N-sphere

From Wikipedia for FEVERv2
Jump to navigation Jump to search

In mathematics, an n-sphere is a topological space that is homeomorphic to a standard n-sphere, which is the set of points in (n + 1)-dimensional Euclidean space that are situated at a constant distance r from a fixed point, called the center. N-sphere_sentence_0

It is the generalization of an ordinary sphere in the ordinary three-dimensional space. N-sphere_sentence_1

The "radius" of a sphere is the constant distance of its points to the center. N-sphere_sentence_2

When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity. N-sphere_sentence_3

In terms of the standard norm, the n-sphere is defined as N-sphere_sentence_4

and an n-sphere of radius r can be defined as N-sphere_sentence_5

The dimension of n-sphere is n, and must not be confused with the dimension (n + 1) of the Euclidean space in which it is naturally embedded. N-sphere_sentence_6

An n-sphere is the surface or boundary of an (n + 1)-dimensional ball. N-sphere_sentence_7

In particular: N-sphere_sentence_8

N-sphere_unordered_list_0

  • the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere,N-sphere_item_0_0
  • a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere,N-sphere_item_0_1
  • the two-dimensional surface of a (three-dimensional) ball in three-dimensional space is a 2-sphere, often simply called a sphere,N-sphere_item_0_2
  • the three-dimensional boundary of a (four-dimensional) 4-ball in four-dimensional Euclidean is a 3-sphere, also known as a glome.N-sphere_item_0_3
  • the n – 1 dimensional boundary of a (n-dimensional) n-ball is an (n – 1)-sphere.N-sphere_item_0_4

For n ≥ 2, the n-spheres that are differential manifolds can be characterized (up to a diffeomorphism) as the simply connected n-dimensional manifolds of constant, positive curvature. N-sphere_sentence_9

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. N-sphere_sentence_10

The 1-sphere is the 1-manifold that is a circle, which is not simply connected. N-sphere_sentence_11

The 0-sphere is the 0-manifold consisting of two points, which is not even connected. N-sphere_sentence_12

Description N-sphere_section_0

For any natural number n, an n-sphere 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. N-sphere_sentence_13

In particular: N-sphere_sentence_14

N-sphere_unordered_list_1

  • a 0-sphere is a pair of points {c − r, c + r}, and is the boundary of a line segment (1-ball).N-sphere_item_1_5
  • a 1-sphere is a circle of radius r centered at c, and is the boundary of a disk (2-ball).N-sphere_item_1_6
  • a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).N-sphere_item_1_7
  • a 3-sphere is a 3-dimensional sphere in 4-dimensional Euclidean space.N-sphere_item_1_8

Euclidean coordinates in (n + 1)-space N-sphere_section_1

where c = (c1, c2, ..., cn+1) is a center point, and r is the radius. N-sphere_sentence_15

The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold. N-sphere_sentence_16

The volume form ω of an n-sphere of radius r is given by N-sphere_sentence_17

where ∗ is the Hodge star operator; see , §6.1) for a discussion and proof of this formula in the case r = 1. N-sphere_sentence_18

As a result, N-sphere_sentence_19

n-ball N-sphere_section_2

Main article: Ball (mathematics) N-sphere_sentence_20

The space enclosed by an n-sphere is called an (n + 1)-ball. N-sphere_sentence_21

An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere. N-sphere_sentence_22

Specifically: N-sphere_sentence_23

N-sphere_unordered_list_2

  • A 1-ball, a line segment, is the interior of a 0-sphere.N-sphere_item_2_9
  • A 2-ball, a disk, is the interior of a circle (1-sphere).N-sphere_item_2_10
  • A 3-ball, an ordinary ball, is the interior of a sphere (2-sphere).N-sphere_item_2_11
  • A 4-ball is the interior of a 3-sphere, etc.N-sphere_item_2_12

