Editing Möbius transformation (section)

== Classification ==
[[File:Apollonian circles.svg|thumb|A hyperbolic transformation is shown.  Pre-images of the unit circle are [[circles of Apollonius]] with distance ratio ''c''/''a'' and foci at −''b''/''a'' and −''d''/''c'' . {{paragraph break}} For the same foci −''b''/''a'' and −''d''/''c'' the red circles map to rays through the origin.]]

In the following discussion we will always assume that the representing matrix <math> \mathfrak{H}</math> is normalized such that {{tmath|1= \det{\mathfrak{H} }=ad-bc=1 }}.

Non-identity Möbius transformations are commonly classified into four types, '''parabolic''', '''elliptic''', '''hyperbolic''' and '''loxodromic''', with the hyperbolic ones being a subclass of the loxodromic ones. The classification has both algebraic and geometric significance. Geometrically, the different types result in different transformations of the complex plane, as the figures below illustrate.

The four types can be distinguished by looking at the [[trace (matrix)|trace]] <math>\operatorname{tr} \mathfrak{H}=a+d</math>. The trace is invariant under [[conjugacy class|conjugation]], that is,
<math display="block">\operatorname{tr}\,\mathfrak{GHG}^{-1} = \operatorname{tr}\,\mathfrak{H},</math>
and so every member of a conjugacy class will have the same trace. Every Möbius transformation can be written such that its representing matrix <math>\mathfrak{H}</math> has determinant one (by multiplying the entries with a suitable scalar). Two Möbius transformations <math> \mathfrak{H}, \mathfrak{H}'</math> (both not equal to the identity transform) with <math> \det \mathfrak{H} = \det\mathfrak{H}' = 1 </math> are conjugate if and only if <math> \operatorname{tr}^2 \mathfrak{H} = \operatorname{tr}^2 \mathfrak{H}'.</math>

=== Parabolic transforms ===
A non-identity Möbius transformation defined by a matrix <math>\mathfrak{H}</math> of determinant one is said to be ''parabolic'' if
<math display="block">\operatorname{tr}^2\mathfrak{H} = (a+d)^2 = 4</math>
(so the trace is plus or minus 2; either can occur for a given transformation since <math>\mathfrak{H}</math> is determined only up to sign). In fact one of the choices for <math>\mathfrak{H}</math> has the same [[characteristic polynomial]] {{nowrap|''X''<sup>2</sup> − 2''X'' + 1}} as the identity matrix, and is therefore [[unipotent]]. A Möbius transform is parabolic if and only if it has exactly one fixed point in the [[Riemann sphere|extended complex plane]] <math>\widehat{\Complex} = \Complex\cup\{\infty\}</math>, which happens if and only if it can be defined by a matrix [[conjugacy class|conjugate to]]
<math display="block">\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}</math>
which describes a translation in the complex plane.

The set of all parabolic Möbius transformations with a ''given'' fixed point in <math>\widehat{\Complex}</math>, together with the identity, forms a [[group (mathematics)|subgroup]] isomorphic to the group of matrices
<math display="block">\left\{\begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} \mid b\in\Complex\right\};</math>
this is an example of the [[unipotent|unipotent radical]] of a [[Borel subgroup]] (of the Möbius group, or of {{nowrap|SL(2, '''C''')}} for the matrix group; the notion is defined for any [[reductive Lie group]]).

=== Characteristic constant ===
All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to
<math display="block">\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}</math>
with the complex number ''λ'' not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number {{nowrap|1=''k'' = ''λ''<sup>2</sup>}}, called the '''characteristic constant''' or '''multiplier''' of the transformation.

=== Elliptic transforms ===
[[File:Smith chart explanation.svg|thumb|upright=2.2|The [[Smith chart]], used by [[electrical engineering|electrical engineers]] for analyzing [[transmission line]]s, is a visual depiction of the elliptic Möbius transformation {{nowrap|1=Γ = (''z'' − 1)/(''z'' + 1)}}. Each point on the Smith chart simultaneously represents both a value of ''z'' (bottom left), and the corresponding value of Γ (bottom right), for {{pipe}}Γ{{pipe}}<1.]]
The transformation is said to be ''elliptic'' if it can be represented by a  matrix <math>\mathfrak H</math> of determinant 1 such that 
<math display="block">0 \le \operatorname{tr}^2\mathfrak{H} < 4.</math>

A transform is elliptic if and only if {{nowrap|1={{abs|''λ''}} = 1}} and {{nowrap|''λ'' ≠ ±1}}. Writing <math>\lambda=e^{i\alpha}</math>, an elliptic transform is conjugate to
<math display="block">\begin{pmatrix} \cos\alpha & -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}</math>
with ''α'' real.

