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  "body": "\n## Statement\n\nFor $\\theta \\in [0, \\pi)$ and $p \\in \\R$, let $L(\\theta, p)$ be the [[line|0MP9BCF6DP]] $\\set{x \\in \\R^2 : x_1 \\cos \\theta + x_2 \\sin \\theta = p}$.\nLet $\\gamma \\colon [a, b] \\to \\R^2$ be a rectifiable [[plane curve|EWS4WFTA7W]] of [[length|0TH78S7GBZ]] $\\ell(\\gamma)$.\nLet $n_\\gamma(\\theta, p) \\in \\set{0, 1, 2, \\ldots} \\cup \\set{\\infty}$ be the number of $t \\in [a, b]$ with $\\gamma(t) \\in L(\\theta, p)$.\nThen $n_\\gamma$ is measurable and\n$$\\ell(\\gamma) = \\frac{1}{2} \\int_0^\\pi \\int_{-\\infty}^{\\infty} n_\\gamma(\\theta, p) \\, dp \\, d\\theta.$$\n\n## Remarks\n\nBoth sides are additive when the curve is cut into pieces, and both are invariant under rigid motions, since $dp \\, d\\theta$ is the invariant measure of [[integral geometry|TAN5G5EQ96]].\nTake a segment of length $\\ell$ in the direction $\\varphi$, and fix $\\theta$.\nThe lines $L(\\theta, p)$ meeting it form an interval of $p$ of length $\\ell \\abs{\\cos(\\theta - \\varphi)}$, and $\\int_0^\\pi \\ell \\abs{\\cos(\\theta - \\varphi)} \\, d\\theta = 2 \\ell$.\nSo the formula holds for polygonal curves, and the general case follows by approximating $\\gamma$ by inscribed polygons and passing to the limit.\nExample: a circle of radius $r$ meets $L(\\theta, p)$ twice when $\\abs{p} \u003c r$, so the right side is $\\frac{1}{2} \\cdot \\pi \\cdot 2r \\cdot 2 = 2 \\pi r$.\n\n## Explanations\n\n### L1 human\n\nThe length of a curve equals, up to a fixed factor, the average number of times a random line crosses it.\nLong or wiggly curves are crossed more often, and the formula makes this exact.\n\n### L2 human\n\nFor the boundary of a plane convex body, almost every line that meets the body crosses the boundary twice.\nSo the perimeter equals $\\int_0^\\pi w(\\theta) \\, d\\theta$, where $w(\\theta)$ is the width in the direction $\\theta$.\nThis is Cauchy's formula, and it gives Barbier's theorem for curves of constant width.\nThe formula explains [[Buffon's needle problem|06X5PF6YFP]].\nIt is the model for Crofton formulas in higher dimensions, which recover intrinsic volumes of convex bodies from intersections with random affine subspaces.\n",
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