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  "body": "\n## Example\n\nFix $c \u003e 0$.\nFor $u \\in \\R$ and $v \\in \\R$, let\n$$x(u, v) = (v \\cos u, \\; v \\sin u, \\; c u).$$\nIts image, the helicoid, is swept out by a horizontal line that turns about the $z$-axis at a constant rate while it moves up.\nIt is a [[minimal surface|41X0YW2R9N]].\n\nCheck.\nThe partial derivatives are $x_u = (-v \\sin u, v \\cos u, c)$ and $x_v = (\\cos u, \\sin u, 0)$.\nSo $E = v^2 + c^2$, $F = 0$ and $G = 1$.\nThe cross product is $x_u \\times x_v = (-c \\sin u, c \\cos u, -v)$, with length $W = \\sqrt{v^2 + c^2}$, and $N = (x_u \\times x_v)/W$.\nThe second derivatives are $x_{uu} = (-v \\cos u, -v \\sin u, 0)$, $x_{uv} = (-\\sin u, \\cos u, 0)$ and $x_{vv} = 0$.\nSo the coefficients of the [[second fundamental form|9KHQR16P1B]] are $e = 0$, $f = c/W$ and $g = 0$.\nThe [[mean curvature|PA78NFEPCF]] is $H = \\dfrac{eG - 2fF + gE}{2(EG - F^2)} = 0$.\n\n## Remarks\n\nThe Gaussian curvature is $K = -f^2/(EG) = -c^2/(v^2 + c^2)^2$.\nThe helicoid is a ruled surface, and together with the plane it is the only ruled minimal surface in $\\R^3$.\nIt is locally isometric to the [[catenoid|6NN1HP54GB]].\n\n## Explanations\n\n### L1 human\n\nPicture a spiral staircase whose steps are thin lines that extend to infinity on both sides of the central pole.\nThat surface is a helicoid.\nThe points at a fixed distance from the pole trace out helices, and a soap film can take this shape.\n",
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