well to a realization by a linkage with few links and joints.

The constructive proof of in [@li15a] can be turned into a short algorithm to actually compute minimal motions. We make a few technical assumptions:

-   The reduced rational curve $x = x_0+x_1{\mathbf{i}}+ x_2{\mathbf{j}}+ x_3{\mathbf{k}}$ satisfies $x(\infty) \coloneqq \lim_{t \to \infty} t^{-d}x = 1$. This can always be accomplished by a suitable translation of the coordinate frame.

-   We want to find a motion polynomial $C = P + {\varepsilon}Q$ of minimal degree $d - c$ such that $P{\overline{P}} + 2P{\overline{Q}} = x$. In view of this means that $x$ is the trajectory of the affine origin in ${\mathbb{R}}^3$.

-   We assume that $C$ is monic which is consistent with the assumption $x(\infty) = 1$ and entails that $C(\infty)$ is the identity. This can always be accomplished by a suitable rotation of the coordinate frame about its origin.

An important ingredient in the computation of minimal motions is right division of quaternion polynomials. Given $F$, $G \in {\mathbb{H}}[t]$, there exist unique polynomials $Q, R \in {\mathbb{H}}[t]$, called right quotient and right remainder, with $F = GQ + R$ and $\deg R < \deg G$. In case of monic $G$, 