![]() However, accuracy is low for judgments requiring veridical perception of Euclidean metric structure, such as judgments of lengths or angles. Their findings indicate that observers are quite accurate and reliable at judging an object's topological, ordinal, or affine properties and that the perception of rigid motion occurs when these properties remain invariant over time. ![]() This hypothesis has been developed and tested by others ( Domini & Caudek, 2003 Domini & Braunstein, 1998 Domini, Caudek & Richman, 1998 Norman & Todd, 1992, 1993 Tittle et al., 1995 Todd & Bressan, 1990 Todd & Norman, 1991 Todd & Reichel, 1989). Siding with more primitive geometries, Gibson (1979) suggested that Euclidean metric distances in 3-dimensional space are not a primary component of an observer's perceptual experience. It is unclear at this time whether and how the conclusions drawn from such studies would be affected if the Euclidean assumption were relaxed. For instance, in studies of structure from motion ( Domini & Caudek, 2003 Domini & Braunstein, 1998 Domini, Caudek & Richman, 1998) and motion-stereo depth cue combination ( Domini, Caudek & Tassinari, 2006), it is often assumed that the relationship between the perceived slant of a surface and the perceived relative depth between two points on that surface satisfies Euclidean geometry. By “indirect method” we mean estimating the variable of interest by measuring a different variable and linking the two variables through standard Euclidean geometry. The conclusions from any psychophysical experiment in which a variable is measured by an indirect method can potentially be affected by erroneously assuming Euclidean geometry as valid. The interpretation of many psychophysical studies relies on the assumption of a Euclidean percept. The fact that perceived structure could be non-Euclidean 2 is not trivial. An instance of this is Riemannian geometry. The second way is to keep a metric structure but to define this metric in a different way than in Euclidean geometry. The first way is for it to stay within the realm of more primitive geometries, this is, geometries that do not have an internal metric structure. There are fundamentally two different ways in which intrinsic geometry could depart from being Euclidean. Of course, we are assuming that there is a geometry of internal perceptual space-in other words, that perception is stable and consistent enough to support such a geometry. That is, we are dealing with the internal geometry of a subject's perceptual space and not at all with the relationship between perceived and actual shape. Our topic here is the structure of the intrinsic geometry of perception 1. Intrinsic geometry, by contrast, provides a global set of constraints by which the judgments of a given observer are formally related to one another, irrespective of their relation to the external environment. For example, perceived distance is compressed over a large range ( Gilinsky, 1951), apparent parallel alleys and equidistance alleys are not physically parallel and equidistant ( Blumenfeld, 1913 Indow & Watanabe, 1984), apparent frontoparallel planes are not physically frontoparallel ( Helmholtz, 1962 Ogle, 1964), and lines perceived as curved might be straight in the physical environment ( Todd et al., 2001). It has long been known that many geometric relations are distorted in perception. ![]() If such were the case, then the extrinsic geometry would be affine. Let us give an example: assume that both physical space and our internal representation of space are Euclidean, and that our percepts are distorted so that perceived objects are veridical only up to a scaling factor in depth (e.g., a circle is perceived as an ellipse). It gives the geometrical transformations necessary to map physical space onto perceptual space. Extrinsic geometry refers to the relationship between the structure of the observer's perception and the actual structure of physical space. Kant seems to be referring to what we today call the “intrinsic” geometry of perceptual space, rather than its “extrinsic” geometry. That is, three-dimensional Euclidean space is a necessary, but not tautological, presupposed form underlying all human spatial experience. In his Critique of Pure Reason, Kant (1902) argued that the truths of geometry are synthetic a priori truths.
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