Perception of Wheel-Generated Motions - Cornell Universitypeople.psych.cornell.edu/~jec7//pubs/wheel_motions.pdfaccurate pendulum clock. Although the period of a pendulum, swinging

Journal of Experimental Psychology:Human Perception and Performance1979, Vol. 5, No. 2, 289-302

Perception of Wheel-Generated Motions

Dennis R. Proffitt, James E. Cutting, and David M. StierWesleyan University

Data are presented on an old and familiar Gestalt demonstration—perceivingwheel-generated motions—in which the perceived motions of a rolling wheelare shown not to be obviously derived from the motions of the parts. Thehistory of study of this phenomenon is presented, and contradictions in theliterature are noted. The focus for experimentation is on the contrastingapproaches found in Johansson's perceptual vector analysis and Wallach'sarguments for the priority of object-relative displacement in the extraction ofinvariants. Johansson's approach asserts that common vectors are extractedfrom moving events first, whereas Wallach asserts that the motion of objectsrelative to each other is first. These two approaches yield different predictionsabout what ought to be seen when different configurations are viewed in rota-tion. In five experiments viewers rated how wheellike the movement of variouspoint-light systems attached to a rolling wheel appeared to be. Results supportWallach's views over Johansson's. Viewer judgments of goodness in wheellikemotion correspond highly with a mathematical description of the parametersof cycloidal motion for the geometric center of any system of lights on a roll-ing wheel. This specification can be made only after the extraction of object-relative displacement information. Number of lights and order of symmetryinfluence viewer judgments to a much lesser degree, and placement of a lightat the wheel's center matters not at all.

Several Gestalt phenomena have becomeclassic examples of the extraction of higherorder invariants by the 'perceptual system.The perception of wheel-generated motionsis one of the most familiar. This particularphenomenon has drawn attention because theperceived motions of a rolling wheel are notobviously manifested in the motions of thewheel's individual parts. Except for the

This research was supported by grants to authorsProffitt and Cutting from the Wesleyan University.We thank C. Stewart Gillmor of the WesleyanUniversity Department of History and HowardBernstein of the Yale University Department ofHistory of Science and Medicine for their assist-ance on the mathematical history of the cycloid,and Ethan M. Coven of the Wesleyan UniversityDepartment of Mathematics for his help with sym-metry theory. Robert J. White provided invaluabletechnical assistance.

Requests for reprints should be sent to Dennis R.Proffitt, who is now at the Department of Psychol-ogy, Gilmer Hall, University of Virginia, Char-lottesville, Virginia 22901.

linear motion of the center, all points on arolling wheel move along paths describingcycloids or prolate cycloids. Rather than per-ceiving these curves, we see points on awheel revolving about the center and thewhole moving with a linear motion. In thetop panel of Figure 1 is a diagram of themotion of a rolling wheel with the centerrepresented as a small dot and a perimeterlight as a large dot. With surround dark-ened, the path of the moving light is seen todescribe a cycloid. Our typical perception ofrolling wheels is shown in the lower panel ofFigure 1, which represents any perimeterpoint revolving about the center that moveslinearly.

Psychologists have been fascinated withthis phenomenon for over a half century;however, some of their accounts possess in-compatibilities. For this reason, we initiateda series of five experiments, and in partic-ular, focused our investigation by constrast-ing Johansson's (1973) perceptual vector

Copyright 1979 by the American Psychological Association, Inc. 0096-1523/79/0502-0289$00.75

289

290 D. PROFFITT, J. CUTTING, AND D. STIER

cycloid

circularmotion

translatory motion

Figure 1. The upper panel shows the trace of alight mounted on the perimeter of a rolling wheel.The lower panel shows the circular and translatorycomponents of motion. One light mounted on awheel is insufficient to perceive wheellike motion.

analysis with Wallach's (1965/1976) discus-sion of the phenomenon.

The nonobvious nature of cycloidal curvesis evidenced by the fact that they were notdiscovered by mathematicians until the sev-enteenth century. A brief sketch is informa-tive, demonstrating the enormous significancethat the cycloid was found to have by themathematicians of this period.

The Helen of the Geometers

It does not appear possible to determinewho first discovered the cycloid. The Greeksseem to have been completely unaware of it.Boyer (1968, p. 173) writes,

Aesthetically one of the most gifted people of alltimes, the only curves that they [the Greeks] foundin the heavens and on earth were combinations ofcircles and straight lines . . . Even the cycloid,generated by a point on a circle that rolls along astraight line, seems to have escaped their notice.That Apollonius, the greatest geometer of antiquity,failed to develop analytic geometry, was probablythe result of a poverty of curves rather than ofthought.

Although there is some controversy aboutwho should get credit for discovering thecycloid, there is agreement that Galileo knewof it, and he is credited with giving it itsname. Rubin (1927) reports being told thatGalileo noticed the cycloid at a peasant fes-tival. The peasants were rolling wagonwheels down a hill at night, and attached toeach wheel was a torch.1 About 1630 Mer-senne brought the cycloid to the attention ofFrench mathematicians and suggested its

importance. "It soon became one of the mostdiscussed curves of the period, the discus-sions occasionally leading to acrimoniousremarks, so that the curve has been com-pared to the apple of discord or called theHelen of the geometers" (Struik, 1969, p.232). At this time the cycloid was frequentlycalled a roulette, or trochoid after the Greektrochos, wheel. Roberval effected the quad-rature of the cycloid in 1634, but mathemat-ical descriptions of the curve culminated withLeibniz, who wrote its equation in 1686.(See Whitman, 1943, for an account of thehistory of the mathematical study of thecycloid.)2

Perceiving Rotary Motions

Among psychologists, Rubin (1927) wasfirst to propose that one does not see the

1 This is such a delightful story that it ought tobe true; however, we have thus far been unable toconfirm it. Galileo tried to determine the area underone arch of the cycloid, its quadrature, but failedbecause the required mathematics had not beenformulated. Besides the Galileo story, Rubin (1927)noted that the philosopher Sigwart (1895, p. 64)also discussed the perception of wheel-generatedmotions.

