计算几何全家桶
一、准备工作
#define LD double #define Vector Point #define Re register int const LP eps=1e-8;//据说:出题的大学生基本上用的这个值 inline int dcmp(LD a){ return a<eps?-1:(a>eps?1:0); } inline LD Abs(LD a){ return a*dcmp(a); }//绝对值 struct Point{ LD x,y;Point(LD X=0,LD Y=0){ x=X,y=Y; } inline void in(){ cin>>x>>y; } inline void out(){ printf("%.2lf %.2lf\n",x,y); } };
二、向量
1.模长
对于
inline LD len(Vector a){ return sqrt(Dot(a,a)); }
2.向量加减
inline Vector operator+(Vector a,Vector b){ return Vector(a.x+b.x,a.y+b.y);} inline Vector operator- (Vector a,Vector b){ return Vector(a.x -b.x,a.y -b.y); }
3.向量数乘
对于
除法也可以理解为数乘:
inline Vector operator* (Vector a,LD b){ return Vector(a.x*b,a.y*b); }
4.点积(内积)(数量积)
inline LD Dot(Vector a,Vector b){ return a.x*b.x+a.y*b.y; }
5.叉积(外积)(向量积)
inline LD Cro(Vector a,Vector b){ return a.x*b.y -a.y*b.x; }
三、点,向量的位置变换
1.点、向量的旋转
(1).对于点 P=(x,y) 或向量 ,将其顺时针旋转 θ 角度(点:关于原点,向量:关于起点):
inline Point turn_P(Point a,LD theta){ LD x=a.x*cos(theta)+a.y*sin(theta); LD y=-a.x*sin(theta)+a.y*cos(theta); return Point(x,y); }
(2).将点A(x,y)绕点B(x0,y0)顺时针旋转θ角度:
inline Point turn_PP(Point a,Point b,LD theta){ LD x=(a.x-b.x)*cos(theta)+(a.y-b.y)*sin(theta)+b.x; LD y=-(a.x-b.x)*sin(theta)+(a.y-b.y)*cos(theta)+b.y; return Point(x,y); }
四、图形与图形之间的关系
1.点与线段
(1).判断点P是否在线段AB上:
inline int pan_PL(Point p,Point a,Point b){ return !dcmp(Cro(p-a,b-a))&&dcmp(Dot(p-a,p-b))<=0; }
(2).点P到线段AB的距离:
inline bool operator==(Point a,Point b){ return !dcmp(a.x-b.x)&&!dcmp(a.y-b.y); }//两点重合则坐标相等 inline LD dis_PL(Point p,Point a,Point b){ if(a==b) return Len(p-a); Vector x=p-a,y=p-b,z=b-a; if(dcmp(Dot(x,z))<0) return Len(x);//P距离A更近 if(dcmp(Dot(y,z))<0) return Len(y);//P距离B更近 return Abs(Cro(x,z)/Len(z));//面积除以底边长 }
2.点与直线
(1).判断点P是否在直线AB上:
inline int pan_PL_(Point p,Point a,Point b){//判断点P是否在直线AB上 return !dcmp(Cro(p-a,b-a)); }
(2).点P到直线AB的垂足F:
inline Point FootPoint(Point p,Point a,Point b){//点P到直线AB的垂足 Vector x=p-a,y=p-b,z=b-a; LD len1=Dot(x,z)/Len(z),len2=-1.0*Dot(y,z)/Len(z);//分别计算AP,BP在AB,BA上的投影 return a+z*(len1/(len1+len2));//点A加上向量AF }
(3).点P关于直线AB的对称点:
inline Point Symmetry_PL(Point p,Point a,Point b){//点P关于直线AB的对称点 return p+(FootPoint(p,a,b)-p)*2;//将PF延迟一倍即可 }
3.线与线
(1).两直线AB,CD的交点Q:
inline Point cross_LL(Point a,Point b,Point c,Point d){//两直线AB,CD的交点 Vector x=b-a,y=d-c,z=a-c; return a+x*(Cro(y,z)/Cro(x,y)); }
(2).判断直线AB与线段CD是否相交:
inline int pan_cross_L_L(Point a,Point b,Point c,Point d){//判断直线AB与线段CD是否相交 return pan_PL(cross_LL(a,b,c,d),c,d);//直线AB与直线CD的交点在线段CD上 }
(3).判断两线段AB,CD是否相交:
inline int pan_cross_LL(Point a,Point b,Point c,Point d){//判断两线段AB,CD是否相交 LD c1=Cro(b-a,c-a),c2=Cro(b-a,d-a); LD d1=Cro(d-c,a-c),d2=Cro(d-c,b-c); return dcmp(c1)*dcmp(c2)<0&&dcmp(d1)*dcmp(d2)<0;//分别在两侧 }
4.点与多边形
(1).判断点A是否在任意多边形Poly以内(射线法):
inline int PIP(Point *P,Re n,Point a){//【射线法】判断点A是否在任意多边形Poly以内 Re cnt=0;LD tmp; for(Re i=1;i<=n;++i){ Re j=i<n?i+1:1; if(pan_PL(a,P[i],P[j])) return 2;//点在多边形上 if(a.y>=min(P[i].y,P[j].y)&&a.y<max(P[i].y,P[j].y))//纵坐标在该线段两端点之间 tmp=P[i].x+(a.y-P[i].y)/(P[j].y-P[i].y)*(P[j].x-P[i].x),cnt+=dcmp(tmp-a.x)>0;//交点在A右方 } return cnt&1;//穿过奇数则在多边形以内 }
5.线与多边形
(1).判断线段AB是否任意多边形Poly以内:不相交且两端点A,B均在多边形以内。
(2).判断线段AB是否在凸多边形Poly以内:两端点A,B均在多边形以内。
6.多边形与多边形
(1).判断任意两个多边形是否相离:属于不同多边形的任意两边都不相交 且 一个多边形上的任意点都不被另一个多边形所包含。
inline int judge_PP(Point *A,Re n,Point *B,Re m){//判断多边形A与多边形B是否相离 for(Re i1=1;i1<=n;++i1){ Re j1=i1<n?i1+1:1; for(Re i2=1;i2<=m;++i2){ Re j2=i2<m?i2+1:1; if(pan_cross_LL(A[i1],A[j1],B[i2],B[j2])) return 0;//两线段相交 if(PIP(B,m,A[i1])||PIP(A,n,B[i2])) return 0;//点包含在内 } } return 1; }
五、图形面积
1.任意多边形面积
inline LD PolyArea(Point *P,Re n){ LD S=0; for(Re i=1;i<=n;++i) S+=Cro(P[i],P[i<n?i+1:1]); return S/2.0; }
2.圆的面积并
自适应辛普森法乱搞
3.三角形面积并
自适应辛普森法乱搞
或者扫描线?
