3D Worlds — Matrices Behind Every Pixel
3D Computer Graphics
Every 3D video game, Pixar movie, and architectural visualization is powered by linear algebra. When you move through a game world, your GPU is multiplying millions of matrices per second.
Matrices Move the World
Every object in a 3D scene is made of triangles, and every triangle is made of three points (vertices). To move, rotate, or scale an object, you multiply each vertex by a transformation matrix.
Translation (move):
$$T = \begin{pmatrix} 1 & 0 & 0 & t_x \\ 0 & 1 & 0 & t_y \\ 0 & 0 & 1 & t_z \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
Rotation around Z-axis (uses trig!):
$$R_z = \begin{pmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
Scaling:
$$S = \begin{pmatrix} s_x & 0 & 0 & 0 \\ 0 & s_y & 0 & 0 \\ 0 & 0 & s_z & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
Notice: rotation uses sine and cosine — the same trig functions from your math class. Every time a character turns in a game, $\sin\theta$ and $\cos\theta$ are calculated.
Rotate the point $(3, 4, 0)$ by $90°$ around the Z-axis.
$\cos 90° = 0$, $\sin 90° = 1$:
$$\begin{pmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 3 \\ 4 \\ 0 \end{pmatrix} = \begin{pmatrix} 0(3) + (-1)(4) \\ 1(3) + 0(4) \\ 0 \end{pmatrix} = \begin{pmatrix} -4 \\ 3 \\ 0 \end{pmatrix}$$
The point moved from $(3, 4)$ to $(-4, 3)$ — a perfect 90° rotation. Matrix multiplication and trig made it happen.
Lighting — The Dot Product
How bright a surface appears depends on the angle between the surface and the light source. This is calculated with the dot product:
$$\text{brightness} = \max(0,\; \mathbf{n} \cdot \mathbf{l}) = \max(0,\; n_x l_x + n_y l_y + n_z l_z)$$
where $\mathbf{n}$ is the surface normal and $\mathbf{l}$ is the light direction. This is multiplication and addition — applied billions of times per second.
The Numbers
| What Happens | Math Used | Times Per Frame |
|---|---|---|
| Transform vertices | Matrix × vector | ~10 million |
| Calculate lighting | Dot products | ~50 million |
| Texture mapping | Interpolation (linear algebra) | ~2 billion |
| Perspective projection | Matrix division | ~10 million |
At 60 frames per second, your GPU does this 60 times every second.
Every frame of every game is millions of matrix multiplications, dot products, and trig evaluations. When you learn linear algebra and trigonometry, you're learning the language of computer graphics.