2 Interest in cycloids was generated from morethan just the acknowledged beauty of the curve.The cycloid proved to be the solution for a numberof the most pressing mathematical problems of thetime. One of these was the attempt to make a moreaccurate pendulum clock. Although the period of apendulum, swinging in a circular arch is relativelyconstant, it is affected by the height from whichthe bob is dropped. A search was begun to discoverthe curve that would allow a mass to fall from anyposition on the curve and reach the bottom in thesame length of time. Huygens, in 1673, found thiscurve to be the inverted cycloid, and clocks wereconstructed that made use of cycloidal pendulums.Unfortunately Huygen's clocks with cycloidal jawsproved to be no more accurate in operation thanthose using ordinary simple pendulums. Anotherpuzzle for which the cycloid was found to be thesolution was the brachistochrone (Greek for brief-est time) problem, proposed by Jean Bernoulli in1696. It entailed finding the curve that describedthe fastest path of descent for an object falling ina nonvertical trajectory. The solution is the in-verted cycloid, and this principle may be discoveredin the tryworks of whaling vessels (Melville, 1851,chap. 46) and also in nature for rapidly erodinghills (Bridge & Beckman, 1977).

WHEEL-GENERATED MOTIONS 291

cycloidal paths of points on the perimeter ofa rolling wheel. Although he did not experi-ment with simple rolling wheels,3 he arguedfrom similar demonstrations that one doesnot perceive the motion of a uniform wholeby perceiving individually the movement ofits various parts. Duncker (1929/1937) per-formed the first experiments on the percep-tion of simple wheel-generated motions. Hisobservers saw a wheel roll along a level trackin a dark room. When a single light was at-tached to the perimeter of the wheel, ob-servers saw the light describe a cycloidalpath. However, when a second light wasplaced at the center of the wheel, they per-ceived a very different phenomenon. De-pending in part on the velocity of the wheel,some saw the perimeter point revolvingcircularly about the center moving in a linearfashion, some saw the perimeter describeloops about the center which again movedwith a translator}' motion, and others sawthe two points of light move like a "tumblingstick" with lights attached to each end.

Most of us, however, became acquaintedwith studies on the perception of wheel-generated motions only through readingKoffka's (1935) Principles of Gestalt Psy-chology. Koffka drew conclusions aboutDuncker's (1929/1937) work basing hisgeneralizations on the experiments of Rubin:

If instead of adding the centre one adds a point onthe same concentric circle as the first point, then,to judge from one of Rubin's experiments per-formed with a somewhat different motion pattern,one can see two such cycloidal motions. If one in-creases the number of such points, one soon reachesthe normal wheel effect, i.e., one sees all pointsrotating round an invisible centre, and at the sametime a translatory motion, (p. 284)

Wallach and Johansson

More recently, Wallach (1965/1976) as-serted that if two lights are present on arolling wheel, diametrically opposite one an-other on the perimeter, then an observersees these lights revolving about each otherand also moving together with a translatorymotion. Wallach did not relate his observa-tion to Koffka's (1935) generalization, butscrutiny of their two accounts reveals anobvious contradiction. Wallach proposed

two components to motion perception. First,the revolution of the two points about eachother is due to object-relative displacement.That is, each point of light moves relative tothe other. This object-relative displacementis the rotational component of the observer'smotion perception. Second, the translatorymotion is determined by angular displace-ment. That is, the pair of lights moves lin-early relative to the observer. This angulardisplacement accounts for the perception oftranslatory movement.

Johansson (1973), like Duncker (1929/1937) before him, commented on the differ-ent perceptions that observers report whenviewing (a) a single point of light on theperimeter of a rolling wheel and (b) thecase in which another point of light is placedat the wheel's center. In the first case, ac-cording to Johansson, the perception is thatof a point moving on a cycloidal path,whereas in the latter case, the perception isof the perimeter point following a circularpath about the center point which describesa linear motion. Johansson proposed a vec-tor-analytic interpretation for the two com-ponents of motion perceived in the lattercase. Linear motion is a common vector de-scription of the motion of both points oflight. Extracting this common component ofmotion from the moving lights, the rotationalcomponent is a derived residual. Johanssonspecified the essential principle of his per-ceptual vector analysis as follows: "When, inthe motion of a set of proximal elements,equal simultaneous motion vectors can bemathematically abstracted (according tosome simple rules), these components areperceptually isolated and perceived as oneunitary motion" (p. 205).

