Constant errors occur in matched reproduction of angles even when likely biases are eliminated

The literature on the reproduction of visually perceived angles often seems to contain an explicit or implicit assumption that systematic errors are caused by misperception of the sizes of the angles. We argue that in the absence of procedural artifacts no consistent error should arise from purely perceptual causes. Accordingly, an experimental procedure was designed to eliminate or evaluate a number of artifacts that probably affected earlier experiments. A computer-controlled display generated both a standard angle and a variable one that could be adjusted by a subject to match the standard. The data analysis followed Cox’s (1970) procedure for the analysis of binary data to focus on the numbers of positive and negative errors rather than on their magnitudes and so eliminate the numerically distorting effects of asymmetrical error distributions. Nevertheless, the results showed what has usually been reported before: acute angles were matched with too large a comparison angle, and obtuse ones with too small an angle. Thus, there is a real effect to be explained, although the explanation must invoke response processes.     

Constant errors donot occur in the matched reproduction of angles

MacRae and Loh (1981) reported that when observers attempt to match angles, whether simultaneously or successively, and regardless of whether the angles have the same orientation or not, acute angles generally are set too large and obtuse angles, too small. Without presenting data, they argued indirectly that starting-position effects in their adjustment procedure could not have accounted for the errors. In four experiments, we have demonstrated that effects similar to MacRae and Loh’s do occur when their starting positions are used and that no such effects occur when psychophysical methods that minimize or eradicate starting position effects are used.     

Visual illusions with acute and obtuse angles: Configurational effects and observer strategies

In three experiments, it is demonstrated that the negative Poggendorff illusions that Restle (1969) and others have reported are a special case, that, in general, acute- and obtuse-angle effects are the same, that there are additional special cases which modulate the illusion, but only slightly, and that these special cases derive from particular configurations which impose other effects, including observer strategies, on the basic illusion. These additional effects do not bear necessarily upon the fundamental explanation of the illusion. The experiments also show that effects obtained with amputated illusory figures are not linearly additive. Recent experiments that have measured apparent oblique separation in the Poggendorff figure and have found large obtuse-angle but small, or negative, acute-angle effects are also discussed. It is argued that these experiments have not demonstrated that the two dependent variables-alignment and length judgments-are correlated, or that one error causes the other; and it is argued that uinaHolland’s theory of parallel attraction to explain the illusion is not tenable.     

Length illusions in conventional and single-wing Müller-Lyer stimuli

Warren and Bashford (1977) reported that eliminating one of the wing components from the conventional (i.e., two-wing) Müller-Lyer figures had no appreciable effect on the magnitude of the acute-angle (contraction) illusion but substantially reduced the magnitude of the obtuse-angle (expansion) illusion. In addition, they found that whereas the contractionary effects of the acute-angle components tended to be confined to the region of the shaft adjacent to the angles, the expansionary effects of the obtuse-angle components were more uniformly distributed across the shaft. Since these findings challenge many theories of the Müller-Lyer illusion, the purpose of the present investigation was to evaluate further Warren and Bashford's work with four experiments. Experiments 1 and 2 assessed length illusion magnitudes by requiring subjects to adjust either the length of a plain comparison line to match the length of the Müller-Lyer test figures (Experiment 1) or the length of comparison Müller-Lyer figures to match the length of plain test lines (Experiment 2). Experiments 3 and 4 used a bisection task to assess whether the illusory effects of the angle components are confined mainly to regions of the shaft adjacent to the angles. Consistent with most theories of the Müller-Lyer illusion, eliminating one of the wing components reduced both forms of the Müller-Lyer length illusion to a similar extent. In addition, the acute- and obtuse-angle forms yielded similar patterns of bisection errors, with substantial errors for regions of the shaft adjacent to the angles and negligible errors for regions of the shaft distant from the angles.     

The effects of angle-arm length on judgments of angle magnitude and orientation contrast

Acute angles frequently are overestimated, an effect that has been attributed to lateral inhibitory mechanisms. It can be derived from such theories that lengthening the arms of an angle should either reduce or not affect its judged subtense. However, in Experiments 1 and 2, it was found that angles with longer arms were judged larger. In Experiment 3, direct measures of anglearm orientation indicated that orientation contrast does not increase with angle-arm length, that it decreases with distance from the angle vertex, and that these effects are not averaged along the entire length of the arm. It follows that inhibitory mechanisms alone cannot explain why longer armed angles appear larger or why angle arms appear straight rather than bowed. It is suggested that a distinction is required between the orientation and the angular domains and that the latter judgments depend, at least partially, on the space or area within an angle. The fact that angles appear unbowed may reflect the fact that perceived orientation differences occur along an angle’s arm only when matches are made to small segments of the arm.     

