Abstract

The aim of this paper is to show that the so-called visual space has a uniquely determined non-Euclidean metric, or psychometric distance function, the numerical parameters of which depend on the individual observer. Certain well-known phenomena of space perception, such as the horopter, the alley experiment and size constancy, are explained on the basis of the distance function. Methods of measuring the personal parameters of the metric are developed, and applications of the theory to the field of binocular instruments and pictorial representation of space are suggested.

© 1950 Optical Society of America

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Corrections

R. K. Luneburg, "Errata:* The Metric of Binocular Visual Space.," J. Opt. Soc. Am. 41, 1071-1071 (1951)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-41-12-1071

References

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  1. H. V. Helmholtz, Treatise on Psychological Optics, J. P. C. Southall, Editor (Optical Society of America, 1925), Vol. 3, pp. 482 f.
  2. F. Hillebrand, Denkschr. Akad. Wiss. Wien, math.-nat. Kl. 72, 255 (1902).
  3. W. Blumenfeld, Zeits. f. Physiol. d. Sinnesorgane 65, 241 (1913).
  4. R. K. Luneburg, Mathematical Analysis of Binocular Vision (Princeton University Press, Princeton, 1947); Metric Methods in Binocular Visual Perception, Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).

1913 (1)

W. Blumenfeld, Zeits. f. Physiol. d. Sinnesorgane 65, 241 (1913).

1902 (1)

F. Hillebrand, Denkschr. Akad. Wiss. Wien, math.-nat. Kl. 72, 255 (1902).

Blumenfeld, W.

W. Blumenfeld, Zeits. f. Physiol. d. Sinnesorgane 65, 241 (1913).

Helmholtz, H. V.

H. V. Helmholtz, Treatise on Psychological Optics, J. P. C. Southall, Editor (Optical Society of America, 1925), Vol. 3, pp. 482 f.

Hillebrand, F.

F. Hillebrand, Denkschr. Akad. Wiss. Wien, math.-nat. Kl. 72, 255 (1902).

Luneburg, R. K.

R. K. Luneburg, Mathematical Analysis of Binocular Vision (Princeton University Press, Princeton, 1947); Metric Methods in Binocular Visual Perception, Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).

Denkschr. Akad. Wiss. Wien, math.-nat. Kl. (1)

F. Hillebrand, Denkschr. Akad. Wiss. Wien, math.-nat. Kl. 72, 255 (1902).

Zeits. f. Physiol. d. Sinnesorgane (1)

W. Blumenfeld, Zeits. f. Physiol. d. Sinnesorgane 65, 241 (1913).

Other (2)

R. K. Luneburg, Mathematical Analysis of Binocular Vision (Princeton University Press, Princeton, 1947); Metric Methods in Binocular Visual Perception, Studies and Essays, Courant Anniversary Volume (Interscience Publishers, Inc., New York, 1948).

H. V. Helmholtz, Treatise on Psychological Optics, J. P. C. Southall, Editor (Optical Society of America, 1925), Vol. 3, pp. 482 f.

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Figures (29)

Fig. 1
Fig. 1

Cartesian coordinate system for physical space; L and R represent centers of rotation of left and right eyes.

Fig. 2
Fig. 2

Subjective coordinate system for visual space.

Fig. 3
Fig. 3

Horopter curves for different fixations.

Fig. 4
Fig. 4

Distance and parallel alleys.

Fig. 5
Fig. 5

Equivalent configurations.

Fig. 6
Fig. 6

A visually straight line represented in hyperbolic space.

Fig. 7
Fig. 7

Bipolar coordinates of a physical point.

Fig. 8
Fig. 8

Bipolar parallax and bipolar latitude of a physical point.

Fig. 9
Fig. 9

Curves of constant bipolar parallax and latitude.

Fig. 10
Fig. 10

Testing correlation between physical and visual spaces on Vieth-Mueller circle by changing convergence.

Fig. 11
Fig. 11

Torus of constant binocular parallax, showing curves of constant elevation and constant binocular latitude.

Fig. 12
Fig. 12

Boundaries of the horizontal half-plane in physical space.

Fig. 13
Fig. 13

Boundaries of the horizontal half-plane of visual space and a frontal plane horopter, S, mapped on a ξ, η-diagram.

Fig. 14
Fig. 14

Frontal plane horopters.

Fig. 15
Fig. 15

Distance and parallel alleys in map of subjective horizontal plane.

Fig. 16
Fig. 16

Distance and parallel alleys in physical horizontal plane.

Fig. 17
Fig. 17

Distance and parallel alleys with a common point in a system of Vieth-Mueller circles.

Fig. 18
Fig. 18

Tangents to distance and parallel alleys at the common point.

Fig. 19
Fig. 19

Points on two Vieth-Mueller circles.

Fig. 20
Fig. 20

The subjective correlation of Fig. 19.

Fig. 21
Fig. 21

Horizontal cross-section of Ames’ experimental room.

Fig. 22
Fig. 22

Rectangle of Fig. 21 mapped in the subjective plane.

