Four experiments in which observers judged the apparent “rubberiness” of a line segment undergoing different types of rigid motion are reported. The results reveal that observers perceive illusory bending when the motion involves certain combinations of translational and rotational components and that the illusion is maximized when these components are presented at a frequency of approximately 3 Hz with a relative phase angle of approximately 120°. Smooth pursuit eye movements can amplify or attenuate the illusion, which is consistent with other results reported in the literature that show effects of eye movements on perceived image motion. The illusion is unaffected by background motion that is in counterphase with the motion of the line segment but is significantly attenuated by background motion that is in-phase. This is consistent with the idea that human observers integrate motion signals within a local frame of reference, and it provides strong evidence that visual persistency cannot be the sole cause of the illusion as was suggested by J. R. Pomerantz (1983). An analysis of the motion patterns suggests that the illusory bending motion may be due to an inability of observers to accurately track the motions of features whose image displacements undergo rapid simultaneous changes in both space and time. A measure of these changes is presented, which is highly correlated with observers' numerical ratings of rubberiness.

^{2}). The length of these lines subtended 4.04° of visual angle, and each dot subtended 0.1° of visual angle. There were eight possible patterns of motion, generated by different combinations of translational and rotational oscillatory motion. The amplitude of the rotational component refers to the angle through which the line is rotated around its midpoint. It could be either 0° or 90°. The amplitude (total vertical excursion) of the translatory motion component could be 0°, 2.02°, or 4.04° of visual angle. When translational and rotational motions were combined, the relative phase angle between the two components was 125°. Examples of four different motion patterns are illustrated in Panels A–D of Figure 3. Each panel in this figure depicts a superposition of all of the discrete frames of a particular motion sequence. In the actual experiment, each 40-frame sequence was presented over time at a rate of 3 Hz. Half of the displays contained an additional 360° linear rotation at a constant angular velocity around the center of the moving line segment at a rate of 0.33 Hz to increase the complexity of the motion traces. Motion traces for the four displays with linear rotation are presented in Figures 3E– 3H. The equations that describe the stimulus motion are shown in the 1.

*F*(1, 7) = 337.02,

*p*< .001, and this one comparison accounted for 96% of the between-display variance. No other comparisons among the different conditions were statistically significant.

^{2}). Observers saw motion displays in two different experimental conditions. In the “speed” condition, the translational and rotational motion was presented with nine different relative phase angles (0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5°, and 180°). For a given phase angle, observers could vary the speed of the motion such that the overall pattern of oscillation could occur at 1.5, 2, 2.5, 3, or 3.75 Hz. Similarly, in the “phase” condition, the motion pattern was presented at five different speeds (1.5, 2, 2.5, 3, or 3.75 Hz), and for any given speed, observers could adjust the relative phase angle of the translational and rotational components with possible values of 0°, 22.5°, 45°, 67.5°, 90°, 112.5°, 135°, 157.5°, and 180°.

*d*=

*f*(

*p,*

*t*) at each position (

*p*) along a moving line at each moment in time (

*t*) for selected conditions from Experiments 1 and 2. The text above each plot shows experimental condition and average judged rubberiness for that condition. It is interesting to note when evaluating these plots that the appearance of rubberiness was larger for displays in which the displacements at each point varied simultaneously in both space and time. These can be identified in Figure 6 by the preponderance of diagonally oriented contours.

*n*= 780) and multiplied the spatial and temporal components of the gradient at each point. The absolute values of the products obtained over an entire sample were then averaged together to provide an overall measure (

*C*) of the extent to which the displacement in each display varied simultaneously in both space and time:

*C*and then reach an asymptote as the gradients become larger. These findings suggest that the illusory bending motion may be due to an inability of observers to accurately track the motions of features whose image displacements undergo rapid simultaneous changes in both space and time.

^{2}). The length of this line was 4.04° and its width was 0.1°. All of the displays included a rotary oscillation, in which the line segment rotated back and forth through an angle of 90° at a rate of 2.5 Hz. On half the trials, this rotary oscillation was the only source of distal motion. On the remaining trials, an additional component of motion was added, in which the center of the line segment was translated along an elliptical orbit around the center of the display screen at a rate of 0.83 Hz. The horizontal and vertical axes of the elliptical trajectory subtended 8.08° and 4.84°, respectively. Both of these distal motion conditions were observed with the eyes fixated on a stationary point and with the eyes tracking a moving fixation point along an 8.08° × 4.84° elliptical trajectory. These different combinations of distal and eye motion resulted in four basic experimental conditions that are illustrated in Figure 8. The equations used to generate these motions are given in the 1.

