Editing Rodrigues' rotation formula (section)

== Matrix notation ==

The linear transformation on <math>\mathbf{v}\isin\mathbb{R}^3 </math> defined by the cross product <math>\mathbf{v} \mapsto \mathbf{k} \times \mathbf{v} </math> is given in coordinates by representing {{math|'''v'''}} and {{math|'''k''' × '''v'''}} as [[row and column vectors|column matrices]]:

:<math>\begin{bmatrix} (\mathbf{k}\times\mathbf{v})_x \\ (\mathbf{k}\times\mathbf{v})_y 
\\ (\mathbf{k}\times\mathbf{v})_z \end{bmatrix} 
= \begin{bmatrix} k_y v_z - k_z v_y \\ k_z v_x - k_x v_z \\ k_x v_y - k_y v_x \end{bmatrix} 
= \left[\begin{array}{rrr}
0\ \,  & -k_z & k_y \\
k_z & 0\ \, & -k_x \\
-k_y & k_x & 0\ \,  
\end{array}\right]
\begin{bmatrix} v_x \\ v_y \\ v_z \end{bmatrix} \,. </math>

That is, the [[Transformation matrix|matrix of this linear transformation]] (with respect to standard coordinates) is the [[Cross product#Conversion to matrix multiplication|cross-product matrix]]:
: <math>\mathbf{K}= 
\left[\begin{array}{rrr}
0\ \,  & -k_z & k_y \\
k_z & 0\ \, & -k_x \\
-k_y & k_x & 0\ \,  
\end{array}\right]\,.
</math>
That is to say,
: <math>\mathbf{k}\times\mathbf{v}=\mathbf{K}\mathbf{v}, 
\qquad\qquad
\mathbf{k}\times(\mathbf{k}\times\mathbf{v})=\mathbf{K}(\mathbf{K}\mathbf{v}) = \mathbf{K}^2\mathbf{v} \,. </math>
The last formula in the previous section can therefore be written as:

:<math>\mathbf{v}_{\mathrm{rot}} = \mathbf{v} + (\sin\theta) \mathbf{K}\mathbf{v} + (1 - \cos\theta)\mathbf{K}^2\mathbf{v}\,.</math>
Collecting terms allows the compact expression
:<math>\mathbf{v}_\mathrm{rot} = \mathbf{R}\mathbf{v}</math>

where
{{Equation box 1
|indent =:
|equation = <math>\mathbf{R} = \mathbf{I} + (\sin\theta) \mathbf{K} + (1-\cos\theta)\mathbf{K}^2</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7
}}
is the [[rotation matrix]] through an angle {{mvar|θ}} counterclockwise about the axis {{math|'''k'''}}, and {{math|'''I'''}} the {{nowrap|3 × 3}} [[identity matrix]].<ref>{{Cite web|last=Belongie|first=Serge|title=Rodrigues' Rotation Formula|url=https://mathworld.wolfram.com/RodriguesRotationFormula.html|access-date=2021-04-07|website=mathworld.wolfram.com|language=en}}</ref> This matrix {{math|'''R'''}} is an element of the rotation group {{math|SO(3)}} of {{math|ℝ<sup>3</sup>}}, and {{math|'''K'''}} is an element of the [[Lie algebra]] <math>\mathfrak{so}(3)</math> generating that Lie group (note that {{math|'''K'''}} is skew-symmetric, which characterizes <math>\mathfrak{so}(3)</math>).

In terms of the matrix exponential,
:<math>\mathbf{R} = \exp (\theta\mathbf{K})\,.</math>

To see that the last identity holds, one notes that
:<math>\mathbf{R}(\theta) \mathbf{R}(\phi) = \mathbf{R} (\theta+\phi), \quad \mathbf{R}(0) = \mathbf{I}\,, </math>
characteristic of a [[one-parameter subgroup]], i.e. exponential, and that the formulas match for infinitesimal {{mvar|θ}}.

For an alternative derivation based on this exponential relationship, see [[Axis–angle representation#Exponential map from so(3) to SO(3)|exponential map from <math>\mathfrak{so}(3)</math> to {{math|SO(3)}}]]. For the inverse mapping, see [[Axis–angle representation#Log map from SO(3) to so(3)|log map from {{math|SO(3)}} to <math>\mathfrak{so}(3)</math>]].

The above result can be written in index notation as follows. The elements of the matrix for an active rotation by an angle <math>\theta</math> about an axis {{math|'''n'''}} are given by
:<math>
R_{ij} = \cos\theta\, \delta_{ij} + (1 - \cos\theta) n_i n_j - \sin\theta\, \epsilon_{ijk} n_k.
</math>
Here, i, j, and k label the Cartesian components (x, y, z) or (1, 2, 3), <math>\delta_{ij}</math> and <math>\epsilon_{ijk}</math> are the Kronecker and Levi-Civita symbols, and there is an implicit sum on repeated indices.

The [[Hodge dual]] of the rotation <math>\mathbf{R}</math> is just <math>\mathbf{R}^* = -\sin(\theta)\mathbf{k}</math> which  enables the extraction of both the axis of rotation and the sine of the angle of the rotation from the rotation matrix itself, with the usual ambiguity,
:<math>\begin{align}
  \sin(\theta) &=  \sigma \left|\mathbf{R}^*\right| \\[3pt]
    \mathbf{k} &= -\frac{\sigma\mathbf{R}^*}{\left|\mathbf{R}^*\right|}
\end{align}</math>
where <math>\sigma = \pm 1</math>. The above simple expression results from the fact that the Hodge duals of <math>\mathbf{I}</math> and <math>\mathbf{K}^2</math> are zero, and <math>\mathbf{K}^* = -\mathbf{k}</math>.