Abstract

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.

© 1947 Optical Society of America

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References

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  1. A. Ames, “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.

1946 (1)

A. Ames, “Binocular vision as affected by relations between uniocular stimulus patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).

Ames, A.

A. Ames, “Binocular vision as affected by relations between uniocular stimulus patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).

Courant,

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.

Luneburg, R.

For further information refer to R. Luneburg, “Mathematical analysis of binocular vision,” published by the Dartmouth Eye Institute, 1947.

Robbins,

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.

Am. J. Psychol. (1)

A. Ames, “Binocular vision as affected by relations between uniocular stimulus patterns in commonplace environments,” Am. J. Psychol. 59, 333 (1946).

Other (3)

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.

For further information refer to R. Luneburg, “Mathematical analysis of binocular vision,” published by the Dartmouth Eye Institute, 1947.

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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Figures (22)

F. 1
F. 1

Binocular displacement of vertical targets from actual asymmetrical to apparent symmetrical position to median plane.

F. 2
F. 2

Binocular displacement of vertical targets from actual symmetrical to apparent asymmetrical position to median plane.

F. 3
F. 3

Intersection of a vertical target with the horizontal plane at curve C which is subdivided into line elements ds.

F. 4
F. 4

L and R represent the left and right eyes, P(x, y) an arbitrary point of the horizontal plane. The bipolar coordinates belonging to P(x, y) are ∢LRP=α and ∢RLP = β.

F. 5
F. 5

Illustrating the condition for equivalence of two lines, l0 and l′, and two coordinated points, P0 and P′.

F. 6
F. 6

The straight lines l1 and l2 equivalent to the parallel y′ = D, are tangents to the ellipse with the eyes as focal points. A special relation holds for the angles of l1 and l2 with the x axis, ω and Ω, respectively.

F. 7
F. 7

All tangents to the ellipse with the eyes L and R as focal points represent equivalent lines.

F. 8
F. 8

Two arbitrary equivalent straight lines with the equations y0 = cx0+d and y′ = c′x′+d′ are given. On these lines P0(x0, y0) and P′(x′, y′) are two coordinated points, as indicated in Fig. 5.

F. 9
F. 9

The points P′, P0, P1, and P2 indicate coordinated points on equivalent straight lines. The problem is to find the curve on which these coordinated points lie.

F. 10
F. 10

l2, l3, and l4 represent three arbitrarily chosen equivalent straight lines. P2, P3, P4 are the points of intersection of l2 with y′ = D, l3, l4, respectively. All points of intersection of the tangents to the ellipse are shown to be coordinated points.

F. 11
F. 11

A figure used to illustrate the cross-ratio theorem of projective geometry concerning tangents to conic sections. Two given tangents to an ellipse, a and a′, are intersected by a third tangent, t, which changes its position around the ellipse from AA′, BB′, CC′, to DD′.

F. 12
F. 12

Illustrating that the cross-ratios of two sets of four coordinated points are equal: ( P 1 P 2 P 3 P 4 ) = ( P 1 P 2 P 3 P 4 ) .

F. 13
F. 13

From the projection centers L and R, the left and right eyes, two pencils of lines, p1 and p2, intersect in the points A, B, C, D on the straight line l. Rotation of the pencils of lines, p1 and p2, around R and L with the constant angles δ and , respectively, results in the projectively corresponding intersection points A′, B′, C′, D′ on a conic section.

F. 14
F. 14

Illustrating a theorem from projective geometry concerning the cross-ratios of line segments between points on a conic.

F. 15
F. 15

Illustrating a certain group of equivalent conic sections. Note that all non-degenerated conic sections go through the eyes L and R. The ellipse E with L and R as focal points does not belong to the group of equivalent conic sections.

F. 16
F. 16

The construction for a suggested experiment. The points P1, P2Pn on the straight line l0 in front and to the right of the observer represent the intersections of the illuminated vertical lines of the actual target with the horizontal plane. The equally distanced points P1′, P2′, ⋯Pn on the parallel l′ to the left of the observer, represent the intersections of the vertical lines of the apparent target with the horizontal plane.