六、凸包
1.求凸包
(1).Graham 扫描法
inline bool cmp1(Vector a,Vector b){ return a.x==b.x?a.y<b.y:a.x<b.x; }//按坐标排序 inline int ConvexHull(Point *P,Re n,Point *cp){//【水平序Graham扫描法(Andrew算法)】求凸包 sort(P+1,P+n+1,cmp1); Re t=0; for(Re i=1;i<=n;++i){//下凸包 while(t>1&&dcmp(Cro(cp[t]-cp[t-1],P[i]-cp[t-1]))<=0) --t; cp[++t]=P[i]; } Re St=t; for(Re i=n-1;i>=1;--i){//上凸包 while(t>St&&dcmp(Cro(cp[t]-cp[t-1],P[i]-cp[t-1]))<=0) --t; cp[++t]=P[i]; } return --t;//要减一 }
2.旋转卡壳
Rd Ans=Len(cp[2]-cp[1]); for(Re i=1,j=3;i<=cnt;++i){ while(dcmp(Cro(cp[i+1]-cp[i],cp[j]-cp[i])-Cro(cp[i+1]-cp[i],cp[j+1]-cp[i]))<0) j=j<cnt?j+1:1; Ans=max(Ans,max(Len(cp[j]-cp[i]),Len(cp[j]-cp[i+1]))); }
3.半平面交
struct Line{ Point a,b;LD k;Line(Point A=Point(0,0),Point B=Point(0,0)){ a=A,b=B,k=atan2(b.y-a.y,b.x-a.x); } inline bool operator<(const Line &O) const{ return dcmp(k-O.k)?dcmp(k-O.k)<0:judge(O.a,O.b,a); }//如果角度相等则取左边的 }L[N],Q[N]; inline Point cross(Line L1,Line L2){ return cross_LL(L1.a,L1.b,L2.a,L2.b); }//获取直线L1,L2的交点 inline int judge(Line L,Point a){ return dcmp(Cro(a-L.a,L.b-L.a))>0; }//判断点a是否在直线L的右边 inline int halfcut(Line *L,Re n,Point *P){//【半平面交】 sort(L+1,L+n+1); Re m=n;n=0; for(Re i=1;i<=m;++i) if(i==1||dcmp(L[i].k-L[i-1].k)) L[++n]=L[i]; Re h=1,t=0; for(Re i=1;i<=n;++i){ while(h<t&&judge(L[i],cross(Q[t],Q[t-1]))) --t;//当队尾两个直线交点不是在直线L[i]上或者左边时就出队 while(h<t&&judge(L[i],cross(q[h],Q[h+1]))) ++h;//当队头两个直线交点不是在直线L[i]上或者左边时就出队 Q[++t]=L[i]; } while(h<t&&judge(Q[h],cross(Q[t],Q[t-1]))) --t; while(h<t&&judge(Q[t],cross(Q[h],Q[h+1]))) ++h; n=0; for(Re i=h;i<=t;++i) P[++n]=cross(Q[i],Q[i<t?i+1:h]); return n; }
4.闵科夫斯基和
Vector V1[N],V2[N]; inline int Mincowki(Point *P1,Re n,Point *P2,Re m,Vector *V){//【闵科夫斯基和】求两个凸包{P1},{P2}的向量集合{V}={P1+P2}构成的凸包 for(Re i=1;i<=n;++i) V1[i]=P1[i<n?i+1:1]-P1[i]; for(Re i=1;i<=m;++i) V2[i]=P2[i<m?i+1:1]-p2[i]; Re t=0,i=1,j=1;V[++t]=P1[1]+P2[1]; while(i<=n&&j<=m) ++t,V[t]=V[t-1]+(dcmp(Cro(V1[i],V2[j]))>0?V1[i++]:V2[j++]); while(i<=n) ++t,V[t]=V[t-1]+V1[i++]; while(j<=m) ++t,V[t]=V[t-1]+V2[j++]; return t; }
5.动态凸包
