Abstract

The Luneburg theory states that the visual space can be represented by one of the Riemannian spaces of constant curvature. This implies that the visual space is Desarguesian. The Desarguesian property may be stated as follows: If two vertices of a triangle are jointed by segments to the opposite sides, these segments intersect. This proposition was tested by having observers move small lights so as to construct apparent segments on a given triangle. Six such experiments were performed using two triangles and five observers. In all cases the minimum physical distance between the two curves determined by the average settings of the lights was between 20 and 220 sec. In two of the six cases this distance was not significantly different from zero. Four determinations of the threshold for discriminating the nonintersection of two segments yielded values in the range of 59 to 92 sec. It was concluded that the visual spaces of a significant proportion of the observers are Desarguesian; those of others may be non-Desarguesian.

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  1. R. K. Luneburg, Mathematical A nalysis of Binocular Vision (Princeton University Press, Princeton, New Jersey, 1947). (a) Pp. 89–94.
  2. R. K. Luneburg, "Metric Methods in Binocular Visual Perception" in Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).
  3. Luneburg used the phrase "visual space" to refer to the three-dimensional space in which are localized the colors, brightnesses, and objects of which we are immediately aware (Ref. 1, p. 1). This phenomenological definition, however, is not essential to his development, since only the mathematical properties of the space enter into his argument. Blank (Ref. 13, p. 329) refers to the visual space as a class of elements called points, having among themn two relations designated as length ordering and alignment. These relations are defined mathematically only in terms of the axioms. In any experiment a finite number of these points are interpreted as points of the stimulus and the relations are interpreted as the relations among the stimulus points after they have been arranged to satisfy some instructional criterion.
  4. H. von Helmholtz and J. Southhall (translator), Physiological Optics (Optical Society of America, Rochester, New York, 1925), Vol. 3, p. 318.
  5. W. Blumenfeld, Z. Psychol. Physiol. Sinnesorg. 65, 241 (1913).
  6. W. H. Ittelson, The Ames Demnonstrations in Perception (Princeton University Press, Princeton, New Jersey, 1952), pp. 50–52.
  7. Luneburg went on to argue, on the basis of the results of the Blumenfeld alley experiment, that the visual space is hyperbolic, that is, of negative constant curvature. Whereas there remains some uncertainty about the constancy of the curvature, considlerable evidence has been amassed to indicate that the curvature is negative for a large majority of observers. Particularly convincing in this respect is the experiment of Blank [J. Opt. Soc. Am. 51, 335 (1961)], which avoids any assumptions either about the curvature of the space or the relation between physical and visual space.
  8. A. A. Blank, J. Opt. Soc. Am. 43, 721 (1953).
  9. T. Shipley, J. Opt. Soc. Am. 47, 795 (1957).
  10. A. Zajaczkowska, J. Opt. Soc. Am. 46, 523 (1956), Table IV.
  11. A. A. Blank,tJ. Opt. Soc. Am. 48, 919–21 (1958).
  12. A. A. Blank, Brit. J. Physiol. Opt. 14, 154, 222 (1957).
  13. A. A. Blank, J. Opt. Soc. Am. 48, 328 (1958). (a) P. 333.
  14. H. Busemann, Metric Methods in Finsler Spaces and in the Foundations of Geometry (Princeton University Press, Princeton, New Jersey, 1942), p. 115.
  15. H. Busernann, The Geometry of Geodesies (Academic Press Inc., New York, 1955).
  16. C. J. Campbell, "An Experimental Investigation of the Size Constancy Phenomenon" (unpublished dissertation, Columbia University, 1952).
  17. Prior to this study a pilot study was carried out. It differed from the present study in that it was done with a slightly less precise apparatus, involved 20 rather than 24 settings of each segment, and had some minor differences in procedure. Two observers, 01 and 02, were used; 02 was the same observer as 02 of the present study. The mean shortest join length was found to be 0.002 cm for 01 and 0.067 cm for 02; the first of these was not significantly different from 0 at the 0.05 level of significance; the second was significantly different.
  18. Since significantly large correlations exist among the components of the shortest join, the three tests for each observer are not independent. This weakness could be overcome by calculating the sum of the three components for each setting and testing the mean of these sums against an estimate of its standard deviation. However, since in all cases the D component is considerably larger than the other two, it is evident that the result of such a test would be essentially the same as the result given in Table II for the D component.

Blank, A. A.

A. A. Blank,tJ. Opt. Soc. Am. 48, 919–21 (1958).

A. A. Blank, Brit. J. Physiol. Opt. 14, 154, 222 (1957).

A. A. Blank, J. Opt. Soc. Am. 43, 721 (1953).

A. A. Blank, J. Opt. Soc. Am. 48, 328 (1958). (a) P. 333.

Blumenfeld, W.

W. Blumenfeld, Z. Psychol. Physiol. Sinnesorg. 65, 241 (1913).

Busemann, H.

