Airport UVA - 11168(凸包+解析几何(向量转直线解析式))

题意：传送门
题解：要求所有点在直线同侧，因此直线不能穿过凸包，选择凸包的边所在的直线，比选择和凸包相离的直线更划算。但是凸包上的边有$n$条，如果每个点对应都算一次，那么$O(n^2)$岂不是$T$，如果能$O(1)$算出来，岂不是很好，那么改用解析几何的方式，直线的一般方程式为$Ax+By+C=0$，点$(x_0,y_0)$到该直线的距离为$$\frac{|A{x}_0+B{y}_0+C|}{\sqrt{A^2+B^2}}$$，所有点都是直线的同一侧，那么所有的$|A{x}_0+B{y}_0+C|$的正负号相同，这样就可以预处理出所有$x$和$y$的坐标之和，$O(1)$时间内算出总距离。
$code:$

#include<bits/stdc++.h>
#define mm(a,b) memset(a,b,sizeof(a))
#define ACCELERATE (ios::sync_with_stdio(false),cin.tie(0))
#define pii pair<int,int>
#define pdd pair<double,double>
#define pll pair<long long,long long>
#define mp make_pair
#define pb push_back
#define fi first
#define se second
#define rep(i,a,b) for(int i=(a);i<=(b);i++)
#define per(i,a,b) for(int i=(a);i>=(b);i--)
#define rush() int T;scanf("%d",&T);while(T--)
#define debug(x) cout<<#x<<": "<<x<<endl
#define sc(a) scanf("%d",&a)
#define sc2(a,b) scanf("%d%d",&a,&b)
#define sc3(a,b,c) scanf("%d%d%d",&a,&b,&c)
#define pf(x) printf("%d\n",x)
#define pf2(x,y) printf("%d %d\n",x,y)
#define pf3(x,y,z) printf("%d %d %d\n",x,y,z)
#define all(x) (x).begin(),(x).end()
#define PI acos(-1.0)
#define E exp(1.0)
#define db double
#define ll long long
#define ld long double
#define ull unsigned long long
//#define io
using namespace std;
const int inf=0x3f3f3f3f;
const ll INF=0x3f3f3f3f3f3f3f3f;
//#define gc getchar
//char buf[1<<21],*p1=buf,*p2=buf;
//inline int gc(){return p1==p2&&(p2=(p1=buf)+fread(buf,1,1<<21,stdin),p1==p2)?EOF:*p1++;}
//inline int read()
//{
    
    
//    int ret=0,f=0;char c=gc();
//    while(!isdigit(c)){if(c=='-')f=1;c=gc();}
//    while(isdigit(c)){ret=ret*10+c-48;c=gc();}
//    if(f)return -ret;return ret;
//}
const int N=10000+5;
const db EPS=1e-8;
inline int sign(db a){
    
    return a<-EPS?-1:a>EPS;}
inline int cmp(db a,db b){
    
    return sign(a-b);}
inline db sqr(db x){
    
    return x*x;}
struct Point{
    
    
    db x,y;
    Point(){
    
    }
    Point(db _x,db _y):x(_x),y(_y){
    
    }
    void input(){
    
    scanf("%lf%lf",&x,&y);}
    void output(){
    
    printf("%.10f %.10f\n",x,y);}
    bool operator == (Point b)const{
    
    
        return sign(x-b.x)==0&&sign(y-b.y)==0;
    }
    bool operator < (Point b)const{
    
    
        return sign(x-b.x)==0?sign(y-b.y)<0:x<b.x;
    }
    Point operator + (const Point &b)const{
    
