In fact, any Möbius transformation can be written as a composition of these.

These pictures illustrate the effect of a single Möbius transformation.

Indeed, every such map is by necessity a Möbius transformation.

The set of all Möbius transformations forms a group under composition.

This decomposition makes many properties of the Möbius transformation obvious.

For the rational functions defined on the complex numbers, see Möbius transformation.

Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles.

One important example of such functions is the group of Möbius transformations.

In the complex plane, a Möbius transformation is frequently called a homography.

This is another way to show that Möbius transformations preserve generalized circles.

A Möbius transformation is equivalent to a sequence of simpler transformations.

Note that the connection coefficients become Möbius transformations on the triangle maps.

The group G acts by Möbius transformations on the extended complex plane.

Möbius transformations, which are compositions of inversions, inherit that property.

Möbius transformations are also sometimes written in terms of their fixed points in so-called normal form.

All isometries within these models are therefore Möbius transformations.

On the extended complex plane one has the class of functions called Möbius transformations:

This is precisely the set of Möbius transformations that preserve the upper half-plane.

Mathematically speaking and are related via a Möbius transformation.

Otherwise f can be conjugated by a Möbius transformation so that the fixed point is zero.

These homeomorphisms can be expressed explicitly, as Möbius transformations.

As such, Möbius transformations play an important role in the theory of Riemann surfaces.

See Chapter 3 for a superbly illustrated discussion of Möbius transformations.

The group of Möbius transformations which preserve cross-ratios.

The family of conformal maps in three dimensions is very poor, and essentially contains only Möbius transformations.