## improved combination of color symmetry and rotation

As mentioned in the last post using two unrelated anamorphic mappings, one for reading the input image and another one for choosing color variants, makes it difficult to create interesting images. From the mapping that determines the color variant we get a related mapping to the input image. This simplifies image creation and gives better results.

For n-fold color symmetry and r-fold rotational symmetry the complex valued mapping W(x,y) changes upon a rotation by an angle of 2π/r  as:

The mapping Z(x,y)=X(x,y)+iY(x,y) to the input image should not change upon this rotation and it should depend on W. This has a simple solution:

where the n-th power of W eliminates the phase factor that appears at rotations of W. We can use the scalar function f to reduce the power of the zero at W=0. This method effectively maps the output image to an anamorphic rosette image of the input and comes close what Farris has presented in “Creating symmetry”.

Here are some results I got using this method for three-color symmetry. A periodic image with six-fold rotational symmetry:

and a quasi-periodic image with 9-fold rotational symmetry and some accidental approximate symmetry:

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