Perception, 1979, volume 8, pages 565-575
Angle-matching illusions and perceived line orientation
Peter Wenderoth, Dennis White Department of Psychology, University of Sydney, Sydney, Australia 2006 Received 23 June 1978, in revised form 12 April 1979
Abstract. Spatial illusions which occur in angle-matching tasks were examined in six experiments, using two different kinds of display. In experiments 1 and 2, illusory errors generally were in the direction predicted by Lennie's hypothesis which states that angle arms are attracted perceptually towards the oblique axes of space, although the display used in these experiments differed from Lennie's. However, experiment 3 showed that these errors might equally be explained by the addition of interactive effects between angle arms (tilt illusions). Parametric investigation of Lennie's figures (experiments 4 and 5) showed that the largest angular illusion occurred with the largest angle used (45°) and an intermediate line length (2 deg 7 min). These angular illusions were not explicable by the addition of tilt illusions (experiment 6), suggesting that different judgmental processes may underlie orientation and angle estimations.
1 Introduction Many investigators have attempted to measure the perceived absolute orientation of line segments. The techniques used have varied from the most direct, such as absolute angle or line tilt naming, in degrees (e.g. Fisher 1969; 1974), to extremely indirect, such as matching pairs of angles at different orientations (e.g. Lennie 1971). In the latter case, the validity of inferences concerning the absolute perceived positions of the individual angle arms in the display rests upon the assumption that interactive effects between the lines, effects such as tilt illusions, are either negligible or cancel. Only then can it be assumed that angle-matching errors reflect the perceived absolute orientations of the individual line segments. The broad rationale of this study is as follows. Fisher's (1974) absolute-naming technique is suspect as a method of orientation measurement because it is impossible to ascertain the degree to which errors merely reflect the use of verbal labels. By analogy, in distance-estimation tasks, observers typically give verbal estimates (in metres) which underestimate the true distance by 30% or so. However, this does not imply that the distance is perceived as less than it is and, hence, researchers in this area seek to use calibration equations to correct for response errors, often with little success (see Mershon et al 1977). As another example, if the orientation-naming method were adequate, then a single line tilted 10° (horizontal: 0°) counterclockwise ought to be named as tilted more when a horizontal line is added to the display, because the two lines (angle arms) should repel each other, as in the usual tilt illusion. However, in unpublished experiments, the first author found the opposite: the addition of the horizontal acted as an anchor so that the 10° tilt was named as less when the anchor was provided. Clearly, many factors other than perceived line orientation enter into absolute-naming tasks. Other techniques also suffer from difficulties of interpretation. For example, dotto-line alignment has been used to index perceived line orientation (e.g. Bouma and Andriessen 1968, 1970; Matin 1974) but it is doubtful whether this method reflects a pure perceived line-tilt effect (see Lennie 1971, 1972; Emerson et al 1975; Wenderoth et al 1978a, 1978b). Consequently, given this plethora of doubtful methods of measuring perceived orientation; it seemed appropriate to examine, in more detail, the angle-matching method since this conceivably could provide adequate
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indirect measures of perceived orientation and it is also an extremely simple task to perform. 2 Experiments 2.1 Experiment 1 Using a line-dot alignment task, Bouma and Andriessen (1968) obtained data which they interpreted to indicate that single line segments appear closer to the nearest major spatial axis (vertical or horizontal) than they really are. This suggests that orientation-selective 'channels' in the visual system might be either more numerous around the major axes or more broadly tuned, compared to channels selective to oblique stimuli. Lennie (1971), on the other hand, pointed to evidence that it is the oblique channels which are less selectively tuned, so that single lines between vertical (or horizontal) and the oblique meridian ought to be seen closer to the oblique axis. Lennie used an angle-matching task to test his hypothesis and obtained clear evidence in support of it. The display which he used comprised one angle between 20° and 40° flanking vertical or horizontal and a second angle which varied in orientation, with the two angles joined at their vertices. Some experiments reported below used this display but in experiment 1 a simpler display was used. This consisted of three line segments: one line was on a major axis (vertical or horizontal); one line was at the oblique, thus creating a 45° angle with the first line; and the third line bisected the angle formed by the others (see figure 1). Initially, it seemed reasonable to assume that interactive effects (tilt illusions) in this display would cancel: since angles AOB and COB are equal, lines OA and OB should have about the same total effect on line OC as lines OC and OB have on line OA; and lines OA and OC should exert roughly equal but opposite effects on line OB. At any rate, these interactive effects were expected to be small because when three lines lie within 45° of each other, disinhibitory effects should reduce orientation illusions (Carpenter and Blakemore 1973). On this reasoning, the major influence on the lines should derive from differential channel selectivity and, from Lennie's hypothesis, line OB should appear nearer the oblique but lines OA and OC should be seen veridically. Hence, angle COB should appear smaller than angle AOB. Observers asked to equate the angles by setting line OB's orientation should therefore set line OB too close to vertical (or horizontal).
o Figure 1. Display used in experiment 1 (see text). 2.1.1 Method. Stimuli were displayed on the flat face of a Tektronix 604 display screen (P4 phosphor) by an Alpha 16 minicomputer, interfaced with a PDP-11/20 computer. The luminance of the lines was about 10 cd m"2 (SEI photometer) and contrast was close to 1-0.
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Observers were positioned in a dental cement bite bar 54 cm from the display so that each of the 20 mm x 0-5 mm lines subtended 2 deg 7 min x 31 min at the eye in an otherwise dark cubicle. (To avoid confusion, the degree symbol '°' is used to refer to degrees of orientation whereas 'deg' refers to distances subtended at the eye.) A pair of response buttons enabled the observer to indicate whether line OB should move clockwise (CW) or counterclockwise (CCW) in order to equate angles AOB and COB. The display terminated for 1 s after each response, and testing continued with a double randomly interleaved staircase technique, until four reversals occurred with a 1° step size. Step size was then halved and testing was terminated when eight additional reversals had occurred. The point of subjective equality (PSE) was taken as the mean of these last eight reversals (see Wenderoth et al 1978a, 1978b). Subjects judged the display in each of four orientations, so that line OB's orientation initially was either 22-5°, 67-5°, 112-5°, or 157-5°. There were eleven subjects, including the first author, four graduate students, and six volunteers from an introductory course. 2.1.2 Results. The predictions from Lennie's hypothesis were that line OB should be set CW (-) when initially oriented 22-5°, CCW (+) when initially 67-5°, CW (-) from 112-5°, and CCW (+) from 157-5°. These predictions, and the results, are shown in table 1. Although the directions of errors were as predicted in three of the four conditions, the errors were small and only one, when line OB was oriented 22-5°, was significantly different from zero, with tl0 = 5-00; p < 0 001. However, postexperimental questioning of observers revealed that all but two had used the strategy of comparing the distances across the line endpoints (i.e. distance AB versus distance BC in figure 1). Lennie (1971) noted that his observers confined their free inspections to the centre of his display, to the angle vertices, and he also noted elsewhere (Lennie 1972) that when the angles were perceptually equated in this way, they nevertheless appeared unequal when the endpoint distances were compared. Table 1. Means (X) and standard errors (in brackets) of errors in settings of line OB, experiment 1. Initial orientation of line OB 22-5° X -0-85° (0-17°) Predicted
67-5° -0-15° (0-19°)
+
112-5° -0-40° (0-22°)
157-5° + 0-33° (0-26°) 4-
2.2 Experiment 2 Experiment 2 was identical to the first experiment except that eight volunteer subjects from an introductory course completed the task under two instruction conditions. These were endpoint (i.e. equate the distances AB and BC) and central (i.e. confine attention to the angle vertices and equate the 'spaces' between the angle arms). Half of the observers completed the central condition first (although order in which the two conditions were run made no obvious difference to the results). 2.2.1 Results. The results are shown in table 2. The errors under endpoint instructions were very similar to those of experiment 1, as expected, and again only that which occurred when line OB was at 22-5° was significant (tn = 3-65; p < 0 0 1 ) . Under central instructions, however, all errors were in the predicted direction and three of the four were significant, those for initial line OB orientations of 22-5° (f7 = 3-74), 112-5° (t7= 3-79), and 157-5° (r7 = 5-04), with p < 0 - 0 1 in every case.
