Abstract

The apparent distortion of stereoscopic space perception, caused by unequal magnifications of the images of the two eyes, as described by Ames, is studied from a mathematical point of view. It is shown that, at least qualitatively, the spatial distortions can be predicted on a geometric basis.

© 1948 Optical Society of America

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References

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  1. K. N. Ogle, “The correction of aniseikonia with ophthalmic lenses,” J. Opt. Soc. Am. 26, 323 (1936).
    [Crossref]
  2. O. F. Wadsworth, “On the effect of a cylindrical lens, with vertical axis placed before one eye,” Tr. Am. Ophth. Soc.342 (1876).
  3. H. Culbertson, “Binocular astigmatism,” J. Am. Med. Assn. 11, 622 (1888); Am. J. Ophth. 5, 117 (1888).
    [Crossref]
  4. J. A. Lippincott, “On the binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 18, 18 (1889).
  5. J. Green, “On certain stereoscopical illusions evoked by prismatic and cylindrical glasses,” Tr. Am. Ophth. Soc.449 (1889).
  6. Harry Friedenwald, “Binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 21, 204 (1892).
  7. C. Koller, “Ueber eine eigenthümliche Sorte dioptrischer Bilder: Ein Beitrag zur Theorie der Cylinderlinsen,” Arch. f. Ophth. 32, 169 (1886), Part 3; “The form of retinal images in the astigmatic eye,” Tr. Am. Ophth. Soc. 6, 425 (1892).
  8. J. A. Lippincott, “On the binocular metamorphopsia produced by optical means,” Arch. Ophth. 46, 397 (1917); also J. A. Lippincott, “New tests for binocular vision,” Tr. Am. Ophth. Soc.560 (1890).
  9. M. von Rohr, “Die Brille als optische Instrument,” (Verlag Julius Springer, Berlin, 1921), third edition, pp. 195.
  10. Franz Hillebrand, “Die Stabilität der Raumwerte auf der Netzhaut,” Zeits. Psychol. 5, 1 (1893).
  11. Adelbert Ames, “Binocular vision as affected by relations between uniocular stimulus-patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).
    [Crossref] [PubMed]
  12. Adelbert Ames and K. N. Ogle, “Size and shape of ocular images: III. Visual sensitivity to differences in the relative sizes of the ocular images of the two eyes,” Arch. Ophth. 7, 904 (1932).
    [Crossref]
  13. Adelbert Ames, “Aniseikonia—a factor in the functioning of vision,” Am. J. Ophth. 18, 1014 (1935).
  14. K. N. Ogle, “Association between aniseikonia and anomalous binocular space perception,” Arch. Ophth. 30, 54 (1943).
    [Crossref]
  15. K. N. Ogle and V. J. Ellerbrock, “Stereoscopic sensitivity in the space eikonometer,” Arch. Ophth. 34, 303 (1945).
    [Crossref]
  16. K. N. Ogle, “Induced size effect: III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,” Arch. Ophth. 22, 613 (1939).
    [Crossref]
  17. K. N. Ogle, “Induced size effect: I. A new phenomenon in binocular space perception associated with the relative sizes of the images of the two eyes,” Arch. Ophth. 20, 604 (1938); “II. An experimental study of the phenomenon with restricted fusion stimuli,”  21, 604 (1939); “III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,”  22, 613 (1939); K. N. Ogle, “Induced size effect,” J. Opt. Soc. Am. 30, 145 (1940); K. N. Ogle, “Induced size effect with eyes in asymmetric convergence,” Arch. Ophth. 23, 1023 (1940); A. S. Householder, “A theory of the induced size effect,” Bull. Math. Biophys. 5, 155 (1943).
    [Crossref]
  18. K. N. Ogle and L. F. Madigan, “Astigmatism at oblique axes and binocular stereoscopic spatial localization,” Arch. Ophth. 33, 116 (1945).
    [Crossref]
  19. K. N. Ogle, “Meridional magnifying lens systems in the measurement and correction of aniseikonia,” J. Opt. Soc. Am. 34, 302 (1944).
    [Crossref]
  20. A. Ames, K. N. Ogle, and G. H. Glidden, “Corresponding retinal points, the horopter and size and shape of ocular images,” J. Opt. Soc. Am. 22, 576 (1932).
  21. H. von Helmholtz, Physiological Optics (English translation) (The Optical Society of America, 1925), Vol. 3, p. 349.

