4 types of transformation

1. Translation

   point vector + translation vector → new targeted point vector

   $\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} T_x \\ T_y \end{bmatrix} +\begin{bmatrix} x\\ y \end{bmatrix}$

   this can also be written in multiplication form so we can combine the translation along with other operations

   $\begin{bmatrix} x'\\ y'\\ w' \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x\\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y\\ 1 \end{bmatrix}$

   and the point is in homogeneous form

2. Scaling with respect to origin

   $\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} S_x & 0\\ 0 & S_y \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}$

3. Rotation in counter-clockwise

   $\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} cos\theta & -sin\theta\\ sin\theta & cos\theta \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}$

4. Shearing

   • Horizontal shear:

     $\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} 1 & k\\ 0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}$

   • Vertical shear

     $\begin{bmatrix} x'\\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0\\k & 1 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix}$

   • Share in 3D homogeneous form

     $\begin{bmatrix} x'\\ y'\\ z'\\ w' \end{bmatrix} = \begin{bmatrix} 1 & 0 & a & 0\\ 0 & 1 &b & 0 \\ 0 & 0 & 1& 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y\\ z\\ 1 \end{bmatrix}$

Transform 2-4 to homogeneous form is trivial, just make it 3x3 matrix and fill the rest slot with 0 except the bottom-right one.

• 3D rotation

• around z $\begin{bmatrix} cos\theta & -sin\theta & 0 & 0\\ sin\theta & cos\theta &0 & 0 \\ 0 & 0 & 1& 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$ around x $\begin{bmatrix} 1 & 0 & 0 & 0\\ 0& cos\theta & -sin\theta & 0 \\ 0 & sin\theta & cos\theta& 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

  around y $\begin{bmatrix} cos\theta & 0 & sin\theta & 0\\ 0 & 1 &0 & 0 \\ -sin\theta & 0 & cos\theta& 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$

orthonormal matrices preserve lengths, angles, ration of side lengths

Affine Transformation(shear, scale): parallelism of lines and planes, ration of edge lengths are preserved

rigid body transformation: a mapping from “this” set to another subset of the Euclidean space, such that the Euclidean distances between points are preserved.

Transforming a vector/normal:

• If we have a matrix M to transform the vertices/point, then for vector/normals, the same transformation will be represented by $(M^T)^{-1}$

To rotate around an arbitrary axis in the space, we just need to rotate that axis to any existing axis and perform the rotation, then rotate the axis back

An orthonormal vector:

each vector has the length of 1 and they are all perpendicular to each other(dot product between any 2 of them is 0)

Change of Basis

To represent a point in a different coordinate system, we can first transform the original basis to the new basis, call it $T$, and the new point in new coordinate system would be $T^{-1}P$ where $P$ is the old coordinate for the point.

Rotation’s inverse → transpose