is real. This is the case if both $h + {\overline{h}}$ and $h{\overline{h}}$ are real. It is well-known that the motion parameterized by $C$ is either a rotation about a fixed axis or, if the primal part of $h$ is real, a translation in fixed direction. In either case, the parameter value $t = \infty$ corresponds to the identity transformation, that is, zero rotation angle or translation distance.

In this article, we often assume that a given motion polynomial has no nontrivial real factor. We call those motion polynomials *reduced.* From a kinematic viewpoint, this is no restriction as $C$ and $CR$ with $R \in {\mathbb{R}}[t] \setminus \{0\}$ parameterize the same motion. Note, however, that multiplication with a real polynomial is a useful technique to ensure existence of a factorization (see [@gallet15; @li15b] and ).

Proof of main theorem, linkage construction {#sec:construction}
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In this section we prove . For each step in our constructive proof we provide a theoretical justification, often with references to existing literature, and an algorithmic description. Given is a bounded rational curve by a reduced rational parametric equation $X = x_0^{-1}(x_1,x_2,x_3)$. We also encode it as polynomial $x = x_0 + x_1{\mathbf{i}}+x_2{\mathbf{j}}+ x_3{\mathbf{k}}\in {\mathbb{H}}[t]$. The 