Total curvature and simple pursuit on domains of curvature bounded above

TL;DR

This paper investigates how total curvature influences the behavior of curves in curved spaces and applies these insights to pursuit-evasion games, extending known results to more general geometric domains.

Contribution

It introduces new bounds relating total curvature to curve behavior in CAT(0) and CAT(K) spaces and generalizes pursuit game results beyond convex Euclidean domains.

Findings

Total curvature controls asymptotic behavior of curves in curved spaces.

Generalized pursuit-evasion game results to broader geometric domains.

Established bounds linking curvature growth rates to pursuit dynamics.

Abstract

We show how circumradius and asymptotic behavior of curves in CAT(0) and CAT(K) spaces(K>0) are controlled by growth rates of total curvature. We apply our results to pursuit and evasion games of capture type with simple pursuit motion, generalizing results that are known for convex Euclidean domains, and obtaining results that are new for convex Euclidean domains and hold on playing fields vastly more general than these.