  In an infinite-dimensional Teichm\"uller space, it is known that the geodesic
connecting two points can be unique or not. In this paper, we study the
situation on the geodesic in the universal asymptotic Teichm\"uller space
$AT(\Delta)$. We introduce the notion of substantial point and non-substantial
point in $AT(\Delta)$. The set of all non-substantial points is open and dense
in $AT(\Delta)$. It is shown that there are infinitely many geodesics joining a
non-substantial point to the basepoint. Although we have difficulty in dealing
with the substantial points, we give an example to show that there are
infinitely many geodesics connecting certain substantial point and the
basepoint. It is also shown that there are always infinitely many straight
lines containing two points in $AT(\Delta)$. Moreover, with the help of the
Finsler structure on the asymptotic Teichm\"uller space, a variation formula
for the asymptotic Teichm\"uller metric is obtained.