struct ConvexHull{ int op;set<Point>s; inline int PIP(Point P){ IT it=s.lower_bound(Point(P.x,-inf));//找到第一个横坐标大于p的点 if(it==s.end()) return 0; if(it->x==P.x) return (P.y-it->y)*op<=0;//比较纵坐标大小 if(it==s.begin()) return 0; IT j=it,k=it;--j;return Cro(P-*j,*k-*j)*op>=0;//看叉积 } inline int judge(IT it){ IT j=it,k=it; if(j==s.begin()) return 0;--j; if(++k==s.end()) return 0; return Cro(*it-*j,*k-*j)*op>=0;//看叉积 } inline void insert(Point p){ if(PIP(P)) return;//如果点p已经在凸壳上或凸包里就不插入了 IT tmp=s.lower_bound(Point(P.x,-inf));if(tmp!=s.end()&&tmp->==P.x) s.erase(tmp); s.insert(P);IT it=s.find(P),p=it; if(p!=s.begin()){ --p;while(judge(p)){ IT tmp=p--;s.erase(tmp); } } if((p=++it)!=s.end()) while(judge(p)){ IT tmp=p++;s.erase(tmp); } } }up,down; int x,y,T,op; int main(){ in(T),up.op=1,down.op=-1; while(T--){ in(op).P.In(); if(op<2) up.insert(P),down.insert(P);//插入点P else puts((up.PIP(P)&&down.PIP(P))?"YES":"NO");//判断点P是否在凸包内 } }
七、圆
1.三点确定一圆
(1).暴力解方程:
设 ,圆心为 O,半径为 r,带入三点
A(x1,y1),B(x2,y2),C(x3,y3),解得:
#define S(a) ((a)*(a)) struct Circle{ Point O;LD r;Circle(Point P,LD R=0;){ O=P,r=R; } }; inline Circle getCircle(Point A,Point B,Point C){//【三点确定一圆】暴力解方程 LD x1=A.x,y1=A.y,x2=B.x,y2=B.y,x3=C.x,y3=C.y; LD D=((S(x2)+S(y2)-S(x3)-S(y3))*(y1-y2)-(S(x1)+S(y1)-S(x2)-S(y2))*(y2-y3))/((x1-x2)*(y2-y3)-(x2-x3)*(y1-y2)); LD E=(S(x1)+S(y1)-S(x2)-S(y2)+D*(x1-x2))/(y2-y1); LD F=-(S(x1)+S(y1)+D*x1+E*y1); return Circle(Point(-D/2.0,-E/2.0),sqrt((S(D)+S(E)-4.0*F)/4.0)); }
(2).向量求三角形垂心:
inline Circle getcircle(Point A,Point B,Point C){//【三点确定一圆】向量垂心法 Point P1=(A+B)*0.5+P2=(A+C)*0.5; Point O=cross_LL(P1,P1+Normal(B-A),P2,P2+Normal(C-A)); return Circle(O,Len(A-O)); }
2.最小覆盖圆
【定理】如果点 p 不在集合 {S} 的最小覆盖圆里,则 p 一定在 {S}∪p 的最小覆盖圆上
inline int PIC(Circle C,Point a){ return dcmp(Len(a-C.O)-C.r)<=0; } //判断点A是否在圆C内 inline void Random(Point *P,Re n){ for(Re i=1;i<=n;++i) swap(P[i],P[rand()%n+1]); } //随机一个排列 inline Circle Min_Circle(Point *P,Re n){ //【求点集P的最小覆盖圆】 Random(P,n);Circle C=Circle(P[1],0); for(Re i=2;i<=n;++i) if(!PIC(C,P[i])){ C=Circle(P[i],0); for(Re j=1;j<i;++j) if(!PIC(C,P[j])){ C.O=(P[i]+P[j])*0.5,C.r=Len(P[j]-C.O); for(Re k=1;k<j;++k) if(!PIC(C,P[k])) C=getcircle(P[i],P[j],P[k]); } } return C; }
3.三角剖分
