However, we have to distinguish between this trajectory degree and the degree of $C$ as polynomial in ${{\mathbb{D}}{\mathbb{H}}}[t]$. The latter was called the motion’s “quaternion degree” in [@juettler93] and we follow this convention. The spherical degree defect accounts for a difference in the respective trajectory degrees of $C$ and its spherical motion component $P$. If the motion polynomial is monic (or at least its leading coefficient is invertible), the spherical degree defect can be computed as degree of $\operatorname{{mrpf}}P \coloneqq \gcd(P,{\overline{P}})$ where $\gcd$ denotes the monic real polynomial factor of maximal degree.

\[th:2\] The rational motion of minimal quaternion degree with a prescribed rational trajectory is *unique.* If the trajectory is of degree $d$ and circularity $c$, this minimal motion is of degree $d-c$ and has a spherical degree defect of $s = d - 2c$.

Lets call the motion of the trajectory’s *minimal motion.* tells us that in general (if the trajectory is of circularity zero), the translation along the trajectory is minimal. But for curves with positive circularity, we can do better. This has effects on our linkage construction: Curves of high circularity lead to minimal motions of low degree and low spherical degree defect and lend themselves 