Topological description N-sphere_section_3

Topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. N-sphere_sentence_24

Briefly, the n-sphere can be described as S = R ∪ {∞}, which is n-dimensional Euclidean space plus a single point representing infinity in all directions. N-sphere_sentence_25

In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to R. This forms the basis for stereographic projection. N-sphere_sentence_26

Volume and surface area N-sphere_section_4

See also: Volume of an n-ball N-sphere_sentence_27

In theory, one could compare the values of Sn(R) and Sm(R) for n ≠ m. However, this is not well-defined. N-sphere_sentence_28

For example, if n = 2 and m = 3 then the comparison is like comparing a number of square meters to a different number of cubic meters. N-sphere_sentence_29

The same applies to a comparison of Vn(R) and Vm(R) for n ≠ m. N-sphere_sentence_30

Examples N-sphere_section_5

The 0-ball consists of a single point. N-sphere_sentence_31

The 0-dimensional Hausdorff measure is the number of points in a set. N-sphere_sentence_32

So, N-sphere_sentence_33

The 0-sphere consists of its two end-points, {−1,1}. N-sphere_sentence_34

So, N-sphere_sentence_35

The unit 1-ball is the interval [−1,1] of length 2. N-sphere_sentence_36

So, N-sphere_sentence_37

The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference (1-dimensional measure) N-sphere_sentence_38

The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure) N-sphere_sentence_39

Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by N-sphere_sentence_40

and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by N-sphere_sentence_41

Recurrences N-sphere_section_6

The surface area, or properly the n-dimensional volume, of the n-sphere at the boundary of the (n + 1)-ball of radius R is related to the volume of the ball by the differential equation N-sphere_sentence_42

or, equivalently, representing the unit n-ball as a union of concentric (n − 1)-sphere shells, N-sphere_sentence_43

So, N-sphere_sentence_44

We can also represent the unit (n + 2)-sphere as a union of tori, each the product of a circle (1-sphere) with an n-sphere. N-sphere_sentence_45

Let r = cos θ and r + R = 1, so that R = sin θ and dR = cos θ dθ. N-sphere_sentence_46

Then, N-sphere_sentence_47

Since S1 = 2π V0, the equation N-sphere_sentence_48

holds for all n. N-sphere_sentence_49

This completes the derivation of the recurrences: N-sphere_sentence_50

Closed forms N-sphere_section_7

Combining the recurrences, we see that N-sphere_sentence_51

So it is simple to show by induction on k that, N-sphere_sentence_52

where !! N-sphere_sentence_53

denotes the double factorial, defined for odd natural numbers 2k + 1 by (2k + 1)!! N-sphere_sentence_54

= 1 × 3 × 5 × ... × (2k − 1) × (2k + 1) and similarly for even numbers (2k)!! N-sphere_sentence_55

= 2 × 4 × 6 × ... × (2k − 2) × (2k). N-sphere_sentence_56

In general, the volume, in n-dimensional Euclidean space, of the unit n-ball, is given by N-sphere_sentence_57

where Γ is the gamma function, which satisfies Γ(1/2) = √π, Γ(1) = 1, and Γ(x + 1) = xΓ(x), and so Γ(x + 1) = x!, and where we conversely define x! N-sphere_sentence_58

= Γ(x + 1) for any x. N-sphere_sentence_59

By multiplying Vn by R, differentiating with respect to R, and then setting R = 1, we get the closed form N-sphere_sentence_60

for the (n-1)-dimensional volume of the sphere S. N-sphere_sentence_61

Other relations N-sphere_section_8

The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram: N-sphere_sentence_62

Index-shifting n to n − 2 then yields the recurrence relations: N-sphere_sentence_63

where S0 = 2, V1 = 2, S1 = 2π and V2 = π. N-sphere_sentence_64

The recurrence relation for Vn can also be proved via integration with 2-dimensional polar coordinates: N-sphere_sentence_65