Exposure-time and spatial-frequency effects in the tilt illusion

The exposure durations of a vertical test line and a tilted inducing grating were varied and the tilt illusion thus generated was found to change as a function of this variation. Significant direct effects (acute-angle expansion) and indirect effects (acute-angle contraction) were found to occur at times consistent with Andrew's estimate of the time course of inhibition in the visual system when the inducing grating had a spatial frequency of 10 cycles deg-1. However, a 2.71 cycles deg-1 grating gave significant effects at exposure durations of 10 as well as 1000 ms, while in a further experiment a 10.91 cycles deg-1 grating gave significant effects at 1000 ms only. These results seem to suggest that orientation interactions thought to be due to inhibition (direct effect) and disinhibition (indirect effect) may occur within both sustained and transient channels with concomitant differences in time constants.     

On the common stimulus condition and explanation of the Müller‐Lyer,Poggendorff and Zöllner illusions: The basis for a class of geometrical illusions

Although the Müller‐Lyer, Poggendorff and Zöllner figures and the illusions with which they are associated– those of length, misalignment and non‐parallelism, respectively – are quite different, all three are attributable to the same basic effect, that of the difference in the apparent length of equal lines forming acute and obtuse angles. The role of this basic affect, reported originally by Müller‐Lyer but overlooked by him and others as a possible cause of the three illusions, is identified and discussed. It is suggested that the demonstration of a basic stimulus condition (acute and obtuse angles) and its associated illusion (difference in apparent length) generating a class of “higher order” geometric illusions has implications not only for a closer understanding of the three illusions under consideration but for other classes of illusory phenomena that might also share a common basis.     

Observational drawing biases are predicted by biases in perception: Empirical support of the misperception hypothesis of drawing accuracy with respect to two angle illusions

We tested the misperception hypothesis of drawing errors, which states that drawing accuracy is strongly influenced by the perceptual encoding of a to-be-drawn stimulus. We used a highly controlled experimental paradigm in which nonartist participants made perceptual judgements and drawings of angles under identical stimulus exposure conditions. Experiment 1 examined the isosceles/scalene triangle angle illusion; congruent patterns of bias in the perception and drawing tasks were found for 40 and 60° angles, but not for 20 or 80° angles, providing mixed support for the misperception hypothesis. Experiment 2 examined shape constancy effects with respect to reproductions of single acute or obtuse angles; congruent patterns of bias in the perception and drawing tasks were found across a range of angles from 29 to 151°, providing strong support for the misperception hypothesis. In both experiments, perceptual and drawing biases were positively correlated. These results are largely consistent with the misperception hypothesis, suggesting that inaccurate perceptual encoding of angles is an important reason that nonartists err in drawing angles from observation.     

Errors towards the perpendicular in children's copies of angular figures: a test of the bisection interpretation

Children distort angular figures so that the constituent angles are nearer 90 degrees than they should be. This could be due to a perpendicular bias or a bisection bias, or to both. A study is reported which was designed to establish whether a perpendicular bias would appear independently of bisection. Twenty four-year-old children were tested on two types of angular figure: (i) a baseline with another line joining at the end at 45 degrees, 90 degrees, or 135 degrees and (ii) a baseline with another line joining at the middle at 45 degrees or 90 degrees. Perpendicular errors were obtained both for 'end' and for 'middle' figures, but overall more strongly for 'middle' figures. However, while 90 degrees 'middle' figures were copied more accurately than 45 degrees/135 degrees figures, this effect was only obtained for vertical and horizontal presentations of 'end' figures, and was reversed for oblique presentations. Also, for 'end' figures, directional errors varied with subtended-line orientation, whereas for 'middle' figures they varied with baseline orientation. It is concluded that although errors towards the perpendicular do occur with single-angle figures, angle equalisation may take place when there are two adjacent angles in the figure. One interpretation of the differing orientation effects is that in 'middle' figures strong internal relational forces produce a distortion that varies with the angle at which the figure is viewed, whereas in 'end' figures the absence of relational forces within the figure leads to a stronger influence from external cues.     

