Perspective Projection

How Perspective Works

Objects farther away appear smaller. Parallel lines converge to a vanishing point.

Geometry

For a camera at origin looking down $-z$, with image plane at distance $d$:

$x' = \frac{x \cdot d}{z}, \quad y' = \frac{y \cdot d}{z}$

Matrix Form

Perspective can be expressed with a matrix:

$\begin{bmatrix} x \cdot d \\ y \cdot d \\ z \cdot d \\ z \end{bmatrix} = \begin{bmatrix} d & 0 & 0 & 0 \\ 0 & d & 0 & 0 \\ 0 & 0 & d & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}$

After dividing by $w = z$: correct projected coordinates.

Why Preserve $1/z$?

The $w$ component stores $z$, enabling:

• Z-buffer interpolation (interpolate $1/z$, not $z$)
• Perspective-correct texture mapping

Properties

• Realistic appearance
• Distant objects smaller
• Parallel lines converge
• Used in games, film, photography

Summary

• Perspective divides by $z$: $x' = xd/z$, $y' = yd/z$
• Preserves $1/z$ for correct depth interpolation
• Matches how cameras and eyes work