THE GEOMETRY OF BINOCULAR SPACE PERCEPTION LsGRAND H. HARDY, M.D. GERTRUDE RAND, Ph.D. M. CATHERINE RITTLER, B.A. wtd ALBERT A. BUNK, PI..D. MoA«M««tlc«l Analyst PAUL BOEDER, Ph.D. THIS REPORT HAS BEEN DELIMITED AND CLEARED FOR PUBLIC RELEASE UNDER DOD DIRECTIVE 5200,20 AND NO RESTRICTIONS ARE IMPOSED UPON ITS USE AND DISCLOSURE, DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED, THE GEOMETRY BINOCULAR SPACE PERCEPTION fcy LEGRAND H. HARDY, M.D. GERTRUDE R*ND, Ph.D, M. CATHERINE RITTLER, B.A. and ALBERT A. BLANK, Ph.D. Mathematical Analyst PAUL BOEDER, Ph.D. Mathematical Consultant From the Knapp Memorial Laboratories, institute of Ophthalmology, Columbia University College of Physicians and Surgeons, New York 'This work was supported by a contract bwtw«*»n the Office of Naval Research and Columbia University FOREWORD We think that this is an important report because here, for the first time, extensive experimental results and analytical details are presented which strongly support the Lüneburg Theory of the Geometry of Binocular Visual Space. This is a terminal report on research done under contract with the Office of Naval Research (N6onril7119; NR 143-638). The work is being terminated because of our inability to acquire and retain adequate personnel with the highly technical skills necessary for such work. A very high degree of mathematical analytical ability must be in constant and harmonious rapport with an equally high degree of laboratory experimental skills in order to carry out these investigations. In the untimely death of Rudolph Lüneburg we suffered an extremely severe loss. After a lapse of two years we were fortunate in acquiring through Professor Richard Courant one of Dr. Luneburg's associates, Dr. Albert A. Blank, who has shown brilliance in his mathematical attack. All the new mathematical analysis herein described and most of the formulation of this report are due to his effcrts. Our mathematical consultant, Dr. Paul Boeder, has given much time and enthusiastic encouragement to our working staff. Professor H.S.M. Coxeter, as a specialist in the non-euclidean geometries, contributed important suggestions which were partly carried out in the ancillary investigations of Dr. Charles Campbell who earned tb* D. Sc. degree for his part in this research. Dr. Bernard Altschuler and Dr. Anna Stein spent respectively one year and two years in the mathematical analyses during the early part of the study. The largest part of the actual experimentation was carried out by Dr. Gertrude Rand and Miss M. Catherine Rirtler. LEGRAND H. HARDY Principal Invesiigaior PREFACE This is a report of progress, theoretical and experimental, in the study of binocular space perception based on the theory of R. K. Lüneburg. ' The experimental evidence definitely supports Lüneburg's major conclusion that the darkroom visual space has a determinate non-euclidean metric or psychometric distance function which is a personal characteristic of the observer. In this report the metric has been developed in terms of coordinates closely related to, but different from those of Lüneburg. Much the same methods are used for determining the form of the metric as were suggested by Lüneburg. The theory gives an explanation of several well-known perceptual space phenomena such as the frontal geodesies, Blumenfeld alleys, and size constancy. We have not attempted here to present a review of all our work of the past five years, but only that portion of it which, still appears relevant and cogent. It would be fruitless to describe all the false clues and blind alleys that as a rule accompany the formation of any new theory. On the other hand, we are conscious that there are many gaps in our testing program. We employed a very limited number of observers because so few were available for experimentation extending over so long a period. We did not investigate every open door because so many doors were open. The greatest setback to our research was the untimely loss of ouV beloved friend and colleague Dr. Rudolph K. Lüneburg. To Lüneburg we owe the basic concepts and formulation of the theory. His was t h«; guiding hand for more than half of our experimental P3) - (P,, P3) where, in general, for any pair of points P., P., the symbol (P., P.) denotes the sensed distance between the points P. and P.. By employing a sufficient variety and number of specific initial conditions upon constructions of diverse kinds we may hope to establish statistically a functional** dependence of perception upon stimulus which may be considered a constant characteristic of the observer. In this way, given the mathematical description of the stimulus, it is possible to describe some constants of the observer's visual responses; that is, to give a mathematical description of the impressions of localization and form with respect to the observer's personal mental frame of reference - the observer's visual space. •WE SHALL ADHERE THROUGHOUT TO THE CONVENTION OF DESIGNATING A PHYSICAL POINT BY 4. TUE PERCEIVED POINT BY P. ••THE WORD FUNCTION IS USED HERE IN THE MATHEMATICIAN'S SENSE AND IT MAY BE WELL TO REPEAT THE DEFINITION FOR THE NON-MATHEMATICAL READER: LET S AND T DENOTE TWO AGGREGATES (OR SETS OR CLASSES) CONSISTING Or ANY ELEMENTS WHATEVER. A (SINGLEVALUED) FUNCTION DEFINED UPON THE SET S WITH VALUES IN THE SET T IS h MEANS OF ASSOCIATING WITH EACH ELEMENT OF S A UNIQUE ELEMENT OF T. WE ALSO SAY THAT S IS MAPPED INTO T. A FUNCTION MAY ALSO BE CALLED A CORRESPONDENCE (TO EVERY ELEMENT OF S THERE CORRESPONDS A UNIQUE ELEMENT OF T). This characterization of the relations between spatial response and stimulus configuration will be given in terms of two mathematical functions: (1) a mapping function which define« the correspondence between points of the stimulus and points of the visual space, and (2) a metric which characterizes the internal geometry of the visual space. The constants of this geometry may vary from observer to observer, but repeated and varied experiments strongly indicate that its general character is that of the three-dimensional hyperbolic space of Lobachevski and Bolyai. A note should be added concerning both the conditions under which the experiments were performed and the method of observation used in viewing the stimulus configurations. All experiments were carried out in a darkroom, thus reducing monocular clues to a minimum. The intensities of the points of light were adjusted to appear equal to the observer but so low that there was no perceptible surrounding illumination. Thfi observer's head was fixed in a headrest and he viewed a static configuration (perception of motion is not considered). The observations were made binocularly and always by allowing the eyes to vary fixation at will over the entire range of the pnysical configuration until a stable perception of the geometry of the situation was achieved. Work has been done by other investigators on perceptions arrived at by keeping the eyes in constant fixation on a single point. It is impossible to state a priori what relationship, if any, exists between visual space as determined by the "fixed eyes" condition and visual space as determined by using freely roving eyes. However, it seems reasonable to suppose that the fixed eyes condition, owing to the very limited field of distinct vision, would permit only the discovery of local properties of visual space. It is not unlikely that a theory obtained under the fixed eyes condition could be completely subsumed in Lüneburg's theory as a theory of th^ local properties of visual space. On the other hand it is highly probable that the use of the restriction of constant fixation would prevent an understanding of the phenomena associated with the free use of the eyes.* * INTHIS CONNECTION IT MAY BE WELL TO MENTION THAT CERTAIN OBSERVATIONS CITED IN OGLE AND IN FRY5, ARE MADE WITH THE EYES IN CONSTANT FIXATION. ThESE AUTHORS APPARENTLY BELIEVE THAT THEIR RESULTS ARE IN CONTRADICTION TO THE LÜNEBURG THEORY. SUCH A CONCLUSION IS NOT WARRANTED BECAUSE OF THE DIFFERENCE IN CONDITIONS. NEITHER AUTHOR HAS CONSIDERED THE POSSIBILITY THAT HIS RESULTS COULD BE CONNECTED TO THE LÜNEBURG THEORY THROUGH LOCAL PROPERTIES AND NEITHER HAS ATTEMPTED TO ACCOUNT FOR THE PHEN0MRN4 ASSOCIATED SITH THE FREE USE OF THE EYES. SPECIFICATION OF THE STIMULUS CONFIGURATION PHYSICAL COORDINATES In order to give a numerical characterization of a stimulus configuration, its points are located by referring them to a suitable coordinate system,- carte- sian, polar or other. The observer's head is assumed to be fixed in normal eiect position. The observer's eyes are assumed to be located at points, the rotation centers of the eyes.* A cartesian system is chosen with the origin placed at the point midway between the rotation centers. The y-axis runs laterally through the rotation centers and is oriented positively to «*-)( the left. The unit of length is fixed by setting the eyes at ± 1 along the y-axis. The x-axis is taken positive in the frontal direction of the median plane. The upward vertical direction is assigned to the z-axis (fig. 1). In this frame- FIG. 1. CARTESIAN COORDINATE SYSTEM FOR PHYSICAL SPACE. L AND R REPRESENT CGNTEP.S OF ROTATION OF LEFT AND RIliHT EYES. work we can assign cartesian coordinates (x, y,z) to any point 0 in physical space, and so determine its position relative to the observer. b'vyc^ A coordinate system better adapted to our purpose is the bipolar system. Let Q be a physical point anywhere in space and let R denote the right eye and L the left eye (fig. 2). The angle which the plane QLR makes with the horizontal is called the ele.ation 6 of the point Q. The angle subtended at 0 by the two eyes is called the bipolar parallax y of 0- To completely specify the position of Q we now define a third coordinate, approximates the average of the inclinations of the two visual axes with respect to the median plane (See Part II, Section 1). For this reason the coordinate y will often be called the convergence. The angle

