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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]].\nA function $f \\colon X \\to Y$ is an isometry when\n$$d_Y\\bigl(f(x), f(y)\\bigr) = d_X(x, y) \\quad \\text{for all } x, y \\in X.$$\n\n## Remarks\n\nEach isometry is [[injective|02J4G4J4TJ]] and [[Lipschitz|0TMZ21WKCN]] with constant $1$.\nSome authors require an isometry to be surjective. Here a surjective isometry is an [[isometric isomorphism|0VRD8GC5BM]].\nThe map $x \\mapsto (x, 0)$ from $\\mathbb{R}$ to $\\mathbb{R}^2$ with the Euclidean metric is an isometry, but it is not surjective.\nRotations, reflections and translations of $\\mathbb{R}^2$ are isometries for the Euclidean metric.\n\n## Explanations\n\n### L1 human\n\nAn isometry moves points without stretching or shrinking any distance.\nIt can put a space into a bigger space, as the real line is put into the plane as the horizontal axis.\n",
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