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(8)　A Riemannian-Geometrical Model of Simultaneous
　　　　Contrast Illusions and the Figural After-Effects on the Basis of Depth Effects
　　　　北海学園大学工学部研究報告,第２６号,pp.401-426,1999-2.

　A study was undertaken to construct a descriptive model of Both simultaneous contrast illusions and the figural after-effects in terms of a Riemannian space with Riemann-Christoffel curvature tensor. This differential-geometrical model was applied to the illusion of concentric circles (the Delboeuf illusion), the figural after-effect and the bent line illusion (the Orbison square illusion).
　The curvature tensor Rkijl, which can be calculated from the metric tensor specifying the perceived distortion in these illusions, implies another distortion effect which cannot be described within two-dimensional plane (depth effect).
　This effect occurs in any figural after-effects, while the simultaneous contrast and only the figural after- effect for small inspection time does not evoke this effect. Therefore, the later phenomenon can be formulated by Rhijl=0, and the former by Rhijl≠0. Assuming that an indicatrix is elliptic in the local perceptual field of parallel line and solving Rhijl=0 uniquely determined the magnitudes of the illusions as a fractional function in which a parameter is involved. The sign of the parameter determines whether the simultaneous illusion or the figural after-effect with small inspection time is evoked. Moreover, extending the present model to the summation of the effect of each circle led us to simulate a visually straight line in the Orbison square illusion.