Perception &: Psychophysics 1990. 48 (2), 105-123 A theory of phenomenal geometry and its applications

WALTER C. GOGEL University of California, Santa Barbara, California

The geometry of perceived space (phenomenal geometry) is specified in terms of three basic factors: the perception of direction, the perception ofdistance or depth, and the perception of the observer's own position or motion. The apparent spatial locations of stimulus points resulting from these three factors thereupon determine the derived perceptions of size, orientation, shape, and motion. Phenomenal geometry is expected to apply to both veridical and illusory perceptions. It is applied here to explain a number of representative illusions, including the illusory rotation of an inverted mask (Gregory, 1970), a trapezoidal window (Ames, 1952), and any single or mul­ tiple point stimuli in which errors in one or more of the three basic factors are present. It is con­ cluded from phenomenal geometry that the size-distance and motion-distance invariance hypotheses are special cases of the head motion paradigm, and that proposed explanations in terms of compensation, expectation, or logical processes often are unnecessary for predicting responses to single or multiple stimuli involving head or stimulus motion. Two hypotheses are identified in applying phenomenal geometry. It is assumed that the perceptual localization of stimulus points determines the same derived perceptions, regardless of the source of perceptual information supporting the localizations. This assumption of cue equivalence or cue substitution provides considerable parsimony to the geometry. Also, it is assumed that the perceptions speci­ fied by the geometry are internally consistent. Departures from this internal consistency, such as those which occur in the size-distance paradox, are considered to often reflect the intrusion of nonperceptual (cognitive) processes into the responses. Some theoretical implications of this analysis of phenomenal geometry are discussed.

An observer, upon viewing a three-dimensional visual Examples of a partial phenomenal geometry are pro­ scene, acquires an internal spatial representation, which vided by hypotheses of invariance. These are equations can then contribute to or determine responses to that that express the interrelation of several perceptions. For scene. This internal representation, usually supported by example, the size-distance invariance hypothesis and the sensory information, is an instance of phenomenal space. shape-slant invariance hypothesis state that a given reti­ The geometrical description of this phenomenal space will nal (proximal) stimulus specifies a ratio of two percep­ be called phenomenal geometry. The purpose of this paper tions-perceived size and perceiveddistance in the one case is to summarize and provide examples of a theory of and perceived shape and perceived slant in the phenomenal geometry, various portions of which have other. Such equations contain terms for proximal stimuli been described in previous articles (Gogel, 1981a, 1982, (retinal size and retinal shape in the above examples), as 1983; Gogel & DaSilva, 1987a, 1987b; Gogel & Tietz, well as perceptual terms. A complete phenomenal geom­ 1974). Two hypotheses underlying the theory and some etry that would reflect all aspects of the phenomenal theoretical consequences of the theory will be discussed. world, however, would involve only perceptual terms. It will be shown that this theory can explain a number Such a complete phenomenal geometry, if it is to be of representative illusions, obviating the need to describe general, must be able to derive more complex perceptions these phenomena in other, usually more complex, terms. from the perceptual variables postulated to be basic. The The theory is expected to apply whether the observer basic variables of the theory described here are (1) per­ and/or the stimulus is physically moving or physically sta­ ceived direction, (2) perceived distance or depth, and tionary and whether the perceptions are accurate or are (3) the observer's awareness (perception) of his or her in error. own position or motion. This theory asserts that these three basic perceptual variables together determine the perceived positions of points in perceptual three­ The preparation of this article was supported by Research Grant MH dimensional space. Numbers of such points at different 39457 from the United States National Institute of Mental Health. I thank apparent spatial locations, or movement between these John M. Foley, Tarow Indow, Michael T. Swanston, and two anony­ different apparent locations, provide the derived percep­ mous reviewers for their helpful comments on an earlier draft of this manuscript. Correspondence should be addressed to Walter C. Gogel, tions of size, shape, orientation, and motion. It is useful Department of Psychology, University of California, Santa Barbara, experimentally that each of the three basic variables can CA 93106. be manipulated independently of the others. For exam-

105 Copyright 1990 Psychonomic Society, Inc. 106 GOGEL

ple, the position of the observer and the stimulus can be rectangles in Figure 1C) and primed letters identify per­ changed simultaneously in perceived space, keeping the ceived characteristics of the test objects. Filled circles or perceived direction and perceived distance of the stimu­ filled rectangles and unprimed letters identify physical lus from the observer constant. Or, the accommodation characteristics of the test objects. Figure 1A represents or convergence to an object can be changed so as to change a situation in which a stationary test point of light at the its apparent distance while leaving its perceived direction physical distance D m from the observer is viewed as the and the perceived position of the observer unmodified. observer's head moves left and right repetitively through The theory, however, does not assert that perceived direc­ a perceived lateral extent, K', which mayor may not be tion, perceived distance, and the perception of self-motion the same as the physical motion K of the head. Ifthe test or -position necessarily are independent of the matrix of point is made to appear nearer, D~, or farther, Dr, than stimuli in which the point is embedded. It is proposed, its physical distance Dm (perhaps by changing the con­ instead, that when such context interactions occur, their vergence to the test point using prisms), the point will effects on perception can be identified as modifications appear to move concomitantly with the head in the same of perceived direction, perceived distance, and/or the ob­ direction as the head motion or opposite to the direction server's perception of his or her own location or motion. of the head motion, respectively. The physical distance, Dm , of the physically stationary test point is the distance APPLICATION OF PHENOMENAL around which the direction from the observer to the test GEOMETRY TO A SINGLE TEST OBJECT point intersects or pivots as the head is moved laterally. This is called the pivot distance, and it is labeled Dp in Considerable evidence for the validity of the theory of Figure l A. It should be noted, however, that if the test phenomenal geometry is found in a method of measuring point had been physically moved laterally, concomitantly apparent distance called the head motion procedure or with the head, in the same direction as, or opposite to, head motion paradigm (see Gogel, 1981a, for a review). the physical direction of the head motion, the pivot dis­ An application of this procedure consists of using the il­ tance would have been greater than or less than the phys­ lusory motion of the test object (or the nulling of this il­ ical distance of the test point, respectively (also see Gogel, lusory motion by an opposite physical motion) that can 1982, Figure 1). As predicted from phenomenal geome­ occur concomitantly with a lateral motion of the head, try, for a given perceived motion of the head, both the to provide an indirect measure of the perceived distance perceived distance and perceived direction ofthe test ob­ of the test object. The kind of situation in which this is ject, with the latter modifiable by changing the pivot dis­ accomplished is shown in Figure l A. In Figure 1, and tance to the test object, will change the apparent concomi­ throughout this article (except in Figure 3), the open ob­ tant motion of the test object. The equation relating the jects (e.g., open circles in Figures lA and IB and open perceived (illusory) motion (W') of the test object, the o o

Dp Dm 1 1

A B c

Figure 1. As is illustrated in Figures 18 and Ie, respectively, the apparent size-distance [5' = 2D' tan (0'/2)] and the apparent motion-distance invariance hypothesis [M' = 2D' tan(O'/2)] are predicted from the head motion paradigm [W' = K' - 2D' tan ( '/2)] shown in Figure lA, in this case for the condition in which ' = 0'. PHENOMINAL GEOMETRY 107

perceived lateral motion (K') of the head, the change in The equation for the SDIH, expressed in completely per­ the perceived direction (c/>') of the point relative to the ceptual terms consistent with Figure 1C (see Gogel & head, and the perceived distance (D') of the test point is DaSilva, 1987b; McCready, 1985, 1986), is W' = K' -2D' tan (c/>'I2). (I) S' = 2D' tan(0'/2). (6)

If tane' = tan e, and if K' = K, then Equation 1can be Since 0' in Figure IB is equal to 0' in Figure lC and equal written as to C/>' in Figure lA, it follows that

W' = K(I-D'lDp ), (2) M' = S' = K'-W'. (7)

