A bijective fractional linear map of the extended complex plane of the form z ↦ (a z + b)/(c z + d) with a d − b c ≠ 0; it acts on the Riemann sphere and sends generalized circles (circles or lines) to generalized circles.

Principle

The set of these maps forms a group of conformal automorphisms of the Riemann sphere; they preserve cross-ratio and local angles and compose via 2×2 complex matrix multiplication up to scalar factors.

Demonstration

The map z ↦ (z−i)/(z+i) sends the upper half-plane to the unit disk and carries real-axis boundary points to the unit circle; any three distinct points on the sphere can be sent to any other three distinct points by a suitable Möbius map.

Misapplication

Treating an expression (a z + b)/(c z + d) as a global holomorphic automorphism when a d − b c = 0 (matrix singular) or forgetting the single point where the denominator vanishes, thereby miscounting poles and bijectivity.

Consequence

Under correct use, Möbius transformations linearize many geometric constraints on the sphere, provide a parameterization of all conformal self-maps, and allow reduction of boundary-value problems by mapping domains to canonical shapes.

Reversal

Restricting to affine maps (c = 0) yields translations, dilations and rotations on the plane — a proper subset that loses the freedom to map infinity to a finite point; conversely, general rational maps of higher degree are not invertible and do not preserve cross-ratio.

Boundary

Defined as maps of the extended complex plane C∪{∞}; not all rational functions are Möbius, and Möbius maps are not defined as holomorphic self-maps of C without including the point at infinity when c ≠ 0.

Semantic Tension

Often confused with arbitrary degree-1 rational functions in formal algebra; the tension is that only those with nonzero determinant are bijective conformal automorphisms of the sphere, not merely linear-fractional expressions.

Synthesis

A Möbius transformation is the class of nondegenerate linear-fractional maps of the Riemann sphere that preserve angles and cross-ratios and form a matrix-representable group of conformal automorphisms.