inline LD calc(Point A,Point B,Point O,LD R){//【三角剖分】 if(A==O||B==O) return 0; Re op=dcmp(Cro(A-O,B-O))>0?1:-1;LD ans=0; Vector x=A-O,y=B-O; Re flag1=dcmp(Len(x)-R)>0,flag2=dcmp(Len(y)-R)>0; if(!flag&&!flag2) ans=Abs(Cro(A-O,B-O))/2.0;//两个点都在里面 else if(flag1&&flag2){ if(dcmp(dis_PL(O,A,B)-R)>=0) ans=R*R*Angle(x,y)/2.0;//完全包含了圆弧 else{//分三段处理,△+圆弧+△ if(dcmp(Cro(A-O,B-O))>0) swap(A,B);//把A换到左边 Point F=FootPoint(O,A,B);LD lenx=Len(F-O),len=sqrt(R*R-lenx*lenx); Vector z=turn_P(F-O,Pi/2.0)*(len/lenx);Point B_=F+z,A_=F-z; ans=R*R(Angle(A-O,A_-O)+Angle(B-O,B_-O))/2.0+Cro(B_-O,A_-O)/2.0; } } else{//一个点在里面,一个点在外面 if(flag1) swap(A,B);//使A为里面的点,B为外面的点 Point F=FootPoint(O,A,B);LD lenx=Len(F-O),len=sqrt(R*R-lenx*lenx); Vector z=turn_P(F-O,Pi/2.0)*(len/lenx);Point C=dcmp(Cro(A-O,B-O))>0?F-z:F+z; ans=Abs(Cro(A-O,C-O))/2.0+R*R*Angle(C-O,B-O)/2.0; } return ans*op; }
计算几何全家桶Code
#include<algorithm> #include<cstdio> #include<cmath> /*一:【准备工作】*/ #define LD double #define LL long long #define Re register int #define Vector Point using namespace std; const int N=262144+3; const LD eps=1e-8,Pi=acos(-1.0); inline int dcmp(LD a){return a<-eps?-1:(a>eps?1:0);}//处理精度 inline LD Abs(LD a){return a*dcmp(a);}//取绝对值 struct Point{ LD x,y;Point(LD X=0,LD Y=0){x=X,y=Y;} inline void in(){scanf("%lf%lf",&x,&y);} inline void out(){printf("%.2lf %.2lf\n",x,y);} }; /*二:【向量】*/ inline LD Dot(Vector a,Vector b){return a.x*b.x+a.y*b.y;}//【点积】 inline LD Cro(Vector a,Vector b){return a.x*b.y-a.y*b.x;}//【叉积】 inline LD Len(Vector a){return sqrt(Dot(a,a));}//【模长】 inline LD Angle(Vector a,Vector b){return acos(Dot(a,b)/Len(a)/Len(b));}//【两向量夹角】 inline Vector Normal(Vector a){return Vector(-a.y,a.x);}//【法向量】 inline Vector operator+(Vector a,Vector b){return Vector(a.x+b.x,a.y+b.y);} inline Vector operator-(Vector a,Vector b){return Vector(a.x-b.x,a.y-b.y);} inline Vector operator*(Vector a,LD b){return Vector(a.x*b,a.y*b);} inline bool operator==(Point a,Point b){return !dcmp(a.x-b.x)&&!dcmp(a.y-b.y);}//两点坐标重合则相等 /*三:【点、向量的位置变换】*/ /*1.【点、向量的旋转】*/ inline Point turn_P(Point a,LD theta){//【点A\向量A顺时针旋转theta(弧度)】 LD x=a.x*cos(theta)+a.y*sin(theta); LD y=-a.x*sin(theta)+a.y*cos(theta); return Point(x,y); } inline Point turn_PP(Point a,Point b,LD theta){//【将点A绕点B顺时针旋转theta(弧度)】 LD x=(a.x-b.x)*cos(theta)+(a.y-b.y)*sin(theta)+b.x; LD y=-(a.x-b.x)*sin(theta)+(a.y-b.y)*cos(theta)+b.y; return Point(x,y); } /*四:【图形与图形之间的关系】*/ /*1.