  The mapping class group invariant ideal cell decomposition of the
Teichmueller space of a punctured surface times an open simplex has been used
in a number of computations. This paper answers a question about the
asymptotics of this decomposition, namely, in a given cell of the
decomposition, which curves can be short? Screens are a new combinatorial
structure which provide an answer to this question. The heart of the
calculation here involves Ptolemy transformations and the triangle inequalities
on lambda lengths.
