已知多边形各点平面坐标，Python实现求面积

版权声明：本文为博主原创文章，未经博主允许不得转载。 https://blog.csdn.net/xiaoyw/article/details/79952520

　　在物联网环境中，或者工业自动化生产过程中，把获取到数据结果放置到坐标系中，按规则生成图表，为技术人员、管理人员提供可供分析的图表，例如下面平面坐标（x，y）：

x = [0.00,0.00,0.01,0.01,0.02,0.04,0.05,0.07,0.10,0.12,0.15,0.18,0.21,0.24,0.28,0.32,0.37,0.42,0.46,0.52,0.57,0.62,0.68,0.74,0.80,0.86,0.92,0.99,1.06,1.12,1.19,1.26,1.33,1.40,1.48,1.68,1.75,1.82,1.88,1.95,2.01,2.08,2.15,2.21,2.28,2.35,2.41,2.48,2.55,2.61,2.68,2.75,2.81,2.88,2.95,3.01,3.08,3.15,3.21,3.27,3.34,3.39,3.46,3.51,3.58,3.64,3.69,3.75,3.81,3.86,3.92,3.97,4.02,4.08,4.13,4.17,4.22,4.27,4.31,4.36,4.41,4.44,4.49,4.52,4.56,4.60,4.64,4.67,4.71,4.74,4.77,4.80,4.82,4.85,4.87,4.89,4.91,4.93,4.94,4.96,4.97,4.98,4.99,4.99,4.99,4.99,4.99,4.99,4.98,4.97,4.96,4.94,4.93,4.91,4.88,4.86,4.83,4.80,4.77,4.73,4.70,4.66,4.62,4.57,4.52,4.46,4.42,4.36,4.29,4.24,4.18,4.11,4.06,3.99,3.92,3.85,3.78,3.70,3.63,3.55,3.48,3.41,3.33,3.26,3.18,3.09,3.02,2.94,2.85,2.78,2.69,2.61,2.54,2.45,2.37,2.30,2.21,2.13,2.06,1.98,1.89,1.82,1.74,1.67,1.59,1.52,1.45,1.37,1.30,1.23,1.16,1.09,1.03,0.96,0.90,0.84,0.78,0.72,0.67,0.61,0.55,0.51,0.45,0.41,0.36,0.32,0.28,0.24,0.21,0.18,0.14,0.12,0.09,0.07,0.05,0.04,0.02,0.01,0.01,0.00]
y = [35.01,35.30,35.32,35.22,37.23,38.91,40.61,41.66,43.01,45.78,49.20,51.85,53.81,56.15,58.65,57.61,55.97,54.22,52.13,50.91,51.01,51.65,52.28,53.65,54.56,54.53,54.43,53.75,52.45,51.85,51.76,51.75,51.80,52.42,52.42,52.47,52.60,52.75,52.83,52.55,52.35,52.25,52.01,51.82,51.82,51.81,51.85,51.88,51.88,51.81,51.80,51.75,51.53,51.49,51.54,51.51,51.51,51.52,51.51,51.48,51.52,51.26,51.09,51.05,50.92,50.93,50.97,50.97,50.95,51.02,50.99,51.04,51.04,50.92,50.65,50.64,50.61,50.61,50.66,50.67,50.64,50.67,50.58,50.47,50.45,50.24,50.07,50.10,50.07,50.05,50.11,50.10,50.07,49.97,49.70,49.67,49.68,49.50,49.50,49.49,49.47,49.50,49.46,49.48,49.21,48.11,47.81,47.37,47.32,46.85,45.77,44.54,43.09,41.66,40.29,38.49,36.54,33.99,31.23,28.23,25.26,23.25,24.20,26.10,29.01,31.74,33.24,33.20,32.61,30.41,27.65,26.16,25.95,25.98,27.61,29.39,31.12,31.89,31.97,30.75,29.65,28.33,27.31,27.00,27.47,28.33,29.30,30.26,30.96,30.99,30.31,29.17,28.83,28.18,28.16,28.18,28.94,29.49,30.08,30.34,30.43,30.24,29.58,29.15,29.08,29.08,29.41,29.76,30.36,30.48,30.55,30.48,30.47,30.14,29.80,29.80,30.17,30.39,30.85,31.42,31.55,31.53,31.54,31.48,31.43,31.40,31.41,31.57,32.01,32.66,33.24,33.25,33.24,33.24,32.80,32.25,32.25,32.40,32.61,33.04]

