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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an interval and let $\\gamma \\colon I \\to \\R^3$ be a curve of class $C^3$ parametrised by [[arc length|00R1JDDV5X]] $s$, with [[curvature|7JQS6VKFEZ]] $\\kappa(s) \u003e 0$ for all $s \\in I$.\nIts unit tangent is $T = \\gamma'$, its unit normal is $N = T' / \\kappa$, and its binormal is $B = T \\times N$, where $\\times$ is the [[cross product|0SEDGXG9S3]].\nThe vector $B'(s)$ is a multiple of $N(s)$.\nThe *torsion* of $\\gamma$ at $s$ is the number $\\tau(s)$ with\n$$B'(s) = -\\tau(s) \\, N(s).$$\n\n## Remarks\n\nFor a regular $C^3$ curve with any parameter $t$ and $\\gamma' \\times \\gamma'' \\ne 0$, the torsion is\n$$\\tau = \\frac{\\det(\\gamma', \\gamma'', \\gamma''')}{\\norm{\\gamma' \\times \\gamma''}^2}.$$\nExample: take the helix $\\gamma(t) = (a \\cos t, a \\sin t, bt)$ with $a \u003e 0$ and $b \\in \\R$.\nThen $\\gamma' \\times \\gamma'' = (ab \\sin t, -ab \\cos t, a^2)$ and $\\det(\\gamma', \\gamma'', \\gamma''') = a^2 b$, so $\\tau = b / (a^2 + b^2)$.\nNon-example: on the curve $t \\mapsto (t, t^3, 0)$ the curvature vanishes at $t = 0$, so $N$ and $B$ are not defined there and the torsion is not defined at that point.\nSome books use the opposite sign convention $B' = \\tau N$.\nA curve with $\\kappa \u003e 0$ lies in a plane if and only if $\\tau = 0$ everywhere.\n",
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