Möbius transformation

A Möbius transformation (or linear fractional transformation) is a map

$f(z)=\frac{az+b}{cz+d},$

with $a,b,c,d\in \mathbb{C}$ and $ad-bc\ne 0$. The simplest non-trivial family of analytic functions on the extended complex plane.

The condition $ad-bc\ne 0$ is necessary; otherwise $f$ is constant.

Basic structure

When $c\ne 0$, $f$ has a pole at $z=-d/c$ (denominator vanishes). Extend to the Riemann sphere $\hat{\mathbb{C}}=\mathbb{C}\cup \{\infty \}$, with $f(-d/c)=\infty$ and $f(\infty )=a/c$.

Special cases:

• $c=0$: affine $f(z)=(a/d)z+b/d$.
• Translation: $f(z)=z+b$.
• Rotation/scaling: $f(z)=az$.
• Inversion: $f(z)=1/z$.

Every Möbius transformation factors as a composition of these elementary types.

Bijectivity and inverse

Every Möbius transformation has an inverse, also a Möbius transformation:

$f^{-1}(w)=\frac{dw-b}{-cw+a}.$

So Möbius transformations are bijections of $\hat{\mathbb{C}}$.

Composition as matrix multiplication

Associate $f(z)=(az+b)/(cz+d)$ with the matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$. Composition of Möbius transformations corresponds to matrix multiplication, inversion to matrix inversion, non-degeneracy to invertibility ($ad-bc=\det \ne 0$).

The group of Möbius transformations is $\mathrm{PGL}(2,\mathbb{C})=\mathrm{GL}(2,\mathbb{C})/\{\lambda I\}$, the matrices up to overall scalar.

Derivative and conformality

$f^{\prime }(z)=\frac{ad-bc}{(cz+d{)}^2}.$

Nonzero everywhere except at the pole, so Möbius transformations are conformal everywhere except at the pole.

Circles map to circles

Theorem. A Möbius transformation maps the family of “circles and lines” to itself.

Treating lines as “circles passing through $\infty$” on the Riemann sphere, this reads: Möbius transformations map circles to circles. A circle in $\mathbb{C}$ can map to a line, but on the sphere both are circles. The family is invariant.

Proof: every Möbius transformation factors into translations, rotations/scalings, and inversion. Translations and rotations/scalings obviously preserve circles and lines. Inversion $z\mapsto 1/z$ does too (verify directly).

Fixed points

A fixed point of $f$ satisfies $f(z)=z$:

$c{z}^2+(d-a)z-b=0.$

A Möbius transformation has at most two fixed points on the finite plane (counting the one at $\infty$ if $c=0$).

One fixing three or more distinct points must be the identity. So three points determine a Möbius transformation completely.

Three points to three points

Given three distinct points $z_1,z_2,z_3$ and three distinct target points $w_1,w_2,w_3$, there is a unique Möbius transformation sending $z_j\mapsto w_j$.

Construction via the cross-ratio:

$[z,z_1,z_2,z_3]=\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}.$

Set $[w,w_1,w_2,w_3]=[z,z_1,z_2,z_3]$ and solve for $w$.

The cross-ratio is invariant under Möbius transformations, which is why this works.

Standard examples

• Identity: $f(z)=z$.
• Inversion: $f(z)=1/z$. Unit circle fixed as a set; $0$ and $\infty$ swapped.
• Cayley transform: $f(z)=(z-i)/(z+i)$. Maps the upper half-plane bijectively to the unit disk.
• Reflection-impedance: $\Gamma (z)=(z-1)/(z+1)$. Maps the right half-plane (positive resistance) to the unit disk, which is the Smith chart in transmission-line theory.

In context

Möbius transformations are the simplest conformal maps. Their three-points-to-three-points property gives full control: pick any three points and their images, and the transformation is determined.

The Riemann mapping theorem says any simply connected proper open subset of $\mathbb{C}$ is conformally equivalent to the unit disk. Explicit Möbius transformations handle the common cases (half-planes, disks, exterior regions) in transmission-line theory, electrostatics, fluid mechanics. The Smith chart in RF engineering is exactly a Möbius transformation visualized.