Editing Möbius transformation

{{Short description|Rational function of the form (az + b)/(cz + d)}}
{{distinguish|Möbius transform|Möbius function}}

In [[geometry]] and [[complex analysis]], a '''Möbius transformation''' of the [[complex plane]] is a [[rational function]] of the form
<math display="block">f(z) = \frac{a z + b}{c z + d}</math>
of one [[complex number|complex]] variable {{mvar|z}}; here the coefficients {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are complex numbers satisfying {{math|''ad'' − ''bc'' ≠ 0}}.

Geometrically, a Möbius transformation can be obtained by first applying the inverse [[stereographic projection]] from the plane to the unit [[sphere]], moving and rotating the sphere to a new location and orientation in space, and then applying a stereographic projection to map from the sphere back to the plane.{{sfn|Arnold|Rogness|2008|loc=Theorem 1}} These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle.

The Möbius transformations are the [[projective transformation]]s of the [[complex projective line]]. They form a [[group (mathematics)|group]] called the '''Möbius group''', which is the [[projective linear group]] {{nowrap|PGL(2, '''C''')}}. Together with its [[subgroup]]s, it has numerous applications in mathematics and physics.

[[Möbius geometry|Möbius geometries]] and their transformations generalize this case to any number of dimensions over other fields.

Möbius transformations are named in honor of [[August Ferdinand Möbius]]; they are an example of [[homography|homographies]], [[linear fractional transformation]]s, bilinear transformations, and spin transformations (in relativity theory).<ref>
{{Cite book|last=Needham|first=Tristan|url=https://press.princeton.edu/books/hardcover/9780691203690/visual-differential-geometry-and-forms|title=Differential Geometry and Forms; A Mathematical Drama in Five Acts|publisher=Princeton University Press|year=2021|isbn=9780691203690 |at=p. 77, footnote 16}}</ref>

== Overview ==
Möbius transformations are defined on the [[Riemann sphere|extended complex plane]] <math>\widehat{\Complex} = \Complex \cup \{\infty\}</math> (i.e., the [[complex plane]] augmented by the [[point at infinity]]).

[[Stereographic projection]] identifies <math>\widehat{\Complex}</math> with a sphere, which is then called the [[Riemann sphere]]; alternatively, <math>\widehat{\Complex}</math> can be thought of as the complex [[projective line]] <math>\Complex\mathbb{P}^1</math>.  The Möbius transformations are exactly the [[bijective]] [[conformal map|conformal]] maps from the Riemann sphere to itself, i.e., the [[automorphism group|automorphisms]] of the Riemann sphere as a [[complex manifold]]; alternatively, they are the automorphisms of <math>\Complex\mathbb{P}^1</math> as an algebraic variety.  Therefore, the set of all Möbius transformations forms a [[group (mathematics)|group]] under [[function composition|composition]].  This group is called the Möbius group, and is sometimes denoted <math>\operatorname{Aut}(\widehat{\Complex})</math>.

The Möbius group is [[group isomorphism|isomorphic]] to the group of orientation-preserving [[isometry|isometries]] of [[hyperbolic space|hyperbolic 3-space]] and therefore plays an important role when studying [[hyperbolic 3-manifold]]s.

In [[physics]], the [[identity component]] of the [[Lorentz group]] acts on the [[celestial sphere]] in the same way that the Möbius group acts on the Riemann sphere. In fact, these two groups are isomorphic. An observer who accelerates to relativistic velocities will see the pattern of constellations as seen near the Earth continuously transform according to infinitesimal Möbius transformations. This observation is often taken as the starting point of [[twistor theory]].

Certain [[subgroup]]s of the Möbius group form the automorphism groups of the other [[simply-connected]] Riemann surfaces (the [[complex plane]] and the [[Hyperbolic space|hyperbolic plane]]). As such, Möbius transformations play an important role in the theory of [[Riemann surface]]s. The [[fundamental group]] of every Riemann surface is a [[discrete subgroup]] of the Möbius group (see [[Fuchsian group]] and [[Kleinian group]]). A particularly important discrete subgroup of the Möbius group is the [[modular group]]; it is central to the theory of many [[fractal]]s, [[modular form]]s, [[elliptic curve]]s and [[Pellian equation]]s.

Möbius transformations can be more generally defined in spaces of dimension {{math|''n'' > 2}} as the bijective conformal orientation-preserving maps from the [[n-sphere|{{nowrap|{{mvar|n}}-sphere}}]] to the {{mvar|n}}-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to [[Liouville's theorem (conformal mappings)|Liouville's theorem]] a Möbius transformation can be expressed as a composition of translations, [[Similarity (geometry)|similarities]], orthogonal transformations and inversions.

== Definition ==
The general form of a Möbius transformation is given by
<math display="block">f(z) = \frac{a z + b}{c z + d},</math>
where {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|d}} are any [[complex number]]s that satisfy {{math|''ad'' − ''bc'' ≠ 0}}.

In case {{math|''c'' ≠ 0}}, this definition is extended to the whole [[Riemann sphere]] by defining
<math display="block">\begin{align}
f\left(\frac{-d}{c}\right) &= \infin, \\
f(\infin) &= \frac{a}{c}.\end{align}</math>

If {{math|1=''c'' = 0}}, we define
<math display="block">f(\infin) = \infin.</math>

Thus a Möbius transformation is always a bijective [[holomorphic function]] from the Riemann sphere to the Riemann sphere.

The set of all Möbius transformations forms a [[group (mathematics)|group]] under [[function composition|composition]]. This group can be given the structure of a [[complex manifold]] in such a way that composition and inversion are [[holomorphic map]]s. The Möbius group is then a [[complex Lie group]]. The Möbius group is usually denoted <math>\operatorname{Aut}(\widehat{\Complex})</math> as it is the [[automorphism group]] of the Riemann sphere.

If {{math|1=''ad'' = ''bc''}}, the rational function defined above is a constant (unless {{math|1=''c'' = ''d'' = 0}}, when it is undefined):
<math display="block">\frac{a z + b}{c z + d} = \frac{a}{c} = \frac{b}{d},</math>
where a fraction with a zero denominator is ignored.  A constant function is not bijective and is thus not considered a Möbius transformation.

An alternative definition is given as the kernel of the [[Schwarzian derivative]].

