最小覆盖圆

问题

给定一系列点，求最小的覆盖圆

解法

点类

const double eps = 1e-8;
const double pi = acos(-1.0);

inline double sqr(double x){
    
    
    return x*x;
}

//浮点数修正
int cmp(double x){
    
    
    if(fabs(x)<eps) return 0;
    if(x>0) return 1;
    return -1;
}

struct point{
    
    
    double x, y;
    point() {
    
    }
    point(double a, double b): x(a), y(b) {
    
    }
    void input(){
    
    
        scanf("%lf%lf", &x, &y);
    }
    friend bool operator == (const point &a, const point &b){
    
    
        return cmp(a.x-b.x)==0 && cmp(a.y-b.y)==0;
    }
    friend point operator + (const point &a, const point &b){
    
    
        return point(a.x+b.x, a.y+b.y);
    }
    friend point operator - (const point &a, const point &b){
    
    
        return point(a.x-b.x, a.y-b.y);
    }
    friend point operator * (const point &a, const double &b){
    
    
        return point(a.x*b, a.y*b);
    }
    friend point operator * (const double &a, const point &b){
    
    
        return point(a*b.x, a*b.y);
    }
    friend point operator / (const point &a, const double &b){
    
    
        return point(a.x/b, a.y/b);
    }
    //求圆心到此点的距离，向量模
    double norm(){
    
    
        return sqrt(sqr(x)+sqr(y));
    }
    //向量积
    double dot(const point &a, const point &b){
    
    
        return a.x*b.x+a.y*b.y;
    }
    //两点间距离
    double dist(const point &a, const point &b){
    
    
        return (a-b).norm();
    }
};

两点定最小圆
以两点直线为直径，中点为圆心

void circleCenter(point p0, point p1, point &cp){
    
    
    cp.x = (p0.x + p1.x)/2;
    cp.y = (p0.y + p1.y)/2;
    return;
}

三点定最小圆
构建三角形，任取两边作垂直平分线，交点为圆心

$$\begin{gathered} p_0(x_0,y_0),p_1(x_1,y_1),p_2(x_2,y_2) \\ {p}_0{p}_1垂直平分线点斜式：y-\frac{y_0+y_1}{2}=-\frac{x_1-x_0}{y_1-y_0}(x-\frac{x_0+x_1}{2}) \\ 化简：(x_1-x_0)x+(y_1-y_0)y=\frac{(x_1^2-x_0^2)+(y_1^2-y_0^2)}{2} \\ {p}_0{p}_2垂直平分线点斜式：y-\frac{y_0+y_2}{2}=-\frac{x_2-x_0}{y_2-y_0}(x-\frac{x_0+x_2}{2}) \\ 化简：(x_2-x_0)x+(y_2-y_0)y=\frac{(x_2^2-x_0^2)+(y_2^2-y_0^2)}{2} \\ 令\{\begin{array}{l}a=x_1-x_0\\ b=y_1-y_0\\ c=x_2-x_0\\ d=y_2-y_0\\ e=\frac{(x_1^2-x_0^2)+(y_1^2-y_0^2)}{2}=\frac{a^2+b^2}{2}\\ f=\frac{(x_2^2-x_0^2)+(y_2^2-y_0^2)}{2}=\frac{c^2+d^2}{2}\end{array} \\ 联立\{\begin{array}{l}ax+by=e\\ cx+dy=f\end{array} \\ 得交点cp(\frac{de-bf}{ad-bc},\frac{af-ce}{ad-bc}) \end{gathered}$$

void circleCenter(point p0, point p1, point p2, point &cp){
    
    
    double a = p1.x - p0.x, b = p1.y - p0.y, e = (a*a+b*b)/2;
    double c = p2.x - p0.x, d = p2.y - p0.y, f = (c*c+d*d)/2;
    double down = a*d - b*c;
    cp.x = (d*e - b*f)/down;
    cp.y = (a*f - c*e)/down;
    return;
}

判断一点是否在圆内

point center;
double radius;

bool in_circle(const point &p){
    
    
    return cmp((p-center).norm() - radius) < 0;
}

随机增量算法

point center;
double radius;

//给定n个点a[]
void min_circle_cover(point a[], int n){
    
    
    //保证效率
    random_shuffle(a, a+n);
    //初始化
    center = a[0];
    radius = 0;
    //以每一个点为重，3层循环检查
    for(int i=1; i<n; i++){
    
    
        if(in_circle(a[i])) continue;
        center = a[i];
        radius = 0;
        //覆盖a[i]与a[0~i-1]
        for(int j=0; j<i; j++){
    
    
            if(in_circle(a[j])) continue;
            //2点定圆
            circleCenter(a[i], a[j], center);
            radius = (a[j]-center).norm();
            //覆盖a[i],a[j]与a[0~j-1]
            for(int k=0; k<j; k++){
    
    
                if(in_circle(a[k])) continue;
                //3点定圆
                circleCenter(a[i], a[j], a[k], center);
                radius = (a[k]-center).norm();
            }
        }
    }
}