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Mobius Transformations

Mobius Transformations

A common type of transformation when dealing with transformation of the complex plane are those of the form $$w = \frac{az + b}{cz + d}.$$ Such transformations are called Mobius transformations. They have a number of interesting properties, which at A-Level are predominantly concerned with their effect on lines and circles. Namely,

  1. A line maps to either a line or a circle
  2. A circle maps to either a line or a circle

Which of the options occurs is dictate by whether the point $z = -\frac{d}{c}$ is contained within original locus of points. If it is, then this point is mapped to infinity; forcing the resulting locus to be a line. This is the point at which A-level study of these functions broadly stops. Examples of involving manipulation of these functions can be found in the complex transformations questions in the question bank (links at the bottom of the article

 

As mentioned previously, these functions are much more interesting than their limited study at A-Level perhaps suggests. Each Mobius function can be thought of as a reverse stereographic projection onto the surface of a sphere, a translation, rotation and enlargement of said sphere and then a stereographic projection back onto the plane. Hence the image!

Stereographic projection is the achieved through taking the North Pole as the start of a line segment, drawing down to the desired point on the plane. The line will intersect the sphere at some point en route; this will be the point to which the point on the plane is mapped. This is easier seen in 2D:

stereographic Projection

In two dimensions, we see that this sets up a 1-1 mapping (bijection) between an infinite line and a punctured circle (the North Pole doesn't map to a point on the line). In three dimensions, we have a bijection between a punctured sphere and a plane.