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Difference between revisions of "Weierstrass coordinates"

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A type of coordinates in an elliptic space. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974101.png" /> be an elliptic space obtained by the identification of diametrically-opposite points of the unit sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974103.png" />-dimensional Euclidean space. The Weierstrass coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974104.png" /> of a point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974105.png" /> are the orthogonal Cartesian coordinates of the point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974106.png" /> that corresponds to it. Since the isometric mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974107.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974108.png" /> is not single-valued, Weierstrass coordinates are defined up to sign. A hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097410/w0974109.png" /> is given by a homogeneous linear equation
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A type of coordinates in an elliptic space. Let $M^n$ be an elliptic space obtained by the identification of diametrically-opposite points of the unit sphere $S^n$ in $(n+1)$-dimensional Euclidean space. The Weierstrass coordinates $(x_0 ,\ldots, x_n)$ of a point in $M^n$ are the orthogonal Cartesian coordinates of the point of $S^n$ that corresponds to it. Since the isometric mapping of $M^n$ into $S^n$ is not single-valued, Weierstrass coordinates are defined up to sign. A hyperplane in $M^n$ is given by a homogeneous linear equation
  
 
<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/w/w097/w097410/w09741010.png" /></td> </tr></table>
 
<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/w/w097/w097410/w09741010.png" /></td> </tr></table>

Revision as of 10:06, 9 May 2012

A type of coordinates in an elliptic space. Let $M^n$ be an elliptic space obtained by the identification of diametrically-opposite points of the unit sphere $S^n$ in $(n+1)$-dimensional Euclidean space. The Weierstrass coordinates $(x_0 ,\ldots, x_n)$ of a point in $M^n$ are the orthogonal Cartesian coordinates of the point of $S^n$ that corresponds to it. Since the isometric mapping of $M^n$ into $S^n$ is not single-valued, Weierstrass coordinates are defined up to sign. A hyperplane in $M^n$ is given by a homogeneous linear equation

Named after K. Weierstrass, who used these coordinates in his courses on Lobachevskii geometry in 1872.


Comments

These coordinates for elliptic space can be normalized so that

The analogous Weierstrass coordinates for hyperbolic space satisfy

with the same equation for a hyperplane.

References

[a1] H. Liebmann, "Nichteuklidische Geometrie" , Göschen (1912) pp. 114–119
[a2] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 121, 281
How to Cite This Entry:
Weierstrass coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_coordinates&oldid=15072
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article