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2-sphere wireframe as an orthogonal projection
2-sphere wireframe as an orthogonal projection
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere's surface into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).
Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere's surface into 3-space. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). This article is about the Perimetry concept For other uses of the word see Meridian. Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. In Mathematics, a conformal map is a function which preserves Angles In the most common case the function is between domains in the Complex plane All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line).

In mathematics, an n-sphere is a generalization of an ordinary sphere to arbitrary dimension. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it For any natural number n, an n-sphere of radius r is defined 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 Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the real numbers may be described informally in several different ways It is an n-dimensional manifold in Euclidean (n + 1)-space. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In particular, a 0-sphere is a pair of points on a line, a 1-sphere is a circle in the plane, and a 2-sphere is an ordinary sphere in three dimensional space. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Spheres of dimension n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centered at the origin is called the unit n-sphere, denoted Sn. The unit n-sphere is often referred to as the n-sphere. In symbols:

S^n = \left\{ x \in \mathbb{R}^{n+1} : \|x\| = 1\right\}.

An n-sphere is the surface or boundary of an (n + 1)-dimensional ball, and is an n-dimensional manifold. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be For n ≥ 2, the n-spheres are the simply connected n-dimensional manifold of constant, positive curvature. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere. In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3 In Topology, the suspension SX of a Topological space X is the Quotient space: SX = (X \times I/\{(x_10\sim(x_20\mbox{

Contents

Description

For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space which are at distance r from a fixed point, where r may be any positive real number. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, the real numbers may be described informally in several different ways In particular:

Euclidean coordinates in (n + 1)-space

The set of points in (n + 1)-space: (x_1,x_2,x_3,\dots,x_{n+1}) that define an n-sphere, (\mathbf S^n) is represented by the equation:

r^2=\sum_{i=1}^{n+1} (x_i - C_i)^2.\,

where C is a center point, and r is the radius.

The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

The volume element ω of n-sphere of radius r is given by

\omega = {1 \over r} \sum_{j=1}^{n+1} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_j \cdots \wedge dx_{n+1}

In fact, dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n+1}

n-ball

The space enclosed by an n-sphere is called an (n + 1)-ball. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric An (n + 1)-ball is closed if it included the equality, and open otherwise. In Topology and related branches of Mathematics, a closed set is a set whose complement is open. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in

Specifically:

Notation

Labelling n-spheres with the dimensionality of the surface (as used in this article) is the convention common in mathematical use. Potentially confusingly, some authors use the dimensionality of the containing space to label n-spheres. [1] Thus what most call a 1-sphere (a regular circle in a plane), others term a 2-sphere (reflecting the dimensionality of the plane in which it lies). Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

Volume of the n-ball

The hyperdimensional volume of the space which a (n − 1)-sphere encloses (the n-ball) is given by

V_n={\pi^\frac{n}{2}R^n\over\Gamma(\frac{n}{2} + 1)}={C_n R^n},

where Γ is the gamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function (For even n, \Gamma\left(\frac{n}{2}+1\right)= \left(\frac{n}{2}\right)!; for odd n, \Gamma\left(\frac{n}{2}+1\right)= \sqrt{\pi} \frac{n!!}{2^{(n+1)/2}}, where n!! denotes the double factorial. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k )

From this, it follows that the value of the constant Cn for a given n is:

C_n={\frac{\pi^k}{k!}}, for even n=2k, and
C_n=C_{2k+1}=\frac{2^{2k+1} k!\, \pi^{k}}{(2k+1)!} for odd n=2k+1.

The "surface area" of this (n-1)-sphere is

S_{n-1}=\frac{dV_n}{dR}=\frac{nV_n}{R}={2\pi^\frac{n}{2}R^{n-1}\over\Gamma(\frac{n}{2})}={n C_n R^{n-1}}

The following relationships hold between the n-spherical surface area and volume:

V_n/S_{n-1} = R/n\,
S_{n+1}/V_n = 2\pi R\,

This leads to the recurrence relation:

V_n = \frac{2 \pi R^2}{n} V_{n-2}\,

The interior of an n-sphere, the set of all points whose distance from the center is less than R, is called a hyperball, or if the n-sphere itself is included (that is, the set of all points whose distance from the center is less than or equal to R), a closed hyperball.

