UVA 11168 Airport 凸包+直线的一般式

   给N个点，找到一条直线，使得所有点到这条直线的距离之和最小。
   首先可以确定，这条直线一定是凸包上的一条直线，那么可以求出凸包，然后枚举每条直线，求出所有点到该直线的距离和，点到直线的距离公式为fabs(Ax+By+C)/sqrt(A^2+B^2)，而且所有点都在直线的一个方向，那么我们可以预先存下来所有x和y的和，这样给一个直线的一般式，我们就可以O（1）求出所有点到该直线的距离之和。

#include <iostream>#include <cstdio>#include <algorithm>#include <cstring>#include <string>#include <cmath>#include <vector>typedef double type;using namespace std;const double PI=acos(-1.0);const double eps=1e-10;struct Point{    type x,y;    Point(){}    Point(type a,type b)    {        x=a;        y=b;    }    void read()    {        scanf("%lf%lf",&x,&y);    }    void print()    {        printf("%.6lf %.6lf",x,y);    }};typedef Point Vector;Vector operator + (Vector A,Vector 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 * (Vector A,type p){    return Vector(A.x*p,A.y*p);}Vector operator / (Vector A,type p){    return Vector(A.x/p,A.y/p);}bool operator < (const Point &a,const Point &b){    return a.x<b.x || (a.x==b.x && a.y<b.y);}int dcmp(double x){    if (fabs(x)<eps) return 0;    else return x<0?-1:1;}bool operator == (const Point& a,const Point b){    return dcmp(a.x-b.x)==0 && dcmp(a.y-b.y)==0;}//atan2(x,y) :向量(x,y)的极角，即从x轴正半轴旋转到该向量方向所需要的角度。type Dot(Vector A,Vector B){    return A.x*B.x+A.y*B.y;}type Cross(Vector A,Vector B){    return A.x*B.y-A.y*B.x;}type Length(Vector A){    return sqrt(Dot(A,A));}type Angle(Vector A,Vector B){    return acos(Dot(A,B))/Length(A)/Length(B);}type 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)//单位法线,左转90度，长度归一{    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=B-A,v2=P-A;    return fabs(Cross(v1,v2))/Length(v1);}double DistanceToSegment(Point P,Point A,Point B){    if (A==B) return Length(P-A);    Vector v1=B-A,v2=P-A,v3=P-B;    if (dcmp(Dot(v1,v2))<0) return Length(v2);    else if (dcmp(Dot(v1,v3))>0) return Length(v3);    else return fabs(Cross(v1,v2))/Length(v1);}Point GetLineProjection(Point P,Point A,Point B)//P在AB上的投影{    Vector v=B-A;    return A+v*(Dot(v,P-A)/Dot(v,v));}bool SegmentProperIntersection(Point a1,Point a2,Point b1,Point b2){    double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1),    c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);    return dcmp(c1)*dcmp(c2)<0 && dcmp(c3)*dcmp(c4)<0;}bool OnSegment(Point p,Point a1,Point a2){    return dcmp(Cross(a1-p,a2-p))==0 && dcmp(Dot(a1-p,a2-p))<0;}double ConvexPolygonArea(Point* p,int n)//多边形面积{    double area=0;    for (int i=1; i<n-1; i++)    area+=Cross(p[i]-p[0],p[i+1]-p[0]);    return area/2.0;}double PolygonArea(Point* p,int n)//有向面积{    double area=0;    for (int i=1; i<n-1; i++)    area+=Cross(p[i]-p[0],p[i+1]-p[0]);    return area/2.0;}struct Line{    Point p;    Vector v;    double ang;    Line(){};    Line(Point PP,Vector vv)    {        p=PP;        v=vv;        ang=atan2(v.y,v.x);    }    bool operator< (const Line& L)const    {        return ang<L.ang;    }    Point point(double t)    {        return p+v*t;    }};struct Circle{    Point c;    double r;    Circle()    {    }    Circle(Point cc,double rr)    {        c=cc;        r=rr;    }    Point point(double a)    {        return Point(c.x+cos(a)*r,c.y+sin(a)*r);    }};int getLineCircleIntersection(Line L,Circle C,double& t1,double &t2,vector<Point>& sol){    double a=L.v.x, b=L.p.x-C.c.x, c=L.v.y, d=L.p.y-C.c.y;    double e=a*a+c*c,f=2*(a*b+c*d), g=b*b+d*d-C.r*C.r;    double