: 
.
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:
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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