In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form $${\displaystyle f(z)={\frac {az+b}{cz+d}}}$$ of one complex variable z; here the coefficients a, b, c, d are complex numbers satisfying 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.^[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 transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2, C). Together with its subgroups, it has numerous applications in mathematics and physics.

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 homographies, linear fractional transformations, bilinear transformations, and spin transformations (in relativity theory).^[2]

Möbius transformations are defined on the extended complex plane ${\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$ (i.e., the complex plane augmented by the point at infinity).

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

The Möbius group is isomorphic to the group of orientation-preserving isometries of hyperbolic 3-space and therefore plays an important role when studying hyperbolic 3-manifolds.

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 subgroups of the Möbius group form the automorphism groups of the other simply-connected Riemann surfaces (the complex plane and the hyperbolic plane). As such, Möbius transformations play an important role in the theory of Riemann surfaces. 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 fractals, modular forms, elliptic curves and Pellian equations.

Möbius transformations can be more generally defined in spaces of dimension n > 2 as the bijective conformal orientation-preserving maps from the n-sphere to the n-sphere. Such a transformation is the most general form of conformal mapping of a domain. According to Liouville's theorem a Möbius transformation can be expressed as a composition of translations, similarities, orthogonal transformations and inversions.

The general form of a Möbius transformation is given by $${\displaystyle f(z)={\frac {az+b}{cz+d}},}$$ where a, b, c, d are any complex numbers that satisfy ad − bc ≠ 0.

In case c ≠ 0, this definition is extended to the whole Riemann sphere by defining $${\displaystyle f\left({\frac {-d}{c}}\right)=\infty {\text{ and }}f(\infty )={\frac {a}{c}}.}$$

If c = 0, we define $${\displaystyle f(\infty )=\infty .}$$

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 under composition. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps. The Möbius group is then a complex Lie group. The Möbius group is usually denoted ${\displaystyle \operatorname {Aut} ({\widehat {\mathbb {C} }})}$ as it is the automorphism group of the Riemann sphere.

If ad = bc, the rational function defined above is a constant (unless c = d = 0, when it is undefined): $${\displaystyle {\frac {az+b}{cz+d}}={\frac {a}{c}}={\frac {b}{d}},}$$ 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.

Every non-identity Möbius transformation has two fixed points ${\displaystyle \gamma _{1},\gamma _{2}}$ on the Riemann sphere. The fixed points are counted here with multiplicity; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity.

The fixed points of the transformation $${\displaystyle f(z)={\frac {az+b}{cz+d}}}$$ are obtained by solving the fixed point equation f(γ) = γ. For c ≠ 0, this has two roots obtained by expanding this equation to $${\displaystyle c\gamma ^{2}-(a-d)\gamma -b=0\ ,}$$ and applying the quadratic formula. The roots are $${\displaystyle \gamma _{1,2}={\frac {(a-d)\pm {\sqrt {(a-d)^{2}+4bc}}}{2c}}={\frac {(a-d)\pm {\sqrt {\Delta }}}{2c}}}$$ with discriminant $${\displaystyle \Delta =(\operatorname {tr} {\mathfrak {H}})^{2}-4\det {\mathfrak {H}}=(a+d)^{2}-4(ad-bc),}$$ where the matrix $${\displaystyle {\mathfrak {H}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$$ 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 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 a ≠ d the second fixed point is finite and is given by $${\displaystyle \gamma =-{\frac {b}{a-d}}.}$$

In this case the transformation will be a simple transformation composed of translations, rotations, and dilations: $${\displaystyle z\mapsto \alpha z+\beta .}$$

If c = 0 and a = d, then both fixed points are at infinity, and the Möbius transformation corresponds to a pure translation: $${\displaystyle z\mapsto z+\beta .}$$

Topologically, the fact that (non-identity) Möbius transformations fix 2 points (with multiplicity) corresponds to the Euler characteristic of the sphere being 2: $${\displaystyle \chi ({\hat {\mathbb {C} }})=2.}$$

Firstly, the projective linear group PGL(2, K) is sharply 3-transitive – 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 ${\displaystyle \mathrm {PGL} (2,\mathbb {C} )}$ 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 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, PGL(2, R) need not fix any points – for example ${\displaystyle (1+x)/(1-x)}$ has no (real) fixed points: as a complex transformation it fixes ±i^{[note 1]} – while the map 2x 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.