Fig. 23
Fig. 23

Horizontal cross sections of visually congruent rooms.

Fig. 24
Fig. 24

A domain in the physical plane and the correlated domain in the subjective plane.

Fig. 25
Fig. 25

The domain of Fig. 24 after a congruency-transformation.

Fig. 26
Fig. 26

The domain of Fig. 25 in the physical plane.

Fig. 27
Fig. 27

Illustrating difference between psychometric representation and central projection.

Fig. 28
Fig. 28

Showing a line segment in the median plane.

Fig. 29
Fig. 29

Showing relative apparent size of line segment of Fig. 28 as a function of its position along the x axis.

Equations (50)

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( 1 ) D ( P 1 , P 2 ) = D ( P 2 , P 1 ) > 0 , if P 1 , P 2 are sensed as different . ( 2 ) D ( P 1 , P 2 ) = 0 if P 1 = P 2 . ( 3 ) D ( P 1 , P 2 ) + D ( P 2 , P 3 ) D ( P 1 , P 3 ) for any 3 points P 1 , P 2 , P 3 .
( 1 ) If ( P 1 , P 2 ) and ( P 3 , P 4 ) are any two pairs of sensed points and ( P 1 , P 2 ) > ( P 3 , P 4 ) then D ( P 1 , P 2 ) > D ( P 3 , P 4 ) . ( 2 ) If P 1 , P 2 , P 3 are sensed on an apparently straight line , then D ( P 1 , P 2 ) + D ( P 2 , P 3 ) = D ( P 1 , P 3 ) and vice versa .
2 ( - K ) 1 2 sin [ 1 2 ( - K ) 1 2 · D C ] = [ ( ξ 1 - ξ 2 ) 2 + ( η 1 - η 2 ) 2 + ( ζ 1 - ζ 2 ) 2 ] 1 2 [ ( 1 + K 4 ρ 1 2 ) ( 1 + K 4 ρ 2 2 ) ] 1 2
γ = π - α - β ϕ = 1 2 ( β - α )
ξ = f ( γ ) cos ϕ cos θ η = f ( γ ) sin ϕ ζ = f ( γ ) cos ϕ sin θ
ξ = f ( γ ) cos ϕ η = f ( γ ) sin ϕ .
A : γ = 0 , ϕ = π / 2 , or x = 0 , y > 1 L : γ = π - 2 ϕ , or x = 0 , y = + 1 ( left eye ) B : γ = π , ϕ = 0 , or x = 0 , - 1 < y < 1 R : γ = π + 2 ϕ , or x = 0 , y = - 1 ( right eye ) C : γ = 0 , ϕ = - π / 2 , or x = 0 , y < - 1.
K 4 ( ξ 2 + η 2 ) - 1 = C ξ .
K 4 ( ξ 2 + η 2 ) + 1 = C η .
K 4 ( ξ 2 + η 2 ) - 1 = - C η
ξ 0 = f ( γ 0 ) cos ϕ 0 η 0 = f ( γ 0 ) sin ϕ 0 .
K 4 f 2 ( γ ) + 1 f ( γ ) sin ϕ 1 = K 4 f 2 ( γ 0 ) + 1 f ( γ 0 ) sin ϕ 0             ( distance curve )
K 4 f 2 ( γ ) - 1 f ( γ ) sin ϕ 2 = K 4 f 2 ( γ 0 ) - 1 f ( γ 0 ) sin ϕ 0             ( parallel curve ) ,
K 4 f 2 ( γ 0 ) ( f ( γ ) f ( γ 0 ) - sin ϕ 1 sin ϕ 0 ) = - f ( γ 0 ) f ( γ ) + sin ϕ 1 sin ϕ 0 K 4 f 2 ( γ 0 ) ( f ( γ ) f ( γ 0 ) - sin ϕ 2 sin ϕ 0 ) = - sin ϕ 2 sin ϕ 0 + f ( γ 0 ) f ( γ ) .
S 1 = sin ϕ 1 sin ϕ 0             and             S 2 = sin ϕ 2 sin ϕ 0