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These data seemed to provide additional evidence consistent with Lennie's (1971) hypothesis regarding oblique-axis attraction and to confirm his view that endpoint judgments differ from central judgments. However, to provide a partial check on the absence of significant interactive effects between the lines, we conducted experiment 3. Table 2. Means and standard errors (in brackets) of errors in settings of line OB, experiment 2. Instruction
Central Endpoint
Initial orientation of line OB 22-5° 67-5° -1-16° (0-31°) -0-84° (0-23°)
112-5°
+ 0-03° (0-24°) -0-32° (0-23°)
157-5°
-1-06° (0-28°) -0-09° (0-18°)
+ 1-41° (0-28°) + 0-71° (0-33°)
2.3 Experiment 3 The basic displays in experiment 3 were the same as those used previously except that line OB was fixed at 22-5°, 67-5°, 112-5°, or 157-5°. When line OB was at one of these four orientations, the observer made two sets of parallel matches, by means of a fourth line alongside either line OA or line OC and 1 deg from it, a distance which is sufficient to exclude interactive effects of the other lines on the variable line (Carpenter and Blakemore 1973). The task was to judge whether the variable line appeared tilted CW or CCW relative to line OA or line OC and the usual staircase procedure was employed. When matches were made to line OA, the task was completed both when line OA only was present (pretest) and again when lines OA, OB, and OC were all present (posttest). As was discussed above, it was expected that both line OA and line OC would be apparently displaced in the direction describable as angle expansion, but that these illusions would be small and roughly equal for the two arms (1) . If this were the case, it could be inferred that interactive effects were unlikely to account for the results of experiments 1 and 2. Nine volunteer observers completed the experiment, including the eight observers used in experiment 2. 2.3.1 Results. The results are shown in table 3. In every case, the matching error was in the direction indicative of angle expansion but, in addition, the oblique arm always was affected more than the arm aligned with vertical and horizontal. In fact, this pattern of results is consistent with data of Carpenter and Blakemore (1973), who concluded that lateral inhibition in the orientation domain is propagated more extensively from vertical or horizontal channels than it is from more oblique channels. Moreover, the result obtained in experiments 1 and 2 can now be explained not only in terms of Lennie's hypothesis but alternatively in terms of interactive effects. If, in Table 3. Means (X) and standard errors (s) of parallel-matching errors, posttests minus pretests, experiment 3. Positive (+) errors indicate effects in the direction of angle expansion. Orientations of standard line are horizontal (H), vertical (V), or oblique (O). Orientation of line OB 67-5 22-5° H V O X s (1)
+0-61° 0-19°
+ 1-60° 0-38°
+ 1-28° 0-19°
112-5° O + 1-40° 0-44°
V + 1-34° 0-23°
157-5° O +2-01° 0-35°
H +0-98° 0-27°
O +2-29° 0-49°
It should be noted that the parallel-matching technique provides a pure measure of interactive effects. If single-line effects such as oblique-axis attraction do occur, then they would occur equally with the standard line (OA or OC) and with the variable, matching line and so would be factored out by this procedure.
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figure 1, line OA exerts a stronger influence than line OC, then both line OC and line OB will be shifted perceptually CW. On the other hand, line OA will be shifted CCW, resulting in angle COB's appearing too small, as if line OB were attracted perceptually to the oblique. Although this result led us to discontinue experiments with the display shown in figure 1, it became clear that there were a number of possible explanations of Lennie's original result apart from oblique attraction and one of these is angle expansion. The remainder of the experiments used Lennie-type displays. 2.4 Experiment 4 An example of the kind of display used by Lennie (1971) is shown in figure 2; in this case, angles AOB and COD are equal to 45° so that all angle arms are 22*5° from the vertical, horizontal, and oblique axes. Clearly, angle COD appears smaller than angle AOB (Lennie's effect) and the illusion seems to be tied to retinal rather than gravitational coordinates, since tilting the head 45° reverses the direction of the apparent inequality. Experiments 4 and 5 were designed to examine some of the parametric attributes of the angle illusion, specifically the effects of angle size and line length. Lennie (1971) merely stated that errors were qualitatively the same for angles of 20°, 30°, and 40°; and that similar results occur with lines 3 deg and 1 deg long. The latter statement is relevant to the possible occurrence of interactive effects since, elsewhere, Lennie (1972) suggested that tilt illusions are minimal between lines 3 deg long, so that such effects are unlikely to be involved in the illusion. In experiment 4, the display was as in figure 2 (45° angles) but the arm lengths were either 5 mm (31-5 min), 20 mm (2 deg 7 min), or 45 mm (4 deg 46 min).
