Difference between revisions of "Weierstrass coordinates"
From Encyclopedia of Mathematics
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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 | 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 | ||
| − | + | $$a_0x_0+\ldots+a_nx_n=0.$$ | |
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 | ||
| − | + | $$x_0^2+\ldots+x_n^2=1.$$ | |
The analogous Weierstrass coordinates for hyperbolic space satisfy | The analogous Weierstrass coordinates for hyperbolic space satisfy | ||
| − | + | $$x_0^2-\ldots-x_n^2=1,$$ | |
| − | with the same equation | + | with the same equation $\sum a_\nu x_\nu=0$ for a hyperplane. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Liebmann, "Nichteuklidische Geometrie" , Göschen (1912) pp. 114–119</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 121, 281</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Liebmann, "Nichteuklidische Geometrie" , Göschen (1912) pp. 114–119</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 121, 281</TD></TR></table> | ||
Revision as of 15:03, 17 July 2014
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
$$a_0x_0+\ldots+a_nx_n=0.$$
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
$$x_0^2+\ldots+x_n^2=1.$$
The analogous Weierstrass coordinates for hyperbolic space satisfy
$$x_0^2-\ldots-x_n^2=1,$$
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=32471
Weierstrass coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_coordinates&oldid=32471
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article