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  "body": "\n## Definition\n\nLet $I \\subseteq \\mathbb{R}$ be an interval with more than one point.\nA parametrised curve in $\\mathbb{R}^n$ is a [[continuous|0S9EK5MQPE]] map $\\gamma \\colon I \\to \\mathbb{R}^n$, $\\gamma(t) = (\\gamma_1(t), \\dots, \\gamma_n(t))$.\nIts image, or trace, is the set $\\gamma(I) = \\{\\gamma(t) : t \\in I\\}$.\nFor $r \\ge 1$, the curve is a $C^r$ curve if each component $\\gamma_i$ is $r$ times continuously differentiable on $I$.\nWhen $I = [a, b]$, the points $\\gamma(a)$ and $\\gamma(b)$ are the initial point and the end point of $\\gamma$.\n\n## Remarks\n\nA curve is different from its image. The maps $t \\mapsto (\\cos t, \\sin t)$ on $[0, 2\\pi]$ and $t \\mapsto (\\cos 2t, \\sin 2t)$ on $[0, 2\\pi]$ have the same image, the unit circle, but the second goes around it twice.\nThe helix $\\gamma(t) = (\\cos t, \\sin t, t)$, $t \\in \\mathbb{R}$, is a $C^\\infty$ curve in $\\mathbb{R}^3$.\nThe curve is simple if $\\gamma$ is injective, and it is closed if $I = [a, b]$ and $\\gamma(a) = \\gamma(b)$.\n\n## Explanations\n\n### L1 human\n\nA parametrised curve tells where a moving point is at each time $t$.\nIt records both the path and the way the point travels along it, including its speed and direction.\n",
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