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
All points $(x)$ which satisfy a linear equation
are said to form a sphere, with coordinates $(y)$. Two spheres $(y)$ and $(z)$ are orthogonal if $(y,z)=0$, tangent if
If two spheres $(y)$ and $(z)$ intersect, the expression
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.
|||F. Klein, "Vorlesungen über höhere Geometrie" , Springer (1926)|
|||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)|
Pentaspherical coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pentaspherical_coordinates&oldid=31752