[NOI2011 智能车比赛]

[关键字]：计算几何

[题目大意]：http://221.192.240.123:8586/JudgeOnline/showproblem?problem_id=1668

//=====================================================================================================

[分析]：计算几何题，需要有清晰的思维（本人就是一开始没想清楚）。从每一个矩形的顶点包括起点开始检查能不能到其它点，并记录长度。因为最短路径上的点一定是在矩形的顶点上，而且因为是不断往前走，所以可以用dp来做最短路。检查方法就是先假定一个视野的上下界，然后检查每个顶点是否在视野范围内（叉积），在就更新距离。然后从每个矩形交界处开始判断是否会阻挡视野，会就更新上下界，直到视野为0或检查到终点所在矩形（此时更新答案）。当所有点都做完后就可以找到答案。

[代码]：

#include<iostream>
#include<cmath>
#include<cstdio>
#include<cstdlib>
#include<algorithm>
using namespace std;
#define sqr(x) (double(x) * (x))
#define MAXN 10000
#define INF 1e100

struct point
{
       int x,y;
}a,b,c,d,S,T;
struct rectangle
{
       point a,b,c,d;
}rec[MAXN];
int n;
double v,ans=INF;
double f[MAXN];

void Init()
{
     int t1,t2,t3,t4;
     scanf("%d",&n);
     for (int i=0;i<n;i++)
     {
         scanf("%d%d%d%d",&t1,&t2,&t3,&t4);         
         rec[i].a.x=t1,rec[i].a.y=t2;
         rec[i].b.x=t1,rec[i].b.y=t4;
         rec[i].c.x=t3,rec[i].c.y=t2;
         rec[i].d.x=t3,rec[i].d.y=t4;
         //printf("%d %d %d %d %d %d %d %d\n",rec[i].a.x,rec[i].a.y,rec[i].b.x,rec[i].b.y,rec[i].c.x,rec[i].c.y,rec[i].d.x,rec[i].d.y);
     }
     scanf("%d%d",&S.x,&S.y);
     scanf("%d%d",&T.x,&T.y);
     scanf("%lf",&v);
     if (S.x>T.x) swap(S,T);
     for (int i=0;i<n*4;i++) f[i]=INF;
     return;
}

double dis(point a,point b)
{
       return sqrt(sqr(a.x-b.x)+sqr(a.y-b.y));
}

long long cross(point a,point b,point c,point d)
{
     return (long long)(b.x-a.x)*(d.y-c.y)-(long long)(d.x-c.x)*(b.y-a.y);
} 

bool cleck(point a,point b,point upper,point lower)
{
     if (b.x<a.x) return 0;
     if (cross(a,lower,a,b)<0) return 0;
     if (cross(a,upper,a,b)>0) return 0;
     return 1;
}

void Work(point start,int now,double val,double &ans)
{
     if (val>=INF) return;
     point lower,upper;
     lower.x=start.x,lower.y=start.y-1;
     upper.x=start.x,upper.y=start.y+1;
     for (int i=now;i<n;i++)
     {
         if (cleck(start,rec[i].a,upper,lower))
            f[i*4]=min(f[i*4],val+dis(start,rec[i].a));
            
         if (cleck(start,rec[i].b,upper,lower))
            f[i*4+1]=min(f[i*4+1],val+dis(start,rec[i].b));
            
         if (cleck(start,rec[i].c,upper,lower))
            f[i*4+2]=min(f[i*4+2],val+dis(start,rec[i].c));
            
         if (cleck(start,rec[i].d,upper,lower))
            f[i*4+3]=min(f[i*4+3],val+dis(start,rec[i].d));
            
         if (rec[i].a.x<=T.x && T.x<=rec[i].d.x)
            if (rec[i].a.y<=T.y && T.y<=rec[i].d.y)
               if (cleck(start,T,upper,lower))
                  ans=min(ans,val+dis(start,T));
         if (i+1<n)
         {
                   point l,r;
                   l.x=rec[i].c.x,l.y=max(rec[i].c.y,rec[i+1].a.y);
                   r.x=rec[i].d.x,r.y=min(rec[i].d.y,rec[i+1].b.y);
                   //printf("%d %d %d %d\n",l.x,l.y,r.x,r.y);
                   if (rec[i].c.x==start.x)
                   {
                      if (l.y>start.y || start.y>r.y)
                      {
                                    f[(i+1)*4]=min(f[(i+1)*4],val+dis(start,rec[i+1].a));
                                    f[(i+1)*4+1]=min(f[(i+1)*4+1],val+dis(start,rec[i+1].b));
                                    return;
                      }
                   }
                   else
                      {
                          if (cross(start,lower,start,l)>0) lower.x=l.x,lower.y=l.y;
                          if (cross(start,upper,start,r)<0) upper.x=r.x,upper.y=r.y;
                          if (cross(start,upper,start,lower)>0) return;
                      }
         }
     }
     return;
}

void Solve()
{
     for (int i=0;i<n;i++)
     {
         if (rec[i].a.x<=S.x && S.x<=rec[i].d.x)
            if (rec[i].a.y<=S.y && S.y<=rec[i].d.y)
               Work(S,i,0,ans);
         Work(rec[i].a,i,f[i*4],ans);
         Work(rec[i].b,i,f[i*4+1],ans);
         Work(rec[i].c,i,f[i*4+2],ans);
         Work(rec[i].d,i,f[i*4+3],ans);         
     }
     //for (int i=0;i<n*4;i++) printf("%d %f\n",i,f[i]);
     printf("%0.10lf\n",ans/v);
     return;
}

int main()
{
    Init();
    Solve();
    system("pause");
    return 0;
}