= Cone singularity of a surface metric
{title2=$\Theta\ne2\pi$}
A <surface> metric has a cone singularity of total <angle> $\Theta>0$ if a punctured <neighbourhood> has the metric obtained by joining the radial boundaries of a sector of <angle> $\Theta$. For constant <Gaussian curvature> $-1$ away from the vertex, its polar expression is $dr^2+(\Theta/2\pi)^2\sinh^2r\,d\theta^2$, where $\theta$ has period $2\pi$. A circle of small <radius> $r$ has <length> asymptotic to $\Theta r$. A smooth <Riemannian surface> instead has circumference asymptotic to $2\pi r$, so the metric extends smoothly at the vertex exactly when $\Theta=2\pi$.
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