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N-sphere

In mathematics, an $⟨0,0,0,1⟩$-sphere or hypersphere is an $n$-dimensional generalization of the $n$-dimensional circle and $1$-dimensional sphere to any non-negative integer $2$. The $n$-sphere is the setting for $n$-dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in $n$-dimensional Euclidean space, an $(n+1)$-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an $n$-dimensional ball. In particular:

 * The $(n+1)$-sphere is the pair of points at the ends of a line segment ($0$-ball).
 * The $1$-sphere is a circle, the circumference of a disk ($1$-ball) in the two-dimensional plane.
 * The $2$-sphere, often simply called a sphere, is the boundary of a $2$-ball in three-dimensional space.
 * The $3$-sphere is the boundary of a $3$-ball in four-dimensional space.
 * The $4$-sphere is the boundary of an $(n-1)$-ball.

Given a Cartesian coordinate system, the unit $n$-sphere of radius $n$ can be defined as:

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

Considered intrinsically, when $1$, the $n \geq 1$-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the $n$-sphere are called great circles.

The stereographic projection maps the $n$-sphere onto $n$-space with a single adjoined point at infinity; under the metric thereby defined, $$\R^n \cup \{\infty\}$$ is a model for the $n$-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit $n$-sphere is called an $n$-sphere. Under inverse stereographic projection, the $n$-sphere is the one-point compactification of $n$-space. The $n$-spheres admit several other topological descriptions: for example, they can be constructed by gluing two $n$-dimensional spaces together, by identifying the boundary of an $n$-cube with a point, or (inductively) by forming the suspension of an $n$-sphere. When $(n-1)$ it is simply connected; the $n \geq 2$-sphere (circle) is not simply connected; the $1$-sphere is not even connected, consisting of two discrete points.

Description
For any natural number $0$, an $n$-sphere of radius $n$ is defined as the set of points in $r$-dimensional Euclidean space that are at distance $(n+1)$ from some fixed point $r$, where $\mathbf{c}$ may be any positive real number and where $r$ may be any point in $\mathbf{c}$-dimensional space. In particular:
 * a 0-sphere is a pair of points $(n+1)$, and is the boundary of a line segment ($\{ c - r, c + r \}$-ball).
 * a $3$-sphere is a circle of radius $1$ centered at $r$, and is the boundary of a disk ($\mathbf{c}$-ball).
 * a $3$-sphere is an ordinary $2$-dimensional sphere in $2$-dimensional Euclidean space, and is the boundary of an ordinary ball ($3$-ball).
 * a $3$-sphere is a $3$-dimensional sphere in $3$-dimensional Euclidean space.

Cartesian coordinates
The set of points in $4$-space, $(n+1)$, that define an $(x_1, x_2, \ldots, x_n+1)$-sphere, $n$, is represented by the equation:

 * $$r^2=\sum_{i=1}^{n+1} (x_i - c_i)^2 ,$$

where $S^n(r)$ is a center point, and $\mathbf{c} = (c_1, c_2, \ldots, c_{n+1})$ is the radius.

The above $r$-sphere exists in $n$-dimensional Euclidean space and is an example of an $(n+1)$-manifold. The volume form $n$ of an $\omega$-sphere of radius $n$ is given by

 * $$\omega = \frac{1}{r} \sum_{j=1}^{n+1} (-1)^{j-1} x_j \,dx_1 \wedge \cdots \wedge dx_{j-1} \wedge dx_{j+1}\wedge \cdots \wedge dx_{n+1} = {\star} dr$$

where $${\star}$$ is the Hodge star operator; see for a discussion and proof of this formula in the case $r$. As a result,
 * $$dr \wedge \omega = dx_1 \wedge \cdots \wedge dx_{n+1}.$$

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

Specifically:
 * A $n$-ball, a line segment, is the interior of a 0-sphere.
 * A $1$-ball, a disk, is the interior of a circle ($2$-sphere).
 * A $1$-ball, an ordinary ball, is the interior of a sphere ($3$-sphere).
 * A $2$-ball is the interior of a $1$-sphere, etc.

Topological description
Topologically, an $4$-sphere can be constructed as a one-point compactification of $n$-dimensional Euclidean space. Briefly, the $n$-sphere can be described as $n$, which is $S^n = \R^n \cup \{ \infty \}$-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 $$\R^n$$. This forms the basis for stereographic projection.

Volume and area
Let $n$ be the surface area of the unit $S_{n-1}$-sphere of radius $(n-1)$ embedded in $1$-dimensional Euclidean space, and let $n$ be the volume of its interior, the unit $V_n$-ball. The surface area of an arbitrary $n$-sphere is proportional to the $(n-1)$st power of the radius, and the volume of an arbitrary $(n-1)$-ball is proportional to the $n$th power of the radius.

The $n$-ball is sometimes defined as a single point. The $n$-dimensional Hausdorff measure is the number of points in a set. So
 * $$V_0=1.$$

A unit $0$-ball is a line segment whose points have a single coordinate in the interval $0$ of length $1$, and the $[-1, 1]$-sphere consists of its two end-points, with coordinate $2$.
 * $$S_0 = 2, \quad V_1 = 2.$$

A unit $0$-sphere is the unit circle in the Euclidean plane, and its interior is the unit disk ($\{ -1, 1 \}$-ball).
 * $$S_1 = 2\pi, \quad V_2 = \pi .$$

The interior of a 2-sphere in three-dimensional space is the unit $1$-ball.
 * $$S_2 = 4\pi, \quad V_3 = \tfrac{4}{3} \pi.$$

In general, $2$ and $3$ are given in closed form by the expressions

S_{n-1} = \frac{2 \pi^{n/2}}{\Gamma\bigl(\frac{n}{2}\bigr)}, \quad V_n    = \frac{\pi^{n/2}}{\Gamma\bigl(\frac{n}{2} + 1\bigr)} $$

where $S_{n-1}$ is the gamma function.

