DANIEL J. WEINTRAUBt, University of Michigan, Ann Arbor, Michigan 48104and

VEIJO VIRSU, University of Helsinki, Helsinki, Finland

Estimating the vertex of converging lines:Angle misperception?*

The angle between two converging but nonintersecting line segments and theorientation of the display were varied; 0 was asked to determine the vertex. Adescriptive model based on four additive factors was devised to explain thesystematic errors: (1) perceptual tilt of individual line segments toward the moreclosely aligned axis of the visual field, either horizontal or vertical, (2) tendencyto place the intersection too high in the visual field, (3) orientation-<lependentcomponent of misperception of the angle between the line segments, and(4) orientation-independent component of angle misperception. A regressionanalysis showed that each of the four components was highly significantstatistically; together, they accounted for 89% of the variance of the meanerrors.

Considerable psychophysicalevidence is available to lead one to theconclusion that acute angles areoverestimated. A brief discussion canbe found in the introduction to ourprevious report (Weintraub & Virsu,1971 ). Recent neurophysiologicalevidence (Burns & Pritchard, 1971)was interpreted as showing that, forthe cat, an acute angle is representedneurally as too large. A relativelyunexploited psychophysical approachwith high face validity mightcontribute converging evidence. Let ano be asked to place a dot at theapparent intersection of two linesegments. If he places the dot too faraway from the ends of the convergingsegments, then a strong inference isthat the angle subtended by thesegments is being underestimated. Bysimilar reasoning, vertex placementtoo close is evidence for angleoverestimation. Unfortunately, Osplace the point of apparentintersection consistently off to the leftor right of the imaginary line bisectingthe angle if such errors are permitted,and what began as a morestraightforward question becomescomplicated in light of the data.

Our angle-estimation technique has

*Supported by a U.S. Public HealthService ResearOO Scientist DevelopmentAward (K2-MH-35.253) to D.J.W. and byKamarineuvos H. Rosenberg's travel grantfrom the UDivenity of HelsiDki to V.V.Research funds were provided by NationalScience Foundation Gru;at GB 8181 and bythe UDiversity of Michipn's BiomedicalSciences Institutional Support Grant PHSFR 0705()'·03. We thank Lillian Tong forservinl as E. We also thllllk threeperspicacious critic....-two anonymousreviewers and Irwin Polladt-whose incisivecomments on earUer drafts were invaluable.

tAddreBB: Human Performance Center,Perry Building, 330 Packard Rd., Univenityof Michigan, Ann Arbo~, Michigan 48104.

definite methodological advantages.Two-dimensional freedom for 0 indetermining the vertex permitsmaximum freedom to err, improvingthe opportunities to detect importantsources of variation. There is nocomparison stimulus required so thatmisperceiving the comparison is not aproblem. The dot marking the vertexis removed from the immediatevicinity of the segments themselves,reducing the possibility of interactionbetween the dot, which marks thejudgment, and the segments, which arethe features to be judged. Since thelines do not form an intersection,theories concerned with retinalblurredness in the region of anintersection (Chiang, 1968) should notapply.

In the prior study (Weintraub &Virsu, 1971), the two-dimensionalityof the errors compelled us to proposethat a salient component of theconverging-line anomaly is theperceptual tilt of each line segmenttoward the more closely aligned visualaxis, either horizontal or vertical,relative to the O. (As yet, 0 himselfhas not been tilted; only displays havebeen tilted relative to 0.) Theproposition is not novel and issupported by other evidence (cf.Obonai, 1931; Bouma & Andriessen,1968; Weintraub & Krantz, 1971).However, the principle of assimilationtoward a cardinal viewing axis appliesindividually to each line segment of apair compriaing an angle and not tothe angle subtended. It is logicallypossible that the outcome of thedot-setting task can be predicted byadding the tilt-assimilation errors ofeach line. However, the previous study(Weintraub & Virsu, 1971) showedthat when the tilt bias for a single lineat a specified inclination in the visual

field was computed, the inclination ofthe other line of the pair was notirrelevant. Angle subtended betweenlines is the logical construct forcharacterizing the interaction. Thus,the hope at the outset of the presentstudy was to gather sufficientparametric data to make it possible toremove, via regression analysis, thetilt-assimilation component ofindividual lines, and then to attributethe remaining systematic bias to anglemisperception.

