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  "body": "\n## Example\n\nThe *umbilic catastrophes* are the three elementary catastrophes with two state variables $(x, y) \\in \\R^2$.\nIn their standard forms they are:\n\n1. the *hyperbolic umbilic* $V = x^3 + y^3 + a xy + b x + c y$, with controls $(a, b, c) \\in \\R^3$;\n2. the *elliptic umbilic* $V = \\tfrac{1}{3}x^3 - x y^2 + a(x^2 + y^2) + b x + c y$, with controls $(a, b, c) \\in \\R^3$;\n3. the *parabolic umbilic* $V = x^2 y + y^4 + a x^2 + b y^2 + c x + d y$, with controls $(a, b, c, d) \\in \\R^4$.\n\nWhen all controls vanish, each $V$ has a [[critical point|0YTCMFB4HC]] at the origin where the whole [[Hessian matrix|0693Y2METT]] is zero.\nFor the hyperbolic umbilic, the Hessian of $x^3 + y^3$ is $\\begin{pmatrix} 6x \u0026 0 \\\\ 0 \u0026 6y \\end{pmatrix}$, which is $0$ at the origin.\nFor the elliptic umbilic, the Hessian of $\\tfrac13 x^3 - x y^2$ is $\\begin{pmatrix} 2x \u0026 -2y \\\\ -2y \u0026 -2x \\end{pmatrix}$, again $0$ at the origin.\nFor the parabolic umbilic, the Hessian of $x^2 y + y^4$ is $\\begin{pmatrix} 2y \u0026 2x \\\\ 2x \u0026 12y^2 \\end{pmatrix}$, again $0$ at the origin.\nSo the degeneracy has corank $2$: no change of coordinates can split off a nondegenerate square in one of the variables.\nThis separates the umbilics from the fold, cusp, swallowtail and butterfly, which have corank $1$ and need only one state variable.\n\nFor the elliptic umbilic, the slices $a = \\text{const} \\ne 0$ of the [[bifurcation set|JV45S0KQE6]] are curves with three cusps.\n\n## Explanations\n\n### L1 human\n\nThe fold, cusp, swallowtail and butterfly describe a ball rolling along a line.\nThe umbilics describe a ball rolling on a surface, so the potential has two state variables.\nAt the most degenerate setting the surface is flat to second order in every direction, like the monkey saddle $\\tfrac13 x^3 - x y^2$ of the elliptic umbilic.\nSmall changes of the controls break this point into several ordinary critical points, in a few typical patterns.\nThom's list of seven elementary catastrophes contains exactly three of these: the hyperbolic, elliptic and parabolic umbilics.\n",
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