The unit circle in the z-plane at the left is mapped by four polynomia of degree 1 to 4 onto the four circles shown in the w-plane at the right. The images of 18 points on the unit circle and in the first quadrant beginning with a green and a red dot are also shown. The images of these dots are phase shifted, and  the distance between them increases proportional to the  degree of the polynomium.
Own work
AstroOgier