*n*< 5). Across the remaining nine observers, 10.4% of all trials (

*n*= 94) were excluded from further analyses, all of them pursuit trials. Average eye-movement gain in the two pursuit conditions was 0.89 and 0.90, respectively (between-observer

*SD*= 0.07 and 0.06, respectively). Average eye-movement gain in both fixation conditions was 0.09 (between-observer

*SD*= 0.02 in both conditions). Only numerical ratings obtained in valid trials were considered for further analysis.

*F*(1, 8) = 10.07,

*p*< .05, and eye motion,

*F*(1, 8) = 11.78,

*p*< .01, and a significant interaction,

*F*(1, 8) = 24.84,

*p*< .001. Additional post hoc paired-sample

*t*tests (two sided) were performed to compare individual pairs of conditions. Significance is indicated in Figure 9 by asterisks.

*n*= 186) of all trials were excluded from further analyses across observers. Eye velocity was comparable across observers and experimental conditions for the remaining trials (average eye velocity = 1.83°/s, median = 1.85°/s,

*SD*= 0.22°/s, range = 0.81°/s). As in Experiment 3, we also used an alternative analysis of eye movements that relied on a position criterion to assess fixation stability, with equivalent results.

Rigid standard | Rectangle in-phase | Texture in-phase | Texture static | Bending standard | Texture off-phase | Rectangle off-phase | |
---|---|---|---|---|---|---|---|

Rigid standard | X | .86 (50) | .98 (51) | .96 (48) | .96 (45) | .98 (50) | .98 (46) |

Rectangle in-phase | X | .82 (50) | .72 (58) | .84 (49) | .85 (54) | .93 (54) | |

Texture in-phase | X | .55 (53) | .70 (53) | .73 (51) | .68 (53) | ||

Texture static | X | .59 (56) | .57 (51) | .63 (51) | |||

Bending standard | X | .45 (49) | .49 (47) | ||||

Texture off-phase | X | .51 (55) | |||||

Rectangle off-phase | X |

*t*test (two sided). Note in the figure that when the background moves in counterphase with the moving line segment, it has no discernable effect on apparent rubberiness relative to the bending standard condition with a homogeneous background. However, if the line segment is presented against a static textured background or if the background motion is identical to that of the line segment, then the perception of rubberiness is intermediate between the rigid standard and bending standard conditions. These results are, in many ways, similar to those obtained with pursuit eye movements in Experiment 3. Both studies suggest that retinal motion is the predominant factor for determining the strength of the rubber pencil illusion but that the effect can be attenuated if the motion of the pencil is the same as its local frame of reference—either the eye or the background.

*x,*

*y*) of a point as a function of time

*t,*measured as an integer number of frames after

*t*= 0. The sinusoidal rotational and translational motion components of this motion are

*A*

_{ θ}is the angular amplitude (in radians) through which a given point is rotated around the origin of the coordinate system,

*A*

_{ τ}is the amplitude of the translational motion component (in degrees of visual angle) relative to the origin,

*ϕ*is the relative phase angle (in radians) between rotational and translational modulation, and

*ω*

_{s}=

*π*/20 in the current experiments. The

*x*and

*y*spatial coordinates of any point as a function of time, then, are sums of rotational and translational components:

*x*and

*y*spatial coordinates of any point after a linear rotation is applied to each of the sinusoidal motion patterns, as it was done in Experiment 1, are

*ω*

_{lin}=

*π*/180 in the current experiments.

*x*and

*y*spatial coordinates of any point after an elliptical translation is applied to the sinusoidal rotation, as it was done in Experiment 3, are

*ω*

_{ell}=

*π*/60,

*a*= 4.04°, and

*b*= 2.42° in the current experiments.

*x*and

*y*spatial coordinates of any point as a function of time

*t*that is moving off-phase with the line segment, as it was done in Experiment 4, are