# Möbius transformation

A **Möbius transformation** (also called a linear fractional or bilinear transformation) is a function of the extended complex plane \( \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\} \) of the form

\[ M_{T}(z) = \frac{az+b}{cz+d}, \qquad \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{GL}_{2}(\mathbb{C}), \]

where the coefficient matrix is required to have nonzero determinant \( ad-bc \neq 0 \). The value is defined as \( \infty \) where the denominator vanishes, and \( M_{T}(\infty) = a/c \) when \( c \neq 0 \) while \( M_{T}(\infty) = \infty \) when \( c = 0 \)<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup><sup> • </sup><sup>[2](https://arxiv.org/abs/2606.20358)</sup>. These maps are conformal bijections of the [Riemann sphere](https://www.edgepedia.org/riemann-sphere) and form a group under composition; multiplying all four coefficients by the same nonzero complex number gives the same transformation, so the group is the projective linear group \( \mathrm{PGL}(2,\mathbb{C}) \), equivalently \( \mathrm{PSL}(2,\mathbb{C}) \) modulo \( \{\pm I\} \)<sup>[2](https://arxiv.org/abs/2606.20358)</sup><sup> • </sup><sup>[3](https://complex-analysis.com/content/linear_fractional_transformations.html)</sup>.

| Key fact | Detail |
|---|---|
| Formula | \( M_{T}(z) = (az+b)/(cz+d) \) with \( a,b,c,d \in \mathbb{C} \) and \( ad-bc \neq 0 \), acting on \( \hat{\mathbb{C}} \)<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup> |
| Group structure | Möbius transformations form a group under composition, identified with \( \mathrm{PGL}(2,\mathbb{C}) \); scaling all coefficients by a nonzero constant leaves the map unchanged<sup>[2](https://arxiv.org/abs/2606.20358)</sup> |
| Building blocks | Every Möbius transformation is a composition of translations, dilations, rotations, and an inversion \( z \mapsto 1/z \), which interchanges 0 and \( \infty \)<sup>[2](https://arxiv.org/abs/2606.20358)</sup> |
| Circle property | Circles and lines map to circles and lines, with every line regarded as passing through \( \infty \)<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup> |
| Cross ratio | Four distinct points have the same cross ratio before and after the map<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup> |
| Three-point theorem | A unique Möbius transformation maps any three ordered distinct points of \( \hat{\mathbb{C}} \) to any other three<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup> |
| Fixed points | One or two fixed points in \( \hat{\mathbb{C}} \); three fixed points force the identity map<sup>[3](https://complex-analysis.com/content/linear_fractional_transformations.html)</sup> |

## Elementary maps and decomposition

The class contains four basic types: translations \( f(z) = z+b \), dilations \( f(z) = az \) with \( a > 0 \), rotations \( f(z) = \omega z \) with \( |\omega| = 1 \), and the inversion \( f(z) = 1/z \)<sup>[2](https://arxiv.org/abs/2606.20358)</sup>. Translations, dilations, and rotations fix the point at infinity, while inversion swaps 0 and \( \infty \)<sup>[2](https://arxiv.org/abs/2606.20358)</sup>. A standard lecture result states that any Möbius transformation is a composition of dilations, translations, and an inversion, which is why the four elementary types generate the whole group<sup>[2](https://arxiv.org/abs/2606.20358)</sup>.

The nondegeneracy condition \( ad-bc \neq 0 \) is exactly the invertibility of the coefficient matrix; if the determinant vanished, the numerator and denominator would be proportional and the map would collapse to a constant<sup>[2](https://arxiv.org/abs/2606.20358)</sup>.

## Circles, lines, and the cross ratio

**The circle property.** Möbius transformations map circles and lines in \( \hat{\mathbb{C}} \) to circles and lines in \( \hat{\mathbb{C}} \), where a line is treated as a circle through infinity<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup>. This follows from the sphere picture: under stereographic projection every line or circle in the plane corresponds to a circle on the Riemann sphere, and Möbius transformations preserve this circle-line geometry<sup>[2](https://arxiv.org/abs/2606.20358)</sup>. The property gives a practical method for finding the image of a region: compute the image of the boundary, check where one interior point goes, and the region bounded by the image curve and containing that point is the image<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup>.