where Dp is the pivot distance and tan(c/>/2) = KI(2Dp ) Several aspects of Figure 1 are of particular interest for (see Gogel, 1983). The perception oflateral motion, W', the present discussion. One is that the relations expressed concomitant with the lateral motion of the head is posi­ by Equations 1,4, 6, and 7 occur entirely between per­ tive if the test object is perceived to move in the same ceptual variables. This means that whenever any of the direction as the lateral motion of the head, and it is nega­ perceptual variables K', c/>', 0', or D' do not equal the cor­ tive if the test object is perceived to move opposite to the responding physical variables K, c/>, 0, or D, it is the per­ direction of the lateral motion of the head. Equation 2 is ceptual and not the physical values that are considered called the apparent distance, pivot distance hypothesis by the theory ofphenomenal geometry to determine W', (Gogel, 1982). It can be applied only if it is assumed that M', and S'. Instances in which K' does not equal K in­ tan C/>' = tan c/> and K' = K. Equation 1, however, because clude an observer's underperception of motion when it is an instance ofcomplete phenomenal geometry, does seated in a moving vehicle or the perception of self-motion not require these assumptions. (vection) that occurs when a stationary observer views Figure IB is similar to Figure lA, except that the head a large field of moving stripes (Wist, Diener, Dichgans, is perceived to be stationary (perhaps because it is sta­ & Brandt, 1975). Substantial errors in the perception of tionary) and the perceived extent of lateral motion of the visual direction can result from prolonged viewing of an stimulus point is M'. It is assumed that the perceived mo­ object moving concomitantly with a lateral motion of the tion ofthe point at Dm is K', which is the same at the per­ head (Hay, 1968; Hay & Goldsmith, 1973; Tietz & Gogel, ceived motion of the head in Figure lA (that is, M' at 1978; Wallach & Floor, 1970). However, two studies in Dm = K'). In this case, it is clear that the change in per­ which the head motion paradigm was used with less con­ ceived direction 0' in Figure IB equals C/>' in Figure lA. tinuous viewing did not reveal significant errors in this It follows that, if the test point in Figure IB is perceived variable (Gogel, 1982; Gogel & Tietz, 1979). Conditions at D~ or D;, its perceived motion M' at these perceived under which D' is not equal to D can occur frequently distances will equal K' - W~ and K' + W;, respectively, in both natural and experimental situations. This can oc­ where W~ and W; in Figure IB are the same as in cur, for example, as a function of the distance of fixation Figure lA. The situation in Figure lA is an instance of (Gogel, 1979a; Gogel & Tietz, 1977), or as a consequence the head motion paradigm; that in Figure IB is an instance of modifying oculomotor cues of distance with physical of what can be called the motion-distance invariance distance constant (Gogel & Tietz, 1979). Experiments in­ hypothesis (MDIH). Evidence for the validity of the volving the latter condition have been used extensively MDIH is found in experiments by Rock, Hill, and Fine­ to examine the validity of Equation 2 (Gogel, 1983). man (1968) and by Wist, Diener, and Dichgans (1976). An aspect of special interest regarding Figure 1 is that An equation expressing the MDIH as a partial perceptual the MDIH (Figure IB) and the variation on the SDIH equation replacing 0'12 by 012 in Figure IB is known as Emmert's ljiW(Figure lC) are special cases of the head motion paradigm applicable to a stationary head. M' = 2D' tan (012). (3) It follows that phenomenal geometry expressed in terms An equation expressing the MDIH in completely percep­ of perceived distance, of perceived direction or changes tual terms consistent with Figure IB is in perceived direction (perceived visual angle), and ofthe observer's perception of the self as stationary or moving M' = 2D' tan (0' /2). (4) can explain phenomena involving either a stationary or Figure 1C represents an instance of the size-distance moving observer viewing either a moving or a stationary invariance hypothesis (SDIH) in which a laterally extended stimulus. The expression of the head motion paradigm test object at a physical distance Dm from the observer in completely perceptual terms (Equation 1) implies that has a perceived width (S') equal to the K' in Figure lA. the MDIH (Equation 4) and SDIH (Equation 6) also must Whenever the test object is perceived at D~ or D;' it is contain only perceptual variables. The latter equation is expected from the SDIH that its perceived width S' will in agreement with McCready's formulation (McCready, equal K' - W~ or K' + W;, respectively. A partially per­ 1965, 1985) of the SDIH in terms of perceived visual ceptual equation replacing 0'12by 0/2 in Figure lC, which angle rather than physical visual angle. It may not seem often is used to express the SDIH, is particularly surprising that perceived direction and per­ ceived distance, together with the contribution of the ob­ S' = 2D' tan (0/2). (5) server's perception of self-motion or -position, can pro- 108 GOGEL vide perceptual localization in space and thus can provide motion when it was perceived at Dr. Figure 2B illustrates a common explanation for a variety ofspatial phenomena. the same conditions as those in Figure 2A, except that As will be discussed, however, this approach can ex­ the test point, rather than remaining physically station­ plain phenomena that have customarily been understood ary, physically moved vertically through a distance M, in terms of higher order factors, such as cognition, com­ concomitant with the lateral motion of the head. In this pensation, or expectation. case, the illusory perceived horizontal motion (W') vec­ tored with the vertical perceived motion (M') resulting Evidence Regarding Figure 1 from the physical vertical motion (M) to produce an ap­ There is evidence from three studies that W', M', or parently tilted path of motion of the test point a~ at D~ 5' can provide similar or highly correlated measures of and ar at Di. The different values of M', illustrated in D' and that they thus are likely to involve the same Figure 2B, resulting from the different values of D' are processes.' In one study (Gogel, 1979a), D' was mea­ consistent with those expected from the MDIH. Since both sured by the physical size adjusted to achieve a particu­ M' and W' determined a', measures of perceived distance lar perceived size, using a variation of the SDIH (Equa­ (D') obtained from a' would equal those obtained from tion 5) to compute D'. Also, D' was computed from W', W' (for the same perceived distance D~ or Dr> only if the using the head motion procedure and a variation of Equa­ MDIH illustrated in Figure 2B is a special case of the head tion 2. The two measures of D' varied in the same direc­ motion paradigm. In other words, the measurement of D' tion, although the measure using the SDIH gave larger from a' in the situation of Figure 2B involves the simul­ values of D' than those obtained from the head motion taneous solution of Equations 2 and 3 (computed in that procedure. In a later study (Gogel, Loomis, Newman, & study using the assumptions that ¢' = ¢, ()' = (), and K' Sharkey, 1985), measures of D' from applying the head = K). If, in Figure 2, the computed values of D' from motion procedure and from size adjustments using the a' and D' from W' are the same or are highly related, SDIH were obtained. As expected, both measures in­ it is likely that the same or similar processes are involved creased with increases in D', but again there was a ten­ in both the head motion paradigm and the MDIH (or dency for the measures obtained from the SDIH to be SDIH). The average D' results from the experiment are larger than those from the head motion procedure. The shown in Figure 3 for two different methods of measuring correlation between the average values resulting from the W' and a'. In one of these, the comparison method, the two kinds of measures in that study was very high (r = observer adjusted the separation between posts or the tilt 0.981). A third study, in which methodological differences ofa rotatable rod to be the same as the perceived horizon­ were minimized in obtaining measures of D' calculated tal motion (W') or perceived path of tilt (a') of the dis­ from W' and M', was done by Gogel and Tietz (1979). play previously seen with a moving head. In the other, The conditions used in that study are illustrated in Fig­ the duplication method, the observer, with the head sta­ ure 2. The situations represented in Figure 2A are basi­ tionary, duplicated on the display monitor the lateral mo­ cally the same as those of Figure l A. In Figure 2A, a tion (W') or tilt (a') ofthe path of motion previously seen physically stationary test point at the physical distance D with a moving head. Three physical distances, 30.0, 55.9, appeared to move laterally in the direction of the head or 96.4 em, were used with either monocular or binocu­ motion when it was perceived at D~ and against the head lar observation of the experimental display. The dashed

Figure 2. Drawings illustrating the vector addition of perceived horizontal motion (W') and perceived vertical motion (M'), concomitant with a perceived horizontal motion (K') of the head, so as to determine the perceived tilt (a') of the apparent path of motion of the test point. Modified from "A Comparison of Oculomotor and Motion Parallax Cues of Egocentric Distance" hy W. C. Gogel and J. D. Tietz, 1979, Vision Research, 19, p. 1162. Copyright 1979 by Pergamon Press. Adapted by permission. PHENOMINAL GEOMETRY 109

100 tions are initiated, which then determine the direction and amount of perceived illusory motion of the test object con­ comitant with the motion of the head. This theory will 80 here be called the VOR theory. A limitation of the VOR theory is that it is incomplete. '7:5 Pursuit eye movements can specify changes in the angu­ 60 lar direction and extent of perceived motion, but they can­ E Dupl rco t ion 0 Method not by themselves determine a linear (metric) extent of u., 0 perceived motion, such as a perception that the test ob­ \ 40 ject is moving laterally for a distance of 2 ft or 2 in. 0 Binocular • • For a linear perception, a perception of distance in ad­ Ccmp o r r so n dition to the angular information is needed. It is clear, Method 20 however, that the perception of illusory motion concomi­ Monocular 0 0 tant with head motion, as illustrated in Figure lA, is Bi nocu t o r • • linear. In a variety of experiments (Gogel, 198Ia), ob­ 0 0 20 40 60 80 100 servers had no difficulty in indicating linear judgments of the lateral extent of the perceived illusory motion, D' f rom v' which they accomplished usually by tactually adjusting the lateral separation of comparison rods. The inability Figure 3. The average results of computing the perceived distance of the VOR theory to specify the linear extent of illusory D' of the test point illustrated in Figure 2 from values of W' and motion is indicated in Figure 4, in which the perception a' obtained using a comparison and duplication method of mea­ surement and monocular or binocular vision. of angular motion 0' from VOR theory is compared with W' from the head motion procedure, as the head moves from Position I to Position 2. Suppose that in Figure 4 line in Figure 3 is the result expected if the D' values mea­ prisms are used to change the convergence distance (Dc) sured from W' were identical to the D' values measured of the test point (physically stationary at Dm) to a distance from a'. The obtained average values are closer to the nearer than its physical distance (Figure 4A) or farther dashed line for the duplication method than for the com­ than its physical distance (Figure 48). Also, suppose for parison method. For either the comparison or duplication convenience that, as a result of this change in conver­ method, however, the results clearly show that D' from gence, the perceived distance of the test point is at D~ W' and D' from a' are highly related. Since the same D' in Figure 4A and at Dr in Figure 48. The apparent angu­ values were being measured by the duplication and com­ lar concomitant motion expected from the angular pur- parison methods, the differences that remain between the data from these two methods probably reflect the sensi­ tivity of measurement of D' to procedural differences. 1 2 Thus, there is both logical and experimental evidence to support the assertion that the MDIH and the SDIH are o 0 special cases of the head motion paradigm. This conclu­ sion also suggests that the perception of extent obtained Dc with a stationary head is a special case of that obtained or with a moving head and that a theory of either must ex­ D~ plain both. ~ D ~:: jm Other Explanations of the Perceptions Illustrated by Figure 1 In a study by Post and Leibowitz (1982), it was con­ cluded that the perception of illusory motion associated -r with a stationary target and a laterally moving head (Fig­ ure IA) is a consequence of two eye movement systems. a As the head is moved laterally, one, the older system, acts to maintain fixation on the test object by means of b c b a c the vestibulo-ocular reflex (VOR), in which the gain of the reflex varies with the convergence of the eyes to the A B test object. If an error in convergence occurs, perhaps because of a tendency for the convergence to be displaced Figure 4. A comparison of the predictions from the vestibulo-ocular in the direction of its resting state, the eyes will tend to reflex theory, and from the head motion paradigm theory for the perceived motion (0' or W', respectively) of a single, physically sta­ tum too much or too little as the head is moved. To coun­ tionary test point, apparently moving concomitantly with a lateral ter this potential or actual loss of fixation, pursuit eye mo- motion of the head. 110 GOGEL

suit eye movement, as postulated by VOR theory, is 0', tude of the oculomotor adjustments according to the which is added because the convergence and physical dis­ reafference principle. This result, using geometrical tance of the test point differ. If the arrow indicating 0' stimuli, was obtained in the above studies by Leibowitz is to represent the perception of illusory motion concomi­ and his associates. Although such results also would be tant with the motion of the head, it is unclear where it expected from the SDIH, the reafference interpretation should be drawn. In their Figure 1, Post and Leibowitz avoids the concept of perceived distance. But, as noted (1982) place the arrow at the physical distance of the test by the authors, the reafference explanation can apply only point D« rather than, as might be expected from the con­ to the near distances over which oculomotor adjustments vergence cue of distance, at D~ or Dr. Ifthe arrow desig­ are effective. A different explanation must be used to ex­ nating angle 0' were placed at D~ or Dr, it is evident from plain the application of the SDIH to distances greater than Figure 4 that the magnitude and direction of the illusory several meters. Nor can the reafference explanation ap­ linear motion could depend upon apparent distance, as is ply to the head motion procedure. Thus, an additional expected from the head motion paradigm but is unspeci­ advantage of the theory of phenomenal geometry, unlike fied from VOR theory. For the perceived linear motion the VOR explanation of W' in Figure IA or the reaffer­ of a single test point viewed with a lateral motion of the ence principle explanation of S' in Figure IC, is that the head, both the VOR and the theory advocated in this paper same theory applies to all of the situations illustrated in could apply equally, but only ifVOR theory incorporated Figure 1. the factor of perceived distance. Perhaps the possibility should be considered that the magnitude of the vestibulo­ APPLICATION OF PHENOMENAL GEOMETRY ocular reflex is determined by the perceived distance of TO MULTIPLE OBJECTS OR TO the test point rather than by convergence per se. OBJECTS EXTENDED IN DEPTH In the situations of Figures 2A and 2B, the D' values computed from W' and ex', particularly when using the The above discussion of applications of phenomenal duplication procedure, are very similar (see Figure 3). geometry have been limited to a single test object. Phe­ But, in the situation of Figure 2B, the VOR was not in­ nomenal geometry is also expected to apply to multiple volved in the component of perceived motion (M') objects located at different perceived distances or to ob­ produced by the vertical motion of the test point. jects extended in depth. Consider the phenomenon of the However, the high correlations between D' measures ob­ illusory direction of rotation of Ames's trapezoidal win­ tained from the head motion procedure alone (Figure 2A), dow, illustrated in Figure 5. This drawing represents top presumably involving the VOR, and the added MDIH sit­ views of the physical (solid lines and filled circles) and uation (Figure 2B), not involving the VOR, suggests that perceived (dashed lines and open circles) orientations of the relation between convergence and the VOR added little the trapezoidal window. In the situation shown, the small that was not also available from perceived distance. In end of the window (n) is physically closer to the observer the case of the illusory motion of a single, physically sta­ than is the large end (f), but it always appears (n') to tionary test object viewed with a laterally moving head, be the more distant end. When the window physically ro­ it is often difficult to distinguish the possible contribu­ tates counterclockwise from n.I, to nsI«. it appears to ro­ tions of these two theories. This is because convergence tate clockwise from n;j; to nU~. The explanation ofthis can determine both the perceived distance of the test ob­ illusion that Ames (1952) proposed is consistent with the ject consistent with the head motion paradigm and the gain "best bet" hypothesis of transactional theory. It is as­ of the VOR consistent with VOR theory. sumed by the "best bet" hypothesis that two factors are Leibowitz (1971), Leibowitz and Moore (1966), and responsible for the illusory perception of direction of ro­ Leibowitz, Shiina, and Hennessy (1972) have interpreted tation in the situation illustrated in Figure 5. One is the the size perception obtained at near distances in terms of illusory perception of the depth orientation of the win­ the reafference principle and oculomotor adjustments, dow. The other is the decrease in the overall angular size instead of perceived distance, as asserted by the SDIH. ofthe window (exl > ex2) as the window physically ro­ According to these authors, it is expected from reaffer­ tates counterclockwise. The logical conclusion for the ob­ ence theory that the oculomotor adjustments initiated by servers from these two sources of information is that the efference commands become associated with the image window is rotating clockwise. According to the present changes expected on the retina when an object of con­ theory of phenomenal geometry, however, logic is not stant physical size is moved to a different distance from required. Ifthe observer correctly senses his or her head the observer. If the change in the size of the retinal im­ as stationary, the perceptual information provided by the age is consistent with that expected from the efference perspective cue to the distance of each end of the win­ command, the efference and reafference will be in agree­ dow, together with substantially accurate perceptions of ment and the apparent size of the object will remain con­ the directions to these ends, localizes the position of the stant (perfect size constancy). Conversely, if the visual window in perceived space at any instant of time and is angle of the object is kept constant, the lack of change sufficient to explain the illusory rotation. The logical con­ in retinal size with changes in efference will result in the sequence of the change in the visual angle of the window perceived size decreasing with the increase in the magni- is superfluous. There is no need to postulate a "best bet" PHENOMINAL GEOMETRY III

