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#include<cstdlib> #include<cmath> #include<cstdio> #include<algorithm> #define max(a,b) (((a)>(b))?(a):(b)) #define min(a,b) (((a)>(b))?(b):(a)) #define sign(x) ((x)>eps?1:((x)<-eps?(-1):(0))) using namespace std; const int MAXN=1000; const double eps=1e-8,inf=1e50 ,Pi=acos(-1.0); struct point { double x,y; point() {} point(double _x,double _y) { x=_x; y=_y; } point operator-(const point &ne)const { return point(x-ne.x,y-ne.y); } point operator+(const point ne)const { return point(x+ne.x,y+ne.y); } point operator*(const double t)const{ return point(x*t,y*t); } point operator/(const double t)const{ if(sign(t)==0)exit(1); return point(x/t,y/t); } }; struct line{ point a,b; line(){} line(point _a,point _b){a=_a;b=_b;} }; struct line2 { double a,b,c; line2() {} line2(double _a,double _b,double _c) { a=_a; b=_b; c=_c; } }; struct circle { point o; double r; circle(){} circle(point _o,double _r){ o=_o; r=_r; } }; struct rectangle{ point a,b,c,d; rectangle(){} rectangle(point _a,point _b,point _c,point _d){ a=_a; b=_b; c=_c; d=_d; } }; struct polygon{ point p[MAXN]; int n; }; inline double xmult(point a,point b){ return a.x*b.y-a.y*b.x; } inline double xmult(point o,point a,point b) { return (a.x-o.x)*(b.y-o.y)-(b.x-o.x)*(a.y-o.y); } inline double xmult(double x1,double y1,double x2,double y2) { return x1*y2-x2*y1; } inline double dmult(point o,point a,point b) { return (a.x-o.x)*(b.x-o.x)+(a.y-o.y)*(b.y-o.y); } inline double dmult(point a,point b) { return a.x*b.x+a.y*b.y; } inline double lenth(point a){ return sqrt(dmult(a,a)); } inline double dist(point a,point b){ return lenth(b-a); } inline double dist2(point a,point b){ return dmult(b-a,b-a); } //直线一般式转两点式 line toline(double a,double b,double c) { if(sign(b)==0)exit(1); point A(0,-c/b),B(-c/a,0); return line(A,B); } //直线两点式转一般式 line2 toline2(point a,point b) { double A=b.y-a.y,B=a.x-b.x,C=-B*a.y-A*a.x; return line2(A,B,C); } //点p绕o逆时针旋转alpha point rotate(point o,point p,double alpha) { point tp; p.x-=o.x; p.y-=o.y; tp.x=p.x*cos(alpha)-p.y*sin(alpha)+o.x; tp.y=p.y*cos(alpha)+p.x*sin(alpha)+o.y; return tp; } //向量u的倾斜角 double angle(point u) { return atan2(u.y,u.x); } //oe与os的夹角，夹角正负满足叉积 double angle(point o,point s,point e) { point os=s-o,oe=e-o; double bot=sqrt(dmult(os,os)*dmult(oe,oe)); double top=dmult(os,oe); double cosfi=top/bot; if (cosfi >= 1.0 ) return 0; if (cosfi <= -1.0 ) return -Pi; double fi=acos(cosfi); if(xmult(o,s,e)>0)return fi; else return -fi; } //p在l上的投影与l关系 double relation(point p,line l) { line tl(l.a,p); return dmult(tl.b-l.a,l.b-l.a)/dist2(l.a,l.b); } //p在l上的垂足 point perpendicular(point p,line l) { double r=relation(p,l); return l.a+((l.b-l.a)*r); } //求点p到线段l的最短距离,并返回线段上距该点最近的点np