AbstractAngles on picture surfaces may be perceived incorrectly. This illusion may be caused by projective information. Alternatively, the angles may only be misperceived if they are depicting very standard angles such as 90°. Two experiments used parallel perspective to test these hypotheses. In Experiment 1, subjects judged angles presented alone (i.e., as V-shapes) or as parts of a right-angled, scalene triangle depicted as the top surface of a double-cube (depicting angles of 26.6°, 63.4°, and 90°). The largest errors were in the double-cube condition. They were biased towards each of the depicted angles of the double-cube. In Experiment 2 the V-shapes were shown as parts of triangles, to test whether triangles alone could create the illusion, perhaps by providing projective 3-D information for a single triangle being depicted at different tilts. The errors were indeed biased towards the depicted angles, but were less than in the double-cube condition. The results support a projective theory of the illusion: that the foreshortening depicting a 3-D object such as a double-cube affects perception of the depicting features on the picture surface.

Introduction

In Figure 1, using parallel projection, the corners of a cube

are depicted by lines that meet at obtuse or acute angles on

the page. What happens when an observer is asked to

judge these “angles-on-the-page”? Errors are made in the

direction of the depicted angle of 90° (Hammad et al., in

press). They may be perceived incorrectly due to the

picture’s perspective (Arnheim, 1977; Gibson, 1979) and

misapplied shape constancy (Gregory, 1972).

What could cause this perceptual illusion?

Projective Theory: the illusion may be caused by projective

information for the angles of the cube.

Good Form Theory: the angles may be misperceived

because they are depicting very standard angles (good

forms), such as 90°.

These two hypotheses can be tested with pictures that

depict acute angles, for example, pictures of a double-cube

presented at various tilts from 5° to 85 ° (see Figure 2).

Experiment 1

Subjects judged angles presented in two conditions:

1) Double-Cube condition: angles presented as parts of a

right-angled, scalene triangle depicted as the top surface of

a double-cube (with depicted angle X=90° and depicted

angles of 26.6° and 63.4°; see Figures 1 and 2).

2) V-Shape condition: angles presented alone (i.e., as V-

shapes)

Projection Biases Perceived Angles on a Picture SurfaceIgor Juricevic, Shazma Rajani, Sherief Hammad, & John M. Kennedy University of Toronto, Scarborough

Figure 1. A double-cube. For angle X, angle-on-the-page = 159° and depicted angle = 90°

Figure 2. Double-cubes at various tilts.

X

5° 15° 25°

35° 45° 55°

65° 75° 85°

Results

Subjects’ responses were converted into Bias scores. If the

response equals the angle-on-the-page, Bias = 0. But if the

response equals the depicted angle, Bias = 1. Intermediate

Bias scores reflect the distance of responses from angle-on-

the-page.

Errors were biased towards each of the depicted angles of

the double-cube, not just 90°. The largest biases were in the

double-cube condition (see Figure 3). Bias in the V-shape

condition can be explained as due to restriction of Bias

scores to values between 0 and 1.

Experiment 2

The V-shapes were shown as parts of triangles, to test

whether triangles alone could create the same magnitude of

illusion, perhaps by providing projective 3-D information for

a single triangle being depicted at different tilts.

Subjects judged angles presented in two conditions:

1) Triangle condition: angles presented as part of a triangle

2) Double-Cube condition: same as in Experiment 1.

Results

The largest biases were again in the double-cube condition

(see Figure 4).

Conclusion

The results from both experiments support a Projective

Theory of the illusion: that the foreshortening depicting a 3-

D object such as a double-cube affects perception of the

depicting features on the picture surface (Arnheim, 1977).

Not only were the errors biased towards “good form” 90°

depicted angles, but also to 26.6° and 63.4° depicted angles

that are not “good form” angles.

DEDICATION: We dedicate this poster to the memory of Rudolf Arnheim (July 15, 1904 — June 9, 2007)

ACKNOWLEDGEMENTS: S. Bernstein and J. Hockin (TSTOP, Ontario) for comments, Sandacre Technology for programming, and I. Abramov, S. Bhasin, S. El Sebae, M. Fazl, B. Haji-Khamneh, E. Hyatt, G. Ilie, S. Moid, and D. Press.

Figure 3. Double-Cube condition Bias scores are higher than V-shape condition.

Figure 4. Double-Cube condition Bias scores are higher than Triangle condition.

0

0.2

0.4

0.6

0.8

1

V-shape Double-cube

Bia

s

26.6°63.4°90°

0

0.2

0.4

0.6

0.8

1

Double-cube Triangle

Bia

s

26.6°63.4°90°

DepictedAngles

DepictedAngles

Cortical Mechanisms of VisionCortical Mechanisms of VisionJune 19 to 23, 2007June 19 to 23, 2007