error in perceived depth combined with substantially ac­ curate perceptions of direction. At Head Position 1, the object appears at eU; and at Head Position 2, it appears at eU~. Thus, point!, appears to move a lateral distance W; in the same direction as the head motion, and point e' appears to move a lateral distance W~ opposite to the direction of the head motion. Ifthe perceived directions to e andjare accurate, as is suggested in this case by the solid direction lines, then the illusory apparent rotation of the extended object, as indicated by the dashed lines and the angle {3', is due to the error in the perception of depth. The same errors of orientation, eU; or eU~, also can occur when the display is viewed statically-that is, with the head physically stationary at different times at head Position 1 or 2, respectively. However, the change in the perceived orientation of ef, occurring in Figure 6A between Position 1 and Position 2, will be the same us­ ing static or dynamic conditions of observation only if the perceived error in depth is the same in the two conditions. When the head moves left and right between Positions I and 2, the situation contains the cue of relative motion parallax. Ifthis depth cue, which is not present when one is observing statically from Position 1 or 2, is effective to a significant degree, it will result in a reduction in the Re 1 I no apparent rotation ({3')of ejfor the dynamic as compared with the static condition. Such a result, however, would Figure 5. A top-view drawing illustrating a perceived rotation of remain consistent with the phenomenal geometry deter­ an example of Ames's trapezoidal window in a direction opposite to that of its physical rotation. This illusory direction of rotation mined by the factors of perceived direction, perceived dis­ is consistent with the correct perception of the relative directions tance, and the observer's awareness of being stationary of the parts of the window but an incorrect perception of their rela­ or moving. tive distances as produced by the illusory perspective of the win­ The drawing of Figure 6B represents the same errors dow. The solid circles and nonprimed notations represent the phys­ ical positionsof the two ends of the windowas the window physically in the perceived orientation, e'[', of an object, ej, extend­ rotates counterclockwise from n.j. to n,!,. The open circles and ing in depth, as in Figure 6A. In this case, however, the primed notations represent the apparent positions of the two ends observer always is physically stationary and the extended ofthe window, which is perceived to rotate clockwise from nU; to object physically moves left and right, repetitively, nU~. The physical visual angles subtended by the window are lXl through a perceived lateral distance, M', with M' equal at n.I. and lX, at n2!2' to the perceived head motion K' in Figure 6A. The near point e appears to move through K' + W~ and the far point process. In addition, the same factors of perceived dis­ j through K' - W; where W~ and W; are the same in tance and perceived direction are expected to determine Figures 6A and 6B. Ifthese perceptual conditions occur, the perceptions of object orientation, whether illusory per­ it follows that the perceived rotation, {3', will be the same ceptions of orientation and rotation using a trapezoidal in the situations ofFfgures 6A and 6B. Again, if the ex­ window, or accurate perceptions of orientation and rota­ tended object in Figure 6B is seen only when it is at the tion using a rectangular window, are produced. right or at the left physical position (static stimulus con­ Other phenomena associated with misperceptions of dition) instead of during the right or left motion of the orientation explained by exclusively considering the per­ points (dynamic stimulus condition), the perception of ceived directions and perceived distances of the points or orientation eU; and eU~ will be the same as in the dy­ parts of the display, together with the observer's infor­ namic condition, but only if the cue of relative motion mation of a constant or changing position of the head, are parallax in the dynamic condition is ineffective. Experi­ illustrated in Figure 6. The drawing of Figure 6A repre­ ments have been conducted to compare the results from sents a situation in which an object physically oriented such static and dynamic conditions (Gogel & Tietz, 1990). at ef but perceptually at an illusory orientation e'[' (for In this case, illusory orientations eU; and eU~ of the ob­ whatever reason) is viewed while the head is physically ject, similar in principle to those of Figures 6A and 6B, moved, repetitively, left and right through a perceived were produced by binocular disparity using a polarizing distance K'. The distance information producing the illu­ stereoscope. It was found, from an analysis of variance sory perception of orientation might be misleading per­ involving the situations of both Figure 6A and Figure 6B, spective cues (e.g., a trapezoidal window), or mislead­ that the illusory rotations {3' were not significantly differ­ ing cues of binocular disparity, or any other source of ent (at the .05 level) when the terminal (right or left) Il2 GOGEL

K't I.J 'e ------;.1

f 1 ,f2 ~m'= K-----7 e'2

2 A B

Figure 6. Top view drawings illustrating the perceived angle (3' between changes in the apparent orientation of an object efas a result of viewing the object at the illusory orientations e; f; and eU~. In Figure 6A, the object is physically station­ ary and is viewed while the head is moved through a sensed lateral distance K' . In Figure 68, the object is moved laterally through a perceived distance 5' equal to K' in Figure 6A. It is likely, under these conditions, that (3' in the situations represented by Figure 6A and Figure 68 will be equal. positions of the display relative to the head were viewed horizontal rotation of the mask also will be seen if the statically as compared with viewing the same terminal po­ observer is stationary and the mask is physically moved sitions during continuous head or stimulus motion. Nor laterally, or an illusory vertical rotation is seen if the mask from this analysis were the {3' values significantly differ­ is slanted physically in depth toward or away from a sta­ ent (at the .05 level) when the results from the situations tionary observer. The latter perceptions also are very of Figures 6A and 6B were compared. It seems that, for likely to be understood in terms of perceived distance, both the static and the dynamic conditions, the illusory perceived direction, and the observer's perception ofself­ rotations obtained in the situations of Figure 6 can be un­ motion or stationariness. derstood (assuming that the observer is aware of his or Additional demonstrations of illusory motions of ex­ her own motion or lack of it) solely in terms of the sen­ tended stimuli that reflect the contribution of the basic fac­ sory cues determining the perceived directions and per­ tors involved in phenomenal geometry are available. Il­ ceived distances of the parts of the stimulus. lusory motions similar in principle to those illustrated in An object that produces illusory motion identical in prin­ Figures 5-7 can be seen when perceived depth is in er­ ciple to the phenomena illustrated in Figures 5 and 6 is ror, using a Necker cube, or a Mach folded card, or a a transparent face mask reversible in depth (see Gregory, stereogram. A colleague, Jack Loomis, has called my at­ 1970). The illusory motion of the face mask is shown in tention to a demonstration likely to be of the same kind. Figure 7 for the case of a physically stationary mask A strip of white paper fashioned into a short tube and viewed monocularly by an observer laterally moving the placed upright on a surface will sometimes be seen as in­ head between Positions 1 and 2. As is shown in Fig­ verted in depth if it is viewed monocularly. When this ure 7A, the mask is physically oriented with the nose occurs, the tube will appear to be tilted at a shallow an­ pointing away from the observer, but the mask appears gle with respect to the surface, rather than appear upright. inverted; for example, the nose appears to be pointing Movement of the head will cause the tube to appear to toward the observer. Thus, the perceived depth (e1') of wobble. Interesting effects also are obtained if the tube the parts ofthe mask is opposite in direction to the physi­ at its illusory orientation is rolled on the surface while cal depth (ef) of these same parts. Figure 7B diagrams viewed with a stationary head. If a stereokinetic display the apparent rotation (3' of this physically stationary mask such as Duncan's "Leaning Tower" (Duncan, 1975) is as the head of the observer moves laterally. The similari­ monocularly viewed during its rotation, it will appear ties between Figures 6A and 7 are obvious. An illusory vividly to be extended in depth, with the motion of the PHENOMINAL GEOMETRY 113

