#### Interpreting the C-space

It is important to consider the topological implications of . Since is multiply connected, and are multiply connected. It is difficult to visualize because it is a 3D manifold; however, there is a nice interpretation using identification. Start with the open unit cube, . Include the boundary points of the form and , and make the identification for all . This means that when traveling in the and directions, there is a frontier'' to the C-space; however, traveling in the direction causes a wraparound.

It is very important for a motion planning algorithm to understand that this wraparound exists. For example, consider because it is easier to visualize. Imagine a path planning problem for which , as depicted in Figure 4.8. Suppose the top and bottom are identified to make a cylinder, and there is an obstacle across the middle. Suppose the task is to find a path from to . If the top and bottom were not identified, then it would not be possible to connect to ; however, if the algorithm realizes it was given a cylinder, the task is straightforward. In general, it is very important to understand the topology of ; otherwise, potential solutions will be lost.

The next section addresses for . The main difficulty is determining the topology of . At least we do not have to consider in this book.

Steven M LaValle 2012-04-20