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  "body": "\n## Example\n\nFor $t \u003e 0$ and $\\theta \\in \\R$, let\n$$x(t, \\theta) = \\left(\\operatorname{sech} t \\cos\\theta, \\; \\operatorname{sech} t \\sin\\theta, \\; t - \\tanh t\\right).$$\nThis is the [[surface of revolution|NE09EDK19V]] of the tractrix $t \\mapsto (\\operatorname{sech} t, \\, t - \\tanh t)$ about the $z$-axis, and its image is called the pseudosphere.\nIts [[Gaussian curvature|SQVQ5DQT6W]] is $-1$ at every point.\n\nCheck.\nWrite $r(t) = \\operatorname{sech} t$ and $z(t) = t - \\tanh t$.\nThen $r'(t) = -\\operatorname{sech} t \\tanh t$ and $z'(t) = \\tanh^2 t$, so $r'^2 + z'^2 = \\tanh^2 t \\,(\\operatorname{sech}^2 t + \\tanh^2 t) = \\tanh^2 t$.\nFor $t \u003e 0$ the speed is $\\tanh t \u003e 0$, and the arc length from $t = 0$ is $s = \\ln\\cosh t$.\nSo in terms of arc length, $r = 1/\\cosh t = e^{-s}$.\nFor a surface of revolution with profile parametrized by arc length, $K = -r''(s)/r(s)$, and here $K = -e^{-s}/e^{-s} = -1$.\n\n## Remarks\n\nThe surface is not complete: as $t \\to 0$ it ends in a sharp circular edge of radius $1$, where the parametrization stops being regular.\nThis agrees with [[Hilbert's theorem|XM772FD0BS]], which says that no complete surface of constant curvature $-1$ sits smoothly and isometrically in $\\R^3$.\nPieces of the pseudosphere are locally isometric to pieces of the hyperbolic plane, so they were used as early models of hyperbolic geometry.\n\n## Explanations\n\n### L1 human\n\nThe pseudosphere looks like a trumpet: a long thin horn that widens to a sharp rim.\nA sphere has the same positive curvature everywhere; the pseudosphere has the same negative curvature everywhere.\nAt each point it bends like a saddle, curving up in one direction and down in the other.\n",
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