# VISIBILITY GRAPH

The algorithm presented by M. de Berg (ie Trapezoidal Map -> central nodes) is refered to as the "cell-decomposition" method of computing a collision-free path.

Another method, known as one-shot uses visibility graph to compute a path in the free space.

Please enable Java for an interactive construction (with Cinderella).

### Yellow path is a shorter path, not THE shortest path.

• Nodes are the vertices of the set S of obstacles.
• There is an edge between v and w if they are visible to each other. Note that this is not clear visibility, hence endpoints of the same obstacle edge can see each other.
• The shortest path of a visibility graph will include 's' and 't', thus 's' and 't' are included in the set S and thought of as vertices. The new set is then, S* = S + {s, t}
• The shortest path is then computed using Dijkstra's Algorithm.

Now that we have computed a road map and shortest path for a robot R, the next question are:
• What happens if R is not a point, but a convex polygon instead?
• Will our selected shortest path provide a collision-free path for R?
• How will R traverse the road map without colliding with any of the obstacles?

We now look at Minkowski Sums for the purpose of answering these questions. We will see that the goal is to "expand" the obstacles in P.

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