Use Eigen to solve the rotation matrices of different coordinate systems

Rigid transformation between different coordinate systems and implementation:

The principle of coordinate system transformation is as follows. Affine transformation is used to realize rotation and translation:
$$\left[\begin{array}{ccc}{r}_{11}& r_{12}& t_x\\ r_{21}& r_{22}& t_y\\ 0& 0& \end{array}\right]\ast \left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=\left[\begin{array}{c}{x}^{{}^{\prime }}\\ y^{{}^{\prime }}\\ 1\end{array}\right]$$
But find the equation A ∗ x = b A*x=b in eigen$A\ast x=b$ , so you can put$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$As A, consider $\left[\begin{array}{c}{x}^{{}^{\prime }}\\ y^{{}^{\prime }}\\ z^{{}^{\prime }}\end{array}\right]$Think of it as $b$ . However, this formula requires a certain amount of deformation to correspond to the previous one, and a transpose is made before and after, as shown below: [ [
$$\begin{gathered} {\left[\left[\begin{array}{ccc}{r}_{11}& r_{12}& t_x\\ r_{21}& r_{22}& t_y\\ 0& 0& \end{array}\right]\ast \\ \left[\begin{array}{c}x\\ y\\ 1\end{array}\right] \\ \right]}^T={\left[\begin{array}{c}{x}^{{}^{\prime }}\\ y^{{}^{\prime }}\\ 1\end{array}\right]}^T \end{gathered}$$
will then be converted into the following:
$$\left[\begin{array}{ccc}x& y& 1\end{array}\right]\ast \left[\begin{array}{ccc}{r}_{11}& r_{21}& 0\\ r_{12}& r_{22}& 0\\ t_x& t_y& 1\end{array}\right]=\left[\begin{array}{ccc}{x}^{{}^{\prime }}& y^{{}^{\prime }}& 1\end{array}\right]$$
So in the end the above formula is used in the program, andA.fullPivLu().solve(B)when the calculation is called, A corresponds to$\left[\begin{array}{ccc}x& y& 1\end{array}\right]$ , B corresponds to$\left[\begin{array}{ccc}{x}^{{}^{\prime }}& y^{{}^{\prime }}& 1\end{array}\right]$ . So the result is also$\left[\begin{array}{ccc}{r}_{11}& r_{21}& 0\\ r_{12}& r_{22}& 0\\ t_x& t_y& 1\end{array}\right]$. See if you need to transpose it

A << -4.93374, -0.056378, 1, 0.397714, -0.126519, 1, 2.63284, -0.0925265, 1;
	B << 109995, 8318, 1, 110066, 13650, 1, 110032, 15885, 1;
Eigen::Matrix3d T = TransMatrix(A, B);
cout << "transform matrix is: " << "\n";
cout << T.transpose() << endl;//显示出A*T=B，但是实际计算的是（T的转置*A的转置）的转置=B的转置
cout << A*T << endl;

The transformation matrix results look like this: