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*n*-DIMENSIONAL SPHERE

Cartesian equation in : .
Cartesian parametrization: 1) with . 2) Stereographic coordinates with center (0, 0,...,- R) :
(the point (0, 0,0,...,-R) is missing).
n+1-dimensional volume of the associated ball:
n-dimensional volume of the sphere: . |

The *n*-dimensional (hyper)sphere with center *O* and radius *R* is the locus of the points in the *n* + 1-dimensional space located at distance *R* from *O*.

It also is a *n*-dimensional manifold homeomorphic to the Alexandroff compactification of the *n*-dimensional space , and is written .
In other words, an *n-*dimensional sphere minus one point is topologically equivalent to .

See here the case of the 3-dimensional sphere.

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© Robert FERRÉOL, 2017