2014/12/02 by Rolf Klein, Klein, Rolf, Elmar Langetepe +3
Computer Science · Mathematics · #05A15 #34H99 #49N75 #Artificial Intelligence in Games #Combinatorics #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Computer science #FOS: Computer and information sciences #Geometry #Mathematics #Mathematics and Applications #Merge (version control) #Physics #cs.CG #msc:05A15 #msc:34H99 #msc:49N75
paper · pdf · doi:10.48550/arxiv.1412.6065
A preliminary version of the paper was presented at SoCG 2015
openalex publication_date 2014/12/02 · arxiv created 2016/04/15 · arxiv updated 2016/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed v>1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve FFv that develops when the fighter keeps building, at speed v, a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function (ew Z - s Z)-1, where w and s are real functions of v. For v>vc=2.6144 … all zeroes are complex conjugate pairs. If ϕ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs Θ( 1/ϕ) rounds before the fire is contained. As v decreases towards vc these two zeroes merge into a real one, so that argument ϕ goes to~0. Thus, curve FFv does not contain the fire if the fighter moves at speed v=vc. (That speed v>vc is sufficient for containing the fire has been proposed before by Bressan et al. [7], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that any curve that visits the four coordinate half-axes in cyclic order, and in inreasing distances from the origin, needs speed v>1.618…, the golden ratio, in order to contain the fire. Keywords: Motion Planning, Dynamic Environments, Spiralling strategies, Lower and upper bounds