Michael Brenner has just pointed out to me this video of the Moebius transformations.
Just wanted to point out an earlier thread on the topic at http://imechanica.org/node/1619
A message from Meghan Purdy:
Hi Professor Suo,
I just found this very useful website. It lets you apply any Mobius transform to a lot of shapes on the z-plane, and it's really useful for checking answers on this homework. You may want to pass this on to the other students.