degenerates to a straight line segment and the minimal motion is the translation $C = t^2 + 1 + a {\varepsilon}{\mathbf{i}}$ (back and forth along this segment). If $a = b > 0$, the parametric curve is a circle. Its circularity is one and the minimal motion polynomial $C = t - {\mathbf{k}}+ a{\varepsilon}{\mathbf{j}}$ is linear, as predicted by . It parameterizes the rotation around the circle axis.

\[ex:2\] As a second example, consider Viviani’s curve $x = x_0 + x_1{\mathbf{i}}+
  x_2{\mathbf{j}}+ x_3{\mathbf{k}}$, given by $$x_0 = (1+t^2)^2,\quad
    x_1 = -4t^2,\quad
    x_2 = 2t(1-t^2),\quad
    x_3 = 2t(1+t^2).$$ It lies on the sphere of radius $1$ with center $(-1,0,0)$ and our parameterization satisfies $x(\infty) = 1$. The minimal motion is $C = t^2 - ({\mathbf{j}}+ {\mathbf{k}}- {\varepsilon}({\mathbf{j}}- {\mathbf{k}}))t - {\mathbf{i}}$. It is a spherical motion because it fixes the sphere center $(-1,0,0)$. We call it *Viviani motion.*

Factorization of the bounded minimal degree motion {#sec:factorization}
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Having constructed a monic motion polynomial $C$ with a trajectory $x$, we have converted our trajectory generation problem to a motion generation problem: We are looking for a linkage with one degree of freedom such that one link follows the motion parameterized by $C$. (Note 