计算集合板子

发表于 2022-07-04 更新于 2026-08-08 分类于 algorithm 本文字数： 4.8k 阅读时长 ≈ 18 分钟
Wordpress的编辑器对大文本的支持也太差了，，，放这备份
jls的计算几何板子
#define mp make_pair
#define fi first
#define se second
#define pb push_back
typedef double db;
const db eps=1e-6;
const db pi=acos(-1);
int sign(db k){
    
    if (k>eps) return 1; else if (k<-eps) return -1; return 0;
}
int cmp(db k1,db k2){
    return sign(k1-k2);}
int inmid(db k1,db k2,db k3){
    return sign(k1-k3)*sign(k2-k3)<=0;}// k3 在 [k1,k2] 内 
struct point{
    
    db x,y;
    point operator + (const point &k1) const{
    return (point){
    k1.x+x,k1.y+y};}
    point operator - (const point &k1) const{
    return (point){
    x-k1.x,y-k1.y};}
    point operator * (db k1) const{
    return (point){
    x*k1,y*k1};}
    point operator / (db k1) const{
    return (point){
    x/k1,y/k1};}
    int operator == (const point &k1) const{
    return cmp(x,k1.x)==0&&cmp(y,k1.y)==0;}
    // 逆时针旋转 
    point turn(db k1){
    return (point){
    x*cos(k1)-y*sin(k1),x*sin(k1)+y*cos(k1)};}
    point turn90(){
    return (point){
    -y,x};}
    bool operator < (const point k1) const{
    
        int a=cmp(x,k1.x);
        if (a==-1) return 1; else if (a==1) return 0; else return cmp(y,k1.y)==-1;
    }
    db abs(){
    return sqrt(x*x+y*y);}
    db abs2(){
    return x*x+y*y;}
    db dis(point k1){
    return ((*this)-k1).abs();}
    point unit(){
    db w=abs(); return (point){
    x/w,y/w};}
    void scan(){
    double k1,k2; scanf("%lf%lf",&k1,&k2); x=k1; y=k2;}
    void print(){
    printf("%.11lf %.11lf\n",x,y);}
    db getw(){
    return atan2(y,x);} 
    point getdel(){
    if (sign(x)==-1||(sign(x)==0&&sign(y)==-1)) return (*this)*(-1); else return (*this);}
	int getP() const{
    return sign(y)==1||(sign(y)==0&&sign(x)==-1);}
};
int inmid(point k1,point k2,point k3){
    return inmid(k1.x,k2.x,k3.x)&&inmid(k1.y,k2.y,k3.y);}
db cross(point k1,point k2){
    return k1.x*k2.y-k1.y*k2.x;}
db dot(point k1,point k2){
    return k1.x*k2.x+k1.y*k2.y;}
db rad(point k1,point k2){
    return atan2(cross(k1,k2),dot(k1,k2));}
// -pi -> pi
int compareangle (point k1,point k2){
    
    return k1.getP()<k2.getP()||(k1.getP()==k2.getP()&&sign(cross(k1,k2))>0);
}
point proj(point k1,point k2,point q){
     // q 到直线 k1,k2 的投影 
    point k=k2-k1; return k1+k*(dot(q-k1,k)/k.abs2());
}
point reflect(point k1,point k2,point q){
    return proj(k1,k2,q)*2-q;}
int clockwise(point k1,point k2,point k3){
    // k1 k2 k3 逆时针 1 顺时针 -1 否则 0  
    return sign(cross(k2-k1,k3-k1));
}
int checkLL(point k1,point k2,point k3,point k4){
    // 求直线 (L) 线段 (S)k1,k2 和 k3,k4 的交点 
    return cmp(cross(k3-k1,k4-k1),cross(k3-k2,k4-k2))!=0;
}
point getLL(point k1,point k2,point k3,point k4){
    
    db w1=cross(k1-k3,k4-k3),w2=cross(k4-k3,k2-k3); return (k1*w2+k2*w1)/(w1+w2);
}
int intersect(db l1,db r1,db l2,db r2){
    
    if (l1>r1) swap(l1,r1); if (l2>r2) swap(l2,r2); return cmp(r1,l2)!=-1&&cmp(r2,l1)!=-1;
}
int checkSS(point k1,point k2,point k3,point k4){
    
