a constant of integration.

(7.46)

.

We P°int °ut again that these "Parallel Curves" are entirely independent

of the function M(y). We also notice that they are identical with the distance

curves (7-191) of the Euclidean geometry.

In the Euclidean geometry Distance Alleys and Parallel Alleys are identical.

We shall find a similar result in the next section where a different mathematical lormulation of parallel alleys is discussed.

By introducing the variables 5, rj by (7-21) in the equation (7-46) we ob¬ tain, of course.
4 const.

80

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

so that the straight line3 of the

r)-plane parallel to the £-axis may be inter¬

preted a3 representing parallel alleys in all three geometries.

This interpretation of the straight lines 7] = const, in the £, 7^-plane leads us to an important conclusion. Let us .assume that a distance alley and a parallel alley of the above type have been constructed, both alleys starting from a fixed pair of end-points P and Q (Fig. 64). The parallel alley, imaged into the £, r]-plane (Fig. 63), is given by the straight line through P parallel to the {--axis, and is identical with the distance curve through P only if the metric is Euclidean. The very fact that distance and parallel alleys are found to differ points out that the visual metric cannot be Euclidean.

Moreover, we see in Fig. 63 that--going from P to the eyes--the distance curve lies above the parallel curve if the geometry is hyperbolic and below this curve if the geometry is elliptic.

This means in the x,y-plane that the distance curve lies outside the parallel curve if the geometry is hyperbolic and inside if the geometry is ellIp! tic.

The alley experiments thus give us the possibility of deciding by experi¬ ment which metric is characteristic for our spatial sensations. Blumenfeld's data show clearly that the distance curves lie outside the parallel curves. Thus if we are allowed to identify his parallel alleys with the above-discussed alleys, we must conclude that the psychometric of space sensations is hyperbolic.

In the next section we shall consider another type of parallel alleys which, if identified with Blumenfeld's alleys, leads to the same conclusion.

7.5. Parallel Curves (2nd type). Neither the distance curves nor the parallel curves of J.h are geodesic lines if the metric is hyperbolic or elliptic. They are geodesic lines, however, in the Euclidean geometry.
Thus these curves will not appear straight if we observe them by paying attention to their total depth extension. In view of this fact, we now change the instruction for the observer: By paying special attention to the walls as a whole he is to arrange the points so that they appear on two straight lines par¬ allel to the x-axis. Consequently the resulting curves must be given mathematical¬
ly by geodesics, i.e., they must be two curves of the set (7.43).

e(c2 + rj2) + 2( $ £0 + rJ No) = 1

(7.51)

in the £, n map of our geometries.
We also know that the second curve must be the mirror image of the first curve with regard to the f-axis.
Obviously there are infinitely many pairs of geodesics of this type; in fact, we may choose the first curve arbitrarily from (7-31) an(l then pair it with its mirror image

s(^2 + r,2)+2(^^o~ N No) = 1

(7*32)

to an alley with walls erected on two straight lines.
In order co pick out among the geodesic lines pairs which give the ap¬ pearance of being parallel to the median plane, we use a sort of principle of

THE ALLEY PROBLEM

81

correspondence. It is clear that in case of the Euclidean geometry (e = 0) the

desired pairs are given by

= 0 in (7*51) 30 that

where r)0 is an arbitrary constant.

?! = +
- 2t)0

(7-53)

We now uphold the same principle in case of the other geometries:

Two alley curves appear to be parallel to the median plane if they are found

from the general set (7 -51) by the same prlnclple (

= 0) which determines paral¬

lel alleys in case of the Euclidean geometry.

This consideration leads us to the result that the special geodesic lines

e U2 + 'n) + 2 7] 7] o = 1

(7-54)

in which rj 0 I3 an arbitrary constant will give the Impression of an apparently straight line "parallel" to the median plane. Obviously these curves are circles in the £, r]-plane symmetrical to the rj-axls if e / 0.
In case of the Hyperbolic Geometry (e = -1), we get the circles

£2 + (71-Tlo)2 = 71o2 - 1
normal to the unit circle and centered around a point rj 0 of the 7^-axis. In case of the Elliptic Geometry we have the circles

(7-55)

£2 + (t] + 7") o ) 2 = Tjo2 + 1

(7.56)

through the points £ = + 1 of the E,-axis and thus also centered around a point -7i0 of the 7>axis. For the graphical illustration of these curves we can use Fig. 63 and Fig. 64, but we have to interchange the legends "Hyperbolic Geometry" and "Elliptic Geometry."
By introducing, instead of £, 7] the variables y, 9, we obtain from (7-55) and (7.56) the following result:

Parallel alleys according to the above instructions are given by the curves

Hyperbolic Geometry: cosh o (y +g) = c sin 9

Euclidean Geometry: eCT(Y + -u) = 2 C sin 9 Elliptic Geometry: si nil <j(y + u) = C sin 9 where C is an arbitrary constant.

