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Möbius Transformations

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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.

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