Rather than diving into the most general forms of combinatorial motion planning, it is helpful to first see several methods explained for a case that is easy to visualize. Several elegant, straightforward algorithms exist for the case in which and is polygonal. Most of these cannot be directly extended to higher dimensions; however, some of the general principles remain the same. Therefore, it is very instructive to see how combinatorial motion planning approaches work in two dimensions. There are also applications where these algorithms may directly apply. One example is planning for a small mobile robot that may be modeled as a point moving in a building that can be modeled with a 2D polygonal floor plan.
After covering representations in Section 6.2.1, Sections 6.2.2-6.2.4 present three different algorithms to solve the same problem. The one in Section 6.2.2 first performs cell decomposition on the way to building the roadmap, and the ones in Sections 6.2.3 and 6.2.4 directly produce a roadmap. The algorithm in Section 6.2.3 computes maximum clearance paths, and the one in Section 6.2.4 computes shortest paths (which consequently have no clearance).