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  "body": "\n## Definition\n\nLet $G$ be a group acting on a space $X$.\n*Integral geometry* is the study of $G$-invariant measures on spaces of geometric objects in $X$.\nIt also studies the integrals of geometric quantities against these measures.\nIn the classical case, $X$ is the [[Euclidean space|0XPR1E85W0]] $\\R^n$ and $G$ is the [[Euclidean group|0H3YY8CPKH]] of rigid motions.\nThe objects are then the affine subspaces of a fixed dimension.\n\nThe basic example is the measure on [[lines|0MP9BCF6DP]] in $\\R^2$.\nFor $\\theta \\in [0, \\pi)$ and $p \\in \\R$, let $L(\\theta, p) = \\set{x \\in \\R^2 : x_1 \\cos \\theta + x_2 \\sin \\theta = p}$; every line is $L(\\theta, p)$ for exactly one pair $(\\theta, p)$.\nTransport the [[measure|PR11BZQQPG]] $dp \\, d\\theta$ from $[0, \\pi) \\times \\R$ to the set of lines by this parametrization.\nThe result is invariant under all rigid motions of $\\R^2$, and up to a constant factor it is the only such measure.\n\n## Remarks\n\nExample: the lines that meet the closed disc of radius $r$ centred at the origin are the $L(\\theta, p)$ with $\\abs{p} \\le r$, a set of measure $2 r \\pi$.\nA translation by a vector $a$ maps $L(\\theta, p)$ to $L(\\theta, p + a_1 \\cos \\theta + a_2 \\sin \\theta)$ and a rotation by $\\alpha$ shifts $\\theta$ by $\\alpha$, so both preserve $dp \\, d\\theta$.\nNon-example: writing non-vertical lines as $y = m x + c$, the measure $dm \\, dc$ is not invariant under rotations.\nIn these coordinates the invariant measure is $(1 + m^2)^{-3/2} \\, dm \\, dc$.\nThe same ideas give invariant measures on affine $k$-planes of $\\R^n$, and lead to the Crofton and kinematic formulas.\n",
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