N-sphere

Last updated

In mathematics, an n-sphere or a hypersphere is a topological space that is homeomorphic to a standardn-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. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit n-sphere or simply the n-sphere for brevity. In terms of the standard norm, the n-sphere is defined as

Contents

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

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

${\displaystyle S^{n}(r)=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=r\right\}.}$

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

In particular:

• the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere,
• a circle, which is the one-dimensional circumference of a (two-dimensional) disk, is a 1-sphere,
• the two-dimensional surface of a three-dimensional ball is a 2-sphere, often simply called a sphere,
• the three-dimensional boundary of a (four-dimensional) 4-ball is a 3-sphere,
• the (n – 1)-dimensional boundary of a (n-dimensional) n-ball is an (n – 1)-sphere.

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. 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. The 1-sphere is the 1-manifold that is a circle, which is not simply connected. The 0-sphere is the 0-manifold, which is not even connected, consisting of two points.

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

Euclidean coordinates in (n + 1)-space

The set of points in (n + 1)-space, (x1, x2, ..., xn+1), that define an n-sphere, Sn(r), is represented by the equation:

${\displaystyle r^{2}=\sum _{i=1}^{n+1}(x_{i}-c_{i})^{2},}$

where c = (c1, c2, ..., cn+1) 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. The volume form ω of an n-sphere of radius r is given by

${\displaystyle \omega ={\frac {1}{r}}\sum _{j=1}^{n+1}(-1)^{j-1}x_{j}\,dx_{1}\wedge \cdots \wedge dx_{j-1}\wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1}={\star }dr}$

where ${\displaystyle {\star }}$ 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,

${\displaystyle dr\wedge \omega =dx_{1}\wedge \cdots \wedge dx_{n+1}.}$

n-ball

The space enclosed by an n-sphere is called an (n + 1)-ball. An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere.

Specifically:

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

Topological description

Topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. Briefly, the n-sphere can be described as Sn = ℝn ∪ {∞}, which is n-dimensional Euclidean space plus a single point representing infinity in all directions. In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to n. This forms the basis for stereographic projection. [1]

Volume and surface area

Vn(R) and Sn(R) are the n-dimensional volume of the n-ball and the surface area of the n-sphere embedded in dimension n + 1, respectively, of radius R.

The constants Vn and Sn (for R = 1, the unit ball and sphere) are related by the recurrences:

{\displaystyle {\begin{aligned}V_{0}&=1&V_{n+1}&={\frac {S_{n}}{n+1}}\\[6pt]S_{0}&=2&S_{n+1}&=2\pi V_{n}\end{aligned}}}

The surfaces and volumes can also be given in closed form:

{\displaystyle {\begin{aligned}S_{n-1}(R)&={\frac {2\,\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}R^{n-1}\\[6pt]V_{n}(R)&={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}\end{aligned}}}

where Γ is the gamma function. Derivations of these equations are given in this section.

In general, the volume of the n-ball in n-dimensional Euclidean space, and the surface area of the n-sphere in (n + 1)-dimensional Euclidean space, of radius R, are proportional to the nth power of the radius, R (with different constants of proportionality that vary with n). We write Vn(R) = VnRn for the volume of the n-ball and Sn(R) = SnRn for the surface area of the n-sphere, both of radius R, where Vn = Vn(1) and Sn = Sn(1) are the values for the unit-radius case.

The volume of the unit n-ball is maximal in dimension five, where it begins to decrease, and tends to zero as n tends to infinity. [2] Furthermore, the sum of the volumes of even-dimensional n-balls of radius R can be expressed in closed form: [2]

${\displaystyle \sum _{n=0}^{\infty }V_{2n}(R)=e^{\pi R^{2}}.}$

For the odd-dimensional analogue,

${\displaystyle \sum _{n=0}^{\infty }V_{2n+1}(R)=e^{\pi R^{2}}\operatorname {erf} ({\sqrt {\pi }}R),}$

where erf is the error function. [3]

