模板 - 几何基础

#include <bits/strc++.h>
using namespace std;

const double eps = 1e-10;
int dcmp(double x){//等于0 0;大于0 1;小于0 -1
if(fabs(x)<eps) return 0;
else return x<0 ? -1 : 1;
}
struct Point{
double x,y;
Point(double x=0,double y=0):x(x),y(y) {}
}

typedef Point Vector;
//向量的+-*/
Vector operator + (Vector A,Vector B){ return Vector(A.x+B.x,A.y+B.y);}
Vector operator -  (Point A,Point 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);}

//坐标的比较
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 (dcmp(a.x-b.x)==0 && dcmp(a.y-b.y)==0);
}

//点积
double Dot(Vector A,Vector B){ return A.x*B.x + A.y*B.y; }
//向量的长度
double Length(Vector A){ return sqrt(Dot(A,A));}
//两个向量的夹角
double Angle(Vector A,Vector B){
return acos(Dot(A,B)/(Length(A)*Length(B)));
}
//叉积
double Cross(Vector A,Vector B){ return A.x*B.y - A.y*B.x; }
//三点组成的三角形的有向面积的两倍
double Area2(Point A,Point B,Point C){return Cross(B-A,C-A);}
//向量旋转 rad:弧度
Vector Rotate(Vector A,double rad){
return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));
}
//获得向量的单位法线(左转90度)
Vector Normal(Vector A){
double L = Length(A);
if(dcmp(L)==0) return Vector(0,0);
retrun Vector(-A.y/L,A.x/L);
}

//得到两条直线的交点
//Point P=P0+tv   可表示在直线上的所有点 v=(B-A) 直线上的两点
//当表示为线段的时候0<=t<=1
//当表示成射线的时候t>0
//需要注意的是：两直线P+vt1,Q+wt2有唯一一个交点。Cross(v,w)!=0
Point GetLineIntersection(Point P,Vector v,Point Q ,Vector w){
Vector 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){
Vector ba = A-B, bc = C-B;
return fabs(Cross(ba,bc)/Length(ba));            //不去绝对值的意思是有向距离
}
//点到线段的距离
//情况一：点的投影在线段上-->点到直线的距离
//情况二：点的投影不在线段上-->点到离它比较近的端点
//用点积判断,用点积和叉积来计算
double DistanceToSegment(Point P,Point A,Point B){
if(A==B) return Length(P,A);
Vector ab = B-A , ap = P-A , bp = P-B;
if(dcmp(Dot(ab,ap))<0) return Length(ap);
else if(dcmp(Dot(ab,bp))>0) return Length(bp);
else return DistanceToLine(P,A,B);
}

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

//判断线段是否相交(不包括端点)
bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){
double c1 = Cross(a2-a1,b1-a1), c2 = Cross(a2-a1,b2-a1);
double c3 = Cross(b2-b1,a1-b1), c4 = Cross(b2-b1,a2-b1);
return (dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0);
}
//判断点是否在线段上(排除在端点上的情况)
//保证p在向量a1a2的方向上 && p不在a1a2或者a2a1的延长线上
bool OnSegment(Point p,Point a1,Point a2){
return (dcmp(Cross(a1-p,a2-p))==0 && dcmp(Dos(a1-p,a2-p))<0);
}

//求凸包面积，p[]里面的点需要根据一定方向排序(顺时针或者逆时针)
double ConvexPolygonArea(Point* p,int n){
double area=0;
for(int i=0;i < n-1;i++){
area += Cross(p[i]-p[0],p[i+1]-p[0]);
}
return area;
}

//返回凸包的顶点个数,ch数组保存了凸包顶点
//输入的点不能有重复
//两个while循环的判定条件里面的<表示允许凸包的边上有点，<=表示凸包的边上不允许有点
//需要的话用dcmp()提高精度
int ConvexHull(Point* p,int n,Point* ch){
sort(p,p+n);
int m=0;
for(int i=0;i<n;i++){
while(m > 1 && Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2]) <=0) m--;
ch[m++] = p[i];
}
int k=m;
for(int i=n-2;i>=0;i--){
while(m > k && Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2]) <=0) m--;
ch[m++] = p[i];
}
if(n > 1) m--;    //去掉起始点
return m;
}