    return intersect(k1.x,k2.x,k3.x,k4.x)&&intersect(k1.y,k2.y,k3.y,k4.y)&&
    sign(cross(k3-k1,k4-k1))*sign(cross(k3-k2,k4-k2))<=0&&
    sign(cross(k1-k3,k2-k3))*sign(cross(k1-k4,k2-k4))<=0;
}
db disSP(point k1,point k2,point q){
    
    point k3=proj(k1,k2,q);
    if (inmid(k1,k2,k3)) return q.dis(k3); else return min(q.dis(k1),q.dis(k2));
}
db disSS(point k1,point k2,point k3,point k4){
    
    if (checkSS(k1,k2,k3,k4)) return 0;
    else return min(min(disSP(k1,k2,k3),disSP(k1,k2,k4)),min(disSP(k3,k4,k1),disSP(k3,k4,k2)));
}
int onS(point k1,point k2,point q){
    return inmid(k1,k2,q)&&sign(cross(k1-q,k2-k1))==0;}
struct circle{
    
    point o; db r;
    void scan(){
    o.scan(); scanf("%lf",&r);}
    int inside(point k){
    return cmp(r,o.dis(k));}
};
struct line{
    
    // p[0]->p[1]
    point p[2];
    line(point k1,point k2){
    p[0]=k1; p[1]=k2;}
    point& operator [] (int k){
    return p[k];}
    int include(point k){
    return sign(cross(p[1]-p[0],k-p[0]))>0;}
    point dir(){
    return p[1]-p[0];}
    line push(){
     // 向外 ( 左手边 ) 平移 eps 
        const db eps = 1e-6;
        point delta=(p[1]-p[0]).turn90().unit()*eps;
        return {
    p[0]-delta,p[1]-delta};
    }
};
point getLL(line k1,line k2){
    return getLL(k1[0],k1[1],k2[0],k2[1]);}
int parallel(line k1,line k2){
    return sign(cross(k1.dir(),k2.dir()))==0;}
int sameDir(line k1,line k2){
    return parallel(k1,k2)&&sign(dot(k1.dir(),k2.dir()))==1;}
int operator < (line k1,line k2){
    
    if (sameDir(k1,k2)) return k2.include(k1[0]); 
    return compareangle(k1.dir(),k2.dir());
}
int checkpos(line k1,line k2,line k3){
    return k3.include(getLL(k1,k2));}
vector<line> getHL(vector<line> &L){
     // 求半平面交 , 半平面是逆时针方向 , 输出按照逆时针
    sort(L.begin(),L.end()); deque<line> q;
    for (int i=0;i<(int)L.size();i++){
    
        if (i&&sameDir(L[i],L[i-1])) continue;
        while (q.size()>1&&!checkpos(q[q.size()-2],q[q.size()-1],L[i])) q.pop_back();
        while (q.size()>1&&!checkpos(q[1],q[0],L[i])) q.pop_front();
        q.push_back(L[i]);
    }
    while (q.size()>2&&!checkpos(q[q.size()-2],q[q.size()-1],q[0])) q.pop_back();
    while (q.size()>2&&!checkpos(q[1],q[0],q[q.size()-1])) q.pop_front();
    vector<line>ans; for (int i=0;i<q.size();i++) ans.push_back(q[i]);
    return ans;
}
db closepoint(vector<point>&A,int l,int r){
     // 最近点对 , 先要按照 x 坐标排序 
    if (r-l<=5){
    