Spherical coordinates N-sphere_section_9

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 φ1, φ2, ... φn−1, where the angles φ1, φ2, ... φn−2 range over [0,π] radians (or over [0,180] degrees) and φn−1 ranges over [0,2π) radians (or over [0,360) degrees). N-sphere_sentence_66

If xi are the Cartesian coordinates, then we may compute x1, ... xn from r, φ1, ... φn−1 with: N-sphere_sentence_67

Except in the special cases described below, the inverse transformation is unique: N-sphere_sentence_68

where if xk ≠ 0 for some k but all of xk+1, ... xn are zero then φk = 0 when xk > 0, and φk = π (180 degrees) when xk < 0. N-sphere_sentence_69

There are some special cases where the inverse transform is not unique; φk for any k will be ambiguous whenever all of xk, xk+1, ... xn are zero; in this case φk may be chosen to be zero. N-sphere_sentence_70

Spherical volume and area elements N-sphere_section_10

To express the volume element of n-dimensional Euclidean space in terms of spherical coordinates, first observe that the Jacobian matrix of the transformation is: N-sphere_sentence_71

The determinant of this matrix can be calculated by induction. N-sphere_sentence_72

When n = 2, a straightforward computation shows that the determinant is r. For larger n, observe that Jn can be constructed from Jn − 1 as follows. N-sphere_sentence_73

Except in column n, rows n − 1 and n of Jn are the same as row n − 1 of Jn − 1, but multiplied by an extra factor of cos φn − 1 in row n − 1 and an extra factor of sin φn − 1 in row n. In column n, rows n − 1 and n of Jn are the same as column n − 1 of row n − 1 of Jn − 1, but multiplied by extra factors of sin φn − 1 in row n − 1 and cos φn − 1 in row n, respectively. N-sphere_sentence_74

The determinant of Jn can be calculated by Laplace expansion in the final column. N-sphere_sentence_75

By the recursive description of Jn, the submatrix formed by deleting the entry at (n − 1, n) and its row and column almost equals Jn − 1, except that its last row is multiplied by sin φn − 1. N-sphere_sentence_76

Similarly, the submatrix formed by deleting the entry at (n, n) and its row and column almost equals Jn − 1, except that its last row is multiplied by cos φn − 1. N-sphere_sentence_77

Therefore the determinant of Jn is N-sphere_sentence_78

Induction then gives a closed-form expression for the volume element in spherical coordinates N-sphere_sentence_79

The formula for the volume of the n-ball can be derived from this by integration. N-sphere_sentence_80

Similarly the surface area element of the (n − 1)-sphere of radius R, which generalizes the area element of the 2-sphere, is given by N-sphere_sentence_81

The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials, N-sphere_sentence_82

for j = 1, 2,... n − 2, and the e for the angle j = n − 1 in concordance with the spherical harmonics. N-sphere_sentence_83

Polyspherical coordinates N-sphere_section_11

The standard spherical coordinate system arises from writing R as the product R × R. These two factors may be related using polar coordinates. N-sphere_sentence_84

For each point x of R, the standard Cartesian coordinates N-sphere_sentence_85

can be transformed into a mixed polar–Cartesian coordinate system: N-sphere_sentence_86

This says that points in R may be expressed by taking the ray starting at the origin and passing through z ∈ R, rotating it towards the first basis vector by θ, and traveling a distance r along the ray. N-sphere_sentence_87

Repeating this decomposition eventually leads to the standard spherical coordinate system. N-sphere_sentence_88

Polyspherical coordinate systems arise from a generalization of this construction. N-sphere_sentence_89

The space R is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. N-sphere_sentence_90

Specifically, suppose that p and q are positive integers such that n = p + q. N-sphere_sentence_91

Then R = R × R. Using this decomposition, a point x ∈ R may be written as N-sphere_sentence_92

This can be transformed into a mixed polar–Cartesian coordinate system by writing: N-sphere_sentence_93

Polyspherical coordinates also have an interpretation in terms of the special orthogonal group. N-sphere_sentence_94