　　对应的平面图像如下图所示：

　　实际工作中的分析图像如下图所示：

　　平面坐标合围的区域，可以看作为多边形，如何求这样多边形的面积呢？

　　多边形面积求解的方法很多，其中比较多见的就是增加一个点P，然后分别连接多边形各个定点与P点，然后计算每个三角形的符号面积（面积有正负之分），求和就可以计算出面积。

　　可以简便些，将P点直接放在A点，分别累加△ABC、△ACD、△ADE……，此处注意按向量计算，判断顺时针还是逆时针，如果我们假设逆时针的三角形（如上图ABC）为正，那么顺时针的面积（如上图AGF）就为负。

　　其中，三角形面积使用海伦公式，海伦公式又译作希伦公式。它是利用三角形的三条边的边长直接求三角形面积的公式。参考代码如下（注：网上参考代码有误，本文已经更正，请读者帮验证）：

'''
Created on 2018年4月15日

@author: XiaoYW
'''
import math

class Point():
    def __init__(self,x,y):
        self.x = x
        self.y = y

def GetAreaOfPolyGon(points):
    # 计算多边形面积
    area = 0
    if(len(points)<3):        
         raise Exception("error")

    p1 = points[0]
    for i in range(1,len(points)-1):
        p2 = points[i]
        p3 = points[i + 1]

        #计算向量
        vecp1p2 = Point(p2.x-p1.x,p2.y-p1.y)
        vecp2p3 = Point(p3.x-p2.x,p3.y-p2.y)

        #判断顺时针还是逆时针，顺时针面积为正，逆时针面积为负
        vecMult = vecp1p2.x*vecp2p3.y - vecp1p2.y*vecp2p3.x   #判断正负方向比较有意思
        sign = 0
        if(vecMult>0):
            sign = 1
        elif(vecMult<0):
            sign = -1

        triArea = GetAreaOfTriangle(p1,p2,p3)*sign
        area += triArea
    return abs(area)

def GetAreaOfTriangle(p1,p2,p3):
    '''计算三角形面积   海伦公式'''
    area = 0
    p1p2 = GetLineLength(p1,p2)
    p2p3 = GetLineLength(p2,p3)
    p3p1 = GetLineLength(p3,p1)
    s = (p1p2 + p2p3 + p3p1)/2
    area = s*(s-p1p2)*(s-p2p3)*(s-p3p1)   #海伦公式
    area = math.sqrt(area)
    return area

def GetLineLength(p1,p2):
    '''计算边长'''
    length = math.pow((p1.x-p2.x),2) + math.pow((p1.y-p2.y),2)  #pow  次方
    length = math.sqrt(length)   
    return length

def main():

