Screw motion (${\Phi }_t(x)=R_{t}x+tve$)

A screw motion combines a rotation in three dimensions about a fixed line with a translation parallel to that line. With unit axis direction $e$, axial speed $v$, and rotation matrix $R_t$ fixing $e$, each map ${\Phi }_t(x)=R_{t}x+tve$ is a proper Euclidean motion of Euclidean three-space, and together these maps form a one-parameter subgroup. Each off-axis group orbit is a circular helix when both rotational speed and axial speed are nonzero. A pure rotation or translation is a limiting case.