poj 1265 (pick定理+求多边形边上的点+多边形面积)

Area
Time Limit: 1000MS Memory Limit: 10000K
Total Submissions: 8478 Accepted: 3507
Description

Being well known for its highly innovative products, Merck would definitely be a good target for industrial espionage. To protect its brand-new research and development facility the company has installed the latest system of surveillance robots patrolling the area. These robots move along the walls of the facility and report suspicious observations to the central security office. The only flaw in the system a competitor抯 agent could find is the fact that the robots radio their movements unencrypted. Not being able to find out more, the agent wants to use that information to calculate the exact size of the area occupied by the new facility. It is public knowledge that all the corners of the building are situated on a rectangular grid and that only straight walls are used. Figure 1 shows the course of a robot around an example area.

Figure 1: Example area.

You are hired to write a program that calculates the area occupied by the new facility from the movements of a robot along its walls. You can assume that this area is a polygon with corners on a rectangular grid. However, your boss insists that you use a formula he is so proud to have found somewhere. The formula relates the number I of grid points inside the polygon, the number E of grid points on the edges, and the total area A of the polygon. Unfortunately, you have lost the sheet on which he had written down that simple formula for you, so your first task is to find the formula yourself.
Input

The first line contains the number of scenarios.
For each scenario, you are given the number m, 3 <= m < 100, of movements of the robot in the first line. The following m lines contain pairs 揹x dy�of integers, separated by a single blank, satisfying .-100 <= dx, dy <= 100 and (dx, dy) != (0, 0). Such a pair means that the robot moves on to a grid point dx units to the right and dy units upwards on the grid (with respect to the current position). You can assume that the curve along which the robot moves is closed and that it does not intersect or even touch itself except for the start and end points. The robot moves anti-clockwise around the building, so the area to be calculated lies to the left of the curve. It is known in advance that the whole polygon would fit into a square on the grid with a side length of 100 units.
Output

The output for every scenario begins with a line containing 揝cenario #i:� where i is the number of the scenario starting at 1. Then print a single line containing I, E, and A, the area A rounded to one digit after the decimal point. Separate the three numbers by two single blanks. Terminate the output for the scenario with a blank line.

Sample Input

2
4
1 0
0 1
-1 0
0 -1
7
5 0
1 3
-2 2
-1 0
0 -3
-3 1
0 -3

Sample Output

Scenario #1:
0 4 1.0

Scenario #2:
12 16 19.0

给你一些点，但是这些点并不是坐标，而是记录路径的点，例如（-1,0）代表沿负方向走了一个单位；最终这些点构成一个多边形，求这个多边形内的点和多边形边上的点和多边形的面积。

设多边形内的点为a，多边形边上的点为b，多边形的面积为S，则：
多边形边上的点可以用gcd来求，gcd(dx,dy)就是多边形边上的点，dx是线段覆盖的在x方向上的点数，dy是线段覆盖在y方向上的点数；
多边形的面积可以用叉积来求：多边形的面积等于按照顺时针或者逆时针的方向上相邻的两个点分别与多边形内一点构成的向量的叉积之和的一半。
pick定理可以求面积：S=a+b/2-1；所以内部的点a=s+1-b/2;

#include
using namespace std;
int t;
struct node{
     
	int x,y;
}p[205];
int area(node a,node b)//计算叉积
{
     
	return a.x*b.y-a.y*b.x;
}
int main()
{
     
	cin>>t;
	int f=1;
	while(t--)
	{
     
		int n,dx,dy;
		cin>>n;
		int b=0,a=0;
		p[0].x=0;p[0].y=0;
		for(int i=1;i<=n;i++)
		{
     
			cin>>dx>>dy;
			p[i].x=dx+p[i-1].x;//计算没个点坐标
			p[i].y=dy+p[i-1].y;
			if(dx<0) dx=-dx;
			if(dy<0) dy=-dy;
			b+=__gcd(dx,dy);//计算边界上点的面积
			a+=area(p[i],p[i-1]); //计算叉积和
		}
		if(a<0) a=-a;
		printf("Scenario #%d:\n%d %d %.1f\n\n",f++,(a+2-b)/2,b,0.5*a);
	}
	return 0;
}