(7-57)

For the interpretation of Blumenfeld's parallel alleys we shall consider the mathematical definition of parallel curves given In this section to be the one which has the greatest probability. We shall justify this point of view in the next section.

We may collect the results of 7.3 and 7.5 with regard to Distance-and Parallel Alleys in the following table.

82

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

Hyperbolic Geometry: Distance Curves Parallel Curves

A: slnhc (y + n) = C sin 9 B: coshu (y + (i.) = C sin 9

Euclidean Geometry: Distance Curves Parallel Curves

<7(y +n)

= 2 C sin 9

Elliptic Geometry: Distance Curves Parallel Curves

B: cosher (y + u) = 6 3^-n T A: sinher (y + n) = C sin 9

By disregarding the Euclidean case, we thus recognize that only two dif¬ ferent types of curves are involved, namely.

Type A: sinher (y + jj.) = C sin 9

(7.58)

Type B: cosher (y + p.) = C sin 9

Curves of the type A represent the Distance Curves of the Hyperbolic Geometry and thT"Parallel Curves of the Elliptic Geometry. Curves of the ^ype B represent the Parallel Curves of the Hyperbolic Geometry and the Distance Curves
of the Elliptic Geometry.

In the

r] -plane these curves A and B are given by the circles shown in

Fig. 67. Starting from a point P we see that the A-curve lies always above the

B-curve in the regxon between P and the eyes. This means in the x,y-plane that

the A-curves lie always outside the B-curves and that both curves approach one of

the eyes (Fig. 64).

This means for our a-lley problem: Consider a Distance Alley and a Parallel Alley both starting at the same fixed point P (and its mirror image Q) .

If the geometry is hyperbolic, then the Distance Alleys lie out side the Parallel Alleys.

If the geometry is .elllptic, the ParalTel Alleys lie outside the Distance Alleys.

then

If the geometry is Euclidean, then Parallel Alleys and Distance Alleys are identical.

Fig. 67

Blumenfeld's data demonstrate that Parallel Alleys and Distance Alleys are different, and that the Distance Alleys lie outside the Parallel Alleys. Hence we are forced again to the conclu sion that the proper geometry for our spatial sensations is the Hyperbolic Geometry.

THE ALLEY PROBLEM

83

7.6. It is not difficult to support the principle of correspondence used

in the previous section hy arguments of a m ore principal nature. Ve may consider

the often used

rj maps of the apparent horizontal plane as a plane map of the

actual sensation of objects in the physical horizontal plane. It is of course not

a perfect map since ve must require from a perfect map that it must give immediate¬

ly the correct linear and angular relations of the sensational manifold. We know

that such a map is impossible. However, the £,, rj map illustrates directly certain

features of our sensations, for example, the fact that the Vieth-Muller circles

y = const, are seen as circles around the observer. Indeed, these circles are

represented by the circles

^ 2 + „ 2 = p = e"1 lY * ^ = const.

(7.61)

in the E,, 7]-plane. Furthermore, the "hyperbolae of sight" 9 = const, are given by the radial line r\/E, = const, in the E,, rj -plane, and thus the E,, q map describes directly the apparent significance of these hyperbolae.

Even more important is the fact that the E,, q map is conformal, i.e., that the Euclidean angles on the map are equal in size to the non-Euclidean apparent angles given to us by the metric of our space perception. Indeed, consider two line elements (d£, dr)) and (S£ , 6 r) ) attached to a point P. The Euclidean angle
on the £ , rj map is given by the formula

\

d^6^ +

d r) 6 7]

cos u = i/as2 + d.,2 /ss'+sn2

(7'62)

The apparent angle co * of the two corresponding line elements (dx, dy) (6x, 6 y) in the physical plane, however, is determined by the metric differential

where

ds2 = n2 (£, rj) (dJ=2 + dr]2)

(7-63)

Fig. 68

as we have seen In 6.2.

By applying the general formula (3-24) for the angle co*, we get

cos CO * =

_h2(d^ 5%+dri 5r?_ /n2 (d£2 + dr]2) \/n2 "(6£2 + Sq2)

=

d q 5 5 + dr)5r)

/d£2 + dr] 2 \/S£2 + 6t] 2 (7-64)

and we see immediately that

cos co* = cos co, i.e., co * = co

This proves our statement, that the E,, T] map is conform, i.e., that the apparent, non-Euclidean angles are directly given by the Euclidean angles of the map.

