 Returning geodesic in a tractoid annulus

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.