4 is the distance from the origin, serve very well to show th~\, the approximations are sufficiently accurate for most practical work. A set cf coordinates, which we shall call the iseikonic coordinates, particularly useful in analyzing binocular space perception, may easily be defined in terms of the bipolar coordinates. Let y^be the least, value of y attributable to any point of the stimulus. If we draw VMC's through all the stimulus points, yo will be the value of y on the outermost VMC. Let 0o and 9 be values associated with suitable directions of reference. To a point having the bipolar coordinates (y, 0, 8) we associate iseikonic coordinates, (4) * « 4> - 4> o 8 = 0.6 o These rnordinates will generally have to be specified anew for each changr«» of stimulus. 3. TH~ METRIC NATURE OF VISUAL SPACE The mathematical characterization of the visual space is founded upon a set of observations in conjunction with a limited number of mathematical assumptions of considerable heuristic appeal. From these fundamentals it is possible to achieve by de- a.!u..t_t..l-T•-C .- Ulu^csoT*-Äil&«13i;ui. G£>_o ~aX1 iffiJ1Ä_G u_Uuaiav.b.C __.l:iii_f»l•b:luu_vf /X tuUi„c g^lfiild4-»-i.y. w-fx ,».-4t —o t,-i m i .u. pu^v-. In fact, he presented strong evidence that the visual space is a metric space, finitely compact, convex and homogeneous. Our further work supports this conclusion. 3a. Visual Orientation One of the curious facts of binocular perception is that the observer is not ordinarily aware of any bipolarity. Sensed distances from the observer are treated as though viewed from a point center of reference. This situation is described by placing the origin of visual coordinates at this "egocenter" of the observer. The observer is, however, aware of the orientations lateral, vertical and frontal. In the visual spece w~ may then take three subjective planes of orientation perpendicular to these axes - the ser.sed median, horizontal and frcntai planes through the origin. The axes in the visual space are the intersections of the three principal subjective planes. Let (£,17, £) be coordinates chosen to represent these subjective orientations. The origin £ = 77 = £ = 0 represents the subjective center of observation. The £ - axis is positive in the frontal direction; the 77 - axis, in the direction left; the £ - axis, in the direction vertically upwards. The subjective horizontal, median and frontal planes are given by the respective equations, £=0, 77=0, £~®- The positional orientation of the observer is generally such that he brings the subjective planes into the proper orientation with respect to objective physical space. This coordination between the visual and proprioceptive senses is not absolute, however. It may easily be disarranged in in airplane or sea-going vessel. We shall see, in fact, that the assumption of the customary correspondence between objective and subjective orientations is not necessary for our theory. 3b. Perception of Distance A configuration consisting of isolated points Qj, Q2, Q3 is sensed as a distribution of points P., P„, P„, in a three-dimensional continuum. An observer obtains rather definite impressions cf the distance oi the points from one another and from the observational center. The sizes of these sensed distances may readiiy be compared. Thus if (Pj, Pj ) denotes the sensed distance between any two points P. and P., we find for any two pairs of points P., P2 and P3, P4 that relations of inequality such as (P,, P2) > (P3, P4) or (P1( P2) < (P3, P4) are easily perceived. The sensed relations of equality and inequality are quite stable for a given observer. In other words, the inequality signs are determined to a high degree of correlation by the physical coordinates of the stimulating points 0,. Q2, %, Q4. 3c. Perception of Straightness A sense of alignment is one of the strong characteristics of visual perception. We quickly perceive whether or not three points lie on a straight line. Furthermore, physical points can be arranged so as to result in the perception of a straight line for every orientation and position in the visual space. Given an arbitrary pair of points, it is possible to arrange others along a curve which will be perceived as the extended straight line joining the points. Perhaps it would be well to emphasize that the perception of straightness may arise from physical curves* which are not physically straight but actually have marked curvature. (See Section 5e) 3d. The Psychometric Distance Function The observations 3b and 3c are a strong indication that the visual space is a mathematical metric space. This means that we can assign positive numerical values «THE WORD CURVE AS USED HERE IS TAKEN IN THE TECHNICAL MATHEMATICAL SENSE. A CURVE IS A ONE-DIMENSIONAL CONTINUOUS MANIFOLD. THUS A STRAIGHT LINE IS A KIND OF CURVE. IT HAS ZERO CURVATURE EVERYWHERE. to sensed distances so that the numbers satisfy inequalities in agreement with the perceived relations of sensed distances. Such a coordination of a number D (Pj, P2) to the sensed distance between a pair of points Pj, P2 is called a distance function or metric if it satisfies the following conditions: (a) D (P, P) = 0. A perceived point has zero distance from itself. (b) D (Pj, P2) = D (P2, Px) > 0, if Pj/ P2. To each perceived pair of distinct points; there is assigned a positive value of distance independent of the order in which the points are considered. (c) D (Pj, P2) + D (P2, P3) > D (P1( P3) for any three points P., P2, P3. We shall say in particular that three points are on a straight line if and only if the equality relationship holds. Whot) we s*»y that the function D (Pj, P2) corresponds to sensed distance we mean that it must satisfy the further conditiors: (d) If Pj, P2 and P3, P4 are any two pairs of perceived points, then D (Pj, r2, z u vr3, r4, according to whether the sensed distances are correspondingly related, (Pi- P2) ? P3. V' (e) If Pj, P2, P3 are perceived as being arranged in that order on a straight line, then D (Pj, P2) + D (P2, P3> = D (Pj, P3), and conversely. A function satisfying conditions (a) to (e) is called a psychometric distance function or simply a metric for visual space. Our problem can be reduced to the determination of such a function in the terms of the physical coordinates of the stimulating point*. Oiiite clearly, the physical distance relations -among the stimulating points will not describe a metric for visual space. Although physical distance satisfies (a) to (c) it can not satisfy (d) or (e) since, for one thing, the physically straight lines are not generally the same as the visually straight lines. To keep these distinctions clear, the curves in physical space which are perceived as straight will be called visual geodesies or siroly geodesies. The function D (Pj, P2) is not completely determinate, for if D (Pj, ?2) satisfies conditions (a) to (e) so does the function C • D (P,, P2) where C is any positive constant whatever. Yet, under certain general mathematical assumptions, this can 10 be proved to be the only indeterminacy possible. These assumptions are: (f) The visual space is finitely compact. E/et; bounded infinite sequence of points has a limit point; i.e. for every infinite sequence cf points P (v = 1, 2, 3, ) satisfying the condition D (P , P ) < M for some point Po and positive constant M, there exists a subsequence Pv (k * 1, 2, 3 ) and a point P of the visual space such that D (P , P^ ) — 0. (6) The visual space is convex. Between every pair of points Pj, P2 (Pj / P2), there is a point P3 on the straight segment joining Pj to P,; i.e., there exists a point P satisfying D (Pj, P3) + D (P3, P2) = D (Pr P2). The proof that, under these assumptions, the metric is completely determinate to within a constant factor i= given in Lüneburg'. Although the assumptions (f), (g) can not be verified by experiment since the proof would require infinitely many tests, they do coincide with our customary convictions about visual perception. Since a distance function may be determined exactly to within a constant factor, it follows for a given stimulus, that the proportions of distance are unique. In other words, the ratio D (P., P9) / D (P., P.) of two sensed distances is a uniquely determined function of the four stimulating points in question and does not depend upon the particular distance function we use. In this way the metric establishes a fixed relationship between the objective physical stimulus and the subjective perception. This relation is a function of no other variables than the coordinates of the stimulus. Any parameters in this relationship which are not physical coordinates must be constant factors of the observer, characterizing his visual reactions to external stimuli. 