This paper studies the time optimal paths of a mobile robot navigating in a planar environment containing an obstacle. The paper considers a point-mass robot that moves with bounded acceleration and limited turn-rate controls in the presence of an obstacle. The optimal control problem yields 12 path primitives that form the time optimal paths of the point-mass robot. The problem is then extended to a disc-robot that moves in the presence of an obstacle with turn in-place capability. The optimality conditions yield 12 modified path primitives that form the time optimal paths of the disc-robot. All path primitives are analytically characterized and examples demonstrate how they form time optimal trajectories in the presence of several obstacle types.