  Geodesics, which play an important role in spray-Finsler geometry, are
integral curves of a spray vector field on a manifold. Some comparison theorems
and rigidity issues are established on the completeness of geodesics of a spray
or a Finsler metric. In this paper, projectively flat sprays with weak Ricci
constant (eps. constant curvature) are classified at the level of geodesics.
Further, a geodesic method is introduced to determine an $n$-dimensional spray
based on a family of curves with $2(n-1)$ free constant parameters as
geodesics. Finally, it shows that a spray is projectively complete under
certain condition satisfied by the domain of geodesic parameter of all
geodesics.
