h y = 0 ,$$ $$\langle x,x \rangle = \langle y,y \rangle = r h$$ (not all equations are needed, for instance $M^t y + h x = 0$ is a consequence of the other equations and the inequality). We define $X_{{\mathbb{R}}}$ as the Zariski closure of ${\mathrm{SE}_3}$, i.e. the zero set of the set of all equations vanishing at ${\mathrm{SE}_3}$. Using computer algebra, a Gröbner basis for this set of equations can be computed[^2]. The degree of $X_{{\mathbb{R}}}$ can also be computed using computer algebra by the leading monomials of the Gröbner basis: it is $40$.

In the first version of this paper we constructed a projectively isomorphic compactification of ${\mathrm{SE}_3}$ using Study parameters. However it turned out that the construction above is more direct, computationally simpler and easier to generalize to higher dimensions.

We consider the spherical conditions $\left\| \sigma(p_i) - P_i \right\| = d_i$ we want to impose to rigid motions in ${\mathbb{R}}^3$; if we set $h = 1$, this can be expressed by: $$\label{equation:sphere_condition}
\begin{aligned}
    d_i^2 & = \langle M p_i + y - P_i,M p_i + y - P_i \rangle \\
    & = \langle M p_i,M p_i\rangle + 2\langle M p_i, y \rangle + r 