A rotation map is a conformal map of the form f(z) = eiθz.
A rotation preserves norms but changes arguments:
This is shown with the polar form of z, z = reiα.
f(z) = eiθz = eiθreiα = rei(θ+α)
⇒ the norm is preserved, namely r, but the argument is increased by the rotation angle, θ.
Use the rockin' applet below to do the following exercises:
Draw a horizontal line in the domain. What happens in the image? Explain.
Draw a square in the domain. How has it changed in the image?
Draw a circle in the domain. How has it changed in the image?
Draw a square in the image. How has it changed in the domain?
Using the applet.