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  "body": "\n## Statement\n\nLet a *regular closed curve* be a $1$-periodic map $\\gamma \\colon \\R \\to \\R^2$ of class $C^1$ with $\\gamma'(t) \\ne 0$ for all $t$.\nIts *turning number* is the degree of the map $t \\mapsto \\gamma'(t) / \\norm{\\gamma'(t)}$ from $\\R / \\Z$ to the unit circle.\nA *regular homotopy* between $\\gamma_0$ and $\\gamma_1$ is a continuous family $(\\gamma_u)_{u \\in [0, 1]}$ of regular closed curves such that $(u, t) \\mapsto \\gamma_u'(t)$ is continuous.\nTwo regular closed curves in the plane are regularly homotopic if and only if they have the same turning number.\n\n## Remarks\n\nThe turning number is the [[winding number|0X7SBS8E0F]] of the derivative $\\gamma'$ around $0$.\nIt is invariant under regular homotopy, because it is an integer that depends continuously on $u$.\nFor the converse, one reparametrises both curves proportionally to arc length.\nTheir unit tangent maps have the same degree, so they are joined by a [[homotopy|9YA6S6HD9V]] of maps to the circle.\nIntegrating these tangent fields and correcting the closing condition gives the required family.\nExample: the circle $t \\mapsto (\\cos 2\\pi t, \\sin 2\\pi t)$ has turning number $1$, and the figure eight $t \\mapsto (\\sin 4\\pi t, \\sin 2\\pi t)$ has turning number $0$.\nSo these two curves are homotopic as maps but not regularly homotopic.\nThe theorem was proved by Hassler Whitney in 1937, who credited the result to William Graustein.\n\n## Explanations\n\n### L1 human\n\nDraw a closed loop on paper, and imagine moving it around smoothly without ever making a sharp corner.\nThe loop may cross itself during the motion, but its velocity must never vanish.\nThe theorem says that one number decides which loops can be turned into each other this way.\nIt is the number of full turns that the direction of the pencil makes in one trip around.\n\n### L2 human\n\nThe theorem classifies immersions of the circle into the plane up to regular homotopy by the integer turning number.\nSmale later extended this classification to immersions of spheres, which led to the eversion of the sphere.\nFor a regular closed curve of class $C^2$, the turning number equals the total signed curvature divided by $2\\pi$.\n",
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