As $V_n$ tends to infinity, the volume of the unit $\Gamma$-ball (ratio between the volume of an $n$-ball of radius $n$ and an $n$-cube of side length $1$) tends to zero.

Recurrences
The surface area, or properly the $n$-dimensional volume, of the $1$-sphere at the boundary of the $n$-ball of radius $n$ is related to the volume of the ball by the differential equation
 * $$S_{n}R^{n}=\frac{dV_{n+1}R^{n+1}}{dR}={(n+1)V_{n+1}R^{n}}.$$

Equivalently, representing the unit $(n+1)$-ball as a union of concentric $R$-sphere shells,
 * $$V_{n+1} = \int_0^1 S_{n}r^{n}\,dr = \frac{1}{n+1}S_n.$$

We can also represent the unit $n$-sphere as a union of products of a circle ($(n-1)$-sphere) with an $(n+2)$-sphere. Then $1$. Since $n$, the equation
 * $$S_{n+1} = 2\pi V_{n}$$

holds for all $S_{n+2} = 2\pi V_{n+1}$. Along with the base cases $$, $n$ 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 $n$, and $3$ angular coordinates $r$, where the angles $n-1$ range over $\varphi_1, \varphi_2, \ldots, \varphi_{n-1}$ radians (or $\varphi_1, \varphi_2, \ldots, \varphi_{n-2}$ degrees) and $[0, \pi]$ ranges over $[0, 180]$ radians (or $\varphi_{n-1}$ degrees). If $[0, 2\pi)$ are the Cartesian coordinates, then we may compute $[0, 360)$ from $x_i$ with:

 * $$\begin{align}

x_1 &= r \cos(\varphi_1), \\[5mu] x_2 &= r \sin(\varphi_1) \cos(\varphi_2), \\[5mu] x_3 &= r \sin(\varphi_1) \sin(\varphi_2) \cos(\varphi_3), \\ &\qquad \vdots\\ x_{n-1} &= r \sin(\varphi_1) \cdots \sin(\varphi_{n-2}) \cos(\varphi_{n-1}), \\[5mu] x_n    &= r \sin(\varphi_1) \cdots \sin(\varphi_{n-2}) \sin(\varphi_{n-1}). \end{align}$$ Except in the special cases described below, the inverse transformation is unique:

\begin{align} r            &=                            {\textstyle \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_2}^2 + {x_1}^2}},     \\[5mu] \varphi_1    &= \operatorname{atan2} \left({\textstyle \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_2}^2}}, x_{1}\right), \\[5mu] \varphi_2    &= \operatorname{atan2} \left({\textstyle \sqrt{{x_n}^2 + {x_{n-1}}^2 + \cdots + {x_3}^2}}, x_{2}\right), \\ &\qquad \vdots\\ \varphi_{n-2} &= \operatorname{atan2} \left({\textstyle \sqrt{{x_n}^2 + {x_{n-1}}^2}}, x_{n-2}\right), \\[5mu] \varphi_{n-1} &= \operatorname{atan2} \left(x_n, x_{n-1}\right). \end{align} $$

where $2$ is the two-argument arctangent function.

There are some special cases where the inverse transform is not unique; $x_1, \ldots, x_n$ for any $r, \varphi_1, \ldots, \varphi_{n-1}$ will be ambiguous whenever all of $n>3$ are zero; in this case $n-3$ may be chosen to be zero. (For example, for the $$-sphere, when the polar angle is $$ or $$ then the point is one of the poles, zenith or nadir, and the choice of azimuthal angle is arbitrary.)

Spherical volume and area elements
To express the volume element of $$-dimensional Euclidean space in terms of spherical coordinates, let $n\ge 2$ and $\textstyle x_n = r\prod_{i=1}^{n-1} \sin \varphi_i$ for concision, then observe that the Jacobian matrix of the transformation is:

J_n = \begin{pmatrix} c_1                       &-rs_1              &0        &0 &\cdots &0      \\ s_1c_2                    &rc_1c_2            &-rs_1s_2 &0 &\cdots &0      \\ \vdots                    &\vdots             & \vdots  &  &\ddots &\vdots \\ &                  &         &  &       &0      \\ s_1\cdots s_{n-2}c_{n-1}   &\cdots             &\cdots   &  &       &-rs_1\cdots s_{n-2}s_{n-1} \\ s_{1}\cdots s_{n-2}s_{n-1} &rc_1\cdots s_{n-1} &\cdots  &  &       &\phantom{-}rs_1\cdots s_{n-2}c_{n-1} \end{pmatrix}. $$

The determinant of this matrix can be calculated by induction. When $\textstyle x_k =r \cos \varphi_k\prod_{i=1}^{k-1} \sin \varphi_i$, a straightforward computation shows that the determinant is $$. For larger $\varphi_k$, observe that $k$ can be constructed from $x_k, x_{k+1}, \ldots x_n$ as follows. Except in column $\varphi_k$, rows $2$ and $0$ of $\pi$ are the same as row $n$ of $$, but multiplied by an extra factor of $$ in row $$ and an extra factor of $r$ in row $n$. In column $J_n$, rows $J_{n-1}$ and $n$ of $n-1$ are the same as column $n$ of row $J_n$ of $n-1$, but multiplied by extra factors of $J_{n-1}$ in row $\cos \varphi_{n-1}$ and $n-1$ in row $\sin \varphi_{n-1}$, respectively. The determinant of $n$ can be calculated by Laplace expansion in the final column. By the recursive description of $n$, the submatrix formed by deleting the entry at $n-1$ and its row and column almost equals $n$, except that its last row is multiplied by $J_n$. Similarly, the submatrix formed by deleting the entry at $n-1$ and its row and column almost equals $n-1$, except that its last row is multiplied by $J_{n-1}$. Therefore the determinant of $\sin \varphi_{n-1}$ is
 * $$\begin{align}

&= (-1)^{(n-1)+n}(-rs_1 \dotsm s_{n-2}s_{n-1})(s_{n-1}|J_{n-1}|) \\ &\qquad {}+ (-1)^{n+n}(rs_1 \dotsm s_{n-2}c_{n-1})(c_{n-1}|J_{n-1}|) \\ &= (rs_1 \dotsm s_{n-2}|J_{n-1}|(s_{n-1}^2 + c_{n-1}^2) \\ &= (rs_1 \dotsm s_{n-2})|J_{n-1}|. \end{align}$$ Induction then gives a closed-form expression for the volume element in spherical coordinates
 * J_n|
 * $$\begin{align}

d^nV &= \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{align}$$ The formula for the volume of the $n-1$-ball can be derived from this by integration.