The previously published study haddivided the orientation of displays intoone of two classes: symmetrical, withthe angle bisector vertical in O's visualfield; asymmetrical, with the moreclockwise line of each subtended angleremaining vertical. For asymmetricaldisplays, the orientation of thebisector of a subtended angle wasconfounded with the size of thesubtended angle. The new designmanipulated orientation and size ofsubtended angle independently. Basedupon the previous data, threesubtended angles were chosen: anacute angle (20 deg) and an obtuseangle (120 deg) that showed largeerrors, and one acute angle (60 deg)that induced very little error.Inclination of individual line segmentswas deemed to be the most criticalvariable, and 13 orientations of eachsubtended angle were selected withparticular attention devoted topositions where single lines fromdifferent angles would match ininclination.

It must be emphasized that we arenot wedded to regression analysis.Rather, the data led us there by virtueof uninterpretable bivariate dotsettings. The resulting model(equation) is not to be viewed as theculmination of a mathematicalpasttime, namely, trying to accountfor maximum variance with the fewestfactors. The experiment was designedto measure and remove the previouslyidentified tilt-assimilation factor inorder to examine the residual as anglemisestimation. This stepwise analysisrevealed a second and unanticipatedsystematic bias: 0 seemed to place thedot representing his judged vertex toohigh in his visual field in addition towhatever else he did. Afterendeavoring to establish the existenceof the second factor by argument andby gathering additional data, thefactor became a post hoc addition tothe model. The ensuing analysis of theresidual bias revealed that it wasunworthy of the title "pure" anglemisperception.

The remainder of the articleattempts to argue convincingly that:(1) misestimation of a subtended anglemust be a function of the orientationof the angle as well as of the size of

Perception & Psychophysics, 1972, Vol. 11 (4) Copyright 1972, Psychonomic Society, Austin, Texas 277

Table 1Meana of Judement. in MiWmeters: The Anll1e Bbector is the Ordinate (Y) With the True Intenection .. Orl8in

Orientation of the Angle Bisector (Decree.)

-90 -60 -60 -45 -30 -10 0 +10 +30 +45 +60 +80 +90

Subtended Angle 20 DegX -.06 -.38 .79 -.26 -2.34 -35 -.05 .83 1.18 .34 -1.16 -.69 .18Y -1.42 1.18 -2.69 .75 1.84 5.19 6.39 5.70 3.52 2.07 3.81 8.27 5.40

Subtended Angle 60 DegX .66 .20 -1.23 -1.07 .23 .12 -.07 -.37 -.46 -.04 .17 -.05 -.12Y -3.50 -3.13 -2.63 -4.04 -1.06 .69 1.02 1.27 -1.63 -1.74 -.72 1.17 .92

Subtended Angle 120 DegX 1.39 1.20 2.17 1.36 1.98 1.03 .18 .07 -.78 -.41 .90 .38 .22Y ~.70 --5.54 -2.52 -.51 -.22 -2.21 -3.36 -2.65 -.44 -.71 -1.54 -4.35 --5.01

Note-Positive values of X signify that the mean wg£ located off the anllie' bisector toward the more counterclockwise line of thepair forming the subtended angle. Positive values of Y signify that the mean 'was too far from the line segments to coincide withthe true intersection.

the angle; (2) the nrst two factors inthe analytic model are more thanintervening variables, that is, morethan mere terms in an equation. Theyare not the polynomials in x of a trendanalysis. Rather, they were chosenpartly on the basis of outside evidence,and emerged after months of worryingthe data to death. We believe thatthese two factors are relevant to otherparadigms and other phenomena.

METHODThe experiment is an elaboration of

the earlier study (Weintraub & Virau,1971), and procedural details areidentical if not discussed below. Thepaid Os were 48 students at theUniversity of Michigan. Twoconverging line segments, each 30 mmlong, .25 mm wide, were drawn withblack India ink on 21.6 x 27.9 cm(81At x 11 in.) rectangular sheets ofwhite cardboard. The distancebetween the true point of intersectionand the near ends of the line segmentswas always 60 mm, With theeye-to-display distance ofapproximately 54 cm used in theexperiment,30 mm is equal to 3.2 degof visual angle.

The angle subtended between theline segments was 20, 60, or 120 deg.The subtended angles were presentedrandomly in 13 different orientationsthroughout the left-hand haIf of thevisual field. (To be more precise, theangle bisector was never located to theright of the vertex.) The O's line ofsight always coincided with the truevertex. The orientations are designatedby the direction of the vertex end ofthe angle bisector: -90 deg (vertextoward the bottom), -80, -60, -45,-30, -10, 0 deg (vertex toward theright), and +10, +30, +45, +60, +80,+90 deg (vertex toward the top). Theorientations for -90, 0, +90 deg areillustrated at the bottom of Fig. 1.Considerable mathematicalsimplifleation in writing equations wasgained by designating orientations inthis manner.

