Lifting the Stone（求任意多边形的重心）_delphi 重心公式-优快云博客

这篇博客探讨了如何求解多边形的重心，通过将多边形划分为多个三角形区域，并利用平面薄板重心公式，将积分转换为求和过程，从而计算出每个小三角形区域的重心，最终得出整个多边形的重心坐标。

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Lifting the Stone

Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 5432 Accepted Submission(s): 2273

Problem Description

There are many secret openings in the floor which are covered by a big heavy stone. When the stone is lifted up, a special mechanism detects this and activates poisoned arrows that are shot near the opening. The only possibility is to lift the stone very slowly and carefully. The ACM team must connect a rope to the stone and then lift it using a pulley. Moreover, the stone must be lifted all at once; no side can rise before another. So it is very important to find the centre of gravity and connect the rope exactly to that point. The stone has a polygonal shape and its height is the same throughout the whole polygonal area. Your task is to find the centre of gravity for the given polygon.

Input

The input consists of T test cases. The number of them (T) is given on the first line of the input file. Each test case begins with a line containing a single integer N (3 <= N <= 1000000) indicating the number of points that form the polygon. This is followed by N lines, each containing two integers Xi and Yi (|Xi|, |Yi| <= 20000). These numbers are the coordinates of the i-th point. When we connect the points in the given order, we get a polygon. You may assume that the edges never touch each other (except the neighboring ones) and that they never cross. The area of the polygon is never zero, i.e. it cannot collapse into a single line.

Output

Print exactly one line for each test case. The line should contain exactly two numbers separated by one space. These numbers are the coordinates of the centre of gravity. Round the coordinates to the nearest number with exactly two digits after the decimal point (0.005 rounds up to 0.01). Note that the centre of gravity may be outside the polygon, if its shape is not convex. If there is such a case in the input data, print the centre anyway.

Sample Input

Sample Output

  
   0.00 0.00
6.00 6.00

Source

Central Europe 1999

题意：求多边形的重心；

意解：
定理1　已知三角形△A1A2A3的顶点坐标Ai ( xi , yi ) ( i =1, 2, 3) 。它的重心坐标为:

　　 xg = (x1+x2+x3) / 3 ; yg = (y1+y2+y3) / 3 ;

定理2: 将多边形划分成n个小三角形区域, 每个小区域面积为σi ,重心为Gi ( . xi , . yi ) ,利用求平面薄板重心公式把积分变
　　成累加和:

AC代码：

#include <iostream>
#include <cstdio>
#include <cmath>

using namespace std;

struct Point
{
    double x,y;
    Point(double x = 0, double y = 0) : x(x),y(y) {};
    void read()
    {
        scanf("%lf %lf",&x,&y);
    }
    Point operator - (const Point &a) const //重载减号；
    {
        return Point(a.x - x, a.y - y);
    }
};

double cross(Point A, Point B) //向量的叉积;
{
    return A.x * B.y - A.y * B.x;
}

double area(Point a, Point b, Point c) //三角型面积公式1/2 * cross(Point A, Point B)
{
    Point A,B;
    A = b - a;
    B = c - a;
    return cross(A,B);//不需要再除以2，因为在最后的结果中抵消了；
}

int main()
{
    int T,n;
    scanf("%d",&T);
    while(T--)
    {
        scanf("%d",&n);
        Point p0,p1,p2;
        double sum_x,sum_y,sum_area;
        p0.read();
        p1.read();
        sum_x = sum_y = sum_area = 0.0;
        for(int i = 2; i < n; i++)
        {
            p2.read();
            double tp = area(p0,p1,p2);
            sum_x += (p0.x + p1.x + p2.x) * tp;
            sum_y += (p0.y + p1.y + p2.y) * tp;
            sum_area += tp;
            p1 = p2;
        }
        printf("%.2lf %.2lf\n",sum_x / sum_area / 3.0, sum_y / sum_area / 3.0);
    }
    return 0;
}