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 ,\ | + | 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 |
| − | $$a_0x_0+\ | + | $$a_0x_0+\dotsb+a_nx_n=0.$$ |
Named after K. Weierstrass, who used these coordinates in his courses on Lobachevskii geometry in 1872. | Named after K. Weierstrass, who used these coordinates in his courses on Lobachevskii geometry in 1872. | ||
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These coordinates for elliptic space can be normalized so that | These coordinates for elliptic space can be normalized so that | ||
| − | $$x_0^2+\ | + | $$x_0^2+\dotsb+x_n^2=1.$$ |
The analogous Weierstrass coordinates for hyperbolic space satisfy | The analogous Weierstrass coordinates for hyperbolic space satisfy | ||
| − | $$x_0^2-\ | + | $$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 |
Weierstrass coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_coordinates&oldid=32471