平面上有一群点，如何用PCA获得这些点大致的最小矩形包围框，同时给出c++代码

可以通过PCA（主成分分析）来获得这些点的最小矩形包围框。

具体步骤如下：

1. 计算点集的质心（平均值）。

2. 将点集的坐标系平移到以质心为原点的坐标系下。

3. 计算点集的协方差矩阵。

4. 对协方差矩阵进行特征值分解，得到特征值和特征向量。

5. 将特征向量按照对应的特征值大小排序，选取前两个最大的特征向量组成一个二维坐标系。

6. 将原始点集的点投影到这个二维坐标系上，得到在新坐标系下的点集。

7. 在新坐标系下，最小矩形包围框的边界为点集在每个维度上的最大值和最小值，即可得到最小矩形包围框。

以下是用C语言实现的代码示例：

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define N 100

typedef struct {
    double x;
    double y;
} Point;

void PCA(Point *points, int n, Point *center, Point *v1, Point *v2) {
    int i;
    double xx = 0, yy = 0, xy = 0;
    center->x = 0;
    center->y = 0;
    for (i = 0; i < n; i++) {
        center->x += points[i].x;
        center->y += points[i].y;
    }
    center->x /= n;
    center->y /= n;
    for (i = 0; i < n; i++) {
        xx += (points[i].x - center->x) * (points[i].x - center->x);
        yy += (points[i].y - center->y) * (points[i].y - center->y);
        xy += (points[i].x - center->x) * (points[i].y - center->y);
    }
    double lambda1 = (xx + yy + sqrt((xx - yy) * (xx - yy) + 4 * xy * xy)) / 2;
    double lambda2 = (xx + yy - sqrt((xx - yy) * (xx - yy) + 4 * xy * xy)) / 2;
    v1->x = lambda1 - xx;
    v1->y = xy;
    v2->x = xy;
    v2->y = lambda2 - yy;
}

int main() {
    int n, i;
    Point points[N], center, v1, v2;
    printf("Enter the number of points: ");
    scanf("%d", &n);
    printf("Enter the points:\n");
    for (i = 0; i < n; i++) {
        scanf("%lf %lf", &points[i].x, &points[i].y);
    }
    PCA(points, n, &center, &v1, &v2);
    printf("Center: (%.2lf, %.2lf)\n", center.x, center.y);
    printf("v1: (%.2lf, %.2lf)\n", v1.x, v1.y);
    printf("v2: (%.2lf, %.2lf)\n", v2.x, v2.y);
    double max_x = -INFINITY, min_x = INFINITY, max_y = -INFINITY, min_y = INFINITY;
    for (i = 0; i < n; i++) {
        double x = (points[i].x - center.x) * v1.x + (points[i].y - center.y) * v1.y;
        double y = (points[i].x - center.x) * v2.x + (points[i].y - center.y) * v2.y;
        if (x > max_x) max_x = x;
        if (x < min_x) min_x = x;
        if (y > max_y) max_y = y;
        if (y < min_y) min_y = y;
    }
    printf("Bounding box:\n");
    printf("x: [%.2lf, %.2lf]\n", min_x, max_x);
    printf("y: [%.2lf, %.2lf]\n", min_y, max_y);
    return 0;
}

输入格式为：

Enter the number of points: 4
Enter the points:
0 0
1 0
0 1
1 1

输出结果为：

Center: (0.50, 0.50)
v1: (0.71, 0.71)
v2: (-0.71, 0.71)
Bounding box:
x: [-0.71, 0.71]
y: [-0.71, 0.71]