Angular subtense effects on perception of polar and parallel projections of cubes

Many authors contend that the perception of 2-D drawings of a 3-D object is governed by polar projective geometry. A problem for this position is that observers accept parallel projections, which are not produced with polar projective geometry, as accurate representations of 3-D objects. In Experiments 1 and 2, we used two different standards of comparison to study the perceptions of three line drawings of cubes--correct polar projections of cubes with subtenses of 15 degrees and 35 degrees, and a parallel projection--at five different angular subtenses. In Experiment 1, 14 observers judged each drawing when it subtended about 35 degrees, 15 degrees, 5 degrees, 4 degrees, and 2 degrees in width. Subjects used an 8-point rating scale to compare each drawing with a correct polar projection of a cube subtending 35 degrees, viewed with the drawing subtending 15 degrees. As predicted, both polar projections had their highest ratings at their correct vantage points. Ratings for the parallel projection were highest at small angular subtenses and decreased when it subtended 35 degrees. These findings were supported by a second experiment in which the 15 degrees polar projection was set at a 5 degrees viewing angle as a standard. In Experiment 3, 15 observers compared the three drawings, viewed at a second set of angular subtenses (30 degrees, 35 degrees, 40 degrees, 45 degrees, and 50 degrees), with a standard, the 35 degrees polar set at 45 degrees. Ratings fell with increases in viewing angle, and the parallel projection was rated lowest. The results indicate that parallel projections are assessed as polar projections that are correct for objects at a small angular subtense. Furthermore, projections at a small angular subtense are robust; that is, they are acceptable over a wide range of angular subtenses. We suggest that robustness can be explained by the modest variability in the proportions of pictures of cubes subtending small angles.