        db ans=1e20;
        for (int i=l;i<=r;i++) for (int j=i+1;j<=r;j++) ans=min(ans,A[i].dis(A[j]));
        return ans;
    }
    int mid=l+r>>1; db ans=min(closepoint(A,l,mid),closepoint(A,mid+1,r));
    vector<point>B; for (int i=l;i<=r;i++) if (abs(A[i].x-A[mid].x)<=ans) B.push_back(A[i]);
    sort(B.begin(),B.end(),[](point k1,point k2){
    return k1.y<k2.y;});
    for (int i=0;i<B.size();i++) for (int j=i+1;j<B.size()&&B[j].y-B[i].y<ans;j++) ans=min(ans,B[i].dis(B[j]));
    return ans;
}
int checkposCC(circle k1,circle k2){
    // 返回两个圆的公切线数量
    if (cmp(k1.r,k2.r)==-1) swap(k1,k2);
    db dis=k1.o.dis(k2.o);  int w1=cmp(dis,k1.r+k2.r),w2=cmp(dis,k1.r-k2.r);
    if (w1>0) return 4; else if (w1==0) return 3; else if (w2>0) return 2; 
    else if (w2==0) return 1; else return 0;
}
vector<point> getCL(circle k1,point k2,point k3){
     // 沿着 k2->k3 方向给出 , 相切给出两个 
    point k=proj(k2,k3,k1.o); db d=k1.r*k1.r-(k-k1.o).abs2();
    if (sign(d)==-1) return {
    };
    point del=(k3-k2).unit()*sqrt(max((db)0.0,d)); return {
    k-del,k+del};
}
vector<point> getCC(circle k1,circle k2){
    // 沿圆 k1 逆时针给出 , 相切给出两个 
    int pd=checkposCC(k1,k2); if (pd==0||pd==4) return {
    };
    db a=(k2.o-k1.o).abs2(),cosA=(k1.r*k1.r+a-k2.r*k2.r)/(2*k1.r*sqrt(max(a,(db)0.0)));
    db b=k1.r*cosA,c=sqrt(max((db)0.0,k1.r*k1.r-b*b));
    point k=(k2.o-k1.o).unit(),m=k1.o+k*b,del=k.turn90()*c;
    return {
    m-del,m+del};
} 
vector<point> TangentCP(circle k1,point k2){
    // 沿圆 k1 逆时针给出 
    db a=(k2-k1.o).abs(),b=k1.r*k1.r/a,c=sqrt(max((db)0.0,k1.r*k1.r-b*b));
    point k=(k2-k1.o).unit(),m=k1.o+k*b,del=k.turn90()*c;
    return {
    m-del,m+del};
} 
vector<line> TangentoutCC(circle k1,circle k2){
    
    int pd=checkposCC(k1,k2); if (pd==0) return {
    }; 
    if (pd==1){
    point k=getCC(k1,k2)[0]; return {
    (line){
    k,k}};}
    if (cmp(k1.r,k2.r)==0){
    
        point del=(k2.o-k1.o).unit().turn90().getdel();
        return {
    (line){
    k1.o-del*k1.r,k2.o-del*k2.r},(line){
    k1.o+del*k1.r,k2.o+del*k2.r}};
    } else {
    
        point p=(k2.o*k1.r-k1.o*k2.r)/(k1.r-k2.r);
        vector<point>A=TangentCP(k1,p),B=TangentCP(k2,p);
        vector<line>ans; for (int i=0;i<A.size();i++) ans.push_back((line){
    A[i],B[i]}); 
        return ans;
    }
}
vector<line> TangentinCC(circle k1,circle k2){
    
    int pd=checkposCC(k1,k2); if (pd<=2) return {
    };
    if (pd==3){
    point k=getCC(k1,k2)[0]; return {
    (line){
    k,k}};} 
    point p=(k2.o*k1.r+k1.o*k2.r)/(k1.r+k2.r);
    vector<point>A=TangentCP(k1,p),B=TangentCP(k2,p);
    vector<line>ans; for (int i=0;i<A.size();i++) ans.push_back((line){
    A[i],B[i]}); 
    return ans;
}
vector<line> TangentCC(circle k1,circle k2){
    
    int flag=0; if (k1.r<k2.r) swap(k1,k2),flag=1;
    vector<line>A=TangentoutCC(k1,k2),B=TangentinCC(k1,k2);
    for (line k:B) A.push_back(k); 
    if (flag) for (line &k:A) swap(k[0],k[1]);
    return A;
}
db getarea(circle k1,point k2,point k3){
    
    // 圆 k1 与三角形 k2 k3 k1.o 的有向面积交
    point k=k1.o; k1.o=k1.o-k; k2=k2-k; k3=k3-k;
    int pd1=k1.inside(k2),pd2=k1.inside(k3); 
    vector<point>A=getCL(k1,k2,k3);
    if (pd1>=0){
    
        if (pd2>=0) return cross(k2,k3)/2;
        return k1.r*k1.r*rad(A[1],k3)/2+cross(k2,A[1])/2;
    } else if (pd2>=0){
     
        return k1.r*k1.r*rad(k2,A[0])/2+cross(A[0],k3)/2;
    }else {
    
        int pd=cmp(k1.r,disSP(k2,k3,k1.o));
        if (pd<=0) return k1.r*k1.r*rad(k2,k3)/2;
        return cross(A[0],A[1])/2+k1.r*k1.r*(rad(k2,A[0])+rad(A[1],k3))/2;
    }
}
circle getcircle(point k1,point k2,point k3){
    