The Poggendorff illusion with obtuse and acute angles

The Poggendorff misalignment illusion with the conventional figure and its obtuse- and acute-angle variants was investigated in five experiments. The method of adjustment was used in Experiments I–IV, and in Experiment V compared with a forced-choice procedure. The first experiment showed that the 45-deg acute-angle illusion is positive but smaller than those from the other two figures, the second that it is the same size as that with two 45-deg oblique lines without parallels, and the third that the 30-deg acute-angle illusion is also positive but smaller than that for the other two patterns. In the fourth experiment, the 30-deg acute-angle illusion was insignificant under the same condition in which it was positive in Experiment III, but significantly positive for a reversed contrast (white on black) figure. The results from the last experiment were paradoxical; the 45-deg acute-angle figure again gave a significantly positive illusion with the method of adjustment, but mainly reports of a negative illusion with a forced-choice technique. A possible basis for this difference is discussed in terms of psychophysical procedures and special features of the acute-angle pattern. The weakness of the 30-deg acute-angle illusion is also considered along with other issues.     

How far can attraction-caused misalignment account for the Morinaga misalignment effect?

Summary When a line (the pointer) is collinear with a dot, the addition of a second line (the induction line) contiguous with the dot or near it may cause the pointer to appear to be collinear with a point further along or nearer to the induction line. The geometrical relations upon which this effect, which we call attraction-caused misalignment, depends have been studied with the Obonai and Wundt-Loeb (Hotopf, 1981; Hotopf & Brown, 1988) figures. Drawing upon the studies of misalignment in the Morinaga figure carried out by Restle (1976), Day, Bellamy, and Norman (1983), and Day and Kasperczyk (1985), as well as upon two new experiments, we show that misalignment in the Morinaga figure is also attraction-caused misalignment, as previously defined. We conclude with a discussion of a number of theories that aim at accounting for attraction misalignment.     

Response frequency equalization: A bias model for psychophysics

A model for response bias in psychophysics is derived. Of the model’s two aspects, one is concerned with the generation of sensory states and the other with rules for transforming sensory states into responses. The model incorporates the bias rule that Ss tend to use available responses with equal frequency. Applications to experiments which use the method of constant stimuli are discussed. Despite the contrary claims of Restle and Levison (1971), the model not only predicts their data quite well but also, under certain conditions, does so better than the theory of adaptation level used by Restle and Levison themselves.     

Context-dependent migrations in visual word perception

In the present study we investigate the effect of context--that is, a prime--on migration errors. Migration errors, or migrations, are caused by perceptual interactions between two words in a visual display, such as line-lace. After postcuing, subjects sometimes report lane or lice, instead of the requested word line. This phenomenon has been demonstrated by previous studies. In the experiments reported here, we replicate this phenomenon. We also find that more migrations of the lane type occur (compared with lice type), when the display line-lace is preceded by a prime related to lane (but not to lice). This shows that higher order word knowledge, in the form of semantic relations between words, influences the migration phenomenon. Further, we show that the number of migration errors are not a result of only a summation of activation from the letters in the display. The method we developed to isolate context-dependent migrations also gives valuable information about the effect of contextual information on word perception.     

The Poggendorff illusion: consider all the angles

In the Poggendorff display, which consists of parallel lines interrupting a transversal, one of the two transversal segments was replaced by a dot lying along the parallel. The angle between the remaining transversal segment and the parallels was varied in 15 degree increments, as was the orientation of the transversal with respect to the subject. Subjects set the dot to appear collinear with the transversal. Judgmental errors can be partitioned into additive components, one linearly related to the size of the obtuse angle between transversal and parallels and the other a sinusoidal function of transversal and parallels and the other a sinusoidal function of transversal orientation (collinearity settings err toward the horizontal or vertical, whichever is closer), plus a meridional effect, an interaction term that magnifies the errors of a given obtuse angle as the transversal approaches an oblique orientation.     

Emotional expression on a profile: Feature height, mouth angle, and tilt

Profiles may vary in expression with changes in the height of the profile’s features, the mouth’s angle at the profile contour, and head tilt. In three experiments, we varied these factors. Features low on the profile, with mouths at an obtuse angle to the profile line and head tilted 15° forehead-forward, appear especially sad. High features with acute mouth angles suit forward, upright, and backward head tilts. Feature height, mouth angle, and head tilt on a profile are physical variables that can specify postures such as downcast heads and turned-down mouths, and thereby psychological factors. We point out alternative accounts of the height and tilt effects.     