Similarly the surface area element of the $\cos \varphi_{n-1}$-sphere of radius $n$, which generalizes the area element of the $J_n$-sphere, is given by

 * $$d_{S^{n-1}}V = R^{n-1}\sin^{n-2}(\varphi_1)\sin^{n-3}(\varphi_2)\cdots \sin(\varphi_{n-2})\, d\varphi_1 \, d\varphi_2\cdots d\varphi_{n-1}.$$

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

 * $$\begin{align}

& {} \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{align}$$

for $J_n$, and the $(n-1, n)$ for the angle $J_{n-1}$ in concordance with the spherical harmonics.

Polyspherical coordinates
The standard spherical coordinate system arises from writing $\sin \varphi_{n-1}$ as the product $(n, n)$. These two factors may be related using polar coordinates. For each point $J_{n-1}$ of $$\R^n$$, the standard Cartesian coordinates
 * $$\mathbf{x} = (x_1, \dots, x_n) = (y_1, z_1, \dots, z_{n-1}) = (y_1, \mathbf{z})$$

can be transformed into a mixed polar–Cartesian coordinate system:
 * $$\mathbf{x} = (r\sin\theta, (r\cos\theta)\hat\mathbf{z}).$$

This says that points in $\cos \varphi_{n-1}$ may be expressed by taking the ray starting at the origin and passing through $$\hat\mathbf{z}=\mathbf{z}/\lVert\mathbf{z}\rVert\in S^{n-2}$$, rotating it towards $$(1,0,\dots,0)$$ by $$\theta=\arcsin y_1/r$$, and traveling a distance $$r=\lVert\mathbf{x}\rVert$$ along the ray. Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction. The space $J_n$ is split as the product of two Euclidean spaces of smaller dimension, but neither space is required to be a line. Specifically, suppose that $n$ and $(n-1)$ are positive integers such that $r$. Then $2$. Using this decomposition, a point $$ may be written as
 * $$\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:
 * $$\mathbf{x} = ((r\sin \theta)\hat\mathbf{y}, (r\cos \theta)\hat\mathbf{z}).$$

Here $$\hat\mathbf{y}$$ and $$\hat\mathbf{z}$$ are the unit vectors associated to $e^{is\varphi_j}$ and $$. This expresses $\R^n$ in terms of $\R \times \R^{n-1}$, $\mathbf x$, $\R^n$, and an angle $\R^n$. It can be shown that the domain of $p$ is $q$ if $$, $\R^n = \R^p \times \R^q$ if exactly one of $x \in \R^n$ and $\mathbf y$ is $\mathbf z$, and $\mathbf x$ if neither $\hat\mathbf{y} \in S^{p-1}$ nor $\hat\mathbf{z} \in S^{q-1}$ are $r \geq 0$. The inverse transformation is
 * $$\begin{align}

r &= \lVert\mathbf{x}\rVert, \\ \theta &= \arcsin\frac{\lVert\mathbf{y}\rVert}{\lVert\mathbf{x}\rVert} = \arccos\frac{\lVert\mathbf{z}\rVert}{\lVert\mathbf{x}\rVert} = \arctan\frac{\lVert\mathbf{y}\rVert}{ \lVert\mathbf{z}\rVert}. \end{align}$$

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 $$\hat\mathbf{y}$$ and $$\hat\mathbf{z}$$ are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and $\theta$ angles. The possible polyspherical coordinate systems correspond to binary trees with $\theta$ 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 $[0, 2\pi)$, and its immediate children represent the first splitting into $p = q = 1$ and $[0, \pi]$. Leaf nodes correspond to Cartesian coordinates for $p$. 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 $q$, taking the left branch introduces a factor of $1$ and taking the right branch introduces a factor of $[0, \pi/2]$. 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 $p$ determines a subgroup
 * $$\operatorname{SO}_p(\R) \times \operatorname{SO}_q(\R) \subseteq \operatorname{SO}_n(\R).$$

This is the subgroup that leaves each of the two factors $$S^{p-1} \times S^{q-1} \subseteq S^{n-1}$$ 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 $q$ and the area measure on $1$ are products. There is one factor for each angle, and the volume measure on $n-1$ also has a factor for the radial coordinate. The area measure has the form:
 * $$dA_{n-1} = \prod_{i=1}^{n-1} F_i(\theta_i)\,d\theta_i,$$

where the factors $n$ are determined by the tree. Similarly, the volume measure is
 * $$dV_n = r^{n-1}\,dr\,\prod_{i=1}^{n-1} F_i(\theta_i)\,d\theta_i.$$

Suppose we have a node of the tree that corresponds to the decomposition $\R^n$ and that has angular coordinate $\R^p$. The corresponding factor $\R^q$ depends on the values of $S^{n-1}$ and $\theta_i$. When the area measure is normalized so that the area of the sphere is $\sin \theta_i$, these factors are as follows. If $\cos \theta_i$, then
 * $$F(\theta) = \frac{d\theta}{2\pi}.$$

If $\R^n = \R^p \times \R^q$ and $\R^n$, and if $S^{n-1}$ denotes the beta function, then
 * $$F(\theta) = \frac{\sin^{n_1 - 1}\theta}{\Beta(\frac{n_1}{2}, \frac{1}{2})}\,d\theta.$$

If $\R^n$ and $F_i$, then
 * $$F(\theta) = \frac{\cos^{n_2 - 1}\theta}{\Beta(\frac{1}{2}, \frac{n_2}{2})}\,d\theta.$$