1946 (1)

Adelbert Ames, “Binocular vision as affected by relations between uniocular stimulus-patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).
[Crossref] [PubMed]

1945 (2)

K. N. Ogle and V. J. Ellerbrock, “Stereoscopic sensitivity in the space eikonometer,” Arch. Ophth. 34, 303 (1945).
[Crossref]

K. N. Ogle and L. F. Madigan, “Astigmatism at oblique axes and binocular stereoscopic spatial localization,” Arch. Ophth. 33, 116 (1945).
[Crossref]

1944 (1)

1943 (1)

K. N. Ogle, “Association between aniseikonia and anomalous binocular space perception,” Arch. Ophth. 30, 54 (1943).
[Crossref]

1939 (1)

K. N. Ogle, “Induced size effect: III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,” Arch. Ophth. 22, 613 (1939).
[Crossref]

1938 (1)

K. N. Ogle, “Induced size effect: I. A new phenomenon in binocular space perception associated with the relative sizes of the images of the two eyes,” Arch. Ophth. 20, 604 (1938); “II. An experimental study of the phenomenon with restricted fusion stimuli,”  21, 604 (1939); “III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,”  22, 613 (1939); K. N. Ogle, “Induced size effect,” J. Opt. Soc. Am. 30, 145 (1940); K. N. Ogle, “Induced size effect with eyes in asymmetric convergence,” Arch. Ophth. 23, 1023 (1940); A. S. Householder, “A theory of the induced size effect,” Bull. Math. Biophys. 5, 155 (1943).
[Crossref]

1936 (1)

1935 (1)

Adelbert Ames, “Aniseikonia—a factor in the functioning of vision,” Am. J. Ophth. 18, 1014 (1935).

1932 (2)

Adelbert Ames and K. N. Ogle, “Size and shape of ocular images: III. Visual sensitivity to differences in the relative sizes of the ocular images of the two eyes,” Arch. Ophth. 7, 904 (1932).
[Crossref]

A. Ames, K. N. Ogle, and G. H. Glidden, “Corresponding retinal points, the horopter and size and shape of ocular images,” J. Opt. Soc. Am. 22, 576 (1932).

1917 (1)

J. A. Lippincott, “On the binocular metamorphopsia produced by optical means,” Arch. Ophth. 46, 397 (1917); also J. A. Lippincott, “New tests for binocular vision,” Tr. Am. Ophth. Soc.560 (1890).

1893 (1)

Franz Hillebrand, “Die Stabilität der Raumwerte auf der Netzhaut,” Zeits. Psychol. 5, 1 (1893).

1892 (1)

Harry Friedenwald, “Binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 21, 204 (1892).

1889 (2)

J. A. Lippincott, “On the binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 18, 18 (1889).

J. Green, “On certain stereoscopical illusions evoked by prismatic and cylindrical glasses,” Tr. Am. Ophth. Soc.449 (1889).

1888 (1)

H. Culbertson, “Binocular astigmatism,” J. Am. Med. Assn. 11, 622 (1888); Am. J. Ophth. 5, 117 (1888).
[Crossref]

1886 (1)

C. Koller, “Ueber eine eigenthümliche Sorte dioptrischer Bilder: Ein Beitrag zur Theorie der Cylinderlinsen,” Arch. f. Ophth. 32, 169 (1886), Part 3; “The form of retinal images in the astigmatic eye,” Tr. Am. Ophth. Soc. 6, 425 (1892).

1876 (1)

O. F. Wadsworth, “On the effect of a cylindrical lens, with vertical axis placed before one eye,” Tr. Am. Ophth. Soc.342 (1876).

Ames, A.

A. Ames, K. N. Ogle, and G. H. Glidden, “Corresponding retinal points, the horopter and size and shape of ocular images,” J. Opt. Soc. Am. 22, 576 (1932).

Ames, Adelbert

Adelbert Ames, “Binocular vision as affected by relations between uniocular stimulus-patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).
[Crossref] [PubMed]

Adelbert Ames, “Aniseikonia—a factor in the functioning of vision,” Am. J. Ophth. 18, 1014 (1935).

Adelbert Ames and K. N. Ogle, “Size and shape of ocular images: III. Visual sensitivity to differences in the relative sizes of the ocular images of the two eyes,” Arch. Ophth. 7, 904 (1932).
[Crossref]

Culbertson, H.

H. Culbertson, “Binocular astigmatism,” J. Am. Med. Assn. 11, 622 (1888); Am. J. Ophth. 5, 117 (1888).
[Crossref]

Ellerbrock, V. J.