【点与线段】*/ inline int pan_PL(Point p,Point a,Point b){//【判断点P是否在线段AB上】 return !dcmp(Cro(p-a,b-a))&&dcmp(Dot(p-a,p-b))<=0;//做法一 // return !dcmp(Cro(p-a,b-a))&&dcmp(min(a.x,b.x)-p.x)<=0&&dcmp(p.x-max(a.x,b.x))<=0&&dcmp(min(a.y,b.y)-p.y)<=0&&dcmp(p.y-max(a.y,b.y))<=0;//做法二 //PA,AB共线且P在AB之间(其实也可以用len(p-a)+len(p-b)==len(a-b)判断,但是精度损失较大) } inline LD dis_PL(Point p,Point a,Point b){//【点P到线段AB距离】 if(a==b)return Len(p-a);//AB重合 Vector x=p-a,y=p-b,z=b-a; if(dcmp(Dot(x,z))<0)return Len(x);//P距离A更近 if(dcmp(Dot(y,z))>0)return Len(y);//P距离B更近 return Abs(Cro(x,z)/Len(z));//面积除以底边长 } /*2.【点与直线】*/ inline int pan_PL_(Point p,Point a,Point b){//【判断点P是否在直线AB上】 return !dcmp(Cro(p-a,b-a));//PA,AB共线 } inline Point FootPoint(Point p,Point a,Point b){//【点P到直线AB的垂足】 Vector x=p-a,y=p-b,z=b-a; LD len1=Dot(x,z)/Len(z),len2=-1.0*Dot(y,z)/Len(z);//分别计算AP,BP在AB,BA上的投影 return a+z*(len1/(len1+len2));//点A加上向量AF } inline Point Symmetry_PL(Point p,Point a,Point b){//【点P关于直线AB的对称点】 return p+(FootPoint(p,a,b)-p)*2;//将PF延长一倍即可 } /*3.【线与线】*/ inline Point cross_LL(Point a,Point b,Point c,Point d){//【两直线AB,CD的交点】 Vector x=b-a,y=d-c,z=a-c; return a+x*(Cro(y,z)/Cro(x,y));//点A加上向量AF } inline int pan_cross_L_L(Point a,Point b,Point c,Point d){//【判断直线AB与线段CD是否相交】 return pan_PL(cross_LL(a,b,c,d),c,d);//直线AB与直线CD的交点在线段CD上 } inline int pan_cross_LL(Point a,Point b,Point c,Point d){//【判断两线段AB,CD是否相交】 LD c1=Cro(b-a,c-a),c2=Cro(b-a,d-a); LD d1=Cro(d-c,a-c),d2=Cro(d-c,b-c); return dcmp(c1)*dcmp(c2)<0&&dcmp(d1)*dcmp(d2)<0;//分别在两侧 } /*4.【点与多边形】*/ inline int PIP(Point *P,Re n,Point a){//【射线法】判断点A是否在任意多边形Poly以内 Re cnt=0;LD tmp; for(Re i=1;i<=n;++i){ Re j=i<n?i+1:1; if(pan_PL(a,P[i],P[j]))return 2;//点在多边形上 if(a.y>=min(P[i].y,P[j].y)&&a.y<max(P[i].y,P[j].y))//纵坐标在该线段两端点之间 tmp=P[i].x+(a.y-P[i].y)/(P[j].y-P[i].y)*(P[j].x-P[i].x),cnt+=dcmp(tmp-a.x)>0;//交点在A右方 } return cnt&1;//穿过奇数次则在多边形以内 } inline int judge(Point a,Point L,Point R){//判断AL是否在AR右边 return dcmp(Cro(L-a,R-a))>0;//必须严格以内 } inline int PIP_(Point *P,Re n,Point a){//【二分法】判断点A是否在凸多边形Poly以内 //点按逆时针给出 if(judge(P[1],a,P[2])||judge(P[1],P[n],a))return 0;//在P[1_2]或P[1_n]外 if(pan_PL(a,P[1],P[2])||pan_PL(a,P[1],P[n]))return 2;//在P[1_2]或P[1_n]上 Re l=2,r=n-1; while(l<r){//二分找到一个位置pos使得P[1]_A在P[1_pos],P[1_(pos+1)]之间 Re mid=l+r+1>>1; if(judge(P[1],P[mid],a))l=mid; else r=mid-1; } if(judge(P[l],a,P[l+1]))return 0;//在P[pos_(pos+1)]外 if(pan_PL(a,P[l],P[l+1]))return 2;//在P[pos_(pos+1)]上 return 1; } /*5.