= Tractoid
{title2=$ds^2=\cot^2\theta\,d\theta^2+\sin^2\theta\,dv^2$}
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 $(dx^2+dy^2)/y^2$, 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>.
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