that we are only interested in bounded motions, that is, motions with only bounded trajectories.) This we accomplish by decomposing the motion into the product of rotations about certain axes. The rotation angles are linked by the common motion parameter $t$. The basic tool for this is the factorization theory for motion polynomials as introduced in [@hegedus13] and its extension to non-generic bounded motion polynomials in [@li15b].

\[def:factorization\] A bounded motion polynomial $C$ is said to *admit a factorization* if there exist bounded linear motion polynomials $t-h_1$, …, $t-h_n$ such that $$\label{eq:2}
    C = (t-h_1) \cdots (t-h_n).$$

We already said that each linear factor $t - h_i$ in parameterizes a rotation with fixed axes or translation in fixed direction. Because $t - h_i$ is bounded, the later cannot occur here. Hence, the product parameterizes the composition of such rotations. Note that a factorization of the form need not exist and if it exists, it need not be unique.

Call a motion polynomial “generic” if its spherical degree defect () is zero. A generic motion polynomial can always be factored [@hegedus13 Theorem 1] but the factorization is in general not unique. Elliptic and circular translation and the Viviani motion serve as 