12. 2D Transformations - 2
Dated: 12-05-2025
Homogeneous Coordinates
All the transformations can be summed up into
\[\vec P^\prime = \textbf A_{2 \times 2} \cdot \vec P + \textbf B_{2 \times 1}\]
We can eliminate \(\textbf B_{2 \times 1}\) by extending \(\textbf A_{2 \times 2}\) into \(\textbf A_{3 \times 3}\).
The coordinates need to be translated to a homogeneous triplet.
\[(x, y) \to (x_h, y_h, h)\]
Where
\[x = \frac{x_h} h, \quad y = \frac{y_h} h\]
For simplicity,
\[h = 1 \implies (x, y, 1)\]
Translation
\[
\begin{bmatrix}
x^\prime \\
y^\prime \\
1
\end{bmatrix}
=
\begin{bmatrix}
1 & 0 & t_x\\
0 & 1 & t_y\\
0 & 0 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
x\\
y\\
1
\end{bmatrix}
\]
\[
\vec P^\prime = \textbf T(t_x, t_y) \cdot \vec P
\]
Rotation
\[
\begin{bmatrix}
x^\prime \\
y^\prime \\
1
\end{bmatrix}
=
\begin{bmatrix}
\cos \theta & -\sin\theta & 0\\
\sin \theta & \cos \theta & 0\\
0 & 0 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
x\\
y\\
1
\end{bmatrix}
\]
\[
\vec P^\prime = \textbf R(\theta) \cdot \vec P
\]
Scaling
\[
\begin{bmatrix}
x^\prime \\
y^\prime \\
1
\end{bmatrix}
=
\begin{bmatrix}
S_x & 0 & 0\\
0 & S_y & 0\\
0 & 0 & 1
\end{bmatrix}
\cdot
\begin{bmatrix}
x\\
y\\
1
\end{bmatrix}
\]
\[
\vec P^\prime = \textbf S(S_x, S_y) \cdot \vec P
\]
Composition
Imagine we have a vector1 \(\vec V\) and 3 transformations \(\textbf A, \textbf B\) and \(\textbf C\).
Although, we can apply transformations as \(\textbf A\), then followed by \(\textbf B\) and then followed by \(\textbf C\).
\[\textbf C (\textbf B (\textbf A \cdot \vec V))\]
Due to associative property of matrices.2
\[\textbf {M} (\vec V)\]
Where
\[\textbf M = \textbf {CBA}\]