The longer edges of the rectangulardisplay sheets were parallel to the90-deg, i.e., vertical, orientation. Alarge transparent acetate overlaycontaining a black dot (1.5 mm diam)near the center covered the display.The 0, standing at the edge of thetable, leaned over to peer downperpendicularly at the display. He wastold to " ... place the dot at the pointwhere the two lines would intersect ifthey were extended." Eye movementswere not restricted; 0 was told not totilt his head to either side. Every 0judged all 39 displays in anindividually randomized order.

Supplementary Experiment 1The original experiment was

replicated for the subtended angle of120 deg, exercising more stringentcontrol of head tilt and eye-to-displaydistance. In the main study, Os wereasked, and observed by E, not to tilttheir heads. Now, a rigid opaqueviewing mask with narrow horizontaleye slits was introduced directly abovethe center of the display so thateye-to-display distance was maintainedat 54 cm and when 0 tilted his headhe could not view the display withboth eyes. In addition, the fieldcontaining the display was changedfrom a rectangular to a large circularfield of 46 cm diam, The Os were anew sample of 30 students.

RESULTSThe bidimensional deviations of

every judgment were recorded initiallyin millimeters, using a Cartesiancoordinate system with the anglebisector as ordinate and the truevertex at zero. The means of thesedeviations are listed in Table 1.Computer plots of the raw dataprovided a scatter diagram for each ofthe 39 stimulus conditions. (Adiscussion and an example of suchdispersions appear in Weintraub &Virau, 1971.) Reducing the data totwo-dimensional mean deviations inmillimeters in order to treat the results

as displacements of a perceived vertexled us to no new insights. Anotherinterpretation of the data can be madein terms of misperceived tilt. It wasassumed that each line segmentcom p r ising an angle rotatesperceptually about its own midpointto point toward the misperceivedvertex (illustrated in Fig. 1, Weintraub& Virsu, 1971); this assumption willbe employed throughout the paper.Note that tilt errors for a pair of linesforming a subtended angle are equal inmagnitude but opposite in directiononly when O's judgment liessomewhere along the angle bisector orits extension. Every judgment of thevertex in Cartesian coordinates wastransformed into a tilt error for theclockwise line and a tilt error for thecounterclockwise line of the pairforming the angle. By averaging thedata across Os, a mean tilt error indegrees was computed for every linesegment of every subtended angle inevery orientation, a total of 78 means.The transformation converted atwo-dimensional variable into twounidimensional measures. Thevariability about the mean provides anestimate for the variance in tilt biasassociated with a line segment. Thesedata, shown in Fig. 2, became the datato be interpreted.

What of angle misperception? Thetilt biases of pairs of line segmentsformin~ a subtended angle can beadded to give the measures of anglemisperception plotted in Fig. 1. Analternate computing procedure fordetermining angle misestimation is theone employed in the previous study(Weintraub & Virsu, 1971), convertingthe mean location of the vertex inmillimeters, the data of Table 1, intoangle J,Disestimation. As a consequenceof the distribution of Os' judgments, aslightly different set of means isthereby obtained: The function for asubtended angle of 120 deg is nearlyidentical to the one plotted (meanswere, on the average, .03 deg higher);for the 60-deg subtended angle, all

278 Perception & Psychophysics, 1972, Vol. 11 (4)

-90 0 +90

TilT (5) OF BISECTOR <DEG)

Fig.1. Empirical anglemiaeatimation as a function ofsubtended angle (z) and orientation ofthe bisector (a). Positive enorsrepresent overestimation.

Z· 20·2

o

2 z·

o

-I

-It------........---:---------------!

s 0

1&.1Q-II: 00II:It: -I1&.1

z -2c1&.1a 4 ANGLE Z • 120·

REPLICATION If

3 REGRESSION- -

2

that random head tilts, small variations measure of angle miaeatimation wasin eye-to-display distance, and the too optimistic. In Fig. 1, there is ashape of the borden of a display exert downward trend in judgmental errora minor influence upon the outcome associated with the orientation of theof the experiment. 'D1ere is a clear display from inverted to upright. Thecyclic component in all six functions bias was dubbed the "elevator factor"consistent with the hypothesis of because 0 seemed to place the dotperceptual tilt of line segments toward representing the vertex too high in histhe more closely aligned visual axis, visual field. For example, with bisectoreither the horizontal or vertical at -90 deg, if the 0 places the dot toodefining O'a visual field. However, the high, then he will place it closer to theaimple plan of removing the cyclic line segments, giving an additionalvariation mathematically in order to component of overestimation to theexamine the residual as a possible subtended angle. With the bisector at

-90 0 +90LINE INCLINATION (DEG)

Fig. 2. Errors in the perceived inclination of individual line segmenta.Counterclockwise errora are designated aa positive.

