# der_R_arbitrary_axis_times_v¶

Linearised rotation vector of the vector v by angle theta about an arbitrary axis u.

The rotation of a vector $$\mathbf{v}$$ about the axis $$\mathbf{u}$$ by an angle $$\boldsymbol{\theta}$$ can be expressed as

$\mathbf{w} = \mathbf{R}(\mathbf{u}, \theta) \mathbf{v},$

where $$\mathbf{R}$$ is a $$\mathbb{R}^{3\times 3}$$ matrix.

This expression can be linearised for it to be included in the linear solver as

$\delta\mathbf{w} = \frac{\partial}{\partial\theta}\left(\mathbf{R}(\mathbf{u}, \theta_0)\right)\delta\theta$

The matrix $$\mathbf{R}$$ is

$\begin{split}\mathbf{R} = \begin{bmatrix}\cos \theta +u_{x}^{2}\left(1-\cos \theta \right) & u_{x}u_{y}\left(1-\cos \theta \right)-u_{z}\sin \theta & u_{x}u_{z}\left(1-\cos \theta \right)+u_{y}\sin \theta \\ u_{y}u_{x}\left(1-\cos \theta \right)+u_{z}\sin \theta & \cos \theta +u_{y}^{2}\left(1-\cos \theta \right)& u_{y}u_{z}\left(1-\cos \theta \right)-u_{x}\sin \theta \\ u_{z}u_{x}\left(1-\cos \theta \right)-u_{y}\sin \theta & u_{z}u_{y}\left(1-\cos \theta \right)+u_{x}\sin \theta & \cos \theta +u_{z}^{2}\left(1-\cos \theta \right)\end{bmatrix},\end{split}$

and its linearised expression becomes

$\begin{split}\frac{\partial}{\partial\theta}\left(\mathbf{R}(\mathbf{u}, \theta_0)\right) = \begin{bmatrix} -\sin \theta +u_{x}^{2}\sin \theta \mathbf{v}_1 + u_{x}u_{y}\sin \theta-u_{z} \cos \theta \mathbf{v}_2 + u_{x}u_{z}\sin \theta +u_{y}\cos \theta \mathbf{v}_3 \\ u_{y}u_{x}\sin \theta+u_{z}\cos \theta\mathbf{v}_1 -\sin \theta +u_{y}^{2}\sin \theta\mathbf{v}_2 + u_{y}u_{z}\sin \theta-u_{x}\cos \theta\mathbf{v}_3 \\ u_{z}u_{x}\sin \theta-u_{y}\cos \theta\mathbf{v}_1 + u_{z}u_{y}\sin \theta+u_{x}\cos \theta\mathbf{v}_2 -\sin \theta +u_{z}^{2}\sin\theta\mathbf{v}_3\end{bmatrix}_{\theta=\theta_0}\end{split}$

and is of dimension $$\mathbb{R}^{3\times 1}$$.

param u

Arbitrary rotation axis

type u

numpy.ndarray

param theta

type theta

float

param v

Vector to rotate

type v

numpy.ndarray

returns

Linearised rotation vector of dimensions $$\mathbb{R}^{3\times 1}$$.

rtype

numpy.ndarray