 nsphere

In mathematics, an nsphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an nsphere of radius r is defined as the set of points in (n + 1)dimensional Euclidean space which are at distance r from a central point, where the radius r may be any positive real number. In symbols:
It is an ndimensional manifold in Euclidean (n + 1)space. In particular, a 0sphere is a pair of points that are the ends of a line segment, a 1sphere is a circle in the plane, and a 2sphere is an ordinary sphere in threedimensional space. Spheres of dimension n > 2 are sometimes called hyperspheres, with 3spheres sometimes known as glomes. The nsphere of unit radius centered at the origin is called the unit nsphere, denoted S^{n}. The unit nsphere is often referred to as the nsphere. An nsphere is the surface or boundary of an (n + 1)dimensional ball, and is an ndimensional manifold. For n ≥ 2, the nspheres are the simply connected ndimensional manifolds of constant, positive curvature. The nspheres admit several other topological descriptions: for example, they can be constructed by gluing two ndimensional Euclidean spaces together, by identifying the boundary of an ncube with a point, or (inductively) by forming the suspension of an (n − 1)sphere.
Contents
Description
For any natural number n, an nsphere of radius r is defined as the set of points in (n + 1)dimensional Euclidean space that are at distance r from some fixed point c, where r may be any positive real number and where c may be any point in (n + 1)dimensional space. In particular:
 a 0sphere is a pair of points {c − r, c + r}, and is the boundary of a line segment (1ball).
 a 1sphere is a circle of radius r centered at c, and is the boundary of a disk (2ball).
 a 2sphere is an ordinary 2dimensional sphere in 3dimensional Euclidean space, and is the boundary of an ordinary ball (3ball).
 a 3sphere is a sphere in 4dimensional Euclidean space.
Euclidean coordinates in (n + 1)space
The set of points in (n + 1)space: (x_{1},x_{1},x_{2},…,x_{n+1}) that define an nsphere, (S^{n}) is represented by the equation:
where c is a center point, and r is the radius.
The above nsphere exists in (n + 1)dimensional Euclidean space and is an example of an nmanifold. The volume form ω of an nsphere of radius r is given by
where * is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case r = 1. As a result,
nball

Main article: ball (mathematics)
The space enclosed by an nsphere is called an (n + 1)ball. An (n + 1)ball is closed if it includes the nsphere, and it is open if it does not include the nsphere.
Specifically:
 A 1ball, a line segment, is the interior of a (0sphere).
 A 2ball, a disk, is the interior of a circle (1sphere).
 A 3ball, an ordinary ball, is the interior of a sphere (2sphere).
 A 4ball, is the interior of a 3sphere, etc.
Topological description
Topologically, an nsphere can be constructed as a onepoint compactification of ndimensional Euclidean space. Briefly, the nsphere can be described as , which is ndimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an nsphere, it becomes homeomorphic to . This forms the basis for stereographic projection. ^{[1]}
Volume and surface area
The nvolume of an nsphere of radius R or, equivalently, the surface area of an (n + 1)ball of radius R is:
The nvolume of a nball of radius R:
The 1sphere of radius R is the circle of radius R in the Euclidean plane, and this has circumference (1dimensional measure)
The region enclosed by the 1sphere is the 2ball, or disk of radius R, and this has area (2dimensional measure)
Analogously, in 3dimensional Euclidean space, the surface area (2dimensional measure) of the 2sphere of radius R is given by
and the volume enclosed is the volume (3dimensional measure) of the 3ball, and is given by
In general, the volume, in ndimensional Euclidean space, of the nball of radius R is proportional to the n^{th} power of the R:
where the constant of proportionality, the volume of the unit nball, is given by
where is the gamma function. For even n, this reduces to
and since
for odd n,
where n!! denotes the double factorial.
The "surface area", or properly the (n − 1)dimensional volume, of the (n−1)sphere at the boundary of the nball is
The following relationships hold between the nspherical surface area and volume:
This leads to the recurrence relations:
The recurrence relation for V_{n}(R) can be proved via integration with 2dimensional polar coordinates:
Hyperspherical coordinates
We may define a coordinate system in an ndimensional Euclidean space which is analogous to the spherical coordinate system defined for 3dimensional Euclidean space, in which the coordinates consist of a radial coordinate, and n − 1 angular coordinates where ranges over radians (or over [0, 360) degrees) and the other angles range over radians (or over [0, 180] degrees). If are the Cartesian coordinates, then we may compute from with:
Except in the special cases described below, the inverse transformation is unique:
where if for some k but all of are zero then ϕ_{k} = 0 when x_{k} > 0, and ϕ_{k} = π radians (180 degrees) when x_{k} < 0.
There are some special cases where the inverse transform is not unique; φ_{k} for any k will be ambiguous whenever all of are zero; in this case φ_{k} may be chosen to be zero.
Note that a halfangle formula is used for ϕ_{n − 1} because the more straightforward is too small by an addend of π when x_{n} < 0.
Hyperspherical volume element
Expressing the angular measures in radians, the volume element in ndimensional Euclidean space will be found from the Jacobian of the transformation:
and the above equation for the volume of the nball can be recovered by integrating:
The volume element of the (n1)–sphere, which generalizes the area element of the 2sphere, is given by
The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,
for j = 1, 2, ..., n − 2, and the e^{ isφj} for the angle j = n − 1 in concordance with the spherical harmonics.
Stereographic projection

