Editing Möbius transformation (section)

== Poles of the transformation ==

The point <math display="inline">z_\infty = - \frac{d}{c}</math> is called the [[Pole (complex analysis)|pole]] of <math>\mathfrak{H}</math>; it is that point which is transformed to the point at infinity under {{tmath|1= \mathfrak{H} }}.

The inverse pole <math display="inline">Z_\infty = \frac{a}{c}</math> is that point to which the point at infinity is transformed. The point midway between the two poles is always the same as the point midway between the two fixed points:
<math display="block">\gamma_1 + \gamma_2 = z_\infty + Z_\infty.</math>

These four points are the vertices of a [[parallelogram]] which is sometimes called the '''characteristic parallelogram''' of the transformation.

A transform <math>\mathfrak{H}</math> can be specified with two fixed points ''γ''<sub>1</sub>, ''γ''<sub>2</sub> and the pole <math>z_\infty</math>.

<math display="block">\mathfrak{H} =
\begin{pmatrix}
Z_\infty & - \gamma_1 \gamma_2 \\
1        & - z_\infty
\end{pmatrix}, \;\;
Z_\infty = \gamma_1 + \gamma_2 - z_\infty.
</math>

This allows us to derive a formula for conversion between ''k'' and <math>z_\infty</math> given <math>\gamma_1, \gamma_2</math>:
<math display="block">z_\infty = \frac{k \gamma_1 - \gamma_2}{1 - k}</math>
<math display="block">k= \frac{\gamma_2 - z_\infty}{\gamma_1 - z_\infty}
= \frac{Z_\infty - \gamma_1}{Z_\infty - \gamma_2}
= \frac {a - c \gamma_1}{a - c \gamma_2},</math>
which reduces down to
<math display="block">k = \frac{(a + d) + \sqrt {(a - d)^2 + 4 b c}}{(a + d) - \sqrt {(a - d)^2 + 4 b c}}.</math>

The last expression coincides with one of the (mutually reciprocal) [[eigenvalue]] ratios <math display="inline">\frac{\lambda_1}{\lambda_2}</math> of <math>\mathfrak{H}</math> (compare the discussion in the preceding section about the characteristic constant of a transformation). Its [[characteristic polynomial]] is equal to
<math display="block">
\det (\lambda I_2- \mathfrak{H})
= \lambda^2-\operatorname{tr} \mathfrak{H}\,\lambda + \det \mathfrak{H}
= \lambda^2-(a+d)\lambda+(ad-bc)
</math>
which has roots
<math display="block"> \lambda_{i} = \frac{(a + d) \pm \sqrt {(a - d)^2 + 4 b c}}{2} = \frac{(a + d) \pm \sqrt {(a + d)^2 - 4(a d - b c)}}{2}=c\gamma_i+d \, .</math>