1\-2 \I

10ANGLE (Z)----.. 120·x- __ • 60·

~6 -- 20·

;4w

~

points were diaplaced slightlydownward from those ahown (anaverage of .24 deg); for the 20-degsubtended angle, all points weredisplaced even farther downward (anaverage of .46 deg). In summary, theahapea of the functions showinconsequential changes. but the 20­and 6O-deg functions are displaceddownward if averaging is performedprior to converting to degreea. Apoaitive error denotea anoverestimation of the angle. Le., thejudged intersection was too close tothe line segment. A subtended angle of120 deg is overestimated in allorientationa, 60 deg is usuallyoverestimated and 20 deg is usuallyunderestimated (contrary to thedictum that acute angles are alwaysoverestimated). Superimposed uponthe cyclic variations, there was a slopeof the functions toward the right, adownward trend in errors 88 aubtendedangles proceed from inverted toupright orientations.

The data of primary interest arethose associated with individual linea.'D1e tilt·assimilation factor applies toline segments. not to aubtended angles.In addition, any representation likethat of Fig. 1 obscur~ the fact thatline pairs forming an angle usuallyshow unequal tilt biasea, and thus acrucial feature of the experimentaloutcome is ignored. Figure 2 showathe errors in perceived orientationassociated with each line segment. Thereplication data for the aubtendedangle of 120 deg are also plotted, andthey agree well with the results of themain experiment. It may be assumed

Perception. Psychophysics, 1972, Vol. 11 (4) 279

Perception &; Psychophysics, 1972, Vol. 11 (4)

+90 deg, a dot too high will put itfarther from the line segments, givingan underestimation component. Whenthe bisector is not oriented vertically(all orientations except ±90 deg),placing the dot too high also moves itoff the angle bisector. Proposingindependent status for the bias wassparked by the observation that asimilar effect had occurred in anotherexperiment. When the Poggendorffdisplay was inverted, an increasederror was found (Weintraub & Krantz,1971). Before we decided toincorporate the apparently systematiceffect into our schema, the bias wasinvestigated in its own right.

Evidence Concerningthe Elevator Factor

An empirical calculation of theelevator factor is possible. For a givensubtended angle, two orientations thatdeviate equally but in oppositedirections from horizontal (i.e., fromzero bisector tilt) can be treated as amatched pair of subtended angles; asubtended angle of 20 deg at +60- and-60-deg bisector tilt form such a pair.For each orientation of the pair, themean vertical (with respect to the 0)component of the error in millimeterscan be computed. The question askedis, disregarding lateral error, how manymillimeters too high or low was theaverage judgment for that orientation?The elevator factor should be the samein magnitude and sign for eachmember of the matched pair.However, the contribution of otherfactors to the vertical error relative too ought to be equal but of oppositesign for each member of the pair.Thus, the reversal of sign will occur fortilt-assimilation errors. Also, othersystematic biases like misperception ofsubtended angle should occur withrespect, not to 0, but to the displays,one of which is inverted relative to theother, providing for reversed signs. Byadding the vertical error componentsfrom matched pairs of subtendedangles and dividing the result by 2, anelevator factor was extracted.(Addition cancels effects withopposite sign and doubles the size ofthe elevator factor.) One can visualizethe process most easily for anupright-inverted (±90 deg) pair, butthe same reasoning applies to allmatched pairs.

The computations revealed that theelevator factor was not a constant. Itwas nearly zero when the bisector of anangle was horizontal, and as an angleapproached either an upright orinverted position, there was anincreasing upward bias to the settings.The slope of the functions in Figs. 1and 2 is a manifestation of the effect.Since the experimental procedurepermitted two-dimensional errors, itwas possible to calculate the standard

280

B

(36lJJe

a::0a:: 4a::lJJ

0lJJ 2I-U "

0 \lJJ 0 '-a::c,

-2L.-90~_-..L_--O~---L--+::'9'='O

TILT (5) OF BISECTOR (DEG)

Fig. 3. Empirical anglemisestimation after removingcomponents that are independent ofthe subtended angle (z). Positive errorsrepresent overestimation. The smoothfunctions were derived from themodel.

