LA 4818 Largest Empty Circle on a Segment 开区间求并集

题目地址：LA4818

先尝试一下求点到线段的距离的最小值，三分圆心的位置，样例能过，不过只是巧合吧，函数不是单峰的，（求和才是，min不是）。

然后二分（所求的）半径，判断一个半径是否大了，就用每一个线段做出一个区域 ，和x轴相交，得到一个“圆心不能取的区域”。N 

然后每一个线段都对应一个“不能取的区域”，然后求并集就可以了。

值得注意的是 1 如果 交点的最小值和最大值是一个点，那么实际上不用对应取不到的区间。

                         2  端点的选取，如果 i<j   s[i].B=s[j].A 的时候，世界上是应该分为两个区间的，因为允许“touch”，那么这个公共点是允许取的。

代码：

#include<iostream>#include<cmath>#include<cstdio>#include<algorithm>#include<vector>#include<cstring>#include<string>const  double INF=0x3fffffff;const  double eps=1e-10;const double PI=acos(-1.0);using namespace std;struct Point{    double x;    double y;    Point(double x=0,double y=0):x(x),y(y){}    };int dcmp(double x)  {return (x>eps)-(x<-eps); }int sgn(double x)  {return (x>eps)-(x<-eps); }typedef  Point  Vector;Vector  operator +(Vector A,Vector B) { return Vector(A.x+B.x,A.y+B.y);}Vector  operator -(Vector A,Vector B) { return Vector(A.x-B.x,A.y-B.y); }Vector  operator *(Vector A,double p) { return Vector(A.x*p,A.y*p);  }Vector  operator /(Vector A,double p) {return Vector(A.x/p,A.y/p);}ostream &operator<<(ostream & out,Point & P) { out<<P.x<<' '<<P.y; return out;}//bool  operator< (const Point &A,const Point &B) { return dcmp(A.x-B.x)<0||(dcmp(A.x-B.x)==0&&dcmp(A.y-B.y)<0); }bool  operator== ( const Point &A,const Point &B) { return dcmp(A.x-B.x)==0&&dcmp(A.y-B.y)==0;}double  Dot(Vector A,Vector B) {return A.x*B.x+A.y*B.y;}double  Cross(Vector A,Vector B)  {return A.x*B.y-B.x*A.y; }double  Length(Vector A)  { return sqrt(Dot(A, A));}double  Angle(Vector A,Vector B) {return acos(Dot(A,B)/Length(A)/Length(B));}double  Area2(Point A,Point B,Point C ) {return Cross(B-A, C-A);}Vector Rotate(Vector A,double rad) { return Vector(A.x*cos(rad)-A.y*sin(rad),A.x*sin(rad)+A.y*cos(rad));}Vector Normal(Vector A) {double L=Length(A);return Vector(-A.y/L,A.x/L);}Point GetLineIntersection(Point P,Vector v,Point Q,Vector w){    Vector u=P-Q;    double t=Cross(w, u)/Cross(v,w);    return P+v*t;    }double DistanceToLine(Point P,Point A,Point B){    Vector v1=P-A; Vector v2=B-A;    return fabs(Cross(v1,v2))/Length(v2);    }double DistanceToSegment(Point P,Point A,Point B){    if(A==B)  return Length(P-A);        Vector v1=B-A;    Vector v2=P-A;    Vector v3=P-B;        if(dcmp(Dot(v1,v2))==-1)    return  Length(v2);    else if(Dot(v1,v3)>0)    return Length(v3);        else return DistanceToLine(P, A, B);    }Point GetLineProjection(Point P,Point A,Point B){    Vector v=B-A;    Vector v1=P-A;    double t=Dot(v,v1)/Dot(v,v);        return  A+v*t;}bool  SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){    double c1=Cross(b1-a1, a2-a1);    double c2=Cross(b2-a1, a2-a1);    double c3=Cross(a1-b1, b2-b1);    double c4=Cross(a2-b1, b2-b1);        return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0 ;    }bool  OnSegment(Point P,Point A,Point B){    return dcmp(Cross(P-A, P-B))==0&&dcmp(Dot(P-A,P-B))<=0;}bool  SegmentCommon(Point a1,Point a2,Point b1,Point b2){    double c1=Cross(b1-a1, a2-a1);    double c2=Cross(b2-a1, a2-a1);    double c3=Cross(a1-b1, b2-b1);    double c4=Cross(a2-b1, b2-b1);        if(c1==0&&c2==0&&c3==0&&c4==0)    {        if(OnSegment(a1, b1, b2))  return 1;        if(OnSegment(a2, b1, b2))  return 1;                return 0;    }        return dcmp(c1)*dcmp(c2)<=0&&dcmp(c3)*dcmp(c4)<=0 ;    }double PolygonArea(Point *p,int n){    double area=0;        for(int i=1;i<n-1;i++)    {        area+=Cross(p[i]-p[0], p[i+1]-p[0]);            }    return area/2;    }Point  read_point(){    Point P;        scanf("%lf%lf",&P.x,&P.y);        return  P;}// ---------------与圆有关的--------struct Circle{    