Suppose you have to build a road to connect two cities on different sides of a lake. How would you plan the road to make it as short as possible?
To simplify the problem statement, a lake is sufficiently well modeled by a polygon, and the cities are just two points. The polygon does not have self-intersections and the endpoints are both outside the polygon. If you have Silverlight installed, you can use drag and drop on the points below to experiment:
A shortest path between two points is a segment that connects them. It’s clear that our route consists of segments (if a part of the path was a curve other than a segment, we could straighten it and get better results). Moreover, those segments (which are part of the route) have their endpoints either on polygon vertices or the start or end point. Again, if this were not the case, we would be able to make the path shorter by routing via the nearest polygon vertex.
Armed with this knowledge, let’s consider all possible segments that connect the start and end point and all polygon vertices that don’t intersect the polygon. Let’s then construct a graph out of these segments. Now we can use Dijkstra’s algorithm (or any other path finding algorithm such as A*) to find the shortest route in the graph between start and endpoints. Note how any shortest path algorithm can essentially boil down to a path finding in a graph, because a graph is a very good representation for a lot of situations.
From the implementation perspective, I used my dynamic geometry library and Silverlight to create a simple demo project that lets you drag the start and end points as well as polygon vertices. You can also drag the polygon and the plane itself. I also added rounded corners to the resulting path and made it avoid polygon vertices to make it look better.
Here is the source code for the sample. Here’s the main algorithm. It defines a data structure to describe a Graph that provides the ShortestPath method, which is the actual implementation of the Dijkstra’s algorithm. ConstructGraph takes care of adding all possible edges to the graph that do not intersect our polygon. SegmentIntersectsPolygon also determines what the name suggests.
I hope to post more about polygon routing in the future and do let me know if you have any questions.