In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The -sphere is the setting for -dimensional spherical geometry.
Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an -dimensional ball. In particular:
Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
Sn=\left\{x\in\Rn+1:\left\|x\right\|=1\right\}.
Considered intrinsically, when, the -sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the -sphere are called great circles.
The stereographic projection maps the -sphere onto -space with a single adjoined point at infinity; under the metric thereby defined,
\Rn\cup\{infty\}
In the more general setting of topology, any topological space that is homeomorphic to the unit -sphere is called an -sphere. Under inverse stereographic projection, the -sphere is the one-point compactification of -space. The -spheres admit several other topological descriptions: for example, they can be constructed by gluing two -dimensional spaces together, by identifying the boundary of an -cube with a point, or (inductively) by forming the suspension of an -sphere. When it is simply connected; the -sphere (circle) is not simply connected; the -sphere is not even connected, consisting of two discrete points.
For any natural number, an -sphere of radius is defined as the set of points in -dimensional Euclidean space that are at distance from some fixed point, where may be any positive real number and where may be any point in -dimensional space. In particular:
The set of points in -space,, that define an -sphere,, is represented by the equation:
| n+1 | |
| r | |
| i=1 |
(xi-
| 2 | |
| c | |
| i) |
,
where is a center point, and is the radius.
The above -sphere exists in -dimensional Euclidean space and is an example of an -manifold. The volume form of an -sphere of radius is given by
\omega=
| 1 | |
| r |
| n+1 | |
| \sum | |
| j=1 |
(-1)j-1xjdx1\wedge … \wedgedxj-1\wedgedxj+1\wedge … \wedgedxn+1={\star}dr
where
{\star}
dr\wedge\omega=dx1\wedge … \wedgedxn+1.
See main article: Ball (mathematics).
The space enclosed by an -sphere is called an -ball. An -ball is closed if it includes the -sphere, and it is open if it does not include the -sphere.
Specifically:
Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as, which is -dimensional Euclidean space plus a single point representing infinity in all directions.In particular, if a single point is removed from an -sphere, it becomes homeomorphic to
\Rn
See also: Volume of an n-ball.
Let be the surface area of the unit -sphere of radius embedded in -dimensional Euclidean space, and let be the volume of its interior, the unit -ball. The surface area of an arbitrary -sphere is proportional to the st power of the radius, and the volume of an arbitrary -ball is proportional to the th power of the radius.