###点的定义app
//考虑偏差的加法 double add(double a, double b) { if (abs(a + b) < EPS*(abs(a) + abs(b)))return 0; return a + b; } struct P { double x, y; P() {} P(double x,double y):x(x),y(y){} P operator+(P p) { return P(add(x,p.x), add(y , p.y)); } P operator-(P p) { return P(add(x, -p.x), add(y, -p.y)); } P operator*(double d) { return P(x*d, y*d); } double dot(P p) {//内积,点积 return add(x*p.x, y*p.y); } double det(P p) {//外积,叉乘 return add(x*p.y, -y * p.x); } }; //判断q点是否在线段p1-p2上 bool on_seg(P p1, P p2, P q) { return (p1 - q).det(p2 - q) == 0 && (p1 - q).dot(p2 - q) <= 0; } //计算直线p1-p2与直线q1-q2的交点 P intersection(P p1, P p2, P q1, P q2) { return p1 + (p2 - p1)*((q2 - q1).det(q1 - p1) / (q2 - q1).det(p2 - p1)); } //判断线段相交的代码,此处是判断n条线段相交的程序 for (int i = 0; i<n; i++) { g[i][i] = 1; for (int j = 0; j<i; j++) { //判断木棍i和木棍j是否有公共点 if ((p[i] - q[i]).det(p[j] - q[j]) == 0) { //平行时 g[i][j] = g[j][i] = on_seg(p[i], q[i], p[j]) || on_seg(p[i], q[i], q[j]) || on_seg(p[j], q[j], p[i]) || on_seg(p[j], q[j], q[i]); } else { //不平行时 P r = intersection(p[i], q[i], p[j], q[j]); g[i][j] = g[j][i] = on_seg(p[i], q[i], r) && on_seg(p[j], q[j], r); } } }
###队友提供spa
#include<stdio.h> #include<math.h> #include<string.h> #define eps 1e-8 #define pi 3.141592653589793 using namespace std; //二维点类 struct Point { double x,y; Point(double a=0,double b=0){x=a;y=b;} }; typedef Point Vector; //二维直线类,通常方程ax+by+c=0 struct Line { double a,b,c,angle; Point p1,p2; Line(Point s,Point e) { a=s.y-e.y; b=e.x-s.x; c=s.x*e.y-e.x*s.y; angle=atan2(e.y-s.y,e.x-s.x); p1=s;p2=e; } Line(){} }; //二维线段类 struct Segment { Point s,e; Segment(Point a,Point b){s=a;e=b;} Segment(double x1,double y1,double x2,double y2) { s=Point(x1,y1); e=Point(x2,y2); } Segment(){} }; //向量的加减及数乘 Vector operator + (Point a,Point b) { return Vector(a.x+b.x,a.y+b.y); } Vector operator - (Point a,Point b) { return Vector(a.x-b.x,a.y-b.y); } Vector operator * (Point a,double k) { return Vector(a.x*k,a.y*k); } Vector operator / (Point a,double k) { return Vector(a.x/k,a.y/k); } //求向量的模(长度) double len(Vector a) { return sqrt(a.x*a.x+a.y*a.y); } //获得sp-op和ep-op的叉积 //>0时ep在opsp的逆时针方向,<0时顺时针,=0时共线 double Cross(Point &sp, Point &ep, Point &op) { return (sp.x-op.x)*(ep.y-op.y)-(ep.x-op.x)*(sp.y-op.y); } //两向量求叉积,求三角形面积须要除以2 double Cross(Vector a,Vector b) { return a.x*b.y-b.x*a.y; } //两向量求点积 double Dot(Vector a,Vector b) { return a.x*b.x+a.y*b.y; } //求最大最小值 double max(double a,double b) { if (a<b) return b; else return a; } double min(double a,double b) { if (a>b) return b; else return a; } //采用eps的精度判断大/小于零 int epssgn(double x) { if (fabs(x)<eps) return 0; else return x<0?-1:1; } //求两点之间的直线距离 double dis(Point a,Point b) { return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y)); } //判断两直线是否平行 int Parallel(Line l1,Line l2) { return (fabs(l1.a*l2.b-l2.a*l1.b)<eps); } //判断两直线是否相等 int LineEqual(Line l1,Line l2) { if (!Parallel(l1,l2)) return 0; else return (fabs(l1.a*l2.c-l2.a*l1.c)<eps && fabs(l1.b*l2.c-l2.b*l1.c)<eps); } //求点C到线段AB的距离 double PointToSegDist(Point A,Point B,Point C) { if (dis(A,B)<eps) return dis(B,C); if (epssgn(Dot(B-A,C-A))<0) return dis(A,C); if (epssgn(Dot(A-B,C-B))<0) return dis(B,C); return fabs(Cross(B-A,C-A))/dis(B,A); } //求线段两端AB到另外一线段CD的距离 double TwoSegMinDist(Point A,Point B,Point C,Point D) { return min(min(PointToSegDist(A,B,C),PointToSegDist(A,B,D)), min(PointToSegDist(C,D,A),PointToSegDist(C,D,B))); } Point SymPoint(Point p,Line l) //求二维平面上点p关于直线p1p2的对称点 { Point result; double a=l.p2.x-l.p1.x; double b=l.p2.y-l.p1.y; double t=((p.x-l.p1.x)*a+(p.y-l.p1.y)*b)/(a*a+b*b); result.x=2*l.p1.x+2*a*t-p.x; result.y=2*l.p1.y+2*b*t-p.y; return result; } //判断线段s1e1与s2e2是否相交(含端点) //不含端点的话将下面的<=改为< int IsSegmentIntersect(Point s1, Point e1, Point s2, Point e2) { if( min(s1.x,e1.x)<= max(s2.x,e2.x) && min(s1.y,e1.y)<= max(s2.y,e2.y) && min(s2.x,e2.x)<= max(s1.x,e1.x) && min(s2.y,e2.y)<= max(s1.y,e1.y) && Cross(s2,e2,s1)*Cross(s2,e2,e1)<=0 && Cross(s1,e1,s2)*Cross(s1,e1,e2)<=0) return 1; return 0; } //知道直线上两点p1p2,判断直线与线段se是否相交,含顶点 int IsLineIntersectSegment(Point p1,Point p2,Point s,Point e) { if (Cross(p1,p2,s)*Cross(p1,p2,e)>eps) return 0; else return 1; } int IsLineIntersectSegment(Line l1,Point s,Point e) { if (Cross(l1.p1,l1.p2,s)*Cross(l1.p1,l1.p2,e)>eps) return 0; else return 1; } //求两条直线l1和l2的交点 Point GetIntersect(Line l1, Line l2) { Point res; res.x=(l1.b*l2.c-l2.b*l1.c)/(l1.a*l2.b-l2.a*l1.b); res.y=(l1.c*l2.a-l2.c*l1.a)/(l1.a*l2.b-l2.a*l1.b); return res; } //求质量均匀分布的多边形的重心 Point BaryCenter(Point *p,int n) { Point res(0,0); double s=0.0,t; int i; for (i=1;i<n-1;i++) { t=Cross(p[i]-p[0],p[i+1]-p[0])/2; //分割成三角形,算面积 s+=t; res.x+=(p[0].x+p[i].x+p[i+1].x)*t; //将面积做为重量放在三角形的重心上 res.y+=(p[0].y+p[i].y+p[i+1].y)*t; //质量*坐标,三角形重心所需的除以3放在后面 } res.x/=(3*s); res.y/=(3*s); return res; } //求多边形面积,要求点集按逆时针顺序 double ConvexPolygonArea(Point *p,int n) { int i; double area=0; for (i=1;i<n-1;i++) area+=Cross(p[i]-p[0],p[i+1]-p[0]); return area/2; } //求以原点为圆心,过a、b,半径为r的扇形面积 double SectorArea(Point a,Point b,double r) { double theta=atan2(a.y,a.x)-atan2(b.y,b.x); while (theta<=0) theta+=2*pi; while (theta>=2*pi) theta-=2*pi; theta=min(theta,2*pi-theta); return r*r*theta/2; } //求以o为圆心r为半径的圆与线段(直线)ab的交点 //返回的ret是交点,num是交点数量 //把判断t范围的两个if去掉能够计算圆与直线的交点 void CircleCrossLine(Point a,Point b,Point o,double r,Point ret[],int &num) { double x0=o.x,y0=o.y; double x1=a.x,y1=a.y; double x2=b.x,y2=b.y; double dx=x2-x1,dy=y2-y1; double A=dx*dx+dy*dy; double B=2*dx*(x1-x0)+2*dy*(y1-y0); double C=(x1-x0)*(x1-x0)+(y1-y0)*(y1-y0)-r*r; double delta=B*B-4*A*C; num=0; if (epssgn(delta)>=0) { double t1=(-B-sqrt(fabs(delta)))/(2*A); double t2=(-B+sqrt(fabs(delta)))/(2*A); if (epssgn(t1-1.0)<=0 && epssgn(t1)>=0) ret[num++]=Point(x1+t1*dx,y1+t1*dy); if (epssgn(t2-1.0)<=0 && epssgn(t2)>=0) ret[num++]=Point(x1+t2*dx,y1+t2*dy); } } //求以原点为圆心半径为r的园,与原点、a、b组成的三角形的重叠面积 double Calc(Point a,Point b,double r) { Point p[2]; int num=0; int ina=(epssgn(len(a)-r)<0); int inb=(epssgn(len(b)-r)<0); if (ina) { if (inb) return fabs(Cross(a,b))/2.0; //情形1:三角形彻底在圆内:直接求三角形面积 else //情形2:一个点在圆内,一个点在圆外:分割成一个彻底在圆内的三角形和一个扇形 { CircleCrossLine(a,b,Point(0,0),r,p,num); return SectorArea(b,p[0],r)+fabs(Cross(a,p[0]))/2.0; } } else { CircleCrossLine(a,b,Point(0,0),r,p,num); //情形2:一个点在圆内,一个点在圆外:分割成一个彻底在圆内的三角形和一个扇形 if (inb) return SectorArea(p[0],a,r)+fabs(Cross(p[0],b))/2.0; else { //情形4:两个点都在圆外,可是两点的连线与圆有两个焦点:分割成一个彻底在圆内的三角形和两个扇形 if (num==2) return SectorArea(a,p[0],r)+SectorArea(p[1],b,r)+fabs(Cross(p[0],p[1]))/2.0; //情形3:两个点都在圆外,而且相交处为一个扇形:直接求扇形面积 else return SectorArea(a,b,r); } } } //求质量均匀分布的三角形的重心 Point TriangleMassCenter(Point a,Point b,Point c) { return (a+b+c)/3.0; } //求三角形的外心 Point TriangleCircumCenter(Point a,Point b,Point c) { Point res; double a1=atan2(b.y-a.y,b.x-a.x)+pi/2; double a2=atan2(c.y-b.y,c.x-b.x)+pi/2; double ax=(a.x+b.x)/2; double ay=(a.y+b.y)/2; double bx=(c.x+b.x)/2; double by=(c.y+b.y)/2; double r1=(sin(a2)*(ax-bx)+cos(a2)*(by-ay))/(sin(a1)*cos(a2)-sin(a2)*cos(a1)); return Point(ax+r1*cos(a1),ay+r1*sin(a1)); } //求三角形的垂心 Point TriangleOrthoCenter(Point a,Point b,Point c) { return TriangleMassCenter(a,b,c)*3.0-TriangleCircumCenter(a,b,c)*2.0; } //求三角形的心里 Point TriangleInnerCenter(Point a,Point b,Point c) { Point res; double la=len(b-c); double lb=len(a-c); double lc=len(a-b); res.x=(la*a.x+lb*b.x+lc*c.x)/(la+lb+lc); res.y=(la*a.y+lb*b.y+lc*c.y)/(la+lb+lc); } /*------Graham求凸包----*/ int cmp(Point &a,Point &b) { if (a.y==b.y) return (a.x<b.x); return (a.y<b.y); } //对全部的点进行一次排序 void QSort(Point p[],int l,int r) { int i=l,j=r; Point mid=p[(l+r)/2],swap; while (i<=j) { while (cmp(p[i],mid)) i++; while (cmp(mid,p[j])) j--; if (i<=j) { swap=p[i]; p[i]=p[j]; p[j]=swap; i++;j--; } } if (i<r) QSort(p,i,r); if (l<j) QSort(p,l,j); } //n为原图的点数,p[]为原图的点(0~n-1),top为凸包点的数量(0~top-1),res[]为凸包点的下标,。 int Graham(Point p[],int n,int res[]) { int i,len,top; top=1; QSort(p,0,n-1); for (i=0;i<3;i++) res[i]=i; for (i=2;i<n;i++) { while (top && epssgn(Cross(p[i],p[res[top]],p[res[top-1]]))>=0) top--; res[++top]=i; } len=top; res[++top]=n-2; for (i=n-3;i>=0;i--) { while (top!=len && epssgn(Cross(p[i], p[res[top]], p[res[top-1]]))>=0) top--; res[++top]=i; } return top; } //判断点x是否在凸包p[res[1~chnum]]中,返回1则为在内部或边界上 int PointInConvexHull(Point p[],int res[],int chnum,Point x) { //找一个凸包内的点 Point g=(p[res[0]]+p[res[chnum/3]]+p[res[2*chnum/3]])/3.0; int l=0,r=chnum,mid; //二分凸包 while (l+1<r) { mid=(l+r)/2; if (epssgn(Cross(p[res[l]]-g,p[res[mid]]-g))>0) { if (epssgn(Cross(p[res[l]]-g,x-g))>=0 && epssgn(Cross(p[res[mid]]-g,x-g))<0) r=mid; else l=mid; } else { if (epssgn(Cross(p[res[l]]-g,x-g))<0 && epssgn(Cross(p[res[mid]]-g,x-g))>=0) l=mid; else r=mid; } } r%=chnum; if (epssgn(Cross(p[res[r]]-x,p[res[l]]-x))==-1) return 1; else return 0; } //旋转卡壳求凸包p[res[1~chnum]]的直径,对踵点数量appnum,存储在app[][2]中 double Diameter(Point p[],int res[],int chnum,int app[][2],int &appnum) //AntiPodal Point { int i,j; double ret=0,nowlen; res[chnum]=res[0]; j=1; appnum=0; for (i=0;i<chnum;i++) { while (Cross(p[res[i]]-p[res[i+1]],p[res[j+1]]-p[res[i+1]]) <Cross(p[res[i]]-p[res[i+1]],p[res[j]]-p[res[i+1]])) { j++; j%=chnum; } app[appnum][0]=res[i]; app[appnum][1]=res[j]; appnum++; nowlen=dis(p[res[i]],p[res[j]]); if (nowlen>ret) ret=nowlen; nowlen=dis(p[res[i+1]],p[res[j+1]]); if (nowlen>ret) ret=nowlen; } return ret; } //旋转卡壳求凸包p[res[1~chnum]]内最大面积三角形 double ConvexHullMaxTriangleArea(Point p[],int res[],int chnum) { int i,j,k; double area=0,tmp; k=2;j=1; res[chnum]=res[0]; for (i=0;i<chnum;i++) { //卡住i,j,若向前旋转k能令面积更大的话就一直转一下去 while (fabs(Cross(p[res[j]]-p[res[i]],p[res[(k+1)%chnum]]-p[res[i]])) >fabs(Cross(p[res[j]]-p[res[i]],p[res[k]] -p[res[i]]))) k=(k+1)%chnum; tmp=fabs(Cross(p[res[j]]-p[res[i]],p[res[k]]-p[res[i]])); if (tmp>area) area=tmp; //卡住i,k,若向前旋转j能令面积更大的话就一直转一下去 while (fabs(Cross(p[res[(j+1)%chnum]]-p[res[i]],p[res[k]]-p[res[i]])) >fabs(Cross(p[res[j]]- p[res[i]],p[res[k]]-p[res[i]]))) j=(j+1)%chnum; tmp=fabs(Cross(p[res[j]]-p[res[i]],p[res[k]]-p[res[i]])); if (tmp>area) area=tmp; //j,k转到最大位置,向前转i(i++) } return area/2; } //求两凸包p、q间的最小距离,注意调用的时候要交换参数位置调用两次 //卡住p(p步进),旋转q,求能获得的最小距离 double TwoConvexHullMinDist(Point P[],Point Q[],int n,int m) { int YMinP=0,YMaxQ=0,i; double tmp,ans=999999999; for (i=0;i<n;i++) if (P[i].y<P[YMinP].y) YMinP=i; for (i=0;i<m;i++) if (Q[i].y>Q[YMaxQ].y) YMaxQ=i; P[n]=P[0]; Q[m]=Q[0]; for (i=0;i<n;i++) { //注意,tmp保存的是>比较的结果 while (tmp=Cross(Q[YMaxQ+1]-P[YMinP+1],P[YMinP]-P[YMinP+1]) >Cross(Q[YMaxQ] -P[YMinP+1],P[YMinP]-P[YMinP+1])) YMaxQ=(YMaxQ+1)%m; if (tmp<0) ans=min(ans,PointToSegDist(P[YMinP],P[YMinP+1],Q[YMaxQ])); else ans=min(ans,TwoSegMinDist (P[YMinP],P[YMinP+1],Q[YMaxQ],Q[YMaxQ+1])); YMinP=(YMinP+1)%n; } return