Research Article Identification of V-Formations and Circular and Doughnut Formations in a Set of Moving Entities with Outliers Francisco Javier Moreno Arboleda, Jaime Alberto Guzmán Luna, and Sebastian Alonso Gomez Arias Universidad Nacional de Colombia, Sede Medell´ ın, Bloque M8A, Medell´ ın, Colombia Correspondence should be addressed to Jaime Alberto Guzm´ an Luna; jaguzman@unal.edu.co Received 1 November 2013; Revised 8 February 2014; Accepted 24 February 2014; Published 10 April 2014 Academic Editor: J.-C. Cort´ es Copyright © 2014 Francisco Javier Moreno Arboleda et al. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Diverse movement patterns may be identiied when we study a set of moving entities. One of these patterns is known as a V- formation for it is shaped like the letter V. Informally, a set of entities shows a V-formation if the entities are located on one of their two characteristic lines. hese lines meet in a position where there is just one entity considered the leader of the formation. Another movement pattern is known as a circular formation for it is shaped like a circle. Informally, circular formations present a set of entities grouped around a center in which the distance from these entities to the center is less than a given threshold. In this paper we present a model to identify V-formations and circular formations with outliers. An outlier is an entity which is part of a formation but is away from it. We also present a model to identify doughnut formations, which are an extension of circular formations. We present formal rules for our models and an algorithm for detecting outliers. he model was validated with NetLogo, a programming and modeling environment for the simulation of natural and social phenomena. 1. Introduction Diverse movement patterns may be identiied when we study a set of moving entities, for example, a lock of birds [1] and a school of ish [2]. One of these patterns is known as a V-formation for it is shaped like the letter V; see Figure 1. Another movement pattern is known as a circular formation for it is shaped like a circle; see Figure 2. Informally, a set of entities shows a V-formation if the entities are located on one of their two characteristic lines. he lines meet in a position where there is just one entity considered the leader of the formation [3]. Several authors have analyzed V-formations. In [4, 5], there is an attempt to explain from a physical point of view the reasons why certain species of birds, such as Canadian geese (Branta canadensis), red knots (Calidris canutus), and plovers (Calidris alpina), tend to ly this way. Other authors try to simulate V-formations at a computa- tional level. For instance, Nathan and Barbosa [6] propose a model based on rules that allows us to generate V-formations depending on speciic parameters. he authors validated their model using NetLogo [7], a programming and modeling environment to simulate natural and social phenomena. On the other hand, a circular formation is a set of entities grouped around a common center and where the entities’ distance to the center is less than a given threshold. Regarding related works with circular formations, we identiied the following. In [8] the authors experimented with a set of data referring to the movement of diferent animal species. It was found that despite being in diferent ecosystems, species follow similar behavioral patterns. he authors also tried to model general grouping behaviors of ish, birds, insects, and even people. One of these behav- iors is circular formation in which they identiied phys- ical forces: attraction, repulsion, alignment, and frontal interaction. On the other hand, researchers in the ield of robotics and in control theory, inspired by social grouping phenomena and by the patterns of birds and ish, have developed applications to coordinate the movement of multivehicle systems. Among these patterns are circular [9] and V-formations; see Figure 3. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 241684, 11 pages http://dx.doi.org/10.1155/2014/241684