group of direct isometries of ${\mathbb{R}}^3$, and in particular we embed and compactify it in a projective space in a way specifically tuned for $n$–pods. The key idea behind all of this is that, if we introduce suitable coordinates in the space of direct isometries and we consider the condition given by Equation (\[equation:spherical\_condition\]) then the latter becomes linear in these coordinates. These coordinates provide the desired compactification, and Section \[compactification\] is devoted to the study of some of its geometric properties, which play an important role in the proofs of the results of the subsequent section. In particular, we focus our attention on the natural action of direct isometries on this compactification and on its boundary. In Section \[bond\_geometric\] we use this information to establish some results which can be framed in the so–called *bond theory* (see [@HegeduesSchichoSchroecker], [@Nawratil2014]): the presence of some boundary point in the projective closure of the set of self–motions of an $n$–pod implies precise geometric constraints on base and platform points. The set of boundary points is a $5$–dimensional complex algebraic variety, but we give geometric interpretation of its complex points, which form a $10$–dimensional real algebraic variety. These results allow us to provide 