Editing 3D rotation group (section)

==Connection between SO(3) and SU(2)==
In this section, we give two different constructions of a two-to-one and [[surjective]] [[homomorphism]] of SU(2) onto SO(3).

===Using quaternions of unit norm ===
{{main|Quaternions and spatial rotation}}
The group {{math|SU(2)}} is [[Group isomorphism|isomorphic]] to the [[quaternion]]s of unit norm via a map given by<ref>{{harvnb|Rossmann|2002}} p. 95.</ref>
<math display="block">q = a\mathbf{1} + b\mathbf{i} + c\mathbf{j} + d\mathbf{k} = \alpha + \beta \mathbf{j} \leftrightarrow \begin{bmatrix}\alpha & \beta \\ -\overline\beta & \overline \alpha\end{bmatrix} = U</math>
restricted to <math display="inline">a^2+ b^2 + c^2 + d^2 = |\alpha|^2 +|\beta|^2 = 1</math> where <math display="inline"> q \in \mathbb{H}</math>, <math display="inline">a, b, c, d \in \R</math>, <math display="inline"> U \in \operatorname{SU}(2)</math>, and <math>\alpha = a+bi \in\mathbb{C}</math>, <math>\beta = c+di \in \mathbb{C}</math>.

Let us now identify <math>\R^3</math> with the span of <math>\mathbf{i},\mathbf{j},\mathbf{k}</math>. One can then verify that if <math>v</math> is in <math>\R^3</math> and <math>q</math> is a unit quaternion, then
<math display="block">qvq^{-1}\in \R^3.</math>

Furthermore, the map <math>v\mapsto qvq^{-1}</math> is a rotation of <math>\R^3.</math> Moreover, <math>(-q)v(-q)^{-1}</math> is the same as <math>qvq^{-1}</math>. This means that there is a {{math|2:1}} homomorphism from quaternions of unit norm to the 3D rotation group {{math|SO(3)}}.

One can work this homomorphism out explicitly: the unit quaternion, {{mvar|q}}, with
<math display="block">\begin{align}
 q &= w + x\mathbf{i} + y\mathbf{j} + z\mathbf{k} , \\
 1 &= w^2 + x^2 + y^2 + z^2 ,
\end{align}</math>
is mapped to the rotation matrix
<math display="block"> Q = \begin{bmatrix}
 1 - 2 y^2 - 2 z^2 & 2 x y - 2 z w & 2 x z + 2 y w \\
 2 x y + 2 z w & 1 - 2 x^2 - 2 z^2 & 2 y z - 2 x w \\
 2 x z - 2 y w & 2 y z + 2 x w & 1 - 2 x^2 - 2 y^2
\end{bmatrix}. </math>

This is a rotation around the vector {{math|(''x'', ''y'', ''z'')}} by an angle {{math|2''θ''}}, where {{math|1=cos ''θ'' = ''w''}} and {{math|1={{!}}sin ''θ''{{!}} = {{norm|(''x'', ''y'', ''z'')}}}}. The proper sign for {{math|sin ''θ''}} is implied, once the signs of the axis components are fixed. The {{nowrap|{{math|2:1}}-nature}} is apparent since both {{math|''q''}} and {{math|−''q''}} map to the same {{math|''Q''}}.

===Using Möbius transformations===
[[Image:Stereoprojnegone.svg|thumb|right|300px|Stereographic projection from the sphere of radius {{math|{{sfrac|1|2}}}} from the north pole {{math|1=(''x'', ''y'', ''z'') = (0, 0, {{sfrac|1|2}})}} onto the plane {{mvar|M}} given by {{math|1=''z'' = −{{sfrac|1|2}}}} coordinatized by {{math|(''ξ'', ''η'')}}, here shown in cross section.]]

The general reference for this section is {{harvtxt|Gelfand|Minlos|Shapiro|1963}}. The points {{math|''P''}} on the sphere

:<math>\mathbf{S} = \left \{(x,y,z)\in\R^3: x^2 +y^2 +z^2 = \frac{1}{4} \right \}</math>

can, barring the north pole {{math|''N''}}, be put into one-to-one bijection with points {{math|1=''S''(''P'') = ''P'''}} on the plane {{math|''M''}} defined by {{math|1=''z'' = −{{sfrac|1|2}}}}, see figure. The map {{math|''S''}} is called [[stereographic projection]].