Figure 2. Display similar to that used by Lennie (1971) with Z_AOB = Z.COD = 45°. 2.4.1 Method. The display techniques, and general procedures were those described previously. Pilot studies indicated that similar results were obtained, as expected, whichever of the four angle arms the observer adjusted. In this experiment, observers completed two sets of judgments, one in which arm OB was the variable and one set for arm OC. These were selected because the response buttons could then be labelled "left angle too small" and "right angle too small" for the left and right buttons, respectively, thus avoiding the more complex judgments of CW and CCW appearances. The two PSEs so obtained were averaged. Lennie (1972) merely noted that 'control' and cendpoint' judgments of the angles differed in his study so to measure the effect, our observers completed the task under both instruction sets. 2.4.1.1 Subjects. Sixty subjects, volunteers from introductory courses, acted as observers, twenty under each line-length condition. Within each group, half of the
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observers completed the 'central' instruction condition first and half of these subjects produced four such PSEs with the variable lines in the order OB, OC, OC, OB; the other half had the order OC, OB, OB, OC. There was no evidence that order had any effect, so the results of the subgroups were averaged. 2.4.2 Results. Table 4 presents the results, and three features of the data are noteworthy. First, all errors were different from zero (p < .0-0.1 in every case) except those with the 0-53 deg line-length. Second, as was the case in experiment 2 with a different display, endpoint judgments exhibited errors in the same direction as central judgments, but the latter effects were always larger. This was true of all twenty subjects in the 2-12 deg line-length group and of fourteen out of twenty in the 4-77 deg line length group. Analysis of variance (Winer 1962, p 306) indicated that mean central errors (+ 3-08°) exceeded mean endpoint errors (+ 1 -58°) with ^ 5 7 = 20-89 and p < 0-001. Third, the effect of line length appears to be nonmonotonic: the largest effects were obtained with the 2-12 deg lines. Analysis of variance showed that the line-length effect was significant CF2}57 = 19-31; p < 0 • 001) and that the length-by-instruction interaction was also significant {F = 4 -08; p < 0-05). The latter result reflects the larger difference between central and endpoint errors at the intermediate line length. However, even if sampling error somehow spuriously reduced the longest line effects, the effect of line length is opposite to that usually found with tilt illusions, where interactive effects decrease as lines increase in length or are separated by greater distances (e.g. Wallace 1969; Robinson 1972; Weale 1978). Table 4. Means and standard errors (in brackets) of errors in setting line OB or line OC to equate 45° angles AOB and COD, experiment 4. (Positive errors indicate angle COD set too large or angle AOB set too small.) Instruction
Angle arm length 0-53°
Central Endpoint
+0-17° (0-33°) -0-43° (0-39°)
2-12° + 3-74° (0-40°) + 1-60° (0-26°)
4-77° 4-2-24° (0-51°) + 1-57° (0-31°)
2.5 Experiment 5 Using the 2 - 1 2 deg line length, which produced largest errors in experiment 4 , we next examined t h e effect of angle size. Thus, t h e display was again as in figure 2, except standard angles AOB and COD were either 10°, 2 0 ° , or 4 5 ° . (The 4 5 ° group was n o t r u n again: t h e data were taken from experiment 4.) T h e m e t h o d s were identical t o those of experiment 4, with twenty observers in each angle group. 2.5.1 Results. All mean errors were significant (p < 0 - 0 1 , at least, in each case) and table 5 shows that errors increased with increasing angles. Again, this result appears not t o implicate interactive effects between t h e lines: tilt illusions are larger at 10° than at 20° than at 4 5 ° . Analysis of variance showed that t h e effects of angle and Table 5. Means and standard errors (in brackets) of errors in setting line OB or line OC to equate angles AOB and COD, experiment 5. (Positive errors indicate < COD set too large or 57 = '5 • 23; p < 0-01), reflecting the fact that the difference between central and endpoint errors increased with angle size. In the next experiment we attempted to measure tilt illusions which might occur in the 10° and 45° displays. 2.6 Experiment 6 Lennie dismissed tilt illusions as a possible basis of the angle illusion by arguing that where the angle effect is largest (i.e. with one angle flanking horizontal or vertical and the other flanking an oblique) "all four arms are at the same orientation with respect to the vertical and horizontal, and interactive effects should therefore cancel" (Lennie 1971, p 156). However, although interactive effects may not be responsible for the effect, this logic is unsound. Large tilt illusions (angle expansions) do occur with 10° angles and smaller, but significant effects occur with 45° angles (e.g. Carpenter and Blakemore 1973). It is conceivable then that, in figure 2, both angles are expanded but that angle AOB expands more than angle COD. A possible basis for this might be that expansion effects (i.e. inhibitory effects) are greater between lines flanking the major axes than between lines flanking the oblique. In fact, there are many ways in which the angle illusion could arise because the net effect is simply that angle COD appears smaller than angle AOB and this implies nothing about the absolute expansion or contraction of either angle. Figure 3 represents ten possible ways in which the angle illusion might occur (i.e. filled cells). Here, - represents contraction, + represents expansion, 0 represents no effect, and ++ or indicate large expansions or contractions. Lennie's explanation of the angle illusion applies only to four of the ten filled cells, those top right cells for which angle AOB is either 4or ++ and for which angle COD is either - or — . The counterexample given above is shown by the lowest filled cell, for which angle AOB is + + and angle COD is +. To test for the occurrence of tilt illusions in the angle displays, we employed a task which is schematically represented in figure 4. Observers completed the same set of judgments for each of the four angle arms but for exemplary purposes consider arm OB in figure 4. First, line OB was presented alone (i.e. lines OA, OC, OD absent) and the only other line present was B\ Using the usual staircase methods, a PSE was obtained such that line B' appeared parallel to line OB (pretest). Next, the complete angle display was presented and the PSE for line B' parallel to line OB was obtained once more (posttest). The posttest minus pretest difference was taken to measure the effect of the other lines on line OB. As noted earlier, this was done Z.AOB
_ •
Q
O O
0
+
0
++
•
•
•
•
©
•
•
•
-J
• + +
Figure 3. Ten different ways to account for the fact the Z_COD appears smaller than LAOB, in figure 2 (see text).
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for all four lines although only one other variable line (D') is shown in the figure (see footnote 1). The dimensions of the display were as scaled in figure 4: angle arms were all 20 mm in length; the variable lines (e.g. line B') were 15 mm long and placed such that their inner ends were 1 mm from the vertex, O, when parallel to the standard lines, and 4 mm from the outer ends of the standard lines. The lateral separation between standard and variable lines was 9-4 mm (1 deg). Twenty-four subjects from the same population as the previous experiment each made matches to all four angle arms with both the 10° and the 45° angle displays. Another group of twelve subjects did the same but this time angle AOB flanked the lower, vertical axis (i.e. its bisector was oriented 270° rather than 0°). Finally, a group of ten subjects made angle matches, as in experiments 4 and 5, but with the standard angles of 10° and 45° flanking the lower vertical (270°).
Figure 4. Display used in experiment 6 (see text). 2.6.1 Results. The parallel-matching results are shown in figure 5, from which it is clear that the expected tilt illusions did occur, with relatively large (l°-2°) expansion effects in the 10° display for all four angle arms and small or negligible effects in the 45° display. Presumably, then, similar effects would have occurred in Lennie's experiments, especially with his 20° and 30° angles. However, it is clear that although tilt effects occurred, they cannot account for the angle illusion. Figure 6 shows both the angle-matching errors from experiment 5 (open circles) and also the summed angle expansions (open squares). By way of explanation,
c
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mi
A
B
i
*4| I
i
i
i_
10°
45°
10°
45°
Angle size
Figure 5. Tilt illusions measured in experiment 6. Angle arms are A (open circles), B (filled circles), C (open squares), and D (filled squares), and inserts show the displayed angles. Left panel: N = 24, right panel: N = 12; vertical bars: 1 standard error.
