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Teng Leng Ooi, Zijiang J. He; Quantitative descriptors of the relationships between physical and perceived distances based on the ground surface representation mechanism. Journal of Vision 2004;4(8):19. doi: 10.1167/4.8.19.
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Various forms of equations serve to describe the relationships between judged and physical distances. To develop meaningful equations, we capitalize on the findings that perceived target distance and shape on the ground are determined by the geometric properties of the represented ground surface and the target's angular declination. When a flat ground is represented as a surface with a slant error (p) the perceived target distance (d) is determined by d = [DH/sin p]/[H cot p + D], where D and H are, respectively, the physical target distance and the subject's eye height (Ooi et al, Nature 01). Grouping H/sin p as A, and when p is small, the equation becomes d = DA/[A+D], identical to the classic Gilinsky equation. It also reveals that A is governed by the eye height and slant error. To verify this, we measured judged distance at various eye heights (H) using Gilinsky's method, and found that A indeed changes with H. We then generalized the analysis to slant surfaces (g) in which judged distance (d) is specified by d = [DH cos g]/[H cos (g + p) + D sin p], where D is the physical distance on the slant surface and p is the perceived slant error. Using Gilinsky's method, we measured judged distance on slant ground surfaces (g = 0, 10, 20 deg). We found our data were fitted well by the equation, reinforcing the validity of the equation tied to a ground-based mechanism. In another study, we derived an equation for perceiving the ratio of aspect ratio (R) of L-shaped targets (Loomis et al, JEP 02, Ooi et al, VSS02) on a slant ground surface. The equation is R = sin (g + a − b)/sin (p + g + a − b), where a and b are, respectively, the angular declination of the L-target and the angular extent of the L's length in depth. Our measurements for g = 0, 10, 20 deg show that R could be accounted for by the equation. Overall, these ground-based equations not only fit the data well but also assign significance to the parameters that describe the distance relationships.
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