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

TL;DR

This paper investigates pursuit-evasion games on a spherical surface, analyzing how equilibrium intercept points relate to the Apollonius domain, and explores strategies for two pursuers against a slower evader with numerical simulations.

Contribution

It extends planar pursuit-evasion concepts to spherical surfaces, identifying conditions for equilibrium points to lie within the Apollonius domain on the sphere.

Findings

Intercept point belongs to the Apollonius domain under specific conditions.

Spherical pursuit-evasion strategies can be characterized similarly to planar cases.

Numerical simulations illustrate the theoretical findings.

Abstract

In classical works on a planar differential pursuit-evasion game with a faster pursuer, the intercept point resulting from the equilibrium strategies lies on the Apollonius circle. This property was exploited for the construction of the equilibrium strategies for two faster pursuers against one evader. Extensions for planar multiple-pursuer single-evader scenarios have been considered. We study a pursuit-evasion game on a sphere and the relation of the equilibrium intercept point to the Apollonius domain on the sphere. The domain is a generalization of the planar Apollonius circle set. We find a condition resulting in the intercept point belonging to the Apollonius domain, which is the characteristic of the planar game solution. Finally, we use this characteristic to discuss pursuit and evasion strategies in the context of two pursuers and a single slower evader on the sphere and illustrate it using numerical simulations.