Problem Description

Matt is a big fan of logo design. Recently he falls in love with logo made up by rings. The following figures are some famous examples you may know.

A ring is a 2-D figure bounded by two circles sharing the common center. The radius for these circles are denoted by r and R (r < R). For more details, refer to the gray part in the illustration below.

Matt just designed a new logo consisting of two rings with the same size in the 2-D plane. For his interests, Matt would like to know the area of the intersection of these two rings.

 

Input

The first line contains only one integer T (T ≤ 10⁵), which indicates the number of test cases. For each test case, the first line contains two integers r, R (0 ≤ r < R ≤ 10).

Each of the following two lines contains two integers x_i, y_i (0 ≤ x_i, y_i ≤ 20) indicating the coordinates of the center of each ring.

 

Output

For each test case, output a single line “Case #x: y”, where x is the case number (starting from 1) and y is the area of intersection rounded to 6 decimal places.

 

Sample Input

 

Sample Output

 

Source

#include<bits/stdc++.h>

using namespace std;

#define LL long long

#define fi first

#define se second

#define mkp make_pair

#define eps 1e-8

const double pi=acos(-);

const int N=2e5+,M=1e6+,inf=1e9+;

const LL INF=1e18+,mod=;

int sgn(double x)

{

    if(fabs(x) < eps)return ;

    if(x < )return -;

    else return ;

}

struct Point

{

    double x,y;

    Point() {}

    Point(double _x,double _y)

    {

        x = _x;

        y = _y;

    }

    Point operator -(const Point &b)const

    {

        return Point(x - b.x,y - b.y);

    }

    //叉积

    double operator ^(const Point &b)const

    {

        return x*b.y - y*b.x;

    }

//点积

    double operator *(const Point &b)const

    {

        return x*b.x + y*b.y;

    }

//绕原点旋转角度B（弧度值），后x,y的变化

    void transXY(double B)

    {

        double tx = x,ty = y;

        x= tx*cos(B) - ty*sin(B);

        y= tx*sin(B) + ty*cos(B);

    }

};

double AREA(Point a, double r1, Point b, double r2)

{

    double d = sqrt((a.x-b.x)*(a.x-b.x) + (a.y-b.y)*(a.y-b.y));

    if (d >= r1+r2)

        return ;

    if (r1>r2)

    {

        double tmp = r1;

        r1 = r2;

        r2 = tmp;

    }

    if(r2 - r1 >= d)

        return pi*r1*r1;

    double ang1=acos((r1*r1+d*d-r2*r2)/(*r1*d));

    double ang2=acos((r2*r2+d*d-r1*r1)/(*r2*d));

    return ang1*r1*r1 + ang2*r2*r2 - r1*d*sin(ang1);

}

int main()

{

    int T,cas=;

    scanf("%d",&T);

    while(T--)

    {

        double r1,r2;

        Point a,b;

        scanf("%lf%lf%lf%lf%lf%lf",&r1,&r2,&a.x,&a.y,&b.x,&b.y);

        printf("Case #%d: %.6f\n",cas++,AREA(a,r2,b,r2)-AREA(a,r1,b,r2)-AREA(a,r2,b,r1)+AREA(a,r1,b,r1));

    }

    return ;

}