< m < n-1$ such that $p_1, \ldots, p_m$ lie on a line $g \subseteq {\mathbb{R}}^3$ and $p_{m+1}, \ldots, p_n$ lie on a line $g'
    \subseteq {\mathbb{R}}^3$ parallel to $g$, and $P_1, \ldots, P_m$ lie on a line $G
    \subseteq {\mathbb{R}}^3$ and $P_{m+1}, \ldots, P_n$ lie on a line $G' \subseteq {\mathbb{R}}^3$ parallel to $G$, up to permutation of indices.

This last result, in particular condition (a), is the starting point of further investigations on pentapods with mobility $2$, which are carried on in [@GalletNawratilSchicho], relying on a new technique called *Möbius Photogrammetry*.

Compactification of ${\mathrm{SE}_3}$ {#compactification}
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We start our discussion in Subsection \[compactification:new\] by introducing a new compactification of the group of direct isometries of ${\mathbb{R}}^3$ in a projective space. Then in Subsection \[compactification:action\] we study the natural action of isometries on this compactification, and we prove that it is given by linear changes of coordinates. At last in Subsection \[compactification:boundary\] we describe the boundary of the compactification.

A new compactification {#compactification:new}
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We study the 6-dimensional algebraic group ${\mathrm{SE}_3}$ of direct isometries of affine $3$–space ${\mathbb{R}}^3$ into itself. One can embed ${\mathrm{SE}_3}$ as an open subset of a quadric hypersurface in ${\mathbb{P}}^7_{{\mathbb{R}}}$, called the *Study quadric*. This 