The Klein View of Geometry

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.

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