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  "body": "\n## Summary\n\nThe Abel transform of a function $f$ of a radius $r \\ge 0$ is $F(y) = 2 \\int_y^\\infty \\frac{f(r)\\, r}{\\sqrt{r^2 - y^2}}\\, dr$.\nIt gives the integral of a circularly symmetric function of the plane along a line at distance $y$ from the centre.\nSo it is the [[Radon transform|VJNZJ352PC]] of a radial function.\nIt can be inverted by a formula of the same type, which solves an integral equation first studied by Abel.\nIt is used to recover radial profiles of emission or density from line-of-sight measurements, for example in plasma physics and astronomy.\n\n## Explanations\n\n### L1 human\n\nPicture a round, glowing cloud that is brighter near the centre.\nLooking at it from the side, each line of sight passes through the cloud and adds up its brightness along the way.\nThe Abel transform turns the brightness at each radius into the total seen along each line.\nFor a uniform disc of radius $R$, the line at distance $y \u003c R$ meets it in a chord of length $2\\sqrt{R^2 - y^2}$, and this is the Abel transform.\n",
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