3e. The Homogeneity of Visual Space* The visual space has two properties which are familiar from common experience but have not been treated experimentally. For this reason these properties are stated here as hypotheses. The first of these properties is: (h) The visual space is locally euclidean. *F0R A FAIRLY COMPLETE ACCOUNT OF THE MATHEMATICAL KNOWLEDGE IN THIS SECTION. SEE BUSEMANN8 In other words, the euclidean laws hold to any desired degree of approximation in sufficiently small regions of space. The earti., considered as a spherical surface, is a familiar example of a space having this kind of property. In surveying a small area it suffices Lo use the euclidean laws of ordinary trigonometry, but for navigating over great distances only spherical trigonometry will do. The locally euclidean property of visual space explains why we may notice no distortion in viewing small geometrical diagrams frontally. The property (h), together witli the properties of finite compactness and convexity, forms a necessary and sufficient condition that the space be riemannian. The second property which we postulate is that sensorially plane surfaces exist in ar.y gi*en orientation and localization. The visual perception of planeness is such that the visual geodesic connecting any two points of a sensory plane does not anywhere depart from that plane. Any three physical stimulus points can be imbedded in one surface, and only one, which gives the impression of planeness. All the statements concerning the nature of the visual planes can be summarized in one: (i) The visual space is a desarguesian geometry. From the propositions (a) to (i) it can be proved that the visual space is homogeneous. The binocular visual space is one of the riemannian spaces of consiani gaussian curvature. A mathematical consequence of the homogeneity of visual space is that the metric must be one of three simpJe kinds. For the (£, 77, £) coordinate system used by Lüneburg, the psychometric distance function D = D(Pj, P2) is given by the formula: 2 . , n-K)/2 _DI <5> 1/ smh (-K) L 2 cJ (f - <5 )2 + (T) - T) )2 + ' r - r *2 (i •£ PS) (i+^22) F(VPlf D? \ where (^,, ^ , ^t) and (^2'7'2,^2^ are tne coordinates of P. and P_ respectively and where pi = £. + 77. + ^2 (i = 1.2)* The constant K may be interpreted as the generalized gaussian curvature of the space. The constant C is the arbitrary constant factor of indeterminacy in the metric. If K is allowed to approach zero from either side, the formula (5) becomes (5a) D — = F (P1# P2) (K = 0) 12 and the relation obtained is simply the familiar euclidean metric. If K is positive, the formula (5) is usually written more conventionally as (5b) 2 TK'' D ! 7< s^ LY —\ - 4 F (Pj. P2) , 0) The metric (5b) is that of elliptic geometry. The two-dimensional case is familiar to us as the geometry on the suriace of a sphere. Negative K gives us the hyperbolic geometry of Lobachevski and Bolyai. The evidence of our experimental studies indicates repeatedly and in a variety of ways that the geometry of visual space is, in fact, just this hyperbolic geometry. If we interpret (£, r/, £,) as cartesian coordinates we can map visual space in a euclidean space. Visual distances could not be represented correctly by distances on the map unless the metric were euclidean, since the three metric» are clearly not proportional. In fact, we know that in making a map of the earth (el lintic case) on a euclidean sheet of paner we cannot avoid distorting distances. The map described by the (£, rj, £) coordinates does, however, have one clear advantage,-it is con formal..This means that perceived angles »ill be exactly represented by angles on the map. As a matter of convenience in formulation we prefer to use an equivalent set of coordinates, polar coordinates (r,(p,?y~; in visual space. With Lüneburg, we set (fi) £, - p cos

0 The radial coordinate r is to be interpreted as a quantity measuring sensed distance from the observer. It is never to be taken as an absolute of sensation, but only as a correct description of relative distance when taken together with other v-ilues. In any case, all points perceived as having the same distance from the observer must be assigned the same value of r. The equation r = constant represents a sphere about the egocenter. The coordinate V" simply represents the perceived angle of elevation from the subjective horizon- tal. Thus on the sphere r = constant, the curves •ft" = constant represent meridans; of longitude passing through poles on the left and right of the egocenter. In the same way, the curves Cp = constant represent parallels of latitude on the visual sphere, ihe visual sphere r = constant can be conceived in this way as the earth with its axis oriented horizontally. By employing the coordinate transformations (6), (7), we obtain the hyperbolic metric in teri"s of the visual polar coordinates in the form D (8) cosh = cosh Tj cosh r2 - sinh rx sinh r2 f (^fU^j di$~z)• ^ < ^ whe r e f fyl'fyzi $v %) = cos ^*fa~ 9fl) ~ cos a > «i^j &~z ), (K > 0) It will be seen that equations (8) and (8b) may be transformed into each other by replacing the sensed radial distance r with its imaginary counterpart ir. Twodimensional hyperbolic space might in this way be interpreted as the geometry on the surface of a sphere of "imaginary radius" .* It is weli-known that there is an absolute measure of length in elliptic geometry. In the two-dimensional case, for example, it is possible to represent the elliptic geometry isometrically on the surface of a sphere. The radius jf the mapping sphere may then be used as an absolute measure of length. If the radius of curvature in this representation is taken as unity, then we must take C = 1 in (8b). By the analogy cited above, it is possible to specify an absolute measure in the hyperbolic geometry, too. Gauss remarked that he wished the physical world were not euclidean for then there would be a priori an absolute measure of length.** We shall, by analogy with the elliptic case, take C = 1 in (8). However, it should be remembered that this particular metric for visual space is only one choice out of a possible oneparameter infinity. * THIS IS NOT TO BE CONSTRUED AS HAVING ANY SIGNIFICANCE DEEPER THAN THAT IMPLIED BY THE SUBSTITUTION OP ir FOR . IN <8b). **GAUSS, LETTER TO F. A.TAURINUS (1824): "ICH HABE DAHER WOHL ZUWEILEN IN SCHERZ DEN WUNSCH GEAUSSERT, DAS DIE EUKLIDISCHE GEOMETRIE NICHT DIE WAHRE WARE. WEIL WIR DANN EIN ABSOLUTES MAASS A PRIORI HABEN WURDEN. " (SEE ENGSL P. AND STACKEL, P., THEORIE DER PARALLELLINIEN. LEIPZIG. 1895. FOR ENTIRE LETTER). 14 3f. Plane Trigonometry of the Visual Space* If we let 7?" = 0 in the formulas (8) and consider the metric relations between the sides and angles of triangles, we shall compile a set of useful relations which may be used to measure the visual space just as we use trigonometry to measure the physical world. Let the scale factor C in (8) be unity. Denote by a, b, c the perceived lengths of the sides of a triangle and let A, B, C denote the perceived sizes of the opposite vertex angles. By employing the metric (8) it is possible to derive the analog to the law of cosines for the hyperbolic case: (9) cosh c = cosh a cosh b - sinh a sinh b cos C The corresponding rules for the euclidean and elliptic cases are (K < 0) (9a) c" = a' + b' • 2ab cos (K = 0) and (9b) cos c = cos a cos b + sin a sin b cos C (X > 0). The "Pythagorean theorem" for hyperbolic right triangles is obtained by setting C = 90° in (9): vtO) cosh c = cosh a cosh b and in the two other cases we have (10a) c2 = a2 + b2 (K <- 0) (K = 0) (10b) cos c = cos a cos b (K > 0) In fact, we may set down the usual laws for the angle functions of right triangles in all three geometries: K <0 K =0 K•> 0 ill) cos tanh b tanh c _b_ tan b c tan c (12) (13) sinh a sin A = sinh c tanh a tan A = sinh b a sin a c sin c a tan a b si n b (14) -£°*4-s cosh a sin B cos a (15) cot A cot B •= cosh 1 cos c For small triangles it is easy to see that the hyperbolic and elliptic rules both approach the euclidean one. * THE READER IS REFERRED TO C0XETER9 AND CARSLAt10 15 The law of sines in hyperbolic trigonometry is especially simple: (16) sinh a sinh b sinh c sin A sin B siin C (K < 0) For the other geometries we have (16a) a sin b siin B siin C (K = 0) (16b) sin a sin A sin1 b lin B sin c sin C (K > 0) RELATION OF VISUAL TO PHYSICAL SPACE At the Dartmouth Eye Institute, Ames succeeded empirically in constructing a sequence of distorted rooms which could hard- ly be distinguished from a given rectangular room with respect to binocular vision. At first Lüneburg suggested that the construc- tion of these rooms could be mathematically derived from the rectangular original by em- ploying a certain kind of transformation which he called an iseikonic transformation (Fig. 4). This transformation was determined by the as- sumption that the rotatory motion of the eyes in looking from point to point of a configura- tion was the sole determining factor in the perception of the relative positions of the points.