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  "body": "\n## Statement\n\nLet $M$ be a smooth surface, and give $C^\\infty(M, \\R^2)$ the [[Whitney $C^\\infty$ topology|7PN9MD1PQB]].\nThe smooth maps $f \\colon M \\to \\R^2$ with the following property form a dense subset of $C^\\infty(M, \\R^2)$.\nAt every point $p \\in M$, there are local coordinates centred at $p$ and at $f(p)$ in which $f$ has one of these forms:\n\n1. $(x, y) \\mapsto (x, y)$, a regular point;\n2. $(x, y) \\mapsto (x, y^2)$, a fold point;\n3. $(x, y) \\mapsto (x, y^3 + x y)$, a cusp point.\n\n## Remarks\n\nAt a fold point the [[singular set|1JG3HFWX3T]] is a smooth curve and the kernel of the differential is transverse to it.\nAt a cusp point the kernel is tangent to the singular curve.\nFor the cusp map $f(x, y) = (x, y^3 + xy)$ the singular set is the parabola $x = -3y^2$, and its image is the cusp curve $(-3y^2, -2y^3)$, which satisfies $27 v^2 = -4 u^3$ in the target coordinates $(u, v)$.\nThe proof uses transversality of the jet of $f$ to the subsets of the jet space defined by these singularities.\nWorse singularities have codimension greater than $2$, so generic maps of a surface avoid them.\nBoth the fold and the cusp are stable germs, so they cannot be removed by small perturbations.\nThe theorem is due to Whitney and was a starting point of the theory of [[stable maps|1991WETZF9]].\n\n## Explanations\n\n### L1 human\n\nProject a smooth surface onto a plane, as when you look at a curved object and see its outline.\nWhitney's theorem says that, after a tiny wiggle of the surface, only two kinds of special points appear.\nAlong fold curves the surface turns over like the edge of a folded sheet of paper.\nAt isolated cusp points a fold curve stops, and its outline forms a sharp spike.\nNothing more complicated is typical.\n\n### L2 human\n\nThis is the first nontrivial case of the classification of generic singularities of maps between manifolds of equal dimension.\nIn the language of [[Thom–Boardman singularities|A9TYDPD5D7]], folds are the $\\Sigma^{1,0}$ points and cusps are the $\\Sigma^{1,1,0}$ points.\nHigher-dimensional analogues are the [[Morin singularities|DF8CJ1A9MF]], and the fold and cusp reappear in [[catastrophe theory|6FKJ6SXP3G]] as the fold and cusp catastrophes.\n",
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