  We compare some natural triangulations of the Teichm\"uller space of
hyperbolic surfaces with geodesic boundary and of some bordifications. We adapt
Scannell-Wolf's proof to show that grafting semi-infinite cylinders at the ends
of hyperbolic surfaces with fixed boundary lengths is a homeomorphism. This
way, we construct a family of triangulations of the Teichm\"uller space of
punctures surfaces that interpolates between Penner-Bowditch-Epstein's (using
the spine construction) and Harer-Mumford-Thurston's (using Strebel's
differentials). Finally, we show (adapting arguments of Dumas) that on a fixed
punctured surface, when the triangulation approaches HMT's, the associated
Strebel differential is well-approximated by the Schwarzian of the associated
projective structure and by the Hopf differential of the collapsing map.
