模板 - 几何基础

  1 #include <bits/strc++.h>
  2 using namespace std;
  3 
  4 const double eps = 1e-10;
  5 int dcmp(double x){//等于0 0;大于0 1;小于0 -1
  6     if(fabs(x)<eps) return 0;
  7     else return x<0 ? -1 : 1;
  8 }
  9 struct Point{
 10     double x,y;
 11     Point(double x=0,double y=0):x(x),y(y) {}
 12 }
 13 
 14 typedef Point Vector;
 15 //向量的+-*/
 16 Vector operator + (Vector A,Vector B){ return Vector(A.x+B.x,A.y+B.y);}
 17 Vector operator -  (Point A,Point B){ return Vector(A.x-B.x,A.y-B.y);}
 18 Vector operator * (Vector A,double p){ return Vector(A.x*p,A.y*p);}
 19 Vector operator / (Vector A,double p){ return Vector(A.x/p,A.y/p);}
 20 
 21 //坐标的比较
 22 bool operator < (const Point& a,const Point& b){
 23     return (a.x<b.x || (a.x == b.x && a.y < b.y));
 24 }
 25 bool operator == (const Point& a,const Point& b){
 26     return (dcmp(a.x-b.x)==0 && dcmp(a.y-b.y)==0);
 27 }
 28 
 29 //点积
 30 double Dot(Vector A,Vector B){ return A.x*B.x + A.y*B.y; }
 31 //向量的长度
 32 double Length(Vector A){ return sqrt(Dot(A,A));}
 33 //两个向量的夹角
 34 double Angle(Vector A,Vector B){
 35     return acos(Dot(A,B)/(Length(A)*Length(B)));
 36 }
 37 //叉积
 38 double Cross(Vector A,Vector B){ return A.x*B.y - A.y*B.x; }
 39 //三点组成的三角形的有向面积的两倍
 40 double Area2(Point A,Point B,Point C){return Cross(B-A,C-A);}
 41 //向量旋转 rad:弧度
 42 Vector Rotate(Vector A,double rad){
 43     return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));
 44 }
 45 //获得向量的单位法线(左转90度)
 46 Vector Normal(Vector A){
 47     double L = Length(A);
 48     if(dcmp(L)==0) return Vector(0,0);
 49     retrun Vector(-A.y/L,A.x/L);
 50 }
 51 
 52 //得到两条直线的交点
 53 //Point P=P0+tv   可表示在直线上的所有点 v=(B-A) 直线上的两点
 54 //当表示为线段的时候0<=t<=1
 55 //当表示成射线的时候t>0
 56 //需要注意的是：两直线P+vt1,Q+wt2有唯一一个交点。Cross(v,w)!=0
 57 Point GetLineIntersection(Point P,Vector v,Point Q ,Vector w){
 58     Vector u = P-Q;
 59     double t = Cross(w,u) / Cross(v,w);
 60     return P+v*t;
 61 }
 62 
 63 //点到直线的距离
 64 double DistanceToLine(Point P,Point A,Point B){
 65     Vector ba = A-B, bc = C-B;
 66     return fabs(Cross(ba,bc)/Length(ba));            //不去绝对值的意思是有向距离
 67 }
 68 //点到线段的距离
 69 //情况一：点的投影在线段上-->点到直线的距离
 70 //情况二：点的投影不在线段上-->点到离它比较近的端点
 71 //用点积判断,用点积和叉积来计算
 72 double DistanceToSegment(Point P,Point A,Point B){
 73     if(A==B) return Length(P,A);
 74     Vector ab = B-A , ap = P-A , bp = P-B;
 75     if(dcmp(Dot(ab,ap))<0) return Length(ap);
 76     else if(dcmp(Dot(ab,bp))>0) return Length(bp);
 77     else return DistanceToLine(P,A,B);
 78 }
 79 
 80 //求点在直线上面的投影
 81 Point GetLineProjectection(Point P ,Point A,Point B){
 82     Vector v = B-A;
 83     return A+v*(Dot(v,P-A)/Dot(v,v));
 84 }
 85 
 86 //判断线段是否相交(不包括端点)
 87 bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){
 88     double c1 = Cross(a2-a1,b1-a1), c2 = Cross(a2-a1,b2-a1);
 89     double c3 = Cross(b2-b1,a1-b1), c4 = Cross(b2-b1,a2-b1);
 90     return (dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0);
 91 }
 92 //判断点是否在线段上(排除在端点上的情况)
 93 //保证p在向量a1a2的方向上 && p不在a1a2或者a2a1的延长线上
 94 bool OnSegment(Point p,Point a1,Point a2){
 95     return (dcmp(Cross(a1-p,a2-p))==0 && dcmp(Dos(a1-p,a2-p))<0);
 96 }
 97 
 98 //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)
 99 double ConvexPolygonArea(Point* p,int n){
100     double area=0;
101     for(int i=0;i < n-1;i++){
102         area += Cross(p[i]-p[0],p[i+1]-p[0]);
103     }
104     return area;
105 }
106 
107 //返回凸包的顶点个数,ch数组保存了凸包顶点
108 //输入的点不能有重复
109 //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
110 //需要的话用dcmp()提高精度
111 int ConvexHull(Point* p,int n,Point* ch){
112     sort(p,p+n);
113     int m=0;
114     for(int i=0;i<n;i++){
115         while(m > 1 && Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2]) <=0) m--;
116         ch[m++] = p[i];
117     }
118     int k=m;
119     for(int i=n-2;i>=0;i--){
120         while(m > k && Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2]) <=0) m--;
121         ch[m++] = p[i];
122     }
123     if(n > 1) m--;    //去掉起始点
124     return m;
125 }