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.