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consider the 10° tilt illusion results in the left panel of figure 5. There, the illusions for arms A, B, C, and D are, respectively, + 0 - 9 9 ° , + 1 -25°, + 1 -47°, and 4-1 -85°. Hence, total expansion of the arms of angle AOB is 0-99° + 1 -25°, or 2 - 2 4 ° . Expansion of angle COD is 3-34°. The difference is - 1 • 10°, as shown for the 10° display in figure 6. So although the tilt-illusion results suggest that angle COD expands more, in the angle-matching task, angle COD is judged almost 2° smaller (figure 6). Similar results occurred when the standard angle was vertically positioned, as figure 6 shows. In this case, angle-matching errors (filled circles) resembled those which occurred when the standard angle was horizontal (although the latter errors were larger) but the summed tilt-illusion effects (filled squares) were negligible (2) . Finally, it might be supposed that the vertical-horizontal illusion is one component of the angle-matching illusion such that the vertical space between a horizontal angle's arms is judged longer than the horizontal space between a vertical angle's arms. This would account for the smaller angle-matching errors with a vertical standard angle (figure 6). However, other data (cf Lennie 1971, 1972) do not always exhibit this difference and, at any rate, if that were the case, large illusions would occur with one angle flanking vertical and the other flanking horizontal but this is not generally found (Lennie 1971, 1972).
Q
O
Angle size
Figure 6. Angle-matching errors obtained in experiments 5 and 6 with standard angle horizontal (open circles) and vertical (filled circles). Also shown are summed tilt illusions from experiment 6 for standard angle horizontal (open squares) and vertical (filled squares). For details, see text. 3 Discussion The results of the last experiment seem to imply one of two conclusions. One possibility is that tilt illusions do affect the perceived orientations of the angle arms in the angle illusion display but other factors, such as directionally opposite attraction of the arms towards the oblique axes, override the tilt effects and result in the judged inequalities initially observed by Lennie and confirmed in the experiments reported here. However, another possibility is that judgments of angles are partly or ^ Subjects who completed parallel matches to the angle arms when the standard angle was oriented 270° were asked, at the end of the experiment, which angle appeared smaller, just to check that the usual angle illusion occurred with subjects who performed the parallel matches. Of the twelve subjects in the 10° angle group, seven judged angle COD smaller, four judged the angles equal, and one said angle AOB was smaller; in the 45° group the corresponding figures were ten, two, and zero.
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wholly independent of the perceived arm orientations. That is, the directionally opposite effects reported here might be likened to effects such as 'Morinaga's displacement paradox' (cf Robinson 1972, pp 25-26). In that effect, the Brentano version of the Muller-Lyer illusion (i.e. no shafts) is arranged so that the outward-going fin component is directly above or below the inward-going fin component. Simultaneously, the space between the fins appears shorter in the latter component, yet the endpoints of its fins appear to lie outside those of the outwardgoing fin component. The observation is said to be paradoxical because, geometrically, it cannot be true that one spatial extent is both longer and shorter than another. By analogy, then, it is conceivable that in the angle display the arms of the angle are expanded in the orientation domain but, simultaneously, the spaces within the angles are differentially contracted, as a function of the angles' orientations. This line of reasoning finds some support in a recent report of Heywood and Chessell (1977), whose observers attempted to match the distance or space across angles by placing a dot on a page which marked off a distance perceptually equivalent to the criterion distance within the standard angle. The general tenor of the data was that the patterns of over- and underestimation frequently were quite different from those which have been reported when tasks requiring judgments of orientation were used (e.g. Bouma and Andriessen 1970; Blakemore et al 1970), although aspects of the results also confirmed some of the previous results. Heywood and Chessell's concluding remark, then, applies equally to the experiments reported here. In discussing differences between their results and those obtained by others using different methods, they observed that "whether the differences in our results are in some obscure way a reflection of this procedural difference, or whether the 'expansions' and 'contractions' of angles, however obtained, can be seen to reflect a process of spatial perception that is not straightforwardly related to Euclidean metrics remains to be discovered" (Heywood and Chessell 1977, p 581). Unfortunately, the precise relevance of the Heywood and Chessell data to those reported here is difficult to assess, if only because the angular illusion we have investigated depends critically on the relative orientations of the two angles; Heywood and Chessell imply (p 580) that their effects were independent of orientation. The above kinds of observations do not, unfortunately, constitute an explanation of apparently paradoxical effects. On the other hand, the experiments reported here, and the above discussion, do demonstrate that the angle-matching task does not necessarily have simple implications about the perceived absolute orientations of individual angle arms. Acknowledgements. This research was supported by the Australian Research Grants Committee (Grant A74/15177). Peripherals and software for the display facilities were developed by John Holden, Alan Parkinson, and Mike Cooper. We gratefully acknowledge Phil Greenwood, Margie Morgan, and Anne Keene for assistance in running the experiments. References Bouma H, Andriessen J J, 1968 "Perceived orientation of isolated line segments" Vision Research 8 493-507 Bouma H, Andriessen J J, 1970 "Induced changes in the perceived orientation of line segments" Vision Research 10 333-349 Blakemore C, Carpenter R H S, Georgeson M A, 1970 "Lateral inhibition between orientation detectors in the human visual system" Nature (London) 228 37-39 Carpenter R H S , Blakemore C B, 1973 "Interactions between orientations in human vision" Experimental Brain Research 18 287-303 Emerson P, Wenderoth P, Curthoys I, Edmonds I, 1975 "Measuring perceived orientation" Vision Research 15 1031-1033 Fisher G H, 1969 "An experimental study of angular subtension" Quarterly Journal of Experimental Psychology 21 356-366
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Fisher G H, 1974 "An experimental study of linear inclination" Quarterly Journal of Experimental Psychology 18 52-62 Heywood S, Chessell K, 1977 "Expanding angles? Systematic distortions of space involving angular figures" Perception 6 571-582 Lennie P, 1971 "Distortions of perceived orientation" Nature (London) New Biology 233 155-156 Lennie P, 1972 Mechanisms Underlying the Perception of Orientation PhD thesis, University of Cambridge, Cambridge, England Matin E, 1974 "Light adaptation and the dynamics of induced tilt" Vision Research 14 255-265 Mershon D H, Kennedy M, Falacara G, 1977 "On the use of 'calibration equations' in perception research" Perception 6 299-311 Robinson J 0,1972 The Psychology of Visual Illusion (London: Hutchinson) Wallace G K, 1969 "The critical distance of interaction in the Zollner illusion" Perception and Psychophysics 5 261-264 Weale R A, 1978 "Experiments on the Zollner and related optical illusions" Vision Research 18 203-208 Wenderoth P, Beh H, White D, 1978a "Perceptual distortion of an oblique line in the presence of an abutting vertical line" Vision Research 18 923-930 Wenderoth P, Beh H, White D, 1978b "Alignment errors to both ends of acute- and obtuse-angle arms" Perception and Psychophysics 23 475 -482 Winer B J, 1962 Statistical Principles in Experimental Design (New York: McGraw-Hill)
Angle-matching illusions and perceived line orientation
Peter Wenderoth, Dennis White Department of Psychology, University of Sydney, Sydney, Australia 2006 Received 23 June 1978, in revised form 12 April 1979
Abstract. Spatial illusions which occur in angle-matching tasks were examined in six experiments, using two different kinds of display. In experiments 1 and 2, illusory errors generally were in the direction predicted by Lennie's hypothesis which states that angle arms are attracted perceptually towards the oblique axes of space, although the display used in these experiments differed from Lennie's. However, experiment 3 showed that these errors might equally be explained by the addition of interactive effects between angle arms (tilt illusions). Parametric investigation of Lennie's figures (experiments 4 and 5) showed that the largest angular illusion occurred with the largest angle used (45°) and an intermediate line length (2 deg 7 min). These angular illusions were not explicable by the addition of tilt illusions (experiment 6), suggesting that different judgmental processes may underlie orientation and angle estimations.
