Euclidean motion of Euclidean three-space ($E(3)$)

A Euclidean motion has the form $E(x)=Ax+b$ with $A\in O(3)$. It preserves Euclidean distance, angles, and all extrinsic geometric quantities transformed with their orientations.

This is the dimension-three case $E(3)=O(3)⋉{\mathbb{R}}^3$ of the Euclidean group.

A proper Euclidean motion has the form $E(x)=Ax+b$ with $A\in SO(3)$. It preserves lengths, dot products, cross products, orientation, curvature, and torsion.

This is the orientation-preserving subgroup of the three-dimensional Euclidean group.

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.

The derivative at the identity of a smooth curve in $\mathrm{SE}(3)$ is a vector fieldAfter shifting the origin perpendicular to $a\ne 0$, it becomes rotation about the axis parallel to $a$ plus translation along that axis.

The integral curves of $V(x)=a\times x+b$ are lines when $a=0$, and helices around an axis when $a\ne 0$. The helix is closed exactly when its axial translation component vanishes, in which case every nonfixed orbit is a circle.

A simple closed curve in ${\mathbb{R}}^3$ preserved by a nonconstant smooth family of proper rigid motions is a circle. Differentiating the normalized family at the identity gives a nonzero infinitesimal proper Euclidean motion tangent to the curve. Compactness rules out its line and nonclosed helical orbits, leaving a circular rotation orbit.