**Useful Links**

Complex Plane ---
Addition ---
Multiplication ---
Division ---
Powers ---
Loxodromic spiral

## Möbius Transformations on the Complex Plane

**Note:** For more information about the complex plane and complex
numbers, visit the Complex Plane tutorial.

A Möbius transformation is a mapping **T** of the complex plane given by

where **a, b, c** and **d** are complex numbers such
that **a*d** is not equal to **b*c**.

This can also be written in the short form

So, given any complex number **z**, we can apply the transformation **T** to **z**,
to get a new complex number newZ. This can be written
.

We can imagine the transformation **T** being applied to all points in the
complex plane simultaneously. This is what we did when we moved,
scaled or rotated Penny.
We then applied **T** again, to the new image creating a sequence of images.

You can try this out in the Complex Plane tutorial

We can also just apply **T** to one point, as we did when constructing the
loxodromic spiral. We applied **T** again to the new point,
creating a path of points which we joined with line segments.

## Special Properties

**Circles and Lines**

A circle will be mapped by a Möbius transformation onto another circle
or onto a straight line. And a straight line will be mapped onto a
straight line or a circle. If we think of a straight line as just a very
big circle that goes through the point at infinity, then we can just say
that circles are mapped to circles.

**Inverses**

Each Möbius transformation has an inverse.

Suppose a Möbius transformation **T** maps a point **p** onto a new point **q**.
Then the Möbius transformation that maps **q** back onto **p** is called the
inverse of **T**, and is written **T**^{-1}. That is, **T**^{-1}
undoes the effect of **T**,
taking the new point back to where it came from.

**Fixed Points**

Each Möbius transformation has one or two fixed points.
These are points that are not moved by the mapping, so **T(z)=z**.

**Next**

A diversion to The Mathematics for those who
really want it, otherwise go straight to the pretty pictures