Tractrix ($r=\sin \theta ,h=\cos \theta +\log \tan (\theta /2)$)

A tractrix is a plane curve whose tangent segment from the point of contact to a fixed axis has constant length. For segment length one and a positive acute tangent angle $\theta$, its height is $r=\sin \theta$. The slope relation $dr/dh=\tan \theta$ gives $dh/d\theta ={\cos }^2\theta /\sin \theta$. Integrating yields the displayed axial coordinate up to an additive constant. Conversely these coordinates have the required slope, and their tangent intercept is at distance $r/\sin \theta =1$. Rotating this profile about its axis gives a tractoid.