delta=f*f-4*e*g;//判别式    if (dcmp(delta)<0) return 0;//相离    if (dcmp(delta)==0)    {        t1=t2=-f/(2*e);        sol.push_back(L.point(t1));        return 1;    }//相切    t1=(-f-sqrt(delta))/(2*e); sol.push_back(L.point(t1));    t2=(-f+sqrt(delta))/(2*e); sol.push_back(L.point(t2));    return 2;}double angle(Vector v)//向量极角{    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;        return 0;    }    if (dcmp(C1.r+C2.r-d)<0) return 0;    if (dcmp(fabs(C1.r-C2.r)-d)>0) return 0;    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),p2=C1.point(a+da);    sol.push_back(p1);    if (p1==p2) return 1;    sol.push_back(p2);    return 2;}int getTangents(Point p,Circle C,Vector* v){    Vector u=C.c-p;    double dist=Length(u);    if (dist<C.r) 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;    }}int getTangents(Circle A,Circle B,Point* a,Point* b){    int cnt=0;    if (A.r<B.r)    {        swap(A,B); swap(a,b);    }    int d2=(A.c.x-B.c.x)*(A.c.x-B.c.x)+(A.c.y-B.c.y)*(A.c.y-B.c.y);    int rdiff=A.r-B.r;    int rsum=A.r+B.r;    if (d2<rdiff*rdiff) return 0;    double base=atan2(B.c.y-A.c.y,B.c.x-A.c.x);    if (d2==0 && A.r==B.r) return -1;    if (d2==rdiff*rdiff)    {        a[cnt]=A.point(base); b[cnt]=B.point(base); cnt++;        return 1;    }    double ang=acos((A.r-B.r)/sqrt(d2));    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 (d2==rsum*rsum)    {        a[cnt]=A.point(base); b[cnt]=B.point(PI+base); cnt++;    }    else if (d2>rsum*rsum)    {        double ang=acos((A.r+B.r)/sqrt(d2));        a[cnt]=A.point(base+ang); b[cnt]=B.point(PI+base+ang); cnt++;        a[cnt]=A.point(base-ang); b[cnt]=B.point(PI+base-ang); cnt++;    }    return cnt;}int ConvexHull(Point *p, int n,Point *ch){    sort(p,p+n);    int m=0;    for (int i=0; i<n; i++)    {        while(m>1 && Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2])<=0) m--;        ch[m++]=p[i];    }    int k=m;    for (int i=n-2; i>=0; i--)    {        while(m>k && Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2])<=0) m--;        ch[m++]=p[i];    }    if (n>1) m--;    return m;}Point poly[24300];Point ch[24300];double sumx,sumy;int main(){    int tt,n;//    freopen("in.txt","r",stdin);    scanf("%d",&tt);    for (int ii=1; ii<=tt; ii++)    {        scanf("%d",&n);        sumx=sumy=0;        for (int i=0; i<n; i++)        {            poly[i].read();            sumx+=poly[i].x;            sumy+=poly[i].y;        }        Point p,q;        int size=ConvexHull(poly,n,ch);        double x2,x1,y2,y1;        double minn=1e10;        for (int i=0; i<size; i++)        {            double A,B,C;            int j=(i+1)%size;            x1=ch[i].x; y1=ch[i].y;            x2=ch[j].x; y2=ch[j].y;            if (dcmp(x1-x2)!=0 && dcmp(y1-y2)!=0)            {                A=(y2-y1);                B=-(x2-x1);                C=-x1*(y2-y1)+y1*(x2-x1);            }            else if (dcmp(x1-x2)==0 && dcmp(y1-y2)==0) continue;            else if (dcmp(x1-x2)==0)            {                A=1;                B=0;                C=-x1;            }            else if (dcmp(y1-y2)==0)            {                A=0;                B=1;                C=-y1;            }            minn=min(minn,fabs(A*sumx+B*sumy+n*C)/sqrt(A*A+B*B));        }        if (n>2)        printf("Case #%d: %.3lf\n",ii,minn/n);        else        printf("Case #%d: 0.000\n",ii);    }}