1. Abstract Angles 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 Surface Igor Juricevic, Shazma Rajani, Sherief Hammad, & John M. KennedyUniversity 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. Depicted Angles Depicted Angles Cortical Mechanisms of Vision June 19 to 23, 2007