deviation of the vertical component ofevery mean setting. From a minimumat the horizontal orientation of theangle, the vertical component of thestandard deviation increased to amaximum as the bisector reachedeither an upright or inverted position.The empirical bias values inmillimeters were correlated with theequivalent vertical component of thestandard deviation in millimeters. Theproduct-moment correlation was +.86,an outcome highly significantstatistically. The correlation indicatesthat the elevator factor for horizontaldisplays is slight and that there is onlysmall vertical variability in these vertexsettings. The elevator factor increasesto a maximum as the bisectorapproaches vertical where the verticalvariability is large. One must becautious in inferring cause and effectfrom a correlation, but it is temptingto conclude that the elevator factor isconstrained to operate in directproportion to the uncertainty in thelocation of the vertex vertically. Theexistence of such a correlation raisesour confidence that the elevator factoris worthy of status as an entity.

Supplementary Experiment 2An experiment was designed to

investigate the possibility that theupward bias is methodological,perhaps in the same class with theknown psychophysical errors ofanticipation and habituation. A variantof the method of constant stimuli, thegroup version of the up-and-downmethod, was employed with 48 newOs, (See Weintraub & Krantz, 1971,for details, which include statisticaltesting.) Only six displays were judgedby each 0, the upright and invertedorientations of the three angles of the

main experiment. Each 0 observed thedot in a fixed position along thebisector of the display and wasrequired to state whether it was placedtoo low or too high to mark the vertexof the angle.

For subtended angles of 20, 60, and120 deg, the differences in errorsbetween inverted and upright settingswere 16.66, 6.53, and 2.78 mm,respectively. The differences indicatethe usual upward bias, and three ztests of these differences were allstatistically significant (in eachinstance at least p < .05, two-tailed).The result indicates that the elevatorfactor is not tied to a particularpsychophysical method but may wellbe a perceptual factor. (No left-rightlateral bias has been found. The leftside of the visual field was selected inthese experiments in order to comparethe data with Weintraub & Virsu,1971, where the right side was used.)

Regression AnalysisThe following four-factor additive

regression model was devised foranalyzing the results:

where y is the tilt error in degrees ofeach individual line .egment; b, , b 2 ,

ba , llz are parameters; x, the angulardeviation of each individual linesegment from the horizontal; z, theangle subtended between linesegments; and s, the orientation of thebisector of angle z, that is, s = x - 1hzfor the counterclockwise line of a pairand s = x + 1hz for the correspondingclockwise line.

Term b , sin4x accounts forperceptual tilt (assimilation) of eachsingle line toward the more closelyaligned visual axis, horizontal orvertical. It has been used as anapproximation for single-line tilt errorsbefore (Bouma & Andriessen, 1968;Virsu, 1971).

The approximation term for theelevator factor is b 2 s. The upwardbias, determined algebraically inmillimeters, was converted to anequivalent tilt error in degrees for eachline segment of each subtended anglein each orientation. The formula forcon version to degrees is amathematically messy trigonometricfunction of the upward millimeterbias. Whenever a line segment ispositioned vertically (x =±90 deg),even a large elevator factor, because italso operates vertically, causes no tilterror. Or, when the bisector (s) ofthesubtended angle is horizontal, theelevator factor was near zero regardlessof the inclination (x) of a linesegment, which depends upon theparticular subtended angle (z). Asstated previously, the upward bias inmillimeters increased from zero as the

orientation of the subtended angledeviated from horizontal (i.e., froms =0), and a simple but effectiveapproximation was a linear b 2 s termwith b , negative for anycounterclockwise line and positive forany clockwise line of a pair.1 Thefunctions in Fig. 2 show the slopetrends superimposed upon othersystematic variations.

Imagine that suitable values forcoefficients b I and b 2 are applied tothe first two terms of the model, thatthese components are then subtractedfrom the six curves of Fig. 2, and that,for lines forming a subtended angle ina given orientation, the remainingcomponents of judgmental error aresummed for each pair and plotted. Theoutcome can be considered as removalof the tilt-assimilation factor and theelevator factor from theangle-misestimation data of Fig. 1. Aneat, easily interpretable "corrected"Fig. 1 would consist of threehorizontal lines, meaning that theremaining error is independent of theorientation of the subtended angle.The distance of each horizontalfunction from a mean tilt error of zerowould achieve the goal, anglemisestimation as a function ofsubtended angle. However, the actualoutcome was a nested set of functionsopening upward, roughly parabolic inshape, symmetrical about s = o.Figure 3 illustrates such correcteddata; the b, and b 2 coefficients werederived, in this instance, from theleast-squares best-fit solution for theentire four-term regression model in itsfinal formulation, as written above.The remaining error had an additionalimportant property: It tended to bedistributed equally between each linesegment of a pair. Therefore, thefour-term model, which is intended toaccount for the tilt error of anindividual line segment. was designedto predict half the remaining error.