Both Wallach (1965/1976) and Johans-son (1973) discuss the same two components

3 Rubin (1927) rolled a wheel within a ring, thediameter of which was twice that of the wheel.When a single light was present on the perimeterof the wheel that rolled within the ring, an observersitting in the dark saw the light describe a straight-line pendulum motion, with the greatest speed inthe middle. When two lights were present on thewheel he saw two such motions, and it was not untilsix lights were placed on the wheel that observerssaw the motion of a wheel rolling within a ring.


of motion—rotational and translatory—intheir interpretations of the perception ofwheel-generated movement. However, theiraccounts differ with respect to the order inwhich these motion components are ex-tracted from the stimulus event. For Wal-lach the object-relative displacement isprior; the rotational motion of the lights isextracted first, leaving the linear motion asa residual, which is perceived as angulardisplacement relative to the observer. ForJohansson, on the other hand, the commonvector is prior; the linear motion is extractedfirst, and the rotational motion becomes theresidual. If the stimulus event is created byrolling a wheel with two lights on its perim-eter, diametrically opposite each other, thenboth interpretations would predict the sameperception, that is, the perception of rota-tional and translatory movement. However,in the case in which the wheel has a light atits center and one on the perimeter, the twointerpretations yield different predictions.Johansson's vector analysis predicts that thelinear translatory motion would be extractedas a common vector and the rotational mo-tion of the perimeter light will be left as theresidual. Wallach's interpretation predictsthat the object-relative displacement of thetwo lights would be extracted first. Thisdisplacement would describe a rotational mo-tion about a center midway between the twopoints. The residual motion, or angular dis-placement, would describe a prolate cycloid.This is the cycloidal motion of the centerbetween the two points of light. Recall thatsome of Duncker's (1929/1937) observersdescribed this stimulus event as a "tumblingstick."

Johansson's (1973) vector analysis pre-dicts that lights placed at any two points on arolling wheel will be perceived in terms oftwo components, linear and rotational mo-tion, the former extracted as a common vec-tor. Wallach's (1965/1976) interpretationplaces a priority on the extraction of object-relative displacement. Thus, the rotationalcomponent of any two lights on a rollingwheel will equal the rotational motion pro-vided by a perceptual vector analysis only inthose cases in which the point midway be-

tween the two lights is at the center of thewheel. Where these centers do not coincide,angular displacement will be cycloidal. Thesetwo views contrast in that if observers ex-tract invariant information in the mannerpredicted by a perceptual vector analysis,then regardless of where lights might beplaced on a rolling wheel, linear translationwill be perceived, since it is common to everylight. On the other hand, if observers extractfirst the invariant component of object-rela-tive displacement, then the perception oftranslational movement will correspond to ametric which describes the invariant param-eters of cycloidal angular displacement.

A Metric for Describing Cycloidal Motions

From our experience with these stimuliand from reports of Verbrugge and Shaw(Note 1), we anticipated that Wallach's(1965/1976) account was correct and thatobservers would judge the movement ofsome stimulus configurations more wheel-like than others. We fashioned a metric thatexpressed these differences in terms of theparameters of cycloidal angular displace-ment remaining after the extraction of theinvariant rotational component of object-relative displacement. The points of lightattached to a wheel form a system of objectswhich move relative to each other. Considerfirst a two-light system. One can easilyimagine a bar, whose end points are the twolights, spinning and traveling across thescreen. Each bar, or system, will have anapparent center of moment, which lies mid-way between the two lights. This is a pointaround which all movement can be said tooccur (Cutting, Proffitt, & Kozlowski,1978). Our purpose here is to show that thisnotion works equally well for wheels as ithas previously for walkers.

Of course, each system also has an actualcenter of moment, which is the center of thewheel. We propose that it is the relative dis-tance between these two, the apparent andthe actual centers of moment, that deter-mines the wheellikeness of these motionsgiven that object-relative displacement is be-ing attended to. More specifically, the nearerthese two centers are to one another the


more the movement will look like a wheelrolling across a flat surface. On the otherhand, the farther apart they are the less itwill look like a wheel and the more it willlook like a hopping system of objects. Wecall this measure Dm/r, the distance fromthe midpoint of the system of lights to thecenter of the wheel, expressed as a propor-tion of the wheel's radius. Figure 2 showshow Dm is determined for two- and three-light systems. The length of the dotted linesequals Dm. Figure 3 is a diagram of twocycloids that describe the motions of thecenters of a one- and a two-light system. Theheight of each curve is equal to twice Dm.The horizontal distance traversed in eachrotation of the wheel is, of course, equal to2trr. Our metric thus describes the variableparameters of cycloidal motion for the centerof moment of any system of lights on a roll-ing wheel. Other possible metrics are con-sidered later, as are the location of centersfor less and more complex light systems.

Experiment 1: Two-Light Systems

Method

The method of generating point-light stimuli wasessentially the same as described in our previouswork. It is a modification of Johansson's (1973)second technique, mounting glass-bead retroreflec-tant tape on moving objects and videorecordingthem. Bright lights are focused on the object, andthe contrast of the image on the television monitorturned to near maximum while the brightness isnear minimum. Barclay, Cutting, and Kozlowski(1978) describe this technique in more detail.

Two circular patches of reflectant tape weremounted on the end of a 2-lb. coffee can that hadbeen painted a dark, nonreflectant color. Its diam-eter was 13 cm. Eleven stimuli were generated byrolling the can across a flat table at a mean of 55revolutions per minute (SD = 3.9). The stimulidiffered only in the location of the two patches.One served as a reference light on the perimeterof the can and was not moved. The other was lo-cated on the perimeter, halfway between the centerand the perimeter, or at the center, as shown in theleft panel of Figure 4. Stimuli 1-5 had second lightsat the perimeter with 180°, 135°, 90°, 60°, or 30°separating each from the reference lights; Stimuli6-10 had second lights halfway at 180°, 145°, 90°,45° or 0° separation; and Stimulus 11 had thesecond light at the center. (No item other thanStimulus 11 had a center light.) Rolling across theviewing field, stimuli were on monitor for 2.5 revo-

two-light system three-light system

Figure 2. The center of a one-light system is easyto determine, it is the center of the one light. Thecenter of two- and three-light systems is somewhatmore difficult to determine. For two lights in rigidrelation, the center of the system is the midpointbetween the two lights, as shown on the left forStimulus 8 of Experiment 1. For a three-light sys-tem it is the center of the triangle formed by thelights, as shown on the right for Stimulus 8 ofExperiment 2. This point is determined by themethod of medians. The distance from the centerof the system of lights is Dm. When divided by theradius, the normalized distance is Dm/r, and rangesfrom .00 to 1.00.

lutions. The diameter of each light was roughly onesixth the diameter of the coffee can. At maximumstimulus height (that for Stimulus 1 verticallyaligned) visual angle was between 1° and 2° for allviewers.