== Fixed points ==
Every non-identity Möbius transformation has two [[fixed point (mathematics)|fixed points]] <math>\gamma_1, \gamma_2</math> on the Riemann sphere. The fixed points are counted here with [[Multiplicity (mathematics)|multiplicity]]; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity.

=== Determining the fixed points ===
The fixed points of the transformation
<math display="block">f(z) = \frac{az + b}{cz + d}</math>
are obtained by solving the fixed point equation {{nowrap|1=''f''(''γ'') = ''γ''}}. For {{nowrap|''c'' ≠ 0}}, this has two roots obtained by expanding this equation to
<math display="block">c \gamma^2 - (a - d) \gamma - b = 0 \ ,</math>
and applying the [[quadratic formula]]. The roots are
<math display="block">\gamma_{1,2} = \frac{(a - d) \pm \sqrt{(a - d)^2 + 4bc}}{2c} = \frac{(a - d) \pm \sqrt{\Delta}}{2c}</math>
with discriminant
<math display="block"> \Delta = (\operatorname{tr}\mathfrak{H})^2 - 4\det\mathfrak{H} = (a+d)^2 - 4(ad-bc),</math>
where the matrix
<math display="block">\mathfrak{H} = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}</math>
represents the transformation.
Parabolic transforms have coincidental fixed points due to zero discriminant. For ''c'' nonzero and nonzero discriminant the transform is elliptic or hyperbolic.

When {{nowrap|1=''c'' = 0}}, the quadratic equation degenerates into a linear equation and the transform is linear. This corresponds to the situation that one of the fixed points is the point at infinity. When {{nowrap|''a'' ≠ ''d''}} the second fixed point is finite and is given by
<math display="block">\gamma = -\frac{b}{a-d}.</math>

In this case the transformation will be a simple transformation composed of [[Translation (geometry)|translation]]s, [[rotation (mathematics)|rotation]]s, and [[dilation (metric space)|dilation]]s:
<math display="block">z \mapsto \alpha z + \beta.</math>

If {{nowrap|1=''c'' = 0}} and {{nowrap|1=''a'' = ''d''}}, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation:
<math display="block">z \mapsto z + \beta.</math>

=== Topological proof ===
Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the [[Euler characteristic]] of the sphere being 2:
<math display="block"> \chi(\hat{\Complex}) = 2.</math>

Firstly, the [[projective linear group]] {{nowrap|PGL(2, ''K'')}} is [[n-transitive|sharply 3-transitive]]&nbsp;– for any two ordered triples of distinct points, there is a unique map that takes one triple to the other, just as for Möbius transforms, and by the same algebraic proof (essentially [[dimension counting]], as the group is 3-dimensional). Thus any map that fixes at least 3 points is the identity.

Next, one can see by identifying the  Möbius group with <math>\mathrm{PGL}(2,\Complex)</math> that any Möbius function is homotopic to the identity. Indeed, any member of the [[general linear group]] can be reduced to the identity map by [[Gaussian elimination|Gauss-Jordan elimination]], this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The [[Lefschetz–Hopf theorem]] states that the sum of the indices (in this context, multiplicity) of the fixed points of a map with finitely many fixed points equals the [[Lefschetz number]] of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.

By contrast, the projective linear group of the real projective line, {{nowrap|PGL(2, '''R''')}} need not fix any points&nbsp;– for example <math>(1+x) / (1-x)</math> has no (real) fixed points: as a complex transformation it fixes ±''i''<ref group="note">Geometrically this map is the [[stereographic projection]] of a rotation by 90° around ±''i'' with period 4, which takes <math>0 \mapsto 1 \mapsto \infty \mapsto -1 \mapsto 0.</math></ref>&nbsp;– while the map 2''x'' fixes the two points of 0 and ∞. This corresponds to the fact that the Euler characteristic of the circle (real projective line) is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more.

=== Normal form ===
Möbius transformations are also sometimes written in terms of their fixed points in so-called '''normal form'''. We first treat the non-parabolic case, for which there are two distinct fixed points.

''Non-parabolic case'':

Every non-parabolic transformation is [[conjugacy class|conjugate]] to a dilation/rotation, i.e., a transformation of the form
<math display="block">z \mapsto k z </math>
{{nowrap|(''k'' ∈ '''C''')}} with fixed points at 0 and ∞. To see this define a map
<math display="block">g(z) = \frac{z - \gamma_1}{z - \gamma_2}</math>
which sends the points (''γ''<sub>1</sub>, ''γ''<sub>2</sub>) to (0, ∞). Here we assume that ''γ''<sub>1</sub> and ''γ''<sub>2</sub> are distinct and finite. If one of them is already at infinity then ''g'' can be modified so as to fix infinity and send the other point to 0.

If ''f'' has distinct fixed points (''γ''<sub>1</sub>, ''γ''<sub>2</sub>) then the transformation <math>gfg^{-1}</math> has fixed points at 0 and ∞ and is therefore a dilation: <math>gfg^{-1}(z) = kz</math>. The fixed point equation for the transformation ''f'' can then be written
<math display="block">\frac{f(z)-\gamma_1}{f(z)-\gamma_2} = k \frac{z-\gamma_1}{z-\gamma_2}.</math>

Solving for ''f'' gives (in matrix form):
<math display="block">\mathfrak{H}(k; \gamma_1, \gamma_2) =
\begin{pmatrix}
\gamma_1 - k\gamma_2 & (k - 1) \gamma_1\gamma_2 \\
1 - k                &  k\gamma_1 - \gamma_2
\end{pmatrix}</math>
or, if one of the fixed points is at infinity:
<math display="block">\mathfrak{H}(k; \gamma, \infty) =
\begin{pmatrix}
k & (1 - k) \gamma \\
0 &  1
\end{pmatrix}.</math>

From the above expressions one can calculate the derivatives of ''f'' at the fixed points:
<math display="block">f'(\gamma_1) = k </math> and <math display="block"> f'(\gamma_2) = 1/k.</math>

Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers (''k'') of ''f'' as the '''characteristic constant''' of ''f''. Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:
<math display="block">\mathfrak{H}(k; \gamma_1, \gamma_2) = \mathfrak{H}(1/k; \gamma_2, \gamma_1).</math>

For loxodromic transformations, whenever {{nowrap|{{abs|''k''}} > 1}}, one says that ''γ''<sub>1</sub> is the '''repulsive''' fixed point, and ''γ''<sub>2</sub> is the '''attractive''' fixed point. For {{nowrap|{{abs|''k''}} < 1}}, the roles are reversed.

