to assume that this parametric equation is *reduced,* that is, $\gcd(x_0,x_1,x_2,x_3) = 1$ because we may always divide by a common factor. The *degree* of the rational curve is the maximum of the polynomials $x_0$, $x_1$, $x_2$, and $x_3$ in reduced form. The *circularity* of a reduced rational parametric equation is $c \coloneqq \frac{1}{2}\deg\gcd(x_0,x_1^2+x_2^2+x_3^2)$. It counts the number of intersection points with the absolute conic of Euclidean geometry, is a positive integer and invariant with respect to rational re-parameterizations and similarity transformations. Finally, the rational curve is called *bounded,* if $x_0$ has no real zeros and $\deg(x_0)\ge\deg(x_1),\deg(x_2),\deg(x_3)$. Note that in this sense bounded segments of unbounded rational curves are bounded rational curves.

A linkage in our context consists of a set of lines in space, called the *joints*, and links (rigid bodies which could have different shapes) that connect two or more axes. Whenever two links are connected by a joint, their relative position is constrained to a rotation about their common joint. More precisely, the relative position is determined by the rotation angle about this joint with respect to a given reference configuration. The set of all tuples of possible rotation angles is called the linkage’s *configuration space.* If 