Let the coordinates on {{mvar|M}} be {{math|(''ξ'', ''η'')}}. The line {{math|''L''}} passing through {{math|''N''}} and {{math|''P''}} can be parametrized as

:<math>L(t) = N + t(N - P) = \left(0,0,\frac{1}{2}\right) + t \left ( \left(0,0,\frac{1}{2}\right) - (x, y, z) \right ), \quad t\in \R.</math>

Demanding that the {{nowrap|{{math|''z''}}-coordinate}} of <math>L(t_0)</math> equals {{math|−{{sfrac|1|2}}}}, one finds

:<math>t_0 = \frac1{z-\frac12}.</math>

We have <math>L(t_0)=(\xi,\eta,-1/2).</math> Hence the map

:<math>\begin{cases} S:\mathbf{S} \to M \\ P = (x,y,z) \longmapsto P'= (\xi, \eta) = \left(\frac{x}{\frac{1}{2} - z}, \frac{y}{\frac{1}{2} - z}\right) \equiv \zeta = \xi + i\eta \end{cases}</math>

where, for later convenience, the plane {{math|''M''}} is identified with the complex plane <math>\Complex.</math>

For the inverse, write {{math|''L''}} as

:<math>L = N + s(P'-N) = \left(0,0,\frac{1}{2}\right) + s\left( \left(\xi, \eta, -\frac{1}{2}\right) - \left(0,0,\frac{1}{2}\right)\right),</math>

and demand {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> + ''z''<sup>2</sup> = {{sfrac|1|4}}}} to find {{math|1=''s'' = {{sfrac|1|1 + ''ξ''<sup>2</sup> + ''η''<sup>2</sup>}}}} and thus

:<math>\begin{cases} S^{-1}:M \to \mathbf{S} \\ P'= (\xi, \eta) \longmapsto P = (x,y,z) = \left(\frac{\xi}{1 + \xi^2 + \eta^2}, \frac{\eta}{1 + \xi^2 + \eta^2}, \frac{-1 + \xi^2 + \eta^2}{2 + 2\xi^2 + 2\eta^2}\right) \end{cases}</math>

If {{math|''g'' ∈ SO(3)}} is a rotation, then it will take points on {{math|'''S'''}} to points on {{math|'''S'''}} by its standard action {{math|Π<sub>''s''</sub>(''g'')}} on the embedding space <math>\R^3.</math> By composing this action with {{math|''S''}} one obtains a transformation {{math|''S'' ∘ Π<sub>''s''</sub>(''g'') ∘ ''S''<sup>−1</sup>}} of {{mvar|M}},

:<math>\zeta=P' \longmapsto P \longmapsto \Pi_s(g)P = gP \longmapsto S(gP) \equiv \Pi_u(g)\zeta = \zeta'.</math>

Thus {{math|Π<sub>''u''</sub>(''g'')}} is a transformation of <math>\Complex</math> associated to the transformation {{math|Π<sub>''s''</sub>(''g'')}} of <math>\R^3</math>.

It turns out that {{math|''g'' ∈ SO(3)}} represented in this way by {{math|Π<sub>''u''</sub>(''g'')}} can be expressed as a matrix {{math|Π<sub>''u''</sub>(''g'') ∈ SU(2)}} (where the notation is recycled to use the same name for the matrix as for the transformation of <math>\Complex</math> it represents). To identify this matrix, consider first a rotation {{math|''g''<sub>''φ''</sub>}} about the {{nowrap|{{math|''z''}}-axis}} through an angle {{mvar|''φ''}},

:<math>\begin{align}
x' &= x\cos \phi - y \sin \phi,\\
y' &= x\sin \phi + y \cos \phi,\\
z' &= z.
\end{align}</math>

Hence

:<math>\zeta' = \frac{x' + iy'}{\frac{1}{2} - z'} = \frac{e^{i\phi}(x + iy)}{\frac{1}{2} - z} = e^{i\phi}\zeta = \frac{e^{\frac{i\phi}{2}} \zeta + 0 }{0 \zeta + e^{-\frac{i\phi}{2}}},</math>

which, unsurprisingly, is a rotation in the complex plane. In an analogous way, if {{math|''g''<sub>''θ''</sub>}} is a rotation about the {{nowrap|{{math|''x''}}-axis}} through an angle {{mvar|θ}}, then

:<math>w' = e^{i\theta}w, \quad w = \frac{y + iz}{\frac{1}{2} - x},</math>

which, after a little algebra, becomes

:<math>\zeta' = \frac{\cos \frac{\theta}{2}\zeta +i\sin \frac{\theta}{2} }{i \sin\frac{\theta}{2}\zeta + \cos\frac{\theta}{2}}.</math>

These two rotations, <math>g_{\phi}, g_{\theta},</math> thus correspond to [[bilinear transform]]s of {{math|'''R'''<sup>2</sup> ≃ '''C''' ≃ ''M''}}, namely, they are examples of [[Möbius transformation]]s.

