Pseudosphere

Example

For $t > 0$ and $\theta \in \R$, let $$x(t, \theta) = \left(\operatorname{sech} t \cos\theta, \; \operatorname{sech} t \sin\theta, \; t - \tanh t\right).$$ This is the surface of revolution of the tractrix $t \mapsto (\operatorname{sech} t, \, t - \tanh t)$ about the $z$-axis, and its image is called the pseudosphere. Its Gaussian curvature is $-1$ at every point.

Check. Write $r(t) = \operatorname{sech} t$ and $z(t) = t - \tanh t$. Then $r'(t) = -\operatorname{sech} t \tanh t$ and $z'(t) = \tanh^2 t$, so $r'^2 + z'^2 = \tanh^2 t \,(\operatorname{sech}^2 t + \tanh^2 t) = \tanh^2 t$. For $t > 0$ the speed is $\tanh t > 0$, and the arc length from $t = 0$ is $s = \ln\cosh t$. So in terms of arc length, $r = 1/\cosh t = e^{-s}$. For a surface of revolution with profile parametrized by arc length, $K = -r''(s)/r(s)$, and here $K = -e^{-s}/e^{-s} = -1$.