Finally, if both $\R^{n_1 + n_2} = \R^{n_1} \times \R^{n_2}$ and $\theta$ are greater than one, then
 * $$F(\theta) = \frac{(\sin^{n_1 - 1}\theta)(\cos^{n_2 - 1}\theta)}{\frac{1}{2}\Beta(\frac{n_1}{2}, \frac{n_2}{2})}\,d\theta.$$

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 $F$-sphere can be mapped onto an $n_1$-dimensional hyperplane by the $n_2$-dimensional version of the stereographic projection. For example, the point $1$ on a two-dimensional sphere of radius $n_1 = n_2 = 1$ maps to the point $n_1 > 1$ on the $$-plane. In other words,
 * $$[x,y,z] \mapsto \left[\frac{x}{1-z},\frac{y}{1-z}\right].$$

Likewise, the stereographic projection of an $\Beta$-sphere $$ of radius $n_2 > 1$ will map to the $n_1$-dimensional hyperplane $n_2$ perpendicular to the $n$-axis as
 * $$[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].$$

Uniformly at random on the $3$-sphere

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

Generate an $[x, y, z]$-dimensional vector of normal deviates (it suffices to use $1$, although in fact the choice of the variance is arbitrary), $\bigl[ \tfrac{x}{1-z}, \tfrac{y}{1-z} \bigr]$. Now calculate the "radius" of this point:

 * $$r=\sqrt{x_1^2+x_2^2+\cdots+x_n^2}.$$

The vector $xy$ is uniformly distributed over the surface of the unit $n$-ball.

An alternative given by Marsaglia is to uniformly randomly select a point $S^n$ in the unit $1$-cube by sampling each $(n-1)$ independently from the uniform distribution over $\R^{n-1}$, computing $x_n$ as above, and rejecting the point and resampling if $(n-1)$ (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 $n$; then again $N(0, 1)$ is uniformly distributed over the surface of the unit $\mathbf x = (x_1, x_2, \ldots, x_n)$-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 $$10^{-24}$$ 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 $\tfrac 1r \mathbf x$-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 $\mathbf x = (x_1, x_2, \ldots, x_n)$ is a number generated uniformly at random from the interval $n$ and $x_i$ is a point selected uniformly at random from the unit $(-1, 1)$-sphere, then $r$ is uniformly distributed within the unit $r \geq 1$-ball.

Alternatively, points may be sampled uniformly from within the unit $n$-ball by a reduction from the unit $\tfrac 1r$-sphere. In particular, if $\tfrac 1r \mathbf x$ is a point selected uniformly from the unit $n$-sphere, then $(n-1)$ is uniformly distributed within the unit $n$-ball (i.e., by simply discarding two coordinates).

If $u$ is sufficiently large, most of the volume of the $[0, 1]$-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.

Distribution of the first coordinate
Let $\mathbf x$ be the square of the first coordinate of a point sampled uniformly at random from the $(n-1)$-sphere, then its probability density function, for $$y\in [0, 1]$$, is

$$ \rho(y) = \frac{\Gamma\bigl(\frac{n}{2} \bigr)}{\sqrt\pi \; \Gamma\bigl(\frac{n-1}{2}\bigr)} (1-y)^{(n-3)/2}y^{-1/2}. $$

Let $$z = y/N$$ be the appropriately scaled version, then at the $$N\to \infty$$ limit, the probability density function of $$z$$ converges to $$ (2\pi ze^z)^{-1/2}$$. This is sometimes called the Porter–Thomas distribution.

Specific spheres

 * $3$-sphere : The pair of points $u^{1/n} \mathbf x$ with the discrete topology for some $n$. The only sphere that is not path-connected. Parallelizable.
 * $V_{n}$-sphere : Commonly called a circle. Has a nontrivial fundamental group. Abelian Lie group structure $1$; the circle group. Homeomorphic to the real projective line.
 * $atan2$-sphere : Commonly simply called a sphere. For its complex structure, see Riemann sphere. Homeomorphic to the complex projective line
 * $(n − 1)$-sphere : Parallelizable, principal $2$-bundle over the $n$-sphere, Lie group structure $0$.
 * $1$-sphere : Homeomorphic to the quaternionic projective line, $(n+1)$. $(x_1, x_2, \ldots, x_{n+2})$.
 * $U(1)$-sphere : Principal $2$-bundle over the complex projective space $(n+1)$. $(x_1, x_2, \ldots, x_n)$. It is undecidable whether a given $n$-dimensional manifold is homeomorphic to $n$ for $n$.
 * $3$-sphere : Possesses an almost complex structure coming from the set of pure unit octonions. $$. The question of whether it has a complex structure is known as the Hopf problem, after Heinz Hopf.
 * $U(1)$-sphere : Topological quasigroup structure as the set of unit octonions. Principal $(n-1)$-bundle over $Sp(1)$. Parallelizable. $\{ \pm R\}$. The $R > 0$-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
 * $4$-sphere : Homeomorphic to the octonionic projective line $2$.
 * $5$-sphere : A highly dense sphere-packing is possible in $\mathbf{HP}^1$-dimensional space, which is related to the unique qualities of the Leech lattice.

Octahedral sphere
The octahedral $\operatorname{SO}(5) / \operatorname{SO}(4)$-sphere is defined similarly to the $\mathbf{CP}^2$-sphere but using the $U(1)$-norm

 * $$ S^n = \left\{ x \in \R^{n+1} : \left\| x \right\|_1 = 1 \right\}$$

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

The octahedral $\operatorname{SO}(6) / \operatorname{SO}(5) = \operatorname{SU}(3) / \operatorname{SU}(2)$-sphere is a square (without its interior). The octahedral $n$-sphere is a regular octahedron; hence the name. The octahedral $S^n$-sphere is the topological join of $n \geq 5$ pairs of isolated points. 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.

title: N-sphere


text:
Hypercube

In geometry, a hypercube is an n-dimensional analogue of a square (n = 2) and a cube (n = 3). It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in n dimensions is equal to $$\sqrt{n}$$.

