Riemannian Geometries of Variable Curvature in Visual Space: Visual Alleys, Horopters, and Triangles in Big Open Fields

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Abstract
Luneburg's model for computation of the curvature K of visual two-dimensional space (horizontal visual surface) was tested with equidistant and parallel alleys in large open spaces. Forty-six subjects used stakes to produce 406 experimental alleys of variable sizes (from 5 × 1 to 240 m × 48 m). The results show that, contrary to results obtained under laboratory conditions with small alleys and light spots, the individual curvature of visual space does not have a negative constant value. K varies in the interval −1 to +1 in ninety computed settings: K ≥ 0 ( N = 38); K < 0 ( N = 52). Therefore the Lobachevskian geometry currently attributed to visual space ought to be replaced by a Riemannian geometry of variable curvature. Moreover K is an individual function dependant on the size of the alley (distance from the subject), and visual perception would be better understood as scale-dependent. Independently of Luneburg's model we have tested the constancy of the curvature hypothesis in experiments with horopters and visual triangles. The results obtained invalidate Luneburg's hypothesis also.

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