F. 17
F. 17

Illustrating the method for finding the lengths of the vertical lines in the subdivision points P1, P2, ⋯Pn of the actual target, if the length h of the vertical lines of the apparent target is given.

F. 18
F. 18

Tables and construction of equivalent line segments for experiment shown in Fig. 1.

F. 19
F. 19

Illustrating the positions of the actual vertical targets, bl and br, and of the apparent vertical targets, bl and br, to the observer. Compare Fig. 2B.

F. 20
F. 20

Tables and construction of equivalent line segments for experiment shown in Fig. 2.

F. 21
F. 21

Illustrating a suggested extreme experiment. The relatively short actual target on the right side of the observer should appear as a long vertical target on the left side of the observer.

F. 22
F. 22

Showing the construction of the subdivision points of the actual and of the apparent targets illustrated in Fig. 21.

Tables (1)

Tables Icon

Table I Conic sections equivalent to a given straight line I with the equation y0 = cx0+d.

Equations (70)

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cot α = ( 1 + y ) / x , cot β = ( 1 y ) / x ,
x = 2 / ( cot α + cot β ) , y = ( cot α cot β ) / ( cot α + cot β ) .
α 0 = α 0 ( t ) ; β 0 = β 0 ( t ) .
d α = d α 0 , d β = d β 0 ,
α = α 0 + δ , β = β 0 + ,
α 0 = O R P 0 , β 0 = O L P 0 , α = O R P , β = O L P .
α = α 0 + δ , β = β 0 + ,
cot α 0 = ( 1 + y 0 ) / x 0 , cot β 0 = ( 1 y 0 ) / x 0 ,
x 0 = 2 / ( cot α 0 + cot β 0 ) , y 0 = ( cot α 0 cot β 0 ) / ( cot α 0 + cot β 0 ) .
cot α = ( 1 + y ) / x , cot β = ( 1 y ) / x
x = 2 / ( cot α + cot β ) , y = ( cot α cot β ) / ( cot α + cot β ) .
y 0 = c x 0 + d
y = D
( cot α cot β ) / ( cot α + cot β ) = D ,
( 1 D ) cot α ( 1 + D ) cot β = 0 .
α = α 0 + δ , β = β 0 + ,
( 1 D ) cot ( α 0 + δ ) ( 1 + D ) cot ( β 0 + ) = 0 ,
( 1 D ) ( cot δ cot α 0 1 ) / ( cot δ + cot α 0 ) ( 1 + D ) ( cot cot β 0 1 ) / ( cot + cot β 0 ) = 0 .
cot α 0 cot β 0 [ cot δ ( 1 D ) cot ( 1 + D ) ] + [ cot ( 1 D ) + cot δ ( 1 + D ) ] + cot α 0 [ cot δ cot ( 1 D ) + ( 1 + D ) ] + cot β 0 [ ( 1 D ) cot δ cot ( 1 + D ) ] } = 0 .
