The geometric approach to the construction of the barrier surface in differential games

In the present work we examine Isaac's geometric method for the construction of the barrier surface in differential games. We are thus led, in a natural way, to a new (geometric) second-order necessary condition that a valid barrier surface must satisfy. This help us, on the one hand, to clarify the appearance of a previously discovered qualitatively new type of barrier surface when the problema parameters enter a certain region of the parameter space, and, on the other hand, our systematic analysis also helps us to identify new interesting regions in parameter space where the barrier surface qualitatively differs from previously observed patterns.