A test sequence of 44 trials was produced byrerecording onto a second videotape (see Cutting& Kozlowski, 1977; Kozlowski & Cutting, 1977).This sequence consisted of four consecutive, ran-domly ordered presentations of all 11 stimuli. Onlythe last three randomizations were scored; thus,the first served as practice to stabilize use of thejudgment scale. The test sequence was played on a

2irr

2-nr

Figure 3. Two cycloidal patterns, one tracing alight at the perimeter and one tracing the center ofmoment for a two-light system are diagrammed.Dm = the distance from the midpoint of the lightsystem to the center of the wheel; r = the radiusof the wheel. Notice that the period of both cycloidsis 2ir r, yet the depth of vertical excursion and thegeneral shape of the curves differ markedly.


90—

6-»—<|sif 4

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Figure 4. In the left panel, a schematic wheel is shown with the locales of lights for the 11stimuli in Experiment 1. The reference light is present for all stimuli, and the second light ofeach stimulus is located at the points designated, with 5 stimuli having the second light on theperimeter, S halfway between the center and the perimeter, and 1 at the center. In the rightpanel are the results of the rating task, with the mean judgment for each stimulus given as afunction of the distance of the center of the system of lights from the center of the wheel, Dm/r.Plus and minus one standard error of the mean are given by vertical hatches.

Sony-Matic solid-state videorecorder (Model AV-3650) attached to a 9-in. (22.86 cm) Sony solid-state television (Model PVJ-51RU) serving as themonitor. An audio track announced each trial.

Twelve Wesleyan University summer schoolstudents were paid to participate in the study. Wetold them to use a 7-point unipolar scale to judge"how wheellike the movement appears to be," with7 indicating the most and 1 the least wheellikemovement.

Results and Discussion

The remarkable correspondence betweenviewers' judgments and the distance mea-sure, Dm/r, is shown in the right panel ofFigure 4. The correlation is striking (r =-.95, p < .001). Moreover, the overall trendis indicative of viewers generally; their co-efficient of concordance for the 11 stimuli isvery strong (W = .75), X

2(10) = 90.0, p <.001 (Siegal, 1956). Thus the distance

between apparent and actual centers ofmoment serves as an excellent index forhow movement of the dynamic point-lightdisplay is perceived. Following the task, thesubjects were asked about how they had usedthe scale and typically reported that the morethe configuration hopped the less wheellikeit was rated.

Particular stimulus comparisons. Severalintriguing comparisons can be made betweencertain stimuli. For example, Duncker(1929/1937) and Johansson (1973; Maas& Johansson, 1971) used Stimulus 11—withthe second light at the center—as a proto-type conveying the most wheellike of rotarymotions in a two-light system. In fact, it isnot the best possible stimulus. Nine of tenviewers (with two ties) rated the movementof Stimulus 1—with lights opposite one an-other—more strongly wheellike than Stim-


ulus 11 (^ = 2.17, p < .03, two-tailed).Stimulus 11, however, appears to be morestrongly wheellike than would be predictedfrom the linear regression on all points. Itsresidual, more than one full judgment point,is 2.41 standard deviations greater than theresidual mean for all stimuli. Stimulus 11,then, may be a special case, even though notprototypic. One account of these results isthat this stimulus can be perceived in twoways. If perceived simply as a two-lightsystem moving about the center, then theDm/r metric would hold, and a judgment of4.62 would be predicted. If, on the otherhand, the second light was noticed not tohave any circular movement component, un-like the second lights of all other stimuli,then it would be perceived as the center,fully specifying rotary movement and sug-gesting a judgment of 7. Its observed scorewas 5.68, nearly midway between the two.The notion that systems with a light at thecenter are better than others is examinedmore carefully in Experiments 2 and 4.

Perhaps more interesting than possiblespecial cases are the two classes of stimuli,one with second lights on the perimeter andthe other with lights halfway out from thecenter. The rank-order correlation betweenDm/r for Stimuli 1-5 and judgments of theirrolling motion is perfect (p = 1.00). It isalso perfect for Stimuli 6-10. The value ofthe distance metric, however, is seen best inthe comparison of members of the two setsof stimuli. Other possible measures are sim-ply not as good.

Other metrics jor rotary motion? We de-rived three other possible measures of thestimuli. This seemed necessary because webelieve that, in advance, no one metricshould be taken as more plausible than an-other. Moreover, they force careful com-parisons between critical pairs of stimuli.Two of these indices employ the relative dis-tance between the two lights. The first ofthese considers distance alone: Perhaps itwould serve as a good predictor of judg-ments; indeed it does (r — .79). As with theprevious index, rank-order correlations forStimuli 1-5 taken as a group and Stimuli6-10 as a group are both perfect. However,

pairwise comparisons across the two setsmake this metric suspect: For example,Stimulus 7 has a shorter distance betweenits two lights than does Stimulus 2, yet itwas judged more wheellike. The same is truefor Stimulus pairs 8 and 3, 9 and 4, and 10and 5. Moreover, Stimulus 11 has exactlythe same length as Stimulus 4, and yet theirscores are quite dissimilar. Our Dm/r metricaccounts for all these relations quite nicely.