double dist_p_to_seg(point p,line l,point &np) { double r=relation(p,l); if(r<0) { np=l.a; return dist(p,l.a); } if(r>1) { np=l.b; return dist(p,l.b); } np=perpendicular(p,l); return dist(p,np); } //求点p到直线l的最短距离 double dist_p_to_line(point p,line l) { return abs(xmult(p-l.a,l.b-l.a))/dist(l.a,l.b); } //线段之间最短距离 inline double dist_seg_to_seg(line p,line q) { return min(min(dist_p_to_seg(p.a,q),dist_p_to_seg(p.b,q)),min(dist_p_to_seg(q.a,p),dist_p_to_seg(q.b,p))); } //求矢量线段夹角的余弦 double cosine(line u,line v) { point pu=u.b-u.a,pv=v.b-v.a; return dmult(pu,pv)/sqrt(dmult(pu,pu)*dmult(pv,pv)); } //求矢量的夹角的余弦 double cosine(point a,point b){ return dmult(a,b)/sqrt(dmult(a,a)*dmult(b,b)); } //求线段夹角 double lsangle(line u,line v) { point o(0,0),a=u.b-u.a,b=v.b-v.a; return angle(o,a,b); } //求直线斜率 double slope(line2 l) { if(abs(l.a) < 1e-20) return 0; if(abs(l.b) < 1e-20) return inf; return -(l.a/l.b); } //直线倾斜角[0,Pi] double alpha(line2 l) { if(abs(l.a)< eps) return 0; if(abs(l.b)< eps) return Pi/2; double k=slope(l); if(k>0) return atan(k); else return Pi+atan(k); } //点关于直线的对称点 point symmetry(line2 l,point p){ point tp; tp.x=((l.b*l.b-l.a*l.a)*p.x-2*l.a*l.b*p.y-2*l.a*l.c)/(l.a*l.a+l.b*l.b); tp.y=((l.a*l.a-l.b*l.b)*p.y-2*l.a*l.b*p.x-2*l.b*l.c)/(l.a*l.a+l.b*l.b); return tp; } //判多边形是否逆时针 bool is_unclock(polygon pg) { int n=pg.n; pg.p[n]=pg.p[0]; double area=0; for(int i=0; i<n; i++) area+=xmult(pg.p[i].x,pg.p[i].y,pg.p[i+1].x,pg.p[i+1].y); return area>-eps; } //改变多边形时针顺序 void to_unclock(polygon &pg) { for(int i=0,j=pg.n-1; i<j; i++,j--) swap(pg.p[i],pg.p[j]); } //判定凸多边形,顶点按顺时针或逆时针给出,允许相邻边共线 int is_convex(point p[],int n) { int i,s[3]= {1,1,1}; for (i=0; i<n&&s[1]|s[2]; i++) s[(sign(xmult(p[(i+1)%n],p[(i+2)%n],p[i]))+3)%3]=0; return s[1]|s[2]; } //判定凸多边形,顶点按顺时针或逆时针给出,不允许相邻边共线 int is_convex_v2(point p[],int n) { int i,s[3]= {1,1,1}; for (i=0; i<n&&s[0]&&s[1]|s[2]; i++) s[(sign(xmult(p[(i+1)%n],p[(i+2)%n],p[i]))+3)%3]=0; return s[0]&&s[1]|s[2]; } //判点在凸多边形内或多边形边上,顶点按顺时针或逆时针给出 int inside_convex(point q,point p[],int n) { int i,s[3]= {1,1,1}; for (i=0; i<n&&s[1]|s[2]; i++) s[(sign(xmult(p[(i+1)%n],q,p[i]))+3)%3]=0; return s[1]|s[2]; } //判点在凸多边形内,顶点按顺时针或逆时针给出,在多边形边上返回0 int inside_convex2(point q,point p[],int n) { int i,s[3]= {1,1,1}; for (i=0; i<n&&s[0]&&s[1]|s[2]; i++) s[(sign(xmult(p[(i+1)%n],q,p[i]))+3)%3]=0; return s[0]&&s[1]|s[2]; } //判点在线段上 inline int p_on_seg(point a,point p1,point p2) { if(fabs(xmult(a,p1,p2))<=eps&&(a.x-p1.x)*(a.x-p2.x)<eps&&(a.y-p1.y)*(a.y-p2.y)<eps) return 1; return 0; } //判点在线段端点左方 inline int p_on_segvex(point s,point p) { return fabs(p.y-s.y)<eps&&(p.x<=s.x+eps); } //判线段相交 <=:不规范相交 inline int seg_inter(line s,line p) { double minx1=min(s.a.x,s.b.x),maxx1=max(s.a.x,s.b.x); double