A $2_{1}$ screw axis is defined as a 180-degree rotation followed by a translation of $\frac{1}{2}$ along a particular unit cell vector. In matrix form:
\begin{equation} \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \begin{bmatrix} 0 \\ \tfrac{1}{2} \\ 0 \end{bmatrix} = \begin{bmatrix} -x \\ \tfrac{1}{2} + y \\ -z \end{bmatrix} \\ \end{equation}
This describes a $2_{1}$ screw axis about $y$. If we connect every point $(x, y, z)$, $(-x, \tfrac{1}{2}+y, -z)$, etc., we can make a spiral-like shape which looks like a screw. Is there some parametric function which describes this curve, and can we derive one for each type of screw axis?
Here is an example for the mirror-image $3_{1}$ and $3_{2}$ screw axes, which form enantiomorphic helices. But I cannot find a parametric equation formally relating them. How would that be derived?