Chasles' theorem

Remarks

Proof idea: by Euler's rotation theorem, $R$ is a rotation by some $\theta$ about an axis with unit direction $u$. Write $b = d u + b_\perp$ with $b_\perp$ orthogonal to $u$. On the plane orthogonal to $u$, the map $x \mapsto R x + b_\perp$ is a plane rotation by $\theta \ne 0$ followed by a translation. It has a fixed point $p$, as $R - I$ is invertible on that plane. Then $g$ is the rotation by $\theta$ about the line $L = p + \R u$ followed by the translation $d u$ along $L$.

Example: $g(x_1, x_2, x_3) = (-x_2 + 2,\ x_1,\ x_3 + 5)$ has $R$ the quarter turn about the $x_3$-axis. Its axis is the vertical line through $p = (1, 1, 0)$, since $(-1 + 2, 1) = (1, 1)$, and $d = 5$.

The orientation hypothesis is needed: the reflection $x \mapsto (x_1, x_2, -x_3)$ is a rigid motion that is not a screw displacement. The analogous statement in the plane says that every orientation-preserving rigid motion is a translation or a rotation about a point.

Going deeper

For a student

The theorem is the starting point of screw theory and describes the elements of the Euclidean group $SE(3)$ geometrically. Applied to infinitesimal motions, it shows that the velocity field of a rigid body is at each instant that of a screw motion, with an instantaneous screw axis. In the plane, the same idea gives the instant centre of rotation. It is used to parametrise displacements in robot kinematics.