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 conjugate to a dilation/rotation, i.e., a transformation of the form $${\displaystyle z\mapsto kz}$$ (k ∈ C) with fixed points at 0 and ∞. To see this define a map $${\displaystyle g(z)={\frac {z-\gamma _{1}}{z-\gamma _{2}}}}$$ which sends the points (γ₁, γ₂) to (0, ∞). Here we assume that γ₁ and γ₂ 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 (γ₁, γ₂) then the transformation ${\displaystyle gfg^{-1}}$ has fixed points at 0 and ∞ and is therefore a dilation: ${\displaystyle gfg^{-1}(z)=kz}$. The fixed point equation for the transformation f can then be written $${\displaystyle {\frac {f(z)-\gamma _{1}}{f(z)-\gamma _{2}}}=k{\frac {z-\gamma _{1}}{z-\gamma _{2}}}.}$$

Solving for f gives (in matrix form): $${\displaystyle {\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\begin{pmatrix}\gamma _{1}-k\gamma _{2}&(k-1)\gamma _{1}\gamma _{2}\\1-k&k\gamma _{1}-\gamma _{2}\end{pmatrix}}}$$ or, if one of the fixed points is at infinity: $${\displaystyle {\mathfrak {H}}(k;\gamma ,\infty )={\begin{pmatrix}k&(1-k)\gamma \\0&1\end{pmatrix}}.}$$

From the above expressions one can calculate the derivatives of f at the fixed points: $${\displaystyle f'(\gamma _{1})=k}$$ and $${\displaystyle f'(\gamma _{2})=1/k.}$$

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: $${\displaystyle {\mathfrak {H}}(k;\gamma _{1},\gamma _{2})={\mathfrak {H}}(1/k;\gamma _{2},\gamma _{1}).}$$

For loxodromic transformations, whenever |k| > 1, one says that γ₁ is the repulsive fixed point, and γ₂ is the attractive fixed point. For |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 $${\displaystyle g(z)={\frac {1}{z-\gamma }}}$$ or the identity if γ is already at infinity. The transformation ${\displaystyle gfg^{-1}}$ fixes infinity and is therefore a translation: $${\displaystyle gfg^{-1}(z)=z+\beta \,.}$$

Here, β is called the translation length. The fixed point formula for a parabolic transformation is then $${\displaystyle {\frac {1}{f(z)-\gamma }}={\frac {1}{z-\gamma }}+\beta .}$$

Solving for f (in matrix form) gives $${\displaystyle {\mathfrak {H}}(\beta ;\gamma )={\begin{pmatrix}1+\gamma \beta &-\beta \gamma ^{2}\\\beta &1-\gamma \beta \end{pmatrix}}}$$Note that ${\displaystyle \det {\mathfrak {H}}(\beta ;\gamma )=|{\mathfrak {H}}(\beta ;\gamma )|=\det {\begin{pmatrix}1+\gamma \beta &-\beta \gamma ^{2}\\\beta &1-\gamma \beta \end{pmatrix}}=1-\gamma ^{2}\beta ^{2}+\gamma ^{2}\beta ^{2}=1}$

If γ = ∞: $${\displaystyle {\mathfrak {H}}(\beta ;\infty )={\begin{pmatrix}1&\beta \\0&1\end{pmatrix}}}$$

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: $${\displaystyle f'(\gamma )=1.}$$

The point ${\textstyle z_{\infty }=-{\frac {d}{c}}}$ is called the pole of ${\displaystyle {\mathfrak {H}}}$; it is that point which is transformed to the point at infinity under ⁠${\displaystyle {\mathfrak {H}}}$⁠.

