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  "body": "\n## Definition\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be a [[regular|06JMR2XWXW]] $C^1$ curve.\nIts arc length function is\n$$s(t) = \\int_a^t \\|\\gamma'(u)\\| \\, du, \\qquad t \\in [a, b],$$\nand its length is $\\ell = s(b)$.\nSince $s'(t) = \\|\\gamma'(t)\\| \u003e 0$, the function $s \\colon [a, b] \\to [0, \\ell]$ is a strictly increasing $C^1$ bijection with a $C^1$ inverse $t(s)$.\nThe arc length parametrisation of $\\gamma$ is the curve $\\tilde\\gamma \\colon [0, \\ell] \\to \\mathbb{R}^n$, $\\tilde\\gamma(s) = \\gamma(t(s))$.\n\n## Remarks\n\nThe curve $\\tilde\\gamma$ has the same image and direction as $\\gamma$, and $\\|\\tilde\\gamma'(s)\\| = 1$ for each $s$. So $\\tilde\\gamma'(s)$ is the unit [[tangent vector|0D98771SM0]].\nThe length of $\\tilde\\gamma$ from $\\tilde\\gamma(s_0)$ to $\\tilde\\gamma(s_1)$ is $s_1 - s_0$.\nFor the helix $\\gamma(t) = (\\cos t, \\sin t, t)$, $\\|\\gamma'(t)\\| = \\sqrt{2}$, so $s = \\sqrt{2}\\, t$ and $\\tilde\\gamma(s) = (\\cos(s / \\sqrt{2}), \\sin(s / \\sqrt{2}), s / \\sqrt{2})$.\nThe arc length parametrisation is unique up to a shift of the parameter. It simplifies the formulas for curvature.\n\n## Explanations\n\n### L1 human\n\nReparametrise the curve so that the parameter is the distance travelled from the start.\nThen the point moves at constant speed $1$, and only the shape of the curve remains.\n",
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        "url": "https://en.wikipedia.org/wiki/Differentiable_curve"
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        "locator": "Finding arc lengths by integration",
        "url": "https://en.wikipedia.org/wiki/Arc_length"
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