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A sphere in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820101.png" /> onto which the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820102.png" /> is conformally and one-to-one transformed under [[Stereographic projection|stereographic projection]]. For example, the unit sphere
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820103.png" /></td> </tr></table>
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can be taken as the Riemann sphere and the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820104.png" /> can be identified with the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820105.png" /> such that the real axis coincides with the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820106.png" /> and the imaginary axis with the axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820107.png" /> (see Fig.).
+
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" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r082010a.gif" />
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Figure: r082010a
 
Figure: r082010a
  
Under stereographic projection to each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820108.png" /> there corresponds the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r0820109.png" /> obtained as the point of intersection of the ray drawn from the north pole of the sphere, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201010.png" />, to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201011.png" /> with the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201012.png" />; the north pole <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201013.png" /> corresponds to the point at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201014.png" />. Analytically this relation can be expressed by the formulas
+
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  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\frac{2z}{| z |  ^ {2} + 1 }
 +
,\ \
 +
=
 +
\frac{| z |  ^ {2} - 1 }{| z |  ^ {2} + 1 }
 +
,\ \
 +
=
 +
\frac{\xi + i \eta }{1-}
 +
t .
 +
$$
  
In other words, the correspondence (*) determines a differentiable imbedding of the one-dimensional complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201016.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201017.png" /> in the form of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201018.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201019.png" /> may be dispensed with if the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201020.png" /> is taken to be the chordal, or spherical, distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201021.png" /> between their images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201022.png" />:
+
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} $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201023.png" /></td> </tr></table>
+
$$
 +
\chi ( z, w)  =
 +
\frac{2| z- w | }{\sqrt {| z |  ^ {2} + 1 } \sqrt {| w |  ^ {2} + 1 } }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201024.png" /></td> </tr></table>
+
,
 +
$$
  
A higher-dimensional complex projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201026.png" />, can be imbedded into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201027.png" /> by a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082010/r08201028.png" />-dimensional stereographic projection, generalizing the formulas (*) (see [[#References|[2]]]).
+
$$
 +
\chi ( z, \infty )  = 
 +
\frac{2}{\sqrt {| z |  ^ {2} + 1 } }
 +
.
 +
$$
 +
 
 +
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====
 
====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></table>
 
<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></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1979)  pp. Chapt. 8</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Ahlfors,  "Complex analysis" , McGraw-Hill  (1979)  pp. Chapt. 8</TD></TR></table>

Revision as of 08:11, 6 June 2020


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. 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.).

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 [2]).

References

[1] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian)
[2] B.A. Fuks, "Introduction to the theory of analytic functions of several complex variables" , Amer. Math. Soc. (1965) (Translated from Russian)

Comments

References

[a1] L.V. Ahlfors, "Complex analysis" , McGraw-Hill (1979) pp. Chapt. 8
How to Cite This Entry:
Riemann sphere. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemann_sphere&oldid=48550
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article