Codeforces Round #339 (Div. 2)-C. Peter and Snow Blower

Peter got a new snow blower as a New Year present. Of course, Peter decided to try it immediately. After reading the instructions he realized that it does not work like regular snow blowing machines. In order to make it work, you need to tie it to some point that it does not cover, and then switch it on. As a result it will go along a circle around this point and will remove all the snow from its path.

Formally, we assume that Peter’s machine is a polygon on a plane. Then, after the machine is switched on, it will make a circle around the point to which Peter tied it (this point lies strictly outside the polygon). That is, each of the points lying within or on the border of the polygon will move along the circular trajectory, with the center of the circle at the point to which Peter tied his machine.

Peter decided to tie his car to point P and now he is wondering what is the area of ​​the region that will be cleared from snow. Help him.
Input

The first line of the input contains three integers — the number of vertices of the polygon n (), and coordinates of point P.

Each of the next n lines contains two integers — coordinates of the vertices of the polygon in the clockwise or counterclockwise order. It is guaranteed that no three consecutive vertices lie on a common straight line.

All the numbers in the input are integers that do not exceed 1 000 000 in their absolute value.
Output

Print a single real value number — the area of the region that will be cleared. Your answer will be considered correct if its absolute or relative error does not exceed 10^- 6.

Namely: let’s assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if .
Sample test(s)
Input

3 0 0
0 1
-1 2
1 2

Output

12.566370614359172464

Input

4 1 -1
0 0
1 2
2 0
1 1

Output

21.991148575128551812

Note

In the first sample snow will be removed from that area:

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题意：给你n个点，算由这n个点围成的多边形绕P点旋转，覆盖的面积是多少？

思路：数学题。先通过坐标变换，转换成绕原点旋转。然后求多边形离原点最远和最近的点，最远的点一定在n个点之中；而最近的点有可能在n个点里面，也有可能在某一条边上。两个圆组成的圆环就是所求的面积

代码

#include 
#include 
#include 
#include 
#include 
#include 
#include 
#include 
#define N 100008
#define ll long long
#define ull unsigned long long
#define pi acos(-1)
using namespace std;
struct point
{
    ll x, y;
    void operator -= (point a)
    {
        this->x -= a.x;
        this->y -= a.y;
    }
}node[N], p;
double len(point a, point b)    //求线段ab到原点的最小距离 
{
    double r = (a.x-b.x)*(a.x) + (a.y-b.y)*a.y;
    double d = (a.x-b.x)*(a.x-b.x) + (a.y-b.y)*(a.y-b.y);
    if (r <= 0) return a.x*a.x + a.y*a.y;
    else if (r >= d)    return b.x*b.x+b.y*b.y;
    r /= d;
    double x = a.x+(b.x-a.x)*r, y = a.y+(b.y-a.y)*r;
    return x*x+y*y;
}
int main()
{
//  freopen("1.txt", "r", stdin);
    ios::sync_with_stdio(false);
    cin.tie(0);
    int i, j, n;
    double r1, r2, t;
    cin >> n >> p.x >> p.y;
    r2 = 0;
    for (i = 0; i < n; i++)
    {
        cin >> node[i].x >> node[i].y;
        node[i] -= p;
        t = node[i].x*node[i].x + node[i].y*node[i].y;
        if (r2 < t) r2 = t;
    }
    r1 = 9999999999999;
    node[n] = node[0];
    for (i = 0; i < n; i++)
        r1 = min(r1, len(node[i+1], node[i]));
    cout.setf(ios::fixed);
    cout << setprecision(18) << pi*(r2-r1) << endl;
    return 0;
}