A rosette is an image with rotational symmetry. For p-fold symmetry we can use the methods of the last post “Anamorphosis and symmetries” with a simple power as a mapping function between output and input images:
Here z=x+iy relates to the position (x,y) of a pixel of the output image. We then have to sample the color of the input image at the position (X,Y) given by Z=X+iY. This gives the color of the pixel. With polar coordinates, where r²=x²+y² and tan(φ)=y/x, we see easily that we have p-fold rotational symmetry. Rotating (x,y) by 2π/p changes φ to φ+2π/p and gives the same Z. Thus we get the same color for the pixel at the rotated position.
With p=5 I got this result:
We can get more interesting images if we use several combinations of z together with its complex conjugate. As proposed by Farris:
where (m-n) has is a multiple of p for p-fold rotational symmetry. For the power of r we can use any integer number resulting in a more detailed center of the rosette. I find it convenient to rearrange these terms and to use real coordinates. Thus
which is convenient for computation and discussing symmetries. Note that k can be any integer and that l is a positive integer. Using fast approximate logarithm and exponential functions we rapidly calculate
We can get more varied images with more detail, such as this one:
Again, I did not use smoothing of pixels.
(Post updated on the 5th June 2017.)