二维几何常用运算

#include <iostream>
#include <cmath>

const double eps = 1e-10;

struct Point{
    double x,y;
    Point(double x = 0, double y = 0):x(x),y(y){} //初始化列表构造函数
};

typedef Point Vector;

//向量+向量 = 向量， 点 +　向量 = 向量
Vector operator + (Vector A, Vector B){return Vector(A.x+B.x, A.y+B.y);}
//点-点=向量
Vector operator - (Point A,Point B){return Vector(A.x-B.x, A.y - B.y);}
//向量*数=向量
Vector operator * (Vector A, double p){return Vector(A.x*p, A.y*p);}
//向量/数 = 向量
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);}
//三态函数
int dcmp(double x){if(fabs(x) < eps ) return 0;else return x <  0 ? -1 : 1;}
//判断两个点是否相等
bool operator == (const Point& a,const Point& b){return dcmp(a.x - b.x) == 0 && dcmp(a.y-b.y)  == 0;}
//求向量的极角
double polar_angle(Vector A){return atan2(A.y,A.x);}
//点积
double Dot(Vector A, Vector B){return A.x*B.x + A.y*B.y;}
//求向量的长度
double Length(Vector A){return sqrt(Dot(A,A));}
//求向量之间的夹角
double Angle(Vector A,Vector B){return acos(Dot(A,B)/Length(A)/Length(B));}
//两个向量的叉积
double Cross(Vector A,Vector B){return A.x*B.y - A.y*B.y;}
//求有向面积
double Area2(Point A, Point B, Point C ){ return Cross(B-A,C-A);}
//向量旋转
Vector Rotate(Vector A,double rad){
    return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));
}
//向量的单位法向量
Vector Normal(Vector A){
    double L = Length(A);
    return Vector(-A.y/L,A.x/L);
}
//求两条直线的交点
Point GetLineIntersection(Point P,Vector v, Point Q, Vector w){
    Vector u = P-Q;
    double t =Cross(w,u)/Cross(v,w);   //注意当Cross(v,w)!=0时才有交点
    return P+v*t;
}
//点到直线的距离
double DistanceToLine(Point P,Point A,Point B){
    Vector v1 = B-A, v2 = P-A;
    return fabs(Cross(v1,v2))/Length(v1);
}
//点到线段的距离
double DistanceToSegment(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);
    else if (dcmp(Dot(v1,v3)) > 0) return Length(v3);
    else return fabs(Cross(v1,v2))/Length(v1);
}

//点在直线上的投影
Point GetLineProjection(Point P, Point A, Point B){
    Vector v = B-A;
    return A+v*(Dot(v,P-A)/Dot(v,v));
}

//线段相交判断
bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){
    double c1 = Cross(a2-a1,b1-a1);
    double c2 = Cross(a2-a1,b2-a1);
    double c3 = Cross(b2-b1,a1-b1);
    double c4 = Cross(b2-b1,a2-b1);
    return dcmp(c1)*dcmp(c2) < 0 && dcmp(c3)*dcmp(c4) < 0;
}

//点是否在线段上
bool OnSegment(Point p,Point a1, Point a2){
    return dcmp(Cross(a1-p,a2-p)) == 0 && dcmp(Dot(a1-p,a2-p)) < 0;
}