    points = []
    x = [0.00,0.00,0.01,0.01,0.02,0.04,0.05,0.07,0.10,0.12,0.15,0.18,0.21,0.24,0.28,0.32,0.37,0.42,0.46,0.52,0.57,0.62,0.68,0.74,0.80,0.86,0.92,0.99,1.06,1.12,1.19,1.26,1.33,1.40,1.48,1.68,1.75,1.82,1.88,1.95,2.01,2.08,2.15,2.21,2.28,2.35,2.41,2.48,2.55,2.61,2.68,2.75,2.81,2.88,2.95,3.01,3.08,3.15,3.21,3.27,3.34,3.39,3.46,3.51,3.58,3.64,3.69,3.75,3.81,3.86,3.92,3.97,4.02,4.08,4.13,4.17,4.22,4.27,4.31,4.36,4.41,4.44,4.49,4.52,4.56,4.60,4.64,4.67,4.71,4.74,4.77,4.80,4.82,4.85,4.87,4.89,4.91,4.93,4.94,4.96,4.97,4.98,4.99,4.99,4.99,4.99,4.99,4.99,4.98,4.97,4.96,4.94,4.93,4.91,4.88,4.86,4.83,4.80,4.77,4.73,4.70,4.66,4.62,4.57,4.52,4.46,4.42,4.36,4.29,4.24,4.18,4.11,4.06,3.99,3.92,3.85,3.78,3.70,3.63,3.55,3.48,3.41,3.33,3.26,3.18,3.09,3.02,2.94,2.85,2.78,2.69,2.61,2.54,2.45,2.37,2.30,2.21,2.13,2.06,1.98,1.89,1.82,1.74,1.67,1.59,1.52,1.45,1.37,1.30,1.23,1.16,1.09,1.03,0.96,0.90,0.84,0.78,0.72,0.67,0.61,0.55,0.51,0.45,0.41,0.36,0.32,0.28,0.24,0.21,0.18,0.14,0.12,0.09,0.07,0.05,0.04,0.02,0.01,0.01,0.00]
    y = [35.01,35.30,35.32,35.22,37.23,38.91,40.61,41.66,43.01,45.78,49.20,51.85,53.81,56.15,58.65,57.61,55.97,54.22,52.13,50.91,51.01,51.65,52.28,53.65,54.56,54.53,54.43,53.75,52.45,51.85,51.76,51.75,51.80,52.42,52.42,52.47,52.60,52.75,52.83,52.55,52.35,52.25,52.01,51.82,51.82,51.81,51.85,51.88,51.88,51.81,51.80,51.75,51.53,51.49,51.54,51.51,51.51,51.52,51.51,51.48,51.52,51.26,51.09,51.05,50.92,50.93,50.97,50.97,50.95,51.02,50.99,51.04,51.04,50.92,50.65,50.64,50.61,50.61,50.66,50.67,50.64,50.67,50.58,50.47,50.45,50.24,50.07,50.10,50.07,50.05,50.11,50.10,50.07,49.97,49.70,49.67,49.68,49.50,49.50,49.49,49.47,49.50,49.46,49.48,49.21,48.11,47.81,47.37,47.32,46.85,45.77,44.54,43.09,41.66,40.29,38.49,36.54,33.99,31.23,28.23,25.26,23.25,24.20,26.10,29.01,31.74,33.24,33.20,32.61,30.41,27.65,26.16,25.95,25.98,27.61,29.39,31.12,31.89,31.97,30.75,29.65,28.33,27.31,27.00,27.47,28.33,29.30,30.26,30.96,30.99,30.31,29.17,28.83,28.18,28.16,28.18,28.94,29.49,30.08,30.34,30.43,30.24,29.58,29.15,29.08,29.08,29.41,29.76,30.36,30.48,30.55,30.48,30.47,30.14,29.80,29.80,30.17,30.39,30.85,31.42,31.55,31.53,31.54,31.48,31.43,31.40,31.41,31.57,32.01,32.66,33.24,33.25,33.24,33.24,32.80,32.25,32.25,32.40,32.61,33.04]

    for index in range(len(x)):
        points.append(Point(x[index],y[index]))

    area = GetAreaOfPolyGon(points)
    print(area)
    print(math.ceil(area))
    assert math.ceil(area)==1

if __name__ == '__main__':
    main()
    print("OK")    

　　另外，仍可继续简化算法，采用向量叉乘计算多边形面积。由图知坐标原点（按P点为原点）与多边形任意相邻的两个顶点构成一个三角形，而三角形的面积可由三个顶点构成的两个平面向量的外积求得。任意多边形的面积公式：

　　基于向量叉乘计算多边形面积参考代码如下：

'''
Created on 2018年4月15日

@author: XiaoYW
'''
import math

class Point():
    def __init__(self,x,y):
        self.x = x
        self.y = y


def GetAreaOfPolyGonbyVector(points):
    # 基于向量叉乘计算多边形面积
    area = 0
    if(len(points)<3):

         raise Exception("error")

    for i in range(0,len(points)-1):
        p1 = points[i]
        p2 = points[i + 1]

        triArea = (p1.x*p2.y - p2.x*p1.y)/2
        area += triArea
    return abs(area)

def main():

    points = []

    x = [1,2,3,4,5,6,5,4,3,2]
    y = [1,2,2,3,3,3,2,1,1,1] 
    for index in range(len(x)):
        points.append(Point(x[index],y[index]))

    area = GetAreaOfPolyGonbyVector(points)
    print(area)
    print(math.ceil(area))
    assert math.ceil(area)==1

if __name__ == '__main__':
    main()
    print("OK") 

　　两种计算多边形面积的算法，对比结果如下：

　　欢迎读者验证代码及算法，请及时反馈！

参考：

1. 《使用Python Matplotlib绘图并输出图像到文件中的实践》 CSDN博客 肖永威 2018年4月
2. 《Python计算任意多边形面积算法》 CSDN博客 zfqhd432 2013年2月
3. 《任意多边形面积的计算》 CSDN博客 _学而时习之_ 2015年10月