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  "body": "\n## Definition\n\nLet $\\gamma \\colon I \\to \\R^n$ be a [[regular|06JMR2XWXW]] curve of class $C^2$, parametrised by [[arc length|00R1JDDV5X]] $s$ on an interval $I$, and let $c \\in \\R$.\nThe *involute* of $\\gamma$ with parameter $c$ is the curve\n$$J(s) = \\gamma(s) + (c - s) \\, \\gamma'(s), \\qquad s \\in I.$$\nHere $\\gamma'(s)$ is the unit [[tangent vector|0D98771SM0]], so $J(s)$ lies on the tangent line at $\\gamma(s)$, at distance $\\abs{c - s}$ from $\\gamma(s)$.\n\n## Remarks\n\nExample: for the unit circle $\\gamma(s) = (\\cos s, \\sin s)$ and $c = 0$, one gets $J(s) = (\\cos s + s \\sin s, \\sin s - s \\cos s)$.\nCheck: $J'(s) = (s \\cos s, s \\sin s)$, which is orthogonal to $\\gamma'(s) = (-\\sin s, \\cos s)$.\nIn general $J'(s) = (c - s) \\gamma''(s)$, so $J$ meets each tangent line of $\\gamma$ at a right angle, and $J$ is singular at $s = c$.\nNon-example: the circle of radius $2$ about the origin is not an involute of the unit circle.\nIndeed, it meets each tangent line of the unit circle at an angle that is not a right angle.\nFor a plane curve with nonzero curvature, the [[evolute|Y3ZNV24V2C]] of each involute is the original curve, on the part where $s \\ne c$.\n",
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