''Parabolic case'':

In the parabolic case there is only one fixed point ''γ''. The transformation sending that point to ∞ is
<math display="block">g(z) = \frac{1}{z - \gamma}</math>
or the identity if ''γ'' is already at infinity. The transformation <math>gfg^{-1}</math> fixes infinity and is therefore a translation:
<math display="block">gfg^{-1}(z) = z + \beta\,.</math>

Here, ''β'' is called the '''translation length'''. The fixed point formula for a parabolic transformation is then
<math display="block">\frac{1}{f(z)-\gamma} = \frac{1}{z-\gamma} + \beta.</math>

Solving for ''f'' (in matrix form) gives
<math display="block">\mathfrak{H}(\beta; \gamma) =
\begin{pmatrix}
1+\gamma\beta    & - \beta \gamma^2  \\
\beta            &   1-\gamma \beta
\end{pmatrix}</math>Note that
<math>\det\mathfrak{H}(\beta;\gamma)=|\mathfrak{H}(\beta;\gamma) |
=\det
\begin{pmatrix}
1+\gamma\beta & -\beta\gamma^2\\
\beta & 1-\gamma\beta
\end{pmatrix}
=1-\gamma^2\beta^2+\gamma^2\beta^2=1

</math>

If {{nowrap|1=''γ'' = ∞}}:
<math display="block">\mathfrak{H}(\beta; \infty) =
\begin{pmatrix}
1 & \beta  \\
0 & 1
\end{pmatrix}</math>

Note that ''β'' is ''not'' the characteristic constant of ''f'', which is always 1 for a parabolic transformation. From the above expressions one can calculate:
<math display="block">f'(\gamma) = 1.</math>

== 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>

== Simple  Möbius transformations and composition ==

A Möbius transformation can be [[Function composition|composed]] as a sequence of simple transformations.

The following simple transformations are also Möbius transformations:
* <math>f(z) = z+b\quad (a=1, c=0, d=1)</math> is a [[translation (geometry)|translation]].
* <math>f(z) = az \quad (b=0, c=0, d=1)</math> is a combination of a [[homothety]] (uniform [[Scaling (geometry)|scaling]]) and a [[Rotation (mathematics)|rotation]]. If <math>|a| = 1</math> then it is a rotation, if <math>a \in \R</math> then it is a homothety.
* <math>f(z)= 1/z \quad (a=0, b=1, c=1, d=0) </math> ([[inversive geometry|inversion]] and [[Reflection (mathematics)|reflection]] with respect to the real axis)

=== Composition of simple transformations ===

If <math>c \neq 0</math>, let:
* <math>f_1(z)= z+d/c \quad</math> (translation by ''d''/''c'')
* <math>f_2(z)= 1/z \quad</math> (inversion and reflection with respect to the real axis)
* <math>f_3(z)= \frac{bc-ad}{c^2} z \quad</math> (homothety and rotation)
* <math>f_4(z)= z+a/c \quad</math> (translation by ''a''/''c'')

Then these functions can be [[Function composition|composed]], showing that, if  
<math display="block"> f(z) = \frac{az+b}{cz+d}, </math>
one has
<math display="block"> f=f_4\circ f_3\circ f_2\circ f_1 . </math>
In other terms, one has
<math display="block">\frac{az+b}{cz+d} = \frac ac + \frac e{z+\frac dc},</math>
with
<math display="block">e= \frac{bc-ad}{c^2}. </math>

This decomposition makes many properties of the Möbius transformation obvious.

== 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.

== 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 }}.

== Subgroups of the Möbius group ==
If we require the coefficients <math>a,b,c,d</math> of a Möbius transformation to be real numbers with {{tmath|1= ad-bc=1 }}, we obtain a subgroup of the Möbius group denoted as [[PSL2(R)|{{nowrap|PSL(2, '''R''')}}]]. This is the group of those Möbius transformations that map the [[upper half-plane]] {{nowrap|1=''H'' = {''x'' + i''y'' : ''y'' > 0} }} to itself, and is equal to the group of all [[biholomorphic]] (or equivalently: [[bijective]], [[conformal map|conformal]] and orientation-preserving) maps {{nowrap|''H'' → ''H''}}. If a proper [[Riemannian metric|metric]] is introduced, the upper half-plane becomes a model of the [[hyperbolic geometry|hyperbolic plane]] ''H''{{i sup|2}}, the [[Poincaré half-plane model]], and {{nowrap|PSL(2, '''R''')}} is the group of all orientation-preserving isometries of ''H''{{i sup|2}} in this model.

The subgroup of all Möbius transformations that map the open disk {{nowrap|1=''D'' = {''z'' : {{abs|''z''}} < 1} }} to itself consists of all transformations of the form
<math display="block">f(z) = e^{i\phi} \frac{z + b}{\bar{b} z + 1}</math>
with {{nowrap|<math>\phi</math> ∈ '''R''', ''b'' ∈ '''C'''}} and {{nowrap|{{abs|''b''}} < 1}}. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps {{nowrap|''D'' → ''D''}}. By introducing a suitable metric, the open disk turns into another model of the hyperbolic plane, the [[Poincaré disk model]], and this group is the group of all orientation-preserving isometries of ''H''{{i sup|2}} in this model.

Since both of the above subgroups serve as isometry groups of ''H''{{i sup|2}}, they are isomorphic. A concrete isomorphism is given by [[inner automorphism|conjugation]] with the transformation
<math display="block">f(z)=\frac{z+i}{iz+1}</math>
which bijectively maps the open unit disk to the upper half plane.

Alternatively, consider an open disk with radius ''r'', centered at ''r''{{hsp}}''i''. The Poincaré disk model in this disk becomes identical to the upper-half-plane model as ''r'' approaches&nbsp;∞.