A set-theoretic random utility model of choice behavior

A random utility model of choice was developed by combining the basic ideas of the well-known theories of Thurstone and Restle. The new model has exactly the same number of free parameters as Tversky's Elimination-by-Aspects model. Furthermore, both models were found to fit, with equal accuracy, the data reported by Rumelhart and Greeno, and Tversky. It was concluded that although the two theories are not identical, they may be difficult to discriminate empirically.     

Reorientation in diamond-shaped environments: encoding of features and angles in enclosures versus arrays by adult humans and pigeons (Columbia livia)

Although geometric reorientation has been extensively studied in numerous species, most research has been conducted in enclosed environments and has focused on use of the geometric property of relative wall length. The current studies investigated how angular information is used by adult humans and pigeons to orient and find a goal in enclosures or arrays that did not provide relative wall length information. In enclosed conditions, the angles formed a diamond shape connected by walls, whereas in array conditions, free-standing angles defined the diamond shape. Adult humans and pigeons were trained to locate two geometrically equivalent corners, either the 60° or 120° angles. Blue feature panels were located in the goal corners so that participants could use either the features or the local angular information to orient. Subsequent tests in manipulated environments isolated the individual cues from training or placed them in conflict with one another. In both enclosed and array environments, humans and pigeons were able to orient when either the angles or the features from training were removed. On conflict tests, female, but not male, adult humans weighted features more heavily than angular geometry. For pigeons, angles were weighted more heavily than features for birds that were trained to go to acute corners, but no difference in weighting was seen for birds trained to go to obtuse corners. These conflict test results were not affected by environment type. A subsequent test with pigeons ruled out an interpretation based on exclusive use of a principal axis rather than angle. Overall, the results indicate that, for both adult humans and pigeons, angular amplitude is a salient orientation cue in both enclosures and arrays of free-standing angles.     

Serial position functions for letter identification at brief and extended exposure durations

The properties of serial position functions for tachistoscopic report were investigated over a wide range of viewing times. Four-letter strings of random consonants were presented in varying display locations relative to the fixation point with the observers’ eye movements monitored to limit them to a single fixation for each display. Salient properties of the serial position curves include an overall central-peripheral gradient, higher performance at the ends than the interior of letter strings regardless of absolute location, and left-right asymmetry in the visual field, all of these being largely independent of viewing time. Errors reflecting loss of positional information are prominent even at extended viewing times, are more nearly symmetrical in the left and right visual fields than other types of errors, and, in contrast to item errors, occur less frequently in letter sequences that have high frequencies in English. Further, transposition errors exhibit a pronounced peripheral-to-central drift, possibly reflecting gradients of positional uncertainty. Such gradients may be implicated in the peripheral-central asymmetry of the lateral interference effects exerted by other letters on a target letter in a nonfoveal location.     

A comparison of visual tilt illusions measured by the techniques of verticle setting, parallel matching, and dot alignment.

The tilt illusion (TI) was investigated by using both short (19 min) and long (2 deg 6 min) test lines, at three angles of test line-inducing line separation (15 degrees, 45 degrees, and 75 degrees). Three groups of ten observers each provided data under one of three task conditions: vertical judgment, parallel matching, and dot alignment on a common visual display. The main result was that both the vertical judgment and the parallel matching task provided similar, classic TI angular functions with the means ordered 15 degrees greater than 45 degrees greater than 75 degrees and with small attraction effects at 75 degrees in three of the four relevant functions. The third task, dot alignment, yielded results different from the average of the other two: no attraction effects occurred and, with the short test line, the obtained mean illusion at 45 degrees exceeded those at the other intersect angles. These results are consistent with alignment data reported by others. One explanation is that the inducing line produces an apparent bowing of the test line which would be reflected in dot alignments but not in vertical setting or in parallel matching. However, direct evidence does not support this hypothesis. An alternate hypothesis, for which independent evidence exists, is that alignment errors reflect perceptual mistracking but that the origin of these errors is not the tip of the test line but within it. Although this does not explain dot alignment errors, it highlights their complexity and the need to interpret them with caution.     

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

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