opengl坐标变换 整个流程的计算

之前用opencv求得某个物体位姿后，结合 相机 内参数矩阵， 使用 opengl 将其渲染出来
使用内参数矩阵计算出投影矩阵， 然后 思考了下 opengl 环境下 某个3D点最终在渲染窗口 的像素坐标是如何计算出来的
最终得到了如下代码， opengl 的坐标计算过程 参考如下链接
http://www.songho.ca/opengl/gl_transform.html
这个人 应该是某个大牛， 文章写得特别好，正确！

代码如下，包含opengl的计算方式， 和 直接使用 位姿变换流程的计算方式

void calcPixelCoor(const cv::Mat & poseMat){    //cv::Point3f ptx(2, 3, 3);    cv::Mat point3d = cv::Mat(4, 1, CV_64FC1);    //41.1071, 8.41076, -36.0432    point3d.at<double>(0) = 41.1071;    point3d.at<double>(1) = 8.41076;    point3d.at<double>(2) = 36.0432;    point3d.at<double>(3) = 1;    cv::Mat _poseMat = cv::Mat::zeros(4, 4, CV_64FC1);    for(int i = 0; i < 3; i++)    {        for(int j = 0; j < 4; j++)        {            _poseMat.at<double>(i, j) = poseMat.at<double>(i, j);        }    }    _poseMat.at<double>(3, 3) = 1;    // 先进行模型视图变换 4*4, 第4行为0，0，0，1    cv::Mat mvPt = _poseMat * point3d;    // 投影变换    double farPlane = 2500.0;    double nearPlane = 1;    double width = 640;    double height = 480;    double alpha = 747.72457662;    double beta = 744.939228;    double cx = 332.835123;    double cy = 252.998865;    cv::Mat project = cv::Mat::zeros(4, 4, CV_64FC1);    project.at<double>(0,0) = -2 * alpha / width;    project.at<double>(0,2) = -(2 * (cx / width) - 1);    project.at<double>(1,1) = -2 * beta / height;    project.at<double>(1,2) = 1 - 2 * cy / height;    project.at<double>(2,2) = (farPlane + nearPlane)/(farPlane-nearPlane);;    project.at<double>(2,3) = -2*farPlane*nearPlane/(farPlane-nearPlane);    project.at<double>(3,2) = -1;    cv::Mat proPt = project * mvPt;    // 透视除法    double x_ndc = proPt.at<double>(0) / proPt.at<double>(3);    double y_ndc = proPt.at<double>(1) / proPt.at<double>(3);    // 视口变换    int xw = (x_ndc + 1) * (width / 2) + 0;    int yw = (y_ndc + 1) * (height / 2) + 0;    std::cout << "xw " << xw << " yw " << yw << std::endl;    // 内参数    double params_LOGIC[] = { 747.72457662,   // fx        744.939228,  // fy        332.835123,      // cx        252.998865};    // cy    cv::Mat A_matrix = cv::Mat::zeros(3, 3, CV_64FC1);    A_matrix.at<double>(0, 0) = params_LOGIC[0];       //      [ fx   0  cx ]    A_matrix.at<double>(1, 1) = params_LOGIC[1];       //      [  0  fy  cy ]    A_matrix.at<double>(0, 2) = params_LOGIC[2];       //      [  0   0   1 ]    A_matrix.at<double>(1, 2) = params_LOGIC[3];    A_matrix.at<double>(2, 2) = 1;    // 2D point vector [u v 1]'    cv::Mat point2d_vec = cv::Mat(4, 1, CV_64FC1);    point2d_vec = A_matrix * poseMat * point3d;    // Normalization of [u v]'    cv::Point2f point2d;    point2d.x = (float)(point2d_vec.at<double>(0) / point2d_vec.at<double>(2));    point2d.y = (float)(point2d_vec.at<double>(1) / point2d_vec.at<double>(2));    std::cout << "xw2 " << point2d.x << " yw2 " << point2d.y << std::endl;}

glViewPort即按照上述公式进行运算， x和y默认取0， 即视口左下角作为即在原点
使用 内参数生成Projection矩阵的代码为

    // 投影变换    double farPlane = 2500.0;    double nearPlane = 1;    double width = 640;    double height = 480;    double alpha = 747.72457662;    double beta = 744.939228;    double cx = 332.835123;    double cy = 252.998865;    cv::Mat project = cv::Mat::zeros(4, 4, CV_64FC1);    project.at<double>(0,0) = -2 * alpha / width;    project.at<double>(0,2) = -(2 * (cx / width) - 1);    project.at<double>(1,1) = -2 * beta / height;    project.at<double>(1,2) = 1 - 2 * cy / height;    project.at<double>(2,2) = (farPlane + nearPlane)/(farPlane-nearPlane);;    project.at<double>(2,3) = -2*farPlane*nearPlane/(farPlane-nearPlane);    project.at<double>(3,2) = -1;

这种计算方法在网上 找了很久都没找到， 只找到一些相近的， 最终 偶然试出来的