Pentaspherical coordinates

A kind of homogeneous coordinates $x_0:x_1:x_2:x_3:x_4$ for a point $(x)$ in complex inversive space. The numbers $x_\nu$, not all zero, are connected by the relation

$$(x,x)\equiv x_0^2+x_1^2+x_2^2+x_3^2+x_4^2=0.$$

All points $(x)$ which satisfy a linear equation

$$(y,x)\equiv y_0x_0+y_1x_1+y_2x_2+y_3x_3+y_4x_4=0$$

are said to form a sphere, with coordinates $(y)$. Two spheres $(y)$ and $(z)$ are orthogonal if $(y,z)=0$, tangent if

$$(y,y)(z,z)-(y,z)^2=0.$$

If two spheres $(y)$ and $(z)$ intersect, the expression

$$\frac{(y,z)}{\sqrt{(y,y)}\sqrt{(z,z)}}$$

measures the cosine of their angle (or the hyperbolic cosine of their inverse distance).

Setting $x_4=0$, one obtains the analogous tetracyclic coordinates, which lead to circles instead of spheres.

Completely analogous constructions can be performed for spaces of higher dimensions, which give polyspherical coordinates. In the $4$-dimensional case they are called hexaspherical coordinates. Polyspherical coordinates are used in conformal geometry in examining manifolds of figures.

References

[1]	F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)
[2]	G.V. Bushmanova, A.P. Norden, "Elements of conformal geometry" , Kazan' (1972) (In Russian)
[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2]	J.L. Coolidge, "A treatise on the circle and the sphere" , Chelsea, reprint (1971)