The second alternative is a combination ofthe distance metric and Dm/r. If Dm, thedistance from the midpoint of the two-lightsystem to the center, is divided 'by the lengthof the chord or partial chord formed by thatsystem, then the correlation between thismetric and viewer judgments is quite high(r — —.89). However, in addition to beingless parsimonious than Dm/r, this measurepredicts identical judgments for Stimuli 3,8, and 11, when in fact they differ markedly.Again, our previous index accounts for theirrelationships well.

Finally, a third metric was considered thatmight support the relationship between thesystems of lights and the wheel's actual cen-ter. A perpendicular could be dropped to thewheel's center from the line through the two-point system. Its length, as a function of thelength of the radius, might also serve to pre-dict viewer judgments. Crucial to this mea-sure is that it does not matter whether theperpendicular is dropped from the midpointbetween the lights (as it would be for Stim-uli 1-5), from elsewhere between the lights(for Stimuli 6-8 and Stimulus 11), or froman extension of the line beyond them (as itwould be for Stimuli 9 and 10). The correla-tion between the length of this perpendicularline and viewer judgments, however, is notsignificant (r = —.28). This is primarily dueto the fact that it fails to distinguish betweenStimuli 1, 6, 10, and 11, which all have zerodistance from the center by this method ofmeasurement. Again, our Dm/r index suitsthese stimuli well.

Thus, in all cases our original metric issuperior in its predictions of viewer judg-ments of wheel-generated motions. It is sogood that, barring the special case of Stim-ulus 11, there is only one reversal—Stimulus


onelight

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Figure 5. In the left panel are shown the 11 stimuli rated by viewers in Experiment 2, and inthe right panel the mean (and standard error) for each stimulus as a function of its systemicdistance from the center of the wheel (Dm/r).

3 with Stimulus 9—in an otherwise perfectrank-order correlation. Indeed, given thecrude manner with which we fashioned thesestimuli, this may not actually be a reversalbut may reflect measurement error in mak-ing the stimuli.

We next sought to confirm the predictivevalue of the Dm/r metric with both more andless complex dynamic point-light stimuli.

Experiment 2: One-, Two-, andThree-Light Systems

Method

Procedures were identical to those of the previ-ous experiment. Three old and eight new stimuliwere generated, as shown in the left panel of Fig-ure 5. Two were one-light systems, with Stimulus1 having the light at the center and Stimulus 2 atthe perimeter. The distance measures for thesetwo, of course, would simply be their distancefrom the center as a function of the radius. Thus,they would be .00 and 1.00, respectively. Threestimuli were two-light systems seen previously:Stimulus 3 of this set had perimeter lights 180°apart; Stimulus 4 had lights 180° apart but withone on the perimeter and the other halfway out fromthe center; Stimulus S had perimeter lights 90°apart. (These were Stimuli 1, 6, and 3, respectively,

in Experiment 1.) The six other stimuli were three-light systems: Stimulus 6 had lights on the perim-eter 120° apart ; Stimulus 7 had one perimeter lightand two halfway lights at 135° and 225°; Stimulus 8had two halfway lights at 45° and 225°; Stimulus 9was identical to Stimulus 3, but with the addition ofanother light at 90° on the perimeter; Stimulus10 had two halfway lights at 45° and 315°; andStimulus 11 had perimeter lights each separatedby 30°. The Dm/r for these stimuli was determinedfrom the center of their triangular systems, as mea-sured by the method of medians (see Figure 2).

Twelve Wesleyan University summer school stu-dents were paid to view a test sequence of 41 trials.This sequence consisted of four randomizations ofStimuli 2-11, with the 41st trial consisting of theonly presentation of Stimulus 1. Viewers used thesame 7-point scale as before, and Stimulus 1 waspresented last and only once so as not to affect itsoverall use. Again, the first presentation of theother stimuli served to stabilize judgments andwas not counted.


Shown in the right panel of Figure 5 is aplot of the judgments of wheellike motionfor each stimulus as a function of its Dm/rindex. Again, the correlation is quite strik-ing (ignoring the special case of Stimulus 1for a moment, r=-.92, p < .001). This


value is marginally lower than that for theprevious study, but it is perhaps even moreremarkable because of the more widely vary-ing stimuli used here. Viewer concordancein judgments was again quite high (W —.60), X

2(9)=64.8, / > < .001. Thus, ourmetric appears to have repeated as a goodpredictor of viewer performance. With onlyone reversal—Stimuli 9 and 10—the rank-order correlation for all stimuli is perfect.

Particular stimulus comparisons. Stim-ulus 1 was judged to be the least wheellikeof the 11 stimuli. At first this may appearodd, since its index is zero and since its onlylight source is directly at the actual center ofmoment. Its importance here is to demon-strate that a light at the center of the mo-ment is unnecessary to a dynamic point-lightdisplay and by itself is utterly insufficient.Coherent movement in a system occurs aboutthe center, not at the center. This center maymove—laterally for the coffee-can stimulihere and both laterally and slightly up anddown for the human walkers of Cutting et al.(1978)—but its own movement is not whatis crucial to the percept. The viewer per-ceives the systematic movement about thispoint in determining the identity of the ob-ject-event.

Stimulus 3 and Stimulus 5 yielded judg-ments similar to those they had in Experi-ment 1 (where they were Stimuli 1 and 3,respectively). Stimulus 4 of the presentstudy, however, yielded significantly lowerjudgments than before (as Stimulus 6). Thiseffect is almost certainly due to the fact thatin the present study there were three stimuliwith lower Dm/r indices, whereas in theprevious study there was only one. Givenmore stimuli that garner more judgments ofnearly perfect wheellike motion, it is likelythat viewers would adjust their criterion up-ward as to what type of movements merithigh scores. In pneumatic fashion, then,Stimulus 4 of the present study would beassigned lower judgments.