For ''any'' <math>\mathfrak{H}</math> with characteristic constant ''k'', the characteristic constant of <math>\mathfrak{H}^n</math> is ''k<sup>n</sup>''. Thus, all Möbius transformations of finite [[order (group theory)|order]] are elliptic transformations, namely exactly those where ''λ'' is a [[root of unity]], or, equivalently, where ''α'' is a [[Rational number|rational]] multiple of [[pi|{{pi}}]]. The simplest possibility of a fractional multiple means {{nowrap|1=''α'' = {{pi}}/2}}, which is also the unique case of <math>\operatorname{tr}\mathfrak{H} = 0</math>, is also denoted as a '''{{visible anchor|circular transform}}'''; this corresponds geometrically to rotation by 180° about two fixed points. This class is represented in matrix form as:
<math display="block">\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.</math>
There are 3 representatives fixing {0, 1, ∞}, which are the three transpositions in the symmetry group of these 3 points: <math>1/z,</math> which fixes 1 and swaps 0 with ''∞'' (rotation by 180° about the points 1 and −1), <math>1-z</math>, which fixes ''∞'' and swaps 0 with 1 (rotation by 180° about the points 1/2 and ''∞''), and <math>z/(z - 1)</math> which fixes 0 and swaps 1 with ''∞'' (rotation by 180° about the points 0 and 2).

=== Hyperbolic transforms ===
The transform is said to be ''hyperbolic'' if it can be represented by a matrix <math>\mathfrak H</math> whose trace is [[real number|real]] with
<math display="block">\operatorname{tr}^2\mathfrak{H} > 4.</math>

A transform is hyperbolic if and only if ''λ'' is real and {{nowrap|''λ'' ≠ ±1}}.

=== Loxodromic transforms ===
The transform is said to be ''loxodromic'' if <math>\operatorname{tr}^2\mathfrak{H}</math> is not in {{nowrap|[0, 4]}}. A transformation is loxodromic if and only if <math>|\lambda|\ne 1</math>.

Historically, [[navigation]] by [[loxodrome]] or [[rhumb line]] refers to a path of constant [[bearing (navigation)|bearing]]; the resulting path is a [[logarithmic spiral]], similar in shape to the transformations of the complex plane that a loxodromic Möbius transformation makes. See the geometric figures below.

=== General classification ===

{| border=1 cellpadding=2 style="margin: auto; border-collapse: collapse; text-align: center;"
|- style="background:#ddf;"
! Transformation || Trace squared || Multipliers
! colspan=2 | Class representative
|-
! Circular
| ''σ'' = 0
| ''k'' = −1
| <math>\begin{pmatrix}i & 0 \\ 0 & -i\end{pmatrix}</math>
| ''z'' ↦ −''z''
|-
! Elliptic
| 0 ≤ ''σ'' < 4 <br/> <math>\sigma = 2+2\cos(\theta)</math>
|  | |''k''| = 1{{Clear}}<math>k = e^{\pm i\theta} \neq 1</math>
| <math>\begin{pmatrix} e^{i\theta/2} & 0 \\ 0 & e^{-i\theta/2}\end{pmatrix}</math>
| ''z'' ↦ ''e''<sup>''iθ''</sup> ''z''
|-
! Parabolic
| ''σ'' = 4
| ''k'' = 1
| <math>\begin{pmatrix}1 & a \\ 0 & 1\end{pmatrix}</math>
| ''z'' ↦ ''z'' + ''a''
|-
! Hyperbolic
| 4 < ''σ'' < ∞ <br/> <math>\sigma = 2+2\cosh(\theta)</math>
| <math>k \in \R^{+}</math>{{Clear}}<math>k = e^{\pm \theta} \neq 1</math>
| <math>\begin{pmatrix}e^{\theta/2} & 0 \\ 0 & e^{-\theta/2}\end{pmatrix}</math>
| ''z'' ↦ ''e''<sup>''θ''</sup> ''z''
|-
! Loxodromic
| ''σ'' ∈ '''C''' \ [0,4] <br/> <math>\sigma = (\lambda + \lambda^{-1})^2</math>
| <math>|k| \neq 1</math>{{Clear}}<math>k = \lambda^{2}, \lambda^{-2}</math>
| <math>\begin{pmatrix}\lambda & 0 \\ 0 & \lambda^{-1}\end{pmatrix}</math>
| ''z'' ↦ ''kz''
|}

=== The real case and a note on terminology ===
Over the real numbers (if the coefficients must be real), there are no non-hyperbolic loxodromic transformations, and the classification is into elliptic, parabolic, and hyperbolic, as for real [[conic]]s. The terminology is due to considering half the absolute value of the trace, |tr|/2, as the [[Eccentricity (mathematics)|eccentricity]] of the transformation&nbsp;– division by 2 corrects for the dimension, so the identity has eccentricity 1 (tr/''n'' is sometimes used as an alternative for the trace for this reason), and absolute value corrects for the trace only being defined up to a factor of ±1 due to working in PSL. Alternatively one may use half the trace ''squared'' as a proxy for the eccentricity squared, as was done above; these classifications (but not the exact eccentricity values, since squaring and absolute values are different) agree for real traces but not complex traces. The same terminology is used for the [[SL2(R)#Classification of elements|classification of elements of {{nowrap|SL(2, '''R''')}}]] (the 2-fold cover), and [[Eccentricity (mathematics)#Analogous classifications|analogous classifications]] are used elsewhere. Loxodromic transformations are an essentially complex phenomenon, and correspond to complex eccentricities.