    
        return Point(x+b.x,y+b.y);
    }
    Point operator - (const Point &b)const{
    
    
        return Point(x-b.x,y-b.y);
    }
    db operator ^ (const Point &b)const{
    
    
        return x*b.y-y*b.x;
    }
    db operator * (const Point &b)const{
    
    
        return x*b.x+y*b.y;
    }
    db len(){
    
    return hypot(x,y);}
    db len2(){
    
    return x*x+y*y;}
    db distance(Point p){
    
    return hypot(x-p.x,y-p.y);}
    Point operator * (const db &k)const{
    
    
        return Point(x*k,y*k);
    }
    Point operator / (const db &k)const{
    
    
        return Point(x/k,y/k);
    }
    db rad(Point a,Point b){
    
    
        Point p=*this;
        return fabs(atan2(fabs((a-p)^(b-p)),(a-p)*(b-p)));
    }
    Point unit(){
    
    return *this/len();}
    Point trunc(db r){
    
    
        db l=len();
        if(!sign(l))return *this;
        r/=l;
        return Point(x*r,y*r);
    }
    Point rotleft(){
    
    return Point(-y,x);}
    Point rotright(){
    
    return Point(y,-x);}
    Point rotate(Point p,db angle){
    
    
        Point v=(*this)-p;
        db c=cos(angle),s=sin(angle);
        return Point(p.x+v.x*c-v.y*s,p.y+v.x*s+v.y*c);
    }
};
#define dot(p1,p2,p3) ((p2.x-p1.x)*(p3.x-p1.x)+(p2.y-p1.y)*(p3.y-p1.y))
#define cross(p1,p2,p3) ((p2.x-p1.x)*(p3.y-p1.y)-(p3.x-p1.x)*(p2.y-p1.y))
#define crossOp(p1,p2,p3) sign(cross(p1,p2,p3))
struct Line{
    
    
    Point s,e;
    Line(){
    
    }
    Line(Point _s,Point _e):s(_s),e(_e){
    
    }
    bool operator == (Line v){
    
    
        return (s==v.s)&&(e==v.e);
    }
    Line(Point p,db angle){
    
    
        s=p;
        if(sign(angle-PI/2)==0)e=(s+Point(0,1));
        else e=(s+Point(1,tan(angle)));
    }
    Line(db a,db b,db c){
    
    
        if(sign(a)==0){
    
    
            s=Point(0,-c/b);e=Point(1,-c/b);
        }else if(sign(b)==0){
    
    
            s=Point(-c/a,0);e=Point(-c/a,1);
        }else{
    
    
            s=Point(0,-c/b);e=Point(1,(-c-a)/b);
        }
    }
    void input(){
    
    s.input();e.input();}
    void adjust(){
    
    if(e<s)swap(s,e);}
    db length(){
    
    return s.distance(e);}
    //[0,PI)
    db angle(){
    
    
        db k=atan2(e.y-s.y,e.x-s.x);
        if(sign(k)<0)k+=PI;
        if(sign(k-PI)==0)k-=PI;
        return k;
    }
    //line1 on the left|2 on the right|3 on line|
    int relation(Point p){
    
    
        int c=sign((p-s)^(e-s));
        if(c<0)return 1;
        else if(c>0)return 2;
        else return 3;
    }
    bool pointonseg(Point p){
    
    
        return sign((p-s)^(e-s))==0&&sign((p-s)*(p-e))<=0;
    }
    bool parallel(Line v){
    
    
        return sign((e-s)^(v.e-v.s))==0;
    }
    //|0 not intersect|1 non-standard|2 intersect|
    int segcrossseg(Line v){
    
    
        int d1=sign((e-s)^(v.s-s));
        int d2=sign((e-s)^(v.e-s));
        int d3=sign((v.e-v.s)^(s-v.s));
        int d4=sign((v.e-v.s)^(e-v.s));
        if((d1^d2)==-2&&(d3^d4)==-2)return 2;
        return (d1==0&&sign((v.s-s)*(v.s-e))<=0)||
            (d2==0&&sign((v.e-s)*(v.e-e))<=0)||
            (d3==0&&sign((s-v.s)*(s-v.e))<=0)||
            (d4==0&&sign((e-v.s)*(e-v.e))<=0);
    }
    //|0 not intersect|1 non-standard|2 intersect|
    int linecrossseg(Line v){
    
