In this paper, we address a visibility-based target tracking game for the scenario when the environment contains a circular obstacle. The game is originally formulated in four dimensions, but due to the symmetry of the environment, the dimension of the state space can be reduced to three. The control policies of the players on possible barrier surfaces are computed on the basis of semipermeability of the barriers. A standard surface, that can be a barrier, is constructed using Isaacs' techniques. It is shown that the surface lies outside the game space. This opens up the possibility that the evader might be able to win the underlying game of kind for all initial positions in the game space or that the set of such win positions does not coincide with the game space and is determined by some barrier surfaces, construction of which may represent an independent difficult problem. We present the construction of the optimal control policies, and trajectories for the players near the usable part on the terminal manifold by analyzing a related game of degree.
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