The inverse pole ${\textstyle Z_{\infty }={\frac {a}{c}}}$ 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: $${\displaystyle \gamma _{1}+\gamma _{2}=z_{\infty }+Z_{\infty }.}$$

These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation.

A transform ${\displaystyle {\mathfrak {H}}}$ can be specified with two fixed points γ₁, γ₂ and the pole ${\displaystyle z_{\infty }}$.

$${\displaystyle {\mathfrak {H}}={\begin{pmatrix}Z_{\infty }&-\gamma _{1}\gamma _{2}\\1&-z_{\infty }\end{pmatrix}},\;\;Z_{\infty }=\gamma _{1}+\gamma _{2}-z_{\infty }.}$$

This allows us to derive a formula for conversion between k and ${\displaystyle z_{\infty }}$ given ${\displaystyle \gamma _{1},\gamma _{2}}$: $${\displaystyle z_{\infty }={\frac {k\gamma _{1}-\gamma _{2}}{1-k}}}$$ $${\displaystyle k={\frac {\gamma _{2}-z_{\infty }}{\gamma _{1}-z_{\infty }}}={\frac {Z_{\infty }-\gamma _{1}}{Z_{\infty }-\gamma _{2}}}={\frac {a-c\gamma _{1}}{a-c\gamma _{2}}},}$$ which reduces down to $${\displaystyle k={\frac {(a+d)+{\sqrt {(a-d)^{2}+4bc}}}{(a+d)-{\sqrt {(a-d)^{2}+4bc}}}}.}$$

The last expression coincides with one of the (mutually reciprocal) eigenvalue ratios ${\textstyle {\frac {\lambda _{1}}{\lambda _{2}}}}$ of ${\displaystyle {\mathfrak {H}}}$ (compare the discussion in the preceding section about the characteristic constant of a transformation). Its characteristic polynomial is equal to $${\displaystyle \det(\lambda I_{2}-{\mathfrak {H}})=\lambda ^{2}-\operatorname {tr} {\mathfrak {H}}\,\lambda +\det {\mathfrak {H}}=\lambda ^{2}-(a+d)\lambda +(ad-bc)}$$ which has roots $${\displaystyle \lambda _{i}={\frac {(a+d)\pm {\sqrt {(a-d)^{2}+4bc}}}{2}}={\frac {(a+d)\pm {\sqrt {(a+d)^{2}-4(ad-bc)}}}{2}}=c\gamma _{i}+d\,.}$$

A Möbius transformation can be composed as a sequence of simple transformations.

The following simple transformations are also Möbius transformations:

• ${\displaystyle f(z)=z+b\quad (a=1,c=0,d=1)}$ is a translation.
• ${\displaystyle f(z)=az\quad (b=0,c=0,d=1)}$ is a combination of a homothety (uniform scaling) and a rotation. If ${\displaystyle |a|=1}$ then it is a rotation, if ${\displaystyle a\in \mathbb {R} }$ then it is a homothety.
• ${\displaystyle f(z)=1/z\quad (a=0,b=1,c=1,d=0)}$ (inversion and reflection with respect to the real axis)

If ${\displaystyle c\neq 0}$, let:

• ${\displaystyle f_{1}(z)=z+d/c\quad }$ (translation by d/c)
• ${\displaystyle f_{2}(z)=1/z\quad }$ (inversion and reflection with respect to the real axis)
• ${\displaystyle f_{3}(z)={\frac {bc-ad}{c^{2}}}z\quad }$ (homothety and rotation)
• ${\displaystyle f_{4}(z)=z+a/c\quad }$ (translation by a/c)

Then these functions can be composed, showing that, if $${\displaystyle f(z)={\frac {az+b}{cz+d}},}$$ one has $${\displaystyle f=f_{4}\circ f_{3}\circ f_{2}\circ f_{1}.}$$ In other terms, one has $${\displaystyle {\frac {az+b}{cz+d}}={\frac {a}{c}}+{\frac {e}{z+{\frac {d}{c}}}},}$$ with $${\displaystyle e={\frac {bc-ad}{c^{2}}}.}$$

This decomposition makes many properties of the Möbius transformation obvious.

