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  "body": "\n## Summary\n\nThe Legendre transformation turns a strictly [[convex function|07Q7CNM5WF]] $f$ into a new function $f^*$ of the slope $p = f'(x)$.\nIt records the same information as $f$, but with slopes as the variable instead of points.\nApplied twice it gives back $f$, and the derivatives of $f$ and $f^*$ are inverse functions of each other.\nIt is the classical case of the convex conjugate and is used in mechanics, thermodynamics and optimization; the [[definition|HKPHRSN3ER]] gives the precise formula.\n\n## Explanations\n\n### L1 human\n\nA convex curve can be described by its points, or by its tangent lines.\nEach tangent line has a slope $p$ and meets the vertical axis at some height; the Legendre transform $f^*(p)$ is minus that height.\nSo $f^*$ lists the tangent lines of the graph, one for each slope.\nFor $f(x) = x^2 / 2$ the tangent at $x$ has slope $p = x$, and one finds $f^*(p) = p^2 / 2$, the same function.\n",
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