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  "body": "\n## Definition\n\nLet $J \\subseteq \\R$ be an open interval, let $U \\subseteq \\R^2$ be open, and let $F : U \\times J \\to \\R$ be continuously differentiable.\nFor $c \\in J$, let $C_c = \\set{(x, y) \\in U : F(x, y, c) = 0}$, a one-parameter family of curves.\nAn *envelope* of the family is a [[regular curve|06JMR2XWXW]] $\\gamma : J \\to U$ such that for each $c \\in J$:\n\n1. $\\gamma(c)$ lies on $C_c$, that is, $F(\\gamma(c), c) = 0$;\n2. the gradient $\\nabla_{x,y} F(\\gamma(c), c)$ is nonzero;\n3. $\\gamma$ is tangent to $C_c$ at $\\gamma(c)$, that is, $\\nabla_{x,y} F(\\gamma(c), c) \\cdot \\gamma'(c) = 0$.\n\n## Remarks\n\nDifferentiating $F(\\gamma(c), c) = 0$ and using clause 3 gives $\\partial F / \\partial c\\, (\\gamma(c), c) = 0$, where $\\partial F/\\partial c$ is the [[partial derivative|05AXG3QRKR]] in $c$.\nSo the points of an envelope satisfy the two equations $F = 0$ and $\\partial F / \\partial c = 0$.\nEliminating $c$ gives the discriminant set, which contains the envelope but may also contain singular points of the curves.\nExample: $F(x, y, c) = x \\cos c + y \\sin c - 1$, the lines at distance $1$ from the origin.\nThen $\\gamma(c) = (\\cos c, \\sin c)$ satisfies $F = \\cos^2 c + \\sin^2 c - 1 = 0$, and $\\gamma'(c) = (-\\sin c, \\cos c)$ is orthogonal to $\\nabla F = (\\cos c, \\sin c)$; so the unit circle is the envelope.\nNon-example: the concentric circles $x^2 + y^2 - c^2 = 0$, $c \u003e 0$, have no envelope, since $\\partial F / \\partial c = -2c \\ne 0$ and clause 3 can never hold together with clause 1.\nFor a regular curve $\\beta$, the family of its [[tangent lines|01R1HW3Y42]], with parameter $c$ the parameter of $\\beta$, has $\\beta$ itself as an envelope.\n",
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