Definition (Transform)

Given the Pose of one Reference Frame relative to another, we can transform the Coordinates, $\mathbf{r}_{v}^{pv}$, of a point $P$ expressed in some frame $\vec{\mathcal{F}}_{v}$ to another frame $\vec{\mathcal{F}}_{i}$ in the following fashion: $$\mathbf{r}_{i}^{pi}=\mathbf{r}_{i}^{pv}+\mathbf{r}_{i}^{vi}=\mathbf{C}_{vi}^{\top}\mathbf{r}_{v}^{pv}+\mathbf{r}_{i}^{vi}=\mathbf{C}_{vi}\mathbf{r}_{v}^{pv}+\mathbf{r}_{i}^{vi}$$ or in block matrix form: $$\begin{bmatrix}\mathbf{r}_{i}^{pi} \\ \mathbf{1}\end{bmatrix}=\begin{bmatrix}\mathbf{C}_{vi} & \mathbf{r}_{i}^{vi} \\ \mathbf{0}^{\top} & \mathbf{1}\end{bmatrix}=\mathbf{T}_{iv}\cdot\begin{bmatrix}\mathbf{r}_{v}^{pv} \\ \mathbf{1}\end{bmatrix}$$where $\mathbf{T}_{iv}$ is the transformation matrix to frame $i$ from frame $v$. A transformation matrix is a representation of a Rotation and a translation as a $4\times4$ matrix.