affine 3D transformation reconstruction

Question

How can we get the affine 3D matrix in case we have the 3D rotation matrix, the 3D translation vector, the scale factors and the shearing factors?

A = SHEARING (4,4) * ScaleMatrix (4,4) * RotationMatrix(4,4) + T(4,1)

is that accurate?

Answer 1

For example, let $$R=\begin{pmatrix}r_{11}&r_{12}&r_{13}\\ r_{21}&r_{22}&r_{23}\\ r_{31}&r_{32}&r_{33}\end{pmatrix}$$ be the $3\times 3$ rotation matrix, $$\vec{t}=\begin{pmatrix}t_1\\t_2\\t_3\end{pmatrix}$$ be the $3\times 1$ translation vector, $s$ be the scale factor. If the order is rotation, translation, then scaling, the matrix should be $$A=s\begin{pmatrix}r_{11}&r_{12}&r_{13}&t_1\\ r_{21}&r_{22}&r_{23}&t_2\\ r_{31}&r_{32}&r_{33}&t_3\\ 0&0&0&1\end{pmatrix}$$

If otherwise, the order is rotation, scaling, then translation, the matrix should be

$$A=\begin{pmatrix}sr_{11}&sr_{12}&sr_{13}&t_1\\ sr_{21}&sr_{22}&sr_{23}&t_2\\ sr_{31}&sr_{32}&sr_{33}&t_3\\ 0&0&0&1\end{pmatrix}$$

Then $$\begin{pmatrix}x_1'\\x_2'\\x_3'\\1\end{pmatrix}=A\begin{pmatrix}x_1\\x_2\\x_3\\1\end{pmatrix}$$