## Improved two-color symmetry upon rotation

As discussed in the post “two-color rotational symmetry” we get only a single real color-changing function U(x,y) instead of a mapping W(x,y)=U(x,y)+iV(x,y) to the complex plane. Thus we need a special approach to get a mapping to the input image which is continuous when the color changes and relates to the color-changing function U.

For r-fold rotational symmetry the elementary rotation R(2π/r) simply changes the sign of U and we can use the absolute value of U(x,y) as the x-component for the mapping to the input image X(x,y)=|U(x,y)|. This is an extra imposed mirror symmetry and relates the color change to the anamorphic distortion of the input image. For the y-component we can use any function Y(x,y) that does not change upon the elementary rotation. This is a typical result:

There is another, more sophisticated way. We can use another two-color-changing function V(x,y) to get a full mapping to the W=U+iV plane. An elementary rotation changes the sign of W(x,y) which implies a two-fold rotational symmetry. Thus, similar to the other color symmetries we can use as a mapping to the input image Z(x,y)=f(|W|)*[W(x,y)]². The color is still determined by the sign of U(x,y) and a typical result looks like this:

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