Perspective Projection

In chase-camera mode, a boat sailing away from the drone should dwindle toward the horizon, and one heading toward it should swell until it fills the view. An orthographic camera can't do that; it keeps every size fixed. Perspective projection does it with one division.

The rule

Perspective: divide by depth, so twice as far looks half as big.

Picture the camera as a single point, with a flat screen at distance $f$ in front of it. The straight line from the camera to a point $(x, y, z)$ crosses the screen somewhere, and similar triangles tell you where:

$s_x = \dfrac{f\,x}{z}, \qquad s_y = \dfrac{f\,y}{z}$

With $f = 1$, a crate corner at $(2, 1, 4)$ lands at $(0.5, 0.25)$. Move it back to $z = 8$ and it lands at $(0.25, 0.125)$: twice as far, half the size, and slid toward the centre of the view. That is exactly how an eye or a lens sees.

The field of view sets $f$. If the screen runs from −1 to 1 across, then $f = 1 / \tan(\text{fov}/2)$, so a 90° view gives $f = 1$. Widening the view makes $f$ smaller: more of the world fits on the screen, and everything in it looks smaller.

In a real renderer the GPU does the division. A 4×4 projection matrix copies each point's depth into its $w$ coordinate, and after the multiplication the hardware divides by $w$, a step called the perspective divide. The same $w$ that told points from vectors now carries depth.

The shortcut

On-screen size = f × real size ÷ depth. You rarely need the full coordinates: halve the distance and the size doubles; triple it and the size drops to a third.

The trap is points behind the camera. With z = −5 the division flips the signs, and something behind the drone is drawn, mirrored, in front of it; at z = 0 the division blows up. Renderers therefore clip geometry against a near plane just in front of the camera before they divide.

Drills

Answer fast — trust the rule.

Remember

• Perspective: screen position = f × (x, y) ÷ depth; twice as far looks half as big.
• On-screen size ∝ 1 ÷ distance; a wider field of view means a smaller f and smaller objects.
• Clip at a near plane before dividing, or things behind the camera appear mirrored in front.