The Geometry of Binocular Space Perception

Abstract

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.

A new method for determining the personal constants in the Luneburg theory of binocular visual space

“…Many workers have since concerned themselves with the original or modified theory from theoretical and experimental viewpoints. Some of the results of various experiments carried out to test the validity of the theory confirmed it (e.g., Blank, 1961;Foley, 1964;Hagino, Yoshioka, & Hirabayashi, 1963;Hardy, Rand, Rittler, Blank, & Boeder, 1953; This study was supported by a scientific research grant provided by the Japanese Ministry of Education. The present article is largely based on an earlier article (Hagino & Yoshioka,I%8) and further presents the results and implications of the study in the context of subsequent related research.…”

“…On the other hand, Ishii (1972) proposed that-the geometry of visual space might not be so exclusively hyperbolic as Luneburg claimed but might vary from hyperbolic to elliptical and from individual to individual, that-is, -I~K~+ I, based on the statistics of signs of Ks obtained by the earlier researchers, including himself (Blank, 1961;Hagino et al, 1963;Hardy et al, 1951Hardy et al, , 1953Indow et al, 1962a, b;Ishii, 1962Ishii, , 1964Shipley, 1957;Zajaczkowska, 1956a, b). According to his statistics, Ks were positive in 13 of 52 cases in the alley experiments and in 8 of 68 cases in the 3-and 4-point and in other experiments.…”

“…In his system (Blank, 1953(Blank, , 1958aHardy et al, 1953), r(r), where r 0= Y -Yo, r corresponds to (I, the visual radial distance in the Luneburg system, and co, defined as co = r(O), is used as the measure of visual radial distance of the farthest point, Qo(Yo), in a given stimulus configuration. By definition, co is conceived to be constant for an observer irrespective of the value of r. The value of co is determined by the following equation: (16) where SIT is an index reflecting the degree of the hyperbolic discrepancy between the parallel and the distance alley and is considered to be the same irrespective of the value of r. However, Zajaczkowska (1956a) noted that there were systematic changes in the values of co with the variation of Yo. The values of co became larger as the values of Yo decreased, while Hagino et al (1963) observed an opposite tendency of co, that is, the smaller the values of Yo, the smaller the values of co. But Indow, Inoue, and Matsushima (1963) reported that they found neither tendency consistently.…”

The visual perception of size and distance

“…However, the fact that convergence does have some perceptual consequences with a few observers is also of significance. For example, the assumption that the addition of a constant amount of convergence will not affect the perceived depth in a binocular configuration (BLANK, 1957(BLANK, , 1959HARDY et al, 1953) is neither true nor false. From the above studies by Gogel, it is true with most observers but not with all.…”

Size Constancy and the Intrinsic Geometry of Binocular Visual Space

“…This model has been tested in several ways by many investigators: the 3-and 4-point experiments (Hardy, Rand, Rittler, Blank, & Boeder, 1953;Higashiyama, 1976;Indow, Inoue, & Matsushima, 1962a;Zajaczkowska, 1956b), the alley problem (Hardy, Rand, & Rittler, 1951;Indow, Inoue, & Matsushima, 1962b;Ishii, 1972;Shipley, 1957;Zajaczkowska, 1956a) and examinations concerning the axiom system (Foley, 1964(Foley, , 1972.…”