    db a1=k2.x-k1.x,b1=k2.y-k1.y,c1=(a1*a1+b1*b1)/2;
    db a2=k3.x-k1.x,b2=k3.y-k1.y,c2=(a2*a2+b2*b2)/2;
    db d=a1*b2-a2*b1;
    point o=(point){
    k1.x+(c1*b2-c2*b1)/d,k1.y+(a1*c2-a2*c1)/d};
    return (circle){
    o,k1.dis(o)};
}
circle getScircle(vector<point> A){
    
    random_shuffle(A.begin(),A.end());
    circle ans=(circle){
    A[0],0};
    for (int i=1;i<A.size();i++)
        if (ans.inside(A[i])==-1){
    
            ans=(circle){
    A[i],0};
            for (int j=0;j<i;j++)
                if (ans.inside(A[j])==-1){
    
                    ans.o=(A[i]+A[j])/2; ans.r=ans.o.dis(A[i]);
                    for (int k=0;k<j;k++)
                        if (ans.inside(A[k])==-1)
                            ans=getcircle(A[i],A[j],A[k]);
                }
        }
    return ans;
}
db area(vector<point> A){
     // 多边形用 vector<point> 表示 , 逆时针 
    db ans=0;
    for (int i=0;i<A.size();i++) ans+=cross(A[i],A[(i+1)%A.size()]);
    return ans/2;
}
int checkconvex(vector<point>A){
    
    int n=A.size(); A.push_back(A[0]); A.push_back(A[1]);
    for (int i=0;i<n;i++) if (sign(cross(A[i+1]-A[i],A[i+2]-A[i]))==-1) return 0;
    return 1;
}
int contain(vector<point>A,point q){
     // 2 内部 1 边界 0 外部
    int pd=0; A.push_back(A[0]);
    for (int i=1;i<A.size();i++){
    
        point u=A[i-1],v=A[i];
        if (onS(u,v,q)) return 1; if (cmp(u.y,v.y)>0) swap(u,v);
        if (cmp(u.y,q.y)>=0||cmp(v.y,q.y)<0) continue;
        if (sign(cross(u-v,q-v))<0) pd^=1;
    }
    return pd<<1;
}
vector<point> ConvexHull(vector<point>A,int flag=1){
     // flag=0 不严格 flag=1 严格 
    int n=A.size(); vector<point>ans(n*2); 
    sort(A.begin(),A.end()); int now=-1;
    for (int i=0;i<A.size();i++){
    
        while (now>0&&sign(cross(ans[now]-ans[now-1],A[i]-ans[now-1]))<flag) now--;
        ans[++now]=A[i];
    } int pre=now;
    for (int i=n-2;i>=0;i--){
    
        while (now>pre&&sign(cross(ans[now]-ans[now-1],A[i]-ans[now-1]))<flag) now--;
        ans[++now]=A[i];
    } ans.resize(now); return ans;
}
db convexDiameter(vector<point>A){
    
    int now=0,n=A.size(); db ans=0;
    for (int i=0;i<A.size();i++){
    
        now=max(now,i);
        while (1){
    
            db k1=A[i].dis(A[now%n]),k2=A[i].dis(A[(now+1)%n]);
            ans=max(ans,max(k1,k2)); if (k2>k1) now++; else break;
        }
    }
    return ans;
}
vector<point> convexcut(vector<point>A,point k1,point k2){
    
    // 保留 k1,k2,p 逆时针的所有点
    int n=A.size(); A.push_back(A[0]); vector<point>ans;
    for (int i=0;i<n;i++){
    
        int w1=clockwise(k1,k2,A[i]),w2=clockwise(k1,k2,A[i+1]);
        if (w1>=0) ans.push_back(A[i]);
        if (w1*w2<0) ans.push_back(getLL(k1,k2,A[i],A[i+1]));
    }
    return ans;
}
int checkPoS(vector<point>A,point k1,point k2){
    
