Sixuan Chen¹, Heiko Schütt², Weiji Ma³; ¹Brown University, ²University of Luxembourg, ³New York University

Mathematically, the length of a path is simply the sum of the lengths of its parts. Human perception of path length is more interesting. Previous studies have claimed biases in the perception of the length of zigzag paths, but have been contradictory on the direction of the bias: some claim that zigzag paths are perceived as longer than equally long straight ones, others as shorter. To resolve this contradiction, we conducted a psychophysical experiment in which we systematically varied the length, the number of segments, and the sum of angles of a zigzag path. On each trial, a participant compared the lengths of a zigzag path and a straight line. We analyzed the point of subjective equality (PSE) and the just noticeable difference (JND) as a function of the number of segments, the sum of angles, and other summary statistics. Surprisingly, PSE showed a U-shaped relationship with two different measures of zigzagness: participants overestimated length at low zigzagness but underestimated it at high zigzagness. To account for these results, we constructed a model in which the observer estimates path length using three noisy cues: the true length, the Euclidean distance between the path endpoints, and the square root of the convex hull area of the path. This model qualitatively and quantitatively accounts for the trends in PSE and JND. In a second experiment, we replicated a previous study that used only right angles and further validated the model; the stimuli in this experiment had low zigzagness and consequently, data and model both only show overestimation of length. Together, these findings resolve the contradiction in the literature and suggest that the visual system utilizes global shape statistics for path length estimation.