Editing 3D rotation group (section)

== Baker–Campbell–Hausdorff formula ==
{{main|Baker–Campbell–Hausdorff formula}}
Suppose {{mvar|X}} and {{mvar|Y}} in the Lie algebra are given. Their exponentials, {{math|exp(''X'')}} and {{math|exp(''Y'')}}, are rotation matrices, which can be multiplied. Since the exponential map is a surjection, for some {{mvar|Z}} in the Lie algebra, {{math|1=exp(''Z'') = exp(''X'') exp(''Y'')}}, and one may tentatively write

:<math> Z = C(X, Y),</math>

for {{mvar|C}} some expression in {{math|''X''}} and {{math|''Y''}}. When {{math|exp(''X'')}} and {{math|exp(''Y'')}} commute, then {{math|1=''Z'' = ''X'' + ''Y''}}, mimicking the behavior of complex exponentiation.

The general case is given by the more elaborate [[BCH formula]], a series expansion of nested Lie brackets.<ref>{{Harvnb|Hall|2003|loc=Ch.&nbsp;3}}; {{Harvnb|Varadarajan|1984|loc=§2.15}}</ref> For matrices, the Lie bracket is the same operation as the [[commutator]], which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,<ref group="nb">For a full proof, see [[Derivative of the exponential map]]. Issues of convergence of this series to the correct element of the Lie algebra are here swept under the carpet. Convergence is guaranteed when <math>\|X\| + \|Y\| < \log 2 </math> and <math>\|Z\| < \log 2.</math> The series may still converge even if these conditions are not fulfilled. A solution always exists since {{math|exp}} is onto in the cases under consideration.</ref>

:<math>Z = C(X, Y) = X + Y + \frac{1}{2} [X, Y] + \tfrac{1}{12} [X, [X, Y]] - \frac{1}{12} [Y, [X, Y]] + \cdots.</math>

The infinite expansion in the BCH formula for {{math|SO(3)}} reduces to a compact form,

:<math>Z = \alpha X + \beta Y + \gamma[X, Y],</math>

for suitable trigonometric function coefficients {{math|(''α'', ''β'', ''γ'')}}.
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The {{math|(''α'', ''β'', ''γ'')}} are given by
:<math>\alpha = \phi \cot\left(\frac{\phi}{2}\right) \gamma, \qquad \beta = \theta \cot\left(\frac{\theta}{2}\right)\gamma, \qquad \gamma = \frac{\sin^{-1}d}{d}\frac{c}{\theta \phi},</math>
where

:<math>\begin{align}
  c &= \frac{1}{2}\sin\theta\sin\phi - 2\sin^2\frac{\theta}{2}\sin^2\frac{\phi}{2}\cos(\angle(u, v)),\quad a = c \cot\left(\frac{\phi}{2}\right), \quad b = c \cot\left(\frac{\theta}{2}\right), \\
  d &= \sqrt{a^2 + b^2 + 2ab\cos(\angle(u, v)) + c^2 \sin^2(\angle(u, v))},
\end{align}</math>

for

:<math>\theta = \|X\|,\quad \phi = \|Y\|,\quad \angle(u, v) = \cos^{-1}\frac{\langle X, Y\rangle}{\|X\|\|Y\|}.</math>

The inner product is the [[Hilbert–Schmidt inner product]] and the norm is the associated norm. Under the hat-isomorphism,

:<math>\langle u, v\rangle = \frac{1}{2}\operatorname{Tr}X^\mathrm{T}Y,</math>

which explains the factors for {{mvar|θ}} and {{mvar|φ}}. This drops out in the expression for the angle. {{see also|Rotation formalisms in three dimensions#Rodrigues parameters and Gibbs representation}}
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It is worthwhile to write this composite rotation generator as

:<math>\alpha X + \beta Y + \gamma[X, Y]\underset{\mathfrak{so}(3)}{=} X + Y + \frac{1}{2} [X, Y] + \frac{1}{12} [X, [X, Y]] - \frac{1}{12} [Y, [X, Y]] + \cdots,</math>

to emphasize that this is a ''Lie algebra identity''.

The above identity holds for all [[faithful representation]]s of {{math|𝖘𝖔(3)}}. The [[kernel (algebra)|kernel]] of a Lie algebra homomorphism is an [[ideal (Lie algebra)|ideal]], but {{math|𝖘𝖔(3)}}, being [[simple (abstract algebra)|simple]], has no nontrivial ideals and all nontrivial representations are hence faithful. It holds in particular in the doublet or spinor representation. The same explicit formula thus follows in a simpler way through Pauli matrices, cf. the [[Pauli matrices#Exponential of a Pauli vector|2×2 derivation for SU(2)]].

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=The SU(2) case}}
The [[Pauli matrices#Exponential of a Pauli vector|Pauli vector version]] of the same BCH formula is the somewhat simpler group composition law of SU(2),

:<math>
  e^{i a'\left(\hat{u} \cdot \vec{\sigma}\right)}e^{i b'\left(\hat{v} \cdot \vec{\sigma}\right)} =
  \exp\left(
    \frac{c'}{\sin c'} \sin a' \sin b' \left(\left(i\cot b'\hat{u} + i \cot a' \hat{v}\right)\cdot\vec{\sigma} +
    \frac{1}{2} \left[i \hat{u} \cdot \vec{\sigma}, i \hat{v} \cdot \vec{\sigma}\right]\right)
  \right),
</math>

where

:<math>\cos c' = \cos a' \cos b' - \hat{u} \cdot\hat{v} \sin a' \sin b',</math>

the [[spherical law of cosines]]. (Note {{math| '' a', b', c' ''}} are angles, not the {{math|''a'', ''b'', ''c''}} above.)

