Tractoid ($d{s}^2={\cot }^2\theta d{\theta }^2+{\sin }^2\theta d{v}^2$)

The tractoid is the surface of revolution of a unit tractrix. For $X(\theta ,v)=(h(\theta ),\sin \theta \cos v,\sin \theta \sin v)$ with $0<\theta <\pi /2$, its coordinate tangent vectors are orthogonal and have squared lengths ${\cot }^2\theta ,{\sin }^2\theta$. The coordinates $x=v$, $y=\csc \theta$ transform this first fundamental form into $(d{x}^2+d{y}^2)/y^2$, giving a local isometry to the hyperbolic plane. Thus its Gaussian curvature is minus one. The angular coordinate is periodic globally, while its unwrapped coordinate is a real variable; this distinguishes the local metric identification from a global embedding of the whole hyperbolic plane.

In unwrapped upper-half-plane coordinates, the annulus $b\le y\le a$ on a tractoid identifies $x$ modulo $2\pi$. Connect $(x_0,b)$ and $(x_0+2\pi ,b)$ by the upper circular hyperbolic geodesic with center $(x_0+\pi ,0)$. Its largest height is $\sqrt{b^2+{\pi }^2}$. The displayed condition puts the whole arc in the annulus. Under the angular identification its endpoints coincide, and the resulting geodesic segment winds once before returning. Its terminal tangent generally differs from the initial tangent, so this construction does not give a smooth periodic closed geodesic. It explains how local hyperbolic geometry creates a returning segment on a surface of revolution.