= 1$ and $\operatorname{{mrpf}}(P)=x_0/g$. Moreover, Lemma 3 of [@li15a] is applicable (with $C = D$ and $R = g$) and guarantees that $L \coloneqq \operatorname{{lgcd}}(D, g)$ satisfies $L{\overline{L}} = g$. There exists $D' \in {\mathbb{H}}[t]$ such that $D = LD'$. But then $-D = {\overline{D}} = {\overline{D'}}\,{\overline{L}}$ which implies ${\overline{L}} = \operatorname{{rgcd}}(D,g)$ and also ${\overline{Q}} = \frac{1}{2}\operatorname{{lquo}}(D,{\overline{L}}) = \frac{1}{2}D'$. Thus, $$-4Q{\overline{Q}} = D'{\overline{D'}} = D{\overline{D}}(L{\overline{L}})^{-1} = D{\overline{D}}/g$$ and $\operatorname{{mrpf}}(P)=x_0/g$ and $Q{\overline{Q}}=-(x_1^2+x_2^2+x_3^2)/(4g)$ are relatively prime.

If the rational curve is planar or spherical, then the minimal motion is also planar or spherical, respectively. If a planar curve with a non-planar minimal motion existed, we could reflect the motion in the curve’s plane and obtain a contradiction to uniqueness. For spherical curves, this follows from Folgerung 9 and the proof of Satz 6 of [@juettler93].

Let us look at some examples of minimal motions:

\[ex:1\] Consider the parametric curve $x = t^2 + 1 - 2 a{\mathbf{i}}-2 b{\mathbf{j}}t$ with $a, b \in {\mathbb{R}}$. For $a > b > 0$, it is an ellipse and the minimal motion polynomial is $C = t^2 + 1 + {\varepsilon}(a{\mathbf{i}}+ b{\mathbf{j}}t)$. It parameterizes the translation along the ellipse. If $a > b = 0$, the curve 