A [[maximal compact subgroup]] of the Möbius group <math>\mathcal{M}</math> is given by {{Harvard citation|Tóth|2002}}{{sfn |Tóth|2002|loc=Section 1.2, Rotations and Möbius Transformations, [https://books.google.com/books?id=i76mmyvDHYUC&pg=PA22 p.&nbsp;22]}}
<math display="block">\mathcal{M}_0 := \left\{z \mapsto \frac{uz - \bar v}{vz + \bar u} : |u|^2 + |v|^2 = 1\right\},</math>
and corresponds under the isomorphism <math>\mathcal{M} \cong \operatorname{PSL}(2,\Complex)</math> to the [[projective special unitary group]] {{nowrap|PSU(2, '''C''')}} which is isomorphic to the [[special orthogonal group]] SO(3) of rotations in three dimensions, and can be interpreted as rotations of the Riemann sphere. Every finite subgroup is conjugate into this maximal compact group, and thus these correspond exactly to the polyhedral groups, the [[point groups in three dimensions]].

[[Icosahedral group]]s of Möbius transformations were used by [[Felix Klein]] to give an analytic solution to the [[quintic equation]] in {{Harvard citation|Klein|1913}}; a modern exposition is given in {{Harvard citation|Tóth|2002}}.{{sfn|Tóth|2002|loc=Section 1.6, Additional Topic: Klein's Theory of the Icosahedron, [https://books.google.com/books?id=i76mmyvDHYUC&pg=PA66 p.&nbsp;66]}}

If we require the coefficients ''a'', ''b'', ''c'', ''d'' of a Möbius transformation to be [[integer]]s with {{nowrap|1=''ad'' − ''bc'' = 1}}, we obtain the [[modular group]] {{nowrap|PSL(2, '''Z''')}}, a discrete subgroup of {{nowrap|PSL(2, '''R''')}} important in the study of [[Lattice (order)|lattice]]s in the complex plane, [[elliptic function]]s and [[elliptic curve]]s. The discrete subgroups of {{nowrap|PSL(2, '''R''')}} are known as [[Fuchsian group]]s; they are important in the study of [[Riemann surface]]s.

== 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.

== Geometric interpretation of the characteristic constant ==
The following picture depicts (after stereographic transformation from the sphere to the plane) the two fixed points of a Möbius transformation in the non-parabolic case:

[[File:Mobius Identity.jpeg]]

The characteristic constant can be expressed in terms of its [[Natural logarithm|logarithm]]:
<math display="block">e^{\rho + \alpha i} = k. </math>
When expressed in this way, the real number ''ρ'' becomes an expansion factor. It indicates how repulsive the fixed point ''γ''<sub>1</sub> is, and how attractive ''γ''<sub>2</sub> is. The real number ''α'' is a rotation factor, indicating to what extent the transform rotates the plane anti-clockwise about ''γ''<sub>1</sub> and clockwise about&nbsp;''γ''<sub>2</sub>.

=== Elliptic transformations ===
If {{nowrap|1=''ρ'' = 0}}, then the fixed points are neither attractive nor repulsive but indifferent, and the transformation is said to be ''elliptic''. These transformations tend to move all points in circles around the two fixed points. If one of the fixed points is at infinity, this is equivalent to doing an affine rotation around a point.

If we take the [[one-parameter subgroup]] generated by any elliptic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a family of circles which is nested between the two fixed points on the Riemann sphere. In general, the two fixed points can be any two distinct points.

This has an important physical interpretation.
Imagine that some observer rotates with constant angular velocity about some axis. Then we can take the two fixed points to be the North and South poles of the celestial sphere. The appearance of the night sky is now transformed continuously in exactly the manner described by the one-parameter subgroup of elliptic transformations sharing the fixed points 0, ∞, and with the number ''α'' corresponding to the constant angular velocity of our observer.

Here are some figures illustrating the effect of an elliptic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):

[[File:Mobius Small Neg Elliptical.jpeg]]

[[File:Mobius Large Pos Elliptical.jpeg]]

These pictures illustrate the effect of a single Möbius transformation. The one-parameter subgroup which it generates ''continuously'' moves points along the family of circular arcs suggested by the pictures.

=== Hyperbolic transformations ===

If ''α'' is zero (or a multiple of 2{{pi}}), then the transformation is said to be ''hyperbolic''. These transformations tend to move points along circular paths from one fixed point toward the other.

If we take the [[one-parameter group|one-parameter subgroup]] generated by any hyperbolic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a certain family of circular arcs ''away'' from the first fixed point and ''toward'' the second fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.

This too has an important physical interpretation. Imagine that an observer accelerates (with constant magnitude of acceleration) in the direction of the North pole on his celestial sphere. Then the appearance of the night sky is transformed in exactly the manner described by the one-parameter subgroup of hyperbolic transformations sharing the fixed points 0, ∞, with the real number ''ρ'' corresponding to the magnitude of his acceleration vector. The stars seem to move along longitudes, away from the South pole toward the North pole. (The longitudes appear as circular arcs under stereographic projection from the sphere to the plane.)

Here are some figures illustrating the effect of a hyperbolic Möbius transformation on the Riemann sphere (after stereographic projection to the plane):

[[File:Mobius Small Neg Hyperbolic.jpeg]]

[[File:Mobius Large Pos Hyperbolic.jpeg]]

These pictures resemble the field lines of a positive and a negative electrical charge located at the fixed points, because the circular flow lines subtend a constant angle between the two fixed points.

=== Loxodromic transformations ===

If both ''ρ'' and ''α'' are nonzero, then the transformation is said to be ''loxodromic''. These transformations tend to move all points in S-shaped paths from one fixed point to the other.

The word "[[loxodrome]]" is from the Greek: "λοξος (loxos), ''slanting'' + δρόμος (dromos), ''course''". When [[sailing]] on a constant [[bearing (navigation)|bearing]] – if you maintain a heading of (say) north-east, you will eventually wind up sailing around the [[north pole]] in a [[logarithmic spiral]]. On the [[mercator projection]] such a course is a straight line, as the north and south poles project to infinity. The angle that the loxodrome subtends relative to the lines of longitude (i.e. its slope, the "tightness" of the spiral) is the argument of ''k''. Of course, Möbius transformations may have their two fixed points anywhere, not just at the north and south poles. But any loxodromic transformation will be conjugate to a transform that moves all points along such loxodromes.

