N-sphere

In mathematics, an -sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer.
The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term hypersphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension, which means that they cannot be easily visualized. 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 from a given center point. Its interior, consisting of all points closer to the center than the radius, is an -dimensional ball. In particular:

The -sphere is the pair of points at the ends of a line segment.
The -sphere is a circle, the circumference of a disk in the two-dimensional plane.
The -sphere, often simply called a sphere, is the boundary of a -ball in three-dimensional space.
The -sphere is the boundary of a -ball in four-dimensional space.
The -sphere is the boundary of an -ball.

Given a Cartesian coordinate system, the unit -sphere of radius can be defined as:
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, is a model for the -sphere.
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 by forming the suspension of an -sphere. When it is simply connected; the -sphere is not simply connected; the -sphere is not even connected, consisting of two discrete points.

Description

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:

a 0-sphere is a pair of points, and is the boundary of a line segment.
a -sphere is a circle of radius centered at, and is the boundary of a disk.
a -sphere is an ordinary -dimensional sphere in -dimensional Euclidean space, and is the boundary of an ordinary ball.
a -sphere is a -dimensional sphere in -dimensional Euclidean space.
Cartesian coordinates

The set of points in -space,, that define an -sphere,, is represented by the equation:
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
where is the Hodge star operator; see for a discussion and proof of this formula in the case. As a result,

''n''-ball

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:

A -ball, a line segment, is the interior of a 0-sphere.
A -ball, a disk, is the interior of a circle.
A -ball, an ordinary ball, is the interior of a sphere.
A -ball is the interior of a -sphere, etc.
Topological description

, 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. This forms the basis for stereographic projection.

Volume and area

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.
File:hypersphere_volume_and_surface_area_graphs.svg|thumb|right|Graphs of volumes and surface areas of -balls of radius.
The -ball is sometimes defined as a single point. The -dimensional Hausdorff measure is the number of points in a set. So
A unit -ball is a line segment whose points have a single coordinate in the interval of length, and the -sphere consists of its two end-points, with coordinate.
A unit -sphere is the unit circle in the Euclidean plane, and its interior is the unit disk.
The interior of a 2-sphere in three-dimensional space is the unit -ball.
In general, and are given in closed form by the expressions
where is the gamma function. Note that 's values at half-integers contain a factor of that cancels out the factor in the numerator.
As tends to infinity, the volume of the unit -ball tends to zero.

Recurrences

The surface area, or properly the -dimensional volume, of the -sphere at the boundary of the -ball of radius is related to the volume of the ball by the differential equation
Equivalently, representing the unit -ball as a union of concentric -sphere shells,
We can also represent the unit -sphere as a union of products of a circle with an -sphere. Then. Since, the equation
holds for all. Along with the base cases, from above, these recurrences can be used to compute the surface area of any sphere or volume of any ball.

Spherical coordinates

We may define a coordinate system in an -dimensional Euclidean space which is analogous to the spherical coordinate system defined for -dimensional Euclidean space, in which the coordinates consist of a radial coordinate, and angular coordinates, where the angles range over radians and ranges over radians. If are the Cartesian coordinates, then we may compute from with:
Except in the special cases described below, the inverse transformation is unique:
where is the two-argument arctangent function.
There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case may be chosen to be zero.

Spherical volume and area elements

The arc length element isTo express the volume element of -dimensional Euclidean space in terms of spherical coordinates, let and for concision, then observe that the Jacobian matrix of the transformation is:
The determinant of this matrix can be calculated by induction. When, a straightforward computation shows that the determinant is. For larger, observe that can be constructed from as follows. Except in column, rows and of are the same as row of, but multiplied by an extra factor of in row and an extra factor of in row. In column, rows and of are the same as column of row of, but multiplied by extra factors of in row and in row, respectively. The determinant of can be calculated by Laplace expansion in the final column. By the recursive description of, the submatrix formed by deleting the entry at and its row and column almost equals, except that its last row is multiplied by. Similarly, the submatrix formed by deleting the entry at and its row and column almost equals, except that its last row is multiplied by. Therefore the determinant of is
Induction then gives a closed-form expression for the volume element in spherical coordinates
The formula for the volume of the -ball can be derived from this by integration.
Similarly the surface area element of the -sphere of radius, which generalizes the area element of the -sphere, is given by
The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,
for, and the for the angle in concordance with the spherical harmonics.

Polyspherical coordinates

The standard spherical coordinate system arises from writing as the product. These two factors may be related using polar coordinates. For each point of, the standard Cartesian coordinates
can be transformed into a mixed polar–Cartesian coordinate system:
This says that points in may be expressed by taking the ray starting at the origin and passing through, rotating it towards by, and traveling a distance along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.
Polyspherical coordinate systems arise from a generalization of this construction. The space is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that and are positive integers such that. Then. Using this decomposition, a point may be written as
This can be transformed into a mixed polar–Cartesian coordinate system by writing:
Here and are the unit vectors associated to and. This expresses in terms of,,, and an angle. It can be shown that the domain of is if, if exactly one of and is, and if neither nor are. The inverse transformation is
These splittings may be repeated as long as one of the factors involved has dimension two or greater. A polyspherical coordinate system is the result of repeating these splittings until there are no Cartesian coordinates left. Splittings after the first do not require a radial coordinate because the domains of and are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and angles. The possible polyspherical coordinate systems correspond to binary trees with leaves. Each non-leaf node in the tree corresponds to a splitting and determines an angular coordinate. For instance, the root of the tree represents, and its immediate children represent the first splitting into and. Leaf nodes correspond to Cartesian coordinates for. The formulas for converting from polyspherical coordinates to Cartesian coordinates may be determined by finding the paths from the root to the leaf nodes. These formulas are products with one factor for each branch taken by the path. For a node whose corresponding angular coordinate is, taking the left branch introduces a factor of and taking the right branch introduces a factor of. The inverse transformation, from polyspherical coordinates to Cartesian coordinates, is determined by grouping nodes. Every pair of nodes having a common parent can be converted from a mixed polar–Cartesian coordinate system to a Cartesian coordinate system using the above formulas for a splitting.
Polyspherical coordinates also have an interpretation in terms of the special orthogonal group. A splitting determines a subgroup
This is the subgroup that leaves each of the two factors fixed. Choosing a set of coset representatives for the quotient is the same as choosing representative angles for this step of the polyspherical coordinate decomposition.
In polyspherical coordinates, the volume measure on and the area measure on are products. There is one factor for each angle, and the volume measure on also has a factor for the radial coordinate. The area measure has the form:
where the factors are determined by the tree. Similarly, the volume measure is
Suppose we have a node of the tree that corresponds to the decomposition and that has angular coordinate. The corresponding factor depends on the values of and. When the area measure is normalized so that the area of the sphere is, these factors are as follows. If, then
If and, and if denotes the beta function, then
If and, then
Finally, if both and are greater than one, then