  In this article, based on two case studies, we discuss the role of abnormal
geodesics in planar Zermelo navigation problems. Such curves are limit curves
of the accessibility set, in the domain where the current is strong. The
problem is set in the frame of geometric time optimal control, where the
control is the heading angle of the ship and in this context, abnormal curves
are shown to separate time minimal curves from time maximal curves and are both
small-time minimizing and maximizing. We describe the small-time minimal balls.
For bigger time, a cusp singularity can occur in the abnormal direction, which
corresponds to a conjugate point along the non-smooth image. It is interpreted
in terms of the regularity property of the time minimal value function.
