There are a few cases that are easy to see. First off, if the man is either north ( M y ≥ L y ) or east ( M x ≥ L x ) then the man can escape being caught forever. But what if M x < L x and M y < L y ? It turns out that the man will eventually be caught. However, it is not easy to see exactly why. Here is a winning strategy for the tiger for the above case. First, observe that M lies below and to the left of L , then there exists a point C with the following properties: (1) L lies along the line segment CM , (2) a circle centered at C and passing through L intersects both the positive x - and positive y -axes (Fig. 98(b)). Crowd Motion: Today, we will discuss motion simulation involving a number of intelligent autonomous agents, as arises in crowds of pedestrians. Unlike flocking systems, in which it is assumed that the agents behave homogeneously, in a crowd it is assumed that each agent has its own agenda to pursue (see Fig. 99). For example, we might imagine a group of students walking through a crowded campus on their way to their next classes. Since such agents are acting within a social system, however, it is assumed that they will tend to behave in a manner that is consistent with social conventions. (I don’t want you to bump into me, and so I will act in a manner to avoid bumping into you as well.) Crowd simulation is actually a very broad area of study, ranging from work in game programming and computer graphics, artificial intelligence, social psychology, and urban architecture (e.g., planning evacuation routes). In order to operate in the context of a computer game, such a system needs to Crowds and Pursuit/Evasion 119 CMSC 425
(a) (b) (c) L M L M C M i - 1 L i - 1 to C M i L i Fig. 98: Lion and Man Problem Fig. 99: Crowd simulation. Crowds and Pursuit/Evasion 120 CMSC 425
be decentralized, where each member of the crowd determines its own action based on the perceived actions of other nearby agents. The problem with applying a simple boid-like flocking behavior is that, whereas flocking rules such as alignment naturally produce systems that avoid collisions between agents, the diverse agendas of agents in crowds naturally brings them directly into collisions with other agents (as in pedestrians traversing a crosswalk from both sides). In order to produce more realistic motion, the agents should anticipate where nearby agents are moving, and then plan their future motion accordingly. We discuss two models of crowd behavior, social-force dynamics and (reciprocal) velocity obstacles. Social-Force Dynamics: In our presentation of flocking behavior with Boids, we presented a model in which the motion of each artificial animal is determined by a collection of simple local forces based on the other agents in the system. In much the same manner that a physicist would simulate the motion of a collection of particles in computational fluid dynamics, we can simulate the fluid-like motion of animals in the flock. This can be applied to human crowd behavior as well. At an individual level, human behavior is quite chaotic, and it is not easy to predict future motion based on past motion.
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