  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.
