计算几何的简单模板

const double eps = 1e-8;

struct Point
{
    double x,y;
    Point(double tx = 0,double ty = 0) : x(tx),y(ty){}
};
typedef Point Vtor;
//向量的加减乘除
Vtor operator + (Vtor A,Vtor B) { return Vtor(A.x + B.x,A.y + B.y); }
Vtor operator - (Point A,Point B) { return Vtor(A.x - B.x,A.y - B.y); }
Vtor operator * (Vtor A,double p) { return Vtor(A.x*p,A.y*p); }
Vtor operator / (Vtor A,double p) { return Vtor(A.x/p,A.y/p); }
bool operator < (Point A,Point B) { return A.x < B.x || (A.x == B.x && A.y < B.y);}
int dcmp(double x){ if (fabs(x) < eps) return 0; else return x < 0 ? -1 : 1; }
bool operator == (Point A,Point B) {return dcmp(A.x - B.x) == 0 && dcmp(A.y - B.y) == 0; }
//向量的点积，长度，夹角
double Dot(Vtor A,Vtor B) { return A.x*B.x + A.y*B.y; }
double Length(Vtor A) { return sqrt(Dot(A,A)); }
double Angle(Vtor A,Vtor B) { return acos(Dot(A,B)/Length(A)/Length(B)); }
//叉积，三角形面积
double Cross(Vtor A,Vtor B) { return A.x*B.y - A.y*B.x; }
double Area2(Point A,Point B,Point C) { return Cross(B - A,C - A); }
//向量的旋转,求向量的单位法线（即左转90度，然后长度归一）
Vtor Rotate(Vtor A,double rad){ return Vtor(A.x*cos(rad) - A.y*sin(rad),A.x*sin(rad) + A.y*cos(rad)); }
Vtor Normal(Vtor A)
{
    double L = Length(A);
    return Vtor(-A.y/L, A.x/L);
}
//直线的交点
Point GetLineIntersection(Point P,Vtor v,Point Q,Vtor w)
{
    Vtor u = P - Q;
    double t = Cross(w,u)/Cross(v,w);
    return P + v*t;
}
//点到直线的距离
double DistanceToLine(Point P,Point A,Point B)
{
    Vtor v1 = B - A;
    return fabs(Cross(P - A,v1))/Length(v1);
}
//点到线段的距离
double DistanceToSegment(Point P,Point A,Point B)
{
    if (A == B) return Length(P - A);
    Vtor v1 =  B - A , v2 = P - A, v3 = P - B;
    if (dcmp(Dot(v1,v2)) < 0) return Length(v2);
    else if (dcmp(Dot(v1,v3)) > 0) return Length(v3);
    else return fabs(Cross(v1,v2))/Length(v1);
}
//点到直线的映射
Point GetLineProjection(Point P,Point A,Point B)
{
    Vtor v = B - A;
    return A + v*Dot(v,P - A)/Dot(v,v);
}

//判断线段是否规范相交
bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2)
{
    double c1 = Cross(a2 - a1,b1 - a1), c2 = Cross(a2 - a1,b2 - a1),
           c3 = Cross(b2 - b1,a1 - b1), c4 = Cross(b2 - b1,a2 - b1);
    return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3)*dcmp(c4) < 0;
}
//判断点是否在一条线段上
bool OnSegment(Point P,Point a1,Point a2)
{
    return dcmp(Cross(a1 - P,a2 - P)) == 0 && dcmp(Dot(a1 - P,a2 - P)) < 0;
}
//多边形面积
double PolgonArea(Point *p,int n)
{
    double area = 0;
    for (int i = 1; i < n - 1; ++i)
    area += Cross(p[i] - p[0],p[i + 1] - p[0]);
    return area/2;
}

　　

和圆有关的计算

struct Line
{
    Point p;
    Vtor v;
    Line(Point p,Vtor v) : p(p),v(v){}
    Point point(double t) { return p + v*t; }
};
struct Circle
{
    Point c;
    double r;
    Circle(Point tc,double tr) : c(tc),r(tr){}
    Point point(double a)
    {
        return Point(c.x + cos(a)*r + c.y + sin(a)*r);
    }
};
//判断圆与直线是否相交以及求出交点
int getLineCircleIntersection(Line L,Circle C,double t1,double t2,vector<Point> &sol)
{
    //注意sol没有清空哦
    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;
    double delta = f*f - 4*e*g;
    if (dcmp(delta) < 0) return 0;
    else if (dcmp(delta) == 0)
    {
        t1 = t2 = -f/(2*e);
        sol.push_back(L.point(t1));
        return 1;
    }
    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;
}
//判断并求出两圆的交点
double angle(Vtor v) { return atan2(v.y, v.x); }
int getCircleIntersection(Circle C1,Circle C2,vector<Point> &sol)
{
    double d = Length(C2.c - C1.c);
    // 圆心重合
    if (dcmp(d) == 0)
    {
        if (dcmp(C1.r - C2.r) == 0) return -1; // 两圆重合
        return 0; // 包含
    }

    // 圆心不重合
    if (dcmp(C1.r + C2.r - d) < 0) return 0; // 相离
    if (dcmp(fabs(C1.r - C2.r) - d) > 0) return 0; // 包含

    double a = angle(C2.c - C1.c);
    double da = acos(C1.r*C1.r + d*d - C2.r*C2.r) / (2*C1.r*d);
    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;
}
//求点到圆的切线
int getTangents(Point p,Circle C,Vtor *v)
{
    Vtor u = C.c - p;
    double dis = Length(u);
    if (dis < C.r)  return 0;
    else if (dcmp(dis - C.r) == 0)
    {
        v[0] = Rotate(u,PI/2);
        return 1;
    }
    else
    {
        double ang = asin(C.r / dis);
        v[0] = Rotate(u, -ang);
        v[1] = Rotate(u, ang);
        return 2;
    }
}
//求两圆的切线
int getCircleTangents(Circle A,Circle B,Point *a,Point *b)
{
    int cnt = 0;
    if (A.r < B.r) { swap(A,B); swap(a, b) ; }
    //圆心距的平方
    double d2 = (A.c.x - B.c.x)*(A.c.x - B.c.x) + (A.c.y - B.c.y)*(A.c.y - B.c.y);
    double rdiff = A.r - B.r;
    double rsum = A.r + B.r;
    double base = angle(B.c - A.c);
    //重合有无限多条
    if (d2 == 0 && dcmp(A.r - B.r) == 0) return -1;
    //内切
    if (dcmp(d2 - rdiff*rdiff) == 0)
    {
        a[cnt] = A.point(base);
        b[cnt] = B.point(base); cnt++;
        return 1;
    }
    //有外公切线
    double ang = acos((A.r - B.r) / sqrt(d2));
    a[cnt] = A.point(base + ang); b[cnt] = B.point(base + ang); cnt++;
    a[cnt] = A.point(base - ang); b[cnt] = B.point(base - ang); cnt++;

    //一条内切线
    if (dcmp(d2 - rsum*rsum) == 0)
    {
        a[cnt] = A.point(base); b[cnt] = B.point(PI + base); cnt++;
    }//两条内切线
    else if (dcmp(d2 - rsum*rsum) > 0)
    {
        double ang = acos((A.r + B.r) / sqrt(d2));
        a[cnt] = A.point(base + ang); b[cnt] = B.point(base + ang); cnt++;
        a[cnt] = A.point(base - ang); b[cnt] = B.point(base - ang); cnt++;
    }
    return cnt;
}