Transform Matrix

Structure of the Transforms Matrix

• The upper left corner is the rotation matrix
• The right side is the translation vector
• The lower left corner is $0$ vector
• The lower right corner is $1$

This set of transformation matrices forms the special Euclidean group:

$$\text{SE}(3)=\left\{\begin{array}{c}\mathbf{T}=\left[\begin{array}{cc}\mathbf{R}& t\\ 0^T& 1\end{array}\right]\in {\mathbb{R}}^{4\times 4}|\mathbf{R}\in \text{SO}(3),\mathbf{t}\in {\mathbb{R}}^3\end{array}\right\}$$

Here, the $\mathbf{R}$ rotation matrix is using the same special orthogonal group definition from Rotation Matrix.

Inverse

The inverse of the transform matrix also represents an inverse transformation:

$${\mathbf{T}}^{-1}=\left[\begin{array}{cc}{\mathbf{R}}^T& -{\mathbf{R}}^T\mathbf{t}\\ 0^T& 1\end{array}\right]$$

Notation

• We use ${\mathbf{T}}_{12}$ to represent the transformation from $2$ to $1$.
• When using $\mathbf{Ta}$, we use homogeneous coordinates.
• When using $\mathbf{Ra}$, we use non-homogeneous coordinates.
• If they are written in the same equation, it is assumed that the conversion from homogeneous coordinates to normal coordinates is already done