If we take the [[one-parameter group|one-parameter subgroup]] generated by any loxodromic Möbius transformation, we obtain a continuous transformation, such that every transformation in the subgroup fixes the ''same'' two points. All other points flow along a certain family of curves, ''away'' from the first fixed point and ''toward'' the second fixed point. Unlike the hyperbolic case, these curves are not circular arcs, but certain curves which under stereographic projection from the sphere to the plane appear as spiral curves which twist counterclockwise infinitely often around one fixed point and twist clockwise infinitely often around the other fixed point. In general, the two fixed points may be any two distinct points on the Riemann sphere.

You can probably guess the physical interpretation in the case when the two fixed points are 0,&nbsp;∞: an observer who is both rotating (with constant angular velocity) about some axis and moving along the ''same'' axis, will see the appearance of the night sky transform according to the one-parameter subgroup of loxodromic transformations with fixed points 0, ∞, and with ''ρ'', ''α'' determined respectively by the magnitude of the actual linear and angular velocities.

=== Stereographic projection ===
These images show Möbius transformations [[stereographic projection|stereographically projected]] onto the [[Riemann sphere]]. Note in particular that when projected onto a sphere, the special case of a fixed point at infinity looks no different from having the fixed points in an arbitrary location.
{|
|-
| colspan="3" style="text-align:center;"| '''One fixed point at infinity'''
|-
| style="text-align:center;"| [[File:Mob3d-elip-inf-480.png|thumb|Elliptic]]
| style="text-align:center;"| [[File:Mob3d-hyp-inf-480.png|thumb|Hyperbolic]]
| style="text-align:center;"| [[File:Mob3d-lox-inf-480.png|thumb|Loxodromic]]
|-
| colspan="3" style="text-align:center;"| '''Fixed points diametrically opposite'''
|-
| style="text-align:center;"| [[File:Mob3d-elip-opp-480.png|thumb|Elliptic]]
| style="text-align:center;"| [[File:Mob3d-hyp-opp-480.png|thumb|Hyperbolic]]
| style="text-align:center;"| [[File:Mob3d-lox-opp-480.png|thumb|Loxodromic]]
|-
| colspan="3" style="text-align:center;"| '''Fixed points in an arbitrary location'''
|-
| style="text-align:center;"| [[File:Mob3d-elip-arb-480.png|thumb|Elliptic]]
| style="text-align:center;"| [[File:Mob3d-hyp-arb-480.png|thumb|Hyperbolic]]
| style="text-align:center;"| [[File:Mob3d-lox-arb-480.png|thumb|Loxodromic]]
|}

== Iterating a transformation ==
{{see also|Iterated function system}}

If a transformation <math>\mathfrak{H}</math> has fixed points ''γ''<sub>1</sub>, ''γ''<sub>2</sub>, and characteristic constant ''k'', then <math>\mathfrak{H}' = \mathfrak{H}^n</math> will have <math>\gamma_1' = \gamma_1, \gamma_2' = \gamma_2, k' = k^n</math>.

This can be used to [[iterate (math)|iterate]] a transformation, or to animate one by breaking it up into steps.

These images show three points (red, blue and black) continuously iterated under transformations with various characteristic constants.

{| style="margin: auto;"
|-
| [[File:Mobius23621.jpeg]] || [[File:Mobius23622.jpeg]] || [[File:Mobius23623.jpeg]] ||
|-
|
|}

And these images demonstrate what happens when you transform a circle under Hyperbolic, Elliptical, and Loxodromic transforms. In the elliptical and loxodromic images, the value of ''α'' is 1/10.

[[File:IteratedHyperbolicTsfm.png]] [[File:IteratedEllipticalTsfm.png]] [[File:IteratedLoxodromicTsfm.png]]
<!-- TODO. Mention the idea of a pencil of transformations. Link here
[[User:Pmurray_bigpond.com/Geometry of Complex Numbers]] -->

== Higher dimensions ==
In higher dimensions, a '''Möbius transformation''' is a [[homeomorphism]] of {{tmath|1= \overline{\mathbb R^n} }}, the [[one-point compactification]] of {{tmath|1= \mathbb R^n }}, which is a finite composition of [[Inversion in a sphere|inversions in spheres]] and [[Reflection (mathematics)|reflections]] in [[hyperplanes]].<ref>Iwaniec, Tadeusz and Martin, Gaven, The Liouville theorem, Analysis and topology, 339–361, World Sci. Publ., River Edge, NJ, 1998</ref> [[Liouville's theorem (conformal mappings)|Liouville's theorem in conformal geometry]] states that in dimension at least three, all [[Conformal map|conformal]] transformations are Möbius transformations. Every Möbius transformation can be put in the form
<math display="block">f(x) = b + \frac{\alpha A(x - a)}{|x - a|^\varepsilon} ,</math>
where <math>a,b\in \mathbb R^n</math>, <math>\alpha\in\mathbb R</math>, <math>A</math> is an [[orthogonal matrix]], and <math>\varepsilon</math> is 0 or 2. The group of Möbius transformations is also called the '''Möbius group'''.<ref>J.B. Wilker (1981) "Inversive Geometry", {{mr|id=0661793}}</ref>

The orientation-preserving Möbius transformations form the connected component of the identity in the Möbius group.  In dimension {{nowrap|1=''n'' = 2}}, the orientation-preserving Möbius transformations are exactly the maps of the Riemann sphere covered here.  The orientation-reversing ones are obtained from these by complex conjugation.<ref>{{citation | first=Marcel| last=Berger| author-link=Marcel Berger| title=Geometry II| page=18.10|year=1987|publisher=Springer (Universitext)}}</ref>

The domain of Möbius transformations, i.e. {{tmath|1= \overline{\R^n} }}, is homeomorphic to the ''n''-dimensional sphere <math>S^n</math>. The canonical isomorphism between these two spaces is the [[Cayley transform]], which is itself a Möbius transformation of {{tmath|1= \overline{\R^{n + 1} } }}. This identification means that Möbius transformations can also be thought of as conformal isomorphisms of <math>S^n</math>. The ''n''-sphere, together with action of the Möbius group, is a geometric structure (in the sense of Klein's [[Erlangen program]]) called [[Conformal geometry#Möbius geometry|Möbius geometry]].<ref>{{citation|first1=Maks|last1=Akivis|first2=Vladislav|last2=Goldberg|title=Conformal differential geometry and its generalizations| publisher=Wiley-Interscience|year=1992}}</ref>