cot α 0 = ( 1 + y 0 ) / x 0 , cot β 0 = ( 1 y 0 ) / x 0 ,
cot δ ( 1 D ) cot ( 1 + D ) = 0 .
cot δ = λ ( 1 + D ) , cot = λ ( 1 D ) .
y 0 = 2 λ D λ 2 ( D 2 1 ) 1 · x 0 + D · λ 2 ( D 2 1 ) + 1 λ 2 ( D 2 1 ) 1 ,
c ( λ ) = 2 λ D λ 2 ( D 2 1 ) 1 , d ( λ ) = D λ 2 ( D 2 1 ) + 1 λ 2 ( D 2 1 ) 1 .
y 0 = c ( λ ) x 0 + d ( λ ) , y 0 = [ c ( λ ) + c ( λ ) d λ ] x 0 + [ d ( λ ) + d ( λ ) d λ ] .
x 0 = d ( λ ) c ( λ ) .
c ( λ ) = 2 D λ 2 ( D 2 1 ) + 1 [ λ 2 ( D 2 1 ) 1 ] 2 , d ( λ ) = 4 D λ ( D 2 1 ) [ λ ( D 2 1 ) 1 ] 2 ,
x 0 = 2 λ ( D 2 1 ) λ 2 ( D 2 1 ) + 1 .
x 0 = ( D 2 1 ) 2 λ λ 2 ( D 2 1 ) + 1 , y 0 = D · λ 2 ( D 2 1 ) 1 λ 2 ( D 2 1 ) + 1 .
x 2 / ( D 2 1 ) + y 2 / D 2 = 1 ,
( cot α cot β ) / ( cot α + cot β ) = 2 c / ( cot α + cot β ) + d ,
( 1 d ) cot α ( 1 + d ) cot β 2 c = 0 .
α = α 0 + δ , β = β 0 + ,
( 1 d ) cot ( α 0 + δ ) ( 1 + d ) cot ( β 0 + ) 2 c = 0
cot α 0 cot β 0 [ cot δ ( 1 d ) cot ( 1 + d ) 2 c ] + [ cot ( 1 d ) + cot δ ( 1 + d ) 2 c cot δ cot ] + cot α 0 [ cot δ × cot ( 1 d ) + ( 1 + d ) 2 c cot ] + cot β 0 [ ( 1 d ) cot δ × cot ( 1 + d ) 2 c cot δ ] } = 0 .
( cot δ cot ) ( cot δ + cot ) d = 2 c ,
y 0 = c · cot ( δ ) 1 cot ( δ ) + c · x 0 + ( cot δ + cot ) [ d cot ( + δ ) + c ] ( cot δ cot ) [ cot ( δ ) + c ] .
Ω ω = δ ,
[ x 2 / ( D 2 1 ) ] + [ y 2 / D 2 ] = 1 .
x 2 / ( d 2 1 ) + y 2 / ( d 2 + c 2 ) = 1 / ( 1 + c 2 ) .
x 2 + y 2 = d 2 / ( 1 + c 2 ) ,
x 0 = 2 / ( cot α 0 + cot β 0 ) = 2 / [ cot ( α δ ) + cot ( β ) ] , y 0 = ( cot α 0 cot β 0 ) / ( cot α 0 + cot β 0 ) = [ cot ( α δ ) cot ( β ) ] / [ cot ( α δ ) + cot ( β ) ] .
cot α = ( 1 + y ) / x and cot β = ( 1 y ) / x ,
x 0 = 2 x [ y ( cot δ cot ) ( cot δ + cot ) ] + [ ( 1 y 2 ) + x 2 ] cot δ cot 2 x ( cot δ cot 1 ) [ ( 1 y 2 ) x 2 ] ( cot δ + cot ) , y 0 = 2 x y ( cot δ cot + 1 ) [ ( 1 y 2 ) + x 2 ] ( cot δ cot ) 2 x ( cot δ cot 1 ) [ ( 1 y 2 ) x 2 ] ( cot δ + cot ) .
2 c = ( cot δ cot ) ( cot δ + cot ) d ,
x 0 = 2 ( cot β cot ) · ( cot α cot δ ) [ cot α cot δ + 1 ] ( cot β cot ) + [ cot β cot + 1 ] ( cot α cot δ ) , y 0 = [ cot α cot δ + 1 ) ( cot β cot ) [ cot β cot + 1 ] ( cot α cot δ ) [ cot α cot δ + 1 ) ( cot β cot ) + [ cot β cot + 1 ] ( cot α cot δ ) .
