A question about the standard euclidean group $\mathbb{SE}(3)$

I have a little perplexity about the following fact:

If we consider the standard euclidean group $\mathbb{SE}(3)$, an element $g$ can be represented by a matrix

$$\mathbb{SE}(n) \ni g=\begin{pmatrix}R & v \\ 0 & 1\end{pmatrix} \, \, , \, \, R \in \mathbb{SO}(3) \, , \, v \in \mathbb{R}^3.$$

So $g$ can be thought as a mapping $g : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ parametrized by $R,v$.

What I don't understand is, isn't the $\mathbb{SE}(3)$ group originally used to describe a rotation + translation in a $3$-dimensional space? Or should we use $\mathbb{SE}(2)$ to describe such transformations in $\mathbb{R}^3$?

Solution 1:

The additional last row is only to write it conveniently as a matrix group. Then we can also take the vector $w\in \Bbb R^3$ with one component more and have the action on $\Bbb R^4$ as $$ \begin{pmatrix} R & v \cr 0 & 1 \end{pmatrix} \begin{pmatrix} w \cr 1 \end{pmatrix}= \begin{pmatrix} Rw+v \cr 1\end{pmatrix}. $$ But of course we can still view this just as the action on $\Bbb R^3$.