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Stable polygons of cyclic pursuit

Stable polygons of cyclic pursuit In a companion paper [5] we resolved the question of whether cyclic pursuits can exhibit ‘non-mutual’ captures. Although, as we showed, non-mutual captures can occur, the set of initial conditions which lead to them has Lebesgue measure zero. Thus, generically, cyclic pursuits collapse into a mutual capture. In this paper we consider whether the pursuit configuration can asymptotically approach a regular one for a non-trivial set of initial conditions. More precisely, we study the stability of regular geometries of cyclic pursuit. We show that in all dimensions the only stable regular n-bug shapes are the regular two dimensional n-gons, n≥7, in which each vertex chases its neighboring vertex in some fixed orientation. We also analize the three bug cyclic pursuit in detail, proving that, except for the equilateral initial position, the triangle formed is asymptotically degenerate with the minimum interior angle tending to zero while the vertex at which the minimum is located rotates among the vertices infinitely often. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Mathematics and Artificial Intelligence Springer Journals

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References (4)

Publisher
Springer Journals
Copyright
Copyright © 2001 by Kluwer Academic Publishers
Subject
Computer Science; Computer Science, general; Artificial Intelligence (incl. Robotics); Mathematics, general; Complexity
ISSN
1012-2443
eISSN
1573-7470
DOI
10.1023/A:1016678406688
Publisher site
See Article on Publisher Site

Abstract

In a companion paper [5] we resolved the question of whether cyclic pursuits can exhibit ‘non-mutual’ captures. Although, as we showed, non-mutual captures can occur, the set of initial conditions which lead to them has Lebesgue measure zero. Thus, generically, cyclic pursuits collapse into a mutual capture. In this paper we consider whether the pursuit configuration can asymptotically approach a regular one for a non-trivial set of initial conditions. More precisely, we study the stability of regular geometries of cyclic pursuit. We show that in all dimensions the only stable regular n-bug shapes are the regular two dimensional n-gons, n≥7, in which each vertex chases its neighboring vertex in some fixed orientation. We also analize the three bug cyclic pursuit in detail, proving that, except for the equilateral initial position, the triangle formed is asymptotically degenerate with the minimum interior angle tending to zero while the vertex at which the minimum is located rotates among the vertices infinitely often.

Journal

Annals of Mathematics and Artificial IntelligenceSpringer Journals

Published: Oct 19, 2004

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