The apparent place of the observer- Is the center E, = r) = 0 from which the lines of sight 9 = const, seem to emerge and around which the Vieth-Muller circles y = const, are apparently located. He observes the horizontal half-plane, i.e., the often illustrated sickle-shaped part of the E, , q map, through the receiving set represented by the logarithmic spirals

P =

p e i2<J9 o

(7-63) 2

84

MATHEMATICAL ANALYSIS OF BINOCULAR VISION

The ^-axle of the map represents the median line of the observer and, since the map is conform, the r)-axi3 must represent a direction apparently normal to the median line. This direction, of course, no longer lies in the field of view, but, without doubt, the observer is conscious of it. An alley with straight walls thus will be interpreted as parallel to the median line if the imagined ex¬ tensions of the walls approach the apparent Y-axis of the observer at right angles. In the conformal f;, r) map such alleys thus must be given by geodes 1 c
ne3 wh 1 ch are normal to the r) -ax is. By (7-51) it follows that such geodesic lines must have the form (7-54), namely

£ (5 2 + n 2) + 2 TJ T} o = 1

(7-66)

which leads U3 back to the parallel curves of the preceding section.

[ .7 • The imperfection of the £ , 7] map has to be seen in the fact that it gives directly only the correct apparent angles. We know that an isometric Euclidean map which gives directly the 3ize (and thus also angles) is impossible. However, we can easily construct another significant map which, at least, repre¬ sents the quality of "straightness" directly. This means that the Euclidean straight lines of the map represent the geodesic lines of the non-Euclidean mani1 olds in question. This so-called projective map (Klein's Model) is obtained by submitting the 5, r\ map to a "radial distortion."

R

(7-71)

In case e = 0 this new map is a similarity transformation of the £, q map.

In case e = -1 (Hyperbolic geometry) the unit circle p < 1 is transformed Into the unit circle R t 1.

However, for e = 1 (Elliptic geometry), the unit circle p < 1 is expanded over the whole plane.

By introducing

X = R cos <p Y = R sin 9

(7-72)

we may express the transformation (7-71) as a transformation of the physical x,y-plane into the X,Y-piane, namely, in terms of bipolar coordinates y, 9:

_

2 cos 9__

X ~ ToTy “+ M J~1 e :<r (Y + H )

(7-73)

Y

eff (y + H ) -

e-<jTy + H)

(7-73)

which takes the place of the transformation (7.21).
It can be seen that the geodesic lines in all three geometries are given by the straight lines of the X, Y map. Indeed, in £, q coordinates, we have found in (6.62) the equation for geodesic lines

1 - e p' = 2 k cos (9 - 90)

(7-74)

THE ALLEY PROBLEM

85

Hence, It follows with the aid of (7.71) that their equation in the X,Y-plane is

R cos (9 -90) = const. or

X cos 90 + Y sin 9 o = const, which is the equation of a straight line.

(7.75)

The metric quadratic differential in these new coordinates X, Y or R, 9 has now the form

ds = L dX_ + dYr2 + e (YdX - XdY) 4 [1 + e (X* + Y^)]^
as one easily verifies.

dR*

R*

(1 + e Rrr)"? + 1 + e R‘ d? (7-76)

If we consider the X,Y map as another attempt to interpret our sensations in a Euclidean realization, we may say that it represents truly the quality of ap¬ parent straightness. However, it is not in the least conform and thus does not allow us to determine apparent angles from the angles of the map.

Except for the angles 9 of the lines of sight at the origin! Indeed, the hyperbolae of sight 9 = const, are still given by the straight lines through the origin in the X,Y-plane, and their angles with each other are equal to the appar¬ ent, non-Euclidean angles. Also the property of Vieth-Muller circles to appear as circles around the observer is truly represented on our map. Thus if any visual curve includes an apparent right angle with a radial line, then its image on the X,Y map will do the same. As before, the "egocenter^ of the observer has to be identilied with the point X = Y = 0 on the map. He observes a section of the

X,Y-piane through a "receiving set" of a

y

similar form as in the case of the

7}

map. The physical half-plane x l 0 lies

between two circles of radius R0 and Ri

given by

Ro = yn 71 + (X ) and

e e 7^(ri + u)

Ri =

- £ e -ap

the latter corresponding to the infinity of the physical space. The two eyes are represented by the two curves of spiral type

e + 2CT9

R

Po

1 - £p02 e- 4 a T

(7.77)

Fig. 69

as shown in Fig. 69.

The X-axis determines the median line of the observer and the Y-axis since the map is coniorn at the point 0, a direction in the consciousness of the observer normal to the median line.