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I know that under the arc length parametrization for the curve $t\mapsto (x(t),0,z(t))$, the Gaussian curvature of the surface of revolution is $K = -\frac{x''}{x}$. So, $K = -1\iff x'' = x$. Now, I know that the pseudosphere $\gamma(t) = (sin(t),0,cos(t)+ln(tan(t/2)))$ for $t\in (\pi/2,\pi)$ has -1 Gaussian curvature. As the arc length of $\gamma(t)$ is $Arclength(\gamma) =cot(s)$, $\gamma(cot^{-1}(s))$ is an arc length parameterization of it. But, $\frac{d^2}{ds^2}sin(cot^{-1}(s))\neq sin(cot^{-1}(s))$. What's wrong in here?

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First, the arclength $s$ satisfies $ds/dt =-\cot t$. You lost a sign and an integral. So, in fact, $s=-\log\sin t$. Now finish correctly.

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