Editing Möbius transformation (section)

== Projective matrix representations ==
=== Isomorphism between the Möbius group and {{nowrap|PGL(2, C)}} ===
The natural [[Group action (mathematics)|action]] of {{nowrap|PGL(2, '''C''')}} on the [[complex projective line]] '''CP'''<sup>1</sup> is exactly the natural action of the Möbius group on the Riemann sphere
==== Correspondance between the complex projective line and the Riemann sphere ====
Here, the projective line '''CP'''<sup>1</sup> and the Riemann sphere are identified as follows:
<math display="block">[z_1 : z_2]\ \thicksim \frac{z_1}{z_2}.</math>

Here [''z''<sub>1</sub>:''z''<sub>2</sub>] are [[homogeneous coordinates]] on '''CP'''<sup>1</sup>; the point [1:0] corresponds to the point {{math|∞}} of the Riemann sphere. By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with {{math|∞}} are required.

==== Action of PGL(2,&nbsp;C) on the complex projective line ====
Every [[invertible matrix|invertible]] complex 2×2 matrix
<math display="block">\mathfrak H = \begin{pmatrix} a & b \\ c & d \end{pmatrix}</math>
acts on the projective line as
<math display="block">z = [z_1:z_2]\mapsto w = [w_1:w_2],</math>
where
<math display="block"> \begin{pmatrix}w_1\\w_2\end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}= \begin{pmatrix}az_1 + bz_2\\ cz_1 + dz_2\end{pmatrix}.</math>

The result is therefore 
<math display="block"> w = [w_1:w_2] = [az_1 + bz_2 : cz_1 + dz_2] </math>

Which, using the above identification, corresponds to the following point on the Riemann sphere : 

<math display="block"> w = [az_1 + bz_2 : cz_1 + dz_2] \thicksim \frac{az_1 + bz_2}{cz_1 + dz_2} = \frac{a\frac{z_1}{z_2} + b}{c\frac{z_1}{z_2} + d}. </math>

==== Equivalence with a Möbius transformation on the Riemann sphere ====
Since the above matrix is invertible if and only if its [[determinant]] {{math|''ad'' − ''bc''}} is not zero, this induces an identification of the action of the group of Möbius transformations with the action of {{nowrap|PGL(2, '''C''')}} on the complex projective line. In this identification, the above matrix <math>\mathfrak H</math> corresponds to the Möbius transformation <math>z\mapsto \frac{az+b}{cz+d}.</math>

This identification is a [[group isomorphism]], since the multiplication of <math>\mathfrak H</math> by a non zero scalar <math>\lambda</math> does not change the element of {{nowrap|PGL(2, '''C''')}}, and, as this multiplication consists of multiplying all matrix entries by <math>\lambda,</math> this does not change the corresponding Möbius transformation.

=== Other groups ===

For any [[field (mathematics)|field]] ''K'', one can similarly identify the group {{nowrap|PGL(2, ''K'')}} of the projective linear automorphisms with the group of fractional linear transformations. This is widely used; for example in the study of [[homography|homographies]] of the [[real line]] and its applications in [[optics]]. 

If one divides <math>\mathfrak{H}</math> by a square root of its determinant, one gets a matrix of determinant one. This induces a surjective group homomorphism from the [[special linear group]] {{nowrap|SL(2, '''C''')}} to {{nowrap|PGL(2, '''C''')}}, with <math>\pm I</math> as its kernel.

This allows showing that the Möbius group is a 3-dimensional complex [[Lie group]] (or a 6-dimensional real Lie group), which is a [[Semisimple Lie group|semisimple]] and  non-[[Compact group|compact]], and that SL(2,'''C''') is a [[Double covering group|double cover]] of {{nowrap|PSL(2, '''C''')}}.  Since {{nowrap|SL(2, '''C''')}} is [[simply-connected]], it is the [[universal cover]] of the Möbius group, and the [[fundamental group]] of the Möbius group is&nbsp;'''Z'''<sub>2</sub>.