minx2=min(p.a.x,p.b.x),maxx2=max(p.a.x,p.b.x); double miny1=min(s.a.y,s.b.y),maxy1=max(s.a.y,s.b.y); double miny2=min(p.a.y,p.b.y),maxy2=max(p.a.y,p.b.y); if((minx1>maxx2+eps)||(minx2>maxx1+eps)|| (miny1>maxy2+eps)||(miny2>maxy1+eps)) return 0; else return sign(xmult(s.a,s.b,p.a)*xmult(s.a,s.b,p.b))<=0&& sign(xmult(p.a,p.b,s.a)*xmult(p.a,p.b,s.b))<=0; } //判点在多边形内部 inline int p_in_polygon(point a,point p[],int n) { int count = 0; line s,ps; ps.a = a,ps.b = a; ps.b.x = inf; for(int i = 0; i < n; i++) { s.a = p[i]; if(i + 1 < n)s.b = p[i+1]; else s.b = p[0]; if (s.a.y > s.b.y)swap(s.a,s.b); if (p_on_seg(a,s.a,s.b))return 2; if ((fabs(s.a.y-s.b.y)>eps)) { if (p_on_segvex(s.b,a)) count++; else if (seg_inter(ps,s))count++; } } if (count%2)return 1; return 0; } //多边形内部最长线段 point stk[MAXN]; double seg_max_len(line u,polygon &pg){ double ans=0.0,tmp=inf; pg.p[pg.n]=pg.p[0]; int n=pg.n,top=0; for(int i=0;i<n;i++){ line v(pg.p[i],pg.p[i+1]); int s1=sign(xmult(u.a,u.b,v.a)),s2=sign(xmult(u.a,u.b,v.b)); if(s1*s2<=0&&(s1!=0||s2!=0)){ stk[top++]=line_intersection(u,v); } } stk[top++]=u.a; stk[top++]=u.b; sort(stk,stk+top); top=unique(stk,stk+top)-stk; point mp,lp=stk[0]; for(int i=1;i<top;i++){ mp=(lp+stk[i])*0.5; if(!p_in_polygon(mp,pg))lp=stk[i]; ans=max(ans,dist2(lp,stk[i])); } return sqrt(ans); } double maxlenth(polygon &pg){ double ans=0.0; for(int i=0;i<pg.n-1;i++){ for(int j=i+1;j<pg.n;j++){ ans=max(ans,seg_max_len(line(pg.p[i],pg.p[j]),pg)); ans=max(ans,seg_max_len(line(pg.p[j],pg.p[i]),pg)); } } return ans; } //凸包对踵点长度 double opposite_lenth(polygon &pg){ double ans=inf; int a,b,c; pg.p[pg.n]=pg.p[0]; for(a=0,b=1,c=2;a<pg.n;a++,b++){ while(getarea(pg.p[a],pg.p[b],pg.p[c])<getarea(pg.p[a],pg.p[b],pg.p[(c+1)%pg.n]))c=(c+1)%pg.n; ans=min(ans,dist_p_to_line(pg.p[c],line(pg.p[a],pg.p[b]))); } return ans; } //判p1,p2是否在l1,l2两侧 inline int opposite_side(point p1,point p2,point l1,point l2) { return xmult(l1,l2,p1)*xmult(l1,l2,p2)<-eps; } //判线段在任意多边形内,顶点按顺时针或逆时针给出,与边界相交返回1 int seg_in_polygon(point l1,point l2,point p[],int n) { point t[MAXN],tt; int i,j,k=0; if (!p_in_polygon(l1,p,n)||!p_in_polygon(l2,p,n))//不在内部 return 0; for (i=0; i<n; i++) if (opposite_side(l1,l2,p[i],p[(i+1)%n])&&opposite_side(p[i],p[(i+1)%n],l1,l2)) return 0; else if (p_on_seg(l1,p[i],p[(i+1)%n])) t[k++]=l1; else if (p_on_seg(l2,p[i],p[(i+1)%n])) t[k++]=l2; else if (p_on_seg(p[i],l1,l2)) t[k++]=p[i]; for (i=0; i<k; i++) for (j=i+1; j<k; j++) { tt.x=(t[i].x+t[j].x)/2; tt.y=(t[i].y+t[j].y)/2; if (!p_in_polygon(tt,p,n)) return 0; } return 1; } //求直线交点，必须存在交点，或者预判断【解析几何方法】 point line_intersection(line u,line v) { double a1=u.b.y-u.a.y,b1=u.a.x-u.b.x; double c1=u.b.y*(-b1)-u.b.x*a1; double a2=v.b.y-v.a.y,b2=v.a.x-v.b.x; double c2=v.b.y*(-b2)-v.b.x*a2; double D=xmult(a1,b1,a2,b2); return point(xmult(b1,c1,b2,c2)/D,xmult(c1,a1,c2,a2)/D); } //求线段交点，必须存在交点，或者预判断【平面几何方法】 point line_intersection2(line u,line v) { point ret=u.a; double t=xmult(u.a-v.a,v.b-v.a)/xmult(u.b-u.a,v.b-v.a); t=fabs(t); ret.x+=(u.b.x-u.a.x)*t; ret.y+=(u.b.y-u.a.y)*t; return ret; } //三角形重心 point barycenter(point a,point b,point c){ return (a+b+c)/3.0; } //多边形重心 point barycenter(point p[],int n) { point ret,t; double t1=0,t2; int i; ret.x=ret.y=0; for (i=1; i<n-1; i++) if (fabs(t2=xmult(p[i+1],p[0],p[i]))>eps) { t=barycenter(p[0],p[i],p[i+1]); ret.x+=t.x*t2; ret.y+=t.y*t2; t1+=t2; } if (fabs(t1)>eps) ret.x/=t1,ret.y/=t1; return ret; } //求多边形面积 inline double getarea(point pg[],int n) { double area=0; pg[n]=pg[0]; for(int i=0; i<n; i++) area+=xmult(pg[i].x,pg[i].y,pg[i+1].x,pg[i+1].y); return fabs(area)/2.0; } //解方程ax^2+bx+c=0 int equaltion(double a,double b,double c,double &x1,double &x2) { double der=b*b-4*a*c; switch(sign(der)) { case -1: return 0; case 0: x1=-b/(2*a); return 1; case 1: der=sqrt(der); x1=(-b-der)/(2*a); x2=(-b+der)/(2*a); return 2; } } //线段与圆交点 int line_circle_intersection(line u,circle c,point &p1,point &p2) { double dis=lenth(u.b-u.a); point d=(u.b-u.a)/dis; point E=c.o-u.a; double a=dmult(E,d); double a2=a*a; double e2=dmult(E,E); double r2=c.r*c.r; if((r2-e2+a2)<0){ return 0; } else { double f=sqrt(r2 - e2 + a2); double t=a-f; int cnt=0; if(t>-eps&&t-dis<eps){//去掉后面变成射线 p1=u.a+(d*t); cnt++; } t=a+f; if(t>-eps&&t-dis<eps) { p2=u.a+(d*t); cnt++; } return cnt; } } //给出在任意多边形内部的一个点 point a_point_in_polygon(polygon pg) { point v,a,b,r; int i,index; v=pg.p[0]; index=0; for(i=1; i<pg.n; i++) { if(pg.p[i].y<v.y) { v=pg.p[i]; index=i; } } a=pg.p[(index-1+pg.n)%pg.n]; b=pg.p[(index+1)%pg.n]; point q; polygon tri; tri.n=3; tri.p[0]=a; tri.p[1]=v; tri.p[2]=b; double md=inf; int in1=index; bool bin=false; for(i=0; i<pg.n; i++) { if(i == index)continue; if(i == (index-1+pg.n)%pg.n)continue; if(i == (index+1)%pg.n)continue; if(!inside_convex2(pg.p[i],tri.p,3))continue; bin=true; if(dist(v,pg.p[i])<md) { q=pg.p[i]; md=dist(v,q); } } if(!bin) { r.x=(a.x+b.x)/2; r.y=(a.y+b.y)/2; return r; } r.x=(v.x+q.x)/2; r.y=(v.y+q.y)/2; return r; } //求在多边形外面的点到凸包的切点 void p_cut_polygon(point p,polygon pg,point &rp,point &lp) { line ep,en; bool blp,bln; rp=pg.p[0]; lp=pg.p[0]; for(int i=1; i<pg.n; i++) { ep.a=pg.p[(i+pg.n-1)%pg.n]; ep.b=pg.p[i]; en.a=pg.p[i]; en.b=pg.p[(i+1)%pg.n]; blp=xmult(ep.b-ep.a,p-ep.a)>=0; bln=xmult(en.b-en.a,p-en.a)>=0; if(!blp&&bln) { if(xmult(pg.p[i]-p,rp-p)>0) rp=pg.p[i]; } if(blp&&!bln) { if(xmult(lp-p,pg.p[i]-p)>0) lp=pg.p[i]; } } return ; } //判断点p在圆c内 bool p_in_circle(point p,circle c) { return