On boundaries of highly visible spaces and applications

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The purpose of this paper is to investigate the properties of a certain class of highly visible spaces. For a given geometric space C containing obstacles specified by disjoint subsets of C , the free space F is defined to be the portion of C not occupied by these obstacles. The space is said to be highly visible if at each point in F a viewer can see at least an ε fraction of the entire F . This assumption has been used for robotic motion planning in the analysis of random sampling of points in the robot's configuration space, as well as the upper bound of the minimum number of guards needed for art gallery problems. However, there is no prior result on the implication of this assumption to the geometry of the space under study. For the two-dimensional case, with the additional assumptions that C is bounded within a rectangle of constant aspect ratio and that the volume ratio between F and C is a constant, we use the proof technique of “charging” each obstacle boundary segment by a certain portion of C to show that the total length of all obstacle boundaries in C is O ( n μ ( F ) / ε ) , if C contains polygonal obstacles with a total of n boundary edges; or O ( n μ ( F ) / ε ) , if C contains n convex obstacles that are piecewise smooth. In both cases, μ ( F ) is the volume of F . For the polygonal case, this bound is tight as we can construct a space whose boundary size is Θ ( n μ ( F ) / ε ) . These results can be partially extended to three dimensions. We show that these results can be applied to the analysis of certain probabilistic roadmap planners, as well as a variant of the art gallery problem. We also propose a number of conjectures on the properties of these highly visible spaces.