games101作业04：Bézier 曲线

内容

Bézier 曲线是一种用于计算机图形学的参数曲线。在本次作业中，你需要实 de Casteljau 算法来绘制由 4 个控制点表示的 Bézier 曲线 (当你正确实现该算法时，你可以支持绘制由更多点来控制的 Bézier 曲线)。你需要修改的函数在提供的 main.cpp 文件中。

1. bezier：该函数实现绘制 Bézier 曲线的功能。它使用一个控制点序列和一个OpenCV::Mat 对象作为输入，没有返回值。它会使 t 在 0 到 1 的范围内进行迭代，并在每次迭代中使 t 增加一个微小值。对于每个需要计算的 t，将调用另一个recursive_bezier，然后该函数将返回在Bézier 曲线上 t处的点。最后，将返回的点绘制在 OpenCV::Mat 对象上。
2. recursive_bezier：该函数使用一个控制点序列和一个浮点数 t 作为输入，实现 de Casteljau 算法来返回 Bézier 曲线上对应点的坐标。、

基础内容

参考

代码

#include <chrono>
#include <iostream>
#include <opencv2/opencv.hpp>

std::vector<cv::Point2f> control_points;

void mouse_handler(int event, int x, int y, int flags, void *userdata) 
{
    if (event == cv::EVENT_LBUTTONDOWN && control_points.size() < 4) 
    {
        std::cout << "Left button of the mouse is clicked - position (" << x << ", "
        << y << ")" << '\n';
        control_points.emplace_back(x, y);
    }     
}

void naive_bezier(const std::vector<cv::Point2f> &points, cv::Mat &window) 
{
    auto &p_0 = points[0];
    auto &p_1 = points[1];
    auto &p_2 = points[2];
    auto &p_3 = points[3];

    for (double t = 0.0; t <= 1.0; t += 0.001) 
    {
        auto point = std::pow(1 - t, 3) * p_0 + 3 * t * std::pow(1 - t, 2) * p_1 +
                 3 * std::pow(t, 2) * (1 - t) * p_2 + std::pow(t, 3) * p_3;

        window.at<cv::Vec3b>(point.y, point.x)[2] = 255;
    }
}


int factorial(int x)
{
    int f;

    if (x == 0 || x == 1)
        f = 1;
    else
        f = factorial(x - 1) * x;

    return f;
}

cv::Point2f recursive_bezier(const std::vector<cv::Point2f> &control_points, float t) 
{
    // TODO: Implement de Casteljau's algorithm

    // 数学方法
    // int n = control_points.size();
    // cv::Point2f point = {0.0f, 0.0f};
    // for (int i = 0; i < n; i++)
    // {
    //     int c = factorial(n - 1) / (factorial(i) * factorial(n - 1 - i));
    //     point += c * std::pow(t, i) * std::pow(1 - t, n - 1 - i) * control_points[i];
    // }

    // 递归方法
    
    if(control_points.size() == 2)
    {
        return control_points[0] + t * (control_points[1] - control_points[0]);
    }

    std::vector<cv::Point2f> vec;
    for (int i = 0; i < control_points.size() - 1; i++)
    {
        vec.push_back(control_points[i] + t * (control_points[i + 1] - control_points[i]));
    }

    return recursive_bezier(vec, t);
}

//将点显示在屏幕
void bezier(const std::vector<cv::Point2f> &control_points, cv::Mat &window) 
{
    // TODO: Iterate through all t = 0 to t = 1 with small steps, and call de Casteljau's 
    // recursive Bezier algorithm.
    for (double t = 0.0; t<=1.0; t+=0.001){
        cv::Point2f point = recursive_bezier(control_points, t);
        window.at<cv::Vec3b>(point.y,point.x)[1] = 255;
    }

}
//对BezierCurve进行插值
/*
void bezier(const std::vector<cv::Point2f> &control_points, cv::Mat &window) 
{
    // TODO: Iterate through all t = 0 to t = 1 with small steps, and call de Casteljau's 
    // recursive Bezier algorithm.
    for (double t = 0.0; t <= 1.0; t += 0.001)
    {
        cv::Point2f point = recursive_bezier(control_points, t);
        window.at<cv::Vec3b>(point.y, point.x)[1] = 255;

        //anti-aliasing
        float x = point.x - std::floor(point.x);
        float y = point.y - std::floor(point.y);
        int x_flag = x < 0.5f ? -1 : 1;
        int y_flag = y < 0.5f ? -1 : 1;

        // 距离采样点最近的4个坐标点
        cv::Point2f p00 = cv::Point2f(std::floor(point.x) + 0.5f, std::floor(point.y) + 0.5f);
        cv::Point2f p01 = cv::Point2f(std::floor(point.x + x_flag * 1.0f) + 0.5f, std::floor(point.y) + 0.5f);
        cv::Point2f p10 = cv::Point2f(std::floor(point.x) + 0.5f, std::floor(point.y + y_flag * 1.0f) + 0.5f);
        cv::Point2f p11 = cv::Point2f(std::floor(point.x + x_flag * 1.0f) + 0.5f, std::floor(point.y + y_flag * 1.0f) + 0.5f);

        std::vector<cv::Point2f> vec;
        vec.push_back(p01);
        vec.push_back(p10);
        vec.push_back(p11);

        // 计算最近的坐标点与采样点距离
        cv::Point2f distance = p00 - point;
        float len = sqrt(distance.x * distance.x + distance.y * distance.y);

        // 对边缘点进行着色
        for(auto p:vec)
        {
            // 根据距离比, 计算边缘点影响系数 
            cv::Point2f d = p - point;
            float l = sqrt(d.x * d.x + d.y * d.y);
            float percnet = len / l;

            cv::Vec3d color = window.at<cv::Vec3b>(p.y, p.x);
            // 此处简单粗暴取最大值
            color[1] = std::max(color[1], (double)255 * percnet);
            window.at<cv::Vec3b>(p.y, p.x) = color;
        }
    }
}*/


int main() 
{
    cv::Mat window = cv::Mat(700, 700, CV_8UC3, cv::Scalar(0));
    cv::cvtColor(window, window, cv::COLOR_BGR2RGB);
    cv::namedWindow("Bezier Curve", cv::WINDOW_AUTOSIZE);

    cv::setMouseCallback("Bezier Curve", mouse_handler, nullptr);

    int key = -1;
    while (key != 27) 
    {
        for (auto &point : control_points) 
        {
            cv::circle(window, point, 3, {255, 255, 255}, 3);
        }

        if (control_points.size() == 4) 
        {
            //naive_bezier(control_points, window);
            bezier(control_points, window);

            cv::imshow("Bezier Curve", window);
            cv::imwrite("my_bezier_curve.png", window);
            key = cv::waitKey(0);

            return 0;
        }

        cv::imshow("Bezier Curve", window);
        key = cv::waitKey(20);
    }

return 0;
}