[n sphere] spherical coordinates

n-sphere  [https://en.wikipedia.org/wiki/N-sphere]

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, ${\displaystyle r\,,}$ and n − 1 angular coordinates ${\displaystyle \phi _{1},\phi _{2},\dots ,\phi _{n-1}\,}$ where ${\displaystyle \phi _{n-1}\,}$ ranges over ${\displaystyle [0,2\pi )\,}$radians (or over [0, 360) degrees) and the other angles range over ${\displaystyle [0,\pi ]\,}$ radians (or over [0, 180] degrees). If ${\displaystyle \ x_{i}}$ are the Cartesian coordinates, then we may compute ${\displaystyle x_{1},\ldots ,x_{n}}$ from ${\displaystyle r,\phi _{1},\ldots ,\phi _{n-1}}$ with:

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

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

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

where if ${\displaystyle x_{k}\neq 0}$ for some ${\displaystyle k}$ but all of ${\displaystyle x_{k+1},\ldots ,x_{n}}$ are zero then ${\displaystyle \phi _{k}=0}$ when ${\displaystyle x_{k}>0}$, and ${\displaystyle \phi _{k}=\pi }$ radians (180 degrees) when ${\displaystyle x_{k}<0}$.

There are some special cases where the inverse transform is not unique; ${\displaystyle \phi _{k}}$ for any ${\displaystyle k}$ will be ambiguous whenever all of ${\displaystyle x_{k},x_{k+1},\ldots ,x_{n}}$ are zero; in this case ${\displaystyle \phi _{k}}$ may be chosen to be zero.