Möbius transform

Möbious transformation is a rational map on the complex plane prescribed by ζ ↦ (aζ + b)/(cζ + d) for complex coefficients a, b, c, d forming some regular matrix M. It maps circles to circles again or to lines (corresponding to the limit of infinite radius). It is conformal, bijective, orientation-preserving and different transforms form a group in which composition corresponds to matrix multiplication.

Explanation

• If the matrix M is unitary (¹), the transformation acts in stereographic projection as a rotation.
• If it is real with a positive determinant (²), it preserves the upper and lower complex half-planes.
• If it is a member of the generalized rotation group U(1,1) (³), the transformation maps the unit disk to itself and, if interpreted as the Poincaré disk, has a meaning of rotation in hyperbolic space.
• A complex prefactor of M does not affect its associated transform. That is why for example σ_y induces a transform appropriate to SL(2,R) matrices even though it is not one.

See also

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