HDU3982 Harry Potter and J.K.Rowling

Harry Potter and J.K.Rowling

In July 31st, last month, the author of the famous novel series J.K.Rowling celebrated her 46th birthday. Many friends gave their best wishes. They gathered together and shared one large beautiful cake.

Rowling had lots of friends, and she had a knife to cut the cake into many pieces. On the cake there was a cherry. After several cuts, the piece with the cherry was left for Rowling. Before she enjoyed it, she wondered how large this piece was, i.e., she wondered how much percentage of the cake the piece with the only cherry has.

\(n\leq 2000\)

题解

半平面交+凸多边形与圆的面积交。

然后被精度卡烂了。

时间复杂度\(O(n\log n)\)。

struct point {double x,y;};

IN point operator+(CO point&a,CO point&b){
	return (point){a.x+b.x,a.y+b.y};
}
IN point operator-(CO point&a,CO point&b){
	return (point){a.x-b.x,a.y-b.y};
}
IN point operator*(CO point&a,double b){
	return (point){a.x*b,a.y*b};
}
IN point operator/(CO point&a,double b){
	return (point){a.x/b,a.y/b};
}
IN double dot(CO point&a,CO point&b){
	return a.x*b.x+a.y*b.y;
}
IN double cross(CO point&a,CO point&b){
	return a.x*b.y-a.y*b.x;
}
IN double len(CO point&a){
	return sqrt(dot(a,a));
}
IN double angle(CO point&a,CO point&b){
	double w=dot(a,b)/len(a)/len(b);
	w=max(w,-1.0),w=min(w,1.0); // edit 3
	return acos(w);
}

struct line {point x,y;};

IN bool on_left(CO point&a,CO line&b){
	return cross(a-b.x,b.y)<0;
}
IN int quad(CO point&a){
	if(a.x>0 and a.y>=0) return 1;
	else if(a.x<=0 and a.y>0) return 2;
	else if(a.x<0 and a.y<=0) return 3;
	else return 4;
}
IN bool operator<(CO line&a,CO line&b){
	if(quad(a.y)!=quad(b.y)) return quad(a.y)<quad(b.y);
	return cross(a.y,b.y)==0?on_left(a.x,b):cross(a.y,b.y)>0; // edit 2
}
IN point intersect(CO line&a,CO line&b){
	return b.x+b.y*cross(b.x-a.x,a.y)/cross(a.y,b.y);
}

CO double pi=acos(-1),eps=1e-6;

vector<point> seg_cross_circle(CO point&a,CO point&b,double r){
	double dx=b.x-a.x,dy=b.y-a.y;
	double A=dx*dx+dy*dy;
	double B=2*dx*a.x+2*dy*a.y; // edit 1
	double C=a.x*a.x+a.y*a.y-r*r;
	double delta=B*B-4*A*C;
	vector<point> ans;
	if(delta<-eps) return ans; // edit 2
	delta=sqrt(max(delta,0.0));
	double t1=(-B+delta)/(2*A);
	double t2=(-B-delta)/(2*A);
	if(-t1<eps and t1-1<eps) // edit 3
		ans.push_back((point){a.x+t1*dx,a.y+t1*dy});
	if(-t2<eps and t2-1<eps)
		ans.push_back((point){a.x+t2*dx,a.y+t2*dy});
	return ans;
}
double triangle_cross_circle(point a,point b,double r){
//	cerr<<"a="<<a.x<<" "<<a.y<<" b="<<b.x<<" "<<b.y<<endl;
	bool AinC=dot(a,a)<r*r,BinC=dot(b,b)<r*r;
	double sign=0.5*(cross(a,b)>0?1:-1),ans=0;
	if(AinC and BinC) ans=abs(cross(a,b)); // edit 4
	else if(AinC or BinC){
		if(BinC) swap(a,b);
		vector<point> tmp=seg_cross_circle(a,b,r);
		ans=abs(cross(a,tmp[0]))+r*r*angle(tmp[0],b);
	}
	else{
		vector<point> p=seg_cross_circle(a,b,r);
		ans=r*r*angle(a,b);
		if(p.size()==2){
			ans-=r*r*angle(p[0],p[1]);
			ans+=abs(cross(p[0],p[1]));
		}
	}
//	cerr<<"ans="<<ans<<endl;
	return ans*=sign;
}

CO int N=2e3+10;
point p[N];
line ln[N];

int halfplane(int n){
	sort(ln+1,ln+n+1);
	int m=1;
	for(int i=2;i<=n;++i){
		if(cross(ln[i].y,ln[m].y)==0) continue;
		ln[++m]=ln[i];
	}
	int l=1,r=1;
	for(int i=2;i<=m;++i){
		for(;l<r and !on_left(p[r],ln[i]);--r);
		for(;l<r and !on_left(p[l+1],ln[i]);++l);
		ln[++r]=ln[i];
		if(l<r) p[r]=intersect(ln[r-1],ln[r]);
	}
	for(;l<r and !on_left(p[r],ln[l]);--r);
	if(l<r) p[l]=p[r+1]=intersect(ln[l],ln[r]);
	for(int i=1;i<=r-l+1;++i) p[i]=p[l+i-1];
	return r-l+1;
}
void real_main(){
	double r;scanf("%lf",&r);
	int n=4+read<int>();
	ln[1].x=(point){0,-r},ln[1].y=(point){1,0};
	ln[2].x=(point){r,0},ln[2].y=(point){0,1};
	ln[3].x=(point){0,r},ln[3].y=(point){-1,0};
	ln[4].x=(point){-r,0},ln[4].y=(point){0,-1};
	for(int i=5;i<=n;++i){
		point a,b;scanf("%lf%lf%lf%lf",&a.x,&a.y,&b.x,&b.y);
		ln[i].x=a,ln[i].y=b-a;
	}
	point o;scanf("%lf%lf",&o.x,&o.y);
	for(int i=5;i<=n;++i)
		if(!on_left(o,ln[i])) ln[i].y=ln[i].y*-1;
	n=halfplane(n),p[n+1]=p[1];
//	cerr<<"p=";
//	for(int i=1;i<=n+1;++i)
//		cerr<<" ("<<p[i].x<<","<<p[i].y<<")"<<endl;
	double ans=0;
	for(int i=1;i<=n;++i)
		ans+=triangle_cross_circle(p[i],p[i+1],r);
	ans=ans/(pi*r*r);
	printf("%.5lf%%\n",100*ans);
}
int main(){
	int T=read<int>();
	for(int i=1;i<=T;++i){
		printf("Case %d: ",i);
		real_main();
	}
	return 0;
}