Hypersphere - Spherical Coordinates

Spherical Coordinates

We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate, and n − 1 angular coordinates where ranges over radians (or over degrees). If are the Cartesian coordinates, then we may compute from with:

Except in the special cases described below, the inverse transformation is unique:

where if for some but all of are zero then when, and radians (180 degrees) when .

There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case may be chosen to be zero.

Note that a half-angle formula is used for because the more straightforward is too small by an addend of π when < 0.