Examples

The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set. So,

${\displaystyle V_{0}=1.}$

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

${\displaystyle S_{0}=2.}$

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

${\displaystyle V_{1}=2.}$

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

${\displaystyle S_{1}=2\pi .}$

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

${\displaystyle V_{2}=\pi .}$

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

${\displaystyle S_{2}=4\pi .}$

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

${\displaystyle V_{3}={\tfrac {4}{3}}\pi .}$

Recurrences

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

${\displaystyle S_{n}R^{n}={\frac {dV_{n+1}R^{n+1}}{dR}}={(n+1)V_{n+1}R^{n}},}$

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

${\displaystyle V_{n+1}=\int _{0}^{1}S_{n}r^{n}\,dr.}$

So,

${\displaystyle V_{n+1}={\frac {S_{n}}{n+1}}.}$

We can also represent the unit (n + 2)-sphere as a union of products of a circle (1-sphere) with an n-sphere. Let r = cos θ and r2 + R2 = 1, so that R = sin θ and dR = cos θ. Then,

{\displaystyle {\begin{aligned}S_{n+2}&=\int _{0}^{\frac {\pi }{2}}S_{1}r\cdot S_{n}R^{n}\,d\theta \\[6pt]&=\int _{0}^{\frac {\pi }{2}}S_{1}\cdot S_{n}R^{n}\cos \theta \,d\theta \\[6pt]&=\int _{0}^{1}S_{1}\cdot S_{n}R^{n}\,dR\\[6pt]&=S_{1}\int _{0}^{1}S_{n}R^{n}\,dR\\[6pt]&=2\pi V_{n+1}.\end{aligned}}}

Since S1 = 2π V0, the equation

${\displaystyle S_{n+1}=2\pi V_{n}}$

holds for all n.

This completes the derivation of the recurrences:

{\displaystyle {\begin{aligned}V_{0}&=1&V_{n+1}&={\frac {S_{n}}{n+1}}\\[6pt]S_{0}&=2&S_{n+1}&=2\pi V_{n}\end{aligned}}}

Closed forms

Combining the recurrences, we see that

${\displaystyle V_{n+2}=2\pi {\frac {V_{n}}{n+2}}.}$

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

{\displaystyle {\begin{aligned}V_{2k}&={\frac {\left(2\pi \right)^{k}}{(2k)!!}}={\frac {\pi ^{k}}{k!}}\\[6pt]V_{2k+1}&={\frac {2\left(2\pi \right)^{k}}{(2k+1)!!}}={\frac {2k!\left(4\pi \right)^{k}}{(2k+1)!}}\end{aligned}}}

where !! denotes the double factorial, defined for odd natural numbers 2k + 1 by (2k + 1)!! = 1 × 3 × 5 × ... × (2k − 1) × (2k + 1) and similarly for even numbers (2k)!! = 2 × 4 × 6 × ... × (2k − 2) × (2k).

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

${\displaystyle V_{n}={\frac {\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}={\frac {\pi ^{\frac {n}{2}}}{\left({\frac {n}{2}}\right)!}}}$

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! = Γ(x + 1) for every x.

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

${\displaystyle S_{n-1}={\frac {n\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}+1\right)}}={\frac {2\pi ^{\frac {n}{2}}}{\Gamma \left({\frac {n}{2}}\right)}}.}$

for the (n− 1)-dimensional surface of the sphere Sn−1.

Other relations

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

${\displaystyle S_{n-1}={\frac {n}{2\pi }}S_{n+1}}$

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

{\displaystyle {\begin{aligned}V_{n}&={\frac {2\pi }{n}}V_{n-2}\\[6pt]S_{n-1}&={\frac {2\pi }{n-2}}S_{n-3}\end{aligned}}}

where S0 = 2, V1 = 2, S1 = 2π and V2 = π.

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

{\displaystyle {\begin{aligned}V_{n}&=\int _{0}^{1}\int _{0}^{2\pi }V_{n-2}\left({\sqrt {1-r^{2}}}\right)^{n-2}\,r\,d\theta \,dr\\[6pt]&=\int _{0}^{1}\int _{0}^{2\pi }V_{n-2}\left(1-r^{2}\right)^{{\frac {n}{2}}-1}\,r\,d\theta \,dr\\[6pt]&=2\pi V_{n-2}\int _{0}^{1}\left(1-r^{2}\right)^{{\frac {n}{2}}-1}\,r\,dr\\[6pt]&=2\pi V_{n-2}\left[-{\frac {1}{n}}\left(1-r^{2}\right)^{\frac {n}{2}}\right]_{r=0}^{r=1}\\[6pt]&=2\pi V_{n-2}{\frac {1}{n}}={\frac {2\pi }{n}}V_{n-2}.\end{aligned}}}

Spherical 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 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). If xi are the Cartesian coordinates, then we may compute x1, ..., xn from r, φ1, ..., φn−1 with: [4]