== Applications ==

=== Lorentz transformation ===
{{main|Lorentz transformation}}
An isomorphism of the Möbius group with the [[Lorentz group]] was noted by several authors: Based on previous work of [[Felix Klein]] (1893, 1897)<ref>Felix Klein (1893), [https://archive.org/details/nichteuklidisch00liebgoog Nicht-Euklidische Geometrie], Autogr. Vorl., Göttingen;<br />[[Robert Fricke]] & Felix Klein (1897), [https://archive.org/details/vorlesungenber01fricuoft Autormorphe Funktionen I.], Teubner, Leipzig</ref> on [[automorphic function]]s related to hyperbolic geometry and Möbius geometry, [[Gustav Herglotz]] (1909)<ref>{{Citation|author=Herglotz, Gustav|year=1910|orig-year=1909|trans-title=On bodies that are to be designated as 'rigid' from the relativity principle standpoint|title=Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper|journal=Annalen der Physik|volume=336|issue=2|language=de |pages=393–415|doi=10.1002/andp.19103360208|bibcode = 1910AnP...336..393H |url=https://zenodo.org/record/1424161}}</ref> showed that [[hyperbolic motion]]s (i.e. [[isometry|isometric]] [[automorphism]]s of a [[hyperbolic space]]) transforming the [[unit sphere]] into itself correspond to Lorentz transformations, by which Herglotz was able to classify the one-parameter Lorentz transformations into loxodromic, elliptic, hyperbolic, and parabolic groups. Other authors include [[Emil Artin]] (1957),<ref>Emil Artin (1957) ''[[Geometric Algebra (book)|Geometric Algebra]]'', page 204</ref> [[H. S. M. Coxeter]] (1965),<ref>[[H. S. M. Coxeter]] (1967) "The Lorentz group and the group of homographies", in L. G. Kovacs & B. H. Neumann (editors) ''Proceedings of the International Conference on The Theory of Groups held at Australian National University, Canberra, 10—20 August 1965'', [[Gordon and Breach]] Science Publishers</ref> and [[Roger Penrose]], [[Wolfgang Rindler]] (1984),{{sfn|Penrose|Rindler|1984|pp=8–31}} [[Tristan Needham]] (1997)<ref>{{Cite book|last=Needham|first=Tristan|url=https://umv.science.upjs.sk/hutnik/NeedhamVCA.pdf|title=Visual Complex Analysis|publisher=Oxford University Press|year=1997|location=Oxford|pages=122–124}}</ref> and W. M. Olivia (2002).<ref>{{cite book|first=Waldyr Muniz|last=Olivia|date=2002|title=Geometric Mechanics|chapter=Appendix B: Möbius transformations and the Lorentz group|pages=195–221|publisher=Springer|mr=1990795 |isbn=3-540-44242-1}}</ref> 

[[Minkowski space]] consists of the four-dimensional real coordinate space '''R'''<sup>4</sup> consisting of the space of ordered quadruples {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} of real numbers, together with a [[quadratic form]]
<math display="block">Q(x_0,x_1,x_2,x_3) = x_0^2-x_1^2-x_2^2-x_3^2.</math>

Borrowing terminology from [[special relativity]], points with {{math|''Q'' > 0}} are considered ''timelike''; in addition, if {{math|''x''<sub>0</sub> > 0}}, then the point is called ''future-pointing''. Points with {{math|''Q'' < 0}} are called ''spacelike''. The [[null cone]] ''S'' consists of those points where {{math|1=''Q'' = 0}}; the ''future null cone'' ''N''<sup>+</sup> are those points on the null cone with {{math|''x''<sub>0</sub> > 0}}. The [[celestial sphere]] is then identified with the collection of rays in ''N''<sup>+</sup> whose initial point is the origin of '''R'''<sup>4</sup>. The collection of [[linear transformation]]s on '''R'''<sup>4</sup> with positive [[determinant]] preserving the quadratic form ''Q'' and preserving the time direction form the [[restricted Lorentz group]] {{nowrap|SO<sup>+</sup>(1, 3)}}.

In connection with the geometry of the celestial sphere, the group of transformations {{nowrap|SO<sup>+</sup>(1, 3)}} is identified with the group {{nowrap|PSL(2, '''C''')}} of Möbius transformations of the sphere. To each {{math|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) ∈ '''R'''<sup>4</sup>}}, associate the [[hermitian matrix]]
<math display="block">X=\begin{bmatrix}
x_0+x_1 & x_2+ix_3\\
x_2-ix_3 & x_0-x_1
\end{bmatrix}.</math>

The [[determinant]] of the matrix ''X'' is equal to {{math|''Q''(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}}. The [[special linear group]] acts on the space of such matrices via

{{NumBlk2|:|<math>X \mapsto AXA^*</math>|1}}

for each {{nowrap|''A'' ∈ SL(2, '''C''')}}, and this action of {{nowrap|SL(2, '''C''')}} preserves the determinant of ''X'' because {{math|1=det ''A'' = 1}}. Since the determinant of ''X'' is identified with the quadratic form ''Q'', {{nowrap|SL(2, '''C''')}} acts by Lorentz transformations. On dimensional grounds, {{nowrap|SL(2, '''C''')}} covers a neighborhood of the identity of {{nowrap|SO(1, 3)}}. Since {{nowrap|SL(2, '''C''')}} is connected, it covers the entire restricted Lorentz group {{nowrap|SO<sup>+</sup>(1, 3)}}. Furthermore, since the [[kernel (group theory)|kernel]] of the action ({{EquationNote|1}}) is the subgroup {{mset|±''I''}}, then passing to the [[quotient group]] gives the [[group isomorphism]]

{{NumBlk2|:|<math>\operatorname{PSL}(2,\Complex)\cong \operatorname{SO}^+(1,3).</math>|2}}

Focusing now attention on the case when {{nowrap|(''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} is null, the matrix ''X'' has zero determinant, and therefore splits as the [[outer product]] of a complex two-vector ''ξ'' with its complex conjugate:

{{NumBlk2|:|<math>X = \xi\bar{\xi}^\text{T}=\xi\xi^*.</math>|3}}

The two-component vector ''ξ'' is acted upon by {{nowrap|SL(2, '''C''')}} in a manner compatible with ({{EquationNote|1}}). It is now clear that the kernel of the representation of {{nowrap|SL(2, '''C''')}} on hermitian matrices is {{mset|±''I''}}.