* Subsequently, he discarded this notion in favor of the idea that the fixation angles themselves, rather than only the changes in fixation angles, were significant in binocular perceptions. The distorted rooms could then be FIG. 4. BINOCULARLY INDISTINGUISHABLE accounted for by translatory displacements in CONFIGURATIONS, the hyperbolic visual space. In «ach case he obtained a one-parameter family of dis- torted rioms which would account for the characteristic shape of the Ames constructions (See Lüneburg11). *WHETHER IT IS THE SEQUENCE OF RETINAL IMAGES, OR THE MUSCULAR ACTION OR BOTH TOGETHER WHICH INFORM US IN THIS WAY. IS IRRELEVANT HERE. 16 The two hypotheses do give measurable differences and it would be possible to discover by experiment which is correct. However, experimental evidence obtained in other ways has led us to utilize the earlier point of view. 4a. The iseikonic Transformations According to Lüneburg's earlier hypothesis, if the bipolar coordinates of all points in a given stimulus were changed by constant amounts K, /J., V by means of the transformation y' - y + K (17) + M e' = e + v then, to any one observer, the new stimulating configuration would yield the same perceptions as the original configuration. In particular the Ames rooms could be constructed by employing the special transformations y' = y + k (17a) 4>' = 4> 6' = 6 One reason Lüneburg gave for discarding this hypothesis was the fact that two segments having the same disparities Ay, A = •d~= 9 The function r (D is a constant characteristic of the observer. In particular, so is the special value (19a) co = r (0) Under the assumptions of the foregoing analysis we have,reduced the problem of determining the coordination between visual and physical space to the determination of the 20 single function r = r (D. The function r (D is a personal characteristic (i.e., a constant such as Lüneburg predicated) of the observer. If our assumptions are correct, a complete description of the observer's binocular visual space can be supplied nnrc rh*» function r (P) is determined. 5. EXPERIMENTAL METHODS FOR DETERMINING r (T) AND RELATED EXPERIMENTS The rules of trigonometry given in Section 3f may be used to measure the visual space. In Part II we shall discuss several relevant experiments which have been performed in this laboratory, together with a detailed account of the technics, apparatus and results obtained. In the present section only a general description ox various experiments related to the theory will be presented. Convenience has led us to restrict our work to the use of stimulus configurations in the horizontal plane, 8=0. Although it would be desirable to complete the evidence by performing experiments in all three dimensions, there is some foundation, in theory, for the hope that conclusions based on results obtained in the horizontal plane may have validity also for the three-dimensional case. As a matter of consistent notation, points of the stimulus configuration will be denoted by the letters Qj, Q2, Q3 and the corresponding perceived points by the letters Pj, P2, P3. . . . 5a. Parallel and Distance Alleys The most striking evidence that visual space is non-euclidean lies in the distinction in visual perception between apparently parallel straight, lines and curves of apparent equidistance. This difference was first reported by Blumenfeld12. The experiment is quite simple. Two lights are fixed at the points Q. = (%, sin Co. (K < 0) For the other two geometries the same method gives (20a) (20b) r si n <2> = co sin G>. tan r sin (p = tan to sin(ö. (K = 0) (K > 0) The distance alleys, on the other hand, may be characterized as the loci of constant perceived distance d from the median (Fig. 8.) For a variable point 23 PIG. 8. REPRESENTATION Of A DISTANCE ALLEY IN VISUAL COORDINATES. n. P - (r, (p ) on the left-hand alley, we obtain from (12) sin G - sinh d / sinh r in the hyperbolic case. From the condition that the alley go through P = {co, sin co si n(p3 (K = 0) (K > 0) In the euclidean case, as we know, the parallel and distance alleys are the same and this geometry does not account for the experimental observation. Now, if we let (pp be the angular coordinate on the parallel alley and Co^ be that on the distance alley for a given value of r = co (Fig. 9), we find from (20) and (21), tanh r sind = sinh co and TI tan r sin (p tan co (K>0) sin r sin, the above equation for the hyparh^lic case yields (22) smCpp cosh r sin (öj cosh £> (K < 0) This implies that Cr>n < Co^ smA the parallel alley must be inside the distance alley. For the elliptic case on the other hand we find sin 0) Consequently, ^Pd. an(^ tne parallel alley lies outside the distance alley. Clearly, the hyperbolic case is the only one that can fit this experimental evidence. We shall find that other experimental tests of the question lead to the same conclusion. For this reason we shall no longer follow this parallel presentation of the three cases, and we shall employ only the hyperbolic geometry. The reader will find it not difficult to carry out the analogous reasoning for the other cases if he wishes to do so. The ?»iley experiments may be used not only as a means of indicating the hyperbolic character of the geometry, but also to calculate the function r ( T ). Consider the VMC corresponding to *; given value of I"1 (y is already specified) and let r be the perceived radial distance corresponding to T. The point (r, CP,^) on the distance alleys satisfies equation (21) and, hence, si.nn,2 r = . , sinh 2co sin2i - cos20 The visual coordinates (r, ) are related to these by r = r ( T ) , . Thus, with the understanding that co • r (0) , the visual Pe - (co, 0) . Pl - (^(fc) , P2 - (r, «fa ) where tfi • (P0 , Px) - o (P0 , P2) - d and with the use of the cosine law, equation (9), we obtain cosh2d • cos\\2co- sinh2&> cos wh ence cos Cp j £a sinh r cos cosh co - cosh co cosh r f sinh co sinh2o; 26 FIG. 11. REPRESENTATION OF THE SENSORY SITUATION IN THE THREE-POINT DVMC EXPERIMENT. II l,I*C CWU( *- ^*-jc»fc*wxO»» puts inh r m = sinh co (24) Y = cos

0 and determine the corresponding values of Qfx = $2 ~ ^o • the plot of cosCöj as ordinate against cos&j as abscissa will in theory be a straight line, (25) Y = mX + b . It is an experimental fact that tliis graph is very nearly linear. Lüneburg has shown that if this result holds for each pair of Vieth-Mviller Circles, ther. the space has constant curvature. The values of m and b are easily determined from the plotted graph. The value of co may then be found by eliminating r from the equations for m and b. Thus sinh2r = m sinh2w = cosh2r - 1 b2sinh4o; rosh2r = cosh2o) - 2b sinh^x) + cosh2ct> 27 Combining these equations and setting" cosh OJ = sinh co + 1 , we get sinh *co 1 + m2sinh2co = 1 + (i - 2b)s:inh2co + b2 cosh2co whence. „ sinh^co . „ 1 m2 - 1 - 2b + b< — ' 1 - 2b +b2 (1 -z~) cosh co cosh to and (26) cosh'-co b2 (1 - b)2 - m2 Having determined a> in this fashion, the value of r is easily found from the equation (24) for m. It is clear that the quantity on the left in (25) must be greater than 1 if CO is to be a real quantity. The fact that this is experimentally true is further evidence that the yeometry is hypezooli1:. it can be seen that the geometry is hyperbolic, euclidean, or elliptic, according to whether m is greater than, equal to, or less than l-2b. (ii) The Four-Point Experiment. The three-point method is found to be somewhat insensitive since the values of X and Y in equation (25) are plotted upon points much nearer to each other than to the intercept of the line (see Part II, rig. 25). It follows that the intercept b, depends rather critically on the determination of the slope m. In order to surmount this difficulty, Lüneburg suggested a method of determining m by the use of four points. Let Qj • (y0, j) and Q2 • (yo, <£,,) be two points fixed on the circle y = yo and let Q_ « (%, c/^) and Q, » (^j, i ..*^:t) (i * 1, 2, 3,). Let us suppose that y, > J2 > y3. The two lights Q^* are fixed in position. "(Ire lights Q, , are restricted to motion en the VMC y - y. . The lights Qj* are fr*-ei^ movable is the horizontal plane (Fig- 13). The observer adjusts the lights Q, - and Q.-1 sa that the corresponding sensed lights P, and P* appear to be lined ?s> with P3* in a parallel alley. line iignts V*^ ®r« then further adjusted so that the observer perceives the points P2 as feeing exactly midway in distance between P. and P« « *hea this has beea done we say the alley has been equipartitiaried, or siinply partitioned, and we refer to the points Q2" as the partition points. For this experiment the appropriate iseikonic coordinates are r • "f - % and • • $\ -O- L R FT8. 13. PBISICAL. ÜOA.«(SSHEJtT 0? Ai BQDXPMtllTIONES FAÄILLEL .