二维几何基础模板(三)

一、通过圆心角求坐标的函数

struct Circle
{
    Point c;
    double r;
    Circle(Point c,double r):c(c),r(r) {}
    Point point(double a)
    {
        return Point(c.x+cos(a)*r,c.y+sin(s)*r);
    }
};

二、直线和圆的交点

int getLineCircleIntersection(Line L,Circle C,double& t1,double& t2,vector<Point>& sol){  
    double a = L.v.x,b = L.p.x-C.c.x,c = L.v.y,d = L.p.y-C.c.y;  
    double e = a*a+c*c,f = 2*(a*b+c*d),g = b*b+d*d-C.r*C.r;  
    double delta = f*f-4*e*g;    //判别式  
    if(dcmp(delta) < 0)           //相离  
        return 0;               //相切  
    if(dcmp(delta) == 0){  
        t1 = t2 = -f/(2*e);  
        sol.push_back(L.point(t1));  
        return 1;  
    }  
    //相交  
    t1 = (-f-sqrt(delta))/(2*e);  
    sol.push_back(L.point(t1));  
    t2 = (-f+sqrt(delta))/(2*e);  
    sol.push_back(L.point(t2));  
    return 2;  
}
//函数返回的是交点的个数，参数sol存放的是交点本身。   注意上述代码并没有清空sol

三、计算向量极角的方法

double angle(Vector v)
{
    return atan2(v.y,v.x);
}

四、两圆相交的代码

int getCircleCircleIntersection(Circle C1,Circle C2,vector<Point>& sol)
{
    double d=Length(C1.c-C2.c);
    if(dcmp(d)==0)
    {
        if(dcmp(C1.r-C2.r)==0)      //两圆重合
            return -1;
        return 0;
    }
    if(dcmp(C1.r+C2.r-d)<0)
        return 0;
    if(dcmp(fabs(C1.r-C2.r)-d)>0)
        return 0;
    double a=angle(C2.c-C1.c);      //向量C1C2的极角
    double da=acos((C1.r*C1.r+d*d-C2.r*C2.r)/(2*C1.r*d));   //C1C2到C1P1的角
    Point p1=C1.point(a-da),p2=C1.point(a+da);

    sol.push_back(p1);
    if(p1==p2)
        return 1;
    sol.push_back(p2);
    return 2;
}

五、两圆相交的面积

double area(Point c1,double r1,Point c2,double r2){  
    double d = dis(c1,c2);  
    if (r1+r2 < d+exs)
        return 0;//相离  
    if (d < fabs(r1-r2)+exp){//内含 
        double r = min(r1,r2);  
        return PI*r*r;  
    }  
    double x = (d*d+r1*r1-r2*r2)/(2*d);  
    double t1 = acos(x/r1);  
    double t2 = acos((d-x)/r2);  
    return r1*r1*t1+r2*r2*t2-d*r1*sin(t1);//相交  
}

//过点p到圆C的切线。v[i]是第i条切线的向量。返回切线条数
int getTangents(Point p,Circle C,Vector* v)
{
    Vector u=C.c-p;
    double dist=Length(u);
    if(dist<C.r)
        return 0;
    else if(dcmp(dist-C.r)==0)
    {
        //p在圆上，只有一条切线
         v[0]=Rotate(u,PI/2);
         return 1;
    }
    else
    {
        double ang=asin(C.r/dist);
        v[0]=Rotate(u,-ang);
        v[1]=Rotate(u,+ang);
        trturn 2;
    }
}

七、外公切线（假定r1>=r2）

//返回切线的条数。-1表示无穷条切线
//a[i]和b[i]分别是第i条切线在圆A和圆B上的切点
int getTangents(Circle A,Circle B,Point* a,Point* b)
{
    int cnt=0;
    if(A.r<B.r)
    {
        swap(A,B);
        swap(a,b);
    }
    int d2=(A.x-B.x)*(A.x-B.x)+(A.y-B.y)*(A.y-B.y);
    int rdiff=A.r-B.r;
    int rsum=A.r+B.r;
    if(d2<rdiff*rdiff)
        return 0;       //内含
    double base=atan2(B.y-A.y,B.x-A.x);
    if(d2==0&&A.r==B.r)
        return -1;      //无限条切线
    if(d2==rdiff*rdiff)
    {
        //内切，1条切线
        a[cnt]=A.getPoint(base);
        b[cnt]=B.getPoint(base);
        cnt++;
        return 1;
    }
    //有外公切线
    double ang=acos((A.r-B.r)sqrt(d2));
    a[cnt]=A.getPoint(base+ang);
    b[cnt]=B.getPoint(base+ang);
    cnt++;
    a[cnt]=A.getPoint(base-ang);
    b[cnt]=B.getPoint(base-ang);
    cnt++;
    if(d2==rsum*rsum)
    {
        //一条公切线
        a[cnt]=A.getPoint(base);
        b[cnt]=B.getPoint(PI+base);
        cnt++;
    }
    else if(d2>rsum*rsum)
    {
        //两条公切线
        double ang=acos((A.r+b.r)/sqrt(d2));
        a[cnt]=A.getPoint(base+ang);
        b[cnt]=B.getPoint(PI+base+ang);
        cnt++;
        a[cnt]=A.getPoint(base-ang);
        b[cnt]=B.getPoint(PI+base-ang);
        cnt++;
    }
    return cnt;
}