1 Introduction Many investigators have attempted to measure the perceived absolute orientation of line segments. The techniques used have varied from the most direct, such as absolute angle or line tilt naming, in degrees (e.g. Fisher 1969; 1974), to extremely indirect, such as matching pairs of angles at different orientations (e.g. Lennie 1971). In the latter case, the validity of inferences concerning the absolute perceived positions of the individual angle arms in the display rests upon the assumption that interactive effects between the lines, effects such as tilt illusions, are either negligible or cancel. Only then can it be assumed that angle-matching errors reflect the perceived absolute orientations of the individual line segments. The broad rationale of this study is as follows. Fisher's (1974) absolute-naming technique is suspect as a method of orientation measurement because it is impossible to ascertain the degree to which errors merely reflect the use of verbal labels. By analogy, in distance-estimation tasks, observers typically give verbal estimates (in metres) which underestimate the true distance by 30% or so. However, this does not imply that the distance is perceived as less than it is and, hence, researchers in this area seek to use calibration equations to correct for response errors, often with little success (see Mershon et al 1977). As another example, if the orientation-naming method were adequate, then a single line tilted 10° (horizontal: 0°) counterclockwise ought to be named as tilted more when a horizontal line is added to the display, because the two lines (angle arms) should repel each other, as in the usual tilt illusion. However, in unpublished experiments, the first author found the opposite: the addition of the horizontal acted as an anchor so that the 10° tilt was named as less when the anchor was provided. Clearly, many factors other than perceived line orientation enter into absolute-naming tasks. Other techniques also suffer from difficulties of interpretation. For example, dotto-line alignment has been used to index perceived line orientation (e.g. Bouma and Andriessen 1968, 1970; Matin 1974) but it is doubtful whether this method reflects a pure perceived line-tilt effect (see Lennie 1971, 1972; Emerson et al 1975; Wenderoth et al 1978a, 1978b). Consequently, given this plethora of doubtful methods of measuring perceived orientation; it seemed appropriate to examine, in more detail, the angle-matching method since this conceivably could provide adequate
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indirect measures of perceived orientation and it is also an extremely simple task to perform. 2 Experiments 2.1 Experiment 1 Using a line-dot alignment task, Bouma and Andriessen (1968) obtained data which they interpreted to indicate that single line segments appear closer to the nearest major spatial axis (vertical or horizontal) than they really are. This suggests that orientation-selective 'channels' in the visual system might be either more numerous around the major axes or more broadly tuned, compared to channels selective to oblique stimuli. Lennie (1971), on the other hand, pointed to evidence that it is the oblique channels which are less selectively tuned, so that single lines between vertical (or horizontal) and the oblique meridian ought to be seen closer to the oblique axis. Lennie used an angle-matching task to test his hypothesis and obtained clear evidence in support of it. The display which he used comprised one angle between 20° and 40° flanking vertical or horizontal and a second angle which varied in orientation, with the two angles joined at their vertices. Some experiments reported below used this display but in experiment 1 a simpler display was used. This consisted of three line segments: one line was on a major axis (vertical or horizontal); one line was at the oblique, thus creating a 45° angle with the first line; and the third line bisected the angle formed by the others (see figure 1). Initially, it seemed reasonable to assume that interactive effects (tilt illusions) in this display would cancel: since angles AOB and COB are equal, lines OA and OB should have about the same total effect on line OC as lines OC and OB have on line OA; and lines OA and OC should exert roughly equal but opposite effects on line OB. At any rate, these interactive effects were expected to be small because when three lines lie within 45° of each other, disinhibitory effects should reduce orientation illusions (Carpenter and Blakemore 1973). On this reasoning, the major influence on the lines should derive from differential channel selectivity and, from Lennie's hypothesis, line OB should appear nearer the oblique but lines OA and OC should be seen veridically. Hence, angle COB should appear smaller than angle AOB. Observers asked to equate the angles by setting line OB's orientation should therefore set line OB too close to vertical (or horizontal).
o Figure 1. Display used in experiment 1 (see text). 2.1.1 Method. Stimuli were displayed on the flat face of a Tektronix 604 display screen (P4 phosphor) by an Alpha 16 minicomputer, interfaced with a PDP-11/20 computer. The luminance of the lines was about 10 cd m"2 (SEI photometer) and contrast was close to 1-0.
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Observers were positioned in a dental cement bite bar 54 cm from the display so that each of the 20 mm x 0-5 mm lines subtended 2 deg 7 min x 31 min at the eye in an otherwise dark cubicle. (To avoid confusion, the degree symbol '°' is used to refer to degrees of orientation whereas 'deg' refers to distances subtended at the eye.) A pair of response buttons enabled the observer to indicate whether line OB should move clockwise (CW) or counterclockwise (CCW) in order to equate angles AOB and COB. The display terminated for 1 s after each response, and testing continued with a double randomly interleaved staircase technique, until four reversals occurred with a 1° step size. Step size was then halved and testing was terminated when eight additional reversals had occurred. The point of subjective equality (PSE) was taken as the mean of these last eight reversals (see Wenderoth et al 1978a, 1978b). Subjects judged the display in each of four orientations, so that line OB's orientation initially was either 22-5°, 67-5°, 112-5°, or 157-5°. There were eleven subjects, including the first author, four graduate students, and six volunteers from an introductory course. 2.1.2 Results. The predictions from Lennie's hypothesis were that line OB should be set CW (-) when initially oriented 22-5°, CCW (+) when initially 67-5°, CW (-) from 112-5°, and CCW (+) from 157-5°. These predictions, and the results, are shown in table 1. Although the directions of errors were as predicted in three of the four conditions, the errors were small and only one, when line OB was oriented 22-5°, was significantly different from zero, with tl0 = 5-00; p < 0 001. However, postexperimental questioning of observers revealed that all but two had used the strategy of comparing the distances across the line endpoints (i.e. distance AB versus distance BC in figure 1). Lennie (1971) noted that his observers confined their free inspections to the centre of his display, to the angle vertices, and he also noted elsewhere (Lennie 1972) that when the angles were perceptually equated in this way, they nevertheless appeared unequal when the endpoint distances were compared. Table 1. Means (X) and standard errors (in brackets) of errors in settings of line OB, experiment 1. Initial orientation of line OB 22-5° X -0-85° (0-17°) Predicted
67-5° -0-15° (0-19°)
+
112-5° -0-40° (0-22°)
157-5° + 0-33° (0-26°) 4-
2.2 Experiment 2 Experiment 2 was identical to the first experiment except that eight volunteer subjects from an introductory course completed the task under two instruction conditions. These were endpoint (i.e. equate the distances AB and BC) and central (i.e. confine attention to the angle vertices and equate the 'spaces' between the angle arms). Half of the observers completed the central condition first (although order in which the two conditions were run made no obvious difference to the results). 2.2.1 Results. The results are shown in table 2. The errors under endpoint instructions were very similar to those of experiment 1, as expected, and again only that which occurred when line OB was at 22-5° was significant (tn = 3-65; p < 0 0 1 ) . Under central instructions, however, all errors were in the predicted direction and three of the four were significant, those for initial line OB orientations of 22-5° (f7 = 3-74), 112-5° (t7= 3-79), and 157-5° (r7 = 5-04), with p < 0 - 0 1 in every case.
