Editing Möbius transformation (section)

=== 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).