Yesterday I was confronted with a seemingly simple problem: how can one find out if the rectangles or, more general, polygons on a surface are overlapping or not? Surprisingly, because they have an amazing collection of tools, the matplotlib library used in this context to actually draw the rectangles, doesn’t seem to have this kind of functionality. Of course there are loads of examples on the web for 2D collision detection, but I couldn’t find one written in Python. But nevertheless, I found lecture notes on Robert Pless’s website, which taught me the quadrant method to check if one point lies inside a polygon. Then it is only a small step to find if polygons are intersecting.
After writing a Python module providing the functionality, it naturally turned out to be rather slow on big numbers of polygons to be checked. So I continued to write a function which checks which polygons in a given set are overlapping or touching. To use the resources on my machine efficiently, I made two versions of this later function: one for conservative, serial processing and a second which uses Parallel Python to distribute the workload among the CPUs found in the system.
If someone else needs this kind of functionaltiy as well or simply is interested in how I did it, here is the interesting part. The whole file with unit tests and documentation can be downloaded as well: polygons_overlapping.py
import pylab class PolygonsTouching( Exception ): """ This exception is triggered when two polygons touch at one point. This is for internal use only and will be caught before returning. """ def __init__( self, x=0, y=0 ): self.x, self.y = x, y def __str__( self ): return 'The tested polygons at least touch each other at (%f,%f)'\ % ( self.x, self.y ) def shift( self, dx, dy ): self.x += dx self.y += dy def pair_overlapping( polygon1, polygon2, digits = None ): """ Find out if polygons are overlapping or touching. The function makes use of the quadrant method to find out if a point is inside a given polygon. polygon1, polygon2 -- Two arrays of [x,y] pairs where the last and the first pair is the same, because the polygon has to be closed. digits -- The number of digits relevant for the decision between separate and touching or touching and overlapping Returns 0 if the given polygons are neither overlapping nor touching, returns 1 if they are not overlapping, but touching and returns 2 if they are overlapping """ def calc_walk_summand( r1, r2, digits = None ): """ Calculates the summand along one edge depending on axis crossings. Follows the edge between two points and checks if one or both axes are being crossed. If They are crossed in clockwise sense, it returns +1 otherwise -1. Going through the origin raises the PolygonsTouching exception. Returns one of -2, -1, 0, +1, +2 or raises PolygonsTouching """ x, y = 0, 1 # indices for better readability summand = 0 # the return value tx, ty = None, None # on division by zero, set parameters to None if r1[x] != r2[x]: ty = r1[x] / ( r1[x] - r2[x] ) # where it crosses the y axis if r1[y] != r2[y]: tx = r1[y] / ( r1[y] - r2[y] ) # where it crosses the x axis if tx == None: tx = ty if ty == None: ty = tx rsign = pylab.sign if digits != None: rsign = lambda x: pylab.sign( round( x, digits ) ) sign_x = rsign( r1[x] + tx * ( r2[x] - r1[x] ) ) sign_y = rsign( r1[y] + ty * ( r2[y] - r1[y] ) ) if ( tx >= 0 ) and ( tx < 1 ): if ( sign_x == 0 ) and ( sign_y == 0 ): raise PolygonsTouching() summand += sign_x * pylab.sign( r2[y] - r1[y] ) if ( ty >= 0 ) and ( ty < 1 ): if ( sign_x == 0 ) and ( sign_y == 0 ): raise PolygonsTouching() summand += sign_y * pylab.sign( r1[x] - r2[x] ) return summand def current_and_next( iterable ): """ Returns an iterator for each element and its following element. """ iterator = iter( iterable ) item = iterator.next() for next in iterator: yield ( item, next ) item = next def point_in_polygon( xy, xyarray, digits = None ): """ Checks if a point lies inside a polygon using the quadrant method. This moves the given point to the origin and shifts the polygon accordingly. Then