Classification of isometries of the plane

Statement

Let $f$ be a Euclidean isometry of the plane $\mathbb{R}^2$. Then $f$ is of exactly one of these five types:

• the identity,
• a translation by a nonzero vector,
• a rotation about a point through an angle $\theta \notin 2\pi\mathbb{Z}$,
• a reflection in a line,
• a glide reflection with a nonzero glide vector.

Remarks

Write $f(x) = A x + b$ with $A$ orthogonal. If $\det A = 1$, then $f$ is the identity, a translation or a rotation. If $\det A = -1$, then $f$ is a reflection or a glide reflection. The fixed points tell the types apart. A translation and a glide reflection have no fixed point. A rotation has one, and a reflection fixes a whole line. Each isometry of the plane is the composite of at most three reflections in lines.

Explanations

L1 human

Each rigid motion of the plane is a shift, a turn, a mirror image, or a mirror image with a shift along the mirror. A turn and a shift keep the clockwise order of points. A mirror image and a glide reverse it.