The Poggendorff illusion: Amputations,rotations, and other perturbations

Studies of the Poggendorff illusion (a transversal interrupted by parallel lines) showed that illusory effects increased linearly with increasing separation between the parallels, increased in inverse proportion to the tangent of the angle of intersection between transversal and parallels, decreased whenever line segments (other than a transversal segment) were omitted, decreasing to zero when the segment of a parallel forming the obtuse angle with the transversal was omitted, and varied systematically with the tilt of the whole display, approaching zero when the transversal was oriented in a horizontal or vertical position. Hypothesis: The Poggendorff illusion involves at least three kinds of effects on the perceived orientation of a segment: distortion by other segments (especially a segment intersecting at an obtuse angle), stability of vertical and Horizontal orientations, and assimilation towards vertical or horizontal.     

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.     

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 perceivedlocation conditions) 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.     

Contour displacements and tracking errors: Probing ’twixt Poggendorff parallels

Explanations of the Poggendorff effect were tested by varying the separation between outer parallels and by adding interior parallels. Error decreased with the addition of interior parallels, which can be explained by repulsion of parallels. A strong linear trend existed for judgmental error in millimeters plotted against separation between outer parallels. The nonzero intercept of a best-fit line and the slight nonlinearity of the data suggest a hypothesis of contour repulsion between parallels at moderate separations coupled with mistracking of the transversal across the region between parallels. Since the Poggendorff effect was independent of viewing distance, perceptual errors cannot be explained by purely peripheral mechanisms. A true intersection between transversal and parallel was the most critical feature of a display. Inverting a display increased the mean error.     

Factors affecting perceived orientation of the Poggendorff transversal

The Poggendorff figure was simplified by removing the right transversal segment. When Ss judged the distance between a dot located on the right parallel and the imagined point where the left transversal, if extended, would intersect the right parallel, the error was independent of dot location. This result is consistent with the idea that the Poggendorff figure is processed asymmetrically by (mis)projecting one of the transversals across to the opposite parallel. Systematically omitting line segments reduced (and sometimes reversed) illusory effects. The most critical Poggendorff feature was the obtuse angle formed between transversal and parallel. Ss vertically adjusted one of two dots to apparent collinearity with an implied transversal, the tip of an intervening vertical line and the other (stationary) dot. Which dot was stationary proved critical. This primitive Poggendorff display generated no illusion unless the implied transversal, defined by the stationary dot, was the one that formed an obtuse angle with the vertical line. This result also strongly supports asymmetric active processing ideas. Perceived orientation is a property of directed line segments.     

The parallel-less Poggendorff: Virtual contours put the illusion down but not out

Four versions of Poggendorff displays were compared: (1) the conventional display of a transversal interrupted by parallel lines, (2) a transversals-only display, (3) a virtual-contour display where the parallels were suggested (but not drawn) via “good continuation” and “c1osure, ” and (4) the profile display, a degraded version of the virtual-contour display that retained many of its borders but did not suggest parallels. Each display was drawn with transversal tilted at 45 and 51 deg from vertical. All displays produced significant illusions at both tilts. By comparison with the illusion magnitudes of the conventional display, the effects for the other displays were small. The outcome supports the existence of two salient illusion-producing factors: the necessity for actual contours to form the subtended angle and assimilation of the transversal toward O’s horizontal.     

The Poggendorff illusion: Apparent misalignment which is not attributable to apparent orientation of the transversals

Subjects while looking down were required to adjust a horizontal field of parallel lines (Experiment I) or a single line (Experiment II) to the apparent sagittal direction with and without a superimposed rectangle in the centre of a circular field. The rectangle was tilted at 20, 30 or 40° to the parallels and at 20° to the line. For the 20° condition the parallel lines were apparently oriented at about half a degree compared with the field without a rectangle but in the direction opposite to that necessary to account for the Poggendorff misalignment effect. For the 30 and 40° conditions the lines did not change in apparent orientation. The orientation of the single line did not change. Almost all subjects readily reported an apparent misalignment between the collinear parallels and line separated by the oblique rectangle. It is concluded that the Poggendorff misalignment illusion occurs without apparent regression of the lines to right angles with the figure.     

Alignment errors in Poggendorff-like displays when the variable segment is a dot,a dot series,or a line

In the standard Poggendorff figure, the abutting tip of the upper right transversal appears more misaligned than its distal tip. This appears paradoxical, since dot alignment errors increase with line-to-dot separation. To attempt a resolution of the paradox, four experiments were conducted in which single dots, series of dots, and lines were adjusted to be apparently collinear with a standard line segment, with neither, one, or both vertical inducing lines present. The results, taken together, suggested an explanation of the paradox and also that the standard Poggendorff display may result in alignment errors which represent a compromise judgment, based on conflicting cues to collinearity. In particular, the fourth experiment showed that line-to-line alignment errors could be made to resemble dot-to-line alignment errors by instructional variables. It was suggested that the effect was produced by forcing observers to process asymmetrically rather than symmetrically (Krantz & Weintraub, 1973).     