Most satisfying to us is that three-lightsystems are not necessarily better stimulithan two-light systems. Indeed, the multiplecorrelation of Dm/r and the number of lightsin the stimulus showed no contributions in

accounting for overall variance by the latterindependent variable. For example, therewas no significant difference between Stimuli6 and 3, both with indices of .00. Indeed,two-light systems can be superior: Stimulus3 is judged more wheellike than Stimulus 9,even though the latter stimulus is exactly likethe former except that one light is added tothe perimeter. Thus, this is a clear examplewhere more information yields a less potentstimulus. In Experiment 3, we hold Dm/rconstant and make direct comparisons be-tween two- and three-light systems.

Experiment 3: Two- and Three-LightSystems With Equal Dm/r Indices

The results of the previous experimentsuggested that the number of lights in thesystem mattered little in wheellikeness judg-ments. Since this is contrary to that assumedtrue by Koffka (1935), we offer this studyas a more direct test.

Method

Procedures were the same here as in the previousstudies. Five pairs of stimuli, one of each pair withthree lights and the other two, were fashioned sothat they had equal Dm/r indices. They are shownin the left panel of Figure 6. Stimuli 1, 2, 3, 5, and9 were used in the previous study, and Stimulus 8in the first. Stimuli 4, 6, 7, and 10 were matched tothem. Twelve viewers from the same populationused the scale while viewing a 40-item tape, 4complete randomizations of the 10 stimuli. Only thelast 30 trials were scored.


Again, and as shown in the right panel ofFigure 6, correspondence between the dis-tance measures and viewer judgments wasquite high (r = —.97, p < .001). Viewer con-cordance followed suit (W = .66), x2(9) =

71.3, p < .001. Three-light systems garneredmarginally higher ratings than two-light sys-tems, F(\, 11) = 4.65, p < .07, but only thecomparisons between Stimuli 1 and 2, and9 and 10 significantly favored three-lightsystems, correlated f s ( l l ) = 3.04 and 2.64,respectively, ps < .05. These effects areminuscule when compared with that for the


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Figure 6. In the left panel are shown the 10 stimuli rated by viewers in Experiment 3, and inthe right panel the mean (and standard error) for each stimulus as a function of its systemicdistance from the center of the wheel (Dm/r).

Dm/r index, F(4, 44) = 32.8, p < .001.Thus, perhaps to some small degree Koffka(1935) was correct that, other things beingequal, the more lights the better. However,in the present study by multiple correlation,the distance measure accounts for 47 timesmore variance than does the number of lightsin the dynamic system.

Experiment 4: Systems With and WithoutCenter Lights

The results of Experiment 1 suggestedthat a stimulus with a light at the centermight be more wheellike than would be pre-dicted by its distance measure alone. Experi-ment 2 demonstrated that for a single lightthis was not true, but we should regard thatas a special case. Here, we test the notiondirectly, holding number of lights and Dm/rconstant for pairs of dynamic stimuli with orwithout central lights.

Method

Three pairs of stimuli were fashioned. All areshown in the left panel of Figure 7. Stimulus 1 wasidentical to Stimulus 1 of Experiment 1, but with

a light at the center. Stimulus 2 was used in Ex-periments 2 and 3, and Stimulus 3 in Experiment 1.Stimuli 5 and 6 shared common locales for twolights on the perimeter. The 12 viewers of Experi-ment 3 viewed 18 trials (3 randomizations of the6 stimuli) immediately following those of the pre-vious study. No trials were discounted.


As seen in the right panel of Figure 7,the correlation between Dm/r indices andviewer judgments was high (r — —.94, p <.001), as was viewer concordance (W —

X"(5) = 28.2, p < .001. There was no.47),effect of having a light at the center of thesystem, F(l, 11) = .01, ns, although therewas the strong effect of distance, F(2, 22) =24.6, p < .001. Thus, even in conjunctionwith other lights, the center position is notspecial in its contribution to wheellikenessjudgments, disconfirming the suspicionraised by the result of Stimulus 11 in Ex-periment 1.

Experiment 5 : Order of Symmetry

The stimuli used in the previous four ex-periments differed in degree of symmetry.


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Figure 7. In the left panel are the six stimuli rated by viewers in Experiment 4, and in the rightpanel the mean (and standard error) for each as a function of its systemic distance from thecenter of the wheel (Dm/r).

Verbrugge and Shaw (Note 1) suggestedthat symmetry considerations are importantin the perception of simple wheel-generatedmotions. We offer this last experiment as anexamination of symmetry influences.

Rosen (1975, p. 12) defines symmetry asfollows: "The general term symmetry meansinvariance under one or more transforma-tions. The more different transformations asystem is invariant under, the higher itsdegree of symmetry." With respect to thetransformation of rotation, our stimuli dif-fered in degree of symmetry. Consider thestimulus with three lights on the perimeter,each 120°apart. Rotations by multiples of120° are symmetry transformations of thisstimulus. The degree of symmetry of thisstimulus is higher than any other stimulusused. The stimulus with two lights diametri-cally opposite to each other can be rotated bymultiples of 180° and remain invariant;however, all other stimuli used thus far haveonly multiples of 360° as their symmetrytransformations.