【线与多边形】*/ /*6.【多边形与多边形】*/ inline int judge_PP(Point *A,Re n,Point *B,Re m){//【判断多边形A与多边形B是否相离】 for(Re i1=1;i1<=n;++i1){ Re j1=i1<n?i1+1:1; for(Re i2=1;i2<=m;++i2){ Re j2=i2<m?i2+1:1; if(pan_cross_LL(A[i1],A[j1],B[i2],B[j2]))return 0;//两线段相交 if(PIP(B,m,A[i1])||PIP(A,n,B[i2]))return 0;//点包含在内 } } return 1; } /*五:【图形面积】*/ /*1.【任意多边形面积】*/ inline LD PolyArea(Point *P,Re n){//【任意多边形P的面积】 LD S=0; for(Re i=1;i<=n;++i)S+=Cro(P[i],P[i<n?i+1:1]); return S/2.0; } /*2.【圆的面积并】*/ /*3.【三角形面积并】*/ /*六:【凸包】*/ /*1.【求凸包】*/ inline bool cmp1(Vector a,Vector b){return a.x==b.x?a.y<b.y:a.x<b.x;};//按坐标排序 inline int ConvexHull(Point *P,Re n,Point *cp){//【水平序Graham扫描法(Andrew算法)】求凸包 sort(P+1,P+n+1,cmp1); Re t=0; for(Re i=1;i<=n;++i){//下凸包 while(t>1&&dcmp(Cro(cp[t]-cp[t-1],P[i]-cp[t-1]))<=0)--t; cp[++t]=P[i]; } Re St=t; for(Re i=n-1;i>=1;--i){//上凸包 while(t>St&&dcmp(Cro(cp[t]-cp[t-1],P[i]-cp[t-1]))<=0)--t; cp[++t]=P[i]; } return --t;//要减一 } /*2.【旋转卡壳】*/ /*3.【半平面交】*/ struct Line{ Point a,b;LD k;Line(Point A=Point(0,0),Point B=Point(0,0)){a=A,b=B,k=atan2(b.y-a.y,b.x-a.x);} inline bool operator<(const Line &O)const{return dcmp(k-O.k)?dcmp(k-O.k)<0:judge(O.a,O.b,a);}//如果角度相等则取左边的 }L[N],Q[N]; inline Point cross(Line L1,Line L2){return cross_LL(L1.a,L1.b,L2.a,L2.b);}//获取直线L1,L2的交点 inline int judge(Line L,Point a){return dcmp(Cro(a-L.a,L.b-L.a))>0;}//判断点a是否在直线L的右边 inline int halfcut(Line *L,Re n,Point *P){//【半平面交】 sort(L+1,L+n+1);Re m=n;n=0; for(Re i=1;i<=m;++i)if(i==1||dcmp(L[i].k-L[i-1].k))L[++n]=L[i]; Re h=1,t=0; for(Re i=1;i<=n;++i){ while(h<t&&judge(L[i],cross(Q[t],Q[t-1])))--t;//当队尾两个直线交点不是在直线L[i]上或者左边时就出队 while(h<t&&judge(L[i],cross(Q[h],Q[h+1])))++h;//当队头两个直线交点不是在直线L[i]上或者左边时就出队 Q[++t]=L[i]; } while(h<t&&judge(Q[h],cross(Q[t],Q[t-1])))--t; while(h<t&&judge(Q[t],cross(Q[h],Q[h+1])))++h; n=0; for(Re i=h;i<=t;++i)P[++n]=cross(Q[i],Q[i<t?i+1:h]); return n; } /*4.【闵可夫斯基和】*/ Vector V1[N],V2[N]; inline int Mincowski(Point *P1,Re n,Point *P2,Re m,Vector *V){//【闵可夫斯基和】求两个凸包{P1},{P2}的向量集合{V}={P1+P2}构成的凸包 for(Re i=1;i<=n;++i)V1[i]=P1[i<n?i+1:1]-P1[i]; for(Re i=1;i<=m;++i)V2[i]=P2[i<m?i+1:1]-P2[i]; Re t=0,i=1,j=1;V[++t]=P1[1]+P2[1]; while(i<=n&&j<=m)++t,V[t]=V[t-1]+(dcmp(Cro(V1[i],V2[j]))>0?V1[i++]:V2[j++]); while(i<=n)++t,V[t]=V[t-1]+V1[i++]; while(j<=m)++t,V[t]=V[t-1]+V2[j++]; return t; } /*5.