A splitting R = R × R determines a subgroup N-sphere_sentence_95

In polyspherical coordinates, the volume measure on R and the area measure on S are products. N-sphere_sentence_96

There is one factor for each angle, and the volume measure on R also has a factor for the radial coordinate. N-sphere_sentence_97

The area measure has the form: N-sphere_sentence_98

where the factors Fi are determined by the tree. N-sphere_sentence_99

Similarly, the volume measure is N-sphere_sentence_100

Suppose we have a node of the tree that corresponds to the decomposition R = R × R and that has angular coordinate θ. N-sphere_sentence_101

The corresponding factor F depends on the values of n1 and n2. N-sphere_sentence_102

When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. N-sphere_sentence_103

If n1 = n2 = 1, then N-sphere_sentence_104

If n1 > 1 and n2 = 1, and if B denotes the beta function, then N-sphere_sentence_105

If n1 = 1 and n2 > 1, then N-sphere_sentence_106

Finally, if both n1 and n2 are greater than one, then N-sphere_sentence_107

Stereographic projection N-sphere_section_12

Main article: Stereographic projection N-sphere_sentence_108

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. N-sphere_sentence_109

For example, the point [x,y,z] on a two-dimensional sphere of radius 1 maps to the point [x/1 − z,y/1 − z] on the xy-plane. N-sphere_sentence_110

In other words, N-sphere_sentence_111

Likewise, the stereographic projection of an n-sphere S of radius 1 will map to the (n − 1)-dimensional hyperplane R perpendicular to the xn-axis as N-sphere_sentence_112

Generating random points N-sphere_section_13

Uniformly at random on the (n − 1)-sphere N-sphere_section_14

To generate uniformly distributed random points on the unit (n − 1)-sphere (that is, the surface of the unit n-ball), gives the following algorithm. N-sphere_sentence_113

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), x = (x1, x2,... xn). N-sphere_sentence_114

Now calculate the "radius" of this point: N-sphere_sentence_115

The vector 1/rx is uniformly distributed over the surface of the unit n-ball. N-sphere_sentence_116

Uniformly at random within the n-ball N-sphere_section_15

With a point selected uniformly at random from the surface of the unit (n - 1)-sphere (e.g., by using Marsaglia's algorithm), one needs only a radius to obtain a point uniformly at random from within the unit n-ball. N-sphere_sentence_117

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 unit (n - 1)-sphere, then ux is uniformly distributed within the unit n-ball. N-sphere_sentence_118

Alternatively, points may be sampled uniformly from within the unit n-ball by a reduction from the unit (n + 1)-sphere. N-sphere_sentence_119

In particular, if (x1,x2,...,xn+2) is a point selected uniformly from the unit (n + 1)-sphere, then (x1,x2,...,xn) is uniformly distributed within the unit n-ball (i.e., by simply discarding two coordinates). N-sphere_sentence_120

If n is sufficiently large, most of the volume of the n-ball will be contained in the region very close to its surface, so a point selected from that volume will also probably be close to the surface. N-sphere_sentence_121

This is one of the phenomena leading to the so-called curse of dimensionality that arises in some numerical and other applications. N-sphere_sentence_122

Specific spheres N-sphere_section_16

Octahedral sphere N-sphere_section_17

The octahedral n-sphere is defined similarly to the n-sphere but using the 1-norm N-sphere_sentence_123

The octahedral 1-sphere is a square (without its interior). N-sphere_sentence_124

The octahedral 2-sphere is a regular octahedron; hence the name. N-sphere_sentence_125

The octahedral n-sphere is the topological join of n+1 pairs of isolated points. N-sphere_sentence_126

Intuitively, the topological join of two pairs is generated by drawing a segment between each point in one pair and each point in the other pair; this yields a square. N-sphere_sentence_127

To join this with a third pair, draw a segment between each point on the square and each point in the third pair; this gives a octahedron. N-sphere_sentence_128

See also N-sphere_section_18

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/N-sphere.