Point c;    double r;        Circle(Point c=Point(0,0),double r=0):c(c),r(r) {}        Point point(double a)    {        return Point(c.x+r*cos(a),c.y+r*sin(a));    }        };struct  Line{    Point p;    Vector v;    Line(Point p=Point(0,0),Vector v=Vector(0,1)):p(p),v(v) {}        Point point(double t)    {        return Point(p+v*t);    }    };int getLineCircleIntersection(Line L,Circle C,double &t1,double &t2,vector<Point> &sol){    double a=L.v.x;    double b=L.p.x-C.c.x;    double c=L.v.y;    double d=L.p.y-C.c.y;        double e=a*a+c*c;    double f=2*(a*b+c*d);    double g=b*b+d*d-C.r*C.r;        double delta=f*f-4*e*g;        if(dcmp(delta)<0) return 0;        if(dcmp(delta)==0)    {        t1=t2=-f/(2*e);        sol.push_back(L.point(t1));        return 1;    }        else    {        t1=(-f-sqrt(delta))/(2*e);        t2=(-f+sqrt(delta))/(2*e);                sol.push_back(L.point(t1));        sol.push_back(L.point(t2));                return 2;    }    }// 向量极角公式double angle(Vector v)  {return atan2(v.y,v.x);}int getCircleCircleIntersection(Circle C1,Circle C2,vector<Point> &sol){    double d=Length(C1.c-C2.c);        if(dcmp(d)==0)    {        if(dcmp(C1.r-C2.r)==0)  return -1;  // 重合        else return 0;    //  内含  0 个公共点    }        if(dcmp(C1.r+C2.r-d)<0)  return 0;  // 外离    if(dcmp(fabs(C1.r-C2.r)-d)>0)  return 0;  // 内含        double a=angle(C2.c-C1.c);    double da=acos((C1.r*C1.r+d*d-C2.r*C2.r)/(2*C1.r*d));        Point p1=C1.point(a-da);    Point p2=C1.point(a+da);        sol.push_back(p1);        if(p1==p2)  return 1; // 相切    else    {        sol.push_back(p2);        return 2;    }}//  求点到圆的切线int getTangents(Point p,Circle C,Vector *v){    Vector u=C.c-p;        double dist=Length(u);        if(dcmp(dist-C.r)<0)  return 0;        else if(dcmp(dist-C.r)==0)    {        v[0]=Rotate(u,PI/2);        return 1;    }        else    {                double ang=asin(C.r/dist);        v[0]=Rotate(u,-ang);        v[1]=Rotate(u,+ang);        return 2;    }    }//  求两圆公切线int getTangents(Circle A,Circle B,Point *a,Point *b){    int cnt=0;        if(A.r<B.r)    {        swap(A,B); swap(a, b);  //  有时需标记    }        double d=Length(A.c-B.c);        double rdiff=A.r-B.r;    double rsum=A.r+B.r;        if(dcmp(d-rdiff)<0)  return 0;   // 内含        double base=angle(B.c-A.c);        if(dcmp(d)==0&&dcmp(rdiff)==0)   return -1 ;  // 重合 无穷多条切线        if(dcmp(d-rdiff)==0)             // 内切   外公切线    {        a[cnt]=A.point(base);        b[cnt]=B.point(base);        cnt++;        return 1;    }        // 有外公切线的情形        double ang=acos(rdiff/d);    a[cnt]=A.point(base+ang);    b[cnt]=B.point(base+ang);    cnt++;    a[cnt]=A.point(base-ang);    b[cnt]=B.point(base-ang);    cnt++;        if(dcmp(d-rsum)==0)     // 外切 有内公切线    {        a[cnt]=A.point(base);        b[cnt]=B.point(base+PI);        cnt++;    }        else  if(dcmp(d-rsum)>0)   // 外离   又有两条外公切线    {        double  ang_in=acos(rsum/d);        a[cnt]=A.point(base+ang_in);        b[cnt]=B.point(base+ang_in+PI);        cnt++;        a[cnt]=A.point(base-ang_in);        b[cnt]=B.point(base-ang_in+PI);        cnt++;    }        return cnt;}//  几何算法模板int  isPointInPolygon(Point p,Point * poly,int n){    int wn=0;    for(int i=0;i<n;i++)    {        if(OnSegment(p, poly[i], poly[(i+1)%n]))  return -1;        int k=dcmp(Cross(poly[(i+1)%n]-poly[i], p-poly[i]));        int d1=dcmp(poly[i].y-p.y);        int d2=dcmp(poly[(i+1)%n].y-p.y);                if(k>0&&d1<=0&&d2>0) wn++;        if(k<0&&d2<=0&&d1>0) wn--;            }        if(wn!