# Riemann sphere
## English
### Etymology

Named after German mathematician Bernhard Riemann.

### Noun

Riemann sphere (plural Riemann spheres)
* (topology, complex analysis) The complex numbers extended with the number ∞; the complex plane (representation of the complex numbers as a Euclidean plane) extended with a single idealised point at infinity and consequently homeomorphic to a sphere in 3-dimensional Euclidean space.
** 2002, Wenhua Zhao, “Some Generalizations of Genus Zero Two-Dimensional Conformal Field Theory”, in Stephen Berman, Paul Fendley, Yi-Zhi Huang, Kailash Misra, Brian Parshall, editors, Recent Developments in Infinite-dimensional Lie Algebras and Conformal Field Theory, American Mathematical Society, page 306:We use ${\hat {\mathbb {C} }}^{\times }$ (resp. $\mathbb {C}$) to denote the subset $\{z\in {\hat {\mathbb {C} }}:z\neq 0\}$ (resp. $\{z\in {\hat {\mathbb {C} }}:z\neq \infty \}$) in the Riemann sphere ${\hat {\mathbb {C} }}$.
* (topology, complex analysis) The 2-sphere embedded in Euclidean three-dimensional space and often represented as a unit sphere, regarded as a homeomorphic representation of the extended complex plane and thus the extended complex numbers.
** 1967 [Prentice-Hall], Richard A. Silverman, Introductory Complex Analysis, Dover, 1972, page 22,
    Every circle $\gamma$ on the Riemann sphere $\Sigma$ which does not go through a given point $P^{*}\in \Sigma$ divides $\Sigma$ into two parts, such that one part contains $P^{*}$ and the other does not.
** 2002, Yue Kuen Kwok, Applied Complex Variables for Scientists and Engineers, Cambridge University Press, page 27:In order to visualize the point at infinity, we consider the Riemann sphere that has radius $1/2$ and is tangent to the complex plane at the origin (see Figure 1.8). We call the point of contact the south pole (denoted by $S$) and the point diametrically opposite $S$ the north pole (denoted by $N$). Let $z$ be an arbitrary point in the complex plane, represented by the point $P$. We draw the line $PN$ which intersects the Riemann sphere at the unique point $P'$, distinct from $N$. Conversely, to each point $P'$ on the sphere, other than the north pole $N$, we draw the line $P'N$ which cuts uniquely one point $P$ in the complex plane. Clearly, there exists a one-to-one correspondence between points on the Riemann sphere, except $N$, and all the finite points on the complex plane. We assign the north pole $N$ as the point at infinity. With such an assignment, we then establish a one-to-one correspondence between all the points on the Riemann sphere and all the points in the extended complex plane. This correspondence is known as the stereographic projection.
** 2003, Philip L. Bowers, Monica K. Hurdal, “Planar Conformal Mappings of Piecewise Flat Surfaces”, in Hans-Christian Hege, Konrad Polthier, editors, Visualization and Mathematics III, Springer, page 6:We find that many mathematicians, even those who specialize in complex analysis and conformal geometry, are not familiar with the inversive distance between pairs of circles in the Riemann sphere.

#### Usage notes

* A suitable (and often cited) homeomorphism is the one represented geometrically as a stereographic projection. In graphic representations of the projection, the Riemann sphere is an object in Euclidean space, while the projective plane (i.e., the complex plane) is itself a Euclidean representation of the complex numbers.

#### Synonyms

* (complex numbers with infinity): extended complex numbers
* (complex plane with point at infinity): closed complex plane, extended complex plane

#### Translations
### See also

* Argand plane
* complex manifold
* complex plane
* complex projective line (differently constructed set homeomorphic to the Riemann sphere)
* Riemann surface

### Further reading

* Complex plane on Wikipedia.Wikipedia
* Riemann surface on Wikipedia.Wikipedia