ABSTRACT
The study shows that Weber’s ratio ΔS/S = const, for the just-noticeable difference (i) constitutes a projective geometric invariant and (ii) can be interpreted as a representation of an empirical bisection operation. These conditions permit the extension of the theory of projective ordinal scaling (Suppes, Krantz, Luce, Tversky, 1989) to a projective interval scaling. The invariance of Weber’s ratio (iii) is traced in its consequences for perceptual spaces of higher dimensions. There, the invariants of projective transformations are a line, a plane or, respectively, a hyper-plane as compared with a point in the unidimensional case. Besides projective invariance as a consequence of Weber’s law, additional projective invariants of perceptual spaces are generated by various natural laws. With monocular vision in perspective, such invariants are the phenomenal line of the horizon and constancy of area. They imply a projective unimodular structure of the visual horizontal plane. In the two-dimensional case of the binocular visual horizontal plane, invariance of the greatest circle around the observer—corresponding to threshold disparity—generates a projective metric hyperbolic structure. This result generalizes Luneburg’s (1947) theory of a hyperbolic structure. In the three-dimensional perceptual space of colors, invariance of the extremal of the convex hull of colors—the surface of the color cone—with respect to color adaptation, results in a new formula for color difference. It is related to the “center of gravity rule.” The three-dimensional generalization of the invariance of Weber’s ratio, different from Helmholtz (1891), leads to a projective metric hyperbolic structure of color space. Analysis of the set of qualitative empirical assumptions sufficient for the derivation of this result yields a representation of color vision which is independent of the classical Grassmann approach. The present study tries to detect the causes for the occurrence of markedly different structures in perceptual spaces. The question 70whether for instance binocular visual space is of Euclidean, of hyperbolic, or of parabolic structure has been vigorously discussed in the past. Possible causes, however, were not stated. The present study shows that certain well-known psychophysical invariants endow perceptual spaces with structure, each in its own specific way.
