Weierstrass coordinates

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