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  "body": "\n## Example\n\nFor $u \\in \\R$ and $v \\in \\R$, let\n$$x(u, v) = (\\cosh v \\cos u, \\; \\cosh v \\sin u, \\; v).$$\nIts image, the catenoid, is the [[surface of revolution|NE09EDK19V]] obtained by rotating the catenary $r = \\cosh z$ about the $z$-axis.\nIt is a [[minimal surface|41X0YW2R9N]].\n\nCheck.\nThe partial derivatives are $x_u = (-\\cosh v \\sin u, \\cosh v \\cos u, 0)$ and $x_v = (\\sinh v \\cos u, \\sinh v \\sin u, 1)$.\nSo $E = \\cosh^2 v$, $F = 0$ and $G = \\sinh^2 v + 1 = \\cosh^2 v$, and the parametrization is conformal.\nFor a conformal parametrization, $x_{uu} + x_{vv} = 2 E H N$, where $H$ is the mean curvature and $N$ the unit normal.\nHere $x_{uu} = (-\\cosh v \\cos u, -\\cosh v \\sin u, 0)$ and $x_{vv} = (\\cosh v \\cos u, \\cosh v \\sin u, 0)$, so $x_{uu} + x_{vv} = 0$ and $H = 0$.\n\n## Remarks\n\nThe principal curvatures are $\\pm 1/\\cosh^2 v$, and the Gaussian curvature is $K = -1/\\cosh^4 v$.\nApart from pieces of planes, the catenoids are the only minimal surfaces of revolution.\nThe catenoid is locally isometric to the [[helicoid|EQ5JSYGWG2]], and one can be bent into the other through a family of minimal surfaces.\n\n## Explanations\n\n### L1 human\n\nDip two parallel circular rings into soapy water and pull them a little apart.\nThe soap film between them forms a tube with a narrow waist: that is a catenoid.\nIts profile is the catenary, the curve of a hanging chain.\n",
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