This relationship led us to hypothesize that the profiles of illusion strength at different eccentricities might overlap, if they were appropriately shifted in an oblique direction. Scaling of visual angle at an eccentricity in question results in translation of a profile in a log-log plot. For example, consider a hypothetical case in which the stimulus occupying 1 [deg] of visual angle at
x [deg] eccentricity makes the same impact as the stimulus occupying 2 [deg] at 13 [deg] eccentricity. This relationship can be written as 1 [deg
x] = 2 [deg
13], where “deg
x” denotes the unit “deg” at
x deg eccentricity. Equivalently, 1 [deg
13] = 0.5 [deg
x], and this value, 0.5, is called the scaling factor,
f, for
x deg eccentricity. Accordingly, a stimulus size
z [deg
x] will be converted to
z [1/0.5 deg
13] = 2
z [deg
13] (more generally,
Display Formula\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\unicode[Times]{x1D6C2}}\)\(\def\bupbeta{\unicode[Times]{x1D6C3}}\)\(\def\bupgamma{\unicode[Times]{x1D6C4}}\)\(\def\bupdelta{\unicode[Times]{x1D6C5}}\)\(\def\bupepsilon{\unicode[Times]{x1D6C6}}\)\(\def\bupvarepsilon{\unicode[Times]{x1D6DC}}\)\(\def\bupzeta{\unicode[Times]{x1D6C7}}\)\(\def\bupeta{\unicode[Times]{x1D6C8}}\)\(\def\buptheta{\unicode[Times]{x1D6C9}}\)\(\def\bupiota{\unicode[Times]{x1D6CA}}\)\(\def\bupkappa{\unicode[Times]{x1D6CB}}\)\(\def\buplambda{\unicode[Times]{x1D6CC}}\)\(\def\bupmu{\unicode[Times]{x1D6CD}}\)\(\def\bupnu{\unicode[Times]{x1D6CE}}\)\(\def\bupxi{\unicode[Times]{x1D6CF}}\)\(\def\bupomicron{\unicode[Times]{x1D6D0}}\)\(\def\buppi{\unicode[Times]{x1D6D1}}\)\(\def\buprho{\unicode[Times]{x1D6D2}}\)\(\def\bupsigma{\unicode[Times]{x1D6D4}}\)\(\def\buptau{\unicode[Times]{x1D6D5}}\)\(\def\bupupsilon{\unicode[Times]{x1D6D6}}\)\(\def\bupphi{\unicode[Times]{x1D6D7}}\)\(\def\bupchi{\unicode[Times]{x1D6D8}}\)\(\def\buppsy{\unicode[Times]{x1D6D9}}\)\(\def\bupomega{\unicode[Times]{x1D6DA}}\)\(\def\bupvartheta{\unicode[Times]{x1D6DD}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bUpsilon{\bf{\Upsilon}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\(\def\iGamma{\unicode[Times]{x1D6E4}}\)\(\def\iDelta{\unicode[Times]{x1D6E5}}\)\(\def\iTheta{\unicode[Times]{x1D6E9}}\)\(\def\iLambda{\unicode[Times]{x1D6EC}}\)\(\def\iXi{\unicode[Times]{x1D6EF}}\)\(\def\iPi{\unicode[Times]{x1D6F1}}\)\(\def\iSigma{\unicode[Times]{x1D6F4}}\)\(\def\iUpsilon{\unicode[Times]{x1D6F6}}\)\(\def\iPhi{\unicode[Times]{x1D6F7}}\)\(\def\iPsi{\unicode[Times]{x1D6F9}}\)\(\def\iOmega{\unicode[Times]{x1D6FA}}\)\(\def\biGamma{\unicode[Times]{x1D71E}}\)\(\def\biDelta{\unicode[Times]{x1D71F}}\)\(\def\biTheta{\unicode[Times]{x1D723}}\)\(\def\biLambda{\unicode[Times]{x1D726}}\)\(\def\biXi{\unicode[Times]{x1D729}}\)\(\def\biPi{\unicode[Times]{x1D72B}}\)\(\def\biSigma{\unicode[Times]{x1D72E}}\)\(\def\biUpsilon{\unicode[Times]{x1D730}}\)\(\def\biPhi{\unicode[Times]{x1D731}}\)\(\def\biPsi{\unicode[Times]{x1D733}}\)\(\def\biOmega{\unicode[Times]{x1D734}}\)\({f^{ - 1}}z\) [deg
13]), and the log plot will be shifted rightward by one log
2 unit after scaling. Similarly, a spatial frequency of
q [cycles/deg
x] will be converted to
q [cycles/{1/0.5 deg
13}] = 0.5
q [cycles/deg
13], and the log plot will be shifted left by one log
2 unit, after scaling. Therefore, spatial scaling of fundamental spatial frequency is equivalent to a horizontal shift of the plot. Illusion strength,
m, is similarly altered by spatial scaling. For rotation speed
s [rpm] at eccentricity
x [deg], angular change per second
θ =
Display Formula\({\pi \over {30}}s\) [rad]; arc length per second
m =
x tan (
θ) [deg
x]. By the same logic as above, illusion strength as represented by
m [deg
x] will be converted to
m [1/0.5 deg
13] = 2
m [deg
13], and be converted back again to rotation speed,
Display Formula\({{30} \over \pi }{\tan ^{ - 1}}{{2m} \over x}\) [rpm]. Since tan (
θ) ≈
kθ [rad], where
k ≈ 0.0175, for small
θ, the above unit conversion through scaling is reduced to linear scaling:
Display Formula\({\tan ^{ - 1}}\left( {{f^{ - 1}}\tan \theta } \right) \approx {{{f^{ - 1}}k\theta } \over k} = {f^{ - 1}}\theta \). Therefore, spatial scaling of illusion strength is equivalent to a vertical shift of the plot. We assumed that a common scaling factor should be used to scale the abscissa and ordinate, and thus the translation of each profile was constrained in the diagonal direction with a slope of −1 in the log-log plot. Factors greater than unity would shift the profile diagonally to the right and down, whereas factors less than unity would shift it to the left and up. What scaling factor would best characterize the behavior at each eccentricity? To answer this question, we applied the assumption-free method for spatial scaling (Whitaker et al.,
1992). Graphically, the procedure was equivalent to shifting the profiles for different eccentricities along the diagonal line so that all profiles collapsed into a single function. More specifically, we stabilized the profile at 13 deg eccentricity as standard and attempted to find the scaling factors for the three other eccentricities that would deliver all the profiles over the profile at 13 deg eccentricity, and that would minimize the residual error of curve fitting applied to the scaled dataset. The model curve we chose was the parabola,
y =
a (
x –
c)
2 +
b, the simplest formulation depicting U-shaped nonlinearity.