Abstract

A new visual illusion is predicted from the assumption that the perceived distance along any path depends on the discriminability for position along the path. A disk is placed between two dots, so that the straight path between the dots is nearly tangent to the disk. It is predicted that, for the perceived straight path between the dots to be tangent to the disk, the disk must overlap the physically straight path between the dots by an amount proportional to its radius. Furthermore, certain patternings of the disk are predicted to reduce the amount of illusion. All predictions are confirmed in detail. The results are compared with those obtained in an experiment on the filled-space illusion.

© 1962 Optical Society of America

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References

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  1. , “Popular Scientific Recreations, a storehouse of instruction and amusement: in which the Marvels of Natural Philosphy, Chemistry, Geology, Astronomy, etc., are explained and illustrated, mainly by means of pleasing experiments and attractive pastimes.” Translated and enlarged from Gaston Tissandier, Les Recreations Scientifiques, New and Enlarged Edition [Ward, Lock, Bowden and co., London, 1882 (?)], p. 113. The statement does not appear in the original.
  2. W. R. Garner, J. Exptl. Psychol. 43, 232–238 (1952); J. Acoust. Soc. Am. 30, 1005–1012 (1958); M. M. Taylor, Can. J. Psychol. (to be published).
    [Crossref]
  3. S. S. Stevens and E. H. Galanter, J. Exptl. Psychol. 54, 377–411 (1957); S. S. Stevens, Am. J. Psychol. 71, 633–646 (1958).
    [Crossref] [PubMed]
  4. W. P. Tanner, J. Acoust. Soc. Am. 30, 922–928 (1958); see also D. M. Green, J. Acoust. Soc. Am. 32, 1189–1202 (1960).
    [Crossref]
  5. W. R. Garner, reference 2 (1952).
  6. M. M. Taylor, Perceptual and Motor Skills 12, 203–30 (1961).
    [Crossref]
  7. This expansion may be obtained from the basic equation by using M instead of M2, so that the term (∂D/∂M2)(dM2/dA) is replaced by(∂D/∂M)(dM/dA)=kA/L(∂D/∂M). ∂D/∂M is always evaluated for M≐ L, so that it is a constant, and this constant will probably not be far from unity, compared with the range of uncertainty in the value of k. For purposes of calculation in this section, it is assumed that ∂D/∂M= 1.
  8. H. G. Spiegel, Psychol. Forsch. 21, 327–83 (1937). A summary in English of his main result is given by Taylor, reference 6.
    [Crossref]

1961 (1)

M. M. Taylor, Perceptual and Motor Skills 12, 203–30 (1961).
[Crossref]

1958 (1)

W. P. Tanner, J. Acoust. Soc. Am. 30, 922–928 (1958); see also D. M. Green, J. Acoust. Soc. Am. 32, 1189–1202 (1960).
[Crossref]

1957 (1)

S. S. Stevens and E. H. Galanter, J. Exptl. Psychol. 54, 377–411 (1957); S. S. Stevens, Am. J. Psychol. 71, 633–646 (1958).
[Crossref] [PubMed]

1952 (1)

W. R. Garner, J. Exptl. Psychol. 43, 232–238 (1952); J. Acoust. Soc. Am. 30, 1005–1012 (1958); M. M. Taylor, Can. J. Psychol. (to be published).
[Crossref]

1937 (1)

H. G. Spiegel, Psychol. Forsch. 21, 327–83 (1937). A summary in English of his main result is given by Taylor, reference 6.
[Crossref]

Galanter, E. H.

S. S. Stevens and E. H. Galanter, J. Exptl. Psychol. 54, 377–411 (1957); S. S. Stevens, Am. J. Psychol. 71, 633–646 (1958).
[Crossref] [PubMed]

Garner, W. R.

W. R. Garner, J. Exptl. Psychol. 43, 232–238 (1952); J. Acoust. Soc. Am. 30, 1005–1012 (1958); M. M. Taylor, Can. J. Psychol. (to be published).
[Crossref]

W. R. Garner, reference 2 (1952).

Spiegel, H. G.

H. G. Spiegel, Psychol. Forsch. 21, 327–83 (1937). A summary in English of his main result is given by Taylor, reference 6.
[Crossref]

Stevens, S. S.

S. S. Stevens and E. H. Galanter, J. Exptl. Psychol. 54, 377–411 (1957); S. S. Stevens, Am. J. Psychol. 71, 633–646 (1958).
[Crossref] [PubMed]

Tanner, W. P.

