compactification of ${\mathrm{SE}_3}$ turns out to be extremely useful in the study of mobility properties of objects coming from robotics and kinematics (see for example [@HegeduesSchichoSchroecker], [@Nawratil2014]). However, in our situation we will see that a different compactification will lead us to a better comprehension of the phenomena which can arise.

Any isometry of ${\mathbb{R}}^3$ can be written as a pair $(M,y)$, where $M \in {\mathrm{SO}_3}$ is the linear contribution and $y \in {\mathbb{R}}^3$ is the image of the origin $o \in {\mathbb{R}}^3$. We define $x := -M^t y = -M^{-1} y$ and $r := \langle
x,x \rangle = \langle y,y \rangle$, where $\langle \cdot, \cdot \rangle$ is the Euclidean scalar product. The isometry $(M, y)$ is considered as a point in ${\mathbb{P}}^{16}_{{\mathbb{R}}}$ with coordinates

-   $m_{11}, \dots, m_{33}$ (the entries of the matrix),

-   $x_1,\dots,x_3$ (the coordinates of $x$),

-   $y_1,\dots,y_3$ (the coordinates of $y$),

-   $r$ and $h$ (a homogenization coordinate; for group elements we can assume that it is equal to $1$).

The group ${\mathrm{SE}_3}$ is defined by the inequality $h \ne 0$ and equations $$M M^t = M^t M = h^2 \cdot \mathrm{id}_{{\mathbb{R}}^3}, \;\; \det(M)=h^3,$$ $$M^t y + h x = 0, \;\; M x + 