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 ?

  • $\begingroup$ I don't really follow what you are asking - what do you believe the difference between (1) and (2) is? $\endgroup$ May 2, 2016 at 14:13
  • $\begingroup$ @Siddharth Bhat, thanks your reply. In my point of view, (1) is a position transformation within the same xoy coordinate system, while (2)(3) are transformation btween different coordinate system using basis to do the computation.Am i right ? My puzzle is in 3D graphics, when rotate the object by matrix (1) or (3) , then the coordinate transformed from object space to world space, does (1) or (3) the so called model matrix ? Is this the right way to understand and get the model matrix ? $\endgroup$
    – wangdq
    May 2, 2016 at 14:27
  • 2
    $\begingroup$ Once upon a time, this was known as the “alias” vs. “alibi” interpretation of a transformation. In the “alias” view, the transformation relabels points with new coordinates, i.e., it transforms the coordinate system. In the “alibi” view, it maps points to new points in the same coordinate system, i.e., it “physically” alters the objects. In terms of linear transformation matrices, these differing points of view often result in the two matrices being transposes of each other, as you have here. $\endgroup$
    – amd
    May 2, 2016 at 18:42
  • $\begingroup$ @amd,you are right, but this 'alibi' and 'alias' transformation is rarely mentioned in the textbook I read. Thanks! $\endgroup$
    – wangdq
    May 3, 2016 at 3:21

1 Answer 1


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

  • $\begingroup$ thanks a lot ! You broad my math sight(I am not a student from math background), I have understood the matrix above.Now how to understand question(2) about 3D graphics , that is after the model transformation(like rotation), the coordinate transformed from local object space to global world space ? I tried to interpret it with passive transformation, but feel hard to figure it out. $\endgroup$
    – wangdq
    May 3, 2016 at 3:07

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