    // 多边形 A 和直线 ( 线段 )k1->k2 严格相交 , 注释部分为线段
    struct ins{
    
        point m,u,v;
        int operator < (const ins& k) const {
    return m<k.m;}
    }; vector<ins>B;
    //if (contain(A,k1)==2||contain(A,k2)==2) return 1;
    vector<point>poly=A; A.push_back(A[0]); 
    for (int i=1;i<A.size();i++) if (checkLL(A[i-1],A[i],k1,k2)){
    
        point m=getLL(A[i-1],A[i],k1,k2); 
        if (inmid(A[i-1],A[i],m)/*&&inmid(k1,k2,m)*/) B.push_back((ins){
    m,A[i-1],A[i]});
    }
    if (B.size()==0) return 0; sort(B.begin(),B.end()); 
    int now=1; while (now<B.size()&&B[now].m==B[0].m) now++; 
    if (now==B.size()) return 0;
    int flag=contain(poly,(B[0].m+B[now].m)/2);
    if (flag==2) return 1;
    point d=B[now].m-B[0].m;
    for (int i=now;i<B.size();i++){
    
        if (!(B[i].m==B[i-1].m)&&flag==2) return 1;
        int tag=sign(cross(B[i].v-B[i].u,B[i].m+d-B[i].u));
        if (B[i].m==B[i].u||B[i].m==B[i].v) flag+=tag; else flag+=tag*2;
    }
    //return 0;
    return flag==2;
}
int checkinp(point r,point l,point m){
    
	if (compareangle(l,r)){
    return compareangle(l,m)&&compareangle(m,r);}
	return compareangle(l,m)||compareangle(m,r);
}
int checkPosFast(vector<point>A,point k1,point k2){
     // 快速检查线段是否和多边形严格相交
	if (contain(A,k1)==2||contain(A,k2)==2) return 1; if (k1==k2) return 0;
	A.push_back(A[0]); A.push_back(A[1]);
	for (int i=1;i+1<A.size();i++)
		if (checkLL(A[i-1],A[i],k1,k2)){
    
			point now=getLL(A[i-1],A[i],k1,k2);
			if (inmid(A[i-1],A[i],now)==0||inmid(k1,k2,now)==0) continue;
			if (now==A[i]){
    
				if (A[i]==k2) continue;
				point pre=A[i-1],ne=A[i+1];
				if (checkinp(pre-now,ne-now,k2-now)) return 1;
			} else if (now==k1){
    
				if (k1==A[i-1]||k1==A[i]) continue;
				if (checkinp(A[i-1]-k1,A[i]-k1,k2-k1)) return 1;
			} else if (now==k2||now==A[i-1]) continue;
			else return 1;
		}
	return 0;
}
// 拆分凸包成上下凸壳 凸包尽量都随机旋转一个角度来避免出现相同横坐标 
// 尽量特判只有一个点的情况 凸包逆时针
void getUDP(vector<point>A,vector<point>&U,vector<point>&D){
    
    db l=1e100,r=-1e100;
    for (int i=0;i<A.size();i++) l=min(l,A[i].x),r=max(r,A[i].x);
    int wherel,wherer;
    for (int i=0;i<A.size();i++) if (cmp(A[i].x,l)==0) wherel=i;
    for (int i=A.size();i;i--) if (cmp(A[i-1].x,r)==0) wherer=i-1;
    U.clear(); D.clear(); int now=wherel;
    while (1){
    D.push_back(A[now]); if (now==wherer) break; now++; if (now>=A.size()) now=0;}
    now=wherel;
    while (1){
    U.push_back(A[now]); if (now==wherer) break; now--; if (now<0) now=A.size()-1;}
}
// 需要保证凸包点数大于等于 3,2 内部 ,1 边界 ,0 外部
int containCoP(const vector<point>&U,const vector<point>&D,point k){
    
    db lx=U[0].x,rx=U[U.size()-1].x;
    if (k==U[0]||k==U[U.size()-1]) return 1;
    if (cmp(k.x,lx)==-1||cmp(k.x,rx)==1) return 0;
    int where1=lower_bound(U.begin(),U.end(),(point){
    k.x,-1e100})-U.begin();
    int where2=lower_bound(D.begin(),D.end(),(point){
    k.x,-1e100})-D.begin();
    int w1=clockwise(U[where1-1],U[where1],k),w2=clockwise(D[where2-1],D[where2],k);
    if (w1==1||w2==-1) return 0; else if (w1==0||w2==0) return 1; return 2;
}
// d 是方向 , 输出上方切点和下方切点
pair<point,point> getTangentCow(const vector<point> &U,const vector<point> &D,point d){
    