The role of angles in inducing misalignment in the poggendorff figure

Much experimental evidence has been put forward against the idea that angles are necessary for the occurrence of the Poggendorff illusion. We show that five separate alignment illusions can be demonstrated in the Poggendorff figure according to its orientation, length of the parallels, and so on. In one of these (angle-caused misalignment) angles are a necessary component. The main source of the belief that angles are not necessary is the alignment illusion (attraction-caused misalignment), which is due to the action of the distant parallel on the transversal that does not abut it. We show finally that it is unlikely that the angle-caused misalignment illusion is due to a change in the apparent size of the angle.     

Assessment of apparent length and angle explanations of the Poggendorff effect

Explanations of the Poggendorff effect were assessed by comparing the degree of angular distortion induced by modified and traditional configurations. Assimilation theory predicted that the traditional effect would be reversed in modified configurations. Analysis showed that the effect, although reduced in magnitude, was not reversed. Comparison of the degree of the effect induced by modified and traditional configurations indicated that a substantial portion of the Poggendorff effect was due to processing of areas between the long vertical lines of the display. This finding is not consistent with theories based on subjective distortion of angles. It was concluded that a theory of the Poggendorff effect must include processing of internal areas of the configuration.     

A depth processing theory of the Poggendorff illusion

The Poggendorff illusion is attributed to the processing of the oblique lines of the Poggendorff figure as receding horizontal lines with their inner ends equidistant because of attachment to a frontal plane (defined by the parallel lines of the figure). Collinearity in three-dimensional space is inconsistent with such equidistance; one line must lie on a higher horizontal plane than the other. This necessarily noncollinear resolution of the lines in depth processing (which is inferred irrespective of the O’s consciousness of depth) is assumed to influence apparent projective relationships within the figure, thus accounting for the illusion. Predictions from the theory, involving manipulations of the plane defined by the parallels, were confirmed experimentally. In addition, the theory is shown to account very well for the effects of amputations and rotations of the figure, which other theories of the illusion cannot handle.     

The orientation of a parallel-line texture between the verticals can modify the strength of the Poggendorff illusion

In the present experiments, we attempted to evaluate the modification of the strength of the Poggendorff illusion as a function of the different orientation of a parallel-line texture filling the space between the vertical lines. In Experiment 1, the standard version of the Poggendorff configuration was tested against four different parallel-line textures oriented at 0°, 45°,90°, and 135° with respect to the obliques. The results showed that the illusory effect was a linear function of the progressive discrepancy between the angle of the lines of the texture and that of the obliques. In Experiment 2, we tested the same textures used in Experiment 1 after the elimination of the two vertical lines. The data obtained approximated a linear function, as in the previous experiment, but the alignment errors were consistently lower. The statistical analysis performed on the data of all eight experimental conditions shows that both factors—texture and presence/absence of verticals—were significant, but most of the effect was due to the texture factor. The results may be interpreted through the “perceptual compromise hypothesis,” originally proposed for the bisection forms of the Poggendorff illusion, but with important modifications. The data are also discussed in terms of their implications for other theories proposed for the Poggendorff illusion.     

The oblique line illusion: The poggendorff effect without parallels

The apparent misalignment of two oblique collinear lines was investigated in two experiments. In the first the effect with the lines at 45° to the median plane was compared with that for the same two lines separated by the conventional parallels of the Poggendorff figure. The illusion with the two lines was consistent and significant but about one-third the magnitude of that with the parallels. The two illusions were significantly correlated. In the second experiment the angle of the two oblique, collinear lines was varied in 15° steps. The misalignment illusion was maximal at 45° and smaller but significant at 60 and 75°. There was no significant effect at 15 and 30°.     

The orientation of a parallel-line texture between the verticals can modify the strength of the Poggendorff illusion.

In the present experiments, we attempted to evaluate the modification of the strength of the Poggendorff illusion as a function of the different orientation of a parallel-line texture filling the space between the vertical lines. In Experiment 1, the standard version of the Poggendorff configuration was tested against four different parallel-line textures oriented at 0 degrees, 45 degrees, 90 degrees, and 135 degrees with respect to the obliques. The results showed that the illusory effect was a linear function of the progressive discrepancy between the angle of the lines of the texture and that of the obliques. In Experiment 2, we tested the same textures used in Experiment 1 after the elimination of the two vertical lines. The data obtained approximated a linear function, as in the previous experiment, but the alignment errors were consistently lower. The statistical analysis performed on the data of all eight experimental conditions shows that both factors--texture and presence/absence of verticals--were significant, but most of the effect was due to the texture factor. The results may be interpreted through the "perceptual compromise hypothesis," originally proposed for the bisection forms of the Poggendorff illusion, but with important modifications. The data are also discussed in terms of their implications for other theories proposed for the Poggendorff illusion.     

