Distance between two sets

For the Boolean-valued function , there is no information about how far the robot is from hitting the obstacles. Such information is very important in planning algorithms. A distance function provides this information and is defined as , in which the real value in the range of indicates the distance in the world, , between the closest pair of points over all pairs from and . In general, for two closed, bounded subsets, and , of , the distance is defined as

 (5.28)

in which is the Euclidean norm. Clearly, if , then . The methods described in this section may be used to either compute distance or only determine whether . In the latter case, the computation is often much faster because less information is required.

Steven M LaValle 2012-04-20