give them geometric meaning, since their presence as limits of self–motion determine geometric conditions the base and platform points have to satisfy.

Recall from Section \[compactification\] that $\Phi({\mathrm{SE}_3})$ is an embedding of ${\mathrm{SE}_3}$ in ${\mathbb{P}}^{16}_{{\mathbb{C}}}$, that we denoted by ${\mathrm{SE}_{3,{\mathbb{C}}}}$ its complexification and that we defined $X$ as the Zariski closure of ${\mathrm{SE}_{3,{\mathbb{C}}}}$. Moreover, recall that we think of an $n$–pod as a triple $$\Pi = \Big( (p_1, \dots, p_n),(P_1, \dots, P_n),(d_1, \dots, d_n) \Big)$$ Eventually, recall that in the new coordinates of ${\mathbb{P}}^{16}_{{\mathbb{C}}}$ the spherical condition given by Equation reads as $$\label{equation:leg_condition}
    d_i^2 {\thinspace}h \; = \; \big( \langle p_i, p_i\rangle {\thinspace}- \langle P_i, P_i
\rangle \big) {\thinspace}h + r - 2 \langle p_i, x\rangle - 2 \langle y, P_i \rangle
-2 \langle M p_i, P_i \rangle,$$ which gives a linear form $l_i$ on ${\mathbb{P}}^{16}_{{\mathbb{C}}}$.

Let $\Pi$ be an $n$–pod, then the real points of the intersection $${\mathrm{SE}_{3,{\mathbb{C}}}}\cap \big\{ l_i \, = \, 0 \big\}$$ are in bijective correspondence with the set of all $\sigma \in {\mathrm{SE}_3}$ such that the distance between $\sigma(p_i)$ and $P_i$ is $d_i$, namely the set of direct isometries satisfying the spherical condition for $p_i$ and $P_i$.

Let $\Pi$ be an $n$–pod, then the intersection 