Orientation-dependent biases in length judgments of isolated stimuli

Vertical line segments tend to be perceived as longer than horizontal ones of the same length, but this may in part be due to configuration effects. To minimize such effects, we used isolated line segments in a two-interval, forced choice paradigm, not limiting ourselves to horizontal and vertical. We fitted psychometric curves using a Bayesian method that assumes that, for a given subject, the lapse rate is the same across all conditions. The closer a line segment’s orientation was to vertical, the longer it was perceived to be. Moreover, subjects tended to report the standard line (in the second interval) as longer. The data were well described by a model that contains both an orientation-dependent and an interval-dependent multiplicative bias. Using this model, we estimated that a vertical line was on average perceived as 9.2% 6 2.1% longer than a horizontal line, and a second-interval line was on average perceived as 2.4% 6 0.9% longer than a first-interval line. Moving from a descriptive to an explanatory model, we hypothesized that anisotropy in the polar angle of lines in three dimensions underlies the horizontal–vertical illusion, specifically, that line segments more often have a polar angle of 908 (corresponding to the ground plane) than any other polar angle. This model qualitatively accounts not only for the empirical relationship between projected length and projected orientation that predicts the horizontal–vertical illusion, but also for the empirical distribution of projected orientation in photographs of natural scenes and for paradoxical results reported earlier for slanted surfaces.