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Pentaspherical coordinates

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A kind of homogeneous coordinates for a point in complex inversive space. The numbers , not all zero, are connected by the relation

All points which satisfy a linear equation

are said to form a sphere, with coordinates . Two spheres and are orthogonal if , tangent if

If two spheres and intersect, the expression

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

Setting , 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 -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)


Comments

References

[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)
How to Cite This Entry:
Pentaspherical coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pentaspherical_coordinates&oldid=31752
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article