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  "body": "\n## Definition\n\nLet $v$ be a [[vector|0QN3D80AWP]] of $\\mathbb{R}^n$. The translation by $v$ is the map\n$$T_v \\colon \\mathbb{R}^n \\to \\mathbb{R}^n, \\qquad T_v(p) = p + v.$$\n\n## Remarks\n\nEach translation is a [[Euclidean isometry|0J1N3WTKM4]], and it is direct. For $v \\neq 0$, it has no fixed point.\n$T_v \\circ T_w = T_{v + w}$ and $T_v^{-1} = T_{-v}$. So the translations form an abelian group, isomorphic to $\\mathbb{R}^n$.\nIn the plane, the composite of the reflections in two parallel lines is a translation.\nMathlib's `AffineEquiv.constVAdd` is the translation by a vector in an affine space.\n\n## Explanations\n\n### L1 human\n\nA translation slides every point by the same arrow. Shapes keep their size, their form and their direction. The translation by $(2, 1)$ sends the point $(0, 0)$ to $(2, 1)$ and the point $(3, -1)$ to $(5, 0)$.\n",
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