Hypersphere

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In mathematics, a hypersphere is a sphere which has dimension 3 or higher. The term [n]-sphere is used for a sphere of dimension [n], for any positive integer [n]. An origin-centered sphere of radius [R] consists of all points in n-dimensional Euclidean space for which the sum of the squares of every coordinate is constant. The constant is [R^2], and its square root is the Euclidean distance of every point on the sphere from the origin. The set of all points on this sphere has dimension [n-1], so it is called the [(n-1)]-sphere and is denoted [\mathbb S^]. It may be written as [(x_1,x_2,...,x_n)] where

[R^2=\sum_^n x_i^2.\,]

The above hypersphere in [n]-dimensional Euclidean space is an example of an [(n-1)]-manifold. For example, an ordinary sphere in three dimensions is a 2-sphere, denoted by [\mathbb^2]; the 1-sphere being a circle, and the 0-sphere is the end points of an interval. Of course, translating the coordinates (i.e. moving the center around) doesn't change the analytic or geometric properties of the sphere.

Contents

1 Hyperspherical volume
2 Hyperspherical volume - some examples
3 Hyperspherical coordinates
4 Stereographic projection
5 See also

Hyperspherical volume

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

[V_n=R^n\over\Gamma(n/2+1)}]

where [\Gamma] is the gamma function. (For even [n], [].)

The "surface area" of this sphere is

[S_n=\frac=\frac=R^\over\Gamma(n/2)}]

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:

[V_1\,]	=	[2\,]
[V_2\,]	=	[\pi\,]	=	[3.14159\ldots\,]
[V_3\,]	=	[\frac\,]	=	[4.18879\ldots\,]
[V_4\,]	=	[\frac\,]	=	[4.93480\ldots\,]
[V_5\,]	=	[\frac\,]	=	[5.26379\ldots\,]
[V_6\,]	=	[\frac\,]	=	[5.16771\ldots\,]
[V_7\,]	=	[\frac\,]	=	[4.72478\ldots\,]
[V_8\,]	=	[\frac\,]	=	[4.05871\ldots\,]
[\lim_ V_n\,]			=	[0\,]

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...

The hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2ⁿ; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically 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 _]. 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_=r\sin(\phi_1)\cdots\sin(\phi_)\cos(\phi_)\,]

[x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_)\sin(\phi_)\,]

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

[d^nr = \left|\det\frac\right|dr\,d\phi_1 \, d\phi_2\ldots d\phi_]

[=r^\sin^(\phi_1)\sin^(\phi_2)\cdots \sin(\phi_)\,dr\,d\phi_1 \, d\phi_2\cdots d\phi_]

and the above equation for the volume of the hypersphere can be recovered by integrating:

[V_n=\int_^R \int_^\pi\cdots \int_=0}^\pi\int_=0}^d^nr. \,]

Stereographic projection

Just as a two dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-dimensional hypersphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. For example, the point [\ [x,y,z]] on a two-dimensional sphere of radius 1 maps to the point [\ [x,y,z] \mapsto \left[frac,fracright]] on the [\ xy] plane. In other words:

[\ [x,y,z] \mapsto \left[frac,fracright]]

Likewise, the stereographic projection of a hypersphere [\mathbb^] of radius 1 will map to the n-1 dimensional hyperplane [\mathbb^] perpendicular to the [\ x_n] axis as:

[[x_1,x_2,ldots,x_n] \mapsto \left[frac,frac,ldots,frac}right]]

Hypersphere

Hyperspherical volume

Hyperspherical volume - some examples

Hyperspherical coordinates

Stereographic projection

See also