Motions in the plane

Source: bonn-3D-cs

Rotation of $\mathcal{X}$ to $\mathcal{X}^\prime$
$$ \begin{align} \left[ \begin{array} ~x \ y \end{array} \right] &= \left[ \begin{array} ~r\cos \alpha \ y\cos \alpha \end{array} \right]\

\mathcal{X}^\prime = \mathcal{R}(\mathcal{X}) : ~ \left[ \begin{array} ~x^\prime \ y^\prime \end{array} \right] &= \left[ \begin{array} ~\cos\gamma & -\sin\gamma \
\sin\gamma & \cos\gamma \end{array} \right] \left[ \begin{array} ~x \ y \end{array} \right] \end{align} $$

Basic homogeneous transformations

Translation of $\mathbf{t}$

$$ \begin{align} \mathbf{x}^\prime &= \mathbf{T} \mathbf{x} \
\text{with } \mathbf{T} &= \left[ \begin{array} \mathbf{I} & \mathbf{t}\
\mathbf{0}^\text{T} & 1 \end{array} \right] \end{align} $$

Rotation around origin by $\varphi$

$$ \begin{align} \mathbf{x}^\prime &= \mathbf{T} \mathbf{x} \
\text{with } \mathbf{T} &= \left[ \begin{array} \mathbf{R} & \mathbf{0}\
\mathbf{0}^\text{T} & 1 \end{array} \right]

\left[ \begin{array} \cos\varphi & -\sin\varphi & 0\
\sin\varphi & \cos\varphi & 0\
0 & 0 & 1 \end{array} \right] \end{align} $$

Inverse homogeneous transformations

Translation

$$ \mathbf{T}^{-1}

\left[ \begin{array} \mathbf{I} & -\mathbf{t}\
\mathbf{0}^\text{T} & 1 \end{array} \right] $$

Rotation

$$ \mathbf{T}^{-1}

\left[ \begin{array} \mathbf{R}^{-1} & \mathbf{0}\
\mathbf{0}^\text{T} & 1 \end{array} \right]

\left[ \begin{array} \mathbf{R}^{\text{T}} & \mathbf{0}\
\mathbf{0}^\text{T} & 1 \end{array} \right] $$