#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 ",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)>=0);} }; 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，小技巧：极角排序前，总是会碰到要扫两次的情况(要把数组扩展一倍)，其实可以从左到右进行枚举，把所有点的角度约束在[-pi/2,pi/2]内 int comp(point k1,point k2){
if(k1.getP()==k2.getP())return sign(cross(k1,k2))>0;
return k1.getP()<k2.getP();
} 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){ // 求直线和圆的交点，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){// 求两圆交点，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){// 求点到圆的切点，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){ //求两圆外切线 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){ //求两圆内切线 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){//求两圆切线 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){//通过三点求元 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){ //最小圆覆盖 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){ //判断是否是凸包 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){ // 求凸包：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; } vector<point> convexcut(vector<point>A,point k1,point k2){ // 用向量割出凸包左侧，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;} } // 判断点和凸包的关系，要将凸包拆成上下凸包，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); } // 点和凸包的切点，要提前检查 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; }
补充以及一些错误模板的修订
点集直径(q是凸包)
db RC(vector<point> q) { if(q.size()==2)return (q[1]-q[0]).abs2(); q.push_back(q[0]); int now=1; db ans=0; for(int i=0;i<q.size()-1;i++)//逆时针枚举凸包上的边(i,i+1)，找到对应这条边的对踵点 { while(cross((q[i+1]-q[i]),(q[now]-q[i]))<cross((q[i+1]-q[i]),(q[now+1]-q[i]))) { now++; if(now==q.size()-1)now=0; } ans=max(ans,(q[now]-q[i]).abs2()); } return ans; }
半平面交
如果一点也算交，那么有个地方需要修改 struct line{}里面的一个函数
int include(point k){return sign(cross(p[1]-p[0],k-p[0]))>=0;}//k在l左端
判给定的点是逆时针 return 1 还是顺时针 return 0
//输入的点是顺时针：ans<0,逆时针：ans>0 bool judge(vector<point> v){ double ans=0; for(int i=1;i<v.size()-1;i++) ans+=cross(v[i]-v[0],v[i+1]-v[0]); return ans>0; }
半平面交
int 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++){ //cout<<L[i][0].x<<" "<<L[i][0].y<<" "<<L[i][1].x<<" "<<L[i][1].y<<' '; 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]); if(q.size()<3)return 0; return 1; }