Characterizing projective geometry of binocular visual space by Möbius transformation

Publication date: February 2019Source: Journal of Mathematical Psychology, Volume 88Author(s): Jun ZhangAbstractBinocular vision involves the projection of objects in the 3-D visual space onto the two retinae and the comparison of spatial layout of objects in these retinal half-images. Here we characterize the unitary representation of the binocular space as a complex half-plane from the perspective of the cyclopean eye. We then investigate its automorphism group, namely the Möbius transformation group, and the associated invariants when the two eye positions are treated as fixed points of the automorphism. A three-point simple ratio from an object point to both eyes is constructed; as a complex number, its angle measures the difference in azimuth of the projected rays from the object point to each eye (i.e., horizontal disparity) while its modulus measures the ratio of the distances of the object point to the two eyes (i.e., relative vertical magnification or vertical size ratio). The four-point cross-ratio of two such simple ratios, as the only four-point invariant under Möbius transforms, reflects the fact that the relative disparity between any two object points remains unchanged when the eyes change fixation. Since the complex half-plane is biholomorphically equivalent to an open unit disk, both Poincaré model and Klein–Beltrami model give rise to a hyperbolic geometry, consistent with the empirically supported Luneburg’s (1947; 1950) model of binocular geometry (...

Source: Journal of Mathematical Psychology - November 27, 2018 Category: Psychiatry & Psychology Source Type: research