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can project a sphere's surface to a plane, it can also project a 3-sphere's surface into 3-space. This image shows three coordinate directions projected to 3-space:
parallels (red), Meridians (blue) and hypermeridians (green).
Due to the Conformal property of the stereographic projection,
the curves intersect each other orthogonally (in the yellow points) as in 4D.
All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius
(= straight line).]]

In Mathematics , a hypersphere is a higher dimensional analog of the usual 2-sphere . It can be constructed by gluing two Euclidean Space s together and can be embedded in Euclidean space as the locus of points a fixed distance from the origin.

A hypersphere of dimension ''n'' is called an n-sphere and denoted \mathbf{S}^n. It is an ''n''-dimensional Manifold and can be embedded in Euclidean (''n''+1)-space.


EUCLIDEAN COORDINATES IN (''N''+1)-SPACE


The set of points in (''n''+1)-space: (x_1,x_2,x_3,...,x_{n+1}) that define an n-sphere, (\mathbf S^n) is represented by the equation:
:r^2=\sum_{i=1}^{n+1} (x_i - C_i)^2.\,
where ''C'' is a center point, and ''r'' is the radius.

The above hypersphere exists in n+1-dimensional Euclidean space and is an example of an n- Manifold .


''n''-ball


The space enclosed by an ''n''-sphere is called an (''n''+1)- Ball . An (''n''+1)-ball is Closed if it included the equality, and Open otherwise.

Specifically:
  • A ''1-ball'', a Line Segment , is the interior of a (0-sphere).

  • A ''2-ball'', a Disk , is the interior of a Circle (1-sphere).

  • A ''3-ball'', an ordinary Ball , is the interior of a Sphere (2-sphere).

  • A ''4-ball'', is the interior of a 3-sphere , etc.



Notation


Labelling hyperspheres with the dimensionality of the surface (as used in this article) is the convention common in mathematical use. Potentially confusingly, some authors use the dimensionality of the containing space to label hyperspheres. {Link without Title} Thus what most call a 1-sphere (a regular Circle in a Plane ), others term a 2-sphere (reflecting the dimensionality of the plane in which it lies).


HYPERSPHERICAL VOLUME


The hyperdimensional volume of the space which a (n-1)-sphere encloses (the n-ball) is:

:V_n={\pi^ rac{n}{2}R^n\over\Gamma( rac{n}{2} + 1)}

where \Gamma is the Gamma Function . (For even n, \Gamma\left( rac{n}{2}+1 ight)= \left( rac{n}{2} ight)!; for odd n, \Gamma\left( rac{n}{2}+1 ight)= \sqrt{\pi} rac{n!!}{2^{(n+1)/2}}, where n!! denotes the Double Factorial .)

The "surface area" of this (n-1)-sphere is

:S_n= rac{dV_n}{dR}= rac{nV_n}{R}={2\pi^ rac{n}{2}R^{n-1}\over\Gamma( rac{n}{2})}

The following relationships hold between the hyperspherical surface area and volume:

:V_n/S_n = R/n\,

:S_{n+2}/V_n = 2\pi R\,

The interior of a hypersphere, that is the set of all points whose distance from the centre is less than R, is called a hyperball, or if the hypersphere itself is included, a closed hyperball.


HYPERSPHERICAL VOLUME - SOME EXAMPLES


For small values of n, the volumes, V_n , of the unit n-ball (R=1) are:

If the dimension ''n'', is not limited to integral values, the hypersphere volume is a Continuous Function of ''n'' with a Global Maximum for the unit sphere in "dimension" ''n'' = 5.2569464... where the "volume" is 5.277768... It has a hypervolume of 1 when ''n'' = 0 or when ''n''
 = 12.76405...

The hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2''n''; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically as the dimension increases.

This apparently strange behavior of the unit n-sphere volume can be understood by assigning units of length to each dimension. It then becomes clearly meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area. A meaningful comparison is obtained by using a dimensionless measure of the volume, such as the ratio of the hypersphere and its circumscribed hypercube volumes. Using this measure restores the intuitively normal behavior of a monotonic decline in the volume as the dimension increases.


HYPERSPHERICAL 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 \ r, and \ n-1 angular coordinates \ \phi _1 , \phi _2 , ... , \phi _{n-1}. If \ x_i are the
Cartesian coordinates, then we may define

:x_1=r\cos(\phi_1)\,
:x_2=r\sin(\phi_1)\cos(\phi_2)\,
:x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,
:\cdots\,
:x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,
:x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,

While the inverse transformations can be derived from those above:
: an(\phi_{n-1})= rac{x_n}{x_{n-1}}
: an(\phi_{n-2})= rac{\sqrt^2}}{x_{n-2}}
:\cdots\,
: an(\phi_{1})= rac{\sqrt^2+\cdots+{x_2}^2}}{x_{1}}
Note that last angle \phi _{n-1} has a range of 2\pi while the other angles have a range of \pi. This range covers the whole sphere.

The hyperspherical volume element will be found from the Jacobian of the transformation:

:d^nr =

bcgfhgffh''hgfhhhhhh

hhhhhghf
hfgdhgfdhgfhgfhgfhgf

fghgfdhgdhgfhgfhggh
blockqugote>''
hg


REFERENCES