assign to each inversion or butterfly point a pair $(L,R)$ of elements in $S^2$. We would like to do the same for similarity points. There is a glaring obstruction in doing this, namely the fact that for similarity points both the $h$ and the $m_{ij}$–coordinates are zero. On the other hand for all boundary points the two matrices $M$ and $xy^t$ are linear dependent, and in the case of similarity points $x y^t$ is different from zero. Moreover $x$ and $y$ satisfy $\langle x, x \rangle = \langle y, y \rangle = 0$. So we can associate to a similarity point the pair of elements of $S^2$ coming from the vectors $x$ and $y$.

\[definition:left\_right\] Via these identifications we can associate to every inversion, butterfly or similarity point $\beta$ in $B$ a pair $(L,R)$ of elements of $S^2$, which are respectively called the *left* and the *right vector* of $\beta$.

Geometric Interpretation of Bonds {#bond_geometric}
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This section represents an instance of a more general technique called *bond theory*: the goal is to extract information from boundary points which arise as limits of self–motions of an $n$–pod. Boundary points do not represent direct isometries of ${\mathbb{R}}^3$, but nevertheless we can 