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These data seemed to provide additional evidence consistent with Lennie's (1971) hypothesis regarding oblique-axis attraction and to confirm his view that endpoint judgments differ from central judgments. However, to provide a partial check on the absence of significant interactive effects between the lines, we conducted experiment 3. Table 2. Means and standard errors (in brackets) of errors in settings of line OB, experiment 2. Instruction
Central Endpoint
Initial orientation of line OB 22-5° 67-5° -1-16° (0-31°) -0-84° (0-23°)
112-5°
+ 0-03° (0-24°) -0-32° (0-23°)
157-5°
-1-06° (0-28°) -0-09° (0-18°)
+ 1-41° (0-28°) + 0-71° (0-33°)
2.3 Experiment 3 The basic displays in experiment 3 were the same as those used previously except that line OB was fixed at 22-5°, 67-5°, 112-5°, or 157-5°. When line OB was at one of these four orientations, the observer made two sets of parallel matches, by means of a fourth line alongside either line OA or line OC and 1 deg from it, a distance which is sufficient to exclude interactive effects of the other lines on the variable line (Carpenter and Blakemore 1973). The task was to judge whether the variable line appeared tilted CW or CCW relative to line OA or line OC and the usual staircase procedure was employed. When matches were made to line OA, the task was completed both when line OA only was present (pretest) and again when lines OA, OB, and OC were all present (posttest). As was discussed above, it was expected that both line OA and line OC would be apparently displaced in the direction describable as angle expansion, but that these illusions would be small and roughly equal for the two arms (1) . If this were the case, it could be inferred that interactive effects were unlikely to account for the results of experiments 1 and 2. Nine volunteer observers completed the experiment, including the eight observers used in experiment 2. 2.3.1 Results. The results are shown in table 3. In every case, the matching error was in the direction indicative of angle expansion but, in addition, the oblique arm always was affected more than the arm aligned with vertical and horizontal. In fact, this pattern of results is consistent with data of Carpenter and Blakemore (1973), who concluded that lateral inhibition in the orientation domain is propagated more extensively from vertical or horizontal channels than it is from more oblique channels. Moreover, the result obtained in experiments 1 and 2 can now be explained not only in terms of Lennie's hypothesis but alternatively in terms of interactive effects. If, in Table 3. Means (X) and standard errors (s) of parallel-matching errors, posttests minus pretests, experiment 3. Positive (+) errors indicate effects in the direction of angle expansion. Orientations of standard line are horizontal (H), vertical (V), or oblique (O). Orientation of line OB 67-5 22-5° H V O X s (1)
+0-61° 0-19°
+ 1-60° 0-38°
+ 1-28° 0-19°
112-5° O + 1-40° 0-44°
V + 1-34° 0-23°
157-5° O +2-01° 0-35°
H +0-98° 0-27°
O +2-29° 0-49°
It should be noted that the parallel-matching technique provides a pure measure of interactive effects. If single-line effects such as oblique-axis attraction do occur, then they would occur equally with the standard line (OA or OC) and with the variable, matching line and so would be factored out by this procedure.
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figure 1, line OA exerts a stronger influence than line OC, then both line OC and line OB will be shifted perceptually CW. On the other hand, line OA will be shifted CCW, resulting in angle COB's appearing too small, as if line OB were attracted perceptually to the oblique. Although this result led us to discontinue experiments with the display shown in figure 1, it became clear that there were a number of possible explanations of Lennie's original result apart from oblique attraction and one of these is angle expansion. The remainder of the experiments used Lennie-type displays. 2.4 Experiment 4 An example of the kind of display used by Lennie (1971) is shown in figure 2; in this case, angles AOB and COD are equal to 45° so that all angle arms are 22*5° from the vertical, horizontal, and oblique axes. Clearly, angle COD appears smaller than angle AOB (Lennie's effect) and the illusion seems to be tied to retinal rather than gravitational coordinates, since tilting the head 45° reverses the direction of the apparent inequality. Experiments 4 and 5 were designed to examine some of the parametric attributes of the angle illusion, specifically the effects of angle size and line length. Lennie (1971) merely stated that errors were qualitatively the same for angles of 20°, 30°, and 40°; and that similar results occur with lines 3 deg and 1 deg long. The latter statement is relevant to the possible occurrence of interactive effects since, elsewhere, Lennie (1972) suggested that tilt illusions are minimal between lines 3 deg long, so that such effects are unlikely to be involved in the illusion. In experiment 4, the display was as in figure 2 (45° angles) but the arm lengths were either 5 mm (31-5 min), 20 mm (2 deg 7 min), or 45 mm (4 deg 46 min).