for each edge of the polygon, calc_walk_summand is called. If the sum of all returned values from these calls is +4 or -4, the point lies indeed inside the polygon. Otherwise, if a PolygonsTouching exception has been caught, the point lies on ond of the edges of the polygon. Returns the number of nodes of the polygon, if the point lies inside, otherwise 1 if the point lies on the polygon and if not, 0. """ moved = xyarray - xy # move currently checked point to the origin (0,0) touching = False # this is used only if no overlap is found walk_sum = 0 for cnxy in current_and_next( moved ): try: walk_sum += calc_walk_summand( cnxy, cnxy, digits ) except PolygonsTouching, (e): e.shift( *xy ) touching = True if ( abs( walk_sum ) == 4 ): return len( xyarray ) elif touching: return 1 else: return 0 def polygons_overlapping( p1, p2, digits = None ): """ Checks if one of the nodes of p1 lies inside p2. This repeatedly calls point_in_polygon for each point of polygon p1 and immediately returns if it is the case, because then the polygons are obviously overlapping. Returns 2 for overlapping polygons, 1 for touching polygons and 0 otherwise. """ degree_of_contact = 0 xyarrays = [ p1, p2 ] for xy in xyarrays: degree_of_contact += point_in_polygon( xy, xyarrays, digits ) if degree_of_contact >= len( xyarrays ): return 2 if degree_of_contact > 0: return 1 else: return 0 way1 = polygons_overlapping( polygon1, polygon2, digits ) way2 = 0 if way1 < 2: # Only if the polygons are not already found to be overlapping way2 = polygons_overlapping( polygon2, polygon1, digits ) return max( way1, way2 ) def collection_overlapping_serial( polygons, digits = None ): """ Similar to the collection_overlapping function, but forces serial processing. """ result =  pickle_polygons = [p.get_xy() for p in polygons] for i in xrange( len( polygons ) ): for j in xrange( i+1, len( polygons ) ): result.append( ( i, j, \ pair_overlapping( pickle_polygons[i], pickle_polygons[j], \ digits ) ) ) return result def __cop_bigger_job( polygons, index, digits = None ): """ This is a helper to efficiently distribute workload among processors. """ result =  for j in xrange( index + 1, len( polygons ) ): result.append( ( index, j, \ pair_overlapping( polygons[index], polygons[j], digits ) ) ) return result def collection_overlapping_parallel( polygons, digits = None, \ ncpus = 'autodetect' ): """ Like collection_overlapping, but forces parallel processing. This function crashes if Parallel Python is not found on the system. """ import pp ppservers = () job_server = pp.Server( ncpus, ppservers=ppservers ) pickle_polygons = [p.get_xy() for p in polygons] jobs =  for i in xrange( len( polygons ) ): job = job_server.submit( __cop_bigger_job, \ ( pickle_polygons, i, digits, ), \ ( pair_overlapping, PolygonsTouching, ), \ ( "pylab", ) ) jobs.append( job ) result =  for job in jobs: result += job() #job_server.print_stats() return result def collection_overlapping( polygons, digits = None ): """ Look for pair-wise overlaps in a given list of polygons. The function makes use of the quadrant method to find out if a point is inside a given polygon. It invokes the pair_overlapping function for each combination and produces and array of index pairs of these combinations together with the overlap number of that pair. The overlap number is 0 for no overlap, 1 for touching and 2 for overlapping polygons. This function automatically selects between a serial and a parallel implementation of the search depending on whether Parallel Python is installed and can be imported or not. polygons -- A list of arrays of [x,y] pairs where the last and the first pair of each array in the list is the same, because the polygons have to be closed. digits -- The number of digits relevant for the decision between separate and touching or touching and overlapping polygons. Returns a list of 3-tuples """ try: import pp # try if parallel python is installed except ImportError: return collection_overlapping_serial( polygons, digits ) else: return collection_overlapping_parallel( polygons, digits )