【动态凸包】*/ /*七:【圆】*/ /*1.【三点确定一圆】*/ #define S(a) ((a)*(a)) struct Circle{Point O;LD r;Circle(Point P,LD R=0){O=P,r=R;}}; inline Circle getCircle(Point A,Point B,Point C){//【三点确定一圆】暴力解方程 LD x1=A.x,y1=A.y,x2=B.x,y2=B.y,x3=C.x,y3=C.y; LD D=((S(x2)+S(y2)-S(x3)-S(y3))*(y1-y2)-(S(x1)+S(y1)-S(x2)-S(y2))*(y2-y3))/((x1-x2)*(y2-y3)-(x2-x3)*(y1-y2)); LD E=(S(x1)+S(y1)-S(x2)-S(y2)+D*(x1-x2))/(y2-y1); LD F=-(S(x1)+S(y1)+D*x1+E*y1); return Circle(Point(-D/2.0,-E/2.0),sqrt((S(D)+S(E)-4.0*F)/4.0)); } inline Circle getcircle(Point A,Point B,Point C){//【三点确定一圆】向量垂心法 Point P1=(A+B)*0.5,P2=(A+C)*0.5; Point O=cross_LL(P1,P1+Normal(B-A),P2,P2+Normal(C-A)); return Circle(O,Len(A-O)); } /*2.【最小覆盖圆】*/ inline int PIC(Circle C,Point a){return dcmp(Len(a-C.O)-C.r)<=0;}//判断点A是否在圆C内 inline void Random(Point *P,Re n){for(Re i=1;i<=n;++i)swap(P[i],P[rand()%n+1]);}//随机一个排列 inline Circle Min_Circle(Point *P,Re n){//【求点集P的最小覆盖圆】 // random_shuffle(P+1,P+n+1); Random(P,n);Circle C=Circle(P[1],0); for(Re i=2;i<=n;++i)if(!PIC(C,P[i])){ C=Circle(P[i],0); for(Re j=1;j<i;++j)if(!PIC(C,P[j])){ C.O=(P[i]+P[j])*0.5,C.r=Len(P[j]-C.O); for(Re k=1;k<j;++k)if(!PIC(C,P[k]))C=getcircle(P[i],P[j],P[k]); } } return C; } /*3.【三角剖分】*/ inline LD calc(Point A,Point B,Point O,LD R){//【三角剖分】 if(A==O||B==O)return 0; Re op=dcmp(Cro(A-O,B-O))>0?1:-1;LD ans=0; Vector x=A-O,y=B-O; Re flag1=dcmp(Len(x)-R)>0,flag2=dcmp(Len(y)-R)>0; if(!flag1&&!flag2)ans=Abs(Cro(A-O,B-O))/2.0;//两个点都在里面 else if(flag1&&flag2){//两个点都在外面 if(dcmp(dis_PL(O,A,B)-R)>=0)ans=R*R*Angle(x,y)/2.0;//完全包含了圆弧 else{//分三段处理 △+圆弧+△ if(dcmp(Cro(A-O,B-O))>0)swap(A,B);//把A换到左边 Point F=FootPoint(O,A,B);LD lenx=Len(F-O),len=sqrt(R*R-lenx*lenx); Vector z=turn_P(F-O,Pi/2.0)*(len/lenx);Point B_=F+z,A_=F-z; ans=R*R*(Angle(A-O,A_-O)+Angle(B-O,B_-O))/2.0+Cro(B_-O,A_-O)/2.0; } } else{//一个点在里面,一个点在外面 if(flag1)swap(A,B);//使A为里面的点,B为外面的点 Point F=FootPoint(O,A,B);LD lenx=Len(F-O),len=sqrt(R*R-lenx*lenx); Vector z=turn_P(F-O,Pi/2.0)*(len/lenx);Point C=dcmp(Cro(A-O,B-O))>0?F-z:F+z; ans=Abs(Cro(A-O,C-O))/2.0+R*R*Angle(C-O,B-O)/2.0; } return ans*op; } int main(){}