Editing Euler's rotation theorem (section)

===Generators of rotations===
{{Main|Rotation matrix|Rotation group SO(3)|Infinitesimal transformation}}

Suppose we specify an axis of rotation by a unit vector {{math|[''x'', ''y'', ''z'']}}, and suppose we have an [[Infinitesimal rotation|infinitely small rotation]] of angle {{math|Δ''θ''}} about that vector. Expanding the rotation matrix as an infinite addition, and taking the first order approach, the rotation matrix {{math|Δ''R''}} is represented as:

: <math>
 \Delta R =
 \begin{bmatrix}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1
 \end{bmatrix} +
 \begin{bmatrix}
  0 &  z & -y \\
 -z &  0 &  x \\
  y & -x &  0
 \end{bmatrix}\,\Delta \theta =
  \mathbf{I} + \mathbf{A}\,\Delta \theta.
</math>

A finite rotation through angle {{mvar|θ}} about this axis may be seen as a succession of small rotations about the same axis. Approximating {{math|Δ''θ''}} as {{math|{{sfrac|''θ''|''N''}}}} where {{math|''N''}} is a large number, a rotation of {{mvar|θ}} about the axis may be represented as:

:<math>R = \left(\mathbf{1}+\frac{\mathbf{A}\theta}{N}\right)^N \approx e^{\mathbf{A}\theta}.</math>

It can be seen that Euler's theorem essentially states that ''all'' rotations may be represented in this form. The product {{math|'''A'''''θ''}} is the "generator" of the particular rotation, being the vector {{math|(''x'',''y'',''z'')}} associated with the matrix {{math|'''A'''}}. This shows that the rotation matrix and the [[Axis–angle representation|axis–angle]] format are related by the exponential function.

One can derive a simple expression for the generator {{math|'''G'''}}. One starts with an arbitrary plane (in Euclidean space) defined by a pair of perpendicular unit vectors {{math|'''a'''}} and {{math|'''b'''}}. In this plane one can choose an arbitrary vector {{math|'''x'''}} with perpendicular {{math|'''y'''}}. One then solves for {{math|'''y'''}} in terms of {{math|'''x'''}} and substituting into an expression for a rotation in a plane yields the rotation matrix {{math|'''R'''}} which includes the generator {{nowrap|{{math|'''G''' {{=}} '''ba'''}}{{sup|T}}{{math| − '''ab'''}}{{sup|T}}}}.

:<math>\begin{align}
   \mathbf{x} &=  \mathbf{a}\cos\alpha + \mathbf{b}\sin\alpha \\
   \mathbf{y} &= -\mathbf{a}\sin\alpha + \mathbf{b}\cos\alpha \\[8pt]
   \cos\alpha &=  \mathbf{a}^\mathsf{T}\mathbf{x} \\
   \sin\alpha &=  \mathbf{b}^\mathsf{T}\mathbf{x} \\[8px]
  \mathbf{y}  &= -\mathbf{ab}^\mathsf{T}\mathbf{x} + \mathbf{ba}^\mathsf{T}\mathbf{x}
               =  \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right)\mathbf{x} \\[8px]
  \mathbf{x}' &= \mathbf{x}\cos\beta + \mathbf{y}\sin\beta \\ 
              &=  \left( \mathbf{I}\cos\beta + \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right) \sin\beta \right)\mathbf{x} \\[8px] 
   \mathbf{R} &=  \mathbf{I}\cos\beta + \left( \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} \right)\sin\beta \\ 
              &=  \mathbf{I}\cos\beta + \mathbf{G}\sin\beta \\[8px] 
   \mathbf{G} &=  \mathbf{ba}^\mathsf{T} - \mathbf{ab}^\mathsf{T} 
\end{align}</math>

To include vectors outside the plane in the rotation one needs to modify the above expression for {{math|'''R'''}} by including two [[Projection (linear algebra)|projection operators]] that partition the space. This modified rotation matrix can be rewritten as an [[Matrix exponential#Rotation case|exponential function]].

:<math>\begin{align}
  \mathbf{P_{ab}} &= -\mathbf{G}^2 \\ 
       \mathbf{R} &= \mathbf{I} - \mathbf{P_{ab}} + \left( \mathbf{I} \cos \beta + \mathbf{G} \sin \beta \right)\mathbf{P_{ab}} = e^{\mathbf{G}\beta } 
\end{align}</math>

Analysis is often easier in terms of these generators, rather than the full rotation matrix. Analysis in terms of the generators is known as the [[Lie algebra]] of the rotation group.