Viewing transformation

Projection transformation

Orthographic projection

无近大远小

相机离的无限远

具体操作

Camera located at origin, looking at -Z, up at Y (looks familiar?)
Center cuboid by translating
Scale into “canonical” cube *方便之后的计算

Transformation matrix

Caveat

因为我们是从-Z方向看的，所以近的z大，远的z小
FYI: that’s why OpenGL (a Graphics API) uses left hand coords.

Perspective projection

有近大远小

具体操作

First “squish” the frustum into a cuboid (n -> n, f -> f) (Mpersp->ortho)
Do orthographic projection (Mortho, already known!)

Transformation matrix

由相似三角形出发，加上特殊点的坐标推导，可得步骤一的矩阵为：

作业0

Eigen的简单使用

点乘：dot()

叉乘：cross()

头文件声明：

c++

1	#include < eigen3/Eigen/Core >

矩阵声明：

c++

//n阶方阵
Eigen::Matrix2d m_matrix;
m_matrix<< 2,3,2.2,1;
std::cout << "m_matrix = \n"<< m_matrix<std::endl;
//mxn方阵
Eigen::MatrixXd xx(5,2);
xx << 2,3,2.2,1,2,3,2,1,2,3;
//零矩阵
Eigen::MatrixXf m_matrix = Eigen::MatrixXf::Zero(5,5);
//对角矩阵
Eigen::MatrixXf m_matrix = Eigen::MatrixXf::Identity(5,5);

更多基础操作见：https://blog.csdn.net/weixin_42156097/article/details/108011364?utm_medium=distribute.pc_aggpage_search_result.none-task-blog-2~aggregatepage~first_rank_ecpm_v1~rank_v31_ecpm-2-108011364.pc_agg_new_rank&utm_term=eigen%E5%BA%93%E4%B8%AD%E6%80%8E%E4%B9%88%E5%AE%9A%E4%B9%89%E7%9F%A9%E9%98%B5&spm=1000.2123.3001.4430

Eigen库头文件类型：

https://www.jianshu.com/p/931dff3b1b21

作业0

作业描述

给定一个点P=(2,1), 将该点绕原点先逆时针旋转45◦，再平移(1,2), 计算出
变换后点的坐标（要求用齐次坐标进行计算）。

解答

c++

#include < cmath > 
#include < eigen3/Eigen/Core >
#include < eigen3/Eigen/Dense >
#include < iostream >
using namespace std;

int main(){
    Eigen::Vector3f P(2,1,1);
    Eigen::Matrix3f TR;
    //TR << 0.5, -0.5, 1, 0.5, 0.5, 2, 0, 0, 1;
    float a = 45.0/180.0*M_PI;
    TR << cos(a), -sin(a), 1, 
          sin(a),  cos(a), 2, 
             0,      0,    1;
    Eigen::Vector3f P1;
    P1 = TR * P;
    cout << "After transformation, P = (" << P1[0] << ","<< P1[1]<<")" << endl;

    return 0;
}

注意在C++中，cos、sin函数的参数为弧度制。

P = （1.70711， 4.12132）

编译

shell

mkdir build
cd build
cmake ..
make
./Transformation