# Why is the notation (R|RT) for “first apply translation then rotation”?

I'm re-learning math as I go here.

The lecture notes The Pinhole Camera are explaining how to transform between 3D spatial coordinates and pixel coordinates of a camera. I've just run across some notation I don't recognize, namely $$(R|RT)$$ where R and T are matrices for rotation and translation respectively in this particular case.

note 1: The procedure involves homogeneous coordinates, but this matrix will only be 3 × 4 and so no final row of $\begin{matrix} 0 & 0 & 0 & 1 \\ \end{matrix}$. It might look something like the following, but I'm just guessing:

$$\begin{matrix} r_{11} & r_{12} & r_{13} & t_1 \\ r_{21} & r_{22} & r_{23} & t_2 \\ r_{31} & r_{32} & r_{33} & t_3 \\ \end{matrix}$$

note 2: Could it be an augmented matrix?

Today I'm far from a library, and searching for "matrix notation" hasn't helped me find an explanation.

I don't need a long answer, probably a sentence or two and/or a judiciously chosen link is all I need for an "Aha!" moment (provided that it's not at a high level) so I can keep going.

On page 2:

Now if the camera does not have its center of projection at $(0, 0, 0)$ and is oriented in an arbitrary fashion (not necessarily z perpendicular to the image plane), then we need a rotation and translation to make the camera coordinate system coincide with the configuration in Figure 1. Let the camera translation to origin of the $XYZ$ coordinate be given by $T(T_x, T_y, T_z)$. Let the the rotation applied to coincide the principal axis with $Z$ axis be given by a 3 × 3 rotation matrix $R$. Then the matrix formed by first applying the translation followed by the rotation is given by the 3 × 4 matrix

$$E = (R | RT)$$

called the extrinsic parameter matrix. So, the complete camera transformation can now represented as

$$K(R | RT) = (KR | KRT) = KR(I | T)$$

Hence Pc, the projection of P is given by

$$Pc = KR(I | T)P = CP$$

C is a 3 × 4 matrix usually called the complete camera calibration matrix. Note that since C is 3 × 4 we need P to be in 4D homogeneous coordinates and Pc derived by CP will be in 3D homogeneous coordinates. The exact 2D location of the projection on the camera image plane will be obtained by dividing the first two coordinates of Pc by the third.

Note that the vector (or $3\times 1$ column matrix) $RT$ is the translation vector after applying the rotation. In particular it is still a $3\times 1$ column matrix. The notation $(R\vert RT)$ refers to the combination of two matrices into a single one. That is, $(R\vert RT)$ is the $3\times 4$ matrix whose first three columns are given by the matrix $R$ and the last column by $RT$.
Explicitly, for $$R=\begin{bmatrix}r_{11}&r_{12}&r_{13}\\r_{21}&r_{22}&r_{23}\\r_{31}&r_{32}&r_{33}\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}t_1\\t_2\\t_3\end{bmatrix},$$ the matrix is given by $$(R\vert RT) = \begin{bmatrix}r_{11}&r_{12}&r_{13}&r_{11}t_1+r_{12}t_2+r_{13}t_3\\r_{21}&r_{22}&r_{23}&r_{21}t_1+r_{22}t_2+r_{23}t_3\\r_{31}&r_{32}&r_{33}&r_{31}t_1+r_{32}t_2+r_{33}t_3\end{bmatrix}.$$