如何在矩形的并集中找到空洞？

Question

我有一些随机的矩形（黑色），它们在一个单位正方形（红色）内外，我需要找出所有在单位正方形内但没有被任何矩形覆盖的多边形区域。

看起来可以用Shapely这个库来实现这个功能。我已经得到了这些矩形的合并区域（绿色），但我不太确定怎么从单位正方形中减去这些区域，并获取一个多边形的列表。

这是我用来生成测试数据的代码：

import pylab
import random
from matplotlib import pyplot
from shapely.geometry import Point, Polygon
from shapely.ops import cascaded_union
from descartes import PolygonPatch

def make_square(x, y, size1, size2):
    dx = [size1, -size1, -size1, size1, size1]
    dy = [size2, size2, -size2, -size2, size2]
    return [(x+sx, y+sy) for sx, sy in zip(dx, dy)]


pylab.figure()

square = make_square(0.5, 0.5, 1.0, 1.0)
a, b = zip(*square)
pylab.plot(a, b, 'r-')
polygons = []

for i in xrange(10):
    x = random.random()
    y = random.random()
    s1 = random.random()
    s2 = random.random()

    square = make_square(x, y, s1, s2)
    polygons.append(Polygon(square))
    a, b = zip(*square)
    pylab.plot(a, b, 'k-')

u = cascaded_union(polygons)
patch2b = PolygonPatch(u, fc='#00ff00', ec='#00ff00', alpha=0.5, zorder=2)
pylab.gca().add_patch(patch2b)


pylab.show()

Answer 1

基本上，你需要把你合并后的多边形和那个大正方形进行差集运算，然后把结果变成单独的多边形。举个例子：

#-- Get the region not covered by individual squares.
uncovered_region = Polygon(bigsquare).difference(union)

# In some cases, the result will be a single polygon...
if not isinstance(uncovered_region, MultiPolygon):
    uncovered_region = [uncovered_region]

for poly in polygonize(uncovered_region):
    patch = PolygonPatch(poly, fc='purple', alpha=0.5, zorder=2)
    ax.add_patch(patch)

这是一个完整的例子，基于你的情况：

import matplotlib.pyplot as plt
import random
from shapely.geometry import Polygon, MultiPolygon
from shapely.ops import cascaded_union, polygonize
from descartes import PolygonPatch

def make_square(x, y, size1, size2):
    dx = [size1, -size1, -size1, size1, size1]
    dy = [size2, size2, -size2, -size2, size2]
    return [(x+sx, y+sy) for sx, sy in zip(dx, dy)]


fig, ax = plt.subplots()

bigsquare = make_square(0.5, 0.5, 1.0, 1.0)
a, b = zip(*bigsquare)
ax.plot(a, b, 'r-')
polygons = []

for i in xrange(10):
    x = random.random()
    y = random.random()
    s1 = random.random()
    s2 = random.random()

    square = make_square(x, y, s1, s2)
    polygons.append(Polygon(square))
    a, b = zip(*square)
    ax.plot(a, b, 'k-')

union = cascaded_union(polygons)
patch2b = PolygonPatch(union, fc='#00ff00', ec='#00ff00', alpha=0.5, zorder=2)
ax.add_patch(patch2b)

#-- Get the region not covered by individual squares.
uncovered_region = Polygon(bigsquare).difference(union)

# In some cases, the result will be a single polygon...
if not isinstance(uncovered_region, MultiPolygon):
    uncovered_region = [uncovered_region]

for poly in polygonize(uncovered_region):
    print poly
    patch = PolygonPatch(poly, fc='purple', alpha=0.5, zorder=2)
    ax.add_patch(patch)

plt.margins(0.05)
plt.show()