The action of {{nowrap|PSL(2, '''C''')}} on the celestial sphere may also be described geometrically using [[stereographic projection]]. Consider first the hyperplane in '''R'''<sup>4</sup> given by ''x''<sub>0</sub>&nbsp;=&nbsp;1. The celestial sphere may be identified with the sphere ''S''<sup>+</sup> of intersection of the hyperplane with the future null cone ''N''<sup>+</sup>. The stereographic projection from the north pole {{nowrap|(1, 0, 0, 1)}} of this sphere onto the plane {{nowrap|1=''x''<sub>3</sub> = 0}} takes a point with coordinates {{nowrap|(1, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} with
<math display="block">x_1^2+x_2^2+x_3^2=1</math>
to the point
<math display="block">\left(1, \frac{x_1}{1-x_3}, \frac{x_2}{1-x_3},0\right).</math>

Introducing the [[complex number|complex]] coordinate
<math display="block">\zeta = \frac{x_1+ix_2}{1-x_3},</math>
the inverse stereographic projection gives the following formula for a point {{nowrap|(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} on ''S''<sup>+</sup>:

{{NumBlk2|:|<math>
\begin{align}
x_1 &= \frac{\zeta+\bar{\zeta}}{\zeta\bar{\zeta}+1}\\
x_2 &= \frac{\zeta-\bar{\zeta}}{i(\zeta\bar{\zeta}+1)}\\
x_3 &= \frac{\zeta\bar{\zeta}-1}{\zeta\bar{\zeta}+1}.
\end{align}
</math>|4}}

The action of {{nowrap|SO<sup>+</sup>(1, 3)}} on the points of ''N''<sup>+</sup> does not preserve the hyperplane ''S''<sup>+</sup>, but acting on points in ''S''<sup>+</sup> and then rescaling so that the result is again in ''S''<sup>+</sup> gives an action of {{nowrap|SO<sup>+</sup>(1, 3)}} on the sphere which goes over to an action on the complex variable ''ζ''. In fact, this action is by fractional linear transformations, although this is not easily seen from this representation of the celestial sphere. Conversely, for any fractional linear transformation of ''ζ'' variable goes over to a unique Lorentz transformation on ''N''<sup>+</sup>, possibly after a suitable (uniquely determined) rescaling.

A more invariant description of the stereographic projection which allows the action to be more clearly seen is to consider the variable {{nowrap|1=''ζ'' = ''z'':''w''}} as a ratio of a pair of homogeneous coordinates for the complex projective line '''CP'''<sup>1</sup>. The stereographic projection goes over to a transformation from {{nowrap|'''C'''<sup>2</sup> − {{mset|0}}}} to ''N''<sup>+</sup> which is homogeneous of degree two with respect to real scalings

{{NumBlk2|:|<math>(z,w)\mapsto (x_0,x_1,x_2,x_3)=(z\bar{z}+w\bar{w}, z\bar{z}-w\bar{w}, z\bar{w}+w\bar{z}, i^{-1}(z\bar{w}-w\bar{z}))</math>|5}}

which agrees with ({{EquationNote|4}}) upon restriction to scales in which <math>z\bar{z}+w\bar{w}=1.</math> The components of ({{EquationNote|5}}) are precisely those obtained from the outer product
<math display="block">
\begin{bmatrix}
x_0+x_1 & x_2+ix_3 \\
x_2-ix_3 & x_0-x_1
\end{bmatrix} =
2\begin{bmatrix} z \\ w \end{bmatrix}
\begin{bmatrix} \bar{z} & \bar{w} \end{bmatrix}.
</math>

In summary, the action of the restricted Lorentz group SO<sup>+</sup>(1,3) agrees with that of the Möbius group {{nowrap|PSL(2, '''C''')}}. This motivates the following definition. In dimension {{nowrap|''n'' ≥ 2}}, the '''Möbius group''' Möb(''n'') is the group of all orientation-preserving [[conformal geometry|conformal]] [[isometry|isometries]] of the round sphere ''S''<sup>''n''</sup> to itself. By realizing the conformal sphere as the space of future-pointing rays of the null cone in the Minkowski space '''R'''<sup>1,n+1</sup>, there is an isomorphism of Möb(''n'') with the restricted Lorentz group SO<sup>+</sup>(1,''n''+1) of Lorentz transformations with positive determinant, preserving the direction of time.

Coxeter began instead with the equivalent quadratic form {{tmath|1= Q(x_1,\ x_2,\ x_3 \ x_4) = x_1^2 + x_2^2 + x_3^2 - x_4^2 }}.

He identified the Lorentz group with transformations for which {{mset|''x'' | Q(''x'') {{=}} −1}} is [[invariant (mathematics)#Invariant set|stable]]. Then he interpreted the ''x''{{null}}'s as [[homogeneous coordinates]]  and {{mset|''x'' | Q(''x'') {{=}} 0}}, the [[null cone]], as the [[Cayley absolute]] for a hyperbolic space of points {{mset|''x'' | Q(''x'') < 0}}. Next, Coxeter introduced the variables
<math display="block">\xi = \frac {x_1}{x_4} , \ \eta = \frac {x_2}{x_4}, \ \zeta = \frac {x_3}{x_4} </math>
so that the Lorentz-invariant quadric corresponds to the sphere {{tmath|1= \xi^2 + \eta^2 + \zeta^2 = 1 }}. Coxeter notes that [[Felix Klein]] also wrote of this correspondence, applying stereographic projection from {{nowrap|(0, 0, 1)}} to the complex plane <math display="inline">z = \frac{\xi + i \eta}{1 - \zeta}.</math> Coxeter used the fact that circles of the inversive plane represent planes of hyperbolic space, and the general homography is the product of inversions in two or four circles, corresponding to the general hyperbolic displacement which is the product of inversions in two or four planes.

