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  "body": "\n## Example\n\nLet $s \\mapsto (r(s), z(s))$, for $s$ in an open interval $I$, be a smooth regular curve in the $xz$-plane.\nAssume that it is parametrized by [[arc length|0TH78S7GBZ]] and that $r(s) \u003e 0$ for all $s$.\nRotating it about the $z$-axis gives the surface of revolution\n$$x(s, \\theta) = (r(s) \\cos\\theta, \\; r(s) \\sin\\theta, \\; z(s)), \\qquad s \\in I, \\ \\theta \\in \\R.$$\nThe partial derivatives are $x_s = (r' \\cos\\theta, r' \\sin\\theta, z')$ and $x_\\theta = (-r \\sin\\theta, r \\cos\\theta, 0)$.\nThey are orthogonal, $\\abs{x_s} = 1$ because $r'^2 + z'^2 = 1$, and $\\abs{x_\\theta} = r \u003e 0$.\nSo $x$ is an immersion, and if the profile curve is a homeomorphism onto its image then the image of $x$ is a [[regular surface|6080X3FAD4]].\nThe [[first fundamental form|MAJ3DH7MZS]] is $ds^2 + r(s)^2 \\, d\\theta^2$.\nThe meridians ($\\theta$ constant) and the parallels ($s$ constant) are lines of curvature, and the [[Gaussian curvature|SQVQ5DQT6W]] is\n$$K = -\\frac{r''(s)}{r(s)}.$$\n\n## Remarks\n\nFor $r(s) = \\cos s$ and $z(s) = \\sin s$ with $s \\in (-\\pi/2, \\pi/2)$, one gets the unit sphere without its poles, and $K = \\cos s / \\cos s = 1$.\nRotating a circle that does not meet the axis gives a torus.\nRotating a catenary gives the [[catenoid|6NN1HP54GB]], and rotating a tractrix gives the [[pseudosphere|44WEBQSKEY]].\n\n## Explanations\n\n### L1 human\n\nTake a curve drawn in a half-plane and spin it around the edge line of the half-plane.\nIt sweeps out a surface, like a vase on a potter's wheel.\nSpheres, cylinders, cones and doughnuts all arise this way.\nBecause of the symmetry, the curvature of such a surface depends only on the profile curve.\n",
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