  In this paper are studied the nets of principal curvature lines on surfaces
embedded in Euclidean $3-$space near their end points, at which the surfaces
tend to infinity.
  This is a natural complement and extension to smooth surfaces of the work of
Garcia and Sotomayor (1996), devoted to the study of principal curvature nets
which are structurally stable -- do not change topologically-- under small
perturbations on the coefficients of the equations defining algebraic surfaces.
  This paper goes one step further and classifies the patterns of the most
common and stable behaviors at the ends, present also in generic families of
surfaces depending on one-parameter.