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

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 g₁, g₂, g₃, g₄ such that each g_i is the inverse of f_i. Then the composition $${\displaystyle g_{1}\circ g_{2}\circ g_{3}\circ g_{4}(z)=f^{-1}(z)={\frac {dz-b}{-cz+a}}}$$ gives a formula for the inverse.

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 circles 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-ratios are invariant under Möbius transformations. That is, if a Möbius transformation maps four distinct points ${\displaystyle z_{1},z_{2},z_{3},z_{4}}$ to four distinct points ${\displaystyle w_{1},w_{2},w_{3},w_{4}}$ respectively, then $${\displaystyle {\frac {(z_{1}-z_{3})(z_{2}-z_{4})}{(z_{2}-z_{3})(z_{1}-z_{4})}}={\frac {(w_{1}-w_{3})(w_{2}-w_{4})}{(w_{2}-w_{3})(w_{1}-w_{4})}}.}$$

If one of the points ${\displaystyle z_{1},z_{2},z_{3},z_{4}}$ is the point at infinity, then the cross-ratio has to be defined by taking the appropriate limit; e.g. the cross-ratio of ${\displaystyle z_{1},z_{2},z_{3},\infty }$ is $${\displaystyle {\frac {(z_{1}-z_{3})}{(z_{2}-z_{3})}}.}$$

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.

Two points z₁ and z₂ are conjugate with respect to a generalized circle C, if, given a generalized circle D passing through z₁ and z₂ and cutting C in two points a and b, (z₁, z₂; 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.^[3][4]

Two points z, z^∗ are conjugate with respect to a line, if they are symmetric with respect to the line. Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle.

The point z^∗ is conjugate to z when L is the line determined by the vector based upon e^iθ, at the point z₀. This can be explicitly given as $${\displaystyle z^{*}=e^{2i\theta }\,{\overline {z-z_{0}}}+z_{0}.}$$

The point z^∗ is conjugate to z when C is the circle of a radius r, centered about z₀. This can be explicitly given as $${\displaystyle z^{*}={\frac {r^{2}}{\overline {z-z_{0}}}}+z_{0}.}$$

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

The natural action of PGL(2, C) on the complex projective line CP¹ is exactly the natural action of the Möbius group on the Riemann sphere

Here, the projective line CP¹ and the Riemann sphere are identified as follows: $${\displaystyle [z_{1}:z_{2}]\ \thicksim {\frac {z_{1}}{z_{2}}}.}$$

Here [z₁:z₂] are homogeneous coordinates on CP¹; the point [1:0] corresponds to the point ∞ of the Riemann sphere. By using homogeneous coordinates, many calculations involving Möbius transformations can be simplified, since no case distinctions dealing with ∞ are required.

Every invertible complex 2×2 matrix $${\displaystyle {\mathfrak {H}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}}$$ acts on the projective line as $${\displaystyle z=[z_{1}:z_{2}]\mapsto w=[w_{1}:w_{2}],}$$ where $${\displaystyle {\begin{pmatrix}w_{1}\\w_{2}\end{pmatrix}}={\begin{pmatrix}a&b\\c&d\end{pmatrix}}{\begin{pmatrix}z_{1}\\z_{2}\end{pmatrix}}={\begin{pmatrix}az_{1}+bz_{2}\\cz_{1}+dz_{2}\end{pmatrix}}.}$$

