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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 37, Iss. 12 — Dec. 1, 1947
  • pp: 944–956

A Certain Class of Binocularly Equivalent Configurations

ANNA STEIN  »View Author Affiliations

JOSA, Vol. 37, Issue 12, pp. 944-956 (1947)

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Originating in a mathematical study of a series of binocular visual phenomena experienced in certain demonstrations developed by The Hanover Institute (formerly Dartmouth Eye Institute), the following analysis shows that even binocular vision does not provide absolute localization of objects. On the contrary, a series of vertical targets may furnish binocularly the same angular clues, if their location, their inclination, and their shape are suitably chosen. Although the straight lines where the possible equivalent vertical targets intersect the horizontal plane represent only a special, so to speak degenerated subset of the whole set of equivalent conic sections, their mathematical analysis reveals that two theorems of projective geometry dealing with projective properties of conic sections find an interesting application in the field of binocular vision. In the introduction, two actual experiments are described. The geometrical construction and the computations on which these experiments are based are discussed in the last section where two additional theoretically possible experiments are suggested.

ANNA STEIN, "A Certain Class of Binocularly Equivalent Configurations," J. Opt. Soc. Am. 37, 944-956 (1947)

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  1. A. Ames, Jr. "Binocular vision as affected by relations between uniocular stimulus patterns in commonplace environments," Am. J. Psychol. 59, 333 (1946).
  2. While the apparent shift of the configurations in the direction shown is markedly unequivocal for most observers, it does not necessarily reach the mathematically determined amount. This may be due to other conflicting clues as to distance and direction. It has been found that any parallax movement dampens the distance effect, and it is possible that directional clues resulting from eye movement may dampen the directional effect. To experience these phenomena, the observer must have good stereoscopic vision and no aniseikonia or, if he has aniseikonia, it must be accurately corrected. If he has any aniseikonia, he will experience a shift of the targets due to that cause, which may nullify or exaggerate the phenomena.
  3. For further information refer to R. Luneburg, "Mathematical analysis of binocular vision," published by the Dartmouth Eye Institute, 1947.
  4. For a more extended treatment of the applied theorems, we refer the reader to the literature on projective geometry; e.g., Courant and Robbins, What is Mathematics? (Oxford University Press, 1941), Chap. IV, pp. 165–214, whose notation we follow in general.

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