K. N. Ogle and V. J. Ellerbrock, “Stereoscopic sensitivity in the space eikonometer,” Arch. Ophth. 34, 303 (1945).
[Crossref]

Friedenwald, Harry

Harry Friedenwald, “Binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 21, 204 (1892).

Glidden, G. H.

A. Ames, K. N. Ogle, and G. H. Glidden, “Corresponding retinal points, the horopter and size and shape of ocular images,” J. Opt. Soc. Am. 22, 576 (1932).

Green, J.

J. Green, “On certain stereoscopical illusions evoked by prismatic and cylindrical glasses,” Tr. Am. Ophth. Soc.449 (1889).

Hillebrand, Franz

Franz Hillebrand, “Die Stabilität der Raumwerte auf der Netzhaut,” Zeits. Psychol. 5, 1 (1893).

Koller, C.

C. Koller, “Ueber eine eigenthümliche Sorte dioptrischer Bilder: Ein Beitrag zur Theorie der Cylinderlinsen,” Arch. f. Ophth. 32, 169 (1886), Part 3; “The form of retinal images in the astigmatic eye,” Tr. Am. Ophth. Soc. 6, 425 (1892).

Lippincott, J. A.

J. A. Lippincott, “On the binocular metamorphopsia produced by optical means,” Arch. Ophth. 46, 397 (1917); also J. A. Lippincott, “New tests for binocular vision,” Tr. Am. Ophth. Soc.560 (1890).

J. A. Lippincott, “On the binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 18, 18 (1889).

Madigan, L. F.

K. N. Ogle and L. F. Madigan, “Astigmatism at oblique axes and binocular stereoscopic spatial localization,” Arch. Ophth. 33, 116 (1945).
[Crossref]

Ogle, K. N.

K. N. Ogle and L. F. Madigan, “Astigmatism at oblique axes and binocular stereoscopic spatial localization,” Arch. Ophth. 33, 116 (1945).
[Crossref]

K. N. Ogle and V. J. Ellerbrock, “Stereoscopic sensitivity in the space eikonometer,” Arch. Ophth. 34, 303 (1945).
[Crossref]

K. N. Ogle, “Meridional magnifying lens systems in the measurement and correction of aniseikonia,” J. Opt. Soc. Am. 34, 302 (1944).
[Crossref]

K. N. Ogle, “Association between aniseikonia and anomalous binocular space perception,” Arch. Ophth. 30, 54 (1943).
[Crossref]

K. N. Ogle, “Induced size effect: III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,” Arch. Ophth. 22, 613 (1939).
[Crossref]

K. N. Ogle, “Induced size effect: I. A new phenomenon in binocular space perception associated with the relative sizes of the images of the two eyes,” Arch. Ophth. 20, 604 (1938); “II. An experimental study of the phenomenon with restricted fusion stimuli,”  21, 604 (1939); “III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,”  22, 613 (1939); K. N. Ogle, “Induced size effect,” J. Opt. Soc. Am. 30, 145 (1940); K. N. Ogle, “Induced size effect with eyes in asymmetric convergence,” Arch. Ophth. 23, 1023 (1940); A. S. Householder, “A theory of the induced size effect,” Bull. Math. Biophys. 5, 155 (1943).
[Crossref]

K. N. Ogle, “The correction of aniseikonia with ophthalmic lenses,” J. Opt. Soc. Am. 26, 323 (1936).
[Crossref]

A. Ames, K. N. Ogle, and G. H. Glidden, “Corresponding retinal points, the horopter and size and shape of ocular images,” J. Opt. Soc. Am. 22, 576 (1932).

Adelbert Ames and K. N. Ogle, “Size and shape of ocular images: III. Visual sensitivity to differences in the relative sizes of the ocular images of the two eyes,” Arch. Ophth. 7, 904 (1932).
[Crossref]

von Helmholtz, H.

H. von Helmholtz, Physiological Optics (English translation) (The Optical Society of America, 1925), Vol. 3, p. 349.

von Rohr, M.

M. von Rohr, “Die Brille als optische Instrument,” (Verlag Julius Springer, Berlin, 1921), third edition, pp. 195.

Wadsworth, O. F.

O. F. Wadsworth, “On the effect of a cylindrical lens, with vertical axis placed before one eye,” Tr. Am. Ophth. Soc.342 (1876).

Am. J. Ophth. (1)

Adelbert Ames, “Aniseikonia—a factor in the functioning of vision,” Am. J. Ophth. 18, 1014 (1935).

