bzoj4445 小凸想跑步

题目链接

半平面交，注意直线方向！！！

对于凸包上任意一条边$LINE(p_i,p_{i+1})$都有$S_{\Delta{p_i} {p_{i + 1}}p} < S_{\Delta{p_0} {p_1}p}$

如果我们用叉积来算面积：

$P=(x,y)$

$A=p_0=(x_1,y_1)$

$B=p_1=(x_2,y_2)$

$C=p_{i+1}=(x_3,y_3)$(至于为什么是C为i+1而不是D为i+1，画画图就知道了)

$D=p_i=(x_4,y_4)$

就有不等式：

$(x-x_2,y-y_2)(x-x_1,y-y_1)\leqslant(x-x_3,y-y_3)(x-x_4,y-y_4)$

$\Rightarrow$

$(x-x_2)(y-y_1)-(y-y_2)(x-x_1) \leqslant (x-x_3)(y-y_4)-(y-y_3)(x-x_4)$

$\Rightarrow$

$(-y_1+y_2-y_3-y_4)x+(x_1-x_2+x_3-x_4)y+$$(x_2\times y_1-y_2\times x_1)+(x_4\times y_3-y_4\times x_3) \leqslant 0$

$\Rightarrow$

$(-y_1+y_2-y_3-y_4)x+(x_1-x_2+x_3-x_4)y+\overrightarrow B\times\overrightarrow A+\overrightarrow D\times \overrightarrow C\leqslant 0$

就得到了一个半平面

交一交即可

 #include<iostream>
 #include<cstdio>
 #include<cstdlib>
 #include<algorithm>
 #include<cmath>
 #include<string>
 #include<cstring>
 #include<set>
 #include<queue>
 #define op operator
 #define re(i,l,r) for(int i=(l);i<=(r);i++)
 #define rre(i,r,l) for(int i=(r);i>=(l);i--)
 using namespace std;
 template <typename Q>
 void inin(Q &ret)
 {
     ret=;int f=;char ch=getchar();
     while(ch<''||ch>''){if(ch=='-')f=;ch=getchar();}
     while(ch>=''&&ch<='')ret=(ret<<)+(ret<<)+ch-'',ch=getchar();
     ret=f?-ret:ret;
 }
 int shurux,shuruy;
 const double eps=1e-;
 int dcmp(const double &x){return abs(x)<eps?:x<?-:;}
 const double pi=acos(-);
 const double f(const double &a){return a*a;}
 struct xl
 {
     double x,y;
     xl(double x=,double y=):x(x),y(y){}
     void in(){inin(shurux),inin(shuruy);x=shurux,y=shuruy;}
     bool op < (const xl &rhs)const {return x==rhs.x?y<rhs.y:x<rhs.x;}
     xl op + (const double &rhs)const {return xl(x+rhs,y+rhs);}
     xl op - (const double &rhs)const {return xl(x-rhs,y-rhs);}
     xl op * (const double &rhs)const {return xl(x*rhs,y*rhs);}
     xl op / (const double &rhs)const {return xl(x/rhs,y/rhs);}
     xl op + (const xl &rhs)const {return xl(x+rhs.x,y+rhs.y);}
     xl op - (const xl &rhs)const {return xl(x-rhs.x,y-rhs.y);}
     double angle()const {return atan2(y,x);}
     double len()const {return sqrt(f(x)+f(y));}
 };
 double D_(const xl &a,const xl &b){return a.x*b.x+a.y*b.y;}
 double X_(const xl &a,const xl &b){return a.x*b.y-a.y*b.x;}
 double dis(const xl &a,const xl &b){return sqrt(f(a.x-b.x)+f(a.y-b.y));}
 double dis2(const xl &a,const xl &b){return (a.x-b.x)*(a.x+b.x)+f(a.y-b.y);}
 double angle(const xl &a,const xl &b){return acos(D_(a,b)/a.len()/b.len());}
 struct LINE
 {
     xl u,v;double rad;
     LINE(xl u=xl(,),xl v=xl(,)):u(u),v(v){}
     void js(){rad=(v-u).angle();}
     bool op < (const LINE &rhs)const {return rad<rhs.rad;}
 };
 xl p[],ch[];
 LINE a[],b[];
 bool onleft(const xl &a,const LINE &l){return dcmp(X_(l.v-l.u,a-l.u))>;}
 LINE getline2(xl C,xl D)
 {
     xl A=p[],B=p[];
 //    if(D_(B-A,D-C)<0)swap(C,D);
     double a=(-A.y+B.y-C.y+D.y),b=(A.x-B.x+C.x-D.x),c=X_(B,A)+X_(D,C);
     LINE ret(xl(,-c/b),xl(,(-c-a)/b));
     if(!dcmp(b))ret=LINE(xl(-c/a,),xl((-c-b)/a,));
     if(!onleft(p[],ret)&&!onleft(p[],ret))swap(ret.u,ret.v);
     return ret;
 }
 LINE getline(xl C,xl D)
 {
     LINE l=getline2(C,D);
     int wocao1=onleft(p[],l)|onleft(p[],l);
     int wocao2=onleft(C,l)|onleft(D,l);
     if(wocao1^wocao2)return l;
     else return getline2(D,C);
 }
 xl getjd(const LINE &a,const LINE &b)
 {
     xl u=a.u-b.u,v1=a.v-a.u,v2=b.v-b.u;
     double t=X_(v2,u)/X_(v1,v2);
     return a.u+v1*t;
 }
 int n,nn;
 bool halfplanej(LINE *l)
 {
     re(i,,n-)l[i].rad=(l[i].v-l[i].u).angle();
     sort(l,l+n);
     int ll=,rr=;
     xl p[n];LINE b[n];b[]=l[];
     re(i,,n-)
     {
         while(ll<rr&&!onleft(p[rr-],l[i]))rr--;
         while(ll<rr&&!onleft(p[ll],l[i]))ll++;
         b[++rr]=l[i];
         if(!dcmp(X_(b[rr].v-b[rr].u,b[rr-].v-b[rr-].u)))
         {
             rr--;
             if(onleft(l[i].u,b[rr]))b[rr]=l[i];
         }
         if(ll<rr)p[rr-]=getjd(b[rr-],b[rr]);
     }
     while(ll<rr&&!onleft(p[rr-],b[ll]))rr--;
     if(rr-ll<=)return ;
     p[rr]=getjd(b[rr],b[ll]);
     re(i,ll,rr)ch[nn++]=p[i];
     ch[nn]=ch[];return ;
 }
 double zongarea,yaoarea;
 int main()
 {
     inin(n);
     re(i,,n-)p[i].in();
     p[n]=p[];
     xl temp=p[]-p[];
     re(i,,n-)zongarea+=abs(X_(temp,p[i+]-p[i])),temp=temp+(p[i+]-p[i]);
     a[]=LINE(p[],p[]);
     re(i,,n-)
         a[i]=getline2(p[i+],p[i]);
     halfplanej(a);
     temp=ch[]-ch[];
     re(i,,nn-)
         yaoarea+=abs(X_(temp,ch[i+]-ch[i])),temp=temp+(ch[i+]-ch[i]);
     printf("%.4f",yaoarea/zongarea);
     return ;
 }