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Parametrize a nonconstant hyperbolic geodesic by unit speed. Its energy is $({\dot{x}}^2+{\dot{y}}^2)/(2y^2)$, and the Euler-Lagrange equation in the cyclic coordinate $x$ gives $\dot{x}/y^2=c$. Unit speed gives ${\dot{x}}^2+{\dot{y}}^2=y^2$. If $c=0$ the curve is vertical. Otherwise, on either side of a possible highest point,The two branches join smoothly at the top. Conversely these curves with their unit-speed parametrizations satisfy the energy equations. Thus the unparametrized complete geodesics are vertical half-lines and upper semicircles centered on the real axis.

Let the tractrix point have positive height $r(\theta )$. Its tangent segment to the axis has length one and makes angle $\theta$, so $r=\sin \theta$. Since its slope is $\tan \theta$, $h^{\prime }(\theta )={\cos }^2\theta /\sin \theta$ for its axial coordinate. Rotating $X(\theta ,v)=(h(\theta ),\sin \theta \cos v,\sin \theta \sin v)$ gives orthogonal coordinate vectors with squared lengths ${\cot }^2\theta$ and ${\sin }^2\theta$. Hence the stated first fundamental form follows.

Now put $x=v$, $y=\csc \theta$. Direct substitution givesOn the specified intervals this coordinate map is a diffeomorphism onto the stated rectangle, so is an isometry there and a local isometry on the tractoid.

Set $b=\csc \beta$, $a=\csc \alpha$. In the unwrapped coordinate, connect $(x_0,b)$ to $(x_0+2\pi ,b)$ along the semicircle centered at $(x_0+\pi ,0)$ of radius $\sqrt{b^2+{\pi }^2}$. Its height lies between $b$ and that radius. The hypothesis $a^2-b^2>{\pi }^2$ puts the whole arc in the closed height strip $b\le y\le a$. Projecting the angular coordinate modulo $2\pi$ therefore gives a geodesic segment in the closed tractoid annulus which winds once and returns to its starting point. Taking $x_0$ to represent any chosen angular position proves the assertion for every point of its $\theta =\beta$ boundary. The returned tangent need not equal the starting tangent, so this proves the claimed returning segment without asserting a periodically smooth closed geodesic.

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.

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.