Examples

For small values of n, the volumes, Vn , of the n-ball of radius R are:

V_0\, (point) = 1\,    
V_1\, (line segment) = 2\,R    
V_2\, (disk) = \pi\,R^2 = 3.14159\ldots\,R^2
V_3\, (ball) = \frac{4 \pi}{3}\,R^3 = 4.18879\ldots\,R^3
V_4\, = \frac{\pi^2}{2}\,R^4 = 4.93480\ldots\,R^4
V_5\, = \frac{8 \pi^2}{15}\,R^5 = 5.26379\ldots\,R^5
V_6\, = \frac{\pi^3}{6}\,R^6 = 5.16771\ldots\,R^6
V_7\, = \frac{16 \pi^3}{105}\,R^7 = 4.72477\ldots\,R^7
V_8\, = \frac{\pi^4}{24}\,R^8 = 4.05871\ldots\,R^8
\lim_{n\rightarrow\infty} \frac{V_n}{R^n}\, = 0\,

If the dimension n is not limited to integral values, the n-sphere volume is a continuous function of n with a global maximum for the unit sphere in "dimension" n = 5. In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points In Geometry, a disk (also spelled disc) is the region in a plane bounded by a Circle. In Mathematics, a ball is the inside of a Sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions and for metric In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output In Mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum or smallest value (minimum that 2569464. . . where the "volume" is 5. 277768. . . It has a hypervolume of 1 when n = 0 or when n  = 12. 76405. . .

The hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2n; the ratio of the volume of the n-sphere to its circumscribed hypercube decreases monotonically as the dimension increases. In Geometry, a hypercube is an n -dimensional analogue of a square ( n = 2 and a Cube ( n = 3

The non-monotonic behaviour of the numerical value of n-spheres as a function of n may seem strange at first glance. However, by assigning units of length to each dimension one can see it is meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area in other contexts. A meaningful comparison is obtained by using a dimensionless measure of the volume, such as the ratio of the n-sphere and its circumscribed hypercube volumes. Using this measure restores the intuitively normal behavior of a monotonic decline in the volume as the dimension increases.

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 \ \phi _1 , \phi _2 , ... , \phi _{n-1}. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial If \ x_i are the Cartesian coordinates, then we may define

x_1=r\cos(\phi_1)\,
x_2=r\sin(\phi_1)\cos(\phi_2)\,
x_3=r\sin(\phi_1)\sin(\phi_2)\cos(\phi_3)\,
\cdots\,
x_{n-1}=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\cos(\phi_{n-1})\,
x_n~~\,=r\sin(\phi_1)\cdots\sin(\phi_{n-2})\sin(\phi_{n-1})\,

While the inverse transformations can be derived from those above:

\tan(\phi_{n-1})=\frac{x_n}{x_{n-1}}
\tan(\phi_{n-2})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2}}{x_{n-2}}
\cdots\,
\tan(\phi_{1})=\frac{\sqrt{{x_n}^2+{x_{n-1}}^2+\cdots+{x_2}^2}}{x_{1}}

Note that last angle φn − 1 has a range of 2π while the other angles have a range of π. This range covers the whole sphere.

The volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation:

d_{\mathbb{R}^n}V = \left|\det\frac{\partial (x_i)}{\partial(r,\phi_j)}\right| dr\,d\phi_1 \, d\phi_2\ldots d\phi_{n-1}
=r^{n-1}\sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, dr\,d\phi_1 \, d\phi_2\cdots d\phi_{n-1}

and the above equation for the volume of the n-ball can be recovered by integrating:

V_n=\int_{r=0}^R \int_{\phi_1=0}^\pi \cdots \int_{\phi_{n-2}=0}^\pi\int_{\phi_{n-1}=0}^{2\pi}d_{\mathbb{R}^n}V. \,

The volume element of the (n-1)–sphere, which generalizes the area element of the 2-sphere, is given by

d_{S^{n-1}}V = \sin^{n-2}(\phi_1)\sin^{n-3}(\phi_2)\cdots \sin(\phi_{n-2})\, d\phi_1 \, d\phi_2\ldots d\phi_{n-1}

Stereographic projection

Just as a two dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-sphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant. In Mathematics, a volume form is a nowhere zero differential ''n''-form on an n - Manifold. In Geometry, the stereographic projection is a particular mapping ( function) that projects a Sphere onto a plane For example, the point \ [x,y,z] on a two-dimensional sphere of radius 1 maps to the point \ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right] on the \ xy plane. In other words:

\ [x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].

Likewise, the stereographic projection of an n-sphere \mathbf{S}^{n-1} of radius 1 will map to the n-1 dimensional hyperplane \mathbf{R}^{n-1} perpendicular to the \ x_n axis as:

[x_1,x_2,\ldots,x_n] \mapsto \left[\frac{x_1}{1-x_n},\frac{x_2}{1-x_n},\ldots,\frac{x_{n-1}}{1-x_n}\right].

Generating points on the surface of the n-ball

To generate points on the surface of the n ball, Marsaglia (1972) gives the following algorithm.

Generate an n-dimensional vector of Normal deviates (it suffices to use N(0,1), although in fact the choice of the variance is arbitrary), \mathbf{x}=(x_1,x_2,\ldots,x_n).

Now calculate the "radius" of this point, r=\sqrt{x_1^2+x_2^2+\ldots+x_n^2}.

The vector \frac1r \mathbf{x} is uniformly distributed over the surface of the n-ball.

See also

References

External links

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