**The cross ratio.** For four distinct points \( z_{0}, z_{1}, z_{2}, z_{3} \) in \( \hat{\mathbb{C}} \) and any Möbius transformation \( f \), the cross ratio is invariant:

\[ (f(z_{0}), f(z_{1}); f(z_{2}), f(z_{3})) = (z_{0}, z_{1}; z_{2}, z_{3}). \]

The proof uses the three-point theorem: the map \( f \) and a map built from cross-ratio functions agree on three points, so they are equal<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup>. Invariance of the cross ratio among four complex points, not necessarily collinear, was among the properties August Ferdinand Möbius derived in his 1855 paper on what he called the *Kreisverwandtschaft* (circle transformation), without using the modern function definition<sup>[4](https://notes.math.ca/en/article/geometric-transformations-1800-1855/)</sup>.

**Three points determine the map.** Given two ordered triples of distinct points \( \{z_{1}, z_{2}, z_{3}\} \) and \( \{w_{1}, w_{2}, w_{3}\} \) in \( \hat{\mathbb{C}} \), there exists a unique Möbius transformation with \( f(z_{i}) = w_{i} \)<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup>. Existence comes from composing two cross-ratio maps, and uniqueness from the fact that a Möbius transformation with three fixed points must be the identity<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup>. To construct the map in practice, one writes the equality of cross ratios and rearranges for \( f(z) \)<sup>[1](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)</sup>.

## Fixed points and classification

A fixed point satisfies \( M_{T}(z) = z \), a quadratic equation, so a Möbius transformation has either one or two fixed points in \( \hat{\mathbb{C}} \)<sup>[3](https://complex-analysis.com/content/linear_fractional_transformations.html)</sup>. Since scaling the matrix does not change the map, the matrix can be normalized to determinant 1, and the classification then depends on the trace \( \tau = a + d \)<sup>[3](https://complex-analysis.com/content/linear_fractional_transformations.html)</sup>:

- **Elliptic**: \( \tau \in \mathbb{R} \) and \( |\tau| < 2 \); conjugate to a rotation \( z \mapsto kz \) with \( |k| = 1 \), \( k \neq 1 \).
- **Hyperbolic**: \( \tau \in \mathbb{R} \) and \( |\tau| > 2 \); conjugate to a real scaling \( k > 0 \), \( k \neq 1 \).
- **Loxodromic**: \( \tau \notin \mathbb{R} \); conjugate to a \( k \) that is neither real positive nor of unit modulus, combining rotation and dilation.
- **Parabolic**: \( \tau = \pm 2 \), one double fixed point; conjugate to a translation \( z \mapsto z + c \), \( c \neq 0 \)<sup>[3](https://complex-analysis.com/content/linear_fractional_transformations.html)</sup>.

The classification matters in geometry and dynamical systems because it describes the qualitative behavior of iterates: hyperbolic and loxodromic transformations behave chaotically on the Riemann sphere, elliptic ones are periodic, and parabolic ones have a single neutral fixed point<sup>[3](https://complex-analysis.com/content/linear_fractional_transformations.html)</sup>.

## History

The transformation is named for August Ferdinand Möbius (1790–1868), the German mathematician and astronomer born at Schulpforta<sup>[5](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/M%C3%B6bius%2C_August_Ferdinand)</sup>. His 1827 work *Der Barycentrische Calcul* gave the first overview of geometric transformations, and a second paper, published in 1855, treated what has come to be called the Möbius transformation; there he derived the familiar properties, including cross-ratio invariance, and was a pioneer in working in the complex plane<sup>[4](https://notes.math.ca/en/article/geometric-transformations-1800-1855/)</sup>. Möbius's name is attached to several other mathematical objects, including the [Möbius function](https://www.edgepedia.org/mobius-function), the [Möbius inversion formula](https://www.edgepedia.org/mobius-inversion-formula), and the [Möbius strip](https://www.edgepedia.org/mobius-strip), which he discovered in 1858<sup>[6](https://mathshistory.st-andrews.ac.uk/Biographies//Mobius/)</sup>.

## Hyperbolic geometry and automorphism groups

Möbius transformations supply the symmetry groups of the standard models of the hyperbolic plane. The conformal automorphisms of the unit disk \( \mathbb{D} \) are exactly the maps

\[ z \mapsto e^{i\theta} \cdot \frac{z - a}{1 - \bar{a} z}, \qquad \theta \in [0, 2\pi), \ |a| < 1, \]

a three-real-parameter family in which every element is a Möbius transformation; the factor \( e^{i\theta} \) is a rotation and the remaining Blaschke factor sends \( a \) to 0<sup>[7](https://math.hodapp.com/lectures/foundations/analysis/complex-conformal.html)</sup>. The map \( \phi(z) = i \cdot (1+z)/(1-z) \) is a biholomorphism from the disk to the upper half-plane \( \mathbb{H}^{2} \), with inverse \( \phi^{-1}(w) = (w-i)/(w+i) \), and conjugating by it gives