The result is therefore $${\displaystyle w=[w_{1}:w_{2}]=[az_{1}+bz_{2}:cz_{1}+dz_{2}]}$$

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

$${\displaystyle w=[az_{1}+bz_{2}:cz_{1}+dz_{2}]\thicksim {\frac {az_{1}+bz_{2}}{cz_{1}+dz_{2}}}={\frac {a{\frac {z_{1}}{z_{2}}}+b}{c{\frac {z_{1}}{z_{2}}}+d}}.}$$

Since the above matrix is invertible if and only if its determinant ad − bc is not zero, this induces an identification of the action of the group of Möbius transformations with the action of PGL(2, C) on the complex projective line. In this identification, the above matrix ${\displaystyle {\mathfrak {H}}}$ corresponds to the Möbius transformation ${\displaystyle z\mapsto {\frac {az+b}{cz+d}}.}$

This identification is a group isomorphism, since the multiplication of ${\displaystyle {\mathfrak {H}}}$ by a non zero scalar ${\displaystyle \lambda }$ does not change the element of PGL(2, C), and, as this multiplication consists of multiplying all matrix entries by ${\displaystyle \lambda ,}$ this does not change the corresponding Möbius transformation.

For any field K, one can similarly identify the group 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 homographies of the real line and its applications in optics.

If one divides ${\displaystyle {\mathfrak {H}}}$ 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 SL(2, C) to PGL(2, C), with ${\displaystyle \pm I}$ 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 and non-compact, and that SL(2,C) is a double cover of PSL(2, C). Since SL(2, C) is simply-connected, it is the universal cover of the Möbius group, and the fundamental group of the Möbius group is Z₂.

Given a set of three distinct points ${\displaystyle z_{1},z_{2},z_{3}}$ on the Riemann sphere and a second set of distinct points ⁠${\displaystyle w_{1},w_{2},w_{3}}$⁠, there exists precisely one Möbius transformation ${\displaystyle f(z)}$ with ${\displaystyle f(z_{j})=w_{j}}$ for ⁠${\displaystyle j=1,2,3}$⁠. (In other words: the action of the Möbius group on the Riemann sphere is sharply 3-transitive.) There are several ways to determine ${\displaystyle f(z)}$ from the given sets of points.

It is easy to check that the Möbius transformation $${\displaystyle f_{1}(z)={\frac {(z-z_{1})(z_{2}-z_{3})}{(z-z_{3})(z_{2}-z_{1})}}}$$ with matrix $${\displaystyle {\mathfrak {H}}_{1}={\begin{pmatrix}z_{2}-z_{3}&-z_{1}(z_{2}-z_{3})\\z_{2}-z_{1}&-z_{3}(z_{2}-z_{1})\end{pmatrix}}}$$ maps ${\displaystyle z_{1},z_{2}{\text{ and }}z_{3}}$ to ⁠${\displaystyle 0,1,\ {\text{and}}\ \infty }$⁠, respectively. If one of the ${\displaystyle z_{j}}$ is ${\displaystyle \infty }$, then the proper formula for ${\displaystyle {\mathfrak {H}}_{1}}$ is obtained from the above one by first dividing all entries by ${\displaystyle z_{j}}$ and then taking the limit ⁠${\displaystyle z_{j}\to \infty }$⁠.

If ${\displaystyle {\mathfrak {H}}_{2}}$ is similarly defined to map ${\displaystyle w_{1},w_{2},w_{3}}$ to ${\displaystyle 0,1,\ {\text{and}}\ \infty ,}$ then the matrix ${\displaystyle {\mathfrak {H}}}$ which maps ${\displaystyle z_{1,2,3}}$ to ${\displaystyle w_{1,2,3}}$ becomes $${\displaystyle {\mathfrak {H}}={\mathfrak {H}}_{2}^{-1}{\mathfrak {H}}_{1}.}$$

The stabilizer of ${\displaystyle \{0,1,\infty \}}$ (as an unordered set) is a subgroup known as the anharmonic group.

