[BZOJ1185]最小矩形覆盖

省赛之前沉迷于几乎不考的计算几何（我大概是没救了）

这个是写完模板后的第一题，看了之后感觉思路还挺清晰的。

首先因为它的tag是'凸包'，所以我们当然要先求凸包啦~Graham就好了

然后有一个结论是矩形的某一条边一定与凸包的某一条边共线，脑补一下如果不共线，稍微转一下矩形就会更小了。

那么我们枚举每一条凸包上的边$x$，每次找离x最远的点。

这里可以用二分，只有最远的那个点连接的两条边相对于$x$一个左转一个右转，把这条性质作为二分依据，就可以愉快地二分啦。

这题的细节处理较为烦人，像我这种弱智调了几天才调好。

#include<stdio.h>
#include<math.h>
#include<algorithm>
using namespace std;
#define dmax 1.7976931348623158e+308
#define eps 1e-10
struct point{
double x,y,ang;
int num;
}a[100010],t;
struct point2{
double x,y;
point2(){x=y=0;}
}ha[100010];
int n,i,ori,stack[100010];
double miny,minx,orix,oriy,area;
point2 p1,p2,p3,p4;
double bigger(double x,double y){return x-y>=eps;}
double notsm(double x,double y){return bigger(x,y)||abs(x-y)<=eps;}
double angle(double x,double y){
if(abs(x<=eps))return dmax;
return y/x;
}
bool cmp(point a,point b){
if(bigger(b.ang,a.ang))return 1;
if(bigger(a.ang,b.ang))return 0;
return bigger(b.x*b.x+b.y*b.y,a.x*a.x+a.y*b.y);
}
double cross(double ax,double ay,double bx,double by){
return ax*by-ay*bx;
}
int hassium(double vx,double vy,int ll,int rr){
int l,r,m;
l=ll+1;
r=rr-1;
while(l<=r){
m=(l+r)/2;
orix=cross(vx,vy,ha[m].x-ha[m-1].x,ha[m].y-ha[m-1].y);
oriy=cross(vx,vy,ha[m+1].x-ha[m].x,ha[m+1].y-ha[m].y);
if(orix>=eps){
if(oriy>=eps)l=m+1;
if(notsm(-oriy,0))return m;
}
if(abs(orix)<=eps)return m;
if(orix<-eps)r=m-1;
}
}
point2 getpoint(double xa,double ya,double x1,double y1,double xb,double yb,double x2,double y2){
point2 p;
if(abs(x1)<=eps){
p.x=xa;
p.y=yb;
return p;
}
if(abs(y1)<=eps){
p.x=xb;
p.y=ya;
return p;
}
p.y=((xb-xa)/x1/x2-yb/x1/y2+ya/x2/y1)/(1./x2/y1-1./x1/y2);
p.x=(p.y-ya)/y1*x1+xa;
return p;
}
double recta(double ax,double ay,double bx,double by){
return sqrt((ax*ax+ay*ay)*(bx*bx+by*by));
}
void getrect(double vx,double vy,int num){
int farest=hassium(vx,vy,num,num+stack[0]-1);
int leftest=hassium(-vy,vx,farest,num+stack[0]-1),rightest=hassium(vy,-vx,num,farest);
point2 t1=getpoint(ha[num].x-vx,ha[num].y-vy,vx,vy,ha[rightest].x,ha[rightest].y,vy,-vx),
t2=getpoint(ha[rightest].x,ha[rightest].y,-vy,vx,ha[farest].x,ha[farest].y,vx,vy),
t3=getpoint(ha[leftest].x,ha[leftest].y,-vy,vx,ha[farest].x,ha[farest].y,-vx,-vy),
t4=getpoint(ha[leftest].x,ha[leftest].y,vy,-vx,ha[num].x,ha[num].y,-vx,-vy);
double tarea=recta(t2.x-t1.x,t2.y-t1.y,t4.x-t1.x,t4.y-t1.y);
if(bigger(area,tarea)){
area=tarea;
p1=t1;
p2=t2;
p3=t3;
p4=t4;
}
}
bool comp(point2 a,point2 b,point2 c,point2 d){
return((bigger(b.y,a.y))||(abs(a.y-b.y)<=eps&&bigger(b.x,a.x)))
&&((bigger(c.y,a.y))||(abs(a.y-c.y)<=eps&&bigger(c.x,a.x)))
&&((bigger(d.y,a.y))||(abs(a.y-d.y)<=eps&&bigger(d.x,a.x)));
}
void gao(point2&a){
if(a.x<0&&abs(a.x)<=1e-5)a.x=-a.x;
if(a.y<0&&abs(a.y)<=1e-5)a.y=-a.y;
}
int main(){
scanf("%d",&n);
miny=minx=dmax;
for(i=1;i<=n;i++){
scanf("%lf%lf",&a[i].x,&a[i].y);
a[i].num=i;
if((a[i].x==minx&&a[i].y<miny)||a[i].x<minx){
miny=a[i].y;
minx=a[i].x;
ori=i;
}
}
orix=a[ori].x;
oriy=a[ori].y;
for(i=1;i<=n;i++){
a[i].x-=orix;
a[i].y-=oriy;
if(i!=ori)
a[i].ang=angle(a[i].x,a[i].y);
}
t=a[1];
a[1]=a[ori];
a[ori]=t;
sort(a+2,a+n+1,cmp);
stack[0]=2;
stack[1]=1;
stack[2]=2;
for(i=3;i<=n;i++){
while(stack[0]>1&&cross(a[stack[stack[0]]].x-a[stack[stack[0]-1]].x,a[stack[stack[0]]].y-a[stack[stack[0]-1]].y,
a[i].x-a[stack[stack[0]]].x,a[i].y-a[stack[stack[0]]].y)<=0)stack[0]--;
stack[0]++;
stack[stack[0]]=i;
}
for(i=1;i<=n;i++){
a[i].x+=orix;
a[i].y+=oriy;
}
for(i=1;i<=stack[0];i++){
ha[i].x=a[stack[i]].x;
ha[i].y=a[stack[i]].y;
}
area=dmax;
getrect(ha[1].x-ha[stack[0]].x,ha[1].y-ha[stack[0]].y,1);
for(i=1;i<stack[0];i++){
ha[i+stack[0]]=ha[i];
getrect(ha[i+1].x-ha[i].x,ha[i+1].y-ha[i].y,i+1);
}
gao(p1);
gao(p2);
gao(p3);
gao(p4);
printf("%.5lf\n",area);
if(comp(p1,p2,p3,p4))
printf("%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n",p1.x,p1.y,p2.x,p2.y,p3.x,p3.y,p4.x,p4.y);
if(comp(p2,p1,p3,p4))
printf("%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n",p2.x,p2.y,p3.x,p3.y,p4.x,p4.y,p1.x,p1.y);
if(comp(p3,p2,p1,p4))
printf("%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n",p3.x,p3.y,p4.x,p4.y,p1.x,p1.y,p2.x,p2.y);
if(comp(p4,p2,p3,p1))
printf("%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n%.5lf %.5lf\n",p4.x,p4.y,p1.x,p1.y,p2.x,p2.y,p3.x,p3.y);
}