A general Möbius transformation is given by

:<math>\zeta' = \frac{\alpha \zeta + \beta}{\gamma \zeta + \delta}, \quad \alpha\delta - \beta\gamma \ne 0.</math>

The rotations, <math>g_{\phi}, g_{\theta}</math> generate all of {{math|SO(3)}} and the composition rules of the Möbius transformations show that any composition of <math>g_{\phi}, g_{\theta}</math> translates to the corresponding composition of Möbius transformations. The Möbius transformations can be represented by matrices

:<math>\begin{pmatrix}\alpha & \beta\\ \gamma & \delta\end{pmatrix}, \qquad \alpha\delta - \beta\gamma = 1,</math>

since a common factor of {{math|''α'', ''β'', ''γ'', ''δ''}} cancels.

For the same reason, the matrix is ''not'' uniquely defined since multiplication by {{math|−''I''}} has no effect on either the determinant or the Möbius transformation. The composition law of Möbius transformations follow that of the corresponding matrices. The conclusion is that each Möbius transformation corresponds to two matrices {{math|''g'', −''g'' ∈ SL(2, '''C''')}}.

Using this correspondence one may write

:<math>\begin{align}
\Pi_u(g_\phi) &= \Pi_u\left[\begin{pmatrix}
\cos \phi & -\sin \phi & 0\\
\sin \phi & \cos \phi & 0\\
0 & 0 & 1
\end{pmatrix}\right] = \pm
\begin{pmatrix}
e^{i\frac{\phi}{2}} & 0\\
0 & e^{-i\frac{\phi}{2}}
\end{pmatrix},\\
\Pi_u(g_\theta) &= \Pi_u\left[\begin{pmatrix}
1 & 0 & 0\\
0 & \cos \theta & -\sin \theta\\
0 & \sin \theta & \cos \theta
\end{pmatrix}\right] = \pm
\begin{pmatrix}
\cos\frac{\theta}{2} & i\sin\frac{\theta}{2}\\
i\sin\frac{\theta}{2} & \cos\frac{\theta}{2}
\end{pmatrix}.
\end{align}</math>

These matrices are unitary and thus {{math|Π<sub>''u''</sub>(SO(3)) ⊂ SU(2) ⊂ SL(2, '''C''')}}. In terms of [[Euler angles]]<ref group="nb">This is effected by first applying a rotation <math>g_{\theta}</math> through {{mvar|''φ''}} about the {{nowrap|{{math|''z''}}-axis}} to take the {{nowrap|{{math|''x''}}-axis}} to the line {{math|''L''}}, the intersection between the planes {{math|''xy''}} and {{math|''x'y'''}}, the latter being the rotated {{nowrap|{{math|''xy''}}-plane}}. Then rotate with <math>g_{\theta}</math> through {{mvar|θ}} about {{math|''L''}} to obtain the new {{nowrap|{{math|''z''}}-axis}} from the old one, and finally rotate by <math>g_{\psi}</math> through an angle {{mvar|ψ}} about the ''new'' {{nowrap|{{math|''z''}}-axis}}, where {{mvar|ψ}} is the angle between {{mvar|L}} and the new {{nowrap|{{math|''x''}}-axis}}. In the equation, <math>g_{\theta}</math> and <math>g_{\psi}</math> are expressed in a temporary ''rotated basis'' at each step, which is seen from their simple form. To transform these back to the original basis, observe that <math>\mathbf{g}_{\theta} = g_{\phi}g_{\theta}g_{\phi}^{-1}.</math> Here boldface means that the rotation is expressed in the ''original'' basis. Likewise, 
:<math>\mathbf{g}_{\psi} = g_{\phi}g_{\theta}g_{\phi}^{-1} g_{\phi} g_{\psi} \left [ g_{\phi}g_{\theta}g_{\phi}^{-1} g_{\phi} \right ]^{-1}.</math>
Thus 
:<math>\mathbf{g}_{\psi}\mathbf{g}_{\theta}\mathbf{g}_{\phi} = g_{\phi}g_{\theta}g_{\phi}^{-1} g_{\phi}g_{\psi} \left [g_{\phi} g_{\theta} g_{\phi}^{-1} g_{\phi} \right ]^{-1} * g_{\phi}g_{\theta}g_{\phi}^{-1}* g_{\phi} = g_{\phi}g_{\theta}g_{\psi}.</math></ref> one finds for a general rotation

