The logic of misperceived distance (or location) theories of the Poggendorff illusion.

For the Poggendorff display (transversal interrupted by parallel lines), the typical distance-misperception theory postulates that a particular linear distance extending across the empty space between parallels is underestimated; examples are the intertransversal slant distance defined by the closest ends of the transversal segments (a "wings-in Müller-Lyer like" underestimation) or the perpendicular distance between parallels (parallels "attract"). Distance misperception by itself, however, can neither establish that perceived transversal misalignment exists for a Poggendorff display nor specify the perceived-location condition(s) that will produce perceptual collinearity. The perceptual displacement vector is introduced as a means of specifying fully the perceptual mislocation (displacement) of one transversal segment with respect to the other. Given this vector information (direction as well as distance), the logical soundness of theories postulating distance or location misperception were evaluated, and they were compared on the basis of extant data. Such vector information can be used to evaluate other classes of theories as well.

Perceptual plane geometry: collinearity judgments probe the perceived orientation of an angle's sides.

The task was to position a dot to lie one line length beyond the vertex of an angle by mentally extending one line segment forming the angle (and later the other line segment). Eight angles, variously oriented, provided judgmental errors attributable to the size of the subtended angle and line orientation. Collinearity errors are consistent with the hypothesis that the sizes of all subtended angles are underestimated. Horizontal-vertical assimilation describes the line-orientation effect. The largest error component is associated with the interaction of the two independent variables. Judgmental errors were modeled by a set of theoretically meaningful additive terms. Modeling was facilitated by symmetry assumptions (associated with odd and even mathematical functions) about the absolute magnitude and sign of judgmental errors that lead to the comparison of judgments derived from "interesting pairs" of stimuli.

The Poggendorff illusion: consider all the angles.

In the Poggendorff display, which consists of parallel lines interrupting a transversal, one of the two transversal segments was replaced by a dot lying along the parallel. The angle between the remaining transversal segment and the parallels was varied in 15 degree increments, as was the orientation of the transversal with respect to the subject. Subjects set the dot to appear collinear with the transversal. Judgmental errors can be partitioned into additive components, one linearly related to the size of the obtuse angle between transversal and parallels and the other a sinusoidal function of transversal and parallels and the other a sinusoidal function of transversal orientation (collinearity settings err toward the horizontal or vertical, whichever is closer), plus a meridional effect, an interaction term that magnifies the errors of a given obtuse angle as the transversal approaches an oblique orientation.

Ebbinghaus illusion: context, contour, and age influence the judged size of a circle amidst circles.

Subjects (groups of age 6, 8, 10, 12, and 21 years, with 60 or more persons per group) made comparative size judgments between a single circle and a standard circle that was flanked by four context circles. The context circles varied in diameter, proximity to the standard circle, and amount of circumference displayed. (When circumferences are complete, the display is the usual Ebbinghaus configuration. A variant consisted of the Ebbinghaus display with the outer three quarters of each context-circle circumference removed.) A contour-plus-context theory accounts for the data: Contours attract at every age level, and context (size of flanking circles relative to the standard circle) leads to constrast in judged size beyond age 6. Contour and context effects decreased with decreasing proximity between context circles and the standard circle.