A hypersphere is a higher-dimensional analogue of a sphere. A hypersphere of radius R in n-dimensional Euclidean space consists of all points at distance R from a given fixed point (the centre of the hypersphere).
The "volume" it encloses is
where Γ is the gamma function.
The "surface area" of this hypersphere is
The above hypersphere in n-dimensional Euclidean space is an example of an (n−1)-manifold. It is called an (n−1)-sphere and is denoted Sn−1. For example, an ordinary sphere in three dimensions is a 2-sphere.
The interior of a hypersphere, that is the set of all points whose distance from the centre is less than or equal to R, is called an hyperball.
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 {φ1,φ2...φn-1}. If xi are the Cartesian coordinates, then we may define
The hyperspherical volume element will be found from the Jacobian of the transformation:
and the above equation for the volume of the hypersphere can be recovered by integrating:
See also