= (a_i, b_i, c_i) \in {\mathbb{R}}^3$ and of the $n$ base anchor points $P_i = (A_i, B_i, C_i) \in {\mathbb{R}}^3$ in one of their possible configurations. All pairs of points $(p_i, P_i)$ are connected by a rigid body, called *leg*, so that for all possible configurations the distance $d_i = \left\| p_i - P_i \right\|$ is preserved.

[pentapod]{}

(-8,4)[$P_1$]{} (32,-5)[$P_2$]{} (99.5,4)[$P_3$]{} (58,21)[$P_4$]{} (16,16)[$P_5$]{} (-6,72)[$p_1$]{} (41,68)[$p_2$]{} (69.5,61)[$p_3$]{} (96.5,81)[$p_4$]{} (12,87)[$p_5$]{}

We will think of an $n$–pod $\Pi$ as a triple $$\Pi = \Big( (p_1,\dots,p_n),(P_1,\dots,P_n),(d_1,\dots,d_n) \Big)$$ where $p_i$, $P_i$ and $d_i$ are defined as above.

We are interested in describing the *self–motions* of a given $n$–pod $\Pi$, namely which direct isometries $\sigma$ of ${\mathbb{R}}^3$ satisfy the condition (which is called the *spherical condition*) $$\label{equation:spherical_condition}
    \left\| \sigma(p_i) - P_i \right\| \; = \; d_i \quad \quad \textrm{for all }
i \in \{1, \ldots, n\}$$ In particular we want to understand what is the *dimension* of the set of these isometries, namely the *mobility* of $\Pi$ (we will make these concepts precise at the beginning of Section \[bond\_geometric\]), and what conditions we have to impose on the base and platform points to reach a prescribed mobility.

In order to do so we first study the 