bzoj 3564 信号增幅仪

• 一道有意义的好题.启发我们善用坐标变换,如在遇到椭圆时使用坐标变换转化为圆来处理.
• 直接在一个方向上进行伸缩不方便,先将坐标系旋转,使椭圆长轴与 \(x\) 轴平行,点的坐标变换用旋转公式处理.
• 坐标系逆时针转过了 \(\theta\) ,相当于点逆时针转过了 \(-\theta\) .
• 再沿长轴方向伸缩,将点的横坐标除以 \(p\) ,椭圆就转化为了圆,求一个最小圆覆盖即可.

#include<bits/stdc++.h>
using namespace std;
#define ll long long
#define mp make_pair
#define pii pair<int,int>
inline int read()
{
    int x=0;
    bool pos=1;
    char ch=getchar();
    for(;!isdigit(ch);ch=getchar())
        if(ch=='-')
            pos=0;
    for(;isdigit(ch);ch=getchar())
        x=x*10+ch-'0';
    return pos?x:-x;
}
struct v2{
    double x,y;
    v2(double x=0,double y=0):x(x),y(y) {}
    v2 operator + (const v2 &rhs) const
        {
            return v2(x+rhs.x,y+rhs.y);
        }
    v2 operator / (const double &rhs) const
        {
            return v2(x/rhs,y/rhs);
        }
    v2 operator - (const v2 &rhs) const
        {
            return v2(x-rhs.x,y-rhs.y);
        }
    double operator * (const v2 &rhs) const
        {
            return x*rhs.y-y*rhs.x;
        }
    double modulus()
        {
            return sqrt(x*x+y*y);
        }
};
const double PI=acos(-1.0);
v2 rotate(v2 v,double degree)
{
    double rad=degree/180*PI;
    //v2 res=v2(cos(rad)*v.x-sin(rad)*v.y,sin(rad)*v.x+cos(rad)*v.y);
    v2 res=v2(v.x*cos(rad)-v.y*sin(rad),v.x*sin(rad)+v.y*cos(rad));
    return res;
}
const int MAXN=5e4+10;
int n;
v2 a[MAXN];
double theta,p;
void transform()
{
    for(int i=1;i<=n;++i)
        {
            a[i]=rotate(a[i],theta);
            a[i].x/=p;
        }
}
const double eps=1e-9;
int dcmp(double x)
{
    return fabs(x)<eps?0:(x>0?1:-1);
}
v2 getcentre(v2 a,v2 b,v2 c)
{
    v2 centre;
    double a1=b.x-a.x;
    double b1=b.y-a.y;
    double c1=(a1*a1+b1*b1)/2.0;
    double a2=c.x-a.x;
    double b2=c.y-a.y;
    double c2=(a2*a2+b2*b2)/2.0;
    double d=a1*b2-a2*b1;
    centre.x=a.x+(c1*b2-c2*b1)/d;
    centre.y=a.y+(a1*c2-a2*c1)/d;
    return centre;
}
bool incir(v2 centre,double r,v2 p)
{
    return dcmp((p-centre).modulus()-r)<=0;
}
double MinCircleCover()
{
    srand(19260817);
    random_shuffle(a+1,a+1+n);
    v2 centre=a[1];
    double r=0;
    for(int i=2;i<=n;++i)
        {
            if(!incir(centre,r,a[i]))
                {
                    centre=a[i];
                    r=0;
                    for(int j=1;j<i;++j)
                        {
                            if(!incir(centre,r,a[j]))
                                {
                                    centre=(a[i]+a[j])/2.0;
                                    r=(a[i]-centre).modulus();
                                    for(int k=1;k<j;++k)
                                        {
                                            if(!incir(centre,r,a[k]))
                                                {
                                                    centre=getcentre(a[i],a[j],a[k]);
                                                    r=(a[i]-centre).modulus();
                                                }
                                        }
                                }
                        }
                }
        }
    return r;
}
int main()
{
    n=read();
    for(int i=1;i<=n;++i)
        scanf("%lf%lf",&a[i].x,&a[i].y);
    scanf("%lf%lf",&theta,&p);
    theta*=-1;
    transform();
    double ans=MinCircleCover();
    printf("%.3lf\n",ans);
    return 0;
}