Term bz encompasses the residualcomponent of angle rnisperceptionthat is independent of the orientationof the angle. Without assuming anyfunction to explain the differencesamong angles. bz muat be estimatedseparately for each angle z.

The other residual component varieswith the orientation of the angle sothat the differences between the twoerror functions for the pair of linesegments forming an angle becomelarger when the bisector of the angleapproaches a vertical position. i.e.•there is more overestimation (or lessunderestimation) of the angle as itsbisector approaches vertical. Thetendency is stronger. the larger thesubtended angle z. and anapproximation for this errorcomponent is directly proportional toangle z and to the square of thedeviation of the bisector from the

horizontal. The expression b. ZS3 is asimplification. but it is adequate forthe range of subtended angles in thepresent experiment.

A least-squares solution of theparameter values of the regressionmodel was obtained through a series ofsuccessive linear regression solutions.In all, six estimates are required, onevalue each for b l • b 2 • b •• and a bz foreach angle z. All the mean data wereutilized in the estimation of everyparameter. The resulting parametervalues were b , == -.8223,b2 = +.006321, ba = ±.000002622,b 2 0 deg = +.2911, b6 0 deg = ±.1658,and b1 2 0 def = ±.6829. The uppersi gns re er to the morecountercrockwise nne and the lowersigns to the more clockwise line ofeach pair. The standard error of theestimate was .5159. The smoothdotted curves in Fig. 2 show the fit ofthe regression model to the results.The fit is reasonable (multiplecorrelation .945), and 89% of thevariance of the mean errors isexplained by the regression model.Each of the four components yielded astatistically highly significantcontribution to the explanation of thetotal variance.

Each term of the regression modelcan be employed alone by dropping allother terms and using a least-squaresbest-fit criterion to determine theremaining coefficient. Thetilt-assimilation term containing b I

explains 59% of the variance of themean tilt errors when it is the soleterm, the elevator factor (with b 2 )

34%, the orientation-dependentcomponent of angle misperception(b. ) 66%. The bz term. actually threevalues, one for each subtended angle.nonetheless accounts for only 32% ofthe variance. Since the largestproportion of variance is explained bythe orientation-dependent componentof angle misperception, it ismeaningless to refer to themisperception of an angle withoutreference to the orientation of theangle in the visual field. even afterremoving single-line effects.

The data of Fig. 1 stand as empiricalangle misperception. However, thefour-factor model implies that two ofthe components leading to errors inmisperceiving a vertex. tilt ofindividual line segments toward acardinal visual axis, and the elevatorfactor. are independent of thesubtended angle itself. Therefore. byemploying only the last two terms ofthe model. which are dependent uponthe angle subtended, a smoothedprediction of "uncontaminated" anglemisestimation can be obtained. Thepoint at which each parabolic functionhas its minimum occurs when theangle bisector is horizontal (s = 0). andthe error at the minimum is twice the

bz term for any angle z. The smoothfunctions of Fig. 3. then. represent abeat prediction of purified anglemisperception for subtended angles of20. 60. and 120 deg for angles formedby nonintersecting lines when thedefining operation is to determine theapparent vertex. The empirical data ofFig. 3 correspond to residual tilt errorsafter removing the componentscorresponding to the first two terms inthe model, and these data contain allthe nonsystematic (random) error aswell as systematic error introduced bythe inappropriateness of the model.The first two terms of the model dooperate as advertised. accounting forcyclic variations and sloping functions.

DISCUSSIONThe model evolved logically. The b,

and bz terms were planned; the b 2term was added after corollaryevidence was gathered; the b.parabolic term was then selected toapproximate the remaining systematicvariation. Among the infinity ofalternative descriptions. many arereasonable. For examples. greater tiltassimilation may obtain toward one ofthe visual axes than toward the other.or size of subtended angle may beassumed to interact with tiltassimilation. We were unable to devisea formulation that captured theessence of the tilt-assimilation andelevator factors such that only oneadditional factor, which could bemade to be independent of angleorien t at ion , ascribable to anglemisperception, was necessary. Thestumbling block for an economicalmathematical description is that thelogically different factors are not easilyspecified in the same units or by thesame parameters. The regression modelserves as a useful analytic descriptionof the results. Our hope isthat the first two terms representphenomena that exist beyond theconfines of the vertex-locationparadigm. The model has generality atleast beyond the present experiment.In the earlier study (Weintraub &Virsu. 1971). the subtended angle wasvaried systematically. In oneexperiment. the bisector of the angleremained vertical and in the other theleft line segment was held at verticalwhile the angle varied over a largerange of values. Dropping theunknown component b z from themodel. the model explains 71.6% ofthe variance of mean errors of thoseexperiments. using only the threevalues b l • 'b 2 • and b. retained fromthe present experiment. The resultimplies good stability of the parametervalues over a large variation in displayorientations and angles between theline segments.

