Moebius transformation

Moebius transformation of the plane is a rational function of the form
f(z)=(az+b)/(cz+d)
The variable z and the coefficients a, b, c, d are complex numbers satisfying ad − bc ≠ 0. Moebius transformations are named in honor of August Ferdinand Moebius.

Here is a short movie depicting the beauty of Moebius Transformations in mathematics, it shows how moving to a higher dimension can make the transformations easier to understand.