This is manifestly of the same format as above, 
:<math>Z = \alpha' X + \beta' Y + \gamma' [X, Y],</math>
with
:<math>X = i a'\hat{u} \cdot \mathbf{\sigma}, \quad Y = ib'\hat{v} \cdot \mathbf{\sigma} \in \mathfrak{su}(2),</math>
so that

:<math>\begin{align}
  \alpha' &= \frac{c'}{\sin c'}\frac{\sin a'}{a'}\cos b' \\ 
   \beta' &= \frac{c'}{\sin c'}\frac{\sin b'}{b'}\cos a' \\ 
  \gamma' &= \frac{1}{2}\frac{c'}{\sin c'}\frac{\sin a'}{a'}\frac{\sin b'}{b'}.
\end{align}</math>

For uniform normalization of the generators in the Lie algebra involved, express the Pauli matrices in terms of {{mvar|t}}-matrices, {{math|'''''σ'''''  → 2''i'' '''''t'''''}}, so that 
:<math>a' \mapsto -\frac{\theta}{2}, \quad b' \mapsto - \frac{\phi}{2}.</math>

To verify then these are the same coefficients as above, compute the ratios of the coefficients, 
:<math>\begin{align}
  \frac{\alpha'}{\gamma'} &= \theta\cot\frac{\theta}{2} &= \frac{\alpha}{\gamma}\\
   \frac{\beta'}{\gamma'} &= \phi\cot\frac{\phi}{2} &= \frac{\beta}{\gamma}.
\end{align}</math>
Finally, {{math|1= ''γ'' = ''γ' ''}} given the identity {{math|1= ''d'' = sin 2''c'''}}. 
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For the general {{math|''n'' × ''n''}} case, one might use Ref.<ref>{{harvnb|Curtright|Fairlie|Zachos|2014}} Group elements of SU(2) are expressed in closed form as finite polynomials of the Lie algebra generators, for all definite spin representations of the rotation group.</ref>

{{Hidden begin| titlestyle = color:green;background:lightgrey;|title=The quaternion case}}
The [[quaternion]] formulation of the composition of two rotations R<sub>B</sub> and R<sub>A</sub> also yields directly the [[Axis of rotation|rotation axis]] and angle of the composite rotation R<sub>C</sub> = R<sub>B</sub>R<sub>A</sub>.

Let the quaternion associated with a spatial rotation R is constructed from its [[Axis of rotation|rotation axis]] '''S''' and the rotation angle ''φ'' this axis. The associated quaternion is given by,

:<math>S = \cos\frac{\phi}{2} + \sin\frac{\phi}{2} \mathbf{S}.</math>

Then the composition of the rotation R<sub>R</sub> with R<sub>A</sub> is the rotation R<sub>C</sub> = R<sub>B</sub>R<sub>A</sub> with rotation axis and angle defined by the product of the quaternions

:<math>A = \cos\frac{\alpha}{2} + \sin\frac{\alpha}{2}\mathbf{A}\quad\text{ and }\quad B = \cos\frac{\beta}{2} + \sin\frac{\beta}{2}\mathbf{B},</math>

that is

:<math>
  C = \cos\frac{\gamma}{2} + \sin\frac{\gamma}{2}\mathbf{C} =
  \left(\cos\frac{\beta}{2} + \sin\frac{\beta}{2}\mathbf{B}\right)\left(\cos\frac{\alpha}{2} + \sin\frac{\alpha}{2}\mathbf{A}\right).</math>

Expand this product to obtain

:<math>
  \cos\frac{\gamma}{2} + \sin\frac{\gamma}{2} \mathbf{C} =
  \left(
    \cos\frac{\beta}{2}\cos\frac{\alpha}{2} -
    \sin\frac{\beta}{2}\sin\frac{\alpha}{2} \mathbf{B}\cdot \mathbf{A}
  \right) + \left(
    \sin\frac{\beta}{2}\cos\frac{\alpha}{2} \mathbf{B} +
    \sin\frac{\alpha}{2}\cos\frac{\beta}{2} \mathbf{A} +
    \sin\frac{\beta}{2}\sin\frac{\alpha}{2} \mathbf{B} \times \mathbf{A}
  \right).
</math>

Divide both sides of this equation by the identity, which is the [[spherical law of cosines|law of cosines on a sphere]],

:<math>\cos\frac{\gamma}{2} = \cos\frac{\beta}{2}\cos\frac{\alpha}{2} - \sin\frac{\beta}{2}\sin\frac{\alpha}{2} \mathbf{B}\cdot \mathbf{A},</math>

and compute

:<math>\tan\frac{\gamma}{2} \mathbf{C} = \frac{\tan\frac{\beta}{2}\mathbf{B} + \tan\frac{\alpha}{2} \mathbf{A} + \tan\frac{\beta}{2}\tan\frac{\alpha}{2} \mathbf{B} \times \mathbf{A}}{1 - \tan\frac{\beta}{2}\tan\frac{\alpha}{2} \mathbf{B} \cdot \mathbf{A}}.</math>

This is Rodrigues' formula for the axis of a composite rotation defined in terms of the axes of the two rotations. He derived this formula in 1840 (see page 408).<ref>Rodrigues, O. (1840), Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace, et la variation des coordonnées provenant de ses déplacements con- sidérés indépendamment des causes qui peuvent les produire, Journal de Mathématiques Pures et Appliquées de Liouville 5, 380–440.</ref>

The three rotation axes '''A''', '''B''', and '''C''' form a spherical triangle and the dihedral angles between the planes formed by the sides of this triangle are defined by the rotation angles.

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