Editing Möbius transformation (section)

== Elementary properties ==
A Möbius transformation is equivalent to a sequence of simpler transformations. The composition makes many properties of the Möbius transformation obvious.

=== Formula for the inverse transformation ===
The existence of the inverse Möbius transformation and its explicit formula are easily derived by the composition of the inverse functions of the simpler transformations. That is, define functions ''g''<sub>1</sub>, ''g''<sub>2</sub>, ''g''<sub>3</sub>, ''g''<sub>4</sub> such that each ''g<sub>i</sub>'' is the inverse of ''f<sub>i</sub>''. Then the composition
<math display="block">g_1\circ g_2\circ g_3\circ g_4 (z) = f^{-1}(z) = \frac{dz-b}{-cz+a}</math>
gives a formula for the inverse.

=== Preservation of angles and generalized circles ===
From this decomposition, we see that Möbius transformations carry over all non-trivial properties of [[circle inversion]]. For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilations and [[isometries]] (translation, reflection, rotation), which trivially preserve angles.

Furthermore, Möbius transformations map [[generalized circle]]s to generalized circles since circle inversion has this property. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Note that a Möbius transformation does not necessarily map circles to circles and lines to lines: it can mix the two. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center.

=== Cross-ratio preservation ===
[[Cross-ratio]]s are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points <math>z_1, z_2, z_3, z_4</math> to four distinct points <math>w_1, w_2, w_3, w_4</math> respectively, then
<math display="block">\frac{(z_1-z_3)(z_2-z_4)}{(z_2-z_3)(z_1-z_4)} =\frac{(w_1-w_3)(w_2-w_4)}{(w_2-w_3)(w_1-w_4)}. </math>

If one of the points <math>z_1, z_2, z_3, z_4</math> is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of <math>z_1, z_2, z_3, \infin</math> is
<math display="block">\frac{(z_1-z_3)}{(z_2-z_3)}.</math>

The cross ratio of four different points is real if and only if there is a line or a circle passing through them. This is another way to show that Möbius transformations preserve generalized circles.

=== Conjugation ===
Two points ''z''<sub>1</sub> and ''z''<sub>2</sub> are '''conjugate''' with respect to a generalized circle ''C'', if, given a generalized circle ''D'' passing through ''z''<sub>1</sub> and ''z''<sub>2</sub> and cutting ''C'' in two points ''a'' and ''b'', {{nowrap|(''z''<sub>1</sub>, ''z''<sub>2</sub>; ''a'', ''b'')}} are in [[harmonic cross-ratio]] (i.e. their cross ratio is −1). This property does not depend on the choice of the circle ''D''. This property is also sometimes referred to as being '''symmetric''' with respect to a line or circle.<ref>{{citation|last=Olsen|first=John|title=The Geometry of Mobius Transformations |url=http://www.johno.dk/mathematics/moebius.pdf}}</ref><ref>{{mathworld|id=SymmetricPoints|title=Symmetric Points}}</ref>

Two points ''z'', ''z''<sup>∗</sup> are conjugate with respect to a line, if they are [[reflection symmetry|symmetric]] with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the [[inversive geometry|inversion]] with respect to this circle.

The point ''z''<sup>∗</sup> is conjugate to ''z'' when ''L'' is the line determined by the vector based upon ''e<sup>iθ</sup>'', at the point ''z''<sub>0</sub>. This can be explicitly given as
<math display="block">z^* = e^{2i\theta}\, \overline{z - z_0} + z_0.</math>

The point ''z''<sup>∗</sup> is conjugate to ''z'' when ''C'' is the circle of a radius ''r'', centered about ''z''<sub>0</sub>. This can be explicitly given as
<math display="block">z^* = \frac{r^2}{\overline{z - z_0}} + z_0.</math>

Since Möbius transformations preserve generalized circles and cross-ratios, they also preserve the conjugation.