Apparent misalignment of oblique coplanar bars in depth

Following preliminary observations of apparent misalignment of coplanar surfaces in a three-dimensional (3-D) form of the Poggendorff Figure viewed in depth, the effect was investigated in five experiments and compared with the 2-D Poggendorff effect in the same object in a sixth. The effect in depth occurred with the complete object when it was viewed binocularly but did not do so with the oblique bars alone or when the object was viewed monocularly. The effect did not vary with oblique-parallel angles of 30°, 45°, and 60° and was absent when the angle was 90°. It varied as a function of the distance between the parallels, but was unaffected by a regular pattern on the oblique bars. A smaller 2-D Poggendorff effect occurred when the upper edges of the object were viewed from above. Although the depth effect was robust, its variance was high compared with that of the 2-D effect, indicating that acuity for misalignments of oblique elements in depth is poor. An explanation of apparent misalignment of oblique elements in 2-D and 3-D space in terms of a perceptual compromise between alignment in an oblique axis and separation relative to the axes of the parallels is proposed.     

Spatial distortions within the Poggendorff fig are and its variants: A parametric analysis

Angle size and horizontal separation of the Poggendorff figure and three components (acute angles, obtuse angles, and obliques) were varied parametrically. Two adult trained observers judged the distance between the obliques of the test figure relative to a pair of comparison dots during 250-msec tachistoscopic presentations. Results indicated: (1) the Poggendorff effect is created by an underestimation of intercontour distance, resulting in apparent misalignment; (2) only the obtuse angles can serve as a significant predictor of the classical illusion; (3) the parallel lines play an important contributory role in the Poggendorff illusion when they are present. Findings tended to support a theoretical explanation based on contour interactions.     

The corner Poggendorff

With the classic Poggendorff illusion a set of parallel 'induction lines' will cause a set of oblique line segments to look misaligned even though they are collinear. A different kind of misalignment can be produced by placing the induction lines so that they form a corner. Under these conditions the obliques will appear to be angled slightly, one relative to the other. The effects are small, but can be seen and reliably reported by a group of naive subjects. The influence of the induction lines drops sharply as their relative position is moved from parallel to orthogonal, but there is a small residual influence which may be called the corner Poggendorff effect.     

Angular induction as a function of contact and target orientation

The Poggendorff effect is seen as misalignment of two obliques, or misprojection of one, when the obliques are placed outside a set of parallel lines. To understand better the mechanisms behind this effect, the orientation of the lines which are normally parallel was systematically manipulated. The results indicate that projection bias is affected by the orientation of either line, is at a minimum where the line is orthogonal to the oblique, and is maximal at small angles. This is in line with classic theories which attribute the illusion to misperception of angular size. However, such explanations presuppose that in order to be effective the induction line must be proximal to the oblique so that an angle can be formed. Results are reported which show that the angle formed by the oblique and a line placed at a distance from the oblique, serving as the target of the projection, follows an angular rule of effectiveness similar to what is seen when the line is placed directly in contact with the oblique. The underlying process is described as 'angular induction'.     

The effect of a moving dot transversal on the Poggendorff illusion

In attempting to derive the minimal component of the Poggendorff figure which would still produce an illusion, responses to three types of transversal were measured. One was the customary solid line type; a second type presented the two segments sequentially, alternating between them; and the third consisted of a moving dot which traveled the transversal path. Each transversal was shown with and without verticals, for a total of six conditions. Ten subjects in each condition adjusted the luminous transversal until the segments appeared to be collinear. Figures with verticals present showed a greater magnitude of illusion than those without, and discrepancies for moving dot transversals were greater than those for comparable solid line figures. Since alternating transversals were not significantly larger than solid line figures, it was concluded that the magnitude of the moving dot effect could not be attributed to temporal sequence. An eyemovement hypothesis was suggested instead.     

Reduction of the Zöllner illusion with motion of inducing figure

The magnitude of the Zöllner illusion was measured when the inducing lines moved rightward or leftward and were tracked by subjects. Motion of the inducing lines significantly reduced the magnitude of illusion, as in the Poggendorff illusion. Increasing velocity markedly increased the reduction, and, again, this effect was not significantly different from that obtained with the Poggendorff illusion. The current evidence seems to support the suggestion outlined earlier in relation to the Poggendorff illusion, that is, moving and stationary figures are processed by separate channels and, therefore, the interaction between them is reduced.