The symmetry group of a system is de-fined by the set of all symmetry transforma-tions of that system. Rosen (1975, p. 42)

defines the rotational symmetry group asfollows: "The set of transformations con-sisting of the identity transformation androtations about a given rotation center byk X 360°/M, with n a given integer and k =1, 2, . . . , M — 1, and with consecutive rota-tion as composition, forms a commutativegroup called the cyclic group of order n anddenoted Cn." The order of symmetry for eachof our stimuli is defined by the integer, n,of its cyclic group. Generally speaking, it isthe number of rotations that the stimulus cango through, up to 360°, and remain invari-ant. The stimulus with three lights 120° aparthas an order of symmetry of 3, that with twolights 180° apart has an order of symmetryof 2, and all other stimuli used thus far (ex-cept for Stimulus 1 of Experiment 2) havean order of symmetry of 1. This latter caseof 1-fold symmetry is the trivial group con-sisting of only the identity element.

Previously, all of our stimuli with a Dm/r= .00 have had an order of symmetry greaterthan 1, and other stimuli have had 1-foldsymmetry. This experiment introduces stim-uli with Dm/r = .00 and orders of symmetryvarying from 1 to 4.


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Figure 8. In the left panel are the 15 stimuli rated by viewers in Experiment 5, and in the rightpanel the mean (and standard error) for each as a function of its systemic distance from thecenter of the wheel (Dm/r).

Method

Our procedure for generating point-light stimuliwas modified slightly. Rather than using reflectanttape, we fashioned our coffee can with small mov-able lights. The can was rolled across a table andvideorecorded as in the previous experiments. Fif-teen stimuli were created and are shown in the leftpanel of Figure 8. Five of these stimuli had Dm/rindices equal to zero. Stimulus 1 had four lights onthe perimeter, 90° apart, and thus had an order ofsymmetry equal to 4. Stimulus 2 had four lightsand a 2-fold order of symmetry, Stimulus 3 hadthree lights and 3-fold symmetry, and Stimulus Shad two lights and 2-fold symmetry. Stimulus 4had one light on the perimeter, two lights within,and an order of symmetry of 1. The remaining 10stimuli (8 of which were used in Experiment 3)all had 1-fold symmetry, varying Dm/r indices, andeither one, two, or three lights. Note that an orderof symmetry greater than one is possible only whenDm/r is equal to zero.

The test sequence consisted of 60 trials of fourrandomly ordered presentations of the 15 stimuli.Only the last three randomizations were scored.The presentation differed from those of the pre-vious studies only in that a 23-in. (58.42 cm)Setchell Carson (Model 2100SD) television mon-itor was used.

Twelve Wesleyan University undergraduate vol-unteers participated in the study. They were in-structed to use the same 7-point scale as was em-ployed previously.


The correspondence between viewers'judgments and the distance measure, Dm/r,is shown in the right panel of Figure 8. Thecorrespondence is high (r = — .82, p <.001). Viewer concordance was high as well(W = .42), X

3(14) = 70.6, p < .001. Theseresults thus replicate those of our previousstudies.

Order of symmetry cannot, by itself, ac-count for much of the variability in viewerjudgments. This is so primarily becauseorder of symmetry can vary only for thosestimuli that have a Dm/r = .00. If we ex-amine the correspondence between viewerjudgments and Dm/r for only those stimuliwith 1-fold symmetry, also omitting thosestimuli with but one light, we find the corre-spondence to be fairly high (r = —.75, p <.05). Comparing the number of lights foreach of these nine stimuli and viewer judg-ments yields a nonsignificant correlation of.48.

For the first five stimuli, with Dm/r = .00and differing orders of symmetry, there ap-pears to be some effect for order of sym-


metry. The correlation between order ofsymmetry and viewers' judgments for Stim-uli 1-5 is quite high (r=.87, p < .05).However, the correlation between number oflights and judgments is high as well, al-though not significant due to the small num-ber of stimuli compared (r = .69). Numberof lights and order of symmetry are neces-sarily intercorrelated, since the former de-fines the upper bound for the latter. The de-sign of this experiment does not allow us todetermine the independent contribution ofthese variables.

Our data suggest that order of symmetrymay have an effect on viewers' judgments ofwheellike motion; however, this effect is onlypossible for that subset of stimuli with Dm/r= .00. Moreover, the necessary confoundingof order of symmetry with number of lights,within this subset of stimuli, makes an ac-curate estimation of the effect of order ofsymmetry impossible to determine from ourdata. That symmetry considerations are notapplicable to the majority of our stimulicauses us to subordinate its importance toour analysis in terms of Dm/r.

Conclusion

The results of our five experiments pro-vide strong evidence for the priority of ob-ject-relative displacement in the extractionof invariant information in event perception.Observers perceive the movement of a smallnumber of lights attached to a rolling wheelby extracting invariant information in a par-ticular order. First, the movement of thelights relative to each other is extracted. Thisobject-relative displacement is perceived asa rotation of points of light about a centerof moment defined by the system's geometriccenter. Second, the angular displacement ofthe system relative to the observer is ex-tracted. This movement is defined by themotion of the center of moment for the sys-tem of lights. When this center of momentdoes not correspond to the center of thewheel, its movement follows a cycloidal path.The invariant parameters of cycloidal angu-lar displacement for any particular systemare defined by its Dm/r metric. This metric

specifies the distance from the midpoint ofthe system of lights to the center of thewheel, expressed as a proportion of thelength of the wheel's radius. The angular dis-placement for a system of lights follows acycloidal path with a height equal to 2Dm

and a period of 2v r.We asked our viewers to judge how

wheellike the motion of each stimulus ap-peared and correlated these judgments withthe Dm/r metrics. The magnitude of thesecorrelations is striking, and we confess oursurprise at the strength of the results. Wecould find no other metric that accounts forour results as well. Examining the influencesof number of lights, order of symmetry, andplacement of a light at the wheel's center, wefound the effects of the first two to be smalland the latter nonexistent.

