+# Copyright (c) 2009 W. Trevor King and the authors listed at the
+# following URL, and/or the authors of referenced articles or
+# incorporated external code:
+# http://en.literateprograms.org/Quickhull_(Python,_arrays)?action=history&offset=20090125100840
+#
+# Permission is hereby granted, free of charge, to any person obtaining
+# a copy of this software and associated documentation files (the
+# "Software"), to deal in the Software without restriction, including
+# without limitation the rights to use, copy, modify, merge, publish,
+# distribute, sublicense, and/or sell copies of the Software, and to
+# permit persons to whom the Software is furnished to do so, subject to
+# the following conditions:
+#
+# The above copyright notice and this permission notice shall be
+# included in all copies or substantial portions of the Software.
+#
+# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+# EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
+# MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
+# IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
+# CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
+# TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
+# SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+#
+# Retrieved from: http://en.literateprograms.org/Quickhull_(Python,_arrays)?oldid=16036
+# Modified by W. Trevor King to add Simplex and point_inside_hull
+
+from numpy import *
+
+link = lambda a,b: concatenate((a,b[1:]))
+edge = lambda a,b: concatenate(([a],[b]))
+
+def qhull(sample):
+ def dome(sample,base):
+ h, t = base
+ dists = dot(sample-h, dot(((0,-1),(1,0)),(t-h)))
+ outer = repeat(sample, dists>0, 0)
+
+ if len(outer):
+ pivot = sample[argmax(dists)]
+ return link(dome(outer, edge(h, pivot)),
+ dome(outer, edge(pivot, t)))
+ else:
+ return base
+
+ if len(sample) > 2:
+ axis = sample[:,0]
+ base = take(sample, [argmin(axis), argmax(axis)], 0)
+ return link(dome(sample, base),
+ dome(sample, base[::-1]))
+ else:
+ return sample
+
+class Simplex(object): # N+1 points in N dimensions
+ def __init__(self, vertices):
+ self.vertices = vertices
+ self.origin = self.vertices[0]
+ self.basis = array([v-self.origin for v in self.vertices[1:]],
+ dtype=double).transpose()
+ self.inverse_basis = linalg.inv(self.basis)
+ def simplex_coordinates(self, points):
+ rel_coords = (points-self.origin).transpose()
+ return dot(self.inverse_basis, rel_coords).transpose()
+ def inside_single_point_simplex_coords(self, point, tol=1e-15):
+ # check that we're in the positive coordinate
+ for coordinate in point:
+ if coordinate < -tol:
+ return False
+ # check that we're under the (1,1,1,...) plane
+ if point.sum() > 1+tol:
+ return False
+ return True
+ def inside(self, points):
+ ps = self.simplex_coordinates(points)
+ if not hasattr(ps[0], "__len__"): # single point
+ return self.inside_single_point_simplex_coords(ps)
+ else:
+ insides = []
+ for p in ps:
+ insides.append(self.inside_single_point_simplex_coords(p))
+ return insides
+
+def points_inside_hull(hull, points):
+ if not hasattr(points[0], "__len__"): # single point
+ points = numpy.array([points])
+ inside = [False]*len(points)
+ pivot = hull[0]
+ for a,b in zip(hull[1:], hull[2:]):
+ rb = b-pivot
+ if vdot(rb, rb) == 0: # hull is a closed loop
+ continue
+ simplex = Simplex([pivot,a,b])
+ simplex_inside = simplex.inside(points)
+ if any(simplex_inside):
+ for i,value in enumerate(simplex_inside):
+ if value == True:
+ inside[i] = True
+ return inside
+
+def print_postscript(sample, hull):
+ print "%!"
+ print "100 500 translate 2 2 scale 0 0 moveto"
+ print "/tick {moveto 0 2 rlineto 0 -4 rlineto 0 2 rlineto"
+ print " 2 0 rlineto -4 0 rlineto 2 0 rlineto} def"
+ for (x,y) in sample:
+ print x, y, "tick"
+ print "stroke"
+ print hull[0,0], hull[0,1], "moveto"
+ for (x,y) in hull[1:]:
+ print x, y, "lineto"
+ print "closepath stroke showpage"
+
+if __name__ == "__main__":
+ #sample = 10*array([(x,y) for x in arange(10) for y in arange(10)])
+ sample = 100*random.random((32,2))
+ hull = qhull(sample)
+ if all(point_inside_hull(hull, sample)) != True:
+ for point in sample:
+ assert point_inside_hull(hull, point) == True, \
+ "point %s should be in hull\n%s" % (sample, hull)
+ apoints = hull[0:-1]
+ bpoints = hull[1:]
+ if any(point_inside_hull(hull, apoints - 0.1*(bpoints-apoints))) == True:
+ for a,b in zip(apoints, bpoints):
+ assert point_inside_hull(hull, a - 0.1*(b-a)) == False, \
+ "point %s (a=%s, b=%s) should be outside hull\n%s" \
+ % (point, a, b, hull)
+ print_postscript(sample, hull)
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