SE(n)

The set of all transformations in \(n\) dimensional Euclidean space which preserve the Euclidean distance between all points once applied.

The case \(n=3\) is used to represent rigid body transformations. Namely, let \(T = \begin{bmatrix} R & p\\ \textbf{0} & 1 \end{bmatrix}\) where \(R\) is from SO(n) with \(n=3\), \(p \in \mathbb{R}^3\), \(\textbf{0}\) is a row vector of zeros and 1 is simply the scalar. All possible matrices \(T\) define \(SE(3)\), and is closed under matrix multiplication. In the context of robotics they are called homogeneous transforms, and are often used to:

• Represent a configuration \(T_{sb} = \begin{bmatrix}R_{sb} & p\\ \textbf{0} & 1 \end{bmatrix}\)
• Change reference frames \(T_{sc} = T_{sb}T_{bc}\).
• Displace frames and vectors, where points \(\mathbf{p} \in \mathbb{R}^3\) are first converted into $\begin{bmatrix} \mathbf{p}
  1

\end{bmatrix}$