Constrained Pursuit-Evasion Problems in the Plane

Abstract In pursuit-evasion problems, we are presented with one or more pursuers attempt-ing to capture one or more evaders. We consider pursuers and evaders limited bya maximum speed moving in the two-dimensional plane with obstacles. We theninvestigate two problems in this domain. In the first, where we are given the startingconfiguration of pursuers and evaders, we identify all possible paths by the evadersthat are not intercepted by pursuers, and the points reachable by the evaders be-fore the pursuers by following these paths. In the second problem, we consider apursuer forced to maintain visibility with an evader. We construct an example thatdemonstrates there exists, in addition to the two standard outcomes of the pursuercapturing the evader and the evader losing sight of the pursuer, a third tie outcome,where the pursuer never loses sight of the evader, but the evader can also avoid cap-ture indefinitely. We give the conditions under which each of these three outcomesoccur for our specific situation.iii

[1]  Sampath Kannan,et al.  Randomized pursuit-evasion with limited visibility , 2004, SODA '04.

[2]  Kokichi Sugihara,et al.  Crystal Voronoi Diagram and Its Applications to Collision-Free Paths , 2001, International Conference on Computational Science.

[3]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[4]  Joseph O'Rourke,et al.  Worst-case optimal algorithms for constructing visibility polygons with holes , 1986, SCG '86.

[5]  Kenneth L. Clarkson,et al.  Rectilinear shortest paths through polygonal obstacles in O(n(logn)2) time , 1987, SCG '87.

[6]  VoronoiDiagramsWaldir L. Roque,et al.  The Green Island Formation in Forest Fire Modeling with , 2007 .

[7]  Héctor H. González-Baños,et al.  Motion strategies for maintaining visibility of a moving target , 1997, Proceedings of International Conference on Robotics and Automation.

[8]  Tsai-Yen Li,et al.  Planning tracking motions for an intelligent virtual camera , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

[9]  Barry Schaudt,et al.  Multiplicatively weighted crystal growth Voronoi diagrams (extended abstract) , 1991, SCG '91.

[10]  Barry F. Schaudt.  Multiplicatively Weighted Crystal Growth Voronoi Diagrams (Thesis) , 1992 .

[11]  Stephen G. Kobourov,et al.  Optimal strategies to track and capture a predictable target , 2003, 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422).

[12]  Kyung-Yong Chwa,et al.  Visibility-Based Pursuit-Evasion in a Polygonal Region by a Searcher , 2001, ICALP.

[13]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[14]  Varol Akman,et al.  Voronoi diagrams with barriers and on polyhedra for minimal path planning , 1985, The Visual Computer.

[15]  Mark H. Overmars,et al.  Motion Planning for Camera Movements in Virtual Environments , 2003 .

[16]  Franz Aurenhammer,et al.  An optimal algorithm for constructing the weighted voronoi diagram in the plane , 1984, Pattern Recognit..

[17]  B. Schaudt.  Multiplicatively Weighted Crystal Growth Voronoi Diagrams , 1991 .

[18]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[19]  Takao Asano,et al.  Voronoi Diagram for Points in a Simple Polygon , 1987 .

[20]  Franz Aurenhammer,et al.  Quickest Paths, Straight Skeletons, and the City Voronoi Diagram , 2002, SCG '02.

[21]  Leonidas J. Guibas,et al.  Motion Planning with Visibility Constraints: Building Autonomous Observers , 1998 .