The geometrical properties of optimal trajectories in the game of simple pursue on plane in the presence of obstacle

The game problem of pursue on plane in the presence of obstacle is considered. The optimal trajectories of this system are either geodesic lines or their envelopes (singular paths). An algorithm of constructing the domain on which all singular paths end is suggested. It is shown that in the two-dimensional case this domain is always non-empty. The existence of singular dispersal surface is proved. In several examples the domains of unconditional envelope of obstacle by pursuer are constructed (irrespective of evader).