Editing Euler's rotation theorem (section)

==Applications==
===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.

===Quaternions===
{{Main|Three-dimensional rotation operator|Quaternions and spatial rotation}}

It follows from Euler's theorem that the relative orientation of any pair of coordinate systems may be specified by a set of three independent numbers. Sometimes a redundant fourth number is added to simplify operations with quaternion algebra. Three of these numbers are the direction cosines that orient the eigenvector. The fourth is the angle about the eigenvector that separates the two sets of coordinates. Such a set of four numbers is called a '''[[quaternion]]'''.

While the quaternion described above does not involve [[complex number]]s, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative [[quaternion]] algebra derived by [[William Rowan Hamilton]] through the use of imaginary numbers.

Rotation calculation via quaternions has come to replace the use of [[direction cosines]] in aerospace applications through their reduction of the required calculations, and their ability to minimize [[round-off error]]s. Also, in [[computer graphics]] the ability to perform spherical interpolation between quaternions with relative ease is of value.