parametric equation $X$ and the polynomial $x$ are of the same degree $d$.

Motion of minimal degree to given trajectory {#sec:minimal-degree}
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The first step is the construction of a rational motion such that the trajectory of one point equals the parametric curve $X$. An obvious and simple choice is the translation along $X$, given by the motion polynomial $C = x_0 - \frac{1}{2}{\varepsilon}(x_1{\mathbf{i}}+x_2{\mathbf{j}}+ x_3{\mathbf{k}})$. However, in order to keep low the number of links and joints, we try to find a motion polynomial $C$ of minimal degree with trajectory $X$. The main result in this context states uniqueness of this motion and characterizes curves for which the trivial translation along the curve is not optimal [@li15a]. In order to fully appreciate it, we need a definition:

\[def:trajectory-degree\] The *trajectory degree* of a reduced rational motion $C = P + {\varepsilon}Q \in {{\mathbb{D}}{\mathbb{H}}}[t]$ is the maximal degree of its trajectories. The *quaternion degree* is the degree of $C$ as polynomial in ${{\mathbb{D}}{\mathbb{H}}}[t]$. The *spherical degree defect* of $C$ is the degree of the real polynomial factor of maximal degree of the primal part $P$.

Some authors refer to the trajectory degree of a reduced rational motion as just the “degree”. 