  In this paper, we investigate the asymptotic behavior of regular ends of flat
surfaces in the hyperbolic 3-space H^3. Galvez, Martinez and Milan showed that
when the singular set does not accumulate at an end, the end is asymptotic to a
rotationally symmetric flat surface. As a refinement of their result, we show
that the asymptotic order (called "pitch" p) of the end determines the limiting
shape, even when the singular set does accumulate at the end. If the singular
set is bounded away from the end, we have -1<p<=0. If the singular set
accumulates at the end, the pitch p is a positive rational number not equal to
1. Choosing appropriate positive integers n and m so that p=n/m, suitable
slices of the end by horospheres are asymptotic to d-coverings (d-times wrapped
coverings) of epicycloids or d-coverings of hypocycloids with 2n_0 cusps and
whose normal directions have winding number m_0, where n=n_0d, m=m_0d (n_0, m_0
are integers or half-integers) and d is the greatest common divisor of m-n and
m+n. Furthermore, it is known that the caustics of flat surfaces are also flat.
So, as an application, we give a useful explicit formula for the pitch of ends
of caustics of complete flat fronts.