    
        int d1=sign((e-s)^(v.s-s));
        int d2=sign((e-s)^(v.e-s));
        if((d1^d2)==-2)return 2;
        return (d1==0||d2==0);
    }
    //|0 parallel|1 coincides|2 intersect|
    int linecrossline(Line v){
    
    
        if((*this).parallel(v))return v.relation(s)==3;
        return 2;
    }
    //要保证两直线不平行或重合
    Point crosspoint(Line v){
    
    
        db a1=(v.e-v.s)^(s-v.s);
        db a2=(v.e-v.s)^(e-v.s);
        return Point((s.x*a2-e.x*a1)/(a2-a1),(s.y*a2-e.y*a1)/(a2-a1));
    }
    db dispointtoline(Point p){
    
    
        return fabs((p-s)^(e-s))/length();
    }
    db dispointtoseg(Point p){
    
    
        if(sign((p-s)*(e-s))<0||sign((p-e)*(s-e))<0)return min(p.distance(s),p.distance(e));
        return dispointtoline(p);
    }
    //前提是两线段不相交，相交距离就是0了
    db dissegtoseg(Line v){
    
    
        return min(min(dispointtoseg(v.s),dispointtoseg(v.e)),min(v.dispointtoseg(s),v.dispointtoseg(e)));
    }
    Point lineprog(Point p){
    
    
        return s + ( ((e-s)*((e-s)*(p-s)))/((e-s).len2()) );
    }
    Point symmetrypoint(Point p){
    
    
        Point q = lineprog(p);
        return Point(2*q.x-p.x,2*q.y-p.y);
    }
};
struct circle{
    
    
    Point p;
    db r;
    circle(){
    
    }
    circle(Point _p,db _r):p(_p),r(_r){
    
    }
    circle(db _x,db _y,db _r){
    
    
        p=Point(_x,_y);r=_r;
    }
    Point point(db a){
    
    
        return Point(p.x+r*cos(a),p.y+r*sin(a));
    }
    //三角形的外接圆
    circle(Point a,Point b,Point c){
    
    
        Line u=Line((a+b)/2,((a+b)/2)+((b-a).rotleft()));
        Line v=Line((b+c)/2,((b+c)/2)+((c-b).rotleft()));
        p=u.crosspoint(v);
        r=p.distance(a);
    }
    //三角形的内切圆
    circle(Point a,Point b,Point c,bool t){
    
    
        Line u,v;
        db m=atan2(b.y-a.y,b.x-a.x),n=atan2(c.y-a.y,c.x-a.x);
        u.s=a;
        u.e=u.s+Point(cos((n+m)/2),sin((n+m)/2));
        m=atan2(a.y-b.y,a.x-b.x),n=atan2(c.y-b.y,c.x-b.x);
        v.s=b;
        v.e=v.s+Point(cos((n+m)/2),sin((n+m)/2));
        p=u.crosspoint(v);
        r=Line(a,b).dispointtoseg(p);
    }
    void input(){
    
    
        p.input();
        scanf("%lf",&r);
    }
    void output(){
    
    
        printf("%.2f %.2f %.2f\n",p.x,p.y,p,r);
    }
    bool operator == (circle v){
    
    
        return (p==v.p)&&sign(r-v.r)==0;
    }
    bool operator < (circle v)const{
    
    
        return ((p<v.p)||((p==v.p)&&sign(r-v.r)<0));
    }
    db area(){
    
    return PI*r*r;}
    db circumference(){
    
    return 2*PI*r;}
    //|0 not|1 on|2 in|
    int relation(Point b){
    
    
        db dst=b.distance(p);
        if(sign(dst-r)<0)return 2;
        else if(sign(dst-r)==0)return 1;
        return 0;
    }
    //|0 not|1 on|2 in|
    int relationseg(Line v){
    
