python如何计算n个圆面积的并集？

Question

1.有n个圆相互重叠

2.圆心已知（坐标集），半径相同

3.如何用python求得重叠之后的圆面积的并集

4.多圆重叠如图所示（蓝色区域）

Answer 1

自问自答，interesting

from random import uniform
from math import sqrt
from matplotlib import pyplot as plt

# 投点总数
total = 100000

# 统计点数初始值
inn = out = 0.0

# 三个圆心坐标
location = [[123.75, 31.6], [123.71, 32.3], [124.09, 31.66]]
num_ship = len(location)

# 矩形区域坐标
rect = {'left_bottom': [123.5, 31.5], 'right_up': [125, 32.5],
        'line_lon': [123.5, 125], 'line_lat': [31.5, 32.5],
        'line_lat_up': [32.5, 32.5], 'line_lat_down': [31.5, 31.5],
        'line_lon_left': [123.5, 123.5], 'line_lon_right': [125, 125]}

# 船舶节点坐标，圆心坐标
center_x = [i[0] for i in location]
center_y = [i[1] for i in location]

# 圆外坐标列表
x_out_list = []
y_out_list = []

for j in range(total):  # 先取随机点坐标
    x = uniform(rect['left_bottom'][0], rect['right_up'][0])
    y = uniform(rect['left_bottom'][1], rect['right_up'][1])

    for i in range(len(center_x)):  # 再取圆心坐标
        ox = center_x[i]
        oy = center_y[i]
        judge_range = sqrt((x - ox) * (x - ox) + (y - oy) * (y - oy))  # 计算圆半径公式
        if judge_range < 0.5:  # 判断随机点是否在圆范围内，圆半径为0.5
            break  # 如果在，退出循环，继续与下一个圆心对比
        else:  # 如果不在，添加到圆外列表，并计数
            if i == len(center_x) - 1:
                x_out_list.append(x)
                y_out_list.append(y)
                out += 1.0  # 圆外节点计数
            continue
inn = total - out
print(total, inn, out)
print(inn / total)  # 重叠区域投点比例

fig = plt.figure(dpi=200)
ax = fig.add_subplot(111)
plt.xlim(123.5, 125)
plt.ylim(31.5, 32.5)
# plt.scatter(x_in_list, y_in_list, color="r")
# 绘图，蓝色区域为圆外面积
# todo 反向填充颜色，优化绘图
plt.scatter(x_out_list, y_out_list, color="blue")
# plt.savefig("./img/area_proportion")
plt.show()

效果图，白色为多圆重叠区域，蓝色为圆外区域，可用矩形面积*比例得到重叠面积

为对比效果，上其他圆心图形