+\langle
P_i,P_i\rangle
    -2\langle M p_i, P_i \rangle - 2 \langle y, P_i\rangle \\
    & = \langle p_i, p_i \rangle -\langle P_i, P_i\rangle + r + 2\langle p_i,M^t
y \rangle -2\langle M p_i, P_i \rangle - 2 \langle y, P_i \rangle \\
    & = \langle p_i, p_i\rangle -\langle P_i, P_i \rangle + r - 2\langle
p_i, x\rangle - 2 \langle y, P_i \rangle -2\langle M p_i, P_i\rangle.
\end{aligned}$$

After homogenization, Equation (\[equation:sphere\_condition\]) becomes linear in the projective coordinates of ${\mathbb{P}}^{16}_{{\mathbb{R}}}$.

By introducing this compactification of ${\mathrm{SE}_3}$ we reduced the problem of dealing with Equation (\[equation:spherical\_condition\]) to the problem of understanding linear equations on $X_{{\mathbb{R}}} \subseteq {\mathbb{P}}^{16}_{{\mathbb{R}}}$. In order to fully use the techniques from algebraic geometry and to be able to set up bond theory, we need to extend our ground field to the complex numbers.

From now on we work with the complexification of $X_{{\mathbb{R}}}$, denoted by $X$. In order to do this we simply take the equations defining $X$ (which have real coefficients), and we think of them as polynomials over ${\mathbb{C}}$. Hence what we get is a projective variety in ${\mathbb{P}}^{16}_{{\mathbb{C}}}$ of complex dimension $6$ and degree $40$ whose real points are in bijection with the points of 