Mobius Transformations

Introduction

Mobius transformations are the simplest non-trivial conformal maps of the extended complex plane (the Riemann sphere). Despite their simple form—a ratio of linear functions—they possess remarkable geometric properties: they map circles to circles (where lines count as circles through infinity) and preserve angles.

These transformations form a group that acts transitively on the Riemann sphere, meaning any three distinct points can be mapped to any other three distinct points. This flexibility makes Mobius transformations powerful tools for solving problems in complex analysis, conformal mapping, and hyperbolic geometry.

Mobius transformations appear throughout mathematics and physics: in special relativity (Lorentz transformations), in the classification of conic sections, in the study of hyperbolic geometry, and in the conformal field theory of theoretical physics.

Definition

A Mobius transformation (or linear fractional transformation) is a function of the form:

ƒ(z)=(a*z+b)/(c*z+d)

where a,b,c,d∈ℂ and a*d−b*c≠0 The condition a*d−b*c≠0 ensures the transformation is non-constant and invertible.

We extend to the Riemann sphere ℂ∪{∞} by defining:

ƒ*(−d/c)=∞*(if *c≠0*)

ƒ(∞)=a/c*(if *c≠0*),ƒ(∞)=∞*(if *c=0*)

Matrix Representation

Mobius transformations correspond to matrices:

ƒ↔([a,b],[c,d])

Composition of transformations corresponds to matrix multiplication. Matrices M and λ*M (for λ≠0 give the same transformation, so Mobius transformations form the projective linear group PGL(2,ℂ)

Key Properties

Circle-Preserving

Mobius transformations map circles to circles, where a line is considered a circle through ∞. A circle may map to a circle or to a line (if it passes through the pole −d/c.

Conformal

Mobius transformations preserve angles: if two curves meet at angle θ their images meet at the same angle θ This follows from being analytic with nonzero derivative.

Triple Transitivity

Given any two sets of three distinct points {(z_1),(z_2),(z_3)} and {(w_1),(w_2),(w_3)} on the Riemann sphere, there exists a unique Mobius transformation mapping (z_i)↦(w_i)

Building Blocks

Every Mobius transformation is a composition of simpler ones:

Translation: z↦z+b (shifts the plane)

Dilation/Rotation: z↦a*z (scales and rotates about origin)

Inversion: z↦1/z (inverts through the unit circle and reflects)

Any Mobius transformation can be written as a composition of these basic types.

Fixed Points and Classification

A fixed point satisfies ƒ(z)=z For ƒ(z)=(a*z+b)/(c*z+d) with c≠0 this leads to:

c*z2+(d−a)*z−b=0

A non-identity Mobius transformation has one or two fixed points (counting ∞).

Parabolic: One fixed point. Conjugate to z↦z+1 Occurs when (tr *M)2=4*det(M)

Elliptic: Two fixed points, conjugate to a rotation z↦e(i*θ)*z Points rotate around the fixed points.

Hyperbolic: Two fixed points, conjugate to z↦k*z with k>0,k≠1 Points flow from one fixed point toward the other.

Loxodromic: Two fixed points, conjugate to z↦λ*z with |λ|≠1 Combines rotation and scaling—points spiral between fixed points.

The Cross-Ratio

The cross-ratio of four points is defined as:

((z_1),(z_2);(z_3),(z_4))=(((z_1)−(z_3))*((z_2)−(z_4)))/(((z_1)−(z_4))*((z_2)−(z_3)))

The cross-ratio is invariant under Mobius transformations: if (w_i)=ƒ((z_i)) then ((w_1),(w_2);(w_3),(w_4))=((z_1),(z_2);(z_3),(z_4)) This invariant is fundamental for constructing specific Mobius transformations.

Connection to Other Concepts

In hyperbolic geometry, the group of orientation-preserving isometries of the hyperbolic plane (in the upper half-plane model) is exactly the Mobius transformations with real coefficients and a*d−b*c>0

Mobius transformations preserving the unit disk (mapping |z|<1 to itself) have the form ƒ(z)=e(i*θ)(z−a)/(1−a*z) with |a|<1 These are the automorphisms of the disk.

Summary

Mobius transformations ƒ(z)=(a*z+b)/(c*z+d) with a*d−b*c≠0 are conformal bijections of the Riemann sphere. They map circles to circles, preserve angles, and are uniquely determined by their action on three points.

They are classified by fixed points: parabolic (one), elliptic (two, rotation), hyperbolic (two, scaling), or loxodromic (two, spiral). The cross-ratio is invariant under Mobius transformations.

Mobius transformations form a group PGL(2,ℂ) with matrix composition. They connect complex analysis to hyperbolic geometry, projective geometry, and special relativity, making them fundamental across mathematics and physics.