For a point in an irregular polygon, what is the most efficient way to select the edge closest to the point?

Question

Given an irregular polygon and a point within that polygon, how do I determine which edge in the polygon is closest to the point?

Answer 1

The right answer depends on the bigger-picture structure of the problem: what happens when you take into account multiple queries? I assume that each query will deal with a different point. But what about the polygon? Do you expect to receive multiple queries for the same polygon? Or each time the polygon is different?

If each query is applied to a different, unpredictable polygon, then the only solution you have is essentially the total inspection of all polygon edges with point-to-segment distance testing for each one. It can be optimized in various [heuristic] ways (to discard the unnecessary tests early), but in the worst case there's no way around the full test.

However, if you expect some kind of predictability and stability on the polygon side of the problem (sufficiently many point-queries to the same polygon or to a fixed set of polygons), then the situation changes dramatically. The best approach in this case would be to pre-build the edge-based Voronoi diagram inside the polygon(s). Then you can solve the point-location problem (for which there are known efficient algorithms) in order to determine which Voronoi region the query point falls into. That will immediately tell you which edge is the closest.

The latter is incomparably more efficient when you need to handle many point-queries to the same polygon(s), but requires quite an effort to implement. So, it all depends on what kind of solution you need.

P.S. I see that you state in your question that your are going to run it for a large set of points for a single polygon. This immediately makes the Voronoi-diagram-based solution the way to go. Extra nuances of the algorithm might depend on whether that large set of points is entirely known in advance or arrives point-by-point in unpredictable way.