Lifting the Stone
Time Limit: 2000/1000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 6104 Accepted Submission(s):
2546
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.
(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.
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.
算法一：在讲该算法时，先要明白下面几个定理。
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 已知三角形△A1A2A3的顶点坐标Ai ( xi , yi ) ( i =1, 2, 3) 。该三角形的面积为:
S = ( (x2 - x1) * (y3 - y1) - (x3 - x1) * (y2 - y1) ) / 2 ;
△A1A2A3 边界构成逆时针回路时取+ , 顺时针时取 -。
另外在求解的过程中，不需要考虑点的输入顺序是顺时针还是逆时针，相除后就抵消了。
原理:将多边形划分成n个小区域, 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:
由前面所提出的原理和数学定理可以得出求离散数据点所围多边形的一般重心公式:以Ai ( xi , yi ) ( i = 1, 2, ., n) 为顶点的任意N边形A1A2 .An ,将它划 分成N - 2个三角形(如图1) 。每个三角形的重心为Gi ( . xi , . yi ) ,面积为σi。那么多边形的重心坐标G( .x2, .y2) 为:
图1 多边形分解
#include<cstdio>
#include<iostream>
using namespace std;
int main()
{
int T;
double ss,S,SX,SY;
double x,y;
double x0,y0,X,Y;
int n,i;
// freopen("in.txt","r",stdin);
cin>>T;
while(T--)
{
S=,SX=,SY=;
scanf("%d%lf%lf%lf%lf",&n,&x0,&y0,&X,&Y);
for(i=;i<n;i++)
{
scanf("%lf%lf",&x,&y);
ss=( (X - x0) * (y - y0) - (x -x0) * (Y - y0) ) / ;
S+=ss;
SX+=ss*(x0+x+X);
SY+=ss*(y0+Y+y);
X=x,Y=y;
}
printf("%.2f %.2f\n",SX/S/,SY/S/);
}
return ;
}