Visibility-Based Pursuit-Evasion in Three-Dimensional Environments

The problem of visibility-based pursuit-evasion was first introduced in 1992 by Suzuki and Yamashita [19]. Since then, it has attracted considerable attention in the communities of robot motion planning and computational geometry. Researchers have considered many variations of the problem, including curved environments[11]; pursuers equipped with one or two “flashlights”, or rays of visibility[12, 15, 16]; and pursuers guarding “rooms”[13] and “corridors”[2]. The complexity of pursuit-evasion with omnidirectional visibility has remained an unsolved problem for almost a decade, and was settled only recently[16]. With all this effort focused on the twodimensional case, the three-dimensional case has remained completely unexplored. However, many of the tools to reason about 3D visibility already exist in literature from computer vision and graphics. Data structures for maintaining global visibility information have been studied for several decades within the framework of aspect graphs[9, 17, 18], and, more recently, visibility complexes[4]. The present paper leverages this body of work to reason about visibility issues for the pursuit-evasion problem in 3D. We develop a visibility framework similar to the conservative regions decomposition of Guibas et. al.[7] for both polyhedral and curved environments. Part of the reason for the apparent previous “neglect” of pursuit-evasion in 3D is that visibility issues are perceived to be a lot more complex than in 2D. As we will show in our paper, this impression is not entirely unjustified. In 2D, there are only two kinds of critical events that change the state of visibility-based information about the environment, and the changes they