Game Math 2
Matrices for 3D
Umer Noor, PhD
Faculty of Media, Creative Arts, and Design, Humber Polytechnic
Demos built using Claude Code
Matrices
A matrix is just a table of numbers. Here’s an example from the wild:
\[\begin{pmatrix}
3 & 0 & 0 & 4 \\
0 & 3 & 0 & 5 \\
0 & 0 & 1 & -1\\
0 & 0 & 0 & 1
\end{pmatrix}\]
Why do game developers love them so much? They allow us to do useful things to positions and directions!
Scale Matrix in 3D
You can make vectors bigger, smaller or point the other way with the scale matrix:
\[\begin{pmatrix}
scale_x & 0 & 0 & 0\\
0 & scale_y & 0 & 0\\
0 & 0 & scale_z & 0\\
0 & 0 & 0 & 1 \end{pmatrix}\]
Translate Matrix in 3D
You can move positions around using the translation matrix:
\[\begin{pmatrix}
1 & 0 & 0 & \Delta x\\
0 & 1 & 0 & \Delta y\\
0 & 0 & 1 & \Delta z\\
0 & 0 & 0 & 1
\end{pmatrix}\]
Rotation Matrix in 3D (pick your axis of rotation)
\[R_x = \begin{pmatrix}
1 & 0 & 0 & 0\\
0 & \cos\theta & -\sin\theta & 0\\
0 & \sin\theta & \cos\theta & 0\\
0 & 0 & 0 & 1 \end{pmatrix}
\quad
R_y = \begin{pmatrix}
\cos\theta & 0 & \sin\theta & 0\\
0 & 1 & 0 & 0\\
-\sin\theta & 0 & \cos\theta & 0\\
0 & 0 & 0 & 1 \end{pmatrix}
\quad
R_z = \begin{pmatrix}
\cos\theta & -\sin\theta & 0 & 0\\
\sin\theta & \cos\theta & 0 & 0\\
0 & 0 & 1 & 0\\
0 & 0 & 0 & 1 \end{pmatrix}\]
Order Matters
Multiply matrices to combine transformations. But be careful, order matters!
Dumptruck Method
Think of a row vector multiplying a column vector:
\[\begin{pmatrix} 1 & 2 & 3 & 4 \end{pmatrix}
\times
\begin{pmatrix} 10\\20\\30\\40 \end{pmatrix}\]
\[= 1(10) + 2(20) + 3(30) + 4(40)\]
Multiplying a Matrix by a Vector
Use the dumptruck method four times for a 4x4 matrix multiplied with a 4D column vector:
The Model Matrix
Just keep on dumptrucking to build the Model Matrix = \(TRS\)
Order Matters with Rotations too
Directions Don’t Translate
Remember that \(w\) is zero for a direction? That means it won’t translate:
