「SCOI2015」小凸想跑步 解题报告

「SCOI2015」小凸想跑步

最开始以为和多边形的重心有关，后来发现多边形的重心没啥好玩的性质

实际上你把面积小于的不等式列出来，发现是一次的，那么就可以半平面交了

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Code:

#include <cstdio>
#include <cmath>
#include <algorithm>
#define Vector Point
const int N=2e5+10;
const double eps=1e-7;
int n,m,l,r;
struct Point
{
	double x,y;
	Point(){}
	Point(double X,double Y){x=X,y=Y;}
	double angle(){return atan2(y,x);}
	Vector friend operator +(Vector a,Vector b){return Vector(a.x+b.x,a.y+b.y);}
	Vector friend operator -(Vector a,Vector b){return Vector(a.x-b.x,a.y-b.y);}
	Vector friend operator *(Vector a,double b){return Vector(a.x*b,a.y*b);}
}bee[N],q1[N];
struct Line
{
	Point s,t;double ang;
	Line(){}
	Line(Point S,Point T){s=S,t=T,ang=(t-s).angle();}
}yuu[N],q2[N];
double Cross(Vector a,Vector b){return a.x*b.y-a.y*b.x;}
bool isrig(Line a,Point b){return Cross(b-a.s,a.t-a.s)>0;}
bool cmp(Line a,Line b){return fabs(a.ang-b.ang)<eps?isrig(a,b.t):a.ang<b.ang;}
Point jd(Line a,Line b){return a.s+(a.t-a.s)*(Cross(a.s-b.s,b.t-b.s)/Cross(b.t-b.s,a.t-a.s));}
void SI()
{
	std::sort(yuu+1,yuu+1+m,cmp);
	q2[l=r=1]=yuu[1];
	for(int i=2;i<=m;i++)
		if(fabs(yuu[i].ang-yuu[i-1].ang)>eps)
		{
			while(l<r&&isrig(yuu[i],q1[r-1])) --r;
			while(l<=r&&isrig(yuu[i],q1[l])) ++l;
			q1[r]=jd(q2[r],yuu[i]);
			q2[++r]=yuu[i];
		}
	while(l<r&&isrig(q2[l],q1[r-1])) --r;
	q1[r]=jd(q2[l],q2[r]);
}
double area()
{
    if(r-l<2) return 0;
	double ret=0;
	for(int i=l;i<r;i++) ret+=Cross(q1[i],q1[i+1]);
	ret+=Cross(q1[r],q1[l]);
	return ret/2;
}
int main()
{
	scanf("%d",&n);m=n;double sum=0;
	for(int i=1;i<=n;i++)
    {
        scanf("%lf%lf",&bee[i].x,&bee[i].y);
        yuu[i]=Line(bee[i-1],bee[i]);
        if(i!=1) sum+=Cross(bee[i-1],bee[i]);
    }
	yuu[1]=Line(bee[n],bee[1]);
	sum+=Cross(bee[n],bee[1]);
	sum/=2;
	bee[++n]=bee[1];
	for(int i=3;i<=n;i++)
	{
		double A=bee[1].y-bee[2].y-(bee[i-1].y-bee[i].y);
		double B=bee[2].x-bee[1].x-(bee[i].x-bee[i-1].x);
		double C=Cross(bee[1],bee[2])-Cross(bee[i-1],bee[i]);
		if(fabs(A)<eps) yuu[++m]=Line(Point(0,-C/B),Point(0,-C/B)+Vector(-B,A));
		else yuu[++m]=Line(Point(-C/A,0),Point(-C/A,0)+Vector(-B,A));
	}
	SI();
	printf("%.4f\n",area()/sum);
	return 0;
}

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2019.3.4