# The Klein View of Geometry

The Mobius group

Definition

The Mobius group M(2) consists of all bilinear transformations of the form

m(z) = (αz+β)/(γz+δ),

where α, β, γ, δ are in C, with αδ - βγ ≠ 0.

This defines a transformation of E+ once we put
m(z) = {
(αz + β)/(γz + δ) if z ≠ ∞ and γz+δ ≠ 0
α/γ if γ ≠ 0 and z = ∞
if γ = 0 and z = ∞, or γ ≠ 0 and z = -δ/γ
With the transformation m , we associate the matrix M = [
α β
γ δ
]

It is easier to use matrices to calculate composites and inverses in M(2).

Suppose that m and n are the Mobius transformations

m(z) = (αz+β)/(γz+δ), and n(z) = (α'z+β')/(γ'z+δ'),

With the transformations m and n , we associate the matrices
M = [
 α β γ δ
]
and
N = [
 α' β' γ' δ'
]
 Now mon(z) = m(n(z)) = m((α'z+β')/(γ'z+δ')) = (α(α'z+β')/(γ'z+δ') +β)/(γ(α'z+β')/(γ'z+δ') +δ) = ((αα' + βγ')z +(αβ' + βδ')/((γα' + δγ')z + (γβ' + δδ')).

Thus mon has matrix
MN = [
 αα'+βγ' αβ'+βδ' γα'+δγ' γβ'+δδ'
]

It follows easily that the inverse of m has matrix M-1,
since MM-1 = I, and matrix I corresponds to transformation
sending z to (1z+0)/(0z+1) = z, i.e. to the identity transformation.

In fact, since matrices A and λA give rise to the same transformation,
we may use the matrix
[
 δ -β -γ α
]
rather than M-1 to describe the inverse of m.