The equation $${\displaystyle w={\frac {az+b}{cz+d}}}$$ is equivalent to the equation of a standard hyperbola $${\displaystyle cwz-az+dw-b=0}$$ in the ${\displaystyle (z,w)}$-plane. The problem of constructing a Möbius transformation ${\displaystyle {\mathfrak {H}}(z)}$ mapping a triple ${\displaystyle (z_{1},z_{2},z_{3})}$ to another triple ${\displaystyle (w_{1},w_{2},w_{3})}$ is thus equivalent to finding the coefficients ${\displaystyle a,b,c,d}$ of the hyperbola passing through the points ⁠${\displaystyle (z_{i},w_{i})}$⁠. An explicit equation can be found by evaluating the determinant $${\displaystyle {\begin{vmatrix}zw&z&w&1\\z_{1}w_{1}&z_{1}&w_{1}&1\\z_{2}w_{2}&z_{2}&w_{2}&1\\z_{3}w_{3}&z_{3}&w_{3}&1\end{vmatrix}}\,}$$ by means of a Laplace expansion along the first row, resulting in explicit formulae, $${\displaystyle {\begin{aligned}a&=z_{1}w_{1}(w_{2}-w_{3})+z_{2}w_{2}(w_{3}-w_{1})+z_{3}w_{3}(w_{1}-w_{2}),\\[5mu]b&=z_{1}w_{1}(z_{2}w_{3}-z_{3}w_{2})+z_{2}w_{2}(z_{3}w_{1}-z_{1}w_{3})+z_{3}w_{3}(z_{1}w_{2}-z_{2}w_{1}),\\[5mu]c&=w_{1}(z_{3}-z_{2})+w_{2}(z_{1}-z_{3})+w_{3}(z_{2}-z_{1}),\\[5mu]d&=z_{1}w_{1}(z_{2}-z_{3})+z_{2}w_{2}(z_{3}-z_{1})+z_{3}w_{3}(z_{1}-z_{2})\end{aligned}}}$$ for the coefficients ${\displaystyle a,b,c,d}$ of the representing matrix ⁠${\displaystyle {\mathfrak {H}}}$⁠. The constructed matrix ${\displaystyle {\mathfrak {H}}}$ has determinant equal to ⁠${\displaystyle (z_{1}-z_{2})(z_{1}-z_{3})(z_{2}-z_{3})(w_{1}-w_{2})(w_{1}-w_{3})(w_{2}-w_{3})}$⁠, which does not vanish if the ${\displaystyle z_{j}}$ resp. ${\displaystyle w_{j}}$ are pairwise different thus the Möbius transformation is well-defined. If one of the points ${\displaystyle z_{j}}$ or ${\displaystyle w_{j}}$ is ⁠${\displaystyle \infty }$⁠, then we first divide all four determinants by this variable and then take the limit as the variable approaches ⁠${\displaystyle \infty }$⁠.

If we require the coefficients ${\displaystyle a,b,c,d}$ of a Möbius transformation to be real numbers with ⁠${\displaystyle ad-bc=1}$⁠, we obtain a subgroup of the Möbius group denoted as PSL(2, R). This is the group of those Möbius transformations that map the upper half-plane H = {x + iy : y > 0} to itself, and is equal to the group of all biholomorphic (or equivalently: bijective, conformal and orientation-preserving) maps H → H. If a proper metric is introduced, the upper half-plane becomes a model of the hyperbolic plane H², the Poincaré half-plane model, and PSL(2, R) is the group of all orientation-preserving isometries of H² in this model.

The subgroup of all Möbius transformations that map the open disk D = {z : |z| < 1} to itself consists of all transformations of the form $${\displaystyle f(z)=e^{i\phi }{\frac {z+b}{{\bar {b}}z+1}}}$$ with ${\displaystyle \phi }$ ∈ R, b ∈ C and |b| < 1. This is equal to the group of all biholomorphic (or equivalently: bijective, angle-preserving and orientation-preserving) maps 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 H² in this model.

