AbstractIsaacs' formulation of the game of two cars, which assumes players of circular shape, is extended to include the effect of elongated players and elliptical shapes in particular. The geometry is used to model the problem of ship‐collision avoidance as a differential game which suggests the name of ‘game of two ships’. The game of two ships is formulated as a free‐time unbounded game of kind. A general solution for the optimal controls on the terminal manifold is given and is also used to derive a model analytical solution for the primary optimal path emanating from the target set. The occurrence of singular arcs as well as the condition for switching are also discussed. Finally, a capture criterion for the pursuit‐evasion game between a faster elliptical pursuer and a more manoeuvrable circular evader is also derived. This capture criterion is given in the form of a relationship between the geometrical dimensions of the players, their speed and acceleration ratios, and the maximum range that the manoeuvrable, yet slower, evader can indefinitely guarantee without being captured. If both players have circular shapes, our solution reduces to an already‐known capture criterion for the ‘game