Figure 2. Display similar to that used by Lennie (1971) with Z_AOB = Z.COD = 45°. 2.4.1 Method. The display techniques, and general procedures were those described previously. Pilot studies indicated that similar results were obtained, as expected, whichever of the four angle arms the observer adjusted. In this experiment, observers completed two sets of judgments, one in which arm OB was the variable and one set for arm OC. These were selected because the response buttons could then be labelled "left angle too small" and "right angle too small" for the left and right buttons, respectively, thus avoiding the more complex judgments of CW and CCW appearances. The two PSEs so obtained were averaged. Lennie (1972) merely noted that 'control' and cendpoint' judgments of the angles differed in his study so to measure the effect, our observers completed the task under both instruction sets. 2.4.1.1 Subjects. Sixty subjects, volunteers from introductory courses, acted as observers, twenty under each line-length condition. Within each group, half of the
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observers completed the 'central' instruction condition first and half of these subjects produced four such PSEs with the variable lines in the order OB, OC, OC, OB; the other half had the order OC, OB, OB, OC. There was no evidence that order had any effect, so the results of the subgroups were averaged. 2.4.2 Results. Table 4 presents the results, and three features of the data are noteworthy. First, all errors were different from zero (p < .0-0.1 in every case) except those with the 0-53 deg line-length. Second, as was the case in experiment 2 with a different display, endpoint judgments exhibited errors in the same direction as central judgments, but the latter effects were always larger. This was true of all twenty subjects in the 2-12 deg line-length group and of fourteen out of twenty in the 4-77 deg line length group. Analysis of variance (Winer 1962, p 306) indicated that mean central errors (+ 3-08°) exceeded mean endpoint errors (+ 1 -58°) with ^ 5 7 = 20-89 and p < 0-001. Third, the effect of line length appears to be nonmonotonic: the largest effects were obtained with the 2-12 deg lines. Analysis of variance showed that the line-length effect was significant CF2}57 = 19-31; p < 0 • 001) and that the length-by-instruction interaction was also significant {F = 4 -08; p < 0-05). The latter result reflects the larger difference between central and endpoint errors at the intermediate line length. However, even if sampling error somehow spuriously reduced the longest line effects, the effect of line length is opposite to that usually found with tilt illusions, where interactive effects decrease as lines increase in length or are separated by greater distances (e.g. Wallace 1969; Robinson 1972; Weale 1978). Table 4. Means and standard errors (in brackets) of errors in setting line OB or line OC to equate 45° angles AOB and COD, experiment 4. (Positive errors indicate angle COD set too large or angle AOB set too small.) Instruction
Angle arm length 0-53°
Central Endpoint
+0-17° (0-33°) -0-43° (0-39°)
2-12° + 3-74° (0-40°) + 1-60° (0-26°)
4-77° 4-2-24° (0-51°) + 1-57° (0-31°)
2.5 Experiment 5 Using the 2 - 1 2 deg line length, which produced largest errors in experiment 4 , we next examined t h e effect of angle size. Thus, t h e display was again as in figure 2, except standard angles AOB and COD were either 10°, 2 0 ° , or 4 5 ° . (The 4 5 ° group was n o t r u n again: t h e data were taken from experiment 4.) T h e m e t h o d s were identical t o those of experiment 4, with twenty observers in each angle group. 2.5.1 Results. All mean errors were significant (p < 0 - 0 1 , at least, in each case) and table 5 shows that errors increased with increasing angles. Again, this result appears not t o implicate interactive effects between t h e lines: tilt illusions are larger at 10° than at 20° than at 4 5 ° . Analysis of variance showed that t h e effects of angle and Table 5. Means and standard errors (in brackets) of errors in setting line OB or line OC to equate angles AOB and COD, experiment 5. (Positive errors indicate < COD set too large or 57 = '5 • 23; p < 0-01), reflecting the fact that the difference between central and endpoint errors increased with angle size. In the next experiment we attempted to measure tilt illusions which might occur in the 10° and 45° displays. 2.6 Experiment 6 Lennie dismissed tilt illusions as a possible basis of the angle illusion by arguing that where the angle effect is largest (i.e. with one angle flanking horizontal or vertical and the other flanking an oblique) "all four arms are at the same orientation with respect to the vertical and horizontal, and interactive effects should therefore cancel" (Lennie 1971, p 156). However, although interactive effects may not be responsible for the effect, this logic is unsound. Large tilt illusions (angle expansions) do occur with 10° angles and smaller, but significant effects occur with 45° angles (e.g. Carpenter and Blakemore 1973). It is conceivable then that, in figure 2, both angles are expanded but that angle AOB expands more than angle COD. A possible basis for this might be that expansion effects (i.e. inhibitory effects) are greater between lines flanking the major axes than between lines flanking the oblique. In fact, there are many ways in which the angle illusion could arise because the net effect is simply that angle COD appears smaller than angle AOB and this implies nothing about the absolute expansion or contraction of either angle. Figure 3 represents ten possible ways in which the angle illusion might occur (i.e. filled cells). Here, - represents contraction, + represents expansion, 0 represents no effect, and ++ or indicate large expansions or contractions. Lennie's explanation of the angle illusion applies only to four of the ten filled cells, those top right cells for which angle AOB is either 4or ++ and for which angle COD is either - or — . The counterexample given above is shown by the lowest filled cell, for which angle AOB is + + and angle COD is +. To test for the occurrence of tilt illusions in the angle displays, we employed a task which is schematically represented in figure 4. Observers completed the same set of judgments for each of the four angle arms but for exemplary purposes consider arm OB in figure 4. First, line OB was presented alone (i.e. lines OA, OC, OD absent) and the only other line present was B\ Using the usual staircase methods, a PSE was obtained such that line B' appeared parallel to line OB (pretest). Next, the complete angle display was presented and the PSE for line B' parallel to line OB was obtained once more (posttest). The posttest minus pretest difference was taken to measure the effect of the other lines on line OB. As noted earlier, this was done Z.AOB
_ •
Q
O O
0
+
0
++
•
•
•
•
©
•
•
•
-J
• + +
Figure 3. Ten different ways to account for the fact the Z_COD appears smaller than LAOB, in figure 2 (see text).
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for all four lines although only one other variable line (D') is shown in the figure (see footnote 1). The dimensions of the display were as scaled in figure 4: angle arms were all 20 mm in length; the variable lines (e.g. line B') were 15 mm long and placed such that their inner ends were 1 mm from the vertex, O, when parallel to the standard lines, and 4 mm from the outer ends of the standard lines. The lateral separation between standard and variable lines was 9-4 mm (1 deg). Twenty-four subjects from the same population as the previous experiment each made matches to all four angle arms with both the 10° and the 45° angle displays. Another group of twelve subjects did the same but this time angle AOB flanked the lower, vertical axis (i.e. its bisector was oriented 270° rather than 0°). Finally, a group of ten subjects made angle matches, as in experiments 4 and 5, but with the standard angles of 10° and 45° flanking the lower vertical (270°).
Figure 4. Display used in experiment 6 (see text). 2.6.1 Results. The parallel-matching results are shown in figure 5, from which it is clear that the expected tilt illusions did occur, with relatively large (l°-2°) expansion effects in the 10° display for all four angle arms and small or negligible effects in the 45° display. Presumably, then, similar effects would have occurred in Lennie's experiments, especially with his 20° and 30° angles. However, it is clear that although tilt effects occurred, they cannot account for the angle illusion. Figure 6 shows both the angle-matching errors from experiment 5 (open circles) and also the summed angle expansions (open squares). By way of explanation,
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mi
A
B
i
*4| I
i
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i_
10°
45°
10°
45°
Angle size
Figure 5. Tilt illusions measured in experiment 6. Angle arms are A (open circles), B (filled circles), C (open squares), and D (filled squares), and inserts show the displayed angles. Left panel: N = 24, right panel: N = 12; vertical bars: 1 standard error.
