# stereographic projection

The $n$-dimensional Euclidean http://planetmath.org/node/186unit sphere^{} ${S}^{n}$ is
defined as a subset
of ${\mathbb{R}}^{n+1}$:

$${S}^{n}=\{({x}_{1},\mathrm{\dots},{x}_{n+1})\in {\mathbb{R}}^{n+1}\mid \sum _{k=1}^{n+1}{x}_{k}^{2}=1\}.$$ |

The *stereographic projection* maps all points of ${S}^{n}$ to
the $n$-dimensional Euclidean space^{} ${\mathbb{R}}^{n}$ except one. Let
$N:=(0,\mathrm{\dots},0,1)\in {S}^{n}$ be this point (it is usually called the
*north pole*). Then the stereographic projection is defined by

$$\sigma :{S}^{n}\setminus N\to {\mathbb{R}}^{n},({x}_{1},\mathrm{\dots},{x}_{n+1})\mapsto \frac{c-1}{{x}_{n+1}-1}({x}_{1},\mathrm{\dots},{x}_{n}).$$ |

Here, $c$ is an arbitrary real number. If $c=1$, the projection degenerates; in all other cases, however, $\sigma $ is a smooth bijective mapping.

The image ${P}^{\prime}$ of a point $P$ under $\sigma $ can be geometrically
constructed as follows. Embed ${\mathbb{R}}^{n}$ into
${\mathbb{R}}^{n+1}$ as a hyperplane^{} at ${x}_{n+1}=c$. Unless $c=1$, the
straight line defined by $N$ and $P$ intersects with ${\mathbb{R}}^{n}$ in
precisely one point, ${P}^{\prime}$. The most common values for $c$ are $c=-1$
and $c=0$, see figures 1 and 2.

Let $-\mathrm{id}:{\mathbb{R}}^{n+1}\to {\mathbb{R}}^{n+1}$ be the map $x\mapsto -x$, then $\stackrel{~}{\sigma}:=\sigma \circ (-\mathrm{id})$ (a suitably restricted composition^{}) maps all points of ${S}^{n}$
except the *south pole* $S:=(0,\mathrm{\dots},0,-1)$ smoothly and
bijectively to ${\mathbb{R}}^{n}$. Together, $\sigma $ and $\stackrel{~}{\sigma}$
form an atlas of ${S}^{n}$, so ${S}^{n}$ is an $n$-dimensional smooth
manifold^{}.

Title | stereographic projection |

Canonical name | StereographicProjection |

Date of creation | 2013-03-22 15:18:35 |

Last modified on | 2013-03-22 15:18:35 |

Owner | GrafZahl (9234) |

Last modified by | GrafZahl (9234) |

Numerical id | 5 |

Author | GrafZahl (9234) |

Entry type | Definition |

Classification | msc 54E40 |

Classification | msc 54C25 |

Classification | msc 54C05 |

Classification | msc 51M15 |

Related topic | CoordinateSystems |

Related topic | ClosedComplexPlane |

Related topic | RiemannSphere |

Defines | north pole |

Defines | south pole |