An n-dimensional hypercube is more commonly referred to as an n-cube or sometimes as an n-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γ n polytopes.

The hypercube is the special case of a hyperrectangle (also called an n-orthotope).

A unit hypercube is a hypercube whose side has length one unit. Often, the hypercube whose corners (or vertices) are the 2n points in Rn with each coordinate equal to 0 or 1 is called the unit hypercube.

By the number of dimensions

A hypercube can be defined by increasing the numbers of dimensions of a shape:
 * 0 – A point is a hypercube of dimension zero.
 * 1 – If one moves this point one unit length, it will sweep out a line segment, which is a unit hypercube of dimension one.
 * 2 – If one moves this line segment its length in a perpendicular direction from itself; it sweeps out a 2-dimensional square.
 * 3 – If one moves the square one unit length in the direction perpendicular to the plane it lies on, it will generate a 3-dimensional cube.
 * 4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract).

This can be generalized to any number of dimensions. This process of sweeping out volumes can be formalized mathematically as a Minkowski sum: the d-dimensional hypercube is the Minkowski sum of d mutually perpendicular unit-length line segments, and is therefore an example of a zonotope.

The 1-skeleton of a hypercube is a hypercube graph.

Vertex coordinates

A unit hypercube of dimension $$n$$ is the convex hull of all the $$2^n$$ points whose $$n$$ Cartesian coordinates are each equal to either $$0$$ or $$1$$. These points are its vertices. The hypercube with these coordinates is also the cartesian product $$[0,1]^n$$ of $$n$$ copies of the unit interval $$[0,1]$$. Another unit hypercube, centered at the origin of the ambient space, can be obtained from this one by a translation. It is the convex hull of the $$2^n$$ points whose vectors of Cartesian coordinates are

\left(\pm \frac{1}{2}, \pm \frac{1}{2}, \cdots, \pm \frac{1}{2}\right)\!\!. $$

Here the symbol $$\pm$$ means that each coordinate is either equal to $$1/2$$ or to $$-1/2$$. This unit hypercube is also the cartesian product $$[-1/2,1/2]^n$$. Any unit hypercube has an edge length of $$1$$ and an $$n$$-dimensional volume of $$1$$.

The $$n$$-dimensional hypercube obtained as the convex hull of the points with coordinates $$(\pm 1, \pm 1, \cdots, \pm 1)$$ or, equivalently as the Cartesian product $$[-1,1]^n$$ is also often considered due to the simpler form of its vertex coordinates. Its edge length is $$2$$, and its $$n$$-dimensional volume is $$2^n$$.

Faces
Every hypercube admits, as its faces, hypercubes of a lower dimension contained in its boundary. A hypercube of dimension $$n$$ admits $$2n$$ facets, or faces of dimension $$n-1$$: a ($$1$$-dimensional) line segment has $$2$$ endpoints; a ($$2$$-dimensional) square has $$4$$ sides or edges; a $$3$$-dimensional cube has $$6$$ square faces; a ($$4$$-dimensional) tesseract has $$8$$ three-dimensional cubes as its facets. The number of vertices of a hypercube of dimension $$n$$ is $$2^n$$ (a usual, $$3$$-dimensional cube has $$2^3=8$$ vertices, for instance).

The number of the $$m$$-dimensional hypercubes (just referred to as $$m$$-cubes from here on) contained in the boundary of an $$n$$-cube is

 * $$ E_{m,n} = 2^{n-m}{n \choose m} $$,    where $${n \choose m}=\frac{n!}{m!\,(n-m)!}$$ and $$n!$$ denotes the factorial of $$n$$.

For example, the boundary of a $$4$$-cube ($$n=4$$) contains $$8$$ cubes ($$3$$-cubes), $$24$$ squares ($$2$$-cubes), $$32$$ line segments ($$1$$-cubes) and $$16$$ vertices ($$0$$-cubes). This identity can be proven by a simple combinatorial argument: for each of the $$2^n$$ vertices of the hypercube, there are $$\tbinom n m$$ ways to choose a collection of $$m$$ edges incident to that vertex. Each of these collections defines one of the $$m$$-dimensional faces incident to the considered vertex. Doing this for all the vertices of the hypercube, each of the $$m$$-dimensional faces of the hypercube is counted $$2^m$$ times since it has that many vertices, and we need to divide $$2^n\tbinom n m$$ by this number.

The number of facets of the hypercube can be used to compute the $$(n-1)$$-dimensional volume of its boundary: that volume is $$2n$$ times the volume of a $$(n-1)$$-dimensional hypercube; that is, $$2ns^{n-1}$$ where $$s$$ is the length of the edges of the hypercube.

These numbers can also be generated by the linear recurrence relation.

 * $$E_{m,n} = 2E_{m,n-1} + E_{m-1,n-1} \!$$,    with $$E_{0,0}= 1$$, and $$E_{m,n}=0$$ when $$n < m$$, $$n < 0$$, or $$m < 0$$.

For example, extending a square via its 4 vertices adds one extra line segment (edge) per vertex. Adding the opposite square to form a cube provides $$E_{1,3}=12$$ line segments.

The extended f-vector for an n-cube can also be computed by expanding $$(2x+1)^n$$ (concisely, (2,1)n), and reading off the coefficients of the resulting polynomial. For example, the elements of a tesseract is (2,1)4 = (4,4,1)2 = (16,32,24,8,1).

Graphs
An n-cube can be projected inside a regular 2n-gonal polygon by a skew orthogonal projection, shown here from the line segment to the 16-cube.

Related families of polytopes
The hypercubes are one of the few families of regular polytopes that are represented in any number of dimensions.

The hypercube (offset) family is one of three regular polytope families, labeled by Coxeter as γn. The other two are the hypercube dual family, the cross-polytopes, labeled as βn, and the simplices, labeled as αn. A fourth family, the infinite tessellations of hypercubes, is labeled as δn.