{\displaystyle {\begin{aligned}x_{1}&=r\cos(\varphi _{1})\\x_{2}&=r\sin(\varphi _{1})\cos(\varphi _{2})\\x_{3}&=r\sin(\varphi _{1})\sin(\varphi _{2})\cos(\varphi _{3})\\&\,\,\,\vdots \\x_{n-1}&=r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1})\\x_{n}&=r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1}).\end{aligned}}}

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

{\displaystyle {\begin{aligned}r&={\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}+{x_{1}}^{2}}}\\[6pt]\varphi _{1}&=\operatorname {arccot} {\frac {x_{1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}}&&=\arccos {\frac {x_{1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{1}}^{2}}}}\\[6pt]\varphi _{2}&=\operatorname {arccot} {\frac {x_{2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{3}}^{2}}}}&&=\arccos {\frac {x_{2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+\cdots +{x_{2}}^{2}}}}\\[6pt]&\,\,\,\vdots &&\,\,\,\vdots \\[6pt]\varphi _{n-2}&=\operatorname {arccot} {\frac {x_{n-2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&&=\arccos {\frac {x_{n-2}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}+{x_{n-2}}^{2}}}}\\[6pt]\varphi _{n-1}&=2\operatorname {arccot} {\frac {x_{n-1}+{\sqrt {x_{n}^{2}+x_{n-1}^{2}}}}{x_{n}}}&&={\begin{cases}\arccos {\frac {x_{n-1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&x_{n}\geq 0,\\[6pt]2\pi -\arccos {\frac {x_{n-1}}{\sqrt {{x_{n}}^{2}+{x_{n-1}}^{2}}}}&x_{n}<0.\end{cases}}\end{aligned}}}

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.

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.

Spherical volume and area elements

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:

${\displaystyle J_{n}={\begin{pmatrix}\cos(\varphi _{1})&-r\sin(\varphi _{1})&0&0&\cdots &0\\\sin(\varphi _{1})\cos(\varphi _{2})&r\cos(\varphi _{1})\cos(\varphi _{2})&-r\sin(\varphi _{1})\sin(\varphi _{2})&0&\cdots &0\\\vdots &\vdots &\vdots &&\ddots &\vdots \\&&&&&0\\\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1})&\cdots &\cdots &&&-r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1})\\\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\sin(\varphi _{n-1})&r\cos(\varphi _{1})\cdots \sin(\varphi _{n-1})&\cdots &&&r\sin(\varphi _{1})\cdots \sin(\varphi _{n-2})\cos(\varphi _{n-1})\end{pmatrix}}.}$

The determinant of this matrix can be calculated by induction. When n = 2, a straightforward computation shows that the determinant is r. For larger n, observe that Jn can be constructed from Jn1 as follows. Except in column n, rows n 1 and n of Jn are the same as row n 1 of Jn1, but multiplied by an extra factor of cos φn1 in row n 1 and an extra factor of sin φn1 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 Jn1, but multiplied by extra factors of sin φn1 in row n 1 and cos φn1 in row n, respectively. The determinant of Jn can be calculated by Laplace expansion in the final column. By the recursive description of Jn, the submatrix formed by deleting the entry at (n 1, n) and its row and column almost equals Jn1, except that its last row is multiplied by sin φn1. Similarly, the submatrix formed by deleting the entry at (n, n) and its row and column almost equals Jn1, except that its last row is multiplied by cos φn1. Therefore the determinant of Jn is

{\displaystyle {\begin{aligned}|J_{n}|&=(-1)^{(n-1)+n}(-r\sin(\varphi _{1})\dotsm \sin(\varphi _{n-2})\sin(\varphi _{n-1}))(\sin(\varphi _{n-1})|J_{n-1}|)\\&\qquad {}+(-1)^{n+n}(r\sin(\varphi _{1})\dotsm \sin(\varphi _{n-2})\cos(\varphi _{n-1}))(\cos(\varphi _{n-1})|J_{n-1}|)\\&=(r\sin(\varphi _{1})\dotsm \sin(\varphi _{n-2})|J_{n-1}|(\sin ^{2}(\varphi _{n-1})+\cos ^{2}(\varphi _{n-1}))\\&=(r\sin(\varphi _{1})\dotsm \sin(\varphi _{n-2}))|J_{n-1}|.\end{aligned}}}

