#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){} //初始化列表建構函式 }; typedef Point Vector; //向量+向量 = 向量, 點 + 向量 = 向量 Vector operator + (Vector A, Vector B){return Vector(A.x+B.x, A.y+B.y);} //點-點=向量 Vector operator - (Point A,Point B){return Vector(A.x-B.x, A.y - B.y);} //向量*數=向量 Vector operator * (Vector A, double p){return Vector(A.x*p, A.y*p);} //向量/數 = 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);} //三態函式 int dcmp(double x){if(fabs(x) < eps ) return 0;else return x < 0 ? -1 : 1;} //判斷兩個點是否相等 bool operator == (const Point& a,const Point& b){return dcmp(a.x - b.x) == 0 && dcmp(a.y-b.y) == 0;} //求向量的極角 double polar_angle(Vector A){return atan2(A.y,A.x);} //點積 double Dot(Vector A, Vector B){return A.x*B.x + A.y*B.y;} //求向量的長度 double Length(Vector A){return sqrt(Dot(A,A));} //求向量之間的夾角 double Angle(Vector A,Vector B){return acos(Dot(A,B)/Length(A)/Length(B));} //兩個向量的叉積 double Cross(Vector A,Vector B){return A.x*B.y - A.y*B.y;} //求有向面積 double Area2(Point A, Point B, Point C ){ return Cross(B-A,C-A);} //向量旋轉 Vector Rotate(Vector A,double rad){ return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad)); } //向量的單位法向量 Vector Normal(Vector A){ double L = Length(A); return Vector(-A.y/L,A.x/L); } //求兩條直線的交點 Point GetLineIntersection(Point P,Vector v, Point Q, Vector w){ Vector u = P-Q; double t =Cross(w,u)/Cross(v,w); //注意當Cross(v,w)!=0時才有交點 return P+v*t; } //點到直線的距離 double DistanceToLine(Point P,Point A,Point B){ Vector v1 = B-A, v2 = P-A; return fabs(Cross(v1,v2))/Length(v1); } //點到線段的距離 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); } //點在直線上的投影 Point GetLineProjection(Point P, Point A, Point B){ Vector v = B-A; return A+v*(Dot(v,P-A)/Dot(v,v)); } //線段相交判斷 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; } //點是否線上段上 bool OnSegment(Point p,Point a1, Point a2){ return dcmp(Cross(a1-p,a2-p)) == 0 && dcmp(Dot(a1-p,a2-p)) < 0; }