I am puzzled about linear transformation and coordinate transformation, any help will be appreciated.
From wiki rotation matrix, we know rotates points in the xy-Cartesian plane counter-clockwise through an angle θ about the origin, we get matrix:
$$\begin{align}
\begin{pmatrix}x' \\ y' \end{pmatrix}=\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\begin{pmatrix}x\\y\end{pmatrix}(1).
\end{align}$$
From wiki rotation of axes, we know
by rotate xy-Cartesian coordinate system through an angle $\theta$ to an $x^{'}y^{'}$-Cartesian coordinate system, we get:
$$\begin{align}\begin{pmatrix} x' \\ y'\end{pmatrix} = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} x \\ y\end{pmatrix}(2)\end{align}$$
$$\begin{align}\begin{pmatrix} x \\ y\end{pmatrix} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\begin{pmatrix} x' \\ y'\end{pmatrix}(3)\end{align}$$
My question
1. using matrix (2)(3) we can transform coordinate between $xoy$ and $x'oy'$, rotation of axes really involves coordinate transformation between different basis while rotation around the origin point using matrix(1) does not, is that right ?
2. In 3D graphics, often being declared that after model transfromations(rotate,scale, shear), coordinate transformed from local object space to global world space, that is $$\begin{align}Objectspace \xrightarrow{rotate} WorldSpace\end{align}$$
And this matrix called model matrix.
Take the above rotation for example, matrix(1) just equal matrix(3), does this a coincidence or a true fact ? when rotate an object $\theta$ angle, which matrix is the so called model matrix and how to interpret it ?
I think what you're looking at is what's called as Active versus Passive transforms. They're just two ways of looking at the same transform.
In an active transform, you think of the coordinate frame remaining the same, while the "objects" moving.
In a passive transform, you think of the objects staying in the same place, but the coordinate system moving in the opposite direction so the same (net) effect takes place