f ( γ ) f ( γ 0 ) - S 1 f ( γ ) f ( γ 0 ) - S 2 = - f ( γ 0 ) f ( γ ) - S 1 f ( γ 0 ) f ( γ ) - S 2 ,
1 2 ( f ( γ ) f ( γ 0 ) + f ( γ 0 ) f ( γ ) ) = 1 + S 1 S 2 S 1 + S 2 ,
f ( γ 0 ) f ( γ ) = 1 + S 1 S 2 + [ ( S 1 2 - 1 ) ( S 2 2 - 1 ) ] 1 2 S 1 + S 2 .
f ( γ 0 ) f ( γ ) S = sin ϕ sin ϕ 0 ,
- K 4 f 2 ( γ 0 ) + 1 K 4 f 2 ( γ 0 ) + 1 = ( S 1 2 - 1 ) 1 2 ( S 2 2 - 1 ) 1 2 = ( sin 2 ϕ 1 - sin 2 ϕ 0 ) 1 2 ( sin 2 ϕ 2 - sin 2 ϕ 0 ) 1 2 .
( 1 - K 4 f 2 ( γ 0 ) 1 + K 4 f 2 ( γ 0 ) ) 2 = d ϕ 1 d γ d ϕ 2 d γ = b 1 b 2 .
( 1 - K 1 + K ) 2 = b 1 b 2 .
{ 2 ρ 0 2 [ 1 - cos ( ϕ 1 - ϕ 0 ) ] } 1 2 1 + K 4 ρ 0 2 = [ ρ 0 2 + ρ 1 2 - 2 ρ 0 ρ 1 cos ( ϕ 2 - ϕ 1 ) ] 1 2 ( 1 + K 4 ρ 0 2 ) 1 2 ( 1 + K 4 ρ 1 2 ) 1 2 ,
4 sin 2 1 2 ( ϕ 2 - ϕ 0 ) = ρ 0 ρ 1 1 + K 4 ρ 1 2 1 + K 4 ρ 0 2 × 4 sin 2 1 2 ( ϕ 1 - ϕ 0 ) - [ ( ρ 0 ρ 1 ) 1 2 - ( ρ 1 ρ 0 ) 1 2 ] 2 .
A = ρ 0 ρ 1 1 + K 4 ρ 1 2 1 + K 4 ρ 0 2
B = [ ( ρ 0 ρ 1 ) 1 2 - ( ρ 1 ρ 0 ) 1 2 ] 2 = ρ 0 ρ 1 + ρ 1 ρ 0 - 2
X = 4 sin 2 1 2 ( ϕ 1 - ϕ 0 ) ;             Y = 4 sin 2 1 2 ( ϕ 2 - ϕ 0 )
Y = A X - B ,
ρ 1 ρ 0 = f ( γ 1 ) f ( γ 0 ) ,
f ( γ ) = 2 e - σ γ ,
σ = ( B ) 1 2 γ 1 - γ 0 .
D ( P 0 , P 1 ) = D ( P 2 , P 3 ) .
2 sin 1 2 ( ϕ 3 - ϕ 2 ) = ρ 0 ρ 1 1 + K 4 ρ 1 2 1 + K 4 ρ 0 2 2 sin 1 2 ( ϕ 1 - ϕ 0 )
Y = A X
X = 2 sin 1 2 ( ϕ 1 - ϕ 0 )             and             Y = 2 sin 1 2 ( ϕ 3 - ϕ 2 ) .
A = e σ ( γ 1 - γ 0 ) · 1 + K e - 2 σ γ 1 1 + K e - 2 σ γ 0 .
cos Ω 12 = cos ϕ 1 cos ϕ 2 cos ( θ 2 - θ 1 ) + sin ϕ 1 sin ϕ 2 ,
2 ( - K ) 1 2 sin [ 1 2 ( - K ) 1 2 · D C ] = ( ρ 0 2 + ρ 1 2 - 2 ρ 0 ρ 1 cos Ω 01 ) 1 2 ( 1 + K 4 ρ 0 2 ) 1 2 ( 1 + K 4 ρ 1 2 ) 1 2 .
D ( P 0 , P 1 ) = F [ M ( ρ 0 , ρ 1 ) + N ( ρ 0 , ρ 1 ) cos Ω 01 ]
d s 2 = n 2 ( ρ ) ( d ξ 2 + d η 2 + d ζ 2 )
D ξ 2 + D η 2 + D ζ 2 = n 2 ( ρ ) D ξ 0 2 + D η 0 2 + D ζ 0 2 = n 2 ( ρ 0 ) .
n ( ρ ) = C 1 + K 4 ρ 2
ξ = - a - ( a 2 + 4 K ) ξ - a ( ξ - a ) 2 + η 2 + ζ 2 η = ( a 2 + 4 K ) η ( ξ - a ) 2 + η 2 + ζ 2 ζ = ( a 2 + 4 K ) ζ ( ξ - a ) 2 + η 2 + ζ 2 ,
2 ( - K ) 1 2 sin [ 1 2 ( - K ) 1 2 · D C ] = [ 2 ρ 2 ( 1 - cos 2 θ ) ] 1 2 1 + K 4 ρ 2 = 2 ρ sin θ 1 + K 4 ρ 2 .
sin θ = z ( x 2 + z 2 ) 1 2 = z tan γ 2 ;
2 ( - K ) 1 2 sin [ 1 2 ( - K ) 1 2 · D C ] = z 4 tan γ 2 e σ γ + K e - σ γ .
D C = z 4 tan γ 2 e σ γ + K e - σ γ .
w ( v ) = C log c + v c - v ,
A = f ( γ 0 ) f ( γ 1 ) 1 + K 4 f 2 ( γ 1 ) 1 + K 4 f 2 ( γ 0 ) ,
D C = z 2 f ( γ ) 1 + K 4 f 2 ( γ ) .