«.LEI The equation of the parallel alleys, as we have already seen, is {*§) tans r sin are involved. In particular we nave (30b» tanh Y tanh sä = sin (p 3 Clearly, the assunption r = r (PS does not enter in the design of the experimesit in any way. Since perceived distance as measured by this experiment is independent of any hypotheses concerning the aatare of sraised eqoi distance, particularly Hj , ILj and H^ „ it may b»» used to test the validity off these assumptions. To do this in adequate detail vould require in excess off MKS experiments per observer. The equri partitioned alleys also give evidence that the space is hyperbolic. This is the consequence of the fact that the quantity on the right in (30) is found experimentally to be positive. If it were zero or negative we would take the result to mean that the geometry is eoclidean or elliptic in the respective cases. It is easy to see that this condition anounts to saying < K >* O •CCJ to »hether Meat ^j * cotGjl = cot*2 sinh r2 If the size of the retinal irn^gse were the effective criterion, the ratio of the sizes wald Vi T2 The departure from this ratio may be considered am indication of the effectiveness of oar dep»th perception in judging the relative sizes of objects. If the $ angles are sufficiently small we may use the approximations r1 '*• &s and "*" ~(y» ~ >•# to obtain <32a) y2 7,sinh as •/, >2sinh ra 32 PIG. 15. REPRESENTATION OP THE SENSORY SITUATION IN MAKING A SIZE MATCH. n It should be stressed that the size-constancy relationship will depend upon the position of the distant reference object. If we employ small values of cp we may use (32a) to determine r (D once we know the value of ox If we do not restrict cp in this way we should use equation (21) for the distance alleys instead. Some results obtained in our laboratory do give evidence of size constancy, even for darkroom observation (C. J. Campbell16). The size constancy data alone cannot be utilized to demonstrate the curvature of visual space. However if the size constancy experiment were considered an equidistant alley and compared with a corresponding parallel alley, ther the results could be used to determine the nature of the space in the manner of Section 5a. Se. ine phenomenon of the Frontal Geodesies To Helmholtz 7 we attribute the observation that the physically straights lines do not appear straight at all distances. Qirves which do give the impression of straightness are not physically straight but are cor.cnve toward the observer at near distances and convex at 33 far (Fig. 16). For some intermediate distance the frontal geodesic will be straight in the vicinity of :.he median. Although this phenomenon is not Y«ry useful in computing r (D it is X an example of the kind of observation which may be given a quantitative description by means of the theory. The equation of the frontal geodesies is easily written. Let us suppose we are dealing with the geodesic segment between the points Qo = (yo, 'LCt>0) - AS iseikonic coordinates we take The equation is then obtained from (11) (33) tanh r ( F ) cos

#- e where T , - Lines, 0 = 0°, ±5°, ±10°, +15° ±35° (Fig. 17). Thelight on the median, 4> - 0°, was fixed and the observer was asked to adjust the remaining lights according to the instruction: ''The median light is fixed. Adjust the position of the other lights by having them moved toward you or away from you until you have the impression that, together with the median light, the lights form a circle about you with yourself at the center." 37 FIG. 17. SENSED RADIAL EQUIDISTANCE EXPERIMENT. PHYSICAL ARRANGEMENT OF THE LIGHTS. In the limited number of experiments performed on several observers the lights do not always seem to fall on a VMC but, more generally, on a slightly flatter curve. The result of a sample experiment showing this type of deviation is given in Fig. 18. The effect decreases slightly with increasing distance. Occasionally an experimental setting actually fell inside the VMC Since the experimental curve was close enough to the VMC in general to satisfy us with regard to use of the circle y • constant as a first approximation, we did not pursue an extended course of experiments on this question. Furthermore there is a possibility that the flattening of the VMC may be attributable to experimental and theoretical factors such as the following: (a) In the experimental situation, the lights were placed on a horizontal table covered with a sheet of coordinate paper so that their positions could be marked. Ordinarily, a great deal of attention was paid to keeping the illumination of the surroundings sufficiently low that the observer had no idea of the position of the lights with respect to the room. However, in this case with fifteen lights, although very dim, placed above a light reflecting surface, it is conceivable that the observer was able to obtain some shadowy impression of his surroundings and so would modify his setting of the liphts to tend slightly toward the circle of equal physical distance. 38 xcm 200 173 150 FIG. 18. SENSED RADIAL EftUIDISTANCE EXPERIMENT. TYPICAL SAMPLE SETTING. FLOTTED POINTS REPRESENT BILATERAL AVERAGE. OBSERVER G.R. + E x peri menta! Setting 12 5 * Vieth-Müller Circle' • , / ./ Z5 SO 73 z^cra (b) The visual axis of the eye actually makes an angle with the optic axis at the anterior nodal point of approximately 5° temporally. If we define y as the angle of convergence of the visual axes, the angle should presumably be measured with respect to the anterior nodal points. Since the position of the nodal points with respect to the head changes as the eyes shift fixation, this choice of coordinate is not as convenient as that based on rotation centers of the eyes, The use of the nodal points instead of the rotation centers does give a flatter curve than the VMC, but the effect predicted on this basis does not seem to be as great as the empirically determined flattening. The nodal points shift in the eye with accommodation also. Due to the drift of the nodal points in accommodation, the flattening sliould be most marked for the nearer VMC. rbwever; the contribution of accommodation to this effect is minute and, although such an effect is found, it is quite likely attributable to the factor mentioned in (a). If we were to make consistent use of the nodal points in defining the bipolar coordinates in three dimensions, it would be necessary to use Listing's Law to give the bipolar coordinates in terms of fixed physical coordinates. Since our experiments were conducted in the horizontal plane only, we have not felt the use of the nodal points instead of the centers of rotation would give sufficient advantage to justify the inconvenience. For the purposes of the theory it is irrelevant whether or not we take the ocular mechanism into account in characterizing the loci of apparent equidistance. It TOuld be sufficient to determine these loci experimentally and then to devise mathematically a suitable parametric representation for the experimental curves. 2. TESTS OF THE ISEIKONIC TRANSFORMATIONS In this section we .shall exhibit a consi-. able amount of evidence to show that binocular spatial relations are invariant under the iseikonic transformations (Part I, Section 4a). In other words, the perceptions of strajghtness, relative distance, form, etc. among the points of a stimulus configuration are not altered by changing the bipolar coordinates (y + fj, A transformation of this kind may be subdivided into two separate transformations, one of the form (36) y • y+\ and the other of the form (37) y =r 4? •

+ fi , y% » y In this transformation the angles equally while not altering the values of-y, and the observer was asked to repeat the experiment for the new setting of the fixed lights; i.e., to form an oblique geodesic. If the observer placed the lights for the new setting to correspond to the old one through equation (37), the hypothesis of the iseiko-iic transformation would be verified for this special case. This procedure was called the Predicted Oblique Geodesies Experiment. In the laboratory, nine lights Qn were placed so as to be adjustable along the 0-lines, 40 4>n = 5n°, (n = 0. ±1, ±2, ±3, ±4). The fixed lights at 4> = +20° mere pre-set symmetrically to the median at x = 330 cm. The observer adjusted the remaining lights according to the specific instruction: ''The two end lights are fixed. Adjust the remaining lights by having them moved toward you or away from you until they appear to lie o;: a straight line between the end points. The experiment was repeated several times under these conditions. With the mean of the repeated settings taken as the basis for computation, the total stimulus configuration was subjected to the transformation (37) with u = + 10°. The point Q = (y , cß ) of the original configuration was then transformed into the point Q'n = {yn , = + 30° to Q'-4 at rp = - 10°. For lack of space we could utilize for experiment only the part of the configuration stretching from Q'+2 at + 20° to Q'_4 at - 10°. Fixing two lights at Q'_4 and Q' r2 the experiment was repeated using the same instructions and with the lights piaced at 5° intervals between - 10 and + 20 , A similar series of observations was obtained also for \J. = -10°. In Fig. 19 we compare the results of these settings for five observers with the predictions on»the basis of the iseikonic transformation (37). The data are given in tabular form in Table I. In general, the agreement between prediction and experiment is good. Wherever there is a marked deviation r.f r.h« «setting frwi iht ^redictioii there is also a marked asymmetry. For observers who exhibit this asymmetry we might reasonably assume that the two eyes do not play equal roles in binocular vision.. The interesting problem of generalizing the theory for such observers is left open. 'fhis experiment was actually designed to test a somewhat different hypothesis. For the present purpose it would have been desirable not to alter the number of points in the stimulus configuration so that the original and transformed configurations might be complete images of each other under iseikonic transformation. However, it is felt that the conditions were adequate to bring out the point in question. 