Riemann sphere

The complex plane with one point at infinity, represented as a sphere by stereographic projection. It turns the extended complex plane into a compact space.

Stereographic projection: A map that projects points on a sphere from one pole onto a plane. It identifies each finite complex number with a point on the sphere.
Riemann sphere metric: The spherical metric that measures distances between points represented on the Riemann sphere. It gives the sphere’s geometry a precise distance rule for complex values.
Rational function: A quotient of two polynomials, defined wherever its denominator is nonzero. Every rational function extends to a map of the sphere when poles map to infinity.
One-point compactification: A compactification formed by adding one point representing escape beyond every compact subset. The Riemann sphere is the one-point compactification of the complex plane.
Extended complex plane: The complex numbers together with a single point at infinity. It is the algebraic description of the points represented by the sphere.
Möbius transformation: A function of the form (az+b)/(cz+d), with ad−bc nonzero, acting on the extended complex plane. These transformations act naturally on the sphere, including at infinity.
Complex dynamics: The study of iteration and long-term behavior of functions in the complex plane. Rational maps iterate globally on the sphere rather than failing at poles.
Real projective line: The real line together with one point at infinity, topologically a circle. It is the one-dimensional analogue, but its added point has different complex-geometric structure.
Complex infinity: The added point that represents divergence to infinity in the extended complex plane. It corresponds to the sphere’s projection pole and completes the plane.
Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Holomorphic maps can be studied near the added point as well as finite values.

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