ans; } /*------Graham求凸包----*/ /*-----半平面交O(nlogn)模板-----*/ int cmp(const Line & l1,const Line & l2) { int d=epssgn(l1.angle-l2.angle); if (!d) return (epssgn(Cross(l2.p1-l1.p1,l2.p2-l1.p1))>0); //极角相同时,将更靠半平面里面的放在前面 return d<0; } void QSort(Line L[],int l,int r) { int i=l,j=r; Line swap,mid=L[(l+r)/2]; while (i<=j) { while (cmp(L[i],mid)) i++; while (cmp(mid,L[j])) j--; if (i<=j) { swap=L[i]; L[i]=L[j]; L[j]=swap; i++;j--; } } if (i<r) QSort(L,i,r); if (l<j) QSort(L,l,j); } //判断l1与l2的交点是否在半平面hpl外 int IntersectionOutOfHalfPlane(Line &hpl,Line &l1,Line &l2) { Point p=GetIntersect(l1,l2); return (epssgn(Cross(hpl.p1-p,hpl.p2-p))<0); } //半平面hpl向内推动dis长度 Line HalfPlaneMoveIn(Line &hpl,double &dis) { double dx=hpl.p1.x-hpl.p2.x; double dy=hpl.p1.y-hpl.p2.y; double ll=len(hpl.p1-hpl.p2); Point pa=Point(dis*dy/ll+hpl.p1.x,hpl.p1.y-dis*dx/ll); Point pb=Point(dis*dy/ll+hpl.p2.x,hpl.p2.y-dis*dx/ll); return Line(pa,pb); } //求n个半平面l的半平面交,获得的交点储存在p中,交点数目返回到pn //能够将一个多边形每一条边当作半平面,求出来的交就是多边形的核,要求pn>=3 void HalfPlaneIntersect(Line l[],int n,Point p[],int &pn) { int i,j; int dq[MaxN],top,bot; //排序是在知足全部半平面A*x+B*y+C>0或(<,<=,>=), //也就是全部半平面的符号均相同的状况下对极角进行排序。 QSort(l,0,n-1); //极角相同时,只保留最靠里面的那条 for (i=j=0;i<n;i++) if (epssgn(l[i].angle-l[j].angle)>0) l[++j]=l[i]; n=j+1; dq[0]=0; //双端队列 dq[1]=1; top=1; //顶部和底部 bot=0; for (i=2;i<n;i++) { //当栈顶的两条直线交点在当前半平面外部时,弹栈 while (top>bot && IntersectionOutOfHalfPlane(l[i],l[dq[top]],l[dq[top-1]])) top--; //因为求的是一个凸多边形,因此当半平面转过接近一圈时,某个半平面知足上一个while的条件后, //它又会影响到底部的两条直线,当底部的两条直线的交点,在当前的半平面外部时,底部弹栈 while (top>bot && IntersectionOutOfHalfPlane(l[i],l[dq[bot]],l[dq[bot+1]])) bot++; dq[++top]=i; //当前半平面入栈 } //当最顶部的两条直线的交点不在最底部的半平面内时,顶部的那个半平面是多余的,顶部弹栈 while (top>bot && IntersectionOutOfHalfPlane(l[dq[bot]],l[dq[top]],l[dq[top-1]])) top--; //当最底部的两条直线的交点不在最顶部的半平面内时,底部的那个半平面是多余的,底部弹栈 while (top>bot && IntersectionOutOfHalfPlane(l[dq[top]],l[dq[bot]],l[dq[bot+1]])) bot++; dq[++top]=dq[bot]; //将最底部的半平面放到最顶部来,方便下面求顶点 for (pn=0,i=bot;i<top;i++,pn++) p[pn]=GetIntersect(l[dq[i+1]],l[dq[i]]); } /*-----半平面交O(nlogn)模板-----*/ /*-----半平面交O(n^2)模板-----*/ //p是如今切出来的半平面的点,pn是点的数量,须要按顺时针或者逆时针排序 //新的半平面直线为ax+by+c=0 void HalfPlaneCut(Point p[],int &pn,double a,double b,double c) { int cnt=0,i; Point tp[MaxN]; //temp_p //如今交出来的点在新的半平面内,保留 for (i=1;i<=pn;i++) if (epssgn(a*p[i].x+b*p[i].y+c)>=0) tp[++cnt]=p[i]; else //不然若是其先后的点在半平面内,从新切割 { if (epssgn(a*p[i-1].x+b*p[i-1].y+c)>0) tp[++cnt]=GetIntersect(Line(p[i-1],p[i]),Line(a,b,c)); if (epssgn(a*p[i+1].x+b*p[i+1].y+c)>0) tp[++cnt]=GetIntersect(Line(p[i],p[i+1]),Line(a,b,c)); } pn=cnt; for (i=1;i<=cnt;i++) p[i]=tp[i]; p[0]=p[cnt]; p[cnt+1]=p[1]; } /*-----半平面交O(n^2)模板-----*/