C# 编程实现非自相交多边形质心

计算公式公式: http://en.wikipedia.org/wiki/Centroid#Centroid_of_polygon

多边形的质心：

一个非自相交的n个顶点的多边形(x₀,y₀), (x₁,y₁), ..., (x_n−1,y_n−1) 的质心 (C_x, C_y):

A是多边形的有向面积：

.

在这些公式中，顶点被假定为沿多边形周长编号。此外，顶点（xn，yn）与（x0，y0）是相同的，意味着 对最后一次循环的i + 1 会回到i = 0.
注意，如果点按顺时针方向进行编号，按上述方法计算，会出现一个无法预知的结果；但在这种情况下质心坐标也会是正确的。

In these formulas, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter. Furthermore, the vertex ( x_n, y_n ) is assumed to be the same as ( x₀, y₀ ), meaning i + 1 on the last case must loop around to i = 0. Note that if the points are numbered in clockwise order the areaA, computed as above, will have a negative sign; but the centroid coordinates will be correct even in this case.

对于那些难以理解这些公式∑符号，下面给出C#的实现代码：

class Program
{
    static void Main(string[] args)
    {
        List<Point> vertices = new List<Point>();

        vertices.Add(new Point() { X = 1, Y = 1 });
        vertices.Add(new Point() { X = 1, Y = 10 });
        vertices.Add(new Point() { X = 2, Y = 10 });
        vertices.Add(new Point() { X = 2, Y = 2 });
        vertices.Add(new Point() { X = 10, Y = 2 });
        vertices.Add(new Point() { X = 10, Y = 1 });
        vertices.Add(new Point() { X = 1, Y = 1 });

        Point centroid = Compute2DPolygonCentroid(vertices);
    }

    static Point Compute2DPolygonCentroid(List<Point> vertices)
    {
        Point centroid = new Point() { X = 0.0, Y = 0.0 };
        double signedArea = 0.0;
        double x0 = 0.0; // Current vertex X
        double y0 = 0.0; // Current vertex Y
        double x1 = 0.0; // Next vertex X
        double y1 = 0.0; // Next vertex Y
        double a = 0.0;  // Partial signed area

        // For all vertices except last
        int i=0;
        for (i = 0; i < vertices.Count - 1; ++i)
        {
            x0 = vertices[i].X;
            y0 = vertices[i].Y;
            x1 = vertices[i+1].X;
            y1 = vertices[i+1].Y;
            a = x0*y1 - x1*y0;
            signedArea += a;
            centroid.X += (x0 + x1)*a;
            centroid.Y += (y0 + y1)*a;
        }

        // Do last vertex
        x0 = vertices[i].X;
        y0 = vertices[i].Y;
        x1 = vertices[0].X;
        y1 = vertices[0].Y;
        a = x0*y1 - x1*y0;
        signedArea += a;
        centroid.X += (x0 + x1)*a;
        centroid.Y += (y0 + y1)*a;

        signedArea *= 0.5;
        centroid.X /= (6*signedArea);
        centroid.Y /= (6*signedArea);

        return centroid;
    }
}

public class Point
{
    public double X { get; set; }
    public double Y { get; set; }
}