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

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