# Möbius transformations and SO(3)

1. Jan 9, 2010

### csopi

Hi,

I was given the following problem, and i couldn't solve it yet:

Give a bijection between the elements of SO(3) and the fractional linear transformations of the form
$$\varphi_{z,w}\,(u)=\frac{zu+w}{-\bar wu+\bar z}$$, where $$u\in \mathbb C\cup \{\infty\};\, z,w\in \mathbb C$$.

Any ideas would be highly appreciated.

2. Jan 12, 2010

### wofsy

Here are a few things to think about.

- A fractional linear transformation is the projection of a linear transformation of a 2 dimensional complex vector space. The linear map with matrix

a b
c d

projects to the fractional linear transformation az + b/ cz + d.

- two linear transformations of the complex 2 space that differ by a complex constant project to the same fractional linear transformation.

- Some linear transformations carry the unit sphere in complex 2 space i.e. the vectors (a,b) such that |a|^2 + |b|^2 = 1 , into itself. If two of these differ by a sign they project to the same fractional linear transformation

- SO(3) is the same as the unit sphere in complex 2 space modulo multiplication by -1. In other words, SO(3) may be thought of as those fractional linear transformations that are projections of linear maps of C^2 that preserve the unit sphere.

- if w = 0 then the fractional linear transformation just rotates the plane around the origin. As a map of the Riemann sphere into itself is is just a rotation with infinity and zero as the two fixed poles.

The general fractional linear transformation has two fixed points that are complex conjugates of each other.