9.2.6. Transformations¶
9.2.6.1. Transformation matrices¶
To transform a vector, we multiply the vector with a transformation matrix
\[\begin{split}\begin{pmatrix} x' \\ y' \end{pmatrix} = \mathbf{T} \begin{pmatrix} x \\ y \end{pmatrix}\end{split}\]
Usual transformation matrices in 2D are
\[\begin{split}\begin{align}
\mathbf{Id} &= \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} \\[1em]
\mathbf{Scale}(s_x, s_y) &= \begin{bmatrix}
s_x & 0 \\
0 & s_y
\end{bmatrix} \\[1em]
\mathbf{Rotation}(\theta) &= \begin{bmatrix}
\cos\theta & \sin\theta \\
-\sin\theta & \cos\theta
\end{bmatrix} \\[1em]
\end{align}\end{split}\]
For translation and affine transformation, we must introduce the concept of homogeneous coordinate which add a virtual third dimension:
\[\begin{split}\mathbf{V} = \begin{bmatrix}
x \\
y \\
1
\end{bmatrix}\end{split}\]
Then the translation and affine transformation matrix are expressed as:
\[\begin{split}\begin{align}
\mathbf{Translation}(t_x, t_y) &= \begin{bmatrix}
1 & 0 & t_x \\
0 & 1 & t_y \\
0 & 0 & 1
\end{bmatrix} \\[1em]
\mathbf{Generic} &= \begin{bmatrix}
r_{11} & r_{12} & t_x \\
r_{12} & r_{22} & t_y \\
0 & 0 & 1
\end{bmatrix}
\end{align}\end{split}\]
To compose transformations, we must multiply the transformations in this order:
\[\mathbf{T} = \mathbf{T_n} \ldots \mathbf{T_2} \mathbf{T_1}\]
Note the matrix multiplication is not commutative.