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

{\displaystyle {\begin{aligned}d^{n}V&=\left|\det {\frac {\partial (x_{i})}{\partial \left(r,\varphi _{j}\right)}}\right|dr\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1}\\&=r^{n-1}\sin ^{n-2}(\varphi _{1})\sin ^{n-3}(\varphi _{2})\cdots \sin(\varphi _{n-2})\,dr\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1}.\end{aligned}}}

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

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

${\displaystyle d_{S^{n-1}}V=R^{n-1}\sin ^{n-2}(\varphi _{1})\sin ^{n-3}(\varphi _{2})\cdots \sin(\varphi _{n-2})\,d\varphi _{1}\,d\varphi _{2}\cdots d\varphi _{n-1}.}$

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

{\displaystyle {\begin{aligned}&{}\quad \int _{0}^{\pi }\sin ^{n-j-1}\left(\varphi _{j}\right)C_{s}^{\left({\frac {n-j-1}{2}}\right)}\cos \left(\varphi _{j}\right)C_{s'}^{\left({\frac {n-j-1}{2}}\right)}\cos \left(\varphi _{j}\right)\,d\varphi _{j}\\[6pt]&={\frac {2^{3-n+j}\pi \Gamma (s+n-j-1)}{s!(2s+n-j-1)\Gamma ^{2}\left({\frac {n-j-1}{2}}\right)}}\delta _{s,s'}\end{aligned}}}

for j = 1, 2, ..., n − 2, and the eisφj for the angle j = n − 1 in concordance with the spherical harmonics.

Polyspherical coordinates

The standard spherical coordinate system arises from writing n as the product ℝ × ℝn1. These two factors may be related using polar coordinates. For each point x of n, the standard Cartesian coordinates

${\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})=(y_{1},z_{1},\dots ,z_{n-1})=(y_{1},\mathbf {z} )}$

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

${\displaystyle \mathbf {x} =(r\sin \theta ,(r\cos \theta ){\hat {\mathbf {z} }}).}$

This says that points in n may be expressed by taking the ray starting at the origin and passing through ${\displaystyle {\hat {\mathbf {z} }}=\mathbf {z} /\lVert \mathbf {z} \rVert \in S^{n-2}}$, rotating it towards ${\displaystyle (1,0,\dots ,0)}$ by ${\displaystyle \theta =\arcsin y_{1}/r}$, and traveling a distance ${\displaystyle r=\lVert \mathbf {x} \rVert }$ along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction. [5] The space n is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that p and q are positive integers such that n = p + q. Then n = ℝp×q. Using this decomposition, a point x ∈ ℝn may be written as

${\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})=(y_{1},\dots ,y_{p},z_{1},\dots ,z_{q})=(\mathbf {y} ,\mathbf {z} ).}$

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

${\displaystyle \mathbf {x} =((r\sin \theta ){\hat {\mathbf {y} }},(r\cos \theta ){\hat {\mathbf {z} }}).}$

Here ${\displaystyle {\hat {\mathbf {y} }}}$ and ${\displaystyle {\hat {\mathbf {z} }}}$ are the unit vectors associated to y and z. This expresses x in terms of ${\displaystyle {\hat {\mathbf {y} }}\in S^{p-1}}$, ${\displaystyle {\hat {\mathbf {z} }}\in S^{q-1}}$, r ≥ 0, and an angle θ. It can be shown that the domain of θ is [0, 2π) if p = q = 1, [0, π] if exactly one of p and q is 1, and [0, π/2] if neither p nor q are 1. The inverse transformation is

{\displaystyle {\begin{aligned}r&=\lVert \mathbf {x} \rVert ,\\\theta &=\arcsin(\lVert \mathbf {y} \rVert /\lVert \mathbf {x} \rVert )\\&=\arccos(\lVert \mathbf {z} \rVert /\lVert \mathbf {x} \rVert )\\&=\arctan(\lVert \mathbf {y} \rVert /\lVert \mathbf {z} \rVert ).\end{aligned}}}

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 ${\displaystyle {\hat {\mathbf {y} }}}$ and ${\displaystyle {\hat {\mathbf {z} }}}$ are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and n 1 angles. The possible polyspherical coordinate systems correspond to binary trees with n 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 n, and its immediate children represent the first splitting into p and q. Leaf nodes correspond to Cartesian coordinates for Sn1. 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 θi, taking the left branch introduces a factor of sin θi and taking the right branch introduces a factor of cos θi. 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 n = ℝp×q determines a subgroup

