Abnormal Curves in a Zermelo Navigation Problem in the Plane and the Fan Shape of Small Time Balls

In this note, motivated by the Zermelo navigation problem in the flat plane, where the current field is associated to a point vortex, we discuss the role of the abnormal curves in the shape of the small time balls. Abnormal curves are occurring in the strong current domain in the vicinity of the vortex and form with the hyperbolic geodesics the boundaries of the small time accessibility sets and give to the small time balls a fan shape. Implications with the regularity of the value function are discussed.