Linear fractional maps
Definition. A Möbius transformation has the form \(T(z)=(az+b)/(cz+d)\), with \(ad-bc\ne0\), acting on the Riemann sphere \(\widehat{\mathbb C}\). Matrices differing by a nonzero scalar define the same transformation.
Circle-line preservation
Theorem. Möbius transformations map generalized circles, meaning circles and lines in \(\widehat{\mathbb C}\), to generalized circles. This can be checked because they are generated by translations, dilations-rotations, and inversion \(z\mapsto1/z\).
Three-point determination
Proposition. There is a unique Möbius transformation sending any ordered triple of distinct points in \(\widehat{\mathbb C}\) to any other ordered triple. This follows from preservation of the cross-ratio.
Cross-ratio
Definition. The cross-ratio \([z_1,z_2,z_3,z_4]\) is invariant under Möbius transformations. It is the basic projective coordinate on four points of the sphere.
Disk and half-plane
Example. The Cayley map \(z\mapsto (z-i)/(z+i)\) sends the upper half-plane biholomorphically to the unit disc. This makes many disc and half-plane problems equivalent.
Automorphism groups
Remark. Möbius transformations are the conformal automorphisms of the Riemann sphere. Subgroups preserving the disc or half-plane are central in hyperbolic geometry.
Projective linear action
A Möbius transformation is \[ T(z)=\frac{az+b}{cz+d},\qquad ad-bc\ne0, \] acting on the Riemann sphere \(\widehat{\mathbb C}\), with \(T(-d/c)=\infty\) and \(T(\infty)=a/c\) when \(c\ne0\). Multiplying \(\begin{psmallmatrix}a&b\\c&d\end{psmallmatrix}\) by a nonzero scalar does not change \(T\); the group is \(\operatorname{PGL}(2,\mathbb C)\).
Proposition. Möbius transformations are exactly the biholomorphic self-maps of \(\widehat{\mathbb C}\). Their inverse is \[ T^{-1}(w)=\frac{dw-b}{-cw+a}. \]
Composition corresponds to matrix multiplication modulo nonzero scalars. The determinant condition ensures the inverse exists.
Three-point normalization and cross ratios
Theorem. Given distinct \(z_1,z_2,z_3\) and distinct \(w_1,w_2,w_3\) on the sphere, there is a unique Möbius transformation with \(T(z_j)=w_j\).
Proof. The map \[ S_z(\zeta) =\frac{(\zeta-z_1)(z_2-z_3)} {(\zeta-z_3)(z_2-z_1)} \] sends \(z_1,z_2,z_3\) to \(0,1,\infty\). Then \(T=S_w^{-1}\circ S_z\) has the required action. If two such maps existed, their composition would fix \(0,1,\infty\); solving its coefficients makes it the identity.
The cross ratio \[ [z_1,z_2;z_3,z_4] =\frac{(z_1-z_3)(z_2-z_4)} {(z_1-z_4)(z_2-z_3)} \] is Möbius invariant. It is real exactly when the four points lie on a generalized circle.
Circles and lines
Proposition. Möbius transformations send generalized circles, meaning circles or lines in \(\widehat{\mathbb C}\), to generalized circles.
Proof. Every Möbius map is generated by translations, nonzero dilations/rotations, and inversion \(z\mapsto1/z\). The first two visibly preserve generalized circles. Substitution into a circle or line equation shows inversion does as well.
Fixed-point classification
Fixed points solve \[ cz^2+(d-a)z-b=0. \] After conjugating fixed points to \(0,\infty\), a transformation with two fixed points becomes \(z\mapsto\lambda z\). Depending on \(\lambda\), one obtains elliptic, hyperbolic, or loxodromic behavior. A single double fixed point gives a parabolic map conjugate to \(z\mapsto z+1\). This classification is central to iteration and hyperbolic geometry.