= Smooth hyperbolic polygon gluing criterion
{title2=$\sum_{v\in C}\theta_v=2\pi$}
Suppose a finite simple <polygon> in the curvature-minus-one <hyperbolic plane> has <hyperbolic geodesic> sides paired to produce a closed <topological surface>. Its metric descends to a smooth <hyperbolic metric> exactly when paired <polygon sides> have equal <lengths> and the sum of internal <angles> in each vertex <equivalence class> is $2\pi$. Equal <lengths> allow <isometric> gluing at open edges; at vertices the sectors assemble into a hyperbolic disc precisely when their total <angle> is $2\pi$. Other positive total <angles> produce a <cone singularity of a surface metric>. The <area of a hyperbolic geodesic polygon> and the <Gauss-Bonnet theorem> then give area $-2\pi\chi$ for the closed <surface>.
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