Another related family of semiregular and uniform polytopes is the demihypercubes, which are constructed from hypercubes with alternate vertices deleted and simplex facets added in the gaps, labeled as hγn.

n-cubes can be combined with their duals (the cross-polytopes) to form compound polytopes:

 * In two dimensions, we obtain the octagrammic star figure {8/2},
 * In three dimensions we obtain the compound of cube and octahedron,
 * In four dimensions we obtain the compound of tesseract and 16-cell.

Relation to (n−1)-simplices
The graph of the n-hypercube's edges is isomorphic to the Hasse diagram of the (n−1)-simplex's face lattice. This can be seen by orienting the n-hypercube so that two opposite vertices lie vertically, corresponding to the (n−1)-simplex itself and the null polytope, respectively. Each vertex connected to the top vertex then uniquely maps to one of the (n−1)-simplex's facets (n−2 faces), and each vertex connected to those vertices maps to one of the simplex's n−3 faces, and so forth, and the vertices connected to the bottom vertex map to the simplex's vertices.

This relation may be used to generate the face lattice of an (n−1)-simplex efficiently, since face lattice enumeration algorithms applicable to general polytopes are more computationally expensive.

Generalized hypercubes
Regular complex polytopes can be defined in complex Hilbert space called generalized hypercubes, γ = p{4}2{3}...2{3}2, or ... Real solutions exist with p = 2, i.e. γ = γn = 2{4}2{3}...2{3}2 = {4,3,..,3}. For p > 2, they exist in $$\mathbb{C}^n$$. The facets are generalized (n−1)-cube and the vertex figure are regular simplexes.

The regular polygon perimeter seen in these orthogonal projections is called a petrie polygon. The generalized squares (n = 2) are shown with edges outlined as red and blue alternating color p-edges, while the higher n-cubes are drawn with black outlined p-edges.

The number of m-face elements in a p-generalized n-cube are: $$p^{n-m}{n \choose m}$$. This is pn vertices and pn facets.

Relation to exponentiation
Any positive integer raised to another positive integer power will yield a third integer, with this third integer being a specific type of figurate number corresponding to an n-cube with a number of dimensions corresponding to the exponential. For example, the exponent 2 will yield a square number or "perfect square", which can be arranged into a square shape with a side length corresponding to that of the base. Similarly, the exponent 3 will yield a perfect cube, an integer which can be arranged into a cube shape with a side length of the base. As a result, the act of raising a number to 2 or 3 is more commonly referred to as "squaring" and "cubing", respectively. However, the names of higher-order hypercubes do not appear to be in common use for higher powers.

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title: Hypercube


text:
Rotation

Rotation or rotational motion is the circular movement of an object around a central line, known as axis of rotation. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersecting anywhere inside or outside the figure at a center of rotation. A solid figure has an infinite number of possible axes and angles of rotation, including chaotic rotation (between arbitrary orientations), in contrast to rotation around a axis.

The special case of a rotation with an internal axis passing through the body's own center of mass is known as a spin (or autorotation). In that case, the surface intersection of the internal spin axis can be called a pole; for example, Earth's rotation defines the geographical poles. A rotation around an axis completely external to the moving body is called a revolution (or orbit), e.g. Earth's orbit around the Sun. The ends of the external axis of revolution can be called the orbital poles.

Either type of rotation is involved in a corresponding type of angular velocity (spin angular velocity and orbital angular velocity) and angular momentum (spin angular momentum and orbital angular momentum).

Mathematics

Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps at least one point fixed. This definition applies to rotations in two dimensions (in a plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rotation around a fixed axis, as infinite line).

All rigid body movements are rotations, translations, or combinations of the two.

A rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion.

If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. The reverse (inverse) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation.

Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations.

Fixed axis vs. fixed point
The combination of any sequence of rotations of an object in three dimensions about a fixed point is always equivalent to a rotation about an axis (which may be considered to be a rotation in the plane that is perpendicular to that axis). Similarly, the rotation rate of an object in three dimensions at any instant is about some axis, although this axis may be changing over time.

In other than three dimensions, it does not make sense to describe a rotation as being around an axis, since more than one axis through the object may be kept fixed; instead, simple rotations are described as being in a plane. In four or more dimensions, a combination of two or more rotations about in a plane is not in general a rotation in a single plane.

Axis of 2-dimensional rotations
2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2 dimensional rotation, except, of course, the identity.

The question of the existence of such a direction is the question of existence of an eigenvector for the matrix A representing the rotation. Every 2D rotation around the origin through an angle $$\theta$$ in counterclockwise direction can be quite simply represented by the following matrix:

 * $$A =

\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{bmatrix} $$

A standard eigenvalue determination leads to the characteristic equation
 * $$\lambda^2 -2 \lambda \cos \theta + 1 = 0,$$

which has
 * $$\cos \theta \pm i \sin \theta$$

as its eigenvalues. Therefore, there is no real eigenvalue whenever $$\cos \theta \neq \pm 1$$, meaning that no real vector in the plane is kept unchanged by A.

Rotation angle and axis in 3 dimensions
Knowing that the trace is an invariant, the rotation angle $$\alpha$$ for a proper orthogonal 3×3 rotation matrix $$A$$ is found by

 * $$\alpha=\cos^{-1}\left(\frac{A_{11}+A_{22}+A_{33}-1}{2}\right)$$

Using the principal arc-cosine, this formula gives a rotation angle satisfying $$0\le\alpha\le 180^\circ$$. The corresponding rotation axis must be defined to point in a direction that limits the rotation angle to not exceed 180 degrees. (This can always be done because any rotation of more than 180 degrees about an axis $$m$$ can always be written as a rotation having $$0\le\alpha\le 180^\circ$$ if the axis is replaced with $$n=-m$$.)