2D. The Transformation y" = y + K , 'j, with Q0 on the meuiun. Tne light Q2 is restricted to move on the VMC y " y2 with y2 > y, (see Part I, Fig. 10). The observer is asked to make a setting according to the specific instructions: third can be moved.'' (This light and its range of motion are demonstrated.) ''Direct the experimenter to adjust this light so that the distance between it and the middle light appears to be the same as the distance between the pair of fixed lights. Allow the eyes to roam freely both ways over the spatial interval between each pair of lights. Be sure to fixate on each light in turn and to sweep the eyes across the interval between each pair of lights until you are satisfied that the two distances appear to you to be the same.'' A series of these experiments was undertaken with values of y. ranging from zero to .07 and y2 u y, + .01. In each case settings were taken for the same fixed sequence of values of !, the azimuth angle associated with Ql • If the hypothesis of the iseikonic transformations is correct, the values of the azimuth angle 4>2 of Q2 associated with a given 2 f°r *-'ne fcwo series. The value of 2 in each series was computed as the average of three or four experimental settings. Each entry in the table thus represents at least six experimental observations. TABIE II Test of the iseikonic transformation T «T + A/ ,0=0 (Three-Point Double Veith-Miiller Circle Experiment) Average settings of 02 for different values of T and 0 , when T "".01. "0, and 02 are expressed in radians. XT- •00 •01 Value of 02 for different values of T, and 0, .02 .03 •Cl* .05 .06 .07 Average value of 02 •1226 .Has .1583 .173U •187U • 2003 *212$ Observer G. R. .01*1*3 •0629 .0719 .0985 .1130 •1301 .11*28 •ol*o5 •0600 •0796 •0965 .1177 =1311 .1536 •0i*2l* •0620 .0791* .1025 .1116 •1283 .11*88 .01*36 .0696 .0892 •105U •1186 .1211* .1508 .0316 .01*58 .0629 .0827 .0931 .1167 .131*6 .0332 .051*8 .0768 .0919 .!C5i* .1211 .11*70 •cUoo .051*8 .0721 .0858 *10?U .3226 .11*12 •cl*oo .0583 •C800 .1008 •1110 •1238 .1372 • C39)i .0585 •0765 •C955 .11.00 .1257 .11*1*5 •1226 .11*15 .1583 .1731* •187U .2003 .212$ Observer Ü.C.P. •G«58 .0661* .0790 «1002 .1167 .1270 .11421* .0551 •0693 •C803 .09UU *X*A-v *1275 .lii35 .0332 .0520 .0625 .CÖ03 .1002 .1032 .1561* .0387 •C533 .0675 .0831 .0970 .1175 .11*03 .01*12 •0587 .C790 .0922 .1061 .121*6 .11*1*5 .C37U .0562 •0690 .0872 .101*9 .1326 .1510 .01-21* •0578 .0739 •0855 .10% .1213 .3i*uT .Cl*9li •C603 .0617 •c8oo .0851 .1175 .1326 .01*29 .0592 .0716 .0879 .1033 .1211* FIB. 20 Thrt«-Po»*t DVMC Eio«ciiv.«nl. Giitff»«. ol jLfOf • och voiui 0? 9t and r »rh»n r * 01. 45, 2b (ii) The Equipartitioned Parallel Alleys. This experiment is the same as that discussed in part I, Section 5c. The polaroid rack to be described in Section 4 was used. Six lights, Q , (n = 1, 2, 3) are set out in two rows of three on either side of the median (see Fart I, Fig. 13). The lights Q 3 and Q~3 are fixed symmetrically to the median at the respective points (y3, 03) and (73,~<^3). The lights 0 j and Q~ are restricted to move on a line x = Xj. The remaining pair Q 2 and Q 2 may be moved freely in the two dimensions of the horizontal plane. A seventh light is placed on the median in line with Q 3 and Q~3 to aid the observer in establishing his orientation. The observer is asked to set the two rows of three lights in a parallel alley and then to set the middle light in each row exactly half way between the near and far lights. His specific instructions are these: "The three distant lights are fixed. We shall call the central one the median light. (1) Arrange the two rows of three lights on the right and left of the median so that they appear to you straight and parallel. Make sure tr.Pt (a) the two lines of lights have the same direction and that this direction appears parallel to that in which the medi m lights lies; (b) the two lines appear to you perpendicular to the frontal plane, ana (c) the two lines appear to neither converge nor diverge in the distance. (Avoid the effect given by railroad tracks.) (2) In each line of lights place the light intermediate between the near and far lights so that it appears exactly half way between the two.'' Some observers have difficulty in making a distinction between sensory parallelism and the impression given by physically parallel lines which most would agree is not one of sensory parallelism (e.g., the impressiou given by railroad tracks.) In Table III and Fig. 21 we give the results of an experimental series for three observers. The value of does not show variation with y, and depends TABDä IXC Test of the iseikonic transformation T :T+X j 0 Z0 (Bqui- partltioned Parallel Alley Experiment). Values of tan 0a, tan 0, , P2 and T, for different positions of T3 , «hen tan 0£ r .1000. *3 = -.01318 ttannc00,2 r, .1161 •160U .00593 .03928 Observer u.R. T3 = -.00528 r3 = .00792 •H5U .1617 •00#2 •03906 .1169 .1583 •OOU97 •C3891 V= .02112 *1121 .11*51 •oo<85 .03892 Observer M.C.R. Ts = -.01766 tan 02 tan Pi r, .1188 .151*9 .00777 .03931 T3 = -.001*1*6 T3 = .00875 •1155 •11*80 •0071*5 .03920 •1192 •15U5 .00781 .03895 Observer C.J.C, T3 = .02195 •111*2 .U*95 •00620 •03886 T3 = -•0l8lj8 tan 02 tan0 | r, «1127 •3itf5 .00681* •03906 r5 = -.00528 .1119 .31*15 .00678 .03926 r5 = .00792 .1156 •1531 .00685 •03898 T3 = .02312 .1133 .3J*97 •OO802 .03887 ton0 .20 03SERVER G. R. tanV0 OBSERVER M.C.R. 20 - .10 is*' .05 01 -.0i848 -.00528 .00752 + T, * .02112 .02 .03 .04 .15 __^S .(0- ^+S""^~~ .05 • • .01 .02 • T, = -01766 A T'3 = -.004 46 a <-j- = .00875 + T = .02(95 3 1 .03 .04 Fia. 21. Equipartitioned Parallel Alley Experiment. Plot of tan 0 against f for four values of T3 , when tan 03= .1000 48 only upon P. In this experiment also the assumption about the iseikonic transformations is confirmed for the special transformation (36) by the clear agreement among the data obtained under differing initial conditions. 3. DETERMINATION OF r ( r ) In Part I, Section 4, we showed how it was possible to describe an individual's visual metric space in terms of a single function r ( T ). It then becomes a matter of considerable importance to obtain a good estimate of the values of this function. Ihe first three experimental technics given in Part I, Section 5, were used for this determination. Since in each of these experiments small variations in an observer's settings may result in considerable differences in the values of the function, hunan variability becomes an important factor to consider. The responses required of the observer were unusual and difficult. A tendency for the settings to drift, in one direction was now and then noted, particularly when a new type of observation was initiated, but the settings soon stabilized. Random variation from day to day was also noted. Time did not permit us to study this factor in any detail. Ch the whole, considering what was required of _he observer, we were surprised at the consistency of his settings. In a series of observations in which the effect of a progressive modification in the conditions was to be measured, we learned that a presentation of the individual experiments in random order resulted in a more stable; picture of the effect to be measured and minimized any directional trend due to practice. We realize that to obtain a meaningful estimate of a given individual's typical binocular spatial response it would probably be necessary to conduct a detailed statistical study with each experimental technic. Despite the fact that time did not permit us to make such an extended series of observations, the function r ( T ) emerges more clearly than might be expected. In general character, r is a monotonically decreasing function with a monotonically increasing slope. The values of the function for the two observers who have been able to complete the whole series of tests appear to be determined to within one part in five. Furthermore, individual differences are brought out. For one of these observers the values of r (T) are consistently somewhat higher lhic IV. Ins lärtheät distant liguts were iixcu «M, X — juL» cm., y • oo cm. Tni» observer was aslfAH to set the lights in ? r>sr?-llel sllev sud then in a distance alley according to instructions which were in general similar io those used by Blumenfeld. They are as follows: "This is an experiment dealing with space perception. We know one does not. always perceive objects in space where they actually are in physical space, or to be cf their actual physical sizef We want to measure some of these differences. "In the first experiment we shall show you some small lights which wt: shall arrange under your direction so that when you look down between them they appear to you to form straight, parallel lines of light. We wish you to think not of where the lights actually are, but merely of how yoc sense them. When they are all arranged, we want you to be able to say that these straight lines of lights as you see them could never, if extended, meet at any distance in front of you or at any distance behind you; that, is, that they form walls that appear to you as parallel walls that appear neither to converge nor to diverge" To familiarize th'i observer with the observation, a trial run utilizing only stations 1, 3, 5 and 8 was made, no measurements being taken of this trial. The instructions continued: "In the second experiment we shall give you two pairs of lights ut a time. The position of one pair will be fixed. We want you to direct us to move the lights of the other pair so that the lateral distance between them appears to you the same as that between the first pair. We want you to make an immediate, instantaneous judgment of wh'.ther the distance between the lights of the second pair is greater or smaller than or equal to that between the first pair. Do not think in terms of physical units of distance between the lights, for example, inches or centimeters. Just direct us in adjusting them until you immediately sense the two pairs of lights as being the same distance apart.'' Again a trial run utilizing only stations 1 and 3, 1 and 5, and 1 and 8 was made, again without measurements. After these preliminary observations, the experiment continued with the formation of the complete parallel and distance alleys in that sequence. The usual procedure was to give a second trial of each alley on the same day and to repeat the series on a second Hay. The data given in fable IV and Fig. 22 for two observers represent the average of three such settings, averaged again on the left and right. 50 TABI£ IV Blumenfeld Parallel and Distance Alleys. Average setting of y for each value of x. Values of x and y are expressed in centimeters« Observer G.R. Observer M.C.R. Parallel 50 9*95 65 83 10.65 ru55 108 12.90 139 1U.55 180 16.6* 232 19.35 300 23.35 387 28.85 500 35.00 Distance yd 23.75 23.05 23.90 2U.25 23.85 22.95 23.75 26.10 29.95 35.00 Parallel 7 P 13.85 lli.liO i5.Itf 16.75 18.10 19,?5 21.h$ 2U.U0 28.8C 35.00 Distance 20.60 21.50 21.80 21.25 21.05 22.65 2h.50 27.10 29.10 35.X For each point of the setting the values of y and

