Many of us of a certain age will have had their first true, good 3D video game experience with Super Mario 64. Unlike previous 3D games, the camera was an object controllable by the player, rather than a first-person-ony mode or one where the game tries to guess the best placement for the camera. We might take this mechanic for granted today, but 3D was a new technology at the time that took experimentation before settling on the norms we have today. From a programming perspective, 3D graphics can be a bit of a head-scratcher but [Gabriel] shows that perspective and the camera can be as simple as a few lines of math.

When starting out as a programmer, [Gabriel] used various tools that provided a camera somewhat automatically. But after reaching the limits of these types of frameworks, the next step is to learn how that works from scratch. It turns out that it’s a bit of matrix math, with values for foreground and background clipping planes as well as aspect, field of view, and position. This basically replicates a trapezoidal prism which can be thought of as a viewer looking at a scene from the perspective of a camera. To provide the depth effect, the X and Y coordinates are divided by the Z coordinate within this matrix system, making far-away objects smaller and generating the 3D effect.

On [Gabriel]’s site which explains this method, there are a few sliders in several examples that demonstrate how changing values of each of these variables changes the perspective and the object being displayed. For a math lesson it is very interactive and helps intuit these concepts. Cameras aside, the generation of 3D objects has its own unique set of math equations to learn about that are “equally” interesting.