Every proper rotation $$A$$ in 3D space has an axis of rotation, which is defined such that any vector $$v$$ that is aligned with the rotation axis will not be affected by rotation. Accordingly, $$ A v = v $$, and the rotation axis therefore corresponds to an eigenvector of the rotation matrix associated with an eigenvalue of 1. As long as the rotation angle $$\alpha$$ is nonzero (i.e., the rotation is not the identity tensor), there is one and only one such direction. Because A has only real components, there is at least one real eigenvalue, and the remaining two eigenvalues must be complex conjugates of each other (see Eigenvalues and eigenvectors). Knowing that 1 is an eigenvalue, it follows that the remaining two eigenvalues are complex conjugates of each other, but this does not imply that they are complex—they could be real with double multiplicity. In the degenerate case of a rotation angle $$\alpha=180^\circ$$, the remaining two eigenvalues are both equal to −1. In the degenerate case of a zero rotation angle, the rotation matrix is the identity, and all three eigenvalues are 1 (which is the only case for which the rotation axis is arbitrary).

A spectral analysis is not required to find the rotation axis. If $$n$$ denotes the unit eigenvector aligned with the rotation axis, and if $$\alpha$$ denotes the rotation angle, then it can be shown that $$2\sin(\alpha)n=\{A_{32}-A_{23},A_{13}-A_{31},A_{21}-A_{12}\}$$. Consequently, the expense of an eigenvalue analysis can be avoided by simply normalizing this vector if it has a nonzero magnitude. On the other hand, if this vector has a zero magnitude, it means that $$\sin(\alpha)=0$$. In other words, this vector will be zero if and only if the rotation angle is 0 or 180 degrees, and the rotation axis may be assigned in this case by normalizing any column of $$A+I$$ that has a nonzero magnitude.

This discussion applies to a proper rotation, and hence $$\det A = 1$$. Any improper orthogonal 3x3 matrix $$B$$ may be written as $$B=-A$$, in which $$A$$ is proper orthogonal. That is, any improper orthogonal 3x3 matrix may be decomposed as a proper rotation (from which an axis of rotation can be found as described above) followed by an inversion (multiplication by −1). It follows that the rotation axis of $$A$$ is also the eigenvector of $$B$$ corresponding to an eigenvalue of −1.

Rotation plane
As much as every tridimensional rotation has a rotation axis, also every tridimensional rotation has a plane, which is perpendicular to the rotation axis, and which is left invariant by the rotation. The rotation, restricted to this plane, is an ordinary 2D rotation.

The proof proceeds similarly to the above discussion. First, suppose that all eigenvalues of the 3D rotation matrix A are real. This means that there is an orthogonal basis, made by the corresponding eigenvectors (which are necessarily orthogonal), over which the effect of the rotation matrix is just stretching it. If we write A in this basis, it is diagonal; but a diagonal orthogonal matrix is made of just +1s and −1s in the diagonal entries. Therefore, we do not have a proper rotation, but either the identity or the result of a sequence of reflections.

It follows, then, that a proper rotation has some complex eigenvalue. Let v be the corresponding eigenvector. Then, as we showed in the previous topic, $$ \bar{v} $$ is also an eigenvector, and $$ v + \bar{v} $$ and $$ i(v - \bar{v}) $$ are such that their scalar product vanishes:

 * $$ i (v^\text{T} + \bar{v}^\text{T})(v - \bar{v}) = i (v^\text{T} v - \bar{v}^\text{T} \bar{v} + \bar{v}^\text{T} v - v^\text{T} \bar{v} ) = 0 $$

because, since $$ \bar{v}^\text{T} \bar{v} $$ is real, it equals its complex conjugate $$ v^\text{T} v $$, and $$ \bar{v}^\text{T} v $$ and $$ v^\text{T} \bar{v} $$ are both representations of the same scalar product between $$ v $$ and $$ \bar{v} $$.

This means $$ v + \bar{v} $$ and $$ i(v - \bar{v}) $$ are orthogonal vectors. Also, they are both real vectors by construction. These vectors span the same subspace as $$ v $$ and $$ \bar{v} $$, which is an invariant subspace under the application of A. Therefore, they span an invariant plane.

This plane is orthogonal to the invariant axis, which corresponds to the remaining eigenvector of A, with eigenvalue 1, because of the orthogonality of the eigenvectors of A.

Rotation of vectors
A vector is said to be rotating if it changes its orientation. This effect is generally only accompanied when its rate of change vector has non-zero perpendicular component to the original vector. This can be shown to be the case by considering a vector $$ \vec A $$ which is parameterized by some variable $ t $ for which:

$${d|\vec A|^2 \over dt}={d(\vec A\cdot \vec A) \over dt} \Rightarrow  {d|\vec A| \over dt}={d\vec A  \over dt}\cdot \hat{A}$$

Which also gives a relation of rate of change of unit vector by taking A to be such a vector: $$ {d\hat A \over dt}\cdot \hat A = 0 $$showing that $ {d\hat A \over dt} $ vector is perpendicular to the vector $$ \vec A $$.

From:

$${d\vec A \over dt} = {d(|\vec A|\hat A) \over dt} = {d|\vec A| \over dt}\hat{A}+|\vec A|\left({d\hat A \over dt}\right) $$,

since the first term is parallel to $$ \vec A $$ and the second perpendicular to it, we can conclude in general that the parallel and perpendicular components of rate of change of a vector independently influence only the magnitude or orientation of the vector respectively. Hence, a rotating vector always has a non-zero perpendicular component of its rate of change vector against the vector itself.

In higher dimensions
As dimensions increase the number of rotation vectors increases. Along a four dimensional space (a hyper-volume), rotations occur along x, y, z, and w axis. An object rotated on a w axis intersects through various volumes, where each intersection is equal to a self contained volume at an angle. This gives way to a new axis of rotation in a 4d hypervolume, were a 3d object can be rotated perpendicular to the z axis.

Physics
The speed of rotation is given by the angular frequency (rad/s) or frequency (turns per time), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s2), caused by torque. The ratio of torque to the angular acceleration is given by the moment of inertia.

The angular velocity vector (an axial vector) also describes the direction of the axis of rotation. Similarly, the torque is an axial vector.

