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  "body": "\n## Statement\n\nLet $f$ be a [[Euclidean isometry|0J1N3WTKM4]] of the plane $\\mathbb{R}^2$. Then $f$ is of exactly one of these five types:\n- the identity,\n- a [[translation|087VJM5Q77]] by a nonzero vector,\n- a [[rotation|0HFREYRSZP]] about a point through an angle $\\theta \\notin 2\\pi\\mathbb{Z}$,\n- a [[reflection|06BRDF3F0E]] in a line,\n- a [[glide reflection|0167F9B344]] with a nonzero glide vector.\n\n## Remarks\n\nWrite $f(x) = A x + b$ 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.\nThe 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.\nEach isometry of the plane is the composite of at most three reflections in lines.\n\n## Explanations\n\n### L1 human\n\nEach 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.\n",
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