    if (sign(d.x)<0||(sign(d.x)==0&&sign(d.y)<0)) d=d*(-1);
    point whereU,whereD;
    if (sign(d.x)==0) return mp(U[0],U[U.size()-1]);
    int l=0,r=U.size()-1,ans=0;
    while (l<r){
    int mid=l+r>>1; if (sign(cross(U[mid+1]-U[mid],d))<=0) l=mid+1,ans=mid+1; else r=mid;}
    whereU=U[ans]; l=0,r=D.size()-1,ans=0;
    while (l<r){
    int mid=l+r>>1; if (sign(cross(D[mid+1]-D[mid],d))>=0) l=mid+1,ans=mid+1; else r=mid;}
    whereD=D[ans]; return mp(whereU,whereD);
}
// 先检查 contain, 逆时针给出
pair<point,point> getTangentCoP(const vector<point>&U,const vector<point>&D,point k){
    
    db lx=U[0].x,rx=U[U.size()-1].x;
    if (k.x<lx){
    
        int l=0,r=U.size()-1,ans=U.size()-1;
        while (l<r){
    int mid=l+r>>1; if (clockwise(k,U[mid],U[mid+1])==1) l=mid+1; else ans=mid,r=mid;}
        point w1=U[ans]; l=0,r=D.size()-1,ans=D.size()-1;
        while (l<r){
    int mid=l+r>>1; if (clockwise(k,D[mid],D[mid+1])==-1) l=mid+1; else ans=mid,r=mid;}
        point w2=D[ans]; return mp(w1,w2);
    } else if (k.x>rx){
    
        int l=1,r=U.size(),ans=0;
        while (l<r){
    int mid=l+r>>1; if (clockwise(k,U[mid],U[mid-1])==-1) r=mid; else ans=mid,l=mid+1;}
        point w1=U[ans]; l=1,r=D.size(),ans=0;
        while (l<r){
    int mid=l+r>>1; if (clockwise(k,D[mid],D[mid-1])==1) r=mid; else ans=mid,l=mid+1;}
        point w2=D[ans]; return mp(w2,w1);
    } else {
    
        int where1=lower_bound(U.begin(),U.end(),(point){
    k.x,-1e100})-U.begin();
        int where2=lower_bound(D.begin(),D.end(),(point){
    k.x,-1e100})-D.begin();
        if ((k.x==lx&&k.y>U[0].y)||(where1&&clockwise(U[where1-1],U[where1],k)==1)){
    
            int l=1,r=where1+1,ans=0;
            while (l<r){
    int mid=l+r>>1; if (clockwise(k,U[mid],U[mid-1])==1) ans=mid,l=mid+1; else r=mid;}
            point w1=U[ans]; l=where1,r=U.size()-1,ans=U.size()-1;
            while (l<r){
    int mid=l+r>>1; if (clockwise(k,U[mid],U[mid+1])==1) l=mid+1; else ans=mid,r=mid;}
            point w2=U[ans]; return mp(w2,w1);
        } else {
    
            int l=1,r=where2+1,ans=0;
            while (l<r){
    int mid=l+r>>1; if (clockwise(k,D[mid],D[mid-1])==-1) ans=mid,l=mid+1; else r=mid;}
            point w1=D[ans]; l=where2,r=D.size()-1,ans=D.size()-1;
            while (l<r){
    int mid=l+r>>1; if (clockwise(k,D[mid],D[mid+1])==-1) l=mid+1; else ans=mid,r=mid;}
            point w2=D[ans]; return mp(w1,w2);
        }
    }
}
struct P3{
    
    db x,y,z;
    P3 operator + (P3 k1){
    return (P3){
    x+k1.x,y+k1.y,z+k1.z};}
    P3 operator - (P3 k1){
    return (P3){
    x-k1.x,y-k1.y,z-k1.z};}
    P3 operator * (db k1){
    return (P3){
    x*k1,y*k1,z*k1};}
    P3 operator / (db k1){
    return (P3){
    x/k1,y/k1,z/k1};}
    db abs2(){
    return x*x+y*y+z*z;}
    db abs(){
    return sqrt(x*x+y*y+z*z);}
    P3 unit(){
    return (*this)/abs();}
    int operator < (const P3 k1) const{
    
        if (cmp(x,k1.x)!=0) return x<k1.x;
        if (cmp(y,k1.y)!=0) return y<k1.y;
        return cmp(z,k1.z)==-1;
    }
    int operator == (const P3 k1){
    