j tanh r = tanh co sin (p. for the points of the parallel alley; and the relation (21) sinh r • sinh co sin _ 4>x sin 0d • for the points of the distance alley. The values of r (T) found by this method are presented * WEIGHTED AVERAGE S£Tlr = 2 r 53 TABIS V Twenty values of r (V ) computed from the results of the Blumenfeld Parallel and Distance Alley Experiment» Observer G. R. Observer M« C« R. a) = -931 Parallel Alley r r Distance Alley r r Parallel Alley r r Distance Alley r r »13.0$ .0832 •C629 -rii5? •C328 .0225 .01)47 .0085 .0037 .0000 *33U •L12 .h$2 .590 .693 •e33 .986 1.126 1.238 1.1470 •C917 .C7U7 .0585 .Oli3? •C320 •C222 •008U .0037 «0000 .329 .M8 •U99 .617 .772 .975 1.H5 1.280 1,381 T.li70 .1103 .08I4J .C6U2 .Cli69 .0338 .0233 .0152 .0088 .0038 .0000 .19U .2U0 .238 .316 .I1I8 .517 .625 *7Ul •8I*U .931 ,1005 .0790 .0617 .OI46O .0331* .0231 .0151 *G08o .0038 •0000 .236 .290 .3-/6 .1465 .568 •662 .75? .880 .931 in Table V for two observers. The graphic representation of r plotted against F is displayed in Fig. 28 where these results are compared with those of other experiments. 3b. The Double Vieth-Muller Circle Experiments The Ebuble Vieth-Muller Qrcle experiments were performed with the lights viewed directly; i.e., without the tcieostereoscopic device. The theoretical background for these experiments is given in Part I, Sections 5b (i) and 5b (ii). Both the three-point and four-point experiments were performed for the four values of r = y2 - yj • .005, .01, .02r .04. The convergence angle y. was fixed throughout at .025 for observer G.R. and .026 for M. C.R. At any sitting, a mixed order of F and

j) ""»s moved on the circle y = y, , to satisfy the instruction of Section 2b (i). Settings of Qj were taken for five positions of Q2 at 4>2 = 5°, 10°. 15°, 20° and 25°. At least three settings were taken for each position of Q2 at a given time. The entire series of experiments for the four values of T was performed twice. 54 TABUS VI Three-Point Double Vieth-Muiler Circle Experiment» Average values of X s cos 0, „ for given vaiies of X I cos 02 and P • Observer G.Rs Avei-age values of T for given values of X and T r X .005 .01 .02 T, * -C25 »CU *9962 •98U8 •9659 .9063 ID u *>= ©9892 .9778 •960U .93&U .9012 .9613 »0315 .9928 .9799 .9702 .cine .8992 •9002 .06u0 .931* • •-/./»• .9511» .9389 _ 00-50 .8859 .8995 •06U3 .9638 .8812 •6836 .8811 ft*.flrt .e566 .3U86 *5U05 .8891 Observer M.C.R. Average values or T for given values of X and T — .«.cw X^\, .0052 •OIOU •C208 .Chl6 .9962 .981*8 .9659 .9397 .9063 m 2 b = .9899 .9799 -9653 .91-12 .915U *8Utf .1L86 .9933 e9839 .9706 *9562 •9367 •9120 .7737 .2102 .9839 .9U85 .9U22 .9306 .9133 .89L8 .6135 .3378 •9513 .8908 .88U0 •wu£ .8688 .8561; .3668 .5239 .8907 55 X If) O) O if) 0) c c if) CM 2 8 X 0) 5 t> O o a. o eon 0) a. x LJ c CJ a> a> a > a> ja o .<= i .1 a> CVJ o m o If) o en•--) CM <7) o en co in CD ' " 1 so The values of Y = cos „) and Q_2 = (y2 > 0-2^ *ere restricted to motion along a smaller VMC y - y2 (see .23 Fig. 26). The lights Q±2 on the inner circle were left fixed and the observer was asked to set the lights Qj.( on the outer circle according to the specific instructions: ''Four lights are presented .50 .?5 FIG. 25 THREE-POINT DVMC EAPKKIMENT. PLOT EXHIBITING INSENSITIVITY OP EXPERIMENT IN DETERMINING u AND b. 1.0* to you. Two of them are fixed in position and the other two can be moved." (These lights and their range of motion are demonstrated.) ''Direct the experimenter to adjust these lights so 57 that the distance between them appears to ycu to be the same as the distance between the pair of fixed lights. Allow the eyes to roam freely both ways over the spatial interval between each pair of lights. Be sure to fixate on each light in turn and to sweep the eyes across the interval between each pair of lights until you are satisfied that the two distances appear to you to be the same." Settings of Q+1 = {yl , <$>{) and Q-j = (/, , <£_,) were taken for five positions of Q+2 and and Q_2 en the VMC J = 72 symmetric to the median with differences in azimuth A2 = 10°. 30°. 50° (Fie. 26). At least three settings were taken at a £iven time for each position of Q±?. The entire series of experiments for the four values of V was per- formed twice. r-ri The values of Y = sin /4A { were determined for each of the values of X = sin !4A0 at a given value of P, and then Y was plot- ted against X. From equation (27) m = Y/X the slope m was computed from „..v. L; ..^e..—ie ........ in- spect to X, (39) m=--- FIG. 26. PHYSICAL, ARRANGEMENT IN FOUR-POINT DVHC SXPHUKENT. The value of Y for each value of X is given in Table VII for each value of F, and the data are also plotted in Fig, 27, Taking the value of m from (39) and the value of b from (38), the value ofu for T = 0 was obtained from formula (26) b o) = arc cosn [(1-b)2 -m2]* The values of t were ihei» computed according to formula '27) r = arc sinh (m sinh OJ) . The values r are given in Table VIII and plotted in Fig. 28 where they are compared with those obtained in other experiments. 53 'S < m -I«M *oc-> c X c 3 fc > V ö. V Ul y O 2 ok. ä -» c <3 o Q. r. i— 3 O U- 1Ä > •oc u r- C\J 59 TA3IS VII Four-Point Double Vieiih-Jiüller Circle Experiment f Average values of x = sin 1/2 A for given values of X z sin l/2 A2 and I , Observer G»S» T. s .025 Average values of 1 for given values of X and I-1 »oo5 •01 .02 •ok •0872 *258S •U226 m •0685 .1972 •-JO50 •7U25 .060I1 .1805 .2660 .6596 .033li •12U9 .1887 J*5l5 .0160 .0562 .1353 .3220 Observer M.G.R. T. = »026 Average values of T for given values of X and r .0052 •OlOU .0208 •0Ul6 .0872 .2588 •'x226 m - .0809 .2339 .3291 •3Uii2 .07U9 •1923 .2858 .7193 .0732 .1755 .2231 .6137 .0390 .1176 .1586 •Jaoo 3c. The Equipartitioned Parallel Alleys This experiment is chat of 2b (ii). The mean values of ^ and T2, tan <£, and tan cp2 for the four positions of ?3 were taken from Table III and the values of r computed from these data. Hence the values of r given here represent sixteen experimental settings. To compute the values of r we first determine the values of S = tan 4>2 I tan $\ an(^ T • tar. <£o / tan #„. For each of the experimental points r is then computed frc~ formula (28) as ! tanh Y r- = arc tanh | • | sin cp. 60 TABI£ VIII Five values of r ( P ) computed from the results of the Double Vieth-Muller Circle Experiments. observer G»JU T. = .025 ,0000 .0050 •3100 •0200 •31*00 to Value o£ CO 1.1*8 1.22 1.13 0c8U 0.63 Observer M*S«R» P •0000 .0052 .0101; •0208 •3106 T, = .026 .95* •*3 .73 .63 •Ul4 where we use formula (30) for V sinh Y • tan 0, 2 - (S • T) (S • T) - 2 S.F.' L The results of these computations are given in Table IX. In Fig. 28 the results of this experimental series ai« compared with those obtained from the Blumenfeld Alleys and the Double Vieth-Müller Circle Experiment.-, for the two observers who completed the full series of experiments. 3d. The Personal Characteristic, r ( F ), If it be assumed that an individual's metrization of spsce is constant over long periods of life, then by determining the function r (F) #e are able to give a useful and significant description of his space sense. This function r (IP) may be thought of as a personal characteristic of the individual which describes his spatial responses for r'ueless vision in the same sense that his color matrix describes his responses to color mixtures. With respect to th#» function r (F) we have sought to answer the following questions: (a) What are its obvious characteristics? (b) How well do the values obtained from different kinds of experiments agree with each other? 