Main article: Stereographic projection
Just as a two dimensional sphere embedded in three dimensions can be mapped onto a twodimensional plane by a stereographic projection, an nsphere can be mapped onto an ndimensional hyperplane by the ndimensional version of the stereographic projection. For example, the point on a twodimensional sphere of radius 1 maps to the point on the plane. In other words,
Likewise, the stereographic projection of an nsphere of radius 1 will map to the n − 1 dimensional hyperplane perpendicular to the axis as
Generating random points
Uniformly at random from the (n − 1)sphere
To generate uniformly distributed random points on the (n − 1)sphere (i.e., the surface of the nball), Marsaglia (1972) gives the following algorithm.
Generate an ndimensional vector of normal deviates (it suffices to use N(0, 1), although in fact the choice of the variance is arbitrary), .
Now calculate the "radius" of this point, .
The vector is uniformly distributed over the surface of the unit nball.
Examples
For example, when n = 2 the normal distribution exp(−x_{1}^{2}) when expanded over another axis exp(−x_{2}^{2}) after multiplication takes the form exp(−x_{1}^{2}−x_{2}^{2}) or exp(−r^{2}) and so is only dependent on distance from the origin.
Alternatives
Another way to generate a random distribution on a hypersphere is to make a uniform distribution over a hypercube that includes the unit hyperball, exclude those points that are outside the hyperball, then project the remaining interior points outward from the origin onto the surface. This will give a uniform distribution, but it is necessary to remove the exterior points. As the relative volume of the hyperball to the hypercube decreases very rapidly with dimension, this procedure will succeed with high probability only for fairly small numbers of dimensions.
Wendel's theorem gives the probability that all of the points generated will lie in the same half of the hypersphere.
Uniformly at random from the nball
With a point selected from the surface of the nball uniformly at random, one needs only a radius to obtain a point uniformly at random within the nball. If u is a number generated uniformly at random from the interval [0, 1] and x is a point selected uniformly at random from the surface of the nball then u^{1/n}x is uniformly distributed over the entire unit nball.
Specific spheres
 0sphere
 The pair of points {±R} with the discrete topology for some R > 0. The only sphere that is disconnected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
 1sphere
 Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Topologically equivalent to the real projective line, RP^{1}. Parallelizable. SO(2) = U(1).
 2sphere
 Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP^{1}. SO(3)/SO(2).
 3sphere
 Parallelizable, Principal U(1)bundle over the 2sphere, Lie group structure Sp(1), where also
 .
 4sphere
 Equivalent to the quaternionic projective line, HP^{1}. SO(5)/SO(4).
 5sphere
 Principal U(1)bundle over CP^{2}. SO(6)/SO(5) = SU(3)/SU(2).
 6sphere
 Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G_{2}/SU(3).
 7sphere
 Topological quasigroup structure as the set of unit octonions. Principal Sp(1)bundle over S^{4}. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/G_{2} = Spin(6)/SU(3). The 7sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
 8sphere
 Equivalent to the octonionic projective line OP^{1}.
 23sphere
 A highly dense spherepacking is possible in 24 dimensional space, which is related to the unique qualities of the Leech lattice.
See also
 Affine sphere
 Conformal geometry
 Deriving the volume of an nball
 Inversive geometry
 Homology sphere
 Homotopy groups of spheres
 Homotopy sphere
 Hyperbolic group
 Hypercube
 Loop (topology)
 Manifold
 Möbius transformation
 Orthogonal group
 Spherical cap
Notes
 ^ James W. Vick (1994). Homology theory, p. 60. Springer
References
 Flanders, Harley (1989). Differential forms with applications to the physical sciences. New York: Dover Publications. ISBN 9780486661698..
 Moura, Eduarda; Henderson, David G. (1996). Experiencing geometry: on plane and sphere. Prentice Hall. ISBN 9780133737707. http://www.math.cornell.edu/~henderson/books/eg00 (Chapter 20: 3spheres and hyperbolic 3spaces.)
 Weeks, Jeffrey R. (1985). The Shape of Space: how to visualize surfaces and threedimensional manifolds. Marcel Dekker. ISBN 9780824774370 (Chapter 14: The Hypersphere)
 Marsaglia, G. (1972). "Choosing a Point from the Surface of a Sphere". Ann. Math. Stat. 43 (2): 645–646. doi:10.1214/aoms/1177692644.
 Huber, Greg (1982). "Gamma function derivation of nsphere volumes". Am. Math. Monthly 89 (5): 301–302. doi:10.2307/2321716. JSTOR 2321716. MR1539933.
External links
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