Physical Apparent ure 7, the perceived direction in which the portrayal of a face appears to be pointing relative to the Observer in­ e volves the same processes, whether it is a realistic two­ dimensional portrait or actually a three-dimensional 0<::::)' replica in an illusory or a nonillusory orientation. The A magnitude or changes in direction of the perceived orien­ 2 tation in depth of the stimulus in these several cases will , differ as the head or the stimulus is moved, but the pro­ e, cesses underlying these phenomena are expected to be the same, consistent with phenomenal geometry. It is asserted that all of the above phenomena involv­ ing single or multiple test objects or objects extended in depth can be understood in terms of the theory of phe­ nomenal geometry. Also, consistent with the theory, if e' in addition to the illusory rotations, the entire stimulus 2 configuration appears closer to or farther from the ob­ server than its physical distance, the entire configuration B will appear to move laterally with or against the motion of the head. Figure 7. The effect of a perceptual reversal of the depth of a face mask (Figure 7A) on the illusory rotation {J' of the mask concomi­ SUMMARY OF THE EVIDENCE FOR tant with a lateral motion of the head (Figure 78). Similar effects THE THREE BASIC FACTORS are obtained if the head is stationary and the mask is moved (simi­ IN PHENOMENAL GEOMETRY lar to Figure 6B)or the mask is stationary while viewedwith a mov­ ing head (similar to Figure 6A). The perceptions can he explained by the correct perception of the directions to the parts of the mask, The contribution of perceived distance or depth to the the correct perception of the observer's own motion, and the illu­ perceptual localization of stimuli in phenomenal geome­ sory perception of depth within the mask. try is well established. Using the head motion procedure, it has been found that distance or depth cues that would be expected to modify perceived distance will also, as tower tracing an inverted cone around the observer's line predicted from phenomenal geometry, modify the illusory of sight to the base of the tower. If, instead of viewing translation (W') or illusory rotation ({3') concomitant with the display from a position directly above the base of the a motion ofthe head. In practice, the procedure has been tower, the head is located to the right or left, the axis of simplified by assuming that the conditions are such that the cone of perceived rotation will change so as to remain Equation 2 can be used. According to Equation 2, for symmetrical to the observer's line of sight, illustrating known values of K and Dp , D' can be calculated from the effect of varying perceived direction for a constant the observer's indication of W'. Or, Dp can be adjusted error in perceived distance. Whatever the cause of the er­ by moving the test object physically (W) in a direction ror in perceived depth in the stereokinetic phenomenon, (phase) opposite to its apparent motion (W') until W' is that error plus perceived direction and the apparent posi­ nulled (W' = 0). When this is achieved, it follows from tion of the observer will predictably determine what is Equation 2 that Dp = D'; that is, Dp at the null adjust­ perceived, consistent with phenomenal geometry. ment is a measure oftile perceived distance (D'). Results Still further displays for which the theory of phenomenal obtained using the head motion procedure have been ap­ geometry has implications are the changes in the perceived plied to the measurement of perceived depth or distance orientation of pictured objects designed to look three­ from oculomotor cues of convergence and accommoda­ dimensional (Cutting, 1988; Ellis, Smith, & McGreevy, tion (Gogel, 1977, 1982), perspective (Gogel & Tietz, 1987; Goldstein, 1979, 1987, 1988) as a consequence of 1974), relative size (Gogel, 1976), binocular disparity changes in the viewing position of the observer. Such (Gogel, 1980) the specific distance tendency (Gogel & phenomena are obviously related or identical to the orien­ Tietz, 1973), the equidistance tendency (Gogel, 1979a; tation in perceived depth associated with realistic pictures Gogel & Tietz, 1977), familiar size (Gogel, 1976), and ofa face, or to the orientation in perceived depth between absolute motion parallax (Gogel & Tietz, 1979). The im­ parts of a stereogram, or to the description of the orien­ portance of perceived direction for the geometry also is tation of the cone of perceived movement of a stereokinetic supported by the effect on W' in Equation 2 of changing display. As discussed in relation to Figures 5, 6, and 7, the pivot distance independently of the apparent distance, the perceptions associated with distance, direction, and as in the case of the null adjustment (Gogel, 1976, 1977, observer position or motion are sufficient to specify the 1979a, 1982; Gogel & Tietz, 1974, 1979; MacCracken, perceived orientation of either real or simulated stimuli Gogel, & Blum, 1980). The extent of the lateral motion as found either in two-dimensional drawings or pictures, of the head also has been found to be a significant factor or in three-dimensional displays. For example, in Fig- in determining W', as was demonstrated by the change 114 GOGEL in the magnitude of the illusory motion, as a consequence and predictably as the perceived depth is changed (Gogel, of a change in the magnitude of the head motion, for a 1980). Two situations il1ustrating this prediction are constant perceived distance (Gogel & Tietz, 1977). shown in Figure 8. In this figure, a physical1y stationary rod at a constant physical slant in depth is perceived at OTHER EXPLANATIONS OF THE PERCEIVED a different il1usoryslant in Figures 8A and 8B. As a con­ ROTATION OF MULTIPLE OBJECTS sequence of the different errors in perceived depth in the OR OBJECTS EXTENDED IN DEPTH two situations, the apparent rotation shown by the angle {3' is predictably different in the two cases. This is ex­ The VOR theory cannot explain the illusory rotation pected to occur, according to phenomenal geometry, even of stimuli extended in depth concomitant with head mo­ though the retinal motion ofthe top (t) relative to the bot­ tion. An illusory rotation signifies that different parts of tom (b) of the rod, as determined by the physical slant the object appear to move simultaneously in opposite of the rod, is identical in the two instances. In Figure 8A, directions, whereas only a perception of motion in one the direction of the apparent rotation ofthe rod (although direction can occur at anyone time from pursuit eye not the amount) is consistent with that expected exclu­ movements. sively from the retinal motion. In Figure 8B, neither the Another explanation of illusory rotation concomitant direction of apparent rotation of the rod nor its apparent with head motion is that it is a direct response to relative amount is predictable exclusively from the retinal motion. motions on the retina (for discussions of this problem, That results consistent with these predictions from phe­ see Gogel, 1981a, 1983; Peterson & Shyi, 1988; Shebilske nomenal geometry are obtained has been shown in the & Proffitt, 1981, 1983). Whenever stimuli are distributed study by Gogel (1980) of this phenomenon. The predict­ in depth, their images on the eye will undergo different ability from phenomenal geometry of these apparent rota­ motions as the head is moved lateral1y. But these reti­ tions obtained with a moving head provides clear limita­ nal motions are not sufficient to determine the perceived tions of the ability to understand the geometry of perceived illusory rotation, as is shown by keeping the same physi­ space from retinal (or reafference) information without cal stimulus distribution in depth while varying the per­ considering the role of perceived depth or distance. ceived depth. In this case, the perceived rotation, accord­ Stereograms, because they produce an illusory depth ing to phenomenal geometry, will change systematical1y between parts of the display, provide another example of

t'I

t 2

B

Figure 8. Apparent illusory rotation concomitant with a Iate~ motion of the head of a line (tb) slanted in depth, with t physically more distant than b. In Figure 8A, the perceived slant is in the same direction as the physical slant but is greater in magnitude, and the apparent rotation (fJ') of the line is consistent in direction (but not in magnitude) with that expected solelyfrom the relative motion on the retina. In Figure 88, the perceived slant of the line is in a direction oppo­ site to the direction of the physical slant, and the apparent rotation fJ' of the line is opposite to that expected solely from the direction of the relative motion on the retina. (From "The Sensing of Retinal Motion" by W. C. Gogel, 1980, Perception & Psychophysics, 28, p. 157. Copyright 1980 by The Psychonomic Society, Inc. Reprinted by permission.) PHENOMINAL GEOMETRY us

the apparent rotation illustrated in Figures 6A, 7, and 8 ometry, if a physically three-dimensional but perceptu­ and obtained as the head is moved laterally. Rock (\983) ally depth-inverted object had been used instead of the has classified this and the similar apparent rotation ob­ two-dimensional drawing. It is difficult to see how these tained with an apparent depth inversion of a Necker cube phenomena can be explained without considering the mag­ as cases of perceptual intelligence. It is as though the per­ nitude and direction of errors in perceived depth. This ceived depth generates certain expectations of stimulus limitation in the theory remains if the process of com­ changes on the retina as the head is moved. Since these pensation expressed by the reafference principle (von changes do not occur or are inconsistent with the expec­ Holst & Mittelstaedt, 1950) that normally is applied to tations, a reasonable conclusion is that the scene must have eye movements is extended to situations involving a mo­ rotated in the direction required in order to produce the tion of the head. The reafference principle is designed to stimulus changes obtained. As stated by Rock (\983): "It permit the observer to distinguish between proximal would seem that the perceptual system 'knows' certain changes produced by the observer's own motion and laws of optics that normally obtain and then 'interprets' changes produced by motions of the physical stimulus. seeming departures from these laws in such a way as to When a lateral motion of the head is involved, it is obvi­ be compatible with them. In doing so, it invents or con­ ous from the previous discussion that the ability to per­ structs environmental events that logically would have to ceive a stationary object as stationary despite retinal mo­ be occurring to account for the unexpected stimulus tion will depend on the perceived distance of and perceived change or lack of change" (pp. 10-11). According to the depth within the stimulus object, in addition to informa­ phenomenal geometry explanation favored in the present tion about perceived directions and the observer's own article, the only requirement for explaining these phe­ motion. Indeed, it is unlikely, in the absence of a back­ nomena is that the object or parts be localized in percep­ ground environment, that the observer can discriminate tual space by whatever information is effectively avail­ between illusory and real motion or rotation of a test able to determine perceived direction, perceived distance, stimulus. Evidence that real and illusory motion can be and the observer's perception of self-motion or station­ perceptually equivalent is found in (1) studies that show ariness. The same localization processes-and, therefore, that illusory motion concomitant with head motion can the same explanation-apply whether either the observer be nulled out by real motion (Gogel, 1977, 1979a, 1981a, or the stimulus is moving or stationary and whether the 1982; Gogel & Tietz, 1979; MacCracken et al., 1980), perception of direction and/or depth is veridical or grossly (2) studies in which the illusory and real motion add vee­ in error. Knowledge of the laws of optics by the percep­ torially (Gogel, 1979a; Gogel & Tietz, 1979), and (3) a tual system is not needed. study indicating the inability to distinguish perceptually Wallach (1985, 1987) explains the perception of illu­ between a real and an illusory rotation of a Necker cube sory rotation of a stationary object as viewed by a later­ (Peterson & Shyi, 1988). Particularly difficult for either ally moving observer in terms of compensation processes. a compensation theory or a theory according to which per­ Consider the situation in which a stationary object that ception is similar to problem solving is the similarity of is physically three-dimensional is viewed as the observer's perceived differences in stimulus orientation that can oc­ head moves laterally. The changes in the position of the cur with static and dynamic stimulus/observer conditions, head resulting from the lateral motion will produce as has been discussed with respect to Figure 6. In the case changes in the image of the stationary object on the ret­ of the static stimulus/observer conditions, compensation ina. But, because the observer has knowledge of his own from observer motion is not available, nor is there a motion, these changes will not result in the perception that problem to solve. the object has rotated. The information about the motion of the head compensates for the changing proximal stimu­ HYPOTHESES IN APPLYING THE lus, so that the object is perceived to remain stationary. THEORY OJ" PHENOMENAL GEOMETRY Suppose, however, that instead of an actual three-dimen­ sional object, a realistic two-dimensional drawing of the The Hypothesis of Cue Equivalence three-dimensional object is presented such that the object In the interest of predictive parsimony, it is assumed, is perceived to be three-dimensional. In this case, as the in applying the theory of phenomenal geometry, that the observer's head moves laterally, the retinal image of the factors of perceived direction, perceived distance, or the drawing undergoes little, if any, change. But the com­ observer's sensing of self-position will have the same con­ pensation will still be applied, resulting in the perception sequences for the perceived localization of stimulus points that the object has rotated. and, hence, for the derived perceptions of size, shape, A limitation of compensation theory in explaining illu­ orientation, and motion, regardless of which sources of sory or lack of illusory rotation as the head is moved later­ information determine these perceptions. For example, ally is that it does not take into account the role of errors the perceived width (S') of a stimulus extending between in perceived depth in these phenomena. For example, in two perceived directions «(}') will have the same relation the case just cited, a more exaggerated illusory rotation to perceived distance (D'), regardless of whether the per­ of the object for the same perceived depth would have ceived distance is determined by one or another of the been obtained, according to the theory of phenomenal ge- binocular or monocular cues of distance or by any other 116 GOGEL factor capable of modifying perceived distance, such as to measure the perceived distance of the test object (Ittel­ the effect ofadjacent context (Gogel & Mershon, 1977), son, 1952, chap. 7). A second kind of evidence is the abil­ or the allocation of attention (Gogel et al., 1985). Simi­ ity of two motions involving different sources ofperceived larly, the effect of perceived direction on the derived per­ motion (e.g., real and illusory) to add vectorially (Gogel, ceptions will be the same, whether the perceived direc­ 1979a, 1979b; Gogel & Tietz, 1977; Wallach, Bacon, & tion is modified by pursuit eye motions or by the Roelofs' Schulman, 1978; Wallach & Frey, 1969). Under these effect (I935) or by any other factor, as long as the modifi­ conditions, the visual system presumably accepts both cation is indeed perceptual. This hypothesis, called the sources as contributing to the same perception. A third hypothesis ofcue equivalence or cue substitution, has the kind of evidence is found in situations in which the per­ effect of making the phenomenal geometry very par­ ception represents a compromise between conflicting cues simonious. Perception is the final channel through which (Gillam, 1968; Gogel, 1970; Ittelson, 1968; Schriver, a variety ofeffects or cues are funneled to produce a par­ 1925; van der Meer, 1979). In these situations, the differ­ ticular perceptual outcome. But this same perceptual out­ ent sources contribute to the same final perception, which come can be produced by the appropriate values of any quantitatively is often unlike the perception expected from other effective set or mixture of cues or other sources of either source alone. A somewhat more elaborate discus­ perceptual information. sion of the evidence for the assumption of cue equiva­ There are at least three kinds of evidence for the lence is found in Gogel (1984, pp. 32-36). hypothesis of cue equivalence-evidence, that is, that different sources of information or cues can contribute The Hypothesis of Internal Consistency to or determine the same perception. One kind results A second hypothesis involved in the theory of phe­ from a procedure for measuring the perceived distance nomenal geometry is that the geometry is internally con­ ofor depth between parts ofthe visual field. This method sistent. Internal consistency requires that the perceptions consists of adjusting a probe object (often a luminous of size, shape, orientation, and motion that are derived point) to appear at the same distance as the parts of the from perceived spatial locations defined by perceptions visual scene of interest (Deregowski, 1980; Gogel, 1954, of direction, distance, and self-position must be consis­ 1970; Gogel & Harker, 1955; Gregory, 1968). The probe tent with those perceived spatial locations. Examples of object may utilize cues of distance different from those responses (R) obtained from a stationary observer that found in the portion of the visual field to which the probes' would indicate internal consistency or inconsistency, if perceived distance is adjusted. Or a comparison field, they were interpreted as measuring perceptions, are con­ again involving a configuration ofdistance or depth cues trasted in Figure 9. In Figure 9, an observer presented different from those found in the test object, may be used with a rectangular stimulus at some distance is instructed

