線遠近法の数理とリーマン空間？
since May 17, 2012
網膜座標系と錯視
問： 網膜地図（retinal map）は、中心を決め、中心を通る上下・左右の軸を決めると、中心からの距離（視角）で座標を表す。理想的な視野世界はどのように網膜地図に投射されるか？
Figure 1. An instance of retinal coordinates. Suppose that twenty by twenty squares (each being 10 cm x 10 cm) are drawn at the center of an infinitely-extended flat plane and observed from 50 cm apart. (a) When the plane is perpendicular to the visual axis, the mesh appears to bulge out though it really has no curves. (b) When the visual axis is parallel to the plane, the lines parallel to the visual axis appear to converge at the center of the visual field while the lines perpendicular to the visual axis (except those that cross the visual axis) appear to curve outward. (c) When the plane is observed obliquely, both (bulging and converging) appearances coexist. (d) Parallel lines on a plane perpendicular to the visual axis meet at two points on the surrounding circle. The gray circle at the center in each panel represents the foveal vision. The radius of the circle is 90 degree of visual angle.
Given equations
Figure Appendix. The method to obtain the retinal coordinates (ρ,θ) from the visual world. Suppose that the seen object is placed on the x-y plane where a target point T (x, y, 0) is located, and that an observing eye is located on the z-axis with the height of "d". "α" is the angle between the visual axis and the z-axis when the eye sees a point F (foveal point) that is located on the y-axis. The visual angle "ρ" of the target point is the angle between the line from the eye to the foveal point F (vector f) and the line from the eye to the target point T (vector t). The visual angle ρ is obtained from these two vectors. The angle θ is given as the angle between the line parallel to the x-axis through the foveal point F and the line from the foveal point F to another point T´ where the extension of vector t meets the plane that is perpendicular to the visual axis (or vector f) and passes through the foveal point F.
Figure 2. Extended retinal coordinates. The radius of the outer circle represents 180 degrees of “visual angle” while the radius of the inner one shows 90 degrees of visual angle. (a) The image of Figure 1d is placed in these coordinates. (b) and (c) The same image is seen obliquely. (d) The same image is seen in parallel to the plane. (e)-(h) The thick line shows the line at infinity of a plane within which the entire plane is represented.
結論 網膜座標地図は無限を表現することができる。
Kitaoka, A. (2007) Psychological approaches to art. Japanese Journal of Psychonomic Science,26, 97-102. PDF