There are two approaches to thestudy of angle misestimation. One is

Perception & Psychophysics, 1972, Vol. 11 (4) 281

indirect, like using judgments of theapparent vertex to infer that anglemisestimation has occurred. Often, themotive for this type of study is toexplain a visual illusion-thePoggendorff serves as a primeexample-or to support hypothesesabout interactions among the neuralrepresentations of contours.

Corrigan (1970) conducted similarexperiments with a dot-settingtechnique for finding the vertex ofconverging line segments. Themovement of the dot was restricted tothe imaginary bisector of thesubtended angle or its extension sothat no lateral errors were permitted.Since Os make very few lateral errorswhen the angle bisector is orientedvertically, Corrigan's one-dimensionalvertical settings should provide dataequivalent to two-dimensional settings,and they do. Figure 3A of Weintrauband Virsu (1971), which is plotted inmillimeters, serves as a tolerablesubstitute for Corrigan's own graph.Corrigan concluded from displays inthree orientations that angles generallytend to be overestimated, and thatthere is tilt assimilation of lines towardthe vertical axis only. We feelconfident that his conclusions wouldhave more closely matched our own iftwo-dimensional judgments had beenpermitted and/or more orientationshad been assayed. A problem mustarise for the 0 when no setting canquite match his subjective vertex.(Does he minimize the discrepancy?)Information about the subjectivevertex is undoubtedly lost.

Bouma and Andriessen (1970) havealso employed a dot-setting procedure.However, they empirically correctedtheir data on two lines (subtendedan gIe) for single-line effects bysubtracting single-line "control" datafrom the two-line data. The validity ofthe subtraction procedure rests uponthe implicit assumption that single-lineeffects are additive and involve noin t era c tive components. Thefour-factor model lends support to thisassumption. Bouma and Andriessen(1968) have found tilt assimilationtoward both horizontal and verticalfor single lines.

The second approach treats anglemisperception as a performanceproblem to be confronted directly byrequiring Os to produce an angle thatmatches the standard, or to make averbal estimate in degrees. Two recentstudies used the production method(Beery, 1968; MacLean & Stacey,1971), but the comparison matchedthe standard in orientation. Thefour-factor model predicts zerojudgmental error because all effects areacting equally on each display. That is,in a null-match experiment, whereboth standard and comparison arevisible simultaneously and oriented

similarly, they should be perceived asidentical. Location in the visual field,attentional differences, etc., may beoperating, for which the mOder wasnot designed. For predictions of otherthan zero from the model, it isnecessary for the standard andcomparison angles to be presented indifferent orientations. Fisher (1969)and MacLean and Stacey (1971) alsoemployed verbal estimation, aboutwhich the model might makedefensible predictions. The startlingfinding .is that either direct methodyields results diametrically opposed tothose of the vertex-location technique.MacLean and Stacey confirm Fisher inconcluding that an angle whosebisector is oriented vertically(s = ±90 deg) is judged as smaller thanthe same angle oriented horizontally(s = 0 deg); Fig.1 shows just theopposite. Furthermore, theWeintraub-Virsu (1971, Fig. 3A) datafor displays oriented vertically(s = 90 deg) show thatunderestimation occurs for acuteangles less than 60 deg; all other anglesare overestimated. TheMacLean-Stacey (1971, Fig. 3) datashow overestimation for the smalleracute angles and underestimationelsewhere. Beery (1968) concludedthat, rather than tilt assimilation, thereis tilt repulsion away from horizontaland vertical,

Besides the nature of O's task, thereare other methodological differences.The new data are based upon displayswith nonintersecting lines forming theangle. (There is reason to believe thatintersecting lines will not produceequivalent perceptual outcomes. SeeWeintraub and Virsu, 1971, p, 8.)Also, the distances between the linesegments were large, up to 12.7 deg ofvisual angle. Therefore, the interactioneffects between the line segments arenot easily explained on the basis ofinhibitory interactions betweencontour or orientation detectors (seeGanz, 1966; Blakemore, Carpenter, &Georgeson, 1970; Bouma &Andriessen, 1970) because stronginhibitory neural influences cannot beexpected over such large distances.Some other features of the results,such as the overestimation of the120-deg angles in all orientations, arealso difficult to understand asinhibitory interactions.

