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  "body": "\n## Definition\n\nLet $\\gamma \\colon I \\to \\R^2$ be a [[regular|06JMR2XWXW]] curve of class $C^2$ on an interval $I$.\nLet $N(t)$ be its unit tangent rotated by a right angle counterclockwise, and let $k(t)$ be its signed [[curvature|7JQS6VKFEZ]].\nAssume that $k(t) \\ne 0$ for all $t \\in I$.\nThe *evolute* of $\\gamma$ is the curve\n$$E(t) = \\gamma(t) + \\frac{1}{k(t)} N(t), \\qquad t \\in I.$$\nSo $E(t)$ is the centre of the [[osculating circle|NWSFMCRWHJ]] of $\\gamma$ at $t$.\n\n## Remarks\n\nExample: for $\\gamma(t) = (t, t^2)$, one has $k(t) = 2 (1 + 4t^2)^{-3/2}$ and $N(t) = (-2t, 1) / \\sqrt{1 + 4t^2}$.\nThen $E(t) = (-4t^3, 1/2 + 3t^2)$, so the evolute is the semicubical parabola $27 x^2 = 16 (y - 1/2)^3$.\nNon-example: for $\\gamma(t) = (t, t^3)$ on $\\R$, the curvature vanishes at $t = 0$, so the evolute is not defined on the whole interval; near $t = 0$ the centres of curvature go off to infinity.\nThe evolute of a circle of radius $r$ is the constant curve at its centre.\nIf $\\gamma$ is of class $C^3$, then $E'(t)$ is parallel to $N(t)$, and $E'(t) = 0$ exactly where $k'(t) = 0$, that is, at the vertices of $\\gamma$.\n",
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