H. Busemann, Metric Methods in Finsler Spaces and in the Foundations of Geometry (Princeton University Press, Princeton, New Jersey, 1942), p. 115.

Busernann, H.

H. Busernann, The Geometry of Geodesies (Academic Press Inc., New York, 1955).

Campbell, C. J.

C. J. Campbell, "An Experimental Investigation of the Size Constancy Phenomenon" (unpublished dissertation, Columbia University, 1952).

Ittelson, W. H.

W. H. Ittelson, The Ames Demnonstrations in Perception (Princeton University Press, Princeton, New Jersey, 1952), pp. 50–52.

Luneburg, R. K.

R. K. Luneburg, "Metric Methods in Binocular Visual Perception" in Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).

R. K. Luneburg, Mathematical A nalysis of Binocular Vision (Princeton University Press, Princeton, New Jersey, 1947). (a) Pp. 89–94.

Shipley, T.

T. Shipley, J. Opt. Soc. Am. 47, 795 (1957).

Southhall, J.

H. von Helmholtz and J. Southhall (translator), Physiological Optics (Optical Society of America, Rochester, New York, 1925), Vol. 3, p. 318.

von Helmholtz, H.

H. von Helmholtz and J. Southhall (translator), Physiological Optics (Optical Society of America, Rochester, New York, 1925), Vol. 3, p. 318.

Zajaczkowska, A.

A. Zajaczkowska, J. Opt. Soc. Am. 46, 523 (1956), Table IV.

Other (18)

R. K. Luneburg, Mathematical A nalysis of Binocular Vision (Princeton University Press, Princeton, New Jersey, 1947). (a) Pp. 89–94.

R. K. Luneburg, "Metric Methods in Binocular Visual Perception" in Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).

Luneburg used the phrase "visual space" to refer to the three-dimensional space in which are localized the colors, brightnesses, and objects of which we are immediately aware (Ref. 1, p. 1). This phenomenological definition, however, is not essential to his development, since only the mathematical properties of the space enter into his argument. Blank (Ref. 13, p. 329) refers to the visual space as a class of elements called points, having among themn two relations designated as length ordering and alignment. These relations are defined mathematically only in terms of the axioms. In any experiment a finite number of these points are interpreted as points of the stimulus and the relations are interpreted as the relations among the stimulus points after they have been arranged to satisfy some instructional criterion.

H. von Helmholtz and J. Southhall (translator), Physiological Optics (Optical Society of America, Rochester, New York, 1925), Vol. 3, p. 318.

W. Blumenfeld, Z. Psychol. Physiol. Sinnesorg. 65, 241 (1913).

W. H. Ittelson, The Ames Demnonstrations in Perception (Princeton University Press, Princeton, New Jersey, 1952), pp. 50–52.

Luneburg went on to argue, on the basis of the results of the Blumenfeld alley experiment, that the visual space is hyperbolic, that is, of negative constant curvature. Whereas there remains some uncertainty about the constancy of the curvature, considlerable evidence has been amassed to indicate that the curvature is negative for a large majority of observers. Particularly convincing in this respect is the experiment of Blank [J. Opt. Soc. Am. 51, 335 (1961)], which avoids any assumptions either about the curvature of the space or the relation between physical and visual space.

A. A. Blank, J. Opt. Soc. Am. 43, 721 (1953).

T. Shipley, J. Opt. Soc. Am. 47, 795 (1957).

A. Zajaczkowska, J. Opt. Soc. Am. 46, 523 (1956), Table IV.

A. A. Blank,tJ. Opt. Soc. Am. 48, 919–21 (1958).

A. A. Blank, Brit. J. Physiol. Opt. 14, 154, 222 (1957).

A. A. Blank, J. Opt. Soc. Am. 48, 328 (1958). (a) P. 333.

H. Busemann, Metric Methods in Finsler Spaces and in the Foundations of Geometry (Princeton University Press, Princeton, New Jersey, 1942), p. 115.

H. Busernann, The Geometry of Geodesies (Academic Press Inc., New York, 1955).

C. J. Campbell, "An Experimental Investigation of the Size Constancy Phenomenon" (unpublished dissertation, Columbia University, 1952).

Prior to this study a pilot study was carried out. It differed from the present study in that it was done with a slightly less precise apparatus, involved 20 rather than 24 settings of each segment, and had some minor differences in procedure. Two observers, 01 and 02, were used; 02 was the same observer as 02 of the present study. The mean shortest join length was found to be 0.002 cm for 01 and 0.067 cm for 02; the first of these was not significantly different from 0 at the 0.05 level of significance; the second was significantly different.

Since significantly large correlations exist among the components of the shortest join, the three tests for each observer are not independent. This weakness could be overcome by calculating the sum of the three components for each setting and testing the mean of these sums against an estimate of its standard deviation. However, since in all cases the D component is considerably larger than the other two, it is evident that the result of such a test would be essentially the same as the result given in Table II for the D component.

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