=0)  return 1;    else   return 0;    }bool  SegmentLineIntersection(Point a1,Point a2,Point b1,Point b2)  //a1,a2 是直线  规范相交{    double c1=Cross(b1-a1, a2-a1);    double c2=Cross(b2-a1, a2-a1);    return dcmp(c1)*dcmp(c2)<=0;}Point ori(0,0);Point o_X(1,0);int n;double L,R;struct Segment{    Point A;    Point B;    Segment(Point A=ori,Point B=ori):A(A),B(B) {}    };bool seg_cmp(Segment A,Segment B){    return  A.A<B.A;    }Point a[2005],b[2005],a1[2005],a2[2005],b1[2005],b2[2005];Circle C1[2005],C2[2005];Segment s[2005];vector<Point> inter[2005];double  minDist(double x){    Point p=Point(x,0);    bool first=1;        double ans=0;    for(int i=0;i<n;i++)    {            if(first)      {          ans=DistanceToSegment(p, a[i], b[i]);          first=0;      }      else      {          ans=min(ans,DistanceToSegment(p, a[i], b[i]));                }    }        return ans;}vector<Segment> Union;Line ox=Line(ori,Vector(1,0));bool toobig(double x){    // init    //!    for(int i=0;i<n;i++)    {        s[i]=Segment(ori,ori);    }    Union.clear();        for(int i=0;i<n;i++)    {        inter[i].clear();    }            for(int i=0;i<n;i++)    {        C1[i].r=x;        C2[i].r=x;    }        double t1,t2;    for(int i=0;i<n;i++)    {        getLineCircleIntersection(ox,C1[i], t1, t2,inter[i]);        getLineCircleIntersection(ox,C2[i], t1, t2,inter[i]);                Vector norm=Normal(b[i]-a[i]);               a1[i]=a[i]+norm*x;        a2[i]=a[i]-norm*x;                b1[i]=b[i]+norm*x;        b2[i]=b[i]-norm*x;            if(SegmentLineIntersection(ori, o_X, a1[i], b1[i]))        {           if(dcmp(Cross(o_X,b1[i]-a1[i]))!=0)              {                  Point ip=GetLineIntersection(ori, o_X, a1[i], b1[i]-a1[i]);                  inter[i].push_back(ip);              }        }                        if(SegmentLineIntersection(ori, o_X, a2[i], b2[i]))        {            if(dcmp(Cross(o_X,b2[i]-a2[i]))!=0)            {                 Point ip=GetLineIntersection(ori, o_X, a2[i], b2[i]-a2[i]);                 inter[i].push_back(ip);            }        }                        sort(inter[i].begin(),inter[i].end());            int sz=inter[i].size();                if(sz>1)        {            Point i_l=inter[i][0];            Point i_r=inter[i][sz-1];                        s[i].A=i_l;            s[i].B=i_r;                    }                         }        s[n]=Segment(Point(-INF,0),Point(L,0));    s[n+1]=Segment(Point(R,0),Point(INF,0));        sort(s,s+n+2,seg_cmp);        Union.push_back(s[0]);    for(int i=1;i<n+2;i++)    {        if(s[i].A==ori&&s[i].B==ori)  continue;                Segment seg=Union[Union.size()-1];        Union.erase(Union.end()-1);                if(seg.B<s[i].A||seg.B==s[i].A)        {            Union.push_back(seg);            Union.push_back(s[i]);                    }                else        {            Segment newseg=Segment(seg.A,max(seg.B,s[i].B));            Union.push_back(newseg);        }            }            if(Union.size()==1)  return 1;        else  return 0;    }int main(){      int T;    cin>>T;    while(T--)    {        L=0;        cin>>n>>R;                                     for(int i=0;i<n;i++)        {            a[i]=read_point();            b[i]=read_point();            C1[i].c=a[i];            C2[i].c=b[i];        }                                double l=0,r=INF;                double mid;                        for(int i=0;i<50;i++)        {            mid=l+(r-l)/2;                        if(toobig(mid))            {                r=mid;                            }                        else l=mid;                                }                printf("%.3f\n",mid);            }}