Pursuit-Evasion on a Sphere and When It Can Be Considered Flat

In planar pursuit-evasion differential games considering a faster pursuer and slower evader, the interception points resulting from equilibrium strategies lie on the Apollonius circle. This property is instrumental for leveraging geometric approaches for solving multiple pursuit-evasion scenarios in the plane. Here, we study a pursuit-evasion differential game on a sphere and generalize the planar Apollonius set to the spherical domain. We find that the interception point from the equilibrium strategies can leave the Apollonius set boundary and present a condition to keep the intercept point on the boundary. This condition allows for generalizing planar pursuitevasion strategies to the sphere.