What is unique about the Möbius transform?

Question

... is it the only map to accomplish a transformation in 2D and keep certain characteristics invariant? Which?

What else makes it special to be studied so much?

Answer 1

Möbius transformations are bijective

Such a transformation maps the extended complex plane to itself in such a way that every point gets covered, and each point gets covered only once. "Extended" means we also consider the point "at infinity" (To formalize this, we need to give the one point compactification a complex manifold structure, aka the Riemann sphere).

Möbius transformations are conformal

This means they preserve (oriented) angles. If two lines intersect, and their intersection forms an angle $\theta$ before the intersection, the images of these two lines (which are now some curves) still intersect at angle $\theta$. By this I mean that their tangent lines at the point of intersection still have angle $\theta$. "oriented" means that a clockwise path must correspond to a clockwise path.

The big theorem is the Möbius transformations are the only maps with these two properties!

Answer 2

Möbius transformations correspond to rigid motions of the sphere in a very natural way. Möbius Transformations Revealed (also on YouTube) is the definitive illustration of this. Here's simpler illustration:

In this animation, the graticule of lines that we see on the plane and on the sphere form the image of a square in the complex plane. If we project the square to the sphere via stereographic projection, we get the slightly deformed graticule that you see on the sphere. If we then rotate or translate the sphere, the graticule is moved in a very simple way. If we project this back to the sphere, we get the graticule that you see on the plane. This happens to be the image of the original square under a Möbius transformation and every Möbius transformation can be described in this way.

Some other important observations:

• The set of all Möbius transformations forms a group under composition. This is exactly the group of rigid motions (not reflections) of the sphere.

• Möbius transformations preserve certain geometric properties of sets. For example, they map the set of circles and lines to the set of circles and lines. They do so conformally as well - i.e., they preserve angles.

• The previous two comments are easy to see, when viewed via the rigid transformation approach.