${\displaystyle \operatorname {SO} _{p}(\mathbb {R} )\times \operatorname {SO} _{q}(\mathbb {R} )\subseteq \operatorname {SO} _{n}(\mathbb {R} ).}$

This is the subgroup that leaves each of the two factors ${\displaystyle S^{p-1}\times S^{q-1}\subseteq S^{n-1}}$ 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 n and the area measure on Sn1 are products. There is one factor for each angle, and the volume measure on n also has a factor for the radial coordinate. The area measure has the form:

${\displaystyle dA_{n-1}=\prod _{i=1}^{n-1}F_{i}(\theta _{i})\,d\theta _{i},}$

where the factors Fi are determined by the tree. Similarly, the volume measure is

${\displaystyle dV_{n}=r^{n-1}\,dr\,\prod _{i=1}^{n-1}F_{i}(\theta _{i})\,d\theta _{i}.}$

Suppose we have a node of the tree that corresponds to the decomposition n1+n2 = ℝn1 × ℝn2 and that has angular coordinate θ. The corresponding factor F depends on the values of n1 and n2. When the area measure is normalized so that the area of the sphere is 1, these factors are as follows. If n1 = n2 = 1, then

${\displaystyle F(\theta )={\frac {d\theta }{2\pi }}.}$

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

${\displaystyle F(\theta )={\frac {\sin ^{n_{1}-1}\theta }{\mathrm {B} ({\frac {n_{1}}{2}},{\frac {1}{2}})}}\,d\theta .}$

If n1 = 1 and n2 > 1, then

${\displaystyle F(\theta )={\frac {\cos ^{n_{2}-1}\theta }{\mathrm {B} ({\frac {1}{2}},{\frac {n_{2}}{2}})}}\,d\theta .}$

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

${\displaystyle F(\theta )={\frac {(\sin ^{n_{1}-1}\theta )(\cos ^{n_{2}-1}\theta )}{{\frac {1}{2}}\mathrm {B} ({\frac {n_{1}}{2}},{\frac {n_{2}}{2}})}}\,d\theta .}$

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. 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. In other words,

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

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

${\displaystyle [x_{1},x_{2},\ldots ,x_{n}]\mapsto \left[{\frac {x_{1}}{1-x_{n}}},{\frac {x_{2}}{1-x_{n}}},\ldots ,{\frac {x_{n-1}}{1-x_{n}}}\right].}$

Generating random points

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

To generate uniformly distributed random points on the unit (n − 1)-sphere (that is, the surface of the unit 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), x = (x1, x2, ..., xn). Now calculate the "radius" of this point:

${\displaystyle r={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}.}$

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

An alternative given by Marsaglia is to uniformly randomly select a point x = (x1, x2, ..., xn) in the unit n-cube by sampling each xi independently from the uniform distribution over (–1, 1), computing r as above, and rejecting the point and resampling if r ≥ 1 (i.e., if the point is not in the n-ball), and when a point in the ball is obtained scaling it up to the spherical surface by the factor 1/r; then again 1/rx is uniformly distributed over the surface of the unit n-ball. 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. In seventy dimensions, less than ${\displaystyle 10^{-24}}$ of the cube is filled, meaning typically a trillion quadrillion trials will be needed, far more than a computer could ever carry out.

Uniformly at random within the n-ball

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. 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 u1/nx is uniformly distributed within the unit n-ball.

Alternatively, points may be sampled uniformly from within the unit n-ball by a reduction from the unit (n + 1)-sphere. 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). [6]

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

Specific spheres

0-sphere
The pair of points R} with the discrete topology for some R > 0. The only sphere that is not path-connected. Parallelizable.
1-sphere
Commonly called a circle. Has a nontrivial fundamental group. Abelian Lie group structure ; the circle group. Homeomorphic to the real projective line.
2-sphere
Commonly simply called a sphere. For its complex structure, see Riemann sphere . Equivalent[ clarification needed ] to the complex projective line
3-sphere
Parallelizable, principal U(1)-bundle over the 2-sphere, Lie group structure .
4-sphere
Equivalent to the quaternionic projective line, HP1. SO(5) / SO(4).
5-sphere
Principal U(1)-bundle over CP2. SO(6) / SO(5) = SU(3) / SU(2). It is undecidable whether a given n-dimensional manifold is homeomorphic to Sn for n ≥ 5. [7]
6-sphere
Possesses an almost complex structure coming from the set of pure unit octonions. SO(7) / SO(6) = G2 / SU(3). The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf. [8]
7-sphere
Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over S4. Parallelizable. SO(8) / SO(7) = SU(4) / SU(3) = Sp(2) / Sp(1) = Spin(7) / G2 = Spin(6) / SU(3). The 7-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
8-sphere
Equivalent to the octonionic projective line OP1.
23-sphere
A highly dense sphere-packing is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice.

