Editing 3-sphere (section)

===Hyperspherical coordinates===
It is convenient to have some sort of [[N-sphere#Spherical coordinates|hyperspherical coordinates]] on {{math|''S''<sup>3</sup>}} in analogy to the usual [[spherical coordinates]] on {{math|''S''<sup>2</sup>}}. One such choice — by no means unique — is to use {{math|(''ψ'', ''θ'', ''φ'')}}, where
:<math>\begin{align}
x_0 &= r\cos\psi \\
x_1 &= r\sin\psi \cos\theta \\
x_2 &= r\sin\psi \sin\theta \cos \varphi \\
x_3 &= r\sin\psi \sin\theta \sin\varphi \end{align} </math>
where {{mvar|ψ}} and {{mvar|θ}} run over the range 0 to {{pi}}, and {{mvar|φ}} runs over 0 to 2{{pi}}. Note that, for any fixed value of {{mvar|ψ}}, {{mvar|θ}} and {{mvar|φ}} parameterize a 2-sphere of radius <math>r\sin\psi</math>, except for the degenerate cases, when {{mvar|ψ}} equals 0 or {{pi}}, in which case they describe a point.

The [[metric tensor|round metric]] on the 3-sphere in these coordinates is given by<ref>{{Cite book
| last1 = Landau |first1=Lev D. |author1-link=Lev Landau
| first2 = Evgeny M. |last2=Lifshitz |author2-link=Evgeny Lifshitz
| title = Classical Theory of Fields
| edition = 7th
| location = Moscow | publisher=[[Nauka (publisher)|Nauka]]
| year = 1988 |isbn=978-5-02-014420-0
| volume = 2
| page = 385
| series = [[Course of Theoretical Physics]]
| url = https://books.google.com/books?id=X18PF4oKyrUC

}} </ref>
:<math>ds^2 = r^2 \left[ d\psi^2 + \sin^2\psi\left(d\theta^2 + \sin^2\theta\, d\varphi^2\right) \right]</math>
and the [[volume form]] by
:<math>dV =r^3 \left(\sin^2\psi\,\sin\theta\right)\,d\psi\wedge d\theta\wedge d\varphi.</math>

These coordinates have an elegant description in terms of [[quaternion]]s. Any unit quaternion {{mvar|q}} can be written as a [[versor]]:
:<math>q = e^{\tau\psi} = \cos\psi + \tau\sin\psi</math>
where {{mvar|τ}} is a [[Quaternion#Square roots of −1|unit imaginary quaternion]]; that is, a quaternion that satisfies {{math|1=''τ''<sup>2</sup> = −1}}. This is the quaternionic analogue of [[Euler's formula]]. Now the unit imaginary quaternions all lie on the unit 2-sphere in {{math|Im '''H'''}} so any such {{mvar|τ}} can be written:
:<math>\tau = (\cos\theta) i + (\sin\theta\cos\varphi) j + (\sin\theta\sin\varphi) k</math>
With {{mvar|τ}} in this form, the unit quaternion {{mvar|q}} is given by
:<math>q = e^{\tau\psi} = x_0 + x_1 i + x_2 j + x_3 k</math>
where {{math|''x''<sub>0,1,2,3</sub>}} are as above.

When {{mvar|q}} is used to describe spatial rotations (cf. [[quaternions and spatial rotation]]s), it describes a rotation about {{mvar|τ}} through an angle of {{math|2''ψ''}}.