Zusammenfassung
Weber's law of 1834, Delta S/S = c for the just noticeable difference (jnd), can be written as S + Delta S = kS, k = 1 + c. It follows that the stimulus decrement required to elicit one jnd of sensation is S - Delta S* = k(-1)S. If generalized for two stimulus dimensions and two corresponding response dimensions, Weber's law would have to state such equations for all directions of change in the ...
Zusammenfassung
Weber's law of 1834, Delta S/S = c for the just noticeable difference (jnd), can be written as S + Delta S = kS, k = 1 + c. It follows that the stimulus decrement required to elicit one jnd of sensation is S - Delta S* = k(-1)S. If generalized for two stimulus dimensions and two corresponding response dimensions, Weber's law would have to state such equations for all directions of change in the plane. A two-dimensional Weber law with exactly these properties is realized by [S-x + Delta S-x(0), S-y + Delta S-y(0)] = [k(sin(0))S(x), k(cos(0))S(y)] which determines the stimulus coordinates for all stimuli just noticeably different from the stimulus (S-x, S-y) in all directions 0 less than or equal to theta less than or equal to 2 pi. Fechner's problem now is understood as finding a transformation of the plane which maps the set of stimuli one jnd apart from the standard stimulus onto a unit circle around the standard stimulus' image. This transformation (R-+(2) --> R-2) is [x, y] bar right arrow [log(k)(x), log(k)(y)]. The solution is generalized to arbitrarily many dimensions by substituting the sin and cos in the generalized Weber law by the standard coordinates of a unit vector. (C) 2000 Academic Press.
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