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  "body": "\n## Definition\n\nLet $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean space|0XPR1E85W0]] $\\mathbb{R}^n$.\nA Euclidean isometry of $\\mathbb{R}^n$ is a map $f \\colon \\mathbb{R}^n \\to \\mathbb{R}^n$ with\n$$d(f(p), f(q)) = d(p, q) \\quad \\text{for all } p, q \\in \\mathbb{R}^n.$$\n\n## Remarks\n\nEach Euclidean isometry is an [[affine map|0JF0TVYGWT]] $f(x) = A x + b$, with an orthogonal $n \\times n$ matrix $A$ and a vector $b \\in \\mathbb{R}^n$. Thus $f$ is a bijection, and its inverse is also an isometry.\nThe isometry is direct if $\\det A = 1$ and indirect if $\\det A = -1$. The direct isometries keep the orientation of figures.\nMathlib's `AffineIsometryEquiv` is an affine bijection that keeps distances, between normed affine spaces.\n\n## Explanations\n\n### L1 human\n\nAn isometry moves the plane or the space without stretching, bending or tearing it. Lengths and angles do not change. In the plane, examples are a shift, a turn about a point and a mirror image in a line.\n",
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