已知一条边的起点和终点坐标和宽度，求矩形顶点的坐标，从c++代码写出来

假设已知一条边的起点为 $(x_0,y_0)$，终点为 $(x_1,y_1)$，宽度为 $w$，则可以求出两个垂线的方向向量 $\vec{v_1}$ 和 $\vec{v_2}$：

$$\vec{v_1} = \begin{pmatrix} y_1-y_0 \\ -(x_1-x_0) \end{pmatrix}, \quad \vec{v_2} = \begin{pmatrix} -(y_1-y_0) \\ x_1-x_0 \end{pmatrix}$$

然后可以将这两个向量单位化：

$$\vec{u_1} = \frac{\vec{v_1}}{\|\vec{v_1}\|}, \quad \vec{u_2} = \frac{\vec{v_2}}{\|\vec{v_2}\|}$$

矩形的四个顶点坐标分别为：

$$\begin{aligned} (x_0,y_0) + \frac{w}{2}\vec{u_1} + \frac{w}{2}\vec{u_2} \\ (x_0,y_0) + \frac{w}{2}\vec{u_1} - \frac{w}{2}\vec{u_2} \\ (x_1,y_1) - \frac{w}{2}\vec{u_1} + \frac{w}{2}\vec{u_2} \\ (x_1,y_1) - \frac{w}{2}\vec{u_1} - \frac{w}{2}\vec{u_2} \end{aligned}$$

以下是用 C++ 实现的代码：

#include <iostream>
#include <cmath>
using namespace std;

struct Point {
    double x, y;
};

void rectFromLine(Point start, Point end, double width, Point rect[4]) {
    // 计算垂线方向向量
    double vx = end.y - start.y, vy = -(end.x - start.x);
    // 计算单位向量
    double norm = sqrt(vx * vx + vy * vy);
    double ux1 = vx / norm, uy1 = vy / norm;
    double ux2 = -uy1, uy2 = ux1;
    // 计算矩形顶点
    rect[0].x = start.x + 0.5 * width * ux1 + 0.5 * width * ux2;
    rect[0].y = start.y + 0.5 * width * uy1 + 0.5 * width * uy2;
    rect[1].x = start.x + 0.5 * width * ux1 - 0.5 * width * ux2;
    rect[1].y = start.y + 0.5 * width * uy1 - 0.5 * width * uy2;
    rect[2].x = end.x - 0.5 * width * ux1 + 0.5 * width * ux2;
    rect[2].y = end.y - 0.5 * width * uy1 + 0.5 * width * uy2;
    rect[3].x = end.x - 0.5 * width * ux1 - 0.5 * width * ux2;
    rect[3].y = end.y - 0.5 * width * uy1 - 0.5 * width * uy2;
}

int main() {
    Point start = {1, 2}, end = {5, 6};
    double width = 2;
    Point rect[4];
    rectFromLine(start, end, width, rect);
    for (int i = 0; i < 4; i++) {
        cout << "(" << rect[i].x << "," << rect[i].y << ")" << endl;
    }
    return 0;
}