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  "body": "\n## Summary\n\nThe Beltrami equation is the partial differential equation $f_{\\bar z} = \\mu f_z$ for a function $f$ of one complex variable.\nThe coefficient $\\mu$ is a given measurable function with $\\norm{\\mu}_\\infty \u003c 1$.\nFor $\\mu = 0$ it is the Cauchy–Riemann equation, whose solutions are the [[holomorphic functions|0NAHXR1FRH]].\nIts homeomorphic solutions are exactly the [[quasiconformal mappings|SB8KG46NHP]] with complex dilatation $\\mu$.\nThe [[measurable Riemann mapping theorem|TZHD2WKNSD]] says that it always has such a solution.\n\n## Explanations\n\n### L1 human\n\nThe coefficient $\\mu$ prescribes, at each point, an ellipse field: a direction and a stretching ratio.\nA solution $f$ of the Beltrami equation is a map that sends these tiny ellipses to tiny circles.\nSo solving the equation means finding coordinates in which a given distorted geometry looks round.\nFor example, with the constant coefficient $\\mu = 1/3$ the map $f(z) = z + \\frac{1}{3}\\bar z$ solves the equation, since $f_z = 1$ and $f_{\\bar z} = 1/3$.\n",
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