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

Abnormal Geodesics in 2D-Zermelo Navigation Problems in the Case of Revolution and the Fan Shape of the Small Time Balls