A )I~!R

B )1~~rSR c

Figure 9. Drawings contrasting internal consistency (Figure 9A) and a lack of internal consistency (Figures 98 and 9<::) in the observer's reports (R) of the visual angle (JR, the distance DR, and the size SRof the stimulus object. Unlike the situation of Figure 9A, in Figures 98 and 9<::, the reported size of the stimu­ lus is too large or too small, respectively, to be consistent with the reported visual angle of the stimulus at its reported distance. PHENOMINAL GEOMETRY 117

to indicate its perceived distance D', perceived size S', EXPLANAnONS FOR SEEMING FAILURES and the change in perceived direction 8' between its top OF INTERNAL CONSISTENCY IN and bottom. The observer's responses (R) are labeled SR, PHENOMENAL GEOMETRY DR, and 8R. Assuming that these responses are measures of what is actually perceived, in Figure 9A, since SR/DR There are several possible explanations for these seem­ = tan8R, it follows that S' /D' = tan 8', in agreement with ing failures of internal consistency in the phenomenal ge­ a strict form of the SDIH. It can be concluded in this case ometry, as they are illustrated by failures of the SDIH. that the phenomenal geometry is internally consistent. In Using binocularly viewed points of light in an otherwise Figures 9B and 9C, however, SR/DR "* tan8R; that is, dark room, Foley (1972) found that observers adjusted SR does not fit the extent specified by the combined in­ a frontal extent to be about one half as large as its dis­ formation given by DR and 8R. If it can be assumed that tance from the observer in order for the two extents to the responses shown in Figures 9B and 9C also measure appear equal. This implies that the constant of propor­ perceived characteristics, it can be concluded, in disagree­ tionality in the SDIH relating S' ID' and tan 8 is consider­ ment with the strict form of the SDIH, that S'ID' "* tan8'. ably greater than unity. This result, together with that from In these cases, the hypothesis of internal consistency in a previous study (Foley, 1965) in which it was found that the phenomenal geometry is violated. It is seldom that perceived visual angle usually exceeded physical visual responses measuring all three perceptions (S', D', and 8') angle by about only 10%, suggested that the intrinsic ge­ are obtained in experiments. However, it is not unusual ometry of visual space was non-Euclidean. It was found, to find instances often similar to those indicated by Figures however, that subjective space became more Euclidean 9B and 9C, in which the strict form of the SDIH exem­ as distance information was increased (Foley, 1972). plified by Figure 9A and Equation 6 and needed for phe­ Also, it should be noted that the values of the constant nomenal internal consistency is not satisfied (see Epstein of proportionality in the SDIH obtained in various experi­ & Landauer, 1969; Foley, 1968, 1972; Gogel, 1977; ments can vary considerably for different observers and Gogel et al., 1985; Gogel & Sturm, 1971).2 different procedures, with this value possibly becoming A still more extreme case in which the strict form of less than one for quite small values of 8 (Gogel, 1977; the SDIH is not always consistent with the obtained Gogel & Sturm, 1971; Landauer & Epstein, 1969). But, responses is called the size-distance paradox. This para­ whether the phenomenal geometry is described as Euclid­ dox is clearly inconsistent with the requirement of inter­ ean or as non-Euclidean does not remove the need to con­ nal consistency in the phenomenal geometry if, indeed, sider the geometry in terms of the three basic factors of the measurements obtained in this paradox are purely per­ perceived direction, perceived distance, and the perceived ceptual. In the size-distance paradox, objects of the same motion or stationariness of the observer. shape (same possible identification) and subtending the McCready (1965, 1985, 1986) has advocated that the same visual angle are presented at different distances from perceived visual angle (8') be used instead of the physi­ the observer, often with oculomotor cues being the only cal visual angle (8) in the SDIH. Furthermore, McCready source of distance information. It is found that although (1985, 1986) has proposed that the size-distance paradox, the change in the perceived size of the test object is con­ including the paradox of the moon illusion, can be ex­ sistent with its change in physical size, as is expected from plained in terms of variations in 8'. This explanation would the SDIH, the change in its perceived distance is not con­ be plausible if sufficiently large changes in 8' (despite the sistent with the SDIH (Biersdorf, 1966; Heinemann, Tul­ constant 8) occurred in the object misjudged in distance ving, & Nachmias, 1959; Komoda & Ono, 1974; Ono, in the size-distance paradox. Similarly, if the 8' of the Muter, & Mitson, 1974). Another example of this para­ moon on the horizon in-the moon illusion were sufficiently dox is the moon illusion, in which the horizon moon ap­ larger than the 8' ofthe zenith moon, the paradox pro­ pears larger than the zenith moon but often is reported duced by the report that the horizon moon is seen as both as being closer not farther than the zenith moon (as would larger and nearer than the zenith moon would be resolved, be required by the SDIH and by internal consistency). A and in both cases the seeming inconsistency with respect study by Swanston and Gogel (1986) provides another in­ to the SDIH and phenomenal geometry would disappear. stance of results grossly inconsistent with the SDIH. A Unfortunately, there is no evidence that the needed luminous line viewed monocularly in an otherwise dark changes in 8' do occur. environment was made to increase or decrease in length A third explanation of the internal geometrical incon­ so as to represent approach or recession from the ob­ sistencies indicated by failures of the SDIH is the modifi­ server. Although the expected changes in perceived depth cation of the results by cognitive (nonperceptual) factors. were obtained, it also was found that the perceived length One such factor that has been identified is called off-sized ofthe line changed by amounts approximating its change judgments (or Off-sized perceptions). 3 This is the notion in angular length. This result is inconsistent with the that an object seen to be larger or smaller than its assumed SDIH. To the degree that the line appeared to approach or standard (normal) size is judged to be at a nearer or or recede, to that degree its perceived size should have farther distance, respectively, than the distance at which appeared to remain more constant than would be expected it is perceived. Let S, be the remembered size of the stan­ from its changing visual angle. dard (s). Let S: be the perceived size S' of the test object 118 GOGEL

(t) being viewed. Off-sized judgments occur whenever the be seen to be about one third the size of a normal guitar, ratio of the assumed size (Ss) provided by experience with such that Ss/S; is about 3. The observer's report of dis­ a standard size and the perceived size (S;) of the test ob­ tance DR, however, often will deviate from the perceived ject differ-that is, whenever Ss/S; * 1, where Ss/S; is distance toward the physical distance of the guitar as a the inverse of the off-sized judgment. For example, if a result of the off-sized judgment. Ifthe off-sized judgment playing card with a standard size of5.7 x 8.9 em is seen were to completely determine the verbal report (usually to be 2.35 x 4.45 em in size, it is seen as a half-sized the effect is only partial), the report of distance would playing card (Ss/S; = 2). Since the standard or normal­ be accurate; that is, DR = 3 m . 3 = 9 m, according to sized playing card is customarily seen at various distances Equation 8. As is indicated by this example, the useful­ from the observer, it provides only a standard size, not ness of this cognitive process resulting from off-sized a standard distance. Some standard objects provide both. judgments is that the verbal report of distance can be more For example, for a test object presented in the size­ accurate than it would have been had it been based ex­ distance paradox, the standard size (Ss) is the memory of clusively on the erroneously perceived distance. But, it the perceived size of the object of the same shape pre­ seems that the observer, when asked to judge the perceived sented first, and the standard distance (D s) is the memory size of the familiar object, often responds with the size of the perceived distance of this first presentation. The as it is perceived rather than with the familiar size. The standard size S, and the standard distance D s, when ap­ combination of perceived size and cognitively modified plied in determining the observer's response to the test DR produces seeming variations in the constant ofpropor­ object, are considered to be cognitive. They both involve tionality relating S' /D' and tan () in the SDIH. Results of the storage and retrieval of an internal (mental) represen­ this kind are consistent with data obtained by Gogel tation associated with the standard object. (1969a) and Gogel and DaSilva (1987a). The equation for calculating the observer's frequent As is suggested by the discussion of the size-distance response to distance (DR), when the observer is presented paradox, off-sized judgments are not limited to familiar with a test object that is judged to be off-sized and for objects. They can occur in any situation in which a stan­ which a standard distance is not available, is dard size is remembered as different from the size of the test object. Coltheart (1968) and Gogel and Sturm (1971) (8) have proposed that test objects such as a disk or rectangle In Equation 8, D; is the perceived distance of the test that have a range ofphysical sizes nevertheless often tend object and DR is the observer's response of the distance to have an assumed or standard size. The ranges of such ofthe test object reflecting the effect of the off-sized judg­ "rionfamiliar" objects generally are greater than those ment. In a situation such as the size-distance paradox or of more "familiar" objects, such as a guitar, so that the the moon illusion, in which a standard distance (D s) as judgment that the disk or rectangle is off-sized is likely well as a standard size (Ss) is available from memory, the to be less precise than in the case of the guitar. Neverthe­ remembered value of D, is likely to be used by the ob­ less, off-sized judgments with plain geometrical objects server in place of D;, and Equation 8 then becomes seem to occur. This is supported by the study of Gogel and Sturm (1971) in which it was found, even on the first (9) presentation of a rectangle viewed under reduced condi­ The consequences of Equation 8 for the SDIH and thus tions, that substantial deviations from unity occurred for by implication for the hypothesis of internal consistency the proportionality constant relating S'ID' and tan (). These ofphenomenal geometry are discussed in several articles deviations were explained by the effect of off-sized judg­ (Gogel & DaSilva, 1987a, 1987b; Gogel & Sturm, 1971; ments. Consistent with this interpretation, it is sometimes Swanston & Gogel, 1986) and will only be summarized found that the reported distance of the first presentation here. As a hypothetical example of the effect of an off­ ofa large geometrical stimulus is less than that ofa small sized judgment on the response to distance DR that is con­ geometrical stimulus, even though a successive compari­ sistent in principle with the results from several studies son between the stimuli is avoided (Coltheart, 1968, Ex­ (Gogel, 1969a; Gogel & DaSilva, 1987a; Gogel & New­ periment 5; Gogel & Sturm, 1971). If it is possible for ton, 1969), consider the case of a familiar object (e.g., off-sized judgments to occur with any frontal extent, the a guitar of normal size) located 9 m from an observer and results obtained by Foley (1972) might be explained by viewed monocularly in a dark environment. In this case, Equation 8 and by the notion that the frontal extent, be­ it is likely that the guitar will appear to the observer to cause ofits large off-sized appearance, was judged to be be at a closer distance, perhaps at about 3 m. This error closer than it would if it were not judged as off-sized and is considered to be a consequence of the "specific dis­ that this occurred without a proportionate change in its tance tendency," which, in the absence of effective perceived visual angle. Ifthis were found to be the case, egocentric cues of distance, results in an object's appear­ an explanation in terms of non-Euclidean geometry would ing at a relatively near distance from the observer, regard­ be unnecessary. less of its physical distance (Gogel, 1969b; Gogel & Tietz, Off-sized judgments also have been applied as an ex­ 1973). Because the guitar appears at about one third of planation of the failure of the SDIH to predict the DR its physical distance, in agreement with the SDIH, it will results obtained in the optical expansion of a luminous PHENOMINAL GEOMETRY 119