Octahedral sphere

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

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

In general, it takes the shape of a cross-polytope.

The octahedral 1-sphere is a square (without its interior). The octahedral 2-sphere is a regular octahedron; hence the name. The octahedral n-sphere is the topological join of n + 1 pairs of isolated points. [9] 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. 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.

Notes

1. James W. Vick (1994). Homology theory, p. 60. Springer
2. Smith, David J.; Vamanamurthy, Mavina K. (1989). "How Small Is a Unit Ball?". Mathematics Magazine. 62 (2): 101–107. doi:10.1080/0025570X.1989.11977419. JSTOR   2690391.
3. Smith, David J.; Vamanamurthy, Mavina K. (1989). "How Small Is a Unit Ball?". Mathematics Magazine. 62 (2): 106. doi:10.1080/0025570X.1989.11977419. JSTOR   2690391.
4. Blumenson, L. E. (1960). "A Derivation of n-Dimensional Spherical Coordinates". The American Mathematical Monthly. 67 (1): 63–66. doi:10.2307/2308932. JSTOR   2308932.
5. N. Ja. Vilenkin and A. U. Klimyk, Representation of Lie groups and special functions, Vol. 2: Class I representations, special functions, and integral transforms, translated from the Russian by V. A. Groza and A. A. Groza, Math. Appl., vol. 74, Kluwer Acad. Publ., Dordrecht, 1992, ISBN   0-7923-1492-1, pp. 223–226.
6. Voelker, Aaron R.; Gosmann, Jan; Stewart, Terrence C. (2017). Efficiently sampling vectors and coordinates from the n-sphere and n-ball (Report). Centre for Theoretical Neuroscience. doi:10.13140/RG.2.2.15829.01767/1.
7. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72, Springer, p. 247, ISBN   9780387979700 .
8. Agricola, Ilka; Bazzoni, Giovanni; Goertsches, Oliver; Konstantis, Panagiotis; Rollenske, Sönke (2018). "On the history of the Hopf problem". Differential Geometry and Its Applications . 57: 1–9. arXiv:. doi:10.1016/j.difgeo.2017.10.014. S2CID   119297359.
9. Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN   1439-6912. S2CID   207006642.

Related Research Articles

In physics, the cross section is a measure of the probability that a specific process will take place when some kind of radiant excitation intersects a localized phenomenon. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus. Cross section is typically denoted σ (sigma) and is expressed in units of area, more specifically in barns. In a way, it can be thought of as the size of the object that the excitation must hit in order for the process to occur, but more exactly, it is a parameter of a stochastic process.

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians.

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as

In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere, the boundary of a ball in four dimensions is a 3-sphere. A 3-sphere is an example of a 3-manifold and an n-sphere.

In physics, the Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).

An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.

In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.

In classical mechanics, the shell theorem gives gravitational simplifications that can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.

This is a table of orthonormalized spherical harmonics that employ the Condon-Shortley phase up to degree . Some of these formulas are expressed in terms of the Cartesian expansion of the spherical harmonics into polynomials in x, y, z, and r. For purposes of this table, it is useful to express the usual spherical to Cartesian transformations that relate these Cartesian components to and as

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

In mathematics, a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in are called double integrals, and integrals of a function of three variables over a region in are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.

In astronomy, position angle is the convention for measuring angles on the sky. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images, this is a counterclockwise measure relative to the axis into the direction of positive declination.

In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles S1
a
and S1
b
. It is named after William Kingdon Clifford. It resides in R4, as opposed to in R3. To see why R4 is necessary, note that if S1
a
and S1
b
each exists in its own independent embedding space R2
a
and R2
b
, the resulting product space will be R4 rather than R3. The historically popular view that the cartesian product of two circles is an R3 torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis z available to it after the first circle consumes x and y.

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

In four-dimensional geometry, the spherinder, or spherical cylinder or spherical prism, is a geometric object, defined as the Cartesian product of a 3-ball of radius r1 and a line segment of length 2r2: