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);
}

}