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TOTAL CURVATURE AND SIMPLE PURSUIT ON DOMAINS OF CURVATURE BOUNDED ABOVE
 

Summary: TOTAL CURVATURE AND SIMPLE PURSUIT ON DOMAINS OF
CURVATURE BOUNDED ABOVE
S. ALEXANDER, R. BISHOP, AND R. GHRIST
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.
1. INTRODUCTION
The goals of this paper are twofold:
(1) We study total curvature of a curve (the integral of its curvature) in
spaces of curvature bounded above, and relate the total curvature,
the curve's circumradius function, the asymptotic behavior of the
curve, and the domain's curvature bound.
(2) We apply these results to a foundational problem in pursuit-evasion
games, where an evader moves in a domain and is followed by a
pursuer along a pursuit curve. We study the capture problem: whether
the pursuer ever catches (comes sufficiently close to) the evader. Al-

  

Source: Alexander, Stephanie - Department of Mathematics, University of Illinois at Urbana-Champaign

 

Collections: Mathematics