        return cmp(x,k1.x)==0&&cmp(y,k1.y)==0&&cmp(z,k1.z)==0;
    }
    void scan(){
    
        double k1,k2,k3; scanf("%lf%lf%lf",&k1,&k2,&k3);
        x=k1; y=k2; z=k3;
    }
};
P3 cross(P3 k1,P3 k2){
    return (P3){
    k1.y*k2.z-k1.z*k2.y,k1.z*k2.x-k1.x*k2.z,k1.x*k2.y-k1.y*k2.x};}
db dot(P3 k1,P3 k2){
    return k1.x*k2.x+k1.y*k2.y+k1.z*k2.z;}
//p=(3,4,5),l=(13,19,21),theta=85 ans=(2.83,4.62,1.77)
P3 turn3D(db k1,P3 l,P3 p){
    
    l=l.unit(); P3 ans; db c=cos(k1),s=sin(k1);
    ans.x=p.x*(l.x*l.x*(1-c)+c)+p.y*(l.x*l.y*(1-c)-l.z*s)+p.z*(l.x*l.z*(1-c)+l.y*s);
    ans.y=p.x*(l.x*l.y*(1-c)+l.z*s)+p.y*(l.y*l.y*(1-c)+c)+p.z*(l.y*l.z*(1-c)-l.x*s);
    ans.z=p.x*(l.x*l.z*(1-c)-l.y*s)+p.y*(l.y*l.z*(1-c)+l.x*s)+p.z*(l.x*l.x*(1-c)+c);
    return ans;
}
typedef vector<P3> VP;
typedef vector<VP> VVP;
db Acos(db x){
    return acos(max(-(db)1,min(x,(db)1)));}
// 球面距离 , 圆心原点 , 半径 1
db Odist(P3 a,P3 b){
    db r=Acos(dot(a,b)); return r;}
db r; P3 rnd;
vector<db> solve(db a,db b,db c){
    
    db r=sqrt(a*a+b*b),th=atan2(b,a);
    if (cmp(c,-r)==-1) return {
    0};
    else if (cmp(r,c)<=0) return {
    1};
    else {
    
        db tr=pi-Acos(c/r); return {
    th+pi-tr,th+pi+tr};
    }
}
vector<db> jiao(P3 a,P3 b){
    
    // dot(rd+x*cos(t)+y*sin(t),b) >= cos(r)
    if (cmp(Odist(a,b),2*r)>0) return {
    0};
    P3 rd=a*cos(r),z=a.unit(),y=cross(z,rnd).unit(),x=cross(y,z).unit();
    vector<db> ret = solve(-(dot(x,b)*sin(r)),-(dot(y,b)*sin(r)),-(cos(r)-dot(rd,b))); 
    return ret;
}
db norm(db x,db l=0,db r=2*pi){
     // change x into [l,r)
    while (cmp(x,l)==-1) x+=(r-l); while (cmp(x,r)>=0) x-=(r-l);
    return x;
}
db disLP(P3 k1,P3 k2,P3 q){
    
    return (cross(k2-k1,q-k1)).abs()/(k2-k1).abs();
}
db disLL(P3 k1,P3 k2,P3 k3,P3 k4){
    
    P3 dir=cross(k2-k1,k4-k3); if (sign(dir.abs())==0) return disLP(k1,k2,k3);
    return fabs(dot(dir.unit(),k1-k2));
}
VP getFL(P3 p,P3 dir,P3 k1,P3 k2){
    
    db a=dot(k2-p,dir),b=dot(k1-p,dir),d=a-b;
    if (sign(fabs(d))==0) return {
    };
    return {
    (k1*a-k2*b)/d};
}
VP getFF(P3 p1,P3 dir1,P3 p2,P3 dir2){
    // 返回一条线
    P3 e=cross(dir1,dir2),v=cross(dir1,e);
    db d=dot(dir2,v); if (sign(abs(d))==0) return {
    };
    P3 q=p1+v*dot(dir2,p2-p1)/d; return {
    q,q+e};
}
// 3D Covex Hull Template
db getV(P3 k1,P3 k2,P3 k3,P3 k4){
     // get the Volume
    return dot(cross(k2-k1,k3-k1),k4-k1);
}
db rand_db(){
    return 1.0*rand()/RAND_MAX;}
VP convexHull2D(VP A,P3 dir){
    
    P3 x={
    (db)rand(),(db)rand(),(db)rand()}; x=x.unit();
    x=cross(x,dir).unit(); P3 y=cross(x,dir).unit();
    P3 vec=dir.unit()*dot(A[0],dir);
    vector<point>B;
    for (int i=0;i<A.size();i++) B.push_back((point){
    dot(A[i],x),dot(A[i],y)});
    B=ConvexHull(B); A.clear();
    for (int i=0;i<B.size();i++) A.push_back(x*B[i].x+y*B[i].y+vec);
    return A;
}
namespace CH3{
    
