## Riemann Sphere and Möbius Transformation

### Hubert Shutrick

In projective geometry, the *xy*-plane is supplemented by adding an extra line and homogeneous coordinates are introduced to cope with this line at infinity (see also Chasles' Theorem). If points in the plane are described by a complex number *z* = *x* + I*y*,*z*_{1}, *z*_{2})*a**z*_{1}, *a**z*_{2})*a* that is not zero. If *z*_{2} ≠ 0,*z* = *z*_{1}/*z*_{2}*z*_{1}, 0)*projective line*.

Since stereographic projection sets up a one-to-one correspondence between points of the plane and points of the sphere except N - the North pole - and since it has nice properties, it is convenient to take the sphere as a model for the complex *projective line* (see Hopf fibration).

It is natural in projective geometry to consider linear transformations and they turn out to be important in this context:

*z'*_{1} = *az*_{1} + *bz*_{2}

*z'*_{2} = *cz*_{1} + *dz*_{2}

is a linear transformation if the determinant *ad* - *bc**homeomorphisms* (i.e. *1-1* and *onto* transformations continuous in both directions) of the complex line.

Converting to the non-homogeneous coordinate *z*, the linear transformations become

*z'* = (*az* + *b*) / (*cz* + *d*)

and are called *Möbius transformations* and, in this context, the sphere is called the *Riemann sphere*. The condition *ad* - *bc* ≠ 0,*z'* is not constant.

Möbius transformations transform circles and lines into circles and lines which on the sphere become simply circles into circles. To prove this property, divide the fraction in the case *c* ≠ 0

* z'* =

*f*+

*g*/ (

*cz*+

*d*)

The transformation can then be considered as the composite

*z* to *cz* to *cz* + *d* to 1 / (*cz* + *d*) to *g* / (*cz* + *d*) to *z'*.

Multiplying by *c* = *re*^{iθ} rotates a circle about the origin through theta and blows it up away from the origin or shrinks it towards by a factor r, adding a constant merely translates it and inverting, which takes *re*^{iθ} to 1/*re*^{- Iθ}, is inversion with respect to the unit circle at the origin followed by complex conjugation. All these transformations are known to take circles to circles unless the circle goes through -*d/c* when the circle goes to a line. The case *c* = 0

Corresponding to the fact that any three different points determine a change of base on the real projective line, a Möbius transformation can be determined by choosing three points that are to be the images of 0, 1 and infinity, which is N on the sphere.

Möbius transformations have had many important applications, some of which are described in the internet.

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