Here is the picture of the linear fractional transformation
Notice that the preimages of lines through the origin are arcs of circles through 1/3 and 3. Preimages of circles centered at the origin are also circles, orthogonal to the previous family. This particular linear fractional transformation maps the unit circle onto itself, so it must map the unit disk onto itself. Such maps are called Moebius transformations. Groups of Moebius transformations are used to study all surfaces that look locally like a piece of the complex plane (the group operation is composition).
Why is it colored yellow-green between the pole and zero?
Notice that the pre-image of circles and rays are circles. Moreover,
the two families of circles are orthogonal. (may be hard to see in the
online versions)
Why is the color red below 1/3? The function is close to a rotation by
i at 1/3, so 1/3-ci is mapped to the positive reals (red) for c near
0. Look at pictures of other rational functions and see if you can
determine the argument of the derivative at the zeros.
zero.