{{NumBlk
|:| <math>\begin{align}
  g(\phi, \theta, \psi) = g_\phi g_\theta g_\psi
  &= \begin{pmatrix}
    \cos \phi & -\sin \phi & 0\\
    \sin \phi &  \cos \phi & 0\\
            0 &          0 & 1
  \end{pmatrix}\begin{pmatrix}
    1 &           0 &            0\\
    0 & \cos \theta & -\sin \theta\\
    0 & \sin \theta &  \cos \theta
  \end{pmatrix}\begin{pmatrix}
    \cos \psi & -\sin \psi & 0\\
    \sin \psi &  \cos \psi & 0\\
            0 &          0 & 1
  \end{pmatrix}\\
  &= \begin{pmatrix}
    \cos\phi\cos\psi - \cos\theta\sin\phi\sin\psi & -\cos\phi\sin\psi - \cos\theta\sin\phi\cos\psi & \sin\phi\sin\theta \\
    \sin\phi\cos\psi + \cos\theta\cos\phi\sin\psi & -\sin\phi\sin\psi + \cos\theta\cos\phi\cos\psi & -\cos\phi\sin\theta\\
    \sin\psi\sin\theta & \cos\psi\sin\theta & \cos\theta
  \end{pmatrix},
\end{align}</math>
| {{EquationRef|1}}
}}

one has<ref>These expressions were, in fact, seminal in the development of quantum mechanics in the 1930s, cf. Ch III,  § 16, B.L. van der Waerden, 1932/1932</ref>

{{NumBlk|:|<math>\begin{align}\Pi_u(g(\phi, \theta, \psi)) &= \pm
\begin{pmatrix}
e^{i\frac{\phi}{2}} & 0\\
0 & e^{-i\frac{\phi}{2}}
\end{pmatrix}
\begin{pmatrix}
\cos\frac{\theta}{2} & i\sin\frac{\theta}{2}\\
i\sin\frac{\theta}{2} & \cos\frac{\theta}{2}
\end{pmatrix}
\begin{pmatrix}
e^{i\frac{\psi}{2}} & 0\\
0 & e^{-i\frac{\psi}{2}}
\end{pmatrix}\\
&= \pm
\begin{pmatrix}
\cos\frac{\theta}{2}e^{i\frac{\phi + \psi}{2}} & i\sin\frac{\theta}{2}e^{i\frac{\phi - \psi}{2}}\\
i\sin\frac{\theta}{2}e^{-i\frac{\phi - \psi}{2}} & \cos\frac{\theta}{2}e^{-i\frac{\phi + \psi}{2}}
\end{pmatrix}.
\end{align}</math>|{{EquationRef|2}}}}

For the converse, consider a general matrix

:<math>\pm\Pi_u(g_{\alpha,\beta}) = \pm\begin{pmatrix} \alpha & \beta\\ -\overline{\beta} & \overline{\alpha} \end{pmatrix} \in \operatorname{SU}(2).</math>

Make the substitutions

:<math>\begin{align}
   \cos\frac{\theta}{2} &= |\alpha|,    &  \sin\frac{\theta}{2} &= |\beta|,    & (0 \le \theta \le \pi),\\
  \frac{\phi + \psi}{2} &= \arg \alpha, & \frac{\psi - \phi}{2} &= \arg \beta. &
\end{align}</math>

With the substitutions, {{math|Π(''g''<sub>''α'', ''β''</sub>)}} assumes the form of the right hand side ([[Right-hand side|RHS]]) of ({{EquationNote|2}}), which corresponds under {{math|Π<sub>''u''</sub>}} to a matrix on the form of the RHS of ({{EquationNote|1}}) with the same {{math|''φ'', ''θ'', ''ψ''}}. In terms of the complex parameters {{math|''α'', ''β''}},

:<math>g_{\alpha,\beta} = \begin{pmatrix}
   \frac{1}{2}\left( \alpha^2 - \beta^2 + \overline{\alpha^2} - \overline{\beta^2}\right) &
   \frac{i}{2}\left(-\alpha^2 - \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) &
                    -\alpha\beta - \overline{\alpha}\overline{\beta}\\

   \frac{i}{2}\left(\alpha^2 - \beta^2 - \overline{\alpha^2} + \overline{\beta^2}\right) &
   \frac{1}{2}\left(\alpha^2 + \beta^2 + \overline{\alpha^2} + \overline{\beta^2}\right) &
                  -i\left(+\alpha\beta - \overline{\alpha}\overline{\beta}\right)\\

           \alpha\overline{\beta}  + \overline{\alpha}\beta &
   i\left(-\alpha\overline{\beta}  + \overline{\alpha}\beta\right) &
           \alpha\overline{\alpha} - \beta\overline{\beta}
\end{pmatrix}.</math>

To verify this, substitute for {{math|''α''. ''β''}} the elements of the matrix on the RHS of ({{EquationNote|2}}). After some manipulation, the matrix assumes the form of the RHS of ({{EquationNote|1}}).

It is clear from the explicit form in terms of Euler angles that the map

:<math> \begin{cases}
p:\operatorname{SU}(2) \to \operatorname{SO}(3)\\
\pm \Pi_u(g_{\alpha \beta}) \mapsto g_{\alpha \beta}
\end{cases}</math>

just described is a smooth, {{math|2:1}} and surjective [[group homomorphism]]. It is hence an explicit description of the [[universal covering space]] of {{math|SO(3)}} from the [[universal covering group]] {{math|SU(2)}}.