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consider the 10° tilt illusion results in the left panel of figure 5. There, the illusions for arms A, B, C, and D are, respectively, + 0 - 9 9 ° , + 1 -25°, + 1 -47°, and 4-1 -85°. Hence, total expansion of the arms of angle AOB is 0-99° + 1 -25°, or 2 - 2 4 ° . Expansion of angle COD is 3-34°. The difference is - 1 • 10°, as shown for the 10° display in figure 6. So although the tilt-illusion results suggest that angle COD expands more, in the angle-matching task, angle COD is judged almost 2° smaller (figure 6). Similar results occurred when the standard angle was vertically positioned, as figure 6 shows. In this case, angle-matching errors (filled circles) resembled those which occurred when the standard angle was horizontal (although the latter errors were larger) but the summed tilt-illusion effects (filled squares) were negligible (2) . Finally, it might be supposed that the vertical-horizontal illusion is one component of the angle-matching illusion such that the vertical space between a horizontal angle's arms is judged longer than the horizontal space between a vertical angle's arms. This would account for the smaller angle-matching errors with a vertical standard angle (figure 6). However, other data (cf Lennie 1971, 1972) do not always exhibit this difference and, at any rate, if that were the case, large illusions would occur with one angle flanking vertical and the other flanking horizontal but this is not generally found (Lennie 1971, 1972).
Q
O
Angle size
Figure 6. Angle-matching errors obtained in experiments 5 and 6 with standard angle horizontal (open circles) and vertical (filled circles). Also shown are summed tilt illusions from experiment 6 for standard angle horizontal (open squares) and vertical (filled squares). For details, see text. 3 Discussion The results of the last experiment seem to imply one of two conclusions. One possibility is that tilt illusions do affect the perceived orientations of the angle arms in the angle illusion display but other factors, such as directionally opposite attraction of the arms towards the oblique axes, override the tilt effects and result in the judged inequalities initially observed by Lennie and confirmed in the experiments reported here. However, another possibility is that judgments of angles are partly or ^ Subjects who completed parallel matches to the angle arms when the standard angle was oriented 270° were asked, at the end of the experiment, which angle appeared smaller, just to check that the usual angle illusion occurred with subjects who performed the parallel matches. Of the twelve subjects in the 10° angle group, seven judged angle COD smaller, four judged the angles equal, and one said angle AOB was smaller; in the 45° group the corresponding figures were ten, two, and zero.
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wholly independent of the perceived arm orientations. That is, the directionally opposite effects reported here might be likened to effects such as 'Morinaga's displacement paradox' (cf Robinson 1972, pp 25-26). In that effect, the Brentano version of the Muller-Lyer illusion (i.e. no shafts) is arranged so that the outward-going fin component is directly above or below the inward-going fin component. Simultaneously, the space between the fins appears shorter in the latter component, yet the endpoints of its fins appear to lie outside those of the outwardgoing fin component. The observation is said to be paradoxical because, geometrically, it cannot be true that one spatial extent is both longer and shorter than another. By analogy, then, it is conceivable that in the angle display the arms of the angle are expanded in the orientation domain but, simultaneously, the spaces within the angles are differentially contracted, as a function of the angles' orientations. This line of reasoning finds some support in a recent report of Heywood and Chessell (1977), whose observers attempted to match the distance or space across angles by placing a dot on a page which marked off a distance perceptually equivalent to the criterion distance within the standard angle. The general tenor of the data was that the patterns of over- and underestimation frequently were quite different from those which have been reported when tasks requiring judgments of orientation were used (e.g. Bouma and Andriessen 1970; Blakemore et al 1970), although aspects of the results also confirmed some of the previous results. Heywood and Chessell's concluding remark, then, applies equally to the experiments reported here. In discussing differences between their results and those obtained by others using different methods, they observed that "whether the differences in our results are in some obscure way a reflection of this procedural difference, or whether the 'expansions' and 'contractions' of angles, however obtained, can be seen to reflect a process of spatial perception that is not straightforwardly related to Euclidean metrics remains to be discovered" (Heywood and Chessell 1977, p 581). Unfortunately, the precise relevance of the Heywood and Chessell data to those reported here is difficult to assess, if only because the angular illusion we have investigated depends critically on the relative orientations of the two angles; Heywood and Chessell imply (p 580) that their effects were independent of orientation. The above kinds of observations do not, unfortunately, constitute an explanation of apparently paradoxical effects. On the other hand, the experiments reported here, and the above discussion, do demonstrate that the angle-matching task does not necessarily have simple implications about the perceived absolute orientations of individual angle arms. Acknowledgements. This research was supported by the Australian Research Grants Committee (Grant A74/15177). Peripherals and software for the display facilities were developed by John Holden, Alan Parkinson, and Mike Cooper. We gratefully acknowledge Phil Greenwood, Margie Morgan, and Anne Keene for assistance in running the experiments. References Bouma H, Andriessen J J, 1968 "Perceived orientation of isolated line segments" Vision Research 8 493-507 Bouma H, Andriessen J J, 1970 "Induced changes in the perceived orientation of line segments" Vision Research 10 333-349 Blakemore C, Carpenter R H S, Georgeson M A, 1970 "Lateral inhibition between orientation detectors in the human visual system" Nature (London) 228 37-39 Carpenter R H S , Blakemore C B, 1973 "Interactions between orientations in human vision" Experimental Brain Research 18 287-303 Emerson P, Wenderoth P, Curthoys I, Edmonds I, 1975 "Measuring perceived orientation" Vision Research 15 1031-1033 Fisher G H, 1969 "An experimental study of angular subtension" Quarterly Journal of Experimental Psychology 21 356-366
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Fisher G H, 1974 "An experimental study of linear inclination" Quarterly Journal of Experimental Psychology 18 52-62 Heywood S, Chessell K, 1977 "Expanding angles? Systematic distortions of space involving angular figures" Perception 6 571-582 Lennie P, 1971 "Distortions of perceived orientation" Nature (London) New Biology 233 155-156 Lennie P, 1972 Mechanisms Underlying the Perception of Orientation PhD thesis, University of Cambridge, Cambridge, England Matin E, 1974 "Light adaptation and the dynamics of induced tilt" Vision Research 14 255-265 Mershon D H, Kennedy M, Falacara G, 1977 "On the use of 'calibration equations' in perception research" Perception 6 299-311 Robinson J 0,1972 The Psychology of Visual Illusion (London: Hutchinson) Wallace G K, 1969 "The critical distance of interaction in the Zollner illusion" Perception and Psychophysics 5 261-264 Weale R A, 1978 "Experiments on the Zollner and related optical illusions" Vision Research 18 203-208 Wenderoth P, Beh H, White D, 1978a "Perceptual distortion of an oblique line in the presence of an abutting vertical line" Vision Research 18 923-930 Wenderoth P, Beh H, White D, 1978b "Alignment errors to both ends of acute- and obtuse-angle arms" Perception and Psychophysics 23 475 -482 Winer B J, 1962 Statistical Principles in Experimental Design (New York: McGraw-Hill)