W. P. Tanner, J. Acoust. Soc. Am. 30, 922–928 (1958); see also D. M. Green, J. Acoust. Soc. Am. 32, 1189–1202 (1960).
[Crossref]

Taylor, M. M.

M. M. Taylor, Perceptual and Motor Skills 12, 203–30 (1961).
[Crossref]

J. Acoust. Soc. Am. (1)

W. P. Tanner, J. Acoust. Soc. Am. 30, 922–928 (1958); see also D. M. Green, J. Acoust. Soc. Am. 32, 1189–1202 (1960).
[Crossref]

J. Exptl. Psychol. (2)

W. R. Garner, J. Exptl. Psychol. 43, 232–238 (1952); J. Acoust. Soc. Am. 30, 1005–1012 (1958); M. M. Taylor, Can. J. Psychol. (to be published).
[Crossref]

S. S. Stevens and E. H. Galanter, J. Exptl. Psychol. 54, 377–411 (1957); S. S. Stevens, Am. J. Psychol. 71, 633–646 (1958).
[Crossref] [PubMed]

Perceptual and Motor Skills (1)

M. M. Taylor, Perceptual and Motor Skills 12, 203–30 (1961).
[Crossref]

Psychol. Forsch. (1)

H. G. Spiegel, Psychol. Forsch. 21, 327–83 (1937). A summary in English of his main result is given by Taylor, reference 6.
[Crossref]

Other (3)

, “Popular Scientific Recreations, a storehouse of instruction and amusement: in which the Marvels of Natural Philosphy, Chemistry, Geology, Astronomy, etc., are explained and illustrated, mainly by means of pleasing experiments and attractive pastimes.” Translated and enlarged from Gaston Tissandier, Les Recreations Scientifiques, New and Enlarged Edition [Ward, Lock, Bowden and co., London, 1882 (?)], p. 113. The statement does not appear in the original.

This expansion may be obtained from the basic equation by using M instead of M2, so that the term (∂D/∂M2)(dM2/dA) is replaced by(∂D/∂M)(dM/dA)=kA/L(∂D/∂M). ∂D/∂M is always evaluated for M≐ L, so that it is a constant, and this constant will probably not be far from unity, compared with the range of uncertainty in the value of k. For purposes of calculation in this section, it is assumed that ∂D/∂M= 1.

W. R. Garner, reference 2 (1952).

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Figures (4)

Fig. 1
Fig. 1

Constructions for determining the path of the tangent geodesic.

Fig. 2
Fig. 2

Typical test pattern used in the study. The line was 0.015 in. thick, and all dots were about 0.01 in. diam. Test patterns. were on 8 1 2 by 11-in. “Albanene” tracing paper.

Fig. 3
Fig. 3

Relation for solid disks between amount of illusion and disk diameter. Results are also shown for striped and speckled disks.

Fig. 4
Fig. 4

Relation between consistency and amount of illusion for individual Ss. More reversals indicate less consistency.

Equations (23)

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D = D ( A , B , Y 2 , M 2 ) ,
d D / d A = 0
d F / d x = F / x + ( F / y 1 ) ( d y 1 / d x ) + ( F / y 2 ) ( d y 2 / d x ) + ,
d D / d A = D / A + ( D / B ) ( d B / d A ) + ( D / Y 2 ) ( d Y 2 / d A ) + ( D / M 2 ) ( d M 2 / d A ) ,
d B / d A = 1
d Y 2 / d A = ( d Y 2 / d B ) ( d B / d A ) = ( d Y 2 / d B ) .
( R B ) 2 + ( Y / 2 ) 2 = R 2 ,
Y 2 = 8 R B 4 B 2 ,
d Y 2 / d B = 8 ( R B )
d Y 2 / d A = 8 ( R B ) .
Δ L 2 A 2 / L
Δ L 8 A 2 / 3 L .
d Δ L / d A k A / L .
d Δ L / d A = d ( L + Δ L ) / d A = d M / d A
d M 2 / d A = 2 M ( d M / d A ) 2 L ( d M / d A )
= 2 k A .
d D / d A = D / A D / B 8 ( R B ) D / Y 2 + k A ( D / M 2 )
c A = R ,
c = 2 k ( D / M 2 ) M = L / 8 ( D / Y 2 ) Y 2 = 0 ,
A / R = ( 8 / k ) ( L D / Y 2 )
D / Y 2 = k A / 8 R L .
D / Y 2 = 0.7 rad 1 at Y = 0 .
(D/M)(dM/dA)=kA/L(D/M).