line presented in an otherwise dark environment (Swan­ had a clear effect on both indirect measures of perceived ston & Gogel, 1986). It was assumed that the perceived sagittal motion obtained with the head motion procedure size of the initial presentation in the expansion series and direct sagittal measures obtained from adjustments provided the standard size S, for judging the subsequent ofthe separation ofcomparison rods. Ifit can be assumed stimuli in the expansion as continuously larger off-sized that the perceived and physical visual angle were approx­ objects. Equation 9, which involves the memory of both imately equal in Experiment 3 of the Swanston and Gogel a standard size, Ss,and a standard distance, Ds, is designed study, the results suggest that at least under rather spe­ to predict the reports of distance DR obtained from the cial conditions a cognitive process can disturb the inter­ size-distance paradox. In this case, the Ss, to be used in nal consistency of phenomenal geometry. Effects super­ the off-sized judgment of the test object, is the memory posed on the primary (phenomenal) process by cognitive of the perceived size of the identically shaped standard factors, such as off-sized judgments, whether or not they object presented previously; D, is the memory ofthe per­ modify perceived distance, are called the secondary ceived distance of this standard object; and S; is the per­ process. The secondary process normally tends to reduce ceived size of the test object presented second. For the systematic errors found in the primary process and caused moon illusion, in which the horizon moon is reported to by limitations in sensory information. Such limitations oc­ be larger and closer than the zenith moon, in Equation 9, cur in reduced conditions of observation, or at far dis­ S, and D, are the remembered size and distance of the tances, or for isolated stimuli. As suggested by Equations zenith moon, and S; is the perceived size of the moon on 8 and 9, the primary process provides a basis for the the horizon. secondary process. The contribution of the secondary pro­ According to this account, the internal consistency of cess necessarily decreases, however, as the primary pro­ phenomenal geometry can be disturbed by the intrusion cess becomes the main determiner of the response. ofmemories of the perceived size and sometimes the per­ Day and Parks (1989) have recently discussed the ceived distance of standard objects observed, often repeat­ problems presented by a number of paradoxes. It seems edly, in the immediate or more remote past. But, if this difficult to achieve general laws of visual perception in seeming inconsistency is only found in the responses and view of the seeming inconsistencies demonstrated by the not in the perceptions, which the responses are intended paradoxes. For example, the SDIH clearly is necessary to measure, the phenomenal geometry itself cannot be said for the explanation of many phenomena. But, also, it fails to be disturbed. Clearly, off-sized judgments contain a to explain other (e.g., paradoxical) data to which it might cognitive component that makes their possible contribution be expected to apply. Hopefully, the distinction between to the response of size inappropriate for an examination primary processes, as defined by phenomenal geometry, of the SDIH, which is concerned only with perceived and secondary processes, involving cognitive effects, can characteristics. It is of interest, however, whether off­ contribute to the resolution of some of these difficulties. sized perceptions under any conditions can modify the per­ The combination of the two different processes, since they ception of distance or, as in the example of the guitar, sometimes follow different rules, can result in data that can modify only the judgment ofor response to distance, seem paradoxical. The study of the modification or dis­ with the actual perception of distance unchanged. This tortion of responses by cognition is in its early stages. question is considered in three studies discussed in Gogel It is possible that, through investigation of a number of and DaSilva (1987b). To examine this question, measures these paradoxes, the distinction between perceptual and of perceived distance, using the head motion procedure cognitive processes can be sharpened, so that the conse­ ofFigure lA and Equation 2, were compared with more quences of cognitive effects are not confused with the direct measures of perceived distance, such as verbal purely perceptual processes associated with phenomenal reports. Unlike direct measures, the head motion proce­ geometry. dure is assumed to be essentially immune to cognitive in­ trusions based on off-sized judgments. It was found IMPLICATIONS OF (Gogel, 1981b) that information leading to off-sizedjudg­ PHENOMENAL GEOMETRY ments of the test object, provided by presenting large or small examples of familiar objects of the same shape as The Role of Perceived Distance the test object, had no effect on perceived distance as According to phenomenal geometry, the localization of measured by the head motion procedure. But, such infor­ a stimulus in phenomenal space requires a combination mation had a large effect on verbal reports ofthe distance of perceived distance, perceived direction, and the per­ ofthe test object. If, however, the information determin­ ception of the position or motion of the self. Only when ing the familiar size was contained in the test stimulus perceptual values of each of these factors are available (i.e., the test object was a familiar object), it had a small are perceptions of size, motion, orientation, or shape pos­ effect on perceived distance as measured by the head mo­ sible. A size or motion on the retina essentially is speci­ tion procedure, but it had a large effect on the verbal fied by the distribution of visual directions or changes in report of distance (Gogel, 1976). In the case of optical visual direction produced by the parts of the stimuli at expansion (Swanston & Gogel, 1986, Experiment 3), it the nodal point of the eye. Without a perception of dis­ seems that the off-sized judgments hypothesized to occur tance, however, neither the retinal image nor the perceived 120 GOGEL

