= Returning geodesic in a tractoid annulus
{title2=$a^2-b^2>\pi^2$}
In unwrapped upper-half-plane coordinates, the annulus $b\leq y\leq a$ 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>.
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