题意：一个很多个点p构成的多边形，pb <= pa * k时p所占区域与多边形相交面积
设p(x,y), (x - xb)^2+(y - yb)^2 / (x - xa)^2+(y - ya)^2 = k^2
所以可以化成圆的一般式x^2 + y^2 + dx + ey + f = 0;
推公式：
圆心：（-d/2,-e/2） 半径：(sqrt(d*d + e*e - 4*f)) / 2
得出圆后，套模板求圆与多边形的相交
#include <iostream>
#include <cstring>
#include <cstdio>
#include <algorithm>
#include <cmath>
using namespace std;
typedef long long ll;
#define eps 1e-8
#define N 100017 struct Point
{
double x,y;
Point(double x=0, double y=0):x(x),y(y) {}
void get(int x1,int y1)
{
x = x1;
y = y1;
}
};
typedef Point Vector;
struct Circle
{
Point c;
double r;
Circle() {}
Circle(Point c,double r):c(c),r(r) {}
Point point(double a)
{
return Point(c.x + cos(a)*r, c.y + sin(a)*r);
}
void input()
{
scanf("%lf%lf%lf",&c.x,&c.y,&r);
}
};
int dcmp(double x)
{
if(x < -eps) return -1;
if(x > eps) return 1;
return 0;
}
template <class T> T sqr(T x)
{
return x * x;
}
Vector operator + (Vector A, Vector B)
{
return Vector(A.x + B.x, A.y + B.y);
}
Vector operator - (Vector A, Vector B)
{
return Vector(A.x - B.x, A.y - B.y);
}
Vector operator * (Vector A, double p)
{
return Vector(A.x*p, A.y*p);
}
Vector operator / (Vector A, double p)
{
return Vector(A.x/p, A.y/p);
}
bool operator < (const Point& a, const Point& b)
{
return a.x < b.x || (a.x == b.x && a.y < b.y);
}
bool operator >= (const Point& a, const Point& b)
{
return a.x >= b.x && a.y >= b.y;
}
bool operator <= (const Point& a, const Point& b)
{
return a.x <= b.x && a.y <= b.y;
}
bool operator == (const Point& a, const Point& b)
{
return dcmp(a.x-b.x) == 0 && dcmp(a.y-b.y) == 0;
}
double Dot(Vector A, Vector B)
{
return A.x*B.x + A.y*B.y;
}
double Length(Vector A)
{
return sqrt(Dot(A, A));
}
double Angle(Vector A, Vector B)
{
return acos(Dot(A, B) / Length(A) / Length(B));
}
double Cross(Vector A, Vector B)
{
return A.x*B.y - A.y*B.x;
}
Vector VectorUnit(Vector x)
{
return x / Length(x);
}
Vector Normal(Vector x)
{
return Point(-x.y, x.x) / Length(x);
}
double angle(Vector v)
{
return atan2(v.y, v.x);
} bool OnSegment(Point P, Point A, Point B)
{
return dcmp(Cross(A-P,B-P)) == 0 && dcmp(Dot(A-P,B-P)) < 0;
}
double DistanceToSeg(Point P, Point A, Point B)
{
if(A == B) return Length(P-A);
Vector v1 = B-A, v2 = P-A, v3 = P-B;
if(dcmp(Dot(v1, v2)) < 0) return Length(v2);
if(dcmp(Dot(v1, v3)) > 0) return Length(v3);
return fabs(Cross(v1, v2)) / Length(v1);
}
double DistanceToLine(Point P, Point A, Point B)
{
Vector v1 = B-A, v2 = P-A;
return fabs(Cross(v1,v2)) / Length(v1);
}
Point DisP(Point A, Point B)
{
return Length(B-A);
}
bool SegmentIntersection(Point A,Point B,Point C,Point D)
{
return max(A.x,B.x) >= min(C.x,D.x) &&
max(C.x,D.x) >= min(A.x,B.x) &&
max(A.y,B.y) >= min(C.y,D.y) &&
max(C.y,D.y) >= min(A.y,B.y) &&
dcmp(Cross(C-A,B-A)*Cross(D-A,B-A)) <= 0 &&
dcmp(Cross(A-C,D-C)*Cross(B-C,D-C)) <= 0;
} Point Zero = Point(0,0);
double TriAngleCircleInsection(Circle C, Point A, Point B)
{
Vector OA = A-C.c, OB = B-C.c;
Vector BA = A-B, BC = C.c-B;
Vector AB = B-A, AC = C.c-A;
double DOA = Length(OA), DOB = Length(OB),DAB = Length(AB), r = C.r;
if(dcmp(Cross(OA,OB)) == 0) return 0;
if(dcmp(DOA-C.r) < 0 && dcmp(DOB-C.r) < 0) return Cross(OA,OB)*0.5;
else if(DOB < r && DOA >= r)
{
double x = (Dot(BA,BC) + sqrt(r*r*DAB*DAB-Cross(BA,BC)*Cross(BA,BC)))/DAB;
double TS = Cross(OA,OB)*0.5;
return asin(TS*(1-x/DAB)*2/r/DOA)*r*r*0.5+TS*x/DAB;
}
else if(DOB >= r && DOA < r)
{
double y = (Dot(AB,AC)+sqrt(r*r*DAB*DAB-Cross(AB,AC)*Cross(AB,AC)))/DAB;
double TS = Cross(OA,OB)*0.5;
return asin(TS*(1-y/DAB)*2/r/DOB)*r*r*0.5+TS*y/DAB;
}
else if(fabs(Cross(OA,OB)) >= r*DAB || Dot(AB,AC) <= 0 || Dot(BA,BC) <= 0)
{
if(Dot(OA,OB) < 0)
{
if(Cross(OA,OB) < 0) return (-acos(-1.0)-asin(Cross(OA,OB)/DOA/DOB))*r*r*0.5;
else return ( acos(-1.0)-asin(Cross(OA,OB)/DOA/DOB))*r*r*0.5;
}
else return asin(Cross(OA,OB)/DOA/DOB)*r*r*0.5;
}
else
{
double x = (Dot(BA,BC)+sqrt(r*r*DAB*DAB-Cross(BA,BC)*Cross(BA,BC)))/DAB;
double y = (Dot(AB,AC)+sqrt(r*r*DAB*DAB-Cross(AB,AC)*Cross(AB,AC)))/DAB;
double TS = Cross(OA,OB)*0.5;
return (asin(TS*(1-x/DAB)*2/r/DOA)+asin(TS*(1-y/DAB)*2/r/DOB))*r*r*0.5 + TS*((x+y)/DAB-1);
}
} Point p[2000]; int main()
{
int n;
double k;
int cas = 1;
while(scanf("%d%lf",&n,&k) != EOF)
{
for(int i = 1; i <= n; i++)
{
double x,y;
scanf("%lf%lf",&x,&y);
p[i].get(x,y);
}
p[n+1] = p[1];
double x1,x2,y1,y2;
scanf("%lf%lf%lf%lf",&x1,&y1,&x2,&y2); double d = (2.0*k*k*x1-2.0*x2)/(1.0-k*k);
double e = (2.0*k*k*y1-2.0*y2)/(1.0-k*k);
double f = (x2*x2+y2*y2-k*k*(x1*x1+y1*y1))/(1.0-k*k);
Point O = Point(-d*0.5,-e*0.5);
double R= sqrt(d*d+e*e-4.0*f)*0.5;
Circle C = Circle(O,R);
double ans = 0.0;
for(int i=1; i<=n; i++)
ans += TriAngleCircleInsection(C, p[i], p[i+1]);
printf("Case %d: %.10f\n",cas++,fabs(ans));
}
return 0;
}