cot α cot δ = 1 / x · [ ( 1 + d ) + ( c cot δ ) x ] , cot β cot = 1 / x · [ ( 1 d ) ( c + cot ) x ] ,
cot α cot δ = ( 1 + d ) 2 ( cot δ + cot ) x 2 x , cot β cot = ( 1 d ) 2 ( cot δ + cot ) x 2 x .
x 0 = 1 2 ( cot δ + cot ) ( d 2 1 ) x ( d 2 1 ) c y x + [ 1 2 ( cot δ + cot ) ( d 2 1 ) + c d ] , y 0 = [ 1 2 ( cot δ + cot ) ( d 2 1 ) + c d ] y + d x c c y x + [ 1 2 ( cot δ + cot ) ( d 2 1 ) + c d ] ,
x 0 = 1 2 ( cot δ + cot ) ( d 2 1 ) x ( d 2 1 ) 1 2 ( cot δ + cot ) ( d 2 1 ) ( 1 + c 2 ) x , y 0 = [ 1 2 ( cot δ + cot ) ( d 2 1 ) c + ( 1 + c 2 ) d ] x + 1 2 ( cot δ cot ) ( d 2 1 ) 1 2 ( cot δ + cot ) ( d 2 1 ) ( 1 + c 2 ) x .
1 2 ( cot δ + cot ) = λ , 1 2 ( cot δ cot ) = λ D , x 0 = ( D 2 1 ) λ x 1 λ ( D 2 1 ) x , y 0 = D λ ( D 2 1 ) + x λ ( D 2 1 ) x ,
y 0 = 2 λ D λ 2 ( D 2 1 ) 1 · x 0 + D · λ 2 ( D 2 1 ) + 1 λ 2 ( D 2 1 ) 1 .
x = λ ( D 2 1 ) x ( D 2 1 ) λ ( D 2 1 ) x , y = D λ ( D 2 1 ) + x λ ( D 2 1 ) x .
y = 2 D x ( D 2 1 ) x 2 x + D ( D 2 1 ) + x 2 ( D 2 1 ) x 2 .
y = 2 ( 1 / x ) D ( D 2 1 ) ( 1 / x ) 2 1 x + D ( D 2 1 ) ( 1 / x ) 2 + 1 ( D 2 1 ) ( 1 / x ) 2 1 ,
y = 2 λ D λ 2 ( D 2 1 ) 1 x + D λ 2 ( D 2 1 ) + 1 λ 2 ( D 2 1 ) 1 .
( P 1 P 2 P 3 P 4 ) = x 3 x 1 x 3 x 2 ÷ x 4 x 1 x 4 x 2 .
( P 1 P 2 P 3 P 4 ) = x 3 x 1 x 3 x 2 ÷ x 4 x 1 x 4 x 2 = y 3 y 1 y 3 y 2 ÷ y 4 y 1 y 4 y 2 ,
( P 1 P 2 P 3 P 4 ) = ( P 1 P 2 P 3 P 4 )
( r 1 i r 2 i r 3 i r 4 i ) = ( r 1 r 2 r 3 r 4 ) = ( ABCD ) ; ( s 1 i s 2 i s 3 i s 4 i ) = ( s 1 s 2 s 3 s 4 ) = ( ABCD ) ,
δ 5 5 = ω = 5 ° .
tan 2 η = [ tan ( δ ) + cot ( δ ) ] = 2 sin 2 ( δ ) .
( RA RB RC RD ) = ( R A i R B i R C i R D i ) = ( LA LB LC LD ) = ( L A i L B i L C i L D i ) .
ω 1 = δ i i .
[ x 2 / ( D 2 1 ) ] + [ y 2 / D 2 ] = 1 ,
x i = ( D 2 1 ) λ 0 x i 1 λ 0 ( D 2 1 ) x i ,
y i = D λ 0 ( D 2 1 ) + x i λ 0 ( D 2 1 ) x i ,
x 3 x 1 x 3 x 2 ÷ x 4 x 1 x 4 x 2 = 2 s 1 s ÷ 3 s 2 s = 4 3 ,
x 3 x 1 x 3 x 2 ÷ x 4 x 1 x 4 x 2 = y 3 y 1 y 3 y 2 ÷ y 4 y 1 y 4 y 2 = 4 3 .
( P 1 P 2 P 3 P 4 ) = ( P 1 P 2 P 3 P 4 ) .