    VVP ret; set<pair<int,int> >e;
    int n; VP p,q;
    void wrap(int a,int b){
    
        if (e.find({
    a,b})==e.end()){
    
            int c=-1;
            for (int i=0;i<n;i++) if (i!=a&&i!=b){
    
                if (c==-1||sign(getV(q[c],q[a],q[b],q[i]))>0) c=i;
            }
            if (c!=-1){
    
                ret.push_back({
    p[a],p[b],p[c]});
                e.insert({
    a,b}); e.insert({
    b,c}); e.insert({
    c,a});
                wrap(c,b); wrap(a,c);
            }
        }
    }
    VVP ConvexHull3D(VP _p){
    
        p=q=_p; n=p.size();
        ret.clear(); e.clear();
        for (auto &i:q) i=i+(P3){
    rand_db()*1e-4,rand_db()*1e-4,rand_db()*1e-4};
        for (int i=1;i<n;i++) if (q[i].x<q[0].x) swap(p[0],p[i]),swap(q[0],q[i]);
        for (int i=2;i<n;i++) if ((q[i].x-q[0].x)*(q[1].y-q[0].y)>(q[i].y-q[0].y)*(q[1].x-q[0].x)) swap(q[1],q[i]),swap(p[1],p[i]);
        wrap(0,1);
        return ret;
    }
}
VVP reduceCH(VVP A){
    
    VVP ret; map<P3,VP> M;
    for (VP nowF:A){
    
        P3 dir=cross(nowF[1]-nowF[0],nowF[2]-nowF[0]).unit();
        for (P3 k1:nowF) M[dir].pb(k1);
    }
    for (pair<P3,VP> nowF:M) ret.pb(convexHull2D(nowF.se,nowF.fi));
    return ret;
}
//  把一个面变成 ( 点 , 法向量 ) 的形式
pair<P3,P3> getF(VP F){
    
    return mp(F[0],cross(F[1]-F[0],F[2]-F[0]).unit());
}
// 3D Cut 保留 dot(dir,x-p)>=0 的部分
VVP ConvexCut3D(VVP A,P3 p,P3 dir){
    
    VVP ret; VP sec;
    for (VP nowF: A){
    
        int n=nowF.size(); VP ans; int dif=0;
        for (int i=0;i<n;i++){
    
            int d1=sign(dot(dir,nowF[i]-p));
            int d2=sign(dot(dir,nowF[(i+1)%n]-p));
            if (d1>=0) ans.pb(nowF[i]);
            if (d1*d2<0){
    
                P3 q=getFL(p,dir,nowF[i],nowF[(i+1)%n])[0];
                ans.push_back(q); sec.push_back(q);
            }
            if (d1==0) sec.push_back(nowF[i]); else dif=1;
            dif|=(sign(dot(dir,cross(nowF[(i+1)%n]-nowF[i],nowF[(i+1)%n]-nowF[i])))==-1);
        }
        if (ans.size()>0&&dif) ret.push_back(ans);
    }
    if (sec.size()>0) ret.push_back(convexHull2D(sec,dir));
    return ret;
}
db vol(VVP A){
    
    if (A.size()==0) return 0; P3 p=A[0][0]; db ans=0;
    for (VP nowF:A)
        for (int i=2;i<nowF.size();i++)
            ans+=abs(getV(p,nowF[0],nowF[i-1],nowF[i]));
    return ans/6;
}
VVP init(db INF) {
    
    VVP pss(6,VP(4));
    pss[0][0] = pss[1][0] = pss[2][0] = {
    -INF, -INF, -INF};
    pss[0][3] = pss[1][1] = pss[5][2] = {
    -INF, -INF, INF};
    pss[0][1] = pss[2][3] = pss[4][2] = {
    -INF, INF, -INF};
    pss[0][2] = pss[5][3] = pss[4][1] = {
    -INF, INF, INF};
    pss[1][3] = pss[2][1] = pss[3][2] = {
    INF, -INF, -INF};
    pss[1][2] = pss[5][1] = pss[3][3] = {
    INF, -INF, INF};
    pss[2][2] = pss[4][3] = pss[3][1] = {
    INF, INF, -INF};
    pss[5][0] = pss[4][0] = pss[3][0] = {
    INF, INF, INF};
    return pss;
}