\{0\}$ and if its leading coefficient $\operatorname{{lcoeff}}(C)$ is invertible.

Motion polynomials are a central concept in this article. Their introduction is motivated by the fact that for every $t_0 \in {\mathbb{R}}$, the value $C(t_0)$ is an element of ${\mathbb{S}}$ so that $C$ acts on ${\mathbb{R}}^3$. Varying $t_0$, we get a one-parametric set of rigid body displacements, that is, a motion. By virtue of , the orbit of any point is (part of) a rational curve, whence the motion itself is called *rational.* It is possible to extend the parameter range from ${\mathbb{R}}$ to ${\mathbb{R}}\cup \{\infty\}$: with the definition $C(\infty) \coloneqq \operatorname{{lcoeff}}(C)$, the parametrization of the orbit is continuous except in the points $t_1$ such that $C(t_1)$ has norm zero. One could also represent the map as a regular map from the real projective line to real projective space ${\mathbb{P}}^7$, but this would require a second homogeneous variable for the parameter $t$, which complicates the algebraic theory; hence, we prefer to use an affine parameter space and a projective image.

Of particular importance to us are linear motion polynomials. The linear polynomial $C = t - h$ is a motion polynomial if $C{\overline{C}} = t^2 - (h + {\overline{h}})t + h{\overline{h}}$ 