Am. J. Psychol. (1)

Adelbert Ames, “Binocular vision as affected by relations between uniocular stimulus-patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).
[Crossref] [PubMed]

Arch. f. Ophth. (1)

C. Koller, “Ueber eine eigenthümliche Sorte dioptrischer Bilder: Ein Beitrag zur Theorie der Cylinderlinsen,” Arch. f. Ophth. 32, 169 (1886), Part 3; “The form of retinal images in the astigmatic eye,” Tr. Am. Ophth. Soc. 6, 425 (1892).

Arch. Ophth. (9)

J. A. Lippincott, “On the binocular metamorphopsia produced by optical means,” Arch. Ophth. 46, 397 (1917); also J. A. Lippincott, “New tests for binocular vision,” Tr. Am. Ophth. Soc.560 (1890).

Harry Friedenwald, “Binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 21, 204 (1892).

J. A. Lippincott, “On the binocular metamorphopsia produced by correcting glasses,” Arch. Ophth. 18, 18 (1889).

K. N. Ogle, “Association between aniseikonia and anomalous binocular space perception,” Arch. Ophth. 30, 54 (1943).
[Crossref]

K. N. Ogle and V. J. Ellerbrock, “Stereoscopic sensitivity in the space eikonometer,” Arch. Ophth. 34, 303 (1945).
[Crossref]

K. N. Ogle, “Induced size effect: III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,” Arch. Ophth. 22, 613 (1939).
[Crossref]

K. N. Ogle, “Induced size effect: I. A new phenomenon in binocular space perception associated with the relative sizes of the images of the two eyes,” Arch. Ophth. 20, 604 (1938); “II. An experimental study of the phenomenon with restricted fusion stimuli,”  21, 604 (1939); “III. A study of the phenomenon as influenced by horizontal disparity of the fusion contours,”  22, 613 (1939); K. N. Ogle, “Induced size effect,” J. Opt. Soc. Am. 30, 145 (1940); K. N. Ogle, “Induced size effect with eyes in asymmetric convergence,” Arch. Ophth. 23, 1023 (1940); A. S. Householder, “A theory of the induced size effect,” Bull. Math. Biophys. 5, 155 (1943).
[Crossref]

K. N. Ogle and L. F. Madigan, “Astigmatism at oblique axes and binocular stereoscopic spatial localization,” Arch. Ophth. 33, 116 (1945).
[Crossref]

Adelbert Ames and K. N. Ogle, “Size and shape of ocular images: III. Visual sensitivity to differences in the relative sizes of the ocular images of the two eyes,” Arch. Ophth. 7, 904 (1932).
[Crossref]

J. Am. Med. Assn. (1)

H. Culbertson, “Binocular astigmatism,” J. Am. Med. Assn. 11, 622 (1888); Am. J. Ophth. 5, 117 (1888).
[Crossref]

J. Opt. Soc. Am. (3)

Tr. Am. Ophth. Soc. (2)

O. F. Wadsworth, “On the effect of a cylindrical lens, with vertical axis placed before one eye,” Tr. Am. Ophth. Soc.342 (1876).

J. Green, “On certain stereoscopical illusions evoked by prismatic and cylindrical glasses,” Tr. Am. Ophth. Soc.449 (1889).

Zeits. Psychol. (1)

Franz Hillebrand, “Die Stabilität der Raumwerte auf der Netzhaut,” Zeits. Psychol. 5, 1 (1893).

Other (2)

M. von Rohr, “Die Brille als optische Instrument,” (Verlag Julius Springer, Berlin, 1921), third edition, pp. 195.

H. von Helmholtz, Physiological Optics (English translation) (The Optical Society of America, 1925), Vol. 3, p. 349.

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Figures (13)

Fig. 1
Fig. 1

The geometric disparities between images of two objects in space, as the basis for stereoscopic spatial localization.

Fig. 2
Fig. 2

The effect of a meridional magnifying lens on the apparent positions of objects in the binocular visual field.

Fig. 3
Fig. 3

Illustrating the definition of angular magnification.

Fig. 4
Fig. 4

Relationship between objective localization of objects in space and their stereoscopic localization when the image of the right eye is magnified in the horizontal meridian. Objective planes parallel to median plane and planes parallel to the frontal plane.

Fig. 5
Fig. 5

Relationship between objective localization of spatial objects and their stereoscopic localization when the image of the right eye is magnified in the horizontal meridian. Objective planes parallel to visual plane and those parallel to median plane.

Fig. 6
Fig. 6

The increase (or decrease) in the apparent size of patterns observed with stereoscopic vision when the image of the right eye is magnified in horizontal meridian (visual plane).

