Non-Riemannian approach to geometry of visual space: An application of affinely connected geometry to visual alleys and horopter

Abstract

A study was undertaken to give a concrete model to the theory of non-Riemannian visual space developed with reference to Luneburg's Riemannian model, and to apply the model to visual alleys and horopter. The parallel and distance alleys were formulated to be geodesics in the non-Riemannian visual space under the restriction of teleparallelism and Riemannian configuration, respectively. The present model was described in terms of an affinely connected space which was one of the non-Riemannian spaces without the concept of length. On the supposition that the parallel alley and horopter are paths in visual space, an asymmetric connection was obtained under the restriction of teleparallelism. There are two options for carrying out the numerical simulation. In Option 1, to calculate the connection, families of curves were fitted to the data of parallel alley and horopter. By the numerical solutions of the equation of geodesics with Christoffel symbols which were deduced from the asymmetric connection, the distance alley was drawn independently of the experimental data. The resulting curve was found to lie outside the parallel alley as did Blumenfeld's experiment. One of the possibilities causing the discrepancy between parallel and distance alleys was tested in 1.3. Furthermore, two alleys are expected to coincide if points are adjusted in position, while the alleys are being constructed, along the horopter curves rather than the front-parallel lines. It was shown that the numerical solutions of the geodesic with the connection thus obtained lie midway between parallel and distance alleys. In Option 2, the metric tensor was obtained in the explicit form, and it became clear that the Gaussian curvature with respect to the metric is not constant.