= Cone point
{title2=$\theta=2\pi/m$}
A cone point on a metric surface has total angular measure $\theta$ about its tip. The flat model is $dr^2+(\theta/2\pi)^2r^2d\varphi^2$, with $0\le\varphi<2\pi$; the curvature-minus-one model replaces $r$ by $\sinh r$. In these models the tip is smooth when $\theta=2\pi$. For integer $m>1$, angle $2\pi/m$ gives a cyclic <orbifold> point of order $m$. Under a degree-$q$ local branched lift, the total angle is multiplied by $q$, and the residual orbifold order, when defined, is $m/q$.
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