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Difference between revisions of "Weierstrass coordinates"

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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
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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 $(x_0 ,\dotsc, x_n)$ 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
  
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$$a_0x_0+\ldots+a_nx_n=0.$$
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$$a_0x_0+\dotsb+a_nx_n=0.$$
  
 
Named after K. Weierstrass, who used these coordinates in his courses on Lobachevskii geometry in 1872.
 
Named after K. Weierstrass, who used these coordinates in his courses on Lobachevskii geometry in 1872.
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These coordinates for elliptic space can be normalized so that
 
These coordinates for elliptic space can be normalized so that
  
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$$x_0^2+\ldots+x_n^2=1.$$
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$$x_0^2+\dotsb+x_n^2=1.$$
  
 
The analogous Weierstrass coordinates for hyperbolic space satisfy
 
The analogous Weierstrass coordinates for hyperbolic space satisfy
  
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$$x_0^2-\ldots-x_n^2=1,$$
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$$x_0^2-\dotsb-x_n^2=1,$$
  
 
with the same equation $\sum a_\nu x_\nu=0$ for a hyperplane.
 
with the same equation $\sum a_\nu x_\nu=0$ for a hyperplane.

Latest revision as of 12:50, 14 February 2020

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 $(x_0 ,\dotsc, x_n)$ 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

$$a_0x_0+\dotsb+a_nx_n=0.$$

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

$$x_0^2+\dotsb+x_n^2=1.$$

The analogous Weierstrass coordinates for hyperbolic space satisfy

$$x_0^2-\dotsb-x_n^2=1,$$

with the same equation $\sum a_\nu x_\nu=0$ 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=44600
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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