View source for Riemann sphere

<!--
r0820101.png
$#A+1 = 28 n = 0
$#C+1 = 28 : ~/encyclopedia/old_files/data/R082/R.0802010 Riemann sphere
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A sphere in the Euclidean space  $  \mathbf R  ^ {3} ( \xi , \eta , t) $
onto which the extended complex plane  $  \overline{\mathbf C}\; $
is conformally and one-to-one transformed under [[Stereographic projection|stereographic projection]]. For example, the unit sphere

$$ 
S _ {2}  =  \{ {( \xi , \eta , t) \in \mathbf R  ^ {3} } : {
\xi  ^ {2} + \eta  ^ {2} + t  ^ {2} = 1 } \}
$$

can be taken as the Riemann sphere and the plane  $  \overline{\mathbf C}\; $
can be identified with the plane  $  t = 0 $
such that the real axis coincides with the axis  $  \eta = 0, t = 0 $
and the imaginary axis with the axis  $  \xi = 0, t = 0 $ (see Fig.).

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082010a.gif" />

Figure: r082010a

Under stereographic projection to each point  $  z = x + iy \neq \infty $
there corresponds the point  $  M( \xi , \eta , t) \neq P( 0, 0, 1) $
obtained as the point of intersection of the ray drawn from the north pole of the sphere,  $  P( 0, 0, 1) $, 
to the point  $  z $
with the sphere  $  S _ {2} $; 
the north pole  $  P( 0, 0, 1) $
corresponds to the point at infinity,  $  z = \infty $. 
Analytically this relation can be expressed by the formulas

$$ \tag{* }
\xi + i \eta  = \ 

\frac{2z}{| z |  ^ {2} + 1 }
 ,\ \ 
t  =  
\frac{| z |  ^ {2} - 1 }{| z |  ^ {2} + 1 }
 ,\ \ 
z  =  
\frac{\xi + i \eta }{1-t }.
$$

In other words, the correspondence (*) determines a differentiable imbedding of the one-dimensional complex projective space  $  \mathbf C P  ^ {1} $
into the space  $  \mathbf R  ^ {3} $
in the form of the sphere  $  S _ {2} $. 
In many questions of the theory of functions, the extended complex plane is identified with the Riemann sphere. The exclusive role of the point at infinity of the plane  $  \overline{\mathbf C}\; $
may be dispensed with if the distance between two points  $  z, w \in \overline{\mathbf C}\; $
is taken to be the chordal, or spherical, distance  $  \chi ( z, w) $
between their images  $  M, N \in S _ {2} $:

$$ 
\chi ( z, w)  =  
\frac{2| z- w | }{\sqrt {| z |  ^ {2} + 1 } \sqrt {| w |  ^ {2} + 1 } }

,
$$

$$ 
\chi ( z, \infty )  =  
\frac{2}{\sqrt {| z |  ^ {2} + 1 } }
 .
$$

A higher-dimensional complex projective space  $  \mathbf C P  ^ {n} $, 
$  n > 1 $, 
can be imbedded into the space  $  \mathbf R  ^ {n(n+ 2)} $
by a complex  $  n $-dimensional stereographic projection, generalizing the formulas (*) (see [[#References|[2]]]).

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1–2''' , Moscow  (1976)  (In Russian)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  B.A. Fuks,   "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc.  (1965)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,   "Complex analysis" , McGraw-Hill  (1979)  pp. Chapt. 8</TD></TR>
</table>