In summary, as the joint result ofthe earlier and present experiments, itis possible to conclude that, when thevertex of two converging, relativelylong line segments was estimated, thejudgments were consistent with thefollowing generalizations:

(1) A line segment appearsperceptually tilted toward the moreclosely aligned axis either thehorizontal or vertical of the visualfield.

(2) There is a tendency for 0 toestimate the apparent vertex too highin his visual field.

(3) An interaction effect betweenline segments exists suggestingmisperception of the angle subtended.The interaction depends strongly uponthe orientation of the angle.

(4) The overall outcome (Fig. 1) isthat 20 deg is underestimated in mostorientations, 60 deg is usuallyoverestimated, 120 deg is alwaysoverestimated.

The tWO-dimensionalvertex-location technique producesreliable data, possesses methodologicaladvantages, has face validity as ameans of assessing anglemisperception. The fact that itproduces discrepant data raisesquestions concerning all techniques.An encouraging facet of the evidenceis that, although disparate methodsstrongly disagree as to the direction oferrors, they tend to implicate the sameexplanatory variables. Since wealready hold a strong bias that anglemisperception is multiply caused,additional causative agents linked todifferent procedures would be not atall surprising.

In any· event, a four-termmathematical statement presents aspecific target and an open invitationto all parties to commence firing.Given the bumps and grinds in thedata, four factors may prove highlyconstraining when confronted withnew evidence. In our presentoptimism, we believe that, when thesmoke clears, the model will be neitherbadly mutilated nor mortallywounded. Whatever the outcome, itlooks like an exciting shoot-out.

REFERENCESBEERY. K. E. Estimation of angles.

Perceptual & Motor Skills. 1968. 26.11-14.

BLAKEMORE. C•• CARPENTER. R. H. S .•& GEORGESON. M. A. Lateral inhibitionbetween orientation detectors in thehuman visual system. Nature, 1970. 228.37-39.

BOUMA. H., & ANDRIESSEN, J. J.Perceived orientation of isolated linesegments. Vision Research, 1968, 8.493-507.

BOUMA, H •• & ANDRIESSEN, J. J.Induced changes in the perceivedorientation of line segments. VisionResearch, 1970, 10. 333-349.

BURNS. B. D.. & PRITCHARD. R.Geometrical illusions and the response ofneurones in the cat's visual cortex toangle patterns. Journal of Physiology.1971, 213. 599-616.

CHIANG. C. A new theory to explaingeometrical illusions produced bycrossing lines. Perception &Psychophysics, 1968. 3. 174-176.

CORRIGAN. B. Constant error in theperceptual extrapolation of straight lines.Paper presented at the meeting of theWestern Psychological Association. LosAngeles, April 1970.

FISHER. G. H. An experimental study ofangular subtension. Quarterly Journal ofExperimental Psychology, 1969, 21.356-366.

GANZ. L. Mechanism of the figural

282 Perception &; Psychophysics, 1972, Vol. 11 (4)

aftereffect. Psychological Review. 1966.73. 128-150.

MacLEAN, I. E., & STACEY, B. G.Judgment of angle size. Perception &Psychophysics, 1971. 9, 499-504.

OBONAI. T. ExperimentelleUntersuchungen uber den Aufbau desSehraumes. Archiv der gesamtenPsychologie. 1931, 82. 308-328.

VIRSU. V. Tendencies to eye movement.and misperception of curvature.direction. and length. Perception &Psychophysics. 1971. 9. 65·72.

WEINTRAUB. D. J •• & VIRSU. V. The

misperception of angles: Estimating thevertex of converging line segments.Perception & Psychopkysics, 1971. 9.5-8.

WEINTRAUB. D. J .• & KRANTZ. D. H.The Poggendorff illusion: Amputations.rotations, and other perturbations.Perception & Psychophysics. 1971. 10.257-264.

NOTE1. Roughly speaking. as the orientation of

the bisector proceeds from -90 through 0to +90 deg for the counterclockwise line of

a pair, the tilt error component resultingfrom the elevator factor proceeds frompositive (counterclockwise error) throughzero to negative (clockwise error). Theclockwise line. conversely. has a positiveslope. (The reader who sketched out theshape of these functions would find a humpin each that is ignored by the linearapproxtmatton.)

(Received for publication November 8,1971.)

Perception & Psychophysics. 1972. Vol. 11 (4) Copyright 1972, Psychonomic Society, Austin. Texas 283