Consider the tractrix
$$c:(0,\frac{\pi}{2})\rightarrow(0,1)\times(-\infty,0),\quad c(t)=\Big(\sin(t),\cos(t)+\log\circ\tan(\frac{t}{2})\Big)$$
I want to write the parametrisation of its surface of revolution like this:
$$F:(-\infty,0)\times(0,2\pi)\rightarrow \mathbb{R}^3,\quad F(t,\varphi)=(r(t)\cos(\varphi),r(t)\sin(\varphi),t)$$ This means I want to find a function $s:(-\infty,0)\rightarrow(0,\frac{\pi}{2})$, such that $$\cos(s(t))+\log\circ\tan(\frac{s(t)}{2})=t$$
Then I would define $$r:(-\infty,0)\rightarrow(0,1),\quad r(t)=\sin(s(t))$$
I hope this makes sense so far. How could I determine the function $s$?