=== Specifying a transformation by three points ===

Given a set of three distinct points <math>z_1,z_2,z_3</math> on the Riemann sphere and a second set of distinct points {{tmath|1= w_1,w_2,w_3 }}, there exists precisely one Möbius transformation <math>f(z)</math> with <math>f(z_j)=w_j</math> for {{tmath|1= j=1,2,3 }}. (In other words: the [[Group action (mathematics)|action]] of the Möbius group on the Riemann sphere is ''sharply 3-transitive''.) There are several ways to determine <math>f(z)</math> from the given sets of points.

==== Mapping first to 0, 1, {{math|∞}} ====
It is easy to check that the Möbius transformation
<math display="block">f_1(z)= \frac {(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}</math>
with matrix
<math display="block">\mathfrak{H}_1 = \begin{pmatrix}
z_2 - z_3 & -z_1 (z_2 - z_3)\\
z_2-z_1 & -z_3 (z_2-z_1)
\end{pmatrix}</math>
maps <math>z_1,z_2 \text{ and } z_3</math> to {{tmath|1= 0,1,\ \text{and}\ \infty}}, respectively. If one of the ''<math>z_j</math>'' is <math>\infty</math>, then the proper formula for <math>\mathfrak{H}_1</math> is obtained from the above one by first dividing all entries by ''<math>z_j</math>'' and then taking the limit {{tmath|1= z_j\to\infty }}.

If <math>\mathfrak{H}_2</math> is similarly defined to map <math>w_1,w_2,w_3</math> to <math>0,1,\ \text{and}\ \infty,</math> then the matrix <math>\mathfrak{H}</math> which maps <math>z_{1,2,3}</math> to <math>w_{1,2,3}</math> becomes
<math display="block">\mathfrak{H} = \mathfrak{H}_2^{-1} \mathfrak{H}_1.</math>

The stabilizer of <math>\{0,1,\infty\}</math> (as an unordered set) is a subgroup known as the [[anharmonic group]].

==== Explicit determinant formula ====
The equation
<math display="block">w=\frac{az+b}{cz+d}</math>
is equivalent to the equation of a standard [[hyperbola]]
<math display="block"> c wz -az+dw -b=0 </math>
in the <math>(z,w)</math>-plane. The problem of constructing a Möbius transformation <math> \mathfrak{H}(z) </math> mapping a triple <math> (z_1, z_2, z_3 )</math> to another triple <math>(w_1, w_2, w_3 )</math> is thus equivalent to finding the coefficients <math>a,b,c,d</math> of the hyperbola passing through the points {{tmath|1= (z_i, w_i ) }}. An explicit equation can be found by evaluating the [[determinant]]
<math display="block"> \begin{vmatrix} zw & z & w & 1 \\ z_1w_1 & z_1 & w_1 & 1 \\   z_2w_2 & z_2 & w_2 & 1 \\   z_3w_3 & z_3 & w_3 & 1\end{vmatrix}\, </math>
by means of a [[Laplace expansion]] along the first row, resulting in explicit formulae,
<math display="block">\begin{align}
a &= z_1w_1(w_2 - w_3) + z_2w_2(w_3 - w_1) + z_3w_3(w_1 - w_2), \\[5mu]
b &= z_1w_1(z_2w_3-z_3w_2)+z_2w_2(z_3w_1-z_1w_3)+z_3w_3(z_1w_2-z_2w_1), \\[5mu]
c &= w_1(z_3-z_2) + w_2(z_1-z_3) + w_3(z_2-z_1), \\[5mu]
d &= z_1w_1(z_2 - z_3) + z_2w_2(z_3 - z_1) + z_3w_3(z_1 - z_2)
\end{align}</math>
for the coefficients <math>a,b,c,d</math> of the representing matrix {{tmath|\mathfrak H}}. The constructed matrix <math> \mathfrak{H} </math> has determinant equal to {{tmath|1= (z_1-z_2) (z_1-z_3)(z_2-z_3)(w_1-w_2) (w_1-w_3)(w_2-w_3) }}, which does not vanish if the <math>z_j</math> resp. <math>w_j</math> are pairwise different thus the Möbius transformation is well-defined. If one of the points <math>z_j</math> or <math>w_j</math> is {{tmath|1= \infty }}, then we first divide all four determinants by this variable and then take the limit as the variable approaches {{tmath|1= \infty }}.