= Screw motion
{title2=$\Phi_t(x)=R_tx+tve$}
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 $\Phi_t(x)=R_tx+tve$ 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.
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