Since both of the above subgroups serve as isometry groups of H², they are isomorphic. A concrete isomorphism is given by conjugation with the transformation $${\displaystyle f(z)={\frac {z+i}{iz+1}}}$$ which bijectively maps the open unit disk to the upper half plane.

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

A maximal compact subgroup of the Möbius group ${\displaystyle {\mathcal {M}}}$ is given by (Tóth 2002)^[5] $${\displaystyle {\mathcal {M}}_{0}:=\left\{z\mapsto {\frac {uz-{\bar {v}}}{vz+{\bar {u}}}}:|u|^{2}+|v|^{2}=1\right\},}$$ and corresponds under the isomorphism ${\displaystyle {\mathcal {M}}\cong \operatorname {PSL} (2,\mathbb {C} )}$ to the projective special unitary group 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 groups of Möbius transformations were used by Felix Klein to give an analytic solution to the quintic equation in (Klein 1913); a modern exposition is given in (Tóth 2002).^[6]

If we require the coefficients a, b, c, d of a Möbius transformation to be integers with ad − bc = 1, we obtain the modular group PSL(2, Z), a discrete subgroup of PSL(2, R) important in the study of lattices in the complex plane, elliptic functions and elliptic curves. The discrete subgroups of PSL(2, R) are known as Fuchsian groups; they are important in the study of Riemann surfaces.

In the following discussion we will always assume that the representing matrix ${\displaystyle {\mathfrak {H}}}$ is normalized such that ⁠${\displaystyle \det {\mathfrak {H}}=ad-bc=1}$⁠.

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

A non-identity Möbius transformation defined by a matrix ${\displaystyle {\mathfrak {H}}}$ of determinant one is said to be parabolic if $${\displaystyle \operatorname {tr} ^{2}{\mathfrak {H}}=(a+d)^{2}=4}$$ (so the trace is plus or minus 2; either can occur for a given transformation since ${\displaystyle {\mathfrak {H}}}$ is determined only up to sign). In fact one of the choices for ${\displaystyle {\mathfrak {H}}}$ has the same characteristic polynomial X² − 2X + 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 extended complex plane ${\displaystyle {\widehat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}}$, which happens if and only if it can be defined by a matrix conjugate to $${\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$$ which describes a translation in the complex plane.

The set of all parabolic Möbius transformations with a given fixed point in ${\displaystyle {\widehat {\mathbb {C} }}}$, together with the identity, forms a subgroup isomorphic to the group of matrices $${\displaystyle \left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}\mid b\in \mathbb {C} \right\};}$$ this is an example of the unipotent radical of a Borel subgroup (of the Möbius group, or of SL(2, C) for the matrix group; the notion is defined for any reductive Lie group).

All non-parabolic transformations have two fixed points and are defined by a matrix conjugate to $${\displaystyle {\begin{pmatrix}\lambda &0\\0&\lambda ^{-1}\end{pmatrix}}}$$ with the complex number λ not equal to 0, 1 or −1, corresponding to a dilation/rotation through multiplication by the complex number k = λ², called the characteristic constant or multiplier of the transformation.

The transformation is said to be elliptic if it can be represented by a matrix ${\displaystyle {\mathfrak {H}}}$ of determinant 1 such that $${\displaystyle 0\leq \operatorname {tr} ^{2}{\mathfrak {H}}<4.}$$

A transform is elliptic if and only if |λ| = 1 and λ ≠ ±1. Writing ${\displaystyle \lambda =e^{i\alpha }}$, an elliptic transform is conjugate to $${\displaystyle {\begin{pmatrix}\cos \alpha &-\sin \alpha \\\sin \alpha &\cos \alpha \end{pmatrix}}}$$ with α real.