170mm x 244mm Collerton c10. tex V3-11/21/2014 5:28 P. M. Page 235 10. 3 CONDITIONS FOR THE LOSS OF STABILITY OF THE HOMOGENEOUS STATE 235 (a) (b) Figure 10. 15 (a)Contouredevenaxialplanformonthesquarelattice. (b)Contouredevenaxial planform onthehexagonal lattice. [Whenf(x,y)=0,y∕x=tan𝜙. ] Such a line has a constant tangent vector, whose magnitude is 𝑣=cos𝜙𝜕 𝜕x+sin𝜙𝜕 𝜕y The image of such a line in the visual field is obtained by changing to retinal coor-dinates via the inverse of the complex logarithmic map-the complex exponential so that f(x,y)𝜃Rcos𝜙-logr Rsin𝜙 thelevelcurvesofwhichare logarithmicspirals oftheformr R(𝜃R)∝exp(cot(𝜙)𝜃R). Itcanbeshownthatthetangentvectorofsuchacurvehasamagnitudeoftheform ̃𝑣=r Rcos(𝜙+𝜃R)𝜕 𝜕x R+r Rsin(𝜙+𝜃R)𝜕 𝜕y R Sotheretinalvectorfieldinducedbytheconstantvectorfieldin V1twistswiththe retinal angle 𝜃Rand shifts (stretches) with the retinal radius r R. Thus it performs a twist-shift of the cortical vector field. A direct consequence of this is that if 𝜙Ris the orientation of a line in the visual field, then 𝜙=𝜙R-𝜃R (10. 13) that is, local orientation in V1 is relative to the angular coordinate of position in thevisualfield. Consequentlythelocalorientationsofcircles,rays,andlogarithmic