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  "body": "\n## Statement\n\nLet $I \\subseteq \\R$ be an open interval, and let $\\kappa \\colon I \\to \\R$ and $\\tau \\colon I \\to \\R$ be smooth functions with $\\kappa(s) \u003e 0$ for all $s \\in I$.\nThen there is a smooth curve $\\gamma \\colon I \\to \\R^3$ parametrised by [[arc length|00R1JDDV5X]] whose [[curvature|7JQS6VKFEZ]] is $\\kappa$ and whose [[torsion|S4BZ697DSH]] is $\\tau$.\nIf $\\tilde\\gamma \\colon I \\to \\R^3$ is another such curve, then $\\tilde\\gamma = \\rho \\circ \\gamma$ for a unique orientation-preserving rigid motion $\\rho$ of $\\R^3$.\nHere a rigid motion is a map $x \\mapsto Rx + b$ with $R$ a rotation matrix and $b \\in \\R^3$.\n\n## Remarks\n\nFor existence, solve the [[Frenet–Serret formulas|ZM6W89YNS7]] as a linear system of ordinary differential equations for $(T, N, B)$, with an orthonormal positive initial frame at some $s_0 \\in I$.\nThe solution stays orthonormal because the coefficient matrix is skew-symmetric; then set $\\gamma(s) = \\int_{s_0}^s T(u) \\, du$.\nFor uniqueness, move $\\tilde\\gamma$ by a rigid motion so that both curves have the same point and frame at $s_0$.\nThen the uniqueness of solutions of the linear system gives equal frames and equal curves.\nExample: constant $\\kappa = a/(a^2+b^2)$ and $\\tau = b/(a^2+b^2)$ with $a \u003e 0$ give exactly the helices $(a \\cos t, a \\sin t, bt)$, $t = s / \\sqrt{a^2 + b^2}$, up to rigid motion.\nThe hypothesis $\\kappa \u003e 0$ matters: if $\\kappa$ vanishes on an interval, the torsion is not defined there and the curve is not determined by $\\kappa$ alone.\nAn orientation-reversing motion, such as a reflection, changes the sign of $\\tau$, so it does not keep the torsion.\n\n## Explanations\n\n### L1 human\n\nIf you know at each moment how sharply a curve bends and how much it twists, you know the curve.\nThe only freedom left is where you place it in space and which way you point it.\nThis is like following driving instructions: \"turn this much, climb this much\" fixes the route once the start is given.\n\n### L2 human\n\nThe theorem says that $(\\kappa, \\tau)$ is a complete system of invariants of unit-speed space curves with positive curvature.\nThe symmetry group here is the group of orientation-preserving rigid motions, a subgroup of the [[Euclidean group|0H3YY8CPKH]].\nFor plane curves the analogue is simpler: the signed curvature determines a unit-speed curve up to orientation-preserving rigid motion, through $\\theta' = k$ and $\\gamma' = (\\cos \\theta, \\sin \\theta)$.\nThe same idea, a frame solving a linear system, reappears in the theory of surfaces, where the Gauss–Codazzi equations are the integrability conditions.\n",
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