Editing N-sphere (section)

=== Polyspherical coordinates ===
The standard spherical coordinate system arises from writing {{tmath|\R^n}} as the product {{tmath|\R \times \R^{n-1} }}.  These two factors may be related using polar coordinates.  For each point {{tmath|\mathbf x}} of <math>\R^n</math>, the standard Cartesian coordinates
:<math>\mathbf{x} = (x_1, \dots, x_n) = (y_1, z_1, \dots, z_{n-1}) = (y_1, \mathbf{z})</math>
can be transformed into a mixed polar–Cartesian coordinate system:
:<math>\mathbf{x} = (r\sin\theta, (r\cos\theta)\hat\mathbf{z}).</math>
This says that points in {{tmath|\R^n}} may be expressed by taking the ray starting at the origin and passing through <math>\hat\mathbf{z}=\mathbf{z}/\lVert\mathbf{z}\rVert\in S^{n-2}</math>, rotating it towards <math>(1,0,\dots,0)</math> by <math>\theta=\arcsin y_1/r</math>, and traveling a distance <math>r=\lVert\mathbf{x}\rVert</math> along the ray.  Repeating this decomposition eventually leads to the standard spherical coordinate system.

Polyspherical coordinate systems arise from a generalization of this construction.<ref>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.</ref> The space {{tmath|\R^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 {{tmath|p}} and {{tmath|q}} are positive integers such that {{tmath|n {{=}} p + q}}.  Then {{tmath|\R^n {{=}} \R^p \times \R^q}}.  Using this decomposition, a point {{tmath| x \in \R^n }} may be written as
:<math>\mathbf{x} = (x_1, \dots, x_n) = (y_1, \dots, y_p, z_1, \dots, z_q) = (\mathbf{y}, \mathbf{z}).</math>
This can be transformed into a mixed polar–Cartesian coordinate system by writing:
:<math>\mathbf{x} = ((r\sin \theta)\hat\mathbf{y}, (r\cos \theta)\hat\mathbf{z}).</math>
Here <math>\hat\mathbf{y}</math> and <math>\hat\mathbf{z}</math> are the unit vectors associated to {{tmath| \mathbf y}} and {{tmath|\mathbf z}}.  This expresses {{tmath|\mathbf x}} in terms of {{tmath| \hat\mathbf{y} \in S^{p-1} }}, {{tmath| \hat\mathbf{z} \in S^{q-1} }}, {{tmath|r \geq 0}}, and an angle {{tmath|\theta}}.  It can be shown that the domain of {{tmath|\theta}} is {{tmath|[0, 2\pi)}} if {{tmath|1= p = q = 1}}, {{tmath|[0, \pi]}} if exactly one of {{tmath|p}} and {{tmath|q}} is {{tmath|1}}, and {{tmath|[0, \pi/2]}} if neither {{tmath|p}} nor {{tmath|q}} are {{tmath|1}}.  The inverse transformation is
:<math>\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}</math>

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 <math>\hat\mathbf{y}</math> and <math>\hat\mathbf{z}</math> are spheres, so the coordinates of a polyspherical coordinate system are a non-negative radius and {{tmath|n-1}} angles.  The possible polyspherical coordinate systems correspond to binary trees with {{tmath|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 {{tmath|\R^n}}, and its immediate children represent the first splitting into {{tmath|\R^p}} and {{tmath|\R^q}}.  Leaf nodes correspond to Cartesian coordinates for {{tmath|S^{n-1} }}.  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 {{tmath|\theta_i}}, taking the left branch introduces a factor of {{tmath|\sin \theta_i}} and taking the right branch introduces a factor of {{tmath|\cos \theta_i}}.  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 {{tmath|\R^n {{=}} \R^p \times \R^q }} determines a subgroup
:<math>\operatorname{SO}_p(\R) \times \operatorname{SO}_q(\R) \subseteq \operatorname{SO}_n(\R).</math>
This is the subgroup that leaves each of the two factors <math>S^{p-1} \times S^{q-1} \subseteq S^{n-1}</math> 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 {{tmath|\R^n}} and the area measure on {{tmath|S^{n-1} }} are products.  There is one factor for each angle, and the volume measure on {{tmath|\R^n}} also has a factor for the radial coordinate.  The area measure has the form:
:<math>dA_{n-1} = \prod_{i=1}^{n-1} F_i(\theta_i)\,d\theta_i,</math>
where the factors {{tmath|F_i}} are determined by the tree.  Similarly, the volume measure is
:<math>dV_n = r^{n-1}\,dr\,\prod_{i=1}^{n-1} F_i(\theta_i)\,d\theta_i.</math>
Suppose we have a node of the tree that corresponds to the decomposition {{tmath|\R^{n_1 + n_2} {{=}} \R^{n_1} \times \R^{n_2} }} and that has angular coordinate {{tmath|\theta}}.  The corresponding factor {{tmath|F}} depends on the values of {{tmath|n_1}} and {{tmath|n_2}}.  When the area measure is normalized so that the area of the sphere is {{tmath|1}}, these factors are as follows.  If {{tmath|1=n_1 = n_2 = 1}}, then
:<math>F(\theta) = \frac{d\theta}{2\pi}.</math>
If {{tmath|n_1 > 1}} and {{tmath|n_2 {{=}} 1}}, and if {{tmath|\Beta}} denotes the [[beta function]], then
:<math>F(\theta) = \frac{\sin^{n_1 - 1}\theta}{\Beta(\frac{n_1}{2}, \frac{1}{2})}\,d\theta.</math>
If {{tmath|n_1 {{=}} 1}} and {{tmath|n_2 > 1}}, then
:<math>F(\theta) = \frac{\cos^{n_2 - 1}\theta}{\Beta(\frac{1}{2}, \frac{n_2}{2})}\,d\theta.</math>
Finally, if both {{tmath|n_1}} and {{tmath|n_2}} are greater than one, then
:<math>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.</math>