=== Hyperbolic space ===
As seen above, the Möbius group {{nowrap|PSL(2, '''C''')}} acts on Minkowski space as the group of those isometries that preserve the origin, the orientation of space and the direction of time. Restricting to the points where {{nowrap|1=''Q'' = 1}} in the positive light cone, which form a model of [[hyperbolic space|hyperbolic 3-space]] ''H''{{i sup|3}}, we see that the Möbius group acts on ''H''{{i sup|3}} as a group of orientation-preserving isometries. In fact, the Möbius group is equal to the group of orientation-preserving isometries of hyperbolic 3-space. If we use the [[Poincaré disk model|Poincaré ball model]], identifying the unit ball in '''R'''<sup>3</sup> with ''H''{{i sup|3}}, then we can think of the Riemann sphere as the "conformal boundary" of ''H''{{i sup|3}}. Every orientation-preserving isometry of ''H''{{i sup|3}} gives rise to a Möbius transformation on the Riemann sphere and vice versa.

== See also ==
{{div col|colwidth=22em}}
* [[Bilinear transform]]
* [[Conformal geometry]]
* [[Fuchsian group]]
* [[Generalised circle]]
* [[Hyperbolic geometry]]
* [[Infinite compositions of analytic functions]]
* [[Inversion transformation]]
* [[Kleinian group]]
* [[Lie sphere geometry]]
* [[Linear fractional transformation]]
* [[Liouville's theorem (conformal mappings)]]
* [[Lorentz group]]
* [[Modular group]]
* [[Poincaré half-plane model]]
* [[Projective geometry]]
* [[Projective line over a ring]]
* [[Representation theory of the Lorentz group]]
* [[Schottky group]]
* [[Smith chart]]
{{div col end}}

== Notes ==
{{reflist|group=note}}

== References ==
'''Specific'''
{{reflist|40em}}
'''General'''
{{refbegin}}
* {{citation |last1=Arnold |first1=Douglas N. |last2=Rogness |first2=Jonathan |year=2008 |title=Möbius Transformations Revealed | journal=Notices of the AMS |pages=1226–1231 |volume=55 |issue=10 |url=http://www-users.math.umn.edu/~arnold/papers/moebius.pdf }}
* {{citation | first=Alan F.|last=Beardon | title=The Geometry of Discrete Groups | publisher=New York: Springer-Verlag | year=1995 | isbn=978-0-387-90788-8}}
* {{citation | first=G. S. |last=Hall | title=Symmetries and Curvature Structure in General Relativity | publisher=Singapore: World Scientific | year=2004 | isbn=978-981-02-1051-9}} ''(See Chapter 6 for the classification, up to conjugacy, of the Lie subalgebras of the Lie algebra of the Lorentz group.)''
* {{citation | first=Svetlana|last=Katok |author-link=Svetlana Katok| title=Fuchsian Groups | publisher=Chicago:University of Chicago Press | year=1992 | isbn=978-0-226-42583-2}} ''See Chapter 2''.
* {{citation | last = Klein | first = Felix | author-link = Felix Klein | title = Lectures on the icosahedron and the solution of equations of the fifth degree | year = 1913 | edition = 2nd | orig-year = 1st German ed. 1884 | location = London | publisher = Kegan Paul, Trench, Trübner, & Co. | translator-last = Morrice | translator-first = George Gavin | url = https://archive.org/details/in.ernet.dli.2015.217159/ }} translated from {{citation | last = Klein | first = Felix | display-authors = 0 | url = https://archive.org/details/vorlesungenuberd0000feli/ | year = 1884 | title = Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade |language=de |publisher=Teubner }}
* {{citation | first=Konrad | last=Knopp | title=Elements of the Theory of Functions | publisher=New York: Dover | year=1952 | isbn=978-0-486-60154-0 | url-access=registration | url=https://archive.org/details/elementsoftheory00konr }} ''(See Chapters 3–5 of this classic book for a beautiful introduction to the Riemann sphere, stereographic projection, and Möbius transformations.)''
* {{citation | first1= David |last1=Mumford|author-link1=David Mumford|first2=Caroline|last2=Series |first3=David|last3=Wright | title=Indra's Pearls: The Vision of Felix Klein | year=2002 | publisher= Cambridge University Press | isbn= 978-0-521-35253-6|title-link=Indra's Pearls (book)}} ''(Aimed at non-mathematicians, provides an excellent exposition of theory and results, richly illustrated with diagrams.)''
* {{citation | first=Tristan|last=Needham | title=Visual Complex Analysis | publisher=Oxford: Clarendon Press | year=1997 | isbn=978-0-19-853446-4}} ''(See Chapter 3 for a beautifully illustrated introduction to Möbius transformations, including their classification up to conjugacy.)''
* {{citation|first1=Roger|last1=Penrose|author-link1=Roger Penrose|first2=Wolfgang|last2=Rindler|author-link2=Wolfgang Rindler|title=Spinors and space–time, Volume 1: Two-spinor calculus and relativistic fields|publisher=Cambridge University Press|year=1984|isbn=978-0-521-24527-2}}
* {{citation|first1=Hans|last1=Schwerdtfeger|author-link1=Hans Schwerdtfeger|title= Geometry of Complex Numbers |title-link= Geometry of Complex Numbers|publisher=Dover|year=1979|isbn=978-0-486-63830-0}} ''(See Chapter 2 for an introduction to Möbius transformations.)''
* {{citation| title = Finite Möbius groups, minimal immersions of spheres, and moduli| first = Gábor| last = Tóth| year = 2002 }}
{{refend}}

== Further reading ==
* {{cite journal | doi = 10.1006/jabr.1997.7242| title = The Möbius Inverse Monoid| journal = Journal of Algebra| volume = 200| issue = 2| page = 428| year = 1998| last1 = Lawson | first1 = M. V. | doi-access = free}}

== External links ==
{{Commonscat}}
* {{springer|title=Quasi-conformal mapping|id=p/q076430}}
* [http://virtualmathmuseum.org/ConformalMaps/index.html Conformal maps gallery]
* {{MathWorld | urlname= LinearFractionalTransformation | title= Linear Fractional Transformation }}

{{DEFAULTSORT:Mobius transformation}}
[[Category:Projective geometry]]
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[[Category:Lie groups]]
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[[Category:Conformal mappings]]