Möbius transformation

A function of the form \((az+b)/(cz+d)\), with complex coefficients and \(ad-bc\ne0\), acting on the extended complex plane. It is a bijective conformal map that sends circles and lines to circles or lines.

Extended complex plane: The complex plane together with a single point at infinity, forming the Riemann sphere. The point at infinity lets the formula remain a bijection when its denominator vanishes.
Complex number: A number expressed as a real part plus an imaginary part. The input, output, and coefficients of a Möbius transformation are complex numbers.
Affine transformation: A map of the form z ↦ az+b, where a is nonzero. When c=0, a Möbius transformation reduces to an affine map.
Riemann mapping theorem: A theorem stating that every simply connected proper open subset of the complex plane is conformally equivalent to the unit disk. Möbius maps provide explicit conformal equivalences for many basic domains.
Entire function: A complex function holomorphic at every finite point of the complex plane. Most entire functions are not Möbius maps, which are rational and have at most one finite pole.
Projective linear group: The group of invertible matrices modulo multiplication by nonzero scalars. Möbius transformations are represented by invertible two-by-two matrices, with scalar multiples giving the same map.
Complex division: Division of one complex number by another nonzero complex number. The defining formula is a quotient whose denominator may vanish at one point.
Parabolic transformation: A Möbius transformation with one repeated fixed point on the extended complex plane. This canonical type has a single fixed point and is conjugate to a translation.
Poincaré disk model: A model of hyperbolic geometry in which points lie inside a disk and geodesics are circular arcs orthogonal to its boundary. Disk-preserving Möbius transformations act as orientation-preserving hyperbolic isometries.
Polynomial: A finite sum of powers of a variable with fixed coefficients. A polynomial Möbius map must be affine; a nonzero denominator term introduces a pole.

Linked from 15 pages