Via conjugation by φ, the automorphisms of the upper half-plane correspond to those of the disk, giving Aut(ℍ²) as PSL(2,ℝ), the quotient by ±I of the group of maps w ↦ (aw+b)/(cw+d) having real coefficients a, b, c, d and determinant ad − bc = 1.

the Möbius group of the upper half-plane, used in hyperbolic geometry and number theory<sup>[7](https://math.hodapp.com/lectures/foundations/analysis/complex-conformal.html)</sup>. In the upper half-plane model of hyperbolic geometry, the transformation group consists of all Möbius transformations that send the upper half-plane to itself, and hyperbolic distance can be written as a logarithm of a cross ratio of ideal points, which is invariant because Möbius transformations preserve the cross ratio<sup>[8](https://mphitchman.com/geometry/section5-5.html)</sup>. The disk and half-plane models are transferred into each other by a Möbius transformation built from two inversions<sup>[8](https://mphitchman.com/geometry/section5-5.html)</sup>.

## Applications in engineering and physics

**The Smith chart.** [Microwave](https://www.edgepedia.org/microwave) engineers use the [Smith chart](https://www.edgepedia.org/smith-chart), a Möbius transformation that conformally maps the impedance plane onto the unit circle; under the mapping \( \Gamma = (z-1)/(z+1) \), the right half-plane of normalized impedance is carried into the unit disk containing reflection coefficients<sup>[9](https://mathworld.wolfram.com/MoebiusTransformation.html)</sup>. The transformation maps circles to circles and preserves angles, so constant-resistance values appear as circles and constant-reactance values as arcs, and all complex impedance values from 0 to \( \infty \) fit on one chart<sup>[10](https://www.eit.lth.se/fileadmin/eit/home/scd.gkr/Books/MicroWave.pdf)</sup>. Feedback analysis of high-frequency networks likewise produces a conformal Möbius transformation that maps disks of impedance or admittance to disks of the new scattering parameters on the Smith chart<sup>[11](https://doi.org/10.1109/ISCAS.1989.100410)</sup>.

**Conformal mapping in physics.** Because Möbius transformations are conformal, they preserve angles and fit into the broader use of conformal maps to turn boundary-value problems on complicated regions into problems on simple ones, in steady-state heat distribution, two-dimensional electrostatics, and ideal fluid flow governed by [Laplace's equation](https://www.edgepedia.org/laplaces-equation)<sup>[7](https://math.hodapp.com/lectures/foundations/analysis/complex-conformal.html)</sup>. Möbius transformations also appear in conformal field theory and special relativity<sup>[2](https://arxiv.org/abs/2606.20358)</sup>.

## References

1. [Complex Analysis II, Chapter 4: Möbius transformations, Durham University lecture notes (S. Bögli)](https://maths.dur.ac.uk/users/sabine.boegli/CA2.html)
2. [Formalizing Extended Complex Numbers, Möbius Transformations, and Cross Ratio in Lean 4, arXiv:2606.20358](https://arxiv.org/abs/2606.20358)
3. [Linear Fractional Transformations, complex-analysis.com](https://complex-analysis.com/content/linear_fractional_transformations.html)
4. [Christopher Baltus, "Geometric Transformations, 1800–1855," CSHPM Notes, Canadian Mathematical Society (December 2023)](https://notes.math.ca/en/article/geometric-transformations-1800-1855/)
5. ["Möbius, August Ferdinand," 1911 Encyclopædia Britannica, Wikisource](https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/M%C3%B6bius%2C_August_Ferdinand)
6. [August Möbius (1790–1868), MacTutor History of Mathematics, University of St Andrews](https://mathshistory.st-andrews.ac.uk/Biographies//Mobius/)
7. [Conformal maps and Möbius transformations, Math foundations lecture notes](https://math.hodapp.com/lectures/foundations/analysis/complex-conformal.html)
8. [The Upper Half-Plane Model, section 5.5, A Gateway to Modern Geometry (D. Hitchman)](https://mphitchman.com/geometry/section5-5.html)
9. [Möbius Transformation, Wolfram MathWorld](https://mathworld.wolfram.com/MoebiusTransformation.html)
10. [A. Karlsson, Microwave circuits lecture notes, Lund University](https://www.eit.lth.se/fileadmin/eit/home/scd.gkr/Books/MicroWave.pdf)
11. [Conformal mapping of S-parameters of a three-terminal network from its feedback impedance admittance, IEEE ISCAS 1989](https://doi.org/10.1109/ISCAS.1989.100410)

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