c.r*c.r>dist2(p,c.o); } //求矩形第4个点 point rect4th(point a,point b,point c) { point d; if(abs(dmult(a-c,b-c))<eps) { d=a+b-c; } if(abs(dmult(a-b,c-b))<eps) { d=a+c-b; } if(abs(dmult(c-a,b-a))<eps) { d=c+b-a; } return d; } //判两圆关系 int CircleRelation(circle c1,circle c2){ double d=lenth(c1.o-c2.o); if( fabs(d-c1.r-c2.r) < eps ) return 2;//外切 if( fabs(d-fabs(c1.r-c2.r)) < eps ) return 4;//内切 if( d > c1.r+c2.r ) return 1;//相离 if( d < fabs(c1.r-c2.r) ) return 5;//内含 if( fabs(c1.r-c2.r) < d && d < c1.r+c2.r) return 3;//相交 return 0; //error! } //判圆与矩形关系,矩形水平 bool Circle_In_Rec(circle c,rectangle r) { if( r.a.x < c.o.x && c.o.x < r.b.x && r.c.y < c.o.y && c.o.y < r.b.y ) { line line1(r.a, r.b); line line2(r.b, r.c); line line3(r.c, r.d); line line4(r.d, r.a); if(c.r<dist_p_to_line(c.o,line1)&&c.r<dist_p_to_line(c.o,line2)&&c.r<dist_p_to_line(c.o,line3)&&c.r<dist_p_to_line(c.o,line4)) return true; } return false; } //射线关于平面的反射 void reflect(line2 u,line2 v,line2 &l){ double n,m; double tpb,tpa; tpb=u.b*v.b+u.a*v.a; tpa=v.a*u.b-u.a*v.b; m=(tpb*u.b+tpa*u.a)/(u.b*u.b+u.a*u.a); n=(tpa*u.b-tpb*u.a)/(u.b*u.b+u.a*u.a); if(fabs(u.a*v.b-v.a*u.b)<1e-20) { l.a=v.a; l.b=v.b; l.c=v.c; return; } double xx,yy; //(xx,yy)是入射线与镜面的交点。 xx=(u.b*v.c-v.b*u.c)/(u.a*v.b-v.a*u.b); yy=(v.a*u.c-u.a*v.c)/(u.a*v.b-v.a*u.b); l.a=n; l.b=-m; l.c=m*yy-xx*n; } //两圆交点（预判断不相交情况） void c2point(circle c1,circle c2,point &rp1,point &rp2) { double a,b,r; a=c2.o.x-c1.o.x; b=c2.o.y-c1.o.y; r=(a*a+b*b+c1.r*c1.r-c2.r*c2.r)/2; if(a==0&&b!=0) { rp1.y=rp2.y=r/b; rp1.x=sqrt(c1.r*c1.r-rp1.y*rp1.y); rp2.x=-rp1.x; } else if(a!=0&&b==0) { rp1.x=rp2.x=r/a; rp1.y=sqrt(c1.r*c1.r-rp1.x*rp2.x); rp2.y=-rp1.y; } else if(a!=0&&b!=0) { double delta; delta=b*b*r*r-(a*a+b*b)*(r*r-c1.r*c1.r*a*a); rp1.y=(b*r+sqrt(delta))/(a*a+b*b); rp2.y=(b*r-sqrt(delta))/(a*a+b*b); rp1.x=(r-b*rp1.y)/a; rp2.x=(r-b*rp2.y)/a; } rp1=rp1+c1.o; rp2=rp2+c1.o; } //圆外一点引圆的切线 void cutpoint(circle c,point sp,point &rp1,point &rp2) { circle c2; c2.o=(c.o+sp)/2.0; c2.r=lenth(c2.o-sp); c2point(c,c2,rp1,rp2); } //圆c1上，与c2的外切点 void c2cuto(circle c1, circle c2, point &p1, point &p2) { double d = dist(c1.o, c2.o), dr = c1.r - c2.r; double b = acos(dr / d); double a = angle(c2.o-c1.o); double a1 = a - b, a2 = a + b; p1=point(cos(a1) * c1.r, sin(a1) * c1.r) + c1.o; p2=point(cos(a2) * c1.r, sin(a2) * c1.r) + c1.o; } //圆c1上，与c2的内切点 void c2cuti(circle c1,circle c2,point &p1,point &p2){ point dr=c2.o-c1.o; dr=dr/lenth(dr); point a=c1.o-(dr*c1.r),b=c1.o+(dr*c1.r); point c=c2.o-(dr*c2.r),d=c2.o+(dr*c2.r); circle E((a+c)/2.0,lenth(c-a)/2.0),F((b+d)/2.0,lenth(d-b)/2.0); point q1,q2; c2point(E,F,q1,q2); point L=line_intersection2(line(c1.o,c2.o),line(q1,q2)); circle c3((c1.o+L)/2.0,lenth(L-c1.o)/2.0); c2point(c1,c3,p1,p2); }