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This is a question similar to the one here, but I figure that it would be helpful if I can recast it in a more general terms.
I have a set of polygons, these polygons can touch one another, overlap and can take on any shape. My question is, given a list of points, how to devise an efficient algorithm that find which polygons are the points located?
One of the interesting restriction of the location of the points is that, all the points are located at the edges of the polygons, if this helps.
I understand that r-trees can help, but given that I am doing a series of points, is there a more efficient algorithm instead of computing for each point one by one?
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Draw a horizontal line to the right of each point and extend it to infinity. Count the number of times the line intersects with polygon edges. A point is inside the polygon if either count of intersections is odd or point lies on an edge of polygon. If none of the conditions is true, then point lies outside.
One of the simplest and most efficient ways of getting a list of points inside a polygon with Python is to rely on the Geopandas library and perform a spatial join. To do this, we simply create one geodataframe with points and another one with polygons, and join them with 'sjoin'.
One simple way of finding whether the point is inside or outside a simple polygon is to test how many times a ray, starting from the point and going in any fixed direction, intersects the edges of the polygon. If the point is on the outside of the polygon the ray will intersect its edge an even number of times.
The Point in Polygon (PIP) problem is the problem of determining whether a point is any arbitrary polygon. This might sound trivial for a simple polygon like a square or a triangle, but gets more complex with more complex polygons like the one in the example below.
The key search term here is point location. Under that name, there are many algorithms in the computational geometry literature for various cases, from special to general. For example, this link lists various software packages, including my own. (A somewhat out-of-date list now.)
There is a significant tradeoff between speed and program complexity (and therefore implementation effort). The easiest-to-program method is to check each point against each polygon, using standard point-in-polygon code. But this could be slow depending on how many polygons you have. More difficult is to build a point-location data structure by sweeping the plane and finding all the edge-edge intersection points. See the this Wikipedia article to see some of your options.
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I think you are bumping up against intuition about the problem (which is a quasi-analog perception) versus a computational approach which is of necessity O(n).
Given a plane, a degenerate polygon (a line), and an arbitrary set of points on the plane, do the points intersect the line or fall "above" or "below" it? I cannot think of an approach that is smaller than O(n) even for this degenerate case.
Either, each point would have to be checked for its relation to the line, or you'd have to partition the points into some tree-like structure which would require at least O(n) operations but very likely more.
If I were better at computational geometry, I might be able to say with authority that you've just restated Klee's measure problem but as it is I just have to suggest it.
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