\[line:lcoeff\]$\operatorname{{lgcd}}(G, R \operatorname{{lcoeff}}(R)^{-1})$

The computation of the minimal degree rational motion with the prescribed trajectory $x$ is illustrated in . It computes a monic rational motion polynomial $C$ such that the trajectory of the affine origin is parameterized by $x$. The correctness of has been proved in [@li15a]. The function $\operatorname{{quo}}$ in Line \[line:quo\] denotes the quotient of polynomial division for real polynomials. It may also be computed by .

Reduced rational parametric equation $x = x_0 + x_1{\mathbf{i}}+ x_2{\mathbf{j}}+ x_3{\mathbf{k}}$ with $x(\infty) = 1$. Monic motion polynomial $C$ of minimal degree such that $x = P{\overline{P}} + 2P{\overline{Q}}$ (trajectory of affine origin is parameterized by $x$). $g \leftarrow \gcd(x_0, x_1^2+x_2^2+x_3^2)$ \[line:quo\]$w \leftarrow \operatorname{{quo}}(x_0,g)$ quotient of polynomial division in ${\mathbb{R}}[t]$ $D \leftarrow x_1{\mathbf{i}}+ x_2{\mathbf{j}}+ x_3{\mathbf{k}}$ $P' \leftarrow \operatorname{{lgcd}}(D,g)$, $Q' \leftarrow \operatorname{{rquo}}(D,P')$, $C = wP' + \frac{1}{2} {\varepsilon}{\overline{Q'}}$

\[lem:1\] Let $C=P+{\varepsilon}Q$ be a minimal motion to the reduced rational curve $x = x_0 + x_1{\mathbf{i}}+ x_2{\mathbf{j}}+ x_3{\mathbf{k}}$. Then $\operatorname{{mrpf}}(P)$ and $Q{\overline{Q}}$ are relatively prime.

Assume that the trajectory of the origin is $x_0+x_1{\mathbf{i}}+x_2{\mathbf{j}}+x_3{\mathbf{k}}$. Following Algorithm \[alg:minmot\], we define $g \coloneqq \gcd(x_0,x_1^2+x_2^2+x_3^2)$ and $D \coloneqq x_1{\mathbf{i}}+ x_2{\mathbf{j}}+ x_3{\mathbf{k}}$. Then $$P = wP' = \frac{x_0}{g}\operatorname{{lgcd}}(D, g).$$ Because $x$ is reduced, $\gcd(x_1{\mathbf{i}}+ x_2{\mathbf{j}}+ x_3{\mathbf{k}},g) 