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  "body": "\n## Statement\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]] $s$.\nAssume that its [[curvature|7JQS6VKFEZ]] $\\kappa(s)$ is positive for all $s \\in I$.\nLet $T = \\gamma'$, $N = T' / \\kappa$ and $B = T \\times N$, and let $\\tau$ be the [[torsion|S4BZ697DSH]] of $\\gamma$.\nThen $(T(s), N(s), B(s))$ is a positively oriented [[orthonormal basis|0FWJ0VW561]] of $\\R^3$ for each $s$, and\n$$T' = \\kappa N, \\qquad N' = -\\kappa T + \\tau B, \\qquad B' = -\\tau N.$$\n\n## Remarks\n\nThe first formula is the definition of $N$, and the third is the definition of $\\tau$.\nFor the second, differentiate the identities $\\ip{N}{N} = 1$, $\\ip{N}{T} = 0$ and $\\ip{N}{B} = 0$.\nThey give $\\ip{N'}{N} = 0$, $\\ip{N'}{T} = -\\ip{N}{T'} = -\\kappa$ and $\\ip{N'}{B} = -\\ip{N}{B'} = \\tau$.\nIn matrix form the formulas say that $(T, N, B)' = (T, N, B) A$ with a skew-symmetric matrix $A$ built from $\\kappa$ and $\\tau$.\nFor the helix $(a \\cos t, a \\sin t, bt)$ with $a \u003e 0$, both $\\kappa = a / (a^2 + b^2)$ and $\\tau = b / (a^2 + b^2)$ are constant.\nThe hypothesis $\\kappa \u003e 0$ is needed, since the normal $N$ is not defined where $\\kappa = 0$.\nFor example, two plane arcs in different planes, smoothly joined through a straight segment, form a curve with $\\tau = 0$ wherever $\\kappa \u003e 0$ that lies in no plane.\n\n## Explanations\n\n### L1 human\n\nAt each point of a space curve there are three natural directions: forward, the way the curve bends, and the direction perpendicular to both.\nThe Frenet–Serret formulas tell how these three arrows turn as one moves along the curve.\nOnly two numbers control this turning: the curvature and the torsion.\n\n### L2 human\n\nThe formulas form a linear system of ordinary differential equations for the frame $(T, N, B)$ with coefficients $\\kappa$ and $\\tau$.\nThis is why curvature and torsion determine the curve up to a rigid motion, by the [[fundamental theorem of space curves|TNYQ5R0NVM]].\nThey give quick proofs of basic facts.\nA curve with $\\kappa \u003e 0$ and $\\tau = 0$ lies in a plane.\nA curve with constant $\\kappa \u003e 0$ and $\\tau = 0$ is an arc of a circle.\nIn $\\R^n$ there is an analogous frame with $n - 1$ generalised curvatures, under a nondegeneracy hypothesis on the derivatives.\n",
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