 Triangulation of a polygon with paired sides

Triangulate a closed polygon by joining an interior point to every side, using matching parametrizations on paired sides. Orient every radial edge away from the interior point and choose consistent orientations on paired boundary edges. Each triangle then has a local vertex ordering with the centre first and its boundary edge in the chosen direction. These local orderings agree along shared edges. The quotient triangles and their affine face identifications form a finite delta-complex, even if some triangle vertices become identified. The second barycentric subdivision of a delta-complex removes these identifications at the simplicial level without changing the underlying quotient space. Thus every polygonal schema with a complete pairing has a finite triangulation. A single subdivision need not suffice to give an abstract complex.