The physics of the rotation around a fixed axis is mathematically described with the axis–angle representation of rotations. According to the right-hand rule, the direction away from the observer is associated with clockwise rotation and the direction towards the observer with counterclockwise rotation, like a screw.

Circular motion
It is possible for objects to have periodic circular trajectories without changing their orientation. These types of motion are treated under circular motion instead of rotation, more specifically as a curvilinear translation. Since translation involves displacement of rigid bodies while preserving the orientation of the body, in the case of curvilinear translation, all the points have the same instantaneous velocity whereas relative motion can only be observed in motions involving rotation.

In rotation, the orientation of the object changes and the change in orientation is independent of the observers whose frames of reference have constant relative orientation over time. By Euler's theorem, any change in orientation can be described by rotation about an axis through a chosen reference point. Hence, the distinction between rotation and circular motion can be made by requiring an instantaneous axis for rotation, a line passing through instantaneous center of circle and perpendicular to the plane of motion. In the example depicting curvilinear translation, the center of circles for the motion lie on a straight line but it is parallel to the plane of motion and hence does not resolve to an axis of rotation. In contrast, a rotating body will always have its instantaneous axis of zero velocity, perpendicular to the plane of motion.

More generally, due to Chasles' theorem, any motion of rigid bodies can be treated as a composition of rotation and translation, called general plane motion. A simple example of pure rotation is considered in rotation around a fixed axis.

Cosmological principle
The laws of physics are currently believed to be invariant under any fixed rotation. (Although they do appear to change when viewed from a rotating viewpoint: see rotating frame of reference.)

In modern physical cosmology, the cosmological principle is the notion that the distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throughout the universe and have no preferred direction, and should, therefore, produce no observable irregularities in the large scale structuring over the course of evolution of the matter field that was initially laid down by the Big Bang.

In particular, for a system which behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally invariant. According to Noether's theorem, if the action (the integral over time of its Lagrangian) of a physical system is invariant under rotation, then angular momentum is conserved.

Euler rotations
Euler rotations provide an alternative description of a rotation. It is a composition of three rotations defined as the movement obtained by changing one of the Euler angles while leaving the other two constant. Euler rotations are never expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture. They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z, the second rotates around the line of nodes and the third one is an intrinsic rotation around an axis fixed in the body that moves.

These rotations are called precession, nutation, and intrinsic rotation.

Astronomy

In astronomy, rotation is a commonly observed phenomenon; it includes both spin (auto-rotation) and orbital revolution.

Spin
Stars, planets and similar bodies may spin around on their axes. The rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features. An example is sunspots, which rotate around the Sun at the same velocity as the outer gases that make up the Sun.

Under some circumstances orbiting bodies may lock their spin rotation to their orbital rotation around a larger body. This effect is called tidal locking; the Moon is tidal-locked to the Earth.

This rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravitation the closer one is to the equator. Earth's gravity combines both mass effects such that an object weighs slightly less at the equator than at the poles. Another is that over time the Earth is slightly deformed into an oblate spheroid; a similar equatorial bulge develops for other planets.

Another consequence of the rotation of a planet are the phenomena of precession and nutation. Like a gyroscope, the overall effect is a slight "wobble" in the movement of the axis of a planet. Currently the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is 23.44 degrees, but this angle changes slowly (over thousands of years). (See also Precession of the equinoxes and Pole Star.)

Revolution
While revolution is often used as a synonym for rotation, in many fields, particularly astronomy and related fields, revolution, often referred to as orbital revolution for clarity, is used when one body moves around another while rotation is used to mean the movement around an axis. Moons revolve around their planets, planets revolve about their stars (such as the Earth around the Sun); and stars slowly revolve about their galaxial centers. The motion of the components of galaxies is complex, but it usually includes a rotation component.

Retrograde rotation
Most planets in the Solar System, including Earth, spin in the same direction as they orbit the Sun. The exceptions are Venus and Uranus. Venus may be thought of as rotating slowly backward (or being "upside down"). Uranus rotates nearly on its side relative to its orbit. Current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history. The dwarf planet Pluto (formerly considered a planet) is anomalous in several ways, including that it also rotates on its side.

Flight dynamics
In flight dynamics, the principal rotations described with Euler angles above are known as pitch, roll and yaw. The term rotation is also used in aviation to refer to the upward pitch (nose moves up) of an aircraft, particularly when starting the climb after takeoff.

Principal rotations have the advantage of modelling a number of physical systems such as gimbals, and joysticks, so are easily visualised, and are a very compact way of storing a rotation. But they are difficult to use in calculations as even simple operations like combining rotations are expensive to do, and suffer from a form of gimbal lock where the angles cannot be uniquely calculated for certain rotations.

Amusement rides
Many amusement rides provide rotation. A Ferris wheel has a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result, at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centripetal force. In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where inertia keeps people in their seats.

Sports
Rotation of a ball or other object, usually called spin, plays a role in many sports, including topspin and backspin in tennis, English, follow and draw in billiards and pool, curve balls in baseball, spin bowling in cricket, flying disc sports, etc. Table tennis paddles are manufactured with different surface characteristics to allow the player to impart a greater or lesser amount of spin to the ball.

Rotation of a player one or more times around a vertical axis may be called spin in figure skating, twirling (of the baton or the performer) in baton twirling, or 360, 540, 720, etc. in snowboarding, etc. Rotation of a player or performer one or more times around a horizontal axis may be called a flip, roll, somersault, heli, etc. in gymnastics, waterskiing, or many other sports, or a one-and-a-half, two-and-a-half, gainer (starting facing away from the water), etc. in diving, etc. A combination of vertical and horizontal rotation (back flip with 360°) is called a möbius in waterskiing freestyle jumping.

Rotation of a player around a vertical axis, generally between 180 and 360 degrees, may be called a spin move and is used as a deceptive or avoidance manoeuvre, or in an attempt to play, pass, or receive a ball or puck, etc., or to afford a player a view of the goal or other players. It is often seen in hockey, basketball, football of various codes, tennis, etc.

title: Rotation