  We start by describing how ideal triangulations on a surface degenerate under
pinching of a multicurve. We use this process to construct a homomorphism from
the Ptolemy groupoid of a surface to that of a pinched surface which is natural
with respect to the action of the mapping class group. We then apply this
construction to the study of shear coordinates and their extension to the
augmented Teichm\"uller space. In particular, we give an explicit description
of the action of the mapping class group on the augmented Teichm\"uller space
in terms of shear coordinates.
