Editing Möbius transformation (section)

=== Determining the fixed points ===
The fixed points of the transformation
<math display="block">f(z) = \frac{az + b}{cz + d}</math>
are obtained by solving the fixed point equation {{nowrap|1=''f''(''γ'') = ''γ''}}. For {{nowrap|''c'' ≠ 0}}, this has two roots obtained by expanding this equation to
<math display="block">c \gamma^2 - (a - d) \gamma - b = 0 \ ,</math>
and applying the [[quadratic formula]]. The roots are
<math display="block">\gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a - d)^2 + 4bc}}{2c} = \frac{(a - d) \pm \sqrt{\Delta}}{2c}</math>
with discriminant
<math display="block"> \Delta = (\operatorname{tr}\mathfrak{H})^2 - 4\det\mathfrak{H} = (a+d)^2 - 4(ad-bc),</math>
where the matrix
<math display="block">\mathfrak{H} = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}</math>
represents the transformation.
Parabolic transforms have coincidental fixed points due to zero discriminant. For ''c'' nonzero and nonzero discriminant the transform is elliptic or hyperbolic.

When {{nowrap|1=''c'' = 0}}, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When {{nowrap|''a'' ≠ ''d''}} the second fixed point is finite and is given by
<math display="block">\gamma = -\frac{b}{a-d}.</math>

In this case the transformation will be a simple transformation composed of [[Translation (geometry)|translation]]s, [[rotation (mathematics)|rotation]]s, and [[dilation (metric space)|dilation]]s:
<math display="block">z \mapsto \alpha z + \beta.</math>

If {{nowrap|1=''c'' = 0}} and {{nowrap|1=''a'' = ''d''}}, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:
<math display="block">z \mapsto z + \beta.</math>