We are indebted to Johansson's (1973)vector-analytic description of events, but weare in basic disagreement with one of itsmost essential principles. Johansson's ap-proach provides us with a clear set of termsfor describing invariant information ex-tracted in event perception; however, he as-serts that common vectors are always ex-tracted first. If this were the case then all ofour stimuli would have been perceived asbeing equally wheellike and having identicalrotational and translatory vector components.Our results provide no support for thiscontention. Rather, we find overwhelmingsupport for Wallach's (1965/1976) position,which emphasizes the priority of the extrac-tion of invariant information specifying ob-ject-relative displacement.

We also find some support for Koffka's(1935) and Verbrugge and Shaw's (Note1) discussions of simple wheel-generatedmotions. All other things being equal, thegreater the number of lights or order ofsymmetry, the more wheellike will a stimulusappear. These influences probably serve anauxiliary function in specifying centers ofmoment.

The perception of angular displacement forour wheel-generated motions is defined bythe movement of a true abstract entity, thecenter of moment for the system of lights.Once the invariant information specifying


object-relative displacement has been ex-tracted, the motion of the system of objectsrelative to the observer is specified, not bythe motion of any particular object, ratherby the movement of the system's center ofmoment. This center is abstract, since it neednot be concretely specified by any light. Infact only one stimulus (Stimulus 1 in Ex-periment 4) had a light at the system's cen-ter of moment, and it was judged somewhatless wheellike than another stimulus (Stim-ulus 2 of the same experiment) with thesame Dm/r metric and number of lights alllocated on the perimeter.

There are an indefinite number of differ-ent mathematical descriptions which couldadequately describe the movement of ourstimulus lights. The perceptual systemselects a particular description by placing apriority on the extraction of object-relativedisplacement. In so doing, the perceptual sys-tem must specify the residual angular dis-placement in terms of the movement of anabstract entity, the center of moment for thesystem of objects. Although our D:n/r metricspecifies a relationship between two abstractentities—the center of moment for the sys-tem of lights and for the wheel—an observerattends to only the former in perceiving themovement of the system. In fact, it is thenonspecification of the wheel's actual centerof moment that results in the perception ofcycloidal motion.

It would certainly be delightful if futurestudy of the properties of this curve provedas interesting for perceptual psychologistsas its study was for mathematicians of theseventeenth century. In subsequent workwith wheel-generated motions (Proffitt &Cutting, in press), we use a better math-ematical description of geometric centers andcontinue to examine their perceptual utility.

Reference Note

1. Verbrugge, R. R., & Shaw, R. E. How we seeevents: A symmetry analysis of perceptual in-formation for rolling objects. Paper presentedat the 47th Annual Meeting of the MidwesternPsychological Association, Chicago, May 1975.

References

Barclay, C. D., Cutting, J. E., & Kozlowski, L. T.Temporal and spatial factors in gait perceptionthat influence gender recognition. Perception &Psychophysics, 1978, 23, 145-152.

Bridge, B. J., & Beckman, G. G. Slope profiles ofcycloidal form. Science, 1977, 198, 610-612.

Boyer, C. B. A history of mathematics. New York:Wiley, 1968.

Cutting, J. E., & Kozlowski, L. T. Recognizingfriends by their walk: Gait perception withoutfamiliarity cues. Bulletin of the PsychonomicSociety, 1977, 9, 353-356.

Cutting, J. E., Proffitt, D. R., & Kozlowski, L. T.A biomechanical invariant for gait perception.Journal of Experimental Psychology: HumanPerception and Performance, 1978, 4, 357-372.

Duncker, K. Induced motion. In W. D. Ellis (Ed.),A sourcebook of Gestalt psychology. London:Routledge & Kegan Paul, 1937. (Originally pub-lished, 1929).

Johansson, G. Visual perception of biological mo-tion and a model for its analysis. Perception &Psychophysics, 1973, 14, 201-211.

Koffka, K. Principles of Gestalt psychology. NewYork: Harcourt, 1935.

Kozlowski, L. T., & Cutting, J. E. Recognizingthe sex of a walker from a dynamic point-lightdisplay. Perception f? Psychophysics, 1977, 21,575-580.

Maas, J. B., & Johansson, G. (Producers). Motionperception. I: 2-dimensional motion perception.Boston: Houghton Mifflin, 1971. (Film)

Melville, H. Moby Dick. New York: Harper, 1851.Proffitt, D. R., & Cutting, J. E. Perceiving the

centroid of configurations on a rolling wheel.Perception & Psychophysics, in press.

Rosen, J. Symmetry discovered: Concepts and ap-plications in nature and science. Cambridge,England: Cambridge University Press, 1975.

Rubin, E. Visuell wahrgenommene wirkliche Be-wegungen. Zeitshrift fur Psychologic, 1927, 103,384-392.

Siegal, S. Nonparametric statistics for the be-havioral sciences. Tokyo: McGraw-Hill Koga-kusha, 1956.

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Struik, D. J. (Ed.). A source book in mathematics.Cambridge, Mass.: Harvard University Press,1969.

Wallach, H. Visual perception of motion. In G.Kepes (Ed.), The nature and the art of motion.New York: George Braziller, 1965. (Also inH. Wallach, On perception. New York: Quad-rangle, 1976.)

Whitman, E. A. Some historical notes on the cy-cloid. American Mathematical Monthly, 1943, 50,309-315.

Received July 10, 1978

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Perception of Wheel-Generated Motions - Cornell Universitypeople.psych.cornell.edu/~jec7//pubs/wheel_motions.pdfaccurate pendulum clock. Although the period of a pendulum, swinging