3D Projection

A scene lives in three dimensions, but a screen is flat. Projection is the math that flattens 3D points onto the 2D image plane the way a camera or your eye does. The cube below is a set of eight 3D points spinning in space, projected to the screen each frame. Drag the field-of-view and distance sliders to see perspective change.

From world space to the camera

A point starts in world space, the shared coordinate system of the scene. The view transform moves everything so the camera sits at the origin looking down one axis — it is just another matrix multiply, the inverse of the camera’s own position and orientation. After this step a vertex has coordinates $(x, y, z)$ measured relative to the camera, where $z$ is its depth straight ahead.

The perspective divide

Perspective is the rule that distant things look smaller. It comes from one operation: dividing by depth. With a focal length $f$, a camera-space point projects to the image plane as

$x' = \frac{f \, x}{z}, \qquad y' = \frac{f \, y}{z}$

Double the depth $z$ and the projected size halves. That single division produces foreshortening — parallel rails appearing to meet at a horizon, near faces of the cube looming larger than far ones. The focal length ties directly to the field of view: $f = 1 / \tan(\text{fov}/2)$, so a wide field of view means a short focal length and exaggerated, almost fisheye depth, while a narrow one flattens the scene toward an orthographic look where depth no longer changes size.

Why a 4x4 matrix

Just like translation in 2D needed homogeneous coordinates, 3D graphics carries points as four numbers $\begin{bmatrix} x & y & z & w \end{bmatrix}$ and uses $4 \times 4$ matrices. The projection matrix is cleverly arranged to copy $z$ into the $w$ slot; the hardware then divides $x$, $y$, and $z$ by $w$ in a final step called the perspective divide. Packing the divide into $w$ lets one matrix chain — model, view, and projection multiplied together — handle the entire transform from a model’s local space all the way to the screen.

Takeaways

• Projection flattens 3D points onto a 2D image plane via the camera.
• The view transform repositions the world so the camera is at the origin.
• Dividing screen position by depth $z$ creates perspective foreshortening; focal length $f = 1/\tan(\text{fov}/2)$ controls how strong it is.
• Homogeneous $4 \times 4$ matrices store depth in $w$ so the perspective divide is a single uniform step at the end of the pipeline.

References

• Scratchapixel — The perspective and orthographic projection matrix
• Akenine-Moller et al., “Real-Time Rendering” — projection chapter
• LearnOpenGL — Coordinate Systems