directions associated with the parts of the retinal image they will appear to be essentially proportional in size and alone can produce a linear perception of size or motion. shape to the retinal image. But, that is simply a happen­ Conversely, if a change in the linear perception of size stance of their perceived equidistance. or motion occurs with the perceived directions to the A perceptual aspect ofstimuli, termed phenomenal ex­ stimulus and the perceived position of the self held con­ tensity, is described by Rock (1977, 1983) and Rock and stant, the linear change in perceived size or motion must McDermott (1964). Probably phenomenal extensity is be attributed to a change in perceived distance. Thus, in similar to, or the same as, the perceived visual angle (J'. Figure lA, the linear change in the magnitude and direc­ In the context of phenomenal geometry, it is important tion of the perceived motion W~ and W; for the same (J' to be able to distinguish experimentally between the per­ and K' is the result of the difference between D~ and D;. ception of visual angle, (J', and the perception of linear Similarly, the changes in perceived lateral (linear) mo­ size, S'. A method for achieving this is suggested by Gogel tion of stimuli diagrammed in Figures 2, 4, 5, 6, 7, and and Eby (1990). The method involves the tactile adjust­ 8 can be understood in terms of differences in perceived ment of the lateral separation of two unseen posts that, distance for a constant change in perceived direction and as a pair, can be positioned at different distances from for a constant perceived motion of the observer's head. a stationary observer while always remaining in a fronto­ The research associated with Figure 8 clearly demon­ parallel plane. To measure ()' (and probably phenomenal strates the inability of retinal (or directional) changes in­ extensity), the observer is instructed to adjust the lateral dependently of perceived distances to determine the per­ position of the right post until it appears to be on the direc­ ceived rotation, (J', of the stimulus. This is clear, because tion from the observer to the right edge of the stimulus. (J' is opposite in direction in Situations A and B, whereas Similarly, the left post is adjusted laterally until it appears the direction of relative retinal motion on the retina is the to be on the direction to the left edge ofthe stimulus. This same for the two situations. It seems unlikely from the is repeated with the pair of posts physically positioned research associated with these demonstrations that linear at several different distances from the observer while re­ perceptions of size, motion, shape, and orientation can maining in a frontoparallel plane. If()' is being measured, be explained in any situation by perceptions determined the lateral separation between the posts resulting from exclusively by the size, motion, shape, or orientation of their adjustment will increase with increases in the phys­ the retinal image. ical distance of the pair of posts from the observer. To A point of view seemingly at variance with the above measure S', the observer is instructed to separate the posts comments reflects the observation that as distance infor­ laterally until the distance between the posts is the same mation is increasingly reduced, either by removing cues as the perceived lateral extent of the stimulus. IfS' is be­ of distance or by withdrawing attention from the stimu­ ing measured, the lateral separation of the posts should lus, the perceptions of size and shape increasingly resem­ not systematically change with a change in the distance ble the proximal stimulus. This has resulted in the postula­ from the observer of the frontoparalle1 plane containing tion of different modes of perception. One mode involves the posts. The angle obtained from the former adjustments a direct response to the retinal image and is not depen­ is a measure of ()'. The constant lateral extent obtained dent on perceived distance. The other is capable of pro­ from the latter adjustments is a measure of S'. ducing size and shape constancy or, more generally, is consistent with the requirements ofthe size-distance and The Common Occurrence of Errors shape-slant invariance hypotheses (Boring, 1946; Epstein in Perceived Distance & Babler, 1989; Epstein & Broota, 1986; Epstein & The theory of phenomenal geometry asserts that the per­ Lovitts, 1985; Holway & Boring, 1941; Mack, 1978; ceived distance of a stimulus is essential to its location Rock, 1977, 1983). In terms of the theory presented in in phenomenal space. If it can be assumed in a situation this paper, a seeming regression toward the proximal that ()' = (J and K' = K in Equation 1, it follows that an stimulus and, thus, toward independence from perceived illusory lateral motion (W') of the test object concomi­ distance as cue effectiveness is lessened either by cue tant with the lateral motion of the head indicates that the reduction or by withdrawal of attention is misleading perception of the distance of the test object is in error. (Gogel & DaSilva, 1987b; Gogel & Sharkey, 1989). In­ This relationship can be used to identify the kinds of con­ stead of indicating independence from perceived distance, ditions in which errors in perceived distance are likely such results indicate only that the stimuli are perceived to be found. As indicated throughout this paper, the usual as displaced toward a constant perceived distance defined cues of distance or depth, such as convergence, binocu­ by the specific distance tendency and/or the equidistance lar disparity, or perspective, can readily be manipulated tendency. The latter tendency refers to the finding that to produce large errors in D' as signaled by substantial stimuli or parts of stimuli at different physical distances values of W'. Also, errors in perceived distance can oc­ increasingly tend to appear at the same distance (equidis­ cur from less traditional distance or depth factors, such tant) as information about their actual distances is increas­ as the specific distance and equidistance tendencies, the ingly reduced (Gogel, 1965). Whenever stimuli are per­ distance at which the eyes are fixated, and possibly the ceived to be at the same distance from the observer (for distance to which attention is directed (Gogel, 1979a; whatever reason), regardless of their physical distances, Gogel & Tietz, 1973, 1977). Several of these less tradi- PHENOMINAL GEOMETRY 121 tional factors can produce clear values of W' even for sit­ decrease in the perceived depth between the fingers, which uations involving distances from the observer within is produced by the equidistance tendency and augmented which traditional cues are available and are considered by the distance of fixation. Similar results can be obtained to be quite effective. Two factors likely to be involved between other objects in the visual field when one of them in producing errors in perceived distance under these con­ is fixated. Results of this kind present difficulties for the ditions are the equidistance tendency combined with the theory of direct perception (Gibson, 1950, 1966), which distance at which the gaze is fixated. In a study by Gogel assumes that the visual perception of the world is essen­ and Tietz (1977), a vertically moving test point at a con­ tially veridical. In general, as the observer moves through stant distance from the observer was suspended in a visual space, nonfixated objects, although physically stationary, field. The floor of the alley was covered with black cloth often appear to move. This illusory motion usually is dis­ containing white polka dots. The observation was either regarded by the observer, because the nonfixated objects binocular or monocular, and the observers fixated either are part of the surrounding visual field, which the observer the luminous test point or a fixation object resting on the knows to be physically stationary. However, if this sur­ alley floor and physically closer or farther than the dis­ rounding field is absent, the illusory motion becomes in­ tance of the test point. To measure W' in order to calcu­ distinguishable from real motion (Peterson & Shyi, 1988). late D', measures of the apparent tilt (a') of the vertically But, whether or not the illusory motion is disregarded by moving test point, viewed with a lateral motion of the the observer, the errors in perceived depth producing the head, were obtained (see Figure 2B for a similar proce­ illusory motion remain undiminished. dure). The perceived distance of the test point was found to vary directly with changes in the distance of fixation Indirect Measures of Perceived Distance for binocular as well as monocular observation. In other According to phenomenal geometry, errors in the per­ words, the nonfixated test point was perceptually displaced ception of motions, orientations, sizes, or shapes, as the toward the perceived distance of the fixated object, and, observer moves through the environment, reflect the in the case of binocular observation, this occurred despite presence of errors in the perception of distance, direc­ the binocular disparity between the test point and fixa­ tion, or self-motion. The relation between these derived tion object or between the test point and the dots on the and basic characteristics can be measured by indirect alley floor. Attending to the near or far object while fix­ procedures that significantly exclude cognitive processes ating the test point had a similar but smaller effect. The from modifying the observer's responses. The use of per­ robust effect of fixation can be attributed to a tendency ceived size to measure perceived distance, as expressed (the equidistance tendency) for objects closer than or be­ by Equation 5 or 6 of the SDIH, is one such indirect proce­ yond the distance of the fixated object to appear displaced dure. Unfortunately for this procedure, the response to toward the distance of fixation. This result agrees with size can be modified by a remembered standard, as is data obtained in a study by Wist and Summons (1976). found in off-sized judgments, and it seems that this can The effect of the combination of fixation distance and occur even though plain geometrical shapes are used as the equidistance tendency, illustrated in the above experi­ test objects. An indirect procedure that is apt to be very ment, can explain the apparent concomitant motion (W') efficient both in avoiding cognitive effects and in not in­ obtained in most situations in which the head is moved troducing any modifications of the perceived distance to laterally while, for example, two fingers located at differ­ be measured is the null adjustment method used with the ent physical distances are viewed. Usually, the nonfix­ head motion procedure. It avoids the need to judge an ex­ ated finger will appear to move more than the fixated tended W', because at the null setting W' = O. Also, since finger. This is consistent with the presence ofoculomotor at the null setting the pivot distance (Equation 2) equals cues appropriate to the distance of the fixated, but not the the perceived distanceof the test object, any motion paral­ nonfixated, finger. The nonfixated finger will appear to lax that might be introduced by the head motion should move in the same direction as the motion of the head if agree with the perceived distance of the test object in the it is physically more distant than the fixated finger, or absence of head motion. In principle, a variety of indirect opposite to the direction of the head motion if it is the measures applicable to measuring various aspects of phe­ physically closer finger. The values of W' obtained in this nomenal geometry are possible. For example, in Figure 6, case often are cited as an example of the effect of relative the magnitude of {3' is related to the magnitude of errors motion parallax. On the contrary, they represent a failure in the depth perceived between the two test points. of relative motion parallax to determine the correct depth Indirect measures, to the degree that they are insensi­ between the fingers. If it were completely effective, rela­ tive to cognitive effects, are useful, particularly when they tive motion parallax would result in both fingers' appear­ are compared with more direct measures, in distinguishing ing stationary and located at their physical distances as experimentally between primary (perceptual) processes the head moved laterally. Also, although the illusory rela­ and secondary (cognitive) processes. The ability to make tive motions of the fingers produced in this manner are this distinction is important for the separation of the rules consistent in direction with the image motions on the eye, or laws that apply to these different domains. This dis­ as has been discussed, they cannot be explained by only tinction also is likely to be important for the evaluation the retinal motions. Instead, the most likely cause is a of the development of spatial responses as organisms rna- 122 GOGEL ture. It is suggested that phenomenal geometry is a pri­ GOGEL, W. C. (I969a). The effect of object familiarity on the percep­ mary system that is available innately as soon as infor­ tion of size and distance. Quanerly Journal ofExperimental Psychol­ mation regarding the three basic factors is capable of being ogy, 21, 239-247. GOGEL, W. C. (I969b). The sensing of retinal size. Vision Research, used. Secondary processes that require experience can 9, 1079-1094. provide a supplement or correction to spatial responses GOGEL, W. C. (1970). The adjacency principle and three dimensional that is needed because of limitations in the information visual illusions. In 1. C. Baird (Ed.), Human space perception: provided by the primary system. Thus, secondary pro­ Proceedings of the Dartmouth Conference. Psychonomic Monograph Supplement, 3(13, Whole No. 45), 153-169. cesses can contribute to the increased accuracy of spatial GOGEL, W. C. (1971). A validity of the size-distance invariance hypothe­ responses often characteristic of experienced observers. sis with cue reduction. Perception & Psychophysics, 9, 92-94. GOGEL, W. C. (1976). An indirect method of measuring perceived dis­ tance from familiar size. Perception & Psychophysics, 20, 419-429. REFERENCES GOGEL, W. C. (1977). An indirect measure of perceived distance from oculomotor cues. Perception & Psychophysics, 21, 3-11. AMES, A., JR. (1952). The rotating trapezoid (Cont): Explanation of GOGEL, W. C. (I 979a). The common occurrence of errors of perceived visual phenomena. In F. P. Kilpatrick (Ed.), Human behavior from distance. Perception & Psychophysics, 25, 2-11. the transactional point ofview (pp, 75-86). Washington, DC: Office GOGEL, W. C. (I 979b). Induced motion as a function of the speed of of Naval Research, Department of the Navy. the inducing objects, measured by means of two methods. Percep­ BIERSDORF, W. R. (1966). Convergence and apparent distance as corre­ tion, 8, 255-262. lates of size judgments at near distances. Journal ofGeneral Psychol­ GOGEL, W. C. (1980). The sensing of retinal motion. Perception & ogy, 75, 249-262. Psychophysics, 28, 155-163. BORING, E. G. (1946). The perception of objects. American Journal GOGEL, W. C. (I98Ia). Perceived depth is a necessary factor in appar­ of Physics, 14,99-107. ent motion concomitant with head motion: A reply to Shebilske and CUTnNG, J. E. (1988). Affine distortions ofpictoriaJ space: Some predic­ Proffitt. Perception & Psychophysics, 29, 173-177. tions for Goldstein (1987) that La Gournerie (1859) might have made. GOGEL, W. C. (l98Ib). The role of suggested size in distance responses. Journal ofExperimental Psychology: Human Perception & Perfor­ Perception & Psychophysics, 30, 149-155. mance, 14, 305-311. GOGEL, W. C. (1982). Analysis of the perception of motion concomi­ COLTHEART, M. (1968). Size information as a factor in visual judg­ tant with a lateral motion of the head. Perception & Psychophysics, ments ofabsolute distance under reduced conditions. Unpublished doc­ 32, 241-250. toral dissertation, University of Sydney, Sydney, Australia. GOGEL, W. C. (1983). An illusory motion of a stationary target during DAY,R. H., & PARKS, T. E. (1989). To exorcize a ghost from the per­ head motion is unaffected by paradoxical retinal motion: A reply to ceptual machine. In M. Hershenson (Ed.), Themoon illusion (pp. 343­ Shebilske and Proffitt (1983). Perception & Psychophysics, 34, 350). Hillsdale, NJ: Erlbaum. 482-487. DEREGOWSKI, 1. B. (1980). Illusions, patterns, and pictures: A cross­ GOGEL, W. C. (1984). The role of perceptual interrelations in figural cultural perspective. New York: Academic Press. synthesis. In P. Dodwell & T. Caelli (Eds.), Figural synthesis (pp. 31­ DUNCAN, F. S. (1975). Kinetic art: On my psychokinematic objects. 82). Hillsdale, Nl: Erlbaum. Leonardo 8, 97-101. GOGEL, W. c., & DASILVA, 1. A. (I987a). Familiar size and the the­ ELLIS, S. R., SMITH, S., & McGREEVY, M. W. (1987). Distortions of ory of off-sized perceptions. Perception & Psychophysics, 41, 318-328. perceived visual directions out of pictures. Perception & Psycho­ GOGEL, W. C., & DASILVA, 1. A. (I987b). A two-process theory of physics, 42, 535-544. the response to size and distance. Perception & Psychophysics, 41, EPSTEIN, W., & BABLER, T. (1989). Perception of slant-in-depth is auto­ 220-238. matic. Perception & Psychophysics, 45, 31-33. GOGEL, W. C.; & EBY, D. W. (1990). The perception ofvisual angle EPSTEIN, W., & BRooTA, K. D. (1986). Automatic and attentional com­ in optical change. Unpublished manuscript. ponents in perception of size-at-a-distance. Perception & Psycho­ GOGEL, W. c., & HARKER, G. S. (1955). The effectiveness of size cues physics, 40, 256-262. to relative distance as a function of lateral visual separation. Journal EPSTEIN, W., & LANDAUER, A. A. (1969). Size and distance judgments of Experimental Psychology, 50, 309-315. under reduced conditions of viewing. Perception & Psychophysics, GOGEL, W. C., LOOMIS, 1. M., NEWMAN, N. 1., & SHARKEY, T. 1. 6, 269-272. (1985). Agreement between indirect measures of perceived distance. EPSTEIN, W., & LOVITTS, B. E. (1985). Automatic and attentional com­ Perception & Psychophysics, 37, 17-27. ponents in perception of shape-at-a-slant. Journal ofExperimental Psy­ GOGEL, W. C., & MERSHON, D. H. (1977). Local autonomy in visual chology: Human Perception & Performance, 11, 335-366. space. Scandinavian Journal of Psychology, 18, 237-250. FOLEY, 1. M. (1965). Visual space: A scale of perceived visual direc­ GOGEL, W. C., & NEWTON, R. E. (1969). Perception of off-sized ob­ tion. Proceedings ofthe 73rd Annual Convention ofthe American Psy­ jects. Perception & Psychophysics, 5, 7-9. chological Association, I, 49-50. GOGEL, W. C., & SHARKEY, T. 1. (1989). Measuring attention using FOLEY, 1. M. (1968). Depth, size and distance in stereoscopic vision. induced motion. Perception, 18, 303-320. Perception & Psychophysics, 3, 265-274. GOGEL, W. C., & STURM, R. D. (1971). Directional separation and the FOLEY, 1. M. (1972). The size-distance relation and intrinsic geometry size cue to distance. Psychologische Forschung ; 35, 57-80. of visual space: Implications for processing. Vision Research, 12, GOGEL, W. c., & TIETZ, J. D. (1973). Absolute motion parallax and 323-332. the specific distance tendency. Perception & Psychophysics, 13, GIBSON,l.l. (1950). Theperception ofthe visual world. Boston: Hough­ 284-292. ton Mifflin. GOGEL, W. c., & TIETZ, 1. D. (1974). The effect of perceived dis­ GIBSON, 1. 1. (1966). The senses considered as perceptual systems. tance on perceived movement. Perception & Psychophysics, 16, 70-78. Boston: Houghton Mifflin. GOGEL, W. C., & TIETZ, 1. D. (1977). Eye fixation and attention as GILLAM, B.l. (1968). Perception of slant when perspective and stereopsis modifiers of perceived distance. Perceptual & Motor Skills, 45, conflict: Experiments with aniseikonic lenses. Journal ofExperimental 343-362. Psychology, 78, 299-305. GOGEL, W. C., & TIETZ, 1. D. (1979). A comparison of oculomotor GOGEL, W. C. (1954). Perception of the relative position of objects as and motion parallax cues of egocentric distance. Vision Research, 19, a function of other objects in the field. Journal ofExperimental Psy­ 1161-1170. chology, 47, 335-342. GOGEL, W. C., & TIETZ,1. D. (1990). Limitations ofthe role ofcogni­ GOGEL, W. C. (1965). Equidistance tendency and its consequences. Psy­ tive factors in the perception of orientation in depth. Unpublished chological Bulletin, 64, 153-163. manuscript. PHENOMINAL GEOMETRY 123