    
        db dst=v.dispointtoseg(p);
        if(sign(dst-r)<0)return 2;
        else if(sign(dst-r)==0)return 1;
        return 0;
    }
    int relationline(Line v){
    
    
        db dst=v.dispointtoline(p);
        if(sign(dst-r)<0)return 2;
        else if(sign(dst-r)==0)return 1;
        return 0;
    }
    //|1 内含|2 内切|3 相交|4 外切|5 相离|
    int relationcircle(circle v){
    
    
        db d=p.distance(v.p);
        if(sign(d-r-v.r)>0)return 5;
        if(sign(d-r-v.r)==0)return 4;
        db l=fabs(r-v.r);
        if(sign(d-r-v.r)<0&&sign(d-l)>0)return 3;
        if(sign(d-l)==0)return 2;
        if(sign(d-l)<0)return 1;
    }
    //|0 没有交点|1 一个交点|2 两个交点|
    int pointcrosscircle(circle v,Point &p1,Point &p2){
    
    
        int rel=relationcircle(v);
        if(rel==1||rel==5)return 0;
        db d=p.distance(v.p);
        db l=(d*d+r*r-v.r*v.r)/(2*d);
        db h=sqrt(r*r-l*l);
        Point tmp=p+(v.p-p).trunc(l);
        p1 = tmp+((v.p-p).rotleft().trunc(h));
        p2 = tmp+((v.p-p).rotright().trunc(h));
        if(rel==2||rel==4)return 1;
        return 2;
    }
    int pointcrossline(Line v,Point &p1,Point &p2){
    
    
        if(!(*this).relationline(v))return 0;
        Point a=v.lineprog(p);
        db d=v.dispointtoline(p);
        d=sqrt(r*r-d*d);
        if(sign(d)==0){
    
    
            p1=a;p2=a;
            return 1;
        }
        p1=a+(v.e-v.s).trunc(d);
        p2=a-(v.e-v.s).trunc(d);
        return 2;
    }
    int getcircle(Point a,Point b,db r1,circle &c1,circle &c2){
    
    
        circle x(a,r1),y(b,r1);
        int t=x.pointcrosscircle(y,c1.p,c2.p);
        if(!t)return 0;
        c1.r=c2.r=r;
        return t;
    }
    int getcircle(Line u,Point q,db r1,circle &c1,circle &c2){
    
    
        db dis=u.dispointtoline(q);
        if(sign(dis-r1*2)>0)return 0;
        if(sign(dis)==0){
    
    
            c1.p=q+((u.e-u.s).rotleft().trunc(r1));
            c2.p=q+((u.e-u.s).rotright().trunc(r1));
            c1.r=c2.r=r1;
            return 2;
        }
        Line u1=Line((u.s+(u.e-u.s).rotleft().trunc(r1)),(u.e+(u.e-u.s).rotleft().trunc(r1)));
        Line u2=Line((u.s+(u.e-u.s).rotright().trunc(r1)),(u.e+(u.e-u.s).rotright().trunc(r1)));
        circle cc = circle(q,r1);
        Point p1,p2;
        if(!cc.pointcrossline(u1,p1,p2))cc.pointcrossline(u2,p1,p2);
        c1 = circle(p1,r1);
        if(p1 == p2){
    
    
            c2 = c1;
            return 1;
        }
        c2 = circle(p2,r1);
        return 2;
    }
    int getcircle(Line u,Line v,db r1,circle &c1,circle &c2,circle &c3,circle &c4){
    
    
        if(u.parallel(v))return 0;//两直线平行
        Line u1=Line(u.s+(u.e-u.s).rotleft().trunc(r1),u.e+(u.e-u.s).rotleft().trunc(r1));
        Line u2=Line(u.s+(u.e-u.s).rotright().trunc(r1),u.e+(u.e-u.s).rotright().trunc(r1));
        Line v1=Line(v.s+(v.e-v.s).rotleft().trunc(r1),v.e+(v.e-v.s).rotleft().trunc(r1));
        Line v2=Line(v.s+(v.e-v.s).rotright().trunc(r1),v.e+(v.e-v.s).rotright().trunc(r1));
        c1.r=c2.r=c3.r=c4.r=r1;
        c1.p=u1.crosspoint(v1);
        c2.p=u1.crosspoint(v2);
        c3.p=u2.crosspoint(v1);
        c4.p=u2.crosspoint(v2);
        return 4;
    }
    int getcircle(circle cx,circle cy,db r1,circle &c1,circle &c2){
    
    
        circle x(cx.p,r1+cx.r),y(cy.p,r1+cy.r);
        int t=x.pointcrosscircle(y,c1.p,c2.p);
        if(!t)return 0;
        c1.r=c2.r=r1;
        return t;
    }
    int tangentline(Point q,Line &u,Line &v){
    
    
        int x   = relation(q);
        if(x == 2)return 0;
        if(x == 1){
    
    
            u = Line(q,q + (q-p).rotleft());
            v = u;
            return 1;
        }
        db d = p.distance(q);
        db l = r*r/d;
        db h = sqrt(r*r-l*l);
        u = Line(q,p + ((q-p).trunc(l) + (q-p).rotleft().trunc(h)));
        v = Line(q,p + ((q-p).trunc(l) + (q-p).rotright().trunc(h)));
        return 2;
    }
    db areacircle(circle v){
    
    
        int rel = relationcircle(v);
        if(rel >= 4)return 0.0;
        if(rel <= 2)return min(area(),v.area());
        db d = p.distance(v.p);
        db hf = (r+v.r+d)/2.0;
        db ss = 2*sqrt(hf*(hf-r)*(hf-v.r)*(hf-d));
        db a1 = acos((r*r+d*d-v.r*v.r)/(2.0*r*d));
        a1 = a1*r*r;
        db a2 = acos((v.r*v.r+d*d-r*r)/(2.0*v.r*d));
        a2 = a2*v.r*v.r;
        return a1+a2-ss;
    }
    db areatriangle(Point a,Point b){
    
    
        if(sign((p-a)^(p-b)) == 0)return 0.0;
        Point q[5];
        int len = 0;
        q[len++] = a;
        Line l(a,b);
        Point p1,p2;
        if(pointcrossline(l,q[1],q[2])==2){
    
    
            if(sign((a-q[1])*(b-q[1]))<0)q[len++] = q[1];
            if(sign((a-q[2])*(b-q[2]))<0)q[len++] = q[2];
        }
        q[len++] = b;
        if(len == 4 && sign((q[0]-q[1])*(q[2]-q[1]))>0)swap(q[1],q[2]);
        db res = 0;
        for(int i = 0;i < len-1;i++){
    
    
            if(relation(q[i])==0||relation(q[i+1])==0){
    
    
                db arg = p.rad(q[i],q[i+1]);
                res += r*r*arg/2.0;
            }
            else{
    
    
                res += fabs((q[i]-p)^(q[i+1]-p))/2.0;
            }
        }
        return res;
    }
};
int getTangents(circle A,circle B,Point *a,Point *b)
{
    
    
    int cnt=0;        //存切点用
    if(sign(A.r-B.r)<0){
    
    
        swap(A,B);
        swap(a,b);
    }
    db d=sqrt((A.p.x-B.p.x)*(A.p.x-B.p.x)+(A.p.y-B.p.y)*(A.p.y-B.p.y));     //圆心距
    db rdiff=A.r-B.r;      //两圆半径差
    db rsum=A.r+B.r;       //两圆半径和
    if(sign(d-rdiff)<0)return 0;        //1.内含
    db base=atan2(B.p.y-A.p.y,B.p.x-A.p.x);      //向量AB的极角
    if(sign(d)== 0)return -1;        //2.重合
    if(sign(d-rdiff) == 0){
    
           //3.内切
        a[cnt]=b[cnt]=A.point(base);
        cnt++;
        return 1;
    }
    db ang=acos((A.r-B.r)/d);
    a[cnt]=A.point(base+ang);b[cnt]=B.point(base+ang);cnt++;      //4.相交（外切、外离的外公切线也在此求出）
    a[cnt]=A.point(base-ang);b[cnt]=B.point(base-ang);cnt++;      //两条外公切线的切点
    if(sign(d-rsum)==0){
    
            //5.外切
        a[cnt]=b[cnt]=A.point(base);
        cnt++;
    }
    else if(sign(d-rsum)>0){
    
          //6.外离
        db ang=acos((A.r+B.r)/d);
        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;
}
struct polygon{
    
    
    int n;
    Point p[N];
    Line l[N];
    void input(int _n){
    
    
        n=_n;
        rep(i,0,n-1)p[i].input();
    }
    void add(Point q){
    
    p[n++]=q;}
    void getline(){
    
    rep(i,0,n-1)l[i]=Line(p[i],p[(i+1)%n]);}
    struct cmp{
    
    
        Point p;
        cmp(const Point &p0){
    
    p = p0;}
        bool operator()(const Point &aa,const Point &bb){
    
    
            Point a = aa, b = bb;
            int d = sign((a-p)^(b-p));
            if(d == 0){
    
    
                return sign(a.distance(p)-b.distance(p)) < 0;
            }
            return d > 0;
        }
    };
    //极角排序
    void norm(){
    
    
        Point mi=p[0];
        rep(i,1,n-1)mi=min(mi,p[i]);
        sort(p,p+n,cmp(mi));
    }
    //`得到的凸包里面的点编号是[0,n-1]的`
    //`注意如果有影响，要特判下所有点共点，或者共线的特殊情况`
    void getconvex(polygon &convex){
    
    
        sort(p,p+n);
        convex.n=n;
        rep(i,0,min(n,2)-1)convex.p[i]=p[i];
        if(convex.n==2&&(convex.p[0]==convex.p[1]))convex.n--;//特判
        if(n<=2)return;
        int &top=convex.n;
        top=1;
        rep(i,2,n-1){
    
    
            while(top&&sign((convex.p[top]-p[i])^(convex.p[top-1]-p[i]))<=0)top--;
            convex.p[++top] = p[i];
        }
        int temp=top;
        convex.p[++top]=p[n-2];
        per(i,n-3,0){
    
    
            while(top!=temp&&sign((convex.p[top]-p[i])^(convex.p[top-1]-p[i]))<=0)top--;
            convex.p[++top]=p[i];
        }
        if(convex.n==2&&(convex.p[0]==convex.p[1]))convex.n--;//特判
        convex.norm();//`原来得到的是顺时针的点，排序后逆时针`
    }
    //`得到凸包的另外一种方法`
    void Graham(polygon &convex){
    
    
        norm();
        int &top=convex.n;
        top=0;
        if(n==1){
    
    
            top=1;
            convex.p[0]=p[0];
            return;
        }
        if(n==2){
    
    
            top=2;
            convex.p[0]=p[0];
            convex.p[1]=p[1];
            if(convex.p[0]==convex.p[1])top--;
            return;
        }
        convex.p[0]=p[0];
        convex.p[1]=p[1];
        top=2;
        rep(i,2,n-1){
    
    
            while(top>1&&sign((convex.p[top-1]-convex.p[top-2])^(p[i]-convex.p[top-2]))<=0)top--;
            convex.p[top++]=p[i];
        }
        if(convex.n==2&&(convex.p[0]==convex.p[1]))convex.n--;//特判
    }
    //`判断是不是凸的`
    bool isconvex(){
    
    
        bool s[2];
        memset(s,false,sizeof(s));
        rep(i,0,n-1){
    
    
            int j=(i+1)%n;
            int k=(j+1)%n;
            s[sign((p[j]-p[i])^(p[k]-p[i]))+1]=true;
            if(s[0]&&s[2])return false;
        }
        return true;
    }
    //`判断点和任意多边形的关系`
    //|` 0 外部`|` 1 内部`|` 2 边上`|` 3 点上`|
    int relationpoint(Point q){
    
    
        rep(i,0,n-1){
    
    
            if(p[i]==q)return 3;
        }
        getline();
        rep(i,0,n-1){
    
    
            if(l[i].pointonseg(q))return 2;
        }
        int cnt=0;
        rep(i,0,n-1){
    
    
            int j=(i+1)%n;
            int k=sign((q-p[j])^(p[i]-p[j]));
            int u=sign(p[i].y-q.y);
            int v=sign(p[j].y-q.y);
            if(k>0&&u<0&&v>=0)cnt++;//逆时针穿过+1
            if(k<0&&v<0&&u>=0)cnt--;//顺时针穿过-1
        }
        return cnt!=0;
    }
    //`直线u切割凸多边形左侧`
    //`注意直线方向`
    void convexcut(Line u,polygon &po){
    
    
        int &top = po.n;//注意引用
        top = 0;
        for(int i = 0;i < n;i++){
    
    
            int d1 = sign((u.e-u.s)^(p[i]-u.s));
            int d2 = sign((u.e-u.s)^(p[(i+1)%n]-u.s));
            if(d1 >= 0)po.p[top++] = p[i];
            if(d1*d2 < 0)po.p[top++] = u.crosspoint(Line(p[i],p[(i+1)%n]));
        }
    }
    //    //`得到周长`
//    //`测试 LightOJ1239`
    db getcircumference(){
    
    
        db sum = 0;
        for(int i = 0;i < n;i++){
    
    
            sum += p[i].distance(p[(i+1)%n]);
        }
        return sum;
    }
    //`得到面积`
    db getarea(){
    
    
        db sum = 0;
        for(int i = 0;i < n;i++){
    
    
            sum += (p[i]^p[(i+1)%n]);
        }
        return fabs(sum)/2;
    }
    //    //`得到方向`
//    //` 1 表示逆时针，0表示顺时针`
    bool getdir(){
    
    
        db sum = 0;
        for(int i = 0;i < n;i++)
            sum += (p[i]^p[(i+1)%n]);
        if(sign(sum) > 0)return 1;
        return 0;
    }
    //    //`得到重心`
    Point getbarycentre(){
    
    
        Point ret(0,0);
        db area = 0;
        for(int i = 1;i < n-1;i++){
    
    
            db tmp = (p[i]-p[0])^(p[i+1]-p[0]);
            if(sign(tmp) == 0)continue;
            area += tmp;
            ret.x += (p[0].x+p[i].x+p[i+1].x)/3*tmp;
            ret.y += (p[0].y+p[i].y+p[i+1].y)/3*tmp;
        }
        if(sign(area)) ret = ret/area;
        return ret;
    }
    //    //`多边形和圆交的面积`
    db areacircle(circle c){
    
    
        db ans = 0;
        for(int i = 0;i < n;i++){
    
    
            int j = (i+1)%n;
            if(sign( (p[j]-c.p)^(p[i]-c.p) ) >= 0)
                ans += c.areatriangle(p[i],p[j]);
            else ans -= c.areatriangle(p[i],p[j]);
        }
        return fabs(ans);
    }
    //`多边形和圆关系`
    //` 2 圆完全在多边形内`
    //` 1 圆在多边形里面，碰到了多边形边界`
    //` 0 其它`
    int relationcircle(circle c){
    
    
        getline();
        int x = 2;
        if(relationpoint(c.p) != 1)return 0;//圆心不在内部
        for(int i = 0;i < n;i++){
    
    
            if(c.relationseg(l[i])==2)return 0;
            if(c.relationseg(l[i])==1)x = 1;
        }
        return x;
    }
};
//旋转卡壳，计算凸包直径
db RC(polygon v)
{
    
    
    db res=0;
    int n=v.n;
    int k=1;
    rep(i,0,n-1){
    
    
        while(sign(cross(v.p[i],v.p[(i+1)%n],v.p[(k+1)%n])-cross(v.p[i],v.p[(i+1)%n],v.p[k%n]))==1)k=(k+1)%n;
        res=max(res,max((v.p[i]-v.p[k]).len(),(v.p[(i+1)%n]-v.p[k]).len()));
    }
    return res;
}
//`最小矩形面积覆盖`
//` A 必须是凸包(而且是逆时针顺序)`
db minRectangleCover(polygon A){
    
    
    //`要特判A.n < 3的情况`
    if(A.n < 3)return 0.0;
    A.p[A.n] = A.p[0];
    db ans = -1;
    int r = 1, p = 1, q;
    for(int i = 0;i < A.n;i++){
    
    
        //`卡出离边A.p[i] - A.p[i+1]最远的点`
        while( sign( cross(A.p[i],A.p[i+1],A.p[r+1]) - cross(A.p[i],A.p[i+1],A.p[r]) ) >= 0 )
            r = (r+1)%A.n;
        //`卡出A.p[i] - A.p[i+1]方向上正向n最远的点`
        while(sign( dot(A.p[i],A.p[i+1],A.p[p+1]) - dot(A.p[i],A.p[i+1],A.p[p]) ) >= 0 )
            p = (p+1)%A.n;
        if(i == 0)q = p;
        //`卡出A.p[i] - A.p[i+1]方向上负向最远的点`
        while(sign(dot(A.p[i],A.p[i+1],A.p[q+1]) - dot(A.p[i],A.p[i+1],A.p[q])) <= 0)
            q = (q+1)%A.n;
        db d = (A.p[i] - A.p[i+1]).len2();
        db tmp = cross(A.p[i],A.p[i+1],A.p[r]) *
            (dot(A.p[i],A.p[i+1],A.p[p]) - dot(A.p[i],A.p[i+1],A.p[q]))/d;
        if(ans < 0 || ans > tmp)ans = tmp;
    }
    return ans;
}
//`直线切凸多边形`
//`多边形是逆时针的，在q1q2的左侧`
//`测试:HDU3982`
vector<Point> convexCut(const vector<Point> &ps,Point q1,Point q2){
    
    
    vector<Point>qs;
    int n = ps.size();
    for(int i = 0;i < n;i++){
    
    
        Point p1 = ps[i], p2 = ps[(i+1)%n];
        int d1 = sign((q2-q1)^(p1-q1)), d2 = sign((q2-q1)^(p2-q1));
        if(d1 >= 0)
            qs.push_back(p1);
        if(d1 * d2 < 0)
            qs.push_back(Line(p1,p2).crosspoint(Line(q1,q2)));
    }
    return qs;
}
int n,casen=1;
polygon g,r;
db sumx,sumy,a,b,c;
int main()
{
    
    
    #ifdef io
    freopen("in.txt","r",stdin);
    #endif
    rush(){
    
    
        sc(n);
        g.n=0;
        sumx=sumy=0;
        rep(i,1,n){
    
    
            g.p[g.n++].input();
            sumx+=g.p[i-1].x;
            sumy+=g.p[i-1].y;
        }
        g.getconvex(r);
        db ans=inf;
        if(r.n<=2)ans=0;
        else{
    
    
            rep(i,0,r.n-1){
    
    
                Point p1=r.p[i],p2=r.p[(i+1)%(r.n)];
                a=p2.y-p1.y;
                b=p1.x-p2.x;
                c=-a*p1.x-b*p1.y;
                ans=min(ans,fabs(a*sumx+b*sumy+c*n)/sqrt(a*a+b*b));
            }
        }
        printf("Case #%d: %.3f\n",casen++,ans/n);
    }
    return 0;
}