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  • $\begingroup$ The speed will depend linearly on the number of vertices. That is often fine but could be a problem if there are both many vertices and many query points. $\endgroup$ Commented Jan 20, 2014 at 22:53
  • $\begingroup$ @DanielLichtblau: Yes, you are right of course that for a large polygon you want to do something hierarchical along the lines of your answer to get decent scaling. One reason I keep coming back to this implementation is the partitioning guarantee which is critical in much of what I do. $\endgroup$ Commented Jan 21, 2014 at 8:41
  • $\begingroup$ I had a look at "Insignificance Galore" where it mentions the partitioning guarantee. But I still do not understand what it means. Is it for the case of multiple disconnected polygons? Self-intersecting? Or does it also have meaning in the case of one non-self-intersecting polygon. $\endgroup$ Commented Jan 21, 2014 at 15:29
  • $\begingroup$ A partitioning of a set S is a collection of disjoint subsets of S whose union is S mathworld.wolfram.com/SetPartition.html. The practical problem with partitioning (part of) the plane into polygons is to specify what happens to points on the edges and vertices: it's a lot of tedious details which are usually unimportant from a mathematical point of view (since the combined edges have 0 area), but still needs to be done right for some numerical algorithms to work. $\endgroup$ Commented Jan 22, 2014 at 8:45
  • $\begingroup$ Okay, thanks for the explanation. I will add that it is also critical, in polynomial irreducibility testing, to know if an exponent vector is or is not a vertex in the convex hull corresponding to a certain a Newton polytope. I can say that numerical convex hull methods have made such determination much more difficult than I would like. So there is at least one math algorithm where this does matter. $\endgroup$ Commented Jan 22, 2014 at 15:51