n-sphere

The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they exist as shapes in 1- and 2-dimensional space.

Its interior, consisting of all points closer to the center than the radius, is an ⁠

⁠-sphere is a Riemannian manifold of positive constant curvature, and is orientable.

⁠-space with a single adjoined point at infinity; under the metric thereby defined,

is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case ⁠

⁠-dimensional Euclidean space plus a single point representing infinity in all directions.

⁠ is related to the volume of the ball by the differential equation Equivalently, representing the unit ⁠

⁠ from above, these recurrences can be used to compute the surface area of any sphere or volume of any ball.

⁠-dimensional Euclidean space which is analogous to the spherical coordinate system defined for ⁠

⁠ with:[3][a] Except in the special cases described below, the inverse transformation is unique: where atan2 is the two-argument arctangent function.

⁠ then the point is one of the poles, zenith or nadir, and the choice of azimuthal angle is arbitrary.)

⁠-sphere, is given by The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials, for ⁠

⁠ may be expressed by taking the ray starting at the origin and passing through

Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction.

The inverse transformation is These splittings may be repeated as long as one of the factors involved has dimension two or greater.

The possible polyspherical coordinate systems correspond to binary trees with ⁠

Each non-leaf node in the tree corresponds to a splitting and determines an angular coordinate.

Polyspherical coordinates also have an interpretation in terms of the special orthogonal group.

Similarly, the volume measure is Suppose we have a node of the tree that corresponds to the decomposition ⁠

⁠-axis as To generate uniformly distributed random points on the unit ⁠

An alternative given by Marsaglia is to uniformly randomly select a point ⁠

⁠-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor ⁠

This method becomes very inefficient for higher dimensions, as a vanishingly small fraction of the unit cube is contained in the sphere.

In ten dimensions, less than 2% of the cube is filled by the sphere, so that typically more than 50 attempts will be needed.

of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.

With a point selected uniformly at random from the surface of the unit ⁠

⁠-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit ⁠

This is one of the phenomena leading to the so-called curse of dimensionality that arises in some numerical and other applications.

⁠ be the square of the first coordinate of a point sampled uniformly at random from the ⁠