You may or may not have heard about moebius transformations.  To put it simply,  it is a bijective conformal map of the extended complex plane (I be you have no idea what this is,  and if you do I envy you).

They allow for some quite useful transformations and the math behind them is a little mind bending.  Turns out though that you can get a really good idea of what is going on through this demonstration that has taken it to the third dimension.  Much easier to visualise after watching this.