Editing 3-sphere (section)

==Coordinate systems on the 3-sphere==
The four Euclidean coordinates for {{math|''S''<sup>3</sup>}} are redundant since they are subject to the condition that {{math|1=''x''<sub>0</sub><sup>2</sup> + ''x''<sub>1</sub><sup>2</sup> + ''x''<sub>2</sub><sup>2</sup> + ''x''<sub>3</sub><sup>2</sup> = 1}}. As a 3-dimensional manifold one should be able to parameterize {{math|''S''<sup>3</sup>}} by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as [[latitude]] and [[longitude]]). Due to the nontrivial topology of {{math|''S''<sup>3</sup>}} it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use ''at least'' two [[coordinate chart]]s. Some different choices of coordinates are given below.

===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''}}.

===Hopf coordinates===
[[File:Hopf Fibration.png|right|250px|thumb|The Hopf fibration can be visualized using a [[stereographic projection]] of {{math|''S''<sup>3</sup>}} to {{math|'''R'''<sup>3</sup>}} and then compressing {{math|''R''<sup>3</sup>}} to a ball.  This image shows points on {{math|''S''<sup>2</sup>}} and their corresponding fibers with the same color.]]
For unit radius another choice of hyperspherical coordinates, {{math|(''η'', ''ξ''<sub>1</sub>, ''ξ''<sub>2</sub>)}}, makes use of the embedding of {{math|''S''<sup>3</sup>}} in {{math|'''C'''<sup>2</sup>}}. In complex coordinates {{math|(''z''<sub>1</sub>, ''z''<sub>2</sub>) ∈ '''C'''<sup>2</sup>}} we write
:<math>\begin{align}
z_1 &= e^{i\,\xi_1}\sin\eta \\
z_2 &= e^{i\,\xi_2}\cos\eta. \end{align}</math>

This could also be expressed in {{math|'''R'''<sup>4</sup>}} as
:<math>\begin{align}
x_0 &= \cos\xi_1\sin\eta \\
x_1 &= \sin\xi_1\sin\eta \\
x_2 &= \cos\xi_2\cos\eta \\
x_3 &= \sin\xi_2\cos\eta. \end{align}</math>
Here {{mvar|η}} runs over the range 0 to {{sfrac|{{pi}}|2}}, and {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}} can take any values between 0 and 2{{pi}}. These coordinates are useful in the description of the 3-sphere as the [[Hopf bundle]]
:<math>S^1 \to S^3 \to S^2.\,</math>

[[File:Toroidal coord.png|thumb|A diagram depicting the poloidal ({{math|''ξ''<sub>1</sub>}}) direction, represented by the red arrow, and the toroidal ({{math|''ξ''<sub>2</sub>}}) direction, represented by the blue arrow, although the terms ''poloidal'' and ''toroidal'' are arbitrary in this ''[[Flat torus#Flat torus|flat torus]]'' case.]]
For any fixed value of {{mvar|η}} between 0 and {{sfrac|{{pi}}|2}}, the coordinates {{math|(''ξ''<sub>1</sub>, ''ξ''<sub>2</sub>)}} parameterize a 2-dimensional [[torus]]. Rings of constant {{math|''ξ''<sub>1</sub>}} and {{math|''ξ''<sub>2</sub>}} above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when {{mvar|η}} equals 0 or {{sfrac|{{pi}}|2}}, these coordinates describe a [[circle]].

The round metric on the 3-sphere in these coordinates is given by
:<math>ds^2 = d\eta^2 + \sin^2\eta\,d\xi_1^2 + \cos^2\eta\,d\xi_2^2</math>
and the volume form by
:<math>dV = \sin\eta\cos\eta\,d\eta\wedge d\xi_1\wedge d\xi_2.</math>

To get the interlocking circles of the [[Hopf fibration]], make a simple substitution in the equations above<ref>{{cite web|last1=Banchoff|first1=Thomas|title=The Flat Torus in the Three-Sphere|url=http://www.geom.uiuc.edu/~banchoff/script/b3d/hypertorus.html}}</ref>
:<math>\begin{align}
z_1 &= e^{i\,(\xi_1+\xi_2)}\sin\eta \\
z_2 &= e^{i\,(\xi_2-\xi_1)}\cos\eta. \end{align}</math>

In this case {{mvar|η}}, and {{math|''ξ''<sub>1</sub>}} specify which circle, and {{math|''ξ''<sub>2</sub>}} specifies the position along each circle. One round trip (0 to 2{{pi}}) of {{math|''ξ''<sub>1</sub>}} or {{math|''ξ''<sub>2</sub>}} equates to a round trip of the torus in the 2 respective directions.

===Stereographic coordinates===
Another convenient set of coordinates can be obtained via [[stereographic projection]] of {{math|''S''<sup>3</sup>}} from a pole onto the corresponding equatorial {{math|'''R'''<sup>3</sup>}} [[hyperplane]]. For example, if we project from the point {{math|(−1, 0, 0, 0)}} we can write a point {{mvar|p}} in {{math|''S''<sup>3</sup>}} as
:<math>p = \left(\frac{1-\|u\|^2}{1+\|u\|^2}, \frac{2\mathbf{u}}{1+\|u\|^2}\right) = \frac{1+\mathbf{u}}{1-\mathbf{u}}</math>
where {{math|1='''u''' = (''u''<sub>1</sub>, ''u''<sub>2</sub>, ''u''<sub>3</sub>)}} is a vector in {{math|'''R'''<sup>3</sup>}} and {{math|1={{norm|''u''}}<sup>2</sup> = ''u''<sub>1</sub><sup>2</sup> + ''u''<sub>2</sub><sup>2</sup> + ''u''<sub>3</sub><sup>2</sup>}}. In the second equality above, we have identified {{mvar|p}} with a unit quaternion and {{math|1='''u''' = ''u''<sub>1</sub>''i'' + ''u''<sub>2</sub>''j'' + ''u''<sub>3</sub>''k''}} with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes {{math|1=''p'' = (''x''<sub>0</sub>, ''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>)}} in {{math|''S''<sup>3</sup>}} to
:<math>\mathbf{u} = \frac{1}{1+x_0}\left(x_1, x_2, x_3\right).</math>

We could just as well have projected from the point {{math|(1, 0, 0, 0)}}, in which case the point {{mvar|p}} is given by
:<math>p = \left(\frac{-1+\|v\|^2}{1+\|v\|^2}, \frac{2\mathbf{v}}{1+\|v\|^2}\right) = \frac{-1+\mathbf{v}}{1+\mathbf{v}}</math>
where {{math|1='''v''' = (''v''<sub>1</sub>, ''v''<sub>2</sub>, ''v''<sub>3</sub>)}} is another vector in {{math|'''R'''<sup>3</sup>}}. The inverse of this map takes {{mvar|p}} to
:<math>\mathbf{v} = \frac{1}{1-x_0}\left(x_1,x_2,x_3\right).</math>

Note that the {{math|'''u'''}} coordinates are defined everywhere but {{math|(−1, 0, 0, 0)}} and the {{math|'''v'''}} coordinates everywhere but {{math|(1, 0, 0, 0)}}. This defines an [[atlas (topology)|atlas]] on {{math|''S''<sup>3</sup>}} consisting of two [[chart (topology)|coordinate charts]] or "patches", which together cover all of {{math|''S''<sup>3</sup>}}. Note that the transition function between these two charts on their overlap is given by
:<math>\mathbf{v} = \frac{1}{\|u\|^2}\mathbf{u}</math>
and vice versa.