2 u Ul oUi >u U O 15 ffl CoO g i o o • t & « I -*S-+- o c 13 CO g mro o m O g 62 TABIE IX Three values of r ( P ) computed from the results of the Sq*iipartitioned Faxallwi £/J.ey Sxperiment Observer 0» EU r r .00000 •00562 •039ÖU 1*3706 1.0153 Q*bhh$ Observer M.C.R. •OOOOO .00731 .03908 1.0308 0.7993 0.5676 Value of 00 (c) Are there measurable differences between individuals? A partial answer to these questions may he obtained from a perusal of Fig. 28 which summarizes the results of Tables V, VIII and IX. (a) The function r (D is a decreasing function and convex downward. (b) The results of the three different experimental technics employed here do show a measure of agreement in the values of r (F). Whether or not the differences lie within the range of variability of the observer for any given experiment is a problem for further investigation and statistical analysis. (c) Individual differences have been found. The function r (F) for observer G.R. is consistently larger than for observer M.C.R. Another problem that has occupied our attention is that of determining a uniform way of interpolating a curve between the experimentally found values of the function, so that r (T) may be specified in terms of a limited number of real constants. To attempt a solution of this problem for di normal individuals is probabiy premature since it would require the testing of many more observers. Within the range of variability of the two observers employed here, however, the function r (T) can be represented adequately in tne form CO l +ar For observer G.R. we have approximately co - 1,48 , a = 33,2; for observer M.CR. , co = 1.00 , a = 37.2 . 4. INSTRUMENTATION In describing the geometry of binocular visual space we are concerned here not with thresholds and acuities, but with observations in the largn where the eyes rove over extensive regions of space. For the present purpose we are interested in the response to gross stimuli rather than to the barely perceptible. The range of convergence and azimuch supplied by single light points in the laboratory is by no means an adequate domain for testing the range of the binocular responses to gross stimuli. Not only is it desirable to test with distant obi^cts and at large angles of azimuth, but it is instructive to extend the range of observation as far as possible, even into the region of divergence. Such requirements can only be met by the use of special stereoscopic devices^ Since the use of a stereoscopic device of either of the types described here upsets the normal relationship between accommodation and convergence it is necessary to show that accommodation is a negligible or minor factor in binocular responses of the kind measured here. Campbell 6 in his study of size constancy phenomena in clueless vision was able to demonstrate that the substitution of stimuli formed by his stereoscopic device for single physical light points resulted in no appreciable change in the observer's settings. Further, he was able to commingle stimuli of the two kinds, again without appreciable change in the responses of the observer. We have used Campbell's demonstration as adequate justification for the use of our devices.* 4a. The Telestereoscope or ' 'Giant' s Eyes'' Instrument. This device is based on a mirror arrangement. A right-angled first surface mirror is placed symmetrically with respect to the median, apex toward observer (Fig. 2^). Two mirrors are set symmetrically to the median so that the extension of the plane of each of the mirrors meets the extension away from the apex of the corresponding side of the right-angled mirror. The angle of intersection is denoted by c»^ Let Q be a physical light viewed through ehe instrument. Let Q be the position of the •ON THE OTHER HAND. WE HAVE NOT USED STIMULUS POINTS CLOSER THAN 50cm. SO THAT THE RANGE OF ACCOMMODATION IS NOT EXTREMELY LARGE. 64 FIG. 29. THE TELESTEREOSCOPIC OR ' *GI ANT' S EYES' ' INSTRUMENT. binocular image of Q (i.e., the point at which the visual axes must cross to throw the separate images of Q on the respective ioveae of the two eyes). Now consider the path of a ray of light from 0 to the fovea of the right eye at 11. Proceeding from R to Q we see first that the ray which reaches R from its side of the 90° mirror must have been inflected from » ray directed through the image R of R in the mirror. Similarly, the ray directed through R' mu-L iwvc !«ten reflected at the side mirror from a r*y directed through the image R of R' in that mirror, L$y ayawstry '.ve determine *ne corresponding points L and L. for the '.eft eye. Thus, to determine the position oi Q from that of Q we trace the rays to the two eyes and extend tl:•*> terminal segnents at the eyes to their poir^ of intersection. Thus, for the right eye, we draw Q R and find its intersection I with the side mirror. We then draw IR' and determine the intersection J with the right face of the 90° mirror. The point Q will then He on the extension of JR. 65 For algebraic convenience we use bipolar coordinates y , 4> with respect to the points L, R (the ''Giant's Eyes") to specify the position of Q, and ordinary bipolar coordinates y ,

/2(YR +YL). The separation of Q^ and Q^ is S " \-\ The cartesian coordinates (x, y) öf Q are givsn fay (40) x J£ X p +s PY p +S where p is the interpupillary distance of the observer. From equation (40) approximate formulas 56 ii i iiiiiiiiiiii FIG. 30. SCHEMATIC REPRESENTATION OF THE PRINCIPLE OF THE POLAROID RACK. for the bipolar coordinates (y , _ wot Laken into account. To avoid entering into such considerations it would be desirable to employ emmetrcpic observers. 67 5. CONCLUSIONS The experimental work presented here is direct evidence that Luneberg's approach to binocular vision is at once sensible and fe^ible. In fact this evidence strongly supports Lüneburg's major conclusions: (a) The binocular visual space is a determinate metric space with constant characteristics for a given observer. (b) By experiment, it is possible to determin» the metric of an observer's visual space and so to completely characterize the geometry of his binocular visual sen;>e. (c) The metric is that of riemannian space of constant negative curvature, the so-called hyperbolic space. In particular, the experiments show that in all likelihood, the metric of visual space may be written in terms of special coordinates attached to the stimulus configuration; i.e., the iseikonic coordinates, f, 3>, 0. The problem of determining the metric for a given observer is then reduced tc the problem of determining the one function r (T). The function i (T) is to be thought of as a sensory characteristic of the cbserver which describes his geometric visual sense, much as an individual's <. lor matrix describes his sense of hue and saturation. The determination of norms for the function might therefore be useful, especially with regard to our understanding of deviant or abnormal binocular function. As yet it is too early to make predictions concerning the eventual usefulness (clinically or otherwise) of the theory. A great deal of work to set up standards muut first be undertaken. Yet some practical results will undoubtedly follow from the increased understanding we already have. For example, it may be suggested that parallel rows of guide lights be set up along all airplane runways at e uniform standard separation of the rows and at a uniform standard spacing of the lights, so J\at a pilot landing at night can rely on facing the same situation each time he lands at zny field And on any runway. If this were carried into national or international standard patterns, it might do much to reduce hazards of visual landings - particularly at strange airports. Perhaps similar standards would prove useful in other applications where space judgments must be made in a situation providing reduced clues; e.g., it might lead to a consideration and solution of the problem of integrating the magnification of all binocular viewing instruments If the interpupiilary distance of a 6X binocular, for example, were optically magnified by the same factor of 6, the relative changes in convergence required in using the instrument should lead to a more realistic appraisal, by the observer, of frontal distances involved in the field cf view. «* REFERENCES 1. R. K. Lüneburg, Mathematical Analysis of Binocular Vision (Princeton Univ. Press, Princeton, N. J., 1947). 2. R. K. Lüneburg, Metric Methods in binocular Visual Perception, Studies and Essays, Courant Anniversary Volume (Inttrscience Publishers, Inc., New Yorkv 1948). 3. R K. Lüneburg, ''The metric of binocular visual space," J. Opt, Sor, Am. 40, 627 (1950). 4. K. N. Ogle, Researches in binocular Vision (W. B. Saunders Co., Philadelphia, Pa. 1950). 5. G. A. Fry, ''Visual perception of space,'' An. J. Optometry, 27, 531 (1950). 6. G. A. Fry, "Comments on Lüneburg's analysis of binocular vision," An. J. Optometry, 29, 3 (1952). 7. R. K. Lüneburg, See Reference 3, p. 630. 8. H. Busemann, Metric Methods in Finsler Spaces and in the Foundations oj Geometry (Princetot. Urdv. Press, Princeton, N. J., 1942). 9. H. S. M. Goxeter, Non-Euclidean Geometry (Univ. of Toronto Press, Toronto, Can., 1947). 10. rL S. Carslaw, The Elements of Non-Euclidean Plane Geometry and Trigonometry (Longmans Green & Co., London, 1916). 13. R. K. Lüneburg, See Reference 1, pp. 17, 89. 12. W. ßlunenfeld, ' 'Lnterruciiungcn über die scheinbare Grosse ir Sehraume', Z. Psychol. u, Physiol. d. Sinnesorg. 65 (Abt. 1), 241 (1913). 13, L. H. Hardy, G. Rand and M. C. Rittler, 'Investigation of visual space. The Blunenfeld alleys,'' Aich. Cpht.hal. (Chicago) 45. 53 (1951). )4. R. K. Luneb-jrg, See Reference 3, p. 638. 15. C ii. Graham, 'Visual perception". In 3. S. Stevens, Handbook of Experimental Psychology, 868 (John Wiley & Sons, Inc., New York, 1951). 16. C. J, Campbell, An Experimental Investigation of the Size Constancy Phenomenon. Col. linjv. Thesis (1952). 17. ILL.F. Itelmholtz, Physiological Optics, Edited by.]. P. C, Southall, Rochester, N. Y., Opt. Soc. Am. (1925) vol. 3, p, 318. 18. R. K. Lüneburg, See Reference 1, p.. 104 and Reference 3, p. 633.