Fig. 7
Fig. 7

The distortion of details on a frontal wall, when the image of the right eye is magnified in the horizontal meridian.

Fig. 8
Fig. 8

The case of a surface inclined to the frontal (XZ) plane.

Fig. 9
Fig. 9

The positions of the meridional magnifications at oblique axes for maximal effect (case of axes of magnifications converging up), showing the “scissors” or torsional deviations of the images of vertical and horizontal lines.

Fig. 10
Fig. 10

Diagram for finding the displacement of the image of a point, due to a meridional magnification M at meridian θ.

Fig. 11
Fig. 11

The transformation of planes parallel to the median and to the visual planes caused by meridional magnification at oblique meridians.

Fig. 12
Fig. 12

The transformation of planes parallel to the frontal and to the visual planes caused by meridional magnifications at oblique meridians.

Fig. 13
Fig. 13

Relationship between the unique point z=−aC/B, and the vertical declination angles of the two eyes.

Tables (1)

Tables Icon

Table I Calculated values for the unique point on the X axis where the apparent planes of all frontoparallel planes converge for an interpupillary distance of 65 mm.

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

tan α 1 = ( x + a ) / y , tan α 2 = ( x - a ) / y , tan α 1 = ( x + a ) / y tan α 2 = ( x - a ) / y , tan α 1 = tan α 1 , and tan α 2 = M tan α 2 .
x = - a [ ( M + 1 ) x - ( M - 1 ) a ] / [ ( M - 1 ) x - ( M + 1 ) a ] ,
y = - 2 a y / [ ( M - 1 ) x - ( M + 1 ) a ] ,
z = - 2 a z / [ ( M - 1 ) x - ( M + 1 ) a ] .
x = a [ ( M + 1 ) x + ( M - 1 ) a ] / [ ( M - 1 ) x + ( M + 1 ) a ] ,
y = 2 a M y / [ ( M - 1 ) x + ( M + 1 ) a ] ,
z = 2 a M z / [ ( M - 1 ) x + ( M + 1 ) a ] ,
y = [ ( M - 1 ) y 0 / 2 a M ] x + ( M + 1 ) y 0 / 2 M .
tan ψ = [ ( M - 1 ) / 2 M ] [ y 0 / a ] .
z = [ ( M - 1 ) z 0 / 2 a M ] x + [ ( M + 1 ) z 0 / 2 M ] .
y 2 - y 1 = k ( y 2 - y 1 )             and             z 2 - z 1 = k ( z 2 - z 1 ) ,
k = - 2 a / [ ( M - 1 ) x - ( M + 1 ) a ] ;
x [ ( M - 1 ) b ] - y [ 2 M a ] + z [ 2 M a tan i ] + [ ( M + 1 ) a b ] = 0.
tan ψ i = [ ( M - 1 ) / 2 M ] [ b / a ] cos i .
w θ = w cos θ + v sin θ , w = w θ cos θ - v θ sin θ , v θ = v cos θ - w sin θ , v = w θ sin θ + v θ cos θ .
w = A w + B v ,
v = B w + C v ,
A = 1 2 [ ( M + 1 ) + ( M - 1 ) cos 2 θ ] , B = 1 2 [ ( M - 1 ) sin 2 θ ] , C = 1 2 [ ( M + 1 ) - ( M - 1 ) cos 2 θ ] .
( x + a ) / y = A ( x + a ) / y - B z / y , ( x - a ) / y = A ( x - a ) / y + B z / y ,
z 1 / y = - B ( x + a ) / y + C z / y , z 2 / y = B ( x - a ) / y + C z / y .
x = A a x / ( A a - B z ) ,
y = a y / ( A a - B z ) ,
z = a ( C z - B a ) / ( A a - B z ) .
z = [ a ( A C - B 2 ) / A B x ] x - [ C a / B ] ,
z = [ a ( A C - B 2 ) / B y ] y - [ C a / B ] .
M = M 0 + h tan α ,
tan α 1 = tan α 1 ( M 0 - h tan α 1 ) ,
tan α 2 = tan α 2 ( M 0 + h tan α 2 ) .
( x + a ) / y = ( x + a ) / y - h ( x + a ) 2 / y 2 , ( x - a ) / y = ( x - a ) / y + h ( x - a ) 2 / y 2 ,
x = a x [ y - 2 a h ] / [ a y - h ( x 2 + a 2 ) ] , y = a y 2 / [ a y - h ( x 2 + a 2 ) ] , z = a y z / [ a y - h ( x 2 + a 2 ) ] .