8 Trivial infinitesimal motions

The translations and rotations of \({\mathbb {R}}^{d}\) give infinitesimal motions of every framework in \({\mathbb {R}}^{d}\), whatever the graph (section 6). In this section we define the space \(T(p)\) of these trivial motions at a framework \(p\), and we show that it lies in the kernel of the rigidity map \(R(G, p)\) for every graph \(G\). We then show that \(T(p)\) has dimension at least \(d(d+1)/2\) when the points \(p_v\) affinely span \({\mathbb {R}}^{d}\), so that the kernel of \(R(G, p)\) has at least this dimension too. By rank–nullity, the rank of \(R(G, p)\) is then at most \(d|V| - d(d+1)/2\). The proof of Laman’s theorem uses this bound in the plane, where \(d(d+1)/2 = 3\) (lemma 9.10).

We define the translations first, and then the infinitesimal rotations, which apply a skew-symmetric linear map to every point. The trivial motions are the linear combinations of the two. For the dimension bound we exhibit \(d(d+1)/2\) trivial motions that are linearly independent: the translations along the \(d\) coordinate axes, and one infinitesimal rotation in each of the \(d(d-1)/2\) coordinate planes. Whiteley  [ Whi96 ] (§11.1, before Lemma 11.1.2) generates the trivial motions in the same way, and notes that these generators are linearly independent once the points affinely span a hyperplane of \({\mathbb {R}}^{d}\). Jordán  [ Jor16 ] (p. 36) states the resulting dimension, \(d(d+1)/2\). Our proof uses the stronger assumption that the points affinely span \({\mathbb {R}}^{d}\). We prove only the lower bound, since the rank bound needs no more; the upper bound is the count in section 6.

Throughout, \(G\) is a graph on a set \(V\), and \(d \in {\mathbb {N}}\). An element \(x\) of \(({\mathbb {R}}^{d})^{V}\) gives each vertex \(v\) a velocity \(x_v \in {\mathbb {R}}^{d}\). It is an infinitesimal motion of \(G\) at a framework \(p\) when \(R(G, p)(x) = 0\) (definition 6.3). We write \(e_1, \dots , e_d\) for the standard basis of \({\mathbb {R}}^{d}\), and \(y_i\) for the \(i\)-th coordinate of a vector \(y \in {\mathbb {R}}^{d}\). We identify a linear map of \({\mathbb {R}}^{d}\) with its matrix in this basis, and we write \(E_{ij}\) for the matrix with entry \(1\) in row \(i\) and column \(j\) and all other entries \(0\). A linear map \(A\) of \({\mathbb {R}}^{d}\) is skew-symmetric if \(\langle y, A y \rangle = 0\) for every \(y \in {\mathbb {R}}^{d}\). Applying this to \(y + z\) shows that it holds exactly when \(\langle y, A z \rangle = -\langle A y, z \rangle \) for all \(y, z \in {\mathbb {R}}^{d}\), that is, when \(A^{\top } = -A\).

8.1 Translations

A translation of \({\mathbb {R}}^{d}\) moves every point with the same velocity.

8.2 Infinitesimal rotations

Let \(A\) be a skew-symmetric linear map of \({\mathbb {R}}^{d}\). The maps \(e^{tA}\), for \(t \in {\mathbb {R}}\), are rotations of \({\mathbb {R}}^{d}\), and the point \(e^{tA} p_v\) has velocity \(A p_v\) at \(t = 0\). So the velocities \(A p_v\) describe a rotation of the framework to first order.

8.3 The space of trivial motions

In section 6 the trivial motions at \(p\) are the motions \(v \mapsto A p_v + b\), with \(A\) skew-symmetric and \(b \in {\mathbb {R}}^{d}\). We define them as a span.

The translations form a linear subspace of \(({\mathbb {R}}^{d})^{V}\), and so do the infinitesimal rotations at \(p\). So \(T(p)\) is their sum, which consists of the motions \(v \mapsto A p_v + b\) above.

8.4 A lower bound on the dimension

The points \(p_v\) affinely span \({\mathbb {R}}^{d}\) when their affine hull is all of \({\mathbb {R}}^{d}\), or equivalently when \(V\) is nonempty and the differences \(p_v - p_w\), for \(v, w \in V\), span \({\mathbb {R}}^{d}\). Under this condition we exhibit \(d(d+1)/2\) linearly independent trivial motions. The rotations among them come from the following linear maps.

For \(i \ne j\), the map \(E_{ij} - E_{ji}\) sends \(e_j\) to \(e_i\) and \(e_i\) to \(-e_j\), and it vanishes on the other basis vectors. So the maps \(e^{t(E_{ij} - E_{ji})}\) rotate the plane spanned by \(e_i\) and \(e_j\), and they fix the other basis vectors. The map is zero when \(i = j\), and exchanging \(i\) and \(j\) changes its sign.

Since exchanging \(i\) and \(j\) only changes the sign of \(E_{ij} - E_{ji}\), we use the pairs with \(j {\lt} i\).