GOLDSTEIN, E. B. (1979). Rotation of objects in pictures viewed at an ROCK, I., & McDERMOTT, W. (1964). The perception of visual angle. angle: Evidence for different properties of two types of pictorial space. Acta PsychoLogica, 22, 119-134. JournaL of ExperimentaL PsychoLogy: Human Perception & Perfor­ ROELOFS, C. O. (1935). Optische LokaIisation. Archiv fUr Augen­ mance, 5,78-87. heilkunde, 109, 395-415. GOLDSTEIN, E. B. (1987). Spatial layout, orientation relative to the ob­ SCHRIVER, W. (1925). Experimentelle Studien iiber stereoskopisches server, and perceived projection in pictures viewed at an angle. JournaL Sehen. Zeitschrift fur Psychologie, 96, 113-170. ofExperimentaLPsychoLogy:Human Perception & Performance, 13, SHEBILSKE, W. L., & PROffiTT, D. R. (1981). The priority of perceived 256-266. distance for perceiving motion has not been demonstrated: Critical GOLDSTEIN, E. B. (1988). Geometry or not geometry? Perceived orien­ comments on Gogel's "The sensing of retinal motion." Perception tation and spatial layout in pictures viewed at an angle. JournaL of & Psychophysics, 29, 170-172. ExperimentaL PsychoLogy: Human Perception & Performance, 14, SHEBILSKE, W. L., & PROffITT, D. R. (1983). Paradoxical retinal mo­ 312-314. tions during head movements: Apparent motion without equivalent GREGORY, R. L. (1968, November). Visual illusions. Scientific Ameri­ apparent displacement. Perception & Psychophysics, 34, 476-481. can, pp. 66-76. SWANSTON, M. T., & GOGEL, W. C. (1986). Perceived size and mo­ GREGORY, R. L. (1970). Theintelligent eye. New York: McGraw-Hili. tion in depth from optical expansion. Perception & Psychophysics, HAY, J. C. (1968). Visual adaptation to an altered correlation between 39, 309-326. eye movement and hand movement. Science, 160, 429-430. TIETZ, J. D., & GOGEL, W. C. (1978). Adaptation to apparent con­ HAY, J. C., & GOLDSMITH, W. M. (1973). Space-time adaptation of comitant motion in the absence of physical or retinal motion. Bulle­ visual position constancy. JournaL ofExperimentaL PsychoLogy, 99, tin of the Psychonomic Society, 12, 1-4. 1-9. VAN DER MEER, H. C. (1979). Interrelation of the effects of binocular HEINEMANN, E. G., TULVING, E., & NACHMIAS, J. (1959). The effect disparity and perspective cues on judgments of depth and height. Per­ of oculomotor adjustments on apparent size. American JournaLofPsy­ ception & Psychophysics, 26, 481-488. choLogy, 72, 32-45. VON HOLST, E., & MITTELSTAEDT, H. (1950). Das Reafferenz-Prinzip. HOLWAY, A. H., & BORING, E. G. (1941). Determinants of apparent Die Naturwissenschaften, 20, 464-476. visual size with distance variant. American JournaL of Psychology, WALLACH, H. (1985, May). Perceiving a stable environment. Scien­ 54, 21-37. tific American, pp. 1\8-124. ITTELSON, W. H. (1952). The Ames demonstrations in perception WALLACH, H. (1987). Perceiving a stable environment when one moves. (pp. 21-25). Princeton, NJ: Princeton University Press. AnnuaL Review of PsychoLogy, 38, 1-27. KOMODA, M. K., & ONO, H. (1974). Oculomotor adjustments and size­ WALLACH, H., BACON, J., & SCHULMAN, P. (1978). Adaptation in mo­ distance perception. Perception & Psychophysics, 15, 353-360. tion perception: Alteration of induced motion. Perception & Psycho­ LANDAUER, A. A., & EPSTEIN, W. (1969). Does retinal size have a physics, 24, 509-514. unique correlate in perceived size? Perception & Psychophysics, 6, WALLACH, H., & FLOOR, L. (1970). On the relation of adaptation to 273-275. field displacement during head movements to the constancy of visual LEIBOWITZ, H. W. (1971). Sensory, learned, and cognitive mechanisms direction. Perception & Psychophysics, 8, 95-98. of size perception. AnnaLsofthe New York Academy ofSciences, 188, WALLACH, H., & FREY, K. J. (1969). Adaptation in the constancy of 47-62. visual direction measured by a one-trial method. Perception & Psycho­ LEIBOWITZ, H., & MOORE, D. (1966). Role of changes in accommoda­ physics,S, 249-252. tion and convergence in the perception of size. JournaLofthe OpticaL WIST,E. R., DIENER, H. C., & DICHGANS, J. (1976). Motion constancy Society ofAmerica, 56, 1\20-1\23. dependent upon perceived distance and the spatial frequency of the LEIBOWITZ, H. W., SHIINA, K., & HENNESSY, R. T. (1972). Oculo­ stimulus pattern. Perception & Psychophysics, 19, 485-491. motor adjustments and size constancy. Perception & Psychophysics, WIST, E. R., DIENER, H. C., DICHGANS, J., & BRANDT, TH. (1975). 12, 497-500. Perceived distance and the perceived speed of self-motion: Linear vs. MACK, A. (1978). Three modes of visual perception. In H. L. Pick, angular velocity? Perception & Psychophysics, 17, 549-554. Jr., & E. Saltzman (Eds.). Modes ofperceiving and processing in­ WIST, E. R., & SUMMONS, E. (1976). Spatial and fixation conditions formation (pp. 171-186). Hillsdale, NJ: Erlbaum. affecting the temporal course of changes in perceived relative distance. MCCREADY, D. W., JR. (1965). Size-distance perception and PsychoLogicaL Research, 39, 99-1\2. accommodation-convergence micropsia: A critique. Vision Research, 5, 189-206. NOTES MCCREADY, D. (1985). On size, distance, and visual angle perception. Perception & Psychophysics, 37, 323-334. 1. A paper concerning evidence relevant to Figure 1 was presented MCCREADY, D. (1986). Moon illusions redescribed. Perception & at the 27th Annual Meetjog of the Psychonomic Society, New Orleans, Psychophysics, 39, 64-72. LA, November 1986. MACCRACKEN, P. J., GOGEL, W. C., & BLUM, G. S. (1980). Effects 2. Often deviations from the strict form of the SDIH are indicated of posthypnotic suggestion on perceived egocentric distance. Percep­ by values different from unity of a constant of proportionality between tion, 9, 561-568. S'/D' and tan I/' (derived from the responses shown in Figure 9) or, in ONO, H., MUTER, P., & MITSON, L. (1974). Size-distance paradox with the case of Equation 6, between S'/2D' and tan(8'/2) (derived from accommodative micropsia. Perception & Psychophysics, IS, 301-307. Figure 1C). In Figure 9, this constant is unity in A, is greater than unity PETERSON, M. A., & SHYI,G. C.-W. (1988). The detection ofreal and in B, and is less than unity in C. It also is possible to express this devia­ apparent concomitant rotation in a three-dimensional cube: Implica­ tion in terms of a power function in which either the constant of propor­ tions for perceptual interactions. Perception & Psychophysics, 44, tionality or the exponent differs from unity, or in which both of them 31-42. do (Foley, 1968; Gogel, 1971). POST, R. B., & LEIBOWITZ, H. W. (1982). The effect of convergence 3. It is equally appropriate to call the ratio sUS, either an off-sized on the vestibulo-ocular reflex and implications for perceived move­ judgment or an off-sized perception, since it involves both a memory ment. Vision Research, 22, 461-465. S, and a perception S; (see Gogel, 1981b, note 1). ROCK, I. (1977). In defense of unconscious inference. In W. Epstein (Ed.), Stability and constancy in visuaLperception: Mechanisms and processes (pp. 321-373). New York: Wiley. ROCK, I. (1983). TheLogic ofperception. Cambridge, MA: MIT Press. ROCK, I., HILL, A. L., & FINEMAN, M. (1968). Speed constancy as (Manuscript received July 17, 1989; a function of size constancy. Perception & Psychophysics, 4, 37-40. revision accepted for publication March 21, 1990.)