Smooth hyperbolic polygon gluing criterion (${\sum }_{v\in C}{\theta }_v=2\pi$)

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.